{"video_id": "edGzq6CyE6g", "transcript": [{"start": 0.64, "duration": 4.48, "text": "hi everyone"}, {"start": 1.92, "duration": 4.879, "text": "welcome to my youtube channel"}, {"start": 5.12, "duration": 3.84, "text": "first of all i want to thank all of you"}, {"start": 6.799, "duration": 4.321, "text": "who watch my youtube videos"}, {"start": 8.96, "duration": 4.799, "text": "and i definitely appreciate all of the"}, {"start": 11.12, "duration": 4.0, "text": "good comments that you post"}, {"start": 13.759, "duration": 3.44, "text": "it's always good to know that these"}, {"start": 15.12, "duration": 2.88, "text": "videos are helping you to succeed in"}, {"start": 17.199, "duration": 2.321, "text": "school"}, {"start": 18.0, "duration": 2.8, "text": "now for those of you who are new to my"}, {"start": 19.52, "duration": 3.519, "text": "channel"}, {"start": 20.8, "duration": 4.08, "text": "feel free to check out my playlist on"}, {"start": 23.039, "duration": 3.041, "text": "algebra trig"}, {"start": 24.88, "duration": 3.28, "text": "calculus"}, {"start": 26.08, "duration": 4.08, "text": "chemistry physics and even organic"}, {"start": 28.16, "duration": 3.439, "text": "chemistry"}, {"start": 30.16, "duration": 3.52, "text": "you should be able to see the links"}, {"start": 31.599, "duration": 4.161, "text": "posted on this video if you're using the"}, {"start": 33.68, "duration": 3.6, "text": "desktop computer or laptop"}, {"start": 35.76, "duration": 2.479, "text": "if you're using your cell phone or"}, {"start": 37.28, "duration": 2.88, "text": "tablet"}, {"start": 38.239, "duration": 3.121, "text": "you may not see those links so"}, {"start": 40.16, "duration": 2.239, "text": "just check out my channel for those"}, {"start": 41.36, "duration": 2.24, "text": "playlists"}, {"start": 42.399, "duration": 3.68, "text": "and you can find"}, {"start": 43.6, "duration": 5.279, "text": "almost any video that you need"}, {"start": 46.079, "duration": 6.081, "text": "so thanks again for all of your support"}, {"start": 48.879, "duration": 3.281, "text": "and have a good day"}], "title": "The Organic Chemistry Tutor Channel", "description": "This video is the trailer of my organic chemistry tutor channel.", "lengthSeconds": 50, "width": 640, "height": 360}
{"video_id": "yscrBkfPRyc", "transcript": [{"start": 1.68, "duration": 4.159, "text": "in this video we're going to talk about"}, {"start": 3.6, "duration": 4.239, "text": "the law of reflection"}, {"start": 5.839, "duration": 3.201, "text": "so what exactly is that what is its"}, {"start": 7.839, "duration": 2.88, "text": "state"}, {"start": 9.04, "duration": 3.679, "text": "the law of reflection states that the"}, {"start": 10.719, "duration": 5.761, "text": "angle of incidence"}, {"start": 12.719, "duration": 5.601, "text": "is equal to the angle of reflection so"}, {"start": 16.48, "duration": 4.559, "text": "as an example let's say if we have a"}, {"start": 18.32, "duration": 2.719, "text": "flat surface"}, {"start": 21.119, "duration": 6.0, "text": "and let's say this is the normal line"}, {"start": 22.72, "duration": 6.719, "text": "which is perpendicular to the surface"}, {"start": 27.119, "duration": 4.32, "text": "and let's say if we have a light ray"}, {"start": 29.439, "duration": 4.8, "text": "that strikes the surface"}, {"start": 31.439, "duration": 4.64, "text": "at an angle of incidence of 30 degrees"}, {"start": 34.239, "duration": 5.681, "text": "the angle of incidence is between the"}, {"start": 36.079, "duration": 3.841, "text": "light ray and the normal line"}, {"start": 40.559, "duration": 4.081, "text": "according to the law of reflection"}, {"start": 43.04, "duration": 2.96, "text": "the ray is going to bounce back at the"}, {"start": 44.64, "duration": 4.079, "text": "same angle"}, {"start": 46.0, "duration": 4.64, "text": "so the angle of reflection which is also"}, {"start": 48.719, "duration": 4.881, "text": "between the normal line and the light"}, {"start": 50.64, "duration": 6.559, "text": "ray that's going to be 30 degrees"}, {"start": 53.6, "duration": 3.599, "text": "and so that's the law of reflection"}, {"start": 57.36, "duration": 3.92, "text": "now keep in mind these two angles are"}, {"start": 59.44, "duration": 3.599, "text": "complementary"}, {"start": 61.28, "duration": 4.239, "text": "the normal line always forms a right"}, {"start": 63.039, "duration": 4.641, "text": "angle with the surface"}, {"start": 65.519, "duration": 4.481, "text": "so this angle is 60 and this angle is"}, {"start": 67.68, "duration": 4.64, "text": "60."}, {"start": 70.0, "duration": 4.799, "text": "now let's work on this problem a light"}, {"start": 72.32, "duration": 5.04, "text": "ray strikes the mirror at an angle of 30"}, {"start": 74.799, "duration": 4.96, "text": "degrees from the horizontal"}, {"start": 77.36, "duration": 4.399, "text": "what is the angle x where the ray leaves"}, {"start": 79.759, "duration": 3.601, "text": "the second mirror"}, {"start": 81.759, "duration": 4.321, "text": "so here this is the first mirror and"}, {"start": 83.36, "duration": 5.2, "text": "this is the second mirror"}, {"start": 86.08, "duration": 4.24, "text": "so this is the incident ray"}, {"start": 88.56, "duration": 3.44, "text": "and let's start by calculating the angle"}, {"start": 90.32, "duration": 3.52, "text": "of incidence"}, {"start": 92.0, "duration": 3.6, "text": "so we know that the normal line makes a"}, {"start": 93.84, "duration": 3.04, "text": "90 degree angle with"}, {"start": 95.6, "duration": 3.68, "text": "the surface"}, {"start": 96.88, "duration": 5.12, "text": "so these two angles add up to 90."}, {"start": 99.28, "duration": 4.879, "text": "so 90 minus 30 is 60"}, {"start": 102.0, "duration": 4.32, "text": "and so that's the angle of incidence"}, {"start": 104.159, "duration": 4.161, "text": "and based on the law of reflection the"}, {"start": 106.32, "duration": 3.759, "text": "angle of incidence is equal to the angle"}, {"start": 108.32, "duration": 3.439, "text": "of reflection"}, {"start": 110.079, "duration": 3.441, "text": "so the angle of reflection for the first"}, {"start": 111.759, "duration": 4.241, "text": "mirror is also 60"}, {"start": 113.52, "duration": 4.959, "text": "which means this is 30."}, {"start": 116.0, "duration": 5.439, "text": "now the three angles of a triangle"}, {"start": 118.479, "duration": 4.561, "text": "must add to 180"}, {"start": 121.439, "duration": 4.081, "text": "so what is this missing angle which"}, {"start": 123.04, "duration": 6.8, "text": "we'll call y"}, {"start": 125.52, "duration": 4.32, "text": "well 30 plus 130"}, {"start": 130.959, "duration": 3.681, "text": "plus y"}, {"start": 132.64, "duration": 5.2, "text": "is equal to 180"}, {"start": 134.64, "duration": 4.8, "text": "now 30 plus 130 is 160"}, {"start": 137.84, "duration": 2.64, "text": "and so now we need to subtract both"}, {"start": 139.44, "duration": 4.2, "text": "sides"}, {"start": 140.48, "duration": 3.16, "text": "by 160"}, {"start": 145.28, "duration": 5.36, "text": "so 180 minus 160 is 20"}, {"start": 148.16, "duration": 4.56, "text": "and so that's the missing angle here"}, {"start": 150.64, "duration": 5.28, "text": "so the angle of incidence for the second"}, {"start": 152.72, "duration": 4.4, "text": "mirror is going to be 90 minus 20 which"}, {"start": 155.92, "duration": 3.12, "text": "is 70."}, {"start": 157.12, "duration": 4.08, "text": "and the angle of reflection for the"}, {"start": 159.04, "duration": 5.839, "text": "second row is 70 as well"}, {"start": 161.2, "duration": 5.44, "text": "which means x is 90 minus 70 so x"}, {"start": 164.879, "duration": 3.961, "text": "is 20 degrees"}, {"start": 166.64, "duration": 3.84, "text": "so that's the angle"}, {"start": 168.84, "duration": 3.399, "text": "that"}, {"start": 170.48, "duration": 4.88, "text": "the reflected ray"}, {"start": 172.239, "duration": 3.121, "text": "leaves the second mirror"}, {"start": 175.84, "duration": 3.92, "text": "so the angle of reflection for the"}, {"start": 177.36, "duration": 3.92, "text": "second mirror is 70 but the angle at"}, {"start": 179.76, "duration": 5.08, "text": "which it leaves the mirror is 20 which"}, {"start": 181.28, "duration": 3.56, "text": "is what we're looking for"}, {"start": 203.28, "duration": 2.08, "text": "you"}], "title": "Law of Reflection - Geometric Optics - Physics", "description": "This physics video tutorial provides a basic introduction into the law of reflection.  The law of reflection states that the angle of incidence is equal to the angle of reflection.  This video on optics contain 1 practice problem associated with the law of reflection.\n\nRC Circuits - Capacitors & Resistors:\nhttps://www.youtube.com/watch?v=PLQrPqYlPmI\n\nRL Circuits - Inductors & Resistors:\nhttps://www.youtube.com/watch?v=RR0sQeDAxgk\n\nInductive Reactance:\nhttps://www.youtube.com/watch?v=5uftqyvUKSc\n\nCapacitive Reactance:\nhttps://www.youtube.com/watch?v=ZOn0L42cyFE\n\nImpedance:\nhttps://www.youtube.com/watch?v=8D9XPDNY3Mk\n\n_____________________________\nSeries RLC Circuits:\nhttps://www.youtube.com/watch?v=2GvqQvohP2k\n\nParallel RLC Circuits:\nhttps://www.youtube.com/watch?v=8MMzeeHNjIw\n\nResonant Frequency of LC Circuits:\nhttps://www.youtube.com/watch?v=r72f0ZZusT0\n\nAC Circuits - Review:\nhttps://www.youtube.com/watch?v=tcP9c6SnGQs\n\nMaxwell's Equations & EM Waves:\nhttps://www.youtube.com/watch?v=mOEFTX9DAEw\n\n____________________________\nLaw of Reflection - Geometric Optics:\nhttps://www.youtube.com/watch?v=yscrBkfPRyc\n\nThe Mirror Equation:\nhttps://www.youtube.com/watch?v=efPZ5uSDeuI\n\nRefraction of Light:\nhttps://www.youtube.com/watch?v=ON1QGqB6vxg\n\nFinal Exams and Video Playlists:   \nhttps://www.video-tutor.net/\n\nFull-Length Videos and Worksheets:\nhttps://www.patreon.com/MathScienceTutor/collections", "lengthSeconds": 204, "width": 640, "height": 360}
{"video_id": "sZpdS4339wc", "transcript": [{"start": 0.08, "duration": 6.04, "text": "go for it this one's grenard reagent"}, {"start": 2.08, "duration": 6.88, "text": "beautiful I love this one we reduce down"}, {"start": 6.12, "duration": 6.16, "text": "the carbonal group to an alcohol and we"}, {"start": 8.96, "duration": 6.32, "text": "add this group at the beginning"}, {"start": 12.28, "duration": 3.0, "text": "done"}], "title": "Converting a Ketone into an Alcohol Using a Grignard Reaction #organicchemistry", "description": "\ud83d\udcaf Start Understanding Ochem Today for Free! \ud83c\udf89\nhttps://chemmunity.info/youtube\n\n\ud83d\udc81\u200d\u2640\ufe0f HI I'M MELISSA MARIBEL\nI help students pass Chemistry and Organic Chemistry. I used to struggle with this subject, so when I finally graduated with a bachelor's degree in Chemistry, I became a tutor so that you wouldn't have to struggle like I did. I know that with the right help, YOU CAN LEARN ANYTHING!", "lengthSeconds": 13, "width": 360, "height": 640}
{"video_id": "FFm-zaFW_X4", "transcript": [{"start": 0.799, "duration": 2.88, "text": "in this lesson"}, {"start": 2.08, "duration": 5.92, "text": "we're going to talk about evaluating"}, {"start": 3.679, "duration": 7.92, "text": "logs using the change of base formula"}, {"start": 8.0, "duration": 6.0, "text": "so we know that log base 2 of 16"}, {"start": 11.599, "duration": 4.801, "text": "is equal to 4 because 2 to the fourth"}, {"start": 14.0, "duration": 5.76, "text": "power is 16."}, {"start": 16.4, "duration": 4.719, "text": "but let's say if you don't know that"}, {"start": 19.76, "duration": 4.24, "text": "here's how you can use the change of"}, {"start": 21.119, "duration": 5.441, "text": "base formula to get the same answer"}, {"start": 24.0, "duration": 4.64, "text": "but here's the formula log"}, {"start": 26.56, "duration": 4.639, "text": "base a of b"}, {"start": 28.64, "duration": 5.2, "text": "is equal to log of b"}, {"start": 31.199, "duration": 4.001, "text": "divided by log of a"}, {"start": 33.84, "duration": 3.52, "text": "and the reason why it's called the"}, {"start": 35.2, "duration": 4.32, "text": "change of base formula is because"}, {"start": 37.36, "duration": 4.719, "text": "you can change the base"}, {"start": 39.52, "duration": 5.519, "text": "originally the base is a"}, {"start": 42.079, "duration": 4.48, "text": "but you can make it into any new base"}, {"start": 45.039, "duration": 4.961, "text": "we'll call it c"}, {"start": 46.559, "duration": 5.281, "text": "c could be 5 8 12."}, {"start": 50.0, "duration": 4.079, "text": "as long as these two are the same it"}, {"start": 51.84, "duration": 5.28, "text": "doesn't really matter what c is"}, {"start": 54.079, "duration": 3.041, "text": "the equation will work"}, {"start": 57.36, "duration": 4.96, "text": "so log base 2 of 16"}, {"start": 60.559, "duration": 3.521, "text": "you can type it in your calculator as"}, {"start": 62.32, "duration": 4.159, "text": "log 16"}, {"start": 64.08, "duration": 3.84, "text": "divided by log 2."}, {"start": 66.479, "duration": 2.64, "text": "now when you type it in your calculator"}, {"start": 67.92, "duration": 3.04, "text": "like this"}, {"start": 69.119, "duration": 4.721, "text": "it's automatically going to make it base"}, {"start": 70.96, "duration": 2.88, "text": "10."}, {"start": 73.92, "duration": 2.559, "text": "so regardless of whatever the new base"}, {"start": 75.68, "duration": 2.64, "text": "is"}, {"start": 76.479, "duration": 4.161, "text": "you will still get the right answer so"}, {"start": 78.32, "duration": 3.6, "text": "go ahead and type that in log 16 divided"}, {"start": 80.64, "duration": 6.519, "text": "by log 2"}, {"start": 81.92, "duration": 5.239, "text": "and you'll see that is equal to 4."}, {"start": 87.52, "duration": 4.08, "text": "try these two examples"}, {"start": 89.6, "duration": 5.6, "text": "use the change of base formula"}, {"start": 91.6, "duration": 3.6, "text": "to evaluate the two logs"}, {"start": 99.36, "duration": 4.399, "text": "so let's divide log 243"}, {"start": 102.799, "duration": 3.521, "text": "by"}, {"start": 103.759, "duration": 2.561, "text": "log 3."}, {"start": 110.24, "duration": 3.839, "text": "the calculator will give you 5."}, {"start": 112.56, "duration": 5.199, "text": "and it makes sense because 3 to the"}, {"start": 114.079, "duration": 6.961, "text": "fifth power is 243"}, {"start": 117.759, "duration": 7.241, "text": "next we have log of 343"}, {"start": 121.04, "duration": 3.96, "text": "divided by log 7."}, {"start": 129.119, "duration": 3.2, "text": "and this"}, {"start": 130.479, "duration": 3.601, "text": "is equal to three"}, {"start": 132.319, "duration": 4.801, "text": "so that's how you can use the change of"}, {"start": 134.08, "duration": 6.04, "text": "base formula in order to evaluate"}, {"start": 137.12, "duration": 3.0, "text": "logs"}, {"start": 158.72, "duration": 2.08, "text": "you"}], "title": "Change of Base Formula - Logarithms", "description": "This algebra video tutorial explains how to use the change of base formula to evaluate logarithms.  \n\nLogarithms - Free Formula Sheet:       https://bit.ly/4fEwqv0\n\n_______________________________\nLogarithms - The Easy Way!                              \nhttps://www.youtube.com/watch?v=kqVpPSzkTYA\n\nLog to Exponential Form:                                    \nhttps://www.youtube.com/watch?v=f0C1KL7GkqY\n\nChange of Base Formula:                                   \nhttps://www.youtube.com/watch?v=FFm-zaFW_X4\n\nChange of Base Log Problem:                           \nhttps://www.youtube.com/watch?v=p7hD9VdXv9U\n\nProperties of Logarithms:                                   \nhttps://www.youtube.com/watch?v=Jtv9Lnf7Zw8\n\n____________________________________\nExpanding Logarithmic Expressions:               \nhttps://www.youtube.com/watch?v=OIz-5MyJA3g\n\nCondensing Logarithmic Expressions:             \nhttps://www.youtube.com/watch?v=luRrOlsB4cY\n\nNatural Logarithms:                                             \nhttps://www.youtube.com/watch?v=daUlTsnCNRQ\n\nSolving Exponential Equations:                          \nhttps://www.youtube.com/watch?v=9tutJ5xrRwg\n\nExponential Equations - Quadratic Form:         \nhttps://www.youtube.com/watch?v=yNgmVu0R_T8\n\n_______________________________________\nSolving Logarithmic Equations:                         \nhttps://www.youtube.com/watch?v=fnhFneOz6n8\n\nGraphing Logarithmic Functions:                      \nhttps://www.youtube.com/watch?v=-nptxS9rZNA\n\nGraphing Exponential Functions:                       \nhttps://www.youtube.com/watch?v=DASfP8KAyvs\n\nCompound Interest Word Problems:                \nhttps://www.youtube.com/watch?v=Hn0eLcOSQGw\n\nLogarithms Practice Problems:                         \nhttps://www.youtube.com/watch?v=7DVbQKI600k\n\n_______________________________________\nFinal Exams and Video Playlists:   \nhttps://www.video-tutor.net/\n\nFull-Length Videos and Worksheets:\nhttps://www.patreon.com/MathScienceTutor/collections", "lengthSeconds": 159, "width": 640, "height": 360}
{"video_id": "uYNKJWFYh7k", "transcript": [{"start": 1.12, "duration": 3.76, "text": "now let's work on some more examples"}, {"start": 2.72, "duration": 4.4, "text": "graphing quadratic functions in standard"}, {"start": 4.88, "duration": 2.24, "text": "form"}, {"start": 7.839, "duration": 4.161, "text": "now we did one example"}, {"start": 9.519, "duration": 3.681, "text": "using the data table"}, {"start": 12.0, "duration": 3.28, "text": "and there are some other things that you"}, {"start": 13.2, "duration": 3.919, "text": "may need to find such as the x-intercept"}, {"start": 15.28, "duration": 3.6, "text": "and the y-intercept"}, {"start": 17.119, "duration": 3.92, "text": "but i think it's still good to like put"}, {"start": 18.88, "duration": 4.639, "text": "that information in the form of a data"}, {"start": 21.039, "duration": 3.761, "text": "table it's simply helpful"}, {"start": 23.519, "duration": 2.641, "text": "the first thing we should do is find a"}, {"start": 24.8, "duration": 2.559, "text": "vertex"}, {"start": 26.16, "duration": 2.68, "text": "which is"}, {"start": 27.359, "duration": 4.72, "text": "negative b over"}, {"start": 28.84, "duration": 5.32, "text": "2a b is negative 4."}, {"start": 32.079, "duration": 4.721, "text": "a is the number in front of x squared so"}, {"start": 34.16, "duration": 4.719, "text": "a is 1."}, {"start": 36.8, "duration": 3.759, "text": "so this is going to be 2 times 1. so"}, {"start": 38.879, "duration": 2.801, "text": "that's going to be positive 4 divided by"}, {"start": 40.559, "duration": 2.641, "text": "2."}, {"start": 41.68, "duration": 3.92, "text": "so the vertex"}, {"start": 43.2, "duration": 6.48, "text": "has an x value of 2."}, {"start": 45.6, "duration": 4.08, "text": "let's just put that in the data table"}, {"start": 50.879, "duration": 4.16, "text": "now let's find a y value when x is 2."}, {"start": 56.879, "duration": 4.081, "text": "so two squared minus four times two plus"}, {"start": 59.6, "duration": 4.0, "text": "three"}, {"start": 60.96, "duration": 4.8, "text": "so that's four minus eight plus three"}, {"start": 63.6, "duration": 4.24, "text": "four minus eight is negative four plus"}, {"start": 65.76, "duration": 6.359, "text": "three that's negative one"}, {"start": 67.84, "duration": 4.279, "text": "so that's the y value of the vertex"}, {"start": 73.28, "duration": 4.96, "text": "now let's find the x-intercepts in order"}, {"start": 75.84, "duration": 6.16, "text": "to find the x-intercepts"}, {"start": 78.24, "duration": 5.519, "text": "set y equal to zero and factor"}, {"start": 82.0, "duration": 3.36, "text": "two numbers that multiply to three but"}, {"start": 83.759, "duration": 3.921, "text": "add to negative four"}, {"start": 85.36, "duration": 4.56, "text": "are negative one and negative three"}, {"start": 87.68, "duration": 4.32, "text": "so it's going to be x minus one"}, {"start": 89.92, "duration": 4.0, "text": "and x minus three therefore the"}, {"start": 92.0, "duration": 3.36, "text": "x-intercepts are one"}, {"start": 93.92, "duration": 4.48, "text": "and three"}, {"start": 95.36, "duration": 5.759, "text": "which means x is one y is zero"}, {"start": 98.4, "duration": 5.039, "text": "so we have the points one zero and three"}, {"start": 101.119, "duration": 2.32, "text": "zero"}, {"start": 106.159, "duration": 5.6, "text": "now let's find the y intercept"}, {"start": 108.64, "duration": 6.72, "text": "to do that replace x with zero"}, {"start": 111.759, "duration": 3.601, "text": "so you can see that it's going to be 3."}, {"start": 115.68, "duration": 4.56, "text": "so we have the point 0"}, {"start": 117.68, "duration": 3.84, "text": "comma 3."}, {"start": 120.24, "duration": 2.479, "text": "notice that these two are the same"}, {"start": 121.52, "duration": 2.48, "text": "because they're equidistant from the"}, {"start": 122.719, "duration": 5.04, "text": "vertex"}, {"start": 124.0, "duration": 5.44, "text": "so if we have the .03 then when x is 4 y"}, {"start": 127.759, "duration": 4.321, "text": "must be 3 as well"}, {"start": 129.44, "duration": 5.2, "text": "because 0 and 4 are equidistant from the"}, {"start": 132.08, "duration": 2.56, "text": "vertex"}, {"start": 134.959, "duration": 5.28, "text": "so now we have everything that we need"}, {"start": 136.879, "duration": 3.36, "text": "in order to graph this function"}, {"start": 141.599, "duration": 2.801, "text": "so let's plot it"}, {"start": 148.959, "duration": 3.28, "text": "starting with the vertex it's 2 negative"}, {"start": 150.959, "duration": 4.0, "text": "1"}, {"start": 152.239, "duration": 4.481, "text": "and then the x-intercepts are 3 0"}, {"start": 154.959, "duration": 4.961, "text": "and 1 0 that's where it touches the"}, {"start": 156.72, "duration": 5.44, "text": "x-axis the y-intercept is 0 3 that's"}, {"start": 159.92, "duration": 4.959, "text": "where the curve touches the y-axis and"}, {"start": 162.16, "duration": 4.0, "text": "we have the 0.43"}, {"start": 164.879, "duration": 2.64, "text": "and so"}, {"start": 166.16, "duration": 5.68, "text": "the graph"}, {"start": 167.519, "duration": 4.321, "text": "is going to look something like this"}, {"start": 173.599, "duration": 3.761, "text": "we can see that the axis of symmetry"}, {"start": 178.239, "duration": 4.561, "text": "is x is equal to 2"}, {"start": 180.4, "duration": 4.559, "text": "the x coordinate of the vertex"}, {"start": 182.8, "duration": 4.719, "text": "the minimum value"}, {"start": 184.959, "duration": 4.481, "text": "is negative one"}, {"start": 187.519, "duration": 5.041, "text": "and so the range"}, {"start": 189.44, "duration": 4.56, "text": "will vary from negative one to infinity"}, {"start": 192.56, "duration": 2.24, "text": "and the domain is always going to be the"}, {"start": 194.0, "duration": 3.44, "text": "same"}, {"start": 194.8, "duration": 4.96, "text": "all real numbers"}, {"start": 197.44, "duration": 4.0, "text": "and so that's it for"}, {"start": 199.76, "duration": 5.119, "text": "graphing functions"}, {"start": 201.44, "duration": 3.439, "text": "in standard form"}], "title": "Graphing Quadratic Functions In Standard Form Using X & Y Intercepts | Algebra", "description": "This algebra video tutorial explains how to graph quadratic functions in standard form using x and y intercepts and a data table.  it explains how to identify the axis of symmetry and the coordinates of the vertex using a simple process as well as the domain and range of the function.\n\nQuadratic Equations - Free Formula Sheet:     \nhttps://bit.ly/3WZ8v1Z\n\n______________________________\nHow To Solve Simple Quadratic Equations:      \nhttps://www.youtube.com/watch?v=-KWsS2FZVTA\n\nSolving Quadratic Equations by Factoring:       \nhttps://www.youtube.com/watch?v=qeByhTF8WEw\n\nHow To Factor Difficult Quadratic Equations:   \nhttps://www.youtube.com/watch?v=VG1luiL_INQ\n\nSolving QE By Completing the Square:              \nhttps://www.youtube.com/watch?v=WOb25i8FBBw\n\nSolving Equations In Quadratic Form:                \nhttps://www.youtube.com/watch?v=eWjixwGK3YM\n\nSum and Products - Quadratic Equations:\nhttps://www.youtube.com/watch?v=kBSj35rik8w\n\n_______________________________________\nHow To Calculate the Discriminant:                  \nhttps://www.youtube.com/watch?v=6fU9ZgwbU74\n\nStandard Form to Vertex Form:                           \nhttps://www.youtube.com/watch?v=pf9LkX8hpTQ\n\nGraphing Quadratic Functions:                            \nhttps://www.youtube.com/watch?v=H8OQtxSRe_k\n\nHow To Find the Axis of Symmetry:                   \nhttps://www.youtube.com/watch?v=tyzd4VujaqY\n\nGraphing Quadratic Functions - Vertex Form:   \nhttps://www.youtube.com/watch?v=OHH7fX_M8Ns \n\nGraphing QF in Standard Form:                           \nhttps://www.youtube.com/watch?v=uYNKJWFYh7k\n\n___________________________________________\nWriting Quadratic Equations:                               \nhttps://www.youtube.com/watch?v=SB6eCyG4GEA\n\nMaximum and Minimum Word Problems:         \nhttps://www.youtube.com/watch?v=d5YktqR-4FU\n\nSystems of Quadratic Equations:                        \nhttps://www.youtube.com/watch?v=rq5AjFs95qw\n\nSystems of Nonlinear Equations:                        \nhttps://www.youtube.com/watch?v=JPdLEaxlnZQ\n\nQuadratic Inequalities:                                          \nhttps://www.youtube.com/watch?v=_gWjLKsFOPE\n\nQuadratic Equations - Test Review:                    \nhttps://www.youtube.com/watch?v=fFFA7Q4eVuY\n\n___________________________________________\nFinal Exams and Video Playlists:   \nhttps://www.video-tutor.net/\n\nFull-Length Videos and Worksheets:\nhttps://www.patreon.com/MathScienceTutor/collections", "lengthSeconds": 203, "width": 640, "height": 360}
{"video_id": "_-lVFKG5_08", "transcript": [{"start": 1.199, "duration": 3.841, "text": "what is equity"}, {"start": 3.6, "duration": 4.72, "text": "have you ever wondered the answer to"}, {"start": 5.04, "duration": 5.92, "text": "that question what is it perhaps in a"}, {"start": 8.32, "duration": 5.359, "text": "conversation you heard someone say yeah"}, {"start": 10.96, "duration": 3.96, "text": "man i have a 150k in equity"}, {"start": 13.679, "duration": 3.281, "text": "what does that even"}, {"start": 14.92, "duration": 3.48, "text": "mean well in this video we're going to"}, {"start": 16.96, "duration": 4.079, "text": "talk about that"}, {"start": 18.4, "duration": 4.56, "text": "so let's say you decide to buy a house"}, {"start": 21.039, "duration": 3.361, "text": "for 300 000"}, {"start": 22.96, "duration": 3.52, "text": "now we're going to keep this discussion"}, {"start": 24.4, "duration": 4.719, "text": "simple we're not going to go into all of"}, {"start": 26.48, "duration": 4.799, "text": "the the details and the fees of owning a"}, {"start": 29.119, "duration": 4.321, "text": "home but let's say that's the current"}, {"start": 31.279, "duration": 4.001, "text": "value of your home"}, {"start": 33.44, "duration": 5.6, "text": "and you took out a loan or mortgage for"}, {"start": 35.28, "duration": 5.76, "text": "three hundred thousand to buy the home"}, {"start": 39.04, "duration": 4.16, "text": "equity"}, {"start": 41.04, "duration": 3.44, "text": "is the difference"}, {"start": 43.2, "duration": 2.56, "text": "between"}, {"start": 44.48, "duration": 4.239, "text": "assets"}, {"start": 45.76, "duration": 5.119, "text": "and liabilities"}, {"start": 48.719, "duration": 4.401, "text": "your home is an asset because"}, {"start": 50.879, "duration": 5.2, "text": "you can go to the market and sell it in"}, {"start": 53.12, "duration": 4.88, "text": "exchange for cash it has a value"}, {"start": 56.079, "duration": 5.441, "text": "the loan is a liability because it's"}, {"start": 58.0, "duration": 7.68, "text": "something you have to pay back"}, {"start": 61.52, "duration": 4.16, "text": "so right now your equity is zero"}, {"start": 66.159, "duration": 4.96, "text": "because these two are the same"}, {"start": 68.4, "duration": 5.12, "text": "but let's say in five years"}, {"start": 71.119, "duration": 4.481, "text": "the value of your home appreciates"}, {"start": 73.52, "duration": 3.68, "text": "to four hundred thousand"}, {"start": 75.6, "duration": 3.36, "text": "so that means that you've gained a"}, {"start": 77.2, "duration": 4.239, "text": "hundred thousand dollars"}, {"start": 78.96, "duration": 4.479, "text": "in appreciation or in value"}, {"start": 81.439, "duration": 4.401, "text": "now in the course of those five years"}, {"start": 83.439, "duration": 4.561, "text": "let's say that you pay down the mortgage"}, {"start": 85.84, "duration": 4.48, "text": "from three hundred thousand"}, {"start": 88.0, "duration": 4.799, "text": "to two hundred ten thousand"}, {"start": 90.32, "duration": 4.0, "text": "how much equity do you now have in your"}, {"start": 92.799, "duration": 3.36, "text": "home"}, {"start": 94.32, "duration": 2.88, "text": "as was mentioned before"}, {"start": 96.159, "duration": 3.681, "text": "equity"}, {"start": 97.2, "duration": 5.44, "text": "is the difference between assets and"}, {"start": 99.84, "duration": 2.8, "text": "liabilities"}, {"start": 102.96, "duration": 5.199, "text": "you have an asset that is four hundred"}, {"start": 106.0, "duration": 4.399, "text": "thousand dollars in value"}, {"start": 108.159, "duration": 3.6, "text": "your liability is two hundred ten"}, {"start": 110.399, "duration": 2.961, "text": "thousand"}, {"start": 111.759, "duration": 2.881, "text": "equity will be the difference between"}, {"start": 113.36, "duration": 3.28, "text": "those two values"}, {"start": 114.64, "duration": 3.519, "text": "so four hundred thousand minus two"}, {"start": 116.64, "duration": 4.079, "text": "hundred ten thousand"}, {"start": 118.159, "duration": 2.56, "text": "will give us"}, {"start": 120.88, "duration": 4.8, "text": "an equity of a hundred ninety thousand"}, {"start": 123.68, "duration": 4.719, "text": "so that's how much equity you would have"}, {"start": 125.68, "duration": 4.4, "text": "in your home so what does that even mean"}, {"start": 128.399, "duration": 4.081, "text": "well it means that if you liquidate your"}, {"start": 130.08, "duration": 3.68, "text": "asset if you sell your home for four"}, {"start": 132.48, "duration": 2.16, "text": "hundred thousand"}, {"start": 133.76, "duration": 2.72, "text": "and"}, {"start": 134.64, "duration": 3.76, "text": "pay all of your mortgage for two hundred"}, {"start": 136.48, "duration": 3.2, "text": "ten thousand"}, {"start": 138.4, "duration": 2.16, "text": "this is how much cash you're going to"}, {"start": 139.68, "duration": 3.199, "text": "have"}, {"start": 140.56, "duration": 4.96, "text": "at the end of those transactions"}, {"start": 142.879, "duration": 5.36, "text": "so equity tells you how much value you"}, {"start": 145.52, "duration": 4.32, "text": "have in your home after paying off all"}, {"start": 148.239, "duration": 3.441, "text": "of your liabilities"}, {"start": 149.84, "duration": 4.0, "text": "that's the basic concept of it so"}, {"start": 151.68, "duration": 4.32, "text": "hopefully this video helped you to"}, {"start": 153.84, "duration": 5.679, "text": "understand the concept of equity as it"}, {"start": 156.0, "duration": 3.519, "text": "relates to your home"}], "title": "Personal Finance - Assets, Liabilities, & Equity", "description": "This video explains what it means to have equity in your home.  Equity is the difference between assets and liabilities.\n\nLiquidity Ratios:\nhttps://www.youtube.com/watch?v=bmZVPks4yQg\n\nAssets, Liabilities, & Equity:\nhttps://www.youtube.com/watch?v=_-lVFKG5_08\n\nThe Short Ratio:\nhttps://www.youtube.com/watch?v=9kQQ4iEdxiU\n\nDebt to Equity Ratio:\nhttps://www.youtube.com/watch?v=enhdH7Tl5bY\n\nTime Value of Money:\nhttps://www.youtube.com/watch?v=cy4PiY5ERTI\n\n_____________________________\nFuture Value of Annuity:\nhttps://www.youtube.com/watch?v=bcuXY8WkjF4\n\nMore Examples on Annuities:\nhttps://www.youtube.com/watch?v=bIIndFa0RCM\n\nHow To Calculate The Monthly Mortgage Payment:\nhttps://www.youtube.com/watch?v=6bLg_Ex0A-4\n\nThe Present Value of an Annuity:\nhttps://www.youtube.com/watch?v=RU-osjAs6hE\n\nNet Present Value & Internal Rate of Return:\nhttps://www.youtube.com/watch?v=pj5wN7TUdOY\n\n____________________________\nAnnual Percentage Yield:\nhttps://www.youtube.com/watch?v=qGR3kMH0TX0\n\nAmortization Loan Formula:\nhttps://www.youtube.com/watch?v=lkNJvsy0qU8\n\nAmortization Table:\nhttps://www.youtube.com/watch?v=QZfMW203v4U\n\nRule of 72:\nhttps://www.youtube.com/watch?v=91nnN2XwTMg\n\nBond Yields:\nhttps://www.youtube.com/watch?v=OnkmoSTeHuc", "lengthSeconds": 158, "width": 640, "height": 360}
{"video_id": "bpRsuccb6dI", "transcript": [{"start": 1.68, "duration": 4.48, "text": "in this lesson we're going to talk about"}, {"start": 3.36, "duration": 6.36, "text": "how to calculate the volume"}, {"start": 6.16, "duration": 3.56, "text": "of a cube"}, {"start": 20.24, "duration": 3.36, "text": "so let's say the side length is"}, {"start": 22.48, "duration": 4.48, "text": "seven"}, {"start": 23.6, "duration": 3.36, "text": "what is the volume of this cube"}, {"start": 27.359, "duration": 3.601, "text": "now we know that the volume of a"}, {"start": 28.8, "duration": 4.64, "text": "rectangular prism is the length times"}, {"start": 30.96, "duration": 4.24, "text": "the width times the height"}, {"start": 33.44, "duration": 3.36, "text": "however"}, {"start": 35.2, "duration": 3.92, "text": "for a cube"}, {"start": 36.8, "duration": 5.04, "text": "all sides are the same"}, {"start": 39.12, "duration": 4.16, "text": "so this is x x and x"}, {"start": 41.84, "duration": 4.559, "text": "so it turns out that the volume of the"}, {"start": 43.28, "duration": 5.04, "text": "cube is just x cubed so in this example"}, {"start": 46.399, "duration": 4.721, "text": "is going to be seven to the third power"}, {"start": 48.32, "duration": 5.759, "text": "or seven times seven times seven"}, {"start": 51.12, "duration": 4.8, "text": "seven to the third power is 343."}, {"start": 54.079, "duration": 3.441, "text": "so this was seven inches this would be"}, {"start": 55.92, "duration": 2.88, "text": "343"}, {"start": 57.52, "duration": 2.559, "text": "cubic inches"}, {"start": 58.8, "duration": 5.48, "text": "and that's all you need to do in order"}, {"start": 60.079, "duration": 4.201, "text": "to calculate the volume of a cube"}, {"start": 82.96, "duration": 2.08, "text": "you"}], "title": "Volume of a Cube", "description": "This geometry video tutorial explains how to calculate the volume of a cube.\n\n3D Shapes - Faces, Edges, & Vertices:     https://www.youtube.com/watch?v=Wmwe1fmR1SM\n\nRectangular Prism - Volume:                     \nhttps://www.youtube.com/watch?v=rNIKbT0-ywU\n\nRectangular Prism - Surface Area:            \nhttps://www.youtube.com/watch?v=BsAQbJyEF9g\n\nRectangular Prism - Diagonal Length:      \nhttps://www.youtube.com/watch?v=iO7vb3HLHqU\n\nTriangular Prism - Volume:                         \nhttps://www.youtube.com/watch?v=HfDHYFgTbxk\n\nTriangular Prism - Surface Area:               \nhttps://www.youtube.com/watch?v=152tLGR0lms\n\nTriangular Prism - Lateral Area:                 \nhttps://www.youtube.com/watch?v=PljECBkHQpg\n\n_______________________________\nHexagonal Prism - Surface Area:              \nhttps://www.youtube.com/watch?v=JKyF2r-bIC0\n\nCube - Volume:                                             \nhttps://www.youtube.com/watch?v=bpRsuccb6dI\n\nCube - Surface Area:                                   \nhttps://www.youtube.com/watch?v=ZMhkvFwLOR4\n\nCube - Diagonal Length:                             \nhttps://www.youtube.com/watch?v=wr3EvQZzoRI\n\nPyramid - Volume:                                       \nhttps://www.youtube.com/watch?v=8onylJ72ZPQ\n\nPyramid - Surface Area:                              \nhttps://www.youtube.com/watch?v=2g6QuOizzac\n\n________________________________\nSquare Pyramids - Volume:                       \nhttps://www.youtube.com/watch?v=x8wEnG4GURQ\n\nCylinder - Surface Area:                              \nhttps://www.youtube.com/watch?v=LdztqPczYiI\n\nSphere - Volume:                                         \nhttps://www.youtube.com/watch?v=eAwc8rwMRnI\n\nSphere - Surface Area:                                \nhttps://www.youtube.com/watch?v=aXEcU-M_qNA\n\nCone - Volume:                                             \nhttps://www.youtube.com/watch?v=LypmeZ0tnJA\n\nCone - Surface Area:                                   \nhttps://www.youtube.com/watch?v=MeWoIcQNqRA\n\n_________________________________\nFinal Exams and Video Playlists:   \nhttps://www.video-tutor.net/", "lengthSeconds": 83, "width": 640, "height": 360}
{"video_id": "aof9QBID8c4", "transcript": [{"start": 1.5, "duration": 6.06, "text": "number 84 the chemical structure of"}, {"start": 4.799, "duration": 5.281, "text": "inositol formerly known as the vitamin"}, {"start": 7.56, "duration": 4.92, "text": "B8 is shown below"}, {"start": 10.08, "duration": 4.82, "text": "draw a chair conformation of this"}, {"start": 12.48, "duration": 2.42, "text": "molecule"}, {"start": 17.58, "duration": 4.82, "text": "so let's"}, {"start": 19.44, "duration": 2.96, "text": "draw it this way"}, {"start": 27.3, "duration": 8.88, "text": "now let's call this"}, {"start": 30.599, "duration": 7.261, "text": "carbon one two three four five six so"}, {"start": 36.18, "duration": 4.5, "text": "this is going to be one"}, {"start": 37.86, "duration": 5.82, "text": "I'm going to put 2 here"}, {"start": 40.68, "duration": 4.559, "text": "and then 3 4"}, {"start": 43.68, "duration": 4.1, "text": "5."}, {"start": 45.239, "duration": 2.541, "text": "6"}, {"start": 50.1, "duration": 4.86, "text": "so on carbon one we have the o h group"}, {"start": 52.92, "duration": 4.139, "text": "it's on the wedge"}, {"start": 54.96, "duration": 4.5, "text": "it's coming out of the page so"}, {"start": 57.059, "duration": 4.881, "text": "let's put the first group in the axial"}, {"start": 59.46, "duration": 2.48, "text": "up position"}, {"start": 63.84, "duration": 6.42, "text": "now on carbon two"}, {"start": 66.9, "duration": 6.68, "text": "this second oh is on the wedge as well"}, {"start": 70.26, "duration": 3.32, "text": "so it's going to be going up"}, {"start": 74.64, "duration": 5.04, "text": "so this has to be equatorial up"}, {"start": 77.76, "duration": 3.96, "text": "there's only two options equatorial up"}, {"start": 79.68, "duration": 3.72, "text": "or axial down"}, {"start": 81.72, "duration": 5.84, "text": "and these two have to be on the same"}, {"start": 83.4, "duration": 4.16, "text": "side either they're both up or both down"}, {"start": 88.14, "duration": 3.96, "text": "now in carbon 3 we have an oh going into"}, {"start": 90.9, "duration": 2.64, "text": "the page"}, {"start": 92.1, "duration": 3.3, "text": "so that's going to be going in the"}, {"start": 93.54, "duration": 4.5, "text": "downward Direction so it's not going to"}, {"start": 95.4, "duration": 5.179, "text": "be axial up but it has to be equatorial"}, {"start": 98.04, "duration": 2.539, "text": "down"}, {"start": 104.04, "duration": 5.52, "text": "now let's move on to carbon 4."}, {"start": 106.259, "duration": 5.581, "text": "so this is out of the page"}, {"start": 109.56, "duration": 3.48, "text": "and that's going to be up the same as"}, {"start": 111.84, "duration": 3.9, "text": "that one"}, {"start": 113.04, "duration": 3.66, "text": "so either it's axial down or equatorial"}, {"start": 115.74, "duration": 5.059, "text": "up"}, {"start": 116.7, "duration": 4.099, "text": "we're going to go with equatorial up"}, {"start": 122.64, "duration": 4.14, "text": "now for carbon 5 it's down"}, {"start": 125.34, "duration": 4.619, "text": "it's going"}, {"start": 126.78, "duration": 5.099, "text": "into the page so it's either axial down"}, {"start": 129.959, "duration": 3.601, "text": "or equatorial down"}, {"start": 131.879, "duration": 4.821, "text": "this is going to be"}, {"start": 133.56, "duration": 3.14, "text": "equatorial down"}, {"start": 138.9, "duration": 4.199, "text": "now this one is up"}, {"start": 140.879, "duration": 4.561, "text": "so either we have axial down or"}, {"start": 143.099, "duration": 6.081, "text": "equatorial up"}, {"start": 145.44, "duration": 3.74, "text": "so this is going to be equatorial up"}, {"start": 151.14, "duration": 4.26, "text": "so notice that"}, {"start": 152.76, "duration": 4.92, "text": "five of the six hydroxyl groups are in"}, {"start": 155.4, "duration": 4.62, "text": "the equatorial position"}, {"start": 157.68, "duration": 4.44, "text": "so the chair conformation this"}, {"start": 160.02, "duration": 4.14, "text": "particular chair conformation is"}, {"start": 162.12, "duration": 5.58, "text": "relatively stable"}, {"start": 164.16, "duration": 6.299, "text": "now if we were to take this o h"}, {"start": 167.7, "duration": 5.28, "text": "and put it in this Direction all the"}, {"start": 170.459, "duration": 4.621, "text": "other positions will be flipped we would"}, {"start": 172.98, "duration": 3.479, "text": "have five hydroxyl groups in the axle"}, {"start": 175.08, "duration": 3.78, "text": "position"}, {"start": 176.459, "duration": 4.321, "text": "rather than what we see here so this"}, {"start": 178.86, "duration": 3.36, "text": "particular Shear conformation is more"}, {"start": 180.78, "duration": 3.62, "text": "stable than the other one that we can"}, {"start": 182.22, "duration": 2.18, "text": "draw"}, {"start": 186.12, "duration": 4.56, "text": "so that's the chair conformation for"}, {"start": 189.12, "duration": 6.38, "text": "inositol"}, {"start": 190.68, "duration": 4.82, "text": "vitamin B8 or what used to be vitamin B8"}], "title": "How To Draw The Chair Conformation of Inositol", "description": "This organic chemistry video tutorial explains how to draw the chair conformation of inositol.\n\nFull 7 Hour Organic Chemistry Exam 1 Video:               https://www.patreon.com/MathScienceTutor\n\nDirect Link To The Full Video - Part 1:                             \nhttps://bit.ly/3R3KPFi\n\nDirect Link to The Full Video - Part 2:                              \nhttps://bit.ly/3XwUg2u\n\nPDF Worksheet - 90 Test Questions:                               \nhttps://bit.ly/3WFBwMU\n\n__________________________________\nExam 1 - Part 1 on YouTube:                                    \nhttps://www.youtube.com/watch?v=WfXDSOZwYyI\n\nExam 1 - Part 2 on YouTube:                                    \nhttps://www.youtube.com/watch?v=8TKVg0t6jSc\n\nOrganic Chemistry Exam 1 Playlist:                                 \nhttps://bit.ly/3kJnNXU\n\nOrganic Chemistry PDF Worksheets:                               \nhttps://www.video-tutor.net/orgo-chem.html", "lengthSeconds": 195, "width": 640, "height": 360}
{"video_id": "0tHjaOknwIw", "transcript": [{"start": 0.0, "duration": 4.56, "text": "identify which one is the acid which one"}, {"start": 2.159, "duration": 4.74, "text": "is the base"}, {"start": 4.56, "duration": 4.02, "text": "good yeah why well this is a carboxylic"}, {"start": 6.899, "duration": 3.421, "text": "acid right here yep it's added in the"}, {"start": 8.58, "duration": 5.099, "text": "name this is a negative charge perfect"}, {"start": 10.32, "duration": 5.34, "text": "and and this to basically what an acid"}, {"start": 13.679, "duration": 4.321, "text": "is specifically the bronson-lowry acid"}, {"start": 15.66, "duration": 4.379, "text": "we know that this is going to donate a"}, {"start": 18.0, "duration": 3.72, "text": "proton so it has to have a proton so if"}, {"start": 20.039, "duration": 3.541, "text": "it's going to proton again h plus right"}, {"start": 21.72, "duration": 3.84, "text": "hydrogen so it has to have a hydrogen to"}, {"start": 23.58, "duration": 4.019, "text": "give this one I know it's a base because"}, {"start": 25.56, "duration": 3.84, "text": "it doesn't have a hydrogen to give it"}, {"start": 27.599, "duration": 3.84, "text": "wants to accept a proton so that's why"}, {"start": 29.4, "duration": 4.02, "text": "yes bases tend to be negative acids"}, {"start": 31.439, "duration": 4.161, "text": "either have proton or are positively"}, {"start": 33.42, "duration": 2.18, "text": "charged"}], "title": "Identifying the acid and base in an acid-base reaction #acidsandbases #organicchemistry", "description": "", "lengthSeconds": 34, "width": 360, "height": 640}
{"video_id": "8TMcliZOu0M", "transcript": [{"start": 0.12, "duration": 2.7, "text": "this alkane kind of looks like an"}, {"start": 1.62, "duration": 3.12, "text": "octopus"}, {"start": 2.82, "duration": 3.66, "text": "the first step to naming is to find the"}, {"start": 4.74, "duration": 3.3, "text": "parent chain and number the parent"}, {"start": 6.48, "duration": 3.24, "text": "chains so the substituents are at the"}, {"start": 8.04, "duration": 3.059, "text": "lowest numbered position there's seven"}, {"start": 9.72, "duration": 2.94, "text": "carbons in the parent chain so the"}, {"start": 11.099, "duration": 3.181, "text": "ending of the name will be heptane next"}, {"start": 12.66, "duration": 3.539, "text": "means to find the substituents there's"}, {"start": 14.28, "duration": 3.66, "text": "one methyl here at position two and one"}, {"start": 16.199, "duration": 3.901, "text": "at position six and there's an ethyl"}, {"start": 17.94, "duration": 3.72, "text": "group at position four the ethyl comes"}, {"start": 20.1, "duration": 3.48, "text": "before methyl in the name because e"}, {"start": 21.66, "duration": 4.379, "text": "comes before m in the alphabet the final"}, {"start": 23.58, "duration": 4.019, "text": "name is four ethyl 2 6 dimethylheptane"}, {"start": 26.039, "duration": 3.181, "text": "thanks for watching don't forget to like"}, {"start": 27.599, "duration": 3.741, "text": "and also subscribe to the chemistry kit"}, {"start": 29.22, "duration": 2.12, "text": "for more"}], "title": "Alkane Or Octopus? Organic Chemistry Question #chemistry #shorts", "description": "This is a question going over how to name alkanes in organic chemistry.\n\nSubscribe for more chemistry tutor help!", "lengthSeconds": 30, "width": 360, "height": 640}
{"video_id": "gkF9zNccqiE", "transcript": [{"start": 1.52, "duration": 3.04, "text": "okemtutor.org question of the day"}, {"start": 3.37, "duration": 2.57, "text": "[Music]"}, {"start": 4.56, "duration": 5.94, "text": "this question we're going to do"}, {"start": 5.94, "duration": 7.44, "text": "nitration it's with hno3 h2so4 and the"}, {"start": 10.5, "duration": 6.06, "text": "Benzene ring so first let's draw the"}, {"start": 13.38, "duration": 4.62, "text": "Lewis structure of our hno3"}, {"start": 16.56, "duration": 3.78, "text": "you can see that we have a negative sign"}, {"start": 18.0, "duration": 4.859, "text": "on the oxygen neutral oxygen with the"}, {"start": 20.34, "duration": 6.72, "text": "double bond o and an o h with a couple"}, {"start": 22.859, "duration": 6.84, "text": "lone pairs next let's draw our h2so4"}, {"start": 27.06, "duration": 4.379, "text": "it's just a 4 is electrophilic so we can"}, {"start": 29.699, "duration": 4.261, "text": "go attack it"}, {"start": 31.439, "duration": 5.221, "text": "we're going to go for the proton and"}, {"start": 33.96, "duration": 5.279, "text": "redraw our hno3 which now is going to"}, {"start": 36.66, "duration": 4.079, "text": "have an oh2 plus so that's going to be"}, {"start": 39.239, "duration": 3.121, "text": "our leaving group"}, {"start": 40.739, "duration": 3.961, "text": "now that we have our leaving group and"}, {"start": 42.36, "duration": 4.379, "text": "we have our HS of four minus we can have"}, {"start": 44.7, "duration": 7.16, "text": "the H2O leave and that's going to leave"}, {"start": 46.739, "duration": 5.121, "text": "behind an NO2 that's positively charged"}, {"start": 54.059, "duration": 5.34, "text": "onwards we can see that we have our NO2"}, {"start": 57.059, "duration": 4.82, "text": "that's positively charged the nitrogen"}, {"start": 59.399, "duration": 2.48, "text": "is electron"}], "title": "Ochem tutor explains benzene  #chem #ochem #ucdavis #chemistry #premed #mcat #orgo #organicchemistry", "description": "", "lengthSeconds": 60, "width": 360, "height": 640}
{"video_id": "cMoUMcDErJs", "transcript": [{"start": 0.96, "duration": 5.1, "text": "we're going to figure out how many grams"}, {"start": 3.36, "duration": 6.96, "text": "of carbon dioxide are produced when 25"}, {"start": 6.06, "duration": 5.999, "text": "grams of butane burn so I have my"}, {"start": 10.32, "duration": 4.92, "text": "balanced equation here I'm going to"}, {"start": 12.059, "duration": 5.341, "text": "write 25 grams on top of butane and I've"}, {"start": 15.24, "duration": 5.039, "text": "got a blank here I'm trying to find the"}, {"start": 17.4, "duration": 5.82, "text": "grams of carbon dioxide next I need the"}, {"start": 20.279, "duration": 6.0, "text": "molar masses from the periodic table"}, {"start": 23.22, "duration": 5.76, "text": "when I add up 4 times carbon and 10"}, {"start": 26.279, "duration": 5.221, "text": "times hydrogen add that up I get 58 from"}, {"start": 28.98, "duration": 5.4, "text": "the periodic table one carbon and two"}, {"start": 31.5, "duration": 5.579, "text": "times oxygen gives me 44."}, {"start": 34.38, "duration": 8.1, "text": "then I'm going to take my 25 and divide"}, {"start": 37.079, "duration": 5.401, "text": "by 58 and that's going to give me 0.431"}, {"start": 43.02, "duration": 5.84, "text": "then multiply times 8 divided by 2 to"}, {"start": 47.34, "duration": 4.379, "text": "get"}, {"start": 48.86, "duration": 6.039, "text": "1.72 moles finally I'm going to take"}, {"start": 51.719, "duration": 7.641, "text": "that 1.72 times 44."}, {"start": 54.899, "duration": 4.461, "text": "to get 75.9 grams"}], "title": "Stoichiometry in chemistry example problem", "description": "Here\u2019s the best method I know of how to your stoichiometry problems in chemistry!", "lengthSeconds": 58, "width": 360, "height": 640}
{"video_id": "Jy0zwfFAipE", "transcript": [{"start": 0.0, "duration": 4.08, "text": "going to look at every single carbon"}, {"start": 1.92, "duration": 4.68, "text": "right that's what it asked us to find"}, {"start": 4.08, "duration": 4.38, "text": "the hybridization for each carbon in the"}, {"start": 6.6, "duration": 3.66, "text": "structure so let's start off with just"}, {"start": 8.46, "duration": 3.84, "text": "this one and I can kind of even just"}, {"start": 10.26, "duration": 4.32, "text": "cover this right okay I just want to"}, {"start": 12.3, "duration": 4.02, "text": "look at the bonds or lone pairs in this"}, {"start": 14.58, "duration": 3.539, "text": "case just bonds that are surrounding"}, {"start": 16.32, "duration": 3.6, "text": "that atom that would be four that would"}, {"start": 18.119, "duration": 5.16, "text": "be four exactly so that would then be"}, {"start": 19.92, "duration": 6.48, "text": "sp3 yep oh see I really struggled with"}, {"start": 23.279, "duration": 4.561, "text": "hybridization for some reason okay"}, {"start": 26.4, "duration": 3.24, "text": "and then now let's move on to this"}, {"start": 27.84, "duration": 3.72, "text": "carbon so how many bonds again same same"}, {"start": 29.64, "duration": 4.02, "text": "thing four bonds"}, {"start": 31.56, "duration": 3.78, "text": "and now looking at this one how many"}, {"start": 33.66, "duration": 3.6, "text": "electron groups do we have that would be"}, {"start": 35.34, "duration": 4.92, "text": "three three correct because this only"}, {"start": 37.26, "duration": 5.16, "text": "counts as one so then we know it's SP2"}, {"start": 40.26, "duration": 3.54, "text": "what about the next carbon also three"}, {"start": 42.42, "duration": 4.68, "text": "yeah"}, {"start": 43.8, "duration": 5.46, "text": "so SP2 yes yeah yeah three electron"}, {"start": 47.1, "duration": 3.779, "text": "groups good and then moving on to this"}, {"start": 49.26, "duration": 4.02, "text": "carbon that would be four electrons"}, {"start": 50.879, "duration": 3.961, "text": "great what about that one how many"}, {"start": 53.28, "duration": 3.66, "text": "electron groups do we have that one's"}, {"start": 54.84, "duration": 4.559, "text": "two great yeah"}, {"start": 56.94, "duration": 4.639, "text": "and then the last one's also too that's"}, {"start": 59.399, "duration": 2.18, "text": "a"}], "title": "Quick Hybridization Review #organicchemistry #chemistry", "description": "What the full video on my youtube channel", "lengthSeconds": 60, "width": 360, "height": 640}
{"video_id": "wmnQgD5oqQw", "transcript": [{"start": 1.28, "duration": 4.24, "text": "now sometimes you may need to find"}, {"start": 3.6, "duration": 3.36, "text": "the absolute value of"}, {"start": 5.52, "duration": 3.68, "text": "a complex number"}, {"start": 6.96, "duration": 4.88, "text": "what is the absolute value of 3"}, {"start": 9.2, "duration": 4.08, "text": "plus 4i"}, {"start": 11.84, "duration": 4.96, "text": "here's the equation"}, {"start": 13.28, "duration": 5.999, "text": "the absolute value of a plus bi"}, {"start": 16.8, "duration": 5.28, "text": "is equal to the square root"}, {"start": 19.279, "duration": 5.361, "text": "of a squared plus b squared"}, {"start": 22.08, "duration": 4.08, "text": "it's just the way it works out"}, {"start": 24.64, "duration": 4.2, "text": "so this is going to be the square root"}, {"start": 26.16, "duration": 5.439, "text": "of 3 squared plus 4"}, {"start": 28.84, "duration": 5.08, "text": "squared three squared is nine"}, {"start": 31.599, "duration": 4.321, "text": "four squared is sixteen"}, {"start": 33.92, "duration": 3.76, "text": "and the square root of well nine plus"}, {"start": 35.92, "duration": 3.12, "text": "sixteen is twenty-five"}, {"start": 37.68, "duration": 2.559, "text": "and the square root of twenty-five is"}, {"start": 39.04, "duration": 2.96, "text": "five"}, {"start": 40.239, "duration": 3.601, "text": "so that's the absolute value of three"}, {"start": 42.0, "duration": 3.44, "text": "plus four"}, {"start": 43.84, "duration": 4.08, "text": "so now it's your turn"}, {"start": 45.44, "duration": 4.16, "text": "calculate the absolute value of"}, {"start": 47.92, "duration": 3.84, "text": "5"}, {"start": 49.6, "duration": 4.24, "text": "minus 12i"}, {"start": 51.76, "duration": 3.68, "text": "so this is going to be the square root"}, {"start": 53.84, "duration": 3.28, "text": "of 5 squared"}, {"start": 55.44, "duration": 3.68, "text": "plus the square root of negative 12"}, {"start": 57.12, "duration": 2.959, "text": "squared which you could just write 12"}, {"start": 59.12, "duration": 2.959, "text": "squared"}, {"start": 60.079, "duration": 4.961, "text": "5 squared is 25"}, {"start": 62.079, "duration": 5.921, "text": "12 times 12 is 144"}, {"start": 65.04, "duration": 5.8, "text": "and these two add up to 169. the square"}, {"start": 68.0, "duration": 4.4, "text": "root of 169 is"}, {"start": 70.84, "duration": 3.959, "text": "13."}, {"start": 72.4, "duration": 4.399, "text": "and so that's the absolute value of 5"}, {"start": 74.799, "duration": 4.561, "text": "minus 12i"}, {"start": 76.799, "duration": 4.401, "text": "so now why is the absolute value of 3"}, {"start": 79.36, "duration": 4.24, "text": "plus 4i"}, {"start": 81.2, "duration": 5.279, "text": "equal to 5"}, {"start": 83.6, "duration": 4.96, "text": "one way i like to think about it is this"}, {"start": 86.479, "duration": 4.721, "text": "if we plot 3 plus 4i"}, {"start": 88.56, "duration": 5.76, "text": "we need to travel 3 units on the x-axis"}, {"start": 91.2, "duration": 4.32, "text": "and up 4 on the y-axis"}, {"start": 94.32, "duration": 5.24, "text": "so here it is"}, {"start": 95.52, "duration": 4.04, "text": "now let's turn it into a triangle"}, {"start": 101.36, "duration": 3.759, "text": "so this portion of the triangle is three"}, {"start": 103.2, "duration": 3.04, "text": "this part is four"}, {"start": 105.119, "duration": 3.441, "text": "and"}, {"start": 106.24, "duration": 3.68, "text": "let's find the missing side this is a"}, {"start": 108.56, "duration": 3.519, "text": "this is b"}, {"start": 109.92, "duration": 4.72, "text": "the hypotenuse is c we know that a"}, {"start": 112.079, "duration": 4.0, "text": "squared plus b squared is equal to c"}, {"start": 114.64, "duration": 4.56, "text": "squared"}, {"start": 116.079, "duration": 6.481, "text": "so c is simply the square root"}, {"start": 119.2, "duration": 5.36, "text": "of a squared plus b squared"}, {"start": 122.56, "duration": 3.199, "text": "and it turns out that the absolute value"}, {"start": 124.56, "duration": 2.32, "text": "is basically the hypotenuse of the"}, {"start": 125.759, "duration": 2.801, "text": "triangle"}, {"start": 126.88, "duration": 2.96, "text": "which is five"}, {"start": 128.56, "duration": 4.08, "text": "so it helps to know the special"}, {"start": 129.84, "duration": 5.28, "text": "triangles like 3 4 5"}, {"start": 132.64, "duration": 4.319, "text": "5 12 13. notice the answer for the"}, {"start": 135.12, "duration": 5.36, "text": "second example was 13."}, {"start": 136.959, "duration": 8.041, "text": "there's also the 7 24 25 triangle"}, {"start": 140.48, "duration": 4.52, "text": "and the 8 15 17 triangle"}, {"start": 165.36, "duration": 2.08, "text": "you"}], "title": "Absolute Value of Complex Numbers", "description": "This algebra video tutorial explains how to determine the absolute value of complex numbers which is equivalent to the magnitude of the complex number in standard form. \n\nComplex Numbers - Free Formula Sheet:      https://bit.ly/3YAyxZe\n\n_______________________________\nImaginary Numbers - Basic Intro:                      \nhttps://www.youtube.com/watch?v=vRJqPo_zcnQ\n\nGraphing Complex Numbers:                             \nhttps://www.youtube.com/watch?v=BJ6b9t0ryhU\n\nAbsolute Value of Complex Numbers:              \nhttps://www.youtube.com/watch?v=wmnQgD5oqQw\n\nAdding & Subtracting Complex Numbers:       \nhttps://www.youtube.com/watch?v=qNNX79cnFyQ\n\nMultiplying Complex Numbers:                          \nhttps://www.youtube.com/watch?v=MzCS_8Rzja8\n\nDividing Complex Numbers:                               \nhttps://www.youtube.com/watch?v=EfRRpVB62Ko\n\n_____________________________________\nSimplifying Complex Numbers:                         \nhttps://www.youtube.com/watch?v=XyksedEntD4\n\nSAT Math - Complex Numbers:                          \nhttps://www.youtube.com/watch?v=PKO53M0SEPQ\n\nSolving Equations With Complex Numbers:    \nhttps://www.youtube.com/watch?v=nRawKl7jP1M\n\nImaginary Solutions of Polynomials:                \nhttps://www.youtube.com/watch?v=tRrPfxR5h-w\n\nComplex Numbers - Test Review:                      \nhttps://www.youtube.com/watch?v=y2wPAZwZTng\n\n______________________________________\nThe Discriminant:\nhttps://www.youtube.com/watch?v=k-TDe0OP2XY\n\nComplex Numbers In Polar Form:\nhttps://www.youtube.com/watch?v=J6TnZxUUzqU\n\nFinal Exams and Video Playlists:   \nhttps://www.video-tutor.net/\n\nFull-Length Videos and Worksheets:\nhttps://www.patreon.com/MathScienceTutor/collections", "lengthSeconds": 166, "width": 640, "height": 360}
{"video_id": "ZMhkvFwLOR4", "transcript": [{"start": 2.08, "duration": 2.8, "text": "now in this lesson"}, {"start": 3.439, "duration": 3.121, "text": "we're going to calculate the surface"}, {"start": 4.88, "duration": 4.399, "text": "area"}, {"start": 6.56, "duration": 2.719, "text": "of a cube"}, {"start": 9.44, "duration": 4.48, "text": "now what's the formula for that"}, {"start": 11.36, "duration": 5.52, "text": "how can we calculate the surface area"}, {"start": 13.92, "duration": 6.64, "text": "of a cube"}, {"start": 16.88, "duration": 3.68, "text": "so all sides of a cube are the same"}, {"start": 22.64, "duration": 4.399, "text": "now let's say if you want to calculate"}, {"start": 23.92, "duration": 6.56, "text": "the area of one of the six faces"}, {"start": 27.039, "duration": 5.281, "text": "let's say the area of this face"}, {"start": 30.48, "duration": 3.2, "text": "it's going to be the length"}, {"start": 32.32, "duration": 2.72, "text": "times the width"}, {"start": 33.68, "duration": 3.52, "text": "now both the length of the width is"}, {"start": 35.04, "duration": 5.199, "text": "equal to x"}, {"start": 37.2, "duration": 5.039, "text": "so if the area of one face is x times x"}, {"start": 40.239, "duration": 4.881, "text": "or x squared"}, {"start": 42.239, "duration": 5.441, "text": "and each face is the same the area or"}, {"start": 45.12, "duration": 4.72, "text": "the total surface area of all six faces"}, {"start": 47.68, "duration": 4.399, "text": "is six x squared"}, {"start": 49.84, "duration": 5.12, "text": "and so let's say if"}, {"start": 52.079, "duration": 4.881, "text": "the side length of a cube is"}, {"start": 54.96, "duration": 3.52, "text": "eight centimeters"}, {"start": 56.96, "duration": 3.439, "text": "the surface area"}, {"start": 58.48, "duration": 4.559, "text": "is going to be six times eight"}, {"start": 60.399, "duration": 4.961, "text": "centimeters squared"}, {"start": 63.039, "duration": 4.881, "text": "eight squared is 64"}, {"start": 65.36, "duration": 3.64, "text": "and six times 64"}, {"start": 67.92, "duration": 3.92, "text": "is"}, {"start": 69.0, "duration": 4.36, "text": "384 and it's going to be square"}, {"start": 71.84, "duration": 2.959, "text": "centimeters"}, {"start": 73.36, "duration": 2.96, "text": "so that's how you can calculate the"}, {"start": 74.799, "duration": 4.241, "text": "surface area"}, {"start": 76.32, "duration": 6.439, "text": "of a cube it's 6x squared where x is the"}, {"start": 79.04, "duration": 3.719, "text": "side length of the cube"}, {"start": 101.28, "duration": 2.08, "text": "you"}], "title": "Surface Area of a Cube", "description": "This geometry video tutorial explains how to calculate the surface area of a cube.\n\n3D Shapes - Faces, Edges, & Vertices:     https://www.youtube.com/watch?v=Wmwe1fmR1SM\n\nRectangular Prism - Volume:                     \nhttps://www.youtube.com/watch?v=rNIKbT0-ywU\n\nRectangular Prism - Surface Area:            \nhttps://www.youtube.com/watch?v=BsAQbJyEF9g\n\nRectangular Prism - Diagonal Length:      \nhttps://www.youtube.com/watch?v=iO7vb3HLHqU\n\nTriangular Prism - Volume:                         \nhttps://www.youtube.com/watch?v=HfDHYFgTbxk\n\nTriangular Prism - Surface Area:               \nhttps://www.youtube.com/watch?v=152tLGR0lms\n\nTriangular Prism - Lateral Area:                 \nhttps://www.youtube.com/watch?v=PljECBkHQpg\n\n_______________________________\nHexagonal Prism - Surface Area:              \nhttps://www.youtube.com/watch?v=JKyF2r-bIC0\n\nCube - Volume:                                             \nhttps://www.youtube.com/watch?v=bpRsuccb6dI\n\nCube - Surface Area:                                   \nhttps://www.youtube.com/watch?v=ZMhkvFwLOR4\n\nCube - Diagonal Length:                             \nhttps://www.youtube.com/watch?v=wr3EvQZzoRI\n\nPyramid - Volume:                                       \nhttps://www.youtube.com/watch?v=8onylJ72ZPQ\n\nPyramid - Surface Area:                              \nhttps://www.youtube.com/watch?v=2g6QuOizzac\n\n________________________________\nSquare Pyramids - Volume:                       \nhttps://www.youtube.com/watch?v=x8wEnG4GURQ\n\nCylinder - Surface Area:                              \nhttps://www.youtube.com/watch?v=LdztqPczYiI\n\nSphere - Volume:                                         \nhttps://www.youtube.com/watch?v=eAwc8rwMRnI\n\nSphere - Surface Area:                                \nhttps://www.youtube.com/watch?v=aXEcU-M_qNA\n\nCone - Volume:                                             \nhttps://www.youtube.com/watch?v=LypmeZ0tnJA\n\nCone - Surface Area:                                   \nhttps://www.youtube.com/watch?v=MeWoIcQNqRA\n\n_________________________________\nFinal Exams and Video Playlists:   \nhttps://www.video-tutor.net/", "lengthSeconds": 102, "width": 640, "height": 360}
{"video_id": "WWwG2uh4OUE", "transcript": [{"start": 1.04, "duration": 3.04, "text": "in this video we're going to talk about"}, {"start": 2.639, "duration": 3.281, "text": "how to"}, {"start": 4.08, "duration": 4.16, "text": "simplify rational expressions when"}, {"start": 5.92, "duration": 4.639, "text": "adding or subtracting them whenever they"}, {"start": 8.24, "duration": 4.64, "text": "have like denominators"}, {"start": 10.559, "duration": 3.841, "text": "so let's start with this example 3x over"}, {"start": 12.88, "duration": 2.56, "text": "5"}, {"start": 14.4, "duration": 4.0, "text": "plus"}, {"start": 15.44, "duration": 4.88, "text": "four x plus seven over five"}, {"start": 18.4, "duration": 3.76, "text": "because these two fractions share the"}, {"start": 20.32, "duration": 4.16, "text": "same denominator"}, {"start": 22.16, "duration": 4.56, "text": "we can combine it as a single fraction"}, {"start": 24.48, "duration": 4.719, "text": "so we can write it as three x"}, {"start": 26.72, "duration": 4.559, "text": "plus four x plus seven"}, {"start": 29.199, "duration": 3.761, "text": "all divided by the common denominator"}, {"start": 31.279, "duration": 5.201, "text": "which is five"}, {"start": 32.96, "duration": 4.96, "text": "three x plus four x is seven x"}, {"start": 36.48, "duration": 3.599, "text": "and if we want to we can take out the"}, {"start": 37.92, "duration": 5.52, "text": "gcf which is seven"}, {"start": 40.079, "duration": 4.881, "text": "leaving behind x plus one"}, {"start": 43.44, "duration": 3.92, "text": "so that's the final answer for this"}, {"start": 44.96, "duration": 2.4, "text": "example"}, {"start": 49.36, "duration": 3.199, "text": "let's try another one"}, {"start": 50.719, "duration": 4.16, "text": "7 over x"}, {"start": 52.559, "duration": 3.921, "text": "minus 11 over x"}, {"start": 54.879, "duration": 2.561, "text": "so we can combine it as a single"}, {"start": 56.48, "duration": 3.36, "text": "fraction"}, {"start": 57.44, "duration": 5.759, "text": "seven minus eleven over x"}, {"start": 59.84, "duration": 4.8, "text": "and seven minus eleven is negative four"}, {"start": 63.199, "duration": 4.96, "text": "so the answer is going to be negative"}, {"start": 64.64, "duration": 3.519, "text": "four divided by x"}, {"start": 68.479, "duration": 4.561, "text": "try this one"}, {"start": 69.92, "duration": 6.96, "text": "seven x plus four"}, {"start": 73.04, "duration": 6.48, "text": "divided by x plus two"}, {"start": 76.88, "duration": 6.08, "text": "plus five x"}, {"start": 79.52, "duration": 6.08, "text": "minus seven over x plus two"}, {"start": 82.96, "duration": 4.159, "text": "so let's write it as a single fraction"}, {"start": 85.6, "duration": 3.519, "text": "seven x"}, {"start": 87.119, "duration": 3.281, "text": "plus five x i'm gonna put the like terms"}, {"start": 89.119, "duration": 3.601, "text": "together"}, {"start": 90.4, "duration": 4.64, "text": "plus four minus seven"}, {"start": 92.72, "duration": 5.68, "text": "all divided by x plus two"}, {"start": 95.04, "duration": 8.32, "text": "seven x plus five x is twelve x"}, {"start": 98.4, "duration": 4.96, "text": "four minus seven is negative three"}, {"start": 103.92, "duration": 4.08, "text": "so we can take out a three if we want so"}, {"start": 105.92, "duration": 5.68, "text": "it's going to be four x"}, {"start": 108.0, "duration": 7.119, "text": "minus one over x plus two"}, {"start": 111.6, "duration": 5.92, "text": "and so that's it"}, {"start": 115.119, "duration": 5.201, "text": "let's try one more example"}, {"start": 117.52, "duration": 4.08, "text": "three x minus ten"}, {"start": 120.32, "duration": 2.799, "text": "divided by"}, {"start": 121.6, "duration": 3.6, "text": "x minus one"}, {"start": 123.119, "duration": 4.481, "text": "minus"}, {"start": 125.2, "duration": 4.64, "text": "five x minus twelve"}, {"start": 127.6, "duration": 3.84, "text": "over x minus one"}, {"start": 129.84, "duration": 3.039, "text": "now be careful with this one go ahead"}, {"start": 131.44, "duration": 3.12, "text": "and try it"}, {"start": 132.879, "duration": 3.44, "text": "so first let's write it as a single"}, {"start": 134.56, "duration": 5.12, "text": "fraction"}, {"start": 136.319, "duration": 6.321, "text": "so it's going to be three x minus ten"}, {"start": 139.68, "duration": 4.08, "text": "now this negative sign applies to the 5x"}, {"start": 142.64, "duration": 2.8, "text": "and"}, {"start": 143.76, "duration": 3.68, "text": "negative 12."}, {"start": 145.44, "duration": 4.72, "text": "so initially i'm going to write it using"}, {"start": 147.44, "duration": 2.72, "text": "parentheses"}, {"start": 150.64, "duration": 3.92, "text": "before you combine like terms distribute"}, {"start": 152.879, "duration": 4.321, "text": "the negative sign"}, {"start": 154.56, "duration": 4.24, "text": "so it's going to be 3x minus 10"}, {"start": 157.2, "duration": 3.44, "text": "minus 5x"}, {"start": 158.8, "duration": 5.12, "text": "plus 12."}, {"start": 160.64, "duration": 5.28, "text": "3x minus 5x is negative 2x"}, {"start": 163.92, "duration": 4.08, "text": "negative 10 plus 12"}, {"start": 165.92, "duration": 4.319, "text": "is positive 2."}, {"start": 168.0, "duration": 5.36, "text": "now let's take out the gcf which is"}, {"start": 170.239, "duration": 4.241, "text": "negative two negative two x divided by"}, {"start": 173.36, "duration": 2.879, "text": "negative two"}, {"start": 174.48, "duration": 4.64, "text": "that's positive x"}, {"start": 176.239, "duration": 5.36, "text": "and positive two divided by negative two"}, {"start": 179.12, "duration": 5.92, "text": "is negative one"}, {"start": 181.599, "duration": 5.92, "text": "so notice that we can cancel x minus one"}, {"start": 185.04, "duration": 6.919, "text": "which means the final answer"}, {"start": 187.519, "duration": 4.44, "text": "is simply minus two"}], "title": "Adding and Subtracting Rational Expressions With The Same Denominators", "description": "This algebra video tutorial explains how to add and subtract rational expressions with like denominators.  You simply need to add or subtract the numerators of the fractions.  Don't forget to distribute the negative sign.  This video contains plenty of examples and practice problems.\n\nFundamental Theorem of Algebra:              \nhttps://www.youtube.com/watch?v=cAfRgqEOURw\n\nRational Expressions - Basic Intro:                    \nhttps://www.youtube.com/watch?v=0Gq3uw2p6fA\n\nSimplifying Rational Expressions:                      \nhttps://www.youtube.com/watch?v=uVpsz-xpnPo\n\nMultiplying Rational Expressions:                      \nhttps://www.youtube.com/watch?v=RROSgr4oXjU\n\nDividing Rational Expressions:                           \nhttps://www.youtube.com/watch?v=rMVOdIDNjpA\n\n______________________________________\nAdding & Subtracting Rational Expressions:    \nhttps://www.youtube.com/watch?v=WWwG2uh4OUE\n\nRational Expressions - Unlike Denominators:   \nhttps://www.youtube.com/watch?v=1KGVwaUhR-s\n\nSimplifying Complex Rational Expressions:     \nhttps://www.youtube.com/watch?v=ifDEetTq8bM\n\nHow To Solve Rational Equations:                     \nhttps://www.youtube.com/watch?v=1fR_9ke5-n8\n\nRational Equations - Extraneous Solutions:      \nhttps://www.youtube.com/watch?v=PvMYq9JD724\n\n_____________________________________\nHorizontal and Slant Asymptotes:                     \nhttps://www.youtube.com/watch?v=kvhpu1TkSjI\n\nFinding Rational Functions Given 2 Points:      \nhttps://www.youtube.com/watch?v=ODs3t1vX1GY\n\nRational Functions - X and Y Intercepts:           \nhttps://www.youtube.com/watch?v=ebpEDR4-Q3Y\n\nGraphing Advanced Rational Functions:           \nhttps://www.youtube.com/watch?v=GxynsEIvjRA\n\nRational Inequalities:\nhttps://www.youtube.com/watch?v=gfnVHwhEe6U\n\n________________________________________\nFinal Exams and Video Playlists:   \nhttps://www.video-tutor.net/\n\nFull-Length Videos and Worksheets:\nhttps://www.patreon.com/MathScienceTutor/collections", "lengthSeconds": 192, "width": 640, "height": 360}
{"video_id": "FLMrkngIPcE", "transcript": [{"start": 0.0, "duration": 2.53, "text": "[Music]"}, {"start": 0.06, "duration": 4.92, "text": "let's do open in 60 seconds or less"}, {"start": 2.53, "duration": 5.15, "text": "[Music]"}, {"start": 4.98, "duration": 4.26, "text": "today we are doing a Benzene reaction"}, {"start": 7.68, "duration": 3.12, "text": "and we're making this product over here"}, {"start": 9.24, "duration": 4.88, "text": "give it a try let me know what you think"}, {"start": 10.8, "duration": 6.42, "text": "all right so what I see I see a Hera"}, {"start": 14.12, "duration": 5.32, "text": "relationship so they're para to one"}, {"start": 17.22, "duration": 4.68, "text": "another what I need to do then if I want"}, {"start": 19.44, "duration": 5.339, "text": "parent only is I needed but this Groupon"}, {"start": 21.9, "duration": 5.42, "text": "first and then second I need to place my"}, {"start": 24.779, "duration": 2.541, "text": "NO2"}, {"start": 28.019, "duration": 6.06, "text": "after I place my carbon chain here which"}, {"start": 31.619, "duration": 5.341, "text": "is my turbutyl and then what I can do"}, {"start": 34.079, "duration": 3.721, "text": "this is a para directing only functional"}, {"start": 36.96, "duration": 2.88, "text": "group"}, {"start": 37.8, "duration": 3.84, "text": "so it's going to place the next group in"}, {"start": 39.84, "duration": 5.78, "text": "the Paris spot only that's going to be"}, {"start": 41.64, "duration": 3.98, "text": "my hno3 and h2so4"}, {"start": 50.64, "duration": 5.12, "text": "there you have it thanks for tuning in"}, {"start": 52.8, "duration": 2.96, "text": "and we'll see you next time"}], "title": "Synthesis: Benzene Reaction #EricTheTutor OrganicChemistry #OCHEM", "description": "Visit http://OrganicChemistryTutor.org/ for worksheets, videos and more!\n\nFollow Eric The Tutor on Facebook at https://www.Facebook.com/EricRomanTheTutor\n\nFollow Eric The Tutor on Instagram at  http://instagram.com/EricRomanTheTutor", "lengthSeconds": 58, "width": 360, "height": 640}
{"video_id": "5xC_6xM-4tQ", "transcript": [{"start": 0.1, "duration": 3.72, "text": "how many valence electrons do each of"}, {"start": 2.08, "duration": 3.239, "text": "the following atoms have to find the"}, {"start": 3.82, "duration": 3.179, "text": "number of valence electrons for these"}, {"start": 5.319, "duration": 3.481, "text": "representative elements all we have to"}, {"start": 6.999, "duration": 3.241, "text": "do is look at their group number valence"}, {"start": 8.8, "duration": 3.06, "text": "electrons are the electrons in the"}, {"start": 10.24, "duration": 3.9, "text": "highest occupied energy level of an"}, {"start": 11.86, "duration": 3.78, "text": "element's atom the periodic table is"}, {"start": 14.14, "duration": 3.06, "text": "organized to tell us the valence"}, {"start": 15.64, "duration": 3.96, "text": "electrons according to the group numbers"}, {"start": 17.2, "duration": 3.6, "text": "for the representative elements so all"}, {"start": 19.6, "duration": 2.88, "text": "we have to do is look at the group"}, {"start": 20.8, "duration": 5.04, "text": "number and that's the number of valence"}, {"start": 22.48, "duration": 4.82, "text": "electrons calcium CA has two valence"}, {"start": 25.84, "duration": 5.939, "text": "electrons because it's in group two"}, {"start": 27.3, "duration": 6.46, "text": "nitrogen n has five and Xenon XE has"}, {"start": 31.779, "duration": 3.6, "text": "eight thanks for watching if this video"}, {"start": 33.76, "duration": 4.34, "text": "helped you out please give it a like And"}, {"start": 35.379, "duration": 2.721, "text": "subscribe to the channel"}], "title": "How to Determine the Number of Valence Electrons #chemistry #homework FAST", "description": "\ud83c\udd70\ufe0fWant to get an A in Chemistry? Or just pass? Subscribe to the Channel, I'll be your virtual Chemistry tutor!\n\n\u2705Subscribers to this channel do better in their science class!\n\n#chemistry #science #education #stem \n#homework #tutor #stemeducation\n\n#shorts #chemistry #science #education #youtubeshorts", "lengthSeconds": 36, "width": 360, "height": 640}
{"video_id": "LypmeZ0tnJA", "transcript": [{"start": 3.12, "duration": 5.12, "text": "in this lesson we're going to focus on"}, {"start": 4.56, "duration": 3.68, "text": "calculating the volume of a cone"}, {"start": 9.84, "duration": 4.799, "text": "so let's say the radius of the cone"}, {"start": 12.96, "duration": 3.52, "text": "is 12 inches"}, {"start": 14.639, "duration": 3.041, "text": "and the height of the cone"}, {"start": 16.48, "duration": 4.719, "text": "is"}, {"start": 17.68, "duration": 6.16, "text": "16 inches what is the volume"}, {"start": 21.199, "duration": 5.92, "text": "the volume of a cone is 1 3"}, {"start": 23.84, "duration": 6.4, "text": "pi r square times the height"}, {"start": 27.119, "duration": 6.4, "text": "so the radius we said is 12 inches and"}, {"start": 30.24, "duration": 3.279, "text": "the height is 16 inches"}, {"start": 33.92, "duration": 6.799, "text": "12 squared is 144 if we divide that by 3"}, {"start": 36.88, "duration": 5.6, "text": "that's 48 and 48 times 16"}, {"start": 40.719, "duration": 4.401, "text": "is 768."}, {"start": 42.48, "duration": 4.72, "text": "so it's 768 pi"}, {"start": 45.12, "duration": 3.599, "text": "cubic inches"}, {"start": 47.2, "duration": 2.72, "text": "and so that's how you can calculate the"}, {"start": 48.719, "duration": 3.121, "text": "volume"}, {"start": 49.92, "duration": 3.279, "text": "of a cone you can do so using this"}, {"start": 51.84, "duration": 3.12, "text": "formula"}, {"start": 53.199, "duration": 5.321, "text": "so this is the radius"}, {"start": 54.96, "duration": 3.56, "text": "and this is the height"}, {"start": 60.64, "duration": 4.8, "text": "now the decimal answer"}, {"start": 63.12, "duration": 2.32, "text": "is"}, {"start": 65.64, "duration": 6.64, "text": "2412.7 cubic inches for those of you who"}, {"start": 68.96, "duration": 3.32, "text": "have that answer"}, {"start": 90.88, "duration": 2.08, "text": "you"}], "title": "Volume of a Cone", "description": "This geometry video tutorial explains how to calculate the volume of a cone given the height and the radius.\n\n3D Shapes - Faces, Edges, & Vertices:     https://www.youtube.com/watch?v=Wmwe1fmR1SM\n\nRectangular Prism - Volume:                     \nhttps://www.youtube.com/watch?v=rNIKbT0-ywU\n\nRectangular Prism - Surface Area:            \nhttps://www.youtube.com/watch?v=BsAQbJyEF9g\n\nRectangular Prism - Diagonal Length:      \nhttps://www.youtube.com/watch?v=iO7vb3HLHqU\n\nTriangular Prism - Volume:                         \nhttps://www.youtube.com/watch?v=HfDHYFgTbxk\n\nTriangular Prism - Surface Area:               \nhttps://www.youtube.com/watch?v=152tLGR0lms\n\nTriangular Prism - Lateral Area:                 \nhttps://www.youtube.com/watch?v=PljECBkHQpg\n\n_______________________________\nHexagonal Prism - Surface Area:              \nhttps://www.youtube.com/watch?v=JKyF2r-bIC0\n\nCube - Volume:                                             \nhttps://www.youtube.com/watch?v=bpRsuccb6dI\n\nCube - Surface Area:                                   \nhttps://www.youtube.com/watch?v=ZMhkvFwLOR4\n\nCube - Diagonal Length:                             \nhttps://www.youtube.com/watch?v=wr3EvQZzoRI\n\nPyramid - Volume:                                       \nhttps://www.youtube.com/watch?v=8onylJ72ZPQ\n\nPyramid - Surface Area:                              \nhttps://www.youtube.com/watch?v=2g6QuOizzac\n\n________________________________\nSquare Pyramids - Volume:                       \nhttps://www.youtube.com/watch?v=x8wEnG4GURQ\n\nCylinder - Surface Area:                              \nhttps://www.youtube.com/watch?v=LdztqPczYiI\n\nSphere - Volume:                                         \nhttps://www.youtube.com/watch?v=eAwc8rwMRnI\n\nSphere - Surface Area:                                \nhttps://www.youtube.com/watch?v=aXEcU-M_qNA\n\nCone - Volume:                                             \nhttps://www.youtube.com/watch?v=LypmeZ0tnJA\n\nCone - Surface Area:                                   \nhttps://www.youtube.com/watch?v=MeWoIcQNqRA\n\n_________________________________\nFinal Exams and Video Playlists:   \nhttps://www.video-tutor.net/", "lengthSeconds": 91, "width": 640, "height": 360}
{"video_id": "AWH5icK7htU", "transcript": [{"start": 0.08, "duration": 3.44, "text": "go for it what do we think it's got a"}, {"start": 1.88, "duration": 3.879, "text": "charge so I know it's a strong"}, {"start": 3.52, "duration": 6.4, "text": "nucleophile good and then I look at this"}, {"start": 5.759, "duration": 7.641, "text": "and this alkal halide is primary so I"}, {"start": 9.92, "duration": 6.119, "text": "know it it will have to be once again"}, {"start": 13.4, "duration": 5.16, "text": "sn2 yep no trick here I know that this"}, {"start": 16.039, "duration": 4.441, "text": "is sn2 because Yep this is primary I"}, {"start": 18.56, "duration": 6.0, "text": "know this is a strong nucleophile that"}, {"start": 20.48, "duration": 4.08, "text": "checks out that's it yay"}], "title": "Question 2: Is this an SN1 or SN2 reaction? #organicchemistry #chemistry", "description": "", "lengthSeconds": 23, "width": 360, "height": 640}
{"video_id": "WnULTOrs2Ds", "transcript": [{"start": 0.56, "duration": 3.92, "text": "in this video we're going to talk about"}, {"start": 1.839, "duration": 3.92, "text": "the formation of a cyanohydrin from a"}, {"start": 4.48, "duration": 3.52, "text": "ketone"}, {"start": 5.759, "duration": 4.321, "text": "so let's begin with the overall"}, {"start": 8.0, "duration": 4.88, "text": "chemical reaction"}, {"start": 10.08, "duration": 4.479, "text": "so here we have a ketone specifically"}, {"start": 12.88, "duration": 4.159, "text": "acetone"}, {"start": 14.559, "duration": 5.601, "text": "and we're going to react it with hcn"}, {"start": 17.039, "duration": 4.961, "text": "hydrogen cyanide"}, {"start": 20.16, "duration": 5.119, "text": "the ketone functional group will be"}, {"start": 22.0, "duration": 6.16, "text": "converted into a hydroxyl group"}, {"start": 25.279, "duration": 5.121, "text": "with the addition of a nitrile group"}, {"start": 28.16, "duration": 3.68, "text": "and so this molecule"}, {"start": 30.4, "duration": 4.679, "text": "is called"}, {"start": 31.84, "duration": 3.239, "text": "a cyanohydrin"}, {"start": 36.64, "duration": 3.04, "text": "now let's talk about the mechanism by"}, {"start": 38.8, "duration": 3.04, "text": "which"}, {"start": 39.68, "duration": 3.44, "text": "we can get this product"}, {"start": 41.84, "duration": 2.719, "text": "so the first thing we're going to do is"}, {"start": 43.12, "duration": 3.2, "text": "we're going to use"}, {"start": 44.559, "duration": 5.361, "text": "excess cyanide"}, {"start": 46.32, "duration": 3.6, "text": "and a small amount of hcl"}, {"start": 55.68, "duration": 4.48, "text": "some of the cyanide ions will pick up a"}, {"start": 57.76, "duration": 3.76, "text": "proton from hcl"}, {"start": 60.16, "duration": 3.039, "text": "turning into"}, {"start": 61.52, "duration": 3.359, "text": "hydrogen cyanide"}, {"start": 63.199, "duration": 3.361, "text": "and we're going to get the chloride ion"}, {"start": 64.879, "duration": 3.201, "text": "so keep in mind we have a large amount"}, {"start": 66.56, "duration": 3.919, "text": "of cyanide"}, {"start": 68.08, "duration": 6.96, "text": "a small amount of hcl and so we're going"}, {"start": 70.479, "duration": 4.561, "text": "to get a small amount of hcn"}, {"start": 75.52, "duration": 3.76, "text": "so starting with the ketone functional"}, {"start": 77.36, "duration": 3.52, "text": "group"}, {"start": 79.28, "duration": 4.08, "text": "which"}, {"start": 80.88, "duration": 4.559, "text": "the oxygen has two lone pairs"}, {"start": 83.36, "duration": 4.799, "text": "we're going to take"}, {"start": 85.439, "duration": 5.121, "text": "the cyanide ion"}, {"start": 88.159, "duration": 3.521, "text": "and as the nucleophile it's going to be"}, {"start": 90.56, "duration": 2.96, "text": "attracted"}, {"start": 91.68, "duration": 3.92, "text": "to the carbonyl group"}, {"start": 93.52, "duration": 4.0, "text": "keep in mind the carbon atom has a"}, {"start": 95.6, "duration": 4.159, "text": "partial positive charge"}, {"start": 97.52, "duration": 3.44, "text": "whereas the cyanide ion has a negative"}, {"start": 99.759, "duration": 2.0, "text": "charge so they're attracted to each"}, {"start": 100.96, "duration": 2.88, "text": "other"}, {"start": 101.759, "duration": 3.601, "text": "so as the nucleophile attacks the"}, {"start": 103.84, "duration": 4.08, "text": "carbonyl carbon"}, {"start": 105.36, "duration": 4.0, "text": "the pi electrons in this double bond"}, {"start": 107.92, "duration": 3.76, "text": "are going to break and they're going to"}, {"start": 109.36, "duration": 6.48, "text": "be pulled back to oxygen because oxygen"}, {"start": 111.68, "duration": 4.16, "text": "is more electronegative than carbon"}, {"start": 117.92, "duration": 3.04, "text": "and so what we're going to get right now"}, {"start": 120.0, "duration": 4.159, "text": "is"}, {"start": 120.96, "duration": 3.199, "text": "an alkoxide ion"}, {"start": 124.32, "duration": 5.52, "text": "with"}, {"start": 125.759, "duration": 4.081, "text": "a nitro group attached to it"}, {"start": 130.08, "duration": 4.48, "text": "now in the next step"}, {"start": 132.319, "duration": 4.321, "text": "the alkoxide ion is going to be a prot"}, {"start": 134.56, "duration": 5.44, "text": "let me say that again the alkoxide ion"}, {"start": 136.64, "duration": 6.48, "text": "will be protonated by hcn"}, {"start": 140.0, "duration": 6.0, "text": "so it's going to grab a hydrogen"}, {"start": 143.12, "duration": 4.72, "text": "regenerating the cyanide ion"}, {"start": 146.0, "duration": 2.879, "text": "and so the end result"}, {"start": 147.84, "duration": 4.16, "text": "is"}, {"start": 148.879, "duration": 5.201, "text": "a cyanohydrin molecule"}, {"start": 152.0, "duration": 4.64, "text": "and so that's the mechanism for this"}, {"start": 154.08, "duration": 2.56, "text": "reaction"}], "title": "Cyanohydrin Formation Reaction Mechanism", "description": "This organic chemistry video tutorial discusses the reaction mechanism of the formation of a cyanohydrin molecule using a ketone and HCN.\n\nMy Website:  https://www.video-tutor.net\nPatreon Donations:  https://www.patreon.com/MathScienceTutor\nAmazon Store:  https://www.amazon.com/shop/theorganicchemistrytutor\n\nSubscribe:\nhttps://www.youtube.com/channel/UCEWpbFLzoYGPfuWUMFPSaoA?sub_confirmation=1\n\nDisclaimer:  Some of the links associated with this video may generate affiliate commissions on my behalf.  As an amazon associate, I earn from qualifying purchases that you may make through such affiliate links.", "lengthSeconds": 155, "width": 640, "height": 360}
{"video_id": "daUlTsnCNRQ", "transcript": [{"start": 0.96, "duration": 3.92, "text": "in this lesson we're going to focus on"}, {"start": 2.8, "duration": 4.16, "text": "simplifying natural logs"}, {"start": 4.88, "duration": 3.6, "text": "with e"}, {"start": 6.96, "duration": 4.8, "text": "so let's start with the basics what is"}, {"start": 8.48, "duration": 3.28, "text": "the natural log of one"}, {"start": 12.0, "duration": 3.599, "text": "log of one is always zero"}, {"start": 15.839, "duration": 4.321, "text": "a log"}, {"start": 17.119, "duration": 3.041, "text": "can have any base"}, {"start": 20.24, "duration": 4.0, "text": "it could be base 4 it can be base 10 it"}, {"start": 22.64, "duration": 2.799, "text": "could be anything"}, {"start": 24.24, "duration": 3.6, "text": "however"}, {"start": 25.439, "duration": 5.361, "text": "when it has the base e"}, {"start": 27.84, "duration": 4.8, "text": "a log becomes a natural log"}, {"start": 30.8, "duration": 4.88, "text": "so natural logs"}, {"start": 32.64, "duration": 3.88, "text": "only have base e and e is a number it's"}, {"start": 35.68, "duration": 3.28, "text": "like"}, {"start": 36.52, "duration": 5.08, "text": "2.718 something"}, {"start": 38.96, "duration": 2.64, "text": "but that's e"}, {"start": 42.239, "duration": 5.281, "text": "so now what is the natural log of e"}, {"start": 45.92, "duration": 3.6, "text": "because the base"}, {"start": 47.52, "duration": 4.4, "text": "is the same as e"}, {"start": 49.52, "duration": 4.559, "text": "ln base e of e is one"}, {"start": 51.92, "duration": 4.0, "text": "so the natural log of e is always equal"}, {"start": 54.079, "duration": 4.841, "text": "to one make sure you know that by the"}, {"start": 55.92, "duration": 3.0, "text": "way"}, {"start": 59.84, "duration": 5.039, "text": "now what is the natural log of e raised"}, {"start": 62.399, "duration": 4.4, "text": "to the seventh"}, {"start": 64.879, "duration": 3.201, "text": "we could take the seven and move it to"}, {"start": 66.799, "duration": 2.801, "text": "the front"}, {"start": 68.08, "duration": 2.719, "text": "so this is seven"}, {"start": 69.6, "duration": 4.0, "text": "lne"}, {"start": 70.799, "duration": 5.521, "text": "and we know that lne is one so this is"}, {"start": 73.6, "duration": 2.72, "text": "equal to seven"}, {"start": 77.68, "duration": 2.24, "text": "now what about"}, {"start": 79.04, "duration": 3.28, "text": "e"}, {"start": 79.92, "duration": 5.04, "text": "raised to the ln 8"}, {"start": 82.32, "duration": 4.479, "text": "what is that equal to"}, {"start": 84.96, "duration": 3.92, "text": "do you recall what 5"}, {"start": 86.799, "duration": 4.881, "text": "log base 5"}, {"start": 88.88, "duration": 4.879, "text": "of 9 is equal to"}, {"start": 91.68, "duration": 4.72, "text": "since these are the same"}, {"start": 93.759, "duration": 4.961, "text": "the answer is simply not"}, {"start": 96.4, "duration": 4.0, "text": "the natural log has the base e and so"}, {"start": 98.72, "duration": 5.28, "text": "these are the same"}, {"start": 100.4, "duration": 3.6, "text": "and the answer is simply 8."}, {"start": 104.32, "duration": 4.96, "text": "now what about e raised to the natural"}, {"start": 106.64, "duration": 4.799, "text": "log of x cubed"}, {"start": 109.28, "duration": 3.56, "text": "so once again the base is e"}, {"start": 111.439, "duration": 4.32, "text": "and so these will"}, {"start": 112.84, "duration": 5.239, "text": "cancel and this is simply equal to x"}, {"start": 115.759, "duration": 2.32, "text": "cubed"}, {"start": 118.64, "duration": 4.64, "text": "let's try one more example"}, {"start": 120.479, "duration": 5.841, "text": "e raised to the seven l and y"}, {"start": 123.28, "duration": 3.04, "text": "what is that equal to"}, {"start": 127.04, "duration": 3.04, "text": "we can't just say it's y because of the"}, {"start": 129.2, "duration": 2.8, "text": "seven"}, {"start": 130.08, "duration": 4.239, "text": "however what we can do"}, {"start": 132.0, "duration": 4.879, "text": "is take the seven and move it here"}, {"start": 134.319, "duration": 4.801, "text": "so this is equivalent to e"}, {"start": 136.879, "duration": 3.841, "text": "ln y to the seventh"}, {"start": 139.12, "duration": 3.52, "text": "and the base is e"}, {"start": 140.72, "duration": 3.84, "text": "so we can get rid of these two"}, {"start": 142.64, "duration": 5.959, "text": "and the final answer"}, {"start": 144.56, "duration": 4.039, "text": "is simply y to the seventh"}, {"start": 168.56, "duration": 2.08, "text": "you"}], "title": "Natural Logarithms", "description": "This algebra video tutorial provides a basic introduction into natural logarithms.  It explains how to evaluate natural logarithmic expressions with the natural base e and how to evaluate exponential expressions with natural logs in on the exponent of the natural base e using properties of logarithms.  \n\nLogarithms - Free Formula Sheet:                \nhttps://bit.ly/4fEwqv0\n\nAlgebra Final Exam Review:\nhttps://www.youtube.com/watch?v=U0Y8nSmEpNM\n\n________________________________\nLogarithms - The Easy Way!                              \nhttps://www.youtube.com/watch?v=kqVpPSzkTYA\n\nLog to Exponential Form:                                    \nhttps://www.youtube.com/watch?v=f0C1KL7GkqY\n\nChange of Base Formula:                                   \nhttps://www.youtube.com/watch?v=FFm-zaFW_X4\n\nChange of Base Log Problem:                           \nhttps://www.youtube.com/watch?v=p7hD9VdXv9U\n\nProperties of Logarithms:                                   \nhttps://www.youtube.com/watch?v=Jtv9Lnf7Zw8\n\n____________________________________\nExpanding Logarithmic Expressions:               \nhttps://www.youtube.com/watch?v=OIz-5MyJA3g\n\nCondensing Logarithmic Expressions:             \nhttps://www.youtube.com/watch?v=luRrOlsB4cY\n\nNatural Logarithms:                                             \nhttps://www.youtube.com/watch?v=daUlTsnCNRQ\n\nSolving Exponential Equations:                          \nhttps://www.youtube.com/watch?v=9tutJ5xrRwg\n\nExponential Equations - Quadratic Form:         \nhttps://www.youtube.com/watch?v=yNgmVu0R_T8\n\n_______________________________________\nSolving Logarithmic Equations:                         \nhttps://www.youtube.com/watch?v=fnhFneOz6n8\n\nGraphing Logarithmic Functions:                      \nhttps://www.youtube.com/watch?v=-nptxS9rZNA\n\nGraphing Exponential Functions:                       \nhttps://www.youtube.com/watch?v=DASfP8KAyvs\n\nCompound Interest Word Problems:                \nhttps://www.youtube.com/watch?v=Hn0eLcOSQGw\n\nLogarithms Practice Problems:                         \nhttps://www.youtube.com/watch?v=7DVbQKI600k\n\n_______________________________________\nFinal Exams and Video Playlists:   \nhttps://www.video-tutor.net/\n\nFull-Length Videos and Worksheets:\nhttps://www.patreon.com/MathScienceTutor/collections", "lengthSeconds": 169, "width": 640, "height": 360}
{"video_id": "pf9LkX8hpTQ", "transcript": [{"start": 1.04, "duration": 3.84, "text": "in this lesson we're going to talk about"}, {"start": 2.48, "duration": 4.08, "text": "how to convert a quadratic function in"}, {"start": 4.88, "duration": 5.04, "text": "standard form"}, {"start": 6.56, "duration": 4.959, "text": "to vertex form and vice versa"}, {"start": 9.92, "duration": 2.32, "text": "sometimes you may need to be able to do"}, {"start": 11.519, "duration": 2.641, "text": "this"}, {"start": 12.24, "duration": 4.959, "text": "so let's cover it in this video"}, {"start": 14.16, "duration": 7.039, "text": "and a way to do it is to use the"}, {"start": 17.199, "duration": 4.0, "text": "we need to complete the square basically"}, {"start": 22.32, "duration": 3.92, "text": "half of six is three"}, {"start": 24.64, "duration": 4.0, "text": "so we're going to add three squared to"}, {"start": 26.24, "duration": 4.32, "text": "both sides"}, {"start": 28.64, "duration": 3.439, "text": "adding it to the left side"}, {"start": 30.56, "duration": 2.8, "text": "is the same as subtracting it from the"}, {"start": 32.079, "duration": 3.681, "text": "right side"}, {"start": 33.36, "duration": 5.359, "text": "so we can add 3 squared to both sides or"}, {"start": 35.76, "duration": 5.2, "text": "we can add 9 to the right side and also"}, {"start": 38.719, "duration": 3.121, "text": "simultaneously subtract 9 from the right"}, {"start": 40.96, "duration": 3.2, "text": "side"}, {"start": 41.84, "duration": 4.64, "text": "9 and negative 9 is zero so we're not"}, {"start": 44.16, "duration": 4.32, "text": "changing the value of the right side"}, {"start": 46.48, "duration": 4.48, "text": "so let's do it that way"}, {"start": 48.48, "duration": 4.399, "text": "now let's factor x squared plus six x"}, {"start": 50.96, "duration": 3.84, "text": "plus three squared every time you"}, {"start": 52.879, "duration": 3.84, "text": "complete the square you're creating a"}, {"start": 54.8, "duration": 3.759, "text": "perfect square trinomial"}, {"start": 56.719, "duration": 3.52, "text": "and to factor it you can see everything"}, {"start": 58.559, "duration": 4.32, "text": "you need here"}, {"start": 60.239, "duration": 4.56, "text": "first it's x"}, {"start": 62.879, "duration": 3.041, "text": "and then whatever this sign is"}, {"start": 64.799, "duration": 2.561, "text": "it's plus"}, {"start": 65.92, "duration": 2.48, "text": "and then it's this number before you"}, {"start": 67.36, "duration": 1.84, "text": "square it"}, {"start": 68.4, "duration": 2.0, "text": "three"}, {"start": 69.2, "duration": 4.72, "text": "squared"}, {"start": 70.4, "duration": 6.24, "text": "outside we have negative five minus nine"}, {"start": 73.92, "duration": 4.0, "text": "so in vertex form it's x plus three"}, {"start": 76.64, "duration": 2.64, "text": "squared"}, {"start": 77.92, "duration": 4.16, "text": "minus fourteen"}, {"start": 79.28, "duration": 6.72, "text": "so we can clearly see that the vertex"}, {"start": 82.08, "duration": 5.44, "text": "is negative three negative fourteen"}, {"start": 86.0, "duration": 4.479, "text": "and we can also use this equation to"}, {"start": 87.52, "duration": 5.04, "text": "find vertex negative b over 2a"}, {"start": 90.479, "duration": 4.241, "text": "b is six"}, {"start": 92.56, "duration": 3.919, "text": "a is one negative six over two is"}, {"start": 94.72, "duration": 5.88, "text": "negative three which"}, {"start": 96.479, "duration": 4.121, "text": "confirms the x value here"}, {"start": 100.64, "duration": 4.08, "text": "now let's talk about how we can"}, {"start": 103.2, "duration": 4.4, "text": "go back"}, {"start": 104.72, "duration": 2.88, "text": "to standard form"}, {"start": 113.04, "duration": 4.8, "text": "so how can we convert this expression"}, {"start": 115.439, "duration": 3.521, "text": "which is in vertex form back to standard"}, {"start": 117.84, "duration": 4.0, "text": "form"}, {"start": 118.96, "duration": 5.199, "text": "all you need to do is expand x plus 3"}, {"start": 121.84, "duration": 2.319, "text": "squared"}, {"start": 125.2, "duration": 5.759, "text": "let's foil it x times x is x squared"}, {"start": 128.399, "duration": 4.801, "text": "x times 3 is 3x"}, {"start": 130.959, "duration": 5.36, "text": "3 times x is 3x"}, {"start": 133.2, "duration": 3.119, "text": "3 times 3 is 9."}, {"start": 137.12, "duration": 4.64, "text": "and then combine like terms 3x plus 3x"}, {"start": 140.0, "duration": 4.4, "text": "is 6x"}, {"start": 141.76, "duration": 4.24, "text": "minus 14 is negative 5."}, {"start": 144.4, "duration": 4.32, "text": "so this gives us"}, {"start": 146.0, "duration": 4.56, "text": "the original function in standard form"}, {"start": 148.72, "duration": 4.159, "text": "so now you know how to convert from"}, {"start": 150.56, "duration": 4.0, "text": "standard form to vertex form by"}, {"start": 152.879, "duration": 4.241, "text": "completing the square"}, {"start": 154.56, "duration": 5.84, "text": "and you know how to convert from vertex"}, {"start": 157.12, "duration": 6.8, "text": "form to standard form by expanding and"}, {"start": 160.4, "duration": 3.52, "text": "using the foil method"}], "title": "Standard Form to Vertex Form - Quadratic Equations", "description": "This algebra video tutorial explains how to convert a quadratic equation from standard form to vertex form and from vertex form to standard form.  \n\nQuadratic Equations - Free Formula Sheet:     https://bit.ly/3WZ8v1Z\n\nAlgebra Final Exam Review:\nhttps://www.youtube.com/watch?v=U0Y8nSmEpNM\n\n________________________________\nHow To Solve Simple Quadratic Equations:      \nhttps://www.youtube.com/watch?v=-KWsS2FZVTA\n\nSolving Quadratic Equations by Factoring:       \nhttps://www.youtube.com/watch?v=qeByhTF8WEw\n\nHow To Factor Difficult Quadratic Equations:   \nhttps://www.youtube.com/watch?v=VG1luiL_INQ\n\nSolving QE By Completing the Square:              \nhttps://www.youtube.com/watch?v=WOb25i8FBBw\n\nSolving Equations in Quadratic Form:                \nhttps://www.youtube.com/watch?v=eWjixwGK3YM\n\nSum and Products - Quadratic Equations:\nhttps://www.youtube.com/watch?v=kBSj35rik8w\n\n_______________________________________\nHow To Calculate the Discriminant:                  \nhttps://www.youtube.com/watch?v=6fU9ZgwbU74\n\nStandard Form to Vertex Form:                           \nhttps://www.youtube.com/watch?v=pf9LkX8hpTQ\n\nGraphing Quadratic Functions:                            \nhttps://www.youtube.com/watch?v=H8OQtxSRe_k\n\nHow To Find The Axis of Symmetry:                   \nhttps://www.youtube.com/watch?v=tyzd4VujaqY\n\nGraphing Quadratic Functions - Vertex Form:   \nhttps://www.youtube.com/watch?v=OHH7fX_M8Ns \n\nGraphing QF in Standard Form:                           \nhttps://www.youtube.com/watch?v=uYNKJWFYh7k\n\n___________________________________________\nWriting Quadratic Equations:                               \nhttps://www.youtube.com/watch?v=SB6eCyG4GEA\n\nMaximum and Minimum Word Problems:         \nhttps://www.youtube.com/watch?v=d5YktqR-4FU\n\nSystems of Quadratic Equations:                        \nhttps://www.youtube.com/watch?v=rq5AjFs95qw\n\nSystems of Nonlinear Equations:                        \nhttps://www.youtube.com/watch?v=JPdLEaxlnZQ\n\nQuadratic Inequalities:                                          \nhttps://www.youtube.com/watch?v=_gWjLKsFOPE\n\nQuadratic Equations - Test Review:                    \nhttps://www.youtube.com/watch?v=fFFA7Q4eVuY\n\n___________________________________________\nFinal Exams and Video Playlists:   \nhttps://www.video-tutor.net/\n\nFull-Length Videos and Worksheets:\nhttps://www.patreon.com/MathScienceTutor/collections", "lengthSeconds": 162, "width": 640, "height": 360}
{"video_id": "f0C1KL7GkqY", "transcript": [{"start": 1.04, "duration": 3.44, "text": "now in this lesson we're going to focus"}, {"start": 3.6, "duration": 4.0, "text": "on"}, {"start": 4.48, "duration": 5.92, "text": "converting logarithmic expressions"}, {"start": 7.6, "duration": 5.6, "text": "into exponential equations"}, {"start": 10.4, "duration": 5.04, "text": "so consider this logarithmic expression"}, {"start": 13.2, "duration": 4.4, "text": "log base 3 of 9"}, {"start": 15.44, "duration": 4.32, "text": "is equal to 2."}, {"start": 17.6, "duration": 6.56, "text": "how can you convert it from logarithmic"}, {"start": 19.76, "duration": 4.4, "text": "form to exponential form"}, {"start": 24.96, "duration": 4.479, "text": "well here is the generic equation let's"}, {"start": 27.199, "duration": 5.36, "text": "say if we have log base a"}, {"start": 29.439, "duration": 4.001, "text": "of b is equal to c"}, {"start": 32.559, "duration": 2.721, "text": "a"}, {"start": 33.44, "duration": 3.76, "text": "raised to the c"}, {"start": 35.28, "duration": 4.48, "text": "is equal to b"}, {"start": 37.2, "duration": 4.56, "text": "so this is the equivalent expression in"}, {"start": 39.76, "duration": 4.319, "text": "exponential form"}, {"start": 41.76, "duration": 4.639, "text": "so likewise three"}, {"start": 44.079, "duration": 3.441, "text": "raised to the second power is equal to"}, {"start": 46.399, "duration": 3.601, "text": "nine"}, {"start": 47.52, "duration": 4.0, "text": "so we can write it as three squared"}, {"start": 50.0, "duration": 3.199, "text": "is equal to nine"}, {"start": 51.52, "duration": 3.519, "text": "now it's your turn"}, {"start": 53.199, "duration": 4.84, "text": "convert the following logarithmic"}, {"start": 55.039, "duration": 3.0, "text": "expression"}, {"start": 58.079, "duration": 4.96, "text": "into its exponential form and you can"}, {"start": 60.16, "duration": 2.879, "text": "try this one as well"}, {"start": 67.52, "duration": 5.2, "text": "4 raised to the third power"}, {"start": 69.84, "duration": 4.48, "text": "is equal to 64."}, {"start": 72.72, "duration": 4.079, "text": "so that's how you can write it in"}, {"start": 74.32, "duration": 4.799, "text": "exponential form"}, {"start": 76.799, "duration": 5.441, "text": "eight raised to the second power is"}, {"start": 79.119, "duration": 3.121, "text": "equal to 64."}, {"start": 82.4, "duration": 4.12, "text": "and so we can write it like this"}, {"start": 88.08, "duration": 5.76, "text": "now what about the reverse"}, {"start": 90.32, "duration": 5.68, "text": "let's say if we have an equation"}, {"start": 93.84, "duration": 4.8, "text": "in exponential form"}, {"start": 96.0, "duration": 5.84, "text": "convert it to logarithmic form"}, {"start": 98.64, "duration": 3.2, "text": "try these three examples"}, {"start": 106.64, "duration": 3.119, "text": "feel free to pause the video"}, {"start": 110.399, "duration": 5.04, "text": "so the base is two the base is the one"}, {"start": 113.52, "duration": 4.08, "text": "with the exponent"}, {"start": 115.439, "duration": 4.561, "text": "so log base 2"}, {"start": 117.6, "duration": 3.839, "text": "and what is across the equal side"}, {"start": 120.0, "duration": 2.719, "text": "will go here"}, {"start": 121.439, "duration": 3.441, "text": "and"}, {"start": 122.719, "duration": 4.561, "text": "the logarithmic function always equals"}, {"start": 124.88, "duration": 5.2, "text": "the exponent"}, {"start": 127.28, "duration": 4.4, "text": "so to check it you can see that 2 raised"}, {"start": 130.08, "duration": 4.4, "text": "to the third power"}, {"start": 131.68, "duration": 5.68, "text": "is equal to 8."}, {"start": 134.48, "duration": 4.16, "text": "the next one is going to be log base 4"}, {"start": 137.36, "duration": 3.28, "text": "of 16"}, {"start": 138.64, "duration": 4.64, "text": "is equal to the exponent"}, {"start": 140.64, "duration": 2.64, "text": "which is two"}, {"start": 143.36, "duration": 4.56, "text": "four raised to the second power is equal"}, {"start": 146.08, "duration": 4.159, "text": "to sixteen"}, {"start": 147.92, "duration": 4.399, "text": "for the last one log"}, {"start": 150.239, "duration": 3.28, "text": "base five the base is the one with the"}, {"start": 152.319, "duration": 3.121, "text": "exponent"}, {"start": 153.519, "duration": 6.161, "text": "of 125"}, {"start": 155.44, "duration": 4.24, "text": "is equal to the exponent of three"}, {"start": 160.4, "duration": 5.72, "text": "five"}, {"start": 161.28, "duration": 4.84, "text": "to the third power is equal to 125"}, {"start": 186.08, "duration": 2.079, "text": "you"}], "title": "Writing Logarithmic Equations In Exponential Form", "description": "This algebra video tutorial explains how to write logarithmic equations in exponential form.  It also explains how to convert exponential equations to logarithmic form.  \n\nLogarithms - Free Formula Sheet:       https://bit.ly/4fEwqv0\n\nAlgebra Final Exam Review:\nhttps://www.youtube.com/watch?v=U0Y8nSmEpNM\n\n_________________________________\nLogarithms - The Easy Way!                              \nhttps://www.youtube.com/watch?v=kqVpPSzkTYA\n\nLog to Exponential Form:                                    \nhttps://www.youtube.com/watch?v=f0C1KL7GkqY\n\nChange of Base Formula:                                   \nhttps://www.youtube.com/watch?v=FFm-zaFW_X4\n\nChange of Base Log Problem:                           \nhttps://www.youtube.com/watch?v=p7hD9VdXv9U\n\nProperties of Logarithms:                                   \nhttps://www.youtube.com/watch?v=Jtv9Lnf7Zw8\n\n____________________________________\nExpanding Logarithmic Expressions:               \nhttps://www.youtube.com/watch?v=OIz-5MyJA3g\n\nCondensing Logarithmic Expressions:             \nhttps://www.youtube.com/watch?v=luRrOlsB4cY\n\nNatural Logarithms:                                             \nhttps://www.youtube.com/watch?v=daUlTsnCNRQ\n\nSolving Exponential Equations:                          \nhttps://www.youtube.com/watch?v=9tutJ5xrRwg\n\nExponential Equations - Quadratic Form:         \nhttps://www.youtube.com/watch?v=yNgmVu0R_T8\n\n_______________________________________\nSolving Logarithmic Equations:                         \nhttps://www.youtube.com/watch?v=fnhFneOz6n8\n\nGraphing Logarithmic Functions:                      \nhttps://www.youtube.com/watch?v=-nptxS9rZNA\n\nGraphing Exponential Functions:                       \nhttps://www.youtube.com/watch?v=DASfP8KAyvs\n\nCompound Interest Word Problems:                \nhttps://www.youtube.com/watch?v=Hn0eLcOSQGw\n\nLogarithms Practice Problems:                         \nhttps://www.youtube.com/watch?v=7DVbQKI600k\n\n_______________________________________\nFinal Exams and Video Playlists:   \nhttps://www.video-tutor.net/\n\nFull-Length Videos and Worksheets:\nhttps://www.patreon.com/MathScienceTutor/collections", "lengthSeconds": 186, "width": 640, "height": 360}
{"video_id": "foooNlsXhYI", "transcript": [{"start": 0.0, "duration": 3.76, "text": "seven things i wish i remembered before"}, {"start": 1.76, "duration": 4.079, "text": "starting oak chem number one"}, {"start": 3.76, "duration": 4.879, "text": "drawing lewis structures it's already"}, {"start": 5.839, "duration": 4.001, "text": "giving you the order in which uh like"}, {"start": 8.639, "duration": 2.88, "text": "how to actually start drawing the"}, {"start": 9.84, "duration": 4.24, "text": "structure so i'm going to start off with"}, {"start": 11.519, "duration": 4.721, "text": "ch3 knowing that carbon is here it's"}, {"start": 14.08, "duration": 3.92, "text": "surrounded by"}, {"start": 16.24, "duration": 4.4, "text": "three hydrogens"}, {"start": 18.0, "duration": 4.88, "text": "next this carbon is bonded to the next"}, {"start": 20.64, "duration": 4.16, "text": "carbon that follows"}, {"start": 22.88, "duration": 4.639, "text": "and then this carbon is bonded to"}, {"start": 24.8, "duration": 5.76, "text": "hydrogen and to oxygen so i'm going to"}, {"start": 27.519, "duration": 4.481, "text": "place oxygen here and hydrogen here"}, {"start": 30.56, "duration": 3.519, "text": "all right this is not telling us that"}, {"start": 32.0, "duration": 4.88, "text": "they're bonded to each other if that"}, {"start": 34.079, "duration": 5.921, "text": "were the case it would have said oh"}, {"start": 36.88, "duration": 4.8, "text": "now in this case carbon is not happy it"}, {"start": 40.0, "duration": 3.92, "text": "doesn't follow the octet rule right now"}, {"start": 41.68, "duration": 3.92, "text": "it wants to it always needs to follow"}, {"start": 43.92, "duration": 4.88, "text": "that octet rule when we're drawing these"}, {"start": 45.6, "duration": 5.84, "text": "structures so what we need to do is see"}, {"start": 48.8, "duration": 5.04, "text": "this would be incorrect instead one pair"}, {"start": 51.44, "duration": 5.279, "text": "of lone pairs would actually"}, {"start": 53.84, "duration": 7.199, "text": "form a double bond"}, {"start": 56.719, "duration": 4.32, "text": "and this would be the correct structure"}], "title": "What to remember from General Chemistry for Organic Chemistry #shorts", "description": "7 main things to remember from General Chemistry before starting Organic Chemistry.", "lengthSeconds": 59, "width": 360, "height": 640}
{"video_id": "F2WNkATR4Es", "transcript": [{"start": 1.28, "duration": 4.4, "text": "in this lesson we're going to talk about"}, {"start": 2.8, "duration": 5.76, "text": "how to calculate the volume"}, {"start": 5.68, "duration": 2.88, "text": "of a cylinder"}, {"start": 11.519, "duration": 5.361, "text": "so let's say the radius of the cylinder"}, {"start": 15.12, "duration": 5.28, "text": "is 5 inches"}, {"start": 16.88, "duration": 3.52, "text": "and the height of the cylinder"}, {"start": 20.8, "duration": 5.36, "text": "let's say it's 10 inches"}, {"start": 23.519, "duration": 5.201, "text": "so using this information"}, {"start": 26.16, "duration": 6.559, "text": "what is the volume of the cylinder"}, {"start": 28.72, "duration": 6.8, "text": "the volume is pi r squared times height"}, {"start": 32.719, "duration": 5.84, "text": "it's basically the area of the base"}, {"start": 35.52, "duration": 4.879, "text": "times the height of the cylinder"}, {"start": 38.559, "duration": 3.361, "text": "so for any prism the volume is base"}, {"start": 40.399, "duration": 3.041, "text": "times height"}, {"start": 41.92, "duration": 4.0, "text": "the area of the base which is the area"}, {"start": 43.44, "duration": 4.959, "text": "of a circle that's pi r squared and so"}, {"start": 45.92, "duration": 5.279, "text": "it's pi r squared times h"}, {"start": 48.399, "duration": 5.601, "text": "in this example the radius is 5"}, {"start": 51.199, "duration": 7.921, "text": "and the height is 10."}, {"start": 54.0, "duration": 7.44, "text": "5 squared is 25 and 25 times 10 is 250."}, {"start": 59.12, "duration": 3.52, "text": "so the volume for this cylinder is 250"}, {"start": 61.44, "duration": 3.92, "text": "pi"}, {"start": 62.64, "duration": 4.88, "text": "and it's cubic inches"}, {"start": 65.36, "duration": 3.6, "text": "so that is the answer"}, {"start": 67.52, "duration": 4.76, "text": "now if you want to get the decimal value"}, {"start": 68.96, "duration": 3.32, "text": "of that answer"}, {"start": 73.439, "duration": 5.881, "text": "it's approximately"}, {"start": 76.32, "duration": 3.0, "text": "785.4"}, {"start": 79.36, "duration": 3.399, "text": "cubic inches"}, {"start": 101.759, "duration": 2.081, "text": "you"}], "title": "Volume of a Cylinder", "description": "This geometry video tutorial explains how to calculate the volume of a cylinder.\n\n3D Shapes - Faces, Edges, & Vertices:     https://www.youtube.com/watch?v=Wmwe1fmR1SM\n\nRectangular Prism - Volume:                     \nhttps://www.youtube.com/watch?v=rNIKbT0-ywU\n\nRectangular Prism - Surface Area:            \nhttps://www.youtube.com/watch?v=BsAQbJyEF9g\n\nRectangular Prism - Diagonal Length:      \nhttps://www.youtube.com/watch?v=iO7vb3HLHqU\n\nTriangular Prism - Volume:                         \nhttps://www.youtube.com/watch?v=HfDHYFgTbxk\n\nTriangular Prism - Surface Area:               \nhttps://www.youtube.com/watch?v=152tLGR0lms\n\nTriangular Prism - Lateral Area:                 \nhttps://www.youtube.com/watch?v=PljECBkHQpg\n\n_______________________________\nHexagonal Prism - Surface Area:              \nhttps://www.youtube.com/watch?v=JKyF2r-bIC0\n\nCube - Volume:                                             \nhttps://www.youtube.com/watch?v=bpRsuccb6dI\n\nCube - Surface Area:                                   \nhttps://www.youtube.com/watch?v=ZMhkvFwLOR4\n\nCube - Diagonal Length:                             \nhttps://www.youtube.com/watch?v=wr3EvQZzoRI\n\nPyramid - Volume:                                       \nhttps://www.youtube.com/watch?v=8onylJ72ZPQ\n\nPyramid - Surface Area:                              \nhttps://www.youtube.com/watch?v=2g6QuOizzac\n\n________________________________\nSquare Pyramids - Volume:                       \nhttps://www.youtube.com/watch?v=x8wEnG4GURQ\n\nCylinder - Surface Area:                              \nhttps://www.youtube.com/watch?v=LdztqPczYiI\n\nSphere - Volume:                                         \nhttps://www.youtube.com/watch?v=eAwc8rwMRnI\n\nSphere - Surface Area:                                \nhttps://www.youtube.com/watch?v=aXEcU-M_qNA\n\nCone - Volume:                                             \nhttps://www.youtube.com/watch?v=LypmeZ0tnJA\n\nCone - Surface Area:                                   \nhttps://www.youtube.com/watch?v=MeWoIcQNqRA\n\n_________________________________\nFinal Exams and Video Playlists:   \nhttps://www.video-tutor.net/", "lengthSeconds": 102, "width": 640, "height": 360}
{"video_id": "IFHZQ6MaG6w", "transcript": [{"start": 0.56, "duration": 2.639, "text": "now there's one more lesson that we need"}, {"start": 2.0, "duration": 2.799, "text": "to talk about"}, {"start": 3.199, "duration": 4.881, "text": "and that is"}, {"start": 4.799, "duration": 5.84, "text": "a lesson on recursive formulas so let's"}, {"start": 8.08, "duration": 7.12, "text": "say if we have the formula"}, {"start": 10.639, "duration": 7.761, "text": "a of n plus one is equal to a of n"}, {"start": 15.2, "duration": 4.88, "text": "or rather three times a of n plus two"}, {"start": 18.4, "duration": 3.76, "text": "you need to know that a of n is the"}, {"start": 20.08, "duration": 2.88, "text": "previous term a of n plus one is an x"}, {"start": 22.16, "duration": 3.52, "text": "term"}, {"start": 22.96, "duration": 4.159, "text": "so let's say if the first term"}, {"start": 25.68, "duration": 3.28, "text": "is one"}, {"start": 27.119, "duration": 5.121, "text": "what is the value of"}, {"start": 28.96, "duration": 3.28, "text": "the next four terms"}, {"start": 32.559, "duration": 5.201, "text": "so if n is one n plus one is two"}, {"start": 36.16, "duration": 4.48, "text": "so the second term is three times the"}, {"start": 37.76, "duration": 5.04, "text": "value of the first term plus two"}, {"start": 40.64, "duration": 4.399, "text": "so three times one plus two which is"}, {"start": 42.8, "duration": 4.079, "text": "five"}, {"start": 45.039, "duration": 3.601, "text": "now if you want to find the third term"}, {"start": 46.879, "duration": 3.68, "text": "it's three times the second term plus"}, {"start": 48.64, "duration": 4.16, "text": "two"}, {"start": 50.559, "duration": 4.401, "text": "the second term is five"}, {"start": 52.8, "duration": 3.2, "text": "three times five is fifteen plus two is"}, {"start": 54.96, "duration": 3.04, "text": "seventeen"}, {"start": 56.0, "duration": 4.079, "text": "now if we wanna find the fourth term"}, {"start": 58.0, "duration": 3.52, "text": "it's three times the third term"}, {"start": 60.079, "duration": 4.08, "text": "plus two"}, {"start": 61.52, "duration": 4.959, "text": "the third term is seventeen"}, {"start": 64.159, "duration": 4.801, "text": "seventeen times three is fifty-one plus"}, {"start": 66.479, "duration": 4.401, "text": "two that's fifty-three so that's the"}, {"start": 68.96, "duration": 3.68, "text": "value of the fourth term"}, {"start": 70.88, "duration": 3.68, "text": "let's try another example"}, {"start": 72.64, "duration": 4.159, "text": "so let's say that a of n"}, {"start": 74.56, "duration": 4.559, "text": "is equal to two"}, {"start": 76.799, "duration": 5.601, "text": "times a of n minus one"}, {"start": 79.119, "duration": 3.281, "text": "and let's go ahead and square it"}, {"start": 82.96, "duration": 6.24, "text": "and then we'll subtract it by five"}, {"start": 86.4, "duration": 5.2, "text": "so a of n plus 1 was the next term a of"}, {"start": 89.2, "duration": 4.48, "text": "n was the previous term in this case"}, {"start": 91.6, "duration": 3.92, "text": "a of n minus 1 is the previous term a of"}, {"start": 93.68, "duration": 4.24, "text": "n is the next term"}, {"start": 95.52, "duration": 5.04, "text": "and let's say the first term"}, {"start": 97.92, "duration": 3.92, "text": "is equal to"}, {"start": 100.56, "duration": 4.96, "text": "two"}, {"start": 101.84, "duration": 6.239, "text": "so a of two is two times the square"}, {"start": 105.52, "duration": 4.72, "text": "of a of one minus five"}, {"start": 108.079, "duration": 4.4, "text": "so it's two times two squared minus five"}, {"start": 110.24, "duration": 4.4, "text": "two squared is four times two is eight"}, {"start": 112.479, "duration": 3.68, "text": "minus five that's three"}, {"start": 114.64, "duration": 4.079, "text": "a of three"}, {"start": 116.159, "duration": 5.041, "text": "is two times a of two"}, {"start": 118.719, "duration": 4.561, "text": "squared minus five"}, {"start": 121.2, "duration": 3.76, "text": "so a of two is three"}, {"start": 123.28, "duration": 3.6, "text": "three squared is nine"}, {"start": 124.96, "duration": 4.639, "text": "two times nine is eighteen eighteen"}, {"start": 126.88, "duration": 4.32, "text": "minus 5 is 13."}, {"start": 129.599, "duration": 4.161, "text": "a sub 4"}, {"start": 131.2, "duration": 4.56, "text": "is 2 times the square of a sub 3 minus"}, {"start": 133.76, "duration": 4.72, "text": "5."}, {"start": 135.76, "duration": 5.04, "text": "a sub 3 is 13"}, {"start": 138.48, "duration": 4.96, "text": "13 squared is 169"}, {"start": 140.8, "duration": 5.2, "text": "169 times 2"}, {"start": 143.44, "duration": 5.519, "text": "that's 338 minus 5"}, {"start": 146.0, "duration": 4.56, "text": "so this is equal to 333"}, {"start": 148.959, "duration": 5.801, "text": "and so that's how you can work with a"}, {"start": 150.56, "duration": 4.2, "text": "recursively defined formula"}, {"start": 173.36, "duration": 2.08, "text": "you"}], "title": "Recursive Formulas For Sequences", "description": "This algebra video tutorial provides a basic introduction into recursive formulas and how to use it to find the first four terms or the nth term of a sequence.  This video contains 2 examples on recursive sequences.  This topic is taught within the chapter of arithmetic sequences.\n\nArithmetic Sequences - Free Formula Sheet:\nhttps://bit.ly/4eau2KQ\n\n_____________________________________________\nAlgebra For Beginners:                                         \nhttps://www.youtube.com/watch?v=MHeirBPOI6w\n\nArithmetic Sequences and Series:\nhttps://www.youtube.com/watch?v=XZJdyPkCxuE&t=2s\n\nSigma Notation and Summation:\nhttps://www.youtube.com/watch?v=xavgv1m9feE\n\nWriting a General Formula of an Arithmetic Sequence:\nhttps://www.youtube.com/watch?v=UPvyQV6wZQw\n\n___________________________\nRecursive Formulas:\nhttps://www.youtube.com/watch?v=IFHZQ6MaG6w\n\nFibonacci Sequence:\nhttps://www.youtube.com/watch?v=mVO2dcuR7P0&t=1s\n\nGeometric Sequences and Series:\nhttps://www.youtube.com/watch?v=zRKZ0-kOUZM&t=2s\n\nAlgebra and Precalculus Videos:\nhttps://www.video-tutor.net/", "lengthSeconds": 174, "width": 640, "height": 360}
{"video_id": "GaltHJF6WyI", "transcript": [{"start": 1.12, "duration": 3.44, "text": "in this video we're going to talk about"}, {"start": 2.72, "duration": 4.8, "text": "how to identify"}, {"start": 4.56, "duration": 5.999, "text": "the relative maximum and minimum values"}, {"start": 7.52, "duration": 4.56, "text": "of the function f of x"}, {"start": 10.559, "duration": 6.241, "text": "so"}, {"start": 12.08, "duration": 7.199, "text": "here we have a maximum a relative max"}, {"start": 16.8, "duration": 6.239, "text": "it's the highest point in that region"}, {"start": 19.279, "duration": 5.681, "text": "and over here we have a relative minimum"}, {"start": 23.039, "duration": 4.4, "text": "now the location"}, {"start": 24.96, "duration": 4.319, "text": "of these values is associated with the x"}, {"start": 27.439, "duration": 4.401, "text": "values"}, {"start": 29.279, "duration": 5.44, "text": "so we have a relative max"}, {"start": 31.84, "duration": 4.239, "text": "at x equals negative three and we have a"}, {"start": 34.719, "duration": 4.0, "text": "relative min"}, {"start": 36.079, "duration": 5.041, "text": "at x equals one"}, {"start": 38.719, "duration": 4.081, "text": "now the actual relative"}, {"start": 41.12, "duration": 3.68, "text": "maximum value"}, {"start": 42.8, "duration": 3.68, "text": "is the y value that corresponds to this"}, {"start": 44.8, "duration": 4.0, "text": "point"}, {"start": 46.48, "duration": 5.36, "text": "so the y value is positive three"}, {"start": 48.8, "duration": 4.399, "text": "and here the y value is negative three"}, {"start": 51.84, "duration": 3.52, "text": "so"}, {"start": 53.199, "duration": 4.321, "text": "let's say if we were asked to find the"}, {"start": 55.36, "duration": 4.32, "text": "location of the relative maximum we"}, {"start": 57.52, "duration": 3.28, "text": "could say that it's located"}, {"start": 59.68, "duration": 3.6, "text": "at x"}, {"start": 60.8, "duration": 3.999, "text": "equals negative three"}, {"start": 63.28, "duration": 4.48, "text": "and the location"}, {"start": 64.799, "duration": 5.68, "text": "of the relative minimum"}, {"start": 67.76, "duration": 4.48, "text": "we could say it's located at"}, {"start": 70.479, "duration": 4.161, "text": "x equals 1."}, {"start": 72.24, "duration": 6.16, "text": "and then for this part what is"}, {"start": 74.64, "duration": 6.24, "text": "the relative minimum value of f of x"}, {"start": 78.4, "duration": 4.32, "text": "so the relative minimum value"}, {"start": 80.88, "duration": 3.84, "text": "that's equal to the y value that"}, {"start": 82.72, "duration": 3.6, "text": "corresponds to that point so that's"}, {"start": 84.72, "duration": 4.24, "text": "negative three"}, {"start": 86.32, "duration": 4.64, "text": "and the relative maximum value"}, {"start": 88.96, "duration": 2.799, "text": "is the y value that corresponds to that"}, {"start": 90.96, "duration": 3.92, "text": "point"}, {"start": 91.759, "duration": 5.36, "text": "which is positive three"}, {"start": 94.88, "duration": 3.84, "text": "let's try another example"}, {"start": 97.119, "duration": 3.761, "text": "so identify"}, {"start": 98.72, "duration": 5.28, "text": "all of the relative extreme values the"}, {"start": 100.88, "duration": 5.519, "text": "minimums the maximums the location and"}, {"start": 104.0, "duration": 4.88, "text": "their actual values"}, {"start": 106.399, "duration": 5.441, "text": "so here we have a minimum"}, {"start": 108.88, "duration": 4.879, "text": "and we have a relative maximum"}, {"start": 111.84, "duration": 3.52, "text": "a minimum and another"}, {"start": 113.759, "duration": 3.68, "text": "maximum"}, {"start": 115.36, "duration": 4.719, "text": "so the first relative minimum is located"}, {"start": 117.439, "duration": 4.561, "text": "at an x value of negative four"}, {"start": 120.079, "duration": 3.68, "text": "and the second one is located at an x"}, {"start": 122.0, "duration": 4.799, "text": "value of one"}, {"start": 123.759, "duration": 6.401, "text": "the first relative maximum is located at"}, {"start": 126.799, "duration": 6.08, "text": "approximately x equals negative two"}, {"start": 130.16, "duration": 4.64, "text": "and the second relative max is at"}, {"start": 132.879, "duration": 4.481, "text": "x equals four"}, {"start": 134.8, "duration": 4.64, "text": "so that's the location of these extreme"}, {"start": 137.36, "duration": 5.2, "text": "values"}, {"start": 139.44, "duration": 4.159, "text": "now the first actual relative minimum"}, {"start": 142.56, "duration": 4.16, "text": "value"}, {"start": 143.599, "duration": 4.801, "text": "has a y value of"}, {"start": 146.72, "duration": 5.12, "text": "two"}, {"start": 148.4, "duration": 3.44, "text": "and the second minimum value"}, {"start": 151.92, "duration": 5.16, "text": "the y value is negative one"}, {"start": 157.36, "duration": 7.44, "text": "now the first maximum value on the left"}, {"start": 161.2, "duration": 6.399, "text": "it has a y value of five"}, {"start": 164.8, "duration": 2.799, "text": "and the second one"}, {"start": 167.76, "duration": 5.36, "text": "has a y value of"}, {"start": 170.959, "duration": 3.841, "text": "three"}, {"start": 173.12, "duration": 4.08, "text": "now sometimes depending on what type of"}, {"start": 174.8, "duration": 4.56, "text": "problem you're working on"}, {"start": 177.2, "duration": 4.56, "text": "the problem may ask for the entire"}, {"start": 179.36, "duration": 4.56, "text": "coordinate of each of these relative"}, {"start": 181.76, "duration": 4.16, "text": "maximums and minimums so for this one"}, {"start": 183.92, "duration": 4.08, "text": "you could say it's located at negative"}, {"start": 185.92, "duration": 4.56, "text": "four comma two"}, {"start": 188.0, "duration": 4.4, "text": "and then this maximum have the point"}, {"start": 190.48, "duration": 2.96, "text": "let's put it here negative two comma"}, {"start": 192.4, "duration": 3.6, "text": "five"}, {"start": 193.44, "duration": 4.32, "text": "and for this one it's"}, {"start": 196.0, "duration": 4.239, "text": "four three"}, {"start": 197.76, "duration": 3.44, "text": "and for this minimum it's one negative"}, {"start": 200.239, "duration": 2.561, "text": "one"}, {"start": 201.2, "duration": 2.56, "text": "so you can actually write the entire"}, {"start": 202.8, "duration": 4.359, "text": "point"}, {"start": 203.76, "duration": 3.399, "text": "if you want to"}, {"start": 225.84, "duration": 2.08, "text": "you"}], "title": "Identifying The Relative Maximum and Minimum Values of a Function", "description": "This precalculus video tutorial provides a basic introduction into the relative maximum and minimum values of a function.  It explains how to identify where the relative maxima and relative minima are located as well as determining the relative maximum and minimum values.\n\nFunctions - Free Formula Sheet:               https://bit.ly/3Ai8NZv\n________________________________________________________________________________________\nIntroduction to Functions:                          https://www.youtube.com/watch?v=DrEXTC6mIO8\nEvaluating Functions:                                  https://www.youtube.com/watch?v=HyNie_PYgsY\nFunction Operations:                                   https://www.youtube.com/watch?v=3gaxVHVI4cI\nIncreasing and Decreasing Functions:     https://www.youtube.com/watch?v=Dyl7jPlJXOM\nRelative Maximum & Minimum Values:   https://www.youtube.com/watch?v=GaltHJF6WyI\nRelations and Functions:                            https://www.youtube.com/watch?v=wbBY2tTqXDA\nDomain and Range from a Graph:            https://www.youtube.com/watch?v=KirGQOwjBVI\nHow To Find the Domain of a Function:  https://www.youtube.com/watch?v=djT6-YamHaA\nAverage Rate of Change of a Function:    https://www.youtube.com/watch?v=f_Q0qlwlclQ\nThe Difference Quotient of a Function:    https://www.youtube.com/watch?v=qQgVomi8lCc\n________________________________________________________________________________________\nEvaluating Piecewise Functions:               https://www.youtube.com/watch?v=OYOXMyFKotc\nGraphing Piecewise Functions:                 https://www.youtube.com/watch?v=Uzw9tsGq2Pw\nComposite Functions:                                 https://www.youtube.com/watch?v=ZFPkQkURSxk\nHow To Evaluate Composite Functions:  https://www.youtube.com/watch?v=6vQxIG4lJ3w\nComposite Functions - Data Table:           https://www.youtube.com/watch?v=6vQxIG4lJ3w\nComp. of 3 Functions - Data Table:           https://www.youtube.com/watch?v=Ote1KvW6BXw\nDecomposing Functions:                            https://www.youtube.com/watch?v=NmfaC7etZms\nIntroduction to Inverse Functions:            https://www.youtube.com/watch?v=TN4ybFiuV3k\nHow To Find The Inverse of a Function:   https://www.youtube.com/watch?v=2zeYEx4eTdc\nVerifying Inverse Functions:                       https://www.youtube.com/watch?v=6QamgSPprjA\n________________________________________________________________________________________\nHorizontal Line Test - 1 to 1 Functions:    https://www.youtube.com/watch?v=6u9Qh1Vi_Qw\nVertical Line Test:                                         https://www.youtube.com/watch?v=Mxe2lX1htNk\nGraphing Inverse Functions:                       https://www.youtube.com/watch?v=ukEtad_aml4\nComposite and Inverse Functions:            https://www.youtube.com/watch?v=GEgFmxxOJno\nEven and Odd Functions:                            https://www.youtube.com/watch?v=fKyBOLsqRlo\nTransformations of Functions:                  https://www.youtube.com/watch?v=Tmdrjs9xufc\nLinear Functions:                                          https://www.youtube.com/watch?v=BtcKotD6Ni8\nFunctions and Graphs - Problems:            https://www.youtube.com/watch?v=kvU9sOzT2mk\nFunctions - Test Review:                             https://www.youtube.com/watch?v=1xATmTI-YY8\n________________________________________________________________________________________\nLogarithms Practice Problems:                 https://www.youtube.com/watch?v=7DVbQKI600k\nWriting Equations of Ellipses:                    https://www.youtube.com/watch?v=OPSCKXXvWiM\nHyperbolas - Conic Sections:                     https://www.youtube.com/watch?v=Iu-4-fizlD4\nArithmetic Sequences:                                https://www.youtube.com/watch?v=XZJdyPkCxuE\nGeometric Sequences:                                https://www.youtube.com/watch?v=zRKZ0-kOUZM\nIntro to Probability:                                       https://www.youtube.com/watch?v=SkidyDQuupA\nIntro to Statistics:                                         https://www.youtube.com/watch?v=XZo4xyJXCak\nTrigonometry For Beginners:                      https://www.youtube.com/watch?v=PUB0TaZ7bhA\nSAT Math Test Prep:                                    https://www.youtube.com/watch?v=fTGuTEQCsZY\nACT Math Test Prep:                                    https://www.youtube.com/watch?v=SsA7rZ8kczM\n________________________________________________________________________________________\nAlgebra For Beginners:                                https://www.youtube.com/watch?v=MHeirBPOI6w\nAlgebra 2 Introduction:                                https://www.youtube.com/watch?v=i6sbjtJjJ-A\nAlgebra 1 Review Study Guide:                   https://www.youtube.com/watch?v=TbJ5gqLRpeM\nAlgebra Final Exam Review:                        https://www.youtube.com/watch?v=U0Y8nSmEpNM\nPre-Calculus Final Exam Review:               https://www.youtube.com/watch?v=Tj-V6KnwM5w\nFull Length Exam Videos + Worksheets:  https://bit.ly/4990rzU", "lengthSeconds": 226, "width": 640, "height": 360}
{"video_id": "6X6JwgqlhgE", "transcript": [{"start": 0.04, "duration": 5.0, "text": "what about this one okay so I have an"}, {"start": 1.8, "duration": 6.32, "text": "alcohol which is neutral and then on"}, {"start": 5.04, "duration": 4.559, "text": "this guy right here it is secondary well"}, {"start": 8.12, "duration": 3.24, "text": "the indicator would be the nucleophile"}, {"start": 9.599, "duration": 4.24, "text": "because it's neutral it have to be sn1"}, {"start": 11.36, "duration": 4.999, "text": "this is a secondary which works for both"}, {"start": 13.839, "duration": 6.041, "text": "sn1 and sn2 it does how else can we"}, {"start": 16.359, "duration": 6.441, "text": "classify that alal haly as um alyc good"}, {"start": 19.88, "duration": 4.719, "text": "so it is secondary and it's also alyc"}, {"start": 22.8, "duration": 4.479, "text": "which makes it even faster when it is"}, {"start": 24.599, "duration": 5.481, "text": "ailc that's true so okay you're right"}, {"start": 27.279, "duration": 3.721, "text": "it's SN one because this is neutral this"}, {"start": 30.08, "duration": 2.359, "text": "still works with something that's"}, {"start": 31.0, "duration": 5.559, "text": "secondary and especially works with"}, {"start": 32.439, "duration": 4.12, "text": "something that's ayic yeah okay"}], "title": "Question 3: Is this an SN1 or SN2 reaction? #chemistry #organicchemistry #chemtube", "description": "", "lengthSeconds": 35, "width": 360, "height": 640}
{"video_id": "2_1PYcdUWy0", "transcript": [{"start": 0.53, "duration": 5.67, "text": "[Music]"}, {"start": 4.04, "duration": 4.16, "text": "hi everyone this is a video tutorial"}, {"start": 6.2, "duration": 3.319, "text": "about aromatic compounds so we're"}, {"start": 8.2, "duration": 3.0, "text": "basically going to look at the three"}, {"start": 9.519, "duration": 3.881, "text": "characteristics that make a compound"}, {"start": 11.2, "duration": 4.519, "text": "aromatic and how you can apply that to"}, {"start": 13.4, "duration": 4.48, "text": "an example to determine whether or not a"}, {"start": 15.719, "duration": 4.32, "text": "specific compound does fall in the"}, {"start": 17.88, "duration": 3.8, "text": "aromatic bracket so if we take a look"}, {"start": 20.039, "duration": 3.961, "text": "here you'll see that we have three basic"}, {"start": 21.68, "duration": 4.4, "text": "rules for an aromatic compound the first"}, {"start": 24.0, "duration": 3.32, "text": "one is it has to be cyclic so it could"}, {"start": 26.08, "duration": 3.279, "text": "be a three-membered ring or a"}, {"start": 27.32, "duration": 3.52, "text": "six-membered ring but as long as it is a"}, {"start": 29.359, "duration": 3.681, "text": "closed Cy"}, {"start": 30.84, "duration": 4.879, "text": "the second thing is it has to be planer"}, {"start": 33.04, "duration": 4.679, "text": "so any and all atoms involved in that"}, {"start": 35.719, "duration": 4.081, "text": "cyclic component of your structure even"}, {"start": 37.719, "duration": 5.441, "text": "if it's a bigger structure have to be"}, {"start": 39.8, "duration": 4.84, "text": "SP2 hybridized the third and final piece"}, {"start": 43.16, "duration": 5.039, "text": "is that you're going to have to have an"}, {"start": 44.64, "duration": 5.0, "text": "odd number of P Cloud electron pairs so"}, {"start": 48.199, "duration": 3.161, "text": "let's see what this looks like in"}, {"start": 49.64, "duration": 3.52, "text": "practice okay so now let's"}, {"start": 51.36, "duration": 3.719, "text": "systematically apply these three rules"}, {"start": 53.16, "duration": 3.239, "text": "to each of these four compounds to"}, {"start": 55.079, "duration": 3.64, "text": "determine whether or not they're going"}, {"start": 56.399, "duration": 4.041, "text": "to be aromatic so if we take a look at"}, {"start": 58.719, "duration": 4.041, "text": "this first compound here we see that it"}, {"start": 60.44, "duration": 4.679, "text": "meets the first criteria it does have a"}, {"start": 62.76, "duration": 4.48, "text": "cycle in it the second criteria is that"}, {"start": 65.119, "duration": 3.561, "text": "it has to be planer and if we take a"}, {"start": 67.24, "duration": 4.16, "text": "look all of the carbons that are"}, {"start": 68.68, "duration": 4.079, "text": "participating in this cycle here are SP2"}, {"start": 71.4, "duration": 3.96, "text": "hybridized which would mean they would"}, {"start": 72.759, "duration": 5.161, "text": "be planer remember that this carbon here"}, {"start": 75.36, "duration": 4.88, "text": "even though it's sp3 doesn't impact our"}, {"start": 77.92, "duration": 4.8, "text": "aromaticity because it's not directly"}, {"start": 80.24, "duration": 4.32, "text": "part of this ring the third rule is that"}, {"start": 82.72, "duration": 5.68, "text": "we need to have an odd number of py"}, {"start": 84.56, "duration": 6.44, "text": "cloud electron pairs so here we have 1 2"}, {"start": 88.4, "duration": 4.679, "text": "3 so this one meets all three criteria"}, {"start": 91.0, "duration": 4.88, "text": "so this here would be an aromatic"}, {"start": 93.079, "duration": 4.881, "text": "compound a really stable compound let's"}, {"start": 95.88, "duration": 3.84, "text": "take a look at the second one so it's"}, {"start": 97.96, "duration": 4.68, "text": "got the cycle so good on the first"}, {"start": 99.72, "duration": 4.719, "text": "criteria the second one has to be planer"}, {"start": 102.64, "duration": 3.479, "text": "so we can see that all of the carbons"}, {"start": 104.439, "duration": 4.96, "text": "that are participating in those double"}, {"start": 106.119, "duration": 5.96, "text": "bonds would be SP2 hybridized these ones"}, {"start": 109.399, "duration": 5.68, "text": "here though would not they both have two"}, {"start": 112.079, "duration": 5.881, "text": "hydrogens implied coming off of them so"}, {"start": 115.079, "duration": 5.441, "text": "they would be sp3 hybridized and because"}, {"start": 117.96, "duration": 5.159, "text": "of that this would not be a plain cyclic"}, {"start": 120.52, "duration": 4.8, "text": "compound so this one here would not be"}, {"start": 123.119, "duration": 5.28, "text": "aromatic let's try again on the third"}, {"start": 125.32, "duration": 5.12, "text": "one so here cyclic it is it may look"}, {"start": 128.399, "duration": 5.321, "text": "like a square but that's still a cyclic"}, {"start": 130.44, "duration": 5.799, "text": "compound the second criteria planer all"}, {"start": 133.72, "duration": 4.72, "text": "of these carbons are SP2 hybridized so"}, {"start": 136.239, "duration": 4.161, "text": "we're doing well so far the third"}, {"start": 138.44, "duration": 4.439, "text": "criteria an odd number of py cloud"}, {"start": 140.4, "duration": 4.919, "text": "electrons in this case we actually have"}, {"start": 142.879, "duration": 5.881, "text": "two so this would be a different kind of"}, {"start": 145.319, "duration": 5.64, "text": "compound and it would be called"}, {"start": 148.76, "duration": 4.199, "text": "anti-aromatic so whereas aromatic"}, {"start": 150.959, "duration": 4.161, "text": "compounds are particularly stable"}, {"start": 152.959, "duration": 4.481, "text": "anti-aromatic compounds are particularly"}, {"start": 155.12, "duration": 5.0, "text": "unstable so this would definitely not"}, {"start": 157.44, "duration": 5.92, "text": "fall in the aromatic category and the"}, {"start": 160.12, "duration": 5.199, "text": "fourth and final one here we see cyclic"}, {"start": 163.36, "duration": 4.599, "text": "so we're meeting that criteria the"}, {"start": 165.319, "duration": 7.321, "text": "second one SP2 hybridize so this would"}, {"start": 167.959, "duration": 7.2, "text": "be SP2 SP2 SP2 and SP2 this one here"}, {"start": 172.64, "duration": 4.599, "text": "kind of defies expectation so this would"}, {"start": 175.159, "duration": 5.681, "text": "have a hydrogen coming off and if you"}, {"start": 177.239, "duration": 5.561, "text": "counted you have 1 2 3 four areas of"}, {"start": 180.84, "duration": 4.8, "text": "electron density which is typically"}, {"start": 182.8, "duration": 4.56, "text": "going to be labeled sp3 but now this is"}, {"start": 185.64, "duration": 3.92, "text": "the thing right we have all these rules"}, {"start": 187.36, "duration": 4.36, "text": "to help us understand molecules but they"}, {"start": 189.56, "duration": 4.039, "text": "don't follow our rules and they do"}, {"start": 191.72, "duration": 4.68, "text": "whatever leads to the greatest stability"}, {"start": 193.599, "duration": 6.161, "text": "most often so in this case here if this"}, {"start": 196.4, "duration": 6.199, "text": "one adopted an SP2 hybridization then it"}, {"start": 199.76, "duration": 4.8, "text": "would have one 2 three pie Cloud"}, {"start": 202.599, "duration": 4.28, "text": "electron pairs meaning that would make"}, {"start": 204.56, "duration": 4.52, "text": "it aromatic and that actually is what"}, {"start": 206.879, "duration": 3.44, "text": "this compound does so this one may not"}, {"start": 209.08, "duration": 3.159, "text": "look at initial"}, {"start": 210.319, "duration": 4.081, "text": "but this one would fall under an"}, {"start": 212.239, "duration": 4.041, "text": "aromatic heading so that's pretty much"}, {"start": 214.4, "duration": 5.039, "text": "how you apply the rules to understand"}, {"start": 216.28, "duration": 3.159, "text": "whether or not a compound is"}, {"start": 222.81, "duration": 16.269, "text": "[Music]"}, {"start": 236.079, "duration": 3.0, "text": "aromatic"}], "title": "What Is An Aromatic Compound?", "description": "This video discusses the criteria for aromaticity and how to apply it to a compound to determine if it is aromatic.", "lengthSeconds": 237, "width": 640, "height": 360}
{"video_id": "XzG-fi__hsU", "transcript": [{"start": 0.88, "duration": 5.8, "text": "now let's say if we're given an equation"}, {"start": 3.8, "duration": 2.88, "text": "in point slope"}, {"start": 6.759, "duration": 5.88, "text": "form how can we graph this equation on a"}, {"start": 10.2, "duration": 2.439, "text": "coordinate"}, {"start": 14.599, "duration": 4.52, "text": "plane what would you do in order to"}, {"start": 16.88, "duration": 2.239, "text": "graph"}, {"start": 24.359, "duration": 3.961, "text": "it now the first thing you should do is"}, {"start": 26.679, "duration": 4.121, "text": "identify the"}, {"start": 28.32, "duration": 4.96, "text": "slope and the point"}, {"start": 30.8, "duration": 4.88, "text": "the XY ordered pair so here's the"}, {"start": 33.28, "duration": 2.4, "text": "original"}, {"start": 36.559, "duration": 5.32, "text": "equation so we can see that the slope is"}, {"start": 40.0, "duration": 5.239, "text": "equal to"}, {"start": 41.879, "duration": 5.921, "text": "two so m is"}, {"start": 45.239, "duration": 6.441, "text": "two and then we can see that"}, {"start": 47.8, "duration": 8.32, "text": "X1 is positive 1 not negative 1 but"}, {"start": 51.68, "duration": 5.199, "text": "positive 1 and y1 is posi 3 so we have"}, {"start": 56.12, "duration": 7.079, "text": "the"}, {"start": 56.879, "duration": 8.121, "text": "point 1 comma 3 X is 1 and Y is 3 so"}, {"start": 63.199, "duration": 5.441, "text": "what you want to do is plot the point"}, {"start": 65.0, "duration": 5.64, "text": "first 1 comma 3 is over here and the"}, {"start": 68.64, "duration": 4.519, "text": "slope is equal to"}, {"start": 70.64, "duration": 4.32, "text": "two that means as you travel one unit to"}, {"start": 73.159, "duration": 5.081, "text": "the right you need to go up two"}, {"start": 74.96, "duration": 6.519, "text": "units so the next point is going to be"}, {"start": 78.24, "duration": 5.839, "text": "at 2 comma 5 so if you go one to the"}, {"start": 81.479, "duration": 5.401, "text": "left you need to go down to"}, {"start": 84.079, "duration": 7.841, "text": "units now granted this Line's not going"}, {"start": 86.88, "duration": 5.04, "text": "to be perfectly straight but you get the"}, {"start": 93.759, "duration": 4.201, "text": "picture and then just connect them with"}, {"start": 95.72, "duration": 5.96, "text": "a straight"}, {"start": 97.96, "duration": 3.72, "text": "line let's try another"}, {"start": 102.32, "duration": 8.079, "text": "example so let's say if we have this"}, {"start": 104.399, "duration": 11.241, "text": "equation y + 3 is equal to 32 * x +"}, {"start": 110.399, "duration": 8.241, "text": "4 so identify the slope and the point we"}, {"start": 115.64, "duration": 7.439, "text": "can see that the slope is equal to 3"}, {"start": 118.64, "duration": 4.439, "text": "over2 now what's X1 and what's"}, {"start": 123.24, "duration": 8.96, "text": "y1 X1 is -4 y1 is3 basically you just"}, {"start": 130.16, "duration": 4.92, "text": "got to change the sign of these numbers"}, {"start": 132.2, "duration": 4.92, "text": "if you see a plus4 make it a Nega four"}, {"start": 135.08, "duration": 3.96, "text": "if you see a positive three make it a"}, {"start": 137.12, "duration": 4.44, "text": "negative3 if you see a negative3 make it"}, {"start": 139.04, "duration": 4.68, "text": "a positive three so just reverse the"}, {"start": 141.56, "duration": 6.12, "text": "signs and you're going to get the point"}, {"start": 143.72, "duration": 3.96, "text": "which is4 comm3"}, {"start": 151.239, "duration": 3.321, "text": "so now we can go ahead and plot"}, {"start": 160.64, "duration": 8.239, "text": "it the first point is at"}, {"start": 165.48, "duration": 8.08, "text": "-43 and the slope is 3 over"}, {"start": 168.879, "duration": 7.64, "text": "two three is the rise two is the"}, {"start": 173.56, "duration": 5.16, "text": "run so we're going to travel two units"}, {"start": 176.519, "duration": 5.321, "text": "to the right and we're going to go up"}, {"start": 178.72, "duration": 3.12, "text": "three units"}, {"start": 183.0, "duration": 7.319, "text": "so the next point that we're going to"}, {"start": 185.28, "duration": 7.72, "text": "get is -2 Comm 0 x is -2 Y is"}, {"start": 190.319, "duration": 5.0, "text": "z so now let's use the slope again let's"}, {"start": 193.0, "duration": 5.0, "text": "travel two units to the right and let's"}, {"start": 195.319, "duration": 4.361, "text": "go up three units so this will take us"}, {"start": 198.0, "duration": 4.879, "text": "to the"}, {"start": 199.68, "duration": 7.0, "text": "point0 comma"}, {"start": 202.879, "duration": 8.841, "text": "3 which is a y intercept of"}, {"start": 206.68, "duration": 5.04, "text": "three and now let's connect these points"}, {"start": 211.799, "duration": 5.44, "text": "so now you know how to graph a linear"}, {"start": 214.0, "duration": 6.72, "text": "equation in point slope form so all you"}, {"start": 217.239, "duration": 5.161, "text": "need to do is find the X and Y values of"}, {"start": 220.72, "duration": 4.079, "text": "the point the first point that you need"}, {"start": 222.4, "duration": 3.399, "text": "to plot and then use the slope to get"}, {"start": 224.799, "duration": 4.0, "text": "the next"}, {"start": 225.799, "duration": 4.401, "text": "Point all you need is two points if you"}, {"start": 228.799, "duration": 3.241, "text": "have two points you can draw a straight"}, {"start": 230.2, "duration": 4.759, "text": "line if you want at the third Point"}, {"start": 232.04, "duration": 4.559, "text": "that's fine too but you just need two"}, {"start": 234.959, "duration": 4.56, "text": "points at minimum to graph a straight"}, {"start": 236.599, "duration": 2.92, "text": "line"}], "title": "How To Graph Linear Equations In Point Slope Form | Algebra", "description": "This algebra video tutorial provides a basic introduction on how to graph linear equations in point slope form.  It contains a few examples and practice problems.  \n\nLinear Equations - Free Formula Sheet:       https://bit.ly/3WxIvJF\n\n______________________________\nAlgebra For Beginners:                                          \nhttps://www.youtube.com/watch?v=MHeirBPOI6w\n\nHow To Solve Linear Equations:                         \nhttps://www.youtube.com/watch?v=7DPWeBszNSM\n\nSolving Linear Equations - More Problems:      \nhttps://www.youtube.com/watch?v=Z-ZkmpQBIFo\n\nTranslating Words to Algebraic Expressions:   \nhttps://www.youtube.com/watch?v=QEnFIgN8UBw\n\nGraphing Ordered Pairs:                                       \nhttps://www.youtube.com/watch?v=yaOf7ioCgFQ\n\nHow To Find The Slope of a Line:                       \nhttps://www.youtube.com/watch?v=jlkE4VCnhdE\n\n________________________________\nFinding The Slope Given 2 Points:                      \nhttps://www.youtube.com/watch?v=Id_UqMLAXzY\n\nUndefined Slope:                                                    \nhttps://www.youtube.com/watch?v=4eYlTOy-IXw\n\nSlope-Intercept Form:                                           \nhttps://www.youtube.com/watch?v=qXX47hS2KLw\n\nSlope-Intercept Form to Standard Form:           \nhttps://www.youtube.com/watch?v=mCzVZptDhvU\n\nHow To Find the X and Y Intercepts:                  \nhttps://www.youtube.com/watch?v=zL1E5w2sMxc\n\nPoint-Slope Form:                                                  \nhttps://www.youtube.com/watch?v=yoHs1h5qtuQ\n\n____________________________________\nGraphing Equations In Point Slope Form:          \nhttps://www.youtube.com/watch?v=XzG-fi__hsU\n\nGraphing Horizontal and Vertical Lines:            \nhttps://www.youtube.com/watch?v=ADGduYvdFmc\n\nWriting Equations Given a Point & Slope:          \nhttps://www.youtube.com/watch?v=Ms5ojdI6UuM\n\nWriting Linear Equations Given 2 Points:          \nhttps://www.youtube.com/watch?v=lzqTD0JWwhY\n\nWriting Linear Equations From Graphs:             \nhttps://www.youtube.com/watch?v=JaX_dIDUYBg\n\nFinding The Point of Intersection:                      \nhttps://www.youtube.com/watch?v=R0bGxNzgL2o\n\n______________________________________\nWriting Equations Given X & Y Intercepts:         \nhttps://www.youtube.com/watch?v=ri1bagtR08A\n\nWriting Equations In Standard Form:                 \nhttps://www.youtube.com/watch?v=CN4-ty9e5Jk\n\nWriting Equations Using a Function Table:        \nhttps://www.youtube.com/watch?v=DffvEuX1v60\n\nParallel and Perpendicular Lines:                       \nhttps://www.youtube.com/watch?v=Uq8pSFaXPmQ\n\nWriting Equations of Perpendicular Lines:        \nhttps://www.youtube.com/watch?v=Nj6OKT-zKWg\n\nLinear Equations - Test Review:                          \nhttps://www.youtube.com/watch?v=Ft2_QtXAnh8\n\nFinal Exams and Video Playlists:\nhttps://www.video-tutor.net/", "lengthSeconds": 238, "width": 640, "height": 360}
{"video_id": "9_Y8P3oHwL0", "transcript": [{"start": 1.04, "duration": 5.92, "text": "in this video we're going to talk about"}, {"start": 4.16, "duration": 4.399, "text": "polygons a polygon with three sides is"}, {"start": 6.96, "duration": 5.96, "text": "known as a"}, {"start": 8.559, "duration": 6.441, "text": "triangle a polygon with four sides is a"}, {"start": 12.92, "duration": 6.32, "text": "quadrilateral now if we're dealing with"}, {"start": 15.0, "duration": 7.199, "text": "regular polygons all four sides are"}, {"start": 19.24, "duration": 4.119, "text": "congruent and all angles are the same"}, {"start": 22.199, "duration": 4.041, "text": "which makes this particular"}, {"start": 23.359, "duration": 2.881, "text": "quadrilateral a"}, {"start": 26.72, "duration": 5.92, "text": "square a five-sided polygon"}, {"start": 30.56, "duration": 6.04, "text": "is known as a"}, {"start": 32.64, "duration": 9.28, "text": "pentagon and a polygon with six"}, {"start": 36.6, "duration": 9.799, "text": "sides is known as a"}, {"start": 41.92, "duration": 4.479, "text": "hexagon a polygon with eight"}, {"start": 47.28, "duration": 5.599, "text": "sides is known as an"}, {"start": 50.559, "duration": 4.761, "text": "octagon so those are some common names"}, {"start": 52.879, "duration": 6.48, "text": "that you want to be familiar"}, {"start": 55.32, "duration": 5.919, "text": "with now let's say if we have a regular"}, {"start": 59.359, "duration": 3.561, "text": "pentagon"}, {"start": 61.239, "duration": 6.601, "text": "what is the"}, {"start": 62.92, "duration": 7.08, "text": "measure of an interior angle inside a"}, {"start": 67.84, "duration": 5.2, "text": "pentagon the first thing you need to do"}, {"start": 70.0, "duration": 5.64, "text": "is find the sum of all the interior"}, {"start": 73.04, "duration": 7.52, "text": "angles in a polygon and you could use"}, {"start": 75.64, "duration": 6.96, "text": "this formula it's N - 2 * 180 where n"}, {"start": 80.56, "duration": 5.239, "text": "represents the number of sides in a"}, {"start": 82.6, "duration": 5.08, "text": "polygon in a pentagon there are five"}, {"start": 85.799, "duration": 3.881, "text": "sides so n is"}, {"start": 87.68, "duration": 4.92, "text": "five so the first thing you want to do"}, {"start": 89.68, "duration": 6.799, "text": "is subtract 5 by two which is three and"}, {"start": 92.6, "duration": 8.92, "text": "then take that number multiply it by"}, {"start": 96.479, "duration": 8.041, "text": "180\u00b0 so the sum of all five angles in a"}, {"start": 101.52, "duration": 5.72, "text": "pentagon is"}, {"start": 104.52, "duration": 7.239, "text": "540\u00b0 now to find the interior angle"}, {"start": 107.24, "duration": 7.28, "text": "which we can call X take the total angle"}, {"start": 111.759, "duration": 4.68, "text": "and divide it by the number of sides so"}, {"start": 114.52, "duration": 6.04, "text": "we're going to divide it by"}, {"start": 116.439, "duration": 10.921, "text": "five 500 ID 5 is 100"}, {"start": 120.56, "duration": 9.28, "text": "40 / 5 is 8 so 540 / 5 is 100 + 8 or"}, {"start": 127.36, "duration": 8.0, "text": "108 so that's going to be the measure of"}, {"start": 129.84, "duration": 7.759, "text": "each interior angle inside a"}, {"start": 135.36, "duration": 5.68, "text": "pentagon so every angle that you see"}, {"start": 137.599, "duration": 3.441, "text": "inside of it is"}, {"start": 142.239, "duration": 6.241, "text": "108\u00b0 with a total angle of"}, {"start": 145.4, "duration": 5.4, "text": "540 so now it's your turn find the"}, {"start": 148.48, "duration": 6.039, "text": "measure of each interior"}, {"start": 150.8, "duration": 3.719, "text": "angle inside a"}, {"start": 156.239, "duration": 5.92, "text": "hexagon so first we need to find the sum"}, {"start": 159.959, "duration": 5.36, "text": "of all interior"}, {"start": 162.159, "duration": 4.481, "text": "angles so let's use the same Formula nus"}, {"start": 165.319, "duration": 4.321, "text": "2 *"}, {"start": 166.64, "duration": 9.2, "text": "180 a hexagon has six"}, {"start": 169.64, "duration": 7.599, "text": "sides 6 - 2 is 4 so then this is 4 * 180"}, {"start": 175.84, "duration": 4.52, "text": "which is"}, {"start": 177.239, "duration": 5.201, "text": "720 so now we're going to take 720"}, {"start": 180.36, "duration": 4.12, "text": "and divid by the number of sides of a"}, {"start": 182.44, "duration": 8.96, "text": "hexagon which is"}, {"start": 184.48, "duration": 9.88, "text": "6 72 / 6 is 12 so 720 ID 6 is 120 so"}, {"start": 191.4, "duration": 5.44, "text": "that's the measure of each interior"}, {"start": 194.36, "duration": 7.84, "text": "angle"}, {"start": 196.84, "duration": 5.36, "text": "inside the hexagon every angle is"}, {"start": 208.76, "duration": 3.0, "text": "120\u00b0"}, {"start": 221.28, "duration": 3.0, "text": "for"}], "title": "Interior Angles of a Polygon - Geometry", "description": "This geometry video tutorial focuses on polygons and explains how to calculate the interior angle of a polygon such as hexagons, pentagons, and octagons.\n\nPre-Algebra Video Playlist:\nhttps://www.youtube.com/watch?v=WJqw-cxvKgo&list=PL0o_zxa4K1BVoTlaXWFcFZ7fU3RvmFMMG", "lengthSeconds": 222, "width": 640, "height": 360}
{"video_id": "DuZtXgSUfdM", "transcript": [{"start": 0.0, "duration": 4.56, "text": "in a chemical reaction the reactants are"}, {"start": 2.34, "duration": 3.72, "text": "consumed in specific ratios based on the"}, {"start": 4.56, "duration": 3.359, "text": "Stoichiometry of the balanced chemical"}, {"start": 6.06, "duration": 4.02, "text": "equation in this equation for every one"}, {"start": 7.919, "duration": 3.841, "text": "mole of ethene three moles of oxygen gas"}, {"start": 10.08, "duration": 3.54, "text": "react we could say that the reactants"}, {"start": 11.76, "duration": 3.48, "text": "react in a one to three ratio when"}, {"start": 13.62, "duration": 3.419, "text": "there's a limiting reactant it means the"}, {"start": 15.24, "duration": 3.9, "text": "correct ratio isn't met and there will"}, {"start": 17.039, "duration": 3.9, "text": "be some leftover reactant and excess"}, {"start": 19.14, "duration": 3.24, "text": "reactant the limiting reactant is the"}, {"start": 20.939, "duration": 3.0, "text": "reactant that runs out first and"}, {"start": 22.38, "duration": 3.42, "text": "determines the maximum amount of"}, {"start": 23.939, "duration": 4.561, "text": "products that can be produced in this"}, {"start": 25.8, "duration": 5.52, "text": "problem we have two reactants 2.7 moles"}, {"start": 28.5, "duration": 4.32, "text": "of ethene reacts with 6.3 moles of o2 to"}, {"start": 31.32, "duration": 2.94, "text": "determine the limiting reactant we need"}, {"start": 32.82, "duration": 3.3, "text": "to compare these amounts to the mole"}, {"start": 34.26, "duration": 3.9, "text": "ratio of the reactants in the ballast"}, {"start": 36.12, "duration": 5.34, "text": "equation so we can calculate the number"}, {"start": 38.16, "duration": 6.0, "text": "of moles of o2 needed if all 2.7 moles"}, {"start": 41.46, "duration": 5.46, "text": "of ethylene reacts we just multiply 2.7"}, {"start": 44.16, "duration": 5.46, "text": "moles by the ratio of o2 to ethene which"}, {"start": 46.92, "duration": 5.459, "text": "is three over one so if all 2.7 moles of"}, {"start": 49.62, "duration": 5.22, "text": "ethene reacts 8.1 moles of o2 is"}, {"start": 52.379, "duration": 5.041, "text": "required however we only have 6.3 moles"}, {"start": 54.84, "duration": 4.379, "text": "of o2 not 8.1 this means that O2 is a"}, {"start": 57.42, "duration": 4.159, "text": "limiting reactant and ethine is in"}, {"start": 59.219, "duration": 2.36, "text": "excess"}], "title": "Limiting Reactant made SIMPLE \ud83d\ude03\ud83d\udcaf Chemistry Tutoring #chemistry #shorts #education #homework", "description": "The equation for the complete combustion of ethene (C2H4) is\nC2H4(g) + 3O2(g)    =    2CO2(g) + 2H2O(g)\nIf 2.70 mol C2H4 reacts with 6.30 mole of O2, what is the limiting reactant? \n\n\nchemistry tutor\nchemistry homework \n\n#science #chemistry #education #homework #youtubeshorts #stem #stemeducation", "lengthSeconds": 60, "width": 360, "height": 640}
{"video_id": "eBTNzScLUg4", "transcript": [{"start": 1.12, "duration": 3.12, "text": "in this video we're going to talk about"}, {"start": 2.72, "duration": 4.559, "text": "the law"}, {"start": 4.24, "duration": 6.319, "text": "of conservation of mass"}, {"start": 7.279, "duration": 5.121, "text": "so what is the main idea behind this law"}, {"start": 10.559, "duration": 4.16, "text": "the main idea is that"}, {"start": 12.4, "duration": 4.16, "text": "matter cannot be created or destroyed in"}, {"start": 14.719, "duration": 4.001, "text": "a chemical reaction"}, {"start": 16.56, "duration": 5.52, "text": "so let me give you an example"}, {"start": 18.72, "duration": 4.799, "text": "let's say if we react methane gas"}, {"start": 22.08, "duration": 2.959, "text": "with oxygen"}, {"start": 23.519, "duration": 3.361, "text": "so let's say if we have a closed"}, {"start": 25.039, "duration": 2.881, "text": "container"}, {"start": 26.88, "duration": 3.84, "text": "and"}, {"start": 27.92, "duration": 5.599, "text": "we're going to make 16 grams of methane"}, {"start": 30.72, "duration": 2.799, "text": "in this container"}, {"start": 33.52, "duration": 5.6, "text": "we're also going to place 64 grams of"}, {"start": 36.399, "duration": 2.721, "text": "oxygen gas"}, {"start": 39.76, "duration": 2.479, "text": "so"}, {"start": 40.64, "duration": 2.56, "text": "the methane and the oxygen gas they"}, {"start": 42.239, "duration": 3.921, "text": "cannot"}, {"start": 43.2, "duration": 5.359, "text": "leave the container they're stuck there"}, {"start": 46.16, "duration": 3.919, "text": "now as the reaction proceeds it's a"}, {"start": 48.559, "duration": 3.52, "text": "combustion reaction it's going to"}, {"start": 50.079, "duration": 4.401, "text": "produce carbon dioxide"}, {"start": 52.079, "duration": 6.12, "text": "and water vapor"}, {"start": 54.48, "duration": 3.719, "text": "depending on the temperature"}, {"start": 58.96, "duration": 4.64, "text": "now let's say if you collect 44 grams of"}, {"start": 61.84, "duration": 3.679, "text": "carbon dioxide"}, {"start": 63.6, "duration": 2.8, "text": "according to the law of conservation of"}, {"start": 65.519, "duration": 3.041, "text": "mass"}, {"start": 66.4, "duration": 3.68, "text": "how many grams of water"}, {"start": 68.56, "duration": 2.96, "text": "should you expect"}, {"start": 70.08, "duration": 3.68, "text": "to be present at the end of this"}, {"start": 71.52, "duration": 3.84, "text": "reaction"}, {"start": 73.76, "duration": 4.08, "text": "now think of the total mass of the"}, {"start": 75.36, "duration": 5.759, "text": "reactants the reactants"}, {"start": 77.84, "duration": 3.279, "text": "are the stuff on the left side"}, {"start": 81.2, "duration": 5.52, "text": "we have 16 grams of methane"}, {"start": 84.159, "duration": 3.841, "text": "and 64 grams of oxygen"}, {"start": 86.72, "duration": 3.84, "text": "so therefore"}, {"start": 88.0, "duration": 5.04, "text": "we're starting with 80 grams"}, {"start": 90.56, "duration": 4.559, "text": "of reactants"}, {"start": 93.04, "duration": 5.119, "text": "so before the reaction"}, {"start": 95.119, "duration": 5.441, "text": "we had 80 grams of matter"}, {"start": 98.159, "duration": 4.561, "text": "at the end of the reaction according to"}, {"start": 100.56, "duration": 3.44, "text": "the law of conservation of mass in a"}, {"start": 102.72, "duration": 5.12, "text": "closed system"}, {"start": 104.0, "duration": 7.6, "text": "you should still have 80 grams of mass"}, {"start": 107.84, "duration": 5.36, "text": "so if we have 44 grams of carbon dioxide"}, {"start": 111.6, "duration": 4.32, "text": "the remaining amount of mass should be"}, {"start": 113.2, "duration": 4.4, "text": "water 80 minus 44"}, {"start": 115.92, "duration": 3.199, "text": "is 36"}, {"start": 117.6, "duration": 4.24, "text": "so therefore"}, {"start": 119.119, "duration": 3.68, "text": "we should expect to collect 36 grams of"}, {"start": 121.84, "duration": 2.559, "text": "water"}, {"start": 122.799, "duration": 3.68, "text": "and that's the main idea behind the law"}, {"start": 124.399, "duration": 4.801, "text": "of conservation of mass"}, {"start": 126.479, "duration": 5.76, "text": "if you have 80 grams of material inside"}, {"start": 129.2, "duration": 5.28, "text": "the closed container before the reaction"}, {"start": 132.239, "duration": 5.521, "text": "then when the reaction is complete you"}, {"start": 134.48, "duration": 5.28, "text": "should still have 80 grams of matter"}, {"start": 137.76, "duration": 3.52, "text": "inside that container"}, {"start": 139.76, "duration": 2.8, "text": "in a chemical reaction the only thing"}, {"start": 141.28, "duration": 3.039, "text": "that's happening"}, {"start": 142.56, "duration": 3.36, "text": "is that the atoms are rearranging"}, {"start": 144.319, "duration": 4.0, "text": "themselves"}, {"start": 145.92, "duration": 3.44, "text": "here carbon is bonded to hydrogen but"}, {"start": 148.319, "duration": 3.121, "text": "now"}, {"start": 149.36, "duration": 4.08, "text": "it's bonded to oxygen"}, {"start": 151.44, "duration": 3.519, "text": "so therefore the total mass"}, {"start": 153.44, "duration": 4.24, "text": "should remain the same"}, {"start": 154.959, "duration": 4.881, "text": "the atoms are simply switching partners"}, {"start": 157.68, "duration": 3.6, "text": "they just rearrange themselves in its"}, {"start": 159.84, "duration": 3.36, "text": "chemical reaction"}, {"start": 161.28, "duration": 4.72, "text": "but the amount of mass the quantity of"}, {"start": 163.2, "duration": 4.88, "text": "matter should still remain the same"}, {"start": 166.0, "duration": 3.92, "text": "the only way you can change it is if you"}, {"start": 168.08, "duration": 6.6, "text": "add more mass into the"}, {"start": 169.92, "duration": 4.76, "text": "chamber or the reaction container"}, {"start": 193.2, "duration": 2.08, "text": "you"}], "title": "Law of Conservation of Mass - Fundamental Chemical Laws, Chemistry", "description": "This chemistry video tutorial discusses the law of conservation of mass and provides examples associated with chemical reactions.  The conservation of mass in a chemical reaction is one of the fundamental laws in chemistry.  The total mass of the reactants must equal the total mass of the products in a chemical reaction.\n\nChemistry - Basic Introduction:                            \nhttps://www.youtube.com/watch?v=-KfG8kH-r3Y\n\nScientific Notation Review:                                   \nhttps://www.youtube.com/watch?v=ZtB0vJMGve4\n\nSignificant Figures Review:                                   \nhttps://www.youtube.com/watch?v=l2yuDvwYq5g\n\nUnit Conversion Problems:                                   \nhttps://www.youtube.com/watch?v=eK8gXP3pImU\n\nAccuracy and Precision:                                        \nhttps://www.youtube.com/watch?v=0IiHPKAvo7g\n\nDensity Practice Problems:                                  \nhttps://www.youtube.com/watch?v=9CKDQE35qXQ\n\n________________________________\nPure Substances & Mixtures:                               \nhttps://www.youtube.com/watch?v=OHhnm2p5G3o\n\nHomogeneous & Heterogeneous Mixtures:       \nhttps://www.youtube.com/watch?v=eI-tmv4DLEk  \n    \nPhysical and Chemical Changes:                         \nhttps://www.youtube.com/watch?v=YE2xaMsoGFU   \n\nSolids, Liquids, Gases, & Plasma:                         \nhttps://www.youtube.com/watch?v=9TVOlTolKFA   \n\nPhysical Vs Chemical Properties:                        \nhttps://www.youtube.com/watch?v=gH1R87ahFvA    \n\n__________________________________\nLaw of Conservation of Mass:                             \nhttps://www.youtube.com/watch?v=eBTNzScLUg4\n\nLaw of Definite Proportions:                                 \nhttps://www.youtube.com/watch?v=ly0ywRdVG_M\n\nLaw of Multiple Proportions:                                \nhttps://www.youtube.com/watch?v=sxE95VOY-YY\n\nRutherford's Gold Foil Experiment:                      \nhttps://www.youtube.com/watch?v=sNQsdrqsD_s\n\nCathode Ray Tube Experiment:                            \nhttps://www.youtube.com/watch?v=i6zyPOSreCg\n\n_________________________________\nAtoms - Basic Introduction:                                  \nhttps://www.youtube.com/watch?v=acdkMeEKCNQ\n\nCations and Anions Explained:                            \nhttps://www.youtube.com/watch?v=aAV2DMAI5f8\n\nDiatomic Elements & Molecules:                         \nhttps://www.youtube.com/watch?v=gi337Mx7wTc\n\nElements, Atoms, & Molecules:                            \nhttps://www.youtube.com/watch?v=pSJeMJaCkVU\n\nProtons, Neutrons, & Electrons:                           \nhttps://www.youtube.com/watch?v=65dDZulPhtg \n\n_______________________________\nFinal Exams and Video Playlists:   \nhttps://www.video-tutor.net/\n\nFull-Length Videos and Worksheets:\nhttps://www.patreon.com/MathScienceTutor/collections", "lengthSeconds": 193, "width": 640, "height": 360}
{"video_id": "p6-Cut8Z3uk", "transcript": [{"start": 0.199, "duration": 5.68, "text": "so how many stereoisomers can you have"}, {"start": 2.879, "duration": 6.401, "text": "for this molecule well let's investigate"}, {"start": 5.879, "duration": 5.561, "text": "we have two chyal atoms so that means"}, {"start": 9.28, "duration": 5.239, "text": "that we can have a maximum number of"}, {"start": 11.44, "duration": 5.52, "text": "stereoisomers here being four but we"}, {"start": 14.519, "duration": 5.001, "text": "also have a plane of symmetry making"}, {"start": 16.96, "duration": 5.28, "text": "things a tad more complicated so how"}, {"start": 19.52, "duration": 5.96, "text": "many stere isomers do we have here then"}, {"start": 22.24, "duration": 5.92, "text": "well we have a miso isomer and we have a"}, {"start": 25.48, "duration": 5.24, "text": "pair of Anan tumers which means that we"}, {"start": 28.16, "duration": 5.56, "text": "have a total of three stereo isomers for"}, {"start": 30.72, "duration": 3.0, "text": "this molecule"}], "title": "How many stereoisomers can we have for this molecule? #organicchemistry #stereochemistry", "description": "\ud83e\uddea More tutorials, practice questions, and organic chemistry workbooks \ud83e\uddea\nhttps://www.organicchemistrytutor.com/courses/organic-chemistry/\n\n\ud83d\udcdd Download the Organic Chemistry Study Notes \ud83d\udcdd \nhttps://www.organicchemistrytutor.com/shop/\n\n\ud83d\udc4b Connect with me on social media:\nInstagram: https://www.instagram.com/ochemtutor/\nDiscord: https://discord.com/invite/EJwkJf2mrg", "lengthSeconds": 32, "width": 270, "height": 480}
{"video_id": "1YDjVoOinU4", "transcript": [{"start": 0.0, "duration": 3.78, "text": "let's double check your work so I'm just"}, {"start": 1.86, "duration": 3.899, "text": "going to start off here so we started"}, {"start": 3.78, "duration": 6.68, "text": "with a ch3 at the end which is correct"}, {"start": 5.759, "duration": 7.141, "text": "then we move on to CH so we have that CH"}, {"start": 10.46, "duration": 4.36, "text": "this is in parentheses telling us that"}, {"start": 12.9, "duration": 2.58, "text": "this belongs to the carbon prior which"}, {"start": 14.82, "duration": 3.66, "text": "is"}, {"start": 15.48, "duration": 7.139, "text": "good so we have that ch3 then continuing"}, {"start": 18.48, "duration": 7.26, "text": "we're at ch2 ch2 next that's bonded to"}, {"start": 22.619, "duration": 6.781, "text": "this other carbon which has technically"}, {"start": 25.74, "duration": 5.64, "text": "three of chch3s but at least looking at"}, {"start": 29.4, "duration": 4.74, "text": "this guy here it tells us that there are"}, {"start": 31.38, "duration": 5.339, "text": "two ch3s specifically bonded to that"}, {"start": 34.14, "duration": 5.419, "text": "carbon and this is the end has to"}, {"start": 36.719, "duration": 2.84, "text": "continue perfect"}], "title": "Condensed Structural Formula to Full Lewis Structure #organicchemistry #studywithme", "description": "Watch the full video on my youtube channel", "lengthSeconds": 38, "width": 360, "height": 640}
{"video_id": "PG0HEQq5Z8o", "transcript": [{"start": 0.0, "duration": 4.02, "text": "now this next one is going to be going"}, {"start": 2.1, "duration": 4.14, "text": "over two different types of structures"}, {"start": 4.02, "duration": 5.34, "text": "that you're going to see so you might"}, {"start": 6.24, "duration": 5.46, "text": "run into the term acyclic versus cyclic"}, {"start": 9.36, "duration": 4.26, "text": "and it's not as scary as it sounds it"}, {"start": 11.7, "duration": 4.26, "text": "really just means there's no ring for"}, {"start": 13.62, "duration": 4.919, "text": "acyclic so we have that zigzag sort of"}, {"start": 15.96, "duration": 5.899, "text": "pattern and then if it's cyclic it has a"}, {"start": 18.539, "duration": 3.32, "text": "ring okay that's it"}], "title": "Acyclic Compounds vs Cyclic Compounds #organicchemistry #study", "description": "Watch the full video on my youtube channel", "lengthSeconds": 20, "width": 360, "height": 640}
{"video_id": "0m25oTKMv1Y", "transcript": [{"start": 1.36, "duration": 2.72, "text": "in this video"}, {"start": 2.639, "duration": 3.281, "text": "we're going to talk about two ways in"}, {"start": 4.08, "duration": 4.479, "text": "which we could calculate the root mean"}, {"start": 5.92, "duration": 5.599, "text": "square molecular speed of nitrogen gas"}, {"start": 8.559, "duration": 2.96, "text": "at 300 kelvin"}, {"start": 11.759, "duration": 3.84, "text": "so let's start with this formula it's"}, {"start": 13.92, "duration": 4.88, "text": "equal to 3"}, {"start": 15.599, "duration": 4.321, "text": "kt divided by the mass"}, {"start": 18.8, "duration": 3.76, "text": "now k"}, {"start": 19.92, "duration": 5.519, "text": "is boltzmann's constant and m is the"}, {"start": 22.56, "duration": 5.76, "text": "mass of a single nitrogen gas molecule"}, {"start": 25.439, "duration": 2.881, "text": "in kilograms"}, {"start": 28.64, "duration": 5.12, "text": "now using the periodic table the atomic"}, {"start": 30.96, "duration": 5.279, "text": "mass of nitrogen is 14.01"}, {"start": 33.76, "duration": 4.799, "text": "but in this problem we have two of them"}, {"start": 36.239, "duration": 4.401, "text": "so it's 28.02"}, {"start": 38.559, "duration": 4.081, "text": "atomic mass units"}, {"start": 40.64, "duration": 4.48, "text": "now we need to convert atomic mass units"}, {"start": 42.64, "duration": 3.599, "text": "to kilograms and the conversion is"}, {"start": 45.12, "duration": 4.16, "text": "pretty simple"}, {"start": 46.239, "duration": 5.201, "text": "and so here's what you need to do"}, {"start": 49.28, "duration": 3.2, "text": "you need to know that one atomic mass"}, {"start": 51.44, "duration": 1.72, "text": "unit"}, {"start": 52.48, "duration": 2.559, "text": "is"}, {"start": 53.16, "duration": 6.68, "text": "1.66"}, {"start": 55.039, "duration": 4.801, "text": "times 10 to the negative 27 kilograms"}, {"start": 64.08, "duration": 3.52, "text": "so let's multiply those two numbers"}, {"start": 71.36, "duration": 5.04, "text": "so you should get 4.651"}, {"start": 75.2, "duration": 4.0, "text": "times 10"}, {"start": 76.4, "duration": 4.24, "text": "to the negative 26 kilograms"}, {"start": 79.2, "duration": 5.559, "text": "so what we're going to do is take this"}, {"start": 80.64, "duration": 4.119, "text": "value plug it into this equation"}, {"start": 87.84, "duration": 4.8, "text": "boltzmann's constant is 1.38"}, {"start": 91.439, "duration": 3.281, "text": "times 10"}, {"start": 92.64, "duration": 6.08, "text": "to the negative 23"}, {"start": 94.72, "duration": 5.92, "text": "and the temperature is 300 kelvin"}, {"start": 98.72, "duration": 4.16, "text": "and now let's plug in the mass that we"}, {"start": 100.64, "duration": 2.24, "text": "have"}, {"start": 104.56, "duration": 5.0, "text": "so go ahead and type this in to your"}, {"start": 106.56, "duration": 3.0, "text": "calculator"}, {"start": 118.96, "duration": 5.36, "text": "so this will give you about 517"}, {"start": 122.32, "duration": 4.56, "text": "meters per second"}, {"start": 124.32, "duration": 4.48, "text": "and so that's the answer"}, {"start": 126.88, "duration": 3.84, "text": "now in chemistry"}, {"start": 128.8, "duration": 3.2, "text": "you might see this equation used more"}, {"start": 130.72, "duration": 4.32, "text": "often"}, {"start": 132.0, "duration": 6.4, "text": "3rt divided by capital m"}, {"start": 135.04, "duration": 7.12, "text": "which is the molar mass of n2"}, {"start": 138.4, "duration": 5.44, "text": "in kilograms per mole"}, {"start": 142.16, "duration": 4.4, "text": "now the atomic mass"}, {"start": 143.84, "duration": 5.759, "text": "we said was 14.01 times two"}, {"start": 146.56, "duration": 6.16, "text": "so that gave us the molar mass of 28.02"}, {"start": 149.599, "duration": 5.521, "text": "atomic mass units which is the same as"}, {"start": 152.72, "duration": 3.92, "text": "grams per mole"}, {"start": 155.12, "duration": 2.8, "text": "but we need to convert grams to"}, {"start": 156.64, "duration": 3.36, "text": "kilograms"}, {"start": 157.92, "duration": 6.0, "text": "and you need to know that one kilogram"}, {"start": 160.0, "duration": 3.92, "text": "is equal to a thousand grams"}, {"start": 165.44, "duration": 3.84, "text": "and so"}, {"start": 166.519, "duration": 6.0, "text": "28.02 divided by a thousand that's going"}, {"start": 169.28, "duration": 3.239, "text": "to be .02802"}, {"start": 172.72, "duration": 3.44, "text": "kilograms per mole"}, {"start": 174.8, "duration": 2.32, "text": "so this is the value that we're going to"}, {"start": 176.16, "duration": 3.6, "text": "plug in"}, {"start": 177.12, "duration": 4.8, "text": "to this formula"}, {"start": 179.76, "duration": 4.96, "text": "so the root mean square velocity is"}, {"start": 181.92, "duration": 6.319, "text": "going to be the square root of 3"}, {"start": 184.72, "duration": 5.04, "text": "and r is going to be 8.3145"}, {"start": 188.239, "duration": 4.241, "text": "and the temperature"}, {"start": 189.76, "duration": 4.08, "text": "is still 300 kelvin"}, {"start": 192.48, "duration": 5.399, "text": "divided by"}, {"start": 193.84, "duration": 4.039, "text": "the molar mass of 0.02802"}, {"start": 206.319, "duration": 4.321, "text": "so this will give you the same answer"}, {"start": 208.48, "duration": 5.36, "text": "of 517"}, {"start": 210.64, "duration": 3.2, "text": "meters per second"}, {"start": 232.799, "duration": 2.08, "text": "you"}], "title": "Molecular Speed of Gases Formula With Boltzmann's Constant", "description": "This physics video tutorial provides a basic introduction into the calculation of the molecular speed of gases.  It provides two formulas.  It explains how to calculate the root mean square velocity of a gas using a formula containing boltzmann's constant and one without it.\n\nBoyle's Law Practice Problems:           \nhttps://www.youtube.com/watch?v=v8r_AU_TlPg\n\nHow Does a Bike Pump Work?\nhttps://www.youtube.com/watch?v=I0b_i6-HLBA\n\nCharles Law:\nhttps://www.youtube.com/watch?v=uZ2d79NFx2w\n\nGay Lussac's Law:\nhttps://www.youtube.com/watch?v=RszgzH_2A9k\n\nAvogadro's Law:\nhttps://www.youtube.com/watch?v=Czo2rIai5u0\n\nIdeal Gas Law Problems:\nhttps://www.youtube.com/watch?v=LZUs6nawHAk\n\n_____________________________\nTranslational Kinetic Energy:\nhttps://www.youtube.com/watch?v=bg5d5BFANVU\n\nMean Free Path and Free Time:\nhttps://www.youtube.com/watch?v=ypK7h3NKe1E\n\nRMS vs Average Speed:\nhttps://www.youtube.com/watch?v=hBlwfG6IBLg\n\nMolar Heat Capacities of Gases:\nhttps://www.youtube.com/watch?v=SwMODT8m4Ls\n\nPhase Diagrams of H2O & CO2:\nhttps://www.youtube.com/watch?v=QrHlwgmMTq4\n\n____________________________\nRelative Humidity & Dew Point:\nhttps://www.youtube.com/watch?v=BqFVtlQa-2w\n\nFick's Law of Diffusion:\nhttps://www.youtube.com/watch?v=JgAKv1Zlgcw\n\nOpen Vs Closed Vs Isolated System:\nhttps://www.youtube.com/watch?v=TFP6SvWPOQc\n\nFirst Law of Thermodynamics:\nhttps://www.youtube.com/watch?v=7Siv2NNCFag\n\nFinal Exams and Video Playlists:   \nhttps://www.video-tutor.net/\n\nPhysics PDF Worksheets:\nhttps://www.video-tutor.net/physics-basic-introduction.html", "lengthSeconds": 233, "width": 640, "height": 360}
{"video_id": "NmfaC7etZms", "transcript": [{"start": 1.12, "duration": 4.96, "text": "in this video we're going to talk about"}, {"start": 3.12, "duration": 5.439, "text": "how to decompose a function"}, {"start": 6.08, "duration": 6.16, "text": "into two functions"}, {"start": 8.559, "duration": 5.12, "text": "so for instance let's say that"}, {"start": 12.24, "duration": 4.799, "text": "h of x"}, {"start": 13.679, "duration": 5.76, "text": "is equal to the square root of seven x"}, {"start": 17.039, "duration": 4.721, "text": "plus five"}, {"start": 19.439, "duration": 4.161, "text": "express the function h of x"}, {"start": 21.76, "duration": 4.64, "text": "as a composition"}, {"start": 23.6, "duration": 4.08, "text": "of two functions f of x and g of x"}, {"start": 26.4, "duration": 5.36, "text": "such that"}, {"start": 27.68, "duration": 6.399, "text": "h of x is f of g of x"}, {"start": 31.76, "duration": 4.72, "text": "so how can we determine f of x and g of"}, {"start": 34.079, "duration": 4.721, "text": "x given h of x"}, {"start": 36.48, "duration": 4.64, "text": "so notice that f is the function on the"}, {"start": 38.8, "duration": 3.919, "text": "outside and g is the function on the"}, {"start": 41.12, "duration": 3.84, "text": "inside"}, {"start": 42.719, "duration": 3.281, "text": "the stuff on the inside is seven x plus"}, {"start": 44.96, "duration": 4.48, "text": "five"}, {"start": 46.0, "duration": 3.44, "text": "so that's going to be g of x"}, {"start": 50.079, "duration": 4.48, "text": "this stuff on the outside is the square"}, {"start": 51.92, "duration": 6.479, "text": "root symbol and that's gonna be f of x"}, {"start": 54.559, "duration": 6.16, "text": "so f of x is the square root of x so if"}, {"start": 58.399, "duration": 5.441, "text": "we want to determine h of x"}, {"start": 60.719, "duration": 5.041, "text": "which is the composition of f and g"}, {"start": 63.84, "duration": 2.72, "text": "notice that if we take"}, {"start": 65.76, "duration": 2.48, "text": "g"}, {"start": 66.56, "duration": 2.96, "text": "and insert it"}, {"start": 68.24, "duration": 3.919, "text": "into f"}, {"start": 69.52, "duration": 4.16, "text": "that is if we replace this x with seven"}, {"start": 72.159, "duration": 5.041, "text": "x plus five"}, {"start": 73.68, "duration": 3.52, "text": "that will give us h of x"}, {"start": 77.92, "duration": 4.72, "text": "so here's the square root symbol but"}, {"start": 79.92, "duration": 5.28, "text": "without the x so replace the x with"}, {"start": 82.64, "duration": 3.92, "text": "seven x plus five"}, {"start": 85.2, "duration": 3.68, "text": "and that will give us the original"}, {"start": 86.56, "duration": 2.32, "text": "function"}, {"start": 89.119, "duration": 4.64, "text": "but these are the answers that we're"}, {"start": 90.24, "duration": 7.68, "text": "looking for g of x is seven x plus five"}, {"start": 93.759, "duration": 6.32, "text": "and f of x is the square root of x"}, {"start": 97.92, "duration": 4.879, "text": "so let me give you another example that"}, {"start": 100.079, "duration": 2.72, "text": "you can work on"}, {"start": 103.52, "duration": 5.12, "text": "so let's say that"}, {"start": 106.0, "duration": 2.64, "text": "h of x"}, {"start": 109.2, "duration": 5.68, "text": "is one divided by x plus three"}, {"start": 113.04, "duration": 3.039, "text": "go ahead and determine the outside"}, {"start": 114.88, "duration": 2.96, "text": "function"}, {"start": 116.079, "duration": 3.921, "text": "f of x"}, {"start": 117.84, "duration": 4.72, "text": "and also the inside function"}, {"start": 120.0, "duration": 2.56, "text": "g of x"}, {"start": 123.36, "duration": 4.32, "text": "the outside function is going to be one"}, {"start": 125.92, "duration": 3.92, "text": "over x"}, {"start": 127.68, "duration": 4.72, "text": "the inside function is simply x plus"}, {"start": 129.84, "duration": 4.24, "text": "three because if you replace x with x"}, {"start": 132.4, "duration": 4.32, "text": "plus three you're going to get one over"}, {"start": 134.08, "duration": 4.4, "text": "x plus three"}, {"start": 136.72, "duration": 4.72, "text": "and that's pretty straightforward so"}, {"start": 138.48, "duration": 6.08, "text": "these problems are not too bad"}, {"start": 141.44, "duration": 3.12, "text": "let's try another example"}, {"start": 147.44, "duration": 4.24, "text": "so what about this one let's say that"}, {"start": 149.76, "duration": 4.08, "text": "h of x"}, {"start": 151.68, "duration": 4.96, "text": "is the absolute value"}, {"start": 153.84, "duration": 6.08, "text": "of three x plus two"}, {"start": 156.64, "duration": 6.72, "text": "what's the outside function f of x"}, {"start": 159.92, "duration": 5.2, "text": "and the inside function g of x"}, {"start": 163.36, "duration": 4.4, "text": "we can see that on the inside we have"}, {"start": 165.12, "duration": 6.0, "text": "three x plus two so that's g of x"}, {"start": 167.76, "duration": 5.92, "text": "the outside is the absolute value of x"}, {"start": 171.12, "duration": 2.56, "text": "and that's it"}, {"start": 174.239, "duration": 6.801, "text": "and let's do one more example"}, {"start": 177.44, "duration": 3.6, "text": "let's say that h of x"}, {"start": 181.68, "duration": 4.88, "text": "is 2x"}, {"start": 183.599, "duration": 4.881, "text": "cubed plus 7x minus 3"}, {"start": 186.56, "duration": 4.72, "text": "raised to the fourth power"}, {"start": 188.48, "duration": 5.52, "text": "so what's the outside function f of x"}, {"start": 191.28, "duration": 4.72, "text": "and the inside function g of x"}, {"start": 194.0, "duration": 3.04, "text": "on the outside it's x to the fourth"}, {"start": 196.0, "duration": 4.319, "text": "power"}, {"start": 197.04, "duration": 5.199, "text": "on the inside two x cubed plus seven x"}, {"start": 200.319, "duration": 3.2, "text": "minus three"}, {"start": 202.239, "duration": 3.201, "text": "and they go"}, {"start": 203.519, "duration": 3.921, "text": "so that's g of x"}, {"start": 205.44, "duration": 3.76, "text": "and f of x"}, {"start": 207.44, "duration": 6.04, "text": "so that's it for this video thanks again"}, {"start": 209.2, "duration": 4.28, "text": "for watching and have a good day"}, {"start": 231.84, "duration": 2.08, "text": "you"}], "title": "Decomposing Functions - Composition of Functions", "description": "This precalculus video tutorial explains how to decompose a composite function into two functions f(x) and g(x).  This video contains plenty examples and practice problems on decomposing a composition of functions.  \n\nFunctions - Free Formula Sheet:    \nhttps://bit.ly/3Ai8NZv\n\n__________________________________\nIntroduction to Functions:                          \nhttps://www.youtube.com/watch?v=DrEXTC6mIO8\n\nEvaluating Functions:                                  \nhttps://www.youtube.com/watch?v=HyNie_PYgsY\n\nFunction Operations:                                   \nhttps://www.youtube.com/watch?v=3gaxVHVI4cI\n\nIncreasing and Decreasing Functions:     \nhttps://www.youtube.com/watch?v=Dyl7jPlJXOM\n\nRelative Maximum & Minimum Values:   \nhttps://www.youtube.com/watch?v=GaltHJF6WyI\n\n_____________________________________\nRelations and Functions:                            \nhttps://www.youtube.com/watch?v=wbBY2tTqXDA\n\nDomain and Range from a Graph:            \nhttps://www.youtube.com/watch?v=KirGQOwjBVI\n\nHow To Find the Domain of a Function:  \nhttps://www.youtube.com/watch?v=djT6-YamHaA\n\nAverage Rate of Change of a Function:    \nhttps://www.youtube.com/watch?v=f_Q0qlwlclQ\n\nThe Difference Quotient of a Function:    \nhttps://www.youtube.com/watch?v=qQgVomi8lCc\n\n_____________________________________\nEvaluating Piecewise Functions:               \nhttps://www.youtube.com/watch?v=OYOXMyFKotc\n\nGraphing Piecewise Functions:                 \nhttps://www.youtube.com/watch?v=Uzw9tsGq2Pw\n\nComposite Functions:                                 \nhttps://www.youtube.com/watch?v=ZFPkQkURSxk\n\nHow To Evaluate Composite Functions:  \nhttps://www.youtube.com/watch?v=6vQxIG4lJ3w\n\nComposite Functions - Data Table:           \nhttps://www.youtube.com/watch?v=6vQxIG4lJ3w\n\nComp. of 3 Functions - Data Table:           \nhttps://www.youtube.com/watch?v=Ote1KvW6BXw\n\nDecomposing Functions:                            \nhttps://www.youtube.com/watch?v=NmfaC7etZms\n\n_________________________________\nIntroduction to Inverse Functions:            \nhttps://www.youtube.com/watch?v=TN4ybFiuV3k\n\nHow To Find The Inverse of a Function:   \nhttps://www.youtube.com/watch?v=2zeYEx4eTdc\n\nVerifying Inverse Functions:                       \nhttps://www.youtube.com/watch?v=6QamgSPprjA\n\nHorizontal Line Test - 1 to 1 Functions:    \nhttps://www.youtube.com/watch?v=6u9Qh1Vi_Qw\n\nVertical Line Test:                                         \nhttps://www.youtube.com/watch?v=Mxe2lX1htNk\n\nGraphing Inverse Functions:                       \nhttps://www.youtube.com/watch?v=ukEtad_aml4\n\nComposite and Inverse Functions:            \nhttps://www.youtube.com/watch?v=GEgFmxxOJno\n\n__________________________________\nEven and Odd Functions:                            \nhttps://www.youtube.com/watch?v=fKyBOLsqRlo\n\nTransformations of Functions:                  \nhttps://www.youtube.com/watch?v=Tmdrjs9xufc\n\nLinear Functions:                                          \nhttps://www.youtube.com/watch?v=BtcKotD6Ni8\n\nFunctions and Graphs - Problems:            \nhttps://www.youtube.com/watch?v=kvU9sOzT2mk\n\nFunctions - Test Review:                             \nhttps://www.youtube.com/watch?v=1xATmTI-YY8\n\n______________________________________\nFinal Exams and Video Playlists:   \nhttps://www.video-tutor.net/\n\nFull-Length Videos and Worksheets:\nhttps://www.patreon.com/MathScienceTutor/collections", "lengthSeconds": 232, "width": 640, "height": 360}
{"video_id": "WvoFgL4P_rw", "transcript": [{"start": 1.36, "duration": 3.519, "text": "now let's go back to the unit circle"}, {"start": 5.52, "duration": 5.68, "text": "in the unit circle we said that"}, {"start": 7.52, "duration": 7.76, "text": "it's a circle that has a radius of one"}, {"start": 11.2, "duration": 4.08, "text": "so if we were to draw a triangle"}, {"start": 15.759, "duration": 4.241, "text": "the hypotenuse of the triangle will have"}, {"start": 17.52, "duration": 3.679, "text": "a value of 1 because that's the radius"}, {"start": 20.0, "duration": 4.8, "text": "that's the distance between the center"}, {"start": 21.199, "duration": 3.601, "text": "of the circle and a point on a circle"}, {"start": 25.84, "duration": 6.8, "text": "now this is x and this is y"}, {"start": 29.119, "duration": 5.921, "text": "and on the inside we have the angle"}, {"start": 32.64, "duration": 2.4, "text": "theta"}, {"start": 35.2, "duration": 5.28, "text": "now let's see if that angle corresponds"}, {"start": 37.52, "duration": 6.559, "text": "to an angle of 45 degrees"}, {"start": 40.48, "duration": 6.0, "text": "well actually let's make it 30."}, {"start": 44.079, "duration": 5.041, "text": "let's say theta is 30."}, {"start": 46.48, "duration": 4.88, "text": "so that would correspond to a point"}, {"start": 49.12, "duration": 5.119, "text": "of"}, {"start": 51.36, "duration": 4.64, "text": "root 3 over 2 that's the x value and the"}, {"start": 54.239, "duration": 2.601, "text": "y value would be"}, {"start": 56.0, "duration": 5.359, "text": "one"}, {"start": 56.84, "duration": 7.0, "text": "half now we said that sine of theta"}, {"start": 61.359, "duration": 3.921, "text": "when we have a unit circle"}, {"start": 63.84, "duration": 4.16, "text": "and keep in mind this will be true when"}, {"start": 65.28, "duration": 4.8, "text": "r is one if r is not one the equation"}, {"start": 68.0, "duration": 3.68, "text": "will change which we'll talk about later"}, {"start": 70.08, "duration": 4.0, "text": "but for a unit circle"}, {"start": 71.68, "duration": 4.56, "text": "sine theta is simply y"}, {"start": 74.08, "duration": 4.24, "text": "so in this case sine 30 is the y"}, {"start": 76.24, "duration": 3.44, "text": "coordinate one half"}, {"start": 78.32, "duration": 3.68, "text": "cosine theta"}, {"start": 79.68, "duration": 4.079, "text": "we said it's x"}, {"start": 82.0, "duration": 3.2, "text": "now there are some other trigonometric"}, {"start": 83.759, "duration": 3.68, "text": "functions they need to know there's six"}, {"start": 85.2, "duration": 5.04, "text": "of them these are the first two"}, {"start": 87.439, "duration": 4.161, "text": "tangent theta"}, {"start": 90.24, "duration": 4.48, "text": "is sine"}, {"start": 91.6, "duration": 8.36, "text": "divided by cosine"}, {"start": 94.72, "duration": 5.24, "text": "so tangent theta is y divided by x"}, {"start": 100.56, "duration": 6.32, "text": "cosecant theta"}, {"start": 103.36, "duration": 6.719, "text": "is one over y"}, {"start": 106.88, "duration": 5.279, "text": "and secant theta"}, {"start": 110.079, "duration": 6.72, "text": "is one over x"}, {"start": 112.159, "duration": 6.401, "text": "cotangent theta is x divided by y"}, {"start": 116.799, "duration": 2.881, "text": "so those are the six trigonometric"}, {"start": 118.56, "duration": 4.8, "text": "functions"}, {"start": 119.68, "duration": 3.68, "text": "that you need to be aware of"}], "title": "The Six Trigonometric Functions, Basic Introduction, Trigonometry", "description": "This trigonometry video tutorial provides a basic introduction into the six trigonometric functions such as sine, cosine, tangent, cotangent, secant, and cosecant.  It provides the equations and formulas associated with these trigonometric functions as they relate to the unit circle.  \n\nTrigonometry Final Exam Review:\nhttps://www.youtube.com/watch?v=OAsbx4TEnL0\n\nTrigonometry - Free Formula Sheet:         \nhttps://bit.ly/47diggI\n\n_______________________________\nTrigonometry - Basic Introduction:                  \nhttps://www.youtube.com/watch?v=g8VCHoSk5_o\n\nThe Unit Circle:                                                    \nhttps://www.youtube.com/watch?v=57VrEiEPD1I\n\nHow To Remember the Unit Circle Fast:        \nhttps://www.youtube.com/watch?v=gdJq1QunN-o\n\nReference Angles:                                              \nhttps://www.youtube.com/watch?v=E-xFXpVo14o\n\nHow To Solve Right Triangles:                         \nhttps://www.youtube.com/watch?v=i3bjEOA5_zc\n\n_____________________________\nReciprocal Identities:                                         \nhttps://www.youtube.com/watch?v=UeuuFqbTYgA\n\nQuotient Identities:                                            \nhttps://www.youtube.com/watch?v=7PVKbyLFY5U\n\nEven and Odd Trigonometric Identities:         \nhttps://www.youtube.com/watch?v=5p8hokJ3Cqo\n\nPythagorean Identities:                                     \nhttps://www.youtube.com/watch?v=TpADRvW8zm8\n\nThe Pythagorean Theorem:                              \nhttps://www.youtube.com/watch?v=pbf4lcJhIfI\n\nTrig Functions - Periodic Properties:               \nhttps://www.youtube.com/watch?v=8Z60_yXX4xA\n\n________________________________\nThe Exact Value of Trig Functions:                  \nhttps://www.youtube.com/watch?v=bTdjcvm6pPo\n\nSpecial Right Triangles - 30 60 90:                  \nhttps://www.youtube.com/watch?v=p70UBGCHZrQ\n\nSpecial Patterns - Pythagorean Theorem:     \nhttps://www.youtube.com/watch?v=MeBcJPdjeI4\n\nCofunction Identities:                                        \nhttps://www.youtube.com/watch?v=35fxto48HZY\n\nFinal Exams and Video Playlists:   \nhttps://www.video-tutor.net/\n\nFull-Length Videos and Worksheets:\nhttps://www.patreon.com/MathScienceTutor/collections", "lengthSeconds": 122, "width": 640, "height": 360}
{"video_id": "3yimRzW2o4A", "transcript": [{"start": 0.04, "duration": 3.879, "text": "so in this case we'd be looking at"}, {"start": 1.68, "duration": 3.96, "text": "getting like cuz The Spectator will"}, {"start": 3.919, "duration": 3.721, "text": "leave the spect ad will leave and then"}, {"start": 5.64, "duration": 4.8, "text": "we'll just have the ether left good"}, {"start": 7.64, "duration": 4.64, "text": "which would be this one right good and"}, {"start": 10.44, "duration": 4.04, "text": "actually I'm going to let's switch it"}, {"start": 12.28, "duration": 3.8, "text": "just cuz this is going true the other"}, {"start": 14.48, "duration": 4.4, "text": "way I know it's upside down that's fine"}, {"start": 16.08, "duration": 2.8, "text": "like that good"}], "title": "Going from an Alkyl Halide to an Ether #organicchemistry #studywithme", "description": "\ud83d\udcaf Start Understanding Ochem Today for Free! \ud83c\udf89\nhttps://chemmunity.info/youtube\n\n\ud83d\udc81\u200d\u2640\ufe0f HI I'M MELISSA MARIBEL\nI help students pass Chemistry and Organic Chemistry. I used to struggle with this subject, so when I finally graduated with a bachelor's degree in Chemistry, I became a tutor so that you wouldn't have to struggle like I did. I know that with the right help, YOU CAN LEARN ANYTHING!", "lengthSeconds": 17, "width": 270, "height": 480}
{"video_id": "OqBZCycIYfw", "transcript": [{"start": 1.04, "duration": 3.36, "text": "in this video we're going to talk about"}, {"start": 2.399, "duration": 5.281, "text": "the harmonic series and if it's"}, {"start": 4.4, "duration": 3.28, "text": "convergent or divergent"}, {"start": 8.96, "duration": 4.719, "text": "so consider the series which starts from"}, {"start": 11.2, "duration": 4.88, "text": "1 and goes to infinity of the sequence 1"}, {"start": 13.679, "duration": 4.721, "text": "over n so if we list out the terms when"}, {"start": 16.08, "duration": 4.0, "text": "n is one it's just going to be one when"}, {"start": 18.4, "duration": 3.6, "text": "n is two it's one half"}, {"start": 20.08, "duration": 3.439, "text": "and then plus a third"}, {"start": 22.0, "duration": 4.96, "text": "plus a fourth"}, {"start": 23.519, "duration": 6.0, "text": "and plus one fifth and so forth"}, {"start": 26.96, "duration": 5.2, "text": "now if you keep adding"}, {"start": 29.519, "duration": 4.481, "text": "these terms you'll realize that the sum"}, {"start": 32.16, "duration": 3.52, "text": "will continue to increase it's going to"}, {"start": 34.0, "duration": 4.0, "text": "get bigger and bigger and bigger and"}, {"start": 35.68, "duration": 5.12, "text": "bigger it's not going to converge to a"}, {"start": 38.0, "duration": 5.04, "text": "single value if you just add it"}, {"start": 40.8, "duration": 4.64, "text": "and so just by doing that it tells you"}, {"start": 43.04, "duration": 5.359, "text": "that this series"}, {"start": 45.44, "duration": 2.959, "text": "is divergent"}, {"start": 50.079, "duration": 3.12, "text": "now if you're thinking about doing the"}, {"start": 51.68, "duration": 3.68, "text": "divergence test"}, {"start": 53.199, "duration": 3.52, "text": "to prove that it's divergent"}, {"start": 55.36, "duration": 2.879, "text": "it's not really going to work out too"}, {"start": 56.719, "duration": 3.52, "text": "well"}, {"start": 58.239, "duration": 4.16, "text": "so if we take the limit as n approaches"}, {"start": 60.239, "duration": 5.841, "text": "infinity for a sub n"}, {"start": 62.399, "duration": 6.001, "text": "so a sub n is one over n"}, {"start": 66.08, "duration": 5.039, "text": "this will equal zero"}, {"start": 68.4, "duration": 5.2, "text": "now for the divergence test to work"}, {"start": 71.119, "duration": 3.521, "text": "if it doesn't equal zero the limit"}, {"start": 73.6, "duration": 2.96, "text": "then"}, {"start": 74.64, "duration": 3.28, "text": "the series diverges but if it equals"}, {"start": 76.56, "duration": 4.559, "text": "zero"}, {"start": 77.92, "duration": 4.4, "text": "well it's inconclusive the series it may"}, {"start": 81.119, "duration": 2.961, "text": "converge"}, {"start": 82.32, "duration": 4.479, "text": "or it may"}, {"start": 84.08, "duration": 5.44, "text": "diverge"}, {"start": 86.799, "duration": 4.561, "text": "so the divergence test won't help us for"}, {"start": 89.52, "duration": 4.0, "text": "this example"}, {"start": 91.36, "duration": 5.2, "text": "but another test that we could use"}, {"start": 93.52, "duration": 3.04, "text": "is the integral test"}, {"start": 97.119, "duration": 3.521, "text": "now if we take the integral let's say"}, {"start": 99.439, "duration": 2.801, "text": "from"}, {"start": 100.64, "duration": 5.2, "text": "one to infinity"}, {"start": 102.24, "duration": 3.6, "text": "of f of x dx"}, {"start": 106.0, "duration": 3.759, "text": "if this integral is convergent"}, {"start": 108.72, "duration": 2.88, "text": "then"}, {"start": 109.759, "duration": 4.561, "text": "the series"}, {"start": 111.6, "duration": 4.479, "text": "the corresponding series"}, {"start": 114.32, "duration": 4.32, "text": "is also"}, {"start": 116.079, "duration": 2.561, "text": "convergent"}, {"start": 119.68, "duration": 4.64, "text": "likewise if the integral test fails if"}, {"start": 121.84, "duration": 5.48, "text": "it's divergent then the series will be"}, {"start": 124.32, "duration": 3.0, "text": "divergent"}, {"start": 129.599, "duration": 5.36, "text": "so the corresponding function for f of x"}, {"start": 132.16, "duration": 7.52, "text": "is going to be 1 over x"}, {"start": 134.959, "duration": 4.721, "text": "so let's integrate it from 1 to infinity"}, {"start": 139.92, "duration": 5.12, "text": "the anti-derivative of one over x is ln"}, {"start": 143.28, "duration": 4.0, "text": "x"}, {"start": 145.04, "duration": 2.24, "text": "and"}, {"start": 147.76, "duration": 5.199, "text": "we can't really use the infinity symbol"}, {"start": 150.72, "duration": 3.28, "text": "so i'm going to use some constant a with"}, {"start": 152.959, "duration": 2.321, "text": "the limit"}, {"start": 154.0, "duration": 3.519, "text": "as"}, {"start": 155.28, "duration": 4.72, "text": "a approaches infinity since we have an"}, {"start": 157.519, "duration": 5.041, "text": "improper integral"}, {"start": 160.0, "duration": 5.599, "text": "and so this is going to be equal to"}, {"start": 162.56, "duration": 4.959, "text": "the limit as a approaches infinity"}, {"start": 165.599, "duration": 5.121, "text": "the natural log of a"}, {"start": 167.519, "duration": 5.44, "text": "minus the natural log of 1."}, {"start": 170.72, "duration": 3.36, "text": "so the limit as a approaches infinity of"}, {"start": 172.959, "duration": 3.201, "text": "ln a"}, {"start": 174.08, "duration": 4.48, "text": "what is that equal to"}, {"start": 176.16, "duration": 3.76, "text": "well the natural log of infinity is"}, {"start": 178.56, "duration": 3.36, "text": "going to be a large number that's still"}, {"start": 179.92, "duration": 4.8, "text": "going to be infinity"}, {"start": 181.92, "duration": 5.52, "text": "and ln 1 is zero"}, {"start": 184.72, "duration": 4.08, "text": "so the limit i mean not the limit but"}, {"start": 187.44, "duration": 3.6, "text": "the integral"}, {"start": 188.8, "duration": 3.359, "text": "of one over x from one to infinity is"}, {"start": 191.04, "duration": 2.24, "text": "divergent"}, {"start": 192.159, "duration": 3.281, "text": "because"}, {"start": 193.28, "duration": 4.0, "text": "we didn't get a finite answer"}, {"start": 195.44, "duration": 3.68, "text": "our answer is infinity"}, {"start": 197.28, "duration": 3.2, "text": "so because"}, {"start": 199.12, "duration": 2.72, "text": "according to the integral test the"}, {"start": 200.48, "duration": 3.839, "text": "integral from one to infinity is"}, {"start": 201.84, "duration": 4.72, "text": "divergent then the series must also be"}, {"start": 204.319, "duration": 3.361, "text": "divergent so that's another way in which"}, {"start": 206.56, "duration": 5.72, "text": "you could show"}, {"start": 207.68, "duration": 4.6, "text": "that the harmonic series is diverging"}, {"start": 231.2, "duration": 2.08, "text": "you"}], "title": "Harmonic Series", "description": "This calculus 2 video provides a basic introduction into the harmonic series.  It explains why the harmonic series diverges using the integral test for series.\n\nImproper Integrals:                                    https://www.youtube.com/watch?v=ND9cEdfCFr0\n\nConverging & Diverging Sequences:      \nhttps://www.youtube.com/watch?v=XdkoTb8PEG0\n\nMonotonic & Bounded Sequences:        \nhttps://www.youtube.com/watch?v=tHy3TXmZpF0\n\nAbsolute Value Theorem - Sequences:  \nhttps://www.youtube.com/watch?v=kH5Zqg5j-3M\n\nSqueeze Theorem - Sequences:              \nhttps://www.youtube.com/watch?v=21loY9MBK3A\n\n________________________________\nGeometric Series & Sequences:              \nhttps://www.youtube.com/watch?v=zRKZ0-kOUZM\n\nIntroduction to Series - Convergence:    \nhttps://www.youtube.com/watch?v=zg9N2gAf6a4  \n\nDivergence Test For Series:                     \nhttps://www.youtube.com/watch?v=nC_IU1IzzS4\n\nHarmonic Series:                                       \nhttps://www.youtube.com/watch?v=OqBZCycIYfw\n\nTelescoping Series:                                   \nhttps://www.youtube.com/watch?v=XVkdhU6nJbo\n\n__________________________________\nIntegral Test For Divergence:                   \nhttps://www.youtube.com/watch?v=f9SJz4-UaQQ\n\nRemainder Estimate - Integral Test:       \nhttps://www.youtube.com/watch?v=vaU5DDcTfZk\n\nP-Series:                                                      \nhttps://www.youtube.com/watch?v=x621rtIwb_4\n\nDirect Comparison Test:                           \nhttps://www.youtube.com/watch?v=oZtAgihok5s\n\nLimit Comparison Test:                            \nhttps://www.youtube.com/watch?v=LBxYQ0TJxYM\n\n___________________________________\nCalculus Final Exam and Video Playlists:   \nhttps://www.video-tutor.net/\n\nFull-Length Videos and Worksheets:\nhttps://www.patreon.com/MathScienceTutor/collections", "lengthSeconds": 231, "width": 640, "height": 360}
{"video_id": "eAwc8rwMRnI", "transcript": [{"start": 3.12, "duration": 5.84, "text": "now let's say if we have a sphere"}, {"start": 5.839, "duration": 5.441, "text": "with a radius of"}, {"start": 8.96, "duration": 2.32, "text": "10."}, {"start": 12.32, "duration": 4.12, "text": "what is the volume of the sphere"}, {"start": 19.6, "duration": 5.28, "text": "the volume of a sphere is four thirds"}, {"start": 22.72, "duration": 4.08, "text": "pi r cubed"}, {"start": 24.88, "duration": 4.88, "text": "so let's say this is ten inches it's"}, {"start": 26.8, "duration": 4.559, "text": "going to be four over three times pi"}, {"start": 29.76, "duration": 3.68, "text": "times 10 inches raised to the third"}, {"start": 31.359, "duration": 4.321, "text": "power"}, {"start": 33.44, "duration": 4.639, "text": "10 to the third that's 10 times 10 times"}, {"start": 35.68, "duration": 4.8, "text": "10 which is a thousand"}, {"start": 38.079, "duration": 3.361, "text": "and four times a thousand is four"}, {"start": 40.48, "duration": 2.64, "text": "thousand"}, {"start": 41.44, "duration": 4.0, "text": "so it's going to be four thousand over"}, {"start": 43.12, "duration": 3.759, "text": "three times pi"}, {"start": 45.44, "duration": 2.88, "text": "cubic inches"}, {"start": 46.879, "duration": 4.321, "text": "and so that's the volume of this"}, {"start": 48.32, "duration": 2.88, "text": "particular sphere"}, {"start": 51.6, "duration": 3.599, "text": "and let's see what the answer is as a"}, {"start": 52.8, "duration": 2.399, "text": "decimal"}, {"start": 55.28, "duration": 4.56, "text": "so this is equal to four thousand"}, {"start": 57.12, "duration": 2.72, "text": "hundred"}, {"start": 60.399, "duration": 7.16, "text": "cubic inches"}, {"start": 62.16, "duration": 5.399, "text": "so that's the rounded answer for volume"}, {"start": 85.84, "duration": 2.08, "text": "you"}], "title": "Volume of a Sphere", "description": "This geometry video tutorial explains how to calculate the volume of a sphere.\n\n3D Shapes - Faces, Edges, & Vertices:     https://www.youtube.com/watch?v=Wmwe1fmR1SM\n\nRectangular Prism - Volume:                     \nhttps://www.youtube.com/watch?v=rNIKbT0-ywU\n\nRectangular Prism - Surface Area:            \nhttps://www.youtube.com/watch?v=BsAQbJyEF9g\n\nRectangular Prism - Diagonal Length:      \nhttps://www.youtube.com/watch?v=iO7vb3HLHqU\n\nTriangular Prism - Volume:                         \nhttps://www.youtube.com/watch?v=HfDHYFgTbxk\n\nTriangular Prism - Surface Area:               \nhttps://www.youtube.com/watch?v=152tLGR0lms\n\nTriangular Prism - Lateral Area:                 \nhttps://www.youtube.com/watch?v=PljECBkHQpg\n\n_______________________________\nHexagonal Prism - Surface Area:              \nhttps://www.youtube.com/watch?v=JKyF2r-bIC0\n\nCube - Volume:                                             \nhttps://www.youtube.com/watch?v=bpRsuccb6dI\n\nCube - Surface Area:                                   \nhttps://www.youtube.com/watch?v=ZMhkvFwLOR4\n\nCube - Diagonal Length:                             \nhttps://www.youtube.com/watch?v=wr3EvQZzoRI\n\nPyramid - Volume:                                       \nhttps://www.youtube.com/watch?v=8onylJ72ZPQ\n\nPyramid - Surface Area:                              \nhttps://www.youtube.com/watch?v=2g6QuOizzac\n\n________________________________\nSquare Pyramids - Volume:                       \nhttps://www.youtube.com/watch?v=x8wEnG4GURQ\n\nCylinder - Surface Area:                              \nhttps://www.youtube.com/watch?v=LdztqPczYiI\n\nSphere - Volume:                                         \nhttps://www.youtube.com/watch?v=eAwc8rwMRnI\n\nSphere - Surface Area:                                \nhttps://www.youtube.com/watch?v=aXEcU-M_qNA\n\nCone - Volume:                                             \nhttps://www.youtube.com/watch?v=LypmeZ0tnJA\n\nCone - Surface Area:                                   \nhttps://www.youtube.com/watch?v=MeWoIcQNqRA\n\n_________________________________\nFinal Exams and Video Playlists:   \nhttps://www.video-tutor.net/", "lengthSeconds": 86, "width": 640, "height": 360}
{"video_id": "30PrH3OIiqk", "transcript": [{"start": 1.36, "duration": 4.32, "text": "now what if we have an absolute value"}, {"start": 3.28, "duration": 4.4, "text": "expression"}, {"start": 5.68, "duration": 3.839, "text": "with an inequality"}, {"start": 7.68, "duration": 3.039, "text": "what do we need to do"}, {"start": 9.519, "duration": 3.04, "text": "well we still need to write two"}, {"start": 10.719, "duration": 3.121, "text": "equations"}, {"start": 12.559, "duration": 2.321, "text": "the first one is going to remain the"}, {"start": 13.84, "duration": 2.88, "text": "same"}, {"start": 14.88, "duration": 2.72, "text": "now for the second one we need to do two"}, {"start": 16.72, "duration": 2.8, "text": "things"}, {"start": 17.6, "duration": 4.4, "text": "we need to change the positive five to a"}, {"start": 19.52, "duration": 4.08, "text": "negative five and"}, {"start": 22.0, "duration": 3.199, "text": "we need to change the direction of the"}, {"start": 23.6, "duration": 3.04, "text": "inequality"}, {"start": 25.199, "duration": 2.721, "text": "so that's important make sure you"}, {"start": 26.64, "duration": 4.559, "text": "understand that"}, {"start": 27.92, "duration": 3.279, "text": "and then after that just it"}, {"start": 31.679, "duration": 3.681, "text": "five plus one is six and then let's"}, {"start": 33.76, "duration": 4.319, "text": "divide by three"}, {"start": 35.36, "duration": 4.8, "text": "so x is greater than two"}, {"start": 38.079, "duration": 3.841, "text": "now for the other one negative five plus"}, {"start": 40.16, "duration": 4.64, "text": "one"}, {"start": 41.92, "duration": 4.88, "text": "is negative four and if we divided by"}, {"start": 44.8, "duration": 5.12, "text": "three we're going to get a fraction x is"}, {"start": 46.8, "duration": 4.16, "text": "less than negative four thirds"}, {"start": 49.92, "duration": 2.4, "text": "now because we're dealing with"}, {"start": 50.96, "duration": 4.48, "text": "inequalities we need to plot the"}, {"start": 52.32, "duration": 3.12, "text": "solution on a number line"}, {"start": 56.32, "duration": 4.719, "text": "so we need two negative four thirds is"}, {"start": 58.8, "duration": 5.52, "text": "about negative one point three which is"}, {"start": 61.039, "duration": 3.281, "text": "between negative one and two"}, {"start": 64.4, "duration": 3.039, "text": "so x"}, {"start": 65.68, "duration": 3.6, "text": "is greater than two"}, {"start": 67.439, "duration": 5.601, "text": "which means that we have an open circle"}, {"start": 69.28, "duration": 5.839, "text": "at two shaded towards the right and x is"}, {"start": 73.04, "duration": 4.48, "text": "less than negative four thirds which is"}, {"start": 75.119, "duration": 4.161, "text": "about negative one point three"}, {"start": 77.52, "duration": 4.239, "text": "so it's an open circle"}, {"start": 79.28, "duration": 4.08, "text": "shaded towards the left"}, {"start": 81.759, "duration": 4.081, "text": "now the solution"}, {"start": 83.36, "duration": 3.759, "text": "using interval"}, {"start": 85.84, "duration": 3.04, "text": "let's write it up here it's going to be"}, {"start": 87.119, "duration": 4.081, "text": "negative infinity"}, {"start": 88.88, "duration": 4.16, "text": "to negative 4 over 3"}, {"start": 91.2, "duration": 3.52, "text": "and then union"}, {"start": 93.04, "duration": 4.0, "text": "2 to infinity"}, {"start": 94.72, "duration": 4.0, "text": "so here's the 2 to infinity part"}, {"start": 97.04, "duration": 3.68, "text": "and here's the negative infinity to"}, {"start": 98.72, "duration": 3.439, "text": "negative four thirds"}, {"start": 100.72, "duration": 4.64, "text": "the open circle is at negative four"}, {"start": 102.159, "duration": 3.201, "text": "thirds just keep that in mind"}, {"start": 105.68, "duration": 3.719, "text": "let's try another example"}, {"start": 114.479, "duration": 2.96, "text": "feel free to pause the video and work on"}, {"start": 116.159, "duration": 2.96, "text": "it"}, {"start": 117.439, "duration": 3.521, "text": "now right now we can't"}, {"start": 119.119, "duration": 3.441, "text": "write two equations yet"}, {"start": 120.96, "duration": 3.28, "text": "we need to get the absolute value"}, {"start": 122.56, "duration": 3.599, "text": "expression on one side of the equation"}, {"start": 124.24, "duration": 4.239, "text": "first"}, {"start": 126.159, "duration": 3.921, "text": "so first let's subtract both sides by"}, {"start": 128.479, "duration": 3.041, "text": "3."}, {"start": 130.08, "duration": 4.56, "text": "so what we're going to have is negative"}, {"start": 131.52, "duration": 6.24, "text": "2 times the absolute value of 2x minus 1"}, {"start": 134.64, "duration": 4.239, "text": "is less than or equal to negative 10."}, {"start": 137.76, "duration": 3.6, "text": "next"}, {"start": 138.879, "duration": 4.961, "text": "let's divide both sides"}, {"start": 141.36, "duration": 4.239, "text": "by negative two"}, {"start": 143.84, "duration": 4.56, "text": "so this will change the direction of the"}, {"start": 145.599, "duration": 5.601, "text": "inequality just keep that in mind"}, {"start": 148.4, "duration": 5.119, "text": "so the absolute value of two x minus one"}, {"start": 151.2, "duration": 4.319, "text": "is equal to or greater than"}, {"start": 153.519, "duration": 3.761, "text": "positive five"}, {"start": 155.519, "duration": 4.161, "text": "now since we have the absolute value"}, {"start": 157.28, "duration": 5.2, "text": "expression on one side by itself"}, {"start": 159.68, "duration": 5.52, "text": "with no other numbers outside of it"}, {"start": 162.48, "duration": 6.0, "text": "this is when we can write two equations"}, {"start": 165.2, "duration": 5.44, "text": "2x minus 1 is greater than or equal to 5"}, {"start": 168.48, "duration": 3.68, "text": "and 2x minus one is less than or equal"}, {"start": 170.64, "duration": 2.959, "text": "to negative five"}, {"start": 172.16, "duration": 3.2, "text": "don't forget to change the direction of"}, {"start": 173.599, "duration": 4.0, "text": "the inequality"}, {"start": 175.36, "duration": 5.44, "text": "so now let's add one"}, {"start": 177.599, "duration": 5.841, "text": "so two x is greater than or equal to six"}, {"start": 180.8, "duration": 4.719, "text": "and next let's divide by two so x is"}, {"start": 183.44, "duration": 4.4, "text": "equal to or greater than 3."}, {"start": 185.519, "duration": 4.321, "text": "here we're going to add 1"}, {"start": 187.84, "duration": 4.56, "text": "so 2x is less than or equal to negative"}, {"start": 189.84, "duration": 4.72, "text": "4 and then divide by 2."}, {"start": 192.4, "duration": 4.4, "text": "so x is less than or equal to negative"}, {"start": 194.56, "duration": 2.24, "text": "2."}, {"start": 197.44, "duration": 4.32, "text": "so we have x is less than or equal to"}, {"start": 199.36, "duration": 4.879, "text": "negative two and x is equal to a greater"}, {"start": 201.76, "duration": 2.479, "text": "than three"}, {"start": 204.319, "duration": 4.401, "text": "so here's zero"}, {"start": 206.64, "duration": 4.8, "text": "here's negative two"}, {"start": 208.72, "duration": 2.72, "text": "and here's three"}, {"start": 211.68, "duration": 5.919, "text": "so x is equal to or greater than three"}, {"start": 215.2, "duration": 4.24, "text": "so we have a closed circle by three"}, {"start": 217.599, "duration": 4.481, "text": "shaded towards the right"}, {"start": 219.44, "duration": 4.0, "text": "x is equal to or less than negative two"}, {"start": 222.08, "duration": 3.439, "text": "so we have a closed circle shaded"}, {"start": 223.44, "duration": 3.92, "text": "towards left"}, {"start": 225.519, "duration": 4.481, "text": "so therefore the solution"}, {"start": 227.36, "duration": 3.84, "text": "is negative infinity to negative 2 with"}, {"start": 230.0, "duration": 2.319, "text": "a bracket"}, {"start": 231.2, "duration": 4.599, "text": "union"}, {"start": 232.319, "duration": 3.48, "text": "3 to infinity"}], "title": "How To Solve Absolute Value Inequalities, Basic Introduction, Algebra", "description": "This Algebra video tutorial provides a basic introduction into absolute value inequalities.  it explains how to solve absolute value inequalities and graphing the solution using a number line and using interval notation.  This tutorial contains many examples and practice problems.  \n\nPlotting Inequalities on a Number Line:            \nhttps://www.youtube.com/watch?v=KkiYqww4eg0\n\nHow To Solve Linear Inequalities:                      \nhttps://www.youtube.com/watch?v=rIl2USa8XPY\n\nSolving Absolute Value Inequalities:                 \nhttps://www.youtube.com/watch?v=ZJvOGkykJ44\n\nGraphing Linear Inequalities In 2 Variables:     \nhttps://www.youtube.com/watch?v=cskHdgwB5k8\n\nGraphing Systems of Linear Inequalities:         \nhttps://www.youtube.com/watch?v=FWbcKade3rw\n\n__________________________________\nHow To Solve Compound Inequalities:\nhttps://www.youtube.com/watch?v=msiceXvwriE\n\nQuadratic Inequalities:\nhttps://www.youtube.com/watch?v=_gWjLKsFOPE\n\nPolynomial Inequalities:\nhttps://www.youtube.com/watch?v=Fd5ys4PQ-aM\n\nRational Inequalities:\nhttps://www.youtube.com/watch?v=gfnVHwhEe6U\n\nSAT Math - Inequalities:\n\n_______________________________________\nFinal Exams and Video Playlists:   \nhttps://www.video-tutor.net/\n\nFull-Length Videos and Worksheets:\nhttps://www.patreon.com/MathScienceTutor/collections", "lengthSeconds": 236, "width": 640, "height": 360}
{"video_id": "luRrOlsB4cY", "transcript": [{"start": 0.799, "duration": 3.921, "text": "in this lesson"}, {"start": 2.08, "duration": 4.48, "text": "instead of expanding a single log into"}, {"start": 4.72, "duration": 3.919, "text": "multiple logarithms"}, {"start": 6.56, "duration": 5.52, "text": "we're going to condense"}, {"start": 8.639, "duration": 5.521, "text": "multiple logs into a single log"}, {"start": 12.08, "duration": 2.96, "text": "so this is the opposite of the last"}, {"start": 14.16, "duration": 2.48, "text": "lesson"}, {"start": 15.04, "duration": 4.56, "text": "so hopefully you've watched the last"}, {"start": 16.64, "duration": 2.96, "text": "lesson before this one"}, {"start": 21.039, "duration": 3.921, "text": "so we're going to write this as a single"}, {"start": 22.72, "duration": 2.24, "text": "log"}, {"start": 26.32, "duration": 3.199, "text": "now the ones that contain a positive"}, {"start": 28.32, "duration": 4.16, "text": "sign"}, {"start": 29.519, "duration": 4.801, "text": "will go on top so that's a and c so it's"}, {"start": 32.48, "duration": 2.96, "text": "going to be a times c"}, {"start": 34.32, "duration": 3.6, "text": "on top"}, {"start": 35.44, "duration": 4.0, "text": "now b has a negative in front of it so"}, {"start": 37.92, "duration": 3.6, "text": "it's going to go on the bottom"}, {"start": 39.44, "duration": 4.959, "text": "so it's going to be log"}, {"start": 41.52, "duration": 5.44, "text": "ac over b"}, {"start": 44.399, "duration": 5.521, "text": "let's try another example"}, {"start": 46.96, "duration": 2.96, "text": "2 log a"}, {"start": 50.8, "duration": 4.399, "text": "plus"}, {"start": 52.16, "duration": 3.039, "text": "5 log b"}, {"start": 55.52, "duration": 5.679, "text": "minus 7 log c"}, {"start": 58.399, "duration": 5.281, "text": "so how can we express"}, {"start": 61.199, "duration": 4.24, "text": "this expression as a single log"}, {"start": 63.68, "duration": 3.759, "text": "the first thing we need to do"}, {"start": 65.439, "duration": 5.881, "text": "is move the coefficient"}, {"start": 67.439, "duration": 3.881, "text": "to the exponent position"}, {"start": 72.479, "duration": 5.441, "text": "so this is going to be log"}, {"start": 75.2, "duration": 4.64, "text": "a squared plus"}, {"start": 77.92, "duration": 3.44, "text": "log b to the fifth"}, {"start": 79.84, "duration": 3.44, "text": "minus"}, {"start": 81.36, "duration": 5.28, "text": "log c to the seventh"}, {"start": 83.28, "duration": 5.519, "text": "now we can write a single log"}, {"start": 86.64, "duration": 3.839, "text": "a and b both have a positive sign in"}, {"start": 88.799, "duration": 3.761, "text": "front so they're going to be on top a"}, {"start": 90.479, "duration": 4.401, "text": "squared b to the fifth"}, {"start": 92.56, "duration": 4.4, "text": "c has a negative sign in front of it"}, {"start": 94.88, "duration": 3.36, "text": "so c to the seventh is going to be on"}, {"start": 96.96, "duration": 4.88, "text": "the bottom"}, {"start": 98.24, "duration": 3.6, "text": "therefore this is the answer"}, {"start": 103.52, "duration": 2.639, "text": "try this one"}, {"start": 106.32, "duration": 2.64, "text": "one third"}, {"start": 109.439, "duration": 5.201, "text": "log a"}, {"start": 111.28, "duration": 5.199, "text": "minus two thirds"}, {"start": 114.64, "duration": 2.799, "text": "log b"}, {"start": 116.479, "duration": 3.6, "text": "plus"}, {"start": 117.439, "duration": 2.64, "text": "one fourth"}, {"start": 120.32, "duration": 5.52, "text": "log c"}, {"start": 122.88, "duration": 4.48, "text": "minus four over five"}, {"start": 125.84, "duration": 3.68, "text": "log d"}, {"start": 127.36, "duration": 4.08, "text": "so first so let's move all of the"}, {"start": 129.52, "duration": 5.2, "text": "coefficients"}, {"start": 131.44, "duration": 3.28, "text": "to the exponent position"}, {"start": 135.599, "duration": 5.441, "text": "so this is equal to log a"}, {"start": 138.959, "duration": 4.881, "text": "raised to the 1 3"}, {"start": 141.04, "duration": 6.16, "text": "minus log b"}, {"start": 143.84, "duration": 4.56, "text": "to the two thirds plus"}, {"start": 147.2, "duration": 3.119, "text": "log c"}, {"start": 148.4, "duration": 3.68, "text": "to the one fourth"}, {"start": 150.319, "duration": 4.481, "text": "minus log d"}, {"start": 152.08, "duration": 6.32, "text": "to the four fifths"}, {"start": 154.8, "duration": 7.12, "text": "and now let's write it as a single log"}, {"start": 158.4, "duration": 5.919, "text": "so a and c both have a positive sign"}, {"start": 161.92, "duration": 5.36, "text": "so it's going to be a to the one-third"}, {"start": 164.319, "duration": 7.041, "text": "c to the one-fourth on top"}, {"start": 167.28, "duration": 5.92, "text": "now b and d both contain a negative sign"}, {"start": 171.36, "duration": 3.44, "text": "in front of the log"}, {"start": 173.2, "duration": 3.52, "text": "so they're going to go on the bottom b"}, {"start": 174.8, "duration": 4.719, "text": "to the two thirds"}, {"start": 176.72, "duration": 4.72, "text": "and then d to the four fifths"}, {"start": 179.519, "duration": 4.161, "text": "so you can leave your answer like this"}, {"start": 181.44, "duration": 4.64, "text": "or you can convert it back"}, {"start": 183.68, "duration": 4.08, "text": "into radical form"}, {"start": 186.08, "duration": 4.0, "text": "so you can write it like this as well"}, {"start": 187.76, "duration": 5.119, "text": "the cube root of a"}, {"start": 190.08, "duration": 6.48, "text": "times the fourth root of c"}, {"start": 192.879, "duration": 9.0, "text": "divided by the cube root of b squared"}, {"start": 196.56, "duration": 5.319, "text": "times the fifth root of d to the fourth"}, {"start": 221.84, "duration": 2.08, "text": "you"}], "title": "Condensing Logarithmic Expressions", "description": "This algebra video tutorial explains how to condense logarithmic expressions into a single logarithm using properties of logarithmic functions.  \n\nLogarithms - Free Formula Sheet:  https://bit.ly/4fEwqv0\n\n_______________________________\nLogarithms - The Easy Way!                              \nhttps://www.youtube.com/watch?v=kqVpPSzkTYA\n\nLog to Exponential Form:                                    \nhttps://www.youtube.com/watch?v=f0C1KL7GkqY\n\nChange of Base Formula:                                   \nhttps://www.youtube.com/watch?v=FFm-zaFW_X4\n\nChange of Base Log Problem:                           \nhttps://www.youtube.com/watch?v=p7hD9VdXv9U\n\nProperties of Logarithms:                                   \nhttps://www.youtube.com/watch?v=Jtv9Lnf7Zw8\n\n____________________________________\nExpanding Logarithmic Expressions:               \nhttps://www.youtube.com/watch?v=OIz-5MyJA3g\n\nCondensing Logarithmic Expressions:             \nhttps://www.youtube.com/watch?v=luRrOlsB4cY\n\nNatural Logarithms:                                             \nhttps://www.youtube.com/watch?v=daUlTsnCNRQ\n\nSolving Exponential Equations:                          \nhttps://www.youtube.com/watch?v=9tutJ5xrRwg\n\nExponential Equations - Quadratic Form:         \nhttps://www.youtube.com/watch?v=yNgmVu0R_T8\n\n_______________________________________\nSolving Logarithmic Equations:                         \nhttps://www.youtube.com/watch?v=fnhFneOz6n8\n\nGraphing Logarithmic Functions:                      \nhttps://www.youtube.com/watch?v=-nptxS9rZNA\n\nGraphing Exponential Functions:                       \nhttps://www.youtube.com/watch?v=DASfP8KAyvs\n\nCompound Interest Word Problems:                \nhttps://www.youtube.com/watch?v=Hn0eLcOSQGw\n\nLogarithms Practice Problems:                         \nhttps://www.youtube.com/watch?v=7DVbQKI600k\n\n_______________________________________\nFinal Exams and Video Playlists:   \nhttps://www.video-tutor.net/\n\nFull-Length Videos and Worksheets:\nhttps://www.patreon.com/MathScienceTutor/collections", "lengthSeconds": 222, "width": 640, "height": 360}
{"video_id": "iH-hil58FzQ", "transcript": [{"start": 0.0, "duration": 3.72, "text": "I would say it would move from here to"}, {"start": 1.86, "duration": 4.02, "text": "here you're right is that right okay so"}, {"start": 3.72, "duration": 3.96, "text": "yes the tail does have to go from"}, {"start": 5.88, "duration": 3.9, "text": "whatever electrons we're moving and then"}, {"start": 7.68, "duration": 3.72, "text": "from there that's going to move towards"}, {"start": 9.78, "duration": 3.72, "text": "the bond a lot of times students want to"}, {"start": 11.4, "duration": 4.439, "text": "point this to the positive charge but"}, {"start": 13.5, "duration": 3.96, "text": "it's not going there necessarily like we"}, {"start": 15.839, "duration": 3.481, "text": "want to point this to then show that"}, {"start": 17.46, "duration": 3.72, "text": "there's going to now be a double bond at"}, {"start": 19.32, "duration": 3.36, "text": "this end that's correct next let's draw"}, {"start": 21.18, "duration": 3.06, "text": "the resonance structure and then my"}, {"start": 22.68, "duration": 4.7, "text": "positive charge would be here now yes"}, {"start": 24.24, "duration": 3.14, "text": "that is exactly correct"}], "title": "Drawing resonance structures when there\u2019s a positive charge #organicchemistry #resonance", "description": "", "lengthSeconds": 26, "width": 360, "height": 640}
{"video_id": "y0R54UeqClo", "transcript": [{"start": 0.03, "duration": 3.45, "text": "alright here I'm going to do another"}, {"start": 1.62, "duration": 3.63, "text": "example of solving a compound inequality"}, {"start": 3.48, "duration": 4.26, "text": "and then we'll write the solution and"}, {"start": 5.25, "duration": 5.91, "text": "interval notation so here it says"}, {"start": 7.74, "duration": 7.41, "text": "negative 4 has to be less than 3w plus 5"}, {"start": 11.16, "duration": 6.06, "text": "which has to be less than 12 when we"}, {"start": 15.15, "duration": 4.709, "text": "write this in one fell swoop really what"}, {"start": 17.22, "duration": 9.51, "text": "they mean is negative 4 has to be less"}, {"start": 19.859, "duration": 10.231, "text": "than 3w plus 5 and also 3w plus 5 has to"}, {"start": 26.73, "duration": 6.54, "text": "be less than 12 okay so it's really just"}, {"start": 30.09, "duration": 6.33, "text": "a shorthand way of writing both of these"}, {"start": 33.27, "duration": 5.58, "text": "statements at once ok so let me go back"}, {"start": 36.42, "duration": 4.38, "text": "to the original problem here so just"}, {"start": 38.85, "duration": 5.31, "text": "something that you should think about"}, {"start": 40.8, "duration": 4.71, "text": "what the heck do they mean here so again"}, {"start": 44.16, "duration": 3.09, "text": "another way to think about it is you"}, {"start": 45.51, "duration": 3.72, "text": "know we're going to pick some number W"}, {"start": 47.25, "duration": 3.75, "text": "multiplied by 3 add 5 at the end of the"}, {"start": 49.23, "duration": 3.75, "text": "day it's got to be between negative 4"}, {"start": 51.0, "duration": 5.07, "text": "and 12 those are the numbers that we are"}, {"start": 52.98, "duration": 4.53, "text": "looking for the way that you do this"}, {"start": 56.07, "duration": 4.259, "text": "again you know imagine if there was only"}, {"start": 57.51, "duration": 5.279, "text": "one inequality here you know negative 4"}, {"start": 60.329, "duration": 4.141, "text": "less than 3w + 5 I mean to me the first"}, {"start": 62.789, "duration": 5.851, "text": "thing I would do would be to subtract 5"}, {"start": 64.47, "duration": 6.54, "text": "from both sides I would get negative 4"}, {"start": 68.64, "duration": 5.4, "text": "less than 5 which is negative 9 and then"}, {"start": 71.01, "duration": 5.82, "text": "I would have the 3w left over likewise"}, {"start": 74.04, "duration": 4.71, "text": "if we just had you know 3w plus 5 less"}, {"start": 76.83, "duration": 3.359, "text": "than 12 you know to solve that"}, {"start": 78.75, "duration": 4.35, "text": "inequality I would do the exact same"}, {"start": 80.189, "duration": 5.25, "text": "thing which is subtract 5 so typically"}, {"start": 83.1, "duration": 4.23, "text": "you know what you'll I'm going to do it"}, {"start": 85.439, "duration": 4.621, "text": "all at once I think you know I've got"}, {"start": 87.33, "duration": 5.579, "text": "this negative for less than 3w plus 5"}, {"start": 90.06, "duration": 4.8, "text": "less than 12 all when I see that I'm"}, {"start": 92.909, "duration": 4.741, "text": "trying to get the W by itself in the"}, {"start": 94.86, "duration": 5.219, "text": "middle so to me it says just subtract 5"}, {"start": 97.65, "duration": 5.249, "text": "from everybody so on the left we'll have"}, {"start": 100.079, "duration": 6.601, "text": "negative 9 we'll be left with 3w in the"}, {"start": 102.899, "duration": 5.851, "text": "middle let's see 12 minus 5 would be 7"}, {"start": 106.68, "duration": 4.17, "text": "and again the same thing I would do here"}, {"start": 108.75, "duration": 5.52, "text": "if only one of the inequalities were"}, {"start": 110.85, "duration": 4.83, "text": "there I would divide everything by 3 and"}, {"start": 114.27, "duration": 2.55, "text": "again that's what I'm going to do"}, {"start": 115.68, "duration": 4.92, "text": "everywhere I'm just going to divide"}, {"start": 116.82, "duration": 5.31, "text": "everything by 3 so negative 9 over 3 is"}, {"start": 120.6, "duration": 5.04, "text": "going to be negative 3"}, {"start": 122.13, "duration": 6.9, "text": "so it says we'll get W in the middle"}, {"start": 125.64, "duration": 4.89, "text": "just one W and then less than 7/3 so"}, {"start": 129.03, "duration": 3.45, "text": "basically it says the numbers that work"}, {"start": 130.53, "duration": 6.84, "text": "it says W has to be strictly greater"}, {"start": 132.48, "duration": 6.24, "text": "than negative 3 but less than 7/3 so if"}, {"start": 137.37, "duration": 3.06, "text": "we want to write that again I'm going to"}, {"start": 138.72, "duration": 5.61, "text": "make my little number line I always like"}, {"start": 140.43, "duration": 7.95, "text": "to do that first so negative 3 we don't"}, {"start": 144.33, "duration": 6.93, "text": "use it so we make an open circle and it"}, {"start": 148.38, "duration": 4.82, "text": "says 7/3 that's something positive over"}, {"start": 151.26, "duration": 4.32, "text": "there I'm going to put an open circle"}, {"start": 153.2, "duration": 3.97, "text": "sometimes people always have a tendency"}, {"start": 155.58, "duration": 3.36, "text": "to want to put 0 in the number line I"}, {"start": 157.17, "duration": 4.22, "text": "think is just a point of reference I"}, {"start": 158.94, "duration": 5.1, "text": "just put the numbers that I'm using"}, {"start": 161.39, "duration": 4.3, "text": "because if I put 0 down there to me that"}, {"start": 164.04, "duration": 3.09, "text": "thinks maybe I you know I have to use"}, {"start": 165.69, "duration": 3.93, "text": "that in my interval somewhere and I"}, {"start": 167.13, "duration": 3.9, "text": "don't so I just put these numbers down"}, {"start": 169.62, "duration": 6.27, "text": "it says it has to be bigger than"}, {"start": 171.03, "duration": 7.019, "text": "negative 3 but smaller than 7/3 so the"}, {"start": 175.89, "duration": 4.29, "text": "numbers in between there would be the"}, {"start": 178.049, "duration": 6.211, "text": "numbers that work so to write our"}, {"start": 180.18, "duration": 5.49, "text": "solution and interval notation okay so"}, {"start": 184.26, "duration": 3.06, "text": "if you kind of think about the left"}, {"start": 185.67, "duration": 4.2, "text": "number that's going to be the number"}, {"start": 187.32, "duration": 5.16, "text": "that goes on the left with interval"}, {"start": 189.87, "duration": 3.899, "text": "notation since it's less than we don't"}, {"start": 192.48, "duration": 4.2, "text": "include it so we just use our"}, {"start": 193.769, "duration": 6.181, "text": "parentheses we put a comma and then it"}, {"start": 196.68, "duration": 5.399, "text": "says we go up to 7/3 and again we use a"}, {"start": 199.95, "duration": 5.94, "text": "parenthesis because we don't include it"}, {"start": 202.079, "duration": 7.97, "text": "okay so this is the interval notation to"}, {"start": 205.89, "duration": 4.159, "text": "describe this set of solutions"}], "title": "Solving Linear Compound Inequalities - Example 2", "description": "Thanks to all of you who support me on Patreon. You da real mvps!   $1 per month helps!! :)  https://www.patreon.com/patrickjmt !! Solving Linear Compound Inequalities - Example 2", "lengthSeconds": 209, "width": 480, "height": 360}
{"video_id": "oVYG6MvW-ig", "transcript": [{"start": 0.88, "duration": 3.12, "text": "all right so here we're going to look at"}, {"start": 2.32, "duration": 2.72, "text": "one more example using the order of"}, {"start": 4.0, "duration": 2.48, "text": "operations"}, {"start": 5.04, "duration": 4.639, "text": "and here we're going to evaluate the"}, {"start": 6.48, "duration": 5.76, "text": "expression negative x minus y squared"}, {"start": 9.679, "duration": 4.561, "text": "all divided by x times y minus the"}, {"start": 12.24, "duration": 3.04, "text": "quantity 3 minus x"}, {"start": 14.24, "duration": 3.92, "text": "but what we're going to do is we're"}, {"start": 15.28, "duration": 3.68, "text": "going to substitute in x equals negative"}, {"start": 18.16, "duration": 4.24, "text": "2"}, {"start": 18.96, "duration": 6.239, "text": "and y equals positive 3. so"}, {"start": 22.4, "duration": 3.68, "text": "anytime i substitute things in i usually"}, {"start": 25.199, "duration": 4.4, "text": "put"}, {"start": 26.08, "duration": 3.92, "text": "it in parentheses okay so what i mean is"}, {"start": 29.599, "duration": 4.161, "text": "we've got"}, {"start": 30.0, "duration": 4.64, "text": "a negative well x is negative 2. so"}, {"start": 33.76, "duration": 2.88, "text": "notice i'm"}, {"start": 34.64, "duration": 3.84, "text": "replacing i'm putting the x inside of"}, {"start": 36.64, "duration": 5.759, "text": "parentheses"}, {"start": 38.48, "duration": 6.48, "text": "minus the same thing i've got y squared"}, {"start": 42.399, "duration": 4.48, "text": "but again y had the value of positive"}, {"start": 44.96, "duration": 5.119, "text": "three"}, {"start": 46.879, "duration": 6.481, "text": "x times y so x is negative two"}, {"start": 50.079, "duration": 4.721, "text": "times three minus three"}, {"start": 53.36, "duration": 4.64, "text": "minus and again another set of"}, {"start": 54.8, "duration": 6.48, "text": "parentheses negative two"}, {"start": 58.0, "duration": 6.479, "text": "and now i've replaced everything"}, {"start": 61.28, "duration": 5.519, "text": "and this looks uh certainly"}, {"start": 64.479, "duration": 4.32, "text": "a little obnoxious and i guess it is but"}, {"start": 66.799, "duration": 4.721, "text": "it shouldn't be too bad"}, {"start": 68.799, "duration": 4.481, "text": "so i've got multiplication i've got"}, {"start": 71.52, "duration": 4.239, "text": "negative one times negative two that's"}, {"start": 73.28, "duration": 4.159, "text": "going to give me a positive two"}, {"start": 75.759, "duration": 3.441, "text": "but next i'm gonna have to do my"}, {"start": 77.439, "duration": 4.481, "text": "exponents okay so we've got"}, {"start": 79.2, "duration": 3.76, "text": "minus three squared well we do exponents"}, {"start": 81.92, "duration": 3.6, "text": "first so"}, {"start": 82.96, "duration": 5.04, "text": "i'll just leave the minus sign in there"}, {"start": 85.52, "duration": 5.279, "text": "3 squared is 3 times 3 which is"}, {"start": 88.0, "duration": 4.0, "text": "9. in the denominator i see"}, {"start": 90.799, "duration": 4.481, "text": "multiplication"}, {"start": 92.0, "duration": 6.159, "text": "i see subtraction i see some parentheses"}, {"start": 95.28, "duration": 4.799, "text": "i'm going to do negative 2 times 3 which"}, {"start": 98.159, "duration": 4.64, "text": "will be negative 6."}, {"start": 100.079, "duration": 4.161, "text": "there's my minus sign inside the"}, {"start": 102.799, "duration": 4.241, "text": "parentheses we have three"}, {"start": 104.24, "duration": 4.0, "text": "minus negative two well that's the same"}, {"start": 107.04, "duration": 5.119, "text": "thing as three"}, {"start": 108.24, "duration": 6.48, "text": "plus two which is going to give us five"}, {"start": 112.159, "duration": 3.521, "text": "and now we're getting pretty close 2"}, {"start": 114.72, "duration": 3.039, "text": "minus 9"}, {"start": 115.68, "duration": 4.399, "text": "that's going to be negative 7. so again"}, {"start": 117.759, "duration": 4.881, "text": "i'm just thinking 9 minus 2"}, {"start": 120.079, "duration": 3.441, "text": "but changing you know the changing it to"}, {"start": 122.64, "duration": 3.68, "text": "negative"}, {"start": 123.52, "duration": 3.68, "text": "so 2 minus 9 is negative 7 i see"}, {"start": 126.32, "duration": 3.919, "text": "negative 6"}, {"start": 127.2, "duration": 5.92, "text": "minus five negative six minus"}, {"start": 130.239, "duration": 4.161, "text": "five is going to be negative eleven well"}, {"start": 133.12, "duration": 3.52, "text": "the last thing we can do"}, {"start": 134.4, "duration": 3.44, "text": "a negative divided by a negative is a"}, {"start": 136.64, "duration": 3.679, "text": "positive"}, {"start": 137.84, "duration": 5.84, "text": "so we'll be left with the value of seven"}, {"start": 140.319, "duration": 3.361, "text": "over eleven"}], "title": "The Order of Operations, Example 3", "description": "Thanks to all of you who support me on Patreon. You da real mvps!   $1 per month helps!! :)  https://www.patreon.com/patrickjmt !! The Order of Operations, Example 3.  Here we look at using the order of operations to evaluate some expressions!", "lengthSeconds": 142, "width": 480, "height": 360}
{"video_id": "ujcC47TwVjE", "transcript": [{"start": 0.65, "duration": 4.48, "text": "okay in this video we're just going to"}, {"start": 2.79, "duration": 6.21, "text": "do some more derivatives involving the"}, {"start": 5.13, "duration": 7.049, "text": "power rule so in Part A here we've got P"}, {"start": 9.0, "duration": 6.089, "text": "of Q equals negative 3 over pi times the"}, {"start": 12.179, "duration": 4.951, "text": "square root of Q to the 4.1 power and"}, {"start": 15.089, "duration": 4.981, "text": "all I'm gonna do in Part A is I'm just"}, {"start": 17.13, "duration": 4.92, "text": "gonna rewrite this first I'm gonna"}, {"start": 20.07, "duration": 4.32, "text": "rewrite it and again any time I see a"}, {"start": 22.05, "duration": 3.989, "text": "square root or a radical in there the"}, {"start": 24.39, "duration": 5.91, "text": "first thing I always do is just try to"}, {"start": 26.039, "duration": 8.031, "text": "rewrite this with no radical just as a"}, {"start": 30.3, "duration": 7.56, "text": "power so we can write this as P of Q"}, {"start": 34.07, "duration": 5.47, "text": "equals negative 3 over PI and don't be"}, {"start": 37.86, "duration": 4.35, "text": "thrown off by the negative 3 over pi"}, {"start": 39.54, "duration": 4.47, "text": "that's just a constant so nothing's"}, {"start": 42.21, "duration": 4.41, "text": "really gonna happen to it when we take"}, {"start": 44.01, "duration": 4.56, "text": "the derivative we can just kind of take"}, {"start": 46.62, "duration": 3.81, "text": "the derivative of the stuff involving"}, {"start": 48.57, "duration": 4.98, "text": "the variable and then multiply that at"}, {"start": 50.43, "duration": 5.67, "text": "the end so we've got Q raised to the 4"}, {"start": 53.55, "duration": 4.14, "text": "point 1 recall a square root is"}, {"start": 56.1, "duration": 3.47, "text": "equivalent to raising something to the"}, {"start": 57.69, "duration": 3.779, "text": "1/2 power"}, {"start": 59.57, "duration": 4.09, "text": "well again I'm just going to keep"}, {"start": 61.469, "duration": 5.131, "text": "simplifying this so we have negative 3"}, {"start": 63.66, "duration": 4.68, "text": "over pi recall if things are in"}, {"start": 66.6, "duration": 5.79, "text": "parentheses we just multiply the"}, {"start": 68.34, "duration": 7.38, "text": "exponents so we would have Q to the 4.1"}, {"start": 72.39, "duration": 7.95, "text": "times one-half or equivalently 4.1 over"}, {"start": 75.72, "duration": 7.23, "text": "2 so that's negative 3 over PI and then"}, {"start": 80.34, "duration": 5.01, "text": "if we simplify we would have well 4.1"}, {"start": 82.95, "duration": 6.209, "text": "divided by 2 is going to be two point"}, {"start": 85.35, "duration": 6.03, "text": "zero five and now I'm kind of in a good"}, {"start": 89.159, "duration": 4.471, "text": "place where I think well now I'm finally"}, {"start": 91.38, "duration": 7.26, "text": "going to take the derivative so P prime"}, {"start": 93.63, "duration": 7.14, "text": "of Q the negative 3 over pi just kind of"}, {"start": 98.64, "duration": 3.42, "text": "gets carried along and we'll just do the"}, {"start": 100.77, "duration": 4.41, "text": "same thing as before we've got a"}, {"start": 102.06, "duration": 5.04, "text": "variable to a number okay so that in"}, {"start": 105.18, "duration": 5.61, "text": "that number that exponent just comes out"}, {"start": 107.1, "duration": 5.28, "text": "front times our variable Q and then we"}, {"start": 110.79, "duration": 3.75, "text": "do the same thing as before we just take"}, {"start": 112.38, "duration": 4.949, "text": "one away from this number so if we"}, {"start": 114.54, "duration": 6.09, "text": "subtract one from 2.05 we'll just get"}, {"start": 117.329, "duration": 5.25, "text": "one point zero five and if you wanted to"}, {"start": 120.63, "duration": 4.129, "text": "I mean you could always multiply you"}, {"start": 122.579, "duration": 4.531, "text": "could think about this as being over 1 I"}, {"start": 124.759, "duration": 4.301, "text": "guess we could take negative three times"}, {"start": 127.11, "duration": 5.56, "text": "two point zero five we could make that"}, {"start": 129.06, "duration": 6.31, "text": "negative six point one five the"}, {"start": 132.67, "duration": 4.71, "text": "you race to the 1.05 would also go in"}, {"start": 135.37, "duration": 4.86, "text": "the numerator and then we would have our"}, {"start": 137.38, "duration": 4.98, "text": "pie in the denominator so if you wanted"}, {"start": 140.23, "duration": 4.17, "text": "to kind of write it all back together it"}, {"start": 142.36, "duration": 4.31, "text": "certainly would be okay to write it like"}, {"start": 144.4, "duration": 5.01, "text": "this but that's all there is to it so"}, {"start": 146.67, "duration": 5.05, "text": "again you'll see derivative problems all"}, {"start": 149.41, "duration": 4.67, "text": "the time that involve square roots cube"}, {"start": 151.72, "duration": 4.8, "text": "roots fourth roots other types of roots"}, {"start": 154.08, "duration": 5.62, "text": "always first thing I do is just"}, {"start": 156.52, "duration": 5.37, "text": "immediately rewrite those as exponents"}, {"start": 159.7, "duration": 5.63, "text": "and if I can I simplify it before I"}, {"start": 161.89, "duration": 3.44, "text": "start taking the derivative"}], "title": "Power Rule and Derivatives, A Basic Example #1", "description": "Thanks to all of you who support me on Patreon. You da real mvps!   $1 per month helps!! :)  https://www.patreon.com/patrickjmt !! Power Rule and Derivatives, A Basic Example. In this video I use the power rule to find the derivative of a function.  There is a touch of algebra involved, but nothing too heavy! : )", "lengthSeconds": 174, "width": 480, "height": 360}
{"video_id": "XDC53pvATcs", "transcript": [{"start": 0.96, "duration": 4.04, "text": "all right so one more example here using"}, {"start": 2.52, "duration": 3.399, "text": "alternating series so here we've got1 to"}, {"start": 5.0, "duration": 5.12, "text": "the N"}, {"start": 5.919, "duration": 5.88, "text": "+1 all divided by the square < TK of n^2"}, {"start": 10.12, "duration": 5.519, "text": "+ 6 N -"}, {"start": 11.799, "duration": 6.48, "text": "2 so in this case uh again if we get rid"}, {"start": 15.639, "duration": 4.241, "text": "of the the alternating the 1 to the n +"}, {"start": 18.279, "duration": 4.721, "text": "1 you can think about there as being a"}, {"start": 19.88, "duration": 8.559, "text": "times one on top so my B subn is going"}, {"start": 23.0, "duration": 10.079, "text": "to be 1 over the square Ro T of n^ 2 + 6"}, {"start": 28.439, "duration": 6.881, "text": "N - 2 so again same two questions um is"}, {"start": 33.079, "duration": 2.241, "text": "it"}, {"start": 37.719, "duration": 4.281, "text": "decreasing and does the limit go to"}, {"start": 44.28, "duration": 7.279, "text": "zero I think this question is"}, {"start": 48.12, "duration": 5.4, "text": "uh easy to address because as n goes to"}, {"start": 51.559, "duration": 4.201, "text": "Infinity we're certainly going to get a"}, {"start": 53.52, "duration": 4.679, "text": "very large number in the denominator"}, {"start": 55.76, "duration": 4.84, "text": "we'll have one over a large"}, {"start": 58.199, "duration": 4.761, "text": "number and that will get arbitrarily"}, {"start": 60.6, "duration": 5.0, "text": "close to zero so that condition"}, {"start": 62.96, "duration": 4.44, "text": "satisfied to do decreasing again you"}, {"start": 65.6, "duration": 2.96, "text": "know our series I think we started it at"}, {"start": 67.4, "duration": 3.64, "text": "nals"}, {"start": 68.56, "duration": 4.68, "text": "one you can always again just check"}, {"start": 71.04, "duration": 3.92, "text": "values so let's see if we plug in one I"}, {"start": 73.24, "duration": 4.0, "text": "think you know this one it's pretty"}, {"start": 74.96, "duration": 7.119, "text": "clear if you plug in one you'll get 1 +"}, {"start": 77.24, "duration": 9.559, "text": "6 that's 7 7 - 2 is going to be"}, {"start": 82.079, "duration": 8.0, "text": "5 when we plug in two well 2^ 2 is 4 6 *"}, {"start": 86.799, "duration": 6.801, "text": "4 or excuse me 6 * 2 is 12 so we have 12"}, {"start": 90.079, "duration": 6.32, "text": "+ 4 12 + 4 is 16 - 2 that's going to"}, {"start": 93.6, "duration": 4.64, "text": "give us the sare < TK of 14 and then you"}, {"start": 96.399, "duration": 6.241, "text": "know you can keep plugging in values and"}, {"start": 98.24, "duration": 7.36, "text": "you'll see that you know n^2 + 6 nus 2"}, {"start": 102.64, "duration": 5.799, "text": "larger and larger and larger so"}, {"start": 105.6, "duration": 5.44, "text": "absolutely it looks like our our our B"}, {"start": 108.439, "duration": 5.04, "text": "subn is decreasing so since both"}, {"start": 111.04, "duration": 7.48, "text": "conditions are satisfied we would say"}, {"start": 113.479, "duration": 5.041, "text": "that this alternating series converges"}], "title": "Alternating Series - Another Example 3", "description": "Thanks to all of you who support me on Patreon. You da real mvps!   $1 per month helps!! :)  https://www.patreon.com/patrickjmt !! Alternating Series - Another Example 3.  Just another example showing an alternating series converges or diverges.", "lengthSeconds": 120, "width": 480, "height": 360}
{"video_id": "MS_vs4IsGN0", "transcript": [{"start": 1.199, "duration": 4.921, "text": "okay in this example we're going to find"}, {"start": 2.52, "duration": 4.76, "text": "the domain of the function x / x^2 - 10"}, {"start": 6.12, "duration": 4.399, "text": "x -"}, {"start": 7.28, "duration": 5.479, "text": "24 so the only restrictions the only"}, {"start": 10.519, "duration": 4.16, "text": "values of X that we'll have to Omit will"}, {"start": 12.759, "duration": 4.721, "text": "be those values of X that make the"}, {"start": 14.679, "duration": 5.561, "text": "denominator equal to zero so I'm going"}, {"start": 17.48, "duration": 6.92, "text": "to find those values so I'm going to"}, {"start": 20.24, "duration": 7.119, "text": "take the denominator x^2 - 10 x - 24 and"}, {"start": 24.4, "duration": 5.199, "text": "set that equal to zero and to solve this"}, {"start": 27.359, "duration": 4.921, "text": "we've got a quadratic equation I'm going"}, {"start": 29.599, "duration": 5.001, "text": "to try to"}, {"start": 32.28, "duration": 5.36, "text": "factor so in this case again we want two"}, {"start": 34.6, "duration": 8.4, "text": "numbers that multiply to"}, {"start": 37.64, "duration": 7.32, "text": "-4 but add up to -10 so let's see uh"}, {"start": 43.0, "duration": 4.879, "text": "let's see if any if we can find any that"}, {"start": 44.96, "duration": 4.32, "text": "work um I think some combination of 2"}, {"start": 47.879, "duration": 4.561, "text": "and"}, {"start": 49.28, "duration": 5.64, "text": "12 right that's going to multiply to 24"}, {"start": 52.44, "duration": 6.2, "text": "and I think let's see if we make the the"}, {"start": 54.92, "duration": 6.52, "text": "12 negative and the two positive those"}, {"start": 58.64, "duration": 5.559, "text": "two numbers will multiply to -4 but add"}, {"start": 61.44, "duration": 4.999, "text": "up to -10 so that's a correct"}, {"start": 64.199, "duration": 5.001, "text": "factorization and then we just take each"}, {"start": 66.439, "duration": 6.441, "text": "factor set it equal to Z and solve so"}, {"start": 69.2, "duration": 6.919, "text": "from the first Factor we'll get that x ="}, {"start": 72.88, "duration": 7.52, "text": "-2 is a solution to our quadratic and"}, {"start": 76.119, "duration": 7.561, "text": "then we'll get x - 12 also equal 0 well"}, {"start": 80.4, "duration": 7.719, "text": "we can add 12 to both"}, {"start": 83.68, "duration": 4.439, "text": "sides so the domain"}, {"start": 90.36, "duration": 8.92, "text": "it'll be all values of X such that X"}, {"start": 95.0, "duration": 6.2, "text": "does not equal -2 and also X does not"}, {"start": 99.28, "duration": 4.4, "text": "equal"}, {"start": 101.2, "duration": 5.76, "text": "pos2 and you could also certainly write"}, {"start": 103.68, "duration": 7.28, "text": "this using interval notation um but"}, {"start": 106.96, "duration": 4.0, "text": "however you prefer to write it"}], "title": "Finding the Domain of a Rational Function / Fraction with Variable in Denominator", "description": "Thanks to all of you who support me on Patreon. You da real mvps!   $1 per month helps!! :)  https://www.patreon.com/patrickjmt !! Finding the Domain of a Rational Function / Fraction with Variable in Denominator.\nIn this example, I find the domain of a rational function by factoring the denominator.", "lengthSeconds": 109, "width": 450, "height": 360}
{"video_id": "J6LZK2azoeQ", "transcript": [{"start": 0.799, "duration": 3.681, "text": "okay in an act of uh Shameless"}, {"start": 2.48, "duration": 5.359, "text": "self-promotion here uh I'm thinking"}, {"start": 4.48, "duration": 6.48, "text": "about making some T-shirts after people"}, {"start": 7.839, "duration": 5.641, "text": "can consistently keep uh asking well not"}, {"start": 10.96, "duration": 5.76, "text": "consistently periodically often enough"}, {"start": 13.48, "duration": 5.16, "text": "so um basically got a couple t-shirt"}, {"start": 16.72, "duration": 4.559, "text": "designs uh I'm just kind of wondering if"}, {"start": 18.64, "duration": 4.12, "text": "anybody's actually interested so um I've"}, {"start": 21.279, "duration": 3.08, "text": "thought about doing this for a while so"}, {"start": 22.76, "duration": 4.16, "text": "I've got a couple different ideas you"}, {"start": 24.359, "duration": 3.92, "text": "can vote for your favorite t-shirt idea"}, {"start": 26.92, "duration": 4.6, "text": "I'll put a link too somewhere in the"}, {"start": 28.279, "duration": 5.641, "text": "comments at just math tutoring"}, {"start": 31.52, "duration": 6.64, "text": "blogspot.com so a couple different ideas"}, {"start": 33.92, "duration": 6.68, "text": "so the first idea is iart Patrick JMT"}, {"start": 38.16, "duration": 6.12, "text": "right a little less than three uh so"}, {"start": 40.6, "duration": 5.68, "text": "iHeart Patrick JMT um okay so again very"}, {"start": 44.28, "duration": 6.68, "text": "uh self-gratifying"}, {"start": 46.28, "duration": 6.36, "text": "here um the classic smart equals sexy"}, {"start": 50.96, "duration": 4.16, "text": "okay so would you wear a shirt that says"}, {"start": 52.64, "duration": 6.239, "text": "that I would um I would wear either one"}, {"start": 55.12, "duration": 8.16, "text": "of them um number three girls love a guy"}, {"start": 58.879, "duration": 7.521, "text": "with a big dot dot dot brain okay so hey"}, {"start": 63.28, "duration": 5.32, "text": "a little racy right how exciting um last"}, {"start": 66.4, "duration": 5.24, "text": "but not least hey the classic the Geeks"}, {"start": 68.6, "duration": 4.92, "text": "shell inherit the earth um so if you go"}, {"start": 71.64, "duration": 3.76, "text": "there I think I've got those listed if"}, {"start": 73.52, "duration": 3.919, "text": "you've got a great idea for a shirt feel"}, {"start": 75.4, "duration": 3.68, "text": "free to post it as a comment uh maybe"}, {"start": 77.439, "duration": 4.761, "text": "I'll add it into the survey I think I"}, {"start": 79.08, "duration": 5.28, "text": "can still add things into the survey um"}, {"start": 82.2, "duration": 4.239, "text": "and uh who knows maybe your your uh"}, {"start": 84.36, "duration": 3.96, "text": "T-shirt design will be will be picked if"}, {"start": 86.439, "duration": 4.32, "text": "it is and I make them I'll definitely"}, {"start": 88.32, "duration": 5.04, "text": "send you one for free so"}, {"start": 90.759, "duration": 4.481, "text": "um you can even go on there and say hey"}, {"start": 93.36, "duration": 4.16, "text": "I would never ever buy one of your"}, {"start": 95.24, "duration": 4.519, "text": "shirts so that's useful information to"}, {"start": 97.52, "duration": 5.239, "text": "you so uh just a little idea tell me"}, {"start": 99.759, "duration": 4.841, "text": "what you think um be uh brutally honest"}, {"start": 102.759, "duration": 3.561, "text": "it's not going to hurt my feelings so"}, {"start": 104.6, "duration": 4.119, "text": "all right everybody um I know this is"}, {"start": 106.32, "duration": 4.32, "text": "the last week of uh finals finals week"}, {"start": 108.719, "duration": 3.481, "text": "for a lot of you um those of you in high"}, {"start": 110.64, "duration": 3.6, "text": "school still have a couple weeks I know"}, {"start": 112.2, "duration": 3.839, "text": "so hang in there it's almost over hope"}, {"start": 114.24, "duration": 4.64, "text": "your semester's going well it's been"}, {"start": 116.039, "duration": 5.801, "text": "busy for me got a few more days be happy"}, {"start": 118.88, "duration": 7.159, "text": "when it's over so all right um take care"}, {"start": 121.84, "duration": 4.199, "text": "I appreciate your feedback out there"}], "title": "PatrickJMT T-Shirts?!?!", "description": "Thanks to all of you who support me on Patreon. You da real mvps!   $1 per month helps!! :)  https://www.patreon.com/patrickjmt !! Buy a shirt at http://www.printmojo.com/PatrickJMT/\n\nWould you waste your hard earned (or borrowed) cash to look like a total geek?! I would! Let me know if you like any of the T-shirt ideas.\nYou can vote at : http://JustMathTutoring.blogspot.com", "lengthSeconds": 124, "width": 480, "height": 360}
{"video_id": "2W7XqbaV3s0", "transcript": [{"start": 1.0, "duration": 4.52, "text": "okay suppose we know that the sum of"}, {"start": 2.48, "duration": 5.12, "text": "three consecutive even integers is 180"}, {"start": 5.52, "duration": 3.96, "text": "we want to find the largest of those"}, {"start": 7.6, "duration": 5.959, "text": "three"}, {"start": 9.48, "duration": 5.56, "text": "integers so the idea is we've got three"}, {"start": 13.559, "duration": 6.241, "text": "even integers we don't know what they"}, {"start": 15.04, "duration": 4.76, "text": "are so I'm going to let X denote the"}, {"start": 20.279, "duration": 3.321, "text": "smallest even"}, {"start": 25.48, "duration": 5.959, "text": "integer well to get to the next even"}, {"start": 27.88, "duration": 6.519, "text": "integer we would just add two to the"}, {"start": 31.439, "duration": 4.841, "text": "original number well to get to the next"}, {"start": 34.399, "duration": 5.121, "text": "even integer after that we would have to"}, {"start": 36.28, "duration": 8.52, "text": "add yet another two so we would get x +"}, {"start": 39.52, "duration": 7.359, "text": "2 + 2 or x + 4 well we know that if we"}, {"start": 44.8, "duration": 6.64, "text": "add these"}, {"start": 46.879, "duration": 4.561, "text": "up if we add these three numbers"}, {"start": 53.719, "duration": 3.921, "text": "up we get"}, {"start": 55.6, "duration": 4.439, "text": "180 and that's now going to be the"}, {"start": 57.64, "duration": 3.48, "text": "equation that we have to solve you you"}, {"start": 60.039, "duration": 3.281, "text": "can certainly just remove the"}, {"start": 61.12, "duration": 5.16, "text": "parenthesis here and combine like terms"}, {"start": 63.32, "duration": 5.839, "text": "so we would have X Plus X Plus X which"}, {"start": 66.28, "duration": 5.36, "text": "is 3x we would have 2 +"}, {"start": 69.159, "duration": 6.6, "text": "4 which is"}, {"start": 71.64, "duration": 6.159, "text": "six that equals 180 we can subtract six"}, {"start": 75.759, "duration": 5.801, "text": "from both"}, {"start": 77.799, "duration": 6.081, "text": "sides we'll have 3x ="}, {"start": 81.56, "duration": 5.84, "text": "174 and now the last thing we have to do"}, {"start": 83.88, "duration": 6.4, "text": "is divide both sides by"}, {"start": 87.4, "duration": 4.64, "text": "three so 3 divid 3 will will just leave"}, {"start": 90.28, "duration": 6.92, "text": "us with"}, {"start": 92.04, "duration": 7.6, "text": "1X let's see uh 174 / 3 that looks to me"}, {"start": 97.2, "duration": 5.52, "text": "like that's going to give us uh"}, {"start": 99.64, "duration": 4.839, "text": "58 okay so x equals 58 so we have to be"}, {"start": 102.72, "duration": 4.679, "text": "careful right that's not our the"}, {"start": 104.479, "duration": 5.561, "text": "solution to the question uh but we now"}, {"start": 107.399, "duration": 5.841, "text": "know what the smallest integer is uh we"}, {"start": 110.04, "duration": 5.96, "text": "know that it is equal to 58 well that"}, {"start": 113.24, "duration": 4.12, "text": "means the next one would have to be 60"}, {"start": 116.0, "duration": 2.84, "text": "and then the one after that would have"}, {"start": 117.36, "duration": 5.32, "text": "to be"}, {"start": 118.84, "duration": 6.199, "text": "62 so the solution to our our question"}, {"start": 122.68, "duration": 4.84, "text": "here when we want to find the largest of"}, {"start": 125.039, "duration": 3.401, "text": "the three integers we would say we would"}, {"start": 127.52, "duration": 3.92, "text": "get"}, {"start": 128.44, "duration": 3.0, "text": "62"}], "title": "Linear Equation, Word Problem #2 : Consecutive Even Integers", "description": "Thanks to all of you who support me on Patreon. You da real mvps!   $1 per month helps!! :)  https://www.patreon.com/patrickjmt !! Linear Equation, Word Problem #2 : Consecutive Even Integers.\r\n Here, we are given that three consecutive even integers add up to 180.  We find the largest of these three integers using a linear equation.", "lengthSeconds": 130, "width": 480, "height": 360}
{"video_id": "lAAEnE9nk1U", "transcript": [{"start": 0.84, "duration": 3.919, "text": "okay I mentioned this in the uh last"}, {"start": 2.72, "duration": 3.679, "text": "part of one of my videos but uh I was"}, {"start": 4.759, "duration": 3.721, "text": "going to see if uh anyone was interested"}, {"start": 6.399, "duration": 4.041, "text": "in doing a little video response problem"}, {"start": 8.48, "duration": 3.64, "text": "and uh what this has to do with changing"}, {"start": 10.44, "duration": 4.8, "text": "the order of Integrations for triple"}, {"start": 12.12, "duration": 5.36, "text": "integral so if you want to uh uh show"}, {"start": 15.24, "duration": 4.0, "text": "how to work this problem I would love if"}, {"start": 17.48, "duration": 3.879, "text": "you would uh like to post a video"}, {"start": 19.24, "duration": 4.359, "text": "response I'll absolutely uh let people"}, {"start": 21.359, "duration": 3.92, "text": "see it this way too again you know I"}, {"start": 23.599, "duration": 3.641, "text": "would encourage if you're having issues"}, {"start": 25.279, "duration": 3.521, "text": "with this stuff practice this you know"}, {"start": 27.24, "duration": 3.64, "text": "hopefully you watched the other videos"}, {"start": 28.8, "duration": 4.599, "text": "they were helped a little bit practice"}, {"start": 30.88, "duration": 6.48, "text": "this example on your own um you know"}, {"start": 33.399, "duration": 9.041, "text": "again what we want to do is rewrite"}, {"start": 37.36, "duration": 5.08, "text": "it using the five other orders of"}, {"start": 45.559, "duration": 5.401, "text": "integration so my advice to you um if"}, {"start": 49.28, "duration": 3.119, "text": "you're having issues with it you know"}, {"start": 50.96, "duration": 4.119, "text": "practice it on your own check out the"}, {"start": 52.399, "duration": 4.8, "text": "video responses see where you went wrong"}, {"start": 55.079, "duration": 4.44, "text": "if uh if you feel like it's finally"}, {"start": 57.199, "duration": 5.601, "text": "starting to click for you help others I"}, {"start": 59.519, "duration": 4.6, "text": "think you know know again um there's no"}, {"start": 62.8, "duration": 3.639, "text": "better way to learn something than to"}, {"start": 64.119, "duration": 5.0, "text": "try to explain it to someone I think uh"}, {"start": 66.439, "duration": 5.68, "text": "it really uh verbalizing things makes"}, {"start": 69.119, "duration": 8.121, "text": "you really um"}, {"start": 72.119, "duration": 6.68, "text": "uh I guess uh uh it makes you really"}, {"start": 77.24, "duration": 3.239, "text": "again I look I can't say at it I'm"}, {"start": 78.799, "duration": 4.68, "text": "having a hard time it makes you really"}, {"start": 80.479, "duration": 4.64, "text": "clarify your thoughts I guess right um"}, {"start": 83.479, "duration": 4.0, "text": "you know if you can explain something"}, {"start": 85.119, "duration": 4.281, "text": "simply you probably don't know it well"}, {"start": 87.479, "duration": 3.801, "text": "enough I think uh Einstein said"}, {"start": 89.4, "duration": 4.079, "text": "something like that that so sorry to"}, {"start": 91.28, "duration": 4.28, "text": "throw the Einstein quotes out there but"}, {"start": 93.479, "duration": 3.32, "text": "uh I think it's a good one so anyway"}, {"start": 95.56, "duration": 2.919, "text": "what we're doing is we've got this"}, {"start": 96.799, "duration": 3.161, "text": "little uh three-dimensional object that"}, {"start": 98.479, "duration": 3.761, "text": "we're integrating over kind of like a"}, {"start": 99.96, "duration": 5.199, "text": "little almost like a little uh wedge of"}, {"start": 102.24, "duration": 5.28, "text": "cheese here or something so kind of the"}, {"start": 105.159, "duration": 5.32, "text": "uh sort of the roof and the front part"}, {"start": 107.52, "duration": 6.239, "text": "of this uh um of this little wedge of"}, {"start": 110.479, "duration": 5.761, "text": "cheese is the equation zal 1 - x^2 the"}, {"start": 113.759, "duration": 6.72, "text": "side of it can be represented by the"}, {"start": 116.24, "duration": 6.44, "text": "plane yal 1 - x um"}, {"start": 120.479, "duration": 5.161, "text": "the back of it looks like the YZ plane"}, {"start": 122.68, "duration": 5.039, "text": "the side of it is just the XZ plane they"}, {"start": 125.64, "duration": 4.959, "text": "uh give you one correct way of setting"}, {"start": 127.719, "duration": 5.041, "text": "up the uh the orders of integration so"}, {"start": 130.599, "duration": 3.801, "text": "again uh if you're interested try to"}, {"start": 132.76, "duration": 5.199, "text": "show people how to switch these orders"}, {"start": 134.4, "duration": 5.52, "text": "of integration do Dy dxdz and then do"}, {"start": 137.959, "duration": 4.481, "text": "the other ones first do X first and then"}, {"start": 139.92, "duration": 4.599, "text": "uh do Z first so if you uh if you're"}, {"start": 142.44, "duration": 3.879, "text": "interested that'd be great if uh if"}, {"start": 144.519, "duration": 4.0, "text": "people like it maybe I can uh send you a"}, {"start": 146.319, "duration": 6.2, "text": "free shirt or uh you know do something"}, {"start": 148.519, "duration": 5.841, "text": "to uh uh um I don't know to uh to reward"}, {"start": 152.519, "duration": 3.601, "text": "your hard work U definitely would love"}, {"start": 154.36, "duration": 4.2, "text": "to put it up on my channel as a featured"}, {"start": 156.12, "duration": 4.16, "text": "video for for at least a few days so if"}, {"start": 158.56, "duration": 5.039, "text": "you're interested great if you need some"}, {"start": 160.28, "duration": 5.319, "text": "uh pointers or some hints um post"}, {"start": 163.599, "duration": 4.92, "text": "comments and uh you know try to help"}, {"start": 165.599, "duration": 5.72, "text": "each other out so all right um good luck"}, {"start": 168.519, "duration": 4.841, "text": "and uh I don't know hopefully uh"}, {"start": 171.319, "duration": 6.241, "text": "hopefully somebody will uh will will"}, {"start": 173.36, "duration": 4.2, "text": "post some good some good Solutions"}], "title": "Video Response Challenge for Triple Integrals!", "description": "Thanks to all of you who support me on Patreon. You da real mvps!   $1 per month helps!! :)  https://www.patreon.com/patrickjmt !! Video Response Challenge for Triple Integrals!\nWant to show your stuff? Or maybe you are just bored?! :)\nPost a video response about how to change the order of integration on the given problem!  Show how to set up the other 5 orders of integration!", "lengthSeconds": 176, "width": 480, "height": 360}
{"video_id": "i8YJ9ZIkvTE", "transcript": [{"start": 0.03, "duration": 4.07, "text": "all right in this video I want to solve"}, {"start": 2.28, "duration": 4.74, "text": "another quadratic equation by factoring"}, {"start": 4.1, "duration": 6.61, "text": "so here we want to solve the quantity y"}, {"start": 7.02, "duration": 5.249, "text": "plus 6 squared equals 81 and for those"}, {"start": 10.71, "duration": 3.839, "text": "of you who have seen the square root"}, {"start": 12.269, "duration": 4.59, "text": "property and you're saying hey Patrick"}, {"start": 14.549, "duration": 5.941, "text": "the square root property is much easier"}, {"start": 16.859, "duration": 4.951, "text": "to use I say I absolutely agree but"}, {"start": 20.49, "duration": 4.73, "text": "again you know we're just trying to"}, {"start": 21.81, "duration": 5.879, "text": "illustrate factoring I don't you know"}, {"start": 25.22, "duration": 5.83, "text": "the procedure is what's important here"}, {"start": 27.689, "duration": 4.441, "text": "so okay in this case the first thing I'm"}, {"start": 31.05, "duration": 5.31, "text": "going to do since I'm going to try to"}, {"start": 32.13, "duration": 6.98, "text": "factor it again we want one side of our"}, {"start": 36.36, "duration": 7.08, "text": "quadratic equation to equal zero"}, {"start": 39.11, "duration": 7.929, "text": "so I'm going to subtract 81 from both"}, {"start": 43.44, "duration": 7.68, "text": "sides so well we've just got the"}, {"start": 47.039, "duration": 7.711, "text": "quantity y plus 6 squared now minus 81"}, {"start": 51.12, "duration": 7.259, "text": "equals 0 there's two things you could do"}, {"start": 54.75, "duration": 5.73, "text": "here you could now oil out the Y plus 6"}, {"start": 58.379, "duration": 3.77, "text": "you know take Y plus 6 times y plus 6"}, {"start": 60.48, "duration": 4.53, "text": "collect your like terms"}, {"start": 62.149, "duration": 6.131, "text": "subtract 81 and then kind of try to"}, {"start": 65.01, "duration": 4.8, "text": "refactor that but to me when I see this"}, {"start": 68.28, "duration": 5.879, "text": "this is really a difference of perfect"}, {"start": 69.81, "duration": 6.809, "text": "squares I see something squared and"}, {"start": 74.159, "duration": 6.111, "text": "there's my minus sign and then I see"}, {"start": 76.619, "duration": 6.871, "text": "something else squared namely 9 squared"}, {"start": 80.27, "duration": 6.82, "text": "ok and remember a squared minus B"}, {"start": 83.49, "duration": 8.55, "text": "squared that factors nicely so a squared"}, {"start": 87.09, "duration": 7.529, "text": "minus B squared is just a minus B a plus"}, {"start": 92.04, "duration": 6.17, "text": "B so that's what I'm gonna do in this"}, {"start": 94.619, "duration": 8.191, "text": "case so to me it says you're gonna get"}, {"start": 98.21, "duration": 8.11, "text": "okay so my a is the Y plus 6 so you get"}, {"start": 102.81, "duration": 5.58, "text": "y plus 6 minus B and again the number"}, {"start": 106.32, "duration": 4.89, "text": "being squared is 9"}, {"start": 108.39, "duration": 4.38, "text": "and then we multiply that by basically"}, {"start": 111.21, "duration": 4.68, "text": "the exact same thing but we just put a"}, {"start": 112.77, "duration": 9.18, "text": "plus sign in the middle so y plus six"}, {"start": 115.89, "duration": 10.86, "text": "plus nine equals zero and now inside the"}, {"start": 121.95, "duration": 6.54, "text": "parentheses we could really think about"}, {"start": 126.75, "duration": 3.24, "text": "there's being a positive one so if you"}, {"start": 128.49, "duration": 4.8, "text": "distribute that it would just get rid of"}, {"start": 129.99, "duration": 5.85, "text": "the parentheses on the y plus six so we"}, {"start": 133.29, "duration": 5.7, "text": "have y plus six minus 9 which is going"}, {"start": 135.84, "duration": 5.22, "text": "to be Y minus three and then we have Y"}, {"start": 138.99, "duration": 7.14, "text": "plus six plus nine which is going to be"}, {"start": 141.06, "duration": 6.81, "text": "Y plus 15 equals zero and now that we"}, {"start": 146.13, "duration": 5.19, "text": "have it nice and factored we just do the"}, {"start": 147.87, "duration": 7.14, "text": "same thing we set the first factor equal"}, {"start": 151.32, "duration": 7.05, "text": "to zero we set the second factor equal"}, {"start": 155.01, "duration": 5.64, "text": "to zero I'll add three to both sides on"}, {"start": 158.37, "duration": 5.64, "text": "the first equation and get y equals"}, {"start": 160.65, "duration": 6.29, "text": "three and I'll subtract fifteen on my"}, {"start": 164.01, "duration": 5.28, "text": "second one to get y equals negative"}, {"start": 166.94, "duration": 4.96, "text": "fifteen and now we've got our two"}, {"start": 169.29, "duration": 4.44, "text": "solutions and notice it's pretty easy to"}, {"start": 171.9, "duration": 4.23, "text": "check these if you plug three in you"}, {"start": 173.73, "duration": 4.65, "text": "would get three plus six or nine squared"}, {"start": 176.13, "duration": 3.69, "text": "which is certainly 81 if you plug"}, {"start": 178.38, "duration": 3.03, "text": "negative fifteen and you would get"}, {"start": 179.82, "duration": 2.85, "text": "negative fifteen plus six which would"}, {"start": 181.41, "duration": 5.63, "text": "give you negative nine"}, {"start": 182.67, "duration": 4.37, "text": "but that squared is still equal to 81"}], "title": "Solving Quadratic Equations by Factoring - Another Example", "description": "Thanks to all of you who support me on Patreon. You da real mvps!   $1 per month helps!! :)  https://www.patreon.com/patrickjmt !! Solving Quadratic Equations by Factoring - Another Example", "lengthSeconds": 186, "width": 480, "height": 360}
{"video_id": "Y3uu3Hsk4QU", "transcript": [{"start": 0.709, "duration": 4.331, "text": "okay so this is another SAT question"}, {"start": 3.51, "duration": 3.09, "text": "taken from the Princeton Review says"}, {"start": 5.04, "duration": 3.45, "text": "which of the following points is not on"}, {"start": 6.6, "duration": 4.709, "text": "the graph of absolute value of negative"}, {"start": 8.49, "duration": 4.89, "text": "x minus the absolute value of negative y"}, {"start": 11.309, "duration": 4.711, "text": "equals one and then they give you a"}, {"start": 13.38, "duration": 5.22, "text": "bunch of points this again is another"}, {"start": 16.02, "duration": 4.47, "text": "very typical you know don't don't try to"}, {"start": 18.6, "duration": 8.82, "text": "graph this that's not at all what you"}, {"start": 20.49, "duration": 8.609, "text": "want to do I mean you could I guess you"}, {"start": 27.42, "duration": 2.88, "text": "could do a couple simplifications you"}, {"start": 29.099, "duration": 4.261, "text": "know basically just start plugging"}, {"start": 30.3, "duration": 6.27, "text": "things in that's what I would do so"}, {"start": 33.36, "duration": 4.89, "text": "let's see it says again we have to make"}, {"start": 36.57, "duration": 3.84, "text": "sure we're finding which one is not on"}, {"start": 38.25, "duration": 3.66, "text": "the graph so if we plug negative 2 in"}, {"start": 40.41, "duration": 5.43, "text": "the absolute if we take negative"}, {"start": 41.91, "duration": 6.09, "text": "negative 2 will get positive 2 minus if"}, {"start": 45.84, "duration": 7.05, "text": "we plug in negative 3 we'll get positive"}, {"start": 48.0, "duration": 8.94, "text": "3 well does that equal 1 well we get 2"}, {"start": 52.89, "duration": 7.2, "text": "minus 3 which is negative 1 so hey 1"}, {"start": 56.94, "duration": 5.759, "text": "negative 1 does not equal 1 so tada"}, {"start": 60.09, "duration": 4.469, "text": "there's our answer negative 2 negative 3"}, {"start": 62.699, "duration": 6.001, "text": "would be the solution you can check the"}, {"start": 64.559, "duration": 7.85, "text": "others make sure that they work but this"}, {"start": 68.7, "duration": 3.709, "text": "is one that will not be on the graph"}], "title": "SAT Math Question #7", "description": "Thanks to all of you who support me on Patreon. You da real mvps!   $1 per month helps!! :)  https://www.patreon.com/patrickjmt !! A sample math SAT question taken from the Princeton Review.", "lengthSeconds": 71, "width": 480, "height": 360}
{"video_id": "3g92tS6UtUM", "transcript": [{"start": 1.079, "duration": 4.401, "text": "okay in this video I just want to do a"}, {"start": 2.879, "duration": 5.04, "text": "few examples of simplifying some"}, {"start": 5.48, "duration": 5.96, "text": "Expressions involving absolute value so"}, {"start": 7.919, "duration": 6.76, "text": "I've got three examples um and let's do"}, {"start": 11.44, "duration": 7.04, "text": "uh the first one first um so we're going"}, {"start": 14.679, "duration": 8.001, "text": "to take the absolute value of the of 8 -"}, {"start": 18.48, "duration": 6.44, "text": "12 and again on these problems typically"}, {"start": 22.68, "duration": 7.8, "text": "you know I just try to work inside out"}, {"start": 24.92, "duration": 8.28, "text": "so um if we do 8 - 12 we get -4 and then"}, {"start": 30.48, "duration": 5.68, "text": "remember if we take the absolute value"}, {"start": 33.2, "duration": 6.08, "text": "of uh a number negative a negative"}, {"start": 36.16, "duration": 6.079, "text": "number it simply turns into the positive"}, {"start": 39.28, "duration": 6.919, "text": "of that number so the absolute value of"}, {"start": 42.239, "duration": 7.241, "text": "-4 is simply four and again be careful"}, {"start": 46.199, "duration": 5.2, "text": "um because again a very one of the"}, {"start": 49.48, "duration": 5.399, "text": "classic mistakes you don't want to do"}, {"start": 51.399, "duration": 5.921, "text": "this um it would be easy to make this"}, {"start": 54.879, "duration": 6.241, "text": "the absolute value of 8 minus the"}, {"start": 57.32, "duration": 8.24, "text": "absolute value of 12 which would be 8 -"}, {"start": 61.12, "duration": 7.679, "text": "12 which would be -4 which is not"}, {"start": 65.56, "duration": 6.559, "text": "correct okay so do not do any of these"}, {"start": 68.799, "duration": 5.36, "text": "steps here in Red so don't you can't"}, {"start": 72.119, "duration": 3.561, "text": "break up the absolute value when you're"}, {"start": 74.159, "duration": 4.32, "text": "adding and subtracting you have to be"}, {"start": 75.68, "duration": 4.88, "text": "real careful about that okay so just a"}, {"start": 78.479, "duration": 4.6, "text": "little uh word of warning I'm sure I've"}, {"start": 80.56, "duration": 4.96, "text": "done it myself uh more than"}, {"start": 83.079, "duration": 5.4, "text": "once okay so in the next one here we"}, {"start": 85.52, "duration": 5.44, "text": "have the absolute value of -4 time the"}, {"start": 88.479, "duration": 4.561, "text": "absolute value of three and again I"}, {"start": 90.96, "duration": 6.199, "text": "think well what's the absolute value of"}, {"start": 93.04, "duration": 8.28, "text": "-4 that's simply pos4 the absolute value"}, {"start": 97.159, "duration": 6.841, "text": "of positive3 is just itself positive3"}, {"start": 101.32, "duration": 4.28, "text": "and then we get 4 * 3 and that gives us"}, {"start": 104.0, "duration": 4.52, "text": "our solution of"}, {"start": 105.6, "duration": 6.24, "text": "12 and in the last one here the same"}, {"start": 108.52, "duration": 8.12, "text": "thing we have the absolute value of -25"}, {"start": 111.84, "duration": 8.879, "text": "over 5 uh we could simplify inside First"}, {"start": 116.64, "duration": 7.68, "text": "- 255 over 5 and make that ne5 and the"}, {"start": 120.719, "duration": 6.161, "text": "absolute value of5 is pos5 another way"}, {"start": 124.32, "duration": 4.439, "text": "to do this uh just to illustrate some of"}, {"start": 126.88, "duration": 4.719, "text": "the properties remember if you have a"}, {"start": 128.759, "duration": 6.12, "text": "fraction inside of absolute value we can"}, {"start": 131.599, "duration": 7.161, "text": "write that as the absolute value of the"}, {"start": 134.879, "duration": 6.72, "text": "numerator over the absolute value of the"}, {"start": 138.76, "duration": 7.04, "text": "denominator and the absolute value of -"}, {"start": 141.599, "duration": 7.801, "text": "255 would simply be posi 25 the absolute"}, {"start": 145.8, "duration": 7.24, "text": "value of five is just five and if we"}, {"start": 149.4, "duration": 7.559, "text": "divide 25 over 5 again gives us our"}, {"start": 153.04, "duration": 5.479, "text": "solution of positive five so um and just"}, {"start": 156.959, "duration": 3.601, "text": "one little you know one little thing to"}, {"start": 158.519, "duration": 3.881, "text": "point out if you accidentally did this"}, {"start": 160.56, "duration": 3.92, "text": "mistake you know back in the first"}, {"start": 162.4, "duration": 4.4, "text": "example where you got a negative number"}, {"start": 164.48, "duration": 5.119, "text": "to me you know I'm taking the absolute"}, {"start": 166.8, "duration": 5.079, "text": "value of something uh for example in"}, {"start": 169.599, "duration": 4.961, "text": "this first problem the absolute value of"}, {"start": 171.879, "duration": 4.961, "text": "anything either has to be zero or a"}, {"start": 174.56, "duration": 5.679, "text": "positive number so at the end if I did"}, {"start": 176.84, "duration": 5.119, "text": "get some negative number out I I I I"}, {"start": 180.239, "duration": 5.0, "text": "would just recognize that something had"}, {"start": 181.959, "duration": 4.721, "text": "to be wrong there and uh maybe I would"}, {"start": 185.239, "duration": 3.56, "text": "you know figure out that this was the"}, {"start": 186.68, "duration": 4.479, "text": "incorrect step so just think about that"}, {"start": 188.799, "duration": 4.0, "text": "too when you're doing absolute value so"}, {"start": 191.159, "duration": 5.44, "text": "um all right I hope these few uh"}, {"start": 192.799, "duration": 3.8, "text": "examples help"}], "title": "Evaluating Expressions Involving Absolute Value - Example 1", "description": "Thanks to all of you who support me on Patreon. You da real mvps!   $1 per month helps!! :)  https://www.patreon.com/patrickjmt !! Evaluating Expressions Involving Absolute Value - Example 1\nJust a few very basic problems involving simplifications with absolute value.", "lengthSeconds": 195, "width": 480, "height": 360}
{"video_id": "nq95m-X_lp0", "transcript": [{"start": 0.03, "duration": 3.599, "text": "okay here we're going to solve one last"}, {"start": 2.01, "duration": 4.769, "text": "equation where we have variables on both"}, {"start": 3.629, "duration": 5.76, "text": "sides here we're going to solve 6 minus"}, {"start": 6.779, "duration": 9.211, "text": "x over 5 equals 3 times the quantity X"}, {"start": 9.389, "duration": 9.261, "text": "plus 2 over 10 so we could start you"}, {"start": 15.99, "duration": 6.84, "text": "know distributing out in the numerator"}, {"start": 18.65, "duration": 7.0, "text": "we could break up the fraction we could"}, {"start": 22.83, "duration": 5.16, "text": "do lots of things here typically I think"}, {"start": 25.65, "duration": 5.19, "text": "the easiest thing to do if you have"}, {"start": 27.99, "duration": 5.85, "text": "fractions a lot of times it's it's"}, {"start": 30.84, "duration": 5.01, "text": "easiest just to try to multiply both"}, {"start": 33.84, "duration": 4.649, "text": "sides of the equation by something that"}, {"start": 35.85, "duration": 6.33, "text": "will get rid of the fractions so in this"}, {"start": 38.489, "duration": 6.931, "text": "case if we think about the least common"}, {"start": 42.18, "duration": 6.84, "text": "multiple of 1 5 and 10 the least common"}, {"start": 45.42, "duration": 4.95, "text": "multiple of 1 5 and 10 would be 10 so"}, {"start": 49.02, "duration": 5.37, "text": "what's the smallest number that's"}, {"start": 50.37, "duration": 6.48, "text": "divisible by 10 5 and 1 well I'm going"}, {"start": 54.39, "duration": 7.079, "text": "to multiply both sides by that number"}, {"start": 56.85, "duration": 7.29, "text": "again 10 in this case so on the left"}, {"start": 61.469, "duration": 5.791, "text": "side if we distribute well 10 times 6"}, {"start": 64.14, "duration": 7.83, "text": "will be 60 I'll write it out this is 10"}, {"start": 67.26, "duration": 6.48, "text": "over 1 we would get 10 x over 5 but"}, {"start": 71.97, "duration": 5.43, "text": "again 10 x over 5 that's just going to"}, {"start": 73.74, "duration": 7.949, "text": "be 2x right so we're getting rid of the"}, {"start": 77.4, "duration": 6.39, "text": "fractions by multiplying by 10 so on the"}, {"start": 81.689, "duration": 4.801, "text": "right side the tens will just cancel out"}, {"start": 83.79, "duration": 5.67, "text": "and we'll have 3 times the quantity X"}, {"start": 86.49, "duration": 5.489, "text": "plus 2 I'm going to go ahead and"}, {"start": 89.46, "duration": 7.769, "text": "distribute the 3 so we'll have 3 times X"}, {"start": 91.979, "duration": 8.25, "text": "which is 3x 3 times 2 will be 6 and now"}, {"start": 97.229, "duration": 4.921, "text": "it's a matter of putting all the terms"}, {"start": 100.229, "duration": 4.5, "text": "involving X on one side and all the"}, {"start": 102.15, "duration": 4.8, "text": "constants on the other so the first"}, {"start": 104.729, "duration": 7.71, "text": "thing I'm going to do to both sides is"}, {"start": 106.95, "duration": 7.32, "text": "add 2x again you could move the"}, {"start": 112.439, "duration": 4.021, "text": "constants around first as well it"}, {"start": 114.27, "duration": 8.489, "text": "doesn't matter on the left side we'll be"}, {"start": 116.46, "duration": 8.58, "text": "left with 60 3x plus 2x is 5x and again"}, {"start": 122.759, "duration": 4.86, "text": "I chose to add 2x I could have"}, {"start": 125.04, "duration": 5.1, "text": "subtracted 3x but again I just kind of"}, {"start": 127.619, "duration": 4.521, "text": "observe that by adding 2x I'm going to"}, {"start": 130.14, "duration": 4.79, "text": "keep everything positive"}, {"start": 132.14, "duration": 4.59, "text": "to me I like just having things positive"}, {"start": 134.93, "duration": 4.59, "text": "it keeps me from making sign mistakes"}, {"start": 136.73, "duration": 3.71, "text": "but again nothing says you have to do it"}, {"start": 139.52, "duration": 5.52, "text": "that way"}, {"start": 140.44, "duration": 9.4, "text": "so 60 equals 5x plus 6 will subtract 6"}, {"start": 145.04, "duration": 8.97, "text": "from both sides 60 minus 6 will be 54"}, {"start": 149.84, "duration": 6.21, "text": "equals 5x and the last thing we'll do"}, {"start": 154.01, "duration": 6.0, "text": "here since we're multiplying by five is"}, {"start": 156.05, "duration": 6.689, "text": "we'll divide both sides by five and then"}, {"start": 160.01, "duration": 5.79, "text": "we'll have our solution x equals 54 over"}, {"start": 162.739, "duration": 5.431, "text": "five that doesn't reduce so I'm going to"}, {"start": 165.8, "duration": 5.18, "text": "leave our solution just like that as a"}, {"start": 168.17, "duration": 2.81, "text": "fraction"}], "title": "Solving a Linear Equation with Variables on Both Sides of the Equation, Ex 3", "description": "Thanks to all of you who support me on Patreon. You da real mvps!   $1 per month helps!! :)  https://www.patreon.com/patrickjmt !! Solving a Linear Equation with Variables on Both Sides of the Equation, Ex 3", "lengthSeconds": 170, "width": 480, "height": 360}
{"video_id": "v5gALD9kSP0", "transcript": [{"start": 0.5, "duration": 5.379, "text": "okay suppose that the graph of a linear"}, {"start": 2.939, "duration": 5.281, "text": "equation has the following values it"}, {"start": 5.879, "duration": 4.531, "text": "says if we put in x equals zero we get y"}, {"start": 8.22, "duration": 4.62, "text": "equals one if we put in x equals one we"}, {"start": 10.41, "duration": 4.74, "text": "get y equals negative one and if we"}, {"start": 12.84, "duration": 4.38, "text": "substitute in x equals three we get y"}, {"start": 15.15, "duration": 3.9, "text": "equals negative five we want to know"}, {"start": 17.22, "duration": 4.68, "text": "what's the slope of the line is it"}, {"start": 19.05, "duration": 8.399, "text": "negative 4 negative 2 positive 2 or"}, {"start": 21.9, "duration": 8.16, "text": "positive 4 so this again is just kind of"}, {"start": 27.449, "duration": 8.841, "text": "an application of using that slope"}, {"start": 30.06, "duration": 10.53, "text": "formula M equals y2 minus y1 over x2"}, {"start": 36.29, "duration": 6.28, "text": "minus x1 and to use this formula all you"}, {"start": 40.59, "duration": 3.87, "text": "need you only need two points notice"}, {"start": 42.57, "duration": 4.219, "text": "we've actually been given three points"}, {"start": 44.46, "duration": 5.34, "text": "here we've got the point zero comma one"}, {"start": 46.789, "duration": 5.68, "text": "we've got the point 1 comma negative one"}, {"start": 49.8, "duration": 7.259, "text": "and then we also have the point 3 comma"}, {"start": 52.469, "duration": 8.64, "text": "negative 5 we can use any two of these"}, {"start": 57.059, "duration": 6.151, "text": "that we that we wish so I don't know it"}, {"start": 61.109, "duration": 4.2, "text": "doesn't really matter I'll use the 1"}, {"start": 63.21, "duration": 5.79, "text": "comma negative 1 and the 3 comma"}, {"start": 65.309, "duration": 7.291, "text": "negative 5 for no particular reason just"}, {"start": 69.0, "duration": 6.57, "text": "picking two at random so okay so there's"}, {"start": 72.6, "duration": 4.82, "text": "going to be my x1 there that's going to"}, {"start": 75.57, "duration": 8.159, "text": "be my value of y1"}, {"start": 77.42, "duration": 9.879, "text": "there's x2 and there is y2 okay so let's"}, {"start": 83.729, "duration": 7.921, "text": "do the change in Y so we'll do negative"}, {"start": 87.299, "duration": 7.5, "text": "5 minus negative 1 so I'm going to put"}, {"start": 91.65, "duration": 4.859, "text": "that in parentheses just to make it look"}, {"start": 94.799, "duration": 8.761, "text": "a little cleaner here and then we've got"}, {"start": 96.509, "duration": 10.14, "text": "3 minus positive 1 well negative 5 minus"}, {"start": 103.56, "duration": 6.9, "text": "negative 1 a negative and a negative"}, {"start": 106.649, "duration": 5.911, "text": "will make a positive well in the"}, {"start": 110.46, "duration": 5.85, "text": "denominator 3 minus 1 that's going to be"}, {"start": 112.56, "duration": 7.86, "text": "positive 2 negative 5 plus 1 that's"}, {"start": 116.31, "duration": 5.669, "text": "negative 4 and a negative divided by a"}, {"start": 120.42, "duration": 6.13, "text": "positive that's going to be a negative"}, {"start": 121.979, "duration": 6.041, "text": "number 4 divided by 2 is equal to 2"}, {"start": 126.55, "duration": 4.5, "text": "that's going to be the slope of our line"}, {"start": 128.02, "duration": 4.53, "text": "negative two and you can check it"}, {"start": 131.05, "duration": 3.21, "text": "doesn't matter you know you could have"}, {"start": 132.55, "duration": 3.69, "text": "used the first point in the second point"}, {"start": 134.26, "duration": 5.82, "text": "or the first point in the third point"}, {"start": 136.24, "duration": 5.97, "text": "and have went through the same you know"}, {"start": 140.08, "duration": 3.75, "text": "calculation and you would have found in"}, {"start": 142.21, "duration": 3.99, "text": "all cases that the slope is equal to"}, {"start": 143.83, "duration": 5.04, "text": "negative two so it doesn't matter you"}, {"start": 146.2, "duration": 5.3, "text": "know which which of those two points"}, {"start": 148.87, "duration": 2.63, "text": "that you pick"}], "title": "GED 'Most Missed' Math Practice Problems , Code A.5.b,  #3", "description": "Thanks to all of you who support me on Patreon. You da real mvps!   $1 per month helps!! :)  https://www.patreon.com/patrickjmt !! The GED Testing Service keeps a list of the most missed math topics and can be seen here: http://www.gedtestingservice.com/uploads/files/126de0283c94ff6323b3b5b3f310da5d.pdf\nThis video shows how to work a sample problem using their coding system that is shown on the left using the link above.", "lengthSeconds": 151, "width": 480, "height": 360}
{"video_id": "4FwqRASFmJo", "transcript": [{"start": 0.599, "duration": 4.841, "text": "okay here we're going to evaluate some"}, {"start": 3.4, "duration": 3.519, "text": "factorials uh just to get a feel for"}, {"start": 5.44, "duration": 2.92, "text": "factorials since you have to use"}, {"start": 6.919, "duration": 3.521, "text": "factorials when you calculate"}, {"start": 8.36, "duration": 5.199, "text": "permutations and"}, {"start": 10.44, "duration": 9.48, "text": "combinations well five factorial that's"}, {"start": 13.559, "duration": 11.441, "text": "just shorthand for 5 * 4 * 3 * 2 * 1 so"}, {"start": 19.92, "duration": 8.32, "text": "let's see 5 * 4 is going to be 20 3 * 2"}, {"start": 25.0, "duration": 8.32, "text": "is going to be 6 so if we multiply 20 *"}, {"start": 28.24, "duration": 5.999, "text": "6 we get 1 120 so 5 factorial is equal"}, {"start": 33.32, "duration": 4.72, "text": "to"}, {"start": 34.239, "duration": 6.16, "text": "120 um factorials get very large very"}, {"start": 38.04, "duration": 5.28, "text": "quick you know five factorial is already"}, {"start": 40.399, "duration": 6.281, "text": "bounced up to"}, {"start": 43.32, "duration": 5.759, "text": "120 suppose we want to do four time five"}, {"start": 46.68, "duration": 4.16, "text": "factorial well you need to do the"}, {"start": 49.079, "duration": 3.28, "text": "factorial first or you know it"}, {"start": 50.84, "duration": 4.68, "text": "definitely doesn't hurt to write it all"}, {"start": 52.359, "duration": 6.561, "text": "the way out so this is four * again well"}, {"start": 55.52, "duration": 4.92, "text": "5 * 4 * 3 * 2 *"}, {"start": 58.92, "duration": 3.68, "text": "1"}, {"start": 60.44, "duration": 4.36, "text": "well we've already seen that 5 factorial"}, {"start": 62.6, "duration": 2.2, "text": "is"}, {"start": 64.88, "duration": 5.36, "text": "120 so 4 * 120 if we simplify that we"}, {"start": 69.52, "duration": 6.44, "text": "get"}, {"start": 70.24, "duration": 8.519, "text": "480 so 4 * 5 factorial is 480 uh do not"}, {"start": 75.96, "duration": 4.72, "text": "multiply you know something uh that that"}, {"start": 78.759, "duration": 4.201, "text": "might be tempting to do that's"}, {"start": 80.68, "duration": 5.759, "text": "definitely not"}, {"start": 82.96, "duration": 5.36, "text": "correct don't say it's 20 factorial you"}, {"start": 86.439, "duration": 4.961, "text": "know don't somehow multiply the four and"}, {"start": 88.32, "duration": 5.32, "text": "the five 20 Factor is going to be a huge"}, {"start": 91.4, "duration": 5.88, "text": "big number definitely not going to be"}, {"start": 93.64, "duration": 5.519, "text": "480 so do not do it that way if you're"}, {"start": 97.28, "duration": 4.28, "text": "not sure whether or not you can sort of"}, {"start": 99.159, "duration": 4.361, "text": "do a process just write it out and then"}, {"start": 101.56, "duration": 4.28, "text": "you'll sort of see everything you know"}, {"start": 103.52, "duration": 5.199, "text": "written out and you can just"}, {"start": 105.84, "duration": 6.2, "text": "multiply okay so let's see um in part C"}, {"start": 108.719, "duration": 7.561, "text": "we've got six factorial so that's 6 * 5"}, {"start": 112.04, "duration": 7.64, "text": "* 4 * 3 * 2 *"}, {"start": 116.28, "duration": 7.759, "text": "1us we've got 7 - 3 factorial so I'm"}, {"start": 119.68, "duration": 8.119, "text": "going to going to uh simplify 7 - 3 is 4"}, {"start": 124.039, "duration": 5.72, "text": "factorial over 3 factorial well let's"}, {"start": 127.799, "duration": 5.641, "text": "see we can always uh simplify this we've"}, {"start": 129.759, "duration": 4.441, "text": "seen that 5 * 4 * 3 * 2 * 1 that's equal"}, {"start": 133.44, "duration": 6.36, "text": "to"}, {"start": 134.2, "duration": 8.119, "text": "120 so if we take 6 * 120 uh 6 * 120 is"}, {"start": 139.8, "duration": 5.04, "text": "going to be 720 so again if you multiply"}, {"start": 142.319, "duration": 5.0, "text": "those all out you'll get 720 you could"}, {"start": 144.84, "duration": 6.039, "text": "calculate four factorial calculate 3"}, {"start": 147.319, "duration": 6.28, "text": "factorial and then uh you know simpli it"}, {"start": 150.879, "duration": 3.841, "text": "uh notice though 4 factorial is 4 * 3 *"}, {"start": 153.599, "duration": 5.401, "text": "2 *"}, {"start": 154.72, "duration": 6.68, "text": "1 3 factorial is just 3 * 2 * one so if"}, {"start": 159.0, "duration": 3.56, "text": "we cancel out those common factors"}, {"start": 161.4, "duration": 4.04, "text": "really we're just going to be left with"}, {"start": 162.56, "duration": 7.12, "text": "a four in the numerator so we have"}, {"start": 165.44, "duration": 7.12, "text": "720 minus 4 if we simplify that we get"}, {"start": 169.68, "duration": 5.88, "text": "seven uh"}, {"start": 172.56, "duration": 3.0, "text": "716"}], "title": "Factorials ; Evaluating Expressions With Factorials", "description": "Thanks to all of you who support me on Patreon. You da real mvps!   $1 per month helps!! :)  https://www.patreon.com/patrickjmt !! Just a few basic examples of evaluating numerical expressions involving factorials.", "lengthSeconds": 174, "width": 480, "height": 360}
{"video_id": "zTAZiBMSlBE", "transcript": [{"start": 0.68, "duration": 3.44, "text": "all right here we're going to look at"}, {"start": 1.8, "duration": 4.76, "text": "some examples related to subtraction of"}, {"start": 4.12, "duration": 3.639, "text": "integers and what we're going to do is"}, {"start": 6.56, "duration": 3.84, "text": "we're just going to make some number"}, {"start": 7.759, "duration": 7.12, "text": "lines to illustrate these subtraction"}, {"start": 10.4, "duration": 7.959, "text": "equations so in part A we've got 2 - 3 ="}, {"start": 14.879, "duration": 6.361, "text": "-1 well in terms of our number line the"}, {"start": 18.359, "duration": 5.881, "text": "first number we have is positive two so"}, {"start": 21.24, "duration": 6.16, "text": "I'm going to go two units to the right"}, {"start": 24.24, "duration": 6.0, "text": "and then we're subtracting away positive"}, {"start": 27.4, "duration": 5.6, "text": "three units so you can think about again"}, {"start": 30.24, "duration": 8.12, "text": "subtracting a positive as going to the"}, {"start": 33.0, "duration": 7.48, "text": "left so we'll go left one 2 three units"}, {"start": 38.36, "duration": 5.199, "text": "and again if we if we subtract three"}, {"start": 40.48, "duration": 4.88, "text": "units um we'll end up at the value of"}, {"start": 43.559, "duration": 5.0, "text": "negative one so that could be our number"}, {"start": 45.36, "duration": 5.92, "text": "line two to the right three to the"}, {"start": 48.559, "duration": 4.881, "text": "left part B is a little tricky here a"}, {"start": 51.28, "duration": 3.759, "text": "little bit trickier just because of uh"}, {"start": 53.44, "duration": 3.88, "text": "you there's a lot of minus signs"}, {"start": 55.039, "duration": 4.561, "text": "floating around so the first thing I'm"}, {"start": 57.32, "duration": 6.12, "text": "going to do is -2 so that's going to"}, {"start": 59.6, "duration": 3.84, "text": "move us two units to the"}, {"start": 63.559, "duration": 6.24, "text": "left so there's our two units to the"}, {"start": 66.479, "duration": 5.521, "text": "left now if we were subtracting away a"}, {"start": 69.799, "duration": 5.68, "text": "positive five we would keep going to the"}, {"start": 72.0, "duration": 5.799, "text": "left but since we're subtracting a Nega"}, {"start": 75.479, "duration": 5.561, "text": "five what we're going to do is now move"}, {"start": 77.799, "duration": 7.0, "text": "to the right so let's see we're at -2"}, {"start": 81.04, "duration": 7.719, "text": "then we move one unit two units three"}, {"start": 84.799, "duration": 7.081, "text": "units four units and five units and"}, {"start": 88.759, "duration": 5.64, "text": "again you can count this would 0 1 2 and"}, {"start": 91.88, "duration": 5.0, "text": "this would put us at posi3 so that would"}, {"start": 94.399, "duration": 6.4, "text": "be a diagram that goes with or a number"}, {"start": 96.88, "duration": 3.919, "text": "line that goes with Part B"}], "title": "Subtracting / Subtraction of Integers", "description": "Thanks to all of you who support me on Patreon. You da real mvps!   $1 per month helps!! :)  https://www.patreon.com/patrickjmt !! Subtracting / Subtraction of Integers.\nHere we look at some examples of subtracting integers. In these examples, we create a subtraction statement from a number line.", "lengthSeconds": 99, "width": 480, "height": 360}
{"video_id": "QfyTKuaCcT4", "transcript": [{"start": 0.08, "duration": 4.44, "text": "suppose a fountain has a circular base"}, {"start": 1.8, "duration": 4.32, "text": "with a diameter of 12 ft we want to know"}, {"start": 4.52, "duration": 4.4, "text": "which of the following Expressions could"}, {"start": 6.12, "duration": 4.16, "text": "be used to find the area of the bottom"}, {"start": 8.92, "duration": 3.92, "text": "surface of the"}, {"start": 10.28, "duration": 5.32, "text": "fountain so this is just an area"}, {"start": 12.84, "duration": 5.04, "text": "question here so again let me draw a"}, {"start": 15.6, "duration": 4.88, "text": "picture there's our Fountain if we were"}, {"start": 17.88, "duration": 6.44, "text": "looking at it from above we said that it"}, {"start": 20.48, "duration": 5.92, "text": "has a diameter of 12T okay so a diameter"}, {"start": 24.32, "duration": 2.84, "text": "of 12 ft that means the distance across"}, {"start": 26.4, "duration": 3.92, "text": "is"}, {"start": 27.16, "duration": 5.079, "text": "12 or equivalently"}, {"start": 30.32, "duration": 4.96, "text": "if the diameter is 12T that means that"}, {"start": 32.239, "duration": 8.361, "text": "it has a radius of 6t so we're just"}, {"start": 35.28, "duration": 8.959, "text": "wanting to find that that area of the"}, {"start": 40.6, "duration": 7.92, "text": "fountain well the area of a circle"}, {"start": 44.239, "duration": 6.601, "text": "recall that that's Pi * the radius"}, {"start": 48.52, "duration": 3.76, "text": "squared well we're going to approximate"}, {"start": 50.84, "duration": 3.68, "text": "the radius or excuse me we're going to"}, {"start": 52.28, "duration": 4.64, "text": "approximate Pi by using"}, {"start": 54.52, "duration": 5.76, "text": "3.14 okay so we'll approximate Pi by"}, {"start": 56.92, "duration": 6.24, "text": "using 3.14 we know the radius exactly"}, {"start": 60.28, "duration": 5.36, "text": "right we said the diameter is 12 so half"}, {"start": 63.16, "duration": 5.04, "text": "of that is going to be six so our radius"}, {"start": 65.64, "duration": 3.76, "text": "is going to have a value of six so we'll"}, {"start": 68.2, "duration": 4.84, "text": "just take"}, {"start": 69.4, "duration": 6.079, "text": "3.14 and we would multiply that by 6^"}, {"start": 73.04, "duration": 4.68, "text": "squared and that would give us our"}, {"start": 75.479, "duration": 3.801, "text": "solution so this is just one of those"}, {"start": 77.72, "duration": 4.719, "text": "questions it's just a matter of really"}, {"start": 79.28, "duration": 6.64, "text": "knowing uh knowing the formula so in"}, {"start": 82.439, "duration": 7.801, "text": "this case our answer let's see so 3.14 *"}, {"start": 85.92, "duration": 8.239, "text": "6^ 2 it looks like answer Choice B"}, {"start": 90.24, "duration": 7.239, "text": "would be our solution we've got 3.14 *"}, {"start": 94.159, "duration": 5.92, "text": "6^ s and again if I saw a question like"}, {"start": 97.479, "duration": 5.401, "text": "this on the test and I wasn't really"}, {"start": 100.079, "duration": 5.161, "text": "sure if you at least you know I don't"}, {"start": 102.88, "duration": 4.199, "text": "know I mean if you knew the area formula"}, {"start": 105.24, "duration": 4.96, "text": "again the thing that gets squared in the"}, {"start": 107.079, "duration": 6.4, "text": "area formula is the radius so if I saw"}, {"start": 110.2, "duration": 6.0, "text": "this 3.14 squ and this 3.14 squ I would"}, {"start": 113.479, "duration": 4.521, "text": "know that C and D are not correct um and"}, {"start": 116.2, "duration": 3.879, "text": "it would be one of the two choices A and"}, {"start": 118.0, "duration": 4.719, "text": "B so at least at that point I would have"}, {"start": 120.079, "duration": 5.64, "text": "it narrowed down to two possible"}, {"start": 122.719, "duration": 3.0, "text": "solutions"}], "title": "GED 'Most Missed' Math Practice Problems , Code Q.4.b,  #3", "description": "Thanks to all of you who support me on Patreon. You da real mvps!   $1 per month helps!! :)  https://www.patreon.com/patrickjmt !! The GED Testing Service keeps a list of the most missed math topics and can be seen here: http://www.gedtestingservice.com/uploads/files/126de0283c94ff6323b3b5b3f310da5d.pdf\nThis video shows how to work a sample problem using their coding system that is shown on the left using the link above.", "lengthSeconds": 125, "width": 480, "height": 360}
{"video_id": "I2ZWuTWoBkU", "transcript": [{"start": 0.68, "duration": 5.0, "text": "all right in this example we're going to"}, {"start": 2.679, "duration": 5.761, "text": "start uh looking at some derivatives of"}, {"start": 5.68, "duration": 3.879, "text": "the inverse trig functions so part A"}, {"start": 8.44, "duration": 6.92, "text": "here we're going to find the derivative"}, {"start": 9.559, "duration": 11.161, "text": "of x * AR cosine of < TK X so recall the"}, {"start": 15.36, "duration": 8.2, "text": "derivative of Arc cosine of some"}, {"start": 20.72, "duration": 7.68, "text": "function what we get is we get negative"}, {"start": 23.56, "duration": 7.16, "text": "uh 1 over theun of 1 minus the"}, {"start": 28.4, "duration": 4.92, "text": "function squar then we have to do the"}, {"start": 30.72, "duration": 5.44, "text": "chain Rule and take the derivative of"}, {"start": 33.32, "duration": 5.2, "text": "that function so uh we'll definitely"}, {"start": 36.16, "duration": 4.12, "text": "have to use uh this formula and we're"}, {"start": 38.52, "duration": 3.44, "text": "also going to have to use the product uh"}, {"start": 40.28, "duration": 4.4, "text": "the product rule because hey well we've"}, {"start": 41.96, "duration": 4.36, "text": "got a product so uh we'll definitely"}, {"start": 44.68, "duration": 3.28, "text": "come back to this formula so I'm going"}, {"start": 46.32, "duration": 4.04, "text": "to use it in just a"}, {"start": 47.96, "duration": 7.2, "text": "second so maybe we can rewrite this real"}, {"start": 50.36, "duration": 6.519, "text": "quick we've got y = x * AR cosine of x"}, {"start": 55.16, "duration": 5.239, "text": "to the"}, {"start": 56.879, "duration": 5.32, "text": "12 well okay the derivative let's see so"}, {"start": 60.399, "duration": 4.921, "text": "the derivative of 1x that's easy we'll"}, {"start": 62.199, "duration": 6.361, "text": "just get one so we'll leave the uh Arc"}, {"start": 65.32, "duration": 6.28, "text": "cosine of the X to the 12 or the square"}, {"start": 68.56, "duration": 6.12, "text": "otk of X we'll leave that all alone plus"}, {"start": 71.6, "duration": 5.96, "text": "and now we'll leave the X alone when we"}, {"start": 74.68, "duration": 5.399, "text": "take the derivative of AR cosine of x to"}, {"start": 77.56, "duration": 5.72, "text": "the 12 this is where we'll use our"}, {"start": 80.079, "duration": 6.481, "text": "formula so we'll get -1 over the square"}, {"start": 83.28, "duration": 6.32, "text": "< TK of 1 minus the function which is x^"}, {"start": 86.56, "duration": 4.68, "text": "12 all of that's going to get squared"}, {"start": 89.6, "duration": 3.76, "text": "and then we have to multiply by the"}, {"start": 91.24, "duration": 5.76, "text": "derivative of the inside"}, {"start": 93.36, "duration": 6.16, "text": "function so that'll give us 12 x raised"}, {"start": 97.0, "duration": 5.32, "text": "to thega"}, {"start": 99.52, "duration": 5.72, "text": "one2 so I'm going to try to clean this"}, {"start": 102.32, "duration": 5.479, "text": "up a little bit so we've got AR cosine"}, {"start": 105.24, "duration": 4.12, "text": "of squ < TK of X let's see we've got a"}, {"start": 107.799, "duration": 4.801, "text": "positive and a negative so that would"}, {"start": 109.36, "duration": 6.24, "text": "give us a negative um the X could go on"}, {"start": 112.6, "duration": 5.199, "text": "top um the two would be in the"}, {"start": 115.6, "duration": 5.559, "text": "denominator um underneath the square"}, {"start": 117.799, "duration": 6.041, "text": "root we would just be left with 1us x uh"}, {"start": 121.159, "duration": 5.64, "text": "when we take x to the 1/2 and square it"}, {"start": 123.84, "duration": 5.96, "text": "and then our x to the -"}, {"start": 126.799, "duration": 5.681, "text": "one2 um you know or excuse me I think"}, {"start": 129.8, "duration": 4.88, "text": "it's our x to the negative one2 uh you"}, {"start": 132.48, "duration": 5.6, "text": "know we could even take our x to the 1"}, {"start": 134.68, "duration": 5.08, "text": "and multiply that by x to the one2 and"}, {"start": 138.08, "duration": 3.48, "text": "that would leave us with X to the"}, {"start": 139.76, "duration": 3.72, "text": "positive 1/2 again we're just adding the"}, {"start": 141.56, "duration": 4.52, "text": "exponents so really this would be an x"}, {"start": 143.48, "duration": 5.72, "text": "to the positive2 which we can rewrite"}, {"start": 146.08, "duration": 4.519, "text": "simply as the square root of x so you"}, {"start": 149.2, "duration": 3.2, "text": "know I don't see a lot lot of other good"}, {"start": 150.599, "duration": 4.521, "text": "meaningful ways to combine this you"}, {"start": 152.4, "duration": 6.08, "text": "could write this as a single square root"}, {"start": 155.12, "duration": 5.16, "text": "the square root of x over 1 - x but I"}, {"start": 158.48, "duration": 4.0, "text": "say let's leave it right there and that"}, {"start": 160.28, "duration": 3.92, "text": "would be our derivative so you know all"}, {"start": 162.48, "duration": 3.8, "text": "the same rules still apply chain rule"}, {"start": 164.2, "duration": 4.399, "text": "quotient rule product rule you just have"}, {"start": 166.28, "duration": 3.8, "text": "kind of these new formulas now you know"}, {"start": 168.599, "duration": 4.241, "text": "just these new formulas now that you're"}, {"start": 170.08, "duration": 4.48, "text": "going to have to remember and uh just"}, {"start": 172.84, "duration": 4.319, "text": "have to have these uh you know kind of"}, {"start": 174.56, "duration": 4.28, "text": "ready uh just to uh they're kind of easy"}, {"start": 177.159, "duration": 3.881, "text": "to forget they're very similar to some"}, {"start": 178.84, "duration": 4.64, "text": "of you know like a the inverse sign is"}, {"start": 181.04, "duration": 4.199, "text": "very similar to this um so again just"}, {"start": 183.48, "duration": 5.44, "text": "kind of new formulas you got to remember"}, {"start": 185.239, "duration": 3.681, "text": "and uh be ready to use"}], "title": "Derivatives of Inverse Trigonometric Functions - Another Example!", "description": "Thanks to all of you who support me on Patreon. You da real mvps!   $1 per month helps!! :)  https://www.patreon.com/patrickjmt !! Derivatives of Inverse Trigonometric Functions - Another Example!.  Just another example involving a derivative of an inverse trig function.", "lengthSeconds": 187, "width": 480, "height": 360}
{"video_id": "Fjv_iP-wQqs", "transcript": [{"start": 0.17, "duration": 3.85, "text": "okay in this video we're gonna do"}, {"start": 2.28, "duration": 4.05, "text": "another example related to the law of"}, {"start": 4.02, "duration": 3.899, "text": "cosines and what we want to do here is"}, {"start": 6.33, "duration": 5.189, "text": "just find the measure of the smallest"}, {"start": 7.919, "duration": 5.84, "text": "angle in our given triangle here so we"}, {"start": 11.519, "duration": 5.971, "text": "have side lengths of four six and nine"}, {"start": 13.759, "duration": 6.041, "text": "the first observation is the smallest"}, {"start": 17.49, "duration": 4.5, "text": "angle is going to be opposite the side"}, {"start": 19.8, "duration": 4.559, "text": "of the triangle that has shortest length"}, {"start": 21.99, "duration": 5.039, "text": "so the side with shortest length would"}, {"start": 24.359, "duration": 4.83, "text": "be our side here a B that has length 4"}, {"start": 27.029, "duration": 5.46, "text": "so that means again we're trying to"}, {"start": 29.189, "duration": 6.991, "text": "figure out the angle here the the"}, {"start": 32.489, "duration": 7.07, "text": "measure for angle C well we can use law"}, {"start": 36.18, "duration": 8.46, "text": "of cosines so 4 squared that would equal"}, {"start": 39.559, "duration": 9.581, "text": "9 squared plus 6 squared minus 2 times"}, {"start": 44.64, "duration": 6.78, "text": "well 9 and 6 and then that's going to"}, {"start": 49.14, "duration": 5.099, "text": "help us bring in our cosine of the angle"}, {"start": 51.42, "duration": 9.119, "text": "C so now just a little bit of rithmetic"}, {"start": 54.239, "duration": 10.56, "text": "4 squared 16 9 squared - 81 81 plus 36"}, {"start": 60.539, "duration": 8.731, "text": "is going to give us 117 2 times 9 times"}, {"start": 64.799, "duration": 7.531, "text": "6 let's see so 9 times 6 is 54 if you"}, {"start": 69.27, "duration": 5.13, "text": "double that you'll get 108 which will be"}, {"start": 72.33, "duration": 4.98, "text": "cosine of C"}, {"start": 74.4, "duration": 5.13, "text": "we can subtract the 117 from both sides"}, {"start": 77.31, "duration": 6.41, "text": "that's gonna leave us with negative what"}, {"start": 79.53, "duration": 7.86, "text": "I want on the left equals negative 108"}, {"start": 83.72, "duration": 7.39, "text": "times cosine of C we can divide both"}, {"start": 87.39, "duration": 6.66, "text": "sides by the negative 108 so that'll"}, {"start": 91.11, "duration": 9.18, "text": "just leave us with positive 101 over 108"}, {"start": 94.05, "duration": 9.3, "text": "equals cosine of C squeeze this in here"}, {"start": 100.29, "duration": 9.48, "text": "101 over 108 I'm getting that to be"}, {"start": 103.35, "duration": 8.25, "text": "point 9 3 5 equals cosine of C and now"}, {"start": 109.77, "duration": 8.19, "text": "if we just take the inverse cosine of"}, {"start": 111.6, "duration": 9.96, "text": "both sides that's going to give us our"}, {"start": 117.96, "duration": 5.58, "text": "angle C and let's see here I think this"}, {"start": 121.56, "duration": 5.43, "text": "should be let me double check one more"}, {"start": 123.54, "duration": 6.03, "text": "time so point nine three five if we do"}, {"start": 126.99, "duration": 5.01, "text": "the inverse cosine of that I'm getting"}, {"start": 129.57, "duration": 5.19, "text": "this to be twenty point seven seven"}, {"start": 132.0, "duration": 6.68, "text": "degrees after rounding here a little bit"}, {"start": 134.76, "duration": 6.54, "text": "so that's all there is to it"}, {"start": 138.68, "duration": 4.42, "text": "and again you know we need the four"}, {"start": 141.3, "duration": 3.15, "text": "isolated at the beginning on the left"}, {"start": 143.1, "duration": 4.109, "text": "side again because that's what's going"}, {"start": 144.45, "duration": 5.16, "text": "to give us the cosine of this angle that"}, {"start": 147.209, "duration": 5.701, "text": "we want here so nothing worse than that"}, {"start": 149.61, "duration": 5.25, "text": "just again just noticing that the"}, {"start": 152.91, "duration": 4.68, "text": "smallest angle will be opposite that"}, {"start": 154.86, "duration": 4.29, "text": "side of length 4 using law of cosines"}, {"start": 157.59, "duration": 6.29, "text": "just being a little careful with the"}, {"start": 159.15, "duration": 4.73, "text": "arithmetic and that's all there is to it"}], "title": "\u2756 The Law of Cosines - SSS Example \u2756", "description": "In this video we look at an example involving the Law of Cosines.\nIn this triangle we are given the 3 sides of the triangle and asked to find the measure of the smallest angle.  In order to solve the question we merely use the Law of Cosines.\nThis is just a quick example to refresh or quiz yourself!", "lengthSeconds": 163, "width": 480, "height": 360}
{"video_id": "R4xqCBV07xg", "transcript": [{"start": 1.079, "duration": 4.121, "text": "okay in this example we've got a right"}, {"start": 2.919, "duration": 5.201, "text": "triangle with side lengths Three Square"}, {"start": 5.2, "duration": 5.0, "text": "7 and then a hypotenuse with length four"}, {"start": 8.12, "duration": 4.519, "text": "we're going to find the S cosine tangent"}, {"start": 10.2, "duration": 4.599, "text": "cosecant secant and cent of that angle"}, {"start": 12.639, "duration": 4.961, "text": "Theta so I've jotted down the"}, {"start": 14.799, "duration": 5.801, "text": "definitions quickly here um so you can"}, {"start": 17.6, "duration": 5.04, "text": "take a look at them and again notice my"}, {"start": 20.6, "duration": 3.8, "text": "triangle the Theta is in a uh a"}, {"start": 22.64, "duration": 4.639, "text": "different corner so be careful about"}, {"start": 24.4, "duration": 6.879, "text": "that when you read off the angles or the"}, {"start": 27.279, "duration": 6.8, "text": "side lengths from the angle and again s"}, {"start": 31.279, "duration": 4.721, "text": "of theta is defined to be the the length"}, {"start": 34.079, "duration": 4.241, "text": "of the opposite side divided by the"}, {"start": 36.0, "duration": 4.48, "text": "length of the hypotenuse cosin's the"}, {"start": 38.32, "duration": 5.0, "text": "adjacent over the hypotenuse tangent the"}, {"start": 40.48, "duration": 4.919, "text": "opposite over adjacent um Etc there's"}, {"start": 43.32, "duration": 5.84, "text": "the other one so really to compute these"}, {"start": 45.399, "duration": 7.96, "text": "it's just a matter it's just a matter of"}, {"start": 49.16, "duration": 7.44, "text": "knowing those definitions"}, {"start": 53.359, "duration": 6.321, "text": "and maybe you have to rationalize a"}, {"start": 56.6, "duration": 5.88, "text": "denominator at some point so s of theta"}, {"start": 59.68, "duration": 4.96, "text": "okay that's the opposite side divided by"}, {"start": 62.48, "duration": 4.44, "text": "the hypotenuse again the length of the"}, {"start": 64.64, "duration": 5.2, "text": "opposite side well in this case the"}, {"start": 66.92, "duration": 6.16, "text": "opposite side it's it's the"}, {"start": 69.84, "duration": 5.2, "text": "side furthest away from our angle Theta"}, {"start": 73.08, "duration": 4.079, "text": "the opposite side is going to be theun"}, {"start": 75.04, "duration": 2.119, "text": "of"}, {"start": 77.2, "duration": 5.16, "text": "7 and then we divide that by the length"}, {"start": 80.479, "duration": 3.841, "text": "of the hypotenuse which in this case"}, {"start": 82.36, "duration": 4.56, "text": "will be"}, {"start": 84.32, "duration": 5.6, "text": "pos4 cosine of"}, {"start": 86.92, "duration": 4.519, "text": "theta well that's the adjacent side and"}, {"start": 89.92, "duration": 2.96, "text": "in this case three is going to be the"}, {"start": 91.439, "duration": 4.241, "text": "adjacent"}, {"start": 92.88, "duration": 4.08, "text": "side divided by the hypotenuse again"}, {"start": 95.68, "duration": 4.36, "text": "which is"}, {"start": 96.96, "duration": 5.28, "text": "four and tangent of theta that's the"}, {"start": 100.04, "duration": 6.119, "text": "opposite side squ of"}, {"start": 102.24, "duration": 6.919, "text": "7 over the adjacent side which is"}, {"start": 106.159, "duration": 4.92, "text": "three now we could go through here"}, {"start": 109.159, "duration": 4.121, "text": "cosecant secant cotangent you know you"}, {"start": 111.079, "duration": 4.36, "text": "could read everything off notice that"}, {"start": 113.28, "duration": 3.799, "text": "cosecant you just take the value you got"}, {"start": 115.439, "duration": 3.881, "text": "for sign and you just take the"}, {"start": 117.079, "duration": 4.761, "text": "reciprocal just flip the fraction over"}, {"start": 119.32, "duration": 4.24, "text": "so the same thing with cosine and secant"}, {"start": 121.84, "duration": 6.44, "text": "and tangent and"}, {"start": 123.56, "duration": 4.72, "text": "Cent so let's see so cosecant of"}, {"start": 128.72, "duration": 6.96, "text": "theta that's going to be 4 / the < TK of"}, {"start": 132.52, "duration": 5.48, "text": "7 here we're going to rationalize the"}, {"start": 135.68, "duration": 4.16, "text": "denominator so multiply top and bottom"}, {"start": 138.0, "duration": 5.2, "text": "by thek of"}, {"start": 139.84, "duration": 8.479, "text": "7 that'll leave you with 4 * the of 7 in"}, {"start": 143.2, "duration": 7.72, "text": "the numerator the 7 * the 7 is just 7"}, {"start": 148.319, "duration": 4.401, "text": "the fraction 4 7 doesn't reduce so"}, {"start": 150.92, "duration": 4.76, "text": "that'll be our"}, {"start": 152.72, "duration": 4.28, "text": "solution then we have secant of theta"}, {"start": 155.68, "duration": 2.88, "text": "same thing I'm just going to flip the"}, {"start": 157.0, "duration": 4.08, "text": "fraction that'll be"}, {"start": 158.56, "duration": 4.88, "text": "4/3 okay so we don't have to rationalize"}, {"start": 161.08, "duration": 5.799, "text": "anything there it's already"}, {"start": 163.44, "duration": 6.439, "text": "rationalized and also cotangent that'll"}, {"start": 166.879, "duration": 5.801, "text": "be 3 / the < TK"}, {"start": 169.879, "duration": 7.0, "text": "7 and you can multiply top and bottom by"}, {"start": 172.68, "duration": 6.559, "text": "the < TK of 7 that'll give us 3un 7 over"}, {"start": 176.879, "duration": 3.961, "text": "7 and now we've got all our Solutions"}, {"start": 179.239, "duration": 3.321, "text": "again you can also so of course you can"}, {"start": 180.84, "duration": 5.959, "text": "think about cosecant secant and"}, {"start": 182.56, "duration": 6.28, "text": "cotangent in terms of ratios of of side"}, {"start": 186.799, "duration": 4.08, "text": "lengths of the triangle but to me it's"}, {"start": 188.84, "duration": 5.319, "text": "much easier just if I can remember"}, {"start": 190.879, "duration": 5.561, "text": "cosecant goes with s uh secant goes with"}, {"start": 194.159, "duration": 4.44, "text": "cosine and cotangent goes with Tangent"}, {"start": 196.44, "duration": 5.0, "text": "it to me it's easier just to take the"}, {"start": 198.599, "duration": 7.521, "text": "flip of the fraction and simplify it or"}, {"start": 201.44, "duration": 4.68, "text": "rationalize the denominator if I need to"}], "title": "All the Trigonometric Values Given a Right Triangle", "description": "Thanks to all of you who support me on Patreon. You da real mvps!   $1 per month helps!! :)  https://www.patreon.com/patrickjmt !! All the Trigonometric Values Given a Right Triangle.  Just a quick little example of finding all of the trigonometric values given a right triangle.", "lengthSeconds": 204, "width": 450, "height": 360}
{"video_id": "BAoMp7SImVc", "transcript": [{"start": 1.28, "duration": 3.579, "text": "all right in this video we're going to"}, {"start": 3.24, "duration": 4.23, "text": "do another example of permutations"}, {"start": 4.859, "duration": 3.841, "text": "involving repeated symbols and we saw in"}, {"start": 7.47, "duration": 4.08, "text": "the first example how many different"}, {"start": 8.7, "duration": 5.01, "text": "linear Arrangements there are using the"}, {"start": 11.55, "duration": 4.14, "text": "letters from the word Mississippi so"}, {"start": 13.71, "duration": 4.59, "text": "slight variation on this problem at"}, {"start": 15.69, "duration": 4.71, "text": "first may not seem like it's related but"}, {"start": 18.3, "duration": 4.29, "text": "then I think it's pretty clear it's"}, {"start": 20.4, "duration": 4.26, "text": "really the exact same type of problem so"}, {"start": 22.59, "duration": 5.099, "text": "suppose we want to look at the number of"}, {"start": 24.66, "duration": 6.75, "text": "staircase paths in the XY plane that"}, {"start": 27.689, "duration": 4.741, "text": "start from the origin 0 0 and so we're"}, {"start": 31.41, "duration": 3.84, "text": "going to start at the origin and move"}, {"start": 32.43, "duration": 5.16, "text": "over to the point 5 comma 3 and each"}, {"start": 35.25, "duration": 5.579, "text": "path is made of steps moving one unit to"}, {"start": 37.59, "duration": 5.46, "text": "the right or one unit upwards ok so"}, {"start": 40.829, "duration": 4.951, "text": "again that's how our paths are made we"}, {"start": 43.05, "duration": 4.8, "text": "can't go diagonally or we're you know"}, {"start": 45.78, "duration": 3.66, "text": "other things like that so for example"}, {"start": 47.85, "duration": 4.17, "text": "one little path you know you could go"}, {"start": 49.44, "duration": 5.279, "text": "over you know maybe I go over twice and"}, {"start": 52.02, "duration": 5.34, "text": "then I go up once and then I go over one"}, {"start": 54.719, "duration": 5.011, "text": "so then I go up once then I go over once"}, {"start": 57.36, "duration": 5.37, "text": "and I go up once and hey then I go over"}, {"start": 59.73, "duration": 5.55, "text": "again ok that would be an example of a"}, {"start": 62.73, "duration": 5.1, "text": "path where I'm either moving to the"}, {"start": 65.28, "duration": 6.71, "text": "right or moving upwards so we want to"}, {"start": 67.83, "duration": 7.44, "text": "count all of those different paths ok"}, {"start": 71.99, "duration": 6.82, "text": "well if you think about it you know each"}, {"start": 75.27, "duration": 5.52, "text": "path is going to be made up of well to"}, {"start": 78.81, "duration": 5.22, "text": "go all the way over we're going to have"}, {"start": 80.79, "duration": 5.85, "text": "to go to the right 1 2 3 for a total of"}, {"start": 84.03, "duration": 6.9, "text": "5 times ok so we're going to really move"}, {"start": 86.64, "duration": 9.119, "text": "right 5 times and notice we would have"}, {"start": 90.93, "duration": 7.049, "text": "to go up 1 2 3 times and if you think"}, {"start": 95.759, "duration": 4.981, "text": "about the path I just did I could really"}, {"start": 97.979, "duration": 5.011, "text": "write it so maybe you know we go right"}, {"start": 100.74, "duration": 4.86, "text": "one unit so I'm going to list that as"}, {"start": 102.99, "duration": 5.309, "text": "you know move right and then we moved"}, {"start": 105.6, "duration": 6.18, "text": "right again and then we moved up and"}, {"start": 108.299, "duration": 7.14, "text": "then we moved right and then we moved up"}, {"start": 111.78, "duration": 6.299, "text": "and then we move to the right and then"}, {"start": 115.439, "duration": 3.081, "text": "we moved up and hey then we move to the"}, {"start": 118.079, "duration": 3.211, "text": "right"}, {"start": 118.52, "duration": 4.69, "text": "really what what's going on to make our"}, {"start": 121.29, "duration": 4.38, "text": "our paths you can really think about it"}, {"start": 123.21, "duration": 5.249, "text": "as being a word kind of like our"}, {"start": 125.67, "duration": 4.28, "text": "Mississippi example where we've got five"}, {"start": 128.459, "duration": 3.621, "text": "R's to two"}, {"start": 129.95, "duration": 6.72, "text": "from we've got a you know go right five"}, {"start": 132.08, "duration": 7.56, "text": "times and we have three use okay so the"}, {"start": 136.67, "duration": 4.74, "text": "total number of you know letters in our"}, {"start": 139.64, "duration": 6.12, "text": "word if you think about this as being a"}, {"start": 141.41, "duration": 6.659, "text": "word we've got eight objects total again"}, {"start": 145.76, "duration": 4.29, "text": "there's five hours three use so now"}, {"start": 148.069, "duration": 4.561, "text": "we're just using it's very much like the"}, {"start": 150.05, "duration": 6.39, "text": "Mississippi example so again our"}, {"start": 152.63, "duration": 7.59, "text": "original formula it says we look at the"}, {"start": 156.44, "duration": 6.24, "text": "total number of objects in this case"}, {"start": 160.22, "duration": 6.0, "text": "there's eight factorial objects and then"}, {"start": 162.68, "duration": 5.73, "text": "we just multiply so we've got eight"}, {"start": 166.22, "duration": 4.109, "text": "factorial on top and then in the"}, {"start": 168.41, "duration": 3.75, "text": "denominator we would have five factorial"}, {"start": 170.329, "duration": 4.41, "text": "because eventually we have to go right"}, {"start": 172.16, "duration": 4.71, "text": "five times so our word would have five"}, {"start": 174.739, "duration": 4.201, "text": "hours and it's somewhere and then we"}, {"start": 176.87, "duration": 4.44, "text": "have to go up three times so our word"}, {"start": 178.94, "duration": 4.29, "text": "would have three use in it and this is"}, {"start": 181.31, "duration": 4.5, "text": "going to be the total number of little"}, {"start": 183.23, "duration": 4.56, "text": "staircase patterns that we can make so"}, {"start": 185.81, "duration": 4.64, "text": "again this is just 8 times 7 times 6"}, {"start": 187.79, "duration": 6.15, "text": "times 5 times 4 times 3 times 2 times 1"}, {"start": 190.45, "duration": 8.02, "text": "5 factorials 5 times 4 times 3 times 2"}, {"start": 193.94, "duration": 6.48, "text": "times 1 3 factorial 3 times 2 times 1 so"}, {"start": 198.47, "duration": 5.82, "text": "let's see we can cancel out the 5 4 3 2"}, {"start": 200.42, "duration": 6.03, "text": "1 from the top and bottom 3 + 2 is 6 so"}, {"start": 204.29, "duration": 4.979, "text": "it looks like we're left with 8 times 7"}, {"start": 206.45, "duration": 6.39, "text": "or it says there's a total of 56"}, {"start": 209.269, "duration": 7.461, "text": "different paths to go from the origin to"}, {"start": 212.84, "duration": 3.89, "text": "that point 5 comma 3"}], "title": "Permutations Involving Repeated Symbols - Example 2", "description": "Thanks to all of you who support me on Patreon. You da real mvps!   $1 per month helps!! :)  https://www.patreon.com/patrickjmt !! Permutations Involving Repeated Symbols - Example 2. In this example, we count how many ' stair case ' paths there are from the origin to the point (5,3).", "lengthSeconds": 216, "width": 480, "height": 360}
{"video_id": "qxjs3NHI5T8", "transcript": [{"start": 0.03, "duration": 3.81, "text": "all right in this example we're gonna"}, {"start": 2.01, "duration": 3.93, "text": "write the standard form equation of a"}, {"start": 3.84, "duration": 7.2, "text": "parabola with a vertex at the origin and"}, {"start": 5.94, "duration": 9.359, "text": "a focus at the point 1 8 comma zero so"}, {"start": 11.04, "duration": 6.27, "text": "we don't have to sketch anything at all"}, {"start": 15.299, "duration": 6.601, "text": "but I always like to have a little"}, {"start": 17.31, "duration": 6.809, "text": "sketch it helps it helps me just to"}, {"start": 21.9, "duration": 3.629, "text": "visualize and to kind of a double check"}, {"start": 24.119, "duration": 4.83, "text": "and make sure that I'm doing things"}, {"start": 25.529, "duration": 7.411, "text": "correctly so we know the focus here is"}, {"start": 28.949, "duration": 6.961, "text": "at the point 1 8 comma 0 that's given"}, {"start": 32.94, "duration": 6.119, "text": "and we're assuming that the vertex is at"}, {"start": 35.91, "duration": 7.43, "text": "the origin so I know that my parabola in"}, {"start": 39.059, "duration": 7.32, "text": "this case is gonna open to the right"}, {"start": 43.34, "duration": 5.559, "text": "well recall if our parabola opens to the"}, {"start": 46.379, "duration": 6.901, "text": "right it's gonna be of the form Y"}, {"start": 48.899, "duration": 6.18, "text": "squared equals 4p times X so all we"}, {"start": 53.28, "duration": 4.41, "text": "really have to somehow is we just need"}, {"start": 55.079, "duration": 4.23, "text": "to deduce the value of P do the"}, {"start": 57.69, "duration": 6.359, "text": "arithmetic and hey we'll have the"}, {"start": 59.309, "duration": 10.501, "text": "standard form recall though that our"}, {"start": 64.049, "duration": 9.511, "text": "focus our focus is gonna be of the form"}, {"start": 69.81, "duration": 7.62, "text": "P comma 0 but we know what our focus is"}, {"start": 73.56, "duration": 7.08, "text": "is 1/8 comma 0 so we're actually already"}, {"start": 77.43, "duration": 5.64, "text": "being told our value of P in this case"}, {"start": 80.64, "duration": 5.369, "text": "we know that P in this case is going to"}, {"start": 83.07, "duration": 6.839, "text": "equal 1/8 so that's all we have to do is"}, {"start": 86.009, "duration": 7.381, "text": "just substitute in the 1/8 and 4p so we"}, {"start": 89.909, "duration": 10.311, "text": "have Y squared equals 4 times our p"}, {"start": 93.39, "duration": 10.86, "text": "value of 1/8 simply multiply that by X"}, {"start": 100.22, "duration": 5.41, "text": "and we get Y squared equals well for you"}, {"start": 104.25, "duration": 4.229, "text": "can think about this as being 4 over 1"}, {"start": 105.63, "duration": 6.75, "text": "so I have 4 over 8 which will reduce to"}, {"start": 108.479, "duration": 6.451, "text": "1/2 times X or if you wanted to you"}, {"start": 112.38, "duration": 7.08, "text": "could write that as Y squared equals x"}, {"start": 114.93, "duration": 7.56, "text": "over 2 either one looks good to me and"}, {"start": 119.46, "duration": 5.449, "text": "now we've got the standard form of our"}, {"start": 122.49, "duration": 2.419, "text": "parabola"}], "title": "Conic Sections, Parabola : Find Equation of Parabola Given the Focus", "description": "Thanks to all of you who support me on Patreon. You da real mvps!   $1 per month helps!! :)  https://www.patreon.com/patrickjmt !! Conic Sections, Parabola : Find Equation of Parabola Given the Focus.  In this example, we are given that a parabola has a vertex at the origin and we are given the coordinates of the focus.  From that, we find the equation of the parabola.", "lengthSeconds": 124, "width": 480, "height": 360}
{"video_id": "RoAz-OqhqF0", "transcript": [{"start": 0.919, "duration": 4.88, "text": "okay here we're going to graph the"}, {"start": 2.0, "duration": 5.44, "text": "function y = -23 * tangent X we'll pick"}, {"start": 5.799, "duration": 5.521, "text": "out the vertical asmp tootes and the"}, {"start": 7.44, "duration": 6.64, "text": "zeros as well so tangent of X normally"}, {"start": 11.32, "duration": 5.6, "text": "has ASM tootes its first ASM toote"}, {"start": 14.08, "duration": 2.84, "text": "occurs"}, {"start": 17.48, "duration": 5.44, "text": "at the first positive ASM toote occurs"}, {"start": 21.119, "duration": 3.001, "text": "at Pi /"}, {"start": 22.92, "duration": 4.559, "text": "2"}, {"start": 24.12, "duration": 3.359, "text": "[Music]"}, {"start": 27.92, "duration": 8.12, "text": "and I guess you could say the large just"}, {"start": 30.519, "duration": 5.521, "text": "negative ASM toote is atunk / 2"}, {"start": 36.48, "duration": 5.559, "text": "and the the graph of tangent tangent has"}, {"start": 40.16, "duration": 4.0, "text": "a period of Pi so every time we move"}, {"start": 42.039, "duration": 6.081, "text": "over Pi units we're going to get another"}, {"start": 44.16, "duration": 6.719, "text": "ASM toote so let's see uh 1 2 3 4 so 1 2"}, {"start": 48.12, "duration": 6.32, "text": "3 4 1 2 3 4 let me try to keep at the"}, {"start": 50.879, "duration": 7.281, "text": "scale here that would be another ASM"}, {"start": 54.44, "duration": 6.52, "text": "toote that would be at 3 Pi / 2 and"}, {"start": 58.16, "duration": 6.079, "text": "likewise I'm not going to count exactly"}, {"start": 60.96, "duration": 6.839, "text": "but over here at x"}, {"start": 64.239, "duration": 8.2, "text": "= 3i / 2 again we'll have another"}, {"start": 67.799, "duration": 8.0, "text": "vertical ASM toote so tangent normally"}, {"start": 72.439, "duration": 5.881, "text": "if will go through 0 0 and this graph is"}, {"start": 75.799, "duration": 5.761, "text": "certainly doing the same as you move to"}, {"start": 78.32, "duration": 5.92, "text": "the right it increases and it decreases"}, {"start": 81.56, "duration": 4.84, "text": "but since we have -2/3 it's going to"}, {"start": 84.24, "duration": 4.12, "text": "reflect that so the graph is now going"}, {"start": 86.4, "duration": 5.48, "text": "to go"}, {"start": 88.36, "duration": 7.6, "text": "down and we'll go go"}, {"start": 91.88, "duration": 8.599, "text": "up if you move over Pi"}, {"start": 95.96, "duration": 7.88, "text": "units so here's xal Pi again this graph"}, {"start": 100.479, "duration": 6.881, "text": "is just going to repeat itself and"}, {"start": 103.84, "duration": 3.52, "text": "likewise at NE"}, {"start": 108.96, "duration": 5.68, "text": "Pi okay so let's see so the zeros in"}, {"start": 112.36, "duration": 2.28, "text": "this"}, {"start": 117.6, "duration": 5.72, "text": "case the zeros"}, {"start": 121.119, "duration": 6.681, "text": "those are going to occur at multiples of"}, {"start": 123.32, "duration": 8.279, "text": "Pi so we'll say Pi * n where n is an"}, {"start": 127.8, "duration": 6.2, "text": "integer and the vertical ASM tootes"}, {"start": 131.599, "duration": 8.601, "text": "those are going to occur"}, {"start": 134.0, "duration": 9.72, "text": "at xal Pi / 2 plus multiples of Pi so"}, {"start": 140.2, "duration": 4.52, "text": "we'll say Pi / 2 plus pi n again where n"}, {"start": 143.72, "duration": 3.239, "text": "is an"}, {"start": 144.72, "duration": 5.32, "text": "integer if you wanted to you could plot"}, {"start": 146.959, "duration": 5.56, "text": "maybe even an extra point so tangent at"}, {"start": 150.04, "duration": 4.88, "text": "Pi over4 normally goes through positive"}, {"start": 152.519, "duration": 4.8, "text": "one but in this case it's going to go"}, {"start": 154.92, "duration": 5.679, "text": "through at Pi over4 due to the"}, {"start": 157.319, "duration": 5.361, "text": "reflection it'll go through"}, {"start": 160.599, "duration": 5.64, "text": "-2/3 and"}, {"start": 162.68, "duration": 6.479, "text": "at Pi over4 so let's see if I can fit"}, {"start": 166.239, "duration": 8.0, "text": "that in there attive Pi over4 it would"}, {"start": 169.159, "duration": 7.761, "text": "be going through the POS 23 so likewise"}, {"start": 174.239, "duration": 5.08, "text": "piun over4 comma 23 would be a point on"}, {"start": 176.92, "duration": 7.16, "text": "the graph and this just keeps repeating"}, {"start": 179.319, "duration": 4.761, "text": "forever and ever and ever"}], "title": "Graphing a Tangent Function - EX 1", "description": "Thanks to all of you who support me on Patreon. You da real mvps!   $1 per month helps!! :)  https://www.patreon.com/patrickjmt !! Graphing a Tangent Function - EX 1", "lengthSeconds": 182, "width": 450, "height": 360}
{"video_id": "cngzQ5xIOBY", "transcript": [{"start": 2.679, "duration": 3.761, "text": "all right here's another example of"}, {"start": 4.12, "duration": 4.96, "text": "finding derivatives using the product"}, {"start": 6.44, "duration": 5.76, "text": "rule along with the chain rule so again"}, {"start": 9.08, "duration": 6.759, "text": "I see two separate pieces here being"}, {"start": 12.2, "duration": 6.2, "text": "multiplied so the derivative of this if"}, {"start": 15.839, "duration": 7.761, "text": "I take the derivative of x 4th I'll get"}, {"start": 18.4, "duration": 5.2, "text": "4X 3r you leave the other piece"}, {"start": 25.4, "duration": 6.16, "text": "alone and then from that I will leave"}, {"start": 29.039, "duration": 3.801, "text": "the X to the 4th alone and I'll take the"}, {"start": 31.56, "duration": 3.56, "text": "derivative of the other piece"}, {"start": 32.84, "duration": 4.92, "text": "remembering I have to use the chain rule"}, {"start": 35.12, "duration": 6.48, "text": "so the three will come out front I'll"}, {"start": 37.76, "duration": 6.16, "text": "leave the inside part alone so 5x minus"}, {"start": 41.6, "duration": 6.04, "text": "one I'll take one away and get to the"}, {"start": 43.92, "duration": 6.24, "text": "second power and then multiply that by"}, {"start": 47.64, "duration": 4.599, "text": "five so at this point we have to"}, {"start": 50.16, "duration": 4.44, "text": "basically clean things up a little bit"}, {"start": 52.239, "duration": 3.8, "text": "and this can be the tricky part so"}, {"start": 54.6, "duration": 2.959, "text": "there's really nothing to simplify in"}, {"start": 56.039, "duration": 4.68, "text": "the first part so I'm going to leave"}, {"start": 57.559, "duration": 5.121, "text": "that part alone"}, {"start": 60.719, "duration": 3.8, "text": "and I'm going to multiply things out"}, {"start": 62.68, "duration": 3.799, "text": "over on the right side so notice I have"}, {"start": 64.519, "duration": 5.121, "text": "a three and a five that's going to give"}, {"start": 66.479, "duration": 7.441, "text": "me the 15 I have x to the 4th still"}, {"start": 69.64, "duration": 6.72, "text": "hanging out and then I have a 5x -1"}, {"start": 73.92, "duration": 4.64, "text": "sared now at this point we can keep"}, {"start": 76.36, "duration": 4.56, "text": "factoring even a little"}, {"start": 78.56, "duration": 4.519, "text": "further okay so I look at my numbers"}, {"start": 80.92, "duration": 4.32, "text": "there's a four and a 15 well the largest"}, {"start": 83.079, "duration": 3.841, "text": "number that goes into four and 15 is"}, {"start": 85.24, "duration": 4.199, "text": "just simply one so you can't really"}, {"start": 86.92, "duration": 5.08, "text": "factor a number out notice I have an x"}, {"start": 89.439, "duration": 5.72, "text": "to the 3 thir power and an x to the 4th"}, {"start": 92.0, "duration": 7.479, "text": "power well I can factor out an x to the"}, {"start": 95.159, "duration": 7.361, "text": "3 power notice also I have a 5x -1 I"}, {"start": 99.479, "duration": 4.881, "text": "have another 5x - one just like with the"}, {"start": 102.52, "duration": 4.44, "text": "X's you have to pull out the smaller"}, {"start": 104.36, "duration": 4.64, "text": "power I'll pull out the smaller power so"}, {"start": 106.96, "duration": 5.36, "text": "I can pull out 5x"}, {"start": 109.0, "duration": 5.119, "text": "-1 squared and then I'm just going to"}, {"start": 112.32, "duration": 4.079, "text": "fill things back in on the"}, {"start": 114.119, "duration": 3.96, "text": "parentheses so to get the first term"}, {"start": 116.399, "duration": 5.161, "text": "I'll need my four because I'm missing"}, {"start": 118.079, "duration": 6.161, "text": "the four I'm going to need an extra 5x -"}, {"start": 121.56, "duration": 4.76, "text": "one term because again I've got two of"}, {"start": 124.24, "duration": 5.12, "text": "them outside I need an extra one on the"}, {"start": 126.32, "duration": 5.04, "text": "inside so notice notice if I multiply"}, {"start": 129.36, "duration": 4.56, "text": "the first thing to the first thing here"}, {"start": 131.36, "duration": 4.76, "text": "I will get my first term well I'm going"}, {"start": 133.92, "duration": 6.84, "text": "to need a"}, {"start": 136.12, "duration": 8.32, "text": "15 and an X to get back the second"}, {"start": 140.76, "duration": 3.68, "text": "term and now you can"}, {"start": 145.16, "duration": 5.719, "text": "simply reduce things down inside of the"}, {"start": 148.56, "duration": 6.0, "text": "parentheses so if you distribute you'll"}, {"start": 150.879, "duration": 8.961, "text": "get 20x + 15x that's"}, {"start": 154.56, "duration": 9.08, "text": "35x you'll get 4 * -1 which is -4 and"}, {"start": 159.84, "duration": 5.36, "text": "that's now simplified and factored so"}, {"start": 163.64, "duration": 2.84, "text": "these problems are a little confusing I"}, {"start": 165.2, "duration": 3.52, "text": "think people know how to do the"}, {"start": 166.48, "duration": 4.32, "text": "derivatives some of the factoring can"}, {"start": 168.72, "duration": 5.76, "text": "get a little tricky but here's a nice"}, {"start": 170.8, "duration": 3.68, "text": "basic example on what you need"}], "title": "\u2756 Product Rule and Chain Rule Example \u2756", "description": "\ud83d\udcda Mastering the Product Rule and Chain Rule: Essential Derivative Techniques with Algebra Simplification \ud83e\uddee\n\nIn this video, we dive into two fundamental derivative techniques: the Product Rule and the Chain Rule. These rules are crucial for differentiating functions that are products of other functions, especially when one or both involve compositions. You\u2019ll learn how to apply both rules together effectively, followed by simplifying the result using algebra.\n\n\ud83d\ude80 What you\u2019ll learn:\n\nThe formulas for both the Product Rule and Chain Rule, and how they work together.\nStep-by-step examples applying these rules to solve real derivative problems.\nA walk-through of algebraic simplification at the end of the process to tidy up your final answer.\nKey tips to avoid common mistakes when using both rules on complex functions.\nWhether you're prepping for AP Calculus, refreshing your derivative techniques, or looking for clear examples, this video will help you gain confidence in using these powerful rules. Stick around for the final step, where I simplify the derivative using algebra for a cleaner solution!\n\nFor more advanced calculus tutorials, be sure to check out my other videos on implicit differentiation, quotient rule, and more.\n\n\ud83d\udc4d Don\u2019t forget to like, subscribe, and hit the notification bell for more math tutorials! \ud83d\udd14\u2728\n\nWatch now to master both the Product Rule and Chain Rule and take your calculus skills to the next level!\n\n#ProductRule #ChainRule #Calculus #Derivatives #MathTutorial #LearningMath #APCalculus #Mathematics #StudyTips #DerivativeTechniques #AlgebraSimplification #Differentiation #ImplicitDifferentiation #MathHelp #STEM #STEMEducation #MathLessons #MathConcepts #DerivativesExplained #MathSkills #CompositeFunctions #DerivativeExamples #ChainRuleExplained", "lengthSeconds": 173, "width": 320, "height": 240}
{"video_id": "3m9glYPAfD4", "transcript": [{"start": 0.76, "duration": 4.16, "text": "okay here we're going to do a limit"}, {"start": 2.96, "duration": 4.92, "text": "problem involving the absolute value"}, {"start": 4.92, "duration": 7.12, "text": "theorem so here we're going to find the"}, {"start": 7.88, "duration": 8.04, "text": "limit of the sequence x sub n = -1 n +"}, {"start": 12.04, "duration": 6.399, "text": "n^2 over n^2 and again intuitively to me"}, {"start": 15.92, "duration": 5.04, "text": "what's going to happen you know if you"}, {"start": 18.439, "duration": 4.121, "text": "plug in larger and larger values for n"}, {"start": 20.96, "duration": 4.04, "text": "let's forget about the first part for a"}, {"start": 22.56, "duration": 4.68, "text": "moment as we plug in larger and larger"}, {"start": 25.0, "duration": 4.32, "text": "values for n you know we're going to get"}, {"start": 27.24, "duration": 4.999, "text": "some big number squared over some big"}, {"start": 29.32, "duration": 4.759, "text": "number squared it's the same number so"}, {"start": 32.239, "duration": 3.84, "text": "this ratio you know if it wasn't for the"}, {"start": 34.079, "duration": 4.8, "text": "negative 1 to the end we would just get"}, {"start": 36.079, "duration": 5.041, "text": "one well what's going to happen um you"}, {"start": 38.879, "duration": 4.0, "text": "know the negative 1 to the N all that's"}, {"start": 41.12, "duration": 4.759, "text": "either going to do to the numerator is"}, {"start": 42.879, "duration": 6.041, "text": "either subtract one or add one so again"}, {"start": 45.879, "duration": 5.441, "text": "to me for large values of n the the"}, {"start": 48.92, "duration": 4.799, "text": "numerator and the denominator are almost"}, {"start": 51.32, "duration": 4.719, "text": "going to be exactly the same value which"}, {"start": 53.719, "duration": 4.601, "text": "makes me think this limit should equal"}, {"start": 56.039, "duration": 4.641, "text": "zero to justify this again we're going"}, {"start": 58.32, "duration": 4.76, "text": "to use the absolute value theorem"}, {"start": 60.68, "duration": 6.08, "text": "so remember that says if the"}, {"start": 63.08, "duration": 7.399, "text": "limit of the absolute value of a subn"}, {"start": 66.76, "duration": 7.56, "text": "equals 0 or in this case X ofn if that"}, {"start": 70.479, "duration": 6.361, "text": "happens that tells us that the sequence"}, {"start": 74.32, "duration": 5.08, "text": "uh without the absolute value also"}, {"start": 76.84, "duration": 4.639, "text": "equals zero so we're going to use this"}, {"start": 79.4, "duration": 3.399, "text": "little result and the way that we're"}, {"start": 81.479, "duration": 4.121, "text": "going to do this is well we're going to"}, {"start": 82.799, "duration": 7.721, "text": "take the limit as n goes to Infinity of"}, {"start": 85.6, "duration": 7.159, "text": "-1 n + n 2 all over n 2 and all we're"}, {"start": 90.52, "duration": 4.559, "text": "going to do is just break this up using"}, {"start": 92.759, "duration": 5.081, "text": "some algebra so we can write this as the"}, {"start": 95.079, "duration": 6.08, "text": "limit as goes the limit as n goes to"}, {"start": 97.84, "duration": 7.639, "text": "Infinity of 1 the N over n"}, {"start": 101.159, "duration": 8.801, "text": "2 plus n^ 2 over n"}, {"start": 105.479, "duration": 9.041, "text": "2 so we'll have the limit as n goes to"}, {"start": 109.96, "duration": 6.68, "text": "Infinity again of -1 to the N over n^2"}, {"start": 114.52, "duration": 4.32, "text": "well n^2 over n^2 that's just going to"}, {"start": 116.64, "duration": 4.72, "text": "give us"}, {"start": 118.84, "duration": 5.879, "text": "one"}, {"start": 121.36, "duration": 3.359, "text": "so again we can kind of look at these"}, {"start": 125.399, "duration": 6.56, "text": "individually and then the limit as n"}, {"start": 128.2, "duration": 6.08, "text": "goes to Infinity of just one well the"}, {"start": 131.959, "duration": 4.401, "text": "limit as n goes to an Infinity of one is"}, {"start": 134.28, "duration": 3.84, "text": "just one the limit of a constant is the"}, {"start": 136.36, "duration": 3.12, "text": "constant so the only thing we really"}, {"start": 138.12, "duration": 3.56, "text": "have to think about now is What's"}, {"start": 139.48, "duration": 4.16, "text": "Happening Here well again I think it's"}, {"start": 141.68, "duration": 3.68, "text": "pretty easy to argue the top is always"}, {"start": 143.64, "duration": 3.72, "text": "going to bounce back and forth between"}, {"start": 145.36, "duration": 3.84, "text": "negative one and positive one the"}, {"start": 147.36, "duration": 4.239, "text": "denominator is going to get arbitrarily"}, {"start": 149.2, "duration": 4.679, "text": "large so to me this this first limit"}, {"start": 151.599, "duration": 5.241, "text": "should certainly go to zero but again to"}, {"start": 153.879, "duration": 6.28, "text": "bring in this absolute value"}, {"start": 156.84, "duration": 6.44, "text": "theorem notice"}, {"start": 160.159, "duration": 5.481, "text": "um so notice that the limit as n goes to"}, {"start": 163.28, "duration": 5.48, "text": "Infinity if we take the absolute value"}, {"start": 165.64, "duration": 6.64, "text": "of1 to the N over n^2 so I'm just"}, {"start": 168.76, "duration": 5.28, "text": "looking at the first part well that's"}, {"start": 172.28, "duration": 3.92, "text": "just going to equal the limit as in goes"}, {"start": 174.04, "duration": 5.0, "text": "to Infinity if we take the absolute"}, {"start": 176.2, "duration": 4.28, "text": "value of the numerator U again you know"}, {"start": 179.04, "duration": 3.16, "text": "without the absolute abute value we're"}, {"start": 180.48, "duration": 4.039, "text": "either getting positive one or negative"}, {"start": 182.2, "duration": 4.319, "text": "1 uh when we raise it when we take the"}, {"start": 184.519, "duration": 6.481, "text": "absolute value of it it'll just become"}, {"start": 186.519, "duration": 7.08, "text": "postive one over N2 but again uh this"}, {"start": 191.0, "duration": 5.0, "text": "limit as n goes to Infinity the"}, {"start": 193.599, "duration": 4.121, "text": "denominator will get arbitrarily large"}, {"start": 196.0, "duration": 4.48, "text": "one over a big number is going to be"}, {"start": 197.72, "duration": 4.879, "text": "zero so hey in fact by the absolute"}, {"start": 200.48, "duration": 4.319, "text": "value theorem this first limit is"}, {"start": 202.599, "duration": 5.72, "text": "actually going to be zero so really"}, {"start": 204.799, "duration": 9.481, "text": "we're left with 0 + one so again in this"}, {"start": 208.319, "duration": 5.961, "text": "case our limit will simp converge to one"}], "title": "The Squeeze Theorem and Absolute Value Theorem, #2", "description": "Thanks to all of you who support me on Patreon. You da real mvps!   $1 per month helps!! :)  https://www.patreon.com/patrickjmt !! The Squeeze Theorem and Absolute Value Theorem, #2. Just another example of finding the limit of a sequence by using the squeeze and / or absolute value theorem.", "lengthSeconds": 212, "width": 480, "height": 360}
{"video_id": "HGIHD7zee0Q", "transcript": [{"start": 0.599, "duration": 3.72, "text": "all right in this video I just want to"}, {"start": 1.92, "duration": 5.56, "text": "start looking at some examples of just"}, {"start": 4.319, "duration": 6.921, "text": "simplifying trig functions involving uh"}, {"start": 7.48, "duration": 5.56, "text": "fractions so nothing too crazy here so"}, {"start": 11.24, "duration": 4.959, "text": "uh maybe we can do at least one or two"}, {"start": 13.04, "duration": 6.2, "text": "so uh in part A we've got s time secant"}, {"start": 16.199, "duration": 5.761, "text": "times tangent a lot of times what I will"}, {"start": 19.24, "duration": 4.72, "text": "do um you know there's really not much"}, {"start": 21.96, "duration": 4.239, "text": "going on here there's nothing to add or"}, {"start": 23.96, "duration": 4.04, "text": "subtract uh a lot of times what I'll do"}, {"start": 26.199, "duration": 4.721, "text": "is I'll just rewrite everything in terms"}, {"start": 28.0, "duration": 5.879, "text": "of s and cosine and see see what if"}, {"start": 30.92, "duration": 6.08, "text": "anything I can cancel out so I'll leave"}, {"start": 33.879, "duration": 5.36, "text": "the sin x alone see of X that's the same"}, {"start": 37.0, "duration": 8.2, "text": "thing as 1 cosine of"}, {"start": 39.239, "duration": 9.16, "text": "x well tangent of X that's sin x over"}, {"start": 45.2, "duration": 5.24, "text": "cosine X well um I think you'll notice"}, {"start": 48.399, "duration": 4.521, "text": "in the top you can really put sinx over"}, {"start": 50.44, "duration": 6.119, "text": "one so really in the numerator we have"}, {"start": 52.92, "duration": 3.639, "text": "sinx over cosine"}, {"start": 57.359, "duration": 8.521, "text": "X and again that's simply being divided"}, {"start": 60.48, "duration": 8.8, "text": "sin x over cosine X well you have S you"}, {"start": 65.88, "duration": 5.4, "text": "have sin / cosine over sinid by cosine"}, {"start": 69.28, "duration": 4.08, "text": "or equivalently we have tangent of X"}, {"start": 71.28, "duration": 4.32, "text": "over tangent of X however you want to"}, {"start": 73.36, "duration": 3.759, "text": "look at it or you know last but not"}, {"start": 75.6, "duration": 4.159, "text": "least you could even think about it just"}, {"start": 77.119, "duration": 5.521, "text": "in terms of fractions uh we've got a"}, {"start": 79.759, "duration": 4.881, "text": "fraction divided by a fraction flip the"}, {"start": 82.64, "duration": 3.839, "text": "denominator and make it into"}, {"start": 84.64, "duration": 5.0, "text": "multiplication and again basically"}, {"start": 86.479, "duration": 4.761, "text": "everything's just going to cancel out um"}, {"start": 89.64, "duration": 3.08, "text": "we still have you know you can think"}, {"start": 91.24, "duration": 3.8, "text": "about there as being ones in the"}, {"start": 92.72, "duration": 4.64, "text": "numerator so this first expression is"}, {"start": 95.04, "duration": 5.56, "text": "simply going to simp is just going to uh"}, {"start": 97.36, "duration": 5.719, "text": "just reduce down to the number one so s"}, {"start": 100.6, "duration": 4.879, "text": "* secant over tangent is just another"}, {"start": 103.079, "duration": 4.961, "text": "way of writing"}, {"start": 105.479, "duration": 6.201, "text": "one um let's look at Part B we've got"}, {"start": 108.04, "duration": 5.6, "text": "cosecant time cotangent divided tangent"}, {"start": 111.68, "duration": 4.64, "text": "time secant so I think I'm going to do"}, {"start": 113.64, "duration": 5.24, "text": "the same thing in this example um let's"}, {"start": 116.32, "duration": 4.839, "text": "see so cosecant that's going to be 1/"}, {"start": 118.88, "duration": 4.68, "text": "sin Theta"}, {"start": 121.159, "duration": 5.32, "text": "coent is cine"}, {"start": 123.56, "duration": 5.559, "text": "thet over sin"}, {"start": 126.479, "duration": 6.961, "text": "thet let's see tangent of theta that's"}, {"start": 129.119, "duration": 4.321, "text": "going to give us sin Theta over cosine"}, {"start": 134.12, "duration": 7.36, "text": "Theta and then secant of theta is going"}, {"start": 136.64, "duration": 6.84, "text": "to be 1 over cosine of theta okay so"}, {"start": 141.48, "duration": 5.44, "text": "let's try to uh clean all this stuff up"}, {"start": 143.48, "duration": 6.52, "text": "here a little bit so again you"}, {"start": 146.92, "duration": 5.12, "text": "know couple different uh sort of ways"}, {"start": 150.0, "duration": 5.04, "text": "can regroup things we we really have"}, {"start": 152.04, "duration": 5.12, "text": "cosine of theta on top um we have sin"}, {"start": 155.04, "duration": 4.96, "text": "squar in the"}, {"start": 157.16, "duration": 7.359, "text": "denominator of the numerator and then in"}, {"start": 160.0, "duration": 8.12, "text": "the denominator we have S of theta over"}, {"start": 164.519, "duration": 6.8, "text": "cosine squar of theta and what I'm going"}, {"start": 168.12, "duration": 6.56, "text": "to do now is just simply take this"}, {"start": 171.319, "duration": 6.2, "text": "fraction these two fractions and we can"}, {"start": 174.68, "duration": 5.279, "text": "multiply so I'm going to take cosine"}, {"start": 177.519, "duration": 3.881, "text": "over sin squ"}, {"start": 179.959, "duration": 4.241, "text": "I'm going to multiply that by the"}, {"start": 181.4, "duration": 4.52, "text": "reciprocal of the denominator so uh"}, {"start": 184.2, "duration": 6.16, "text": "we're just going to flip the the bottom"}, {"start": 185.92, "duration": 6.959, "text": "fraction so we'll get cosine s Theta"}, {"start": 190.36, "duration": 5.519, "text": "over s of"}, {"start": 192.879, "duration": 7.241, "text": "theta well cosine * cosine squ is going"}, {"start": 195.879, "duration": 6.881, "text": "to be cosine cubed of theta over sin"}, {"start": 200.12, "duration": 5.399, "text": "cubed of theta but again we could really"}, {"start": 202.76, "duration": 7.44, "text": "just factor out the uh the exponent we"}, {"start": 205.519, "duration": 7.601, "text": "could write this as cosine over s all"}, {"start": 210.2, "duration": 4.2, "text": "cubed and remember cosine over s that's"}, {"start": 213.12, "duration": 6.16, "text": "simply"}, {"start": 214.4, "duration": 8.199, "text": "coent so really we have OFA cubed or"}, {"start": 219.28, "duration": 6.319, "text": "again we can rewrite that asent cubed"}, {"start": 222.599, "duration": 3.0, "text": "Theta"}], "title": "Simplifying Trigonometric Expressions Involving Fractions, Ex 1", "description": "Thanks to all of you who support me on Patreon. You da real mvps!   $1 per month helps!! :)  https://www.patreon.com/patrickjmt !! Simplifying Trigonometric Expressions Involving Fractions, Ex 1", "lengthSeconds": 227, "width": 480, "height": 360}
{"video_id": "f3Y3mXI8KHk", "transcript": [{"start": 1.46, "duration": 1.08, "text": "So in this question it says,"}, {"start": 2.54, "duration": 5.3, "text": "For what value of n is the absolute value of the quantity n minus one"}, {"start": 8.1, "duration": 1.52, "text": "plus one equal to zero?"}, {"start": 9.62, "duration": 5.86, "text": "So the answer choices as are zero, positive one, positive two, or there is no such value of n."}, {"start": 15.48, "duration": 2.32, "text": "So, just recall that the absolute,"}, {"start": 17.8, "duration": 3.76, "text": "I mean, to me there is not much to do in this problem as long as you remember absolute values."}, {"start": 21.56, "duration": 3.02, "text": "So remember if you take the absolute value of a number"}, {"start": 24.58, "duration": 2.54, "text": "you know, so the absolute value of zero is zero."}, {"start": 27.12, "duration": 2.1, "text": "You can think about this being a distance from a number,"}, {"start": 29.22, "duration": 1.9, "text": "you know, the distance from zero in the number line,"}, {"start": 31.12, "duration": 2.04, "text": "that's someway people think about it."}, {"start": 33.16, "duration": 3.44, "text": "I always just think of it, you know, if the number's already positive,"}, {"start": 36.6, "duration": 1.0, "text": "you leave it alone."}, {"start": 37.84, "duration": 2.88, "text": "So, for example the absolute value of two would be two."}, {"start": 40.72, "duration": 5.76, "text": "If the number's negative, the absolute value of negative four, again we just drop the negative sign."}, {"start": 46.48, "duration": 2.7, "text": "So the absolute value of negative four is positive four."}, {"start": 49.48, "duration": 5.2, "text": "The main thing you remember is that the absolute value is always gonna give you, either,"}, {"start": 54.68, "duration": 3.32, "text": "if you simplify, its either gonna be a number that's zero or larger."}, {"start": 58.0, "duration": 4.28, "text": "So, just based on that, I know that the absolute value n minus one,"}, {"start": 62.42, "duration": 2.84, "text": "no matter what value I substitute in for n,"}, {"start": 65.56, "duration": 2.66, "text": "this absolute value is always gonna give me a number"}, {"start": 68.22, "duration": 1.94, "text": "that is greater than or equal to zero."}, {"start": 70.42, "duration": 2.38, "text": "Okay, so I'm gonna get something that is zero or larger."}, {"start": 73.62, "duration": 4.24, "text": "Well, if I add one to that, again if I add one to that value,"}, {"start": 77.86, "duration": 2.4, "text": "it's gonna be, its gonna give me a number"}, {"start": 80.26, "duration": 1.6, "text": "that's greater than or equal to one."}, {"start": 82.3, "duration": 2.8, "text": "So in this case, there would be no such value of n."}, {"start": 85.1, "duration": 1.86, "text": "And again, you could check, you know,"}, {"start": 86.96, "duration": 4.08, "text": "if you substitute in zero, you would have zero minus one which would be negative one,"}, {"start": 91.22, "duration": 3.16, "text": "the absolute value of negative one is positive one,"}, {"start": 94.38, "duration": 1.88, "text": "so you have positive one plus one."}, {"start": 96.54, "duration": 4.3, "text": "If you substitute in zero, you actually get the value of two."}, {"start": 101.3, "duration": 3.92, "text": "If you substitute in n equals one, one minus one is zero,"}, {"start": 105.22, "duration": 4.72, "text": "the absolute value of zero is zero, plus one would give you one as value."}, {"start": 110.46, "duration": 4.38, "text": "And if you substitute in the value of two, well, two minus one is one,"}, {"start": 114.84, "duration": 3.96, "text": "the absolute value is still one, plus one, it wll actually gonna give you the value of two."}, {"start": 118.94, "duration": 2.28, "text": "So if you substitute in zero, you get two."}, {"start": 121.22, "duration": 1.74, "text": "If you substitute in one, you get one."}, {"start": 122.96, "duration": 2.24, "text": "If you substitute in two, you get two."}, {"start": 125.5, "duration": 0.5, "text": "Again,"}, {"start": 126.0, "duration": 4.52, "text": "we're trying to get zero out, so there is no such value of n in this case."}, {"start": 130.52, "duration": 2.22, "text": "You know, they could give you some really wonky problem,"}, {"start": 132.74, "duration": 1.26, "text": "you know, they could make it like"}, {"start": 134.0, "duration": 5.62, "text": "n squared plus n raised to the four hundredth power minus the square root of pi"}, {"start": 139.62, "duration": 3.18, "text": "plus, you know, eight to the eighth to the eighth"}, {"start": 143.96, "duration": 0.64, "text": "plus one."}, {"start": 144.6, "duration": 2.78, "text": "You know, what value of n, they could ask you the same question,"}, {"start": 147.38, "duration": 3.28, "text": "for what value of n is this quantity equal to zero?"}, {"start": 151.6, "duration": 3.68, "text": "Same thing, it doesn't, they'll be no solution because"}, {"start": 155.36, "duration": 3.46, "text": "no matter what you substitute in for n,"}, {"start": 159.66, "duration": 2.94, "text": "the absolute value, again your gonna get something that's zero or larger."}, {"start": 162.72, "duration": 3.42, "text": "If you add one to that, it's gonna give you someting that's one or larger."}, {"start": 166.14, "duration": 3.84, "text": "So, that's all there is to it, just making sure that you remember absolute values."}, {"start": 169.98, "duration": 2.32, "text": "So, you know, hopefully if you saw this in the SAT,"}, {"start": 172.3, "duration": 1.82, "text": "this would be pretty quick question and you could just"}, {"start": 174.28, "duration": 0.64, "text": "you know,"}, {"start": 175.82, "duration": 2.4, "text": "circle answer choice D and keep on truckin'."}], "title": "SAT Math - Question 8, Section 2, 2016 Edition and Beyond", "description": "Thanks to all of you who support me on Patreon. You da real mvps!   $1 per month helps!! :)  https://www.patreon.com/patrickjmt !! SAT Math - Question 8, 2016 Edition and Beyond.\nSection 2 - A Calculator IS allowed!", "lengthSeconds": 179, "width": 480, "height": 360}
{"video_id": "L_ZANhk70-k", "transcript": [{"start": 0.44, "duration": 4.04, "text": "all right so just one more example here"}, {"start": 1.88, "duration": 6.6, "text": "of solving a basic trig equation suppose"}, {"start": 4.48, "duration": 6.8, "text": "we want to solve see of theta equal"}, {"start": 8.48, "duration": 4.079, "text": "2 well all I would do to solve secant"}, {"start": 11.28, "duration": 5.12, "text": "theta equals"}, {"start": 12.559, "duration": 6.48, "text": "2 um you know I have values memorized"}, {"start": 16.4, "duration": 6.6, "text": "about s and cosine so I'm going to turn"}, {"start": 19.039, "duration": 5.721, "text": "secant back into one/"}, {"start": 23.0, "duration": 3.519, "text": "cosine and now I'm just going to do a"}, {"start": 24.76, "duration": 6.04, "text": "little bit of algebra we can multiply"}, {"start": 26.519, "duration": 4.281, "text": "both sides by cosine Theta"}, {"start": 31.759, "duration": 7.361, "text": "well the cosine thetas uh would cancel"}, {"start": 34.399, "duration": 6.081, "text": "out we would have 1 = 2 * cosine Theta"}, {"start": 39.12, "duration": 4.599, "text": "and now I'm simply just going to divide"}, {"start": 40.48, "duration": 7.2, "text": "both sides by two by two and that'll"}, {"start": 43.719, "duration": 3.961, "text": "give us cosine Theta equal"}, {"start": 48.64, "duration": 7.28, "text": "12 all right so I do know angles where"}, {"start": 52.0, "duration": 6.879, "text": "cosine equals 12 let's see where would"}, {"start": 55.92, "duration": 4.72, "text": "those be um I think actually again at"}, {"start": 58.879, "duration": 4.921, "text": "the angle pi"}, {"start": 60.64, "duration": 6.519, "text": "3 again that's where the x coordinat is"}, {"start": 63.8, "duration": 7.24, "text": "12 and the y coordinate is < tk3 over2"}, {"start": 67.159, "duration": 7.121, "text": "well so that means that Theta = pi over"}, {"start": 71.04, "duration": 6.04, "text": "3 would be a solution um if we kind of"}, {"start": 74.28, "duration": 4.72, "text": "make one revolution 0 to 2 pi I think"}, {"start": 77.08, "duration": 5.399, "text": "the next place where the x coordinate is"}, {"start": 79.0, "duration": 6.96, "text": "going to be positive uh2 would"}, {"start": 82.479, "duration": 8.32, "text": "be down here in Quadrant 4 that's where"}, {"start": 85.96, "duration": 6.159, "text": "we would get 12 < tk3 over 2"}, {"start": 90.799, "duration": 4.32, "text": "let's see again if we go all the way"}, {"start": 92.119, "duration": 5.121, "text": "around that's 2 pi radians equivalently"}, {"start": 95.119, "duration": 5.64, "text": "if we multiply top and bottom by three"}, {"start": 97.24, "duration": 6.68, "text": "that'll be 6 pi over 3 radians but we"}, {"start": 100.759, "duration": 5.96, "text": "didn't go 6 pi over 3 we went one less"}, {"start": 103.92, "duration": 7.68, "text": "so if you subtract 1 pi over 3 we'll get"}, {"start": 106.719, "duration": 8.641, "text": "5 pi over 3 so that would be another"}, {"start": 111.6, "duration": 4.92, "text": "uh another solution that works um and"}, {"start": 115.36, "duration": 5.48, "text": "again you know sometimes there's"}, {"start": 116.52, "duration": 6.239, "text": "definitely uh you know more uh maybe"}, {"start": 120.84, "duration": 3.44, "text": "more succinct ways of writing it I'm"}, {"start": 122.759, "duration": 4.761, "text": "just going to say our solutions would be"}, {"start": 124.28, "duration": 5.399, "text": "Theta = pi over 3 again to end back up"}, {"start": 127.52, "duration": 4.96, "text": "at the same place we can add or subtract"}, {"start": 129.679, "duration": 5.881, "text": "multiples of 2 pi so I'm going to write"}, {"start": 132.48, "duration": 6.32, "text": "plus 2 pi *"}, {"start": 135.56, "duration": 6.16, "text": "n same thing for our angle 5 pi over 3"}, {"start": 138.8, "duration": 6.2, "text": "we can add or subtract multiples of 2 pi"}, {"start": 141.72, "duration": 8.159, "text": "so I'm going to write plus 2 pi * n and"}, {"start": 145.0, "duration": 7.879, "text": "again in both cases um n is an integer"}, {"start": 149.879, "duration": 3.0, "text": "for"}], "title": "Solving a Basic Trigonometric Equation, Example 3", "description": "Thanks to all of you who support me on Patreon. You da real mvps!   $1 per month helps!! :)  https://www.patreon.com/patrickjmt !! Solving a Basic Trigonometric Equation, Example 3", "lengthSeconds": 157, "width": 480, "height": 360}
{"video_id": "vBVK2fkciaE", "transcript": [{"start": 0.53, "duration": 5.289, "text": "okay and these two examples I just want"}, {"start": 3.51, "duration": 5.49, "text": "to just do some evaluating of these"}, {"start": 5.819, "duration": 7.2, "text": "exponential expressions and we've got"}, {"start": 9.0, "duration": 5.88, "text": "two problems here so in the first one we"}, {"start": 13.019, "duration": 4.651, "text": "have negative 2 raised to the third"}, {"start": 14.88, "duration": 6.0, "text": "power times 3 raised to the second power"}, {"start": 17.67, "duration": 5.099, "text": "and again you can just you know you can"}, {"start": 20.88, "duration": 5.96, "text": "simply write it all out so negative 2"}, {"start": 22.769, "duration": 7.77, "text": "times negative 2 times negative 2 and"}, {"start": 26.84, "duration": 9.85, "text": "then we have 3 squared which means we"}, {"start": 30.539, "duration": 8.811, "text": "have 3 times 3 well negative 2 times"}, {"start": 36.69, "duration": 5.7, "text": "negative 2 will give us positive for"}, {"start": 39.35, "duration": 7.75, "text": "positive 4 times negative 2 is going to"}, {"start": 42.39, "duration": 8.55, "text": "give us negative 8 if we do 3 times 3"}, {"start": 47.1, "duration": 5.849, "text": "that's going to give us 9 we have a"}, {"start": 50.94, "duration": 4.529, "text": "negative times a positive which is going"}, {"start": 52.949, "duration": 6.301, "text": "to give us a negative and then 8 times 9"}, {"start": 55.469, "duration": 5.971, "text": "will simply give us 72 so negative 2 to"}, {"start": 59.25, "duration": 5.78, "text": "the third power times 3 to the second"}, {"start": 61.44, "duration": 7.469, "text": "power will equal the number negative 72"}, {"start": 65.03, "duration": 5.62, "text": "and our last one here on the second one"}, {"start": 68.909, "duration": 4.681, "text": "we have negative 3 squared and negative"}, {"start": 70.65, "duration": 7.24, "text": "2 squared so again that's just negative"}, {"start": 73.59, "duration": 8.41, "text": "3 times negative 3"}, {"start": 77.89, "duration": 7.98, "text": "and then we have two negative twos so we"}, {"start": 82.0, "duration": 6.06, "text": "have negative 2 times negative 2 well"}, {"start": 85.87, "duration": 5.46, "text": "negative 3 times negative 3 it's going"}, {"start": 88.06, "duration": 4.92, "text": "to give us a positive 9 negative 2 times"}, {"start": 91.33, "duration": 8.0, "text": "negative 2 is going to give us a"}, {"start": 92.98, "duration": 9.45, "text": "positive 4 and 9 times 4 will give us 36"}, {"start": 99.33, "duration": 5.649, "text": "so negative 3 to the second power times"}, {"start": 102.43, "duration": 5.81, "text": "negative 2 to the second power will"}, {"start": 104.979, "duration": 3.261, "text": "equal 36"}], "title": "Evaluating Exponential Expressions (Numbers Only, No Variables) - Ex 1", "description": "Thanks to all of you who support me on Patreon. You da real mvps!   $1 per month helps!! :)  https://www.patreon.com/patrickjmt !! Evaluating Exponential Expressions (Numbers Only, No Variables)", "lengthSeconds": 108, "width": 480, "height": 360}
{"video_id": "bcp9pJxaAOk", "transcript": [{"start": 1.0, "duration": 3.799, "text": "okay in this video I just want to talk"}, {"start": 2.28, "duration": 4.639, "text": "about the midpoint formula and all the"}, {"start": 4.799, "duration": 4.281, "text": "midpoint formula says it says that the"}, {"start": 6.919, "duration": 6.201, "text": "midpoint between two points x sub 1"}, {"start": 9.08, "duration": 5.639, "text": "comma y sub 1 and x sub 2 comma y sub 2"}, {"start": 13.12, "duration": 3.52, "text": "it says all you do is add the x"}, {"start": 14.719, "duration": 5.121, "text": "coordinates divide by two add the"}, {"start": 16.64, "duration": 5.08, "text": "y-coordinates and also divide by two um"}, {"start": 19.84, "duration": 3.48, "text": "I've uh in my experience I've noticed"}, {"start": 21.72, "duration": 4.44, "text": "some people sometimes forget you know"}, {"start": 23.32, "duration": 4.879, "text": "the signs they'll sometimes subtract and"}, {"start": 26.16, "duration": 4.439, "text": "one way to maybe think about it is you"}, {"start": 28.199, "duration": 4.24, "text": "know take a kind of a concrete example"}, {"start": 30.599, "duration": 5.12, "text": "suppose you took two tests and you made"}, {"start": 32.439, "duration": 5.761, "text": "a 50% on the first one and 100% on the"}, {"start": 35.719, "duration": 4.0, "text": "second one and I asked you I said well"}, {"start": 38.2, "duration": 3.96, "text": "if those are your test scores what's"}, {"start": 39.719, "duration": 4.481, "text": "your test average most people would say"}, {"start": 42.16, "duration": 4.84, "text": "your test average is"}, {"start": 44.2, "duration": 4.839, "text": "75% and I normally when I ask them how"}, {"start": 47.0, "duration": 4.0, "text": "you get it they say well to average you"}, {"start": 49.039, "duration": 4.721, "text": "add the things together and you divide"}, {"start": 51.0, "duration": 7.84, "text": "by the number of them so you have 150"}, {"start": 53.76, "duration": 8.04, "text": "over two or simply 75 um% so you know"}, {"start": 58.84, "duration": 6.12, "text": "notice a mid point is just an average of"}, {"start": 61.8, "duration": 4.92, "text": "two things so if you can remember it in"}, {"start": 64.96, "duration": 4.199, "text": "that context maybe it'll help you"}, {"start": 66.72, "duration": 4.68, "text": "remember that um you should be adding"}, {"start": 69.159, "duration": 5.081, "text": "the x coordinates divide by two adding"}, {"start": 71.4, "duration": 5.0, "text": "the y-coordinates divide by two so in"}, {"start": 74.24, "duration": 4.16, "text": "this example um I'm just going to Simply"}, {"start": 76.4, "duration": 6.079, "text": "find the midpoint between these two"}, {"start": 78.4, "duration": 7.359, "text": "points I've got the point -5 comma -6"}, {"start": 82.479, "duration": 6.761, "text": "and then I've got the point um 5 comma"}, {"start": 85.759, "duration": 5.481, "text": "8 so let me try to you know maybe"}, {"start": 89.24, "duration": 4.72, "text": "connect those with with a line if I can"}, {"start": 91.24, "duration": 5.36, "text": "so not the straightest of lines but a"}, {"start": 93.96, "duration": 4.36, "text": "line nonetheless so to me intuitively"}, {"start": 96.6, "duration": 5.799, "text": "the midpoint should be you know"}, {"start": 98.32, "duration": 5.96, "text": "somewhere around there so you know based"}, {"start": 102.399, "duration": 3.801, "text": "on my solution at least I can even get a"}, {"start": 104.28, "duration": 5.04, "text": "quick little conceptual guess about what"}, {"start": 106.2, "duration": 4.279, "text": "it should be so it simply says again to"}, {"start": 109.32, "duration": 3.799, "text": "find the"}, {"start": 110.479, "duration": 5.401, "text": "midpoint piece of cake all we have to do"}, {"start": 113.119, "duration": 5.28, "text": "is add the x coordinates so maybe we can"}, {"start": 115.88, "duration": 6.08, "text": "label them as well so this will be X"}, {"start": 118.399, "duration": 5.281, "text": "sub1 this will be the point Y sub one"}, {"start": 121.96, "duration": 4.199, "text": "again these are kind of generic labels"}, {"start": 123.68, "duration": 6.079, "text": "it doesn't matter um which one you call"}, {"start": 126.159, "duration": 5.961, "text": "X1 y1 which one you call X2"}, {"start": 129.759, "duration": 6.961, "text": "Y2 so it says if we add the x"}, {"start": 132.12, "duration": 7.119, "text": "coordinates we'll get -5 + 5 / by two"}, {"start": 136.72, "duration": 6.599, "text": "and then it says uh the midpoint for the"}, {"start": 139.239, "duration": 7.72, "text": "y-coordinates will get -6 + 8 and divide"}, {"start": 143.319, "duration": 7.521, "text": "that by 2 and if we simplify that down"}, {"start": 146.959, "duration": 9.321, "text": "-5 + 5 will be 0 / 2"}, {"start": 150.84, "duration": 9.88, "text": "um -6 + 8 is 2 / 2 and that simply"}, {"start": 156.28, "duration": 6.76, "text": "reduces to the point 0 comma 1 and to me"}, {"start": 160.72, "duration": 4.56, "text": "that seems reasonable 01 hey that would"}, {"start": 163.04, "duration": 4.68, "text": "be a point right on the y axis so that"}, {"start": 165.28, "duration": 5.0, "text": "seems correct and notice in this case as"}, {"start": 167.72, "duration": 5.12, "text": "well things are kind of symmetric um"}, {"start": 170.28, "duration": 4.4, "text": "we're at NE five we're at positive five"}, {"start": 172.84, "duration": 4.399, "text": "so we're five units to the right of the"}, {"start": 174.68, "duration": 5.479, "text": "y- axis five units to the left of the y"}, {"start": 177.239, "duration": 5.241, "text": "axis the midpoint should be you know"}, {"start": 180.159, "duration": 4.841, "text": "zero units away from the y- axis which"}, {"start": 182.48, "duration": 6.399, "text": "would correspond to an x coordinate of"}, {"start": 185.0, "duration": 6.599, "text": "zero um you know so"}, {"start": 188.879, "duration": 4.161, "text": "again it's good to know formulas but I"}, {"start": 191.599, "duration": 3.121, "text": "think if you sketch a little picture on"}, {"start": 193.04, "duration": 4.119, "text": "these two if you're taking an exam or"}, {"start": 194.72, "duration": 4.48, "text": "something like that um you can quickly"}, {"start": 197.159, "duration": 5.041, "text": "verify that whether it's at least in the"}, {"start": 199.2, "duration": 4.679, "text": "ballpark or not so um all right I hope"}, {"start": 202.2, "duration": 3.119, "text": "this video makes some sense and helps"}, {"start": 203.879, "duration": 3.681, "text": "you out if you have any questions about"}, {"start": 205.319, "duration": 4.601, "text": "the midpoint or any questions or"}, {"start": 207.56, "duration": 4.08, "text": "comments about this video as always"}, {"start": 209.92, "duration": 3.319, "text": "always please feel free to post them"}, {"start": 211.64, "duration": 5.12, "text": "hopefully me or somebody else can point"}, {"start": 213.239, "duration": 3.521, "text": "you in the right direction"}], "title": "The Midpoint Formula", "description": "Thanks to all of you who support me on Patreon. You da real mvps!   $1 per month helps!! :)  https://www.patreon.com/patrickjmt !! The Midpoint Formula - In this video I give the formula for finding the midpoint of two points and do one simple example to find the midpoint.\r\nFor more free math videos, visit http://PatrickJMT.com", "lengthSeconds": 215, "width": 480, "height": 360}
{"video_id": "hDNAVh0VtTc", "transcript": [{"start": 0.599, "duration": 2.921, "text": "all right in this video I'm going to do"}, {"start": 1.839, "duration": 5.201, "text": "one more example of graphing a"}, {"start": 3.52, "duration": 5.48, "text": "piece-wise defined function so here in"}, {"start": 7.04, "duration": 4.36, "text": "in this particular case notice we've got"}, {"start": 9.0, "duration": 5.839, "text": "a function that sort of has three parts"}, {"start": 11.4, "duration": 7.2, "text": "it looks like the line y = -2 if x is"}, {"start": 14.839, "duration": 6.401, "text": "less than or equal to3 looks like 4 - x"}, {"start": 18.6, "duration": 4.919, "text": "for X values in between -3 and 0 not"}, {"start": 21.24, "duration": 4.92, "text": "including either and it looks like the"}, {"start": 23.519, "duration": 6.121, "text": "line y equals x if x is greater than or"}, {"start": 26.16, "duration": 3.48, "text": "equal to zero"}, {"start": 35.2, "duration": 3.72, "text": "okay so here is uh here's my little"}, {"start": 37.68, "duration": 3.84, "text": "graph so I'm just going to do them kind"}, {"start": 38.92, "duration": 7.959, "text": "of in order here uh if you think about"}, {"start": 41.52, "duration": 10.0, "text": "the first equation y = -2 y = -2 is just"}, {"start": 46.879, "duration": 8.081, "text": "a horizontal line at the yalue of -2 so"}, {"start": 51.52, "duration": 6.839, "text": "let me drop down here -1 -2"}, {"start": 54.96, "duration": 7.279, "text": "units but it says it only looks like uh"}, {"start": 58.359, "duration": 7.001, "text": "this line Y = 2 for x coordinates that"}, {"start": 62.239, "duration": 4.721, "text": "are less than or equal to -3 okay so it"}, {"start": 65.36, "duration": 6.32, "text": "looks like that line but we don't get"}, {"start": 66.96, "duration": 7.76, "text": "the whole line -1 -2 -3 it says it only"}, {"start": 71.68, "duration": 7.32, "text": "looks like that for x coordinates uh"}, {"start": 74.72, "duration": 6.759, "text": "starting at -3 and then it says it has"}, {"start": 79.0, "duration": 4.84, "text": "uh it has that shape forever and ever"}, {"start": 81.479, "duration": 5.561, "text": "and ever for all x coordinates less than"}, {"start": 83.84, "duration": 5.4, "text": "or equal to -3 so it says we're getting"}, {"start": 87.04, "duration": 4.84, "text": "that that segment of the line up to an"}, {"start": 89.24, "duration": 5.519, "text": "including 3 but then it"}, {"start": 91.88, "duration": 6.84, "text": "stops okay I recognize the second"}, {"start": 94.759, "duration": 6.761, "text": "equation Y = 4 -"}, {"start": 98.72, "duration": 4.399, "text": "x that's simply a line we could actually"}, {"start": 101.52, "duration": 3.76, "text": "rewrite this"}, {"start": 103.119, "duration": 5.161, "text": "ASX +"}, {"start": 105.28, "duration": 8.159, "text": "4 that's a line with a Y intercept of"}, {"start": 108.28, "duration": 7.519, "text": "positive4 so 1 2 3 4 um and a slope of"}, {"start": 113.439, "duration": 4.561, "text": "negative 1 so I know it's pointing"}, {"start": 115.799, "duration": 4.481, "text": "downwards um Let me give myself a little"}, {"start": 118.0, "duration": 5.64, "text": "more room here five six we'll say sevens"}, {"start": 120.28, "duration": 5.56, "text": "up here again what I do I always just"}, {"start": 123.64, "duration": 5.8, "text": "plug in the end points so if I were to"}, {"start": 125.84, "duration": 7.36, "text": "plug -3 into this I would get four - -3"}, {"start": 129.44, "duration": 7.079, "text": "or 4 + 3 which would give me 7 but since"}, {"start": 133.2, "duration": 6.679, "text": "it's not including -3 I'm going to put"}, {"start": 136.519, "duration": 6.841, "text": "an open circle up"}, {"start": 139.879, "duration": 5.801, "text": "there okay if I plug in the value if I"}, {"start": 143.36, "duration": 5.239, "text": "plug in the other end point 0 I would"}, {"start": 145.68, "duration": 4.919, "text": "get 4 minus 0 which would give me four"}, {"start": 148.599, "duration": 3.801, "text": "but again since it doesn't include the"}, {"start": 150.599, "duration": 4.201, "text": "value zero I'm going to make an open"}, {"start": 152.4, "duration": 4.52, "text": "circle there so I'm kind of finding the"}, {"start": 154.8, "duration": 4.4, "text": "end points and again since this is a"}, {"start": 156.92, "duration": 4.52, "text": "line and it it it does look like the"}, {"start": 159.2, "duration": 5.319, "text": "line for all x coordinates in between I"}, {"start": 161.44, "duration": 6.28, "text": "can just play Connect the Dots uh with a"}, {"start": 164.519, "duration": 7.201, "text": "straight line and that'll now be the the"}, {"start": 167.72, "duration": 8.879, "text": "portion of the graph y = 4 - x so again"}, {"start": 171.72, "duration": 6.76, "text": "here was a y = -2 down here and last but"}, {"start": 176.599, "duration": 4.0, "text": "not least if we graph the last little"}, {"start": 178.48, "duration": 5.24, "text": "segment y equal x"}, {"start": 180.599, "duration": 5.601, "text": "X well again I just plug in the value if"}, {"start": 183.72, "duration": 6.519, "text": "I plug zero in well I get zero out so"}, {"start": 186.2, "duration": 6.16, "text": "that means 0 Z's on the graph and yal X"}, {"start": 190.239, "duration": 3.56, "text": "is just our kind of a line that goes"}, {"start": 192.36, "duration": 5.519, "text": "through the origin kind of makes it like"}, {"start": 193.799, "duration": 7.8, "text": "a 45\u00b0 angle um but again it only looks"}, {"start": 197.879, "duration": 5.681, "text": "like this line yal X for X values"}, {"start": 201.599, "duration": 4.121, "text": "greater than or equal to zero so to me"}, {"start": 203.56, "duration": 5.039, "text": "that's only the right portion of the"}, {"start": 205.72, "duration": 4.879, "text": "graph so now I'll just make it extend on"}, {"start": 208.599, "duration": 6.081, "text": "forever and ever and ever and there's"}, {"start": 210.599, "duration": 6.36, "text": "our line Y = X and hey now we're in"}, {"start": 214.68, "duration": 5.8, "text": "business we've got our nice piecewise"}, {"start": 216.959, "duration": 3.521, "text": "defined function"}], "title": "Graphing a Piece-Wise Defined Function - Another Example", "description": "Thanks to all of you who support me on Patreon. You da real mvps!   $1 per month helps!! :)  https://www.patreon.com/patrickjmt !! Graphing a Piece-Wise Defined Function - Another Example", "lengthSeconds": 219, "width": 480, "height": 360}
{"video_id": "DXK0kbrUujM", "transcript": [{"start": 0.84, "duration": 3.4, "text": "all right uh in this video that I'm not"}, {"start": 2.44, "duration": 3.359, "text": "really going to do anything uh other"}, {"start": 4.24, "duration": 3.119, "text": "than just put up a table of llao"}, {"start": 5.799, "duration": 3.241, "text": "transforms because we'll use this in"}, {"start": 7.359, "duration": 4.001, "text": "some of the other examples so just as a"}, {"start": 9.04, "duration": 4.4, "text": "reference so um on the left we have our"}, {"start": 11.36, "duration": 4.359, "text": "function f of T and then um on the right"}, {"start": 13.44, "duration": 4.96, "text": "we have the Lao transform associated"}, {"start": 15.719, "duration": 5.201, "text": "with that so uh the Lao transform of the"}, {"start": 18.4, "duration": 4.719, "text": "function one is just going to be 1 / s"}, {"start": 20.92, "duration": 6.199, "text": "um if you have t to a power n you end up"}, {"start": 23.119, "duration": 8.761, "text": "getting in factorial over s the n + 1 if"}, {"start": 27.119, "duration": 6.96, "text": "it's s of a we get a/ s\u00b2 + a 2 again a"}, {"start": 31.88, "duration": 4.76, "text": "lot of these are just Justified uh"}, {"start": 34.079, "duration": 5.96, "text": "straight using the definition um if you"}, {"start": 36.64, "duration": 7.16, "text": "have cosine of a t you get S over s^2 +"}, {"start": 40.039, "duration": 8.881, "text": "a 2 if it's e to the a t we get 1 / s -"}, {"start": 43.8, "duration": 8.72, "text": "a e to the a t * sin of BT we get b/ S -"}, {"start": 48.92, "duration": 8.88, "text": "a^ 2 + b^ 2 if it's e to the a t cosine"}, {"start": 52.52, "duration": 7.839, "text": "of BT we get S - A over s - a^ 2 + b^ 2"}, {"start": 57.8, "duration": 5.64, "text": "and if we have t to the power of n time"}, {"start": 60.359, "duration": 6.201, "text": "e to the a t we get n factorial over the"}, {"start": 63.44, "duration": 5.16, "text": "quantity s minus a raised to the n + one"}, {"start": 66.56, "duration": 4.199, "text": "so we'll use these in some other"}, {"start": 68.6, "duration": 6.199, "text": "examples to actually calculate Lelo"}, {"start": 70.759, "duration": 4.04, "text": "transforms of functions"}], "title": "Table of Laplace Transforms", "description": "Thanks to all of you who support me on Patreon. You da real mvps!   $1 per month helps!! :)  https://www.patreon.com/patrickjmt !! Table of Laplace Transforms.  In this video, I show the Laplace transform for a few common functions.  No theory or derivations here, just a quick reference that I will refer to in other videos.", "lengthSeconds": 73, "width": 450, "height": 360}
{"video_id": "RKfGMX0pn2k", "transcript": [{"start": 0.88, "duration": 4.24, "text": "okay so just one more uh example here"}, {"start": 2.879, "duration": 5.281, "text": "finding the derivative of an inverse"}, {"start": 5.12, "duration": 6.16, "text": "function so uh in this problem we have"}, {"start": 8.16, "duration": 6.599, "text": "FX = cosine x + 3x and we're given that"}, {"start": 11.28, "duration": 6.079, "text": "F inverse of 3 piun / 2 = piun / 2 we"}, {"start": 14.759, "duration": 7.001, "text": "want to find the derivative of f inverse"}, {"start": 17.359, "duration": 6.721, "text": "at x = 3i / 2 um again notice that this"}, {"start": 21.76, "duration": 4.679, "text": "function does have an inverse uh the"}, {"start": 24.08, "duration": 3.4, "text": "derivative of cosine is just going to be"}, {"start": 26.439, "duration": 3.721, "text": "Nega"}, {"start": 27.48, "duration": 5.96, "text": "sinx uh and then the derivative of 3 X"}, {"start": 30.16, "duration": 6.68, "text": "will be positive 3 um if you think about"}, {"start": 33.44, "duration": 5.959, "text": "negative sinx you know s normally looks"}, {"start": 36.84, "duration": 4.68, "text": "basically sort of like that um if we"}, {"start": 39.399, "duration": 5.561, "text": "flip it you know it's just going to"}, {"start": 41.52, "duration": 6.76, "text": "instead of going up it'll go down"}, {"start": 44.96, "duration": 5.32, "text": "um but what I'm trying to say is uh"}, {"start": 48.28, "duration": 5.36, "text": "negative s of x if you think about the"}, {"start": 50.28, "duration": 5.279, "text": "range of negative sinx for any value of"}, {"start": 53.64, "duration": 4.64, "text": "x this is going to be between negative"}, {"start": 55.559, "duration": 7.921, "text": "one and positive one well if we add"}, {"start": 58.28, "duration": 7.879, "text": "three to it so sinx + 3 if we add three"}, {"start": 63.48, "duration": 5.319, "text": "everywhere we'll get two on the left"}, {"start": 66.159, "duration": 4.64, "text": "side and four on the right side it says"}, {"start": 68.799, "duration": 4.201, "text": "the derivative is always between two and"}, {"start": 70.799, "duration": 4.801, "text": "four so the main thing is this"}, {"start": 73.0, "duration": 4.04, "text": "derivative is always greater than zero"}, {"start": 75.6, "duration": 3.8, "text": "and again that tells us that our"}, {"start": 77.04, "duration": 5.16, "text": "original function f is always strictly"}, {"start": 79.4, "duration": 5.12, "text": "increasing so it passes the horizontal"}, {"start": 82.2, "duration": 4.76, "text": "line test so it does have a"}, {"start": 84.52, "duration": 3.84, "text": "derivative well again if we use our"}, {"start": 86.96, "duration": 3.799, "text": "original"}, {"start": 88.36, "duration": 4.48, "text": "formula it says we basically have to do"}, {"start": 90.759, "duration": 3.921, "text": "is take the derivative and plug in F"}, {"start": 92.84, "duration": 4.16, "text": "inverse of our"}, {"start": 94.68, "duration": 6.96, "text": "value well again we're given that F"}, {"start": 97.0, "duration": 8.88, "text": "inverse of 3 pi/ 2 is um going to equal"}, {"start": 101.64, "duration": 4.24, "text": "pi/ 2 so really we're just left"}, {"start": 106.159, "duration": 4.161, "text": "evaluating so when we kind of fill in"}, {"start": 108.119, "duration": 6.241, "text": "our"}, {"start": 110.32, "duration": 8.0, "text": "formula our F inverse of 3 pi/ 2 uh"}, {"start": 114.36, "duration": 5.759, "text": "we're going to plug in our value Pi / 2"}, {"start": 118.32, "duration": 3.399, "text": "so now again uh all we have to do is"}, {"start": 120.119, "duration": 3.96, "text": "just take the derivative and plug in Pi"}, {"start": 121.719, "duration": 5.121, "text": "/ 2 but that's what we did just a second"}, {"start": 124.079, "duration": 6.001, "text": "ago so again we said frime of X was"}, {"start": 126.84, "duration": 7.56, "text": "going to equal sinx and then we had"}, {"start": 130.08, "duration": 7.72, "text": "positive3 left over well F Prime of pi/"}, {"start": 134.4, "duration": 6.199, "text": "2 that's going to give us negative S pi/"}, {"start": 137.8, "duration": 6.4, "text": "2 +"}, {"start": 140.599, "duration": 7.28, "text": "3 S of pi over 2 is equal to 1 so we've"}, {"start": 144.2, "duration": 5.039, "text": "got -1 + 3 which will give us pos2 and"}, {"start": 147.879, "duration": 3.961, "text": "now it's just a matter of plugging all"}, {"start": 149.239, "duration": 4.841, "text": "of that back again so it says uh our"}, {"start": 151.84, "duration": 5.24, "text": "solution in this case will be positive"}, {"start": 154.08, "duration": 3.0, "text": "one2"}], "title": "Derivative of an Inverse Function , Ex 2", "description": "In this video I work a quick example of where we need to evaluate the derivative of an inverse function but we are only given the original function, not the inverse.\nThis is a typical topic in a calculus course.  In order to evaluate the derivative at a point we have a clever work around that uses the derivative of the original function, evaluated at the inverse.\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n#math, #algebra, #khanacademy, #khan, #geometry, #education, #montessori, #calculus, #homeschooling, #functions, #notation, #organicchemistry, #organicchemistrytutor, #calculus, #derivative, #powerrule, #limits,  #limit, #graph, #graphing, #conjugate, #infinity, #patrickjmt, #rational, #shortcut, #trick, #commondenominator, #fraction, #university, #admissions, #test #continuity #onesided #infinite #absolutevalue #tangents #tangentsandnormals #tangentline #definition #hindi #india", "lengthSeconds": 156, "width": 480, "height": 360}
{"video_id": "Ja2Suuuqjvs", "transcript": [{"start": 0.96, "duration": 3.96, "text": "all right so one more example here of"}, {"start": 2.48, "duration": 3.6, "text": "finding a linear approximation so here"}, {"start": 4.92, "duration": 3.96, "text": "we're going to find the linear"}, {"start": 6.08, "duration": 6.16, "text": "approximation for cosine X at x = pi"}, {"start": 8.88, "duration": 6.32, "text": "over 4 all right so again if we plug pi"}, {"start": 12.24, "duration": 6.32, "text": "over 4 into our function well we get"}, {"start": 15.2, "duration": 7.4, "text": "cosine of pi over 4 and that's just <"}, {"start": 18.56, "duration": 5.92, "text": "tk2 over 2 so I know when we do y minus"}, {"start": 22.6, "duration": 4.679, "text": "the y coordinate the Y coordinat is"}, {"start": 24.48, "duration": 5.799, "text": "going to be < tk2 over two our slope of"}, {"start": 27.279, "duration": 6.161, "text": "our line x minus the x coordinate well"}, {"start": 30.279, "duration": 5.361, "text": "the x coordinate is given to us as Pi"}, {"start": 33.44, "duration": 4.76, "text": "over4 to get our slope we'll have to"}, {"start": 35.64, "duration": 5.64, "text": "take the derivative of our function well"}, {"start": 38.2, "duration": 7.24, "text": "the derivative of cosine would be"}, {"start": 41.28, "duration": 6.759, "text": "sinx if we plug in pi over 4 well we'll"}, {"start": 45.44, "duration": 7.24, "text": "get ne s of pi over"}, {"start": 48.039, "duration": 7.801, "text": "4 and that's going to be < tk2 over 2 so"}, {"start": 52.68, "duration": 7.44, "text": "now all we have to do is just plug that"}, {"start": 55.84, "duration": 6.519, "text": "in so we have Yus < tk2 over2 the slope"}, {"start": 60.12, "duration": 7.16, "text": "of our tangent line our linearization"}, {"start": 62.359, "duration": 7.76, "text": "will be < tk2 / 2 then we'll have x - <"}, {"start": 67.28, "duration": 4.72, "text": "over 4 and now I'm just going to you"}, {"start": 70.119, "duration": 5.201, "text": "know uh just multiply things out a"}, {"start": 72.0, "duration": 6.119, "text": "little bit just uh get y by itself so if"}, {"start": 75.32, "duration": 6.4, "text": "we distribute we'll have < tk2 over"}, {"start": 78.119, "duration": 6.0, "text": "2x um I guess we'll get a positive uh"}, {"start": 81.72, "duration": 7.48, "text": "when we multiply so we have positive <"}, {"start": 84.119, "duration": 8.161, "text": "tk2 * piun over 2 * 4 which would be 8"}, {"start": 89.2, "duration": 6.599, "text": "we can add add the uh positive we can"}, {"start": 92.28, "duration": 5.24, "text": "add over < tk2 over two and again you"}, {"start": 95.799, "duration": 3.241, "text": "can get common denominators whatever you"}, {"start": 97.52, "duration": 3.919, "text": "want to do if you want to write your"}, {"start": 99.04, "duration": 5.28, "text": "constant as a single term but again this"}, {"start": 101.439, "duration": 5.881, "text": "is now our line um so this would be our"}, {"start": 104.32, "duration": 3.0, "text": "linearization"}], "title": "Finding a Linear Approximation (Linearization, Tangent Line Approx), Another Ex 2", "description": "Thanks to all of you who support me on Patreon. You da real mvps!   $1 per month helps!! :)  https://www.patreon.com/patrickjmt !! Finding a Linear Approximation (Linearization, Tangent Line Approx), Another Ex 2.  Just another example of finding a linear approximation for a function at a particular point.", "lengthSeconds": 106, "width": 480, "height": 360}
{"video_id": "osUEjDZ2fg0", "transcript": [{"start": 0.68, "duration": 4.45, "text": "okay in this video I want to talk about"}, {"start": 2.73, "duration": 4.339, "text": "integration using inverse trig functions"}, {"start": 5.13, "duration": 4.77, "text": "and this will be the first example so"}, {"start": 7.069, "duration": 5.141, "text": "kind of the two most useful formulas are"}, {"start": 9.9, "duration": 4.08, "text": "here at the bottom so recall that the"}, {"start": 12.21, "duration": 4.05, "text": "derivative of arcsin is one over the"}, {"start": 13.98, "duration": 3.809, "text": "square root of 1 minus x squared well"}, {"start": 16.26, "duration": 3.39, "text": "that means if we integrate that function"}, {"start": 17.789, "duration": 3.57, "text": "1 over the square root of 1 minus x"}, {"start": 19.65, "duration": 4.02, "text": "squared will just get arc sine plus C"}, {"start": 21.359, "duration": 7.471, "text": "back and the same thing for arctangent"}, {"start": 23.67, "duration": 7.47, "text": "okay so in our first problem here let's"}, {"start": 28.83, "duration": 5.28, "text": "integrate 1 over the square root of 1"}, {"start": 31.14, "duration": 4.38, "text": "minus 4t squared so typically when I see"}, {"start": 34.11, "duration": 6.06, "text": "something like this you know I'm just"}, {"start": 35.52, "duration": 6.51, "text": "thinking well lots of things really but"}, {"start": 40.17, "duration": 4.229, "text": "if I you know if I know it's an inverse"}, {"start": 42.03, "duration": 3.84, "text": "trig function well I'm just kind of"}, {"start": 44.399, "duration": 4.41, "text": "thinking about which one it looks like"}, {"start": 45.87, "duration": 5.25, "text": "the most well certainly of the two that"}, {"start": 48.809, "duration": 6.061, "text": "we just had listed it looks the most"}, {"start": 51.12, "duration": 5.64, "text": "like the integral of 1 over the square"}, {"start": 54.87, "duration": 6.869, "text": "root of 1 minus x squared which we said"}, {"start": 56.76, "duration": 7.17, "text": "was arc sine of X ok so basically what I"}, {"start": 61.739, "duration": 5.521, "text": "have to do is I have to make my integral"}, {"start": 63.93, "duration": 6.229, "text": "you know look like this formula and the"}, {"start": 67.26, "duration": 5.37, "text": "way to do it is usually just going to be"}, {"start": 70.159, "duration": 4.421, "text": "usually just a little bit of algebra and"}, {"start": 72.63, "duration": 4.83, "text": "I mean basically it boils down to a u"}, {"start": 74.58, "duration": 5.0, "text": "substitution is really you'll see the"}, {"start": 77.46, "duration": 4.949, "text": "the gist of a lot of the problems so"}, {"start": 79.58, "duration": 6.039, "text": "really it says you want one - some"}, {"start": 82.409, "duration": 5.1, "text": "something squared well I do have that in"}, {"start": 85.619, "duration": 5.161, "text": "a sense I have 4t squared but I'm going"}, {"start": 87.509, "duration": 5.491, "text": "to rewrite 4t squared as some quantity"}, {"start": 90.78, "duration": 5.31, "text": "being squared well I guess I could"}, {"start": 93.0, "duration": 5.009, "text": "simply rewrite that as 2t squared right"}, {"start": 96.09, "duration": 7.319, "text": "square the 2 you'll get the four square"}, {"start": 98.009, "duration": 7.14, "text": "the T certainly you get the T squared so"}, {"start": 103.409, "duration": 3.6, "text": "what we're going to do at this point is"}, {"start": 105.149, "duration": 6.83, "text": "this is where the U substitution comes"}, {"start": 107.009, "duration": 9.091, "text": "into it so we'll simply let u equal to T"}, {"start": 111.979, "duration": 7.481, "text": "what's inside the parentheses so D u"}, {"start": 116.1, "duration": 8.58, "text": "would be 2 DT divide both sides by 2"}, {"start": 119.46, "duration": 7.5, "text": "we'll get 1/2 D u equals DT so now if we"}, {"start": 124.68, "duration": 4.439, "text": "just simply rewrite everything the DT"}, {"start": 126.96, "duration": 7.109, "text": "again oh or place that with the"}, {"start": 129.119, "duration": 5.881, "text": "one half D you and now underneath the"}, {"start": 134.069, "duration": 5.03, "text": "square root"}, {"start": 135.0, "duration": 6.87, "text": "we simply have one minus u squared and"}, {"start": 139.099, "duration": 4.841, "text": "now we've made our integral through u"}, {"start": 141.87, "duration": 5.25, "text": "substitution look just like the formula"}, {"start": 143.94, "duration": 8.67, "text": "so it says we'll simply get one-half arc"}, {"start": 147.12, "duration": 7.47, "text": "sine of U plus C but again since we"}, {"start": 152.61, "duration": 4.62, "text": "started with T's in this problem let's"}, {"start": 154.59, "duration": 9.42, "text": "finish out with T's so we'll get one"}, {"start": 157.23, "duration": 7.349, "text": "half arc sine of 2t plus C and that's"}, {"start": 164.01, "duration": 2.58, "text": "your answer"}, {"start": 164.579, "duration": 3.601, "text": "so that's all there is to it in this"}, {"start": 166.59, "duration": 3.42, "text": "case it's just basically knowing a"}, {"start": 168.18, "duration": 4.71, "text": "formula and just doing a little bit of"}, {"start": 170.01, "duration": 4.949, "text": "algebra you know I guess you could"}, {"start": 172.89, "duration": 4.019, "text": "certainly skip this algebra step but to"}, {"start": 174.959, "duration": 4.11, "text": "me it makes it clear you know what"}, {"start": 176.909, "duration": 4.291, "text": "you're substituting in so that's why I"}, {"start": 179.069, "duration": 3.601, "text": "like to put that step in there but all"}, {"start": 181.2, "duration": 4.02, "text": "right I'm going to do some other ones"}, {"start": 182.67, "duration": 6.39, "text": "you know a couple other ones definitely"}, {"start": 185.22, "duration": 6.75, "text": "at least one with a definite integral so"}, {"start": 189.06, "duration": 4.98, "text": "the follow up example too should be"}, {"start": 191.97, "duration": 4.32, "text": "posted as a video response at the bottom"}, {"start": 194.04, "duration": 3.84, "text": "if you scroll down just a little bit and"}, {"start": 196.29, "duration": 3.74, "text": "you can check that one out if you're"}, {"start": 197.88, "duration": 2.15, "text": "interested"}], "title": "Integration Using Inverse Trigonometric Functions - Ex 1", "description": "Thanks to all of you who support me on Patreon. You da real mvps!   $1 per month helps!! :)  https://www.patreon.com/patrickjmt !! Integration Using Inverse Trigonometric Functions - Ex 1.  In this video, I give two formulas and show how to solve a problem with a bit of algebra and a u-substitution.  Example 2 (the other video) is a definite integral.\r\nFor more free math videos, visit http://PatrickJMT.com", "lengthSeconds": 199, "width": 480, "height": 360}
{"video_id": "IHieGivL8Cw", "transcript": [{"start": 0.719, "duration": 4.12, "text": "all right so one more example here"}, {"start": 2.44, "duration": 5.359, "text": "factoring some trig Expressions so here"}, {"start": 4.839, "duration": 5.281, "text": "we've got two 2 cosine ^ 2 X plus 2"}, {"start": 7.799, "duration": 4.72, "text": "coinx minus"}, {"start": 10.12, "duration": 4.679, "text": "24 okay the first thing I always do in"}, {"start": 12.519, "duration": 4.76, "text": "factoring is look for you know the"}, {"start": 14.799, "duration": 4.361, "text": "greatest common factor well two will go"}, {"start": 17.279, "duration": 3.881, "text": "into all of those so we can certainly"}, {"start": 19.16, "duration": 7.48, "text": "Factor the two out and then we would"}, {"start": 21.16, "duration": 10.4, "text": "have cosine 2 X plus cosine x minus well"}, {"start": 26.64, "duration": 6.4, "text": "I guess 2 * -12 will give us our 24 back"}, {"start": 31.56, "duration": 3.999, "text": "but I think we can factor a little"}, {"start": 33.04, "duration": 4.92, "text": "further um at this point and to me what"}, {"start": 35.559, "duration": 5.081, "text": "I see at this point is um to me I see"}, {"start": 37.96, "duration": 4.919, "text": "something that looks very much quadratic"}, {"start": 40.64, "duration": 5.52, "text": "okay so when we think about the cosine 2"}, {"start": 42.879, "duration": 5.961, "text": "X plus cosine x - 12 that to me really"}, {"start": 46.16, "duration": 5.44, "text": "says hey you've got something squared"}, {"start": 48.84, "duration": 5.44, "text": "plus that same thing minus 12 and I"}, {"start": 51.6, "duration": 6.16, "text": "would think you know how does x^2 + x -"}, {"start": 54.28, "duration": 6.0, "text": "12 Factor well that's going to"}, {"start": 57.76, "duration": 4.92, "text": "factor I think rather nicely so we need"}, {"start": 60.28, "duration": 5.72, "text": "two numbers that multiply to -2 but add"}, {"start": 62.68, "duration": 8.799, "text": "up to pos1 I think we could use"}, {"start": 66.0, "duration": 8.159, "text": "pos4 and -3 well that's how x^2 + x -2"}, {"start": 71.479, "duration": 6.881, "text": "would Factor well we don't have x^2 + x"}, {"start": 74.159, "duration": 5.841, "text": "- 12 we have cosine 2 X plus cosine X so"}, {"start": 78.36, "duration": 3.56, "text": "all I'm going to do is just replace my"}, {"start": 80.0, "duration": 5.96, "text": "x's with cosine"}, {"start": 81.92, "duration": 8.879, "text": "X's so the same way to factor"}, {"start": 85.96, "duration": 4.839, "text": "um to factor this expression"}, {"start": 91.079, "duration": 4.4, "text": "instead of just X we'll have cosine x +"}, {"start": 95.84, "duration": 6.919, "text": "4 and then we'll have cosine"}, {"start": 99.36, "duration": 6.32, "text": "x uh minus 3 and notice again if you"}, {"start": 102.759, "duration": 5.121, "text": "distribute all this stuff out um if you"}, {"start": 105.68, "duration": 4.96, "text": "do distribute out we'll get cosine x *"}, {"start": 107.88, "duration": 5.919, "text": "cosine X which will be cosine s we'll"}, {"start": 110.64, "duration": 6.92, "text": "get -3 cosine x + 4 cosine X which would"}, {"start": 113.799, "duration": 6.6, "text": "give us our positive cosine X pos4 and"}, {"start": 117.56, "duration": 4.479, "text": "-3 will leave us with our -12"}, {"start": 120.399, "duration": 3.64, "text": "so to me this looks like a nice"}, {"start": 122.039, "duration": 5.481, "text": "factorization for this original"}, {"start": 124.039, "duration": 5.92, "text": "expression so I don't see much else we"}, {"start": 127.52, "duration": 6.2, "text": "can do with it so um that to me looks"}, {"start": 129.959, "duration": 3.761, "text": "like a good place to stop"}], "title": "Factoring and Simplifying Trigonometric Expressions - Example 2", "description": "Thanks to all of you who support me on Patreon. You da real mvps!   $1 per month helps!! :)  https://www.patreon.com/patrickjmt !! Factoring and Simplifying Trigonometric Expressions - Example 2", "lengthSeconds": 132, "width": 480, "height": 360}
{"video_id": "6z6fzPXUbSQ", "transcript": [{"start": 0.719, "duration": 3.16, "text": "all right in this video we're going to"}, {"start": 1.88, "duration": 4.2, "text": "do an example of expressing a complex"}, {"start": 3.879, "duration": 5.081, "text": "number in trigonometric or Polar form"}, {"start": 6.08, "duration": 5.559, "text": "and the idea is okay we've got some"}, {"start": 8.96, "duration": 6.12, "text": "complex number Z we can write that as uh"}, {"start": 11.639, "duration": 8.321, "text": "X+ Yi if the real part is X and the"}, {"start": 15.08, "duration": 8.08, "text": "imaginary part is y so this uh this this"}, {"start": 19.96, "duration": 5.04, "text": "complex number uh we can make an angle"}, {"start": 23.16, "duration": 4.32, "text": "with the Positive x-axis and that'll"}, {"start": 25.0, "duration": 4.48, "text": "simply be our angle Theta if you think"}, {"start": 27.48, "duration": 4.119, "text": "about the distance out to that point the"}, {"start": 29.48, "duration": 4.04, "text": "distance to the point is the modulus but"}, {"start": 31.599, "duration": 3.521, "text": "we're going to call that R you can kind"}, {"start": 33.52, "duration": 5.039, "text": "of think about that as being a radius in"}, {"start": 35.12, "duration": 5.959, "text": "a sense well uh we could come up with"}, {"start": 38.559, "duration": 4.601, "text": "some relationships we could do cosine so"}, {"start": 41.079, "duration": 5.521, "text": "cosine would be cosine of theta would be"}, {"start": 43.16, "duration": 6.239, "text": "adjacent X over H the hypotenuse which"}, {"start": 46.6, "duration": 5.599, "text": "is R if you solve that for X we'll get x"}, {"start": 49.399, "duration": 6.0, "text": "= r cosine Theta the same way we could"}, {"start": 52.199, "duration": 6.241, "text": "solve for y and get yal R sin"}, {"start": 55.399, "duration": 5.84, "text": "Theta if you replace the X with r cosine"}, {"start": 58.44, "duration": 5.96, "text": "Theta and the Y with r sin Theta you can"}, {"start": 61.239, "duration": 6.801, "text": "Factor the r out and it basically says"}, {"start": 64.4, "duration": 8.16, "text": "we can rewrite our complex number Z as R"}, {"start": 68.04, "duration": 7.84, "text": "* the quantity cosine theta plus I sin"}, {"start": 72.56, "duration": 5.0, "text": "Theta so um let's try to make some sense"}, {"start": 75.88, "duration": 5.12, "text": "out of one of"}, {"start": 77.56, "duration": 6.599, "text": "these suppose we want to uh we're going"}, {"start": 81.0, "duration": 6.159, "text": "to convert this complex number 2 + 2i"}, {"start": 84.159, "duration": 4.64, "text": "into Polar form okay so the first thing"}, {"start": 87.159, "duration": 7.521, "text": "I'm going to do is just make a little"}, {"start": 88.799, "duration": 8.96, "text": "sketch here so over two up"}, {"start": 94.68, "duration": 10.2, "text": "two okay so there's"}, {"start": 97.759, "duration": 10.64, "text": "our there's our complex number Z = 2 + 2"}, {"start": 104.88, "duration": 4.64, "text": "I so now to go back to our formulas um"}, {"start": 108.399, "duration": 4.68, "text": "again the first thing I'm going to"}, {"start": 109.52, "duration": 5.239, "text": "figure out is R which is our modulus so"}, {"start": 113.079, "duration": 4.201, "text": "R is just going to be the"}, {"start": 114.759, "duration": 5.761, "text": "modulus and to get the modulus we just"}, {"start": 117.28, "duration": 6.04, "text": "take uh each part the real part and the"}, {"start": 120.52, "duration": 4.559, "text": "imaginary part and we Square them add"}, {"start": 123.32, "duration": 4.28, "text": "them together in square root so we'll"}, {"start": 125.079, "duration": 5.8, "text": "get 4 + 4 which is8 so we'll get the"}, {"start": 127.6, "duration": 6.24, "text": "square root of 8 uh we can rewrite that"}, {"start": 130.879, "duration": 6.201, "text": "though as the < TK of 4 * thek two which"}, {"start": 133.84, "duration": 4.56, "text": "will give us 2 < tk2 so now we've got"}, {"start": 137.08, "duration": 4.12, "text": "our r"}, {"start": 138.4, "duration": 6.72, "text": "value well we don't know our angle Theta"}, {"start": 141.2, "duration": 7.039, "text": "here just yet uh but we can figure that"}, {"start": 145.12, "duration": 5.72, "text": "out just using again a little bit of"}, {"start": 148.239, "duration": 6.401, "text": "trigonometry so the ad Adent side has"}, {"start": 150.84, "duration": 7.92, "text": "length two um the opposite side also has"}, {"start": 154.64, "duration": 6.76, "text": "a length of two if we do tangent OFA we"}, {"start": 158.76, "duration": 6.199, "text": "would get opposite over adjacent so 2"}, {"start": 161.4, "duration": 5.4, "text": "over two which is going to be one well"}, {"start": 164.959, "duration": 4.64, "text": "tangent of Pi"}, {"start": 166.8, "duration": 5.96, "text": "over4 or"}, {"start": 169.599, "duration": 4.881, "text": "45\u00b0 that'll give us one so now we know"}, {"start": 172.76, "duration": 4.119, "text": "our angle"}, {"start": 174.48, "duration": 4.679, "text": "Theta and now we're in business we can"}, {"start": 176.879, "duration": 7.481, "text": "rewrite our complex number it says we"}, {"start": 179.159, "duration": 9.121, "text": "write r R so the R value we said was 2 <"}, {"start": 184.36, "duration": 7.599, "text": "tk2 um and then we simply take cosine of"}, {"start": 188.28, "duration": 7.64, "text": "theta which is cosine of"}, {"start": 191.959, "duration": 8.441, "text": "45\u00b0 plus I * s"}, {"start": 195.92, "duration": 5.92, "text": "of 45 degrees and now uh this is what"}, {"start": 200.4, "duration": 3.8, "text": "we've got what we want this would be our"}, {"start": 201.84, "duration": 6.039, "text": "final solution"}, {"start": 204.2, "duration": 8.36, "text": "here okay we've now uh taken our complex"}, {"start": 207.879, "duration": 4.681, "text": "number and put it in polar form"}], "title": "Expressing a Complex Number in Trigonometric or Polar Form, Ex 1", "description": "Thanks to all of you who support me on Patreon. You da real mvps!   $1 per month helps!! :)  https://www.patreon.com/patrickjmt !! Expressing a Complex Number in Trigonometric or Polar Form, Ex 1.  In this video, I show how to write a complex number in polar form. Example 1.", "lengthSeconds": 211, "width": 480, "height": 360}
{"video_id": "6VbIv43gNto", "transcript": [{"start": 1.079, "duration": 3.441, "text": "okay in this example we have two"}, {"start": 2.48, "duration": 4.399, "text": "rectangles that are similar and we want"}, {"start": 4.52, "duration": 4.039, "text": "to find the value of x so our small"}, {"start": 6.879, "duration": 4.481, "text": "rectangle has a height of six and a"}, {"start": 8.559, "duration": 5.321, "text": "width of 10 and our larger rectangle has"}, {"start": 11.36, "duration": 3.84, "text": "a height of 21 and a width of X and the"}, {"start": 13.88, "duration": 4.84, "text": "way I'm going to do this is I'm going to"}, {"start": 15.2, "duration": 6.12, "text": "set up a proportion so what I'm going to"}, {"start": 18.72, "duration": 6.319, "text": "do is I'm going to set up my proportion"}, {"start": 21.32, "duration": 3.719, "text": "by looking at the small"}, {"start": 25.92, "duration": 6.04, "text": "length over the longer"}, {"start": 28.96, "duration": 3.0, "text": "length"}, {"start": 32.16, "duration": 5.88, "text": "and I'm going to make that equal to the"}, {"start": 35.04, "duration": 3.0, "text": "same thing the small"}, {"start": 38.52, "duration": 6.64, "text": "length over the long length and I'm"}, {"start": 42.68, "duration": 4.519, "text": "going to do this for each rectangle so"}, {"start": 45.16, "duration": 5.919, "text": "okay so for my smaller rectangle well"}, {"start": 47.199, "duration": 6.921, "text": "the small side small length is six the"}, {"start": 51.079, "duration": 5.521, "text": "long one is 10 so there's my"}, {"start": 54.12, "duration": 4.72, "text": "fraction my proportion and I'm going to"}, {"start": 56.6, "duration": 5.56, "text": "do the same thing with the other side so"}, {"start": 58.84, "duration": 5.599, "text": "the smaller length L is 21 and the"}, {"start": 62.16, "duration": 4.599, "text": "longer one is X and that's what we don't"}, {"start": 64.439, "duration": 5.921, "text": "know so we want to determine that value"}, {"start": 66.759, "duration": 5.601, "text": "X okay so now we've got an algebra"}, {"start": 70.36, "duration": 6.04, "text": "equation to work with and what I'm going"}, {"start": 72.36, "duration": 4.04, "text": "to do is I'm going to do cross"}, {"start": 76.439, "duration": 4.72, "text": "multiplication so what cross"}, {"start": 78.24, "duration": 2.919, "text": "multiplication"}, {"start": 81.52, "duration": 5.36, "text": "says it says if you have one fraction"}, {"start": 84.36, "duration": 4.92, "text": "we'll call it A over B equals another"}, {"start": 86.88, "duration": 5.68, "text": "fraction C over D it says you can turn"}, {"start": 89.28, "duration": 6.24, "text": "this into equivalent uh equation by"}, {"start": 92.56, "duration": 5.48, "text": "cross multiplying we'll take a *"}, {"start": 95.52, "duration": 5.599, "text": "D and then on the other side we'll do B"}, {"start": 98.04, "duration": 4.399, "text": "* C so that's why they call it cross"}, {"start": 101.119, "duration": 3.201, "text": "right it looks like a little cross and"}, {"start": 102.439, "duration": 3.72, "text": "you're multiplying so I'm going to do"}, {"start": 104.32, "duration": 4.88, "text": "the same thing I'm going to take six and"}, {"start": 106.159, "duration": 5.32, "text": "multiply that by X well that's just"}, {"start": 109.2, "duration": 4.48, "text": "going to give me 6X and then I'm going"}, {"start": 111.479, "duration": 5.32, "text": "to have 10 ultip"}, {"start": 113.68, "duration": 7.359, "text": "21 now you could go ahead and multiply"}, {"start": 116.799, "duration": 6.92, "text": "10 * 21 and get 210 I'm not going to do"}, {"start": 121.039, "duration": 5.0, "text": "that and the reason is is um it's to me"}, {"start": 123.719, "duration": 4.921, "text": "it's going to be easier just to go ahead"}, {"start": 126.039, "duration": 4.241, "text": "and simplify without multiplying so I'm"}, {"start": 128.64, "duration": 4.2, "text": "trying to solve for x since I'm"}, {"start": 130.28, "duration": 6.599, "text": "multiplying by six I'm going to start"}, {"start": 132.84, "duration": 8.24, "text": "off by dividing both sides by six well 6"}, {"start": 136.879, "duration": 7.521, "text": "/ 6 is just 1 x and now let's see we can"}, {"start": 141.08, "duration": 6.0, "text": "we can uh cancel out some things so I'm"}, {"start": 144.4, "duration": 6.08, "text": "going to write uh so first off I'm going"}, {"start": 147.08, "duration": 7.04, "text": "to write 6 as 2 * 3"}, {"start": 150.48, "duration": 9.32, "text": "notice we could write 10 as 2 * 5 that's"}, {"start": 154.12, "duration": 8.199, "text": "still 10 and 21 is 3 * 7 so what I'm"}, {"start": 159.8, "duration": 5.079, "text": "trying to do is cancel out common"}, {"start": 162.319, "duration": 5.28, "text": "factors and so notice we could cancel"}, {"start": 164.879, "duration": 5.481, "text": "out a two and a two we could cancel out"}, {"start": 167.599, "duration": 4.161, "text": "a three and a three now be careful it"}, {"start": 170.36, "duration": 4.239, "text": "looks like we've crossed everything out"}, {"start": 171.76, "duration": 4.96, "text": "in the denominator notice if we multiply"}, {"start": 174.599, "duration": 5.0, "text": "by one that's not going to change the"}, {"start": 176.72, "duration": 6.2, "text": "value right 2 * 3 * 1 is still the same"}, {"start": 179.599, "duration": 5.961, "text": "thing thing is 2 * 3 so there is still"}, {"start": 182.92, "duration": 7.399, "text": "an imaginary one there if you will so"}, {"start": 185.56, "duration": 10.8, "text": "what we're left with is 5 * 7 which is"}, {"start": 190.319, "duration": 8.761, "text": "35 over 1 well 35 over 1 is just 35 and"}, {"start": 196.36, "duration": 5.2, "text": "that's going to be our value for x and"}, {"start": 199.08, "duration": 5.879, "text": "again that seems uh you know intuitively"}, {"start": 201.56, "duration": 5.599, "text": "that seems kind of reasonable to me um"}, {"start": 204.959, "duration": 4.84, "text": "you know if it has a height of six and a"}, {"start": 207.159, "duration": 4.8, "text": "width of 10 well the width isn't quite"}, {"start": 209.799, "duration": 4.481, "text": "double right it isn't quite double if"}, {"start": 211.959, "duration": 5.241, "text": "you double six you would get 12 so if"}, {"start": 214.28, "duration": 4.64, "text": "you double 21 you would get I guess 42"}, {"start": 217.2, "duration": 3.28, "text": "so to me intuitively this length should"}, {"start": 218.92, "duration": 4.48, "text": "be something a little bit smaller than"}, {"start": 220.48, "duration": 4.92, "text": "42 and hey that's what we're finding so"}, {"start": 223.4, "duration": 5.0, "text": "just a quick little intuitive check as"}, {"start": 225.4, "duration": 3.0, "text": "well"}], "title": "GED 'Most Missed' Math Practice Problems , Code Q.3.b,  #3", "description": "Thanks to all of you who support me on Patreon. You da real mvps!   $1 per month helps!! :)  https://www.patreon.com/patrickjmt !! The GED Testing Service keeps a list of the most missed math topics and can be seen here: http://www.gedtestingservice.com/uploads/files/126de0283c94ff6323b3b5b3f310da5d.pdf\nThis video shows how to work a sample problem using their coding system that is shown on the left using the link above.", "lengthSeconds": 226, "width": 480, "height": 360}
{"video_id": "Ympm-AxmJ14", "transcript": [{"start": 0.64, "duration": 3.639, "text": "all right in this video we're going to"}, {"start": 1.839, "duration": 5.56, "text": "do a few examples related to finding 2x"}, {"start": 4.279, "duration": 4.561, "text": "two determinants of a matrix so the idea"}, {"start": 7.399, "duration": 3.641, "text": "is if your Matrix a if you have the"}, {"start": 8.84, "duration": 4.08, "text": "entries a b c d to calculate the"}, {"start": 11.04, "duration": 3.16, "text": "determinant of that Matrix we simply"}, {"start": 12.92, "duration": 4.16, "text": "sort of do like a little cross"}, {"start": 14.2, "duration": 6.48, "text": "multiplication we take a * D and then"}, {"start": 17.08, "duration": 6.24, "text": "subtract away the product of B * C if"}, {"start": 20.68, "duration": 5.04, "text": "this determinant equals zero we say that"}, {"start": 23.32, "duration": 3.76, "text": "Matrix a is a singular Matrix so just"}, {"start": 25.72, "duration": 3.36, "text": "some"}, {"start": 27.08, "duration": 4.039, "text": "terminology um so a couple questions"}, {"start": 29.08, "duration": 8.08, "text": "here true or suppose we want to know if"}, {"start": 31.119, "duration": 8.681, "text": "this Matrix -2 -10 4 -20 um is that a"}, {"start": 37.16, "duration": 4.64, "text": "singular Matrix or not well again all we"}, {"start": 39.8, "duration": 4.68, "text": "have to do is just simply calculate the"}, {"start": 41.8, "duration": 8.079, "text": "determinant so the determinant of a here"}, {"start": 44.48, "duration": 8.68, "text": "will take -2 * -20 we subtract away the"}, {"start": 49.879, "duration": 8.041, "text": "product of 4 *"}, {"start": 53.16, "duration": 7.399, "text": "-10 well -2 * -20 is pos4 it looks like"}, {"start": 57.92, "duration": 5.119, "text": "we have a -4 and a -10 which will be"}, {"start": 60.559, "duration": 5.321, "text": "another positive 40 so since we're"}, {"start": 63.039, "duration": 5.041, "text": "getting 80 out for the determinant and"}, {"start": 65.88, "duration": 6.16, "text": "80 well that's certainly not equal to"}, {"start": 68.08, "duration": 5.399, "text": "zero this Matrix is not singular so by"}, {"start": 72.04, "duration": 5.36, "text": "saying it's singular we would say that's"}, {"start": 73.479, "duration": 6.081, "text": "a false statement okay definitely not"}, {"start": 77.4, "duration": 4.32, "text": "singular suppose we want to calculate a"}, {"start": 79.56, "duration": 6.36, "text": "determinant here generically we've got a"}, {"start": 81.72, "duration": 7.12, "text": "= 4X -3 and Y well again to calculate"}, {"start": 85.92, "duration": 4.64, "text": "the determinant of this we just sort of"}, {"start": 88.84, "duration": 3.279, "text": "cross multiply here here so I'm going to"}, {"start": 90.56, "duration": 5.28, "text": "get 4 *"}, {"start": 92.119, "duration": 6.721, "text": "Yus I'm going to take -3 *"}, {"start": 95.84, "duration": 6.559, "text": "X well we can simplify that and say it's"}, {"start": 98.84, "duration": 6.639, "text": "4 y uh +"}, {"start": 102.399, "duration": 4.961, "text": "3x one last Quick one here suppose we"}, {"start": 105.479, "duration": 7.361, "text": "know that the value of the determinant"}, {"start": 107.36, "duration": 8.759, "text": "of this Matrix is 24 um we've got -2 6 W"}, {"start": 112.84, "duration": 5.36, "text": "-2 6 3 and W if we know this determinant"}, {"start": 116.119, "duration": 4.801, "text": "of this Matrix is 24 we want to figure"}, {"start": 118.2, "duration": 4.959, "text": "out the value for w"}, {"start": 120.92, "duration": 3.64, "text": "well again to calculate the determinant"}, {"start": 123.159, "duration": 5.401, "text": "we'll take"}, {"start": 124.56, "duration": 8.319, "text": "-2 * W and then we'll subtract away the"}, {"start": 128.56, "duration": 7.0, "text": "product of 3 * 6 and again we know that"}, {"start": 132.879, "duration": 4.281, "text": "this determinant has to equal 24 so now"}, {"start": 135.56, "duration": 7.8, "text": "we just got a nice little equation we"}, {"start": 137.16, "duration": 9.76, "text": "can solve this is -2 W - 18 = 24 well we"}, {"start": 143.36, "duration": 6.56, "text": "can add 18 to both sides if we add 18"}, {"start": 146.92, "duration": 5.76, "text": "we'll get postive 42 on the right if we"}, {"start": 149.92, "duration": 6.44, "text": "divide both sides by -2 will be left"}, {"start": 152.68, "duration": 6.44, "text": "with W = -21 so we would basically have"}, {"start": 156.36, "duration": 5.2, "text": "to have a -21 hanging out in the bottom"}, {"start": 159.12, "duration": 6.6, "text": "right corner of our Matrix so that this"}, {"start": 161.56, "duration": 4.16, "text": "determinant equals 24"}], "title": "Determinant of a 2 x 2 Matrix - A Few Basic Questions", "description": "Thanks to all of you who support me on Patreon. You da real mvps!   $1 per month helps!! :)  https://www.patreon.com/patrickjmt !! Determinant of a 2 x 2 Matrix - A Few Basic Questions. In this video, I show how to find the determinant of a 2 x 2 matrix, and do a few related problems.\r\nFor more free math videos, visit http://PatrickJMT.com", "lengthSeconds": 164, "width": 480, "height": 360}
{"video_id": "QxboPshf__g", "transcript": [{"start": 0.64, "duration": 3.16, "text": "okay in this video we'll do one more"}, {"start": 1.959, "duration": 3.721, "text": "example of multiplying and dividing"}, {"start": 3.8, "duration": 4.28, "text": "complex numbers that are given to us in"}, {"start": 5.68, "duration": 3.839, "text": "polar form and actually not quite"}, {"start": 8.08, "duration": 3.8, "text": "because the bottom one isn't going to be"}, {"start": 9.519, "duration": 4.08, "text": "given to us in polar form uh well the"}, {"start": 11.88, "duration": 5.52, "text": "numerator of the bottom problem so we've"}, {"start": 13.599, "duration": 7.881, "text": "got 3 + 3 I over cosine of 90 Dees plus"}, {"start": 17.4, "duration": 7.24, "text": "I sin of 90\u00b0 so recall uh we can write a"}, {"start": 21.48, "duration": 6.68, "text": "number in polar form we can write zal R"}, {"start": 24.64, "duration": 6.24, "text": "cosine of theta plus I sin Theta um to"}, {"start": 28.16, "duration": 4.64, "text": "get R remember we just take the real"}, {"start": 30.88, "duration": 4.04, "text": "part and square it and add to that the"}, {"start": 32.8, "duration": 5.4, "text": "IM imaginary part squared and then we"}, {"start": 34.92, "duration": 5.479, "text": "square root that so for 3 + 3 I went"}, {"start": 38.2, "duration": 4.76, "text": "ahead and uh graphed that real quick so"}, {"start": 40.399, "duration": 5.32, "text": "there's a our 3 + 3"}, {"start": 42.96, "duration": 4.4, "text": "I well to get our r value we would"}, {"start": 45.719, "duration": 6.201, "text": "simply take 3^"}, {"start": 47.36, "duration": 11.039, "text": "2ar + 3 2ar well that's going to be 9 +"}, {"start": 51.92, "duration": 8.279, "text": "9 or 18 we can always write 18 as 9 * 2"}, {"start": 58.399, "duration": 5.041, "text": "we can pull the square root of 9 out is"}, {"start": 60.199, "duration": 6.481, "text": "a three so we would be left with 3 < tk2"}, {"start": 63.44, "duration": 4.88, "text": "for our value for R to figure out our"}, {"start": 66.68, "duration": 3.96, "text": "value for Theta"}, {"start": 68.32, "duration": 5.36, "text": "here I just think about making a little"}, {"start": 70.64, "duration": 5.72, "text": "right triangle well the adjacent side is"}, {"start": 73.68, "duration": 5.24, "text": "length three the height is also uh"}, {"start": 76.36, "duration": 4.36, "text": "length three the opposite side so it"}, {"start": 78.92, "duration": 5.68, "text": "says tangent of theta would is be the"}, {"start": 80.72, "duration": 6.0, "text": "opposite over the adjacent or 3 over one"}, {"start": 84.6, "duration": 6.36, "text": "Excuse Me 3 over three which gives us"}, {"start": 86.72, "duration": 7.679, "text": "one and tangent of 45 degrees is going"}, {"start": 90.96, "duration": 5.96, "text": "to equal one so let's see um I think we"}, {"start": 94.399, "duration": 4.72, "text": "can rewrite the numerator now it says"}, {"start": 96.92, "duration": 5.64, "text": "this would be the same thing as our r"}, {"start": 99.119, "duration": 5.761, "text": "value again which we said is 3 < tk2 so"}, {"start": 102.56, "duration": 4.879, "text": "we' have 3 < tk2"}, {"start": 104.88, "duration": 3.44, "text": "time cosine of the angle which we said"}, {"start": 107.439, "duration": 3.68, "text": "is"}, {"start": 108.32, "duration": 4.839, "text": "45\u00b0 so cosine of"}, {"start": 111.119, "duration": 7.401, "text": "45\u00b0 plus"}, {"start": 113.159, "duration": 5.361, "text": "I uh times s of 45\u00b0"}, {"start": 121.64, "duration": 6.8, "text": "okay that's all over uh we can even"}, {"start": 123.92, "duration": 8.6, "text": "write this as 1 time cosine of"}, {"start": 128.44, "duration": 7.2, "text": "90\u00b0 plus I of"}, {"start": 132.52, "duration": 5.84, "text": "90\u00b0 but now we've seen uh from the other"}, {"start": 135.64, "duration": 5.76, "text": "examples all we do again uh when we're"}, {"start": 138.36, "duration": 6.04, "text": "dividing we simply divide the R1 by the"}, {"start": 141.4, "duration": 5.4, "text": "R2 and then we subtract the"}, {"start": 144.4, "duration": 4.839, "text": "angles so in this case if we take 3 <"}, {"start": 146.8, "duration": 5.48, "text": "tk2 over 1 we'll just be left with 3 *"}, {"start": 149.239, "duration": 4.921, "text": "the root of two and then we would just"}, {"start": 152.28, "duration": 4.679, "text": "have cosine of"}, {"start": 154.16, "duration": 7.88, "text": "45\u00b0 -"}, {"start": 156.959, "duration": 8.64, "text": "90\u00b0 + I * s of"}, {"start": 162.04, "duration": 3.559, "text": "45\u00b0 -"}, {"start": 165.92, "duration": 4.56, "text": "90\u00b0 and uh at that point we've basically"}, {"start": 168.76, "duration": 5.559, "text": "got our solution so we can always clean"}, {"start": 170.48, "duration": 7.16, "text": "it up a little bit more 3 < tk2 this"}, {"start": 174.319, "duration": 8.601, "text": "will be cosine of"}, {"start": 177.64, "duration": 7.959, "text": "-45\u00b0 plus I s of"}, {"start": 182.92, "duration": 4.879, "text": "-45\u00b0 and I think I would stop leave it"}, {"start": 185.599, "duration": 4.681, "text": "there need one more set of parentheses"}, {"start": 187.799, "duration": 5.921, "text": "now I'll leave it there um and now we've"}, {"start": 190.28, "duration": 8.319, "text": "got uh the polar form for our our new"}, {"start": 193.72, "duration": 4.879, "text": "number after we perform the division"}], "title": "Complex Numbers: Multiplying and Dividing in Polar Form, Ex 2", "description": "Thanks to all of you who support me on Patreon. You da real mvps!   $1 per month helps!! :)  https://www.patreon.com/patrickjmt !! Complex Numbers: Multiplying and Dividing in Polar Form, Ex 2. In this video, I give the formula for multiplication and division of two complex numbers that are in polar form.", "lengthSeconds": 197, "width": 480, "height": 360}
{"video_id": "7__EoR-7D3o", "transcript": [{"start": 0.56, "duration": 3.32, "text": "all right so just another example of"}, {"start": 2.159, "duration": 4.76, "text": "estimating the sum of an alternating"}, {"start": 3.88, "duration": 5.759, "text": "series so here suppose we uh estimate"}, {"start": 6.919, "duration": 6.161, "text": "the the series Nega 1 the n + 1 over 3"}, {"start": 9.639, "duration": 8.4, "text": "to the N by using uh this this expansion"}, {"start": 13.08, "duration": 9.359, "text": "so notice this is n = 1 2 3 4 five so in"}, {"start": 18.039, "duration": 6.4, "text": "this case the air is at most uh the size"}, {"start": 22.439, "duration": 4.961, "text": "of the absolute value of the first"}, {"start": 24.439, "duration": 6.08, "text": "neglected term well again if we filled"}, {"start": 27.4, "duration": 7.76, "text": "in the next term um you know so any = 1"}, {"start": 30.519, "duration": 7.921, "text": "2 3 4 5 if we plug in Nal 6 into our"}, {"start": 35.16, "duration": 5.2, "text": "into our formula we would get well let's"}, {"start": 38.44, "duration": 5.959, "text": "see negative - 1 to the 7th that's just"}, {"start": 40.36, "duration": 8.56, "text": "Nega 1 then we would get 3 to the 6th so"}, {"start": 44.399, "duration": 7.401, "text": "3 raised to the 6 power uh"}, {"start": 48.92, "duration": 4.56, "text": "729 and let's see if we take one and"}, {"start": 51.8, "duration": 5.04, "text": "divide that by"}, {"start": 53.48, "duration": 4.96, "text": "729 I'm getting"}, {"start": 56.84, "duration": 5.64, "text": "0.1"}, {"start": 58.44, "duration": 9.24, "text": "37 1 74 when I throw that into a"}, {"start": 62.48, "duration": 5.2, "text": "calculator so it says our error is at"}, {"start": 69.24, "duration": 5.04, "text": "most the value 0.001"}, {"start": 74.96, "duration": 3.0, "text": "37174"}], "title": "Alternating Series - Error Estimation #2", "description": "Thanks to all of you who support me on Patreon. You da real mvps!   $1 per month helps!! :)  https://www.patreon.com/patrickjmt !! Alternating Series - Error Estimation #2.  In this video, I sum up the first few terms of an alternating series and then I find the maximum error involved.", "lengthSeconds": 76, "width": 480, "height": 360}
{"video_id": "_R0NEXjoqvE", "transcript": [{"start": 0.52, "duration": 3.6, "text": "all right in this video I'm going to do"}, {"start": 1.8, "duration": 4.84, "text": "another example of finding a polom that"}, {"start": 4.12, "duration": 4.68, "text": "satisfies certain conditions so in this"}, {"start": 6.64, "duration": 3.76, "text": "case we want a polinomial f ofx of"}, {"start": 8.8, "duration": 5.719, "text": "degree"}, {"start": 10.4, "duration": 7.2, "text": "3 okay it's got zeros at 2 and -2 so"}, {"start": 14.519, "duration": 6.401, "text": "that means I know that xus 2 is going to"}, {"start": 17.6, "duration": 7.72, "text": "be a factor and x + 2 is also going to"}, {"start": 20.92, "duration": 6.88, "text": "be a factor the leading coefficient is"}, {"start": 25.32, "duration": 4.76, "text": "pos1 we have this extra bit of"}, {"start": 27.8, "duration": 5.24, "text": "information that if you plug -4 in we"}, {"start": 30.08, "duration": 5.12, "text": "get 30 out right now if we multiply this"}, {"start": 33.04, "duration": 5.16, "text": "out this would only be degree 2 so that"}, {"start": 35.2, "duration": 8.16, "text": "means there's another factor in there um"}, {"start": 38.2, "duration": 7.08, "text": "let's call it xus C so this the C value"}, {"start": 43.36, "duration": 4.519, "text": "that's the number if I knew that I would"}, {"start": 45.28, "duration": 4.959, "text": "basically have my polom so I have to"}, {"start": 47.879, "duration": 4.84, "text": "work to figure out this value C and"}, {"start": 50.239, "duration": 4.921, "text": "again I'm just going to"}, {"start": 52.719, "duration": 5.721, "text": "use um this condition it says if you"}, {"start": 55.16, "duration": 6.079, "text": "plug -4 in we get 30 out so I'm going to"}, {"start": 58.44, "duration": 4.119, "text": "put 30 on the left side"}, {"start": 61.239, "duration": 4.64, "text": "and then everywhere there's an X I'm"}, {"start": 62.559, "duration": 3.32, "text": "just going to plug in"}, {"start": 68.96, "duration": 4.64, "text": "-4 and then I'm going to do a bit of"}, {"start": 70.96, "duration": 4.72, "text": "arithmetic again to solve for my value C"}, {"start": 73.6, "duration": 4.28, "text": "and then I'll I'll know my"}, {"start": 75.68, "duration": 4.079, "text": "polom so let's see I'm going to get rid"}, {"start": 77.88, "duration": 7.04, "text": "of the one I mean one times anything"}, {"start": 79.759, "duration": 8.241, "text": "doesn't change it so -4 and -2 is -6 -4"}, {"start": 84.92, "duration": 7.92, "text": "+ 2 is going to give us a -2 we still"}, {"start": 88.0, "duration": 9.96, "text": "have our -4 - C left over so let's see"}, {"start": 92.84, "duration": 9.16, "text": "this is 30 -6 and -2 is pos2 we still"}, {"start": 97.96, "duration": 5.68, "text": "have4 minus c um a couple different ways"}, {"start": 102.0, "duration": 4.079, "text": "we could go about doing it here I'm"}, {"start": 103.64, "duration": 4.36, "text": "going to just divide both sides by 12"}, {"start": 106.079, "duration": 4.64, "text": "you could distribute out first it'll all"}, {"start": 108.0, "duration": 5.799, "text": "get you to the same place so if we"}, {"start": 110.719, "duration": 6.201, "text": "divide by 12 um well the TW will cancel"}, {"start": 113.799, "duration": 4.841, "text": "on the right side 30 over 12 what's the"}, {"start": 116.92, "duration": 4.4, "text": "biggest number that goes into both of"}, {"start": 118.64, "duration": 6.68, "text": "those um I think think six will go into"}, {"start": 121.32, "duration": 8.04, "text": "both of those so 30 / 6 would be 5 30"}, {"start": 125.32, "duration": 5.88, "text": "div excuse me 12 / 6 would be 2 on the"}, {"start": 129.36, "duration": 5.879, "text": "right side we would simply be left with"}, {"start": 131.2, "duration": 6.48, "text": "this value -4 minus"}, {"start": 135.239, "duration": 4.441, "text": "C well okay I'm going to add four to"}, {"start": 137.68, "duration": 7.76, "text": "both sides so that'll give me negative"}, {"start": 139.68, "duration": 8.839, "text": "excuse me 5 Hales + 4 over 1 = C we can"}, {"start": 145.44, "duration": 6.519, "text": "multiply top and bottom by two so"}, {"start": 148.519, "duration": 6.321, "text": "that'll give us 8 + 5 which is 13 / 2"}, {"start": 151.959, "duration": 6.961, "text": "and that's going to give us C well if we"}, {"start": 154.84, "duration": 6.52, "text": "multiply both sides by 1 it says -3/2 is"}, {"start": 158.92, "duration": 4.56, "text": "going to be our C value and now we know"}, {"start": 161.36, "duration": 5.239, "text": "our polinomial I can basically just go"}, {"start": 163.48, "duration": 10.16, "text": "back up to my f ofx and fill it in so it"}, {"start": 166.599, "duration": 12.961, "text": "says the polom would be x - 2 * x + 2 x"}, {"start": 173.64, "duration": 8.8, "text": "- -3 would give us x + 13 and now we've"}, {"start": 179.56, "duration": 6.399, "text": "got our polom uh that satisfies the"}, {"start": 182.44, "duration": 3.519, "text": "given conditions"}], "title": "Finding the Formula for a Polynomial Given:  Zeros/Roots, Degree, and One Point - Example 2", "description": "Thanks to all of you who support me on Patreon. You da real mvps!   $1 per month helps!! :)  https://www.patreon.com/patrickjmt !! Finding the Formula for a Polynomial Given:  Zeros/Roots, Degree, and One Point - Example 2.  If you know the roots of a polynomial, its degree and one point that the polynomial goes through, you can sometimes find the equation of the polynomial.  In this video, I show how!", "lengthSeconds": 184, "width": 480, "height": 360}
{"video_id": "KwLWAyDVAV4", "transcript": [{"start": 1.079, "duration": 4.24, "text": "okay in this video we'll do one more"}, {"start": 2.56, "duration": 4.44, "text": "example of solving a linear equation so"}, {"start": 5.319, "duration": 4.161, "text": "again the first thing I want to do is I"}, {"start": 7.0, "duration": 3.759, "text": "tend to work sort of inside out so I'm"}, {"start": 9.48, "duration": 3.76, "text": "going to try to clean up the stuff"}, {"start": 10.759, "duration": 3.88, "text": "inside the brackets um basically try to"}, {"start": 13.24, "duration": 4.039, "text": "get rid of all the parentheses and all"}, {"start": 14.639, "duration": 5.281, "text": "the brackets so our negative is out"}, {"start": 17.279, "duration": 4.361, "text": "front our six will leave alone you could"}, {"start": 19.92, "duration": 6.4, "text": "think about this as being negative 1 so"}, {"start": 21.64, "duration": 8.68, "text": "if we distribute 1 * positive 7 is -7 -1"}, {"start": 26.32, "duration": 5.759, "text": "* POS X ISX"}, {"start": 30.32, "duration": 6.12, "text": "on the right side if we distribute our"}, {"start": 32.079, "duration": 8.361, "text": "positive 3 3 * 2x is going to give us 6X"}, {"start": 36.44, "duration": 6.799, "text": "3 * -5 will give us"}, {"start": 40.44, "duration": 6.799, "text": "-5 um I'm going to keep simplifying"}, {"start": 43.239, "duration": 6.121, "text": "inside the brackets so 6 - 7 that's ne1"}, {"start": 47.239, "duration": 3.761, "text": "and then we still have our negx and"}, {"start": 49.36, "duration": 3.839, "text": "again brackets are the same thing as"}, {"start": 51.0, "duration": 5.239, "text": "parentheses it's just sort of a you know"}, {"start": 53.199, "duration": 4.88, "text": "a way to help us visually break it up um"}, {"start": 56.239, "duration": 5.721, "text": "if you have 80 sets of parentheses it"}, {"start": 58.079, "duration": 6.561, "text": "can be confusing to look at 10 - 15 is"}, {"start": 61.96, "duration": 5.159, "text": "going to give us -5 so I'm combining my"}, {"start": 64.64, "duration": 6.32, "text": "like terms on the right side we still"}, {"start": 67.119, "duration": 5.241, "text": "have our positive 6X left over if I"}, {"start": 70.96, "duration": 5.839, "text": "distribute again you can think about"}, {"start": 72.36, "duration": 10.0, "text": "this as being Nega - 1 so -1 * 1 is pos1"}, {"start": 76.799, "duration": 8.161, "text": "-1 * X is postive X we've got --5 + 6x"}, {"start": 82.36, "duration": 4.799, "text": "still on the right side and again I'm"}, {"start": 84.96, "duration": 4.36, "text": "just going to try to put um all my"}, {"start": 87.159, "duration": 3.92, "text": "numbers on one side all my variables on"}, {"start": 89.32, "duration": 4.2, "text": "the other"}, {"start": 91.079, "duration": 6.04, "text": "I'm going to start by subtracting"}, {"start": 93.52, "duration": 7.72, "text": "X from both sides that'll leave us with"}, {"start": 97.119, "duration": 7.64, "text": "positive one on the left we've got -5"}, {"start": 101.24, "duration": 6.839, "text": "plus okay so 6x - x is going to be"}, {"start": 104.759, "duration": 7.481, "text": "5x and last but not least to move the"}, {"start": 108.079, "duration": 8.72, "text": "neg five over I'll have to add five to"}, {"start": 112.24, "duration": 7.76, "text": "both sides and 1 + 5 is going to give us"}, {"start": 116.799, "duration": 5.721, "text": "six we still just have our 5x left over"}, {"start": 120.0, "duration": 4.56, "text": "on the right side and since we're"}, {"start": 122.52, "duration": 5.08, "text": "multiplying by five to get rid of that"}, {"start": 124.56, "duration": 7.92, "text": "we'll divide both sides by five and then"}, {"start": 127.6, "duration": 8.76, "text": "we'll get X = 65s as our solution to the"}, {"start": 132.48, "duration": 3.88, "text": "original equation"}], "title": "Solving a Basic Linear Equation - Example 3", "description": "Thanks to all of you who support me on Patreon. You da real mvps!   $1 per month helps!! :)  https://www.patreon.com/patrickjmt !! Solving a Basic Linear Equation - Example 3", "lengthSeconds": 135, "width": 480, "height": 360}
{"video_id": "ENzhUj1kxo8", "transcript": [{"start": 0.42, "duration": 2.08, "text": "Okay, here we have the question"}, {"start": 2.64, "duration": 2.4, "text": "it says, "If a divided by b equals two,"}, {"start": 5.18, "duration": 3.04, "text": "what is the value of four times b divided by a?""}, {"start": 8.22, "duration": 2.24, "text": "So we have the answer choices  zero, one, two, and four."}, {"start": 10.46, "duration": 3.16, "text": "Now, we can certainly do some algebraic manipulation,"}, {"start": 13.62, "duration": 5.0, "text": "and, you know, solve for a or b, and then substitute it in here, and do stuff like that."}, {"start": 18.62, "duration": 3.26, "text": "Don't do that. I definitely wouldn't do that on this question."}, {"start": 21.88, "duration": 4.4, "text": "I would just pick two that-I would just pick a value of a and b, so that when I divide I get two,"}, {"start": 26.28, "duration": 1.56, "text": "and then I would plug them in here."}, {"start": 28.74, "duration": 0.8, "text": "So, notice,"}, {"start": 32.9, "duration": 4.24, "text": "You know, if, well a equals two, and b equals one."}, {"start": 37.8, "duration": 4.08, "text": "Right, we would have a divided by b, that would be two over one..."}, {"start": 42.18, "duration": 1.12, "text": "that equals two."}, {"start": 43.9, "duration": 6.7, "text": "Okay, so these values of a and b satisfy this original, you know, requirement."}, {"start": 50.74, "duration": 3.44, "text": "Just substitute those values now into four b over a."}, {"start": 55.24, "duration": 4.26, "text": "So, four times b over a, that would equal four times,"}, {"start": 59.5, "duration": 1.98, "text": "Again, we said b is equal to one,"}, {"start": 62.48, "duration": 1.12, "text": "a is equal to two,"}, {"start": 63.96, "duration": 1.84, "text": "that would be four over two,"}, {"start": 65.8, "duration": 3.4, "text": "which would give you two, and boom, there's your answer."}, {"start": 69.2, "duration": 1.14, "text": "So again super quick,"}, {"start": 71.18, "duration": 1.84, "text": "nothing tricky in that one,"}, {"start": 73.02, "duration": 4.58, "text": "Again, you know, you can certainly do some algebraic manipulation to come up with this solution."}, {"start": 77.6, "duration": 1.92, "text": "But again, to me that would be overkill,"}, {"start": 79.52, "duration": 5.5, "text": "It would be so much faster just to pick some values that work, and you've done this, you know,"}, {"start": 85.02, "duration": 2.6, "text": "in probably less than ten seconds, and you're off to the next one."}], "title": "SAT Math - Question 8, 2016 Edition and Beyond", "description": "Thanks to all of you who support me on Patreon. You da real mvps!   $1 per month helps!! :)  https://www.patreon.com/patrickjmt !! SAT Math - Question 8, 2016 Edition and Beyond", "lengthSeconds": 89, "width": 480, "height": 360}
{"video_id": "TZAmpPZ4DH4", "transcript": [{"start": 0.56, "duration": 3.159, "text": "all right in this video we're going to"}, {"start": 1.8, "duration": 5.519, "text": "start looking at some examples related"}, {"start": 3.719, "duration": 5.721, "text": "to the integral test so in part A here U"}, {"start": 7.319, "duration": 5.32, "text": "we've got the series from Nal 1 to"}, {"start": 9.44, "duration": 4.56, "text": "Infinity of 1 over 3 to the n and again"}, {"start": 12.639, "duration": 6.48, "text": "we're just going to figure out whether"}, {"start": 14.0, "duration": 6.88, "text": "these series converge or diverge so um"}, {"start": 19.119, "duration": 3.841, "text": "certainly different ways to do this one"}, {"start": 20.88, "duration": 5.319, "text": "other than the integral test but uh"}, {"start": 22.96, "duration": 5.36, "text": "again just so we can talk about it using"}, {"start": 26.199, "duration": 4.801, "text": "the integral test so if you think about"}, {"start": 28.32, "duration": 7.52, "text": "the function 1 over 3 X that's the same"}, {"start": 31.0, "duration": 9.0, "text": "thing as well 13 all raised to the X and"}, {"start": 35.84, "duration": 4.16, "text": "recall an exponential function"}, {"start": 41.879, "duration": 7.68, "text": "um y = 1 3 to X would look uh you know"}, {"start": 47.0, "duration": 4.399, "text": "like our uh the graph I have here so"}, {"start": 49.559, "duration": 4.52, "text": "certainly we can use the integral test"}, {"start": 51.399, "duration": 5.48, "text": "on this because it's continuous it's"}, {"start": 54.079, "duration": 5.081, "text": "positive and it is decreasing so this is"}, {"start": 56.879, "duration": 3.561, "text": "certainly a a problem that we can use"}, {"start": 59.16, "duration": 3.719, "text": "the integral test"}, {"start": 60.44, "duration": 4.719, "text": "on so we've got the series from 1 to"}, {"start": 62.879, "duration": 3.761, "text": "Infinity of 1 over 3 to the N uh"}, {"start": 65.159, "duration": 3.721, "text": "correspondingly I'm going to think about"}, {"start": 66.64, "duration": 5.96, "text": "the integral from 1 to Infinity of 1"}, {"start": 68.88, "duration": 5.96, "text": "over 3 to X DX and then I'm going to uh"}, {"start": 72.6, "duration": 5.36, "text": "decide whether this improper integral"}, {"start": 74.84, "duration": 5.2, "text": "converges or diverges so I'm going to"}, {"start": 77.96, "duration": 5.76, "text": "replace my Infinity with a t we'll do"}, {"start": 80.04, "duration": 4.96, "text": "the limit as T goes to Infinity um the 3"}, {"start": 83.72, "duration": 4.759, "text": "to the X I'm going to bring that"}, {"start": 85.0, "duration": 6.159, "text": "upstairs and make it 3 to"}, {"start": 88.479, "duration": 4.481, "text": "thex so when when we integrate this"}, {"start": 91.159, "duration": 5.24, "text": "again you could just do a u substitution"}, {"start": 92.96, "duration": 3.439, "text": "let U equal"}, {"start": 96.72, "duration": 5.64, "text": "x after we do our U substitution we'll"}, {"start": 100.439, "duration": 4.28, "text": "get 3 to"}, {"start": 102.36, "duration": 3.719, "text": "thex um we'll have to divide by the"}, {"start": 104.719, "duration": 3.921, "text": "coefficient we'll end up getting a"}, {"start": 106.079, "duration": 4.4, "text": "negative one and recall for exponentials"}, {"start": 108.64, "duration": 4.92, "text": "you also have to divide by the natural"}, {"start": 110.479, "duration": 7.0, "text": "logarithm of the"}, {"start": 113.56, "duration": 3.919, "text": "base so from one to"}, {"start": 117.68, "duration": 5.64, "text": "T so here we've got the limit as T goes"}, {"start": 120.719, "duration": 5.521, "text": "to Infinity um so let's see I'm just"}, {"start": 123.32, "duration": 5.96, "text": "going to pull the negative out front so"}, {"start": 126.24, "duration": 5.84, "text": "there's my minus sign um let's see we"}, {"start": 129.28, "duration": 5.4, "text": "would have one over so we would have 3"}, {"start": 132.08, "duration": 5.239, "text": "to the X but that would turn into a t"}, {"start": 134.68, "duration": 5.4, "text": "times the natural logarithm of three"}, {"start": 137.319, "duration": 5.64, "text": "minus we would have one over so we would"}, {"start": 140.08, "duration": 6.6, "text": "have 3 to the 1 when we put it uh back"}, {"start": 142.959, "duration": 5.881, "text": "in the denominator 3 thex again times Ln"}, {"start": 146.68, "duration": 4.559, "text": "of"}, {"start": 148.84, "duration": 5.72, "text": "3 the only thing I'm really interested"}, {"start": 151.239, "duration": 5.481, "text": "in is whether this is a convergent or"}, {"start": 154.56, "duration": 5.679, "text": "Divergent improper"}, {"start": 156.72, "duration": 6.519, "text": "integral well you know 1 over 3 * Ln of"}, {"start": 160.239, "duration": 5.36, "text": "3 this is definitely some finite number"}, {"start": 163.239, "duration": 3.841, "text": "so no problem there uh the only thing"}, {"start": 165.599, "duration": 5.28, "text": "I'm thinking about is what happens with"}, {"start": 167.08, "duration": 6.96, "text": "the limit well as T goes to Infinity 3"}, {"start": 170.879, "duration": 4.761, "text": "to the T would get very very very large"}, {"start": 174.04, "duration": 3.559, "text": "so then we would have one over a very"}, {"start": 175.64, "duration": 5.72, "text": "very big number well one over a big"}, {"start": 177.599, "duration": 6.881, "text": "number goes to zero so we've got 0 minus"}, {"start": 181.36, "duration": 4.92, "text": "some finite number this uh improper"}, {"start": 184.48, "duration": 4.479, "text": "integral is definitely going to equal"}, {"start": 186.28, "duration": 4.959, "text": "some finite number and that tells us"}, {"start": 188.959, "duration": 2.28, "text": "it's"}, {"start": 191.44, "duration": 5.439, "text": "convergent and then so also since the"}, {"start": 194.08, "duration": 6.799, "text": "improper integral is convergent we can"}, {"start": 196.879, "duration": 6.401, "text": "also immediately conclude that uh the"}, {"start": 200.879, "duration": 5.841, "text": "series from 1 to Infinity of one 1 over"}, {"start": 203.28, "duration": 3.44, "text": "3 to the N is also"}, {"start": 206.799, "duration": 3.8, "text": "convergent again you know probably the"}, {"start": 208.76, "duration": 4.119, "text": "easiest way to adjust this one just to"}, {"start": 210.599, "duration": 3.881, "text": "point it out real quick in case uh maybe"}, {"start": 212.879, "duration": 4.28, "text": "you didn't think about it again this is"}, {"start": 214.48, "duration": 4.64, "text": "just a geometric series as well 1 over3"}, {"start": 217.159, "duration": 4.481, "text": "raised to the N so certainly you can"}, {"start": 219.12, "duration": 4.08, "text": "recognize it as being a geometric series"}, {"start": 221.64, "duration": 3.64, "text": "um which is what I would do and say it"}, {"start": 223.2, "duration": 4.2, "text": "converges because the r values between"}, {"start": 225.28, "duration": 4.08, "text": "negative 1 and one um and then I would"}, {"start": 227.4, "duration": 3.559, "text": "be off to the next problem but again"}, {"start": 229.36, "duration": 7.36, "text": "just to talk about it just to illustrate"}, {"start": 230.959, "duration": 5.761, "text": "the use of the uh the integral test here"}], "title": "Integral Test to Evaluate Series, Ex 1", "description": "Thanks to all of you who support me on Patreon. You da real mvps!   $1 per month helps!! :)  https://www.patreon.com/patrickjmt !! Integral Test to Evaluate Series, Ex 1.  In this video, I show how to evaluate a series by using the integral test.", "lengthSeconds": 235, "width": 480, "height": 360}
{"video_id": "w81y25anEOM", "transcript": [{"start": 0.03, "duration": 3.359, "text": "all right in this video I want to talk"}, {"start": 1.709, "duration": 3.3, "text": "about satisfying the domain of a"}, {"start": 3.389, "duration": 4.831, "text": "function or equivalently we're going to"}, {"start": 5.009, "duration": 5.071, "text": "find the domain typically anytime I see"}, {"start": 8.22, "duration": 4.169, "text": "a domain question there's two big things"}, {"start": 10.08, "duration": 4.469, "text": "I think I have to avoid these two big"}, {"start": 12.389, "duration": 4.741, "text": "restrictions if there's any fractions"}, {"start": 14.549, "duration": 4.831, "text": "floating around we have to omit any"}, {"start": 17.13, "duration": 4.26, "text": "values of X that give us zero in the"}, {"start": 19.38, "duration": 5.28, "text": "denominator so what to solve that"}, {"start": 21.39, "duration": 8.039, "text": "equation and if there's any square roots"}, {"start": 24.66, "duration": 6.959, "text": "or really any even powered root we don't"}, {"start": 29.429, "duration": 4.501, "text": "want any negative numbers to appear"}, {"start": 31.619, "duration": 7.831, "text": "underneath that radical after we kind of"}, {"start": 33.93, "duration": 7.11, "text": "simplifying do all the arithmetic so two"}, {"start": 39.45, "duration": 5.28, "text": "questions let's find the domain of these"}, {"start": 41.04, "duration": 5.999, "text": "two one over x squared minus 7x minus 30"}, {"start": 44.73, "duration": 6.21, "text": "and then B we've got the square root of"}, {"start": 47.039, "duration": 6.871, "text": "2x plus 3 so to me kind of the idea the"}, {"start": 50.94, "duration": 5.4, "text": "values that are bad or the values that I"}, {"start": 53.91, "duration": 5.399, "text": "must omit for this first equation I"}, {"start": 56.34, "duration": 6.42, "text": "don't want any value of x that makes the"}, {"start": 59.309, "duration": 6.06, "text": "denominator 0 so I'm basically thinking"}, {"start": 62.76, "duration": 5.49, "text": "if I said you know what makes the"}, {"start": 65.369, "duration": 6.5, "text": "denominator x squared minus 7x minus 30"}, {"start": 68.25, "duration": 7.95, "text": "I'm thinking what makes that equal zero"}, {"start": 71.869, "duration": 6.371, "text": "well we just hopefully factor this so"}, {"start": 76.2, "duration": 4.41, "text": "let's see I need to know I need an X and"}, {"start": 78.24, "duration": 4.68, "text": "an X to get my x squared I need two"}, {"start": 80.61, "duration": 3.72, "text": "numbers that multiply to negative 30 so"}, {"start": 82.92, "duration": 4.32, "text": "I know I'm going to need a negative and"}, {"start": 84.33, "duration": 4.98, "text": "a positive those two numbers when I"}, {"start": 87.24, "duration": 5.129, "text": "multiply have to get negative 30 but I"}, {"start": 89.31, "duration": 5.099, "text": "have to add up to negative 7 so I think"}, {"start": 92.369, "duration": 4.38, "text": "if we use negative 10"}, {"start": 94.409, "duration": 5.041, "text": "and positive three those two numbers"}, {"start": 96.749, "duration": 5.16, "text": "will fit the bill and again I want this"}, {"start": 99.45, "duration": 5.279, "text": "to equal zero so if we set X minus ten"}, {"start": 101.909, "duration": 5.341, "text": "equals zero we'll get x equals ten and"}, {"start": 104.729, "duration": 6.09, "text": "we'll get x equals negative 3 is"}, {"start": 107.25, "duration": 9.09, "text": "solutions so our domain is going to be"}, {"start": 110.819, "duration": 12.181, "text": "all values of x all values of x except"}, {"start": 116.34, "duration": 9.33, "text": "for x equals 10 and x equals negative 3"}, {"start": 123.0, "duration": 5.16, "text": "we could write that in interval notation"}, {"start": 125.67, "duration": 6.3, "text": "by simply saying it's from negative"}, {"start": 128.16, "duration": 6.75, "text": "infinity up to negative 3 then we have"}, {"start": 131.97, "duration": 6.18, "text": "to skip over negative 3 and go up to 10"}, {"start": 134.91, "duration": 5.79, "text": "and then we have to skip over 10 and go"}, {"start": 138.15, "duration": 5.25, "text": "all the way to positive infinity so sort"}, {"start": 140.7, "duration": 4.349, "text": "of two different equivalent ways the"}, {"start": 143.4, "duration": 6.059, "text": "other one here we have the square root"}, {"start": 145.049, "duration": 6.961, "text": "of 2x plus 3 so again to me the values"}, {"start": 149.459, "duration": 5.191, "text": "that are good in this case is the way I"}, {"start": 152.01, "duration": 5.28, "text": "think about it it's good if whatever"}, {"start": 154.65, "duration": 4.23, "text": "underneath the square root is greater"}, {"start": 157.29, "duration": 3.599, "text": "than or equal to zero that's what we"}, {"start": 158.88, "duration": 3.81, "text": "need to happen we only want numbers that"}, {"start": 160.889, "duration": 4.311, "text": "are greater than or equal to zero"}, {"start": 162.69, "duration": 5.519, "text": "whatever is under the square root well"}, {"start": 165.2, "duration": 5.619, "text": "2x plus 3 that's what's underneath the"}, {"start": 168.209, "duration": 5.31, "text": "square root so this now produces an"}, {"start": 170.819, "duration": 4.771, "text": "inequality I have to solve unfortunately"}, {"start": 173.519, "duration": 5.37, "text": "this is just a linear inequality so it's"}, {"start": 175.59, "duration": 5.519, "text": "not too bad I can subtract 3 from both"}, {"start": 178.889, "duration": 6.811, "text": "sides and get 2x greater than or equal"}, {"start": 181.109, "duration": 7.321, "text": "to negative 3/2 and it says X needs to"}, {"start": 185.7, "duration": 5.55, "text": "be greater than or equal to negative 3"}, {"start": 188.43, "duration": 5.1, "text": "halves so again this is my domain in"}, {"start": 191.25, "duration": 5.01, "text": "this case and an interval notation we"}, {"start": 193.53, "duration": 6.539, "text": "can write that as brackets negative 3"}, {"start": 196.26, "duration": 7.11, "text": "halves up to infinity and that would be"}, {"start": 200.069, "duration": 6.26, "text": "the domain for our our radical or square"}, {"start": 203.37, "duration": 2.959, "text": "root function"}], "title": "\u2756 Finding the Domain of a Function Algebraically (No graph!) \u2756", "description": "Finding the Domain of a Function Algebraically - Step-by-Step Examples!\n\nDescription: In this video, I walk through finding the domain of a function algebraically\u2014without using any graphs! We'll cover two key examples.\n\nIn the first example, we analyze a rational function where we need to exclude values of \ud835\udc65 that result in a zero denominator. This helps us determine the valid domain of the function.\n\nIn the second example, we solve a linear inequality to ensure that we don\u2019t end up with negative numbers under a square root. \n\nEach of these examples is fundamental and widely applicable, appearing throughout algebra, precalculus, calculus, and more. Whether you're learning about domains for the first time or need a refresher, these examples will help you master finding domains algebraically.\n\nWhat You Will Learn:\n\nHow to find the domain of rational functions by excluding zero denominators\nHow to solve linear inequalities to ensure non-negative square root values\nThe foundational skills for determining the domain of a function without graphing\n\nSupport my work on Patreon for more math content:\nhttps://www.patreon.com/patrickjmt?ty=c\n\n#maths #algebra #domain #squareroot #rationalfunctions #roots #graphing #domainofafunction #mathhelp #algebrabasics #functiondomain #rationalexpressions #findingdomain #inequalities #functionanalysis #algebratutorial", "lengthSeconds": 206, "width": 480, "height": 360}
{"video_id": "FBpg1wwNGJU", "transcript": [{"start": 0.88, "duration": 2.72, "text": "all right in this video i'm going to do"}, {"start": 2.32, "duration": 2.88, "text": "another problem"}, {"start": 3.6, "duration": 3.52, "text": "uh we're going to solve x to the two"}, {"start": 5.2, "duration": 4.08, "text": "thirds minus nine equals zero"}, {"start": 7.12, "duration": 3.28, "text": "and you can kind of think about this as"}, {"start": 9.28, "duration": 4.56, "text": "in a sense is"}, {"start": 10.4, "duration": 5.6, "text": "sort of like being a quadratic um"}, {"start": 13.84, "duration": 3.68, "text": "and the idea is we can really rewrite x"}, {"start": 16.0, "duration": 4.96, "text": "to the two-thirds as"}, {"start": 17.52, "duration": 5.28, "text": "x to the one-third quantity"}, {"start": 20.96, "duration": 3.84, "text": "squared because again if it's in"}, {"start": 22.8, "duration": 4.559, "text": "parenthesis you would multiply these and"}, {"start": 24.8, "duration": 5.52, "text": "get to the two-thirds power"}, {"start": 27.359, "duration": 4.32, "text": "minus nine equals zero and again you"}, {"start": 30.32, "duration": 3.28, "text": "know i don't know if it makes it any"}, {"start": 31.679, "duration": 3.361, "text": "easier or not a lot of times people will"}, {"start": 33.6, "duration": 4.88, "text": "do a little substitution"}, {"start": 35.04, "duration": 4.0, "text": "maybe we can let y equal x to the one"}, {"start": 38.48, "duration": 2.079, "text": "third"}, {"start": 39.04, "duration": 3.199, "text": "and then it does look very much like a"}, {"start": 40.559, "duration": 2.881, "text": "quadratic we could actually say this"}, {"start": 42.239, "duration": 4.961, "text": "original equation"}, {"start": 43.44, "duration": 6.639, "text": "is quadratic in x to the one third"}, {"start": 47.2, "duration": 4.96, "text": "but if we rewrite our x to the one third"}, {"start": 50.079, "duration": 4.721, "text": "squared minus nine equals zero using the"}, {"start": 52.16, "duration": 6.399, "text": "substitution we would simply get well"}, {"start": 54.8, "duration": 6.16, "text": "y squared minus nine equals 0"}, {"start": 58.559, "duration": 3.441, "text": "and again we could factor both sides or"}, {"start": 60.96, "duration": 3.279, "text": "we could add 9"}, {"start": 62.0, "duration": 3.439, "text": "to both sides in this case y squared"}, {"start": 64.239, "duration": 3.121, "text": "equals 9"}, {"start": 65.439, "duration": 4.0, "text": "and then to get our solution we would"}, {"start": 67.36, "duration": 3.68, "text": "take the square root of both sides"}, {"start": 69.439, "duration": 4.321, "text": "remember you have to put a plus or a"}, {"start": 71.04, "duration": 5.52, "text": "minus a plus and a minus on one side"}, {"start": 73.76, "duration": 5.44, "text": "whoops i'm jumping ahead of myself"}, {"start": 76.56, "duration": 4.48, "text": "we still have the square root of 9 which"}, {"start": 79.2, "duration": 4.88, "text": "is simply"}, {"start": 81.04, "duration": 4.8, "text": "going to give us 3 so it says y has to"}, {"start": 84.08, "duration": 2.96, "text": "equal positive or negative three but"}, {"start": 85.84, "duration": 2.8, "text": "again we have to remember our"}, {"start": 87.04, "duration": 4.56, "text": "substitution that we did here"}, {"start": 88.64, "duration": 4.479, "text": "because we let y equal x to the"}, {"start": 91.6, "duration": 4.4, "text": "one-third"}, {"start": 93.119, "duration": 5.04, "text": "so what that would tell us then it says"}, {"start": 96.0, "duration": 3.68, "text": "x to the one-third either has to equal"}, {"start": 98.159, "duration": 5.361, "text": "positive three"}, {"start": 99.68, "duration": 5.439, "text": "or x to the one third has to equal"}, {"start": 103.52, "duration": 3.76, "text": "negative three"}, {"start": 105.119, "duration": 4.561, "text": "and since we're raising the left side to"}, {"start": 107.28, "duration": 3.36, "text": "the one third power we're taking a cube"}, {"start": 109.68, "duration": 2.719, "text": "root really"}, {"start": 110.64, "duration": 3.36, "text": "the way that we get rid of that is we"}, {"start": 112.399, "duration": 3.921, "text": "cube both sides"}, {"start": 114.0, "duration": 3.439, "text": "so we'll raise both sides to the third"}, {"start": 116.32, "duration": 3.6, "text": "power"}, {"start": 117.439, "duration": 3.921, "text": "x to the one third cubed is simply x to"}, {"start": 119.92, "duration": 3.12, "text": "the first"}, {"start": 121.36, "duration": 3.68, "text": "three cubed is going to be three times"}, {"start": 123.04, "duration": 3.759, "text": "three which is nine times another three"}, {"start": 125.04, "duration": 4.4, "text": "which is 27"}, {"start": 126.799, "duration": 3.201, "text": "and likewise we have x to the one third"}, {"start": 129.44, "duration": 3.439, "text": "so if we"}, {"start": 130.0, "duration": 4.56, "text": "cube that and we also cube the right"}, {"start": 132.879, "duration": 3.521, "text": "side as well"}, {"start": 134.56, "duration": 3.679, "text": "negative 3 times negative 3 times"}, {"start": 136.4, "duration": 4.4, "text": "negative 3 is going to give us"}, {"start": 138.239, "duration": 4.401, "text": "negative 27. so now we have our"}, {"start": 140.8, "duration": 5.68, "text": "solutions to our"}, {"start": 142.64, "duration": 3.84, "text": "original equation"}], "title": "Solving Fancy Quadratics - Example 2", "description": "Thanks to all of you who support me on Patreon. You da real mvps!   $1 per month helps!! :)  https://www.patreon.com/patrickjmt !! Solving Fancy Quadratics - Example 2.  In this video, I solve problems using the same techniques (after a relabeling) that we use for 'regular' quadratic equations.", "lengthSeconds": 145, "width": 480, "height": 360}
{"video_id": "YHcAO6Obglk", "transcript": [{"start": 0.0, "duration": 5.37, "text": "all right so just one more problem here"}, {"start": 2.28, "duration": 5.16, "text": "using the arc length formula so in this"}, {"start": 5.37, "duration": 4.47, "text": "problem we've got a tree that's 1,500"}, {"start": 7.44, "duration": 4.47, "text": "yards from an observer and that subtends"}, {"start": 9.84, "duration": 4.679, "text": "an angle of 2 degrees we're gonna use"}, {"start": 11.91, "duration": 6.619, "text": "the arc length formula to estimate the"}, {"start": 14.519, "duration": 7.711, "text": "height of the tree okay so here's a"}, {"start": 18.529, "duration": 4.871, "text": "here's you or somebody and so suppose"}, {"start": 22.23, "duration": 6.15, "text": "you know suppose we could almost"}, {"start": 23.4, "duration": 6.57, "text": "represent you as a little dot and a tree"}, {"start": 28.38, "duration": 3.719, "text": "we've got this tree that's way far away"}, {"start": 29.97, "duration": 4.62, "text": "or pretty far away you know fifteen"}, {"start": 32.099, "duration": 6.511, "text": "hundred yards is a good little ways and"}, {"start": 34.59, "duration": 6.0, "text": "to get to the top of the tree so let's"}, {"start": 38.61, "duration": 5.64, "text": "make it a little Christmas tree here"}, {"start": 40.59, "duration": 7.17, "text": "we'll holiday festivities so there's our"}, {"start": 44.25, "duration": 6.05, "text": "little tree or a little pine tree and"}, {"start": 47.76, "duration": 6.689, "text": "the idea is to the top of the tree"}, {"start": 50.3, "duration": 9.399, "text": "you're gonna make a little a little"}, {"start": 54.449, "duration": 7.29, "text": "angle here of two degrees okay so the"}, {"start": 59.699, "duration": 4.141, "text": "idea is you know the tree assumedly is"}, {"start": 61.739, "duration": 4.081, "text": "standing straight up and down you know"}, {"start": 63.84, "duration": 4.11, "text": "the the circle that you're really"}, {"start": 65.82, "duration": 3.96, "text": "thinking about is definitely curved so"}, {"start": 67.95, "duration": 3.69, "text": "that's why we're approximating the"}, {"start": 69.78, "duration": 4.44, "text": "height of the tree here but the idea is"}, {"start": 71.64, "duration": 4.159, "text": "if we can just find that arc length that"}, {"start": 74.22, "duration": 4.079, "text": "arc length is going to be approximately"}, {"start": 75.799, "duration": 5.5, "text": "equal to the height of the tree so"}, {"start": 78.299, "duration": 4.651, "text": "that's the basic idea we just have to do"}, {"start": 81.299, "duration": 5.851, "text": "the same thing as before just use our"}, {"start": 82.95, "duration": 9.18, "text": "arc length formula so let's see we said"}, {"start": 87.15, "duration": 7.05, "text": "that s was equal to R times theta R is"}, {"start": 92.13, "duration": 5.309, "text": "the radius well in this case this is"}, {"start": 94.2, "duration": 5.669, "text": "1,500 yards away you can imagine that's"}, {"start": 97.439, "duration": 5.07, "text": "just sitting on a circle so the radius"}, {"start": 99.869, "duration": 6.661, "text": "will be 1500 again we've got to convert"}, {"start": 102.509, "duration": 6.961, "text": "theta into radians so let's see pi"}, {"start": 106.53, "duration": 6.15, "text": "radians is equivalent is equal to 180"}, {"start": 109.47, "duration": 6.87, "text": "degrees if we divide both sides by 180"}, {"start": 112.68, "duration": 5.7, "text": "we'll get PI over 180 radians is"}, {"start": 116.34, "duration": 5.67, "text": "equivalent to one degree and now if we"}, {"start": 118.38, "duration": 7.91, "text": "just simply multiply both sides by two"}, {"start": 122.01, "duration": 8.28, "text": "we'll get two pi over 180 radians equals"}, {"start": 126.29, "duration": 7.84, "text": "two degrees so that's simply going to be"}, {"start": 130.29, "duration": 6.33, "text": "PI over 90 radians is going to be the"}, {"start": 134.13, "duration": 5.85, "text": "same thing as equivalent to two degrees"}, {"start": 136.62, "duration": 7.17, "text": "so for our angle theta we'll plug in PI"}, {"start": 139.98, "duration": 6.33, "text": "over 90 and now I'm just going to do the"}, {"start": 143.79, "duration": 5.58, "text": "arithmetic so let's see we'll do 1500"}, {"start": 146.31, "duration": 8.28, "text": "over 90 and I'm going to approximate pi"}, {"start": 149.37, "duration": 11.55, "text": "by using 3.14 let's see here let me get"}, {"start": 154.59, "duration": 10.68, "text": "my calculator so we've got 1500 divided"}, {"start": 160.92, "duration": 11.97, "text": "by 90 I'm getting that to be sixteen"}, {"start": 165.27, "duration": 11.94, "text": "point six repeating times 3.14 so let's"}, {"start": 172.89, "duration": 8.34, "text": "see when I multiply that by 3.14 I'm"}, {"start": 177.21, "duration": 6.87, "text": "getting this to equal 52 point three"}, {"start": 181.23, "duration": 5.76, "text": "repeating but again we wanted to round"}, {"start": 184.08, "duration": 8.1, "text": "this to the nearest yard so this is"}, {"start": 186.99, "duration": 6.99, "text": "pretty close to 52 yards so we would say"}, {"start": 192.18, "duration": 5.04, "text": "the height of that tree give or take a"}, {"start": 193.98, "duration": 5.63, "text": "little bit is 52 yards using our arc"}, {"start": 197.22, "duration": 2.39, "text": "length formula"}], "title": "Arc Length Formula - Example 2", "description": "Thanks to all of you who support me on Patreon. You da real mvps!   $1 per month helps!! :)  https://www.patreon.com/patrickjmt !! Arc Length Formula - Example 2. In this video, I use the arc length formula to estimate the height of a tree.", "lengthSeconds": 199, "width": 480, "height": 360}
{"video_id": "9RjI_so9oSM", "transcript": [{"start": 0.88, "duration": 2.96, "text": "okay in this video I just want to do a"}, {"start": 2.28, "duration": 3.68, "text": "quick little discussion about Newton's"}, {"start": 3.84, "duration": 4.16, "text": "method again and how it you know may not"}, {"start": 5.96, "duration": 4.52, "text": "work so again that was the Newton's"}, {"start": 8.0, "duration": 4.24, "text": "method formula um hopefully you've seen"}, {"start": 10.48, "duration": 4.6, "text": "the other examples on how to compute"}, {"start": 12.24, "duration": 4.72, "text": "things so uh again this is kind of the"}, {"start": 15.08, "duration": 3.6, "text": "basic geometric idea so given the"}, {"start": 16.96, "duration": 3.28, "text": "following equation an initial guess we"}, {"start": 18.68, "duration": 4.359, "text": "want to know why Newton's method would"}, {"start": 20.24, "duration": 7.24, "text": "be bad so suppose we want to find the"}, {"start": 23.039, "duration": 7.48, "text": "Zer or the roots to uh this equation x -"}, {"start": 27.48, "duration": 5.119, "text": "2^ 2 - 1 and we start with the first"}, {"start": 30.519, "duration": 3.521, "text": "guess of two and again you want to plug"}, {"start": 32.599, "duration": 4.241, "text": "in a number that gives you something"}, {"start": 34.04, "duration": 5.12, "text": "close to zero well if we plug two into"}, {"start": 36.84, "duration": 5.16, "text": "this um if we plug two into this we'll"}, {"start": 39.16, "duration": 5.52, "text": "get well 2 - 2 we'll get 0^ 2 - 1 we'll"}, {"start": 42.0, "duration": 5.039, "text": "get - 1 and in one of the other videos"}, {"start": 44.68, "duration": 5.24, "text": "negative-1 was actually uh you know kind"}, {"start": 47.039, "duration": 4.441, "text": "of uh would have been our so we started"}, {"start": 49.92, "duration": 3.2, "text": "with an initial guess and we got"}, {"start": 51.48, "duration": 3.64, "text": "negative one out and again that's close"}, {"start": 53.12, "duration": 5.04, "text": "to zero and we started using it and"}, {"start": 55.12, "duration": 5.959, "text": "everything worked out great so um what's"}, {"start": 58.16, "duration": 5.079, "text": "the issue why would uh x equal to be bad"}, {"start": 61.079, "duration": 5.08, "text": "here so I'm just going to graph this"}, {"start": 63.239, "duration": 5.88, "text": "function real quick uh remember x^2 is a"}, {"start": 66.159, "duration": 6.361, "text": "parabola um xus two on the inside"}, {"start": 69.119, "duration": 6.081, "text": "actually shifts it two to the right okay"}, {"start": 72.52, "duration": 5.44, "text": "so there's xal 2 and then the minus one"}, {"start": 75.2, "duration": 6.0, "text": "would move it down uh one unit so that"}, {"start": 77.96, "duration": 5.88, "text": "would be the vertex of our Parabola and"}, {"start": 81.2, "duration": 4.44, "text": "it's going to cross the uh you know it's"}, {"start": 83.84, "duration": 3.919, "text": "going to cross the x-axis somewhere I"}, {"start": 85.64, "duration": 6.28, "text": "think we can almost uh maybe you can"}, {"start": 87.759, "duration": 4.161, "text": "almost figure out the answers here uh"}, {"start": 93.159, "duration": 5.401, "text": "um but suppose again you know um just"}, {"start": 96.88, "duration": 5.68, "text": "talking about geometrically why I can go"}, {"start": 98.56, "duration": 6.879, "text": "wrong what Newton's method does is"}, {"start": 102.56, "duration": 5.4, "text": "somehow when you start at your first"}, {"start": 105.439, "duration": 6.0, "text": "guess what it does is it looks at a"}, {"start": 107.96, "duration": 5.68, "text": "tangent line at that point and the"}, {"start": 111.439, "duration": 3.241, "text": "tangent line you know hopefully you pick"}, {"start": 113.64, "duration": 3.32, "text": "a"}, {"start": 114.68, "duration": 4.92, "text": "point where the tangent line will"}, {"start": 116.96, "duration": 5.799, "text": "eventually hit the X AIS because if that"}, {"start": 119.6, "duration": 4.839, "text": "t line hits the X AIS actually that"}, {"start": 122.759, "duration": 3.801, "text": "place where it hits so suppose instead"}, {"start": 124.439, "duration": 3.8, "text": "of using two we use some other value"}, {"start": 126.56, "duration": 5.36, "text": "maybe like"}, {"start": 128.239, "duration": 6.401, "text": "2.6 um so 2.6 I'm going to get a tangent"}, {"start": 131.92, "duration": 7.12, "text": "line that looks something like that so"}, {"start": 134.64, "duration": 6.679, "text": "if 2.6 if that was my first guess this x"}, {"start": 139.04, "duration": 4.72, "text": "intercept uh Newton's method that's what"}, {"start": 141.319, "duration": 5.441, "text": "it actually does it picks out the x"}, {"start": 143.76, "duration": 4.759, "text": "intercept so whatever this x intercept"}, {"start": 146.76, "duration": 3.88, "text": "is when we go through Newton's method"}, {"start": 148.519, "duration": 5.321, "text": "and we do this iteration and compute our"}, {"start": 150.64, "duration": 5.36, "text": "formula for little x sub 2 xub 2 is"}, {"start": 153.84, "duration": 3.759, "text": "magically going to be that point and"}, {"start": 156.0, "duration": 3.519, "text": "then the idea is well maybe I could go"}, {"start": 157.599, "duration": 4.681, "text": "up here and find that point on the graph"}, {"start": 159.519, "duration": 4.561, "text": "and I could find a new tangent line and"}, {"start": 162.28, "duration": 4.56, "text": "again we're trying to get close to X"}, {"start": 164.08, "duration": 5.079, "text": "intercepts on the real graph so notice"}, {"start": 166.84, "duration": 5.2, "text": "by looking at this new value I'm getting"}, {"start": 169.159, "duration": 6.0, "text": "even closer to the true x intercept and"}, {"start": 172.04, "duration": 6.24, "text": "that would be our third guess etc etc"}, {"start": 175.159, "duration": 4.841, "text": "etc well the point is if we take uh if"}, {"start": 178.28, "duration": 3.84, "text": "we try to use two as our guess if if you"}, {"start": 180.0, "duration": 3.76, "text": "think about the tangent line there the"}, {"start": 182.12, "duration": 4.36, "text": "tangent line there is just going to be"}, {"start": 183.76, "duration": 5.0, "text": "horizontal it's going to be a flat line"}, {"start": 186.48, "duration": 4.72, "text": "so the issue is the tangent line in this"}, {"start": 188.76, "duration": 5.399, "text": "case well it never ever actually hits"}, {"start": 191.2, "duration": 9.0, "text": "the x- axis so you never would actually"}, {"start": 194.159, "duration": 7.64, "text": "get yet a new um a new a new guess okay"}, {"start": 200.2, "duration": 3.08, "text": "um and you can try you know going"}, {"start": 201.799, "duration": 2.921, "text": "through the formula and plugging it into"}, {"start": 203.28, "duration": 3.72, "text": "the derivative and kind of see what"}, {"start": 204.72, "duration": 5.239, "text": "happens but geometrically this is the"}, {"start": 207.0, "duration": 5.28, "text": "issue if we picked it at x equals 2"}, {"start": 209.959, "duration": 4.84, "text": "unfortunately our tangent line again"}, {"start": 212.28, "duration": 5.64, "text": "never hits the x- axis and that means we"}, {"start": 214.799, "duration": 4.881, "text": "never get our second iteration guess and"}, {"start": 217.92, "duration": 3.879, "text": "our or I guess technically we would do"}, {"start": 219.68, "duration": 4.08, "text": "one iteration to get x sub 2 but we"}, {"start": 221.799, "duration": 4.561, "text": "never get our new guesses because the"}, {"start": 223.76, "duration": 5.6, "text": "tangent line never hits the"}, {"start": 226.36, "duration": 3.0, "text": "xaxis"}], "title": "Newton's Method - How it Can FAIL - More Examples Part 3 of 3", "description": "Thanks to all of you who support me on Patreon. You da real mvps!   $1 per month helps!! :)  https://www.patreon.com/patrickjmt !! Newton's Method - More Examples Part 3 of 3.  Here I give the geometric idea behind Newton's Method and show how it can go wrong and fail to yield an approximation.  Newton's method does not always work!", "lengthSeconds": 227, "width": 480, "height": 360}
{"video_id": "-NjMiHFV-yQ", "transcript": [{"start": 0.5, "duration": 5.08, "text": "okay this is a sample SAT problem taken"}, {"start": 3.84, "duration": 4.11, "text": "from the Princeton Review and it says if"}, {"start": 5.58, "duration": 5.849, "text": "X over 2 is less than 12 and y equals X"}, {"start": 7.95, "duration": 5.16, "text": "minus 4 then X X Y could equal which of"}, {"start": 11.429, "duration": 3.721, "text": "the following and this is a pretty"}, {"start": 13.11, "duration": 3.87, "text": "typical just plug and chug problem so"}, {"start": 15.15, "duration": 3.389, "text": "again the first coordinates are x"}, {"start": 16.98, "duration": 4.049, "text": "coordinate the second ones are y"}, {"start": 18.539, "duration": 6.391, "text": "coordinate we just have to have to test"}, {"start": 21.029, "duration": 6.57, "text": "which one of these pairs works so if we"}, {"start": 24.93, "duration": 5.22, "text": "plug X into the first one certainly 5"}, {"start": 27.599, "duration": 6.031, "text": "over 2 is less than 12 so that that is"}, {"start": 30.15, "duration": 5.49, "text": "satisfied well if you plug these same"}, {"start": 33.63, "duration": 7.62, "text": "values into the second part it says y is"}, {"start": 35.64, "duration": 9.329, "text": "9 and if we plug in x equals 5 well 9 -"}, {"start": 41.25, "duration": 5.7, "text": "excuse me 5 - 4 does not equal 9 so it"}, {"start": 44.969, "duration": 5.581, "text": "says that this solution when in fact did"}, {"start": 46.95, "duration": 7.71, "text": "not work if we plug in the second one it"}, {"start": 50.55, "duration": 5.669, "text": "says well it has to satisfy 19/2 being"}, {"start": 54.66, "duration": 4.739, "text": "less than 12"}, {"start": 56.219, "duration": 5.541, "text": "well 19 is smaller than 20 20 divided by"}, {"start": 59.399, "duration": 5.821, "text": "2 is certainly less than 12 so that"}, {"start": 61.76, "duration": 6.25, "text": "criteria is satisfied and let's see so"}, {"start": 65.22, "duration": 6.899, "text": "it says if we plug in y equals 23 does"}, {"start": 68.01, "duration": 6.57, "text": "that equal X which is 19 minus 4 and"}, {"start": 72.119, "duration": 5.971, "text": "again we're subtracting so this"}, {"start": 74.58, "duration": 6.84, "text": "condition is not satisfied so the second"}, {"start": 78.09, "duration": 6.18, "text": "one couldn't be an answer let's plug in"}, {"start": 81.42, "duration": 6.12, "text": "20 over 17 so we have to ask ourselves"}, {"start": 84.27, "duration": 5.97, "text": "this 20 over 2 less than 12 well"}, {"start": 87.54, "duration": 5.88, "text": "certainly it's 10 so that works let's"}, {"start": 90.24, "duration": 5.82, "text": "see if we plug 17 in for y does that"}, {"start": 93.42, "duration": 5.94, "text": "equal 20 minus 4"}, {"start": 96.06, "duration": 6.629, "text": "well again 20 minus 4 is 16 so that"}, {"start": 99.36, "duration": 6.99, "text": "doesn't work let's see if we plug in 22"}, {"start": 102.689, "duration": 9.061, "text": "we have to ask ourselves is 22 divided"}, {"start": 106.35, "duration": 9.629, "text": "by 2 less than 12 well 22 divided by 2"}, {"start": 111.75, "duration": 6.75, "text": "is 11 so that condition is satisfied and"}, {"start": 115.979, "duration": 5.311, "text": "then we have to plug in 18 for Y and we"}, {"start": 118.5, "duration": 8.07, "text": "have to ask ourselves does 18 equal"}, {"start": 121.29, "duration": 8.91, "text": "let's plug in 22 minus 4 well 22-4"}, {"start": 126.57, "duration": 6.21, "text": "does equal 18 so that will be"}, {"start": 130.2, "duration": 6.74, "text": "solution for this problem just a very"}, {"start": 132.78, "duration": 4.16, "text": "typical plug and chug type of problem"}], "title": "SAT Math Question #4", "description": "Thanks to all of you who support me on Patreon. You da real mvps!   $1 per month helps!! :)  https://www.patreon.com/patrickjmt !! A sample SAT math question taken from a practice exam of the Princeton Review.\r\nFor more free math videos visit http://JustMathTutoring.com", "lengthSeconds": 136, "width": 480, "height": 360}
{"video_id": "mrfvKe7X5y4", "transcript": [{"start": 0.11, "duration": 4.99, "text": "alright as promised here's gonna be one"}, {"start": 3.21, "duration": 5.099, "text": "more example about solving some absolute"}, {"start": 5.1, "duration": 5.76, "text": "value inequalities so in this case we've"}, {"start": 8.309, "duration": 4.681, "text": "got twelve minus two times the absolute"}, {"start": 10.86, "duration": 4.439, "text": "value of two minus and then in"}, {"start": 12.99, "duration": 5.85, "text": "parentheses five X minus six all of that"}, {"start": 15.299, "duration": 5.551, "text": "greater than or equal to four yea so the"}, {"start": 18.84, "duration": 4.05, "text": "first thing again I typically try to get"}, {"start": 20.85, "duration": 4.29, "text": "the absolute value by itself I'm gonna"}, {"start": 22.89, "duration": 5.01, "text": "do I'm gonna subtract 12 from both sides"}, {"start": 25.14, "duration": 5.25, "text": "so that's gonna leave me with negative 2"}, {"start": 27.9, "duration": 4.35, "text": "and then absolute value inside the"}, {"start": 30.39, "duration": 4.02, "text": "absolute value I'm going to distribute"}, {"start": 32.25, "duration": 4.68, "text": "the negative so I would have negative 5x"}, {"start": 34.41, "duration": 6.059, "text": "and then a negative and a negative would"}, {"start": 36.93, "duration": 5.309, "text": "make positive 6 if I do 4 minus 12"}, {"start": 40.469, "duration": 5.281, "text": "that's going to give me negative 8 on"}, {"start": 42.239, "duration": 7.291, "text": "the right side the next thing I'm gonna"}, {"start": 45.75, "duration": 5.129, "text": "do is divide both sides by negative 2 so"}, {"start": 49.53, "duration": 3.9, "text": "on the left the negative twos will"}, {"start": 50.879, "duration": 4.68, "text": "cancel out inside the absolute value"}, {"start": 53.43, "duration": 5.399, "text": "again I'll have a positive 2 and a"}, {"start": 55.559, "duration": 6.27, "text": "positive 6 which is a positive 8 minus"}, {"start": 58.829, "duration": 5.181, "text": "5x and again since we divided by a"}, {"start": 61.829, "duration": 5.761, "text": "negative number our inequality flips"}, {"start": 64.01, "duration": 6.219, "text": "negative 8 over negative 2 is positive 4"}, {"start": 67.59, "duration": 7.59, "text": "and again for this type of inequality I"}, {"start": 70.229, "duration": 6.96, "text": "just remove the absolute value and also"}, {"start": 75.18, "duration": 5.009, "text": "make the quantity inside the absolute"}, {"start": 77.189, "duration": 5.701, "text": "value greater than the negative of the"}, {"start": 80.189, "duration": 5.551, "text": "number that we had okay so now I just"}, {"start": 82.89, "duration": 6.99, "text": "subtract 8 from each side so on the Left"}, {"start": 85.74, "duration": 7.559, "text": "negative 4 minus 8 is negative 12 we've"}, {"start": 89.88, "duration": 6.9, "text": "got negative 5x left in the middle 4"}, {"start": 93.299, "duration": 5.64, "text": "minus 8 is going to be negative 4 and"}, {"start": 96.78, "duration": 3.9, "text": "again now simply to get the X by itself"}, {"start": 98.939, "duration": 4.89, "text": "I'm just going to divide everything by"}, {"start": 100.68, "duration": 6.11, "text": "negative 5 again remembering that we"}, {"start": 103.829, "duration": 5.371, "text": "have to flip our inequality in this case"}, {"start": 106.79, "duration": 5.05, "text": "so I'm going to divide by negative 5"}, {"start": 109.2, "duration": 6.18, "text": "divided by negative 5 divided by"}, {"start": 111.84, "duration": 5.819, "text": "negative 5 okay so we've got a positive"}, {"start": 115.38, "duration": 5.25, "text": "12 fifths on the left again my"}, {"start": 117.659, "duration": 5.651, "text": "inequality flips direction the negative"}, {"start": 120.63, "duration": 4.96, "text": "fives will cancel leaving me with just X"}, {"start": 123.31, "duration": 4.309, "text": "again the inequality flips and then we"}, {"start": 125.59, "duration": 5.03, "text": "would have positive four-fifths"}, {"start": 127.619, "duration": 6.761, "text": "equivalently we could write this as"}, {"start": 130.62, "duration": 7.18, "text": "four-fifths less than or equal to X less"}, {"start": 134.38, "duration": 4.77, "text": "than or equal to 12 fifths and again if"}, {"start": 137.8, "duration": 6.21, "text": "you want to write this using interval"}, {"start": 139.15, "duration": 10.19, "text": "notation we'll use brackets from 4/5 up"}, {"start": 144.01, "duration": 5.33, "text": "to and including twelve fifths"}], "title": "Solving Absolute Value Inequalities, MORE Examples - Example 2", "description": "Thanks to all of you who support me on Patreon. You da real mvps!   $1 per month helps!! :)  https://www.patreon.com/patrickjmt !! Solving Absolute Value Inequalities, MORE Examples - Example 2", "lengthSeconds": 149, "width": 480, "height": 360}
{"video_id": "eCQupcOGkRg", "transcript": [{"start": 0.0, "duration": 6.96, "text": "Hey and in this limit example again"}, {"start": 4.98, "duration": 4.47, "text": "we'll try to plug in H equals zero"}, {"start": 6.96, "duration": 6.33, "text": "notice if we do plug in H equals zero"}, {"start": 9.45, "duration": 6.81, "text": "we'll get 4 squared which is 16 minus 16"}, {"start": 13.29, "duration": 6.329, "text": "so again we'll get 0 on top and clearly"}, {"start": 16.26, "duration": 5.64, "text": "we're getting 0 on the bottom so we have"}, {"start": 19.619, "duration": 6.561, "text": "to do something algebraic to try to"}, {"start": 21.9, "duration": 8.07, "text": "figure out what's going on here notice"}, {"start": 26.18, "duration": 6.25, "text": "the only thing that I see right off the"}, {"start": 29.97, "duration": 5.79, "text": "bat at least is there's this 4 plus h"}, {"start": 32.43, "duration": 6.21, "text": "squared so we could always cancel or"}, {"start": 35.76, "duration": 6.84, "text": "excuse me expand that out and see if"}, {"start": 38.64, "duration": 7.59, "text": "something cancels out so again remember"}, {"start": 42.6, "duration": 7.65, "text": "this is 4 plus h times 4 plus h so we'll"}, {"start": 46.23, "duration": 10.77, "text": "get 16 plus 4 h plus 4 h that'll give us"}, {"start": 50.25, "duration": 13.129, "text": "plus 8 h plus h squared minus 16 all"}, {"start": 57.0, "duration": 10.1, "text": "over H well the 16s will cancel out and"}, {"start": 63.379, "duration": 3.721, "text": "I'm left with"}, {"start": 68.69, "duration": 6.19, "text": "8h + H in the numerator but since they"}, {"start": 72.21, "duration": 6.83, "text": "both have an H I can factor an H out of"}, {"start": 74.88, "duration": 8.01, "text": "there then I have 8 plus h leftover"}, {"start": 79.04, "duration": 6.88, "text": "divided by H and again now I can cancel"}, {"start": 82.89, "duration": 4.77, "text": "out my H's and this H in the bottom is"}, {"start": 85.92, "duration": 3.57, "text": "what's you know certainly giving me some"}, {"start": 87.66, "duration": 5.4, "text": "problems anyway because I'm dividing by"}, {"start": 89.49, "duration": 9.93, "text": "0 so I'm left with the limit as H"}, {"start": 93.06, "duration": 8.16, "text": "approaches 0 of 8 + H and again we go"}, {"start": 99.42, "duration": 4.5, "text": "back to our old trick of plugging and"}, {"start": 101.22, "duration": 6.24, "text": "chugging if you plug in H equals 0"}, {"start": 103.92, "duration": 5.87, "text": "we're left with 8 and that is our"}, {"start": 107.46, "duration": 2.33, "text": "solution"}], "title": "Calculating a Limit by Expanding and Simplifying", "description": "Thanks to all of you who support me on Patreon. You da real mvps!   $1 per month helps!! :)  https://www.patreon.com/patrickjmt !! Calculating a Limit by Expanding and Simplifying", "lengthSeconds": 110, "width": 320, "height": 240}
{"video_id": "C7de7f2LER0", "transcript": [{"start": 0.76, "duration": 3.199, "text": "all right in this video I'm going to do"}, {"start": 2.28, "duration": 4.76, "text": "an example of solving an equation that"}, {"start": 3.959, "duration": 4.68, "text": "involves two absolute value Expressions"}, {"start": 7.04, "duration": 3.28, "text": "so at first it's not going to look like"}, {"start": 8.639, "duration": 5.241, "text": "there's two but wait just a second here"}, {"start": 10.32, "duration": 8.92, "text": "so suppose we have the absolute value of"}, {"start": 13.88, "duration": 7.08, "text": "the fraction p + 1 / 3 - P = 12 well"}, {"start": 19.24, "duration": 3.92, "text": "there's only one absolute value in there"}, {"start": 20.96, "duration": 3.88, "text": "but to get started if you have the"}, {"start": 23.16, "duration": 3.6, "text": "absolute value of a fraction you can"}, {"start": 24.84, "duration": 5.199, "text": "break this up as the absolute value of"}, {"start": 26.76, "duration": 5.28, "text": "the numerator over the absolute value of"}, {"start": 30.039, "duration": 4.481, "text": "of the denominator this is an equ"}, {"start": 32.04, "duration": 4.12, "text": "equivalent algebraic way to break up"}, {"start": 34.52, "duration": 4.359, "text": "absolute value of"}, {"start": 36.16, "duration": 4.44, "text": "fractions equals 12 and what I'm going"}, {"start": 38.879, "duration": 3.84, "text": "to do is I'm going to multiply both"}, {"start": 40.6, "duration": 4.639, "text": "sides by this quantity the absolute"}, {"start": 42.719, "duration": 5.081, "text": "value of 3 minus P so that's just going"}, {"start": 45.239, "duration": 7.401, "text": "to move over so I've got the absolute"}, {"start": 47.8, "duration": 7.919, "text": "value of p + 1 equal 12 * the absolute"}, {"start": 52.64, "duration": 5.96, "text": "value of 3 minus"}, {"start": 55.719, "duration": 6.121, "text": "P at this point what we do is we create"}, {"start": 58.6, "duration": 7.44, "text": "two separate equations to solve we take"}, {"start": 61.84, "duration": 7.08, "text": "p + 1 = 12 * the quantity we just"}, {"start": 66.04, "duration": 5.28, "text": "basically uh remove the bars on both of"}, {"start": 68.92, "duration": 4.4, "text": "them you can imagine they're both in"}, {"start": 71.32, "duration": 3.24, "text": "parentheses the other equation that"}, {"start": 73.32, "duration": 4.88, "text": "we're going to solve is we're going to"}, {"start": 74.56, "duration": 6.68, "text": "remove the bars p + 1 ="}, {"start": 78.2, "duration": 5.68, "text": "12 but on one of these um we're going to"}, {"start": 81.24, "duration": 5.8, "text": "stick a negative in front of"}, {"start": 83.88, "duration": 5.04, "text": "it so instead of just writing 3 minus P"}, {"start": 87.04, "duration": 4.36, "text": "like we did the first time I'm going to"}, {"start": 88.92, "duration": 4.479, "text": "write 3 - p but there's a negative"}, {"start": 91.4, "duration": 4.2, "text": "associated with it and now this is going"}, {"start": 93.399, "duration": 6.4, "text": "to give me my two equations that I have"}, {"start": 95.6, "duration": 10.159, "text": "to solve so on the left we have p + 1 12"}, {"start": 99.799, "duration": 7.28, "text": "* 3 is 36 12 * P will be -2 p and then"}, {"start": 105.759, "duration": 4.4, "text": "what I'm going to do is I'm going to add"}, {"start": 107.079, "duration": 3.08, "text": "12 P to both"}, {"start": 111.159, "duration": 5.081, "text": "sides so that'll give me 13 P at the"}, {"start": 114.32, "duration": 6.32, "text": "same time I think I'll subtract one from"}, {"start": 116.24, "duration": 7.879, "text": "both sides so that'll give me 13 P = 35"}, {"start": 120.64, "duration": 8.36, "text": "and if I divide both sides by 13 that'll"}, {"start": 124.119, "duration": 8.48, "text": "give me that P = 35 13 as one of my"}, {"start": 129.0, "duration": 7.319, "text": "Solutions and the other solution here"}, {"start": 132.599, "duration": 7.681, "text": "this is p + 1 we could write that as -2"}, {"start": 136.319, "duration": 9.92, "text": "* 3 - P so now if I distribute things"}, {"start": 140.28, "duration": 10.679, "text": "out we have p + 1 -12 and 3 is - 36 -12"}, {"start": 146.239, "duration": 7.08, "text": "and P is POS 12 p and the same thing so"}, {"start": 150.959, "duration": 6.56, "text": "let's add 36 to both sides that'll give"}, {"start": 153.319, "duration": 7.441, "text": "me positive 37 I'm going to subtract 1 P"}, {"start": 157.519, "duration": 6.08, "text": "from both sides that will give me 11 P"}, {"start": 160.76, "duration": 8.24, "text": "and now if I divide both sides by 11"}, {"start": 163.599, "duration": 11.28, "text": "I'll get 37 / 11 = P as my second"}, {"start": 169.0, "duration": 5.879, "text": "solution to this absolute value equation"}], "title": "Solving Absolute Value Equations Containing TWO Absolute Value Expressions - Example 1", "description": "Thanks to all of you who support me on Patreon. You da real mvps!   $1 per month helps!! :)  https://www.patreon.com/patrickjmt !! Solving Absolute Value Equations Containing TWO Absolute Value Expressions - Example 1", "lengthSeconds": 173, "width": 480, "height": 360}
{"video_id": "jbxM8CSOOLg", "transcript": [{"start": 0.0, "duration": 4.259, "text": "all right so one more example here of"}, {"start": 1.89, "duration": 5.04, "text": "simplifying products of binomials"}, {"start": 4.259, "duration": 5.581, "text": "involving trig functions and the one"}, {"start": 6.93, "duration": 5.669, "text": "that we I think left undone from the"}, {"start": 9.84, "duration": 4.83, "text": "previous video was tangent of theta plus"}, {"start": 12.599, "duration": 4.981, "text": "cosine squared theta plus sine squared"}, {"start": 14.67, "duration": 5.43, "text": "theta multiplied by tangent theta minus"}, {"start": 17.58, "duration": 4.5, "text": "cosine squared theta minus sine squared"}, {"start": 20.1, "duration": 3.81, "text": "theta so of course you can start"}, {"start": 22.08, "duration": 2.16, "text": "multiplying this all out but hey don't"}, {"start": 23.91, "duration": 2.73, "text": "do that"}, {"start": 24.24, "duration": 4.529, "text": "anytime you see cosine squared and sine"}, {"start": 26.64, "duration": 3.78, "text": "Squared's again I always think is there"}, {"start": 28.769, "duration": 5.011, "text": "any way to sort of bring this identity"}, {"start": 30.42, "duration": 6.6, "text": "into it so cosine squared theta plus"}, {"start": 33.78, "duration": 5.01, "text": "sine squared theta equals 1 well hey"}, {"start": 37.02, "duration": 4.379, "text": "that's exactly what we have sitting here"}, {"start": 38.79, "duration": 6.42, "text": "the first factor so certainly we can do"}, {"start": 41.399, "duration": 5.611, "text": "that notice in the second term we can"}, {"start": 45.21, "duration": 7.529, "text": "actually I think factor and negative out"}, {"start": 47.01, "duration": 10.04, "text": "of there so let's let's maybe do that so"}, {"start": 52.739, "duration": 8.37, "text": "I'm going to rewrite this whole"}, {"start": 57.05, "duration": 6.52, "text": "expression so we have tangent of theta"}, {"start": 61.109, "duration": 4.291, "text": "maybe we can already replace our cosine"}, {"start": 63.57, "duration": 3.96, "text": "squared plus sine squared we'll go ahead"}, {"start": 65.4, "duration": 5.64, "text": "and replace that with our positive 1"}, {"start": 67.53, "duration": 5.55, "text": "okay so just regrouping all that stuff"}, {"start": 71.04, "duration": 4.469, "text": "and replacing it with a plus one and"}, {"start": 73.08, "duration": 5.219, "text": "then on the right side we have tangent"}, {"start": 75.509, "duration": 5.281, "text": "of theta notice we could really factor"}, {"start": 78.299, "duration": 6.21, "text": "the negative out of there just out of"}, {"start": 80.79, "duration": 6.3, "text": "the cosine squared in the sine squared"}, {"start": 84.509, "duration": 3.691, "text": "if i factor the negative out I would"}, {"start": 87.09, "duration": 5.849, "text": "actually have to make that a positive"}, {"start": 88.2, "duration": 6.48, "text": "sine squared theta but again the same"}, {"start": 92.939, "duration": 4.841, "text": "thing as before now we've got cosine"}, {"start": 94.68, "duration": 5.57, "text": "squared theta plus sine squared theta"}, {"start": 97.78, "duration": 4.9, "text": "so I'm gonna make my brackets into"}, {"start": 100.25, "duration": 5.58, "text": "parenthesis it doesn't really matter if"}, {"start": 102.68, "duration": 6.479, "text": "they're all the same here so then we've"}, {"start": 105.83, "duration": 7.23, "text": "got tangent of theta minus well cosine"}, {"start": 109.159, "duration": 5.641, "text": "squared plus sine squared equals 1 again"}, {"start": 113.06, "duration": 3.8, "text": "we're gonna use negative 1 because well"}, {"start": 114.8, "duration": 4.26, "text": "there's a negative you know from before"}, {"start": 116.86, "duration": 5.47, "text": "and now what I'm going to do is just"}, {"start": 119.06, "duration": 5.4, "text": "simply multiply this stuff out so we're"}, {"start": 122.33, "duration": 3.899, "text": "gonna have tangent of theta times"}, {"start": 124.46, "duration": 5.159, "text": "tangent of theta which is going to give"}, {"start": 126.229, "duration": 5.611, "text": "us tangent squared theta will have"}, {"start": 129.619, "duration": 5.941, "text": "tangent times negative 1 that's gonna"}, {"start": 131.84, "duration": 6.21, "text": "give me a negative tangent theta will"}, {"start": 135.56, "duration": 5.64, "text": "have a positive 1 times tangent theta so"}, {"start": 138.05, "duration": 5.1, "text": "that's just plus tangent theta and then"}, {"start": 141.2, "duration": 4.59, "text": "we have plus 1 times negative 1 so"}, {"start": 143.15, "duration": 4.08, "text": "that's just gonna be a negative 1 well"}, {"start": 145.79, "duration": 3.15, "text": "the negative tangent theta and the"}, {"start": 147.23, "duration": 4.56, "text": "positive tangent theta are just gonna"}, {"start": 148.94, "duration": 6.39, "text": "cancel out we're left with tangent"}, {"start": 151.79, "duration": 5.61, "text": "squared theta minus 1 and again now I"}, {"start": 155.33, "duration": 5.76, "text": "think about this identity that relates"}, {"start": 157.4, "duration": 5.22, "text": "tangent that involves tangent squared so"}, {"start": 161.09, "duration": 5.369, "text": "there's one at least that says tangent"}, {"start": 162.62, "duration": 7.11, "text": "squared of theta plus 1 equals secant"}, {"start": 166.459, "duration": 5.551, "text": "squared theta and I think well you know"}, {"start": 169.73, "duration": 6.33, "text": "is there really much of anything to do"}, {"start": 172.01, "duration": 8.55, "text": "you know can I somehow can I somehow"}, {"start": 176.06, "duration": 6.09, "text": "manipulate this tangent squared theta is"}, {"start": 180.56, "duration": 3.51, "text": "there any way to what's a bit of a says"}, {"start": 182.15, "duration": 4.29, "text": "is there any way can I manipulate this"}, {"start": 184.07, "duration": 4.32, "text": "identity tangent squared theta plus 1"}, {"start": 186.44, "duration": 3.45, "text": "equals secant squared theta is there any"}, {"start": 188.39, "duration": 3.72, "text": "way I can sort of manipulate that to"}, {"start": 189.89, "duration": 4.8, "text": "make it say you know tangent squared"}, {"start": 192.11, "duration": 5.04, "text": "theta minus 1 well if you multiply both"}, {"start": 194.69, "duration": 5.16, "text": "sides by negative one that's going to"}, {"start": 197.15, "duration": 5.16, "text": "make the negative tangent squared minus"}, {"start": 199.85, "duration": 4.59, "text": "one so that's not what we need"}, {"start": 202.31, "duration": 5.459, "text": "you could always you know replace"}, {"start": 204.44, "duration": 6.299, "text": "tangent squared maybe with you know"}, {"start": 207.769, "duration": 4.621, "text": "secant squared minus one but again I"}, {"start": 210.739, "duration": 3.87, "text": "don't see how that's really gonna help"}, {"start": 212.39, "duration": 4.14, "text": "things simplify down very much at all so"}, {"start": 214.609, "duration": 3.961, "text": "I think I would probably just leave it"}, {"start": 216.53, "duration": 4.79, "text": "right here and I would consider that to"}, {"start": 218.57, "duration": 2.75, "text": "be simplified"}], "title": "Simplifying Products of Binomials Involving Trigonometric Functions, Ex 2", "description": "Thanks to all of you who support me on Patreon. You da real mvps!   $1 per month helps!! :)  https://www.patreon.com/patrickjmt !! Simplifying Products of Binomials Involving Trigonometric Functions, Ex 2", "lengthSeconds": 221, "width": 480, "height": 360}
{"video_id": "29VW-NAd3lA", "transcript": [{"start": 0.68, "duration": 3.679, "text": "okay in this video we're going to start"}, {"start": 2.0, "duration": 4.64, "text": "looking at converting between polar and"}, {"start": 4.359, "duration": 4.521, "text": "rectangular equations and again these"}, {"start": 6.64, "duration": 4.32, "text": "are kind of the uh the equations that"}, {"start": 8.88, "duration": 4.879, "text": "relate our polar coordinates to our"}, {"start": 10.96, "duration": 5.319, "text": "rectangular coordinates so we're going"}, {"start": 13.759, "duration": 4.481, "text": "to find the rectangular form of each of"}, {"start": 16.279, "duration": 3.801, "text": "these given polar"}, {"start": 18.24, "duration": 4.68, "text": "equations"}, {"start": 20.08, "duration": 7.88, "text": "so kind of the idea is so okay so here"}, {"start": 22.92, "duration": 8.48, "text": "we've got R = 3 * cosine of"}, {"start": 27.96, "duration": 5.32, "text": "theta well to me uh um we could always"}, {"start": 31.4, "duration": 3.159, "text": "on the left side we could even solve for"}, {"start": 33.28, "duration": 4.0, "text": "r on the"}, {"start": 34.559, "duration": 5.84, "text": "left we could have that R equals the"}, {"start": 37.28, "duration": 5.72, "text": "square root of x^2 +"}, {"start": 40.399, "duration": 4.081, "text": "y^2 but the problem is if I plug that on"}, {"start": 43.0, "duration": 4.199, "text": "the left I still don't know what to plug"}, {"start": 44.48, "duration": 4.599, "text": "in on the right so if only there was an"}, {"start": 47.199, "duration": 3.921, "text": "extra R hanging out on the right we"}, {"start": 49.079, "duration": 4.681, "text": "would have R cosine Theta and then I"}, {"start": 51.12, "duration": 3.8, "text": "could uh replace that with an X well"}, {"start": 53.76, "duration": 4.68, "text": "that's what I'm going to do I'll just"}, {"start": 54.92, "duration": 6.0, "text": "multiply both sides by R so by"}, {"start": 58.44, "duration": 5.96, "text": "R then we'll m multiply the right side"}, {"start": 60.92, "duration": 5.28, "text": "by R so we have uh r^2 equals I'm going"}, {"start": 64.4, "duration": 5.28, "text": "to write it as 3"}, {"start": 66.2, "duration": 5.72, "text": "R cosine Theta but now we can just start"}, {"start": 69.68, "duration": 5.0, "text": "using our formula so R 2 is the same"}, {"start": 71.92, "duration": 5.92, "text": "thing as x^2 +"}, {"start": 74.68, "duration": 5.84, "text": "y^2 r cosine Theta that's the same thing"}, {"start": 77.84, "duration": 6.68, "text": "as X so if you wanted to we could write"}, {"start": 80.52, "duration": 8.08, "text": "this as x^2 - 3x +"}, {"start": 84.52, "duration": 6.959, "text": "y^2 equal 0 and hey uh now we''ve turned"}, {"start": 88.6, "duration": 4.559, "text": "it into uh an equation involving"}, {"start": 91.479, "duration": 4.361, "text": "rectangular coordinates or cartisian"}, {"start": 93.159, "duration": 4.201, "text": "coordinates so that's all there is to it"}, {"start": 95.84, "duration": 4.48, "text": "um I would just"}, {"start": 97.36, "duration": 5.079, "text": "say so the solution here would just be"}, {"start": 100.32, "duration": 6.0, "text": "x^2 -"}, {"start": 102.439, "duration": 6.881, "text": "3x + y^2 ="}, {"start": 106.32, "duration": 3.0, "text": "0"}], "title": "Converting Between Polar and Rectangular Equations, Ex 1", "description": "Thanks to all of you who support me on Patreon. You da real mvps!   $1 per month helps!! :)  https://www.patreon.com/patrickjmt !! Converting Between Polar and Rectangular Equations, Ex 1.  In this video, I take one polar equation and write it in rectangular (cartesian) form.", "lengthSeconds": 107, "width": 480, "height": 360}
{"video_id": "hw8aSrlJ-UQ", "transcript": [{"start": 0.5, "duration": 3.0, "text": "Okay, in this video, I want to do another example of working  with fractions and,"}, {"start": 3.5, "duration": 4.9, "text": "what we're gonna do is we're gonna add or subtract some fractions that have different denominators,"}, {"start": 8.4, "duration": 5.3, "text": "and also have some variables present. So I think there's two main things that you have to remember"}, {"start": 13.7, "duration": 4.32, "text": "when you're adding or subtracting fractions. Uh, those two things would be"}, {"start": 18.08, "duration": 3.12, "text": "that you have to common denominators; that's the first thing."}, {"start": 21.2, "duration": 4.64, "text": "So notice neither of my examples we have common denominators, so we'll have to fix that a little bit."}, {"start": 25.84, "duration": 5.22, "text": "The other thing is once you do have common denominators, the arithmetic occurs in the numerator."}, {"start": 31.06, "duration": 4.5, "text": "It happens in the top part of the fraction, so let's do our first example here."}, {"start": 35.56, "duration": 7.22, "text": "We have 3x/5 minus 2x/15, so definitely different numbers in the denominator."}, {"start": 42.78, "duration": 5.9, "text": "And, what I first think of is I think what's the greatest, excuse me, what's the least common multiple?"}, {"start": 48.68, "duration": 5.68, "text": "What's the least common multiple of 5 and 15? What's the first number that both of those divide into evenly?"}, {"start": 54.36, "duration": 4.14, "text": "And the least common multiple of those two numbers would be 15."}, {"start": 58.5, "duration": 4.62, "text": "So, I think well, I want a 15 here in my bottom fraction on the first side, and I think"}, {"start": 63.12, "duration": 1.92, "text": "well, how do I do that?"}, {"start": 65.04, "duration": 5.26, "text": "The only real way to, to change the denominator is to multiply it by something."}, {"start": 70.3, "duration": 3.68, "text": "So, I think in this case, we would have to multiply the denominator by 3,"}, {"start": 73.98, "duration": 5.34, "text": "but remember if you do it to the denominator, you also have to do it to the numerator."}, {"start": 79.32, "duration": 6.72, "text": "And this is all valid because really we're just multiplying by the number 3 over 3 which is 1."}, {"start": 86.04, "duration": 3.52, "text": "If you multiply something by 1, it doesn't change its value."}, {"start": 89.56, "duration": 6.12, "text": "So we may change how it looks, but numerically we would still, we would still get the same value."}, {"start": 95.68, "duration": 10.7, "text": "Okay, so 3 times 3x would be 9x, and hey now I've got my 15 on the bottom.; that's why we did it, minus 2x/15."}, {"start": 106.38, "duration": 9.48, "text": "And now I can just do the arithmetic in the numerator. I do 9x minus 2x. 9x minus 2x would be 7x/15."}, {"start": 115.86, "duration": 6.22, "text": "And if you wanted to, you could, uh, pull the x out to the side and write it as (7/15)x."}, {"start": 122.08, "duration": 8.48, "text": "Alrighty, in my next example here we've got 3y/4 and 5y/24,"}, {"start": 130.56, "duration": 6.72, "text": "so the first thing I'm gonna have to do again is get common denominators.  So I think"}, {"start": 137.28, "duration": 2.9, "text": "what's the least common multiple of 4 and 24?"}, {"start": 140.18, "duration": 4.12, "text": "and in this case I believe the least common multiple would be 24."}, {"start": 144.3, "duration": 4.54, "text": "So we would have to multiply the numerator and the denominator by 6."}, {"start": 148.84, "duration": 5.14, "text": "Again, the important thing is you gotta multiply the denominator by 6 to get the 24, but as a result"}, {"start": 153.98, "duration": 1.8, "text": "we also have to do it to the numerator."}, {"start": 155.78, "duration": 9.66, "text": "So on top, we get, uh, 18y/24 plus 5y/24"}, {"start": 165.44, "duration": 6.72, "text": "If we add 18y and 5y, we simply get 23y/24."}, {"start": 172.16, "duration": 7.98, "text": "And again, we could write this as, we could pull the y out to the side 23/24 times y."}, {"start": 180.16, "duration": 3.98, "text": "And now we have combined our fractions."}], "title": "Fractions:  Adding and Subtracting Fractions with Unlike Denominators", "description": "Thanks to all of you who support me on Patreon. You da real mvps!   $1 per month helps!! :)  https://www.patreon.com/patrickjmt !! Fractions:  Adding and Subtracting Fractions with Different Denominators and containing variables!\r\nTwo examples!", "lengthSeconds": 185, "width": 480, "height": 360}
{"video_id": "ILjgHfT58U8", "transcript": [{"start": 0.88, "duration": 2.32, "text": "Okay, so here we've got a little graph above that displays"}, {"start": 3.2, "duration": 4.56, "text": "the total cost, C, in dollars of renting a boat for h hours."}, {"start": 8.08, "duration": 1.52, "text": "So, my graph's a little rough, and I've labeled"}, {"start": 9.6, "duration": 4.62, "text": "so this would be the point (0,5), there on the vertical axis."}, {"start": 14.66, "duration": 4.96, "text": "It goes through (1,8), (2,11), (3,14), (4,17)."}, {"start": 19.7, "duration": 1.46, "text": "Just to give you some more points."}, {"start": 21.16, "duration": 4.2, "text": "So, the first question here is, what does the C-intercept represent in the graph?"}, {"start": 25.98, "duration": 1.6, "text": "Is it 'the initial cost of renting the boat',"}, {"start": 28.1, "duration": 2.0, "text": "'the total number of boats rented',"}, {"start": 30.1, "duration": 2.58, "text": "'the total number of hours the boat is rented',"}, {"start": 32.68, "duration": 3.92, "text": "or 'the increase in cost to rent the boat for each additional hours'?"}, {"start": 37.14, "duration": 5.52, "text": "So, again, it tells us that the vertical axis represents the total cost in dollars."}, {"start": 43.28, "duration": 4.44, "text": "So, the idea is if you go, you know, if you rent a boat  for zero hours,"}, {"start": 48.18, "duration": 2.62, "text": "maybe had plans to go and you don't get out there."}, {"start": 51.3, "duration": 7.52, "text": "It says, it's still gonna cost you five dollars, after being on the boat for one hour, it's jumped up to eight dollars."}, {"start": 59.34, "duration": 6.36, "text": "So, the idea is the initial cost is gonna be this five dollars."}, {"start": 65.7, "duration": 5.32, "text": "So, answer choice A is clearly correct here. It's 'the initial cost of renting the boat'."}, {"start": 71.02, "duration": 2.22, "text": "It's not 'the total number of boats rented', right?"}, {"start": 73.24, "duration": 4.12, "text": "It has nothing to do with the vertical axis, it has nothing to do with the number of boats."}, {"start": 77.36, "duration": 1.42, "text": "It tells you the cost."}, {"start": 79.26, "duration": 5.28, "text": "Again, 'the total number of hours the boat is rented', that's answer choice C,"}, {"start": 84.64, "duration": 6.08, "text": "again, the vertical axis has nothing to do with the number of hours it has to do with the cost."}, {"start": 91.36, "duration": 5.18, "text": "And D says 'the increase in cost to rent the boat for each additional hour'."}, {"start": 96.54, "duration": 3.44, "text": "Well, no 'cause, again, the C-intercept has a value of five"}, {"start": 99.98, "duration": 2.92, "text": "and if we go an additional hour,  it's only jumping up to eight,"}, {"start": 103.38, "duration": 2.34, "text": "two hours, it's only jumping up to eleven."}, {"start": 105.72, "duration": 3.56, "text": "So, it's only jumping up by three dollars per hour."}, {"start": 110.76, "duration": 2.52, "text": "Definitely wouldn't be answer choice D either."}, {"start": 113.72, "duration": 4.28, "text": "Again, if I saw this on the test, answer choice A. Boom and I would be on to the next question."}], "title": "SAT Math - Question 15, Section 2, 2016 Edition and Beyond", "description": "Thanks to all of you who support me on Patreon. You da real mvps!   $1 per month helps!! :)  https://www.patreon.com/patrickjmt !! SAT Math - Question 14, 2016 Edition and Beyond.\nSection 2 - A Calculator IS allowed!", "lengthSeconds": 118, "width": 480, "height": 360}
{"video_id": "mH6JRjE7Egk", "transcript": [{"start": 0.199, "duration": 5.261, "text": "okay here we're gonna look at solving a"}, {"start": 2.929, "duration": 3.91, "text": "few different equations more"}, {"start": 5.46, "duration": 3.66, "text": "specifically what are known as linear"}, {"start": 6.839, "duration": 5.88, "text": "equations and we'll have to do a couple"}, {"start": 9.12, "duration": 6.03, "text": "a couple steps to solve them so in Part"}, {"start": 12.719, "duration": 6.031, "text": "A here we have negative 4x minus 14"}, {"start": 15.15, "duration": 6.03, "text": "equals 10 typically the way I always"}, {"start": 18.75, "duration": 4.949, "text": "thought about it is first I want to get"}, {"start": 21.18, "duration": 4.32, "text": "for an equation where there's an X to"}, {"start": 23.699, "duration": 4.41, "text": "the first power or variable to the first"}, {"start": 25.5, "duration": 5.279, "text": "power I want to get the term involving"}, {"start": 28.109, "duration": 5.851, "text": "just X all by itself so if there are"}, {"start": 30.779, "duration": 5.221, "text": "other X's I'll try to combine them so on"}, {"start": 33.96, "duration": 4.14, "text": "the left side I see two terms I see one"}, {"start": 36.0, "duration": 4.14, "text": "involving X and then I see just a number"}, {"start": 38.1, "duration": 6.029, "text": "and then on the other side I see just a"}, {"start": 40.14, "duration": 7.2, "text": "number so to get the the X's basically"}, {"start": 44.129, "duration": 6.84, "text": "by itself since I've subtracting 14 I'm"}, {"start": 47.34, "duration": 5.1, "text": "going to add 14 to both sides so on the"}, {"start": 50.969, "duration": 4.261, "text": "left side that's gonna leave me with"}, {"start": 52.44, "duration": 6.63, "text": "negative 4x equals well let's see 10"}, {"start": 55.23, "duration": 5.16, "text": "plus 14 is going to be 24 and now what I"}, {"start": 59.07, "duration": 6.33, "text": "need to do at this point since I'm"}, {"start": 60.39, "duration": 6.93, "text": "multiplying by negative 4 to get the X"}, {"start": 65.4, "duration": 6.21, "text": "by itself I'm just simply going to"}, {"start": 67.32, "duration": 7.369, "text": "divide both sides by negative 4 so"}, {"start": 71.61, "duration": 6.42, "text": "negative 4 divided by negative 4 is 1x"}, {"start": 74.689, "duration": 4.121, "text": "let's see a positive over a negative is"}, {"start": 78.03, "duration": 4.89, "text": "a negative"}, {"start": 78.81, "duration": 6.54, "text": "24 divided by 4 will give us 6 so the"}, {"start": 82.92, "duration": 5.839, "text": "solution to our first equation will be x"}, {"start": 85.35, "duration": 3.409, "text": "equals negative 6"}], "title": "Solving Two - Step Linear Equations", "description": "Thanks to all of you who support me on Patreon. You da real mvps!   $1 per month helps!! :)  https://www.patreon.com/patrickjmt !! Solving Two - Step Linear Equations.  Here we look at solving linear equations where we have to perform addition / subtraction as well as division / multiplication to solve.", "lengthSeconds": 88, "width": 480, "height": 360}
{"video_id": "V6YoXOEq-Ng", "transcript": [{"start": 0.02, "duration": 4.6, "text": "all right in this video we're gonna do a"}, {"start": 2.58, "duration": 6.269, "text": "couple questions related to finding the"}, {"start": 4.62, "duration": 6.75, "text": "period of a tangent and a secant"}, {"start": 8.849, "duration": 4.741, "text": "function things have been changed here a"}, {"start": 11.37, "duration": 5.04, "text": "little bit so we've got three tangent of"}, {"start": 13.59, "duration": 5.1, "text": "2 times the quantity X minus 1 plus a"}, {"start": 16.41, "duration": 5.07, "text": "and then in Part B we have negative 4"}, {"start": 18.69, "duration": 6.87, "text": "secant of 4 times the quantity X plus PI"}, {"start": 21.48, "duration": 5.91, "text": "minus the square root of 2 so recall the"}, {"start": 25.56, "duration": 4.2, "text": "formula before we have been using for"}, {"start": 27.39, "duration": 5.91, "text": "the period when we were doing sine and"}, {"start": 29.76, "duration": 7.02, "text": "cosine we were using 2pi divided by the"}, {"start": 33.3, "duration": 5.25, "text": "absolute value of b but that's because"}, {"start": 36.78, "duration": 5.13, "text": "sine and cosine had a period of 2pi"}, {"start": 38.55, "duration": 6.89, "text": "recall the tangent just plain old"}, {"start": 41.91, "duration": 3.53, "text": "tangent X has a period"}, {"start": 47.6, "duration": 5.55, "text": "of just pie okay so okay the next has a"}, {"start": 50.96, "duration": 4.17, "text": "period of just pie so instead of taking"}, {"start": 53.15, "duration": 4.2, "text": "two pie and dividing it by the absolute"}, {"start": 55.13, "duration": 4.86, "text": "value of B all we have to do is just"}, {"start": 57.35, "duration": 6.78, "text": "take PI and divide it by our absolute"}, {"start": 59.99, "duration": 6.48, "text": "value of B well again our B value is"}, {"start": 64.13, "duration": 6.03, "text": "just whatever is the coefficient on the"}, {"start": 66.47, "duration": 5.19, "text": "X inside the brackets so you can imagine"}, {"start": 70.16, "duration": 3.75, "text": "if you distribute it you would have 2x"}, {"start": 71.66, "duration": 4.35, "text": "or again equivalently this is gonna be"}, {"start": 73.91, "duration": 5.88, "text": "our B value so we would have the"}, {"start": 76.01, "duration": 5.76, "text": "absolute value of 2 which is just 2 so"}, {"start": 79.79, "duration": 6.39, "text": "our period would actually be just PI"}, {"start": 81.77, "duration": 5.43, "text": "over 2 in this case and again just kind"}, {"start": 86.18, "duration": 3.72, "text": "of the effect that would have"}, {"start": 87.2, "duration": 5.25, "text": "remember the the asymptotes normally for"}, {"start": 89.9, "duration": 5.19, "text": "tangent or at PI over 2 and negative PI"}, {"start": 92.45, "duration": 5.01, "text": "over 2 and then of course there's others"}, {"start": 95.09, "duration": 8.52, "text": "now our asymptotes will get pulled into"}, {"start": 97.46, "duration": 9.24, "text": "PI over 4 and also negative PI over 4 so"}, {"start": 103.61, "duration": 5.28, "text": "this is definitely not the graph of this"}, {"start": 106.7, "duration": 6.15, "text": "one but this would be the graph of"}, {"start": 108.89, "duration": 5.97, "text": "tangent of 2x again just a little a"}, {"start": 112.85, "duration": 3.78, "text": "little segment of it there so again"}, {"start": 114.86, "duration": 4.619, "text": "that's the effect it's just compressing"}, {"start": 116.63, "duration": 6.419, "text": "the graph by a factor of 2 to get the"}, {"start": 119.479, "duration": 7.981, "text": "period for our second one here"}, {"start": 123.049, "duration": 8.401, "text": "well secant is really just 1 over cosine"}, {"start": 127.46, "duration": 7.23, "text": "cosine does have period 2pi secant also"}, {"start": 131.45, "duration": 5.25, "text": "has period 2pi so now we go do you go"}, {"start": 134.69, "duration": 5.1, "text": "back to our original formula that we saw"}, {"start": 136.7, "duration": 7.47, "text": "before 2pi divided by the absolute value"}, {"start": 139.79, "duration": 7.68, "text": "of B so we'll get 2pi over in this case"}, {"start": 144.17, "duration": 7.16, "text": "4 well 2 over 4 it looks like again hey"}, {"start": 147.47, "duration": 6.63, "text": "we're getting a period of PI over 2 so"}, {"start": 151.33, "duration": 5.8, "text": "only thing to remember again tangent and"}, {"start": 154.1, "duration": 5.88, "text": "cotangent have a period of Pi sine"}, {"start": 157.13, "duration": 5.37, "text": "cosine secant cosecant all have a period"}, {"start": 159.98, "duration": 7.039, "text": "of 2pi and then it's the same thing just"}, {"start": 162.5, "duration": 4.519, "text": "dividing by the absolute value of your B"}], "title": "Basic Questions Related to Tangent, Cotangent, Secant, Cosecant, Ex 3", "description": "Thanks to all of you who support me on Patreon. You da real mvps!   $1 per month helps!! :)  https://www.patreon.com/patrickjmt !! Basic Questions Related to Tangent, Cotangent, Secant, Cosecant, Ex 3. In this video, I just address a few basic questions about Tangent, Cotangent, Secant, Cosecant.", "lengthSeconds": 166, "width": 480, "height": 360}
{"video_id": "4U68TZhBgVA", "transcript": [{"start": 0.03, "duration": 3.24, "text": "all right in this video I'm gonna do"}, {"start": 1.68, "duration": 4.949, "text": "another example of solving an equation"}, {"start": 3.27, "duration": 4.799, "text": "that involves rational exponents so I"}, {"start": 6.629, "duration": 4.171, "text": "should have probably a little lowercase"}, {"start": 8.069, "duration": 5.551, "text": "Y so let's solve for y here so we've got"}, {"start": 10.8, "duration": 5.88, "text": "4 times the quantity y squared minus y"}, {"start": 13.62, "duration": 5.19, "text": "plus 1 to the 1/4 the exponent it's only"}, {"start": 16.68, "duration": 4.439, "text": "applying to the stuff in parentheses not"}, {"start": 18.81, "duration": 6.6, "text": "to the 4 and on the right side we have"}, {"start": 21.119, "duration": 6.0, "text": "16 y all being raised to the 1/2 so what"}, {"start": 25.41, "duration": 4.199, "text": "I do is again I basically look at the"}, {"start": 27.119, "duration": 5.16, "text": "denominators so the denominator is a 2"}, {"start": 29.609, "duration": 4.86, "text": "and a 4 and what I try to do is I try to"}, {"start": 32.279, "duration": 5.251, "text": "find the least common multiple of those"}, {"start": 34.469, "duration": 6.121, "text": "numbers that is I'm trying to find the"}, {"start": 37.53, "duration": 5.61, "text": "first number that both 2 and 4 divide"}, {"start": 40.59, "duration": 5.43, "text": "into evenly and the least common"}, {"start": 43.14, "duration": 5.099, "text": "multiple of the numbers 2 & 4 that is"}, {"start": 46.02, "duration": 4.19, "text": "the number 4 that's the first number"}, {"start": 48.239, "duration": 4.561, "text": "that both of them divide into evenly"}, {"start": 50.21, "duration": 7.57, "text": "when I determine this number what I do"}, {"start": 52.8, "duration": 8.22, "text": "is I raise both sides to exactly that"}, {"start": 57.78, "duration": 5.22, "text": "power so I'm just exactly rewriting the"}, {"start": 61.02, "duration": 5.22, "text": "left side but now I'm gonna write it all"}, {"start": 63.0, "duration": 6.99, "text": "to the fourth power and on the right"}, {"start": 66.24, "duration": 7.02, "text": "side I've got 16 Y to the 1/2 again I'm"}, {"start": 69.99, "duration": 6.6, "text": "gonna raise all of that to the fourth"}, {"start": 73.26, "duration": 6.6, "text": "power okay so on the left side again we"}, {"start": 76.59, "duration": 6.18, "text": "have to be a little careful because this"}, {"start": 79.86, "duration": 6.96, "text": "exponent has to get applied to both the"}, {"start": 82.77, "duration": 7.05, "text": "4 and the factor y squared minus y plus"}, {"start": 86.82, "duration": 5.25, "text": "1 so we'll actually get 4 to the fourth"}, {"start": 89.82, "duration": 6.689, "text": "so again if I just multiply the 1 in the"}, {"start": 92.07, "duration": 7.38, "text": "4 but on the y squared minus y plus 1 if"}, {"start": 96.509, "duration": 6.351, "text": "I take the 1/4 times the 4 that's just"}, {"start": 99.45, "duration": 3.41, "text": "going to give me to the first power"}, {"start": 103.39, "duration": 6.24, "text": "on the right side notice we're gonna get"}, {"start": 106.42, "duration": 7.16, "text": "okay so now notice the 16 why's in"}, {"start": 109.63, "duration": 8.19, "text": "parentheses okay so I'm just gonna take"}, {"start": 113.58, "duration": 7.02, "text": "16 Y and I'm gonna take 1/2 times 4 and"}, {"start": 117.82, "duration": 5.52, "text": "that's gonna give me to the second power"}, {"start": 120.6, "duration": 5.71, "text": "okay so now we can keep simplifying here"}, {"start": 123.34, "duration": 7.32, "text": "a little bit more so 4 to the fourth is"}, {"start": 126.31, "duration": 7.53, "text": "actually 16 squared which is 256 then we"}, {"start": 130.66, "duration": 5.22, "text": "have Y squared minus y plus 1 left over"}, {"start": 133.84, "duration": 5.85, "text": "well conveniently on the right we have"}, {"start": 135.88, "duration": 6.99, "text": "16 squared or 256 we also have to square"}, {"start": 139.69, "duration": 5.94, "text": "the Y so the first thing I'm going to do"}, {"start": 142.87, "duration": 4.41, "text": "here this is starting to type well this"}, {"start": 145.63, "duration": 4.59, "text": "is now certainly a quadratic equation"}, {"start": 147.28, "duration": 8.34, "text": "I'm going to divide both sides by 256"}, {"start": 150.22, "duration": 7.53, "text": "just to get rid of those coefficients so"}, {"start": 155.62, "duration": 6.06, "text": "on the left side we've got W squared"}, {"start": 157.75, "duration": 7.29, "text": "minus y plus 1 on the right we're simply"}, {"start": 161.68, "duration": 5.309, "text": "going to be left with a W squared and"}, {"start": 165.04, "duration": 5.91, "text": "now this is nice because if we subtract"}, {"start": 166.989, "duration": 7.141, "text": "I keep saying W if we subtract Y squared"}, {"start": 170.95, "duration": 6.12, "text": "from both sides so minus y squared minus"}, {"start": 174.13, "duration": 6.66, "text": "y squared we'll be left with negative y"}, {"start": 177.07, "duration": 7.11, "text": "plus 1 equals 0 and now I can simply add"}, {"start": 180.79, "duration": 4.919, "text": "Y to both sides and we're going to give"}, {"start": 184.18, "duration": 4.05, "text": "our solution again you should"}, {"start": 185.709, "duration": 4.441, "text": "technically whenever you're dealing with"}, {"start": 188.23, "duration": 3.39, "text": "fractional exponents it's a good rule of"}, {"start": 190.15, "duration": 4.11, "text": "thumb to go back and check these"}, {"start": 191.62, "duration": 4.98, "text": "solutions but I think if you go back and"}, {"start": 194.26, "duration": 6.289, "text": "check you will in fact find that y"}, {"start": 196.6, "duration": 3.949, "text": "equals 1 is a solution"}], "title": "Solving an Equation Involving Rational Exponents - Example 2", "description": "Thanks to all of you who support me on Patreon. You da real mvps!   $1 per month helps!! :)  https://www.patreon.com/patrickjmt !! Solving an Equation Involving Rational Exponents - Example 2", "lengthSeconds": 199, "width": 480, "height": 360}
{"video_id": "Hj3pBcf_ZfA", "transcript": [{"start": 0.56, "duration": 2.56, "text": "all right in this video we're going to"}, {"start": 1.599, "duration": 4.2, "text": "do an example of solving a trigger"}, {"start": 3.12, "duration": 4.56, "text": "equation using the quadratic formula so"}, {"start": 5.799, "duration": 4.401, "text": "we're going to uh do got three different"}, {"start": 7.68, "duration": 4.959, "text": "examples here let's try to find uh all"}, {"start": 10.2, "duration": 3.96, "text": "Solutions in the interval from 0 to 2 pi"}, {"start": 12.639, "duration": 5.881, "text": "for these and the first one will be"}, {"start": 14.16, "duration": 6.24, "text": "cosine ^ 2 x - cosine x + 1 equals z and"}, {"start": 18.52, "duration": 4.04, "text": "you know you can certainly try to uh"}, {"start": 20.4, "duration": 3.799, "text": "sort of factor this out I think you'll"}, {"start": 22.56, "duration": 4.84, "text": "convince yourself that you can't really"}, {"start": 24.199, "duration": 8.24, "text": "Factor it out um I'm just going to"}, {"start": 27.4, "duration": 6.44, "text": "rewrite this as cosine x uh 2 - 1 *"}, {"start": 32.439, "duration": 6.12, "text": "cosine"}, {"start": 33.84, "duration": 7.039, "text": "x + 1 = 0 and the way the reason I'm I'm"}, {"start": 38.559, "duration": 5.601, "text": "writing this is this is what we could"}, {"start": 40.879, "duration": 5.881, "text": "call a quadratic equation uh not an X"}, {"start": 44.16, "duration": 4.879, "text": "but an cosine X so kind of think you"}, {"start": 46.76, "duration": 8.479, "text": "know sort of similarly uh suppose this"}, {"start": 49.039, "duration": 9.121, "text": "just said x^2 - 1 * x + 1 equals"}, {"start": 55.239, "duration": 5.64, "text": "0 this is called a quadratic equation in"}, {"start": 58.16, "duration": 6.92, "text": "X this would be a quadratic equation in"}, {"start": 60.879, "duration": 6.041, "text": "cosine X now normally uh you know since"}, {"start": 65.08, "duration": 3.399, "text": "we can't find solutions by factoring at"}, {"start": 66.92, "duration": 3.6, "text": "least I can't figure out how to factor"}, {"start": 68.479, "duration": 6.68, "text": "it we can always use the quadratic"}, {"start": 70.52, "duration": 12.08, "text": "formula so remember X = B plus or minus"}, {"start": 75.159, "duration": 10.28, "text": "the square < TK of b^2 - 4 a c over 2 a"}, {"start": 82.6, "duration": 5.64, "text": "now that's what we would do um if we had"}, {"start": 85.439, "duration": 5.601, "text": "X here and again remember a is the"}, {"start": 88.24, "duration": 6.36, "text": "coefficient in front of the x\u00b2"}, {"start": 91.04, "duration": 4.8, "text": "B is the number in front of the X and C"}, {"start": 94.6, "duration": 4.199, "text": "is just the"}, {"start": 95.84, "duration": 5.84, "text": "constant the exact same thing uh holds"}, {"start": 98.799, "duration": 6.6, "text": "true for this equation uh this quadratic"}, {"start": 101.68, "duration": 5.479, "text": "equation in cosine x uh the a the B and"}, {"start": 105.399, "duration": 3.881, "text": "the C are still going to be the same the"}, {"start": 107.159, "duration": 4.881, "text": "only difference is instead of Simply"}, {"start": 109.28, "duration": 5.36, "text": "writing x equals blah blah blah we would"}, {"start": 112.04, "duration": 5.719, "text": "have well now the thing being squared"}, {"start": 114.64, "duration": 4.96, "text": "isn't just X but it's cosine X so"}, {"start": 117.759, "duration": 3.68, "text": "equivalently what we would do is we"}, {"start": 119.6, "duration": 4.439, "text": "would plug in cosine"}, {"start": 121.439, "duration": 5.72, "text": "X but now we get the coefficients in the"}, {"start": 124.039, "duration": 5.401, "text": "same way I look at uh sort of the cosine"}, {"start": 127.159, "duration": 4.761, "text": "to the first Power the number in front"}, {"start": 129.44, "duration": 5.439, "text": "of that would be my B value so I'll get"}, {"start": 131.92, "duration": 6.0, "text": "negative"}, {"start": 134.879, "duration": 6.681, "text": "um1 plus or minus the square root of"}, {"start": 137.92, "duration": 7.28, "text": "again b^ 2 which would be -1"}, {"start": 141.56, "duration": 4.56, "text": "2ar - 4 * a which again would just be"}, {"start": 145.2, "duration": 3.72, "text": "positive"}, {"start": 146.12, "duration": 5.0, "text": "1 times C which is going to be the"}, {"start": 148.92, "duration": 6.16, "text": "constant which is just POS"}, {"start": 151.12, "duration": 6.759, "text": "1 all over 2 * a which we said is"}, {"start": 155.08, "duration": 4.92, "text": "1 and now what we do is we just have a"}, {"start": 157.879, "duration": 5.161, "text": "new equation involving cosine X that we"}, {"start": 160.0, "duration": 4.76, "text": "would have to solve but I think in this"}, {"start": 163.04, "duration": 4.24, "text": "case at least uh so we get a negative"}, {"start": 164.76, "duration": 4.72, "text": "and negative which will make positive 1"}, {"start": 167.28, "duration": 6.48, "text": "I think underneath the radical Well -"}, {"start": 169.48, "duration": 7.6, "text": "1^2 is 1 we'll get 1 - 4 which would be"}, {"start": 173.76, "duration": 6.08, "text": "-3 well we can't uh I mean we can use"}, {"start": 177.08, "duration": 5.32, "text": "complex numbers uh you know to find Sol"}, {"start": 179.84, "duration": 5.36, "text": "Solutions but what we want to do is find"}, {"start": 182.4, "duration": 4.479, "text": "real valued Solutions so since we have a"}, {"start": 185.2, "duration": 3.959, "text": "negative number underneath the square"}, {"start": 186.879, "duration": 7.08, "text": "root uh we would simply say that this"}, {"start": 189.159, "duration": 4.8, "text": "equation has no real uh"}, {"start": 194.56, "duration": 5.16, "text": "Solutions we'll look at an example uh I"}, {"start": 197.36, "duration": 4.12, "text": "believe the next one certainly for sure"}, {"start": 199.72, "duration": 3.879, "text": "um so we we'll definitely see at least"}, {"start": 201.48, "duration": 3.64, "text": "an example somewhere where you know you"}, {"start": 203.599, "duration": 3.041, "text": "do get a solution and hey what do you"}, {"start": 205.12, "duration": 2.8, "text": "have to do from there because in that"}, {"start": 206.64, "duration": 3.44, "text": "case you're going to have to usually"}, {"start": 207.92, "duration": 3.72, "text": "typically use inverse cosine or invers"}, {"start": 210.08, "duration": 3.48, "text": "sign the inverse of whatever your trig"}, {"start": 211.64, "duration": 4.2, "text": "function is uh you'll have to think"}, {"start": 213.56, "duration": 3.72, "text": "about you know all solutions so a little"}, {"start": 215.84, "duration": 3.759, "text": "trick here but again this is kind of the"}, {"start": 217.28, "duration": 5.319, "text": "basic idea um that I want to point out"}, {"start": 219.599, "duration": 4.84, "text": "in this example it's still a quadratic"}, {"start": 222.599, "duration": 3.84, "text": "the coefficients you know still think"}, {"start": 224.439, "duration": 3.961, "text": "about the coefficients the same way"}, {"start": 226.439, "duration": 3.961, "text": "again just instead of putting x equals"}, {"start": 228.4, "duration": 4.24, "text": "blah blah blah you'll have to put you"}, {"start": 230.4, "duration": 3.8, "text": "know whatever trig function is being"}, {"start": 232.64, "duration": 4.799, "text": "squared you'll have to put that on the"}, {"start": 234.2, "duration": 3.239, "text": "left side"}], "title": "Solving Trigonometric Equations Using the Quadratic Formula - Example 1", "description": "Thanks to all of you who support me on Patreon. You da real mvps!   $1 per month helps!! :)  https://www.patreon.com/patrickjmt !! Solving Trigonometric Equations Using the Quadratic Formula - Example 1", "lengthSeconds": 236, "width": 480, "height": 360}
{"video_id": "Bq7Jd7vXC08", "transcript": [{"start": 0.599, "duration": 3.161, "text": "all right in this video I want to do an"}, {"start": 2.04, "duration": 4.279, "text": "example of finding the inverse of a"}, {"start": 3.76, "duration": 5.08, "text": "function if one exists and if it doesn't"}, {"start": 6.319, "duration": 5.121, "text": "exist we'll justify that as well so in"}, {"start": 8.84, "duration": 6.839, "text": "our first example here we'll use f ofx ="}, {"start": 11.44, "duration": 6.36, "text": "8 - x^2 over 5 so the first thing uh"}, {"start": 15.679, "duration": 4.081, "text": "that we'll do when we find an inverse I"}, {"start": 17.8, "duration": 6.12, "text": "always just write it you know instead of"}, {"start": 19.76, "duration": 9.359, "text": "F ofx I write y so we have 8 - x^2 over"}, {"start": 23.92, "duration": 7.92, "text": "5 = Y and then um the Y I replace that"}, {"start": 29.119, "duration": 5.321, "text": "simply with an X and everywhere that"}, {"start": 31.84, "duration": 5.399, "text": "there's an x uh in the original function"}, {"start": 34.44, "duration": 5.4, "text": "I'll replace that with those with y's so"}, {"start": 37.239, "duration": 4.721, "text": "here there's obviously only uh One X to"}, {"start": 39.84, "duration": 4.239, "text": "replace but you could end up with"}, {"start": 41.96, "duration": 3.84, "text": "certainly more than one X that turns"}, {"start": 44.079, "duration": 4.041, "text": "into multiple"}, {"start": 45.8, "duration": 4.96, "text": "y's okay so all we're going to do here"}, {"start": 48.12, "duration": 5.16, "text": "now is simply try to solve for y so I"}, {"start": 50.76, "duration": 5.4, "text": "can multiply both sides by five to get"}, {"start": 53.28, "duration": 7.72, "text": "rid of my denominator that'll give me 5x"}, {"start": 56.16, "duration": 6.68, "text": "on the left = 8 - y^2"}, {"start": 61.0, "duration": 4.88, "text": "um I'm going to do two steps at once"}, {"start": 62.84, "duration": 7.44, "text": "here I'm going to add the y^2 to the"}, {"start": 65.88, "duration": 7.16, "text": "left side so now we would have positive"}, {"start": 70.28, "duration": 5.76, "text": "y^2 the 8 he's still hanging out on the"}, {"start": 73.04, "duration": 4.48, "text": "right and the 5x I'm going to subtract"}, {"start": 76.04, "duration": 6.48, "text": "that from both"}, {"start": 77.52, "duration": 7.559, "text": "sides so I'll have y^2 = 8 - 5x and now"}, {"start": 82.52, "duration": 4.4, "text": "what I have to do to get rid of the uh"}, {"start": 85.079, "duration": 3.121, "text": "the square is I'll take the square root"}, {"start": 86.92, "duration": 3.6, "text": "of both"}, {"start": 88.2, "duration": 5.08, "text": "sides but remember when you take the"}, {"start": 90.52, "duration": 6.84, "text": "square root you have to put a positive"}, {"start": 93.28, "duration": 6.6, "text": "and negative on um one of the sides and"}, {"start": 97.36, "duration": 5.399, "text": "that's true for any even powered root"}, {"start": 99.88, "duration": 6.16, "text": "we'll get y on the left equal"}, {"start": 102.759, "duration": 6.921, "text": "posi 8 -"}, {"start": 106.04, "duration": 5.8, "text": "5x okay and this immediately tells me"}, {"start": 109.68, "duration": 5.2, "text": "that in fact there is no"}, {"start": 111.84, "duration": 5.76, "text": "inverse and again kind of the idea is it"}, {"start": 114.88, "duration": 5.4, "text": "says for two distinct inputs you know if"}, {"start": 117.6, "duration": 5.08, "text": "you plug in something for x you're going"}, {"start": 120.28, "duration": 4.199, "text": "to get two different uh basically two"}, {"start": 122.68, "duration": 3.479, "text": "different values you know if you plug in"}, {"start": 124.479, "duration": 3.201, "text": "zero you would get the square OT of 8"}, {"start": 126.159, "duration": 4.001, "text": "you would get positive negative square"}, {"start": 127.68, "duration": 4.68, "text": "root of 8 and that says for one input"}, {"start": 130.16, "duration": 4.52, "text": "you're getting multiple outputs and that"}, {"start": 132.36, "duration": 4.04, "text": "means you don't have a function and uh"}, {"start": 134.68, "duration": 3.199, "text": "that's what we want we want when we find"}, {"start": 136.4, "duration": 6.28, "text": "the inverse we want that to be a"}, {"start": 137.879, "duration": 4.801, "text": "function so we'll say no inverse"}, {"start": 143.12, "duration": 4.199, "text": "function and another way to see this"}, {"start": 145.319, "duration": 3.961, "text": "really quickly um if you think about it"}, {"start": 147.319, "duration": 6.761, "text": "you could always break this fraction up"}, {"start": 149.28, "duration": 6.36, "text": "as 8 fths - x^2 over 5 or I can write"}, {"start": 154.08, "duration": 4.68, "text": "that as 1/5"}, {"start": 155.64, "duration": 6.08, "text": "x^2 and if you remember your graphs here"}, {"start": 158.76, "duration": 6.52, "text": "um the graph of x^2 is just a"}, {"start": 161.72, "duration": 7.28, "text": "parabola um netive x^2 would actually uh"}, {"start": 165.28, "duration": 5.959, "text": "flip it upside down the positive 8 fths"}, {"start": 169.0, "duration": 4.239, "text": "would move it up so actually the graph"}, {"start": 171.239, "duration": 3.161, "text": "would look something like that unless"}, {"start": 173.239, "duration": 3.841, "text": "I'm"}, {"start": 174.4, "duration": 4.559, "text": "mistaken and again now the idea is this"}, {"start": 177.08, "duration": 4.48, "text": "function is going to fail that"}, {"start": 178.959, "duration": 4.761, "text": "horizontal line test and remember if you"}, {"start": 181.56, "duration": 4.44, "text": "draw a horizontal line and it hits your"}, {"start": 183.72, "duration": 4.519, "text": "graph in more than one place that again"}, {"start": 186.0, "duration": 4.64, "text": "is another uh way to determine that your"}, {"start": 188.239, "duration": 5.321, "text": "function has no inverse so kind of a"}, {"start": 190.64, "duration": 7.48, "text": "quick graphical way uh but here also you"}, {"start": 193.56, "duration": 4.56, "text": "know the important algebraic way"}], "title": "Finding the Inverse of a Function or Showing One Does not Exist (Without a Graph!), Ex 2", "description": "Thanks to all of you who support me on Patreon. You da real mvps!   $1 per month helps!! :)  https://www.patreon.com/patrickjmt !! Finding the Inverse of a Function or Showing One Does not Exist, Ex 2.\nIn this video, I justify algebraically that a function does not have an inverse.", "lengthSeconds": 196, "width": 480, "height": 360}
{"video_id": "BmjbDINGZGg", "transcript": [{"start": 0.88, "duration": 2.759, "text": "okay in this video I'm going to do"}, {"start": 2.04, "duration": 4.08, "text": "another example of finding the inverse"}, {"start": 3.639, "duration": 8.721, "text": "of a function so in this case we have G"}, {"start": 6.12, "duration": 12.8, "text": "ofx = -5 + 5x over -3 - 3x so I'm going"}, {"start": 12.36, "duration": 9.839, "text": "to write y = -5 + 5x over -3 -"}, {"start": 18.92, "duration": 5.199, "text": "3x um and now to find the inverse I'm"}, {"start": 22.199, "duration": 5.08, "text": "going to switch out I'm going to turn my"}, {"start": 24.119, "duration": 7.361, "text": "Y into an X and everywhere I have an X"}, {"start": 27.279, "duration": 4.201, "text": "those are going to become y's"}, {"start": 31.56, "duration": 5.319, "text": "and again now my goal is to try to solve"}, {"start": 34.32, "duration": 4.239, "text": "for y and hopefully get a single uh"}, {"start": 36.879, "duration": 4.16, "text": "unique function here which will tell me"}, {"start": 38.559, "duration": 4.68, "text": "it has an inverse the first thing I'm"}, {"start": 41.039, "duration": 4.281, "text": "going to do is I'm going to get rid of"}, {"start": 43.239, "duration": 6.681, "text": "the denominator I'm going to multiply"}, {"start": 45.32, "duration": 9.52, "text": "both sides by -3 - 3 y so on the left"}, {"start": 49.92, "duration": 9.2, "text": "side we'll have well -3 - 3 y * X on the"}, {"start": 54.84, "duration": 9.039, "text": "right side we've got our -5 + 5 y over"}, {"start": 59.12, "duration": 7.2, "text": "-3 - 3 Y and I'm going to"}, {"start": 63.879, "duration": 4.521, "text": "multiply well again I'm multiplying by"}, {"start": 66.32, "duration": 3.24, "text": "that denominator so that it'll simply"}, {"start": 68.4, "duration": 4.24, "text": "cancel"}, {"start": 69.56, "duration": 4.84, "text": "out um at this point you can think about"}, {"start": 72.64, "duration": 3.6, "text": "the X's being out front if you'd rather"}, {"start": 74.4, "duration": 3.359, "text": "move it out front it's multiplication so"}, {"start": 76.24, "duration": 3.76, "text": "you can kind of put it in any order you"}, {"start": 77.759, "duration": 3.561, "text": "want I'm going to distribute that X so"}, {"start": 80.0, "duration": 9.479, "text": "I've got"}, {"start": 81.32, "duration": 11.56, "text": "-3x - 3x y = -5 + 5 y so again our goal"}, {"start": 89.479, "duration": 4.68, "text": "here is to solve for y um what I'm going"}, {"start": 92.88, "duration": 3.68, "text": "to do is I'm going to put everything"}, {"start": 94.159, "duration": 7.0, "text": "that has a y on the same side so we have"}, {"start": 96.56, "duration": 6.72, "text": "our -3 XY notice we have a y on a term"}, {"start": 101.159, "duration": 5.401, "text": "involving y on the right side I'm going"}, {"start": 103.28, "duration": 6.08, "text": "to add 3xy to both"}, {"start": 106.56, "duration": 4.36, "text": "sides at the same time this neg five"}, {"start": 109.36, "duration": 5.24, "text": "that's on the right since he doesn't"}, {"start": 110.92, "duration": 6.4, "text": "have a y I'm going to add five to both"}, {"start": 114.6, "duration": 7.6, "text": "sides so on the left side we would have"}, {"start": 117.32, "duration": 8.28, "text": "5 - 3x the 3x y would be canel out with"}, {"start": 122.2, "duration": 6.6, "text": "the uh positive 3x y on the right side"}, {"start": 125.6, "duration": 6.68, "text": "the 5 and the POS 5 cancel and now we"}, {"start": 128.8, "duration": 7.6, "text": "have 5 y +"}, {"start": 132.28, "duration": 6.56, "text": "3xy so last but not least uh what we're"}, {"start": 136.4, "duration": 6.24, "text": "going to do here um to try to get y by"}, {"start": 138.84, "duration": 7.72, "text": "itself is I can Factor the Y out both"}, {"start": 142.64, "duration": 4.879, "text": "terms have a y I would need 5 + 3x in"}, {"start": 146.56, "duration": 3.64, "text": "the"}, {"start": 147.519, "duration": 6.321, "text": "parentheses um and okay at this point we"}, {"start": 150.2, "duration": 6.84, "text": "simply just divide out the 5 +"}, {"start": 153.84, "duration": 5.679, "text": "3x 5 +"}, {"start": 157.04, "duration": 5.559, "text": "3x and then that's then going to give us"}, {"start": 159.519, "duration": 6.561, "text": "our inverse function so it says on the"}, {"start": 162.599, "duration": 6.761, "text": "right side now we would be left with one"}, {"start": 166.08, "duration": 4.96, "text": "y um I think I said on the right side we"}, {"start": 169.36, "duration": 3.28, "text": "would have just one y on the left side"}, {"start": 171.04, "duration": 6.119, "text": "we have 5 -"}, {"start": 172.64, "duration": 6.56, "text": "3x um over 5 + 3x this is going to be"}, {"start": 177.159, "duration": 3.72, "text": "our inverse function notice we're not"}, {"start": 179.2, "duration": 3.319, "text": "getting like any positive negative"}, {"start": 180.879, "duration": 4.481, "text": "square roots like in our previous"}, {"start": 182.519, "duration": 3.881, "text": "example so it says the inverse function"}, {"start": 185.36, "duration": 5.76, "text": "of that"}, {"start": 186.4, "duration": 11.52, "text": "original would have the formula uh 5 -"}, {"start": 191.12, "duration": 6.8, "text": "3x over uh 5 + 3x"}], "title": "Finding the Inverse of a Function or Showing One Does not Exist, Ex 3", "description": "Thanks to all of you who support me on Patreon. You da real mvps!   $1 per month helps!! :)  https://www.patreon.com/patrickjmt !! Finding the Inverse of a Function or Showing One Does not Exist, Ex 3. In this video, I find the inverse of a rational function.", "lengthSeconds": 197, "width": 480, "height": 360}
{"video_id": "lWAsS82pyYA", "transcript": [{"start": 0.03, "duration": 6.99, "text": "in these examples I want to simplify"}, {"start": 3.51, "duration": 6.36, "text": "some expressions with fractional"}, {"start": 7.02, "duration": 3.9, "text": "exponents and notice in these examples"}, {"start": 9.87, "duration": 3.149, "text": "at least they all have common"}, {"start": 10.92, "duration": 5.31, "text": "denominators I'm also going to do the"}, {"start": 13.019, "duration": 5.701, "text": "extra step of converting those into"}, {"start": 16.23, "duration": 3.69, "text": "radical notation as well just so"}, {"start": 18.72, "duration": 5.79, "text": "hopefully you'll see how to do both"}, {"start": 19.92, "duration": 9.08, "text": "things alright so here's my examples X"}, {"start": 24.51, "duration": 7.2, "text": "to the 1/4 times X to the 3/4 well"}, {"start": 29.0, "duration": 5.829, "text": "remember our algebraic property that if"}, {"start": 31.71, "duration": 5.82, "text": "you have like bases being multiplied all"}, {"start": 34.829, "duration": 8.25, "text": "we simply do is add the exponents so we"}, {"start": 37.53, "duration": 9.66, "text": "have to take 1/4 plus 3/4 but that just"}, {"start": 43.079, "duration": 7.23, "text": "gives us X to the 4 over 4 which we can"}, {"start": 47.19, "duration": 5.4, "text": "simply write as X to the first or just X"}, {"start": 50.309, "duration": 5.011, "text": "and since its X to the first you know I"}, {"start": 52.59, "duration": 5.309, "text": "wouldn't write that as a radical at all"}, {"start": 55.32, "duration": 5.64, "text": "I will just leave that alone but the"}, {"start": 57.899, "duration": 8.431, "text": "next one okay the same idea we have X to"}, {"start": 60.96, "duration": 7.55, "text": "the 2/10 multiplied by X to the 7/10 so"}, {"start": 66.33, "duration": 5.88, "text": "again we'll just add two tenths and"}, {"start": 68.51, "duration": 8.59, "text": "seven tenths and that will give us X to"}, {"start": 72.21, "duration": 10.589, "text": "the 9 over 10 and if you wanted this as"}, {"start": 77.1, "duration": 7.86, "text": "a simplified rational exponent well then"}, {"start": 82.799, "duration": 4.801, "text": "we have at 9 over 10 doesn't reduce as a"}, {"start": 84.96, "duration": 5.13, "text": "fraction but if we also wanted to write"}, {"start": 87.6, "duration": 7.71, "text": "it say in radical notation we could do"}, {"start": 90.09, "duration": 7.349, "text": "that so again X goes underneath the the"}, {"start": 95.31, "duration": 3.7, "text": "denominator which is the 10 that becomes"}, {"start": 97.439, "duration": 4.031, "text": "the root"}, {"start": 99.01, "duration": 6.09, "text": "and then we simply raise all of that to"}, {"start": 101.47, "duration": 5.7, "text": "the ninth power ok so again my my nines"}, {"start": 105.1, "duration": 3.72, "text": "have a tendency to look like A's but"}, {"start": 107.17, "duration": 4.77, "text": "that's supposed to be a nine there so"}, {"start": 108.82, "duration": 6.42, "text": "alright last one here we have X to the"}, {"start": 111.94, "duration": 7.05, "text": "3/8 times X to the two eighths well if"}, {"start": 115.24, "duration": 8.27, "text": "we just again add three eighths plus two"}, {"start": 118.99, "duration": 7.29, "text": "eighths will give us X to the 5/8 and"}, {"start": 123.51, "duration": 6.82, "text": "I'm going to rewrite that one more time"}, {"start": 126.28, "duration": 9.33, "text": "again using radical notation so it"}, {"start": 130.33, "duration": 6.65, "text": "becomes the eighth root of x all raised"}, {"start": 135.61, "duration": 4.26, "text": "to the fifth power"}, {"start": 136.98, "duration": 5.64, "text": "ok so again a useful trick I think for"}, {"start": 139.87, "duration": 6.72, "text": "sure being able to convert from rational"}, {"start": 142.62, "duration": 5.79, "text": "exponents into radical notation a lot of"}, {"start": 146.59, "duration": 4.74, "text": "times when you have to simplify again"}, {"start": 148.41, "duration": 5.29, "text": "you know numbers to fractional exponents"}, {"start": 151.33, "duration": 3.96, "text": "this at least is how I think about them"}, {"start": 153.7, "duration": 4.26, "text": "this is how I compute them I think about"}, {"start": 155.29, "duration": 4.65, "text": "roots and then exponents so alright I"}, {"start": 157.96, "duration": 4.34, "text": "hope these examples make some sense and"}, {"start": 159.94, "duration": 2.36, "text": "help you out"}], "title": "Multiplying Variables with Rational Exponents - Basic Example 1", "description": "Thanks to all of you who support me on Patreon. You da real mvps!   $1 per month helps!! :)  https://www.patreon.com/patrickjmt !! Multiplying Variables with Rational Exponents - Basic Example 1", "lengthSeconds": 161, "width": 480, "height": 360}
{"video_id": "VIfjMkQNRuw", "transcript": [{"start": 0.599, "duration": 5.601, "text": "so just another quick uh sequence"}, {"start": 3.04, "duration": 6.639, "text": "question here so does the sequence uh Ln"}, {"start": 6.2, "duration": 6.319, "text": "of 1/ n Cub uh does that converge is"}, {"start": 9.679, "duration": 5.841, "text": "that true or false well again I think"}, {"start": 12.519, "duration": 8.0, "text": "this is you know not not too crazy to"}, {"start": 15.52, "duration": 8.519, "text": "evaluate you know basically as you know"}, {"start": 20.519, "duration": 6.08, "text": "um as n"}, {"start": 24.039, "duration": 5.08, "text": "increases and again"}, {"start": 26.599, "duration": 4.121, "text": "uh we're technically only using uh you"}, {"start": 29.119, "duration": 3.521, "text": "know like a"}, {"start": 30.72, "duration": 5.2, "text": "whole numbers when we plug into our"}, {"start": 32.64, "duration": 6.2, "text": "sequence so but the idea is n increases"}, {"start": 35.92, "duration": 7.0, "text": "certainly 1 over n the 3 this is going"}, {"start": 38.84, "duration": 7.559, "text": "to approach zero so as n increases 1"}, {"start": 42.92, "duration": 5.36, "text": "over n the 3 that will approach zero um"}, {"start": 46.399, "duration": 4.521, "text": "and if you think about the graph uh you"}, {"start": 48.28, "duration": 2.64, "text": "know natural"}, {"start": 53.8, "duration": 5.439, "text": "logarithm here's our natural logarithm"}, {"start": 56.239, "duration": 5.241, "text": "graph um you know as as as the input"}, {"start": 59.239, "duration": 3.681, "text": "gets closer closer to zero if you just"}, {"start": 61.48, "duration": 4.679, "text": "want to think about it in terms of"}, {"start": 62.92, "duration": 6.12, "text": "limits uh almost you know as as one over"}, {"start": 66.159, "duration": 5.64, "text": "n the3 approaches uh zero we should say"}, {"start": 69.04, "duration": 4.68, "text": "from the right side um the natural"}, {"start": 71.799, "duration": 5.281, "text": "logarithm simply goes down to negative"}, {"start": 73.72, "duration": 6.48, "text": "Infinity so I would say false uh the"}, {"start": 77.08, "duration": 5.84, "text": "sequence actually"}, {"start": 80.2, "duration": 4.64, "text": "diverges and the sequence would uh"}, {"start": 82.92, "duration": 4.92, "text": "diverge it would just approach negative"}, {"start": 84.84, "duration": 3.0, "text": "Infinity"}], "title": "Sequence Example : Converge or Diverge", "description": "Thanks to all of you who support me on Patreon. You da real mvps!   $1 per month helps!! :)  https://www.patreon.com/patrickjmt !! Sequence Example : Converge or Diverge.\nA simple example of deciding whether a sequence converges or diverges.", "lengthSeconds": 89, "width": 480, "height": 360}
{"video_id": "oTfs7foidf0", "transcript": [{"start": 0.06, "duration": 6.15, "text": "alright one more example here of writing"}, {"start": 2.909, "duration": 6.091, "text": "a fraction as a decimal so here we're"}, {"start": 6.21, "duration": 4.08, "text": "gonna do negative 13 over 5 if you"}, {"start": 9.0, "duration": 6.839, "text": "wanted to you could make this into a"}, {"start": 10.29, "duration": 7.38, "text": "mixed number and then take the the you"}, {"start": 15.839, "duration": 3.721, "text": "know the fraction that's left over and"}, {"start": 17.67, "duration": 6.39, "text": "write that as a decimal again I'm gonna"}, {"start": 19.56, "duration": 7.229, "text": "do it just exactly like we did our last"}, {"start": 24.06, "duration": 4.74, "text": "example and I'm simply going to just do"}, {"start": 26.789, "duration": 4.021, "text": "long division I know that there's a"}, {"start": 28.8, "duration": 4.29, "text": "negative in there so I'm gonna tack that"}, {"start": 30.81, "duration": 3.929, "text": "you know I'm gonna include that back in"}, {"start": 33.09, "duration": 5.489, "text": "here in a moment but for now I'm just"}, {"start": 34.739, "duration": 5.551, "text": "gonna say well 13 divided by 5"}, {"start": 38.579, "duration": 4.921, "text": "I'm just gonna worry about you know the"}, {"start": 40.29, "duration": 8.699, "text": "actual digits I'm gonna do the same"}, {"start": 43.5, "duration": 8.489, "text": "thing I'm gonna add in some zeros and in"}, {"start": 48.989, "duration": 4.681, "text": "a few zeros so now I'll simply say well"}, {"start": 51.989, "duration": 5.131, "text": "let's see you know five won't go into"}, {"start": 53.67, "duration": 6.299, "text": "one but five will go into 13 it'll go in"}, {"start": 57.12, "duration": 5.599, "text": "there twice without going over two times"}, {"start": 59.969, "duration": 5.52, "text": "five is ten and again we just subtract"}, {"start": 62.719, "duration": 6.311, "text": "13 minus 10 is gonna leave this with"}, {"start": 65.489, "duration": 6.181, "text": "three we just dropped down our zero so"}, {"start": 69.03, "duration": 4.65, "text": "now I'm looking at the number 30 and"}, {"start": 71.67, "duration": 5.03, "text": "they say well five will go into 30 how"}, {"start": 73.68, "duration": 7.14, "text": "many times or again 30 divided by 5"}, {"start": 76.7, "duration": 6.669, "text": "would give us exactly in this case 6 6"}, {"start": 80.82, "duration": 5.25, "text": "times 5 is 30 hey that's perfect"}, {"start": 83.369, "duration": 5.161, "text": "we've got no remainder that tells us"}, {"start": 86.07, "duration": 5.88, "text": "that our decimal place now terminates"}, {"start": 88.53, "duration": 6.57, "text": "and we've got our solution so it says we"}, {"start": 91.95, "duration": 5.01, "text": "can write negative 13 over 5 as again"}, {"start": 95.1, "duration": 6.69, "text": "let's not forget our negative that's"}, {"start": 96.96, "duration": 6.299, "text": "negative 2.6 so that one I think was"}, {"start": 101.79, "duration": 3.57, "text": "even a little bit easier than the last"}, {"start": 103.259, "duration": 4.82, "text": "one since we only had to do a couple a"}, {"start": 105.36, "duration": 2.719, "text": "couple steps"}], "title": "Writing a Fraction as a Decimal, Example 2", "description": "Thanks to all of you who support me on Patreon. You da real mvps!   $1 per month helps!! :)  https://www.patreon.com/patrickjmt !! Writing a Fraction as a Decimal, Example 2.  In this video, we use long division to write a fraction as a decimal.", "lengthSeconds": 108, "width": 480, "height": 360}
{"video_id": "Tj_tQxtI3gs", "transcript": [{"start": 0.0, "duration": 5.13, "text": "all right in this video I'm gonna do"}, {"start": 2.01, "duration": 5.369, "text": "another equation involving two absolute"}, {"start": 5.13, "duration": 4.02, "text": "value expressions so again it first"}, {"start": 7.379, "duration": 4.291, "text": "initially certainly look like there's"}, {"start": 9.15, "duration": 4.649, "text": "only one but algebraically again we can"}, {"start": 11.67, "duration": 5.119, "text": "break up this fraction on the left as"}, {"start": 13.799, "duration": 9.711, "text": "the absolute value of 2x plus five over"}, {"start": 16.789, "duration": 8.621, "text": "the absolute value of 3x plus 9 equals 6"}, {"start": 23.51, "duration": 4.359, "text": "what I'm going to do is I'm going to"}, {"start": 25.41, "duration": 6.17, "text": "multiply both sides by this absolute"}, {"start": 27.869, "duration": 3.711, "text": "value of 3x plus 9"}, {"start": 34.25, "duration": 5.24, "text": "so it cancels on the left so we're left"}, {"start": 36.68, "duration": 6.48, "text": "with the absolute value of 2x plus 5"}, {"start": 39.49, "duration": 6.25, "text": "equals 6 times the absolute value of 3x"}, {"start": 43.16, "duration": 4.5, "text": "plus 9 and again what we do at this"}, {"start": 45.74, "duration": 4.5, "text": "point to find our solutions we just"}, {"start": 47.66, "duration": 5.61, "text": "create two equations the first equation"}, {"start": 50.24, "duration": 7.65, "text": "I just basically imagined the bars turn"}, {"start": 53.27, "duration": 7.08, "text": "into parenthesis so I'm just replacing"}, {"start": 57.89, "duration": 4.86, "text": "the absolute values with parentheses"}, {"start": 60.35, "duration": 5.25, "text": "I'll have to solve that equation the"}, {"start": 62.75, "duration": 10.68, "text": "other equation what I do is I remove the"}, {"start": 65.6, "duration": 10.29, "text": "bars and on one of the absolute values"}, {"start": 73.43, "duration": 5.16, "text": "okay so this is not correct just yet I"}, {"start": 75.89, "duration": 5.16, "text": "have to stick a negative on one of the"}, {"start": 78.59, "duration": 3.96, "text": "absolute values or the other I think it"}, {"start": 81.05, "duration": 3.3, "text": "was sticking on the one on the left just"}, {"start": 82.55, "duration": 2.34, "text": "so it looks a little bit better when I"}, {"start": 84.35, "duration": 3.75, "text": "write it"}, {"start": 84.89, "duration": 5.28, "text": "so notice the only distinction we remove"}, {"start": 88.1, "duration": 4.159, "text": "the absolute values in both of them one"}, {"start": 90.17, "duration": 4.41, "text": "of the equations has a negative present"}, {"start": 92.259, "duration": 3.671, "text": "okay so to solve the first one we can"}, {"start": 94.58, "duration": 3.87, "text": "just get removed removing the"}, {"start": 95.93, "duration": 6.18, "text": "parentheses 2x plus 5 if we distribute"}, {"start": 98.45, "duration": 7.41, "text": "the 6 we'll get 18 X 6 and 9 is positive"}, {"start": 102.11, "duration": 6.21, "text": "54 let's see I can subtract 2x from both"}, {"start": 105.86, "duration": 8.01, "text": "sides that would give me 16x on the"}, {"start": 108.32, "duration": 10.47, "text": "right plus 54 now I can subtract 54 from"}, {"start": 113.87, "duration": 8.97, "text": "both sides let's see I think 55 minus 54"}, {"start": 118.79, "duration": 6.3, "text": "would be negative 49 equals 16x"}, {"start": 122.84, "duration": 5.85, "text": "and if we want to simply get the X by"}, {"start": 125.09, "duration": 7.08, "text": "itself we can divide both sides by 16 so"}, {"start": 128.69, "duration": 6.86, "text": "that'll tell us that negative 49 over 16"}, {"start": 132.17, "duration": 5.48, "text": "is one of our solutions"}, {"start": 135.55, "duration": 3.69, "text": "to do the other solution we're going to"}, {"start": 137.65, "duration": 3.45, "text": "do the same thing the first thing I"}, {"start": 139.24, "duration": 6.78, "text": "would do is distribute my negative so"}, {"start": 141.1, "duration": 11.22, "text": "I'll get negative 2x minus 5 equals 18x"}, {"start": 146.02, "duration": 11.27, "text": "again plus 54 let's see if I add 2x I'm"}, {"start": 152.32, "duration": 7.98, "text": "going to get 20 X on the right plus 54"}, {"start": 157.29, "duration": 7.63, "text": "and again if I subtract 54 from both"}, {"start": 160.3, "duration": 9.63, "text": "sides now negative 5 minus 54 is"}, {"start": 164.92, "duration": 6.45, "text": "negative 59 equals 20 times X again if I"}, {"start": 169.93, "duration": 3.75, "text": "want to solve for X I'll just divide"}, {"start": 171.37, "duration": 7.17, "text": "both sides by 20 and that'll say"}, {"start": 173.68, "duration": 7.85, "text": "negative 59 divided by 20 equals x is my"}, {"start": 178.54, "duration": 2.99, "text": "other solution"}], "title": "Solving Absolute Value Equations Containing TWO Absolute Value Expressions - Example 2", "description": "Thanks to all of you who support me on Patreon. You da real mvps!   $1 per month helps!! :)  https://www.patreon.com/patrickjmt !! Solving Absolute Value Equations Containing TWO Absolute Value Expressions - Example 2", "lengthSeconds": 181, "width": 480, "height": 360}
{"video_id": "1ryTz3mA3qc", "transcript": [{"start": 0.76, "duration": 2.959, "text": "all right in this video we're going to"}, {"start": 1.839, "duration": 4.321, "text": "do an example of finding the height of"}, {"start": 3.719, "duration": 4.561, "text": "an object um and we're just simply going"}, {"start": 6.16, "duration": 5.639, "text": "to use some basic trigonometry to help"}, {"start": 8.28, "duration": 5.08, "text": "us do this so suppose um suppose you're"}, {"start": 11.799, "duration": 3.08, "text": "you're just chilling out you know"}, {"start": 13.36, "duration": 3.52, "text": "watching a balloon go up in the air"}, {"start": 14.879, "duration": 3.881, "text": "because you got nothing else to do"}, {"start": 16.88, "duration": 3.76, "text": "suppose uh that you're able to measure"}, {"start": 18.76, "duration": 3.2, "text": "this distance from you this horizontal"}, {"start": 20.64, "duration": 4.399, "text": "distance from you to the balloon and"}, {"start": 21.96, "duration": 4.28, "text": "that's 2,000 ft suppose also you've got"}, {"start": 25.039, "duration": 2.841, "text": "you know you happen to have some"}, {"start": 26.24, "duration": 4.76, "text": "surveying equipment laying around and"}, {"start": 27.88, "duration": 4.839, "text": "you get a little angle of 30\u00b0 to the"}, {"start": 31.0, "duration": 3.52, "text": "balloon we want to know what's the"}, {"start": 32.719, "duration": 5.881, "text": "height of the balloon so this is"}, {"start": 34.52, "duration": 5.879, "text": "certainly very useful practical um"}, {"start": 38.6, "duration": 5.2, "text": "normal types of questions out there in"}, {"start": 40.399, "duration": 5.561, "text": "engineering um you know you know anytime"}, {"start": 43.8, "duration": 3.32, "text": "people are making buildings or roads or"}, {"start": 45.96, "duration": 2.72, "text": "definitely doing all this stuff just"}, {"start": 47.12, "duration": 5.0, "text": "making measurements and angles so"}, {"start": 48.68, "duration": 6.12, "text": "certainly very practical math here so um"}, {"start": 52.12, "duration": 6.32, "text": "okay again to me we're trying to figure"}, {"start": 54.8, "duration": 6.2, "text": "out um this height so here's my little"}, {"start": 58.44, "duration": 5.92, "text": "angle that I'm given in this problem so"}, {"start": 61.0, "duration": 7.6, "text": "2,000 ft that would represent um stuff"}, {"start": 64.36, "duration": 5.799, "text": "about the adjacent side um and this Ang"}, {"start": 68.6, "duration": 4.04, "text": "the the height would represent sort of"}, {"start": 70.159, "duration": 5.441, "text": "the uh the opposite"}, {"start": 72.64, "duration": 5.68, "text": "side so I'm thinking which of the trig"}, {"start": 75.6, "duration": 5.0, "text": "uh trig functions relates opposite sides"}, {"start": 78.32, "duration": 4.0, "text": "and adjacent sides well there's actually"}, {"start": 80.6, "duration": 4.8, "text": "two of them uh but the one I'm going to"}, {"start": 82.32, "duration": 6.6, "text": "use here is just tangent so remember"}, {"start": 85.4, "duration": 6.16, "text": "tangent um of an angle is defined to be"}, {"start": 88.92, "duration": 5.519, "text": "the the ratio the opposite"}, {"start": 91.56, "duration": 5.4, "text": "side to the adjacent"}, {"start": 94.439, "duration": 5.281, "text": "side so in this case we're going to have"}, {"start": 96.96, "duration": 5.28, "text": "that well tangent of"}, {"start": 99.72, "duration": 4.24, "text": "30\u00b0 that's going to equal again the"}, {"start": 102.24, "duration": 4.519, "text": "opposite side so we'll just call it"}, {"start": 103.96, "duration": 5.199, "text": "maybe o for opposite um over the"}, {"start": 106.759, "duration": 4.64, "text": "adjacent side but we know that's 2,000"}, {"start": 109.159, "duration": 4.521, "text": "actually let's don't use o for opposite"}, {"start": 111.399, "duration": 5.201, "text": "because it looks uh it looks like a zero"}, {"start": 113.68, "duration": 5.2, "text": "to me so let's just call this height Y"}, {"start": 116.6, "duration": 4.519, "text": "and we'll make this a y and everything"}, {"start": 118.88, "duration": 4.0, "text": "will be a little less confusing so try"}, {"start": 121.119, "duration": 6.401, "text": "to use some better"}, {"start": 122.88, "duration": 7.96, "text": "notation okay so we have tangent of 30\u00b0"}, {"start": 127.52, "duration": 6.16, "text": "that equal y over"}, {"start": 130.84, "duration": 7.64, "text": "2000 again simply to get the Y by itself"}, {"start": 133.68, "duration": 4.8, "text": "we can just multiply both sides uh by"}, {"start": 142.4, "duration": 5.36, "text": "2000 so if we multiply both sides by"}, {"start": 145.12, "duration": 5.6, "text": "2,000 uh we'll get the 2,000 on the left"}, {"start": 147.76, "duration": 4.4, "text": "just like we did but if we multiply y um"}, {"start": 150.72, "duration": 4.48, "text": "on the right they're just going to"}, {"start": 152.16, "duration": 5.04, "text": "cancel out so we get the Y by itself so"}, {"start": 155.2, "duration": 3.36, "text": "that's all we're doing there and now"}, {"start": 157.2, "duration": 4.24, "text": "again this is something that I would"}, {"start": 158.56, "duration": 5.2, "text": "just want to uh plug into a calculator"}, {"start": 161.44, "duration": 5.36, "text": "so again at this point make sure your"}, {"start": 163.76, "duration": 4.839, "text": "calculator is in degree mode so let's"}, {"start": 166.8, "duration": 5.88, "text": "see tangent of"}, {"start": 168.599, "duration": 5.92, "text": "30\u00b0 um let's see so I'm going to"}, {"start": 172.68, "duration": 6.08, "text": "approximate this"}, {"start": 174.519, "duration": 7.201, "text": "now so tangent of 30\u00b0 uh is"}, {"start": 178.76, "duration": 6.479, "text": "approximately"}, {"start": 181.72, "duration": 6.439, "text": "577 so that's roughly going to be the"}, {"start": 185.239, "duration": 8.521, "text": "height of our uh balloon here so now if"}, {"start": 188.159, "duration": 9.881, "text": "we simply take 577 and multiply that by"}, {"start": 193.76, "duration": 6.6, "text": "2,000 I'm getting"}, {"start": 198.04, "duration": 7.119, "text": "1154 so it says the height of this"}, {"start": 200.36, "duration": 4.799, "text": "balloon is going to be roughly equal to"}, {"start": 205.959, "duration": 5.721, "text": "1,154"}, {"start": 208.68, "duration": 3.0, "text": "Ft"}], "title": "Finding the Height of an Object Using Trigonometry, Example 1", "description": "Thanks to all of you who support me on Patreon. You da real mvps!   $1 per month helps!! :)  https://www.patreon.com/patrickjmt !! Finding the Height of an Object Using Trigonometry, Example 1. In this video, I find the height of a balloon by knowing a horizontal distance and an angle.", "lengthSeconds": 211, "width": 480, "height": 360}
{"video_id": "uFbbF-IYFjM", "transcript": [{"start": 0.269, "duration": 3.721, "text": "all right in this example we're going to"}, {"start": 2.19, "duration": 4.919, "text": "use some half angle identities to"}, {"start": 3.99, "duration": 4.74, "text": "evaluate some trig expressions so these"}, {"start": 7.109, "duration": 3.78, "text": "are the identities the half angle"}, {"start": 8.73, "duration": 3.659, "text": "identities notice for cosine and sine"}, {"start": 10.889, "duration": 3.391, "text": "there's kind of a choice positive or"}, {"start": 12.389, "duration": 3.031, "text": "negative basically that's just you're"}, {"start": 14.28, "duration": 3.089, "text": "going to have to think about the"}, {"start": 15.42, "duration": 3.869, "text": "quadrant that your angle is in and then"}, {"start": 17.369, "duration": 5.25, "text": "determine if it's positive or negative"}, {"start": 19.289, "duration": 5.4, "text": "so for example suppose we want to figure"}, {"start": 22.619, "duration": 6.361, "text": "out sine of twenty two point five"}, {"start": 24.689, "duration": 8.161, "text": "degrees well all I'm going to do is I'm"}, {"start": 28.98, "duration": 6.81, "text": "simply going to rewrite a sine of twenty"}, {"start": 32.85, "duration": 5.13, "text": "two point five notice we could rewrite"}, {"start": 35.79, "duration": 6.51, "text": "that if you double twenty two point five"}, {"start": 37.98, "duration": 7.59, "text": "you get 45 degrees well so I could write"}, {"start": 42.3, "duration": 6.45, "text": "twenty two point five degrees as 45"}, {"start": 45.57, "duration": 5.219, "text": "degrees divided by two okay"}, {"start": 48.75, "duration": 3.539, "text": "well twenty two point five degrees this"}, {"start": 50.789, "duration": 5.101, "text": "is definitely going to be in quadrant"}, {"start": 52.289, "duration": 11.04, "text": "one now I'm just going to go back to our"}, {"start": 55.89, "duration": 12.12, "text": "identity it says well once you have sine"}, {"start": 63.329, "duration": 6.061, "text": "of your angle divided by two it says"}, {"start": 68.01, "duration": 3.149, "text": "again you have to determine whether it's"}, {"start": 69.39, "duration": 3.9, "text": "positive or negative but again since"}, {"start": 71.159, "duration": 4.471, "text": "twenty two point five degrees is in"}, {"start": 73.29, "duration": 5.189, "text": "quadrant one I know that we'll use the"}, {"start": 75.63, "duration": 5.01, "text": "positive square root it says all you"}, {"start": 78.479, "duration": 5.521, "text": "have to do is you take one minus cosine"}, {"start": 80.64, "duration": 5.71, "text": "of whatever value is on top so that will"}, {"start": 84.0, "duration": 3.939, "text": "be 45 degrees"}, {"start": 86.35, "duration": 4.74, "text": "then it says we simply divide all of"}, {"start": 87.939, "duration": 5.341, "text": "that by two and that will give us our"}, {"start": 91.09, "duration": 5.7, "text": "value so I think we can clean this up a"}, {"start": 93.28, "duration": 6.29, "text": "little bit so we've got the square root"}, {"start": 96.79, "duration": 6.84, "text": "of 1 minus well cosine of 45 degrees"}, {"start": 99.57, "duration": 7.329, "text": "that's square root of 2 over 2 all over"}, {"start": 103.63, "duration": 6.419, "text": "2 and I think what I'm going to do just"}, {"start": 106.899, "duration": 6.871, "text": "to get rid of the fractions I'm going to"}, {"start": 110.049, "duration": 6.091, "text": "multiply you know my the top of my"}, {"start": 113.77, "duration": 6.139, "text": "fraction by 2 and the bottom of my"}, {"start": 116.14, "duration": 6.03, "text": "fraction also by 2 and if we do that"}, {"start": 119.909, "duration": 3.401, "text": "again I don't know not the end of the"}, {"start": 122.17, "duration": 4.17, "text": "world probably if you don't do that"}, {"start": 123.31, "duration": 8.07, "text": "we'll get 2 times 1 which is 2 minus the"}, {"start": 126.34, "duration": 6.81, "text": "square root of 2 all over 4 couldn't"}, {"start": 131.38, "duration": 3.09, "text": "even rewrite this a little bit more we"}, {"start": 133.15, "duration": 4.86, "text": "could take the square root of the"}, {"start": 134.47, "duration": 5.19, "text": "numerator so we would have 2 minus the"}, {"start": 138.01, "duration": 4.18, "text": "square root of 2 again all of that's"}, {"start": 139.66, "duration": 4.27, "text": "being square rooted"}, {"start": 142.19, "duration": 4.14, "text": "and then we could take the square root"}, {"start": 143.93, "duration": 3.96, "text": "of four individually and the square root"}, {"start": 146.33, "duration": 4.53, "text": "of four is simply going to give us two"}, {"start": 147.89, "duration": 4.86, "text": "and to me I would say you know I would"}, {"start": 150.86, "duration": 3.36, "text": "probably leave it at this at this place"}, {"start": 152.75, "duration": 5.03, "text": "there's not a lot else you can do in"}, {"start": 154.22, "duration": 3.56, "text": "terms of the simplification here"}], "title": "Half Angle Identities to Evaluate Trigonometric Expressions, Example 1", "description": "Thanks to all of you who support me on Patreon. You da real mvps!   $1 per month helps!! :)  https://www.patreon.com/patrickjmt !! Half Angle Identities to Evaluate Trigonometric Expressions, Example 1.  In this video, I give some half angle identities and show how they can be used to solve some trigonometric equations.", "lengthSeconds": 156, "width": 480, "height": 360}
{"video_id": "bi8-qKOWbnI", "transcript": [{"start": 0.56, "duration": 3.84, "text": "all right so a couple more examples here"}, {"start": 2.08, "duration": 4.48, "text": "of adding whole numbers and here I've"}, {"start": 4.4, "duration": 4.6, "text": "got everything written uh vertically"}, {"start": 6.56, "duration": 7.48, "text": "already so in part A we're going to add"}, {"start": 9.0, "duration": 8.6, "text": "546 and 85 546 and 85 and in Part B"}, {"start": 14.04, "duration": 6.56, "text": "We'll add 289 and 76"}, {"start": 17.6, "duration": 6.2, "text": "together so again I just move right to"}, {"start": 20.6, "duration": 5.519, "text": "left let's see 6 + 5 that's going to"}, {"start": 23.8, "duration": 5.479, "text": "equal"}, {"start": 26.119, "duration": 5.841, "text": "11 so again the one's digit I just leave"}, {"start": 29.279, "duration": 5.161, "text": "in the one's column and then the 10's"}, {"start": 31.96, "duration": 6.919, "text": "digit which is well also one that's what"}, {"start": 34.44, "duration": 7.16, "text": "I carry to the next column so there's my"}, {"start": 38.879, "duration": 7.36, "text": "one getting carried to the next column"}, {"start": 41.6, "duration": 6.2, "text": "um so let's see 1+ 4 that's five 5 + 8"}, {"start": 46.239, "duration": 5.201, "text": "well that's going to give us"}, {"start": 47.8, "duration": 5.84, "text": "13 so in this case I'm going to drop my"}, {"start": 51.44, "duration": 5.16, "text": "three down and now I'm going to carry"}, {"start": 53.64, "duration": 6.719, "text": "again the digit the one over to the next"}, {"start": 56.6, "duration": 9.64, "text": "column and 1 + 5 is going to be six so"}, {"start": 60.359, "duration": 8.8, "text": "it says 546 and 85 gives us 631"}, {"start": 66.24, "duration": 6.84, "text": "631 let's see 9 plus six that's going to"}, {"start": 69.159, "duration": 9.361, "text": "be 15 so again I'm going to put the five"}, {"start": 73.08, "duration": 9.8, "text": "down carry the one let's see 1 + 8 is 9"}, {"start": 78.52, "duration": 8.279, "text": "9 + 7 that's going to give us 16 so"}, {"start": 82.88, "duration": 8.279, "text": "again I'm going to drop my six down and"}, {"start": 86.799, "duration": 7.081, "text": "again carry yet another one and then 1 +"}, {"start": 91.159, "duration": 4.361, "text": "2 is going to be three you can think"}, {"start": 93.88, "duration": 5.519, "text": "about the you know the space as just"}, {"start": 95.52, "duration": 8.16, "text": "being a zero 1 plus 2 is 3 so it says if"}, {"start": 99.399, "duration": 7.481, "text": "we take 289 and add 76 we're going to"}, {"start": 103.68, "duration": 3.2, "text": "get 365"}], "title": "Adding Whole Numbers , Example 2.  Three Digit Plus Two Digit", "description": "Thanks to all of you who support me on Patreon. You da real mvps!   $1 per month helps!! :)  https://www.patreon.com/patrickjmt !! Adding Whole Numbers , Example 2.  Here we look at two examples of adding a two digit and three digit number together.", "lengthSeconds": 107, "width": 480, "height": 360}
{"video_id": "30n1eiu8lFs", "transcript": [{"start": 0.599, "duration": 4.561, "text": "all right so one more little example"}, {"start": 2.159, "duration": 6.52, "text": "here so suppose we approximate uh theare"}, {"start": 5.16, "duration": 8.04, "text": "otk of 1 + x^2 near x = 1 with the"}, {"start": 8.679, "duration": 7.641, "text": "polinomial < TK 2 +22 * X - 1 again we"}, {"start": 13.2, "duration": 5.839, "text": "want to know uh what the remainder"}, {"start": 16.32, "duration": 5.039, "text": "is okay so it says the remainder"}, {"start": 19.039, "duration": 5.4, "text": "associated with a degree one"}, {"start": 21.359, "duration": 6.641, "text": "polom it says well that's going to be uh"}, {"start": 24.439, "duration": 8.361, "text": "we'll have to take the second derivative"}, {"start": 28.0, "duration": 8.28, "text": "evaluated at Z over 2 factorial and then"}, {"start": 32.8, "duration": 6.84, "text": "we'll have x minus our C value which is"}, {"start": 36.28, "duration": 5.88, "text": "positive1 again raised to the second"}, {"start": 39.64, "duration": 4.68, "text": "power well all we have to do is take a"}, {"start": 42.16, "duration": 5.32, "text": "couple derivatives"}, {"start": 44.32, "duration": 6.239, "text": "here and then uh replace our x with our"}, {"start": 47.48, "duration": 7.32, "text": "Z and uh that's it"}, {"start": 50.559, "duration": 8.64, "text": "so F ofx okay so that's 1 +"}, {"start": 54.8, "duration": 9.92, "text": "x^2 uh to the 12 power so the first"}, {"start": 59.199, "duration": 7.96, "text": "derivative will get 12 * 1 + x^2 to the"}, {"start": 64.72, "duration": 4.24, "text": "-2 power and then we have to do the"}, {"start": 67.159, "duration": 4.721, "text": "chain rule so *"}, {"start": 68.96, "duration": 7.199, "text": "2x so the 1/2 and the two will cancel"}, {"start": 71.88, "duration": 8.4, "text": "and we'll be left with x * 1 + x^2 to"}, {"start": 76.159, "duration": 6.481, "text": "the - one2 power well we can do yet"}, {"start": 80.28, "duration": 3.96, "text": "another derivative so our second"}, {"start": 82.64, "duration": 4.4, "text": "derivative we'll have to use the product"}, {"start": 84.24, "duration": 5.68, "text": "rule so uh when we take the derivative"}, {"start": 87.04, "duration": 5.439, "text": "of 1x we'll get one we'll have 1 plus"}, {"start": 89.92, "duration": 4.72, "text": "plus x^2 -"}, {"start": 92.479, "duration": 5.201, "text": "12 plus we'll have to take another"}, {"start": 94.64, "duration": 5.159, "text": "derivative so let's leave the X Alone um"}, {"start": 97.68, "duration": 5.479, "text": "we'll have to do the chain rule so the -"}, {"start": 99.799, "duration": 5.32, "text": "one2 will come out front we'll have 1 +"}, {"start": 103.159, "duration": 4.521, "text": "x^2 to the"}, {"start": 105.119, "duration": 3.881, "text": "-3 and then the derivative of the inside"}, {"start": 107.68, "duration": 3.719, "text": "will give us"}, {"start": 109.0, "duration": 6.84, "text": "2x so let's see if we can't clean this"}, {"start": 111.399, "duration": 6.32, "text": "up a little bit so 1 + x^2 to the - one2"}, {"start": 115.84, "duration": 4.559, "text": "um it looks like the 1/2 and the two"}, {"start": 117.719, "duration": 4.76, "text": "will cancel we'll still have a we have"}, {"start": 120.399, "duration": 7.041, "text": "an X and an X so"}, {"start": 122.479, "duration": 7.48, "text": "X2 then we have 1 + x^2 -3"}, {"start": 127.44, "duration": 7.799, "text": "power I'm going to factor this I'm going"}, {"start": 129.959, "duration": 7.92, "text": "to pull out the 1 + x^2 to the -3"}, {"start": 135.239, "duration": 5.041, "text": "power pulling out the smaller of the two"}, {"start": 137.879, "duration": 6.041, "text": "exponents on the 1 plus"}, {"start": 140.28, "duration": 5.76, "text": "X2 well if we multiply that by 1+ x^2 to"}, {"start": 143.92, "duration": 3.76, "text": "the first when we distribute We'll add"}, {"start": 146.04, "duration": 4.16, "text": "the exponents and that'll give us our"}, {"start": 147.68, "duration": 5.04, "text": "first term back and then we'll just Min"}, {"start": 150.2, "duration": 3.96, "text": "- x^2 when we distribute to get our"}, {"start": 152.72, "duration": 4.96, "text": "second term"}, {"start": 154.16, "duration": 5.76, "text": "back so let's see we can put the 1 + x^2"}, {"start": 157.68, "duration": 5.04, "text": "to the three"}, {"start": 159.92, "duration": 5.2, "text": "Hales in the denominator then we would"}, {"start": 162.72, "duration": 4.92, "text": "have 1 + x^2 - x^2 which is just going"}, {"start": 165.12, "duration": 2.52, "text": "to leave us"}, {"start": 167.76, "duration": 5.88, "text": "one all right so now it says um"}, {"start": 171.72, "duration": 4.239, "text": "therefore our second derivative"}, {"start": 173.64, "duration": 6.84, "text": "evaluated at"}, {"start": 175.959, "duration": 7.481, "text": "Z would be 1 over 1 +"}, {"start": 180.48, "duration": 5.28, "text": "z^ 2 to the 3es"}, {"start": 183.44, "duration": 6.2, "text": "power and now I'm just going to replace"}, {"start": 185.76, "duration": 3.88, "text": "that back in my uh"}, {"start": 191.159, "duration": 9.281, "text": "expression so if we plug that in it says"}, {"start": 194.0, "duration": 9.56, "text": "we'll have one over 1 + z^2 to the 3"}, {"start": 200.44, "duration": 5.56, "text": "power we would have a 2 factorial in the"}, {"start": 203.56, "duration": 5.36, "text": "denominator um so two factorial is just"}, {"start": 206.0, "duration": 5.799, "text": "the same thing as multiplying by two and"}, {"start": 208.92, "duration": 5.319, "text": "then we would be left with our x -1 2"}, {"start": 211.799, "duration": 4.921, "text": "and now we have our expression so again"}, {"start": 214.239, "duration": 4.681, "text": "it's really just a matter of just taking"}, {"start": 216.72, "duration": 4.879, "text": "derivatives um plugging in that"}, {"start": 218.92, "duration": 6.36, "text": "expression simplifying the factorial and"}, {"start": 221.599, "duration": 3.681, "text": "that's really all you have to do"}], "title": "Taylor's Remainder Theorem - Finding the Remainder, Ex 2", "description": "Thanks to all of you who support me on Patreon. You da real mvps!   $1 per month helps!! :)  https://www.patreon.com/patrickjmt !! Taylor's Remainder Theorem - Finding the Remainder, Ex 2. In this example, I use Taylor's Remainder Theorem to find an expression for the remainder.", "lengthSeconds": 223, "width": 480, "height": 360}
{"video_id": "PoydouDNidQ", "transcript": [{"start": 0.799, "duration": 4.601, "text": "so in this question we're given the"}, {"start": 2.0, "duration": 5.16, "text": "system of equations x + y = 9 and x + 2"}, {"start": 5.4, "duration": 4.199, "text": "y ="}, {"start": 7.16, "duration": 4.599, "text": "-25 and we want to know according to the"}, {"start": 9.599, "duration": 4.12, "text": "system of equations what is the value of"}, {"start": 11.759, "duration": 4.121, "text": "x so we don't have to figure out you"}, {"start": 13.719, "duration": 5.961, "text": "know the complete solution X and Y just"}, {"start": 15.88, "duration": 7.08, "text": "X so what I'm going to do in this one is"}, {"start": 19.68, "duration": 6.32, "text": "again you could use uh substitution to"}, {"start": 22.96, "duration": 3.04, "text": "solve this if you want"}, {"start": 26.64, "duration": 5.039, "text": "to I think I've done at least a couple"}, {"start": 29.4, "duration": 4.36, "text": "examples um you know of these sat"}, {"start": 31.679, "duration": 5.001, "text": "questions where we've used substitution"}, {"start": 33.76, "duration": 4.88, "text": "let's use elimination by addition and"}, {"start": 36.68, "duration": 4.8, "text": "since I want to solve for x to me that"}, {"start": 38.64, "duration": 6.36, "text": "tells me well let's"}, {"start": 41.48, "duration": 4.96, "text": "eliminate let's eliminate the variable Y"}, {"start": 45.0, "duration": 3.36, "text": "and that way we'll be left with an"}, {"start": 46.44, "duration": 3.72, "text": "equation involving only X and then we'll"}, {"start": 48.36, "duration": 5.719, "text": "be able to solve"}, {"start": 50.16, "duration": 6.12, "text": "it okay so I want to do elimination by"}, {"start": 54.079, "duration": 5.041, "text": "addition so I notice I have a positive Y"}, {"start": 56.28, "duration": 6.32, "text": "and a positive 2 Y and I think well if I"}, {"start": 59.12, "duration": 6.44, "text": "had a NE 2 y here if I add --2 Y and"}, {"start": 62.6, "duration": 4.72, "text": "posi 2 y I would be left with 0 Y and"}, {"start": 65.56, "duration": 4.08, "text": "then I would have an equation involving"}, {"start": 67.32, "duration": 6.28, "text": "only X so to do that I'm going to"}, {"start": 69.64, "duration": 7.36, "text": "multiply both sides of my first equation"}, {"start": 73.6, "duration": 5.64, "text": "by -2 so I've got to do the left side"}, {"start": 77.0, "duration": 4.92, "text": "and also the right side that's going to"}, {"start": 79.24, "duration": 4.8, "text": "give me um a new system of equations"}, {"start": 81.92, "duration": 5.96, "text": "that I can then solve using elimination"}, {"start": 84.04, "duration": 6.16, "text": "by addition okay"}, {"start": 87.88, "duration": 3.519, "text": "so okay so if I distribute on the left"}, {"start": 90.2, "duration": 5.84, "text": "I'll have"}, {"start": 91.399, "duration": 9.4, "text": "-2X um -2 * POS Y is -2"}, {"start": 96.04, "duration": 9.24, "text": "y 9 * by -2 will give us POS 18 I'm not"}, {"start": 100.799, "duration": 6.401, "text": "doing anything to the second"}, {"start": 105.28, "duration": 4.439, "text": "equation and now I'm going to do my"}, {"start": 107.2, "duration": 6.48, "text": "elimination by addition"}, {"start": 109.719, "duration": 8.04, "text": "so -2x + x that's going to leave me with"}, {"start": 113.68, "duration": 7.719, "text": "x -2 y + 2 y well that's 0 Y and that's"}, {"start": 117.759, "duration": 6.72, "text": "what we wanted um 18 + -25 that's going"}, {"start": 121.399, "duration": 5.4, "text": "to be -7 and now if I just multiply both"}, {"start": 124.479, "duration": 5.801, "text": "sides by"}, {"start": 126.799, "duration": 5.481, "text": "-1 or equivalently I could divide by-1"}, {"start": 130.28, "duration": 6.0, "text": "I'll be left with positive X on the left"}, {"start": 132.28, "duration": 6.679, "text": "side and POS 7 on the right side so it"}, {"start": 136.28, "duration": 4.64, "text": "says the value of x that would you know"}, {"start": 138.959, "duration": 6.801, "text": "um part of the solution to this equation"}, {"start": 140.92, "duration": 4.84, "text": "the x value would be equal to pos7"}], "title": "SAT Math - Question 18, 2016 Edition and Beyond", "description": "Thanks to all of you who support me on Patreon. You da real mvps!   $1 per month helps!! :)  https://www.patreon.com/patrickjmt !! SAT Math - Question 18, 2016 Edition and Beyond.\nSection 1 - No Calculator allowed!", "lengthSeconds": 145, "width": 480, "height": 360}
{"video_id": "7v8RXiswGJI", "transcript": [{"start": 0.68, "duration": 4.119, "text": "all right in this video I want to do"}, {"start": 2.12, "duration": 4.96, "text": "another uh factoring of a perfect square"}, {"start": 4.799, "duration": 5.321, "text": "trinomial and in this video I'm going to"}, {"start": 7.08, "duration": 5.639, "text": "end up using the second video a^ 2 - 2 a"}, {"start": 10.12, "duration": 6.36, "text": "+ b^ 2 it says we can factor that as the"}, {"start": 12.719, "duration": 5.001, "text": "quantity A- B * Aus B again multiply out"}, {"start": 16.48, "duration": 3.76, "text": "the bottom one you'll see you'll get"}, {"start": 17.72, "duration": 7.52, "text": "this right back and in this video I'm"}, {"start": 20.24, "duration": 8.68, "text": "going to factor 81 x^2 - 36x y + 4"}, {"start": 25.24, "duration": 6.039, "text": "y^2 again what I do in these is uh I'm"}, {"start": 28.92, "duration": 4.52, "text": "not so much looking at that and just"}, {"start": 31.279, "duration": 5.361, "text": "going wow that is a that is a perfect"}, {"start": 33.44, "duration": 5.84, "text": "square trinomial um to me in my head I'm"}, {"start": 36.64, "duration": 4.2, "text": "thinking well I got to factor it so uh"}, {"start": 39.28, "duration": 4.64, "text": "there's three terms so I know I'm going"}, {"start": 40.84, "duration": 6.879, "text": "to make two sets of parentheses here and"}, {"start": 43.92, "duration": 7.159, "text": "again at least you know one thing I'm"}, {"start": 47.719, "duration": 7.84, "text": "hoping that would work so to get the 81x"}, {"start": 51.079, "duration": 5.521, "text": "squ um I could use a 9x and a 9x you"}, {"start": 55.559, "duration": 4.721, "text": "know again there's certainly other"}, {"start": 56.6, "duration": 6.68, "text": "numbers that multiply to give us 881 but"}, {"start": 60.28, "duration": 5.32, "text": "this is a lot of times my first guess to"}, {"start": 63.28, "duration": 6.199, "text": "get the 4 y^2 I'm either going to need a"}, {"start": 65.6, "duration": 5.72, "text": "positive 2 Y and a positive 2 y or"}, {"start": 69.479, "duration": 4.721, "text": "they'll both have to be negative but"}, {"start": 71.32, "duration": 4.839, "text": "since the term in the middle is"}, {"start": 74.2, "duration": 4.879, "text": "negative they'll both have to be"}, {"start": 76.159, "duration": 5.92, "text": "negative and in my head I'm thinking you"}, {"start": 79.079, "duration": 7.521, "text": "know is that right well if you multiply"}, {"start": 82.079, "duration": 9.561, "text": "it out 9x and 9x is 81 x^2 which works"}, {"start": 86.6, "duration": 8.6, "text": "9x and -2 y will give us a -8 18 XY on"}, {"start": 91.64, "duration": 7.119, "text": "the inside we'll get another -8x y if we"}, {"start": 95.2, "duration": 7.04, "text": "combine those hey we'll get our - 36x Y"}, {"start": 98.759, "duration": 5.881, "text": "and then on the outside we'll get a pos4"}, {"start": 102.24, "duration": 3.32, "text": "y^2 so again you can really check that"}, {"start": 104.64, "duration": 3.68, "text": "this"}, {"start": 105.56, "duration": 6.76, "text": "original if we think about our a as"}, {"start": 108.32, "duration": 6.36, "text": "being uh 9x and if we think about our b"}, {"start": 112.32, "duration": 5.52, "text": "as being"}, {"start": 114.68, "duration": 3.16, "text": "um the"}, {"start": 118.079, "duration": 4.32, "text": "2y it says really you can think about"}, {"start": 120.399, "duration": 8.92, "text": "this original problem of being in the"}, {"start": 122.399, "duration": 11.121, "text": "form a^ 2 - 2 * a *"}, {"start": 129.319, "duration": 6.401, "text": "B plus b^2 and if you rewrite uh you"}, {"start": 133.52, "duration": 4.76, "text": "know if you were to fill this in using"}, {"start": 135.72, "duration": 4.56, "text": "these values we would get this original"}, {"start": 138.28, "duration": 6.36, "text": "back and again it says it factors as a"}, {"start": 140.28, "duration": 7.56, "text": "minus B * a minus B and that's what we"}, {"start": 144.64, "duration": 6.12, "text": "just did so again you know I I'm not"}, {"start": 147.84, "duration": 5.64, "text": "really noticing this thing I I don't"}, {"start": 150.76, "duration": 4.72, "text": "notice that if I double you know the"}, {"start": 153.48, "duration": 4.28, "text": "first and the last and hey everything"}, {"start": 155.48, "duration": 4.16, "text": "works out to be the middle to me I'm"}, {"start": 157.76, "duration": 4.6, "text": "just thinking you know again one thing"}, {"start": 159.64, "duration": 6.239, "text": "that kind of Clues me in to try this is"}, {"start": 162.36, "duration": 5.959, "text": "you know 81 is a nice is is a is a"}, {"start": 165.879, "duration": 5.201, "text": "perfect square four is also a perfect"}, {"start": 168.319, "duration": 4.92, "text": "square and again it just makes me think"}, {"start": 171.08, "duration": 4.32, "text": "probably you know not always but at"}, {"start": 173.239, "duration": 5.161, "text": "least that's a good first place to"}, {"start": 175.4, "duration": 3.0, "text": "check"}], "title": "Factoring Perfect Square Trinomials - Ex 2", "description": "Thanks to all of you who support me on Patreon. You da real mvps!   $1 per month helps!! :)  https://www.patreon.com/patrickjmt !! Factoring Perfect Square Trinomials - Ex 2", "lengthSeconds": 176, "width": 480, "height": 360}
{"video_id": "lZRmOcBOht4", "transcript": [{"start": 0.44, "duration": 4.24, "text": "all right here we're going to look at an"}, {"start": 1.719, "duration": 5.2, "text": "example of dividing a whole number by a"}, {"start": 4.68, "duration": 3.76, "text": "mixed number and again what I'm going to"}, {"start": 6.919, "duration": 4.6, "text": "do is I'm just going to make both of"}, {"start": 8.44, "duration": 4.68, "text": "these into single fractions and then I'm"}, {"start": 11.519, "duration": 2.681, "text": "going to take my division and write it"}, {"start": 13.12, "duration": 3.079, "text": "as"}, {"start": 14.2, "duration": 4.839, "text": "multiplication so I'm going to write 10"}, {"start": 16.199, "duration": 5.441, "text": "is 10 over 1 we'll leave the division"}, {"start": 19.039, "duration": 7.121, "text": "alone for just a moment well we'll make"}, {"start": 21.64, "duration": 7.52, "text": "this into a fraction 5 * 2 is 10 + 3 is"}, {"start": 26.16, "duration": 5.8, "text": "13 we still have the five in the"}, {"start": 29.16, "duration": 2.8, "text": "denominator"}, {"start": 32.119, "duration": 4.361, "text": "well I leave my first fraction alone"}, {"start": 35.079, "duration": 3.441, "text": "division becomes"}, {"start": 36.48, "duration": 5.599, "text": "multiplication and then we"}, {"start": 38.52, "duration": 5.8, "text": "flip our fraction over and at this point"}, {"start": 42.079, "duration": 4.841, "text": "well I think are there any common"}, {"start": 44.32, "duration": 4.719, "text": "factors well 10 and five in the"}, {"start": 46.92, "duration": 2.84, "text": "numerator there's only a one and a 13 in"}, {"start": 49.039, "duration": 3.761, "text": "the"}, {"start": 49.76, "duration": 4.599, "text": "denominator uh nothing has uh uh we"}, {"start": 52.8, "duration": 3.36, "text": "can't reduce any of these fractions"}, {"start": 54.359, "duration": 7.121, "text": "immediately so I'm just going to go"}, {"start": 56.16, "duration": 10.08, "text": "ahead and multiply so 10 * 5 will be 50"}, {"start": 61.48, "duration": 7.08, "text": "1 * 13 is just 13 and of course we could"}, {"start": 66.24, "duration": 6.64, "text": "leave it as a fraction we could write it"}, {"start": 68.56, "duration": 8.32, "text": "as a mixed number again so let's see uh"}, {"start": 72.88, "duration": 6.76, "text": "so if we do 50 divided by 13 let's see I"}, {"start": 76.88, "duration": 6.08, "text": "guess 13 * 3 would be"}, {"start": 79.64, "duration": 6.92, "text": "39 we could subtract that would leave us"}, {"start": 82.96, "duration": 5.159, "text": "with 11 so we could write it as a mixed"}, {"start": 86.56, "duration": 3.84, "text": "number by saying it's going to equal"}, {"start": 88.119, "duration": 5.801, "text": "three 11 is our"}, {"start": 90.4, "duration": 6.2, "text": "remainder divided by 13"}, {"start": 93.92, "duration": 5.4, "text": "so either one of those would be correct"}, {"start": 96.6, "duration": 6.159, "text": "again just the improper fraction or as a"}, {"start": 99.32, "duration": 3.439, "text": "mixed number"}], "title": "Dividing Fractions and Mixed Numbers, Ex 2", "description": "Thanks to all of you who support me on Patreon. You da real mvps!   $1 per month helps!! :)  https://www.patreon.com/patrickjmt !! Dividing Fractions and Mixed Numbers, Ex 2", "lengthSeconds": 101, "width": 480, "height": 360}
{"video_id": "EM14y1pf3V4", "transcript": [{"start": 1.199, "duration": 3.401, "text": "okay here we're going to identify we've"}, {"start": 3.159, "duration": 3.921, "text": "got eight functions we're going to"}, {"start": 4.6, "duration": 6.039, "text": "determine which of those are exponential"}, {"start": 7.08, "duration": 7.24, "text": "functions so recall an exponential"}, {"start": 10.639, "duration": 5.8, "text": "function is a function of the form a"}, {"start": 14.32, "duration": 6.28, "text": "raised to the power of"}, {"start": 16.439, "duration": 6.721, "text": "X and A has to satisfy a"}, {"start": 20.6, "duration": 6.439, "text": "couple a couple conditions either it has"}, {"start": 23.16, "duration": 5.4, "text": "to be between Z and one or a has to be"}, {"start": 27.039, "duration": 6.001, "text": "strictly greater than"}, {"start": 28.56, "duration": 7.04, "text": "one we don't want a to equal one nor do"}, {"start": 33.04, "duration": 4.519, "text": "we want it to be a negative number okay"}, {"start": 35.6, "duration": 3.04, "text": "we don't want a negative number raised"}, {"start": 37.559, "duration": 5.201, "text": "to a"}, {"start": 38.64, "duration": 6.12, "text": "power so okay so which one of these"}, {"start": 42.76, "duration": 4.639, "text": "maybe all of them maybe none of them can"}, {"start": 44.76, "duration": 4.92, "text": "we write in this form well certainly the"}, {"start": 47.399, "duration": 4.0, "text": "very first function that looks exactly"}, {"start": 49.68, "duration": 5.76, "text": "like an exponential to me we've got a"}, {"start": 51.399, "duration": 6.96, "text": "number raised to a a an exponent a a"}, {"start": 55.44, "duration": 5.759, "text": "variable and our number is certainly"}, {"start": 58.359, "duration": 5.241, "text": "greater than one so G of X works that"}, {"start": 61.199, "duration": 5.001, "text": "would certainly be an"}, {"start": 63.6, "duration": 6.559, "text": "exponential notice we have this next one"}, {"start": 66.2, "duration": 6.8, "text": "H ofx equals 2 raed to the^ of"}, {"start": 70.159, "duration": 6.0, "text": "3x we could write that as"}, {"start": 73.0, "duration": 6.2, "text": "2 cubed raised to the power of X that"}, {"start": 76.159, "duration": 6.28, "text": "still gives us the exact same thing 2"}, {"start": 79.2, "duration": 6.279, "text": "the 3 Power 8 so really this function is"}, {"start": 82.439, "duration": 6.521, "text": "the same thing as 8 raised to the power"}, {"start": 85.479, "duration": 5.64, "text": "of X so that one is also an exponential"}, {"start": 88.96, "duration": 5.119, "text": "function"}, {"start": 91.119, "duration": 3.96, "text": "well P ofx = -4 that's just a constant"}, {"start": 94.079, "duration": 5.281, "text": "that doesn't"}, {"start": 95.079, "duration": 7.641, "text": "work now f ofx = 1 raised to the^ of 3x"}, {"start": 99.36, "duration": 5.2, "text": "+ 4 that sort of right we've got this"}, {"start": 102.72, "duration": 3.84, "text": "number we've got some variables in we"}, {"start": 104.56, "duration": 5.559, "text": "got a variable in the exponent but one"}, {"start": 106.56, "duration": 7.239, "text": "to any power just equals one so that's a"}, {"start": 110.119, "duration": 3.68, "text": "constant so that one doesn't"}, {"start": 114.079, "duration": 7.761, "text": "work m ofx = 50 + x that's a linear"}, {"start": 119.079, "duration": 8.801, "text": "function that doesn't"}, {"start": 121.84, "duration": 8.759, "text": "work Q ofx = 8x - 3 well again 8 to the"}, {"start": 127.88, "duration": 4.719, "text": "power of X we said that's exponential"}, {"start": 130.599, "duration": 6.321, "text": "the minus three is just going to shift"}, {"start": 132.599, "duration": 8.64, "text": "our function down so 8x - 3 that also"}, {"start": 136.92, "duration": 6.959, "text": "looks like an exponential function to me"}, {"start": 141.239, "duration": 6.64, "text": "X2 well that's quadratic that doesn't"}, {"start": 143.879, "duration": 6.521, "text": "work again and N ofx = 5x that's just a"}, {"start": 147.879, "duration": 4.761, "text": "linear a linear function"}, {"start": 150.4, "duration": 5.96, "text": "so it looks like we have 1 2 3"}, {"start": 152.64, "duration": 3.72, "text": "exponential functions"}], "title": "Can You Identify the Exponential Functions?  Which of the Following is an Exponential Function?", "description": "Can you identify which of the following functions are exponential functions? Some algebra may be required to make it perfectly clear!  In this video I show why each of the given functions is an exponential function or explain why it is not an exponential function.\nThese questions would be appropriate for someone who has already seen a quick introduction to exponential functions.\nRecall that an exponential function is a function whose base is a number raised to a variable, with the base being larger than 0 and not equal to 1.\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n#math, #algebra, #khanacademy, #khan, #geometry, #education, #montessori, #calculus, #homeschooling, #functions, #notation, #organicchemistry, #organicchemistrytutor, #calculus, #derivative, #powerrule, #limits,  #limit, #graph, #graphing, #conjugate, #infinity, #patrickjmt, #rational, #shortcut, #trick, #commondenominator, #fraction, #university, #admissions, #test #continuity #onesided #infinite #absolutevalue #tangents #tangentsandnormals #tangentline #definition #hindi #india #exponential #exponents #exponentialfunctions #exponentialform #exponentialproblemsexamples", "lengthSeconds": 155, "width": 450, "height": 360}
{"video_id": "sukb8ORaHr0", "transcript": [{"start": 0.439, "duration": 5.291, "text": "okay in this video I just want to talk"}, {"start": 2.85, "duration": 4.23, "text": "about quadratic inequalities and I just"}, {"start": 5.73, "duration": 5.789, "text": "want to talk about a really common"}, {"start": 7.08, "duration": 10.47, "text": "mistake people make so don't do what I'm"}, {"start": 11.519, "duration": 8.071, "text": "about to do so a second ago or in my"}, {"start": 17.55, "duration": 4.5, "text": "other video we actually solved this"}, {"start": 19.59, "duration": 5.279, "text": "quadratic equation x squared plus 2x"}, {"start": 22.05, "duration": 4.469, "text": "minus 8 and we came up with the"}, {"start": 24.869, "duration": 6.091, "text": "solutions negative infinity to negative"}, {"start": 26.519, "duration": 6.991, "text": "4 and then 2 to infinity a lot of times"}, {"start": 30.96, "duration": 7.02, "text": "what people will do so again what I'm"}, {"start": 33.51, "duration": 8.31, "text": "about to do is very much incorrect so"}, {"start": 37.98, "duration": 5.52, "text": "don't do what I'm about to do but it's"}, {"start": 41.82, "duration": 4.53, "text": "basically I see it"}, {"start": 43.5, "duration": 5.399, "text": "I see it all the time okay so you can"}, {"start": 46.35, "duration": 6.57, "text": "factor x squared plus 2x minus 8 we"}, {"start": 48.899, "duration": 9.121, "text": "factored that a second ago as X plus 4"}, {"start": 52.92, "duration": 7.049, "text": "and then X minus 2 so our inequality has"}, {"start": 58.02, "duration": 5.19, "text": "to be greater than or equal to 0"}, {"start": 59.969, "duration": 5.461, "text": "and now what people will do is they'll"}, {"start": 63.21, "duration": 5.61, "text": "just you know if it was an equation you"}, {"start": 65.43, "duration": 6.03, "text": "set each piece equal to 0 but if it's an"}, {"start": 68.82, "duration": 4.32, "text": "inequality you can't do the following"}, {"start": 71.46, "duration": 5.76, "text": "people still want to do the same thing"}, {"start": 73.14, "duration": 6.54, "text": "they want to take X plus 4 and make that"}, {"start": 77.22, "duration": 5.64, "text": "greater than or equal to 0 and then they"}, {"start": 79.68, "duration": 6.96, "text": "want to take X minus 2 and make that"}, {"start": 82.86, "duration": 5.43, "text": "greater than or equal to 0 so the stuff"}, {"start": 86.64, "duration": 4.25, "text": "down here again was the actual correct"}, {"start": 88.29, "duration": 2.6, "text": "solution"}, {"start": 92.24, "duration": 5.73, "text": "before I lose my focus here and then"}, {"start": 96.229, "duration": 3.961, "text": "they'll subtract four and they say well"}, {"start": 97.97, "duration": 5.009, "text": "X has to be greater than or equal to"}, {"start": 100.19, "duration": 5.58, "text": "negative four and then it says if you"}, {"start": 102.979, "duration": 6.75, "text": "solve this by adding two it says X has"}, {"start": 105.77, "duration": 5.79, "text": "to be greater than or equal to two so"}, {"start": 109.729, "duration": 3.691, "text": "what does that mean does it have okay so"}, {"start": 111.56, "duration": 5.36, "text": "it has to be greater than negative four"}, {"start": 113.42, "duration": 7.17, "text": "or two or is it have to do both of these"}, {"start": 116.92, "duration": 5.26, "text": "in either case I think you're gonna see"}, {"start": 120.59, "duration": 3.419, "text": "that definitely you're not going to get"}, {"start": 122.18, "duration": 7.2, "text": "this part of the the solution that we"}, {"start": 124.009, "duration": 7.23, "text": "found before so this parts okay writing"}, {"start": 129.38, "duration": 4.079, "text": "it but doing this step where you take"}, {"start": 131.239, "duration": 4.741, "text": "each piece and make it greater than or"}, {"start": 133.459, "duration": 4.681, "text": "equal to or less than or equal to or"}, {"start": 135.98, "duration": 4.8, "text": "whatever it is that is very much"}, {"start": 138.14, "duration": 4.83, "text": "incorrect again you have to make this"}, {"start": 140.78, "duration": 6.39, "text": "number line and check all the points so"}, {"start": 142.97, "duration": 6.63, "text": "I just want to point this out so that"}, {"start": 147.17, "duration": 4.46, "text": "hopefully you won't do it alright good"}, {"start": 149.6, "duration": 2.03, "text": "luck"}], "title": "Solving Quadratic Inequalities - A Common Mistake", "description": "Thanks to all of you who support me on Patreon. You da real mvps!   $1 per month helps!! :)  https://www.patreon.com/patrickjmt !! Solving Quadratic Inequalities - A Common Mistake", "lengthSeconds": 150, "width": 480, "height": 360}
{"video_id": "rYwSiJPerF4", "transcript": [{"start": 0.65, "duration": 4.99, "text": "so in this question we were given that a"}, {"start": 3.3, "duration": 5.04, "text": "line and the XY plane passes through the"}, {"start": 5.64, "duration": 4.2, "text": "origin and has a slope of 1 over 7 we"}, {"start": 8.34, "duration": 4.2, "text": "want to know which of the phone points"}, {"start": 9.84, "duration": 5.879, "text": "lies on the line is it 0 comma 7 1 comma"}, {"start": 12.54, "duration": 5.34, "text": "7 7 comma 7 or 14 comma 2 again"}, {"start": 15.719, "duration": 3.81, "text": "different ways to do this one way again"}, {"start": 17.88, "duration": 2.85, "text": "I'm not a huge fan of graphing and"}, {"start": 19.529, "duration": 5.25, "text": "general you know it depends on the"}, {"start": 20.73, "duration": 5.73, "text": "question because graphing can be a bit"}, {"start": 24.779, "duration": 3.24, "text": "sloppy but i think this is one where"}, {"start": 26.46, "duration": 3.36, "text": "graphing would work we'll also just talk"}, {"start": 28.019, "duration": 3.721, "text": "about the equation of a line here real"}, {"start": 29.82, "duration": 3.419, "text": "quick as well okay so i know that it"}, {"start": 31.74, "duration": 4.86, "text": "goes through the origin so there's one"}, {"start": 33.239, "duration": 4.98, "text": "point it has a slope of 1 over 7 so"}, {"start": 36.6, "duration": 3.6, "text": "remember the top part that's the change"}, {"start": 38.219, "duration": 4.831, "text": "in Y we sometimes write a little"}, {"start": 40.2, "duration": 7.98, "text": "triangle that triangle Delta means"}, {"start": 43.05, "duration": 8.31, "text": "change so change in Y over the change in"}, {"start": 48.18, "duration": 5.149, "text": "X so it says from the origin we would go"}, {"start": 51.36, "duration": 5.609, "text": "up one unit and then I would go over"}, {"start": 53.329, "duration": 7.601, "text": "seven units to get another point okay"}, {"start": 56.969, "duration": 5.401, "text": "well let's see so you know 0 comma 7 0"}, {"start": 60.93, "duration": 5.25, "text": "comma 7 would be up here on the y-axis"}, {"start": 62.37, "duration": 5.31, "text": "that's definitely not correct 1 comma 7"}, {"start": 66.18, "duration": 5.04, "text": "well let's be careful because this is"}, {"start": 67.68, "duration": 5.369, "text": "the point this is the point 7 comma 1 1"}, {"start": 71.22, "duration": 4.8, "text": "comma 7 would be up here somewhere so"}, {"start": 73.049, "duration": 5.881, "text": "it's not answer choice B 7 comma 7 would"}, {"start": 76.02, "duration": 6.0, "text": "be up here so that's not right oh so by"}, {"start": 78.93, "duration": 4.47, "text": "deduction okay 14 / 2 seems to be"}, {"start": 82.02, "duration": 3.54, "text": "corrected again that makes sense because"}, {"start": 83.4, "duration": 4.59, "text": "from this point if we go up another one"}, {"start": 85.56, "duration": 6.419, "text": "unit will be at two and then we go over"}, {"start": 87.99, "duration": 6.809, "text": "another seven units will be at 14 so"}, {"start": 91.979, "duration": 4.951, "text": "yeah so it looks like 14 over to excuse"}, {"start": 94.799, "duration": 5.371, "text": "me 14 comma 2 it's going to be the"}, {"start": 96.93, "duration": 5.28, "text": "correct answer so answer choice D also"}, {"start": 100.17, "duration": 5.4, "text": "remember that the equation of a line we"}, {"start": 102.21, "duration": 10.949, "text": "can use y equals MX plus B where B"}, {"start": 105.57, "duration": 9.07, "text": "represents the y-intercept and n is the"}, {"start": 113.159, "duration": 3.861, "text": "slope"}, {"start": 114.64, "duration": 8.38, "text": "so we could write the equation of our"}, {"start": 117.02, "duration": 10.139, "text": "line as y equals 1 over 7x plus 0 or y"}, {"start": 123.02, "duration": 6.15, "text": "equals 1 over 7x and you could also"}, {"start": 127.159, "duration": 4.261, "text": "another way to do it so if you didn't"}, {"start": 129.17, "duration": 3.81, "text": "want to graph it you know maybe they"}, {"start": 131.42, "duration": 3.21, "text": "gave you you know some really kind of"}, {"start": 132.98, "duration": 3.9, "text": "weird numbers something not so easy"}, {"start": 134.63, "duration": 4.95, "text": "again you could start substituting these"}, {"start": 136.88, "duration": 4.89, "text": "points into the equation and see which"}, {"start": 139.58, "duration": 3.84, "text": "one result in a true statement so for"}, {"start": 141.77, "duration": 5.88, "text": "example I could ask myself answered"}, {"start": 143.42, "duration": 8.22, "text": "choice a 0 comma 7 work well ok if I put"}, {"start": 147.65, "duration": 9.15, "text": "0 in for X and seven in for y I would"}, {"start": 151.64, "duration": 7.98, "text": "ask myself does 7 equal 1 7th x 0 again"}, {"start": 156.8, "duration": 9.06, "text": "I'm substituting in y equals 7 and 0 for"}, {"start": 159.62, "duration": 7.86, "text": "X well one over 7 x 0 0 does not equal 7"}, {"start": 165.86, "duration": 3.69, "text": "so I would know that answer choice a is"}, {"start": 167.48, "duration": 3.66, "text": "not correct and I could go through your"}, {"start": 169.55, "duration": 3.17, "text": "answer choice B and C that doesn't work"}, {"start": 171.14, "duration": 4.11, "text": "answer choice see that doesn't work"}, {"start": 172.72, "duration": 7.63, "text": "again answer choice D just to show it"}, {"start": 175.25, "duration": 8.18, "text": "works so okay so i'm going to put two in"}, {"start": 180.35, "duration": 7.65, "text": "for y and I'm thinking does that equal"}, {"start": 183.43, "duration": 8.2, "text": "one over 7 x 14 well this is the same"}, {"start": 188.0, "duration": 5.28, "text": "thing as 14 / 7 which gives you two yeah"}, {"start": 191.63, "duration": 3.48, "text": "two equals two so that would be another"}, {"start": 193.28, "duration": 3.45, "text": "way to do it that would be sort of a"}, {"start": 195.11, "duration": 4.32, "text": "more algebraic way to do it without"}, {"start": 196.73, "duration": 4.44, "text": "graphing so again two ways to do it you"}, {"start": 199.43, "duration": 3.21, "text": "know whichever way makes you happy again"}, {"start": 201.17, "duration": 3.3, "text": "if you peeked over at me on the test I"}, {"start": 202.64, "duration": 4.349, "text": "would definitely be doing a rough sketch"}, {"start": 204.47, "duration": 4.62, "text": "because you know the numbers are pretty"}, {"start": 206.989, "duration": 4.381, "text": "small and I could do that pretty quickly"}, {"start": 209.09, "duration": 4.08, "text": "but again you know notice they're trying"}, {"start": 211.37, "duration": 3.959, "text": "to trip you up there giving you a slope"}, {"start": 213.17, "duration": 4.62, "text": "of 1 over 7 they're giving you 1 comma 7"}, {"start": 215.329, "duration": 4.44, "text": "as a solution so make sure you're doing"}, {"start": 217.79, "duration": 5.39, "text": "your change in Y over your change in X"}, {"start": 219.769, "duration": 3.411, "text": "correctly when you graph it"}], "title": "SAT Math - Question 12, 2016 Edition and Beyond", "description": "Thanks to all of you who support me on Patreon. You da real mvps!   $1 per month helps!! :)  https://www.patreon.com/patrickjmt !! SAT Math - Question 12, 2016 Edition and Beyond.\nSection 1 - No Calculator allowed!", "lengthSeconds": 222, "width": 480, "height": 360}
{"video_id": "PIr6NMbfTCM", "transcript": [{"start": 0.64, "duration": 3.2, "text": "all right here we're going to do one"}, {"start": 1.68, "duration": 6.199, "text": "more example using the limit comparison"}, {"start": 3.84, "duration": 6.839, "text": "test so here we've got s of 1 over n and"}, {"start": 7.879, "duration": 5.96, "text": "you know the series from one to Infinity"}, {"start": 10.679, "duration": 5.52, "text": "so you know there's not a lot going on"}, {"start": 13.839, "duration": 5.44, "text": "here with this series uh you might first"}, {"start": 16.199, "duration": 4.92, "text": "try to do the test for Divergence and if"}, {"start": 19.279, "duration": 3.76, "text": "you do the test for Divergence well we"}, {"start": 21.119, "duration": 3.16, "text": "can do the limit as in goes to Infinity"}, {"start": 23.039, "duration": 4.041, "text": "so this is what I do if I'm kind of"}, {"start": 24.279, "duration": 5.961, "text": "completely unsure about a series well as"}, {"start": 27.08, "duration": 4.72, "text": "n goes to Infinity you know uh inside"}, {"start": 30.24, "duration": 4.8, "text": "the parenthesis we'll get one over a"}, {"start": 31.8, "duration": 6.24, "text": "large number so that'll approach zero so"}, {"start": 35.04, "duration": 5.64, "text": "it says it approaches sign of zero but"}, {"start": 38.04, "duration": 5.6, "text": "sign of zero is zero so the test for"}, {"start": 40.68, "duration": 2.96, "text": "Divergence would be"}, {"start": 44.44, "duration": 5.599, "text": "inconclusive well I see this series uh"}, {"start": 47.399, "duration": 5.8, "text": "or I see this this this one over n so"}, {"start": 50.039, "duration": 3.16, "text": "maybe we can compare it"}, {"start": 53.68, "duration": 6.359, "text": "to one over n and see what happens and"}, {"start": 57.6, "duration": 5.52, "text": "recall this is the harmonic series again"}, {"start": 60.039, "duration": 3.081, "text": "a Divergent P"}, {"start": 64.84, "duration": 4.16, "text": "series okay so we can do the limit as n"}, {"start": 67.28, "duration": 8.08, "text": "goes to"}, {"start": 69.0, "duration": 9.6, "text": "Infinity S of 1 n / 1"}, {"start": 75.36, "duration": 5.04, "text": "n but again uh as n goes to Infinity we"}, {"start": 78.6, "duration": 4.76, "text": "set sign of 1 over n is going to go to"}, {"start": 80.4, "duration": 6.759, "text": "zero 1 over n is going to go to zero so"}, {"start": 83.36, "duration": 3.799, "text": "we can use lal's rule on"}, {"start": 88.92, "duration": 5.519, "text": "this so we'll have to take the"}, {"start": 90.84, "duration": 5.0, "text": "derivative of the uh numerator so we'll"}, {"start": 94.439, "duration": 4.081, "text": "have to use the chain rule so the"}, {"start": 95.84, "duration": 4.599, "text": "derivative of s is cosine we'll leave"}, {"start": 98.52, "duration": 4.32, "text": "the 1 over n"}, {"start": 100.439, "duration": 4.441, "text": "alone and then we would have to multiply"}, {"start": 102.84, "duration": 4.04, "text": "by the derivative of 1 over n that would"}, {"start": 104.88, "duration": 3.96, "text": "be the chain Rule and I'm going to be"}, {"start": 106.88, "duration": 3.68, "text": "lazy and not even do it because I"}, {"start": 108.84, "duration": 3.48, "text": "recognize in the"}, {"start": 110.56, "duration": 3.96, "text": "denominator well we'll get the"}, {"start": 112.32, "duration": 4.56, "text": "derivative of 1 over n well we would"}, {"start": 114.52, "duration": 4.8, "text": "have to do the exact same thing well"}, {"start": 116.88, "duration": 3.839, "text": "those two terms would simply uh you"}, {"start": 119.32, "duration": 4.079, "text": "those two fact factors would simply"}, {"start": 120.719, "duration": 4.641, "text": "cancel out so I mean you can always"}, {"start": 123.399, "duration": 3.561, "text": "actually compute them but it's going to"}, {"start": 125.36, "duration": 4.0, "text": "be the exact same thing so they're going"}, {"start": 126.96, "duration": 4.639, "text": "to go away well now we're left with the"}, {"start": 129.36, "duration": 5.92, "text": "limit as n goes to Infinity of cosine 1"}, {"start": 131.599, "duration": 6.321, "text": "/ n that's going to give us cosine of 0"}, {"start": 135.28, "duration": 5.0, "text": "cosine of 0 is one again we get a"}, {"start": 137.92, "duration": 5.76, "text": "positive finite number since we're"}, {"start": 140.28, "duration": 5.76, "text": "comparing uh to a Divergent P series"}, {"start": 143.68, "duration": 6.8, "text": "that would tell us that our Series in"}, {"start": 146.04, "duration": 4.44, "text": "question also diverges"}], "title": "Limit Comparison Test for Series - Another Example 8", "description": "Thanks to all of you who support me on Patreon. You da real mvps!   $1 per month helps!! :)  https://www.patreon.com/patrickjmt !! Limit Comparison Test for Series - Another Example 8. In this video, I use the limit comparison test to determine whether or not a given series converges or diverges. I do not delve heavily into the theory in this video (but do in another video if that is what you need to see!)", "lengthSeconds": 153, "width": 480, "height": 360}
{"video_id": "8eOXAfgipJ0", "transcript": [{"start": 1.12, "duration": 4.08, "text": "all right this is going to be an example"}, {"start": 2.8, "duration": 4.72, "text": "of solving a quadratic equation by"}, {"start": 5.2, "duration": 4.439, "text": "factoring and this is kind of one of the"}, {"start": 7.52, "duration": 4.84, "text": "more to me it's one of the it's kind of"}, {"start": 9.639, "duration": 6.961, "text": "the fundamental case simply because the"}, {"start": 12.36, "duration": 7.64, "text": "coefficient on the m s is a one and in"}, {"start": 16.6, "duration": 5.839, "text": "that case what we need is we need two"}, {"start": 20.0, "duration": 4.68, "text": "numbers so we're looking for two numbers"}, {"start": 22.439, "duration": 2.241, "text": "that"}, {"start": 27.56, "duration": 5.36, "text": "multiply to -35"}, {"start": 31.679, "duration": 5.361, "text": "but"}, {"start": 32.92, "duration": 6.799, "text": "add to the value they have to add up to"}, {"start": 37.04, "duration": 4.8, "text": "the coefficient in front of the linear"}, {"start": 39.719, "duration": 5.441, "text": "term the term to the first Power which"}, {"start": 41.84, "duration": 6.039, "text": "is just uh in this case positive two so"}, {"start": 45.16, "duration": 5.559, "text": "we need two numbers that multiply to35"}, {"start": 47.879, "duration": 2.84, "text": "but add up to"}, {"start": 52.16, "duration": 6.719, "text": "positive2 so to get M SED we're going to"}, {"start": 54.6, "duration": 6.36, "text": "use an m and an M an m and an M um and"}, {"start": 58.879, "duration": 3.48, "text": "whatever numbers uh satisfy this"}, {"start": 60.96, "duration": 3.4, "text": "requirement that's what we're going to"}, {"start": 62.359, "duration": 3.8, "text": "put in each we'll put one in each set of"}, {"start": 64.36, "duration": 6.2, "text": "parentheses"}, {"start": 66.159, "duration": 8.201, "text": "so okay to get uh"}, {"start": 70.56, "duration": 5.599, "text": "35 and I guess to get Negative 35 um you"}, {"start": 74.36, "duration": 3.799, "text": "know we could use some combination of 1"}, {"start": 76.159, "duration": 4.201, "text": "and 35 you know one of them will have to"}, {"start": 78.159, "duration": 3.681, "text": "be positive one will have to be negative"}, {"start": 80.36, "duration": 3.96, "text": "but in either case I don't see how"}, {"start": 81.84, "duration": 5.239, "text": "that's possibly going to add up to"}, {"start": 84.32, "duration": 5.08, "text": "pos2 um how about some combination of"}, {"start": 87.079, "duration": 5.08, "text": "five and seven well I guess if we made"}, {"start": 89.4, "duration": 5.48, "text": "the seven positive and the five negative"}, {"start": 92.159, "duration": 6.081, "text": "those would certainly multiply to -35"}, {"start": 94.88, "duration": 5.32, "text": "but add up to positive2 so those are"}, {"start": 98.24, "duration": 6.519, "text": "going to be our candidates we'll have M"}, {"start": 100.2, "duration": 7.559, "text": "minus 5 m + 7 that's how the left side"}, {"start": 104.759, "duration": 5.64, "text": "factors so we'll set that equal to"}, {"start": 107.759, "duration": 4.521, "text": "zero and now once we once we have our"}, {"start": 110.399, "duration": 5.68, "text": "equation factored we just take each"}, {"start": 112.28, "duration": 6.159, "text": "factor set it equal to zero and"}, {"start": 116.079, "duration": 4.201, "text": "solve so the first equation we can"}, {"start": 118.439, "duration": 6.0, "text": "simply add five to both sides sides"}, {"start": 120.28, "duration": 6.879, "text": "we'll get M = 5 and if we subtract"}, {"start": 124.439, "duration": 6.601, "text": "s from the second equation we'll simply"}, {"start": 127.159, "duration": 6.881, "text": "get M -7 and that will be our other"}, {"start": 131.04, "duration": 8.6, "text": "solution so our two solutions here are M"}, {"start": 134.04, "duration": 5.6, "text": "= 5 and M = -7"}], "title": "Solving a Quadratic Equation: Factoring a Simple Example", "description": "Thanks to all of you who support me on Patreon. You da real mvps!   $1 per month helps!! :)  https://www.patreon.com/patrickjmt !! Solving a Quadratic Equation: Factoring a Simple Example.\r\n In this video, I show an example of solving a basic quadratic equation using factoring.", "lengthSeconds": 138, "width": 480, "height": 360}
{"video_id": "9xnhLTXZi50", "transcript": [{"start": 1.04, "duration": 5.52, "text": "okay in this example I want to evaluate"}, {"start": 4.0, "duration": 4.92, "text": "again a a fractional expression"}, {"start": 6.56, "duration": 3.92, "text": "involving absolute value and here we're"}, {"start": 8.92, "duration": 4.04, "text": "going to have to just substitute in some"}, {"start": 10.48, "duration": 6.12, "text": "values so we want to evaluate the"}, {"start": 12.96, "duration": 7.0, "text": "absolute value of B minus the absolute"}, {"start": 16.6, "duration": 6.28, "text": "value of positive a all divided byga a"}, {"start": 19.96, "duration": 7.559, "text": "and we want to do this when a = 3 and b"}, {"start": 22.88, "duration": 6.28, "text": "= -5 so all right let's just basically"}, {"start": 27.519, "duration": 3.52, "text": "fill everything in and hopefully"}, {"start": 29.16, "duration": 5.52, "text": "simplify it down so we've got the"}, {"start": 31.039, "duration": 6.601, "text": "absolute value of"}, {"start": 34.68, "duration": 5.16, "text": "B in general when I substitute I like to"}, {"start": 37.64, "duration": 5.439, "text": "put things in parentheses I I found it"}, {"start": 39.84, "duration": 5.16, "text": "keeps me uh from making mistakes it's it"}, {"start": 43.079, "duration": 4.761, "text": "ends up being good form especially if"}, {"start": 45.0, "duration": 4.52, "text": "you have lots of things to substitute uh"}, {"start": 47.84, "duration": 4.239, "text": "because you might have to distribute"}, {"start": 49.52, "duration": 5.32, "text": "this negative sign so you'll see that as"}, {"start": 52.079, "duration": 5.16, "text": "you go along um but just in general I"}, {"start": 54.84, "duration": 6.48, "text": "like to put things in parentheses so I'm"}, {"start": 57.239, "duration": 6.401, "text": "simply replacing my B with 5 there's our"}, {"start": 61.32, "duration": 5.479, "text": "minus sign we have the absolute value of"}, {"start": 63.64, "duration": 7.44, "text": "a which is simply"}, {"start": 66.799, "duration": 5.68, "text": "pos3 we're dividing all that by a which"}, {"start": 71.08, "duration": 4.359, "text": "is"}, {"start": 72.479, "duration": 5.761, "text": "-3 and if we keep simplifying this we've"}, {"start": 75.439, "duration": 5.36, "text": "got the negative of5 which inside the"}, {"start": 78.24, "duration": 6.199, "text": "bars inside the absolute value will be"}, {"start": 80.799, "duration": 5.241, "text": "positive five minus the absolute value"}, {"start": 84.439, "duration": 6.161, "text": "of"}, {"start": 86.04, "duration": 7.32, "text": "positive3 again being divided by -3"}, {"start": 90.6, "duration": 6.12, "text": "well the absolute value of postive 5 is"}, {"start": 93.36, "duration": 5.92, "text": "five the absolute value of posi3 is"}, {"start": 96.72, "duration": 6.52, "text": "three so in the numerator we're really"}, {"start": 99.28, "duration": 3.96, "text": "getting the value 5 minus"}, {"start": 105.2, "duration": 9.16, "text": "3 again still over -3 and 5 - 3 that"}, {"start": 111.079, "duration": 7.281, "text": "simply gives us pos2"}, {"start": 114.36, "duration": 6.0, "text": "over3 and that will be our solution"}, {"start": 118.36, "duration": 4.52, "text": "-2/3 um and again you could put the"}, {"start": 120.36, "duration": 5.359, "text": "negative on the on the top or out front"}, {"start": 122.88, "duration": 4.559, "text": "however you want to write it um so all"}, {"start": 125.719, "duration": 4.32, "text": "right just a little bit of substitution"}, {"start": 127.439, "duration": 4.241, "text": "here again you know being careful I"}, {"start": 130.039, "duration": 4.041, "text": "think it's a good rule of thumb to put"}, {"start": 131.68, "duration": 4.68, "text": "things in parentheses um although I"}, {"start": 134.08, "duration": 5.4, "text": "clearly didn't do it on the a part maybe"}, {"start": 136.36, "duration": 4.959, "text": "I I could have um but especially when"}, {"start": 139.48, "duration": 3.56, "text": "you have these negatives because you"}, {"start": 141.319, "duration": 3.961, "text": "have to be again really careful to"}, {"start": 143.04, "duration": 4.199, "text": "distribute things so uh that's the"}, {"start": 145.28, "duration": 4.959, "text": "reason why I'm doing that"}, {"start": 147.239, "duration": 3.0, "text": "there"}], "title": "Evaluating Expressions Involving Absolute Value - Example 3", "description": "Thanks to all of you who support me on Patreon. You da real mvps!   $1 per month helps!! :)  https://www.patreon.com/patrickjmt !! Evaluating Expressions Involving Absolute Value - Example 3\nA basic problem involving absolute value, a bit of substitution and some simplifying!", "lengthSeconds": 148, "width": 480, "height": 360}
{"video_id": "HaaGRnlAonQ", "transcript": [{"start": 1.319, "duration": 3.881, "text": "okay in this example we've got this"}, {"start": 2.96, "duration": 6.759, "text": "rather you know long formula here m"}, {"start": 5.2, "duration": 8.16, "text": "equal uh R / 1200 multili by 1 + r/ 1200"}, {"start": 9.719, "duration": 6.841, "text": "ra to^ of n / 1 + r over 1200 ra to^ of"}, {"start": 13.36, "duration": 5.6, "text": "nus 1 multiplied P they give you some"}, {"start": 16.56, "duration": 5.24, "text": "context this this has to do with um a"}, {"start": 18.96, "duration": 4.52, "text": "loan payment formula the important part"}, {"start": 21.8, "duration": 3.12, "text": "says which of the following gives the"}, {"start": 23.48, "duration": 5.32, "text": "question is which of the following gives"}, {"start": 24.92, "duration": 6.119, "text": "p in terms of m r and N so again what"}, {"start": 28.8, "duration": 4.68, "text": "that means is it says we want P isolated"}, {"start": 31.039, "duration": 4.601, "text": "we want some equation that says P equals"}, {"start": 33.48, "duration": 3.56, "text": "something which is you know so there's"}, {"start": 35.64, "duration": 4.12, "text": "the answer choices as you can see"}, {"start": 37.04, "duration": 4.28, "text": "they've got P all by itself so the only"}, {"start": 39.76, "duration": 3.479, "text": "thing to me that seems really daunting"}, {"start": 41.32, "duration": 4.079, "text": "in this problem is you know this is kind"}, {"start": 43.239, "duration": 3.441, "text": "of a long formula when you first look at"}, {"start": 45.399, "duration": 4.64, "text": "it you know maybe your eyes kind of"}, {"start": 46.68, "duration": 5.76, "text": "glaze over and you say oh I don't know"}, {"start": 50.039, "duration": 5.641, "text": "so let me show you though that this"}, {"start": 52.44, "duration": 3.24, "text": "really isn't very bad at"}, {"start": 56.039, "duration": 6.561, "text": "all so to Simply solve for p"}, {"start": 61.32, "duration": 3.64, "text": "okay so I'm going to rewrite the"}, {"start": 62.6, "duration": 3.559, "text": "original expression Okay so we've got M"}, {"start": 64.96, "duration": 3.799, "text": "there on the left I'm going to give"}, {"start": 66.159, "duration": 4.801, "text": "myself a little space um before so"}, {"start": 68.759, "duration": 6.68, "text": "here's the original expression there's"}, {"start": 70.96, "duration": 8.76, "text": "our multiplied by P so we've got R over"}, {"start": 75.439, "duration": 7.881, "text": "1200 ultip 1 + r over"}, {"start": 79.72, "duration": 5.759, "text": "1200 raised to the power of n we've got"}, {"start": 83.32, "duration": 5.6, "text": "1 + r over"}, {"start": 85.479, "duration": 5.081, "text": "1200 raised to the^ of n-1 and I can"}, {"start": 88.92, "duration": 4.04, "text": "stick that in Brackets it doesn't change"}, {"start": 90.56, "duration": 6.04, "text": "its value so there's the original"}, {"start": 92.96, "duration": 6.68, "text": "expression that I have right there well"}, {"start": 96.6, "duration": 5.08, "text": "all I want to do is just multiply so"}, {"start": 99.64, "duration": 4.2, "text": "that I can get rid of some of this stuff"}, {"start": 101.68, "duration": 5.2, "text": "well I've got a r over 1200 in the"}, {"start": 103.84, "duration": 5.52, "text": "numerator if I put one in the"}, {"start": 106.88, "duration": 5.0, "text": "denominator when I multiply I would be"}, {"start": 109.36, "duration": 5.16, "text": "able to cancel cancel the same thing to"}, {"start": 111.88, "duration": 6.599, "text": "get rid of this 1+ R over 1200 raised to"}, {"start": 114.52, "duration": 7.32, "text": "the power of n I would need a 1 + r over"}, {"start": 118.479, "duration": 4.881, "text": "1200 raised to the power of n and to get"}, {"start": 121.84, "duration": 5.639, "text": "rid of the"}, {"start": 123.36, "duration": 6.92, "text": "denominator I would need 1 + r/"}, {"start": 127.479, "duration": 6.241, "text": "1200 raised to the^ of"}, {"start": 130.28, "duration": 5.28, "text": "nus1 and notice now everything on the"}, {"start": 133.72, "duration": 4.72, "text": "right side is going to cancel except for"}, {"start": 135.56, "duration": 4.36, "text": "p the the numerator and denominator"}, {"start": 138.44, "duration": 3.32, "text": "would cancel and again now I've gotten"}, {"start": 139.92, "duration": 4.2, "text": "rid of each of these factors that were"}, {"start": 141.76, "duration": 4.08, "text": "originally in the numerator as well well"}, {"start": 144.12, "duration": 3.8, "text": "again if I do it on the right I've got"}, {"start": 145.84, "duration": 6.0, "text": "to do it on the left"}, {"start": 147.92, "duration": 6.319, "text": "so there's 1+ r R 1200 raised to the^ of"}, {"start": 151.84, "duration": 2.399, "text": "n"}, {"start": 155.04, "duration": 7.68, "text": "-1 and then R over"}, {"start": 159.04, "duration": 6.52, "text": "1200 ultip 1 + r over"}, {"start": 162.72, "duration": 4.36, "text": "1200 raised to the power of n and again"}, {"start": 165.56, "duration": 3.8, "text": "you can think about this as being M over"}, {"start": 167.08, "duration": 3.799, "text": "one if you want to again everything's"}, {"start": 169.36, "duration": 6.08, "text": "going to cancel"}, {"start": 170.879, "duration": 7.401, "text": "out on the right side except for"}, {"start": 175.44, "duration": 4.6, "text": "p so there's our P equals and again"}, {"start": 178.28, "duration": 3.92, "text": "there's nothing really to do on the left"}, {"start": 180.04, "duration": 4.919, "text": "you just you know I think the answer"}, {"start": 182.2, "duration": 5.28, "text": "Choice had it just written like this so"}, {"start": 184.959, "duration": 8.041, "text": "I'm just going to rewrite it one more"}, {"start": 187.48, "duration": 5.52, "text": "time the exact same thing that we just"}, {"start": 193.959, "duration": 5.161, "text": "had raised to the power of n and then"}, {"start": 197.12, "duration": 3.039, "text": "they had the M sticking off to the side"}, {"start": 199.12, "duration": 4.119, "text": "and again that just means you're"}, {"start": 200.159, "duration": 5.241, "text": "multiplying by M over one so that's it"}, {"start": 203.239, "duration": 4.801, "text": "so I believe that was answer Choice B"}, {"start": 205.4, "duration": 5.8, "text": "would be the correct answer so again to"}, {"start": 208.04, "duration": 5.88, "text": "make a long story short all that you"}, {"start": 211.2, "duration": 4.759, "text": "have to do for this one is you just take"}, {"start": 213.92, "duration": 3.84, "text": "you know this original okay forget about"}, {"start": 215.959, "duration": 4.761, "text": "the P for a second take this original"}, {"start": 217.76, "duration": 5.199, "text": "fraction and you just multiply uh by the"}, {"start": 220.72, "duration": 6.239, "text": "flip of it on the left and that gives"}, {"start": 222.959, "duration": 4.0, "text": "you your solution"}], "title": "SAT Math - Question 7, 2016 Edition and Beyond", "description": "Thanks to all of you who support me on Patreon. You da real mvps!   $1 per month helps!! :)  https://www.patreon.com/patrickjmt !! SAT Math - Question 7, 2016 Edition and Beyond", "lengthSeconds": 225, "width": 480, "height": 360}
{"video_id": "1O5NEI8UuHM", "transcript": [{"start": 0.64, "duration": 3.279, "text": "all right in this video we're going to"}, {"start": 1.88, "duration": 4.36, "text": "do a couple examples related to finding"}, {"start": 3.919, "duration": 3.52, "text": "the difference quotient of a function"}, {"start": 6.24, "duration": 3.399, "text": "and the idea is if you have some"}, {"start": 7.439, "duration": 4.641, "text": "function f ofx the difference quotient"}, {"start": 9.639, "duration": 6.361, "text": "is what's known as the quantity f ofx"}, {"start": 12.08, "duration": 5.359, "text": "plus h minus F ofx over H um if it's the"}, {"start": 16.0, "duration": 2.88, "text": "first time you've ever seen the"}, {"start": 17.439, "duration": 3.521, "text": "difference quotient it feels like kind"}, {"start": 18.88, "duration": 3.96, "text": "of an arbitrary you know like where the"}, {"start": 20.96, "duration": 3.96, "text": "heck is this coming from sort of thing"}, {"start": 22.84, "duration": 3.64, "text": "it actually has a lot of applications um"}, {"start": 24.92, "duration": 5.199, "text": "it has to do with sort of average rates"}, {"start": 26.48, "duration": 6.44, "text": "of change um slopes of tangent lines and"}, {"start": 30.119, "duration": 4.801, "text": "those actually tie into uh surprisingly"}, {"start": 32.92, "duration": 3.4, "text": "maybe maybe not surprisingly it actually"}, {"start": 34.92, "duration": 3.52, "text": "ties into some important ideas and"}, {"start": 36.32, "duration": 4.68, "text": "calculus so it's one of these things it"}, {"start": 38.44, "duration": 4.32, "text": "looks arbitrary but uh who whoever's"}, {"start": 41.0, "duration": 4.64, "text": "throwing these sorts of problems at you"}, {"start": 42.76, "duration": 5.959, "text": "knows knows that uh that there's a a"}, {"start": 45.64, "duration": 4.88, "text": "reason for it for sure so um okay in"}, {"start": 48.719, "duration": 4.961, "text": "part A we've got this function f ofx is"}, {"start": 50.52, "duration": 4.44, "text": "2x - 5 we're not going to do all the"}, {"start": 53.68, "duration": 3.32, "text": "difference quotient we're just going to"}, {"start": 54.96, "duration": 4.96, "text": "kind of find the numerator f ofx plus h"}, {"start": 57.0, "duration": 6.6, "text": "minus F ofx well we've already got our"}, {"start": 59.92, "duration": 6.0, "text": "our F ofx to get f ofx plus h what do we"}, {"start": 63.6, "duration": 8.44, "text": "do we just well everywhere there's an X"}, {"start": 65.92, "duration": 10.44, "text": "we replace it with X+ H so x + H - 5 so"}, {"start": 72.04, "duration": 8.96, "text": "it says this difference um f of x + H -"}, {"start": 76.36, "duration": 9.04, "text": "F ofx we would get 2 * the quantity x +"}, {"start": 81.0, "duration": 6.28, "text": "H - 5 there's my f ofx plus h and then"}, {"start": 85.4, "duration": 3.64, "text": "it says subtract away the original"}, {"start": 87.28, "duration": 5.0, "text": "function which is f ofx I'm going to put"}, {"start": 89.04, "duration": 5.48, "text": "that in parentheses 2x - 5 to make sure"}, {"start": 92.28, "duration": 4.56, "text": "I distribute the negative and now we've"}, {"start": 94.52, "duration": 4.84, "text": "got it I'm going to simplify but this is"}, {"start": 96.84, "duration": 6.12, "text": "our F ofx +"}, {"start": 99.36, "duration": 7.439, "text": "H minus our F"}, {"start": 102.96, "duration": 6.96, "text": "ofx so let's see um notice if we if we"}, {"start": 106.799, "duration": 6.081, "text": "multiply this out we would get 2 * X and"}, {"start": 109.92, "duration": 5.12, "text": "A minus 2 * X those would cancel it"}, {"start": 112.88, "duration": 6.159, "text": "looks like we would get a 2 * H left in"}, {"start": 115.04, "duration": 8.759, "text": "there um and notice we have a -5 minus"}, {"start": 119.039, "duration": 6.641, "text": "A5 well that's -5 + 5 so that would also"}, {"start": 123.799, "duration": 4.121, "text": "cancel out so I claim that the"}, {"start": 125.68, "duration": 5.36, "text": "difference quotient here is just the"}, {"start": 127.92, "duration": 6.28, "text": "quantity uh 2 H and that's"}, {"start": 131.04, "duration": 5.839, "text": "it okay so suppose we've got um suppose"}, {"start": 134.2, "duration": 5.36, "text": "f ofx is 3x + 2 and we actually want to"}, {"start": 136.879, "duration": 5.0, "text": "find this difference quotient um f ofx"}, {"start": 139.56, "duration": 5.56, "text": "plus h minus F ofx over"}, {"start": 141.879, "duration": 5.72, "text": "H so I'm going to squeeze it all in here"}, {"start": 145.12, "duration": 5.479, "text": "okay so again if we find f ofx plus h"}, {"start": 147.599, "duration": 8.201, "text": "that just says replace the X with"}, {"start": 150.599, "duration": 7.761, "text": "x + H we've got our + 2 and then it says"}, {"start": 155.8, "duration": 6.84, "text": "subtract away the original function f"}, {"start": 158.36, "duration": 6.799, "text": "ofx which is 3x + 2 and then it says"}, {"start": 162.64, "duration": 5.36, "text": "divide all of that by"}, {"start": 165.159, "duration": 6.201, "text": "H so if we distribute out the numerator"}, {"start": 168.0, "duration": 4.64, "text": "again notice we would get a 3X plus 3 H"}, {"start": 171.36, "duration": 4.519, "text": "in the first set of"}, {"start": 172.64, "duration": 6.28, "text": "parentheses um if I take the 3x minus"}, {"start": 175.879, "duration": 6.401, "text": "the 3x that would cancel so I would be"}, {"start": 178.92, "duration": 5.56, "text": "left with 3 H notice we have a plus two"}, {"start": 182.28, "duration": 5.0, "text": "and we would get a -2 when we distribute"}, {"start": 184.48, "duration": 4.839, "text": "so those would also cancel out we've"}, {"start": 187.28, "duration": 4.599, "text": "still got the H in the denominator but"}, {"start": 189.319, "duration": 5.081, "text": "now we can simply cancel out our H's and"}, {"start": 191.879, "duration": 6.161, "text": "it says the difference quotient of the"}, {"start": 194.4, "duration": 7.16, "text": "function 3x + 2 is just the value the"}, {"start": 198.04, "duration": 3.52, "text": "number three"}], "title": "\u2756 The Difference Quotient - Example 1 \u2756", "description": "The Difference Quotient - Formula & Examples\n\nDescription:\nIn this video, we explore the concept of the difference quotient, sometimes referred to as the subtraction fraction (but only by me). You'll learn the formula for the difference quotient and see how it's applied through several examples with different functions. Understanding the difference quotient is a fundamental step in calculus, as it serves as the basis for the derivative and provides insight into the rate of change of a function.\n\nThis video will guide you through:\n\nThe formula for the difference quotient:  ( f ( x + h ) \u2212 f( x )) / h.\nApplying the difference quotient to various functions.\nSimplifying expressions to reveal the structure of the difference quotient.\nGaining foundational knowledge for future topics in calculus, including derivatives and instantaneous rate of change.\nIf you find this video helpful, please subscribe, comment, like, and share! I appreciate it if you tell your friends, teachers, or parents about my videos :)\n\nFor more support and exclusive content, visit my Patreon:\nhttps://www.patreon.com/patrickjmt?ty=c\n\n\n#differencequotient #calculus #math #rateofchange #function #mathhelp #differencequotientformula #subtractionfraction #mathvideos #patrickjmt #learncalculus #limitdefinition #functionanalysis #derivativeintro #differentiation #functionproperties #subscribe #patreon #algebra #precalculus #findthedifferencequotient", "lengthSeconds": 200, "width": 480, "height": 360}
{"video_id": "vyQopC7dMgA", "transcript": [{"start": 0.35, "duration": 3.64, "text": "okay in this example we're going to plot"}, {"start": 2.58, "duration": 4.199, "text": "three more points we're going to plot"}, {"start": 3.99, "duration": 7.26, "text": "the point negative 3 comma negative 1 4"}, {"start": 6.779, "duration": 6.451, "text": "comma 0 and 0 comma negative 4 so define"}, {"start": 11.25, "duration": 4.679, "text": "net the point negative 3 comma negative"}, {"start": 13.23, "duration": 6.57, "text": "1 well from the origin I'm going to move"}, {"start": 15.929, "duration": 5.281, "text": "3 units to the left so 1 2 3 units to"}, {"start": 19.8, "duration": 3.87, "text": "the left so now I'm sitting at the x"}, {"start": 21.21, "duration": 4.2, "text": "coordinate of negative 3 and then the y"}, {"start": 23.67, "duration": 5.13, "text": "coordinate is negative 1 so we'll go"}, {"start": 25.41, "duration": 7.169, "text": "down one unit so that will be the point"}, {"start": 28.8, "duration": 6.96, "text": "negative 3 comma negative 1 the next"}, {"start": 32.579, "duration": 5.041, "text": "point is going to be 4 comma 0 so from"}, {"start": 35.76, "duration": 4.77, "text": "the origin will go to the x coordinate"}, {"start": 37.62, "duration": 4.74, "text": "of 4 or 4 units to the right and then"}, {"start": 40.53, "duration": 5.25, "text": "for the Y value it says don't go"}, {"start": 42.36, "duration": 5.25, "text": "anywhere don't go up or down so the"}, {"start": 45.78, "duration": 5.4, "text": "point 4 comma 0 that's going to be"}, {"start": 47.61, "duration": 6.48, "text": "sitting on the x axis and last but not"}, {"start": 51.18, "duration": 5.3, "text": "least we have 0 comma negative 4 so I'll"}, {"start": 54.09, "duration": 5.489, "text": "move 0 units away from the origin"}, {"start": 56.48, "duration": 8.17, "text": "horizontally and then we'll go down 1 2"}, {"start": 59.579, "duration": 7.83, "text": "3 4 units so there's the point 0 comma"}, {"start": 64.65, "duration": 5.69, "text": "negative 4 that's going to be sitting on"}, {"start": 67.409, "duration": 2.931, "text": "the y axis"}], "title": "GED 'Most Missed' Math Practice Problems , Code A.5.a,  #2", "description": "Thanks to all of you who support me on Patreon. You da real mvps!   $1 per month helps!! :)  https://www.patreon.com/patrickjmt !! The GED Testing Service keeps a list of the most missed math topics and can be seen here: http://www.gedtestingservice.com/uploads/files/126de0283c94ff6323b3b5b3f310da5d.pdf\nThis video shows how to work a sample problem using their coding system that is shown on the left using the link above.", "lengthSeconds": 71, "width": 480, "height": 360}
{"video_id": "MO6qU3SCLbM", "transcript": [{"start": 0.799, "duration": 4.081, "text": "all right in this video I want to do an"}, {"start": 2.12, "duration": 5.279, "text": "example using deor theorem to raise a"}, {"start": 4.88, "duration": 4.92, "text": "complex number to a power and it's"}, {"start": 7.399, "duration": 5.801, "text": "actually extremely easy um if you have a"}, {"start": 9.8, "duration": 5.799, "text": "complex number Z in polar form so R *"}, {"start": 13.2, "duration": 5.28, "text": "the quantity cosine theta plus I sin"}, {"start": 15.599, "duration": 5.201, "text": "Theta it says if you take your uh"}, {"start": 18.48, "duration": 4.76, "text": "complex number and raise it to a power"}, {"start": 20.8, "duration": 6.04, "text": "we just take R and raise it to that"}, {"start": 23.24, "duration": 8.359, "text": "power of n and then our angles Theta we"}, {"start": 26.84, "duration": 7.52, "text": "simply multiply that by n so"}, {"start": 31.599, "duration": 5.48, "text": "actually uh not too bad at all a lot of"}, {"start": 34.36, "duration": 4.199, "text": "times just you know uh the time"}, {"start": 37.079, "duration": 3.401, "text": "consuming part is going to be just"}, {"start": 38.559, "duration": 4.801, "text": "putting it in polar form in the first"}, {"start": 40.48, "duration": 5.079, "text": "place so let's see if we can squeeze it"}, {"start": 43.36, "duration": 6.16, "text": "all in here so here we're going to find"}, {"start": 45.559, "duration": 7.201, "text": "the value of 2 * cosine of 60\u00b0 plus I"}, {"start": 49.52, "duration": 6.559, "text": "sin of 60\u00b0 all of that to the second"}, {"start": 52.76, "duration": 6.24, "text": "power okay so um all we have to do in"}, {"start": 56.079, "duration": 6.081, "text": "this case so here to the second power"}, {"start": 59.0, "duration": 6.64, "text": "that's going to be our value of"}, {"start": 62.16, "duration": 7.12, "text": "n so here's our R here's our Theta and"}, {"start": 65.64, "duration": 4.92, "text": "our Theta this will be our value of n so"}, {"start": 69.28, "duration": 3.199, "text": "all we have to do is just plug"}, {"start": 70.56, "duration": 6.76, "text": "everything into our formula it says we"}, {"start": 72.479, "duration": 9.921, "text": "would get well 2 to the second power and"}, {"start": 77.32, "duration": 6.6, "text": "then we would have cosine of 2 * Theta"}, {"start": 82.4, "duration": 9.52, "text": "which is"}, {"start": 83.92, "duration": 8.0, "text": "60\u00b0 plus I * s of 2 * 60\u00b0"}, {"start": 92.52, "duration": 5.279, "text": "have enough parentheses here I think one"}, {"start": 95.04, "duration": 6.6, "text": "more um and now you can always clean it"}, {"start": 97.799, "duration": 6.841, "text": "up a little bit so 2^2 is 4 this will be"}, {"start": 101.64, "duration": 7.68, "text": "cosine of"}, {"start": 104.64, "duration": 8.24, "text": "120\u00b0 plus I of"}, {"start": 109.32, "duration": 5.96, "text": "120\u00b0 and you know you can always keep"}, {"start": 112.88, "duration": 5.839, "text": "going you can evaluate cosine of 120 s"}, {"start": 115.28, "duration": 4.879, "text": "of 120 um simplify this a little further"}, {"start": 118.719, "duration": 3.44, "text": "or we can always just leave it in the"}, {"start": 120.159, "duration": 5.841, "text": "polar form uh which is what I'm going to"}, {"start": 122.159, "duration": 5.801, "text": "do here so again very simple uh you know"}, {"start": 126.0, "duration": 4.92, "text": "kind of a one more formula to remember"}, {"start": 127.96, "duration": 6.48, "text": "but it's easy to use uh just raise your"}, {"start": 130.92, "duration": 8.399, "text": "your R to the power multiply your thetas"}, {"start": 134.44, "duration": 4.879, "text": "by that power and you get your solution"}], "title": "DeMoivre's Theorem: Raising a Complex Number to a Power, Ex 1", "description": "Thanks to all of you who support me on Patreon. You da real mvps!   $1 per month helps!! :)  https://www.patreon.com/patrickjmt !! DeMoivre's Theorem: Raising a Complex Number to a Power, Ex 1.  In this video I give DeMoivre's theorem and use it to raise a complex number to a power.  Note that our number must be in polar form!", "lengthSeconds": 137, "width": 480, "height": 360}
{"video_id": "eMAg1o6_0MI", "transcript": [{"start": 1.199, "duration": 2.801, "text": "okay in this video we're going to talk"}, {"start": 2.48, "duration": 4.24, "text": "about vector addition and scalar"}, {"start": 4.0, "duration": 4.32, "text": "multiplication and all it really says is"}, {"start": 6.72, "duration": 3.2, "text": "if you have two vectors if you want to"}, {"start": 8.32, "duration": 3.6, "text": "add or subtract them"}, {"start": 9.92, "duration": 4.48, "text": "you you either add or subtract their"}, {"start": 11.92, "duration": 4.88, "text": "corresponding components it says if you"}, {"start": 14.4, "duration": 5.12, "text": "multiply a vector by a real number it"}, {"start": 16.8, "duration": 5.12, "text": "says you just multiply each component by"}, {"start": 19.52, "duration": 4.16, "text": "that real number so nothing worse than"}, {"start": 21.92, "duration": 5.519, "text": "that"}, {"start": 23.68, "duration": 6.32, "text": "let's look at a couple examples here so"}, {"start": 27.439, "duration": 4.8, "text": "here we've got three vectors v has"}, {"start": 30.0, "duration": 3.76, "text": "components zero two u has components"}, {"start": 32.239, "duration": 3.921, "text": "negative one four"}, {"start": 33.76, "duration": 4.639, "text": "w has components five comma negative"}, {"start": 36.16, "duration": 4.32, "text": "three and we wanna find the components"}, {"start": 38.399, "duration": 4.961, "text": "of a couple different vectors so let's"}, {"start": 40.48, "duration": 5.68, "text": "do part a here first we'll do vector"}, {"start": 43.36, "duration": 5.44, "text": "w minus vector u"}, {"start": 46.16, "duration": 6.239, "text": "well the components for vector w we said"}, {"start": 48.8, "duration": 4.96, "text": "that was five comma negative three"}, {"start": 52.399, "duration": 3.84, "text": "and then we're going to subtract away"}, {"start": 53.76, "duration": 5.84, "text": "the components of vector u which are"}, {"start": 56.239, "duration": 6.881, "text": "negative one and four"}, {"start": 59.6, "duration": 5.519, "text": "all we do is we take five minus"}, {"start": 63.12, "duration": 4.72, "text": "well negative one so i'm just"}, {"start": 65.119, "duration": 4.801, "text": "subtracting uh the first components and"}, {"start": 67.84, "duration": 4.88, "text": "then we'll take negative three"}, {"start": 69.92, "duration": 4.8, "text": "uh minus four"}, {"start": 72.72, "duration": 4.24, "text": "and well after we simplify we'll get"}, {"start": 74.72, "duration": 4.399, "text": "five minus negative one that's five plus"}, {"start": 76.96, "duration": 3.68, "text": "one so we'll get our first component to"}, {"start": 79.119, "duration": 3.68, "text": "be six"}, {"start": 80.64, "duration": 3.76, "text": "negative three minus four will give us"}, {"start": 82.799, "duration": 4.0, "text": "negative seven"}, {"start": 84.4, "duration": 3.359, "text": "so those would be the components of the"}, {"start": 86.799, "duration": 4.561, "text": "vector"}, {"start": 87.759, "duration": 3.601, "text": "w minus u"}, {"start": 92.799, "duration": 3.921, "text": "okay so"}, {"start": 94.24, "duration": 5.28, "text": "part b kind of combining both ideas so"}, {"start": 96.72, "duration": 4.88, "text": "it says take 3 times vector w"}, {"start": 99.52, "duration": 3.68, "text": "which is five has components of five and"}, {"start": 101.6, "duration": 4.479, "text": "negative three"}, {"start": 103.2, "duration": 5.52, "text": "plus two times vector v which are has"}, {"start": 106.079, "duration": 5.68, "text": "components zero and two"}, {"start": 108.72, "duration": 6.8, "text": "minus four times vector u which has"}, {"start": 111.759, "duration": 7.201, "text": "components negative one and 4."}, {"start": 115.52, "duration": 6.239, "text": "well if we distribute the 3 we'll get 15"}, {"start": 118.96, "duration": 5.04, "text": "and negative 9 for our first"}, {"start": 121.759, "duration": 4.32, "text": "the components of our first vector here"}, {"start": 124.0, "duration": 3.68, "text": "after multiplying"}, {"start": 126.079, "duration": 3.281, "text": "i'll distribute you can fill you know"}, {"start": 127.68, "duration": 5.68, "text": "it's basically feels like distributing"}, {"start": 129.36, "duration": 7.2, "text": "so 2 times 0 will be just still 0. 2"}, {"start": 133.36, "duration": 4.56, "text": "times 2 will give us 4."}, {"start": 136.56, "duration": 4.399, "text": "if you want to you can imagine"}, {"start": 137.92, "duration": 4.24, "text": "distributing the negative 4 so"}, {"start": 140.959, "duration": 3.841, "text": "you could always think about this as"}, {"start": 142.16, "duration": 4.64, "text": "being plus negative four"}, {"start": 144.8, "duration": 3.84, "text": "so if we distribute that negative four"}, {"start": 146.8, "duration": 3.2, "text": "negative four times negative one will be"}, {"start": 148.64, "duration": 3.679, "text": "positive four"}, {"start": 150.0, "duration": 4.8, "text": "negative four times four will be"}, {"start": 152.319, "duration": 4.401, "text": "negative sixteen"}, {"start": 154.8, "duration": 3.92, "text": "and now it's addition um you could do"}, {"start": 156.72, "duration": 4.32, "text": "these pairwise you could do them two at"}, {"start": 158.72, "duration": 4.72, "text": "a time if you want to but the basic idea"}, {"start": 161.04, "duration": 4.479, "text": "is if you have a bunch of addition"}, {"start": 163.44, "duration": 4.799, "text": "you just add"}, {"start": 165.519, "duration": 6.241, "text": "all of the respective first components"}, {"start": 168.239, "duration": 5.441, "text": "so 15 plus 0 plus 4"}, {"start": 171.76, "duration": 5.44, "text": "and then our next component would just"}, {"start": 173.68, "duration": 7.44, "text": "be negative 9 plus 4"}, {"start": 177.2, "duration": 5.44, "text": "plus well a negative 16."}, {"start": 181.12, "duration": 4.08, "text": "and let's see i think we can clean this"}, {"start": 182.64, "duration": 5.76, "text": "up without too much trouble 15 plus 4 is"}, {"start": 185.2, "duration": 3.2, "text": "going to give us 19."}, {"start": 188.48, "duration": 5.92, "text": "negative 9 plus 4 will be negative 5."}, {"start": 191.92, "duration": 4.239, "text": "negative 5 plus negative 16 will be"}, {"start": 194.4, "duration": 3.68, "text": "negative 21."}, {"start": 196.159, "duration": 3.921, "text": "and now we have the components of this"}, {"start": 198.08, "duration": 7.56, "text": "new vector"}, {"start": 200.08, "duration": 5.56, "text": "3w plus 2v minus 4u"}], "title": "408 Vector addition and scalar multiplication Patrickjmt", "description": "", "lengthSeconds": 204, "width": 480, "height": 360}
{"video_id": "ZhvvkCa_60w", "transcript": [{"start": 0.599, "duration": 2.721, "text": "all right in this video I want to do"}, {"start": 1.839, "duration": 4.121, "text": "another example of using sum and"}, {"start": 3.32, "duration": 4.88, "text": "difference identities to find a value of"}, {"start": 5.96, "duration": 4.799, "text": "a trig function so suppose we want to"}, {"start": 8.2, "duration": 5.399, "text": "evaluate s of"}, {"start": 10.759, "duration": 5.241, "text": "195\u00b0 so sometimes kind of the tricky"}, {"start": 13.599, "duration": 4.76, "text": "part is just finding out you know um how"}, {"start": 16.0, "duration": 6.32, "text": "to write this as a sum of uh angles that"}, {"start": 18.359, "duration": 7.481, "text": "we already know well we know 45\u00b0 we know"}, {"start": 22.32, "duration": 6.92, "text": "how to evaluate s and cosine at 45\u00b0"}, {"start": 25.84, "duration": 5.279, "text": "90\u00b0 135\u00b0 I'm just basically taking"}, {"start": 29.24, "duration": 6.76, "text": "multiples of"}, {"start": 31.119, "duration": 8.401, "text": "45\u00b0 180\u00b0 Etc um let's see let's maybe"}, {"start": 36.0, "duration": 7.879, "text": "stick 30\u00b0 up there we can do 30 60 we"}, {"start": 39.52, "duration": 8.359, "text": "know 90 120 um"}, {"start": 43.879, "duration": 5.041, "text": "Etc so uh what I'm going to try to do um"}, {"start": 47.879, "duration": 5.921, "text": "you know you could do the same thing"}, {"start": 48.92, "duration": 6.76, "text": "with 60\u00b0 60\u00b0 120\u00b0 we know 180\u00b0 again"}, {"start": 53.8, "duration": 4.439, "text": "just multiples of"}, {"start": 55.68, "duration": 4.039, "text": "60\u00b0 I'm either looking for sort of two"}, {"start": 58.239, "duration": 4.48, "text": "numbers in here that either add up to"}, {"start": 59.719, "duration": 5.72, "text": "one 195 or whose difference is"}, {"start": 62.719, "duration": 4.041, "text": "195 well there's definitely uh more than"}, {"start": 65.439, "duration": 5.36, "text": "one way to do this but certainly it"}, {"start": 66.76, "duration": 6.6, "text": "looks like we could use uh 60\u00b0 and"}, {"start": 70.799, "duration": 5.36, "text": "135\u00b0 so s of"}, {"start": 73.36, "duration": 5.079, "text": "195\u00b0 is the same thing well that's going"}, {"start": 76.159, "duration": 4.881, "text": "to be 60\u00b0"}, {"start": 78.439, "duration": 4.881, "text": "plus"}, {"start": 81.04, "duration": 5.2, "text": "35\u00b0 and now we just have to use our"}, {"start": 83.32, "duration": 5.4, "text": "identity um and recall our sum identity"}, {"start": 86.24, "duration": 4.239, "text": "for sign here it says we just take sign"}, {"start": 88.72, "duration": 6.16, "text": "of the first one"}, {"start": 90.479, "duration": 7.161, "text": "60\u00b0 we take cosine of the second one"}, {"start": 94.88, "duration": 5.96, "text": "135\u00b0 since there's a A plus in between"}, {"start": 97.64, "duration": 9.479, "text": "we keep that and then it says we use"}, {"start": 100.84, "duration": 9.44, "text": "cosine of 60\u00b0 and then we'll use S of"}, {"start": 107.119, "duration": 5.721, "text": "135\u00b0 so I'm just using my uh very top"}, {"start": 110.28, "duration": 6.119, "text": "identity and now hopefully these are uh"}, {"start": 112.84, "duration": 7.0, "text": "values that we know so s of 60\u00b0 that's"}, {"start": 116.399, "duration": 4.961, "text": "the same as s of Pi 3 um that's going to"}, {"start": 119.84, "duration": 3.559, "text": "going to give us < tk3"}, {"start": 121.36, "duration": 4.24, "text": "over2 cosine of"}, {"start": 123.399, "duration": 4.761, "text": "135\u00b0 that's going to be in the second"}, {"start": 125.6, "duration": 4.92, "text": "quadrant um that's going to give us <"}, {"start": 128.16, "duration": 5.84, "text": "tk2 over"}, {"start": 130.52, "duration": 4.76, "text": "2 cosine of 60\u00b0 um that's going to give"}, {"start": 134.0, "duration": 3.76, "text": "us"}, {"start": 135.28, "duration": 5.52, "text": "12 s of"}, {"start": 137.76, "duration": 5.6, "text": "135\u00b0 that's going to be positive < tk2"}, {"start": 140.8, "duration": 5.64, "text": "over 2 and now we can just simplify this"}, {"start": 143.36, "duration": 7.44, "text": "down if we multiply we'll get negative 3"}, {"start": 146.44, "duration": 7.48, "text": "* two we can write that Asun 6 uh 1 * Ro"}, {"start": 150.8, "duration": 5.6, "text": "< tk2 is simply < tk2 uh we would have"}, {"start": 153.92, "duration": 5.039, "text": "our common denominator of 2 * 2 which is"}, {"start": 156.4, "duration": 6.16, "text": "four so to me that looks like a nice"}, {"start": 158.959, "duration": 6.601, "text": "simplified solution for S of"}, {"start": 162.56, "duration": 3.0, "text": "195\u00b0"}], "title": "Sum and Difference Identities for Sine and Cosine, More Examples #1", "description": "Thanks to all of you who support me on Patreon. You da real mvps!   $1 per month helps!! :)  https://www.patreon.com/patrickjmt !! Sum and Difference Identities for Sine and Cosine, More Examples #1.  More examples of using the sum and difference identities to find value other trig values. For more free math videos, visit http://PatrickJMT.com", "lengthSeconds": 164, "width": 480, "height": 360}
{"video_id": "gkDDEbzR9MA", "transcript": [{"start": 1.8, "duration": 4.24, "text": "so in this example we have our friend"}, {"start": 3.6, "duration": 5.039, "text": "Patrick who's building a rectangular"}, {"start": 6.04, "duration": 4.92, "text": "patio and that's going to be centered on"}, {"start": 8.639, "duration": 5.04, "text": "one side of his yard and the rest of the"}, {"start": 10.96, "duration": 4.919, "text": "yard is grass okay so I've kind of got"}, {"start": 13.679, "duration": 3.961, "text": "it a little uh you know kind of marked"}, {"start": 15.879, "duration": 4.761, "text": "here the outside part's going to be"}, {"start": 17.64, "duration": 4.76, "text": "grass and if the measurements in the"}, {"start": 20.64, "duration": 4.0, "text": "diagram are in feet we want to know"}, {"start": 22.4, "duration": 4.119, "text": "what's the square footage of the grass"}, {"start": 24.64, "duration": 4.28, "text": "portion of the yard so we want to know"}, {"start": 26.519, "duration": 5.08, "text": "you know how much basically how much uh"}, {"start": 28.92, "duration": 4.319, "text": "uh area is the grass covering now"}, {"start": 31.599, "duration": 3.521, "text": "there's a couple different ways you"}, {"start": 33.239, "duration": 3.921, "text": "could do this you could start trying to"}, {"start": 35.12, "duration": 3.72, "text": "you know say okay well this is nine and"}, {"start": 37.16, "duration": 3.04, "text": "you could start you know saying okay"}, {"start": 38.84, "duration": 2.92, "text": "this is centered this is 15 you could"}, {"start": 40.2, "duration": 3.44, "text": "start trying to chop this up into a"}, {"start": 41.76, "duration": 4.56, "text": "bunch of little rectangles but that"}, {"start": 43.64, "duration": 6.28, "text": "would definitely be Overkill um that"}, {"start": 46.32, "duration": 5.879, "text": "would be much more uh uh be a lot a lot"}, {"start": 49.92, "duration": 3.76, "text": "more trouble than what you need to do so"}, {"start": 52.199, "duration": 3.721, "text": "the idea that we're going to use in this"}, {"start": 53.68, "duration": 3.12, "text": "one is well we're going to figure out"}, {"start": 55.92, "duration": 4.799, "text": "the"}, {"start": 56.8, "duration": 6.559, "text": "area of the entire uh rectangle"}, {"start": 60.719, "duration": 2.64, "text": "and then we're going to"}, {"start": 64.199, "duration": 4.92, "text": "subtract we're going to subtract away"}, {"start": 66.28, "duration": 2.839, "text": "the area of the"}, {"start": 70.0, "duration": 4.479, "text": "patio and if we take the entire"}, {"start": 72.439, "duration": 4.081, "text": "rectangle and subtract away the area of"}, {"start": 74.479, "duration": 2.881, "text": "the patio that's just going to leave us"}, {"start": 76.52, "duration": 3.639, "text": "with"}, {"start": 77.36, "duration": 4.719, "text": "the with the area that the grass is"}, {"start": 80.159, "duration": 3.96, "text": "covering so that's the idea we're going"}, {"start": 82.079, "duration": 4.68, "text": "to figure out the area of the entire"}, {"start": 84.119, "duration": 5.04, "text": "rectangle we'll subtract away the area"}, {"start": 86.759, "duration": 4.921, "text": "of the that the patio covers and that's"}, {"start": 89.159, "duration": 5.64, "text": "going to give us the area of the"}, {"start": 91.68, "duration": 5.04, "text": "grass all right so really we're just"}, {"start": 94.799, "duration": 4.761, "text": "finding areas of rectangles here and"}, {"start": 96.72, "duration": 7.719, "text": "just doing a little bit of arithmetic"}, {"start": 99.56, "duration": 7.12, "text": "so the area of the of the large"}, {"start": 104.439, "duration": 5.481, "text": "rectangle well that's going to be 30"}, {"start": 106.68, "duration": 9.64, "text": "multiplied by 18 so let's do that here"}, {"start": 109.92, "duration": 10.6, "text": "so um 30 multiplied by 18 well 8 * 0 is"}, {"start": 116.32, "duration": 5.439, "text": "0 8 * 3 is 24"}, {"start": 120.52, "duration": 4.04, "text": "and then I'm going to drop down some"}, {"start": 121.759, "duration": 9.161, "text": "people put a zero I always uh we always"}, {"start": 124.56, "duration": 9.08, "text": "did little X's so 1 * 0 is 0 1 * 3 is 3"}, {"start": 130.92, "duration": 7.08, "text": "and now we just add those up so we've"}, {"start": 133.64, "duration": 7.599, "text": "got 0 4 + 0 is going to be 4 and then 2"}, {"start": 138.0, "duration": 7.48, "text": "+ 3 is going to be 5 so the entire"}, {"start": 141.239, "duration": 7.041, "text": "rectangle has an area of 540 square ft"}, {"start": 145.48, "duration": 5.16, "text": "let's figure out the area of the"}, {"start": 148.28, "duration": 7.2, "text": "patio well again that's is going to be"}, {"start": 150.64, "duration": 4.84, "text": "the length time the width so 9 *"}, {"start": 156.28, "duration": 5.599, "text": "15 well 9 multiplied by 5 that's 45 so"}, {"start": 160.239, "duration": 6.36, "text": "we'll drop down the five and carry the"}, {"start": 161.879, "duration": 7.44, "text": "four 9 * 1 is 9 if we take 9 + 4 that's"}, {"start": 166.599, "duration": 6.56, "text": "going to give us 13 so the area of the"}, {"start": 169.319, "duration": 5.761, "text": "patio is 135 squ ft so now we just need"}, {"start": 173.159, "duration": 5.561, "text": "to do the subtraction to get the area of"}, {"start": 175.08, "duration": 8.439, "text": "the grassy part so we'll take 540 we'll"}, {"start": 178.72, "duration": 7.56, "text": "subtract away 13 5 let's set this up"}, {"start": 183.519, "duration": 5.36, "text": "um"}, {"start": 186.28, "duration": 4.679, "text": "vertically all right 0 mult or excuse me"}, {"start": 188.879, "duration": 4.64, "text": "zero subtract five can't do that so"}, {"start": 190.959, "duration": 6.081, "text": "we'll take one from the four we'll carry"}, {"start": 193.519, "duration": 6.72, "text": "that so that'll give us 10 10 - 5 is"}, {"start": 197.04, "duration": 6.96, "text": "going to give us 5 3 - 3 that's just"}, {"start": 200.239, "duration": 6.441, "text": "going to leave us with 0 and 5 - 1"}, {"start": 204.0, "duration": 7.76, "text": "that's going to leave us with four so it"}, {"start": 206.68, "duration": 5.08, "text": "says that the area of the grassy part"}, {"start": 213.239, "duration": 2.601, "text": "that's going to be"}, {"start": 215.959, "duration": 4.48, "text": "405 Square ft"}], "title": "GED 'Most Missed' Math Practice Problems , Code Q.4.c,  #3", "description": "Thanks to all of you who support me on Patreon. You da real mvps!   $1 per month helps!! :)  https://www.patreon.com/patrickjmt !! The GED Testing Service keeps a list of the most missed math topics and can be seen here: http://www.gedtestingservice.com/uploads/files/126de0283c94ff6323b3b5b3f310da5d.pdf\nThis video shows how to work a sample problem using their coding system that is shown on the left using the link above.", "lengthSeconds": 223, "width": 480, "height": 360}
{"video_id": "JKWpI2EuIDA", "transcript": [{"start": 0.84, "duration": 4.16, "text": "all right um in this video I'm going to"}, {"start": 2.76, "duration": 4.4, "text": "just do some uh math calculation tricks"}, {"start": 5.0, "duration": 4.24, "text": "nothing maybe that mind-blowing I picked"}, {"start": 7.16, "duration": 3.2, "text": "up a book the other day and I don't know"}, {"start": 9.24, "duration": 2.96, "text": "I thought some of the tricks were kind"}, {"start": 10.36, "duration": 4.279, "text": "of interesting so this is just going to"}, {"start": 12.2, "duration": 6.079, "text": "deal with squaring numbers close to 100"}, {"start": 14.639, "duration": 6.601, "text": "so I'm going to do 106 squar 94 squ 112"}, {"start": 18.279, "duration": 6.361, "text": "squar and 88 squar"}, {"start": 21.24, "duration": 6.24, "text": "um and the idea to do these uh maybe"}, {"start": 24.64, "duration": 5.479, "text": "let's look at 106 squar"}, {"start": 27.48, "duration": 5.0, "text": "First notice all of these numbers again"}, {"start": 30.119, "duration": 5.321, "text": "are close to 100 and that's basically"}, {"start": 32.48, "duration": 5.32, "text": "the trick um on all of them that's the"}, {"start": 35.44, "duration": 3.4, "text": "that's the important part if you think"}, {"start": 37.8, "duration": 4.759, "text": "about"}, {"start": 38.84, "duration": 7.32, "text": "106 notice that 106 is actually six"}, {"start": 42.559, "duration": 7.961, "text": "bigger than 100 what we do is we take"}, {"start": 46.16, "duration": 7.12, "text": "the number six and we add it to 106 that"}, {"start": 50.52, "duration": 5.16, "text": "gives us 112 so that's how we get our"}, {"start": 53.28, "duration": 5.52, "text": "first three digits we think well it's"}, {"start": 55.68, "duration": 5.32, "text": "six away so just add them and then to"}, {"start": 58.8, "duration": 5.04, "text": "get the last two digits all you do is"}, {"start": 61.0, "duration": 7.4, "text": "take this number you square it and you"}, {"start": 63.84, "duration": 7.08, "text": "just tack it on okay so 106 squared"}, {"start": 68.4, "duration": 2.52, "text": "would be"}, {"start": 71.08, "duration": 5.96, "text": "11,23 same idea for 94 squared the only"}, {"start": 74.36, "duration": 6.84, "text": "thing that's different now is instead of"}, {"start": 77.04, "duration": 6.96, "text": "being um six units above 100 it's six"}, {"start": 81.2, "duration": 6.64, "text": "units below so we do the same thing we"}, {"start": 84.0, "duration": 6.479, "text": "add these numbers so -6 and 94 that's"}, {"start": 87.84, "duration": 4.599, "text": "going to give us 88"}, {"start": 90.479, "duration": 5.521, "text": "but then we do the same thing we Square"}, {"start": 92.439, "duration": 8.241, "text": "the number so -6 squared would be the"}, {"start": 96.0, "duration": 4.68, "text": "number 36 and we get our value"}, {"start": 100.72, "duration": 3.8, "text": "8836 um maybe two more that are just a"}, {"start": 103.56, "duration": 3.04, "text": "little bit"}, {"start": 104.52, "duration": 4.879, "text": "trickier um we're going to do the same"}, {"start": 106.6, "duration": 7.96, "text": "thing here if we look at the number"}, {"start": 109.399, "duration": 11.801, "text": "112 well 112 is 12 bigger than 100 so if"}, {"start": 114.56, "duration": 8.64, "text": "we add 112 + 12 that'll give us um 124"}, {"start": 121.2, "duration": 5.199, "text": "but notice now if we Square the number"}, {"start": 123.2, "duration": 5.559, "text": "12 12 squar is"}, {"start": 126.399, "duration": 4.84, "text": "144 we're going to get too many digits"}, {"start": 128.759, "duration": 5.601, "text": "if we tack on the"}, {"start": 131.239, "duration": 7.201, "text": "144 what you do is you carry the one to"}, {"start": 134.36, "duration": 6.48, "text": "the 100 spot drop your 44 um at the end"}, {"start": 138.44, "duration": 7.32, "text": "and then you just add them together so 1"}, {"start": 140.84, "duration": 7.399, "text": "two 5 44 that's going to be 112 squared"}, {"start": 145.76, "duration": 4.72, "text": "and you do the same thing for 88 that's"}, {"start": 148.239, "duration": 5.08, "text": "going to be 12 units"}, {"start": 150.48, "duration": 6.96, "text": "um smaller than the number 100 so if we"}, {"start": 153.319, "duration": 6.84, "text": "take -12 and 88 that's going to give us"}, {"start": 157.44, "duration": 6.28, "text": "76 um and again if we do the same thing"}, {"start": 160.159, "duration": 6.561, "text": "if you square -12 you'll get 1 44 we've"}, {"start": 163.72, "duration": 6.56, "text": "got to carry the one we'll get"}, {"start": 166.72, "duration": 6.48, "text": "77 uh 44 so"}, {"start": 170.28, "duration": 5.959, "text": "7,744 is our final answer so I don't"}, {"start": 173.2, "duration": 4.56, "text": "know I thought it was kind of neat um"}, {"start": 176.239, "duration": 3.28, "text": "don't get rid of your calculator just"}, {"start": 177.76, "duration": 3.479, "text": "yet but I'll probably post some other"}, {"start": 179.519, "duration": 3.761, "text": "videos on some neat little math tricks"}, {"start": 181.239, "duration": 4.201, "text": "if uh if people find these interesting"}, {"start": 183.28, "duration": 4.599, "text": "so all right um I hope this was mildly"}, {"start": 185.44, "duration": 5.799, "text": "entertaining and um feel free to post"}, {"start": 187.879, "duration": 3.36, "text": "comments or questions"}], "title": "Fun Math Trick! Squaring Numbers Near 100", "description": "Thanks to all of you who support me on Patreon. You da real mvps!   $1 per month helps!! :)  https://www.patreon.com/patrickjmt !! Fun Math Trick! Squaring Numbers Near 100.\r\nA neat way to square numbers near 100 without having to multiply it out the long way!\r\nFor more free math videos, visit http://PatrickJMT.com", "lengthSeconds": 189, "width": 480, "height": 360}
{"video_id": "B3WFIzNf-Mo", "transcript": [{"start": 0.919, "duration": 5.4, "text": "okay here we're going to find the"}, {"start": 2.04, "duration": 8.88, "text": "vertices of the hyperbola y - 20 2 / 4 -"}, {"start": 6.319, "duration": 7.641, "text": "x + 2 2 / 36 = 1 so the first thing I'm"}, {"start": 10.92, "duration": 6.4, "text": "going to do is pick out the center so I"}, {"start": 13.96, "duration": 6.52, "text": "see x + 2^2 but instead of positive2 we"}, {"start": 17.32, "duration": 7.32, "text": "use the opposite sign or -2 and then we"}, {"start": 20.48, "duration": 5.559, "text": "have y - 20 2ar instead of -20 we'll use"}, {"start": 24.64, "duration": 4.479, "text": "positive"}, {"start": 26.039, "duration": 5.08, "text": "20 so I know that my Center again you"}, {"start": 29.119, "duration": 4.041, "text": "don't have to sketch this you could just"}, {"start": 31.119, "duration": 3.561, "text": "use some formulas but I always like to"}, {"start": 33.16, "duration": 4.919, "text": "think about it in terms of a graph it"}, {"start": 34.68, "duration": 7.039, "text": "helps me remember so there I've"}, {"start": 38.079, "duration": 5.8, "text": "got -2 comma 20 I'll put a little X"}, {"start": 41.719, "duration": 6.801, "text": "there that's where it's centered at"}, {"start": 43.879, "duration": 8.121, "text": "since it's of the form y^2 - x^2 = 1 I"}, {"start": 48.52, "duration": 3.48, "text": "know it's going to open"}, {"start": 53.0, "duration": 5.079, "text": "upwards I know my hyperbola is going to"}, {"start": 55.359, "duration": 5.401, "text": "open upwards and"}, {"start": 58.079, "duration": 6.281, "text": "downwards so I'm looking for the"}, {"start": 60.76, "duration": 7.2, "text": "vertices well to get the"}, {"start": 64.36, "duration": 6.68, "text": "vertices what we do is for our hyperbola"}, {"start": 67.96, "duration": 5.4, "text": "we look at the the the first term when"}, {"start": 71.04, "duration": 5.84, "text": "we have it in standard form Okay so"}, {"start": 73.36, "duration": 5.84, "text": "we've got y - 202 / 4 if we take the"}, {"start": 76.88, "duration": 3.76, "text": "square of four we'll get two and that's"}, {"start": 79.2, "duration": 4.72, "text": "going to be our a"}, {"start": 80.64, "duration": 7.08, "text": "value and that a value tells us from the"}, {"start": 83.92, "duration": 6.6, "text": "center how far how many units you move"}, {"start": 87.72, "duration": 4.8, "text": "to get to the vertices"}, {"start": 90.52, "duration": 3.919, "text": "so in this case since it opens upwards"}, {"start": 92.52, "duration": 3.959, "text": "and downwards I know from the center"}, {"start": 94.439, "duration": 6.401, "text": "that I'm going to move up and down to"}, {"start": 96.479, "duration": 6.081, "text": "get my vertices and again I just take"}, {"start": 100.84, "duration": 4.239, "text": "the square root of that first term the"}, {"start": 102.56, "duration": 5.519, "text": "square root of the pos4 which is two and"}, {"start": 105.079, "duration": 3.0, "text": "that's going to give me my"}, {"start": 108.719, "duration": 5.4, "text": "vertices okay so since the x coordinate"}, {"start": 112.2, "duration": 5.559, "text": "doesn't"}, {"start": 114.119, "duration": 7.521, "text": "change that'll just remain -2 it was"}, {"start": 117.759, "duration": 6.561, "text": "originally centered at -2a 20 so if we"}, {"start": 121.64, "duration": 6.0, "text": "move up two units we'll be at -2 comma"}, {"start": 124.32, "duration": 9.68, "text": "22 and if we move down we'll be at -2"}, {"start": 127.64, "duration": 6.36, "text": "comma 18 and those will be our vertices"}], "title": "Vertices of a Hyperbola - Easy, Basic Example", "description": "Thanks to all of you who support me on Patreon. You da real mvps!   $1 per month helps!! :)  https://www.patreon.com/patrickjmt !! Vertices of a Hyperbola - Easy, Basic Example\nIn this video I show how to find the vertices of a hyperbola by working an example.", "lengthSeconds": 133, "width": 450, "height": 360}
{"video_id": "RJnGEi2KDLI", "transcript": [{"start": 1.28, "duration": 4.88, "text": "okay here we're going to identify the"}, {"start": 3.24, "duration": 6.84, "text": "quadrant or axis that contains the point"}, {"start": 6.16, "duration": 6.32, "text": "8 comma 6 so the first coordinate of"}, {"start": 10.08, "duration": 6.16, "text": "eight that tells us how far we travel"}, {"start": 12.48, "duration": 9.52, "text": "along the x axis so we'll travel let's"}, {"start": 16.24, "duration": 7.799, "text": "see so uh 1 2 3 4 five six 7even 8 so"}, {"start": 22.0, "duration": 7.56, "text": "we've moved over eight units on the"}, {"start": 24.039, "duration": 9.52, "text": "xaxis and 1 2 3 4 five six uh units on"}, {"start": 29.56, "duration": 6.92, "text": "the y AIS and we'll put our DOT right"}, {"start": 33.559, "duration": 6.201, "text": "there our point right there so our point"}, {"start": 36.48, "duration": 5.599, "text": "in this case is going to be in quadrant"}, {"start": 39.76, "duration": 3.959, "text": "one so just to remind you remember this"}, {"start": 42.079, "duration": 8.841, "text": "is quadrant"}, {"start": 43.719, "duration": 11.36, "text": "one quadrant 2 quadrant 3 and Quadrant 4"}, {"start": 50.92, "duration": 7.72, "text": "so so our Point here will belong to"}, {"start": 55.079, "duration": 3.561, "text": "quadrant one"}], "title": "How to Plot a Point in the X-Y Plane", "description": "Thanks to all of you who support me on Patreon. You da real mvps!   $1 per month helps!! :)  https://www.patreon.com/patrickjmt !! How to Plot a Point in the X-Y Plane.\nJust a quick discussion for those beginning algebra on plotting the point (8,6).", "lengthSeconds": 61, "width": 450, "height": 360}
{"video_id": "pGt5kAU96gE", "transcript": [{"start": 0.05, "duration": 4.84, "text": "all right here we've got a question"}, {"start": 2.399, "duration": 5.46, "text": "related to vector functions and velocity"}, {"start": 4.89, "duration": 4.89, "text": "acceleration and position so here we've"}, {"start": 7.859, "duration": 4.291, "text": "got a car and it travels with a velocity"}, {"start": 9.78, "duration": 5.069, "text": "vector given by V of T equals T squared"}, {"start": 12.15, "duration": 5.34, "text": "times e to the T plus 1 T's measured"}, {"start": 14.849, "duration": 4.921, "text": "measured in seconds and the vector"}, {"start": 17.49, "duration": 4.35, "text": "components are measured in feet if we"}, {"start": 19.77, "duration": 5.55, "text": "know the initial position of the car is"}, {"start": 21.84, "duration": 5.97, "text": "R of 0 equals 1 comma 3 we want to find"}, {"start": 25.32, "duration": 7.32, "text": "the position of the car say one second"}, {"start": 27.81, "duration": 7.71, "text": "later well to do this if we're given"}, {"start": 32.64, "duration": 5.55, "text": "velocity to get back to our position"}, {"start": 35.52, "duration": 6.15, "text": "vector R of T all we're going to have to"}, {"start": 38.19, "duration": 7.529, "text": "do is we're going to integrate our"}, {"start": 41.67, "duration": 6.6, "text": "velocity vector with respect to T okay"}, {"start": 45.719, "duration": 4.5, "text": "so we can write this as the integral"}, {"start": 48.27, "duration": 5.609, "text": "we're just going to integrate each part"}, {"start": 50.219, "duration": 6.11, "text": "so the integral of T squared DT and then"}, {"start": 53.879, "duration": 7.171, "text": "we'll have to integrate e to the T plus"}, {"start": 56.329, "duration": 6.851, "text": "one DT well let's see if we integrate"}, {"start": 61.05, "duration": 4.859, "text": "the first part the antiderivative of T"}, {"start": 63.18, "duration": 5.04, "text": "squared will be T cubed over 3 we have"}, {"start": 65.909, "duration": 4.621, "text": "to add our arbitrary constant here see"}, {"start": 68.22, "duration": 5.04, "text": "and then when we integrate the second"}, {"start": 70.53, "duration": 5.22, "text": "part well we'll just get e to the T plus"}, {"start": 73.26, "duration": 4.649, "text": "T again we have to introduce a new"}, {"start": 75.75, "duration": 6.75, "text": "constant let's call it D since we've"}, {"start": 77.909, "duration": 7.5, "text": "already used C all right so but we are"}, {"start": 82.5, "duration": 10.439, "text": "given an initial condition we're given"}, {"start": 85.409, "duration": 10.35, "text": "that R of 0 equals 1 the vector with"}, {"start": 92.939, "duration": 6.691, "text": "components 1 and 3 so now what we can do"}, {"start": 95.759, "duration": 9.241, "text": "is simply plug that in ok so so it says"}, {"start": 99.63, "duration": 9.419, "text": "if T equals 0 well the first component"}, {"start": 105.0, "duration": 5.88, "text": "we would get 0 cubed over 3 plus C that"}, {"start": 109.049, "duration": 4.68, "text": "would have to equal the first component"}, {"start": 110.88, "duration": 7.919, "text": "of 1 and likewise if we plug in T equals"}, {"start": 113.729, "duration": 8.07, "text": "0 well we would get e to the 0 plus 0"}, {"start": 118.799, "duration": 5.93, "text": "plus D that would have to equal the"}, {"start": 121.799, "duration": 6.21, "text": "second component which is going to be 3"}, {"start": 124.729, "duration": 8.081, "text": "well 0 cubed over 3 is just 0 so we'll"}, {"start": 128.009, "duration": 8.31, "text": "get C equals 1 e to the 0 well that's 1"}, {"start": 132.81, "duration": 7.5, "text": "plus D equals three if we subtract we'll"}, {"start": 136.319, "duration": 7.861, "text": "get D equals two all right so really we"}, {"start": 140.31, "duration": 8.7, "text": "know that we can describe our position"}, {"start": 144.18, "duration": 7.94, "text": "vector R of T as T cubed over 3 plus C"}, {"start": 149.01, "duration": 5.88, "text": "which again is just going to be 1 and"}, {"start": 152.12, "duration": 7.449, "text": "then the second component will be e to"}, {"start": 154.89, "duration": 9.45, "text": "the T plus T plus D again which we said"}, {"start": 159.569, "duration": 6.781, "text": "had a value of 2 all right so we wanted"}, {"start": 164.34, "duration": 5.19, "text": "to figure out we wanted to find the"}, {"start": 166.35, "duration": 5.58, "text": "position of the car after 1 second so"}, {"start": 169.53, "duration": 6.42, "text": "really what we want to find is we just"}, {"start": 171.93, "duration": 6.24, "text": "want to find R of 1 well all we have to"}, {"start": 175.95, "duration": 6.66, "text": "do is plug that in we'll get 1 cubed"}, {"start": 178.17, "duration": 11.069, "text": "over 3 plus 1 comma e to the first plus"}, {"start": 182.61, "duration": 10.14, "text": "1 plus 2 well 1 third plus 1 is going to"}, {"start": 189.239, "duration": 6.36, "text": "be 4/3 and then we would have E Plus 1"}, {"start": 192.75, "duration": 5.34, "text": "plus 2 which would give us a positive 3"}, {"start": 195.599, "duration": 7.911, "text": "and now we've got the position of the"}, {"start": 198.09, "duration": 5.42, "text": "car one second later"}], "title": "Vector Functions: Position, Velocity, Acceleration Word Problem, Ex 1", "description": "Thanks to all of you who support me on Patreon. You da real mvps!   $1 per month helps!! :)  https://www.patreon.com/patrickjmt !! Vector Functions: Position, Velocity, Acceleration Word Problem, Ex .  In this video, we are given the velocity vector of a car as well as its initial displacement.  We have to find the position of the car after 1 second.", "lengthSeconds": 203, "width": 480, "height": 360}
{"video_id": "nyCTm9Ff0E4", "transcript": [{"start": 0.88, "duration": 3.28, "text": "all right in this video I'm going to do"}, {"start": 2.2, "duration": 3.8, "text": "one more example of multiplying complex"}, {"start": 4.16, "duration": 3.52, "text": "numbers and I'm going to finish off the"}, {"start": 6.0, "duration": 4.679, "text": "example that I didn't finish a second"}, {"start": 7.68, "duration": 9.28, "text": "ago which was to multiply I * the"}, {"start": 10.679, "duration": 6.281, "text": "quantity 6 + 4 I by the quantity 2 - 7"}, {"start": 20.4, "duration": 6.0, "text": "I all right so again since you're"}, {"start": 24.0, "duration": 5.48, "text": "multiplying we can either multiply the I"}, {"start": 26.4, "duration": 5.119, "text": "by the first uh the first factor or we"}, {"start": 29.48, "duration": 4.0, "text": "can multip the I by the second Factor"}, {"start": 31.519, "duration": 4.681, "text": "you definitely don't distribute because"}, {"start": 33.48, "duration": 4.64, "text": "there's not a plus or a minus in there"}, {"start": 36.2, "duration": 3.16, "text": "and I think what I'm going to do first"}, {"start": 38.12, "duration": 4.84, "text": "is I'm actually going to do the"}, {"start": 39.36, "duration": 7.12, "text": "multiplication between the 6 + 4 I and"}, {"start": 42.96, "duration": 5.16, "text": "the 2 - 7 I again a couple different"}, {"start": 46.48, "duration": 3.8, "text": "ways that you could simplify this stuff"}, {"start": 48.12, "duration": 4.68, "text": "down for sure but this is the way that"}, {"start": 50.28, "duration": 4.68, "text": "I'm going to do it so again we we still"}, {"start": 52.8, "duration": 6.12, "text": "have to distribute as in the previous"}, {"start": 54.96, "duration": 8.72, "text": "example um so 6 * 2 is going to give us"}, {"start": 58.92, "duration": 9.519, "text": "12 we'll say we'll get 6 * -7 I that'll"}, {"start": 63.68, "duration": 7.72, "text": "leave us with a - 42 I on the inside of"}, {"start": 68.439, "duration": 8.081, "text": "when we take 4 I * pos2 that'll give us"}, {"start": 71.4, "duration": 7.2, "text": "positive 8 I and then I think pos4 * -7"}, {"start": 76.52, "duration": 7.84, "text": "is going to give us"}, {"start": 78.6, "duration": 7.839, "text": "-28 then we have I * I which is I"}, {"start": 84.36, "duration": 5.759, "text": "2ar okay so I'm going to continue"}, {"start": 86.439, "duration": 5.801, "text": "simplifying inside the brackets"}, {"start": 90.119, "duration": 7.68, "text": "so the I is just hanging out front we"}, {"start": 92.24, "duration": 8.519, "text": "have 12 let's see we've got -42 I + 8 I"}, {"start": 97.799, "duration": 7.0, "text": "I believe that's going to give us a -34"}, {"start": 100.759, "duration": 8.281, "text": "I we still have our minus 28 but we can"}, {"start": 104.799, "duration": 4.241, "text": "replace our I squared again with"}, {"start": 110.6, "duration": 7.879, "text": "-1 so really on the inside we have 12 we"}, {"start": 113.719, "duration": 9.32, "text": "have a -28 * a -1 so that's really 12 +"}, {"start": 118.479, "duration": 9.24, "text": "28 12 + 28 is going to give us POS"}, {"start": 123.039, "duration": 6.681, "text": "40 - 34 I so I think I've done about all"}, {"start": 127.719, "duration": 3.801, "text": "the simplification I can inside the"}, {"start": 129.72, "duration": 3.96, "text": "brackets so now I'm going to distribute"}, {"start": 131.52, "duration": 6.04, "text": "the I to both"}, {"start": 133.68, "duration": 8.88, "text": "terms so uh what we have I * 40 which is"}, {"start": 137.56, "duration": 8.6, "text": "simply 40 I and then our -34 we have an"}, {"start": 142.56, "duration": 6.039, "text": "i * an i which is I"}, {"start": 146.16, "duration": 6.439, "text": "squared so again we can use this trick"}, {"start": 148.599, "duration": 6.881, "text": "one more time that i^ 2 is equivalent to"}, {"start": 152.599, "duration": 7.881, "text": "1 so really we have 40"}, {"start": 155.48, "duration": 7.72, "text": "I + 34 and again typically you'll write"}, {"start": 160.48, "duration": 3.96, "text": "uh the convention is to write the real"}, {"start": 163.2, "duration": 4.2, "text": "part"}, {"start": 164.44, "duration": 6.76, "text": "first and the imaginary part which is"}, {"start": 167.4, "duration": 7.08, "text": "positive 40i we'll write that second and"}, {"start": 171.2, "duration": 5.0, "text": "now we have multiplied out this original"}, {"start": 174.48, "duration": 4.08, "text": "expression you could have certainly"}, {"start": 176.2, "duration": 5.959, "text": "multiplied the I you know for example to"}, {"start": 178.56, "duration": 6.52, "text": "the six and the 4 I I and get 6 I + 4 i^"}, {"start": 182.159, "duration": 5.961, "text": "2 simplify that down and then distribute"}, {"start": 185.08, "duration": 4.799, "text": "all that to the 2 - 7 I collect"}, {"start": 188.12, "duration": 3.8, "text": "everything and again at the end of the"}, {"start": 189.879, "duration": 5.241, "text": "day you'll end up with this expression"}, {"start": 191.92, "duration": 6.16, "text": "this complex number 34 + 40i is your"}, {"start": 195.12, "duration": 2.96, "text": "answer"}], "title": "Complex Numbers:  Multiplying - Ex 2", "description": "Thanks to all of you who support me on Patreon. You da real mvps!   $1 per month helps!! :)  https://www.patreon.com/patrickjmt !! Complex Numbers:  Multiplying - Ex 2.  In this video, I do an example of multiplying three complex numbers and simplifying!", "lengthSeconds": 196, "width": 480, "height": 360}
{"video_id": "rCGsxYnSoF0", "transcript": [{"start": 0.03, "duration": 3.929, "text": "alright so another example here for"}, {"start": 1.979, "duration": 4.711, "text": "finding the equation of a parabola in"}, {"start": 3.959, "duration": 5.671, "text": "this case with a focus of five six and a"}, {"start": 6.69, "duration": 7.529, "text": "vertex in one six so I'm going to make a"}, {"start": 9.63, "duration": 8.28, "text": "little sketch here so there's one"}, {"start": 14.219, "duration": 7.411, "text": "positive six there's the vertex it's got"}, {"start": 17.91, "duration": 7.05, "text": "to focus over at two three four five two"}, {"start": 21.63, "duration": 4.77, "text": "three four five comma six so again"}, {"start": 24.96, "duration": 2.94, "text": "that's not a point on the graph but I'm"}, {"start": 26.4, "duration": 4.02, "text": "just going to put a little X or a little"}, {"start": 27.9, "duration": 4.379, "text": "mark there since the focus sits inside"}, {"start": 30.42, "duration": 6.96, "text": "the parabola I know that the parabola"}, {"start": 32.279, "duration": 7.221, "text": "would have to be opening to the right so"}, {"start": 37.38, "duration": 5.999, "text": "to me this tells me that this is uh"}, {"start": 39.5, "duration": 7.57, "text": "the form the equation of this parabola"}, {"start": 43.379, "duration": 6.781, "text": "is going to be I'm going to take X minus"}, {"start": 47.07, "duration": 6.93, "text": "the x-coordinate of the vertex which in"}, {"start": 50.16, "duration": 6.03, "text": "this case will be one equals y minus the"}, {"start": 54.0, "duration": 5.43, "text": "y coordinate of the vertex which is"}, {"start": 56.19, "duration": 5.81, "text": "going to be six squared but in front of"}, {"start": 59.43, "duration": 5.759, "text": "this I'm going to put my value for P and"}, {"start": 62.0, "duration": 7.57, "text": "again all P represents as P is the"}, {"start": 65.189, "duration": 6.241, "text": "distance from my vertex to my focus well"}, {"start": 69.57, "duration": 4.409, "text": "since we went from the x coordinate of 1"}, {"start": 71.43, "duration": 6.81, "text": "to the x coordinate of five in this case"}, {"start": 73.979, "duration": 6.75, "text": "my value for P would be P equals four we"}, {"start": 78.24, "duration": 4.5, "text": "would have to move over four units so it"}, {"start": 80.729, "duration": 5.011, "text": "says well the equation of this parabola"}, {"start": 82.74, "duration": 7.58, "text": "then would just be X minus one equals"}, {"start": 85.74, "duration": 8.94, "text": "four times four times y minus 6 squared"}, {"start": 90.32, "duration": 7.6, "text": "so again X minus one equals 16 times y"}, {"start": 94.68, "duration": 8.78, "text": "minus 6 squared or we could write this"}, {"start": 97.92, "duration": 9.059, "text": "as x equals 16 times y minus 6 squared"}, {"start": 103.46, "duration": 6.449, "text": "plus one and now we've got the equation"}, {"start": 106.979, "duration": 2.93, "text": "of our parabola"}], "title": "Conic Sections: Parabolas, Part 5  (Focus and Directrix)", "description": "Thanks to all of you who support me on Patreon. You da real mvps!   $1 per month helps!! :)  https://www.patreon.com/patrickjmt !! Conic Sections: Parabolas, Part 5  (Focus and Directrix).  In this video, I find the equation for a parabola given the vertex and given the  focus and/or directrix.", "lengthSeconds": 109, "width": 480, "height": 360}
{"video_id": "J9qsjYBWWHc", "transcript": [{"start": 0.76, "duration": 4.039, "text": "all right so one more example here of uh"}, {"start": 2.72, "duration": 4.719, "text": "doing a one-sided limit and this one I'm"}, {"start": 4.799, "duration": 5.88, "text": "not going to quite graph all the way so"}, {"start": 7.439, "duration": 5.401, "text": "we've got H ofx = the < TK of x^2 - 25"}, {"start": 10.679, "duration": 4.84, "text": "and we're doing uh X is approaching five"}, {"start": 12.84, "duration": 6.08, "text": "from the right so again we're looking at"}, {"start": 15.519, "duration": 8.0, "text": "the limit as X approaches 5 from the"}, {"start": 18.92, "duration": 8.16, "text": "right the square < TK of x^2 minus"}, {"start": 23.519, "duration": 5.401, "text": "25 well again intuitively um I'm going"}, {"start": 27.08, "duration": 4.6, "text": "to take numbers a little bit bigger than"}, {"start": 28.92, "duration": 4.44, "text": "five so maybe you know"}, {"start": 31.68, "duration": 3.32, "text": "5.1"}, {"start": 33.36, "duration": 4.68, "text": "5.01"}, {"start": 35.0, "duration": 5.32, "text": "5.01 so I'm getting closer and closer"}, {"start": 38.04, "duration": 4.72, "text": "and closer to the number uh to the"}, {"start": 40.32, "duration": 5.84, "text": "number five so I'm bringing it in closer"}, {"start": 42.76, "duration": 5.439, "text": "and closer and closer okay so again"}, {"start": 46.16, "duration": 4.28, "text": "we're we're approaching from the"}, {"start": 48.199, "duration": 5.401, "text": "right well if I take numbers a little"}, {"start": 50.44, "duration": 5.48, "text": "bit bigger than five and square them um"}, {"start": 53.6, "duration": 3.959, "text": "as they get closer and closer to five"}, {"start": 55.92, "duration": 4.88, "text": "the number that I'm squaring is going to"}, {"start": 57.559, "duration": 5.52, "text": "get closer and closer to 25"}, {"start": 60.8, "duration": 5.2, "text": "and if you subtract 25 well it's going"}, {"start": 63.079, "duration": 5.4, "text": "to get closer and closer to zero so"}, {"start": 66.0, "duration": 4.84, "text": "basically the moral of the story here is"}, {"start": 68.479, "duration": 4.361, "text": "um this function would be defined for"}, {"start": 70.84, "duration": 4.08, "text": "numbers a little bit bigger than five"}, {"start": 72.84, "duration": 5.279, "text": "and we can just plug it in uh we would"}, {"start": 74.92, "duration": 6.239, "text": "get 5 squar -"}, {"start": 78.119, "duration": 7.0, "text": "25 which would be the square < TK of 25"}, {"start": 81.159, "duration": 7.081, "text": "- 25 which would be the square < TK of0"}, {"start": 85.119, "duration": 5.68, "text": "which is just zero and that's all we've"}, {"start": 88.24, "duration": 4.6, "text": "got to do is just plug it in notice uh"}, {"start": 90.799, "duration": 4.041, "text": "you'd have to be a little"}, {"start": 92.84, "duration": 7.0, "text": "careful if somebody said what's the"}, {"start": 94.84, "duration": 5.0, "text": "limit as X approaches 5 from the left"}, {"start": 100.399, "duration": 4.561, "text": "um here I would say this does not exist"}, {"start": 103.0, "duration": 3.799, "text": "it doesn't really make sense to ask this"}, {"start": 104.96, "duration": 4.72, "text": "question um because notice if you take"}, {"start": 106.799, "duration": 5.401, "text": "numbers a little bit smaller than five"}, {"start": 109.68, "duration": 5.84, "text": "well if you square them um it's going to"}, {"start": 112.2, "duration": 5.72, "text": "be a little bit smaller than 25 so that"}, {"start": 115.52, "duration": 4.04, "text": "when I subtract 25 away I would"}, {"start": 117.92, "duration": 3.72, "text": "basically end up getting a negative"}, {"start": 119.56, "duration": 3.8, "text": "number"}, {"start": 121.64, "duration": 3.64, "text": "underneath my square root and we don't"}, {"start": 123.36, "duration": 3.48, "text": "take square roots of negative numbers uh"}, {"start": 125.28, "duration": 3.479, "text": "we're going to restrict ourselves to"}, {"start": 126.84, "duration": 4.16, "text": "real numbers and this would give us an"}, {"start": 128.759, "duration": 5.681, "text": "imaginary number so it wouldn't really"}, {"start": 131.0, "duration": 5.08, "text": "make sense to plug in five from the left"}, {"start": 134.44, "duration": 3.12, "text": "again algebraically if you plug it in"}, {"start": 136.08, "duration": 3.799, "text": "youd just get zero and you'd say hey"}, {"start": 137.56, "duration": 4.84, "text": "it's zero so um these are things you"}, {"start": 139.879, "duration": 4.0, "text": "have to again kind of be careful about"}, {"start": 142.4, "duration": 4.04, "text": "um you have to really think when can I"}, {"start": 143.879, "duration": 3.961, "text": "just plug it in and when can I not so"}, {"start": 146.44, "duration": 3.48, "text": "again here if I take something bigger"}, {"start": 147.84, "duration": 4.96, "text": "than five it's going to be bigger than"}, {"start": 149.92, "duration": 4.92, "text": "25 after I Square it well if I subtract"}, {"start": 152.8, "duration": 3.68, "text": "25 I'm basically going to get something"}, {"start": 154.84, "duration": 3.759, "text": "close to zero but it's going to be a"}, {"start": 156.48, "duration": 3.8, "text": "little bit bigger than zero but again if"}, {"start": 158.599, "duration": 3.961, "text": "you take the square root of that number"}, {"start": 160.28, "duration": 5.28, "text": "it's going to get arbitrarily close to"}, {"start": 162.56, "duration": 3.0, "text": "zero"}], "title": "One Sided Limits, Example 3", "description": "In this video we look at finding a one sided limit given a function involving a square root.  In order to find the limit we must choose the appropriate 'piece' of the function depending on whether our limit approaches the given value from the left or from the right.\nThis is a typical limit type of question in calculus and relates to continuity.\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n#math #algebra #khanacademy #khan #geometry #education #montessori #homeschooling #functions #notation #organicchemistry #organicchemistrytutor #calculus #derivative #powerrule #limits  #limit #graph #graphing #conjugate #onesided #continuity", "lengthSeconds": 164, "width": 480, "height": 360}
{"video_id": "O3l0wDegKmM", "transcript": [{"start": 1.439, "duration": 3.721, "text": "okay here we're going to look at a"}, {"start": 2.44, "duration": 4.72, "text": "question related to proportions and all"}, {"start": 5.16, "duration": 4.92, "text": "we want to figure out is which of the"}, {"start": 7.16, "duration": 6.24, "text": "following does not form a proportion"}, {"start": 10.08, "duration": 5.639, "text": "with the value 3 over five so we have"}, {"start": 13.4, "duration": 6.16, "text": "four different choices here all you"}, {"start": 15.719, "duration": 5.681, "text": "basically have to do in this case is uh"}, {"start": 19.56, "duration": 4.68, "text": "basically think what fractions would be"}, {"start": 21.4, "duration": 5.08, "text": "equivalent to the fraction 3 /"}, {"start": 24.24, "duration": 4.0, "text": "5 well there's a couple different ways"}, {"start": 26.48, "duration": 4.16, "text": "to really think about this notice 3"}, {"start": 28.24, "duration": 4.519, "text": "fifths is in reduced form"}, {"start": 30.64, "duration": 3.88, "text": "so another thing we could do is just"}, {"start": 32.759, "duration": 5.001, "text": "reduce these other fractions and see if"}, {"start": 34.52, "duration": 5.64, "text": "you get 3 over five at the end so uh 6"}, {"start": 37.76, "duration": 5.68, "text": "over 10 well if we multiply the"}, {"start": 40.16, "duration": 7.52, "text": "numerator by 1/2 and the denominator"}, {"start": 43.44, "duration": 8.639, "text": "also by 1/2 we would get 3 over five so"}, {"start": 47.68, "duration": 7.28, "text": "certainly uh part A does check out the"}, {"start": 52.079, "duration": 5.16, "text": "same thing with part C 30 over 50 well"}, {"start": 54.96, "duration": 4.759, "text": "we could just cancel out the zeros and"}, {"start": 57.239, "duration": 5.521, "text": "be left with 3 over five so that one's"}, {"start": 59.719, "duration": 6.561, "text": "okay okay let's see uh are there any"}, {"start": 62.76, "duration": 6.399, "text": "common factors between uh 24 and"}, {"start": 66.28, "duration": 7.479, "text": "35 and I think there's not this is"}, {"start": 69.159, "duration": 7.561, "text": "actually uh already reduced well 24 over"}, {"start": 73.759, "duration": 6.961, "text": "35 doesn't reduce to 3 over"}, {"start": 76.72, "duration": 7.079, "text": "five so uh answer Choice B would be our"}, {"start": 80.72, "duration": 5.8, "text": "our our value that doesn't form a"}, {"start": 83.799, "duration": 6.721, "text": "proportion with 3 over5 and just to be"}, {"start": 86.52, "duration": 6.639, "text": "complete notice 12 over 20 um if we"}, {"start": 90.52, "duration": 5.36, "text": "multiply the numerator by 1/4 and the"}, {"start": 93.159, "duration": 5.121, "text": "denominator also by 1/4 equivalently"}, {"start": 95.88, "duration": 5.36, "text": "you're dividing top and bottom by four"}, {"start": 98.28, "duration": 5.96, "text": "we would get 3 over five so again D does"}, {"start": 101.24, "duration": 3.0, "text": "work"}], "title": "Equivalent Proportions / Fractions", "description": "Thanks to all of you who support me on Patreon. You da real mvps!   $1 per month helps!! :)  https://www.patreon.com/patrickjmt !! Equivalent Proportions / Fractions.\n Here I do a quick example of finding an equivalent proportion to a given proportion.", "lengthSeconds": 103, "width": 480, "height": 360}
{"video_id": "rd_9t_QRhsI", "transcript": [{"start": 0.079, "duration": 4.121, "text": "okay so here we're gonna do another"}, {"start": 2.25, "duration": 4.35, "text": "example involving implicit"}, {"start": 4.2, "duration": 5.37, "text": "differentiation so suppose we've got the"}, {"start": 6.6, "duration": 5.1, "text": "curve given by x over X minus y equals y"}, {"start": 9.57, "duration": 7.65, "text": "squared minus one and we want to find"}, {"start": 11.7, "duration": 7.079, "text": "dy/dx so this one same ideas before it's"}, {"start": 17.22, "duration": 2.88, "text": "just going to be a little more tedious I"}, {"start": 18.779, "duration": 4.26, "text": "think really because of this fraction"}, {"start": 20.1, "duration": 4.14, "text": "more than anything so alright we're"}, {"start": 23.039, "duration": 3.75, "text": "going to have to take the derivative"}, {"start": 24.24, "duration": 3.959, "text": "with respect to X of both sides well on"}, {"start": 26.789, "duration": 5.31, "text": "the left side we'll have to use the"}, {"start": 28.199, "duration": 5.431, "text": "quotient rule so the quotient rule says"}, {"start": 32.099, "duration": 4.561, "text": "you get whatever's in the denominator"}, {"start": 33.63, "duration": 4.53, "text": "and then we take the derivative of"}, {"start": 36.66, "duration": 5.04, "text": "what's in the numerator so the"}, {"start": 38.16, "duration": 5.25, "text": "derivative of 1/x will just be one minus"}, {"start": 41.7, "duration": 5.339, "text": "then we leave the stuff in the numerator"}, {"start": 43.41, "duration": 4.8, "text": "alone and then we take the derivative of"}, {"start": 47.039, "duration": 4.02, "text": "the denominator"}, {"start": 48.21, "duration": 6.18, "text": "so the derivative of One X is just 1 and"}, {"start": 51.059, "duration": 8.25, "text": "the derivative of negative 1 Y will be"}, {"start": 54.39, "duration": 9.469, "text": "negative 1 dy over DX and then that's"}, {"start": 59.309, "duration": 4.55, "text": "all over the denominator has squared"}, {"start": 64.159, "duration": 5.171, "text": "okay on the right side we'll have to"}, {"start": 67.2, "duration": 4.97, "text": "again take the derivative using implicit"}, {"start": 69.33, "duration": 6.899, "text": "differentiation so we'll just get 2 y"}, {"start": 72.17, "duration": 9.42, "text": "times dy over DX the derivative of"}, {"start": 76.229, "duration": 7.861, "text": "negative 1 is just 0 so all right now"}, {"start": 81.59, "duration": 5.59, "text": "what's probably the easiest way to solve"}, {"start": 84.09, "duration": 5.52, "text": "for dy DX well I think I'm gonna do a"}, {"start": 87.18, "duration": 4.649, "text": "couple steps at once in the numerator we"}, {"start": 89.61, "duration": 5.13, "text": "would be left with just X minus y and"}, {"start": 91.829, "duration": 5.04, "text": "then we would have minus so if we"}, {"start": 94.74, "duration": 5.309, "text": "distribute the negative x we would have"}, {"start": 96.869, "duration": 4.771, "text": "negative x times 1 or negative x and"}, {"start": 100.049, "duration": 8.881, "text": "then it looks like to me we would have a"}, {"start": 101.64, "duration": 9.21, "text": "positive x times dy over DX well I guess"}, {"start": 108.93, "duration": 7.41, "text": "I won't do so many steps at once let's"}, {"start": 110.85, "duration": 7.26, "text": "slow it down a little bit here all right"}, {"start": 116.34, "duration": 4.62, "text": "well okay I guess we can do a couple"}, {"start": 118.11, "duration": 4.14, "text": "things here now I am going to do a"}, {"start": 120.96, "duration": 2.61, "text": "couple things to get rid of this"}, {"start": 122.25, "duration": 5.43, "text": "denominator I'm just going to multiply"}, {"start": 123.57, "duration": 8.07, "text": "both sides by X minus y squared and by x"}, {"start": 127.68, "duration": 6.49, "text": "minus y squared so those will cancel on"}, {"start": 131.64, "duration": 4.51, "text": "the left notice we have an X my"}, {"start": 134.17, "duration": 4.2, "text": "sex so that would actually those would"}, {"start": 136.15, "duration": 6.05, "text": "just cancel out so really we would be"}, {"start": 138.37, "duration": 7.65, "text": "left with negative y plus x times dy"}, {"start": 142.2, "duration": 5.38, "text": "over DX on the right side I'm gonna pull"}, {"start": 146.02, "duration": 4.83, "text": "kind of all the variables stuff out"}, {"start": 147.58, "duration": 9.39, "text": "front so we have 2y times X minus y"}, {"start": 150.85, "duration": 9.09, "text": "squared times dy over DX getting pretty"}, {"start": 156.97, "duration": 6.0, "text": "close now I want to put all the dy DX's"}, {"start": 159.94, "duration": 6.6, "text": "on one side so maybe I'll just pull all"}, {"start": 162.97, "duration": 5.88, "text": "those dy DX's over to the right side so"}, {"start": 166.54, "duration": 5.09, "text": "I'll leave the original stuff over here"}, {"start": 168.85, "duration": 7.52, "text": "just like it was but then I'll subtract"}, {"start": 171.63, "duration": 7.51, "text": "the x times dy over DX from both sides"}, {"start": 176.37, "duration": 5.89, "text": "and then we'll have negative Y leftover"}, {"start": 179.14, "duration": 4.89, "text": "on the left and now we're almost there"}, {"start": 182.26, "duration": 5.22, "text": "all we have to do on the right side is"}, {"start": 184.03, "duration": 7.08, "text": "just factor out that dy DX so if we"}, {"start": 187.48, "duration": 6.72, "text": "factor out the dy over DX and the"}, {"start": 191.11, "duration": 5.76, "text": "brackets or parentheses here brackets we"}, {"start": 194.2, "duration": 8.789, "text": "would be left with the two Y times X"}, {"start": 196.87, "duration": 9.179, "text": "minus y squared and then minus X on the"}, {"start": 202.989, "duration": 7.801, "text": "right side we're left with negative Y so"}, {"start": 206.049, "duration": 7.56, "text": "to solve for dy DX we just divide so our"}, {"start": 210.79, "duration": 6.65, "text": "derivative will be negative y over all"}, {"start": 213.609, "duration": 8.481, "text": "of this stuff to Y times X minus y"}, {"start": 217.44, "duration": 7.65, "text": "squared minus X and now we have our"}, {"start": 222.09, "duration": 3.0, "text": "derivative"}], "title": "Implicit Differentiation - More Examples #4", "description": "Thanks to all of you who support me on Patreon. You da real mvps!   $1 per month helps!! :)  https://www.patreon.com/patrickjmt !! Implicit Differentiation - More Examples #4\nYet another implicit differentiation example!", "lengthSeconds": 224, "width": 480, "height": 360}
{"video_id": "JVGSKnKwIsc", "transcript": [{"start": 0.32, "duration": 3.04, "text": "all right in this video i'm going to do"}, {"start": 1.599, "duration": 3.841, "text": "another example of predicting the types"}, {"start": 3.36, "duration": 3.439, "text": "of solutions to a quadratic equation"}, {"start": 5.44, "duration": 3.04, "text": "using the discriminant"}, {"start": 6.799, "duration": 3.441, "text": "so again the discriminant is just the"}, {"start": 8.48, "duration": 3.52, "text": "part under the square root the b squared"}, {"start": 10.24, "duration": 4.64, "text": "minus 4ac"}, {"start": 12.0, "duration": 4.16, "text": "so in this case we've got the problem 2x"}, {"start": 14.88, "duration": 4.319, "text": "squared plus six x"}, {"start": 16.16, "duration": 4.879, "text": "minus six equals negative eight"}, {"start": 19.199, "duration": 4.481, "text": "uh again before you start plugging"}, {"start": 21.039, "duration": 5.041, "text": "things in you know the quadratic formula"}, {"start": 23.68, "duration": 4.64, "text": "or to the discriminant you always need"}, {"start": 26.08, "duration": 3.119, "text": "one side of the equation to be equal to"}, {"start": 28.32, "duration": 2.16, "text": "zero"}, {"start": 29.199, "duration": 3.441, "text": "so the first thing i'm going to have to"}, {"start": 30.48, "duration": 3.28, "text": "do in this case is simply add 8 to both"}, {"start": 32.64, "duration": 3.759, "text": "sides"}, {"start": 33.76, "duration": 3.28, "text": "so then we're going to have 2x squared"}, {"start": 36.399, "duration": 4.241, "text": "plus"}, {"start": 37.04, "duration": 5.12, "text": "6x negative 6 plus 8 is going to give us"}, {"start": 40.64, "duration": 5.04, "text": "positive 2"}, {"start": 42.16, "duration": 5.12, "text": "equals 0 and again the positive 2 that's"}, {"start": 45.68, "duration": 3.92, "text": "going to be our a value"}, {"start": 47.28, "duration": 3.52, "text": "the positive 6 that's going to be our b"}, {"start": 49.6, "duration": 2.959, "text": "value"}, {"start": 50.8, "duration": 3.919, "text": "and the positive 2 that's going to be"}, {"start": 52.559, "duration": 4.64, "text": "our c value"}, {"start": 54.719, "duration": 3.52, "text": "so when we fill in the b squared minus"}, {"start": 57.199, "duration": 3.2, "text": "4ac"}, {"start": 58.239, "duration": 3.521, "text": "the discriminant we're going to end up"}, {"start": 60.399, "duration": 5.281, "text": "getting"}, {"start": 61.76, "duration": 7.2, "text": "so we'll have b which is 6 squared"}, {"start": 65.68, "duration": 6.32, "text": "minus 4 times a which is 2"}, {"start": 68.96, "duration": 5.12, "text": "times c which is also 2."}, {"start": 72.0, "duration": 4.0, "text": "so for the discriminant 6 squared is"}, {"start": 74.08, "duration": 5.12, "text": "going to be 36"}, {"start": 76.0, "duration": 6.88, "text": "4 times 2 is 8 8 times 2 is"}, {"start": 79.2, "duration": 6.32, "text": "16. so we're going to have 36 minus 16"}, {"start": 82.88, "duration": 5.52, "text": "which will give us a positive 20."}, {"start": 85.52, "duration": 5.04, "text": "so since the discriminant is a positive"}, {"start": 88.4, "duration": 6.399, "text": "number what that tells us in this case"}, {"start": 90.56, "duration": 6.48, "text": "is that we have two real solutions"}, {"start": 94.799, "duration": 4.041, "text": "i should say two real you can even"}, {"start": 97.04, "duration": 4.8, "text": "clarify distinct"}, {"start": 98.84, "duration": 3.0, "text": "solutions"}, {"start": 102.24, "duration": 5.6, "text": "and therefore we have zero complex"}, {"start": 110.52, "duration": 3.0, "text": "solutions"}], "title": "Using the Discriminant to Predict the Types of Solutions to a Quadratic Equation - Example 2", "description": "Thanks to all of you who support me on Patreon. You da real mvps!   $1 per month helps!! :)  https://www.patreon.com/patrickjmt !! Using the Discriminant to Predict the Types of Solutions to a Quadratic Equation - Example 2", "lengthSeconds": 112, "width": 480, "height": 360}
{"video_id": "EzyhMYgAlWI", "transcript": [{"start": 1.48, "duration": 7.52, "text": "okay so this is going to be question"}, {"start": 3.28, "duration": 8.92, "text": "number one for the 2016 version of the S"}, {"start": 9.0, "duration": 5.08, "text": "and again no calculators in this part so"}, {"start": 12.2, "duration": 4.76, "text": "this one's not hopefully too terribly"}, {"start": 14.08, "duration": 6.16, "text": "bad it says if the quantity X - 1 all"}, {"start": 16.96, "duration": 5.559, "text": "divid 3 equal K and K = 3 we want to"}, {"start": 20.24, "duration": 5.84, "text": "know what's the value of x is it 2 4 9"}, {"start": 22.519, "duration": 4.881, "text": "or 10 so this is just a basic just a"}, {"start": 26.08, "duration": 6.84, "text": "basic linear"}, {"start": 27.4, "duration": 7.72, "text": "equation so we have x - 1 over 3 equals"}, {"start": 32.92, "duration": 5.36, "text": "well they give us that k equals 3 so we"}, {"start": 35.12, "duration": 6.72, "text": "can substitute that in and now it's just"}, {"start": 38.28, "duration": 6.84, "text": "going to be a matter of solving for x so"}, {"start": 41.84, "duration": 6.16, "text": "I've got x -1 / 3 we can simply multiply"}, {"start": 45.12, "duration": 5.119, "text": "both sides by 3 on the left side the"}, {"start": 48.0, "duration": 3.44, "text": "threes will cancel out and we'll be left"}, {"start": 50.239, "duration": 3.441, "text": "with x"}, {"start": 51.44, "duration": 5.16, "text": "-1 on the right side we'll three"}, {"start": 53.68, "duration": 5.44, "text": "multiplied by 3 is going to give us 9"}, {"start": 56.6, "duration": 5.32, "text": "and now to solve for x we can just add"}, {"start": 59.12, "duration": 2.8, "text": "one to both sides"}, {"start": 62.16, "duration": 8.999, "text": "and that'll leave us with the value of x"}, {"start": 64.92, "duration": 6.239, "text": "= 10 which was answer Choice d"}], "title": "SAT Math - Question 1, 2016 Edition", "description": "Thanks to all of you who support me on Patreon. You da real mvps!   $1 per month helps!! :)  https://www.patreon.com/patrickjmt !! SAT Math - Question 1, 2016 Edition and Beyond\nSection 1 - No calculator allowed\nThis video is part of a playlist https://www.youtube.com/playlist?list=PLANMHOrJaFxPm2HZBsDfs1_ie3cmyremr", "lengthSeconds": 70, "width": 480, "height": 360}
{"video_id": "8ZyeHtgMBjk", "transcript": [{"start": 1.079, "duration": 4.361, "text": "okay in this video I just want to"}, {"start": 2.36, "duration": 6.239, "text": "quickly look at inner products and"}, {"start": 5.44, "duration": 5.279, "text": "orthogonal functions so uh the"}, {"start": 8.599, "duration": 4.16, "text": "definition of an inner product uh we say"}, {"start": 10.719, "duration": 4.84, "text": "the inner product of two functions f ofx"}, {"start": 12.759, "duration": 4.881, "text": "and g ofx on the interval A to B is a"}, {"start": 15.559, "duration": 4.401, "text": "number denoted by we just put them in"}, {"start": 17.64, "duration": 5.32, "text": "parentheses so parentheses F comma G"}, {"start": 19.96, "duration": 5.76, "text": "close the parenthesis and to find the"}, {"start": 22.96, "duration": 6.2, "text": "inner product all we do is we multiply"}, {"start": 25.72, "duration": 5.36, "text": "those two functions f ofx * G ofx and"}, {"start": 29.16, "duration": 4.719, "text": "then we calculate that definite integral"}, {"start": 31.08, "duration": 5.479, "text": "from A to B uh integrating with respect"}, {"start": 33.879, "duration": 4.761, "text": "to X so all we do to find the inner"}, {"start": 36.559, "duration": 4.561, "text": "product just multiply them integrate"}, {"start": 38.64, "duration": 5.28, "text": "over the desired"}, {"start": 41.12, "duration": 5.0, "text": "interval if this inner product ends up"}, {"start": 43.92, "duration": 4.68, "text": "equaling zero so when we calculate the"}, {"start": 46.12, "duration": 6.079, "text": "definite integral if we get zero we say"}, {"start": 48.6, "duration": 7.36, "text": "that those functions f and g are"}, {"start": 52.199, "duration": 5.801, "text": "orthogonal and functions being"}, {"start": 55.96, "duration": 4.16, "text": "orthogonal uh that ends up being"}, {"start": 58.0, "duration": 4.64, "text": "something something important and we'll"}, {"start": 60.12, "duration": 6.4, "text": "see that in some some following videos"}, {"start": 62.64, "duration": 7.88, "text": "so very quick simple little"}, {"start": 66.52, "duration": 6.88, "text": "example show that the functions 1 + x"}, {"start": 70.52, "duration": 4.639, "text": "and x - x^2 we're going to Simply show"}, {"start": 73.4, "duration": 3.52, "text": "that those are orthogonal on the"}, {"start": 75.159, "duration": 4.441, "text": "interval neg -2 to"}, {"start": 76.92, "duration": 5.36, "text": "two well all we have to do in this case"}, {"start": 79.6, "duration": 4.44, "text": "we just integrate from -2 to two all"}, {"start": 82.28, "duration": 3.6, "text": "we're going to do is just multiply those"}, {"start": 84.04, "duration": 5.32, "text": "functions"}, {"start": 85.88, "duration": 5.76, "text": "together calculate the definite integral"}, {"start": 89.36, "duration": 4.119, "text": "and we better get zero if they are in"}, {"start": 91.64, "duration": 4.36, "text": "fact"}, {"start": 93.479, "duration": 3.881, "text": "orthogonal okay well this shouldn't be"}, {"start": 96.0, "duration": 3.88, "text": "too terrible it's just a matter of"}, {"start": 97.36, "duration": 6.079, "text": "multiplying it out so if we distribute 1"}, {"start": 99.88, "duration": 6.16, "text": "* X is X we'll have 1 *"}, {"start": 103.439, "duration": 4.68, "text": "x^2 we'll have positive x and x which"}, {"start": 106.04, "duration": 5.88, "text": "will be positive x^2 and then it looks"}, {"start": 108.119, "duration": 3.801, "text": "like we'll havex the"}, {"start": 112.24, "duration": 5.72, "text": "3 okay so we can certainly simplify that"}, {"start": 115.52, "duration": 4.599, "text": "our Nega x^2 and positive x^2 are going"}, {"start": 117.96, "duration": 7.56, "text": "to cancel so now we're we're just"}, {"start": 120.119, "duration": 9.601, "text": "calculating from -2 to 2 of x - x"}, {"start": 125.52, "duration": 6.24, "text": "3r DX well this isn't too bad at all"}, {"start": 129.72, "duration": 5.56, "text": "when we integrate x we'll have x^2 over"}, {"start": 131.76, "duration": 6.88, "text": "2 when we integrate x the 3r we'll have"}, {"start": 135.28, "duration": 6.36, "text": "X to 4 over"}, {"start": 138.64, "duration": 7.08, "text": "4 and now we simply have to evaluate"}, {"start": 141.64, "duration": 6.08, "text": "that from -2 to 2 well both of these are"}, {"start": 145.72, "duration": 4.159, "text": "even functions so they're clearly we're"}, {"start": 147.72, "duration": 3.879, "text": "going to get the same value from -2 to 2"}, {"start": 149.879, "duration": 3.281, "text": "so maybe it's obvious that we get zero"}, {"start": 151.599, "duration": 5.64, "text": "but let's I guess let's go ahead and"}, {"start": 153.16, "duration": 10.04, "text": "evaluate we'll have 2^2 which is 4 4 / 2"}, {"start": 157.239, "duration": 9.36, "text": "is 2 uh 2 4th is 16 16 / 4 is going to"}, {"start": 163.2, "duration": 5.56, "text": "be 4 so that's our uh upper limit of"}, {"start": 166.599, "duration": 4.56, "text": "integration and again since we're"}, {"start": 168.76, "duration": 4.32, "text": "squaring and raising to the fourth power"}, {"start": 171.159, "duration": 4.561, "text": "when we substitute in -2 we're going to"}, {"start": 173.08, "duration": 8.48, "text": "get the exact same"}, {"start": 175.72, "duration": 9.84, "text": "thing okay so we have -2 minus -2 -2 +"}, {"start": 181.56, "duration": 6.759, "text": "2 well that certainly equals zero so yes"}, {"start": 185.56, "duration": 4.92, "text": "in fact those two functions are"}, {"start": 188.319, "duration": 4.401, "text": "orthogonal what I'm going to do in my"}, {"start": 190.48, "duration": 5.36, "text": "next video is actually look at an"}, {"start": 192.72, "duration": 5.439, "text": "orthogonal set of functions and in that"}, {"start": 195.84, "duration": 6.319, "text": "case we'll justify that that entire set"}, {"start": 198.159, "duration": 6.281, "text": "of functions um are orthogonal and uh"}, {"start": 202.159, "duration": 4.121, "text": "we'll also give a I'll give a exact"}, {"start": 204.44, "duration": 6.359, "text": "definition of what we mean when we talk"}, {"start": 206.28, "duration": 4.519, "text": "about orthogonal sets of functions"}], "title": "Inner Product and Orthogonal Functions , Quick Example", "description": "Thanks to all of you who support me on Patreon. You da real mvps!   $1 per month helps!! :)  https://www.patreon.com/patrickjmt !! Inner Product and Orthogonal Functions , Quick Example.\nIn this video, I give the definition of the inner product of two functions and what it means for those functions to be orthogonal.  I work a quick example showing that two functions are orthogonal.", "lengthSeconds": 209, "width": 450, "height": 360}
{"video_id": "42KNe2KgE2Q", "transcript": [{"start": 1.0, "duration": 4.04, "text": "okay here we're going to identify the"}, {"start": 3.12, "duration": 6.559, "text": "conic that's given by the polar equation"}, {"start": 5.04, "duration": 6.519, "text": "R = 5 / 10 - 15 * sin Theta to do that"}, {"start": 9.679, "duration": 4.12, "text": "to determine what type of of conic"}, {"start": 11.559, "duration": 4.761, "text": "section it is we'll first find the"}, {"start": 13.799, "duration": 5.761, "text": "eccentricity and using the excentricity"}, {"start": 16.32, "duration": 5.44, "text": "we can find the directrix so this is"}, {"start": 19.56, "duration": 8.639, "text": "almost in the form"}, {"start": 21.76, "duration": 8.599, "text": "Ral e * P / 1 - e * sin thet so to put"}, {"start": 28.199, "duration": 6.4, "text": "our equation in that form we'll we'll do"}, {"start": 30.359, "duration": 4.24, "text": "is we'll Factor the 10 out of the"}, {"start": 34.76, "duration": 6.28, "text": "denominator well 10 * -15 over 10 would"}, {"start": 38.399, "duration": 2.641, "text": "give us"}, {"start": 41.28, "duration": 5.84, "text": "-15 and we can reduce 5 over 10 will be2"}, {"start": 44.879, "duration": 4.441, "text": "we can just put that in the numerator"}, {"start": 47.12, "duration": 6.919, "text": "then we have 1 minus well 15 over 10"}, {"start": 49.32, "duration": 7.64, "text": "will be 3 * sin Theta so already we get"}, {"start": 54.039, "duration": 5.0, "text": "that our eccentricity here our"}, {"start": 56.96, "duration": 4.759, "text": "eccentricity is going to equal three"}, {"start": 59.039, "duration": 5.52, "text": "Hales and since that's greater than one"}, {"start": 61.719, "duration": 2.84, "text": "we know that we have a"}, {"start": 65.479, "duration": 7.041, "text": "hyperbola and we want the numerator to"}, {"start": 68.56, "duration": 7.28, "text": "be in the form e * P so e * P we want"}, {"start": 72.52, "duration": 5.44, "text": "that to equal the numerator which is 12"}, {"start": 75.84, "duration": 3.959, "text": "well again we know that e is equal to"}, {"start": 77.96, "duration": 4.4, "text": "three"}, {"start": 79.799, "duration": 4.541, "text": "Hales so to solve for p we can just"}, {"start": 82.36, "duration": 3.119, "text": "simply multiply both sides by"}, {"start": 84.34, "duration": 5.819, "text": "[Music]"}, {"start": 85.479, "duration": 7.881, "text": "2/3 so we'll have that P = 1/3 and and"}, {"start": 90.159, "duration": 5.801, "text": "for this type of polar equation we know"}, {"start": 93.36, "duration": 5.759, "text": "that the directrix will be P units below"}, {"start": 95.96, "duration": 6.439, "text": "and parallel to the x-axis so there's"}, {"start": 99.119, "duration": 6.68, "text": "our x axis we go P units down which"}, {"start": 102.399, "duration": 5.32, "text": "would be 1/3 units down it's going to be"}, {"start": 105.799, "duration": 4.841, "text": "parallel to the x-axis that's going to"}, {"start": 107.719, "duration": 5.481, "text": "be the directrix D well since we're 1/3"}, {"start": 110.64, "duration": 4.72, "text": "units down and we have a horizontal line"}, {"start": 113.2, "duration": 2.16, "text": "the"}, {"start": 115.64, "duration": 6.2, "text": "directrix would have an equation Y ="}, {"start": 122.32, "duration": 3.0, "text": "1/3"}], "title": "Polar Form of a Conic: Finding the Type, the Directrix and Eccentricity", "description": "Thanks to all of you who support me on Patreon. You da real mvps!   $1 per month helps!! :)  https://www.patreon.com/patrickjmt !! Polar Form of a Conic: Finding the Type, the Directrix and Eccentricity.\nHere we have the polar form of a conic.  I determine what type of conic section it is and then find the directrix and eccentricity.", "lengthSeconds": 123, "width": 450, "height": 360}
{"video_id": "TNcNvSCBD30", "transcript": [{"start": 0.599, "duration": 2.961, "text": "all right in this video we're going to"}, {"start": 1.76, "duration": 3.599, "text": "do an example of using some sum and"}, {"start": 3.56, "duration": 4.319, "text": "difference identities to simplify an"}, {"start": 5.359, "duration": 4.36, "text": "expression and uh here are kind of the"}, {"start": 7.879, "duration": 4.481, "text": "uh the six identities that we'll make"}, {"start": 9.719, "duration": 4.08, "text": "use of again each one you know has a"}, {"start": 12.36, "duration": 3.8, "text": "positive or A negative giving you a"}, {"start": 13.799, "duration": 4.121, "text": "different identity in there so the first"}, {"start": 16.16, "duration": 5.8, "text": "one here uh we're just going to simplify"}, {"start": 17.92, "duration": 8.359, "text": "s of 20\u00b0 time cosine of 40\u00b0 plus cosine"}, {"start": 21.96, "duration": 6.0, "text": "of 20\u00b0 Time s of 40\u00b0 well it's just"}, {"start": 26.279, "duration": 3.76, "text": "picking out the identity that it looks"}, {"start": 27.96, "duration": 4.599, "text": "like well there's signs and cosin mixed"}, {"start": 30.039, "duration": 5.801, "text": "up so that has to do with the first"}, {"start": 32.559, "duration": 6.201, "text": "one um if there's a positive sign in"}, {"start": 35.84, "duration": 5.16, "text": "between again for our sign identity it"}, {"start": 38.76, "duration": 4.72, "text": "says if there's a positive sign uh we"}, {"start": 41.0, "duration": 6.399, "text": "keep a positive in between so it says"}, {"start": 43.48, "duration": 6.04, "text": "this is simply going to equal s of well"}, {"start": 47.399, "duration": 6.761, "text": "20\u00b0 plus"}, {"start": 49.52, "duration": 8.199, "text": "40\u00b0 and that simply gives us well s of"}, {"start": 54.16, "duration": 5.44, "text": "60\u00b0 and let's see s of 60\u00b0 again on the"}, {"start": 57.719, "duration": 4.201, "text": "unit circle that's the same thing as pi"}, {"start": 59.6, "duration": 2.32, "text": "over"}, {"start": 62.16, "duration": 6.319, "text": "3 and sign at that uh at that angle uh"}, {"start": 65.6, "duration": 8.96, "text": "the unit circle s of 60\u00b0 is going to"}, {"start": 68.479, "duration": 6.081, "text": "give us simply the value < TK 3 over 2"}], "title": "Sum and Difference Identities to Simplify an Expression, Example 1", "description": "Thanks to all of you who support me on Patreon. You da real mvps!   $1 per month helps!! :)  https://www.patreon.com/patrickjmt !! Sum and Difference Identities to Simplify an Expression, Example 1.  In this video, I simply show how to simplify some expressions by using sum and difference identities.  Nothing crazy, just showing how one must know identities to make things more manageable!  For more free math videos, visit http://PatrickJMT.com", "lengthSeconds": 73, "width": 480, "height": 360}
{"video_id": "UPC2bPKFEZs", "transcript": [{"start": 0.919, "duration": 3.4, "text": "all right in this video I'm going to"}, {"start": 2.24, "duration": 4.48, "text": "look at some examples related to the"}, {"start": 4.319, "duration": 4.161, "text": "product rule and the product rule again"}, {"start": 6.72, "duration": 3.32, "text": "says if you have a function times"}, {"start": 8.48, "duration": 3.56, "text": "another function and we want to take the"}, {"start": 10.04, "duration": 4.16, "text": "derivative what we do is we take the"}, {"start": 12.04, "duration": 5.159, "text": "derivative of one of the functions"}, {"start": 14.2, "duration": 4.839, "text": "multiply it by the original of the other"}, {"start": 17.199, "duration": 4.0, "text": "then we stick a plus sign in there and"}, {"start": 19.039, "duration": 4.32, "text": "then the function we took the derivative"}, {"start": 21.199, "duration": 4.041, "text": "of the first time we leave it alone and"}, {"start": 23.359, "duration": 4.801, "text": "then we take the derivative of the other"}, {"start": 25.24, "duration": 5.92, "text": "function um notice a couple things here"}, {"start": 28.16, "duration": 5.6, "text": "uh you know multiplication and addition"}, {"start": 31.16, "duration": 5.919, "text": "are commutative so we can you know we"}, {"start": 33.76, "duration": 6.119, "text": "could have done um you know FG prime"}, {"start": 37.079, "duration": 4.48, "text": "plus frime G um it doesn't really matter"}, {"start": 39.879, "duration": 2.801, "text": "the order that you do these things as"}, {"start": 41.559, "duration": 3.68, "text": "long as you remember to take the"}, {"start": 42.68, "duration": 4.0, "text": "derivative of one part leave one alone"}, {"start": 45.239, "duration": 4.881, "text": "plus and then just kind of switch them"}, {"start": 46.68, "duration": 5.24, "text": "out so people uh you know sometimes are"}, {"start": 50.12, "duration": 3.919, "text": "worried about that a little bit so here"}, {"start": 51.92, "duration": 4.279, "text": "we're going to find the derivative of uh"}, {"start": 54.039, "duration": 5.641, "text": "a couple different functions and we'll"}, {"start": 56.199, "duration": 8.32, "text": "do these uh maybe uh one at a time here"}, {"start": 59.68, "duration": 8.759, "text": "so so uh in part A we have f ofx = x - 2"}, {"start": 64.519, "duration": 5.561, "text": "uh * x + 3 now normally what I would do"}, {"start": 68.439, "duration": 3.561, "text": "before if I were going to take the"}, {"start": 70.08, "duration": 4.92, "text": "derivative of this function I would"}, {"start": 72.0, "duration": 5.28, "text": "actually probably multiply it out first"}, {"start": 75.0, "duration": 4.64, "text": "um combine my like terms and then take"}, {"start": 77.28, "duration": 4.08, "text": "the derivative but just to illustrate"}, {"start": 79.64, "duration": 4.96, "text": "this product"}, {"start": 81.36, "duration": 5.0, "text": "rule so again the the product rule says"}, {"start": 84.6, "duration": 4.12, "text": "uh take the derivative of one of these"}, {"start": 86.36, "duration": 5.439, "text": "two factors so I'm going to take the"}, {"start": 88.72, "duration": 5.6, "text": "derivative of the X x - 2 well the"}, {"start": 91.799, "duration": 5.32, "text": "derivative of 1x is just one the"}, {"start": 94.32, "duration": 4.799, "text": "derivative of a constant is just zero so"}, {"start": 97.119, "duration": 4.121, "text": "I'm not even going to write it in there"}, {"start": 99.119, "duration": 3.921, "text": "and then we'll have the x + 3 so we're"}, {"start": 101.24, "duration": 4.12, "text": "leaving that part"}, {"start": 103.04, "duration": 4.679, "text": "alone and then we'll stick our plus sign"}, {"start": 105.36, "duration": 6.039, "text": "in between there and now we'll leave the"}, {"start": 107.719, "duration": 7.161, "text": "x - 2 alone and then we'll take the"}, {"start": 111.399, "duration": 7.08, "text": "derivative of x + 3 well the derivative"}, {"start": 114.88, "duration": 6.64, "text": "again derivative of 1 x is just one the"}, {"start": 118.479, "duration": 5.361, "text": "derivative of plus three is just 0 so"}, {"start": 121.52, "duration": 4.48, "text": "that would be our derivative and I'm"}, {"start": 123.84, "duration": 5.639, "text": "going to go ahead and combine these we"}, {"start": 126.0, "duration": 6.08, "text": "would get x + 3 plus"}, {"start": 129.479, "duration": 7.361, "text": "X+ oh almost did it wrong so we would"}, {"start": 132.08, "duration": 9.64, "text": "get x + 3 + x - 2 and when we combine"}, {"start": 136.84, "duration": 7.96, "text": "those we have x + x which is 2x pos3 - 2"}, {"start": 141.72, "duration": 5.519, "text": "will give us pos1 and that's our"}, {"start": 144.8, "duration": 5.159, "text": "derivative notice if you had distributed"}, {"start": 147.239, "duration": 6.08, "text": "this out at the very beginning um if you"}, {"start": 149.959, "duration": 6.28, "text": "had distributed out we would have x * X"}, {"start": 153.319, "duration": 12.2, "text": "which is x^2 we would have a posi"}, {"start": 156.239, "duration": 12.401, "text": "3x -2X - 6 so we would have x^2 + x - 6"}, {"start": 165.519, "duration": 4.961, "text": "um if we did the multiplication uh first"}, {"start": 168.64, "duration": 4.72, "text": "now if we just use our actual product"}, {"start": 170.48, "duration": 5.8, "text": "rule U well the derivative of x^2 is"}, {"start": 173.36, "duration": 5.959, "text": "going to be 2x the 1st the derivative of"}, {"start": 176.28, "duration": 6.039, "text": "POS 1X is just going to be POS 1 uh the"}, {"start": 179.319, "duration": 4.84, "text": "derivative of -6 is just zero so again"}, {"start": 182.319, "duration": 4.161, "text": "just illustrating hey you do get the"}, {"start": 184.159, "duration": 4.401, "text": "same thing um but again just in this"}, {"start": 186.48, "duration": 4.08, "text": "video uh just trying to illustrate the"}, {"start": 188.56, "duration": 4.52, "text": "use of the product rule here a little"}, {"start": 190.56, "duration": 5.12, "text": "bit so all right I think I'm going to uh"}, {"start": 193.08, "duration": 4.68, "text": "do the other ones in separate videos and"}, {"start": 195.68, "duration": 3.96, "text": "we'll also talk about maybe trying to"}, {"start": 197.76, "duration": 3.44, "text": "simplify these down and clean them up a"}, {"start": 199.64, "duration": 4.84, "text": "little bit because sometimes that can be"}, {"start": 201.2, "duration": 3.28, "text": "pretty tedious"}], "title": "Basic Product Rule Example #1", "description": "Thanks to all of you who support me on Patreon. You da real mvps!   $1 per month helps!! :)  https://www.patreon.com/patrickjmt !! Please consider supporting PJMT on Patreon! https://www.patreon.com/patrickjmt?ty=c\nBasic Product Rule Example #1", "lengthSeconds": 203, "width": 480, "height": 360}
{"video_id": "2e9pcjD0tzU", "transcript": [{"start": 1.48, "duration": 5.48, "text": "okay in this example we're going to"}, {"start": 2.879, "duration": 6.601, "text": "graph the hyperbola x^2 over 2 = 1 + y^2"}, {"start": 6.96, "duration": 4.36, "text": "over 2 and what I'm going to do is sort"}, {"start": 9.48, "duration": 4.96, "text": "of write this in a you know a more"}, {"start": 11.32, "duration": 6.039, "text": "standard form so the first thing I would"}, {"start": 14.44, "duration": 5.32, "text": "do is you know I'm just going to"}, {"start": 17.359, "duration": 4.881, "text": "multiply both sides by two uh just to"}, {"start": 19.76, "duration": 4.599, "text": "get rid of those denominators to me it"}, {"start": 22.24, "duration": 4.359, "text": "just kind of is something to uh kind of"}, {"start": 24.359, "duration": 4.68, "text": "clutter up the problem and just uh just"}, {"start": 26.599, "duration": 4.561, "text": "maybe confuse us a little bit so uh the"}, {"start": 29.039, "duration": 3.52, "text": "twos will cancel out leave us with X2 on"}, {"start": 31.16, "duration": 4.919, "text": "the left same thing on the right the"}, {"start": 32.559, "duration": 5.801, "text": "twos will cancel out we'll have 1 + y^2"}, {"start": 36.079, "duration": 7.841, "text": "and now if we subtract y^2 from both"}, {"start": 38.36, "duration": 10.6, "text": "sides we'll get x^2 - y^2 = 1 and we"}, {"start": 43.92, "duration": 8.92, "text": "could even write this as x^2 over 1 -"}, {"start": 48.96, "duration": 6.599, "text": "y^2 over 1 ="}, {"start": 52.84, "duration": 5.6, "text": "1 and well you know really we could even"}, {"start": 55.559, "duration": 6.281, "text": "write one as one squared and again one"}, {"start": 58.44, "duration": 8.88, "text": "is one squared so now we have it in this"}, {"start": 61.84, "duration": 9.44, "text": "form you know x^2 over a 2 - y^2 over b^"}, {"start": 67.32, "duration": 5.92, "text": "2al one and to me you can already start"}, {"start": 71.28, "duration": 4.28, "text": "reading off you know a lot of uh a lot"}, {"start": 73.24, "duration": 3.96, "text": "of information about this so for one"}, {"start": 75.56, "duration": 2.599, "text": "thing this is going to be centered this"}, {"start": 77.2, "duration": 3.959, "text": "is going to"}, {"start": 78.159, "duration": 6.28, "text": "be centered at the"}, {"start": 81.159, "duration": 6.441, "text": "origin okay since it's of the form X2"}, {"start": 84.439, "duration": 4.761, "text": "over a number minus y^2 over a number"}, {"start": 87.6, "duration": 5.04, "text": "that tells me that the parabola or"}, {"start": 89.2, "duration": 6.199, "text": "excuse me the hyperbola the hyperbola is"}, {"start": 92.64, "duration": 7.119, "text": "going to open you know to the left and"}, {"start": 95.399, "duration": 8.161, "text": "to the right okay so um so we know that"}, {"start": 99.759, "duration": 6.121, "text": "much of it as well um the value"}, {"start": 103.56, "duration": 5.44, "text": "underneath x s since it's 1 squ that"}, {"start": 105.88, "duration": 5.559, "text": "tells me that a equals 1 and again a"}, {"start": 109.0, "duration": 5.24, "text": "equals 1 that's simply the distance from"}, {"start": 111.439, "duration": 5.0, "text": "wherever it's centered out to one of our"}, {"start": 114.24, "duration": 5.28, "text": "vertices so that tells me that the"}, {"start": 116.439, "duration": 8.761, "text": "vertices are in fact going to be at 1"}, {"start": 119.52, "duration": 5.68, "text": "comma 0 and also at 1 comma"}, {"start": 127.28, "duration": 5.599, "text": "0 okay so uh we've got that much and I"}, {"start": 131.039, "duration": 4.081, "text": "guess the last little bit that we could"}, {"start": 132.879, "duration": 3.881, "text": "use here maybe to help sketch our graph"}, {"start": 135.12, "duration": 6.08, "text": "we know that the ASM tootes are going to"}, {"start": 136.76, "duration": 9.119, "text": "be of the form y = positive negative A"}, {"start": 141.2, "duration": 6.399, "text": "over B * X well again a is equal to one"}, {"start": 145.879, "duration": 4.801, "text": "it's the number that's being squared"}, {"start": 147.599, "duration": 5.481, "text": "underneath X and B is also going to be"}, {"start": 150.68, "duration": 4.16, "text": "equal to one again it's the you know the"}, {"start": 153.08, "duration": 3.799, "text": "number that's being squared underneath"}, {"start": 154.84, "duration": 4.56, "text": "the uh the Y squ"}, {"start": 156.879, "duration": 5.801, "text": "term so our asmp tootes will be of the"}, {"start": 159.4, "duration": 6.8, "text": "form positive negative 1 over"}, {"start": 162.68, "duration": 8.12, "text": "1 x which will just be you know positive"}, {"start": 166.2, "duration": 6.16, "text": "and negative 1X so yal positive X will"}, {"start": 170.8, "duration": 5.6, "text": "be one of our ASM"}, {"start": 172.36, "duration": 9.36, "text": "tootes so there's y = postive x and then"}, {"start": 176.4, "duration": 7.119, "text": "y = x will be one of our other asmt"}, {"start": 181.72, "duration": 4.2, "text": "so of course we should make the graph"}, {"start": 183.519, "duration": 5.36, "text": "kind of uh you know start to approach"}, {"start": 185.92, "duration": 5.92, "text": "those ASM tootes but that would be a"}, {"start": 188.879, "duration": 6.44, "text": "very you know a very rough sketch now of"}, {"start": 191.84, "duration": 5.36, "text": "the of the hyperbola okay it's just uh"}, {"start": 195.319, "duration": 4.48, "text": "kind of getting it you know in that"}, {"start": 197.2, "duration": 4.64, "text": "certain recognizable form again that"}, {"start": 199.799, "duration": 4.961, "text": "tells me if it opens left and right or"}, {"start": 201.84, "duration": 5.28, "text": "up and down um once I figure out my A"}, {"start": 204.76, "duration": 4.479, "text": "and B I can get my ASM tootes I can get"}, {"start": 207.12, "duration": 3.08, "text": "my vertices and again I think that's"}, {"start": 209.239, "duration": 5.521, "text": "enough to make"}, {"start": 210.2, "duration": 4.56, "text": "a good decent rough sketch"}], "title": "Conic Sections, Hyperbola : Graphing a Hyperbola", "description": "Thanks to all of you who support me on Patreon. You da real mvps!   $1 per month helps!! :)  https://www.patreon.com/patrickjmt !! Conic Sections, Hyperbola : Graphing a Hyperbola.\nIn this example, we are given the equation of a hyperbola; I show how to do a rough graph.", "lengthSeconds": 213, "width": 480, "height": 360}
{"video_id": "aP769v8PrZY", "transcript": [{"start": 0.84, "duration": 4.479, "text": "okay here we're going to look at finding"}, {"start": 2.6, "duration": 5.72, "text": "the volume of a cone so we're going to"}, {"start": 5.319, "duration": 6.561, "text": "find the volume of a right cone that has"}, {"start": 8.32, "duration": 7.56, "text": "a circular base with a radius of five"}, {"start": 11.88, "duration": 6.64, "text": "and a height of 15 and the units are in"}, {"start": 15.88, "duration": 5.36, "text": "centimet we're going to approximate Pi"}, {"start": 18.52, "duration": 2.72, "text": "by using"}, {"start": 21.76, "duration": 7.2, "text": "3.14 so the volume is 13 multipli the"}, {"start": 26.119, "duration": 4.721, "text": "base times the"}, {"start": 28.96, "duration": 4.279, "text": "height"}, {"start": 30.84, "duration": 5.16, "text": "well the base in this case is a circle"}, {"start": 33.239, "duration": 5.921, "text": "so recall to find the area of a circle"}, {"start": 36.0, "duration": 6.2, "text": "we use pi times the radius which in this"}, {"start": 39.16, "duration": 6.719, "text": "case is five squared and then we"}, {"start": 42.2, "duration": 7.28, "text": "multiply that by the height which is 15"}, {"start": 45.879, "duration": 8.32, "text": "okay so now I'm going to approximate"}, {"start": 49.48, "duration": 9.16, "text": "here I'm going to replace Pi with"}, {"start": 54.199, "duration": 7.441, "text": "3.14 well 5 SAR will be"}, {"start": 58.64, "duration": 9.88, "text": "25 and then we simply multiply that by"}, {"start": 61.64, "duration": 10.839, "text": "15 so let's see here um 1/3 of 15 so 1/3"}, {"start": 68.52, "duration": 6.72, "text": "of 15 would be five still multiplying by"}, {"start": 72.479, "duration": 6.761, "text": "3.14 *"}, {"start": 75.24, "duration": 6.32, "text": "25 5 * 25 would be"}, {"start": 79.24, "duration": 3.48, "text": "125 and then we still have to multiply"}, {"start": 81.56, "duration": 4.879, "text": "that by"}, {"start": 82.72, "duration": 5.6, "text": "3.14 so I'm going to use a calculator"}, {"start": 86.439, "duration": 4.64, "text": "here so"}, {"start": 88.32, "duration": 6.88, "text": "one so"}, {"start": 91.079, "duration": 4.121, "text": "25 multiplied"}, {"start": 97.24, "duration": 5.72, "text": "3.14 I'm getting this to be equal to"}, {"start": 113.24, "duration": 4.919, "text": "392.2c"}, {"start": 115.159, "duration": 3.0, "text": "cm"}], "title": "Volume of a Cone", "description": "Thanks to all of you who support me on Patreon. You da real mvps!   $1 per month helps!! :)  https://www.patreon.com/patrickjmt !! Volume of a Cone.  Here we look at an example of finding the volume of a cone.", "lengthSeconds": 116, "width": 480, "height": 360}
{"video_id": "iuaCbyoNq_Y", "transcript": [{"start": 0.599, "duration": 3.68, "text": "all right here we're going to do one"}, {"start": 1.88, "duration": 3.999, "text": "more limit problem um here we're going"}, {"start": 4.279, "duration": 3.841, "text": "to look at the limit as n goes to"}, {"start": 5.879, "duration": 3.88, "text": "Infinity of n raised to the negative n"}, {"start": 8.12, "duration": 3.72, "text": "to the thir"}, {"start": 9.759, "duration": 3.8, "text": "power the first thing I'm going to do is"}, {"start": 11.84, "duration": 5.84, "text": "just rewrite"}, {"start": 13.559, "duration": 6.401, "text": "this so n raised to negative n the 3"}, {"start": 17.68, "duration": 4.28, "text": "okay we can always put that over one"}, {"start": 19.96, "duration": 4.76, "text": "well we can make negative exponents"}, {"start": 21.96, "duration": 5.84, "text": "positive by sticking that exponent in"}, {"start": 24.72, "duration": 6.319, "text": "the denominator so now we have 1 over n"}, {"start": 27.8, "duration": 6.2, "text": "raised to positive n the 3"}, {"start": 31.039, "duration": 5.481, "text": "and you know if you think about it again"}, {"start": 34.0, "duration": 4.6, "text": "we're letting n Go to Infinity so"}, {"start": 36.52, "duration": 6.84, "text": "certainly for n values greater than or"}, {"start": 38.6, "duration": 8.0, "text": "equal to one I claim that well n raised"}, {"start": 43.36, "duration": 5.039, "text": "to n the 3 power that's always going to"}, {"start": 46.6, "duration": 5.36, "text": "be whoops I almost got my inequality"}, {"start": 48.399, "duration": 5.16, "text": "backwards n raised to n the 3 power"}, {"start": 51.96, "duration": 4.48, "text": "that's always going to be greater than"}, {"start": 53.559, "duration": 5.32, "text": "or equal to n the 1 I mean imagine if"}, {"start": 56.44, "duration": 4.759, "text": "you plug in one you'll have one to the 1"}, {"start": 58.879, "duration": 4.68, "text": "and on the left and one to the first on"}, {"start": 61.199, "duration": 3.96, "text": "the right if you plug in two for example"}, {"start": 63.559, "duration": 4.481, "text": "though you're going to have 2 raised to"}, {"start": 65.159, "duration": 5.401, "text": "the power of 2 Cub so 2 to the 8 is"}, {"start": 68.04, "duration": 5.0, "text": "certainly greater than or equal to two"}, {"start": 70.56, "duration": 5.48, "text": "again if you plug in three you know 3"}, {"start": 73.04, "duration": 5.36, "text": "raised to the 3 cubed or 3 raised to the"}, {"start": 76.04, "duration": 4.88, "text": "27 power certainly greater than or equal"}, {"start": 78.4, "duration": 3.92, "text": "to three so I think this is uh pretty"}, {"start": 80.92, "duration": 2.76, "text": "clear I don't think many people would"}, {"start": 82.32, "duration": 4.08, "text": "argue with"}, {"start": 83.68, "duration": 4.36, "text": "that well that tells us that if we take"}, {"start": 86.4, "duration": 3.84, "text": "one over this"}, {"start": 88.04, "duration": 5.079, "text": "number that's going to be less than or"}, {"start": 90.24, "duration": 4.8, "text": "equal to 1/ n again you know our"}, {"start": 93.119, "duration": 4.36, "text": "fraction now the fraction on the left"}, {"start": 95.04, "duration": 4.399, "text": "has a bigger denominator well if a"}, {"start": 97.479, "duration": 4.721, "text": "fraction has a bigger denominator and"}, {"start": 99.439, "duration": 6.761, "text": "the same uh numerator that fraction is"}, {"start": 102.2, "duration": 6.32, "text": "going to be smaller well uh again since"}, {"start": 106.2, "duration": 5.68, "text": "we're using positive numbers n is some"}, {"start": 108.52, "duration": 6.0, "text": "positive number likewise 1 over n ra to"}, {"start": 111.88, "duration": 4.68, "text": "n 3r that's always going to be greater"}, {"start": 114.52, "duration": 4.72, "text": "than or equal to"}, {"start": 116.56, "duration": 4.8, "text": "zero so now we can bring in our squeeze"}, {"start": 119.24, "duration": 5.72, "text": "theorem and we can simply look at the"}, {"start": 121.36, "duration": 6.079, "text": "limit as n goes to Infinity of the left"}, {"start": 124.96, "duration": 5.24, "text": "side which is zero well that limit"}, {"start": 127.439, "duration": 5.081, "text": "certainly is zero likewise if we look at"}, {"start": 130.2, "duration": 4.759, "text": "the limit as n goes to Infinity of the"}, {"start": 132.52, "duration": 4.799, "text": "right side well the limit as n goes to"}, {"start": 134.959, "duration": 5.081, "text": "Infinity of 1 over n that's also"}, {"start": 137.319, "duration": 6.721, "text": "definitely zero so what this now tells"}, {"start": 140.04, "duration": 4.0, "text": "us is that by the squeeze"}, {"start": 144.64, "duration": 5.56, "text": "theorem so by The Squeeze theorem the"}, {"start": 146.959, "duration": 7.601, "text": "limit as n goes to Infinity of 1 and"}, {"start": 150.2, "duration": 7.36, "text": "raed to n 3 must also equal"}, {"start": 154.56, "duration": 3.0, "text": "0"}], "title": "Squeeze / Absolute Value Theorem for Sequence : Ex 3", "description": "Thanks to all of you who support me on Patreon. You da real mvps!   $1 per month helps!! :)  https://www.patreon.com/patrickjmt !! Squeeze / Absolute Value Theorem for Sequence : Ex 3\nHere I look at an example where I use the squeeze and or absolute value theorem to find the limit of a sequence.", "lengthSeconds": 156, "width": 480, "height": 360}
{"video_id": "GjbRnAjVlXM", "transcript": [{"start": 1.0, "duration": 5.96, "text": "okay here we're going to solve the"}, {"start": 2.159, "duration": 8.761, "text": "system of equations -2x + y + z = 5 3 y"}, {"start": 6.96, "duration": 7.559, "text": "+ z = 13 and Z equals 4 so this one's"}, {"start": 10.92, "duration": 5.439, "text": "basically uh almost almost done for us"}, {"start": 14.519, "duration": 3.6, "text": "and the idea is eventually what we're"}, {"start": 16.359, "duration": 3.92, "text": "going to try to do is we're going to"}, {"start": 18.119, "duration": 4.721, "text": "make more complicated systems of"}, {"start": 20.279, "duration": 3.601, "text": "equations eventually resemble something"}, {"start": 22.84, "duration": 3.439, "text": "like"}, {"start": 23.88, "duration": 7.399, "text": "this well we're already told that Z"}, {"start": 26.279, "duration": 7.081, "text": "equals 4 So eventually our solution"}, {"start": 31.279, "duration": 4.561, "text": "in this"}, {"start": 33.36, "duration": 4.44, "text": "case it will be a point it will be an"}, {"start": 35.84, "duration": 5.039, "text": "ordered triple so we'll have an x a y"}, {"start": 37.8, "duration": 6.72, "text": "and a z value we already know that the Z"}, {"start": 40.879, "duration": 5.52, "text": "value is four well to get the Y value we"}, {"start": 44.52, "duration": 8.16, "text": "can just substitute that back into the"}, {"start": 46.399, "duration": 9.561, "text": "second equation so we have 3 y + z = 13"}, {"start": 52.68, "duration": 3.28, "text": "but again Z equal"}, {"start": 56.079, "duration": 6.241, "text": "four well if we subtract four from both"}, {"start": 58.84, "duration": 6.12, "text": "sides we'll get we'll have 13 minus 4 is"}, {"start": 62.32, "duration": 6.6, "text": "9 and then if we divide both sides by"}, {"start": 64.96, "duration": 6.36, "text": "three we'll have simply that y equal 3"}, {"start": 68.92, "duration": 5.72, "text": "well now we know the Y value and the Z"}, {"start": 71.32, "duration": 8.0, "text": "value so we can solve for the x value as"}, {"start": 74.64, "duration": 8.479, "text": "well we'll have -2x + Y which is"}, {"start": 79.32, "duration": 7.799, "text": "3 plus Z which is"}, {"start": 83.119, "duration": 8.801, "text": "4 that equals"}, {"start": 87.119, "duration": 4.801, "text": "-5 well 3 + 4 that's 7"}, {"start": 92.96, "duration": 6.159, "text": "if we subtract s from both sides we'll"}, {"start": 95.119, "duration": 7.36, "text": "have -5 - 7 which will be -12 and then"}, {"start": 99.119, "duration": 7.401, "text": "if we divide both sides by -2 we'll have"}, {"start": 102.479, "duration": 8.041, "text": "that xal 6 so our solution to our system"}, {"start": 106.52, "duration": 9.16, "text": "of equations is when the value of x is 6"}, {"start": 110.52, "duration": 5.16, "text": "Y is 3 and Z is four"}], "title": "System of 3 Equations, 3 Unknowns Using Substitution - Basic Example", "description": "Thanks to all of you who support me on Patreon. You da real mvps!   $1 per month helps!! :)  https://www.patreon.com/patrickjmt !! Solving a System of 3 Equations, 3 Unknowns Using Substitution - Basic Example.  In this example, most of the work is already done for us.  All we have to do is substitute in our known value to find the other values.  In general, we would like to take a system of equations and algebraically manipulate it so that we create a system as shown at the beginning of the video and proceed as in this example.", "lengthSeconds": 114, "width": 480, "height": 360}
{"video_id": "oIVpjrD4YvQ", "transcript": [{"start": 1.19, "duration": 5.35, "text": "okay in this video we're gonna start"}, {"start": 3.419, "duration": 5.76, "text": "looking at solving some multi-step"}, {"start": 6.54, "duration": 5.58, "text": "linear equations just where we're gonna"}, {"start": 9.179, "duration": 4.531, "text": "have to combine some like terms maybe"}, {"start": 12.12, "duration": 4.499, "text": "like in Part C get rid of some"}, {"start": 13.71, "duration": 7.11, "text": "parentheses by distributing so just uh"}, {"start": 16.619, "duration": 7.291, "text": "just problems that'll take a few more"}, {"start": 20.82, "duration": 6.57, "text": "operations to to to get to the solution"}, {"start": 23.91, "duration": 6.689, "text": "so Part A here we have 21 equals 3w"}, {"start": 27.39, "duration": 4.65, "text": "minus 12 plus 5w the first thing I'm"}, {"start": 30.599, "duration": 4.291, "text": "gonna do is I see that we have like"}, {"start": 32.04, "duration": 5.94, "text": "terms on the right side of my equation"}, {"start": 34.89, "duration": 5.79, "text": "the 21 I'll just drop that down but I"}, {"start": 37.98, "duration": 5.79, "text": "see a term involving W and another term"}, {"start": 40.68, "duration": 4.59, "text": "involving W so we can simplify those so"}, {"start": 43.77, "duration": 5.25, "text": "again I just kind of almost pretend like"}, {"start": 45.27, "duration": 8.82, "text": "the negative 12s not there well 3w plus"}, {"start": 49.02, "duration": 6.99, "text": "5w would be 8w well I do need to"}, {"start": 54.09, "duration": 4.92, "text": "remember that the negative 12 is there"}, {"start": 56.01, "duration": 7.139, "text": "so I just drop that down and again I'm"}, {"start": 59.01, "duration": 6.06, "text": "trying to get the W by itself so what"}, {"start": 63.149, "duration": 3.771, "text": "I'm gonna do is I'm gonna add 12 to both"}, {"start": 65.07, "duration": 7.22, "text": "sides"}, {"start": 66.92, "duration": 9.55, "text": "well 21 + 12 is gonna we'll get a 33"}, {"start": 72.29, "duration": 6.219, "text": "we're left with 8w on the right side and"}, {"start": 76.47, "duration": 4.56, "text": "again to solve for W since we're"}, {"start": 78.509, "duration": 5.671, "text": "multiplying by 8 I'm gonna divide both"}, {"start": 81.03, "duration": 7.86, "text": "sides by 8 so on the left side we're"}, {"start": 84.18, "duration": 7.079, "text": "left with 1 w 33 divided by 8 you know"}, {"start": 88.89, "duration": 4.92, "text": "that doesn't reduce as a fraction you"}, {"start": 91.259, "duration": 5.011, "text": "can always turn it into a decimal or a"}, {"start": 93.81, "duration": 4.559, "text": "mixed number but I'm gonna leave it"}, {"start": 96.27, "duration": 4.41, "text": "right there that's typically how I would"}, {"start": 98.369, "duration": 3.871, "text": "I think that's how you would typically"}, {"start": 100.68, "duration": 5.009, "text": "probably see the solution in any event"}, {"start": 102.24, "duration": 6.089, "text": "so doesn't have to be a nice whole"}, {"start": 105.689, "duration": 5.22, "text": "number obviously that happens but again"}, {"start": 108.329, "duration": 5.371, "text": "nothing too crazy just a matter of"}, {"start": 110.909, "duration": 6.181, "text": "combining your like terms getting the"}, {"start": 113.7, "duration": 5.099, "text": "term involving W by itself and then just"}, {"start": 117.09, "duration": 4.489, "text": "getting rid of the coefficient by"}, {"start": 118.799, "duration": 2.78, "text": "dividing"}], "title": "Solving Multi - Step Linear Equations", "description": "Thanks to all of you who support me on Patreon. You da real mvps!   $1 per month helps!! :)  https://www.patreon.com/patrickjmt !! Solving Multi - Step Linear Equations.  Here we solve some linear equations that require a few steps to solve.", "lengthSeconds": 120, "width": 480, "height": 360}
{"video_id": "IaSzgJnc5wg", "transcript": [{"start": 0.399, "duration": 2.921, "text": "all right in this video we're going to"}, {"start": 1.48, "duration": 4.68, "text": "do an example of actually just trying to"}, {"start": 3.32, "duration": 8.04, "text": "find an identity uh for this expression"}, {"start": 6.16, "duration": 6.96, "text": "Cent x * tangent xus cosine squ and uh"}, {"start": 11.36, "duration": 3.72, "text": "this one looks maybe a little tricky at"}, {"start": 13.12, "duration": 3.76, "text": "first I don't know it's definitely uh"}, {"start": 15.08, "duration": 4.0, "text": "turns out to be not too bad and again"}, {"start": 16.88, "duration": 4.84, "text": "the idea I'm going to use usually if I'm"}, {"start": 19.08, "duration": 5.64, "text": "not sure what to do um I don't see any"}, {"start": 21.72, "duration": 4.44, "text": "identities that are readily apparent um"}, {"start": 24.72, "duration": 5.319, "text": "I just turn everything into SS and"}, {"start": 26.16, "duration": 4.72, "text": "cosiness so Cent X that's cosine X over"}, {"start": 30.039, "duration": 7.121, "text": "sin"}, {"start": 30.88, "duration": 8.16, "text": "x tangent of X is just sin x over cine X"}, {"start": 37.16, "duration": 4.0, "text": "and then we still have our minus cosine"}, {"start": 39.04, "duration": 5.359, "text": "2 x left"}, {"start": 41.16, "duration": 5.2, "text": "over well we can simply just start"}, {"start": 44.399, "duration": 5.281, "text": "canceling out factors we've got a cosine"}, {"start": 46.36, "duration": 5.64, "text": "over cosine we've got s over s again you"}, {"start": 49.68, "duration": 4.16, "text": "can think about there's being uh ones on"}, {"start": 52.0, "duration": 4.28, "text": "the top and bottom again that wouldn't"}, {"start": 53.84, "duration": 7.6, "text": "change their value so we're really left"}, {"start": 56.28, "duration": 8.239, "text": "with 1 over one or 1 - cosine s X but"}, {"start": 61.44, "duration": 5.319, "text": "again recall our identity that sin^2 X"}, {"start": 64.519, "duration": 7.081, "text": "plus cosine 2"}, {"start": 66.759, "duration": 7.121, "text": "x equals 1 whoops uh so if we simply"}, {"start": 71.6, "duration": 7.6, "text": "subtract the cosine squ from both sides"}, {"start": 73.88, "duration": 6.599, "text": "we'll get that sin^2 xal 1 - cosine 2 x"}, {"start": 79.2, "duration": 5.64, "text": "so that's simply what I'm going to plug"}, {"start": 80.479, "duration": 6.401, "text": "in uh it says 1 - sin^2 is sin^2 X and"}, {"start": 84.84, "duration": 5.2, "text": "that would be a nice you know sort of"}, {"start": 86.88, "duration": 4.76, "text": "much more uh reduced certainly less"}, {"start": 90.04, "duration": 5.039, "text": "cluttered expression than what we"}, {"start": 91.64, "duration": 3.439, "text": "started with"}], "title": "Proving an Identity - Other Examples, Example 2", "description": "Thanks to all of you who support me on Patreon. You da real mvps!   $1 per month helps!! :)  https://www.patreon.com/patrickjmt !! Proving an Identity - Other Examples, Example 2", "lengthSeconds": 93, "width": 480, "height": 360}
{"video_id": "97RasvYIvl4", "transcript": [{"start": 2.399, "duration": 5.281, "text": "all right in this video we're going to"}, {"start": 4.12, "duration": 5.2, "text": "do an integration by parts problem and"}, {"start": 7.68, "duration": 5.48, "text": "again recall that the integration by"}, {"start": 9.32, "duration": 7.76, "text": "parts formula says if you integrate udv"}, {"start": 13.16, "duration": 7.08, "text": "that equals UV minus the integral of vdu"}, {"start": 17.08, "duration": 6.56, "text": "and we'll talk about exactly what that"}, {"start": 20.24, "duration": 7.039, "text": "means so the idea is you're starting"}, {"start": 23.64, "duration": 5.84, "text": "with the left side of this integral"}, {"start": 27.279, "duration": 5.6, "text": "we're going to label something to be U"}, {"start": 29.48, "duration": 6.84, "text": "and we're going to label something to be"}, {"start": 32.879, "duration": 8.081, "text": "DV so in this problem I see two things I"}, {"start": 36.32, "duration": 8.079, "text": "see an X and then I see the e to the X"}, {"start": 40.96, "duration": 6.56, "text": "and we'll attach the DX to one of those"}, {"start": 44.399, "duration": 7.521, "text": "so I have to pick one of those to be U I"}, {"start": 47.52, "duration": 4.4, "text": "have to pick one of those to be"}, {"start": 52.0, "duration": 5.6, "text": "DV and as a general rule of thumb you'll"}, {"start": 55.079, "duration": 4.32, "text": "pick U to be the thing that gets more"}, {"start": 57.6, "duration": 5.52, "text": "simple when you take the derivative of"}, {"start": 59.399, "duration": 7.76, "text": "it so so if I let U equal x the"}, {"start": 63.12, "duration": 6.56, "text": "derivative of that will be one if I let"}, {"start": 67.159, "duration": 4.361, "text": "U be e to the X the derivative of that"}, {"start": 69.68, "duration": 5.16, "text": "is just e to the X which really doesn't"}, {"start": 71.52, "duration": 6.88, "text": "get any better so I'm going to let U"}, {"start": 74.84, "duration": 7.919, "text": "equal x and then from that I'll get that"}, {"start": 78.4, "duration": 8.44, "text": "du equal 1"}, {"start": 82.759, "duration": 8.801, "text": "DX okay I'm going to pick DV to be well"}, {"start": 86.84, "duration": 8.76, "text": "the rest of the problem e to the x"}, {"start": 91.56, "duration": 6.12, "text": "DX and to get back to PL old V well"}, {"start": 95.6, "duration": 4.4, "text": "since I have the derivative of V to get"}, {"start": 97.68, "duration": 4.399, "text": "back to V I would have to find the"}, {"start": 100.0, "duration": 5.64, "text": "anti-derivative but the anti-derivative"}, {"start": 102.079, "duration": 5.481, "text": "of e to the x is just e to the X and on"}, {"start": 105.64, "duration": 3.92, "text": "these problems you don't have to worry"}, {"start": 107.56, "duration": 5.839, "text": "about the plus C on this"}, {"start": 109.56, "duration": 7.919, "text": "part okay so I've now picked"}, {"start": 113.399, "duration": 7.241, "text": "U I've now picked DV in my problem so"}, {"start": 117.479, "duration": 4.32, "text": "the idea is I've used the left side of"}, {"start": 120.64, "duration": 4.64, "text": "this"}, {"start": 121.799, "duration": 6.561, "text": "equation I've rewrote this as udv and"}, {"start": 125.28, "duration": 5.759, "text": "now I'm going to rewrite it using the"}, {"start": 128.36, "duration": 6.239, "text": "right side of the equation so it says"}, {"start": 131.039, "duration": 6.321, "text": "you take whatever U is in this case X"}, {"start": 134.599, "duration": 6.441, "text": "you multiply that by V which in this"}, {"start": 137.36, "duration": 7.92, "text": "case is e to the X and then it says from"}, {"start": 141.04, "duration": 6.12, "text": "that we subtract away the integral of V"}, {"start": 145.28, "duration": 9.84, "text": "okay that's e to the"}, {"start": 147.16, "duration": 11.68, "text": "X time du and again du in this case is 1"}, {"start": 155.12, "duration": 6.199, "text": "DX so the idea in general is hopefully"}, {"start": 158.84, "duration": 5.16, "text": "you've went you've basically turned one"}, {"start": 161.319, "duration": 5.961, "text": "integration problem into a new"}, {"start": 164.0, "duration": 5.76, "text": "integration problem and"}, {"start": 167.28, "duration": 4.92, "text": "notice part of the problem has no"}, {"start": 169.76, "duration": 5.0, "text": "integral in it at all so you're done"}, {"start": 172.2, "duration": 4.48, "text": "with that part of it hopefully the new"}, {"start": 174.76, "duration": 3.839, "text": "thing that you have left that you still"}, {"start": 176.68, "duration": 3.52, "text": "have to integrate has somehow gotten"}, {"start": 178.599, "duration": 3.56, "text": "better than what you originally L"}, {"start": 180.2, "duration": 4.8, "text": "started with if you're getting something"}, {"start": 182.159, "duration": 5.321, "text": "more complicated chances are you picked"}, {"start": 185.0, "duration": 4.879, "text": "the wrong U and the wrong"}, {"start": 187.48, "duration": 5.96, "text": "DV but in this case I can certainly"}, {"start": 189.879, "duration": 6.481, "text": "integrate e to the X so my x e to the X"}, {"start": 193.44, "duration": 5.48, "text": "from the first part just drops right"}, {"start": 196.36, "duration": 4.599, "text": "down minus and now I find the"}, {"start": 198.92, "duration": 5.36, "text": "anti-derivative of e to the X which is"}, {"start": 200.959, "duration": 6.84, "text": "just e to the X now that I've completed"}, {"start": 204.28, "duration": 6.679, "text": "all the anti-derivatives I tag on my"}, {"start": 207.799, "duration": 5.241, "text": "plus C and that's it I'm done I found"}, {"start": 210.959, "duration": 5.081, "text": "the"}, {"start": 213.04, "duration": 3.0, "text": "anti-derivative"}], "title": "\u2756 Integration by Parts - Integrate: xe^x \u2756", "description": "\ud83d\udcda Master Integration by Parts with a Classic Example! \ud83d\udcda\n\nIn this video, we delve into the powerful technique of Integration by Parts in calculus. Join me as I demonstrate how to effectively integrate the function xe^x, a classic example that showcases the method's utility.\n\nWhat You\u2019ll Learn:\n\nUnderstanding Integration by Parts: A brief overview of the formula and its applications in calculus.\nStep-by-Step Example: Follow along as I walk you through the integration process for xe^x, breaking it down into manageable steps.\n\nTips and Tricks: Discover useful strategies for selecting  u and dv to simplify your integration tasks.\n\nWhy Watch This Video?\n\nIdeal for Students: Perfect for high school and college students studying calculus.\nClear Explanations: Easy-to-follow instructions that demystify a key integration technique.\nEnhance Your Calculus Skills: Build confidence in your ability to tackle a variety of integration problems.\n\n\ud83d\udcc8 Don\u2019t Forget to:\n\nLIKE this video if you find it helpful!\nSHARE with classmates or friends looking to improve their calculus skills!\nSUBSCRIBE for more math tutorials, calculus insights, and educational content!", "lengthSeconds": 214, "width": 480, "height": 360}
{"video_id": "iFxV5UZxCjQ", "transcript": [{"start": 0.24, "duration": 3.06, "text": "all right in this video we're going to"}, {"start": 1.949, "duration": 3.781, "text": "look at an example of finding the"}, {"start": 3.3, "duration": 3.42, "text": "interval of convergence so here we're"}, {"start": 5.73, "duration": 4.829, "text": "going to find the interval of"}, {"start": 6.72, "duration": 5.549, "text": "convergence for e to the x squared the"}, {"start": 10.559, "duration": 3.511, "text": "series representation for e to the x"}, {"start": 12.269, "duration": 7.291, "text": "squared which is N equals 0 to infinity"}, {"start": 14.07, "duration": 7.92, "text": "X to the 2n over N factorial what we're"}, {"start": 19.56, "duration": 3.09, "text": "going to do to find this interval of"}, {"start": 21.99, "duration": 1.92, "text": "convergence"}, {"start": 22.65, "duration": 6.629, "text": "is we're just going to use the ratio"}, {"start": 23.91, "duration": 7.32, "text": "test so the ratio test against as we"}, {"start": 29.279, "duration": 9.03, "text": "look at the limit as n goes to infinity"}, {"start": 31.23, "duration": 8.7, "text": "of a sub n plus 1 over a sub n so in"}, {"start": 38.309, "duration": 3.721, "text": "this case we've got the limit as n goes"}, {"start": 39.93, "duration": 5.879, "text": "to infinity so everywhere there's an N"}, {"start": 42.03, "duration": 7.32, "text": "I'm going to replace it with n plus 1 so"}, {"start": 45.809, "duration": 10.171, "text": "we'll have X raised to the second times"}, {"start": 49.35, "duration": 8.939, "text": "n plus 1 over n plus 1 factorial and"}, {"start": 55.98, "duration": 4.32, "text": "again normally you well you know we"}, {"start": 58.289, "duration": 4.651, "text": "divide by the original but again we can"}, {"start": 60.3, "duration": 6.74, "text": "just multiply by the reciprocal so we'll"}, {"start": 62.94, "duration": 6.99, "text": "multiply by n factorial over X to the 2n"}, {"start": 67.04, "duration": 6.189, "text": "so let's see I'm going to just clean"}, {"start": 69.93, "duration": 7.049, "text": "this up a little bit this would be X to"}, {"start": 73.229, "duration": 6.331, "text": "the 2n plus 2 I'm going to put the X to"}, {"start": 76.979, "duration": 8.791, "text": "the 2n underneath that then we've got n"}, {"start": 79.56, "duration": 8.309, "text": "factorial over n plus 1 factorial well"}, {"start": 85.77, "duration": 4.83, "text": "if we simplify we have the limit as n"}, {"start": 87.869, "duration": 4.531, "text": "goes to infinity X to the 2n plus 2 over"}, {"start": 90.6, "duration": 3.479, "text": "X to the 2n we're just subtracting"}, {"start": 92.4, "duration": 4.77, "text": "exponents that will leave us with x"}, {"start": 94.079, "duration": 5.781, "text": "squared recall you can just expand out"}, {"start": 97.17, "duration": 5.159, "text": "in factorial over n plus 1 factorial"}, {"start": 99.86, "duration": 7.869, "text": "that's going to leave us with 1 over n"}, {"start": 102.329, "duration": 7.5, "text": "plus 1 and in this case you can treat"}, {"start": 107.729, "duration": 6.121, "text": "the X like a constant so we can pull it"}, {"start": 109.829, "duration": 6.18, "text": "out technically you need to keep things"}, {"start": 113.85, "duration": 3.72, "text": "in absolute value here you're squaring"}, {"start": 116.009, "duration": 4.261, "text": "it so it really doesn't matter but just"}, {"start": 117.57, "duration": 4.71, "text": "as a force of habit I would say when you"}, {"start": 120.27, "duration": 5.07, "text": "bring constants out keep them an"}, {"start": 122.28, "duration": 5.67, "text": "absolute value so we've got the absolute"}, {"start": 125.34, "duration": 7.64, "text": "value of x squared times the limit as n"}, {"start": 127.95, "duration": 7.49, "text": "goes to infinity of 1 over n plus 1"}, {"start": 132.98, "duration": 5.819, "text": "well this is going to leave us with x"}, {"start": 135.44, "duration": 5.64, "text": "squared times well let's see the limit"}, {"start": 138.799, "duration": 3.931, "text": "as n goes to infinity of 1 over n plus 1"}, {"start": 141.08, "duration": 3.049, "text": "that's going to be 1 over a large number"}, {"start": 142.73, "duration": 7.819, "text": "which is 0"}, {"start": 144.129, "duration": 9.22, "text": "well and recall what we want so we want"}, {"start": 150.549, "duration": 5.951, "text": "you know we want our ratio to be less"}, {"start": 153.349, "duration": 6.901, "text": "than 1 remember for the ratio test for"}, {"start": 156.5, "duration": 5.97, "text": "the series to converge the ratio has to"}, {"start": 160.25, "duration": 4.409, "text": "be less than 1 so I'm kind of thinking"}, {"start": 162.47, "duration": 5.46, "text": "you know what values could we plug in"}, {"start": 164.659, "duration": 6.66, "text": "for X and multiply it by 0 and get"}, {"start": 167.93, "duration": 5.1, "text": "something less than 1 well any number"}, {"start": 171.319, "duration": 4.681, "text": "that you plug in and multiply it by 0"}, {"start": 173.03, "duration": 5.039, "text": "you'll get something less than 1 so what"}, {"start": 176.0, "duration": 5.819, "text": "that tells us is it says the interval of"}, {"start": 178.069, "duration": 5.73, "text": "convergence it says the interval of"}, {"start": 181.819, "duration": 5.73, "text": "convergence is just going to be all real"}, {"start": 183.799, "duration": 7.851, "text": "numbers so everything from negative"}, {"start": 187.549, "duration": 4.101, "text": "infinity up to positive infinity"}], "title": "Finding Interval of Convergence for a Given Power Series Representation", "description": "Thanks to all of you who support me on Patreon. You da real mvps!   $1 per month helps!! :)  https://www.patreon.com/patrickjmt !! Finding Interval of Convergence for a Given Power Series Representation. In this video, we find the interval of convergence for a given power series.", "lengthSeconds": 194, "width": 480, "height": 360}
{"video_id": "YuHJ_pO0Lko", "transcript": [{"start": 0.64, "duration": 3.76, "text": "all right in this video i want to do an"}, {"start": 2.56, "duration": 3.44, "text": "example of a word problem involving"}, {"start": 4.4, "duration": 3.359, "text": "consecutive integers"}, {"start": 6.0, "duration": 4.32, "text": "and here we want to find two consecutive"}, {"start": 7.759, "duration": 6.161, "text": "integers whose product is nine less than"}, {"start": 10.32, "duration": 3.6, "text": "the square of the larger"}, {"start": 14.719, "duration": 5.921, "text": "okay so again i'm gonna try to turn this"}, {"start": 18.72, "duration": 3.68, "text": "this restriction into an equation that"}, {"start": 20.64, "duration": 3.92, "text": "we can do some math on"}, {"start": 22.4, "duration": 3.92, "text": "so again my two consecutive integers"}, {"start": 24.56, "duration": 4.24, "text": "i'll call one of our integers"}, {"start": 26.32, "duration": 4.4, "text": "x and to get to an integer right after"}, {"start": 28.8, "duration": 3.599, "text": "that we would just add one to it"}, {"start": 30.72, "duration": 3.28, "text": "so those are certainly consecutive"}, {"start": 32.399, "duration": 4.961, "text": "integers again assuming"}, {"start": 34.0, "duration": 5.6, "text": "x is an integer okay now it says we"}, {"start": 37.36, "duration": 4.24, "text": "their product is nine less than the"}, {"start": 39.6, "duration": 4.479, "text": "square of the larger"}, {"start": 41.6, "duration": 3.04, "text": "so okay their product is simply going to"}, {"start": 44.079, "duration": 3.601, "text": "be"}, {"start": 44.64, "duration": 4.399, "text": "x times x plus 1."}, {"start": 47.68, "duration": 2.879, "text": "again make sure you put things in"}, {"start": 49.039, "duration": 3.04, "text": "parentheses because we're going to"}, {"start": 50.559, "duration": 5.441, "text": "distribute things out"}, {"start": 52.079, "duration": 6.401, "text": "so that's our product it says it's 9"}, {"start": 56.0, "duration": 4.559, "text": "less than the square of the larger so"}, {"start": 58.48, "duration": 3.599, "text": "this nine less i usually kind of do that"}, {"start": 60.559, "duration": 5.041, "text": "at the end so"}, {"start": 62.079, "duration": 6.641, "text": "whose product is okay"}, {"start": 65.6, "duration": 5.12, "text": "nine less than the square of the larger"}, {"start": 68.72, "duration": 3.52, "text": "well x plus one would be the bigger of"}, {"start": 70.72, "duration": 4.96, "text": "these two numbers"}, {"start": 72.24, "duration": 6.239, "text": "it's square would simply be x plus one"}, {"start": 75.68, "duration": 4.64, "text": "squared and now i think okay this nine"}, {"start": 78.479, "duration": 4.081, "text": "in here i'm either going to put a minus"}, {"start": 80.32, "duration": 3.119, "text": "nine or a plus nine on the left side to"}, {"start": 82.56, "duration": 4.559, "text": "make everything"}, {"start": 83.439, "duration": 4.241, "text": "work out okay so it says this product is"}, {"start": 87.119, "duration": 2.721, "text": "actually"}, {"start": 87.68, "duration": 3.2, "text": "nine smaller than what we have on the"}, {"start": 89.84, "duration": 4.319, "text": "right side"}, {"start": 90.88, "duration": 5.44, "text": "so to make it equal i would have to"}, {"start": 94.159, "duration": 5.521, "text": "add 9 to the left side and now"}, {"start": 96.32, "duration": 4.72, "text": "everything is balanced out"}, {"start": 99.68, "duration": 3.84, "text": "okay so again we've got a little bit of"}, {"start": 101.04, "duration": 4.88, "text": "algebra to do here now so"}, {"start": 103.52, "duration": 4.239, "text": "i'm going to distribute the x to my"}, {"start": 105.92, "duration": 2.08, "text": "first set of parentheses so we would get"}, {"start": 107.759, "duration": 3.36, "text": "x"}, {"start": 108.0, "duration": 6.24, "text": "times x which is x squared x"}, {"start": 111.119, "duration": 3.521, "text": "times positive 1 is positive 1x we have"}, {"start": 114.24, "duration": 3.519, "text": "our"}, {"start": 114.64, "duration": 3.92, "text": "plus 9 left over now on the right side"}, {"start": 117.759, "duration": 3.841, "text": "be careful"}, {"start": 118.56, "duration": 4.0, "text": "because again x plus 1 squared is really"}, {"start": 121.6, "duration": 4.96, "text": "the same thing as"}, {"start": 122.56, "duration": 5.839, "text": "x plus 1 times x plus 1. so we're going"}, {"start": 126.56, "duration": 2.64, "text": "to distribute that out don't you know"}, {"start": 128.399, "duration": 3.281, "text": "you don't want to put"}, {"start": 129.2, "duration": 4.24, "text": "x squared plus 1 down here that would be"}, {"start": 131.68, "duration": 4.16, "text": "incorrect so"}, {"start": 133.44, "duration": 3.28, "text": "but if we multiply it out x times x"}, {"start": 135.84, "duration": 3.92, "text": "would be well"}, {"start": 136.72, "duration": 5.2, "text": "x squared x times positive 1 would be"}, {"start": 139.76, "duration": 5.76, "text": "positive 1x"}, {"start": 141.92, "duration": 5.84, "text": "1 times x again would be a positive 1x"}, {"start": 145.52, "duration": 5.28, "text": "and then positive 1 times positive 1"}, {"start": 147.76, "duration": 6.32, "text": "will give us a positive 1."}, {"start": 150.8, "duration": 5.519, "text": "so now we're left with x squared plus x"}, {"start": 154.08, "duration": 3.2, "text": "plus 9 on the left not doing anything"}, {"start": 156.319, "duration": 3.28, "text": "there"}, {"start": 157.28, "duration": 4.16, "text": "on the right i can collect my like terms"}, {"start": 159.599, "duration": 4.081, "text": "we've got x squared"}, {"start": 161.44, "duration": 4.879, "text": "we've got a plus x and a plus x which is"}, {"start": 163.68, "duration": 5.36, "text": "positive 2x"}, {"start": 166.319, "duration": 3.28, "text": "plus 1 and again i'm trying to solve for"}, {"start": 169.04, "duration": 2.96, "text": "x"}, {"start": 169.599, "duration": 4.081, "text": "so the first thing i'm going to do is"}, {"start": 172.0, "duration": 3.68, "text": "i'm kind of recognizing"}, {"start": 173.68, "duration": 3.199, "text": "that i could subtract x squared from"}, {"start": 175.68, "duration": 3.199, "text": "both sides"}, {"start": 176.879, "duration": 3.041, "text": "and that would just cancel out my x"}, {"start": 178.879, "duration": 5.28, "text": "squared"}, {"start": 179.92, "duration": 7.84, "text": "then i would be left with x plus 9."}, {"start": 184.159, "duration": 5.521, "text": "equals two x plus one"}, {"start": 187.76, "duration": 4.32, "text": "i think i'm going to subtract x from"}, {"start": 189.68, "duration": 4.0, "text": "both sides"}, {"start": 192.08, "duration": 3.84, "text": "so on the left side i still have"}, {"start": 193.68, "duration": 3.12, "text": "positive nine on the right side i have"}, {"start": 195.92, "duration": 4.08, "text": "two x minus"}, {"start": 196.8, "duration": 3.2, "text": "x which is 1x"}, {"start": 200.319, "duration": 4.56, "text": "plus 1. and again now to simply get x by"}, {"start": 203.28, "duration": 5.039, "text": "itself we'll subtract 1"}, {"start": 204.879, "duration": 5.521, "text": "subtract 1. so it says the smaller of"}, {"start": 208.319, "duration": 3.121, "text": "the two integers is going to be equal to"}, {"start": 210.4, "duration": 3.52, "text": "8"}, {"start": 211.44, "duration": 3.04, "text": "and again if we add one to get to the"}, {"start": 213.92, "duration": 3.92, "text": "other one"}, {"start": 214.48, "duration": 3.36, "text": "it says the other integer"}, {"start": 219.2, "duration": 3.039, "text": "it says the other integer would have to"}, {"start": 220.799, "duration": 4.241, "text": "be the number nine"}, {"start": 222.239, "duration": 3.36, "text": "so it says our two consecutive integers"}, {"start": 225.04, "duration": 2.479, "text": "that"}, {"start": 225.599, "duration": 3.041, "text": "that fulfill this requirement are the"}, {"start": 227.519, "duration": 4.64, "text": "integers 8"}, {"start": 228.64, "duration": 3.519, "text": "and 9."}], "title": "Word Problem: Finding Consecutive Numbers That Satisfy a Given Requirement - Ex 3", "description": "Thanks to all of you who support me on Patreon. You da real mvps!   $1 per month helps!! :)  https://www.patreon.com/patrickjmt !! Word Problem: Finding Consecutive Numbers That Satisfy a Given Requirement - Ex 3", "lengthSeconds": 230, "width": 480, "height": 360}
{"video_id": "gjI2QEWUymo", "transcript": [{"start": 2.679, "duration": 4.16, "text": "in this video I'm going to do a very"}, {"start": 5.12, "duration": 5.12, "text": "simple definite"}, {"start": 6.839, "duration": 5.521, "text": "integral and a definite integral is"}, {"start": 10.24, "duration": 4.2, "text": "different than an indefinite integral"}, {"start": 12.36, "duration": 4.04, "text": "first off notice we have these numbers"}, {"start": 14.44, "duration": 5.24, "text": "in the top and bottom of our integral"}, {"start": 16.4, "duration": 6.68, "text": "sign these numbers are aptly named the"}, {"start": 19.68, "duration": 5.88, "text": "bottom one is called the lower limit of"}, {"start": 23.08, "duration": 6.68, "text": "integration lower"}, {"start": 25.56, "duration": 4.2, "text": "limit of integration"}, {"start": 31.48, "duration": 5.28, "text": "and the top one as you might be able to"}, {"start": 33.559, "duration": 3.201, "text": "guess is called the upper"}, {"start": 39.12, "duration": 5.759, "text": "limit of integration so just some"}, {"start": 42.039, "duration": 2.84, "text": "terminology"}, {"start": 46.36, "duration": 5.6, "text": "there and basically what you do in these"}, {"start": 49.879, "duration": 4.881, "text": "problems is you find the"}, {"start": 51.96, "duration": 5.52, "text": "anti-derivative normally it without the"}, {"start": 54.76, "duration": 5.36, "text": "numbers you're finding the most General"}, {"start": 57.48, "duration": 5.559, "text": "anti-derivative a definite integral is"}, {"start": 60.12, "duration": 6.92, "text": "going to turn into a number so long as"}, {"start": 63.039, "duration": 6.72, "text": "this function is continuous over this"}, {"start": 67.04, "duration": 5.56, "text": "interval but basically you just find an"}, {"start": 69.759, "duration": 7.801, "text": "anti-derivative just like normal so the"}, {"start": 72.6, "duration": 8.6, "text": "anti-derivative of x^2 is X cubed /"}, {"start": 77.56, "duration": 5.32, "text": "3 the anti-derivative of x to the 1st"}, {"start": 81.2, "duration": 3.72, "text": "will get"}, {"start": 82.88, "duration": 5.72, "text": "x^2 over"}, {"start": 84.92, "duration": 5.8, "text": "two the anti-derivative of -2 will be"}, {"start": 88.6, "duration": 4.159, "text": "-2X"}, {"start": 90.72, "duration": 4.12, "text": "and in these problems we'll leave off"}, {"start": 92.759, "duration": 3.601, "text": "the plus C and we'll talk about why in"}, {"start": 94.84, "duration": 5.48, "text": "just a"}, {"start": 96.36, "duration": 7.24, "text": "second now we make a little bar over"}, {"start": 100.32, "duration": 5.56, "text": "here four and three are upper and lower"}, {"start": 103.6, "duration": 5.799, "text": "limits of integration and this is just"}, {"start": 105.88, "duration": 5.96, "text": "reminding us now to plug them in so the"}, {"start": 109.399, "duration": 4.481, "text": "first thing you do is everywhere you"}, {"start": 111.84, "duration": 2.959, "text": "have a variable you take your upper"}, {"start": 113.88, "duration": 4.279, "text": "limit of"}, {"start": 114.799, "duration": 7.561, "text": "integration and you plug it in so I plug"}, {"start": 118.159, "duration": 5.521, "text": "it in here here and here all right well"}, {"start": 122.36, "duration": 4.039, "text": "let me do"}, {"start": 123.68, "duration": 4.639, "text": "that so if I substitute that in I'm"}, {"start": 126.399, "duration": 6.761, "text": "going to get 4"}, {"start": 128.319, "duration": 10.801, "text": "cubed over 3 + 4^"}, {"start": 133.16, "duration": 5.96, "text": "2/ 2 - 2 *"}, {"start": 139.2, "duration": 7.52, "text": "2 and then what we do is we subtract"}, {"start": 142.879, "duration": 6.121, "text": "off when we plug in the lower limit and"}, {"start": 146.72, "duration": 4.08, "text": "just like we plugged the upper limit in"}, {"start": 149.0, "duration": 4.08, "text": "we're going to plug the lower limit in"}, {"start": 150.8, "duration": 4.2, "text": "everywhere as well it's a good idea to"}, {"start": 153.08, "duration": 3.799, "text": "use parentheses because you're going to"}, {"start": 155.0, "duration": 5.36, "text": "have to distribute the negative to every"}, {"start": 156.879, "duration": 5.36, "text": "term if you don't simplify first so I'll"}, {"start": 160.36, "duration": 4.76, "text": "get 3"}, {"start": 162.239, "duration": 7.201, "text": "cubed divided by 3 so now I'm plugging"}, {"start": 165.12, "duration": 6.88, "text": "threes in + 3^"}, {"start": 169.44, "duration": 5.439, "text": "2ar /"}, {"start": 172.0, "duration": 6.959, "text": "2 - 2"}, {"start": 174.879, "duration": 6.601, "text": "* 3 and I realize I just made a mistake"}, {"start": 178.959, "duration": 4.881, "text": "here I I plugged a two in here I should"}, {"start": 181.48, "duration": 3.839, "text": "be plugging a four in here so sorry"}, {"start": 183.84, "duration": 4.959, "text": "about that in case you're wondering what"}, {"start": 185.319, "duration": 7.2, "text": "the heck's going on absolutely right so"}, {"start": 188.799, "duration": 5.44, "text": "I've got 4 cubed 4^ 2ar 2 * 4 and then"}, {"start": 192.519, "duration": 3.64, "text": "I'm going to subtract off plugging a"}, {"start": 194.239, "duration": 4.401, "text": "three in"}, {"start": 196.159, "duration": 4.8, "text": "everywhere and we could simplify this"}, {"start": 198.64, "duration": 3.959, "text": "down you could definitely simplify"}, {"start": 200.959, "duration": 4.161, "text": "everything thing down in the parentheses"}, {"start": 202.599, "duration": 4.241, "text": "in the first part simplify everything"}, {"start": 205.12, "duration": 4.16, "text": "down in the parentheses in the second"}, {"start": 206.84, "duration": 3.52, "text": "part take that number minus the new"}, {"start": 209.28, "duration": 3.72, "text": "number"}, {"start": 210.36, "duration": 4.48, "text": "and that'll be your solution so this is"}, {"start": 213.0, "duration": 4.2, "text": "the basic idea with the definite"}, {"start": 214.84, "duration": 4.759, "text": "integral this number is not something"}, {"start": 217.2, "duration": 4.52, "text": "completely just out of the blue it does"}, {"start": 219.599, "duration": 3.521, "text": "have a geometric interpretation and"}, {"start": 221.72, "duration": 3.64, "text": "we'll talk about the geometric"}, {"start": 223.12, "duration": 5.64, "text": "interpretation of a definite integral in"}, {"start": 225.36, "duration": 3.4, "text": "another video"}], "title": "Basic Definite Integrals", "description": "Thanks to all of you who support me on Patreon. You da real mvps!   $1 per month helps!! :)  https://www.patreon.com/patrickjmt !! This video has a couple of examples of calculating relatively simple definite integrals.\r\n\r\nFor more FREE MATH VIDEOS visit:\r\n\r\nhttp://justmathtutoring.com\r\n\r\nand click on the 'Free Video Lessons' tab on the left!\n\nJust Math Tutoring", "lengthSeconds": 228, "width": 480, "height": 360}
{"video_id": "Z_wQvCyKjwE", "transcript": [{"start": 1.079, "duration": 4.28, "text": "all right this video is going to be the"}, {"start": 2.679, "duration": 4.801, "text": "first video um in kind of a series of"}, {"start": 5.359, "duration": 5.561, "text": "videos here talking about the lass"}, {"start": 7.48, "duration": 5.36, "text": "transform and the llas transform is used"}, {"start": 10.92, "duration": 5.56, "text": "to solve certain types of uh"}, {"start": 12.84, "duration": 4.68, "text": "differential equations so the basic idea"}, {"start": 16.48, "duration": 3.959, "text": "in this case we're not going to do"}, {"start": 17.52, "duration": 5.48, "text": "anything in this case um in this video"}, {"start": 20.439, "duration": 3.721, "text": "the basic idea is is as follows so what"}, {"start": 23.0, "duration": 3.039, "text": "we're going to do is we're simply going"}, {"start": 24.16, "duration": 3.76, "text": "to start with some you know a type of"}, {"start": 26.039, "duration": 3.801, "text": "differential equation what we're going"}, {"start": 27.92, "duration": 4.639, "text": "to do is we'll then use this notion of"}, {"start": 29.84, "duration": 4.76, "text": "what it's called the Lao transform okay"}, {"start": 32.559, "duration": 4.84, "text": "so the Lelo transform is just what's"}, {"start": 34.6, "duration": 5.4, "text": "known as an operator um it transforms"}, {"start": 37.399, "duration": 4.721, "text": "one function into a different type of"}, {"start": 40.0, "duration": 5.0, "text": "function okay so what we're going to end"}, {"start": 42.12, "duration": 5.52, "text": "up getting is a nice U algebraic"}, {"start": 45.0, "duration": 5.039, "text": "equation uh after we do our lass"}, {"start": 47.64, "duration": 5.88, "text": "transform what we'll then do is we solve"}, {"start": 50.039, "duration": 4.84, "text": "this algebraic equation and what that"}, {"start": 53.52, "duration": 4.8, "text": "solution is going to do it's going to"}, {"start": 54.879, "duration": 6.0, "text": "help us recover um information about our"}, {"start": 58.32, "duration": 4.68, "text": "original um about solution to the"}, {"start": 60.879, "duration": 5.24, "text": "original differential equation and once"}, {"start": 63.0, "duration": 4.84, "text": "we solve this algebraic equation uh we"}, {"start": 66.119, "duration": 3.841, "text": "use What's called the inverse Lao"}, {"start": 67.84, "duration": 4.8, "text": "transform and that's going to somehow"}, {"start": 69.96, "duration": 5.479, "text": "get us back to the solution to the uh"}, {"start": 72.64, "duration": 4.72, "text": "original differential equation so uh"}, {"start": 75.439, "duration": 4.481, "text": "obviously a lot of little uh things"}, {"start": 77.36, "duration": 4.0, "text": "involved in the process here so um I"}, {"start": 79.92, "duration": 2.6, "text": "don't know hopefully uh we can talk"}, {"start": 81.36, "duration": 3.52, "text": "about them all and try to make some"}, {"start": 82.52, "duration": 3.48, "text": "sense out of it so uh the next video"}, {"start": 84.88, "duration": 3.68, "text": "what I'm actually going to do is give"}, {"start": 86.0, "duration": 4.68, "text": "the definition of the Lelo transform and"}, {"start": 88.56, "duration": 5.96, "text": "actually talk about calculating the llao"}, {"start": 90.68, "duration": 3.84, "text": "transform for a function"}], "title": "The Laplace Transform - The Basic Idea of How We Use It", "description": "Thanks to all of you who support me on Patreon. You da real mvps!   $1 per month helps!! :)  https://www.patreon.com/patrickjmt !! The Laplace Transform - The Basic Idea of How We Use It.\r\nThis video is the first of (what will be) many videos about the Laplace transform.  In this video, I discuss the basic idea of how we will use the Laplace transform.  There are no computations or anything heavy in this video! Just an outline of 'where we are going'.", "lengthSeconds": 93, "width": 480, "height": 360}
{"video_id": "t6SPpIOP3ic", "transcript": [{"start": 0.44, "duration": 5.52, "text": "OK, so this will be the last problem from the first section of the SAT practice test."}, {"start": 5.96, "duration": 4.26, "text": "This is the last problem where you are not allowed to use a calculator."}, {"start": 10.22, "duration": 3.18, "text": "OK, here we are given that a equals five multiplied by the square root of 2"}, {"start": 13.4, "duration": 2.6, "text": "and two times a equals the square root of two times x."}, {"start": 16.0, "duration": 2.52, "text": "We want to know what's the value of x."}, {"start": 18.78, "duration": 3.14, "text": "So, this one is just a matter of doing a substitution."}, {"start": 22.12, "duration": 5.56, "text": "So, again we got two times a equals the square root of two times x."}, {"start": 27.88, "duration": 3.74, "text": "Well we were told that a equals five times the square root of two."}, {"start": 31.62, "duration": 4.58, "text": "So I'm going to replace the a with five times the square root of two."}, {"start": 36.62, "duration": 3.24, "text": "and I am going to simplify the left side so two times five is ten"}, {"start": 39.86, "duration": 4.74, "text": "multiplied by the square root of two, we have the square root of two x on the right side"}, {"start": 44.6, "duration": 4.9, "text": "I can break that up as square root of two multiplied by the square root of x"}, {"start": 50.1, "duration": 3.24, "text": "so again I am trying to solve for x"}, {"start": 54.2, "duration": 5.3, "text": "so the first thing I am going to do is now I am going to divide both sides by the square root of two"}, {"start": 59.94, "duration": 5.12, "text": "So, the square root of two will cancel and I am left with ten equals the square root of x"}, {"start": 65.22, "duration": 2.48, "text": "and now I can simply square both sides"}, {"start": 70.18, "duration": 4.08, "text": "so the square root of x squared is just going to leave me with  x"}, {"start": 74.26, "duration": 5.84, "text": "10 squared is 10 times 10 which will be one hundred and that will be our solution in this case."}, {"start": 80.22, "duration": 3.68, "text": "so not to bad, again, just a little bit of a substitution and"}, {"start": 84.06, "duration": 5.6, "text": "you know, just solving this basic equation that involves a square root."}], "title": "SAT Math - Question 20, 2016 Edition and Beyond", "description": "Thanks to all of you who support me on Patreon. You da real mvps!   $1 per month helps!! :)  https://www.patreon.com/patrickjmt !! SAT Math - Question 20, 2016 Edition and Beyond.\nSection 1 - No Calculator allowed!", "lengthSeconds": 91, "width": 480, "height": 360}
{"video_id": "02bjw4PUh88", "transcript": [{"start": 0.399, "duration": 3.24, "text": "all right in this video we're going to"}, {"start": 1.64, "duration": 3.52, "text": "look at another example involving trig"}, {"start": 3.639, "duration": 4.641, "text": "functions so here we're going to"}, {"start": 5.16, "duration": 7.399, "text": "integrate tangent cubed x * cosecant SAR"}, {"start": 8.28, "duration": 5.399, "text": "x minus one and you know to get started"}, {"start": 12.559, "duration": 2.721, "text": "on this one the first thing I would"}, {"start": 13.679, "duration": 4.36, "text": "think you know if I distribute it out is"}, {"start": 15.28, "duration": 5.079, "text": "that going to make life any easier um"}, {"start": 18.039, "duration": 3.561, "text": "and you know maybe we could there's you"}, {"start": 20.359, "duration": 3.201, "text": "know kind of different ways to get"}, {"start": 21.6, "duration": 4.72, "text": "started I guess but the main thing I"}, {"start": 23.56, "duration": 4.44, "text": "think about is I see this cosecant squar"}, {"start": 26.32, "duration": 4.64, "text": "and I remember there's an identity that"}, {"start": 28.0, "duration": 9.76, "text": "involves cosecant squ one of them recall"}, {"start": 30.96, "duration": 9.599, "text": "is that 1 +^ 2 X = cose 2"}, {"start": 37.76, "duration": 6.04, "text": "X well if we just subtract the one over"}, {"start": 40.559, "duration": 7.081, "text": "we'll getent 2 xal"}, {"start": 43.8, "duration": 8.559, "text": "cose cose 2 X - 1 so I'm just going to"}, {"start": 47.64, "duration": 7.32, "text": "replace the cose 2 X - 1 with Cent s so"}, {"start": 52.359, "duration": 6.561, "text": "really we would be left with Tangent"}, {"start": 54.96, "duration": 6.0, "text": "cubed times ENT"}, {"start": 58.92, "duration": 3.4, "text": "squared"}, {"start": 60.96, "duration": 3.919, "text": "but if you want to kind of break this"}, {"start": 62.32, "duration": 3.92, "text": "apart you know tangent is sin cubed over"}, {"start": 64.879, "duration": 5.56, "text": "cosine"}, {"start": 66.24, "duration": 9.36, "text": "cubed um Cent squ that would be cosine 2"}, {"start": 70.439, "duration": 8.201, "text": "X over sin^2 X and well we've got sin"}, {"start": 75.6, "duration": 5.28, "text": "cubed over sin squar that would leave us"}, {"start": 78.64, "duration": 4.92, "text": "with a sin x in the"}, {"start": 80.88, "duration": 5.52, "text": "numerator we would have cosine squar"}, {"start": 83.56, "duration": 6.919, "text": "over cosine cubed um that would give us"}, {"start": 86.4, "duration": 7.48, "text": "cosine X DX and this is the same thing"}, {"start": 90.479, "duration": 5.68, "text": "as the integral of well tangent of X so"}, {"start": 93.88, "duration": 5.04, "text": "haha um and we already saw how to do"}, {"start": 96.159, "duration": 5.161, "text": "that in the previous example so the"}, {"start": 98.92, "duration": 7.32, "text": "anti-derivative of tangent of X we get"}, {"start": 101.32, "duration": 7.079, "text": "the natural logarithm of see X plus C"}, {"start": 106.24, "duration": 4.4, "text": "and again we found our anti-derivative"}, {"start": 108.399, "duration": 5.161, "text": "so really uh you know just being a"}, {"start": 110.64, "duration": 4.64, "text": "little sneaky this original function is"}, {"start": 113.56, "duration": 6.879, "text": "just a you know an equivalent way of"}, {"start": 115.28, "duration": 5.159, "text": "writing the function tangent X"}], "title": "Integrating Trig Functions , More Examples #2", "description": "Thanks to all of you who support me on Patreon. You da real mvps!   $1 per month helps!! :)  https://www.patreon.com/patrickjmt !! Integrating Trig Functions , More Examples #2", "lengthSeconds": 119, "width": 480, "height": 360}
{"video_id": "8OPg4zTMeMo", "transcript": [{"start": 1.04, "duration": 4.799, "text": "okay here we're going to look at an"}, {"start": 2.08, "duration": 6.639, "text": "example of writing a fraction in uh the"}, {"start": 5.839, "duration": 5.281, "text": "simplest form or most reduced"}, {"start": 8.719, "duration": 4.84, "text": "form and being able to go from a"}, {"start": 11.12, "duration": 4.519, "text": "percentage to a fraction to a decimal is"}, {"start": 13.559, "duration": 5.361, "text": "certainly useful uh to be able to switch"}, {"start": 15.639, "duration": 6.8, "text": "interchangeably between those three so"}, {"start": 18.92, "duration": 5.8, "text": "to write a percent as a fraction we take"}, {"start": 22.439, "duration": 5.121, "text": "whatever the uh the number is in this"}, {"start": 24.72, "duration": 6.0, "text": "case it's 32 and we just simply put that"}, {"start": 27.56, "duration": 4.92, "text": "over 100 and that's always what we do"}, {"start": 30.72, "duration": 4.44, "text": "and at that point it's just a matter of"}, {"start": 32.48, "duration": 7.559, "text": "reducing it so let's see what number"}, {"start": 35.16, "duration": 7.8, "text": "goes into both 32 and 100 well it's even"}, {"start": 40.039, "duration": 6.121, "text": "so I know two does at least let's see 32"}, {"start": 42.96, "duration": 6.48, "text": "divided two would be 16 100 divided by"}, {"start": 46.16, "duration": 6.52, "text": "two would be 50 but those are both still"}, {"start": 49.44, "duration": 8.04, "text": "even so let's see uh 16 divided two"}, {"start": 52.68, "duration": 6.359, "text": "would be 8 50 ided two would be 25 so we"}, {"start": 57.48, "duration": 3.84, "text": "could have just divided top and bottom"}, {"start": 59.039, "duration": 4.561, "text": "by four at the very beginning but I"}, {"start": 61.32, "duration": 5.4, "text": "usually just always just start dividing"}, {"start": 63.6, "duration": 5.96, "text": "by two if I see they're even 8 over 25"}, {"start": 66.72, "duration": 5.32, "text": "well let's see one 2 four and eight go"}, {"start": 69.56, "duration": 5.32, "text": "into eight none of those numbers other"}, {"start": 72.04, "duration": 8.28, "text": "than one go into 25 so this would now be"}, {"start": 74.88, "duration": 5.44, "text": "our fraction in simplest form"}], "title": "Writing a Percent as a Fraction in Reduced Form", "description": "Thanks to all of you who support me on Patreon. You da real mvps!   $1 per month helps!! :)  https://www.patreon.com/patrickjmt !! Writing a Percent as a Fraction in Reduced Form.  Here we write 32% as a fraction in reduced form.", "lengthSeconds": 78, "width": 480, "height": 360}
{"video_id": "KCFl1OZj_48", "transcript": [{"start": 0.56, "duration": 3.52, "text": "all right in this video we're going to"}, {"start": 1.76, "duration": 4.559, "text": "do an application using the law of signs"}, {"start": 4.08, "duration": 4.92, "text": "just a little word problem so suppose"}, {"start": 6.319, "duration": 5.36, "text": "we've got a person um that H that has a"}, {"start": 9.0, "duration": 6.44, "text": "kite out on uh"}, {"start": 11.679, "duration": 5.6, "text": "1,750 ft of string and uh we've got an"}, {"start": 15.44, "duration": 4.4, "text": "observer who's watching the person with"}, {"start": 17.279, "duration": 5.281, "text": "the kite and the Observer notes that the"}, {"start": 19.84, "duration": 3.679, "text": "angle formed by the kite and the flyer"}, {"start": 22.56, "duration": 3.68, "text": "is"}, {"start": 23.519, "duration": 6.6, "text": "102\u00b0 we just want to figure out how far"}, {"start": 26.24, "duration": 5.999, "text": "is the Kite from The Observer so what we"}, {"start": 30.119, "duration": 4.841, "text": "really need to figure out is uh the"}, {"start": 32.239, "duration": 5.761, "text": "distance uh along the right side of our"}, {"start": 34.96, "duration": 5.96, "text": "triangle here so maybe I'll call that D"}, {"start": 38.0, "duration": 4.96, "text": "for distance well this is one where we"}, {"start": 40.92, "duration": 7.6, "text": "can simply use the law of"}, {"start": 42.96, "duration": 7.56, "text": "signs because we can say um so D over"}, {"start": 48.52, "duration": 5.359, "text": "s of"}, {"start": 50.52, "duration": 6.679, "text": "75\u00b0 the angle in the bottom left that's"}, {"start": 53.879, "duration": 6.32, "text": "going to equal um the distance the the"}, {"start": 57.199, "duration": 7.161, "text": "long distance the 1,750"}, {"start": 60.199, "duration": 6.521, "text": "ft over the S of"}, {"start": 64.36, "duration": 5.64, "text": "102\u00b0 and now it's just a simple little"}, {"start": 66.72, "duration": 5.8, "text": "equation uh that we have to solve so um"}, {"start": 70.0, "duration": 5.92, "text": "to do this what I would do again is just"}, {"start": 72.52, "duration": 6.76, "text": "multiply both sides by S of"}, {"start": 75.92, "duration": 6.0, "text": "75\u00b0 so s of"}, {"start": 79.28, "duration": 4.56, "text": "75\u00b0 on the left side uh we'll just be"}, {"start": 81.92, "duration": 4.76, "text": "left with our D the distance that we"}, {"start": 83.84, "duration": 4.279, "text": "want and now it's just simply uh again"}, {"start": 86.68, "duration": 3.88, "text": "just kind of plugging some stuff into a"}, {"start": 88.119, "duration": 5.441, "text": "calculator so 17"}, {"start": 90.56, "duration": 8.96, "text": "75 let's do s of"}, {"start": 93.56, "duration": 5.96, "text": "102 s of 102 I'm going to round that to"}, {"start": 100.0, "duration": 7.799, "text": "978 uh we'll do s of 75 Dees s of 75 I"}, {"start": 105.36, "duration": 6.799, "text": "get that to be"}, {"start": 107.799, "duration": 8.64, "text": "966 after rounding so let's see if we do"}, {"start": 112.159, "duration": 4.28, "text": "1750 multiply that by"}, {"start": 116.759, "duration": 8.761, "text": "966 um I'm getting that to be"}, {"start": 121.479, "duration": 9.001, "text": "69.5 we'll divide that by"}, {"start": 125.52, "duration": 9.76, "text": "978 so if I divide that by"}, {"start": 130.48, "duration": 4.8, "text": "978 um I'm getting that to equal"}, {"start": 135.36, "duration": 5.599, "text": "1,728 uh."}, {"start": 138.319, "duration": 5.56, "text": "53t um after rounding again just a"}, {"start": 140.959, "duration": 5.201, "text": "little bit so um certainly that makes"}, {"start": 143.879, "duration": 5.401, "text": "sense that the uh you know clearly The"}, {"start": 146.16, "duration": 5.32, "text": "Observer uh you know should be uh a"}, {"start": 149.28, "duration": 4.2, "text": "Little Closer closer to the kite than"}, {"start": 151.48, "duration": 4.2, "text": "the person who's flying the Kite at"}, {"start": 153.48, "duration": 4.72, "text": "least based on our picture so um"}, {"start": 155.68, "duration": 4.76, "text": "certainly The Observer is a little close"}, {"start": 158.2, "duration": 4.28, "text": "a little bit closer uh but not by too"}, {"start": 160.44, "duration": 4.48, "text": "much so again that's all there is to it"}, {"start": 162.48, "duration": 6.0, "text": "just a a nice little application using"}, {"start": 164.92, "duration": 3.56, "text": "the law of signs"}], "title": "Law of Sines - Application/Word Problem, Ex 1", "description": "Thanks to all of you who support me on Patreon. You da real mvps!   $1 per month helps!! :)  https://www.patreon.com/patrickjmt !! Law of Sines - Application/Word Problem, Ex 1.  In this video, we do a word/application problem that involves using the law of sines.", "lengthSeconds": 167, "width": 480, "height": 360}
{"video_id": "oYomNM91cms", "transcript": [{"start": 0.199, "duration": 3.121, "text": "all right here we're going to look at"}, {"start": 1.199, "duration": 5.041, "text": "some examples of ordering decimals from"}, {"start": 3.32, "duration": 4.0, "text": "smallest to largest um in part A we've"}, {"start": 6.24, "duration": 3.319, "text": "got"}, {"start": 7.32, "duration": 4.6, "text": "3149"}, {"start": 9.559, "duration": 6.601, "text": "3201 and"}, {"start": 11.92, "duration": 5.96, "text": "3.88 6 um so and I think you'll often"}, {"start": 16.16, "duration": 3.64, "text": "find that's how people actually say"}, {"start": 17.88, "duration": 3.76, "text": "decimals you know people don't typically"}, {"start": 19.8, "duration": 3.479, "text": "say we've got three"}, {"start": 21.64, "duration": 4.44, "text": "and"}, {"start": 23.279, "duration": 5.521, "text": "149,000 people will typically just say"}, {"start": 26.08, "duration": 5.199, "text": "3.49 because it's much easier to say"}, {"start": 28.8, "duration": 4.64, "text": "right so but anyway ways from least to"}, {"start": 31.279, "duration": 4.6, "text": "greatest so you know I look at the whole"}, {"start": 33.44, "duration": 5.6, "text": "number part well they've all got a three"}, {"start": 35.879, "duration": 5.441, "text": "okay so they all start with three uh no"}, {"start": 39.04, "duration": 5.72, "text": "problem there and then I just kind of"}, {"start": 41.32, "duration": 5.32, "text": "look at them you know sort of a uh I"}, {"start": 44.76, "duration": 3.2, "text": "just sort of look at one digit at a time"}, {"start": 46.64, "duration": 3.84, "text": "but I'm looking"}, {"start": 47.96, "duration": 4.68, "text": "for you know to see if there's any I'm"}, {"start": 50.48, "duration": 4.759, "text": "basically looking for the smallest"}, {"start": 52.64, "duration": 4.399, "text": "one okay so they've all got a three so"}, {"start": 55.239, "duration": 3.721, "text": "in that sense to me they're all equal"}, {"start": 57.039, "duration": 7.281, "text": "let's look at the next digit well the"}, {"start": 58.96, "duration": 6.839, "text": "next one uh um we've got oh 3.1 3.2 and"}, {"start": 64.32, "duration": 4.479, "text": "then we would have let's see if I can"}, {"start": 65.799, "duration": 6.36, "text": "make my paper here a little"}, {"start": 68.799, "duration": 6.761, "text": "smaller then we've got a 3.0 so I'm just"}, {"start": 72.159, "duration": 7.521, "text": "looking at all of these next digits so"}, {"start": 75.56, "duration": 6.8, "text": "one two um and a zero and I'm thinking"}, {"start": 79.68, "duration": 6.64, "text": "which of those digits is the smallest"}, {"start": 82.36, "duration": 6.399, "text": "well this is certainly the"}, {"start": 86.32, "duration": 4.28, "text": "smallest that's absolutely the smallest"}, {"start": 88.759, "duration": 4.601, "text": "one and then that would be be the second"}, {"start": 90.6, "duration": 4.96, "text": "smallest and that this uh 3.2 would be"}, {"start": 93.36, "duration": 4.36, "text": "the"}, {"start": 95.56, "duration": 6.519, "text": "largest so all right I've got them"}, {"start": 97.72, "duration": 7.16, "text": "listed now so um we would have"}, {"start": 102.079, "duration": 5.521, "text": "3.88 then"}, {"start": 104.88, "duration": 6.48, "text": "3149 and then"}, {"start": 107.6, "duration": 6.96, "text": "3.21 so let's do that uh with our"}, {"start": 111.36, "duration": 7.84, "text": "second second list here okay so I look"}, {"start": 114.56, "duration": 6.159, "text": "at the very first digit one one one one"}, {"start": 119.2, "duration": 3.559, "text": "those are all the same so again it"}, {"start": 120.719, "duration": 3.241, "text": "doesn't really tell me anything then I"}, {"start": 122.759, "duration": 3.161, "text": "look at the next"}, {"start": 123.96, "duration": 4.24, "text": "digit okay so I'm going to look at the"}, {"start": 125.92, "duration": 2.28, "text": "next"}, {"start": 129.08, "duration": 5.159, "text": "digit okay so my next digit this one has"}, {"start": 132.04, "duration": 5.559, "text": "a one this one has a one this one has a"}, {"start": 134.239, "duration": 5.36, "text": "four this one has a zero so zero is the"}, {"start": 137.599, "duration": 3.161, "text": "smallest of all of those so I know that"}, {"start": 139.599, "duration": 4.72, "text": "the"}, {"start": 140.76, "duration": 6.92, "text": "1.1 is going to be the very smallest"}, {"start": 144.319, "duration": 6.241, "text": "number uh the then we have ones next so"}, {"start": 147.68, "duration": 4.36, "text": "the 1. On's should come next I don't"}, {"start": 150.56, "duration": 3.759, "text": "really know which one of those comes"}, {"start": 152.04, "duration": 4.68, "text": "next yet but I know that the largest one"}, {"start": 154.319, "duration": 5.361, "text": "will be 1.4"}, {"start": 156.72, "duration": 5.08, "text": "31 so now if we move over one more digit"}, {"start": 159.68, "duration": 5.04, "text": "to kind of break the the the stal made"}, {"start": 161.8, "duration": 6.24, "text": "here again we've got a one this one is a"}, {"start": 164.72, "duration": 5.4, "text": "zero well zero is"}, {"start": 168.04, "duration": 4.76, "text": "smaller and now we've got everything"}, {"start": 170.12, "duration": 3.8, "text": "listed from smallest to greatest again"}, {"start": 172.8, "duration": 2.48, "text": "you could just kind of look at the"}, {"start": 173.92, "duration": 4.92, "text": "decimal"}, {"start": 175.28, "duration": 5.84, "text": "part and order them that way as"}, {"start": 178.84, "duration": 5.2, "text": "well so this to me kind of looks like"}, {"start": 181.12, "duration": 5.52, "text": "the number one this looks like 101 this"}, {"start": 184.04, "duration": 9.24, "text": "looks like 110 this looks like"}, {"start": 186.64, "duration": 8.36, "text": "431 well 1 101 110 431 so another way to"}, {"start": 193.28, "duration": 6.12, "text": "just kind of decide which one's the"}, {"start": 195.0, "duration": 4.4, "text": "smallest and which one's the largest"}], "title": "Ordering Decimals from Smallest to Largest", "description": "Thanks to all of you who support me on Patreon. You da real mvps!   $1 per month helps!! :)  https://www.patreon.com/patrickjmt !! Ordering Decimals from Smallest to Largest.  Just a couple of examples of ordering decimals from smallest to largest.", "lengthSeconds": 198, "width": 480, "height": 360}
{"video_id": "iAvF1Rn7How", "transcript": [{"start": 0.03, "duration": 3.629, "text": "okay here we're going to look at another"}, {"start": 1.709, "duration": 4.471, "text": "example of integrating with respect to Y"}, {"start": 3.659, "duration": 5.041, "text": "so here we're going to find the area of"}, {"start": 6.18, "duration": 5.579, "text": "the shaded region so it's bounded by y"}, {"start": 8.7, "duration": 6.18, "text": "equals x plus 2y equals square root of x"}, {"start": 11.759, "duration": 7.201, "text": "and then bounded below by the x-axis so"}, {"start": 14.88, "duration": 8.489, "text": "y equals 0 and then looks like this line"}, {"start": 18.96, "duration": 5.64, "text": "y equals 2 alright so again I think if"}, {"start": 23.369, "duration": 3.451, "text": "we were going to integrate this with"}, {"start": 24.6, "duration": 3.63, "text": "respect to X notice we would have to"}, {"start": 26.82, "duration": 4.08, "text": "break this up into two separate"}, {"start": 28.23, "duration": 4.5, "text": "integrals again so to get the area of"}, {"start": 30.9, "duration": 3.929, "text": "the triangle we could really just kind"}, {"start": 32.73, "duration": 3.63, "text": "of use geometry but if we wanted to you"}, {"start": 34.829, "duration": 3.961, "text": "know think about it in terms of an"}, {"start": 36.36, "duration": 5.46, "text": "integration we would take the top"}, {"start": 38.79, "duration": 4.5, "text": "function which is well X plus 2 and then"}, {"start": 41.82, "duration": 3.84, "text": "we would subtract the bottom one which"}, {"start": 43.29, "duration": 4.38, "text": "would just be y equals 0 that would be"}, {"start": 45.66, "duration": 4.77, "text": "and then we would go it looks like from"}, {"start": 47.67, "duration": 4.65, "text": "negative 2 up to 0 but then we would"}, {"start": 50.43, "duration": 4.44, "text": "have to break it up again and go from 0"}, {"start": 52.32, "duration": 4.71, "text": "up to 4 and we would have to take the"}, {"start": 54.87, "duration": 4.89, "text": "the function on top which is y equals 2"}, {"start": 57.03, "duration": 6.66, "text": "and subtract the bottom one which would"}, {"start": 59.76, "duration": 5.369, "text": "be square root of x so we could do this"}, {"start": 63.69, "duration": 3.66, "text": "again in two pieces let's write it down"}, {"start": 65.129, "duration": 5.191, "text": "so negative 2 to 0 the function on top"}, {"start": 67.35, "duration": 7.589, "text": "is X plus 2 the bottom one would just be"}, {"start": 70.32, "duration": 6.839, "text": "y equals well zero so all of that DX and"}, {"start": 74.939, "duration": 5.43, "text": "then we would add to that the integral"}, {"start": 77.159, "duration": 5.911, "text": "from 0 to 4 again the function on top is"}, {"start": 80.369, "duration": 4.981, "text": "y equals 2 minus the function on the"}, {"start": 83.07, "duration": 3.9, "text": "bottom would be square root of x and"}, {"start": 85.35, "duration": 2.82, "text": "then we could add those together you"}, {"start": 86.97, "duration": 4.53, "text": "know it wouldn't be the end of the world"}, {"start": 88.17, "duration": 7.8, "text": "to compute it that way if we look at it"}, {"start": 91.5, "duration": 6.63, "text": "with respect to Y you know if I were to"}, {"start": 95.97, "duration": 4.71, "text": "integrate this with respect to Y to me"}, {"start": 98.13, "duration": 5.46, "text": "it's now it's always the same curve on"}, {"start": 100.68, "duration": 5.28, "text": "the top always the same curve on the"}, {"start": 103.59, "duration": 3.989, "text": "bottom and that to me suggests hey we"}, {"start": 105.96, "duration": 4.32, "text": "could actually set it up with respect to"}, {"start": 107.579, "duration": 3.871, "text": "Y and make it a single integral but you"}, {"start": 110.28, "duration": 4.049, "text": "might make it hopefully a little bit"}, {"start": 111.45, "duration": 7.199, "text": "easier to compute so again to do that we"}, {"start": 114.329, "duration": 7.65, "text": "do the right - the left curve and as we"}, {"start": 118.649, "duration": 5.371, "text": "said before we have to make these curves"}, {"start": 121.979, "duration": 5.881, "text": "we have to write them in the form x"}, {"start": 124.02, "duration": 5.939, "text": "equals x equals okay so you know we've"}, {"start": 127.86, "duration": 3.98, "text": "got y equals square root of x well we"}, {"start": 129.959, "duration": 4.881, "text": "could just square both sides so that we"}, {"start": 131.84, "duration": 6.33, "text": "give us x equals y squared so we'll use"}, {"start": 134.84, "duration": 5.28, "text": "that version and then y equals x plus 2"}, {"start": 138.17, "duration": 6.42, "text": "well equivalently we can just rewrite"}, {"start": 140.12, "duration": 8.009, "text": "that and say y minus 2 that would equal"}, {"start": 144.59, "duration": 5.88, "text": "X all right so now if we do the"}, {"start": 148.129, "duration": 5.64, "text": "rightmost curve well the rightmost curve"}, {"start": 150.47, "duration": 6.54, "text": "would be x equals y squared minus the"}, {"start": 153.769, "duration": 7.021, "text": "leftmost curve which would be x equals y"}, {"start": 157.01, "duration": 5.759, "text": "minus 2 so this is what we're"}, {"start": 160.79, "duration": 4.05, "text": "integrating with respect to Y the"}, {"start": 162.769, "duration": 4.591, "text": "smallest y-coordinate that gets used is"}, {"start": 164.84, "duration": 5.28, "text": "0 the largest y-coordinate that gets"}, {"start": 167.36, "duration": 6.36, "text": "used is 2 so now this is the function"}, {"start": 170.12, "duration": 5.22, "text": "that we're integrating so 0 2 2 y"}, {"start": 173.72, "duration": 6.359, "text": "squared M is going to distribute that'll"}, {"start": 175.34, "duration": 7.44, "text": "be minus y plus 2 dy so now if we"}, {"start": 180.079, "duration": 6.841, "text": "calculate our antiderivative we'll get Y"}, {"start": 182.78, "duration": 8.459, "text": "to the third over 3 minus y squared over"}, {"start": 186.92, "duration": 8.819, "text": "2 plus 2y and then again we're"}, {"start": 191.239, "duration": 8.131, "text": "evaluating all of this from 0 to 2 so if"}, {"start": 195.739, "duration": 6.541, "text": "we plug in 2 2 cubed is 8 over 32"}, {"start": 199.37, "duration": 5.94, "text": "squared is 4 over 2 so we'll get 2 2"}, {"start": 202.28, "duration": 4.769, "text": "times 2 will give us a positive 4 the"}, {"start": 205.31, "duration": 4.97, "text": "lower limit of integration when we plug"}, {"start": 207.049, "duration": 5.671, "text": "in our 0 will just be a bunch of zeros"}, {"start": 210.28, "duration": 5.59, "text": "so it looks like we're left with 8 over"}, {"start": 212.72, "duration": 6.21, "text": "3 so negative 2 plus 4 will give us"}, {"start": 215.87, "duration": 6.089, "text": "positive 2 over 1 well we could multiply"}, {"start": 218.93, "duration": 6.72, "text": "top and bottom by 3 so that looks like"}, {"start": 221.959, "duration": 7.291, "text": "that would give us 8 plus 6 so 8 plus 6"}, {"start": 225.65, "duration": 8.75, "text": "would be 14 over 3 and that would be the"}, {"start": 229.25, "duration": 5.15, "text": "value for our area in this case"}], "title": "Finding Area by Integrating with Respect to Y", "description": "Thanks to all of you who support me on Patreon. You da real mvps!   $1 per month helps!! :)  https://www.patreon.com/patrickjmt !! Finding Area by Integrating with Respect to Y.  In this video, I find the area of a shaded region by integrating with respect to y.  We could have also integrated with respect to x to find the area, but it would involve more than one integral to do so.", "lengthSeconds": 234, "width": 480, "height": 360}
{"video_id": "eVbTmjdMqXM", "transcript": [{"start": 0.079, "duration": 4.061, "text": "so in this question we want to know"}, {"start": 2.25, "duration": 5.19, "text": "which of the Fallen grass best shows a"}, {"start": 4.14, "duration": 6.749, "text": "strong negative association between D"}, {"start": 7.44, "duration": 5.369, "text": "and T and to have a strong negative"}, {"start": 10.889, "duration": 3.75, "text": "association it basically says is one"}, {"start": 12.809, "duration": 4.621, "text": "quantity goes up the other wing goes"}, {"start": 14.639, "duration": 5.191, "text": "down and for it to be strong we want the"}, {"start": 17.43, "duration": 4.38, "text": "dots to be bunched close together and"}, {"start": 19.83, "duration": 4.14, "text": "again you can think about negative as"}, {"start": 21.81, "duration": 4.02, "text": "being like a negative slope you know so"}, {"start": 23.97, "duration": 4.62, "text": "a line that has negative slope is"}, {"start": 25.83, "duration": 5.699, "text": "pointing down as you move left to right"}, {"start": 28.59, "duration": 4.71, "text": "so answer choice a they're all kind of"}, {"start": 31.529, "duration": 4.351, "text": "spread out there's not really you know"}, {"start": 33.3, "duration": 3.9, "text": "any sort of strong association at all"}, {"start": 35.88, "duration": 5.82, "text": "between the dots everything is kind of"}, {"start": 37.2, "duration": 6.69, "text": "all over the place answer choice B you"}, {"start": 41.7, "duration": 4.32, "text": "know that has a positive association but"}, {"start": 43.89, "duration": 4.829, "text": "again the dots are slowly trending"}, {"start": 46.02, "duration": 4.59, "text": "upwards same thing with answer choice C"}, {"start": 48.719, "duration": 3.57, "text": "answer choice D would be the correct"}, {"start": 50.61, "duration": 5.82, "text": "answer because again the dots are all"}, {"start": 52.289, "duration": 6.631, "text": "very close together and as the value on"}, {"start": 56.43, "duration": 6.6, "text": "the horizontal axis as the values of"}, {"start": 58.92, "duration": 5.699, "text": "little D get bigger the vertical axis T"}, {"start": 63.03, "duration": 3.69, "text": "those values are getting smaller smaller"}, {"start": 64.619, "duration": 5.401, "text": "smaller and again you can thank you know"}, {"start": 66.72, "duration": 5.73, "text": "if I tried to put a line you know if I"}, {"start": 70.02, "duration": 3.99, "text": "tried to fit a line somehow through"}, {"start": 72.45, "duration": 3.66, "text": "those dots that I'd probably make it"}, {"start": 74.01, "duration": 4.679, "text": "look something like that you know"}, {"start": 76.11, "duration": 5.009, "text": "roughly and again this line has negative"}, {"start": 78.689, "duration": 4.231, "text": "slope and the dots are very close to it"}, {"start": 81.119, "duration": 4.261, "text": "so i probably could have made a slightly"}, {"start": 82.92, "duration": 4.5, "text": "better looking line there but again"}, {"start": 85.38, "duration": 4.199, "text": "that's the basic idea you want them to"}, {"start": 87.42, "duration": 6.15, "text": "be bunched close together that gives it"}, {"start": 89.579, "duration": 6.271, "text": "a strong association and again you want"}, {"start": 93.57, "duration": 4.589, "text": "them to the dots to basically move down"}, {"start": 95.85, "duration": 5.93, "text": "as you move to the right so that gives"}, {"start": 98.159, "duration": 3.621, "text": "you the negative association"}], "title": "SAT Math - Question 5, Section 2, 2016 Edition and Beyond", "description": "Thanks to all of you who support me on Patreon. You da real mvps!   $1 per month helps!! :)  https://www.patreon.com/patrickjmt !! SAT Math - Question 5, 2016 Edition and Beyond.\nSection 2 - A Calculator IS allowed!", "lengthSeconds": 101, "width": 480, "height": 360}
{"video_id": "7v1koVqwXK4", "transcript": [{"start": 0.03, "duration": 6.42, "text": "All right, in this video I'm going to do an example\u00a0\nof finding the equation for a hyperbola, and again,\u00a0\u00a0"}, {"start": 6.45, "duration": 4.98, "text": "kind of our two, sort of, standard forms. We can\u00a0\nactually write this a little more generically.\u00a0\u00a0"}, {"start": 11.43, "duration": 4.17, "text": "So I'll point that out here in this example.\u00a0\nSo we're going to find the equation for this\u00a0\u00a0"}, {"start": 15.6, "duration": 6.15, "text": "hyperbola here, you know, in the bottom left. We\u00a0\ncan write the equation of a hyperbola a little\u00a0\u00a0"}, {"start": 21.75, "duration": 7.98, "text": "more generically by saying x minus h squared over\u00a0\na squared, minus y minus k squared over b squared. H\u00a0\u00a0"}, {"start": 29.73, "duration": 8.22, "text": "and k is going to be the point where your \nasymptotes are intersecting, okay? It's going to be\u00a0\u00a0"}, {"start": 37.95, "duration": 6.45, "text": "the point equidistant from the vertices. So to me\u00a0\nit looks like the vertices here are at 1, 2, 3, it\u00a0\u00a0"}, {"start": 44.4, "duration": 6.45, "text": "looks like we have-- one of our vertices will be\u00a0\nat 4, 0, and the other one will be at negative 4, 0.\u00a0\u00a0"}, {"start": 50.85, "duration": 7.38, "text": "Okay, it looks like our asymptotes here are\u00a0\ncrossing then at the point 0, 0. So that means\u00a0\u00a0"}, {"start": 58.23, "duration": 8.52, "text": "my h, k is going to be equal to 0, 0 so I can fill\u00a0\nthat in at any point. Remember, a, that's going to\u00a0\u00a0"}, {"start": 66.75, "duration": 11.37, "text": "be the distance from your. . . from where the\u00a0\nasymptotes cross, over to one of the vertices. So to\u00a0\u00a0"}, {"start": 78.12, "duration": 5.43, "text": "me, in this case, it looks like our a value is going\u00a0\nto be an a value of positive 4. Again, we're just\u00a0\u00a0"}, {"start": 83.55, "duration": 7.32, "text": "simply moving 4 units to the right to get to\u00a0\nthat vertice. So a is four, and the last thing\u00a0\u00a0"}, {"start": 90.87, "duration": 6.54, "text": "I'm going to use is remember our asymptotes are\u00a0at\npositive and negative for this type, the ones that\u00a0\u00a0"}, {"start": 97.41, "duration": 8.73, "text": "open to the left and to the right, we use positive\u00a0\nand negative b over a, x. But we know our a value\u00a0\u00a0"}, {"start": 106.14, "duration": 8.07, "text": "in this case, we've already figured it out, that's\u00a0\n4. We just need to figure out our b value. Well,\u00a0\u00a0"}, {"start": 114.21, "duration": 10.59, "text": "notice in this-- our asymptote goes through\u00a0\nthe points 2, 1 and 4, 2. Okay, so you go up 1 over\u00a0\u00a0"}, {"start": 124.8, "duration": 8.91, "text": "2, up 1 over 2, so to me it looks like the slope\u00a0\nof this line, at least this, this asymptote with\u00a0\u00a0"}, {"start": 133.71, "duration": 9.48, "text": "a positive slope, to me it looks like this line\u00a0\nhas a slope of positive 1 over 2. Ok. So again,\u00a0\u00a0"}, {"start": 143.19, "duration": 9.69, "text": "up 1 over 2, up 1 over 2. So our slope has-- is a\u00a0\npositive 1/2, so our asymptote is going to be\u00a0\u00a0"}, {"start": 152.88, "duration": 8.55, "text": "1 over 2x. Well, I can rewrite this asymptote y\u00a0\nequals 1 over 2x, I could get a denominator of\u00a0\u00a0"}, {"start": 161.43, "duration": 8.97, "text": "4, and if I multiply top and bottom by 2, I would\u00a0\nget, well, 2 over 4, x, as again this asymptote. Since\u00a0\u00a0"}, {"start": 170.4, "duration": 6.87, "text": "I know that my a value has-- my a value is\u00a0\npositive 4, well then now this numerator must be\u00a0\u00a0"}, {"start": 177.27, "duration": 6.57, "text": "our b value. So it says our B value is going\u00a0\nto equal 2, and now we've got everything we\u00a0\u00a0"}, {"start": 183.84, "duration": 9.93, "text": "need to fill in the equation for our hyperbola.\u00a0\nSo it says we'll have x minus 0 squared over a\u00a0\u00a0"}, {"start": 193.77, "duration": 12.3, "text": "squared, minus y minus 0 squared over b squared,\u00a0\nwhich is 2 squared, equals 1. And you could always\u00a0\u00a0"}, {"start": 206.07, "duration": 7.8, "text": "multiply this out. It says our equation would be\u00a0\nx squared over 16, minus y squared over 4, equals 1."}], "title": "Finding the Equation for a Hyperbola Given the Graph - Example 1", "description": "Thanks to all of you who support me on Patreon. You da real mvps!   $1 per month helps!! :)  https://www.patreon.com/patrickjmt !! Finding the Equation for a Hyperbola Given the Graph - Example 1.", "lengthSeconds": 215, "width": 480, "height": 360}
{"video_id": "ALVijKml6iw", "transcript": [{"start": 1.04, "duration": 3.239, "text": "all right in this video we're going to"}, {"start": 2.28, "duration": 3.88, "text": "look at an example related to The"}, {"start": 4.279, "duration": 3.56, "text": "Squeeze theorem and uh we're going to"}, {"start": 6.16, "duration": 4.24, "text": "use the squeeze theorem to evaluate"}, {"start": 7.839, "duration": 4.401, "text": "these two limits so we'll do U part A"}, {"start": 10.4, "duration": 4.68, "text": "here first so we want to find the limit"}, {"start": 12.24, "duration": 5.119, "text": "as X approaches one of F ofx and we're"}, {"start": 15.08, "duration": 4.959, "text": "given the information that f ofx is"}, {"start": 17.359, "duration": 6.0, "text": "always greater than or equal to 4 and is"}, {"start": 20.039, "duration": 6.0, "text": "always less than or equal to x^2 + 6 x -"}, {"start": 23.359, "duration": 5.401, "text": "3 for all values of"}, {"start": 26.039, "duration": 4.48, "text": "X um usually kind of the tricky part of"}, {"start": 28.76, "duration": 4.28, "text": "the squeeze theorem is taking your"}, {"start": 30.519, "duration": 4.04, "text": "function and bounding it below by some"}, {"start": 33.04, "duration": 3.839, "text": "function in this case a constant"}, {"start": 34.559, "duration": 4.16, "text": "function and bounding it above by"}, {"start": 36.879, "duration": 3.641, "text": "another function here that's given to us"}, {"start": 38.719, "duration": 4.041, "text": "so it's going to make life a little bit"}, {"start": 40.52, "duration": 4.32, "text": "easier in Part B for example though"}, {"start": 42.76, "duration": 4.119, "text": "we're going to have to have to come up"}, {"start": 44.84, "duration": 3.399, "text": "with the uh kind of the smaller and the"}, {"start": 46.879, "duration": 4.041, "text": "bigger and that'll make it a little bit"}, {"start": 48.239, "duration": 5.041, "text": "trickier but um in this case if you're"}, {"start": 50.92, "duration": 4.92, "text": "given all this information the idea is"}, {"start": 53.28, "duration": 4.36, "text": "we can just take the limit of each part"}, {"start": 55.84, "duration": 4.6, "text": "so if we take the limit as X approaches"}, {"start": 57.64, "duration": 5.719, "text": "one of the number four that's going to"}, {"start": 60.44, "duration": 5.76, "text": "be well less than or equal to the Limit"}, {"start": 63.359, "duration": 5.681, "text": "as X approaches one of our function f"}, {"start": 66.2, "duration": 7.16, "text": "ofx and that's going to be less than or"}, {"start": 69.04, "duration": 8.28, "text": "equal to the Limit as X approaches 1 of"}, {"start": 73.36, "duration": 7.24, "text": "x^2 + 6 x -"}, {"start": 77.32, "duration": 5.04, "text": "3 well the limit of a constant is just"}, {"start": 80.6, "duration": 3.68, "text": "the constant so the limit as X"}, {"start": 82.36, "duration": 3.119, "text": "approaches one of the number four is"}, {"start": 84.28, "duration": 3.519, "text": "going to be"}, {"start": 85.479, "duration": 4.081, "text": "four um again so that's going to be less"}, {"start": 87.799, "duration": 4.921, "text": "than or equal to the Limit as X"}, {"start": 89.56, "duration": 6.4, "text": "approaches which is 1 of f"}, {"start": 92.72, "duration": 6.56, "text": "ofx well again to evaluate this limit um"}, {"start": 95.96, "duration": 5.88, "text": "x^2 + 6x - 3 again that's a nice little"}, {"start": 99.28, "duration": 4.28, "text": "Parabola it's continuous everywhere so"}, {"start": 101.84, "duration": 5.239, "text": "all we have to do is simply plug in our"}, {"start": 103.56, "duration": 6.8, "text": "value if we plug in one we would get 1 2"}, {"start": 107.079, "duration": 6.0, "text": "+ 6 * 1 -"}, {"start": 110.36, "duration": 4.48, "text": "3 so again we've got four is less than"}, {"start": 113.079, "duration": 5.201, "text": "or equal to the Limit as X approaches"}, {"start": 114.84, "duration": 6.68, "text": "one of F ofx and well let's see so 1^ 2"}, {"start": 118.28, "duration": 8.68, "text": "is 1 6 * 1 is going to get us uh so 1 +"}, {"start": 121.52, "duration": 7.48, "text": "6 is 7 7 - 3 is 4 okay well it says the"}, {"start": 126.96, "duration": 4.24, "text": "limit as X approaches one of f ofx has"}, {"start": 129.0, "duration": 5.08, "text": "to be greater than or equal to 4 but it"}, {"start": 131.2, "duration": 5.52, "text": "also has to be less than or equal to 4"}, {"start": 134.08, "duration": 5.32, "text": "well that would imply that the limit as"}, {"start": 136.72, "duration": 5.239, "text": "X approaches one of our function is"}, {"start": 139.4, "duration": 4.839, "text": "simply going to equal the number four"}, {"start": 141.959, "duration": 4.881, "text": "okay so again what you're kind of doing"}, {"start": 144.239, "duration": 4.961, "text": "is you're deducing information about uh"}, {"start": 146.84, "duration": 4.56, "text": "the limit of some function by knowing"}, {"start": 149.2, "duration": 4.0, "text": "information about you know a function"}, {"start": 151.4, "duration": 5.52, "text": "that's smaller than it and a function"}, {"start": 153.2, "duration": 6.0, "text": "that's larger than it so uh kind of an a"}, {"start": 156.92, "duration": 6.08, "text": "indirect way to get information about"}, {"start": 159.2, "duration": 3.8, "text": "the original function"}], "title": "The Squeeze Theorem for Limits, Example 2", "description": "Thanks to all of you who support me on Patreon. You da real mvps!   $1 per month helps!! :)  https://www.patreon.com/patrickjmt !! The Squeeze Theorem for Limits, Example 2. In this video, I work a problem involving limits using the squeeze theorem.  This is an 'easy' squeeze theorem problem since the 'small' and 'large' function are both given.", "lengthSeconds": 163, "width": 480, "height": 360}
{"video_id": "TQ_tpKZ2PN0", "transcript": [{"start": 0.96, "duration": 3.52, "text": "okay here we're going to find the volume"}, {"start": 2.919, "duration": 4.241, "text": "of a"}, {"start": 4.48, "duration": 4.96, "text": "cylinder so to figure out the volume"}, {"start": 7.16, "duration": 4.96, "text": "again what I'm going to do is find the"}, {"start": 9.44, "duration": 5.6, "text": "area of one of the bases and then simply"}, {"start": 12.12, "duration": 6.76, "text": "multiply it by the height well to figure"}, {"start": 15.04, "duration": 5.479, "text": "out the the area of one of the bases"}, {"start": 18.88, "duration": 5.68, "text": "well since it's a cylinder that will be"}, {"start": 20.519, "duration": 6.801, "text": "a circle so the the area of one of the"}, {"start": 24.56, "duration": 6.24, "text": "faces would be pi times"}, {"start": 27.32, "duration": 5.84, "text": "the pi * the radius"}, {"start": 30.8, "duration": 5.68, "text": "squared and then we would multiply that"}, {"start": 33.16, "duration": 6.48, "text": "by the height which would be 15 but in"}, {"start": 36.48, "duration": 5.52, "text": "this case again we're approximating Pi"}, {"start": 39.64, "duration": 2.36, "text": "with"}, {"start": 45.399, "duration": 4.48, "text": "3.14 so now just a little bit of"}, {"start": 47.76, "duration": 6.04, "text": "arithmetic so"}, {"start": 49.879, "duration": 5.921, "text": "3.14 4^ SAR will be 16 and then we have"}, {"start": 53.8, "duration": 7.88, "text": "to multiply that by"}, {"start": 55.8, "duration": 5.88, "text": "15 so let's see here uh 3.14"}, {"start": 62.6, "duration": 6.76, "text": "multiplied by"}, {"start": 64.76, "duration": 5.84, "text": "16 multiplied by 15 I'm getting this to"}, {"start": 69.36, "duration": 5.88, "text": "equal"}, {"start": 70.6, "duration": 4.64, "text": "75 7 7 let me say that again"}, {"start": 75.68, "duration": 6.04, "text": "7536 and again we're using centimet so"}, {"start": 78.72, "duration": 6.439, "text": "our units will be centimet cubed or"}, {"start": 81.72, "duration": 3.439, "text": "cubic cenim"}], "title": "Volume of a Cylinder", "description": "Thanks to all of you who support me on Patreon. You da real mvps!   $1 per month helps!! :)  https://www.patreon.com/patrickjmt !! Volume of a Cylinder.  Here we look at a simple example of finding the volume of a cylinder.", "lengthSeconds": 84, "width": 480, "height": 360}
{"video_id": "qReHmJOuYNE", "transcript": [{"start": 0.079, "duration": 5.411, "text": "okay here we're gonna look at some"}, {"start": 2.399, "duration": 6.691, "text": "examples involving integrals again with"}, {"start": 5.49, "duration": 6.359, "text": "secant and tangent and actually you know"}, {"start": 9.09, "duration": 5.429, "text": "Part A I already you know take that back"}, {"start": 11.849, "duration": 6.421, "text": "because Part A here doesn't have you"}, {"start": 14.519, "duration": 6.27, "text": "know tangent or secant in there but to"}, {"start": 18.27, "duration": 5.849, "text": "integrate this notice you know we do"}, {"start": 20.789, "duration": 6.931, "text": "have the cube root of okay so cotangent"}, {"start": 24.119, "duration": 7.141, "text": "X we'll leave that alone we really have"}, {"start": 27.72, "duration": 5.999, "text": "a 1 over sine squared and 1 over sine"}, {"start": 31.26, "duration": 5.04, "text": "that's cosecant so really we have a"}, {"start": 33.719, "duration": 5.071, "text": "cosecant squared X hanging out here as"}, {"start": 36.3, "duration": 3.96, "text": "well so kind of the moral of the story a"}, {"start": 38.79, "duration": 3.06, "text": "lot of the tricks you know that work"}, {"start": 40.26, "duration": 4.47, "text": "with the secant and tangent will work"}, {"start": 41.85, "duration": 5.369, "text": "with cosecant and also cotangent so just"}, {"start": 44.73, "duration": 3.84, "text": "keep that in mind I think this one's one"}, {"start": 47.219, "duration": 4.11, "text": "where we can do with just a nice little"}, {"start": 48.57, "duration": 6.899, "text": "substitution here we can let u equal"}, {"start": 51.329, "duration": 5.761, "text": "cotangent of X our D u will be negative"}, {"start": 55.469, "duration": 3.98, "text": "so let's see the derivative of tangent"}, {"start": 57.09, "duration": 6.109, "text": "secant squared cotangent is negative"}, {"start": 59.449, "duration": 5.801, "text": "cosecant squared X will tack on our DX"}, {"start": 63.199, "duration": 3.821, "text": "again I kind of just want a positive"}, {"start": 65.25, "duration": 4.369, "text": "cosecant squared so I'm going to"}, {"start": 67.02, "duration": 5.48, "text": "multiply both sides by negative 1 and"}, {"start": 69.619, "duration": 5.201, "text": "now I think we've got what we want so"}, {"start": 72.5, "duration": 6.909, "text": "I'm just gonna start plugging everything"}, {"start": 74.82, "duration": 7.35, "text": "in here so our cosecant squared a"}, {"start": 79.409, "duration": 5.611, "text": "cosecant squared X DX that's gonna be"}, {"start": 82.17, "duration": 4.14, "text": "our negative 1 D U and then the only"}, {"start": 85.02, "duration": 4.02, "text": "thing we would have left would be the"}, {"start": 86.31, "duration": 5.519, "text": "cube root of cotangent so we would have"}, {"start": 89.04, "duration": 3.75, "text": "you you know underneath the radical and"}, {"start": 91.829, "duration": 4.621, "text": "I'm going to go ahead and write that"}, {"start": 92.79, "duration": 6.09, "text": "it's just you to the 1/3 power alright"}, {"start": 96.45, "duration": 4.62, "text": "so if we integrate we'll add 1 to the"}, {"start": 98.88, "duration": 5.279, "text": "exponent so that'll give us 4/3"}, {"start": 101.07, "duration": 7.589, "text": "we'll divide by 4/3 or equivalently"}, {"start": 104.159, "duration": 7.381, "text": "multiplied by 3/4 plus C and again now"}, {"start": 108.659, "duration": 5.49, "text": "all we have to do is just plug in our"}, {"start": 111.54, "duration": 5.429, "text": "value so let's see let's go ahead and"}, {"start": 114.149, "duration": 7.32, "text": "just make that negative 3/4 and then our"}, {"start": 116.969, "duration": 8.97, "text": "value for u is cotangent of X raised to"}, {"start": 121.469, "duration": 7.491, "text": "the 4/3 power plus C and that will be"}, {"start": 125.939, "duration": 3.021, "text": "the antiderivative"}], "title": "Integrating Even and Odd Powers of Cosecant and Cotangent, Ex 1", "description": "Thanks to all of you who support me on Patreon. You da real mvps!   $1 per month helps!! :)  https://www.patreon.com/patrickjmt !! Integrating Even and Odd Powers of Cosecant and Cotangent, Ex 1.  Here I look at another random example of integrating a function involving different powers of the trigonometric functions.", "lengthSeconds": 128, "width": 480, "height": 360}
{"video_id": "7fy2U0Sm1Vc", "transcript": [{"start": 0.44, "duration": 3.6, "text": "all right so another example using sum"}, {"start": 2.12, "duration": 4.08, "text": "and difference identities to simplify an"}, {"start": 4.04, "duration": 5.92, "text": "expression and here we're going to"}, {"start": 6.2, "duration": 5.64, "text": "evaluate tangent of x + 4 Pi so we're"}, {"start": 9.96, "duration": 4.16, "text": "just going to uh you know simplify that"}, {"start": 11.84, "duration": 5.64, "text": "down a little bit so uh we've got our"}, {"start": 14.12, "duration": 5.12, "text": "identity here it says tangent of x + y"}, {"start": 17.48, "duration": 3.76, "text": "it says if there's a plus sign we use"}, {"start": 19.24, "duration": 4.4, "text": "tangent of X Plus tangent of Y in the"}, {"start": 21.24, "duration": 4.08, "text": "numerator but then in the denominator"}, {"start": 23.64, "duration": 5.959, "text": "you kind of take the opposite sign we'll"}, {"start": 25.32, "duration": 6.72, "text": "use 1us tangent x * tangent y so in this"}, {"start": 29.599, "duration": 4.8, "text": "case well we would simply get tangent of"}, {"start": 32.04, "duration": 6.08, "text": "X Plus"}, {"start": 34.399, "duration": 3.721, "text": "tangent of 4"}, {"start": 38.2, "duration": 5.64, "text": "pi over one but let's see so since it"}, {"start": 41.32, "duration": 4.68, "text": "was a plus originally it switches to a"}, {"start": 43.84, "duration": 7.68, "text": "negative in the denominator and then"}, {"start": 46.0, "duration": 11.519, "text": "we'll get tangent of X time tangent of 4"}, {"start": 51.52, "duration": 10.96, "text": "Pi but tangent of 4 Pi um tangent of 4"}, {"start": 57.519, "duration": 4.961, "text": "Pi that's just going to be S of 4"}, {"start": 63.199, "duration": 8.96, "text": "pi over cosine of 4 Pi well s of 4 Pi is"}, {"start": 69.36, "duration": 5.56, "text": "going to equal 0er cosine of 4 Pi is"}, {"start": 72.159, "duration": 6.041, "text": "going to equal 1 so tangent of 4 Pi"}, {"start": 74.92, "duration": 6.839, "text": "simply reduces to zero so really we're"}, {"start": 78.2, "duration": 4.8, "text": "just left with Tangent X Plus 0 in the"}, {"start": 81.759, "duration": 3.641, "text": "numerator so I'm just going to leave"}, {"start": 83.0, "duration": 5.24, "text": "that out uh well let's stick it in there"}, {"start": 85.4, "duration": 5.8, "text": "one time uh plus Z then we'll have 1us"}, {"start": 88.24, "duration": 5.08, "text": "tangent X"}, {"start": 91.2, "duration": 4.64, "text": "times tangent of 4 Pi which again is"}, {"start": 93.32, "duration": 5.839, "text": "going to give us zero well all we're"}, {"start": 95.84, "duration": 5.279, "text": "left with in the numerator is tangent X"}, {"start": 99.159, "duration": 4.721, "text": "in the denominator we're simply left"}, {"start": 101.119, "duration": 8.401, "text": "with one so that expression actually"}, {"start": 103.88, "duration": 5.64, "text": "simply reduces down to just tangent of x"}], "title": "Sum and Difference Identities to Simplify an Expression, Example 2", "description": "Thanks to all of you who support me on Patreon. You da real mvps!   $1 per month helps!! :)  https://www.patreon.com/patrickjmt !! Sum and Difference Identities to Simplify an Expression, Example 2.  In this video, I simply show how to simplify some expressions by using sum and difference identities.  Nothing crazy, just showing how one must know identities to make things more manageable!  For more free math videos, visit http://PatrickJMT.com", "lengthSeconds": 108, "width": 480, "height": 360}
{"video_id": "4Q-qwRmcN8E", "transcript": [{"start": 0.719, "duration": 3.16, "text": "all right in this video we're going to"}, {"start": 1.68, "duration": 4.28, "text": "do an example of solving an investment"}, {"start": 3.879, "duration": 4.161, "text": "problem so pretty basic problem here"}, {"start": 5.96, "duration": 4.04, "text": "just involving simple interest so"}, {"start": 8.04, "duration": 4.0, "text": "suppose our friend Abby has 600 bucks"}, {"start": 10.0, "duration": 4.16, "text": "and she divides them between two savings"}, {"start": 12.04, "duration": 4.44, "text": "account suppose one of them ended up"}, {"start": 14.16, "duration": 4.52, "text": "earning 5% simple interest and the other"}, {"start": 16.48, "duration": 6.52, "text": "earned 3% simple"}, {"start": 18.68, "duration": 5.96, "text": "interest if she earned a $25 total um in"}, {"start": 23.0, "duration": 3.32, "text": "interest maybe over the course of that"}, {"start": 24.64, "duration": 4.2, "text": "year we want to know how much she earned"}, {"start": 26.32, "duration": 5.039, "text": "from each account okay so we don't know"}, {"start": 28.84, "duration": 5.16, "text": "how much she invested but we can easily"}, {"start": 31.359, "duration": 5.921, "text": "um you know we've got the 5% account and"}, {"start": 34.0, "duration": 5.559, "text": "we've got the 3% account we can assume"}, {"start": 37.28, "duration": 2.279, "text": "she"}, {"start": 39.64, "duration": 7.96, "text": "invested maybe X in the 5% account um"}, {"start": 44.28, "duration": 5.04, "text": "since she had 600 total if we take 600"}, {"start": 47.6, "duration": 3.16, "text": "minus X that would be the remaining"}, {"start": 49.32, "duration": 5.8, "text": "amount of money and that would be how"}, {"start": 50.76, "duration": 8.0, "text": "much she invested in the 3% account um"}, {"start": 55.12, "duration": 3.64, "text": "the simple interest uh"}, {"start": 58.92, "duration": 3.24, "text": "earned"}, {"start": 60.96, "duration": 7.159, "text": "so if you think about the interest"}, {"start": 62.16, "duration": 10.08, "text": "earned 5% is 05 so it'd be 05 * x 3% is"}, {"start": 68.119, "duration": 6.201, "text": "03 and it would be 3% of this 600 minus"}, {"start": 72.24, "duration": 3.919, "text": "X and this is going to give us our"}, {"start": 74.32, "duration": 3.64, "text": "equation because we know if we take the"}, {"start": 76.159, "duration": 5.6, "text": "interest earned"}, {"start": 77.96, "duration": 7.92, "text": "from the 5% account and add to that the"}, {"start": 81.759, "duration": 6.961, "text": "interest earned from the uh 3% account"}, {"start": 85.88, "duration": 5.559, "text": "we know that the total amount earned was"}, {"start": 88.72, "duration": 4.16, "text": "$25 and now we've got our equation to"}, {"start": 91.439, "duration": 4.201, "text": "work"}, {"start": 92.88, "duration": 4.919, "text": "with Okay so we've got"}, {"start": 95.64, "duration": 7.2, "text": "0.05x um if we take"}, {"start": 97.799, "duration": 8.64, "text": "0.003 * 600 I believe that should be"}, {"start": 102.84, "duration": 9.44, "text": "18 minus 03"}, {"start": 106.439, "duration": 10.441, "text": "x = 25 now we can combine 0 3x minus"}, {"start": 112.28, "duration": 7.759, "text": "excuse me 05x -3x will be"}, {"start": 116.88, "duration": 5.48, "text": "0.02x we can also subtract 18 from both"}, {"start": 120.039, "duration": 4.641, "text": "sides which will give us seven and now"}, {"start": 122.36, "duration": 2.32, "text": "if we"}, {"start": 128.72, "duration": 7.879, "text": "divide we get that x equal 350 and 350"}, {"start": 133.239, "duration": 5.72, "text": "was the amount in the 5% account because"}, {"start": 136.599, "duration": 5.801, "text": "again that's what we let X represent um"}, {"start": 138.959, "duration": 6.121, "text": "so since she had 350 in the 5% account"}, {"start": 142.4, "duration": 6.44, "text": "that would mean that she has 250 she had"}, {"start": 145.08, "duration": 5.519, "text": "invested $250 in the whoops the"}, {"start": 148.84, "duration": 4.759, "text": "3%"}, {"start": 150.599, "duration": 3.0, "text": "account"}], "title": "Solving an Investment Problem", "description": "Thanks to all of you who support me on Patreon. You da real mvps!   $1 per month helps!! :)  https://www.patreon.com/patrickjmt !! Solving an Investment Problem.  If you deposit money into two different accounts and earn so much from each one given a certain interest rate, how much did you deposit in each account?", "lengthSeconds": 152, "width": 480, "height": 360}
{"video_id": "uS_5bmRUYEI", "transcript": [{"start": 0.03, "duration": 2.73, "text": "all right here we're going to start"}, {"start": 1.469, "duration": 3.481, "text": "looking at solving separable"}, {"start": 2.76, "duration": 3.9, "text": "differential equations and the way that"}, {"start": 4.95, "duration": 4.74, "text": "I always reminded myself of this"}, {"start": 6.66, "duration": 7.439, "text": "procedure is we just separate and we"}, {"start": 9.69, "duration": 6.21, "text": "integrate so if I separate what we're"}, {"start": 14.099, "duration": 3.661, "text": "going to try to do is we're going to try"}, {"start": 15.9, "duration": 3.959, "text": "to put all the Y's on one side of the"}, {"start": 17.76, "duration": 4.23, "text": "equal sign and all the X's on the other"}, {"start": 19.859, "duration": 3.93, "text": "side of the equal sign so that's the"}, {"start": 21.99, "duration": 3.69, "text": "idea of sort of separating them I'm"}, {"start": 23.789, "duration": 5.011, "text": "going to separate the variables from"}, {"start": 25.68, "duration": 4.679, "text": "each other and we want the dy to be in"}, {"start": 28.8, "duration": 6.0, "text": "the numerator and we want the DX to be"}, {"start": 30.359, "duration": 8.871, "text": "in the numerator as well so Part A we've"}, {"start": 34.8, "duration": 6.63, "text": "got dy over DX equals x over y squared"}, {"start": 39.23, "duration": 4.509, "text": "well what we're going to do is we're"}, {"start": 41.43, "duration": 5.16, "text": "going to multiply the DX over to the"}, {"start": 43.739, "duration": 5.101, "text": "right side so we can multiply both sides"}, {"start": 46.59, "duration": 3.989, "text": "by DX based on cross multiplying that's"}, {"start": 48.84, "duration": 4.02, "text": "what I'm doing maybe that's the easier"}, {"start": 50.579, "duration": 4.591, "text": "way to think about it so x times DX"}, {"start": 52.86, "duration": 6.539, "text": "that's we're getting on the right side"}, {"start": 55.17, "duration": 7.529, "text": "we'll get dy times y squared or Y"}, {"start": 59.399, "duration": 5.521, "text": "squared dy so now we've separated and"}, {"start": 62.699, "duration": 5.071, "text": "the next thing I'm going to do is to"}, {"start": 64.92, "duration": 4.379, "text": "integrate okay so if we start with two"}, {"start": 67.77, "duration": 4.529, "text": "things that are equal we can integrate"}, {"start": 69.299, "duration": 6.121, "text": "both sides well the antiderivative of Y"}, {"start": 72.299, "duration": 6.421, "text": "squared will be Y cubed over three we"}, {"start": 75.42, "duration": 6.0, "text": "can certainly put the plus C wherever we"}, {"start": 78.72, "duration": 5.219, "text": "want to the antiderivative of the right"}, {"start": 81.42, "duration": 3.9, "text": "side will be x squared over two I'm just"}, {"start": 83.939, "duration": 3.36, "text": "going to stick the plus C on the right"}, {"start": 85.32, "duration": 7.009, "text": "side you don't have to put constants on"}, {"start": 87.299, "duration": 7.771, "text": "both sides if you wanted to as well now"}, {"start": 92.329, "duration": 6.4, "text": "we can even solve for y in this case so"}, {"start": 95.07, "duration": 9.869, "text": "we can multiply both sides by three so"}, {"start": 98.729, "duration": 9.871, "text": "three times x squared over two plus C so"}, {"start": 104.939, "duration": 6.511, "text": "three halves x squared plus three C"}, {"start": 108.6, "duration": 4.559, "text": "again that's going to be Y cubed and now"}, {"start": 111.45, "duration": 5.669, "text": "to get the Y by itself we could simply"}, {"start": 113.159, "duration": 8.511, "text": "take the cube root of both sides so if"}, {"start": 117.119, "duration": 8.43, "text": "we have Y cubed equals 3 over 2x squared"}, {"start": 121.67, "duration": 8.83, "text": "plus three C I'm just going to take the"}, {"start": 125.549, "duration": 7.08, "text": "cube root of both sides so on the left"}, {"start": 130.5, "duration": 2.94, "text": "side we'll be left with Y on the right"}, {"start": 132.629, "duration": 5.071, "text": "side we've got our"}, {"start": 133.44, "duration": 10.5, "text": "cube root so with the cube root of 3"}, {"start": 137.7, "duration": 10.14, "text": "over 2x squared plus 3c and that would"}, {"start": 143.94, "duration": 6.09, "text": "definitely be a solution now sometimes"}, {"start": 147.84, "duration": 3.51, "text": "you won't see like this 3c and I think"}, {"start": 150.03, "duration": 4.86, "text": "this is something that can be a little"}, {"start": 151.35, "duration": 11.31, "text": "misleading sometimes what you'll see"}, {"start": 154.89, "duration": 9.96, "text": "instead I equally valid okay also valid"}, {"start": 162.66, "duration": 5.04, "text": "would be to say you know suppose instead"}, {"start": 164.85, "duration": 5.58, "text": "of labeling this as a you know see"}, {"start": 167.7, "duration": 6.45, "text": "suppose I call it this little C sub 1 C"}, {"start": 170.43, "duration": 6.24, "text": "sub 1 and C sub 1 C sub 1 and C sub 1 so"}, {"start": 174.15, "duration": 6.6, "text": "okay it's just some generic constant I"}, {"start": 176.67, "duration": 7.65, "text": "could say okay let's let C equal 3 times"}, {"start": 180.75, "duration": 10.47, "text": "C sub 1 so I'm just replacing this"}, {"start": 184.32, "duration": 9.99, "text": "generic constant 3 times C sub 1 with"}, {"start": 191.22, "duration": 5.67, "text": "just the value C and this would also be"}, {"start": 194.31, "duration": 4.62, "text": "equally valid okay we could say either"}, {"start": 196.89, "duration": 5.28, "text": "one of these are going to be solutions"}, {"start": 198.93, "duration": 6.42, "text": "to the original the original"}, {"start": 202.17, "duration": 4.68, "text": "differential equation so sometimes"}, {"start": 205.35, "duration": 4.62, "text": "you'll see it written either way so"}, {"start": 206.85, "duration": 4.92, "text": "don't be thrown off by that this is"}, {"start": 209.97, "duration": 3.3, "text": "basically all that's happening probably"}, {"start": 211.77, "duration": 4.58, "text": "sometimes I'll leave it alone and"}, {"start": 213.27, "duration": 7.01, "text": "sometimes I'll change it but again it's"}, {"start": 216.35, "duration": 3.93, "text": "correct to say either one"}], "title": "Solving a Separable Differential Equation, Another Example #1", "description": "(https://youtu.be/nNHlSB6b1HU)\n(https://youtu.be/XExEixAPK6s)\n(http://www.youtube.com/watch?v=XExEixAPK6s)\nSeparable Differential Equations | Solving Step-by-Step\nIn this video, we tackle the problem of solving a separable differential equation:   dy/dx = x/y^2\nWe walk through the process of rearranging the equation to separate the variables and then integrate to find the general solution. Separable differential equations are a fundamental concept in calculus, and this lesson provides clear steps to help you solve these types of problems with confidence.\n\nWhat You Will Learn:\nHow to recognize and solve a separable differential equation.\nTechniques for separating variables in the equation.\nIntegrating both sides to find the general solution.\nGreat for anyone studying differential equations or looking to improve their problem-solving skills in calculus!\n\nSupport me on Patreon: https://www.patreon.com/patrickjmt?ty=c\n\n#SeparableDifferentialEquations #Calculus #DifferentialEquations #PatrickJMT #SeparationOfVariables #MathTutorial #CalculusHelp #MathSolutions #AdvancedCalculus #YouTubeMath\n\nKeywords: separable differential equations, solving differential equations, calculus differential equations, separation of variables, integrating differential equations, PatrickJMT, calculus tutorial, differential equations examples, math problem solving\n\n\n\n\n\n\nYou said:", "lengthSeconds": 219, "width": 480, "height": 360}
{"video_id": "BI1LDeNruxw", "transcript": [{"start": 0.48, "duration": 3.159, "text": "all right so one more example here of"}, {"start": 2.0, "duration": 5.08, "text": "solving a separable differential"}, {"start": 3.639, "duration": 6.441, "text": "equation so part C we've got Dy DX ="}, {"start": 7.08, "duration": 5.16, "text": "cosine x - one I'm going to do the same"}, {"start": 10.08, "duration": 3.479, "text": "thing um as on the other ones I'm just"}, {"start": 12.24, "duration": 4.32, "text": "going to cross"}, {"start": 13.559, "duration": 5.841, "text": "multiply so let's see we would have Dy *"}, {"start": 16.56, "duration": 5.08, "text": "the quantity Yus one and I'm going to"}, {"start": 19.4, "duration": 4.68, "text": "put the Dy on the right"}, {"start": 21.64, "duration": 6.0, "text": "side or excuse me well to the right of"}, {"start": 24.08, "duration": 7.84, "text": "the y- one um on the actual right side"}, {"start": 27.64, "duration": 6.64, "text": "of the equal sign we'll have DX * cosine"}, {"start": 31.92, "duration": 4.159, "text": "X well again now what we're going to do"}, {"start": 34.28, "duration": 4.24, "text": "we've got all the Y's on one side all"}, {"start": 36.079, "duration": 5.96, "text": "the x's on the other we can just find"}, {"start": 38.52, "duration": 7.44, "text": "the anti-derivative of both sides so on"}, {"start": 42.039, "duration": 7.04, "text": "the left we'll be left with y^2 over 2 -"}, {"start": 45.96, "duration": 6.079, "text": "y on the right side the anti-derivative"}, {"start": 49.079, "duration": 5.96, "text": "of cine X will be sinx again I'm going"}, {"start": 52.039, "duration": 5.881, "text": "to stick our plus C on that side and"}, {"start": 55.039, "duration": 4.52, "text": "again you know to solve this for y um"}, {"start": 57.92, "duration": 3.72, "text": "you know our very first example we were"}, {"start": 59.559, "duration": 4.441, "text": "able to take a cube root eventually and"}, {"start": 61.64, "duration": 4.96, "text": "solve for y again here I'm not sure"}, {"start": 64.0, "duration": 5.64, "text": "exactly how to solve for y and again"}, {"start": 66.6, "duration": 4.8, "text": "sometimes you simply can't do it um so"}, {"start": 69.64, "duration": 3.799, "text": "here I'm just going to leave it alone"}, {"start": 71.4, "duration": 5.039, "text": "and say our solution to the original"}, {"start": 73.439, "duration": 9.201, "text": "differential equation would be y^2 uh"}, {"start": 76.439, "duration": 6.201, "text": "y^2 2 - y = sinx + C"}], "title": "Solving a Separable Differential Equation, Another Example #3", "description": "Thanks to all of you who support me on Patreon. You da real mvps!   $1 per month helps!! :)  https://www.patreon.com/patrickjmt !! Solving a Separable Differential Equation, Another Example #3.  In this video, I solve an example of a separable differential equation.", "lengthSeconds": 81, "width": 480, "height": 360}
{"video_id": "oU6xW78wnXc", "transcript": [{"start": 0.03, "duration": 4.65, "text": "all right in this video I just want to"}, {"start": 2.159, "duration": 4.981, "text": "do another example of solving a linear"}, {"start": 4.68, "duration": 6.18, "text": "inequality so in this case we have X"}, {"start": 7.14, "duration": 7.44, "text": "minus 6 divided X minus 6 divided by 2"}, {"start": 10.86, "duration": 4.949, "text": "greater than 4x divided by 7 so a couple"}, {"start": 14.58, "duration": 5.1, "text": "of different ways we can start about"}, {"start": 15.809, "duration": 5.821, "text": "going about doing this problem to me"}, {"start": 19.68, "duration": 4.259, "text": "usually one thing I like to do again my"}, {"start": 21.63, "duration": 3.75, "text": "own personal preference you know the"}, {"start": 23.939, "duration": 4.26, "text": "thing I don't like about this problem or"}, {"start": 25.38, "duration": 4.2, "text": "the fractions is a - there's a 7 the way"}, {"start": 28.199, "duration": 3.841, "text": "that we can get rid of that is we can"}, {"start": 29.58, "duration": 6.06, "text": "multiply by the least common multiple of"}, {"start": 32.04, "duration": 5.64, "text": "those numbers 2 & 7 so again the least"}, {"start": 35.64, "duration": 6.439, "text": "common multiple says what's the smallest"}, {"start": 37.68, "duration": 7.949, "text": "number that is divisible evenly by both"}, {"start": 42.079, "duration": 5.261, "text": "positive 2 and positive 7 well I think"}, {"start": 45.629, "duration": 5.7, "text": "the first number that fits that bill"}, {"start": 47.34, "duration": 5.43, "text": "would be the number 14 so what I'm going"}, {"start": 51.329, "duration": 4.741, "text": "to do is I'm going to multiply the left"}, {"start": 52.77, "duration": 5.4, "text": "side by 14 and the right side also by 14"}, {"start": 56.07, "duration": 6.989, "text": "so you can really think about that as"}, {"start": 58.17, "duration": 6.599, "text": "being 14 over 1 so be careful 14 over 2"}, {"start": 63.059, "duration": 4.35, "text": "is 7 we're still going to have to"}, {"start": 64.769, "duration": 5.28, "text": "distribute that number to everything in"}, {"start": 67.409, "duration": 6.181, "text": "the parenthesis but the 14 over the 2"}, {"start": 70.049, "duration": 7.801, "text": "becomes a 7 we've got our 4x on the"}, {"start": 73.59, "duration": 8.069, "text": "right and again 14 divided by 7 is going"}, {"start": 77.85, "duration": 5.01, "text": "to be 2 and now this to me is a little"}, {"start": 81.659, "duration": 4.89, "text": "bit better I'm just going to distribute"}, {"start": 82.86, "duration": 7.829, "text": "things out like normal so I'll get 7 X 7"}, {"start": 86.549, "duration": 8.07, "text": "times negative 6 is negative 42 4x times"}, {"start": 90.689, "duration": 5.43, "text": "2 is 8 X and what I'm gonna do in this"}, {"start": 94.619, "duration": 2.82, "text": "case again I like to keep the X as"}, {"start": 96.119, "duration": 4.531, "text": "positive so I think I'm going to"}, {"start": 97.439, "duration": 7.741, "text": "subtract 7x from both sides and then"}, {"start": 100.65, "duration": 8.07, "text": "I've got negative 42 is greater than 8x"}, {"start": 105.18, "duration": 5.52, "text": "minus 7x is just positive 1x or again"}, {"start": 108.72, "duration": 5.34, "text": "equivalently I could write that as X"}, {"start": 110.7, "duration": 5.11, "text": "less than negative 42 says the exact"}, {"start": 114.06, "duration": 7.03, "text": "these inequalities are the"}, {"start": 115.81, "duration": 6.87, "text": "same say the exact same thing all my"}, {"start": 121.09, "duration": 3.48, "text": "inequalities all my problems have been"}, {"start": 122.68, "duration": 4.26, "text": "less than some negative numbers so just"}, {"start": 124.57, "duration": 4.56, "text": "a coincidence there so since it's got to"}, {"start": 126.94, "duration": 6.86, "text": "be strictly less than negative 42 I've"}, {"start": 129.13, "duration": 8.28, "text": "put a little open circle at negative 42"}, {"start": 133.8, "duration": 5.92, "text": "if I want to write this this solution"}, {"start": 137.41, "duration": 4.92, "text": "and interval notation we can write that"}, {"start": 139.72, "duration": 6.0, "text": "as parentheses negative infinity up to"}, {"start": 142.33, "duration": 7.159, "text": "but not including negative 42 and we'll"}, {"start": 145.72, "duration": 3.769, "text": "put our parentheses around it"}], "title": "An Introduction To Solving Linear Inequalities - Example 3", "description": "Thanks to all of you who support me on Patreon. You da real mvps!   $1 per month helps!! :)  https://www.patreon.com/patrickjmt !! An Introduction To Solving Inequalities - Example 3", "lengthSeconds": 148, "width": 480, "height": 360}
{"video_id": "8ZyGwbYoWZ4", "transcript": [{"start": 0.56, "duration": 5.239, "text": "okay here I'm going to graph two periods"}, {"start": 2.12, "duration": 6.76, "text": "of y = 3 * cent of"}, {"start": 5.799, "duration": 5.401, "text": "12x and we'll pick out the vertical ASM"}, {"start": 8.88, "duration": 4.12, "text": "tootes and zeros the only thing that's"}, {"start": 11.2, "duration": 4.519, "text": "changed on this compared to regular"}, {"start": 13.0, "duration": 4.72, "text": "cotangent is the period has changed so"}, {"start": 15.719, "duration": 3.921, "text": "to figure out the new period we'll take"}, {"start": 17.72, "duration": 6.12, "text": "pi and divide it by the absolute value"}, {"start": 19.64, "duration": 6.52, "text": "of B which is 12 that'll give us 2 pi"}, {"start": 23.84, "duration": 4.48, "text": "and since it's multiplied by three it's"}, {"start": 26.16, "duration": 4.32, "text": "stretched vertically by this factor of"}, {"start": 28.32, "duration": 6.12, "text": "three but otherwise it looks"}, {"start": 30.48, "duration": 6.84, "text": "just like regular Cent soent has a"}, {"start": 34.44, "duration": 4.36, "text": "vertical ASM toote at the Y AIS or x"}, {"start": 37.32, "duration": 4.88, "text": "equals"}, {"start": 38.8, "duration": 3.4, "text": "0 to get"}, {"start": 45.44, "duration": 5.88, "text": "to another vertical ASM toote will'll"}, {"start": 48.52, "duration": 5.12, "text": "move over the period of 2 pi and"}, {"start": 51.32, "duration": 6.12, "text": "likewise if we go to the"}, {"start": 53.64, "duration": 9.2, "text": "left that'll be the line x"}, {"start": 57.44, "duration": 8.039, "text": "= -2 pi if you look between say 0 and 2"}, {"start": 62.84, "duration": 5.76, "text": "pi the point right in the middle at Pi"}, {"start": 65.479, "duration": 5.081, "text": "that's where Co cotangent will hit zero"}, {"start": 68.6, "duration": 4.08, "text": "and since it's multiplied by three it's"}, {"start": 70.56, "duration": 5.64, "text": "just going to get stretched vertically"}, {"start": 72.68, "duration": 6.799, "text": "by that factor of three likewise at Nega"}, {"start": 76.2, "duration": 6.76, "text": "Pi it's going to have the exact same"}, {"start": 79.479, "duration": 6.96, "text": "shape and there will be two periods so"}, {"start": 82.96, "duration": 8.32, "text": "the vertical ASM tootes occur"}, {"start": 86.439, "duration": 7.441, "text": "at well at x = 2 pi and then we add 2 pi"}, {"start": 91.28, "duration": 5.0, "text": "to get to the next one so we're really"}, {"start": 93.88, "duration": 5.0, "text": "just looking at multiples of 2 pi so the"}, {"start": 96.28, "duration": 5.28, "text": "vertical ASM tootes occur when x = 2 pi"}, {"start": 98.88, "duration": 6.04, "text": "n where n is an"}, {"start": 101.56, "duration": 6.32, "text": "integer and the zeros"}, {"start": 104.92, "duration": 5.4, "text": "occur when x equals well we can pick one"}, {"start": 107.88, "duration": 4.559, "text": "of them pi to get to the next one we"}, {"start": 110.32, "duration": 4.88, "text": "would have to add 2 pi because that's"}, {"start": 112.439, "duration": 8.04, "text": "the period so the zeros will occur when"}, {"start": 115.2, "duration": 8.279, "text": "x = < + 2 piun N again where n is is an"}, {"start": 120.479, "duration": 3.0, "text": "integer"}], "title": "Graphing a Cotangent Function, EX 1", "description": "Thanks to all of you who support me on Patreon. You da real mvps!   $1 per month helps!! :)  https://www.patreon.com/patrickjmt !! Graphing a Cotangent Function, EX 1\nI graph two periods of the function but it obviously repeats in both directions!", "lengthSeconds": 122, "width": 450, "height": 360}
{"video_id": "IId4Iydg_0A", "transcript": [{"start": 1.199, "duration": 3.761, "text": "okay in this example we're going to look"}, {"start": 2.52, "duration": 5.12, "text": "at finding the volume of a sphere and a"}, {"start": 4.96, "duration": 6.52, "text": "separate one we'll find the surface area"}, {"start": 7.64, "duration": 6.76, "text": "so the volume is simply 4/3 * piun * the"}, {"start": 11.48, "duration": 7.36, "text": "cube of the radius the surface area is"}, {"start": 14.4, "duration": 7.24, "text": "going to be 4 Pi * the square of the"}, {"start": 18.84, "duration": 4.72, "text": "radius so nothing too"}, {"start": 21.64, "duration": 5.719, "text": "terrible so we're going to find the"}, {"start": 23.56, "duration": 7.559, "text": "volume of a sphere with a radius of 6.2"}, {"start": 27.359, "duration": 7.321, "text": "M both in terms of Pi and also so by"}, {"start": 31.119, "duration": 8.401, "text": "approximating pi with"}, {"start": 34.68, "duration": 8.0, "text": "3.14 so it says the volume is 4/3 time"}, {"start": 39.52, "duration": 4.24, "text": "Pi time the cube of the radius so we'll"}, {"start": 42.68, "duration": 4.08, "text": "have to take"}, {"start": 43.76, "duration": 3.0, "text": "6.2"}, {"start": 46.84, "duration": 7.199, "text": "cubed so let's see"}, {"start": 49.6, "duration": 4.439, "text": "um 6.2"}, {"start": 55.199, "duration": 4.921, "text": "cubed I'm getting that to be"}, {"start": 61.44, "duration": 3.56, "text": "238"}, {"start": 63.08, "duration": 5.84, "text": "328 and I think I'm going to start"}, {"start": 65.0, "duration": 7.24, "text": "rounding this to the nearest 100th 43 is"}, {"start": 68.92, "duration": 6.04, "text": "1.333 repeating so this is now going to"}, {"start": 72.24, "duration": 4.76, "text": "be approximately equal to so 4/3 we'll"}, {"start": 74.96, "duration": 5.44, "text": "approximate that as"}, {"start": 77.0, "duration": 6.759, "text": "1.33 we still have Pi in there and then"}, {"start": 80.4, "duration": 5.8, "text": "we've got 238 point if we round to the"}, {"start": 83.759, "duration": 4.601, "text": "nearest 100th we'll have"}, {"start": 86.2, "duration": 6.12, "text": "2.38"}, {"start": 88.36, "duration": 7.92, "text": "33 so let's see um"}, {"start": 92.32, "duration": 3.96, "text": "1.33 *"}, {"start": 97.52, "duration": 5.76, "text": "23833 I'm getting again I'm going to"}, {"start": 100.479, "duration": 2.801, "text": "round so"}, {"start": 103.799, "duration": 6.041, "text": "3169 I'm getting 978 so I'm going to"}, {"start": 107.159, "duration": 5.201, "text": "make that 98 and then we can multiply"}, {"start": 109.84, "duration": 4.68, "text": "that by pi so I'm going to make my"}, {"start": 112.36, "duration": 4.52, "text": "little equal signs a little curvy just"}, {"start": 114.52, "duration": 4.68, "text": "to indicate that we're approximating so"}, {"start": 116.88, "duration": 4.239, "text": "that would be our solution in terms of"}, {"start": 119.2, "duration": 4.839, "text": "Pi"}, {"start": 121.119, "duration": 7.04, "text": "well if we actually uh change that if we"}, {"start": 124.039, "duration": 4.12, "text": "change Pi into uh"}, {"start": 134.36, "duration": 8.36, "text": "3.14 let's see so"}, {"start": 138.68, "duration": 4.04, "text": "31698 multiplied"}, {"start": 147.36, "duration": 6.48, "text": "3.14 I'm getting this to equal"}, {"start": 151.36, "duration": 4.76, "text": "95."}, {"start": 153.84, "duration": 4.679, "text": "3172 and again if we round to the"}, {"start": 156.12, "duration": 5.399, "text": "nearest hundredth because of the seven"}, {"start": 158.519, "duration": 7.08, "text": "we'll have to round the uh the hundreds"}, {"start": 161.519, "duration": 4.08, "text": "digit up one so we'll have"}, {"start": 167.28, "duration": 5.52, "text": "99532 and again since we started our"}, {"start": 169.76, "duration": 5.96, "text": "original units were in terms of"}, {"start": 172.8, "duration": 6.68, "text": "meters it says our units will be in"}, {"start": 175.72, "duration": 7.08, "text": "terms of cubic meters since we're"}, {"start": 179.48, "duration": 7.52, "text": "finding a volume a threedimensional a"}, {"start": 182.8, "duration": 4.2, "text": "threedimensional volume here"}], "title": "Volume of a Sphere", "description": "Thanks to all of you who support me on Patreon. You da real mvps!   $1 per month helps!! :)  https://www.patreon.com/patrickjmt !! Volume of a Sphere.  Here we look at a quick example of finding the surface area of a sphere.", "lengthSeconds": 185, "width": 480, "height": 360}
{"video_id": "qILCTT6wPt4", "transcript": [{"start": 0.03, "duration": 5.13, "text": "all right so one more example here about"}, {"start": 2.399, "duration": 4.201, "text": "solving logarithmic equations and in"}, {"start": 5.16, "duration": 4.859, "text": "this case we're going to solve log base"}, {"start": 6.6, "duration": 7.86, "text": "7 of X plus 4 minus log base 7 of X"}, {"start": 10.019, "duration": 5.521, "text": "minus 4 equals log base 7 of 5 so the"}, {"start": 14.46, "duration": 3.48, "text": "first thing I'm going to do is rewrite"}, {"start": 15.54, "duration": 4.739, "text": "the left side is a single logarithm and"}, {"start": 17.94, "duration": 5.13, "text": "we can write that by saying log base 7"}, {"start": 20.279, "duration": 5.781, "text": "of the first part and since it's"}, {"start": 23.07, "duration": 5.82, "text": "subtraction this will give us division"}, {"start": 26.06, "duration": 6.73, "text": "on the right side we just have log base"}, {"start": 28.89, "duration": 6.689, "text": "7 of 5 and at this point the thing I"}, {"start": 32.79, "duration": 6.48, "text": "recognize is hey you know that we've got"}, {"start": 35.579, "duration": 6.331, "text": "log base 7 of something log base 7 of"}, {"start": 39.27, "duration": 4.65, "text": "something the idea is we can just set"}, {"start": 41.91, "duration": 6.68, "text": "the stuff inside equal to each other so"}, {"start": 43.92, "duration": 9.45, "text": "this says X plus 4 over X minus 4 equals"}, {"start": 48.59, "duration": 7.09, "text": "5 and to simply get rid of the X minus 4"}, {"start": 53.37, "duration": 6.06, "text": "I'm going to multiply both sides by X"}, {"start": 55.68, "duration": 6.6, "text": "minus 4 so on the left side we would be"}, {"start": 59.43, "duration": 4.26, "text": "left with our numerator x plus 4 on the"}, {"start": 62.28, "duration": 4.29, "text": "right side I'm going to distribute that"}, {"start": 63.69, "duration": 5.28, "text": "would give 5x minus 20 and now I'm"}, {"start": 66.57, "duration": 4.619, "text": "simply trying to solve for X we can"}, {"start": 68.97, "duration": 5.13, "text": "subtract X from both sides that would"}, {"start": 71.189, "duration": 6.75, "text": "leave me with 4 on the left and now for"}, {"start": 74.1, "duration": 7.5, "text": "X on the right minus 20 then we can"}, {"start": 77.939, "duration": 6.391, "text": "simply add 20 to both sides so if we add"}, {"start": 81.6, "duration": 7.29, "text": "20 to both sides that will give us 24"}, {"start": 84.33, "duration": 8.039, "text": "equals 4x and if well if we divide both"}, {"start": 88.89, "duration": 6.0, "text": "sides by 4 we'll get x equals 6 and now"}, {"start": 92.369, "duration": 4.801, "text": "again we have to go back and check so I"}, {"start": 94.89, "duration": 4.47, "text": "go back to the very very very original"}, {"start": 97.17, "duration": 4.11, "text": "problem I was looking at and I think"}, {"start": 99.36, "duration": 4.77, "text": "well if I plug 6 inside of here I'm"}, {"start": 101.28, "duration": 4.129, "text": "going to get the log base 7 of 10 so"}, {"start": 104.13, "duration": 3.919, "text": "that's okay"}, {"start": 105.409, "duration": 4.231, "text": "log base 7 of 10 that's a positive"}, {"start": 108.049, "duration": 4.921, "text": "number so that that number we can"}, {"start": 109.64, "duration": 6.269, "text": "evaluate we would get log base 7 of well"}, {"start": 112.97, "duration": 5.429, "text": "6 minus 4 would be 2 so that's still"}, {"start": 115.909, "duration": 5.011, "text": "good because we have a logarithm of a"}, {"start": 118.399, "duration": 5.671, "text": "positive number and well does that equal"}, {"start": 120.92, "duration": 5.55, "text": "log base 7 of 5 yea because remember"}, {"start": 124.07, "duration": 5.579, "text": "we've got subtraction so we can turn the"}, {"start": 126.47, "duration": 6.12, "text": "left side into division we can make that"}, {"start": 129.649, "duration": 6.271, "text": "log base 7 of 10 over 2"}, {"start": 132.59, "duration": 6.179, "text": "which is what we want that would reduce"}, {"start": 135.92, "duration": 4.53, "text": "to log base 7 of 5 so in fact we have"}, {"start": 138.769, "duration": 5.811, "text": "found the one solution to this"}, {"start": 140.45, "duration": 4.13, "text": "logarithmic logarithmic equation"}], "title": "Solving Logarithmic Equations - Example 2", "description": "Solving Logarithmic Equations Using Properties of Logarithms\nIn this video, I'll walk you through solving a logarithmic equation by applying properties of logarithms and using some algebra techniques. We'll specifically use the formula logA - logB = log(A/B) to combine and simplify the logarithmic terms, allowing us to solve the equation effectively. This example will reinforce how to manipulate and solve equations involving logarithms.\n\nWhat You Will Learn:\n\nHow to use properties of logarithms to combine and simplify terms.\nApplying algebraic techniques to isolate variables and solve the equation.\nUnderstanding the role of cancellation properties in solving logarithmic equations.\nA practical example that demonstrates each step clearly.\n\ud83d\udcda Check out my book: 1001 Calculus Problems for Dummies for more practice!\n\n\ud83d\udc4d **If you find this video helpful, please like, share, and subscribe for more math tutorials!\n\nSupport My Work:\n\nIf you'd like to support the creation of more math content, consider becoming a patron on Patreon: https://www.patreon.com/patrickjmt\n\n:#Logarithms #SolvingEquations #MathTutorial #PatrickJMT #Algebra #Mathematics #LogProperties #LogarithmicEquations #MathHelp #Precalculus #AlgebraTechniques", "lengthSeconds": 143, "width": 480, "height": 360}
{"video_id": "CtW74iasNEU", "transcript": [{"start": 0.799, "duration": 4.081, "text": "all right in this video I want to talk"}, {"start": 2.04, "duration": 6.599, "text": "about finding the X intercepts and the Y"}, {"start": 4.88, "duration": 7.44, "text": "intercepts of an equation so"}, {"start": 8.639, "duration": 6.04, "text": "um a graph may or may not have any X"}, {"start": 12.32, "duration": 4.16, "text": "intercepts it may or may not have any y"}, {"start": 14.679, "duration": 3.281, "text": "intercepts if it's going to be a"}, {"start": 16.48, "duration": 4.52, "text": "function technically we should say"}, {"start": 17.96, "duration": 5.079, "text": "there's just a y intercept okay the"}, {"start": 21.0, "duration": 4.56, "text": "basic idea to find the Y intercept what"}, {"start": 23.039, "duration": 4.921, "text": "we do is we just plug in xals 0 and then"}, {"start": 25.56, "duration": 4.959, "text": "solve for y this is usually pretty easy"}, {"start": 27.96, "duration": 3.96, "text": "I think it's just arithmetic"}, {"start": 30.519, "duration": 3.36, "text": "second part's a little harder if you"}, {"start": 31.92, "duration": 5.0, "text": "want to find the X"}, {"start": 33.879, "duration": 5.2, "text": "intercepts we then plug in yal 0 and"}, {"start": 36.92, "duration": 3.84, "text": "solve for x and this usually you know"}, {"start": 39.079, "duration": 4.64, "text": "provides some sort of equation that you"}, {"start": 40.76, "duration": 5.72, "text": "have to work with so let's do two"}, {"start": 43.719, "duration": 4.041, "text": "examples here real quick um suppose we"}, {"start": 46.48, "duration": 4.44, "text": "want to know what are the X and Y"}, {"start": 47.76, "duration": 5.92, "text": "intercepts of the function y = 2 * x -1"}, {"start": 50.92, "duration": 5.479, "text": "* x + 2 * x -"}, {"start": 53.68, "duration": 7.0, "text": "3 and then in our other example down"}, {"start": 56.399, "duration": 6.64, "text": "here we'll have y = x^2 - 3x - 18 so to"}, {"start": 60.68, "duration": 6.72, "text": "find the Y intercept again we just plug"}, {"start": 63.039, "duration": 8.76, "text": "in X = 0 and if we plug in X = 0 we'll"}, {"start": 67.4, "duration": 9.32, "text": "get Y = 2 * well 0 - 1 would leave us"}, {"start": 71.799, "duration": 8.561, "text": "with - 1 0 + 2 would leave us with pos2"}, {"start": 76.72, "duration": 10.52, "text": "0 - 3 would leave us with -3 so let's"}, {"start": 80.36, "duration": 9.799, "text": "see 2 * -1 is -2 -2 * 2 is -4 -4 * -3 is"}, {"start": 87.24, "duration": 5.519, "text": "pos2 so it says the Y intercept is at at"}, {"start": 90.159, "duration": 4.96, "text": "positive 12 and we could also simply"}, {"start": 92.759, "duration": 6.68, "text": "conclude that the point"}, {"start": 95.119, "duration": 4.32, "text": "012 is on the"}, {"start": 100.24, "duration": 8.44, "text": "graph okay so not too bad so to find the"}, {"start": 105.079, "duration": 8.08, "text": "X intercepts do it down"}, {"start": 108.68, "duration": 7.68, "text": "here well now we substitute in y equal 0"}, {"start": 113.159, "duration": 6.28, "text": "and if we plug in y equal 0 in our"}, {"start": 116.36, "duration": 10.879, "text": "equation we'll have zero on the left we"}, {"start": 119.439, "duration": 10.96, "text": "have 2 on the right * x -1 * x + 2 * x -"}, {"start": 127.239, "duration": 5.761, "text": "3 the good thing is is that all of uh"}, {"start": 130.399, "duration": 4.681, "text": "the right side is factored Okay so we've"}, {"start": 133.0, "duration": 3.76, "text": "got two * some number time some number"}, {"start": 135.08, "duration": 3.36, "text": "time some number if we're going to get"}, {"start": 136.76, "duration": 5.479, "text": "zero on the right what would have to be"}, {"start": 138.44, "duration": 8.879, "text": "true is either x -1 would have to equal"}, {"start": 142.239, "duration": 8.121, "text": "zero or x + 2 would have to equal Z or x"}, {"start": 147.319, "duration": 5.121, "text": "- 3 would have to equal 0 well if I"}, {"start": 150.36, "duration": 5.08, "text": "solve the first equation I can add one"}, {"start": 152.44, "duration": 7.0, "text": "and get x = 1 the second equation I'll"}, {"start": 155.44, "duration": 7.519, "text": "get x = -2 and for the third equation"}, {"start": 159.44, "duration": 6.56, "text": "I'll get x = 3 so it actually says this"}, {"start": 162.959, "duration": 6.041, "text": "original function has three x intercepts"}, {"start": 166.0, "duration": 8.68, "text": "at 1 -2 and 3 or we could write them as"}, {"start": 169.0, "duration": 6.92, "text": "points as 1 comma 0 -2 comma 0 and 3"}, {"start": 174.68, "duration": 6.04, "text": "comma"}, {"start": 175.92, "duration": 8.16, "text": "0 so um and this other problem yal x^2 -"}, {"start": 180.72, "duration": 5.159, "text": "3x - 18 I'll do that in another video"}, {"start": 184.08, "duration": 6.0, "text": "same idea um we're going to have to do a"}, {"start": 185.879, "duration": 4.201, "text": "little extra step of factoring though"}], "title": "X-Intercepts and Y-Intercepts of a Functions and Finding Them! Example 1", "description": "Thanks to all of you who support me on Patreon. You da real mvps!   $1 per month helps!! :)  https://www.patreon.com/patrickjmt !! X-Intercepts and Y-Intercepts and Finding Them! Example 1", "lengthSeconds": 188, "width": 480, "height": 360}
{"video_id": "FvcHKhdaAyw", "transcript": [{"start": 0.0, "duration": 6.0, "text": "Alright in this video we are going to look at finding components of vectors along another vector"}, {"start": 6.0, "duration": 2.0, "text": "and projections of vectors onto other vectors"}, {"start": 8.0, "duration": 3.44, "text": "so to figure out the component of b onto a"}, {"start": 11.62, "duration": 3.8, "text": "we take the dot product a dot b and we divide that"}, {"start": 15.42, "duration": 3.14, "text": "by the magnitude of vector a"}, {"start": 18.7, "duration": 2.0, "text": "and then"}, {"start": 21.08, "duration": 2.44, "text": "to figure out the projection of vector b along"}, {"start": 23.52, "duration": 3.88, "text": "vector a, we take a dot b over a dot a"}, {"start": 27.4, "duration": 2.0, "text": "and multiply that by vector a"}, {"start": 29.4, "duration": 2.98, "text": "equivalently we can write that as a dot b over the"}, {"start": 32.38, "duration": 3.22, "text": "magnitude of a squared again times vector a"}, {"start": 35.6, "duration": 4.5, "text": "so if I leave off any of the little lines again those indicate vectors"}, {"start": 40.1, "duration": 5.64, "text": "everything I am talking about here is vectors so hopefully I wont accidentally do that."}, {"start": 45.74, "duration": 5.58, "text": "Okay so what we want to do is we are going to find both the component of b along a"}, {"start": 51.32, "duration": 5.08, "text": "and also the vector projection of b along a for these given vectors."}, {"start": 56.4, "duration": 4.16, "text": "alright so again to do the component"}, {"start": 62.24, "duration": 3.94, "text": "so to figure out the component of vector b along a"}, {"start": 66.18, "duration": 3.12, "text": "again we take the dot product a dot b"}, {"start": 69.3, "duration": 4.02, "text": "and then we divide that by the magnitude of vector a"}, {"start": 73.32, "duration": 3.9, "text": "so to do the dot product again we multiply respective"}, {"start": 77.22, "duration": 2.0, "text": "components so 1 times 2 will be 2"}, {"start": 79.22, "duration": 2.66, "text": "We will take 2 times -1"}, {"start": 81.88, "duration": 2.0, "text": "that will be -2"}, {"start": 83.88, "duration": 2.32, "text": "then we will add to that 3 times 4"}, {"start": 86.2, "duration": 2.0, "text": "which is going to be 12"}, {"start": 88.2, "duration": 2.46, "text": "We will divide that by the magnitude of"}, {"start": 90.66, "duration": 3.66, "text": "vector a, well the magnitude we just square the components"}, {"start": 94.32, "duration": 2.0, "text": "so 1 squared is 1"}, {"start": 96.32, "duration": 2.0, "text": "2 squared is 4"}, {"start": 98.32, "duration": 2.68, "text": "3 squared is going to be 9"}, {"start": 101.0, "duration": 2.52, "text": "and now we can just simplify this so"}, {"start": 103.52, "duration": 3.38, "text": "2 plus -2 is 0, we will have 12 on top"}, {"start": 106.9, "duration": 3.44, "text": "in the denominator looks like we are going to have"}, {"start": 110.34, "duration": 2.84, "text": "the square root of 14 so that is our component"}, {"start": 113.18, "duration": 3.08, "text": "of vector b along vector a"}, {"start": 116.26, "duration": 6.42, "text": "to do the projection of b along a"}, {"start": 122.68, "duration": 3.22, "text": "again the formula for that we just said was"}, {"start": 125.9, "duration": 5.6, "text": "a dot b and I am going to use this slightly different instead of doing a dot a"}, {"start": 131.5, "duration": 4.74, "text": "I am going to take the magnitude squared because we have already figured out the magnitude"}, {"start": 137.66, "duration": 2.58, "text": "and then we will multiply that by vector a"}, {"start": 141.64, "duration": 4.2, "text": "well again we figured out what a dot b was that is what we had in the numerator a second ago"}, {"start": 145.84, "duration": 2.0, "text": "thats just going to be 12"}, {"start": 147.84, "duration": 4.54, "text": "we can take the magnitude of vector a and square it. Well again the denominator"}, {"start": 152.38, "duration": 5.1, "text": "was the magnitude so we will just take the square root of 14 and square it."}, {"start": 157.48, "duration": 2.0, "text": "and that will just leave us with 14"}, {"start": 159.48, "duration": 3.4, "text": "and then we multiply that by our vector a"}, {"start": 162.88, "duration": 4.76, "text": "so again vector a had components 1, 2, 3"}, {"start": 167.64, "duration": 4.7, "text": "well lets see I guess this reduces to 6 over 7"}, {"start": 172.34, "duration": 2.0, "text": "and again we are just going to multiply"}, {"start": 175.78, "duration": 4.62, "text": "so recall when you have a constant out front we just multiply each component by that number"}, {"start": 180.4, "duration": 3.96, "text": "so we will get 6 over 7 times 1 which will just be 6 over 7"}, {"start": 184.36, "duration": 5.36, "text": "we will get 6 over 7 times 2 which will give us 12 over 7"}, {"start": 189.72, "duration": 3.8, "text": "and then  we will multiply 6 over 7 times 3"}, {"start": 193.52, "duration": 2.0, "text": "that will give us 18 over 7"}, {"start": 195.52, "duration": 4.38, "text": "and now we have our vector projection."}], "title": "Orthogonal Projections - Scalar and Vector Projections", "description": "Scalar and Orthogonal Projections: Quick Example\n\nIn this video, we work through a quick example of finding both the scalar and orthogonal projections of vectors. Specifically, we'll cover:\n\nHow to compute the scalar projection.\nHow to find the vector projection.\n\nUsing the vectors: a=\u27e81,2,3\u27e9andb=\u27e82,\u22121,4\u27e9\n\nI will walk you through the calculations step-by-step, explaining the key concepts involved in projections and how they relate to geometry. By the end of the video, you'll have a solid understanding of how to compute both the scalar and vector projections of one vector onto another.\n\nWhat You Will Learn:\n\nHow to compute scalar projections\nHow to compute vector projections (orthogonal projections)\nUnderstanding of the geometric significance of projections\n\nMake sure to leave a comment if you find the video helpful, and don't forget to like, share, and subscribe! Also, let your friends, teachers, or parents know about this channel if you think they would benefit from these lessons.\n\nSupport me on Patreon if you'd like to see more content: https://www.patreon.com/patrickjmt?ty=c\n\n#VectorProjections #ScalarProjections #OrthogonalProjections #MathHelp #LinearAlgebra #DotProduct #Projections #PatrickJMT #MathTutorials", "lengthSeconds": 200, "width": 480, "height": 360}
{"video_id": "eiQcqZWgojY", "transcript": [{"start": 0.0, "duration": 5.19, "text": "all right so just one more problem here"}, {"start": 2.85, "duration": 4.11, "text": "involving fractions and in this case"}, {"start": 5.19, "duration": 4.05, "text": "just a little word problem nothing too"}, {"start": 6.96, "duration": 4.74, "text": "crazy hopefully so one day we've got a"}, {"start": 9.24, "duration": 4.979, "text": "plant and it's three and three-fifths of"}, {"start": 11.7, "duration": 4.589, "text": "a centimeter tall two weeks later it's"}, {"start": 14.219, "duration": 4.14, "text": "three and seven tenths of a centimeter"}, {"start": 16.289, "duration": 5.311, "text": "tall we want to know how much did the"}, {"start": 18.359, "duration": 5.491, "text": "plant grow during those two weeks well"}, {"start": 21.6, "duration": 5.519, "text": "the growth is going to be the final"}, {"start": 23.85, "duration": 5.49, "text": "height which is three and seven tenths"}, {"start": 27.119, "duration": 6.001, "text": "and we would have to subtract away the"}, {"start": 29.34, "duration": 6.35, "text": "initial height three and three-fifths"}, {"start": 33.12, "duration": 5.04, "text": "again of a of a centimeter in both cases"}, {"start": 35.69, "duration": 4.45, "text": "well again certainly the 3s are gonna"}, {"start": 38.16, "duration": 5.969, "text": "cancel out so this is gonna be the same"}, {"start": 40.14, "duration": 6.78, "text": "thing as seven tenths minus three-fifths"}, {"start": 44.129, "duration": 5.311, "text": "and again now we have to find a common"}, {"start": 46.92, "duration": 5.369, "text": "denominator so I look at my largest"}, {"start": 49.44, "duration": 5.389, "text": "denominator which is ten I think does"}, {"start": 52.289, "duration": 5.761, "text": "the other denominator is the other"}, {"start": 54.829, "duration": 5.741, "text": "number divisible will that go into ten"}, {"start": 58.05, "duration": 5.759, "text": "as well let me say it that way well ten"}, {"start": 60.57, "duration": 6.45, "text": "is divisible by five and if we multiply"}, {"start": 63.809, "duration": 6.541, "text": "by two we're gonna get our same"}, {"start": 67.02, "duration": 8.04, "text": "denominator of ten so we have seven"}, {"start": 70.35, "duration": 7.65, "text": "tenths - it looks like six tenths well"}, {"start": 75.06, "duration": 6.93, "text": "seven minus six is gonna be one over ten"}, {"start": 78.0, "duration": 7.29, "text": "so it says the plant has grown one tenth"}, {"start": 81.99, "duration": 8.059, "text": "of a centimeter so not too much a very"}, {"start": 85.29, "duration": 4.759, "text": "very slow growing little plant there"}], "title": "Adding and Subtracting Fractions / Mixed Numbers: Example 4", "description": "Thanks to all of you who support me on Patreon. You da real mvps!   $1 per month helps!! :)  https://www.patreon.com/patrickjmt !! Adding and Subtracting Fractions / Mixed Numbers: Example 4.  Here we look at another subtraction problem involving fractions.", "lengthSeconds": 89, "width": 480, "height": 360}
{"video_id": "eXHD2lXYEOw", "transcript": [{"start": 0.53, "duration": 5.499, "text": "okay and this question we're given the"}, {"start": 3.03, "duration": 5.159, "text": "following so it says if 3x minus y"}, {"start": 6.029, "duration": 4.231, "text": "equals 12 we want to know what's the"}, {"start": 8.189, "duration": 6.691, "text": "value of 8 raised to the power of X"}, {"start": 10.26, "duration": 6.15, "text": "divided by 2 raised to the power of Y so"}, {"start": 14.88, "duration": 4.17, "text": "we have the answer choices 2 raised to"}, {"start": 16.41, "duration": 6.06, "text": "the power of 12 4 raised to the power of"}, {"start": 19.05, "duration": 5.1, "text": "4 8 raised to the power of 2 or lastly"}, {"start": 22.47, "duration": 4.02, "text": "Part D it says the value cannot be"}, {"start": 24.15, "duration": 4.17, "text": "determined from the given information so"}, {"start": 26.49, "duration": 4.049, "text": "this is one that I think you could"}, {"start": 28.32, "duration": 4.2, "text": "always start plugging in value no make"}, {"start": 30.539, "duration": 4.321, "text": "some values of x and y that give you 12"}, {"start": 32.52, "duration": 4.71, "text": "and simplify it let's do it the"}, {"start": 34.86, "duration": 4.109, "text": "algebraic way though so it's kind of"}, {"start": 37.23, "duration": 4.5, "text": "show I think what they're what they're"}, {"start": 38.969, "duration": 4.981, "text": "going for here so if we look at 8 raised"}, {"start": 41.73, "duration": 4.95, "text": "to the power of X divided by 2 raised to"}, {"start": 43.95, "duration": 4.8, "text": "the power of Y the first thing I think"}, {"start": 46.68, "duration": 5.25, "text": "the first key observation is noticing"}, {"start": 48.75, "duration": 8.329, "text": "that 8 is a power of 2 right we can"}, {"start": 51.93, "duration": 9.12, "text": "write 8 as 2 raised to the third power"}, {"start": 57.079, "duration": 6.881, "text": "because 2 times 2 times 2 times 2 is 4"}, {"start": 61.05, "duration": 6.179, "text": "times 2 will give us 8 so again 8 is"}, {"start": 63.96, "duration": 6.089, "text": "equal to 2 raised to the power of 3 so"}, {"start": 67.229, "duration": 5.46, "text": "I'm going to substitute that in so I can"}, {"start": 70.049, "duration": 5.0, "text": "have 2 to the third raised to the power"}, {"start": 72.689, "duration": 6.54, "text": "of X that's what I'm replacing my 8 with"}, {"start": 75.049, "duration": 6.191, "text": "divided by 2 raised to the power of Y so"}, {"start": 79.229, "duration": 6.331, "text": "again this is 2 raised to the power of"}, {"start": 81.24, "duration": 7.019, "text": "3x over 2 to the Y okay so hey you know"}, {"start": 85.56, "duration": 5.64, "text": "I'm seeing this 3x and this Y and that"}, {"start": 88.259, "duration": 5.79, "text": "looks you know I see this 3x minus y in"}, {"start": 91.2, "duration": 6.09, "text": "my original this original equation and"}, {"start": 94.049, "duration": 6.061, "text": "now we can use properties of exponents"}, {"start": 97.29, "duration": 4.95, "text": "and we can just subtract we can write 2"}, {"start": 100.11, "duration": 5.579, "text": "to the power of 3x divided by 2 to the"}, {"start": 102.24, "duration": 8.64, "text": "power of y as 2 raised to the power of"}, {"start": 105.689, "duration": 7.051, "text": "3x minus y and well now we know we've"}, {"start": 110.88, "duration": 5.489, "text": "got we're given this information that 3x"}, {"start": 112.74, "duration": 6.33, "text": "minus y equals 12 so we're simply left"}, {"start": 116.369, "duration": 4.621, "text": "with 2 raised to the power of 12 and"}, {"start": 119.07, "duration": 4.95, "text": "that will be our solution so answer"}, {"start": 120.99, "duration": 4.229, "text": "choice a is the correct solution you"}, {"start": 124.02, "duration": 2.849, "text": "could also do something like the"}, {"start": 125.219, "duration": 4.15, "text": "following you know again if you had to"}, {"start": 126.869, "duration": 4.541, "text": "if you weren't sure about this"}, {"start": 129.369, "duration": 6.661, "text": "you could always for example you know if"}, {"start": 131.41, "duration": 7.23, "text": "you let X equal 4 and y equals 0 right 3"}, {"start": 136.03, "duration": 3.989, "text": "times 4 minus 0 will give you 12 and"}, {"start": 138.64, "duration": 4.08, "text": "then you can substitute in you would"}, {"start": 140.019, "duration": 5.371, "text": "have 8 to the 4th over 2 to the 0 and"}, {"start": 142.72, "duration": 4.44, "text": "again you can reduce 8 to the 4th again"}, {"start": 145.39, "duration": 4.379, "text": "you have to make this observation that"}, {"start": 147.16, "duration": 4.17, "text": "you can write 8 is 2 to the power 3 but"}, {"start": 149.769, "duration": 3.75, "text": "again you could arrive at this same"}, {"start": 151.33, "duration": 4.219, "text": "exact answer by doing it that way as"}, {"start": 153.519, "duration": 2.03, "text": "well"}], "title": "SAT Math - Question 14, 2016 Edition and Beyond", "description": "Thanks to all of you who support me on Patreon. You da real mvps!   $1 per month helps!! :)  https://www.patreon.com/patrickjmt !! SAT Math - Question 14, 2016 Edition and Beyond.\nSection 1 - No Calculator allowed!", "lengthSeconds": 155, "width": 480, "height": 360}
{"video_id": "HO9oBpqaqDU", "transcript": [{"start": 0.0, "duration": 4.71, "text": "all right in this example we're gonna"}, {"start": 1.92, "duration": 5.84, "text": "start looking at some some problems"}, {"start": 4.71, "duration": 5.91, "text": "involving division of two numbers so"}, {"start": 7.76, "duration": 4.24, "text": "kind of the basic idea is in all of"}, {"start": 10.62, "duration": 3.929, "text": "these as well what we're gonna do is"}, {"start": 12.0, "duration": 4.77, "text": "we're going to rewrite things and we're"}, {"start": 14.549, "duration": 5.611, "text": "gonna get rid of the decimal place and"}, {"start": 16.77, "duration": 5.759, "text": "whatever we're dividing by so to start"}, {"start": 20.16, "duration": 8.49, "text": "off by doing that well we can write 94"}, {"start": 22.529, "duration": 7.861, "text": "divided by 0.25 as you know just with a"}, {"start": 28.65, "duration": 3.029, "text": "division instead of a you know the"}, {"start": 30.39, "duration": 3.689, "text": "division sign we can use a little"}, {"start": 31.679, "duration": 3.931, "text": "division bar what we're gonna do is"}, {"start": 34.079, "duration": 3.721, "text": "we're gonna get rid of the decimal place"}, {"start": 35.61, "duration": 4.199, "text": "in the denominator so what I would like"}, {"start": 37.8, "duration": 4.11, "text": "to do is move it two places to the right"}, {"start": 39.809, "duration": 5.191, "text": "and then in a sense it would be gone"}, {"start": 41.91, "duration": 6.39, "text": "well I'm gonna do that by multiplying by"}, {"start": 45.0, "duration": 4.289, "text": "an appropriate number so to move it two"}, {"start": 48.3, "duration": 3.66, "text": "decimal places"}, {"start": 49.289, "duration": 4.41, "text": "well let me say if you multiply by 10 it"}, {"start": 51.96, "duration": 5.579, "text": "moves at one decimal place if we"}, {"start": 53.699, "duration": 6.601, "text": "multiply by 100 it'll move the decimal"}, {"start": 57.539, "duration": 4.981, "text": "place twice but of course if we do it in"}, {"start": 60.3, "duration": 4.62, "text": "the denominator we also have to do it in"}, {"start": 62.52, "duration": 3.15, "text": "the numerator so when we do our"}, {"start": 64.92, "duration": 4.23, "text": "arithmetic"}, {"start": 65.67, "duration": 6.59, "text": "well 94 times 100 is ninety four hundred"}, {"start": 69.15, "duration": 5.37, "text": "and then again we said when we multiply"}, {"start": 72.26, "duration": 5.35, "text": "0.25 by a hundred that's just going to"}, {"start": 74.52, "duration": 4.83, "text": "move the decimal place two places so"}, {"start": 77.61, "duration": 4.77, "text": "okay equivalently we have ninety four"}, {"start": 79.35, "duration": 7.07, "text": "hundred divided by 25 and now I'm just"}, {"start": 82.38, "duration": 6.48, "text": "going to do my old long division so"}, {"start": 86.42, "duration": 5.769, "text": "let's see I always think about quarters"}, {"start": 88.86, "duration": 6.899, "text": "and money so let's see so I guess that"}, {"start": 92.189, "duration": 9.331, "text": "we could do three times 25 that would"}, {"start": 95.759, "duration": 8.461, "text": "give us 75 so let's see 94 minus 75"}, {"start": 101.52, "duration": 4.26, "text": "let's see if it was 95 we would get 20"}, {"start": 104.22, "duration": 6.719, "text": "but it's one less so I guess we'll get"}, {"start": 105.78, "duration": 7.589, "text": "19 then we'll drop down our zero so"}, {"start": 110.939, "duration": 6.47, "text": "let's see I think if we had seven"}, {"start": 113.369, "duration": 5.971, "text": "quarters seven times 25 would be 175"}, {"start": 117.409, "duration": 6.1, "text": "okay and again we can just do our"}, {"start": 119.34, "duration": 7.17, "text": "subtraction so okay we've got 190 minus"}, {"start": 123.509, "duration": 6.841, "text": "175 that'll leave us with 15"}, {"start": 126.51, "duration": 8.52, "text": "and again if we drop down our 0:25 will"}, {"start": 130.35, "duration": 8.82, "text": "go into 150 exactly six times six times"}, {"start": 135.03, "duration": 7.29, "text": "twenty-five again is 150 so we don't"}, {"start": 139.17, "duration": 10.91, "text": "have any remainder so it says if we do"}, {"start": 142.32, "duration": 10.76, "text": "94 divided by 0.25 we'll be left with"}, {"start": 150.08, "duration": 3.0, "text": "376"}], "title": "Dividing Decimals, Ex 1", "description": "Thanks to all of you who support me on Patreon. You da real mvps!   $1 per month helps!! :)  https://www.patreon.com/patrickjmt !! Dividing Decimals, Ex 1", "lengthSeconds": 153, "width": 480, "height": 360}
{"video_id": "TkL0Iqs9mpY", "transcript": [{"start": 0.799, "duration": 3.681, "text": "all right in this video I'm going to do"}, {"start": 2.2, "duration": 3.8, "text": "an example of solving a linear equation"}, {"start": 4.48, "duration": 2.879, "text": "um we've actually got three here but"}, {"start": 6.0, "duration": 4.0, "text": "we'll I think I'm going to break these"}, {"start": 7.359, "duration": 5.28, "text": "up into three separate parts uh so in"}, {"start": 10.0, "duration": 7.32, "text": "the first one we have negative uh the"}, {"start": 12.639, "duration": 8.041, "text": "quantity -3 + x + 4 2 * the quantity x +"}, {"start": 17.32, "duration": 5.36, "text": "3 - 6 x + 7 to me you know when you"}, {"start": 20.68, "duration": 3.599, "text": "solve uh linear inequalities the first"}, {"start": 22.68, "duration": 3.839, "text": "thing I do is get rid of all the"}, {"start": 24.279, "duration": 4.361, "text": "parentheses um so I'm going to"}, {"start": 26.519, "duration": 4.6, "text": "distribute the negative so negative ofg"}, {"start": 28.64, "duration": 6.48, "text": "-3 would be positive 3 and then we would"}, {"start": 31.119, "duration": 7.28, "text": "get our X plus 4 is just hanging out"}, {"start": 35.12, "duration": 7.759, "text": "we'll have to distribute the two to both"}, {"start": 38.399, "duration": 8.441, "text": "terms so we'll get 2 * X and then um 2 *"}, {"start": 42.879, "duration": 4.961, "text": "3 will be POS 6 we've got our NE -6x and"}, {"start": 46.84, "duration": 4.08, "text": "our"}, {"start": 47.84, "duration": 6.519, "text": "pos7 and usually I try to just simplify"}, {"start": 50.92, "duration": 5.52, "text": "everything on each side first so 3 + 4"}, {"start": 54.359, "duration": 5.961, "text": "will give us a postive"}, {"start": 56.44, "duration": 8.079, "text": "7x is still just sitting there um on the"}, {"start": 60.32, "duration": 8.2, "text": "right side we have 2x - 6X which is"}, {"start": 64.519, "duration": 5.561, "text": "-4x we have POS 6 + 7 which is going to"}, {"start": 68.52, "duration": 3.88, "text": "give me"}, {"start": 70.08, "duration": 4.6, "text": "positive3 and now my goal is I want to"}, {"start": 72.4, "duration": 4.48, "text": "put all the x's on the same side and all"}, {"start": 74.68, "duration": 4.24, "text": "the numbers on the other so I think what"}, {"start": 76.88, "duration": 4.919, "text": "I'm going to do here is I'm going to add"}, {"start": 78.92, "duration": 4.32, "text": "to move my negative -4x over I'll add 4X"}, {"start": 81.799, "duration": 5.32, "text": "to both"}, {"start": 83.24, "duration": 7.919, "text": "sides so then I've got seven on the left"}, {"start": 87.119, "duration": 8.561, "text": "x + 4x will give us positive"}, {"start": 91.159, "duration": 7.161, "text": "3x -4x + 4x is 0x and that's why we did"}, {"start": 95.68, "duration": 5.52, "text": "it we still have our positive 13 left"}, {"start": 98.32, "duration": 5.159, "text": "over and now um I'm going to move the"}, {"start": 101.2, "duration": 4.64, "text": "the seven over by simply subtracting"}, {"start": 103.479, "duration": 5.361, "text": "seven from both"}, {"start": 105.84, "duration": 7.8, "text": "sides um we'll be left with just our 3x"}, {"start": 108.84, "duration": 7.16, "text": "on the left 13 - 7 is going to give us 6"}, {"start": 113.64, "duration": 4.399, "text": "and now last but not least to get rid of"}, {"start": 116.0, "duration": 4.399, "text": "the the three since we're multiplying by"}, {"start": 118.039, "duration": 4.881, "text": "three to undo that I'm going to divide"}, {"start": 120.399, "duration": 6.601, "text": "both sides by 3 and that'll leave me"}, {"start": 122.92, "duration": 5.88, "text": "with 1X = pos2 and that'll be our"}, {"start": 127.0, "duration": 4.16, "text": "solution and again you can always check"}, {"start": 128.8, "duration": 4.439, "text": "that this is a solution by plugging two"}, {"start": 131.16, "duration": 4.24, "text": "into the original equation doing all the"}, {"start": 133.239, "duration": 4.201, "text": "arithmetic on the left all the"}, {"start": 135.4, "duration": 4.76, "text": "arithmetic on the right and verifying"}, {"start": 137.44, "duration": 4.2, "text": "that you do in fact get uh a number uh"}, {"start": 140.16, "duration": 4.84, "text": "the same number on each side of the"}, {"start": 141.64, "duration": 3.36, "text": "equal sign"}], "title": "Solving Linear Equations - Example 1", "description": "Thanks to all of you who support me on Patreon. You da real mvps!   $1 per month helps!! :)  https://www.patreon.com/patrickjmt !! Solving Linear Equations - Some Basic Examples", "lengthSeconds": 143, "width": 480, "height": 360}
{"video_id": "iPhZRbhJAds", "transcript": [{"start": 0.03, "duration": 3.96, "text": "okay in this video we're going to look"}, {"start": 2.07, "duration": 4.86, "text": "at some more examples involving implicit"}, {"start": 3.99, "duration": 6.0, "text": "differentiation so in Part A here we're"}, {"start": 6.93, "duration": 6.089, "text": "going to find the derivative 4x to the"}, {"start": 9.99, "duration": 4.79, "text": "negative 2nd minus Ln of Y to the third"}, {"start": 13.019, "duration": 4.231, "text": "equals 7"}, {"start": 14.78, "duration": 7.72, "text": "so okay I'm just going to rewrite it"}, {"start": 17.25, "duration": 7.41, "text": "down here one time I think one thing I'm"}, {"start": 22.5, "duration": 3.81, "text": "actually going to do first in order just"}, {"start": 24.66, "duration": 3.75, "text": "to make the derivative a little bit"}, {"start": 26.31, "duration": 4.799, "text": "easier I'm going to use these properties"}, {"start": 28.41, "duration": 4.59, "text": "of logarithms to bring the exponent out"}, {"start": 31.109, "duration": 5.761, "text": "front so we can write this as 3 times"}, {"start": 33.0, "duration": 6.09, "text": "the natural logarithm of y equals 7 and"}, {"start": 36.87, "duration": 5.64, "text": "now I will take the derivative of both"}, {"start": 39.09, "duration": 4.89, "text": "sides with respect to X so if we take"}, {"start": 42.51, "duration": 4.02, "text": "the derivative of x to the negative"}, {"start": 43.98, "duration": 5.489, "text": "second we'll get negative 2x to the"}, {"start": 46.53, "duration": 6.119, "text": "negative 3 and then when we take the"}, {"start": 49.469, "duration": 6.27, "text": "derivative of the natural logarithm of Y"}, {"start": 52.649, "duration": 6.631, "text": "we'll just get 1 over Y but then we have"}, {"start": 55.739, "duration": 5.64, "text": "to multiply by our dy/dx and then on the"}, {"start": 59.28, "duration": 4.97, "text": "right side the derivative of 7 is just"}, {"start": 61.379, "duration": 2.871, "text": "going to give us 0"}, {"start": 64.92, "duration": 5.68, "text": "so now all we have to do is just solve"}, {"start": 67.66, "duration": 4.32, "text": "for dy/dx I think I'm going to do one"}, {"start": 70.6, "duration": 5.01, "text": "thing first I'm going to write this as"}, {"start": 71.98, "duration": 10.11, "text": "negative 2 over X to the third minus 3"}, {"start": 75.61, "duration": 8.91, "text": "over Y dy over DX equals 0 and what we"}, {"start": 82.09, "duration": 5.85, "text": "can do at this point is add the negative"}, {"start": 84.52, "duration": 6.12, "text": "2 over X to the 3rd we'll add a 2 over X"}, {"start": 87.94, "duration": 6.36, "text": "to the 3rd to both sides so that'll give"}, {"start": 90.64, "duration": 6.6, "text": "us 2 over X to the 3rd on the right side"}, {"start": 94.3, "duration": 7.5, "text": "and then we're just left with negative 3"}, {"start": 97.24, "duration": 6.45, "text": "over Y dy over DX and now all we have to"}, {"start": 101.8, "duration": 5.04, "text": "do to get rid of the negative 3 over Y"}, {"start": 103.69, "duration": 8.46, "text": "we can just multiply both sides by Y"}, {"start": 106.84, "duration": 7.59, "text": "over negative 3 so Y over negative 3 and"}, {"start": 112.15, "duration": 4.62, "text": "then on the left side all that stuff a"}, {"start": 114.43, "duration": 6.469, "text": "just cancels out we'll be left with our"}, {"start": 116.77, "duration": 9.3, "text": "dy over DX and then on the right side"}, {"start": 120.899, "duration": 9.72, "text": "we've got 2y over negative 3 X to the"}, {"start": 126.07, "duration": 4.549, "text": "3rd and now we found our derivative"}], "title": "Implicit Differentiation - Basic Example 1 / 3", "description": "Thanks to all of you who support me on Patreon. You da real mvps!   $1 per month helps!! :)  https://www.patreon.com/patrickjmt !! Implicit Differentiation - Basic Example 1 / 3 .  Just a fairly straight forward example of finding a derivative using implicit differentiation.", "lengthSeconds": 130, "width": 480, "height": 360}
{"video_id": "HSpSQQNVosA", "transcript": [{"start": 0.96, "duration": 4.04, "text": "all right in this video I want to do an"}, {"start": 2.28, "duration": 6.04, "text": "example of actually uh determining"}, {"start": 5.0, "duration": 4.84, "text": "whether sort of a a more complicated uh"}, {"start": 8.32, "duration": 6.96, "text": "product of trig"}, {"start": 9.84, "duration": 5.44, "text": "functions uh is going to be even OD or"}, {"start": 15.8, "duration": 5.2, "text": "neither so remember we gave this"}, {"start": 17.92, "duration": 6.279, "text": "definition of even and odd um and I've"}, {"start": 21.0, "duration": 5.599, "text": "made a little handy dandy uh kind of"}, {"start": 24.199, "duration": 5.041, "text": "reference guide for me uh kind of"}, {"start": 26.599, "duration": 4.241, "text": "listing you know basically plugging in"}, {"start": 29.24, "duration": 3.12, "text": "Negative X and whether you get the"}, {"start": 30.84, "duration": 2.92, "text": "negative of the original or just the"}, {"start": 32.36, "duration": 3.879, "text": "original and that tells us if they're"}, {"start": 33.76, "duration": 4.44, "text": "even or odd so uh I'm going to you know"}, {"start": 36.239, "duration": 4.0, "text": "kind of refer to this and this is what"}, {"start": 38.2, "duration": 4.24, "text": "we just justified in the other"}, {"start": 40.239, "duration": 4.8, "text": "video so suppose we want to figure out"}, {"start": 42.44, "duration": 4.4, "text": "if see X time tangent X whether that's"}, {"start": 45.039, "duration": 3.721, "text": "even odd or"}, {"start": 46.84, "duration": 4.039, "text": "neither well okay we're just going to"}, {"start": 48.76, "duration": 4.439, "text": "use our definition and say we'll have to"}, {"start": 50.879, "duration": 5.32, "text": "look at F"}, {"start": 53.199, "duration": 8.721, "text": "ofx Well that's going to give us"}, {"start": 56.199, "duration": 5.721, "text": "secant ofx time tangent"}, {"start": 62.16, "duration": 5.12, "text": "ofx but if you remember in terms of"}, {"start": 64.4, "duration": 5.359, "text": "these being even odd or neither um we"}, {"start": 67.28, "duration": 4.96, "text": "said that secant ofx is really we can"}, {"start": 69.759, "duration": 5.201, "text": "replace that with just secant x all"}, {"start": 72.24, "duration": 6.6, "text": "right I'll do that and it says tangent"}, {"start": 74.96, "duration": 7.28, "text": "ofx it says we can replace tangent ofx"}, {"start": 78.84, "duration": 3.4, "text": "with just the negative of"}, {"start": 82.4, "duration": 5.8, "text": "tangent of"}, {"start": 85.159, "duration": 4.28, "text": "X well okay uh things everything's just"}, {"start": 88.2, "duration": 3.8, "text": "being multiplied you can pull the"}, {"start": 89.439, "duration": 6.841, "text": "negative back out front so we have"}, {"start": 92.0, "duration": 7.64, "text": "netive secant x time tangent of"}, {"start": 96.28, "duration": 6.519, "text": "X well uh the very original function was"}, {"start": 99.64, "duration": 6.72, "text": "see x * tangent X so really we have the"}, {"start": 102.799, "duration": 5.561, "text": "negative of that function so we started"}, {"start": 106.36, "duration": 6.16, "text": "with uh you know so we plugged in F of"}, {"start": 108.36, "duration": 6.32, "text": "negx we got out negative f ofx and again"}, {"start": 112.52, "duration": 5.16, "text": "in terms of our definitions it says if"}, {"start": 114.68, "duration": 4.88, "text": "you get F of negx if you start with that"}, {"start": 117.68, "duration": 5.039, "text": "and get out negative f ofx we would say"}, {"start": 119.56, "duration": 7.919, "text": "that function is odd so it turns out"}, {"start": 122.719, "duration": 4.76, "text": "that if you graph see x * tangent"}, {"start": 128.52, "duration": 5.04, "text": "X that's going to be an odd"}, {"start": 131.36, "duration": 4.12, "text": "function and again that just means when"}, {"start": 133.56, "duration": 5.64, "text": "you graph it it's going to be symmetric"}, {"start": 135.48, "duration": 3.72, "text": "about the origin"}], "title": "Examples with Trigonometric Functions: Even, Odd or Neither, Example 2", "description": "Thanks to all of you who support me on Patreon. You da real mvps!   $1 per month helps!! :)  https://www.patreon.com/patrickjmt !! Examples with Trigonometric Functions: Even, Odd or Neither, Example 2", "lengthSeconds": 138, "width": 480, "height": 360}
{"video_id": "Q_Mj2oZ4M1M", "transcript": [{"start": 1.36, "duration": 3.12, "text": "okay in this video we're going to look"}, {"start": 2.8, "duration": 3.68, "text": "at an example related to finding"}, {"start": 4.48, "duration": 4.119, "text": "critical points based on a"}, {"start": 6.48, "duration": 5.64, "text": "graph and the main thing to remember"}, {"start": 8.599, "duration": 5.08, "text": "critical points happen uh there's a a"}, {"start": 12.12, "duration": 4.84, "text": "couple different conditions that we look"}, {"start": 13.679, "duration": 6.641, "text": "for so we either think about where the"}, {"start": 16.96, "duration": 5.72, "text": "derivative equals zero or where the"}, {"start": 20.32, "duration": 5.719, "text": "derivative is"}, {"start": 22.68, "duration": 4.999, "text": "undefined and the idea also we have to"}, {"start": 26.039, "duration": 3.56, "text": "keep in mind is that these points have"}, {"start": 27.679, "duration": 4.08, "text": "to be in the domain of the original"}, {"start": 29.599, "duration": 4.161, "text": "function so here our original graph"}, {"start": 31.759, "duration": 4.281, "text": "looks like a nice smooth continuous"}, {"start": 33.76, "duration": 4.959, "text": "function so uh All Points would be in"}, {"start": 36.04, "duration": 4.4, "text": "the domain and what I'm going to look at"}, {"start": 38.719, "duration": 5.201, "text": "now so we have kind of a few different x"}, {"start": 40.44, "duration": 5.84, "text": "coordinates you know -6 -1 posi 5 I've"}, {"start": 43.92, "duration": 4.0, "text": "got 10 listed here we want to know at"}, {"start": 46.28, "duration": 3.64, "text": "which of those x coordinates again do we"}, {"start": 47.92, "duration": 3.92, "text": "have a critical point well the"}, {"start": 49.92, "duration": 3.84, "text": "derivative is going to equal zero if you"}, {"start": 51.84, "duration": 3.84, "text": "think about the derivative equaling zero"}, {"start": 53.76, "duration": 5.72, "text": "that means the slope of the tangent line"}, {"start": 55.68, "duration": 5.48, "text": "has to equal zero well at -1 um it looks"}, {"start": 59.48, "duration": 4.44, "text": "like the slope of that tangent line"}, {"start": 61.16, "duration": 5.12, "text": "could certainly equal zero so I would"}, {"start": 63.92, "duration": 6.64, "text": "say at x = -1 that's where the"}, {"start": 66.28, "duration": 5.879, "text": "derivative equals zero so in that case"}, {"start": 70.56, "duration": 4.08, "text": "uh that would be one of my critical"}, {"start": 72.159, "duration": 4.6, "text": "points the other place you have critical"}, {"start": 74.64, "duration": 4.32, "text": "points are where the function there any"}, {"start": 76.759, "duration": 5.081, "text": "we call them cusps anywhere there's kind"}, {"start": 78.96, "duration": 5.08, "text": "of any sharp points any abrupt changes"}, {"start": 81.84, "duration": 5.04, "text": "in Direction so notice here there's kind"}, {"start": 84.04, "duration": 3.92, "text": "of a Sharp little point on the graph and"}, {"start": 86.88, "duration": 3.52, "text": "that would be a place where the"}, {"start": 87.96, "duration": 5.439, "text": "derivative would actually be undefined"}, {"start": 90.4, "duration": 5.88, "text": "so we would say at x = 5 the derivative"}, {"start": 93.399, "duration": 6.72, "text": "is undefined and those two points uh we"}, {"start": 96.28, "duration": 7.28, "text": "would say at x = -1 and at xal 5 those"}, {"start": 100.119, "duration": 4.761, "text": "would be the x coordinates uh those"}, {"start": 103.56, "duration": 3.519, "text": "would be the x coordinates that"}, {"start": 104.88, "duration": 6.199, "text": "correspond to points where a function"}, {"start": 107.079, "duration": 4.0, "text": "does have a critical point"}], "title": "Critical Points from a Graph", "description": "Thanks to all of you who support me on Patreon. You da real mvps!   $1 per month helps!! :)  https://www.patreon.com/patrickjmt !! Just a quick example of finding critical points from a given graph.\nRecall that if f ' (x) = 0 or f ' (x) is undefined, there is a critical point.", "lengthSeconds": 109, "width": 480, "height": 360}
{"video_id": "VTAsbx5wBaI", "transcript": [{"start": 0.68, "duration": 4.52, "text": "all right in this video I'm going to"}, {"start": 1.839, "duration": 6.401, "text": "talk about rewriting some powers of I"}, {"start": 5.2, "duration": 5.479, "text": "and the main thing to remember kind of"}, {"start": 8.24, "duration": 4.519, "text": "is a cyclical nature about powers of I"}, {"start": 10.679, "duration": 4.16, "text": "so I went ahead and wrote some down here"}, {"start": 12.759, "duration": 5.921, "text": "so I to the 1 that's just I I mean"}, {"start": 14.839, "duration": 7.761, "text": "anything raised to itself is itself I 2"}, {"start": 18.68, "duration": 5.72, "text": "by definition we say that I * I is -1"}, {"start": 22.6, "duration": 5.64, "text": "well if you think about I to the 3D"}, {"start": 24.4, "duration": 6.84, "text": "power we can write that as i^ 2 * I we"}, {"start": 28.24, "duration": 5.44, "text": "know that i^ 2 is-1 so if we multiply"}, {"start": 31.24, "duration": 5.4, "text": "that by I we'll just get negative"}, {"start": 33.68, "duration": 4.52, "text": "I I 4th this is going to end up being"}, {"start": 36.64, "duration": 4.68, "text": "kind of one of the ones that I use"}, {"start": 38.2, "duration": 5.92, "text": "that's important to me so I to 4th we"}, {"start": 41.32, "duration": 6.36, "text": "can write that as i^ 2 * i^ 2 but that's"}, {"start": 44.12, "duration": 7.52, "text": "just - 1 * 1 and that gives me positive"}, {"start": 47.68, "duration": 6.76, "text": "1 and likewise notice if you do I to 5th"}, {"start": 51.64, "duration": 5.719, "text": "that's I 4th * I to the 1st which is 1"}, {"start": 54.44, "duration": 6.48, "text": "time I you get I hey we're back where we"}, {"start": 57.359, "duration": 5.961, "text": "started Okay so"}, {"start": 60.92, "duration": 5.0, "text": "again one of the things that's important"}, {"start": 63.32, "duration": 4.479, "text": "to me is if if the exponent is a"}, {"start": 65.92, "duration": 6.16, "text": "multiple of"}, {"start": 67.799, "duration": 7.121, "text": "four so for example I to the 8th or I to"}, {"start": 72.08, "duration": 5.32, "text": "the 12th or to the 16th or to the 20th"}, {"start": 74.92, "duration": 4.68, "text": "if the exponent is a multiple of four"}, {"start": 77.4, "duration": 5.6, "text": "that's always going to equal the number"}, {"start": 79.6, "duration": 5.6, "text": "one okay so that's one little trick I"}, {"start": 83.0, "duration": 4.88, "text": "use when I when I try to simplify these"}, {"start": 85.2, "duration": 5.32, "text": "down so all right so three little or"}, {"start": 87.88, "duration": 6.44, "text": "excuse me three five examples here"}, {"start": 90.52, "duration": 5.959, "text": "um again just I to kind of some big"}, {"start": 94.32, "duration": 5.64, "text": "exponents so in the first one we have I"}, {"start": 96.479, "duration": 5.201, "text": "to the -42 I'm going to write that as -1"}, {"start": 99.96, "duration": 6.36, "text": "* I to"}, {"start": 101.68, "duration": 8.32, "text": "42nd and again I'm thinking first off um"}, {"start": 106.32, "duration": 6.92, "text": "is 42 a multiple of four and well I"}, {"start": 110.0, "duration": 6.479, "text": "don't think four goes into 42 evenly but"}, {"start": 113.24, "duration": 7.839, "text": "it will go into 40 evenly it would go 10"}, {"start": 116.479, "duration": 7.68, "text": "times and then I would need an i squared"}, {"start": 121.079, "duration": 5.881, "text": "okay so what do we have here we've got"}, {"start": 124.159, "duration": 5.761, "text": "-1 notice I to the 40th that's a"}, {"start": 126.96, "duration": 4.359, "text": "multiple of four So based on what we"}, {"start": 129.92, "duration": 4.319, "text": "said a second ago that's just going to"}, {"start": 131.319, "duration": 6.441, "text": "equal the number one we still have our I"}, {"start": 134.239, "duration": 7.441, "text": "squ left over but i^ 2 is equal to the"}, {"start": 137.76, "duration": 6.839, "text": "number Nega 1 so again our I to the 40th"}, {"start": 141.68, "duration": 5.44, "text": "is turning into positive 1 and our i^ 2"}, {"start": 144.599, "duration": 2.521, "text": "is turning"}, {"start": 147.519, "duration": 6.681, "text": "into1 um and now we can just simplify it"}, {"start": 150.04, "duration": 6.199, "text": "says we have - 1 * 1 * - 1 that's simply"}, {"start": 154.2, "duration": 5.72, "text": "going to give us positive 1 as our"}, {"start": 156.239, "duration": 6.441, "text": "solution so negative I to the 42nd power"}, {"start": 159.92, "duration": 5.52, "text": "is a weird way of writing the number"}, {"start": 162.68, "duration": 5.96, "text": "one okay so I to the 31st I'm just going"}, {"start": 165.44, "duration": 7.56, "text": "to do the same trick I think does uh"}, {"start": 168.64, "duration": 7.239, "text": "four go into 31 evenly I say well no um"}, {"start": 173.0, "duration": 6.48, "text": "but I think it would go into it uh seven"}, {"start": 175.879, "duration": 7.08, "text": "time excuse me seven times which means"}, {"start": 179.48, "duration": 7.6, "text": "we could use I to the 28th power and"}, {"start": 182.959, "duration": 7.881, "text": "then if we write that as I to the 3 so"}, {"start": 187.08, "duration": 8.48, "text": "again 28 is a multiple of four it's 7 *"}, {"start": 190.84, "duration": 7.759, "text": "4 so that's just going to give us 1 * I"}, {"start": 195.56, "duration": 5.12, "text": "3r and again the way I really write this"}, {"start": 198.599, "duration": 4.441, "text": "is now this is kind of a small exponent"}, {"start": 200.68, "duration": 5.96, "text": "I just break it down so I think I to the"}, {"start": 203.04, "duration": 9.44, "text": "3 that's i^ 2 *"}, {"start": 206.64, "duration": 7.84, "text": "I but i^ 2 we said that's 1 1 * I and"}, {"start": 212.48, "duration": 6.119, "text": "that'll simply leave us with a negative"}, {"start": 214.48, "duration": 6.119, "text": "I so okay so two quick examples there um"}, {"start": 218.599, "duration": 3.801, "text": "I'm going to do the other four examples"}, {"start": 220.599, "duration": 4.84, "text": "or excuse me the other three examples if"}, {"start": 222.4, "duration": 6.24, "text": "I can count um in another video uh"}, {"start": 225.439, "duration": 3.201, "text": "really quickly"}], "title": "Rewriting Powers of  ' i  '  - Ex 1", "description": "Thanks to all of you who support me on Patreon. You da real mvps!   $1 per month helps!! :)  https://www.patreon.com/patrickjmt !! Rewriting Powers of  ' i  '  - In this video, I take the complex number ' i ', raise it to some different powers, and simplify!", "lengthSeconds": 227, "width": 480, "height": 360}
{"video_id": "YyiCC3EDs98", "transcript": [{"start": 1.12, "duration": 5.679, "text": "okay in this example we want to identify"}, {"start": 3.639, "duration": 5.361, "text": "two opposite rays and the definition"}, {"start": 6.799, "duration": 5.88, "text": "here we're looking for two rays with a"}, {"start": 9.0, "duration": 7.52, "text": "common endpoint that point in opposite"}, {"start": 12.679, "duration": 6.881, "text": "directions and they form a straight line"}, {"start": 16.52, "duration": 5.599, "text": "so recall the definition of array array"}, {"start": 19.56, "duration": 5.4, "text": "is a portion of a line which just starts"}, {"start": 22.119, "duration": 5.721, "text": "at a single point and then goes off in"}, {"start": 24.96, "duration": 5.319, "text": "some particular direction to Infinity"}, {"start": 27.84, "duration": 3.399, "text": "okay so that's what we're looking for in"}, {"start": 30.279, "duration": 3.641, "text": "this"}, {"start": 31.239, "duration": 7.761, "text": "case well let's see here we've got the"}, {"start": 33.92, "duration": 8.319, "text": "ray OA and OC well OA would be this Ray"}, {"start": 39.0, "duration": 6.28, "text": "that goes uh directly directly up points"}, {"start": 42.239, "duration": 5.521, "text": "in the north Direction here and"}, {"start": 45.28, "duration": 4.52, "text": "OC um it's definitely in a different"}, {"start": 47.76, "duration": 4.56, "text": "direction but again these two rays do"}, {"start": 49.8, "duration": 4.279, "text": "not form a straight line so answer"}, {"start": 52.32, "duration": 4.2, "text": "Choice one would not be"}, {"start": 54.079, "duration": 7.241, "text": "correct uh we've got"}, {"start": 56.52, "duration": 7.679, "text": "OB and O well those share a point and"}, {"start": 61.32, "duration": 5.2, "text": "they do form a straight line going an"}, {"start": 64.199, "duration": 5.121, "text": "opposite direction so certainly it looks"}, {"start": 66.52, "duration": 5.08, "text": "like answer Choice number two here is"}, {"start": 69.32, "duration": 5.0, "text": "correct let's look at the other ones as"}, {"start": 71.6, "duration": 6.64, "text": "well just to make sure we've got the ray"}, {"start": 74.32, "duration": 6.479, "text": "OE so that would be this Ray and OB"}, {"start": 78.24, "duration": 5.16, "text": "again well those do not form a straight"}, {"start": 80.799, "duration": 5.041, "text": "line so that's why answer Choice number"}, {"start": 83.4, "duration": 6.679, "text": "three wouldn't work and number four"}, {"start": 85.84, "duration": 7.08, "text": "we've got OA and OD again the same idea"}, {"start": 90.079, "duration": 5.241, "text": "those rays do share a point but again"}, {"start": 92.92, "duration": 4.96, "text": "they don't form a straight line so"}, {"start": 95.32, "duration": 4.68, "text": "answer Choice number two would make our"}, {"start": 97.88, "duration": 4.96, "text": "our little straight line here and that"}, {"start": 100.0, "duration": 2.84, "text": "would be the correct"}, {"start": 103.88, "duration": 3.0, "text": "choice"}], "title": "Rays : Identify Opposite Rays ( Geometry )", "description": "Thanks to all of you who support me on Patreon. You da real mvps!   $1 per month helps!! :)  https://www.patreon.com/patrickjmt !! Rays : Identify Opposite Rays.\nHere we identify a pair of opposite rays that all start from the same point.", "lengthSeconds": 105, "width": 480, "height": 360}
{"video_id": "oGwNLitgbqY", "transcript": [{"start": 0.0, "duration": 3.78, "text": "all right this video we're going to talk"}, {"start": 1.86, "duration": 4.26, "text": "about inconsistent systems of linear"}, {"start": 3.78, "duration": 3.39, "text": "equations and what we're doing really is"}, {"start": 6.12, "duration": 4.2, "text": "we're just trying to determine if a"}, {"start": 7.17, "duration": 4.53, "text": "system has no solutions again what we're"}, {"start": 10.32, "duration": 5.01, "text": "trying to do is we're going to use"}, {"start": 11.7, "duration": 6.45, "text": "elimination by addition to approach"}, {"start": 15.33, "duration": 6.42, "text": "these again there's certainly other"}, {"start": 18.15, "duration": 5.91, "text": "techniques to solve these systems of"}, {"start": 21.75, "duration": 4.89, "text": "equations but this is what we'll pick on"}, {"start": 24.06, "duration": 4.08, "text": "for this problem so true or false the"}, {"start": 26.64, "duration": 3.719, "text": "following system of equations is"}, {"start": 28.14, "duration": 8.13, "text": "inconsistent we've got 2x minus 2y plus"}, {"start": 30.359, "duration": 10.38, "text": "6z equals 10 3x minus 5y plus 12 Z"}, {"start": 36.27, "duration": 7.92, "text": "equals negative 6 for X minus 4y plus 12"}, {"start": 40.739, "duration": 5.311, "text": "Z equals 18 so one thing you can do is"}, {"start": 44.19, "duration": 5.49, "text": "just simply start trying to eliminate"}, {"start": 46.05, "duration": 4.95, "text": "one of your variables just you know the"}, {"start": 49.68, "duration": 3.149, "text": "normal way you'd go about doing"}, {"start": 51.0, "duration": 5.16, "text": "elimination by addition I'm going to"}, {"start": 52.829, "duration": 8.271, "text": "look at the first and third equation so"}, {"start": 56.16, "duration": 6.859, "text": "notice we've got 2x minus 2y plus 6z"}, {"start": 61.1, "duration": 8.519, "text": "equals 10"}, {"start": 63.019, "duration": 10.42, "text": "we've got 4x minus 4y plus 12 Z equals"}, {"start": 69.619, "duration": 7.511, "text": "18 well for example if I wanted to try"}, {"start": 73.439, "duration": 7.141, "text": "to get the X's to cancel out I could"}, {"start": 77.13, "duration": 5.309, "text": "multiply my first row by negative 2 and"}, {"start": 80.58, "duration": 5.55, "text": "we would have to do that to both sides"}, {"start": 82.439, "duration": 8.811, "text": "and then what will we get well our first"}, {"start": 86.13, "duration": 8.07, "text": "row would become negative 4x positive 4y"}, {"start": 91.25, "duration": 5.979, "text": "negative 12 Z and then on the right side"}, {"start": 94.2, "duration": 6.83, "text": "we would have negative 20 if we take our"}, {"start": 97.229, "duration": 10.481, "text": "original second equation we've got 4x"}, {"start": 101.03, "duration": 8.66, "text": "minus 4y plus 12 Z equals 18 and"}, {"start": 107.71, "duration": 4.769, "text": "this if we now do our elimination by"}, {"start": 109.69, "duration": 8.88, "text": "addition if we add those together we get"}, {"start": 112.479, "duration": 8.011, "text": "zero X plus zero y plus zero Z which"}, {"start": 118.57, "duration": 5.01, "text": "says you're basically just left with"}, {"start": 120.49, "duration": 7.5, "text": "zero on the left side equals well"}, {"start": 123.58, "duration": 7.71, "text": "negative 20 plus 18 is negative two so"}, {"start": 127.99, "duration": 6.99, "text": "this says hey zero equals negative two"}, {"start": 131.29, "duration": 6.15, "text": "which is clearly not the case so this"}, {"start": 134.98, "duration": 4.62, "text": "kind of a this statement that's nonsense"}, {"start": 137.44, "duration": 3.69, "text": "is clearly not true statement it's"}, {"start": 139.6, "duration": 3.389, "text": "trying to tell you something and what"}, {"start": 141.13, "duration": 6.3, "text": "it's trying to tell you is it says there"}, {"start": 142.989, "duration": 7.311, "text": "is no solution is what that means and if"}, {"start": 147.43, "duration": 6.02, "text": "there's no solution to your system of"}, {"start": 150.3, "duration": 5.469, "text": "equations we do call that an"}, {"start": 153.45, "duration": 5.38, "text": "inconsistent system it doesn't make"}, {"start": 155.769, "duration": 6.571, "text": "sense somehow okay so so I would say yes"}, {"start": 158.83, "duration": 7.56, "text": "this is true this this following system"}, {"start": 162.34, "duration": 6.45, "text": "of equations is inconsistent one way in"}, {"start": 166.39, "duration": 3.929, "text": "this you know this this isn't exclusive"}, {"start": 168.79, "duration": 3.57, "text": "but one way you can look at it and"}, {"start": 170.319, "duration": 4.401, "text": "determine that a system is inconsistent"}, {"start": 172.36, "duration": 6.24, "text": "and this is basically the trick I used"}, {"start": 174.72, "duration": 6.579, "text": "notice that the third row is a multiple"}, {"start": 178.6, "duration": 5.13, "text": "almost of the first row I mean if you"}, {"start": 181.299, "duration": 4.41, "text": "multiply the first number by two we'll"}, {"start": 183.73, "duration": 4.89, "text": "get the you know we'll get the 4x if we"}, {"start": 185.709, "duration": 6.691, "text": "multiply the negative 2y by two we get"}, {"start": 188.62, "duration": 6.209, "text": "negative 4y if we multiply 6z by two we"}, {"start": 192.4, "duration": 4.53, "text": "get 12 Z so if I multiply the first"}, {"start": 194.829, "duration": 4.651, "text": "equation by two everything matches up"}, {"start": 196.93, "duration": 5.94, "text": "until on the right side if we multiply"}, {"start": 199.48, "duration": 5.49, "text": "10 by two we don't get 18 and if"}, {"start": 202.87, "duration": 4.35, "text": "everything matches up except for the"}, {"start": 204.97, "duration": 3.989, "text": "numbers basically what's going to happen"}, {"start": 207.22, "duration": 3.51, "text": "when you do your elimination by addition"}, {"start": 208.959, "duration": 4.141, "text": "you're going to get some nonsense"}, {"start": 210.73, "duration": 4.53, "text": "statement so something like this I could"}, {"start": 213.1, "duration": 3.72, "text": "just look at make that realization to be"}, {"start": 215.26, "duration": 3.74, "text": "like oh there's not going to be any"}, {"start": 216.82, "duration": 6.65, "text": "solutions to this system of equations"}, {"start": 219.0, "duration": 4.47, "text": "therefore it would be inconsistent"}], "title": "Systems of Linear Equations - Inconsistent Systems Using Elimination by Addition - Example 1", "description": "Thanks to all of you who support me on Patreon. You da real mvps!   $1 per month helps!! :)  https://www.patreon.com/patrickjmt !! Systems of Linear Equations - Inconsistent Systems Using Elimination by Addition - Example 1", "lengthSeconds": 222, "width": 480, "height": 360}
{"video_id": "pP4VWFGMHNg", "transcript": [{"start": 0.26, "duration": 4.99, "text": "all right so a couple more examples here"}, {"start": 2.73, "duration": 8.76, "text": "dividing whole numbers so in part a"}, {"start": 5.25, "duration": 10.65, "text": "we're gonna have three 37,200 4/6 Part B"}, {"start": 11.49, "duration": 6.36, "text": "we'll have 13,000 267 divided by 18 and"}, {"start": 15.9, "duration": 5.73, "text": "I think I'll do these in separate"}, {"start": 17.85, "duration": 5.7, "text": "examples here just because mainly"}, {"start": 21.63, "duration": 7.76, "text": "because Part B can be a little bit long"}, {"start": 23.55, "duration": 9.39, "text": "so let me rewrite it real quick"}, {"start": 29.39, "duration": 6.759, "text": "so dividing by a single digit such as"}, {"start": 32.94, "duration": 6.51, "text": "six really makes life a bit easier for"}, {"start": 36.149, "duration": 7.621, "text": "sure you'll see Part B certainly is a"}, {"start": 39.45, "duration": 7.32, "text": "bit longer so six will go into three"}, {"start": 43.77, "duration": 6.03, "text": "well zero times so I'm gonna move it"}, {"start": 46.77, "duration": 5.58, "text": "over one place also in this example like"}, {"start": 49.8, "duration": 5.31, "text": "I made them up I'm pretty sure we should"}, {"start": 52.35, "duration": 6.27, "text": "have a remainder somewhere so definitely"}, {"start": 55.11, "duration": 8.67, "text": "in Part B so six will go into 37 let's"}, {"start": 58.62, "duration": 7.819, "text": "see so I guess six times so 6 times 6 is"}, {"start": 63.78, "duration": 7.949, "text": "36 really gonna know your multiplication"}, {"start": 66.439, "duration": 8.141, "text": "tables here really 37 minus 36 will be 1"}, {"start": 71.729, "duration": 7.441, "text": "I'll drop down my next digit which is a"}, {"start": 74.58, "duration": 8.34, "text": "2 well I'll say 6 will go into 12 two"}, {"start": 79.17, "duration": 7.65, "text": "times so 2 times 6 is 12 you can"}, {"start": 82.92, "duration": 6.629, "text": "subtract get our 0 I'm gonna drop down"}, {"start": 86.82, "duration": 6.619, "text": "my next digit which pay happens to be a"}, {"start": 89.549, "duration": 9.061, "text": "0 so I would say well 6 will go into 0"}, {"start": 93.439, "duration": 8.441, "text": "zero times so 0 times 6 zero again we"}, {"start": 98.61, "duration": 9.689, "text": "can drop down our last digit which is a"}, {"start": 101.88, "duration": 10.559, "text": "4 again 6 will go into 4 zero times zero"}, {"start": 108.299, "duration": 6.721, "text": "times six zero again if we subtract the"}, {"start": 112.439, "duration": 6.121, "text": "left with four we've kind of exhausted"}, {"start": 115.02, "duration": 5.52, "text": "all the digits and in this case after"}, {"start": 118.56, "duration": 7.8, "text": "we've done that this number is what's"}, {"start": 120.54, "duration": 7.98, "text": "known as the remainder so it says if we"}, {"start": 126.36, "duration": 6.57, "text": "take three seven two"}, {"start": 128.52, "duration": 9.87, "text": "Oh for and divide it by 6 it says we'll"}, {"start": 132.93, "duration": 10.19, "text": "get 6,200 and a remainder of 4 so it"}, {"start": 138.39, "duration": 8.31, "text": "says 6 will go into 37,200 for exactly"}, {"start": 143.12, "duration": 6.81, "text": "6200 times with a little bit left over"}, {"start": 146.7, "duration": 3.23, "text": "with 4 left over"}], "title": "Long Division of Numbers - Arithmetic Basics, Ex 3", "description": "Thanks to all of you who support me on Patreon. You da real mvps!   $1 per month helps!! :)  https://www.patreon.com/patrickjmt !! Example 4 : https://www.youtube.com/watch?v=So9YQbSh5PI\nLong Division of Numbers - Arithmetic Basics, Ex 2. In this video, I show how to make a fraction into a decimal doing long division.\nFor more free math videos, visit http://PatrickJMT.com", "lengthSeconds": 150, "width": 480, "height": 360}
{"video_id": "LkvRmBnoAt4", "transcript": [{"start": 2.159, "duration": 3.361, "text": "all right in this video we're going to"}, {"start": 3.439, "duration": 4.561, "text": "look at integrating some composite"}, {"start": 5.52, "duration": 4.119, "text": "exponential uh functions and the"}, {"start": 8.0, "duration": 5.12, "text": "separate video we'll look at some"}, {"start": 9.639, "duration": 5.04, "text": "rational functions so um in this one in"}, {"start": 13.12, "duration": 5.36, "text": "part A we're going to integrate here sin"}, {"start": 14.679, "duration": 5.44, "text": "of x * e to the cosine X and again when"}, {"start": 18.48, "duration": 4.6, "text": "I see something like this you know kind"}, {"start": 20.119, "duration": 4.521, "text": "of the first thing I think I I notice a"}, {"start": 23.08, "duration": 3.32, "text": "couple things I see the sign and the"}, {"start": 24.64, "duration": 4.039, "text": "cosine in there and those are"}, {"start": 26.4, "duration": 4.16, "text": "derivatives you know of each other up to"}, {"start": 28.679, "duration": 3.521, "text": "you know a constant obviously when you"}, {"start": 30.56, "duration": 3.839, "text": "take the derivative of cosine you get"}, {"start": 32.2, "duration": 3.44, "text": "negative sign but you know they're"}, {"start": 34.399, "duration": 3.081, "text": "they're they're close enough to where I"}, {"start": 35.64, "duration": 4.56, "text": "could potentially make a U substitution"}, {"start": 37.48, "duration": 5.079, "text": "work and also the only formula I really"}, {"start": 40.2, "duration": 4.6, "text": "have to integrate e is if I just have e"}, {"start": 42.559, "duration": 4.721, "text": "to a single variable you know e to the x"}, {"start": 44.8, "duration": 4.36, "text": "or e to the U or e to the whatever so"}, {"start": 47.28, "duration": 4.4, "text": "that to me suggests maybe we should let"}, {"start": 49.16, "duration": 3.8, "text": "U equal the exponent here which is"}, {"start": 51.68, "duration": 4.679, "text": "cosine"}, {"start": 52.96, "duration": 8.16, "text": "X our DU would be"}, {"start": 56.359, "duration": 6.52, "text": "sinx DX well we have a positive sinx DX"}, {"start": 61.12, "duration": 4.96, "text": "so I'm just going to multiply both sides"}, {"start": 62.879, "duration": 6.401, "text": "of this by -1 which will make this now"}, {"start": 66.08, "duration": 5.679, "text": "positive sinx DX and now we can replace"}, {"start": 69.28, "duration": 6.68, "text": "everything so it says again our sinx DX"}, {"start": 71.759, "duration": 6.601, "text": "we'll replace that with -1 du and then"}, {"start": 75.96, "duration": 4.119, "text": "we would be left with e but instead of"}, {"start": 78.36, "duration": 5.0, "text": "cosine X again that's what we're calling"}, {"start": 80.079, "duration": 6.481, "text": "U so now we're just integrating netive e"}, {"start": 83.36, "duration": 6.079, "text": "to the U well uh the negative one comes"}, {"start": 86.56, "duration": 4.599, "text": "along the anti-derivative of e to the U"}, {"start": 89.439, "duration": 5.601, "text": "is just e to the"}, {"start": 91.159, "duration": 6.92, "text": "U let's stick on our plus C and now I'm"}, {"start": 95.04, "duration": 6.16, "text": "going to replace the U with our"}, {"start": 98.079, "duration": 5.801, "text": "substitution which was cosine X and now"}, {"start": 101.2, "duration": 2.68, "text": "we've got the"}, {"start": 104.04, "duration": 5.16, "text": "anti-derivative okay so let's look at uh"}, {"start": 107.0, "duration": 4.36, "text": "the second example here as well kind of"}, {"start": 109.2, "duration": 4.879, "text": "almost the same idea you know I've got e"}, {"start": 111.36, "duration": 5.28, "text": "to some stuff um I've got to make that"}, {"start": 114.079, "duration": 4.72, "text": "into a single variable so and again I"}, {"start": 116.64, "duration": 4.079, "text": "see stuff cubed and stuff squared when I"}, {"start": 118.799, "duration": 3.801, "text": "take the derivative of that I'll get"}, {"start": 120.719, "duration": 4.281, "text": "stuff squared and stuff to the first"}, {"start": 122.6, "duration": 4.36, "text": "Power which is what we have out front"}, {"start": 125.0, "duration": 3.08, "text": "oops I left off my DX again so let's"}, {"start": 126.96, "duration": 3.4, "text": "stick that in"}, {"start": 128.08, "duration": 7.08, "text": "there so I'm going to pick youu to be"}, {"start": 130.36, "duration": 7.36, "text": "the uh exponent 2x 3r +"}, {"start": 135.16, "duration": 4.12, "text": "3x^2 that would make our DU well let's"}, {"start": 137.72, "duration": 2.879, "text": "see the three would come out so that"}, {"start": 139.28, "duration": 6.0, "text": "would give us"}, {"start": 140.599, "duration": 7.961, "text": "6x2 plus 2 * 3 would give us 6X and then"}, {"start": 145.28, "duration": 5.239, "text": "all of that times DX uh notice we don't"}, {"start": 148.56, "duration": 3.8, "text": "want you know really any Co efficients"}, {"start": 150.519, "duration": 4.641, "text": "we don't want the six in front of the x"}, {"start": 152.36, "duration": 5.879, "text": "s to the X so what I can do is we can"}, {"start": 155.16, "duration": 3.079, "text": "just Factor the six"}, {"start": 160.12, "duration": 4.72, "text": "out and well to get rid of the six you"}, {"start": 162.64, "duration": 3.72, "text": "can basically just multiply both sides"}, {"start": 164.84, "duration": 4.8, "text": "by one over"}, {"start": 166.36, "duration": 7.28, "text": "six and that that will leave us with x^2"}, {"start": 169.64, "duration": 5.4, "text": "+ x DX on the right side and now again I"}, {"start": 173.64, "duration": 4.0, "text": "think we can go ahead and just you know"}, {"start": 175.04, "duration": 7.04, "text": "drop everything into there"}, {"start": 177.64, "duration": 6.56, "text": "so so our X2 + x DX it says all of that"}, {"start": 182.08, "duration": 3.68, "text": "is going to be replaced with the 1 over"}, {"start": 184.2, "duration": 4.16, "text": "6"}, {"start": 185.76, "duration": 5.96, "text": "du and then again we're just left with e"}, {"start": 188.36, "duration": 7.2, "text": "to the well again our exponent was 2x 3r"}, {"start": 191.72, "duration": 5.32, "text": "+ 3x^2 that's what we're we're calling U"}, {"start": 195.56, "duration": 3.319, "text": "so again we've just turned it into a"}, {"start": 197.04, "duration": 5.559, "text": "substitution into a problem where we"}, {"start": 198.879, "duration": 6.601, "text": "just have the integral of e to the U"}, {"start": 202.599, "duration": 6.681, "text": "du so if we integrate e to the U again"}, {"start": 205.48, "duration": 5.52, "text": "we just get e to the U plus C and now"}, {"start": 209.28, "duration": 7.08, "text": "the same thing before let's just plug"}, {"start": 211.0, "duration": 9.84, "text": "our u in which is 2 x 3r +"}, {"start": 216.36, "duration": 7.48, "text": "3x^2 + C and again now we have our"}, {"start": 220.84, "duration": 3.0, "text": "anti-derivative"}], "title": "\u2756 Substitution Rule / U-sub Basic Example 1 \u2756", "description": "Thanks to all of you who support me on Patreon. You da real mvps!   $1 per month helps!! :)  https://www.patreon.com/patrickjmt !! Substitution Rule / U-sub Basic Example 1.\nJust a quick example of calculating an integral using a substitution.", "lengthSeconds": 223, "width": 480, "height": 360}
{"video_id": "XTZl7Kn6u4Y", "transcript": [{"start": 1.04, "duration": 4.08, "text": "okay in this video I want to do an"}, {"start": 2.52, "duration": 5.64, "text": "example of adding and subtracting some"}, {"start": 5.12, "duration": 5.36, "text": "rational expressions and as with you"}, {"start": 8.16, "duration": 3.92, "text": "know normal addition and subtraction of"}, {"start": 10.48, "duration": 3.6, "text": "fractions it's it's actually a little"}, {"start": 12.08, "duration": 3.92, "text": "more tedious than multiplication because"}, {"start": 14.08, "duration": 4.359, "text": "we're going to have to go through the"}, {"start": 16.0, "duration": 4.88, "text": "extra effort of getting common"}, {"start": 18.439, "duration": 4.961, "text": "denominators and uh typically that just"}, {"start": 20.88, "duration": 4.319, "text": "makes things a little tedious so the"}, {"start": 23.4, "duration": 5.799, "text": "first thing that you always have to do"}, {"start": 25.199, "duration": 6.121, "text": "is well get a common denominator and to"}, {"start": 29.199, "duration": 4.481, "text": "go about doing that typically you're"}, {"start": 31.32, "duration": 6.559, "text": "going to want to factor the denominators"}, {"start": 33.68, "duration": 9.32, "text": "of your rational expression so x^2 - 64"}, {"start": 37.879, "duration": 9.0, "text": "that factors as x + 8 * x -"}, {"start": 43.0, "duration": 5.48, "text": "8 and then uh in our our other fraction"}, {"start": 46.879, "duration": 3.52, "text": "uh that doesn't really Factor at all so"}, {"start": 48.48, "duration": 5.48, "text": "we'll just leave that"}, {"start": 50.399, "duration": 5.401, "text": "alone and now what we have to do is we"}, {"start": 53.96, "duration": 4.88, "text": "have to think of the least common"}, {"start": 55.8, "duration": 5.88, "text": "multiple of the denominators and what I"}, {"start": 58.84, "duration": 4.399, "text": "basically do when I do this is you know"}, {"start": 61.68, "duration": 4.0, "text": "I kind of look at the very first"}, {"start": 63.239, "duration": 7.121, "text": "fraction and I look at his denominator x"}, {"start": 65.68, "duration": 7.2, "text": "+ 8 x - 8 and then I look at the other"}, {"start": 70.36, "duration": 5.36, "text": "any other fractions that are present and"}, {"start": 72.88, "duration": 4.32, "text": "I look at their denominators and I think"}, {"start": 75.72, "duration": 5.359, "text": "well what are the factors in their"}, {"start": 77.2, "duration": 7.279, "text": "denominators so here there's an x + 8"}, {"start": 81.079, "duration": 5.72, "text": "and if there was not an x + 8 originally"}, {"start": 84.479, "duration": 4.841, "text": "in my first fraction I would have to"}, {"start": 86.799, "duration": 4.28, "text": "multiply top and bottom to make that X+"}, {"start": 89.32, "duration": 5.159, "text": "8 Factor"}, {"start": 91.079, "duration": 5.881, "text": "appear well the X+ 8 is there so we're"}, {"start": 94.479, "duration": 4.841, "text": "fine then I look at my next fraction I"}, {"start": 96.96, "duration": 5.119, "text": "look at his denominator x + 8 and I"}, {"start": 99.32, "duration": 5.24, "text": "think what does the other denominators"}, {"start": 102.079, "duration": 5.441, "text": "have that this one doesn't well notice"}, {"start": 104.56, "duration": 5.96, "text": "this one has an x - 8 that's not present"}, {"start": 107.52, "duration": 5.16, "text": "in this denominator so to get our common"}, {"start": 110.52, "duration": 3.919, "text": "denominator you basically multiply by"}, {"start": 112.68, "duration": 6.64, "text": "whatever is missing so I'm going to have"}, {"start": 114.439, "duration": 6.441, "text": "to multiply top and bottom by x - 8 in"}, {"start": 119.32, "duration": 4.64, "text": "our second second"}, {"start": 120.88, "duration": 6.44, "text": "fraction and now we do have our common"}, {"start": 123.96, "duration": 5.079, "text": "denominator x + 8 x - 8 we could go"}, {"start": 127.32, "duration": 4.999, "text": "ahead and stick that back together as"}, {"start": 129.039, "duration": 5.121, "text": "x^2 - 64 and then we simply do the"}, {"start": 132.319, "duration": 7.441, "text": "arithmetic in the numerator so it says"}, {"start": 134.16, "duration": 7.76, "text": "we have 7 + 3 * the quantity x - 8 and"}, {"start": 139.76, "duration": 4.96, "text": "all we'll do is just distribute this out"}, {"start": 141.92, "duration": 5.88, "text": "and just simplify it down a little"}, {"start": 144.72, "duration": 7.159, "text": "bit so it looks like in the"}, {"start": 147.8, "duration": 7.159, "text": "numerator we have seven +"}, {"start": 151.879, "duration": 4.521, "text": "3x um if we distribute the 3 to the 8"}, {"start": 154.959, "duration": 4.2, "text": "we'll get"}, {"start": 156.4, "duration": 4.64, "text": "-4 and again that's all being divided by"}, {"start": 159.159, "duration": 4.761, "text": "x^2 -"}, {"start": 161.04, "duration": 5.16, "text": "64 and again uh we can just simplify the"}, {"start": 163.92, "duration": 5.959, "text": "numerator one more time it looks like we"}, {"start": 166.2, "duration": 5.319, "text": "have 7 - 24 since the 3x is positive I'm"}, {"start": 169.879, "duration": 5.241, "text": "going to write that term"}, {"start": 171.519, "duration": 5.64, "text": "first and then if you do 7 - 24 I"}, {"start": 175.12, "duration": 5.759, "text": "believe that would give us"}, {"start": 177.159, "duration": 6.761, "text": "-17 uh the denominator he's just hanging"}, {"start": 180.879, "duration": 4.841, "text": "out and now we have simplified our"}, {"start": 183.92, "duration": 3.959, "text": "rational expression we've combined them"}, {"start": 185.72, "duration": 5.519, "text": "into a single fraction and that's what"}, {"start": 187.879, "duration": 3.36, "text": "we wanted to do"}], "title": "Rational Expressions:  Adding and Subtracting.  Ex 1", "description": "Thanks to all of you who support me on Patreon. You da real mvps!   $1 per month helps!! :)  https://www.patreon.com/patrickjmt !! Rational Expressions:  Adding and Subtracting.  Ex 1.  In this video, I do a relatively simply example of finding a common denominator and adding two rational expressions.", "lengthSeconds": 189, "width": 480, "height": 360}
{"video_id": "H5AM1bzqCQw", "transcript": [{"start": 1.319, "duration": 3.321, "text": "all right in this video I want to do"}, {"start": 2.72, "duration": 4.639, "text": "another example of solving a quadratic"}, {"start": 4.64, "duration": 5.039, "text": "equation using the quadratic formula and"}, {"start": 7.359, "duration": 5.36, "text": "in this case we're going to solve 3x^2 +"}, {"start": 9.679, "duration": 5.801, "text": "5x = 7 one thing that's important before"}, {"start": 12.719, "duration": 4.601, "text": "you use the quadratic formula is you"}, {"start": 15.48, "duration": 5.24, "text": "always need one side of the equation to"}, {"start": 17.32, "duration": 5.6, "text": "be zero so some people would just rush"}, {"start": 20.72, "duration": 4.799, "text": "in and start trying to label a b and c"}, {"start": 22.92, "duration": 4.439, "text": "but be careful um again the first thing"}, {"start": 25.519, "duration": 3.6, "text": "I always do is you want one side to be"}, {"start": 27.359, "duration": 3.92, "text": "zero so in this case we'll subtract"}, {"start": 29.119, "duration": 9.12, "text": "seven from both sides sides which will"}, {"start": 31.279, "duration": 9.76, "text": "give us 3x^2 + 5x - 7 = 0 and now I can"}, {"start": 38.239, "duration": 4.521, "text": "read off my values it says a is going to"}, {"start": 41.039, "duration": 4.921, "text": "have the value"}, {"start": 42.76, "duration": 8.2, "text": "pos3 B is going to have the value"}, {"start": 45.96, "duration": 8.16, "text": "pos5 and C is going to have the value"}, {"start": 50.96, "duration": 6.16, "text": "-7 so when we go to fill in our"}, {"start": 54.12, "duration": 6.04, "text": "quadratic formula again our netive B"}, {"start": 57.12, "duration": 5.8, "text": "plus or minus the < TK of b^2 - 4 A C"}, {"start": 60.16, "duration": 6.2, "text": "over 2 a when we do"}, {"start": 62.92, "duration": 8.199, "text": "that we'll end up getting our Solutions"}, {"start": 66.36, "duration": 7.799, "text": "as so negative of the B value which is"}, {"start": 71.119, "duration": 6.36, "text": "five plus or minus the square root it"}, {"start": 74.159, "duration": 5.081, "text": "says we take B and we Square it so we'll"}, {"start": 77.479, "duration": 7.32, "text": "have 5^"}, {"start": 79.24, "duration": 10.68, "text": "2ar - 4 * our a value which is 3 times"}, {"start": 84.799, "duration": 5.121, "text": "our C value which we said was -7"}, {"start": 90.479, "duration": 7.561, "text": "and then that's all going to be divided"}, {"start": 92.32, "duration": 7.479, "text": "by 2 * a or uh in this case 2 * 3 so now"}, {"start": 98.04, "duration": 4.399, "text": "it's just a matter again of simplifying"}, {"start": 99.799, "duration": 6.0, "text": "down this expression um and you know"}, {"start": 102.439, "duration": 3.36, "text": "just trying to reduce it if you"}, {"start": 106.04, "duration": 10.28, "text": "can so we have X = we'll have -5 plus or"}, {"start": 110.88, "duration": 9.839, "text": "minus the Square t let's see so 5 * 5 is"}, {"start": 116.32, "duration": 11.32, "text": "25 and then we have -4 * 3 *"}, {"start": 120.719, "duration": 10.52, "text": "-7 so -4 * pos3 is going to be -12 -12 *"}, {"start": 127.64, "duration": 7.56, "text": "excuse me * -7 so we have -12 * -7 will"}, {"start": 131.239, "duration": 5.881, "text": "give us a positive and then 12 * 7 is"}, {"start": 135.2, "duration": 8.2, "text": "going to give us"}, {"start": 137.12, "duration": 6.28, "text": "84 and that's all divided by 2 * a or"}, {"start": 143.72, "duration": 8.4, "text": "6 so we have -5 plus or minus the square"}, {"start": 148.04, "duration": 5.96, "text": "Ro T so let's see 20 5 and 84 so I"}, {"start": 152.12, "duration": 5.56, "text": "believe that's going to give us a"}, {"start": 154.0, "duration": 5.84, "text": "postive 109 when we add those together"}, {"start": 157.68, "duration": 5.199, "text": "and all of that's going to be divided"}, {"start": 159.84, "duration": 3.039, "text": "Again by"}, {"start": 164.319, "duration": 4.241, "text": "six and if you wanted to again we can"}, {"start": 166.879, "duration": 3.72, "text": "kind of break this down into our two"}, {"start": 168.56, "duration": 6.0, "text": "separate"}, {"start": 170.599, "duration": 5.241, "text": "statements so we've got -5 + the sare <"}, {"start": 174.56, "duration": 5.88, "text": "TK of"}, {"start": 175.84, "duration": 8.28, "text": "109 / 6 as one solution our other"}, {"start": 180.44, "duration": 4.76, "text": "solution will be -5 minus the square <"}, {"start": 184.12, "duration": 5.56, "text": "TK of"}, {"start": 185.2, "duration": 7.44, "text": "109 divided 6 and here we could start"}, {"start": 189.68, "duration": 5.32, "text": "trying to break down um 109 trying to"}, {"start": 192.64, "duration": 4.599, "text": "reduce it a little bit but I think if"}, {"start": 195.0, "duration": 3.64, "text": "you play with it a little bit um you"}, {"start": 197.239, "duration": 3.92, "text": "know maybe you'll find something that"}, {"start": 198.64, "duration": 5.44, "text": "divides into it but to me I'm pretty"}, {"start": 201.159, "duration": 4.72, "text": "sure that 109 is actually a prime number"}, {"start": 204.08, "duration": 4.04, "text": "so this isn't going to factor any"}, {"start": 205.879, "duration": 4.401, "text": "further um we're not going to be able to"}, {"start": 208.12, "duration": 4.319, "text": "reduce it in any kind of meaningful way"}, {"start": 210.28, "duration": 8.36, "text": "so to me these would simply be our two"}, {"start": 212.439, "duration": 12.0, "text": "solutions -5 + < TK 109 6 and -5 minus"}, {"start": 218.64, "duration": 5.799, "text": "the < TK of 109 over 6"}], "title": "Solving Quadratic Equations using the Quadratic Formula - Example 3", "description": "Thanks to all of you who support me on Patreon. You da real mvps!   $1 per month helps!! :)  https://www.patreon.com/patrickjmt !! Solving Quadratic Equations using the Quadratic Formula - Example 3", "lengthSeconds": 224, "width": 480, "height": 360}
{"video_id": "_U8Y5z-kVvI", "transcript": [{"start": 1.319, "duration": 2.56, "text": "okay in this video I'm going to talk"}, {"start": 2.56, "duration": 4.44, "text": "about using What's called the method of"}, {"start": 3.879, "duration": 5.401, "text": "undetermined coefficients to solve"}, {"start": 7.0, "duration": 4.719, "text": "non-homogeneous second order linear"}, {"start": 9.28, "duration": 6.439, "text": "differential equations and this is the"}, {"start": 11.719, "duration": 6.961, "text": "basic form of such an equation a y prime"}, {"start": 15.719, "duration": 5.441, "text": "plus b y Prime + C yals some function of"}, {"start": 18.68, "duration": 4.72, "text": "x to do this we're first going to solve"}, {"start": 21.16, "duration": 3.56, "text": "What's called the complimentary equation"}, {"start": 23.4, "duration": 4.48, "text": "and basically we just replace our"}, {"start": 24.72, "duration": 5.799, "text": "function G ofx with a zero and solve"}, {"start": 27.88, "duration": 6.24, "text": "that first so this is is going to be"}, {"start": 30.519, "duration": 5.841, "text": "part one um so cuz I think it can be a"}, {"start": 34.12, "duration": 4.2, "text": "little bit of a long process um so the"}, {"start": 36.36, "duration": 3.76, "text": "first part is just the theorem here it"}, {"start": 38.32, "duration": 5.079, "text": "says basically okay we're trying to"}, {"start": 40.12, "duration": 5.32, "text": "solve um one of these non-homogeneous"}, {"start": 43.399, "duration": 4.401, "text": "second order linear"}, {"start": 45.44, "duration": 4.52, "text": "equations it says basically to get the"}, {"start": 47.8, "duration": 5.12, "text": "solution to that you have to look at two"}, {"start": 49.96, "duration": 5.16, "text": "different things it says um you can look"}, {"start": 52.92, "duration": 3.279, "text": "at what's called so it says the solution"}, {"start": 55.12, "duration": 4.919, "text": "is going to be What's called the"}, {"start": 56.199, "duration": 6.081, "text": "particular solution y of P um and also"}, {"start": 60.039, "duration": 5.641, "text": "kind of What's called the uh the general"}, {"start": 62.28, "duration": 6.76, "text": "solution so it says y of p is going to"}, {"start": 65.68, "duration": 6.16, "text": "be a particular solution of this um"}, {"start": 69.04, "duration": 5.0, "text": "original equation we're trying to solve"}, {"start": 71.84, "duration": 4.279, "text": "and Y if C is going to be a solution to"}, {"start": 74.04, "duration": 4.92, "text": "the complementary equation that we're"}, {"start": 76.119, "duration": 5.121, "text": "trying to solve okay so let's try to"}, {"start": 78.96, "duration": 4.72, "text": "make some sense out of all this um"}, {"start": 81.24, "duration": 4.559, "text": "solving this last part isn't too bad I"}, {"start": 83.68, "duration": 5.24, "text": "mean you just have to remember some of"}, {"start": 85.799, "duration": 4.96, "text": "the other formulas for um differential"}, {"start": 88.92, "duration": 5.519, "text": "equations but"}, {"start": 90.759, "duration": 6.0, "text": "hopefully not not too bad to start off"}, {"start": 94.439, "duration": 5.601, "text": "with Okay so let's solve this"}, {"start": 96.759, "duration": 6.64, "text": "corresponding um let's try to solve"}, {"start": 100.04, "duration": 6.439, "text": "this so we're going to have y"}, {"start": 103.399, "duration": 7.68, "text": "prime plus 3 y"}, {"start": 106.479, "duration": 7.081, "text": "Prime + 2 y ="}, {"start": 111.079, "duration": 4.121, "text": "X2 okay so the first thing what we're"}, {"start": 113.56, "duration": 3.68, "text": "going to do again is just turn this into"}, {"start": 115.2, "duration": 6.68, "text": "our complementary equation and we're"}, {"start": 117.24, "duration": 9.479, "text": "going to solve first off y Prime + 3 y"}, {"start": 121.88, "duration": 8.48, "text": "Prime + 2 y ="}, {"start": 126.719, "duration": 8.52, "text": "0 okay now we turn this into What's"}, {"start": 130.36, "duration": 9.12, "text": "called the auxiliary equation R 2 + 3"}, {"start": 135.239, "duration": 6.64, "text": "R um+ 2al 0 and then based on the"}, {"start": 139.48, "duration": 5.0, "text": "solutions to this auxiliary equation"}, {"start": 141.879, "duration": 4.761, "text": "it's going to tell us Solutions um to"}, {"start": 144.48, "duration": 4.839, "text": "our homogeneous second order"}, {"start": 146.64, "duration": 4.2, "text": "differential equation that we have here"}, {"start": 149.319, "duration": 4.041, "text": "okay so I think this is going to factor"}, {"start": 150.84, "duration": 2.52, "text": "relatively"}, {"start": 156.0, "duration": 6.959, "text": "easily looks like we'll just get r + 1"}, {"start": 159.8, "duration": 5.76, "text": "um * r + 2 = 0 so that means our"}, {"start": 162.959, "duration": 7.161, "text": "solution Solutions are going to be"}, {"start": 165.56, "duration": 7.08, "text": "ral1 and also Ral -2 so this is a case"}, {"start": 170.12, "duration": 4.56, "text": "where we have two distinct real roots"}, {"start": 172.64, "duration": 5.0, "text": "and remember if you have two distinct"}, {"start": 174.68, "duration": 4.919, "text": "real Roots it says basically your"}, {"start": 177.64, "duration": 4.16, "text": "solution is going to look something like"}, {"start": 179.599, "duration": 5.961, "text": "just going to be some uh"}, {"start": 181.8, "duration": 5.24, "text": "coefficient e to the first root um it"}, {"start": 185.56, "duration": 4.959, "text": "doesn't matter which one so one of the"}, {"start": 187.04, "duration": 8.479, "text": "roots time X Plus again this coefficient"}, {"start": 190.519, "duration": 6.601, "text": "C2 e to the 2 root * X so okay well we"}, {"start": 195.519, "duration": 4.921, "text": "know our Roots here all we have to do is"}, {"start": 197.12, "duration": 7.119, "text": "plug those in so our general solution is"}, {"start": 200.44, "duration": 8.12, "text": "going to be um C1 e to"}, {"start": 204.239, "duration": 7.161, "text": "thex plus C2 e to"}, {"start": 208.56, "duration": 4.679, "text": "-2X okay so this is going to incorporate"}, {"start": 211.4, "duration": 5.039, "text": "be incorporated into part of our"}, {"start": 213.239, "duration": 4.521, "text": "solution at the end so um and I think"}, {"start": 216.439, "duration": 4.16, "text": "I'm going to stop this part here in the"}, {"start": 217.76, "duration": 5.679, "text": "second part I'm going to solve the"}, {"start": 220.599, "duration": 5.84, "text": "um the particular solution part using"}, {"start": 223.439, "duration": 5.52, "text": "the method of um of undetermined"}, {"start": 226.439, "duration": 4.321, "text": "coefficients and we'll try to put it all"}, {"start": 228.959, "duration": 3.241, "text": "together into some coherent solution so"}, {"start": 230.76, "duration": 5.64, "text": "feel free to take a look at that it"}, {"start": 232.2, "duration": 4.2, "text": "should be um posted nearby"}], "title": "Method of Undetermined Coefficients/ 2nd Order Linear DE", "description": "Solving Second Order Linear Non-Homogeneous Differential Equations\nIn this video, you'll learn how to apply the method of undetermined coefficients to find the general solution of second-order linear non-homogeneous differential equations. We'll break down the key concepts of these types of equations, walk through an example, and introduce the theorem for finding general solutions. Perfect for students learning about differential equations and their applications!\n\n(https://www.youtube.com/watch?v=XPwSR3cqGi4&feature=youtu.be)\n\nWhat You Will Learn\n\nThe basics of second-order linear non-homogeneous differential equations.\nHow to solve the complementary (homogeneous) equation.\nHow to use the method of undetermined coefficients to find a particular solution.\nSteps to combine the complementary and particular solutions for the general solution.\nThe method of undetermined coefficients is a useful technique for finding particular solutions to certain types of non-homogeneous differential equations. \n\nThis video covers the procedure step by step, beginning with the associated homogeneous equation, then applying the method to find a particular solution for the given non-homogeneous term, and finally combining them to form the general solution. By the end, you'll have a solid understanding of how to tackle these problems and find their solutions.\n\nIf you find this content helpful, please like, comment, and subscribe! Share it with friends, classmates, or teachers studying differential equations.\n\nSupport my work on Patreon: https://www.patreon.com/patrickjmt?ty=c\n\n#DifferentialEquations #UndeterminedCoefficients #SecondOrderDE #NonHomogeneousEquations #MathTutorial #PatrickJMT #Calculus #SolvingDEs #Mathematics #ComplementarySolution #GeneralSolution #MathForStudents #MathHelp #MathematicsLearning #DEExamples #CalculusConcepts #MathEducation #CalculusProblems #LearnDifferentialEquations #Education", "lengthSeconds": 235, "width": 480, "height": 360}
{"video_id": "_gX1LOYpR8o", "transcript": [{"start": 0.76, "duration": 2.84, "text": "all right in this video we're going to"}, {"start": 2.04, "duration": 3.96, "text": "start looking at some examples of"}, {"start": 3.6, "duration": 4.239, "text": "solving trigonometric equations by"}, {"start": 6.0, "duration": 4.12, "text": "factoring and we're just going to find"}, {"start": 7.839, "duration": 6.481, "text": "the Solutions in the interval 0 to 2 pi"}, {"start": 10.12, "duration": 7.96, "text": "for these so uh let's look at 2 sin^2 ="}, {"start": 14.32, "duration": 5.64, "text": "1 + cosine X here first so um you know"}, {"start": 18.08, "duration": 4.359, "text": "initially if I see something like this"}, {"start": 19.96, "duration": 5.28, "text": "uh you know I see signs and cosiness I"}, {"start": 22.439, "duration": 4.401, "text": "don't really see you know much to factor"}, {"start": 25.24, "duration": 4.879, "text": "but we can certainly use our trig"}, {"start": 26.84, "duration": 6.68, "text": "identity so recall that s^ squ plus Co"}, {"start": 30.119, "duration": 7.321, "text": "cosine s = 1 so if we solve for sin s we"}, {"start": 33.52, "duration": 8.28, "text": "would get 1 - cosine 2"}, {"start": 37.44, "duration": 6.84, "text": "x so I'm just going to basically try to"}, {"start": 41.8, "duration": 4.96, "text": "turn this problem into I just want"}, {"start": 44.28, "duration": 4.599, "text": "everything to involve cosiness is all"}, {"start": 46.76, "duration": 6.2, "text": "I'm doing so I'm going to distribute the"}, {"start": 48.879, "duration": 6.281, "text": "two so 2 * 1 will be two we'll get -2"}, {"start": 52.96, "duration": 4.84, "text": "cosine 2"}, {"start": 55.16, "duration": 5.919, "text": "x = 1"}, {"start": 57.8, "duration": 5.48, "text": "+ cosine X"}, {"start": 61.079, "duration": 5.4, "text": "and I think what I'm going to do is um"}, {"start": 63.28, "duration": 5.48, "text": "I'm going to add the 2 cosine squ to the"}, {"start": 66.479, "duration": 2.281, "text": "right"}, {"start": 69.04, "duration": 4.68, "text": "side we've already got our positive"}, {"start": 71.36, "duration": 5.32, "text": "cosine"}, {"start": 73.72, "duration": 5.52, "text": "X I'm also going to subtract two from"}, {"start": 76.68, "duration": 5.119, "text": "both sides so 1 minus 2 would simply"}, {"start": 79.24, "duration": 5.4, "text": "give us -1 and then we'll be left with"}, {"start": 81.799, "duration": 6.241, "text": "zero um on the left side and now"}, {"start": 84.64, "duration": 7.2, "text": "hopefully um the right side factors"}, {"start": 88.04, "duration": 3.8, "text": "without too much trouble so"}, {"start": 92.32, "duration": 4.479, "text": "so I think if we use 2 cosine X and"}, {"start": 95.36, "duration": 4.48, "text": "cosine X that's certainly going to give"}, {"start": 96.799, "duration": 5.161, "text": "us our 2 cosine squ X um to get a"}, {"start": 99.84, "duration": 4.76, "text": "negative one I know I've got to use uh a"}, {"start": 101.96, "duration": 5.199, "text": "positive one and a negative one um I"}, {"start": 104.6, "duration": 5.839, "text": "think the positive would have to go uh"}, {"start": 107.159, "duration": 5.6, "text": "with our second Factor since uh when we"}, {"start": 110.439, "duration": 4.96, "text": "multiply it out we'll get 2 cosine ^ 2"}, {"start": 112.759, "duration": 4.96, "text": "plus 2 cosine x minus cosine X that"}, {"start": 115.399, "duration": 5.161, "text": "would give us a positive cosine X and"}, {"start": 117.719, "duration": 4.841, "text": "then we've got negative 1 left over so"}, {"start": 120.56, "duration": 6.44, "text": "now once we have it factored we just set"}, {"start": 122.56, "duration": 8.44, "text": "each uh each factor equal to"}, {"start": 127.0, "duration": 6.2, "text": "zero well here I can add one and divide"}, {"start": 131.0, "duration": 5.519, "text": "by two and then we'll be left simply"}, {"start": 133.2, "duration": 7.84, "text": "with cosine X um if we set our second"}, {"start": 136.519, "duration": 9.0, "text": "Factor cosine x + 1 = 0 we'll simply get"}, {"start": 141.04, "duration": 7.88, "text": "cine x = -1 and now I have to remember"}, {"start": 145.519, "duration": 7.281, "text": "um where cosine x = 12 let's see cosine"}, {"start": 148.92, "duration": 5.24, "text": "x = 1 I believe at pi over 3 that would"}, {"start": 152.8, "duration": 4.48, "text": "be one of our"}, {"start": 154.16, "duration": 5.88, "text": "Solutions and also in Quadrant 4 cosine"}, {"start": 157.28, "duration": 7.64, "text": "X would be 12 and that would be at the"}, {"start": 160.04, "duration": 8.8, "text": "angle 5 pi over 3 where does cosine x"}, {"start": 164.92, "duration": 7.239, "text": "equal ne1 well I believe cosine x equals"}, {"start": 168.84, "duration": 6.319, "text": "1 at the angle of"}, {"start": 172.159, "duration": 5.241, "text": "Pi so those would be our solutions for"}, {"start": 175.159, "duration": 5.321, "text": "uh the first equation we've now found"}, {"start": 177.4, "duration": 6.119, "text": "all Solutions in the interval from 0"}, {"start": 180.48, "duration": 3.039, "text": "to 2 pi"}], "title": "Solve Trigonometric Equation by Factoring, Example 1", "description": "Thanks to all of you who support me on Patreon. You da real mvps!   $1 per month helps!! :)  https://www.patreon.com/patrickjmt !! Solving a Trigonometric Equation by Factoring, Example 1", "lengthSeconds": 182, "width": 480, "height": 360}
{"video_id": "yJWFAG1g4Jw", "transcript": [{"start": 1.12, "duration": 3.239, "text": "okay here we're going to look at finding"}, {"start": 2.6, "duration": 4.92, "text": "some expressions for the remainder of a"}, {"start": 4.359, "duration": 5.401, "text": "tailor polinomial and these uh are"}, {"start": 7.52, "duration": 4.199, "text": "actually not too bad at all all we're"}, {"start": 9.76, "duration": 3.999, "text": "really going to have to do is just uh"}, {"start": 11.719, "duration": 5.121, "text": "fill in this expression with the N plus"}, {"start": 13.759, "duration": 4.961, "text": "first derivative um at Z and then just"}, {"start": 16.84, "duration": 4.0, "text": "plug in where we're centering it and"}, {"start": 18.72, "duration": 4.76, "text": "calculate a a few things so let's look"}, {"start": 20.84, "duration": 5.16, "text": "at an example here so suppose we want to"}, {"start": 23.48, "duration": 5.28, "text": "approximate uh the function e to the X"}, {"start": 26.0, "duration": 9.76, "text": "near xal 1 with the polom the tailor"}, {"start": 28.76, "duration": 12.639, "text": "polom e + e * x -1 + e over 2 * x -1"}, {"start": 35.76, "duration": 8.88, "text": "2 plus e over 6 * x -1 Cub + e over 24 *"}, {"start": 41.399, "duration": 3.241, "text": "x -1"}, {"start": 45.44, "duration": 7.24, "text": "4 okay so um in our case um so we're"}, {"start": 50.28, "duration": 5.32, "text": "figuring out the remainder associated"}, {"start": 52.68, "duration": 6.359, "text": "with a fourth degree polinomial Okay so"}, {"start": 55.6, "duration": 5.599, "text": "we've gone out to uh to degree 4 so it"}, {"start": 59.039, "duration": 5.2, "text": "says the remainder and we need to put"}, {"start": 61.199, "duration": 6.321, "text": "this in absolute value here it says"}, {"start": 64.239, "duration": 7.281, "text": "it'll be the well the n + one so the"}, {"start": 67.52, "duration": 6.72, "text": "fifth derivative evaluated at Z over"}, {"start": 71.52, "duration": 4.44, "text": "well again we'll have five factorial and"}, {"start": 74.24, "duration": 3.919, "text": "then we'll take x minus where we're"}, {"start": 75.96, "duration": 4.24, "text": "centering it which for us is at"}, {"start": 78.159, "duration": 4.201, "text": "positive1 and then all of that will be"}, {"start": 80.2, "duration": 4.919, "text": "raised to the fifth"}, {"start": 82.36, "duration": 6.88, "text": "power well in this case our function f"}, {"start": 85.119, "duration": 7.921, "text": "ofx is just e to the X well all the"}, {"start": 89.24, "duration": 6.519, "text": "derivatives for f ofx or E to X So"}, {"start": 93.04, "duration": 6.32, "text": "eventually the fifth derivative for X"}, {"start": 95.759, "duration": 6.4, "text": "well that's also going to be e to the"}, {"start": 99.36, "duration": 6.079, "text": "X well that means the fifth derivative"}, {"start": 102.159, "duration": 6.361, "text": "evaluated at Z well that's simply going"}, {"start": 105.439, "duration": 5.441, "text": "to equal e to the Z and now that's all"}, {"start": 108.52, "duration": 4.72, "text": "we have to do is is just simplify that"}, {"start": 110.88, "duration": 7.64, "text": "uh and uh plug that in and simplify the"}, {"start": 113.24, "duration": 6.799, "text": "expression so um so again the uh F the"}, {"start": 118.52, "duration": 3.599, "text": "fifth derivative of our function"}, {"start": 120.039, "duration": 5.841, "text": "evaluate it at Z that's going to be e to"}, {"start": 122.119, "duration": 7.721, "text": "the Z let's see 5 factorial I believe"}, {"start": 125.88, "duration": 7.2, "text": "that's 120 so 5 * 4 * 3 * 2 * 1 that's"}, {"start": 129.84, "duration": 6.72, "text": "20 and another six that's 120 then we'll"}, {"start": 133.08, "duration": 7.08, "text": "have x -1 raised to the 5th power and"}, {"start": 136.56, "duration": 6.6, "text": "that will now be an expression for our"}, {"start": 140.16, "duration": 3.0, "text": "remainder"}], "title": "\ud83d\udcda Finding the Remainder of a Taylor Polynomial \u2013 Example Using Taylor\u2019s Remainder Theorem \ud83d\udcda", "description": "\ud83d\udcda Finding the Remainder of a Taylor Polynomial \u2013 Example Using Taylor\u2019s Remainder Theorem \ud83d\udcda\n\nIn this video, I walk through Taylor's Remainder Theorem and show how to find an expression for the remainder when approximating f(x) = e^x using a fourth-degree Taylor polynomial centered at x = 1. We show how the remainder formula applies to this specific example.\n\n\ud83d\ude80 What\u2019s covered:\n\nExplanation of Taylor's Remainder Theorem and its importance in approximating functions.\nStep-by-step calculation of the Taylor polynomial for f(x) = e^x at x = 1.\nHow to find the remainder using the general remainder formula and a detailed breakdown of the process.\nThis is an essential technique for students studying calculus and higher-level math, helping you understand the accuracy of Taylor series approximations and how to handle errors in your estimates.\n\n\ud83d\udc4d Don\u2019t forget to like, subscribe, and hit the notification bell for more calculus tutorials and advanced math tips! \ud83d\udd14\u2728\n\n\n#TaylorPolynomial #RemainderTheorem #Calculus #TaylorSeries #MathTutorial #AdvancedMath #Mathematics #TaylorSeriesRemainder #MathHelp #LearningMath #STEM #HigherMath #SeriesApproximation", "lengthSeconds": 142, "width": 480, "height": 360}
{"video_id": "6o_S7u7Ddx4", "transcript": [{"start": 0.64, "duration": 3.319, "text": "all right in this video we're going to"}, {"start": 1.68, "duration": 5.079, "text": "talk about finding a unit vector and the"}, {"start": 3.959, "duration": 5.72, "text": "unit the unit Vector for a vector"}, {"start": 6.759, "duration": 5.481, "text": "v all we do is we take our vector v and"}, {"start": 9.679, "duration": 5.12, "text": "we divide it by its magnitude or its"}, {"start": 12.24, "duration": 4.48, "text": "length and what a unit Vector is it's"}, {"start": 14.799, "duration": 4.881, "text": "simply a vector that points in the same"}, {"start": 16.72, "duration": 5.08, "text": "direction as the original Vector but uh"}, {"start": 19.68, "duration": 4.12, "text": "we're going to kind of uh alter its"}, {"start": 21.8, "duration": 4.96, "text": "magnitude or length so that it has a"}, {"start": 23.8, "duration": 8.72, "text": "magnitude or length of"}, {"start": 26.76, "duration": 8.44, "text": "one so nothing too table here let's uh"}, {"start": 32.52, "duration": 5.0, "text": "we got three examples here let's find uh"}, {"start": 35.2, "duration": 4.24, "text": "the components of a unit Vector in the"}, {"start": 37.52, "duration": 5.12, "text": "direction of each of these given vectors"}, {"start": 39.44, "duration": 6.639, "text": "so just a unit Vector here for for each"}, {"start": 42.64, "duration": 5.84, "text": "one well so we've got the vector uh 3"}, {"start": 46.079, "duration": 3.441, "text": "comma 4 with components 3 comma 4 let's"}, {"start": 48.48, "duration": 3.239, "text": "find its"}, {"start": 49.52, "duration": 3.6, "text": "magnitude um you know maybe we'll call"}, {"start": 51.719, "duration": 5.36, "text": "this again vector"}, {"start": 53.12, "duration": 5.8, "text": "v so the magnitude of vector v again we"}, {"start": 57.079, "duration": 5.881, "text": "just take each"}, {"start": 58.92, "duration": 8.519, "text": "component and we Square it we add them"}, {"start": 62.96, "duration": 4.479, "text": "together and square root it so 3^ squ is"}, {"start": 68.159, "duration": 6.081, "text": "n 4 squ is 16 that's going to give us"}, {"start": 71.92, "duration": 4.12, "text": "the square root of 25 which is simply"}, {"start": 74.24, "duration": 2.96, "text": "going to give us five when we take the"}, {"start": 76.04, "duration": 6.24, "text": "square root of"}, {"start": 77.2, "duration": 5.08, "text": "it so really um it says the unit"}, {"start": 82.479, "duration": 6.481, "text": "Vector would be this Vector uh three"}, {"start": 86.96, "duration": 4.44, "text": "with components three four but now we're"}, {"start": 88.96, "duration": 4.04, "text": "dividing by five"}, {"start": 91.4, "duration": 3.2, "text": "and remember all that happens here you"}, {"start": 93.0, "duration": 3.119, "text": "can really think we're multiplying by a"}, {"start": 94.6, "duration": 2.839, "text": "scalar which we are we're just"}, {"start": 96.119, "duration": 4.201, "text": "multiplying by"}, {"start": 97.439, "duration": 5.561, "text": "1 so we'll have"}, {"start": 100.32, "duration": 5.88, "text": "1 uh being multiplied by the vector with"}, {"start": 103.0, "duration": 5.479, "text": "components 3 comma 4 and recall uh if"}, {"start": 106.2, "duration": 4.8, "text": "you have a a real number out front all"}, {"start": 108.479, "duration": 5.801, "text": "we do is just multiply it in so we'll"}, {"start": 111.0, "duration": 7.159, "text": "get three fifths and four fifths and"}, {"start": 114.28, "duration": 5.159, "text": "that'll give us our unit Vector U again"}, {"start": 118.159, "duration": 4.481, "text": "it's going to be pointing off in the"}, {"start": 119.439, "duration": 4.921, "text": "exact same direction as the original"}, {"start": 122.64, "duration": 5.88, "text": "again but now it simply is going to have"}, {"start": 124.36, "duration": 4.16, "text": "a length of one"}], "title": "Finding a Unit Vector, Ex 1", "description": "Thanks to all of you who support me on Patreon. You da real mvps!   $1 per month helps!! :)  https://www.patreon.com/patrickjmt !! Finding a Unit Vector, Ex 1.  In this video I discuss the idea of a unit vector and show how to find it (divide the vector by its magnitude!).", "lengthSeconds": 127, "width": 480, "height": 360}
{"video_id": "MJaGrODQr1U", "transcript": [{"start": 0.96, "duration": 4.16, "text": "all right in this video I want to do"}, {"start": 2.399, "duration": 4.88, "text": "another example of multiplying a couple"}, {"start": 5.12, "duration": 4.439, "text": "rational expressions in this case we"}, {"start": 7.279, "duration": 3.961, "text": "have quadratic trinomials in both the"}, {"start": 9.559, "duration": 4.321, "text": "numerators and the"}, {"start": 11.24, "duration": 4.359, "text": "denominators so again the main idea on"}, {"start": 13.88, "duration": 3.84, "text": "these types of problems is just to"}, {"start": 15.599, "duration": 3.76, "text": "factor and cancel that's usually that's"}, {"start": 17.72, "duration": 4.399, "text": "what people that's usually the gist of"}, {"start": 19.359, "duration": 4.641, "text": "the problems so I'm going to give myself"}, {"start": 22.119, "duration": 4.08, "text": "a little room and try to factor and"}, {"start": 24.0, "duration": 3.92, "text": "cancel again you know keep in mind too"}, {"start": 26.199, "duration": 5.201, "text": "that usually these problems are set up"}, {"start": 27.92, "duration": 6.72, "text": "so that things will cancel so"}, {"start": 31.4, "duration": 5.999, "text": "so for example okay uh let me Factor"}, {"start": 34.64, "duration": 4.32, "text": "Okay so we've got 3x^2 + 7 x + 2 to me"}, {"start": 37.399, "duration": 3.561, "text": "that one's the worst one I'll come back"}, {"start": 38.96, "duration": 5.96, "text": "to that one in the denominator we have"}, {"start": 40.96, "duration": 6.64, "text": "x^2 - 2x - 8 so since the coefficient on"}, {"start": 44.92, "duration": 4.279, "text": "the x^2 is a 1 remember the rule is we"}, {"start": 47.6, "duration": 2.639, "text": "basically look for two numbers that"}, {"start": 49.199, "duration": 6.88, "text": "multiply"}, {"start": 50.239, "duration": 8.241, "text": "to8 but add up to -2 well I think -4 and"}, {"start": 56.079, "duration": 4.96, "text": "pos2 would do that and then to get the"}, {"start": 58.48, "duration": 6.24, "text": "X2 we simply use an X and an"}, {"start": 61.039, "duration": 6.12, "text": "X okay so I've got an x - 4 and an x + 2"}, {"start": 64.72, "duration": 4.039, "text": "in the denominator I'm already thinking"}, {"start": 67.159, "duration": 4.241, "text": "you know there's a good chance maybe"}, {"start": 68.759, "duration": 4.4, "text": "there's an x - 4 or an x + 2 somewhere"}, {"start": 71.4, "duration": 4.88, "text": "in the numerator if things are going to"}, {"start": 73.159, "duration": 5.081, "text": "cancel out so at least it helps me you"}, {"start": 76.28, "duration": 5.159, "text": "know sometimes guess what the factors"}, {"start": 78.24, "duration": 6.839, "text": "might be so I'm going to do my x^2 - 3x"}, {"start": 81.439, "duration": 5.761, "text": "- 4 I believe that would Factor as well"}, {"start": 85.079, "duration": 5.281, "text": "so to get the X squ we need an X and an"}, {"start": 87.2, "duration": 8.959, "text": "x two numbers that multiply to4 4 but"}, {"start": 90.36, "duration": 9.759, "text": "add up to -3 I think we could use -4 and"}, {"start": 96.159, "duration": 7.361, "text": "pos1 hey there's our x - 4S showing up"}, {"start": 100.119, "duration": 7.761, "text": "x^2 + 3x + 2 I think that one's not too"}, {"start": 103.52, "duration": 7.08, "text": "bad either x + 1 x + 2 again if we"}, {"start": 107.88, "duration": 4.279, "text": "multiply we'll get pos2 and those will"}, {"start": 110.6, "duration": 5.24, "text": "add up to"}, {"start": 112.159, "duration": 5.92, "text": "positive3 um and then 3x^2 + 7 x + 2 you"}, {"start": 115.84, "duration": 3.639, "text": "could do this by factoring by grouping"}, {"start": 118.079, "duration": 3.32, "text": "this is one that I would certainly just"}, {"start": 119.479, "duration": 5.041, "text": "kind of do by trial and error assuming"}, {"start": 121.399, "duration": 5.881, "text": "it factors nicely so to get 3x^2 we'll"}, {"start": 124.52, "duration": 5.12, "text": "need 3x and X really the only good way"}, {"start": 127.28, "duration": 5.52, "text": "to get positive2 is to have a positive"}, {"start": 129.64, "duration": 5.12, "text": "one and a positive two but it definitely"}, {"start": 132.8, "duration": 4.56, "text": "matters in which set of parentheses you"}, {"start": 134.76, "duration": 4.8, "text": "put the positive one or the positive2 I"}, {"start": 137.36, "duration": 4.48, "text": "think we would need the positive2 to go"}, {"start": 139.56, "duration": 4.039, "text": "on the second Factor the positive one to"}, {"start": 141.84, "duration": 3.0, "text": "go on the first because if you multiply"}, {"start": 143.599, "duration": 6.401, "text": "that out we'll get"}, {"start": 144.84, "duration": 6.56, "text": "3x^2 + 6x + 1 x which gives us our 7 x"}, {"start": 150.0, "duration": 3.76, "text": "and then our plus"}, {"start": 151.4, "duration": 4.199, "text": "two all right and now I'm just going to"}, {"start": 153.76, "duration": 5.08, "text": "cancel everything out that I can so it"}, {"start": 155.599, "duration": 6.28, "text": "looks like we have an X+ 2 and x + 2 it"}, {"start": 158.84, "duration": 6.52, "text": "looks like we have an x - 4 and x - 4"}, {"start": 161.879, "duration": 7.36, "text": "will cancel we've got an x + 1 and an x"}, {"start": 165.36, "duration": 6.36, "text": "+ 1 and what we're left with we've got"}, {"start": 169.239, "duration": 5.64, "text": "3x + 1 in the"}, {"start": 171.72, "duration": 5.439, "text": "numerator and we've got an X +2 left in"}, {"start": 174.879, "duration": 4.36, "text": "the denominator and we have now"}, {"start": 177.159, "duration": 3.681, "text": "simplified down our original"}, {"start": 179.239, "duration": 4.601, "text": "multiplication of our rational"}, {"start": 180.84, "duration": 3.0, "text": "expressions"}], "title": "Rational Expressions:  Multiplying and Dividing.  Ex 2", "description": "Thanks to all of you who support me on Patreon. You da real mvps!   $1 per month helps!! :)  https://www.patreon.com/patrickjmt !! Rational Expressions:  Multiplying and Dividing.  Ex 2.  In this video, I divide two rational expressions containing quadratic trinomials.", "lengthSeconds": 182, "width": 480, "height": 360}
{"video_id": "XDxOqwzomLE", "transcript": [{"start": 1.36, "duration": 5.88, "text": "so on a Saturday afternoon Armand sent M"}, {"start": 4.6, "duration": 5.28, "text": "text messages each hour for five hours"}, {"start": 7.24, "duration": 5.12, "text": "and Tyrone sent P text messages each"}, {"start": 9.88, "duration": 4.12, "text": "hour for 4 hours we want to know which"}, {"start": 12.36, "duration": 4.2, "text": "of the following represents the total"}, {"start": 14.0, "duration": 5.6, "text": "number of messages sent by arand and"}, {"start": 16.56, "duration": 6.0, "text": "Tyrone on Saturday afternoon so we have"}, {"start": 19.6, "duration": 7.48, "text": "the choices 9 multiplied by m multiplied"}, {"start": 22.56, "duration": 9.16, "text": "by P 20 multiplied by m multiplied P 5 m"}, {"start": 27.08, "duration": 5.84, "text": "+ 4 p and then 4 m + 5 p"}, {"start": 31.72, "duration": 3.56, "text": "so you"}, {"start": 32.92, "duration": 5.24, "text": "know you could probably you know maybe"}, {"start": 35.28, "duration": 5.88, "text": "you can reason this one out without um"}, {"start": 38.16, "duration": 5.2, "text": "you know even even using some real"}, {"start": 41.16, "duration": 5.399, "text": "numbers but let's suppose that Arman"}, {"start": 43.36, "duration": 6.0, "text": "just for to give some concrete numbers"}, {"start": 46.559, "duration": 6.48, "text": "suppose Armand sent um suppose he sent"}, {"start": 49.36, "duration": 3.679, "text": "three text messages each"}, {"start": 56.6, "duration": 5.24, "text": "hour well he did that for 5 hours total"}, {"start": 59.559, "duration": 4.041, "text": "so he sent three plus another three plus"}, {"start": 61.84, "duration": 5.88, "text": "another three plus another three plus"}, {"start": 63.6, "duration": 6.159, "text": "another three for a total of 15 messages"}, {"start": 67.72, "duration": 3.6, "text": "well the way that we got that is we"}, {"start": 69.759, "duration": 4.441, "text": "would say again if he sent three"}, {"start": 71.32, "duration": 7.159, "text": "messages he sent three messages"}, {"start": 74.2, "duration": 5.879, "text": "multiplied by 5 hours so 3 multiplied 5"}, {"start": 78.479, "duration": 5.96, "text": "is going to give us"}, {"start": 80.079, "duration": 7.72, "text": "15 okay so again instead of three though"}, {"start": 84.439, "duration": 8.121, "text": "that was our the value we used for m so"}, {"start": 87.799, "duration": 8.401, "text": "we would have M multiplied 5 or 5 m"}, {"start": 92.56, "duration": 7.16, "text": "message messages total sent by Armand so"}, {"start": 96.2, "duration": 5.72, "text": "Armand sent 5 multipli by m messages and"}, {"start": 99.72, "duration": 4.8, "text": "the same reasoning Tyrone sent P"}, {"start": 101.92, "duration": 6.32, "text": "messages for 4"}, {"start": 104.52, "duration": 6.32, "text": "hours so Tyrone sent P messages for 4"}, {"start": 108.24, "duration": 4.8, "text": "hours so that means he we would take"}, {"start": 110.84, "duration": 5.16, "text": "four and multiply that by P to get the"}, {"start": 113.04, "duration": 4.96, "text": "number of messages that Tyrone sent so"}, {"start": 116.0, "duration": 4.479, "text": "the total in this case we would just add"}, {"start": 118.0, "duration": 7.079, "text": "those two values together we would have"}, {"start": 120.479, "duration": 6.32, "text": "5 m + 4 p and those are not like terms"}, {"start": 125.079, "duration": 4.481, "text": "because we have the variable M and we"}, {"start": 126.799, "duration": 4.481, "text": "have the variable P um so we can't"}, {"start": 129.56, "duration": 5.039, "text": "combine that any further but that would"}, {"start": 131.28, "duration": 5.959, "text": "be our solution in this case so it looks"}, {"start": 134.599, "duration": 6.081, "text": "like answer Choice C here would be the"}, {"start": 137.239, "duration": 3.441, "text": "correct choice"}], "title": "SAT Math - Question 3, 2016 Edition", "description": "Thanks to all of you who support me on Patreon. You da real mvps!   $1 per month helps!! :)  https://www.patreon.com/patrickjmt !! SAT Math - Question 3, 2016 Edition\nSection 1 - No calculator allowed", "lengthSeconds": 140, "width": 480, "height": 360}
{"video_id": "5s75_WtolRg", "transcript": [{"start": 0.599, "duration": 3.12, "text": "all right so another couple examples"}, {"start": 2.12, "duration": 3.32, "text": "here involving reading and writing whole"}, {"start": 3.719, "duration": 3.881, "text": "numbers and what we're going to do is"}, {"start": 5.44, "duration": 4.48, "text": "we're just going to take some uh you"}, {"start": 7.6, "duration": 5.44, "text": "know kind of some English and translate"}, {"start": 9.92, "duration": 5.68, "text": "it into actual you know actual numbers"}, {"start": 13.04, "duration": 6.48, "text": "so we're going to write uh"}, {"start": 15.6, "duration": 6.04, "text": "2,500 and four in standard form so just"}, {"start": 19.52, "duration": 4.8, "text": "you know write that out"}, {"start": 21.64, "duration": 4.32, "text": "numerically so again just as a little"}, {"start": 24.32, "duration": 4.08, "text": "reminder"}, {"start": 25.96, "duration": 5.36, "text": "here remember if we have digits I'm just"}, {"start": 28.4, "duration": 5.64, "text": "going to write things at random here"}, {"start": 31.32, "duration": 4.759, "text": "uh uh"}, {"start": 34.04, "duration": 5.519, "text": "three and of course you could make"}, {"start": 36.079, "duration": 5.521, "text": "numbers bigger than this but um again"}, {"start": 39.559, "duration": 5.041, "text": "just to remind you about the the the"}, {"start": 41.6, "duration": 5.68, "text": "names so the eight here is in the one's"}, {"start": 44.6, "duration": 6.56, "text": "place the seven is in the"}, {"start": 47.28, "duration": 3.88, "text": "tens the six is in the"}, {"start": 51.64, "duration": 4.439, "text": "hundreds let's see the five is in the"}, {"start": 56.76, "duration": 5.16, "text": "thousands the four would be in the"}, {"start": 65.2, "duration": 5.919, "text": "10,000s uh our three next would be in"}, {"start": 68.0, "duration": 3.119, "text": "the 100"}, {"start": 75.6, "duration": 7.24, "text": "THS uh and if we kept going uh this uh"}, {"start": 80.24, "duration": 5.519, "text": "our two will be in the millions"}, {"start": 82.84, "duration": 5.919, "text": "place and then our next digit would be"}, {"start": 85.759, "duration": 4.68, "text": "in the 10 Millions okay so I'm just"}, {"start": 88.759, "duration": 4.241, "text": "going to kind of reference this this on"}, {"start": 90.439, "duration": 6.521, "text": "my next two examples here real"}, {"start": 93.0, "duration": 7.52, "text": "quickly so it says we have well"}, {"start": 96.96, "duration": 5.479, "text": "254 well 2,000 you know I find the"}, {"start": 100.52, "duration": 5.16, "text": "thousand's place which would basically"}, {"start": 102.439, "duration": 5.36, "text": "be uh right there and the thousand's"}, {"start": 105.68, "duration": 4.039, "text": "place it says we have well 2,000 so"}, {"start": 107.799, "duration": 3.761, "text": "instead of my five I'm simply going to"}, {"start": 109.719, "duration": 4.72, "text": "have a"}, {"start": 111.56, "duration": 4.68, "text": "two and then the number uh you know the"}, {"start": 114.439, "duration": 3.36, "text": "next part is kind of smaller than the"}, {"start": 116.24, "duration": 5.12, "text": "Thousand so we have"}, {"start": 117.799, "duration": 5.881, "text": "500 so I think well the hundreds uh"}, {"start": 121.36, "duration": 5.24, "text": "instead of a six since it's 500 we're"}, {"start": 123.68, "duration": 5.639, "text": "going to put a five right there and then"}, {"start": 126.6, "duration": 6.24, "text": "the last part of the statement it says"}, {"start": 129.319, "duration": 6.0, "text": "uh we also have the number four well the"}, {"start": 132.84, "duration": 4.64, "text": "number four there's no tens right you"}, {"start": 135.319, "duration": 5.041, "text": "know it's not 1 four or two four or 3"}, {"start": 137.48, "duration": 6.56, "text": "four because that would be 14 24"}, {"start": 140.36, "duration": 7.48, "text": "34 so since it's just Plano four we can"}, {"start": 144.04, "duration": 6.44, "text": "write four as well 04 and now we've got"}, {"start": 147.84, "duration": 4.56, "text": "the number 2,500"}, {"start": 150.48, "duration": 5.2, "text": "4 in standard"}, {"start": 152.4, "duration": 5.52, "text": "form so one more example"}, {"start": 155.68, "duration": 4.16, "text": "here um and this is why I went out to"}, {"start": 157.92, "duration": 4.679, "text": "Millions because I knew my next example"}, {"start": 159.84, "duration": 5.28, "text": "did go out to Millions so it says uh"}, {"start": 162.599, "duration": 5.201, "text": "what is uh what's the following number"}, {"start": 165.12, "duration": 5.399, "text": "in standard form"}, {"start": 167.8, "duration": 4.88, "text": "6,200"}, {"start": 170.519, "duration": 4.681, "text": "386 so"}, {"start": 172.68, "duration": 4.199, "text": "again kind of the same thing you know"}, {"start": 175.2, "duration": 3.36, "text": "I'm going to think well it says six"}, {"start": 176.879, "duration": 3.841, "text": "million so I'm going to find the the"}, {"start": 178.56, "duration": 4.12, "text": "million's place"}, {"start": 180.72, "duration": 3.48, "text": "well there's the millions place so we"}, {"start": 182.68, "duration": 3.08, "text": "have six millions so I'm just going to"}, {"start": 184.2, "duration": 4.2, "text": "put a six"}, {"start": 185.76, "duration": 5.36, "text": "there it says we have 2"}, {"start": 188.4, "duration": 5.32, "text": "100,000 well there's the 100, thousand's"}, {"start": 191.12, "duration": 4.8, "text": "place uh the 100, thousand's place is"}, {"start": 193.72, "duration": 3.84, "text": "you know the very next one it says we"}, {"start": 195.92, "duration": 4.64, "text": "have"}, {"start": 197.56, "duration": 6.8, "text": "200,000 so I'm going to put a two in the"}, {"start": 200.56, "duration": 6.239, "text": "100,000 place um it says we have"}, {"start": 204.36, "duration": 4.0, "text": "200,000 and then"}, {"start": 206.799, "duration": 4.881, "text": "386"}, {"start": 208.36, "duration": 5.48, "text": "so going to put it a couple zeros so"}, {"start": 211.68, "duration": 5.479, "text": "there's our"}, {"start": 213.84, "duration": 5.52, "text": "200,000 and now it says we have 386 so"}, {"start": 217.159, "duration": 5.921, "text": "again in the 100 spot is our"}, {"start": 219.36, "duration": 6.56, "text": "three and then our 86 we can just write"}, {"start": 223.08, "duration": 8.159, "text": "that as 8 and six and now we've got our"}, {"start": 225.92, "duration": 5.319, "text": "number again written in standard form"}], "title": "Writing Whole Numbers in Standard Form (English to Number!)", "description": "Thanks to all of you who support me on Patreon. You da real mvps!   $1 per month helps!! :)  https://www.patreon.com/patrickjmt !! Writing Whole Numbers in Standard Form (English to Number!).  Here we have a number written out in English and we write it in standard form.  Two examples are shown.", "lengthSeconds": 229, "width": 480, "height": 360}
{"video_id": "SFs2ys2gpoE", "transcript": [{"start": 0.68, "duration": 4.48, "text": "all right so another example here using"}, {"start": 2.399, "duration": 5.801, "text": "the alternating series test so we've got"}, {"start": 5.16, "duration": 7.32, "text": "the series from Nal 1 to infinity1 to"}, {"start": 8.2, "duration": 6.0, "text": "the n * n/ 5n + 6 so in this case again"}, {"start": 12.48, "duration": 4.879, "text": "if we kind of get rid of the alternating"}, {"start": 14.2, "duration": 7.28, "text": "alternating part we're left with n over"}, {"start": 17.359, "duration": 6.24, "text": "5n + 6 so two things we have to look at"}, {"start": 21.48, "duration": 4.52, "text": "for this case we have to think you know"}, {"start": 23.599, "duration": 2.401, "text": "is it"}, {"start": 27.279, "duration": 5.44, "text": "decreasing and the other one is does the"}, {"start": 29.679, "duration": 6.56, "text": "limit as n goes to Infinity of this"}, {"start": 32.719, "duration": 5.241, "text": "function does that equal"}, {"start": 36.239, "duration": 4.441, "text": "zero"}, {"start": 37.96, "duration": 5.0, "text": "well I'm going to answer the second part"}, {"start": 40.68, "duration": 5.039, "text": "first um you know if we take the limit"}, {"start": 42.96, "duration": 4.4, "text": "as n goes to Infinity uh of this"}, {"start": 45.719, "duration": 3.721, "text": "function since the degree of the"}, {"start": 47.36, "duration": 4.719, "text": "numerator equals the degree of the"}, {"start": 49.44, "duration": 4.599, "text": "denominator again we've seen these cases"}, {"start": 52.079, "duration": 3.721, "text": "you can always just do lal's rule on"}, {"start": 54.039, "duration": 4.121, "text": "this problem as well because we have"}, {"start": 55.8, "duration": 4.16, "text": "Infinity over infinity but again if the"}, {"start": 58.16, "duration": 3.32, "text": "degree of the numerator equals the"}, {"start": 59.96, "duration": 4.32, "text": "degree of the denominator we can just"}, {"start": 61.48, "duration": 5.639, "text": "use the ratio of those coefficients so"}, {"start": 64.28, "duration": 5.6, "text": "our limit's simply going to equal 1/5"}, {"start": 67.119, "duration": 5.081, "text": "well that's not zero so in this case the"}, {"start": 69.88, "duration": 5.4, "text": "conditions aren't satisfied and we would"}, {"start": 72.2, "duration": 4.32, "text": "simply say that this series diverges and"}, {"start": 75.28, "duration": 2.76, "text": "you can do the same thing at the very"}, {"start": 76.52, "duration": 4.36, "text": "beginning you could simply just take the"}, {"start": 78.04, "duration": 4.399, "text": "you know use the test for Divergence and"}, {"start": 80.88, "duration": 3.96, "text": "uh when you take the limit of this stuff"}, {"start": 82.439, "duration": 7.281, "text": "it doesn't equal zero so again that"}, {"start": 84.84, "duration": 4.88, "text": "automatically says our series diverges"}], "title": "Alternating Series - Another Example 2", "description": "Thanks to all of you who support me on Patreon. You da real mvps!   $1 per month helps!! :)  https://www.patreon.com/patrickjmt !! Alternating Series - Another Example 2.  In this video, I show that an alternating series converges or diverges.", "lengthSeconds": 88, "width": 480, "height": 360}
{"video_id": "HflALiqXJDo", "transcript": [{"start": 0.74, "duration": 3.66, "text": "Howdy out there Youtube I am PatrickJMT and this is my math channel"}, {"start": 4.78, "duration": 1.66, "text": "What do I do on my math channel?"}, {"start": 6.46, "duration": 2.0, "text": "Well I make math videos"}, {"start": 8.52, "duration": 1.2, "text": "How many math videos?"}, {"start": 10.0, "duration": 6.34, "text": "a lot of math videos (chuckle) over a ton of topics and I plan on expanding on this just like I do every school year"}, {"start": 17.56, "duration": 3.62, "text": "got a decent, certainly a fairly robust I think math background"}, {"start": 21.4, "duration": 5.32, "text": "lots of experience teaching, a masters degree in math, did a lot of private tutoring as well"}, {"start": 27.96, "duration": 1.14, "text": "What's my goal?"}, {"start": 29.18, "duration": 2.0, "text": "Just to make clear and effective videos"}, {"start": 31.24, "duration": 2.94, "text": "I feel like people are probably looking at these at 11 o'clock at night"}, {"start": 34.2, "duration": 1.66, "text": "they're trying to get their homework done"}, {"start": 35.92, "duration": 5.16, "text": "maybe not always 11 o'clock at night, but just some extra supplements for when you're not in the classroom"}, {"start": 41.18, "duration": 2.96, "text": "and you have those questions at night and you don't know where to turn"}, {"start": 44.94, "duration": 4.26, "text": "I appreciate all of the love and support through the years from everybody out there on Youtube"}, {"start": 49.52, "duration": 4.2, "text": "So I hope you guys keep watching and tell your friends and teachers as well"}, {"start": 53.92, "duration": 3.32, "text": "and again thanks for all the support, it means the world to me"}], "title": "Welcome to PatrickJMT's Math Channel!", "description": "Thanks to all of you who support me on Patreon. You da real mvps!   $1 per month helps!! :)  https://www.patreon.com/patrickjmt !! Please Subscribe! Thanks!!\n** WANT TO WATCH PATRICKJMT ADVERTISEMENT FREE? VISIT: http://www.youtube.com/channel/UCLEu9VwxySzltDuOb2f6BVQ ********\nTons of supplemental mathematics videos for your viewing pleasure.  I try and explain the mechanics and technical side of mathematics as clearly as possible.\n  Thanks for all of the super nice comments.  I sincerely appreciate them!\nNote: These videos are intended to be supplements to your own studies and not stand-alone lectures.", "lengthSeconds": 57, "width": 450, "height": 360}
{"video_id": "7rzZQJtwiIk", "transcript": [{"start": 0.96, "duration": 3.08, "text": "okay in this example I want to express"}, {"start": 3.32, "duration": 3.28, "text": "the"}, {"start": 4.04, "duration": 7.24, "text": "inequalities um X greater than or equal"}, {"start": 6.6, "duration": 7.36, "text": "to -6 and X less than 10 using interval"}, {"start": 11.28, "duration": 4.399, "text": "notation and the way again I like to do"}, {"start": 13.96, "duration": 5.04, "text": "these is I like to make a little number"}, {"start": 15.679, "duration": 6.321, "text": "line so the first condition it says X is"}, {"start": 19.0, "duration": 6.119, "text": "greater than or equal to"}, {"start": 22.0, "duration": 5.32, "text": "-6 okay so I'm just going to put -6 um"}, {"start": 25.119, "duration": 4.92, "text": "on my number line again I could use a"}, {"start": 27.32, "duration": 6.12, "text": "solid dot or we can use a bracket to"}, {"start": 30.039, "duration": 5.081, "text": "indicate that it gets uh used and for"}, {"start": 33.44, "duration": 3.52, "text": "this inequality we would just shade"}, {"start": 35.12, "duration": 4.439, "text": "everything to the right forever and ever"}, {"start": 36.96, "duration": 4.24, "text": "and ever and ever and ever and ever but"}, {"start": 39.559, "duration": 2.721, "text": "that's not the only inequality we're"}, {"start": 41.2, "duration": 3.92, "text": "trying to"}, {"start": 42.28, "duration": 3.959, "text": "satisfy we want to simultaneously"}, {"start": 45.12, "duration": 4.4, "text": "satisfy the"}, {"start": 46.239, "duration": 4.521, "text": "inequality that uh X also has to be less"}, {"start": 49.52, "duration": 3.719, "text": "than"}, {"start": 50.76, "duration": 5.0, "text": "10 so I'm going to put 10 over here in"}, {"start": 53.239, "duration": 6.0, "text": "my number line and we again we could use"}, {"start": 55.76, "duration": 6.72, "text": "an open circle or we can use parentheses"}, {"start": 59.239, "duration": 5.761, "text": "to indicate that we get arbitrarily"}, {"start": 62.48, "duration": 6.279, "text": "close to 10 but we don't include it and"}, {"start": 65.0, "duration": 6.56, "text": "then if you shade to the left for that"}, {"start": 68.759, "duration": 4.801, "text": "one well everything less than 10 would"}, {"start": 71.56, "duration": 5.199, "text": "work for that one but again we're trying"}, {"start": 73.56, "duration": 5.48, "text": "to satisfy both inequalities at the same"}, {"start": 76.759, "duration": 5.04, "text": "time so if you look at the region in the"}, {"start": 79.04, "duration": 5.96, "text": "Middle where they overlap that's going"}, {"start": 81.799, "duration": 6.32, "text": "to be the set of x's that satisfy both"}, {"start": 85.0, "duration": 4.92, "text": "inequalities at the same time so the way"}, {"start": 88.119, "duration": 4.32, "text": "that we can express these two in"}, {"start": 89.92, "duration": 6.68, "text": "inequalities using interval notation we"}, {"start": 92.439, "duration": 6.881, "text": "can say that we include -6 and then we"}, {"start": 96.6, "duration": 7.199, "text": "get up to but not including the number"}, {"start": 99.32, "duration": 6.799, "text": "10 so -6 in Brackets up to positive 10"}, {"start": 103.799, "duration": 4.801, "text": "in parentheses would be an equivalent"}, {"start": 106.119, "duration": 3.921, "text": "way of writing those inequalities using"}, {"start": 108.6, "duration": 4.44, "text": "interval"}, {"start": 110.04, "duration": 3.0, "text": "notation"}], "title": "Using Interval Notation to Express Inequalities - Example 1", "description": "Thanks to all of you who support me on Patreon. You da real mvps!   $1 per month helps!! :)  https://www.patreon.com/patrickjmt !! Using Interval Notation to Express Inequalities - Example 1", "lengthSeconds": 111, "width": 480, "height": 360}
{"video_id": "EZnMfD3UTCA", "transcript": [{"start": 0.88, "duration": 5.879, "text": "okay here we're going to graph two"}, {"start": 2.44, "duration": 8.6, "text": "periods of the function y = 4 * cosin X"}, {"start": 6.759, "duration": 7.241, "text": "and the period for cosine X is equal to"}, {"start": 11.04, "duration": 5.4, "text": "2 pi so what I like to do when I graph"}, {"start": 14.0, "duration": 4.84, "text": "trig functions is first off I like to"}, {"start": 16.44, "duration": 5.04, "text": "find the period I should say for sign"}, {"start": 18.84, "duration": 6.16, "text": "and cosine I like to find the period and"}, {"start": 21.48, "duration": 8.6, "text": "then what I like to do is to divide that"}, {"start": 25.0, "duration": 10.0, "text": "into four equal parts so 2 Pi / 4 will"}, {"start": 30.08, "duration": 7.6, "text": "Al Pi / 2 and what I do is I I use these"}, {"start": 35.0, "duration": 5.879, "text": "values as my increments when I Mark"}, {"start": 37.68, "duration": 6.719, "text": "values on the x- axis so okay so what I"}, {"start": 40.879, "duration": 6.281, "text": "mean is here's uh you know x equals 0"}, {"start": 44.399, "duration": 5.48, "text": "I'll move over there's pi over two I'll"}, {"start": 47.16, "duration": 4.239, "text": "move over another pi/ 2 which would put"}, {"start": 49.879, "duration": 4.52, "text": "me at"}, {"start": 51.399, "duration": 6.96, "text": "pi and let's see if I go over another"}, {"start": 54.399, "duration": 8.48, "text": "I'll have 3 Pi / 2 and then I'll have 4"}, {"start": 58.359, "duration": 7.041, "text": "Pi / 2 or 2 pi let's see so 1 piun / 2 2"}, {"start": 62.879, "duration": 7.761, "text": "piun over 2 3 piun over 2 4 piun / 2"}, {"start": 65.4, "duration": 7.8, "text": "we'll have 5 piun / 2 6 pi over 2 um"}, {"start": 70.64, "duration": 5.32, "text": "let's see so 6 pi over"}, {"start": 73.2, "duration": 6.8, "text": "2 well 6 pi over 2 is just going to be"}, {"start": 75.96, "duration": 6.72, "text": "equal to 3 Pi then we would have 7 pi"}, {"start": 80.0, "duration": 5.88, "text": "over 2 and then we would have 8 pi over"}, {"start": 82.68, "duration": 5.759, "text": "2 which would be equal to 4 Pi so"}, {"start": 85.88, "duration": 4.8, "text": "that'll give us two complete periods"}, {"start": 88.439, "duration": 6.761, "text": "okay and now you"}, {"start": 90.68, "duration": 4.52, "text": "know what you can always plot a couple"}, {"start": 95.759, "duration": 9.481, "text": "points so if x is 0 cosine of 0 is equal"}, {"start": 100.36, "duration": 9.039, "text": "to 1 so you'll have 4 * 1 or you'll get"}, {"start": 105.24, "duration": 7.199, "text": "four so here we are up here at 1 2 3 4"}, {"start": 109.399, "duration": 4.881, "text": "again uh the X the coordinates on the x-"}, {"start": 112.439, "duration": 4.0, "text": "axis are not going to be the same as the"}, {"start": 114.28, "duration": 5.759, "text": "coordinates on the Y AIS here so not"}, {"start": 116.439, "duration": 4.36, "text": "exactly to scale but uh you know reason"}, {"start": 120.039, "duration": 4.0, "text": "ly"}, {"start": 120.799, "duration": 4.881, "text": "close and then what's going to happen is"}, {"start": 124.039, "duration": 4.481, "text": "so that's going to be the highest point"}, {"start": 125.68, "duration": 5.919, "text": "that cosine touches so the reason why I"}, {"start": 128.52, "duration": 4.92, "text": "like to divide these into four parts is"}, {"start": 131.599, "duration": 5.28, "text": "when I go to the next increment that's"}, {"start": 133.44, "duration": 5.2, "text": "where cosine is going to hit the xaxis"}, {"start": 136.879, "duration": 5.64, "text": "and then at the next value that's when"}, {"start": 138.64, "duration": 8.08, "text": "we'll be at - 1 2 3 we'll be down at -4"}, {"start": 142.519, "duration": 7.521, "text": "so I'm going from positive4 to 0 to"}, {"start": 146.72, "duration": 5.0, "text": "-4 back up to zero and then I'll be back"}, {"start": 150.04, "duration": 4.12, "text": "up at"}, {"start": 151.72, "duration": 5.599, "text": "pos4"}, {"start": 154.16, "duration": 6.439, "text": "and that'll roughly be a little sketch"}, {"start": 157.319, "duration": 5.241, "text": "of one period of cosine and now I just"}, {"start": 160.599, "duration": 4.161, "text": "repeat this again so again this is why I"}, {"start": 162.56, "duration": 5.12, "text": "mark them off in fours I kind of get the"}, {"start": 164.76, "duration": 5.839, "text": "the high point the midpoint the low"}, {"start": 167.68, "duration": 4.32, "text": "Point uh back to the midpoint and then"}, {"start": 170.599, "duration": 5.241, "text": "back to the high"}, {"start": 172.0, "duration": 9.28, "text": "point so it'll go at 5 pi/ 2 it'll be"}, {"start": 175.84, "duration": 8.88, "text": "zero at 3 Pi again we'll be back at -4"}, {"start": 181.28, "duration": 8.16, "text": "7 piun / 2 I'm back at zero and then at"}, {"start": 184.72, "duration": 8.12, "text": "4 Pi you'll be back at positive 4 so"}, {"start": 189.44, "duration": 8.359, "text": "that would be two periods"}, {"start": 192.84, "duration": 7.2, "text": "of Y = 4 * cosine X and again you can"}, {"start": 197.799, "duration": 4.36, "text": "always plot points uh do other things to"}, {"start": 200.04, "duration": 5.199, "text": "make sure that the graph is exactly what"}, {"start": 202.159, "duration": 5.561, "text": "you want but again sign and cosine I"}, {"start": 205.239, "duration": 4.681, "text": "take the period divide it into four"}, {"start": 207.72, "duration": 4.28, "text": "that's what I use as my increments and"}, {"start": 209.92, "duration": 4.48, "text": "and then I just start uh you know"}, {"start": 212.0, "duration": 5.08, "text": "putting some points down and again uh"}, {"start": 214.4, "duration": 7.44, "text": "just connecting them with those curvy"}, {"start": 217.08, "duration": 4.76, "text": "curvy coine uh lines"}], "title": "Graphing a Cosine Function  EX 2", "description": "Thanks to all of you who support me on Patreon. You da real mvps!   $1 per month helps!! :)  https://www.patreon.com/patrickjmt !! Just another example of graphing a trigonometric function", "lengthSeconds": 220, "width": 450, "height": 360}
{"video_id": "tquUDsduwys", "transcript": [{"start": 0.03, "duration": 4.59, "text": "okay here we're going to solve the"}, {"start": 1.86, "duration": 5.34, "text": "system of equations for X minus y equals"}, {"start": 4.62, "duration": 4.95, "text": "two and negative 12x plus 3y equals"}, {"start": 7.2, "duration": 4.92, "text": "negative six so I'm going to use"}, {"start": 9.57, "duration": 3.84, "text": "elimination by addition and the thing"}, {"start": 12.12, "duration": 4.89, "text": "that I'm going to try to do is to"}, {"start": 13.41, "duration": 4.83, "text": "eliminate the the terms containing X so"}, {"start": 17.01, "duration": 5.22, "text": "I think you know if I just had a"}, {"start": 18.24, "duration": 6.09, "text": "positive 12x in my as my first term in"}, {"start": 22.23, "duration": 4.83, "text": "my first equation well then I could add"}, {"start": 24.33, "duration": 5.76, "text": "I would have 12x minus 12x the terms"}, {"start": 27.06, "duration": 5.73, "text": "involving X would cancel so to make a"}, {"start": 30.09, "duration": 7.17, "text": "12x appear what I can do to my first"}, {"start": 32.79, "duration": 9.3, "text": "equation is I can simply multiply both"}, {"start": 37.26, "duration": 10.619, "text": "sides by 3 well if I do that I'll get"}, {"start": 42.09, "duration": 9.449, "text": "12x minus 3y equals 6 and then we have"}, {"start": 47.879, "duration": 6.721, "text": "negative 12x plus 3y equals negative 6"}, {"start": 51.539, "duration": 4.95, "text": "just our original second equation notice"}, {"start": 54.6, "duration": 5.04, "text": "if I add these two equations"}, {"start": 56.489, "duration": 7.831, "text": "I'll get 12x minus 12x I'll get 0 X"}, {"start": 59.64, "duration": 7.17, "text": "negative 3y plus 3y is 0 Y so we're left"}, {"start": 64.32, "duration": 5.4, "text": "with 0 on the left side on the right"}, {"start": 66.81, "duration": 7.76, "text": "side we have 6 minus 6 which is 0 and"}, {"start": 69.72, "duration": 4.85, "text": "what this tells me is we have infinitely"}, {"start": 74.6, "duration": 10.659, "text": "we have infinitely many solutions"}, {"start": 79.729, "duration": 9.271, "text": "so if you actually want to write your"}, {"start": 85.259, "duration": 6.57, "text": "solutions down to sort of give a a"}, {"start": 89.0, "duration": 4.84, "text": "generic sort of form people sometimes"}, {"start": 91.829, "duration": 4.201, "text": "ask for this what you can do is take any"}, {"start": 93.84, "duration": 5.069, "text": "one of your two equations I started with"}, {"start": 96.03, "duration": 5.33, "text": "the 4x minus y equals 2 I started with"}, {"start": 98.909, "duration": 4.951, "text": "my equation before we you know before we"}, {"start": 101.36, "duration": 6.939, "text": "multiplied everything out 4x minus y"}, {"start": 103.86, "duration": 6.09, "text": "equals 2 I'm going to solve for y so I'm"}, {"start": 108.299, "duration": 5.161, "text": "going to subtract the 2 to the left side"}, {"start": 109.95, "duration": 7.14, "text": "I'm going to add Y over and now the idea"}, {"start": 113.46, "duration": 8.22, "text": "is we can say let's let X equal some"}, {"start": 117.09, "duration": 9.9, "text": "constant K well if we do that then Y is"}, {"start": 121.68, "duration": 7.799, "text": "going to equal 4 K minus 2 so we can say"}, {"start": 126.99, "duration": 6.34, "text": "our solutions that are of the form K"}, {"start": 129.479, "duration": 9.041, "text": "comma 4 K minus 2"}, {"start": 133.33, "duration": 9.84, "text": "so the idea the idea is pick your"}, {"start": 138.52, "duration": 6.54, "text": "favorite value for X and then do four"}, {"start": 143.17, "duration": 4.02, "text": "times that number minus two that will"}, {"start": 145.06, "duration": 4.02, "text": "give you the Y value and that's the way"}, {"start": 147.19, "duration": 4.47, "text": "that you can actually find specific"}, {"start": 149.08, "duration": 6.86, "text": "specific pairs of numbers that satisfy"}, {"start": 151.66, "duration": 4.28, "text": "the original system of equations"}], "title": "A Linear System of Equations with Infinitely Many Solutions: Quick Example", "description": "(https://youtu.be/rnyLW2-lL7o)\n(https://youtu.be/CsTOUbeMPUo)\n(https://youtu.be/cJg2AuSFdjw)\n(https://youtu.be/hu6B1d3vvqU)\n(https://youtu.be/9LYVi-n-6Jw)\n(https://youtu.be/iMQRo0tHORw)\n(https://youtu.be/Re1F4d24Fxc)\n(https://youtu.be/tquUDsduwys)I\n\nn this video I work a system of two linear equations and show that they are linearly dependent, meaning that there are infinitely many solutions to the system.  For two linear equations, this simply implies that the two lines are exactly the same and would cover each other if you were to graph them.\nThis is a typical topic seen in algebra, linear algebra, linear programming, and other areas.\nFor more detailed info about these types of systems, check out my other videos!\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n#math #algebra #khanacademy #khan #geometry #education #montessori #homeschooling #functions #notation #organicchemistry #organicchemistrytutor\n#logarithm #naturallog #exponential #system #linearsystems #linearlyindependent #linearlydependent #linearequations #graphing", "lengthSeconds": 155, "width": 450, "height": 360}
{"video_id": "Q86iLeaOpkY", "transcript": [{"start": 0.68, "duration": 3.039, "text": "all right in this example we're going to"}, {"start": 1.959, "duration": 6.201, "text": "arrange some integers from the least to"}, {"start": 3.719, "duration": 7.441, "text": "the greatest um we've got -7 pos4 -10 0"}, {"start": 8.16, "duration": 5.16, "text": "18 and ne1 and again maybe it's easiest"}, {"start": 11.16, "duration": 5.64, "text": "to think about a number line so the"}, {"start": 13.32, "duration": 5.879, "text": "first thing I do is pick out my negative"}, {"start": 16.8, "duration": 5.12, "text": "values so I know those are going to be"}, {"start": 19.199, "duration": 5.24, "text": "the smallest numbers well let's see if"}, {"start": 21.92, "duration": 4.279, "text": "here if there's zero negative 1 would be"}, {"start": 24.439, "duration": 5.721, "text": "right about there I'm not going to fill"}, {"start": 26.199, "duration": 6.961, "text": "in every digit but there's -7 and 10"}, {"start": 30.16, "duration": 6.36, "text": "would be the furthest one to the left so"}, {"start": 33.16, "duration": 5.64, "text": "that means -10 would be our smallest"}, {"start": 36.52, "duration": 5.4, "text": "number let's see I guess the next one"}, {"start": 38.8, "duration": 6.0, "text": "would be -7 and then we would have ne1"}, {"start": 41.92, "duration": 5.76, "text": "so that takes care of those zero would"}, {"start": 44.8, "duration": 4.96, "text": "be the next number and then last but not"}, {"start": 47.68, "duration": 5.719, "text": "least we've got four and 18 well"}, {"start": 49.76, "duration": 5.76, "text": "certainly four is smaller than"}, {"start": 53.399, "duration": 7.041, "text": "18 and now we've got our numbers"}, {"start": 55.52, "duration": 4.92, "text": "arranged from least to greatest"}], "title": "Integers : Arranging Integers from Smallest to Largest on Number Line", "description": "Thanks to all of you who support me on Patreon. You da real mvps!   $1 per month helps!! :)  https://www.patreon.com/patrickjmt !! Integers : Arranging Integers from Smallest to Largest on Number Line.\nHere we look at arranging some integers from smallest to largest on a number line.", "lengthSeconds": 61, "width": 480, "height": 360}
{"video_id": "5p2859SCzdM", "transcript": [{"start": 0.24, "duration": 3.12, "text": "all right in this video I want to talk"}, {"start": 1.76, "duration": 4.039, "text": "about functions and the vertical line"}, {"start": 3.36, "duration": 4.16, "text": "test and the basic idea is if you have a"}, {"start": 5.799, "duration": 5.041, "text": "graph and you want to decide if that"}, {"start": 7.52, "duration": 5.199, "text": "graph represents a function the idea is"}, {"start": 10.84, "duration": 3.679, "text": "if you were to draw any vertical line"}, {"start": 12.719, "duration": 4.241, "text": "and if it touches the graph in two or"}, {"start": 14.519, "duration": 5.441, "text": "more places then the graph does not"}, {"start": 16.96, "duration": 4.12, "text": "represent a function uh nothing nothing"}, {"start": 19.96, "duration": 4.84, "text": "worse than"}, {"start": 21.08, "duration": 5.32, "text": "that so here's a three different graphs"}, {"start": 24.8, "duration": 4.36, "text": "um so let's look at the one here on the"}, {"start": 26.4, "duration": 6.039, "text": "left um kind of looks like a parabola or"}, {"start": 29.16, "duration": 6.44, "text": "sort of like a lazy U we want to decide"}, {"start": 32.439, "duration": 5.96, "text": "if that would represent a function okay"}, {"start": 35.6, "duration": 4.88, "text": "well again the idea is to me you know"}, {"start": 38.399, "duration": 4.32, "text": "the arrows it's kind of always opening"}, {"start": 40.48, "duration": 4.48, "text": "to the left and always opening a little"}, {"start": 42.719, "duration": 4.281, "text": "bit and moving to the right well to me"}, {"start": 44.96, "duration": 3.4, "text": "that says no matter where I were you"}, {"start": 47.0, "duration": 4.96, "text": "know no matter where I was to put a"}, {"start": 48.36, "duration": 6.6, "text": "vertical line it would hit the graph in"}, {"start": 51.96, "duration": 5.439, "text": "at most one"}, {"start": 54.96, "duration": 5.0, "text": "place and since it doesn't hit the graph"}, {"start": 57.399, "duration": 4.48, "text": "in two or more places so since it only"}, {"start": 59.96, "duration": 8.04, "text": "hits the graph and at most one place we"}, {"start": 61.879, "duration": 6.121, "text": "would say yes that graph uh is a"}, {"start": 69.4, "duration": 4.44, "text": "function okay so let's look at this"}, {"start": 71.92, "duration": 3.559, "text": "graph in the middle notice on the left"}, {"start": 73.84, "duration": 4.08, "text": "side if we were to draw vertical lines"}, {"start": 75.479, "duration": 5.201, "text": "it would only hit it in one place but"}, {"start": 77.92, "duration": 4.8, "text": "notice for example the Y AIS hits the"}, {"start": 80.68, "duration": 4.72, "text": "graph in two places and there's"}, {"start": 82.72, "duration": 4.92, "text": "definitely some uh some lines to the"}, {"start": 85.4, "duration": 5.28, "text": "right of the y- AIS vertical lines where"}, {"start": 87.64, "duration": 8.159, "text": "it would hit the graph in two places"}, {"start": 90.68, "duration": 5.119, "text": "so in that case we would say no not a"}, {"start": 97.439, "duration": 4.161, "text": "function okay and then we have a graph"}, {"start": 99.64, "duration": 5.28, "text": "over here on the right uh notice it's"}, {"start": 101.6, "duration": 5.92, "text": "not one smooth graph there's a break in"}, {"start": 104.92, "duration": 4.879, "text": "it okay if it's not kind of one smooth"}, {"start": 107.52, "duration": 4.76, "text": "piece we say it's not"}, {"start": 109.799, "duration": 4.161, "text": "continuous okay so that's okay a graph"}, {"start": 112.28, "duration": 3.839, "text": "could be not continuous and be a"}, {"start": 113.96, "duration": 4.799, "text": "function and I think this one would be a"}, {"start": 116.119, "duration": 4.92, "text": "function because no matter where I put a"}, {"start": 118.759, "duration": 5.72, "text": "vertical line to me it's going to hit"}, {"start": 121.039, "duration": 5.12, "text": "the graph in at most one place notice"}, {"start": 124.479, "duration": 4.28, "text": "there's there's some places where the"}, {"start": 126.159, "duration": 4.761, "text": "vertical line won't hit the graph at all"}, {"start": 128.759, "duration": 5.081, "text": "but that's okay as long as vertical"}, {"start": 130.92, "duration": 4.599, "text": "lines don't hit it in two or more places"}, {"start": 133.84, "duration": 4.32, "text": "likewise any vertical line on the right"}, {"start": 135.519, "duration": 8.601, "text": "would hit it in at most one place so we"}, {"start": 138.16, "duration": 5.96, "text": "would say yes this graph um is a"}, {"start": 146.0, "duration": 3.0, "text": "function"}], "title": "The Vertical Line Test", "description": "Thanks to all of you who support me on Patreon. You da real mvps!   $1 per month helps!! :)  https://www.patreon.com/patrickjmt !! The Vertical Line Test.  Using the Vertical Line Test to determine if a given graph represents a function.", "lengthSeconds": 147, "width": 480, "height": 360}
{"video_id": "PuD61nMOyzM", "transcript": [{"start": 0.48, "duration": 2.76, "text": "all right in this video we're going to"}, {"start": 1.64, "duration": 4.84, "text": "look at finding some Arc lengths of"}, {"start": 3.24, "duration": 7.6, "text": "parametric curves so in part A we've got"}, {"start": 6.48, "duration": 7.0, "text": "the absolute value of six - t uhal x and"}, {"start": 10.84, "duration": 5.72, "text": "then yal T and we're going to evaluate"}, {"start": 13.48, "duration": 6.76, "text": "this from uh 0 to"}, {"start": 16.56, "duration": 5.639, "text": "3 so we'll have to integrate from 0 to 3"}, {"start": 20.24, "duration": 3.56, "text": "we'll take the derivative of x with"}, {"start": 22.199, "duration": 4.441, "text": "respect to"}, {"start": 23.8, "duration": 4.239, "text": "t^2 plus the derivative of y with"}, {"start": 26.64, "duration": 4.959, "text": "respect to"}, {"start": 28.039, "duration": 6.161, "text": "t^2 DT"}, {"start": 31.599, "duration": 4.8, "text": "so notice uh for x equals the absolute"}, {"start": 34.2, "duration": 4.64, "text": "value of 6 minus"}, {"start": 36.399, "duration": 5.961, "text": "t we can actually just say this is the"}, {"start": 38.84, "duration": 3.52, "text": "same thing as 6 minus"}, {"start": 43.16, "duration": 5.919, "text": "t for t on the interval 0 to three"}, {"start": 46.879, "duration": 5.921, "text": "because uh if you take any value between"}, {"start": 49.079, "duration": 5.241, "text": "0o and three uh the 6 minus t will stay"}, {"start": 52.8, "duration": 3.919, "text": "positive so we can just remove the"}, {"start": 54.32, "duration": 4.32, "text": "absolute value well that says the"}, {"start": 56.719, "duration": 4.201, "text": "derivative of x with respect to T then"}, {"start": 58.64, "duration": 4.2, "text": "is just going to be1"}, {"start": 60.92, "duration": 5.0, "text": "and that tells us the derivative of x"}, {"start": 62.84, "duration": 4.52, "text": "with respect to t^ S is just going to"}, {"start": 65.92, "duration": 4.199, "text": "equal"}, {"start": 67.36, "duration": 5.48, "text": "pos1 the derivative of y with respect to"}, {"start": 70.119, "duration": 5.64, "text": "T that's also just going to be positive"}, {"start": 72.84, "duration": 6.639, "text": "one so the derivative of y with respect"}, {"start": 75.759, "duration": 5.641, "text": "to t^2 will also just be positive one"}, {"start": 79.479, "duration": 7.241, "text": "easy enough so we've got the integral"}, {"start": 81.4, "duration": 7.88, "text": "from 0 to 3 we would just have 1 + one"}, {"start": 86.72, "duration": 8.399, "text": "DT well that's the integral from 0 to"}, {"start": 89.28, "duration": 8.76, "text": "three of just theun of 2 * uh or of 2 DT"}, {"start": 95.119, "duration": 4.761, "text": "so when we integrate we'll get of 2 * T"}, {"start": 98.04, "duration": 3.92, "text": "we'll have to evaluate that from 0 to"}, {"start": 99.88, "duration": 5.839, "text": "three and that's just going to leave us"}, {"start": 101.96, "duration": 9.119, "text": "with 3 < tk2 minus 0 so our Arc Length"}, {"start": 105.719, "duration": 5.36, "text": "will simply be 3 * the < TK of two"}], "title": "Find Arc Length of a Curve Given Parametric Equations , x(t) , y(t) , Ex 1", "description": "Thanks to all of you who support me on Patreon. You da real mvps!   $1 per month helps!! :)  https://www.patreon.com/patrickjmt !! Find Arc Length of a Curve Given Parametric Equations , x(t) , y(t)", "lengthSeconds": 109, "width": 480, "height": 360}
{"video_id": "2tiv14iok0k", "transcript": [{"start": 0.48, "duration": 3.36, "text": "all right in this video I'm going to do"}, {"start": 1.719, "duration": 5.16, "text": "some examples of factoring and or"}, {"start": 3.84, "duration": 6.2, "text": "simplifying uh some Expressions so in"}, {"start": 6.879, "duration": 5.961, "text": "part A here we have S S Plus Cent s *"}, {"start": 10.04, "duration": 5.96, "text": "sin s now I think we could certainly"}, {"start": 12.84, "duration": 4.76, "text": "factor a sin squ out maybe even um I"}, {"start": 16.0, "duration": 3.92, "text": "don't know if it's any quicker or any"}, {"start": 17.6, "duration": 3.72, "text": "easier um you could even rewrite"}, {"start": 19.92, "duration": 4.4, "text": "cotangent"}, {"start": 21.32, "duration": 5.359, "text": "squared as cosine squar over sin squar"}, {"start": 24.32, "duration": 4.879, "text": "cancel and use an identity but let's use"}, {"start": 26.679, "duration": 3.92, "text": "this idea of factoring so we can factor"}, {"start": 29.199, "duration": 3.561, "text": "out the sin squ"}, {"start": 30.599, "duration": 6.841, "text": "and then we would have 1"}, {"start": 32.76, "duration": 11.76, "text": "+ S th but recall there's an identity"}, {"start": 37.44, "duration": 7.08, "text": "for 1 + Cent s thet that is cose squ of"}, {"start": 44.6, "duration": 4.799, "text": "theta but what is cose s Theta that's"}, {"start": 47.719, "duration": 5.84, "text": "the same thing as 1"}, {"start": 49.399, "duration": 7.521, "text": "over sin 2 Theta so really uh we have"}, {"start": 53.559, "duration": 6.721, "text": "sin s thet over sin s thet and that's"}, {"start": 56.92, "duration": 6.599, "text": "simply going to equal pos1 so this first"}, {"start": 60.28, "duration": 8.12, "text": "expression simply reduces to positive"}, {"start": 63.519, "duration": 4.881, "text": "one um let's maybe do B here as"}, {"start": 70.08, "duration": 8.399, "text": "well so let's see we've got 2 - cosine 2"}, {"start": 73.759, "duration": 7.521, "text": "over 1 - sin x so um couple different"}, {"start": 78.479, "duration": 4.121, "text": "things that we could do here uh again"}, {"start": 81.28, "duration": 3.159, "text": "the first thing that kind you know we"}, {"start": 82.6, "duration": 4.04, "text": "could try to get common denominators and"}, {"start": 84.439, "duration": 5.121, "text": "group things together uh maybe that'll"}, {"start": 86.64, "duration": 4.28, "text": "make things work out nicely um I also"}, {"start": 89.56, "duration": 3.599, "text": "remember remember though there's an a"}, {"start": 90.92, "duration": 4.76, "text": "relationship between cosine squ and sin"}, {"start": 93.159, "duration": 8.401, "text": "squ so again we can use our identity"}, {"start": 95.68, "duration": 8.399, "text": "that cosine 2 x + sin^2 x = 1 so if we"}, {"start": 101.56, "duration": 6.04, "text": "solve this for cosine s we'll get that"}, {"start": 104.079, "duration": 6.761, "text": "cine 2 x = 1"}, {"start": 107.6, "duration": 4.04, "text": "- sin^2 x that's what I'm going to plug"}, {"start": 110.84, "duration": 5.48, "text": "in the"}, {"start": 111.64, "duration": 10.28, "text": "numerator so we would have 2 - well 1 -"}, {"start": 116.32, "duration": 8.64, "text": "sin^2 x over 1 - sin X and you know why"}, {"start": 121.92, "duration": 4.879, "text": "why does this help well um I recognize"}, {"start": 124.96, "duration": 4.68, "text": "that this numerator is the difference of"}, {"start": 126.799, "duration": 6.401, "text": "perfect squares we can now simply Factor"}, {"start": 129.64, "duration": 5.72, "text": "this numerator um a little"}, {"start": 133.2, "duration": 3.84, "text": "bit you know and kind of think about you"}, {"start": 135.36, "duration": 2.92, "text": "know sort of equivalently if this was"}, {"start": 137.04, "duration": 4.44, "text": "like 1"}, {"start": 138.28, "duration": 5.2, "text": "minus x^2 you've got one minus something"}, {"start": 141.48, "duration": 5.6, "text": "squared how does one minus something"}, {"start": 143.48, "duration": 5.759, "text": "squared Factor it factors as well 1 - x"}, {"start": 147.08, "duration": 3.84, "text": "and 1 + x again it's a difference of"}, {"start": 149.239, "duration": 4.08, "text": "perfect squar"}, {"start": 150.92, "duration": 7.2, "text": "so instead of just plal X we've got sin"}, {"start": 153.319, "duration": 8.0, "text": "x so 1 - sin^2 is going to factor as 1 -"}, {"start": 158.12, "duration": 6.52, "text": "sinx and then 1 +"}, {"start": 161.319, "duration": 5.761, "text": "sinx but hey we've got that 1us sinx"}, {"start": 164.64, "duration": 5.64, "text": "just hanging out in the bottom so really"}, {"start": 167.08, "duration": 4.879, "text": "we can cancel those uh those factors out"}, {"start": 170.28, "duration": 7.2, "text": "and then we'll be left with"}, {"start": 171.959, "duration": 7.64, "text": "2us 1 + sinx but now I can simply"}, {"start": 177.48, "duration": 2.88, "text": "distribute out this negative I will get"}, {"start": 179.599, "duration": 4.321, "text": "2"}, {"start": 180.36, "duration": 8.92, "text": "2 - 1 and again if we distribute the"}, {"start": 183.92, "duration": 9.44, "text": "negative - sinx well 2 - 1 is one so"}, {"start": 189.28, "duration": 6.28, "text": "we'll simply be left with 1 minus sinx"}, {"start": 193.36, "duration": 3.76, "text": "so you know again a little trickier"}, {"start": 195.56, "duration": 3.319, "text": "again this is what makes these problems"}, {"start": 197.12, "duration": 4.6, "text": "hard you know I I know it has something"}, {"start": 198.879, "duration": 4.161, "text": "to do with an identity um it's got"}, {"start": 201.72, "duration": 2.96, "text": "something to do with an identity but"}, {"start": 203.04, "duration": 4.559, "text": "it's just sort of"}, {"start": 204.68, "duration": 4.759, "text": "uh you know where do you use it and when"}, {"start": 207.599, "duration": 4.521, "text": "do you use it and sometimes once you use"}, {"start": 209.439, "duration": 4.16, "text": "it you know maybe it's not readily uh"}, {"start": 212.12, "duration": 4.199, "text": "maybe you don't see that this factors"}, {"start": 213.599, "duration": 3.72, "text": "immediately but uh you know again you"}, {"start": 216.319, "duration": 3.0, "text": "just kind of have to get your hands"}, {"start": 217.319, "duration": 3.721, "text": "dirty play with things a little bit"}, {"start": 219.319, "duration": 3.28, "text": "certainly as you do more examples I"}, {"start": 221.04, "duration": 6.8, "text": "think they'll start becoming a little"}, {"start": 222.599, "duration": 5.241, "text": "more intuitive and a little bit easier"}], "title": "Factoring Trigonometric Expressions, Example 1", "description": "Thanks to all of you who support me on Patreon. You da real mvps!   $1 per month helps!! :)  https://www.patreon.com/patrickjmt !! Factoring Trigonometric Expressions, Example 1", "lengthSeconds": 226, "width": 480, "height": 360}
{"video_id": "59j0ALU3N7k", "transcript": [{"start": 0.64, "duration": 3.44, "text": "all right in this video I'm going to do"}, {"start": 1.88, "duration": 4.12, "text": "a couple examples of solving logarithmic"}, {"start": 4.08, "duration": 3.92, "text": "equations and anytime you solve"}, {"start": 6.0, "duration": 3.679, "text": "logarithmic equations once you get your"}, {"start": 8.0, "duration": 4.16, "text": "answers you always have to go back and"}, {"start": 9.679, "duration": 4.441, "text": "check for extraneous solutions remember"}, {"start": 12.16, "duration": 4.56, "text": "we can't take logarithms of any"}, {"start": 14.12, "duration": 4.239, "text": "nonpositive number so this is something"}, {"start": 16.72, "duration": 4.319, "text": "you just have to worry about at the end"}, {"start": 18.359, "duration": 6.92, "text": "so uh in in a here we're going to solve"}, {"start": 21.039, "duration": 7.24, "text": "2 * log base 3 of x minus log base 3 of"}, {"start": 25.279, "duration": 4.801, "text": "x + 6 = 1 uh the first thing I'm going"}, {"start": 28.279, "duration": 4.361, "text": "to try to do on the left side is rewrite"}, {"start": 30.08, "duration": 6.479, "text": "this all as a single logarithm so I can"}, {"start": 32.64, "duration": 7.16, "text": "rewrite uh 2 * log base 3 of X as log"}, {"start": 36.559, "duration": 6.921, "text": "base 3 of x squared we can move the"}, {"start": 39.8, "duration": 6.48, "text": "coefficient um up as an exponent not"}, {"start": 43.48, "duration": 3.84, "text": "going to do anything with uh my second"}, {"start": 46.28, "duration": 4.88, "text": "term"}, {"start": 47.32, "duration": 6.12, "text": "here but remember if we have like bases"}, {"start": 51.16, "duration": 6.52, "text": "if we have subtraction this becomes"}, {"start": 53.44, "duration": 8.68, "text": "division so we'll have log base 3 of x^2"}, {"start": 57.68, "duration": 6.16, "text": "over x + 6 thatal = 1 and now what I'm"}, {"start": 62.12, "duration": 4.2, "text": "going to do is just rewrite this using"}, {"start": 63.84, "duration": 3.36, "text": "properties of exponents so it says 3 to"}, {"start": 66.32, "duration": 3.76, "text": "the"}, {"start": 67.2, "duration": 6.599, "text": "1st um would equal"}, {"start": 70.08, "duration": 6.079, "text": "x^2 over x + 6 so okay now we've gotten"}, {"start": 73.799, "duration": 6.32, "text": "rid of the the logarithms and now I'm"}, {"start": 76.159, "duration": 6.441, "text": "just going to multiply both sides by x +"}, {"start": 80.119, "duration": 4.721, "text": "6 on the left side we would have to"}, {"start": 82.6, "duration": 8.08, "text": "distribute our positive3 that would give"}, {"start": 84.84, "duration": 7.559, "text": "us 3x + 18 = x^2 on the right um and now"}, {"start": 90.68, "duration": 5.439, "text": "this is a quadratic I'm going to make"}, {"start": 92.399, "duration": 6.881, "text": "the left side Zero by subtracting 3x"}, {"start": 96.119, "duration": 5.521, "text": "from both sides and by subtracting 18"}, {"start": 99.28, "duration": 4.36, "text": "from both sides and I think this is a"}, {"start": 101.64, "duration": 5.96, "text": "quadratic that's going to factor so I"}, {"start": 103.64, "duration": 9.04, "text": "believe this factors as um how about x -"}, {"start": 107.6, "duration": 7.839, "text": "6 and x + 3 uh that looks okay so if we"}, {"start": 112.68, "duration": 5.24, "text": "said each factor equal to zero we'll get"}, {"start": 115.439, "duration": 6.96, "text": "our Solutions our potential solutions to"}, {"start": 117.92, "duration": 6.879, "text": "be xal 6 and x ="}, {"start": 122.399, "duration": 3.921, "text": "-3 now let's go back and look at the"}, {"start": 124.799, "duration": 4.761, "text": "original"}, {"start": 126.32, "duration": 6.88, "text": "problem um we were trying to solve 2 *"}, {"start": 129.56, "duration": 6.44, "text": "log base 3 of x - log base 3 of x + 6 I"}, {"start": 133.2, "duration": 5.96, "text": "kind of check notice if we"}, {"start": 136.0, "duration": 5.2, "text": "plug if we put in the value six we'll"}, {"start": 139.16, "duration": 3.84, "text": "get a six here which is fine I'll get a"}, {"start": 141.2, "duration": 4.52, "text": "logarithm of a positive"}, {"start": 143.0, "duration": 4.44, "text": "number if I plug six into the other"}, {"start": 145.72, "duration": 4.56, "text": "logarithm I'm going to get a logarithm"}, {"start": 147.44, "duration": 4.36, "text": "of a positive number and so far so good"}, {"start": 150.28, "duration": 4.319, "text": "and you can check that if you plug six"}, {"start": 151.8, "duration": 5.76, "text": "in this is going to work but notice if"}, {"start": 154.599, "duration": 6.041, "text": "we plug -3 in we're going to have a"}, {"start": 157.56, "duration": 4.84, "text": "logarithm of a negative number um on the"}, {"start": 160.64, "duration": 4.679, "text": "second term we would end up still if we"}, {"start": 162.4, "duration": 4.68, "text": "plug in neg3 we' have positive three but"}, {"start": 165.319, "duration": 7.92, "text": "on the first part we would end up with 2"}, {"start": 167.08, "duration": 7.84, "text": "* log base 3 of -3 uh minus some other"}, {"start": 173.239, "duration": 4.201, "text": "stuff but already because we have this"}, {"start": 174.92, "duration": 5.599, "text": "negative inside of a logarithm that is"}, {"start": 177.44, "duration": 6.48, "text": "bad and that means X = three is what's"}, {"start": 180.519, "duration": 5.881, "text": "called an extraneous solution so xal 6"}, {"start": 183.92, "duration": 5.48, "text": "would be our only solution in this"}, {"start": 186.4, "duration": 3.0, "text": "problem"}], "title": "\u2756 Solving Logarithmic Equations - Example 1 \u2756", "description": "Solving Logarithmic Equations: Step-by-Step Using Log Properties and Algebra!\nIn this video, we'll solve a logarithmic equation by applying properties of logarithms and some essential algebra techniques. You'll learn how to use logarithmic properties to combine logarithmic terms and rewrite the equation using exponentials. From there, we'll produce a quadratic equation that needs to be solved. Be careful\u2014we'll also discuss how to identify and handle any extraneous solutions that may arise when solving logarithmic equations.\n\nWhat You Will Learn:\n\nHow to use properties of logarithms to combine and simplify terms.\nTechniques for rewriting logarithmic equations using exponentials.\nSolving the resulting quadratic equations.\nTips for identifying and avoiding extraneous solutions.\nStep-by-step examples to deepen your understanding.\n\ud83d\udcda Check out my book: 1001 Calculus Problems for Dummies for more practice!\n\n\ud83d\udc4d **If you find this video helpful, please like, share, and subscribe for more math tutorials!\n\nSupport My Work:\n\nIf you'd like to support the creation of more math content, consider becoming a patron on Patreon: https://www.patreon.com/PatrickJMT\n\n\n\n#Logarithms #Algebra #LogarithmicEquations #MathTutorial #PatrickJMT #SolvingEquations #QuadraticEquations #Exponentials #ExtraneousSolutions #Mathematics #MathHelp #EquationSolving #MathProblems #LogProperties #AlgebraTechniques", "lengthSeconds": 187, "width": 480, "height": 360}
{"video_id": "zrGC3nawUJA", "transcript": [{"start": 0.0, "duration": 4.62, "text": "all right so one more example here using"}, {"start": 2.34, "duration": 4.05, "text": "the direct comparison test part see"}, {"start": 4.62, "duration": 9.689, "text": "we've got the series from N equals 1 to"}, {"start": 6.39, "duration": 9.3, "text": "infinity arctangent of n over N cubed so"}, {"start": 14.309, "duration": 3.091, "text": "in this one again I think you know I'm"}, {"start": 15.69, "duration": 3.96, "text": "not sure I think a lot of people though"}, {"start": 17.4, "duration": 4.26, "text": "would be thrown off by the arctangent of"}, {"start": 19.65, "duration": 5.549, "text": "n you know sort of sort of what's going"}, {"start": 21.66, "duration": 4.289, "text": "on in you know in this case as in"}, {"start": 25.199, "duration": 2.58, "text": "increases"}, {"start": 25.949, "duration": 8.161, "text": "well let's recall the graph of"}, {"start": 27.779, "duration": 8.761, "text": "arctangent of X so recall arctangent of"}, {"start": 34.11, "duration": 4.83, "text": "X looks roughly like my little graph"}, {"start": 36.54, "duration": 7.109, "text": "here and again this has an asymptote at"}, {"start": 38.94, "duration": 6.99, "text": "PI over 2 and at negative PI over 2 well"}, {"start": 43.649, "duration": 4.771, "text": "you know we're plugging in you know N"}, {"start": 45.93, "duration": 4.98, "text": "equals 1 N equals 2 N equals 3 or"}, {"start": 48.42, "duration": 5.549, "text": "equivalently x equals 1 x equals 2 x"}, {"start": 50.91, "duration": 4.59, "text": "equals 3 x equals 4 etc and those are"}, {"start": 53.969, "duration": 3.991, "text": "going to be the values we get in the"}, {"start": 55.5, "duration": 6.98, "text": "numerator but you know again if you"}, {"start": 57.96, "duration": 6.809, "text": "think about it our tangent of n is"}, {"start": 62.48, "duration": 5.05, "text": "always going to be well it's always"}, {"start": 64.769, "duration": 5.521, "text": "smaller than this asymptote PI over 2"}, {"start": 67.53, "duration": 6.06, "text": "and it's again always greater than or"}, {"start": 70.29, "duration": 6.75, "text": "equal to 0 again for values of n that"}, {"start": 73.59, "duration": 6.9, "text": "are greater than or equal to 1 well if"}, {"start": 77.04, "duration": 6.93, "text": "this is true we can simply say well our"}, {"start": 80.49, "duration": 4.62, "text": "tangent of n divided by what do we have"}, {"start": 83.97, "duration": 4.289, "text": "here in cubed"}, {"start": 85.11, "duration": 5.82, "text": "well if divided by n cubed that's going"}, {"start": 88.259, "duration": 7.191, "text": "to be less than or equal to PI over N"}, {"start": 90.93, "duration": 6.93, "text": "excuse-me PI over 2 divided by n cubed"}, {"start": 95.45, "duration": 5.2, "text": "and again these are all positive so I'm"}, {"start": 97.86, "duration": 4.77, "text": "kind of leaving out the zeros so that"}, {"start": 100.65, "duration": 6.57, "text": "means the series from N equals 1 to"}, {"start": 102.63, "duration": 6.809, "text": "infinity of arctangent of n over n cubed"}, {"start": 107.22, "duration": 5.52, "text": "that would be less than or equal to the"}, {"start": 109.439, "duration": 5.43, "text": "series you know PI over 2 divided by n"}, {"start": 112.74, "duration": 3.51, "text": "cubed I'm just going to pull the PI over"}, {"start": 114.869, "duration": 4.11, "text": "2 out front though because that's a"}, {"start": 116.25, "duration": 5.64, "text": "constant and we are certainly allowed to"}, {"start": 118.979, "duration": 4.861, "text": "do that well if you think about the"}, {"start": 121.89, "duration": 6.239, "text": "series N equals 1 to infinity again this"}, {"start": 123.84, "duration": 6.449, "text": "is our familiar you know p series and"}, {"start": 128.129, "duration": 2.85, "text": "again since our p-value is greater than"}, {"start": 130.289, "duration": 3.03, "text": "1"}, {"start": 130.979, "duration": 8.191, "text": "our power is greater than one in this"}, {"start": 133.319, "duration": 8.371, "text": "case we have a convergent P series and"}, {"start": 139.17, "duration": 4.44, "text": "again since our series that we're trying"}, {"start": 141.69, "duration": 4.56, "text": "to decide whether it converges or"}, {"start": 143.61, "duration": 5.82, "text": "diverges since it's smaller than this"}, {"start": 146.25, "duration": 4.709, "text": "convergent series and again we always"}, {"start": 149.43, "duration": 6.3, "text": "know that this is greater than or equal"}, {"start": 150.959, "duration": 7.31, "text": "to zero by the direct comparison test we"}, {"start": 155.73, "duration": 8.16, "text": "can say that our series must also"}, {"start": 158.269, "duration": 7.421, "text": "converge so again nothing too crazy and"}, {"start": 163.89, "duration": 4.5, "text": "again you know if I just looked at this"}, {"start": 165.69, "duration": 4.769, "text": "problem and sort of had no clue you know"}, {"start": 168.39, "duration": 3.39, "text": "I tell people you know again maybe get"}, {"start": 170.459, "duration": 3.631, "text": "rid of the stuff you don't like maybe"}, {"start": 171.78, "duration": 4.31, "text": "you don't like the arctangent of n if"}, {"start": 174.09, "duration": 3.989, "text": "you're just left with 1 over N cubed"}, {"start": 176.09, "duration": 3.85, "text": "hopefully you remember you know you say"}, {"start": 178.079, "duration": 3.66, "text": "oh well that's a convergent P series"}, {"start": 179.94, "duration": 3.99, "text": "again at that point you know I'm"}, {"start": 181.739, "duration": 5.28, "text": "thinking maybe this is also a convergent"}, {"start": 183.93, "duration": 5.82, "text": "series and somehow I can use you know"}, {"start": 187.019, "duration": 5.731, "text": "some sort of comparison test with that"}, {"start": 189.75, "duration": 4.62, "text": "series 1 over N cubed and you know"}, {"start": 192.75, "duration": 3.09, "text": "that's exactly what we did we kind of"}, {"start": 194.37, "duration": 3.839, "text": "had to justify you know the correct"}, {"start": 195.84, "duration": 3.989, "text": "inequalities a little bit but to do that"}, {"start": 198.209, "duration": 4.141, "text": "we just had to really remember the graph"}, {"start": 199.829, "duration": 4.47, "text": "of arctangent and the fact that has an"}, {"start": 202.35, "duration": 4.02, "text": "asymptote and I think if you can"}, {"start": 204.299, "duration": 5.75, "text": "remember this part you know the rest of"}, {"start": 206.37, "duration": 3.679, "text": "it isn't isn't too terribly bad"}], "title": "Direct Comparison Test - Another Example 5", "description": "Thanks to all of you who support me on Patreon. You da real mvps!   $1 per month helps!! :)  https://www.patreon.com/patrickjmt !! Direct Comparison Test - Another Example 5. In this video I show that another series converges or diverges using the direct comparison theorem. I do not run over the formula/theory in this video (but do in another video, so look around if that is what you need!).", "lengthSeconds": 211, "width": 480, "height": 360}
{"video_id": "nkWpUbXu4hU", "transcript": [{"start": 0.03, "duration": 5.4, "text": "okay here I'm going to graph the"}, {"start": 2.19, "duration": 5.85, "text": "function cotangent of 2x and also we'll"}, {"start": 5.43, "duration": 8.15, "text": "pick out any vertical asymptotes and"}, {"start": 8.04, "duration": 5.54, "text": "zeros so this still has the same basic"}, {"start": 13.79, "duration": 6.22, "text": "basic resemblance to cotangent of X the"}, {"start": 18.84, "duration": 3.18, "text": "only thing that's going to change here"}, {"start": 20.01, "duration": 6.089, "text": "is going to be the period so in this"}, {"start": 22.02, "duration": 6.829, "text": "case the period will take PI and divide"}, {"start": 26.099, "duration": 9.001, "text": "it by the absolute value of B which is 2"}, {"start": 28.849, "duration": 9.611, "text": "it'll give us PI over 2 cotangent has a"}, {"start": 35.1, "duration": 5.31, "text": "vertical asymptote at x equals 0 so to"}, {"start": 38.46, "duration": 7.009, "text": "get to the next vertical asymptote we'll"}, {"start": 40.41, "duration": 5.059, "text": "go over here to x equals PI over 2 and"}, {"start": 45.86, "duration": 7.449, "text": "likewise if we want to do maybe a couple"}, {"start": 48.8, "duration": 7.029, "text": "two periods we can put an asymptote"}, {"start": 53.309, "duration": 8.401, "text": "there as well at x equals negative PI"}, {"start": 55.829, "duration": 9.001, "text": "over 2 and now cotangent if we take the"}, {"start": 61.71, "duration": 5.909, "text": "point in between the asymptotes so this"}, {"start": 64.83, "duration": 6.329, "text": "would be PI over 4 it still has that"}, {"start": 67.619, "duration": 6.031, "text": "same basic shape as cotangent so it'll"}, {"start": 71.159, "duration": 4.921, "text": "increase as we get close to the y axis"}, {"start": 73.65, "duration": 5.55, "text": "and then it will start decreasing down"}, {"start": 76.08, "duration": 5.579, "text": "to negative infinity as we approach PI"}, {"start": 79.2, "duration": 6.419, "text": "over 2 likewise this is the point"}, {"start": 81.659, "duration": 8.85, "text": "negative PI over 4 it's going to"}, {"start": 85.619, "duration": 6.631, "text": "increase and then decrease so I say"}, {"start": 90.509, "duration": 3.451, "text": "increasing as we move to the left the"}, {"start": 92.25, "duration": 4.86, "text": "graph is definitely decreasing"}, {"start": 93.96, "duration": 5.97, "text": "everywhere so just to hopefully clear up"}, {"start": 97.11, "duration": 4.17, "text": "any confusion there so that's all there"}, {"start": 99.93, "duration": 3.509, "text": "is to it it's just the graph of"}, {"start": 101.28, "duration": 8.46, "text": "cotangent the only thing that has"}, {"start": 103.439, "duration": 8.851, "text": "changed is the period so the zeros the"}, {"start": 109.74, "duration": 5.82, "text": "zeros the first positive zeros at PI"}, {"start": 112.29, "duration": 4.14, "text": "over 4 and then we said the period is PI"}, {"start": 115.56, "duration": 4.29, "text": "over 2"}, {"start": 116.43, "duration": 9.03, "text": "so we'll move over multiples of PI over"}, {"start": 119.85, "duration": 9.09, "text": "2 to get the zeros and the vertical"}, {"start": 125.46, "duration": 6.359, "text": "asymptotes those occur at multiples of"}, {"start": 128.94, "duration": 4.3, "text": "PI over 2 so we can write that as PI"}, {"start": 131.819, "duration": 5.461, "text": "over 2 times n"}, {"start": 133.24, "duration": 4.04, "text": "again wherein is an integer"}], "title": "Graphing a Cotangent Function, EX 3", "description": "Thanks to all of you who support me on Patreon. You da real mvps!   $1 per month helps!! :)  https://www.patreon.com/patrickjmt !! Graphing a Cotangent Function, EX 3", "lengthSeconds": 139, "width": 450, "height": 360}
{"video_id": "avfZ9P7ILGo", "transcript": [{"start": 0.56, "duration": 2.76, "text": "all right in this video we're going to"}, {"start": 1.719, "duration": 5.04, "text": "look at some examples of adding and"}, {"start": 3.32, "duration": 5.399, "text": "subtracting fractions and to add or"}, {"start": 6.759, "duration": 4.401, "text": "subtract fractions we have to have"}, {"start": 8.719, "duration": 4.561, "text": "common denominators we need to have the"}, {"start": 11.16, "duration": 5.4, "text": "same number in the bottom part of the"}, {"start": 13.28, "duration": 5.04, "text": "fraction and if that happens well then"}, {"start": 16.56, "duration": 4.32, "text": "we just do the the arithmetic in the"}, {"start": 18.32, "duration": 4.719, "text": "numerator so a over C plus b over C we"}, {"start": 20.88, "duration": 4.76, "text": "just do the arithmetic a plus b all over"}, {"start": 23.039, "duration": 5.361, "text": "C and likewise if it's subtraction we"}, {"start": 25.64, "duration": 5.119, "text": "just do a minus B all over C if you"}, {"start": 28.4, "duration": 4.76, "text": "don't have a common denominator"}, {"start": 30.759, "duration": 5.12, "text": "ators you'll have to make common"}, {"start": 33.16, "duration": 4.239, "text": "denominators appear but in these first"}, {"start": 35.879, "duration": 4.281, "text": "examples here I think everything should"}, {"start": 37.399, "duration": 6.68, "text": "work out without too much trouble so we"}, {"start": 40.16, "duration": 6.559, "text": "have 4 over 11us 3 over 11 again we have"}, {"start": 44.079, "duration": 4.201, "text": "the same number in the denominator uh so"}, {"start": 46.719, "duration": 4.601, "text": "we just do the arithmetic in the"}, {"start": 48.28, "duration": 5.32, "text": "numerator so 4 Min - 3 will leave us"}, {"start": 51.32, "duration": 4.759, "text": "with 1 over"}, {"start": 53.6, "duration": 4.56, "text": "11 the same thing happens if you have"}, {"start": 56.079, "duration": 4.521, "text": "multiple fractions again as long as you"}, {"start": 58.16, "duration": 7.84, "text": "have the same denominator so in this"}, {"start": 60.6, "duration": 9.92, "text": "case we would get 7 + 4 + 3 again all"}, {"start": 66.0, "duration": 9.08, "text": "being divided by 10 so let's see 7 + 4"}, {"start": 70.52, "duration": 6.239, "text": "is 11 + 3 is going to be 14 over 10 and"}, {"start": 75.08, "duration": 3.679, "text": "I can reduce that so I'm going to"}, {"start": 76.759, "duration": 5.081, "text": "they're both even so if we divide top"}, {"start": 78.759, "duration": 6.201, "text": "and bottom by two that'll leave us with"}, {"start": 81.84, "duration": 4.599, "text": "7 over five and I'm going to leave it uh"}, {"start": 84.96, "duration": 2.72, "text": "as a fraction I'm not going to write it"}, {"start": 86.439, "duration": 5.841, "text": "as a mixed"}, {"start": 87.68, "duration": 6.759, "text": "number here we have 8 and 34 - 3 and 1/4"}, {"start": 92.28, "duration": 3.76, "text": "different ways you can do this um you"}, {"start": 94.439, "duration": 5.161, "text": "could just subtract the whole numbers"}, {"start": 96.04, "duration": 5.399, "text": "and subtract the fractions um in general"}, {"start": 99.6, "duration": 5.119, "text": "uh I tend to you know and this is"}, {"start": 101.439, "duration": 5.28, "text": "probably a bit um you know too much for"}, {"start": 104.719, "duration": 6.841, "text": "this problem again we could just do 8 -"}, {"start": 106.719, "duration": 8.521, "text": "3 which would be 5 34s - 1/4 well that"}, {"start": 111.56, "duration": 3.68, "text": "would be 3 - 1 or"}, {"start": 115.36, "duration": 4.2, "text": "24s so that would be one way to do it I"}, {"start": 117.92, "duration": 5.32, "text": "have a tendency to always just turn them"}, {"start": 119.56, "duration": 10.039, "text": "back and into fractions so 4 * 8 is 32 +"}, {"start": 123.24, "duration": 9.6, "text": "3 is 35 over 4 uh 4 * 3 is 12 + 1 is"}, {"start": 129.599, "duration": 5.681, "text": "13 and now we can do the subtraction uh"}, {"start": 132.84, "duration": 4.56, "text": "35 minus 13 that's going to leave us"}, {"start": 135.28, "duration": 5.16, "text": "with 22 over"}, {"start": 137.4, "duration": 6.08, "text": "4 which again reduces they're both even"}, {"start": 140.44, "duration": 6.12, "text": "into 11 over two which if we made it"}, {"start": 143.48, "duration": 7.32, "text": "into a mixed number that is five and2"}, {"start": 146.56, "duration": 6.679, "text": "so um in general um I do this method"}, {"start": 150.8, "duration": 3.84, "text": "just because uh you know some weird"}, {"start": 153.239, "duration": 4.0, "text": "things can happen with the fractions"}, {"start": 154.64, "duration": 4.4, "text": "I'll put it that way um you know you may"}, {"start": 157.239, "duration": 3.801, "text": "end up with a fraction over one and then"}, {"start": 159.04, "duration": 5.6, "text": "you have to carry a digit to the whole"}, {"start": 161.04, "duration": 5.559, "text": "number um things of this nature so in"}, {"start": 164.64, "duration": 4.16, "text": "general um you're going to see me do"}, {"start": 166.599, "duration": 3.36, "text": "this method although in some cases you"}, {"start": 168.8, "duration": 3.48, "text": "can"}, {"start": 169.959, "duration": 3.961, "text": "just well really in all cases you can"}, {"start": 172.28, "duration": 3.4, "text": "just do this subtraction or addition"}, {"start": 173.92, "duration": 3.2, "text": "normally but you have to be a little"}, {"start": 175.68, "duration": 4.8, "text": "careful I guess what I'm trying to say"}, {"start": 177.12, "duration": 6.479, "text": "is maybe if you had eight and 34s"}, {"start": 180.48, "duration": 5.64, "text": "plus you know five and another"}, {"start": 183.599, "duration": 4.321, "text": "34s in this case you know if you add the"}, {"start": 186.12, "duration": 4.759, "text": "whole numbers eight and five is going to"}, {"start": 187.92, "duration": 7.679, "text": "give you 13 and then you would do 34s"}, {"start": 190.879, "duration": 7.08, "text": "and 34s which would give you um 64s so"}, {"start": 195.599, "duration": 5.481, "text": "that would be you know so you would have"}, {"start": 197.959, "duration": 4.961, "text": "13 and really You' have another 64s"}, {"start": 201.08, "duration": 5.12, "text": "which you'd have to turn into one and a"}, {"start": 202.92, "duration": 7.44, "text": "half Etc okay so maybe more than you"}, {"start": 206.2, "duration": 5.599, "text": "wanted to hear but I'm going to uh to uh"}, {"start": 210.36, "duration": 4.159, "text": "kind of deal with these situations"}, {"start": 211.799, "duration": 4.681, "text": "you'll see me just write things um as"}, {"start": 214.519, "duration": 4.0, "text": "fractions from the beginning it may take"}, {"start": 216.48, "duration": 5.2, "text": "an extra step or two but not the end of"}, {"start": 218.519, "duration": 3.161, "text": "the world"}], "title": "Adding and Subtracting Fractions / Mixed Numbers: Example 1", "description": "Thanks to all of you who support me on Patreon. You da real mvps!   $1 per month helps!! :)  https://www.patreon.com/patrickjmt !! Adding and Subtracting Fractions / Mixed Numbers: Example 1", "lengthSeconds": 220, "width": 480, "height": 360}
{"video_id": "rF36a8K_3QM", "transcript": [{"start": 0.11, "duration": 4.419, "text": "okay so another example here of using a"}, {"start": 2.58, "duration": 4.529, "text": "double angle identity to solve a trig"}, {"start": 4.529, "duration": 6.15, "text": "equation so here we're gonna solve sine"}, {"start": 7.109, "duration": 5.341, "text": "of 2t equals sine of T and again we're"}, {"start": 10.679, "duration": 4.41, "text": "gonna find inner solutions on the"}, {"start": 12.45, "duration": 4.499, "text": "interval 0 to 2pi this one will actually"}, {"start": 15.089, "duration": 5.19, "text": "be I think a little less involved than"}, {"start": 16.949, "duration": 8.281, "text": "the first one sine of 2t we can simply"}, {"start": 20.279, "duration": 9.33, "text": "write that as 2 sine T times cosine T"}, {"start": 25.23, "duration": 6.48, "text": "equals sine T and again be careful you"}, {"start": 29.609, "duration": 4.11, "text": "don't want to divide both sides by sine"}, {"start": 31.71, "duration": 4.68, "text": "T you would end up throwing some"}, {"start": 33.719, "duration": 6.061, "text": "solutions out the window so what we're"}, {"start": 36.39, "duration": 7.38, "text": "gonna do is we're simply gonna subtract"}, {"start": 39.78, "duration": 7.529, "text": "sine T from both sides set that equal to"}, {"start": 43.77, "duration": 6.87, "text": "zero and now we can factor the sine of T"}, {"start": 47.309, "duration": 8.881, "text": "out that'll leave us with two cosine t"}, {"start": 50.64, "duration": 7.98, "text": "minus 1 equals 0 and now all we have to"}, {"start": 56.19, "duration": 4.889, "text": "do is simply just take you know each"}, {"start": 58.62, "duration": 4.819, "text": "factor to set it equal to zero so we'll"}, {"start": 61.079, "duration": 8.551, "text": "set sine of T equal to zero"}, {"start": 63.439, "duration": 9.61, "text": "we'll take two cosine t minus 1 and set"}, {"start": 69.63, "duration": 6.87, "text": "that equal to zero well we can add 1 and"}, {"start": 73.049, "duration": 5.581, "text": "divide by 2 so now we're just thinking"}, {"start": 76.5, "duration": 4.979, "text": "about solutions I'm thinking okay where"}, {"start": 78.63, "duration": 6.12, "text": "does sine of T equals zero on the"}, {"start": 81.479, "duration": 7.201, "text": "interval 0 to 2pi well that would happen"}, {"start": 84.75, "duration": 7.07, "text": "at the angle zero at the angle PI and at"}, {"start": 88.68, "duration": 3.14, "text": "the angle 2 pi"}, {"start": 92.86, "duration": 7.17, "text": "cosine of T equals 1/2 let's see cosine"}, {"start": 96.909, "duration": 6.181, "text": "of the angle let's see I believe PI over"}, {"start": 100.03, "duration": 6.629, "text": "3 cosine of PI over 3 that gives us 1/2"}, {"start": 103.09, "duration": 7.11, "text": "and down in Quadrant 4 will also find"}, {"start": 106.659, "duration": 6.481, "text": "the angle 5 PI over 3 as our other"}, {"start": 110.2, "duration": 6.419, "text": "solution so in this case we've got 5"}, {"start": 113.14, "duration": 9.35, "text": "solutions to our original equation sine"}, {"start": 116.619, "duration": 12.42, "text": "of 2t equals sine of T get 0 PI over 3"}, {"start": 122.49, "duration": 9.93, "text": "PI 5 PI over 3 and our last solution we"}, {"start": 129.039, "duration": 3.381, "text": "found was 2 pi"}], "title": "Using Double Angle Identities to Solve Equations, Example 2", "description": "Thanks to all of you who support me on Patreon. You da real mvps!   $1 per month helps!! :)  https://www.patreon.com/patrickjmt !! Using Double Angle Identities to Solve Equations, Example 2.\nIn this video, I use some double angle identities for sine and/or cosine to solve some equations. Great fun!!  For more free math videos, visit http://PatrickJMT.com", "lengthSeconds": 131, "width": 480, "height": 360}
{"video_id": "yVpWJguWulc", "transcript": [{"start": 0.03, "duration": 4.53, "text": "okay so one last example here using"}, {"start": 2.669, "duration": 4.59, "text": "these sum and difference identities so"}, {"start": 4.56, "duration": 6.659, "text": "suppose we want to find the exact value"}, {"start": 7.259, "duration": 6.811, "text": "of tangent of 75 degrees minus tangent"}, {"start": 11.219, "duration": 6.091, "text": "at 15 degrees divided by 1 plus tangent"}, {"start": 14.07, "duration": 3.9, "text": "of 75 degrees times tangent of 15"}, {"start": 17.31, "duration": 3.479, "text": "degrees"}, {"start": 17.97, "duration": 6.03, "text": "well okay certainly this looks like you"}, {"start": 20.789, "duration": 4.771, "text": "know our tangent identity here and again"}, {"start": 24.0, "duration": 4.08, "text": "it says if there's a negative sign in"}, {"start": 25.56, "duration": 4.91, "text": "the numerator it says we're going to use"}, {"start": 28.08, "duration": 5.88, "text": "the the difference identity"}, {"start": 30.47, "duration": 6.31, "text": "it says we'll take tangent of X minus y"}, {"start": 33.96, "duration": 10.05, "text": "and in this case our X is just going to"}, {"start": 36.78, "duration": 11.61, "text": "be the 75 degrees and the Y is just"}, {"start": 44.01, "duration": 10.77, "text": "gonna be our 15 degrees here so it says"}, {"start": 48.39, "duration": 8.73, "text": "simply we'll get tangent of the first"}, {"start": 54.78, "duration": 5.579, "text": "one 75 degrees again if there's a"}, {"start": 57.12, "duration": 4.919, "text": "negative in the numerator we we preserve"}, {"start": 60.359, "duration": 4.641, "text": "that sign in between so we're going to"}, {"start": 62.039, "duration": 5.641, "text": "keep it negative in between here so"}, {"start": 65.0, "duration": 5.799, "text": "really we're just evaluating tangent of"}, {"start": 67.68, "duration": 7.17, "text": "75 degrees minus 15 degrees which is"}, {"start": 70.799, "duration": 7.191, "text": "going to be tangent of 60 degrees but"}, {"start": 74.85, "duration": 8.52, "text": "hey that's just sine of 60 degrees"}, {"start": 77.99, "duration": 8.86, "text": "over cosine of 60 degrees sine of 60"}, {"start": 83.37, "duration": 7.05, "text": "degrees is gonna be root 3 over 2 cosine"}, {"start": 86.85, "duration": 5.7, "text": "of 60 degrees is gonna be 1/2 you can"}, {"start": 90.42, "duration": 4.71, "text": "always simplify this a little bit we can"}, {"start": 92.55, "duration": 5.52, "text": "leave the numerator alone we can flip"}, {"start": 95.13, "duration": 5.97, "text": "the denominator and multiply so in this"}, {"start": 98.07, "duration": 5.1, "text": "case the twos will simply cancel out and"}, {"start": 101.1, "duration": 5.33, "text": "we'll be left with our solution of"}, {"start": 103.17, "duration": 3.26, "text": "square root of 3"}], "title": "Sum and Difference Identities to Simplify an Expression, Example 3", "description": "Thanks to all of you who support me on Patreon. You da real mvps!   $1 per month helps!! :)  https://www.patreon.com/patrickjmt !! Sum and Difference Identities to Simplify an Expression, Example 3.  In this video, I simply show how to simplify some expressions by using sum and difference identities.  Nothing crazy, just showing how one must know identities to make things more manageable!  For more free math videos, visit http://PatrickJMT.com", "lengthSeconds": 106, "width": 480, "height": 360}
{"video_id": "MVW9EoEiLgc", "transcript": [{"start": 0.84, "duration": 3.84, "text": "all right in this video I'm going to do"}, {"start": 2.48, "duration": 4.839, "text": "another example of solving a quadratic"}, {"start": 4.68, "duration": 4.879, "text": "equation by completing the square so in"}, {"start": 7.319, "duration": 6.041, "text": "this case we're going to solve x^2 + 4x"}, {"start": 9.559, "duration": 6.28, "text": "- 20 = 12 and again there's probably you"}, {"start": 13.36, "duration": 4.56, "text": "know slightly easier ways to to solve"}, {"start": 15.839, "duration": 4.321, "text": "this quadratic other than completing the"}, {"start": 17.92, "duration": 6.24, "text": "square but again we're just trying to"}, {"start": 20.16, "duration": 7.32, "text": "illustrate the procedure here um so in"}, {"start": 24.16, "duration": 5.039, "text": "this case notice we have you know x's on"}, {"start": 27.48, "duration": 3.44, "text": "one side we've got a number and then we"}, {"start": 29.199, "duration": 4.2, "text": "also have a number floating around on"}, {"start": 30.92, "duration": 4.0, "text": "the right side as well a lot there's"}, {"start": 33.399, "duration": 3.441, "text": "sort of different ways that you can do"}, {"start": 34.92, "duration": 3.88, "text": "these they're all equivalent but one way"}, {"start": 36.84, "duration": 5.239, "text": "to go about doing this would be to get"}, {"start": 38.8, "duration": 5.88, "text": "all the numbers on the right side so the"}, {"start": 42.079, "duration": 4.201, "text": "first thing I'm going to do is add 20 to"}, {"start": 44.68, "duration": 3.76, "text": "both sides of this"}, {"start": 46.28, "duration": 3.279, "text": "equation so that's going to leave me"}, {"start": 48.44, "duration": 5.88, "text": "with"}, {"start": 49.559, "duration": 7.52, "text": "X2 + 4x ="}, {"start": 54.32, "duration": 6.32, "text": "32 and now kind of the the completing"}, {"start": 57.079, "duration": 5.201, "text": "the square step uh the idea is we're"}, {"start": 60.64, "duration": 3.919, "text": "going to make the left side into a"}, {"start": 62.28, "duration": 3.8, "text": "perfect square by doing this following"}, {"start": 64.559, "duration": 4.281, "text": "procedure and you'll you'll see that"}, {"start": 66.08, "duration": 5.44, "text": "happen so whatever numbers in front of"}, {"start": 68.84, "duration": 6.279, "text": "the X the coefficient on the X what we"}, {"start": 71.52, "duration": 6.8, "text": "do is we take 1/2 always 1/2 of that"}, {"start": 75.119, "duration": 5.64, "text": "number so 1/2 of four is simply going to"}, {"start": 78.32, "duration": 5.92, "text": "give us two and then what we do with"}, {"start": 80.759, "duration": 4.68, "text": "that number we take that number and we"}, {"start": 84.24, "duration": 4.68, "text": "Square"}, {"start": 85.439, "duration": 7.32, "text": "it so 2^2 is going to equal"}, {"start": 88.92, "duration": 8.08, "text": "pos4 what you do is on the left side"}, {"start": 92.759, "duration": 7.961, "text": "we're going to write this as x^2 +"}, {"start": 97.0, "duration": 5.6, "text": "4x and there's our equal 32 equals 32"}, {"start": 100.72, "duration": 3.8, "text": "that was there from before what we're"}, {"start": 102.6, "duration": 3.799, "text": "going to do is whatever this whenever"}, {"start": 104.52, "duration": 3.639, "text": "you take 1/2 of the number and square it"}, {"start": 106.399, "duration": 5.241, "text": "whatever that new number is we're just"}, {"start": 108.159, "duration": 5.64, "text": "going to add that to the left side but"}, {"start": 111.64, "duration": 5.119, "text": "if we add it to the left side you also"}, {"start": 113.799, "duration": 7.481, "text": "have to add it to the right side to keep"}, {"start": 116.759, "duration": 8.521, "text": "everything uh equal and the I is now x^2"}, {"start": 121.28, "duration": 6.319, "text": "+ 4x + 4 factors it's a perfect square"}, {"start": 125.28, "duration": 5.48, "text": "it factors as x +"}, {"start": 127.599, "duration": 6.64, "text": "2 * x +"}, {"start": 130.76, "duration": 4.96, "text": "2 on the right side we have 32 + 4 which"}, {"start": 134.239, "duration": 6.401, "text": "is"}, {"start": 135.72, "duration": 7.48, "text": "36 and again if you take x + 2 um we can"}, {"start": 140.64, "duration": 7.879, "text": "X+ 2 * X+ 2 we can simply rewrite this"}, {"start": 143.2, "duration": 7.759, "text": "as x + 2 quantity squared equal 36 okay"}, {"start": 148.519, "duration": 3.841, "text": "so this is kind of a to me sort of phase"}, {"start": 150.959, "duration": 3.161, "text": "one this is the first thing you want to"}, {"start": 152.36, "duration": 4.08, "text": "get done is you want to make the left"}, {"start": 154.12, "duration": 3.8, "text": "side into a perfect square and yet"}, {"start": 156.44, "duration": 3.079, "text": "you'll have some number hanging around"}, {"start": 157.92, "duration": 4.36, "text": "on hanging around on the other side"}, {"start": 159.519, "duration": 4.36, "text": "that's fine so now the idea is to get"}, {"start": 162.28, "duration": 4.76, "text": "rid of the square we're going to take"}, {"start": 163.879, "duration": 4.561, "text": "the square root of both sides okay so I"}, {"start": 167.04, "duration": 4.479, "text": "should be a little careful here with my"}, {"start": 168.44, "duration": 6.799, "text": "notation because when you take um square"}, {"start": 171.519, "duration": 5.601, "text": "roots you get positive and negative the"}, {"start": 175.239, "duration": 4.08, "text": "square root of that number so I should"}, {"start": 177.12, "duration": 3.8, "text": "put a little plus minus you know in the"}, {"start": 179.319, "duration": 3.64, "text": "previous step since I already threw my"}, {"start": 180.92, "duration": 5.039, "text": "square root in"}, {"start": 182.959, "duration": 3.0, "text": "there"}, {"start": 186.36, "duration": 5.959, "text": "um what we're going to do is well we can"}, {"start": 190.2, "duration": 6.0, "text": "take the square root of 36 that's just"}, {"start": 192.319, "duration": 6.241, "text": "going to be 6 so we have um x + 2 ="}, {"start": 196.2, "duration": 4.8, "text": "positive or -6 is what we're trying to"}, {"start": 198.56, "duration": 5.599, "text": "say here we basically break this off"}, {"start": 201.0, "duration": 7.959, "text": "into two little equations now we're"}, {"start": 204.159, "duration": 5.761, "text": "going to make the equation x + 2 = POS 6"}, {"start": 208.959, "duration": 5.521, "text": "and then we're going to make the"}, {"start": 209.92, "duration": 6.599, "text": "equation x + 2 = -6 and we'll solve"}, {"start": 214.48, "duration": 4.119, "text": "those individual equations and that will"}, {"start": 216.519, "duration": 4.681, "text": "give us our solution to the original"}, {"start": 218.599, "duration": 4.321, "text": "quadratic equation so in the first one"}, {"start": 221.2, "duration": 4.72, "text": "we can simply subtract two from both"}, {"start": 222.92, "duration": 4.72, "text": "sides we would get X = pos4 we could"}, {"start": 225.92, "duration": 5.64, "text": "subtract two from both sides and we"}, {"start": 227.64, "duration": 8.439, "text": "would get x = 8 and now we've got our"}, {"start": 231.56, "duration": 4.519, "text": "two solutions to our quadratic equation"}], "title": "Completing the Square to Solve Quadratic Equations: More Examples - 1", "description": "Thanks to all of you who support me on Patreon. You da real mvps!   $1 per month helps!! :)  https://www.patreon.com/patrickjmt !! Completing the Square to Solve Quadratic Equations: More Examples - 1", "lengthSeconds": 234, "width": 480, "height": 360}
{"video_id": "FG1itU-D3oo", "transcript": [{"start": 0.719, "duration": 5.04, "text": "all right so just one more example here"}, {"start": 2.48, "duration": 5.16, "text": "finding uh components of of vectors"}, {"start": 5.759, "duration": 4.121, "text": "along other vectors and projections of"}, {"start": 7.64, "duration": 5.2, "text": "vectors onto other"}, {"start": 9.88, "duration": 4.08, "text": "vectors so we did part A here before so"}, {"start": 12.84, "duration": 3.56, "text": "we're going to do the same thing we'll"}, {"start": 13.96, "duration": 5.6, "text": "find the component and the projection of"}, {"start": 16.4, "duration": 6.639, "text": "b along a so in this one uh Vector a is"}, {"start": 19.56, "duration": 6.039, "text": "given by 1 0 0 Vector B is -6 comma 2"}, {"start": 23.039, "duration": 7.281, "text": "comma 1 so the first thing we'll do is"}, {"start": 25.599, "duration": 4.721, "text": "just find the the component"}, {"start": 31.64, "duration": 4.04, "text": "so again to find the component of b"}, {"start": 34.32, "duration": 5.16, "text": "along"}, {"start": 35.68, "duration": 5.44, "text": "a we take a. B and we divide that by the"}, {"start": 39.48, "duration": 4.64, "text": "magnitude of vector"}, {"start": 41.12, "duration": 5.439, "text": "a so the dot product is going to be"}, {"start": 44.12, "duration": 3.599, "text": "pretty easy we'll get 1 time ne6 which"}, {"start": 46.559, "duration": 4.121, "text": "will be"}, {"start": 47.719, "duration": 6.721, "text": "-6 um and then we'll just simply"}, {"start": 50.68, "duration": 8.24, "text": "multiply so we'll have 0 * uh 2 which is"}, {"start": 54.44, "duration": 8.759, "text": "0 we have 0 * 1 which is"}, {"start": 58.92, "duration": 8.72, "text": "0 if we take the magnitude of vector um"}, {"start": 63.199, "duration": 6.96, "text": "of vector a we'll do 1 2 plus well 0 2 +"}, {"start": 67.64, "duration": 6.04, "text": "0^ 2quared so that's simply going to"}, {"start": 70.159, "duration": 6.6, "text": "leave us with -6 over"}, {"start": 73.68, "duration": 6.16, "text": "1 or a value"}, {"start": 76.759, "duration": 6.641, "text": "of6 to figure out the"}, {"start": 79.84, "duration": 7.4, "text": "projection of vector b along a again we"}, {"start": 83.4, "duration": 6.48, "text": "just do a b we can divide that by the"}, {"start": 87.24, "duration": 5.76, "text": "magnitude of a s and then we multip"}, {"start": 89.88, "duration": 4.879, "text": "multiply that by Vector a so again we've"}, {"start": 93.0, "duration": 3.32, "text": "already figured out the dot product we"}, {"start": 94.759, "duration": 4.561, "text": "figured that was"}, {"start": 96.32, "duration": 5.6, "text": "-6 um if we take the original magnitude"}, {"start": 99.32, "duration": 5.56, "text": "which was one and we Square it well we"}, {"start": 101.92, "duration": 5.0, "text": "still get one and then we just multiply"}, {"start": 104.88, "duration": 6.72, "text": "that by Vector a which again just had"}, {"start": 106.92, "duration": 9.44, "text": "components of 1 0 0 so if we multiply"}, {"start": 111.6, "duration": 8.68, "text": "we'll be left with -6 comma 0 comma 0 as"}, {"start": 116.36, "duration": 3.92, "text": "our Vector projection"}], "title": "Orthogonal Projections - Scalar and Vector Projections - Example 2", "description": "Thanks to all of you who support me on Patreon. You da real mvps!   $1 per month helps!! :)  https://www.patreon.com/patrickjmt !! Orthogonal Projections - Scalar and Vector Projections. In this video, we look at the idea of a scalar and vector projection of one vector onto another.", "lengthSeconds": 119, "width": 480, "height": 360}
{"video_id": "L9xrmXtmMcA", "transcript": [{"start": 0.399, "duration": 2.961, "text": "all right in this video we're going to"}, {"start": 1.439, "duration": 4.201, "text": "find a power series representation by"}, {"start": 3.36, "duration": 4.28, "text": "doing a littleit of integration so we're"}, {"start": 5.64, "duration": 3.6, "text": "given the power series for cosine X is"}, {"start": 7.64, "duration": 6.159, "text": "1-"}, {"start": 9.24, "duration": 7.48, "text": "x^2 um plus x 4 over 4 factorial - x 6"}, {"start": 13.799, "duration": 4.601, "text": "over 6 factorial Etc we want to find"}, {"start": 16.72, "duration": 6.76, "text": "what function is represented by the"}, {"start": 18.4, "duration": 10.56, "text": "series x^2 - x 4 over 3 factorial + x 6"}, {"start": 23.48, "duration": 8.0, "text": "over 5 factorial - x 8 over 7 factorial"}, {"start": 28.96, "duration": 7.56, "text": "Etc so what I'm going to do is integrate"}, {"start": 31.48, "duration": 5.04, "text": "both sides of our of our function"}, {"start": 44.16, "duration": 4.039, "text": "here okay so on the left side the"}, {"start": 46.36, "duration": 2.92, "text": "anti-derivative of cosine that's just"}, {"start": 48.199, "duration": 3.761, "text": "going to be"}, {"start": 49.28, "duration": 5.599, "text": "sinx on the right side um the"}, {"start": 51.96, "duration": 5.599, "text": "anti-derivative of one is just X then"}, {"start": 54.879, "duration": 6.961, "text": "we'll get x to the 3 notice we'll get 3"}, {"start": 57.559, "duration": 7.64, "text": "* 2 factorial um so I guess let's even"}, {"start": 61.84, "duration": 8.919, "text": "write that and then we'll get x 5 over 5"}, {"start": 65.199, "duration": 7.56, "text": "* 4 factorial minus X 7 over 7 * 6"}, {"start": 70.759, "duration": 5.4, "text": "factorial"}, {"start": 72.759, "duration": 5.161, "text": "Etc now when we integrate um again we"}, {"start": 76.159, "duration": 4.32, "text": "pick up this arbitrary constant I'm"}, {"start": 77.92, "duration": 5.36, "text": "going to call it C so to figure out the"}, {"start": 80.479, "duration": 5.28, "text": "C value uh we can plug in any value that"}, {"start": 83.28, "duration": 4.28, "text": "we want for x uh but it needs to be"}, {"start": 85.759, "duration": 4.601, "text": "something that's easy to compute so"}, {"start": 87.56, "duration": 7.4, "text": "notice if we plug in x equals z we'll"}, {"start": 90.36, "duration": 7.719, "text": "get S of 0 equal C + well we're plugging"}, {"start": 94.96, "duration": 4.88, "text": "in zeros for all the X's so those would"}, {"start": 98.079, "duration": 4.441, "text": "all just be"}, {"start": 99.84, "duration": 6.279, "text": "zeros but sign of zero we know that that"}, {"start": 102.52, "duration": 6.52, "text": "is zero so in this case our C value is"}, {"start": 106.119, "duration": 6.521, "text": "just going to be zero so what we have is"}, {"start": 109.04, "duration": 6.52, "text": "we have sin x is going to be x - x 3r"}, {"start": 112.64, "duration": 4.759, "text": "over well 3 * 2 factorial that's the"}, {"start": 115.56, "duration": 4.68, "text": "same thing as 3"}, {"start": 117.399, "duration": 8.561, "text": "factorial um 5 * 4 factorial faial"}, {"start": 120.24, "duration": 8.159, "text": "that's 5 factorial X 7th 7 factorial"}, {"start": 125.96, "duration": 4.48, "text": "Etc okay so let's let's what we're"}, {"start": 128.399, "duration": 5.681, "text": "actually trying to find a series"}, {"start": 130.44, "duration": 5.879, "text": "for notice we've got um you know X squ"}, {"start": 134.08, "duration": 5.04, "text": "notice the exponents are sort of one"}, {"start": 136.319, "duration": 6.121, "text": "larger um than the uh you know the"}, {"start": 139.12, "duration": 5.0, "text": "factorials in the bottom so the"}, {"start": 142.44, "duration": 4.12, "text": "exponents are a little bit larger than"}, {"start": 144.12, "duration": 3.839, "text": "the factorials okay so one thing we"}, {"start": 146.56, "duration": 4.039, "text": "could try to do is I guess we could"}, {"start": 147.959, "duration": 6.64, "text": "integrate again um but then we're going"}, {"start": 150.599, "duration": 6.521, "text": "to have like x^2 over two and X 4th over"}, {"start": 154.599, "duration": 5.241, "text": "4 factorial and then the factorials are"}, {"start": 157.12, "duration": 4.56, "text": "kind of messed up so I think the easiest"}, {"start": 159.84, "duration": 4.92, "text": "way just to increase the"}, {"start": 161.68, "duration": 6.639, "text": "exponents uh by one we can just simply"}, {"start": 164.76, "duration": 3.559, "text": "multiply both sides by"}, {"start": 168.599, "duration": 7.0, "text": "X and I think if we do that that's going"}, {"start": 171.08, "duration": 4.519, "text": "to give us exactly what we"}, {"start": 175.64, "duration": 4.959, "text": "needed because when we multiply by X"}, {"start": 178.319, "duration": 9.84, "text": "we'll just have well X2"}, {"start": 180.599, "duration": 10.521, "text": "- x 4 3 factorial + x 6 5 factorial - x"}, {"start": 188.159, "duration": 5.841, "text": "to 8 over 7 factorial so we're just"}, {"start": 191.12, "duration": 4.92, "text": "increasing all the exponents by one and"}, {"start": 194.0, "duration": 4.72, "text": "again that's actually the uh Power"}, {"start": 196.04, "duration": 5.919, "text": "series that we wanted to produce so"}, {"start": 198.72, "duration": 5.32, "text": "we've now got it so this original uh"}, {"start": 201.959, "duration": 4.801, "text": "function uh that we were trying to"}, {"start": 204.04, "duration": 4.279, "text": "figure out what well when we were trying"}, {"start": 206.76, "duration": 4.199, "text": "to decide what function is represented"}, {"start": 208.319, "duration": 7.321, "text": "by this power series we've now got it"}, {"start": 210.959, "duration": 4.681, "text": "this is just going to equal x *"}, {"start": 215.68, "duration": 3.0, "text": "sinx"}], "title": "Power Series Representation Using Integration", "description": "Thanks to all of you who support me on Patreon. You da real mvps!   $1 per month helps!! :)  https://www.patreon.com/patrickjmt !! Power Series Representation Using Integration.\nIn this video, I find a power series representation of a new function and identify that function by integrating a known power series.", "lengthSeconds": 217, "width": 480, "height": 360}
{"video_id": "DcduHCJuiWk", "transcript": [{"start": 0.58, "duration": 2.0, "text": "\u062d\u0633\u0646\u0627\u064b , \u0647\u0646\u0627 \u0633\u0646\u0642\u0648\u0645 \u0628\u062d\u0644 \u0645\u062b\u0627\u0644\u064a\u0646 \u0633\u0631\u064a\u0639\u064a\u0646"}, {"start": 2.92, "duration": 2.64, "text": "\u0645\u062a\u0639\u0644\u0642\u0627\u0646 \u0628\u0625\u062e\u062a\u0628\u0627\u0631 \u0627\u0644\u062d\u062f \u0627\u0644\u0646\u0648\u0646\u064a \u0644\u0644\u0645\u062a\u0633\u0644\u0633\u0644\u0629 \u0627\u0644\u0645\u062a\u0628\u0627\u0639\u062f\u0629"}, {"start": 5.72, "duration": 5.44, "text": "\u0648\u0628\u0627\u0644\u0646\u0633\u0628\u0629  \u0644\u064a , \u0647\u0626\u0627 \u0627\u0644\u0625\u062e\u062a\u0628\u0627\u0631 \u0645\u0641\u064a\u062f \u062c\u062f\u0627\u064e  \u0644\u0645\u0639\u0631\u0641\u0629 \u0625\u0630\u0627 \u0645\u0627 \u0643\u0627\u0646\u062a \u0627\u0644\u0645\u062a\u0633\u0644\u0633\u0644\u0629  \u0645\u062a\u0642\u0627\u0631\u0628\u0629 \u0623\u0645 \u0645\u062a\u0628\u0627\u0639\u062f\u0629"}, {"start": 11.38, "duration": 2.0, "text": "\u0625\u0646\u0647\u0627 \u063a\u0627\u0644\u0628\u0627 \u0623\u0648\u0644 \u0645\u0627 \u0623\u0641\u0643\u0631 \u0628\u0629"}, {"start": 13.88, "duration": 2.0, "text": "\u063a\u0627\u0644\u0628\u0627\u064b \u0623\u0648\u0644 \u0645\u0627 \u0623\u0641\u0643\u0631 \u0628\u0629"}, {"start": 16.493, "duration": 3.32, "text": "\u0625\u0630\u0627 , \u0645\u0627 \u064a\u0642\u0648\u0644\u0647 \u0625\u062e\u062a\u0628\u0627\u0631 \u0627\u0644\u062d\u062f \u0627\u0644\u0646\u0648\u0646\u064a \u0644\u0644\u0633\u0644\u0633\u0629 \u0627\u0644\u0645\u062a\u0628\u0627\u0639\u062f\u0629 \u0647\u0648"}, {"start": 20.06, "duration": 2.32, "text": "\u0623\u0646 \u0644\u062f\u064a\u0643 \u0645\u062a\u0633\u0644\u0633\u0644\u0629\u0650 \u0645\u0627 \u0628\u0635\u064a\u063a\u0629\u064d \u0645\u0627"}, {"start": 22.38, "duration": 1.26, "text": "\u0623 \u0646 (\u0623\u0644\u0641 \u0646\u0648\u0646)"}, {"start": 24.08, "duration": 2.0, "text": "\u064a\u0642\u0648\u0644 \u0627\u0644\u0625\u062e\u062a\u0628\u0627\u0631 \u0625\u0630\u0627 \u0623\u062e\u0630\u062a \u0646\u0647\u0627\u064a\u0629 \u0639\u0646\u062f\u0645\u0627 \u062a\u0642\u062a\u0631\u0628 \u0627\u0644(\u0646\u0648\u0646) \u0644\u0645\u0627 \u0644\u0627 \u0646\u0647\u0627\u064a\u0629"}, {"start": 26.08, "duration": 1.36, "text": "\u0644 (\u0623 \u0646)"}, {"start": 27.72, "duration": 2.2, "text": "\u0625\u0630\u0627 \u0644\u0645 \u062a\u0633\u0627\u0648\u064a \u062a\u0644\u0643 \u0627\u0644\u0646\u0647\u0627\u064a\u0629 \u0635\u0641\u0631\u0627\u064b"}, {"start": 29.92, "duration": 2.2, "text": "\u0639\u0646\u062f\u0647\u0627 \u062a\u0643\u0648\u0646 \u0627\u0644\u0645\u062a\u0633\u0644\u0633\u0644\u0629 \u062a\u0644\u0642\u0627\u0626\u064a\u0627\u064b \u0645\u062a\u0628\u0627\u0639\u062f\u0629"}, {"start": 32.34, "duration": 2.18, "text": "\u0648\u0630\u0644\u0643 \u0647\u0648 \u0627\u0644\u0623\u0645\u0631 , \u0642\u062f \u0625\u0646\u062a\u0647\u064a\u062a  \u0645\u0646 \u0647\u0630\u0647 \u0627\u0644\u0645\u0634\u0643\u0644\u0629 \u0641\u062a\u0646\u062a\u0642\u0644 \u0644\u0644\u0645\u0634\u0643\u0644\u0629 \u0627\u0644\u062a\u0627\u0644\u064a\u0629"}, {"start": 35.14, "duration": 2.68, "text": "\u0623\u0645\u0627 \u0625\u0630\u0627 \u0643\u0627\u0646\u062a \u0627\u0644\u0646\u0647\u0627\u064a\u0629 \u062a\u0633\u0627\u0648\u064a \u0635\u0641\u0631\u0627\u064b"}, {"start": 37.82, "duration": 1.5, "text": "\u0633\u064a\u0643\u0648\u0646 \u0630\u0644\u0643 \u063a\u064a\u0631 \u0645\u0624\u062f\u064d \u0644\u0646\u062a\u064a\u062c\u0629"}, {"start": 39.64, "duration": 3.6, "text": "\u062d\u064a\u062b \u0623\u0646 \u0627\u0644\u0633\u0644\u0633\u0644\u0629 \u0642\u062f \u062a\u0643\u0648\u0646 \u0645\u062a\u0642\u0627\u0631\u0628\u0629 \u0623\u0648 \u0645\u062a\u0628\u0627\u0639\u062f\u0629 , \u0646\u062d\u0646 \u0644\u0627\u0646\u0639\u0644\u0645 \u0634\u064a\u0626\u0627\u064e"}, {"start": 43.68, "duration": 2.58, "text": "\u0625\u0630\u0627\u064b , \u0643\u0646 \u062d\u0630\u0631\u0627\u064e \u0644\u0623\u0646\u0629 \u062e\u0637\u0623 \u0634\u0627\u0626\u0639"}, {"start": 46.26, "duration": 1.28, "text": "\u0627\u0644\u0645\u062a\u0633\u0644\u0633\u0644\u0627\u062a \u0645\u0631\u0628\u0643\u0629"}, {"start": 47.7, "duration": 1.04, "text": "\u0644\u0643\u062b\u064a\u0631 \u0645\u0646 \u0627\u0644\u0646\u0627\u0633 \u0643\u0645\u0627 \u0623\u0639\u062a\u0642\u062f"}, {"start": 48.74, "duration": 1.08, "text": "\u0628\u0627\u0644\u062a\u0623\u0643\u064a\u062f \u0643\u0627\u0646\u062a \u0645\u0631\u0628\u0643\u0629 \u0644\u064a"}, {"start": 50.24, "duration": 1.36, "text": "\u0639\u0646\u062f\u0645\u0627 \u0628\u062f\u0623\u062a \u0628\u0627\u0644\u062a\u0639\u0627\u0645\u0644 \u0645\u0639\u0647\u0627 \u0644\u0623\u0648\u0644 \u0645\u0631\u0629"}, {"start": 51.6, "duration": 2.82, "text": "\u0648 \u0645\u0627\u0632\u0627\u0644\u062a \u0627\u0644\u0645\u062a\u0633\u0644\u0633\u0644\u0627\u062a \u0645\u0631\u0628\u0643\u0629 \u0643\u0645\u0627 \u0623\u0639\u062a\u0642\u062f \u0641\u064a \u0643\u062b\u064a\u0631 \u0645\u0646 \u0627\u0644\u0623\u0648\u0642\u0627\u062a"}, {"start": 54.62, "duration": 1.74, "text": "\u0643\u0645\u0627 \u062a\u0639\u0644\u0645\u0648\u0646 , \u0627\u0644\u0646\u0627\u0633 \u064a\u0642\u0639\u0648\u0646 \u0641\u064a \u0627\u0644\u0641\u062e"}, {"start": 56.36, "duration": 1.48, "text": "\u0648\u064a\u0642\u0648\u0644\u0648\u0646 \u0625\u0630\u0627 \u0644\u0645 \u062a\u0633\u0627\u0648\u064a \u0627\u0644\u0646\u0647\u0627\u064a\u0629 \u0635\u0641\u0631\u0627\u064b"}, {"start": 57.84, "duration": 1.82, "text": "\u0641\u062a\u0630\u0643\u0631 \u0625\u0646\u0647\u0627 \u0645\u062a\u0628\u0627\u0639\u062f\u0629"}, {"start": 59.66, "duration": 1.76, "text": "\u0648\u0628\u0639\u062f\u0647\u0627 \u0633\u064a\u0642\u0648\u0644\u0648\u0646 \u0628\u062a\u0641\u0643\u064a\u0631 \u062e\u0627\u0637\u0626"}, {"start": 61.52, "duration": 2.4, "text": "\u0625\u0630\u0627 \u0643\u0627\u0646\u062a \u0627\u0644\u0646\u0647\u0627\u064a\u0629 \u062a\u0633\u0627\u0648\u064a \u0635\u0641\u0631\u0627 . \u0641\u0623\u0639\u062a\u0642\u062f \u0623\u0646 \u0627\u0644\u0633\u0644\u0633\u0629 \u0645\u062a\u0642\u0627\u0631\u0628\u0629"}, {"start": 64.28, "duration": 3.74, "text": "\u0644\u0648 \u0643\u0627\u0646\u062a \u0627\u0644\u0623\u0645\u0648\u0631 \u0628\u062a\u0644\u0643 \u0627\u0644\u0628\u0633\u0627\u0637\u0629 \u0648\u0643\u0627\u0646 \u0630\u0644\u0643 \u0635\u062d\u064a\u062d\u0627\u064b"}, {"start": 68.34, "duration": 3.94, "text": "\u0644\u0645 \u0646\u0643\u0646 \u0644\u0646\u0632\u0639\u062c \u0623\u0646\u0641\u0633\u0646\u0627 \u0641\u064a \u0639\u0645\u0644 \u0643\u0644 \u0647\u0630\u0629 \u0627\u0644\u0625\u062e\u062a\u0628\u0627\u0631\u0627\u062a \u0648\u0627\u0644\u0645\u0637\u0627\u0644\u064a\u0628 \u0627\u0644\u0645\u062e\u062a\u0644\u0641\u0629 \u0627\u0644\u062a\u064a \u0646\u0642\u0648\u0645 \u0628\u0647\u0627"}, {"start": 73.08, "duration": 3.06, "text": "\u0625\u0630\u0627 \u0645\u062c\u062f\u062f\u0627\u064e , \u0627\u0644\u0625\u0641\u0627\u062f\u0629 \u0627\u0644\u0623\u0648\u0644\u0649  \u0647\u064a"}, {"start": 77.88, "duration": 1.94, "text": "\u0647\u064a \u0645\u0627 \u0633\u0646\u0633\u062a\u062e\u062f\u0645\u0629 \u0647\u0646\u0627"}, {"start": 79.98, "duration": 2.0, "text": "\u0625\u0630\u0627\u064b \u0647\u0646\u0627 \u0646\u0631\u064a\u062f \u0627\u0646 \u0646\u0643\u062a\u0634\u0641 \u0645\u0627 \u0625\u0630\u0627 \u0643\u0627\u0646\u062a \u0627\u0644\u0645\u062a\u0633\u0644\u0633\u0644\u0629 \u0627\u0644\u062a\u0627\u0644\u064a\u0629"}, {"start": 81.98, "duration": 1.5, "text": "\u0645\u062a\u0642\u0627\u0631\u0628\u0629 \u0623\u0645 \u0645\u062a\u0628\u0627\u0639\u062f\u0629"}, {"start": 85.62, "duration": 2.0, "text": "\u0625\u0630\u0627\u064e"}, {"start": 88.88, "duration": 1.52, "text": "\u0625\u0630\u0627\u064b  \u0644\u0644\u062c\u0632\u0621 (\u0623) \u0647\u0646\u0627"}, {"start": 90.58, "duration": 2.8, "text": "\u0644\u062f\u064a\u0646\u0627 \u0627\u0644\u0646\u0647\u0627\u064a\u0629.. \u0623\u0642\u0635\u062f \u0644\u062f\u064a\u0646\u0627  \u0627\u0644\u0645\u062a\u0633\u0644\u0633\u0629  \u0645\u0646 \u0646= 1 \u0625\u0644\u0649 \u0645\u0627\u0644\u0627 \u0646\u0647\u0627\u064a\u0629"}, {"start": 93.38, "duration": 4.0, "text": "\u0644 \u0644 \u0646^2+3\u0646+1  \u0639\u0644\u0649 2\u0646^2+4"}, {"start": 97.88, "duration": 2.44, "text": "\u062d\u0633\u0646\u0627,  \u0625\u0630\u0627 \u0623\u062e\u0630\u0646\u0627 \u0627\u0644\u0646\u0647\u0627\u064a\u0629 \u062d\u064a\u062b (\u0646) \u062a\u062a\u062c\u0629 \u0644\u0645\u0627\u0644\u0627\u0646\u0647\u0627\u064a\u0629"}, {"start": 100.32, "duration": 6.88, "text": "\u0644 \u0646^2+3\u0646+1  \u0639\u0644\u0649 2\u0646^2+4"}, {"start": 107.88, "duration": 1.6, "text": "\u0644\u0642\u062f \u0631\u0623\u064a\u0646\u0627 \u0647\u0630\u0647 \u0627\u0644\u0623\u0646\u0648\u0627\u0639 \u0645\u0646 \u0627\u0644\u0646\u0647\u0627\u064a\u0627\u062a"}, {"start": 109.52, "duration": 2.98, "text": "\u0625\u0630\u0627 \u0643\u0627\u0646\u062a \u062f\u0631\u062c\u0629 \u0627\u0644\u0628\u0633\u0637 \u0645\u0633\u0627\u0648\u064a\u0629 \u0644\u062f\u0631\u062c\u0629 \u0627\u0644\u0645\u0642\u0627\u0645"}, {"start": 112.5, "duration": 1.48, "text": "\u062d\u064a\u062b \u064a\u062a\u0633\u0627\u0648\u064a\u0627\u0646 \u0641\u064a \u0647\u0630\u0647 \u0627\u0644\u062d\u0627\u0644\u0629"}, {"start": 115.32, "duration": 2.0, "text": "\u0646\u0633\u062a\u0637\u064a\u0639 \u0623\u0646 \u0646\u0633\u062a\u062e\u062f\u0645 \u0627\u0644\u0646\u0633\u0628\u0629 \u0644\u0644\u0645\u0639\u0627\u0645\u0650\u0644\u0627\u062a \u0627\u0644\u0631\u0626\u064a\u0633\u064a\u0629"}, {"start": 118.16, "duration": 2.0, "text": "\u0625\u0630\u0627 \u0641\u064a \u0647\u0630\u0647 \u0627\u0644\u062d\u0627\u0644\u0629 ,  \u0627\u0644\u0646\u0647\u0627\u064a\u0629 \u062a\u0633\u0627\u0648\u064a \u0646\u0635\u0641"}, {"start": 120.7, "duration": 2.0, "text": "\u0648\u0627\u0644\u0630\u064a \u0642\u0637\u0639\u0627 \u0644\u0627 \u064a\u0633\u0627\u0648\u064a \u0635\u0641\u0631\u0627\u064b"}, {"start": 123.0, "duration": 5.34, "text": "\u0648\u0647\u0630\u0627 \u064a\u062e\u0628\u0631\u0646\u0627 \u0628\u0623\u0646 \u0627\u0644\u0645\u062a\u0633\u0644\u0633\u0644\u0629 \u0645\u062a\u0628\u0627\u0639\u062f\u0629"}, {"start": 128.34, "duration": 1.96, "text": "\u062d\u0633\u0646\u0627\u064b , \u0633\u0647\u0644 \u0628\u0645\u0627 \u0641\u064a\u0629 \u0627\u0644\u0643\u0641\u0627\u064a\u0629"}, {"start": 130.7, "duration": 2.66, "text": "\u0625\u0630\u0627 \u0641\u064a \u0627\u0644\u062c\u0632\u0621 (\u0623) \u0646\u0642\u0648\u0644 \u0628\u0628\u0633\u0627\u0637\u0629 \u0623\u0646 \u0627\u0644\u0645\u062a\u0633\u0644\u0633\u0644\u0629 \u0645\u062a\u0628\u0627\u0639\u062f\u0629"}, {"start": 134.1, "duration": 2.24, "text": "\u0641\u064a \u0627\u0644\u062c\u0632\u0621 (\u0628)  \u0625\u0630\u0627 \u0644\u0627\u062d\u0638\u0646\u0627"}, {"start": 136.52, "duration": 2.0, "text": "\u0643\u0645\u0627 \u062a\u0639\u0644\u0645\u0648\u0646 ,  \u0644\u062f\u064a\u0646\u0627 \u0627\u0644\u0645\u062a\u0633\u0644\u0633\u0644\u0629 \u0645\u0646 \u0646=1 \u0625\u0644\u0649 \u0645\u0627\u0644\u0627\u0646\u0647\u0627\u064a\u0629"}, {"start": 138.52, "duration": 1.3, "text": "\u0644 \u0627\u0644\u0644\u0648\u063a\u0627\u0631\u064a\u062a\u0645 \u0627\u0644\u0637\u0628\u064a\u0639\u064a"}, {"start": 139.82, "duration": 1.08, "text": "\u0644\u0650 \u062c\u062a\u0627 \u0646 (\u062c\u064a\u0628 \u0627\u0644\u062a\u0645\u0627\u0645 \u0644 \u0646)"}, {"start": 141.6, "duration": 2.76, "text": "\u062d\u0633\u0646\u0627 , \u0646\u0623\u062e\u0630 \u0627\u0644\u0646\u0647\u0627\u064a\u0629 \u062d\u064a\u062b \u062a\u062a\u062c\u0629 \u0627\u0644\u0646\u0648\u0646 \u0644\u0645\u0627\u0644\u0627\u0646\u0647\u0627\u064a\u0629"}, {"start": 144.36, "duration": 3.36, "text": "\u0644 \u0627\u0644\u0648\u063a\u0627\u0631\u064a\u062a\u0645 \u0627\u0644\u0637\u0628\u064a\u0639\u064a \u0644 \u062c\u062a\u0627 \u0646"}, {"start": 148.06, "duration": 1.78, "text": "\u0648 \u0642\u0627\u0646\u0648\u0646 \u0627\u0644\u0646\u0647\u0627\u064a\u0627\u062a \u064a\u0642\u0648\u0644 \u0628\u0623\u0646\u0629"}, {"start": 149.96, "duration": 3.28, "text": "\u064a\u0645\u0643\u0646\u0646\u0627  \u0628\u0628\u0633\u0627\u0637\u0629 \u0623\u0646 \u0646\u0636\u0639 \u0627\u0644\u0646\u0647\u0627\u064a\u0629 \u0641\u064a \u0627\u0644\u062f\u0627\u062e\u0644"}, {"start": 153.44, "duration": 3.96, "text": "\u062d\u064a\u062b \u064a\u0643\u0648\u0646 \u0644\u062f\u064a\u0646\u0627 \u0627\u0644\u0644\u0648\u063a\u0627\u0631\u064a\u062a\u0645 \u0627\u0644\u0637\u0628\u064a\u0639\u064a \u0644\u0644\u0646\u0647\u0627\u064a\u0629 \u062d\u064a\u062b (\u0646) \u062a\u062a\u062c\u0629 \u0644\u0645\u0627\u0644\u0627\u0646\u0647\u0627\u064a\u0629"}, {"start": 157.4, "duration": 0.96, "text": "\u0644 \u062c\u062a\u0627 \u0646"}, {"start": 158.72, "duration": 2.0, "text": "\u0644\u0643\u0646 \u0631\u064a\u062b\u0645\u0627 \u062a\u062a\u062c\u0629 \u0627\u0644\u0646\u0648\u0646 \u0644\u0645\u0627\u0644\u0627\u0646\u0647\u0627\u064a\u0629"}, {"start": 160.72, "duration": 1.98, "text": "\u062c\u062a\u0627 \u062a\u0628\u0642\u0649 \u0645\u062a\u0630\u0628\u0630\u0628\u0629"}, {"start": 162.96, "duration": 2.84, "text": "\u0625\u0630\u0627\u064b \u0647\u0630\u0647 \u0627\u0644\u0646\u0647\u0627\u064a\u0629 \u0628\u0628\u0633\u0627\u0637\u0629 \u063a\u064a\u0631 \u0645\u0648\u062c\u0648\u062f\u0629"}, {"start": 166.42, "duration": 2.0, "text": "\u0648\u0625\u0630\u0627 \u0644\u0645 \u062a\u0643\u0646 \u0645\u0648\u062c\u0648\u062f\u0629"}, {"start": 168.66, "duration": 2.58, "text": "\u0647\u064a \u0642\u0637\u0639\u0627\u064b \u0644\u0627 \u062a\u0633\u0627\u0648\u064a \u0635\u0641\u0631\u0627\u064b"}, {"start": 171.3, "duration": 1.36, "text": "\u0625\u0630\u0627 \u0641\u064a \u062a\u0644\u0643 \u0627\u0644\u062d\u0627\u0644\u0629 \u0645\u062c\u062f\u062f\u0627\u064b"}, {"start": 172.96, "duration": 5.8, "text": "\u064a\u0645\u0643\u0646\u0646\u0627 \u0627\u0644\u0642\u0648\u0644 \u0628\u0623\u0646 \u0627\u0644\u0633\u0644\u0633\u0644\u0629 \u0645\u062a\u0628\u0627\u0639\u062f\u0629"}, {"start": 179.28, "duration": 2.84, "text": "\u0637\u0628\u0639\u0627\u064e , \u0643\u0645\u0627 \u062a\u0639\u0644\u0645\u0648\u0646"}, {"start": 182.3, "duration": 2.54, "text": "\u0625\u0630\u0627 \u0643\u0627\u0646\u062a \u0627\u0644\u0646\u0647\u0627\u064a\u0629 \u062a\u0633\u0627\u0648\u064a \u0635\u0641\u0631\u0627\u064b"}, {"start": 184.88, "duration": 2.52, "text": "\u0645\u062c\u062f\u062f\u0627\u064e \u0641\u064a \u062a\u0644\u0643 \u0627\u0644\u062d\u0627\u0644\u0629 \u0644\u0645 \u0646\u0643\u0646 \u0644\u0646\u0633\u062a\u0637\u064a\u0639 \u0642\u0648\u0644 \u0623\u064a \u0634\u064a\u0626"}, {"start": 187.54, "duration": 2.26, "text": "\u0644\u0627 \u0646\u0633\u062a\u0637\u064a\u0639 \u0623\u0646 \u0646\u0642\u0648\u0644 \u0623\u0646 \u0627\u0644\u0633\u0644\u0633\u0644\u0629 \u0645\u062a\u0642\u0627\u0631\u0628\u0629 \u0623\u0648 \u0645\u062a\u0628\u0627\u0639\u062f\u0629"}, {"start": 189.8, "duration": 3.16, "text": "\u0628\u0639\u062f \u0630\u0644\u0643 , \u064a\u062c\u0628 \u0623\u0646 \u062a\u0628\u062f\u0623 \u0628\u0627\u0644\u062a\u0641\u0643\u064a\u0631 \u0628\u0645\u062e\u062a\u0644\u0641 \u0627\u0644\u0625\u062e\u062a\u0628\u0627\u0631\u0627\u062a \u0627\u0644\u062a\u064a \u062a\u0639\u0644\u0645\u062a\u0647\u0627"}, {"start": 192.96, "duration": 1.46, "text": "\u0625\u062e\u062a\u0628\u0627\u0631 \u0627\u0644\u062a\u0643\u0627\u0645\u0644"}, {"start": 194.84, "duration": 3.74, "text": "\u0627\u0644\u0645\u0642\u0627\u0631\u0646\u0629 \u0627\u0644\u0645\u0628\u0627\u0634\u0631\u0629 , \u0645\u0642\u0627\u0631\u0646\u0629 \u0627\u0644\u0646\u0647\u0627\u064a\u0627\u062a , \u0625\u062e\u062a\u0628\u0627\u0631 \u0627\u0644\u0646\u0633\u0628\u0629 , \u0625\u062e\u062a\u0628\u0627\u0631 \u0627\u0644\u062c\u0630\u0631"}, {"start": 198.6, "duration": 2.64, "text": "\u0633\u0648\u0641 \u062a\u0643\u0648\u0646 \u0647\u0646\u0627\u0643 \u0643\u0644 \u0647\u0630\u0647 \u0627\u0644\u0623\u0646\u0648\u0627\u0639 \u0627\u0644\u0645\u062e\u062a\u0644\u0641\u0629 \u0645\u0646 \u0627\u0644\u0623\u062e\u062a\u0628\u0627\u0631\u0627\u062a \u0644\u0644\u062a\u062c\u0631\u0628\u0629"}, {"start": 201.8, "duration": 2.0, "text": "\u0625\u0630\u0627 \u0643\u0627\u0646\u062a \u0627\u0644\u0646\u0647\u0627\u064a\u0629 \u062a\u0633\u0627\u0648\u064a \u0635\u0641\u0631\u0627\u064b"}, {"start": 203.94, "duration": 2.0, "text": "\u0625\u0630\u0627\u064b, \u0625\u0630\u0627 \u0643\u0627\u0646\u062a \u0627\u0644\u0646\u0647\u0627\u064a\u0629 \u0644\u0627\u062a\u0633\u0627\u0648\u064a \u0635\u0641\u0631\u0627\u064b \u0641\u0625\u0646 \u0627\u0644\u0633\u0644\u0633\u0644\u0629 \u0645\u062a\u0628\u0627\u0639\u062f\u0629"}, {"start": 205.94, "duration": 1.02, "text": "\u0648\u0644\u0642\u062f \u0627\u0646\u062a\u0647\u064a\u062a!"}, {"start": 206.96, "duration": 1.32, "text": "\u0625\u0630\u0627 \u0643\u0627\u0646\u062a \u0627\u0644\u0646\u0647\u0627\u064a\u0629 \u062a\u0633\u0627\u0648\u064a \u0635\u0641\u0631\u0627\u064b"}, {"start": 208.74, "duration": 3.04, "text": "\u0633\u0648\u0641 \u062a\u062d\u062a\u0627\u062c \u0644\u0639\u0645\u0644 \u0623\u0643\u062b\u0631 \u0646\u0648\u0639\u0627\u064b \u0645\u0627"}, {"start": 212.02, "duration": 4.24, "text": "\u0644\u0623\u062a\u0643 \u0633\u062a\u062d\u062a\u0627\u062c \u0644\u062a\u0643\u062a\u0634\u0641 \u0637\u0631\u064a\u0642\u0629 \u0644\u062a\u062a\u0639\u0627\u0645\u0644 \u0645\u0639 \u0627\u0644\u0633\u0624\u0627\u0644"}], "title": "Test for Divergence for Series, Two Examples", "description": "Thanks to all of you who support me on Patreon. You da real mvps!   $1 per month helps!! :)  https://www.patreon.com/patrickjmt !! Test for Divergence for Series, Two Examples.  In this video, I discuss the test for divergence and show two examples of series who diverge by using the test for divergence.", "lengthSeconds": 217, "width": 480, "height": 360}
{"video_id": "zvPQljHphjE", "transcript": [{"start": 0.11, "duration": 4.06, "text": "all right in this video I'm going to do"}, {"start": 2.25, "duration": 3.06, "text": "another problem involving the"}, {"start": 4.17, "duration": 2.64, "text": "Pythagorean theorem"}, {"start": 5.31, "duration": 2.91, "text": "so in this one suppose we have two"}, {"start": 6.81, "duration": 2.42, "text": "cyclists and they start from the same"}, {"start": 8.22, "duration": 3.93, "text": "location"}, {"start": 9.23, "duration": 5.92, "text": "suppose one cyclist is going due north"}, {"start": 12.15, "duration": 5.1, "text": "and the other is going due east and"}, {"start": 15.15, "duration": 4.76, "text": "suppose somehow magically they're going"}, {"start": 17.25, "duration": 5.58, "text": "exactly the same speed the whole time"}, {"start": 19.91, "duration": 5.32, "text": "suppose after two hours we know that"}, {"start": 22.83, "duration": 6.24, "text": "there are 17 times the square root of"}, {"start": 25.23, "duration": 6.29, "text": "two models apart and we want to know how"}, {"start": 29.07, "duration": 5.31, "text": "fast each is going in miles per hour"}, {"start": 31.52, "duration": 5.77, "text": "okay so we want the account for this"}, {"start": 34.38, "duration": 4.91, "text": "fact that they've done this two hours so"}, {"start": 37.29, "duration": 4.769, "text": "that'll come into it but at the moment"}, {"start": 39.29, "duration": 4.63, "text": "you know so sometime later the cyclist"}, {"start": 42.059, "duration": 4.23, "text": "going north will be here the cyclist"}, {"start": 43.92, "duration": 4.889, "text": "going east will be here we know that the"}, {"start": 46.289, "duration": 5.011, "text": "distance between them is 17 times the"}, {"start": 48.809, "duration": 4.531, "text": "square root of 2 since they were going"}, {"start": 51.3, "duration": 6.06, "text": "the same speed they went the exact same"}, {"start": 53.34, "duration": 5.969, "text": "distance so we can call each side of the"}, {"start": 57.36, "duration": 5.01, "text": "right triangle we can say that each one"}, {"start": 59.309, "duration": 5.13, "text": "of those distances is X and now we can"}, {"start": 62.37, "duration": 5.85, "text": "just use the Pythagorean theorem it says"}, {"start": 64.439, "duration": 8.941, "text": "x squared plus x squared would have to"}, {"start": 68.22, "duration": 7.23, "text": "equal 17 square root of 2 squared so"}, {"start": 73.38, "duration": 5.97, "text": "we've got x squared plus x squared"}, {"start": 75.45, "duration": 5.73, "text": "that's going to give us 2x squared we've"}, {"start": 79.35, "duration": 4.94, "text": "got we would have to take the square"}, {"start": 81.18, "duration": 5.07, "text": "root of 17 times the square root of 17"}, {"start": 84.29, "duration": 5.38, "text": "excuse me right we would have to take"}, {"start": 86.25, "duration": 4.59, "text": "simply 17 times 17 I'm looking at the"}, {"start": 89.67, "duration": 6.39, "text": "square root part there"}, {"start": 90.84, "duration": 6.66, "text": "so that's 289 so 17 times 17 is 289 we"}, {"start": 96.06, "duration": 3.15, "text": "do have to take the square root of 2"}, {"start": 97.5, "duration": 3.38, "text": "times the square root of 2 which would"}, {"start": 99.21, "duration": 3.799, "text": "be 2"}, {"start": 100.88, "duration": 5.07, "text": "and then we get simply divide both sides"}, {"start": 103.009, "duration": 9.841, "text": "by two and that would give us x squared"}, {"start": 105.95, "duration": 9.27, "text": "equals 289 and again we know well I mean"}, {"start": 112.85, "duration": 4.53, "text": "we squared it to get it so if you square"}, {"start": 115.22, "duration": 4.05, "text": "root both sides again normally we get"}, {"start": 117.38, "duration": 4.98, "text": "positives negatives but these clearly"}, {"start": 119.27, "duration": 7.26, "text": "have to be positive values the square"}, {"start": 122.36, "duration": 7.709, "text": "root of 80 to 89 is just 17 so what it"}, {"start": 126.53, "duration": 7.77, "text": "says again this is in two hours they've"}, {"start": 130.069, "duration": 11.7, "text": "each gone a distance of 17 miles so that"}, {"start": 134.3, "duration": 11.67, "text": "means the speed of each is 17 divided by"}, {"start": 141.769, "duration": 4.741, "text": "2 which would be 8 point 5 miles per"}, {"start": 145.97, "duration": 3.57, "text": "hour"}, {"start": 146.51, "duration": 6.6, "text": "so each cyclist is going 8.5 miles per"}, {"start": 149.54, "duration": 6.36, "text": "hour over 2 hours if they do that though"}, {"start": 153.11, "duration": 6.27, "text": "end up being 17 times the square root of"}, {"start": 155.9, "duration": 5.78, "text": "2 miles apart so nice leisurely bike"}, {"start": 159.38, "duration": 2.3, "text": "ride"}], "title": "Word Problems Using the Pythagorean Theorem - Example 2", "description": "Thanks to all of you who support me on Patreon. You da real mvps!   $1 per month helps!! :)  https://www.patreon.com/patrickjmt !! Word Problems Using the Pythagorean Theorem - Example 2", "lengthSeconds": 161, "width": 480, "height": 360}
{"video_id": "zjSgkoaL_5A", "transcript": [{"start": 0.03, "duration": 3.419, "text": "all right in this video I want to talk a"}, {"start": 1.74, "duration": 4.17, "text": "little bit about graphing sine and"}, {"start": 3.449, "duration": 5.311, "text": "cosine functions with phase shifts and"}, {"start": 5.91, "duration": 6.51, "text": "if you remember horizontal shifts from"}, {"start": 8.76, "duration": 5.43, "text": "you know from just regular algebra I"}, {"start": 12.42, "duration": 2.91, "text": "mean it's the exact same thing all we're"}, {"start": 14.19, "duration": 3.81, "text": "doing is just shifting graphs"}, {"start": 15.33, "duration": 4.89, "text": "horizontally except now assigning cosine"}, {"start": 18.0, "duration": 3.779, "text": "we call them phase shifts so these"}, {"start": 20.22, "duration": 4.71, "text": "definitely have some applications and"}, {"start": 21.779, "duration": 6.451, "text": "physics at least I know and I'm sure in"}, {"start": 24.93, "duration": 4.859, "text": "other places as well so basically what"}, {"start": 28.23, "duration": 2.91, "text": "we're going to have to do is you know we"}, {"start": 29.789, "duration": 3.351, "text": "could have some amplitude out front"}, {"start": 31.14, "duration": 5.37, "text": "whatever we've either got sine or cosine"}, {"start": 33.14, "duration": 5.65, "text": "inside the parenthesis if your variable"}, {"start": 36.51, "duration": 3.33, "text": "X or R is originally has a coefficient"}, {"start": 38.79, "duration": 3.6, "text": "you kind of have to factor that out"}, {"start": 39.84, "duration": 5.61, "text": "within the parenthesis but then whatever"}, {"start": 42.39, "duration": 5.22, "text": "is left over this plus K that's going to"}, {"start": 45.45, "duration": 4.08, "text": "help us figure out the phase shift and"}, {"start": 47.61, "duration": 4.35, "text": "remember if it's on the inside if it's"}, {"start": 49.53, "duration": 6.689, "text": "actually positive K that moves you K"}, {"start": 51.96, "duration": 6.029, "text": "units to the left and if the inside is"}, {"start": 56.219, "duration": 3.961, "text": "actually a negative K its kind of"}, {"start": 57.989, "duration": 6.291, "text": "opposite and we go K units to the right"}, {"start": 60.18, "duration": 4.1, "text": "so let's talk about a couple of these so"}, {"start": 64.76, "duration": 3.64, "text": "we're just going to figure out the phase"}, {"start": 66.689, "duration": 4.411, "text": "shift for each one of these following"}, {"start": 68.4, "duration": 7.74, "text": "functions so all right I've got cosine"}, {"start": 71.1, "duration": 6.839, "text": "of X minus 4 okay I see a negative 4 so"}, {"start": 76.14, "duration": 3.479, "text": "negative normally makes me think kind of"}, {"start": 77.939, "duration": 3.631, "text": "to the left in terms of a number line"}, {"start": 79.619, "duration": 7.161, "text": "but actually this is just going to be"}, {"start": 81.57, "duration": 5.21, "text": "the regular graph of cosine shifted"}, {"start": 86.96, "duration": 9.99, "text": "four units to the to the right again"}, {"start": 94.07, "duration": 4.44, "text": "kind of the opposite direction so notice"}, {"start": 96.95, "duration": 4.02, "text": "the next one we've got sine of two times"}, {"start": 98.51, "duration": 3.66, "text": "the quantity x plus one again I'm just"}, {"start": 100.97, "duration": 3.39, "text": "looking at this number"}, {"start": 102.17, "duration": 4.77, "text": "notice the twos outside the parentheses"}, {"start": 104.36, "duration": 5.01, "text": "and that's what we want we're going to"}, {"start": 106.94, "duration": 4.68, "text": "have to for example modify our next"}, {"start": 109.37, "duration": 5.79, "text": "example here down here but since I see a"}, {"start": 111.62, "duration": 6.6, "text": "plus one again normally positive the"}, {"start": 115.16, "duration": 5.04, "text": "positive makes me think to the right but"}, {"start": 118.22, "duration": 3.899, "text": "in this case again it's sort of opposite"}, {"start": 120.2, "duration": 4.64, "text": "so we're going to shift it will be"}, {"start": 122.119, "duration": 5.311, "text": "shifted in this case one unit"}, {"start": 124.84, "duration": 7.78, "text": "well since it's positive we're going to"}, {"start": 127.43, "duration": 6.99, "text": "go one unit to the left last but not"}, {"start": 132.62, "duration": 4.14, "text": "least here we've got cosine of ten X"}, {"start": 134.42, "duration": 4.83, "text": "plus thirty all I'm going to do is I'm"}, {"start": 136.76, "duration": 4.68, "text": "going to factor the ten out okay it"}, {"start": 139.25, "duration": 3.93, "text": "still has to stay inside the parentheses"}, {"start": 141.44, "duration": 3.78, "text": "though I can't like pull it out front"}, {"start": 143.18, "duration": 4.62, "text": "that's definitely incorrect so be"}, {"start": 145.22, "duration": 5.34, "text": "careful about things like that well if i"}, {"start": 147.8, "duration": 4.65, "text": "factor the ten out we would have so ten"}, {"start": 150.56, "duration": 5.52, "text": "out front and then it looks like X plus"}, {"start": 152.45, "duration": 5.7, "text": "three in the parentheses and again the"}, {"start": 156.08, "duration": 4.17, "text": "same thing I see a positive three but"}, {"start": 158.15, "duration": 3.66, "text": "now I'm really just shifting it's going"}, {"start": 160.25, "duration": 4.05, "text": "to be shifted"}, {"start": 161.81, "duration": 8.1, "text": "the original graph of cosine would be"}, {"start": 164.3, "duration": 8.97, "text": "shifted three units to the in this case"}, {"start": 169.91, "duration": 4.92, "text": "the left okay so again I said the"}, {"start": 173.27, "duration": 4.08, "text": "original graph of cosine actually it"}, {"start": 174.83, "duration": 4.35, "text": "would be different than plain old cosine"}, {"start": 177.35, "duration": 3.78, "text": "because of this this ten in there that"}, {"start": 179.18, "duration": 3.81, "text": "would change the period but yeah this is"}, {"start": 181.13, "duration": 4.47, "text": "the basic idea to figure out just the"}, {"start": 182.99, "duration": 5.16, "text": "horizontal or the phase shift"}, {"start": 185.6, "duration": 5.16, "text": "just make sure within the you know sort"}, {"start": 188.15, "duration": 4.77, "text": "of within the sine or cosine you know if"}, {"start": 190.76, "duration": 3.84, "text": "there's any coefficient factored out and"}, {"start": 192.92, "duration": 3.45, "text": "then just remember this trick that's"}, {"start": 194.6, "duration": 3.6, "text": "sort of if it's on the inside of the"}, {"start": 196.37, "duration": 4.14, "text": "parentheses you kind of have to do the"}, {"start": 198.2, "duration": 4.02, "text": "opposite so I see positive but it"}, {"start": 200.51, "duration": 4.94, "text": "actually moves in the negative direction"}, {"start": 202.22, "duration": 3.23, "text": "which would be to the left"}], "title": "Graphing Sine and Cosine with Phase (Horizontal) Shifts, Example 1", "description": "Thanks to all of you who support me on Patreon. You da real mvps!   $1 per month helps!! :)  https://www.patreon.com/patrickjmt !! In this video, I show how to find the phase shift (aka, the horizontal shift) of a couple of trig functions.  In the next video, I actually graph a trig function by thinking about phase shifts.", "lengthSeconds": 204, "width": 480, "height": 360}
{"video_id": "eWLgWazTc-0", "transcript": [{"start": 0.64, "duration": 3.44, "text": "all right in this video we're going to"}, {"start": 1.88, "duration": 3.64, "text": "look at solving uh another uh"}, {"start": 4.08, "duration": 2.759, "text": "differential equation by using"}, {"start": 5.52, "duration": 4.44, "text": "integrating"}, {"start": 6.839, "duration": 10.44, "text": "factors so here we've got dy over DX +"}, {"start": 9.96, "duration": 9.159, "text": "2x over 1 + x^2 * y = 4 1 + x^2 squared"}, {"start": 17.279, "duration": 4.561, "text": "so the first thing we'll do is calculate"}, {"start": 19.119, "duration": 8.441, "text": "our integrating Factor so we'll do e to"}, {"start": 21.84, "duration": 8.359, "text": "the integral of 2x over 1 + x^2"}, {"start": 27.56, "duration": 7.039, "text": "DX so kind of off to the side here I'm"}, {"start": 30.199, "duration": 4.4, "text": "going to integrate 2x over 1 +"}, {"start": 34.64, "duration": 5.0, "text": "x^2 and to integrate this I think we can"}, {"start": 37.0, "duration": 5.399, "text": "just do a nice little U substitution we"}, {"start": 39.64, "duration": 7.399, "text": "can let u = 1 +"}, {"start": 42.399, "duration": 7.8, "text": "x^2 du is going to give us 2x"}, {"start": 47.039, "duration": 7.0, "text": "DX so after we do our U substitution"}, {"start": 50.199, "duration": 5.761, "text": "we're really just integrating one over U"}, {"start": 54.039, "duration": 4.601, "text": "du and that's going to give us the"}, {"start": 55.96, "duration": 5.119, "text": "natural logarithm of U plus"}, {"start": 58.64, "duration": 5.2, "text": "C so so that's going to be the natural"}, {"start": 61.079, "duration": 6.481, "text": "logarithm of we can replace the uh"}, {"start": 63.84, "duration": 8.319, "text": "absolute value with uh just parentheses"}, {"start": 67.56, "duration": 7.16, "text": "1 plus X2 is always positive plus"}, {"start": 72.159, "duration": 7.6, "text": "C okay so really when we integrate we'll"}, {"start": 74.72, "duration": 6.96, "text": "get e to the natural logarithm of 1 + X2"}, {"start": 79.759, "duration": 4.32, "text": "so again we're trying to find a"}, {"start": 81.68, "duration": 4.32, "text": "particular um integrating Factor doesn't"}, {"start": 84.079, "duration": 3.881, "text": "have to be the most General so again we"}, {"start": 86.0, "duration": 4.4, "text": "don't have to worry about the plus C"}, {"start": 87.96, "duration": 5.0, "text": "here all right so let's see we' got e to"}, {"start": 90.4, "duration": 6.32, "text": "the natural logarithm of 1 + x^2 well"}, {"start": 92.96, "duration": 6.839, "text": "that's just going to leave us with 1 +"}, {"start": 96.72, "duration": 5.439, "text": "x^2 so that'll be our integrating Factor"}, {"start": 99.799, "duration": 4.0, "text": "so we're going to multiply both sides by"}, {"start": 102.159, "duration": 7.521, "text": "1 +"}, {"start": 103.799, "duration": 10.081, "text": "x^2 so we'll have dy over DX + 2x over 1"}, {"start": 109.68, "duration": 7.32, "text": "+ x^2 * y we're going to multiply all of"}, {"start": 113.88, "duration": 7.96, "text": "that by 1 + x^2 and we'll do the same"}, {"start": 117.0, "duration": 4.84, "text": "thing on the right side as well"}, {"start": 123.92, "duration": 7.36, "text": "so again the left side we can write as"}, {"start": 126.2, "duration": 8.679, "text": "the derivative with respect to X of 1 +"}, {"start": 131.28, "duration": 6.76, "text": "X2 all multiplied by"}, {"start": 134.879, "duration": 5.601, "text": "y on the right side notice the uh 1 plus"}, {"start": 138.04, "duration": 5.04, "text": "x^2 will cancel out with one of the 1"}, {"start": 140.48, "duration": 4.44, "text": "plus X SS in the denominator so that's"}, {"start": 143.08, "duration": 5.68, "text": "going to leave us with four time or"}, {"start": 144.92, "duration": 5.44, "text": "excuse me four over the quantity 1 +"}, {"start": 148.76, "duration": 2.88, "text": "x^2"}, {"start": 150.36, "duration": 4.12, "text": "and again what we're going to do is just"}, {"start": 151.64, "duration": 5.12, "text": "integrate simply both sides with respect"}, {"start": 154.48, "duration": 2.28, "text": "to"}, {"start": 166.56, "duration": 7.64, "text": "X all right so again on the left side"}, {"start": 169.239, "duration": 7.481, "text": "we'll just be left with 1 + x^2 * y on"}, {"start": 174.2, "duration": 4.64, "text": "the right side well the anti-derivative"}, {"start": 176.72, "duration": 5.4, "text": "of uh you know so I've pulled the four"}, {"start": 178.84, "duration": 6.84, "text": "out the anti-derivative of 1 1 + x^2"}, {"start": 182.12, "duration": 6.039, "text": "that's just going to be ar tangent of X"}, {"start": 185.68, "duration": 5.4, "text": "I'm going to add on the plus C and now"}, {"start": 188.159, "duration": 6.36, "text": "we can get y by itself by just dividing"}, {"start": 191.08, "duration": 12.519, "text": "so we could write our solution as for"}, {"start": 194.519, "duration": 9.08, "text": "arct tangent of x + C all divided by 1 +"}, {"start": 208.68, "duration": 3.0, "text": "x^2"}], "title": "\u2756 Integrating Factor to Solve a Differential Equation \u2756", "description": "Integrating Factor to Solve a Differential Equation.\nIn this video, I look at an example of using an integrating factor to help us solve a differential equation.\nIn this video our differential equation is already in the proper form so we simply need to multiply both sides by the integrating factor and then integrate.\nNotice that the left side of the equation simplifies nicely as y multiplied by the integrating factor after integrating both sides (this is the magic of the integrating factor!).\n\n\n\n\n\n\n\n\n\n\n\n#differentialequation #maths #integratingfactor #exactdifferentialequation #exactdifferentialequations #ordinarydifferentialequations #patrickjmt", "lengthSeconds": 210, "width": 480, "height": 360}
{"video_id": "z400r4gCoU4", "transcript": [{"start": 0.11, "duration": 4.27, "text": "all right in this example we're going to"}, {"start": 2.31, "duration": 5.31, "text": "sketch the graph of the ellipse y"}, {"start": 4.38, "duration": 6.66, "text": "squared over 4 plus X minus 5 squared"}, {"start": 7.62, "duration": 5.58, "text": "equals 1 so what we'll do is basically"}, {"start": 11.04, "duration": 4.71, "text": "we're going to find the center and we're"}, {"start": 13.2, "duration": 4.8, "text": "going to find some the vertices and also"}, {"start": 15.75, "duration": 5.55, "text": "the co vertices and we'll just make a"}, {"start": 18.0, "duration": 6.449, "text": "rough sketch based on that so we could"}, {"start": 21.3, "duration": 9.03, "text": "write this as Y minus 0 quantity squared"}, {"start": 24.449, "duration": 9.241, "text": "we can write 4 as 2 squared then we have"}, {"start": 30.33, "duration": 6.21, "text": "X minus 5 squared we could put that over"}, {"start": 33.69, "duration": 6.029, "text": "1 but again we could even just write 1"}, {"start": 36.54, "duration": 5.31, "text": "as well 1 squared and again I'm just"}, {"start": 39.719, "duration": 8.131, "text": "putting this you know in the standard"}, {"start": 41.85, "duration": 10.65, "text": "form of an ellipse so the center of our"}, {"start": 47.85, "duration": 8.549, "text": "ellipse is going to be positive 5 comma"}, {"start": 52.5, "duration": 7.41, "text": "0 so we can now read that off from our"}, {"start": 56.399, "duration": 5.221, "text": "standard form and to get the vertices"}, {"start": 59.91, "duration": 3.719, "text": "well I'm even going to go ahead and"}, {"start": 61.62, "duration": 4.05, "text": "start making a rough little sketch based"}, {"start": 63.629, "duration": 7.621, "text": "on this and point out some other things"}, {"start": 65.67, "duration": 9.32, "text": "as well so 1 2 3 4 5 so there's going to"}, {"start": 71.25, "duration": 7.02, "text": "be the center the center of our ellipse"}, {"start": 74.99, "duration": 5.309, "text": "now to get the vertices we can basically"}, {"start": 78.27, "duration": 5.55, "text": "look at our a value and our B value"}, {"start": 80.299, "duration": 5.591, "text": "again a is greater than B so in this"}, {"start": 83.82, "duration": 6.0, "text": "case our a value is going to equal"}, {"start": 85.89, "duration": 7.049, "text": "positive 2 so the way I always again"}, {"start": 89.82, "duration": 6.42, "text": "just kind of thought about things so I'm"}, {"start": 92.939, "duration": 5.07, "text": "going to go in to put 5 0 there okay the"}, {"start": 96.24, "duration": 4.26, "text": "number that's being squared is 2 again"}, {"start": 98.009, "duration": 4.981, "text": "that's our a value and you know that's"}, {"start": 100.5, "duration": 4.89, "text": "underneath the y squared term the term"}, {"start": 102.99, "duration": 6.059, "text": "that involves Y squared that tells me"}, {"start": 105.39, "duration": 6.96, "text": "how many units we go up and down from"}, {"start": 109.049, "duration": 6.57, "text": "our Center so I'm going to go up 1 2"}, {"start": 112.35, "duration": 5.49, "text": "units and then I'm going to go down 1 2"}, {"start": 115.619, "duration": 6.18, "text": "units and those will give us the"}, {"start": 117.84, "duration": 6.809, "text": "vertices of our ellipse so we've got 5"}, {"start": 121.799, "duration": 7.371, "text": "comma 2 and then our other point is"}, {"start": 124.649, "duration": 7.851, "text": "going to be here at 5 comma negative 2"}, {"start": 129.17, "duration": 5.73, "text": "well likewise if we look at our beef"}, {"start": 132.5, "duration": 6.45, "text": "you are B value is going to equal"}, {"start": 134.9, "duration": 7.65, "text": "positive one and that tells me how far I"}, {"start": 138.95, "duration": 7.41, "text": "move left and right from where it's"}, {"start": 142.55, "duration": 5.88, "text": "centered to get my co vertices so from"}, {"start": 146.36, "duration": 5.16, "text": "the center it says move left one unit"}, {"start": 148.43, "duration": 6.33, "text": "okay and then it says move right one"}, {"start": 151.52, "duration": 9.21, "text": "unit as well so the co vertices will be"}, {"start": 154.76, "duration": 7.949, "text": "at 4 comma 0 and also at 6 comma 0 and"}, {"start": 160.73, "duration": 7.35, "text": "now again I'm just going to make a rough"}, {"start": 162.709, "duration": 8.931, "text": "little sketch of my ellipse and not"}, {"start": 168.08, "duration": 6.27, "text": "quite to scale but not too terrible so"}, {"start": 171.64, "duration": 4.75, "text": "all right so again that's what I like to"}, {"start": 174.35, "duration": 4.77, "text": "do I pick out the center and then again"}, {"start": 176.39, "duration": 5.19, "text": "I look at my a value or my B value and"}, {"start": 179.12, "duration": 4.86, "text": "again if it's underneath a you know I"}, {"start": 181.58, "duration": 4.17, "text": "look at the I look at the numerator that"}, {"start": 183.98, "duration": 3.69, "text": "kind of tells me again hey that's how"}, {"start": 185.75, "duration": 4.35, "text": "many units I either go up or down and"}, {"start": 187.67, "duration": 5.789, "text": "again for example if you look underneath"}, {"start": 190.1, "duration": 6.09, "text": "the term involving X the B values 1 well"}, {"start": 193.459, "duration": 4.681, "text": "X I think about the x-axis that's kind"}, {"start": 196.19, "duration": 3.99, "text": "of to the left and to the right so that"}, {"start": 198.14, "duration": 6.26, "text": "helps me remember sort of what direction"}, {"start": 200.18, "duration": 4.22, "text": "to go to get my other points"}], "title": "Conic Sections, Ellipse, Shifted: Sketch Graph Given Equation", "description": "Thanks to all of you who support me on Patreon. You da real mvps!   $1 per month helps!! :)  https://www.patreon.com/patrickjmt !! Conic Sections, Ellipse, Shifted: Sketch Graph Given Equation.  In this example, we sketch the graph of a shifted ellipse.", "lengthSeconds": 207, "width": 480, "height": 360}
{"video_id": "8NCwKUnIbRk", "transcript": [{"start": 1.92, "duration": 3.68, "text": "okay in this video I want to do some"}, {"start": 3.52, "duration": 3.96, "text": "slightly harder uh examples of"}, {"start": 5.6, "duration": 4.24, "text": "evaluating exponential"}, {"start": 7.48, "duration": 7.239, "text": "Expressions so in the first one we have"}, {"start": 9.84, "duration": 7.04, "text": "2 + the quantity 1 - 5^ 2 + 3^ 2"}, {"start": 14.719, "duration": 4.041, "text": "typically what I like to do is just work"}, {"start": 16.88, "duration": 4.36, "text": "inside the parentheses first if there's"}, {"start": 18.76, "duration": 4.72, "text": "any other clear simplifications I'll do"}, {"start": 21.24, "duration": 6.119, "text": "it the two I'm just going to leave him"}, {"start": 23.48, "duration": 7.0, "text": "hanging out front inside we have 1 - 5 1"}, {"start": 27.359, "duration": 5.441, "text": "- 5 is going to give us -4 again that's"}, {"start": 30.48, "duration": 4.36, "text": "being"}, {"start": 32.8, "duration": 5.36, "text": "squared and I'll think I'll go ahead and"}, {"start": 34.84, "duration": 7.28, "text": "simplify we have 3 squar which means 3 *"}, {"start": 38.16, "duration": 10.12, "text": "3 3 * 3 is going to give us the number"}, {"start": 42.12, "duration": 10.0, "text": "n and couple more steps here so ne4 * -4"}, {"start": 48.28, "duration": 6.2, "text": "will give us a positive 16 and then we"}, {"start": 52.12, "duration": 6.88, "text": "still have our 9 hanging out"}, {"start": 54.48, "duration": 9.32, "text": "there well 2 + 16 is going to give us 18"}, {"start": 59.0, "duration": 6.72, "text": "18 + 9 gives us the number 27 so our"}, {"start": 63.8, "duration": 3.04, "text": "first expression will reduce to the"}, {"start": 65.72, "duration": 3.52, "text": "value"}, {"start": 66.84, "duration": 5.919, "text": "27 let's look at our next one here we"}, {"start": 69.24, "duration": 5.919, "text": "have 7 minus the quantity 4 - 2 cubed +"}, {"start": 72.759, "duration": 4.921, "text": "-2 cubed I'm going to do the same thing"}, {"start": 75.159, "duration": 7.401, "text": "I'm going to leave the 7 out front I'm"}, {"start": 77.68, "duration": 9.28, "text": "going to take 4 - 2 which is 2"}, {"start": 82.56, "duration": 7.44, "text": "cubed likewise we have -2 to the 3 power"}, {"start": 86.96, "duration": 8.64, "text": "and again that's -2 * -2 which is which"}, {"start": 90.0, "duration": 5.6, "text": "is pos4 * another -2 will give us"}, {"start": 96.36, "duration": 7.36, "text": "A8 okay"}, {"start": 98.72, "duration": 8.999, "text": "so we then have 7 - 2 the 3 power we can"}, {"start": 103.72, "duration": 7.679, "text": "replace that simply with 8 okay so 2 the"}, {"start": 107.719, "duration": 6.241, "text": "3 power is 8 we have a positive and a"}, {"start": 111.399, "duration": 4.161, "text": "negative so a positive and a negative"}, {"start": 113.96, "duration": 3.88, "text": "remember that turns into a"}, {"start": 115.56, "duration": 3.28, "text": "negative and there's our eight still"}, {"start": 117.84, "duration": 4.76, "text": "just hanging"}, {"start": 118.84, "duration": 8.0, "text": "out and now we're almost there okay so 7"}, {"start": 122.6, "duration": 8.68, "text": "- 8 7 - 8 is going to be -1 we still"}, {"start": 126.84, "duration": 6.28, "text": "have our other - 8 there and -1 - 8 is"}, {"start": 131.28, "duration": 4.76, "text": "going to give us"}, {"start": 133.12, "duration": 5.92, "text": "-9 as our final"}, {"start": 136.04, "duration": 3.0, "text": "solution"}], "title": "Evaluating Exponential Expressions (Numbers Only, No Variables) - Ex 2", "description": "Thanks to all of you who support me on Patreon. You da real mvps!   $1 per month helps!! :)  https://www.patreon.com/patrickjmt !! Evaluating Exponential Expressions (Numbers Only, No Variables) - Ex 2", "lengthSeconds": 137, "width": 480, "height": 360}
{"video_id": "aT1oQxvTIR0", "transcript": [{"start": 0.44, "duration": 3.36, "text": "all right in this video we're just going"}, {"start": 1.599, "duration": 4.481, "text": "to look at a couple examples of adding"}, {"start": 3.8, "duration": 5.4, "text": "whole numbers so in part A here we've"}, {"start": 6.08, "duration": 4.8, "text": "got 61 + 14 and what I like to do uh you"}, {"start": 9.2, "duration": 3.8, "text": "know when I add these is I like to just"}, {"start": 10.88, "duration": 3.679, "text": "write them vertically and I've got"}, {"start": 13.0, "duration": 4.039, "text": "everything lined up in the sense that I"}, {"start": 14.559, "duration": 5.761, "text": "have all the ones and a column and all"}, {"start": 17.039, "duration": 5.16, "text": "the tens in another column and when we"}, {"start": 20.32, "duration": 4.36, "text": "add these you know if we have to carry a"}, {"start": 22.199, "duration": 4.041, "text": "digit to the next column we'll do so but"}, {"start": 24.68, "duration": 4.48, "text": "other than that hopefully you know"}, {"start": 26.24, "duration": 4.879, "text": "relatively straightforward so um well"}, {"start": 29.16, "duration": 5.719, "text": "you know if we add the stuff in the ones"}, {"start": 31.119, "duration": 6.081, "text": "column 1 + 4 is five so I'll just drop"}, {"start": 34.879, "duration": 7.281, "text": "that down and then I just go to the next"}, {"start": 37.2, "duration": 7.92, "text": "column 6 + 1 is 7 so it says Hey 61 + uh"}, {"start": 42.16, "duration": 5.52, "text": "14 that's going to equal"}, {"start": 45.12, "duration": 5.599, "text": "75 we'll do the same thing for our next"}, {"start": 47.68, "duration": 5.559, "text": "example 77 +"}, {"start": 50.719, "duration": 5.961, "text": "18 now here we have to be a little more"}, {"start": 53.239, "duration": 8.32, "text": "careful um you know if we do 7 +"}, {"start": 56.68, "duration": 6.839, "text": "8 well 7 + 8 is 15"}, {"start": 61.559, "duration": 5.481, "text": "so what we're going to do you know 15"}, {"start": 63.519, "duration": 5.401, "text": "would be the uh ones digit so the 15 is"}, {"start": 67.04, "duration": 3.8, "text": "the ones digit that's the digit we're"}, {"start": 68.92, "duration": 4.559, "text": "going to write down so I'm going to"}, {"start": 70.84, "duration": 6.88, "text": "write the one's digit in the ones"}, {"start": 73.479, "duration": 5.96, "text": "column um and then our one would be our"}, {"start": 77.72, "duration": 4.719, "text": "10's digit so that's what I'm going to"}, {"start": 79.439, "duration": 5.921, "text": "carry to the next column so that one is"}, {"start": 82.439, "duration": 4.121, "text": "getting carried to the next column um so"}, {"start": 85.36, "duration": 4.2, "text": "then I'm just going to add again"}, {"start": 86.56, "duration": 9.16, "text": "everything in my column so 1 + 7 is 8"}, {"start": 89.56, "duration": 10.12, "text": "plus 1 is 9 and it says 77 + 18 that's"}, {"start": 95.72, "duration": 3.96, "text": "going to give us 95"}], "title": "Adding Whole Numbers Example 1", "description": "Thanks to all of you who support me on Patreon. You da real mvps!   $1 per month helps!! :)  https://www.patreon.com/patrickjmt !! Adding Whole Numbers ; Adding Two Digit Numbers.  In this video, we look at two examples of adding two digit numbers.", "lengthSeconds": 99, "width": 480, "height": 360}
{"video_id": "yk2TkZQA4Fg", "transcript": [{"start": 0.88, "duration": 3.76, "text": "okay so here we'll have one more example"}, {"start": 3.12, "duration": 3.52, "text": "of solving an equation"}, {"start": 4.64, "duration": 3.199, "text": "that'll take a few steps to uh to"}, {"start": 6.64, "duration": 3.28, "text": "simplify"}, {"start": 7.839, "duration": 3.84, "text": "so we have two times the quantity r"}, {"start": 9.92, "duration": 5.44, "text": "minus three"}, {"start": 11.679, "duration": 6.161, "text": "plus four minus five r equals sixteen"}, {"start": 15.36, "duration": 4.16, "text": "so again my goal is you know basically i"}, {"start": 17.84, "duration": 4.72, "text": "want to get rid of the parentheses"}, {"start": 19.52, "duration": 3.679, "text": "combine all my like terms get the term"}, {"start": 22.56, "duration": 4.0, "text": "involving"}, {"start": 23.199, "duration": 6.16, "text": "r on one side a number on the other"}, {"start": 26.56, "duration": 3.52, "text": "and then divide by whatever the"}, {"start": 29.359, "duration": 2.641, "text": "coefficient"}, {"start": 30.08, "duration": 4.319, "text": "is so it's kind of a rough outline"}, {"start": 32.0, "duration": 3.68, "text": "quickly of what we're going to do"}, {"start": 34.399, "duration": 3.441, "text": "so to start off to get rid of the"}, {"start": 35.68, "duration": 4.48, "text": "parentheses i'm going to distribute so 2"}, {"start": 37.84, "duration": 6.48, "text": "times r will just be 2r"}, {"start": 40.16, "duration": 8.48, "text": "2 times negative 3 will be negative 6."}, {"start": 44.32, "duration": 7.44, "text": "plus 4 minus 5 r"}, {"start": 48.64, "duration": 4.079, "text": "equals 16. and now i'm going to combine"}, {"start": 51.76, "duration": 3.36, "text": "like terms"}, {"start": 52.719, "duration": 4.32, "text": "so i see a term involving r and a term"}, {"start": 55.12, "duration": 4.72, "text": "involving r"}, {"start": 57.039, "duration": 4.481, "text": "what i'm going to do is just take the"}, {"start": 59.84, "duration": 2.559, "text": "coefficients so we have a 2 and a"}, {"start": 61.52, "duration": 5.04, "text": "negative 5."}, {"start": 62.399, "duration": 6.561, "text": "so 2r minus 5r will be negative"}, {"start": 66.56, "duration": 4.8, "text": "3r and then we have constants we have"}, {"start": 68.96, "duration": 5.36, "text": "our negative 6"}, {"start": 71.36, "duration": 3.52, "text": "plus 4 well negative 6 plus 4 will give"}, {"start": 74.32, "duration": 2.799, "text": "us"}, {"start": 74.88, "duration": 3.2, "text": "negative 2. and again i'm going to leave"}, {"start": 77.119, "duration": 3.281, "text": "the 16"}, {"start": 78.08, "duration": 4.16, "text": "on the right side nothing really to do"}, {"start": 80.4, "duration": 4.32, "text": "with it for the moment"}, {"start": 82.24, "duration": 3.44, "text": "i want to get the term involving r by"}, {"start": 84.72, "duration": 3.759, "text": "itself"}, {"start": 85.68, "duration": 3.6, "text": "so what i'm going to do is add 2 to both"}, {"start": 88.479, "duration": 3.041, "text": "sides"}, {"start": 89.28, "duration": 4.72, "text": "since i since i'm subtracting 2 we'll"}, {"start": 91.52, "duration": 5.76, "text": "add 2"}, {"start": 94.0, "duration": 6.479, "text": "we'll have well negative 3r equals"}, {"start": 97.28, "duration": 4.32, "text": "18. and to get the r by itself since"}, {"start": 100.479, "duration": 5.041, "text": "we're multiplying"}, {"start": 101.6, "duration": 7.44, "text": "by negative three what we'll do"}, {"start": 105.52, "duration": 6.32, "text": "we'll just divide uh both sides"}, {"start": 109.04, "duration": 4.16, "text": "by negative three so negative three over"}, {"start": 111.84, "duration": 3.919, "text": "negative three is one"}, {"start": 113.2, "duration": 4.16, "text": "we're left with r on the left and a"}, {"start": 115.759, "duration": 2.561, "text": "positive over a negative will give us a"}, {"start": 117.36, "duration": 4.24, "text": "negative"}, {"start": 118.32, "duration": 6.159, "text": "eighteen over three is six"}, {"start": 121.6, "duration": 3.44, "text": "so our solution will be r equals"}, {"start": 124.479, "duration": 4.321, "text": "negative"}, {"start": 125.04, "duration": 6.32, "text": "six so again kind of the basic outline"}, {"start": 128.8, "duration": 5.84, "text": "get rid of the parentheses"}, {"start": 131.36, "duration": 6.56, "text": "combine like terms get the term"}, {"start": 134.64, "duration": 8.319, "text": "involving your variable isolated"}, {"start": 137.92, "duration": 5.039, "text": "and then just divide by the coefficient"}], "title": "Solving Multi - Step Linear Equations , Another Example", "description": "Thanks to all of you who support me on Patreon. You da real mvps!   $1 per month helps!! :)  https://www.patreon.com/patrickjmt !! Solving Multi - Step Linear Equations.  Here we solve some linear equations that require a few steps to solve.", "lengthSeconds": 141, "width": 480, "height": 360}
{"video_id": "cutdeISlvWU", "transcript": [{"start": 1.12, "duration": 3.56, "text": "all right in this video I'm going to do"}, {"start": 2.44, "duration": 4.56, "text": "another example of finding the center"}, {"start": 4.68, "duration": 4.879, "text": "radius form of a circle so again the"}, {"start": 7.0, "duration": 5.08, "text": "equation of a circle with Center h k and"}, {"start": 9.559, "duration": 2.521, "text": "radius"}, {"start": 12.16, "duration": 7.64, "text": "R is going to be the equation x - h^2 +"}, {"start": 16.199, "duration": 6.041, "text": "y - k^2 = r^ 2 so we basically have to"}, {"start": 19.8, "duration": 4.12, "text": "fill in the value of H the value of K"}, {"start": 22.24, "duration": 3.0, "text": "and we'll have to take R and square it"}, {"start": 23.92, "duration": 4.279, "text": "as what we have to replace in this"}, {"start": 25.24, "duration": 4.879, "text": "formula so in this example suppose we"}, {"start": 28.199, "duration": 5.2, "text": "want to find the equation of a circle"}, {"start": 30.119, "duration": 7.001, "text": "that's centered at -4a 5 if we know that"}, {"start": 33.399, "duration": 8.121, "text": "the point 1A 7 is also a point on the"}, {"start": 37.12, "duration": 8.84, "text": "circle well already when we do our x -"}, {"start": 41.52, "duration": 7.48, "text": "the x coordinate squar + y - the y"}, {"start": 45.96, "duration": 5.84, "text": "coordinate squared equals the radius"}, {"start": 49.0, "duration": 6.199, "text": "squared I already know the center so I"}, {"start": 51.8, "duration": 7.12, "text": "could simply drop in here -4 so if I"}, {"start": 55.199, "duration": 6.481, "text": "plug in my -4 two negatives will make a"}, {"start": 58.92, "duration": 6.96, "text": "positive4 likewise if I plug in my"}, {"start": 61.68, "duration": 5.6, "text": "positive 5 I'm going to have a y - 5^ 2"}, {"start": 65.88, "duration": 2.919, "text": "the only thing we don't really know"}, {"start": 67.28, "duration": 4.72, "text": "still is the"}, {"start": 68.799, "duration": 5.481, "text": "radius but since we know this 17 is on"}, {"start": 72.0, "duration": 5.4, "text": "the circle and it goes from the circle"}, {"start": 74.28, "duration": 6.24, "text": "to the center well the distance between"}, {"start": 77.4, "duration": 6.0, "text": "these two points would uh would be the"}, {"start": 80.52, "duration": 4.36, "text": "the the length of the radius so we need"}, {"start": 83.4, "duration": 5.44, "text": "to find the"}, {"start": 84.88, "duration": 5.879, "text": "distance between our two given"}, {"start": 88.84, "duration": 4.319, "text": "points"}, {"start": 90.759, "duration": 5.64, "text": "-45 and"}, {"start": 93.159, "duration": 6.0, "text": "the17 and the distance formula simply is"}, {"start": 96.399, "duration": 7.961, "text": "just the square root again we take the x"}, {"start": 99.159, "duration": 7.401, "text": "coordinates and subtract them so -4 -1 2"}, {"start": 104.36, "duration": 5.079, "text": "plus then we subtract the Y coordinates"}, {"start": 106.56, "duration": 6.72, "text": "5 - 7^"}, {"start": 109.439, "duration": 10.801, "text": "SAR and if we simplify"}, {"start": 113.28, "duration": 9.68, "text": "here -4 - 1 is -55 * 5 is POS 255 and"}, {"start": 120.24, "duration": 7.799, "text": "then if we have 5 - 7 that's going to be"}, {"start": 122.96, "duration": 6.84, "text": "-2 -2 * -2 is pos4 so the distance"}, {"start": 128.039, "duration": 4.761, "text": "between those two points is going to be"}, {"start": 129.8, "duration": 4.799, "text": "the square < TK of 29 but again we said"}, {"start": 132.8, "duration": 4.2, "text": "in this particular case this this"}, {"start": 134.599, "duration": 4.041, "text": "distance between these two points that's"}, {"start": 137.0, "duration": 3.879, "text": "actually going to be the value of the"}, {"start": 138.64, "duration": 4.52, "text": "radius so now we're pretty much there"}, {"start": 140.879, "duration": 4.521, "text": "all we have to do is just simply replace"}, {"start": 143.16, "duration": 5.439, "text": "our r with the sare < TK of"}, {"start": 145.4, "duration": 6.199, "text": "29 so it says our final answer it says"}, {"start": 148.599, "duration": 4.521, "text": "the equation of the circle will be x +"}, {"start": 151.599, "duration": 4.801, "text": "4^"}, {"start": 153.12, "duration": 7.119, "text": "2ar + y -"}, {"start": 156.4, "duration": 5.119, "text": "5^ 2 and then it says the radius is the"}, {"start": 160.239, "duration": 4.72, "text": "square < TK of"}, {"start": 161.519, "duration": 7.0, "text": "29 so we'll have the square < TK of"}, {"start": 164.959, "duration": 5.28, "text": "29 quantity squared but obviously uh we"}, {"start": 168.519, "duration": 6.241, "text": "can simplify the right side pretty"}, {"start": 170.239, "duration": 9.121, "text": "easily thek of 29 s is just posi 29 and"}, {"start": 174.76, "duration": 8.559, "text": "now we've got our solution x + 4 2 + y -"}, {"start": 179.36, "duration": 6.84, "text": "5 2 = 29 is going to give us the"}, {"start": 183.319, "duration": 6.161, "text": "equation of the circle that satisfies"}, {"start": 186.2, "duration": 3.28, "text": "the original given"}, {"start": 189.56, "duration": 3.0, "text": "conditions"}], "title": "The Center-Radius Form for a Circle - A few Basic Questions, Example 2", "description": "Thanks to all of you who support me on Patreon. You da real mvps!   $1 per month helps!! :)  https://www.patreon.com/patrickjmt !! The Center-Radius Form for a Circle - A few Basic Questions, Example 2", "lengthSeconds": 191, "width": 480, "height": 360}
{"video_id": "mUMdjTZfGpI", "transcript": [{"start": 0.56, "duration": 4.64, "text": "okay so one more example here finding an"}, {"start": 3.04, "duration": 3.359, "text": "interest rate to match certain"}, {"start": 5.2, "duration": 3.92, "text": "conditions so"}, {"start": 6.399, "duration": 3.361, "text": "suppose you deposit nine grand the"}, {"start": 9.12, "duration": 3.12, "text": "account"}, {"start": 9.76, "duration": 4.4, "text": "compounds quarterly at the end of 10"}, {"start": 12.24, "duration": 5.68, "text": "years the balance has doubled we want to"}, {"start": 14.16, "duration": 3.76, "text": "know what the interest rate is"}, {"start": 20.48, "duration": 3.6, "text": "okay so i'm going to start off just by"}, {"start": 22.24, "duration": 3.76, "text": "filling in what i know"}, {"start": 24.08, "duration": 3.199, "text": "so okay so we know the starting"}, {"start": 26.0, "duration": 4.72, "text": "principle is nine"}, {"start": 27.279, "duration": 4.481, "text": "thousand we know that the final amount"}, {"start": 30.72, "duration": 4.4, "text": "has to be double"}, {"start": 31.76, "duration": 5.6, "text": "which would make that eighteen thousand"}, {"start": 35.12, "duration": 3.92, "text": "one plus the rate well we don't know"}, {"start": 37.36, "duration": 3.84, "text": "that it compounds"}, {"start": 39.04, "duration": 3.76, "text": "quarterly so i'm going to divide by four"}, {"start": 41.2, "duration": 4.32, "text": "four times per year"}, {"start": 42.8, "duration": 4.72, "text": "and then 4 times 10 because we're doing"}, {"start": 45.52, "duration": 4.16, "text": "this for 10 years"}, {"start": 47.52, "duration": 6.32, "text": "so a couple things i'm going to do i'm"}, {"start": 49.68, "duration": 4.16, "text": "going to divide both sides by 9000"}, {"start": 55.6, "duration": 4.479, "text": "so 18 000 over nine thousand you can"}, {"start": 57.92, "duration": 3.2, "text": "cancel out the zeros eighteen over nine"}, {"start": 60.079, "duration": 3.841, "text": "is two"}, {"start": 61.12, "duration": 3.2, "text": "the nine thousands cancel then i have"}, {"start": 63.92, "duration": 3.76, "text": "one"}, {"start": 64.32, "duration": 4.64, "text": "plus r over four well four times ten"}, {"start": 67.68, "duration": 4.479, "text": "will be to the fortieth"}, {"start": 68.96, "duration": 4.08, "text": "power and what i want to do now again is"}, {"start": 72.159, "duration": 4.081, "text": "i want to get the"}, {"start": 73.04, "duration": 5.2, "text": "get rid of this exponent of 40."}, {"start": 76.24, "duration": 4.32, "text": "so what i can do is i can raise both"}, {"start": 78.24, "duration": 5.44, "text": "sides to the"}, {"start": 80.56, "duration": 4.879, "text": "1 over 40 power or equivalently i'm"}, {"start": 83.68, "duration": 3.759, "text": "taking a 40th root"}, {"start": 85.439, "duration": 4.961, "text": "so i'm going to have to take 2 to the 1"}, {"start": 87.439, "duration": 5.761, "text": "over 40th power"}, {"start": 90.4, "duration": 2.8, "text": "let's see so"}, {"start": 93.439, "duration": 5.401, "text": "so 1 divided by 40 that looks about"}, {"start": 96.799, "duration": 3.601, "text": "right"}, {"start": 98.84, "duration": 4.279, "text": ".025"}, {"start": 100.4, "duration": 4.399, "text": "on the right side the if we multiply"}, {"start": 103.119, "duration": 3.601, "text": "we'll just get to the first power so"}, {"start": 104.799, "duration": 5.68, "text": "we'll just have 1"}, {"start": 106.72, "duration": 5.359, "text": "plus r over 4. and now i'm going to take"}, {"start": 110.479, "duration": 4.96, "text": "2 and raise it"}, {"start": 112.079, "duration": 11.441, "text": "to the point .025"}, {"start": 115.439, "duration": 11.121, "text": "so i'm getting uh 1.017479692"}, {"start": 123.52, "duration": 6.799, "text": "that's going to equal 1 plus r"}, {"start": 126.56, "duration": 5.759, "text": "over four okay well we can simply"}, {"start": 130.319, "duration": 3.041, "text": "subtract one from both sides in this"}, {"start": 132.319, "duration": 2.961, "text": "case"}, {"start": 133.36, "duration": 3.84, "text": "so if we subtract one from both sides"}, {"start": 135.28, "duration": 4.959, "text": "we'll just be left with zero point"}, {"start": 137.2, "duration": 7.28, "text": "zero one seven four seven nine"}, {"start": 140.239, "duration": 6.241, "text": "six nine two equals r over four"}, {"start": 144.48, "duration": 4.88, "text": "now i'm simply going to multiply both"}, {"start": 146.48, "duration": 2.88, "text": "sides by four"}, {"start": 150.319, "duration": 9.2, "text": "and let's see here so if we multiply"}, {"start": 153.76, "duration": 5.759, "text": "all of that by 4 i'm getting the number"}, {"start": 161.8, "duration": 5.24, "text": "0.069918768"}, {"start": 163.92, "duration": 4.64, "text": "on the right side the 4s would cancel so"}, {"start": 167.04, "duration": 3.919, "text": "that's going to be our interest rate as"}, {"start": 168.56, "duration": 4.08, "text": "a decimal and that tells me as a"}, {"start": 170.959, "duration": 3.601, "text": "percentage"}, {"start": 172.64, "duration": 5.319, "text": "it says that the interest rate is"}, {"start": 174.56, "duration": 5.36, "text": "roughly so we move it over twice"}, {"start": 177.959, "duration": 4.041, "text": "6.99"}, {"start": 179.92, "duration": 3.36, "text": "percent we'll round it off to two more"}, {"start": 182.0, "duration": 3.12, "text": "spots so"}, {"start": 183.28, "duration": 4.0, "text": "it says if you basically have almost a"}, {"start": 185.12, "duration": 4.479, "text": "seven percent interest rate"}, {"start": 187.28, "duration": 3.28, "text": "under those given uh under these given"}, {"start": 189.599, "duration": 4.56, "text": "conditions"}, {"start": 190.56, "duration": 7.2, "text": "you'll end up having a"}, {"start": 194.159, "duration": 6.281, "text": "an accumulated amount of 18 000"}, {"start": 197.76, "duration": 5.68, "text": "if you deposit an initial amount of 9"}, {"start": 200.44, "duration": 3.0, "text": "000."}], "title": "Finding an Interest Rate to Match Certain Financial Goals, Ex 3", "description": "Thanks to all of you who support me on Patreon. You da real mvps!   $1 per month helps!! :)  https://www.patreon.com/patrickjmt !! Finding an Interest Rate to Match Certain Financial Goals, Ex 3.  In this video, I do an example of where one has to find the correct interest rate when given other requirements on time, amount earned, etc.!", "lengthSeconds": 202, "width": 480, "height": 360}
{"video_id": "4HaJc2Y3e94", "transcript": [{"start": 1.4, "duration": 3.12, "text": "okay in this video I just want to look"}, {"start": 2.639, "duration": 4.72, "text": "at an example of finding the inverse"}, {"start": 4.52, "duration": 5.56, "text": "lass transform of a function involving a"}, {"start": 7.359, "duration": 5.001, "text": "factor of the form e to the"}, {"start": 10.08, "duration": 7.48, "text": "Cs so here we we're going to be given"}, {"start": 12.36, "duration": 6.36, "text": "that capital Y of s = e to3 * s / S2"}, {"start": 17.56, "duration": 3.639, "text": "sorry the little two kind of got"}, {"start": 18.72, "duration": 8.0, "text": "squeezed in there we want to compute the"}, {"start": 21.199, "duration": 10.08, "text": "inverse Lelo transform of e to3s /"}, {"start": 26.72, "duration": 10.2, "text": "S2 so I've jotted down a couple things"}, {"start": 31.279, "duration": 7.561, "text": "here so again just the the formula which"}, {"start": 36.92, "duration": 4.84, "text": "we proved in a different video the Lelo"}, {"start": 38.84, "duration": 5.44, "text": "transform of f of T minus C multiplied"}, {"start": 41.76, "duration": 5.76, "text": "Capital H of T minus C we said that"}, {"start": 44.28, "duration": 5.439, "text": "that's equal to e the Cs multiplied by"}, {"start": 47.52, "duration": 3.679, "text": "capital F of s and again H is our he"}, {"start": 49.719, "duration": 4.16, "text": "aide"}, {"start": 51.199, "duration": 5.721, "text": "function so okay so what have we got"}, {"start": 53.879, "duration": 4.761, "text": "here if you take the inverse lass"}, {"start": 56.92, "duration": 3.24, "text": "transform of the right side so that's"}, {"start": 58.64, "duration": 3.12, "text": "what I'm doing I'm just saying hey you"}, {"start": 60.16, "duration": 4.359, "text": "know we've got this original expression"}, {"start": 61.76, "duration": 4.719, "text": "let's take the inverse lla transform if"}, {"start": 64.519, "duration": 4.121, "text": "you take the inverse lass transform of"}, {"start": 66.479, "duration": 3.64, "text": "the left side as well they just undo"}, {"start": 68.64, "duration": 3.159, "text": "each other and you get back with that"}, {"start": 70.119, "duration": 5.36, "text": "that original function that you started"}, {"start": 71.799, "duration": 7.441, "text": "with so just to again point that"}, {"start": 75.479, "duration": 4.481, "text": "out so we've got the form so we had e to"}, {"start": 79.24, "duration": 4.28, "text": "the"}, {"start": 79.96, "duration": 6.24, "text": "ne you know e to the --3s over s^2 I'm"}, {"start": 83.52, "duration": 5.88, "text": "just breaking that up to make it crystal"}, {"start": 86.2, "duration": 8.76, "text": "clear so we were trying to find the"}, {"start": 89.4, "duration": 8.759, "text": "inverse plus transform of E-3 s * 1 s^2"}, {"start": 94.96, "duration": 8.759, "text": "well that means that our C value is"}, {"start": 98.159, "duration": 8.361, "text": "going to be equal to 3 so c ="}, {"start": 103.719, "duration": 6.68, "text": "3 so I can already write you know we've"}, {"start": 106.52, "duration": 6.36, "text": "got h of T minus 3 and then we're going"}, {"start": 110.399, "duration": 4.68, "text": "to have okay we've got this F of T minus"}, {"start": 112.88, "duration": 6.239, "text": "3 and this is the part that we still you"}, {"start": 115.079, "duration": 6.241, "text": "know have to have to figure out so once"}, {"start": 119.119, "duration": 4.081, "text": "we can figure out F of T minus 3 we're"}, {"start": 121.32, "duration": 6.439, "text": "done okay that's the last thing we have"}, {"start": 123.2, "duration": 9.16, "text": "to do well recall I mean to get F of"}, {"start": 127.759, "duration": 7.56, "text": "t f of T is the inverse lass transform"}, {"start": 132.36, "duration": 2.959, "text": "of f of"}, {"start": 135.519, "duration": 5.761, "text": "s so to get F of T we just take the"}, {"start": 138.04, "duration": 6.6, "text": "inverse lass transform of f of s well F"}, {"start": 141.28, "duration": 5.0, "text": "of s is our function here 1 / s^2 so"}, {"start": 144.64, "duration": 4.2, "text": "we're just using our inverse lass"}, {"start": 146.28, "duration": 5.88, "text": "transform of 1 /"}, {"start": 148.84, "duration": 6.24, "text": "s^2 and again just by using a table of"}, {"start": 152.16, "duration": 5.88, "text": "values so if you've got a table of llao"}, {"start": 155.08, "duration": 5.84, "text": "transforms the inverse Lao transform of"}, {"start": 158.04, "duration": 6.76, "text": "1 / s^2 that's just going to be equal to"}, {"start": 160.92, "duration": 8.0, "text": "T so again to get this I'm just using a"}, {"start": 164.8, "duration": 4.12, "text": "table of lus"}, {"start": 173.159, "duration": 4.16, "text": "transforms okay so it says we're really"}, {"start": 175.48, "duration": 4.36, "text": "almost done we're pretty much there"}, {"start": 177.319, "duration": 4.441, "text": "these ones uh aren't too bad so our fun"}, {"start": 179.84, "duration": 5.679, "text": "function f of"}, {"start": 181.76, "duration": 6.08, "text": "t f of T is just actually equal to T"}, {"start": 185.519, "duration": 8.64, "text": "well if F of T equals to"}, {"start": 187.84, "duration": 9.319, "text": "T if we replace our t with tus 3 well F"}, {"start": 194.159, "duration": 6.8, "text": "of T minus 3 is just going to equal T"}, {"start": 197.159, "duration": 5.761, "text": "minus 3 and now we have our solution it"}, {"start": 200.959, "duration": 5.241, "text": "says that"}, {"start": 202.92, "duration": 6.56, "text": "the inverse lla"}, {"start": 206.2, "duration": 4.2, "text": "transform of that original function e to"}, {"start": 209.48, "duration": 5.8, "text": "the"}, {"start": 210.4, "duration": 7.559, "text": "3 S / s^ 2 it says that's going to be"}, {"start": 215.28, "duration": 4.679, "text": "equal to F of tus 3 which we said in"}, {"start": 217.959, "duration": 5.56, "text": "this case turned out to be exactly just"}, {"start": 219.959, "duration": 7.081, "text": "T minus 3 and then we multiply that by h"}, {"start": 223.519, "duration": 5.72, "text": "of T minus C so we would have H of T"}, {"start": 227.04, "duration": 6.399, "text": "minus 3 because again our C value is"}, {"start": 229.239, "duration": 5.72, "text": "equal to 3 and that would be our answer"}, {"start": 233.439, "duration": 4.52, "text": "in this"}, {"start": 234.959, "duration": 3.0, "text": "case"}], "title": "Inverse Laplace Transform Involving Heaviside Functions", "description": "Thanks to all of you who support me on Patreon. You da real mvps!   $1 per month helps!! :)  https://www.patreon.com/patrickjmt !! In this video I find an inverse Laplace transform of a function involving the term e^(-cs) using Heaviside functions.", "lengthSeconds": 236, "width": 480, "height": 360}
{"video_id": "XWwrvMZcSwo", "transcript": [{"start": 1.56, "duration": 3.88, "text": "okay here we're going to use our"}, {"start": 2.8, "duration": 4.64, "text": "compound interest formula to find the"}, {"start": 5.44, "duration": 4.88, "text": "the starting or the principal amount"}, {"start": 7.44, "duration": 4.159, "text": "invested suppose we know that the okay"}, {"start": 10.32, "duration": 4.04, "text": "so we're going to find the principal"}, {"start": 11.599, "duration": 6.641, "text": "amount invested if we know the amount in"}, {"start": 14.36, "duration": 6.64, "text": "a semiannually compounded account with a"}, {"start": 18.24, "duration": 6.119, "text": "4.25% interest rate is"}, {"start": 21.0, "duration": 5.24, "text": "$6,000 after 90 months so you you've"}, {"start": 24.359, "duration": 5.041, "text": "accumulated a total of"}, {"start": 26.24, "duration": 6.32, "text": "$6,000 after 90 months compounded"}, {"start": 29.4, "duration": 3.16, "text": "semiannual at"}, {"start": 33.36, "duration": 4.76, "text": "4.25%"}, {"start": 36.04, "duration": 4.199, "text": "well again we can just fill in our"}, {"start": 38.12, "duration": 3.48, "text": "formula the accumulated amount is going"}, {"start": 40.239, "duration": 3.601, "text": "to be"}, {"start": 41.6, "duration": 4.16, "text": "6,000 the principle is what we don't"}, {"start": 43.84, "duration": 5.48, "text": "know that's what we're going to figure"}, {"start": 45.76, "duration": 6.279, "text": "out one plus our rate so our rate is"}, {"start": 49.32, "duration": 2.719, "text": "going to be"}, {"start": 54.68, "duration": 4.0, "text": "zero 0.0425"}, {"start": 61.199, "duration": 4.801, "text": "we're going to divide that by the number"}, {"start": 63.079, "duration": 6.601, "text": "of compoundings per year which is"}, {"start": 66.0, "duration": 6.799, "text": "two and then we'll do two multiplied by"}, {"start": 69.68, "duration": 6.84, "text": "the number of years well 90 months if we"}, {"start": 72.799, "duration": 6.281, "text": "take 90 and divide that by 12 let's see"}, {"start": 76.52, "duration": 5.959, "text": "that's going to give us so 12 will go"}, {"start": 79.08, "duration": 7.6, "text": "into 97 times that'll leave us with six"}, {"start": 82.479, "duration": 7.96, "text": "months out of 12 left over or 7.5"}, {"start": 86.68, "duration": 5.96, "text": "that'll be the number of years"}, {"start": 90.439, "duration": 5.0, "text": "so now I'm going to simplify the right"}, {"start": 92.64, "duration": 5.08, "text": "side down a little bit we've got"}, {"start": 95.439, "duration": 5.601, "text": "6,000 equals the"}, {"start": 97.72, "duration": 3.32, "text": "principal if we"}, {"start": 101.479, "duration": 6.761, "text": "do if we do"}, {"start": 104.6, "duration": 4.96, "text": "0425 divided by two if we add that to"}, {"start": 108.24, "duration": 3.76, "text": "one we'll get"}, {"start": 109.56, "duration": 9.199, "text": "1.02"}, {"start": 112.0, "duration": 6.759, "text": "one25 well 2 * 7.5 is going to be 15"}, {"start": 120.2, "duration": 3.72, "text": "so again I'm going to simplify we've"}, {"start": 124.88, "duration": 6.68, "text": "got we've got 1.02 one25 I'm going to"}, {"start": 129.08, "duration": 4.32, "text": "raise that to the 15th power again I'm"}, {"start": 131.56, "duration": 4.0, "text": "just using a calculator here off to the"}, {"start": 133.4, "duration": 5.16, "text": "side I'm getting"}, {"start": 135.56, "duration": 3.0, "text": "1.37"}, {"start": 138.72, "duration": 7.12, "text": "08 22 after rounding so to solve for p"}, {"start": 143.879, "duration": 7.041, "text": "the principal or starting amount I'm"}, {"start": 145.84, "duration": 5.08, "text": "just going to divide both sides by 1.37"}, {"start": 151.16, "duration": 3.68, "text": "0822 1."}, {"start": 155.72, "duration": 7.48, "text": "37822 and on the left side let's see so"}, {"start": 159.72, "duration": 6.48, "text": "6,000 divided by"}, {"start": 163.2, "duration": 3.0, "text": "1.37"}, {"start": 166.56, "duration": 5.2, "text": "822 I'm getting that to be"}, {"start": 172.68, "duration": 8.839, "text": "4,376 and 94 cents after rounding so it"}, {"start": 177.84, "duration": 7.96, "text": "says the starting amount was close to"}, {"start": 181.519, "duration": 4.281, "text": "but not quite equal to $4,400"}], "title": "Compound Interest Example - Find Starting Principal", "description": "Thanks to all of you who support me on Patreon. You da real mvps!   $1 per month helps!! :)  https://www.patreon.com/patrickjmt !! Compound Interest Example - Find Starting Principal.\n  Here we are told that an account accrues interest semiannually at a rate of 4.25%.  If the account has a total of $6,000 after 90 months, what was the starting principal?", "lengthSeconds": 185, "width": 480, "height": 360}
{"video_id": "K8IaH3adKz8", "transcript": [{"start": 0.56, "duration": 3.64, "text": "all right here we're going to look at a"}, {"start": 2.24, "duration": 5.76, "text": "little question related to radioactive"}, {"start": 4.2, "duration": 6.16, "text": "decay so suppose a radioactive material"}, {"start": 8.0, "duration": 5.04, "text": "is known to Decay at a yearly rate of"}, {"start": 10.36, "duration": 6.2, "text": "0.2 times the amount of each"}, {"start": 13.04, "duration": 6.04, "text": "moment and suppose uh so in part A there"}, {"start": 16.56, "duration": 4.879, "text": "are a th000 grams of the material right"}, {"start": 19.08, "duration": 4.08, "text": "now we want to know what's the amount"}, {"start": 21.439, "duration": 5.121, "text": "after 10"}, {"start": 23.16, "duration": 6.439, "text": "years so again when things change at a"}, {"start": 26.56, "duration": 5.24, "text": "rate proportional to the amount uh that"}, {"start": 29.599, "duration": 6.321, "text": "tells say that the the original the"}, {"start": 31.8, "duration": 7.48, "text": "amount at uh sometime T equals a this is"}, {"start": 35.92, "duration": 3.36, "text": "going to represent the initial"}, {"start": 40.84, "duration": 10.039, "text": "amount times e to the K T so in part A"}, {"start": 48.12, "duration": 5.079, "text": "here we're given that the initial amount"}, {"start": 50.879, "duration": 6.36, "text": "is"}, {"start": 53.199, "duration": 9.04, "text": "1,000 it's uh decaying at a rate of"}, {"start": 57.239, "duration": 7.92, "text": "0.2 uh K represents the K rate or the uh"}, {"start": 62.239, "duration": 5.081, "text": "growth rate we'll even call it"}, {"start": 65.159, "duration": 5.041, "text": "that sometimes called a growth rate"}, {"start": 67.32, "duration": 5.04, "text": "sometimes also called a Decay rate the"}, {"start": 70.2, "duration": 4.08, "text": "main thing is since it's decaying uh"}, {"start": 72.36, "duration": 6.56, "text": "we'll use"}, {"start": 74.28, "duration": 8.56, "text": "0.2 * T and uh that's that's going to"}, {"start": 78.92, "duration": 5.4, "text": "describe the amount at any time T so in"}, {"start": 82.84, "duration": 4.36, "text": "part A we want to know how much of the"}, {"start": 84.32, "duration": 4.799, "text": "material is there after 10 years well"}, {"start": 87.2, "duration": 3.36, "text": "all we have to do is just plug 10 into"}, {"start": 89.119, "duration": 6.081, "text": "this"}, {"start": 90.56, "duration": 6.76, "text": "so we have 1,00 * e to the"}, {"start": 95.2, "duration": 5.8, "text": "0.2 *"}, {"start": 97.32, "duration": 7.439, "text": "10 so that's going to give us 1,00 * e"}, {"start": 101.0, "duration": 8.04, "text": "to the ne uh I guess just"}, {"start": 104.759, "duration": 4.281, "text": "-2 and now we can evaluate"}, {"start": 109.84, "duration": 5.16, "text": "that so let's"}, {"start": 111.96, "duration": 5.56, "text": "see I'm getting e to the -2 to be"}, {"start": 115.0, "duration": 5.96, "text": "roughly"}, {"start": 117.52, "duration": 7.279, "text": "135 so if we multiply that by ,000 we'll"}, {"start": 120.96, "duration": 7.159, "text": "have well 135 so it says the amount uh"}, {"start": 124.799, "duration": 4.761, "text": "left after 10 years will be roughly 135"}, {"start": 128.119, "duration": 4.641, "text": "I don't know if we had grams at the"}, {"start": 129.56, "duration": 8.0, "text": "beginning yes it will be roughly 135"}, {"start": 132.76, "duration": 4.8, "text": "gram of this material left over"}], "title": "Radioactive Decay and Exponential Growth: Quick Example Involving Exponential Decay", "description": "Question: A radioactive substance decays at a yearly rate of 0.2 times the amount each moment.  If we start with 1000 grams, how much remains after 10 years?\nThis is a basic question related to exponential growth / decay.   Here we know how quickly the material decays and want to know how much remains after ten years.\nWe use the relative decay rate, the initial amount, and time to compute the solution.\n\n\n\n\n\n\n\n\n\n\n\n\n#math #algebra #khanacademy #khan #geometry #education #montessori #homeschooling #functions #notation #organicchemistry #organicchemistrytutor\n#logarithm #naturallog #exponential #decay #growth #exponentialdecayproblem", "lengthSeconds": 137, "width": 480, "height": 360}
{"video_id": "m3YJ4eAwmfc", "transcript": [{"start": 0.03, "duration": 4.019, "text": "all right in this video I want to do one"}, {"start": 2.46, "duration": 4.919, "text": "more example of finding the x and"}, {"start": 4.049, "duration": 5.55, "text": "y-intercepts of an equation so in the"}, {"start": 7.379, "duration": 4.291, "text": "other video I did the first part we"}, {"start": 9.599, "duration": 4.381, "text": "found the X and y intercepts of this"}, {"start": 11.67, "duration": 3.719, "text": "function now we're going to do the same"}, {"start": 13.98, "duration": 3.299, "text": "thing we'll just find the x and y"}, {"start": 15.389, "duration": 5.611, "text": "intercept of this function y equals x"}, {"start": 17.279, "duration": 7.291, "text": "squared minus 3x minus 18 again the idea"}, {"start": 21.0, "duration": 6.689, "text": "is to find the y-intercept we simply let"}, {"start": 24.57, "duration": 6.81, "text": "X equal 0 and we just do the arithmetic"}, {"start": 27.689, "duration": 9.241, "text": "so we would get y equals well 0 squared"}, {"start": 31.38, "duration": 8.22, "text": "minus 3 times 0 minus 18 which just"}, {"start": 36.93, "duration": 5.129, "text": "simplifies to negative 18 so it says the"}, {"start": 39.6, "duration": 5.58, "text": "y-intercept is at negative 18 or we can"}, {"start": 42.059, "duration": 6.901, "text": "say that 0 comma negative 18 is on the"}, {"start": 45.18, "duration": 7.17, "text": "graph ok that would be our y-intercept"}, {"start": 48.96, "duration": 7.4, "text": "now to find the x-intercept again or"}, {"start": 52.35, "duration": 4.01, "text": "intercepts I should say if there are any"}, {"start": 58.64, "duration": 6.99, "text": "well now we substitute in y equals zero"}, {"start": 62.09, "duration": 6.6, "text": "and we simply solve for X so we have"}, {"start": 65.63, "duration": 8.73, "text": "zero now on the left side equals x"}, {"start": 68.69, "duration": 6.9, "text": "squared minus 3x minus 18 so again now"}, {"start": 74.36, "duration": 3.619, "text": "basically I think well I've got a"}, {"start": 75.59, "duration": 7.47, "text": "quadratic equation I have to solve"}, {"start": 77.979, "duration": 6.521, "text": "let's see hopefully this factors so I"}, {"start": 83.06, "duration": 4.71, "text": "need two numbers that multiply to"}, {"start": 84.5, "duration": 9.39, "text": "negative 18 but add up to negative 3"}, {"start": 87.77, "duration": 7.86, "text": "I think negative 6 and positive 3 would"}, {"start": 93.89, "duration": 4.05, "text": "be two numbers that have the two numbers"}, {"start": 95.63, "duration": 3.989, "text": "that have that property so now we've got"}, {"start": 97.94, "duration": 5.61, "text": "a factored so it says we just said X"}, {"start": 99.619, "duration": 6.421, "text": "minus 6 equal to 0 X plus 3 equal to 0"}, {"start": 103.55, "duration": 6.81, "text": "if we solve the first equation we just"}, {"start": 106.04, "duration": 6.45, "text": "add 6x equals 6 is one of them cracked 3"}, {"start": 110.36, "duration": 4.59, "text": "and we'll get x equals negative 3 is our"}, {"start": 112.49, "duration": 5.37, "text": "other solution so now we found our two x"}, {"start": 114.95, "duration": 5.46, "text": "intercepts at the point 6 comma 0 and"}, {"start": 117.86, "duration": 6.08, "text": "there's a second x intercept at negative"}, {"start": 120.41, "duration": 3.53, "text": "3 comma 0"}], "title": "X-Intercepts and Y-Intercepts of a Functions and Finding Them! Example 2", "description": "Thanks to all of you who support me on Patreon. You da real mvps!   $1 per month helps!! :)  https://www.patreon.com/patrickjmt !! X-Intercepts and Y-Intercepts of a Functions and Finding Them! Example 2", "lengthSeconds": 123, "width": 480, "height": 360}
{"video_id": "zpKJQAQ4SyI", "transcript": [{"start": 0.079, "duration": 5.231, "text": "okay here we're going to find the"}, {"start": 1.979, "duration": 4.831, "text": "equations for the hyperbolas asymptotes"}, {"start": 5.31, "duration": 4.14, "text": "and we're going to write these in"}, {"start": 6.81, "duration": 5.31, "text": "point-slope form so we have the"}, {"start": 9.45, "duration": 6.78, "text": "hyperbola X plus 16 quantity squared"}, {"start": 12.12, "duration": 9.84, "text": "over 100 minus y the quantity Y minus 9"}, {"start": 16.23, "duration": 7.139, "text": "squared over 400 equals 1 so this is to"}, {"start": 21.96, "duration": 5.25, "text": "me just kind of writing things in a"}, {"start": 23.369, "duration": 7.021, "text": "proper form and just knowing some some"}, {"start": 27.21, "duration": 5.91, "text": "corresponding formulas so kind of the"}, {"start": 30.39, "duration": 5.04, "text": "formulas that we're going to use we're"}, {"start": 33.12, "duration": 6.599, "text": "going to write our at the hyperbola in"}, {"start": 35.43, "duration": 11.129, "text": "the form X minus H quantity squared over"}, {"start": 39.719, "duration": 10.59, "text": "a squared minus y minus K squared over B"}, {"start": 46.559, "duration": 6.541, "text": "squared equals 1 and the idea is once we"}, {"start": 50.309, "duration": 9.061, "text": "write it in this form this standard form"}, {"start": 53.1, "duration": 9.599, "text": "it says the asymptotes it says the"}, {"start": 59.37, "duration": 8.249, "text": "asymptotes are going to be of the form Y"}, {"start": 62.699, "duration": 10.651, "text": "minus K equals positive and negative B"}, {"start": 67.619, "duration": 7.471, "text": "over a times X minus H okay so again"}, {"start": 73.35, "duration": 4.32, "text": "what I need to do is I need to figure"}, {"start": 75.09, "duration": 5.25, "text": "out a value for K I need to figure out a"}, {"start": 77.67, "duration": 5.43, "text": "value for B I need to figure out a value"}, {"start": 80.34, "duration": 7.889, "text": "for a and I need to figure out a value"}, {"start": 83.1, "duration": 7.92, "text": "for H all right well let's do that okay"}, {"start": 88.229, "duration": 5.03, "text": "so I can rewrite this so it says we're"}, {"start": 91.02, "duration": 5.43, "text": "going to write it as X minus something"}, {"start": 93.259, "duration": 5.951, "text": "so X minus well we want it to be"}, {"start": 96.45, "duration": 5.4, "text": "equivalent to X plus 16 so we can write"}, {"start": 99.21, "duration": 5.519, "text": "it as X minus negative 16"}, {"start": 101.85, "duration": 6.57, "text": "well squared I want to write the"}, {"start": 104.729, "duration": 7.051, "text": "denominator as some number squared well"}, {"start": 108.42, "duration": 6.33, "text": "I want that number to equal 100 well"}, {"start": 111.78, "duration": 5.549, "text": "what number squared is going to be 100"}, {"start": 114.75, "duration": 4.799, "text": "and you could say well hey could be"}, {"start": 117.329, "duration": 4.65, "text": "negative 10 but we always let our a and"}, {"start": 119.549, "duration": 6.151, "text": "B values be positive so we're going to"}, {"start": 121.979, "duration": 6.48, "text": "use positive 10 squared and then we've"}, {"start": 125.7, "duration": 6.41, "text": "got well we've got Y minus 9 we can"}, {"start": 128.459, "duration": 5.961, "text": "leave that part alone this is all over 4"}, {"start": 132.11, "duration": 5.49, "text": "100 again I want to write that as a"}, {"start": 134.42, "duration": 6.66, "text": "number squared a positive number being"}, {"start": 137.6, "duration": 6.39, "text": "squared well we can write 400 as 20"}, {"start": 141.08, "duration": 5.34, "text": "squared so now I've got all my values"}, {"start": 143.99, "duration": 4.5, "text": "okay it says this is going to be our"}, {"start": 146.42, "duration": 6.39, "text": "value for H H is going to equal negative"}, {"start": 148.49, "duration": 10.38, "text": "16 it says 10 is going to be our a value"}, {"start": 152.81, "duration": 9.09, "text": "it says positive 9 again we kind of"}, {"start": 158.87, "duration": 9.12, "text": "forget about the the negative so we"}, {"start": 161.9, "duration": 9.24, "text": "could write this as Y minus 9 okay so"}, {"start": 167.99, "duration": 5.85, "text": "that's going to be our K value and then"}, {"start": 171.14, "duration": 4.319, "text": "20 is going to be our B value so I'm"}, {"start": 173.84, "duration": 4.95, "text": "just going to fill that in and we'll"}, {"start": 175.459, "duration": 6.951, "text": "have our asymptotes so it says our"}, {"start": 178.79, "duration": 7.62, "text": "asymptotes will be Y minus K which is 9"}, {"start": 182.41, "duration": 8.38, "text": "equals positive or negative B over a so"}, {"start": 186.41, "duration": 9.18, "text": "again we said B had value 20 a was equal"}, {"start": 190.79, "duration": 9.479, "text": "to 10 then we do X minus the H value so"}, {"start": 195.59, "duration": 6.09, "text": "the H value is negative 16 and of course"}, {"start": 200.269, "duration": 4.291, "text": "we could simplify this down just a"}, {"start": 201.68, "duration": 5.58, "text": "little bit but now we've done everything"}, {"start": 204.56, "duration": 5.459, "text": "so it's just kind of again just writing"}, {"start": 207.26, "duration": 6.21, "text": "things in the proper form and just"}, {"start": 210.019, "duration": 8.241, "text": "knowing formulas so 20 over 10 is going"}, {"start": 213.47, "duration": 10.71, "text": "to be 2x minus negative 16 will be X"}, {"start": 218.26, "duration": 8.46, "text": "plus 16 and now we have we have our"}, {"start": 224.18, "duration": 2.54, "text": "asymptotes"}], "title": "Conic Sections, Hyperbola , Shifted : Find Asymptotes", "description": "Thanks to all of you who support me on Patreon. You da real mvps!   $1 per month helps!! :)  https://www.patreon.com/patrickjmt !! Conic Sections, Hyperbola , Shifted : Find Asymptotes.  Here we find the point-slope form of the asymptotes of a hyperbola.", "lengthSeconds": 229, "width": 480, "height": 360}
{"video_id": "y3lB4MrXMQI", "transcript": [{"start": 0.03, "duration": 4.729, "text": "all right here we're gonna look at some"}, {"start": 1.86, "duration": 5.58, "text": "examples of rewriting fractions as"}, {"start": 4.759, "duration": 7.811, "text": "decimals and if we have to we'll round"}, {"start": 7.44, "duration": 7.23, "text": "off all right basically to write a"}, {"start": 12.57, "duration": 8.459, "text": "fraction as a decimal you just do long"}, {"start": 14.67, "duration": 10.83, "text": "division that's kind of the whole the"}, {"start": 21.029, "duration": 8.281, "text": "whole trick okay so we've got seven over"}, {"start": 25.5, "duration": 6.42, "text": "eight seven divided by eight so again we"}, {"start": 29.31, "duration": 6.06, "text": "can write that as well"}, {"start": 31.92, "duration": 5.069, "text": "7 divided by 8 and I'm gonna go ahead"}, {"start": 35.37, "duration": 4.709, "text": "and write seven I'm gonna add in some"}, {"start": 36.989, "duration": 6.901, "text": "some decimal places so I'm gonna write"}, {"start": 40.079, "duration": 7.081, "text": "this as seven point zero zero zero tens"}, {"start": 43.89, "duration": 7.079, "text": "hundreds thousands maybe one more place"}, {"start": 47.16, "duration": 6.27, "text": "in case we have to round okay so usually"}, {"start": 50.969, "duration": 4.651, "text": "if somebody doesn't say round to a"}, {"start": 53.43, "duration": 3.93, "text": "certain place I'll just put some number"}, {"start": 55.62, "duration": 5.22, "text": "of zeros down and if I need more I'll"}, {"start": 57.36, "duration": 5.76, "text": "add more so okay so eight will go into"}, {"start": 60.84, "duration": 5.13, "text": "seven zero times"}, {"start": 63.12, "duration": 6.6, "text": "so zero times eight zero again we can"}, {"start": 65.97, "duration": 7.469, "text": "just subtract 7 - 0 7 and we'll drop"}, {"start": 69.72, "duration": 5.82, "text": "down our zero so now I think eight will"}, {"start": 73.439, "duration": 5.161, "text": "go into seventy how many you know how"}, {"start": 75.54, "duration": 6.329, "text": "many times without going over let's see"}, {"start": 78.6, "duration": 6.93, "text": "I guess 8 times 8 is 64 I think that's"}, {"start": 81.869, "duration": 7.651, "text": "gonna work and again we just subtract 70"}, {"start": 85.53, "duration": 7.979, "text": "minus 64 will leave us with six we can"}, {"start": 89.52, "duration": 6.72, "text": "drop down our zero so now let's see"}, {"start": 93.509, "duration": 8.511, "text": "eight will go into 60 I guess"}, {"start": 96.24, "duration": 9.51, "text": "seven times seven times eight will be 56"}, {"start": 102.02, "duration": 7.12, "text": "again we can subtract 60 minus 56 will"}, {"start": 105.75, "duration": 7.56, "text": "leave us with four drop down our next"}, {"start": 109.14, "duration": 7.799, "text": "zero well eight will go into 40 exactly"}, {"start": 113.31, "duration": 6.3, "text": "five times and now five times eight"}, {"start": 116.939, "duration": 4.261, "text": "would be 40 we could subtract we would"}, {"start": 119.61, "duration": 4.859, "text": "get zero but then we would just simply"}, {"start": 121.2, "duration": 4.919, "text": "drop down you know our neck Siro and we"}, {"start": 124.469, "duration": 3.66, "text": "would say well eight goes into 0 zero"}, {"start": 126.119, "duration": 3.721, "text": "times and we would just now start"}, {"start": 128.129, "duration": 2.721, "text": "getting zeros forever and ever and ever"}, {"start": 129.84, "duration": 3.77, "text": "or"}, {"start": 130.85, "duration": 4.74, "text": "since there's no remainder that tells us"}, {"start": 133.61, "duration": 4.65, "text": "that we have a fraction that we can"}, {"start": 135.59, "duration": 5.04, "text": "write using a decimal representation"}, {"start": 138.26, "duration": 3.93, "text": "where the decimal terminates so in this"}, {"start": 140.63, "duration": 4.98, "text": "case again we just keep the decimal"}, {"start": 142.19, "duration": 8.81, "text": "place exactly lined up so it says we can"}, {"start": 145.61, "duration": 5.39, "text": "write seven over eight as the decimal"}, {"start": 151.18, "duration": 3.0, "text": "0.875"}], "title": "Writing a Fraction as a Decimal, Example 1", "description": "Thanks to all of you who support me on Patreon. You da real mvps!   $1 per month helps!! :)  https://www.patreon.com/patrickjmt !! Writing a Fraction as a Decimal, Example 1", "lengthSeconds": 156, "width": 480, "height": 360}
{"video_id": "0y3BhOOB4JY", "transcript": [{"start": 0.64, "duration": 3.84, "text": "all right here we're going to look at uh"}, {"start": 2.639, "duration": 3.24, "text": "some mixed numbers and what we're going"}, {"start": 4.48, "duration": 6.0, "text": "to do is we're going to write the number"}, {"start": 5.879, "duration": 6.8, "text": "31 / 4 as a mixed number so to do this"}, {"start": 10.48, "duration": 6.879, "text": "um I'm just going to write it as long"}, {"start": 12.679, "duration": 6.881, "text": "division we've got 31 / 4 and to do this"}, {"start": 17.359, "duration": 4.92, "text": "I think well okay so let's see four"}, {"start": 19.56, "duration": 7.2, "text": "certainly won't go into three uh four"}, {"start": 22.279, "duration": 6.32, "text": "will go into 31 I guess seven times so 7"}, {"start": 26.76, "duration": 6.04, "text": "* 4 is"}, {"start": 28.599, "duration": 7.401, "text": "28 and then we subtract uh 31 - 28 is"}, {"start": 32.8, "duration": 5.439, "text": "going to be 3 so this is our"}, {"start": 36.0, "duration": 6.0, "text": "remainder all we have to do is we take"}, {"start": 38.239, "duration": 6.281, "text": "whatever numbers on top which is"}, {"start": 42.0, "duration": 5.36, "text": "seven and then we take our remainder"}, {"start": 44.52, "duration": 6.519, "text": "which is three and we divide it by the"}, {"start": 47.36, "duration": 8.12, "text": "same original divisor so it says 31"}, {"start": 51.039, "duration": 7.081, "text": "over4 says we can write that as 7 and"}, {"start": 55.48, "duration": 5.399, "text": "34s so that's the way to write it as a"}, {"start": 58.12, "duration": 4.119, "text": "mixed number"}, {"start": 60.879, "duration": 3.841, "text": "I'm going to go back sort of the"}, {"start": 62.239, "duration": 4.88, "text": "opposite direction suppose we started"}, {"start": 64.72, "duration": 4.92, "text": "with the uh you know the right side and"}, {"start": 67.119, "duration": 4.36, "text": "we wanted to write it just as a a"}, {"start": 69.64, "duration": 3.08, "text": "fraction a single fraction with no whole"}, {"start": 71.479, "duration": 4.081, "text": "number out front that's what I'm going"}, {"start": 72.72, "duration": 5.64, "text": "to do next so I'm going to convert 6 and"}, {"start": 75.56, "duration": 6.4, "text": "23 to what we call an improper"}, {"start": 78.36, "duration": 6.32, "text": "fraction the way that we do"}, {"start": 81.96, "duration": 5.04, "text": "this is pretty easy what we do is we"}, {"start": 84.68, "duration": 5.759, "text": "take the bottom number and we multiply"}, {"start": 87.0, "duration": 5.439, "text": "so we say 6 * 3 and then we add it to"}, {"start": 90.439, "duration": 6.04, "text": "the top number so we're going to"}, {"start": 92.439, "duration": 8.881, "text": "multiply and then add okay so let's see"}, {"start": 96.479, "duration": 8.121, "text": "6 * 3 plus two is what we're doing so"}, {"start": 101.32, "duration": 5.92, "text": "that's going to give us 18 + 2 which is"}, {"start": 104.6, "duration": 4.839, "text": "20 and what we do is when we rewrite"}, {"start": 107.24, "duration": 4.0, "text": "this number this number that we have"}, {"start": 109.439, "duration": 5.121, "text": "here is what goes in the numerator of"}, {"start": 111.24, "duration": 5.559, "text": "the fraction and again we keep the same"}, {"start": 114.56, "duration": 6.76, "text": "divisor okay so when I look at this I"}, {"start": 116.799, "duration": 7.92, "text": "say well 3 * 6 is 18 + 2 is 20 I get 20"}, {"start": 121.32, "duration": 7.6, "text": "over3 and now we have our mixed number"}, {"start": 124.719, "duration": 4.201, "text": "written as an improper fraction"}], "title": "Improper Fractions and Mixed Numbers : Switching Between Them", "description": "Thanks to all of you who support me on Patreon. You da real mvps!   $1 per month helps!! :)  https://www.patreon.com/patrickjmt !! Improper Fractions and Mixed Numbers : Switching Between Them.  Here we take an improper fraction and write it as a mixed number and then take a mixed number and write it as an improper fraction.", "lengthSeconds": 127, "width": 480, "height": 360}
{"video_id": "fmeIBQZTBg0", "transcript": [{"start": 0.03, "duration": 4.41, "text": "okay so one more example here of"}, {"start": 2.19, "duration": 5.22, "text": "integrating a trig function so here"}, {"start": 4.44, "duration": 6.77, "text": "we're gonna integrate cosine x times"}, {"start": 7.41, "duration": 6.42, "text": "sine of cosecant X all over sine squared"}, {"start": 11.21, "duration": 4.57, "text": "so you know kind of when I look at a"}, {"start": 13.83, "duration": 3.869, "text": "problem like this again the only thing"}, {"start": 15.78, "duration": 3.839, "text": "you know the only way I can integrate"}, {"start": 17.699, "duration": 3.481, "text": "sort of sine is just to get it to be"}, {"start": 19.619, "duration": 3.871, "text": "sine of a single variable"}, {"start": 21.18, "duration": 3.839, "text": "you know the sine of cosecant X I'm not"}, {"start": 23.49, "duration": 3.93, "text": "really sure what to do with it"}, {"start": 25.019, "duration": 3.871, "text": "so in a sense I think well the only way"}, {"start": 27.42, "duration": 4.17, "text": "to get rid of it would be to use au"}, {"start": 28.89, "duration": 4.919, "text": "substitution so I'm gonna let u equal"}, {"start": 31.59, "duration": 4.86, "text": "cosecant X and we'll see what happens"}, {"start": 33.809, "duration": 5.761, "text": "here so recall the derivative of"}, {"start": 36.45, "duration": 7.17, "text": "cosecant X is negative cosecant x times"}, {"start": 39.57, "duration": 6.509, "text": "cotangent X well I don't see that in the"}, {"start": 43.62, "duration": 5.24, "text": "problem immediately but recall you know"}, {"start": 46.079, "duration": 6.931, "text": "cosecant again that's just 1 over sine X"}, {"start": 48.86, "duration": 7.719, "text": "cotangent X that's cosine X over sine X"}, {"start": 53.01, "duration": 6.45, "text": "and well I guess let's not forget our DX"}, {"start": 56.579, "duration": 7.341, "text": "on here if you multiply this will get"}, {"start": 59.46, "duration": 7.229, "text": "negative cosine X over sine squared X DX"}, {"start": 63.92, "duration": 6.67, "text": "okay that is in the problem we have"}, {"start": 66.689, "duration": 5.551, "text": "cosine X over sine squared x times DX we"}, {"start": 70.59, "duration": 3.54, "text": "don't want the negative so we can just"}, {"start": 72.24, "duration": 3.78, "text": "multiply both sides by negative one and"}, {"start": 74.13, "duration": 3.51, "text": "now it looks like to me are you"}, {"start": 76.02, "duration": 2.4, "text": "substitutions gonna work out rather"}, {"start": 77.64, "duration": 5.089, "text": "nicely"}, {"start": 78.42, "duration": 7.62, "text": "so we would have the integral of sine of"}, {"start": 82.729, "duration": 5.68, "text": "U so we replacing the cosecant X with"}, {"start": 86.04, "duration": 5.759, "text": "you and again the cosine x over sine"}, {"start": 88.409, "duration": 6.691, "text": "squared DX all of that's just gonna"}, {"start": 91.799, "duration": 5.64, "text": "equal negative 1 D U and now we can"}, {"start": 95.1, "duration": 5.4, "text": "integrate so the antiderivative of sine"}, {"start": 97.439, "duration": 4.89, "text": "is negative cosine but what the extra"}, {"start": 100.5, "duration": 5.88, "text": "negative will just get positive cosine"}, {"start": 102.329, "duration": 6.631, "text": "of U plus C and now all we have to do is"}, {"start": 106.38, "duration": 6.12, "text": "just sort of free substitute so again u"}, {"start": 108.96, "duration": 5.58, "text": "was equal to cosecant X so it says our"}, {"start": 112.5, "duration": 6.2, "text": "antiderivative will be cosine of"}, {"start": 114.54, "duration": 4.16, "text": "cosecant X plus C"}], "title": "Integrating Trig Functions , More Examples #3", "description": "Thanks to all of you who support me on Patreon. You da real mvps!   $1 per month helps!! :)  https://www.patreon.com/patrickjmt !! Integrating Trig Functions , More Examples #3.  Here we integrate a trig function using a u-substitution.", "lengthSeconds": 118, "width": 480, "height": 360}
{"video_id": "0piu2uP4zr4", "transcript": [{"start": 0.52, "duration": 2.919, "text": "all right in this video we're going to"}, {"start": 1.719, "duration": 4.6, "text": "look at determining the limit of a few"}, {"start": 3.439, "duration": 5.001, "text": "different sequences so in part A we have"}, {"start": 6.319, "duration": 5.28, "text": "uh the sequence associated with one over"}, {"start": 8.44, "duration": 5.72, "text": "n cubed so what we'll do is again we"}, {"start": 11.599, "duration": 6.481, "text": "just take the limit as n goes to"}, {"start": 14.16, "duration": 6.039, "text": "Infinity of uh one over n cubed well as"}, {"start": 18.08, "duration": 3.4, "text": "n gets really really really big the"}, {"start": 20.199, "duration": 3.16, "text": "denominator is going to get really"}, {"start": 21.48, "duration": 4.36, "text": "really really"}, {"start": 23.359, "duration": 5.24, "text": "big we're going to have one over a very"}, {"start": 25.84, "duration": 4.599, "text": "very big number so one over a large"}, {"start": 28.599, "duration": 4.441, "text": "number gets arbitrarily close close to"}, {"start": 30.439, "duration": 5.44, "text": "zero so the limit of the following uh"}, {"start": 33.04, "duration": 5.4, "text": "would just simply equal"}, {"start": 35.879, "duration": 5.0, "text": "zero um Part"}, {"start": 38.44, "duration": 5.24, "text": "B same thing we'll just take the limit"}, {"start": 40.879, "duration": 2.801, "text": "as n goes to"}, {"start": 43.76, "duration": 6.24, "text": "Infinity so recall the shortcut um if"}, {"start": 47.84, "duration": 4.039, "text": "the degree of the numerator is equal to"}, {"start": 50.0, "duration": 3.559, "text": "the degree of the denominator we just do"}, {"start": 51.879, "duration": 3.68, "text": "the ratio of the"}, {"start": 53.559, "duration": 4.16, "text": "coefficients um so I can just simply"}, {"start": 55.559, "duration": 4.921, "text": "look at this and say well this is 1 n^ 2"}, {"start": 57.719, "duration": 5.16, "text": "and 3 N2 the degree of the top equals"}, {"start": 60.48, "duration": 4.12, "text": "the degree of the bottom we'll get 1/3"}, {"start": 62.879, "duration": 4.081, "text": "it's a very useful little shortcut to"}, {"start": 64.6, "duration": 4.28, "text": "remember when you do sequences in series"}, {"start": 66.96, "duration": 3.56, "text": "but again kind of to show it with a"}, {"start": 68.88, "duration": 6.12, "text": "little bit more work we can divide"}, {"start": 70.52, "duration": 8.16, "text": "everything by n^ 2 so n^2 over n^2 is 1"}, {"start": 75.0, "duration": 9.24, "text": "2 N over n 2 will be 2 over n then we'll"}, {"start": 78.68, "duration": 8.799, "text": "have 4 over n 2 3 n^ 2 over n^2 is 3"}, {"start": 84.24, "duration": 5.879, "text": "then 1 over n^2 and again the idea is"}, {"start": 87.479, "duration": 5.68, "text": "now it's more clear because as in goes"}, {"start": 90.119, "duration": 6.401, "text": "to Infinity this term will go to zero"}, {"start": 93.159, "duration": 5.6, "text": "the 4 over n^2 will go to zero the 1"}, {"start": 96.52, "duration": 6.2, "text": "over n^2 will get arbitrarily close to"}, {"start": 98.759, "duration": 8.96, "text": "zero so again we're just left with"}, {"start": 102.72, "duration": 7.759, "text": "1/3 as the limit uh for the second"}, {"start": 107.719, "duration": 6.201, "text": "sequence last but not least here we have"}, {"start": 110.479, "duration": 8.24, "text": "n the 3 * e the"}, {"start": 113.92, "duration": 4.799, "text": "N if we do a limit as n goes to"}, {"start": 118.84, "duration": 5.08, "text": "Infinity"}, {"start": 121.079, "duration": 5.841, "text": "notice we'll get uh you know as n goes"}, {"start": 123.92, "duration": 6.0, "text": "to Infinity n the 3 will go to Infinity"}, {"start": 126.92, "duration": 5.28, "text": "but we'll get e to the negative infinity"}, {"start": 129.92, "duration": 4.64, "text": "and if you take e and raise it to a uh"}, {"start": 132.2, "duration": 5.24, "text": "quote unquote big negative number that"}, {"start": 134.56, "duration": 5.759, "text": "gets close to zero so this is going to"}, {"start": 137.44, "duration": 5.92, "text": "be an indeterminant product so you may"}, {"start": 140.319, "duration": 5.321, "text": "have to use lal's rule on these"}, {"start": 143.36, "duration": 4.56, "text": "problems I'm going to leave the N cubed"}, {"start": 145.64, "duration": 4.84, "text": "in the numerator I'll put the e to the"}, {"start": 147.92, "duration": 4.24, "text": "negative n in the denominator but then"}, {"start": 150.48, "duration": 5.399, "text": "that would uh be written as just e to"}, {"start": 152.16, "duration": 7.439, "text": "the N so now if we use lal's"}, {"start": 155.879, "duration": 5.881, "text": "rule well we'll get 3 n^2 over e to the"}, {"start": 159.599, "duration": 4.56, "text": "N so the derivative of the top and the"}, {"start": 161.76, "duration": 5.16, "text": "derivative of the bottom this is still"}, {"start": 164.159, "duration": 6.041, "text": "Infinity over infinity an indeterminate"}, {"start": 166.92, "duration": 6.88, "text": "form we can use lal's rule"}, {"start": 170.2, "duration": 6.399, "text": "again we'll get 6n over e to the N again"}, {"start": 173.8, "duration": 2.799, "text": "as n goes to"}, {"start": 176.64, "duration": 3.959, "text": "Infinity both top and bottom will go to"}, {"start": 179.2, "duration": 4.36, "text": "Infinity"}, {"start": 180.599, "duration": 5.401, "text": "so we can use lal's rule one more"}, {"start": 183.56, "duration": 5.0, "text": "time then we'll be left with six in the"}, {"start": 186.0, "duration": 4.799, "text": "numerator we'll still have e to the n in"}, {"start": 188.56, "duration": 5.08, "text": "the denominator but now this is going to"}, {"start": 190.799, "duration": 4.481, "text": "be six over a really really big number"}, {"start": 193.64, "duration": 4.36, "text": "which is going to get arbitrarily close"}, {"start": 195.28, "duration": 6.959, "text": "to zero so we would say this last"}, {"start": 198.0, "duration": 7.239, "text": "sequence uh converges to the number"}, {"start": 202.239, "duration": 3.0, "text": "zero"}], "title": "\u2756 Finding the Limit of a Sequence, 3 more examples \u2756", "description": "Finding the Limit of a Sequence, 3 more examples, #1.  Just another example of finding the limit of a sequence by taking a limit as n approaches infinity.\nOne often will encounter sequences in a Calculus course prior to dealing with Series.  Make sure you can find limits of sequences as it will be an important skill to have when working with series.\n\n\n\n\n\n\n\n\n#maths #sequence #series #limits #converge #convergent #diverge #diverges #divergent #pseries #exponential #rationalfunctions", "lengthSeconds": 203, "width": 480, "height": 360}
{"video_id": "WHnRyzXXT1U", "transcript": [{"start": 1.319, "duration": 3.601, "text": "okay in this video we're going to look"}, {"start": 2.6, "duration": 5.84, "text": "at an example related to the slope of a"}, {"start": 4.92, "duration": 7.4, "text": "tangent line and we were given the"}, {"start": 8.44, "duration": 5.68, "text": "function um f ofx = x Cub - 2x and"}, {"start": 12.32, "duration": 4.959, "text": "already given that the derivative is"}, {"start": 14.12, "duration": 5.04, "text": "3x^2 - 2 we want to answer a couple"}, {"start": 17.279, "duration": 5.721, "text": "questions what's the slope of the"}, {"start": 19.16, "duration": 5.439, "text": "tangent line um at the point 2 comma 4"}, {"start": 23.0, "duration": 4.08, "text": "and also what would actually be the"}, {"start": 24.599, "duration": 6.121, "text": "equation of that tangent"}, {"start": 27.08, "duration": 7.84, "text": "line well again to get the slope um at"}, {"start": 30.72, "duration": 7.48, "text": "the point uh in this case uh at at xal 2"}, {"start": 34.92, "duration": 5.119, "text": "and yal 4 our derivative you know the"}, {"start": 38.2, "duration": 4.0, "text": "derivative formula in a sense it says"}, {"start": 40.039, "duration": 4.081, "text": "all you need to know uh to figure out"}, {"start": 42.2, "duration": 4.72, "text": "the slope of the tangent line is the x"}, {"start": 44.12, "duration": 5.56, "text": "coordinate so in this case the slope is"}, {"start": 46.92, "duration": 5.36, "text": "going to be F Prime of 2 we're plugging"}, {"start": 49.68, "duration": 4.0, "text": "in the x coordinate of 2 into our"}, {"start": 52.28, "duration": 3.2, "text": "derivative formula again that's what the"}, {"start": 53.68, "duration": 4.879, "text": "derivative will do it tells you the"}, {"start": 55.48, "duration": 4.8, "text": "instantaneous rate of change or um in"}, {"start": 58.559, "duration": 4.441, "text": "terms of a graph it tells you the slope"}, {"start": 60.28, "duration": 6.479, "text": "of the tangent line so we'll just get 3"}, {"start": 63.0, "duration": 7.119, "text": "* 2^ 2 -"}, {"start": 66.759, "duration": 6.601, "text": "2 well 2^ 2 is"}, {"start": 70.119, "duration": 7.241, "text": "4 3 * 4 is going to be"}, {"start": 73.36, "duration": 6.04, "text": "12 12 - 2 is 10 and again that's going"}, {"start": 77.36, "duration": 4.84, "text": "to be the slope of the tangent line at"}, {"start": 79.4, "duration": 4.719, "text": "that point well to get the equation of"}, {"start": 82.2, "duration": 6.68, "text": "the line remember uh we can just use"}, {"start": 84.119, "duration": 9.601, "text": "point slope formula here so y - y1 equal"}, {"start": 88.88, "duration": 7.0, "text": "the slope x - X1 recall this a point"}, {"start": 93.72, "duration": 4.56, "text": "slope formula for a"}, {"start": 95.88, "duration": 5.199, "text": "line well now all we have to do is just"}, {"start": 98.28, "duration": 5.199, "text": "fill in our information um the point was"}, {"start": 101.079, "duration": 5.561, "text": "given to us again we're given the point"}, {"start": 103.479, "duration": 5.801, "text": "2 comma 4 so if we fill that in we'll"}, {"start": 106.64, "duration": 3.96, "text": "just take Yus the y coordinate which is"}, {"start": 109.28, "duration": 3.92, "text": "well"}, {"start": 110.6, "duration": 4.159, "text": "four equals the slope of the tangent"}, {"start": 113.2, "duration": 4.44, "text": "line that's what we kind of had to work"}, {"start": 114.759, "duration": 6.121, "text": "to figure out and then we'll take x"}, {"start": 117.64, "duration": 6.04, "text": "minus the x coordinate which is a"}, {"start": 120.88, "duration": 4.599, "text": "positive2 and we've now got the equation"}, {"start": 123.68, "duration": 4.88, "text": "of our line if you want to we can always"}, {"start": 125.479, "duration": 6.321, "text": "put this in slope intercept form so if"}, {"start": 128.56, "duration": 6.12, "text": "we distribute the 10 we'll get 10 x 10"}, {"start": 131.8, "duration": 6.799, "text": "and -2 would be"}, {"start": 134.68, "duration": 7.639, "text": "-20 we can add four to both sides and"}, {"start": 138.599, "duration": 5.481, "text": "then we'll simply be left with y = 10 x"}, {"start": 142.319, "duration": 5.161, "text": "-"}, {"start": 144.08, "duration": 5.159, "text": "16 um as the equation of our tangent"}, {"start": 147.48, "duration": 2.92, "text": "line so again both of these are"}, {"start": 149.239, "duration": 4.08, "text": "equivalent"}, {"start": 150.4, "duration": 5.4, "text": "um sometimes again people uh like to see"}, {"start": 153.319, "duration": 5.241, "text": "things in slope intercept form so again"}, {"start": 155.8, "duration": 5.359, "text": "that's it okay so just need to remember"}, {"start": 158.56, "duration": 4.319, "text": "the equation of a line um point slope"}, {"start": 161.159, "duration": 3.121, "text": "formula again you're given the point for"}, {"start": 162.879, "duration": 3.72, "text": "free the only thing you really have to"}, {"start": 164.28, "duration": 4.599, "text": "figure out is the slope but to get the"}, {"start": 166.599, "duration": 3.841, "text": "slope you just plug in the uh you know"}, {"start": 168.879, "duration": 3.321, "text": "plug the point into your derivative"}, {"start": 170.44, "duration": 5.28, "text": "formula and that gives you the slope of"}, {"start": 172.2, "duration": 3.52, "text": "the tangent line"}], "title": "Finding the Equation of a Tangent Line Using a Derivative", "description": "Thanks to all of you who support me on Patreon. You da real mvps!   $1 per month helps!! :)  https://www.patreon.com/patrickjmt !! Finding the Equation of a Tangent Line Using a Derivative.  In this video, we find the slope and the equation of a tangent line simply by using the derivative!", "lengthSeconds": 174, "width": 480, "height": 360}
{"video_id": "tQBN11d1ROQ", "transcript": [{"start": 0.77, "duration": 2.98, "text": "alright in this video I want to talk about\nfinding limits graphically"}, {"start": 3.75, "duration": 3.62, "text": "and so here we've got our function f(x) we've got a little"}, {"start": 7.37, "duration": 4.7, "text": "removable discontinuity at -4 we have a\njump discontinuity here"}, {"start": 12.07, "duration": 5.58, "text": "at 5 so part A we just wanna ask yourself\ndoes the limit as X approaches -4"}, {"start": 17.65, "duration": 4.629, "text": "exist for this function: well again as X\napproaches -4"}, {"start": 22.279, "duration": 3.57, "text": "I'm thinking about what's happening to\nthe y values do they approach"}, {"start": 25.849, "duration": 4.69, "text": "some specific single y value so\nnotice that we take"}, {"start": 30.539, "duration": 3.081, "text": "X values  a little bit smaller than -4"}, {"start": 33.62, "duration": 3.63, "text": "I think about on the graph what are the\ny values getting closer to"}, {"start": 37.25, "duration": 4.44, "text": "well the y values are getting closer and\ncloser to the"}, {"start": 41.69, "duration": 4.88, "text": "the value positive 2 likewise as I come\nfrom the right"}, {"start": 46.57, "duration": 3.27, "text": "ok take numbers a little bit bigger than\n-4"}, {"start": 49.84, "duration": 3.379, "text": "if I think about what's happening to\nthe Y values on the graph"}, {"start": 53.219, "duration": 3.241, "text": "again those are getting closer and\ncloser to 2"}, {"start": 56.46, "duration": 3.18, "text": "so I would say yes and we can write"}, {"start": 59.64, "duration": 3.829, "text": "the limit as X approaches -4"}, {"start": 63.469, "duration": 3.141, "text": "of our function we would say that that\nequals"}, {"start": 66.61, "duration": 4.189, "text": "positive 2 again the y values are\ngetting closer and closer to 2"}, {"start": 70.799, "duration": 3.031, "text": "so again notice the actual the y\nvalue"}, {"start": 73.83, "duration": 3.01, "text": "at -4 is something totally different"}, {"start": 76.84, "duration": 3.069, "text": "but again it doesn't matter what\nactually happens at that point"}, {"start": 79.909, "duration": 3.931, "text": "we just think as you get close to -4 on\nthe left"}, {"start": 83.84, "duration": 3.889, "text": "and on the right are the y values\nthey're getting close to some"}, {"start": 87.729, "duration": 3.68, "text": "specific number yes and in this case\nthat answer"}, {"start": 91.409, "duration": 3.041, "text": "that number is 2. Same question"}, {"start": 94.45, "duration": 3.75, "text": "does the limit as X approaches 5 exist\nfor f(x)?"}, {"start": 98.2, "duration": 5.77, "text": "well if I come in from the left okay so\nhere the y values are getting closer and"}, {"start": 103.97, "duration": 1.27, "text": "closer to some"}, {"start": 105.24, "duration": 3.4, "text": "you know whatever it is let's pretend\nit's up here at +4 just to give"}, {"start": 108.64, "duration": 1.33, "text": "it a specific value"}, {"start": 109.97, "duration": 4.52, "text": "so here as we approach from the left the\ny values are getting closer and closer"}, {"start": 114.49, "duration": 1.01, "text": "to 4"}, {"start": 115.5, "duration": 3.6, "text": "but then when we come from the right\nwell"}, {"start": 119.1, "duration": 3.119, "text": "if I look at x values a little bit\nbigger than 5"}, {"start": 122.219, "duration": 3.661, "text": "well now the y values are getting closer\nand closer to something"}, {"start": 125.88, "duration": 3.219, "text": "different let's maybe pretend that\nthat's +1"}, {"start": 129.099, "duration": 3.401, "text": "well since when you come from the left"}, {"start": 132.5, "duration": 5.29, "text": "the y values on one side get close to\n4 and as when you come from the right"}, {"start": 137.79, "duration": 3.73, "text": "the y values get closer and closer to\nsomething different"}, {"start": 141.52, "duration": 4.51, "text": "in this case +1 if the limit\nas you approach from the left and from"}, {"start": 146.03, "duration": 1.459, "text": "the right is not the same"}, {"start": 147.489, "duration": 4.431, "text": "we say no that limit does not exist in\nkind of another way I always thought"}, {"start": 151.92, "duration": 4.399, "text": "for a limit to exist for example at -4\nnotice if I can't put my fingers on the"}, {"start": 156.319, "duration": 3.42, "text": "graph and bring them closer and closer\nand closer to -4"}, {"start": 159.739, "duration": 3.341, "text": "they're gonna basically touch ok\nthey're gonna get closer and closer and"}, {"start": 163.08, "duration": 0.57, "text": "closer"}, {"start": 163.65, "duration": 3.729, "text": "and that y value where they would sort\nof be touching would be the y value"}, {"start": 167.379, "duration": 0.541, "text": "of 2"}, {"start": 167.92, "duration": 3.7, "text": "and a limit exists if I do the same\nthing at x = 5 if"}, {"start": 171.62, "duration": 3.27, "text": "I put my finger you know on the graph\nand try to you know"}, {"start": 174.89, "duration": 4.28, "text": "move them closer you know take x values\ncloser and closer to 5 I think well"}, {"start": 179.17, "duration": 3.539, "text": "are my fingers gonna be touching are they\ngetting closer and closer together?"}, {"start": 182.709, "duration": 4.81, "text": "no cuz one's up here one's down here if\nthat happens we say the limit simply"}, {"start": 187.519, "duration": 0.5, "text": "does not exist"}], "title": "Finding Limits from a Graph", "description": "Thanks to all of you who support me on Patreon. You da real mvps!   $1 per month helps!! :)  https://www.patreon.com/patrickjmt !! Finding Limits from a Graph.  In this video, I show how to find limits using a graph.", "lengthSeconds": 189, "width": 480, "height": 360}
{"video_id": "RAjlZXGcwQI", "transcript": [{"start": 0.84, "duration": 5.039, "text": "okay so one more example here uh about a"}, {"start": 3.84, "duration": 4.24, "text": "little introduction to Polar coordinates"}, {"start": 5.879, "duration": 4.401, "text": "so suppose we want to figure out what"}, {"start": 8.08, "duration": 4.76, "text": "are the polar coordinates for the point"}, {"start": 10.28, "duration": 6.28, "text": "with a rectangular coordinates of 3"}, {"start": 12.84, "duration": 5.96, "text": "comma 3 so hey we just move three units"}, {"start": 16.56, "duration": 5.639, "text": "to the right three units up that's our"}, {"start": 18.8, "duration": 5.08, "text": "good old regular uh rectangular"}, {"start": 22.199, "duration": 3.481, "text": "coordinates well we'll have to figure"}, {"start": 23.88, "duration": 2.799, "text": "out two things we'll have to figure out"}, {"start": 25.68, "duration": 4.599, "text": "the"}, {"start": 26.679, "duration": 6.121, "text": "angle uh that's made by the uh the posi"}, {"start": 30.279, "duration": 4.721, "text": "xais and that point and we'll also have"}, {"start": 32.8, "duration": 4.4, "text": "to figure out our r"}, {"start": 35.0, "duration": 4.039, "text": "value um both of these are pretty"}, {"start": 37.2, "duration": 3.92, "text": "straightforward to figure out"}, {"start": 39.039, "duration": 4.761, "text": "though because really all we're going to"}, {"start": 41.12, "duration": 5.68, "text": "do is just use a"}, {"start": 43.8, "duration": 5.52, "text": "little little trigonometry so we've"}, {"start": 46.8, "duration": 4.919, "text": "moved three units to the right three"}, {"start": 49.32, "duration": 7.28, "text": "units up to get our r value we can just"}, {"start": 51.719, "duration": 8.84, "text": "use Pythagorean theorem so 3^ 2 + 3^ 2"}, {"start": 56.6, "duration": 7.919, "text": "would equal R 2 so that's what 9+ + 9"}, {"start": 60.559, "duration": 5.681, "text": "which will be 18 = R 2 uh so normally"}, {"start": 64.519, "duration": 3.841, "text": "when we take the square root we get"}, {"start": 66.24, "duration": 4.8, "text": "positive and"}, {"start": 68.36, "duration": 4.799, "text": "negative but I'm going to use the"}, {"start": 71.04, "duration": 4.16, "text": "positive square root in this"}, {"start": 73.159, "duration": 4.041, "text": "case and if you want to you can even"}, {"start": 75.2, "duration": 4.239, "text": "simplify the square root of 18 I guess"}, {"start": 77.2, "duration": 5.0, "text": "that's the sare of two times the square"}, {"start": 79.439, "duration": 6.04, "text": "< TK of 9 uh the square of N9 is just"}, {"start": 82.2, "duration": 6.8, "text": "three so we could write this as 3 < tk2"}, {"start": 85.479, "duration": 5.0, "text": "as our value for R so again we write R"}, {"start": 89.0, "duration": 5.84, "text": "and then we write"}, {"start": 90.479, "duration": 7.401, "text": "so our R would be 3un"}, {"start": 94.84, "duration": 7.84, "text": "2 um to figure out this anglea well we"}, {"start": 97.88, "duration": 4.8, "text": "can just use tangent so tangent of"}, {"start": 103.92, "duration": 6.0, "text": "theta would be the opposite over the"}, {"start": 106.52, "duration": 5.919, "text": "adjacent so 3 over 3 so that's just"}, {"start": 109.92, "duration": 6.36, "text": "going to give us tangent of theta equal"}, {"start": 112.439, "duration": 6.481, "text": "1 well we're sitting here in a quadrant"}, {"start": 116.28, "duration": 5.36, "text": "one this is definitely going to be Theta"}, {"start": 118.92, "duration": 6.119, "text": "= pi 4 or"}, {"start": 121.64, "duration": 5.64, "text": "45\u00b0 so uh one way to describe this point"}, {"start": 125.039, "duration": 6.241, "text": "would be to use the polar coordinates 3"}, {"start": 127.28, "duration": 6.679, "text": "< tk2 comma pi over"}, {"start": 131.28, "duration": 4.16, "text": "4 uh again notice there's uh lots of"}, {"start": 133.959, "duration": 4.681, "text": "different ways that we could describe"}, {"start": 135.44, "duration": 6.6, "text": "this point as well so uh notice we could"}, {"start": 138.64, "duration": 8.92, "text": "have even used 3 < tk2 comma say how"}, {"start": 142.04, "duration": 8.36, "text": "about -7 pi over 4 because -7 Pi"}, {"start": 147.56, "duration": 4.88, "text": "over4 um"}, {"start": 150.4, "duration": 5.72, "text": "we would go around that would put us at"}, {"start": 152.44, "duration": 5.72, "text": "the exact same angle okay -7 Pi over4"}, {"start": 156.12, "duration": 3.759, "text": "would put us at the same angle and again"}, {"start": 158.16, "duration": 4.92, "text": "we would just go at a distance of 3 <"}, {"start": 159.879, "duration": 5.761, "text": "tk2 so this would also be an equivalent"}, {"start": 163.08, "duration": 6.72, "text": "solution um so you can always add or"}, {"start": 165.64, "duration": 7.64, "text": "subtract multiples of 2 pi from your uh"}, {"start": 169.8, "duration": 5.64, "text": "your value your your your angle we could"}, {"start": 173.28, "duration": 5.56, "text": "have even used uh say a different one"}, {"start": 175.44, "duration": 7.2, "text": "suppose we use the angle negative"}, {"start": 178.84, "duration": 9.84, "text": "uh how about the angle um let's"}, {"start": 182.64, "duration": 8.159, "text": "see -3 pi over 4 so normally positive 3"}, {"start": 188.68, "duration": 4.88, "text": "Pi over4 would put us you know up in the"}, {"start": 190.799, "duration": 6.72, "text": "second quadrant negative 3 Pi over4"}, {"start": 193.56, "duration": 5.92, "text": "would put us down in the third quadrant"}, {"start": 197.519, "duration": 4.201, "text": "um if we use a positive radius though it"}, {"start": 199.48, "duration": 3.92, "text": "would keep us in the third quadrant well"}, {"start": 201.72, "duration": 4.92, "text": "to reflect things back we would just"}, {"start": 203.4, "duration": 5.36, "text": "take the negative of that so we could"}, {"start": 206.64, "duration": 7.2, "text": "use <"}, {"start": 208.76, "duration": 7.92, "text": "tk33 < tk2 comma -3i over4 as another"}, {"start": 213.84, "duration": 5.119, "text": "way to describe that uh that point using"}, {"start": 216.68, "duration": 4.16, "text": "polar coordinates so maybe more than you"}, {"start": 218.959, "duration": 3.601, "text": "wanted to see again I think little"}, {"start": 220.84, "duration": 3.44, "text": "things that are definitely uh good to"}, {"start": 222.56, "duration": 4.0, "text": "think about though but certainly this"}, {"start": 224.28, "duration": 5.8, "text": "would be the the answer choice that I"}, {"start": 226.56, "duration": 3.52, "text": "would go with"}], "title": "Intro to Polar Coordinates, Ex 1", "description": "Thanks to all of you who support me on Patreon. You da real mvps!   $1 per month helps!! :)  https://www.patreon.com/patrickjmt !! Intro to Polar Coordinates.  Just a few quick questions related to the basics of polar coordinates.", "lengthSeconds": 229, "width": 480, "height": 360}
{"video_id": "s7d7Go42iNw", "transcript": [{"start": 0.23, "duration": 6.01, "text": "okay in this example we're gonna solve"}, {"start": 2.52, "duration": 8.43, "text": "the inequality X minus 2 less than 2x"}, {"start": 6.24, "duration": 6.93, "text": "plus 6 being divided by 4 so so here's"}, {"start": 10.95, "duration": 3.599, "text": "my little recipe on when I see this kind"}, {"start": 13.17, "duration": 3.18, "text": "of problem I think you know there's a"}, {"start": 14.549, "duration": 5.251, "text": "procedure to do it it's just like being"}, {"start": 16.35, "duration": 4.74, "text": "a baker you know there's an order in"}, {"start": 19.8, "duration": 3.63, "text": "what you do things and the same thing"}, {"start": 21.09, "duration": 4.38, "text": "for this type of inequality so the first"}, {"start": 23.43, "duration": 3.509, "text": "thing I'm thinking is well it would be"}, {"start": 25.47, "duration": 4.02, "text": "nice to get rid of that fraction I've"}, {"start": 26.939, "duration": 4.111, "text": "got this fraction on the right side you"}, {"start": 29.49, "duration": 3.36, "text": "know fractions can sometimes be a little"}, {"start": 31.05, "duration": 3.0, "text": "just tedious to deal with so maybe if we"}, {"start": 32.85, "duration": 2.869, "text": "can get rid of the fraction that'll be"}, {"start": 34.05, "duration": 4.2, "text": "that'll be helpful"}, {"start": 35.719, "duration": 4.36, "text": "the next thing I'm gonna do is I say"}, {"start": 38.25, "duration": 3.87, "text": "well you know I've got terms involving X"}, {"start": 40.079, "duration": 4.681, "text": "I'm gonna have constants floating around"}, {"start": 42.12, "duration": 6.36, "text": "let's get all the X's combined and let's"}, {"start": 44.76, "duration": 5.819, "text": "also combine all the constants the last"}, {"start": 48.48, "duration": 6.68, "text": "thing I want to do is get one X all by"}, {"start": 50.579, "duration": 6.841, "text": "itself and then I'll have my my solution"}, {"start": 55.16, "duration": 6.219, "text": "okay so the first thing we said was"}, {"start": 57.42, "duration": 7.709, "text": "let's get rid of this fraction so I've"}, {"start": 61.379, "duration": 7.951, "text": "got X minus 2 less than 2x plus 6"}, {"start": 65.129, "duration": 5.85, "text": "divided by 4 well since I'm dividing by"}, {"start": 69.33, "duration": 3.51, "text": "4 that's kind of the thing I want to go"}, {"start": 70.979, "duration": 3.991, "text": "away I wish that wasn't there right I"}, {"start": 72.84, "duration": 4.62, "text": "wish I just had 2 X plus 6 left over I"}, {"start": 74.97, "duration": 3.81, "text": "would feel a little bit better in that"}, {"start": 77.46, "duration": 4.26, "text": "I'm getting closer to solving the"}, {"start": 78.78, "duration": 5.64, "text": "problems since I'm dividing by 4 I'm"}, {"start": 81.72, "duration": 4.259, "text": "gonna multiply both sides by 4 so make"}, {"start": 84.42, "duration": 2.79, "text": "sure you add parentheses on the left"}, {"start": 85.979, "duration": 3.841, "text": "side because you want to use this"}, {"start": 87.21, "duration": 6.96, "text": "distributive property if not you'll"}, {"start": 89.82, "duration": 7.35, "text": "certainly make a mistake so 4 times X"}, {"start": 94.17, "duration": 6.449, "text": "will be 4 X 4 times negative 2 will be"}, {"start": 97.17, "duration": 6.12, "text": "negative 8 on the left side the fours"}, {"start": 100.619, "duration": 6.741, "text": "will just cancel out and we'll be left"}, {"start": 103.29, "duration": 6.3, "text": "with 2x plus 6 okay so now I've gotten"}, {"start": 107.36, "duration": 3.66, "text": "gotten rid of the fraction the next"}, {"start": 109.59, "duration": 4.739, "text": "thing I want to do is I want to combine"}, {"start": 111.02, "duration": 6.58, "text": "the terms involving X and I want to also"}, {"start": 114.329, "duration": 4.5, "text": "combine the just the constants so it"}, {"start": 117.6, "duration": 3.96, "text": "doesn't sort of matter the order in"}, {"start": 118.829, "duration": 5.941, "text": "which you do it first let's get the"}, {"start": 121.56, "duration": 4.83, "text": "constants together so I'm gonna put all"}, {"start": 124.77, "duration": 5.099, "text": "the X's on the left side and all the"}, {"start": 126.39, "duration": 4.92, "text": "constants on the right side just just to"}, {"start": 129.869, "duration": 2.691, "text": "pick an order I mean you could do an"}, {"start": 131.31, "duration": 3.8, "text": "opposite it doesn't matter"}, {"start": 132.56, "duration": 4.17, "text": "so you could put all the the terms"}, {"start": 135.11, "duration": 4.349, "text": "involving X on the right side and all"}, {"start": 136.73, "duration": 4.38, "text": "the constants on the left so okay so if"}, {"start": 139.459, "duration": 2.901, "text": "I want to keep the X's on the left that"}, {"start": 141.11, "duration": 2.4, "text": "means I'm going to move this constant"}, {"start": 142.36, "duration": 3.61, "text": "over"}, {"start": 143.51, "duration": 6.03, "text": "so since I'm subtracting eight I'm gonna"}, {"start": 145.97, "duration": 6.769, "text": "add 8 to both sides well I'll just be"}, {"start": 149.54, "duration": 8.16, "text": "left with 4 X on the left side Oh"}, {"start": 152.739, "duration": 6.701, "text": "that'll be less than 2x plus 14 and now"}, {"start": 157.7, "duration": 4.38, "text": "the same thing I want to combine all the"}, {"start": 159.44, "duration": 4.29, "text": "terms involving X well I said I'm gonna"}, {"start": 162.08, "duration": 3.96, "text": "put all the terms involving X on the"}, {"start": 163.73, "duration": 4.92, "text": "left side so let's let's move this"}, {"start": 166.04, "duration": 5.0, "text": "positive 2x by subtracting 2x from both"}, {"start": 168.65, "duration": 2.39, "text": "sides"}, {"start": 171.34, "duration": 7.119, "text": "well 4x minus 2x that's gonna leave me"}, {"start": 174.62, "duration": 6.839, "text": "with 2x 2x minus 2x well hey that's 0x"}, {"start": 178.459, "duration": 5.491, "text": "we still have our positive 14 on the"}, {"start": 181.459, "duration": 5.071, "text": "right side and now the last thing I want"}, {"start": 183.95, "duration": 5.52, "text": "to do is get X all by itself so since"}, {"start": 186.53, "duration": 6.12, "text": "I'm multiplying by positive 2 I'm gonna"}, {"start": 189.47, "duration": 5.94, "text": "divide both sides by positive 2 so the"}, {"start": 192.65, "duration": 5.73, "text": "twos will cancel we'll be left with X on"}, {"start": 195.41, "duration": 6.48, "text": "the left side that's less than well 14"}, {"start": 198.38, "duration": 6.99, "text": "divided by 2 is positive 7 and that's"}, {"start": 201.89, "duration": 6.15, "text": "our solution recall that if you multiply"}, {"start": 205.37, "duration": 5.19, "text": "or divide by a negative number you have"}, {"start": 208.04, "duration": 4.47, "text": "to flip the inequality nowhere did we do"}, {"start": 210.56, "duration": 3.899, "text": "that in this example but certainly"}, {"start": 212.51, "duration": 4.44, "text": "something I want you to keep in mind and"}, {"start": 214.459, "duration": 4.761, "text": "just to remember and to be aware of that"}, {"start": 216.95, "duration": 2.27, "text": "fact"}], "title": "GED 'Most Missed' Math Practice Problems , Code A.3.a,  #3", "description": "Thanks to all of you who support me on Patreon. You da real mvps!   $1 per month helps!! :)  https://www.patreon.com/patrickjmt !! The GED Testing Service keeps a list of the most missed math topics and can be seen here: http://www.gedtestingservice.com/uploads/files/126de0283c94ff6323b3b5b3f310da5d.pdf\nThis video shows how to work a sample problem using their coding system that is shown on the left using the link above.", "lengthSeconds": 219, "width": 480, "height": 360}
{"video_id": "_kLWcuJzYh8", "transcript": [{"start": 0.96, "duration": 2.879, "text": "okay so here we're going to do some more"}, {"start": 2.24, "duration": 4.24, "text": "examples related to implicit"}, {"start": 3.839, "duration": 4.601, "text": "differentiation and these equations are"}, {"start": 6.48, "duration": 4.6, "text": "now implicit equations we kind of have"}, {"start": 8.44, "duration": 4.6, "text": "the X and the Y um on the same side of"}, {"start": 11.08, "duration": 4.24, "text": "the equal sign so that's kind of what I"}, {"start": 13.04, "duration": 3.84, "text": "look for what sort of Clues me in so the"}, {"start": 15.32, "duration": 4.039, "text": "main thing I remember when we're doing"}, {"start": 16.88, "duration": 6.04, "text": "implicit differentiation so uh let's"}, {"start": 19.359, "duration": 5.401, "text": "look at x^2 + y^2 = 25 mechanically what"}, {"start": 22.92, "duration": 4.64, "text": "you have to remember okay we're taking"}, {"start": 24.76, "duration": 4.96, "text": "the derivative with respect to X of both"}, {"start": 27.56, "duration": 4.28, "text": "sides whenever you take the derivative"}, {"start": 29.72, "duration": 4.359, "text": "of you know the variable X we just do it"}, {"start": 31.84, "duration": 4.84, "text": "like normal so the derivative of x^2 is"}, {"start": 34.079, "duration": 5.281, "text": "just 2x now we're going to take the"}, {"start": 36.68, "duration": 6.76, "text": "derivative of y^2 and we do it again"}, {"start": 39.36, "duration": 5.92, "text": "quote unquote like normal um we get 2 y"}, {"start": 43.44, "duration": 3.279, "text": "um but what we have to do anytime you"}, {"start": 45.28, "duration": 4.68, "text": "take the derivative of something"}, {"start": 46.719, "duration": 5.081, "text": "involving y we have to multiply that by"}, {"start": 49.96, "duration": 4.2, "text": "Dy"}, {"start": 51.8, "duration": 3.919, "text": "DX and on the right side when we take"}, {"start": 54.16, "duration": 4.64, "text": "the derivative with respect to X we'll"}, {"start": 55.719, "duration": 6.281, "text": "just get zero so normally we have the"}, {"start": 58.8, "duration": 4.96, "text": "dydx all by itself it's all isolated and"}, {"start": 62.0, "duration": 3.439, "text": "that's what we want to do in this case"}, {"start": 63.76, "duration": 4.64, "text": "so what I'm going to do is I can just"}, {"start": 65.439, "duration": 5.521, "text": "subtract 2x from both"}, {"start": 68.4, "duration": 5.64, "text": "sides so if we subtract 2x from both"}, {"start": 70.96, "duration": 7.159, "text": "sides we'll just get -2X and then we can"}, {"start": 74.04, "duration": 4.079, "text": "just divide both sides by 2"}, {"start": 78.32, "duration": 6.28, "text": "Y and then we'll have our solution it"}, {"start": 80.6, "duration": 6.64, "text": "says Dy DX is going to equal well we can"}, {"start": 84.6, "duration": 8.32, "text": "simplify um -2 over 2 is just going to"}, {"start": 87.24, "duration": 7.96, "text": "be -1 uh X over y and now we've got our"}, {"start": 92.92, "duration": 4.6, "text": "derivative so let's look at uh at least"}, {"start": 95.2, "duration": 4.64, "text": "one other example here uh maybe one"}, {"start": 97.52, "duration": 4.68, "text": "that's uh a touch more"}, {"start": 99.84, "duration": 4.88, "text": "complicated again hopefully nothing"}, {"start": 102.2, "duration": 2.52, "text": "nothing too"}, {"start": 105.079, "duration": 5.961, "text": "crazy so let's figure out uh let's do"}, {"start": 107.64, "duration": 6.519, "text": "the derivative of cosine of x^2 + S of Y"}, {"start": 111.04, "duration": 5.24, "text": "Cub = 6 we'll take the derivative of"}, {"start": 114.159, "duration": 3.841, "text": "this all right well we'll have to use"}, {"start": 116.28, "duration": 4.199, "text": "the chain rule so the derivative of"}, {"start": 118.0, "duration": 5.119, "text": "cosine of x^2 when we do the derivative"}, {"start": 120.479, "duration": 4.92, "text": "of cosine we'll get Negative s we'll"}, {"start": 123.119, "duration": 3.96, "text": "leave the inside alone and then we have"}, {"start": 125.399, "duration": 3.761, "text": "to take the derivative of the inside"}, {"start": 127.079, "duration": 4.681, "text": "which will just be"}, {"start": 129.16, "duration": 7.159, "text": "2x then we have to do the derivative of"}, {"start": 131.76, "duration": 7.839, "text": "the s y cubed so the derivative of s is"}, {"start": 136.319, "duration": 4.601, "text": "cosine we'll leave the inside alone and"}, {"start": 139.599, "duration": 4.601, "text": "now we're going to take the derivative"}, {"start": 140.92, "duration": 5.0, "text": "of something involving y so we get uh"}, {"start": 144.2, "duration": 4.6, "text": "when we take the derivative of y cubed"}, {"start": 145.92, "duration": 5.24, "text": "we just get 3 y^2 but this again is when"}, {"start": 148.8, "duration": 4.6, "text": "I have to remember to tack on the"}, {"start": 151.16, "duration": 5.439, "text": "dydx so I took the derivative of"}, {"start": 153.4, "duration": 5.479, "text": "something involving y um right here I"}, {"start": 156.599, "duration": 6.28, "text": "tack on the Dy"}, {"start": 158.879, "duration": 5.481, "text": "DX well on the right side uh 6 uh the"}, {"start": 162.879, "duration": 4.161, "text": "derivative of that's just going to be"}, {"start": 164.36, "duration": 6.12, "text": "zero and now uh again our goal is to"}, {"start": 167.04, "duration": 5.119, "text": "solve for dydx so this first term I'm"}, {"start": 170.48, "duration": 4.88, "text": "just going to add it over to the right"}, {"start": 172.159, "duration": 5.961, "text": "side so instead of having um you know"}, {"start": 175.36, "duration": 4.92, "text": "Nega sin x^2 * 2x I'm going to bring"}, {"start": 178.12, "duration": 3.199, "text": "that over and I'm going to rewrite it as"}, {"start": 180.28, "duration": 4.76, "text": "positive"}, {"start": 181.319, "duration": 6.761, "text": "2x sin x^2 so I'm just adding this term"}, {"start": 185.04, "duration": 4.68, "text": "to both sides and we'll be left with um"}, {"start": 188.08, "duration": 8.12, "text": "our second term and I'm going to rewrite"}, {"start": 189.72, "duration": 8.72, "text": "that as 3 y^ 2 * cosine of y 3 power *"}, {"start": 196.2, "duration": 5.08, "text": "dy over"}, {"start": 198.44, "duration": 6.0, "text": "DX and now just the same thing to get Dy"}, {"start": 201.28, "duration": 6.4, "text": "DX all by itself we can just divide so"}, {"start": 204.44, "duration": 6.879, "text": "dy over DX we'll divide both sides by"}, {"start": 207.68, "duration": 8.199, "text": "the 3 y^ 2 cosine y cubed so on the"}, {"start": 211.319, "duration": 6.2, "text": "right side we'll just be left with 2x *"}, {"start": 215.879, "duration": 8.481, "text": "sin of"}, {"start": 217.519, "duration": 10.761, "text": "x^2 again all over 3 y^ 2 * cine of Y"}, {"start": 224.36, "duration": 6.92, "text": "the 3 power and again now we've got our"}, {"start": 228.28, "duration": 3.0, "text": "derivative"}], "title": "\u2756 Implicit Differentiation for Calculus - More Examples,  #1 \u2756", "description": "Implicit Differentiation for Calculus - More Examples.\nIn this video I look at using implicit differentiation in order to find the derivative of an implicit function.  This is a common technique used and taught in any calculus course.\nRecall that implicit differentiation is really us just using the chain rule on a function defined implicitly by x.\nIn this video I show solutions for the first two examples.\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n#maths #calculus #calculusbc #implicitdifferentiation #implicit_differentiation #derivative #differentiation #implicitfunction #apcalculus #apcalc", "lengthSeconds": 231, "width": 480, "height": 360}
{"video_id": "O2LC5uoZzXg", "transcript": [{"start": 0.68, "duration": 3.4, "text": "all right in this video I'm going to do"}, {"start": 2.159, "duration": 3.64, "text": "another example related to the product"}, {"start": 4.08, "duration": 6.84, "text": "rule so here we're going to take the"}, {"start": 5.799, "duration": 8.121, "text": "derivative of G ofx = 3x to 4 + 2x -1 *"}, {"start": 10.92, "duration": 5.72, "text": "the quantity X 5th -"}, {"start": 13.92, "duration": 5.08, "text": "2x^2 so all right well if we take our"}, {"start": 16.64, "duration": 4.36, "text": "derivative here if I do the derivative"}, {"start": 19.0, "duration": 4.4, "text": "of the first"}, {"start": 21.0, "duration": 4.599, "text": "Factor okay so the four would come out"}, {"start": 23.4, "duration": 5.4, "text": "front and multiply by the three and that"}, {"start": 25.599, "duration": 5.481, "text": "would give us 12 x and then we take one"}, {"start": 28.8, "duration": 5.84, "text": "away from our EXP which will give us to"}, {"start": 31.08, "duration": 6.799, "text": "the thir power um the derivative of POS"}, {"start": 34.64, "duration": 5.04, "text": "2x is just going to be pos2 and then the"}, {"start": 37.879, "duration": 4.961, "text": "derivative of our constant negative-1"}, {"start": 39.68, "duration": 5.399, "text": "just turns into a zero um so we took the"}, {"start": 42.84, "duration": 5.039, "text": "derivative of the first Factor we'll"}, {"start": 45.079, "duration": 2.8, "text": "leave the second one"}, {"start": 50.199, "duration": 7.761, "text": "alone and then we'll take uh we'll leave"}, {"start": 54.16, "duration": 7.84, "text": "the first Factor alone so we'll have 3x"}, {"start": 57.96, "duration": 8.36, "text": "4th + 2x - 1 and then when we take the"}, {"start": 62.0, "duration": 6.36, "text": "derivative of uh X to 5th well the five"}, {"start": 66.32, "duration": 5.159, "text": "will come out front we'll take one away"}, {"start": 68.36, "duration": 6.56, "text": "from the exponent and then"}, {"start": 71.479, "duration": 7.241, "text": "uh when we take the derivative of -2X"}, {"start": 74.92, "duration": 5.76, "text": "^2 okay so the -2 comes along and then"}, {"start": 78.72, "duration": 4.32, "text": "uh the two will multiply by the two"}, {"start": 80.68, "duration": 4.6, "text": "which will give us four and then we'll"}, {"start": 83.04, "duration": 5.36, "text": "have X we take one away from the"}, {"start": 85.28, "duration": 6.72, "text": "exponent so that'll be the second"}, {"start": 88.4, "duration": 5.999, "text": "Factor um and now all right well uh we"}, {"start": 92.0, "duration": 5.88, "text": "could always clean this up a little bit"}, {"start": 94.399, "duration": 6.04, "text": "um probably what I usually start looking"}, {"start": 97.88, "duration": 4.239, "text": "for at this point are common factors uh"}, {"start": 100.439, "duration": 2.921, "text": "notice there's not anything there's not"}, {"start": 102.119, "duration": 4.161, "text": "really the same thing inside the"}, {"start": 103.36, "duration": 4.52, "text": "parentheses on any of these so I think"}, {"start": 106.28, "duration": 3.519, "text": "probably what I would do in this case is"}, {"start": 107.88, "duration": 3.76, "text": "just leave this one alone and that's our"}, {"start": 109.799, "duration": 3.521, "text": "derivative you could always multiply all"}, {"start": 111.64, "duration": 4.0, "text": "this stuff back out collect all your"}, {"start": 113.32, "duration": 4.24, "text": "like terms but um again just to"}, {"start": 115.64, "duration": 5.56, "text": "illustrate the product rule I think I'm"}, {"start": 117.56, "duration": 3.64, "text": "going to leave it there"}], "title": "Product Rule to Find a Derivative : Basic Example , No Simplifying", "description": "In this video I look at a quick example of finding the derivative of a function by using the product rule.  Notice that we could perform some algebra on this function before taking the derivative but I just jump into things in order to illustrate the procedure.\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n#math, #algebra, #khanacademy, #khan, #geometry, #education, #montessori, #calculus, #homeschooling, #functions, #notation, #organicchemistry, #organicchemistrytutor, #calculus, #derivative, #powerrule, #limits,  #limit, #graph, #graphing, #conjugate, #infinity, #patrickjmt, #rational, #shortcut, #trick, #commondenominator, #fraction, #university, #admissions, #test #continuity #onesided #infinite #absolutevalue derivative derivative derivative derivative derivative derivative derivative derivative product rule product rule product rule product rule product rule product rule product rule product rule product rule product rule product rule product rule product rule product rule product rule product rule product rule product rule product rule product rule", "lengthSeconds": 120, "width": 480, "height": 360}
{"video_id": "FMWd8yXCMTw", "transcript": [{"start": 0.68, "duration": 3.44, "text": "all right in this video I just want to"}, {"start": 2.0, "duration": 5.279, "text": "do a couple examples about finding the"}, {"start": 4.12, "duration": 5.559, "text": "maximum and minimum values for uh"}, {"start": 7.279, "duration": 5.32, "text": "different signing cosine functions and"}, {"start": 9.679, "duration": 6.401, "text": "in this case we've got 6 s of 7x and"}, {"start": 12.599, "duration": 6.281, "text": "then -2 cosine of 3"}, {"start": 16.08, "duration": 4.4, "text": "pix okay a couple different ways uh I"}, {"start": 18.88, "duration": 5.399, "text": "guess you can go by thinking about this"}, {"start": 20.48, "duration": 5.44, "text": "to me you know when I look at this um"}, {"start": 24.279, "duration": 4.041, "text": "the first thing I see for example is the"}, {"start": 25.92, "duration": 2.4, "text": "sign"}, {"start": 28.96, "duration": 6.2, "text": "function so s if you think about you"}, {"start": 32.559, "duration": 4.081, "text": "know if you think about the graph of it"}, {"start": 35.16, "duration": 5.879, "text": "you know a couple different arguments I"}, {"start": 36.64, "duration": 6.439, "text": "guess here um Remember by multiplying by"}, {"start": 41.039, "duration": 4.761, "text": "the seven all that's really going to do"}, {"start": 43.079, "duration": 4.201, "text": "is just change the uh the period so I'm"}, {"start": 45.8, "duration": 3.88, "text": "forgetting about the six for a second"}, {"start": 47.28, "duration": 4.599, "text": "here um all that's going to do is change"}, {"start": 49.68, "duration": 2.199, "text": "the"}, {"start": 53.039, "duration": 5.48, "text": "period so okay maybe it's going to uh do"}, {"start": 55.92, "duration": 6.319, "text": "things a little"}, {"start": 58.519, "duration": 6.6, "text": "faster but still sign uh sign of some"}, {"start": 62.239, "duration": 6.281, "text": "value is always between is at most"}, {"start": 65.119, "duration": 6.04, "text": "positive one and uh the smallest that"}, {"start": 68.52, "duration": 5.279, "text": "ever is is negative 1 so really I'm just"}, {"start": 71.159, "duration": 5.721, "text": "thinking about the range of 7 sinx it's"}, {"start": 73.799, "duration": 5.68, "text": "always between negative 1 and positive 1"}, {"start": 76.88, "duration": 8.48, "text": "well what that means is if we multiply"}, {"start": 79.479, "duration": 7.561, "text": "the inside by six so it says 6 s of 7x"}, {"start": 85.36, "duration": 5.079, "text": "well I would have to multiply the left"}, {"start": 87.04, "duration": 5.64, "text": "side by six as well which is -6 I would"}, {"start": 90.439, "duration": 5.32, "text": "have to multiply the right side by six"}, {"start": 92.68, "duration": 4.799, "text": "as well which would give me positive six"}, {"start": 95.759, "duration": 5.04, "text": "and these are going to be your maximum"}, {"start": 97.479, "duration": 7.041, "text": "and minimum values the maximum value is"}, {"start": 100.799, "duration": 6.28, "text": "going to be uh just six and the minimum"}, {"start": 104.52, "duration": 6.12, "text": "value again is going to be"}, {"start": 107.079, "duration": 5.161, "text": "-6 so um again definitely taking this"}, {"start": 110.64, "duration": 3.4, "text": "this coefficient into account kind of"}, {"start": 112.24, "duration": 4.199, "text": "thinking about the amplitude here how"}, {"start": 114.04, "duration": 4.399, "text": "this coefficient changes the amplitude"}, {"start": 116.439, "duration": 4.921, "text": "instead of being a positive one and"}, {"start": 118.439, "duration": 6.04, "text": "negative 1 it's not going to go up to 6"}, {"start": 121.36, "duration": 4.719, "text": "and -6 when we graph"}, {"start": 124.479, "duration": 4.56, "text": "it"}, {"start": 126.079, "duration": 6.88, "text": "okay so the same thing for uh our next"}, {"start": 129.039, "duration": 6.361, "text": "one here -2 * cosine of 3 pix well again"}, {"start": 132.959, "duration": 2.441, "text": "the same"}, {"start": 136.84, "duration": 6.08, "text": "argument cosine of anything is in"}, {"start": 139.36, "duration": 5.36, "text": "between positive one and negative 1 and"}, {"start": 142.92, "duration": 5.44, "text": "again notice there's no restrictions on"}, {"start": 144.72, "duration": 6.12, "text": "the domains so you know we can let X be"}, {"start": 148.36, "duration": 4.76, "text": "whatever number we want from Infinity to"}, {"start": 150.84, "duration": 5.36, "text": "positive infinity and plug it inside of"}, {"start": 153.12, "duration": 5.32, "text": "this cosine function so uh"}, {"start": 156.2, "duration": 4.679, "text": "correspondingly cosine is going to hit"}, {"start": 158.44, "duration": 5.0, "text": "every value between negative 1 and"}, {"start": 160.879, "duration": 4.201, "text": "positive one well we have to be a little"}, {"start": 163.44, "duration": 3.64, "text": "careful if I do this little argument"}, {"start": 165.08, "duration": 5.239, "text": "that I did a second ago if I multiply"}, {"start": 167.08, "duration": 3.239, "text": "the inside by -"}, {"start": 171.159, "duration": 5.641, "text": "one2 remember if you multiply uh by a"}, {"start": 174.8, "duration": 7.519, "text": "negative number you have to flip the"}, {"start": 176.8, "duration": 7.32, "text": "inequalities so -1 * 1 12 would be"}, {"start": 182.319, "duration": 3.761, "text": "positive2 again we'll flip the"}, {"start": 184.12, "duration": 6.839, "text": "inequality because we're multiplying Now"}, {"start": 186.08, "duration": 7.64, "text": "by -2 -2 * 1 is going to give us just a"}, {"start": 190.959, "duration": 5.92, "text": "- one2 and again we've got our maximum"}, {"start": 193.72, "duration": 5.96, "text": "and our minimum it says our function -2"}, {"start": 196.879, "duration": 7.401, "text": "cosine of 3 pix is less than or equal to"}, {"start": 199.68, "duration": 8.88, "text": "1 12 but it's greater than or equal to"}, {"start": 204.28, "duration": 7.36, "text": "-2 so in this case our max value is just"}, {"start": 208.56, "duration": 6.28, "text": "positive2 and our minimum value will be"}, {"start": 211.64, "duration": 3.2, "text": "- one2"}], "title": "Maximum and Minimum Values of Sine and Cosine Functions, Ex 1", "description": "Thanks to all of you who support me on Patreon. You da real mvps!   $1 per month helps!! :)  https://www.patreon.com/patrickjmt !! Maximum and Minimum Values of Sine and Cosine Functions, Ex 1. In this video, I talk about how to find the maximum and minimum values of sine and cosine functions with different coefficients thrown in!", "lengthSeconds": 213, "width": 480, "height": 360}
{"video_id": "DXgafNXZke8", "transcript": [{"start": 0.96, "duration": 3.16, "text": "all right so here we're going to look at"}, {"start": 2.28, "duration": 4.24, "text": "one more example related to the quotient"}, {"start": 4.12, "duration": 6.76, "text": "rule so um here we've got the function f"}, {"start": 6.52, "duration": 6.6, "text": "ofx = x^2 - x + 1 over 3x + 2 we want to"}, {"start": 10.88, "duration": 5.76, "text": "find the equation of the tangent line at"}, {"start": 13.12, "duration": 7.84, "text": "the point 0 comma 1 12 so using point"}, {"start": 16.64, "duration": 8.479, "text": "slope formula y - y1 um equals the slope"}, {"start": 20.96, "duration": 6.04, "text": "* x - x sub1 um we've used this a few"}, {"start": 25.119, "duration": 4.201, "text": "times now to find equations of tangent"}, {"start": 27.0, "duration": 3.679, "text": "lines well the y coordinate is just"}, {"start": 29.32, "duration": 3.759, "text": "going to be 1"}, {"start": 30.679, "duration": 4.72, "text": "the x coordinate is zero that's given to"}, {"start": 33.079, "duration": 5.401, "text": "us again the slope is what we have to"}, {"start": 35.399, "duration": 4.961, "text": "figure out but the slope comes from the"}, {"start": 38.48, "duration": 4.44, "text": "derivative and in this case we'll have"}, {"start": 40.36, "duration": 5.16, "text": "to plug zero into the derivative to get"}, {"start": 42.92, "duration": 4.44, "text": "the slope of our tangent line well to"}, {"start": 45.52, "duration": 4.12, "text": "take the derivative of this function we"}, {"start": 47.36, "duration": 4.879, "text": "can just use the quotient"}, {"start": 49.64, "duration": 5.8, "text": "rule so it says you get whatever's in"}, {"start": 52.239, "duration": 6.441, "text": "the the uh denominator 3x + 2 in this"}, {"start": 55.44, "duration": 5.2, "text": "case we'll multiply that by um the"}, {"start": 58.68, "duration": 3.839, "text": "derivative of the numerator"}, {"start": 60.64, "duration": 4.239, "text": "so the derivative of x to the second"}, {"start": 62.519, "duration": 3.96, "text": "power the two would come out front and"}, {"start": 64.879, "duration": 5.121, "text": "then we would take one away from the"}, {"start": 66.479, "duration": 7.121, "text": "exponent the derivative of -1x will just"}, {"start": 70.0, "duration": 5.72, "text": "be -1 and then um the derivative of the"}, {"start": 73.6, "duration": 4.4, "text": "constant is just zero so that'll be kind"}, {"start": 75.72, "duration": 4.68, "text": "of the first part of the product rule"}, {"start": 78.0, "duration": 5.64, "text": "excuse me the quotient"}, {"start": 80.4, "duration": 5.64, "text": "rule um and then we'll put a minus sign"}, {"start": 83.64, "duration": 4.68, "text": "in between so at the bottom times the"}, {"start": 86.04, "duration": 7.88, "text": "derivative of the top minus the stuff in"}, {"start": 88.32, "duration": 8.4, "text": "the top so x^2 - x + 1 and then we have"}, {"start": 93.92, "duration": 5.839, "text": "to multiply by the derivative of the"}, {"start": 96.72, "duration": 6.2, "text": "denominator so the derivative of 3x is"}, {"start": 99.759, "duration": 4.921, "text": "just three um the constant again just"}, {"start": 102.92, "duration": 4.879, "text": "turns into a zero when we take the"}, {"start": 104.68, "duration": 5.479, "text": "derivative and that'll give us the uh"}, {"start": 107.799, "duration": 3.801, "text": "the numerator of our quotient Rule and"}, {"start": 110.159, "duration": 5.96, "text": "then it says we take whatever is"}, {"start": 111.6, "duration": 6.519, "text": "downstairs 3x + 2 and we Square it all"}, {"start": 116.119, "duration": 4.04, "text": "right so again um in one of the quotient"}, {"start": 118.119, "duration": 3.881, "text": "rule problems I kind of simplified all"}, {"start": 120.159, "duration": 5.64, "text": "this but to me all we're trying to do is"}, {"start": 122.0, "duration": 6.36, "text": "plug in xal 0 so I don't know to you"}, {"start": 125.799, "duration": 4.16, "text": "could simplify it but um in terms of"}, {"start": 128.36, "duration": 4.0, "text": "getting the slope of the tangent line we"}, {"start": 129.959, "duration": 5.64, "text": "certainly don't need to do that here so"}, {"start": 132.36, "duration": 7.12, "text": "notice if we plug in xal 0 we'll get 3 *"}, {"start": 135.599, "duration": 7.041, "text": "0 or zero then we'll just be left with"}, {"start": 139.48, "duration": 7.119, "text": "two if we plug in zero for the X in our"}, {"start": 142.64, "duration": 5.679, "text": "second uh Factor we'll get 2 * 0 -1 or"}, {"start": 146.599, "duration": 4.601, "text": "just we'll be left with the"}, {"start": 148.319, "duration": 4.92, "text": "-1 um and the second part let's see if"}, {"start": 151.2, "duration": 5.64, "text": "we plug in 0er so there's our minus in"}, {"start": 153.239, "duration": 6.521, "text": "between we'll just be left with 0 - 0 +"}, {"start": 156.84, "duration": 6.56, "text": "1 which is 1 and then we'll multiply"}, {"start": 159.76, "duration": 8.6, "text": "that by 3 and then in the denominator"}, {"start": 163.4, "duration": 6.24, "text": "well 3 * 0 is 0 + 2^ 2ar so we've got 2^"}, {"start": 168.36, "duration": 3.68, "text": "2 left"}, {"start": 169.64, "duration": 8.0, "text": "over well let's see this is going to be"}, {"start": 172.04, "duration": 9.199, "text": "-2 - 3 which uh looks like -5 to me and"}, {"start": 177.64, "duration": 6.4, "text": "then 2^2 is 4 so now we've got our slope"}, {"start": 181.239, "duration": 5.041, "text": "of our our tangent line so it says the"}, {"start": 184.04, "duration": 4.4, "text": "slope of the line um excuse me the"}, {"start": 186.28, "duration": 4.64, "text": "equation of the line we can just write"}, {"start": 188.44, "duration": 5.719, "text": "that as y -"}, {"start": 190.92, "duration": 6.56, "text": "12 equals okay we now know the slope by"}, {"start": 194.159, "duration": 6.481, "text": "using the derivative so we've got -5"}, {"start": 197.48, "duration": 6.479, "text": "over4 and x - 0 we can just write that"}, {"start": 200.64, "duration": 5.599, "text": "as X and that's now the equation of our"}, {"start": 203.959, "duration": 6.881, "text": "tangent line of course you can also add"}, {"start": 206.239, "duration": 6.961, "text": "the 1/2 over you could make it y equals"}, {"start": 210.84, "duration": 6.399, "text": "negative -5 over"}, {"start": 213.2, "duration": 6.48, "text": "4x + 12"}, {"start": 217.239, "duration": 6.36, "text": "and again now we've got the equation of"}, {"start": 219.68, "duration": 5.72, "text": "our line either way so um again finding"}, {"start": 223.599, "duration": 3.36, "text": "slopes of lines what do you got to do"}, {"start": 225.4, "duration": 3.6, "text": "you got to take the derivative and"}, {"start": 226.959, "duration": 4.521, "text": "basically just plug the point into the"}, {"start": 229.0, "duration": 4.239, "text": "formula um so again in this case to take"}, {"start": 231.48, "duration": 5.319, "text": "the derivative well we just had to use"}, {"start": 233.239, "duration": 3.56, "text": "the quotient rule"}], "title": "Calculus : Equation of the Tangent Line at Point using the Quotient Rule", "description": "Thanks to all of you who support me on Patreon. You da real mvps!   $1 per month helps!! :)  https://www.patreon.com/patrickjmt !! Calculus : Equation of the Tangent Line at Point using the Quotient Rule", "lengthSeconds": 236, "width": 480, "height": 360}
{"video_id": "HSzFkChaVXs", "transcript": [{"start": 1.0, "duration": 3.64, "text": "here we're going to find the S cosine"}, {"start": 2.52, "duration": 4.08, "text": "tangent cosecant secant and cent of"}, {"start": 4.64, "duration": 4.039, "text": "theta given that Theta is an angle in"}, {"start": 6.6, "duration": 4.44, "text": "standard position and the given point is"}, {"start": 8.679, "duration": 6.601, "text": "on the terminal side of theta so in this"}, {"start": 11.04, "duration": 8.88, "text": "example we're going to use -1 comma -2"}, {"start": 15.28, "duration": 9.52, "text": "our x value will be -1 our y value is -2"}, {"start": 19.92, "duration": 6.96, "text": "to get our r value we'll use the r^2 ="}, {"start": 24.8, "duration": 3.879, "text": "x^2"}, {"start": 26.88, "duration": 4.639, "text": "+ y^"}, {"start": 28.679, "duration": 4.081, "text": "2 well this will give us 1 + 4 which"}, {"start": 31.519, "duration": 3.641, "text": "will be"}, {"start": 32.76, "duration": 4.72, "text": "5 and if we take the square root of both"}, {"start": 35.16, "duration": 5.8, "text": "sides keeping the positive solution"}, {"start": 37.48, "duration": 6.88, "text": "we'll get that R equals the sare < TK"}, {"start": 40.96, "duration": 6.439, "text": "5 so let's see s of"}, {"start": 44.36, "duration": 5.6, "text": "theta cosine of"}, {"start": 47.399, "duration": 7.16, "text": "theta tangent of theta we can just"}, {"start": 49.96, "duration": 11.439, "text": "simply use our formulas s of theta is y"}, {"start": 54.559, "duration": 8.881, "text": "/ R again Y is -2 R is theare < TK of 5"}, {"start": 61.399, "duration": 6.121, "text": "and if we multiply the numerator and"}, {"start": 63.44, "duration": 7.359, "text": "denominator by < TK 5 we'll get -2 * the"}, {"start": 67.52, "duration": 7.959, "text": "> 5 /"}, {"start": 70.799, "duration": 8.081, "text": "5 cosine of theta is going to be x / R"}, {"start": 75.479, "duration": 6.841, "text": "so again X in this case is1 r is the"}, {"start": 78.88, "duration": 5.279, "text": "sare < TK of 5 multiply numerator and"}, {"start": 82.32, "duration": 5.799, "text": "denominator by the square < TK of 5"}, {"start": 84.159, "duration": 11.121, "text": "we'll get < TK 5 /"}, {"start": 88.119, "duration": 10.441, "text": "5 tangent is y / X the Y value was -2"}, {"start": 95.28, "duration": 4.64, "text": "the x value is -1 so that'll leave us"}, {"start": 98.56, "duration": 5.0, "text": "with"}, {"start": 99.92, "duration": 6.12, "text": "positive2 and now to get the other three"}, {"start": 103.56, "duration": 6.96, "text": "I'm just going to use reciprocals so"}, {"start": 106.04, "duration": 8.359, "text": "cosecant of theta that's 1/ sin Theta"}, {"start": 110.52, "duration": 6.12, "text": "and I use the value before we we"}, {"start": 114.399, "duration": 5.04, "text": "rationalize I just flip that I'll get"}, {"start": 116.64, "duration": 6.04, "text": "theun 5 over -2 I'm going to put the"}, {"start": 119.439, "duration": 6.161, "text": "Nega up front and the reason I use it uh"}, {"start": 122.68, "duration": 5.0, "text": "before we rationalize because if I use"}, {"start": 125.6, "duration": 4.24, "text": "the other form I'm going to actually end"}, {"start": 127.68, "duration": 3.68, "text": "up making a radical appear in the"}, {"start": 129.84, "duration": 3.84, "text": "denominator and then we'll have to"}, {"start": 131.36, "duration": 6.04, "text": "rationalize all over so to me it's"}, {"start": 133.68, "duration": 6.4, "text": "easier to do it this way secant of theta"}, {"start": 137.4, "duration": 6.88, "text": "well that's 1 over cosine"}, {"start": 140.08, "duration": 6.72, "text": "Theta again cosine of thet was- 1 overun"}, {"start": 144.28, "duration": 5.92, "text": "5 if we take the reciprocal we'll just"}, {"start": 146.8, "duration": 6.36, "text": "simply getun 5"}, {"start": 150.2, "duration": 5.759, "text": "and last but not least cotangent of"}, {"start": 153.16, "duration": 8.24, "text": "theta hey that's one over"}, {"start": 155.959, "duration": 8.121, "text": "tangent of theta well we had a positive2"}, {"start": 161.4, "duration": 4.4, "text": "or 2 over one if we take the reciprocal"}, {"start": 164.08, "duration": 4.96, "text": "of that we'll simply get"}, {"start": 165.8, "duration": 6.04, "text": "1/2 and we've now got all six of our"}, {"start": 169.04, "duration": 4.88, "text": "values so again I always just find S"}, {"start": 171.84, "duration": 4.0, "text": "cosine tangent and then I just take the"}, {"start": 173.92, "duration": 4.92, "text": "reciprocals of those to get the other"}, {"start": 175.84, "duration": 3.0, "text": "three"}], "title": "Finding the Trigonometric Values Given a Point in the Plane", "description": "Thanks to all of you who support me on Patreon. You da real mvps!   $1 per month helps!! :)  https://www.patreon.com/patrickjmt !! Finding the Trigonometric Values Given a Point in the Plane.  \n In this example, we find the sine, cosine, tangent, cotangent, secant and cosecent given a point in a plane and an angle in standard position.", "lengthSeconds": 179, "width": 450, "height": 360}
{"video_id": "-gQ2E33V77o", "transcript": [{"start": 0.48, "duration": 4.2, "text": "all right here we're going to start uh"}, {"start": 2.24, "duration": 5.199, "text": "looking at some examples of finding the"}, {"start": 4.68, "duration": 4.4, "text": "radius and interval of convergence so in"}, {"start": 7.439, "duration": 5.721, "text": "part A here we've got the series from 1"}, {"start": 9.08, "duration": 6.12, "text": "to infinity x to the n * 2 the N Over N"}, {"start": 13.16, "duration": 5.0, "text": "factorial so again to do these we're"}, {"start": 15.2, "duration": 6.239, "text": "going to use the ratio test so we'll"}, {"start": 18.16, "duration": 4.879, "text": "take the limit as n goes to Infinity"}, {"start": 21.439, "duration": 4.92, "text": "everywhere there's an N I'm going to"}, {"start": 23.039, "duration": 3.32, "text": "replace it with n +"}, {"start": 26.76, "duration": 4.12, "text": "one so then we would have n + one"}, {"start": 29.48, "duration": 3.56, "text": "factorial in the"}, {"start": 30.88, "duration": 4.96, "text": "denominator okay then we have to flip"}, {"start": 33.04, "duration": 7.519, "text": "and multiply by the reciprocal so n"}, {"start": 35.84, "duration": 6.52, "text": "factorial over x n * 2 the N so let's"}, {"start": 40.559, "duration": 4.721, "text": "see if we simplify here we've got the"}, {"start": 42.36, "duration": 6.16, "text": "limit as n goes to"}, {"start": 45.28, "duration": 5.119, "text": "infinity x the n + 1 over X the N is"}, {"start": 48.52, "duration": 5.4, "text": "going to leave us with X to the"}, {"start": 50.399, "duration": 4.921, "text": "1st um 2 the n + 1 over 2 the N is going"}, {"start": 53.92, "duration": 5.119, "text": "to leave us with a"}, {"start": 55.32, "duration": 7.52, "text": "two and then n factorial Over N +1"}, {"start": 59.039, "duration": 6.921, "text": "factorial is going to leave us with n +"}, {"start": 62.84, "duration": 5.76, "text": "one you could pull the 2x out and keep"}, {"start": 65.96, "duration": 5.0, "text": "that an absolute value and then we have"}, {"start": 68.6, "duration": 6.36, "text": "the limit as n goes to"}, {"start": 70.96, "duration": 7.08, "text": "Infinity of 1/ n +"}, {"start": 74.96, "duration": 7.0, "text": "1 but this limit is simply going to"}, {"start": 78.04, "duration": 6.52, "text": "equal zero and again we"}, {"start": 81.96, "duration": 4.72, "text": "want we want to figure out what values"}, {"start": 84.56, "duration": 4.36, "text": "of X we can plug in so that at the end"}, {"start": 86.68, "duration": 4.32, "text": "of the day when we multiply we get"}, {"start": 88.92, "duration": 3.879, "text": "something less than one one well no"}, {"start": 91.0, "duration": 3.96, "text": "matter what you plug in for x and"}, {"start": 92.799, "duration": 4.081, "text": "multiply by zero you'll always get"}, {"start": 94.96, "duration": 4.96, "text": "something less than"}, {"start": 96.88, "duration": 5.0, "text": "one so that means the solution um the"}, {"start": 99.92, "duration": 4.32, "text": "interval of convergence is going to be"}, {"start": 101.88, "duration": 6.72, "text": "from negative Infinity to positive"}, {"start": 104.24, "duration": 4.36, "text": "Infinity so this will be our interval of"}, {"start": 111.92, "duration": 3.879, "text": "convergence well in this case if you"}, {"start": 113.88, "duration": 3.919, "text": "think about the uh you know the length"}, {"start": 115.799, "duration": 4.201, "text": "of this interval from negative Infinity"}, {"start": 117.799, "duration": 4.721, "text": "to positive Infinity that's just"}, {"start": 120.0, "duration": 5.32, "text": "Infinity so we would say the radius is"}, {"start": 122.52, "duration": 5.799, "text": "also Infinity so the interval of"}, {"start": 125.32, "duration": 5.159, "text": "convergence all real numbers the radius"}, {"start": 128.319, "duration": 4.761, "text": "uh again the radius is of a number or in"}, {"start": 130.479, "duration": 5.601, "text": "this case we would just say positive"}, {"start": 133.08, "duration": 3.0, "text": "Infinity"}], "title": "Interval and Radius of Convergence for a Series, Ex 3", "description": "Thanks to all of you who support me on Patreon. You da real mvps!   $1 per month helps!! :)  https://www.patreon.com/patrickjmt !! Interval and Radius of Convergence for a Series, Ex 3. In this video, I show another example of finding the interval and radius of convergence for a series.", "lengthSeconds": 134, "width": 480, "height": 360}
{"video_id": "Xv_7Pyl09xQ", "transcript": [{"start": 0.11, "duration": 3.7, "text": "all right so here we're going to look at"}, {"start": 2.159, "duration": 3.87, "text": "some more examples of multiplying"}, {"start": 3.81, "duration": 4.17, "text": "fractions and again I'm going to use"}, {"start": 6.029, "duration": 4.711, "text": "this this idea of getting rid of common"}, {"start": 7.98, "duration": 6.0, "text": "factors immediately so we have 7 over 5"}, {"start": 10.74, "duration": 8.4, "text": "times 5 over 6 again we could write that"}, {"start": 13.98, "duration": 6.9, "text": "as 7 times 5 over 5 times 6 well we"}, {"start": 19.14, "duration": 3.059, "text": "could multiply immediately but then we"}, {"start": 20.88, "duration": 4.35, "text": "would just immediately have to get rid"}, {"start": 22.199, "duration": 5.16, "text": "of common factors so I notice you know"}, {"start": 25.23, "duration": 4.379, "text": "there's a multiplication by 5 in the"}, {"start": 27.359, "duration": 4.711, "text": "numerator and a multiplication by 5 in"}, {"start": 29.609, "duration": 6.751, "text": "the denominator I can cancel those out"}, {"start": 32.07, "duration": 5.94, "text": "again 5 divided by 5 is just 1 so we're"}, {"start": 36.36, "duration": 4.8, "text": "left with you know if you want to you"}, {"start": 38.01, "duration": 5.549, "text": "can put a 1 back in there so you have 7"}, {"start": 41.16, "duration": 4.8, "text": "times 1 in the numerator which is 7 and"}, {"start": 43.559, "duration": 7.311, "text": "then 1 times 6 in the denominator which"}, {"start": 45.96, "duration": 9.48, "text": "is 6 and again we have our solution"}, {"start": 50.87, "duration": 6.849, "text": "suppose we have 12 times 7 over 15 well"}, {"start": 55.44, "duration": 3.99, "text": "the first thing I do is usually I'll"}, {"start": 57.719, "duration": 4.98, "text": "write my numbers I'll write everything"}, {"start": 59.43, "duration": 4.08, "text": "as a fraction I can write 12 as just 12"}, {"start": 62.699, "duration": 3.721, "text": "over 1"}, {"start": 63.51, "duration": 7.859, "text": "okay so we're multiplying by again 7"}, {"start": 66.42, "duration": 8.07, "text": "over 15 well again I recognize that 12"}, {"start": 71.369, "duration": 6.661, "text": "and 15 both have a common factor of 3 so"}, {"start": 74.49, "duration": 6.87, "text": "I say well 12 divided by 3 that's going"}, {"start": 78.03, "duration": 6.96, "text": "to leave me with 4 15 divided by 3"}, {"start": 81.36, "duration": 5.46, "text": "that's going to leave me with a 5 so all"}, {"start": 84.99, "duration": 5.1, "text": "I'm really reducing all I'm really doing"}, {"start": 86.82, "duration": 5.85, "text": "is I'm reducing 12 over 15 and reducing"}, {"start": 90.09, "duration": 5.22, "text": "that to 4 over 5 and again we just"}, {"start": 92.67, "duration": 5.659, "text": "multiply what's left over so 4 times 7"}, {"start": 95.31, "duration": 6.84, "text": "in the numerator will leave us with 28"}, {"start": 98.329, "duration": 9.82, "text": "and in the denominator we have 1 times 5"}, {"start": 102.15, "duration": 10.44, "text": "which is 5 so again got our solution so"}, {"start": 108.149, "duration": 7.141, "text": "maybe do one more here so 6 over 45"}, {"start": 112.59, "duration": 5.849, "text": "times 9 again I'm going to write 9 as"}, {"start": 115.29, "duration": 4.02, "text": "just 9 over 1 but now I can do the same"}, {"start": 118.439, "duration": 5.82, "text": "thing as before"}, {"start": 119.31, "duration": 9.869, "text": "again just cancelling out factors okay"}, {"start": 124.259, "duration": 8.801, "text": "so 9 and 45 have a common factor namely"}, {"start": 129.179, "duration": 8.591, "text": "9 well let's see 9 divided by 9"}, {"start": 133.06, "duration": 8.759, "text": "you would leave me with 140 5/9 would"}, {"start": 137.77, "duration": 7.02, "text": "leave me with a 5 so now I've got a 6 in"}, {"start": 141.819, "duration": 5.371, "text": "the numerator a 1 in the numerator 5 in"}, {"start": 144.79, "duration": 3.12, "text": "the denominator and again just a 1 in"}, {"start": 147.19, "duration": 3.03, "text": "the denominator"}, {"start": 147.91, "duration": 5.73, "text": "there's no more common factors so I'm"}, {"start": 150.22, "duration": 7.11, "text": "just going to multiply 6 times 1 is 6 5"}, {"start": 153.64, "duration": 8.15, "text": "times 1 is 5 and again now we've got our"}, {"start": 157.33, "duration": 4.46, "text": "solution and it's nice and reduced"}], "title": "Multiplying Fractions and Mixed Numbers, Example 2", "description": "Thanks to all of you who support me on Patreon. You da real mvps!   $1 per month helps!! :)  https://www.patreon.com/patrickjmt !! Multiplying Fractions and Mixed Numbers, Example 2", "lengthSeconds": 161, "width": 480, "height": 360}
{"video_id": "qcGQIMRCvsM", "transcript": [{"start": 0.76, "duration": 4.09, "text": "Alright: in this video I'm gonna do an  example of \nrewriting complex fractions"}, {"start": 4.85, "duration": 4.37, "text": "and there's two different ways that you can\ngo about simplifying complex fractions"}, {"start": 9.22, "duration": 3.99, "text": "and again it's just a complex fraction\nall that is is you know you got a big"}, {"start": 13.21, "duration": 0.809, "text": "fraction"}, {"start": 14.019, "duration": 4.161, "text": "and then within that you have even more\nfractions is the basic idea so"}, {"start": 18.18, "duration": 5.269, "text": "again two ways to do 'em one way is to\nlook at all the denominators"}, {"start": 23.449, "duration": 4.881, "text": "and you can multiply the top and bottom\nby the least common multiple"}, {"start": 28.33, "duration": 4.63, "text": "of the of the denominators and the ideas\nthat'll get rid of"}, {"start": 32.96, "duration": 3.4, "text": "everything except for it you know the\none big attraction"}, {"start": 36.36, "duration": 4.57, "text": "another thing is to simply write the\nnumerator as a single fraction"}, {"start": 40.93, "duration": 4.03, "text": "and the denominators a single fraction\nand then we're gonna do you is just use"}, {"start": 44.96, "duration": 0.27, "text": "that"}, {"start": 45.23, "duration": 4.21, "text": "the trick that we can flip and multiply\nand that's how I've always done so"}, {"start": 49.44, "duration": 1.119, "text": "that's how I'm gonna do"}, {"start": 50.559, "duration": 6.051, "text": "this example and the others as well\nso we have 2 over "x" plus three divided"}, {"start": 56.61, "duration": 1.83, "text": "by 1 plus 3 divided by "x""}, {"start": 58.44, "duration": 3.58, "text": "you can think about the wine is being\none divided by one"}, {"start": 62.02, "duration": 5.08, "text": "and again what I want to do is the\nnumerator is already a single fraction"}, {"start": 67.1, "duration": 3.43, "text": "I'm gonna write the denominator also is\na single fraction"}, {"start": 70.53, "duration": 5.36, "text": "so in this case I look at the denominators\nwe have 1 and "x""}, {"start": 75.89, "duration": 4.299, "text": "so to get a common denominator I could\nmultiply the first term"}, {"start": 80.189, "duration": 5.911, "text": "a by "x" over "x" and in the numerator\nwill simply leave that alone"}, {"start": 86.1, "duration": 4.659, "text": "2 over x+3 and"}, {"start": 90.759, "duration": 3.911, "text": "then and the denominator we would have\nx over x"}, {"start": 94.67, "duration": 3.769, "text": "plus three over x and again the"}, {"start": 98.439, "duration": 3.981, "text": "the whole reason we we did that is so\nthat we could write the denominators a"}, {"start": 102.42, "duration": 1.11, "text": "single fraction"}, {"start": 103.53, "duration": 3.9, "text": "so often times all you can skip this step\nso"}, {"start": 107.43, "duration": 3.979, "text": "in the numerator of the denominator\nwe would have  x +3"}, {"start": 111.409, "duration": 5.121, "text": "which we can't really do much with\ndivided by our common denominator"}, {"start": 116.53, "duration": 3.019, "text": "of x so this is kinda"}, {"start": 119.549, "duration": 3.871, "text": "the first half but getting the numerator\nsingle fraction"}, {"start": 123.42, "duration": 4.1, "text": "and now the denominator as a single\nfraction and"}, {"start": 127.52, "duration": 3.71, "text": "remember the trick is if you have a\nfraction divided by a fraction"}, {"start": 131.23, "duration": 4.589, "text": "we leave the top fraction alone and we\ntake the denominator"}, {"start": 135.819, "duration": 3.941, "text": "we flip it over but we turn it into\nmultiplication"}, {"start": 139.76, "duration": 3.11, "text": "okay so instead of divided by"}, {"start": 142.87, "duration": 3.8, "text": "X plus three over X we're multiplying by\nx-over"}, {"start": 146.67, "duration": 3.289, "text": "X plus three and in this case on"}, {"start": 149.959, "duration": 3.621, "text": "you know if there is anything to you to\nfactor and cancel"}, {"start": 153.58, "duration": 3.39, "text": "I would certainly try to factor in\ncancel"}, {"start": 156.97, "duration": 3.43, "text": "but in this case in the numerator we\nhave a 2 times an x"}, {"start": 160.4, "duration": 3.38, "text": "in the denominator we have x plus three in\nx plus three"}, {"start": 163.78, "duration": 3.9, "text": "I would probably rewrite this simply is\n2 x"}, {"start": 167.68, "duration": 3.4, "text": "and then X plus 3 quantity squared"}, {"start": 171.08, "duration": 3.739, "text": "again you can multiply denominator out\nyou could FOIL it out"}, {"start": 174.819, "duration": 4.5, "text": "x squared plus 6x plus 9 but I think"}, {"start": 179.319, "duration": 3.53, "text": "my own personal preference would again\njust to be leave it"}, {"start": 182.849, "duration": 1.42, "text": "in this nice factored form"}], "title": "\u2756 Simplifying Complex Fractions - Ex 1 \u2756", "description": "Simplifying Complex Fractions - Example 1\nIn this video, I break down the process of simplifying a complex fraction. Starting from the basics of finding a common denominator for the terms in the original denominator, I then demonstrate how to flip, multiply, and cancel to simplify the fraction effectively. This method is essential for handling more complicated expressions and is a foundational skill in algebra and calculus.\n\nWhat You Will Learn:\nFinding a Common Denominator: Learn the first step in simplifying complex fractions by finding a common denominator for the terms in the original denominator.\nRewriting the Fraction: Understand how to rewrite the complex fraction by flipping and multiplying, a key process in simplification.\nCanceling Terms to Simplify: Watch how to cancel terms to reduce the fraction to its simplest form.\nStep-by-Step Example: Follow a clear, guided example that covers each part of the process, making it easy to understand and apply to other problems.\nIf you find this video helpful, please subscribe for more tutorials on algebra, calculus, and other math topics! Don't forget to like, comment, and share the video. Support me on Patreon to access more exclusive content:\nhttps://www.patreon.com/patrickjmt?ty=c\n\n#complexfractions #algebra #fractionmultiplication #simplifyingfractions #commondenominator #patrickjmt #mathhelp #mathvideos #math #rationalexpressions #fractionreduction #algebratips #mathtutorial #fractionsimplification #mathpractice #mathstepbystep #maths", "lengthSeconds": 185, "width": 480, "height": 360}
{"video_id": "jZgwkECLKRQ", "transcript": [{"start": 0.06, "duration": 4.949, "text": "okay here we're gonna look at some anti"}, {"start": 2.129, "duration": 5.071, "text": "derivatives of some trig and some"}, {"start": 5.009, "duration": 4.981, "text": "exponential functions and the only rule"}, {"start": 7.2, "duration": 5.01, "text": "of anti derivative formula that we'll"}, {"start": 9.99, "duration": 5.059, "text": "need here for the exponential is just"}, {"start": 12.21, "duration": 4.86, "text": "the antiderivative of e to the X so"}, {"start": 15.049, "duration": 4.57, "text": "again we could have a different"}, {"start": 17.07, "duration": 5.16, "text": "exponential instead of e we could have 2"}, {"start": 19.619, "duration": 5.041, "text": "or 3 or 1/2 or some different number but"}, {"start": 22.23, "duration": 5.01, "text": "for now we'll just stick with e to the X"}, {"start": 24.66, "duration": 8.04, "text": "and again the antiderivative of e to the"}, {"start": 27.24, "duration": 9.21, "text": "X is just e to the X and again plus C so"}, {"start": 32.7, "duration": 6.69, "text": "we'll make use of this formula so when"}, {"start": 36.45, "duration": 7.05, "text": "we do the first integral here so the"}, {"start": 39.39, "duration": 9.989, "text": "integral with negative 5 e to the X plus"}, {"start": 43.5, "duration": 8.489, "text": "7 sine X well the negative 5 I can just"}, {"start": 49.379, "duration": 5.641, "text": "you know carry that along we just saw"}, {"start": 51.989, "duration": 7.37, "text": "the antiderivative of e to the X is e to"}, {"start": 55.02, "duration": 6.929, "text": "the X let's see then I think whose"}, {"start": 59.359, "duration": 5.77, "text": "derivative is sine X or equivalently"}, {"start": 61.949, "duration": 6.061, "text": "what's the antiderivative of sine X and"}, {"start": 65.129, "duration": 5.161, "text": "at first maybe you say hey cosine X but"}, {"start": 68.01, "duration": 4.649, "text": "recall the derivative of cosine X would"}, {"start": 70.29, "duration": 4.53, "text": "actually give us a negative sign so to"}, {"start": 72.659, "duration": 4.89, "text": "cancel out that negative we'll use a"}, {"start": 74.82, "duration": 4.2, "text": "negative cosine X and now I think we've"}, {"start": 77.549, "duration": 4.231, "text": "done everything all we have to do is"}, {"start": 79.02, "duration": 4.23, "text": "just include our plus C the last thing"}, {"start": 81.78, "duration": 4.35, "text": "I'm gonna do is just sort of clean it up"}, {"start": 83.25, "duration": 4.799, "text": "that's negative 5 e to the X we could"}, {"start": 86.13, "duration": 6.69, "text": "write you know positive 7 and negative"}, {"start": 88.049, "duration": 8.271, "text": "cosine is just negative 7 cosine X plus"}, {"start": 92.82, "duration": 7.89, "text": "C so that would be our antiderivative"}, {"start": 96.32, "duration": 5.799, "text": "all right let's look at a let's look at"}, {"start": 100.71, "duration": 3.479, "text": "the other one as well here I think you"}, {"start": 102.119, "duration": 4.921, "text": "can knock them both out ok so the first"}, {"start": 104.189, "duration": 5.011, "text": "part let's see we've got 2 e to the X I"}, {"start": 107.04, "duration": 5.25, "text": "say hey no problem there we've got a"}, {"start": 109.2, "duration": 5.61, "text": "negative 1 and hey no problem there but"}, {"start": 112.29, "duration": 6.42, "text": "then I see the sine times cosecant and I"}, {"start": 114.81, "duration": 5.4, "text": "think this is a product so I have to be"}, {"start": 118.71, "duration": 3.03, "text": "a little more careful when I go to you"}, {"start": 120.21, "duration": 3.39, "text": "know find the antiderivative of this"}, {"start": 121.74, "duration": 3.9, "text": "function and I don't think that's what I"}, {"start": 123.6, "duration": 5.76, "text": "know just immediately off the top of my"}, {"start": 125.64, "duration": 4.739, "text": "head you know again you can't do them"}, {"start": 129.36, "duration": 3.32, "text": "individually when they're being"}, {"start": 130.379, "duration": 8.661, "text": "multiplied but notice kind of"}, {"start": 132.68, "duration": 9.179, "text": "you know the trick on this I mean what"}, {"start": 139.04, "duration": 6.179, "text": "is cosecant X well cosecant X is the"}, {"start": 141.859, "duration": 7.25, "text": "same thing as 1 over sine X so really"}, {"start": 145.219, "duration": 3.89, "text": "just kind of a sneaky little problem"}, {"start": 150.43, "duration": 5.529, "text": "because really when we take positive"}, {"start": 152.87, "duration": 7.229, "text": "sign and divide it by sine X that's just"}, {"start": 155.959, "duration": 5.64, "text": "gonna give us a positive 1 well we can"}, {"start": 160.099, "duration": 3.39, "text": "just cancel out the negative 1 and the"}, {"start": 161.599, "duration": 4.711, "text": "positive 1 so now we're really just"}, {"start": 163.489, "duration": 4.351, "text": "finding the antiderivative of 2 e to the"}, {"start": 166.31, "duration": 4.259, "text": "X DX"}, {"start": 167.84, "duration": 6.149, "text": "well again the antiderivative of e to"}, {"start": 170.569, "duration": 5.34, "text": "the X is just e to the X and then we'll"}, {"start": 173.989, "duration": 3.511, "text": "be left with our plus C we have to tack"}, {"start": 175.909, "duration": 3.961, "text": "that on so that would be the"}, {"start": 177.5, "duration": 4.349, "text": "antiderivative now of our second"}, {"start": 179.87, "duration": 3.179, "text": "function here so again you know"}, {"start": 181.849, "duration": 4.86, "text": "sometimes people will give you these"}, {"start": 183.049, "duration": 4.741, "text": "sort of you know contrived problems and"}, {"start": 186.709, "duration": 3.51, "text": "again what they're just trying to"}, {"start": 187.79, "duration": 4.02, "text": "illustrate to is a you know make sure"}, {"start": 190.219, "duration": 3.931, "text": "you remember trig identities this is"}, {"start": 191.81, "duration": 3.959, "text": "important and just the idea that you do"}, {"start": 194.15, "duration": 4.35, "text": "a lot of times have to do some sort of"}, {"start": 195.769, "duration": 5.34, "text": "algebraic manipulation to make things"}, {"start": 198.5, "duration": 6.44, "text": "work out so rarely do we get things"}, {"start": 201.109, "duration": 3.831, "text": "where it just sort of immediately works"}], "title": "Calculus : Antiderivative / Integration Problems with Trig and Exponential Functions", "description": "In this video we find the antiderivatives / integrate two different functions, each of which involve an exponential function as well as a trigonometric function.\nNotice that it is often good to look for any algebra simplifications before starting, such as in the second example.\n\n\n\n\n\nIntegrate e^x\nIntegrate sinx\n\n\n\n#math, #algebra, #khanacademy, #khan, #geometry, #education, #montessori, #calculus, #homeschooling, #functions, #notation, #organicchemistry, , #calculus, #derivative, #powerrule, #limits,  #limit, #graph, #graphing, #conjugate, #infinity, #patrickjmt, #rational, #shortcut, #trick, #commondenominator, #fraction, #university, #admissions, #test #continuity #onesided #infinite #absolutevalue #trigonometry #trigonometric #integration #integrate", "lengthSeconds": 204, "width": 480, "height": 360}
{"video_id": "vzPzSS2heK8", "transcript": [{"start": 0.0, "duration": 4.02, "text": "all right in this video I'm going to do"}, {"start": 2.22, "duration": 4.619, "text": "one more example of solving an equation"}, {"start": 4.02, "duration": 4.59, "text": "containing a radical so in this case we"}, {"start": 6.839, "duration": 4.711, "text": "have the square root of 3x plus 9 minus"}, {"start": 8.61, "duration": 5.909, "text": "x equals negative 3 so the first thing"}, {"start": 11.55, "duration": 5.37, "text": "that I do is i isolate the radical I get"}, {"start": 14.519, "duration": 5.881, "text": "that all by itself so in this case to do"}, {"start": 16.92, "duration": 5.04, "text": "that I'm going to add X to both sides so"}, {"start": 20.4, "duration": 5.25, "text": "we would be left with the square root of"}, {"start": 21.96, "duration": 6.479, "text": "3x plus 9 on the right side I could"}, {"start": 25.65, "duration": 3.69, "text": "write negative 3 plus X but I think I'm"}, {"start": 28.439, "duration": 4.831, "text": "just gonna go ahead and write that"}, {"start": 29.34, "duration": 5.73, "text": "equivalently as X minus 3 now again to"}, {"start": 33.27, "duration": 8.25, "text": "get rid of the square root we simply"}, {"start": 35.07, "duration": 8.46, "text": "square both sides the square root of 3x"}, {"start": 41.52, "duration": 7.289, "text": "plus 9 squared is just going to give us"}, {"start": 43.53, "duration": 7.29, "text": "3x plus 9x minus 3 squared we have to"}, {"start": 48.809, "duration": 5.461, "text": "foil that out or distribute everything"}, {"start": 50.82, "duration": 6.54, "text": "out so on the Left again we just have 3x"}, {"start": 54.27, "duration": 5.16, "text": "plus 9 on the right when I multiply it"}, {"start": 57.36, "duration": 4.56, "text": "out I'll get x times X which is x"}, {"start": 59.43, "duration": 4.5, "text": "squared I'm going to get a negative 3x"}, {"start": 61.92, "duration": 5.55, "text": "and another negative 3x which is"}, {"start": 63.93, "duration": 6.42, "text": "negative 6x negative 3 and negative 3 is"}, {"start": 67.47, "duration": 5.1, "text": "going to give us a positive 9 so now"}, {"start": 70.35, "duration": 4.62, "text": "again I've got a more familiar quadratic"}, {"start": 72.57, "duration": 4.86, "text": "equation notice we could simply subtract"}, {"start": 74.97, "duration": 4.38, "text": "9 from both sides and that would get rid"}, {"start": 77.43, "duration": 4.14, "text": "of the nines the other thing I'm going"}, {"start": 79.35, "duration": 5.43, "text": "to do is I'm going to subtract 3x from"}, {"start": 81.57, "duration": 5.189, "text": "both sides so that would leave me with 0"}, {"start": 84.78, "duration": 4.519, "text": "on the left and then I would have x"}, {"start": 86.759, "duration": 5.481, "text": "squared minus 9x"}, {"start": 89.299, "duration": 4.53, "text": "and again I like the quadratic equations"}, {"start": 92.24, "duration": 3.659, "text": "where there's no constant because"}, {"start": 93.829, "duration": 7.381, "text": "basically all you have to do in this"}, {"start": 95.899, "duration": 6.9, "text": "case is we have to factor out the X and"}, {"start": 101.21, "duration": 6.269, "text": "then in parentheses I think we would"}, {"start": 102.799, "duration": 6.66, "text": "have X minus 9 left over if we set the"}, {"start": 107.479, "duration": 5.161, "text": "first factor equal to zero we'll simply"}, {"start": 109.459, "duration": 5.34, "text": "get x equals zero as a solution if we"}, {"start": 112.64, "duration": 4.86, "text": "set X minus nine equal to zero we can"}, {"start": 114.799, "duration": 6.27, "text": "add 9 to both sides and then simply get"}, {"start": 117.5, "duration": 6.03, "text": "x equals positive 9 as our other"}, {"start": 121.069, "duration": 4.44, "text": "solution I should say potential"}, {"start": 123.53, "duration": 6.0, "text": "solutions again you always need to go"}, {"start": 125.509, "duration": 6.421, "text": "back and check notice if you plug 0 into"}, {"start": 129.53, "duration": 5.91, "text": "the original equation we'll just have"}, {"start": 131.93, "duration": 6.45, "text": "the square root of 9 which is 3 minus 0"}, {"start": 135.44, "duration": 5.359, "text": "that would give us 3 but 3 is not equal"}, {"start": 138.38, "duration": 5.31, "text": "to negative 3 slip solution doesn't work"}, {"start": 140.799, "duration": 6.851, "text": "notice if we plug 9 in we would get 3"}, {"start": 143.69, "duration": 7.049, "text": "times 9 which is 27 plus 9 is gonna be"}, {"start": 147.65, "duration": 8.01, "text": "what 36 so then we would have the square"}, {"start": 150.739, "duration": 6.961, "text": "root of 36 minus 9 and we're asking"}, {"start": 155.66, "duration": 5.579, "text": "ourselves does that give us negative 3"}, {"start": 157.7, "duration": 5.159, "text": "well the square root of 36 is 6 6 minus"}, {"start": 161.239, "duration": 5.011, "text": "9 does equal negative 3"}, {"start": 162.859, "duration": 7.22, "text": "yea that all works so it says x equals 9"}, {"start": 166.25, "duration": 7.43, "text": "is a solution to our original equation"}, {"start": 170.079, "duration": 3.601, "text": "involving the radical"}], "title": "Solving an Equation Involving a Single Radical (Square Root) - Example 3", "description": "Thanks to all of you who support me on Patreon. You da real mvps!   $1 per month helps!! :)  https://www.patreon.com/patrickjmt !! Solving an Equation Involving a Single Radical (Square Root) - Example 3", "lengthSeconds": 173, "width": 480, "height": 360}
{"video_id": "OKHIiD1OeMg", "transcript": [{"start": 0.719, "duration": 3.921, "text": "okay so one quick question here uh here"}, {"start": 2.84, "duration": 7.0, "text": "we want to find the X and Y intercepts"}, {"start": 4.64, "duration": 8.56, "text": "of this graph Ral 5 - 4 cosine Theta and"}, {"start": 9.84, "duration": 4.56, "text": "the idea is for these types of graphs"}, {"start": 13.2, "duration": 2.52, "text": "what they're going to turn out to be is"}, {"start": 14.4, "duration": 4.12, "text": "what's called a"}, {"start": 15.72, "duration": 5.719, "text": "lemon and if your a value is greater"}, {"start": 18.52, "duration": 4.919, "text": "than your B value so here our a value"}, {"start": 21.439, "duration": 5.641, "text": "would be five kind of forget about the"}, {"start": 23.439, "duration": 5.24, "text": "negative your B value would be four um"}, {"start": 27.08, "duration": 6.2, "text": "and if both of those are greater than"}, {"start": 28.679, "duration": 7.2, "text": "zero okay so a is 5 b is 4 it says then"}, {"start": 33.28, "duration": 5.08, "text": "to get the X intercepts we simply add a"}, {"start": 35.879, "duration": 4.041, "text": "plus b and we take the negative of it"}, {"start": 38.36, "duration": 3.679, "text": "and then so that'll be one of our X"}, {"start": 39.92, "duration": 5.119, "text": "intercepts and then we take a minus B"}, {"start": 42.039, "duration": 4.601, "text": "that'll be our other x intercept the Y"}, {"start": 45.039, "duration": 4.641, "text": "intercepts are just negative a and"}, {"start": 46.64, "duration": 4.439, "text": "positive a if your B value is bigger"}, {"start": 49.68, "duration": 3.48, "text": "than your a"}, {"start": 51.079, "duration": 4.401, "text": "value it's going to turn out that the X"}, {"start": 53.16, "duration": 4.48, "text": "and Y intercepts also includes zero but"}, {"start": 55.48, "duration": 5.719, "text": "we don't have that in this"}, {"start": 57.64, "duration": 7.04, "text": "case so simply um we we've got five is"}, {"start": 61.199, "duration": 6.841, "text": "greater than four is greater than zero"}, {"start": 64.68, "duration": 6.68, "text": "this is our a value this is our B value"}, {"start": 68.04, "duration": 5.32, "text": "again it says take negative of a + b and"}, {"start": 71.36, "duration": 6.52, "text": "then take a minus"}, {"start": 73.36, "duration": 6.96, "text": "B so a plus b would just be 5 + 4 which"}, {"start": 77.88, "duration": 5.919, "text": "would give us -9 that's going to be one"}, {"start": 80.32, "duration": 7.64, "text": "of our X intercepts if we do a minus B"}, {"start": 83.799, "duration": 6.36, "text": "that'll be 5 - 4 or pos1 so that'll be"}, {"start": 87.96, "duration": 3.96, "text": "our other x intercept and and then it"}, {"start": 90.159, "duration": 5.161, "text": "says the Y intercepts we just take"}, {"start": 91.92, "duration": 7.239, "text": "negative a and positive a so the Y"}, {"start": 95.32, "duration": 6.04, "text": "intercepts will be at5 and at posi five"}, {"start": 99.159, "duration": 4.64, "text": "and that's it now we're done so we've"}, {"start": 101.36, "duration": 4.96, "text": "got our X intercepts over here we've got"}, {"start": 103.799, "duration": 5.36, "text": "our two y intercepts here and that's it"}, {"start": 106.32, "duration": 4.52, "text": "so again uh you'll see polar curves here"}, {"start": 109.159, "duration": 3.481, "text": "and there uh if you end up taking a"}, {"start": 110.84, "duration": 4.76, "text": "Calculus class kind of sprinkled"}, {"start": 112.64, "duration": 4.439, "text": "throughout the course um uh these are"}, {"start": 115.6, "duration": 3.32, "text": "formulas that definitely I kind of"}, {"start": 117.079, "duration": 4.161, "text": "forget I am aware in fact that there are"}, {"start": 118.92, "duration": 4.28, "text": "formulas out there so you know when I do"}, {"start": 121.24, "duration": 3.32, "text": "start doing polar stuff again I say oh"}, {"start": 123.2, "duration": 3.0, "text": "yeah there's these formulas and I dig"}, {"start": 124.56, "duration": 3.4, "text": "out my pre-calculus book or my"}, {"start": 126.2, "duration": 4.52, "text": "trigonometry book and I look it up and I"}, {"start": 127.96, "duration": 4.6, "text": "say oh yeah that was it so um you know"}, {"start": 130.72, "duration": 3.879, "text": "definitely stuff that's probably easy to"}, {"start": 132.56, "duration": 3.72, "text": "forget but at least be aware that"}, {"start": 134.599, "duration": 3.441, "text": "they're out there because sometimes"}, {"start": 136.28, "duration": 4.0, "text": "graphing these polar curves can be a bit"}, {"start": 138.04, "duration": 5.24, "text": "of a pain so some of these little"}, {"start": 140.28, "duration": 7.84, "text": "shortcuts will definitely make uh make"}, {"start": 143.28, "duration": 4.84, "text": "producing a good graph a lot easier"}], "title": "Basic Info About a Limacon", "description": "Thanks to all of you who support me on Patreon. You da real mvps!   $1 per month helps!! :)  https://www.patreon.com/patrickjmt !! Graphing a Limacon - Just some info about how to find x and y intercepts for a limacon (I do not actually graph it, sorry!).", "lengthSeconds": 146, "width": 480, "height": 360}
{"video_id": "uMgK000XFhA", "transcript": [{"start": 0.84, "duration": 2.96, "text": "all right in this video I want to talk"}, {"start": 2.08, "duration": 3.96, "text": "about what's called chbby shev's theorem"}, {"start": 3.8, "duration": 4.52, "text": "and chbby shev's Theorem is pretty"}, {"start": 6.04, "duration": 4.0, "text": "simple and easy to use and what it says"}, {"start": 8.32, "duration": 4.279, "text": "it says the proportion of any"}, {"start": 10.04, "duration": 5.479, "text": "distribution that lies within K standard"}, {"start": 12.599, "duration": 6.361, "text": "deviations of the mean is at least 1 - 1"}, {"start": 15.519, "duration": 5.961, "text": "k s where K is any positive number"}, {"start": 18.96, "duration": 3.84, "text": "greater than one the big use is that"}, {"start": 21.48, "duration": 4.44, "text": "this theorem applies to all"}, {"start": 22.8, "duration": 4.28, "text": "distributions distributions of data um"}, {"start": 25.92, "duration": 2.679, "text": "you know so if you know your data is"}, {"start": 27.08, "duration": 3.599, "text": "normally distributed you can come up"}, {"start": 28.599, "duration": 4.241, "text": "with a stronger result than but often"}, {"start": 30.679, "duration": 5.681, "text": "times you don't know"}, {"start": 32.84, "duration": 6.039, "text": "that so for example"}, {"start": 36.36, "duration": 4.679, "text": "um this theorem says that within two"}, {"start": 38.879, "duration": 5.481, "text": "standard deviations of the mean you'll"}, {"start": 41.039, "duration": 5.761, "text": "always find at least well since in this"}, {"start": 44.36, "duration": 7.519, "text": "case k is going to equal two we'll get 1"}, {"start": 46.8, "duration": 7.52, "text": "- 1 over 2^ 2 which is 1 - a 4 or 34s"}, {"start": 51.879, "duration": 5.881, "text": "which is 75 or it says you're going to"}, {"start": 54.32, "duration": 7.559, "text": "get at least 75% of your"}, {"start": 57.76, "duration": 6.72, "text": "data let's do uh kind of a specific"}, {"start": 61.879, "duration": 2.601, "text": "example"}, {"start": 64.92, "duration": 6.879, "text": "here so suppose that the average score"}, {"start": 69.24, "duration": 4.96, "text": "on a math test is an 84 with a standard"}, {"start": 71.799, "duration": 4.481, "text": "deviation of Four Points according to"}, {"start": 74.2, "duration": 5.68, "text": "Chevy Chef's theorem at least what per"}, {"start": 76.28, "duration": 5.36, "text": "of the tests have a grade of a 72 and at"}, {"start": 79.88, "duration": 6.36, "text": "most a"}, {"start": 81.64, "duration": 7.799, "text": "96 well okay so the idea"}, {"start": 86.24, "duration": 7.159, "text": "again so here's our average 84 that's"}, {"start": 89.439, "duration": 6.04, "text": "our mean mu notice if we count over um"}, {"start": 93.399, "duration": 3.601, "text": "one standard deviation so if we go over"}, {"start": 95.479, "duration": 4.161, "text": "a standard deviation again we're"}, {"start": 97.0, "duration": 5.439, "text": "basically adding four points that would"}, {"start": 99.64, "duration": 4.64, "text": "put us at a grade of an 88 if we go over"}, {"start": 102.439, "duration": 4.201, "text": "another standard deviation or another"}, {"start": 104.28, "duration": 5.0, "text": "four points that's going to put us at a"}, {"start": 106.64, "duration": 5.04, "text": "92 and if we go over another standard"}, {"start": 109.28, "duration": 5.56, "text": "deviation that's going to put us at our"}, {"start": 111.68, "duration": 5.32, "text": "uh our grade of a 96 which we wanted"}, {"start": 114.84, "duration": 4.639, "text": "likewise notice if you go down three"}, {"start": 117.0, "duration": 4.68, "text": "standard deviations you'll be at an 80"}, {"start": 119.479, "duration": 4.68, "text": "if you subtract Four Points will be at a"}, {"start": 121.68, "duration": 4.719, "text": "76 if you subtract Four Points you'll be"}, {"start": 124.159, "duration": 5.32, "text": "at a"}, {"start": 126.399, "duration": 5.401, "text": "72 so what we're doing is we're looks to"}, {"start": 129.479, "duration": 5.4, "text": "me like we're going three standard"}, {"start": 131.8, "duration": 5.4, "text": "deviations away from the mean okay so in"}, {"start": 134.879, "duration": 6.161, "text": "this case our K value is simply going to"}, {"start": 137.2, "duration": 7.48, "text": "equal three so according to chbby shev's"}, {"start": 141.04, "duration": 8.64, "text": "uh theorem we'll get 1 - 1 over K which"}, {"start": 144.68, "duration": 8.68, "text": "is 3^ sared that's going to be 1 - 1 9th"}, {"start": 149.68, "duration": 6.52, "text": "well we can write that as 9 over 99 - 1"}, {"start": 153.36, "duration": 6.519, "text": "over9 do the arithmetic on the top we"}, {"start": 156.2, "duration": 5.2, "text": "get 8 9 8 9 which is going to be uh"}, {"start": 159.879, "duration": 8.72, "text": "roughly equal"}, {"start": 161.4, "duration": 11.16, "text": "to 89 if we round and 89 equals hey"}, {"start": 168.599, "duration": 6.841, "text": "89% so it says if uh in your class if if"}, {"start": 172.56, "duration": 4.679, "text": "the teacher told you you know hey the"}, {"start": 175.44, "duration": 4.48, "text": "the you know the the test scores were an"}, {"start": 177.239, "duration": 5.201, "text": "84 the average was an 84 and a standard"}, {"start": 179.92, "duration": 6.44, "text": "deviation of Four Points um it says that"}, {"start": 182.44, "duration": 8.439, "text": "89% of the class made a grade uh between"}, {"start": 186.36, "duration": 4.519, "text": "72 and 96"}], "title": "Chebyshev's Theorem", "description": "Thanks to all of you who support me on Patreon. You da real mvps!   $1 per month helps!! :)  https://www.patreon.com/patrickjmt !! Chebyshev's Theorem - In this video, I state Chebyshev's Theorem and use it in a 'real life' problem.", "lengthSeconds": 190, "width": 480, "height": 360}
{"video_id": "w4P_CteN6V8", "transcript": [{"start": 0.0, "duration": 6.33, "text": "all right in this video I'm gonna graph"}, {"start": 2.46, "duration": 8.55, "text": "the equation y equals 4x minus 1 by"}, {"start": 6.33, "duration": 6.389, "text": "plotting points ok it turns out down the"}, {"start": 11.01, "duration": 4.35, "text": "road you'll see this if you have a"}, {"start": 12.719, "duration": 4.951, "text": "problem just involving a Y to the first"}, {"start": 15.36, "duration": 5.37, "text": "power X to the first power and some"}, {"start": 17.67, "duration": 4.59, "text": "numbers you know kind of a few more"}, {"start": 20.73, "duration": 2.85, "text": "restrictions on that but this is going"}, {"start": 22.26, "duration": 5.189, "text": "to turn out to just make a straight line"}, {"start": 23.58, "duration": 5.609, "text": "when you graph it but you know assuming"}, {"start": 27.449, "duration": 3.271, "text": "we weren't aware of that at this point"}, {"start": 29.189, "duration": 3.03, "text": "let's just do the same thing of plots"}, {"start": 30.72, "duration": 3.63, "text": "and points"}, {"start": 32.219, "duration": 6.951, "text": "maybe we can plug in x equals negative 2"}, {"start": 34.35, "duration": 7.639, "text": "negative 1 0 positive 1 and positive 2"}, {"start": 39.17, "duration": 5.26, "text": "again you can use any numbers you want"}, {"start": 41.989, "duration": 4.871, "text": "typically I try to graph stuff around"}, {"start": 44.43, "duration": 5.25, "text": "the origin and just you know smaller"}, {"start": 46.86, "duration": 4.83, "text": "numbers to make the arithmetic easier so"}, {"start": 49.68, "duration": 4.41, "text": "if we use x equals negative 2"}, {"start": 51.69, "duration": 6.509, "text": "we'll get that y equals 4 times negative"}, {"start": 54.09, "duration": 6.51, "text": "2 minus 1 4 times negative 2 is negative"}, {"start": 58.199, "duration": 5.61, "text": "8 negative 8 minus 1 is going to give us"}, {"start": 60.6, "duration": 6.03, "text": "negative 9 so it says the Y value of"}, {"start": 63.809, "duration": 6.981, "text": "negative 9 goes with X the x value of"}, {"start": 66.63, "duration": 7.44, "text": "negative 2 let's see x equals negative 1"}, {"start": 70.79, "duration": 7.45, "text": "we'll get that y equals 4 times negative"}, {"start": 74.07, "duration": 7.08, "text": "1 minus 1 so negative 4 minus 1 is going"}, {"start": 78.24, "duration": 6.58, "text": "to give us negative 5"}, {"start": 81.15, "duration": 5.88, "text": "if we plug 0 in we'll just be left with"}, {"start": 84.82, "duration": 6.11, "text": "negative one so that one's not too bad"}, {"start": 87.03, "duration": 7.06, "text": "if we plug in x equals positive one"}, {"start": 90.93, "duration": 6.34, "text": "that'll give y equals four times one"}, {"start": 94.09, "duration": 4.35, "text": "minus one well four minus one equals"}, {"start": 97.27, "duration": 4.77, "text": "three"}, {"start": 98.44, "duration": 6.39, "text": "and then lastly one more point if I plug"}, {"start": 102.04, "duration": 6.18, "text": "in two it looks like we get y equals"}, {"start": 104.83, "duration": 8.07, "text": "four times 2 minus 1 which is 8 minus 1"}, {"start": 108.22, "duration": 6.81, "text": "or 7 notice as we jump up as we increase"}, {"start": 112.9, "duration": 4.38, "text": "the x-values by one unit by one unit by"}, {"start": 115.03, "duration": 4.32, "text": "one unit on the left notice the"}, {"start": 117.28, "duration": 4.26, "text": "y-coordinates are jumping up by four"}, {"start": 119.35, "duration": 4.68, "text": "units by four units by four units by"}, {"start": 121.54, "duration": 4.02, "text": "four units and I think if you think"}, {"start": 124.03, "duration": 4.32, "text": "about it you can kind of convince"}, {"start": 125.56, "duration": 5.04, "text": "yourself of that that trend is going to"}, {"start": 128.35, "duration": 5.16, "text": "continue so now I'm just gonna plot"}, {"start": 130.6, "duration": 5.1, "text": "these points here again you know kind of"}, {"start": 133.51, "duration": 6.72, "text": "maybe a little rough by hand so negative"}, {"start": 135.7, "duration": 7.59, "text": "1 negative 2 negative 1 2 3 4 5 6 7 8"}, {"start": 140.23, "duration": 6.09, "text": "negative 9 so let's see if I can make"}, {"start": 143.29, "duration": 4.41, "text": "this look like the line by hand so"}, {"start": 146.32, "duration": 4.71, "text": "they're supposed to be negative 2"}, {"start": 147.7, "duration": 8.099, "text": "negative 9 here's negative 1 we go down"}, {"start": 151.03, "duration": 7.19, "text": "1 2 3 4 5 units there's another point it"}, {"start": 155.799, "duration": 5.461, "text": "says at 0 we're getting negative 1 out"}, {"start": 158.22, "duration": 6.31, "text": "it says when you plug one end we get 1 2"}, {"start": 161.26, "duration": 6.69, "text": "3 out and then it says when you plug 2"}, {"start": 164.53, "duration": 6.81, "text": "in we're up here at positive 7 and again"}, {"start": 167.95, "duration": 5.34, "text": "the idea is if you were to graph this"}, {"start": 171.34, "duration": 4.89, "text": "and then maybe if you had some graph"}, {"start": 173.29, "duration": 4.43, "text": "paper to produce a slightly better graph"}, {"start": 176.23, "duration": 3.65, "text": "than what I have"}, {"start": 177.72, "duration": 5.49, "text": "the idea is if you were to kind of play"}, {"start": 179.88, "duration": 5.16, "text": "connect the dots if you had a better"}, {"start": 183.21, "duration": 4.44, "text": "graph you would see that this line goes"}, {"start": 185.04, "duration": 3.69, "text": "through all the points and it makes a"}, {"start": 187.65, "duration": 4.47, "text": "straight line"}, {"start": 188.73, "duration": 5.91, "text": "okay so again eventually you're kind of"}, {"start": 192.12, "duration": 4.11, "text": "learned shortcuts to say well depending"}, {"start": 194.64, "duration": 3.24, "text": "on the type of equation you'll be able"}, {"start": 196.23, "duration": 3.54, "text": "so obviously I know this isn't a perfect"}, {"start": 197.88, "duration": 3.12, "text": "line it's a little a little broken there"}, {"start": 199.77, "duration": 3.75, "text": "but again it's supposed to just be a"}, {"start": 201.0, "duration": 4.98, "text": "straight line eventually you'll be able"}, {"start": 203.52, "duration": 5.88, "text": "to look at you know the exponents on the"}, {"start": 205.98, "duration": 4.92, "text": "X's and Y's and be able to say you know"}, {"start": 209.4, "duration": 3.36, "text": "you'll be able to recognize immediately"}, {"start": 210.9, "duration": 4.59, "text": "oh that's the equation of a line or"}, {"start": 212.76, "duration": 5.33, "text": "that's the equation of a circle that's"}, {"start": 215.49, "duration": 4.47, "text": "the equation of a parabola etc"}, {"start": 218.09, "duration": 4.36, "text": "eventually those are things that you'll"}, {"start": 219.96, "duration": 5.31, "text": "want to know just because graphing ends"}, {"start": 222.45, "duration": 4.47, "text": "up being a very useful it's a use of the"}, {"start": 225.27, "duration": 3.45, "text": "picture is worth a thousand words a good"}, {"start": 226.92, "duration": 3.74, "text": "graph a lot a lot of times give you a"}, {"start": 228.72, "duration": 6.71, "text": "lot of information about what's actually"}, {"start": 230.66, "duration": 4.77, "text": "how the equation is behaving"}], "title": "Graph a Line by Plotting Points - Example 2", "description": "Thanks to all of you who support me on Patreon. You da real mvps!   $1 per month helps!! :)  https://www.patreon.com/patrickjmt !! Graph a Line by Plotting Points - Example 2\nHere look at graphing an equation by plotting points.", "lengthSeconds": 235, "width": 480, "height": 360}
{"video_id": "WuJL0b-vZHg", "transcript": [{"start": 1.52, "duration": 3.64, "text": "okay so here we're going to do another"}, {"start": 3.199, "duration": 3.6, "text": "uh problem related to the equation of a"}, {"start": 5.16, "duration": 4.76, "text": "tangent line and this is kind of a very"}, {"start": 6.799, "duration": 4.96, "text": "common type of problem so uh you've got"}, {"start": 9.92, "duration": 4.32, "text": "some function in this case square otk of"}, {"start": 11.759, "duration": 5.36, "text": "x uh it turns out that the derivative of"}, {"start": 14.24, "duration": 5.32, "text": "square otk of X is 1 over 2 * the < TK"}, {"start": 17.119, "duration": 5.0, "text": "of X and what we want to do is we want"}, {"start": 19.56, "duration": 4.08, "text": "to find the point on the graph and maybe"}, {"start": 22.119, "duration": 3.92, "text": "there's more than one in this case"}, {"start": 23.64, "duration": 5.2, "text": "they're actually won't be but uh we're"}, {"start": 26.039, "duration": 4.64, "text": "going to find the point uh XY so that"}, {"start": 28.84, "duration": 4.719, "text": "the tangent line to the gra graph has a"}, {"start": 30.679, "duration": 5.081, "text": "slope equal to 1 18 so again we're"}, {"start": 33.559, "duration": 4.081, "text": "trying to figure out again you know"}, {"start": 35.76, "duration": 5.24, "text": "where on the original graph would the"}, {"start": 37.64, "duration": 6.599, "text": "tangent line have some particular slope"}, {"start": 41.0, "duration": 5.28, "text": "and to do this it's easy um in a sense"}, {"start": 44.239, "duration": 3.64, "text": "and once you have the derivative uh"}, {"start": 46.28, "duration": 3.759, "text": "because again the derivative basically"}, {"start": 47.879, "duration": 4.68, "text": "tells you for tells you a formula for"}, {"start": 50.039, "duration": 4.84, "text": "the slope of the tangent line and it"}, {"start": 52.559, "duration": 5.561, "text": "says well our formula for the slope of"}, {"start": 54.879, "duration": 6.801, "text": "the tangent line uh at any particular x"}, {"start": 58.12, "duration": 8.56, "text": "coordinate is given by 1 1 2 * the < TK"}, {"start": 61.68, "duration": 6.6, "text": "X and well we want that to equal 1 18 so"}, {"start": 66.68, "duration": 7.2, "text": "all we have to do in this case is just"}, {"start": 68.28, "duration": 9.64, "text": "solve the uh equation 1/ 2 < tkx = 1"}, {"start": 73.88, "duration": 6.04, "text": "18 so uh to do this uh to solve this"}, {"start": 77.92, "duration": 4.32, "text": "equation this is one that's not not too"}, {"start": 79.92, "duration": 4.239, "text": "bad um typically anytime I have a"}, {"start": 82.24, "duration": 5.68, "text": "fraction on both sides I just cross"}, {"start": 84.159, "duration": 6.761, "text": "multiply so uh we can do 1 * 18 that'll"}, {"start": 87.92, "duration": 7.0, "text": "give us 18 and then we can do well uh 1"}, {"start": 90.92, "duration": 5.04, "text": "* 2 < TK X that'll just give us well 2 <"}, {"start": 94.92, "duration": 4.96, "text": "TK"}, {"start": 95.96, "duration": 8.32, "text": "X we can divide both sides by two so by"}, {"start": 99.88, "duration": 7.12, "text": "two 18 over 2 is 9 on the right we still"}, {"start": 104.28, "duration": 4.839, "text": "are just left with square otk of X well"}, {"start": 107.0, "duration": 5.399, "text": "to solve for x all we would need to do"}, {"start": 109.119, "duration": 5.841, "text": "now is just Square both"}, {"start": 112.399, "duration": 7.76, "text": "sides so on the right side we're left"}, {"start": 114.96, "duration": 6.839, "text": "with x and 9 * 9 is going to be 81 so uh"}, {"start": 120.159, "duration": 3.801, "text": "if you graph the original function"}, {"start": 121.799, "duration": 4.561, "text": "square otk of X recall square otk of X"}, {"start": 123.96, "duration": 4.799, "text": "looks something like"}, {"start": 126.36, "duration": 5.48, "text": "that and what we've done is we've gone"}, {"start": 128.759, "duration": 6.12, "text": "over to the x coordinate of 81 so it"}, {"start": 131.84, "duration": 5.8, "text": "says at the x coordinate of 81 uh it"}, {"start": 134.879, "duration": 6.36, "text": "says the slope of that tangent line is"}, {"start": 137.64, "duration": 6.28, "text": "going to be uh 1 over"}, {"start": 141.239, "duration": 5.08, "text": "18 okay so we're almost there um the"}, {"start": 143.92, "duration": 4.84, "text": "question uh actually asks for we want"}, {"start": 146.319, "duration": 5.161, "text": "the the X and the y coordinate on the"}, {"start": 148.76, "duration": 4.759, "text": "original uh function function well again"}, {"start": 151.48, "duration": 4.88, "text": "if you if you're using the square root"}, {"start": 153.519, "duration": 6.201, "text": "function if you plug 81 in the square"}, {"start": 156.36, "duration": 7.56, "text": "root of 81 is just 9 so this point will"}, {"start": 159.72, "duration": 6.519, "text": "be 81 comma 9 so again all we're doing"}, {"start": 163.92, "duration": 3.959, "text": "is we're taking our x coordinate we're"}, {"start": 166.239, "duration": 3.201, "text": "not putting it into the derivative but"}, {"start": 167.879, "duration": 3.681, "text": "we're putting it into the original"}, {"start": 169.44, "duration": 5.079, "text": "function so again if you take the square"}, {"start": 171.56, "duration": 5.599, "text": "root of 81 again we get n so it says the"}, {"start": 174.519, "duration": 6.08, "text": "point that we're looking for would be"}, {"start": 177.159, "duration": 5.481, "text": "the point 81 comma 9 so again it's"}, {"start": 180.599, "duration": 2.92, "text": "common uh they'll you'll try to figure"}, {"start": 182.64, "duration": 4.159, "text": "out"}, {"start": 183.519, "duration": 4.841, "text": "where um you know you'll try to figure"}, {"start": 186.799, "duration": 4.201, "text": "out points on the graph where the"}, {"start": 188.36, "duration": 4.2, "text": "tangent line equals some number A lot of"}, {"start": 191.0, "duration": 3.959, "text": "times we'll we'll try to figure out"}, {"start": 192.56, "duration": 4.84, "text": "where the tangent line equals zero and"}, {"start": 194.959, "duration": 3.601, "text": "you'll see that um in a few sections"}, {"start": 197.4, "duration": 3.16, "text": "that turns out to kind of be an"}, {"start": 198.56, "duration": 3.84, "text": "important question but again that's all"}, {"start": 200.56, "duration": 3.759, "text": "there is to it in this case just take"}, {"start": 202.4, "duration": 3.8, "text": "the derivative set it equal to the slope"}, {"start": 204.319, "duration": 3.881, "text": "that you want that'll give you the x"}, {"start": 206.2, "duration": 4.0, "text": "coordinate plug the x coordinate back"}, {"start": 208.2, "duration": 6.28, "text": "into the original function to find the"}, {"start": 210.2, "duration": 4.28, "text": "corresponding y value"}], "title": "Find Where a Tangent Line to a Curve Has a Desired Slope", "description": "Thanks to all of you who support me on Patreon. You da real mvps!   $1 per month helps!! :)  https://www.patreon.com/patrickjmt !! Find Where a Tangent Line to a Curve Has a Desired Slope. In this video, we are given a curve and its derivative.  We have to determine at which point on the curve has a specified slope.", "lengthSeconds": 213, "width": 480, "height": 360}
{"video_id": "B4G0qoHiMMQ", "transcript": [{"start": 0.52, "duration": 3.28, "text": "all right here we're going to look at an"}, {"start": 1.76, "duration": 3.36, "text": "example related to addition of integers"}, {"start": 3.8, "duration": 2.999, "text": "and what we're going to do is we're"}, {"start": 5.12, "duration": 3.84, "text": "going to write an addition statement for"}, {"start": 6.799, "duration": 4.081, "text": "each of these number lines and again all"}, {"start": 8.96, "duration": 4.08, "text": "the number lines are it's just kind of a"}, {"start": 10.88, "duration": 4.6, "text": "visual way of sort of a you know"}, {"start": 13.04, "duration": 4.84, "text": "thinking about adding positive numbers"}, {"start": 15.48, "duration": 5.36, "text": "and negative numbers and uh just kind of"}, {"start": 17.88, "duration": 6.12, "text": "a way to to to uh to see it you know"}, {"start": 20.84, "duration": 4.88, "text": "again on a number line so in part A here"}, {"start": 24.0, "duration": 4.4, "text": "you know we we're starting at zero and"}, {"start": 25.72, "duration": 5.24, "text": "then we're going uh four units to the"}, {"start": 28.4, "duration": 6.12, "text": "right so we're going off in the positive"}, {"start": 30.96, "duration": 5.439, "text": "direction positive 1 2 3 4 so our"}, {"start": 34.52, "duration": 4.4, "text": "addition statement for this would be"}, {"start": 36.399, "duration": 5.401, "text": "well positive 4 and then we're going to"}, {"start": 38.92, "duration": 5.68, "text": "add um something to it well it looks"}, {"start": 41.8, "duration": 4.599, "text": "like from four now what we're interested"}, {"start": 44.6, "duration": 3.76, "text": "in is the length of the arrow so be"}, {"start": 46.399, "duration": 4.121, "text": "careful so okay so we go from four back"}, {"start": 48.36, "duration": 4.44, "text": "to three but what we're really doing is"}, {"start": 50.52, "duration": 6.199, "text": "we're kind of going uh you know sort of"}, {"start": 52.8, "duration": 5.8, "text": "one unit to the left to the left in the"}, {"start": 56.719, "duration": 6.8, "text": "negative Direction you can think about"}, {"start": 58.6, "duration": 6.44, "text": "it that way so we we've got pos4 + -1"}, {"start": 63.519, "duration": 3.521, "text": "well and from the number line you can"}, {"start": 65.04, "duration": 4.32, "text": "see that if we go four units to the"}, {"start": 67.04, "duration": 4.32, "text": "right and then if we add to that"}, {"start": 69.36, "duration": 4.36, "text": "negative one which would give us take us"}, {"start": 71.36, "duration": 4.84, "text": "back one unit to the left this would be"}, {"start": 73.72, "duration": 5.079, "text": "our addition statement um and we could"}, {"start": 76.2, "duration": 4.64, "text": "say well hey that equals"}, {"start": 78.799, "duration": 5.881, "text": "positive3 uh we'll do the same thing for"}, {"start": 80.84, "duration": 6.56, "text": "part uh Part B here do both of these at"}, {"start": 84.68, "duration": 5.52, "text": "once okay so on the top part it looks"}, {"start": 87.4, "duration": 4.719, "text": "like we're going two units to the left"}, {"start": 90.2, "duration": 5.0, "text": "well two units to the left is going to"}, {"start": 92.119, "duration": 4.801, "text": "give us well -2 and then we're going to"}, {"start": 95.2, "duration": 4.04, "text": "add to that and again let's count so"}, {"start": 96.92, "duration": 4.96, "text": "we're starting here we're going over uh"}, {"start": 99.24, "duration": 6.28, "text": "we're going one unit two units three"}, {"start": 101.88, "duration": 6.8, "text": "units four units five units six units so"}, {"start": 105.52, "duration": 8.16, "text": "we're adding six units by moving to the"}, {"start": 108.68, "duration": 7.039, "text": "right so we have -2 plus a positive 6"}, {"start": 113.68, "duration": 4.92, "text": "and as you can see from the number line"}, {"start": 115.719, "duration": 5.641, "text": "if we go two units to the left and then"}, {"start": 118.6, "duration": 5.879, "text": "we add six unit to that that's going to"}, {"start": 121.36, "duration": 6.119, "text": "put us at the value of"}, {"start": 124.479, "duration": 3.0, "text": "pos4"}], "title": "Using a Number Line to Write an Addition of Integer Statement", "description": "Thanks to all of you who support me on Patreon. You da real mvps!   $1 per month helps!! :)  https://www.patreon.com/patrickjmt !! Using a Numberline to Write an Addition of Integer Statement.\nHere we look at an example of using a number line and the notion of direction to help visualize what is happening when adding integers.", "lengthSeconds": 126, "width": 480, "height": 360}
{"video_id": "kyDOnV0XYI8", "transcript": [{"start": 0.359, "duration": 4.041, "text": "all right here I'm going to take a"}, {"start": 1.48, "duration": 6.2, "text": "fraction uh 24 over 32 and I'm going to"}, {"start": 4.4, "duration": 5.84, "text": "write it in lowest terms so what that"}, {"start": 7.68, "duration": 3.879, "text": "means is uh basically get rid of any"}, {"start": 10.24, "duration": 3.72, "text": "common"}, {"start": 11.559, "duration": 5.64, "text": "factors so one thing we could do is we"}, {"start": 13.96, "duration": 6.52, "text": "could write 24 over 32 I could say well"}, {"start": 17.199, "duration": 6.481, "text": "24 is the same thing as 2 *"}, {"start": 20.48, "duration": 5.879, "text": "12 and then in the denominator 32 that's"}, {"start": 23.68, "duration": 5.759, "text": "really the same thing as 2 * 16 there's"}, {"start": 26.359, "duration": 5.121, "text": "certainly different uh uh uh ways we can"}, {"start": 29.439, "duration": 4.12, "text": "do it besid besides mine the way I'm"}, {"start": 31.48, "duration": 3.84, "text": "doing it right now um we could have"}, {"start": 33.559, "duration": 4.201, "text": "factored for example a four out of the"}, {"start": 35.32, "duration": 4.759, "text": "numerator and denominator you could have"}, {"start": 37.76, "duration": 3.84, "text": "factor that an eight as well but a lot"}, {"start": 40.079, "duration": 3.681, "text": "of times when I look at these I say hey"}, {"start": 41.6, "duration": 4.599, "text": "they're both even I can certainly reduce"}, {"start": 43.76, "duration": 5.92, "text": "them okay so now the idea is we can"}, {"start": 46.199, "duration": 5.801, "text": "cancel out common factors and the idea"}, {"start": 49.68, "duration": 5.24, "text": "is really what we have is okay so 2 * 12"}, {"start": 52.0, "duration": 8.239, "text": "over 2 * 16 uh we could really write"}, {"start": 54.92, "duration": 9.44, "text": "that as 2 over2 * 12 over 16 but but 2"}, {"start": 60.239, "duration": 8.761, "text": "divided two is the same thing as 1 so 1"}, {"start": 64.36, "duration": 7.48, "text": "* 12 16 that'll just give us 12 over 16"}, {"start": 69.0, "duration": 6.4, "text": "okay so I've definitely reduced 24 over"}, {"start": 71.84, "duration": 6.959, "text": "32 I've reduced it to 12 over 16 but now"}, {"start": 75.4, "duration": 5.0, "text": "I ask myself does it reduce any anymore"}, {"start": 78.799, "duration": 4.0, "text": "well again it certainly could they're"}, {"start": 80.4, "duration": 4.719, "text": "still both even I could factor a two out"}, {"start": 82.799, "duration": 4.081, "text": "just like I did but I say oh I think"}, {"start": 85.119, "duration": 4.441, "text": "there's a bigger number that goes into"}, {"start": 86.88, "duration": 8.8, "text": "both of these so notice I can write 12"}, {"start": 89.56, "duration": 8.8, "text": "as 4 * 3 16 I can write that as 4 *"}, {"start": 95.68, "duration": 4.6, "text": "4 and again people typically we don't"}, {"start": 98.36, "duration": 3.92, "text": "typically write this step but that's you"}, {"start": 100.28, "duration": 3.92, "text": "know the reason why you can just cancel"}, {"start": 102.28, "duration": 4.36, "text": "these out okay that's one way of"}, {"start": 104.2, "duration": 5.08, "text": "thinking about it I'm just going to"}, {"start": 106.64, "duration": 5.0, "text": "cancel out my common factors of uh you"}, {"start": 109.28, "duration": 6.879, "text": "know the four and the four so we would"}, {"start": 111.64, "duration": 7.079, "text": "be left with 3 over four and now there's"}, {"start": 116.159, "duration": 4.92, "text": "nothing else we can't really Factor them"}, {"start": 118.719, "duration": 4.881, "text": "they're no more common factors other"}, {"start": 121.079, "duration": 4.601, "text": "than one we could write three as 3 * 1"}, {"start": 123.6, "duration": 3.439, "text": "we could write four as 2 * 2 there would"}, {"start": 125.68, "duration": 3.96, "text": "be nothing else that we could really"}, {"start": 127.039, "duration": 5.721, "text": "cancel out so we've now reduced our"}, {"start": 129.64, "duration": 5.959, "text": "fraction uh to lowest terms and again"}, {"start": 132.76, "duration": 6.559, "text": "the idea is if you make 24 over 32 into"}, {"start": 135.599, "duration": 7.321, "text": "a decimal you'll get the same thing as 3"}, {"start": 139.319, "duration": 6.601, "text": "over4 if you make that into a decimal"}, {"start": 142.92, "duration": 3.0, "text": "um"}], "title": "Reducing a Fraction to Lowest Terms", "description": "Reducing a Fraction to Lowest Terms\nLearn how to simplify fractions to their lowest terms by factoring and canceling common factors. This video walks you through reducing fractions step-by-step, making it easy to understand and apply to any fraction problem.", "lengthSeconds": 144, "width": 480, "height": 360}
{"video_id": "On89CSH0Pb8", "transcript": [{"start": 1.16, "duration": 5.759, "text": "all right in this video we're going to"}, {"start": 2.52, "duration": 6.84, "text": "do an example of uh finding an angle the"}, {"start": 6.919, "duration": 5.081, "text": "measure of an angle if we know a little"}, {"start": 9.36, "duration": 5.319, "text": "more information so in this case suppose"}, {"start": 12.0, "duration": 7.92, "text": "we know that cosine of theta is equal to"}, {"start": 14.679, "duration": 5.241, "text": "< TK 3/2 we want to find the angle"}, {"start": 20.119, "duration": 4.16, "text": "um what the measure is for our angle"}, {"start": 22.88, "duration": 5.6, "text": "here"}, {"start": 24.279, "duration": 7.96, "text": "Theta so remember cosine of an angle is"}, {"start": 28.48, "duration": 6.2, "text": "defined to be the adjacent side over the"}, {"start": 32.239, "duration": 4.241, "text": "hypotenuse and again for our angle Theta"}, {"start": 34.68, "duration": 4.559, "text": "this would be the adjacent so I'm simply"}, {"start": 36.48, "duration": 3.72, "text": "going to label the adjacent side as squ"}, {"start": 39.239, "duration": 3.681, "text": "< TK"}, {"start": 40.2, "duration": 4.12, "text": "3 I'm going to label the hypotenuse here"}, {"start": 42.92, "duration": 3.84, "text": "with length"}, {"start": 44.32, "duration": 4.6, "text": "two and now we would have to figure out"}, {"start": 46.76, "duration": 5.88, "text": "you know this missing side you may know"}, {"start": 48.92, "duration": 4.92, "text": "this uh you know if you uh from geometry"}, {"start": 52.64, "duration": 2.88, "text": "but we can always figure it out just"}, {"start": 53.84, "duration": 5.039, "text": "using Pythagorean"}, {"start": 55.52, "duration": 6.44, "text": "theorem so we'll take the < TK of 3 S"}, {"start": 58.879, "duration": 5.801, "text": "Plus this missing side we'll call it b^"}, {"start": 61.96, "duration": 6.28, "text": "2 that equals 2^"}, {"start": 64.68, "duration": 7.64, "text": "2 well uh the < TK of 3 S is just going"}, {"start": 68.24, "duration": 6.519, "text": "to be 3 plus b^ 2 uh 2^ s is going to"}, {"start": 72.32, "duration": 5.28, "text": "give us four and now if we subtract"}, {"start": 74.759, "duration": 6.32, "text": "three from both sides we'll just get"}, {"start": 77.6, "duration": 6.32, "text": "that b ^2 = 1 and when we take the"}, {"start": 81.079, "duration": 4.601, "text": "square root we'll get positive Nega 1"}, {"start": 83.92, "duration": 3.72, "text": "but again it's got to be the positive"}, {"start": 85.68, "duration": 5.84, "text": "one uh because we're talking about a"}, {"start": 87.64, "duration": 5.32, "text": "length here so this simply has a length"}, {"start": 91.52, "duration": 5.2, "text": "of"}, {"start": 92.96, "duration": 6.799, "text": "one so again what we're trying to do is"}, {"start": 96.72, "duration": 6.28, "text": "we're trying to figure out uh our our"}, {"start": 99.759, "duration": 5.521, "text": "our angle Theta so now uh we have to be"}, {"start": 103.0, "duration": 5.399, "text": "a little bit clever here and what we're"}, {"start": 105.28, "duration": 4.76, "text": "going to do is I'm going to kind of uh"}, {"start": 108.399, "duration": 4.441, "text": "take this triangle and I'm going to"}, {"start": 110.04, "duration": 5.24, "text": "reflect it Okay so we've got our"}, {"start": 112.84, "duration": 6.0, "text": "triangle here this has length"}, {"start": 115.28, "duration": 6.36, "text": "two um this has length two this has"}, {"start": 118.84, "duration": 4.72, "text": "length one this has length square > of"}, {"start": 121.64, "duration": 4.36, "text": "three but I don't really care about that"}, {"start": 123.56, "duration": 4.64, "text": "uh I'm going to imagine sort of uh just"}, {"start": 126.0, "duration": 5.84, "text": "reflecting this triangle over to the"}, {"start": 128.2, "duration": 6.28, "text": "other side okay so just imagine it sort"}, {"start": 131.84, "duration": 4.479, "text": "of getting reflected over um it's going"}, {"start": 134.48, "duration": 4.92, "text": "to go down"}, {"start": 136.319, "duration": 5.121, "text": "here so we reflect it over it's going to"}, {"start": 139.4, "duration": 4.72, "text": "come down"}, {"start": 141.44, "duration": 4.48, "text": "here okay so this was length two okay so"}, {"start": 144.12, "duration": 3.479, "text": "this is not great Artistry here but if"}, {"start": 145.92, "duration": 4.36, "text": "it's the same triangle this head length"}, {"start": 147.599, "duration": 5.201, "text": "two this head length one that means the"}, {"start": 150.28, "duration": 4.16, "text": "hypotenuse here would have length two"}, {"start": 152.8, "duration": 4.24, "text": "and this other side again would have"}, {"start": 154.44, "duration": 5.0, "text": "length one but if you think about it now"}, {"start": 157.04, "duration": 3.919, "text": "what we really have is we've got um an"}, {"start": 159.44, "duration": 4.48, "text": "equilateral"}, {"start": 160.959, "duration": 6.161, "text": "triangle because this is length two this"}, {"start": 163.92, "duration": 6.28, "text": "is length two this entire length adds up"}, {"start": 167.12, "duration": 5.16, "text": "to two so that means uh since all the"}, {"start": 170.2, "duration": 5.08, "text": "side lengths are the same that means all"}, {"start": 172.28, "duration": 6.319, "text": "the angles inside are the same and since"}, {"start": 175.28, "duration": 5.72, "text": "all the angles have to add up to"}, {"start": 178.599, "duration": 5.041, "text": "180\u00b0 we we know that each one of those"}, {"start": 181.0, "duration": 5.44, "text": "angles has to be 60\u00b0"}, {"start": 183.64, "duration": 5.319, "text": "individually so okay so that means again"}, {"start": 186.44, "duration": 5.719, "text": "this angle is"}, {"start": 188.959, "duration": 4.92, "text": "60\u00b0 but notice our little angle Theta"}, {"start": 192.159, "duration": 4.601, "text": "here that was kind of"}, {"start": 193.879, "duration": 5.961, "text": "sitting you know kind of sitting in this"}, {"start": 196.76, "duration": 5.199, "text": "uh this upper region right this whole"}, {"start": 199.84, "duration": 3.24, "text": "angle if we trace it out that's what's"}, {"start": 201.959, "duration": 4.161, "text": "going to be"}, {"start": 203.08, "duration": 4.92, "text": "60\u00b0 so in this case Theta is only going"}, {"start": 206.12, "duration": 4.0, "text": "to be half of that so we would have to"}, {"start": 208.0, "duration": 4.48, "text": "take our 60\u00b0"}, {"start": 210.12, "duration": 5.28, "text": "and divide it by two and that would"}, {"start": 212.48, "duration": 6.479, "text": "simply give us that Theta has uh the"}, {"start": 215.4, "duration": 10.6, "text": "measure of uh 30\u00b0 or we could put that"}, {"start": 218.959, "duration": 7.041, "text": "in radians and say that's pi/ 6 radians"}], "title": "Finding an Angle Given the Value of a Trigonometric Function - Example 1", "description": "Thanks to all of you who support me on Patreon. You da real mvps!   $1 per month helps!! :)  https://www.patreon.com/patrickjmt !! Finding an Angle Given the Value of a Trigonometric Function - Example 1.  In this video, I deduce the angle associated with a given trigonometric value.", "lengthSeconds": 225, "width": 480, "height": 360}
{"video_id": "cQXSXkBMsXU", "transcript": [{"start": 0.599, "duration": 4.441, "text": "all right so another example here using"}, {"start": 2.36, "duration": 5.16, "text": "the law of signs so suppose we've got"}, {"start": 5.04, "duration": 7.4, "text": "our a triangle here we've got angle of"}, {"start": 7.52, "duration": 7.239, "text": "35\u00b0 120\u00b0 we don't know our angle beta um"}, {"start": 12.44, "duration": 5.679, "text": "the side opposite 120 is"}, {"start": 14.759, "duration": 5.6, "text": "10.45 we want to find the side length B"}, {"start": 18.119, "duration": 5.32, "text": "well kind of the same idea U we know"}, {"start": 20.359, "duration": 4.201, "text": "that the angles have to add up to 180 so"}, {"start": 23.439, "duration": 3.401, "text": "we've got"}, {"start": 24.56, "duration": 6.4, "text": "120\u00b0 plus"}, {"start": 26.84, "duration": 7.559, "text": "35\u00b0 well that would give us 155\u00b0"}, {"start": 30.96, "duration": 6.16, "text": "well if we take 180 and uh subtract our"}, {"start": 34.399, "duration": 7.521, "text": "155 that's simply going to leave us with"}, {"start": 37.12, "duration": 8.16, "text": "25 degrees so our angle beta must be uh"}, {"start": 41.92, "duration": 6.36, "text": "25\u00b0 down here and now we can simply just"}, {"start": 45.28, "duration": 10.04, "text": "use our law of signs so I'm going to"}, {"start": 48.28, "duration": 10.959, "text": "take uh Little B over s of 25 Dees so"}, {"start": 55.32, "duration": 5.8, "text": "little B over s of 25\u00b0 the only uh side"}, {"start": 59.239, "duration": 4.88, "text": "length we're given is the"}, {"start": 61.12, "duration": 3.64, "text": "10.45 so I'm going to use the 10.45 and"}, {"start": 64.119, "duration": 4.441, "text": "the"}, {"start": 64.76, "duration": 5.32, "text": "120\u00b0 so we have"}, {"start": 68.56, "duration": 4.96, "text": "1045"}, {"start": 70.08, "duration": 6.359, "text": "over s of"}, {"start": 73.52, "duration": 6.16, "text": "120\u00b0 and now all we have to do uh is"}, {"start": 76.439, "duration": 5.121, "text": "simply solve for b we can multiply both"}, {"start": 79.68, "duration": 6.68, "text": "sides by S of"}, {"start": 81.56, "duration": 9.559, "text": "25\u00b0 so that'll leave us with B ="}, {"start": 86.36, "duration": 9.16, "text": "10.45 * s of 20"}, {"start": 91.119, "duration": 7.721, "text": "25\u00b0 all over s of"}, {"start": 95.52, "duration": 6.04, "text": "120\u00b0 so again now I think this is"}, {"start": 98.84, "duration": 4.639, "text": "probably good place to start plugging"}, {"start": 101.56, "duration": 7.12, "text": "things into a"}, {"start": 103.479, "duration": 5.201, "text": "calculator so let's see um sign of 25"}, {"start": 113.64, "duration": 6.6, "text": "degrees so I'm getting s of 25\u00b0 to be"}, {"start": 117.24, "duration": 3.0, "text": "0.423"}, {"start": 120.719, "duration": 6.281, "text": "um after"}, {"start": 123.2, "duration": 3.8, "text": "rounding and S of"}, {"start": 128.399, "duration": 5.681, "text": "120\u00b0 um a sign of 120 degrees I'm"}, {"start": 131.36, "duration": 2.72, "text": "getting that to be"}, {"start": 134.28, "duration": 6.44, "text": "0.866 after rounding a little bit so now"}, {"start": 137.64, "duration": 3.08, "text": "we'll simply take"}, {"start": 141.2, "duration": 5.52, "text": "10.45 multiply that"}, {"start": 144.28, "duration": 7.56, "text": "by"}, {"start": 146.72, "duration": 5.12, "text": "423 I'm getting that to be 4"}, {"start": 152.48, "duration": 4.56, "text": "42035 again we're going to divide that"}, {"start": 154.72, "duration": 5.12, "text": "by"}, {"start": 157.04, "duration": 6.279, "text": "0.866 so when I divide"}, {"start": 159.84, "duration": 3.479, "text": "this so"}, {"start": 163.92, "duration": 5.399, "text": "0.866 I'm getting this to be uh"}, {"start": 169.879, "duration": 8.72, "text": "5.14 after rounding so uh We've now got"}, {"start": 173.879, "duration": 6.601, "text": "our our missing length it simply equals"}, {"start": 178.599, "duration": 4.081, "text": "5104 looks like like we were using"}, {"start": 180.48, "duration": 5.399, "text": "centimeters so those will still be our"}, {"start": 182.68, "duration": 5.44, "text": "our units um so again that's all there"}, {"start": 185.879, "duration": 4.881, "text": "is to it just figure out the missing"}, {"start": 188.12, "duration": 5.08, "text": "angle we know they add up to 180 set up"}, {"start": 190.76, "duration": 4.72, "text": "your uh your equation using the law of"}, {"start": 193.2, "duration": 4.759, "text": "signs and then it's just a matter of you"}, {"start": 195.48, "duration": 4.24, "text": "know to again to make sense out of it I"}, {"start": 197.959, "duration": 5.761, "text": "would definitely want a calculator to"}, {"start": 199.72, "duration": 4.0, "text": "get some decimal values"}], "title": "The Law of Sines, Example 2", "description": "Thanks to all of you who support me on Patreon. You da real mvps!   $1 per month helps!! :)  https://www.patreon.com/patrickjmt !! The Law of Sines, Example 2.\n  This video talks a bit about the Law of Sines.  I give the formula and proceed to do an example using it.", "lengthSeconds": 203, "width": 480, "height": 360}
{"video_id": "c3T2Po9TH0c", "transcript": [{"start": 0.05, "duration": 4.09, "text": "all right so here I'm gonna look at"}, {"start": 2.1, "duration": 4.92, "text": "another example of subtracting whole"}, {"start": 4.14, "duration": 4.65, "text": "numbers so in Part A we have four"}, {"start": 7.02, "duration": 4.47, "text": "hundred twelve thousand three hundred"}, {"start": 8.79, "duration": 4.92, "text": "forty seven minus two hundred sixteen"}, {"start": 11.49, "duration": 4.34, "text": "thousand five hundred seventeen I'm"}, {"start": 13.71, "duration": 6.42, "text": "gonna do that one first"}, {"start": 15.83, "duration": 7.119, "text": "so my first column there's no problem I"}, {"start": 20.13, "duration": 5.04, "text": "don't have to borrow a digit from the"}, {"start": 22.949, "duration": 4.051, "text": "next column or anything as long as the"}, {"start": 25.17, "duration": 4.32, "text": "number on top is greater than or equal"}, {"start": 27.0, "duration": 5.009, "text": "to the number on the bottom we can just"}, {"start": 29.49, "duration": 5.52, "text": "do our subtraction so seven minus seven"}, {"start": 32.009, "duration": 6.931, "text": "is gonna leave us with zero if I look at"}, {"start": 35.01, "duration": 7.11, "text": "my next column well four minus one is"}, {"start": 38.94, "duration": 5.61, "text": "gonna leave us with three well now this"}, {"start": 42.12, "duration": 4.2, "text": "is our first kind of against spot where"}, {"start": 44.55, "duration": 5.939, "text": "we have to be careful because now the"}, {"start": 46.32, "duration": 6.54, "text": "number on top is larger than the number"}, {"start": 50.489, "duration": 3.871, "text": "on the bottom so if you think about it"}, {"start": 52.86, "duration": 3.75, "text": "if you've got three in a sense I guess"}, {"start": 54.36, "duration": 3.96, "text": "somebody couldn't take away five well"}, {"start": 56.61, "duration": 3.51, "text": "you can you've got negative numbers of"}, {"start": 58.32, "duration": 4.919, "text": "course but again when we do subtraction"}, {"start": 60.12, "duration": 6.12, "text": "that's not how we do it so if the number"}, {"start": 63.239, "duration": 5.281, "text": "on top is smaller than the number on the"}, {"start": 66.24, "duration": 6.27, "text": "bottom what we do again is we borrow a"}, {"start": 68.52, "duration": 6.0, "text": "digit from the next the next column so"}, {"start": 72.51, "duration": 4.859, "text": "instead of a two I'm gonna bump this"}, {"start": 74.52, "duration": 5.43, "text": "down by one and well that would make it"}, {"start": 77.369, "duration": 4.531, "text": "into a 1 and instead of a three I'm"}, {"start": 79.95, "duration": 6.36, "text": "gonna now pretend this is a thirteen"}, {"start": 81.9, "duration": 7.56, "text": "okay so I'm kind of borrowing you know"}, {"start": 86.31, "duration": 5.28, "text": "one digit from that next column so 13"}, {"start": 89.46, "duration": 5.159, "text": "minus five well we can do that that's"}, {"start": 91.59, "duration": 4.68, "text": "gonna leave us with eight all right but"}, {"start": 94.619, "duration": 3.901, "text": "now we've really got the same problem"}, {"start": 96.27, "duration": 4.95, "text": "again because now I'm thinking about one"}, {"start": 98.52, "duration": 5.19, "text": "minus six well the number on top is"}, {"start": 101.22, "duration": 5.039, "text": "smaller so what I'm gonna have to do"}, {"start": 103.71, "duration": 5.01, "text": "again is borrow a digit so instead of a"}, {"start": 106.259, "duration": 5.521, "text": "one I'm gonna make this into one smaller"}, {"start": 108.72, "duration": 5.55, "text": "which again is a zero and again then we"}, {"start": 111.78, "duration": 6.689, "text": "just tack a one in front of our original"}, {"start": 114.27, "duration": 6.15, "text": "number so now we have 11 minus six which"}, {"start": 118.469, "duration": 4.321, "text": "is gonna give us five you can just drop"}, {"start": 120.42, "duration": 5.85, "text": "down or comma again we've got this"}, {"start": 122.79, "duration": 6.66, "text": "problem again because now we have zero"}, {"start": 126.27, "duration": 4.82, "text": "minus one well I'm gonna have to borrow"}, {"start": 129.45, "duration": 4.25, "text": "a digit so instead of a"}, {"start": 131.09, "duration": 5.869, "text": "or this is gonna become a three again we"}, {"start": 133.7, "duration": 8.789, "text": "just put a 1 out front 10 minus 1 is 9"}, {"start": 136.959, "duration": 7.661, "text": "and then 3 minus 2 is 1 so it says if we"}, {"start": 142.489, "duration": 6.541, "text": "do our subtraction it says our final"}, {"start": 144.62, "duration": 7.83, "text": "answer will be 190 5830"}, {"start": 149.03, "duration": 6.17, "text": "and I think I'll do Part B here just as"}, {"start": 152.45, "duration": 2.75, "text": "a separate example"}], "title": "Subtracting Whole Numbers , Example 3, Longer Examples", "description": "Thanks to all of you who support me on Patreon. You da real mvps!   $1 per month helps!! :)  https://www.patreon.com/patrickjmt !! Subtracting Whole Numbers , Example 3.\r\n  Here we look at two examples of subtracting whole numbers.", "lengthSeconds": 155, "width": 480, "height": 360}
{"video_id": "R35BycA-RgI", "transcript": [{"start": 0.399, "duration": 3.361, "text": "all right so another example here"}, {"start": 1.839, "duration": 3.761, "text": "figuring out whether a function is even"}, {"start": 3.76, "duration": 6.799, "text": "odd or neither and here we're going to"}, {"start": 5.6, "duration": 7.6, "text": "do x 4 * sinx * cine 2"}, {"start": 10.559, "duration": 5.681, "text": "x um I'm going to rewrite this just a"}, {"start": 13.2, "duration": 8.96, "text": "little bit first so G of X that's going"}, {"start": 16.24, "duration": 8.4, "text": "to be X 4th we've got s of X and cosine"}, {"start": 22.16, "duration": 3.359, "text": "2 x we can just pull that out so we'll"}, {"start": 24.64, "duration": 5.6, "text": "have"}, {"start": 25.519, "duration": 7.281, "text": "cosine of x all squared Okay so that all"}, {"start": 30.24, "duration": 5.2, "text": "says the same thing so now I'm going to"}, {"start": 32.8, "duration": 6.12, "text": "think about G"}, {"start": 35.44, "duration": 7.0, "text": "ofx well that's just X to"}, {"start": 38.92, "duration": 3.52, "text": "4th sin"}, {"start": 42.48, "duration": 6.96, "text": "ofx cine"}, {"start": 46.199, "duration": 7.16, "text": "ofx squared and now um I just have to"}, {"start": 49.44, "duration": 7.08, "text": "simplify things down so okay wellx to"}, {"start": 53.359, "duration": 4.88, "text": "4th uh if you take a negative one and"}, {"start": 56.52, "duration": 4.64, "text": "raise it to an even power that'll give"}, {"start": 58.239, "duration": 7.361, "text": "us positive one we be left with the X to"}, {"start": 61.16, "duration": 8.04, "text": "4th we said s ofx in terms of being even"}, {"start": 65.6, "duration": 7.24, "text": "and odd we said sign was odd therefore s"}, {"start": 69.2, "duration": 3.64, "text": "ofx ="}, {"start": 75.08, "duration": 6.32, "text": "sinx okay cosine ofx we said cosine ofx"}, {"start": 79.64, "duration": 5.4, "text": "is even so that's just going to equal"}, {"start": 81.4, "duration": 5.24, "text": "cosine X so this is just cosine"}, {"start": 85.04, "duration": 3.759, "text": "x"}, {"start": 86.64, "duration": 3.76, "text": "squared and now what I'm going to do"}, {"start": 88.799, "duration": 4.201, "text": "again is just pull the sign all the way"}, {"start": 90.4, "duration": 6.12, "text": "out front so we'll pull the negative um"}, {"start": 93.0, "duration": 6.799, "text": "out front we've got X 4th we would still"}, {"start": 96.52, "duration": 7.52, "text": "have our sinx we can put the square back"}, {"start": 99.799, "duration": 7.241, "text": "uh cine 2 x and now let's see the"}, {"start": 104.04, "duration": 6.359, "text": "original function we started with was x"}, {"start": 107.04, "duration": 5.679, "text": "4 sin x cine"}, {"start": 110.399, "duration": 4.841, "text": "SAR well that just means we have the"}, {"start": 112.719, "duration": 5.32, "text": "negative of that and again in terms of"}, {"start": 115.24, "duration": 6.6, "text": "being even or odd um this is just simply"}, {"start": 118.039, "duration": 3.801, "text": "going to be an odd function"}, {"start": 123.479, "duration": 5.2, "text": "so our very original"}, {"start": 125.88, "duration": 5.799, "text": "function we would say that's an odd"}, {"start": 128.679, "duration": 3.0, "text": "function"}], "title": "Examples with Trigonometric Functions: Even, Odd or Neither, Example 3", "description": "Thanks to all of you who support me on Patreon. You da real mvps!   $1 per month helps!! :)  https://www.patreon.com/patrickjmt !! Examples with Trigonometric Functions: Even, Odd or Neither, Example 3", "lengthSeconds": 131, "width": 480, "height": 360}
{"video_id": "-uU5QbQALcM", "transcript": [{"start": 0.03, "duration": 5.309, "text": "some some more examples here using the"}, {"start": 2.76, "duration": 4.11, "text": "limit comparison test and this is kind"}, {"start": 5.339, "duration": 2.701, "text": "of similar to what we've been doing so"}, {"start": 6.87, "duration": 4.14, "text": "here we're going to use the limit"}, {"start": 8.04, "duration": 5.639, "text": "comparison test on the series N equals 1"}, {"start": 11.01, "duration": 4.499, "text": "to infinity of n cubed minus n plus 14"}, {"start": 13.679, "duration": 5.791, "text": "over N to the fifth plus n to the third"}, {"start": 15.509, "duration": 6.121, "text": "plus 1 by comparing it to the series 1"}, {"start": 19.47, "duration": 7.77, "text": "over N squared and again we know that"}, {"start": 21.63, "duration": 9.569, "text": "this is just a convergent P series so if"}, {"start": 27.24, "duration": 7.95, "text": "we do the limit as n goes to infinity so"}, {"start": 31.199, "duration": 6.901, "text": "we've got n cubed minus n plus 14 over N"}, {"start": 35.19, "duration": 5.81, "text": "to the fifth plus n cubed plus 1 again"}, {"start": 38.1, "duration": 5.1, "text": "we would divide by 1 over N squared or"}, {"start": 41.0, "duration": 6.879, "text": "equivalently multiplied by n squared"}, {"start": 43.2, "duration": 6.179, "text": "over 1 well notice in the numerator when"}, {"start": 47.879, "duration": 3.541, "text": "we distribute we're going to get an end"}, {"start": 49.379, "duration": 4.231, "text": "to the fifth and then the rest of it's"}, {"start": 51.42, "duration": 4.02, "text": "going to be smaller degree will have an"}, {"start": 53.61, "duration": 3.99, "text": "end to the fifth over N to the fifth"}, {"start": 55.44, "duration": 4.2, "text": "since the degree of the numerator equals"}, {"start": 57.6, "duration": 4.049, "text": "the degree of the denominator the limit"}, {"start": 59.64, "duration": 4.23, "text": "exists and the coefficients would just"}, {"start": 61.649, "duration": 4.83, "text": "be 1 so in this case the limit is going"}, {"start": 63.87, "duration": 6.74, "text": "to equal 1 again a positive finite"}, {"start": 66.479, "duration": 7.621, "text": "number says they both do the same thing"}, {"start": 70.61, "duration": 6.189, "text": "well since we're using a convergent p"}, {"start": 74.1, "duration": 8.0, "text": "series to compare it to that means our"}, {"start": 76.799, "duration": 5.301, "text": "original series again also converges"}], "title": "Limit Comparison Test for Series - Another Example 6", "description": "Thanks to all of you who support me on Patreon. You da real mvps!   $1 per month helps!! :)  https://www.patreon.com/patrickjmt !! Limit Comparison Test for Series - Another Example 6. In this video, I use the limit comparison test to determine whether or not a given series converges or diverges. I do not delve heavily into the theory in this video (but do in another video if that is what you need to see!).", "lengthSeconds": 86, "width": 480, "height": 360}
{"video_id": "6P3HIXKjsKU", "transcript": [{"start": 1.319, "duration": 4.361, "text": "okay here's another sample sat problem"}, {"start": 3.399, "duration": 6.961, "text": "taken from the prins and review it says"}, {"start": 5.68, "duration": 7.8, "text": "if 20% of X is q and 70% of X is"}, {"start": 10.36, "duration": 5.8, "text": "T it says then in terms of X what is the"}, {"start": 13.48, "duration": 7.28, "text": "value of T minus"}, {"start": 16.16, "duration": 10.08, "text": "q and the solutions they give are"}, {"start": 20.76, "duration": 8.64, "text": "2x 3x 4X 5x or"}, {"start": 26.24, "duration": 7.319, "text": "6X um so again on this type of problem I"}, {"start": 29.4, "duration": 5.679, "text": "always ask myself um is there a for any"}, {"start": 33.559, "duration": 3.52, "text": "sat problem you should always ask"}, {"start": 35.079, "duration": 5.32, "text": "yourself should I do it the mathy way or"}, {"start": 37.079, "duration": 4.841, "text": "should I do it a plug and chug way I"}, {"start": 40.399, "duration": 3.16, "text": "don't really see a good way to do this"}, {"start": 41.92, "duration": 2.88, "text": "one with plug and chug I mean this is"}, {"start": 43.559, "duration": 4.52, "text": "something we're just going to actually"}, {"start": 44.8, "duration": 6.919, "text": "have to have to work out here so let's"}, {"start": 48.079, "duration": 9.561, "text": "check here so it says if 20% of X is Q"}, {"start": 51.719, "duration": 13.08, "text": "so mathematically 20% of X is Q well"}, {"start": 57.64, "duration": 9.76, "text": "remember 20% is 02 so it's says 2 * x ="}, {"start": 64.799, "duration": 10.881, "text": "q and then it says"}, {"start": 67.4, "duration": 10.32, "text": "70% of X is T well that means 7 * x = t"}, {"start": 75.68, "duration": 5.64, "text": "it says in terms of X what is the value"}, {"start": 77.72, "duration": 6.32, "text": "of T minus Q well T minus Q is going to"}, {"start": 81.32, "duration": 6.439, "text": "equal we said t is"}, {"start": 84.04, "duration": 6.719, "text": "7x and Q is"}, {"start": 87.759, "duration": 5.921, "text": "2x so this is simply"}, {"start": 90.759, "duration": 5.161, "text": ".5x so our solution here would be answer"}, {"start": 93.68, "duration": 5.079, "text": "D so I think they're just trying to get"}, {"start": 95.92, "duration": 8.64, "text": "across um can you basically set up these"}, {"start": 98.759, "duration": 5.801, "text": "kind of simple U equations from words"}], "title": "SAT Math Question #5", "description": "Thanks to all of you who support me on Patreon. You da real mvps!   $1 per month helps!! :)  https://www.patreon.com/patrickjmt !! A sample SAT math question taken from a practice exam of the Princeton Review.\r\nFor more free math videos visit http://JustMathTutoring.com", "lengthSeconds": 103, "width": 480, "height": 360}
{"video_id": "13tPLqGUSvg", "transcript": [{"start": 0.68, "duration": 3.44, "text": "all right in this video we're going to"}, {"start": 1.8, "duration": 5.599, "text": "do one more example of integrating uh a"}, {"start": 4.12, "duration": 7.04, "text": "function using substitution so um in"}, {"start": 7.399, "duration": 9.001, "text": "this case we've got the function 12x + 8"}, {"start": 11.16, "duration": 7.48, "text": "* 3x^2 + 4x + 1 all raised to the 9th so"}, {"start": 16.4, "duration": 4.959, "text": "again the thing that makes me think you"}, {"start": 18.64, "duration": 5.36, "text": "substitution um is I see something in"}, {"start": 21.359, "duration": 3.68, "text": "parentheses that's to the second power"}, {"start": 24.0, "duration": 2.32, "text": "and then you know if I take the"}, {"start": 25.039, "duration": 3.24, "text": "derivative of that I'm going to get"}, {"start": 26.32, "duration": 4.0, "text": "something to the first Power and"}, {"start": 28.279, "duration": 4.561, "text": "hopefully it's set up so that everything"}, {"start": 30.32, "duration": 5.04, "text": "will work out correctly so I'm going to"}, {"start": 32.84, "duration": 6.44, "text": "pick U to be the"}, {"start": 35.36, "duration": 5.519, "text": "3x^2 + 4x + 1 so the stuff that's being"}, {"start": 39.28, "duration": 4.92, "text": "raised to the"}, {"start": 40.879, "duration": 6.081, "text": "power our DU well uh the derivative of"}, {"start": 44.2, "duration": 6.72, "text": "3x^2 we would do 3 * 2 which would give"}, {"start": 46.96, "duration": 5.439, "text": "us 6X to the 1 and then our positive 4X"}, {"start": 50.92, "duration": 2.88, "text": "the derivative of that will just turn"}, {"start": 52.399, "duration": 4.0, "text": "into"}, {"start": 53.8, "duration": 6.32, "text": "pos4 um and then we still have our DX"}, {"start": 56.399, "duration": 7.32, "text": "left over well I don't want just 6X Plus"}, {"start": 60.12, "duration": 7.24, "text": "4 I want 12x + 8 but notice if you"}, {"start": 63.719, "duration": 3.641, "text": "multiply the right side by"}, {"start": 67.68, "duration": 4.84, "text": "two well we would have to multiply the"}, {"start": 69.92, "duration": 7.64, "text": "left side by two as well well then if we"}, {"start": 72.52, "duration": 7.44, "text": "distribute we do get 12x + 8 DX so"}, {"start": 77.56, "duration": 4.64, "text": "everything is set up nicely to work out"}, {"start": 79.96, "duration": 3.96, "text": "so it says I'm going to the stuff in"}, {"start": 82.2, "duration": 2.919, "text": "parentheses I'm going to replace that"}, {"start": 83.92, "duration": 3.6, "text": "with"}, {"start": 85.119, "duration": 5.721, "text": "u and then we still have to replace the"}, {"start": 87.52, "duration": 6.919, "text": "12x + 8 DX"}, {"start": 90.84, "duration": 7.68, "text": "so the 12x + 8 DX it says we can replace"}, {"start": 94.439, "duration": 6.081, "text": "that with 2 * du and now we're in"}, {"start": 98.52, "duration": 4.68, "text": "business okay we can just"}, {"start": 100.52, "duration": 7.16, "text": "integrate so the anti-derivative of U to"}, {"start": 103.2, "duration": 7.08, "text": "the 9 will be well U to the 10th over 10"}, {"start": 107.68, "duration": 6.799, "text": "plus C well 2 over 10 that's going to"}, {"start": 110.28, "duration": 9.119, "text": "give us 1 U is our"}, {"start": 114.479, "duration": 7.481, "text": "3x^2 + 4x + 1 so now I'm just replacing"}, {"start": 119.399, "duration": 7.64, "text": "our U with what we originally picked it"}, {"start": 121.96, "duration": 7.6, "text": "as and plus C and now again we found our"}, {"start": 127.039, "duration": 4.641, "text": "anti-derivative so again it's it's"}, {"start": 129.56, "duration": 5.28, "text": "certainly common when you calculate your"}, {"start": 131.68, "duration": 4.6, "text": "du for it to not work out exactly um you"}, {"start": 134.84, "duration": 3.36, "text": "know so just kind of look especially"}, {"start": 136.28, "duration": 4.4, "text": "maybe you can just multiply or divide to"}, {"start": 138.2, "duration": 4.64, "text": "get you know the the the numbers that"}, {"start": 140.68, "duration": 2.16, "text": "you"}, {"start": 143.08, "duration": 3.0, "text": "need"}], "title": "Calculus : U-substitution Involving Polynomial Functions - Ex 3", "description": "Thanks to all of you who support me on Patreon. You da real mvps!   $1 per month helps!! :)  https://www.patreon.com/patrickjmt !! Calculus : U-substitution Involving Polynomial Functions - Ex 3", "lengthSeconds": 144, "width": 480, "height": 360}
{"video_id": "ifYSOMwNsTA", "transcript": [{"start": 0.52, "duration": 4.08, "text": "okay viewers we want to simplify this"}, {"start": 2.5, "duration": 3.72, "text": "you're going to make expression the very"}, {"start": 4.6, "duration": 6.24, "text": "first thing I want to do is to write"}, {"start": 6.22, "duration": 7.2, "text": "this decimal in a index form as log 10"}, {"start": 10.84, "duration": 4.62, "text": "to the power negative 3 base 10 because"}, {"start": 13.42, "duration": 4.8, "text": "10 to the power negative 3 is same as"}, {"start": 15.46, "duration": 4.5, "text": "this expression we have here so of"}, {"start": 18.22, "duration": 3.6, "text": "course technology will affect the power"}, {"start": 19.96, "duration": 5.1, "text": "negative three so I have negative three"}, {"start": 21.82, "duration": 7.5, "text": "you know 10 this 10 and log 10 to this"}, {"start": 25.06, "duration": 7.4, "text": "10 is just one so I have minus 3 as my"}, {"start": 29.32, "duration": 3.14, "text": "final answer thank you"}], "title": "Maths: Logarithmic Expression #mathematics #students #shorts", "description": "#mathematics #shorts #online #onlineclasses #logarithm #education", "lengthSeconds": 31, "width": 360, "height": 640}
{"video_id": "znN3Qys8uS4", "transcript": [{"start": 0.03, "duration": 5.309, "text": "okay so one more example of graphing a"}, {"start": 2.399, "duration": 6.271, "text": "complex number and finding its absolute"}, {"start": 5.339, "duration": 6.751, "text": "value or its modulus so here we have 1"}, {"start": 8.67, "duration": 6.839, "text": "minus I over 2 plus I've to write this"}, {"start": 12.09, "duration": 6.15, "text": "in the more familiar a plus bi form what"}, {"start": 15.509, "duration": 8.131, "text": "we do is we multiply by the conjugate of"}, {"start": 18.24, "duration": 7.74, "text": "the denominator which means we just flip"}, {"start": 23.64, "duration": 4.799, "text": "the sign in the middle so I'm going to"}, {"start": 25.98, "duration": 4.41, "text": "multiply the denominator by 2 minus I"}, {"start": 28.439, "duration": 6.571, "text": "but that also means we have to multiply"}, {"start": 30.39, "duration": 7.47, "text": "the numerator by 2 minus I so let's see"}, {"start": 35.01, "duration": 5.819, "text": "if we distribute things out we'll get 1"}, {"start": 37.86, "duration": 4.92, "text": "times 2 which will be 2 so I'm working"}, {"start": 40.829, "duration": 4.71, "text": "at the top we'll get 1 times negative I"}, {"start": 42.78, "duration": 6.779, "text": "which will be well negative I will get a"}, {"start": 45.539, "duration": 5.851, "text": "negative 2i then we have negative I and"}, {"start": 49.559, "duration": 6.541, "text": "negative I which is just going to give"}, {"start": 51.39, "duration": 6.66, "text": "us a positive I squared okay in the"}, {"start": 56.1, "duration": 5.849, "text": "denominator we'll have 2 times 2 which"}, {"start": 58.05, "duration": 6.719, "text": "is 4 we'll get 2 times negative I which"}, {"start": 61.949, "duration": 5.941, "text": "will be well negative 2i on the inside"}, {"start": 64.769, "duration": 5.611, "text": "we'll get a positive 2 I"}, {"start": 67.89, "duration": 6.66, "text": "then we have positive I times negative I"}, {"start": 70.38, "duration": 6.45, "text": "which will give us negative I squared so"}, {"start": 74.55, "duration": 4.53, "text": "let's see in the top we have two we've"}, {"start": 76.83, "duration": 4.41, "text": "got negative I and a negative 2i"}, {"start": 79.08, "duration": 3.69, "text": "that will give us negative 3i we have"}, {"start": 81.24, "duration": 3.72, "text": "plus I squared"}, {"start": 82.77, "duration": 4.86, "text": "but that's plus negative 1 which we can"}, {"start": 84.96, "duration": 5.31, "text": "write as minus 1 notice in the"}, {"start": 87.63, "duration": 6.54, "text": "denominator the negative 2i and the"}, {"start": 90.27, "duration": 7.73, "text": "positive 2 I cancel we would have 4"}, {"start": 94.17, "duration": 6.839, "text": "minus I squared which is negative 1 so"}, {"start": 98.0, "duration": 9.91, "text": "in the numerator we've got 2 minus 1"}, {"start": 101.009, "duration": 9.991, "text": "which is 1 minus 3i all over 4 plus 1"}, {"start": 107.91, "duration": 10.89, "text": "which is 5 and we can now rewrite this"}, {"start": 111.0, "duration": 10.53, "text": "as 1/5 minus 3/5 times I so now by doing"}, {"start": 118.8, "duration": 8.76, "text": "all this I've got it written as a more"}, {"start": 121.53, "duration": 10.32, "text": "recognizable complex number well all you"}, {"start": 127.56, "duration": 7.64, "text": "would do so maybe this is 1 and we'll"}, {"start": 131.85, "duration": 6.44, "text": "stick down here negative 1"}, {"start": 135.2, "duration": 4.53, "text": "it says we're graphing well how about"}, {"start": 138.29, "duration": 3.33, "text": "negative one how about we put positive"}, {"start": 139.73, "duration": 4.47, "text": "one on the right side"}, {"start": 141.62, "duration": 4.41, "text": "so here's positive one I was already"}, {"start": 144.2, "duration": 3.72, "text": "thinking of the negative so here's a"}, {"start": 146.03, "duration": 3.989, "text": "positive one and negative one"}, {"start": 147.92, "duration": 5.55, "text": "well it says for the real part you're"}, {"start": 150.019, "duration": 5.58, "text": "only going over to one fifth and then we"}, {"start": 153.47, "duration": 5.04, "text": "have to go down so let's see one two"}, {"start": 155.599, "duration": 6.061, "text": "three four five so here would be"}, {"start": 158.51, "duration": 5.22, "text": "negative three-fifths we can put a dot"}, {"start": 161.66, "duration": 4.799, "text": "right there so the real part is positive"}, {"start": 163.73, "duration": 4.8, "text": "one-fifth the imaginary part is negative"}, {"start": 166.459, "duration": 5.011, "text": "three-fifths again we can just make a"}, {"start": 168.53, "duration": 4.35, "text": "vector pointing at that little dot and"}, {"start": 171.47, "duration": 6.57, "text": "that would now be the graph of our"}, {"start": 172.88, "duration": 7.05, "text": "complex number to find the modulus again"}, {"start": 178.04, "duration": 6.57, "text": "to find the modulus we just take each"}, {"start": 179.93, "duration": 8.839, "text": "component the real part and the"}, {"start": 184.61, "duration": 8.959, "text": "imaginary part and square it so plus"}, {"start": 188.769, "duration": 4.8, "text": "negative 3/5 squared"}, {"start": 193.739, "duration": 7.56, "text": "and now just a little bit of a"}, {"start": 195.659, "duration": 10.25, "text": "arithmetic so we'll get 1 over 25 plus 9"}, {"start": 201.299, "duration": 8.851, "text": "over 25 when we square everything"}, {"start": 205.909, "duration": 5.92, "text": "that'll give us 10 over 25 we can"}, {"start": 210.15, "duration": 5.13, "text": "rewrite this as the square root of 10"}, {"start": 211.829, "duration": 6.0, "text": "over the square root of 25 which would"}, {"start": 215.28, "duration": 5.76, "text": "just simply I think reduce down to the"}, {"start": 217.829, "duration": 4.681, "text": "square root of 10 over 5 and we can't"}, {"start": 221.04, "duration": 3.509, "text": "really simplify the square root of 10"}, {"start": 222.51, "duration": 4.349, "text": "any further so I think I would leave it"}, {"start": 224.549, "duration": 7.28, "text": "right there so we would say the modulus"}, {"start": 226.859, "duration": 4.97, "text": "of this complex number is root 10 over 5"}], "title": "Complex Numbers: Graphing and Finding the Modulus, Ex 2", "description": "Thanks to all of you who support me on Patreon. You da real mvps!   $1 per month helps!! :)  https://www.patreon.com/patrickjmt !! Complex Numbers: Graphing and Finding the Modulus, Ex 2. In this video, I show how to graph another complex number and how to find its modulus.", "lengthSeconds": 231, "width": 480, "height": 360}
{"video_id": "2lMnP89qeMU", "transcript": [{"start": 0.88, "duration": 3.959, "text": "all right in this video I just want to"}, {"start": 2.2, "duration": 4.32, "text": "do another another problem where we're"}, {"start": 4.839, "duration": 4.76, "text": "going to solve an equation containing a"}, {"start": 6.52, "duration": 7.119, "text": "radical so here we have the problem < TK"}, {"start": 9.599, "duration": 5.681, "text": "of 4 - W - 6 = -2 again the first thing"}, {"start": 13.639, "duration": 4.761, "text": "you want to do is just get the radical"}, {"start": 15.28, "duration": 4.68, "text": "by itself so you don't well I mean"}, {"start": 18.4, "duration": 4.639, "text": "that's really almost always the best"}, {"start": 19.96, "duration": 5.399, "text": "thing to do so I'm going to add six to"}, {"start": 23.039, "duration": 6.961, "text": "both sides on the left I'll have the"}, {"start": 25.359, "duration": 6.601, "text": "sare < TK of 4 - W on the right side -2"}, {"start": 30.0, "duration": 4.399, "text": "+ 6 is going to give me"}, {"start": 31.96, "duration": 5.119, "text": "pos4 and now again to get rid of the"}, {"start": 34.399, "duration": 6.441, "text": "square root I simply"}, {"start": 37.079, "duration": 6.681, "text": "Square both sides if I square the square"}, {"start": 40.84, "duration": 4.039, "text": "root of 4 minus W I'm just left with 4"}, {"start": 43.76, "duration": 4.52, "text": "minus"}, {"start": 44.879, "duration": 6.52, "text": "W and on the right side 4 squar is going"}, {"start": 48.28, "duration": 4.439, "text": "to give me 16 well I can subtract four"}, {"start": 51.399, "duration": 5.881, "text": "from each"}, {"start": 52.719, "duration": 9.201, "text": "side and I'll get that- W = 12 and then"}, {"start": 57.28, "duration": 4.64, "text": "simply If I multiply both sides by -1"}, {"start": 63.879, "duration": 5.88, "text": "we'll get W = -12 and we can always"}, {"start": 68.24, "duration": 4.08, "text": "again you should always go back and"}, {"start": 69.759, "duration": 3.961, "text": "check your answers to see if they work"}, {"start": 72.32, "duration": 2.799, "text": "so notice if we plug it into the"}, {"start": 73.72, "duration": 5.32, "text": "original"}, {"start": 75.119, "duration": 9.241, "text": "problem we would have the square < TK of"}, {"start": 79.04, "duration": 7.0, "text": "4 - -12 which would make 4 + 12 - 6 and"}, {"start": 84.36, "duration": 5.24, "text": "again we're we're asking ourselves does"}, {"start": 86.04, "duration": 9.32, "text": "that equal -2 well notice 4 + 12 is 16"}, {"start": 89.6, "duration": 9.879, "text": "the 16 is 4 4 - 6 does in fact equal -2"}, {"start": 95.36, "duration": 8.68, "text": "so this solution W = -12 is correct and"}, {"start": 99.479, "duration": 4.561, "text": "we found the the answer"}], "title": "Solving an Equation Involving a Single Radical (Square Root) - Example 2", "description": "Thanks to all of you who support me on Patreon. You da real mvps!   $1 per month helps!! :)  https://www.patreon.com/patrickjmt !! Solving an Equation Involving a Single Radical (Square Root) - Example 2", "lengthSeconds": 102, "width": 480, "height": 360}
{"video_id": "rA9QLzXyAwM", "transcript": [{"start": 1.88, "duration": 6.6, "text": "So, if 16 + 4x is 10 more than 14, we want to know what is the value of 8x?"}, {"start": 8.48, "duration": 6.72, "text": "So, we've got answer choices of 2, 6, 16 or 80, so I guess what I would do on this one is just turn this into"}, {"start": 15.2, "duration": 4.4, "text": "an equation, and then solve for x, and then you could multiply by 8 and get the solution."}, {"start": 19.6, "duration": 2.46, "text": "So that's going to be the approach I'm going to use here."}, {"start": 22.96, "duration": 4.34, "text": "Okay, so 16 + 4x is 10 more than 14. We have to be careful and make sure"}, {"start": 27.3, "duration": 4.66, "text": "we set up the correct equation there, so 16 + 4x."}, {"start": 33.32, "duration": 4.62, "text": "Okay, I know it's not equal to 14, right? This quantity is 10 more,"}, {"start": 37.94, "duration": 9.86, "text": "so to make this side equal 14, since right now 16 + 4x is 10 more, I would have to subtract 10 from that quantity"}, {"start": 47.8, "duration": 1.8, "text": "to make it equal to 14."}, {"start": 50.2, "duration": 3.72, "text": "Okay, so be careful. I think this would be the one place to make a mistake, maybe you could, you know,"}, {"start": 53.92, "duration": 4.02, "text": "make this a plus sign and then, you know, maybe something weird would happen."}, {"start": 58.12, "duration": 7.08, "text": "But, since again, 6 + 4x, since that's 10 larger, I would have to subtract 10 away from it to make it equal 14."}, {"start": 65.56, "duration": 6.48, "text": "And again, now we just have a nice simple equation. So, 16 minus 10, that's going to leave us with 6."}, {"start": 73.22, "duration": 5.26, "text": "Plus 4x equals 14. We could subtract 6 from both sides."}, {"start": 78.78, "duration": 7.78, "text": "So we'll have 4x equals 14 minus 6, which is going to be 8. And I said I was going to solve for x,"}, {"start": 86.56, "duration": 3.84, "text": "but I'm not going to do that, actually, because I want the value- I mean, you could solve for x,"}, {"start": 90.4, "duration": 6.9, "text": "you could divide both sides by 4 and then get the value of x. I want the value of 8x."}, {"start": 97.7, "duration": 3.9, "text": "Well I've already got 4x on the left side, so what I'm going to do is I'm just going to multiply"}, {"start": 101.6, "duration": 8.06, "text": "both sides here by 2. I'm going to multiply the right side also by 2. And I'm multiplying the left side by 2"}, {"start": 109.66, "duration": 4.62, "text": "because hey, I recognize that that's going to give me- 2 times 4x is going to give me 8x."}, {"start": 115.14, "duration": 8.08, "text": "And 8 multiplied by 2 is going to give me 16. So the answer choice of C, of 16 is going to be the correct answer."}], "title": "SAT Math - Question 4, Section 2, 2016 Edition and Beyond", "description": "Thanks to all of you who support me on Patreon. You da real mvps!   $1 per month helps!! :)  https://www.patreon.com/patrickjmt !! SAT Math - Question 4, 2016 Edition and Beyond.\nSection 2 - A Calculator IS allowed!", "lengthSeconds": 124, "width": 480, "height": 360}
{"video_id": "iMKv6jOYut0", "transcript": [{"start": 0.35, "duration": 3.94, "text": "all right in this video and I do another"}, {"start": 2.79, "duration": 4.46, "text": "example of solving a word problem"}, {"start": 4.29, "duration": 5.34, "text": "involving inequality so in this problem"}, {"start": 7.25, "duration": 4.0, "text": "there's suppose you have two options"}, {"start": 9.63, "duration": 3.27, "text": "you're going to rent a car and either"}, {"start": 11.25, "duration": 4.889, "text": "option a charges you have twenty five"}, {"start": 12.9, "duration": 6.12, "text": "bucks a day and five cents a mile or"}, {"start": 16.139, "duration": 4.771, "text": "option B charges you zero dollars a day"}, {"start": 19.02, "duration": 4.499, "text": "but they charge you a lot more per mile"}, {"start": 20.91, "duration": 4.35, "text": "they charge you 70 cents per mile and we"}, {"start": 23.519, "duration": 4.35, "text": "want to know how many miles must one"}, {"start": 25.26, "duration": 4.589, "text": "drive and one day for option a to be"}, {"start": 27.869, "duration": 3.42, "text": "cheaper because certainly right"}, {"start": 29.849, "duration": 3.451, "text": "originally freely gonna drive a few"}, {"start": 31.289, "duration": 3.991, "text": "miles better to pay the seventy cents a"}, {"start": 33.3, "duration": 6.48, "text": "mile and not have to pay this twenty"}, {"start": 35.28, "duration": 8.13, "text": "five dollars a day so we're trying to"}, {"start": 39.78, "duration": 8.04, "text": "figure out here when is the cost of"}, {"start": 43.41, "duration": 9.419, "text": "option a we want that to be less than"}, {"start": 47.82, "duration": 7.23, "text": "the cost of option B okay well let's"}, {"start": 52.829, "duration": 4.41, "text": "think about option A suppose so we'll"}, {"start": 55.05, "duration": 5.579, "text": "let X here denote the number of miles"}, {"start": 57.239, "duration": 5.941, "text": "driven in one day so if you drive X"}, {"start": 60.629, "duration": 4.651, "text": "miles how much would it cost to use"}, {"start": 63.18, "duration": 5.49, "text": "option A well you've got to pay that"}, {"start": 65.28, "duration": 7.17, "text": "initial twenty five dollars plus you pay"}, {"start": 68.67, "duration": 6.42, "text": "five cents per mile so we'll just take"}, {"start": 72.45, "duration": 5.25, "text": "point O five times X and that will give"}, {"start": 75.09, "duration": 6.59, "text": "us the you know the the total number of"}, {"start": 77.7, "duration": 7.05, "text": "the cost incurred by driving X miles"}, {"start": 81.68, "duration": 6.1, "text": "okay option B though okay we don't have"}, {"start": 84.75, "duration": 7.52, "text": "any upfront cost like we did here but it"}, {"start": 87.78, "duration": 6.839, "text": "is gonna cost seventy cents per mile"}, {"start": 92.27, "duration": 6.33, "text": "okay and now we've got a nice little"}, {"start": 94.619, "duration": 3.981, "text": "just simply a nice little"}, {"start": 98.77, "duration": 5.7, "text": "linear inequality to solve so alright"}, {"start": 101.619, "duration": 4.68, "text": "the first thing I'm going to do is I'm"}, {"start": 104.47, "duration": 4.59, "text": "simply going to I think I'm gonna put"}, {"start": 106.299, "duration": 6.201, "text": "the X's on the left side well let's put"}, {"start": 109.06, "duration": 8.82, "text": "them on the right so minus 0.05 X minus"}, {"start": 112.5, "duration": 12.07, "text": "0.05 X so we have 25 less than okay so"}, {"start": 117.88, "duration": 9.96, "text": "0.7 zero minus 0.05 is 0.65 X and now"}, {"start": 124.57, "duration": 10.26, "text": "all we're simply gonna do is we're going"}, {"start": 127.84, "duration": 8.67, "text": "to divide both sides by 0.65 and that"}, {"start": 134.83, "duration": 8.55, "text": "it's gonna give us our answer so if I"}, {"start": 136.51, "duration": 9.03, "text": "take 25 and divide it by 0.65 I'm"}, {"start": 143.38, "duration": 4.52, "text": "getting that to be thirty eight point"}, {"start": 145.54, "duration": 6.96, "text": "four six miles"}, {"start": 147.9, "duration": 6.04, "text": "so maybe assuming you have to drive I"}, {"start": 152.5, "duration": 2.19, "text": "don't know well let's just leave it at"}, {"start": 153.94, "duration": 3.42, "text": "that"}, {"start": 154.69, "duration": 4.29, "text": "so suppose they charge you based on you"}, {"start": 157.36, "duration": 4.909, "text": "know the fraction of the mile that you"}, {"start": 158.98, "duration": 6.24, "text": "drive it says basically if you drive a"}, {"start": 162.269, "duration": 5.681, "text": "little bit over 38 miles almost thirty"}, {"start": 165.22, "duration": 4.59, "text": "eight and a half miles if you drive more"}, {"start": 167.95, "duration": 5.39, "text": "than that in one day it's going to be"}, {"start": 169.81, "duration": 3.53, "text": "cheaper to use option a"}], "title": "Solving Word Problems Involving Inequalities - Example 3", "description": "Thanks to all of you who support me on Patreon. You da real mvps!   $1 per month helps!! :)  https://www.patreon.com/patrickjmt !! Solving Word Problems Involving Inequalities - Example 3", "lengthSeconds": 172, "width": 480, "height": 360}
{"video_id": "wE3Us-d5KJo", "transcript": [{"start": 0.0, "duration": 3.84, "text": "all right in this video I want to do a"}, {"start": 1.709, "duration": 4.771, "text": "couple examples of solving absolute"}, {"start": 3.84, "duration": 4.44, "text": "value equations and to solve the"}, {"start": 6.48, "duration": 4.26, "text": "absolute value equation if we have the"}, {"start": 8.28, "duration": 3.989, "text": "absolute value of some quantity a so"}, {"start": 10.74, "duration": 4.74, "text": "this could be a lot of stuff in between"}, {"start": 12.269, "duration": 5.791, "text": "the absolute value bars if that if we"}, {"start": 15.48, "duration": 4.98, "text": "want that to equal B for this equation"}, {"start": 18.06, "duration": 4.26, "text": "to have solutions B is going to have to"}, {"start": 20.46, "duration": 3.989, "text": "be greater than or equal to 0 otherwise"}, {"start": 22.32, "duration": 4.65, "text": "there won't be solutions so that's one"}, {"start": 24.449, "duration": 4.08, "text": "thing to kind of keep in mind to solve"}, {"start": 26.97, "duration": 3.24, "text": "this equation we basically have to solve"}, {"start": 28.529, "duration": 4.351, "text": "two different equations we just remove"}, {"start": 30.21, "duration": 4.77, "text": "the bars and say you know solve a equals"}, {"start": 32.88, "duration": 6.12, "text": "B this equation and the other equation"}, {"start": 34.98, "duration": 7.68, "text": "we solve is a equals negative B so let's"}, {"start": 39.0, "duration": 5.489, "text": "solve the absolute value of 3x plus 2"}, {"start": 42.66, "duration": 4.05, "text": "equals 8 this will have a solution"}, {"start": 44.489, "duration": 4.081, "text": "because we've got a number that's"}, {"start": 46.71, "duration": 5.15, "text": "greater than or equal to 0 on the right"}, {"start": 48.57, "duration": 8.16, "text": "it basically says just remove the bars"}, {"start": 51.86, "duration": 9.07, "text": "solve that equation also we have to just"}, {"start": 56.73, "duration": 6.059, "text": "remove the bars and solve the"}, {"start": 60.93, "duration": 4.47, "text": "corresponding equation where we make the"}, {"start": 62.789, "duration": 4.711, "text": "value negative ok well this is pretty"}, {"start": 65.4, "duration": 4.56, "text": "easy so now we just have two simple"}, {"start": 67.5, "duration": 6.09, "text": "linear equations I can subtract 2 from"}, {"start": 69.96, "duration": 7.35, "text": "both sides that will give me 3x equals"}, {"start": 73.59, "duration": 6.74, "text": "positive 6 if I divide both sides by 3"}, {"start": 77.31, "duration": 5.669, "text": "I'll get x equals 2 is a solution and"}, {"start": 80.33, "duration": 5.649, "text": "for the other equation I'll simply"}, {"start": 82.979, "duration": 5.911, "text": "subtract 2 from both sides and get 3x"}, {"start": 85.979, "duration": 6.0, "text": "equals negative 10 and then again I'll"}, {"start": 88.89, "duration": 6.15, "text": "just divide both sides by 3 and get my"}, {"start": 91.979, "duration": 5.761, "text": "second solution of x equals negative 10"}, {"start": 95.04, "duration": 4.17, "text": "thirds and again I definitely encourage"}, {"start": 97.74, "duration": 3.54, "text": "you go back and check both of these"}, {"start": 99.21, "duration": 4.199, "text": "numbers do the arithmetic on the inside"}, {"start": 101.28, "duration": 3.15, "text": "take the absolute value at the end and"}, {"start": 103.409, "duration": 2.591, "text": "you'll see that you get something that"}, {"start": 104.43, "duration": 4.58, "text": "equals 8"}, {"start": 106.0, "duration": 5.62, "text": "so another one here real quick suppose"}, {"start": 109.01, "duration": 6.66, "text": "we have the absolute value of 7 minus 2x"}, {"start": 111.62, "duration": 8.91, "text": "equals 10 ok so again we're just going"}, {"start": 115.67, "duration": 7.92, "text": "to do the exact same thing here I just"}, {"start": 120.53, "duration": 5.82, "text": "remove the bars 7 minus 2x I set that"}, {"start": 123.59, "duration": 5.91, "text": "equal to 10 and then again I do the same"}, {"start": 126.35, "duration": 7.14, "text": "thing I just removed the bars but I set"}, {"start": 129.5, "duration": 5.7, "text": "it equal to negative 10 ok so to solve"}, {"start": 133.49, "duration": 4.23, "text": "the first equation I'm going to subtract"}, {"start": 135.2, "duration": 5.27, "text": "7 from both sides that will leave me"}, {"start": 137.72, "duration": 5.54, "text": "with negative 2x equals positive 3"}, {"start": 140.47, "duration": 4.78, "text": "simply divide both sides by negative 2"}, {"start": 143.26, "duration": 6.91, "text": "and that will give me one of my"}, {"start": 145.25, "duration": 6.78, "text": "solutions of negative 3 halves and for"}, {"start": 150.17, "duration": 5.1, "text": "the other one I can subtract 7 from both"}, {"start": 152.03, "duration": 6.6, "text": "sides that will give me negative 2x"}, {"start": 155.27, "duration": 5.07, "text": "equals negative 17 if I divide both"}, {"start": 158.63, "duration": 5.55, "text": "sides by negative 2"}, {"start": 160.34, "duration": 6.02, "text": "I'll get my second solution of x equals"}, {"start": 164.18, "duration": 5.51, "text": "so two negatives make a positive"}, {"start": 166.36, "duration": 3.33, "text": "17 halves"}], "title": "Solving Absolute Value Equations (2 Quick Examples)", "description": "(https://www.youtube.com/watch?v=wE3Us-d5KJo)\n(https://youtu.be/qRFX_HK-6Ik)\n(https://youtu.be/OzmqAZZd88g)\n(https://youtu.be/C7de7f2LER0)\n(https://youtu.be/HlQZ6XkRIb0)\n(https://youtu.be/rC4bRSQetvQ)\n\n\nIn this video, we explore how to solve absolute value equations focusing on two examples:\n\nSolving an absolute value equation: \u2223 3x + 2 \u2223 = 8\nSolving an absolute value equation: \u2223 7 \u2212 2x \u2223 = 10\n\nWe go over the step-by-step process for breaking down these equations, including isolating the absolute value expression, setting up two separate cases (positive and negative), and solving for the variable. By the end of the video, you'll understand how to approach both simple and complex absolute value equations and inequalities.\n\nWhat You Will Learn:\n\nHow to solve absolute value equations by setting up two cases\nHow to solve absolute value inequalities\nTechniques for checking solutions using substitution\nMake sure to leave a comment if you find the video helpful, and don\u2019t forget to like, share, and subscribe! Also, let your friends, teachers, or parents know about this channel if you think they would benefit from these lessons.\n\nSupport me on Patreon if you\u2019d like to see more content: https://www.patreon.com/patrickjmt?ty=c\n\n#AbsoluteValue #Equations #Inequalities #MathHelp #SolvingEquations #Algebra #PatrickJMT", "lengthSeconds": 170, "width": 480, "height": 360}
{"video_id": "V5dBFw0Y0pE", "transcript": [{"start": 1.04, "duration": 3.96, "text": "all right in this example we're going to"}, {"start": 2.56, "duration": 6.8, "text": "find the equation of a hyperbola that"}, {"start": 5.0, "duration": 9.799, "text": "has vertices at 0 comma positive ne5 and"}, {"start": 9.36, "duration": 5.439, "text": "the fosi are located at 0 comma positive"}, {"start": 15.0, "duration": 5.199, "text": "-20 so again uh I'm just going to make a"}, {"start": 18.32, "duration": 6.4, "text": "little rough"}, {"start": 20.199, "duration": 8.681, "text": "sketch uh of my my hyperbola so the"}, {"start": 24.72, "duration": 6.32, "text": "vertices are at 0 comma 15 and 0 comma"}, {"start": 28.88, "duration": 8.96, "text": "15 and again the"}, {"start": 31.04, "duration": 10.44, "text": "fosi are at 0 uh 0 comma 20 and 0 comma"}, {"start": 37.84, "duration": 6.84, "text": "-20 so okay so that tells me that"}, {"start": 41.48, "duration": 5.8, "text": "my hyperbola will be opening"}, {"start": 44.68, "duration": 4.719, "text": "upwards and"}, {"start": 47.28, "duration": 4.04, "text": "downwards So based on that I've got a"}, {"start": 49.399, "duration": 5.441, "text": "little bit of information already I know"}, {"start": 51.32, "duration": 9.039, "text": "that I'm going to use the equation y^ 2"}, {"start": 54.84, "duration": 6.8, "text": "over a 2 - x^2 over b^2 = 1 one so"}, {"start": 60.359, "duration": 2.441, "text": "that's kind of the the first thing I"}, {"start": 61.64, "duration": 5.479, "text": "think"}, {"start": 62.8, "duration": 5.959, "text": "about and again to get the value uh you"}, {"start": 67.119, "duration": 5.121, "text": "know the the"}, {"start": 68.759, "duration": 6.561, "text": "value for a that's just the distance"}, {"start": 72.24, "duration": 6.0, "text": "from wherever our hyperbola is centered"}, {"start": 75.32, "duration": 5.6, "text": "which is at the origin okay so uh it's"}, {"start": 78.24, "duration": 5.32, "text": "centered at the origin we look at the"}, {"start": 80.92, "duration": 4.839, "text": "distance to one of our vertices and"}, {"start": 83.56, "duration": 5.4, "text": "that's going to be our a value so in"}, {"start": 85.759, "duration": 5.68, "text": "this case well a simply equals the value"}, {"start": 88.96, "duration": 4.199, "text": "15 okay so the only thing that we're"}, {"start": 91.439, "duration": 4.0, "text": "really missing is we need to know the"}, {"start": 93.159, "duration": 5.28, "text": "value for B once we know the value for B"}, {"start": 95.439, "duration": 6.72, "text": "hey we're done we don't quite know that"}, {"start": 98.439, "duration": 5.32, "text": "though um we do have this distance from"}, {"start": 102.159, "duration": 5.92, "text": "again where it's centered to one of the"}, {"start": 103.759, "duration": 7.081, "text": "fosi and since that goes from0 up to 20"}, {"start": 108.079, "duration": 6.161, "text": "uh the distance to this the fosi is 20"}, {"start": 110.84, "duration": 6.199, "text": "and we label that uh as C and the"}, {"start": 114.24, "duration": 5.159, "text": "relationship the equation that we'll use"}, {"start": 117.039, "duration": 7.601, "text": "the relationship is the fact that this"}, {"start": 119.399, "duration": 7.68, "text": "distance this c^2 = a^2 +"}, {"start": 124.64, "duration": 4.8, "text": "b^2 okay so now we can use this equation"}, {"start": 127.079, "duration": 6.561, "text": "to solve for b squared and hey we'll be"}, {"start": 129.44, "duration": 8.6, "text": "finished okay so C is 20 we're squaring"}, {"start": 133.64, "duration": 7.04, "text": "that a is 15 we're squaring that and"}, {"start": 138.04, "duration": 7.52, "text": "then we'll figure out the value of"}, {"start": 140.68, "duration": 7.32, "text": "b^2 so 20 s is 400 let's see 15 squar is"}, {"start": 145.56, "duration": 2.44, "text": "going to be"}, {"start": 148.84, "duration": 8.0, "text": "225 + B ^2 well we can subt subtract 225"}, {"start": 153.64, "duration": 5.239, "text": "from both sides 400 minus uh 225 will"}, {"start": 156.84, "duration": 7.0, "text": "leave us with"}, {"start": 158.879, "duration": 7.521, "text": "175 = B ^2 and now I'm finished we can"}, {"start": 163.84, "duration": 7.2, "text": "uh just fill in our equation"}, {"start": 166.4, "duration": 6.559, "text": "so we have y^2 over a 2 and again that's"}, {"start": 171.04, "duration": 5.44, "text": "going to be 15"}, {"start": 172.959, "duration": 7.801, "text": "s 15 2 is"}, {"start": 176.48, "duration": 6.44, "text": "225 - x^2 over B squ which we just said"}, {"start": 180.76, "duration": 2.16, "text": "is"}, {"start": 183.12, "duration": 8.08, "text": "175 equal pos1 and that will be the"}, {"start": 186.799, "duration": 4.401, "text": "equation of our hyperbola"}], "title": "Conic Sections, Hyperbola : Find Equation Given Foci and Vertices", "description": "Thanks to all of you who support me on Patreon. You da real mvps!   $1 per month helps!! :)  https://www.patreon.com/patrickjmt !! Conic Sections, Hyperbola: Find, Equation Given Foci and Vertices.  In this example, we are given the vertices and the foci of an ellipse.  We then find the equation of the ellipse.", "lengthSeconds": 192, "width": 480, "height": 360}
{"video_id": "QY15VEK9slo", "transcript": [{"start": 0.719, "duration": 3.041, "text": "all right here we're going to find the"}, {"start": 1.839, "duration": 5.04, "text": "parametric equations of a line that"}, {"start": 3.76, "duration": 6.0, "text": "passes through a point -1 comma 2 comma"}, {"start": 6.879, "duration": 6.121, "text": "3 and it's parallel to the vector with"}, {"start": 9.76, "duration": 5.28, "text": "components 4 five six so this is just"}, {"start": 13.0, "duration": 5.279, "text": "kind of one of those knowing a formula"}, {"start": 15.04, "duration": 6.28, "text": "type of thing so the equation of a line"}, {"start": 18.279, "duration": 8.16, "text": "we can write that as r equal"}, {"start": 21.32, "duration": 9.08, "text": "rb0 + T * uh we'll use a little V here"}, {"start": 26.439, "duration": 6.28, "text": "so R sub z um you can think about this I"}, {"start": 30.4, "duration": 5.999, "text": "guess is being um it's really going to"}, {"start": 32.719, "duration": 8.68, "text": "be um the position Vector whose tip lies"}, {"start": 36.399, "duration": 7.081, "text": "on the plane okay so uh so if you think"}, {"start": 41.399, "duration": 5.32, "text": "about the uh position Vector pointing at"}, {"start": 43.48, "duration": 4.96, "text": "this point 1 comma 2 comma 3 that's what"}, {"start": 46.719, "duration": 4.84, "text": "we're going to plug in for our little r"}, {"start": 48.44, "duration": 7.279, "text": "sub Z we're going to use -1 comma 2"}, {"start": 51.559, "duration": 6.881, "text": "comma 3 and then we take T * uh"}, {"start": 55.719, "duration": 6.601, "text": "V and again this is just the direction"}, {"start": 58.44, "duration": 6.719, "text": "Vector of the line L the direction"}, {"start": 62.32, "duration": 6.08, "text": "Vector so we know it's parallel to this"}, {"start": 65.159, "duration": 5.521, "text": "Vector so we can use"}, {"start": 68.4, "duration": 5.92, "text": "that and now we've basically got the"}, {"start": 70.68, "duration": 6.079, "text": "equation of our line so we can"}, {"start": 74.32, "duration": 3.439, "text": "uh simplify this a little bit we can"}, {"start": 76.759, "duration": 4.841, "text": "write it as"}, {"start": 77.759, "duration": 6.921, "text": "4T 5T and"}, {"start": 81.6, "duration": 5.6, "text": "6t when I distribute the T and now we"}, {"start": 84.68, "duration": 7.56, "text": "can just add respective components so we"}, {"start": 87.2, "duration": 10.48, "text": "would have negative 1 + 4T comma 2 + 5 T"}, {"start": 92.24, "duration": 7.96, "text": "comma 3 + 6 T and now uh to get the"}, {"start": 97.68, "duration": 5.479, "text": "parametric equations uh we just look at"}, {"start": 100.2, "duration": 5.44, "text": "each individual component so the"}, {"start": 103.159, "duration": 6.441, "text": "parametric equation for x or sometimes"}, {"start": 105.64, "duration": 6.56, "text": "we'll write X of T is going to be -1 +"}, {"start": 109.6, "duration": 9.839, "text": "4T uh y of"}, {"start": 112.2, "duration": 11.199, "text": "T would be 2 + 5 T and then our Z of T"}, {"start": 119.439, "duration": 8.64, "text": "is going to be 3 + 6t and these are now"}, {"start": 123.399, "duration": 4.68, "text": "going to be the uh parametric equations"}], "title": "Parametric Equations of Line Passing Through a Point", "description": "Thanks to all of you who support me on Patreon. You da real mvps!   $1 per month helps!! :)  https://www.patreon.com/patrickjmt !! Parametric Equations of Line Passing Through a Point.  In this video, I show how to find the parametric equations of a line that passes through a given point with a given direction vector.", "lengthSeconds": 127, "width": 480, "height": 360}
{"video_id": "7-7hpHo0Fcw", "transcript": [{"start": 1.0, "duration": 3.24, "text": "all right in this video I'm going to do"}, {"start": 2.32, "duration": 4.039, "text": "another example of graphing a rational"}, {"start": 4.24, "duration": 7.6, "text": "function so here we're going to graph F"}, {"start": 6.359, "duration": 10.641, "text": "ofx equal x^2 - 9/ x - 3 um I'm going to"}, {"start": 11.84, "duration": 6.16, "text": "just rewrite this as y = x^2 - 9 over x"}, {"start": 17.0, "duration": 5.32, "text": "-"}, {"start": 18.0, "duration": 8.48, "text": "3 uh so at this point um notice a couple"}, {"start": 22.32, "duration": 10.0, "text": "things um notice this actually factors"}, {"start": 26.48, "duration": 8.2, "text": "the numerator factors is x - 3 over x +"}, {"start": 32.32, "duration": 6.239, "text": "3 and then in the denominator we've got"}, {"start": 34.68, "duration": 6.44, "text": "our xus 3 so notice we could actually"}, {"start": 38.559, "duration": 7.081, "text": "cancel out the x - 3 factors and we"}, {"start": 41.12, "duration": 6.68, "text": "would be left with Y = X + 3 which is"}, {"start": 45.64, "duration": 4.919, "text": "nice Rec uh nice and recognizable"}, {"start": 47.8, "duration": 5.04, "text": "hopefully um so yal MX plus b this is"}, {"start": 50.559, "duration": 5.041, "text": "just an equation of a line but we have"}, {"start": 52.84, "duration": 5.359, "text": "to be careful before we graph the line Y"}, {"start": 55.6, "duration": 4.84, "text": "= X + 3 and that's what it's going to"}, {"start": 58.199, "duration": 4.601, "text": "look like almost you know identically"}, {"start": 60.44, "duration": 5.28, "text": "there's going to be one small"}, {"start": 62.8, "duration": 5.48, "text": "change notice the domain of the original"}, {"start": 65.72, "duration": 5.16, "text": "function that we start with remember if"}, {"start": 68.28, "duration": 4.32, "text": "you have a fraction um the only thing"}, {"start": 70.88, "duration": 4.08, "text": "that's kind of bad for a fraction is we"}, {"start": 72.6, "duration": 5.24, "text": "don't want to divide by zero so the"}, {"start": 74.96, "duration": 6.4, "text": "domain of this function would be"}, {"start": 77.84, "duration": 5.72, "text": "um we could say it's all reals all real"}, {"start": 81.36, "duration": 7.079, "text": "numbers"}, {"start": 83.56, "duration": 6.96, "text": "except X = pos3 we'll have to leave that"}, {"start": 88.439, "duration": 3.64, "text": "value out of our domain"}, {"start": 90.52, "duration": 3.72, "text": "okay so even though kind of"}, {"start": 92.079, "duration": 5.04, "text": "algebraically we simplify it down to"}, {"start": 94.24, "duration": 4.96, "text": "this Y = X + 3 which is the equation of"}, {"start": 97.119, "duration": 3.96, "text": "a line you still have to take the"}, {"start": 99.2, "duration": 3.52, "text": "original you know the original function"}, {"start": 101.079, "duration": 3.4, "text": "into consideration you have to take its"}, {"start": 102.72, "duration": 5.52, "text": "domain into"}, {"start": 104.479, "duration": 9.761, "text": "account so okay so same thing down here"}, {"start": 108.24, "duration": 6.96, "text": "we've got the graph of y = x + 3 um and"}, {"start": 114.24, "duration": 7.6, "text": "again the"}, {"start": 115.2, "duration": 6.64, "text": "domain is all X or all reals"}, {"start": 122.119, "duration": 5.561, "text": "except x ="}, {"start": 124.88, "duration": 6.6, "text": "3 so let's see the Y intercept of this"}, {"start": 127.68, "duration": 5.76, "text": "is positive three 1 2 3 the way I kind"}, {"start": 131.48, "duration": 5.0, "text": "of think about it is if I plugged three"}, {"start": 133.44, "duration": 5.64, "text": "in we would get six"}, {"start": 136.48, "duration": 4.44, "text": "out but since it's undefined there's"}, {"start": 139.08, "duration": 4.32, "text": "going to be a little circle"}, {"start": 140.92, "duration": 3.36, "text": "there and notice if you plug three into"}, {"start": 143.4, "duration": 3.96, "text": "the"}, {"start": 144.28, "duration": 5.16, "text": "denominator we get zero but we also get"}, {"start": 147.36, "duration": 3.48, "text": "zero in the numerator if we plug three"}, {"start": 149.44, "duration": 4.28, "text": "in"}, {"start": 150.84, "duration": 4.92, "text": "since we get 0 over 0 what that means is"}, {"start": 153.72, "duration": 5.879, "text": "you just have a hole in the graph at"}, {"start": 155.76, "duration": 5.839, "text": "that point so basically our our our"}, {"start": 159.599, "duration": 6.161, "text": "rational function looks exactly like the"}, {"start": 161.599, "duration": 4.161, "text": "line Y = X +"}, {"start": 166.239, "duration": 5.841, "text": "3 um except for this big hole in the"}, {"start": 169.2, "duration": 4.88, "text": "graph otherwise it would be"}, {"start": 172.08, "duration": 3.64, "text": "identical okay and that's all there is"}, {"start": 174.08, "duration": 4.64, "text": "to it in this"}, {"start": 175.72, "duration": 3.0, "text": "case"}], "title": "Graphing a Rational Function - Example 2", "description": "Thanks to all of you who support me on Patreon. You da real mvps!   $1 per month helps!! :)  https://www.patreon.com/patrickjmt !! Graphing a Rational Function - Example 2", "lengthSeconds": 177, "width": 480, "height": 360}
{"video_id": "-fbCI2qcgRk", "transcript": [{"start": 0.799, "duration": 3.52, "text": "all right in this video I want to talk"}, {"start": 2.2, "duration": 4.639, "text": "about finding the components of a vector"}, {"start": 4.319, "duration": 5.001, "text": "and what we'll do in this case is we'll"}, {"start": 6.839, "duration": 4.88, "text": "be given the magnitude and the direction"}, {"start": 9.32, "duration": 5.399, "text": "angle and uh from there we'll have to"}, {"start": 11.719, "duration": 5.241, "text": "figure out the components so the way"}, {"start": 14.719, "duration": 4.681, "text": "typically I like to do these is just use"}, {"start": 16.96, "duration": 4.6, "text": "a little bit of you know trigonometry"}, {"start": 19.4, "duration": 5.6, "text": "here um"}, {"start": 21.56, "duration": 5.44, "text": "so the idea is okay we've got a a vector"}, {"start": 25.0, "duration": 5.32, "text": "uh it makes an angle of"}, {"start": 27.0, "duration": 4.52, "text": "60\u00b0 uh from the origin so here's our"}, {"start": 30.32, "duration": 5.079, "text": "angle of"}, {"start": 31.52, "duration": 7.44, "text": "60\u00b0 and our Vector has a magnitude of 8"}, {"start": 35.399, "duration": 5.16, "text": "so again magnitude uh in in these cases"}, {"start": 38.96, "duration": 4.84, "text": "just correspond to the length of the"}, {"start": 40.559, "duration": 5.48, "text": "vector so really what I imagine is uh"}, {"start": 43.8, "duration": 3.84, "text": "just making like a little right triangle"}, {"start": 46.039, "duration": 4.281, "text": "and what I'm trying to find is I'm"}, {"start": 47.64, "duration": 5.88, "text": "trying to find the value for x and I'm"}, {"start": 50.32, "duration": 5.64, "text": "trying to find the value for y um the"}, {"start": 53.52, "duration": 5.24, "text": "value for x will be the first uh the"}, {"start": 55.96, "duration": 7.8, "text": "first component and the value for y will"}, {"start": 58.76, "duration": 8.32, "text": "be our second component so recall for a"}, {"start": 63.76, "duration": 6.48, "text": "triangle so let's see well we've got 60\u00b0"}, {"start": 67.08, "duration": 5.52, "text": "we've got8 we've got y we've got X we're"}, {"start": 70.24, "duration": 4.32, "text": "trying to solve for x and y so again"}, {"start": 72.6, "duration": 5.24, "text": "recall for a triangle I think we can use"}, {"start": 74.56, "duration": 4.4, "text": "S and cosine so let's see notice cosine"}, {"start": 77.84, "duration": 4.959, "text": "of"}, {"start": 78.96, "duration": 6.64, "text": "60\u00b0 remember cosine is the adjacent over"}, {"start": 82.799, "duration": 6.481, "text": "the hypotenuse so that'll give us X over"}, {"start": 85.6, "duration": 4.8, "text": "8 if we multiply both sides by8 we'll"}, {"start": 89.28, "duration": 6.159, "text": "get 8"}, {"start": 90.4, "duration": 9.079, "text": "cosine of 60\u00b0 is X well let's see cosine"}, {"start": 95.439, "duration": 6.401, "text": "of 60\u00b0 um I believe cine of 60\u00b0 is going"}, {"start": 99.479, "duration": 5.801, "text": "to give us"}, {"start": 101.84, "duration": 6.12, "text": "12 so 8 * a half is going to be"}, {"start": 105.28, "duration": 5.28, "text": "four so that's going to be the first uh"}, {"start": 107.96, "duration": 4.92, "text": "our first component"}, {"start": 110.56, "duration": 4.559, "text": "um so the first component is simply"}, {"start": 112.88, "duration": 3.839, "text": "going to have value four and we can do"}, {"start": 115.119, "duration": 4.04, "text": "the same thing to figure out the Y"}, {"start": 116.719, "duration": 5.601, "text": "component um the second component we can"}, {"start": 119.159, "duration": 6.161, "text": "just do well well s of"}, {"start": 122.32, "duration": 7.079, "text": "60\u00b0 s is the opposite over the"}, {"start": 125.32, "duration": 7.919, "text": "hypotenuse so y/ 8 again we can multiply"}, {"start": 129.399, "duration": 5.641, "text": "both sides by 8 so we'll get 8 * s of 60"}, {"start": 133.239, "duration": 6.28, "text": "="}, {"start": 135.04, "duration": 10.76, "text": "y s of 60\u00b0 that's just < tk3 over"}, {"start": 139.519, "duration": 9.201, "text": "2 well 8 / 2 is 4 we'll get 4 < tk3 as"}, {"start": 145.8, "duration": 4.92, "text": "our value for y so that that'll simply"}, {"start": 148.72, "duration": 4.48, "text": "be our second comp component for our"}, {"start": 150.72, "duration": 5.76, "text": "Vector here so that's it this original"}, {"start": 153.2, "duration": 6.319, "text": "Vector would have components 4 comma 4 <"}, {"start": 156.48, "duration": 3.039, "text": "TK 3"}], "title": "Finding the Components of a Vector, Ex 1", "description": "Thanks to all of you who support me on Patreon. You da real mvps!   $1 per month helps!! :)  https://www.patreon.com/patrickjmt !! Finding the Components of a Vector, Ex 1.  In this video, we are given the magnitude and direction angle for the vector and want to express the vector in component form.", "lengthSeconds": 158, "width": 480, "height": 360}
{"video_id": "6mxmfn8HX0c", "transcript": [{"start": 1.52, "duration": 4.799, "text": "here we're going to determine the"}, {"start": 2.84, "duration": 7.839, "text": "midpoint between the points 4 comma 7"}, {"start": 6.319, "duration": 9.121, "text": "and -6 comma 6 so recall the midpoint if"}, {"start": 10.679, "duration": 8.52, "text": "we have two points X1 comma y1 and X2"}, {"start": 15.44, "duration": 6.24, "text": "comma Y2 to get the midpoint we simply"}, {"start": 19.199, "duration": 4.961, "text": "add the x coordinates together and"}, {"start": 21.68, "duration": 4.28, "text": "divide by two and we do the same thing"}, {"start": 24.16, "duration": 4.32, "text": "for the y-coordinates we add those"}, {"start": 25.96, "duration": 4.559, "text": "together and divide by two you're"}, {"start": 28.48, "duration": 3.88, "text": "basically averaging the X coordinates"}, {"start": 30.519, "duration": 4.88, "text": "and you're averaging the y-coordinates"}, {"start": 32.36, "duration": 6.92, "text": "it doesn't matter what point you call X1"}, {"start": 35.399, "duration": 8.361, "text": "y1 and X2 Y2 so let's assume that you"}, {"start": 39.28, "duration": 7.72, "text": "know X1 y1 is 4 comma 7 and X2 comma Y 2"}, {"start": 43.76, "duration": 7.119, "text": "is -6 comma 6 so if we add the x"}, {"start": 47.0, "duration": 9.16, "text": "coordinates we'll have 4 +"}, {"start": 50.879, "duration": 7.081, "text": "-6 / 2 and then to get the midpoint the"}, {"start": 56.16, "duration": 6.359, "text": "the y-coordinate of the"}, {"start": 57.96, "duration": 7.399, "text": "midpoint We'll add 7 + 6 and then divide"}, {"start": 62.519, "duration": 2.84, "text": "that by"}, {"start": 65.479, "duration": 10.64, "text": "pos2 so let's see 4 + -6 that's going to"}, {"start": 69.119, "duration": 9.201, "text": "be -2 -2 / 2 will be -1 7 + 6 is 13"}, {"start": 76.119, "duration": 5.441, "text": "divided by two that doesn't really"}, {"start": 78.32, "duration": 6.68, "text": "reduce so I'm going to leave it as 13"}, {"start": 81.56, "duration": 6.44, "text": "over2 and that would be our"}, {"start": 85.0, "duration": 3.0, "text": "midpoint"}], "title": "Midpoint of Two Points - Another Example", "description": "Thanks to all of you who support me on Patreon. You da real mvps!   $1 per month helps!! :)  https://www.patreon.com/patrickjmt !! Midpoint of Two Points - Another Example.  \nIn this video I find the midpoint of two points", "lengthSeconds": 86, "width": 450, "height": 360}
{"video_id": "vwGTTJfEiA8", "transcript": [{"start": 1.88, "duration": 5.039, "text": "okay in this example we're going to find"}, {"start": 3.719, "duration": 7.88, "text": "the area and perimeter of a"}, {"start": 6.919, "duration": 7.241, "text": "parallelogram and to find the um the"}, {"start": 11.599, "duration": 6.241, "text": "area all you have to remember is that"}, {"start": 14.16, "duration": 5.76, "text": "the area formula for a parallelogram we"}, {"start": 17.84, "duration": 6.48, "text": "just take the length of the"}, {"start": 19.92, "duration": 7.16, "text": "base and multiply that by the height so"}, {"start": 24.32, "duration": 5.199, "text": "the length of the base multiplied by the"}, {"start": 27.08, "duration": 4.279, "text": "height and fortunately we are already"}, {"start": 29.519, "duration": 5.161, "text": "given both of those things we're given"}, {"start": 31.359, "duration": 6.88, "text": "that it has a base of length eight and"}, {"start": 34.68, "duration": 5.039, "text": "the height is the um this uh this dash"}, {"start": 38.239, "duration": 4.0, "text": "line here of"}, {"start": 39.719, "duration": 5.0, "text": "3.6 so we have to multiply those two"}, {"start": 42.239, "duration": 4.16, "text": "values together so it says the area is"}, {"start": 44.719, "duration": 4.52, "text": "going to equal"}, {"start": 46.399, "duration": 5.84, "text": "8 multiplied by"}, {"start": 49.239, "duration": 6.0, "text": "3.6 and I'm just going to do this by"}, {"start": 52.239, "duration": 6.921, "text": "hand so let's see I've got 3.6 I'm going"}, {"start": 55.239, "duration": 6.64, "text": "to multiply that by 8 so let's see 8 * 6"}, {"start": 59.16, "duration": 9.24, "text": "that's 48 8 so we'll drop down the 8 and"}, {"start": 61.879, "duration": 7.801, "text": "carry the 4 8 * 3 is 24 24 + 4 is going"}, {"start": 68.4, "duration": 4.2, "text": "to give us"}, {"start": 69.68, "duration": 5.32, "text": "28 and if we count we've got uh one"}, {"start": 72.6, "duration": 5.08, "text": "decimal place so I'll put the decimal"}, {"start": 75.0, "duration": 4.92, "text": "place right there and it says that we're"}, {"start": 77.68, "duration": 4.88, "text": "going to get an area of"}, {"start": 79.92, "duration": 5.36, "text": "28.8"}, {"start": 82.56, "duration": 5.68, "text": "units all right so not too terrible"}, {"start": 85.28, "duration": 5.56, "text": "again just just formulas to"}, {"start": 88.24, "duration": 4.36, "text": "know the other part is is the perimeter"}, {"start": 90.84, "duration": 4.959, "text": "and again the the perimeter is just the"}, {"start": 92.6, "duration": 6.44, "text": "distance around and since we have a"}, {"start": 95.799, "duration": 6.241, "text": "parallelogram the left side has a length"}, {"start": 99.04, "duration": 4.679, "text": "of 4.2 that means the right side is also"}, {"start": 102.04, "duration": 4.719, "text": "going to have a length of"}, {"start": 103.719, "duration": 5.0, "text": "4.2 the base we said has a length of 8"}, {"start": 106.759, "duration": 4.281, "text": "that means the top part is also going to"}, {"start": 108.719, "duration": 7.04, "text": "have a length of eight so we just have"}, {"start": 111.04, "duration": 6.079, "text": "to add up those values 4.2 + 8 + 4.2 + 8"}, {"start": 115.759, "duration": 7.32, "text": "and that's going to give us the"}, {"start": 117.119, "duration": 8.28, "text": "perimeter so we said 4.2 + 8 +"}, {"start": 123.079, "duration": 5.4, "text": "4.2 +"}, {"start": 125.399, "duration": 6.48, "text": "8 well let's see"}, {"start": 128.479, "duration": 5.641, "text": "4.2 and 4.2 if we add those together"}, {"start": 131.879, "duration": 5.881, "text": "that's going to give us"}, {"start": 134.12, "duration": 6.36, "text": "8.4 if we add the 8 and"}, {"start": 137.76, "duration": 7.08, "text": "8 that's going to give us"}, {"start": 140.48, "duration": 6.36, "text": "16 and let's see so 8 + 16 that would be"}, {"start": 144.84, "duration": 5.44, "text": "24 and then we still have to remember"}, {"start": 146.84, "duration": 5.28, "text": "the point4 so 8.4 + 16 that's going to"}, {"start": 150.28, "duration": 4.239, "text": "give us a value of"}, {"start": 152.12, "duration": 7.0, "text": "24.4 and that's going to be the"}, {"start": 154.519, "duration": 4.601, "text": "perimeter of our parallelogram"}], "title": "GED 'Most Missed' Math Practice Problems , Code Q.4.c,  #2", "description": "Thanks to all of you who support me on Patreon. You da real mvps!   $1 per month helps!! :)  https://www.patreon.com/patrickjmt !! The GED Testing Service keeps a list of the most missed math topics and can be seen here: http://www.gedtestingservice.com/uploads/files/126de0283c94ff6323b3b5b3f310da5d.pdf\nThis video shows how to work a sample problem using their coding system that is shown on the left using the link above.", "lengthSeconds": 159, "width": 480, "height": 360}
{"video_id": "3QpHH4xtc0E", "transcript": [{"start": 0.84, "duration": 3.919, "text": "all right in this video I want to do"}, {"start": 2.24, "duration": 4.84, "text": "another example of solving a word"}, {"start": 4.759, "duration": 4.041, "text": "problem involving some inequalities so"}, {"start": 7.08, "duration": 3.679, "text": "in this example suppose we've got our"}, {"start": 8.8, "duration": 4.24, "text": "buddy here Jason and suppose he makes a"}, {"start": 10.759, "duration": 6.001, "text": "living uh by making and selling fishing"}, {"start": 13.04, "duration": 8.159, "text": "poles suppose he he it costs him $12,000"}, {"start": 16.76, "duration": 6.56, "text": "a year um plus a cost of $4 per each"}, {"start": 21.199, "duration": 4.601, "text": "poll that he makes so maybe he pays you"}, {"start": 23.32, "duration": 4.76, "text": "know $1,000 rent a month in his little"}, {"start": 25.8, "duration": 3.68, "text": "Factory um and then he has costs"}, {"start": 28.08, "duration": 2.519, "text": "occurred to make each one of these"}, {"start": 29.48, "duration": 2.84, "text": "fishing po"}, {"start": 30.599, "duration": 3.601, "text": "we want to know how many fishing poles"}, {"start": 32.32, "duration": 4.399, "text": "he must make assuming he can sell them"}, {"start": 34.2, "duration": 5.24, "text": "all um if he wants to have a profit of"}, {"start": 36.719, "duration": 6.881, "text": "$48,000 in one year and if we know that"}, {"start": 39.44, "duration": 7.92, "text": "he sells the fishing poles for $28"}, {"start": 43.6, "duration": 6.799, "text": "each okay so to me to kind of really get"}, {"start": 47.36, "duration": 4.719, "text": "me started uh you know intuitively what"}, {"start": 50.399, "duration": 4.281, "text": "I'm thinking is kind of you know the"}, {"start": 52.079, "duration": 6.041, "text": "money that he brings"}, {"start": 54.68, "duration": 6.559, "text": "in minus his"}, {"start": 58.12, "duration": 6.64, "text": "costs right so whatever selling minus"}, {"start": 61.239, "duration": 5.601, "text": "his costs that's going to be his profit"}, {"start": 64.76, "duration": 6.24, "text": "um that's how much money he gets to keep"}, {"start": 66.84, "duration": 5.919, "text": "and we want that to be um so suppose we"}, {"start": 71.0, "duration": 5.64, "text": "want him to make at least"}, {"start": 72.759, "duration": 5.081, "text": "$488,000 here so we said uh to make a"}, {"start": 76.64, "duration": 3.839, "text": "profit of"}, {"start": 77.84, "duration": 5.88, "text": "$448,000 um let's just make let's let's"}, {"start": 80.479, "duration": 6.68, "text": "modify this a little bit and say uh to"}, {"start": 83.72, "duration": 3.439, "text": "make a at least a"}, {"start": 88.92, "duration": 4.44, "text": "profit cuz most people probably uh you"}, {"start": 91.439, "duration": 4.761, "text": "know once they hit some number as long"}, {"start": 93.36, "duration": 4.92, "text": "as they're over that they're happy so"}, {"start": 96.2, "duration": 5.4, "text": "okay so well how much money does he make"}, {"start": 98.28, "duration": 6.159, "text": "so let's let X represent the number of"}, {"start": 101.6, "duration": 5.28, "text": "fishing poles that he sells um if he"}, {"start": 104.439, "duration": 6.201, "text": "sells X of these fishing poles well he"}, {"start": 106.88, "duration": 5.239, "text": "gets $28 per each one okay so that's how"}, {"start": 110.64, "duration": 3.64, "text": "much money he would make if he sells"}, {"start": 112.119, "duration": 5.841, "text": "them but then we have to subtract away"}, {"start": 114.28, "duration": 8.36, "text": "his costs well his costs are"}, {"start": 117.96, "duration": 6.96, "text": "12,000 plus he got to Shell out $4 per"}, {"start": 122.64, "duration": 5.36, "text": "each one that he"}, {"start": 124.92, "duration": 5.319, "text": "makes greater than or equal to"}, {"start": 128.0, "duration": 5.28, "text": "48,000 and hey now we've got our"}, {"start": 130.239, "duration": 6.08, "text": "inequality set up that we need to solve"}, {"start": 133.28, "duration": 7.08, "text": "okay so this is 28x if we distribute the"}, {"start": 136.319, "duration": 6.881, "text": "negative we'll get -1 12,000 we'll get"}, {"start": 140.36, "duration": 3.92, "text": "-4x that has to be greater than or equal"}, {"start": 143.2, "duration": 3.2, "text": "to"}, {"start": 144.28, "duration": 4.4, "text": "48,000 and again now I've just got a"}, {"start": 146.4, "duration": 6.0, "text": "nice little in uh linear inequality so"}, {"start": 148.68, "duration": 8.279, "text": "28x - 4X is going to be"}, {"start": 152.4, "duration": 6.839, "text": "24x I can add the 12,000 to both sides"}, {"start": 156.959, "duration": 6.041, "text": "so that'll give me 24x is greater than"}, {"start": 159.239, "duration": 6.961, "text": "or equal to 48,000 + 12,000 is going to"}, {"start": 163.0, "duration": 5.319, "text": "be 60,000 and now the last thing we have"}, {"start": 166.2, "duration": 4.319, "text": "to do is just simply divide both sides"}, {"start": 168.319, "duration": 2.2, "text": "by"}, {"start": 171.319, "duration": 4.56, "text": "24 so it says the number of fishing"}, {"start": 173.879, "duration": 3.921, "text": "polls that he has to make if he wants to"}, {"start": 175.879, "duration": 5.801, "text": "make"}, {"start": 177.8, "duration": 7.4, "text": "$48,000 per year or more I'm just taking"}, {"start": 181.68, "duration": 5.6, "text": "60,000 dividing by 24 and I'm getting"}, {"start": 185.2, "duration": 5.08, "text": "the number"}, {"start": 187.28, "duration": 6.4, "text": "2500 so it says he's got to make and"}, {"start": 190.28, "duration": 6.76, "text": "sell 2500 fishing poles a year if he"}, {"start": 193.68, "duration": 6.36, "text": "wants to make um at least"}, {"start": 197.04, "duration": 3.0, "text": "$48,000"}], "title": "Solving Word Problems Involving Inequalities - Example 2", "description": "Thanks to all of you who support me on Patreon. You da real mvps!   $1 per month helps!! :)  https://www.patreon.com/patrickjmt !! Solving Word Problems Involving Inequalities - Example 2", "lengthSeconds": 198, "width": 480, "height": 360}
{"video_id": "4BRXaq9tYcQ", "transcript": [{"start": 0.919, "duration": 4.241, "text": "okay so just one more example here of uh"}, {"start": 3.04, "duration": 3.759, "text": "finding a derivative using implicit"}, {"start": 5.16, "duration": 3.24, "text": "differentiation so here we're going to"}, {"start": 6.799, "duration": 2.76, "text": "find the derivative uh we're going to"}, {"start": 8.4, "duration": 8.279, "text": "find"}, {"start": 9.559, "duration": 7.12, "text": "dydt of Y Cub + sin of y * t = t"}, {"start": 18.76, "duration": 4.88, "text": "5 so again the idea in this video is"}, {"start": 21.96, "duration": 3.76, "text": "we're taking the derivative with respect"}, {"start": 23.64, "duration": 4.6, "text": "to T so again we're going to kind of"}, {"start": 25.72, "duration": 3.959, "text": "treat the T kind of like the x that we"}, {"start": 28.24, "duration": 3.64, "text": "you know the way we did it in the first"}, {"start": 29.679, "duration": 4.161, "text": "video so think about T and X as being"}, {"start": 31.88, "duration": 4.56, "text": "your uh you know the independent"}, {"start": 33.84, "duration": 6.239, "text": "variable in this"}, {"start": 36.44, "duration": 6.08, "text": "case so all we have to do in this case"}, {"start": 40.079, "duration": 6.521, "text": "again so when we take the derivative of"}, {"start": 42.52, "duration": 8.48, "text": "y 3 well we'll get 3 y^2 but then we"}, {"start": 46.6, "duration": 7.72, "text": "have to tack on our dy over"}, {"start": 51.0, "duration": 5.039, "text": "DT plus okay we have to use the uh the"}, {"start": 54.32, "duration": 5.12, "text": "chain Rule and we take the derivative of"}, {"start": 56.039, "duration": 4.441, "text": "s of YT so the derivative of s is just"}, {"start": 59.44, "duration": 3.439, "text": "cosine"}, {"start": 60.48, "duration": 3.639, "text": "we'll leave the inside alone but then"}, {"start": 62.879, "duration": 5.001, "text": "when we take the derivative of the"}, {"start": 64.119, "duration": 3.761, "text": "inside we'll have to use the product"}, {"start": 68.64, "duration": 5.56, "text": "rule so let's see when we take the"}, {"start": 70.88, "duration": 6.32, "text": "derivative of one y we'll just get one"}, {"start": 74.2, "duration": 6.4, "text": "but we have to tack out tack on the dydt"}, {"start": 77.2, "duration": 5.68, "text": "term um and then we'll leave the T alone"}, {"start": 80.6, "duration": 4.839, "text": "and then plus okay so now uh we'll leave"}, {"start": 82.88, "duration": 5.4, "text": "the y alone when we take the derivative"}, {"start": 85.439, "duration": 4.881, "text": "of 1T we'll just get"}, {"start": 88.28, "duration": 4.28, "text": "one and then on the right right side"}, {"start": 90.32, "duration": 6.36, "text": "when we take the derivative of T 5th"}, {"start": 92.56, "duration": 8.4, "text": "we'll get 5 T 4th so now what we need to"}, {"start": 96.68, "duration": 6.28, "text": "do is just solve for our dydt term so um"}, {"start": 100.96, "duration": 3.759, "text": "all I'm going to do is just get rid of"}, {"start": 102.96, "duration": 4.08, "text": "the brackets"}, {"start": 104.719, "duration": 5.921, "text": "here and I'm going to have to distribute"}, {"start": 107.04, "duration": 6.32, "text": "this cosine of YT to you know both both"}, {"start": 110.64, "duration": 8.2, "text": "terms so let's see we'll get plus so I'm"}, {"start": 113.36, "duration": 10.2, "text": "going to write this as T time cine of"}, {"start": 118.84, "duration": 6.84, "text": "YT * dy over DT and then when we"}, {"start": 123.56, "duration": 6.04, "text": "distribute again we'll get plus I guess"}, {"start": 125.68, "duration": 3.92, "text": "we'll just get y * cosine of"}, {"start": 130.72, "duration": 5.32, "text": "YT on the right we still have our 5T to"}, {"start": 134.0, "duration": 3.92, "text": "the 4th just hanging out and now I'm"}, {"start": 136.04, "duration": 4.88, "text": "going to move everything that doesn't"}, {"start": 137.92, "duration": 5.56, "text": "have a dydt attached to it I'm going to"}, {"start": 140.92, "duration": 5.56, "text": "move that stuff over to the other side"}, {"start": 143.48, "duration": 8.36, "text": "so let's see here there's our 5T"}, {"start": 146.48, "duration": 5.36, "text": "4th um I could subtract the Y"}, {"start": 152.0, "duration": 6.16, "text": "cosine of YT from both sides so we've"}, {"start": 156.48, "duration": 4.28, "text": "basically moved that over to the other"}, {"start": 158.16, "duration": 5.079, "text": "side notice on the left all we would be"}, {"start": 160.76, "duration": 6.6, "text": "left with would be the 3 y^2"}, {"start": 163.239, "duration": 6.36, "text": "dydt plus T * cosine of YT *"}, {"start": 167.36, "duration": 6.0, "text": "dydt and what I'm going to do now is"}, {"start": 169.599, "duration": 7.36, "text": "just factor that dy over DT"}, {"start": 173.36, "duration": 6.2, "text": "out well let's see if we pull the dyt DT"}, {"start": 176.959, "duration": 5.441, "text": "out we would need a 3 y^2 and then if we"}, {"start": 179.56, "duration": 7.48, "text": "pull it out from the um second term we"}, {"start": 182.4, "duration": 6.96, "text": "would need the positive T * cine of"}, {"start": 187.04, "duration": 5.68, "text": "YT and now all we have to do is just"}, {"start": 189.36, "duration": 6.28, "text": "divide both sides by the 3 y^2 + T"}, {"start": 192.72, "duration": 7.72, "text": "cosine YT and we'll get our solution so"}, {"start": 195.64, "duration": 10.04, "text": "it says Dy DT is going to be 5 T"}, {"start": 200.44, "duration": 13.32, "text": "4 - y * cosine of"}, {"start": 205.68, "duration": 12.36, "text": "YT all over 3 y^ 2 + T * cine of"}, {"start": 213.76, "duration": 7.28, "text": "YT and we've now got our"}, {"start": 218.04, "duration": 3.0, "text": "derivative"}], "title": "Implicit Differentiation - Basic Example 3 / 3", "description": "Thanks to all of you who support me on Patreon. You da real mvps!   $1 per month helps!! :)  https://www.patreon.com/patrickjmt !! Implicit Differentiation - Basic Example 3 / 3 .  Just a fairly straight forward example of finding a derivative using implicit differentiation.", "lengthSeconds": 219, "width": 480, "height": 360}
{"video_id": "tl56MgKA3Z8", "transcript": [{"start": 0.73, "duration": 3.529, "text": "All right, in this video we're going to do some examples of going from,"}, {"start": 5.259, "duration": 5.87, "text": "basically, our polar form back to our complex form. So a couple examples here..."}, {"start": 11.13, "duration": 2.099, "text": "And I think these are certainly a little bit easier"}, {"start": 13.509, "duration": 6.08, "text": "to do than putting them into polar form because really all you have to do is just kind of do the arithmetic. So"}, {"start": 20.23, "duration": 3.469, "text": "part a) here. We've got 3 times cosine of 135 degrees."}, {"start": 24.88, "duration": 3.77, "text": "Well, 135 degrees is going to be in quadrant two."}, {"start": 30.67, "duration": 4.399, "text": "Cosine of 135 degrees is negative square root of 2 over 2."}, {"start": 36.969, "duration": 4.34, "text": "Sine of 135 degrees is positive square root of 2 over 2."}, {"start": 42.28, "duration": 3.38, "text": "So now all we have to do is just distribute out the 3. So we'll get negative"}, {"start": 46.48, "duration": 2.599, "text": "3 root 2 over 2"}, {"start": 51.16, "duration": 7.489, "text": "plus 3 root 2 over 2 ... I'm going to stick my i out to the right side... And now we've taken our"}, {"start": 59.32, "duration": 4.16, "text": "number that was in our polar number and put it in complex form. So"}, {"start": 63.64, "duration": 2.93, "text": "that's all there is to do. You really just have to evaluate whatever"}, {"start": 66.729, "duration": 4.34, "text": "cosine of the angle is, whatever the sine of the angle is, and just multiply it out."}, {"start": 72.04, "duration": 3.949, "text": "So we can do our other example here, I think, real quick as well."}, {"start": 78.04, "duration": 5.54, "text": "So, let's see, 270 degrees. If you think about 270 degrees, well,"}, {"start": 84.88, "duration": 4.279, "text": "270 degrees is just going to put us at the negative y-axis."}, {"start": 91.02, "duration": 6.05, "text": "And, well, cosine of 270 degrees ... the x-coordinate on the unit circle there would be zero."}, {"start": 100.14, "duration": 6.409, "text": "Sine of 270 degrees... that would be the value of the y-coordinate on the unit circle, which would just be negative one."}, {"start": 107.1, "duration": 7.099, "text": "Well, if we distribute 4 times zero, we'll get zero. A positive 4 times a negative one will be negative 4i."}, {"start": 114.869, "duration": 1.14, "text": "So,"}, {"start": 116.009, "duration": 4.55, "text": "typically, we'll just write it as negative 4i. We could write it as zero minus 4i"}, {"start": 120.56, "duration": 4.349, "text": "if you also want to see the real part, but that's all there is to it. Again,"}, {"start": 124.909, "duration": 2.94, "text": "just simply evaluate cosine of the angle, sine of the angle."}, {"start": 128.369, "duration": 3.98, "text": "Distributed out, in this case, there wasn't a lot to cancel or simplify,"}, {"start": 132.569, "duration": 4.37, "text": "But if there was that would be all you'd have to do, and you would get your solution."}], "title": "Complex Numbers: Convert From Polar to Complex Form, Ex 1", "description": "Thanks to all of you who support me on Patreon. You da real mvps!   $1 per month helps!! :)  https://www.patreon.com/patrickjmt !! Complex Numbers: Convert From Polar to Complex Form, Ex 1. In this video, we take a number in polar form and write it in complex form.", "lengthSeconds": 138, "width": 480, "height": 360}
{"video_id": "hOR8xXbdG4k", "transcript": [{"start": 0.29, "duration": 4.589, "text": "okay in this video I just want to do"}, {"start": 2.669, "duration": 5.13, "text": "another example of factoring by grouping"}, {"start": 4.879, "duration": 4.84, "text": "so again I usually start thinking about"}, {"start": 7.799, "duration": 5.431, "text": "factoring by grouping when I see four"}, {"start": 9.719, "duration": 6.361, "text": "terms which we have in this case the"}, {"start": 13.23, "duration": 5.549, "text": "first thing I always ask myself is is"}, {"start": 16.08, "duration": 7.279, "text": "what's the greatest common factor of all"}, {"start": 18.779, "duration": 7.441, "text": "the terms that are present in this case"}, {"start": 23.359, "duration": 6.281, "text": "notice there's no variables in common"}, {"start": 26.22, "duration": 6.149, "text": "between all four terms the first one has"}, {"start": 29.64, "duration": 4.56, "text": "a u the second one has a U and a V the"}, {"start": 32.369, "duration": 3.78, "text": "third one has U and V s and the fourth"}, {"start": 34.2, "duration": 4.74, "text": "one only has V so we can't factor out"}, {"start": 36.149, "duration": 5.491, "text": "any variables but I can think about the"}, {"start": 38.94, "duration": 4.799, "text": "greatest common factor of 12 15 24 and"}, {"start": 41.64, "duration": 3.75, "text": "30 and I think the biggest number that"}, {"start": 43.739, "duration": 4.23, "text": "would go into all of those would be"}, {"start": 45.39, "duration": 5.73, "text": "three so I'm just going to factor that"}, {"start": 47.969, "duration": 4.77, "text": "out so I guess we would need a four u"}, {"start": 51.12, "duration": 5.459, "text": "squared when we distribute to get our"}, {"start": 52.739, "duration": 8.84, "text": "first term we would need a 5 u v to get"}, {"start": 56.579, "duration": 8.22, "text": "back the second term we would need 8u v"}, {"start": 61.579, "duration": 5.97, "text": "squared to get our third term and then"}, {"start": 64.799, "duration": 6.991, "text": "it looks like we would need a 10 v cubed"}, {"start": 67.549, "duration": 5.831, "text": "to get back the fourth term and now I'm"}, {"start": 71.79, "duration": 3.84, "text": "just looking inside the brackets and I'm"}, {"start": 73.38, "duration": 3.93, "text": "going to do the same thing as before I'm"}, {"start": 75.63, "duration": 3.69, "text": "going to look at the first two terms try"}, {"start": 77.31, "duration": 4.5, "text": "to factor out the greatest common factor"}, {"start": 79.32, "duration": 4.799, "text": "I'll look at the third and fourth terms"}, {"start": 81.81, "duration": 5.03, "text": "try to factor out the greatest common"}, {"start": 84.119, "duration": 2.721, "text": "factor"}, {"start": 87.49, "duration": 5.19, "text": "so for the first two terms it looks like"}, {"start": 90.67, "duration": 3.39, "text": "there's nothing in term if you look at"}, {"start": 92.68, "duration": 3.299, "text": "the four and the five the greatest"}, {"start": 94.06, "duration": 3.809, "text": "common factor of that is just one so we"}, {"start": 95.979, "duration": 4.771, "text": "won't factor that out unless there's"}, {"start": 97.869, "duration": 5.191, "text": "nothing else to pull out"}, {"start": 100.75, "duration": 4.35, "text": "they both do have used and again we"}, {"start": 103.06, "duration": 4.41, "text": "factor out the smaller exponent so we"}, {"start": 105.1, "duration": 4.83, "text": "have au to the first so I'm simply going"}, {"start": 107.47, "duration": 4.11, "text": "to factor out a you and I think that's"}, {"start": 109.93, "duration": 3.81, "text": "all we can factor out of the first two"}, {"start": 111.58, "duration": 5.069, "text": "terms and then it looks like we would"}, {"start": 113.74, "duration": 6.03, "text": "need for you so when we distribute to"}, {"start": 116.649, "duration": 6.03, "text": "get the 4u squared and it looks like we"}, {"start": 119.77, "duration": 6.78, "text": "would need a 5v to get the second term"}, {"start": 122.679, "duration": 6.03, "text": "back when we distribute so let's do the"}, {"start": 126.55, "duration": 4.469, "text": "same thing with the third and fourth"}, {"start": 128.709, "duration": 3.841, "text": "terms inside the brackets so I think"}, {"start": 131.019, "duration": 3.181, "text": "what's the biggest number that goes into"}, {"start": 132.55, "duration": 3.39, "text": "eight and four the greatest common"}, {"start": 134.2, "duration": 5.46, "text": "factor I think that would be a positive"}, {"start": 135.94, "duration": 5.4, "text": "two notice they both have V's present so"}, {"start": 139.66, "duration": 3.9, "text": "we can factor out the smaller of the"}, {"start": 141.34, "duration": 4.92, "text": "exponents which would be to the second"}, {"start": 143.56, "duration": 5.61, "text": "power and then again I think well what"}, {"start": 146.26, "duration": 5.43, "text": "would I need back inside so 2 times 4"}, {"start": 149.17, "duration": 7.069, "text": "would give me the 8 and it looks like we"}, {"start": 151.69, "duration": 7.829, "text": "would need a you to get 8 UV squared and"}, {"start": 156.239, "duration": 5.951, "text": "then it looks like 2 times 5 would give"}, {"start": 159.519, "duration": 5.31, "text": "me back with the 10 and I think we would"}, {"start": 162.19, "duration": 8.57, "text": "need an extra 5 or excuse me an extra V"}, {"start": 164.829, "duration": 8.91, "text": "as well so at this case at this point"}, {"start": 170.76, "duration": 5.88, "text": "we've got this this case where we've got"}, {"start": 173.739, "duration": 5.911, "text": "the same thing in parentheses and"}, {"start": 176.64, "duration": 4.81, "text": "remember the trick on these is so 3"}, {"start": 179.65, "duration": 6.42, "text": "still hanging out front we just take the"}, {"start": 181.45, "duration": 6.629, "text": "stuff out front bu and the positive 2 v"}, {"start": 186.07, "duration": 4.41, "text": "squared we'll stick that in one set of"}, {"start": 188.079, "duration": 7.951, "text": "parentheses and then we put our common"}, {"start": 190.48, "duration": 8.3, "text": "factor the 4 u + 5 V in the other set of"}, {"start": 196.03, "duration": 5.74, "text": "parentheses and now we're done we have"}, {"start": 198.78, "duration": 7.88, "text": "completely factored"}, {"start": 201.77, "duration": 4.89, "text": "this original this original expression"}], "title": "Factoring By Grouping - Ex 2", "description": "Thanks to all of you who support me on Patreon. You da real mvps!   $1 per month helps!! :)  https://www.patreon.com/patrickjmt !! Factoring By Grouping - Ex 2.  Another example of factoring by grouping!", "lengthSeconds": 206, "width": 480, "height": 360}
{"video_id": "IE8q4WRubC4", "transcript": [{"start": 1.199, "duration": 3.481, "text": "all right in this video I want to look"}, {"start": 2.44, "duration": 4.28, "text": "at trying to prove some identities so"}, {"start": 4.68, "duration": 3.64, "text": "suppose we have uh these couple of"}, {"start": 6.72, "duration": 4.799, "text": "identities and we would try to try to"}, {"start": 8.32, "duration": 6.04, "text": "figure out if they're true or false so"}, {"start": 11.519, "duration": 6.121, "text": "um let's look at part A here first uh 1"}, {"start": 14.36, "duration": 7.8, "text": "minus sin T over cosine t plus cosine T"}, {"start": 17.64, "duration": 6.559, "text": "over 1 - sin t = 2ant t so I typically"}, {"start": 22.16, "duration": 4.8, "text": "you know in general start with the more"}, {"start": 24.199, "duration": 4.721, "text": "complicated side uh and try to work on"}, {"start": 26.96, "duration": 3.52, "text": "that side you know and to me when I look"}, {"start": 28.92, "duration": 4.36, "text": "at this sometimes getting started is the"}, {"start": 30.48, "duration": 5.68, "text": "hard part um you know I see two terms"}, {"start": 33.28, "duration": 4.92, "text": "here uh and I only see a single thing on"}, {"start": 36.16, "duration": 4.079, "text": "the right side so to me it says well you"}, {"start": 38.2, "duration": 3.8, "text": "know I'm going to have to have at least"}, {"start": 40.239, "duration": 3.521, "text": "only one thing on the left side so I'm"}, {"start": 42.0, "duration": 3.76, "text": "going to try to combine these which"}, {"start": 43.76, "duration": 5.72, "text": "tells me let's get common denominators"}, {"start": 45.76, "duration": 6.439, "text": "and uh so we can combine the"}, {"start": 49.48, "duration": 4.719, "text": "fractions okay so for our first fraction"}, {"start": 52.199, "duration": 4.961, "text": "it looks like we would have to multiply"}, {"start": 54.199, "duration": 7.721, "text": "the numerator and the denominator by"}, {"start": 57.16, "duration": 4.76, "text": "this uh one minus sin t"}, {"start": 65.56, "duration": 4.8, "text": "so plus and then our second"}, {"start": 73.4, "duration": 4.44, "text": "term it looks like we just have to"}, {"start": 75.439, "duration": 5.121, "text": "multiply the numerator and denominator"}, {"start": 77.84, "duration": 2.72, "text": "by cosine"}, {"start": 82.52, "duration": 6.4, "text": "T and again I'm kind of wondering does"}, {"start": 85.24, "duration": 6.76, "text": "this equal to secant T at the"}, {"start": 88.92, "duration": 6.239, "text": "end so okay"}, {"start": 92.0, "duration": 5.479, "text": "um so let's simplify the numerators so"}, {"start": 95.159, "duration": 5.041, "text": "we've got 1 * 1 which will be one we"}, {"start": 97.479, "duration": 7.68, "text": "have 1 * sin T which will be negative"}, {"start": 100.2, "duration": 6.919, "text": "sin T then we have negative sin T * 1"}, {"start": 105.159, "duration": 4.32, "text": "then we have a negative sin T and A"}, {"start": 107.119, "duration": 5.521, "text": "negative uh sin T which will give us"}, {"start": 109.479, "duration": 5.64, "text": "positive sin SAR"}, {"start": 112.64, "duration": 4.88, "text": "t uh looks like over here we'll get"}, {"start": 115.119, "duration": 5.881, "text": "cosine time cosine so we'll get"}, {"start": 117.52, "duration": 5.919, "text": "cosine squar of t"}, {"start": 121.0, "duration": 4.479, "text": "in the denominator we still have our uh"}, {"start": 123.439, "duration": 6.88, "text": "we would have our our single denominator"}, {"start": 125.479, "duration": 8.161, "text": "our common denominator of cosine T * 1"}, {"start": 130.319, "duration": 3.321, "text": "- sin"}, {"start": 135.44, "duration": 5.32, "text": "T all right so we can do a couple things"}, {"start": 138.36, "duration": 5.519, "text": "here uh notice we've got sin squar plus"}, {"start": 140.76, "duration": 4.64, "text": "cosine squar and sin squ uh plus cosine"}, {"start": 143.879, "duration": 4.0, "text": "squ remember there's an identity that"}, {"start": 145.4, "duration": 4.96, "text": "simply says that's equal to one so"}, {"start": 147.879, "duration": 4.681, "text": "really we have uh a negative sign T and"}, {"start": 150.36, "duration": 6.36, "text": "A negative sin T that's going to give us"}, {"start": 152.56, "duration": 5.88, "text": "-2 sin T we have a positive one and by"}, {"start": 156.72, "duration": 4.439, "text": "our identity we have another positive"}, {"start": 158.44, "duration": 9.2, "text": "one so we would have"}, {"start": 161.159, "duration": 6.481, "text": "positive2 over cine T * 1 - sin"}, {"start": 170.64, "duration": 4.519, "text": "T well looks like we could factor a two"}, {"start": 173.4, "duration": 7.8, "text": "out of the numerator then we would have"}, {"start": 175.159, "duration": 11.241, "text": "two uh we' have 1 minus sin t over cine"}, {"start": 181.2, "duration": 8.52, "text": "T * 1 - sin T and hey now we can cancel"}, {"start": 186.4, "duration": 7.119, "text": "out our factors the 1us sin T's we're"}, {"start": 189.72, "duration": 6.48, "text": "left with 2 over cosine t uh you can"}, {"start": 193.519, "duration": 8.36, "text": "think about this as being 2 * 1/"}, {"start": 196.2, "duration": 8.319, "text": "cosine of T and 1/ cosine is secant so"}, {"start": 201.879, "duration": 4.28, "text": "this is in fact an identity um it is"}, {"start": 204.519, "duration": 6.0, "text": "correct to say that that original"}, {"start": 206.159, "duration": 9.08, "text": "expression does equal 2 secant t so this"}, {"start": 210.519, "duration": 4.72, "text": "I would say is true"}], "title": "Proving an Identity, Example 1", "description": "Thanks to all of you who support me on Patreon. You da real mvps!   $1 per month helps!! :)  https://www.patreon.com/patrickjmt !! Proving an Identity, Example 1", "lengthSeconds": 215, "width": 480, "height": 360}
{"video_id": "xjtixDiBz3Q", "transcript": [{"start": 1.04, "duration": 2.8, "text": "okay here we're going to find an"}, {"start": 2.32, "duration": 3.2, "text": "expression for the perimeter of our"}, {"start": 3.84, "duration": 3.36, "text": "rectangle so"}, {"start": 5.52, "duration": 3.52, "text": "suppose we don't have exact numerical"}, {"start": 7.2, "duration": 4.0, "text": "values but we do have these algebraic"}, {"start": 9.04, "duration": 5.599, "text": "expressions so two of the sides are"}, {"start": 11.2, "duration": 5.2, "text": "going to be x plus 4y x plus 4y"}, {"start": 14.639, "duration": 5.521, "text": "and then to the the other two sides will"}, {"start": 16.4, "duration": 5.84, "text": "be 2x plus 3y and 2x plus 3y"}, {"start": 20.16, "duration": 4.959, "text": "so remember to get the perimeter we just"}, {"start": 22.24, "duration": 4.4, "text": "add up each side okay"}, {"start": 25.119, "duration": 3.721, "text": "so that's what we're going to have to do"}, {"start": 26.64, "duration": 5.28, "text": "in this case we'll simply have"}, {"start": 28.84, "duration": 4.12, "text": "to add up the lengths of these four"}, {"start": 31.92, "duration": 3.68, "text": "sides"}, {"start": 32.96, "duration": 4.56, "text": "and of course we'll try to simplify this"}, {"start": 35.6, "duration": 4.08, "text": "expression down"}, {"start": 37.52, "duration": 5.84, "text": "so the perimeter p"}, {"start": 39.68, "duration": 3.68, "text": "that'll be x plus 4y"}, {"start": 43.6, "duration": 4.4, "text": "plus 2x plus 3y"}, {"start": 48.48, "duration": 4.0, "text": "plus x plus 2y"}, {"start": 53.039, "duration": 4.801, "text": "plus"}, {"start": 54.48, "duration": 4.96, "text": "2x plus 3y"}, {"start": 57.84, "duration": 3.519, "text": "again you could have also said this a"}, {"start": 59.44, "duration": 4.799, "text": "little differently"}, {"start": 61.359, "duration": 5.76, "text": "we could say we have two times the side"}, {"start": 64.239, "duration": 5.681, "text": "x plus four y"}, {"start": 67.119, "duration": 6.161, "text": "likewise we have two of the sides two x"}, {"start": 69.92, "duration": 6.559, "text": "plus three y so either one of those"}, {"start": 73.28, "duration": 5.199, "text": "expressions are absolutely correct"}, {"start": 76.479, "duration": 4.801, "text": "probably the second one here is even a"}, {"start": 78.479, "duration": 4.481, "text": "little bit easier to uh well maybe not i"}, {"start": 81.28, "duration": 3.12, "text": "guess it's all it's all the same"}, {"start": 82.96, "duration": 3.44, "text": "difference really"}, {"start": 84.4, "duration": 3.84, "text": "but i'm going to go ahead and do do the"}, {"start": 86.4, "duration": 3.84, "text": "work on this expression"}, {"start": 88.24, "duration": 5.519, "text": "so we have 2 times x"}, {"start": 90.24, "duration": 5.519, "text": "2 times 4y which will be 8y"}, {"start": 93.759, "duration": 4.32, "text": "2 times 2x"}, {"start": 95.759, "duration": 5.761, "text": "which will be 4x"}, {"start": 98.079, "duration": 5.04, "text": "2 times 3y which will be 6y"}, {"start": 101.52, "duration": 3.919, "text": "and now it's just a matter of combining"}, {"start": 103.119, "duration": 4.32, "text": "like terms i see something involving x"}, {"start": 105.439, "duration": 2.96, "text": "to the first power and x to the first"}, {"start": 107.439, "duration": 2.64, "text": "power"}, {"start": 108.399, "duration": 3.841, "text": "so we can combine"}, {"start": 110.079, "duration": 5.121, "text": "those two terms"}, {"start": 112.24, "duration": 4.239, "text": "2 plus 4 will give us 6. again we keep"}, {"start": 115.2, "duration": 2.72, "text": "the x"}, {"start": 116.479, "duration": 5.041, "text": "and then again we have some more like"}, {"start": 117.92, "duration": 6.32, "text": "terms we have a positive 8y"}, {"start": 121.52, "duration": 6.48, "text": "and a positive 6y"}, {"start": 124.24, "duration": 5.999, "text": "well positive 8y plus 6y will give us a"}, {"start": 128.0, "duration": 5.04, "text": "positive 14y"}, {"start": 130.239, "duration": 5.601, "text": "and again at this point x and y are not"}, {"start": 133.04, "duration": 5.04, "text": "like terms so we can't combine those any"}, {"start": 135.84, "duration": 5.24, "text": "further and that would be our simplified"}, {"start": 138.08, "duration": 3.0, "text": "expression"}], "title": "Finding an Algebraic Expression for the Perimeter of a Given Rectangle", "description": "Thanks to all of you who support me on Patreon. You da real mvps!   $1 per month helps!! :)  https://www.patreon.com/patrickjmt !! Finding an Algebraic Expression for the Perimeter of a Given Rectangle.   Here we find a simplified algebraic expression for the perimeter of a rectangle", "lengthSeconds": 140, "width": 480, "height": 360}
{"video_id": "HXAsyRDrYTk", "transcript": [{"start": 1.079, "duration": 4.041, "text": "all right so one more example here"}, {"start": 2.56, "duration": 5.52, "text": "involving multiplication of decimals so"}, {"start": 5.12, "duration": 4.479, "text": "suppose in this town Lake View uh"}, {"start": 8.08, "duration": 4.76, "text": "there's"}, {"start": 9.599, "duration": 5.2, "text": "23.15 in of rain that uh falls on"}, {"start": 12.84, "duration": 5.76, "text": "average each year and we want to know"}, {"start": 14.799, "duration": 5.881, "text": "how much rainfall we could expect in 4.5"}, {"start": 18.6, "duration": 4.12, "text": "years well we could start you know"}, {"start": 20.68, "duration": 3.919, "text": "adding this up multiple times and then"}, {"start": 22.72, "duration": 3.719, "text": "trying to figure out what half of it is"}, {"start": 24.599, "duration": 6.201, "text": "or equivalently you know kind of the the"}, {"start": 26.439, "duration": 6.841, "text": "shortcut way uh we'll just take 23"}, {"start": 30.8, "duration": 4.8, "text": ".15 we're going to have to multiply that"}, {"start": 33.28, "duration": 2.32, "text": "by"}, {"start": 37.079, "duration": 5.081, "text": "4.5 all right so now just the"}, {"start": 39.2, "duration": 7.519, "text": "multiplication 5 * 4 is 20 so we'll keep"}, {"start": 42.16, "duration": 7.919, "text": "the zero carry the two 5 * 1 is 5 + 2 is"}, {"start": 46.719, "duration": 7.52, "text": "7 uh 5 * 3 is going to be 15 so we'll"}, {"start": 50.079, "duration": 5.601, "text": "keep the five carry the one 5 * 2 is 10+"}, {"start": 54.239, "duration": 5.881, "text": "1 will give us"}, {"start": 55.68, "duration": 8.12, "text": "11 drop down my Zero 4 * 4 is 16 so"}, {"start": 60.12, "duration": 10.6, "text": "we'll keep the six and we'll carry a 1 4"}, {"start": 63.8, "duration": 10.6, "text": "* 1 is 4 + 1 is 5 4 * 3 is 12 so we'll"}, {"start": 70.72, "duration": 9.079, "text": "keep the two and again carry a one 4 * 2"}, {"start": 74.4, "duration": 10.44, "text": "is 8 + 1 is 9 now we'll add these up so"}, {"start": 79.799, "duration": 8.881, "text": "0 + 0 is 0 7 + 6 is 13 so I'll keep the"}, {"start": 84.84, "duration": 6.44, "text": "three carry the one 1 + 5 + 5 is going"}, {"start": 88.68, "duration": 7.64, "text": "to be 11 so we keep the one carry the"}, {"start": 91.28, "duration": 6.4, "text": "one 1 + 1 + 2 is 4 and then 9 + 1 will"}, {"start": 96.32, "duration": 4.72, "text": "be"}, {"start": 97.68, "duration": 6.119, "text": "10 so again I count uh the number of"}, {"start": 101.04, "duration": 4.84, "text": "digits to the right of the decimal sign"}, {"start": 103.799, "duration": 3.881, "text": "we've got three digits total to the"}, {"start": 105.88, "duration": 6.279, "text": "right of the decimal signs so I'm going"}, {"start": 107.68, "duration": 7.64, "text": "to move the decimal place one two three"}, {"start": 112.159, "duration": 6.161, "text": "places over so it says we could expect"}, {"start": 115.32, "duration": 5.839, "text": "uh roughly you know says we could expect"}, {"start": 118.32, "duration": 5.839, "text": "104"}, {"start": 121.159, "duration": 3.0, "text": "13"}, {"start": 124.439, "duration": 7.201, "text": "in of rain to fall over the course of"}, {"start": 128.0, "duration": 5.44, "text": "4.5 years and again you know uh counting"}, {"start": 131.64, "duration": 4.319, "text": "decimals is good but also you know think"}, {"start": 133.44, "duration": 5.28, "text": "about it you know this is pretty close"}, {"start": 135.959, "duration": 4.36, "text": "to 20 right it's close to 20 inch you're"}, {"start": 138.72, "duration": 4.12, "text": "doing it for you know if you did it for"}, {"start": 140.319, "duration": 5.841, "text": "four years 20 in Time 4 years is you"}, {"start": 142.84, "duration": 6.44, "text": "know would be 80 in 20 in * 5 years"}, {"start": 146.16, "duration": 4.76, "text": "would be 100 in well this is a little"}, {"start": 149.28, "duration": 3.959, "text": "bit bigger than 20 so it should be a"}, {"start": 150.92, "duration": 5.16, "text": "little bit you know it should be around"}, {"start": 153.239, "duration": 4.841, "text": "you know 100 or so give or take um and"}, {"start": 156.08, "duration": 4.28, "text": "that's exactly what we're getting so not"}, {"start": 158.08, "duration": 3.879, "text": "only does counting you know clearly work"}, {"start": 160.36, "duration": 3.68, "text": "but you know just check the answer with"}, {"start": 161.959, "duration": 4.041, "text": "your intuition and make sure that uh it"}, {"start": 164.04, "duration": 4.96, "text": "seems like a reasonable a reasonable"}, {"start": 166.0, "duration": 3.0, "text": "solution"}], "title": "Multiplying Decimals, Ex 3", "description": "Thanks to all of you who support me on Patreon. You da real mvps!   $1 per month helps!! :)  https://www.patreon.com/patrickjmt !! Multiplying Decimals, Ex 3", "lengthSeconds": 167, "width": 480, "height": 360}
{"video_id": "d0uyVT3whao", "transcript": [{"start": 0.35, "duration": 7.66, "text": "okay here we're going to convert the"}, {"start": 3.33, "duration": 7.979, "text": "polar point negative 5 comma 315 degrees"}, {"start": 8.01, "duration": 7.65, "text": "to rectangular coordinates so we use the"}, {"start": 11.309, "duration": 9.771, "text": "formula x equals R times cosine of theta"}, {"start": 15.66, "duration": 8.34, "text": "and that y equals R times sine of theta"}, {"start": 21.08, "duration": 6.369, "text": "so again in this case our R value will"}, {"start": 24.0, "duration": 7.5, "text": "be negative 5 our theta about theta"}, {"start": 27.449, "duration": 7.681, "text": "value will be 315 degrees so that will"}, {"start": 31.5, "duration": 9.48, "text": "give us x equals negative 5 times cosine"}, {"start": 35.13, "duration": 13.92, "text": "of 315 degrees and Y will be negative 5"}, {"start": 40.98, "duration": 9.989, "text": "times sine of 315 degrees again I always"}, {"start": 49.05, "duration": 6.18, "text": "forget degrees I'm so used to using"}, {"start": 50.969, "duration": 6.84, "text": "radians so I'm gonna convert well let's"}, {"start": 55.23, "duration": 7.5, "text": "see if we're at the negative y-axis"}, {"start": 57.809, "duration": 7.981, "text": "that's 270 degrees which is 3 PI over 2"}, {"start": 62.73, "duration": 6.72, "text": "I guess we would need to add another 45"}, {"start": 65.79, "duration": 9.149, "text": "degrees which would be adding another PI"}, {"start": 69.45, "duration": 7.59, "text": "over 4 radians and if we get common"}, {"start": 74.939, "duration": 6.871, "text": "denominators we would have 6 PI over 4"}, {"start": 77.04, "duration": 10.079, "text": "plus PI over 4 so 7 PI over 4 will be"}, {"start": 81.81, "duration": 7.32, "text": "our our angle in radians so equivalently"}, {"start": 87.119, "duration": 7.771, "text": "we can write this as negative 5 times"}, {"start": 89.13, "duration": 8.84, "text": "cosine of 7 PI over 4 and we can write"}, {"start": 94.89, "duration": 8.85, "text": "the Y value as negative 5 times sine of"}, {"start": 97.97, "duration": 10.27, "text": "7 PI over 4 and to evaluate each of"}, {"start": 103.74, "duration": 8.61, "text": "these recall that at the angle 7 PI over"}, {"start": 108.24, "duration": 8.01, "text": "4 if you're on the unit circle at 7 PI"}, {"start": 112.35, "duration": 5.729, "text": "over 4 on the unit circle you're hitting"}, {"start": 116.25, "duration": 3.84, "text": "the ordered pair what would that be I"}, {"start": 118.079, "duration": 4.89, "text": "guess that would be square root of 2"}, {"start": 120.09, "duration": 6.54, "text": "over 2 comma negative square root of 2"}, {"start": 122.969, "duration": 5.201, "text": "over 2 so just reminding myself of the"}, {"start": 126.63, "duration": 4.97, "text": "values here"}, {"start": 128.17, "duration": 7.84, "text": "so let's see here we'll have you get"}, {"start": 131.6, "duration": 6.15, "text": "another piece of paper so for cosine or"}, {"start": 136.01, "duration": 6.87, "text": "for the x-coordinate we'll have negative"}, {"start": 137.75, "duration": 9.09, "text": "five x cosine of 7 PI over 4 7 PI over 4"}, {"start": 142.88, "duration": 5.49, "text": "will be the square root of 2 over 2 it's"}, {"start": 146.84, "duration": 3.24, "text": "really not a lot of simplification you"}, {"start": 148.37, "duration": 5.61, "text": "can do there so I'm just gonna leave it"}, {"start": 150.08, "duration": 7.95, "text": "negative 5 times root 2 over 2 and for"}, {"start": 153.98, "duration": 8.1, "text": "the Y value we'll have negative 5 times"}, {"start": 158.03, "duration": 8.19, "text": "sine of 7 PI over 4 but 7 PI over 4 is"}, {"start": 162.08, "duration": 5.28, "text": "negative square root of 2 over 2 well"}, {"start": 166.22, "duration": 4.14, "text": "there are two negatives we'll make it"}, {"start": 167.36, "duration": 6.99, "text": "positive so I'll have 5 times root 2"}, {"start": 170.36, "duration": 6.09, "text": "over 2 so in rectangular coordinates we"}, {"start": 174.35, "duration": 5.94, "text": "would use the point negative 5 times the"}, {"start": 176.45, "duration": 7.759, "text": "square root of 2 over 2 comma 5 times"}, {"start": 180.29, "duration": 3.919, "text": "the square root of 2 over 2"}], "title": "Polar Coordinates to Rectangular Coordinates ; A Quick Example #2", "description": "Thanks to all of you who support me on Patreon. You da real mvps!   $1 per month helps!! :)  https://www.patreon.com/patrickjmt !! Polar Coordinates to Rectangular Coordinates ; A Quick Example #2", "lengthSeconds": 185, "width": 450, "height": 360}
{"video_id": "O-Quebw2e-8", "transcript": [{"start": 1.079, "duration": 3.481, "text": "okay here we're going to look at a"}, {"start": 2.2, "duration": 3.559, "text": "couple examples related to the direct"}, {"start": 4.56, "duration": 3.92, "text": "comparison"}, {"start": 5.759, "duration": 5.521, "text": "test so in part A here we're going to"}, {"start": 8.48, "duration": 6.199, "text": "figure out whether the series 1 over n"}, {"start": 11.28, "duration": 5.6, "text": "cubed + n + 4 whether or not that series"}, {"start": 14.679, "duration": 5.041, "text": "converges or diverges again and we'll"}, {"start": 16.88, "duration": 8.12, "text": "compare it to uh a series that we know"}, {"start": 19.72, "duration": 7.12, "text": "converges or diverges so in this case uh"}, {"start": 25.0, "duration": 4.279, "text": "I usually kind of pick out dominant"}, {"start": 26.84, "duration": 4.439, "text": "terms and to me you know again just the"}, {"start": 29.279, "duration": 3.921, "text": "powers that are getting the the biggest"}, {"start": 31.279, "duration": 4.041, "text": "Powers well in the top there's just a"}, {"start": 33.2, "duration": 4.76, "text": "one in the denominator though there's n"}, {"start": 35.32, "duration": 4.64, "text": "cubed and if you think about the series"}, {"start": 37.96, "duration": 5.52, "text": "one over n cubed that would be a"}, {"start": 39.96, "duration": 5.72, "text": "convergent um P Series so let's see if"}, {"start": 43.48, "duration": 6.0, "text": "we think about it you know 1 over n"}, {"start": 45.68, "duration": 5.32, "text": "cubed well if we add n to it I claim"}, {"start": 49.48, "duration": 5.12, "text": "that that's going to be less than or"}, {"start": 51.0, "duration": 6.079, "text": "equal to 1/ n cub and again this is for"}, {"start": 54.6, "duration": 4.84, "text": "n values greater than or equal to one"}, {"start": 57.079, "duration": 4.96, "text": "this would be true and again this makes"}, {"start": 59.44, "duration": 4.439, "text": "sense because because you know um since"}, {"start": 62.039, "duration": 5.921, "text": "we're adding in the"}, {"start": 63.879, "duration": 6.361, "text": "denominator of the left side um must be"}, {"start": 67.96, "duration": 4.72, "text": "bigger than the denominator on the right"}, {"start": 70.24, "duration": 6.239, "text": "side well a bigger denominator makes for"}, {"start": 72.68, "duration": 7.28, "text": "an overall smaller fraction well okay I"}, {"start": 76.479, "duration": 7.68, "text": "think this is true likewise 1 over n Cub"}, {"start": 79.96, "duration": 8.0, "text": "+ n well plus 4 by the same reasoning"}, {"start": 84.159, "duration": 5.32, "text": "would be smaller than uh 1/ n cubed"}, {"start": 87.96, "duration": 6.199, "text": "notice these are certainly greater than"}, {"start": 89.479, "duration": 8.841, "text": "or equal equ Al to uh Z but we know"}, {"start": 94.159, "duration": 9.801, "text": "since you know the series 1 n Cub from"}, {"start": 98.32, "duration": 5.64, "text": "Nal 1 to Infinity that's a convergent P"}, {"start": 104.56, "duration": 6.08, "text": "series that's a convergent P series well"}, {"start": 108.32, "duration": 5.88, "text": "since that's convergent and we know now"}, {"start": 110.64, "duration": 10.64, "text": "that the series associated with one over"}, {"start": 114.2, "duration": 7.08, "text": "n Cub + n + 4 is smaller than that one"}, {"start": 122.0, "duration": 5.96, "text": "and again greater than or equal to zero"}, {"start": 124.56, "duration": 6.28, "text": "since the uh the uh the series on the"}, {"start": 127.96, "duration": 5.08, "text": "right converges and the values uh that"}, {"start": 130.84, "duration": 5.24, "text": "we get out you know the numerical values"}, {"start": 133.04, "duration": 5.04, "text": "we get out are going to be even smaller"}, {"start": 136.08, "duration": 4.48, "text": "uh that would imply by the direct"}, {"start": 138.08, "duration": 6.72, "text": "comparison theorem that our original"}, {"start": 140.56, "duration": 10.52, "text": "series Nal 1 to Infinity of 1 over n Cub"}, {"start": 144.8, "duration": 6.28, "text": "+ n + 4 that also converges"}], "title": "Direct Comparison Test - Another Example 1", "description": "Thanks to all of you who support me on Patreon. You da real mvps!   $1 per month helps!! :)  https://www.patreon.com/patrickjmt !! Direct Comparison Test - Another Example 1.  In this video I show that another series converges or diverges using the direct comparison theorem.  I do not run over the formula/theory in this video (but do in another video, so look around if that is what you need!).", "lengthSeconds": 154, "width": 480, "height": 360}
{"video_id": "xzEjYuUzsU4", "transcript": [{"start": 0.05, "duration": 4.33, "text": "okay so here we're gonna look at some"}, {"start": 2.129, "duration": 4.17, "text": "more examples involving integration of"}, {"start": 4.38, "duration": 5.49, "text": "some trig functions where we first have"}, {"start": 6.299, "duration": 6.181, "text": "to do some substitution so in Part A"}, {"start": 9.87, "duration": 7.68, "text": "here we have secant of cosine times"}, {"start": 12.48, "duration": 6.57, "text": "tangent of cosine times sine X so the"}, {"start": 17.55, "duration": 5.58, "text": "first thing you know again kind of the"}, {"start": 19.05, "duration": 5.639, "text": "things that I see here the first thing"}, {"start": 23.13, "duration": 3.719, "text": "that kind of makes me nervous is I see"}, {"start": 24.689, "duration": 3.93, "text": "the secant of cosine tangent of cosine"}, {"start": 26.849, "duration": 5.1, "text": "and that's kind of a quote unquote"}, {"start": 28.619, "duration": 5.011, "text": "I guess strange function but I know if"}, {"start": 31.949, "duration": 3.661, "text": "it was just secant of something and"}, {"start": 33.63, "duration": 4.199, "text": "tangent of something just a single"}, {"start": 35.61, "duration": 5.4, "text": "variable I know the antiderivative of"}, {"start": 37.829, "duration": 4.8, "text": "secant x times tangent X for example so"}, {"start": 41.01, "duration": 3.869, "text": "that tells me hey maybe if I make this"}, {"start": 42.629, "duration": 5.851, "text": "into a single variable by doing a u"}, {"start": 44.879, "duration": 5.401, "text": "substitution as long as it's D U is in"}, {"start": 48.48, "duration": 4.739, "text": "the problem and well the derivative of"}, {"start": 50.28, "duration": 6.829, "text": "cosine is sine it looks like everything"}, {"start": 53.219, "duration": 9.061, "text": "would work so if we let u equal cosine X"}, {"start": 57.109, "duration": 6.88, "text": "D u would be negative sine X DX and to"}, {"start": 62.28, "duration": 4.56, "text": "get a positive sine X we can just"}, {"start": 63.989, "duration": 4.591, "text": "multiply both sides by negative one and"}, {"start": 66.84, "duration": 5.459, "text": "now we can just fill everything in it"}, {"start": 68.58, "duration": 8.49, "text": "says we would have secant of U times"}, {"start": 72.299, "duration": 6.18, "text": "tangent of U and then our sine X DX we"}, {"start": 77.07, "duration": 5.88, "text": "can replace all of that with our"}, {"start": 78.479, "duration": 6.71, "text": "negative one D U and well the"}, {"start": 82.95, "duration": 9.419, "text": "antiderivative of secant times tangent"}, {"start": 85.189, "duration": 8.771, "text": "is just secant of U plus C and now again"}, {"start": 92.369, "duration": 5.721, "text": "we can just plug our substitution back"}, {"start": 93.96, "duration": 9.35, "text": "in so we would get secant of cosine X"}, {"start": 98.09, "duration": 5.22, "text": "all of that plus C"}], "title": "Substitution Rule / U-sub Basic Example 3", "description": "Thanks to all of you who support me on Patreon. You da real mvps!   $1 per month helps!! :)  https://www.patreon.com/patrickjmt !! Substitution Rule / U-sub Basic Example 3", "lengthSeconds": 103, "width": 480, "height": 360}
{"video_id": "HASYgn1Q6Fc", "transcript": [{"start": 0.68, "duration": 3.4, "text": "all right here we're going to look at a"}, {"start": 1.8, "duration": 3.92, "text": "quick little example of finding distance"}, {"start": 4.08, "duration": 4.479, "text": "between three points where the points"}, {"start": 5.72, "duration": 7.2, "text": "are uh located in three dimensions so"}, {"start": 8.559, "duration": 6.721, "text": "we've got the point 1 35 and then4 07 so"}, {"start": 12.92, "duration": 5.84, "text": "if we label these as x sub 1 y sub 1 Z"}, {"start": 15.28, "duration": 7.239, "text": "sub 1 X sub2 Y sub 2 and Z sub 2 uh the"}, {"start": 18.76, "duration": 6.359, "text": "distance formula is going to be uh the"}, {"start": 22.519, "duration": 4.401, "text": "formula at the bottom again it's just uh"}, {"start": 25.119, "duration": 4.0, "text": "what we normally do I think a lot of"}, {"start": 26.92, "duration": 5.04, "text": "people uh you know hopefully you've seen"}, {"start": 29.119, "duration": 4.201, "text": "the distance form fora and algebra um"}, {"start": 31.96, "duration": 3.279, "text": "and there you typically just deal with"}, {"start": 33.32, "duration": 6.2, "text": "two Dimensions so you see just the"}, {"start": 35.239, "duration": 7.201, "text": "formula X2 - X1 2 + Y2 - y12 but now we"}, {"start": 39.52, "duration": 5.12, "text": "have to add on the Z components as"}, {"start": 42.44, "duration": 4.2, "text": "well all right so we can do this I think"}, {"start": 44.64, "duration": 4.239, "text": "without too much"}, {"start": 46.64, "duration": 7.919, "text": "trouble so we've got the square root"}, {"start": 48.879, "duration": 9.84, "text": "Let's see we would do -4 - 1^ 2ar Y 2 -"}, {"start": 54.559, "duration": 6.361, "text": "y1 would be 0 - 3^ 2 and then we would"}, {"start": 58.719, "duration": 5.881, "text": "have 7"}, {"start": 60.92, "duration": 7.16, "text": "-5 or 7 + 5^"}, {"start": 64.6, "duration": 7.44, "text": "2 so this is going to be -5^ 2 which is"}, {"start": 68.08, "duration": 5.56, "text": "25 uh -3 s is going to be posi 9 and"}, {"start": 72.04, "duration": 3.759, "text": "then we'll have 12 SAR which is going to"}, {"start": 73.64, "duration": 4.4, "text": "be"}, {"start": 75.799, "duration": 4.841, "text": "144 so let's see if we add these"}, {"start": 78.04, "duration": 4.64, "text": "together I guess we get 34 and"}, {"start": 80.64, "duration": 4.72, "text": "144 and let's see that's going to give"}, {"start": 82.68, "duration": 5.2, "text": "us the square root of"}, {"start": 85.36, "duration": 4.88, "text": "178 so I guess you could always see if"}, {"start": 87.88, "duration": 6.68, "text": "this uh simplifies I guess guess we can"}, {"start": 90.24, "duration": 8.519, "text": "write 78 as 2 * um well 2 * 90 would be"}, {"start": 94.56, "duration": 7.519, "text": "180 so 2 * 89 should give us uh"}, {"start": 98.759, "duration": 5.801, "text": "178 and 89 I do believe is a prime"}, {"start": 102.079, "duration": 4.761, "text": "number um three certainly doesn't go"}, {"start": 104.56, "duration": 3.599, "text": "into it what else might go into it um"}, {"start": 106.84, "duration": 3.639, "text": "seven definitely"}, {"start": 108.159, "duration": 4.56, "text": "doesn't um let's see is there anything"}, {"start": 110.479, "duration": 4.801, "text": "much bigger that could potentially work"}, {"start": 112.719, "duration": 4.201, "text": "um so 89 to me looks like it's Prime"}, {"start": 115.28, "duration": 3.199, "text": "it's not going to factor anymore we"}, {"start": 116.92, "duration": 3.559, "text": "can't really take the square root of"}, {"start": 118.479, "duration": 4.0, "text": "either of those two numbers so I would"}, {"start": 120.479, "duration": 6.121, "text": "simply say that the distance is the"}, {"start": 122.479, "duration": 4.121, "text": "square OT of 178"}], "title": "The Distance Formula in 3-Dimensions", "description": "Thanks to all of you who support me on Patreon. You da real mvps!   $1 per month helps!! :)  https://www.patreon.com/patrickjmt !! The Distance Formula in 3-Dimensions. In this video, I show the formula to find the distance between two points in 3-dimension space.  It is just a slight extension of the 2d case, so it will be hopefully easy to remember!", "lengthSeconds": 127, "width": 480, "height": 360}
{"video_id": "xMIIBfrrxHs", "transcript": [{"start": 0.38, "duration": 3.79, "text": "all right in this video I'm gonna do"}, {"start": 2.639, "duration": 4.081, "text": "another example of solving a quadratic"}, {"start": 4.17, "duration": 4.56, "text": "equation by completing the square and"}, {"start": 6.72, "duration": 3.99, "text": "again the first thing I like to do is"}, {"start": 8.73, "duration": 4.619, "text": "put all my X's on one side and all my"}, {"start": 10.71, "duration": 5.61, "text": "numbers on the other so I'm going to"}, {"start": 13.349, "duration": 6.451, "text": "start off by subtracting 12 X from both"}, {"start": 16.32, "duration": 7.65, "text": "sides so I'm going to stick the negative"}, {"start": 19.8, "duration": 6.27, "text": "12 X in the middle there's my -1 so if"}, {"start": 23.97, "duration": 4.319, "text": "we subtract 12x on the right side we"}, {"start": 26.07, "duration": 4.44, "text": "would be left with 0 and then I'm just"}, {"start": 28.289, "duration": 6.241, "text": "going to add 1 to both sides so we would"}, {"start": 30.51, "duration": 8.959, "text": "get 3x squared minus 12x equals positive"}, {"start": 34.53, "duration": 7.38, "text": "1 and again what I like to do on these"}, {"start": 39.469, "duration": 4.811, "text": "the first thing I like to do is always"}, {"start": 41.91, "duration": 6.27, "text": "make the coefficient on the x squared"}, {"start": 44.28, "duration": 6.119, "text": "into a 1 again you kind of need the"}, {"start": 48.18, "duration": 4.559, "text": "coefficient to be a 1 before we can take"}, {"start": 50.399, "duration": 4.8, "text": "1/2 of the coefficient in front of the X"}, {"start": 52.739, "duration": 5.011, "text": "so it's kind of a common mistake to try"}, {"start": 55.199, "duration": 5.04, "text": "to do that step you know to do 1/2 a"}, {"start": 57.75, "duration": 5.789, "text": "little too early so I'm simply going to"}, {"start": 60.239, "duration": 5.28, "text": "divide both sides by 3 if I divide the"}, {"start": 63.539, "duration": 6.091, "text": "left side by 3 I would have to divide"}, {"start": 65.519, "duration": 6.991, "text": "each term by 3 so there I am dividing"}, {"start": 69.63, "duration": 5.489, "text": "everything by 3 on the left side we"}, {"start": 72.51, "duration": 5.1, "text": "would be left with x squared we have"}, {"start": 75.119, "duration": 4.5, "text": "negative 12 over 3 which is simply going"}, {"start": 77.61, "duration": 5.13, "text": "to leave us with positive excuse me a"}, {"start": 79.619, "duration": 5.25, "text": "negative 4 we still have our X leftover"}, {"start": 82.74, "duration": 5.37, "text": "and on the right we have our positive"}, {"start": 84.869, "duration": 5.46, "text": "1/3 so again I'm going to do the step"}, {"start": 88.11, "duration": 4.439, "text": "now of taking one half of the"}, {"start": 90.329, "duration": 5.25, "text": "coefficient in front of the X which is"}, {"start": 92.549, "duration": 5.731, "text": "going to be negative 4 if I take 1/2 of"}, {"start": 95.579, "duration": 6.451, "text": "that I get negative 2 and again I take"}, {"start": 98.28, "duration": 7.25, "text": "this number and I square it so negative"}, {"start": 102.03, "duration": 5.33, "text": "2 squared is going to be positive 4"}, {"start": 105.53, "duration": 4.98, "text": "and this is the number that I add to"}, {"start": 107.36, "duration": 7.05, "text": "both sides so I have x squared minus 4x"}, {"start": 110.51, "duration": 7.83, "text": "I'm going to throw in a positive 4 on"}, {"start": 114.41, "duration": 6.569, "text": "the right side we had our 1/3 and again"}, {"start": 118.34, "duration": 4.8, "text": "I'm going to add a positive 4 to that"}, {"start": 120.979, "duration": 4.951, "text": "side as well so I'm going to add 4 to"}, {"start": 123.14, "duration": 6.03, "text": "both sides again the left side is now a"}, {"start": 125.93, "duration": 6.48, "text": "perfect square x squared minus 4x plus 4"}, {"start": 129.17, "duration": 4.62, "text": "is going to factor as X again whenever"}, {"start": 132.41, "duration": 5.04, "text": "you take one half of the coefficient"}, {"start": 133.79, "duration": 7.29, "text": "whatever that number is that's how the"}, {"start": 137.45, "duration": 5.759, "text": "left side will be written as a perfect"}, {"start": 141.08, "duration": 4.62, "text": "square so you can check if you multiply"}, {"start": 143.209, "duration": 5.391, "text": "X minus 2 times X minus 2"}, {"start": 145.7, "duration": 5.19, "text": "you'll get x squared minus 4x plus 4 on"}, {"start": 148.6, "duration": 3.91, "text": "the right side I'm going to get common"}, {"start": 150.89, "duration": 5.069, "text": "denominators so you can think about"}, {"start": 152.51, "duration": 7.979, "text": "fours being 4 over 1 we can multiply top"}, {"start": 155.959, "duration": 6.571, "text": "and bottom by 3 and get 12 over 3 so on"}, {"start": 160.489, "duration": 7.051, "text": "the left side we're now left with X"}, {"start": 162.53, "duration": 7.86, "text": "minus 2 quantity squared and on the"}, {"start": 167.54, "duration": 6.72, "text": "right side left 1 plus 12 which will be"}, {"start": 170.39, "duration": 7.11, "text": "13 thirds and again now the idea is we"}, {"start": 174.26, "duration": 4.8, "text": "take the square root of both sides we"}, {"start": 177.5, "duration": 4.35, "text": "have to stick our positive and negative"}, {"start": 179.06, "duration": 6.03, "text": "on the right side and we've got X minus"}, {"start": 181.85, "duration": 7.08, "text": "2 equals will have a positive square"}, {"start": 185.09, "duration": 6.78, "text": "root of 13 over 3 and I kind of think is"}, {"start": 188.93, "duration": 4.86, "text": "this fraction simplify or could I kind"}, {"start": 191.87, "duration": 4.53, "text": "of break up the square root make either"}, {"start": 193.79, "duration": 4.59, "text": "part simplify and it doesn't look like"}, {"start": 196.4, "duration": 4.53, "text": "to me if that it simplifies at all so"}, {"start": 198.38, "duration": 4.32, "text": "I'm just gonna leave it alone that'll be"}, {"start": 200.93, "duration": 4.529, "text": "one of the equations we'll have to solve"}, {"start": 202.7, "duration": 6.71, "text": "and the other one will be X minus 2"}, {"start": 205.459, "duration": 6.691, "text": "equals negative square root of 13 over 3"}, {"start": 209.41, "duration": 5.889, "text": "and now all we have to do is add two to"}, {"start": 212.15, "duration": 7.35, "text": "both sides so we'll get positive square"}, {"start": 215.299, "duration": 6.811, "text": "root of 13 over 3 plus 2 as one of our"}, {"start": 219.5, "duration": 5.239, "text": "solutions and then our other solution"}, {"start": 222.11, "duration": 8.189, "text": "will be negative square root of 13 over"}, {"start": 224.739, "duration": 8.59, "text": "3 plus 2 and again now we've got our two"}, {"start": 230.299, "duration": 5.781, "text": "solutions to the original quadratic"}, {"start": 233.329, "duration": 2.751, "text": "equation"}], "title": "Completing the Square to Solve Quadratic Equations: More Examples - 5", "description": "Thanks to all of you who support me on Patreon. You da real mvps!   $1 per month helps!! :)  https://www.patreon.com/patrickjmt !! Completing the Square to Solve Quadratic Equations: More Examples - 5", "lengthSeconds": 237, "width": 480, "height": 360}
{"video_id": "-K4EHzS8VQo", "transcript": [{"start": 0.68, "duration": 4.56, "text": "okay here we're going to do some"}, {"start": 1.76, "duration": 5.999, "text": "examples of uh writing decimals as mixed"}, {"start": 5.24, "duration": 4.64, "text": "numbers and we'll reduce those uh the"}, {"start": 7.759, "duration": 3.92, "text": "fractions that are left over and again"}, {"start": 9.88, "duration": 5.96, "text": "we do this the exact same way as simply"}, {"start": 11.679, "duration": 7.041, "text": "writing you know any decimal uh as a"}, {"start": 15.84, "duration": 6.0, "text": "fraction so I'm going to write 2.8 over"}, {"start": 18.72, "duration": 5.76, "text": "one again we we like to not see decimals"}, {"start": 21.84, "duration": 5.4, "text": "in our fractions uh that's how we tend"}, {"start": 24.48, "duration": 5.639, "text": "to write them so to get rid of the"}, {"start": 27.24, "duration": 7.12, "text": "decimal place in the numerator we'll"}, {"start": 30.119, "duration": 7.161, "text": "have to multiply by 10 over 10 well 2.8"}, {"start": 34.36, "duration": 4.56, "text": "* 10 again that just shifts the decimal"}, {"start": 37.28, "duration": 4.0, "text": "place that'll give us 28 in the"}, {"start": 38.92, "duration": 5.959, "text": "numerator and then in the denominator 1"}, {"start": 41.28, "duration": 7.24, "text": "* 10 is 10 we could start saying well 10"}, {"start": 44.879, "duration": 6.281, "text": "goes into 28 how many times etc etc um"}, {"start": 48.52, "duration": 6.4, "text": "or we can reduce first okay I think I'm"}, {"start": 51.16, "duration": 6.48, "text": "going to go ahead and reduce my"}, {"start": 54.92, "duration": 6.319, "text": "fraction is what I'm going to do again"}, {"start": 57.64, "duration": 6.32, "text": "these are both even and so so when I see"}, {"start": 61.239, "duration": 4.681, "text": "a a fraction where the numerator and the"}, {"start": 63.96, "duration": 4.76, "text": "denominator are both even I just"}, {"start": 65.92, "duration": 4.6, "text": "immediately divide by the top by two"}, {"start": 68.72, "duration": 4.439, "text": "which will give us 14 and the Bottom by"}, {"start": 70.52, "duration": 5.84, "text": "two which will leave us with five there"}, {"start": 73.159, "duration": 5.121, "text": "could have even been a factor uh you"}, {"start": 76.36, "duration": 4.56, "text": "know some factor in common larger than"}, {"start": 78.28, "duration": 4.08, "text": "two in this case it wasn't but again to"}, {"start": 80.92, "duration": 3.159, "text": "me if I see they're even I just start"}, {"start": 82.36, "duration": 3.719, "text": "chopping it down by two because it's"}, {"start": 84.079, "duration": 3.521, "text": "kind of mindless and I can do it quickly"}, {"start": 86.079, "duration": 4.961, "text": "and it just gives me smaller numbers to"}, {"start": 87.6, "duration": 5.76, "text": "deal with okay well we're left 14 over5"}, {"start": 91.04, "duration": 5.2, "text": "I don't think that's going to reduce uh"}, {"start": 93.36, "duration": 4.719, "text": "as far as our fraction goes anymore and"}, {"start": 96.24, "duration": 3.64, "text": "now we can just do you know we can do"}, {"start": 98.079, "duration": 3.601, "text": "our our long division and find the"}, {"start": 99.88, "duration": 4.4, "text": "remainder I think we can do this one"}, {"start": 101.68, "duration": 4.84, "text": "hopefully uh I think hopefully we can do"}, {"start": 104.28, "duration": 4.72, "text": "it in our head let's see so five goes"}, {"start": 106.52, "duration": 4.76, "text": "into 14 I guess two whole times without"}, {"start": 109.0, "duration": 5.84, "text": "going over right because 2 * 5 would"}, {"start": 111.28, "duration": 6.32, "text": "give us 10 we would need an extra four"}, {"start": 114.84, "duration": 5.84, "text": "to get up to 14 and then we just have"}, {"start": 117.6, "duration": 6.199, "text": "the five left over so 14 over five is"}, {"start": 120.68, "duration": 5.68, "text": "going to leave us with 2 and 4 fths as"}, {"start": 123.799, "duration": 4.6, "text": "our mixed number and again notice by"}, {"start": 126.36, "duration": 5.48, "text": "reducing at the beginning our fraction"}, {"start": 128.399, "duration": 6.441, "text": "will already be reduced here at the"}, {"start": 131.84, "duration": 3.0, "text": "end"}], "title": "Writing a Decimal as a Fraction, Example 3", "description": "Thanks to all of you who support me on Patreon. You da real mvps!   $1 per month helps!! :)  https://www.patreon.com/patrickjmt !! Writing a Decimal as a Fraction, Example 3", "lengthSeconds": 133, "width": 480, "height": 360}
{"video_id": "v_Nz6UUQ4HQ", "transcript": [{"start": 0.03, "duration": 4.05, "text": "okay in this video we're going to look"}, {"start": 1.92, "duration": 4.95, "text": "at an example related to the limit laws"}, {"start": 4.08, "duration": 4.559, "text": "so just a couple problems here and"}, {"start": 6.87, "duration": 3.78, "text": "suppose that we're given that the limit"}, {"start": 8.639, "duration": 4.591, "text": "as X approaches two of our function f of"}, {"start": 10.65, "duration": 4.14, "text": "X that equals four and the limit as X"}, {"start": 13.23, "duration": 4.35, "text": "approaches two G of X that equals"}, {"start": 14.79, "duration": 5.399, "text": "negative five we want to compute each of"}, {"start": 17.58, "duration": 4.44, "text": "these following limits so Part A we've"}, {"start": 20.189, "duration": 3.901, "text": "got the limit as X approaches two of"}, {"start": 22.02, "duration": 4.86, "text": "three times f of X minus two times G of"}, {"start": 24.09, "duration": 5.58, "text": "X over f of X Part B again we'll do the"}, {"start": 26.88, "duration": 6.96, "text": "limit as X approaches 2 of f of X times"}, {"start": 29.67, "duration": 7.11, "text": "G of X plus G of X so let's do Part A"}, {"start": 33.84, "duration": 4.59, "text": "here first for the limit laws maybe"}, {"start": 36.78, "duration": 3.81, "text": "we'll go through this one and bits and"}, {"start": 38.43, "duration": 3.809, "text": "pieces if you have a limit of a fraction"}, {"start": 40.59, "duration": 12.0, "text": "it basically says you can take the limit"}, {"start": 42.239, "duration": 14.07, "text": "of the numerator over the limit as X"}, {"start": 52.59, "duration": 6.93, "text": "approaches two of the denominator okay"}, {"start": 56.309, "duration": 4.77, "text": "well there's also some more rules that"}, {"start": 59.52, "duration": 3.39, "text": "say okay if you have addition and"}, {"start": 61.079, "duration": 3.48, "text": "subtraction it says you can look at the"}, {"start": 62.91, "duration": 4.8, "text": "limit individually so we'll have the"}, {"start": 64.559, "duration": 8.31, "text": "limit as X approaches two of three times"}, {"start": 67.71, "duration": 9.51, "text": "f of X minus the limit as X approaches 2"}, {"start": 72.869, "duration": 5.671, "text": "of two times G of X again in the"}, {"start": 77.22, "duration": 3.39, "text": "denominator I'm going to write it a"}, {"start": 78.54, "duration": 5.509, "text": "little closer we've got the limit as X"}, {"start": 80.61, "duration": 5.97, "text": "approaches two of our function f of X"}, {"start": 84.049, "duration": 4.241, "text": "another one of the limit law simply says"}, {"start": 86.58, "duration": 3.719, "text": "if you have constants being multiplied"}, {"start": 88.29, "duration": 4.71, "text": "by your function you can factor those"}, {"start": 90.299, "duration": 6.201, "text": "out so we have three times the limit as"}, {"start": 93.0, "duration": 5.52, "text": "X approaches two of the function f of X"}, {"start": 96.5, "duration": 5.229, "text": "minus I'm going to pull the two out"}, {"start": 98.52, "duration": 8.279, "text": "front two times the limit as X"}, {"start": 101.729, "duration": 6.18, "text": "approaches two of G of X well again all"}, {"start": 106.799, "duration": 4.32, "text": "of that is over"}, {"start": 107.909, "duration": 6.151, "text": "the limit as X approaches 2 of f of X"}, {"start": 111.119, "duration": 6.39, "text": "and now we're in a position to start"}, {"start": 114.06, "duration": 4.86, "text": "plugging in the given values so the"}, {"start": 117.509, "duration": 4.051, "text": "moral of the story is you can basically"}, {"start": 118.92, "duration": 4.65, "text": "just plug things in a lot of people"}, {"start": 121.56, "duration": 5.37, "text": "nobody I mean very few people write all"}, {"start": 123.57, "duration": 5.249, "text": "this stuff out when doing limits so"}, {"start": 126.93, "duration": 4.44, "text": "let's see we said the limit as X"}, {"start": 128.819, "duration": 3.471, "text": "approaches 2 of f of X that was given to"}, {"start": 131.37, "duration": 3.14, "text": "us as"}, {"start": 132.29, "duration": 5.009, "text": "for so I'm just gonna plug that in so"}, {"start": 134.51, "duration": 5.13, "text": "we'll have three times four minus two"}, {"start": 137.299, "duration": 5.431, "text": "times well when we do the limit as X"}, {"start": 139.64, "duration": 5.04, "text": "approaches to G of X we said the limit"}, {"start": 142.73, "duration": 4.95, "text": "as X approaches two of G of X is just"}, {"start": 144.68, "duration": 4.949, "text": "negative five so let me plug that in"}, {"start": 147.68, "duration": 4.139, "text": "there as well and again we just said the"}, {"start": 149.629, "duration": 5.131, "text": "limit as X approaches 2 of f of X that's"}, {"start": 151.819, "duration": 4.911, "text": "giving us the value 4 and now we're in"}, {"start": 154.76, "duration": 5.55, "text": "business we can just do our arithmetic"}, {"start": 156.73, "duration": 6.61, "text": "so 3 times 4 is 12 negative 2 and"}, {"start": 160.31, "duration": 7.62, "text": "negative 5 is positive 10 all over 4"}, {"start": 163.34, "duration": 7.11, "text": "let's see that's 22 over 4 which we"}, {"start": 167.93, "duration": 6.899, "text": "could reduce we could make that 11 over"}, {"start": 170.45, "duration": 6.659, "text": "2 or I guess that's what 5 and 1/2 so"}, {"start": 174.829, "duration": 6.38, "text": "that would be our limit the the answer"}, {"start": 177.109, "duration": 4.1, "text": "to the limit question in Part A"}], "title": "Limit Laws to Evaluate a Limit , Example 1", "description": "Thanks to all of you who support me on Patreon. You da real mvps!   $1 per month helps!! :)  https://www.patreon.com/patrickjmt !! Limit Laws to Evaluate a Limit , Example 1.  In this video, I go through all of the steps using the limit laws to evaluate a limit.", "lengthSeconds": 181, "width": 480, "height": 360}
{"video_id": "-uQbvcHEfWY", "transcript": [{"start": 0.799, "duration": 3.441, "text": "all right in this video I'm going to do"}, {"start": 2.399, "duration": 4.081, "text": "another example involving direct"}, {"start": 4.24, "duration": 5.96, "text": "proportion so again if we say y varies"}, {"start": 6.48, "duration": 7.279, "text": "directly as X we can write that as yal"}, {"start": 10.2, "duration": 5.559, "text": "KX so in this problem uh the way simple"}, {"start": 13.759, "duration": 4.36, "text": "interest works on a bank account is it"}, {"start": 15.759, "duration": 4.921, "text": "varies directly as the amount of"}, {"start": 18.119, "duration": 5.16, "text": "principal in the account so I'm going to"}, {"start": 20.68, "duration": 5.679, "text": "use I for the interest and maybe P for"}, {"start": 23.279, "duration": 5.0, "text": "the principle so it says the interest"}, {"start": 26.359, "duration": 4.281, "text": "varies"}, {"start": 28.279, "duration": 5.401, "text": "directly um and is the amount of"}, {"start": 30.64, "duration": 7.16, "text": "principal so we can write that statement"}, {"start": 33.68, "duration": 6.84, "text": "as i = k * P they tell us a little bit"}, {"start": 37.8, "duration": 5.68, "text": "of information here they tell us if"}, {"start": 40.52, "duration": 6.44, "text": "$1,400 earns 84 in interest we want to"}, {"start": 43.48, "duration": 5.559, "text": "know how much money a 5,000 uh if you"}, {"start": 46.96, "duration": 4.599, "text": "had $5,000 in the bank how much would"}, {"start": 49.039, "duration": 4.481, "text": "that earn an interest so okay the first"}, {"start": 51.559, "duration": 4.881, "text": "thing we're going to do is solve for K"}, {"start": 53.52, "duration": 5.4, "text": "so they tell me that um so the interest"}, {"start": 56.44, "duration": 5.599, "text": "is $84 I'm going to plug that on the"}, {"start": 58.92, "duration": 6.559, "text": "left K is what we're trying to solve for"}, {"start": 62.039, "duration": 7.001, "text": "and that was on an initial principle of"}, {"start": 65.479, "duration": 5.561, "text": "1,400 so to solve for K I'll divide both"}, {"start": 69.04, "duration": 4.759, "text": "sides by"}, {"start": 71.04, "duration": 4.52, "text": "1,400 and you know this is one I think"}, {"start": 73.799, "duration": 4.921, "text": "I'm just uh going to go ahead and use a"}, {"start": 75.56, "duration": 6.64, "text": "calculator on it so 84"}, {"start": 78.72, "duration": 7.2, "text": "/400 I'm getting that to be"}, {"start": 82.2, "duration": 7.44, "text": "0.06 so it really says you're earning 6%"}, {"start": 85.92, "duration": 6.04, "text": "um in in regards to simple interest but"}, {"start": 89.64, "duration": 6.6, "text": "okay um so really it says the equation"}, {"start": 91.96, "duration": 6.4, "text": "we should be using it says i = k which"}, {"start": 96.24, "duration": 2.12, "text": "is"}, {"start": 98.84, "duration": 6.36, "text": "06 times the principal so okay well to"}, {"start": 102.52, "duration": 5.52, "text": "answer the question is to figure out the"}, {"start": 105.2, "duration": 6.0, "text": "interest we'll just simply take 06 and"}, {"start": 108.04, "duration": 5.039, "text": "multiply that by the new principle which"}, {"start": 111.2, "duration": 6.16, "text": "is"}, {"start": 113.079, "duration": 7.801, "text": "5000 and 006 * 5000 uh should simply"}, {"start": 117.36, "duration": 6.28, "text": "just be 300 which it is"}, {"start": 120.88, "duration": 4.36, "text": "so it says if you're earning obviously"}, {"start": 123.64, "duration": 4.44, "text": "the same amount of Interest simple"}, {"start": 125.24, "duration": 5.84, "text": "interest it says whereas $1,400 would"}, {"start": 128.08, "duration": 5.6, "text": "make you $84 in interest this tells us"}, {"start": 131.08, "duration": 4.48, "text": "that uh if you put $5,000 in the bank"}, {"start": 133.68, "duration": 7.6, "text": "you're going to earn uh under this"}, {"start": 135.56, "duration": 5.72, "text": "scenario $300 in interest"}], "title": "Direct Variation / Direct Proportion - Example 2", "description": "Thanks to all of you who support me on Patreon. You da real mvps!   $1 per month helps!! :)  https://www.patreon.com/patrickjmt !! Direct Variation / Direct Proportion - Example 2", "lengthSeconds": 139, "width": 480, "height": 360}
{"video_id": "rM4VLXOc4DE", "transcript": [{"start": 0.03, "duration": 3.149, "text": "all right in this video we're going to"}, {"start": 1.709, "duration": 4.201, "text": "look at finding the derivative of a"}, {"start": 3.179, "duration": 4.14, "text": "vector function and to do this all we"}, {"start": 5.91, "duration": 5.1, "text": "have to do is simply take the derivative"}, {"start": 7.319, "duration": 5.671, "text": "of all of this stuff basically involving"}, {"start": 11.01, "duration": 4.049, "text": "T so we'll take the derivative of the"}, {"start": 12.99, "duration": 5.07, "text": "stuff in front of I the stuff in front"}, {"start": 15.059, "duration": 4.861, "text": "of J and the stuff in front of K so when"}, {"start": 18.06, "duration": 6.03, "text": "we take the derivative of cosine PI T"}, {"start": 19.92, "duration": 5.88, "text": "we'll get negative sine times pi T but"}, {"start": 24.09, "duration": 4.38, "text": "then we'll have to take the derivative"}, {"start": 25.8, "duration": 5.85, "text": "of the inside so we'll pick up a pie so"}, {"start": 28.47, "duration": 8.28, "text": "we'll have negative pi times sine of pi"}, {"start": 31.65, "duration": 6.99, "text": "times T all of that times I plus for the"}, {"start": 36.75, "duration": 4.379, "text": "second part we can use the quotient rule"}, {"start": 38.64, "duration": 4.2, "text": "so we'll get the denominator the"}, {"start": 41.129, "duration": 5.43, "text": "derivative of the numerator will just be"}, {"start": 42.84, "duration": 5.73, "text": "e to the t minus the numerator times the"}, {"start": 46.559, "duration": 4.77, "text": "derivative of the denominator which will"}, {"start": 48.57, "duration": 6.3, "text": "be 2t all over the denominator squared"}, {"start": 51.329, "duration": 5.881, "text": "which is T to the fourth times J and"}, {"start": 54.87, "duration": 3.029, "text": "then if we take the derivative of 4t"}, {"start": 57.21, "duration": 6.119, "text": "cubed"}, {"start": 57.899, "duration": 8.101, "text": "we'll get twelve T squared times K and"}, {"start": 63.329, "duration": 5.671, "text": "that's now our derivative so I think I'm"}, {"start": 66.0, "duration": 5.07, "text": "going to do maybe just a little bit of"}, {"start": 69.0, "duration": 8.1, "text": "simplification so we have negative pi"}, {"start": 71.07, "duration": 11.61, "text": "sine of PI T times I I guess we could"}, {"start": 77.1, "duration": 9.57, "text": "factor we could factor a T out of the"}, {"start": 82.68, "duration": 6.119, "text": "numerator which would cancel out with a"}, {"start": 86.67, "duration": 4.32, "text": "T in the denominator we could even"}, {"start": 88.799, "duration": 5.57, "text": "factor in e to the T out as well so we"}, {"start": 90.99, "duration": 6.15, "text": "could factor out e to the T and a T and"}, {"start": 94.369, "duration": 5.051, "text": "then in parentheses we would have a T"}, {"start": 97.14, "duration": 5.24, "text": "and then I guess we would need a minus"}, {"start": 99.42, "duration": 7.53, "text": "two that's all over t to the fourth"}, {"start": 102.38, "duration": 7.419, "text": "times J and then our 12 T squared I'm"}, {"start": 106.95, "duration": 4.16, "text": "just going to leave that alone so the"}, {"start": 109.799, "duration": 5.85, "text": "last thing I'm going to do is just"}, {"start": 111.11, "duration": 10.93, "text": "simplify this part so negative pi times"}, {"start": 115.649, "duration": 8.97, "text": "sine of PI T times I plus okay so we can"}, {"start": 122.04, "duration": 4.92, "text": "cancel out the T that will give us T to"}, {"start": 124.619, "duration": 5.521, "text": "the third in the bottom we still have e"}, {"start": 126.96, "duration": 5.16, "text": "to the T times t minus two all of that"}, {"start": 130.14, "duration": 8.709, "text": "is being multiplied by j"}, {"start": 132.12, "duration": 9.009, "text": "plus our 12 t squared times K and now we"}, {"start": 138.849, "duration": 4.26, "text": "have our derivative so again nothing"}, {"start": 141.129, "duration": 4.2, "text": "crazy all we're doing is just simply"}, {"start": 143.109, "duration": 4.981, "text": "taking the derivative of each little"}, {"start": 145.329, "duration": 7.011, "text": "part so kind of three derivative"}, {"start": 148.09, "duration": 4.25, "text": "problems combined here into one"}], "title": "Derivative of a Vector Function - Another Ex 1", "description": "Thanks to all of you who support me on Patreon. You da real mvps!   $1 per month helps!! :)  https://www.patreon.com/patrickjmt !! Derivative of a Vector Function - Another Ex 1. In this video, I show an example of finding the derivative of a vector function.  It is pretty simple: just take the derivative of each part!", "lengthSeconds": 154, "width": 480, "height": 360}
{"video_id": "v4wXzl4Uks8", "transcript": [{"start": 0.03, "duration": 5.61, "text": "in this example we're gonna take five"}, {"start": 3.3, "duration": 5.219, "text": "numbers that are written in percent form"}, {"start": 5.64, "duration": 7.499, "text": "and we're gonna turn those into decimals"}, {"start": 8.519, "duration": 7.29, "text": "and this is easiest pie if you remember"}, {"start": 13.139, "duration": 6.751, "text": "how to do it and all we have to do when"}, {"start": 15.809, "duration": 6.931, "text": "we rewrite a number as we rewrite a"}, {"start": 19.89, "duration": 5.91, "text": "percentage as a decimal all you do is"}, {"start": 22.74, "duration": 4.68, "text": "just replace the percent sign with a"}, {"start": 25.8, "duration": 3.389, "text": "think about putting your decimal there"}, {"start": 27.42, "duration": 4.38, "text": "but oh actually it's not in the correct"}, {"start": 29.189, "duration": 7.981, "text": "spot we always move it two places to the"}, {"start": 31.8, "duration": 9.12, "text": "left so 26 percent I'm gonna put as the"}, {"start": 37.17, "duration": 5.94, "text": "decimal places point two six and I'm"}, {"start": 40.92, "duration": 6.15, "text": "even just gonna put a zero in front so"}, {"start": 43.11, "duration": 7.26, "text": "26 percent is zero point two six and"}, {"start": 47.07, "duration": 7.07, "text": "again the reason why we're doing that we"}, {"start": 50.37, "duration": 7.29, "text": "saw that we could write a percentage as"}, {"start": 54.14, "duration": 5.95, "text": "a fraction by just dividing it by 100"}, {"start": 57.66, "duration": 4.59, "text": "and if you divide something by 100 you"}, {"start": 60.09, "duration": 5.13, "text": "in effect move the decimal place two"}, {"start": 62.25, "duration": 5.24, "text": "places but as a shortcut that's all"}, {"start": 65.22, "duration": 5.04, "text": "we're doing so okay"}, {"start": 67.49, "duration": 6.58, "text": "125 percent let's get rid of the percent"}, {"start": 70.26, "duration": 9.84, "text": "move the decimal place two spots so it"}, {"start": 74.07, "duration": 9.96, "text": "says we would get 1.25 so 125 percent is"}, {"start": 80.1, "duration": 5.97, "text": "equivalent to one point two five two"}, {"start": 84.03, "duration": 4.14, "text": "percent well it feels like there's only"}, {"start": 86.07, "duration": 5.61, "text": "one place to move it let's put some"}, {"start": 88.17, "duration": 5.879, "text": "zeros in front so if we get rid of the"}, {"start": 91.68, "duration": 5.13, "text": "percentage move it two places to the"}, {"start": 94.049, "duration": 11.161, "text": "left put our decimal sign right there"}, {"start": 96.81, "duration": 12.839, "text": "we would have 0.02 fourteen thirty five"}, {"start": 105.21, "duration": 6.979, "text": "percent 1435 same thing get rid of the"}, {"start": 109.649, "duration": 7.951, "text": "percentage and move it two places"}, {"start": 112.189, "duration": 9.671, "text": "that'll be 14 point three five and last"}, {"start": 117.6, "duration": 5.46, "text": "but not least if we have 0.34% same"}, {"start": 121.86, "duration": 4.049, "text": "thing we still have to move with two"}, {"start": 123.06, "duration": 5.339, "text": "places to the left so let's see if we"}, {"start": 125.909, "duration": 4.77, "text": "took the decimal place and moved it two"}, {"start": 128.399, "duration": 3.661, "text": "places to the left we would now have"}, {"start": 130.679, "duration": 3.64, "text": "zero point"}, {"start": 132.06, "duration": 4.849, "text": "let's see still two zero"}, {"start": 134.319, "duration": 6.03, "text": "three four that would be the equivalent"}, {"start": 136.909, "duration": 3.44, "text": "decimal form"}], "title": "Writing a Percentage as a Decimal ( 5 examples)", "description": "Thanks to all of you who support me on Patreon. You da real mvps!   $1 per month helps!! :)  https://www.patreon.com/patrickjmt !! Writing a Percentage as a Decimal (5 examples)", "lengthSeconds": 140, "width": 480, "height": 360}
{"video_id": "9Z1uNz__2xA", "transcript": [{"start": 0.32, "duration": 3.599, "text": "all right in this video we're going to"}, {"start": 1.439, "duration": 3.84, "text": "start talking about roots of unity which"}, {"start": 3.919, "duration": 4.361, "text": "just basically means we're taking"}, {"start": 5.279, "duration": 5.24, "text": "complex roots of the number one so we've"}, {"start": 8.28, "duration": 3.56, "text": "seen how to take uh complex Roots the"}, {"start": 10.519, "duration": 4.08, "text": "first thing we can do is start off by"}, {"start": 11.84, "duration": 4.76, "text": "writing our complex number in polar form"}, {"start": 14.599, "duration": 5.92, "text": "and we can write one as just one *"}, {"start": 16.6, "duration": 8.2, "text": "cosine of 0\u00b0 plus I * s of"}, {"start": 20.519, "duration": 7.241, "text": "0\u00b0 uh we can then use our our formula"}, {"start": 24.8, "duration": 5.639, "text": "that says we would get cosine of really"}, {"start": 27.76, "duration": 7.119, "text": "uh the original which would be 0 deg +"}, {"start": 30.439, "duration": 8.721, "text": "2i K N the same thing we'll get S of 0 +"}, {"start": 34.879, "duration": 6.761, "text": "2 piun * k n so we can use this formula"}, {"start": 39.16, "duration": 5.16, "text": "to calculate the roots of unity and"}, {"start": 41.64, "duration": 5.559, "text": "another thing geometrically uh roots of"}, {"start": 44.32, "duration": 5.32, "text": "unity are spread evenly around the unit"}, {"start": 47.199, "duration": 6.401, "text": "circle so that'll kind of make it easy"}, {"start": 49.64, "duration": 6.399, "text": "geometrically to find them so uh suppose"}, {"start": 53.6, "duration": 4.52, "text": "we've got this example here so uh so"}, {"start": 56.039, "duration": 3.961, "text": "again we know that the the roots of"}, {"start": 58.12, "duration": 5.119, "text": "unity are spread evenly around the"}, {"start": 60.0, "duration": 5.32, "text": "Circle we want to calculate the value of"}, {"start": 63.239, "duration": 4.92, "text": "the root to which the arrow is pointing"}, {"start": 65.32, "duration": 4.52, "text": "at okay so if you think about going back"}, {"start": 68.159, "duration": 4.841, "text": "to this"}, {"start": 69.84, "duration": 6.319, "text": "formula um if you plug in notice if you"}, {"start": 73.0, "duration": 6.84, "text": "plugged in k equals 0 you would get 0"}, {"start": 76.159, "duration": 7.96, "text": "degrees um so you'd have cosine of zero"}, {"start": 79.84, "duration": 7.4, "text": "the uh the point um at at on the unit"}, {"start": 84.119, "duration": 6.281, "text": "circle uh on the positive x-axis that's"}, {"start": 87.24, "duration": 4.96, "text": "always going to correspond to K = 0 zero"}, {"start": 90.4, "duration": 4.32, "text": "and there'll always be a point right"}, {"start": 92.2, "duration": 4.64, "text": "there and then uh everything's spread"}, {"start": 94.72, "duration": 5.96, "text": "evenly uh hopefully it looks spread"}, {"start": 96.84, "duration": 9.0, "text": "evenly this would correspond to K = 1 k"}, {"start": 100.68, "duration": 7.24, "text": "= 2 ah okay well this is K = 3 k ="}, {"start": 105.84, "duration": 6.48, "text": "4"}, {"start": 107.92, "duration": 6.879, "text": "five six and seven notice if you count"}, {"start": 112.32, "duration": 5.52, "text": "the number of roots there's one two 3"}, {"start": 114.799, "duration": 5.881, "text": "four five six seven eight so what that"}, {"start": 117.84, "duration": 5.04, "text": "really means is uh we need to use this"}, {"start": 120.68, "duration": 6.0, "text": "formula we just need to calculate the"}, {"start": 122.88, "duration": 6.999, "text": "value of this formula where Nal"}, {"start": 126.68, "duration": 5.44, "text": "8 and again that's the number of roots"}, {"start": 129.879, "duration": 6.521, "text": "if you count and then we said where k"}, {"start": 132.12, "duration": 5.759, "text": "equals 3 so K = 3 so that's all we're"}, {"start": 136.4, "duration": 4.8, "text": "going to do now we'll just plug it into"}, {"start": 137.879, "duration": 10.281, "text": "this formula it says we would get 1 *"}, {"start": 141.2, "duration": 11.84, "text": "cine of 2 pi * K which is 3 over n which"}, {"start": 148.16, "duration": 10.719, "text": "is 8 plus I sin of the same thing we'll"}, {"start": 153.04, "duration": 5.839, "text": "get 2 piun * 3 over"}, {"start": 159.72, "duration": 4.599, "text": "8 and if we clean this up a little bit"}, {"start": 162.239, "duration": 5.401, "text": "that's just going to be cosine let's see"}, {"start": 164.319, "duration": 5.961, "text": "2 over 8 would be uh pi over 4 so we"}, {"start": 167.64, "duration": 4.28, "text": "would really have 3 pi over"}, {"start": 170.28, "duration": 6.4, "text": "4"}, {"start": 171.92, "duration": 8.16, "text": "plus I of 3 pi over"}, {"start": 176.68, "duration": 5.88, "text": "4 well cosine of 3 pi over 4 is is going"}, {"start": 180.08, "duration": 6.159, "text": "to be < tk2 over"}, {"start": 182.56, "duration": 7.28, "text": "2 s of uh 3 piun over 4 is going to be"}, {"start": 186.239, "duration": 6.72, "text": "positive < tk2 over 2 so now we've got"}, {"start": 189.84, "duration": 8.0, "text": "our answer it's going to be < tk2 over"}, {"start": 192.959, "duration": 7.881, "text": "2+ < tk2 over 2 *"}, {"start": 197.84, "duration": 3.0, "text": "I"}], "title": "Roots of Unity, Example 1", "description": "Thanks to all of you who support me on Patreon. You da real mvps!   $1 per month helps!! :)  https://www.patreon.com/patrickjmt !! Roots of Unity, Example 1.  Here is give a little discussion about finding roots of unity and the unit circle.", "lengthSeconds": 199, "width": 480, "height": 360}
{"video_id": "azMnzLcdPPo", "transcript": [{"start": 0.44, "duration": 3.04, "text": "all right so here we're going to do an"}, {"start": 1.56, "duration": 3.88, "text": "example of finding a second derivative"}, {"start": 3.48, "duration": 3.04, "text": "so we'll use the derivative formula at"}, {"start": 5.44, "duration": 3.0, "text": "the"}, {"start": 6.52, "duration": 5.079, "text": "bottom um so here we're going to find"}, {"start": 8.44, "duration": 7.2, "text": "the second derivative uh for x = t minus"}, {"start": 11.599, "duration": 6.6, "text": "t cubed and then y = 2T +"}, {"start": 15.64, "duration": 5.2, "text": "5 so the first thing we'll have to do is"}, {"start": 18.199, "duration": 6.041, "text": "well find the first derivative so the"}, {"start": 20.84, "duration": 9.759, "text": "first derivative dy over DX that's going"}, {"start": 24.24, "duration": 8.479, "text": "to be dy over DT all over DX over DT so"}, {"start": 30.599, "duration": 5.041, "text": "the derivative of y with respect to T is"}, {"start": 32.719, "duration": 5.16, "text": "just going to be two the derivative of x"}, {"start": 35.64, "duration": 5.36, "text": "with respect to t uh so the derivative"}, {"start": 37.879, "duration": 6.961, "text": "of 1 T would be 1 minus uh the"}, {"start": 41.0, "duration": 8.16, "text": "derivative of T 3 will be 3"}, {"start": 44.84, "duration": 4.32, "text": "t^2 all right so now to get the second"}, {"start": 50.92, "duration": 4.799, "text": "derivative it says we have to take the"}, {"start": 53.0, "duration": 5.079, "text": "derivative with respect to T of our"}, {"start": 55.719, "duration": 2.36, "text": "first"}, {"start": 58.92, "duration": 6.0, "text": "derivative and then we divide that by"}, {"start": 61.519, "duration": 8.64, "text": "the original DX over DT we said DX over"}, {"start": 64.92, "duration": 6.48, "text": "DT uh was just 1 - 3T squared so notice"}, {"start": 70.159, "duration": 2.921, "text": "nothing really happens in the"}, {"start": 71.4, "duration": 3.6, "text": "denominator when you take the second"}, {"start": 73.08, "duration": 4.96, "text": "derivative you really just kind of uh"}, {"start": 75.0, "duration": 5.28, "text": "you know uh taking the derivative of the"}, {"start": 78.04, "duration": 5.24, "text": "first derivative in the numerator and"}, {"start": 80.28, "duration": 5.72, "text": "then again just leaving the dxdt part"}, {"start": 83.28, "duration": 4.72, "text": "alone all right so um if we take the"}, {"start": 86.0, "duration": 3.6, "text": "derivative of uh so I'm just going to"}, {"start": 88.0, "duration": 2.759, "text": "kind of do the the numerator off to the"}, {"start": 89.6, "duration": 6.44, "text": "side"}, {"start": 90.759, "duration": 7.521, "text": "so we could write this as 2 * 1 - 3t^"}, {"start": 96.04, "duration": 4.16, "text": "2st again I'm just focusing on the"}, {"start": 98.28, "duration": 4.4, "text": "numerator for the moment and just"}, {"start": 100.2, "duration": 4.599, "text": "bringing the denominator"}, {"start": 102.68, "duration": 6.439, "text": "upstairs well we would use the chain"}, {"start": 104.799, "duration": 8.721, "text": "rule in this case so uh we would get -2"}, {"start": 109.119, "duration": 6.841, "text": "leave the inside Alone 1 - 3t^ 2 uh"}, {"start": 113.52, "duration": 3.959, "text": "subtract one from the exponent and then"}, {"start": 115.96, "duration": 5.799, "text": "we would have to take the derivative of"}, {"start": 117.479, "duration": 8.32, "text": "the inside which is going to give us-6 6"}, {"start": 121.759, "duration": 7.321, "text": "T all right so when we simplify -2 and -"}, {"start": 125.799, "duration": 5.561, "text": "6t will give us POS 12T and then we can"}, {"start": 129.08, "duration": 6.32, "text": "put the 1 -"}, {"start": 131.36, "duration": 4.04, "text": "3t^2 squared back in the"}, {"start": 135.68, "duration": 5.16, "text": "denominator so this is just our"}, {"start": 137.72, "duration": 9.519, "text": "numerator after taking the derivative so"}, {"start": 140.84, "duration": 11.72, "text": "12T over 1 - 3t^ 2^ squared well all of"}, {"start": 147.239, "duration": 6.881, "text": "that is being divided by 1 - 3t^ 2ar so"}, {"start": 152.56, "duration": 3.64, "text": "again if you think about uh the"}, {"start": 154.12, "duration": 4.08, "text": "denominators being over one if you flip"}, {"start": 156.2, "duration": 4.52, "text": "and"}, {"start": 158.2, "duration": 4.16, "text": "multiply when we flip and multiply we'll"}, {"start": 160.72, "duration": 10.36, "text": "just be left with"}, {"start": 162.36, "duration": 10.92, "text": "12T over 1 - 3t^ 2^ 2 * 1 over 1 - 3t^ 2"}, {"start": 171.08, "duration": 8.519, "text": "so I would probably write The Final"}, {"start": 173.28, "duration": 7.0, "text": "Answer here is 12T over 1 - 3 t^ 2 but"}, {"start": 179.599, "duration": 2.72, "text": "uh"}, {"start": 180.28, "duration": 4.16, "text": "so we've got a squared one and a one to"}, {"start": 182.319, "duration": 5.041, "text": "the first power so we can write all of"}, {"start": 184.44, "duration": 4.6, "text": "that to the third power and now we have"}, {"start": 187.36, "duration": 4.68, "text": "our second"}, {"start": 189.04, "duration": 3.0, "text": "derivative"}], "title": "Derivatives of Parametric Equations, Another Example #2 - Second Derivative", "description": "Thanks to all of you who support me on Patreon. You da real mvps!   $1 per month helps!! :)  https://www.patreon.com/patrickjmt !! Derivatives of Parametric Equations, Another Example #2.  In this video, I give the formulas to find the first and second derivative of a parametric curve, and then I find the second derivative.", "lengthSeconds": 190, "width": 480, "height": 360}
{"video_id": "nHnZVFeQvNQ", "transcript": [{"start": 0.179, "duration": 2.551, "text": "all right in this video we're going to"}, {"start": 1.62, "duration": 2.82, "text": "look at the fundamental theorem of"}, {"start": 2.73, "duration": 4.169, "text": "calculus part two and I think this is"}, {"start": 4.44, "duration": 4.199, "text": "the part probably a you know most people"}, {"start": 6.899, "duration": 3.99, "text": "use it maybe they don't recall that it"}, {"start": 8.639, "duration": 4.951, "text": "was called this so what the fundamental"}, {"start": 10.889, "duration": 4.441, "text": "theorem of calculus part two tells us it"}, {"start": 13.59, "duration": 3.54, "text": "tells us how to calculate a definite"}, {"start": 15.33, "duration": 5.49, "text": "integral so if we want to integrate"}, {"start": 17.13, "duration": 7.05, "text": "little f of X DX from A to B it says"}, {"start": 20.82, "duration": 5.73, "text": "what we do is we we find capital f of X"}, {"start": 24.18, "duration": 3.54, "text": "which is an antiderivative and then a"}, {"start": 26.55, "duration": 3.0, "text": "lot of times we'll just make a little"}, {"start": 27.72, "duration": 3.569, "text": "bar here that's the notation kind of"}, {"start": 29.55, "duration": 3.75, "text": "saying well now we've found an"}, {"start": 31.289, "duration": 4.741, "text": "antiderivative we still have to evaluate"}, {"start": 33.3, "duration": 4.89, "text": "that from A to B and to evaluate this"}, {"start": 36.03, "duration": 5.07, "text": "what we do is we calculate F of B minus"}, {"start": 38.19, "duration": 4.59, "text": "F of a so we take the upper limit of"}, {"start": 41.1, "duration": 4.34, "text": "integration and subtract away what's"}, {"start": 42.78, "duration": 5.009, "text": "called the lower limit of integration so"}, {"start": 45.44, "duration": 4.869, "text": "again all this condition says is that"}, {"start": 47.789, "duration": 5.52, "text": "capital F is an anti derivative so"}, {"start": 50.309, "duration": 5.451, "text": "pretty straightforward you know now you"}, {"start": 53.309, "duration": 4.741, "text": "just have to just know antiderivative so"}, {"start": 55.76, "duration": 5.41, "text": "I've picked one here completely at"}, {"start": 58.05, "duration": 5.61, "text": "random sort certainly a pretty basic one"}, {"start": 61.17, "duration": 5.519, "text": "so we're going to calculate the definite"}, {"start": 63.66, "duration": 4.74, "text": "integral from 1 to 3 of X plus 2 well"}, {"start": 66.689, "duration": 4.171, "text": "the first thing we have to do is find an"}, {"start": 68.4, "duration": 5.899, "text": "antiderivative of X to the first and"}, {"start": 70.86, "duration": 6.57, "text": "what we do when we have X to a power is"}, {"start": 74.299, "duration": 6.131, "text": "we add 1 to that power so I'm going to"}, {"start": 77.43, "duration": 5.34, "text": "get X to the second and then divide that"}, {"start": 80.43, "duration": 6.63, "text": "by 2 and then I need to find an"}, {"start": 82.77, "duration": 8.279, "text": "antiderivative of the constant 2 well"}, {"start": 87.06, "duration": 6.36, "text": "2x would be an antiderivative of the"}, {"start": 91.049, "duration": 4.441, "text": "constant 2 and you can always check you"}, {"start": 93.42, "duration": 4.17, "text": "have a correct antiderivative by simply"}, {"start": 95.49, "duration": 4.5, "text": "taking the derivative and you should get"}, {"start": 97.59, "duration": 5.52, "text": "back your your original function back"}, {"start": 99.99, "duration": 4.71, "text": "and what I'm going to do now since we"}, {"start": 103.11, "duration": 5.28, "text": "found an antiderivative I'm just going"}, {"start": 104.7, "duration": 5.43, "text": "to write well from 1 to 3 and what I'm"}, {"start": 108.39, "duration": 3.119, "text": "going to do now is just simply"}, {"start": 110.13, "duration": 3.3, "text": "everywhere there's an X I'm going to"}, {"start": 111.509, "duration": 7.831, "text": "plug in a 3 so I'm going to have 3"}, {"start": 113.43, "duration": 8.85, "text": "squared over 2 plus 2 times 3 so this is"}, {"start": 119.34, "duration": 5.849, "text": "like my F of B part or in this case it's"}, {"start": 122.28, "duration": 5.339, "text": "going to be F of 3 and then what we do"}, {"start": 125.189, "duration": 4.591, "text": "is we subtract away okay so now"}, {"start": 127.619, "duration": 4.821, "text": "everywhere there's an X I simply plug in"}, {"start": 129.78, "duration": 5.84, "text": "1 so we would have 1 squared over 2"}, {"start": 132.44, "duration": 5.79, "text": "two plus two times one so here we're"}, {"start": 135.62, "duration": 6.03, "text": "subtracting away our F of a which in"}, {"start": 138.23, "duration": 7.08, "text": "this case again is just F of one and now"}, {"start": 141.65, "duration": 5.7, "text": "we just have to do the arithmetic so I"}, {"start": 145.31, "duration": 4.71, "text": "don't think this one is a too terribly"}, {"start": 147.35, "duration": 6.39, "text": "bad to compute let's see so 3 squared is"}, {"start": 150.02, "duration": 6.6, "text": "9 so we have 9 over 2 plus 2 times 3"}, {"start": 153.74, "duration": 7.08, "text": "which is 6 minus well let's see we would"}, {"start": 156.62, "duration": 9.21, "text": "have 1/2 plus 2 so let's see that's 9"}, {"start": 160.82, "duration": 7.38, "text": "halves plus 6 minus 1/2 minus 2 when we"}, {"start": 165.83, "duration": 4.71, "text": "distribute the negative let's see 9"}, {"start": 168.2, "duration": 6.15, "text": "halves minus 1/2 that's going to give us"}, {"start": 170.54, "duration": 6.15, "text": "8 halves positive 6 minus 2"}, {"start": 174.35, "duration": 5.28, "text": "let's see positive 6 minus 2 that's"}, {"start": 176.69, "duration": 7.05, "text": "going to give us positive 4 well 8 over"}, {"start": 179.63, "duration": 7.17, "text": "2 is just 4 so we have 4 plus 4 4 we get"}, {"start": 183.74, "duration": 5.37, "text": "the value positive 8 so in this case 8"}, {"start": 186.8, "duration": 5.07, "text": "would be the value of our definite"}, {"start": 189.11, "duration": 4.47, "text": "integral but again all it says all the"}, {"start": 191.87, "duration": 3.36, "text": "fundamental theorem of calculus part 2"}, {"start": 193.58, "duration": 5.1, "text": "says I shouldn't say all it says it's"}, {"start": 195.23, "duration": 6.24, "text": "super super super important it says to"}, {"start": 198.68, "duration": 4.86, "text": "calculate a definite integral find an"}, {"start": 201.47, "duration": 4.32, "text": "antiderivative plug in your limits of"}, {"start": 203.54, "duration": 4.73, "text": "integration subtract and you'll have"}, {"start": 205.79, "duration": 2.48, "text": "your answer"}], "title": "The Fundamental Theorem of Calculus. Part 2", "description": "Thanks to all of you who support me on Patreon. You da real mvps!   $1 per month helps!! :)  https://www.patreon.com/patrickjmt !! The Fundamental Theorem of Calculus. Part 2.  The FTC part 2 simply tells that to evaluate a definite integral, we find an antiderivative, plug in the limits of integration and subtract.  Simple! (Which is what makes it soooooo amazing!)", "lengthSeconds": 208, "width": 480, "height": 360}
{"video_id": "kcRHid3tXWE", "transcript": [{"start": 0.96, "duration": 3.76, "text": "okay here we're going to graph the"}, {"start": 2.24, "duration": 6.399, "text": "system of equations y equals negative 2x"}, {"start": 4.72, "duration": 5.44, "text": "plus 7 and y equals negative 2x minus 5."}, {"start": 8.639, "duration": 3.681, "text": "so notice both of these are linear"}, {"start": 10.16, "duration": 4.24, "text": "equations they both have the same slope"}, {"start": 12.32, "duration": 4.08, "text": "of negative 2. the only thing that's"}, {"start": 14.4, "duration": 4.32, "text": "different is the y-intercept one has a"}, {"start": 16.4, "duration": 4.4, "text": "y-intercept of positive seven and the"}, {"start": 18.72, "duration": 4.719, "text": "other one has a y-intercept of negative"}, {"start": 20.8, "duration": 5.6, "text": "five so i'll graph negative two x plus"}, {"start": 23.439, "duration": 4.641, "text": "seven first so let's see one two three"}, {"start": 26.4, "duration": 4.0, "text": "four five six"}, {"start": 28.08, "duration": 4.08, "text": "seven there's our y-intercept"}, {"start": 30.4, "duration": 6.72, "text": "and now the slope again is negative two"}, {"start": 32.16, "duration": 6.16, "text": "so down two over one down two over one"}, {"start": 37.12, "duration": 4.24, "text": "so again this was going to make"}, {"start": 38.32, "duration": 7.52, "text": "obviously a nice rough little sketch"}, {"start": 41.36, "duration": 4.48, "text": "so if we play connect the dots here"}, {"start": 48.559, "duration": 6.0, "text": "you can play connect the dots and get"}, {"start": 52.079, "duration": 4.081, "text": "one of our lines"}, {"start": 54.559, "duration": 3.84, "text": "you could always even find the"}, {"start": 56.16, "duration": 5.28, "text": "x-intercept if you wanted to to make an"}, {"start": 58.399, "duration": 3.041, "text": "even better graph"}, {"start": 62.239, "duration": 4.801, "text": "so there's y equals negative two x plus"}, {"start": 66.159, "duration": 2.161, "text": "seven"}, {"start": 67.04, "duration": 3.68, "text": "and now the only thing that'll be"}, {"start": 68.32, "duration": 5.36, "text": "different for our other line is that"}, {"start": 70.72, "duration": 3.92, "text": "it'll have a y-intercept"}, {"start": 73.68, "duration": 3.119, "text": "of"}, {"start": 74.64, "duration": 4.72, "text": "negative five but then the same thing"}, {"start": 76.799, "duration": 5.36, "text": "the slope is negative two so down two to"}, {"start": 79.36, "duration": 7.68, "text": "the right one down two to the right one"}, {"start": 82.159, "duration": 7.761, "text": "these should obviously be parallel lines"}, {"start": 87.04, "duration": 2.88, "text": "let's see here"}, {"start": 92.4, "duration": 4.32, "text": "hopefully they'll look relatively"}, {"start": 94.079, "duration": 2.641, "text": "parallel"}, {"start": 99.119, "duration": 4.64, "text": "i think pretty close so looks good to me"}, {"start": 101.68, "duration": 3.04, "text": "so there's our other line y equals"}, {"start": 103.759, "duration": 2.72, "text": "negative"}, {"start": 104.72, "duration": 4.399, "text": "2x"}, {"start": 106.479, "duration": 4.881, "text": "minus 5. so in terms you know the"}, {"start": 109.119, "duration": 4.241, "text": "question obviously doesn't ask this but"}, {"start": 111.36, "duration": 4.64, "text": "if we tried to find a solution to the"}, {"start": 113.36, "duration": 4.799, "text": "system of equations we could immediately"}, {"start": 116.0, "duration": 4.88, "text": "say that there are no solutions and we"}, {"start": 118.159, "duration": 5.201, "text": "can deduce that from the graph"}, {"start": 120.88, "duration": 4.64, "text": "since the lines never intersect we can"}, {"start": 123.36, "duration": 4.959, "text": "simply say that there's no solution to"}, {"start": 125.52, "duration": 4.799, "text": "our original system of equations"}, {"start": 128.319, "duration": 3.841, "text": "hey it didn't ask that"}, {"start": 130.319, "duration": 4.401, "text": "but a lot of times that's why we do"}, {"start": 132.16, "duration": 4.48, "text": "graph our system of equations to try to"}, {"start": 134.72, "duration": 5.4, "text": "see if there is a point of intersection"}, {"start": 136.64, "duration": 3.48, "text": "or not"}], "title": "Graphing a Simple System of Equations", "description": "Thanks to all of you who support me on Patreon. You da real mvps!   $1 per month helps!! :)  https://www.patreon.com/patrickjmt !! Graphing a Simple System of Equations", "lengthSeconds": 138, "width": 450, "height": 360}
{"video_id": "ErVTgggAQjs", "transcript": [{"start": 0.599, "duration": 3.641, "text": "all right so just another applied trig"}, {"start": 2.56, "duration": 4.239, "text": "problem here suppose you've got a hiker"}, {"start": 4.24, "duration": 4.96, "text": "and uh he's hiking up a 12 degree slope"}, {"start": 6.799, "duration": 4.561, "text": "he hikes at a constant speed of 3 miles"}, {"start": 9.2, "duration": 6.319, "text": "hour we want to know how much altitude"}, {"start": 11.36, "duration": 4.159, "text": "he gains over the course of 5 hours of"}, {"start": 17.68, "duration": 4.16, "text": "hiking all right so I'm just going to"}, {"start": 19.56, "duration": 4.08, "text": "make a little right triangle here's our"}, {"start": 21.84, "duration": 5.04, "text": "uh angle of"}, {"start": 23.64, "duration": 6.399, "text": "12\u00b0 um if the guy is walking 3 miles per"}, {"start": 26.88, "duration": 6.12, "text": "hour for 5 hours he's walked a total of"}, {"start": 30.039, "duration": 5.561, "text": "15 miles so that means the hypotenuse is"}, {"start": 33.0, "duration": 4.16, "text": "just going to have length 15 all we're"}, {"start": 35.6, "duration": 2.799, "text": "really trying to figure out is this"}, {"start": 37.16, "duration": 4.239, "text": "change in"}, {"start": 38.399, "duration": 4.121, "text": "elevation U maybe we'll represent that"}, {"start": 41.399, "duration": 3.68, "text": "with"}, {"start": 42.52, "duration": 4.6, "text": "Y so now all we have to do is just use a"}, {"start": 45.079, "duration": 4.601, "text": "trig function so let's see um in"}, {"start": 47.12, "duration": 4.88, "text": "relation to our angle 12\u00b0 we have the"}, {"start": 49.68, "duration": 4.519, "text": "opposite side we know about the"}, {"start": 52.0, "duration": 4.359, "text": "hypotenuse so the trig function that"}, {"start": 54.199, "duration": 6.32, "text": "relates the opposite to the hypotenuse"}, {"start": 56.359, "duration": 7.52, "text": "would be sign so simply s of"}, {"start": 60.519, "duration": 7.121, "text": "12\u00b0 is going to equal the opposite which"}, {"start": 63.879, "duration": 5.28, "text": "is y over the hypotenuse which is 15 and"}, {"start": 67.64, "duration": 4.159, "text": "now all we have to do is simply solve"}, {"start": 69.159, "duration": 4.361, "text": "for y so I'm going to multiply both"}, {"start": 71.799, "duration": 5.761, "text": "sides by"}, {"start": 73.52, "duration": 7.639, "text": "15 so it says 15 s of"}, {"start": 77.56, "duration": 5.84, "text": "12\u00b0 is going to be uh his elevation"}, {"start": 81.159, "duration": 3.92, "text": "change uh let's throw this in a"}, {"start": 83.4, "duration": 4.039, "text": "calculator just to make some more sense"}, {"start": 85.079, "duration": 4.961, "text": "out of these numbers uh again make sure"}, {"start": 87.439, "duration": 4.801, "text": "your calculator's in degree mode I'm I'm"}, {"start": 90.04, "duration": 7.399, "text": "getting s of"}, {"start": 92.24, "duration": 8.239, "text": "12\u00b0 to be roughly equal to uh"}, {"start": 97.439, "duration": 7.96, "text": "208 after I"}, {"start": 100.479, "duration": 4.92, "text": "round so if we simply multiply that by"}, {"start": 106.28, "duration": 3.72, "text": "15 um I'm getting"}, {"start": 110.439, "duration": 4.921, "text": "3.12 so we can simply say that his"}, {"start": 112.92, "duration": 6.04, "text": "elevation gain is roughly"}, {"start": 115.36, "duration": 6.079, "text": "3.12 miles so a lot of elevation gain"}, {"start": 118.96, "duration": 4.199, "text": "there so um but again you know no big"}, {"start": 121.439, "duration": 3.761, "text": "deal here again all you're doing is just"}, {"start": 123.159, "duration": 3.56, "text": "making a little right triangle and just"}, {"start": 125.2, "duration": 3.24, "text": "thinking about you know which trig"}, {"start": 126.719, "duration": 3.121, "text": "function you need and then it's just a"}, {"start": 128.44, "duration": 2.799, "text": "matter of punching in a couple things"}, {"start": 129.84, "duration": 4.399, "text": "into the"}, {"start": 131.239, "duration": 3.0, "text": "calculator"}], "title": "Trigonometry Word Problem, Example 2", "description": "Thanks to all of you who support me on Patreon. You da real mvps!   $1 per month helps!! :)  https://www.patreon.com/patrickjmt !! Trigonometry Word Problem, Example 2. In this video, I use a bit of trigonometry to find the elevation gain of a hiker going up a slope.", "lengthSeconds": 132, "width": 480, "height": 360}
{"video_id": "BQ4QzTcG90g", "transcript": [{"start": 0.719, "duration": 3.441, "text": "okay in this example we're going to find"}, {"start": 2.28, "duration": 3.519, "text": "the amplitude and period of the given"}, {"start": 4.16, "duration": 3.96, "text": "function and then we're also going to"}, {"start": 5.799, "duration": 4.641, "text": "describe any phase shifts or vertical"}, {"start": 8.12, "duration": 5.28, "text": "shifts when compared to the"}, {"start": 10.44, "duration": 7.839, "text": "corresponding basic trig function so"}, {"start": 13.4, "duration": 8.16, "text": "we've got y = /4 cosine of 12x + < / 2 -"}, {"start": 18.279, "duration": 4.721, "text": "1 so to get the amplitude probably the"}, {"start": 21.56, "duration": 2.559, "text": "easiest part we'll just take the"}, {"start": 23.0, "duration": 4.56, "text": "absolute"}, {"start": 24.119, "duration": 6.721, "text": "value of the number out front - 1/4 and"}, {"start": 27.56, "duration": 4.679, "text": "that'll give us 1/4 as our amplitude"}, {"start": 30.84, "duration": 3.76, "text": "now to get the period we're actually"}, {"start": 32.239, "duration": 5.0, "text": "going to have to rewrite this a little"}, {"start": 34.6, "duration": 4.4, "text": "bit and inside the parentheses we want"}, {"start": 37.239, "duration": 3.441, "text": "the coefficient on the X to be a"}, {"start": 39.0, "duration": 5.52, "text": "positive"}, {"start": 40.68, "duration": 5.76, "text": "one so what we can do well we can simply"}, {"start": 44.52, "duration": 5.76, "text": "Factor the 1/2"}, {"start": 46.44, "duration": 9.119, "text": "out so then 1 12 * X would give us well"}, {"start": 50.28, "duration": 8.439, "text": "12 x and then 12 multiplied Pi would"}, {"start": 55.559, "duration": 5.441, "text": "give us our Pi / 2 and the minus one is"}, {"start": 58.719, "duration": 5.121, "text": "still out there"}, {"start": 61.0, "duration": 6.68, "text": "this is going to be our B value when we"}, {"start": 63.84, "duration": 7.319, "text": "do the period so 2 pi ided the absolute"}, {"start": 67.68, "duration": 7.68, "text": "value of B so in this case we'll have 2"}, {"start": 71.159, "duration": 7.841, "text": "Pi / the absolute value of2 well that is"}, {"start": 75.36, "duration": 7.28, "text": "just 1/2 2 pi ID a half that's going to"}, {"start": 79.0, "duration": 6.32, "text": "give us 4 Pi same thing as 2 pi multipli"}, {"start": 82.64, "duration": 4.799, "text": "2 so there's the period and the"}, {"start": 85.32, "duration": 5.479, "text": "amplitude what else did we want U we"}, {"start": 87.439, "duration": 6.441, "text": "wanted the vertical and the horizontal"}, {"start": 90.799, "duration": 6.241, "text": "or the phase shift as well if we look"}, {"start": 93.88, "duration": 4.4, "text": "inside the I used brackets here if we"}, {"start": 97.04, "duration": 2.439, "text": "look inside the brackets or the"}, {"start": 98.28, "duration": 5.4, "text": "parentheses that's where we're going to"}, {"start": 99.479, "duration": 4.201, "text": "get the phase shift so in this"}, {"start": 104.479, "duration": 7.32, "text": "case we have X+ Pi that's me that means"}, {"start": 108.68, "duration": 5.32, "text": "uh if we compare it to the graph of just"}, {"start": 111.799, "duration": 6.041, "text": "cosine uh well really we should be"}, {"start": 114.0, "duration": 7.56, "text": "actually comparing it to cosine of 12x"}, {"start": 117.84, "duration": 8.16, "text": "it's going to be shifted to the left"}, {"start": 121.56, "duration": 6.879, "text": "um to the left Pi units and then the"}, {"start": 126.0, "duration": 2.439, "text": "vertical"}, {"start": 129.44, "duration": 5.48, "text": "shift in this case since we have a minus"}, {"start": 131.879, "duration": 5.44, "text": "one out there the the vertical shift"}, {"start": 134.92, "duration": 6.399, "text": "will equal negative 1 or it'll be one"}, {"start": 137.319, "duration": 4.0, "text": "unit downwards"}], "title": "Finding Amplitude, Period, Horizontal and Vertical Shifts of a Trig Function EX 1", "description": "Thanks to all of you who support me on Patreon. You da real mvps!   $1 per month helps!! :)  https://www.patreon.com/patrickjmt !! Finding Amplitude, Period, Horizontal and Vertical Shifts of a Trig Function EX 1", "lengthSeconds": 141, "width": 450, "height": 360}
{"video_id": "oT6EcJkgFAg", "transcript": [{"start": 0.199, "duration": 4.451, "text": "all right in this video we're going to"}, {"start": 2.73, "duration": 3.42, "text": "look at an example related to the"}, {"start": 4.65, "duration": 4.529, "text": "derivative of the square root function"}, {"start": 6.15, "duration": 5.699, "text": "so suppose we are given that the"}, {"start": 9.179, "duration": 4.411, "text": "derivative of square root of x is going"}, {"start": 11.849, "duration": 4.051, "text": "to be one over two times the square root"}, {"start": 13.59, "duration": 5.01, "text": "of x we want to know what's the"}, {"start": 15.9, "duration": 7.95, "text": "derivative of square root of x over"}, {"start": 18.6, "duration": 6.45, "text": "eight so just to kind of justify this a"}, {"start": 23.85, "duration": 3.12, "text": "little bit again if we use the"}, {"start": 25.05, "duration": 4.979, "text": "definition of the derivative on our new"}, {"start": 26.97, "duration": 5.46, "text": "function we would have the limit as H"}, {"start": 30.029, "duration": 6.96, "text": "goes to zero and then we would have the"}, {"start": 32.43, "duration": 8.52, "text": "square root of x plus h over eight minus"}, {"start": 36.989, "duration": 6.151, "text": "the square root of x over eight all over"}, {"start": 40.95, "duration": 6.72, "text": "H so again I'm just using our definition"}, {"start": 43.14, "duration": 6.599, "text": "f of X plus h minus f of X all over H"}, {"start": 47.67, "duration": 5.22, "text": "and again we're just taking the limit of"}, {"start": 49.739, "duration": 5.91, "text": "that as H goes to zero so I'm just using"}, {"start": 52.89, "duration": 4.95, "text": "the definition of the derivative on our"}, {"start": 55.649, "duration": 4.59, "text": "new function square root of square root"}, {"start": 57.84, "duration": 5.699, "text": "of x over eight well notice what we"}, {"start": 60.239, "duration": 6.781, "text": "could do is in the numerator we could"}, {"start": 63.539, "duration": 5.19, "text": "factor that one-eighth out right we"}, {"start": 67.02, "duration": 4.47, "text": "could just pull the 1/8 out and then we"}, {"start": 68.729, "duration": 5.971, "text": "would have well one eighth times the"}, {"start": 71.49, "duration": 7.83, "text": "square root of x plus h minus the square"}, {"start": 74.7, "duration": 7.98, "text": "root of x all over H and the idea here"}, {"start": 79.32, "duration": 6.75, "text": "is is that the limit depends on H okay"}, {"start": 82.68, "duration": 6.66, "text": "so this constant 1/8 we can actually"}, {"start": 86.07, "duration": 7.32, "text": "just factor that out front so we would"}, {"start": 89.34, "duration": 8.13, "text": "have 1/8 times the limit as H approaches"}, {"start": 93.39, "duration": 7.799, "text": "0 of the square root of x plus h minus"}, {"start": 97.47, "duration": 7.16, "text": "the square root of x all over H but if"}, {"start": 101.189, "duration": 5.731, "text": "you think about it now this last part"}, {"start": 104.63, "duration": 5.05, "text": "this last part would just be the"}, {"start": 106.92, "duration": 6.18, "text": "definition of the derivative for the"}, {"start": 109.68, "duration": 5.7, "text": "square root of x function okay so you"}, {"start": 113.1, "duration": 4.26, "text": "know if you think about if we started"}, {"start": 115.38, "duration": 3.839, "text": "with the function say f of X equals"}, {"start": 117.36, "duration": 4.2, "text": "square root of x if we wanted to find"}, {"start": 119.219, "duration": 5.071, "text": "the derivative we would get the limit as"}, {"start": 121.56, "duration": 7.14, "text": "H goes to 0 square root of x plus h"}, {"start": 124.29, "duration": 7.1, "text": "minus square root of x all over H and"}, {"start": 128.7, "duration": 5.42, "text": "that's what we have and you know in"}, {"start": 131.39, "duration": 5.069, "text": "new expression so what I'm trying to say"}, {"start": 134.12, "duration": 6.0, "text": "is we can actually rewrite this this"}, {"start": 136.459, "duration": 8.551, "text": "thing this would just simply be equal to"}, {"start": 140.12, "duration": 6.57, "text": "1/8 times the derivative of the square"}, {"start": 145.01, "duration": 3.63, "text": "root of x function but we're already"}, {"start": 146.69, "duration": 5.61, "text": "given that the derivative of the square"}, {"start": 148.64, "duration": 6.78, "text": "root of x is going to be 1 over 2 square"}, {"start": 152.3, "duration": 5.22, "text": "root of x so now all we have to do is"}, {"start": 155.42, "duration": 5.69, "text": "just multiply the 8 and the 2 and we'll"}, {"start": 157.52, "duration": 8.189, "text": "get 1 over 16 times the square root of x"}, {"start": 161.11, "duration": 6.67, "text": "so the moral of the story is when you're"}, {"start": 165.709, "duration": 4.111, "text": "kind of multiplying or dividing by a"}, {"start": 167.78, "duration": 4.11, "text": "constant if you know the derivative of"}, {"start": 169.82, "duration": 3.96, "text": "one of them so in this case we knew the"}, {"start": 171.89, "duration": 4.379, "text": "derivative of square root of x is 1 over"}, {"start": 173.78, "duration": 4.65, "text": "2 root X it turns out that the"}, {"start": 176.269, "duration": 3.841, "text": "derivative of square root of x over 8 we"}, {"start": 178.43, "duration": 4.23, "text": "just take the original one and we just"}, {"start": 180.11, "duration": 6.629, "text": "multiplied it by 1/8 that's all we did"}, {"start": 182.66, "duration": 7.74, "text": "in this case so so that's all there is"}, {"start": 186.739, "duration": 4.891, "text": "to it okay so again we could have sort"}, {"start": 190.4, "duration": 2.88, "text": "of just jumped and I could have said hey"}, {"start": 191.63, "duration": 3.72, "text": "just take this one and multiply it by"}, {"start": 193.28, "duration": 3.57, "text": "1/8 but again just trying to give you a"}, {"start": 195.35, "duration": 4.83, "text": "little bit of an argument as to why it"}, {"start": 196.85, "duration": 5.01, "text": "is so we can just factor the 1/8 out and"}, {"start": 200.18, "duration": 3.36, "text": "again what we're left with would just be"}, {"start": 201.86, "duration": 4.29, "text": "the definition of the derivative of"}, {"start": 203.54, "duration": 4.86, "text": "square root of x but again we know it so"}, {"start": 206.15, "duration": 5.33, "text": "we can plug that in and we can get our"}, {"start": 208.4, "duration": 3.08, "text": "new derivative"}], "title": "Limit Definition of Derivative, Square Root Example", "description": "Thanks to all of you who support me on Patreon. You da real mvps!   $1 per month helps!! :)  https://www.patreon.com/patrickjmt !! Limit Definition of Derivative, Square Root Example.  In this video I use the limit definition of a derivative to find the derivative of a function involving a square root.", "lengthSeconds": 210, "width": 480, "height": 360}
{"video_id": "_qyVzH3e1dY", "transcript": [{"start": 0.96, "duration": 3.2, "text": "okay in this video I'm going to do some"}, {"start": 2.56, "duration": 4.04, "text": "examples about factoring the difference"}, {"start": 4.16, "duration": 4.479, "text": "of two squares sometimes people will say"}, {"start": 6.6, "duration": 5.4, "text": "it's factoring uh the difference of two"}, {"start": 8.639, "duration": 5.04, "text": "perfect squares um and the formula that"}, {"start": 12.0, "duration": 3.519, "text": "we're going to use it basically says if"}, {"start": 13.679, "duration": 4.401, "text": "you have something squared minus"}, {"start": 15.519, "duration": 4.6, "text": "something squared you can put that in"}, {"start": 18.08, "duration": 3.32, "text": "parentheses by just taking the things"}, {"start": 20.119, "duration": 3.681, "text": "that are being"}, {"start": 21.4, "duration": 4.52, "text": "squared and one set of parentheses make"}, {"start": 23.8, "duration": 4.12, "text": "it a minus and the other set make it a"}, {"start": 25.92, "duration": 5.599, "text": "plus that's all you got to"}, {"start": 27.92, "duration": 7.12, "text": "do um be very careful F if you have a^2"}, {"start": 31.519, "duration": 5.36, "text": "plus B2 uh if the exponents are squares"}, {"start": 35.04, "duration": 4.679, "text": "this doesn't Factor you can't do"}, {"start": 36.879, "duration": 5.52, "text": "anything at all to this so people will"}, {"start": 39.719, "duration": 4.961, "text": "often try to make it do something"}, {"start": 42.399, "duration": 4.521, "text": "somehow Factor it but uh it it's it's"}, {"start": 44.68, "duration": 6.519, "text": "not correct so be very careful about"}, {"start": 46.92, "duration": 7.799, "text": "that if the exponents are both threes it"}, {"start": 51.199, "duration": 5.84, "text": "does factor in that case so just some"}, {"start": 54.719, "duration": 4.881, "text": "you know little nuances that to uh be"}, {"start": 57.039, "duration": 7.12, "text": "aware of but anyway okay so just a"}, {"start": 59.6, "duration": 9.6, "text": "couple basic examples here uh 9 x^2 - 49"}, {"start": 64.159, "duration": 5.041, "text": "and then uh 36 y 4 -"}, {"start": 70.799, "duration": 5.401, "text": "100 all you have to do and we can even"}, {"start": 73.24, "duration": 4.96, "text": "rewrite it first um we want some we want"}, {"start": 76.2, "duration": 4.879, "text": "to rewrite the first term as something"}, {"start": 78.2, "duration": 5.2, "text": "squared well to write nine as something"}, {"start": 81.079, "duration": 4.961, "text": "squared we would need a three and then"}, {"start": 83.4, "duration": 6.52, "text": "to get x2 we just need an X so if you"}, {"start": 86.04, "duration": 8.039, "text": "multiply 3x by 3x we get 9^ 2 and then 4"}, {"start": 89.92, "duration": 6.64, "text": "49 we can write that as 7^ s i I"}, {"start": 94.079, "duration": 6.121, "text": "typically skip this step honestly but uh"}, {"start": 96.56, "duration": 7.4, "text": "just to illustrate so this is our"}, {"start": 100.2, "duration": 6.44, "text": "a this is our B and hey it says we have"}, {"start": 103.96, "duration": 6.199, "text": "a 2 minus b^ 2 well that factors it says"}, {"start": 106.64, "duration": 5.04, "text": "we take a which is 3x and it doesn't"}, {"start": 110.159, "duration": 3.881, "text": "matter the order in which you put the"}, {"start": 111.68, "duration": 6.84, "text": "negative and the positive uh in the"}, {"start": 114.04, "duration": 7.64, "text": "parentheses so we'll take 3x - 7 and 3x"}, {"start": 118.52, "duration": 5.4, "text": "+ 7 and now we have have it factored so"}, {"start": 121.68, "duration": 4.079, "text": "typically what I always do um kind of"}, {"start": 123.92, "duration": 3.64, "text": "the shortcut for me at least the way I"}, {"start": 125.759, "duration": 3.84, "text": "think about it anytime I see something"}, {"start": 127.56, "duration": 4.08, "text": "with two terms so here's one term here's"}, {"start": 129.599, "duration": 5.321, "text": "the other term if there's a a minus in"}, {"start": 131.64, "duration": 5.0, "text": "between them it always factors basically"}, {"start": 134.92, "duration": 4.48, "text": "I just take the square root of the first"}, {"start": 136.64, "duration": 4.4, "text": "term which um I I guess you know maybe"}, {"start": 139.4, "duration": 2.8, "text": "maybe you've seen it square roots a"}, {"start": 141.04, "duration": 4.8, "text": "little bit maybe you haven't worked with"}, {"start": 142.2, "duration": 6.759, "text": "them a lot but the sare < TK of 9 uh x^2"}, {"start": 145.84, "duration": 5.2, "text": "would just be 3x and 3X and the square <"}, {"start": 148.959, "duration": 3.881, "text": "TK of 49 will just be seven and seven"}, {"start": 151.04, "duration": 3.76, "text": "and then I just stick a minus and a plus"}, {"start": 152.84, "duration": 6.56, "text": "in between them so maybe we can do kind"}, {"start": 154.8, "duration": 7.359, "text": "of the same idea uh on this one 36 y 4us"}, {"start": 159.4, "duration": 4.839, "text": "100 so I recognize there's two terms"}, {"start": 162.159, "duration": 3.72, "text": "there's a minus in between so I say hey"}, {"start": 164.239, "duration": 4.161, "text": "that's going to"}, {"start": 165.879, "duration": 5.121, "text": "factor uh I think what number times"}, {"start": 168.4, "duration": 5.6, "text": "itself is 36 well I would need a six and"}, {"start": 171.0, "duration": 6.56, "text": "a six what would I have to multiply by"}, {"start": 174.0, "duration": 4.48, "text": "itself to get y 4th we would need y^2"}, {"start": 177.56, "duration": 3.48, "text": "and"}, {"start": 178.48, "duration": 5.08, "text": "y^2 and then I think well what number"}, {"start": 181.04, "duration": 5.0, "text": "multiply what's the square root of 100"}, {"start": 183.56, "duration": 4.239, "text": "uh it's just going to be 10 and then I"}, {"start": 186.04, "duration": 3.759, "text": "just stick a negative and a positive"}, {"start": 187.799, "duration": 4.841, "text": "inside of there and again now I have it"}, {"start": 189.799, "duration": 4.321, "text": "factored okay so nothing nothing worse"}, {"start": 192.64, "duration": 3.28, "text": "than that and again you can distribute"}, {"start": 194.12, "duration": 3.92, "text": "these out and make sure in fact that"}, {"start": 195.92, "duration": 5.16, "text": "everything cancels out and you do get"}, {"start": 198.04, "duration": 6.04, "text": "this original expression"}, {"start": 201.08, "duration": 3.0, "text": "back"}], "title": "Factoring the Difference of Two Squares - Ex 1", "description": "Factoring the Difference of Two Squares - Step-by-Step Guide\nIn this video, we'll explore how to factor expressions using the difference of two squares formula. We introduce the formula: A^2 \u2212 B^2 = (A\u2212B) (A+B), show how to identify the terms A and B, and work through example problems to demonstrate how to apply the formula effectively.\n\nWhat You Will Learn\nThe formula for factoring the difference of two squares.\nHow to rewrite expressions in the form A^2 \u2212 B^2\n\n Step-by-step examples to illustrate the factoring process.\n\nTechniques for recognizing and factoring more complex differences of squares.\nFactoring the difference of squares is an essential skill in algebra that can simplify polynomial expressions and solve equations more easily. By the end of this video, you'll understand how to break down and factor these types of expressions using the formula effectively.\n\nIf you find this video helpful, please like, comment, and subscribe! Share it with classmates, teachers, or anyone studying algebra and factoring techniques.\n\nSupport my work on Patreon: https://www.patreon.com/patrickjmt?ty=c\n\n#FactoringPolynomials #DifferenceOfSquares #Algebra #MathTutorial #PatrickJMT #PolynomialFactoring #MathHelp #Mathematics #AlgebraConcepts #MathForStudents #FactoringDifferenceOfSquares #LearnAlgebra #AlgebraProblems #MathConcepts #AlgebraStepByStep #MathEducation #PolynomialExpressions #MathLearning #Education", "lengthSeconds": 202, "width": 480, "height": 360}
{"video_id": "s2ox6IuKiX0", "transcript": [{"start": 0.709, "duration": 4.9, "text": "all right so here we're going to look at"}, {"start": 2.669, "duration": 4.23, "text": "another example using the root test so"}, {"start": 5.609, "duration": 3.541, "text": "in Part B we've got from 1 to infinity"}, {"start": 6.899, "duration": 7.591, "text": "negative 1 to the n times n to the N"}, {"start": 9.15, "duration": 8.37, "text": "over 3 n to the third plus 1 the first"}, {"start": 14.49, "duration": 7.59, "text": "thing I'm going to do is rewrite this a"}, {"start": 17.52, "duration": 7.62, "text": "little bit we can break up the"}, {"start": 22.08, "duration": 5.49, "text": "denominator instead of 3 raised to the"}, {"start": 25.14, "duration": 4.559, "text": "end of the third plus 1 power we can"}, {"start": 27.57, "duration": 6.03, "text": "write that as 3 raised to n to the third"}, {"start": 29.699, "duration": 5.311, "text": "times 3 to the first power ok again if"}, {"start": 33.6, "duration": 3.389, "text": "we have like bases we would add the"}, {"start": 35.01, "duration": 4.709, "text": "exponents so we would certainly get all"}, {"start": 36.989, "duration": 4.201, "text": "of this stuff right back and the 1/3"}, {"start": 39.719, "duration": 3.421, "text": "you know the over 3 you can think about"}, {"start": 41.19, "duration": 3.54, "text": "that as being a 1/3 which is just a"}, {"start": 43.14, "duration": 5.189, "text": "constant so I'm going to factor that out"}, {"start": 44.73, "duration": 8.43, "text": "and again just the reason for doing that"}, {"start": 48.329, "duration": 6.39, "text": "is it's nice if the powers or just if"}, {"start": 53.16, "duration": 4.5, "text": "the exponents are sort of just to the"}, {"start": 54.719, "duration": 5.131, "text": "nth power or into an exponent it'll be"}, {"start": 57.66, "duration": 3.87, "text": "much easier I think to simplify things"}, {"start": 59.85, "duration": 3.15, "text": "down yeah I don't think it'd be the end"}, {"start": 61.53, "duration": 5.19, "text": "of the world if you didn't do this but"}, {"start": 63.0, "duration": 5.43, "text": "to me it'll be a little cleaner so I'm"}, {"start": 66.72, "duration": 3.509, "text": "just going to look at this part now you"}, {"start": 68.43, "duration": 4.32, "text": "know if this series converges when you"}, {"start": 70.229, "duration": 4.171, "text": "multiply it by 1/3 it converges if it"}, {"start": 72.75, "duration": 4.17, "text": "diverges when you multiply it by"}, {"start": 74.4, "duration": 3.96, "text": "one-third it's going to diverge so we'll"}, {"start": 76.92, "duration": 3.75, "text": "take the limit as n goes to infinity"}, {"start": 78.36, "duration": 5.939, "text": "we'll take the absolute value of"}, {"start": 80.67, "duration": 7.94, "text": "negative 1 to the N in race to the N and"}, {"start": 84.299, "duration": 7.53, "text": "then 3 raised to the N to the 3rd power"}, {"start": 88.61, "duration": 7.36, "text": "again well the nth root we'll take that"}, {"start": 91.829, "duration": 6.36, "text": "to the 1 over N power well when we take"}, {"start": 95.97, "duration": 3.75, "text": "the absolute value the only thing that"}, {"start": 98.189, "duration": 3.961, "text": "would happen is the negative 1 to the N"}, {"start": 99.72, "duration": 6.42, "text": "would be gone so we would have N to the"}, {"start": 102.15, "duration": 9.48, "text": "N over 3 to the N cubed raised to the 1"}, {"start": 106.14, "duration": 8.49, "text": "over N but let's see if we take N to the"}, {"start": 111.63, "duration": 4.98, "text": "N power and raise that to the 1 over N"}, {"start": 114.63, "duration": 4.83, "text": "power again we just multiply exponents"}, {"start": 116.61, "duration": 5.039, "text": "so n times 1 over N is going to give us"}, {"start": 119.46, "duration": 4.549, "text": "an end to the first in the numerator and"}, {"start": 121.649, "duration": 5.491, "text": "the denominator we're going to have"}, {"start": 124.009, "duration": 5.59, "text": "let's see so we would have n cubed"}, {"start": 127.14, "duration": 5.51, "text": "divided by n so that would leave this"}, {"start": 129.599, "duration": 7.17, "text": "with 3 N squared in the denominator"}, {"start": 132.65, "duration": 7.39, "text": "well the derivative of this is going to"}, {"start": 136.769, "duration": 4.741, "text": "be is going to be well I guess we should"}, {"start": 140.04, "duration": 3.99, "text": "say first the limit of this is going to"}, {"start": 141.51, "duration": 7.14, "text": "be infinity over infinity so we can"}, {"start": 144.03, "duration": 6.42, "text": "certainly use l'hopital's rule here so"}, {"start": 148.65, "duration": 8.64, "text": "if we take the derivative of this if we"}, {"start": 150.45, "duration": 8.52, "text": "think about this as being you know just"}, {"start": 157.29, "duration": 3.24, "text": "in terms of derivatives well the"}, {"start": 158.97, "duration": 4.95, "text": "derivative of end of the first would be"}, {"start": 160.53, "duration": 5.849, "text": "one this is like an exponent or an"}, {"start": 163.92, "duration": 4.23, "text": "exponential function 3 raised to some"}, {"start": 166.379, "duration": 3.601, "text": "function well when we take the"}, {"start": 168.15, "duration": 4.169, "text": "derivative it'll repeat itself so we'll"}, {"start": 169.98, "duration": 5.039, "text": "get 3 to the n squared if we get the"}, {"start": 172.319, "duration": 4.681, "text": "natural logarithm of the base but then"}, {"start": 175.019, "duration": 3.661, "text": "we also have to by the chain rule take"}, {"start": 177.0, "duration": 4.53, "text": "the derivative of the exponent so we'll"}, {"start": 178.68, "duration": 5.279, "text": "get 2 to the N well now as n goes to"}, {"start": 181.53, "duration": 5.549, "text": "infinity the denominator is definitely"}, {"start": 183.959, "duration": 6.241, "text": "going to infinity the numerator is just"}, {"start": 187.079, "duration": 6.511, "text": "1 so this limit is going to equal 0 so"}, {"start": 190.2, "duration": 5.55, "text": "now we've justified that our limits 0"}, {"start": 193.59, "duration": 3.89, "text": "well that's certainly less than 1 and"}, {"start": 195.75, "duration": 8.37, "text": "that would tell us that our series"}, {"start": 197.48, "duration": 9.039, "text": "converges so again down in the bottom"}, {"start": 204.12, "duration": 6.869, "text": "all I'm using is the the formula a to"}, {"start": 206.519, "duration": 6.601, "text": "the f of X if we have you know a raise"}, {"start": 210.989, "duration": 4.83, "text": "to something the derivative of that is a"}, {"start": 213.12, "duration": 6.66, "text": "raise to the f of X times the natural"}, {"start": 215.819, "duration": 5.461, "text": "logarithm of a times F prime of X that's"}, {"start": 219.78, "duration": 2.9, "text": "how I'm getting again the derivative of"}, {"start": 221.28, "duration": 5.179, "text": "the bottom just in case you've forgotten"}, {"start": 222.68, "duration": 3.779, "text": "this little formula"}], "title": "The Root Test - Another Example, #2", "description": "Thanks to all of you who support me on Patreon. You da real mvps!   $1 per month helps!! :)  https://www.patreon.com/patrickjmt !! The Root Test - Another Example, #2.  Just another example showing that a series converges or diverges using the root test.", "lengthSeconds": 225, "width": 480, "height": 360}
{"video_id": "cZSbjtzfxNk", "transcript": [{"start": 0.149, "duration": 3.601, "text": "all right here we're going to look at"}, {"start": 1.77, "duration": 5.52, "text": "some derivatives involving the number pi"}, {"start": 3.75, "duration": 5.58, "text": "so sorry about my little my little"}, {"start": 7.29, "duration": 3.929, "text": "handwriting there did not like my a so"}, {"start": 9.33, "duration": 5.37, "text": "in Part A we're going to find the"}, {"start": 11.219, "duration": 5.941, "text": "derivative of 5 sine of 8 pi X and the"}, {"start": 14.7, "duration": 4.589, "text": "thing to remember is you know pi is just"}, {"start": 17.16, "duration": 4.59, "text": "a number don't do anything different"}, {"start": 19.289, "duration": 4.681, "text": "than you would any other times people"}, {"start": 21.75, "duration": 4.55, "text": "sometimes see PI and they want to kind"}, {"start": 23.97, "duration": 5.04, "text": "of do I don't know different things so"}, {"start": 26.3, "duration": 4.809, "text": "so here we'll leave the five that just"}, {"start": 29.01, "duration": 4.47, "text": "just gets carried along so the"}, {"start": 31.109, "duration": 4.231, "text": "derivative of sine is going to be cosine"}, {"start": 33.48, "duration": 4.5, "text": "but here we're going to have to use the"}, {"start": 35.34, "duration": 4.98, "text": "chain rule so we'll leave the inside"}, {"start": 37.98, "duration": 5.489, "text": "alone and then we have to take the"}, {"start": 40.32, "duration": 4.86, "text": "derivative of the inside well again if"}, {"start": 43.469, "duration": 4.801, "text": "you know if it just said 8 X the"}, {"start": 45.18, "duration": 5.309, "text": "derivative of 8x would just be 8 but"}, {"start": 48.27, "duration": 5.34, "text": "that's not our number the derivative the"}, {"start": 50.489, "duration": 5.041, "text": "number we do have is 8 times pi so the"}, {"start": 53.61, "duration": 5.16, "text": "derivative of that will just be well 8"}, {"start": 55.53, "duration": 7.59, "text": "times pi and now if we multiply 5 times"}, {"start": 58.77, "duration": 8.07, "text": "8 pi will be 40 times pi and then we"}, {"start": 63.12, "duration": 8.49, "text": "have cosine of 8 times pi times X"}, {"start": 66.84, "duration": 8.01, "text": "leftover and that's our derivative so"}, {"start": 71.61, "duration": 9.84, "text": "maybe let's do Part A Part B here as"}, {"start": 74.85, "duration": 11.22, "text": "well so here we've got 4 pi times cosine"}, {"start": 81.45, "duration": 6.39, "text": "of 3 PI x times sine of 3 PI X again"}, {"start": 86.07, "duration": 3.9, "text": "we've got a constant out front"}, {"start": 87.84, "duration": 4.59, "text": "we'll just sort of leave that alone and"}, {"start": 89.97, "duration": 5.49, "text": "multiply it back at the end so let's"}, {"start": 92.43, "duration": 4.979, "text": "leave the 4 pi alone well now I've got"}, {"start": 95.46, "duration": 4.89, "text": "to use both the product rule and the"}, {"start": 97.409, "duration": 5.011, "text": "chain rule so the derivative of cosine"}, {"start": 100.35, "duration": 5.79, "text": "will be negative sine"}, {"start": 102.42, "duration": 5.1, "text": "let's leave the 3 PI X alone then we"}, {"start": 106.14, "duration": 5.25, "text": "have to take the derivative of the"}, {"start": 107.52, "duration": 6.809, "text": "inside which would be 3 times pi and now"}, {"start": 111.39, "duration": 5.46, "text": "we'll leave the sine of 3 PI X we'll"}, {"start": 114.329, "duration": 4.561, "text": "leave that part alone then we'll put the"}, {"start": 116.85, "duration": 4.11, "text": "plus sign in between there and now we'll"}, {"start": 118.89, "duration": 7.2, "text": "kind of do the same thing so now we can"}, {"start": 120.96, "duration": 7.32, "text": "leave the cosine of 3 PI X alone well if"}, {"start": 126.09, "duration": 3.169, "text": "we do the derivative of sine 3 PI X"}, {"start": 128.28, "duration": 3.14, "text": "again the"}, {"start": 129.259, "duration": 6.211, "text": "we're what use the chain rule so we'll"}, {"start": 131.42, "duration": 6.36, "text": "get cosine of 3 PI X and then the"}, {"start": 135.47, "duration": 6.0, "text": "derivative of the inside will just be 3"}, {"start": 137.78, "duration": 7.23, "text": "times pi all right so I think we've got"}, {"start": 141.47, "duration": 8.489, "text": "it all in there so let's see in terms of"}, {"start": 145.01, "duration": 7.74, "text": "simplification now we've got 4 pi okay"}, {"start": 149.959, "duration": 3.901, "text": "we've got 3 pi in here I'm going to pull"}, {"start": 152.75, "duration": 3.18, "text": "that out front"}, {"start": 153.86, "duration": 5.07, "text": "we've even got a negative so let me make"}, {"start": 155.93, "duration": 6.72, "text": "that negative 3 PI then we have sine of"}, {"start": 158.93, "duration": 9.0, "text": "3 PI X times sine of 3 PI X I'm going to"}, {"start": 162.65, "duration": 7.89, "text": "write that as sine squared 3 PI X then"}, {"start": 167.93, "duration": 5.91, "text": "on the second our second term again we"}, {"start": 170.54, "duration": 6.24, "text": "can pull the 3 pi we have cosine of 3 PI"}, {"start": 173.84, "duration": 7.89, "text": "x times cosine of 3 PI X we can write"}, {"start": 176.78, "duration": 6.629, "text": "that as cosine squared of 3 PI X and I"}, {"start": 181.73, "duration": 4.71, "text": "actually think we could factor out the 3"}, {"start": 183.409, "duration": 5.64, "text": "pi and maybe use an identity on this as"}, {"start": 186.44, "duration": 5.639, "text": "well but for now let's just leave it"}, {"start": 189.049, "duration": 5.851, "text": "like that we'll keep that as our"}, {"start": 192.079, "duration": 4.321, "text": "derivative so again just remember pi is"}, {"start": 194.9, "duration": 3.36, "text": "a number and that's all I'm doing is"}, {"start": 196.4, "duration": 4.5, "text": "just treating it like an another number"}, {"start": 198.26, "duration": 6.77, "text": "so nothing different still product rule"}, {"start": 200.9, "duration": 4.13, "text": "chain rule type problems"}], "title": "More Derivatives Involving Trigonometric Functions, Ex 1", "description": "Thanks to all of you who support me on Patreon. You da real mvps!   $1 per month helps!! :)  https://www.patreon.com/patrickjmt !! More Derivatives Involving Trigonometric Functions, Ex 1.  Just some more derivative problems involving trigonometric functions.", "lengthSeconds": 204, "width": 480, "height": 360}
{"video_id": "XeI8EOZVwf0", "transcript": [{"start": 1.36, "duration": 2.68, "text": "all right in this video we're just going"}, {"start": 2.639, "duration": 4.76, "text": "to do a little introduction to the"}, {"start": 4.04, "duration": 5.519, "text": "trigonometric functions and uh basically"}, {"start": 7.399, "duration": 5.041, "text": "the trigonometric functions are defined"}, {"start": 9.559, "duration": 4.801, "text": "in terms of a right triangle and the"}, {"start": 12.44, "duration": 4.4, "text": "idea is if you have some little angle"}, {"start": 14.36, "duration": 4.8, "text": "here Theta uh this is the little symbol"}, {"start": 16.84, "duration": 3.72, "text": "Theta kind of across from that that's"}, {"start": 19.16, "duration": 3.119, "text": "what we're going to consider that to be"}, {"start": 20.56, "duration": 4.12, "text": "the opposite side and that's going to"}, {"start": 22.279, "duration": 4.92, "text": "have some length associated with it this"}, {"start": 24.68, "duration": 5.079, "text": "will be the adjacent side at the bottom"}, {"start": 27.199, "duration": 4.88, "text": "and the uh the the longest side is the"}, {"start": 29.759, "duration": 5.001, "text": "high hypotenuse of the right triangle"}, {"start": 32.079, "duration": 4.16, "text": "the way that we Define s of theta is"}, {"start": 34.76, "duration": 3.52, "text": "it's just going to be whatever the"}, {"start": 36.239, "duration": 5.0, "text": "opposite length is over the length of"}, {"start": 38.28, "duration": 5.24, "text": "the hypotenuse cosine of theta is the"}, {"start": 41.239, "duration": 5.401, "text": "adjacent length over the hypotenuse"}, {"start": 43.52, "duration": 6.12, "text": "length tangent of theta is the opposite"}, {"start": 46.64, "duration": 6.04, "text": "length over the adjacent length and then"}, {"start": 49.64, "duration": 5.559, "text": "cosecant is basically one over s which"}, {"start": 52.68, "duration": 4.44, "text": "is going to flip it so cosecant of theta"}, {"start": 55.199, "duration": 3.441, "text": "will be the hypotenuse over the opposite"}, {"start": 57.12, "duration": 5.16, "text": "secant will be the hypotenuse over the"}, {"start": 58.64, "duration": 7.799, "text": "adjacent C will be the adjacent over the"}, {"start": 62.28, "duration": 5.76, "text": "opposite so what do we mean here um so"}, {"start": 66.439, "duration": 4.561, "text": "we've got a little triangle we want to"}, {"start": 68.04, "duration": 6.439, "text": "find s cosine and tangent for this angle"}, {"start": 71.0, "duration": 7.72, "text": "D okay so s of"}, {"start": 74.479, "duration": 7.761, "text": "D we've got cosine of angle d uh s s of"}, {"start": 78.72, "duration": 6.68, "text": "angle D cosine of angle D and tangent of"}, {"start": 82.24, "duration": 5.76, "text": "angle D well it says simply to get the"}, {"start": 85.4, "duration": 4.52, "text": "value of sign so again here's our our"}, {"start": 88.0, "duration": 4.439, "text": "angle D kind of in the bottom left Cor"}, {"start": 89.92, "duration": 5.36, "text": "corner it says sign we just simply take"}, {"start": 92.439, "duration": 7.601, "text": "the opposite length over the hypotenuse"}, {"start": 95.28, "duration": 6.799, "text": "length well the opposite length is six"}, {"start": 100.04, "duration": 5.48, "text": "the length of the hypotenuse goes in the"}, {"start": 102.079, "duration": 5.08, "text": "denominator so that's 2 < TK 13 and"}, {"start": 105.52, "duration": 4.279, "text": "that's that's basically it that's your"}, {"start": 107.159, "duration": 8.041, "text": "answer um you can always reduce this uh"}, {"start": 109.799, "duration": 7.841, "text": "6 / 2 is 3 over the < TK of 13 um you"}, {"start": 115.2, "duration": 5.04, "text": "could rationalize this denominator by"}, {"start": 117.64, "duration": 3.36, "text": "multiplying top and bottom by the s < TK"}, {"start": 120.24, "duration": 3.519, "text": "of"}, {"start": 121.0, "duration": 6.36, "text": "13 and if you do that in the numerator"}, {"start": 123.759, "duration": 5.84, "text": "we be we'll be left with 3 < TK 13 and"}, {"start": 127.36, "duration": 4.68, "text": "then in the denominator Ro < tk3 over <"}, {"start": 129.599, "duration": 5.481, "text": "tk3 will just leave us with"}, {"start": 132.04, "duration": 4.76, "text": "13 okay so sometimes people like you to"}, {"start": 135.08, "duration": 4.76, "text": "rationalize the"}, {"start": 136.8, "duration": 5.12, "text": "denominator okay cosine of this angle D"}, {"start": 139.84, "duration": 3.679, "text": "it says to figure out cosine all we do"}, {"start": 141.92, "duration": 4.44, "text": "is just take the adjacent length over"}, {"start": 143.519, "duration": 5.881, "text": "the hypotenuse length well the adjacent"}, {"start": 146.36, "duration": 6.8, "text": "length length is four the hypotenuse"}, {"start": 149.4, "duration": 7.88, "text": "length is 2 < TK 13 again we could"}, {"start": 153.16, "duration": 6.68, "text": "divide uh 4 over 2 is going to be 2 over"}, {"start": 157.28, "duration": 4.16, "text": "the < TK of 13 and then we could do the"}, {"start": 159.84, "duration": 3.6, "text": "same thing as we did before we could"}, {"start": 161.44, "duration": 6.2, "text": "multiply top and bottom by the square"}, {"start": 163.44, "duration": 5.64, "text": "root of 13 and that would give 2 < TK 13"}, {"start": 167.64, "duration": 5.12, "text": "over"}, {"start": 169.08, "duration": 6.04, "text": "13 okay last but not least tangent it"}, {"start": 172.76, "duration": 6.119, "text": "says we use the opposite side which"}, {"start": 175.12, "duration": 6.8, "text": "would be six over the adjacent side"}, {"start": 178.879, "duration": 6.041, "text": "which is four and 6 over 4 just reduces"}, {"start": 181.92, "duration": 7.0, "text": "to three Hales so we've now got our"}, {"start": 184.92, "duration": 8.28, "text": "answers um s of the angle D is 3 < tk3"}, {"start": 188.92, "duration": 7.44, "text": "over 13 cosine of the angle D is 2K3"}, {"start": 193.2, "duration": 7.88, "text": "over 13 and tangent of the angle D is"}, {"start": 196.36, "duration": 4.72, "text": "simply going to be equal to three Hales"}], "title": "The Trigonometric Functions: The Basics! Example 1", "description": "Thanks to all of you who support me on Patreon. You da real mvps!   $1 per month helps!! :)  https://www.patreon.com/patrickjmt !! The Trigonometric Functions: The Basics! Example 1. In this video, I define all of the trigonometric functions in terms of a right triangle and then do a few specific examples where I find some different trig values.", "lengthSeconds": 199, "width": 480, "height": 360}
{"video_id": "818YQOVzZYQ", "transcript": [{"start": 0.02, "duration": 4.82, "text": "Alright, so just a couple more examples here on multiplication of integers."}, {"start": 5.48, "duration": 4.56, "text": "And here, I'm going to do negative three times four times negative five."}, {"start": 10.36, "duration": 2.46, "text": "And then, five times negative one times eight."}, {"start": 13.14, "duration": 2.6, "text": "So,  again, I'm just going to multiply these pair-wise."}, {"start": 15.74, "duration": 1.86, "text": "So, two at a time."}, {"start": 18.08, "duration": 3.96, "text": "Let's see, I've got a- So if I do negative three times four..."}, {"start": 22.2, "duration": 3.76, "text": "Let's see, I've got a negative times a positive, that's definitely going to give me a negative..."}, {"start": 26.32, "duration": 2.64, "text": "Three times four would be twelve."}, {"start": 29.26, "duration": 5.78, "text": "So negative three times negative four will be negative twelve, and then we'll just drop down our negative five."}, {"start": 35.4, "duration": 4.22, "text": "Well, now we have a negative times a negative which is going to be a positive."}, {"start": 39.88, "duration": 3.8, "text": "And then, twelve times five will be sixty."}, {"start": 43.96, "duration": 3.3, "text": "So, it says our final solution here will be positive sixty."}, {"start": 47.6, "duration": 3.94, "text": "Part B: We've got five times negative one."}, {"start": 51.62, "duration": 2.18, "text": "Well, that's going to give us negative five."}, {"start": 53.9, "duration": 2.96, "text": "We have a positive times a negative so that will be a negative."}, {"start": 57.06, "duration": 4.82, "text": "Five and one is five. Again, we'll just drop down our positive eight."}, {"start": 62.66, "duration": 3.36, "text": "Now we have a negative times a positive, which is a negative."}, {"start": 66.88, "duration": 2.0, "text": "Five times eight is going to be forty."}, {"start": 69.54, "duration": 4.0, "text": "So our solution, five times negative one times eight..."}, {"start": 74.04, "duration": 3.2, "text": "When we multiply those together, we'll be left with negative forty."}, {"start": 77.8, "duration": 3.9, "text": "And, again, you know- It's certainly one of the most common mistakes to make..."}, {"start": 81.92, "duration": 2.46, "text": "Is our signs, you know, positive and negatives."}, {"start": 84.38, "duration": 1.8, "text": "And, it's easy to do, you know."}, {"start": 86.22, "duration": 2.0, "text": "But, obviously, it's a big difference, right?"}, {"start": 88.48, "duration": 3.0, "text": "Negative forty is a much diffrent number than positive forty."}, {"start": 91.56, "duration": 4.7, "text": "So, one thing I always try to be careful about is - you know - it's easy to mix up signs mistakes."}, {"start": 96.68, "duration": 3.44, "text": "So, a lot of times if I have a - you know -  problem on homework..."}, {"start": 100.22, "duration": 2.24, "text": "If I was messing it up, a lot of times that would be the first place I would check."}], "title": "Multiplication / Multiplying Integers", "description": "Thanks to all of you who support me on Patreon. You da real mvps!   $1 per month helps!! :)  https://www.patreon.com/patrickjmt !! Multiplication / Multiplying Integers.  Here I look at a couple of quick examples dealing with multiplication of integers.  The only things we have be really careful of is the sign we get after multiplying.  Otherwise, it is the same as multiplying all positive numbers!", "lengthSeconds": 105, "width": 480, "height": 360}
{"video_id": "vxqps5lO9Tw", "transcript": [{"start": 0.399, "duration": 2.8, "text": "all right so another example here of"}, {"start": 2.399, "duration": 4.48, "text": "writing"}, {"start": 3.199, "duration": 4.961, "text": "a decimal negative 4.145"}, {"start": 6.879, "duration": 3.201, "text": "we're going to write that as a mixed"}, {"start": 8.16, "duration": 5.28, "text": "number in simplest or"}, {"start": 10.08, "duration": 3.36, "text": "the most reduced form so"}, {"start": 14.4, "duration": 4.639, "text": "this one will be a little bit more"}, {"start": 15.599, "duration": 6.401, "text": "obnoxious just simply because"}, {"start": 19.039, "duration": 3.361, "text": "there are more digits after the decimal"}, {"start": 22.0, "duration": 3.359, "text": "place"}, {"start": 22.4, "duration": 4.48, "text": "nothing worse than that okay same idea"}, {"start": 25.359, "duration": 4.721, "text": "let's get rid of the old"}, {"start": 26.88, "duration": 4.96, "text": "decimal uh sign so it looks like we'd"}, {"start": 30.08, "duration": 2.88, "text": "have to move it three times so i'm going"}, {"start": 31.84, "duration": 6.239, "text": "to multiply by"}, {"start": 32.96, "duration": 5.119, "text": "1000 over 1000."}, {"start": 38.16, "duration": 4.16, "text": "all righty let's see so that's going to"}, {"start": 40.96, "duration": 4.0, "text": "leave us with negative four"}, {"start": 42.32, "duration": 5.04, "text": "one four five negative four thousand one"}, {"start": 44.96, "duration": 5.68, "text": "hundred forty five in the numerator"}, {"start": 47.36, "duration": 5.6, "text": "over one thousand"}, {"start": 50.64, "duration": 4.16, "text": "well let's see so again lots of"}, {"start": 52.96, "duration": 5.599, "text": "different ways that we could go about"}, {"start": 54.8, "duration": 5.599, "text": "doing this in my other example i reduced"}, {"start": 58.559, "duration": 3.121, "text": "first here i think i'm going to go ahead"}, {"start": 60.399, "duration": 3.601, "text": "and make it into a mixed"}, {"start": 61.68, "duration": 2.32, "text": "number"}, {"start": 65.119, "duration": 4.561, "text": "and the reason why i'm going to do that"}, {"start": 68.0, "duration": 3.119, "text": "well i'll show you"}, {"start": 69.68, "duration": 3.2, "text": "i mean we could immediately i know that"}, {"start": 71.119, "duration": 3.521, "text": "five goes into the top and the bottom i"}, {"start": 72.88, "duration": 2.879, "text": "could start dividing by five again there"}, {"start": 74.64, "duration": 3.76, "text": "could be some factor"}, {"start": 75.759, "duration": 4.881, "text": "larger than five that works but i know"}, {"start": 78.4, "duration": 4.64, "text": "for sure five works and that can help me"}, {"start": 80.64, "duration": 4.159, "text": "make the numbers smaller another way i'm"}, {"start": 83.04, "duration": 3.6, "text": "recognizing to kind of make you know the"}, {"start": 84.799, "duration": 3.201, "text": "fraction sort of smaller"}, {"start": 86.64, "duration": 3.36, "text": "well again i know we have a negative"}, {"start": 88.0, "duration": 5.52, "text": "over a positive so it's a negative"}, {"start": 90.0, "duration": 6.0, "text": "a thousand would go into 4145 exactly"}, {"start": 93.52, "duration": 5.52, "text": "four times without going over"}, {"start": 96.0, "duration": 5.759, "text": "well we would have 145"}, {"start": 99.04, "duration": 4.96, "text": "over a thousand left so as a mixed"}, {"start": 101.759, "duration": 6.801, "text": "number we can write this fraction"}, {"start": 104.0, "duration": 6.399, "text": "as negative four and 145 over a thousand"}, {"start": 108.56, "duration": 4.4, "text": "well again i know that this fraction"}, {"start": 110.399, "duration": 4.801, "text": "reduces 145 over a thousand"}, {"start": 112.96, "duration": 4.08, "text": "but again it's easier to me to reduce"}, {"start": 115.2, "duration": 4.32, "text": "it's easier to me to reduce"}, {"start": 117.04, "duration": 4.0, "text": "than this original thing so again kind"}, {"start": 119.52, "duration": 2.959, "text": "of you know play with it it doesn't"}, {"start": 121.04, "duration": 3.119, "text": "matter how you do it as long as you"}, {"start": 122.479, "duration": 4.801, "text": "simplify eventually you'll"}, {"start": 124.159, "duration": 5.52, "text": "get the same answer so kind of off to"}, {"start": 127.28, "duration": 5.28, "text": "the side i'm just going to reduce 145"}, {"start": 129.679, "duration": 2.881, "text": "over a thousand"}, {"start": 133.2, "duration": 4.08, "text": "again i know that 5 will go into both of"}, {"start": 135.12, "duration": 4.88, "text": "these numbers"}, {"start": 137.28, "duration": 5.76, "text": "so let's see 5 times 2 is 10 again times"}, {"start": 140.0, "duration": 4.16, "text": "20 would be a hundred times 200 will"}, {"start": 143.04, "duration": 4.88, "text": "give us a thousand"}, {"start": 144.16, "duration": 6.64, "text": "in the denominator let's see five times"}, {"start": 147.92, "duration": 3.92, "text": "30 would be 150 but that's kind of one"}, {"start": 150.8, "duration": 3.519, "text": "multiple"}, {"start": 151.84, "duration": 4.24, "text": "too big so instead of 30 we'll need one"}, {"start": 154.319, "duration": 4.721, "text": "smaller which i think would be"}, {"start": 156.08, "duration": 4.64, "text": "29 all right so again we can just cancel"}, {"start": 159.04, "duration": 4.32, "text": "out our fives"}, {"start": 160.72, "duration": 3.76, "text": "so 145 over a thousand we can write that"}, {"start": 163.36, "duration": 5.04, "text": "as 29"}, {"start": 164.48, "duration": 5.52, "text": "over 200 again now you could maybe think"}, {"start": 168.4, "duration": 3.52, "text": "about factors uh let's see"}, {"start": 170.0, "duration": 3.68, "text": "anything in common that will reduce well"}, {"start": 171.92, "duration": 5.599, "text": "29 is a prime number"}, {"start": 173.68, "duration": 6.639, "text": "so again this will be a reduced fraction"}, {"start": 177.519, "duration": 4.481, "text": "so we can write our final mixed number"}, {"start": 180.319, "duration": 5.521, "text": "as negative four"}, {"start": 182.0, "duration": 3.84, "text": "and 29 over"}, {"start": 187.4, "duration": 3.0, "text": "200."}], "title": "Writing a Decimal as a Fraction, Example 4", "description": "Thanks to all of you who support me on Patreon. You da real mvps!   $1 per month helps!! :)  https://www.patreon.com/patrickjmt !! Writing a Decimal as a Fraction, Example 4", "lengthSeconds": 189, "width": 480, "height": 360}
{"video_id": "bAgCeNUvu0E", "transcript": [{"start": 0.88, "duration": 5.24, "text": "okay so here we're just going to look at"}, {"start": 2.08, "duration": 6.08, "text": "a little review of uh some series"}, {"start": 6.12, "duration": 3.8, "text": "notation and in this case what we're"}, {"start": 8.16, "duration": 6.28, "text": "going to do is we're going to sum up"}, {"start": 9.92, "duration": 7.88, "text": "this finite series from i = 1: 4 of -1"}, {"start": 14.44, "duration": 6.16, "text": "to the I * I sared so again all the"}, {"start": 17.8, "duration": 4.96, "text": "summation notation says you know"}, {"start": 20.6, "duration": 4.16, "text": "wherever the the bottom number starts"}, {"start": 22.76, "duration": 5.279, "text": "everywhere you know there's an i we plug"}, {"start": 24.76, "duration": 6.919, "text": "in that number so we'll have -1 to the 1"}, {"start": 28.039, "duration": 5.121, "text": "* 1^ 2 then we just put our plus sign"}, {"start": 31.679, "duration": 5.281, "text": "and then we just increase to the next"}, {"start": 33.16, "duration": 7.04, "text": "whole number so positive2 so then -1"}, {"start": 36.96, "duration": 5.88, "text": "sared and then we're going to have 2^"}, {"start": 40.2, "duration": 6.039, "text": "squared and then again we increase to"}, {"start": 42.84, "duration": 7.16, "text": "another uh the next whole number so1 to"}, {"start": 46.239, "duration": 5.601, "text": "the 3 then we'll have 3^ SAR and then"}, {"start": 50.0, "duration": 3.48, "text": "another plus sign for again our"}, {"start": 51.84, "duration": 3.64, "text": "summation notation and then the last"}, {"start": 53.48, "duration": 4.0, "text": "thing you do uh you plug in the last"}, {"start": 55.48, "duration": 6.44, "text": "number that's like your stopping point"}, {"start": 57.48, "duration": 4.44, "text": "so we'll have1 to 4th"}, {"start": 62.16, "duration": 5.56, "text": "time 4^ 2 and now this is what we have"}, {"start": 65.119, "duration": 5.921, "text": "to evaluate so 1 s is 1 it looks like we"}, {"start": 67.72, "duration": 6.28, "text": "get negative 1 um we'll have 2^2 which"}, {"start": 71.04, "duration": 5.439, "text": "is four and it'll stay positive then it"}, {"start": 74.0, "duration": 7.119, "text": "looks like we'll have I guess"}, {"start": 76.479, "duration": 9.441, "text": "A9 and then it looks like a positive 16"}, {"start": 81.119, "duration": 8.121, "text": "so let's see um so negative 1 + 4 that's"}, {"start": 85.92, "duration": 6.44, "text": "three uh 9 and positive 16 that's going"}, {"start": 89.24, "duration": 5.36, "text": "to leave us POS 7 so to me it looks like"}, {"start": 92.36, "duration": 4.0, "text": "our finite sum in this case is just"}, {"start": 94.6, "duration": 4.76, "text": "going to be"}, {"start": 96.36, "duration": 3.0, "text": "positive1"}], "title": "Intro to Summation Notation and Infinite Series, Ex 1", "description": "Thanks to all of you who support me on Patreon. You da real mvps!   $1 per month helps!! :)  https://www.patreon.com/patrickjmt !! Intro to Summation Notation and Infinite Series, Ex 1. Forgot about summation notation? Here is a little refresher example! Notice the ' i ' has NOTHING to do with complex numbers. : )", "lengthSeconds": 97, "width": 480, "height": 360}
{"video_id": "QXWfVurC5vk", "transcript": [{"start": 0.68, "duration": 4.76, "text": "all right so here a little word problem"}, {"start": 2.159, "duration": 6.681, "text": "to go with our uh integer examples so"}, {"start": 5.44, "duration": 6.079, "text": "suppose it's uh -4\u00b0 at 600 a.m. at your"}, {"start": 8.84, "duration": 5.2, "text": "home so nice and nice and chilly and"}, {"start": 11.519, "duration": 6.041, "text": "then let's suppose uh it goes from -4"}, {"start": 14.04, "duration": 8.68, "text": "degrees and the temperature increases"}, {"start": 17.56, "duration": 6.639, "text": "20\u00b0 but then by 6: p.m. it falls by 14\u00b0"}, {"start": 22.72, "duration": 3.04, "text": "so kind of makes sense right it's cold"}, {"start": 24.199, "duration": 4.121, "text": "in the morning gets warmer and then it"}, {"start": 25.76, "duration": 4.839, "text": "cools off towards the end of the day we"}, {"start": 28.32, "duration": 5.36, "text": "want to know what's the temperature at 6"}, {"start": 30.599, "duration": 5.201, "text": "p.m. so let's see it was -4\u00b0 at the"}, {"start": 33.68, "duration": 5.0, "text": "start of the day or at least at 6:00"}, {"start": 35.8, "duration": 5.56, "text": "a.m. um and then well we would have to"}, {"start": 38.68, "duration": 4.32, "text": "add 20 because we're going to increase"}, {"start": 41.36, "duration": 4.08, "text": "the temperature by"}, {"start": 43.0, "duration": 6.64, "text": "20\u00b0 but then we would have to subtract"}, {"start": 45.44, "duration": 7.119, "text": "away 14\u00b0 because well the temperature"}, {"start": 49.64, "duration": 6.439, "text": "Falls so what I'm going to do here um"}, {"start": 52.559, "duration": 6.041, "text": "we've got -4 + 20 - 14 I'm just going to"}, {"start": 56.079, "duration": 6.721, "text": "combine my first two numbers here -4 and"}, {"start": 58.6, "duration": 6.559, "text": "positive 20 and again um these are uh"}, {"start": 62.8, "duration": 7.28, "text": "opposite signs equivalently you can"}, {"start": 65.159, "duration": 6.521, "text": "think about this as being positive 20 -"}, {"start": 70.08, "duration": 5.24, "text": "4"}, {"start": 71.68, "duration": 5.4, "text": "-4 so again remember our uh these are"}, {"start": 75.32, "duration": 3.24, "text": "commutative we can just switch the order"}, {"start": 77.08, "duration": 3.8, "text": "on these and we'll still get the same"}, {"start": 78.56, "duration": 5.76, "text": "thing so I'm going to use that little"}, {"start": 80.88, "duration": 5.239, "text": "property so 20 minus 4 that's the same"}, {"start": 84.32, "duration": 4.2, "text": "thing as"}, {"start": 86.119, "duration": 6.401, "text": "16 then we'll have to still subtract"}, {"start": 88.52, "duration": 7.2, "text": "away the 14 well 16 minus 14 is going to"}, {"start": 92.52, "duration": 5.08, "text": "leave us with two so it says at 6: p.m."}, {"start": 95.72, "duration": 6.12, "text": "it looks like the temperature is going"}, {"start": 97.6, "duration": 7.36, "text": "to be uh 2\u00b0 C so a little bit warmer"}, {"start": 101.84, "duration": 4.8, "text": "than it was at 6: a.m. but uh still not"}, {"start": 104.96, "duration": 3.479, "text": "weather you'd probably want to be at the"}, {"start": 106.64, "duration": 3.0, "text": "uh you know if it was two degrees"}, {"start": 108.439, "duration": 4.441, "text": "Celsius you probably wouldn't want to be"}, {"start": 109.64, "duration": 7.159, "text": "at the beach right so okay so again it"}, {"start": 112.88, "duration": 6.68, "text": "increases you know so from -4 degrees if"}, {"start": 116.799, "duration": 7.801, "text": "we add 20\u00b0 that means you know at 3 p.m."}, {"start": 119.56, "duration": 10.159, "text": "it's 16\u00b0 C but then it drops by 14\u00b0 so"}, {"start": 124.6, "duration": 5.119, "text": "that's going to leave us with 2\u00b0 C total"}], "title": "Subtraction / Subtracting Integers - Word Problem", "description": "Thanks to all of you who support me on Patreon. You da real mvps!   $1 per month helps!! :)  https://www.patreon.com/patrickjmt !! Subtraction / Subtracting Integers - Word Problem.\nHere I look at a little word problem dealing with adding and subtracting integers.", "lengthSeconds": 128, "width": 480, "height": 360}
{"video_id": "hg3_XgKUzU4", "transcript": [{"start": 0.8, "duration": 3.44, "text": "all right here we're just going to do a"}, {"start": 2.24, "duration": 4.559, "text": "few more examples related to the power"}, {"start": 4.24, "duration": 4.48, "text": "rule so um just a couple different"}, {"start": 6.799, "duration": 3.601, "text": "derivatives so i think most of the"}, {"start": 8.72, "duration": 4.799, "text": "derivatives we saw in the other examples"}, {"start": 10.4, "duration": 3.119, "text": "were kind of like part a"}, {"start": 13.679, "duration": 4.321, "text": "and in part a we've got x to the 11th"}, {"start": 15.839, "duration": 4.321, "text": "well when we take the derivative"}, {"start": 18.0, "duration": 2.88, "text": "again that number just comes out front"}, {"start": 20.16, "duration": 3.039, "text": "so"}, {"start": 20.88, "duration": 3.44, "text": "we'll get 11 times x and then we"}, {"start": 23.199, "duration": 4.16, "text": "subtract"}, {"start": 24.32, "duration": 5.6, "text": "1 from the exponent well 11 minus 1"}, {"start": 27.359, "duration": 4.16, "text": "would just give us 10 and hey we get 11"}, {"start": 29.92, "duration": 4.159, "text": "times x to the 10th that's our"}, {"start": 31.519, "duration": 4.72, "text": "derivative and that's all we have to do"}, {"start": 34.079, "duration": 3.841, "text": "for part a"}, {"start": 36.239, "duration": 3.84, "text": "this is true you know it doesn't have to"}, {"start": 37.92, "duration": 5.12, "text": "be a nice whole number"}, {"start": 40.079, "duration": 4.121, "text": "so in part b we have g of x equals x"}, {"start": 43.04, "duration": 3.359, "text": "raised to the"}, {"start": 44.2, "duration": 4.28, "text": "2.137 well"}, {"start": 46.399, "duration": 4.241, "text": "same rule still applies so if we take"}, {"start": 48.48, "duration": 3.84, "text": "the derivative"}, {"start": 50.64, "duration": 4.28, "text": "whatever the exponent is that's what"}, {"start": 52.32, "duration": 6.559, "text": "comes out front so we have"}, {"start": 54.92, "duration": 7.959, "text": "2.137 times x and then we subtract 1"}, {"start": 58.879, "duration": 7.441, "text": "from the exponent well 2.137 minus 1"}, {"start": 62.879, "duration": 6.001, "text": "would just give us 1.137"}, {"start": 66.32, "duration": 4.4, "text": "and there's our derivative so same idea"}, {"start": 68.88, "duration": 4.239, "text": "hey you got decimals"}, {"start": 70.72, "duration": 5.2, "text": "to me i say who cares you know you still"}, {"start": 73.119, "duration": 5.521, "text": "do the exact same thing"}, {"start": 75.92, "duration": 4.64, "text": "so for part c we have h of x equals x"}, {"start": 78.64, "duration": 3.519, "text": "raised to the square root of five so"}, {"start": 80.56, "duration": 3.44, "text": "again when i look at this i say well"}, {"start": 82.159, "duration": 4.32, "text": "i've got a variable to a number the"}, {"start": 84.0, "duration": 5.52, "text": "exact same rule"}, {"start": 86.479, "duration": 5.441, "text": "gets used as before so the square root"}, {"start": 89.52, "duration": 4.559, "text": "of 5 will come out front"}, {"start": 91.92, "duration": 5.76, "text": "and then we have x raised to the new"}, {"start": 94.079, "duration": 4.881, "text": "power well you know it's kind of"}, {"start": 97.68, "duration": 3.04, "text": "you know in the others you can take one"}, {"start": 98.96, "duration": 4.4, "text": "away and kind of get a nice new clean"}, {"start": 100.72, "duration": 4.24, "text": "number um you know and in this case well"}, {"start": 103.36, "duration": 4.079, "text": "the new exponent would just be square"}, {"start": 104.96, "duration": 4.4, "text": "root of five minus one and i would leave"}, {"start": 107.439, "duration": 3.441, "text": "it like that you know i can't really you"}, {"start": 109.36, "duration": 3.84, "text": "could always approximate this with a"}, {"start": 110.88, "duration": 4.64, "text": "calculator but then you're doing exactly"}, {"start": 113.2, "duration": 4.48, "text": "that you're approximating so if you want"}, {"start": 115.52, "duration": 4.32, "text": "to keep the exponent exactly like it"}, {"start": 117.68, "duration": 4.479, "text": "should be we can just write it again"}, {"start": 119.84, "duration": 4.48, "text": "just a square root of five minus one and"}, {"start": 122.159, "duration": 4.32, "text": "that's all there is to it so sometimes"}, {"start": 124.32, "duration": 3.28, "text": "people get thrown off by the uh you know"}, {"start": 126.479, "duration": 3.2, "text": "the"}, {"start": 127.6, "duration": 3.6, "text": "not whole number exponents i guess but"}, {"start": 129.679, "duration": 3.121, "text": "again"}, {"start": 131.2, "duration": 4.0, "text": "it's still the same type of function you"}, {"start": 132.8, "duration": 4.799, "text": "still just have a variable to a number"}, {"start": 135.2, "duration": 5.56, "text": "so this power rule still applies just"}, {"start": 137.599, "duration": 3.161, "text": "like before"}], "title": "Calculus : Basic Power Rule Examples", "description": "Thanks to all of you who support me on Patreon. You da real mvps!   $1 per month helps!! :)  https://www.patreon.com/patrickjmt !! Calculus : Basic Power Rule Examples.\nDisregard the intro about the proof; these are videos that are licensed to a company so that title goes with their chapter!", "lengthSeconds": 139, "width": 480, "height": 360}
{"video_id": "ok2dQiL09_0", "transcript": [{"start": 0.269, "duration": 3.63, "text": "all right here we're going to start"}, {"start": 2.04, "duration": 4.759, "text": "looking at some examples related to the"}, {"start": 3.899, "duration": 5.071, "text": "chain rule and the chain rule says"}, {"start": 6.799, "duration": 5.8, "text": "basically when you have a composite a"}, {"start": 8.97, "duration": 5.31, "text": "composition of functions F of G of X it"}, {"start": 12.599, "duration": 4.35, "text": "says to take the derivative of a"}, {"start": 14.28, "duration": 4.74, "text": "function like this you take the"}, {"start": 16.949, "duration": 4.561, "text": "derivative of the outside function and"}, {"start": 19.02, "duration": 5.16, "text": "you leave the inside when alone and then"}, {"start": 21.51, "duration": 7.89, "text": "the chain rule part you multiply by the"}, {"start": 24.18, "duration": 8.94, "text": "derivative of the inside part so that's"}, {"start": 29.4, "duration": 6.05, "text": "the chain rule so this is probably"}, {"start": 33.12, "duration": 4.8, "text": "certainly I would say the most used"}, {"start": 35.45, "duration": 3.82, "text": "differentiation rule around it just"}, {"start": 37.92, "duration": 2.79, "text": "seems like every problem you use the"}, {"start": 39.27, "duration": 4.68, "text": "chain rule on it at some point"}, {"start": 40.71, "duration": 5.88, "text": "so not literally every problem but a lot"}, {"start": 43.95, "duration": 5.67, "text": "of problems so let's look at a few"}, {"start": 46.59, "duration": 5.58, "text": "different examples here so we've got f"}, {"start": 49.62, "duration": 5.34, "text": "of X equals 6 times X cubed plus 2x"}, {"start": 52.17, "duration": 9.529, "text": "raised to the 100 power let's take the"}, {"start": 54.96, "duration": 9.15, "text": "derivative of that one first alright so"}, {"start": 61.699, "duration": 6.46, "text": "so it's kind of like before you can kind"}, {"start": 64.11, "duration": 5.97, "text": "of a the 6 when we take the derivative"}, {"start": 68.159, "duration": 3.721, "text": "is just going to come along and then we"}, {"start": 70.08, "duration": 4.649, "text": "can multiply by the derivative at the"}, {"start": 71.88, "duration": 5.55, "text": "end so to me the way I think about it is"}, {"start": 74.729, "duration": 4.951, "text": "I kind of start from the most outside so"}, {"start": 77.43, "duration": 4.829, "text": "to me the thing on the most out side is"}, {"start": 79.68, "duration": 5.64, "text": "this exponent of 100 and I think if I"}, {"start": 82.259, "duration": 7.711, "text": "just had like X to the 100 power I would"}, {"start": 85.32, "duration": 7.32, "text": "get well 100 X to the 99 power well we"}, {"start": 89.97, "duration": 7.259, "text": "don't just have X but we just leave all"}, {"start": 92.64, "duration": 6.36, "text": "of that part alone okay so the 100 will"}, {"start": 97.229, "duration": 4.021, "text": "come out front we'll leave all the stuff"}, {"start": 99.0, "duration": 6.06, "text": "on the inside alone we'll take 1 away"}, {"start": 101.25, "duration": 5.97, "text": "from the exponent and now we have to"}, {"start": 105.06, "duration": 5.849, "text": "take the derivative of the inside part"}, {"start": 107.22, "duration": 5.399, "text": "so again that's kind of with a chain"}, {"start": 110.909, "duration": 4.081, "text": "rule saying here it says whatever sort"}, {"start": 112.619, "duration": 6.54, "text": "of on the most inside at the end we take"}, {"start": 114.99, "duration": 7.589, "text": "the derivative of that stuff so the most"}, {"start": 119.159, "duration": 6.06, "text": "inside part to me is the X cubed plus 2x"}, {"start": 122.579, "duration": 6.061, "text": "so when we take the derivative there the"}, {"start": 125.219, "duration": 5.52, "text": "X to the 3rd will give us 3x squared and"}, {"start": 128.64, "duration": 3.79, "text": "then the derivative of positive 2x will"}, {"start": 130.739, "duration": 5.86, "text": "just be positive 2"}, {"start": 132.43, "duration": 6.419, "text": "and that's it that's now our derivative"}, {"start": 136.599, "duration": 2.761, "text": "so again you can clean this up a little"}, {"start": 138.849, "duration": 4.711, "text": "bit"}, {"start": 139.36, "duration": 7.65, "text": "six times 100 is 600 I'm going to write"}, {"start": 143.56, "duration": 7.41, "text": "the 3x squared plus 2 part next and then"}, {"start": 147.01, "duration": 8.42, "text": "the X to the third plus 2x race to the"}, {"start": 150.97, "duration": 4.46, "text": "99th power will be left over"}], "title": "\u2756 More Chain Rule Examples \u2756", "description": "\ud83d\udcda Understanding the Chain Rule: A Basic Example \ud83e\uddee\n\nIn this video, we explore the Chain Rule, one of the fundamental techniques in calculus for finding derivatives of composite functions. This tutorial is designed for beginners and provides a clear, step-by-step approach to using the Chain Rule in a simple example.\n\n\ud83d\ude80 What you\u2019ll learn:\n\nThe definition and purpose of the Chain Rule in calculus.\nHow to identify composite functions and when to apply the Chain Rule.\nA detailed walkthrough of a basic example, highlighting each step in the differentiation process.\nTips for simplifying your derivative and understanding the relationship between the functions involved.\nWhether you\u2019re new to calculus or brushing up on your skills, this video will help you gain confidence in using the Chain Rule effectively.\n\nBe sure to check out my other videos for additional insights into calculus concepts, including implicit differentiation, product rule, and more!\n\n\ud83d\udc4d Don\u2019t forget to like, subscribe, and hit the notification bell for more math tutorials! \ud83d\udd14\u2728\n\nWatch now to master the basics of the Chain Rule and enhance your calculus skills!\n\nHashtags:\n#ChainRule #Calculus #Derivatives #MathTutorial #LearningMath #BasicCalculus #EducationalContent #STEM #Mathematics #CalculusExamples #MathSkills #APCalculus #DerivativeTechniques", "lengthSeconds": 156, "width": 480, "height": 360}
{"video_id": "LawqeP1u7Fw", "transcript": [{"start": 1.04, "duration": 4.44, "text": "all right so just a little word problem"}, {"start": 2.639, "duration": 5.361, "text": "here involving fractions so suppose to"}, {"start": 5.48, "duration": 4.68, "text": "make a one pound of cookies to make a"}, {"start": 8.0, "duration": 4.799, "text": "pound of cookies one of the ingredients"}, {"start": 10.16, "duration": 5.76, "text": "that you need is 3 and 1 12 ounces of"}, {"start": 12.799, "duration": 4.841, "text": "sugar so suppose we want to make uh not"}, {"start": 15.92, "duration": 3.8, "text": "so many cookies we only want to make a"}, {"start": 17.64, "duration": 5.6, "text": "four of a pound of cookies and we want"}, {"start": 19.72, "duration": 5.52, "text": "to know how much sugar should we use so"}, {"start": 23.24, "duration": 3.039, "text": "we have to kind of scale things down a"}, {"start": 25.24, "duration": 4.08, "text": "little"}, {"start": 26.279, "duration": 5.241, "text": "bit well the idea is what we're going to"}, {"start": 29.32, "duration": 4.919, "text": "need in terms of"}, {"start": 31.52, "duration": 5.68, "text": "sugar you know so for one pound of"}, {"start": 34.239, "duration": 5.961, "text": "cookies we needed 3 and2 oz well now"}, {"start": 37.2, "duration": 3.0, "text": "we're only going to need"}, {"start": 40.879, "duration": 5.32, "text": "1/4 we're only going to need 1/4 of the"}, {"start": 43.64, "duration": 7.64, "text": "original amount of sugar so we're going"}, {"start": 46.199, "duration": 5.081, "text": "to need a fourth of the 3 and 1 half"}, {"start": 52.76, "duration": 6.959, "text": "ounces okay so we need uh a quarter of"}, {"start": 57.0, "duration": 6.0, "text": "the three and A2 so anytime you see the"}, {"start": 59.719, "duration": 4.921, "text": "word of in a problem that almost always"}, {"start": 63.0, "duration": 2.56, "text": "uh stands for multiplication and that's"}, {"start": 64.64, "duration": 4.6, "text": "what we'll have to do we're going to"}, {"start": 65.56, "duration": 5.68, "text": "have to take 1/4 of 3 and a half ounces"}, {"start": 69.24, "duration": 4.16, "text": "okay so one quarter so again we've seen"}, {"start": 71.24, "duration": 6.32, "text": "how to make uh mixed numbers into"}, {"start": 73.4, "duration": 8.0, "text": "fractions so 3 * 2 is 6 plus one is"}, {"start": 77.56, "duration": 6.4, "text": "7 over two we leave the same denominator"}, {"start": 81.4, "duration": 5.759, "text": "and now we just multiply there's no uh"}, {"start": 83.96, "duration": 9.519, "text": "no common factors to get rid of well 1 *"}, {"start": 87.159, "duration": 10.081, "text": "7 is 7 uh 4 * 2 is 8 so let's see uh"}, {"start": 93.479, "duration": 5.441, "text": "before we needed uh 3 and2 o so now"}, {"start": 97.24, "duration": 5.96, "text": "we'll need"}, {"start": 98.92, "duration": 4.28, "text": "78 um of an"}, {"start": 104.52, "duration": 5.84, "text": "ounce okay so we'll need a little bit"}, {"start": 107.2, "duration": 8.599, "text": "less than 1 ounce of sugar uh to make"}, {"start": 110.36, "duration": 5.439, "text": "the this this quarter pound of cookies"}], "title": "Multiplying Fractions and Mixed Numbers, Example 4", "description": "Thanks to all of you who support me on Patreon. You da real mvps!   $1 per month helps!! :)  https://www.patreon.com/patrickjmt !! Multiplying Fractions and Mixed Numbers, Example 4", "lengthSeconds": 114, "width": 480, "height": 360}
{"video_id": "uKTAh6hNqNs", "transcript": [{"start": 0.439, "duration": 4.781, "text": "alright so here another example we're"}, {"start": 3.0, "duration": 5.04, "text": "going to find an angle given the value"}, {"start": 5.22, "duration": 4.23, "text": "of a trig function and in this case"}, {"start": 8.04, "duration": 5.07, "text": "suppose we know the tangent of theta"}, {"start": 9.45, "duration": 5.189, "text": "equals 1 and if fate is again an acute"}, {"start": 13.11, "duration": 5.009, "text": "angle in a right triangle we want to"}, {"start": 14.639, "duration": 7.411, "text": "figure out what the angle theta is equal"}, {"start": 18.119, "duration": 7.341, "text": "to so again I'm going to put theta right"}, {"start": 22.05, "duration": 7.229, "text": "down there so remember tangent of theta"}, {"start": 25.46, "duration": 6.58, "text": "that's the opposite the the the ratio of"}, {"start": 29.279, "duration": 4.701, "text": "the length of the opposite side to the"}, {"start": 32.04, "duration": 5.55, "text": "ratio of the length of the adjacent side"}, {"start": 33.98, "duration": 5.59, "text": "so the opposite side here you know you"}, {"start": 37.59, "duration": 3.87, "text": "can think about since we know tangent"}, {"start": 39.57, "duration": 4.89, "text": "theta equals 1 we can just write that as"}, {"start": 41.46, "duration": 5.46, "text": "1 over 1 so I'm simply going to make the"}, {"start": 44.46, "duration": 6.3, "text": "opposite side have length 1 and the"}, {"start": 46.92, "duration": 5.549, "text": "adjacent side also have length 1 but at"}, {"start": 50.76, "duration": 3.959, "text": "this point we really actually have"}, {"start": 52.469, "duration": 7.381, "text": "enough information already to figure out"}, {"start": 54.719, "duration": 6.93, "text": "theta okay so the idea is if you know if"}, {"start": 59.85, "duration": 5.459, "text": "this is let's maybe think about degrees"}, {"start": 61.649, "duration": 7.351, "text": "if this is theta degrees so opposite the"}, {"start": 65.309, "duration": 6.121, "text": "side theta degrees it has length 1 well"}, {"start": 69.0, "duration": 4.68, "text": "that means since this side has length 1"}, {"start": 71.43, "duration": 6.09, "text": "that means this angle up here would also"}, {"start": 73.68, "duration": 7.95, "text": "have to be equal to theta well if we"}, {"start": 77.52, "duration": 6.15, "text": "take theta plus theta well we know that"}, {"start": 81.63, "duration": 5.279, "text": "this other angle by default that has to"}, {"start": 83.67, "duration": 5.339, "text": "equal 90 degrees and we know if we add"}, {"start": 86.909, "duration": 6.511, "text": "up all the angles inside of a triangle"}, {"start": 89.009, "duration": 7.051, "text": "that equals 180 degrees well one theta"}, {"start": 93.42, "duration": 6.51, "text": "plus 1 theta is going to give us 2 theta"}, {"start": 96.06, "duration": 6.18, "text": "we can also just simply subtract 90"}, {"start": 99.93, "duration": 4.79, "text": "degrees from both sides so if we"}, {"start": 102.24, "duration": 6.54, "text": "subtract 90 degrees we'll get 2 theta"}, {"start": 104.72, "duration": 5.939, "text": "equals 90 degrees and now if we simply"}, {"start": 108.78, "duration": 8.659, "text": "divide both sides by 2"}, {"start": 110.659, "duration": 6.78, "text": "we'll get that theta equals 45 degree"}], "title": "Finding an Angle Given the Value of a Trigonometric Function - Example 2", "description": "Thanks to all of you who support me on Patreon. You da real mvps!   $1 per month helps!! :)  https://www.patreon.com/patrickjmt !! Finding an Angle Given the Value of a Trigonometric Function - Example 2.  In this video, I deduce the angle associated with a given trigonometric value.", "lengthSeconds": 118, "width": 480, "height": 360}
{"video_id": "62Lj-g5owgo", "transcript": [{"start": 1.12, "duration": 3.28, "text": "okay in this video I'm going to look at"}, {"start": 2.52, "duration": 4.6, "text": "finding the average value of a function"}, {"start": 4.4, "duration": 4.68, "text": "on an interval and as far as stuff that"}, {"start": 7.12, "duration": 3.68, "text": "you see dealing with integration and"}, {"start": 9.08, "duration": 3.76, "text": "calculus course I would say this is"}, {"start": 10.8, "duration": 4.52, "text": "probably one of the more straightforward"}, {"start": 12.84, "duration": 5.359, "text": "things to do here so the formula to find"}, {"start": 15.32, "duration": 5.48, "text": "the average value of a function f ofx on"}, {"start": 18.199, "duration": 5.361, "text": "the interval A to B what we do is we"}, {"start": 20.8, "duration": 4.76, "text": "just take 1 over B minus a so one over"}, {"start": 23.56, "duration": 4.28, "text": "the length of the interval and then we"}, {"start": 25.56, "duration": 5.6, "text": "multiply that by the integral from A to"}, {"start": 27.84, "duration": 5.8, "text": "B of f ofx DX so typically when you see"}, {"start": 31.16, "duration": 5.16, "text": "these problems at least when I saw them"}, {"start": 33.64, "duration": 4.84, "text": "there really wasn't a lot more to doing"}, {"start": 36.32, "duration": 4.68, "text": "these problems and just using that"}, {"start": 38.48, "duration": 4.28, "text": "formula so just a quick example here so"}, {"start": 41.0, "duration": 6.079, "text": "we're going to find the average value of"}, {"start": 42.76, "duration": 6.479, "text": "the function f ofx = x^2 + x + 1 on the"}, {"start": 47.079, "duration": 4.721, "text": "interval from 1 to"}, {"start": 49.239, "duration": 5.64, "text": "3 so a lot of times they just write it"}, {"start": 51.8, "duration": 5.84, "text": "abbreviated as FV so you know obviously"}, {"start": 54.879, "duration": 6.64, "text": "the average value of F and what we do is"}, {"start": 57.64, "duration": 8.0, "text": "we just take one over well and this"}, {"start": 61.519, "duration": 7.28, "text": "case a our a value is going to be one"}, {"start": 65.64, "duration": 6.08, "text": "our B value is simply going to be three"}, {"start": 68.799, "duration": 6.281, "text": "so we'll take one over the difference 3"}, {"start": 71.72, "duration": 7.88, "text": "minus 1 and then we integrate from 1 to"}, {"start": 75.08, "duration": 7.719, "text": "3 of our function so we've got x^2 + x +"}, {"start": 79.6, "duration": 6.199, "text": "1 DX and that's now all we have to"}, {"start": 82.799, "duration": 3.0, "text": "calculate here"}, {"start": 87.36, "duration": 8.2, "text": "so well 1 over 3 - 1 that's just 12"}, {"start": 92.24, "duration": 6.159, "text": "let's integrate the function x^2 + x + 1"}, {"start": 95.56, "duration": 5.76, "text": "so the anti-derivative of x^2 well we'll"}, {"start": 98.399, "duration": 6.201, "text": "get x to the 3 over 3 the"}, {"start": 101.32, "duration": 6.079, "text": "anti-derivative of X is simply x^2 over"}, {"start": 104.6, "duration": 5.119, "text": "2 and then the anti-derivative of one"}, {"start": 107.399, "duration": 5.0, "text": "will simply be X and again we have to"}, {"start": 109.719, "duration": 4.961, "text": "evaluate this from 1 to"}, {"start": 112.399, "duration": 5.0, "text": "3 okay so now it's just sort of plugging"}, {"start": 114.68, "duration": 8.439, "text": "and chugging so if we substitute 3n"}, {"start": 117.399, "duration": 7.601, "text": "we'll have 3 3r / 3 which is 3^ 2 or 9"}, {"start": 123.119, "duration": 4.321, "text": "plus again if we substitute 3 in we'll"}, {"start": 125.0, "duration": 5.44, "text": "get 3^ 2 or 9 /"}, {"start": 127.44, "duration": 5.519, "text": "2 then we'll add to that three so"}, {"start": 130.44, "duration": 4.72, "text": "there's our upper limit of integration"}, {"start": 132.959, "duration": 4.401, "text": "minus let's just substitute in the lower"}, {"start": 135.16, "duration": 6.56, "text": "limit so if we plug in ones we'll just"}, {"start": 137.36, "duration": 8.92, "text": "have 1/3 + 12 +"}, {"start": 141.72, "duration": 7.96, "text": "1 and again now it's just a matter of"}, {"start": 146.28, "duration": 7.12, "text": "simplification so let's see um if we"}, {"start": 149.68, "duration": 7.52, "text": "take 9 + 3 that's 12 and if we subtract"}, {"start": 153.4, "duration": 7.4, "text": "1 12 - 1 is going to leave us with 11 so"}, {"start": 157.2, "duration": 6.6, "text": "that takes care of those terms 9 Hales"}, {"start": 160.8, "duration": 5.64, "text": "minus 12 that's going to give us eight"}, {"start": 163.8, "duration": 5.359, "text": "halves and well eight halves is just the"}, {"start": 166.44, "duration": 5.48, "text": "same thing as four so that'll take care"}, {"start": 169.159, "duration": 5.481, "text": "of those two terms and then we simply"}, {"start": 171.92, "duration": 2.72, "text": "have minus"}, {"start": 175.68, "duration": 9.199, "text": "1/3 okay so let's see 11 + 4 is 15 so we"}, {"start": 180.56, "duration": 6.08, "text": "would have 15 over 1 - 1/3 so if we get"}, {"start": 184.879, "duration": 3.841, "text": "common denominators that's going to be"}, {"start": 186.64, "duration": 4.28, "text": "45 over"}, {"start": 188.72, "duration": 6.12, "text": "3 -"}, {"start": 190.92, "duration": 6.56, "text": "1/3 so that's going to be 12 * well 45/3"}, {"start": 194.84, "duration": 7.88, "text": "- 1/3 that's just uh"}, {"start": 197.48, "duration": 9.119, "text": "44 over 3 and 44 ided 2 is going to"}, {"start": 202.72, "duration": 6.439, "text": "leave us with 22 over 3 and that's our"}, {"start": 206.599, "duration": 4.0, "text": "average value so again nothing too"}, {"start": 209.159, "duration": 3.561, "text": "terrible at all"}, {"start": 210.599, "duration": 4.84, "text": "um you know to me the only thing that"}, {"start": 212.72, "duration": 5.519, "text": "could really make a problem like this at"}, {"start": 215.439, "duration": 4.961, "text": "all too terribly difficult which just be"}, {"start": 218.239, "duration": 4.481, "text": "you know if if the function that you're"}, {"start": 220.4, "duration": 4.399, "text": "calculating the anti-derivative of is"}, {"start": 222.72, "duration": 3.799, "text": "just more messy but other than that it's"}, {"start": 224.799, "duration": 3.64, "text": "just one over uh the length of the"}, {"start": 226.519, "duration": 4.561, "text": "interval calculate your definite"}, {"start": 228.439, "duration": 4.401, "text": "integral and multiply those values"}, {"start": 231.08, "duration": 4.999, "text": "together and that'll give you the"}, {"start": 232.84, "duration": 3.239, "text": "average value"}], "title": "Average Value of a Function on an Interval Using Calculus", "description": "Thanks to all of you who support me on Patreon. You da real mvps!   $1 per month helps!! :)  https://www.patreon.com/patrickjmt !! Average Value of a Function on an Interval Using Calculus.\nIn this video, I give the formula to find the average value of a function on a closed interval and work out on simple example.", "lengthSeconds": 234, "width": 450, "height": 360}
{"video_id": "lMXT83M1ET8", "transcript": [{"start": 0.0, "duration": 4.44, "text": "all right here we're going to do one"}, {"start": 1.89, "duration": 4.05, "text": "more limit problem here we're going to"}, {"start": 4.44, "duration": 3.9, "text": "look at the limit as n goes to infinity"}, {"start": 5.94, "duration": 5.67, "text": "of n raised to the negative n to the"}, {"start": 8.34, "duration": 7.44, "text": "third power the first thing I'm going to"}, {"start": 11.61, "duration": 6.509, "text": "do is just rewrite this so n raised to"}, {"start": 15.78, "duration": 4.71, "text": "negative n to the third okay we can"}, {"start": 18.119, "duration": 4.951, "text": "always put that over one well if we can"}, {"start": 20.49, "duration": 4.35, "text": "make negative exponents positive by"}, {"start": 23.07, "duration": 4.56, "text": "sticking that exponent in the"}, {"start": 24.84, "duration": 5.9, "text": "denominator so now we have 1 over N"}, {"start": 27.63, "duration": 5.63, "text": "raised to positive n to the third and"}, {"start": 30.74, "duration": 5.62, "text": "you know if you think about it again"}, {"start": 33.26, "duration": 5.319, "text": "we're letting n go to infinity so"}, {"start": 36.36, "duration": 6.96, "text": "certainly for n values greater than or"}, {"start": 38.579, "duration": 7.681, "text": "equal to one I claim that well in raise"}, {"start": 43.32, "duration": 4.44, "text": "to n to the third power that's always"}, {"start": 46.26, "duration": 4.44, "text": "going to be whoops almost got my"}, {"start": 47.76, "duration": 5.22, "text": "inequality backwards and raised to n to"}, {"start": 50.7, "duration": 4.17, "text": "the third power that's always going to"}, {"start": 52.98, "duration": 4.23, "text": "be greater than or equal to n to the"}, {"start": 54.87, "duration": 5.009, "text": "first I mean imagine if you plug in one"}, {"start": 57.21, "duration": 5.16, "text": "you'll have one to the first on the left"}, {"start": 59.879, "duration": 4.02, "text": "and one to the first on the right if you"}, {"start": 62.37, "duration": 3.359, "text": "plug in two for example though you're"}, {"start": 63.899, "duration": 4.501, "text": "going to have 2 raised to the power of"}, {"start": 65.729, "duration": 4.32, "text": "two cubes so 2 to the 8th is certainly"}, {"start": 68.4, "duration": 4.95, "text": "greater than or equal to 2"}, {"start": 70.049, "duration": 6.721, "text": "again if you plug in 3 you know 3 raised"}, {"start": 73.35, "duration": 5.22, "text": "to the 3 cubed or 3 raised to the 27"}, {"start": 76.77, "duration": 4.5, "text": "power certainly greater than or equal to"}, {"start": 78.57, "duration": 4.26, "text": "3 so I think this is a pretty clear I"}, {"start": 81.27, "duration": 5.07, "text": "don't think many people would argue with"}, {"start": 82.83, "duration": 6.78, "text": "that well that tells us that if we take"}, {"start": 86.34, "duration": 6.3, "text": "1 over this number that's going to be"}, {"start": 89.61, "duration": 5.1, "text": "less than or equal to 1 over N again you"}, {"start": 92.64, "duration": 4.409, "text": "know our fraction now the fraction on"}, {"start": 94.71, "duration": 4.47, "text": "the left has a bigger denominator well"}, {"start": 97.049, "duration": 5.25, "text": "if a fraction has a bigger denominator"}, {"start": 99.18, "duration": 4.049, "text": "in the same numerator that fraction is"}, {"start": 102.299, "duration": 5.131, "text": "going to be smaller"}, {"start": 103.229, "duration": 6.061, "text": "well again since we're using positive"}, {"start": 107.43, "duration": 4.979, "text": "numbers and a some positive number"}, {"start": 109.29, "duration": 5.219, "text": "lifeWise went over n raised to n to the"}, {"start": 112.409, "duration": 5.82, "text": "third that's always going to be greater"}, {"start": 114.509, "duration": 5.61, "text": "than or equal to zero so now we can"}, {"start": 118.229, "duration": 4.951, "text": "bring in our squeeze theorem and we can"}, {"start": 120.119, "duration": 6.0, "text": "simply look at the limit as n goes to"}, {"start": 123.18, "duration": 6.48, "text": "infinity of the left side which is 0"}, {"start": 126.119, "duration": 5.251, "text": "well that limit certainly is 0 likewise"}, {"start": 129.66, "duration": 2.6, "text": "if we look at the limit as n goes to"}, {"start": 131.37, "duration": 3.41, "text": "infinity"}, {"start": 132.26, "duration": 5.009, "text": "the right side well the limit as n goes"}, {"start": 134.78, "duration": 5.459, "text": "to infinity of 1 over N that's also"}, {"start": 137.269, "duration": 8.821, "text": "definitely 0 so what this now tells us"}, {"start": 140.239, "duration": 7.741, "text": "is that by the squeeze theorem so by the"}, {"start": 146.09, "duration": 5.22, "text": "squeeze theorem the limit as n goes to"}, {"start": 147.98, "duration": 8.33, "text": "infinity of 1 over N raised to n to the"}, {"start": 151.31, "duration": 5.0, "text": "third must also equal 0"}], "title": "The Squeeze Theorem and Absolute Value Theorem, #3", "description": "Thanks to all of you who support me on Patreon. You da real mvps!   $1 per month helps!! :)  https://www.patreon.com/patrickjmt !! The Squeeze Theorem and Absolute Value Theorem, #3.  Here we find the limit of another sequence by using the squeeze theorem.", "lengthSeconds": 156, "width": 480, "height": 360}
{"video_id": "J_a9z1ibkSo", "transcript": [{"start": 1.04, "duration": 4.719, "text": "okay in this video I just want to make a"}, {"start": 2.44, "duration": 5.08, "text": "quick comment on some uh notation and uh"}, {"start": 5.759, "duration": 4.161, "text": "talk about a definition of What's called"}, {"start": 7.52, "duration": 4.84, "text": "the gradient vector and I did a couple"}, {"start": 9.92, "duration": 4.0, "text": "examples of directional derivatives and"}, {"start": 12.36, "duration": 3.999, "text": "recall basically the directional"}, {"start": 13.92, "duration": 4.6, "text": "derivative um in the direction so if you"}, {"start": 16.359, "duration": 3.92, "text": "take um to compute the directional"}, {"start": 18.52, "duration": 4.2, "text": "derivative of a function that's"}, {"start": 20.279, "duration": 4.281, "text": "differentiable in two variables um in"}, {"start": 22.72, "duration": 4.639, "text": "the direction of the vector the unit"}, {"start": 24.56, "duration": 5.719, "text": "Vector AB it says we take the partial of"}, {"start": 27.359, "duration": 5.121, "text": "X multiply it by a the partial of Y"}, {"start": 30.279, "duration": 4.8, "text": "multiply partial with respect to Y"}, {"start": 32.48, "duration": 4.52, "text": "multiply that by B um what I want to"}, {"start": 35.079, "duration": 4.721, "text": "point out is well we can actually simply"}, {"start": 37.0, "duration": 5.879, "text": "write this as the product of two vectors"}, {"start": 39.8, "duration": 6.919, "text": "I can take the partial of"}, {"start": 42.879, "duration": 5.68, "text": "X um the partial of Y as my"}, {"start": 46.719, "duration": 5.121, "text": "components and then I'm just going to"}, {"start": 48.559, "duration": 8.121, "text": "dot this by the vector"}, {"start": 51.84, "duration": 7.039, "text": "AB okay so no big deal but we actually"}, {"start": 56.68, "duration": 4.96, "text": "give this first Vector a name and it"}, {"start": 58.879, "duration": 5.36, "text": "ends up uh being important and it's"}, {"start": 61.64, "duration": 5.08, "text": "What's called the"}, {"start": 64.239, "duration": 5.841, "text": "gradient okay so we could call this the"}, {"start": 66.72, "duration": 7.0, "text": "gradient of the function f and this is"}, {"start": 70.08, "duration": 6.6, "text": "often times abbreviated as a little"}, {"start": 73.72, "duration": 5.759, "text": "upside down triangle with a with the F"}, {"start": 76.68, "duration": 6.439, "text": "attached to it so that's it this is"}, {"start": 79.479, "duration": 6.68, "text": "what's called the gradient so um it says"}, {"start": 83.119, "duration": 3.04, "text": "the gradient of"}, {"start": 87.36, "duration": 6.92, "text": "f again it's just simply the partial of"}, {"start": 91.36, "duration": 7.0, "text": "um with respect to X the partial with"}, {"start": 94.28, "duration": 7.879, "text": "respect to Y and again we can simply um"}, {"start": 98.36, "duration": 5.24, "text": "even rewrite this as the partial of f"}, {"start": 102.159, "duration": 6.161, "text": "with respect to"}, {"start": 103.6, "duration": 9.0, "text": "X um I plus the partial of f with"}, {"start": 108.32, "duration": 7.479, "text": "respect to y j um and that's the"}, {"start": 112.6, "duration": 4.559, "text": "definition of our gradient so I just"}, {"start": 115.799, "duration": 4.201, "text": "want to point this out because I'm going"}, {"start": 117.159, "duration": 5.96, "text": "to be using this notation um talking"}, {"start": 120.0, "duration": 5.36, "text": "about the gradient in some other videos"}, {"start": 123.119, "duration": 3.561, "text": "okay so all right nothing nothing deep"}, {"start": 125.36, "duration": 3.16, "text": "but I just want to point that out before"}, {"start": 126.68, "duration": 3.84, "text": "I moved along so again if you have any"}, {"start": 128.52, "duration": 5.92, "text": "questions about the notation or anything"}, {"start": 130.52, "duration": 3.92, "text": "feel free to post comments"}], "title": "The Gradient Vector - Notation and Definition", "description": "Thanks to all of you who support me on Patreon. You da real mvps!   $1 per month helps!! :)  https://www.patreon.com/patrickjmt !! The Gradient Vector - Notation and Definition. In this video, I talk about the directional derivative, and the corresponding definition of the gradient vector.  Nothing deep here!! Just a small observation and some notation!\r\nFor more free math videos, visit http://JustMathTutoring.com", "lengthSeconds": 133, "width": 480, "height": 360}
{"video_id": "f-QBn6FlmZ0", "transcript": [{"start": 0.05, "duration": 3.67, "text": "all right in this video I'm going to do"}, {"start": 2.159, "duration": 3.93, "text": "another example solving an absolute"}, {"start": 3.72, "duration": 3.84, "text": "value inequality it's going to be at the"}, {"start": 6.089, "duration": 3.331, "text": "type where we have a quantity greater"}, {"start": 7.56, "duration": 3.51, "text": "than or equal to some number where that"}, {"start": 9.42, "duration": 3.629, "text": "number is greater than or equal to zero"}, {"start": 11.07, "duration": 3.75, "text": "and what we're going to do again is"}, {"start": 13.049, "duration": 4.161, "text": "we're basically just going to set up two"}, {"start": 14.82, "duration": 5.01, "text": "inequalities that we have to solve"}, {"start": 17.21, "duration": 5.729, "text": "so the first inequality basically just"}, {"start": 19.83, "duration": 8.13, "text": "says remove the absolute value so 8"}, {"start": 22.939, "duration": 8.68, "text": "minus 2 D over 9 greater than or equal"}, {"start": 27.96, "duration": 5.97, "text": "to 5 or it's going to turn out the other"}, {"start": 31.619, "duration": 7.801, "text": "solution will be when the quantity"}, {"start": 33.93, "duration": 8.129, "text": "inside 8 minus 2 D over 9 is less than"}, {"start": 39.42, "duration": 5.79, "text": "or equal to the negative of that value"}, {"start": 42.059, "duration": 4.77, "text": "and that's going to give us our two"}, {"start": 45.21, "duration": 3.81, "text": "inequalities now that we have to solve"}, {"start": 46.829, "duration": 6.091, "text": "so the first thing I'm going to do is"}, {"start": 49.02, "duration": 5.969, "text": "multiply by 9 on each side of the first"}, {"start": 52.92, "duration": 3.9, "text": "inequality that will cancel out the"}, {"start": 54.989, "duration": 5.881, "text": "nines on the left and just leave me with"}, {"start": 56.82, "duration": 8.61, "text": "8 minus 2 D greater than or equal to 45"}, {"start": 60.87, "duration": 6.81, "text": "I can subtract 8 from both sides so I'll"}, {"start": 65.43, "duration": 6.299, "text": "have negative 2 D greater than or equal"}, {"start": 67.68, "duration": 7.049, "text": "to 45 minus 8 I believe that's going to"}, {"start": 71.729, "duration": 6.14, "text": "give us a positive 37 and now we can"}, {"start": 74.729, "duration": 5.25, "text": "divide both sides by negative 2 again"}, {"start": 77.869, "duration": 4.721, "text": "remembering that we have to flip the end"}, {"start": 79.979, "duration": 5.311, "text": "of quality now because we've divided by"}, {"start": 82.59, "duration": 6.54, "text": "a negative number so we'll have 37"}, {"start": 85.29, "duration": 5.369, "text": "basically negative 37 over 2 we'll do"}, {"start": 89.13, "duration": 5.46, "text": "the same thing on the other inequality"}, {"start": 90.659, "duration": 6.03, "text": "just multiply both sides by 9 that again"}, {"start": 94.59, "duration": 5.279, "text": "will cancel out the nines on the Left we"}, {"start": 96.689, "duration": 6.811, "text": "have 8 minus 2 D less than or equal to"}, {"start": 99.869, "duration": 6.691, "text": "negative 45 and again if we just"}, {"start": 103.5, "duration": 6.659, "text": "subtract 8 and subtract 8 I'll have"}, {"start": 106.56, "duration": 6.9, "text": "negative 2 D less than or equal to in"}, {"start": 110.159, "duration": 6.81, "text": "this case I'm going to get negative 50 3"}, {"start": 113.46, "duration": 6.15, "text": "so negative 45 minus 8 and then again"}, {"start": 116.969, "duration": 5.731, "text": "I'm simply going to divide by negative 2"}, {"start": 119.61, "duration": 9.03, "text": "by negative two and it says D has to be"}, {"start": 122.7, "duration": 8.88, "text": "greater than or equal to 53 over two so"}, {"start": 128.64, "duration": 5.67, "text": "it says as long as D falls in one of"}, {"start": 131.58, "duration": 3.84, "text": "these two intervals of numbers that will"}, {"start": 134.31, "duration": 3.48, "text": "give us our solution"}, {"start": 135.42, "duration": 5.0, "text": "so again graphically just just maybe see"}, {"start": 137.79, "duration": 6.059, "text": "it on a number line so here's negative"}, {"start": 140.42, "duration": 10.45, "text": "37 over two it can be less than or equal"}, {"start": 143.849, "duration": 9.871, "text": "to that or 53 over 2 that's positive it"}, {"start": 150.87, "duration": 4.89, "text": "can be greater than or equal to that so"}, {"start": 153.72, "duration": 5.4, "text": "in terms of writing this using interval"}, {"start": 155.76, "duration": 5.729, "text": "notation we can say our solution is the"}, {"start": 159.12, "duration": 6.66, "text": "interval negative infinity up to"}, {"start": 161.489, "duration": 6.81, "text": "negative 37 over two parentheses again"}, {"start": 165.78, "duration": 4.64, "text": "excuse me brackets brackets we've used"}, {"start": 168.299, "duration": 8.16, "text": "brackets here because we can include it"}, {"start": 170.42, "duration": 8.319, "text": "or again brackets 53 over two to"}, {"start": 176.459, "duration": 5.431, "text": "positive infinity again we always put"}, {"start": 178.739, "duration": 6.151, "text": "parentheses around the infinities but"}, {"start": 181.89, "duration": 4.74, "text": "again certainly brackets here and now"}, {"start": 184.89, "duration": 7.25, "text": "we've got our solution to our very"}, {"start": 186.63, "duration": 5.51, "text": "original absolute value inequality"}], "title": "Solving Absolute Value Inequalities - Example 3", "description": "Thanks to all of you who support me on Patreon. You da real mvps!   $1 per month helps!! :)  https://www.patreon.com/patrickjmt !! math algebra absolute value inequality solve example", "lengthSeconds": 192, "width": 480, "height": 360}
{"video_id": "jNv3RcTilrU", "transcript": [{"start": 0.48, "duration": 3.96, "text": "all right so in this example I just want"}, {"start": 1.88, "duration": 4.959, "text": "to do uh another another problem of"}, {"start": 4.44, "duration": 4.279, "text": "writing some compound inequalities and"}, {"start": 6.839, "duration": 3.961, "text": "again all compound inequality typically"}, {"start": 8.719, "duration": 3.281, "text": "means as you have multiple inequalities"}, {"start": 10.8, "duration": 4.64, "text": "we just want to write it in interval"}, {"start": 12.0, "duration": 5.119, "text": "notation so again when I think about or"}, {"start": 15.44, "duration": 4.679, "text": "um I'm going to use everything that I"}, {"start": 17.119, "duration": 4.681, "text": "shade so okay so it says X has to be"}, {"start": 20.119, "duration": 3.521, "text": "less than five so I'm going to put an"}, {"start": 21.8, "duration": 4.479, "text": "open circle there because we don't use"}, {"start": 23.64, "duration": 5.719, "text": "it but everything less than five would"}, {"start": 26.279, "duration": 4.361, "text": "be well everything to the left all the"}, {"start": 29.359, "duration": 4.401, "text": "way out to"}, {"start": 30.64, "duration": 5.439, "text": "Infinity the other inequality says X has"}, {"start": 33.76, "duration": 4.6, "text": "to be greater than 5 but again not"}, {"start": 36.079, "duration": 4.081, "text": "including five so that would give me"}, {"start": 38.36, "duration": 5.24, "text": "everything to"}, {"start": 40.16, "duration": 5.879, "text": "the to the right of five with uh again"}, {"start": 43.6, "duration": 5.08, "text": "all the way out to Infinity so if I want"}, {"start": 46.039, "duration": 4.801, "text": "to describe every you know everything I"}, {"start": 48.68, "duration": 5.359, "text": "have shaded using interval"}, {"start": 50.84, "duration": 7.28, "text": "notation it says we'll go from negative"}, {"start": 54.039, "duration": 7.0, "text": "Infinity up to but not including"}, {"start": 58.12, "duration": 5.32, "text": "five and we can use use this interval"}, {"start": 61.039, "duration": 3.721, "text": "sometimes people will write the word or"}, {"start": 63.44, "duration": 3.88, "text": "sometimes you'll see people write this"}, {"start": 64.76, "duration": 4.92, "text": "symbol this Union so it says we can go"}, {"start": 67.32, "duration": 5.92, "text": "from negative Infinity to five that"}, {"start": 69.68, "duration": 6.36, "text": "takes care of everything on the left or"}, {"start": 73.24, "duration": 4.239, "text": "it says we can kind of skip over five so"}, {"start": 76.04, "duration": 4.079, "text": "again I'm going to put parentheses to"}, {"start": 77.479, "duration": 3.881, "text": "indicate that we don't use it all the"}, {"start": 80.119, "duration": 3.841, "text": "way up to"}, {"start": 81.36, "duration": 4.439, "text": "Infinity so when we write this interval"}, {"start": 83.96, "duration": 4.159, "text": "notation it's really just a fancy way of"}, {"start": 85.799, "duration": 7.28, "text": "saying use every every single number"}, {"start": 88.119, "duration": 4.96, "text": "works except for the number five live"}], "title": "Writing Compound Inequalities Using Interval Notation - Example 2", "description": "Thanks to all of you who support me on Patreon. You da real mvps!   $1 per month helps!! :)  https://www.patreon.com/patrickjmt !! Writing Compound Inequalities Using Interval Notation - Example 1", "lengthSeconds": 91, "width": 480, "height": 360}
{"video_id": "U2iEXP6oqt8", "transcript": [{"start": 0.76, "duration": 4.16, "text": "okay here we're going to do a limit"}, {"start": 2.96, "duration": 4.88, "text": "problem involving the absolute value"}, {"start": 4.92, "duration": 6.879, "text": "theorem so here we're going to find the"}, {"start": 7.84, "duration": 7.799, "text": "limit of the sequence x sub n = -1 the n"}, {"start": 11.799, "duration": 6.441, "text": "+ n^2 over n^2 and again intuitively to"}, {"start": 15.639, "duration": 4.72, "text": "me what's going to happen you know if"}, {"start": 18.24, "duration": 4.16, "text": "you plug in larger and larger values for"}, {"start": 20.359, "duration": 4.641, "text": "n let's forget about the first part for"}, {"start": 22.4, "duration": 4.84, "text": "a moment as we plug in larger and larger"}, {"start": 25.0, "duration": 4.279, "text": "values for n you know we're going to get"}, {"start": 27.24, "duration": 4.96, "text": "some big number squared over some big"}, {"start": 29.279, "duration": 4.761, "text": "number squared it's the same number so"}, {"start": 32.2, "duration": 3.84, "text": "this ratio you know if it wasn't for the"}, {"start": 34.04, "duration": 4.8, "text": "negative 1 to the end we would just get"}, {"start": 36.04, "duration": 4.839, "text": "one well what's going to happen um you"}, {"start": 38.84, "duration": 3.199, "text": "know the negative one to the N all"}, {"start": 40.879, "duration": 3.68, "text": "that's either going to do to the"}, {"start": 42.039, "duration": 6.52, "text": "numerator is either subtract one or add"}, {"start": 44.559, "duration": 6.041, "text": "one so again to me for large values of n"}, {"start": 48.559, "duration": 4.241, "text": "the the numerator and the denominator"}, {"start": 50.6, "duration": 4.24, "text": "are almost going to be exactly the same"}, {"start": 52.8, "duration": 5.279, "text": "value which makes me think this limit"}, {"start": 54.84, "duration": 4.76, "text": "should equal zero to justify this again"}, {"start": 58.079, "duration": 2.601, "text": "we're going to use the absolute value"}, {"start": 59.6, "duration": 3.48, "text": "theorem"}, {"start": 60.68, "duration": 6.08, "text": "so remember that says if the"}, {"start": 63.08, "duration": 7.399, "text": "limit of the absolute value of a subn"}, {"start": 66.76, "duration": 7.56, "text": "equals 0 or in this case X ofn if that"}, {"start": 70.479, "duration": 6.32, "text": "happens that tells us that the sequence"}, {"start": 74.32, "duration": 5.04, "text": "uh without the absolute value also"}, {"start": 76.799, "duration": 4.64, "text": "equals zero so we're going to use this"}, {"start": 79.36, "duration": 3.439, "text": "little result and the way that we're"}, {"start": 81.439, "duration": 4.121, "text": "going to do this is well we're going to"}, {"start": 82.799, "duration": 7.68, "text": "take the limit as in goes to Infinity of"}, {"start": 85.56, "duration": 7.199, "text": "-1 n + n^2 all over n^ 2 and all we're"}, {"start": 90.479, "duration": 4.561, "text": "going to do is just break this up using"}, {"start": 92.759, "duration": 5.081, "text": "some algebra so we can write this as the"}, {"start": 95.04, "duration": 6.079, "text": "limit as goes the limit as n goes to"}, {"start": 97.84, "duration": 7.639, "text": "Infinity of 1 n over n^"}, {"start": 101.119, "duration": 8.841, "text": "2 plus n^ 2 over n"}, {"start": 105.479, "duration": 9.041, "text": "2 so we'll have the limit as n goes to"}, {"start": 109.96, "duration": 6.64, "text": "Infinity again of -1 to the N over n^2"}, {"start": 114.52, "duration": 4.32, "text": "well n^2 over N2 that's just going to"}, {"start": 116.6, "duration": 4.72, "text": "give us"}, {"start": 118.84, "duration": 5.879, "text": "one"}, {"start": 121.32, "duration": 3.399, "text": "so again we can kind of look at these"}, {"start": 125.399, "duration": 6.56, "text": "individually and then the limit as n"}, {"start": 128.16, "duration": 6.12, "text": "goes to Infinity of just one well the"}, {"start": 131.959, "duration": 4.401, "text": "limit as n goes to an Infinity of one is"}, {"start": 134.28, "duration": 3.8, "text": "just one the limit of a constant is the"}, {"start": 136.36, "duration": 3.12, "text": "constant so the only thing we really"}, {"start": 138.08, "duration": 3.6, "text": "have to think about now is What's"}, {"start": 139.48, "duration": 4.119, "text": "Happening Here well again I think it's"}, {"start": 141.68, "duration": 3.639, "text": "pretty easy to argue the top is always"}, {"start": 143.599, "duration": 3.761, "text": "going to bounce back and forth between"}, {"start": 145.319, "duration": 3.841, "text": "negative one and positive one the"}, {"start": 147.36, "duration": 4.239, "text": "denominator is going to get arbitrarily"}, {"start": 149.16, "duration": 4.719, "text": "large so to me this this first limit"}, {"start": 151.599, "duration": 5.201, "text": "should certainly go to zero but again to"}, {"start": 153.879, "duration": 6.241, "text": "bring in this absolute value"}, {"start": 156.8, "duration": 6.439, "text": "theorem notice"}, {"start": 160.12, "duration": 5.479, "text": "um so notice that the limit as n goes to"}, {"start": 163.239, "duration": 5.521, "text": "Infinity if we take the absolute value"}, {"start": 165.599, "duration": 6.681, "text": "of1 to the N over n^2 so I'm just"}, {"start": 168.76, "duration": 5.24, "text": "looking at the first part well that's"}, {"start": 172.28, "duration": 3.92, "text": "just going to equal the limit as in goes"}, {"start": 174.0, "duration": 5.04, "text": "to Infinity if we take the absolute"}, {"start": 176.2, "duration": 4.44, "text": "value of the numerator uh again you know"}, {"start": 179.04, "duration": 4.32, "text": "without the absolute value we're either"}, {"start": 180.64, "duration": 4.36, "text": "getting positive 1 or negative 1 uh when"}, {"start": 183.36, "duration": 3.92, "text": "we raise it when we take the absolute"}, {"start": 185.0, "duration": 7.12, "text": "value of it it'll just become postive"}, {"start": 187.28, "duration": 7.12, "text": "one over n^ 2 but again uh this limit as"}, {"start": 192.12, "duration": 4.56, "text": "n goes to Infinity the denominator will"}, {"start": 194.4, "duration": 4.96, "text": "get arbitrarily large one over a big"}, {"start": 196.68, "duration": 5.04, "text": "number is going to be zero so hey in"}, {"start": 199.36, "duration": 5.0, "text": "fact by the absolute value theorem this"}, {"start": 201.72, "duration": 5.92, "text": "first limit is actually going to be zero"}, {"start": 204.36, "duration": 5.599, "text": "so really we're left with 0 + one so"}, {"start": 207.64, "duration": 6.599, "text": "again in this case our limit will simp"}, {"start": 209.959, "duration": 4.28, "text": "converge to one"}], "title": "Squeeze / Absolute Value Theorem for Sequence : Ex 2", "description": "Thanks to all of you who support me on Patreon. You da real mvps!   $1 per month helps!! :)  https://www.patreon.com/patrickjmt !! Squeeze / Absolute Value Theorem for Sequence : Ex 2\nHere I look at an example where I use the squeeze and or absolute value theorem to find the limit of a sequence.", "lengthSeconds": 212, "width": 480, "height": 360}
{"video_id": "YGcLdzQeEck", "transcript": [{"start": 0.84, "duration": 3.519, "text": "so here's a little proof for the"}, {"start": 2.52, "duration": 4.919, "text": "existence of God I always thought it was"}, {"start": 4.359, "duration": 5.961, "text": "interesting I was a I was initially a"}, {"start": 7.439, "duration": 6.001, "text": "philosophy major after being undecided"}, {"start": 10.32, "duration": 5.519, "text": "until my favorite philosophy Professor"}, {"start": 13.44, "duration": 5.4, "text": "told me I was crazy for being a"}, {"start": 15.839, "duration": 6.401, "text": "philosophy major major which uh I think"}, {"start": 18.84, "duration": 5.92, "text": "he was absolutely probably correct at"}, {"start": 22.24, "duration": 3.92, "text": "least for me so um anyway here's a"}, {"start": 24.76, "duration": 4.16, "text": "little proof and I thought it was"}, {"start": 26.16, "duration": 4.56, "text": "interesting the guy uh the guy was a you"}, {"start": 28.92, "duration": 4.72, "text": "know a philosopher mathematician and"}, {"start": 30.72, "duration": 6.32, "text": "astronomer but this is how it goes"}, {"start": 33.64, "duration": 5.84, "text": "so okay so this guy Guido he thought"}, {"start": 37.04, "duration": 4.32, "text": "that uh he had a little I don't know"}, {"start": 39.48, "duration": 3.64, "text": "that necessarily he even proved it or"}, {"start": 41.36, "duration": 5.32, "text": "came up with it but he thought that this"}, {"start": 43.12, "duration": 6.079, "text": "Justified the existence of God so um as"}, {"start": 46.68, "duration": 5.08, "text": "he said something had been created from"}, {"start": 49.199, "duration": 5.52, "text": "nothing"}, {"start": 51.76, "duration": 6.4, "text": "but I think he was a bit uh incorrect In"}, {"start": 54.719, "duration": 6.64, "text": "His approach so this is how it"}, {"start": 58.16, "duration": 7.359, "text": "goes nothing too crazy"}, {"start": 61.359, "duration": 8.521, "text": "so we start off with all right well 0 ="}, {"start": 65.519, "duration": 8.201, "text": "0 + 0 + 0 plus whatever um well we can"}, {"start": 69.88, "duration": 8.0, "text": "rewrite the zeros as 1 - one 1 - one 1 -"}, {"start": 73.72, "duration": 4.16, "text": "one well then let's get rid of the"}, {"start": 77.96, "duration": 4.0, "text": "parenthesis well let's regroup the"}, {"start": 80.2, "duration": 3.84, "text": "parentheses let's just kind of shift"}, {"start": 81.96, "duration": 8.36, "text": "them over a little bit well now we have"}, {"start": 84.04, "duration": 10.24, "text": "1 +- 1 + 1 uh plus 1 + 1 Etc and now"}, {"start": 90.32, "duration": 7.24, "text": "what we have is well 1 + 0 + 0 plus 0 +"}, {"start": 94.28, "duration": 5.199, "text": "0 and hey that's going to give us one"}, {"start": 97.56, "duration": 4.44, "text": "so"}, {"start": 99.479, "duration": 4.201, "text": "well probably figured out the just by"}, {"start": 102.0, "duration": 3.56, "text": "now what he has"}, {"start": 103.68, "duration": 7.039, "text": "shown"}, {"start": 105.56, "duration": 9.04, "text": "is Tada 0 equals 1 so again he's shown"}, {"start": 110.719, "duration": 7.72, "text": "that 0 equals one so"}, {"start": 114.6, "duration": 5.439, "text": "um if only life were so easy so maybe"}, {"start": 118.439, "duration": 3.0, "text": "you can figure out where the the the"}, {"start": 120.039, "duration": 3.481, "text": "problem is but again I just think it's"}, {"start": 121.439, "duration": 4.761, "text": "kind of interesting"}, {"start": 123.52, "duration": 4.56, "text": "um a lot of the old mathematicians were"}, {"start": 126.2, "duration": 4.679, "text": "definitely philosophers at the same time"}, {"start": 128.08, "duration": 6.56, "text": "and you know um I think now we kind of"}, {"start": 130.879, "duration": 6.801, "text": "see math as this sort of a dry boring"}, {"start": 134.64, "duration": 5.2, "text": "thing we're taught to memorize but um I"}, {"start": 137.68, "duration": 4.04, "text": "think there's something something much"}, {"start": 139.84, "duration": 3.679, "text": "more interesting you can do with it but"}, {"start": 141.72, "duration": 3.56, "text": "that's just my own opinion so again"}, {"start": 143.519, "duration": 4.321, "text": "maybe not necessarily prove the"}, {"start": 145.28, "duration": 4.2, "text": "existence of God but um it's not as dry"}, {"start": 147.84, "duration": 3.96, "text": "and as boring and as mundane I think"}, {"start": 149.48, "duration": 3.64, "text": "some times as we we think it is so"}, {"start": 151.8, "duration": 4.0, "text": "anyways I think it's an interesting"}, {"start": 153.12, "duration": 6.119, "text": "little argument um hopefully you find it"}, {"start": 155.8, "duration": 6.439, "text": "interesting um I certainly"}, {"start": 159.239, "duration": 3.0, "text": "do"}], "title": "A Proof for the Existence of God", "description": "Thanks to all of you who support me on Patreon. You da real mvps!   $1 per month helps!! :)  https://www.patreon.com/patrickjmt !! A Proof for the Existence of God. Get upset, cause you will.  Ya, but the problem is, the math is wrong.  Just a little video showing the proof that a theologian / philosopher of the past gave. Before you get your panties in a knot, it is not my proof, nor do I endorse it. : )", "lengthSeconds": 160, "width": 480, "height": 360}
{"video_id": "i9oDDriIWY8", "transcript": [{"start": 0.4, "duration": 2.96, "text": "all right here we're going to look at"}, {"start": 1.439, "duration": 4.161, "text": "one more example involving a little"}, {"start": 3.36, "duration": 4.399, "text": "division of decimal so"}, {"start": 5.6, "duration": 4.4, "text": "suppose patrick drives at an average"}, {"start": 7.759, "duration": 4.641, "text": "speed of 46.25"}, {"start": 10.0, "duration": 5.519, "text": "miles per hour we want to know how long"}, {"start": 12.4, "duration": 4.639, "text": "does it take him to travel 185"}, {"start": 15.519, "duration": 3.84, "text": "miles"}, {"start": 17.039, "duration": 4.32, "text": "well all we're going to have to do is"}, {"start": 19.359, "duration": 3.601, "text": "take 185"}, {"start": 21.359, "duration": 4.601, "text": "and divide it by"}, {"start": 22.96, "duration": 3.0, "text": "46.25"}, {"start": 26.8, "duration": 5.879, "text": "or equivalently we'll do 185 over 46.25"}, {"start": 32.8, "duration": 3.84, "text": "and now i'm going to do the same thing"}, {"start": 34.0, "duration": 4.96, "text": "as before i'm going to"}, {"start": 36.64, "duration": 6.64, "text": "multiply top and bottom of my fraction"}, {"start": 38.96, "duration": 6.88, "text": "by 100 to move the decimal"}, {"start": 43.28, "duration": 5.52, "text": "in the denominator we'll be left with uh"}, {"start": 45.84, "duration": 7.199, "text": "four six two five and the numerator will"}, {"start": 48.8, "duration": 7.68, "text": "be left with one eight five zero zero"}, {"start": 53.039, "duration": 5.441, "text": "and now i'm going to do my long division"}, {"start": 56.48, "duration": 4.239, "text": "and again these are kind of numbers that"}, {"start": 58.48, "duration": 2.239, "text": "uh"}, {"start": 62.559, "duration": 3.281, "text": "have the potential to be"}, {"start": 65.92, "duration": 3.92, "text": "um to make life a little miserable you"}, {"start": 68.32, "duration": 4.159, "text": "know if you're dividing this number by"}, {"start": 69.84, "duration": 4.88, "text": "this number by hand but okay well"}, {"start": 72.479, "duration": 4.161, "text": "that's that's how it goes so i'm gonna"}, {"start": 74.72, "duration": 4.48, "text": "check here"}, {"start": 76.64, "duration": 6.08, "text": "you know uh i kind of round things off"}, {"start": 79.2, "duration": 5.279, "text": "you know to me uh 46 that's somewhere"}, {"start": 82.72, "duration": 4.48, "text": "i kind of look basically at the very"}, {"start": 84.479, "duration": 5.441, "text": "first digit you know so four i'm going"}, {"start": 87.2, "duration": 4.959, "text": "to count i've rounded up to five um you"}, {"start": 89.92, "duration": 4.08, "text": "know if i think about this number um the"}, {"start": 92.159, "duration": 3.761, "text": "first number is being five"}, {"start": 94.0, "duration": 3.52, "text": "if i look at the first two numbers"}, {"start": 95.92, "duration": 4.08, "text": "inside of here that's somewhere around"}, {"start": 97.52, "duration": 5.12, "text": "18. okay so i'm kind of thinking you"}, {"start": 100.0, "duration": 5.119, "text": "know roughly what's 18 divided by five"}, {"start": 102.64, "duration": 3.439, "text": "well that's somewhere between three and"}, {"start": 105.119, "duration": 3.121, "text": "four"}, {"start": 106.079, "duration": 4.08, "text": "so i'm guessing maybe three or four"}, {"start": 108.24, "duration": 4.159, "text": "forty six point two five will go into"}, {"start": 110.159, "duration": 4.721, "text": "eighteen thousand five hundred"}, {"start": 112.399, "duration": 5.68, "text": "uh maybe you know four or five times"}, {"start": 114.88, "duration": 4.559, "text": "three or four or five times um"}, {"start": 118.079, "duration": 3.921, "text": "and again all i'm doing is i'm getting"}, {"start": 119.439, "duration": 4.161, "text": "rid of the first three digits underneath"}, {"start": 122.0, "duration": 3.28, "text": "and then i'm getting rid of basically"}, {"start": 123.6, "duration": 2.799, "text": "the first three digits of the number out"}, {"start": 125.28, "duration": 3.28, "text": "front and that's kind of what i'm"}, {"start": 126.399, "duration": 4.881, "text": "estimating i'm saying you know roughly"}, {"start": 128.56, "duration": 4.88, "text": "18 divided by four you know that's"}, {"start": 131.28, "duration": 3.36, "text": "somewhere between four and five"}, {"start": 133.44, "duration": 2.799, "text": "um"}, {"start": 134.64, "duration": 3.2, "text": "again depending on how you estimate"}, {"start": 136.239, "duration": 4.0, "text": "obviously you'll get slightly different"}, {"start": 137.84, "duration": 4.8, "text": "values but i know that 46 25 isn't going"}, {"start": 140.239, "duration": 3.681, "text": "to go into this say seven times"}, {"start": 142.64, "duration": 3.2, "text": "okay so again i'm just getting rid of"}, {"start": 143.92, "duration": 4.0, "text": "three digits getting rid of three digits"}, {"start": 145.84, "duration": 3.36, "text": "and i'm thinking about that division to"}, {"start": 147.92, "duration": 4.8, "text": "give me a guess"}, {"start": 149.2, "duration": 6.24, "text": "well i'm going to check 46 to 5 maybe"}, {"start": 152.72, "duration": 5.68, "text": "maybe let's try four"}, {"start": 155.44, "duration": 5.519, "text": "so let's see 4 times 5 is 20"}, {"start": 158.4, "duration": 6.32, "text": "we'll carry the 2 4 times 8 is 10 or"}, {"start": 160.959, "duration": 5.28, "text": "excuse me 4 times 2 is 8 plus 2 is 10."}, {"start": 164.72, "duration": 4.72, "text": "carry the 1."}, {"start": 166.239, "duration": 4.241, "text": "4 times 6 is 24 plus 1 is 25 we'll carry"}, {"start": 169.44, "duration": 4.96, "text": "the 2"}, {"start": 170.48, "duration": 6.96, "text": "4 times 4 16 plus 2 is 18 hey lo and"}, {"start": 174.4, "duration": 4.559, "text": "behold it goes in there exactly four"}, {"start": 177.44, "duration": 3.68, "text": "times so"}, {"start": 178.959, "duration": 4.961, "text": "things were set up nicely for us again"}, {"start": 181.12, "duration": 6.08, "text": "four times 46 25 will leave us with"}, {"start": 183.92, "duration": 6.0, "text": "eighteen five zero zero exactly"}, {"start": 187.2, "duration": 5.039, "text": "if we subtract we've got no remainder"}, {"start": 189.92, "duration": 4.08, "text": "so it says it'll take a good old patrick"}, {"start": 192.239, "duration": 4.321, "text": "that that wonderful guy"}, {"start": 194.0, "duration": 3.44, "text": "it'll take him uh four hours to make"}, {"start": 196.56, "duration": 5.0, "text": "that"}, {"start": 197.44, "duration": 4.12, "text": "to travel that distance"}], "title": "Dividing Decimals, Ex 3", "description": "Thanks to all of you who support me on Patreon. You da real mvps!   $1 per month helps!! :)  https://www.patreon.com/patrickjmt !! Dividing Decimals, Ex 3", "lengthSeconds": 200, "width": 480, "height": 360}
{"video_id": "votVWz-wKeI", "transcript": [{"start": 0.24, "duration": 3.96, "text": "all right here we've got an example"}, {"start": 2.25, "duration": 5.64, "text": "we're going to use the graph to find the"}, {"start": 4.2, "duration": 6.87, "text": "absolute and local maximum and minimum"}, {"start": 7.89, "duration": 7.97, "text": "value so absolute maximum and minimums"}, {"start": 11.07, "duration": 4.79, "text": "and also local maximum and minimums"}, {"start": 17.21, "duration": 4.36, "text": "oftentimes people just give the Y values"}, {"start": 19.8, "duration": 4.469, "text": "so we can kind of use both notations"}, {"start": 21.57, "duration": 6.93, "text": "here so if we want to figure out sort of"}, {"start": 24.269, "duration": 6.541, "text": "the local minimums and also sort of the"}, {"start": 28.5, "duration": 4.649, "text": "local maximum values"}, {"start": 30.81, "duration": 4.23, "text": "well again I just kind of imagine I'm"}, {"start": 33.149, "duration": 4.711, "text": "looking sort of at a mountain range off"}, {"start": 35.04, "duration": 4.05, "text": "in the distance you know obviously kind"}, {"start": 37.86, "duration": 3.33, "text": "of a weird mountain range because it"}, {"start": 39.09, "duration": 3.93, "text": "keeps going forever but if we're looking"}, {"start": 41.19, "duration": 4.23, "text": "for local minimums I'm kind of looking"}, {"start": 43.02, "duration": 4.8, "text": "at you know sort of imagining like you"}, {"start": 45.42, "duration": 4.47, "text": "know kind of looking at the valley of my"}, {"start": 47.82, "duration": 4.5, "text": "mountain range sort of a you know just"}, {"start": 49.89, "duration": 4.95, "text": "where it bottoms out so at this point"}, {"start": 52.32, "duration": 5.1, "text": "negative 5 comma 3 we would say we have"}, {"start": 54.84, "duration": 4.92, "text": "a local minimum and oftentimes people"}, {"start": 57.42, "duration": 5.13, "text": "will just specify the Y value so we"}, {"start": 59.76, "duration": 5.399, "text": "could say the Y value of 3 corresponds"}, {"start": 62.55, "duration": 5.12, "text": "to a local minimum well let's see when"}, {"start": 65.159, "duration": 4.831, "text": "we get to this point negative 2 comma 5"}, {"start": 67.67, "duration": 4.18, "text": "notice this is kind of like a little top"}, {"start": 69.99, "duration": 4.11, "text": "of the hill that would be considered a"}, {"start": 71.85, "duration": 7.29, "text": "local maximum so we could say we have a"}, {"start": 74.1, "duration": 7.199, "text": "local maximum at the Y value of 5 again"}, {"start": 79.14, "duration": 5.28, "text": "as we go down it kind of bottoms out"}, {"start": 81.299, "duration": 4.801, "text": "here at this point 1 comma negative 4 so"}, {"start": 84.42, "duration": 3.66, "text": "again that would be a local minimum so"}, {"start": 86.1, "duration": 5.28, "text": "we have a local minimum at the Y value"}, {"start": 88.08, "duration": 5.07, "text": "of negative 4 here at the point 4 comma"}, {"start": 91.38, "duration": 4.11, "text": "5 we kind of have the you know the"}, {"start": 93.15, "duration": 4.92, "text": "little top of the hill so again 5 would"}, {"start": 95.49, "duration": 4.199, "text": "correspond to a local maximum value but"}, {"start": 98.07, "duration": 4.02, "text": "we've already got it listed so we don't"}, {"start": 99.689, "duration": 5.521, "text": "have to list it again and again at this"}, {"start": 102.09, "duration": 4.709, "text": "point 6 comma 4 you're at the bottom so"}, {"start": 105.21, "duration": 3.449, "text": "we could say that y equals 4 there's a"}, {"start": 106.799, "duration": 5.341, "text": "local minimum if you actually want to"}, {"start": 108.659, "duration": 5.161, "text": "specify you know the points obviously"}, {"start": 112.14, "duration": 3.99, "text": "you just do the same thing so you could"}, {"start": 113.82, "duration": 6.06, "text": "say you have a local minimum at negative"}, {"start": 116.13, "duration": 7.23, "text": "5 comma 3 at the point 1 comma negative"}, {"start": 119.88, "duration": 6.449, "text": "4 and also at the point 6 comma 4"}, {"start": 123.36, "duration": 4.959, "text": "squeeze that guy back in there"}, {"start": 126.329, "duration": 4.66, "text": "we have local maximums"}, {"start": 128.319, "duration": 7.62, "text": "at the points negative 2 comma 5 and"}, {"start": 130.989, "duration": 6.51, "text": "also at the point 4 comma 5 okay so that"}, {"start": 135.939, "duration": 3.651, "text": "takes care of local maximums and"}, {"start": 137.499, "duration": 6.72, "text": "minimums when we talk about the absolute"}, {"start": 139.59, "duration": 6.519, "text": "maximum and the absolute minimum"}, {"start": 144.219, "duration": 5.22, "text": "well now we're just kind of thinking you"}, {"start": 146.109, "duration": 5.311, "text": "know I just think of all the y-values if"}, {"start": 149.439, "duration": 4.231, "text": "I were to list every single Y value on"}, {"start": 151.42, "duration": 3.899, "text": "this graph all infinitely many of them"}, {"start": 153.67, "duration": 4.56, "text": "I'm just thinking what would be the the"}, {"start": 155.319, "duration": 4.381, "text": "largest Y value that ever gets used if"}, {"start": 158.23, "duration": 4.53, "text": "there is one and what would be the"}, {"start": 159.7, "duration": 5.459, "text": "smallest Y value that ever gets used if"}, {"start": 162.76, "duration": 4.17, "text": "there is one well it looks like to me my"}, {"start": 165.159, "duration": 4.651, "text": "function since it keeps going up forever"}, {"start": 166.93, "duration": 4.979, "text": "and ever and ever on either side"}, {"start": 169.81, "duration": 4.319, "text": "well in that case there's really not an"}, {"start": 171.909, "duration": 4.2, "text": "absolute maximum because you know the Y"}, {"start": 174.129, "duration": 4.89, "text": "values always get bigger bigger bigger"}, {"start": 176.109, "duration": 6.69, "text": "bigger so they sort of know there's no"}, {"start": 179.019, "duration": 6.33, "text": "top no biggest Y value so I would say"}, {"start": 182.799, "duration": 4.47, "text": "there's no absolute maximum but when I"}, {"start": 185.349, "duration": 3.961, "text": "think about the absolute minimum well"}, {"start": 187.269, "duration": 3.78, "text": "this point down here at the bottom at 1"}, {"start": 189.31, "duration": 4.709, "text": "negative 4 to me that looks like the"}, {"start": 191.049, "duration": 5.34, "text": "smallest Y value that ever gets used so"}, {"start": 194.019, "duration": 5.43, "text": "we could say our absolute minimum is the"}, {"start": 196.389, "duration": 5.07, "text": "Y value of negative 4 and again you"}, {"start": 199.449, "duration": 3.63, "text": "could specify you know specify the point"}, {"start": 201.459, "duration": 4.291, "text": "like we did you could say the absolute"}, {"start": 203.079, "duration": 5.18, "text": "minimum is at the point 1 comma negative"}, {"start": 205.75, "duration": 2.509, "text": "4"}], "title": "\ud83d\udcc8 Local and Absolute Maximum and Minimum from a Graph \ud83d\udcc8", "description": "Finding Local and Absolute Maximum and Minimum from a Graph\n\nIn this video, we delve into the world of calculus as we explore how to identify local (relative) maximums and minimums, as well as absolute maximums and minimums from a graph. Understanding these concepts is crucial for analyzing functions and making informed decisions based on graphical data.\n\nWhat You\u2019ll Learn:\n\nIdentifying Local Maximums and Minimums: Discover how to spot relative extrema on a graph and what they signify about the function's behavior.\n\nFinding Absolute Maximums and Minimums: Learn the steps to determine the overall highest and lowest points on the graph.\n\nGraphical Analysis Techniques: Gain insights into how to interpret graphs effectively and apply these techniques to various problems.\n\nWhy Watch This Video?\n\nPerfect for Students: This tutorial is ideal for high school and college students tackling calculus or introductory math courses.\nClear and Engaging Explanations: Our step-by-step approach makes complex topics easy to understand and apply.\nReal-World Applications: Understand how these concepts are used in fields like economics, physics, and engineering.\n\n\ud83d\udd14 Join Our Community!\n\nLIKE this video if you find it helpful!\nSHARE it with friends or classmates who are learning calculus!\nSUBSCRIBE for more in-depth tutorials on calculus, mathematics, and graph analysis!\n\n\n#MaximumAndMinimum #Calculus #GraphAnalysis #LocalMaximum #AbsoluteMinimum #Mathematics #MathTutorial #EducationalContent #LearningCalculus #ProblemSolving #HighSchoolMath #CollegeCalculus #GraphingFunctions #CriticalPoints #MathematicalConcepts", "lengthSeconds": 207, "width": 480, "height": 360}
{"video_id": "Xne6Hv9useE", "transcript": [{"start": 1.28, "duration": 4.88, "text": "okay so let's finish off this integral"}, {"start": 3.439, "duration": 5.561, "text": "from from part one so we said the area"}, {"start": 6.16, "duration": 6.559, "text": "between these two curves y Cube minus y"}, {"start": 9.0, "duration": 5.24, "text": "1 - y 4th we graphed them uh sorry I I"}, {"start": 12.719, "duration": 3.56, "text": "fudged there for a little bit I was"}, {"start": 14.24, "duration": 4.44, "text": "thinking um this graph for sure is of"}, {"start": 16.279, "duration": 4.801, "text": "course it makes sense um and then we had"}, {"start": 18.68, "duration": 7.88, "text": "to calculate this integral from- 1 to 1"}, {"start": 21.08, "duration": 7.119, "text": "1 - y 4 - y Cub - y um so let's do that"}, {"start": 26.56, "duration": 4.84, "text": "so I'm going to get rid of my nice"}, {"start": 28.199, "duration": 6.36, "text": "lovely graph here"}, {"start": 31.4, "duration": 6.679, "text": "and simply calculate this out so let's"}, {"start": 34.559, "duration": 6.361, "text": "write it a little bigger"}, {"start": 38.079, "duration": 4.48, "text": "um so from negative 1 to one looks like"}, {"start": 40.92, "duration": 4.88, "text": "my Pen's starting to die let's get rid"}, {"start": 42.559, "duration": 6.32, "text": "of the parentheses so 1 minus y 4th"}, {"start": 45.8, "duration": 4.84, "text": "distribute we'll have negative y cubed"}, {"start": 48.879, "duration": 3.48, "text": "um plus"}, {"start": 50.64, "duration": 5.52, "text": "y"}, {"start": 52.359, "duration": 5.441, "text": "Dy um I always I don't know just like to"}, {"start": 56.16, "duration": 4.96, "text": "have these in descending order so"}, {"start": 57.8, "duration": 6.399, "text": "negative y to 4th"}, {"start": 61.12, "duration": 3.079, "text": "um minus y"}, {"start": 65.159, "duration": 4.761, "text": "cubed +"}, {"start": 67.72, "duration": 5.6, "text": "y +"}, {"start": 69.92, "duration": 5.44, "text": "one Dy okay so now let's just calculate"}, {"start": 73.32, "duration": 3.88, "text": "this so there's really no way to combine"}, {"start": 75.36, "duration": 6.88, "text": "like terms we'll just do it a bit at a"}, {"start": 77.2, "duration": 9.76, "text": "time so we'll get y 5th over"}, {"start": 82.24, "duration": 8.68, "text": "5 we'll get - y 4 over"}, {"start": 86.96, "duration": 8.159, "text": "4 um we'll get + y^2 over"}, {"start": 90.92, "duration": 5.76, "text": "2 + y and then our limits of integration"}, {"start": 95.119, "duration": 4.0, "text": "just - 1:"}, {"start": 96.68, "duration": 5.68, "text": "1 so if we plug one in it looks like"}, {"start": 99.119, "duration": 9.761, "text": "we'll get -5"}, {"start": 102.36, "duration": 7.719, "text": "-4 + 12 um + one so there's our upper"}, {"start": 108.88, "duration": 3.279, "text": "limit of"}, {"start": 110.079, "duration": 4.761, "text": "integration let's subtract away our"}, {"start": 112.159, "duration": 4.96, "text": "lower limit well negative 1 to the 5th"}, {"start": 114.84, "duration": 5.919, "text": "is negative one5 but we'll get an extra"}, {"start": 117.119, "duration": 6.761, "text": "negative so it'll make it positive one5"}, {"start": 120.759, "duration": 4.441, "text": "-1 to 4th is still 1 so we'll be left"}, {"start": 123.88, "duration": 4.92, "text": "with Min"}, {"start": 125.2, "duration": 6.44, "text": "-4 um we'll also get a positive2 when we"}, {"start": 128.8, "duration": 7.799, "text": "square and then we'll get a"}, {"start": 131.64, "duration": 10.44, "text": "minus1 so it looks like we have -5 -4 +"}, {"start": 136.599, "duration": 11.681, "text": "12 + 1 -5"}, {"start": 142.08, "duration": 9.0, "text": "+4 um - 12 + 1 so what cancels out we've"}, {"start": 148.28, "duration": 6.52, "text": "got a negative 1/4 and a POS"}, {"start": 151.08, "duration": 4.76, "text": "1/4 um we've got a positive 1/2 and a"}, {"start": 154.8, "duration": 4.04, "text": "negative"}, {"start": 155.84, "duration": 6.08, "text": "1/2 um it looks like we have a negative"}, {"start": 158.84, "duration": 5.2, "text": "1 and a negative 1/5 which is negative"}, {"start": 161.92, "duration": 6.319, "text": "whoops two"}, {"start": 164.04, "duration": 8.04, "text": "fths um soga - 2 fths and then we have"}, {"start": 168.239, "duration": 7.08, "text": "plus two so get common denominators -2"}, {"start": 172.08, "duration": 6.6, "text": "fths um top and bottom by five so 10"}, {"start": 175.319, "duration": 5.681, "text": "fths that'll give us positive 8 fths"}, {"start": 178.68, "duration": 5.08, "text": "left over as the"}, {"start": 181.0, "duration": 4.159, "text": "area all right so I hope this video"}, {"start": 183.76, "duration": 3.36, "text": "helps and makes some sense if you have"}, {"start": 185.159, "duration": 4.121, "text": "any questions feel free to post comments"}, {"start": 187.12, "duration": 5.44, "text": "and hopefully um somebody can help you"}, {"start": 189.28, "duration": 3.28, "text": "out if not me"}], "title": "Area Between Curves - Integrating with Respect to y - Part 2", "description": "Thanks to all of you who support me on Patreon. You da real mvps!   $1 per month helps!! :)  https://www.patreon.com/patrickjmt !! Area Between Curves - Integrating with Respect to y - Part 2.\r\nIn this video, I actually do the integration and calculate the area.\r\nFor more free math videos, visit http://PatrickJMT.com", "lengthSeconds": 191, "width": 480, "height": 360}
{"video_id": "fUiRqb52n_Y", "transcript": [{"start": 0.03, "duration": 5.19, "text": "all right in this example we're gonna do"}, {"start": 2.34, "duration": 5.459, "text": "a couple a couple different problems"}, {"start": 5.22, "duration": 5.94, "text": "where we evaluate expressions just given"}, {"start": 7.799, "duration": 6.061, "text": "some different values so in Part A we"}, {"start": 11.16, "duration": 7.08, "text": "want to evaluate the expression X plus y"}, {"start": 13.86, "duration": 7.62, "text": "if x equals negative 5 and also we have"}, {"start": 18.24, "duration": 5.16, "text": "that y equals negative 7 so all we're"}, {"start": 21.48, "duration": 3.48, "text": "gonna do is substitute when I plug"}, {"start": 23.4, "duration": 4.98, "text": "things in I like to put things in"}, {"start": 24.96, "duration": 5.55, "text": "parenthesis so X is gonna be negative 5"}, {"start": 28.38, "duration": 6.449, "text": "and in this case you certainly don't"}, {"start": 30.51, "duration": 5.79, "text": "have to but in some cases it'll help"}, {"start": 34.829, "duration": 4.591, "text": "emphasize that you have to distribute"}, {"start": 36.3, "duration": 4.86, "text": "signs I think it's just a good policy so"}, {"start": 39.42, "duration": 5.1, "text": "you'll see me do this some times but"}, {"start": 41.16, "duration": 5.46, "text": "okay negative 5 plus negative 7 well"}, {"start": 44.52, "duration": 4.08, "text": "that's simply gonna give us negative 12"}, {"start": 46.62, "duration": 7.919, "text": "so that would be our solution here for"}, {"start": 48.6, "duration": 8.52, "text": "Part A for Part B again we'll just do"}, {"start": 54.539, "duration": 5.311, "text": "the same thing Part B we have that X"}, {"start": 57.12, "duration": 6.39, "text": "we're gonna evaluate x divided by y"}, {"start": 59.85, "duration": 8.15, "text": "again if X is negative 75 and Y is"}, {"start": 63.51, "duration": 9.48, "text": "positive 5 so we'll get negative 75"}, {"start": 68.0, "duration": 7.0, "text": "divided by Y which is positive 5 well we"}, {"start": 72.99, "duration": 4.129, "text": "have a negative over a positive which is"}, {"start": 75.0, "duration": 5.7, "text": "a negative let's see"}, {"start": 77.119, "duration": 7.271, "text": "75 divided by 5 that's gonna leave us"}, {"start": 80.7, "duration": 7.25, "text": "with 15 so the solution for Part B will"}, {"start": 84.39, "duration": 3.56, "text": "be negative 15"}], "title": "Evaluating Algebraic Expressions for Given Values of X and Y", "description": "Thanks to all of you who support me on Patreon. You da real mvps!   $1 per month helps!! :)  https://www.patreon.com/patrickjmt !! Evaluating Algebraic Expressions for Given Values of X and Y.  Here we evaluate two basic algebraic expressions for given values of x and y.", "lengthSeconds": 87, "width": 480, "height": 360}
{"video_id": "leNCHdO5Lec", "transcript": [{"start": 0.079, "duration": 4.421, "text": "okay here we're gonna look at another"}, {"start": 2.09, "duration": 4.15, "text": "couple examples of solving some"}, {"start": 4.5, "duration": 4.8, "text": "equations where we have to do a couple"}, {"start": 6.24, "duration": 6.06, "text": "steps and we can do both B and C in this"}, {"start": 9.3, "duration": 6.96, "text": "example I think in this video so we've"}, {"start": 12.3, "duration": 7.77, "text": "got 9x minus 4 equals 77 again to get"}, {"start": 16.26, "duration": 5.67, "text": "the the term involving X isolated I'm"}, {"start": 20.07, "duration": 4.619, "text": "going to start off by adding forward to"}, {"start": 21.93, "duration": 5.91, "text": "both sides so we'll be left with simply"}, {"start": 24.689, "duration": 3.781, "text": "9x on the Left negative 4 plus 4 will be"}, {"start": 27.84, "duration": 5.55, "text": "0"}, {"start": 28.47, "duration": 9.9, "text": "there's our equal sign 77 + 4 will give"}, {"start": 33.39, "duration": 7.89, "text": "us 81 and now 2 to get our solution"}, {"start": 38.37, "duration": 5.939, "text": "again we want to be left with just 1x"}, {"start": 41.28, "duration": 6.779, "text": "well since I'm multiplying by 9 I'm"}, {"start": 44.309, "duration": 8.811, "text": "going to divide both sides by 9 9"}, {"start": 48.059, "duration": 9.09, "text": "divided by 9 is 1 X + 81 divided by 9 is"}, {"start": 53.12, "duration": 8.89, "text": "also 9 so our solution for Part B here"}, {"start": 57.149, "duration": 7.08, "text": "will be x equals 9 and again notice you"}, {"start": 62.01, "duration": 7.08, "text": "can always check these 9 times 9 would"}, {"start": 64.229, "duration": 9.96, "text": "be 81 81 - 4 hey that is 77 so you can"}, {"start": 69.09, "duration": 9.27, "text": "always check them Part C here same idea"}, {"start": 74.189, "duration": 7.441, "text": "okay we've got a fraction no big deal so"}, {"start": 78.36, "duration": 5.549, "text": "I've got an N over 5 we've got negative"}, {"start": 81.63, "duration": 4.02, "text": "4 plus and over 5 equals negative 8 I"}, {"start": 83.909, "duration": 4.021, "text": "want to get the term involving the"}, {"start": 85.65, "duration": 5.73, "text": "variable in this case and I want to get"}, {"start": 87.93, "duration": 8.07, "text": "that by itself since I have a negative 4"}, {"start": 91.38, "duration": 7.89, "text": "I'm going to add 4 to both sides so"}, {"start": 96.0, "duration": 8.009, "text": "negative 4 plus 4 will just be 0 we can"}, {"start": 99.27, "duration": 8.639, "text": "drop down our 10 over 5 and negative 8"}, {"start": 104.009, "duration": 6.36, "text": "plus 4 is going to be negative 4 and to"}, {"start": 107.909, "duration": 6.03, "text": "get end by itself since I've got N and"}, {"start": 110.369, "duration": 8.731, "text": "I'm dividing by 5 I'm going to multiply"}, {"start": 113.939, "duration": 9.121, "text": "both sides by 5 so on the left side 5"}, {"start": 119.1, "duration": 6.449, "text": "divided by 5 is just one in a negative"}, {"start": 123.06, "duration": 5.699, "text": "and a positive as a negative 4 times 5"}, {"start": 125.549, "duration": 5.981, "text": "will be 20 and we have our solution for"}, {"start": 128.759, "duration": 6.151, "text": "Part C so"}, {"start": 131.53, "duration": 3.38, "text": "equal negative 20"}], "title": "Solving Two - Step Linear Equations ,  Another Example", "description": "Thanks to all of you who support me on Patreon. You da real mvps!   $1 per month helps!! :)  https://www.patreon.com/patrickjmt !! Solving Two - Step Linear Equations.  Here we look at solving linear equations where we have to perform addition / subtraction as well as division / multiplication to solve.", "lengthSeconds": 134, "width": 480, "height": 360}
{"video_id": "yYxzq_O18Mg", "transcript": [{"start": 0.56, "duration": 3.48, "text": "all right in this video we're going to"}, {"start": 1.8, "duration": 5.72, "text": "look at finding some sums of geometric"}, {"start": 4.04, "duration": 6.76, "text": "series if they exist so in part A here"}, {"start": 7.52, "duration": 7.64, "text": "we've got the series uh from nals 0 to"}, {"start": 10.8, "duration": 7.6, "text": "Infinity of 1 over5 to the N so in this"}, {"start": 15.16, "duration": 5.52, "text": "case I'm going to rewrite it um Nal 0 to"}, {"start": 18.4, "duration": 6.199, "text": "Infinity I'm just going to write it as 1"}, {"start": 20.68, "duration": 5.64, "text": "over5 all raised to the N power again uh"}, {"start": 24.599, "duration": 4.92, "text": "if we take one to the N power in the"}, {"start": 26.32, "duration": 5.08, "text": "numerator 1 to the N is just one so"}, {"start": 29.519, "duration": 4.641, "text": "again in this case this is going to be a"}, {"start": 31.4, "duration": 4.159, "text": "convergent geometric series remember the"}, {"start": 34.16, "duration": 4.68, "text": "number that's getting raised to the"}, {"start": 35.559, "duration": 4.48, "text": "power that's our r value so in this case"}, {"start": 38.84, "duration": 4.84, "text": "r"}, {"start": 40.039, "duration": 7.401, "text": "is5 and again these"}, {"start": 43.68, "duration": 7.24, "text": "converge if our r value is in between"}, {"start": 47.44, "duration": 6.04, "text": "positive one and negative 1 and they"}, {"start": 50.92, "duration": 5.88, "text": "converge to sometimes people will write"}, {"start": 53.48, "duration": 8.68, "text": "it as a over 1 - R I like to think about"}, {"start": 56.8, "duration": 8.56, "text": "it as being the first term over 1 -"}, {"start": 62.16, "duration": 5.44, "text": "R and in this case well let's see if we"}, {"start": 65.36, "duration": 5.48, "text": "plug in n equals 0 That's How we'll get"}, {"start": 67.6, "duration": 6.8, "text": "our first term we'll have well just 1"}, {"start": 70.84, "duration": 7.04, "text": "raised to the 0o over 1 minus the R"}, {"start": 74.4, "duration": 7.359, "text": "value which again is 1/5 well 1/ to the"}, {"start": 77.88, "duration": 8.72, "text": "0er is just 1 1 minus a fth is going to"}, {"start": 81.759, "duration": 8.32, "text": "be four fths and 1 over four FS gives us"}, {"start": 86.6, "duration": 8.159, "text": "5 over 4 that will be the value of of"}, {"start": 90.079, "duration": 4.68, "text": "our convergent geometric series"}], "title": "Sum of an Infinite Geometric Series, Ex 1", "description": "Thanks to all of you who support me on Patreon. You da real mvps!   $1 per month helps!! :)  https://www.patreon.com/patrickjmt !! \nSum of an Infinite Geometric Series, Ex 1. In this video, I show how to find the sum of a convergent infinite series.", "lengthSeconds": 96, "width": 480, "height": 360}
{"video_id": "FFXaeJYaGVY", "transcript": [{"start": 1.319, "duration": 4.121, "text": "okay so one more example here of using"}, {"start": 3.36, "duration": 5.0, "text": "our half angle identities and here we're"}, {"start": 5.44, "duration": 5.68, "text": "simply going to evaluate s of the angle"}, {"start": 8.36, "duration": 5.84, "text": "a over 2 we're given that cosine of a is"}, {"start": 11.12, "duration": 6.76, "text": "3 over5 and we're also told that angle a"}, {"start": 14.2, "duration": 5.919, "text": "is between 0 and 90\u00b0 so that means s of"}, {"start": 17.88, "duration": 3.52, "text": "a over 2 that would be somewhere in"}, {"start": 20.119, "duration": 3.441, "text": "quadrant"}, {"start": 21.4, "duration": 4.36, "text": "one so now I'm just going to go back and"}, {"start": 23.56, "duration": 5.92, "text": "use our"}, {"start": 25.76, "duration": 6.479, "text": "identity so it says for for sign again"}, {"start": 29.48, "duration": 4.079, "text": "it says either get positive or negative"}, {"start": 32.239, "duration": 5.881, "text": "well in this case again we're going to"}, {"start": 33.559, "duration": 6.761, "text": "use the positive one so s of a over 2"}, {"start": 38.12, "duration": 5.32, "text": "again since our angle was in quadrant"}, {"start": 40.32, "duration": 5.2, "text": "one we'll use the positive square"}, {"start": 43.44, "duration": 6.279, "text": "root and then it says we would simply"}, {"start": 45.52, "duration": 8.16, "text": "get one minus well cosine of"}, {"start": 49.719, "duration": 6.52, "text": "a over two and well now I'm going to"}, {"start": 53.68, "duration": 5.48, "text": "make use of the fact that we were given"}, {"start": 56.239, "duration": 4.241, "text": "so we were given the value of cosine of"}, {"start": 59.16, "duration": 4.039, "text": "a"}, {"start": 60.48, "duration": 5.8, "text": "cosine of a was given to us as just the"}, {"start": 63.199, "duration": 5.6, "text": "value 3 fths so now all I have to do is"}, {"start": 66.28, "duration": 6.44, "text": "just really just plug that in so it says"}, {"start": 68.799, "duration": 5.96, "text": "we'll get the square Ro T of 1 -"}, {"start": 72.72, "duration": 5.039, "text": "35ths over"}, {"start": 74.759, "duration": 8.161, "text": "2 well in the numerator uh you can write"}, {"start": 77.759, "duration": 7.121, "text": "one as 5 over 5 5 over 5 - 3 over 5 and"}, {"start": 82.92, "duration": 5.839, "text": "the numerator would leave us with two"}, {"start": 84.88, "duration": 5.52, "text": "fths or 2 over five the denominator okay"}, {"start": 88.759, "duration": 4.281, "text": "this is all still under the square root"}, {"start": 90.4, "duration": 5.399, "text": "we can write two as simply 2 over"}, {"start": 93.04, "duration": 4.32, "text": "one well so underneath the square root"}, {"start": 95.799, "duration": 3.481, "text": "I've got a fraction divided by a"}, {"start": 97.36, "duration": 4.56, "text": "fraction so we can write that as two"}, {"start": 99.28, "duration": 6.0, "text": "fths times well we could flip and"}, {"start": 101.92, "duration": 5.28, "text": "multiply by the denominator 1 over two"}, {"start": 105.28, "duration": 5.159, "text": "those would simply cancel out and we"}, {"start": 107.2, "duration": 5.68, "text": "would be left with the square root of 1"}, {"start": 110.439, "duration": 5.081, "text": "over5 uh we could also take the square"}, {"start": 112.88, "duration": 4.279, "text": "root of the numerator which is one the"}, {"start": 115.52, "duration": 3.8, "text": "square root of the denominator would"}, {"start": 117.159, "duration": 3.201, "text": "just beunk five we could also"}, {"start": 119.32, "duration": 3.36, "text": "rationalize"}, {"start": 120.36, "duration": 5.92, "text": "that the denominator by multiplying top"}, {"start": 122.68, "duration": 6.04, "text": "and bottom by square root of five and"}, {"start": 126.28, "duration": 4.56, "text": "that'll simply leave us withun 5 over"}, {"start": 128.72, "duration": 4.48, "text": "five again you can always uh plug this"}, {"start": 130.84, "duration": 4.72, "text": "into your calculator and get a decimal"}, {"start": 133.2, "duration": 6.319, "text": "approximation but to me this is now is a"}, {"start": 135.56, "duration": 3.959, "text": "nice exact answer"}], "title": "Half Angle Identities to Evaluate Trigonometric Expressions, Example 3", "description": "Thanks to all of you who support me on Patreon. You da real mvps!   $1 per month helps!! :)  https://www.patreon.com/patrickjmt !! Half Angle Identities to Evaluate Trigonometric Expressions, Example 3. In this video, I give some half angle identities and show how they can be used to evaluate trigonometric expressions.", "lengthSeconds": 138, "width": 480, "height": 360}
{"video_id": "bvPxNhs-yrI", "transcript": [{"start": 0.21, "duration": 3.71, "text": "all right here we're going to look at a"}, {"start": 1.709, "duration": 4.201, "text": "little question related to a sequence"}, {"start": 3.92, "duration": 4.03, "text": "and in this case we're going to ask"}, {"start": 5.91, "duration": 4.35, "text": "ourselves is the sequence negative 1 to"}, {"start": 7.95, "duration": 5.67, "text": "the N times 1 over N squared is an"}, {"start": 10.26, "duration": 5.82, "text": "increasing decreasing or neither also"}, {"start": 13.62, "duration": 5.94, "text": "you know is it bounded above below both"}, {"start": 16.08, "duration": 5.64, "text": "for neither so you know there's lots of"}, {"start": 19.56, "duration": 3.87, "text": "different ways to argue these or you"}, {"start": 21.72, "duration": 3.45, "text": "know address these types of questions I"}, {"start": 23.43, "duration": 3.179, "text": "think this one's kind of easy to just"}, {"start": 25.17, "duration": 3.779, "text": "plug in numbers and see what's going to"}, {"start": 26.609, "duration": 5.551, "text": "happen I mean you know this negative 1"}, {"start": 28.949, "duration": 4.591, "text": "to the N basically as n increases all"}, {"start": 32.16, "duration": 6.11, "text": "that's going to do it's going to make"}, {"start": 33.54, "duration": 4.73, "text": "the sequence just alternate and sign"}, {"start": 39.26, "duration": 4.72, "text": "well that to me makes me automatically"}, {"start": 42.21, "duration": 4.169, "text": "feel like it can't be strictly"}, {"start": 43.98, "duration": 3.809, "text": "increasing or decreasing because it"}, {"start": 46.379, "duration": 3.121, "text": "looks like it's going to bounce back and"}, {"start": 47.789, "duration": 4.711, "text": "forth between you know negative positive"}, {"start": 49.5, "duration": 5.61, "text": "negative positive so it's it's not"}, {"start": 52.5, "duration": 4.17, "text": "strictly getting bigger or smaller the 1"}, {"start": 55.11, "duration": 8.609, "text": "over N squared if you think about that"}, {"start": 56.67, "duration": 11.55, "text": "this just gets closer to 0 as n"}, {"start": 63.719, "duration": 6.961, "text": "increases you know so and you can even"}, {"start": 68.22, "duration": 5.82, "text": "plug in numbers if we plug in 1 you know"}, {"start": 70.68, "duration": 5.579, "text": "we would get negative 1 times 1 over 1"}, {"start": 74.04, "duration": 4.5, "text": "so we would get negative 1 if we plug in"}, {"start": 76.259, "duration": 5.701, "text": "2 we would get a positive 1 over 2"}, {"start": 78.54, "duration": 6.6, "text": "squared or 1/4 if we plug in 3 we would"}, {"start": 81.96, "duration": 6.6, "text": "get a negative 1 over 3 squared which"}, {"start": 85.14, "duration": 5.79, "text": "would give us negative 1 over 9 and then"}, {"start": 88.56, "duration": 7.169, "text": "positive again if you plug in 4 we'll"}, {"start": 90.93, "duration": 9.299, "text": "get 16 etc so it's you know it's clear"}, {"start": 95.729, "duration": 7.221, "text": "to see that this is neither increasing"}, {"start": 100.229, "duration": 2.721, "text": "or decreasing"}, {"start": 108.45, "duration": 3.849, "text": "you know and again you're starting like"}, {"start": 110.409, "duration": 5.85, "text": "at negative one and you're uh put forth"}, {"start": 112.299, "duration": 7.261, "text": "a negative nine at the sixteenth but if"}, {"start": 116.259, "duration": 7.56, "text": "you think about it okay so it certainly"}, {"start": 119.56, "duration": 6.15, "text": "looks like to me it is bounded it's"}, {"start": 123.819, "duration": 9.631, "text": "going to be bounded to me both above and"}, {"start": 125.71, "duration": 10.109, "text": "below and to me it looks like it's going"}, {"start": 133.45, "duration": 4.88, "text": "to be bound bounded below you could say"}, {"start": 135.819, "duration": 4.95, "text": "it's bounded below by negative one and"}, {"start": 138.33, "duration": 4.03, "text": "then it's bounded above it looks like"}, {"start": 140.769, "duration": 4.321, "text": "the biggest positive number you would"}, {"start": 142.36, "duration": 6.06, "text": "ever get out of our sequence would be"}, {"start": 145.09, "duration": 9.57, "text": "positive 1/4 so it's bounded below and"}, {"start": 148.42, "duration": 8.34, "text": "above by both of these numbers again if"}, {"start": 154.66, "duration": 4.079, "text": "a sequence is neither strictly"}, {"start": 156.76, "duration": 5.069, "text": "increasing or decreasing it is bounded"}, {"start": 158.739, "duration": 4.921, "text": "above and below you know in that case"}, {"start": 161.829, "duration": 4.981, "text": "you can't really say for sure that it"}, {"start": 163.66, "duration": 4.95, "text": "converges or not the question that nask"}, {"start": 166.81, "duration": 3.989, "text": "is the sequence converge but we could"}, {"start": 168.61, "duration": 3.93, "text": "always address it I think it's pretty"}, {"start": 170.799, "duration": 4.081, "text": "easy to see that it's going to converge"}, {"start": 172.54, "duration": 4.8, "text": "to zero and again in that case we could"}, {"start": 174.88, "duration": 4.41, "text": "use the absolute value theorem so"}, {"start": 177.34, "duration": 3.57, "text": "remember the absolute value theorem we"}, {"start": 179.29, "duration": 5.4, "text": "just take the absolute value of the"}, {"start": 180.91, "duration": 7.04, "text": "sequence in that case if we take the"}, {"start": 184.69, "duration": 6.03, "text": "absolute value the negative 1 to the N"}, {"start": 187.95, "duration": 5.349, "text": "would go away we would just be left with"}, {"start": 190.72, "duration": 6.03, "text": "1 over N squared if we simplify this and"}, {"start": 193.299, "duration": 6.181, "text": "certainly as n goes to infinity this"}, {"start": 196.75, "duration": 5.43, "text": "limit equals zero and therefore by the"}, {"start": 199.48, "duration": 5.55, "text": "absolute value theorem the original"}, {"start": 202.18, "duration": 4.77, "text": "sequence in this case and you know the"}, {"start": 205.03, "duration": 5.37, "text": "seeking particular sequence it would"}, {"start": 206.95, "duration": 5.099, "text": "also converge to zero so again lots of"}, {"start": 210.4, "duration": 4.41, "text": "different ways to kind of justify these"}, {"start": 212.049, "duration": 5.611, "text": "a lot of times to me with these you know"}, {"start": 214.81, "duration": 4.47, "text": "it's to get a solution a lot of times"}, {"start": 217.66, "duration": 4.82, "text": "you can just plug in values and get a"}, {"start": 219.28, "duration": 3.2, "text": "feel for what's going on"}], "title": "Intro to Monotonic and Bounded Sequences, Ex 1", "description": "Thanks to all of you who support me on Patreon. You da real mvps!   $1 per month helps!! :)  https://www.patreon.com/patrickjmt !! Intro to Monotonic and Bounded Sequences, Ex 1.  Just a little question to decide if a sequence is monotonic and/or bounded.", "lengthSeconds": 222, "width": 480, "height": 360}
{"video_id": "zhrLl2r_MBA", "transcript": [{"start": 0.14, "duration": 4.09, "text": "okay in this video I'm gonna do another"}, {"start": 2.429, "duration": 4.051, "text": "example of simplifying radical"}, {"start": 4.23, "duration": 3.66, "text": "expressions involving variables this"}, {"start": 6.48, "duration": 3.09, "text": "time it'll be I guess a little more"}, {"start": 7.89, "duration": 4.289, "text": "complicated because we have a fraction"}, {"start": 9.57, "duration": 5.16, "text": "in here so we've got the square root of"}, {"start": 12.179, "duration": 5.85, "text": "20 U to the fourth B squared W to the"}, {"start": 14.73, "duration": 6.33, "text": "fifth over divided by 80 U to the sixth"}, {"start": 18.029, "duration": 4.441, "text": "B to the third W to the third again a"}, {"start": 21.06, "duration": 4.469, "text": "couple different ways you can start"}, {"start": 22.47, "duration": 6.21, "text": "about simplifying this I could rewrite"}, {"start": 25.529, "duration": 5.67, "text": "all of this to the one-half power but my"}, {"start": 28.68, "duration": 4.789, "text": "own personal preference is certainly"}, {"start": 31.199, "duration": 5.371, "text": "before I do anything I like to simplify"}, {"start": 33.469, "duration": 4.691, "text": "you know inside the inside the root I'm"}, {"start": 36.57, "duration": 5.85, "text": "gonna simplify this fraction a little"}, {"start": 38.16, "duration": 7.379, "text": "bit so I think you know 20 over 80"}, {"start": 42.42, "duration": 7.92, "text": "well 20 will go over go into 20 one time"}, {"start": 45.539, "duration": 6.991, "text": "and 20 will go into 80 four times so 20"}, {"start": 50.34, "duration": 5.82, "text": "over 80 simply reduces to the fraction"}, {"start": 52.53, "duration": 6.99, "text": "1/4 okay I have U to the fourth and U to"}, {"start": 56.16, "duration": 5.52, "text": "the sixth again the bigger exponents in"}, {"start": 59.52, "duration": 4.199, "text": "the denominator so the use will stay in"}, {"start": 61.68, "duration": 5.1, "text": "the denominator if you take the bigger"}, {"start": 63.719, "duration": 7.111, "text": "number 6-4 we'll get to the second power"}, {"start": 66.78, "duration": 6.06, "text": "I notice hey there's these on top B's on"}, {"start": 70.83, "duration": 3.57, "text": "the bottom again we can do this"}, {"start": 72.84, "duration": 3.779, "text": "cancellation because there's all"}, {"start": 74.4, "duration": 4.53, "text": "multiplication in the numerator and the"}, {"start": 76.619, "duration": 4.441, "text": "denominator just to point that out so"}, {"start": 78.93, "duration": 5.07, "text": "again the bigger exponents in the"}, {"start": 81.06, "duration": 5.58, "text": "denominator so 3 minus 2 will give us B"}, {"start": 84.0, "duration": 5.189, "text": "to the first power and then in the"}, {"start": 86.64, "duration": 5.339, "text": "numerator we have W to the fifth over W"}, {"start": 89.189, "duration": 6.241, "text": "to the third the bigger exponents on top"}, {"start": 91.979, "duration": 5.711, "text": "if you take 5 minus 3 that'll give us to"}, {"start": 95.43, "duration": 5.84, "text": "the second power"}, {"start": 97.69, "duration": 6.67, "text": "okay so now I am gonna do this step of"}, {"start": 101.27, "duration": 5.37, "text": "rewriting this all to the 1/2 power"}, {"start": 104.36, "duration": 5.19, "text": "because that's equivalent notation when"}, {"start": 106.64, "duration": 4.44, "text": "we have a square root we have 1 times W"}, {"start": 109.55, "duration": 5.1, "text": "squared but I'm just gonna simply write"}, {"start": 111.08, "duration": 7.56, "text": "that as W squared then we have 4 to the"}, {"start": 114.65, "duration": 7.86, "text": "first power u squared and V to the first"}, {"start": 118.64, "duration": 8.089, "text": "power in the denominator if you have an"}, {"start": 122.51, "duration": 4.219, "text": "exponent applied to a fraction"}, {"start": 128.16, "duration": 5.829, "text": "equivalently the numerator gets raised"}, {"start": 130.929, "duration": 5.491, "text": "to that to that exponent to the one-half"}, {"start": 133.989, "duration": 6.241, "text": "power and then we'll also take the"}, {"start": 136.42, "duration": 5.85, "text": "denominator for u squared V to the first"}, {"start": 140.23, "duration": 4.68, "text": "and we'll raise that also to the one"}, {"start": 142.27, "duration": 4.68, "text": "half power and now again I'm just going"}, {"start": 144.91, "duration": 7.829, "text": "to multiply the exponents since"}, {"start": 146.95, "duration": 7.5, "text": "everything's in parentheses so in the"}, {"start": 152.739, "duration": 4.801, "text": "numerator I've got two times 1/2 which"}, {"start": 154.45, "duration": 7.38, "text": "will be W to the first power in the"}, {"start": 157.54, "duration": 6.419, "text": "denominator I have for my exponent of 1"}, {"start": 161.83, "duration": 6.629, "text": "times 1/2 that'll give me 4 to the 1/2"}, {"start": 163.959, "duration": 7.321, "text": "power if I take my U squared and raise"}, {"start": 168.459, "duration": 4.771, "text": "that to the 1/2 again I multiply so 2"}, {"start": 171.28, "duration": 4.62, "text": "times 1/2 would just give us u to the"}, {"start": 173.23, "duration": 5.28, "text": "first power and then if I multiply on"}, {"start": 175.9, "duration": 5.49, "text": "the V's I would get V to the one-half"}, {"start": 178.51, "duration": 8.339, "text": "power so let me rewrite this one more"}, {"start": 181.39, "duration": 7.62, "text": "time so I've got W on top remember 4 to"}, {"start": 186.849, "duration": 5.401, "text": "the one-half power is the same thing as"}, {"start": 189.01, "duration": 7.89, "text": "the square root of 4 my use just hanging"}, {"start": 192.25, "duration": 7.1, "text": "out I can rewrite the V as V the square"}, {"start": 196.9, "duration": 2.45, "text": "root of V"}, {"start": 199.89, "duration": 4.59, "text": "and the last thing I would do is I think"}, {"start": 203.43, "duration": 2.82, "text": "you know the kind of the one"}, {"start": 204.48, "duration": 4.4, "text": "simplification I could make is I could"}, {"start": 206.25, "duration": 5.34, "text": "certainly take the square root of four"}, {"start": 208.88, "duration": 5.62, "text": "the square root of four is positive two"}, {"start": 211.59, "duration": 6.51, "text": "will leave the you alone will leave the"}, {"start": 214.5, "duration": 5.55, "text": "square root of V alone and this to me"}, {"start": 218.1, "duration": 5.87, "text": "would look like a nice simplified"}, {"start": 220.05, "duration": 3.92, "text": "version of this original problem"}], "title": "Simplifying Radical Expressions Involving Variables - Example 2", "description": "Thanks to all of you who support me on Patreon. You da real mvps!   $1 per month helps!! :)  https://www.patreon.com/patrickjmt !! Simplifying Radical Expressions Involving Variables - Example 2", "lengthSeconds": 226, "width": 480, "height": 360}
{"video_id": "4q8QyW_cWb8", "transcript": [{"start": 0.88, "duration": 3.56, "text": "all right so one more problem here"}, {"start": 2.72, "duration": 3.32, "text": "related to limits and this is actually"}, {"start": 4.44, "duration": 3.88, "text": "going to have a little bit to do with"}, {"start": 6.04, "duration": 4.88, "text": "continuity so we want to figure out what"}, {"start": 8.32, "duration": 5.6, "text": "value of a is this piecewise function"}, {"start": 10.92, "duration": 5.16, "text": "continuous on the entire real line so"}, {"start": 13.92, "duration": 4.96, "text": "we've got the peie wise function x^2 + 3"}, {"start": 16.08, "duration": 6.32, "text": "if x is less than 1 and then we have ax"}, {"start": 18.88, "duration": 6.36, "text": "+ 6 if x is greater than or equal to 1"}, {"start": 22.4, "duration": 5.039, "text": "so we've seen this result that"}, {"start": 25.24, "duration": 6.08, "text": "polinomial functions are continuous on"}, {"start": 27.439, "duration": 6.28, "text": "their domain x^2 + 3 is a polinomial"}, {"start": 31.32, "duration": 4.6, "text": "whatever value we pick for a when we"}, {"start": 33.719, "duration": 5.081, "text": "multiply that by x + 6 that's going to"}, {"start": 35.92, "duration": 4.6, "text": "be continuous as well so definitely it's"}, {"start": 38.8, "duration": 4.16, "text": "continuous for everything less than one"}, {"start": 40.52, "duration": 4.16, "text": "and everything greater than one again"}, {"start": 42.96, "duration": 3.72, "text": "where they kind of break at this x"}, {"start": 44.68, "duration": 4.68, "text": "coordinate of one they may or may not be"}, {"start": 46.68, "duration": 4.0, "text": "continuous depending on the value of a"}, {"start": 49.36, "duration": 3.76, "text": "but we're trying to again pick that"}, {"start": 50.68, "duration": 3.96, "text": "value of a so that it does happen so"}, {"start": 53.12, "duration": 3.8, "text": "again we're going to use our definition"}, {"start": 54.64, "duration": 4.919, "text": "what we want to occur is we want the"}, {"start": 56.92, "duration": 4.599, "text": "limit as X approaches one of the"}, {"start": 59.559, "duration": 5.92, "text": "function"}, {"start": 61.519, "duration": 6.761, "text": "we want that to equal F of one okay well"}, {"start": 65.479, "duration": 5.201, "text": "for the limit to exist the limit as X"}, {"start": 68.28, "duration": 5.879, "text": "approaches one from the left that's"}, {"start": 70.68, "duration": 5.479, "text": "going to have to equal the limit as X"}, {"start": 74.159, "duration": 5.081, "text": "approaches one from the right so my"}, {"start": 76.159, "duration": 4.96, "text": "little sideways equal sign there um well"}, {"start": 79.24, "duration": 5.239, "text": "if x is approaching one from the left"}, {"start": 81.119, "duration": 6.881, "text": "we're using values of X smaller than one"}, {"start": 84.479, "duration": 6.241, "text": "so we would be using the function x^2 +"}, {"start": 88.0, "duration": 4.56, "text": "3 when we do the limit as X approaches 1"}, {"start": 90.72, "duration": 4.52, "text": "from the right well now we're taking x"}, {"start": 92.56, "duration": 6.72, "text": "coordinates larger than 1 so we would"}, {"start": 95.24, "duration": 6.879, "text": "use the function ax +"}, {"start": 99.28, "duration": 7.56, "text": "6 well for the first one if we plug in x"}, {"start": 102.119, "duration": 5.801, "text": "= 1 we'll get 1^ 2 + 3 or we'll get the"}, {"start": 106.84, "duration": 3.879, "text": "value"}, {"start": 107.92, "duration": 4.6, "text": "four um if we plug one into the second"}, {"start": 110.719, "duration": 7.04, "text": "expression again we're plugging it in"}, {"start": 112.52, "duration": 8.44, "text": "for X we'll get a * 1 + 6 or we'll just"}, {"start": 117.759, "duration": 5.36, "text": "be left with a + 6"}, {"start": 120.96, "duration": 4.0, "text": "but again for the limit to exist the"}, {"start": 123.119, "duration": 4.6, "text": "left hand limit has to equal the right"}, {"start": 124.96, "duration": 5.32, "text": "hand limit so that would tell us that"}, {"start": 127.719, "duration": 6.841, "text": "that means that four would have to equal"}, {"start": 130.28, "duration": 6.08, "text": "well a + 6 and well we can just subtract"}, {"start": 134.56, "duration": 6.6, "text": "six from both sides and that would give"}, {"start": 136.36, "duration": 8.36, "text": "us -2 as our value for a so in this case"}, {"start": 141.16, "duration": 6.64, "text": "we've made the limit um equal okay CU if"}, {"start": 144.72, "duration": 4.799, "text": "we use the a value of -2 the left hand"}, {"start": 147.8, "duration": 5.48, "text": "limit and the right hand limit will both"}, {"start": 149.519, "duration": 6.08, "text": "equ Al 4 and um so the limit"}, {"start": 153.28, "duration": 4.72, "text": "exists and notice you know now if you"}, {"start": 155.599, "duration": 3.881, "text": "plug in the x coordinate of one again"}, {"start": 158.0, "duration": 5.08, "text": "we've now figured out that this is the"}, {"start": 159.48, "duration": 5.399, "text": "function we should be using -2x + 6"}, {"start": 163.08, "duration": 5.84, "text": "notice if you plug one into that again"}, {"start": 164.879, "duration": 6.761, "text": "we'll just get -2 + 6 which again will"}, {"start": 168.92, "duration": 5.44, "text": "give us the value four that we want so"}, {"start": 171.64, "duration": 5.92, "text": "it says if we make a equal to -2 the"}, {"start": 174.36, "duration": 5.04, "text": "limits will exist um and in a sense by"}, {"start": 177.56, "duration": 4.88, "text": "default for this particular function"}, {"start": 179.4, "duration": 4.88, "text": "it'll also give you the correct value so"}, {"start": 182.44, "duration": 4.28, "text": "really kind of the uh you know I think"}, {"start": 184.28, "duration": 4.239, "text": "uh in a sense to do the problem you just"}, {"start": 186.72, "duration": 5.96, "text": "need to set the left-and limit equal to"}, {"start": 188.519, "duration": 4.161, "text": "the right hand limit and solve"}, {"start": 193.0, "duration": 3.0, "text": "it"}], "title": "Continuity : Making a Piecewise Function Continuous", "description": "Thanks to all of you who support me on Patreon. You da real mvps!   $1 per month helps!! :)  https://www.patreon.com/patrickjmt !! Continuity : Making a Piecewise Function Continuous.\nHere we have a piecewise function.  We have to find the value for a missing coefficient of one of the functions to make the whole thing into a continuous function.\nThese problems can be done rather quickly, but a typical college professor would want to see limits and the definition of continuity on an exam ( I would have, at least).", "lengthSeconds": 194, "width": 480, "height": 360}
{"video_id": "NucvnDy4QQo", "transcript": [{"start": 1.28, "duration": 3.76, "text": "so suppose chris made 18 out of 20 free"}, {"start": 4.16, "duration": 4.32, "text": "throws"}, {"start": 5.04, "duration": 5.28, "text": "we want to know what percent did he make"}, {"start": 8.48, "duration": 3.84, "text": "so again if we can write a fraction in"}, {"start": 10.32, "duration": 6.24, "text": "the form n over 100"}, {"start": 12.32, "duration": 8.24, "text": "that's the same thing as in percent so"}, {"start": 16.56, "duration": 6.879, "text": "chris made 18 out of a total of"}, {"start": 20.56, "duration": 4.08, "text": "20. so our fraction is going to be 18"}, {"start": 23.439, "duration": 3.441, "text": "over 20."}, {"start": 24.64, "duration": 3.76, "text": "well if we want to make the denominator"}, {"start": 26.88, "duration": 3.28, "text": "into a 100"}, {"start": 28.4, "duration": 3.039, "text": "we could just multiply the denominator"}, {"start": 30.16, "duration": 4.48, "text": "by five"}, {"start": 31.439, "duration": 4.96, "text": "and so the numerator by five well hey"}, {"start": 34.64, "duration": 2.64, "text": "we've now got that 100 in the"}, {"start": 36.399, "duration": 3.441, "text": "denominator"}, {"start": 37.28, "duration": 3.36, "text": "just like we wanted let's see five times"}, {"start": 39.84, "duration": 2.8, "text": "ten is fifty"}, {"start": 40.64, "duration": 6.16, "text": "five five times eight is going to be"}, {"start": 42.64, "duration": 6.48, "text": "forty 50 and 40 would give us 90."}, {"start": 46.8, "duration": 3.36, "text": "90 over 100 is going to be the same"}, {"start": 49.12, "duration": 4.079, "text": "thing as"}, {"start": 50.16, "duration": 5.919, "text": "90 percent so if chris makes"}, {"start": 53.199, "duration": 4.481, "text": "18 out of 20 of his free throws we could"}, {"start": 56.079, "duration": 5.28, "text": "say he's a 90 percent"}, {"start": 57.68, "duration": 3.679, "text": "free throw shooter"}], "title": "Finding the Percent of Free Throws Made ( Word Problem using Basketball)", "description": "Thanks to all of you who support me on Patreon. You da real mvps!   $1 per month helps!! :)  https://www.patreon.com/patrickjmt !! Finding the Percentage of Free Throws Made ( Word Problem using Basketball).", "lengthSeconds": 60, "width": 480, "height": 360}
{"video_id": "Y6moR4ksAZA", "transcript": [{"start": 0.56, "duration": 3.24, "text": "all right so just a few more quick"}, {"start": 2.08, "duration": 4.36, "text": "questions about uh just kind of the"}, {"start": 3.8, "duration": 4.68, "text": "basics of the trigonometric functions so"}, {"start": 6.44, "duration": 5.68, "text": "here we've got a little angle and we've"}, {"start": 8.48, "duration": 5.48, "text": "got a point uh on there 3 comma 4 and we"}, {"start": 12.12, "duration": 4.44, "text": "want to figure out the value of Cent"}, {"start": 13.96, "duration": 4.8, "text": "Theta and also tangent Theta that'll be"}, {"start": 16.56, "duration": 3.92, "text": "the first thing we'll do here well again"}, {"start": 18.76, "duration": 3.519, "text": "you can really think about a little we"}, {"start": 20.48, "duration": 4.4, "text": "really have a little triangle right here"}, {"start": 22.279, "duration": 4.801, "text": "if we just drop a you know drop a line"}, {"start": 24.88, "duration": 6.399, "text": "down that's perpendicular the x"}, {"start": 27.08, "duration": 4.199, "text": "coordinate has value three"}, {"start": 32.68, "duration": 3.92, "text": "so since the x coordinate has value"}, {"start": 35.0, "duration": 5.68, "text": "three well that really means we moved"}, {"start": 36.6, "duration": 5.6, "text": "over three units the Y value is four so"}, {"start": 40.68, "duration": 4.24, "text": "that means this thing has a height of"}, {"start": 42.2, "duration": 5.08, "text": "four and the hypotenuse you may remember"}, {"start": 44.92, "duration": 6.92, "text": "this is just a 345 triangle we could use"}, {"start": 47.28, "duration": 8.24, "text": "Pythagorean theorem as well 3^ 2 + 4^ 2"}, {"start": 51.84, "duration": 10.12, "text": "maybe I'll call the missing side x s"}, {"start": 55.52, "duration": 8.44, "text": "well that's 9 + 16 = x^2 9 + 16 16 is 25"}, {"start": 61.96, "duration": 3.92, "text": "if we take the square root of both sides"}, {"start": 63.96, "duration": 3.76, "text": "normally we would get positive and Nega"}, {"start": 65.88, "duration": 4.16, "text": "five but here it's clearly got to be you"}, {"start": 67.72, "duration": 5.6, "text": "know a positive answer so again we're"}, {"start": 70.04, "duration": 4.68, "text": "just going to get uh our 345 triangle"}, {"start": 73.32, "duration": 5.24, "text": "and now it just basically comes back to"}, {"start": 74.72, "duration": 5.64, "text": "knowing the definition uh coent of theta"}, {"start": 78.56, "duration": 4.48, "text": "we said coent is going to be the"}, {"start": 80.36, "duration": 4.56, "text": "adjacent side divided by the length of"}, {"start": 83.04, "duration": 6.0, "text": "the opposite"}, {"start": 84.92, "duration": 7.32, "text": "side so coent so one more time adjacent"}, {"start": 89.04, "duration": 6.039, "text": "which is has l length three over the"}, {"start": 92.24, "duration": 4.839, "text": "opposite side which is length four so"}, {"start": 95.079, "duration": 4.801, "text": "cent of theta we would say just has"}, {"start": 97.079, "duration": 6.281, "text": "value"}, {"start": 99.88, "duration": 6.64, "text": "34s tangent of theta so again tangent of"}, {"start": 103.36, "duration": 6.28, "text": "theta we said that's opposite over"}, {"start": 106.52, "duration": 6.08, "text": "adjacent so same thing the opposite side"}, {"start": 109.64, "duration": 5.439, "text": "has length four the adjacent side has"}, {"start": 112.6, "duration": 4.799, "text": "length three and that's going to be a"}, {"start": 115.079, "duration": 4.32, "text": "relationship for tangent and cotangent"}, {"start": 117.399, "duration": 4.281, "text": "they're uh basically the reciprocal of"}, {"start": 119.399, "duration": 4.281, "text": "each other always have the same sign but"}, {"start": 121.68, "duration": 6.0, "text": "one is just going to be the flip of the"}, {"start": 123.68, "duration": 6.52, "text": "other one all right so uh just one more"}, {"start": 127.68, "duration": 4.68, "text": "uh actually two more quick questions"}, {"start": 130.2, "duration": 5.319, "text": "here so we've got our little right"}, {"start": 132.36, "duration": 6.239, "text": "triangle here uh 5 7 and square < TK of"}, {"start": 135.519, "duration": 5.601, "text": "74 you can check that this does in fact"}, {"start": 138.599, "duration": 5.28, "text": "satisfy Pythagorean theorem which does"}, {"start": 141.12, "duration": 6.92, "text": "in fact show it's a right triangle we"}, {"start": 143.879, "duration": 6.64, "text": "want to find secant of the angle p and"}, {"start": 148.04, "duration": 5.68, "text": "we also want to find cosecant of the"}, {"start": 150.519, "duration": 6.881, "text": "angle P okay so Shand for secant we just"}, {"start": 153.72, "duration": 6.12, "text": "write SEC Shand for cosecant we write"}, {"start": 157.4, "duration": 5.52, "text": "CSC and again now it's just a matter of"}, {"start": 159.84, "duration": 5.039, "text": "just knowing uh these definitions secant"}, {"start": 162.92, "duration": 5.52, "text": "of an angle we say that's the length of"}, {"start": 164.879, "duration": 5.681, "text": "the hypotenuse over the adjacent side"}, {"start": 168.44, "duration": 5.6, "text": "well the length of the hypotenuse is"}, {"start": 170.56, "duration": 6.92, "text": "square < TK of 74 and the adjacent side"}, {"start": 174.04, "duration": 6.6, "text": "is simply equal to five and now I've got"}, {"start": 177.48, "duration": 7.92, "text": "the value for secant of of P P the angle"}, {"start": 180.64, "duration": 7.319, "text": "p uh P for Patrick cosecant of the angle"}, {"start": 185.4, "duration": 5.28, "text": "P or Theta when we originally had had it"}, {"start": 187.959, "duration": 4.801, "text": "labeled cosecant of an angle is defined"}, {"start": 190.68, "duration": 4.839, "text": "to be the length of the hypotenuse over"}, {"start": 192.76, "duration": 4.479, "text": "the length of the uh opposite side so"}, {"start": 195.519, "duration": 4.8, "text": "here the length of the hypotenuse again"}, {"start": 197.239, "duration": 5.681, "text": "is square < TK of 74 the opposite side"}, {"start": 200.319, "duration": 6.92, "text": "has length seven so we would simply say"}, {"start": 202.92, "duration": 8.84, "text": "the value of cosecant of the angle p is"}, {"start": 207.239, "duration": 4.521, "text": "< TK of 74 / 7"}], "title": "The Trigonometric Functions: The Basics! Example 2", "description": "Thanks to all of you who support me on Patreon. You da real mvps!   $1 per month helps!! :)  https://www.patreon.com/patrickjmt !! The Trigonometric Functions: The Basics! Example 2. Just a few more examples of finding values for the trig functions given a right triangle.", "lengthSeconds": 211, "width": 480, "height": 360}
{"video_id": "BtlD6vstD7A", "transcript": [{"start": 0.48, "duration": 4.12, "text": "all right so another example here"}, {"start": 2.28, "duration": 3.24, "text": "related to hyperbolas and in this case"}, {"start": 4.6, "duration": 2.839, "text": "what we're going to do is we're going to"}, {"start": 5.52, "duration": 5.4, "text": "find again the vertices ASM tootes and"}, {"start": 7.439, "duration": 7.601, "text": "the fosi of this hyperbola y^2 over 100"}, {"start": 10.92, "duration": 6.68, "text": "- x^2 36 = 1 so kind of the first thing"}, {"start": 15.04, "duration": 4.159, "text": "I usually do is I try to think you know"}, {"start": 17.6, "duration": 3.2, "text": "is it going to look like the one on the"}, {"start": 19.199, "duration": 4.601, "text": "left or the one on the"}, {"start": 20.8, "duration": 7.6, "text": "right well notice in this case we've got"}, {"start": 23.8, "duration": 5.92, "text": "y^ 2 over a number - x^2 over a number"}, {"start": 28.4, "duration": 4.6, "text": "so that's going to be more like the"}, {"start": 29.72, "duration": 6.96, "text": "second type here notice we we've got our"}, {"start": 33.0, "duration": 5.6, "text": "value A2 is going to be equal to um 100"}, {"start": 36.68, "duration": 4.719, "text": "it's the value underneath the"}, {"start": 38.6, "duration": 4.68, "text": "y^2 and if we take the square root of"}, {"start": 41.399, "duration": 4.361, "text": "both sides we would get positive and -"}, {"start": 43.28, "duration": 6.56, "text": "10 but again when I fill in my formulas"}, {"start": 45.76, "duration": 5.799, "text": "I just use the positive one okay the b^2"}, {"start": 49.84, "duration": 6.84, "text": "is going to be the value underneath the"}, {"start": 51.559, "duration": 6.761, "text": "x s term so b^2 equals 36 well again if"}, {"start": 56.68, "duration": 3.719, "text": "I take the square root I'm just going to"}, {"start": 58.32, "duration": 4.48, "text": "keep the the positive Sol solution to"}, {"start": 60.399, "duration": 4.441, "text": "this equation which would be just simply"}, {"start": 62.8, "duration": 6.12, "text": "B ="}, {"start": 64.84, "duration": 6.68, "text": "6 okay to get the vertices in this"}, {"start": 68.92, "duration": 4.64, "text": "case so let's see here give myself a"}, {"start": 71.52, "duration": 5.36, "text": "little more"}, {"start": 73.56, "duration": 5.559, "text": "room so to figure out the vertices in"}, {"start": 76.88, "duration": 2.239, "text": "this"}, {"start": 81.52, "duration": 7.12, "text": "example well it says the vertices occur"}, {"start": 84.439, "duration": 6.121, "text": "at 0 positive a and at 0 negative a well"}, {"start": 88.64, "duration": 6.88, "text": "since a is 10 then that means our"}, {"start": 90.56, "duration": 7.8, "text": "vertices will be at 0 pos1 and at 0"}, {"start": 95.52, "duration": 4.84, "text": "-10 okay if we want to find the fosi"}, {"start": 98.36, "duration": 4.719, "text": "again I need to somehow find this value"}, {"start": 100.36, "duration": 6.0, "text": "C but again C satisfies this"}, {"start": 103.079, "duration": 7.281, "text": "relationship that c^2 = A2 +"}, {"start": 106.36, "duration": 5.0, "text": "B2 so c^2 = A2 which in this case will"}, {"start": 110.36, "duration": 5.24, "text": "be"}, {"start": 111.36, "duration": 7.48, "text": "100 plus b^2 which in this case will be"}, {"start": 115.6, "duration": 7.519, "text": "36 so that says c^2"}, {"start": 118.84, "duration": 5.919, "text": "= 36 and again if we want to uh uh solve"}, {"start": 123.119, "duration": 3.68, "text": "for C here we could just take the square"}, {"start": 124.759, "duration": 4.801, "text": "root of both sides again I'm just going"}, {"start": 126.799, "duration": 5.721, "text": "to keep the positive solution here and"}, {"start": 129.56, "duration": 7.24, "text": "now we could uh start reducing this or"}, {"start": 132.52, "duration": 8.24, "text": "at least try to so definitely 136 uh is"}, {"start": 136.8, "duration": 7.0, "text": "even I think we could make that 2 *"}, {"start": 140.76, "duration": 5.16, "text": "68 uh so I'm going to break this down a"}, {"start": 143.8, "duration": 4.6, "text": "little further usually if I see an even"}, {"start": 145.92, "duration": 5.28, "text": "number I just start dividing by"}, {"start": 148.4, "duration": 5.64, "text": "two so let's see 68 we could write that"}, {"start": 151.2, "duration": 6.399, "text": "as 2 *"}, {"start": 154.04, "duration": 7.96, "text": "34 so I've got my two I've got my two 34"}, {"start": 157.599, "duration": 6.161, "text": "we could write that as 2 * 17 well I"}, {"start": 162.0, "duration": 4.879, "text": "think what we could do here so 2 * 2 is"}, {"start": 163.76, "duration": 5.64, "text": "4 we could pull that out as a two 2 and"}, {"start": 166.879, "duration": 3.921, "text": "17 don't really break down any further"}, {"start": 169.4, "duration": 3.0, "text": "so I'm going to go ahead and multiply"}, {"start": 170.8, "duration": 4.6, "text": "those back together and make that the"}, {"start": 172.4, "duration": 4.88, "text": "square otk of 34 so according to our"}, {"start": 175.4, "duration": 4.919, "text": "diagram here it says in this case the"}, {"start": 177.28, "duration": 5.8, "text": "vertices will be at 0 C and at 0 netive"}, {"start": 180.319, "duration": 6.401, "text": "c um excuse me not the vertices but the"}, {"start": 183.08, "duration": 7.239, "text": "fosi so the fosi will be at 0 positive C"}, {"start": 186.72, "duration": 6.0, "text": "and at 0 neg C so in this case our fosi"}, {"start": 190.319, "duration": 9.681, "text": "will be at"}, {"start": 192.72, "duration": 9.2, "text": "0 comma 2 < TK of 34 and at 0 comma -2 <"}, {"start": 200.0, "duration": 4.64, "text": "TK of"}, {"start": 201.92, "duration": 5.319, "text": "34 last but not least if we want to find"}, {"start": 204.64, "duration": 2.599, "text": "the ASM"}, {"start": 208.04, "duration": 7.399, "text": "tootes to get the as ASM tootes we just"}, {"start": 210.799, "duration": 8.281, "text": "let uh we take y = positive and netive A"}, {"start": 215.439, "duration": 7.08, "text": "over B X well again we said our a value"}, {"start": 219.08, "duration": 7.12, "text": "is 10 our B value is 6 so we would get"}, {"start": 222.519, "duration": 8.08, "text": "positive and - 10 over 6X and we could"}, {"start": 226.2, "duration": 8.52, "text": "simply reduce this to get y = positive"}, {"start": 230.599, "duration": 7.121, "text": "and 5/3 X and that would now be our Asm"}, {"start": 234.72, "duration": 3.0, "text": "tootes"}], "title": "Conic Sections: Hyperbolas, An Introduction", "description": "Thanks to all of you who support me on Patreon. You da real mvps!   $1 per month helps!! :)  https://www.patreon.com/patrickjmt !! Conic Sections: Hyperbolas, An Introduction - In this video, I find the vertices and foci for a couple of different hyperbolas.", "lengthSeconds": 236, "width": 480, "height": 360}
{"video_id": "1raifwKcl5A", "transcript": [{"start": 0.68, "duration": 3.48, "text": "okay in this video I'm going to do an"}, {"start": 2.48, "duration": 2.6, "text": "introduction to solving inequalities so"}, {"start": 4.16, "duration": 3.239, "text": "I'm going to solve some pretty"}, {"start": 5.08, "duration": 4.2, "text": "straightforward uh inequalities here to"}, {"start": 7.399, "duration": 4.561, "text": "me the main rule to remember is"}, {"start": 9.28, "duration": 4.64, "text": "basically for inequalities uh you know"}, {"start": 11.96, "duration": 3.0, "text": "at least for linear inequalities which"}, {"start": 13.92, "duration": 2.88, "text": "is what I'm going to do here at the"}, {"start": 14.96, "duration": 3.68, "text": "beginning uh one of the main rules is"}, {"start": 16.8, "duration": 3.8, "text": "simply if you multiply or divide by a"}, {"start": 18.64, "duration": 4.84, "text": "negative number hey you're going to have"}, {"start": 20.6, "duration": 5.56, "text": "to flip around um the inequality it"}, {"start": 23.48, "duration": 4.639, "text": "flips direction is what happens so if"}, {"start": 26.16, "duration": 4.76, "text": "you don't do that your answers are going"}, {"start": 28.119, "duration": 4.96, "text": "to get all crazy okay okay but okay so"}, {"start": 30.92, "duration": 5.639, "text": "in this problem we have 3x - 6 greater"}, {"start": 33.079, "duration": 5.601, "text": "than 8x + 7 in my head I'm almost just"}, {"start": 36.559, "duration": 4.281, "text": "pretending like this is an equal sign"}, {"start": 38.68, "duration": 5.399, "text": "and I'm going to basically solve it the"}, {"start": 40.84, "duration": 5.64, "text": "same way um again just remembering to"}, {"start": 44.079, "duration": 4.521, "text": "flip the inequality If I multiply or"}, {"start": 46.48, "duration": 4.28, "text": "divide by negatives so I'm going to"}, {"start": 48.6, "duration": 5.84, "text": "start off by subtracting 3x from both"}, {"start": 50.76, "duration": 6.799, "text": "sides so on the left we would have -6 uh"}, {"start": 54.44, "duration": 7.599, "text": "greater than 8x - uh 3x is going to give"}, {"start": 57.559, "duration": 8.88, "text": "us 5x + 7 7 and now I'll simply just"}, {"start": 62.039, "duration": 7.561, "text": "subtract seven from both sides -6 - 7 is"}, {"start": 66.439, "duration": 5.521, "text": "-13 that's greater than"}, {"start": 69.6, "duration": 5.08, "text": "5x and now what I'm going to do is"}, {"start": 71.96, "duration": 4.44, "text": "divide both sides by five um the"}, {"start": 74.68, "duration": 4.24, "text": "inequality is going to keep facing the"}, {"start": 76.4, "duration": 4.359, "text": "same direction I am dividing but I'm not"}, {"start": 78.92, "duration": 3.519, "text": "I'm just dividing by a positive number"}, {"start": 80.759, "duration": 3.521, "text": "so if you multiply or divide by a"}, {"start": 82.439, "duration": 6.081, "text": "positive number the inequality stays"}, {"start": 84.28, "duration": 7.0, "text": "where it was and this says -13 over 5 is"}, {"start": 88.52, "duration": 4.639, "text": "greater than x equivalently sometimes"}, {"start": 91.28, "duration": 3.6, "text": "people don't like to see it this way we"}, {"start": 93.159, "duration": 6.521, "text": "could rewrite this equivalently we could"}, {"start": 94.88, "duration": 7.239, "text": "say x is less than ne-3 over 5 so I'm"}, {"start": 99.68, "duration": 3.479, "text": "writing the X first notice if you if you"}, {"start": 102.119, "duration": 2.841, "text": "think about this is kind of being an"}, {"start": 103.159, "duration": 3.361, "text": "arrow it's still pointing at The X and"}, {"start": 104.96, "duration": 4.56, "text": "hey it's still pointing at The X so this"}, {"start": 106.52, "duration": 4.8, "text": "is the same thing um if we were to write"}, {"start": 109.52, "duration": 3.959, "text": "this in interval notation so I'm going"}, {"start": 111.32, "duration": 5.24, "text": "to make a little graph here first so"}, {"start": 113.479, "duration": 4.32, "text": "here's -3 over5 since we don't include"}, {"start": 116.56, "duration": 3.479, "text": "that number I'm going to make a little"}, {"start": 117.799, "duration": 4.161, "text": "open circle that's usually the kind of"}, {"start": 120.039, "duration": 3.481, "text": "the convention that people use and then"}, {"start": 121.96, "duration": 3.6, "text": "I'm going to shade everything that works"}, {"start": 123.52, "duration": 4.32, "text": "which would be every number to the left"}, {"start": 125.56, "duration": 4.64, "text": "of it and we would go forever and ever"}, {"start": 127.84, "duration": 3.72, "text": "and ever out to negative Infinity we"}, {"start": 130.2, "duration": 3.96, "text": "would never get there but again this"}, {"start": 131.56, "duration": 4.8, "text": "just says goes just go as far left just"}, {"start": 134.16, "duration": 5.0, "text": "keep going the way that we can write"}, {"start": 136.36, "duration": 4.44, "text": "this in interval notation is we'll we'll"}, {"start": 139.16, "duration": 4.159, "text": "put a little parentheses always a"}, {"start": 140.8, "duration": 5.6, "text": "parentheses around negative infinity and"}, {"start": 143.319, "duration": 6.0, "text": "then we go up to -13"}, {"start": 146.4, "duration": 5.479, "text": "over5 um but we don't include it so"}, {"start": 149.319, "duration": 4.481, "text": "again we use parentheses uh to indicate"}, {"start": 151.879, "duration": 3.761, "text": "that we don't include that number so"}, {"start": 153.8, "duration": 4.56, "text": "this would be the interval notation that"}, {"start": 155.64, "duration": 6.72, "text": "would provide the solution to our very"}, {"start": 158.36, "duration": 4.0, "text": "original inequality"}], "title": "An Introduction To Solving Linear Inequalities - Example 1", "description": "Thanks to all of you who support me on Patreon. You da real mvps!   $1 per month helps!! :)  https://www.patreon.com/patrickjmt !! An Introduction To Solving Inequalities - Example 1", "lengthSeconds": 160, "width": 480, "height": 360}
{"video_id": "PJ22lotti7s", "transcript": [{"start": 0.14, "duration": 4.21, "text": "okay in this video I'm going to do one"}, {"start": 2.52, "duration": 5.16, "text": "more example of factoring perfect square"}, {"start": 4.35, "duration": 6.27, "text": "trinomials and I'm going to factor this"}, {"start": 7.68, "duration": 7.02, "text": "expression 2p plus T squared plus 6"}, {"start": 10.62, "duration": 5.85, "text": "times 2 P plus T plus 9 and I think at"}, {"start": 14.7, "duration": 3.419, "text": "first this is one of those that looks a"}, {"start": 16.47, "duration": 5.46, "text": "little confusing if you you know you ran"}, {"start": 18.119, "duration": 6.271, "text": "into it you might be inclined to try to"}, {"start": 21.93, "duration": 5.79, "text": "multiply everything out collect any like"}, {"start": 24.39, "duration": 4.83, "text": "terms and to me then you'd have a big"}, {"start": 27.72, "duration": 5.249, "text": "old mess and you'd be trying to factor"}, {"start": 29.22, "duration": 5.55, "text": "that maybe it would work but uh I think"}, {"start": 32.969, "duration": 3.781, "text": "the thing that we need to recognize here"}, {"start": 34.77, "duration": 3.5, "text": "again is really this is a perfect square"}, {"start": 36.75, "duration": 3.69, "text": "trinomial"}, {"start": 38.27, "duration": 4.359, "text": "okay and I'm going to try to rewrite it"}, {"start": 40.44, "duration": 4.92, "text": "in this you know in this form just to"}, {"start": 42.629, "duration": 5.34, "text": "illustrate it just to really emphasize"}, {"start": 45.36, "duration": 9.269, "text": "that so notice okay"}, {"start": 47.969, "duration": 9.871, "text": "2p plus T squared there that is we have"}, {"start": 54.629, "duration": 4.41, "text": "6 so I'm going to leave it as 6 the only"}, {"start": 57.84, "duration": 4.739, "text": "thing I'm really going to do here at the"}, {"start": 59.039, "duration": 7.071, "text": "first step is rewrite my 9 as well I can"}, {"start": 62.579, "duration": 7.65, "text": "rewrite that as positive 3 squared and"}, {"start": 66.11, "duration": 7.0, "text": "it says in this formula it says this"}, {"start": 70.229, "duration": 6.361, "text": "would be our a value so a would equal to"}, {"start": 73.11, "duration": 7.86, "text": "P plus T that's what's being squared B"}, {"start": 76.59, "duration": 6.93, "text": "would have the value of 3 and notice it"}, {"start": 80.97, "duration": 8.24, "text": "says in the middle we take 2 times a"}, {"start": 83.52, "duration": 7.98, "text": "which is 2 P plus T times B which is 3"}, {"start": 89.21, "duration": 5.65, "text": "notice if you multiply all those"}, {"start": 91.5, "duration": 6.509, "text": "together 2 times 3 is 6 and then we'll"}, {"start": 94.86, "duration": 4.939, "text": "get our 2 P plus T we do get this middle"}, {"start": 98.009, "duration": 3.89, "text": "term back so"}, {"start": 99.799, "duration": 9.091, "text": "Soviet says we do have this form a"}, {"start": 101.899, "duration": 8.82, "text": "squared plus 2 a B plus B squared so it"}, {"start": 108.89, "duration": 6.649, "text": "says well we can simply factor that now"}, {"start": 110.719, "duration": 7.65, "text": "it says we take a which is 2p plus T and"}, {"start": 115.539, "duration": 6.631, "text": "it says we add to that B which is"}, {"start": 118.369, "duration": 8.661, "text": "positive 3 and then it says we just"}, {"start": 122.17, "duration": 4.86, "text": "rewrite it again it's a perfect square"}, {"start": 128.179, "duration": 5.73, "text": "okay and in this case you could drop the"}, {"start": 130.799, "duration": 5.491, "text": "parentheses on the inside 2p plus T plus"}, {"start": 133.909, "duration": 4.3, "text": "three again you can think about there's"}, {"start": 136.29, "duration": 3.449, "text": "being a coefficient of positive one so"}, {"start": 138.209, "duration": 5.36, "text": "if you multiply it out it just gets rid"}, {"start": 139.739, "duration": 6.871, "text": "of it then we have two P plus T plus"}, {"start": 143.569, "duration": 8.681, "text": "three or again even we could rewrite"}, {"start": 146.61, "duration": 8.76, "text": "this as 2 P plus T plus 3 since we have"}, {"start": 152.25, "duration": 4.65, "text": "one and two of them being multiplied we"}, {"start": 155.37, "duration": 4.739, "text": "could rewrite that as that whole"}, {"start": 156.9, "duration": 4.8, "text": "quantity square so again you know it"}, {"start": 160.109, "duration": 4.5, "text": "looks a little tricky at the beginning"}, {"start": 161.7, "duration": 5.52, "text": "but just trying to really emphasize this"}, {"start": 164.609, "duration": 4.591, "text": "form this a squared plus 2 a B plus B"}, {"start": 167.22, "duration": 2.96, "text": "squared hey that's a perfect square"}, {"start": 169.2, "duration": 3.899, "text": "trinomial"}, {"start": 170.18, "duration": 5.29, "text": "and it actually turns out it factors a"}, {"start": 173.099, "duration": 6.711, "text": "little bit a little bit more nicely than"}, {"start": 175.47, "duration": 4.34, "text": "I think you might first believe"}], "title": "Factoring Perfect Square Trinomials - Ex3", "description": "Thanks to all of you who support me on Patreon. You da real mvps!   $1 per month helps!! :)  https://www.patreon.com/patrickjmt !! Factoring Perfect Square Trinomials - Ex3", "lengthSeconds": 179, "width": 480, "height": 360}
{"video_id": "X72qoK6i2B8", "transcript": [{"start": 1.16, "duration": 5.04, "text": "okay in this example I want to do just a"}, {"start": 3.36, "duration": 5.319, "text": "couple more examples of applying rules"}, {"start": 6.2, "duration": 5.319, "text": "of exponents and this time we just have"}, {"start": 8.679, "duration": 6.321, "text": "a few fractions floating around and the"}, {"start": 11.519, "duration": 3.481, "text": "main rule you need to remember with"}, {"start": 19.08, "duration": 5.32, "text": "fractions so suppose I have x to the N"}, {"start": 22.08, "duration": 5.16, "text": "over y to the m all being raised to the"}, {"start": 24.4, "duration": 5.08, "text": "P power what it says is it says we can"}, {"start": 27.24, "duration": 6.68, "text": "basically sort of distribute the"}, {"start": 29.48, "duration": 7.44, "text": "exponent to the top part distribute the"}, {"start": 33.92, "duration": 5.2, "text": "exponent to the bottom part and then we"}, {"start": 36.92, "duration": 4.36, "text": "have the rule that we saw in a previous"}, {"start": 39.12, "duration": 6.4, "text": "video where you just multiply the two"}, {"start": 41.28, "duration": 7.959, "text": "exponents together so n * p and on the"}, {"start": 45.52, "duration": 6.64, "text": "bottom we'll have y raised to the m *"}, {"start": 49.239, "duration": 6.84, "text": "P so that's what we'll do in this case"}, {"start": 52.16, "duration": 5.52, "text": "um so here I've got x^2 over 2 again you"}, {"start": 56.079, "duration": 5.201, "text": "could make this to the first Power if"}, {"start": 57.68, "duration": 5.96, "text": "you want to see an exponent um"}, {"start": 61.28, "duration": 5.6, "text": "never hurts I think so here we would"}, {"start": 63.64, "duration": 6.4, "text": "have X2 being raised to the 4th"}, {"start": 66.88, "duration": 4.44, "text": "power over 2 to the 1st being raised to"}, {"start": 70.04, "duration": 4.24, "text": "the 4th"}, {"start": 71.32, "duration": 5.36, "text": "power okay so now we're at this step and"}, {"start": 74.28, "duration": 6.32, "text": "then it says again we just multiply the"}, {"start": 76.68, "duration": 6.399, "text": "exponents so 2 * 4 will be to the e88th"}, {"start": 80.6, "duration": 4.36, "text": "power um if we take two we have an"}, {"start": 83.079, "duration": 4.241, "text": "exponent of one and then a four we"}, {"start": 84.96, "duration": 5.28, "text": "multiply those that'll give us 2 to the"}, {"start": 87.32, "duration": 4.759, "text": "4th power there's not much you can do in"}, {"start": 90.24, "duration": 4.44, "text": "terms of simplifying the numerator we'll"}, {"start": 92.079, "duration": 8.68, "text": "just leave that X to 8 but we can"}, {"start": 94.68, "duration": 6.079, "text": "rewrite 2 4th that's 2 * 2 * 2 *"}, {"start": 102.159, "duration": 7.56, "text": "2 2 * 2 is 4 * another 2 is 8 * another"}, {"start": 106.32, "duration": 7.56, "text": "two will give us 16 so it says our final"}, {"start": 109.719, "duration": 7.04, "text": "solution is X to 8 over"}, {"start": 113.88, "duration": 6.72, "text": "16 okay so we can do the same thing here"}, {"start": 116.759, "duration": 5.72, "text": "with our second example we have 4 over n"}, {"start": 120.6, "duration": 4.119, "text": "the 6 all being squared I'm going to"}, {"start": 122.479, "duration": 4.801, "text": "stick an exponent in"}, {"start": 124.719, "duration": 4.281, "text": "there so again we can write that as 4"}, {"start": 127.28, "duration": 4.88, "text": "the 1st"}, {"start": 129.0, "duration": 6.2, "text": "squared um into the"}, {"start": 132.16, "duration": 4.68, "text": "6 squared I think when you get a little"}, {"start": 135.2, "duration": 3.0, "text": "more comfortable these let me go down"}, {"start": 136.84, "duration": 2.72, "text": "here so it doesn't run into my other"}, {"start": 138.2, "duration": 2.64, "text": "problem I think when you get more"}, {"start": 139.56, "duration": 3.8, "text": "comfortable you can actually kind of"}, {"start": 140.84, "duration": 4.16, "text": "skip this step I I definitely tend to"}, {"start": 143.36, "duration": 3.28, "text": "but I think at the beginning it's"}, {"start": 145.0, "duration": 3.68, "text": "certainly a good idea to write things"}, {"start": 146.64, "duration": 4.72, "text": "out always write things out when you're"}, {"start": 148.68, "duration": 4.32, "text": "unsure um I think it's a good it's just"}, {"start": 151.36, "duration": 4.72, "text": "good form it keeps you from making so"}, {"start": 153.0, "duration": 4.599, "text": "many mistakes so 1 time 2 that'll give"}, {"start": 156.08, "duration": 4.48, "text": "us to the second"}, {"start": 157.599, "duration": 4.121, "text": "power again on the bottom we have 6 * 2"}, {"start": 160.56, "duration": 6.36, "text": "which is to the"}, {"start": 161.72, "duration": 7.2, "text": "12th 4^ 2ar is 4 * 4 that'll give us 16"}, {"start": 166.92, "duration": 5.44, "text": "and then we have our n the 12th just"}, {"start": 168.92, "duration": 6.679, "text": "hanging out on the bottom okay so"}, {"start": 172.36, "duration": 5.519, "text": "that'll be uh our original 4 over n 6"}, {"start": 175.599, "duration": 4.961, "text": "all squared we can simplify that by"}, {"start": 177.879, "duration": 5.64, "text": "saying that 16 over n into the 12th"}, {"start": 180.56, "duration": 2.959, "text": "power"}], "title": "Applying the Rules of Exponents - Basic Examples #2", "description": "Thanks to all of you who support me on Patreon. You da real mvps!   $1 per month helps!! :)  https://www.patreon.com/patrickjmt !! Applying the Rules of Exponents - Basic Examples #2", "lengthSeconds": 182, "width": 480, "height": 360}
{"video_id": "3oyUeiG_mt0", "transcript": [{"start": 0.96, "duration": 4.6, "text": "all right in this video I'm going to"}, {"start": 2.36, "duration": 7.359, "text": "graph a basic rational function and the"}, {"start": 5.56, "duration": 6.999, "text": "idea is um the graph of 1 /x is the"}, {"start": 9.719, "duration": 6.441, "text": "graph here in the red and it's got a"}, {"start": 12.559, "duration": 6.161, "text": "horizontal ASM toote of yals 0 or"}, {"start": 16.16, "duration": 6.6, "text": "equivalently the Y AIS excuse me the x-"}, {"start": 18.72, "duration": 8.68, "text": "axis it has a a vertical ASM toote of uh"}, {"start": 22.76, "duration": 6.4, "text": "xal 0 which is the Y AIS okay and then"}, {"start": 27.4, "duration": 3.119, "text": "it's kind of sandwiched in you know the"}, {"start": 29.16, "duration": 3.52, "text": "bottom left corner the upper right"}, {"start": 30.519, "duration": 4.001, "text": "corner so I'm going to use some of our"}, {"start": 32.68, "duration": 4.48, "text": "sort of Reflections and shifting to help"}, {"start": 34.52, "duration": 5.76, "text": "us get some new graphs here so suppose"}, {"start": 37.16, "duration": 6.64, "text": "we want to graph 1x +"}, {"start": 40.28, "duration": 5.16, "text": "3 well the negative 1 overx all that"}, {"start": 43.8, "duration": 4.599, "text": "does is remember it's just going to take"}, {"start": 45.44, "duration": 4.799, "text": "our graph and reflected about the x-axis"}, {"start": 48.399, "duration": 5.0, "text": "so the part that had been in the top"}, {"start": 50.239, "duration": 5.521, "text": "right will now be in the in the um the"}, {"start": 53.399, "duration": 2.361, "text": "bottom"}, {"start": 56.0, "duration": 4.64, "text": "right and then the part that had been in"}, {"start": 58.079, "duration": 5.241, "text": "the bottom left will now go"}, {"start": 60.64, "duration": 7.36, "text": "up in the top left so this would be my"}, {"start": 63.32, "duration": 8.24, "text": "graph of f ofx ="}, {"start": 68.0, "duration": 5.84, "text": "-1x again the uh horizontal ASM toote is"}, {"start": 71.56, "duration": 5.12, "text": "still at the same place and the vertical"}, {"start": 73.84, "duration": 4.959, "text": "ASM toote is still at the same place"}, {"start": 76.68, "duration": 4.16, "text": "remember if we add three to a function"}, {"start": 78.799, "duration": 4.68, "text": "though all that does is it just I always"}, {"start": 80.84, "duration": 4.319, "text": "just imagine if the graph you know the X"}, {"start": 83.479, "duration": 4.161, "text": "and Y axis couldn't move but everything"}, {"start": 85.159, "duration": 5.721, "text": "else was rigid I just move it up three"}, {"start": 87.64, "duration": 6.68, "text": "units so this uh horizontal Al Asm toote"}, {"start": 90.88, "duration": 5.919, "text": "that had been at yal Z will now be moved"}, {"start": 94.32, "duration": 2.479, "text": "up here"}, {"start": 98.439, "duration": 6.28, "text": "to to y equal"}, {"start": 101.079, "duration": 7.04, "text": "3 okay and let's see a couple things"}, {"start": 104.719, "duration": 6.08, "text": "here um the the vertical ASM toote is"}, {"start": 108.119, "duration": 6.68, "text": "still in the same place so let's see"}, {"start": 110.799, "duration": 6.841, "text": "this is still our x axis and our y AIS"}, {"start": 114.799, "duration": 4.841, "text": "now the the portion that had been in the"}, {"start": 117.64, "duration": 3.64, "text": "uh the the the bottom right is going to"}, {"start": 119.64, "duration": 4.0, "text": "get moved up a little bit the portion"}, {"start": 121.28, "duration": 4.68, "text": "that had been in the bottom or excuse me"}, {"start": 123.64, "duration": 5.56, "text": "the the top left will get moved up so"}, {"start": 125.96, "duration": 5.6, "text": "I'm going to move that part right up"}, {"start": 129.2, "duration": 6.24, "text": "here and I know this part's also going"}, {"start": 131.56, "duration": 3.88, "text": "to get shifted up so it's going"}, {"start": 135.72, "duration": 3.879, "text": "to it's basically going to look like"}, {"start": 137.92, "duration": 3.8, "text": "that we could even find out this x"}, {"start": 139.599, "duration": 3.561, "text": "intercept if we wanted to um we could"}, {"start": 141.72, "duration": 4.36, "text": "find out the coordinates because again"}, {"start": 143.16, "duration": 7.32, "text": "to find X intercepts we just plug in Zer"}, {"start": 146.08, "duration": 9.36, "text": "for f ofx so we would get 0 ="}, {"start": 150.48, "duration": 8.24, "text": "1x + 3 we can add 1X to both"}, {"start": 155.44, "duration": 7.04, "text": "sides I'm going to write 3 is 3 over 1"}, {"start": 158.72, "duration": 8.0, "text": "if we cross multiply we would get 3x = 1"}, {"start": 162.48, "duration": 8.28, "text": "or x = 1/3 so it's actually crossing the"}, {"start": 166.72, "duration": 6.239, "text": "x axis here at this point um uh at the x"}, {"start": 170.76, "duration": 4.44, "text": "coordinate of 1/3 and now we've got our"}, {"start": 172.959, "duration": 5.56, "text": "basic graph just again kind of reflect"}, {"start": 175.2, "duration": 4.92, "text": "it the 1 overx graph and move it up so"}, {"start": 178.519, "duration": 3.521, "text": "um you know again you could all always"}, {"start": 180.12, "duration": 4.28, "text": "plot points plot points plot points and"}, {"start": 182.04, "duration": 5.52, "text": "get this General shape but this basic"}, {"start": 184.4, "duration": 4.44, "text": "graph 1/x is a useful graph to know so"}, {"start": 187.56, "duration": 4.679, "text": "that you can kind of do all these"}, {"start": 188.84, "duration": 7.88, "text": "shifting and uh uh U moving around of it"}, {"start": 192.239, "duration": 4.481, "text": "to help you get new graphs"}], "title": "Graphing Some Basic Rational Functions - Example 1", "description": "Thanks to all of you who support me on Patreon. You da real mvps!   $1 per month helps!! :)  https://www.patreon.com/patrickjmt !! Graphing Some Basic Rational Functions - Example 1", "lengthSeconds": 195, "width": 480, "height": 360}
{"video_id": "y9IKpjK93rg", "transcript": [{"start": 0.0, "duration": 2.0, "text": " "}, {"start": 3.72, "duration": 3.47, "text": "working with fractions in we're gonna do\nin this case is just you"}, {"start": 7.19, "duration": 4.47, "text": "a little division involving fractions\nand the main thing"}, {"start": 11.66, "duration": 3.08, "text": "you need to remember when dividing\nfractions"}, {"start": 14.74, "duration": 3.299, "text": "is that we turned division into"}, {"start": 18.039, "duration": 3.711, "text": "multiplication so typically the ideas"}, {"start": 21.75, "duration": 5.019, "text": "we have one fraction say a/b divided\nby another fraction"}, {"start": 26.769, "duration": 4.51, "text": "c/d what we do is we leave that the\nfirst alone"}, {"start": 31.279, "duration": 4.37, "text": "that one we're dividing by we we\nactually flipped end over"}, {"start": 35.649, "duration": 3.511, "text": "so the "d" goes on top, the "c" goes on the\nbottom"}, {"start": 39.16, "duration": 4.64, "text": "and that's it you're the good thing\nabout multiplication and as"}, {"start": 43.8, "duration": 3.669, "text": "when you multiply fractions you just\nmultiply whatever's on top"}, {"start": 47.469, "duration": 3.701, "text": "so "a" times "d" and then you just multiply\nwhatever's on the bottom"}, {"start": 51.17, "duration": 4.54, "text": ""b" times "c" so remember if you are adding or\nsubtracting fractions you have to have"}, {"start": 55.71, "duration": 1.64, "text": "common denominators but"}, {"start": 57.35, "duration": 3.849, "text": "multiplications actually a little bit\neasier because you just multiply across"}, {"start": 61.199, "duration": 1.691, "text": "the top and across the bottom in"}, {"start": 62.89, "duration": 3.37, "text": "if you can reduce you simplify you do\nbut I am"}, {"start": 66.26, "duration": 5.74, "text": "added net that's that's it so a case on\nthe first one we have six divided by two"}, {"start": 72.0, "duration": 1.01, "text": "over seven"}, {"start": 73.01, "duration": 3.539, "text": "the first thing I always do if I have\nany"}, {"start": 76.549, "duration": 3.75, "text": "member that's not you know not it's not\nwritten as a fraction we can always"}, {"start": 80.299, "duration": 0.551, "text": "write"}, {"start": 80.85, "duration": 3.74, "text": "any number as a fraction by just dividing\nit by one"}, {"start": 84.59, "duration": 3.34, "text": "6 divided by one is the same thing as\nsix"}, {"start": 87.93, "duration": 3.29, "text": "okay so we were dividing by to over\nseven"}, {"start": 91.22, "duration": 5.09, "text": "and now all I'm gonna do you just apply\nmy little rule here and I'm flip the"}, {"start": 96.31, "duration": 1.04, "text": "second fraction"}, {"start": 97.35, "duration": 3.17, "text": "and make this six over one times"}, {"start": 100.52, "duration": 3.26, "text": "7 over two you and we can"}, {"start": 103.78, "duration": 3.619, "text": "do this we can multiply either cancel\nfirst but"}, {"start": 107.399, "duration": 3.83, "text": "maybe we just multiply six times seven\nis 42"}, {"start": 111.229, "duration": 5.151, "text": "divided by two and forty-two divided\nby two is"}, {"start": 116.38, "duration": 4.379, "text": "21 notice the other thing we could have done is we could have \n canceled we could have said"}, {"start": 120.759, "duration": 3.191, "text": "you know so there's a 6 on top and two on the\nbottom"}, {"start": 123.95, "duration": 3.279, "text": "well 2 goes into 2 one time when times"}, {"start": 127.229, "duration": 5.471, "text": "and two will go into  six 3 times so we simplify\nwe will get  three times seven"}, {"start": 132.7, "duration": 4.89, "text": "which would be twenty-one and then on the\nbottom we have one times one so you can"}, {"start": 137.59, "duration": 1.619, "text": "get 21 over one war"}, {"start": 139.209, "duration": 5.811, "text": "21 so typically I tried to cancel things\nout before and multiply I found"}, {"start": 145.02, "duration": 3.25, "text": "in my own experience and it makes things\na little quick and a little easier but"}, {"start": 148.27, "duration": 5.189, "text": "as long as you simplify at the end it doesn't  really\nmatter okay so what's your other example"}, {"start": 153.459, "duration": 0.451, "text": "here"}, {"start": 153.91, "duration": 3.79, "text": "we have -7/4 divided by 4/7 same idea"}, {"start": 157.7, "duration": 4.2, "text": "we just leave the first fraction 11 so\nnegative 7 number 4"}, {"start": 161.9, "duration": 3.699, "text": "and then i turn my division into\nmultiplication"}, {"start": 165.599, "duration": 4.851, "text": "in my second fraction I just flip it\nover so now the 7 goes on top"}, {"start": 170.45, "duration": 6.92, "text": "and the four goes on the bottom and again now I just\n multiply the numbers on top so I have -7"}, {"start": 177.37, "duration": 0.63, "text": "times"}, {"start": 178.0, "duration": 3.72, "text": "positive 7 that will give me a negative 49"}, {"start": 181.72, "duration": 3.939, "text": "and then 4 times 4 is 16"}, {"start": 185.659, "duration": 3.811, "text": "and in this case nothing will cancel out or\n reduced so this from your"}, {"start": 189.47, "duration": 3.39, "text": "final answer okay so nothing too bad"}, {"start": 192.86, "duration": 3.25, "text": "again the main thing with dividing\nfractions you have to remember"}, {"start": 196.11, "duration": 3.33, "text": "division to do that we turn it into\nmultiplication"}, {"start": 199.44, "duration": 3.6, "text": "and again when multiplying you just do the \ncross the top across the bottom"}, {"start": 203.04, "duration": 4.0, "text": "and you know if you can I would say make\nuse of any canceling that's out there"}], "title": "Basic Math:  Dividing Fractions", "description": "Thanks to all of you who support me on Patreon. You da real mvps!   $1 per month helps!! :)  https://www.patreon.com/patrickjmt !! Basic Math:  Dividing Fractions.  In this video, I do two examples.  In the first I divide a whole number by a fraction and in the second I divide a fraction by a fraction.", "lengthSeconds": 208, "width": 480, "height": 360}
{"video_id": "dnpXhVsTHL4", "transcript": [{"start": 0.88, "duration": 3.6, "text": "all right here we're going to start"}, {"start": 1.959, "duration": 4.161, "text": "looking at some videos uh related to"}, {"start": 4.48, "duration": 4.039, "text": "converting between fractions and"}, {"start": 6.12, "duration": 4.24, "text": "decimals and um kind of in general going"}, {"start": 8.519, "duration": 3.841, "text": "from fractions decimals and even"}, {"start": 10.36, "duration": 5.479, "text": "percentages is something that you know"}, {"start": 12.36, "duration": 5.24, "text": "to me that's uh that's useful math to be"}, {"start": 15.839, "duration": 3.44, "text": "able to to understand those things and"}, {"start": 17.6, "duration": 3.519, "text": "to be able to switch from one to the"}, {"start": 19.279, "duration": 3.881, "text": "other so what we're going to do though"}, {"start": 21.119, "duration": 4.4, "text": "is we're going to take decimals um and"}, {"start": 23.16, "duration": 5.0, "text": "we're going to write them as"}, {"start": 25.519, "duration": 5.281, "text": "fractions so kind of the basic procedure"}, {"start": 28.16, "duration": 5.64, "text": "and this to me is what I do in you know"}, {"start": 30.8, "duration": 4.84, "text": "any type of uh uh problem like this the"}, {"start": 33.8, "duration": 5.64, "text": "first thing I you know I do is just"}, {"start": 35.64, "duration": 5.399, "text": "always put everything over one I mean"}, {"start": 39.44, "duration": 4.48, "text": "you can write any number by dividing it"}, {"start": 41.039, "duration": 4.721, "text": "by one 0.4 divided by one is certainly"}, {"start": 43.92, "duration": 4.08, "text": "still"}, {"start": 45.76, "duration": 3.92, "text": "0.4 and now what we want to do is simply"}, {"start": 48.0, "duration": 4.16, "text": "get rid of decimals and to get rid of"}, {"start": 49.68, "duration": 6.76, "text": "decimals you'll either multiply by you"}, {"start": 52.16, "duration": 7.48, "text": "know 10 over 10 or 100 over 100 or a th"}, {"start": 56.44, "duration": 6.719, "text": "over a thousand Etc and again all that's"}, {"start": 59.64, "duration": 6.04, "text": "going to do is uh move the decimal"}, {"start": 63.159, "duration": 5.361, "text": "place so in this case we're going to"}, {"start": 65.68, "duration": 6.0, "text": "multiply by 10/"}, {"start": 68.52, "duration": 5.04, "text": "10 so again multiplying by 10 basically"}, {"start": 71.68, "duration": 4.479, "text": "has the effect of moving the decimal"}, {"start": 73.56, "duration": 6.239, "text": "place one to the"}, {"start": 76.159, "duration": 7.481, "text": "right well 0.4 * 10 then will be uh"}, {"start": 79.799, "duration": 5.921, "text": "simply equal to 4 1 * 10 is just 10 and"}, {"start": 83.64, "duration": 4.519, "text": "hey now we've got our decimal as a"}, {"start": 85.72, "duration": 4.719, "text": "fraction but of course we can reduce"}, {"start": 88.159, "duration": 5.481, "text": "this one a little bit they're both even"}, {"start": 90.439, "duration": 6.36, "text": "so we could write the numerator as 2 * 2"}, {"start": 93.64, "duration": 5.96, "text": "the denominator as 2 * 5 we could cancel"}, {"start": 96.799, "duration": 8.521, "text": "out those common factors of two and be"}, {"start": 99.6, "duration": 11.04, "text": "left with 2 over 5 or two fths so 0.4 is"}, {"start": 105.32, "duration": 5.32, "text": "the same fraction as 2 over5"}], "title": "Writing a Decimal as a Fraction, Example 1", "description": "Thanks to all of you who support me on Patreon. You da real mvps!   $1 per month helps!! :)  https://www.patreon.com/patrickjmt !! Writing a Decimal as a Fraction, Example 1.  Here I look at how to convert a decimal into a reduced fraction.", "lengthSeconds": 109, "width": 480, "height": 360}
{"video_id": "R7EtMAiKrGA", "transcript": [{"start": 0.799, "duration": 3.361, "text": "all right in this video I'm going to do"}, {"start": 2.12, "duration": 3.92, "text": "another example of solving some inqu an"}, {"start": 4.16, "duration": 4.16, "text": "inequality and again this is just a"}, {"start": 6.04, "duration": 3.96, "text": "linear inequality any linear"}, {"start": 8.32, "duration": 3.52, "text": "inequalities are the the kind of the"}, {"start": 10.0, "duration": 4.0, "text": "nice inequalities you'll see as you go"}, {"start": 11.84, "duration": 4.12, "text": "further that when you have um say for"}, {"start": 14.0, "duration": 4.039, "text": "example quadratic inequalities there's"}, {"start": 15.96, "duration": 4.479, "text": "actually a bit more of a process it gets"}, {"start": 18.039, "duration": 5.201, "text": "a a little more um a little more"}, {"start": 20.439, "duration": 5.441, "text": "cumbersome but so for now all I'm going"}, {"start": 23.24, "duration": 4.24, "text": "to do is again just kind of pretend this"}, {"start": 25.88, "duration": 3.28, "text": "is an equation and just solve it you"}, {"start": 27.48, "duration": 3.92, "text": "know using my same technique so I'm"}, {"start": 29.16, "duration": 6.96, "text": "going to distribute to 5 so we would get"}, {"start": 31.4, "duration": 7.64, "text": "5 * x 5 * 3 would be POS 15 on the right"}, {"start": 36.12, "duration": 5.64, "text": "if I distribute we'll get 2 * X and then"}, {"start": 39.04, "duration": 5.72, "text": "2 * -5 will give us"}, {"start": 41.76, "duration": 4.6, "text": "-10 um I like to keep my x's positive so"}, {"start": 44.76, "duration": 5.319, "text": "since the bigger one's on the left I'm"}, {"start": 46.36, "duration": 5.12, "text": "going to subtract 2x from both sides so"}, {"start": 50.079, "duration": 6.64, "text": "that'll leave me with"}, {"start": 51.48, "duration": 7.399, "text": "3x + 15 less than or equal to"}, {"start": 56.719, "duration": 5.16, "text": "-10 and again now I'm just going to"}, {"start": 58.879, "duration": 5.961, "text": "subtract the 15 from both sides that'll"}, {"start": 61.879, "duration": 4.721, "text": "give me 3x is less than or equal"}, {"start": 64.84, "duration": 4.4, "text": "to"}, {"start": 66.6, "duration": 6.32, "text": "25 and now I'm just going to divide both"}, {"start": 69.24, "duration": 7.04, "text": "sides by three divide by 3 and this says"}, {"start": 72.92, "duration": 7.0, "text": "X has to be less than or equal to"}, {"start": 76.28, "duration": 6.68, "text": "-25 over 3 again if we wanted to write"}, {"start": 79.92, "duration": 4.44, "text": "this in interval notation um first I I"}, {"start": 82.96, "duration": 3.199, "text": "always like to make a little graph I"}, {"start": 84.36, "duration": 4.16, "text": "almost think about it in my head so"}, {"start": 86.159, "duration": 5.241, "text": "here's - 255 over3 I'm just kind of"}, {"start": 88.52, "duration": 5.919, "text": "making a little number line arbitrarily"}, {"start": 91.4, "duration": 4.719, "text": "it can be less than or equal to that's"}, {"start": 94.439, "duration": 3.281, "text": "what the little line means that number"}, {"start": 96.119, "duration": 3.881, "text": "so now we're going to shade it in to"}, {"start": 97.72, "duration": 4.759, "text": "indicate that that number works and then"}, {"start": 100.0, "duration": 4.28, "text": "it says everything less than that also"}, {"start": 102.479, "duration": 3.161, "text": "works so we would just keep going going"}, {"start": 104.28, "duration": 5.08, "text": "going going going"}, {"start": 105.64, "duration": 5.72, "text": "forever negative Infinity uh so again"}, {"start": 109.36, "duration": 5.16, "text": "just to indicate that you go arbitrarily"}, {"start": 111.36, "duration": 5.16, "text": "far to the left um so if we write this"}, {"start": 114.52, "duration": 3.76, "text": "in interval notation we'll write a"}, {"start": 116.52, "duration": 5.639, "text": "little parentheses around our negative"}, {"start": 118.28, "duration": 7.0, "text": "infinity and then we have - 255 to 3"}, {"start": 122.159, "duration": 5.721, "text": "excuse me - 255 over 3 and since we can"}, {"start": 125.28, "duration": 4.479, "text": "use that number the notation is we put a"}, {"start": 127.88, "duration": 4.4, "text": "little bracket around that number to"}, {"start": 129.759, "duration": 3.761, "text": "indicate that we can use it I always you"}, {"start": 132.28, "duration": 2.72, "text": "know kind of thought about the little"}, {"start": 133.52, "duration": 2.96, "text": "bracket you know I always imagine those"}, {"start": 135.0, "duration": 3.04, "text": "were like little hooks or little"}, {"start": 136.48, "duration": 3.16, "text": "barriers and that would catch that"}, {"start": 138.04, "duration": 4.04, "text": "number and keep it from getting away so"}, {"start": 139.64, "duration": 4.239, "text": "that we include it so that was my little"}, {"start": 142.08, "duration": 3.4, "text": "weird way of kind of remembering the"}, {"start": 143.879, "duration": 5.08, "text": "difference in notation and if it's"}, {"start": 145.48, "duration": 7.36, "text": "rounded um you know if we didn't include"}, {"start": 148.959, "duration": 7.0, "text": "um if we didn't include -25"}, {"start": 152.84, "duration": 5.08, "text": "over3 we would use parentheses and those"}, {"start": 155.959, "duration": 4.0, "text": "to me are kind of more rounded and kind"}, {"start": 157.92, "duration": 4.56, "text": "of uh not as tight and the number"}, {"start": 159.959, "duration": 3.721, "text": "somehow sneaks away so all right"}, {"start": 162.48, "duration": 3.08, "text": "probably more information than you"}, {"start": 163.68, "duration": 3.559, "text": "wanted to need uh needed to know but"}, {"start": 165.56, "duration": 3.88, "text": "again just kind of a little way to maybe"}, {"start": 167.239, "duration": 4.401, "text": "I I've seen uh in my experience it seems"}, {"start": 169.44, "duration": 5.079, "text": "like people forget this notation um"}, {"start": 171.64, "duration": 5.0, "text": "quite frequently so just you know um"}, {"start": 174.519, "duration": 3.881, "text": "however you can remember it uh just try"}, {"start": 176.64, "duration": 5.2, "text": "to internalize it and remember it any"}, {"start": 178.4, "duration": 3.44, "text": "way that works for you"}], "title": "An Introduction To Solving Linear Inequalities - Example 2", "description": "Thanks to all of you who support me on Patreon. You da real mvps!   $1 per month helps!! :)  https://www.patreon.com/patrickjmt !! An Introduction To Solving Inequalities - Example 2", "lengthSeconds": 180, "width": 480, "height": 360}
{"video_id": "e0K-gWF9Y5g", "transcript": [{"start": 0.0, "duration": 3.899, "text": "all right in this video I just want to"}, {"start": 1.979, "duration": 4.831, "text": "do a couple examples of simplifying"}, {"start": 3.899, "duration": 6.181, "text": "products of binomials involving trig"}, {"start": 6.81, "duration": 8.55, "text": "functions so just some binomial"}, {"start": 10.08, "duration": 7.74, "text": "expressions I'm just going to clean them"}, {"start": 15.36, "duration": 5.19, "text": "up here a little bit so in the first one"}, {"start": 17.82, "duration": 6.78, "text": "we have cosine X minus one times cosine"}, {"start": 20.55, "duration": 5.549, "text": "X plus one if we were to distribute this"}, {"start": 24.6, "duration": 5.13, "text": "out we would get cosine times cosine"}, {"start": 26.099, "duration": 5.821, "text": "which would be cosine squared X we would"}, {"start": 29.73, "duration": 6.03, "text": "get cosine x positive one which would be"}, {"start": 31.92, "duration": 6.12, "text": "positive cosine X then we get negative"}, {"start": 35.76, "duration": 4.709, "text": "one times cosine X X which would be"}, {"start": 38.04, "duration": 4.17, "text": "negative cosine X then we would get"}, {"start": 40.469, "duration": 4.651, "text": "negative one times positive one which"}, {"start": 42.21, "duration": 5.46, "text": "would be negative one well we've got a"}, {"start": 45.12, "duration": 5.25, "text": "positive positive cosine and a negative"}, {"start": 47.67, "duration": 6.09, "text": "cosine that's just canceled out so we'll"}, {"start": 50.37, "duration": 5.939, "text": "be left with cosine squared X minus one"}, {"start": 53.76, "duration": 6.54, "text": "but recall we've got again this identity"}, {"start": 56.309, "duration": 8.1, "text": "cosine squared X plus sine squared x"}, {"start": 60.3, "duration": 6.21, "text": "equals 1 so we can always I think sort"}, {"start": 64.409, "duration": 4.411, "text": "of manipulate this identity to make it"}, {"start": 66.51, "duration": 5.37, "text": "look like this so notice we could"}, {"start": 68.82, "duration": 5.94, "text": "subtract one from both sides that would"}, {"start": 71.88, "duration": 7.11, "text": "give us cosine squared X plus sine"}, {"start": 74.76, "duration": 5.7, "text": "squared X minus 1 equals 0 and then we"}, {"start": 78.99, "duration": 4.86, "text": "could also subtract the sine squared"}, {"start": 80.46, "duration": 6.55, "text": "from both sides and get cosine squared X"}, {"start": 83.85, "duration": 7.15, "text": "minus 1 equals negative"}, {"start": 87.01, "duration": 6.06, "text": "sine squared X so which one is"}, {"start": 91.0, "duration": 3.9, "text": "preferable I don't know probably the"}, {"start": 93.07, "duration": 6.48, "text": "second one just because there's you know"}, {"start": 94.9, "duration": 7.2, "text": "one fewer term so that's it so I would"}, {"start": 99.55, "duration": 7.29, "text": "say the first one just simply reduces to"}, {"start": 102.1, "duration": 6.39, "text": "negative sine squared X maybe let's look"}, {"start": 106.84, "duration": 4.14, "text": "at the second one real quick as well"}, {"start": 108.49, "duration": 6.21, "text": "we've got secant X plus tangent x times"}, {"start": 110.98, "duration": 5.28, "text": "secant X minus tangent X you're going to"}, {"start": 114.7, "duration": 4.74, "text": "do the same thing which is just simply"}, {"start": 116.26, "duration": 4.98, "text": "to distribute it out so we would have"}, {"start": 119.44, "duration": 6.51, "text": "secant times secant which would be"}, {"start": 121.24, "duration": 6.63, "text": "secant squared X we would have secant"}, {"start": 125.95, "duration": 7.56, "text": "times negative tangent which would be"}, {"start": 127.87, "duration": 7.65, "text": "negative secant X times tangent X on the"}, {"start": 133.51, "duration": 3.84, "text": "inside we would have a positive tangent"}, {"start": 135.52, "duration": 3.84, "text": "times secant but since it's"}, {"start": 137.35, "duration": 3.72, "text": "multiplication I can order it however I"}, {"start": 139.36, "duration": 5.07, "text": "want so I'm going to write secant x"}, {"start": 141.07, "duration": 5.37, "text": "times tangent X and then we would have a"}, {"start": 144.43, "duration": 4.2, "text": "positive tangent x times a negative"}, {"start": 146.44, "duration": 6.42, "text": "tangent X so that's going to give us a"}, {"start": 148.63, "duration": 6.42, "text": "negative tangent squared X so our"}, {"start": 152.86, "duration": 4.23, "text": "negative secant tangent and our positive"}, {"start": 155.05, "duration": 5.13, "text": "secant tangent will cancel"}, {"start": 157.09, "duration": 7.41, "text": "we'll have secant squared X minus"}, {"start": 160.18, "duration": 6.3, "text": "tangent squared X but again there's an"}, {"start": 164.5, "duration": 5.1, "text": "identity relating secant squared and"}, {"start": 166.48, "duration": 7.83, "text": "tangent squared so tangent squared X"}, {"start": 169.6, "duration": 6.87, "text": "plus 1 equals secant squared X so I"}, {"start": 174.31, "duration": 4.86, "text": "think if we subtract the tangent squared"}, {"start": 176.47, "duration": 6.94, "text": "from both sides we'll get that one"}, {"start": 179.17, "duration": 7.84, "text": "equals secant squared X minus"}, {"start": 183.41, "duration": 5.34, "text": "tangent squared X so that's what we're"}, {"start": 187.01, "duration": 3.99, "text": "left with over here on the left so"}, {"start": 188.75, "duration": 4.74, "text": "secant squared minus tangent squared"}, {"start": 191.0, "duration": 4.68, "text": "just by manipulating that one sort of"}, {"start": 193.49, "duration": 3.99, "text": "known identity we can see that secant"}, {"start": 195.68, "duration": 4.79, "text": "squared minus tangent squared is simply"}, {"start": 197.48, "duration": 2.99, "text": "going to equal one"}], "title": "Simplifying Products of Binomials Involving Trigonometric Functions, Ex 1", "description": "Thanks to all of you who support me on Patreon. You da real mvps!   $1 per month helps!! :)  https://www.patreon.com/patrickjmt !! Simplifying Products of Binomials Involving Trigonometric Functions, Ex 1", "lengthSeconds": 204, "width": 480, "height": 360}
{"video_id": "ZQjq-o9ZBek", "transcript": [{"start": 0.76, "duration": 2.72, "text": "okay in this video we're going to talk a"}, {"start": 1.959, "duration": 3.481, "text": "little bit about the conjugate pair"}, {"start": 3.48, "duration": 3.279, "text": "theorem and what the conjugate pair"}, {"start": 5.44, "duration": 3.88, "text": "theorem says it says if you have a"}, {"start": 6.759, "duration": 5.321, "text": "complex number of the form for example"}, {"start": 9.32, "duration": 6.12, "text": "A+ bi if we know that that's a zero of"}, {"start": 12.08, "duration": 7.0, "text": "our polinomial then it says a minus bi"}, {"start": 15.44, "duration": 9.04, "text": "is also a zero conversely if uh a minus"}, {"start": 19.08, "duration": 8.279, "text": "bi is a zero then A+ bi is a zero so"}, {"start": 24.48, "duration": 6.959, "text": "this it's this idea that complex numbers"}, {"start": 27.359, "duration": 7.4, "text": "the zeros appear in pairs okay"}, {"start": 31.439, "duration": 3.32, "text": "so suppose"}, {"start": 35.36, "duration": 5.84, "text": "uh suppose we know that f of x is a"}, {"start": 38.52, "duration": 5.32, "text": "polom and it's got real coefficients and"}, {"start": 41.2, "duration": 8.28, "text": "it has the following zeros Square < TK"}, {"start": 43.84, "duration": 7.28, "text": "of 3 netive I 5 - 4 I and 1 + I / 8"}, {"start": 49.48, "duration": 5.32, "text": "suppose those are all zeros and we want"}, {"start": 51.12, "duration": 6.759, "text": "to know what three other zeros would be"}, {"start": 54.8, "duration": 6.64, "text": "well we can write um negative I as a"}, {"start": 57.879, "duration": 8.201, "text": "complex number by writing it as 0 - well"}, {"start": 61.44, "duration": 9.24, "text": "1 I and since 0 - 1 I is a0 that would"}, {"start": 66.08, "duration": 8.6, "text": "imply that 0 + 1 I which just equals"}, {"start": 70.68, "duration": 6.479, "text": "positive I is also a"}, {"start": 74.68, "duration": 8.079, "text": "zero okay so just flip the sign on and"}, {"start": 77.159, "duration": 7.64, "text": "is all it says since 5 - 4 I is a zero"}, {"start": 82.759, "duration": 4.241, "text": "again the conjugate all we do is just"}, {"start": 84.799, "duration": 5.881, "text": "change the sign in the middle so since 5"}, {"start": 87.0, "duration": 7.96, "text": "- 4 I is a zero that means 5"}, {"start": 90.68, "duration": 4.28, "text": "+ 4 I is also a"}, {"start": 95.56, "duration": 7.28, "text": "zero and 1 + I over 8 since that's a"}, {"start": 101.04, "duration": 3.64, "text": "zero again you can think about this as"}, {"start": 102.84, "duration": 6.239, "text": "you know you don't have to do this we"}, {"start": 104.68, "duration": 6.92, "text": "could write as 1/8 plus I over"}, {"start": 109.079, "duration": 4.281, "text": "8 and since that's a zero again we'll"}, {"start": 111.6, "duration": 5.439, "text": "just change the sign in the"}, {"start": 113.36, "duration": 8.64, "text": "middle we could make it 1/8 - I over 8"}, {"start": 117.039, "duration": 7.961, "text": "or equivalently 1 - i/ rate is also a"}, {"start": 122.0, "duration": 3.0, "text": "zero"}], "title": "The Conjugate Pair Theorem - Example 1", "description": "Thanks to all of you who support me on Patreon. You da real mvps!   $1 per month helps!! :)  https://www.patreon.com/patrickjmt !! The Conjugate Pair Theorem - Example 1.  In this video, I show that if we know a complex root, we can use that to find another complex root using the conjugate pair theorem.  Just a basic idea of the theorem.", "lengthSeconds": 126, "width": 480, "height": 360}
{"video_id": "ES2Lwzrw_UE", "transcript": [{"start": 0.76, "duration": 4.32, "text": "all right in this video I want to find"}, {"start": 2.32, "duration": 4.16, "text": "the Lao transform of a function and here"}, {"start": 5.08, "duration": 4.24, "text": "we're going to use uh we're going to"}, {"start": 6.48, "duration": 7.92, "text": "find the llao transform of 4 * cosine of"}, {"start": 9.32, "duration": 8.719, "text": "2T - 5T + 2 * e"}, {"start": 14.4, "duration": 6.039, "text": "3T so this uh won't be too bad at all so"}, {"start": 18.039, "duration": 3.801, "text": "if we take the llao transform of of all"}, {"start": 20.439, "duration": 5.0, "text": "of"}, {"start": 21.84, "duration": 8.88, "text": "this so we'll have the L"}, {"start": 25.439, "duration": 5.281, "text": "transform of 4 cosine of 2T"}, {"start": 30.759, "duration": 5.48, "text": "remember the lass transform is linear so"}, {"start": 33.04, "duration": 5.32, "text": "we can break all this up minus the L"}, {"start": 36.239, "duration": 7.961, "text": "transform of"}, {"start": 38.36, "duration": 8.64, "text": "5T plus the Lelo transform of 2 e to the"}, {"start": 44.2, "duration": 4.519, "text": "3T but also recall that for uh"}, {"start": 47.0, "duration": 3.28, "text": "coefficients we can simply Factor the"}, {"start": 48.719, "duration": 3.561, "text": "coefficients out"}, {"start": 50.28, "duration": 4.959, "text": "front so really we're just going to be"}, {"start": 52.28, "duration": 5.799, "text": "left with uh four * the LL"}, {"start": 55.239, "duration": 8.6, "text": "transform of cosine"}, {"start": 58.079, "duration": 11.281, "text": "2T okay minus 5 * L transform of"}, {"start": 63.839, "duration": 7.561, "text": "t plus 2 * LL transform of e to the"}, {"start": 69.36, "duration": 4.119, "text": "3T and now we're just going to use our"}, {"start": 71.4, "duration": 4.64, "text": "table much like you would use you know a"}, {"start": 73.479, "duration": 5.561, "text": "table of derivatives when you first"}, {"start": 76.04, "duration": 6.2, "text": "start doing calculus so let's see here"}, {"start": 79.04, "duration": 7.079, "text": "let's find our table so let's see um so"}, {"start": 82.24, "duration": 7.199, "text": "for cosine of 8 we said there you get s/"}, {"start": 86.119, "duration": 4.881, "text": "s^2 + a 2 so in this case our number"}, {"start": 89.439, "duration": 5.121, "text": "being squ is"}, {"start": 91.0, "duration": 7.799, "text": "2 so the luse transform we'll get"}, {"start": 94.56, "duration": 7.48, "text": "s/ s^2 plus again our a is going to be"}, {"start": 98.799, "duration": 8.0, "text": "two so we'll get 2^"}, {"start": 102.04, "duration": 9.119, "text": "2 minus 5 * the Lao transform of t uh"}, {"start": 106.799, "duration": 7.441, "text": "the Lao transform at just T where' It Go"}, {"start": 111.159, "duration": 5.361, "text": "um so the Lelo transform if it's t to"}, {"start": 114.24, "duration": 5.04, "text": "the first Power well you would get uh"}, {"start": 116.52, "duration": 5.12, "text": "one factorial which would just be well"}, {"start": 119.28, "duration": 4.759, "text": "one let's write it out we would get 1"}, {"start": 121.64, "duration": 6.88, "text": "factorial and then we would get S raised"}, {"start": 124.039, "duration": 7.081, "text": "to the 1 + 1 or we would get s^"}, {"start": 128.52, "duration": 6.76, "text": "squar um let's see"}, {"start": 131.12, "duration": 5.8, "text": "here notice uh well excuse me so last"}, {"start": 135.28, "duration": 4.36, "text": "but not least if we just take the Lael"}, {"start": 136.92, "duration": 5.64, "text": "transform of e to the 3T no no noticing"}, {"start": 139.64, "duration": 6.64, "text": "anything uh let's see so we said if you"}, {"start": 142.56, "duration": 8.16, "text": "have e to the a you get 1 / s minus the"}, {"start": 146.28, "duration": 8.12, "text": "a value so in this case we'll get 1/ s -"}, {"start": 150.72, "duration": 5.84, "text": "3 and now this is our lass transform um"}, {"start": 154.4, "duration": 3.72, "text": "you're pretty I mean you you found it at"}, {"start": 156.56, "duration": 3.0, "text": "this point um you can always just"}, {"start": 158.12, "duration": 3.92, "text": "multiply the"}, {"start": 159.56, "duration": 5.12, "text": "constants whoops let's make that a plus"}, {"start": 162.04, "duration": 5.559, "text": "4 instead of 2^ s um you know you can"}, {"start": 164.68, "duration": 5.68, "text": "always just multiply the constant back"}, {"start": 167.599, "duration": 5.041, "text": "in but that's all there is to it uh just"}, {"start": 170.36, "duration": 4.4, "text": "kind of knowing the table and then just"}, {"start": 172.64, "duration": 6.08, "text": "kind of taking the Lao transform of each"}, {"start": 174.76, "duration": 3.96, "text": "function uh individually"}], "title": "Calculating the Laplace Transform of a Function Using Tables", "description": "Thanks to all of you who support me on Patreon. You da real mvps!   $1 per month helps!! :)  https://www.patreon.com/patrickjmt !! Calculating the Laplace Transform of a Function Using Tables.  In this video, we find the Laplace transform of a function using linearity and tables.  This is a very basic example!", "lengthSeconds": 177, "width": 450, "height": 360}
{"video_id": "flD9km7AMPA", "transcript": [{"start": 1.06, "duration": 3.84, "text": "Okay, in this question it says if y = kx where k is a constant"}, {"start": 4.9, "duration": 2.94, "text": "and it says y = 24 when x = 6,"}, {"start": 7.84, "duration": 3.58, "text": "we wanna know what's the value of y when x = 5."}, {"start": 11.42, "duration": 2.24, "text": "So this one, you know, hopefully won't be too bad,"}, {"start": 13.66, "duration": 2.24, "text": "again it's just using this information that y = 24"}, {"start": 15.9, "duration": 2.66, "text": "and x = 6 that's gonna allow us to solve for k."}, {"start": 18.9, "duration": 5.92, "text": "Once we solve for k, we can substitute in also x = 5 and we'll get the value of y."}, {"start": 25.1, "duration": 5.06, "text": "So let's use this info that y = 24 when x = 6.."}, {"start": 30.16, "duration": 2.3, "text": "So I'm gonna put that on the left side of my equation,"}, {"start": 32.56, "duration": 3.52, "text": "so we've got 24 for y, again we're trying to solve for k."}, {"start": 37.2, "duration": 2.56, "text": "So k multiplied by x, which is gonna be 6.."}, {"start": 40.36, "duration": 3.2, "text": "So, k * 6, you could right this as 6k but whatever."}, {"start": 43.6, "duration": 3.12, "text": "So to solve for k I'm gonna divide both sides by 6.."}, {"start": 47.98, "duration": 2.32, "text": "24/6, that's just gonna give us 4.."}, {"start": 50.98, "duration": 1.92, "text": "On the right side, we have k.."}, {"start": 53.76, "duration": 5.46, "text": "So now, we know that our equation is of the form y = 4x,"}, {"start": 59.22, "duration": 1.54, "text": "again we just solved for k,"}, {"start": 60.82, "duration": 2.96, "text": "and now all I have to do is substitute in x = 5.."}, {"start": 64.6, "duration": 0.88, "text": "So not too bad."}, {"start": 65.48, "duration": 7.66, "text": "When x = 5, we'll get that y = 4 * 5, or, y will equal 20."}, {"start": 73.3, "duration": 5.06, "text": "So, answer choice C is the correct choice in this case."}], "title": "SAT Math - Question 2, Section 2, 2016 Edition and Beyond", "description": "Thanks to all of you who support me on Patreon. You da real mvps!   $1 per month helps!! :)  https://www.patreon.com/patrickjmt !! SAT Math - Question 2, 2016 Edition and Beyond.\nSection 2 - A Calculator IS allowed!", "lengthSeconds": 79, "width": 480, "height": 360}
{"video_id": "g8JCneEzkvk", "transcript": [{"start": 0.56, "duration": 3.04, "text": "all right so just a little word problem"}, {"start": 2.72, "duration": 3.599, "text": "here involving"}, {"start": 3.6, "duration": 4.48, "text": "fractions and division of fractions"}, {"start": 6.319, "duration": 5.121, "text": "suppose we've got"}, {"start": 8.08, "duration": 5.439, "text": "a board that's five and uh 1 4"}, {"start": 11.44, "duration": 3.279, "text": "feet long we're going to cut this up"}, {"start": 13.519, "duration": 3.201, "text": "into little"}, {"start": 14.719, "duration": 3.681, "text": "planks that are three-quarters of a foot"}, {"start": 16.72, "duration": 3.2, "text": "long and we want to know how many of"}, {"start": 18.4, "duration": 3.2, "text": "these little planks can we get"}, {"start": 19.92, "duration": 3.68, "text": "so you can probably see it through there"}, {"start": 21.6, "duration": 3.759, "text": "i've got my little picture here so"}, {"start": 23.6, "duration": 3.12, "text": "suppose we cut you know we've got our"}, {"start": 25.359, "duration": 4.481, "text": "board here"}, {"start": 26.72, "duration": 4.559, "text": "it's five and uh one-fourth feet long"}, {"start": 29.84, "duration": 3.04, "text": "and we're going to cut it up into these"}, {"start": 31.279, "duration": 4.641, "text": "little planks that are"}, {"start": 32.88, "duration": 5.44, "text": "uh you know three-fourths of a foot"}, {"start": 35.92, "duration": 3.84, "text": "long each and okay so i just put some"}, {"start": 38.32, "duration": 2.16, "text": "arbitrary number in there don't count"}, {"start": 39.76, "duration": 2.88, "text": "them up and"}, {"start": 40.48, "duration": 3.12, "text": "say oh that's not right because i doubt"}, {"start": 42.64, "duration": 2.56, "text": "it is so"}, {"start": 43.6, "duration": 3.6, "text": "but again that's the idea right we've"}, {"start": 45.2, "duration": 2.96, "text": "got a long piece of wood we want to chop"}, {"start": 47.2, "duration": 3.359, "text": "it up into"}, {"start": 48.16, "duration": 4.32, "text": "a bunch of smaller pieces and we want to"}, {"start": 50.559, "duration": 4.721, "text": "know how many of these smaller pieces"}, {"start": 52.48, "duration": 4.399, "text": "can we get well again what we're doing"}, {"start": 55.28, "duration": 4.72, "text": "is"}, {"start": 56.879, "duration": 6.48, "text": "we're taking five or uh"}, {"start": 60.0, "duration": 5.84, "text": "our piece of wood that's five and one uh"}, {"start": 63.359, "duration": 3.44, "text": "one-fourth feet long and we're going to"}, {"start": 65.84, "duration": 5.04, "text": "divide it"}, {"start": 66.799, "duration": 7.121, "text": "into pieces that are three-fourths"}, {"start": 70.88, "duration": 4.8, "text": "of a foot long each okay so that's what"}, {"start": 73.92, "duration": 3.519, "text": "we have to do we've got to take"}, {"start": 75.68, "duration": 3.28, "text": "the five and one-fourths and divide it"}, {"start": 77.439, "duration": 3.441, "text": "by three-fourths and"}, {"start": 78.96, "duration": 3.68, "text": "again sometimes if you have a problem"}, {"start": 80.88, "duration": 3.04, "text": "like this and maybe the numbers are kind"}, {"start": 82.64, "duration": 3.28, "text": "of throwing you off and you're like do"}, {"start": 83.92, "duration": 2.64, "text": "you have to multiply or divide or i"}, {"start": 85.92, "duration": 2.559, "text": "don't know"}, {"start": 86.56, "duration": 4.559, "text": "you know make it easier numbers suppose"}, {"start": 88.479, "duration": 4.801, "text": "it was 10 feet long"}, {"start": 91.119, "duration": 3.921, "text": "and suppose they were you know two feet"}, {"start": 93.28, "duration": 3.199, "text": "each"}, {"start": 95.04, "duration": 3.2, "text": "you know maybe this is a little bit"}, {"start": 96.479, "duration": 2.32, "text": "easier right you know if it's ten feet"}, {"start": 98.24, "duration": 2.96, "text": "long"}, {"start": 98.799, "duration": 3.68, "text": "and they're two feet each well okay um"}, {"start": 101.2, "duration": 2.72, "text": "yeah you can get five and"}, {"start": 102.479, "duration": 3.441, "text": "well how do you get that you take the"}, {"start": 103.92, "duration": 4.239, "text": "total length and you divide it by"}, {"start": 105.92, "duration": 3.92, "text": "you know the length of each little each"}, {"start": 108.159, "duration": 3.201, "text": "smaller piece"}, {"start": 109.84, "duration": 3.36, "text": "so sometimes i think about these"}, {"start": 111.36, "duration": 2.96, "text": "problems you know when i do these"}, {"start": 113.2, "duration": 2.72, "text": "problems"}, {"start": 114.32, "duration": 3.2, "text": "i think about them with easier numbers"}, {"start": 115.92, "duration": 2.4, "text": "just to sort of make sure that i'm doing"}, {"start": 117.52, "duration": 3.919, "text": "the right thing"}, {"start": 118.32, "duration": 4.079, "text": "in this case division so okay division"}, {"start": 121.439, "duration": 2.801, "text": "makes sense"}, {"start": 122.399, "duration": 4.32, "text": "and now i say oh i've got a mixed number"}, {"start": 124.24, "duration": 3.68, "text": "divided by you know a fraction"}, {"start": 126.719, "duration": 4.721, "text": "i'm going to make these both into"}, {"start": 127.92, "duration": 5.76, "text": "fractions 4 times 5 is 20 plus 1 will be"}, {"start": 131.44, "duration": 6.48, "text": "21 over 4"}, {"start": 133.68, "duration": 7.279, "text": "divided by 3 over 4."}, {"start": 137.92, "duration": 5.28, "text": "well now i'm going to make my"}, {"start": 140.959, "duration": 4.961, "text": "division into multiplication by"}, {"start": 143.2, "duration": 4.8, "text": "multiplying by the reciprocal"}, {"start": 145.92, "duration": 3.039, "text": "but now i can cancel out some factors"}, {"start": 148.0, "duration": 4.0, "text": "immediately"}, {"start": 148.959, "duration": 3.521, "text": "okay so the 4 and the 4 we could just"}, {"start": 152.0, "duration": 3.36, "text": "say well"}, {"start": 152.48, "duration": 4.16, "text": "4 divided by 4 is 1 4 divided by 4"}, {"start": 155.36, "duration": 3.519, "text": "divided by 4 is also"}, {"start": 156.64, "duration": 3.44, "text": "1. we can do the same thing with the 21"}, {"start": 158.879, "duration": 2.561, "text": "over the 3."}, {"start": 160.08, "duration": 4.159, "text": "i'm just going to be lazy and write it"}, {"start": 161.44, "duration": 6.0, "text": "though 21 times 1 is 21 over 3"}, {"start": 164.239, "duration": 6.72, "text": "well 21 divided by 3 is going to give us"}, {"start": 167.44, "duration": 5.04, "text": "7. so it says actually uh if we chop up"}, {"start": 170.959, "duration": 3.28, "text": "this piece of wood"}, {"start": 172.48, "duration": 3.759, "text": "this long piece of wood we should get"}, {"start": 174.239, "duration": 6.0, "text": "exactly one two three four five"}, {"start": 176.239, "duration": 7.36, "text": "six seven so um i've got too many"}, {"start": 180.239, "duration": 4.961, "text": "too many parts so it says you would get"}, {"start": 183.599, "duration": 5.28, "text": "exactly"}, {"start": 185.2, "duration": 5.92, "text": "uh seven little little uh planks of wood"}, {"start": 188.879, "duration": 3.761, "text": "that are three-fourths of a foot each"}, {"start": 191.12, "duration": 3.52, "text": "and again all we're doing"}, {"start": 192.64, "duration": 3.599, "text": "just a little bit of division again you"}, {"start": 194.64, "duration": 5.2, "text": "have to make your mixed number"}, {"start": 196.239, "duration": 5.441, "text": "into a fraction division by a fraction"}, {"start": 199.84, "duration": 3.92, "text": "turns into multiplication by the"}, {"start": 201.68, "duration": 3.6, "text": "reciprocal and then it's just a matter"}, {"start": 203.76, "duration": 5.119, "text": "of doing the arithmetic and"}, {"start": 205.28, "duration": 3.599, "text": "simplifying it down"}], "title": "Dividing Fractions and Mixed Numbers, Ex 3", "description": "Thanks to all of you who support me on Patreon. You da real mvps!   $1 per month helps!! :)  https://www.patreon.com/patrickjmt !! Dividing Fractions and Mixed Numbers, Ex 3", "lengthSeconds": 207, "width": 480, "height": 360}
{"video_id": "6eNE38s9Fv0", "transcript": [{"start": 0.48, "duration": 4.64, "text": "all right here we're going to do some"}, {"start": 1.52, "duration": 5.64, "text": "examples of finding mcloren pols so all"}, {"start": 5.12, "duration": 4.76, "text": "a Muran polinomial is it's just a"}, {"start": 7.16, "duration": 5.599, "text": "special case of a Taylor polinomial and"}, {"start": 9.88, "duration": 5.799, "text": "a mclen polinomial is by assumption"}, {"start": 12.759, "duration": 4.761, "text": "centered at a equals Zer so I just kind"}, {"start": 15.679, "duration": 6.401, "text": "of rewrote our new formula here"}, {"start": 17.52, "duration": 7.16, "text": "emphasizing the fact that a is equal to"}, {"start": 22.08, "duration": 5.4, "text": "zero all right so in this example we're"}, {"start": 24.68, "duration": 7.6, "text": "going to find the third degree mclen"}, {"start": 27.48, "duration": 6.96, "text": "polom for f ofx = e"}, {"start": 32.28, "duration": 3.88, "text": "4X so again I'm just going to write it"}, {"start": 34.44, "duration": 4.76, "text": "out generically because that's what I"}, {"start": 36.16, "duration": 8.16, "text": "like to do so we'll have F of"}, {"start": 39.2, "duration": 10.72, "text": "0 plus FR Prime of 0 over 1 factorial x"}, {"start": 44.32, "duration": 6.96, "text": "to the 1st F Prime of 0 over 2 factorial"}, {"start": 49.92, "duration": 6.319, "text": "*"}, {"start": 51.28, "duration": 8.36, "text": "x^2 and then F triple prime of 0 3"}, {"start": 56.239, "duration": 5.681, "text": "factorial * X cubed so let's see our"}, {"start": 59.64, "duration": 2.28, "text": "function"}, {"start": 62.239, "duration": 4.721, "text": "just e to 4X well these derivatives"}, {"start": 65.239, "duration": 3.88, "text": "aren't too bad we have to use the chain"}, {"start": 66.96, "duration": 4.92, "text": "rule so e to the 4X when we take the"}, {"start": 69.119, "duration": 4.04, "text": "derivative we'll get e to 4X but then we"}, {"start": 71.88, "duration": 4.52, "text": "have to take the derivative of the"}, {"start": 73.159, "duration": 4.96, "text": "exponent so we'll get uh an extra four"}, {"start": 76.4, "duration": 4.399, "text": "so we have 4 e"}, {"start": 78.119, "duration": 5.64, "text": "4X well when we take the second"}, {"start": 80.799, "duration": 6.561, "text": "derivative uh we'll get an e to 4X time"}, {"start": 83.759, "duration": 6.201, "text": "another four so that'll give us 16 * e"}, {"start": 87.36, "duration": 5.039, "text": "4X and then our third derivative"}, {"start": 89.96, "duration": 4.799, "text": "evaluated at zero we're going to have to"}, {"start": 92.399, "duration": 6.321, "text": "multiply by yet another four so that's"}, {"start": 94.759, "duration": 6.481, "text": "going to give us 64 * e"}, {"start": 98.72, "duration": 6.16, "text": "4X but when we plug in"}, {"start": 101.24, "duration": 8.12, "text": "zero f of0 will get e to the 0 which is"}, {"start": 104.88, "duration": 8.919, "text": "one F Prime of 0 we'll get well four * e"}, {"start": 109.36, "duration": 7.56, "text": "to the 0 are just four F Prime of 0"}, {"start": 113.799, "duration": 6.561, "text": "will'll be left with 16 * 1 or"}, {"start": 116.92, "duration": 5.28, "text": "16 and our F triple prime of 0 we'll"}, {"start": 120.36, "duration": 4.84, "text": "just be left with a"}, {"start": 122.2, "duration": 4.72, "text": "64 so now if I fill in you know our very"}, {"start": 125.2, "duration": 6.24, "text": "beginning"}, {"start": 126.92, "duration": 7.8, "text": "formula we will have our degree 3 muren"}, {"start": 131.44, "duration": 7.56, "text": "polinomial so F of0 we said is equal to"}, {"start": 134.72, "duration": 7.239, "text": "one F Prime of 0 is four so 4 over 1"}, {"start": 139.0, "duration": 6.84, "text": "factorial is Just 4 * X the"}, {"start": 141.959, "duration": 9.441, "text": "1 um f Prime of zero that's 16 divided"}, {"start": 145.84, "duration": 8.92, "text": "two which is 8 so 8 x^ SAR"}, {"start": 151.4, "duration": 8.479, "text": "let's see um f triple prime of 0 that's"}, {"start": 154.76, "duration": 8.68, "text": "64 over 3 factorial so 3 * 2 * 1 well"}, {"start": 159.879, "duration": 7.041, "text": "let's see two goes into 64 I guess 32"}, {"start": 163.44, "duration": 6.84, "text": "times so we'll be left with uh so F"}, {"start": 166.92, "duration": 8.08, "text": "triple prime so um 64 divided 6 we'll"}, {"start": 170.28, "duration": 7.8, "text": "reduce that to 32 over3 then we have our"}, {"start": 175.0, "duration": 7.36, "text": "x to the 3 power and this will now be"}, {"start": 178.08, "duration": 7.28, "text": "our degree 3 the chlorin"}, {"start": 182.36, "duration": 3.0, "text": "polom"}], "title": "\ud83d\udcda Finding the Third-Degree Taylor Polynomial for f(x) = e^(4x) \ud83d\udcda", "description": "\ud83d\udcda Finding the Third-Degree Taylor Polynomial for f(x) = e^(4x) \ud83d\udcda\n\nIn this video, I walk through the process of finding the third-degree Taylor polynomial for the function f(x) = e^(4x), centered at x = 0. This example shows how to apply the Taylor series formula to compute a polynomial approximation of an exponential function.\n\n\ud83d\ude80 What\u2019s covered:\n\nStep-by-step derivation of the Taylor series for e^(4x).\nDetailed explanation of how to compute the third-degree Taylor polynomial.\nHow to find higher-order derivatives and use them to build the polynomial approximation.\nUnderstanding Taylor polynomials is essential in calculus for approximating functions, so this is a great exercise to strengthen your skills in series expansions and polynomial approximations.\n\n\ud83d\udc4d Don\u2019t forget to like, subscribe, and hit the notification bell for more calculus tutorials and polynomial approximations! \ud83d\udd14\u2728\n\n\n#TaylorPolynomial #ExponentialFunction #TaylorSeries #MaclaurinSeries #Calculus #MathTutorial #Mathematics #SeriesExpansion #LearningMath #MathHelp #PolynomialApproximation #STEM", "lengthSeconds": 183, "width": 480, "height": 360}
{"video_id": "H4wW2Nl2E_g", "transcript": [{"start": 0.799, "duration": 3.001, "text": "okay so the next question again is going"}, {"start": 2.12, "duration": 3.639, "text": "to involve this graph that we saw in the"}, {"start": 3.8, "duration": 3.039, "text": "previous problem and it says which of"}, {"start": 5.759, "duration": 4.561, "text": "the following represents the"}, {"start": 6.839, "duration": 5.001, "text": "relationship between H and C so again"}, {"start": 10.32, "duration": 4.399, "text": "I've labeled some points on here you"}, {"start": 11.84, "duration": 4.88, "text": "could always start substituting in"}, {"start": 14.719, "duration": 4.521, "text": "values right the first cordinate"}, {"start": 16.72, "duration": 4.2, "text": "represents H the second coordinate"}, {"start": 19.24, "duration": 4.959, "text": "represents"}, {"start": 20.92, "duration": 5.64, "text": "C so again the number of hours rented"}, {"start": 24.199, "duration": 5.121, "text": "comma the total cost"}, {"start": 26.56, "duration": 4.559, "text": "involved so there's two ways to do this"}, {"start": 29.32, "duration": 3.279, "text": "um"}, {"start": 31.119, "duration": 3.041, "text": "I mean if you wanted to the way I would"}, {"start": 32.599, "duration": 4.64, "text": "think about it is just think about it"}, {"start": 34.16, "duration": 7.239, "text": "like in terms of y = mx +"}, {"start": 37.239, "duration": 7.761, "text": "b right this is slope intercept"}, {"start": 41.399, "duration": 3.601, "text": "form slope"}, {"start": 45.879, "duration": 6.961, "text": "intercept well instead of Y the vertical"}, {"start": 49.559, "duration": 6.64, "text": "axis is labeled as capital"}, {"start": 52.84, "duration": 6.12, "text": "c instead of the variable X you know our"}, {"start": 56.199, "duration": 4.52, "text": "x axis is replaced by this variable H"}, {"start": 58.96, "duration": 3.36, "text": "the number of hours"}, {"start": 60.719, "duration": 3.12, "text": "but again the same thing m still"}, {"start": 62.32, "duration": 5.799, "text": "represents the"}, {"start": 63.839, "duration": 7.001, "text": "slope B still represents the you know"}, {"start": 68.119, "duration": 6.881, "text": "the normally we call it the Y intercept"}, {"start": 70.84, "duration": 4.16, "text": "but in this case it would be the C"}, {"start": 75.88, "duration": 5.08, "text": "intercept okay well the C intercept had"}, {"start": 78.28, "duration": 7.56, "text": "a value of five it's going through the"}, {"start": 80.96, "duration": 4.88, "text": "point 0 comma 5 so I know"}, {"start": 86.119, "duration": 5.561, "text": "that the plus b is going to be replaced"}, {"start": 88.92, "duration": 4.92, "text": "with + 5 well what's the slope that's"}, {"start": 91.68, "duration": 3.84, "text": "the rise over the run so how much do you"}, {"start": 93.84, "duration": 4.72, "text": "go up and how much do you go over to get"}, {"start": 95.52, "duration": 7.4, "text": "to the next Point well if we go from 1A"}, {"start": 98.56, "duration": 6.599, "text": "8 up to 2A 11 I'm going up 1 2 3 units"}, {"start": 102.92, "duration": 4.6, "text": "I'm going over one unit again my my"}, {"start": 105.159, "duration": 5.0, "text": "scale is definitely off here but you"}, {"start": 107.52, "duration": 4.639, "text": "could do slope change in again change in"}, {"start": 110.159, "duration": 4.96, "text": "y over change in X or in this case it"}, {"start": 112.159, "duration": 4.88, "text": "would be change in C over change in h"}, {"start": 115.119, "duration": 3.28, "text": "but let's think about it as before you"}, {"start": 117.039, "duration": 3.04, "text": "you do the difference of the second"}, {"start": 118.399, "duration": 5.481, "text": "coordinates over the difference of the"}, {"start": 120.079, "duration": 7.441, "text": "coordinates so you would have 11 - 8 so"}, {"start": 123.88, "duration": 6.04, "text": "I'm doing 11 - 8/ 2 -1 that's going to"}, {"start": 127.52, "duration": 4.96, "text": "give you 3 over 1 or three so that's"}, {"start": 129.92, "duration": 7.08, "text": "going to be the slope so the formula"}, {"start": 132.48, "duration": 7.2, "text": "here would be c = 3 H + 5 which would"}, {"start": 137.0, "duration": 4.08, "text": "also be answer Choice C and again once"}, {"start": 139.68, "duration": 3.279, "text": "you had a couple points on here you"}, {"start": 141.08, "duration": 4.4, "text": "could always start substituting these um"}, {"start": 142.959, "duration": 4.92, "text": "you know for example we know that 2"}, {"start": 145.48, "duration": 5.44, "text": "comma 11 is on there if we substitute in"}, {"start": 147.879, "duration": 6.161, "text": "2 for H and answer Choice a we would"}, {"start": 150.92, "duration": 5.24, "text": "have 5 * 2 which is 10 but we should be"}, {"start": 154.04, "duration": 4.36, "text": "getting the value of 11 so we know"}, {"start": 156.16, "duration": 3.88, "text": "answer Choice a is not correct you could"}, {"start": 158.4, "duration": 4.44, "text": "go through and do elimination and get"}, {"start": 160.04, "duration": 4.559, "text": "rid of B and D as well by doing that but"}, {"start": 162.84, "duration": 3.64, "text": "again you're just doing the slope"}, {"start": 164.599, "duration": 4.121, "text": "intercept method you know finding the"}, {"start": 166.48, "duration": 4.16, "text": "equation of a line it's just switching"}, {"start": 168.72, "duration": 4.04, "text": "out again instead of X and Y you've got"}, {"start": 170.64, "duration": 5.84, "text": "H and C is all your you know the only"}, {"start": 172.76, "duration": 3.72, "text": "kind of difference here"}], "title": "SAT Math - Question 16, Section 2, 2016 Edition and Beyond", "description": "Thanks to all of you who support me on Patreon. You da real mvps!   $1 per month helps!! :)  https://www.patreon.com/patrickjmt !! SAT Math - Question 16, 2016 Edition and Beyond.\nSection 2 - A Calculator IS allowed!", "lengthSeconds": 175, "width": 480, "height": 360}
{"video_id": "ZClpvlkxzWM", "transcript": [{"start": 0.03, "duration": 4.05, "text": "alright so another example here"}, {"start": 2.1, "duration": 3.929, "text": "simplifying one of our expressions so"}, {"start": 4.08, "duration": 5.099, "text": "here we're going to simplify cotangent"}, {"start": 6.029, "duration": 5.011, "text": "of PI over 3 minus X again we can't"}, {"start": 9.179, "duration": 5.67, "text": "quite use our cofunction identity"}, {"start": 11.04, "duration": 5.659, "text": "because again we need a PI over 2 what"}, {"start": 14.849, "duration": 5.34, "text": "I'm going to simply do those rewrite"}, {"start": 16.699, "duration": 6.851, "text": "cotangent I can write cotangent as 1"}, {"start": 20.189, "duration": 8.731, "text": "over tangent so we would have tangent of"}, {"start": 23.55, "duration": 7.049, "text": "PI over 3 minus X and again now we can"}, {"start": 28.92, "duration": 7.02, "text": "do the same thing we can just simply use"}, {"start": 30.599, "duration": 9.091, "text": "our identity for tangent so let's see if"}, {"start": 35.94, "duration": 6.18, "text": "we can squeeze it in here let's see so"}, {"start": 39.69, "duration": 3.72, "text": "it says so we're actually going to get"}, {"start": 42.12, "duration": 3.36, "text": "another little fraction in the"}, {"start": 43.41, "duration": 6.899, "text": "denominator so it says we would get"}, {"start": 45.48, "duration": 6.93, "text": "tangent PI over 3 since there's a minus"}, {"start": 50.309, "duration": 6.93, "text": "sign in between we keep the minus sign"}, {"start": 52.41, "duration": 8.399, "text": "so minus tangent X and that's all gonna"}, {"start": 57.239, "duration": 5.91, "text": "be divided by 1 and again if there's a"}, {"start": 60.809, "duration": 6.151, "text": "minus sign in between it says now we use"}, {"start": 63.149, "duration": 7.881, "text": "the plus sign so we'll get tangent of PI"}, {"start": 66.96, "duration": 10.07, "text": "over 3 times tangent of"}, {"start": 71.03, "duration": 8.1, "text": "X okay so on the right side if you have"}, {"start": 77.03, "duration": 3.78, "text": "one over a fraction all that's going to"}, {"start": 79.13, "duration": 2.76, "text": "do is flip your fraction so I'm just"}, {"start": 80.81, "duration": 5.16, "text": "going to go ahead and do that so we'll"}, {"start": 81.89, "duration": 9.57, "text": "get one plus tangent of PI over three"}, {"start": 85.97, "duration": 8.43, "text": "times tangent of X in our denominator we"}, {"start": 91.46, "duration": 8.13, "text": "would have tangent of PI over three"}, {"start": 94.4, "duration": 7.32, "text": "minus tangent of X okay well tangent of"}, {"start": 99.59, "duration": 4.71, "text": "PI over three to evaluate that again"}, {"start": 101.72, "duration": 7.23, "text": "tangent of PI over three that's just"}, {"start": 104.3, "duration": 7.349, "text": "going to be sine of PI over three over"}, {"start": 108.95, "duration": 5.699, "text": "cosine of PI over three"}, {"start": 111.649, "duration": 6.451, "text": "well sine of PI over three recall that's"}, {"start": 114.649, "duration": 5.58, "text": "the square root of three over two cosine"}, {"start": 118.1, "duration": 4.89, "text": "of PI over three is just going to be one"}, {"start": 120.229, "duration": 4.741, "text": "half and if we flip and multiply this"}, {"start": 122.99, "duration": 4.08, "text": "will simply reduce down to the square"}, {"start": 124.97, "duration": 4.56, "text": "root of three so now I'm just gonna plug"}, {"start": 127.07, "duration": 6.74, "text": "that in it says we would get one plus"}, {"start": 129.53, "duration": 8.1, "text": "square root of three times tangent X and"}, {"start": 133.81, "duration": 6.01, "text": "in our denominator again tangent of PI"}, {"start": 137.63, "duration": 6.62, "text": "over three is going to be square root of"}, {"start": 139.82, "duration": 6.6, "text": "three minus tangent X and to me you know"}, {"start": 144.25, "duration": 3.94, "text": "of course there's other things you can"}, {"start": 146.42, "duration": 4.65, "text": "do to it but I think I would just leave"}, {"start": 148.19, "duration": 5.579, "text": "it there and now we have just come up"}, {"start": 151.07, "duration": 4.97, "text": "with a new expression for that original"}, {"start": 153.769, "duration": 2.271, "text": "expression"}], "title": "Cofunction Identities, Example 2", "description": "Thanks to all of you who support me on Patreon. You da real mvps!   $1 per month helps!! :)  https://www.patreon.com/patrickjmt !! Cofunction Identities, Example 2. In this video I use a cofunction identity to simplify an expression.", "lengthSeconds": 156, "width": 480, "height": 360}
{"video_id": "XEdJetN8pMI", "transcript": [{"start": 1.24, "duration": 4.639, "text": "okay in this video I want to talk about"}, {"start": 3.24, "duration": 4.519, "text": "um just a different example of a type of"}, {"start": 5.879, "duration": 4.481, "text": "homogeneous second order linear"}, {"start": 7.759, "duration": 4.161, "text": "differential equation so um this is the"}, {"start": 10.36, "duration": 3.6, "text": "form of a homogeneous second order"}, {"start": 11.92, "duration": 4.519, "text": "linear differential equation P of x q"}, {"start": 13.96, "duration": 4.72, "text": "ofx r ofx g of X have to be continuous"}, {"start": 16.439, "duration": 3.84, "text": "functions in general these are hard to"}, {"start": 18.68, "duration": 4.16, "text": "solve we're going to look at an easier"}, {"start": 20.279, "duration": 5.481, "text": "case where PQ and R are constants"}, {"start": 22.84, "duration": 5.56, "text": "replaced with by values a b and c we"}, {"start": 25.76, "duration": 3.999, "text": "want a to be non zero the idea is we're"}, {"start": 28.4, "duration": 2.08, "text": "going to solve we're going to we're"}, {"start": 29.759, "duration": 4.161, "text": "going to"}, {"start": 30.48, "duration": 5.399, "text": "turn the um differential equation into"}, {"start": 33.92, "duration": 4.52, "text": "basically a quadratic equation we call"}, {"start": 35.879, "duration": 4.121, "text": "this the characteristic equation solve"}, {"start": 38.44, "duration": 3.92, "text": "the corresponding characteristic"}, {"start": 40.0, "duration": 4.48, "text": "equation based on the types of solutions"}, {"start": 42.36, "duration": 4.56, "text": "to the characteristic equation it tells"}, {"start": 44.48, "duration": 5.68, "text": "us the general solution to the original"}, {"start": 46.92, "duration": 4.959, "text": "differential equation um so obviously we"}, {"start": 50.16, "duration": 3.52, "text": "have a quadratic equation different"}, {"start": 51.879, "duration": 4.16, "text": "things can happen in regards to our"}, {"start": 53.68, "duration": 4.519, "text": "Roots one of the results is the"}, {"start": 56.039, "duration": 5.881, "text": "following so it says if your roots are"}, {"start": 58.199, "duration": 3.721, "text": "complex then"}, {"start": 63.519, "duration": 5.801, "text": "it says if the roots are of the form a"}, {"start": 66.0, "duration": 5.159, "text": "plus or minus um beta I it says then the"}, {"start": 69.32, "duration": 4.24, "text": "general solution of this differential"}, {"start": 71.159, "duration": 5.161, "text": "equation is going to be of the form e to"}, {"start": 73.56, "duration": 6.919, "text": "the alpha x time the quantity C1 cosine"}, {"start": 76.32, "duration": 7.36, "text": "beta X Plus C2 sin beta X so um here"}, {"start": 80.479, "duration": 7.441, "text": "we're going to solve y Prime - 6 y Prime"}, {"start": 83.68, "duration": 11.16, "text": "+ 13 y so again we have to look at the"}, {"start": 87.92, "duration": 8.4, "text": "uh equation R 2 - 6 r + 13 = 0 well I"}, {"start": 94.84, "duration": 4.279, "text": "don't think this is going to factor so"}, {"start": 96.32, "duration": 6.079, "text": "I'm just going to use the quadratic"}, {"start": 99.119, "duration": 9.801, "text": "formula B plus or minus the square < TK"}, {"start": 102.399, "duration": 10.241, "text": "of b^ 2 - 4 * a * C 4 * 1 * 13 is 52"}, {"start": 108.92, "duration": 6.04, "text": "over 2 * a okay so if we simplify this"}, {"start": 112.64, "duration": 5.479, "text": "down we'll get 6 plus or minus the"}, {"start": 114.96, "duration": 5.96, "text": "square < TK of -16 so there's our"}, {"start": 118.119, "duration": 5.881, "text": "complex Solutions cropping up"}, {"start": 120.92, "duration": 7.119, "text": "and we can just simply bust this up as 6"}, {"start": 124.0, "duration": 8.319, "text": "plus or minus < TK of 16 excuse me Nega"}, {"start": 128.039, "duration": 8.48, "text": "16 would be 4 I over 2 well let's just"}, {"start": 132.319, "duration": 6.081, "text": "go ahead and reduce this to 3 plus or"}, {"start": 136.519, "duration": 5.281, "text": "minus 2"}, {"start": 138.4, "duration": 7.76, "text": "I so we're basically there at this point"}, {"start": 141.8, "duration": 6.4, "text": "um Alpha is going to be the value three"}, {"start": 146.16, "duration": 4.96, "text": "beta is going to be the"}, {"start": 148.2, "duration": 5.28, "text": "value two don't worry about the plus or"}, {"start": 151.12, "duration": 4.6, "text": "minus um and now all we have to do is"}, {"start": 153.48, "duration": 5.08, "text": "just drop that into"}, {"start": 155.72, "duration": 7.2, "text": "our basically our little formula up here"}, {"start": 158.56, "duration": 8.16, "text": "so I'm going to fill in Alpha = 3 betaal"}, {"start": 162.92, "duration": 5.44, "text": "2 drop in Alpha Beta And beta and then"}, {"start": 166.72, "duration": 4.48, "text": "we'll have our general solution so it"}, {"start": 168.36, "duration": 5.04, "text": "says our solution will look like e to"}, {"start": 171.2, "duration": 7.679, "text": "the alpha which is"}, {"start": 173.4, "duration": 8.759, "text": "3x C1 cosine of beta Co beta being um"}, {"start": 178.879, "duration": 9.961, "text": "two in this case is what we found so"}, {"start": 182.159, "duration": 10.601, "text": "2x um plus C2 s of beta again which is 2"}, {"start": 188.84, "duration": 7.64, "text": "um time x and Tada we now have our"}, {"start": 192.76, "duration": 5.16, "text": "general solution so nothing too bad um"}, {"start": 196.48, "duration": 3.0, "text": "the only thing that's really tricky I"}, {"start": 197.92, "duration": 4.16, "text": "would say in this case is just"}, {"start": 199.48, "duration": 5.039, "text": "recognizing and remembering the formula"}, {"start": 202.08, "duration": 4.68, "text": "um so other than that it's just"}, {"start": 204.519, "duration": 4.521, "text": "basically just a solving a quadratic"}, {"start": 206.76, "duration": 4.28, "text": "equation so things don't get much better"}, {"start": 209.04, "duration": 3.96, "text": "than that and differential equations"}, {"start": 211.04, "duration": 4.919, "text": "class so if you have any questions or"}, {"start": 213.0, "duration": 5.239, "text": "comments please feel free to post them"}, {"start": 215.959, "duration": 5.881, "text": "um and hopefully me or somebody else can"}, {"start": 218.239, "duration": 3.601, "text": "help you out out there"}], "title": "Homogeneous Second Order Linear DE - Complex Roots Example", "description": "Thanks to all of you who support me on Patreon. You da real mvps!   $1 per month helps!! :)  https://www.patreon.com/patrickjmt !! Homogeneous Second Order Linear DE - Complex Roots Example.\r\nFor more free math videos, visit http://PatrickJMT.com", "lengthSeconds": 220, "width": 480, "height": 360}
{"video_id": "wqpGZp2tzZI", "transcript": [{"start": 0.319, "duration": 5.171, "text": "okay suppose at the area of a circle is"}, {"start": 2.939, "duration": 4.051, "text": "740 square inches we want to know what's"}, {"start": 5.49, "duration": 5.729, "text": "the diameter of the circle to the"}, {"start": 6.99, "duration": 7.8, "text": "nearest inch all right so there's our"}, {"start": 11.219, "duration": 5.941, "text": "circle the area is equal to 740 that's"}, {"start": 14.79, "duration": 5.22, "text": "given to us now one thing we'll want to"}, {"start": 17.16, "duration": 5.55, "text": "use is the area formula for a circle and"}, {"start": 20.01, "duration": 5.97, "text": "that is pi times the radius squared"}, {"start": 22.71, "duration": 5.399, "text": "recall that the radius is the distance"}, {"start": 25.98, "duration": 7.59, "text": "from the center of the circle out to one"}, {"start": 28.109, "duration": 10.621, "text": "edge we're gonna approximate PI by using"}, {"start": 33.57, "duration": 6.899, "text": "the value 3.14 so the idea is we're"}, {"start": 38.73, "duration": 3.899, "text": "gonna be able to use this information"}, {"start": 40.469, "duration": 5.401, "text": "about the area to solve for the radius"}, {"start": 42.629, "duration": 4.711, "text": "and recall that the diameter the"}, {"start": 45.87, "duration": 5.04, "text": "diameter is actually the distance all"}, {"start": 47.34, "duration": 5.64, "text": "the way across the circle so once we get"}, {"start": 50.91, "duration": 4.469, "text": "the value for R will simply multiply"}, {"start": 52.98, "duration": 6.78, "text": "that by 2 and that's gonna give us the"}, {"start": 55.379, "duration": 6.691, "text": "diameter all right so we've got 740"}, {"start": 59.76, "duration": 5.399, "text": "again because that's the area that"}, {"start": 62.07, "duration": 6.9, "text": "equals 3 point 1 4 multiplied by R"}, {"start": 65.159, "duration": 7.581, "text": "squared so the first thing I'm gonna do"}, {"start": 68.97, "duration": 6.45, "text": "is I'm gonna divide both sides by 3.14"}, {"start": 72.74, "duration": 5.949, "text": "that's gonna leave us with r squared on"}, {"start": 75.42, "duration": 6.71, "text": "the right side so here I'm gonna use my"}, {"start": 78.689, "duration": 10.11, "text": "calculator so let's see 740 divided by"}, {"start": 82.13, "duration": 9.07, "text": "3.14 I'm getting that to be 235 point"}, {"start": 88.799, "duration": 6.601, "text": "six six eight so I'm gonna round that to"}, {"start": 91.2, "duration": 6.989, "text": "point six seven and now I have to get R"}, {"start": 95.4, "duration": 6.109, "text": "by itself since R is being squared we're"}, {"start": 98.189, "duration": 6.631, "text": "gonna take the square root of both sides"}, {"start": 101.509, "duration": 4.601, "text": "now normally for an algebra algebraic"}, {"start": 104.82, "duration": 2.67, "text": "equation when you take the square root"}, {"start": 106.11, "duration": 4.56, "text": "of both sides you have to include a"}, {"start": 107.49, "duration": 6.059, "text": "positive and negative on one side of the"}, {"start": 110.67, "duration": 3.989, "text": "equation but in this case you know so"}, {"start": 113.549, "duration": 3.061, "text": "taking the square root of r squared is"}, {"start": 114.659, "duration": 3.511, "text": "going to give us R the radius clearly"}, {"start": 116.61, "duration": 4.68, "text": "has to be positive so I'm only going to"}, {"start": 118.17, "duration": 7.11, "text": "keep the positive solution so let's see"}, {"start": 121.29, "duration": 8.46, "text": "here I'm gonna take the square root of"}, {"start": 125.28, "duration": 7.38, "text": "two thirty five point six seven I'm"}, {"start": 129.75, "duration": 4.67, "text": "getting that to be fifteen point three"}, {"start": 132.66, "duration": 4.65, "text": "five after"}, {"start": 134.42, "duration": 5.86, "text": "so that's the radius okay so we said"}, {"start": 137.31, "duration": 5.789, "text": "that the diameter that's gonna be twice"}, {"start": 140.28, "duration": 5.039, "text": "the radius we have to double that so the"}, {"start": 143.099, "duration": 8.671, "text": "diameter is going to be 2 multiplied by"}, {"start": 145.319, "duration": 10.5, "text": "15 0.35 that's gonna give us 30 point"}, {"start": 151.77, "duration": 7.89, "text": "seven zero after doing the arithmetic so"}, {"start": 155.819, "duration": 6.561, "text": "2 times 15 is 32 times 35 is 70 so"}, {"start": 159.66, "duration": 4.95, "text": "that's where I'm getting the 30 point 70"}, {"start": 162.38, "duration": 5.32, "text": "and since we're rounding that to the"}, {"start": 164.61, "duration": 5.659, "text": "nearest we said to the nearest inch the"}, {"start": 167.7, "duration": 6.17, "text": "diameter is roughly going to be equal to"}, {"start": 170.269, "duration": 3.601, "text": "31 inches"}], "title": "GED 'Most Missed' Math Practice Problems , Code Q.4.b,  #1", "description": "Thanks to all of you who support me on Patreon. You da real mvps!   $1 per month helps!! :)  https://www.patreon.com/patrickjmt !! The GED Testing Service keeps a list of the most missed math topics and can be seen here: http://www.gedtestingservice.com/uploads/files/126de0283c94ff6323b3b5b3f310da5d.pdf\nThis video shows how to work a sample problem using their coding system that is shown on the left using the link above.", "lengthSeconds": 173, "width": 480, "height": 360}
{"video_id": "4N5uFxvNJsM", "transcript": [{"start": 0.009, "duration": 4.91, "text": "All right, in this video I want to talk about solving some trigonometric equations, and the equations"}, {"start": 4.92, "duration": 2.0, "text": "we're going to see are going to be relatively,"}, {"start": 6.94, "duration": 1.32, "text": "you know, not too bad."}, {"start": 8.26, "duration": 4.069, "text": "But the idea is typically you're going to have, you know, more complicated equations"}, {"start": 12.58, "duration": 5.33, "text": "that eventually reduce to, you know, finding solutions to these types of equations."}, {"start": 19.06, "duration": 6.469, "text": "Start here with the basics. So suppose we want to find all solutions to sine x equals root 3 over 2."}, {"start": 26.68, "duration": 4.369, "text": "This is one of those kinds of things, you know, that you want to have memorized."}, {"start": 32.41, "duration": 6.619, "text": "So go back to the unit circle, and recall that on the unit circle, at the angle  Pi/3 radians,"}, {"start": 40.03, "duration": 4.91, "text": "the x coordinate is 1/2, but the y coordinate is root 3 over 2, so"}, {"start": 45.73, "duration": 5.239, "text": "definitely, we can say one of our solutions here is going to be x equals Pi/3."}, {"start": 52.39, "duration": 5.869, "text": "Now there are other places on the unit circle where certainly sine equals root 3 over 2."}, {"start": 59.079, "duration": 2.06, "text": "Likewise, if we go over to the angle"}, {"start": 61.84, "duration": 5.509, "text": "2 Pi/ 3. (There's my bad circle.) So now, at the angle 2Pi/3,"}, {"start": 69.1, "duration": 6.349, "text": "the x coordinate is now going to be negative 1/2, but the y coordinate again is still root 3 over 2."}, {"start": 75.97, "duration": 7.279, "text": "So 2Pi/3 is going to be another solution. So x equals 2Pi/3, and"}, {"start": 83.74, "duration": 2.0, "text": "now, well, you know"}, {"start": 85.75, "duration": 4.579, "text": "certainly those aren't the only solutions. Those would be the only solutions between 0  and 2 Pi,"}, {"start": 91.15, "duration": 5.119, "text": "but we're not given any restrictions here, really, on the, you know, the interval for x."}, {"start": 97.329, "duration": 6.89, "text": "So to me the idea is again since I'm at Pi/3, I can either add or subtract multiples of 2 Pi."}, {"start": 104.649, "duration": 2.0, "text": "That'll put me at the same"}, {"start": 106.99, "duration": 6.08, "text": "equivalent place on the unit circle, so we can simply add multiples of 2 Pi by saying"}, {"start": 114.189, "duration": 3.02, "text": "plus 2 Pi times n, where n is an integer,"}, {"start": 120.22, "duration": 1.41, "text": "and"}, {"start": 121.63, "duration": 3.139, "text": "we'll have to do the same thing for our other solution 2 Pi/3."}, {"start": 125.2, "duration": 5.209, "text": "Because, again, if you add or subtract multiples of 2 Pi, it's going to put you back at that same"}, {"start": 130.78, "duration": 4.97, "text": "equivalent angle, so the other solutions would simply be x equals 2 Pi/3"}, {"start": 137.08, "duration": 3.11, "text": "plus 2 Pi times n.   Again, n is"}, {"start": 141.49, "duration": 2.0, "text": "an integer."}], "title": "Solving a Basic Trigonometric Equation, Example 1", "description": "Thanks to all of you who support me on Patreon. You da real mvps!   $1 per month helps!! :)  https://www.patreon.com/patrickjmt !! Solving a Basic Trigonometric Equation, Example 1", "lengthSeconds": 148, "width": 480, "height": 360}
{"video_id": "MGEz57SVn_w", "transcript": [{"start": 1.16, "duration": 4.04, "text": "all right in this example we're going to"}, {"start": 2.84, "duration": 4.999, "text": "find a position Vector for a vector that"}, {"start": 5.2, "duration": 5.559, "text": "starts at the point 3 comma 7 and ends"}, {"start": 7.839, "duration": 5.121, "text": "at4 comma 2 in Part B we're actually"}, {"start": 10.759, "duration": 3.321, "text": "going to find the length of the vector"}, {"start": 12.96, "duration": 5.44, "text": "as"}, {"start": 14.08, "duration": 7.0, "text": "well so to find this position ve Vector"}, {"start": 18.4, "duration": 5.48, "text": "um so again we start at 37 you don't"}, {"start": 21.08, "duration": 7.72, "text": "have to draw the picture at all"}, {"start": 23.88, "duration": 8.0, "text": "um but I'm going to um it starts at 3"}, {"start": 28.8, "duration": 6.12, "text": "comma 7 and goes to the"}, {"start": 31.88, "duration": 4.359, "text": "point4 comma 2 so this is what our our"}, {"start": 34.92, "duration": 4.0, "text": "Vector would look like if we were to"}, {"start": 36.239, "duration": 6.32, "text": "draw it to find this position"}, {"start": 38.92, "duration": 5.04, "text": "Vector all we have to do is okay so I'm"}, {"start": 42.559, "duration": 6.961, "text": "going to list the coordinates again this"}, {"start": 43.96, "duration": 5.56, "text": "is -4 comma 2 this was 3 comma"}, {"start": 50.039, "duration": 7.0, "text": "7 what we do is we take um our so I'm"}, {"start": 55.359, "duration": 5.081, "text": "going to look at the x coordinates we we"}, {"start": 57.039, "duration": 5.36, "text": "take the stopping point which was -4 and"}, {"start": 60.44, "duration": 4.48, "text": "we subtract away whatever the starting"}, {"start": 62.399, "duration": 5.121, "text": "point was which was the x coordinate of"}, {"start": 64.92, "duration": 5.559, "text": "three and then we do the same thing uh"}, {"start": 67.52, "duration": 5.44, "text": "for the Y coordinates so we started at"}, {"start": 70.479, "duration": 5.721, "text": "the uh we stopped at the y-coordinate of"}, {"start": 72.96, "duration": 4.839, "text": "two and we started at the y-coordinate"}, {"start": 76.2, "duration": 3.68, "text": "of"}, {"start": 77.799, "duration": 6.601, "text": "seven and that's now going to be our"}, {"start": 79.88, "duration": 8.239, "text": "position Vector so4 minus 3 that's -7 2"}, {"start": 84.4, "duration": 5.8, "text": "- 7 that's going to be -5 and again all"}, {"start": 88.119, "duration": 3.761, "text": "this to me really says it says if you're"}, {"start": 90.2, "duration": 4.239, "text": "sitting at some point it says what have"}, {"start": 91.88, "duration": 4.96, "text": "you done you've gone seven units to the"}, {"start": 94.439, "duration": 4.481, "text": "left and then you've gone five units"}, {"start": 96.84, "duration": 3.959, "text": "downwards that's all that's happened so"}, {"start": 98.92, "duration": 4.479, "text": "that's our position"}, {"start": 100.799, "duration": 5.881, "text": "Vector Part B we wanted to know the"}, {"start": 103.399, "duration": 6.121, "text": "length well to get the length all we do"}, {"start": 106.68, "duration": 6.079, "text": "is we just take the square root we take"}, {"start": 109.52, "duration": 7.8, "text": "each component and we Square it we take"}, {"start": 112.759, "duration": 7.201, "text": "-7 squared we take5 squared and then we"}, {"start": 117.32, "duration": 4.0, "text": "add those together"}, {"start": 119.96, "duration": 5.439, "text": "so let's see this is going to be the"}, {"start": 121.32, "duration": 7.76, "text": "square root of uh 49 +"}, {"start": 125.399, "duration": 5.961, "text": "25 so I guess uh 49 + 25 that's going to"}, {"start": 129.08, "duration": 5.2, "text": "give us the sare < TK of"}, {"start": 131.36, "duration": 6.64, "text": "74 um we could factor that I guess as 2"}, {"start": 134.28, "duration": 6.8, "text": "* 37 but 37 isn't going to factor any"}, {"start": 138.0, "duration": 5.599, "text": "further so I would simply uh stop here"}, {"start": 141.08, "duration": 6.48, "text": "and say the length of our Vector is the"}, {"start": 143.599, "duration": 3.961, "text": "square root of 74"}], "title": "\u2756 Position Vector and Magnitude / Length \u2756", "description": "In this video I find the position vector and also the length / magnitude of that position vector.\nIn this example we are given the coordinates of where our vector starts and the coordinates of where the vector ends.  \nOnce we find the components of our vector, we simply use the pythagorean theorem to find the magnitude/length.\n\n\n\n\n\n\n#vector #magnitude #components", "lengthSeconds": 146, "width": 480, "height": 360}
{"video_id": "UvWgxCSypFc", "transcript": [{"start": 0.68, "duration": 3.239, "text": "all right here we're going to look at"}, {"start": 1.839, "duration": 4.161, "text": "Absolute and conditional convergence and"}, {"start": 3.919, "duration": 4.321, "text": "just a few examples in trying to decide"}, {"start": 6.0, "duration": 4.92, "text": "whether they're uh absolutely convergent"}, {"start": 8.24, "duration": 4.76, "text": "conditionally convergent or simply"}, {"start": 10.92, "duration": 7.0, "text": "Divergent so part A here we' got the"}, {"start": 13.0, "duration": 8.56, "text": "series -1 to Infinity of1 to the n * 4"}, {"start": 17.92, "duration": 6.199, "text": "2N over 5n um I'm going to simplify this"}, {"start": 21.56, "duration": 2.559, "text": "a little"}, {"start": 24.16, "duration": 6.8, "text": "bit this is -1 to the N we could write"}, {"start": 27.84, "duration": 6.48, "text": "this as 4^2 raised to to the N but that"}, {"start": 30.96, "duration": 5.84, "text": "would just give us 16 to the N over 5 to"}, {"start": 34.32, "duration": 4.919, "text": "the N okay so now if you want to ask"}, {"start": 36.8, "duration": 4.919, "text": "yourself you know is this uh you know"}, {"start": 39.239, "duration": 5.281, "text": "absolutely"}, {"start": 41.719, "duration": 5.881, "text": "convergent well to look to decide if"}, {"start": 44.52, "duration": 6.92, "text": "it's absolutely convergent or"}, {"start": 47.6, "duration": 3.84, "text": "not we would put our"}, {"start": 51.64, "duration": 4.32, "text": "series we would put the formula inside"}, {"start": 53.76, "duration": 4.04, "text": "of absolute value and all that would do"}, {"start": 55.96, "duration": 4.439, "text": "is just basically get rid of the Nega"}, {"start": 57.8, "duration": 5.12, "text": "one to the n"}, {"start": 60.399, "duration": 5.76, "text": "but then we would be left with 16 over5"}, {"start": 62.92, "duration": 5.239, "text": "all raised to the N power this is a"}, {"start": 66.159, "duration": 6.0, "text": "geometric series but this is going to be"}, {"start": 68.159, "duration": 4.0, "text": "a Divergent geometric"}, {"start": 73.08, "duration": 6.84, "text": "series because the R value is not in"}, {"start": 76.479, "duration": 4.921, "text": "between negative 1 and positive one so"}, {"start": 79.92, "duration": 4.92, "text": "uh when we ask ourselves is it"}, {"start": 81.4, "duration": 4.92, "text": "absolutely convergent no definitely not"}, {"start": 84.84, "duration": 4.919, "text": "we could ask ourselves is it"}, {"start": 86.32, "duration": 3.439, "text": "conditionally convergent"}, {"start": 91.6, "duration": 4.76, "text": "well is it conditionally convergent no"}, {"start": 94.24, "duration": 3.4, "text": "um and basically for the same reason um"}, {"start": 96.36, "duration": 4.759, "text": "you know if we look at it the way it"}, {"start": 97.64, "duration": 6.32, "text": "stands this is a uh an alternating"}, {"start": 101.119, "duration": 5.121, "text": "series our B subn in this case again we"}, {"start": 103.96, "duration": 5.479, "text": "can forget about the alternating part"}, {"start": 106.24, "duration": 5.32, "text": "our B subn would be 16 over5 raised to"}, {"start": 109.439, "duration": 5.841, "text": "the N well again we have to ask"}, {"start": 111.56, "duration": 3.72, "text": "ourselves two questions is it"}, {"start": 115.399, "duration": 5.801, "text": "decreasing and does the limit go to zero"}, {"start": 124.24, "duration": 5.2, "text": "well neither one of these are going to"}, {"start": 126.039, "duration": 5.761, "text": "happen if you plug in Nal 1 2 3 4 five"}, {"start": 129.44, "duration": 4.4, "text": "uh you know 16 over five is a number"}, {"start": 131.8, "duration": 4.24, "text": "larger than one so if we start"}, {"start": 133.84, "duration": 4.039, "text": "multiplying it by itself it's definitely"}, {"start": 136.04, "duration": 4.24, "text": "not going to decrease it's going to"}, {"start": 137.879, "duration": 5.521, "text": "increase so neither one of these things"}, {"start": 140.28, "duration": 4.84, "text": "happen so it's likewise uh when we ask"}, {"start": 143.4, "duration": 5.28, "text": "ourselves is it conditionally convergent"}, {"start": 145.12, "duration": 7.88, "text": "we would say no well uh in this case our"}, {"start": 148.68, "duration": 4.32, "text": "series is simply going to be Divergent"}], "title": "Absolute Convergence, Conditional Convergence, Another Example 1", "description": "Thanks to all of you who support me on Patreon. You da real mvps!   $1 per month helps!! :)  https://www.patreon.com/patrickjmt !! Absolute Convergence, Conditional Convergence, Another Example 1.  Here we looks at some more examples to determine whether a series is absolutely convergent, conditionally convergent or divergent.", "lengthSeconds": 155, "width": 480, "height": 360}
{"video_id": "woU1261q7nY", "transcript": [{"start": 0.03, "duration": 4.17, "text": "okay here we're going to graph y equals"}, {"start": 2.04, "duration": 4.56, "text": "five times tangent of one-third X we'll"}, {"start": 4.2, "duration": 4.47, "text": "pick out vertical asymptotes and zeros"}, {"start": 6.6, "duration": 4.2, "text": "this is going to look like the graph of"}, {"start": 8.67, "duration": 5.49, "text": "tangent X except for it's been stretched"}, {"start": 10.8, "duration": 6.72, "text": "vertically by a factor of five and the"}, {"start": 14.16, "duration": 6.029, "text": "period is changed so the period to get"}, {"start": 17.52, "duration": 5.19, "text": "the period be careful for sine and"}, {"start": 20.189, "duration": 5.58, "text": "cosine we use 2pi but tangent has a"}, {"start": 22.71, "duration": 5.85, "text": "period of Pi so we take pi and divide it"}, {"start": 25.769, "duration": 5.16, "text": "by the absolute value of the B value"}, {"start": 28.56, "duration": 6.83, "text": "which in this case is 1/3 and that's"}, {"start": 30.929, "duration": 9.331, "text": "going to give us a value of 3pi"}, {"start": 35.39, "duration": 7.509, "text": "okay so for tangent if you take the"}, {"start": 40.26, "duration": 6.36, "text": "period and divide it by two so it'll be"}, {"start": 42.899, "duration": 6.48, "text": "well three PI over two if you move over"}, {"start": 46.62, "duration": 4.32, "text": "that many units from the y-axis to the"}, {"start": 49.379, "duration": 4.5, "text": "right and to the left those will be your"}, {"start": 50.94, "duration": 7.23, "text": "first vertical asymptotes so here's"}, {"start": 53.879, "duration": 8.91, "text": "going to be the line x equals 3 PI over"}, {"start": 58.17, "duration": 9.09, "text": "2 this will be the vertical asymptote x"}, {"start": 62.789, "duration": 6.271, "text": "equals negative 3 PI over 2 and we can"}, {"start": 67.26, "duration": 4.35, "text": "fill in one point if we want to at least"}, {"start": 69.06, "duration": 5.34, "text": "so normally at PI over 4 it goes through"}, {"start": 71.61, "duration": 6.6, "text": "1 but now at PI over 4 it's going to go"}, {"start": 74.4, "duration": 7.77, "text": "through 5 but it still has that same"}, {"start": 78.21, "duration": 9.39, "text": "general shape to it so again we said the"}, {"start": 82.17, "duration": 7.14, "text": "period in this case is equal to 3 PI so"}, {"start": 87.6, "duration": 4.32, "text": "again certainly not to scale right"}, {"start": 89.31, "duration": 6.87, "text": "here's PI over 4 my vertical asymptotes"}, {"start": 91.92, "duration": 7.44, "text": "already over here 3 PI over 2 so but if"}, {"start": 96.18, "duration": 8.13, "text": "we move over another 3 PI units so we'll"}, {"start": 99.36, "duration": 10.53, "text": "put it right there there's 3 pi 3 PI"}, {"start": 104.31, "duration": 9.54, "text": "over 2 plus 3 PI well that'll be 6 PI"}, {"start": 109.89, "duration": 6.089, "text": "over to our next vertical asymptote our"}, {"start": 113.85, "duration": 6.93, "text": "next positive vertical asymptote is"}, {"start": 115.979, "duration": 11.46, "text": "going to occur at x equals 9 PI over 2"}, {"start": 120.78, "duration": 10.65, "text": "and again it's got that same that same"}, {"start": 127.439, "duration": 5.321, "text": "shape to it so there's two two pieces of"}, {"start": 131.43, "duration": 5.05, "text": "the graph"}, {"start": 132.76, "duration": 5.699, "text": "the zeroes are going to occur the first"}, {"start": 136.48, "duration": 5.039, "text": "one occurs at zero and then it's at"}, {"start": 138.459, "duration": 5.821, "text": "multiples of three pi so those will"}, {"start": 141.519, "duration": 6.511, "text": "occur at the value three pi n where n is"}, {"start": 144.28, "duration": 9.09, "text": "an integer and the asymptotes the"}, {"start": 148.03, "duration": 9.69, "text": "vertical asymptotes the vertical"}, {"start": 153.37, "duration": 8.22, "text": "asymptotes are going to occur at x"}, {"start": 157.72, "duration": 7.56, "text": "equals 3 PI over 2 and again we're"}, {"start": 161.59, "duration": 6.3, "text": "adding 3 pi to get to the next one so"}, {"start": 165.28, "duration": 4.76, "text": "multiples of 3 PI again where n is an"}, {"start": 167.89, "duration": 2.15, "text": "integer"}], "title": "Graphing a Tangent Function - EX 2", "description": "Thanks to all of you who support me on Patreon. You da real mvps!   $1 per month helps!! :)  https://www.patreon.com/patrickjmt !! Graphing a Tangent Function - EX 2", "lengthSeconds": 171, "width": 450, "height": 360}
{"video_id": "HEybx19WhcE", "transcript": [{"start": 0.24, "duration": 3.439, "text": "all right here we're going to look at"}, {"start": 1.16, "duration": 6.92, "text": "one more example of multiplying decimals"}, {"start": 3.679, "duration": 4.401, "text": "so we're going to do 51.5 *"}, {"start": 9.08, "duration": 6.16, "text": "0.014 so let's see"}, {"start": 11.719, "duration": 3.521, "text": "51.5 *"}, {"start": 19.64, "duration": 4.76, "text": "0.014 all right so let's see four time 5"}, {"start": 22.32, "duration": 5.44, "text": "is going to be 20 so I'm going to drop"}, {"start": 24.4, "duration": 6.199, "text": "down the zero and carry the two 4 * 1"}, {"start": 27.76, "duration": 5.56, "text": "will be 4 + 2 will be 6"}, {"start": 30.599, "duration": 5.721, "text": "and then 4 * 5 will be"}, {"start": 33.32, "duration": 6.919, "text": "20 okay let's see so I'm going to add my"}, {"start": 36.32, "duration": 5.6, "text": "zero and then 1 * 5 is five 1 * 1 is 1 1"}, {"start": 40.239, "duration": 4.881, "text": "Time 5 is"}, {"start": 41.92, "duration": 5.68, "text": "five two zeros and then if we multiply"}, {"start": 45.12, "duration": 5.72, "text": "we would just have 0 Time 5 one and five"}, {"start": 47.6, "duration": 6.799, "text": "which will be Z 0o and zero then we can"}, {"start": 50.84, "duration": 6.16, "text": "add in our next row this time we'll have"}, {"start": 54.399, "duration": 6.68, "text": "three zeros but again if you multiply by"}, {"start": 57.0, "duration": 5.44, "text": "zero you'll just get a bunch of zeros"}, {"start": 61.079, "duration": 3.561, "text": "okay so now we're going to add all this"}, {"start": 62.44, "duration": 5.719, "text": "stuff together the first column we'll"}, {"start": 64.64, "duration": 8.04, "text": "just get zero 6 + 5 is 11 so we'll keep"}, {"start": 68.159, "duration": 7.201, "text": "the one and carry the one 1 + 1 is 2 2 +"}, {"start": 72.68, "duration": 5.64, "text": "5 is 7 and then we were left with a"}, {"start": 75.36, "duration": 4.719, "text": "couple zeros and again now I just count"}, {"start": 78.32, "duration": 5.36, "text": "so to the right of the decimal sign we"}, {"start": 80.079, "duration": 5.68, "text": "have one digit two digits three digits"}, {"start": 83.68, "duration": 4.24, "text": "four digits total to the right of the"}, {"start": 85.759, "duration": 4.601, "text": "decimal sign so that's how many places"}, {"start": 87.92, "duration": 5.68, "text": "I've moved my decimal so one place place"}, {"start": 90.36, "duration": 8.0, "text": "two places three places four"}, {"start": 93.6, "duration": 6.12, "text": "places so it says if we multiply 51.5 *"}, {"start": 98.36, "duration": 5.64, "text": "uh"}, {"start": 99.72, "duration": 4.28, "text": "0.014 will'll be left with"}, {"start": 105.52, "duration": 5.04, "text": "0.721 Z as our solution"}], "title": "Multiplying Decimals, Ex 2", "description": "Thanks to all of you who support me on Patreon. You da real mvps!   $1 per month helps!! :)  https://www.patreon.com/patrickjmt !! Multiplying Decimals, Ex 2", "lengthSeconds": 110, "width": 480, "height": 360}
{"video_id": "nTfJCbqHyxI", "transcript": [{"start": 0.0, "duration": 3.33, "text": "all right in this video we're going to"}, {"start": 1.86, "duration": 3.09, "text": "start looking at the route tests to"}, {"start": 3.33, "duration": 3.929, "text": "determine whether or not a series"}, {"start": 4.95, "duration": 4.47, "text": "converges or diverges and again the"}, {"start": 7.259, "duration": 3.871, "text": "route test says we take our series the"}, {"start": 9.42, "duration": 3.57, "text": "formula and we look at the limit as n"}, {"start": 11.13, "duration": 4.169, "text": "goes to infinity of the nth root of the"}, {"start": 12.99, "duration": 3.959, "text": "absolute value of a sub n just like the"}, {"start": 15.299, "duration": 4.471, "text": "root test or excuse me just like the"}, {"start": 16.949, "duration": 5.941, "text": "ratio test the root test has similar"}, {"start": 19.77, "duration": 5.339, "text": "results of the the limits less than 1"}, {"start": 22.89, "duration": 5.07, "text": "the root test says the series converges"}, {"start": 25.109, "duration": 4.851, "text": "if it's greater than 1 it diverges if an"}, {"start": 27.96, "duration": 4.68, "text": "equals 1 it's inconclusive"}, {"start": 29.96, "duration": 4.3, "text": "so just a few different examples here"}, {"start": 32.64, "duration": 4.919, "text": "we'll decide whether it converges or"}, {"start": 34.26, "duration": 5.52, "text": "diverges so Part A here we've got from 1"}, {"start": 37.559, "duration": 4.711, "text": "to infinity of 3 of it and e to the"}, {"start": 39.78, "duration": 4.41, "text": "negative n so we'll take the limit as n"}, {"start": 42.27, "duration": 5.039, "text": "goes to infinity we can take the"}, {"start": 44.19, "duration": 5.639, "text": "absolute value of the stuff again we're"}, {"start": 47.309, "duration": 4.381, "text": "going to take the nth root but taking"}, {"start": 49.829, "duration": 7.741, "text": "the nth root is equivalent to raising it"}, {"start": 51.69, "duration": 8.279, "text": "to the 1 over N power well both of these"}, {"start": 57.57, "duration": 4.44, "text": "are going to be positive no matter what"}, {"start": 59.969, "duration": 3.961, "text": "value of n we plug in so we can just"}, {"start": 62.01, "duration": 5.58, "text": "simply drop you know get rid of the"}, {"start": 63.93, "duration": 6.78, "text": "absolute value well then we'll raise it"}, {"start": 67.59, "duration": 5.91, "text": "to the 1 over N power well we have to"}, {"start": 70.71, "duration": 4.799, "text": "multiply exponents so we'll get n times"}, {"start": 73.5, "duration": 4.979, "text": "1 over n that will give us 3 to the"}, {"start": 75.509, "duration": 4.47, "text": "first then we'll get negative n times 1"}, {"start": 78.479, "duration": 3.991, "text": "over N and that will give us e to the"}, {"start": 79.979, "duration": 4.591, "text": "negative first power this is just a"}, {"start": 82.47, "duration": 4.89, "text": "constant so the limit will equal 3 times"}, {"start": 84.57, "duration": 5.25, "text": "e to the negative 1 but that's going to"}, {"start": 87.36, "duration": 4.95, "text": "be 3 over e to the positive first power"}, {"start": 89.82, "duration": 5.759, "text": "and well this is greater than 1 because"}, {"start": 92.31, "duration": 6.54, "text": "E is smaller than 3 so that's going to"}, {"start": 95.579, "duration": 4.891, "text": "tell us that our series diverges and"}, {"start": 98.85, "duration": 3.69, "text": "again you could have recognized that at"}, {"start": 100.47, "duration": 5.49, "text": "the very beginning by rewriting 3 to the"}, {"start": 102.54, "duration": 7.109, "text": "N over e to the negative n as 3 over e"}, {"start": 105.96, "duration": 5.699, "text": "raised to the N power so you could have"}, {"start": 109.649, "duration": 4.86, "text": "also recognized the original series as"}, {"start": 111.659, "duration": 5.401, "text": "being a geometric series where the R"}, {"start": 114.509, "duration": 4.801, "text": "value is not between negative 1 and"}, {"start": 117.06, "duration": 4.669, "text": "positive 1 so in this case our series"}, {"start": 119.31, "duration": 2.419, "text": "will diverge"}], "title": "The Root Test - Another Example, #1", "description": "Thanks to all of you who support me on Patreon. You da real mvps!   $1 per month helps!! :)  https://www.patreon.com/patrickjmt !! The Root Test - Another Example, #1.  Just another example showing that a series converges or diverges using the root test.", "lengthSeconds": 121, "width": 480, "height": 360}
{"video_id": "BNEULxileT8", "transcript": [{"start": 0.599, "duration": 4.481, "text": "okay this example is going to be about"}, {"start": 2.36, "duration": 4.999, "text": "estimating um a series using the"}, {"start": 5.08, "duration": 3.84, "text": "alternating series estimation theorem"}, {"start": 7.359, "duration": 3.561, "text": "and here we want to know how many terms"}, {"start": 8.92, "duration": 3.719, "text": "of this series we would need to sum up"}, {"start": 10.92, "duration": 4.44, "text": "to make sure our approximation is"}, {"start": 12.639, "duration": 5.441, "text": "accurate to two decimal places so I'm"}, {"start": 15.36, "duration": 5.679, "text": "just going to plug in Nal 1 2 3 4 Etc"}, {"start": 18.08, "duration": 5.08, "text": "into our Series so when we plug in Nal 1"}, {"start": 21.039, "duration": 4.921, "text": "we'll get 1 over"}, {"start": 23.16, "duration": 6.439, "text": "two when we plug in Nal"}, {"start": 25.96, "duration": 6.04, "text": "2 we'll get 2 the 2 which is 4 * 2^ 2"}, {"start": 29.599, "duration": 3.921, "text": "which is another four then we'll get"}, {"start": 32.0, "duration": 5.44, "text": "positive let's see we're plugging in"}, {"start": 33.52, "duration": 6.24, "text": "three so 2 the 3 is 8 3^2 is going to be"}, {"start": 37.44, "duration": 5.76, "text": "9 and then we'll get a negative so now"}, {"start": 39.76, "duration": 6.72, "text": "if we plug in four 2 to 4th is 16 4 SAR"}, {"start": 43.2, "duration": 6.199, "text": "is another 16 and maybe one more term"}, {"start": 46.48, "duration": 7.04, "text": "we'll plug in N = 5 we would get uh 2"}, {"start": 49.399, "duration": 7.041, "text": "5th which is 32 5^2 is 25 and then our"}, {"start": 53.52, "duration": 5.359, "text": "series is just going to keep going so I"}, {"start": 56.44, "duration": 8.24, "text": "plug these into a calculator so well2"}, {"start": 58.879, "duration": 5.801, "text": "that's just 0.5 5 um 1 over 16 is"}, {"start": 65.64, "duration": 8.519, "text": "0.0625 1 over 72 that's roughly"}, {"start": 70.68, "duration": 7.68, "text": "0138 then we have 1 over 16 sared which"}, {"start": 74.159, "duration": 4.201, "text": "is 1 over 256 that's"}, {"start": 78.56, "duration": 5.04, "text": "0.0039 and now I'm going to"}, {"start": 81.04, "duration": 5.8, "text": "stop because it says if we sum up the"}, {"start": 83.6, "duration": 7.32, "text": "first three terms it says the"}, {"start": 86.84, "duration": 6.319, "text": "error is at most"}, {"start": 90.92, "duration": 4.8, "text": "the absolute value of the size of the"}, {"start": 93.159, "duration": 4.401, "text": "first neglected term and we can even say"}, {"start": 95.72, "duration": 3.399, "text": "it's actually less than but it's going"}, {"start": 97.56, "duration": 5.559, "text": "to be"}, {"start": 99.119, "duration": 6.401, "text": "um less than the absolute value of"}, {"start": 103.119, "duration": 5.161, "text": ".39 which well is"}, {"start": 105.52, "duration": 5.68, "text": "00 or"}, {"start": 108.28, "duration": 5.76, "text": "0.0039 so um in this case we're going to"}, {"start": 111.2, "duration": 5.76, "text": "get accuracy uh correct to two decimal"}, {"start": 114.04, "duration": 4.52, "text": "places so we're basically looking for um"}, {"start": 116.96, "duration": 3.28, "text": "the first number where the first two"}, {"start": 118.56, "duration": 4.04, "text": "decimal the first two numbers after the"}, {"start": 120.24, "duration": 5.64, "text": "decimal place are zero so now we've got"}, {"start": 122.6, "duration": 3.28, "text": "two decimal places of"}, {"start": 128.759, "duration": 4.601, "text": "accuracy two decimal places of accuracy"}, {"start": 131.8, "duration": 4.96, "text": "so in this case it says we would just"}, {"start": 133.36, "duration": 7.92, "text": "have to sum up a total of three"}, {"start": 136.76, "duration": 7.0, "text": "terms and then our um our approximation"}, {"start": 141.28, "duration": 5.48, "text": "would be correct within the desired uh"}, {"start": 143.76, "duration": 3.0, "text": "error"}], "title": "Alternating Series - Error Estimation", "description": "Thanks to all of you who support me on Patreon. You da real mvps!   $1 per month helps!! :)  https://www.patreon.com/patrickjmt !! Alternating Series - Error Estimation.  In this example, I find the number of terms required so that we can estimate the value of our convergent alternating series correct to two decimal places.", "lengthSeconds": 145, "width": 480, "height": 360}
{"video_id": "NK2ngJeD2wA", "transcript": [{"start": 1.04, "duration": 4.12, "text": "all right in this video I'm going to do"}, {"start": 2.879, "duration": 4.44, "text": "another example of solving a rational"}, {"start": 5.16, "duration": 4.92, "text": "equation and these are going to be"}, {"start": 7.319, "duration": 4.881, "text": "pretty nicely set up simply because"}, {"start": 10.08, "duration": 5.4, "text": "notice both sides of the equation we"}, {"start": 12.2, "duration": 7.4, "text": "already have just a single fraction um"}, {"start": 15.48, "duration": 5.639, "text": "and the denominators aren't too crazy so"}, {"start": 19.6, "duration": 4.679, "text": "uh it's going to make life a little bit"}, {"start": 21.119, "duration": 7.641, "text": "easier on us so the first thing I'm"}, {"start": 24.279, "duration": 6.24, "text": "going to do in this example is a lot of"}, {"start": 28.76, "duration": 3.56, "text": "times if you can notice if if anything"}, {"start": 30.519, "duration": 4.121, "text": "factors and cancels at the beginning"}, {"start": 32.32, "duration": 6.559, "text": "it'll make your life a little bit easier"}, {"start": 34.64, "duration": 5.84, "text": "so notice the top left side this 3x -3"}, {"start": 38.879, "duration": 3.241, "text": "you know in my head I I think well we"}, {"start": 40.48, "duration": 4.36, "text": "could factor out a"}, {"start": 42.12, "duration": 5.52, "text": "three then we would need an xus one in"}, {"start": 44.84, "duration": 5.52, "text": "the parentheses but hey lo and behold we"}, {"start": 47.64, "duration": 4.079, "text": "have an xus one in the denominator so"}, {"start": 50.36, "duration": 4.08, "text": "that's good I'm going to be able to"}, {"start": 51.719, "duration": 4.561, "text": "cancel that stuff out and it'll just"}, {"start": 54.44, "duration": 4.2, "text": "kind of reduce you know reduce"}, {"start": 56.28, "duration": 6.52, "text": "everything that's present in the problem"}, {"start": 58.64, "duration": 7.24, "text": "so I'm going to cancel out my x -"}, {"start": 62.8, "duration": 5.04, "text": "1's and on the left side you know you"}, {"start": 65.88, "duration": 4.239, "text": "could think about there's being a one in"}, {"start": 67.84, "duration": 5.52, "text": "the denominator so we still have 3 over"}, {"start": 70.119, "duration": 5.121, "text": "one on the left side on the right side I"}, {"start": 73.36, "duration": 3.48, "text": "I think well I could factor a two out of"}, {"start": 75.24, "duration": 4.68, "text": "the numerator but that's not really"}, {"start": 76.84, "duration": 5.2, "text": "going to cancel out anything so I'm just"}, {"start": 79.92, "duration": 4.559, "text": "going to leave it"}, {"start": 82.04, "duration": 4.64, "text": "alone and now I'm going to do the same"}, {"start": 84.479, "duration": 5.041, "text": "thing as in some of my other examples"}, {"start": 86.68, "duration": 4.16, "text": "I'm just simply going to cross multiply"}, {"start": 89.52, "duration": 3.559, "text": "and then that that's going to turn at"}, {"start": 90.84, "duration": 4.72, "text": "least this problem into a nice little"}, {"start": 93.079, "duration": 7.161, "text": "linear equation that won't be hopefully"}, {"start": 95.56, "duration": 6.36, "text": "too bad to to solve so if we take 3 * 5x"}, {"start": 100.24, "duration": 4.44, "text": "that'll give us"}, {"start": 101.92, "duration": 6.28, "text": "15x we would have to take our positive 1"}, {"start": 104.68, "duration": 5.24, "text": "and multiply it by 2x + 4 and I'm going"}, {"start": 108.2, "duration": 3.599, "text": "to write this step out again just"}, {"start": 109.92, "duration": 4.239, "text": "emphasizing that you should put this in"}, {"start": 111.799, "duration": 5.041, "text": "parentheses and distribute everything"}, {"start": 114.159, "duration": 4.841, "text": "out but obviously if you you know"}, {"start": 116.84, "duration": 5.36, "text": "distribute out the one you just get hey"}, {"start": 119.0, "duration": 5.399, "text": "2x + four and now I want to try to put"}, {"start": 122.2, "duration": 5.32, "text": "all my x's on one side and all the"}, {"start": 124.399, "duration": 5.601, "text": "numbers on the other so in this case um"}, {"start": 127.52, "duration": 5.28, "text": "I'm going to subtract this 2x from both"}, {"start": 130.0, "duration": 5.16, "text": "sides and the reason I do that is well"}, {"start": 132.8, "duration": 5.079, "text": "then I'll have all my x's on one side"}, {"start": 135.16, "duration": 6.4, "text": "I'll just have my numbers or my number"}, {"start": 137.879, "duration": 6.121, "text": "um on the other side so 15x - 2x would"}, {"start": 141.56, "duration": 2.44, "text": "give us"}, {"start": 144.4, "duration": 4.44, "text": "13x and then we've got positive4 left"}, {"start": 147.4, "duration": 3.839, "text": "over on the right"}, {"start": 148.84, "duration": 6.0, "text": "side and now we've basically just got"}, {"start": 151.239, "duration": 6.64, "text": "one little step left left to do um 3x ="}, {"start": 154.84, "duration": 5.2, "text": "excuse me 13x = 4 so since we're"}, {"start": 157.879, "duration": 5.0, "text": "multiplying by 13 I just want to get"}, {"start": 160.04, "duration": 6.16, "text": "positive 1X all by itself so since we're"}, {"start": 162.879, "duration": 6.561, "text": "multiplying by 13 we'll do sort of the"}, {"start": 166.2, "duration": 6.88, "text": "opposite which is to divide by 13 on the"}, {"start": 169.44, "duration": 7.519, "text": "left side 13 over 13 is just 1 X on the"}, {"start": 173.08, "duration": 6.56, "text": "right side we get 4 over 13 and that'll"}, {"start": 176.959, "duration": 4.761, "text": "be our solution again as a rule of thumb"}, {"start": 179.64, "duration": 4.519, "text": "you should always kind of go back and"}, {"start": 181.72, "duration": 5.439, "text": "plug this in and make sure that that it"}, {"start": 184.159, "duration": 6.281, "text": "does work in general as long as it's not"}, {"start": 187.159, "duration": 6.0, "text": "making one of your fractions undefined"}, {"start": 190.44, "duration": 4.92, "text": "you should be okay but you know to be"}, {"start": 193.159, "duration": 5.041, "text": "perfectly sure you can always plug it in"}, {"start": 195.36, "duration": 4.84, "text": "and make sure um it is possible to get"}, {"start": 198.2, "duration": 4.24, "text": "extraneous solutions that actually don't"}, {"start": 200.2, "duration": 3.36, "text": "work at the beginning so just one other"}, {"start": 202.44, "duration": 2.799, "text": "little thing that you have to"}, {"start": 203.56, "duration": 5.679, "text": "unfortunately be worried about when you"}, {"start": 205.239, "duration": 4.0, "text": "solve rational equations"}], "title": "Solving a Basic Rational Equation - Ex 3", "description": "Thanks to all of you who support me on Patreon. You da real mvps!   $1 per month helps!! :)  https://www.patreon.com/patrickjmt !! Solving a Basic Rational Equation - Ex 3.  In this video, I solve a rational equation.", "lengthSeconds": 208, "width": 480, "height": 360}
{"video_id": "6BD1FgjvkB4", "transcript": [{"start": 0.719, "duration": 5.881, "text": "okay here we're going to find the volume"}, {"start": 2.399, "duration": 6.041, "text": "of a box and for a nice uh a nice shape"}, {"start": 6.6, "duration": 4.64, "text": "like this basically what we're going to"}, {"start": 8.44, "duration": 5.96, "text": "do is we're going to"}, {"start": 11.24, "duration": 5.48, "text": "find the area of one of the bases and"}, {"start": 14.4, "duration": 5.12, "text": "then multiply it by the corresponding"}, {"start": 16.72, "duration": 6.28, "text": "height so what I'm going to do here is"}, {"start": 19.52, "duration": 6.4, "text": "I'm going to find the area of this front"}, {"start": 23.0, "duration": 4.599, "text": "face and then multiply it by the height"}, {"start": 25.92, "duration": 4.599, "text": "so if you can kind of imagine if the you"}, {"start": 27.599, "duration": 5.041, "text": "know the Box were on its side or again"}, {"start": 30.519, "duration": 5.921, "text": "you know for a rectangle the area is"}, {"start": 32.64, "duration": 6.0, "text": "basically the length time the width time"}, {"start": 36.44, "duration": 3.56, "text": "the height but again the length times"}, {"start": 38.64, "duration": 3.599, "text": "the width that's going to give us the"}, {"start": 40.0, "duration": 6.92, "text": "area of one of the"}, {"start": 42.239, "duration": 8.48, "text": "bases so we'll multiply 4 * 5 and then"}, {"start": 46.92, "duration": 8.04, "text": "just simply multiply that by 10 so 4 * 5"}, {"start": 50.719, "duration": 7.721, "text": "is 20 * 10 will be 200 the units will be"}, {"start": 54.96, "duration": 7.399, "text": "centime time cenm time cenm so we would"}, {"start": 58.44, "duration": 6.119, "text": "have cm cubed in this case so the answer"}, {"start": 62.359, "duration": 6.401, "text": "of our the answer for the volume will be"}, {"start": 64.559, "duration": 4.201, "text": "200 cm cubed"}], "title": "Volume of a Prism (A Box!)", "description": "Thanks to all of you who support me on Patreon. You da real mvps!   $1 per month helps!! :)  https://www.patreon.com/patrickjmt !! For tons of free videos, visit:  http://PatrickJMT.com\n Volume of a Prism (A Box!).   Here we look at a quick example of finding the volume of a box.  All we need to do is multiply the length, width and height together!", "lengthSeconds": 67, "width": 480, "height": 360}
{"video_id": "zxKLbCWuGSQ", "transcript": [{"start": 0.11, "duration": 3.88, "text": "okay in this video I just want to do"}, {"start": 2.429, "duration": 8.281, "text": "some more examples factoring the"}, {"start": 3.99, "duration": 9.149, "text": "difference of two squares so again I"}, {"start": 10.71, "duration": 5.219, "text": "look at this and I think well hey I've"}, {"start": 13.139, "duration": 5.31, "text": "got two terms 16x squared and my others"}, {"start": 15.929, "duration": 6.331, "text": "my 25 y squared there's a I'm"}, {"start": 18.449, "duration": 5.91, "text": "subtracting them so anytime I have a"}, {"start": 22.26, "duration": 5.4, "text": "negative between two terms that always"}, {"start": 24.359, "duration": 5.221, "text": "factors and again I'm basically just"}, {"start": 27.66, "duration": 4.59, "text": "taking the square root of the first term"}, {"start": 29.58, "duration": 5.52, "text": "so the square root of 16x squared again"}, {"start": 32.25, "duration": 5.19, "text": "what multiplied by itself would be 16x"}, {"start": 35.1, "duration": 5.549, "text": "squared we would just need a 4x and a 4x"}, {"start": 37.44, "duration": 4.619, "text": "and then I do the same thing for the 25"}, {"start": 40.649, "duration": 5.16, "text": "y squared"}, {"start": 42.059, "duration": 6.09, "text": "I would simply need a 5y and 5y and"}, {"start": 45.809, "duration": 5.371, "text": "again the formula says you stick a"}, {"start": 48.149, "duration": 5.041, "text": "positive and a negative in between the"}, {"start": 51.18, "duration": 5.129, "text": "terms and again that's now how it"}, {"start": 53.19, "duration": 8.849, "text": "factors so let's do our other example"}, {"start": 56.309, "duration": 9.081, "text": "here 64 x squared minus 81 Y squared so"}, {"start": 62.039, "duration": 6.031, "text": "here I'm just trying to make a little"}, {"start": 65.39, "duration": 5.049, "text": "barrier here just to separate them so"}, {"start": 68.07, "duration": 5.45, "text": "let's see so I'm going to make my two"}, {"start": 70.439, "duration": 3.081, "text": "sets of parentheses"}, {"start": 74.05, "duration": 4.96, "text": "let's see okay it's the same thing I did"}, {"start": 76.88, "duration": 6.12, "text": "I just take the square root of 64 x"}, {"start": 79.01, "duration": 6.48, "text": "squared I would get 8 X + 8 X I do the"}, {"start": 83.0, "duration": 6.84, "text": "same thing with 81 y squared I would get"}, {"start": 85.49, "duration": 6.51, "text": "9 y + 9 Y and again to stick a positive"}, {"start": 89.84, "duration": 4.89, "text": "and a negative in between them and you"}, {"start": 92.0, "duration": 4.62, "text": "now have it factored so again notice if"}, {"start": 94.73, "duration": 5.31, "text": "you do multiply it out we'll get 64 x"}, {"start": 96.62, "duration": 5.76, "text": "squared we'll get a negative 72 X Y but"}, {"start": 100.04, "duration": 4.83, "text": "we'll also get a positive 72 X Y which"}, {"start": 102.38, "duration": 6.62, "text": "cancels and then we get our negative 9y"}, {"start": 104.87, "duration": 4.13, "text": "squared just like we want"}], "title": "Factoring the Difference of Two Squares - Ex 2", "description": "Thanks to all of you who support me on Patreon. You da real mvps!   $1 per month helps!! :)  https://www.patreon.com/patrickjmt !! Factoring the Difference of Two Squares - Ex 2", "lengthSeconds": 108, "width": 480, "height": 360}
{"video_id": "ISb7s_nkdk0", "transcript": [{"start": 0.719, "duration": 3.2, "text": "all right in this video i'm going to do"}, {"start": 2.159, "duration": 4.081, "text": "another example of finding real number"}, {"start": 3.919, "duration": 4.561, "text": "restrictions and in this problem we're"}, {"start": 6.24, "duration": 4.0, "text": "going to have another fraction so again"}, {"start": 8.48, "duration": 3.92, "text": "the values that are going to be that"}, {"start": 10.24, "duration": 4.479, "text": "would have to be omitted are any values"}, {"start": 12.4, "duration": 3.359, "text": "that give 0 in the denominator"}, {"start": 14.719, "duration": 3.841, "text": "so basically i'm going to take the"}, {"start": 15.759, "duration": 5.201, "text": "denominator set it equal to zero and"}, {"start": 18.56, "duration": 2.4, "text": "solve"}, {"start": 21.119, "duration": 5.601, "text": "so notice this is a quadratic equation i"}, {"start": 23.92, "duration": 4.4, "text": "don't think it factors too nicely"}, {"start": 26.72, "duration": 3.44, "text": "at least just looking at it off the top"}, {"start": 28.32, "duration": 3.199, "text": "of my head maybe we can play with it"}, {"start": 30.16, "duration": 2.96, "text": "here and maybe"}, {"start": 31.519, "duration": 3.601, "text": "maybe actually it will work out without"}, {"start": 33.12, "duration": 3.84, "text": "too much trouble"}, {"start": 35.12, "duration": 4.16, "text": "so to get the 2x squared i'm going to"}, {"start": 36.96, "duration": 4.48, "text": "need a 2x and an x"}, {"start": 39.28, "duration": 3.36, "text": "and then i need two numbers in this case"}, {"start": 41.44, "duration": 2.959, "text": "you've got to be a little tricky you"}, {"start": 42.64, "duration": 3.68, "text": "can't just say two numbers that multiply"}, {"start": 44.399, "duration": 4.0, "text": "to negative five and add up to negative"}, {"start": 46.32, "duration": 3.36, "text": "seven like i almost said that's not"}, {"start": 48.399, "duration": 3.761, "text": "correct in this case because of the"}, {"start": 49.68, "duration": 4.96, "text": "coefficient of the two but i know on the"}, {"start": 52.16, "duration": 5.28, "text": "outsides of my parentheses i am going to"}, {"start": 54.64, "duration": 5.12, "text": "need two numbers that multiply together"}, {"start": 57.44, "duration": 5.84, "text": "to give me a negative five"}, {"start": 59.76, "duration": 5.2, "text": "so let's see what if we put um"}, {"start": 63.28, "duration": 3.36, "text": "we could try putting you know and again"}, {"start": 64.96, "duration": 3.44, "text": "one of them has to be positive one of"}, {"start": 66.64, "duration": 3.68, "text": "them has to be negative i think you"}, {"start": 68.4, "duration": 3.84, "text": "could try with different possibilities"}, {"start": 70.32, "duration": 4.799, "text": "of positive and negative 5 and i'm"}, {"start": 72.24, "duration": 4.72, "text": "simply not seeing how it factors at all"}, {"start": 75.119, "duration": 3.921, "text": "so what i do in these cases is i simply"}, {"start": 76.96, "duration": 4.64, "text": "use the quadratic formula"}, {"start": 79.04, "duration": 5.2, "text": "so x equals negative b plus or minus the"}, {"start": 81.6, "duration": 5.76, "text": "square root of b squared"}, {"start": 84.24, "duration": 5.44, "text": "minus 4ac"}, {"start": 87.36, "duration": 4.16, "text": "all over 2a"}, {"start": 89.68, "duration": 4.079, "text": "and again in this case a is going to be"}, {"start": 91.52, "duration": 5.04, "text": "the value positive 2"}, {"start": 93.759, "duration": 5.281, "text": "b is going to be the value negative 7."}, {"start": 96.56, "duration": 5.36, "text": "c is going to be the value negative 5"}, {"start": 99.04, "duration": 5.92, "text": "and then i just plug everything in here"}, {"start": 101.92, "duration": 5.12, "text": "so we have negative of negative 7"}, {"start": 104.96, "duration": 5.04, "text": "plus or minus the square root so we"}, {"start": 107.04, "duration": 4.079, "text": "would have b squared which is negative 7"}, {"start": 110.0, "duration": 4.64, "text": "squared"}, {"start": 111.119, "duration": 6.64, "text": "minus 4 times a which is 2"}, {"start": 114.64, "duration": 4.799, "text": "times c which is negative 5. let me"}, {"start": 117.759, "duration": 3.68, "text": "extend my square root over a little bit"}, {"start": 119.439, "duration": 4.481, "text": "more"}, {"start": 121.439, "duration": 5.761, "text": "and that's all again being divided by 2"}, {"start": 123.92, "duration": 4.959, "text": "times a which is also the value 2."}, {"start": 127.2, "duration": 3.36, "text": "and now it's just a bunch of arithmetic"}, {"start": 128.879, "duration": 3.521, "text": "simplifying this down"}, {"start": 130.56, "duration": 3.759, "text": "so the negative of negative seven is"}, {"start": 132.4, "duration": 3.36, "text": "positive seven"}, {"start": 134.319, "duration": 4.801, "text": "underneath the square root negative"}, {"start": 135.76, "duration": 5.76, "text": "seven times negative seven is forty-nine"}, {"start": 139.12, "duration": 4.479, "text": "let's see negative four times two is"}, {"start": 141.52, "duration": 3.92, "text": "negative eight negative eight"}, {"start": 143.599, "duration": 4.561, "text": "times negative five is going to be"}, {"start": 145.44, "duration": 5.12, "text": "positive forty"}, {"start": 148.16, "duration": 4.48, "text": "and that's all divided by two times two"}, {"start": 150.56, "duration": 3.679, "text": "or four"}, {"start": 152.64, "duration": 4.16, "text": "and we can keep simplifying here a"}, {"start": 154.239, "duration": 4.64, "text": "little bit more we would have seven"}, {"start": 156.8, "duration": 5.04, "text": "plus or minus underneath the square root"}, {"start": 158.879, "duration": 5.36, "text": "we would have 89"}, {"start": 161.84, "duration": 5.2, "text": "and that's divided by four"}, {"start": 164.239, "duration": 4.321, "text": "and i'm pretty positive 89 is actually a"}, {"start": 167.04, "duration": 2.96, "text": "prime number that's at least not one"}, {"start": 168.56, "duration": 3.2, "text": "that i know off the top of my head as"}, {"start": 170.0, "duration": 4.08, "text": "having any factors"}, {"start": 171.76, "duration": 5.119, "text": "so our two restrictions in this case we"}, {"start": 174.08, "duration": 6.159, "text": "would have to omit the values our two"}, {"start": 176.879, "duration": 3.36, "text": "values we would have to omit"}, {"start": 181.599, "duration": 4.881, "text": "we'd have to omit seven plus the square"}, {"start": 184.4, "duration": 4.72, "text": "root of 89"}, {"start": 186.48, "duration": 6.88, "text": "over 4 and then we would also have to"}, {"start": 189.12, "duration": 6.72, "text": "omit 7 minus the square root of 89"}, {"start": 193.36, "duration": 4.48, "text": "over 4. again kind of not nice whole"}, {"start": 195.84, "duration": 3.84, "text": "numbers but that's okay"}, {"start": 197.84, "duration": 3.28, "text": "so those would be the values that we"}, {"start": 199.68, "duration": 4.08, "text": "would have to"}, {"start": 201.12, "duration": 6.199, "text": "we would not be allowed to plug into our"}, {"start": 203.76, "duration": 3.559, "text": "original expression"}], "title": "Finding the Domain of an Expression Involving Fractions - Example 3", "description": "Thanks to all of you who support me on Patreon. You da real mvps!   $1 per month helps!! :)  https://www.patreon.com/patrickjmt !! Finding the Domain of an Expression Involving Fractions - Example 3", "lengthSeconds": 206, "width": 480, "height": 360}
{"video_id": "c5Zx6Ak8Bt4", "transcript": [{"start": 0.399, "duration": 3.761, "text": "all right so another example here of"}, {"start": 1.719, "duration": 3.841, "text": "using our sum and difference identities"}, {"start": 4.16, "duration": 4.68, "text": "um so suppose we just want to Simply"}, {"start": 5.56, "duration": 7.88, "text": "evaluate cosine of 15\u00b0 well we can write"}, {"start": 8.84, "duration": 6.4, "text": "15\u00b0 as 45\u00b0 minus 30\u00b0 and now we can use"}, {"start": 13.44, "duration": 3.96, "text": "this uh bottom identity so it says if"}, {"start": 15.24, "duration": 4.92, "text": "you have cosine of something minus"}, {"start": 17.4, "duration": 4.36, "text": "something it says you take cosine of the"}, {"start": 20.16, "duration": 4.76, "text": "first thing which is"}, {"start": 21.76, "duration": 5.08, "text": "45\u00b0 you multiply it by cosine of the"}, {"start": 24.92, "duration": 5.56, "text": "second thing which is"}, {"start": 26.84, "duration": 5.32, "text": "30\u00b0 um if there's a negative sign in"}, {"start": 30.48, "duration": 3.72, "text": "between a minus sign in between though"}, {"start": 32.16, "duration": 3.28, "text": "it says you turn it into a plus when you"}, {"start": 34.2, "duration": 4.679, "text": "expand it"}, {"start": 35.44, "duration": 5.0, "text": "out and then you use sign of the first"}, {"start": 38.879, "duration": 5.321, "text": "uh quantity which is"}, {"start": 40.44, "duration": 6.48, "text": "45\u00b0 and then we multiply it by S of the"}, {"start": 44.2, "duration": 5.12, "text": "second quantity which is"}, {"start": 46.92, "duration": 4.92, "text": "30\u00b0 okay and now it's just again kind of"}, {"start": 49.32, "duration": 7.36, "text": "knowing uh some trig values so cosine of"}, {"start": 51.84, "duration": 10.039, "text": "45\u00b0 is < tk2 over2 uh cosine of 30\u00b0 is"}, {"start": 56.68, "duration": 11.24, "text": "going to be < tk3 over 2 s of 45\u00b0 again"}, {"start": 61.879, "duration": 7.761, "text": "< tk2 / 2 s of 30\u00b0 is going to be 12 so"}, {"start": 67.92, "duration": 6.32, "text": "okay it looks like we're getting Square"}, {"start": 69.64, "duration": 7.32, "text": "< TK of 6 over 4 + 2 over the square"}, {"start": 74.24, "duration": 4.84, "text": "root excuse me the square < TK of 2 over"}, {"start": 76.96, "duration": 7.0, "text": "4 then again we can write this as a"}, {"start": 79.08, "duration": 7.64, "text": "single fraction > of 6 Plus theun of two"}, {"start": 83.96, "duration": 5.24, "text": "over 4 and I think that's actually the"}, {"start": 86.72, "duration": 4.399, "text": "uh the same value that we figured out a"}, {"start": 89.2, "duration": 3.959, "text": "second ago right we said uh I think in"}, {"start": 91.119, "duration": 6.04, "text": "our first problem we figured out that s"}, {"start": 93.159, "duration": 5.681, "text": "of 75\u00b0 was also this value so does it"}, {"start": 97.159, "duration": 3.92, "text": "make sense from a geometric point of"}, {"start": 98.84, "duration": 4.36, "text": "view so you know just a little side"}, {"start": 101.079, "duration": 4.72, "text": "question does it make sense that s of 75"}, {"start": 103.2, "duration": 5.36, "text": "Dees would in fact be the same as cosine"}, {"start": 105.799, "duration": 4.801, "text": "of 15\u00b0 so maybe think if that seems"}, {"start": 108.56, "duration": 4.159, "text": "reasonable to you but definitely we've"}, {"start": 110.6, "duration": 4.439, "text": "used this identity um in the correct"}, {"start": 112.719, "duration": 5.32, "text": "manner this certainly is the correct"}, {"start": 115.039, "duration": 3.0, "text": "answer"}], "title": "Identities for Sum and Differences of Sine and Cosine, Example 2", "description": "Thanks to all of you who support me on Patreon. You da real mvps!   $1 per month helps!! :)  https://www.patreon.com/patrickjmt !! Identities for Sum and Differences of Sine and Cosine, Example 2.  In this video, I show how we can find other values of sine and cosine (besides the commonly memorized values) using sum and difference identities.  For more free math videos, visit http://PatrickJMT.com", "lengthSeconds": 116, "width": 480, "height": 360}
{"video_id": "0yTQfyiwFT8", "transcript": [{"start": 0.76, "duration": 3.8, "text": "okay so here we're going to do a little"}, {"start": 1.959, "duration": 5.841, "text": "word problem related to the law of"}, {"start": 4.56, "duration": 5.88, "text": "cosiness see if we can squeeze it all in"}, {"start": 7.8, "duration": 4.32, "text": "here okay so the idea is we've got two"}, {"start": 10.44, "duration": 4.8, "text": "people um they're on the same side of"}, {"start": 12.12, "duration": 6.8, "text": "the river and they're 400 meters apart"}, {"start": 15.24, "duration": 6.24, "text": "uh the person on the left side is 700 m"}, {"start": 18.92, "duration": 4.119, "text": "from a house that's across the river so"}, {"start": 21.48, "duration": 5.199, "text": "there's the little house uh that"}, {"start": 23.039, "duration": 6.961, "text": "person's 700 M away and the angle formed"}, {"start": 26.679, "duration": 5.641, "text": "by the people in the house uh is 150"}, {"start": 30.0, "duration": 4.6, "text": "Dees when we use the person on the left"}, {"start": 32.32, "duration": 4.559, "text": "as the vertex we just want to know how"}, {"start": 34.6, "duration": 3.68, "text": "far the person from the right is from"}, {"start": 36.879, "duration": 4.121, "text": "the"}, {"start": 38.28, "duration": 4.16, "text": "house well okay this is really a pretty"}, {"start": 41.0, "duration": 5.16, "text": "straightforward"}, {"start": 42.44, "duration": 3.72, "text": "application of the law of"}, {"start": 51.12, "duration": 5.119, "text": "cosiness okay so this is"}, {"start": 53.32, "duration": 6.32, "text": "700 uh we know that this is"}, {"start": 56.239, "duration": 6.64, "text": "150\u00b0 this is 400 again we can call this"}, {"start": 59.64, "duration": 6.76, "text": "l may be C and now we'll just use our"}, {"start": 62.879, "duration": 7.521, "text": "law of cosin so it says c^2"}, {"start": 66.4, "duration": 5.719, "text": "= 700 2ar +"}, {"start": 70.4, "duration": 5.48, "text": "400"}, {"start": 72.119, "duration": 5.921, "text": "2ar - 2 *"}, {"start": 75.88, "duration": 5.72, "text": "700 time"}, {"start": 78.04, "duration": 7.039, "text": "400 times cosine of the angle but we're"}, {"start": 81.6, "duration": 7.08, "text": "given that that's just 150"}, {"start": 85.079, "duration": 6.761, "text": "degrees so let's see here so 700 s"}, {"start": 88.68, "duration": 3.16, "text": "that'll be 400"}, {"start": 92.24, "duration": 4.64, "text": "90,000 uh let's see so 400"}, {"start": 97.079, "duration": 7.201, "text": "squared that would be"}, {"start": 101.04, "duration": 9.28, "text": "160,000 let's see so 2 *"}, {"start": 104.28, "duration": 10.6, "text": "700 * 400 that's going to be"}, {"start": 110.32, "duration": 6.439, "text": "560,000 let's see cosine of 15\u00b0 I think"}, {"start": 114.88, "duration": 6.08, "text": "we can probably uh we can probably"}, {"start": 116.759, "duration": 6.72, "text": "figure that one out um so let's see 50\u00b0"}, {"start": 120.96, "duration": 5.32, "text": "that would put us in quadrant uh in"}, {"start": 123.479, "duration": 3.84, "text": "quadrant 2 uh the reference angle there"}, {"start": 126.28, "duration": 3.679, "text": "would be"}, {"start": 127.319, "duration": 7.161, "text": "30\u00b0 so let's see if I think about cosine"}, {"start": 129.959, "duration": 6.64, "text": "of 30\u00b0 cosine of 30\u00b0 would be < tk3"}, {"start": 134.48, "duration": 4.0, "text": "over2 but since we're in the second"}, {"start": 136.599, "duration": 4.481, "text": "quadrant uh cosine there would be"}, {"start": 138.48, "duration": 5.36, "text": "negative so we'll get netive < tk3 over2"}, {"start": 141.08, "duration": 6.32, "text": "for cosine of"}, {"start": 143.84, "duration": 8.52, "text": "150 um so let's see here so we've got"}, {"start": 147.4, "duration": 6.44, "text": "490 uh 490,000 and 160,000 that's just"}, {"start": 152.36, "duration": 3.159, "text": "going to be"}, {"start": 153.84, "duration": 4.88, "text": "650,000"}, {"start": 155.519, "duration": 8.401, "text": "two negatives will make a positive let's"}, {"start": 158.72, "duration": 7.48, "text": "see uh 560,000 / 2 that would be"}, {"start": 163.92, "duration": 5.36, "text": "280,000 again we would have to multiply"}, {"start": 166.2, "duration": 7.0, "text": "that still by the square root of"}, {"start": 169.28, "duration": 6.2, "text": "three so let's see two let's see so uh"}, {"start": 173.2, "duration": 6.52, "text": "let's do the square of 3 S of 3 is"}, {"start": 175.48, "duration": 6.36, "text": "roughly 1.73 205 I'll multiply that by 2"}, {"start": 179.72, "duration": 2.12, "text": "2"}, {"start": 182.76, "duration": 10.44, "text": "80 add that to"}, {"start": 186.08, "duration": 9.96, "text": "650,000 I'm getting this to be 1 1 3 4 9"}, {"start": 193.2, "duration": 5.119, "text": "74"}, {"start": 196.04, "duration": 3.88, "text": "226 well the last thing we'll have to do"}, {"start": 198.319, "duration": 4.681, "text": "is just take the square root of this"}, {"start": 199.92, "duration": 7.12, "text": "number if we take the square root of all"}, {"start": 203.0, "duration": 4.04, "text": "this I'm getting this to be"}, {"start": 207.08, "duration": 5.6, "text": "1065 35 after rounding I'm going to"}, {"start": 210.319, "duration": 5.321, "text": "round that down even a little bit more"}, {"start": 212.68, "duration": 6.0, "text": "and just say it's"}, {"start": 215.64, "duration": 6.599, "text": "1,65 I believe our original units were"}, {"start": 218.68, "duration": 5.199, "text": "meters so the second person is going to"}, {"start": 222.239, "duration": 7.041, "text": "be"}, {"start": 223.879, "duration": 5.401, "text": "1,65 M away from the house roughly"}], "title": "Law of Cosines, Example 6", "description": "Thanks to all of you who support me on Patreon. You da real mvps!   $1 per month helps!! :)  https://www.patreon.com/patrickjmt !! Law of Cosines, Example 6.  In this video, we find how far a person is from a house by knowing that person's distance to someone else and the angle formed between them and the house.", "lengthSeconds": 227, "width": 480, "height": 360}
{"video_id": "-P-LYIq50no", "transcript": [{"start": 1.199, "duration": 3.801, "text": "all right in this video I want to take a"}, {"start": 2.8, "duration": 5.6, "text": "couple of equations and just solve for"}, {"start": 5.0, "duration": 7.48, "text": "some specified variable so in the first"}, {"start": 8.4, "duration": 7.319, "text": "example we've got that a = 12 * H * the"}, {"start": 12.48, "duration": 6.2, "text": "quantity B1 + B2 and we want to solve"}, {"start": 15.719, "duration": 4.201, "text": "that for B1 so what that means to me is"}, {"start": 18.68, "duration": 5.599, "text": "eventually we want to do a whole bunch"}, {"start": 19.92, "duration": 6.599, "text": "of algebra and get uh B1 all by"}, {"start": 24.279, "duration": 4.521, "text": "itself so there's certainly a couple"}, {"start": 26.519, "duration": 4.76, "text": "different ways we can go about doing"}, {"start": 28.8, "duration": 5.52, "text": "this um the first thing I notice is that"}, {"start": 31.279, "duration": 4.681, "text": "there's a 1/2 in here and typ typically"}, {"start": 34.32, "duration": 4.04, "text": "if I can get rid of fractions that's the"}, {"start": 35.96, "duration": 5.2, "text": "first thing I like to do so you know"}, {"start": 38.36, "duration": 6.0, "text": "multiplying by 1/2 to me is the same"}, {"start": 41.16, "duration": 5.719, "text": "thing as dividing by two so I think well"}, {"start": 44.36, "duration": 4.719, "text": "if you're dividing by two what undoes"}, {"start": 46.879, "duration": 4.36, "text": "multiplication or excuse me what undoes"}, {"start": 49.079, "duration": 5.441, "text": "division well multiplication undoes"}, {"start": 51.239, "duration": 5.401, "text": "division so since we're dividing by two"}, {"start": 54.52, "duration": 4.4, "text": "to get rid of the 1/2 I could multiply"}, {"start": 56.64, "duration": 3.84, "text": "by two but if I do it to the right side"}, {"start": 58.92, "duration": 4.2, "text": "I have to do it to the left side side as"}, {"start": 60.48, "duration": 5.999, "text": "well so on the left side we're left with"}, {"start": 63.12, "duration": 5.6, "text": "2 a the 2 * a half is just one so we can"}, {"start": 66.479, "duration": 5.721, "text": "basically just get rid of that and then"}, {"start": 68.72, "duration": 7.52, "text": "we have H * B1 +"}, {"start": 72.2, "duration": 5.44, "text": "B2 and now I want to get um there's"}, {"start": 76.24, "duration": 4.36, "text": "again a couple different ways I could do"}, {"start": 77.64, "duration": 4.839, "text": "it since I'm multiplying by H I think"}, {"start": 80.6, "duration": 4.6, "text": "well I would like to get rid of that so"}, {"start": 82.479, "duration": 4.441, "text": "again multiplying by H I'll divide both"}, {"start": 85.2, "duration": 5.52, "text": "sides by"}, {"start": 86.92, "duration": 6.199, "text": "h on the right side H divided by H is"}, {"start": 90.72, "duration": 4.28, "text": "just one on the left side there's really"}, {"start": 93.119, "duration": 5.32, "text": "nothing to simplify down so we still"}, {"start": 95.0, "duration": 5.479, "text": "have the 2 a over H you can think about"}, {"start": 98.439, "duration": 4.68, "text": "in the numerator there being a positive"}, {"start": 100.479, "duration": 5.481, "text": "one in front of the B1 + B2 if you"}, {"start": 103.119, "duration": 6.121, "text": "distribute that we'll be left with B1 +"}, {"start": 105.96, "duration": 6.24, "text": "B2 and now to Simply get B1 all by"}, {"start": 109.24, "duration": 5.799, "text": "itself since I'm adding B2 I would"}, {"start": 112.2, "duration": 5.199, "text": "subtract B2 from both"}, {"start": 115.039, "duration": 4.44, "text": "sides well on the left side they're not"}, {"start": 117.399, "duration": 5.68, "text": "really like terms so there's not a lot"}, {"start": 119.479, "duration": 8.201, "text": "we can do but we do have we could simply"}, {"start": 123.079, "duration": 7.6, "text": "just rewrite it as 2 a h minus B2 and"}, {"start": 127.68, "duration": 4.559, "text": "now we have solved for B1 it's all by"}, {"start": 130.679, "duration": 4.001, "text": "itself"}, {"start": 132.239, "duration": 6.041, "text": "so we are"}, {"start": 134.68, "duration": 6.36, "text": "finished um in the second equation we"}, {"start": 138.28, "duration": 6.72, "text": "have to solve for our variable Z so"}, {"start": 141.04, "duration": 7.32, "text": "we've got 3 Z + x - y = 2x +"}, {"start": 145.0, "duration": 5.08, "text": "4 and since there's only a single Z"}, {"start": 148.36, "duration": 4.4, "text": "floating around on the left side I'm"}, {"start": 150.08, "duration": 4.799, "text": "going to just leave that by itself so"}, {"start": 152.76, "duration": 3.64, "text": "I've really got to do two things um I've"}, {"start": 154.879, "duration": 4.921, "text": "got to get rid of the plus X and the"}, {"start": 156.4, "duration": 6.6, "text": "minus y so I can subtract X from both"}, {"start": 159.8, "duration": 6.519, "text": "sides to get rid of the X on the left"}, {"start": 163.0, "duration": 7.12, "text": "side we would have 3 Z the minus y is"}, {"start": 166.319, "duration": 4.961, "text": "still there well we would have 2x - 1X"}, {"start": 170.12, "duration": 4.8, "text": "which would just leave us with a"}, {"start": 171.28, "duration": 7.76, "text": "positive 1X and then plus"}, {"start": 174.92, "duration": 7.959, "text": "4 and again to get rid of the minus y we"}, {"start": 179.04, "duration": 3.839, "text": "can simply add add y to both"}, {"start": 183.36, "duration": 7.959, "text": "sides so on the left side we have 3"}, {"start": 186.92, "duration": 6.679, "text": "z = x + y +"}, {"start": 191.319, "duration": 3.64, "text": "4 um typically I like to stick my"}, {"start": 193.599, "duration": 4.521, "text": "variables first so I just kind of"}, {"start": 194.959, "duration": 4.681, "text": "arbitrarily put the Y after the X and"}, {"start": 198.12, "duration": 4.199, "text": "again the last thing we want to do now"}, {"start": 199.64, "duration": 5.08, "text": "is get the Z all by itself so since"}, {"start": 202.319, "duration": 7.041, "text": "we're multiplying by three we can divide"}, {"start": 204.72, "duration": 7.519, "text": "by 3 um or equivalently multiply by 1/3"}, {"start": 209.36, "duration": 5.799, "text": "so you have to divide everything on the"}, {"start": 212.239, "duration": 5.2, "text": "right side by three as well so on the"}, {"start": 215.159, "duration": 5.0, "text": "left side we just have one Z and we"}, {"start": 217.439, "duration": 7.921, "text": "could either write the right side as x +"}, {"start": 220.159, "duration": 10.241, "text": "y + 4 over 3 or equivalently we could"}, {"start": 225.36, "duration": 6.439, "text": "rewrite that as 1/3 * the quantity x + y"}, {"start": 230.4, "duration": 5.399, "text": "+"}, {"start": 231.799, "duration": 4.0, "text": "4 and now we've got our"}, {"start": 236.079, "duration": 3.0, "text": "solution"}], "title": "Solving an Equation for a Specified Variable", "description": "Thanks to all of you who support me on Patreon. You da real mvps!   $1 per month helps!! :)  https://www.patreon.com/patrickjmt !! Solving an Equation for a Specified Variable", "lengthSeconds": 237, "width": 480, "height": 360}
{"video_id": "L_gHToviXSk", "transcript": [{"start": 0.84, "duration": 3.24, "text": "okay in this video we're going to do an"}, {"start": 2.28, "duration": 3.44, "text": "example again related to vector addition"}, {"start": 4.08, "duration": 3.279, "text": "and scalar multiplication we're also"}, {"start": 5.72, "duration": 3.799, "text": "going to talk about sketching some"}, {"start": 7.359, "duration": 4.681, "text": "vectors the uh sum of"}, {"start": 9.519, "duration": 5.881, "text": "vectors so we've got vector v that has"}, {"start": 12.04, "duration": 6.88, "text": "components 1 3 2g5 we're going to sketch"}, {"start": 15.4, "duration": 5.68, "text": "the vector v+ W and we'll also find the"}, {"start": 18.92, "duration": 5.32, "text": "components of that"}, {"start": 21.08, "duration": 5.119, "text": "Vector so when you sketch uh the"}, {"start": 24.24, "duration": 3.6, "text": "addition of vectors um so the first"}, {"start": 26.199, "duration": 5.0, "text": "thing I'm going to do is sketch vector v"}, {"start": 27.84, "duration": 4.919, "text": "so it has components of one and three so"}, {"start": 31.199, "duration": 5.001, "text": "just from the origin I'm going to move"}, {"start": 32.759, "duration": 7.001, "text": "one to the right and up one to"}, {"start": 36.2, "duration": 7.56, "text": "three so that's going to be my vector"}, {"start": 39.76, "duration": 7.84, "text": "v now Vector W when we when we sketch"}, {"start": 43.76, "duration": 7.279, "text": "vectors um wherever the first one stops"}, {"start": 47.6, "duration": 6.2, "text": "I just stick the new one onto that so"}, {"start": 51.039, "duration": 5.16, "text": "Vector W is going to start right here"}, {"start": 53.8, "duration": 4.88, "text": "but it says basically you move two units"}, {"start": 56.199, "duration": 5.121, "text": "to the uh to the right and then we go"}, {"start": 58.68, "duration": 5.039, "text": "five units down so I think we were"}, {"start": 61.32, "duration": 3.839, "text": "already up here uh up here uh you can"}, {"start": 63.719, "duration": 3.961, "text": "think about this as being a y value of"}, {"start": 65.159, "duration": 4.561, "text": "three if we go down five units we'll be"}, {"start": 67.68, "duration": 6.32, "text": "down here at"}, {"start": 69.72, "duration": 6.84, "text": "-2 so if we go two units to the um to"}, {"start": 74.0, "duration": 6.68, "text": "the right and then five units down so 1"}, {"start": 76.56, "duration": 9.28, "text": "two 3 four five this would be my"}, {"start": 80.68, "duration": 9.32, "text": "Vector W so again there was vector v"}, {"start": 85.84, "duration": 7.8, "text": "there was Vector W now what we do is um"}, {"start": 90.0, "duration": 7.759, "text": "wherever our original Vector started um"}, {"start": 93.64, "duration": 6.36, "text": "and wherever it stops we connect"}, {"start": 97.759, "duration": 7.281, "text": "those um and that's going to be our"}, {"start": 100.0, "duration": 7.2, "text": "vector v + W okay so it starts uh at the"}, {"start": 105.04, "duration": 4.92, "text": "origin and in this case it's kind of"}, {"start": 107.2, "duration": 5.519, "text": "going a it looks like three units to the"}, {"start": 109.96, "duration": 5.56, "text": "right and then two units down is what it"}, {"start": 112.719, "duration": 5.08, "text": "looks like to me so um so to me I'm"}, {"start": 115.52, "duration": 5.8, "text": "thinking well just based on the graph"}, {"start": 117.799, "duration": 6.521, "text": "shouldn't the components be three comma"}, {"start": 121.32, "duration": 5.159, "text": "-2 uh according to our little"}, {"start": 124.32, "duration": 5.04, "text": "diagram well let's actually find the"}, {"start": 126.479, "duration": 6.561, "text": "components okay so vector v is 1 and"}, {"start": 129.36, "duration": 6.159, "text": "three uh plus W which is 2"}, {"start": 133.04, "duration": 5.279, "text": "and5 again we just add respective"}, {"start": 135.519, "duration": 8.521, "text": "components so 1 + 2 that's going to give"}, {"start": 138.319, "duration": 8.56, "text": "us 3 3 + -5 is going to give"}, {"start": 144.04, "duration": 5.08, "text": "us2 and hey so it says if you add them"}, {"start": 146.879, "duration": 5.241, "text": "the new vectors will have components 3"}, {"start": 149.12, "duration": 4.68, "text": "comma -2 two and that's in fact you know"}, {"start": 152.12, "duration": 5.36, "text": "that's what happens from our from our"}, {"start": 153.8, "duration": 7.079, "text": "graph here so let's do one more here U"}, {"start": 157.48, "duration": 5.839, "text": "got a little little runoff um we're"}, {"start": 160.879, "duration": 5.841, "text": "going to sketch vector 2v and also find"}, {"start": 163.319, "duration": 5.761, "text": "its components so um so vector v again"}, {"start": 166.72, "duration": 6.879, "text": "we said that had components 1 comma 3 so"}, {"start": 169.08, "duration": 4.519, "text": "I'll sketch it down here 1 comma"}, {"start": 173.76, "duration": 6.72, "text": "3 when you multiply a vector by a"}, {"start": 177.48, "duration": 5.0, "text": "positive uh a positive whole number"}, {"start": 180.48, "duration": 4.92, "text": "all it does is it stretches It Out by"}, {"start": 182.48, "duration": 5.679, "text": "that factor so since we're multiplying"}, {"start": 185.4, "duration": 4.32, "text": "by two you basically can stick another"}, {"start": 188.159, "duration": 3.36, "text": "vector v at the end that would have"}, {"start": 189.72, "duration": 4.28, "text": "components again you would move over one"}, {"start": 191.519, "duration": 4.761, "text": "and up three it's just going to stretch"}, {"start": 194.0, "duration": 5.08, "text": "that original Vector out by a factor of"}, {"start": 196.28, "duration": 4.76, "text": "two so again just kind of eyeballing"}, {"start": 199.08, "duration": 4.719, "text": "this new Vector again which we could"}, {"start": 201.04, "duration": 5.72, "text": "call two uh"}, {"start": 203.799, "duration": 4.36, "text": "V it looks like to me it has components"}, {"start": 206.76, "duration": 5.479, "text": "it looks like to me you're moving over"}, {"start": 208.159, "duration": 7.16, "text": "two and up six so I'm thinking well this"}, {"start": 212.239, "duration": 5.0, "text": "should have components of 2 comma 6 and"}, {"start": 215.319, "duration": 3.84, "text": "again if we just use our our rule that"}, {"start": 217.239, "duration": 6.121, "text": "says you just multiply it says you would"}, {"start": 219.159, "duration": 6.841, "text": "have two * vector v which is 1 and three"}, {"start": 223.36, "duration": 6.2, "text": "and if you distribute the two we will"}, {"start": 226.0, "duration": 7.04, "text": "get two and six for the components of"}, {"start": 229.56, "duration": 3.48, "text": "our new vector"}], "title": "Vector Addition and Scalar Multiplication, Example 2", "description": "Thanks to all of you who support me on Patreon. You da real mvps!   $1 per month helps!! :)  https://www.patreon.com/patrickjmt !! Vector Addition and Scalar Multiplication, Example 2.  In this video I add two vectors in component form and also sketch the vectors to illustrate how to add vectors graphically (very useful stuff!).", "lengthSeconds": 231, "width": 480, "height": 360}
{"video_id": "LwRhlW2ZBAw", "transcript": [{"start": 0.799, "duration": 2.761, "text": "all right in this video we're going to"}, {"start": 1.959, "duration": 3.241, "text": "sketch the graph of a polinomial"}, {"start": 3.56, "duration": 3.759, "text": "function and here we're going to sketch"}, {"start": 5.2, "duration": 5.08, "text": "the graph of a polinomial that satisfies"}, {"start": 7.319, "duration": 9.001, "text": "all these following conditions so F of 3"}, {"start": 10.28, "duration": 8.159, "text": "= 2 frime of 1al 0 frime of -1 = -2 uh"}, {"start": 16.32, "duration": 4.48, "text": "it says the first derivative is positive"}, {"start": 18.439, "duration": 4.08, "text": "for all X greater than one the first"}, {"start": 20.8, "duration": 4.319, "text": "derivative is going to be negative for"}, {"start": 22.519, "duration": 4.16, "text": "all X less than one and the second"}, {"start": 25.119, "duration": 2.721, "text": "derivative is going to be positive for"}, {"start": 26.679, "duration": 4.121, "text": "all"}, {"start": 27.84, "duration": 5.559, "text": "X so usually when I sketch these I try"}, {"start": 30.8, "duration": 4.4, "text": "to kind of summarize everything that'll"}, {"start": 33.399, "duration": 4.561, "text": "sort of you know make more sense to me a"}, {"start": 35.2, "duration": 5.199, "text": "little bit so the first thing since F of"}, {"start": 37.96, "duration": 4.88, "text": "3 equals 2 that means that the point 3"}, {"start": 40.399, "duration": 4.201, "text": "comma 2 is on the graph and I usually go"}, {"start": 42.84, "duration": 3.6, "text": "ahead and put those dots on there just"}, {"start": 44.6, "duration": 4.479, "text": "because I know you know this points have"}, {"start": 46.44, "duration": 4.04, "text": "to be there the next thing I'm going to"}, {"start": 49.079, "duration": 4.28, "text": "think about is actually the first"}, {"start": 50.48, "duration": 4.96, "text": "derivative stuff so the first derivative"}, {"start": 53.359, "duration": 4.081, "text": "we said for all x coordinates greater"}, {"start": 55.44, "duration": 3.959, "text": "than one so for all x coordinates"}, {"start": 57.44, "duration": 3.119, "text": "greater than one the derivative is"}, {"start": 59.399, "duration": 2.84, "text": "positive"}, {"start": 60.559, "duration": 4.64, "text": "and that tells us the original function"}, {"start": 62.239, "duration": 4.321, "text": "is increasing and then it said for all x"}, {"start": 65.199, "duration": 3.761, "text": "coordinates less than one it was"}, {"start": 66.56, "duration": 4.559, "text": "negative so that means the function is"}, {"start": 68.96, "duration": 4.44, "text": "decreasing the second derivative was"}, {"start": 71.119, "duration": 4.281, "text": "greater than zero for all X that was our"}, {"start": 73.4, "duration": 4.28, "text": "last condition so that means we know"}, {"start": 75.4, "duration": 4.52, "text": "it's concave up"}, {"start": 77.68, "duration": 3.84, "text": "everywhere and already you know based on"}, {"start": 79.92, "duration": 3.519, "text": "that I'm thinking hey you know maybe a"}, {"start": 81.52, "duration": 3.919, "text": "graph kind of like a parabola would"}, {"start": 83.439, "duration": 5.72, "text": "would work here and I think it almost"}, {"start": 85.439, "duration": 5.241, "text": "will um you know so at xal 1 the slope"}, {"start": 89.159, "duration": 4.841, "text": "of the tangent line line is going to be"}, {"start": 90.68, "duration": 5.04, "text": "zero so since frime of 1 equals 0 so at"}, {"start": 94.0, "duration": 3.479, "text": "xal 1 I'm going to kind of make it you"}, {"start": 95.72, "duration": 4.24, "text": "know sort of bottom out here a little"}, {"start": 97.479, "duration": 4.161, "text": "bit so there at the x coordinate of one"}, {"start": 99.96, "duration": 4.479, "text": "there's the slope of our tangent line"}, {"start": 101.64, "duration": 4.92, "text": "equaling zero um it'll eventually we'll"}, {"start": 104.439, "duration": 5.161, "text": "have to go through that point uh 3 comma"}, {"start": 106.56, "duration": 7.44, "text": "2 and also at negative 1 we said the"}, {"start": 109.6, "duration": 6.04, "text": "slope is -2 well you can kind of imagine"}, {"start": 114.0, "duration": 3.68, "text": "you know having something that's sort of"}, {"start": 115.64, "duration": 4.32, "text": "parabolic in shape and certainly the"}, {"start": 117.68, "duration": 4.799, "text": "slope of that tangent line potentially"}, {"start": 119.96, "duration": 5.759, "text": "could equal -2 again the slope at the"}, {"start": 122.479, "duration": 5.401, "text": "bottom was a a slope of zero and I think"}, {"start": 125.719, "duration": 3.68, "text": "now we've actually got a uh a picture"}, {"start": 127.88, "duration": 4.68, "text": "that would actually agree with all this"}, {"start": 129.399, "duration": 6.48, "text": "so it goes through 3 comma 2 frime of"}, {"start": 132.56, "duration": 7.44, "text": "one that has a tangent line of"}, {"start": 135.879, "duration": 5.761, "text": "zero frime of1 equals -2 so it says at"}, {"start": 140.0, "duration": 4.08, "text": "negative 1 the slope of the tangent line"}, {"start": 141.64, "duration": 5.28, "text": "should be"}, {"start": 144.08, "duration": 5.08, "text": "-2 um we've got it increasing for all x"}, {"start": 146.92, "duration": 3.959, "text": "coordinates greater than one and we've"}, {"start": 149.16, "duration": 4.48, "text": "got it decreasing ining for all x"}, {"start": 150.879, "duration": 5.241, "text": "coordinates less than one and last but"}, {"start": 153.64, "duration": 4.64, "text": "not least it's concave up everywhere so"}, {"start": 156.12, "duration": 4.399, "text": "I think this would be a function f of x"}, {"start": 158.28, "duration": 5.239, "text": "that does satisfy all of those"}, {"start": 160.519, "duration": 3.0, "text": "conditions"}], "title": "Sketching a Graph Given Conditions About Derivative Requirements", "description": "Thanks to all of you who support me on Patreon. You da real mvps!   $1 per month helps!! :)  https://www.patreon.com/patrickjmt !! Sketching a Graph Given Conditions About Derivative Requirements.  In this video, we have a few conditions about points a graph must go through, as well as conditions on the first and second derivative.  I sketch a graph that satisfies the given conditions!", "lengthSeconds": 162, "width": 480, "height": 360}
{"video_id": "oUaIdqhCyTU", "transcript": [{"start": 0.03, "duration": 3.78, "text": "all right in this video I want to do an"}, {"start": 2.1, "duration": 5.31, "text": "example of multiplying and dividing"}, {"start": 3.81, "duration": 5.19, "text": "rational expressions so initially here"}, {"start": 7.41, "duration": 4.17, "text": "we've got a little bit of division of"}, {"start": 9.0, "duration": 5.34, "text": "two fractions containing some binomial"}, {"start": 11.58, "duration": 6.99, "text": "expressions and the basic idea in this"}, {"start": 14.34, "duration": 7.05, "text": "case is to factor and cancel is really"}, {"start": 18.57, "duration": 4.619, "text": "all it comes down to so the first thing"}, {"start": 21.39, "duration": 4.08, "text": "we'll have to do here is remember if you"}, {"start": 23.189, "duration": 5.391, "text": "have a fraction divided by a fraction we"}, {"start": 25.47, "duration": 3.11, "text": "leave the first one alone"}, {"start": 31.27, "duration": 7.5, "text": "and remember we turn our division into"}, {"start": 34.72, "duration": 9.06, "text": "multiplication and then we have to flip"}, {"start": 38.77, "duration": 7.14, "text": "the other fraction so division turns"}, {"start": 43.78, "duration": 5.099, "text": "into multiplication and the one on the"}, {"start": 45.91, "duration": 4.5, "text": "right side flip flops and at this point"}, {"start": 48.879, "duration": 5.271, "text": "again it's just trying to factor things"}, {"start": 50.41, "duration": 8.039, "text": "out so I'm going to make a couple little"}, {"start": 54.15, "duration": 7.15, "text": "observations here notice there's an 8"}, {"start": 58.449, "duration": 5.941, "text": "minus K on top and there's a K minus 8"}, {"start": 61.3, "duration": 6.54, "text": "on the bottom we can't we can't cancel"}, {"start": 64.39, "duration": 6.21, "text": "those out immediately I like to stick"}, {"start": 67.84, "duration": 5.91, "text": "things in parenthesis as well just as a"}, {"start": 70.6, "duration": 4.74, "text": "personal habit doesn't change its value"}, {"start": 73.75, "duration": 2.869, "text": "again because you can imagine there's a"}, {"start": 75.34, "duration": 4.29, "text": "1 out here"}, {"start": 76.619, "duration": 5.021, "text": "notice the signs here are opposite of"}, {"start": 79.63, "duration": 4.11, "text": "what they are down here so whatever I"}, {"start": 81.64, "duration": 4.83, "text": "recognize that the idea is you can"}, {"start": 83.74, "duration": 5.61, "text": "basically just factor a negative out of"}, {"start": 86.47, "duration": 4.98, "text": "one of them so notice if I pull the"}, {"start": 89.35, "duration": 3.93, "text": "negative out I have to change the signs"}, {"start": 91.45, "duration": 5.37, "text": "to make it equivalent I can make it"}, {"start": 93.28, "duration": 5.04, "text": "negative 8 plus K because when I"}, {"start": 96.82, "duration": 2.7, "text": "distribute the negative sign the"}, {"start": 98.32, "duration": 3.479, "text": "negative and the negative will make"}, {"start": 99.52, "duration": 5.07, "text": "positive 8 and the negative times the"}, {"start": 101.799, "duration": 4.71, "text": "positive K will give me negative K so"}, {"start": 104.59, "duration": 4.889, "text": "this is kind of a useful trick factoring"}, {"start": 106.509, "duration": 4.65, "text": "out a negative okay then we have K"}, {"start": 109.479, "duration": 3.57, "text": "squared minus 64"}, {"start": 111.159, "duration": 4.861, "text": "I recognize that as being a difference"}, {"start": 113.049, "duration": 7.89, "text": "of perfect squares so that factors is K"}, {"start": 116.02, "duration": 7.379, "text": "minus 8 K plus 8 ok there's really"}, {"start": 120.939, "duration": 4.441, "text": "nothing to do with the other side k plus"}, {"start": 123.399, "duration": 5.291, "text": "8"}, {"start": 125.38, "duration": 5.26, "text": "k- 8 but now since we have all"}, {"start": 128.69, "duration": 4.74, "text": "multiplication in the numerator all"}, {"start": 130.64, "duration": 5.67, "text": "multiplication in the denominator we can"}, {"start": 133.43, "duration": 4.62, "text": "cancel away so again notice in this"}, {"start": 136.31, "duration": 6.0, "text": "first set of parentheses we could really"}, {"start": 138.05, "duration": 7.71, "text": "rewrite that as k+ k -8 and we have a"}, {"start": 142.31, "duration": 6.69, "text": "positive K minus 8 down here so in this"}, {"start": 145.76, "duration": 5.88, "text": "case we can cancel those out so that"}, {"start": 149.0, "duration": 4.98, "text": "cancels with that and then we have a K"}, {"start": 151.64, "duration": 5.39, "text": "plus 8 in the numerator we have a K plus"}, {"start": 153.98, "duration": 6.15, "text": "8 in the denominator so we can cancel"}, {"start": 157.03, "duration": 7.15, "text": "those out as well and I think we have"}, {"start": 160.13, "duration": 6.39, "text": "now simplified down our original our"}, {"start": 164.18, "duration": 3.96, "text": "original problem here so in the"}, {"start": 166.52, "duration": 2.85, "text": "numerator we have a negative you can"}, {"start": 168.14, "duration": 5.25, "text": "think about that as just being negative"}, {"start": 169.37, "duration": 7.22, "text": "1 and then divided by K minus 8 again"}, {"start": 173.39, "duration": 5.55, "text": "you can put this in parenthesis or not"}, {"start": 176.59, "duration": 4.0, "text": "you know algebraically it's not going to"}, {"start": 178.94, "duration": 5.12, "text": "change its value typically though I"}, {"start": 180.59, "duration": 3.47, "text": "would certainly just write it like this"}], "title": "Rational Expressions:  Multiplying and Dividing.  Ex 1", "description": "Thanks to all of you who support me on Patreon. You da real mvps!   $1 per month helps!! :)  https://www.patreon.com/patrickjmt !! \nRational Expressions:  Multiplying and Dividing.  Ex 1.\nIn this video, I divide two rational expressions containing binomials.", "lengthSeconds": 184, "width": 480, "height": 360}
{"video_id": "y-6FrO65QOI", "transcript": [{"start": 0.03, "duration": 5.07, "text": "okay here we're going to look at an"}, {"start": 1.89, "duration": 6.0, "text": "example involving work and rates to get"}, {"start": 5.1, "duration": 4.38, "text": "a job done so we've got latisha and"}, {"start": 7.89, "duration": 3.84, "text": "Ricky they work for a computer software"}, {"start": 9.48, "duration": 5.52, "text": "company together they can write a"}, {"start": 11.73, "duration": 6.57, "text": "program in 19 hours latisha can do the"}, {"start": 15.0, "duration": 5.039, "text": "program by yourself in 32 hours we want"}, {"start": 18.3, "duration": 4.889, "text": "to know how long it would take Ricky to"}, {"start": 20.039, "duration": 6.15, "text": "write the program all by himself the"}, {"start": 23.189, "duration": 4.801, "text": "equation that we can use here if we look"}, {"start": 26.189, "duration": 4.321, "text": "at the rate per hour of one of the"}, {"start": 27.99, "duration": 4.199, "text": "people if we multiply that by the total"}, {"start": 30.51, "duration": 4.26, "text": "time to get the job done"}, {"start": 32.189, "duration": 4.561, "text": "plus we look at the rate per hour of the"}, {"start": 34.77, "duration": 6.93, "text": "other person multiplied by the total"}, {"start": 36.75, "duration": 7.379, "text": "time that equals one so maybe the the"}, {"start": 41.7, "duration": 5.75, "text": "first person here will let that"}, {"start": 44.129, "duration": 6.331, "text": "represent latisha so by yourself latisha"}, {"start": 47.45, "duration": 6.28, "text": "can write the program in thirty two"}, {"start": 50.46, "duration": 8.04, "text": "hours so that means her rate per hour"}, {"start": 53.73, "duration": 7.919, "text": "she could do well 1 over 32 we said the"}, {"start": 58.5, "duration": 6.479, "text": "total time for them working together to"}, {"start": 61.649, "duration": 8.671, "text": "get the job done the total time was 19"}, {"start": 64.979, "duration": 8.401, "text": "hours we don't know the rate of the"}, {"start": 70.32, "duration": 5.7, "text": "second person of Ricky pull so we'll"}, {"start": 73.38, "duration": 4.65, "text": "call that 1 over X again X will"}, {"start": 76.02, "duration": 4.349, "text": "represent the number of hours it would"}, {"start": 78.03, "duration": 4.949, "text": "take for him to do it alone so just like"}, {"start": 80.369, "duration": 6.331, "text": "it took latisha 32 hours her rate was 1"}, {"start": 82.979, "duration": 6.75, "text": "over 32 his rate will be 1 over X again"}, {"start": 86.7, "duration": 8.34, "text": "we multiply that by the total time which"}, {"start": 89.729, "duration": 6.541, "text": "is 19 and that equals 1 so now we've got"}, {"start": 95.04, "duration": 9.119, "text": "our equation to solve"}, {"start": 96.27, "duration": 10.68, "text": "we have 19 over 32 plus 19 over x equals"}, {"start": 104.159, "duration": 8.25, "text": "1 so now what I'm going to do is just"}, {"start": 106.95, "duration": 7.59, "text": "subtract 19 over 32 from both sides okay"}, {"start": 112.409, "duration": 4.951, "text": "so we would have 19 over X left on the"}, {"start": 114.54, "duration": 7.109, "text": "left okay we've got fractions we can"}, {"start": 117.36, "duration": 7.41, "text": "make 1 into 32 over 32 so we've got 32"}, {"start": 121.649, "duration": 7.2, "text": "over 32 minus 19 over 32 that's going to"}, {"start": 124.77, "duration": 6.23, "text": "leave us with 13 over 32 and now what"}, {"start": 128.849, "duration": 4.891, "text": "we're going to do is cross multiply so"}, {"start": 131.0, "duration": 7.39, "text": "I'm going to take"}, {"start": 133.74, "duration": 8.34, "text": "and multiply that by 32 so 19 times 32"}, {"start": 138.39, "duration": 8.19, "text": "that gives us 608 I'm going to multiply"}, {"start": 142.08, "duration": 8.69, "text": "X times 13 which will give us 13 X and"}, {"start": 146.58, "duration": 6.51, "text": "now simply let's divide both sides by 13"}, {"start": 150.77, "duration": 7.9, "text": "we'll be left with X on the right side"}, {"start": 153.09, "duration": 9.03, "text": "and 608 divided by 13 I'm getting this"}, {"start": 158.67, "duration": 6.15, "text": "to be roughly forty six point seven"}, {"start": 162.12, "duration": 5.76, "text": "seven so it says it's going to take"}, {"start": 164.82, "duration": 8.67, "text": "Ricky about forty six point seven seven"}, {"start": 167.88, "duration": 8.03, "text": "hours if he were to do this task by"}, {"start": 173.49, "duration": 2.42, "text": "himself"}], "title": "Rates and Time Required To Complete a Task", "description": "Thanks to all of you who support me on Patreon. You da real mvps!   $1 per month helps!! :)  https://www.patreon.com/patrickjmt !! Linear Equation , Word Problem #7 :  Work, Rates, Time To Complete a Task.\r\n  Here we are given that a person can complete a task alone in 32 hours and with another person they can finish the task in 19 hours.  We want to know how long it would take the second person working alone.", "lengthSeconds": 175, "width": 480, "height": 360}
{"video_id": "JY5ndajkuww", "transcript": [{"start": 0.64, "duration": 3.88, "text": "okay here we're going to look at solving"}, {"start": 2.04, "duration": 4.799, "text": "some more equations uh in this case"}, {"start": 4.52, "duration": 6.159, "text": "containing some integers and it'll just"}, {"start": 6.839, "duration": 5.041, "text": "be a little uh variety of uh problems uh"}, {"start": 10.679, "duration": 2.721, "text": "you know you may have to add you may"}, {"start": 11.88, "duration": 3.799, "text": "have to subtract you may have to"}, {"start": 13.4, "duration": 4.2, "text": "multiply to solve but uh we'll go"}, {"start": 15.679, "duration": 4.721, "text": "through all three of these uh in this"}, {"start": 17.6, "duration": 7.64, "text": "exam in this video so in part A we have"}, {"start": 20.4, "duration": 7.719, "text": "x - 7 = 10 well to get the X by itself"}, {"start": 25.24, "duration": 5.64, "text": "since we're subtracting seven We'll add"}, {"start": 28.119, "duration": 6.161, "text": "seven to both sides we'll be left with X"}, {"start": 30.88, "duration": 8.08, "text": "on the left side 10 + 7 is"}, {"start": 34.28, "duration": 4.68, "text": "17 and that'll be our solution to part"}, {"start": 42.16, "duration": 11.399, "text": "A okay so for Part B here we have 45 = x"}, {"start": 47.52, "duration": 8.92, "text": "- -3 well we can certainly rewrite"}, {"start": 53.559, "duration": 5.68, "text": "this we have a negative multiplied a"}, {"start": 56.44, "duration": 7.6, "text": "negative that's equivalent to a positive"}, {"start": 59.239, "duration": 7.361, "text": "so we could write x - -3 is just x + 3"}, {"start": 64.04, "duration": 5.439, "text": "and again if I want to solve for x now"}, {"start": 66.6, "duration": 4.76, "text": "since I'm adding three to both sides"}, {"start": 69.479, "duration": 7.601, "text": "what I'm going to do is just subtract"}, {"start": 71.36, "duration": 8.6, "text": "three from both sides so 45 minus 3 is"}, {"start": 77.08, "duration": 6.16, "text": "42 we're left with the X on the right"}, {"start": 79.96, "duration": 8.36, "text": "side and we have our solution to Part B"}, {"start": 83.24, "duration": 5.08, "text": "uh our solution will be the value 42"}, {"start": 91.4, "duration": 4.719, "text": "let's go ahead and look at uh part C"}, {"start": 93.6, "duration": 2.519, "text": "here as"}, {"start": 97.079, "duration": 5.841, "text": "well so we have p over 8 ="}, {"start": 104.88, "duration": 4.76, "text": "-6 I'm going to kind of do this in two"}, {"start": 107.32, "duration": 4.56, "text": "steps you could certainly uh you know do"}, {"start": 109.64, "duration": 4.88, "text": "everything here here in just a one step"}, {"start": 111.88, "duration": 4.8, "text": "but what I'm going to do is first off to"}, {"start": 114.52, "duration": 4.16, "text": "get rid of the negative on the left side"}, {"start": 116.68, "duration": 5.16, "text": "is I'm going to multiply both sides by"}, {"start": 118.68, "duration": 3.16, "text": "negative one"}, {"start": 123.96, "duration": 3.68, "text": "so a negative multiplied by a negative"}, {"start": 125.92, "duration": 5.24, "text": "is a positive so that's going to leave"}, {"start": 127.64, "duration": 5.8, "text": "us with p over 8 on the left side and -6"}, {"start": 131.16, "duration": 5.359, "text": "multiplied -1 it's going to leave us"}, {"start": 133.44, "duration": 4.92, "text": "with positive 6 and now the last thing"}, {"start": 136.519, "duration": 5.72, "text": "we're going to do"}, {"start": 138.36, "duration": 6.2, "text": "here is simply multiply both sides by"}, {"start": 142.239, "duration": 5.401, "text": "eight since we're dividing by eight on"}, {"start": 144.56, "duration": 5.64, "text": "the left side we'll multiply by eight so"}, {"start": 147.64, "duration": 8.599, "text": "the eights will cancel out we be left"}, {"start": 150.2, "duration": 8.48, "text": "with 1 p on the left side 6 * 8 is 48"}, {"start": 156.239, "duration": 5.841, "text": "and that's going to give us our solution"}, {"start": 158.68, "duration": 6.4, "text": "to uh the third equation here so our"}, {"start": 162.08, "duration": 7.0, "text": "solution will be the value P"}, {"start": 165.08, "duration": 4.0, "text": "equals 48"}], "title": "Solving Basic Equations Containing Integers", "description": "Thanks to all of you who support me on Patreon. You da real mvps!   $1 per month helps!! :)  https://www.patreon.com/patrickjmt !! Solving Basic Equations Containing Integers.  Here we look at solving some very basic linear equations containing integers.", "lengthSeconds": 168, "width": 480, "height": 360}
{"video_id": "EZf5U6YaXXc", "transcript": [{"start": 0.84, "duration": 3.8, "text": "all right here we're going to do another"}, {"start": 2.04, "duration": 3.6, "text": "example using the direct comparison test"}, {"start": 4.64, "duration": 2.879, "text": "so here we're going to look at the"}, {"start": 5.64, "duration": 5.119, "text": "series from one to"}, {"start": 7.519, "duration": 7.801, "text": "Infinity from 1 to Infinity of sin^2 n"}, {"start": 10.759, "duration": 7.161, "text": "over 5 raised to the N power so you know"}, {"start": 15.32, "duration": 4.28, "text": "if uh if you have no feel for kind of"}, {"start": 17.92, "duration": 3.599, "text": "what's going on with a series like this"}, {"start": 19.6, "duration": 4.24, "text": "you know a lot of times I'll ask people"}, {"start": 21.519, "duration": 4.321, "text": "you know sort of what part is throwing"}, {"start": 23.84, "duration": 4.519, "text": "you off what part feels the most"}, {"start": 25.84, "duration": 3.88, "text": "confusing um I think as n increases you"}, {"start": 28.359, "duration": 4.561, "text": "know the denominator you're going to get"}, {"start": 29.72, "duration": 5.4, "text": "power of five that's pretty clear uh the"}, {"start": 32.92, "duration": 3.68, "text": "numerator well you know as n increases"}, {"start": 35.12, "duration": 3.599, "text": "you're going to have s squar of all"}, {"start": 36.6, "duration": 3.639, "text": "these different values and well what on"}, {"start": 38.719, "duration": 4.84, "text": "Earth does that really you know what"}, {"start": 40.239, "duration": 4.64, "text": "happens in that case um you know and I"}, {"start": 43.559, "duration": 3.0, "text": "tell people a lot of times just get rid"}, {"start": 44.879, "duration": 3.881, "text": "of the stuff that's throwing you off if"}, {"start": 46.559, "duration": 4.721, "text": "the S squar wasn't in the numerator we"}, {"start": 48.76, "duration": 5.68, "text": "would have 1 over 5 to the n and that's"}, {"start": 51.28, "duration": 6.0, "text": "a convergent geometric Series so I would"}, {"start": 54.44, "duration": 5.24, "text": "think maybe somehow I can somehow relate"}, {"start": 57.28, "duration": 4.32, "text": "this to a convergent geometric series"}, {"start": 59.68, "duration": 4.519, "text": "again metrick certainly uh doesn't"}, {"start": 61.6, "duration": 3.96, "text": "always work but at least it gives me um"}, {"start": 64.199, "duration": 4.56, "text": "something to think about sort of a"}, {"start": 65.56, "duration": 5.32, "text": "direction to start uh to start"}, {"start": 68.759, "duration": 3.4, "text": "going well kind of the trick for this"}, {"start": 70.88, "duration": 2.84, "text": "one is we're actually going to use you"}, {"start": 72.159, "duration": 4.081, "text": "know the direct comparison and really"}, {"start": 73.72, "duration": 5.92, "text": "kind of the squeeze serem as well you"}, {"start": 76.24, "duration": 7.6, "text": "know sign of uh sign of anything is"}, {"start": 79.64, "duration": 7.36, "text": "between positive one and negative one"}, {"start": 83.84, "duration": 4.88, "text": "well sin squar n that's what we have so"}, {"start": 87.0, "duration": 3.36, "text": "I'm going to write it as s of n quantity"}, {"start": 88.72, "duration": 5.079, "text": "squared"}, {"start": 90.36, "duration": 5.16, "text": "what two values is that in between well"}, {"start": 93.799, "duration": 4.881, "text": "don't just mechanically Square both"}, {"start": 95.52, "duration": 5.76, "text": "sides and say one and one um that's"}, {"start": 98.68, "duration": 4.799, "text": "certainly not valid if you take a number"}, {"start": 101.28, "duration": 4.839, "text": "between negative 1 and positive one and"}, {"start": 103.479, "duration": 5.721, "text": "you square it you'll actually just get"}, {"start": 106.119, "duration": 6.201, "text": "values well between zero and positive"}, {"start": 109.2, "duration": 8.4, "text": "one okay so sin squ of n is always going"}, {"start": 112.32, "duration": 8.32, "text": "to be a number between zero and one well"}, {"start": 117.6, "duration": 6.0, "text": "again we're dividing by five to the n"}, {"start": 120.64, "duration": 8.479, "text": "so 0 over 5 n would be less than or"}, {"start": 123.6, "duration": 9.12, "text": "equal to sin n over 5 n Which is less"}, {"start": 129.119, "duration": 6.12, "text": "than or equal to well 1 over 5 to the N"}, {"start": 132.72, "duration": 6.519, "text": "well on the left obviously we just have"}, {"start": 135.239, "duration": 7.0, "text": "zero um so the idea is uh the series"}, {"start": 139.239, "duration": 4.801, "text": "from nals 1 to Infinity of zero that"}, {"start": 142.239, "duration": 6.681, "text": "would be less than or equal to the"}, {"start": 144.04, "duration": 8.0, "text": "series from n = 1 to Infinity of sin^2 n"}, {"start": 148.92, "duration": 7.44, "text": "over 5 n Which is less than or equal to"}, {"start": 152.04, "duration": 7.559, "text": "the series from n = 1 to Infinity of 1 5"}, {"start": 156.36, "duration": 6.239, "text": "the N but again this is just a geometric"}, {"start": 159.599, "duration": 3.0, "text": "uh a geometric"}, {"start": 165.239, "duration": 5.36, "text": "series I guess even more important uh is"}, {"start": 168.08, "duration": 5.96, "text": "that it's a convergent geometric"}, {"start": 170.599, "duration": 7.0, "text": "Series right because here our r value uh"}, {"start": 174.04, "duration": 6.44, "text": "is going to be positive 1/5 and if our R"}, {"start": 177.599, "duration": 5.481, "text": "values between negative 1 and positive 1"}, {"start": 180.48, "duration": 5.28, "text": "uh that's I should say exclusive uh so"}, {"start": 183.08, "duration": 4.76, "text": "if if the r values less than one and"}, {"start": 185.76, "duration": 4.479, "text": "greater than negative one that's when we"}, {"start": 187.84, "duration": 5.92, "text": "have a convergent series well which we"}, {"start": 190.239, "duration": 5.801, "text": "certainly do well since sin s of n over"}, {"start": 193.76, "duration": 4.119, "text": "5 to the N that series it's always"}, {"start": 196.04, "duration": 4.4, "text": "bigger than zero greater than or equal"}, {"start": 197.879, "duration": 6.92, "text": "to zero it's always less than or equal"}, {"start": 200.44, "duration": 9.4, "text": "to this convergent geometric Series so"}, {"start": 204.799, "duration": 7.841, "text": "by the uh by the direct comparison test"}, {"start": 209.84, "duration": 7.2, "text": "our series that we're uh interested in"}, {"start": 212.64, "duration": 4.4, "text": "that must also converge"}], "title": "Direct Comparison Test - Another Example 4", "description": "Thanks to all of you who support me on Patreon. You da real mvps!   $1 per month helps!! :)  https://www.patreon.com/patrickjmt !! Direct Comparison Test - Another Example 4. In this video I show that another series converges or diverges using the direct comparison theorem. I do not run over the formula/theory in this video (but do in another video, so look around if that is what you need!).", "lengthSeconds": 219, "width": 480, "height": 360}
{"video_id": "OPQSWJpdIdI", "transcript": [{"start": 0.96, "duration": 3.879, "text": "okay in this example we just want to"}, {"start": 2.919, "duration": 4.761, "text": "make a a connection between interval"}, {"start": 4.839, "duration": 5.481, "text": "notation and compound inequalities and"}, {"start": 7.68, "duration": 6.52, "text": "the question"}, {"start": 10.32, "duration": 5.28, "text": "is when we write in parentheses -2 comma"}, {"start": 14.2, "duration": 5.079, "text": "to 8 with a"}, {"start": 15.6, "duration": 6.599, "text": "bracket does that mean the same as all"}, {"start": 19.279, "duration": 6.281, "text": "the values of X that are greater than -2"}, {"start": 22.199, "duration": 4.761, "text": "but less than or equal to 8 and you know"}, {"start": 25.56, "duration": 5.559, "text": "sometimes what I do is I like to make a"}, {"start": 26.96, "duration": 5.72, "text": "little number line so here is -2 um and"}, {"start": 31.119, "duration": 3.721, "text": "since it's a parenthesis remember that"}, {"start": 32.68, "duration": 4.32, "text": "means we don't include it so I'm going"}, {"start": 34.84, "duration": 3.48, "text": "to put a parentheses or sometimes people"}, {"start": 37.0, "duration": 4.879, "text": "will put an open"}, {"start": 38.32, "duration": 6.04, "text": "circle and then it says basically to I I"}, {"start": 41.879, "duration": 5.561, "text": "think it it says use every number every"}, {"start": 44.36, "duration": 4.56, "text": "number every number up to and including"}, {"start": 47.44, "duration": 3.24, "text": "eight that's what the little brackets"}, {"start": 48.92, "duration": 3.319, "text": "mean include it and I always thought"}, {"start": 50.68, "duration": 3.679, "text": "about the brackets as being like little"}, {"start": 52.239, "duration": 4.201, "text": "hooks so they catch that number and"}, {"start": 54.359, "duration": 3.72, "text": "bring it along with us uh the"}, {"start": 56.44, "duration": 4.52, "text": "parentheses are kind of rounded so the"}, {"start": 58.079, "duration": 5.241, "text": "number gets away so yeah yes to me this"}, {"start": 60.96, "duration": 5.36, "text": "interval notation from -2 with"}, {"start": 63.32, "duration": 5.479, "text": "parentheses uh up to 8 with a bracket"}, {"start": 66.32, "duration": 6.0, "text": "that does mean use every value of x"}, {"start": 68.799, "duration": 7.521, "text": "that's strictly greater than -2 so X has"}, {"start": 72.32, "duration": 6.28, "text": "to be greater than -2 and X also has to"}, {"start": 76.32, "duration": 4.799, "text": "be less than or equal to 8 so I would"}, {"start": 78.6, "duration": 5.519, "text": "say yes that's definitely a true"}, {"start": 81.119, "duration": 3.0, "text": "statement"}], "title": "Interval Notation - A basic question!", "description": "Thanks to all of you who support me on Patreon. You da real mvps!   $1 per month helps!! :)  https://www.patreon.com/patrickjmt !! A basic true/false question to make sure you understand interval notation!", "lengthSeconds": 83, "width": 480, "height": 360}
{"video_id": "8rHOZvvrNHI", "transcript": [{"start": 1.52, "duration": 5.319, "text": "okay in this example we have the"}, {"start": 2.76, "duration": 5.959, "text": "quadratic equation x^2 - 3x = 4 and"}, {"start": 6.839, "duration": 4.88, "text": "again it's quadratic because the highest"}, {"start": 8.719, "duration": 4.241, "text": "power that we see on X is a power of two"}, {"start": 11.719, "duration": 3.16, "text": "we want to know which of the following"}, {"start": 12.96, "duration": 4.079, "text": "is a first step that you could take to"}, {"start": 14.879, "duration": 5.001, "text": "solve for"}, {"start": 17.039, "duration": 5.441, "text": "x okay so part A says add four to both"}, {"start": 19.88, "duration": 5.559, "text": "sides of the equation B says subtract"}, {"start": 22.48, "duration": 5.4, "text": "four from both sides of the equation C"}, {"start": 25.439, "duration": 5.92, "text": "says factor to the left side is x * the"}, {"start": 27.88, "duration": 5.72, "text": "quantity x - 3 Part D says combine the"}, {"start": 31.359, "duration": 5.841, "text": "terms on the left side of the equation"}, {"start": 33.6, "duration": 5.799, "text": "so the one thing that you really need to"}, {"start": 37.2, "duration": 4.92, "text": "remember um you know if you run across a"}, {"start": 39.399, "duration": 4.64, "text": "quadratic equation you you recognize oh"}, {"start": 42.12, "duration": 3.439, "text": "that's quadratic the first thing you"}, {"start": 44.039, "duration": 2.801, "text": "want to do before you start solving it"}, {"start": 45.559, "duration": 3.32, "text": "is you want to try to make one side"}, {"start": 46.84, "duration": 3.28, "text": "equal to zero so that's that's kind of"}, {"start": 48.879, "duration": 2.881, "text": "the the thing that I'm going to keep in"}, {"start": 50.12, "duration": 4.36, "text": "mind here is I want one side of this"}, {"start": 51.76, "duration": 2.72, "text": "equation to be"}, {"start": 54.64, "duration": 4.28, "text": "zero well part A says let's add four to"}, {"start": 57.44, "duration": 4.52, "text": "both"}, {"start": 58.92, "duration": 8.44, "text": "sides well if I add four to both sides"}, {"start": 61.96, "duration": 8.799, "text": "I've got x^2 - 3x + 4 on the right side"}, {"start": 67.36, "duration": 6.48, "text": "I would have 4 + 4 which is 8 well that"}, {"start": 70.759, "duration": 4.641, "text": "doesn't really do anything to me useful"}, {"start": 73.84, "duration": 3.68, "text": "um right it doesn't help me get make one"}, {"start": 75.4, "duration": 3.8, "text": "side equal to zero all I've really done"}, {"start": 77.52, "duration": 3.72, "text": "is I've got an extra term now on the"}, {"start": 79.2, "duration": 4.559, "text": "left side this positive4 that wasn't"}, {"start": 81.24, "duration": 4.879, "text": "there before and so so that to me"}, {"start": 83.759, "duration": 4.481, "text": "doesn't really seem to be doing anything"}, {"start": 86.119, "duration": 4.481, "text": "productive so part A doesn't seem to"}, {"start": 88.24, "duration": 4.199, "text": "make sense Part B says let's subtract"}, {"start": 90.6, "duration": 4.839, "text": "four from both sides and I would think"}, {"start": 92.439, "duration": 4.72, "text": "aha you know that makes more sense again"}, {"start": 95.439, "duration": 4.0, "text": "if we're wanting to make one side of"}, {"start": 97.159, "duration": 3.92, "text": "this equation equal to zero that seems"}, {"start": 99.439, "duration": 4.121, "text": "like the reasonable thing to do in this"}, {"start": 101.079, "duration": 7.36, "text": "case so on the left side we would have"}, {"start": 103.56, "duration": 7.4, "text": "x^2 - 3x - 4 mean there's no like terms"}, {"start": 108.439, "duration": 4.36, "text": "so we can't combine anything on the"}, {"start": 110.96, "duration": 3.839, "text": "right side we would have 4 Min - 4 which"}, {"start": 112.799, "duration": 4.28, "text": "would give us"}, {"start": 114.799, "duration": 4.721, "text": "zero and now we've we've kind of done"}, {"start": 117.079, "duration": 5.921, "text": "this first step of making one side of"}, {"start": 119.52, "duration": 5.08, "text": "the the equation equal to zero so Part B"}, {"start": 123.0, "duration": 3.479, "text": "certainly looks like it's going to be"}, {"start": 124.6, "duration": 5.159, "text": "the correct answer let's look at the"}, {"start": 126.479, "duration": 7.4, "text": "other two suggestions and we'll see what"}, {"start": 129.759, "duration": 7.321, "text": "happens so part C said Factor the left"}, {"start": 133.879, "duration": 6.0, "text": "side is X multiplied xus"}, {"start": 137.08, "duration": 5.08, "text": "3 okay while it's technically correct"}, {"start": 139.879, "duration": 5.0, "text": "that you could do"}, {"start": 142.16, "duration": 5.84, "text": "that it is correct to factor the left"}, {"start": 144.879, "duration": 5.961, "text": "side is x * x -"}, {"start": 148.0, "duration": 5.84, "text": "3 again that's not doing anything in"}, {"start": 150.84, "duration": 6.679, "text": "terms of making the one side of the"}, {"start": 153.84, "duration": 5.679, "text": "equation equal to zero so this"}, {"start": 157.519, "duration": 3.481, "text": "factorization while it's correct on the"}, {"start": 159.519, "duration": 3.841, "text": "left side it's not going to help you"}, {"start": 161.0, "duration": 6.2, "text": "arrive at a solution at"}, {"start": 163.36, "duration": 7.04, "text": "all so that's also not correct let's"}, {"start": 167.2, "duration": 5.16, "text": "look at d d says combine the terms on"}, {"start": 170.4, "duration": 5.0, "text": "the left side of the"}, {"start": 172.36, "duration": 6.48, "text": "equation well you can't really combine"}, {"start": 175.4, "duration": 6.44, "text": "these terms these are not like"}, {"start": 178.84, "duration": 3.0, "text": "terms"}, {"start": 182.68, "duration": 4.32, "text": "you have one term involving X SAR and"}, {"start": 185.04, "duration": 4.36, "text": "one term involving just x to the first"}, {"start": 187.0, "duration": 4.159, "text": "power so those are not like terms which"}, {"start": 189.4, "duration": 5.08, "text": "means they"}, {"start": 191.159, "duration": 3.321, "text": "can't be"}, {"start": 195.159, "duration": 6.681, "text": "combined so a b and c all you know all"}, {"start": 200.2, "duration": 3.24, "text": "gave you things that were legal to do"}, {"start": 201.84, "duration": 3.44, "text": "it's correct part D is actually"}, {"start": 203.44, "duration": 4.12, "text": "suggesting you know is saying to do"}, {"start": 205.28, "duration": 4.8, "text": "something which isn't even algebraically"}, {"start": 207.56, "duration": 4.599, "text": "correct so all right so again it looks"}, {"start": 210.08, "duration": 4.96, "text": "like uh B was our correct Choice"}, {"start": 212.159, "duration": 5.881, "text": "subtracting four from both sides of the"}, {"start": 215.04, "duration": 3.0, "text": "equation"}], "title": "GED 'Most Missed' Math Practice Problems , Code A.4.a,  #1", "description": "Thanks to all of you who support me on Patreon. You da real mvps!   $1 per month helps!! :)  https://www.patreon.com/patrickjmt !! The GED Testing Service keeps a list of the most missed math topics and can be seen here: http://www.gedtestingservice.com/uploads/files/126de0283c94ff6323b3b5b3f310da5d.pdf\nThis video shows how to work a sample problem using their coding system that is shown on the left using the link above.", "lengthSeconds": 216, "width": 480, "height": 360}
{"video_id": "jO5vIRvRrUU", "transcript": [{"start": 1.719, "duration": 3.12, "text": "all right in this video we're going to"}, {"start": 2.919, "duration": 4.84, "text": "look at some examples related to using"}, {"start": 4.839, "duration": 5.361, "text": "the squeeze Andor absolute value theorem"}, {"start": 7.759, "duration": 4.681, "text": "so in part A here we've got we're going"}, {"start": 10.2, "duration": 5.44, "text": "to find the limit of the sequence 1/ n"}, {"start": 12.44, "duration": 5.64, "text": "Cub Time s of n^"}, {"start": 15.64, "duration": 4.08, "text": "2 so again what we're trying to do is"}, {"start": 18.08, "duration": 3.88, "text": "we're trying to take the limit as n goes"}, {"start": 19.72, "duration": 2.24, "text": "to"}, {"start": 23.8, "duration": 5.52, "text": "Infinity of this function kind of the"}, {"start": 27.199, "duration": 5.88, "text": "problem is though you know as n goes to"}, {"start": 29.32, "duration": 6.759, "text": "Infinity n squ goes to Infinity but as"}, {"start": 33.079, "duration": 5.521, "text": "uh you know the inside uh of s gets"}, {"start": 36.079, "duration": 5.081, "text": "bigger and bigger and bigger S of that"}, {"start": 38.6, "duration": 4.16, "text": "that limit simply does not exist but"}, {"start": 41.16, "duration": 3.84, "text": "that doesn't mean that we can simply say"}, {"start": 42.76, "duration": 4.04, "text": "well this limit doesn't exist and again"}, {"start": 45.0, "duration": 2.84, "text": "what we can use in this cases is the"}, {"start": 46.8, "duration": 3.04, "text": "squeeze"}, {"start": 47.84, "duration": 5.359, "text": "theorem"}, {"start": 49.84, "duration": 7.08, "text": "so s of n s you"}, {"start": 53.199, "duration": 7.121, "text": "know sign of anything uh is always"}, {"start": 56.92, "duration": 7.04, "text": "between positive one and negative one"}, {"start": 60.32, "duration": 6.44, "text": "okay so uh we're going to try to kind of"}, {"start": 63.96, "duration": 5.76, "text": "find a function that that that's smaller"}, {"start": 66.76, "duration": 5.999, "text": "than 1/ n Cub s of n^2 and a function"}, {"start": 69.72, "duration": 4.2, "text": "that's bigger than 1 over n Cub s of n^2"}, {"start": 72.759, "duration": 4.201, "text": "and again we're going to try to show"}, {"start": 73.92, "duration": 5.12, "text": "that those limits um have the same value"}, {"start": 76.96, "duration": 4.4, "text": "and therefore the limit uh that we're"}, {"start": 79.04, "duration": 4.759, "text": "looking for will also have the same"}, {"start": 81.36, "duration": 4.079, "text": "value so this is kind of the first step"}, {"start": 83.799, "duration": 3.601, "text": "I think you know people will agree with"}, {"start": 85.439, "duration": 4.841, "text": "that okay sign of whatever is between"}, {"start": 87.4, "duration": 4.759, "text": "negative 1 and one well again 10 of the"}, {"start": 90.28, "duration": 7.28, "text": "function I want though I don't want just"}, {"start": 92.159, "duration": 7.881, "text": "s of n^ 2 I want s of n^ 2 * 1 n 3 you"}, {"start": 97.56, "duration": 6.44, "text": "know since in some positive number if we"}, {"start": 100.04, "duration": 6.8, "text": "multiply by one over n the"}, {"start": 104.0, "duration": 5.719, "text": "3 so all I'm doing is I'm"}, {"start": 106.84, "duration": 5.36, "text": "multiplying the uh Inside by 1 over n"}, {"start": 109.719, "duration": 4.841, "text": "the 3 well if I multiply the left side"}, {"start": 112.2, "duration": 6.279, "text": "by 1 over n the"}, {"start": 114.56, "duration": 5.479, "text": "3 in the right side by 1 over n the 3r"}, {"start": 118.479, "duration": 4.481, "text": "I'll now kind of get the small small"}, {"start": 120.039, "duration": 5.841, "text": "function and this big function so 1 n"}, {"start": 122.96, "duration": 6.439, "text": "Cub of n^2 is always going to be greater"}, {"start": 125.88, "duration": 6.64, "text": "than or equal to1 n cubed but it's"}, {"start": 129.399, "duration": 6.001, "text": "always smaller than a positive 1 over n"}, {"start": 132.52, "duration": 4.96, "text": "cubed well likewise if we take the limit"}, {"start": 135.4, "duration": 5.68, "text": "as in goes to"}, {"start": 137.48, "duration": 5.96, "text": "Infinity of all of"}, {"start": 141.08, "duration": 4.92, "text": "these so now here we're using the"}, {"start": 143.44, "duration": 2.56, "text": "squeeze"}, {"start": 148.76, "duration": 4.64, "text": "theorem so now we're using the squeeze"}, {"start": 151.2, "duration": 5.28, "text": "theorem and well the limit as n goes to"}, {"start": 153.4, "duration": 6.559, "text": "Infinity of1 Over N cubed that's simply"}, {"start": 156.48, "duration": 5.96, "text": "going to equal 0o likewise the limit as"}, {"start": 159.959, "duration": 6.0, "text": "n goes to Infinity of positive 1 over n"}, {"start": 162.44, "duration": 5.519, "text": "cubed is going to equal zero well we've"}, {"start": 165.959, "duration": 6.721, "text": "now shown that the limit as n goes to"}, {"start": 167.959, "duration": 6.721, "text": "Infinity of 1/ n cubed s of n^ 2 well"}, {"start": 172.68, "duration": 4.44, "text": "that's less than or equal to zero it's"}, {"start": 174.68, "duration": 4.839, "text": "greater than or equal to zero so that"}, {"start": 177.12, "duration": 5.6, "text": "implies that our limit uh our original"}, {"start": 179.519, "duration": 6.8, "text": "limit must also equal zero again using"}, {"start": 182.72, "duration": 3.599, "text": "the squeeze theorem"}], "title": "The Squeeze Theorem and Absolute Value Theorem, #1", "description": "Thanks to all of you who support me on Patreon. You da real mvps!   $1 per month helps!! :)  https://www.patreon.com/patrickjmt !! The Squeeze Theorem and Absolute Value Theorem, #1.  Here we look at finding the limits of some sequences by using the squeeze and / or absolute value theorems.", "lengthSeconds": 184, "width": 480, "height": 360}
{"video_id": "pU0XJndeM4A", "transcript": [{"start": 0.88, "duration": 3.079, "text": "all right in this video we're going to"}, {"start": 2.08, "duration": 4.52, "text": "look at an example related to the"}, {"start": 3.959, "duration": 4.241, "text": "derivative of an inverse function um so"}, {"start": 6.6, "duration": 3.52, "text": "here we've got the uh you know the"}, {"start": 8.2, "duration": 3.76, "text": "formula associated with finding the"}, {"start": 10.12, "duration": 4.8, "text": "derivative of an"}, {"start": 11.96, "duration": 6.399, "text": "inverse so in this one we have a"}, {"start": 14.92, "duration": 6.4, "text": "function X to the3 plus e x and we're"}, {"start": 18.359, "duration": 5.281, "text": "given that F inverse of 1al 0 we want to"}, {"start": 21.32, "duration": 6.119, "text": "find the derivative of the inverse"}, {"start": 23.64, "duration": 5.52, "text": "function at xal 1 you know one thing we"}, {"start": 27.439, "duration": 3.6, "text": "can make sure of is that this function"}, {"start": 29.16, "duration": 5.88, "text": "does have an inverse"}, {"start": 31.039, "duration": 6.36, "text": "so notice that the derivative frime of x"}, {"start": 35.04, "duration": 6.92, "text": "that would equal"}, {"start": 37.399, "duration": 7.401, "text": "3x^2 plus e to X but um notice that this"}, {"start": 41.96, "duration": 6.239, "text": "is always going to be uh greater than"}, {"start": 44.8, "duration": 6.16, "text": "zero for all X and the reason is e to"}, {"start": 48.199, "duration": 4.721, "text": "the x is greater than zero for all X if"}, {"start": 50.96, "duration": 3.96, "text": "you plug any number uh into the E"}, {"start": 52.92, "duration": 3.159, "text": "function uh the exponential function e"}, {"start": 54.92, "duration": 4.159, "text": "to the X you always get something"}, {"start": 56.079, "duration": 6.16, "text": "greater than zero you know 3x^2 is"}, {"start": 59.079, "duration": 5.72, "text": "certainly greater than or equal to"}, {"start": 62.239, "duration": 4.441, "text": "zero and when you add them together well"}, {"start": 64.799, "duration": 4.121, "text": "since this one's already positive if you"}, {"start": 66.68, "duration": 4.32, "text": "add something zero or bigger we'll"}, {"start": 68.92, "duration": 4.199, "text": "always get something greater than"}, {"start": 71.0, "duration": 4.439, "text": "zero and that tells us since our"}, {"start": 73.119, "duration": 4.521, "text": "derivative is always positive it tells"}, {"start": 75.439, "duration": 4.841, "text": "us our function uh is strictly"}, {"start": 77.64, "duration": 6.479, "text": "increasing which means it"}, {"start": 80.28, "duration": 8.479, "text": "passes the horizontal line test and that"}, {"start": 84.119, "duration": 4.64, "text": "tells us the function uh F does have an"}, {"start": 88.84, "duration": 3.76, "text": "inverse"}, {"start": 90.759, "duration": 4.521, "text": "okay so this question is you know"}, {"start": 92.6, "duration": 4.879, "text": "phrased uh properly it does in fact uh"}, {"start": 95.28, "duration": 5.799, "text": "we can ask about the derivative of the"}, {"start": 97.479, "duration": 5.28, "text": "inverse so now using our formula here it"}, {"start": 101.079, "duration": 5.521, "text": "just says well if we want to figure out"}, {"start": 102.759, "duration": 5.96, "text": "the derivative with respect to X of the"}, {"start": 106.6, "duration": 7.839, "text": "uh inverse function"}, {"start": 108.719, "duration": 9.561, "text": "evaluated um at xal 1 we'll just"}, {"start": 114.439, "duration": 3.841, "text": "use our formula"}, {"start": 118.68, "duration": 4.92, "text": "here okay okay so again we know that F"}, {"start": 121.759, "duration": 5.601, "text": "inverse of"}, {"start": 123.6, "duration": 5.359, "text": "one that's given to us as zero so really"}, {"start": 127.36, "duration": 5.959, "text": "all we're doing is we're just figuring"}, {"start": 128.959, "duration": 6.0, "text": "out 1 over F Prime of 0 well all we have"}, {"start": 133.319, "duration": 4.2, "text": "to do now is just take the derivative of"}, {"start": 134.959, "duration": 2.56, "text": "the actual"}, {"start": 137.84, "duration": 4.88, "text": "function which we've already done here"}, {"start": 140.84, "duration": 6.0, "text": "and then all we have to do is simply"}, {"start": 142.72, "duration": 8.599, "text": "plug zero into that so notice F Prime of"}, {"start": 146.84, "duration": 7.64, "text": "0 is just going to give us 3 * 0 S Plus"}, {"start": 151.319, "duration": 5.721, "text": "e the 0 which is going to give us one so"}, {"start": 154.48, "duration": 4.56, "text": "that says uh when we plug it into uh"}, {"start": 157.04, "duration": 5.36, "text": "this part we'll just be left with one"}, {"start": 159.04, "duration": 6.96, "text": "over one or we'll get one as the value"}, {"start": 162.4, "duration": 6.6, "text": "of the derivative um at that"}, {"start": 166.0, "duration": 3.0, "text": "point"}], "title": "Derivative of an Inverse Function , Ex 1", "description": "In this video I work a quick example of where we need to evaluate the derivative of an inverse function but we are only given the original function, not the inverse.\nThis is a typical topic in a calculus course.  In order to evaluate the derivative at a point we have a clever work around that uses the derivative of the original function, evaluated at the inverse.\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n#math, #algebra, #khanacademy, #khan, #geometry, #education, #montessori, #calculus, #homeschooling, #functions, #notation, #organicchemistry, #organicchemistrytutor, #calculus, #derivative, #powerrule, #limits,  #limit, #graph, #graphing, #conjugate, #infinity, #patrickjmt, #rational, #shortcut, #trick, #commondenominator, #fraction, #university, #admissions, #test #continuity #onesided #infinite #absolutevalue #tangents #tangentsandnormals #tangentline #definition #hindi #india", "lengthSeconds": 167, "width": 480, "height": 360}
{"video_id": "MM3BXtVu9eM", "transcript": [{"start": 1.439, "duration": 3.721, "text": "okay um in this video I want to talk"}, {"start": 3.12, "duration": 3.92, "text": "about finding the radius of of"}, {"start": 5.16, "duration": 3.559, "text": "convergence of a power series and I've"}, {"start": 7.04, "duration": 4.88, "text": "had a lot of people ask about this and"}, {"start": 8.719, "duration": 5.04, "text": "it's actually easy after you found the"}, {"start": 11.92, "duration": 3.879, "text": "interval of convergence so to find the"}, {"start": 13.759, "duration": 4.041, "text": "radius of convergence you first"}, {"start": 15.799, "duration": 3.961, "text": "basically have to find the interval of"}, {"start": 17.8, "duration": 4.08, "text": "convergence then you simply find the"}, {"start": 19.76, "duration": 6.24, "text": "length of that interval and divide by"}, {"start": 21.88, "duration": 6.04, "text": "two so um I've got a bunch of intervals"}, {"start": 26.0, "duration": 4.16, "text": "of convergence on the left side and the"}, {"start": 27.92, "duration": 4.72, "text": "associated radius of convergence on on"}, {"start": 30.16, "duration": 2.48, "text": "the right"}, {"start": 32.8, "duration": 5.599, "text": "side if your interval of convergence is"}, {"start": 35.76, "duration": 5.16, "text": "all real numbers negative Infinity to"}, {"start": 38.399, "duration": 4.401, "text": "positive Infinity we simply say that the"}, {"start": 40.92, "duration": 4.24, "text": "radius is equal to"}, {"start": 42.8, "duration": 6.52, "text": "Infinity if it only converges for a"}, {"start": 45.16, "duration": 5.96, "text": "single point so a trivial solution um"}, {"start": 49.32, "duration": 2.84, "text": "then we simply say that the radius is"}, {"start": 51.12, "duration": 3.2, "text": "equal to"}, {"start": 52.16, "duration": 6.199, "text": "zero"}, {"start": 54.32, "duration": 7.68, "text": "um now uh it doesn't matter whether if"}, {"start": 58.359, "duration": 6.041, "text": "you have um an interval say like in this"}, {"start": 62.0, "duration": 5.68, "text": "example negative 1 to 7 it doesn't"}, {"start": 64.4, "duration": 6.2, "text": "matter whether the end points of the"}, {"start": 67.68, "duration": 6.479, "text": "interval produce a convergent series or"}, {"start": 70.6, "duration": 5.68, "text": "not um and all of these cases um kind of"}, {"start": 74.159, "duration": 4.401, "text": "all four possible cases of the end"}, {"start": 76.28, "duration": 4.4, "text": "points either converging or not"}, {"start": 78.56, "duration": 3.72, "text": "converging again all you do is find the"}, {"start": 80.68, "duration": 3.84, "text": "length of the interval take the right"}, {"start": 82.28, "duration": 5.879, "text": "end point minus the left end point and"}, {"start": 84.52, "duration": 5.4, "text": "divide by two so if we go from- 1 to 7"}, {"start": 88.159, "duration": 3.881, "text": "hey that has a length of eight divide"}, {"start": 89.92, "duration": 5.68, "text": "div it by two and we would simply say"}, {"start": 92.04, "duration": 6.88, "text": "the radius of convergence is four so um"}, {"start": 95.6, "duration": 4.479, "text": "very easy to do um but obviously again"}, {"start": 98.92, "duration": 3.76, "text": "you have to know the interval of"}, {"start": 100.079, "duration": 5.0, "text": "convergence and that's the hard part so"}, {"start": 102.68, "duration": 3.6, "text": "all right um uh kind of short and sweet"}, {"start": 105.079, "duration": 4.761, "text": "I hope it makes sense if you have"}, {"start": 106.28, "duration": 3.56, "text": "questions let me know"}], "title": "Radius of Convergence for a Power Series", "description": "Thanks to all of you who support me on Patreon. You da real mvps!   $1 per month helps!! :)  https://www.patreon.com/patrickjmt !! Radius of Convergence for a Power Series - In this video, I discuss how to find the radius of converge.  It is easy after you find the INTERVAL of convergence.\r\nI get tons of questions about it though, so I thought it was time for a video. : )\r\nVisit http://PatrickJMT.com for tons more free math videos.", "lengthSeconds": 108, "width": 480, "height": 360}
{"video_id": "_UdGUULKN-E", "transcript": [{"start": 0.64, "duration": 2.84, "text": "all right in this video I just want to"}, {"start": 1.76, "duration": 3.999, "text": "do a quick little example about the"}, {"start": 3.48, "duration": 3.6, "text": "geometric mean and also to point out you"}, {"start": 5.759, "duration": 3.561, "text": "know what kind of how you can think"}, {"start": 7.08, "duration": 3.84, "text": "about it what it means um so the"}, {"start": 9.32, "duration": 4.04, "text": "definition the geometric mean of two"}, {"start": 10.92, "duration": 4.08, "text": "numbers A and B uh what we do is we"}, {"start": 13.36, "duration": 3.64, "text": "multiply those two numbers together and"}, {"start": 15.0, "duration": 4.48, "text": "then we take the square root of them and"}, {"start": 17.0, "duration": 4.88, "text": "that's the geometric mean so certainly"}, {"start": 19.48, "duration": 4.4, "text": "not the same as the arithmetic mean the"}, {"start": 21.88, "duration": 4.639, "text": "average I think that's the one most of"}, {"start": 23.88, "duration": 5.28, "text": "us are certainly more familiar with so"}, {"start": 26.519, "duration": 5.521, "text": "let's look at a simple example here"}, {"start": 29.16, "duration": 5.8, "text": "let's find the GE metric mean of four"}, {"start": 32.04, "duration": 4.92, "text": "and N so again what we'll do is we'll"}, {"start": 34.96, "duration": 4.04, "text": "multiply those two numbers"}, {"start": 36.96, "duration": 6.88, "text": "together and then we'll simply take the"}, {"start": 39.0, "duration": 7.559, "text": "square root of uh of those so 4 * 9 well"}, {"start": 43.84, "duration": 5.719, "text": "that's going to give us 36 so we have to"}, {"start": 46.559, "duration": 6.16, "text": "take the square root of 36 well that's"}, {"start": 49.559, "duration": 5.68, "text": "simply equal to six so we would say that"}, {"start": 52.719, "duration": 5.0, "text": "the geometric mean of the numbers four"}, {"start": 55.239, "duration": 6.84, "text": "and nine uh the geometric mean is going"}, {"start": 57.719, "duration": 6.16, "text": "to equal six so you know what what is"}, {"start": 62.079, "duration": 3.841, "text": "what's special about the geometric mean"}, {"start": 63.879, "duration": 4.521, "text": "I guess you know uh again I think an"}, {"start": 65.92, "duration": 4.839, "text": "average an arithmetic mean it's kind of"}, {"start": 68.4, "duration": 4.48, "text": "the middle point right and I think uh"}, {"start": 70.759, "duration": 3.841, "text": "you know people you know you find an"}, {"start": 72.88, "duration": 4.4, "text": "arithmetic mean if you want to find for"}, {"start": 74.6, "duration": 5.36, "text": "example your average grade on two tests"}, {"start": 77.28, "duration": 5.199, "text": "so you know what does this number six"}, {"start": 79.96, "duration": 4.32, "text": "represent in relation to four and nine"}, {"start": 82.479, "duration": 3.521, "text": "well what it has to do with it has to do"}, {"start": 84.28, "duration": 5.24, "text": "with"}, {"start": 86.0, "duration": 5.159, "text": "ratios okay so we started with four um"}, {"start": 89.52, "duration": 3.76, "text": "and n those were the numbers that we"}, {"start": 91.159, "duration": 4.721, "text": "were finding the arith excuse me the"}, {"start": 93.28, "duration": 4.879, "text": "geometric mean of and we found that"}, {"start": 95.88, "duration": 5.68, "text": "geometric mean to equal"}, {"start": 98.159, "duration": 8.681, "text": "six well notice if we look at the ratio"}, {"start": 101.56, "duration": 11.76, "text": "4 / 6 and then if we look at the ratio 6"}, {"start": 106.84, "duration": 11.36, "text": "/ 9 well 4 over 6 red reduces to 2/3 hey"}, {"start": 113.32, "duration": 6.839, "text": "6 over9 also reduces to 2/3 so this is"}, {"start": 118.2, "duration": 4.279, "text": "kind of what's special about this number"}, {"start": 120.159, "duration": 4.521, "text": "the geometric mean um if you take the"}, {"start": 122.479, "duration": 4.121, "text": "ratio of the small number to the"}, {"start": 124.68, "duration": 4.439, "text": "geometric mean and then if you look at"}, {"start": 126.6, "duration": 5.24, "text": "the ratio of the geometric mean to the"}, {"start": 129.119, "duration": 4.881, "text": "larger number those ratios or those"}, {"start": 131.84, "duration": 3.88, "text": "fractions are going to be the same okay"}, {"start": 134.0, "duration": 3.44, "text": "so that's that's that's how you can kind"}, {"start": 135.72, "duration": 4.08, "text": "of think about the geometric mean"}, {"start": 137.44, "duration": 5.96, "text": "somehow it's it's giving equal ratios of"}, {"start": 139.8, "duration": 5.24, "text": "those two values so again the arithmetic"}, {"start": 143.4, "duration": 3.76, "text": "mean is the one that I think most of us"}, {"start": 145.04, "duration": 3.919, "text": "a little are a little bit more used to"}, {"start": 147.16, "duration": 3.28, "text": "so again just to refresh you the"}, {"start": 148.959, "duration": 3.121, "text": "arithmetic mean"}, {"start": 150.44, "duration": 4.879, "text": "we would just add those two numbers"}, {"start": 152.08, "duration": 7.76, "text": "together you know 4 + 9 and then divide"}, {"start": 155.319, "duration": 7.961, "text": "by two I guess 4 + 9 that's uh 13 over"}, {"start": 159.84, "duration": 6.0, "text": "two and 13 ided two I believe that's"}, {"start": 163.28, "duration": 6.879, "text": "going to give us what uh"}, {"start": 165.84, "duration": 8.28, "text": "6.5 okay so the arithmetic mean is"}, {"start": 170.159, "duration": 6.841, "text": "6.5 the geometric mean was six and this"}, {"start": 174.12, "duration": 5.16, "text": "leads to one other uh last little remark"}, {"start": 177.0, "duration": 5.159, "text": "uh without a proof but it turns out that"}, {"start": 179.28, "duration": 6.44, "text": "the geom metric mean is always less than"}, {"start": 182.159, "duration": 6.961, "text": "or equal to the arithmetic mean okay so"}, {"start": 185.72, "duration": 4.76, "text": "one other little useful result so again"}, {"start": 189.12, "duration": 3.479, "text": "if you want to find your average grade"}, {"start": 190.48, "duration": 3.96, "text": "on two tests you're not going to use the"}, {"start": 192.599, "duration": 4.801, "text": "geometric mean you're going to use the"}, {"start": 194.44, "duration": 5.159, "text": "more familiar arithmetic mean um but the"}, {"start": 197.4, "duration": 3.919, "text": "geometric mean certainly does show up"}, {"start": 199.599, "duration": 3.521, "text": "and um again I think it's something that"}, {"start": 201.319, "duration": 3.441, "text": "people most people aren't aren't"}, {"start": 203.12, "duration": 4.08, "text": "familiar with I certainly wasn't for a"}, {"start": 204.76, "duration": 5.16, "text": "long time um so just in case you see it"}, {"start": 207.2, "duration": 4.2, "text": "that's all there is to it um and that's"}, {"start": 209.92, "duration": 6.2, "text": "it again it just really has to do with"}, {"start": 211.4, "duration": 4.72, "text": "making ratios equal to each other"}], "title": "The Geometric Mean", "description": "Thanks to all of you who support me on Patreon. You da real mvps!   $1 per month helps!! :)  https://www.patreon.com/patrickjmt !! The Geometric Mean.  In this video, I show how to find the geometric mean of two numbers and show what that number means in relation to the numbers that you started with.", "lengthSeconds": 214, "width": 480, "height": 360}
{"video_id": "hd2Tu0-Qb3k", "transcript": [{"start": 0.0, "duration": 4.38, "text": "all right so one more example here using"}, {"start": 1.979, "duration": 4.501, "text": "sum and difference identities to"}, {"start": 4.38, "duration": 6.33, "text": "evaluate a trig function and here we're"}, {"start": 6.48, "duration": 8.22, "text": "going to find the value for cosine of PI"}, {"start": 10.71, "duration": 5.61, "text": "over 12 and again kind of the tricky"}, {"start": 14.7, "duration": 3.78, "text": "part is how to break this up I think"}, {"start": 16.32, "duration": 4.56, "text": "let's see one way we can break this up"}, {"start": 18.48, "duration": 3.78, "text": "you know and again to me to get PI over"}, {"start": 20.88, "duration": 3.96, "text": "12 that's kind of a you know a small"}, {"start": 22.26, "duration": 7.5, "text": "number I need a couple angles that are"}, {"start": 24.84, "duration": 12.39, "text": "close together let's see four PI over 12"}, {"start": 29.76, "duration": 9.42, "text": "minus three PI over 12 4 PI over 12"}, {"start": 37.23, "duration": 4.91, "text": "minus three PI over 12 that would give"}, {"start": 39.18, "duration": 6.09, "text": "us 1 PI over 12 and well let's see"}, {"start": 42.14, "duration": 6.82, "text": "cosine of 4 PI over 12 that's really the"}, {"start": 45.27, "duration": 7.109, "text": "same thing as PI over 3 3 PI over 12 we"}, {"start": 48.96, "duration": 6.029, "text": "can reduce that to PI over 4 and I know"}, {"start": 52.379, "duration": 5.041, "text": "how to evaluate you know cosine and sine"}, {"start": 54.989, "duration": 4.531, "text": "at the angle PI over 3 and PI over 4 so"}, {"start": 57.42, "duration": 4.229, "text": "this seems like a good a good way to"}, {"start": 59.52, "duration": 3.75, "text": "write it I'm going to use this bottom"}, {"start": 61.649, "duration": 4.351, "text": "identity now it says you have cosine of"}, {"start": 63.27, "duration": 4.29, "text": "something minus something so it says I"}, {"start": 66.0, "duration": 5.43, "text": "can write that as cosine of the first"}, {"start": 67.56, "duration": 6.9, "text": "thing PI over 3 times cosine of the"}, {"start": 71.43, "duration": 4.47, "text": "second one which is PI over 4 again you"}, {"start": 74.46, "duration": 4.92, "text": "just kind of forget all about the plus"}, {"start": 75.9, "duration": 4.5, "text": "of the minus and there excuse me you"}, {"start": 79.38, "duration": 2.879, "text": "forget about the positive a negative"}, {"start": 80.4, "duration": 3.45, "text": "when you're just plugging it in ok so"}, {"start": 82.259, "duration": 4.131, "text": "you just use positive PI over 3 and"}, {"start": 83.85, "duration": 4.89, "text": "positive PI over 4 let me clarify that"}, {"start": 86.39, "duration": 4.24, "text": "and again there's a negative in between"}, {"start": 88.74, "duration": 3.419, "text": "so ok here we are thinking about the"}, {"start": 90.63, "duration": 3.559, "text": "negative but it says if there's a"}, {"start": 92.159, "duration": 5.221, "text": "negative in between we use the positive"}, {"start": 94.189, "duration": 5.621, "text": "so I've got to put plus and then it says"}, {"start": 97.38, "duration": 5.279, "text": "just take sine of the first one sine of"}, {"start": 99.81, "duration": 5.02, "text": "the second one so we'll get sine of PI"}, {"start": 102.659, "duration": 7.841, "text": "over 3"}, {"start": 104.83, "duration": 7.32, "text": "times sine of pi over four and again now"}, {"start": 110.5, "duration": 4.71, "text": "it's just unit circle stuff let's see"}, {"start": 112.15, "duration": 5.85, "text": "cosine of PI over three that's going to"}, {"start": 115.21, "duration": 4.59, "text": "be I believe positive 1/2 because sine"}, {"start": 118.0, "duration": 5.13, "text": "of PI over four is going to be root 2"}, {"start": 119.8, "duration": 7.56, "text": "over 2 sine of PI over 3 that's going to"}, {"start": 123.13, "duration": 9.99, "text": "be root 3 over 2 sine of PI over 4 is"}, {"start": 127.36, "duration": 7.62, "text": "going to be simply root 2 over 2 so it"}, {"start": 133.12, "duration": 4.08, "text": "looks like we get square root of 2 plus"}, {"start": 134.98, "duration": 4.89, "text": "root 3 times root 2 is going to be root"}, {"start": 137.2, "duration": 5.88, "text": "6 again we have our denominator 2 times"}, {"start": 139.87, "duration": 5.57, "text": "2 which is 4 so again we just simply"}, {"start": 143.08, "duration": 6.93, "text": "used this difference identity for cosine"}, {"start": 145.44, "duration": 6.1, "text": "to evaluate you know a trig function to"}, {"start": 150.01, "duration": 2.73, "text": "evaluate cosine of PI over 12 again"}, {"start": 151.54, "duration": 3.9, "text": "that's one I certainly don't have"}, {"start": 152.74, "duration": 4.98, "text": "memorized off the top of my head but I"}, {"start": 155.44, "duration": 5.87, "text": "can deduce it without too much effort so"}, {"start": 157.72, "duration": 3.59, "text": "long as I remember that identity"}], "title": "Sum and Difference Identities for Sine and Cosine, More Examples #2", "description": "Thanks to all of you who support me on Patreon. You da real mvps!   $1 per month helps!! :)  https://www.patreon.com/patrickjmt !! Sum and Difference Identities for Sine and Cosine, More Examples #2. More examples of using the sum and difference identities to find value other trig values. For more free math videos, visit http://PatrickJMT.com", "lengthSeconds": 160, "width": 480, "height": 360}
{"video_id": "aKoWJoO43y8", "transcript": [{"start": 0.919, "duration": 5.48, "text": "so a summer camp counselor wants to find"}, {"start": 3.24, "duration": 6.319, "text": "a length x in feet across a lake as"}, {"start": 6.399, "duration": 10.281, "text": "given in the diagram and the links"}, {"start": 9.559, "duration": 9.72, "text": "AB EB BD and CD measure 1,800 ft 1400 ft"}, {"start": 16.68, "duration": 6.72, "text": "700 ft and 800 ft"}, {"start": 19.279, "duration": 7.16, "text": "respectively um segments a c and"}, {"start": 23.4, "duration": 6.08, "text": "segments D intersect at this point B and"}, {"start": 26.439, "duration": 6.841, "text": "we're given that the angle a e which is"}, {"start": 29.48, "duration": 6.36, "text": "this angle here and cdb which would be"}, {"start": 33.28, "duration": 5.4, "text": "this angle here have the same measure we"}, {"start": 35.84, "duration": 5.239, "text": "want to figure out what's the value of x"}, {"start": 38.68, "duration": 3.84, "text": "okay so I've got my my diagram here and"}, {"start": 41.079, "duration": 2.841, "text": "let me go ahead and fill in the the"}, {"start": 42.52, "duration": 5.0, "text": "values"}, {"start": 43.92, "duration": 8.72, "text": "so okay so we're told that AB measures"}, {"start": 47.52, "duration": 10.519, "text": "1,800 ft we're told that EB measures"}, {"start": 52.64, "duration": 7.8, "text": "1,400 ft BD is 700 ft so that's"}, {"start": 58.039, "duration": 6.241, "text": "700 and then we also"}, {"start": 60.44, "duration": 6.6, "text": "also are given that CD is 800 ft so"}, {"start": 64.28, "duration": 6.04, "text": "again we're told that the angle AE which"}, {"start": 67.04, "duration": 6.84, "text": "is this angle here and cdb which would"}, {"start": 70.32, "duration": 6.0, "text": "be this angle here have the same measure"}, {"start": 73.88, "duration": 6.199, "text": "now notice this angle here the angle"}, {"start": 76.32, "duration": 7.68, "text": "e and CBD those are going to be vertical"}, {"start": 80.079, "duration": 7.4, "text": "angles so those also have similar"}, {"start": 84.0, "duration": 5.4, "text": "measures so the idea is now since um two"}, {"start": 87.479, "duration": 3.881, "text": "of the Interior since two of the angles"}, {"start": 89.4, "duration": 4.16, "text": "are congruent we know that these angles"}, {"start": 91.36, "duration": 3.96, "text": "or these triangles are similar so we can"}, {"start": 93.56, "duration": 6.0, "text": "set up a proportion to solve for the"}, {"start": 95.32, "duration": 7.56, "text": "missing value so we can say that"}, {"start": 99.56, "duration": 5.28, "text": "CD ided X so again I'm kind of looking"}, {"start": 102.88, "duration": 4.64, "text": "at this angle the side opposite that's"}, {"start": 104.84, "duration": 4.8, "text": "CD and for this triangle I'm looking at"}, {"start": 107.52, "duration": 6.919, "text": "the same angle but the opposite side"}, {"start": 109.64, "duration": 8.159, "text": "that's X that's going to equal BD so BD"}, {"start": 114.439, "duration": 6.28, "text": "is the um the side length opposite this"}, {"start": 117.799, "duration": 6.401, "text": "angle that we didn't label"}, {"start": 120.719, "duration": 5.161, "text": "and we can divide that by EB which will"}, {"start": 124.2, "duration": 3.32, "text": "be again if we look at the other"}, {"start": 125.88, "duration": 3.719, "text": "triangle if we look at the angle that we"}, {"start": 127.52, "duration": 5.0, "text": "didn't label if we look at the side"}, {"start": 129.599, "duration": 5.081, "text": "opposite that'll be EB and now we can"}, {"start": 132.52, "duration": 5.04, "text": "simply fill in those values so we said"}, {"start": 134.68, "duration": 8.04, "text": "CD that was equal to"}, {"start": 137.56, "duration": 9.679, "text": "800 / X BD has a length of"}, {"start": 142.72, "duration": 6.2, "text": "700 and E we said had a length of 1400"}, {"start": 147.239, "duration": 3.401, "text": "so now all we have to do is simply solve"}, {"start": 148.92, "duration": 2.959, "text": "this for x and and we'll have our"}, {"start": 150.64, "duration": 4.04, "text": "solution"}, {"start": 151.879, "duration": 5.121, "text": "so let me take uh I guess just one or"}, {"start": 154.68, "duration": 4.479, "text": "two extra steps here for the"}, {"start": 157.0, "duration": 4.48, "text": "algebra one thing we can do is we can"}, {"start": 159.159, "duration": 6.08, "text": "simply flip both fractions so we can"}, {"start": 161.48, "duration": 8.64, "text": "write this as X over 800 ="}, {"start": 165.239, "duration": 8.321, "text": "1400 ID 700 and then to solve for x we"}, {"start": 170.12, "duration": 9.36, "text": "can simply multiply both sides by 800 so"}, {"start": 173.56, "duration": 9.64, "text": "we'll have 1400 over 700 ultip by 800"}, {"start": 179.48, "duration": 8.16, "text": "well 1400 / 700 we can cancel out the"}, {"start": 183.2, "duration": 6.28, "text": "zeros 14 / 7 is 2 so we have 2"}, {"start": 187.64, "duration": 4.48, "text": "multiplied by"}, {"start": 189.48, "duration": 4.8, "text": "800 and that's going to give us a length"}, {"start": 192.12, "duration": 4.56, "text": "of 1,600"}, {"start": 194.28, "duration": 4.56, "text": "ft so again the IDE in this one is just"}, {"start": 196.68, "duration": 4.36, "text": "recognizing that these two triangles are"}, {"start": 198.84, "duration": 6.0, "text": "in fact similar and then setting up the"}, {"start": 201.04, "duration": 3.8, "text": "corre correct proportions"}], "title": "SAT Math - Question 17, 2016 Edition and Beyond", "description": "Thanks to all of you who support me on Patreon. You da real mvps!   $1 per month helps!! :)  https://www.patreon.com/patrickjmt !! SAT Math - Question 17, 2016 Edition and Beyond.\nSection 1 - No Calculator allowed!", "lengthSeconds": 203, "width": 480, "height": 360}
{"video_id": "9ilIY9q7ceg", "transcript": [{"start": 0.399, "duration": 2.841, "text": "all right in this video I'm going to"}, {"start": 1.56, "duration": 3.48, "text": "look at some examples involving"}, {"start": 3.24, "duration": 4.04, "text": "multiplication of"}, {"start": 5.04, "duration": 6.24, "text": "decimals so in part A here we've got"}, {"start": 7.28, "duration": 7.16, "text": "82.4 5 * 16 we basically just do sort of"}, {"start": 11.28, "duration": 3.16, "text": "our normal long"}, {"start": 14.599, "duration": 3.961, "text": "division and then I just count the"}, {"start": 16.64, "duration": 4.559, "text": "number of digits that are to the right"}, {"start": 18.56, "duration": 5.08, "text": "of my decimal places and that'll tell me"}, {"start": 21.199, "duration": 4.16, "text": "how far that basically helps me find"}, {"start": 23.64, "duration": 4.0, "text": "where the decimal place goes at the end"}, {"start": 25.359, "duration": 4.721, "text": "so we'll talk about that in a second so"}, {"start": 27.64, "duration": 4.479, "text": "let's do the multiplication so 6 * 5 is"}, {"start": 30.08, "duration": 7.08, "text": "30 so we'll keep the zero carry the"}, {"start": 32.119, "duration": 8.401, "text": "three 6 * 4 is 24 + 3 will be 27 so"}, {"start": 37.16, "duration": 6.84, "text": "we'll keep the 7 carry the two 6 * 2 is"}, {"start": 40.52, "duration": 8.12, "text": "12 + 2 is 14 so we'll keep the four"}, {"start": 44.0, "duration": 7.239, "text": "carry the one 6 * 8 is 48 + 1 will be"}, {"start": 48.64, "duration": 4.399, "text": "49 okay then I'm going to drop a zero"}, {"start": 51.239, "duration": 4.201, "text": "down and then I'm just going to multiply"}, {"start": 53.039, "duration": 8.881, "text": "everything by one so we'll just get uh 1"}, {"start": 55.44, "duration": 9.999, "text": "* 5 1 * 4 1 * 2 1 * 8 I do my addition 0"}, {"start": 61.92, "duration": 9.6, "text": "and 0 is 0 7 + 5 is 12 so we'll keep the"}, {"start": 65.439, "duration": 7.961, "text": "two carry the one uh 1 + 4 + 4 will be 9"}, {"start": 71.52, "duration": 5.959, "text": "9 + 2 is"}, {"start": 73.4, "duration": 6.8, "text": "11 so let's see uh 1 + 4 is 5 + 8 will"}, {"start": 77.479, "duration": 5.841, "text": "be 13 and now it's just a matter of"}, {"start": 80.2, "duration": 7.279, "text": "finding where the decimal place goes I"}, {"start": 83.32, "duration": 6.2, "text": "basically look to the right of my uh my"}, {"start": 87.479, "duration": 4.521, "text": "decimal sign in each number and I count"}, {"start": 89.52, "duration": 5.32, "text": "the number of digits so in our first"}, {"start": 92.0, "duration": 4.159, "text": "number there's two digits to the right"}, {"start": 94.84, "duration": 3.2, "text": "you could think about adding in a"}, {"start": 96.159, "duration": 3.881, "text": "decimal sign well there's really no"}, {"start": 98.04, "duration": 5.64, "text": "digits to the right of our second number"}, {"start": 100.04, "duration": 5.52, "text": "16 so I count up the total number of uh"}, {"start": 103.68, "duration": 4.64, "text": "digits that are to the right so in this"}, {"start": 105.56, "duration": 5.44, "text": "case there's only two"}, {"start": 108.32, "duration": 5.2, "text": "digits it says that's how many places we"}, {"start": 111.0, "duration": 3.84, "text": "move our decimal sign at the end so I"}, {"start": 113.52, "duration": 4.239, "text": "start all the way to the right of the"}, {"start": 114.84, "duration": 5.68, "text": "number then I just move it one two"}, {"start": 117.759, "duration": 9.28, "text": "places so what it says is it says if we"}, {"start": 120.52, "duration": 6.519, "text": "multiply 8245 * 16 we'll be left with"}, {"start": 127.2, "duration": 5.119, "text": "1,319"}, {"start": 129.319, "duration": 3.0, "text": "uh2"}], "title": "Adding and Subtracting Decimals, Ex 3", "description": "Thanks to all of you who support me on Patreon. You da real mvps!   $1 per month helps!! :)  https://www.patreon.com/patrickjmt !! Adding and Subtracting Decimals, Ex 3", "lengthSeconds": 130, "width": 480, "height": 360}
{"video_id": "lE814ngFt3I", "transcript": [{"start": 0.319, "duration": 4.96, "text": "okay so this is going to be another"}, {"start": 3.06, "duration": 4.319, "text": "volume of revolution using disks and"}, {"start": 5.279, "duration": 4.261, "text": "washers and in this one again I just"}, {"start": 7.379, "duration": 5.13, "text": "want to slightly tweak my first two"}, {"start": 9.54, "duration": 5.34, "text": "examples so again we're doing the region"}, {"start": 12.509, "duration": 3.901, "text": "bounded by X cubed and X where X is"}, {"start": 14.88, "duration": 2.97, "text": "greater than or equal to zero so that's"}, {"start": 16.41, "duration": 3.779, "text": "going to give us the region on the top"}, {"start": 17.85, "duration": 5.099, "text": "right quadrant here's the graph of y"}, {"start": 20.189, "duration": 4.981, "text": "equals x y equals x cubed you can find"}, {"start": 22.949, "duration": 3.9, "text": "the limits of integration the points of"}, {"start": 25.17, "duration": 3.779, "text": "intersection by setting them equal to"}, {"start": 26.849, "duration": 4.77, "text": "each other so that's what I did over"}, {"start": 28.949, "duration": 5.191, "text": "here off to the side and this was"}, {"start": 31.619, "duration": 5.401, "text": "explained in part one and now again we"}, {"start": 34.14, "duration": 5.16, "text": "just want to integrate we want to rotate"}, {"start": 37.02, "duration": 5.64, "text": "this region about the line y equals five"}, {"start": 39.3, "duration": 5.22, "text": "so just like in the last couple our"}, {"start": 42.66, "duration": 4.35, "text": "limits of integration will be from zero"}, {"start": 44.52, "duration": 5.85, "text": "to one and again now we have to take pi"}, {"start": 47.01, "duration": 7.11, "text": "times the outer radius squared minus pi"}, {"start": 50.37, "duration": 8.81, "text": "times the inner radius squared and then"}, {"start": 54.12, "duration": 8.579, "text": "we head agreat that with respect to X"}, {"start": 59.18, "duration": 5.229, "text": "okay so now my outer radius will be this"}, {"start": 62.699, "duration": 7.861, "text": "is usually the ones that that seemed"}, {"start": 64.409, "duration": 7.831, "text": "most tricky to people maybe maybe not so"}, {"start": 70.56, "duration": 3.809, "text": "the line y equals five is what I'm"}, {"start": 72.24, "duration": 3.96, "text": "rotating about we draw a line again"}, {"start": 74.369, "duration": 4.021, "text": "perpendicular to that so I'm going to"}, {"start": 76.2, "duration": 4.5, "text": "draw a line perpendicular to that it"}, {"start": 78.39, "duration": 5.67, "text": "just hits my region and now it's just"}, {"start": 80.7, "duration": 5.94, "text": "about to exit the region so again this"}, {"start": 84.06, "duration": 5.01, "text": "is going to be the outer radius and"}, {"start": 86.64, "duration": 4.769, "text": "that's what I plug in to the first part"}, {"start": 89.07, "duration": 3.57, "text": "this is going to be the outer radius so"}, {"start": 91.409, "duration": 4.651, "text": "again I'm trying to think what's the"}, {"start": 92.64, "duration": 5.909, "text": "length of this arrow well if it went"}, {"start": 96.06, "duration": 5.55, "text": "from y equals five to the x axis it"}, {"start": 98.549, "duration": 6.511, "text": "would have a length of five but I'm not"}, {"start": 101.61, "duration": 5.73, "text": "going all the way there I'm losing a"}, {"start": 105.06, "duration": 3.87, "text": "little bit well this little bit that I'm"}, {"start": 107.34, "duration": 4.2, "text": "leaving off again comes from the"}, {"start": 108.93, "duration": 4.89, "text": "function which is X cubed so it says"}, {"start": 111.54, "duration": 6.96, "text": "this outer radius is actually going to"}, {"start": 113.82, "duration": 8.759, "text": "be five minus that that curve which is X"}, {"start": 118.5, "duration": 7.469, "text": "cubed quantity squared and then my inner"}, {"start": 122.579, "duration": 5.691, "text": "radius I calculate that the same way"}, {"start": 125.969, "duration": 4.68, "text": "so I draw a line that's perpendicular"}, {"start": 128.27, "duration": 4.51, "text": "okay and it's just going to hit the"}, {"start": 130.649, "duration": 4.651, "text": "region but now it's going to hit the"}, {"start": 132.78, "duration": 5.28, "text": "straight line part which is the curve y"}, {"start": 135.3, "duration": 6.54, "text": "equals x so again the entire length"}, {"start": 138.06, "duration": 6.3, "text": "would be five we'd have a length of five"}, {"start": 141.84, "duration": 7.41, "text": "but now we have to subtract off that"}, {"start": 144.36, "duration": 8.0, "text": "height of X and that will be our inner"}, {"start": 149.25, "duration": 6.03, "text": "radius sorry for all the colors here"}, {"start": 152.36, "duration": 6.25, "text": "that'll be our inner radius and we've"}, {"start": 155.28, "duration": 5.13, "text": "now got our problem set up so again to"}, {"start": 158.61, "duration": 4.47, "text": "integrate I would factor out the PI"}, {"start": 160.41, "duration": 4.829, "text": "you'd have to take the five minus X"}, {"start": 163.08, "duration": 4.8, "text": "cubed and foil it out the same thing"}, {"start": 165.239, "duration": 4.441, "text": "with the five minus X collect your like"}, {"start": 167.88, "duration": 5.57, "text": "terms and then it should just be using a"}, {"start": 169.68, "duration": 3.77, "text": "bunch of the power rule stuff"}], "title": "Volumes of Revolution - Disk/Washers Example 3", "description": "Thanks to all of you who support me on Patreon. You da real mvps!   $1 per month helps!! :)  https://www.patreon.com/patrickjmt !! In this video, I use the same region from part 1, but now I rotate the region about the line y = 5.\r\nFor more free math videos, visit http://PatrickJMT.com", "lengthSeconds": 173, "width": 480, "height": 360}
{"video_id": "aaQiNUoZnLE", "transcript": [{"start": 0.719, "duration": 5.481, "text": "so one thing that series are good for"}, {"start": 3.0, "duration": 5.32, "text": "are finding uh polinomial approximations"}, {"start": 6.2, "duration": 4.88, "text": "to certain functions that turns out to"}, {"start": 8.32, "duration": 4.72, "text": "be super super super useful um we're"}, {"start": 11.08, "duration": 3.24, "text": "going to kind of keep it on the easy"}, {"start": 13.04, "duration": 3.84, "text": "level here and we're just going to talk"}, {"start": 14.32, "duration": 4.68, "text": "about finding linear approximations and"}, {"start": 16.88, "duration": 4.319, "text": "this is actually just old first semester"}, {"start": 19.0, "duration": 4.68, "text": "calculus stuff they'll have you find the"}, {"start": 21.199, "duration": 4.4, "text": "equation of the tangent line or a"}, {"start": 23.68, "duration": 3.48, "text": "linearization that's all we're doing at"}, {"start": 25.599, "duration": 3.84, "text": "this case so I'm going to go through"}, {"start": 27.16, "duration": 5.52, "text": "these a little bit a little bit faster"}, {"start": 29.439, "duration": 6.96, "text": "so we're just finding the equation of a"}, {"start": 32.68, "duration": 6.039, "text": "line is all we're doing um so recall the"}, {"start": 36.399, "duration": 4.521, "text": "slope comes from the"}, {"start": 38.719, "duration": 5.441, "text": "derivative um so let's see our function"}, {"start": 40.92, "duration": 7.0, "text": "f ofx that's 2 to the X the derivative"}, {"start": 44.16, "duration": 6.68, "text": "of 2 to the x is 2 x * the natural"}, {"start": 47.92, "duration": 5.08, "text": "logarithm of 2 to get the slope of our"}, {"start": 50.84, "duration": 3.879, "text": "tangent line we just plug in whatever x"}, {"start": 53.0, "duration": 5.0, "text": "coordinate we're interested in which in"}, {"start": 54.719, "duration": 6.241, "text": "this case is three so we'll have 2 3 *"}, {"start": 58.0, "duration": 5.6, "text": "the natural logarithm of 2 two to the 3"}, {"start": 60.96, "duration": 4.479, "text": "is eight so well we're late we're left"}, {"start": 63.6, "duration": 4.92, "text": "with eight times the natural logarithm"}, {"start": 65.439, "duration": 5.0, "text": "of two notice if we plug two in or"}, {"start": 68.52, "duration": 4.44, "text": "excuse me three into our original"}, {"start": 70.439, "duration": 6.521, "text": "function to get the corresponding y"}, {"start": 72.96, "duration": 7.0, "text": "value we would get 2 the 3 and 2 the 3"}, {"start": 76.96, "duration": 5.44, "text": "is just8 so it says we would have y"}, {"start": 79.96, "duration": 6.159, "text": "minus our y coordinate well again if we"}, {"start": 82.4, "duration": 6.32, "text": "plug in three the Y value we get is8 the"}, {"start": 86.119, "duration": 5.521, "text": "slope of the tangent line is 8 * the"}, {"start": 88.72, "duration": 5.92, "text": "natural logarithm of 2 2 then we just"}, {"start": 91.64, "duration": 6.479, "text": "take x minus the x coordinate which was"}, {"start": 94.64, "duration": 5.519, "text": "given to us as three and now this is"}, {"start": 98.119, "duration": 3.841, "text": "going to be our linearization so if we"}, {"start": 100.159, "duration": 3.881, "text": "distribute out on the right side we"}, {"start": 101.96, "duration": 7.28, "text": "would have 8 * the natural logarithm of"}, {"start": 104.04, "duration": 8.119, "text": "2 * X we would get a -4 * the natural"}, {"start": 109.24, "duration": 6.12, "text": "logarithm of 2 then we could add our"}, {"start": 112.159, "duration": 7.24, "text": "postive 8 over and again this is now"}, {"start": 115.36, "duration": 4.039, "text": "this would now be our linearization"}], "title": "Finding a Linear Approximation (Linearization, Tangent Line Approx), Another Ex 1", "description": "Thanks to all of you who support me on Patreon. You da real mvps!   $1 per month helps!! :)  https://www.patreon.com/patrickjmt !! Finding a Linear Approximation (Linearization, Tangent Line Approx), Another Ex 1.  Just another example of finding a linear approximation for a function at a particular point.", "lengthSeconds": 120, "width": 480, "height": 360}
{"video_id": "S-BWQzigAuk", "transcript": [{"start": 0.04, "duration": 4.799, "text": "we're going to graph two periods of Y ="}, {"start": 2.0, "duration": 5.2, "text": "2 * cosine of 2x the first thing I'm"}, {"start": 4.839, "duration": 4.601, "text": "going to do is find the period so we'll"}, {"start": 7.2, "duration": 5.04, "text": "take 2 pi and divide that by the"}, {"start": 9.44, "duration": 5.359, "text": "absolute value of B so we'll take the"}, {"start": 12.24, "duration": 5.64, "text": "absolute value of positive2 which just"}, {"start": 14.799, "duration": 6.961, "text": "is simply equal to two so the period"}, {"start": 17.88, "duration": 7.719, "text": "will equal simply equal to Pi I like to"}, {"start": 21.76, "duration": 5.519, "text": "divide these into uh four equal pieces"}, {"start": 25.599, "duration": 3.361, "text": "and that's what I'm going to use as the"}, {"start": 27.279, "duration": 3.641, "text": "increments on my graph so I'm going to"}, {"start": 28.96, "duration": 4.119, "text": "come back to that I'll use those as the"}, {"start": 30.92, "duration": 6.04, "text": "increments the amplitude is just going"}, {"start": 33.079, "duration": 7.121, "text": "to be the absolute value of two which is"}, {"start": 36.96, "duration": 6.0, "text": "pos2 so here's my graph we've got X"}, {"start": 40.2, "duration": 6.6, "text": "we've got y the amplitude it goes from"}, {"start": 42.96, "duration": 6.759, "text": "positive2 to -2 and again the increments"}, {"start": 46.8, "duration": 5.36, "text": "I'm going to use will be pi over 4 so"}, {"start": 49.719, "duration": 6.64, "text": "again the uh the units on the x axis and"}, {"start": 52.16, "duration": 6.12, "text": "on the Y AIS are not exactly the same so"}, {"start": 56.359, "duration": 4.36, "text": "positive2 will be right"}, {"start": 58.28, "duration": 5.64, "text": "there -2 looks like it's in the right"}, {"start": 60.719, "duration": 7.641, "text": "place so let's see we'll have pi over 4"}, {"start": 63.92, "duration": 8.839, "text": "pi/ 2 3 pi over 4 we'll have"}, {"start": 68.36, "duration": 8.36, "text": "Pi 5 piun over 4 6 piun over 4 will be 3"}, {"start": 72.759, "duration": 7.161, "text": "piun / 2 we'll have 7 piun over 4 and"}, {"start": 76.72, "duration": 7.359, "text": "then 8 pi over 4 will be 2"}, {"start": 79.92, "duration": 6.48, "text": "pi so if we substitute in xals 0 we'll"}, {"start": 84.079, "duration": 4.68, "text": "get cosine of 0 which is one we'll"}, {"start": 86.4, "duration": 4.12, "text": "multiply that by two that'll give us"}, {"start": 88.759, "duration": 3.4, "text": "positive2"}, {"start": 90.52, "duration": 3.8, "text": "and after that again that's why I like"}, {"start": 92.159, "duration": 4.721, "text": "to use these increments at the next"}, {"start": 94.32, "duration": 5.6, "text": "increment it'll be at zero at the next"}, {"start": 96.88, "duration": 3.84, "text": "increment it'll be at -2 and then back"}, {"start": 99.92, "duration": 3.839, "text": "at"}, {"start": 100.72, "duration": 8.039, "text": "zero then back at positive2 then it'll"}, {"start": 103.759, "duration": 7.881, "text": "be at zero it'll be at -2 then at zero"}, {"start": 108.759, "duration": 5.481, "text": "and then back at"}, {"start": 111.64, "duration": 7.119, "text": "positive2 so let"}, {"start": 114.24, "duration": 7.64, "text": "me connect them with some uh curvy"}, {"start": 118.759, "duration": 10.72, "text": "curves and that'll be a rough little"}, {"start": 121.88, "duration": 7.599, "text": "sketch of Y = 2 * cosine of 2x"}], "title": "Graphing a Cosine Function  EX 1", "description": "Thanks to all of you who support me on Patreon. You da real mvps!   $1 per month helps!! :)  https://www.patreon.com/patrickjmt !! Just another example of graphing a trigonometric function", "lengthSeconds": 128, "width": 450, "height": 360}
{"video_id": "WROccOAozXo", "transcript": [{"start": 0.719, "duration": 3.401, "text": "okay here we're going to look at an"}, {"start": 1.8, "duration": 4.28, "text": "example of finding the surface area of a"}, {"start": 4.12, "duration": 5.88, "text": "sphere and the surface area we simply"}, {"start": 6.08, "duration": 5.76, "text": "take 4 * pi * the radius"}, {"start": 10.0, "duration": 4.559, "text": "squared so we're going to find the"}, {"start": 11.84, "duration": 6.439, "text": "surface area of a sphere with a radius"}, {"start": 14.559, "duration": 9.96, "text": "of 7.4 M and we'll use"}, {"start": 18.279, "duration": 9.24, "text": "3.14 for pi so again our formula 4 * pi"}, {"start": 24.519, "duration": 7.161, "text": "* the square of the"}, {"start": 27.519, "duration": 8.04, "text": "radius well we're approximating pi with"}, {"start": 31.68, "duration": 6.24, "text": "3.14 our radius is 7.4 so we simply have"}, {"start": 35.559, "duration": 4.921, "text": "to square"}, {"start": 37.92, "duration": 5.36, "text": "that and again now it's just a"}, {"start": 40.48, "duration": 5.239, "text": "matter of doing a bit of arithmetic so"}, {"start": 43.28, "duration": 7.0, "text": "7.4 *"}, {"start": 45.719, "duration": 4.561, "text": "7.4 I'm getting 54.76%"}, {"start": 54.88, "duration": 3.169, "text": "[Music]"}, {"start": 60.519, "duration": 5.6, "text": "so now I'm going to multiply so we've"}, {"start": 63.96, "duration": 6.479, "text": "got"}, {"start": 66.119, "duration": 4.32, "text": "12.56 we're going to multiply that by"}, {"start": 71.04, "duration": 4.119, "text": "54 multiply that by"}, {"start": 88.92, "duration": 5.0, "text": "54.76% round that digit up one so we'll"}, {"start": 91.68, "duration": 2.24, "text": "get"}, {"start": 94.68, "duration": 5.84, "text": "68779 again our units were in meters"}, {"start": 98.439, "duration": 4.801, "text": "since we're finding a surface area an"}, {"start": 100.52, "duration": 7.08, "text": "area is uh two-dimensional so our units"}, {"start": 103.24, "duration": 4.36, "text": "will be in meters squared"}], "title": "Surface Area of a Sphere", "description": "Thanks to all of you who support me on Patreon. You da real mvps!   $1 per month helps!! :)  https://www.patreon.com/patrickjmt !! Surface Area of a Sphere.  Here I do a quick example of finding the surface area of a sphere.  The formula is given at the beginning of the video.", "lengthSeconds": 106, "width": 480, "height": 360}
{"video_id": "Q48HHoLauyg", "transcript": [{"start": 0.68, "duration": 5.679, "text": "okay so another example here of using"}, {"start": 3.24, "duration": 5.279, "text": "our half angle identities to evaluate a"}, {"start": 6.359, "duration": 4.521, "text": "a trig expression here so we want to"}, {"start": 8.519, "duration": 5.561, "text": "evaluate tangent of"}, {"start": 10.88, "duration": 6.28, "text": "105\u00b0 well let's see again if you double"}, {"start": 14.08, "duration": 7.68, "text": "105\u00b0 that would be 200 and"}, {"start": 17.16, "duration": 6.279, "text": "10\u00b0 so we can write 105 as 210 over 2"}, {"start": 21.76, "duration": 4.4, "text": "and again now all I'm going to do is"}, {"start": 23.439, "duration": 7.121, "text": "just uh use our identity"}, {"start": 26.16, "duration": 7.84, "text": "here so we'll use the bottom identity"}, {"start": 30.56, "duration": 5.8, "text": "and it says we simply get 1 minus cosine"}, {"start": 34.0, "duration": 4.96, "text": "of whatever value is hanging out on top"}, {"start": 36.36, "duration": 2.6, "text": "which is going to be"}, {"start": 39.16, "duration": 7.32, "text": "210\u00b0 all over s of"}, {"start": 44.0, "duration": 7.64, "text": "210\u00b0 and now it's just simply a matter"}, {"start": 46.48, "duration": 7.32, "text": "of evaluating cosine of 210\u00b0 and S of"}, {"start": 51.64, "duration": 4.0, "text": "210\u00b0 well let's see I believe our"}, {"start": 53.8, "duration": 3.239, "text": "reference angle so if we go Halfway"}, {"start": 55.64, "duration": 3.759, "text": "Around that'll be"}, {"start": 57.039, "duration": 6.76, "text": "180\u00b0 so then we would have to go another"}, {"start": 59.399, "duration": 7.601, "text": "30 to get to our angle of"}, {"start": 63.799, "duration": 7.401, "text": "210\u00b0 let's see if we think about um I"}, {"start": 67.0, "duration": 6.68, "text": "guess the angle uh 30\u00b0 in the first"}, {"start": 71.2, "duration": 5.2, "text": "quadrant let's see in the first quadrant"}, {"start": 73.68, "duration": 6.6, "text": "we get < tk3 over2 for the X quinate on"}, {"start": 76.4, "duration": 6.32, "text": "the unit circle and we get2 for the"}, {"start": 80.28, "duration": 6.6, "text": "y-coordinate so that means at"}, {"start": 82.72, "duration": 8.039, "text": "210\u00b0 well uh they would both be negative"}, {"start": 86.88, "duration": 5.879, "text": "in quadrant three so okay so now I've"}, {"start": 90.759, "duration": 5.841, "text": "got my values that I need we'll get 1"}, {"start": 92.759, "duration": 5.481, "text": "minus cosine well cosine would be < tk3"}, {"start": 96.6, "duration": 4.08, "text": "over"}, {"start": 98.24, "duration": 6.199, "text": "2 s is going to be the value of the"}, {"start": 100.68, "duration": 6.28, "text": "y-coordinate which will simply be"}, {"start": 104.439, "duration": 6.081, "text": "one2 so let's see in the numerator we'll"}, {"start": 106.96, "duration": 6.519, "text": "just be left with 1 + < tk3 over 2 in"}, {"start": 110.52, "duration": 4.279, "text": "the denominator we have our - one2 I"}, {"start": 113.479, "duration": 3.92, "text": "think I would do the same thing as"}, {"start": 114.799, "duration": 5.081, "text": "before just uh you know get rid of the"}, {"start": 117.399, "duration": 4.961, "text": "the denominators the twos I would mult"}, {"start": 119.88, "duration": 4.96, "text": "mply the numerator by two and the"}, {"start": 122.36, "duration": 5.0, "text": "denominator also by"}, {"start": 124.84, "duration": 3.44, "text": "two so if we do that in the numerator"}, {"start": 127.36, "duration": 3.44, "text": "we'll get"}, {"start": 128.28, "duration": 4.319, "text": "two uh the twos will cancel out when we"}, {"start": 130.8, "duration": 3.159, "text": "distribute and we'll be left with the"}, {"start": 132.599, "duration": 4.121, "text": "square < TK of"}, {"start": 133.959, "duration": 4.961, "text": "three in the denominator again the twos"}, {"start": 136.72, "duration": 4.879, "text": "will cancel we'll simply be left with"}, {"start": 138.92, "duration": 5.319, "text": "Nega one you can always distribute that"}, {"start": 141.599, "duration": 3.761, "text": "negative um I'll just pull it out front"}, {"start": 144.239, "duration": 2.841, "text": "and say we"}, {"start": 145.36, "duration": 5.64, "text": "have2"}, {"start": 147.08, "duration": 5.4, "text": "+ uh Square < TK of 3 and I think uh"}, {"start": 151.0, "duration": 2.84, "text": "there's not much else you can really do"}, {"start": 152.48, "duration": 4.0, "text": "there you can always turn it into a"}, {"start": 153.84, "duration": 4.119, "text": "decimal but this is a nice exact answer"}, {"start": 156.48, "duration": 4.6, "text": "so I think I'm going to leave it just"}, {"start": 157.959, "duration": 3.121, "text": "like that"}], "title": "Half Angle Identities to Evaluate Trigonometric Expressions, Example 2", "description": "Thanks to all of you who support me on Patreon. You da real mvps!   $1 per month helps!! :)  https://www.patreon.com/patrickjmt !! Half Angle Identities to Evaluate Trigonometric Expressions, Example 2. In this video, I give some half angle identities and show how they can be used to evaluate trigonometric expressions.", "lengthSeconds": 159, "width": 480, "height": 360}
{"video_id": "W3LBeHuetnk", "transcript": [{"start": 0.64, "duration": 3.36, "text": "all right in this video we're going to"}, {"start": 1.92, "duration": 2.72, "text": "look at an example of multiplying whole"}, {"start": 4.0, "duration": 2.399, "text": "numbers"}, {"start": 4.64, "duration": 4.24, "text": "so in part a here we're going to"}, {"start": 6.399, "duration": 6.801, "text": "multiply 234"}, {"start": 8.88, "duration": 6.879, "text": "times 2 and in part b we'll do 1302"}, {"start": 13.2, "duration": 5.12, "text": "times 3. so again just like with"}, {"start": 15.759, "duration": 5.28, "text": "addition and subtraction"}, {"start": 18.32, "duration": 4.4, "text": "i tend and i think most people we'll"}, {"start": 21.039, "duration": 6.641, "text": "just write these vertically so we've got"}, {"start": 22.72, "duration": 6.799, "text": "2 3 4 times the number 2"}, {"start": 27.68, "duration": 4.32, "text": "and again now i'm just going to multiply"}, {"start": 29.519, "duration": 5.761, "text": "a digit at a time so"}, {"start": 32.0, "duration": 4.16, "text": "so i'm going to do 2 times 4 which is"}, {"start": 35.28, "duration": 4.24, "text": "going to be 8."}, {"start": 36.16, "duration": 6.239, "text": "we stick that in the same column okay"}, {"start": 39.52, "duration": 3.44, "text": "we do 2 times 3 which is going to give"}, {"start": 42.399, "duration": 3.121, "text": "us"}, {"start": 42.96, "duration": 3.919, "text": "6 we put that again in the same column"}, {"start": 45.52, "duration": 4.32, "text": "as the 3"}, {"start": 46.879, "duration": 4.401, "text": "and then 2 times 2 is going to be 4 and"}, {"start": 49.84, "duration": 6.239, "text": "that's going to be our solution"}, {"start": 51.28, "duration": 5.919, "text": "it says we have 234 times 2 is 468"}, {"start": 56.079, "duration": 4.081, "text": "and again that makes sense you're just"}, {"start": 57.199, "duration": 6.121, "text": "kind of doubling all of the digits"}, {"start": 60.16, "duration": 4.399, "text": "all right let's do our next example so"}, {"start": 63.32, "duration": 5.72, "text": "1302"}, {"start": 64.559, "duration": 4.481, "text": "we'll multiply that by 3"}, {"start": 69.28, "duration": 4.0, "text": "and again just like with addition and"}, {"start": 70.96, "duration": 2.8, "text": "subtraction i have all of my units lined"}, {"start": 73.28, "duration": 3.839, "text": "up"}, {"start": 73.76, "duration": 6.56, "text": "okay so three times two is going to be"}, {"start": 77.119, "duration": 6.64, "text": "six three times zero"}, {"start": 80.32, "duration": 5.119, "text": "is going to be zero three times three"}, {"start": 83.759, "duration": 3.281, "text": "is going to be nine and again we just"}, {"start": 85.439, "duration": 4.961, "text": "drop it right below"}, {"start": 87.04, "duration": 6.48, "text": "and then 3 times 1 is going to give us"}, {"start": 90.4, "duration": 7.52, "text": "3. so it says"}, {"start": 93.52, "duration": 11.44, "text": "1302 times 3 is going to give us 3"}, {"start": 97.92, "duration": 7.04, "text": "900 and six"}], "title": "Multiplying Whole Numbers, Example 1", "description": "Thanks to all of you who support me on Patreon. You da real mvps!   $1 per month helps!! :)  https://www.patreon.com/patrickjmt !! Multiplying Whole Numbers, Example 1.  Here we look at two examples of multiplying a two digit number by a single digit.", "lengthSeconds": 103, "width": 480, "height": 360}
{"video_id": "h0pfiDJYsvI", "transcript": [{"start": 0.48, "duration": 2.839, "text": "all right in this video I'm going to do"}, {"start": 1.68, "duration": 4.36, "text": "some examples of eliminating the"}, {"start": 3.319, "duration": 5.681, "text": "parameter so we can better describe a"}, {"start": 6.04, "duration": 4.679, "text": "curve given by the parametric equations"}, {"start": 9.0, "duration": 3.92, "text": "um I'm going to do all of these in uh"}, {"start": 10.719, "duration": 4.12, "text": "one video part A and B it's really just"}, {"start": 12.92, "duration": 4.48, "text": "going to be a matter of solving for T"}, {"start": 14.839, "duration": 4.681, "text": "and then plugging in so notice from our"}, {"start": 17.4, "duration": 6.4, "text": "first equation we can just add one to"}, {"start": 19.52, "duration": 7.159, "text": "both sides and then have x + 1 = T well"}, {"start": 23.8, "duration": 7.879, "text": "now in our second equation we'll get y ="}, {"start": 26.679, "duration": 8.88, "text": "4 * t + 1 but again I'm going to replace"}, {"start": 31.679, "duration": 7.04, "text": "the T with our x +"}, {"start": 35.559, "duration": 7.721, "text": "one well we can always distribute we'll"}, {"start": 38.719, "duration": 8.68, "text": "have 4x + 4 +"}, {"start": 43.28, "duration": 7.0, "text": "1 so we'll end up with the equation y ="}, {"start": 47.399, "duration": 4.84, "text": "4x + 5 and that'll be our answer for"}, {"start": 50.28, "duration": 5.52, "text": "part A we've now eliminated the"}, {"start": 52.239, "duration": 5.721, "text": "parameter so nothing at all crazy again"}, {"start": 55.8, "duration": 5.079, "text": "just a graph of a line"}, {"start": 57.96, "duration": 6.839, "text": "here Part B will'll do the exact same"}, {"start": 60.879, "duration": 7.521, "text": "thing um again we've got X = the < TK of"}, {"start": 64.799, "duration": 7.64, "text": "t + 1 well we can square both sides"}, {"start": 68.4, "duration": 8.24, "text": "that'll give us x^2 = t + 1 and then we"}, {"start": 72.439, "duration": 5.441, "text": "can subtract so we'll have x^2 - 1 = T"}, {"start": 76.64, "duration": 7.88, "text": "and the same thing I'm just going to"}, {"start": 77.88, "duration": 8.76, "text": "plug that in so y = 3 * t + 2 well again"}, {"start": 84.52, "duration": 4.32, "text": "we're just going to replace the T with"}, {"start": 86.64, "duration": 4.4, "text": "our x^2"}, {"start": 88.84, "duration": 5.52, "text": "minus1"}, {"start": 91.04, "duration": 7.16, "text": "and that's going to leave us with Y ="}, {"start": 94.36, "duration": 6.52, "text": "3x^2 I guess we'll have -3 + 2 which is"}, {"start": 98.2, "duration": 4.959, "text": "going to leave us with a negative one"}, {"start": 100.88, "duration": 6.199, "text": "and again we've eliminated the"}, {"start": 103.159, "duration": 6.64, "text": "parameter so one more example here part"}, {"start": 107.079, "duration": 6.201, "text": "C again I'm going to do the same thing"}, {"start": 109.799, "duration": 7.68, "text": "here I'm not going to quite solve for T"}, {"start": 113.28, "duration": 5.759, "text": "um I'm going to solve for e to the T So"}, {"start": 117.479, "duration": 5.201, "text": "notice in our first one here I can"}, {"start": 119.039, "duration": 6.241, "text": "divide by 4 x over 4 that would be e to"}, {"start": 122.68, "duration": 6.32, "text": "the T over"}, {"start": 125.28, "duration": 6.52, "text": "4 well what we can do at this point is"}, {"start": 129.0, "duration": 7.959, "text": "we can raise both sides to the fourth"}, {"start": 131.8, "duration": 8.28, "text": "power so we'll have uh X over 4 raised"}, {"start": 136.959, "duration": 6.161, "text": "to the 4th on the right side if we"}, {"start": 140.08, "duration": 5.4, "text": "multiply the exponents you know T over4"}, {"start": 143.12, "duration": 5.24, "text": "* 4 over one that's just going to leave"}, {"start": 145.48, "duration": 5.88, "text": "us with e to the T and again to me"}, {"start": 148.36, "duration": 5.44, "text": "that's good enough because our uh our"}, {"start": 151.36, "duration": 3.519, "text": "parametric equation for y involves e to"}, {"start": 153.8, "duration": 5.12, "text": "the"}, {"start": 154.879, "duration": 6.561, "text": "T so it says Y is going to equal 3 * e"}, {"start": 158.92, "duration": 7.08, "text": "to the T but again we can just replace e"}, {"start": 161.44, "duration": 7.48, "text": "to the T with our X over 4 raised to the"}, {"start": 166.0, "duration": 5.0, "text": "fourth and you can always simplify this"}, {"start": 168.92, "duration": 5.08, "text": "um you know if you want to a little bit"}, {"start": 171.0, "duration": 6.0, "text": "but again we've now eliminated the"}, {"start": 174.0, "duration": 3.0, "text": "parameter"}], "title": "Eliminating the Parameter , A Few Examples", "description": "Thanks to all of you who support me on Patreon. You da real mvps!   $1 per month helps!! :)  https://www.patreon.com/patrickjmt !! You should ALSO think about restrictions on the value of t, based on the parametric equations for x(t) and y(t).  I do not mention that in the video though (I think).  All the same, the curves would lay on the curves I produce.", "lengthSeconds": 175, "width": 480, "height": 360}
{"video_id": "CyaUkZp9LDM", "transcript": [{"start": 1.199, "duration": 4.16, "text": "okay in this video i want to do some"}, {"start": 3.12, "duration": 4.639, "text": "slightly more complicated evaluating"}, {"start": 5.359, "duration": 4.24, "text": "exponential expressions so just some"}, {"start": 7.759, "duration": 3.681, "text": "more numbers floating around"}, {"start": 9.599, "duration": 3.761, "text": "at this point to me i really stop you"}, {"start": 11.44, "duration": 3.52, "text": "know and look at it for a second to"}, {"start": 13.36, "duration": 3.839, "text": "to make sure that i'm doing the order of"}, {"start": 14.96, "duration": 3.76, "text": "operations correctly"}, {"start": 17.199, "duration": 5.361, "text": "but typically what i'll do is i'll try"}, {"start": 18.72, "duration": 3.84, "text": "to work inside the brackets first"}, {"start": 22.8, "duration": 3.12, "text": "so and i think in this case there's"}, {"start": 24.32, "duration": 4.32, "text": "really not much else you can do"}, {"start": 25.92, "duration": 4.88, "text": "except work inside so let's see we've"}, {"start": 28.64, "duration": 4.32, "text": "got to do order of operations"}, {"start": 30.8, "duration": 3.12, "text": "so we've got multiplication but also an"}, {"start": 32.96, "duration": 3.92, "text": "exponent"}, {"start": 33.92, "duration": 3.76, "text": "so we do the exponent first so 4 squared"}, {"start": 36.88, "duration": 4.72, "text": "is 4 times"}, {"start": 37.68, "duration": 3.92, "text": "4 which would give us 16"}, {"start": 42.079, "duration": 3.521, "text": "minus okay so again you know notice that"}, {"start": 44.559, "duration": 3.361, "text": "the negative is not"}, {"start": 45.6, "duration": 4.479, "text": "included it's it's not being squared the"}, {"start": 47.92, "duration": 5.6, "text": "only thing being squared in this case"}, {"start": 50.079, "duration": 5.681, "text": "is the 5 and 5 squared gives us"}, {"start": 53.52, "duration": 4.24, "text": "25 so there is still a negative out"}, {"start": 55.76, "duration": 5.439, "text": "front"}, {"start": 57.76, "duration": 6.64, "text": "that's all being squared so let's see"}, {"start": 61.199, "duration": 4.641, "text": "we have negative 4 still out front 2"}, {"start": 64.4, "duration": 5.12, "text": "times 16 will be"}, {"start": 65.84, "duration": 5.68, "text": "32 minus 25"}, {"start": 69.52, "duration": 3.2, "text": "again that's all being squared so i"}, {"start": 71.52, "duration": 3.2, "text": "think i'm going to run into my next"}, {"start": 72.72, "duration": 4.48, "text": "problem here so let me move it"}, {"start": 74.72, "duration": 4.64, "text": "up here real quick so we still have"}, {"start": 77.2, "duration": 6.88, "text": "negative 4 out front"}, {"start": 79.36, "duration": 8.56, "text": "let's see 32 minus 25 32 minus 25"}, {"start": 84.08, "duration": 7.52, "text": "i believe would give us the number seven"}, {"start": 87.92, "duration": 5.839, "text": "still that's being squared"}, {"start": 91.6, "duration": 3.12, "text": "okay so again we have to do the exponent"}, {"start": 93.759, "duration": 2.881, "text": "first"}, {"start": 94.72, "duration": 3.68, "text": "so the negative four he's just hanging"}, {"start": 96.64, "duration": 6.32, "text": "out there"}, {"start": 98.4, "duration": 6.48, "text": "seven times seven will give us 49"}, {"start": 102.96, "duration": 3.04, "text": "and now we have to do negative four"}, {"start": 104.88, "duration": 5.599, "text": "times uh"}, {"start": 106.0, "duration": 4.799, "text": "49 and you guys can check my arithmetic"}, {"start": 110.479, "duration": 2.64, "text": "here"}, {"start": 110.799, "duration": 3.28, "text": "the way i do stuff like this is to me if"}, {"start": 113.119, "duration": 3.121, "text": "i made it 50"}, {"start": 114.079, "duration": 3.521, "text": "negative 4 times 50 would be negative"}, {"start": 116.24, "duration": 3.919, "text": "200"}, {"start": 117.6, "duration": 4.08, "text": "but since you only have 49 of them it's"}, {"start": 120.159, "duration": 3.521, "text": "one multiple less"}, {"start": 121.68, "duration": 5.68, "text": "so instead of negative 200 it'll give"}, {"start": 123.68, "duration": 5.76, "text": "you negative 196."}, {"start": 127.36, "duration": 3.92, "text": "so again you can use a calculator or do"}, {"start": 129.44, "duration": 3.6, "text": "it out the long way"}, {"start": 131.28, "duration": 3.12, "text": "maybe just another way to think about it"}, {"start": 133.04, "duration": 2.8, "text": "again you know i think it's a little"}, {"start": 134.4, "duration": 3.6, "text": "tricky it takes a little bit"}, {"start": 135.84, "duration": 4.08, "text": "getting used to but that's one way that"}, {"start": 138.0, "duration": 5.2, "text": "i'm able to do the arithmetic in my head"}, {"start": 139.92, "duration": 6.16, "text": "at least all right so that'll be our"}, {"start": 143.2, "duration": 4.96, "text": "final answer for the first example"}, {"start": 146.08, "duration": 3.12, "text": "in the second one very similar problem"}, {"start": 148.16, "duration": 3.2, "text": "we have seven"}, {"start": 149.2, "duration": 3.759, "text": "and then inside of our brackets we have"}, {"start": 151.36, "duration": 4.959, "text": "two times three cubed minus"}, {"start": 152.959, "duration": 4.801, "text": "eight squared all of that being squared"}, {"start": 156.319, "duration": 4.721, "text": "okay so i'm going to do the same thing"}, {"start": 157.76, "duration": 3.28, "text": "i'm going to work on the inside"}, {"start": 161.92, "duration": 6.8, "text": "so 3 cubed that's 3 times 3"}, {"start": 166.08, "duration": 3.12, "text": "times another 3 well 3 times 3 would"}, {"start": 168.72, "duration": 3.04, "text": "give us"}, {"start": 169.2, "duration": 5.28, "text": "9 and then times another 3 would give us"}, {"start": 171.76, "duration": 6.64, "text": "27"}, {"start": 174.48, "duration": 10.56, "text": "minus 8 times 8 would give us"}, {"start": 178.4, "duration": 9.919, "text": "64 being squared"}, {"start": 185.04, "duration": 6.559, "text": "let's see 2 times 27"}, {"start": 188.319, "duration": 7.681, "text": "will give us 54 minus"}, {"start": 191.599, "duration": 8.241, "text": "64 again being squared"}, {"start": 196.0, "duration": 3.84, "text": "if we simplify inside the brackets"}, {"start": 200.959, "duration": 5.441, "text": "54 minus 64 will give us negative 10"}, {"start": 206.56, "duration": 4.72, "text": "squared and now so we have to take"}, {"start": 210.4, "duration": 3.839, "text": "negative 10"}, {"start": 211.28, "duration": 4.64, "text": "times negative 10 which will give us a"}, {"start": 214.239, "duration": 5.041, "text": "positive"}, {"start": 215.92, "duration": 5.92, "text": "100 the 7 is still out front"}, {"start": 219.28, "duration": 4.319, "text": "so since the negative is being included"}, {"start": 221.84, "duration": 2.959, "text": "in the exponent again it will turn"}, {"start": 223.599, "duration": 3.601, "text": "positive in this case"}, {"start": 224.799, "duration": 3.281, "text": "because it's an even exponent and then"}, {"start": 227.2, "duration": 4.0, "text": "we have 7"}, {"start": 228.08, "duration": 6.56, "text": "times 100 which is going to give us"}, {"start": 231.2, "duration": 6.36, "text": "700. so this second expression will"}, {"start": 234.64, "duration": 5.92, "text": "reduce to the value"}, {"start": 237.56, "duration": 3.0, "text": "700."}], "title": "Evaluating Exponential Expressions (Numbers Only, No Variables) - Ex 3", "description": "Thanks to all of you who support me on Patreon. You da real mvps!   $1 per month helps!! :)  https://www.patreon.com/patrickjmt !! Evaluating Exponential Expressions (Numbers Only, No Variables)", "lengthSeconds": 239, "width": 480, "height": 360}
{"video_id": "b52pKwMQZAQ", "transcript": [{"start": 0.84, "duration": 3.079, "text": "all right in this example we're going to"}, {"start": 2.28, "duration": 4.2, "text": "write the standard form equation of a"}, {"start": 3.919, "duration": 6.6, "text": "parabola with a Vertex at the origin and"}, {"start": 6.48, "duration": 6.199, "text": "a directrix of yal ne5 over 6 and again"}, {"start": 10.519, "duration": 4.16, "text": "you don't have to but I have already"}, {"start": 12.679, "duration": 4.041, "text": "sketched a little graph that corresponds"}, {"start": 14.679, "duration": 5.161, "text": "to the information that we're given so"}, {"start": 16.72, "duration": 5.24, "text": "there's our directorx yal 5 over 6 we"}, {"start": 19.84, "duration": 3.439, "text": "know that the vertex is at the origin"}, {"start": 21.96, "duration": 2.92, "text": "and well that would tell me that my"}, {"start": 23.279, "duration": 4.0, "text": "Parabola would have to be opening"}, {"start": 24.88, "duration": 5.159, "text": "upwards in this case so again kind of"}, {"start": 27.279, "duration": 8.8, "text": "the basic form that we are trying to"}, {"start": 30.039, "duration": 9.321, "text": "know fill in we have x^2 = 4 P *"}, {"start": 36.079, "duration": 6.081, "text": "Y and again the only real value we have"}, {"start": 39.36, "duration": 5.24, "text": "to come up with here is the value for p"}, {"start": 42.16, "duration": 5.2, "text": "we just have to deduce the value for p"}, {"start": 44.6, "duration": 7.36, "text": "and then um we plug it in and we're done"}, {"start": 47.36, "duration": 7.44, "text": "really not much else to it so we've seen"}, {"start": 51.96, "duration": 6.2, "text": "um that if the director okay so the"}, {"start": 54.8, "duration": 5.599, "text": "directorx had in this case we can write"}, {"start": 58.16, "duration": 4.6, "text": "it as yal"}, {"start": 60.399, "duration": 6.801, "text": "p and we know that our"}, {"start": 62.76, "duration": 6.08, "text": "focus is at the point 0 comma P okay so"}, {"start": 67.2, "duration": 4.32, "text": "basically it says to get the focus just"}, {"start": 68.84, "duration": 6.68, "text": "kind of take the opposite sign so we've"}, {"start": 71.52, "duration": 6.8, "text": "got uh -5 over 6 so our P value would"}, {"start": 75.52, "duration": 4.88, "text": "just be 5 over 6 so that tells us our"}, {"start": 78.32, "duration": 4.839, "text": "focus is at 0"}, {"start": 80.4, "duration": 4.44, "text": "56 and again you don't maybe even"}, {"start": 83.159, "duration": 4.521, "text": "necessarily have to think about that but"}, {"start": 84.84, "duration": 4.84, "text": "just to point out a couple things and"}, {"start": 87.68, "duration": 4.56, "text": "again now we have our P value mean"}, {"start": 89.68, "duration": 5.52, "text": "that's really the important part here so"}, {"start": 92.24, "duration": 4.68, "text": "our P value will be 5 over 6 again just"}, {"start": 95.2, "duration": 4.599, "text": "based on the directorx you can just take"}, {"start": 96.92, "duration": 4.76, "text": "the opposite sign and that's your P"}, {"start": 99.799, "duration": 5.441, "text": "value and now it's just a little bit of"}, {"start": 101.68, "duration": 8.96, "text": "arithmetic so x^2 = 4 * well let's plug"}, {"start": 105.24, "duration": 7.12, "text": "in 5 over six for our P value times Y"}, {"start": 110.64, "duration": 2.759, "text": "and well we can simplify this down a"}, {"start": 112.36, "duration": 4.24, "text": "little"}, {"start": 113.399, "duration": 7.281, "text": "bit let's see uh 4 over 6 would reduce"}, {"start": 116.6, "duration": 5.64, "text": "to 2 over3 2 * 5 would be 10"}, {"start": 120.68, "duration": 3.52, "text": "we would still have the three in the"}, {"start": 122.24, "duration": 6.96, "text": "denominator so it looks like we would"}, {"start": 124.2, "duration": 8.0, "text": "get x^2 = 10/3 * Y and that would be our"}, {"start": 129.2, "duration": 3.0, "text": "equation"}], "title": "Conic Sections, Parabola : Find Equation of Parabola Given Directrix", "description": "Thanks to all of you who support me on Patreon. You da real mvps!   $1 per month helps!! :)  https://www.patreon.com/patrickjmt !! Conic Sections, Parabola : Find Equation of Parabola Given Directrix.  In this example, we are given that a parabola has a vertex at the origin and we are given the equation of the directrix.  From that, we find the equation of the parabola.", "lengthSeconds": 131, "width": 480, "height": 360}
{"video_id": "zcEWC3LK1sk", "transcript": [{"start": 0.68, "duration": 2.88, "text": "all right here we're going to look at"}, {"start": 1.839, "duration": 3.681, "text": "some examples of multiplication and"}, {"start": 3.56, "duration": 3.4, "text": "division of integers and the only thing"}, {"start": 5.52, "duration": 6.96, "text": "you really have to remember are just the"}, {"start": 6.96, "duration": 8.08, "text": "rules about signs so 8 * -9 we have a"}, {"start": 12.48, "duration": 4.24, "text": "positive time a negative a positive time"}, {"start": 15.04, "duration": 3.72, "text": "a negative is going to give us a"}, {"start": 16.72, "duration": 5.96, "text": "negative number and then I forget all"}, {"start": 18.76, "duration": 9.04, "text": "about the signs I just say well 8 * 9 is"}, {"start": 22.68, "duration": 7.759, "text": "72 therefore 8 * -9 is going to give us"}, {"start": 27.8, "duration": 4.599, "text": "-72 if you have two negatives you know a"}, {"start": 30.439, "duration": 5.481, "text": "negative times a negative that always"}, {"start": 32.399, "duration": 8.16, "text": "gives you a positive so 4 * 7 is going"}, {"start": 35.92, "duration": 6.56, "text": "to give us 28 so it says -4 * -7 hey"}, {"start": 40.559, "duration": 6.241, "text": "that's posi"}, {"start": 42.48, "duration": 7.8, "text": "28 exact same rules with division Okay"}, {"start": 46.8, "duration": 6.2, "text": "so we've got a 48 divided -2 so we have"}, {"start": 50.28, "duration": 5.72, "text": "a positive divided by a negative that's"}, {"start": 53.0, "duration": 7.0, "text": "going to give us a negative and then 48"}, {"start": 56.0, "duration": 8.56, "text": "divided uh 2 is going to give us 24 so"}, {"start": 60.0, "duration": 7.2, "text": "48 / --2 is going to give us"}, {"start": 64.56, "duration": 4.68, "text": "-4 so let's see Part B we've got a"}, {"start": 67.2, "duration": 4.12, "text": "negative divided by a negative it's the"}, {"start": 69.24, "duration": 4.519, "text": "same rule as with multiplication a"}, {"start": 71.32, "duration": 4.119, "text": "negative times a negative is a positive"}, {"start": 73.759, "duration": 3.961, "text": "likewise a negative divided by a"}, {"start": 75.439, "duration": 3.961, "text": "negative is a positive so it's"}, {"start": 77.72, "duration": 3.2, "text": "definitely going to be a positive number"}, {"start": 79.4, "duration": 5.6, "text": "then I just forget again about the the"}, {"start": 80.92, "duration": 9.96, "text": "signs I say well 35 / 5 that's going to"}, {"start": 85.0, "duration": 9.88, "text": "give us 7 so -35 / -5 well that also"}, {"start": 90.88, "duration": 4.0, "text": "gives us positive s"}], "title": "Dividing / Division of Integers", "description": "Thanks to all of you who support me on Patreon. You da real mvps!   $1 per month helps!! :)  https://www.patreon.com/patrickjmt !! Dividing / Division of Integers.\nIn this example, I do some basic examples of dividing integers.", "lengthSeconds": 93, "width": 480, "height": 360}
{"video_id": "DeM5aebVt8g", "transcript": [{"start": 0.48, "duration": 3.68, "text": "all right so one more example here of"}, {"start": 2.0, "duration": 4.359, "text": "solving a trig equation by factoring and"}, {"start": 4.16, "duration": 5.399, "text": "here we're going to solve cosine xal"}, {"start": 6.359, "duration": 7.801, "text": "Cent X and what I'm going to do in this"}, {"start": 9.559, "duration": 4.601, "text": "case uh is simply rewrite"}, {"start": 14.2, "duration": 5.159, "text": "coent so tangent is s over cosine coent"}, {"start": 18.119, "duration": 4.721, "text": "is"}, {"start": 19.359, "duration": 6.08, "text": "cosine over s and again it might be"}, {"start": 22.84, "duration": 4.88, "text": "tempting to sort of uh divide both sides"}, {"start": 25.439, "duration": 4.281, "text": "by cosine of x to get rid of that but"}, {"start": 27.72, "duration": 3.879, "text": "again that would be a mistake as we kind"}, {"start": 29.72, "duration": 3.839, "text": "of pointed out in the last video you"}, {"start": 31.599, "duration": 4.28, "text": "don't want to do that you'll lose some"}, {"start": 33.559, "duration": 6.32, "text": "solutions so what I'm going to do is"}, {"start": 35.879, "duration": 6.481, "text": "simply subtract the cosine X over"}, {"start": 39.879, "duration": 4.16, "text": "sinx from both sides and that'll leave"}, {"start": 42.36, "duration": 5.679, "text": "me with zero on the"}, {"start": 44.039, "duration": 7.04, "text": "right well I can Factor the cosine X out"}, {"start": 48.039, "duration": 5.721, "text": "so cosine x * 1 would give me cosine"}, {"start": 51.079, "duration": 5.921, "text": "xus 1/ sin"}, {"start": 53.76, "duration": 5.16, "text": "x when I distribute these out um we'll"}, {"start": 57.0, "duration": 4.0, "text": "get our cosine over"}, {"start": 58.92, "duration": 4.24, "text": "s"}, {"start": 61.0, "duration": 5.04, "text": "so again now we have to Simply Set uh"}, {"start": 63.16, "duration": 4.92, "text": "each factor equal to zero so we'll get"}, {"start": 66.04, "duration": 6.04, "text": "cosine xal"}, {"start": 68.08, "duration": 8.679, "text": "0 and then we'll get that 1 - 1 /"}, {"start": 72.08, "duration": 7.2, "text": "sinx = 0 and well uh maybe it's pretty"}, {"start": 76.759, "duration": 4.561, "text": "clear what sin x would have to be uh we"}, {"start": 79.28, "duration": 4.32, "text": "can certainly add it over we can add the"}, {"start": 81.32, "duration": 4.24, "text": "one/ sin"}, {"start": 83.6, "duration": 5.32, "text": "x and then what I'm going to do is"}, {"start": 85.56, "duration": 6.48, "text": "simply multiply both sides by"}, {"start": 88.92, "duration": 7.04, "text": "sinx so they'll cancel on the right and"}, {"start": 92.04, "duration": 6.16, "text": "we'll be left simply with sinx = 1 so I"}, {"start": 95.96, "duration": 5.92, "text": "have to think where between 0 and 2 pi"}, {"start": 98.2, "duration": 7.959, "text": "do cosine of xal 0 well again that"}, {"start": 101.88, "duration": 8.4, "text": "happens at x = Pi / 2 and it also uh"}, {"start": 106.159, "duration": 10.081, "text": "happens at 3 pi/ 2 uh where does s of x"}, {"start": 110.28, "duration": 8.4, "text": "equal 1 well s = 1 at pi over 2 uh which"}, {"start": 116.24, "duration": 4.239, "text": "is uh I guess a bit redundant uh we've"}, {"start": 118.68, "duration": 4.32, "text": "already found that solution and that's"}, {"start": 120.479, "duration": 4.761, "text": "the only place where s equals 1 on the"}, {"start": 123.0, "duration": 5.8, "text": "interval 0 to 2 pi so we could just"}, {"start": 125.24, "duration": 7.519, "text": "simply say our Solutions are Pi / 2 and"}, {"start": 128.8, "duration": 3.959, "text": "3 pi/ 2"}], "title": "Solving a Trigonometric Equation by Factoring, Example 3", "description": "Thanks to all of you who support me on Patreon. You da real mvps!   $1 per month helps!! :)  https://www.patreon.com/patrickjmt !! Solving a Trigonometric Equation by Factoring, Example 3", "lengthSeconds": 131, "width": 480, "height": 360}
{"video_id": "nFsoA_dArn4", "transcript": [{"start": 0.079, "duration": 4.811, "text": "okay here we're going to look at some"}, {"start": 1.949, "duration": 4.501, "text": "basic examples of the ratio test and for"}, {"start": 4.89, "duration": 4.11, "text": "the ratio test it says if you have some"}, {"start": 6.45, "duration": 5.16, "text": "formula you know a sub M that describes"}, {"start": 9.0, "duration": 4.83, "text": "the terms of your series it's as we look"}, {"start": 11.61, "duration": 5.46, "text": "at the limit as n goes to infinity of a"}, {"start": 13.83, "duration": 5.119, "text": "sub n plus 1 over a sub n recall if this"}, {"start": 17.07, "duration": 4.95, "text": "is less than 1 the series converges"}, {"start": 18.949, "duration": 5.021, "text": "absolutely if it's greater than 1 it"}, {"start": 22.02, "duration": 4.41, "text": "diverges and if equals 1 it's"}, {"start": 23.97, "duration": 5.85, "text": "inconclusive so here we're going to use"}, {"start": 26.43, "duration": 5.13, "text": "the ratio test and Part A to decide if"}, {"start": 29.82, "duration": 4.77, "text": "the series 2 to the N over N factorial"}, {"start": 31.56, "duration": 6.269, "text": "from N equals 1 to infinity whether or"}, {"start": 34.59, "duration": 6.39, "text": "not that converges so everywhere there's"}, {"start": 37.829, "duration": 9.21, "text": "an N we're going to plug in our we'll"}, {"start": 40.98, "duration": 10.55, "text": "plug in the n plus 1 so here we'll have"}, {"start": 47.039, "duration": 7.801, "text": "2 to the n plus 1 over n plus 1"}, {"start": 51.53, "duration": 6.25, "text": "factorial then we're going to divide"}, {"start": 54.84, "duration": 9.12, "text": "that by the original so 2 to the N over"}, {"start": 57.78, "duration": 9.54, "text": "N factorial all right well this is the"}, {"start": 63.96, "duration": 8.43, "text": "limit as n goes to infinity of 2 to the"}, {"start": 67.32, "duration": 9.299, "text": "n plus 1 over n plus 1 factorial times n"}, {"start": 72.39, "duration": 7.619, "text": "factorial over 2 to the N well let's see"}, {"start": 76.619, "duration": 6.061, "text": "we can simplify this we've got 2 to the"}, {"start": 80.009, "duration": 4.351, "text": "n plus 1 over 2 to the N that's going to"}, {"start": 82.68, "duration": 7.68, "text": "leave us with a 2 to the 1st in the"}, {"start": 84.36, "duration": 10.91, "text": "numerator and then n factorial over n"}, {"start": 90.36, "duration": 4.91, "text": "plus 1 factorial how does that simplify"}, {"start": 96.74, "duration": 4.239, "text": "well we're going to use this trick a lot"}, {"start": 99.21, "duration": 4.83, "text": "so maybe let's talk about it at least"}, {"start": 100.979, "duration": 5.341, "text": "once here so you know for example just"}, {"start": 104.04, "duration": 5.07, "text": "at random say we had 6 factorial"}, {"start": 106.32, "duration": 6.03, "text": "well that's 6 times 5 times 4 times 3"}, {"start": 109.11, "duration": 6.99, "text": "times 2 times 1 we could write it as 6"}, {"start": 112.35, "duration": 6.21, "text": "times you know we'll take 1 away from 6"}, {"start": 116.1, "duration": 4.89, "text": "hey that would give us 5 and then we'll"}, {"start": 118.56, "duration": 4.739, "text": "take two away from 6 hey that would give"}, {"start": 120.99, "duration": 5.339, "text": "us 4 and then I'm going to start putting"}, {"start": 123.299, "duration": 5.07, "text": "kind of a few arbitrary dots assuming"}, {"start": 126.329, "duration": 3.361, "text": "this number is large enough eventually"}, {"start": 128.369, "duration": 2.051, "text": "you know you're going to get every"}, {"start": 129.69, "duration": 2.769, "text": "number in between"}, {"start": 130.42, "duration": 4.74, "text": "main and then at the end we'll end up"}, {"start": 132.459, "duration": 4.651, "text": "being will always multiply by 3 times 2"}, {"start": 135.16, "duration": 5.7, "text": "times 1 you know kind of the tail end of"}, {"start": 137.11, "duration": 5.58, "text": "the sequence so in factorial I'm going"}, {"start": 140.86, "duration": 3.9, "text": "to write that as n you would just get"}, {"start": 142.69, "duration": 5.25, "text": "the very first number then we would take"}, {"start": 144.76, "duration": 5.4, "text": "one away from that then we would take 2"}, {"start": 147.94, "duration": 3.51, "text": "away from that and then eventually you"}, {"start": 150.16, "duration": 3.659, "text": "know assuming ends a big number"}, {"start": 151.45, "duration": 5.039, "text": "eventually we'll get down to 3 times 2"}, {"start": 153.819, "duration": 4.89, "text": "times 1 the same thing in the"}, {"start": 156.489, "duration": 3.121, "text": "denominator if we expand n plus 1"}, {"start": 158.709, "duration": 3.78, "text": "factorial"}, {"start": 159.61, "duration": 4.92, "text": "we'll get n plus 1 but then if we take"}, {"start": 162.489, "duration": 4.141, "text": "one away from that we'll be left with in"}, {"start": 164.53, "duration": 5.19, "text": "and then if we take one away from that"}, {"start": 166.63, "duration": 5.22, "text": "we'll have n minus 1 and minus 2 again"}, {"start": 169.72, "duration": 3.44, "text": "eventually we'll get down to 3 times 2"}, {"start": 171.85, "duration": 3.719, "text": "times 1"}, {"start": 173.16, "duration": 5.68, "text": "so notice everything is going to cancel"}, {"start": 175.569, "duration": 8.011, "text": "out except for the n plus 1 in the"}, {"start": 178.84, "duration": 7.86, "text": "denominator so in this case we've got"}, {"start": 183.58, "duration": 6.629, "text": "the limit as n goes to infinity of 2"}, {"start": 186.7, "duration": 6.0, "text": "over n plus 1 well as n goes to infinity"}, {"start": 190.209, "duration": 6.361, "text": "the denominator is going to get large 2"}, {"start": 192.7, "duration": 6.33, "text": "over a large number is 0 the limit of"}, {"start": 196.57, "duration": 4.139, "text": "that is 0 and since that number is less"}, {"start": 199.03, "duration": 5.07, "text": "than 1 that would tell us that our"}, {"start": 200.709, "duration": 8.581, "text": "series is absolutely convergent in"}, {"start": 204.1, "duration": 7.8, "text": "particular it is also just convergent so"}, {"start": 209.29, "duration": 7.05, "text": "partake we've got a nice example here of"}, {"start": 211.9, "duration": 9.089, "text": "a convergent series and almost any time"}, {"start": 216.34, "duration": 6.33, "text": "I see factorials and a series and they"}, {"start": 220.989, "duration": 4.171, "text": "want me to decide if it converges or"}, {"start": 222.67, "duration": 4.71, "text": "diverges I almost always just jump"}, {"start": 225.16, "duration": 5.15, "text": "immediately to the ratio test very"}, {"start": 227.38, "duration": 2.93, "text": "useful little test"}], "title": "The Ratio Test , Another Example #1", "description": "Thanks to all of you who support me on Patreon. You da real mvps!   $1 per month helps!! :)  https://www.patreon.com/patrickjmt !! The Ratio Test , Another Example #1.  Just another example of using the ratio test to determine if a series converges or diverges.", "lengthSeconds": 229, "width": 480, "height": 360}
{"video_id": "E3tpfIMCO9o", "transcript": [{"start": 0.28, "duration": 3.0, "text": "all right in this video we're going to"}, {"start": 1.36, "duration": 4.24, "text": "talk about graphing some simple polar"}, {"start": 3.28, "duration": 5.68, "text": "equations so here we're going to graph R"}, {"start": 5.6, "duration": 4.919, "text": "= 2 and Theta = pi4 there's two"}, {"start": 8.96, "duration": 4.559, "text": "different ways you can think about these"}, {"start": 10.519, "duration": 5.28, "text": "just to graph R equals 2 um what that"}, {"start": 13.519, "duration": 4.68, "text": "really says to me it says it's kind of"}, {"start": 15.799, "duration": 4.921, "text": "independent of the angle so pick any"}, {"start": 18.199, "duration": 5.4, "text": "angle that you want just go out a"}, {"start": 20.72, "duration": 5.16, "text": "distance of two from the origin well if"}, {"start": 23.599, "duration": 4.161, "text": "you do that you know if you kind of"}, {"start": 25.88, "duration": 4.12, "text": "neglect the angle if you allow it to be"}, {"start": 27.76, "duration": 4.639, "text": "whatever you want it to be but you"}, {"start": 30.0, "duration": 4.64, "text": "always go at a distance of two hey"}, {"start": 32.399, "duration": 5.16, "text": "that's just going to make a nice uh a"}, {"start": 34.64, "duration": 5.28, "text": "nice Circle so I think I put my -2 down"}, {"start": 37.559, "duration": 4.561, "text": "here a little too far let's make -2"}, {"start": 39.92, "duration": 4.44, "text": "right there it's just going to be a"}, {"start": 42.12, "duration": 4.8, "text": "circle of radius two and again you could"}, {"start": 44.36, "duration": 5.24, "text": "see that because we know that R uh we"}, {"start": 46.92, "duration": 4.72, "text": "could rewrite r as the square Ro T of"}, {"start": 49.6, "duration": 5.88, "text": "x^2 +"}, {"start": 51.64, "duration": 9.32, "text": "y^2 = 2 and if you square both sides we"}, {"start": 55.48, "duration": 7.52, "text": "get hey x^2 + y^2 = 4 and again X + y^2"}, {"start": 60.96, "duration": 5.32, "text": "= 4 that's just going to be again a"}, {"start": 63.0, "duration": 7.92, "text": "circle of radius 2 so this would be part"}, {"start": 66.28, "duration": 8.32, "text": "A Part B we got Theta = pi over 4 so"}, {"start": 70.92, "duration": 5.76, "text": "Theta = pi over 4 again you can kind of"}, {"start": 74.6, "duration": 5.879, "text": "think about this in the same way it says"}, {"start": 76.68, "duration": 7.28, "text": "go to an angle pi over 4 so pi over 4"}, {"start": 80.479, "duration": 6.28, "text": "would just be our 45 Dee line here so"}, {"start": 83.96, "duration": 6.04, "text": "there's the angle pi over 4 and then"}, {"start": 86.759, "duration": 5.961, "text": "kind of the same idea it says always let"}, {"start": 90.0, "duration": 6.04, "text": "will be pi4 but pick any r value that"}, {"start": 92.72, "duration": 4.96, "text": "you want well if R is positive we'll get"}, {"start": 96.04, "duration": 4.719, "text": "uh you know the segment of the line that"}, {"start": 97.68, "duration": 4.799, "text": "goes off into the first quadrant if R"}, {"start": 100.759, "duration": 4.481, "text": "was negative though it would reflect"}, {"start": 102.479, "duration": 5.521, "text": "down into the third quadrant um and"}, {"start": 105.24, "duration": 4.879, "text": "basically we would just get a nice"}, {"start": 108.0, "duration": 4.24, "text": "little line and again the way that you"}, {"start": 110.119, "duration": 5.561, "text": "can see that is if you take tangent of"}, {"start": 112.24, "duration": 7.68, "text": "both sides so tangent of theta equals"}, {"start": 115.68, "duration": 6.759, "text": "tangent of pi over 4 well tangent OFA"}, {"start": 119.92, "duration": 5.6, "text": "we've seen that we can write that as"}, {"start": 122.439, "duration": 6.16, "text": "YX tangent of pi over 4 that's just"}, {"start": 125.52, "duration": 6.56, "text": "equal to 1 so if you multiply both sides"}, {"start": 128.599, "duration": 6.161, "text": "you simply get the equation yal x and"}, {"start": 132.08, "duration": 4.2, "text": "again that's what yals x looks like so"}, {"start": 134.76, "duration": 3.72, "text": "two different ways to think about it you"}, {"start": 136.28, "duration": 4.88, "text": "know just uh just to think about the"}, {"start": 138.48, "duration": 4.64, "text": "equation that you're given but uh also a"}, {"start": 141.16, "duration": 4.04, "text": "very very common thing to do is to"}, {"start": 143.12, "duration": 3.96, "text": "convert them back into these uh"}, {"start": 145.2, "duration": 3.319, "text": "rectangular coordinates to where"}, {"start": 147.08, "duration": 4.439, "text": "hopefully it's something a little more"}, {"start": 148.519, "duration": 3.0, "text": "familiar"}], "title": "Graphing Simple Polar Equations, Ex 1", "description": "Thanks to all of you who support me on Patreon. You da real mvps!   $1 per month helps!! :)  https://www.patreon.com/patrickjmt !! Graphing Simple Polar Equations, Ex 1.  Graphing polar curves can be tricky, but here I look at graphing some of the basic ones first!", "lengthSeconds": 150, "width": 480, "height": 360}
{"video_id": "2uH-08hG-Ng", "transcript": [{"start": 0.799, "duration": 2.961, "text": "all right in this video we're going to"}, {"start": 2.0, "duration": 3.44, "text": "look at just some questions related to"}, {"start": 3.76, "duration": 2.88, "text": "the second derivative test and these are"}, {"start": 5.44, "duration": 2.72, "text": "just sort of sort of uh"}, {"start": 6.64, "duration": 4.079, "text": "you know do you know the results type"}, {"start": 8.16, "duration": 4.8, "text": "question so suppose our function f of x"}, {"start": 10.719, "duration": 4.161, "text": "has a critical point at x equals 4"}, {"start": 12.96, "duration": 3.84, "text": "and the second derivative at 4 is"}, {"start": 14.88, "duration": 4.319, "text": "negative we want to know what can we say"}, {"start": 16.8, "duration": 4.239, "text": "about our function at x equals 4"}, {"start": 19.199, "duration": 4.0, "text": "well recall if you have a critical point"}, {"start": 21.039, "duration": 5.441, "text": "and the second derivative is negative"}, {"start": 23.199, "duration": 4.641, "text": "what we can say in that case is the idea"}, {"start": 26.48, "duration": 2.639, "text": "is in that case we're actually going to"}, {"start": 27.84, "duration": 4.48, "text": "have a local maximum"}, {"start": 29.119, "duration": 4.721, "text": "at x equals 4. so since the second"}, {"start": 32.32, "duration": 3.2, "text": "derivative is negative that kind of"}, {"start": 33.84, "duration": 3.68, "text": "is telling us that the function's"}, {"start": 35.52, "duration": 3.92, "text": "concave down"}, {"start": 37.52, "duration": 3.28, "text": "and with that being a critical point"}, {"start": 39.44, "duration": 2.799, "text": "that's going to tell us that we do have"}, {"start": 40.8, "duration": 4.48, "text": "a local maximum"}, {"start": 42.239, "duration": 7.521, "text": "so all right part b here"}, {"start": 45.28, "duration": 6.08, "text": "um so suppose g of x has a critical"}, {"start": 49.76, "duration": 4.08, "text": "point at x equals two"}, {"start": 51.36, "duration": 3.199, "text": "and the second derivative at two equals"}, {"start": 53.84, "duration": 2.64, "text": "zero"}, {"start": 54.559, "duration": 3.041, "text": "we wanna know what can we say about the"}, {"start": 56.48, "duration": 4.0, "text": "function"}, {"start": 57.6, "duration": 4.4, "text": "what's happening at x equals two well"}, {"start": 60.48, "duration": 2.16, "text": "recall in this case if the second"}, {"start": 62.0, "duration": 4.4, "text": "derivative"}, {"start": 62.64, "duration": 6.4, "text": "equals zero in that case actually our"}, {"start": 66.4, "duration": 3.039, "text": "second derivative test is inconclusive"}, {"start": 69.04, "duration": 1.92, "text": "so"}, {"start": 69.439, "duration": 3.521, "text": "what you could do in this case to figure"}, {"start": 70.96, "duration": 3.36, "text": "out if you do potentially have a maximum"}, {"start": 72.96, "duration": 4.08, "text": "or minimum"}, {"start": 74.32, "duration": 4.24, "text": "is you can go back and use like the"}, {"start": 77.04, "duration": 2.0, "text": "first derivative test do you like the"}, {"start": 78.56, "duration": 2.08, "text": "little sign"}, {"start": 79.04, "duration": 3.28, "text": "chart you know we've seen some of these"}, {"start": 80.64, "duration": 2.64, "text": "find your critical points figure out if"}, {"start": 82.32, "duration": 3.439, "text": "it's increasing"}, {"start": 83.28, "duration": 3.36, "text": "decreasing increasing for example and"}, {"start": 85.759, "duration": 3.641, "text": "then you'll know"}, {"start": 86.64, "duration": 5.76, "text": "okay i've got a local maximum or local"}, {"start": 89.4, "duration": 3.0, "text": "minimum"}], "title": "The Second Derivative Test to Find Maximums / Minimums, Quick Example 1", "description": "Thanks to all of you who support me on Patreon. You da real mvps!   $1 per month helps!! :)  https://www.patreon.com/patrickjmt !! The Second Derivative Test to Find Maximums / Minimums.  Suppose we know that a value of x is a critical number and the second derivative is negative there.  What can we say about the point in regards to it being a local maximum or minimum?", "lengthSeconds": 91, "width": 480, "height": 360}
{"video_id": "VguyPrGaXLk", "transcript": [{"start": 0.88, "duration": 3.2, "text": "all right in this video I'm going to do"}, {"start": 2.159, "duration": 5.28, "text": "one more example finding the general"}, {"start": 4.08, "duration": 5.12, "text": "term of an arithmetic sequence and if"}, {"start": 7.439, "duration": 4.12, "text": "you got good eyes you may notice I"}, {"start": 9.2, "duration": 3.519, "text": "changed the subscript on uh one of these"}, {"start": 11.559, "duration": 3.681, "text": "I think it's going to make the values"}, {"start": 12.719, "duration": 3.961, "text": "work out a little bit nicer but anyways"}, {"start": 15.24, "duration": 5.08, "text": "we're going to find the general term of"}, {"start": 16.68, "duration": 8.4, "text": "an arithmetic sequence of a sub 21 ="}, {"start": 20.32, "duration": 6.84, "text": "-14 and a sub 51 ="}, {"start": 25.08, "duration": 5.32, "text": "226 well okay we're just going to use"}, {"start": 27.16, "duration": 9.64, "text": "our formula here that again a subn is a"}, {"start": 30.4, "duration": 8.479, "text": "sub 1 + n -1 * D well a sub 21 we know"}, {"start": 36.8, "duration": 5.16, "text": "that that's going to be well a sub 1"}, {"start": 38.879, "duration": 7.921, "text": "plus we'll take 21 - 1 which will give"}, {"start": 41.96, "duration": 8.4, "text": "us 20 * D we know that has value"}, {"start": 46.8, "duration": 7.72, "text": "-14 likewise a sub 51 that's going to be"}, {"start": 50.36, "duration": 5.719, "text": "a sub 1 plus well we'll take 51 minus 1"}, {"start": 54.52, "duration": 5.879, "text": "which will give us"}, {"start": 56.079, "duration": 6.16, "text": "50d and we know that equals 226"}, {"start": 60.399, "duration": 4.84, "text": "so now I'm just going to uh do some"}, {"start": 62.239, "duration": 5.401, "text": "elimination by addition with the this uh"}, {"start": 65.239, "duration": 3.801, "text": "these two linear equations so what I'm"}, {"start": 67.64, "duration": 4.519, "text": "going to do is I'm going to multiply"}, {"start": 69.04, "duration": 6.96, "text": "both sides of my first equation by -1 so"}, {"start": 72.159, "duration": 6.201, "text": "that's going to give me a sub 1 -"}, {"start": 76.0, "duration": 4.439, "text": "20D ="}, {"start": 78.36, "duration": 6.32, "text": "pos4 I'm going to leave the second"}, {"start": 80.439, "duration": 7.68, "text": "equation alone so we have a sub 1 +"}, {"start": 84.68, "duration": 3.439, "text": "50d ="}, {"start": 88.24, "duration": 5.159, "text": "226 and now if we do our our elimination"}, {"start": 91.079, "duration": 6.201, "text": "by addition negative a sub 1 and"}, {"start": 93.399, "duration": 8.481, "text": "positive a sub 1 will cancel -2d + 50d"}, {"start": 97.28, "duration": 6.199, "text": "is going to give us 30 D 226 + 14 is"}, {"start": 101.88, "duration": 6.879, "text": "going to give us"}, {"start": 103.479, "duration": 9.801, "text": "240 well if we divide both sides by 30"}, {"start": 108.759, "duration": 6.28, "text": "we'll get 24 over 3 which gives us 8 so"}, {"start": 113.28, "duration": 4.4, "text": "now we know our value for"}, {"start": 115.039, "duration": 5.36, "text": "D and at this point we can use either"}, {"start": 117.68, "duration": 6.359, "text": "one of our uh original equations"}, {"start": 120.399, "duration": 5.0, "text": "uh to go back and solve for a sub one"}, {"start": 124.039, "duration": 3.601, "text": "well let's see I don't think it really"}, {"start": 125.399, "duration": 4.121, "text": "matters too much which one we use um I'm"}, {"start": 127.64, "duration": 7.36, "text": "just going to use the first one so we"}, {"start": 129.52, "duration": 7.56, "text": "had a sub 1 + 20 * D but now we know D"}, {"start": 135.0, "duration": 5.44, "text": "is equal to"}, {"start": 137.08, "duration": 5.879, "text": "8 and on the right side we have"}, {"start": 140.44, "duration": 8.799, "text": "-14 well we'll get a sub one let's see"}, {"start": 142.959, "duration": 8.241, "text": "20 * 8 is 160 = -14 if we subtract 160"}, {"start": 149.239, "duration": 3.561, "text": "from both sides"}, {"start": 151.2, "duration": 6.0, "text": "it looks like we're going to get a sub"}, {"start": 152.8, "duration": 6.76, "text": "one the first term to be -"}, {"start": 157.2, "duration": 4.92, "text": "174 well now I can just fill in my"}, {"start": 159.56, "duration": 4.8, "text": "formula so it says a subn the general"}, {"start": 162.12, "duration": 4.759, "text": "term is going to be a sub 1 which we now"}, {"start": 164.36, "duration": 7.28, "text": "know to be"}, {"start": 166.879, "duration": 7.801, "text": "-174 plus n -1 * D which is 8 and now"}, {"start": 171.64, "duration": 5.4, "text": "we've got a formula for the uh for any"}, {"start": 174.68, "duration": 7.04, "text": "uh for the nth term any general term of"}, {"start": 177.04, "duration": 4.68, "text": "our arithmetic sequence"}], "title": "Arithmetic Sequences: Finding a General Formula Given Two Terms", "description": "Thanks to all of you who support me on Patreon. You da real mvps!   $1 per month helps!! :)  https://www.patreon.com/patrickjmt !! Arithmetic Sequences: Finding a General Formula Given Two Terms.  In this video, we have values for a_21 and a_51 and we have to find the general formula for the arithmetic sequence: a_n.", "lengthSeconds": 180, "width": 480, "height": 360}
{"video_id": "AcrSe35s5zE", "transcript": [{"start": 0.719, "duration": 3.16, "text": "all right in this video we're going to"}, {"start": 2.12, "duration": 3.84, "text": "do an example of using determinants to"}, {"start": 3.879, "duration": 3.721, "text": "find the area of a triangle and the"}, {"start": 5.96, "duration": 7.12, "text": "triangle is going to have vertices at"}, {"start": 7.6, "duration": 7.32, "text": "the point 40 72 and 23 the formula we"}, {"start": 13.08, "duration": 3.92, "text": "use is the one here in the middle this a"}, {"start": 14.92, "duration": 4.16, "text": "equal 1 12 and then we have all the"}, {"start": 17.0, "duration": 5.48, "text": "determinants the vertices have to be"}, {"start": 19.08, "duration": 5.8, "text": "listed uh in a a counterclockwise order"}, {"start": 22.48, "duration": 6.92, "text": "around the perimeter so I drew a picture"}, {"start": 24.88, "duration": 6.479, "text": "we've got 40 72 and 23 well I just uh"}, {"start": 29.4, "duration": 3.56, "text": "decided to start at this point 40 it"}, {"start": 31.359, "duration": 4.2, "text": "doesn't matter where you start so that's"}, {"start": 32.96, "duration": 6.24, "text": "going to be my X1 y1 but then if we move"}, {"start": 35.559, "duration": 5.201, "text": "to the right our X2 Y2 will be 7 and two"}, {"start": 39.2, "duration": 4.56, "text": "and then uh if we keep following this"}, {"start": 40.76, "duration": 5.84, "text": "counterclockwise motion X sub3 Y sub3"}, {"start": 43.76, "duration": 4.6, "text": "will be the point 2 comma 3 so at this"}, {"start": 46.6, "duration": 3.599, "text": "point we just have to fill in this"}, {"start": 48.36, "duration": 3.76, "text": "formula and then compute the"}, {"start": 50.199, "duration": 3.0, "text": "determinants and add them together and"}, {"start": 52.12, "duration": 6.599, "text": "multiply"}, {"start": 53.199, "duration": 8.281, "text": "by2 so let's see X1 y1 that was 40 X2 Y2"}, {"start": 58.719, "duration": 7.121, "text": "is 72"}, {"start": 61.48, "duration": 6.16, "text": "uh again X2 Y2 X3 and Y3 that's going to"}, {"start": 65.84, "duration": 4.68, "text": "be 2 and"}, {"start": 67.64, "duration": 6.159, "text": "three and then our last X sub3 and Y"}, {"start": 70.52, "duration": 6.12, "text": "sub3 that'll be 2 and 3 X1 and y1 will"}, {"start": 73.799, "duration": 5.121, "text": "be our 4"}, {"start": 76.64, "duration": 4.72, "text": "and0 all right so let's see what we get"}, {"start": 78.92, "duration": 5.04, "text": "here um we can compute these"}, {"start": 81.36, "duration": 5.96, "text": "determinants so we can do 4 * 2 which is"}, {"start": 83.96, "duration": 6.72, "text": "8 0 * 7 will be"}, {"start": 87.32, "duration": 8.079, "text": "0 our next determinant will get 7 * 3"}, {"start": 90.68, "duration": 7.52, "text": "which is 21 uh - 2 * 2 which is"}, {"start": 95.399, "duration": 7.481, "text": "4 and then our last determinant we get 2"}, {"start": 98.2, "duration": 7.519, "text": "* 0 which is 0 - 3 * 4 which is"}, {"start": 102.88, "duration": 7.12, "text": "12 all right so we get 1 12 it looks"}, {"start": 105.719, "duration": 8.76, "text": "like inside our brackets we have 8 plus"}, {"start": 110.0, "duration": 9.92, "text": "let's see 21 - 4 is going to be 17 um 0"}, {"start": 114.479, "duration": 5.441, "text": "- 12 is going to be uh -12"}, {"start": 121.159, "duration": 7.72, "text": "well 8 + 177 is 25 25 - 12 is going to"}, {"start": 125.36, "duration": 5.239, "text": "be 13 and if we multiply that by 1/2"}, {"start": 128.879, "duration": 5.601, "text": "we're simply going to get that the area"}, {"start": 130.599, "duration": 7.841, "text": "of this triangle uh with these vertices"}, {"start": 134.48, "duration": 6.399, "text": "would be uh the area would be 13 over2"}, {"start": 138.44, "duration": 5.48, "text": "uh units squared whatever your your"}, {"start": 140.879, "duration": 4.561, "text": "units are so um kind of an interesting"}, {"start": 143.92, "duration": 3.76, "text": "little formula and how you can use these"}, {"start": 145.44, "duration": 3.879, "text": "determinants I think to find um the area"}, {"start": 147.68, "duration": 5.76, "text": "of a triangle you can certainly"}, {"start": 149.319, "duration": 7.161, "text": "generalize this to a polygon of any"}, {"start": 153.44, "duration": 5.2, "text": "polygon um with with in for example in"}, {"start": 156.48, "duration": 3.44, "text": "points and we'll do an example with that"}, {"start": 158.64, "duration": 3.239, "text": "where we have something a little more"}, {"start": 159.92, "duration": 5.2, "text": "complicated than just a triangle in"}, {"start": 161.879, "duration": 3.241, "text": "another example"}], "title": "Determinants to Find the Area of a Triangle", "description": "Thanks to all of you who support me on Patreon. You da real mvps!   $1 per month helps!! :)  https://www.patreon.com/patrickjmt !! Determinants to Find the Area of a Triangle - In this video, I show how one can use determinants to find the area of a triangle.\nFor more free math videos, visit http://PatrickJMT.com", "lengthSeconds": 163, "width": 480, "height": 360}
{"video_id": "FS6iQX7jY-s", "transcript": [{"start": 0.68, "duration": 3.12, "text": "all right so in this video I'm going to"}, {"start": 1.839, "duration": 4.201, "text": "do a couple more of the trig identities"}, {"start": 3.8, "duration": 5.32, "text": "simplifying I think I'll do uh B and C"}, {"start": 6.04, "duration": 7.759, "text": "in this example so um in B we want to"}, {"start": 9.12, "duration": 7.2, "text": "simplify see x * cosine x - cosine 2 x"}, {"start": 13.799, "duration": 4.081, "text": "well let's see see of X recall secant is"}, {"start": 16.32, "duration": 3.959, "text": "really just one over"}, {"start": 17.88, "duration": 5.84, "text": "cosine and again that's being multiplied"}, {"start": 20.279, "duration": 6.041, "text": "by cosine X and then our minus cosine 2"}, {"start": 23.72, "duration": 4.44, "text": "x we'll just leave that alone well you"}, {"start": 26.32, "duration": 4.6, "text": "can think about cosine X as being over"}, {"start": 28.16, "duration": 6.04, "text": "one so if we multiply we've got cosine"}, {"start": 30.92, "duration": 7.639, "text": "over cosine which is just one and then"}, {"start": 34.2, "duration": 10.28, "text": "we've got our 1 - cosine 2 x but weall"}, {"start": 38.559, "duration": 7.361, "text": "our uh the identity cosine 2 x + sin^2 x"}, {"start": 44.48, "duration": 5.599, "text": "="}, {"start": 45.92, "duration": 7.0, "text": "1 notice if we subtract the cosine squar"}, {"start": 50.079, "duration": 5.441, "text": "we'll get 1us cosine squar and that's"}, {"start": 52.92, "duration": 5.72, "text": "going to be equal to sin"}, {"start": 55.52, "duration": 5.519, "text": "squ so to me that would be a nice uh"}, {"start": 58.64, "duration": 6.76, "text": "simplification here for Part B to say"}, {"start": 61.039, "duration": 7.241, "text": "that that's going to reduce to sin^2"}, {"start": 65.4, "duration": 6.359, "text": "X let's do"}, {"start": 68.28, "duration": 7.24, "text": "uh let's do our next example"}, {"start": 71.759, "duration": 7.481, "text": "um so let's see uh in in example C we've"}, {"start": 75.52, "duration": 3.72, "text": "got cosecant squ minus"}, {"start": 80.159, "duration": 4.441, "text": "one okay now I would kind of think maybe"}, {"start": 82.759, "duration": 4.561, "text": "about the same idea that we did before"}, {"start": 84.6, "duration": 6.199, "text": "so secant squar though um let's see what"}, {"start": 87.32, "duration": 9.32, "text": "is that that's going to be um"}, {"start": 90.799, "duration": 9.841, "text": "1/ cosine 2 x time sin^2"}, {"start": 96.64, "duration": 5.56, "text": "x all right well so cosecant squ X we'll"}, {"start": 100.64, "duration": 4.96, "text": "come back to that we'll think about that"}, {"start": 102.2, "duration": 8.519, "text": "here in a second um this would be sin"}, {"start": 105.6, "duration": 10.28, "text": "squar of X over cosine 2 X or we could"}, {"start": 110.719, "duration": 7.641, "text": "write that as Sin of X over cosine X all"}, {"start": 115.88, "duration": 4.919, "text": "squared um I'm going to do two things at"}, {"start": 118.36, "duration": 7.56, "text": "once here uh again call one of our"}, {"start": 120.799, "duration": 12.761, "text": "identities um we have that cent"}, {"start": 125.92, "duration": 9.48, "text": "squar of x + 1 that equals cose 2 x what"}, {"start": 133.56, "duration": 8.44, "text": "I'm going to do is I can simply subtract"}, {"start": 135.4, "duration": 9.44, "text": "one and say c^ 2 X = cose 2 X - 1 so I'm"}, {"start": 142.0, "duration": 4.72, "text": "going to plug that in um for the first"}, {"start": 144.84, "duration": 7.0, "text": "uh the first Factor I'll replace"}, {"start": 146.72, "duration": 5.12, "text": "cosecant 2 X - 1 with cotangent"}, {"start": 155.2, "duration": 4.28, "text": "okay well recall again that s over"}, {"start": 157.48, "duration": 4.399, "text": "cosine that's just"}, {"start": 159.48, "duration": 4.72, "text": "tangent but again it's all being squared"}, {"start": 161.879, "duration": 5.801, "text": "so we have tangent s"}, {"start": 164.2, "duration": 6.72, "text": "x well now we're almost there um recall"}, {"start": 167.68, "duration": 5.4, "text": "that uh cent of X that's really one over"}, {"start": 170.92, "duration": 3.959, "text": "tangent of X but again it's being"}, {"start": 173.08, "duration": 4.519, "text": "squared so we would have to square this"}, {"start": 174.879, "duration": 5.561, "text": "as well so cotangent squar is the same"}, {"start": 177.599, "duration": 5.681, "text": "thing as 1 over tangent squar but again"}, {"start": 180.44, "duration": 6.079, "text": "that's being multiplied by tangent s of"}, {"start": 183.28, "duration": 6.48, "text": "X so that'll just give us tangent"}, {"start": 186.519, "duration": 5.321, "text": "SAR over tangent squared which is"}, {"start": 189.76, "duration": 7.24, "text": "finally going to reduce simply all down"}, {"start": 191.84, "duration": 7.28, "text": "to just a one so um part C it says"}, {"start": 197.0, "duration": 6.519, "text": "that's really a complicated way of"}, {"start": 199.12, "duration": 4.399, "text": "writing the number one"}], "title": "Simplifying Trigonometric Expressions Using Identities, Example 2", "description": "Thanks to all of you who support me on Patreon. You da real mvps!   $1 per month helps!! :)  https://www.patreon.com/patrickjmt !! Simplifying Trigonometric Expressions Using Identities, Example 2. In this video, I simplify a trigonometry expression using some trig identities.", "lengthSeconds": 202, "width": 480, "height": 360}
{"video_id": "ZvZ_dLlDS0U", "transcript": [{"start": 0.599, "duration": 3.521, "text": "all right in this video I'm going to do"}, {"start": 2.12, "duration": 4.96, "text": "one more example of dividing complex"}, {"start": 4.12, "duration": 7.36, "text": "numbers and in this case we have 4 - I"}, {"start": 7.08, "duration": 6.639, "text": "over -2 - 7 I so again the basic idea is"}, {"start": 11.48, "duration": 5.0, "text": "I just look at the denominator multiply"}, {"start": 13.719, "duration": 5.041, "text": "by the complex conjugate and again the"}, {"start": 16.48, "duration": 4.6, "text": "only sign that changes is the sign in"}, {"start": 18.76, "duration": 6.04, "text": "the middle so you're still going to have"}, {"start": 21.08, "duration": 6.68, "text": "-2 that'll become a positive 7 I and if"}, {"start": 24.8, "duration": 4.84, "text": "we multiply it to the uh denominator"}, {"start": 27.76, "duration": 4.04, "text": "we've got to do it to the numerator"}, {"start": 29.64, "duration": 3.68, "text": "again I put everything in parentheses"}, {"start": 31.8, "duration": 4.919, "text": "just to remind myself I have to"}, {"start": 33.32, "duration": 3.399, "text": "distribute everything out"}, {"start": 36.96, "duration": 6.36, "text": "here all right so let's see if we can't"}, {"start": 39.879, "duration": 6.52, "text": "uh clean this up nicely so in the"}, {"start": 43.32, "duration": 7.879, "text": "numerator we have pos4 * -2 that'll give"}, {"start": 46.399, "duration": 8.96, "text": "us a 8 we have pos4 * POS 7 I that's"}, {"start": 51.199, "duration": 7.241, "text": "going to give us a positive 28 I we have"}, {"start": 55.359, "duration": 5.36, "text": "netive I * -2 so a negative and a"}, {"start": 58.44, "duration": 5.2, "text": "negative will make a positive we'll get"}, {"start": 60.719, "duration": 4.881, "text": "2 I and then we have a negative * a"}, {"start": 63.64, "duration": 4.08, "text": "positive which is a negative you can"}, {"start": 65.6, "duration": 7.519, "text": "think about there being a 1 * a 7 which"}, {"start": 67.72, "duration": 7.52, "text": "is 7 then we have I * I which is I 2 and"}, {"start": 73.119, "duration": 5.921, "text": "in the denominator okay let me get rid"}, {"start": 75.24, "duration": 8.32, "text": "of my little arrow here we have -2 * -2"}, {"start": 79.04, "duration": 9.96, "text": "which is pos4 we have -2 * POS 7 I which"}, {"start": 83.56, "duration": 9.44, "text": "will give us a -4 I -7 I * -2 will give"}, {"start": 89.0, "duration": 8.0, "text": "us a positive 14 I and then we have -7 I"}, {"start": 93.0, "duration": 6.479, "text": "* POS 7 I that'll give us a"}, {"start": 97.0, "duration": 5.04, "text": "-49 i"}, {"start": 99.479, "duration": 3.561, "text": "2 okay so in the numerator it looks like"}, {"start": 102.04, "duration": 4.719, "text": "we"}, {"start": 103.04, "duration": 9.719, "text": "have8 28 I and 2 I is going to give us a"}, {"start": 106.759, "duration": 10.28, "text": "POS 30 i - 7 i^ 2 is the same thing as"}, {"start": 112.759, "duration": 6.32, "text": "-1 and again the the imaginary all the"}, {"start": 117.039, "duration": 5.641, "text": "imaginary Parts in the denominator are"}, {"start": 119.079, "duration": 5.841, "text": "going to cancel out the4 I is gone um"}, {"start": 122.68, "duration": 7.359, "text": "the positive 14 I will cancel each other"}, {"start": 124.92, "duration": 9.08, "text": "out and then it looks like we have 4 -"}, {"start": 130.039, "duration": 7.48, "text": "49 again i^ 2 is"}, {"start": 134.0, "duration": 6.879, "text": "-1 so in the numerator we have8 we have"}, {"start": 137.519, "duration": 7.44, "text": "a -7 * a -1 those will just turn into"}, {"start": 140.879, "duration": 6.36, "text": "positives when we multiply them so8 + 7"}, {"start": 144.959, "duration": 4.241, "text": "is going to leave us with negative 1"}, {"start": 147.239, "duration": 4.08, "text": "again we still have the 30i just kind of"}, {"start": 149.2, "duration": 5.6, "text": "hanging out"}, {"start": 151.319, "duration": 5.841, "text": "divided by it looks like we have four uh"}, {"start": 154.8, "duration": 5.799, "text": "we have a - 49 * a negative 1 which is"}, {"start": 157.16, "duration": 4.719, "text": "going to again make a positive 49 and 4"}, {"start": 160.599, "duration": 4.601, "text": "is"}, {"start": 161.879, "duration": 7.881, "text": "53 again I'm just going to break this up"}, {"start": 165.2, "duration": 7.44, "text": "-1 over 53 + 30 over"}, {"start": 169.76, "duration": 6.28, "text": "53 I and again you could think you know"}, {"start": 172.64, "duration": 6.72, "text": "does this reduce it all but 53 um I do"}, {"start": 176.04, "duration": 5.199, "text": "believe is a prime number um so there's"}, {"start": 179.36, "duration": 4.12, "text": "really you know even if the top factors"}, {"start": 181.239, "duration": 4.961, "text": "and it certainly does 53 doesn't Factor"}, {"start": 183.48, "duration": 4.039, "text": "so there's nothing to cancel out so um"}, {"start": 186.2, "duration": 2.08, "text": "again at this point we're finished we"}, {"start": 187.519, "duration": 3.841, "text": "have"}, {"start": 188.28, "duration": 7.0, "text": "now divided our two complex numbers we"}, {"start": 191.36, "duration": 7.12, "text": "end up with a new complex number um and"}, {"start": 195.28, "duration": 3.2, "text": "that's it"}], "title": "Complex Numbers: Dividing - Ex 3", "description": "Thanks to all of you who support me on Patreon. You da real mvps!   $1 per month helps!! :)  https://www.patreon.com/patrickjmt !! Complex Numbers: Dividing - Ex 3.  In this video, I show how to divide a complex number by another complex number.", "lengthSeconds": 197, "width": 480, "height": 360}
{"video_id": "JDVRQWA5NZk", "transcript": [{"start": 0.56, "duration": 3.68, "text": "all right in this example I'm going to"}, {"start": 2.28, "duration": 3.68, "text": "do another linear equation and here"}, {"start": 4.24, "duration": 5.76, "text": "we'll solve our our second linear"}, {"start": 5.96, "duration": 6.04, "text": "equation 5 - x over3 = 4 * the quantity"}, {"start": 10.0, "duration": 5.4, "text": "x + 1 over"}, {"start": 12.0, "duration": 5.24, "text": "6 okay definitely lots of different"}, {"start": 15.4, "duration": 3.52, "text": "things that you could do here in terms"}, {"start": 17.24, "duration": 4.64, "text": "of uh getting"}, {"start": 18.92, "duration": 4.96, "text": "started what I'm going to do is I'm"}, {"start": 21.88, "duration": 5.88, "text": "going to uh do a couple things I'm going"}, {"start": 23.88, "duration": 8.199, "text": "to write five as 5 over 1 I've got uh"}, {"start": 27.76, "duration": 6.6, "text": "our X over 3 there here now notice 4"}, {"start": 32.079, "duration": 5.64, "text": "over 6 we could actually reduce that"}, {"start": 34.36, "duration": 7.039, "text": "that would reduce to"}, {"start": 37.719, "duration": 5.641, "text": "23 and then we still have our x + one"}, {"start": 41.399, "duration": 4.201, "text": "now we could start Distributing things"}, {"start": 43.36, "duration": 3.76, "text": "out but eventually I'm going to have"}, {"start": 45.6, "duration": 3.16, "text": "fractions floating around which isn't"}, {"start": 47.12, "duration": 3.68, "text": "the end of the world I mean you've got"}, {"start": 48.76, "duration": 3.92, "text": "to learn to work with fractions but"}, {"start": 50.8, "duration": 4.72, "text": "there's definitely cases where you can"}, {"start": 52.68, "duration": 5.08, "text": "avoid them so what I do is I look at the"}, {"start": 55.52, "duration": 4.519, "text": "denominators we've got a one a three and"}, {"start": 57.76, "duration": 4.84, "text": "a three and I'm thinking what's the"}, {"start": 60.039, "duration": 4.641, "text": "least common multiple of those three"}, {"start": 62.6, "duration": 4.28, "text": "numbers and the first number that they"}, {"start": 64.68, "duration": 4.759, "text": "all would divide into evenly would"}, {"start": 66.88, "duration": 6.36, "text": "simply be three so what I'm going to do"}, {"start": 69.439, "duration": 7.32, "text": "is I'm going to multiply both"}, {"start": 73.24, "duration": 3.519, "text": "sides of my"}, {"start": 78.0, "duration": 3.92, "text": "equation by the number"}, {"start": 82.84, "duration": 4.2, "text": "three and again what this does is it's"}, {"start": 85.4, "duration": 4.56, "text": "simply going to get rid of the fractions"}, {"start": 87.04, "duration": 4.84, "text": "for me and then um to me it makes the"}, {"start": 89.96, "duration": 4.4, "text": "arithmetic a little bit easier keeps me"}, {"start": 91.88, "duration": 6.8, "text": "from making mistakes so we have to"}, {"start": 94.36, "duration": 7.119, "text": "distribute 3 * 5 is 15 if we take uh the"}, {"start": 98.68, "duration": 4.759, "text": "three we'll get -3x over 3 but then the"}, {"start": 101.479, "duration": 3.0, "text": "3's just cancel so we're just left with"}, {"start": 103.439, "duration": 3.28, "text": "our Negative"}, {"start": 104.479, "duration": 4.28, "text": "X again you can think about this as"}, {"start": 106.719, "duration": 4.0, "text": "being 3 over one so I'm just going to"}, {"start": 108.759, "duration": 5.121, "text": "cancel the three and the three would"}, {"start": 110.719, "duration": 5.801, "text": "cancel we would be left with 2 * x + 1"}, {"start": 113.88, "duration": 5.64, "text": "over one which we can just write as well"}, {"start": 116.52, "duration": 3.879, "text": "2 * x + 1 and now to me we're in pretty"}, {"start": 119.52, "duration": 2.0, "text": "good shape"}, {"start": 120.399, "duration": 3.481, "text": "um I'm going to get rid of the"}, {"start": 121.52, "duration": 4.4, "text": "parentheses on the right side"}, {"start": 123.88, "duration": 5.599, "text": "Distributing we'll get"}, {"start": 125.92, "duration": 6.24, "text": "2x + 2 same thing as before I'm going to"}, {"start": 129.479, "duration": 5.441, "text": "put my x's on the same side I think here"}, {"start": 132.16, "duration": 6.04, "text": "I'm going to add X to both sides I like"}, {"start": 134.92, "duration": 7.319, "text": "to keep my x's positive if I can don't"}, {"start": 138.2, "duration": 7.44, "text": "have to um so we'll have 15 on the left"}, {"start": 142.239, "duration": 6.64, "text": "2x + 1X is going to give us"}, {"start": 145.64, "duration": 5.599, "text": "3x we still have our plus two um on the"}, {"start": 148.879, "duration": 4.64, "text": "right side and again to get rid of the"}, {"start": 151.239, "duration": 5.921, "text": "positive two I'm going to subtract two"}, {"start": 153.519, "duration": 7.44, "text": "from both sides so 15 minus 2 is going"}, {"start": 157.16, "duration": 6.84, "text": "to be 13 = 3x again since we're"}, {"start": 160.959, "duration": 6.041, "text": "multiplying by three we'll divide both"}, {"start": 164.0, "duration": 6.64, "text": "sides by three um they cancel on the"}, {"start": 167.0, "duration": 8.68, "text": "right and we're just left with X ="}, {"start": 170.64, "duration": 5.04, "text": "13/3 as our solution"}], "title": "Solving a Basic Linear Equation - Example 2", "description": "Thanks to all of you who support me on Patreon. You da real mvps!   $1 per month helps!! :)  https://www.patreon.com/patrickjmt !! Solving a Basic Linear Equation - Example 2", "lengthSeconds": 175, "width": 480, "height": 360}
{"video_id": "UJgUkLqxiFw", "transcript": [{"start": 1.04, "duration": 3.239, "text": "in this video I want to do two last"}, {"start": 2.639, "duration": 4.961, "text": "examples of factoring the difference of"}, {"start": 4.279, "duration": 5.521, "text": "two squares and the two examples I'm"}, {"start": 7.6, "duration": 3.8, "text": "going to do are here at the bottom and"}, {"start": 9.8, "duration": 3.68, "text": "you could certainly do these problems in"}, {"start": 11.4, "duration": 4.6, "text": "a slightly different way uh we could"}, {"start": 13.48, "duration": 4.559, "text": "always just multiply this out x - 3 * x"}, {"start": 16.0, "duration": 3.56, "text": "- 3 collect your like terms subtract"}, {"start": 18.039, "duration": 3.921, "text": "four and then try to factor it from"}, {"start": 19.56, "duration": 5.479, "text": "there and the same thing on the other"}, {"start": 21.96, "duration": 4.04, "text": "but again really just to emphasize this"}, {"start": 25.039, "duration": 5.041, "text": "this"}, {"start": 26.0, "duration": 5.96, "text": "formula I could again write four is 2^ 2"}, {"start": 30.08, "duration": 3.84, "text": "well again to me I see two terms and I"}, {"start": 31.96, "duration": 4.72, "text": "see a negative in between them so it"}, {"start": 33.92, "duration": 6.04, "text": "says I can Factor them so it says our a"}, {"start": 36.68, "duration": 6.359, "text": "is going to be the x - 3 and then we"}, {"start": 39.96, "duration": 7.32, "text": "have our b as being the two so it says"}, {"start": 43.039, "duration": 4.241, "text": "this is simply going to factor now"}, {"start": 49.399, "duration": 7.361, "text": "as so there's our a minus our B and then"}, {"start": 54.52, "duration": 5.76, "text": "there's our a"}, {"start": 56.76, "duration": 5.68, "text": "term plus our B"}, {"start": 60.28, "duration": 4.599, "text": "and if we simplify this down if we"}, {"start": 62.44, "duration": 5.039, "text": "collect like terms we'll simply get x -"}, {"start": 64.879, "duration": 5.001, "text": "5 in the first set of parentheses and"}, {"start": 67.479, "duration": 4.521, "text": "then it looks like we'll get x uh what"}, {"start": 69.88, "duration": 4.72, "text": "is that going to be a Nega 1 in the"}, {"start": 72.0, "duration": 6.64, "text": "second set of parentheses okay and now"}, {"start": 74.6, "duration": 7.519, "text": "we have it factored so again um maybe be"}, {"start": 78.64, "duration": 5.68, "text": "a little careful you know if uh if we"}, {"start": 82.119, "duration": 4.761, "text": "had something more complicated as our B"}, {"start": 84.32, "duration": 4.28, "text": "value this negative you might you might"}, {"start": 86.88, "duration": 3.12, "text": "want to stick your B in parentheses"}, {"start": 88.6, "duration": 3.519, "text": "because the negative would have to to"}, {"start": 90.0, "duration": 3.88, "text": "get distributed out I'm not putting the"}, {"start": 92.119, "duration": 3.401, "text": "two in parenthesis just because I'm"}, {"start": 93.88, "duration": 4.239, "text": "recognizing there's nothing to"}, {"start": 95.52, "duration": 5.08, "text": "distribute with that negative so one"}, {"start": 98.119, "duration": 4.601, "text": "little place to be"}, {"start": 100.6, "duration": 4.64, "text": "careful okay so we can do the same thing"}, {"start": 102.72, "duration": 4.32, "text": "here in our other example we have x + 7^"}, {"start": 105.24, "duration": 4.239, "text": "2us"}, {"start": 107.04, "duration": 4.439, "text": "25 okay so in the first set of"}, {"start": 109.479, "duration": 5.68, "text": "parentheses I'm going to have x + 7"}, {"start": 111.479, "duration": 5.64, "text": "there's my a term and then um minus our"}, {"start": 115.159, "duration": 5.041, "text": "B term so if you take the square < TK of"}, {"start": 117.119, "duration": 5.0, "text": "25 we just get five and then I just do"}, {"start": 120.2, "duration": 6.88, "text": "the same thing except I change this"}, {"start": 122.119, "duration": 6.64, "text": "negative into a positive so x +"}, {"start": 127.08, "duration": 5.64, "text": "7+"}, {"start": 128.759, "duration": 9.241, "text": "five okay so there's my a term minus our"}, {"start": 132.72, "duration": 7.96, "text": "B term and then our a term plus our B"}, {"start": 138.0, "duration": 5.08, "text": "term and again we can simplify this down"}, {"start": 140.68, "duration": 7.08, "text": "a little bit so we have in the first set"}, {"start": 143.08, "duration": 7.36, "text": "of parentheses x then we have POS 7 - 5"}, {"start": 147.76, "duration": 5.24, "text": "that'll leave us with a positive2 and in"}, {"start": 150.44, "duration": 4.92, "text": "the second set of parentheses we'll have"}, {"start": 153.0, "duration": 4.4, "text": "positive 7 and positive 5 which will"}, {"start": 155.36, "duration": 5.0, "text": "give us uh x +"}, {"start": 157.4, "duration": 5.08, "text": "12 and again now we have it"}, {"start": 160.36, "duration": 4.12, "text": "factored so and you can even try you"}, {"start": 162.48, "duration": 4.8, "text": "know multiply this out collect your like"}, {"start": 164.48, "duration": 4.759, "text": "terms and try to you know try to factor"}, {"start": 167.28, "duration": 6.039, "text": "it uh and see if you don't get the same"}, {"start": 169.239, "duration": 6.121, "text": "thing in both cases so again all roads"}, {"start": 173.319, "duration": 3.761, "text": "lead to Ram different ways to do it but"}, {"start": 175.36, "duration": 3.84, "text": "in this case just trying to again really"}, {"start": 177.08, "duration": 4.76, "text": "emphasize this difference of squares"}, {"start": 179.2, "duration": 2.64, "text": "formula"}], "title": "Factoring the Difference of Two Squares - Ex 3", "description": "Thanks to all of you who support me on Patreon. You da real mvps!   $1 per month helps!! :)  https://www.patreon.com/patrickjmt !! Factoring the Difference of Two Squares - Ex 3", "lengthSeconds": 181, "width": 480, "height": 360}
{"video_id": "4H57ndz-FRA", "transcript": [{"start": 0.599, "duration": 4.361, "text": "all right so here we're going to look at"}, {"start": 1.8, "duration": 4.36, "text": "some examples of the ratio test so part"}, {"start": 4.96, "duration": 3.8, "text": "A we're just going to figure out whether"}, {"start": 6.16, "duration": 7.0, "text": "the series converges or diverges n = 1"}, {"start": 8.76, "duration": 6.8, "text": "to Infinity 4 n * n factorial 2 over n +"}, {"start": 13.16, "duration": 4.44, "text": "2"}, {"start": 15.56, "duration": 3.92, "text": "factorial so I think the thing that'll"}, {"start": 17.6, "duration": 4.519, "text": "probably make this one most confusing is"}, {"start": 19.48, "duration": 4.559, "text": "just the factorials so we'll just have"}, {"start": 22.119, "duration": 4.801, "text": "to be a little careful when we simplify"}, {"start": 24.039, "duration": 6.281, "text": "those so the limit as n goes to Infinity"}, {"start": 26.92, "duration": 5.84, "text": "we would have 4 the n + one um we would"}, {"start": 30.32, "duration": 6.0, "text": "have n +1"}, {"start": 32.76, "duration": 6.0, "text": "factorial squared we would have we would"}, {"start": 36.32, "duration": 6.68, "text": "replace the N with n + one so we would"}, {"start": 38.76, "duration": 6.319, "text": "have n + 1 + 2 or n + 3"}, {"start": 43.0, "duration": 6.079, "text": "factorial we can multiply by the"}, {"start": 45.079, "duration": 4.0, "text": "reciprocal n + 2"}, {"start": 49.32, "duration": 7.84, "text": "factorial over 4 n * n factorial"}, {"start": 54.559, "duration": 5.52, "text": "2 so all right so now just kind of the"}, {"start": 57.16, "duration": 5.52, "text": "simplification part um I'm just going to"}, {"start": 60.079, "duration": 3.921, "text": "sort of reline everything up just to you"}, {"start": 62.68, "duration": 3.08, "text": "know you certainly don't have to do this"}, {"start": 64.0, "duration": 4.28, "text": "but to maybe make things a little"}, {"start": 65.76, "duration": 4.92, "text": "clearer we have 4 the n + 1 I'm going to"}, {"start": 68.28, "duration": 5.479, "text": "put the four to the n in the denominator"}, {"start": 70.68, "duration": 6.16, "text": "we have an N +1 factorial squared so"}, {"start": 73.759, "duration": 5.641, "text": "that means we have n +1 factorial times"}, {"start": 76.84, "duration": 5.239, "text": "another n +1"}, {"start": 79.4, "duration": 4.32, "text": "factorial we've got the N factorial"}, {"start": 82.079, "duration": 4.881, "text": "squared so I'm going to put an N"}, {"start": 83.72, "duration": 6.88, "text": "factorial here and another n factorial"}, {"start": 86.96, "duration": 7.04, "text": "there and then our n + 2 Factor"}, {"start": 90.6, "duration": 6.32, "text": "factorial over n +"}, {"start": 94.0, "duration": 7.0, "text": "3 factorial how often do you get to say"}, {"start": 96.92, "duration": 8.04, "text": "factorial um all right so I think the"}, {"start": 101.0, "duration": 6.24, "text": "first part's easy um we just have 4 the"}, {"start": 104.96, "duration": 5.24, "text": "n + 1 over 4 the N that'll just be a 4"}, {"start": 107.24, "duration": 6.239, "text": "to the 1st in the numerator n + 1"}, {"start": 110.2, "duration": 3.279, "text": "factorial Over N"}, {"start": 114.28, "duration": 7.72, "text": "factorial again if you imagine expanding"}, {"start": 116.68, "duration": 9.84, "text": "this out you would have n + 1 * n * n -1"}, {"start": 122.0, "duration": 9.239, "text": "* N - 2 dot dot dot in the denominator n"}, {"start": 126.52, "duration": 7.2, "text": "factorial would be n * nus1 * nus 2 time"}, {"start": 131.239, "duration": 5.521, "text": "dot dot dot so everything would cancel"}, {"start": 133.72, "duration": 6.32, "text": "out except for an n + one so we would"}, {"start": 136.76, "duration": 5.0, "text": "have an n+ one in the numerator well"}, {"start": 140.04, "duration": 5.24, "text": "same thing is going on with our third"}, {"start": 141.76, "duration": 6.72, "text": "Factor so we would have n + one over 1"}, {"start": 145.28, "duration": 7.08, "text": "and then n + 3 over n + 2 or excuse me n"}, {"start": 148.48, "duration": 3.88, "text": "plus 2 factorial over n+ 3"}, {"start": 154.64, "duration": 5.36, "text": "factorial well again if you imagine"}, {"start": 156.64, "duration": 7.84, "text": "expanding this out we would have n + 2"}, {"start": 160.0, "duration": 7.76, "text": "one smaller would be n + 1 then time n *"}, {"start": 164.48, "duration": 6.24, "text": "n minus one dot dot dot in the"}, {"start": 167.76, "duration": 4.88, "text": "denominator we would start with n + 3"}, {"start": 170.72, "duration": 7.08, "text": "then we would subtract one from that and"}, {"start": 172.64, "duration": 7.48, "text": "have n + 2 n + one dot dot dot notice"}, {"start": 177.8, "duration": 6.12, "text": "everything's going to cancel out except"}, {"start": 180.12, "duration": 5.88, "text": "for the n + 3 in the"}, {"start": 183.92, "duration": 5.039, "text": "denominator okay so now we're taking the"}, {"start": 186.0, "duration": 4.959, "text": "limit of this stuff but we've got the"}, {"start": 188.959, "duration": 5.041, "text": "limit as n goes to"}, {"start": 190.959, "duration": 7.881, "text": "Infinity notice in if we multiply n+ one"}, {"start": 194.0, "duration": 8.0, "text": "and n + one we would get n 2 + 2 n + 1"}, {"start": 198.84, "duration": 4.88, "text": "over n + 3 well this is a case where"}, {"start": 202.0, "duration": 4.2, "text": "we're taking the limit as n goes to"}, {"start": 203.72, "duration": 4.0, "text": "Infinity the degree of the numerator is"}, {"start": 206.2, "duration": 4.08, "text": "larger than the degree of the"}, {"start": 207.72, "duration": 4.96, "text": "denominator so this is just going to go"}, {"start": 210.28, "duration": 4.679, "text": "off to positive Infinity well that's"}, {"start": 212.68, "duration": 6.08, "text": "certainly greater than one and when it's"}, {"start": 214.959, "duration": 8.56, "text": "greater than one the ratio test tells us"}, {"start": 218.76, "duration": 4.759, "text": "that the um original series will"}, {"start": 225.48, "duration": 3.0, "text": "diverge"}], "title": "The Ratio Test , Another Example #3", "description": "Thanks to all of you who support me on Patreon. You da real mvps!   $1 per month helps!! :)  https://www.patreon.com/patrickjmt !! The Ratio Test , Another Example #3.  Just another example showing that series converges or diverges using the ratio test.", "lengthSeconds": 227, "width": 480, "height": 360}
{"video_id": "M9RcX5LMC6o", "transcript": [{"start": 0.48, "duration": 2.8, "text": "all right in this video we're going to"}, {"start": 1.68, "duration": 3.52, "text": "look at finding some arc lengths of"}, {"start": 3.28, "duration": 4.079, "text": "parametric curves"}, {"start": 5.2, "duration": 4.88, "text": "so in part a we've got the absolute"}, {"start": 7.359, "duration": 5.36, "text": "value of 6 minus t"}, {"start": 10.08, "duration": 4.88, "text": "equals x and then y equals t we're going"}, {"start": 12.719, "duration": 5.761, "text": "to evaluate this from"}, {"start": 14.96, "duration": 5.28, "text": "0 to three so we'll have to integrate"}, {"start": 18.48, "duration": 3.6, "text": "from zero to three"}, {"start": 20.24, "duration": 4.48, "text": "we'll take the derivative of x with"}, {"start": 22.08, "duration": 5.199, "text": "respect to t squared"}, {"start": 24.72, "duration": 4.319, "text": "plus the derivative of y with respect to"}, {"start": 27.279, "duration": 5.361, "text": "t squared"}, {"start": 29.039, "duration": 6.001, "text": "dt so notice"}, {"start": 32.64, "duration": 4.72, "text": "for x equals the absolute value of 6"}, {"start": 35.04, "duration": 3.76, "text": "minus t"}, {"start": 37.36, "duration": 6.8, "text": "we can actually just say this is the"}, {"start": 38.8, "duration": 8.96, "text": "same thing as 6 minus t"}, {"start": 44.16, "duration": 6.64, "text": "for t on the interval 0 to 3 because uh"}, {"start": 47.76, "duration": 5.68, "text": "if you take any value between 0 and 3"}, {"start": 50.8, "duration": 5.2, "text": "the 6 minus t will stay positive so we"}, {"start": 53.44, "duration": 4.24, "text": "can just remove the absolute value"}, {"start": 56.0, "duration": 3.28, "text": "well that says the derivative of x with"}, {"start": 57.68, "duration": 3.12, "text": "respect to t then is just going to be"}, {"start": 59.28, "duration": 3.439, "text": "negative 1."}, {"start": 60.8, "duration": 4.079, "text": "and that tells us the derivative of x"}, {"start": 62.719, "duration": 4.321, "text": "with respect to t"}, {"start": 64.879, "duration": 3.441, "text": "squared is just going to equal positive"}, {"start": 67.04, "duration": 3.68, "text": "1."}, {"start": 68.32, "duration": 3.92, "text": "the derivative of y with respect to t"}, {"start": 70.72, "duration": 4.64, "text": "that's also just going to be"}, {"start": 72.24, "duration": 4.08, "text": "positive one so the derivative of y with"}, {"start": 75.36, "duration": 4.079, "text": "respect to t"}, {"start": 76.32, "duration": 5.04, "text": "squared will also just be positive one"}, {"start": 79.439, "duration": 3.68, "text": "easy enough so we've got the integral"}, {"start": 81.36, "duration": 6.24, "text": "from zero to three"}, {"start": 83.119, "duration": 7.201, "text": "we just have one plus one dt"}, {"start": 87.6, "duration": 4.08, "text": "well that's the integral from 0 to 3 of"}, {"start": 90.32, "duration": 4.72, "text": "just the square root of 2"}, {"start": 91.68, "duration": 4.64, "text": "times or square root of 2 dt"}, {"start": 95.04, "duration": 3.439, "text": "so when we integrate we'll get the"}, {"start": 96.32, "duration": 4.56, "text": "square root of 2 times t we'll have to"}, {"start": 98.479, "duration": 3.68, "text": "evaluate that from zero to three"}, {"start": 100.88, "duration": 3.84, "text": "and that's just going to leave us with"}, {"start": 102.159, "duration": 4.721, "text": "three root two minus zero"}, {"start": 104.72, "duration": 6.24, "text": "so our arc length will simply be three"}, {"start": 106.88, "duration": 4.08, "text": "times the square root of two"}], "title": "Arc Length of a Parametric Curve, Another Example #1", "description": "Thanks to all of you who support me on Patreon. You da real mvps!   $1 per month helps!! :)  https://www.patreon.com/patrickjmt !! Arc Length of a Parametric Curve, Another Example #1.  In this video, I find the arc length of a parametric curve.", "lengthSeconds": 109, "width": 480, "height": 360}
{"video_id": "K-ZSeQFeOCg", "transcript": [{"start": 0.64, "duration": 2.799, "text": "all right in this video we're going to"}, {"start": 1.719, "duration": 3.56, "text": "look at finding some limits of vector"}, {"start": 3.439, "duration": 5.921, "text": "functions so in part A we have the"}, {"start": 5.279, "duration": 8.681, "text": "vector function R of tal 4 minus t * I"}, {"start": 9.36, "duration": 8.16, "text": "plus < TK of 12 + T * J minus cosine of"}, {"start": 13.96, "duration": 5.399, "text": "pi T / 8 * K we want to find the limit"}, {"start": 17.52, "duration": 5.32, "text": "as T approaches"}, {"start": 19.359, "duration": 5.961, "text": "4 well all we have to do in this case"}, {"start": 22.84, "duration": 4.96, "text": "kind of like you know with"}, {"start": 25.32, "duration": 5.16, "text": "a you know limits that you see in"}, {"start": 27.8, "duration": 5.12, "text": "calculus one um so we're taking limit as"}, {"start": 30.48, "duration": 7.0, "text": "T approaches 4 all we're going to do is"}, {"start": 32.92, "duration": 4.56, "text": "just plug in um tal"}, {"start": 38.36, "duration": 5.12, "text": "4 and get our solution you know if we"}, {"start": 40.879, "duration": 3.961, "text": "end up with something undefined um you"}, {"start": 43.48, "duration": 2.88, "text": "know maybe we'll have to do something a"}, {"start": 44.84, "duration": 3.64, "text": "little bit"}, {"start": 46.36, "duration": 3.76, "text": "different but in this case you know this"}, {"start": 48.48, "duration": 4.32, "text": "is kind of a very simple one I don't"}, {"start": 50.12, "duration": 5.279, "text": "think we'll run into that issue at all"}, {"start": 52.8, "duration": 6.8, "text": "so notice if we plug in tal"}, {"start": 55.399, "duration": 6.881, "text": "4 well we'll just get 0 * I plus we"}, {"start": 59.6, "duration": 5.8, "text": "would have the < TK of 12 + 4 so we"}, {"start": 62.28, "duration": 6.24, "text": "would get the < TK of 16 * J and then"}, {"start": 65.4, "duration": 6.16, "text": "we'll get minus cosine of I guess we'll"}, {"start": 68.52, "duration": 6.2, "text": "have 4 pi over"}, {"start": 71.56, "duration": 5.8, "text": "8 time"}, {"start": 74.72, "duration": 8.079, "text": "K but if we simplify this this is just"}, {"start": 77.36, "duration": 7.84, "text": "going to be 0 I + 4 * J um let's see"}, {"start": 82.799, "duration": 6.28, "text": "this is going to be cosine of Pi / 2"}, {"start": 85.2, "duration": 5.44, "text": "when we simplify cosine of Pi / 2 is uh"}, {"start": 89.079, "duration": 3.841, "text": "going to give us Z"}, {"start": 90.64, "duration": 5.28, "text": "so 0"}, {"start": 92.92, "duration": 7.32, "text": "K so really uh we could rewrite this as"}, {"start": 95.92, "duration": 7.04, "text": "0 comma 4 comma 0 that will be the limit"}, {"start": 100.24, "duration": 6.519, "text": "of uh the vector valued function as T"}, {"start": 102.96, "duration": 3.799, "text": "approaches pos4"}], "title": "Finding the Limit of a Vector Function - Another Ex 1", "description": "Thanks to all of you who support me on Patreon. You da real mvps!   $1 per month helps!! :)  https://www.patreon.com/patrickjmt !! Finding the Limit of a Vector Function - Another Ex 1. In this video, I find the limit of a vector valued function.", "lengthSeconds": 105, "width": 480, "height": 360}
{"video_id": "me3P5P9rTug", "transcript": [{"start": 0.17, "duration": 5.339, "text": "okay so now we're gonna do another"}, {"start": 2.07, "duration": 6.989, "text": "example using implicit differentiation"}, {"start": 5.509, "duration": 6.821, "text": "so in Part B here we're gonna find DS DT"}, {"start": 9.059, "duration": 6.691, "text": "for the function S to the third minus T"}, {"start": 12.33, "duration": 4.86, "text": "to the third equals eight and sometimes"}, {"start": 15.75, "duration": 2.94, "text": "people get it liking a little confused"}, {"start": 17.19, "duration": 4.53, "text": "when you have different variables"}, {"start": 18.69, "duration": 4.98, "text": "besides x and y so the main idea to"}, {"start": 21.72, "duration": 4.049, "text": "remember you know like in the first"}, {"start": 23.67, "duration": 4.109, "text": "example when we take the derivative with"}, {"start": 25.769, "duration": 4.41, "text": "respect to X that's what's in you know"}, {"start": 27.779, "duration": 4.89, "text": "the bottom part here that's the variable"}, {"start": 30.179, "duration": 4.56, "text": "that we kind of treat like normal so in"}, {"start": 32.669, "duration": 4.05, "text": "this case we're gonna treat the T sort"}, {"start": 34.739, "duration": 6.381, "text": "of like the X in the sense that we'll"}, {"start": 36.719, "duration": 6.781, "text": "treat it like normal so what I mean is"}, {"start": 41.12, "duration": 4.599, "text": "okay so when we take the derivative here"}, {"start": 43.5, "duration": 6.6, "text": "of s to the third minus T to the third"}, {"start": 45.719, "duration": 7.081, "text": "equals eight so s to the third minus T"}, {"start": 50.1, "duration": 4.65, "text": "to the third equals eight again we're"}, {"start": 52.8, "duration": 5.36, "text": "taking the derivative with respect to T"}, {"start": 54.75, "duration": 7.82, "text": "of both sides is what we're doing and"}, {"start": 58.16, "duration": 4.41, "text": "normally people don't write this step"}, {"start": 62.66, "duration": 6.52, "text": "but again the idea is so in this case"}, {"start": 66.33, "duration": 4.89, "text": "when we take the derivative of s to the"}, {"start": 69.18, "duration": 5.13, "text": "third we'll just get well three s"}, {"start": 71.22, "duration": 5.52, "text": "squared but then we have to tack on a DS"}, {"start": 74.31, "duration": 5.64, "text": "over DT and that's what we're trying to"}, {"start": 76.74, "duration": 5.22, "text": "solve for in this problem and then we"}, {"start": 79.95, "duration": 5.669, "text": "would have - ok so the derivative of T"}, {"start": 81.96, "duration": 5.01, "text": "cubed would be 3t squared and in this"}, {"start": 85.619, "duration": 3.691, "text": "case you could think about you're really"}, {"start": 86.97, "duration": 5.039, "text": "tacking on like a DT over DT which is"}, {"start": 89.31, "duration": 5.55, "text": "just 1 so we don't write it so again the"}, {"start": 92.009, "duration": 4.771, "text": "idea is whatever is on the bottom that"}, {"start": 94.86, "duration": 4.59, "text": "variable we just kind of do it like"}, {"start": 96.78, "duration": 7.32, "text": "normal we don't have to tack on any you"}, {"start": 99.45, "duration": 7.379, "text": "know DS DT or dy DT or any other extra"}, {"start": 104.1, "duration": 6.18, "text": "little bit but for the S variable in"}, {"start": 106.829, "duration": 8.93, "text": "this case we do and so the derivative"}, {"start": 110.28, "duration": 5.479, "text": "with respective T of 8 is just 0"}, {"start": 116.18, "duration": 6.12, "text": "and now just the same thing as before we"}, {"start": 118.58, "duration": 6.66, "text": "have to solve for D s DT well we can add"}, {"start": 122.3, "duration": 6.33, "text": "3t squared to both sides that will leave"}, {"start": 125.24, "duration": 5.82, "text": "us with three s squared times DS over DT"}, {"start": 128.63, "duration": 6.57, "text": "and now all we have to do is just divide"}, {"start": 131.06, "duration": 6.54, "text": "both sides by 3s squared so 3s squared"}, {"start": 135.2, "duration": 6.06, "text": "and then we'll be left with our solution"}, {"start": 137.6, "duration": 6.39, "text": "DS DT the 3s will cancel so we can just"}, {"start": 141.26, "duration": 5.3, "text": "simply write that as T squared over s"}, {"start": 143.99, "duration": 2.57, "text": "squared"}], "title": "Implicit Differentiation - Basic Example 2 / 3", "description": "Thanks to all of you who support me on Patreon. You da real mvps!   $1 per month helps!! :)  https://www.patreon.com/patrickjmt !! Implicit Differentiation - Basic Example 2 / 3 .  Just a fairly straight forward example of finding a derivative using implicit differentiation.", "lengthSeconds": 146, "width": 480, "height": 360}
{"video_id": "Iptxk7SlvEM", "transcript": [{"start": 0.719, "duration": 3.961, "text": "all right so one more example here"}, {"start": 2.24, "duration": 3.44, "text": "regarding inconsistent systems and again"}, {"start": 4.68, "duration": 2.639, "text": "what we're trying to do is just"}, {"start": 5.68, "duration": 3.68, "text": "determine if a system of linear"}, {"start": 7.319, "duration": 5.001, "text": "equations and three variables has no"}, {"start": 9.36, "duration": 4.479, "text": "Solutions or if it does have a solution"}, {"start": 12.32, "duration": 4.039, "text": "and again we're just using elimination"}, {"start": 13.839, "duration": 7.921, "text": "by addition so we've got this system of"}, {"start": 16.359, "duration": 8.961, "text": "equations x - 4 y + 2 Z = -2 x + 2 y - 2"}, {"start": 21.76, "duration": 7.2, "text": "Z = -3 and then our last equation just x"}, {"start": 25.32, "duration": 5.64, "text": "- y = 4 so you could write in plus z z"}, {"start": 28.96, "duration": 4.72, "text": "if you if you want uh you know if you"}, {"start": 30.96, "duration": 5.08, "text": "actually want to see that"}, {"start": 33.68, "duration": 3.92, "text": "term so I think what I'm going to do"}, {"start": 36.04, "duration": 3.999, "text": "really again though if you think about"}, {"start": 37.6, "duration": 4.2, "text": "this last equation it it's already uh"}, {"start": 40.039, "duration": 4.441, "text": "down to two variables in the sense that"}, {"start": 41.8, "duration": 4.88, "text": "it's just X and Y so what I'm going to"}, {"start": 44.48, "duration": 4.84, "text": "do is I'm going to take my uh my first"}, {"start": 46.68, "duration": 5.12, "text": "two equations I think and I'm going to"}, {"start": 49.32, "duration": 7.96, "text": "again just eliminate the Z from those"}, {"start": 51.8, "duration": 8.68, "text": "two so let's see so we've got x - 4 y +"}, {"start": 57.28, "duration": 10.12, "text": "2 Z = -2"}, {"start": 60.48, "duration": 11.16, "text": "we've got x + 2 y - 2 Z ="}, {"start": 67.4, "duration": 7.2, "text": "-3 and again if we add these up"}, {"start": 71.64, "duration": 9.76, "text": "immediately we'll get 1 x + 1 x which is"}, {"start": 74.6, "duration": 11.0, "text": "2x we've got -4 y + 2 y which is 2 Y 2 Z"}, {"start": 81.4, "duration": 6.64, "text": "uh plus -2 Z will be 0 Z and then we've"}, {"start": 85.6, "duration": 4.64, "text": "got -2 + -3 which"}, {"start": 88.04, "duration": 3.64, "text": "is5 so now what I'm going to do do is"}, {"start": 90.24, "duration": 4.199, "text": "I'm going to use this equation that"}, {"start": 91.68, "duration": 8.68, "text": "we've just found so"}, {"start": 94.439, "duration": 8.72, "text": "2x - 2 y = -5 and then I'm going to use"}, {"start": 100.36, "duration": 8.92, "text": "my third equation that we had originally"}, {"start": 103.159, "duration": 7.521, "text": "which said uh x - y = 4 and I'm going to"}, {"start": 109.28, "duration": 4.08, "text": "do the same thing I'm going to try to"}, {"start": 110.68, "duration": 5.359, "text": "eliminate one of my variables uh it"}, {"start": 113.36, "duration": 4.24, "text": "looks like to me we could multiply this"}, {"start": 116.039, "duration": 7.68, "text": "uh this bottom equation we could"}, {"start": 117.6, "duration": 8.839, "text": "multiply both sides by -2 so -2 and"}, {"start": 123.719, "duration": 5.201, "text": "-2 so let's see I'm going to copy down"}, {"start": 126.439, "duration": 5.52, "text": "my my 2x - 2"}, {"start": 128.92, "duration": 5.319, "text": "Yus or excuse me equals 5 just like it"}, {"start": 131.959, "duration": 6.601, "text": "was and then our second one if we"}, {"start": 134.239, "duration": 9.321, "text": "distribute we'll get -2X we'll get -2 *"}, {"start": 138.56, "duration": 8.959, "text": "Y which is pos2 y we get 4 * -2 which"}, {"start": 143.56, "duration": 6.88, "text": "is8 and well if we add these together"}, {"start": 147.519, "duration": 5.761, "text": "now we get 0x"}, {"start": 150.44, "duration": 8.4, "text": "we'll get 0 y so we're left with zero on"}, {"start": 153.28, "duration": 8.239, "text": "the left side we have -5 +8 which is -13"}, {"start": 158.84, "duration": 5.08, "text": "again we're at this place where clearly"}, {"start": 161.519, "duration": 5.961, "text": "uh you know 0 doesn't equal"}, {"start": 163.92, "duration": 6.0, "text": "-3 so this tells us that there is no"}, {"start": 167.48, "duration": 7.88, "text": "solution to this system of"}, {"start": 169.92, "duration": 5.44, "text": "equations and we say that it is an"}, {"start": 176.08, "duration": 6.879, "text": "inconsistent we would say it's an"}, {"start": 177.879, "duration": 5.08, "text": "inconsistent system of equation a ations"}], "title": "Systems of Linear Equations - Inconsistent Systems Using Elimination by Addition - Example 3", "description": "Thanks to all of you who support me on Patreon. You da real mvps!   $1 per month helps!! :)  https://www.patreon.com/patrickjmt !! Systems of Linear Equations - Inconsistent Systems Using Elimination by Addition - Example 3", "lengthSeconds": 183, "width": 480, "height": 360}
{"video_id": "HOxqh1sUZ9A", "transcript": [{"start": 0.76, "duration": 4.44, "text": "okay so just another little example of"}, {"start": 2.48, "duration": 5.6, "text": "writing a uh an infinite sum using"}, {"start": 5.2, "duration": 7.559, "text": "series or Sigma notation so here we've"}, {"start": 8.08, "duration": 11.0, "text": "got the series 2 over 13 uh - 4 over 13"}, {"start": 12.759, "duration": 9.481, "text": "2 + 8 over 13 cubed - 16 uh over 13 to"}, {"start": 19.08, "duration": 6.0, "text": "4th Etc so usually to me the first thing"}, {"start": 22.24, "duration": 5.32, "text": "is to kind of you know spot the pattern"}, {"start": 25.08, "duration": 5.08, "text": "and then uh hopefully we can figure out"}, {"start": 27.56, "duration": 4.119, "text": "a nice little formula from that so one"}, {"start": 30.16, "duration": 3.84, "text": "thing I notice immediately is its"}, {"start": 31.679, "duration": 4.72, "text": "alternating sign it's positive negative"}, {"start": 34.0, "duration": 4.8, "text": "positive negative so at some point in"}, {"start": 36.399, "duration": 6.281, "text": "there I'm going to Le like a um you know"}, {"start": 38.8, "duration": 6.52, "text": "a negative one to some power um so I'm"}, {"start": 42.68, "duration": 5.28, "text": "thinking like a negative one to the N"}, {"start": 45.32, "duration": 4.48, "text": "let's don't put a equal sign or -1 to"}, {"start": 47.96, "duration": 4.119, "text": "the n + one we'll need something like"}, {"start": 49.8, "duration": 5.079, "text": "that at some point and again that will"}, {"start": 52.079, "duration": 4.721, "text": "just simply make it alternate signs um"}, {"start": 54.879, "duration": 4.241, "text": "the N or the N plus one just to make it"}, {"start": 56.8, "duration": 4.559, "text": "start positive or negative depending on"}, {"start": 59.12, "duration": 4.0, "text": "our index"}, {"start": 61.359, "duration": 3.721, "text": "okay so I noticed that and then I think"}, {"start": 63.12, "duration": 5.28, "text": "about kind of the numbers on top you"}, {"start": 65.08, "duration": 5.28, "text": "know we've got two to the 1st well four"}, {"start": 68.4, "duration": 4.96, "text": "that's really just 2 to the"}, {"start": 70.36, "duration": 3.88, "text": "2 um and then we've got eight which is 2"}, {"start": 73.36, "duration": 4.0, "text": "to the"}, {"start": 74.24, "duration": 4.84, "text": "3 and then we've got 16 which is 2 to"}, {"start": 77.36, "duration": 3.92, "text": "the 4th so hey I'm just seeing really"}, {"start": 79.08, "duration": 4.719, "text": "powers of two on top again we talked"}, {"start": 81.28, "duration": 4.4, "text": "about the alternating signs and then in"}, {"start": 83.799, "duration": 6.161, "text": "the bottom it's pretty clear we've got"}, {"start": 85.68, "duration": 10.68, "text": "you know powers of of of 13 13 to the 1"}, {"start": 89.96, "duration": 8.96, "text": "13 SAR 13 to the 13th 13 to the 14th Etc"}, {"start": 96.36, "duration": 5.88, "text": "so to me now um it's starting to look"}, {"start": 98.92, "duration": 6.04, "text": "like something where I can put in my my"}, {"start": 102.24, "duration": 2.72, "text": "uh series"}, {"start": 105.04, "duration": 7.48, "text": "notation so let's see so we've got a two"}, {"start": 108.64, "duration": 6.4, "text": "to a power um of n if you want to the"}, {"start": 112.52, "duration": 4.16, "text": "alternating part the negative one uh"}, {"start": 115.04, "duration": 3.2, "text": "instead of having negative 1 to the end"}, {"start": 116.68, "duration": 5.399, "text": "we can just make it"}, {"start": 118.24, "duration": 7.519, "text": "negative -2 to the N because again as n"}, {"start": 122.079, "duration": 5.72, "text": "Al you know increases uh the sign will"}, {"start": 125.759, "duration": 4.681, "text": "still Al alternate between positive and"}, {"start": 127.799, "duration": 4.201, "text": "negative so perfectly valid to instead"}, {"start": 130.44, "duration": 3.6, "text": "of having this negative one term just"}, {"start": 132.0, "duration": 4.68, "text": "include it with the uh you know the"}, {"start": 134.04, "duration": 3.4, "text": "Power of Two And in the bottom we also"}, {"start": 136.68, "duration": 4.32, "text": "have"}, {"start": 137.44, "duration": 6.2, "text": "13 well you know the the uh the"}, {"start": 141.0, "duration": 4.319, "text": "exponents are always the same so again"}, {"start": 143.64, "duration": 3.959, "text": "this the exponent to me is kind of"}, {"start": 145.319, "duration": 4.681, "text": "what's changing so that's where I put an"}, {"start": 147.599, "duration": 5.561, "text": "n generically and notice whatever the"}, {"start": 150.0, "duration": 5.4, "text": "power is on the two The Power of the 13"}, {"start": 153.16, "duration": 4.04, "text": "is exactly the same so if you've got an"}, {"start": 155.4, "duration": 3.08, "text": "n on top we should have an n on the"}, {"start": 157.2, "duration": 4.08, "text": "bottom as"}, {"start": 158.48, "duration": 5.6, "text": "well let's see we want this to start at"}, {"start": 161.28, "duration": 5.599, "text": "an exponent of one so we'll just have"}, {"start": 164.08, "duration": 6.2, "text": "well Nal 1 up to infinity and I'm going"}, {"start": 166.879, "duration": 5.481, "text": "to rewrite this one more time so Nal 1"}, {"start": 170.28, "duration": 5.48, "text": "to Infinity I'm going to write this as"}, {"start": 172.36, "duration": 6.879, "text": "just one big fraction -2 over"}, {"start": 175.76, "duration": 5.32, "text": "13 all raised to the N power and now"}, {"start": 179.239, "duration": 4.761, "text": "we've kind of got a a nice uh clean"}, {"start": 181.08, "duration": 7.159, "text": "little formula to uh to go with our"}, {"start": 184.0, "duration": 4.239, "text": "infinite series here"}], "title": "Writing a Geometric Series using Sigma / Summation Notation, Ex 2", "description": "Thanks to all of you who support me on Patreon. You da real mvps!   $1 per month helps!! :)  https://www.patreon.com/patrickjmt !! Writing a Geometric Series using Sigma / Summation Notation, Ex 2. This video shows how to write an Infinite geometric series .using sigma / summation notation. I do not find the actual sum for this particular convergent geometric series.", "lengthSeconds": 186, "width": 480, "height": 360}
{"video_id": "Hg5RwzLAqRs", "transcript": [{"start": 0.96, "duration": 4.08, "text": "all right here we're going to do uh look"}, {"start": 2.52, "duration": 5.0, "text": "at some examples related to the Limit"}, {"start": 5.04, "duration": 4.96, "text": "comparison test so in part A here we"}, {"start": 7.52, "duration": 5.239, "text": "have the series from 1 to Infinity 5"}, {"start": 10.0, "duration": 5.799, "text": "over 2 n^2 +"}, {"start": 12.759, "duration": 5.841, "text": "3 when I decide what what series to"}, {"start": 15.799, "duration": 5.521, "text": "compare this to again I just kind of"}, {"start": 18.6, "duration": 2.72, "text": "keep"}, {"start": 22.4, "duration": 5.879, "text": "uh I keep again you know what I call"}, {"start": 25.16, "duration": 4.64, "text": "dominant terms so in the numerator you"}, {"start": 28.279, "duration": 4.32, "text": "know kind of the only term that is there"}, {"start": 29.8, "duration": 4.439, "text": "is the five um and when it's a constant"}, {"start": 32.599, "duration": 3.96, "text": "it doesn't even matter whether you keep"}, {"start": 34.239, "duration": 5.241, "text": "the constant or not um in the"}, {"start": 36.559, "duration": 5.081, "text": "denominator as n gets large well again"}, {"start": 39.48, "duration": 4.0, "text": "you know three is a constant the the 2 N"}, {"start": 41.64, "duration": 3.079, "text": "squar is going to be the the dominant"}, {"start": 43.48, "duration": 2.719, "text": "term that's what's going to get the"}, {"start": 44.719, "duration": 5.921, "text": "largest the"}, {"start": 46.199, "duration": 8.04, "text": "fastest um so I'm going to compare um"}, {"start": 50.64, "duration": 6.599, "text": "whoops I'm going to compare um our our"}, {"start": 54.239, "duration": 7.241, "text": "Series 5 over 2 n^ 2 + 3 I'm going to"}, {"start": 57.239, "duration": 6.48, "text": "compare that to the series 5 over 2 n^ 2"}, {"start": 61.48, "duration": 5.679, "text": "and this series again we can Factor just"}, {"start": 63.719, "duration": 3.44, "text": "the 5 over2 out"}, {"start": 67.28, "duration": 6.28, "text": "front and then we would be left with 1"}, {"start": 69.799, "duration": 6.32, "text": "over n^ 2 and again uh 1 over n^2 is a"}, {"start": 73.56, "duration": 4.919, "text": "convergent P"}, {"start": 76.119, "duration": 4.841, "text": "Series so again that simply means it has"}, {"start": 78.479, "duration": 4.521, "text": "some finite value well if you multiply"}, {"start": 80.96, "duration": 5.799, "text": "it by 5 over two it's still going to be"}, {"start": 83.0, "duration": 6.439, "text": "finite so this is a convergent Series so"}, {"start": 86.759, "duration": 5.081, "text": "if we do our limit comparison uh test"}, {"start": 89.439, "duration": 4.481, "text": "we'll do the limit as n goes to Infinity"}, {"start": 91.84, "duration": 7.0, "text": "so I'm going to take our original series"}, {"start": 93.92, "duration": 8.44, "text": "5 2 n^ 2 + 3 and I'm going to divide"}, {"start": 98.84, "duration": 5.0, "text": "that by this new series uh I'm going to"}, {"start": 102.36, "duration": 5.28, "text": "divide that by this series that I'm"}, {"start": 103.84, "duration": 6.16, "text": "comparing it to 5 over 2 n"}, {"start": 107.64, "duration": 5.439, "text": "s well that's going to give us the limit"}, {"start": 110.0, "duration": 6.68, "text": "as n goes to Infinity we've got five"}, {"start": 113.079, "duration": 7.04, "text": "over 2 n^ 2 +"}, {"start": 116.68, "duration": 7.84, "text": "3 if we uh you know flip and multiply we"}, {"start": 120.119, "duration": 6.201, "text": "have 2 n^2 over 5 but what we're doing"}, {"start": 124.52, "duration": 4.0, "text": "here is we're forcing you know the"}, {"start": 126.32, "duration": 5.68, "text": "degree of the numerator to equal the"}, {"start": 128.52, "duration": 4.88, "text": "degree of the denominator so um let's"}, {"start": 132.0, "duration": 3.48, "text": "write this one all the way out we've got"}, {"start": 133.4, "duration": 5.16, "text": "10n SAR usually at this point I just"}, {"start": 135.48, "duration": 5.68, "text": "kind of stop um we would have again a 10"}, {"start": 138.56, "duration": 6.0, "text": "n^ 2ar +"}, {"start": 141.16, "duration": 5.359, "text": "15 well as n goes to Infinity since the"}, {"start": 144.56, "duration": 4.72, "text": "degree of the numerator equals the"}, {"start": 146.519, "duration": 5.281, "text": "degree of the denominator that limit is"}, {"start": 149.28, "duration": 3.48, "text": "simply going to equal the ratio of the"}, {"start": 151.8, "duration": 3.719, "text": "leading"}, {"start": 152.76, "duration": 5.44, "text": "coefficients so the uh limit's going to"}, {"start": 155.519, "duration": 6.041, "text": "equal 10/ 10 or in this case one and"}, {"start": 158.2, "duration": 6.64, "text": "recall the uh the uh limit comparison"}, {"start": 161.56, "duration": 5.959, "text": "test simply says that if you have a"}, {"start": 164.84, "duration": 5.36, "text": "positive finite number both series do"}, {"start": 167.519, "duration": 6.281, "text": "the same thing so in this case it would"}, {"start": 170.2, "duration": 7.28, "text": "mean that our original series must also"}, {"start": 173.8, "duration": 5.519, "text": "converge so this series also converges"}, {"start": 177.48, "duration": 3.0, "text": "again just because we're comparing it to"}, {"start": 179.319, "duration": 4.601, "text": "a series"}, {"start": 180.48, "duration": 3.44, "text": "that converges"}], "title": "Limit Comparison Test for Series - Another Example 1", "description": "(https://youtu.be/Hg5RwzLAqRs)\n(https://youtu.be/xesQnFWw8f8)\n(https://youtu.be/Hg5RwzLAqRs)\n(https://youtu.be/gNSWa3VDx9k)\n(https://youtu.be/dX_bwYp-lVc)\n(https://youtu.be/9kZrKC9ueEE)\n(https://youtu.be/QAZZgGDnnCg)\n\nIn this video, we dive into the Limit Comparison Test for evaluating series, with a specific example to demonstrate the process. \n\nWhat You Will Learn:\nHow to apply the Limit Comparison Test to a given series.\nChoosing a suitable comparison series.\nEvaluating the limit and interpreting the result to conclude about convergence.\nIf you find this video helpful, please consider liking, sharing, and subscribing. Don\u2019t forget to leave a comment if you have any questions or suggestions, and share the video with friends or classmates who might benefit from it!\n\nSupport me on Patreon for more math content:\nhttps://www.patreon.com/patrickjmt?ty=c\n\nHashtags:\n#LimitComparisonTest #Convergence #Divergence #Series #Calculus #InfiniteSeries #PatrickJMT #MathHelp #LearnMath #Mathematics", "lengthSeconds": 183, "width": 480, "height": 360}
{"video_id": "qCScHh_oP-s", "transcript": [{"start": 0.03, "duration": 3.99, "text": "all right here we're going to do an"}, {"start": 1.949, "duration": 6.9, "text": "example of multiplying mixed numbers so"}, {"start": 4.02, "duration": 6.81, "text": "we've got 2 and 4/7 times 1 and 1/2 what"}, {"start": 8.849, "duration": 4.861, "text": "I like to do is just make my mixed"}, {"start": 10.83, "duration": 4.59, "text": "numbers back into fractions and we've"}, {"start": 13.71, "duration": 5.399, "text": "seen how to make a mixed number back"}, {"start": 15.42, "duration": 6.779, "text": "into a fraction so so for example 2 + 4"}, {"start": 19.109, "duration": 6.271, "text": "/ 7 again what we do is we take the"}, {"start": 22.199, "duration": 5.611, "text": "denominator which is 7 we multiply it by"}, {"start": 25.38, "duration": 5.219, "text": "whatever numbers out front which is 2"}, {"start": 27.81, "duration": 5.07, "text": "and then we add whatever numbers in the"}, {"start": 30.599, "duration": 4.771, "text": "numerator of our fraction which is 4 and"}, {"start": 32.88, "duration": 5.37, "text": "we divide that by our original"}, {"start": 35.37, "duration": 7.64, "text": "denominator of 7 so if we do the"}, {"start": 38.25, "duration": 8.129, "text": "arithmetic 7 times 2 is 14 plus 4 is 18"}, {"start": 43.01, "duration": 8.11, "text": "it says 2 and 4/7 we can actually write"}, {"start": 46.379, "duration": 7.61, "text": "that as 18 over 7 well 1 and 1/2 will do"}, {"start": 51.12, "duration": 5.31, "text": "the same thing we'll do 2 times 1 and"}, {"start": 53.989, "duration": 6.761, "text": "then we'll add to it"}, {"start": 56.43, "duration": 7.32, "text": "another one that's all over - 2 times 1"}, {"start": 60.75, "duration": 3.689, "text": "is 2 plus 1 looks like we'll have 3 over"}, {"start": 63.75, "duration": 3.659, "text": "2"}, {"start": 64.439, "duration": 4.741, "text": "so all I'm doing is I'm taking my mixed"}, {"start": 67.409, "duration": 5.131, "text": "numbers and I'm just turning those back"}, {"start": 69.18, "duration": 6.09, "text": "into fractions and then I'm going to"}, {"start": 72.54, "duration": 6.119, "text": "multiply the same way I did a second ago"}, {"start": 75.27, "duration": 11.73, "text": "so it says we're going to do 18 over 7"}, {"start": 78.659, "duration": 10.591, "text": "times 3 over 2 okay well all I'm going"}, {"start": 87.0, "duration": 2.88, "text": "to do now is just cancel just like"}, {"start": 89.25, "duration": 3.81, "text": "before"}, {"start": 89.88, "duration": 7.02, "text": "so let's see 18 and 2 have a common"}, {"start": 93.06, "duration": 7.05, "text": "factor - 18 divided by 2 is going to"}, {"start": 96.9, "duration": 6.6, "text": "leave me with 9 2 divided by 2 is going"}, {"start": 100.11, "duration": 9.06, "text": "to leave me with 1 and now I think we"}, {"start": 103.5, "duration": 8.759, "text": "can just multiply 9 times 3 is 27 7"}, {"start": 109.17, "duration": 5.489, "text": "times 1 is just 7 we could leave it like"}, {"start": 112.259, "duration": 3.811, "text": "this of course if you typically if I"}, {"start": 114.659, "duration": 3.511, "text": "start with mixed numbers you know I'll"}, {"start": 116.07, "duration": 6.149, "text": "finish with mixed numbers so let's see"}, {"start": 118.17, "duration": 8.21, "text": "if we can do this so 7 goes into 27 I"}, {"start": 122.219, "duration": 6.311, "text": "think 3 times let's see 3 times 7 is 21"}, {"start": 126.38, "duration": 5.89, "text": "so we would have 6"}, {"start": 128.53, "duration": 6.42, "text": "as a remainder and our original"}, {"start": 132.27, "duration": 6.13, "text": "denominator which is seven so we can"}, {"start": 134.95, "duration": 7.52, "text": "write 27 over seven as the mixed number"}, {"start": 138.4, "duration": 4.07, "text": "three and six seven"}], "title": "Multiplying Fractions and Mixed Numbers, Example 3", "description": "Thanks to all of you who support me on Patreon. You da real mvps!   $1 per month helps!! :)  https://www.patreon.com/patrickjmt !! Multiplying Fractions and Mixed Numbers, Example 3.", "lengthSeconds": 142, "width": 480, "height": 360}
{"video_id": "in_dAlpjUKk", "transcript": [{"start": 0.03, "duration": 3.78, "text": "okay in this video we're gonna look at"}, {"start": 2.01, "duration": 4.92, "text": "multiplying and dividing complex numbers"}, {"start": 3.81, "duration": 6.09, "text": "in trigonometric or polar form so I've"}, {"start": 6.93, "duration": 6.06, "text": "got two complex numbers here in polar"}, {"start": 9.9, "duration": 5.489, "text": "form it says all you do to multiply them"}, {"start": 12.99, "duration": 5.25, "text": "is we just multiply the r1 times the r2"}, {"start": 15.389, "duration": 5.48, "text": "and then it says we add the angles we"}, {"start": 18.24, "duration": 5.73, "text": "take cosine of the angles added together"}, {"start": 20.869, "duration": 5.861, "text": "plus I times the sine of the angles"}, {"start": 23.97, "duration": 5.969, "text": "added together to divide we just divide"}, {"start": 26.73, "duration": 5.34, "text": "our R sub 1 by R sub 2 and then we"}, {"start": 29.939, "duration": 7.261, "text": "simply subtract the angles instead of"}, {"start": 32.07, "duration": 8.46, "text": "adding them so nothing nothing too bad"}, {"start": 37.2, "duration": 6.42, "text": "at all here so let's do an example here"}, {"start": 40.53, "duration": 7.56, "text": "so we've got 6 cosine of 60 degrees plus"}, {"start": 43.62, "duration": 8.099, "text": "I sine of 60 degrees over 3 times cosine"}, {"start": 48.09, "duration": 7.199, "text": "of 90 degrees plus I sine of 90 degrees"}, {"start": 51.719, "duration": 5.281, "text": "so in this case we're doing division so"}, {"start": 55.289, "duration": 5.07, "text": "it says what we'll simply do is we just"}, {"start": 57.0, "duration": 6.449, "text": "take our r1 over r2 and then it says we"}, {"start": 60.359, "duration": 7.44, "text": "simply subtract we'll take our theta 1"}, {"start": 63.449, "duration": 6.061, "text": "minus our theta 2 so in this case well"}, {"start": 67.799, "duration": 5.311, "text": "if we do the division we'll get 6"}, {"start": 69.51, "duration": 4.649, "text": "divided by 3 whoops I'm already getting"}, {"start": 73.11, "duration": 4.11, "text": "the answer there too"}, {"start": 74.159, "duration": 6.481, "text": "so 6 divided by 3 and then it says we"}, {"start": 77.22, "duration": 5.55, "text": "simply take cosine of 60 degrees minus"}, {"start": 80.64, "duration": 7.74, "text": "90 degrees"}, {"start": 82.77, "duration": 10.05, "text": "plus I times sine of 60 degrees minus 90"}, {"start": 88.38, "duration": 9.779, "text": "degrees and well 6 over 3 is 2 this will"}, {"start": 92.82, "duration": 10.439, "text": "be cosine of negative 30 degrees plus I"}, {"start": 98.159, "duration": 6.27, "text": "sine of negative 30 degrees and boom"}, {"start": 103.259, "duration": 5.701, "text": "that's it we're done"}, {"start": 104.429, "duration": 6.51, "text": "so pretty straightforward I think you"}, {"start": 108.96, "duration": 4.049, "text": "know kind of put another random formula"}, {"start": 110.939, "duration": 3.93, "text": "to remember you know it's always good to"}, {"start": 113.009, "duration": 3.45, "text": "understand the derivations of these two"}, {"start": 114.869, "duration": 3.771, "text": "because then you're not just memorizing"}, {"start": 116.459, "duration": 5.25, "text": "random formulas you're kind of you know"}, {"start": 118.64, "duration": 5.92, "text": "hopefully understanding where they come"}, {"start": 121.709, "duration": 7.291, "text": "from let's go ahead and do Part B here"}, {"start": 124.56, "duration": 6.66, "text": "as well here we're multiplying two two"}, {"start": 129.0, "duration": 4.86, "text": "complex numbers here in polar form and"}, {"start": 131.22, "duration": 4.95, "text": "again it says when you are multiplying"}, {"start": 133.86, "duration": 4.8, "text": "it says all we do is we multiply the r1"}, {"start": 136.17, "duration": 6.51, "text": "and the r2 and then we add the angles"}, {"start": 138.66, "duration": 6.0, "text": "together instead of subtracting them so"}, {"start": 142.68, "duration": 4.8, "text": "in this case we'll simply take our two"}, {"start": 144.66, "duration": 4.74, "text": "times our five which will give us 10 and"}, {"start": 147.48, "duration": 5.849, "text": "then in our parentheses we would have"}, {"start": 149.4, "duration": 8.309, "text": "cosine of 45 degrees and then I would"}, {"start": 153.329, "duration": 8.011, "text": "have to add the angle 30 degrees and"}, {"start": 157.709, "duration": 5.701, "text": "then we just do our plus I sine"}, {"start": 161.34, "duration": 5.97, "text": "we do the same thing we just take 45"}, {"start": 163.41, "duration": 9.27, "text": "degrees and add to that the value of 30"}, {"start": 167.31, "duration": 8.82, "text": "degrees here and in this case we'll just"}, {"start": 172.68, "duration": 8.87, "text": "be left with 10 times cosine of 75"}, {"start": 176.13, "duration": 8.7, "text": "degrees plus I times sine of 75 degrees"}, {"start": 181.55, "duration": 6.46, "text": "and again in this case we'll just leave"}, {"start": 184.83, "duration": 6.32, "text": "them in our polar form again we've now"}, {"start": 188.01, "duration": 5.79, "text": "got our solution here so not a lot to do"}, {"start": 191.15, "duration": 4.54, "text": "Part C which we'll do in another example"}, {"start": 193.8, "duration": 3.9, "text": "that'll be a little a little more"}, {"start": 195.69, "duration": 4.56, "text": "complicated just because the numerators"}, {"start": 197.7, "duration": 5.16, "text": "not in polar form so we'll first have to"}, {"start": 200.25, "duration": 4.38, "text": "put that in polar form and then it'll"}, {"start": 202.86, "duration": 4.82, "text": "just turn back into a problem like we"}, {"start": 204.63, "duration": 3.05, "text": "did here in part eight"}], "title": "Complex Numbers: Multiplying and Dividing in Polar Form, Ex 1", "description": "Thanks to all of you who support me on Patreon. You da real mvps!   $1 per month helps!! :)  https://www.patreon.com/patrickjmt !! Complex Numbers: Multiplying and Dividing in Polar Form, Ex 1. In this video, I give the formula for multiplication and division of two complex numbers that are in polar form.", "lengthSeconds": 206, "width": 480, "height": 360}
{"video_id": "q57NU29KNGw", "transcript": [{"start": 0.03, "duration": 3.599, "text": "all right in this video I'm going to do"}, {"start": 1.89, "duration": 4.409, "text": "another example of using the remainder"}, {"start": 3.629, "duration": 5.731, "text": "theorem so in this case we've got 4x to"}, {"start": 6.299, "duration": 6.331, "text": "the 4th plus 2x squared plus 1 divided"}, {"start": 9.36, "duration": 6.569, "text": "by X minus 3 so the function that we're"}, {"start": 12.63, "duration": 6.12, "text": "that's being divided we'll call that f"}, {"start": 15.929, "duration": 6.661, "text": "of X so that's 4x to the fourth plus 2x"}, {"start": 18.75, "duration": 6.66, "text": "squared plus 1 and again since we're"}, {"start": 22.59, "duration": 4.56, "text": "dividing by X minus 3 again according to"}, {"start": 25.41, "duration": 4.74, "text": "the remainder theorem whatever number"}, {"start": 27.15, "duration": 5.46, "text": "you're subtracting away it says if we"}, {"start": 30.15, "duration": 5.04, "text": "plug that into our function that will"}, {"start": 32.61, "duration": 5.82, "text": "give us the remainder so ok so since"}, {"start": 35.19, "duration": 5.22, "text": "we're subtracting X minus 3 since we're"}, {"start": 38.43, "duration": 3.78, "text": "subtracting the number 3 and says all we"}, {"start": 40.41, "duration": 4.77, "text": "have to do is plug 3 into the function"}, {"start": 42.21, "duration": 7.189, "text": "to get the remainder so that'll be 4"}, {"start": 45.18, "duration": 9.57, "text": "times 3 to the fourth plus 2 times 3"}, {"start": 49.399, "duration": 11.621, "text": "squared plus 1 3 to the fourth is going"}, {"start": 54.75, "duration": 9.39, "text": "to give us 81 3 squared is 9 a little"}, {"start": 61.02, "duration": 7.8, "text": "more arithmetic so 4 times 81 that would"}, {"start": 64.14, "duration": 7.799, "text": "be 324 2 times 9 is 18 plus 1 would be"}, {"start": 68.82, "duration": 10.64, "text": "19 and if we add those together we'll"}, {"start": 71.939, "duration": 7.521, "text": "get our remainder of 343"}], "title": "The Remainder Theorem - Example 2", "description": "Thanks to all of you who support me on Patreon. You da real mvps!   $1 per month helps!! :)  https://www.patreon.com/patrickjmt !! The Remainder Theorem - Example 2", "lengthSeconds": 79, "width": 480, "height": 360}
{"video_id": "ZtPsWriHwa8", "transcript": [{"start": 1.319, "duration": 2.721, "text": "okay in this video I'm going to talk"}, {"start": 2.52, "duration": 4.239, "text": "about what are called arithmetic"}, {"start": 4.04, "duration": 5.88, "text": "sequences and recall a sequence is just"}, {"start": 6.759, "duration": 5.721, "text": "a listing of numbers um a regular just a"}, {"start": 9.92, "duration": 4.16, "text": "planal sequence doesn't have to have any"}, {"start": 12.48, "duration": 4.0, "text": "pattern at all to the numbers that"}, {"start": 14.08, "duration": 4.48, "text": "appear but an arithmetic sequence is"}, {"start": 16.48, "duration": 3.959, "text": "going to be a sequence where succeeding"}, {"start": 18.56, "duration": 4.16, "text": "terms in the sequence differ by a"}, {"start": 20.439, "duration": 3.241, "text": "constant amount so um in this video I"}, {"start": 22.72, "duration": 2.6, "text": "just want to give a real quick"}, {"start": 23.68, "duration": 4.04, "text": "introduction to what arithmetic"}, {"start": 25.32, "duration": 5.52, "text": "sequences are and another video I'll"}, {"start": 27.72, "duration": 6.359, "text": "talk about finding the general term a"}, {"start": 30.84, "duration": 8.92, "text": "and then also about um finding sums of"}, {"start": 34.079, "duration": 8.32, "text": "of partial arithmetic uh uh"}, {"start": 39.76, "duration": 4.08, "text": "series um so the only thing we're going"}, {"start": 42.399, "duration": 4.041, "text": "to do here determine which of the"}, {"start": 43.84, "duration": 4.359, "text": "following sequences are arithmetic if"}, {"start": 46.44, "duration": 4.639, "text": "they are arithmetic we're going to give"}, {"start": 48.199, "duration": 5.321, "text": "the value of D um so we'll talk about"}, {"start": 51.079, "duration": 5.761, "text": "this D here so the idea of an arithmetic"}, {"start": 53.52, "duration": 5.08, "text": "series is is simple um notice okay so to"}, {"start": 56.84, "duration": 5.08, "text": "go from three to eight we would have to"}, {"start": 58.6, "duration": 5.72, "text": "add five to go from 8 to 13 we would"}, {"start": 61.92, "duration": 5.92, "text": "have to add five to go from the number"}, {"start": 64.32, "duration": 5.24, "text": "13 to 18 hey we add five and notice this"}, {"start": 67.84, "duration": 4.12, "text": "pattern"}, {"start": 69.56, "duration": 5.04, "text": "continues um we'll just keep adding five"}, {"start": 71.96, "duration": 4.76, "text": "adding five adding five and you know"}, {"start": 74.6, "duration": 3.48, "text": "assumedly if this pattern continues well"}, {"start": 76.72, "duration": 5.28, "text": "then it would certainly be what we call"}, {"start": 78.08, "duration": 7.359, "text": "an arithmetic sequence because every"}, {"start": 82.0, "duration": 6.68, "text": "term is uh is differing by constant"}, {"start": 85.439, "duration": 5.801, "text": "amount and this value that you have to"}, {"start": 88.68, "duration": 5.079, "text": "add to get to the next term is going to"}, {"start": 91.24, "duration": 3.96, "text": "be positive five so we say our D value"}, {"start": 93.759, "duration": 4.281, "text": "is positive"}, {"start": 95.2, "duration": 9.559, "text": "five notice in my second example here I"}, {"start": 98.04, "duration": 12.0, "text": "start at .7 then we go to - 1.7 -2.7 3.7"}, {"start": 104.759, "duration": 8.68, "text": "-4.7 Etc in this case every term is"}, {"start": 110.04, "duration": 5.8, "text": "differing um you have to subtract one to"}, {"start": 113.439, "duration": 5.081, "text": "get to the next term so in this case we"}, {"start": 115.84, "duration": 4.8, "text": "would say yes this is arithmetic and our"}, {"start": 118.52, "duration": 5.12, "text": "d value equals"}, {"start": 120.64, "duration": 6.479, "text": "1 and our third example okay well let's"}, {"start": 123.64, "duration": 7.399, "text": "just test um one of my numbers here so"}, {"start": 127.119, "duration": 9.321, "text": "um so 1.6 to two uh 2.2 it looks like"}, {"start": 131.039, "duration": 9.92, "text": "we've added on 6 2.2 to 2.8 we've added"}, {"start": 136.44, "duration": 5.6, "text": "on6 oh 2.8 to 3.3 though we've added on"}, {"start": 140.959, "duration": 5.681, "text": "positive"}, {"start": 142.04, "duration": 6.52, "text": ".5 and once you find um two numbers that"}, {"start": 146.64, "duration": 3.959, "text": "are different it doesn't matter what"}, {"start": 148.56, "duration": 3.64, "text": "happens from then on out"}, {"start": 150.599, "duration": 4.841, "text": "um since they're not differing by the"}, {"start": 152.2, "duration": 6.039, "text": "same amount we say it's not"}, {"start": 155.44, "duration": 5.079, "text": "arithmetic okay so you know the only"}, {"start": 158.239, "duration": 4.0, "text": "real thing here to check if a if a"}, {"start": 160.519, "duration": 4.08, "text": "sequence is arithmetic is just to be"}, {"start": 162.239, "duration": 7.041, "text": "careful with your arithmetic and in my"}, {"start": 164.599, "duration": 6.881, "text": "last one here 4/3 5/3 2 7/3 8/3 3 well"}, {"start": 169.28, "duration": 4.52, "text": "notice we could change two if if we make"}, {"start": 171.48, "duration": 5.039, "text": "that into a fraction we could make it 2"}, {"start": 173.8, "duration": 6.439, "text": "over one multiply top and bottom by"}, {"start": 176.519, "duration": 7.161, "text": "three hey we would get 6/3 like wise we"}, {"start": 180.239, "duration": 8.481, "text": "can make three into 9/3 and we go from"}, {"start": 183.68, "duration": 9.16, "text": "1/3 5/3 6/3 7/3 8/3 9/3 so it looks like"}, {"start": 188.72, "duration": 8.599, "text": "we're adding the value of 1/3 um each"}, {"start": 192.84, "duration": 7.039, "text": "time to get to the term after that one"}, {"start": 197.319, "duration": 7.28, "text": "so again we would say this one is"}, {"start": 199.879, "duration": 7.601, "text": "arithmetic and our d value would equal"}, {"start": 204.599, "duration": 5.881, "text": "positive3 so the first one yes it's"}, {"start": 207.48, "duration": 5.839, "text": "arithmetic the second one is AR Matic"}, {"start": 210.48, "duration": 5.319, "text": "the third one we said wasn't um and the"}, {"start": 213.319, "duration": 4.2, "text": "fourth one we also said was arithmatic"}, {"start": 215.799, "duration": 3.961, "text": "so that's the basic idea of an"}, {"start": 217.519, "duration": 5.241, "text": "arithmatic um sequence that's all there"}, {"start": 219.76, "duration": 5.399, "text": "is to it so finding again a formula for"}, {"start": 222.76, "duration": 4.479, "text": "the general term and using that can"}, {"start": 225.159, "duration": 4.72, "text": "sometimes be a little tricky also"}, {"start": 227.239, "duration": 5.521, "text": "sometimes too they'll have you add up um"}, {"start": 229.879, "duration": 4.601, "text": "finitely many terms and we'll in uh some"}, {"start": 232.76, "duration": 5.119, "text": "other videos I'll talk about doing both"}, {"start": 234.48, "duration": 3.399, "text": "of those things"}], "title": "Quick Intro to Arithmetic Sequences", "description": "Thanks to all of you who support me on Patreon. You da real mvps!   $1 per month helps!! :)  https://www.patreon.com/patrickjmt !! Quick Intro to Arithmetic Sequences - Just a quick idea of what an arithmetic sequence is!  I give a few examples!", "lengthSeconds": 236, "width": 480, "height": 360}
{"video_id": "DAwUVqq5vC0", "transcript": [{"start": 0.799, "duration": 3.321, "text": "all right here we're going to look at"}, {"start": 1.92, "duration": 5.16, "text": "finding vertical asmp tootes of rational"}, {"start": 4.12, "duration": 4.92, "text": "functions and kind of the thing that um"}, {"start": 7.08, "duration": 4.24, "text": "you know that you're looking for if you"}, {"start": 9.04, "duration": 4.759, "text": "figure out you know values of X or"}, {"start": 11.32, "duration": 5.32, "text": "whatever your independent variable are"}, {"start": 13.799, "duration": 6.161, "text": "if uh it makes the denominator zero but"}, {"start": 16.64, "duration": 4.84, "text": "the numerator not zero um in that case"}, {"start": 19.96, "duration": 3.84, "text": "you're going to have a vertical ASM"}, {"start": 21.48, "duration": 4.6, "text": "toote at that x coordinate it turns out"}, {"start": 23.8, "duration": 3.639, "text": "if you get 0 over zero what's going to"}, {"start": 26.08, "duration": 4.64, "text": "happen is you would actually just have a"}, {"start": 27.439, "duration": 4.561, "text": "little hole in the graph so"}, {"start": 30.72, "duration": 3.519, "text": "all right so let's find the vertical"}, {"start": 32.0, "duration": 4.96, "text": "asmp tootes of these functions we'll do"}, {"start": 34.239, "duration": 5.721, "text": "uh part A here first um usually what I"}, {"start": 36.96, "duration": 5.68, "text": "try to do is uh I try to factor"}, {"start": 39.96, "duration": 4.64, "text": "everything so um you could just set the"}, {"start": 42.64, "duration": 4.36, "text": "denominator immediately equal to zero"}, {"start": 44.6, "duration": 5.639, "text": "and solve but if you factor let's see so"}, {"start": 47.0, "duration": 6.16, "text": "I think the denominator does Factor"}, {"start": 50.239, "duration": 6.361, "text": "nicely so two numbers that multiply to"}, {"start": 53.16, "duration": 5.16, "text": "-6 but add up to positive one I think we"}, {"start": 56.6, "duration": 3.959, "text": "would need some sort of combination of"}, {"start": 58.32, "duration": 6.08, "text": "three and two and I think"}, {"start": 60.559, "duration": 6.481, "text": "pos3 and -2 would do that for"}, {"start": 64.4, "duration": 5.84, "text": "us so it's kind of easy to see what"}, {"start": 67.04, "duration": 4.96, "text": "makes the denominator equal to zero um"}, {"start": 70.24, "duration": 6.0, "text": "so if we set the denominator equal to"}, {"start": 72.0, "duration": 7.08, "text": "zero we would just get x = -3 and X ="}, {"start": 76.24, "duration": 5.96, "text": "pos2 but notice if we plug uh you know"}, {"start": 79.08, "duration": 5.64, "text": "so if we plug -3 in in the numer"}, {"start": 82.2, "duration": 6.04, "text": "numerator we would have -3 -2 which"}, {"start": 84.72, "duration": 5.8, "text": "would be -5 and if we plug -3 into the"}, {"start": 88.24, "duration": 3.879, "text": "first uh you know the first Factor here"}, {"start": 90.52, "duration": 5.0, "text": "we would get"}, {"start": 92.119, "duration": 5.921, "text": "zero and notice if we plug two in well"}, {"start": 95.52, "duration": 6.12, "text": "clearly we're going to get 0 over 0 in"}, {"start": 98.04, "duration": 5.6, "text": "that case so it turns out at xal 2 we"}, {"start": 101.64, "duration": 5.04, "text": "actually have again a little hole in the"}, {"start": 103.64, "duration": 6.119, "text": "graph but at xal -3 that's where we have"}, {"start": 106.68, "duration": 3.079, "text": "our vertical ASM"}, {"start": 111.799, "duration": 4.28, "text": "toote so sometimes what people will do"}, {"start": 114.36, "duration": 3.799, "text": "is they'll just you know sort of factor"}, {"start": 116.079, "duration": 5.04, "text": "and cancel everything out and here"}, {"start": 118.159, "duration": 5.0, "text": "you're left with 1 /x + three and once"}, {"start": 121.119, "duration": 3.6, "text": "everything's simplified in this case"}, {"start": 123.159, "duration": 4.041, "text": "whatever makes the denominator"}, {"start": 124.719, "duration": 4.16, "text": "denominator equal to zero um that'll be"}, {"start": 127.2, "duration": 3.039, "text": "your vertical ASM toote because again"}, {"start": 128.879, "duration": 3.281, "text": "you you've already kind of gotten rid of"}, {"start": 130.239, "duration": 4.121, "text": "these common factors so you're getting"}, {"start": 132.16, "duration": 6.359, "text": "rid of those uh values of X that would"}, {"start": 134.36, "duration": 4.159, "text": "give you 0 over 0"}], "title": "Vertical Asymptotes of Rational Functions: Quick Way to Find Them, Another Example 1", "description": "Thanks to all of you who support me on Patreon. You da real mvps!   $1 per month helps!! :)  https://www.patreon.com/patrickjmt !! Vertical Asymptotes of Rational Functions: Quick Way to Find Them, Another Example 1.  Just another example of finding vertical asymptotes of rational functions.", "lengthSeconds": 137, "width": 480, "height": 360}
{"video_id": "5QssI9bpTD0", "transcript": [{"start": 1.04, "duration": 3.48, "text": "all right in this video we're going to"}, {"start": 2.36, "duration": 4.8, "text": "look at integrating some rational"}, {"start": 4.52, "duration": 5.56, "text": "functions using"}, {"start": 7.16, "duration": 7.2, "text": "substitution so um to start off with"}, {"start": 10.08, "duration": 5.84, "text": "here we have X over x^2 + 4 so again you"}, {"start": 14.36, "duration": 4.16, "text": "know if I was kind of looking at this"}, {"start": 15.92, "duration": 4.16, "text": "problem just sort of you know U out of"}, {"start": 18.52, "duration": 3.28, "text": "the blue and I didn't know it was U"}, {"start": 20.08, "duration": 3.56, "text": "substitution again the thing that I"}, {"start": 21.8, "duration": 4.96, "text": "would think about is I see something to"}, {"start": 23.64, "duration": 5.399, "text": "the second degree in the denominator and"}, {"start": 26.76, "duration": 4.599, "text": "I see something to the first degree in"}, {"start": 29.039, "duration": 3.801, "text": "the numerator"}, {"start": 31.359, "duration": 3.52, "text": "and that to me makes me think you know"}, {"start": 32.84, "duration": 4.96, "text": "maybe a little U substitution will work"}, {"start": 34.879, "duration": 7.641, "text": "here so if we let U ="}, {"start": 37.8, "duration": 7.64, "text": "X2 + 4 the derivative of X2 is just"}, {"start": 42.52, "duration": 6.48, "text": "going to be 2x uh and then we'll tack on"}, {"start": 45.44, "duration": 5.36, "text": "our DX well notice we have an X DX in"}, {"start": 49.0, "duration": 6.239, "text": "our problem so I'm just going to"}, {"start": 50.8, "duration": 6.04, "text": "multiply both sides by2 so we have2 du"}, {"start": 55.239, "duration": 4.28, "text": "equals x"}, {"start": 56.84, "duration": 4.239, "text": "DX so the X DX that's what I'm going to"}, {"start": 59.519, "duration": 4.121, "text": "replace with"}, {"start": 61.079, "duration": 4.561, "text": "the2 um and then in the denominator"}, {"start": 63.64, "duration": 4.88, "text": "that's what we're calling"}, {"start": 65.64, "duration": 5.76, "text": "U okay so now we've got it down to a"}, {"start": 68.52, "duration": 5.76, "text": "formula that we can uh use so the"}, {"start": 71.4, "duration": 7.64, "text": "anti-derivative of 1/ U is the natural"}, {"start": 74.28, "duration": 6.839, "text": "logarithm of the absolute value of U and"}, {"start": 79.04, "duration": 5.52, "text": "now we just have to plug back in our U"}, {"start": 81.119, "duration": 7.721, "text": "substitution which was x^2 +"}, {"start": 84.56, "duration": 6.559, "text": "4 and now we have our um our"}, {"start": 88.84, "duration": 4.319, "text": "anti-derivative"}, {"start": 91.119, "duration": 4.241, "text": "technically for any value of x this"}, {"start": 93.159, "duration": 3.6, "text": "quantity on the inside would be positive"}, {"start": 95.36, "duration": 4.399, "text": "so you could even write it without"}, {"start": 96.759, "duration": 4.761, "text": "absolute value but in general you know I"}, {"start": 99.759, "duration": 3.161, "text": "just leave it an absolute value and that"}, {"start": 101.52, "duration": 2.279, "text": "way I don't have to really even worry"}, {"start": 102.92, "duration": 3.519, "text": "about"}, {"start": 103.799, "duration": 8.96, "text": "it um let's look at our other example"}, {"start": 106.439, "duration": 9.68, "text": "here so we have 24x 3r - 4 over 3x 4 -"}, {"start": 112.759, "duration": 3.36, "text": "2x +"}, {"start": 116.32, "duration": 4.32, "text": "one so I'm going to do the same thing"}, {"start": 118.64, "duration": 3.96, "text": "here I notice we have an X X to 4th in"}, {"start": 120.64, "duration": 3.68, "text": "the denominator so if I take the"}, {"start": 122.6, "duration": 4.119, "text": "derivative of that I would get something"}, {"start": 124.32, "duration": 4.439, "text": "cubed and then we have an x to the 1st"}, {"start": 126.719, "duration": 4.201, "text": "if I take the derivative of that um I'm"}, {"start": 128.759, "duration": 4.12, "text": "just going to be left with a constant so"}, {"start": 130.92, "duration": 6.679, "text": "hopefully by picking U to be the"}, {"start": 132.879, "duration": 7.841, "text": "denominator we can um manipulate it so"}, {"start": 137.599, "duration": 6.441, "text": "that our U substitution will work so if"}, {"start": 140.72, "duration": 6.2, "text": "we take our DU we would get four * 3"}, {"start": 144.04, "duration": 7.16, "text": "which would be 12 and then X the 3 and"}, {"start": 146.92, "duration": 5.64, "text": "then minus 2 and then DX"}, {"start": 151.2, "duration": 4.759, "text": "well that's almost what we have in the"}, {"start": 152.56, "duration": 6.48, "text": "numerator we have 24x Cub minus 4 well"}, {"start": 155.959, "duration": 6.761, "text": "if I just doubled both sides so 2 du and"}, {"start": 159.04, "duration": 3.68, "text": "If I multiply the right side by"}, {"start": 163.68, "duration": 5.639, "text": "two well now it says that 2 du and if we"}, {"start": 167.04, "duration": 5.16, "text": "distribute hey that would give us our 24"}, {"start": 169.319, "duration": 6.361, "text": "x 3r minus"}, {"start": 172.2, "duration": 6.64, "text": "4 DX just like we want so now we can do"}, {"start": 175.68, "duration": 4.88, "text": "our U substitution okay so it says the"}, {"start": 178.84, "duration": 3.8, "text": "quantity in the denom denominator that's"}, {"start": 180.56, "duration": 4.36, "text": "what we're calling our that's what we're"}, {"start": 182.64, "duration": 5.159, "text": "calling U so we'll just have U"}, {"start": 184.92, "duration": 6.28, "text": "downstairs and then um in the numerator"}, {"start": 187.799, "duration": 6.0, "text": "24x 3r minus 4 DX it says that's going"}, {"start": 191.2, "duration": 4.36, "text": "to be the same thing as 2 du I'm going"}, {"start": 193.799, "duration": 3.0, "text": "to pull the two out front and just leave"}, {"start": 195.56, "duration": 3.88, "text": "the one"}, {"start": 196.799, "duration": 4.481, "text": "inside and again now if we integrate we"}, {"start": 199.44, "duration": 5.2, "text": "just get the natural logarithm of the"}, {"start": 201.28, "duration": 5.599, "text": "absolute value of U but I'm just going"}, {"start": 204.64, "duration": 7.44, "text": "to plug our U substitution back in"}, {"start": 206.879, "duration": 9.961, "text": "that's 3x 4th - 2x +"}, {"start": 212.08, "duration": 7.76, "text": "1 and again we've now found our"}, {"start": 216.84, "duration": 3.0, "text": "anti-derivative"}], "title": "Substitution Rule / U-sub Basic Example 2", "description": "Thanks to all of you who support me on Patreon. You da real mvps!   $1 per month helps!! :)  https://www.patreon.com/patrickjmt !! Substitution Rule / U-sub Basic Example 2", "lengthSeconds": 218, "width": 480, "height": 360}
{"video_id": "RerJTT06SoA", "transcript": [{"start": 1.0, "duration": 4.44, "text": "okay in this example we're going to look"}, {"start": 2.639, "duration": 4.881, "text": "at a problem related to percents so"}, {"start": 5.44, "duration": 3.88, "text": "suppose Patrick's going to put down 20%"}, {"start": 7.52, "duration": 3.88, "text": "of the cost of a bike and that bike"}, {"start": 9.32, "duration": 5.92, "text": "sells for"}, {"start": 11.4, "duration": 7.56, "text": "$179 14 we want to know roughly how much"}, {"start": 15.24, "duration": 5.959, "text": "money he's going to put down so since"}, {"start": 18.96, "duration": 5.559, "text": "we're estimating here"}, {"start": 21.199, "duration": 3.32, "text": "roughly well"}, {"start": 25.0, "duration": 7.519, "text": "1794 that's pretty close to"}, {"start": 28.679, "duration": 3.84, "text": "$180 and we we want to find"}, {"start": 34.0, "duration": 7.2, "text": "20% we want to find 20% of"}, {"start": 38.239, "duration": 5.721, "text": "180 well 20% we can turn that into a"}, {"start": 41.2, "duration": 6.6, "text": "decimal that's"}, {"start": 43.96, "duration": 3.84, "text": "0.20 of turns into"}, {"start": 48.12, "duration": 4.84, "text": "multiplication and then we just multiply"}, {"start": 50.28, "duration": 7.56, "text": "that by"}, {"start": 52.96, "duration": 9.36, "text": "180 so let's see um notice that 10% of1"}, {"start": 57.84, "duration": 6.96, "text": "180 would simply be $18 So 20% of 180"}, {"start": 62.32, "duration": 2.48, "text": "would be"}, {"start": 65.08, "duration": 9.16, "text": "$36 so we can say that Patrick will"}, {"start": 68.28, "duration": 5.96, "text": "roughly put down $36 for the bike"}], "title": "Estimating the Percent of Another Number", "description": "Thanks to all of you who support me on Patreon. You da real mvps!   $1 per month helps!! :)  https://www.patreon.com/patrickjmt !! In this video, I plan on putting down 20% of the cost of a bike that costs $179.14.  How much will I put down approximately?", "lengthSeconds": 73, "width": 480, "height": 360}
{"video_id": "gu9ZkXVsEow", "transcript": [{"start": 0.03, "duration": 5.67, "text": "okay in this example we're going to"}, {"start": 2.52, "duration": 6.96, "text": "simplify some algebraic expressions by"}, {"start": 5.7, "duration": 5.34, "text": "combining like terms so again when I see"}, {"start": 9.48, "duration": 3.96, "text": "an expression like this and it says"}, {"start": 11.04, "duration": 3.54, "text": "simplify okay first you know I basically"}, {"start": 13.44, "duration": 5.04, "text": "see that there's addition and"}, {"start": 14.58, "duration": 6.15, "text": "subtraction at least in Part A I start"}, {"start": 18.48, "duration": 4.5, "text": "looking for like terms that's that's the"}, {"start": 20.73, "duration": 5.19, "text": "first thing I think about so I see 4"}, {"start": 22.98, "duration": 5.16, "text": "times X to the third power X cubed I"}, {"start": 25.92, "duration": 4.22, "text": "start looking for other terms that have"}, {"start": 28.14, "duration": 5.4, "text": "an X to the third power"}, {"start": 30.14, "duration": 5.89, "text": "again terms are separated by pluses and"}, {"start": 33.54, "duration": 4.65, "text": "minuses so we have an x squared well"}, {"start": 36.03, "duration": 5.25, "text": "that's not X to the third power then I"}, {"start": 38.19, "duration": 5.97, "text": "say okay we do have negative 2x to the"}, {"start": 41.28, "duration": 5.549, "text": "third and I don't see anything else"}, {"start": 44.16, "duration": 4.649, "text": "involving X to the third so in my head I"}, {"start": 46.829, "duration": 4.951, "text": "kind of block out the other everything"}, {"start": 48.809, "duration": 6.451, "text": "else and to me I think about well if it"}, {"start": 51.78, "duration": 5.22, "text": "said four X to the third minus two X to"}, {"start": 55.26, "duration": 4.68, "text": "the third what would I get"}, {"start": 57.0, "duration": 7.049, "text": "well four X to the third minus two x to"}, {"start": 59.94, "duration": 9.45, "text": "the third that's just gonna leave us"}, {"start": 64.049, "duration": 8.701, "text": "with two x to the third power so we've"}, {"start": 69.39, "duration": 5.339, "text": "combined those like terms and now there"}, {"start": 72.75, "duration": 4.11, "text": "are no other like terms we have an x"}, {"start": 74.729, "duration": 4.201, "text": "squared and then we have a constant so"}, {"start": 76.86, "duration": 2.399, "text": "typically and it already is in that"}, {"start": 78.93, "duration": 2.79, "text": "order"}, {"start": 79.259, "duration": 4.47, "text": "we write things in descending powers so"}, {"start": 81.72, "duration": 4.89, "text": "I'm just gonna drop those terms right"}, {"start": 83.729, "duration": 5.551, "text": "down and that would now be simplified so"}, {"start": 86.61, "duration": 4.89, "text": "the only thing we could really do we had"}, {"start": 89.28, "duration": 6.12, "text": "a couple like terms that we were able to"}, {"start": 91.5, "duration": 7.2, "text": "condense okay let's go ahead and do Part"}, {"start": 95.4, "duration": 5.91, "text": "B as well so the difference here in Part"}, {"start": 98.7, "duration": 6.87, "text": "B at least kind of the the first extra"}, {"start": 101.31, "duration": 6.87, "text": "step is we're gonna get rid of the"}, {"start": 105.57, "duration": 5.04, "text": "parentheses once we get rid of the"}, {"start": 108.18, "duration": 4.41, "text": "parentheses will basically be back to an"}, {"start": 110.61, "duration": 3.27, "text": "expression like we had in Part A and"}, {"start": 112.59, "duration": 4.529, "text": "then I'm just gonna start thinking what"}, {"start": 113.88, "duration": 5.519, "text": "are the like terms okay so what we're"}, {"start": 117.119, "duration": 3.491, "text": "gonna do here I'm gonna drop my 10 X to"}, {"start": 119.399, "duration": 6.101, "text": "the fifth right"}, {"start": 120.61, "duration": 7.32, "text": "I see a positive 3 positive 3 is going"}, {"start": 125.5, "duration": 8.22, "text": "to get distributed to the 2x to the 5th"}, {"start": 127.93, "duration": 8.85, "text": "and also to the negative B squared so"}, {"start": 133.72, "duration": 6.3, "text": "let's see positive 3 times positive 2 is"}, {"start": 136.78, "duration": 6.12, "text": "gonna get positive 6 then we just stick"}, {"start": 140.02, "duration": 5.3, "text": "on or X to the fifth positive 3 times"}, {"start": 142.9, "duration": 6.21, "text": "negative 4 will give us negative 12"}, {"start": 145.32, "duration": 6.52, "text": "again we have to still multiply by B"}, {"start": 149.11, "duration": 5.61, "text": "squared and now I think okay what are"}, {"start": 151.84, "duration": 6.0, "text": "the like terms well I see something"}, {"start": 154.72, "duration": 6.42, "text": "involving an X to the fifth and again a"}, {"start": 157.84, "duration": 7.41, "text": "number with exactly X to the fifth so we"}, {"start": 161.14, "duration": 6.42, "text": "can combine those two terms well if"}, {"start": 165.25, "duration": 5.01, "text": "you've got 10 X to the fifth and"}, {"start": 167.56, "duration": 4.95, "text": "somebody gives you another 6 X to the"}, {"start": 170.26, "duration": 4.37, "text": "fifth right you're adding them you would"}, {"start": 172.51, "duration": 7.14, "text": "have 16x to the fifth"}, {"start": 174.63, "duration": 7.18, "text": "and again our minus 12 B squared will"}, {"start": 179.65, "duration": 3.39, "text": "just drop down and at this point there's"}, {"start": 181.81, "duration": 3.899, "text": "nothing else we can do"}, {"start": 183.04, "duration": 4.35, "text": "so sometimes when I remember when I"}, {"start": 185.709, "duration": 3.541, "text": "first saw these types of problems I"}, {"start": 187.39, "duration": 3.629, "text": "would want to keep simplifying I would"}, {"start": 189.25, "duration": 4.86, "text": "want to somehow just get a single thing"}, {"start": 191.019, "duration": 6.301, "text": "and definitely that doesn't happen so"}, {"start": 194.11, "duration": 7.159, "text": "there may only be really a minut amount"}, {"start": 197.32, "duration": 3.949, "text": "of simplification that you can do"}], "title": "Simplify an Algebraic Expression by Combining Like Terms", "description": "Thanks to all of you who support me on Patreon. You da real mvps!   $1 per month helps!! :)  https://www.patreon.com/patrickjmt !! Simplify an Algebraic Expression by Combining Like Terms.  Here we simplify a couple of algebraic expressions by combining like terms by adding, subtracting, and using distribution.", "lengthSeconds": 201, "width": 480, "height": 360}
{"video_id": "fkrLkvQSlmk", "transcript": [{"start": 0.03, "duration": 3.81, "text": "all right so just a couple more examples"}, {"start": 1.74, "duration": 4.889, "text": "here of simplifying down some trig"}, {"start": 3.84, "duration": 4.86, "text": "expressions using identities so here"}, {"start": 6.629, "duration": 7.26, "text": "we're gonna simplify cosecant squared X"}, {"start": 8.7, "duration": 7.65, "text": "minus 1 over cosecant squared X and I'm"}, {"start": 13.889, "duration": 5.911, "text": "gonna get make use here I'm gonna write"}, {"start": 16.35, "duration": 6.179, "text": "so we've got a tangent squared X plus 1"}, {"start": 19.8, "duration": 4.38, "text": "equals secant squared X we don't need"}, {"start": 22.529, "duration": 3.84, "text": "that one just yet"}, {"start": 24.18, "duration": 7.829, "text": "the other identity is cotangent squared"}, {"start": 26.369, "duration": 6.78, "text": "X plus 1 equals cosecant squared X and I"}, {"start": 32.009, "duration": 4.081, "text": "think that's the one we're going to need"}, {"start": 33.149, "duration": 6.271, "text": "in this case so notice if you subtract 1"}, {"start": 36.09, "duration": 5.55, "text": "from this second identity will get"}, {"start": 39.42, "duration": 4.799, "text": "cosecant squared X minus 1 equals"}, {"start": 41.64, "duration": 3.989, "text": "cotangent squared so that's what I'm"}, {"start": 44.219, "duration": 3.961, "text": "gonna write in the numerator"}, {"start": 45.629, "duration": 7.081, "text": "it says cosecant squared X minus 1 would"}, {"start": 48.18, "duration": 6.719, "text": "be cotangent squared X and that's all"}, {"start": 52.71, "duration": 5.279, "text": "being divided by the original cosecant"}, {"start": 54.899, "duration": 6.09, "text": "squared X and at this point you know"}, {"start": 57.989, "duration": 5.851, "text": "probably what I would do is I would"}, {"start": 60.989, "duration": 7.471, "text": "rewrite these so let's see cotangent is"}, {"start": 63.84, "duration": 6.48, "text": "gonna be cosine X over sine X but again"}, {"start": 68.46, "duration": 5.82, "text": "they're being squared so I'm gonna put"}, {"start": 70.32, "duration": 8.939, "text": "the squares on both of them cosecant"}, {"start": 74.28, "duration": 6.81, "text": "squared that's 1 over sine squared X so"}, {"start": 79.259, "duration": 5.671, "text": "really I've got cosine squared over sine"}, {"start": 81.09, "duration": 6.919, "text": "squared all over 1 over sine squared so"}, {"start": 84.93, "duration": 7.02, "text": "that's going to give me cosine squared X"}, {"start": 88.009, "duration": 6.72, "text": "over sine squared X and now I can take"}, {"start": 91.95, "duration": 7.5, "text": "the bottom fraction and just flip it"}, {"start": 94.729, "duration": 6.85, "text": "we'll have sine squared X over 1 well we"}, {"start": 99.45, "duration": 5.52, "text": "can cancel the sine Squared's out and"}, {"start": 101.579, "duration": 5.501, "text": "we'll simply be left with cosine squared"}, {"start": 104.97, "duration": 5.18, "text": "X"}, {"start": 107.08, "duration": 8.25, "text": "all right so Part D I would say that"}, {"start": 110.15, "duration": 7.98, "text": "reduce is simply two cosine squared X"}, {"start": 115.33, "duration": 5.26, "text": "alright so here we've got cosecant"}, {"start": 118.13, "duration": 4.77, "text": "squared theta minus cotangent squared"}, {"start": 120.59, "duration": 5.6, "text": "theta and tangent squared theta minus"}, {"start": 122.9, "duration": 3.29, "text": "secant squared theta"}, {"start": 126.94, "duration": 4.15, "text": "notice that looks an awful lot like"}, {"start": 129.05, "duration": 4.86, "text": "these first two identities I put down"}, {"start": 131.09, "duration": 5.01, "text": "that's why I put them down there let's"}, {"start": 133.91, "duration": 4.58, "text": "see if we can't sort of manipulate these"}, {"start": 136.1, "duration": 4.859, "text": "identities to get these expressions"}, {"start": 138.49, "duration": 4.63, "text": "let's maybe deal with the numerator"}, {"start": 140.959, "duration": 4.831, "text": "first we've got tangent squared theta"}, {"start": 143.12, "duration": 5.28, "text": "minus secant squared well I could"}, {"start": 145.79, "duration": 4.31, "text": "subtract the secant squared over and"}, {"start": 148.4, "duration": 7.19, "text": "that would give me tangent squared X"}, {"start": 150.1, "duration": 8.53, "text": "minus secant squared X plus 1 equals 0"}, {"start": 155.59, "duration": 5.08, "text": "well then I can simply subtract the 1"}, {"start": 158.63, "duration": 6.0, "text": "from both sides and I'll get tangent"}, {"start": 160.67, "duration": 9.42, "text": "squared x minus secant squared x equals"}, {"start": 164.63, "duration": 8.25, "text": "well negative 1 so in the denominator it"}, {"start": 170.09, "duration": 5.01, "text": "says we'll be left with a negative 1 and"}, {"start": 172.88, "duration": 4.23, "text": "I'm going to do the same thing to figure"}, {"start": 175.1, "duration": 5.16, "text": "out an expression for cosecant squared"}, {"start": 177.11, "duration": 5.849, "text": "minus cotangent squared well that's our"}, {"start": 180.26, "duration": 5.46, "text": "second identity here so again we want"}, {"start": 182.959, "duration": 4.081, "text": "the cosecant squared to stay positive so"}, {"start": 185.72, "duration": 3.21, "text": "what I'm going to do is just simply"}, {"start": 187.04, "duration": 4.71, "text": "subtract cotangent squared from both"}, {"start": 188.93, "duration": 8.01, "text": "sides and that'll give us cosecant"}, {"start": 191.75, "duration": 8.19, "text": "squared X minus cotangent squared X well"}, {"start": 196.94, "duration": 4.079, "text": "that's the same thing as positive 1 so"}, {"start": 199.94, "duration": 3.42, "text": "that's what I'm going to stick in the"}, {"start": 201.019, "duration": 4.771, "text": "numerator so I really have 1 over"}, {"start": 203.36, "duration": 5.54, "text": "negative 1 which is simply going to give"}, {"start": 205.79, "duration": 3.11, "text": "us negative 1"}], "title": "Simplifying Trigonometric Expressions Using Identities, Example 3", "description": "Thanks to all of you who support me on Patreon. You da real mvps!   $1 per month helps!! :)  https://www.patreon.com/patrickjmt !! Simplifying Trigonometric Expressions Using Identities, Example 3. In this video, I simplify a trigonometry expression using some trig identities.", "lengthSeconds": 208, "width": 480, "height": 360}
{"video_id": "-4aUzPVlx3s", "transcript": [{"start": 0.56, "duration": 2.879, "text": "all right in this video I just want to"}, {"start": 1.64, "duration": 4.48, "text": "do a couple examples about determining"}, {"start": 3.439, "duration": 6.001, "text": "Symmetry and using these definitions so"}, {"start": 6.12, "duration": 6.519, "text": "if f ofx = f ofx we say it's symmetric"}, {"start": 9.44, "duration": 6.159, "text": "about the Y AIS or it's an even function"}, {"start": 12.639, "duration": 4.56, "text": "if f ofx equals the negative of f ofx we"}, {"start": 15.599, "duration": 4.44, "text": "say it's an odd function and it's"}, {"start": 17.199, "duration": 4.281, "text": "symmetric about the origin so we've got"}, {"start": 20.039, "duration": 3.841, "text": "a couple formulas here we're just trying"}, {"start": 21.48, "duration": 5.199, "text": "to figure out what type of symmetry um"}, {"start": 23.88, "duration": 5.36, "text": "if any these functions possess so again"}, {"start": 26.679, "duration": 7.521, "text": "the idea is we just replace all the X's"}, {"start": 29.24, "duration": 4.96, "text": "with negative X X Sox"}, {"start": 35.76, "duration": 7.319, "text": "4x^2 + 5 um well if you take a negative"}, {"start": 41.16, "duration": 4.04, "text": "number and raise it to an even power"}, {"start": 43.079, "duration": 4.64, "text": "it's going to be positive x to the 4th"}, {"start": 45.2, "duration": 7.159, "text": "would still be X to 4th so this would"}, {"start": 47.719, "duration": 6.64, "text": "just simplify to 3x 4th likewise x * X"}, {"start": 52.359, "duration": 5.121, "text": "is postive"}, {"start": 54.359, "duration": 5.561, "text": "x^2 POS 5 still just hanging out there"}, {"start": 57.48, "duration": 4.48, "text": "so notice we plugged in Nega X we got"}, {"start": 59.92, "duration": 5.879, "text": "got back the very original function we"}, {"start": 61.96, "duration": 6.64, "text": "started with so it says F ofx does equal"}, {"start": 65.799, "duration": 6.761, "text": "F ofx so in this case we would say that"}, {"start": 68.6, "duration": 3.96, "text": "this is what's called an even"}, {"start": 73.4, "duration": 4.6, "text": "function okay so it's symmetric about"}, {"start": 75.479, "duration": 6.68, "text": "the Y AIS maybe one other one real quick"}, {"start": 78.0, "duration": 7.72, "text": "suppose we have f ofx = x Cub - x well"}, {"start": 82.159, "duration": 8.64, "text": "if we plug in Negative X we'll"}, {"start": 85.72, "duration": 6.64, "text": "getx cubed minus X"}, {"start": 90.799, "duration": 4.561, "text": "well we can get rid of the parentheses"}, {"start": 92.36, "duration": 6.119, "text": "if you Cube 1 three times will get"}, {"start": 95.36, "duration": 6.68, "text": "Negative we still have X cubed two"}, {"start": 98.479, "duration": 5.92, "text": "negatives make a positive X and notice"}, {"start": 102.04, "duration": 5.88, "text": "we could Factor the negative out we"}, {"start": 104.399, "duration": 5.601, "text": "would have X Cub - x but again that's"}, {"start": 107.92, "duration": 4.0, "text": "the negative of the very original"}, {"start": 110.0, "duration": 4.52, "text": "function that we started with that's the"}, {"start": 111.92, "duration": 5.44, "text": "negative of f ofx so in this case we"}, {"start": 114.52, "duration": 4.48, "text": "would say hey this is an odd function"}, {"start": 117.36, "duration": 4.6, "text": "okay it has the Symmetry about the"}, {"start": 119.0, "duration": 5.88, "text": "origin and usually I kind of stop um"}, {"start": 121.96, "duration": 4.92, "text": "honestly I stop at this step right here"}, {"start": 124.88, "duration": 4.28, "text": "if you look at the signs on the original"}, {"start": 126.88, "duration": 5.04, "text": "notice the original X Cub was positive"}, {"start": 129.16, "duration": 4.64, "text": "and now it's negative the original uh"}, {"start": 131.92, "duration": 4.679, "text": "next sign was a negative now it's a"}, {"start": 133.8, "duration": 4.799, "text": "positive if all the signs flip flop"}, {"start": 136.599, "duration": 3.881, "text": "that's again I mean you can basically do"}, {"start": 138.599, "duration": 4.36, "text": "the step of factoring the negative out"}, {"start": 140.48, "duration": 4.68, "text": "but if you recognize that all the signs"}, {"start": 142.959, "duration": 6.64, "text": "have uh just switched that's going to"}, {"start": 145.16, "duration": 4.439, "text": "imply that it's an odd function"}], "title": "Symmetry - A Quick Discussion for Testing if a Polynomial is Even / Odd", "description": "Thanks to all of you who support me on Patreon. You da real mvps!   $1 per month helps!! :)  https://www.patreon.com/patrickjmt !! Symmetry - A Quick Discussion for Testing if a Polynomial is Even / Odd.  In this video, I give the definition of an even function and an odd function and use it to discuss if two given polynomials possess any symmetry.", "lengthSeconds": 148, "width": 480, "height": 360}
{"video_id": "skAirQrCehY", "transcript": [{"start": 0.76, "duration": 3.96, "text": "okay in this video we're going to do a"}, {"start": 2.04, "duration": 7.32, "text": "rough sketch of the function f ofx equal"}, {"start": 4.72, "duration": 5.879, "text": "the quantity x + 1 4us 1 so the thing"}, {"start": 9.36, "duration": 4.12, "text": "I'm going to do is I'm just going to"}, {"start": 10.599, "duration": 4.841, "text": "think about some graph shifting so x to"}, {"start": 13.48, "duration": 5.68, "text": "the"}, {"start": 15.44, "duration": 7.599, "text": "4th it looks a lot like x to the"}, {"start": 19.16, "duration": 6.6, "text": "2 it has that same U shape to it so"}, {"start": 23.039, "duration": 4.56, "text": "that's going to be the graph of x to 4th"}, {"start": 25.76, "duration": 3.64, "text": "well the plus one inside of the"}, {"start": 27.599, "duration": 4.68, "text": "parentheses that's going to shift our"}, {"start": 29.4, "duration": 5.76, "text": "graph of X to 4th it's going to shift"}, {"start": 32.279, "duration": 6.081, "text": "the graph one"}, {"start": 35.16, "duration": 5.32, "text": "unit left so again when it's in the"}, {"start": 38.36, "duration": 3.96, "text": "parentheses it's I always remembered"}, {"start": 40.48, "duration": 3.399, "text": "it's sort of the opposite right it feels"}, {"start": 42.32, "duration": 3.12, "text": "like it should you know you see a"}, {"start": 43.879, "duration": 3.721, "text": "positive one which makes you think to"}, {"start": 45.44, "duration": 4.52, "text": "the right but it actually goes one unit"}, {"start": 47.6, "duration": 4.2, "text": "to the left the negative one on the"}, {"start": 49.96, "duration": 4.439, "text": "outside is going to have the effect of"}, {"start": 51.8, "duration": 4.439, "text": "Shifting the graph it's going to shift"}, {"start": 54.399, "duration": 5.601, "text": "one"}, {"start": 56.239, "duration": 5.48, "text": "unit down okay so if it's on the outside"}, {"start": 60.0, "duration": 4.08, "text": "you see a negative hey do go down if"}, {"start": 61.719, "duration": 4.241, "text": "it's a positive do go up so it's"}, {"start": 64.08, "duration": 4.44, "text": "basically going to take our graph of x"}, {"start": 65.96, "duration": 4.36, "text": "to the 4th one unit to the left one unit"}, {"start": 68.52, "duration": 4.239, "text": "down and again if you're not sure you"}, {"start": 70.32, "duration": 4.88, "text": "can always plot points that's one thing"}, {"start": 72.759, "duration": 5.481, "text": "that I think people sometimes are forget"}, {"start": 75.2, "duration": 5.599, "text": "about doing so instead of passing"}, {"start": 78.24, "duration": 4.919, "text": "through the origin 0 0 since it's one to"}, {"start": 80.799, "duration": 6.841, "text": "the left and one down it'll pass through"}, {"start": 83.159, "duration": 8.361, "text": "the point -1 comma -1 notice if we plug"}, {"start": 87.64, "duration": 8.759, "text": "in X = 0 we'll get 1 to 4 which is 1"}, {"start": 91.52, "duration": 7.36, "text": "minus 1 will get 0er likewise at -2 by"}, {"start": 96.399, "duration": 6.0, "text": "symmetry it it will equal"}, {"start": 98.88, "duration": 7.879, "text": "Zer notice if we plug in x = 1 we would"}, {"start": 102.399, "duration": 8.441, "text": "have 1 + 1 which is 2 2 4th is going to"}, {"start": 106.759, "duration": 6.4, "text": "be 16 minus 1 it says at at xal 1 we"}, {"start": 110.84, "duration": 5.84, "text": "should actually already be way up here"}, {"start": 113.159, "duration": 7.161, "text": "at yals 15 so my"}, {"start": 116.68, "duration": 7.119, "text": "graph I'm going to make it get pretty"}, {"start": 120.32, "duration": 3.479, "text": "deep uh pretty"}, {"start": 124.68, "duration": 4.88, "text": "quickly and again that's going to be"}, {"start": 126.759, "duration": 9.681, "text": "just a rough sketch of our of our new"}, {"start": 129.56, "duration": 6.88, "text": "graph the quantity x + 1 4 minus 1"}], "title": "\u2756 Horizontal and Vertical Shifts Using f(x) = x^4 - Fast Example! \u2756", "description": "Horizontal and Vertical Shifts Using f(x) = x^4\n\nIn this video, we'll explore how to perform both horizontal and vertical shifts on the parent function f(x) = x^4. This quick example demonstrates how shifting transformations work, moving through the process a bit faster for those who are looking for a quick refresher on graph transformations. Ideal for anyone familiar with basic graphing techniques who wants to brush up on shifting transformations.\n\nWhat You Will Learn:\n\nHow to shift a function vertically and horizontally.\nUnderstanding transformations of the parent function f(x) = x^4\n\n .Quick tips on graphing and interpreting shifted functions\n\nSupport my work on Patreon!\nhttps://www.patreon.com/patrickjmt?ty=c\n\n#Math #Algebra #GraphTransformations #ShiftingFunctions #Graphing #VerticalShift #HorizontalShift #ParentFunctions #Precalculus #Mathematics #FunctionGraphing #GraphShifts #QuadraticGraph #AlgebraReview", "lengthSeconds": 135, "width": 450, "height": 360}
{"video_id": "TjpwzoiSlys", "transcript": [{"start": 0.56, "duration": 2.92, "text": "all right here we're going to look at"}, {"start": 1.439, "duration": 3.92, "text": "some very basic differential equations"}, {"start": 3.48, "duration": 3.919, "text": "and again a differential equation is"}, {"start": 5.359, "duration": 3.961, "text": "simply an equation that involves"}, {"start": 7.399, "duration": 5.081, "text": "derivativ so here we've got the"}, {"start": 9.32, "duration": 5.359, "text": "differential equation dydx = x 4th we"}, {"start": 12.48, "duration": 5.48, "text": "want to know which one of these uh three"}, {"start": 14.679, "duration": 5.76, "text": "functions a b or c are solutions to this"}, {"start": 17.96, "duration": 4.96, "text": "differential equation and maybe there's"}, {"start": 20.439, "duration": 4.961, "text": "uh more than one solution but again all"}, {"start": 22.92, "duration": 5.439, "text": "this really says is is which derivative"}, {"start": 25.4, "duration": 6.8, "text": "is gives us just x to the 4th so we can"}, {"start": 28.359, "duration": 8.481, "text": "check part A Dy DX here will'll get 1/5"}, {"start": 32.2, "duration": 7.679, "text": "* 5 x 4th + 1 well that leaves us with X"}, {"start": 36.84, "duration": 6.039, "text": "4th + 1 but that's certainly not just x"}, {"start": 39.879, "duration": 5.641, "text": "to the 4th so part A is not our solute"}, {"start": 42.879, "duration": 6.52, "text": "the not our solution Part B if we take"}, {"start": 45.52, "duration": 6.879, "text": "the derivative of 4 x + 5 the derivative"}, {"start": 49.399, "duration": 5.761, "text": "of 4 x is going to be 4 x * the natural"}, {"start": 52.399, "duration": 5.361, "text": "logarithm of 4 the plus five just turns"}, {"start": 55.16, "duration": 5.6, "text": "into a constant again that's definitely"}, {"start": 57.76, "duration": 5.88, "text": "not just x to the 4th"}, {"start": 60.76, "duration": 5.08, "text": "and likewise we can check part C here"}, {"start": 63.64, "duration": 6.119, "text": "well the same thing the derivative of y"}, {"start": 65.84, "duration": 6.599, "text": "with respect to x uh we'll get 1/5 * 5x"}, {"start": 69.759, "duration": 6.321, "text": "4th which will give us X 4th the"}, {"start": 72.439, "duration": 7.561, "text": "derivative of Pi is just zero so hey we"}, {"start": 76.08, "duration": 5.76, "text": "found our solution uh uh of these so"}, {"start": 80.0, "duration": 4.04, "text": "part C would be our solution to this"}, {"start": 81.84, "duration": 4.88, "text": "differential"}, {"start": 84.04, "duration": 6.0, "text": "equation let's maybe look at uh one"}, {"start": 86.72, "duration": 5.039, "text": "other very similar idea and now in inad"}, {"start": 90.04, "duration": 3.48, "text": "of just a first derivative we've got a"}, {"start": 91.759, "duration": 4.961, "text": "second derivative so the second"}, {"start": 93.52, "duration": 5.0, "text": "derivative with respect to Y is -4 y we"}, {"start": 96.72, "duration": 5.12, "text": "have these two functions we want to know"}, {"start": 98.52, "duration": 6.48, "text": "which one of these uh again potentially"}, {"start": 101.84, "duration": 6.36, "text": "both uh could be Solutions here so all"}, {"start": 105.0, "duration": 6.88, "text": "right we've got y = cosine of"}, {"start": 108.2, "duration": 7.04, "text": "4X well the first derivative Dy DX the"}, {"start": 111.88, "duration": 5.12, "text": "derivative of cosine is negative s you"}, {"start": 115.24, "duration": 3.32, "text": "leave the inside alone have to chain"}, {"start": 117.0, "duration": 3.6, "text": "rule it so the derivative of the"}, {"start": 118.56, "duration": 6.32, "text": "inside's going to be four"}, {"start": 120.6, "duration": 8.4, "text": "so we'll get --4 sin of"}, {"start": 124.88, "duration": 4.12, "text": "4X well when we take a second"}, {"start": 129.2, "duration": 9.28, "text": "derivative we'll get -4 * cosine of 4X *"}, {"start": 135.16, "duration": 4.96, "text": "4 that's going to give us -16 * cosine"}, {"start": 138.48, "duration": 3.92, "text": "of"}, {"start": 140.12, "duration": 5.28, "text": "4X and you know definitely that doesn't"}, {"start": 142.4, "duration": 5.04, "text": "look like -4 y we can certainly replace"}, {"start": 145.4, "duration": 5.16, "text": "the starting function right we started"}, {"start": 147.44, "duration": 6.84, "text": "with yal cosine of 4X so we could always"}, {"start": 150.56, "duration": 6.759, "text": "plug that back in to the cosine 4X so we"}, {"start": 154.28, "duration": 5.72, "text": "could rewrite this as -16 y but again"}, {"start": 157.319, "duration": 5.161, "text": "that's not the solution that we want so"}, {"start": 160.0, "duration": 7.28, "text": "part A doesn't"}, {"start": 162.48, "duration": 4.8, "text": "work let's try Part B here s of"}, {"start": 169.599, "duration": 6.881, "text": "2x so the first derivative dy over DX I"}, {"start": 174.76, "duration": 3.399, "text": "was going to write y Prime which is"}, {"start": 176.48, "duration": 2.88, "text": "certainly legal but uh you know I'm just"}, {"start": 178.159, "duration": 4.72, "text": "going to stick with the notation that"}, {"start": 179.36, "duration": 6.48, "text": "we've got here the liance notation so s"}, {"start": 182.879, "duration": 8.0, "text": "of 2x the derivative will be cosine of"}, {"start": 185.84, "duration": 9.039, "text": "2x * 2 or hey just 2 cosine of"}, {"start": 190.879, "duration": 6.681, "text": "2x the second derivative with respect to"}, {"start": 194.879, "duration": 5.44, "text": "Y well let's see the derivative of"}, {"start": 197.56, "duration": 4.84, "text": "cosine is going to be negative s we'll"}, {"start": 200.319, "duration": 3.881, "text": "leave the 2x alone but then again the"}, {"start": 202.4, "duration": 4.119, "text": "chain rule so we'll have to multiply by"}, {"start": 204.2, "duration": 8.239, "text": "another two and let's see that's going"}, {"start": 206.519, "duration": 7.881, "text": "to give us -4 * s of 2x but again the"}, {"start": 212.439, "duration": 5.921, "text": "same thing as before we can replace S of"}, {"start": 214.4, "duration": 6.24, "text": "2x uh with just y so we're left with ne4"}, {"start": 218.36, "duration": 5.12, "text": "Y that's exactly what we wanted to be"}, {"start": 220.64, "duration": 4.519, "text": "left with so now we found our solution"}, {"start": 223.48, "duration": 4.679, "text": "here uh Part"}, {"start": 225.159, "duration": 3.0, "text": "B"}], "title": "Differential Equation : What Does It Mean to Be a Solution", "description": "Thanks to all of you who support me on Patreon. You da real mvps!   $1 per month helps!! :)  https://www.patreon.com/patrickjmt !! Differential Equation : What Does It Mean to Be a Solution.\nJust a quick example of what it means when a function is a solution of a differential equation!", "lengthSeconds": 226, "width": 480, "height": 360}
{"video_id": "It_bq5DzQ1Q", "transcript": [{"start": 0.64, "duration": 2.84, "text": "all right in this video we're going to"}, {"start": 1.76, "duration": 4.119, "text": "do an example related to the law of"}, {"start": 3.48, "duration": 5.84, "text": "signs and all the law of signs does it"}, {"start": 5.879, "duration": 6.561, "text": "relates uh angles and the sides opposite"}, {"start": 9.32, "duration": 4.88, "text": "them by this equation so little a little"}, {"start": 12.44, "duration": 4.96, "text": "B little c those will be the side"}, {"start": 14.2, "duration": 5.4, "text": "lengths uh a b and c again capital A B"}, {"start": 17.4, "duration": 3.92, "text": "and C are just going to be the angles so"}, {"start": 19.6, "duration": 3.999, "text": "it says we get little a over sin a"}, {"start": 21.32, "duration": 6.279, "text": "equals little B over sin B equals little"}, {"start": 23.599, "duration": 6.401, "text": "C over sin C so let's do an example here"}, {"start": 27.599, "duration": 5.361, "text": "suppose we're given that one angle of a"}, {"start": 30.0, "duration": 6.2, "text": "triangle is 120\u00b0 and another angle is"}, {"start": 32.96, "duration": 6.24, "text": "40\u00b0 we know the side opposite the 40\u00b0"}, {"start": 36.2, "duration": 6.44, "text": "angle measures 12 M we want to find the"}, {"start": 39.2, "duration": 5.679, "text": "side opposite the unknown angle so just"}, {"start": 42.64, "duration": 4.8, "text": "going to make a little picture"}, {"start": 44.879, "duration": 5.16, "text": "here try to make it somewhat to scale"}, {"start": 47.44, "duration": 6.52, "text": "hopefully so um so here's let's make"}, {"start": 50.039, "duration": 6.921, "text": "this our 120\u00b0 angle this will be our 40\u00b0"}, {"start": 53.96, "duration": 6.36, "text": "angle the side opposite the 40\u00b0 angle"}, {"start": 56.96, "duration": 5.96, "text": "that has length 12 well this would our"}, {"start": 60.32, "duration": 4.559, "text": "missing angle uh in our triangle and"}, {"start": 62.92, "duration": 5.72, "text": "we're simply trying to find the length"}, {"start": 64.879, "duration": 7.481, "text": "opposite that so maybe I'll call that"}, {"start": 68.64, "duration": 5.96, "text": "c all right well we know that the angles"}, {"start": 72.36, "duration": 4.079, "text": "inside of a triangle add up to 180 so"}, {"start": 74.6, "duration": 5.44, "text": "120 + 40"}, {"start": 76.439, "duration": 6.561, "text": "is60 um which means the other angle must"}, {"start": 80.04, "duration": 5.32, "text": "be simply 20\u00b0 those will now all add up"}, {"start": 83.0, "duration": 7.759, "text": "to 180 and now we can just set up our"}, {"start": 85.36, "duration": 11.399, "text": "equation we can do 12 over s of 4"}, {"start": 90.759, "duration": 8.72, "text": "40\u00b0 that's going to equal C over s of"}, {"start": 96.759, "duration": 8.0, "text": "20\u00b0 and now to solve for C we'll simply"}, {"start": 99.479, "duration": 5.28, "text": "just multiply both sides by S of"}, {"start": 105.079, "duration": 6.161, "text": "20\u00b0 so on the left we'll be left with 12"}, {"start": 108.2, "duration": 7.12, "text": "* s of"}, {"start": 111.24, "duration": 7.879, "text": "20\u00b0 over s of"}, {"start": 115.32, "duration": 5.32, "text": "40\u00b0 that's going to be our value for C"}, {"start": 119.119, "duration": 4.04, "text": "and now just imag kind of plugging it"}, {"start": 120.64, "duration": 6.36, "text": "all into a calculator and simplifying it"}, {"start": 123.159, "duration": 9.08, "text": "down here a little bit"}, {"start": 127.0, "duration": 7.76, "text": "so all right so let's do s of 20"}, {"start": 132.239, "duration": 4.481, "text": "degre I'm going to round a three decimal"}, {"start": 134.76, "duration": 4.199, "text": "places just sort of arbitrarily so"}, {"start": 136.72, "duration": 2.239, "text": "that's"}, {"start": 139.2, "duration": 7.119, "text": "0.342 um if we do s of 40\u00b0 I'm getting"}, {"start": 143.2, "duration": 3.119, "text": "that to be"}, {"start": 146.599, "duration": 5.241, "text": "0.643 after rounding so if we take"}, {"start": 153.48, "duration": 8.039, "text": "0.342 and if we multiply that by 12 I'm"}, {"start": 158.0, "duration": 3.519, "text": "getting that to be"}, {"start": 162.0, "duration": 3.879, "text": "4104 uh over our"}, {"start": 166.84, "duration": 6.36, "text": "0.643 so now if we take uh"}, {"start": 170.12, "duration": 7.24, "text": "4104 and simply divide that by"}, {"start": 173.2, "duration": 4.16, "text": "0 whoops uh so"}, {"start": 178.2, "duration": 3.64, "text": "4104 divided by"}, {"start": 181.959, "duration": 4.681, "text": "by sorry I keep punching it in wrong"}, {"start": 184.959, "duration": 6.36, "text": "here so"}, {"start": 186.64, "duration": 4.679, "text": "4104 we'll divide that by"}, {"start": 191.959, "duration": 6.241, "text": "0.643 I'm getting this to be roughly"}, {"start": 194.72, "duration": 5.96, "text": "equal to"}, {"start": 198.2, "duration": 7.56, "text": "6.38 and again after rounding I'm"}, {"start": 200.68, "duration": 7.16, "text": "getting 6. 383 so um just a touch over"}, {"start": 205.76, "duration": 4.479, "text": "uh well a little bit over six meters is"}, {"start": 207.84, "duration": 3.759, "text": "what we're getting"}, {"start": 210.239, "duration": 3.241, "text": "and you know certainly that seems"}, {"start": 211.599, "duration": 4.601, "text": "reasonable um you know if the side"}, {"start": 213.48, "duration": 5.08, "text": "opposite 40\u00b0 is 12 certainly the side"}, {"start": 216.2, "duration": 5.319, "text": "opposite 20\u00b0 would have to be you know"}, {"start": 218.56, "duration": 4.679, "text": "significantly smaller um so certainly"}, {"start": 221.519, "duration": 3.92, "text": "that agrees with my intuition I think"}, {"start": 223.239, "duration": 3.64, "text": "you know that's in the ballpark at least"}, {"start": 225.439, "duration": 4.281, "text": "um so that's all there is to do we're"}, {"start": 226.879, "duration": 5.08, "text": "just simply setting up a little ratio uh"}, {"start": 229.72, "duration": 6.879, "text": "solving for our unknown value by using"}, {"start": 231.959, "duration": 4.64, "text": "these law of sign this law of signs"}], "title": "The Law of Sines, Example 1", "description": "Thanks to all of you who support me on Patreon. You da real mvps!   $1 per month helps!! :)  https://www.patreon.com/patrickjmt !! The Law of Sines, Example 1.\n  This video talks a bit about the Law of Sines.  I give the formula and proceed to do an example using it. Very exciting!! (it actually is quite useful though)", "lengthSeconds": 235, "width": 480, "height": 360}
{"video_id": "URoPcn0fIAg", "transcript": [{"start": 0.0, "duration": 4.23, "text": "all right in this video I'm going to do"}, {"start": 1.92, "duration": 5.33, "text": "an example related to oblique asymptotes"}, {"start": 4.23, "duration": 5.82, "text": "and recall that oblique asymptotes occur"}, {"start": 7.25, "duration": 4.78, "text": "when the degree of the numerator is one"}, {"start": 10.05, "duration": 4.29, "text": "larger than the degree of the"}, {"start": 12.03, "duration": 4.89, "text": "denominator so that's certainly what we"}, {"start": 14.34, "duration": 5.79, "text": "have in this case first off I'm going to"}, {"start": 16.92, "duration": 5.519, "text": "find any vertical asymptotes so vertical"}, {"start": 20.13, "duration": 5.01, "text": "asymptotes basically I just set the"}, {"start": 22.439, "duration": 5.881, "text": "denominator equal to zero that would"}, {"start": 25.14, "duration": 4.59, "text": "give me in this case x equals one and I"}, {"start": 28.32, "duration": 4.02, "text": "know that if you plug one in you get"}, {"start": 29.73, "duration": 4.41, "text": "zero on the bottom I also have to make"}, {"start": 32.34, "duration": 4.77, "text": "sure that if you plug one in we don't"}, {"start": 34.14, "duration": 4.62, "text": "get zero on the top I notice if you plug"}, {"start": 37.11, "duration": 5.55, "text": "one end when we get one plus four which"}, {"start": 38.76, "duration": 6.6, "text": "is 5 plus seven which is 12 over zero we"}, {"start": 42.66, "duration": 4.77, "text": "have something non zero over zero that"}, {"start": 45.36, "duration": 7.23, "text": "tells me that x equals one in this case"}, {"start": 47.43, "duration": 8.399, "text": "is a vertical asymptote again since the"}, {"start": 52.59, "duration": 5.64, "text": "numerator is larger degree than the"}, {"start": 55.829, "duration": 7.491, "text": "denominator that tells me that there are"}, {"start": 58.23, "duration": 5.09, "text": "no horizontal asymptotes"}, {"start": 66.439, "duration": 8.311, "text": "so no horizontal asymptotes since the"}, {"start": 69.439, "duration": 11.6, "text": "degree that of the numerator is larger"}, {"start": 74.75, "duration": 6.289, "text": "than the degree of the denominator"}, {"start": 82.75, "duration": 5.17, "text": "okay so last but not least we said there"}, {"start": 86.0, "duration": 4.259, "text": "is an oblique asymptote since the"}, {"start": 87.92, "duration": 5.1, "text": "numerator is one larger than the"}, {"start": 90.259, "duration": 5.25, "text": "denominator and to find it I basically"}, {"start": 93.02, "duration": 4.68, "text": "just have to do the division well to do"}, {"start": 95.509, "duration": 3.81, "text": "the division since there's just a linear"}, {"start": 97.7, "duration": 4.32, "text": "term on the bottom I think I'm going to"}, {"start": 99.319, "duration": 5.25, "text": "use synthetic division so there's our 1x"}, {"start": 102.02, "duration": 8.12, "text": "squared our positive 4x and our positive"}, {"start": 104.569, "duration": 8.881, "text": "7 again we see X minus 1 but we use +1"}, {"start": 110.14, "duration": 8.259, "text": "so if I drop the 1 down 1 times 1 is 1"}, {"start": 113.45, "duration": 8.159, "text": "we add we get 5 1 times 5 is 5 this will"}, {"start": 118.399, "duration": 4.801, "text": "give us a 12 again we start we started"}, {"start": 121.609, "duration": 3.72, "text": "with something squared so this would be"}, {"start": 123.2, "duration": 4.529, "text": "our X term and this would be our"}, {"start": 125.329, "duration": 5.22, "text": "constant and this would be our remainder"}, {"start": 127.729, "duration": 6.061, "text": "so it says really we can write x squared"}, {"start": 130.549, "duration": 8.931, "text": "plus 4x plus 7 divided by X minus 1 it"}, {"start": 133.79, "duration": 9.69, "text": "says we can rewrite that as 1x plus 5"}, {"start": 139.48, "duration": 7.39, "text": "with a remainder of 12 or we could have"}, {"start": 143.48, "duration": 6.569, "text": "written it as X plus 5 plus 12 over X"}, {"start": 146.87, "duration": 7.53, "text": "minus 1 but the idea is once you've done"}, {"start": 150.049, "duration": 8.581, "text": "the long division forget about the"}, {"start": 154.4, "duration": 5.94, "text": "remainder whatever is out front whatever"}, {"start": 158.63, "duration": 4.439, "text": "you get when you do the long division if"}, {"start": 160.34, "duration": 5.79, "text": "you omit the remainder that's going to"}, {"start": 163.069, "duration": 5.491, "text": "be your oblique asymptote so remember"}, {"start": 166.13, "duration": 4.469, "text": "our oblique asymptote is just a line so"}, {"start": 168.56, "duration": 4.91, "text": "it says in this case our oblique"}, {"start": 170.599, "duration": 2.871, "text": "asymptote"}, {"start": 175.11, "duration": 9.25, "text": "would be the line y equals x plus five"}, {"start": 179.97, "duration": 7.45, "text": "so there is one vertical asymptote at x"}, {"start": 184.36, "duration": 6.45, "text": "equals positive one again we said there"}, {"start": 187.42, "duration": 5.97, "text": "were no no horizontal asymptotes at all"}, {"start": 190.81, "duration": 6.26, "text": "and then we've got our oblique asymptote"}, {"start": 193.39, "duration": 3.68, "text": "of y equals x plus five"}], "title": "Find Asymptotes of a Rational Function (Vertical, Horizontal and Oblique/Slant) - Ex 1", "description": "Thanks to all of you who support me on Patreon. You da real mvps!   $1 per month helps!! :)  https://www.patreon.com/patrickjmt !! Find Asymptotes of a Rational Function (Vertical, Horizontal and Oblique/Slant).  In this video, I find the vertical asymptotes and a slant / oblique asymptotes of a rational function.", "lengthSeconds": 196, "width": 480, "height": 360}
{"video_id": "eXVAU_ExDy4", "transcript": [{"start": 0.86, "duration": 6.14, "text": "[Music]"}, {"start": 3.919, "duration": 3.081, "text": "the square Ro of"}, {"start": 7.25, "duration": 14.789, "text": "[Music]"}, {"start": 16.92, "duration": 5.119, "text": "two the golden racial"}, {"start": 22.279, "duration": 4.201, "text": "fire the square to"}, {"start": 28.32, "duration": 3.64, "text": "three we lose"}, {"start": 34.879, "duration": 4.081, "text": "comen the silver"}, {"start": 37.2, "duration": 3.6, "text": "R"}, {"start": 38.96, "duration": 4.9, "text": "scorpion"}, {"start": 40.8, "duration": 5.96, "text": "sub son elen"}, {"start": 43.86, "duration": 7.9, "text": "[Music]"}, {"start": 46.76, "duration": 5.0, "text": "2 the plastic raal"}, {"start": 54.359, "duration": 6.481, "text": "r l Master on constant Dem"}], "title": "Ten Mortal Math Constants", "description": "In this video, we show visualizations of ten classic mathematical constants:\n\n1) The square root of 2 as the length of the diagonal of a 1 by 1 square.\n2) Pi as the area of the circle with radius 1.\n3) Tau as the circumference of the circle with radius 1.\n4) The golden ratio as the number satisfying (x+1)/x=x/1.\n5) The square root of 3 as the altitude or area of an equilateral triangle with side length 2.\n6) Euler's number e as the ending x-coordinate yielding the area under the curve of 1/x starting at x=1.\n7) The silver ratio as the horizontal length across the regular octagon with side length 1 or the area of the largest rectangle inside the same regular octagon.\n8) The natural logarithm of 2, ln(2),  as the sum of the convergent alternating harmonic series.\n9) The plastic ratio as the limit of the ratio of side lengths in a spiral of equilateral triangles with side lengths dictated by the Padovan sequence.\n10) The Euler-Mascheroni constant as the difference between the harmonic series and the area under the curve y=1/x from 1 to infinity.\n\nIf you like this video, consider subscribing to the channel or consider buying me a coffee: https://www.buymeacoffee.com/VisualProofs. Thanks!\n\n #mathvideo\u200b #math\u200b #numbertheory #mtbos\u200b #manim\u200b #animation\u200b #theorem\u200b #pww\u200b #proofwithoutwords\u200b #visualproof\u200b #proof\u200b #iteachmath #constants #goldenratio #silverratio #plasticratio #squareroot #squareroot2 #squareroot3 #euler #eulerconstant #eulermascheroni #ln2 #logarithm #octagon #square #pi #tau \n\nTo learn more about animating with manim, check out:\nhttps://manim.community", "lengthSeconds": 60, "width": 360, "height": 640}
{"video_id": "uhtv4tRkqYI", "transcript": [{"start": 0.16, "duration": 3.559, "text": "connecting these two line segments of"}, {"start": 1.8, "duration": 4.64, "text": "length one produces a line segment of"}, {"start": 3.719, "duration": 4.521, "text": "length root2 draw a circle of radius one"}, {"start": 6.44, "duration": 3.6, "text": "at the end of that line segment draw"}, {"start": 8.24, "duration": 3.399, "text": "perpendicular to that line segment and"}, {"start": 10.04, "duration": 3.799, "text": "this produces a line segment of length"}, {"start": 11.639, "duration": 4.12, "text": "root three you can continue repeating"}, {"start": 13.839, "duration": 4.001, "text": "this process over and over again drawing"}, {"start": 15.759, "duration": 3.561, "text": "a circle of radius one drawing a"}, {"start": 17.84, "duration": 3.32, "text": "perpendicular to the previous line"}, {"start": 19.32, "duration": 3.4, "text": "segment and then the new line segment"}, {"start": 21.16, "duration": 3.68, "text": "will have a length given by the square"}, {"start": 22.72, "duration": 4.36, "text": "root of the next integer so here we've"}, {"start": 24.84, "duration": 4.759, "text": "constructed the length < TK of two up to"}, {"start": 27.08, "duration": 4.64, "text": "< TK of 7 repeating this process over"}, {"start": 29.599, "duration": 5.241, "text": "and over produces a spiral shape known"}, {"start": 31.72, "duration": 5.4, "text": "as the spiral of theodoris we can use"}, {"start": 34.84, "duration": 4.879, "text": "this spiral to construct lengths of sare"}, {"start": 37.12, "duration": 4.56, "text": "root of n for any n that we want where N"}, {"start": 39.719, "duration": 4.281, "text": "is a positive integer just using a"}, {"start": 41.68, "duration": 4.719, "text": "straight edge and a compass for example"}, {"start": 44.0, "duration": 4.48, "text": "as we keep spiraling out here we'll end"}, {"start": 46.399, "duration": 4.281, "text": "with a line segment whose length is < TK"}, {"start": 48.48, "duration": 5.28, "text": "of 200 do you think you could construct"}, {"start": 50.68, "duration": 3.08, "text": "this by hand"}], "title": "Spiral of Theodorus", "description": "This is a short, animated visual proof demonstrating how to construct square roots of any positive integer using the Spiral of Theodorus\n\nTo buy me a coffee, head over to https://www.buymeacoffee.com/VisualProofs \nThanks!\n\n\n#manim #math\u200b\u200b #mathshorts\u200b #mathvideo\u200b \n #construction #geometry #mtbos\u200b #manim\u200b #animation\u200b #theorem\u200b #pww\u200b #proofwithoutwords\u200b #visualproof\u200b #proof\u200b #iteachmath #spiral #theodorus #squareroot\n\n\nTo learn more about animating with manim, check out:\nhttps://manim.community", "lengthSeconds": 52, "width": 360, "height": 640}
{"video_id": "uIDVRUGkfM8", "transcript": [{"start": 0.0, "duration": 2.84, "text": "if you have a collection of objects that"}, {"start": 1.56, "duration": 3.199, "text": "have five different possible"}, {"start": 2.84, "duration": 4.079, "text": "characteristics you need a proper five-"}, {"start": 4.759, "duration": 4.441, "text": "set vend diagram like this one notice in"}, {"start": 6.919, "duration": 4.8, "text": "this diagram there are 32 possible"}, {"start": 9.2, "duration": 4.96, "text": "regions created by the five sets there"}, {"start": 11.719, "duration": 5.841, "text": "are five regions shown here that lie in"}, {"start": 14.16, "duration": 5.64, "text": "exactly one set after that we see 10"}, {"start": 17.56, "duration": 5.799, "text": "distinct regions pictured here that lie"}, {"start": 19.8, "duration": 6.2, "text": "in exactly two of the sets by symmetry"}, {"start": 23.359, "duration": 5.68, "text": "there are also 10 regions as shown here"}, {"start": 26.0, "duration": 5.16, "text": "that lie in exactly three of the sets"}, {"start": 29.039, "duration": 5.36, "text": "the five regions shown here lie in"}, {"start": 31.16, "duration": 5.559, "text": "exactly four sets and finally we have"}, {"start": 34.399, "duration": 4.441, "text": "one region that lies in all five sets"}, {"start": 36.719, "duration": 4.881, "text": "and one region that lies outside of all"}, {"start": 38.84, "duration": 4.84, "text": "five sets so this is a beautiful proper"}, {"start": 41.6, "duration": 5.24, "text": "five set vend diagram can you draw one"}, {"start": 43.68, "duration": 3.16, "text": "with six sets"}], "title": "Five set Venn diagram", "description": "In this video, we show an example of a five-set Venn Diagram by highlighting all 32 possible regions. \n\nIf you like this video, consider subscribing to the channel or consider buying me a coffee: https://www.buymeacoffee.com/VisualProofs. Thanks!\n\nHere is a longer, wordless video, showing the same thing along with four set Venn diagrams as well : https://youtu.be/WBot4kqc3ro\n\nThe image in this video was inspired by the wonderful article \"Venn Diagrams and Independent Families of Sets\" by Branko Gr\u00fcnbaum  (https://www.maa.org/sites/default/files/pdf/upload_library/22/Ford/BrankoGrunbaum.pdf)\n\n#logic #settheory #union #propositions  #intersection #setdifference  #setminus #setconnectives #subsets #venndiagram #visualproof #foursets #math #manim #discretemathematics #fivesets #complement\n\nTo learn more about animating with manim, check out:\nhttps://manim.community", "lengthSeconds": 45, "width": 360, "height": 640}
{"video_id": "0d59pTzdtaM", "transcript": [{"start": 0.0, "duration": 4.2, "text": "this is one of the coolest things I've"}, {"start": 1.979, "duration": 4.38, "text": "ever seen for two non-negative numbers"}, {"start": 4.2, "duration": 4.38, "text": "their arithmetic mean or simple average"}, {"start": 6.359, "duration": 4.261, "text": "is always greater than or equal to their"}, {"start": 8.58, "duration": 4.38, "text": "geometric mean or square root of their"}, {"start": 10.62, "duration": 4.62, "text": "product equality holds exactly when the"}, {"start": 12.96, "duration": 4.68, "text": "two numbers are equal there's an amazing"}, {"start": 15.24, "duration": 5.22, "text": "way to see this visually construct a"}, {"start": 17.64, "duration": 5.28, "text": "circle with diameter X Plus y the radius"}, {"start": 20.46, "duration": 5.22, "text": "of the circle is half the diameter so it"}, {"start": 22.92, "duration": 5.58, "text": "is the arithmetic mean now construct a"}, {"start": 25.68, "duration": 5.16, "text": "perpendicular chord where X meets y if"}, {"start": 28.5, "duration": 4.98, "text": "each semicard has length G then by the"}, {"start": 30.84, "duration": 5.16, "text": "chord chord power theorem G times G is"}, {"start": 33.48, "duration": 5.16, "text": "equal to x times Y which means G is"}, {"start": 36.0, "duration": 4.739, "text": "equal to the square root of x times y or"}, {"start": 38.64, "duration": 4.259, "text": "the geometric mean we can now"}, {"start": 40.739, "duration": 3.901, "text": "geometrically see the semi chord is"}, {"start": 42.899, "duration": 4.68, "text": "always less than or equal to the radius"}, {"start": 44.64, "duration": 5.579, "text": "with the quality if and only if x equals"}, {"start": 47.579, "duration": 4.98, "text": "y and that is precisely why the GM is"}, {"start": 50.219, "duration": 7.16, "text": "always less than or equal to the am and"}, {"start": 52.559, "duration": 4.82, "text": "thus we have proven the amgm inequality"}], "title": "Visual proof every student should see", "description": "This is one of my favorite visual proofs that uses geometry to demonstrate an algebraic inequality. The arithmetic mean is always greater than or equal to the geometric mean, with equality if and only if the numbers are equal. (Requires the numbers to be non-negative). #shorts \n\nSubscribe: https://www.youtube.com/user/MindYourDecisions?sub_confirmation=1\n\nSend me suggestions by email (address at end of many videos). I may not reply but I do consider all ideas!\n\nIf you purchase through these links, I may be compensated for purchases made on Amazon. As an Amazon Associate I earn from qualifying purchases. This does not affect the price you pay. \n\nBook ratings are from January 2022.\n\nMy Books (worldwide links)\nhttps://mindyourdecisions.com/blog/my-books/#worldwide\n\nMy Books (US links)\nMind Your Decisions: Five Book Compilation\nhttps://amzn.to/2pbJ4wR\nA collection of 5 books:\n\"The Joy of Game Theory\" rated 4.2/5 stars on 224 reviews\nhttp://amzn.to/1uQvA20\n\"The Irrationality Illusion: How To Make Smart Decisions And Overcome Bias\" rated 4/5 stars on 24 reviews\nhttp://amzn.to/1o3FaAg\n\"40 Paradoxes in Logic, Probability, and Game Theory\" rated 4.1/5 stars on 38 reviews\nhttp://amzn.to/1LOCI4U\n\"The Best Mental Math Tricks\" rated 4.2/5 stars on 76 reviews\nhttp://amzn.to/18maAdo\n\"Multiply Numbers By Drawing Lines\" rated 4.3/5 stars on 30 reviews\nhttp://amzn.to/XRm7M4\n\nMind Your Puzzles: Collection Of Volumes 1 To 3\nhttps://amzn.to/2mMdrJr\nA collection of 3 books:\n\"Math Puzzles Volume 1\" rated 4.4/5 stars on 87 reviews\nhttp://amzn.to/1GhUUSH\n\"Math Puzzles Volume 2\" rated 4.1/5 stars on 24 reviews\nhttp://amzn.to/1NKbyCs\n\"Math Puzzles Volume 3\" rated 4.2/5 stars on 22 reviews\nhttp://amzn.to/1NKbGlp\n\n2017 Shorty Awards Nominee. Mind Your Decisions was nominated in the STEM category (Science, Technology, Engineering, and Math) along with eventual winner Bill Nye; finalists Adam Savage, Dr. Sandra Lee, Simone Giertz, Tim Peake, Unbox Therapy; and other nominees Elon Musk, Gizmoslip, Hope Jahren, Life Noggin, and Nerdwriter.\n\nMy Blog\nhttps://mindyourdecisions.com/blog/\n\nTwitter\nhttps://twitter.com/preshtalwalkar\n\nInstagram\nhttps://www.instagram.com/preshtalwalkar/\n\nMerch\nhttps://teespring.com/stores/mind-your-decisions\n\nPatreon\nhttps://www.patreon.com/mindyourdecisions\n\nPress\nhttps://mindyourdecisions.com/blog/press #math #maths #mathematics", "lengthSeconds": 57, "width": 360, "height": 640}
{"video_id": "zRQhjp8lGX0", "transcript": [{"start": 0.08, "duration": 3.28, "text": "here's one of my favorite Pythagorean"}, {"start": 1.68, "duration": 3.56, "text": "theorem proofs start with a right"}, {"start": 3.36, "duration": 4.519, "text": "triangle with legs of length A and B and"}, {"start": 5.24, "duration": 5.279, "text": "a hypotenuse of C use four rotated"}, {"start": 7.879, "duration": 4.561, "text": "copies to create a c by C Square the"}, {"start": 10.519, "duration": 4.441, "text": "region in the middle is an A minus B by"}, {"start": 12.44, "duration": 4.0, "text": "Aus b square as the side length is equal"}, {"start": 14.96, "duration": 3.64, "text": "to the difference of the leg lengths of"}, {"start": 16.44, "duration": 4.64, "text": "the triangle the area enclosed in this"}, {"start": 18.6, "duration": 4.36, "text": "diagram is c^ s but now we can compute"}, {"start": 21.08, "duration": 4.32, "text": "the area of this diagram by moving"}, {"start": 22.96, "duration": 4.399, "text": "around the shapes inside if we place the"}, {"start": 25.4, "duration": 3.639, "text": "triangles next to each other like this"}, {"start": 27.359, "duration": 4.281, "text": "then we get an interesting shape with"}, {"start": 29.039, "duration": 5.121, "text": "Dimensions listed here but we can then"}, {"start": 31.64, "duration": 4.64, "text": "view this shape as a b by b square next"}, {"start": 34.16, "duration": 6.559, "text": "to an a by a square and therefore the"}, {"start": 36.28, "duration": 6.279, "text": "enclosed area is a 2 + b^ 2 because"}, {"start": 40.719, "duration": 5.041, "text": "these areas are the same we conclude"}, {"start": 42.559, "duration": 4.721, "text": "that a 2 + B2 = c^2 where A and B are"}, {"start": 45.76, "duration": 4.0, "text": "the leg lengths of a right triangle with"}, {"start": 47.28, "duration": 4.4, "text": "hypotenuse C this visual proof can be"}, {"start": 49.76, "duration": 5.72, "text": "attributed to bascara an Indian"}, {"start": 51.68, "duration": 3.8, "text": "mathematician from the 12th century"}], "title": "Pythagorean Theorem - Behold!", "description": "This is a short, animated visual proof of the Pythagorean theorem (the right triangle theorem) following essentially Bh\u0101skara's proof (Behold!). This theorem states the square of the hypotenuse of a right triangle is equal to the sum of squares of the two other side lengths. \n\nHere is a bit longer version: https://youtu.be/I75HYC8TRNY\n\nIf you like this video, consider subscribing to the channel or consider buying me a coffee: https://www.buymeacoffee.com/VisualProofs. Thanks!\n\nThis animation is based on a proof due to Bh\u0101skara. For a static version of this proof, see Roger Nelsen's first \"Proof Without Words: Exercises in Visual Thinking\" compendium (page 4). You can also check out Howard Eves' \"Great Moments in Mathematics (Before 1650)\" page 29-32.\n\nCheck out Roger's book on Amazon (affiliate link to follow; I may receive a small commission): https://amzn.to/3SjsqEh\n\nFor other proofs of this same fact check out: \nhttps://youtu.be/q28H5NY5UVU\nhttps://youtu.be/yfGtbNgcrQ8\nhttps://youtu.be/JRjPsuJ7S-g\nhttps://youtu.be/TsKTteGJpZU\nhttps://youtu.be/ZlGaQdNRdqA\nhttps://youtu.be/uqGjs_n-f7U\nhttps://youtu.be/wWEZ-rUlBDQ\n\n#math #manim #pythagoreantheorem #pythagorean #triangle #animation #theorem #pww #proofwithoutwords #visualproof #proof  #mathshorts #mathvideo \n\nTo learn more about animating with manim, check out:\nhttps://manim.community", "lengthSeconds": 54, "width": 360, "height": 640}
{"video_id": "xTN_ZxzPZHs", "transcript": [{"start": 0.12, "duration": 5.239, "text": "the N plus first triangular number t"}, {"start": 2.36, "duration": 5.96, "text": "subn + 1 can be represented by an n + 1"}, {"start": 5.359, "duration": 5.2, "text": "by n +1 triangular array of unit cubes"}, {"start": 8.32, "duration": 4.68, "text": "if we place n such triangular Rays side"}, {"start": 10.559, "duration": 5.441, "text": "by side we end up with an N by T sub n"}, {"start": 13.0, "duration": 5.6, "text": "+1 triangular staircase diagram when we"}, {"start": 16.0, "duration": 5.56, "text": "removed the back n by n +1 rectangle"}, {"start": 18.6, "duration": 5.2, "text": "we're left with an N by T subn staircase"}, {"start": 21.56, "duration": 4.92, "text": "diagram and we can decompose the N by"}, {"start": 23.8, "duration": 4.84, "text": "n+1 rectangle into two more n byn"}, {"start": 26.48, "duration": 3.879, "text": "triangular arrays we place these two"}, {"start": 28.64, "duration": 4.919, "text": "arrays next to the remaining St St case"}, {"start": 30.359, "duration": 5.72, "text": "diagram to produce an n + 2 by T subn"}, {"start": 33.559, "duration": 6.32, "text": "staircase diagram this means that n * t"}, {"start": 36.079, "duration": 5.201, "text": "subn + 1 is equal to n + 2 * T subn from"}, {"start": 39.879, "duration": 5.641, "text": "this we can conclude the recursive"}, {"start": 41.28, "duration": 7.24, "text": "formula that t subn + 1 = n + 2 / n * t"}, {"start": 45.52, "duration": 3.0, "text": "subn"}], "title": "A 3D Triangular Number Recursion", "description": "This is a short, animated visual proof demonstrating how to multiplicatively obtain the (n+1)st triangle number from the nth triangular number.\n\nTo buy me a coffee, head over to https://www.buymeacoffee.com/VisualProofs \nThanks!\n\nTo see a wide screen version of this proof check out A Multiplicative Triangular Number Recurrence (visual proof)\nhttps://youtu.be/69TvfNA0Lxc\n\nThis animation is included in the following sources:\n\nProofs without Words II by Roger B. Nelsen (MAA, 2000) (https://bookstore.ams.org/clrm-14/) page 14\n\n#manim #math #mathvideo #mathshorts #triangularnumbers  #numbertheory  #triangle  #animation #theorem #pww #proofwithoutwords #visualproof #proof\n\nTo learn more about animating with manim, check out:\nhttps://manim.community", "lengthSeconds": 47, "width": 360, "height": 640}
{"video_id": "AzAsz5itOfQ", "transcript": [{"start": 11.76, "duration": 16.739, "text": "[\u0645\u0648\u0633\u064a\u0642\u0649]"}, {"start": 29.96, "duration": 5.78, "text": "\u200fy"}, {"start": 31.64, "duration": 4.1, "text": "[\u0645\u0648\u0633\u064a\u0642\u0649]"}], "title": "#trending #mathematics Visual Proof: Factors of Quadratic Equation Polynomial PART-3 x^2+2x-3", "description": "#trending #mathematics Visual Proof: Factors of Quadratic Polynomial PART-3 x^2+2x-3\n\nOther Videos\nFibonacci Sequence, Golden Ratio, How to draw Golden Spiral:\nhttps://youtu.be/yHBa3EvDPco\nRadius and Tangent Vectors: \nhttps://youtube.com/shorts/KHSafRET3Sk\nDe Morgan's laws: Proof by Venn Diagrams:\nhttps://youtube.com/shorts/pR-REo4aBYk?feature=share\nVisual Proof of Roots of Quadratic Equation\nhttps://youtube.com/shorts/FdE7rHDZRd8\nVisual Proof: Factors of Quadratic Equation PART-1 x^2+3x+2\nhttps://youtube.com/shorts/lIqv5zKAKKE\nVisual Proof: Factors of Quadratic Polynomial PART-2 x^2-3x+2\nhttps://youtube.com/shorts/5pRYbKFWjUM", "lengthSeconds": 34, "width": 360, "height": 640}
{"video_id": "T2Mi0AUIKJY", "transcript": [{"start": 0.199, "duration": 4.521, "text": "start with a pentagon of area one take a"}, {"start": 2.6, "duration": 3.88, "text": "scaled copy that's size 1 six and place"}, {"start": 4.72, "duration": 4.28, "text": "it in the center leaving the outer ring"}, {"start": 6.48, "duration": 4.4, "text": "of size 56 we can cut the outer ring"}, {"start": 9.0, "duration": 4.519, "text": "into five congruent trapezoids each with"}, {"start": 10.88, "duration": 4.959, "text": "Area 1 six when we shade all five we've"}, {"start": 13.519, "duration": 4.481, "text": "shaded 5 six of the Pentagon do this"}, {"start": 15.839, "duration": 4.721, "text": "again on the inner Pentagon of Area 1 16"}, {"start": 18.0, "duration": 5.359, "text": "shading 56 of the 1 16 area therefore"}, {"start": 20.56, "duration": 4.639, "text": "shading a total area of 5 over 6^ SAR we"}, {"start": 23.359, "duration": 4.441, "text": "can repeat this process over and over"}, {"start": 25.199, "duration": 4.801, "text": "again each time shading 5 over 6 to the"}, {"start": 27.8, "duration": 4.12, "text": "N of the outer Pentagon if we can"}, {"start": 30.0, "duration": 4.12, "text": "continue this process indefinitely that"}, {"start": 31.92, "duration": 4.04, "text": "means if we consider a limiting process"}, {"start": 34.12, "duration": 3.599, "text": "we can look at the limiting shape and"}, {"start": 35.96, "duration": 4.24, "text": "the limiting shape is the filled"}, {"start": 37.719, "duration": 4.84, "text": "Pentagon therefore the Shaded area given"}, {"start": 40.2, "duration": 4.24, "text": "by the infinite sum matches the area of"}, {"start": 42.559, "duration": 3.52, "text": "the Pentagon which is one one"}, {"start": 44.44, "duration": 4.599, "text": "interpretation of this is that point"}, {"start": 46.079, "duration": 5.201, "text": "five repeating in base 6 is equal to 1"}, {"start": 49.039, "duration": 4.68, "text": "another one is that 5 * the infinite sum"}, {"start": 51.28, "duration": 4.119, "text": "of the positive powers of 1 six equals 1"}, {"start": 53.719, "duration": 5.961, "text": "or that the infinite sum of the positive"}, {"start": 55.399, "duration": 4.281, "text": "powers of 1 16 is 1/5"}], "title": "0.5555\u2026 = 1 (in base 6)", "description": "This is a short, animated visual proof showing the sum of the infinite geometric series with first term 5/6 and ratio 1/6, which in turn allows us to compute the sum of the series of powers of 1/6 and determine an interesting base 6 representation of 1. \n\nIf you like this video, consider subscribing to the channel or consider buying me a coffee: https://www.buymeacoffee.com/VisualProofs. Thanks! \n\nFor a longer, wordless version (more dramatic) of this animation see\nhttps://youtu.be/o0AyHmRV8RA?si=2fExvp1WqrmZp5SJ\n\nAlso, check out my playlist on geometric sums/series: https://youtube.com/playlist?list=PLZh9gzIvXQUsgw8W5TUVDtF0q4jEJ3iaw\n\nThis animation is based on a proof by James Tanton from the March 2008 issue of The College Mathematics Journal page 106. (https://www.jstor.org/stable/27646594).\n\n#mathshorts\u200b #mathvideo\u200b #math\u200b #calculus #mtbos\u200b #manim\u200b #animation\u200b #theorem\u200b #pww\u200b #proofwithoutwords\u200b #visualproof\u200b #proof\u200b #iteachmath #geometricsums #series #infinitesums #infiniteseries #geometric #geometricseries #pentagon #trapezoid \n\nTo learn more about animating with manim, check out:\nhttps://manim.community", "lengthSeconds": 58, "width": 360, "height": 640}
{"video_id": "4BP_BJjOet0", "transcript": [{"start": 0.56, "duration": 3.52, "text": "let's see a visual proof of an"}, {"start": 1.88, "duration": 5.12, "text": "interesting inequality using the unit"}, {"start": 4.08, "duration": 5.84, "text": "circle in the cartisian plane label the"}, {"start": 7.0, "duration": 5.599, "text": "origin o start with a radius of length"}, {"start": 9.92, "duration": 5.679, "text": "one and draw a unit circle centered at o"}, {"start": 12.599, "duration": 6.801, "text": "like this let's label the point on the"}, {"start": 15.599, "duration": 5.6, "text": "xais by a imagine a point p on the unit"}, {"start": 19.4, "duration": 4.48, "text": "circle in the first quadrant"}, {"start": 21.199, "duration": 5.041, "text": "corresponding to rotating an angle of T"}, {"start": 23.88, "duration": 4.399, "text": "if we drop a perpendicular from P to the"}, {"start": 26.24, "duration": 4.879, "text": "x-axis and label the intersection Point"}, {"start": 28.279, "duration": 6.081, "text": "M then the length of the line p m is the"}, {"start": 31.119, "duration": 5.44, "text": "S of T by the definition of s now use"}, {"start": 34.36, "duration": 5.0, "text": "the line pm to generate a circle"}, {"start": 36.559, "duration": 4.561, "text": "centered at M with radius s of T call"}, {"start": 39.36, "duration": 4.08, "text": "the intersection point of that Circle in"}, {"start": 41.12, "duration": 3.759, "text": "the x-axis B and label the other"}, {"start": 43.44, "duration": 4.24, "text": "intersection point between the two"}, {"start": 44.879, "duration": 4.561, "text": "circles Q notice that the point Q lies"}, {"start": 47.68, "duration": 3.96, "text": "directly below the point P so that"}, {"start": 49.44, "duration": 2.2, "text": "they're"}, {"start": 54.92, "duration": 5.2, "text": "antipaladin segment O M plus the length"}, {"start": 57.52, "duration": 4.96, "text": "of the line segment MP is equal to the"}, {"start": 60.12, "duration": 3.599, "text": "length of the line segment o and this is"}, {"start": 62.48, "duration": 5.04, "text": "greater than or equal to the length of"}, {"start": 63.719, "duration": 5.921, "text": "the line segment O A from these facts we"}, {"start": 67.52, "duration": 4.48, "text": "can see that the arc length of the Ark"}, {"start": 69.64, "duration": 5.6, "text": "pbq is greater than or equal to the arc"}, {"start": 72.0, "duration": 4.759, "text": "length of the Ark P AQ and that in turn"}, {"start": 75.24, "duration": 5.08, "text": "is greater than or equal to the length"}, {"start": 76.759, "duration": 6.121, "text": "of the chord PQ by construction the ark"}, {"start": 80.32, "duration": 5.28, "text": "pbq is the semicircle of a circle with"}, {"start": 82.88, "duration": 7.199, "text": "radius sin T and therefore has a length"}, {"start": 85.6, "duration": 7.44, "text": "of Pi * sin T The Arc P AQ is 2 two"}, {"start": 90.079, "duration": 6.481, "text": "copies of the Ark PA a subtending the"}, {"start": 93.04, "duration": 6.759, "text": "angle T and therefore the ark PA a q has"}, {"start": 96.56, "duration": 5.839, "text": "a length of 2 T and finally the length"}, {"start": 99.799, "duration": 5.6, "text": "of the chord PQ is equal to 2 * the"}, {"start": 102.399, "duration": 5.72, "text": "length of the line p m which is s of T"}, {"start": 105.399, "duration": 5.281, "text": "and therefore the chord length PQ is 2 *"}, {"start": 108.119, "duration": 4.401, "text": "the S of T from this argument we see"}, {"start": 110.68, "duration": 3.68, "text": "that Pi of s of T is greater than or"}, {"start": 112.52, "duration": 4.84, "text": "equal to 2T which is greater than or"}, {"start": 114.36, "duration": 5.28, "text": "equal to 2 * the S of T from the latter"}, {"start": 117.36, "duration": 4.719, "text": "inequality we see that 2 * the S of T is"}, {"start": 119.64, "duration": 4.64, "text": "less or equal to 2 T so that s of T is"}, {"start": 122.079, "duration": 4.441, "text": "less than or equal to T from the first"}, {"start": 124.28, "duration": 4.52, "text": "inequality we see that 2 * T is less"}, {"start": 126.52, "duration": 4.88, "text": "than or equal Tok * the S of T so that"}, {"start": 128.8, "duration": 5.32, "text": "2T / piun is less than or equal to S of"}, {"start": 131.4, "duration": 4.8, "text": "T when we glue these two together we see"}, {"start": 134.12, "duration": 4.88, "text": "that 2T over Pi is less than or equal to"}, {"start": 136.2, "duration": 5.0, "text": "S of T Which is less than or equal to T"}, {"start": 139.0, "duration": 4.44, "text": "notice this is only true if the angle T"}, {"start": 141.2, "duration": 5.6, "text": "is in the first quadrant so if T lies"}, {"start": 143.44, "duration": 6.159, "text": "between 0 and < / 2 this inequality is"}, {"start": 146.8, "duration": 4.519, "text": "known as Jordan's inequality thanks for"}, {"start": 149.599, "duration": 3.321, "text": "watching if you liked this video check"}, {"start": 151.319, "duration": 3.441, "text": "out my playlist on other mathematical"}, {"start": 152.92, "duration": 4.959, "text": "inequalities and consider subscribing to"}, {"start": 154.76, "duration": 3.119, "text": "the channel"}], "title": "Jordan's Inequality Visual Proof", "description": "This is a short, animated visual proof demonstrating Jordan's inequality, which is an equality providing upper and lower bounds for the sine function over the interval [0, pi/2].\n\nIf you like this video, consider subscribing to the channel or consider buying me a coffee: https://www.buymeacoffee.com/VisualProofs. Thanks! \n\nFor other inequality visual proofs, see my inequality playlist:\n\nhttps://youtube.com/playlist?list=PLZh9gzIvXQUuKcFHdtjFkwAVELDxu9nH4&si=R5wj4VJ1DCfI4Wbh\n\nThis animation is based on a visual proof by Feng Yuefeng from the April 1996 issue of Mathematics Magazine (https://www.jstor.org/stable/2690669 page 126 ).\n\n#math\u200b #inequality \u200b #manim\u200b #animation\u200b #theorem\u200b #pww\u200b #proofwithoutwords\u200b #visualproof\u200b #proof\u200b #iteachmath #algebra #areas #mathematics #jordaninequality #jordan #algebraicidentity #mathshorts\u200b #mathvideo\u200b #mtbos #circle #unitcircle \n\nTo learn more about animating with manim, check out:\nhttps://manim.community", "lengthSeconds": 157, "width": 640, "height": 360}
{"video_id": "i_6cH6WvH5A", "transcript": [{"start": 0.0, "duration": 4.259, "text": "what is the sum one plus a half plus a"}, {"start": 2.34, "duration": 3.72, "text": "third plus a quarter plus so on to"}, {"start": 4.259, "duration": 4.38, "text": "figure this out let's plot the function"}, {"start": 6.06, "duration": 4.98, "text": "y equals one over X over The Domain C to"}, {"start": 8.639, "duration": 4.741, "text": "c squared then the area one over C plus"}, {"start": 11.04, "duration": 4.62, "text": "one plus one over C plus two plus one"}, {"start": 13.38, "duration": 4.8, "text": "over C plus three and so on up to one"}, {"start": 15.66, "duration": 5.16, "text": "over c squared can be represented by the"}, {"start": 18.18, "duration": 4.62, "text": "area of the rectangles depicted here but"}, {"start": 20.82, "duration": 4.56, "text": "this geometric region contains a"}, {"start": 22.8, "duration": 4.559, "text": "rectangle with area c squared minus C"}, {"start": 25.38, "duration": 3.96, "text": "divided by c squared pictured here so"}, {"start": 27.359, "duration": 4.08, "text": "the sum of one over C plus one up to one"}, {"start": 29.34, "duration": 4.44, "text": "over c squared is greater than or equal"}, {"start": 31.439, "duration": 5.101, "text": "to c squared minus C divided by c"}, {"start": 33.78, "duration": 4.56, "text": "squared which is 1 minus one over C we"}, {"start": 36.54, "duration": 3.72, "text": "can use this fact to see that the sum of"}, {"start": 38.34, "duration": 3.719, "text": "one over C plus one over C plus one up"}, {"start": 40.26, "duration": 3.18, "text": "to one over c squared is greater than or"}, {"start": 42.059, "duration": 3.241, "text": "equal to one applying this result"}, {"start": 43.44, "duration": 3.24, "text": "repeatedly to the harmonic series shows"}, {"start": 45.3, "duration": 2.7, "text": "us one over one is greater than or equal"}, {"start": 46.68, "duration": 2.58, "text": "to one the sum from one half to"}, {"start": 48.0, "duration": 2.64, "text": "one-fourth is greater than or equal to"}, {"start": 49.26, "duration": 2.58, "text": "one the sum from one-fifth to one"}, {"start": 50.64, "duration": 3.3, "text": "twenty-fifth is greater than or equal to"}, {"start": 51.84, "duration": 3.899, "text": "one and so on continuing this process"}, {"start": 53.94, "duration": 3.959, "text": "indefinitely shows us that the harmonic"}, {"start": 55.739, "duration": 5.361, "text": "series diverges as it's greater than an"}, {"start": 57.899, "duration": 3.201, "text": "infinite sum of ones"}], "title": "Visual Harmonic Series Divergence from Bernoulli!", "description": "This is a short, animated visual proof showing Bernoulli's method of determining that the harmonic series diverges. This technique is similar to, but different from, the technique used by Oresme. #manim #math #mathvideo #series #infiniteseries #mathshorts #geometry  #animation #theorem #pww #proofwithoutwords #visualproof #proof #area #harmonicseries #pww\u200b\u200b \u200b #proof\u200b #areas #mathematics\u200b\u200b #mtbos #harmonic #divergence #bernoulli #oresme\n\nIf you like this video, please click \"like\" and consider subscribing and checking out my other videos.\n\nTo buy me a coffee, head over to https://www.buymeacoffee.com/VisualProofs  \nThanks!\n\nHere is an alternate video showing how to prove the harmonic sum diverges: https://youtu.be/hvIusavrRf8\n\nTo learn more about animating with manim, check out:\nhttps://manim.community", "lengthSeconds": 60, "width": 360, "height": 640}
{"video_id": "ijAZkeXenvE", "transcript": [{"start": 0.719, "duration": 4.001, "text": "let's see a formula's visual proof a"}, {"start": 2.96, "duration": 3.04, "text": "square plus b square"}, {"start": 4.72, "duration": 4.32, "text": "here"}, {"start": 6.0, "duration": 4.08, "text": "a square and b square"}, {"start": 9.04, "duration": 5.759, "text": "then"}, {"start": 10.08, "duration": 4.719, "text": "we form a plus b and a plus b square"}, {"start": 14.88, "duration": 5.12, "text": "now i'm forming two rectangle"}, {"start": 18.16, "duration": 3.92, "text": "a b and a b"}, {"start": 20.0, "duration": 4.88, "text": "two a b"}, {"start": 22.08, "duration": 2.8, "text": "then replace"}, {"start": 25.68, "duration": 5.2, "text": "square"}, {"start": 27.359, "duration": 5.36, "text": "and a square"}, {"start": 30.88, "duration": 3.92, "text": "and subtract"}, {"start": 32.719, "duration": 4.561, "text": "first rectangle a b and"}, {"start": 34.8, "duration": 3.52, "text": "again subtract second rectangle"}, {"start": 37.28, "duration": 4.959, "text": "a b"}, {"start": 38.32, "duration": 3.919, "text": "that is a plus b square minus two b"}, {"start": 43.44, "duration": 2.4, "text": "let's see"}, {"start": 46.32, "duration": 4.32, "text": "the shape is same"}, {"start": 48.0, "duration": 7.48, "text": "a plus whole square minus two a b n"}, {"start": 50.64, "duration": 4.84, "text": "a square plus b square is same"}], "title": "Visual proof of a formula", "description": "some formula of algebra must be memorise for solving general algebra \nbut students sometime dont know the geometrical proof of these formula \nhere you go a visual proof of a formula \nthis called geometrical representation of a algebraic formula\nplease subscribe \nthank you", "lengthSeconds": 54, "width": 324, "height": 360}
{"video_id": "anwT006ZsIw", "transcript": [{"start": 0.11, "duration": 3.079, "text": "[Music]"}, {"start": 4.799, "duration": 3.0, "text": "foreign"}, {"start": 9.29, "duration": 16.849, "text": "[Music]"}, {"start": 30.1, "duration": 9.61, "text": "[Music]"}], "title": "Restricted Lattice Path Enumeration #catalan", "description": "This synthwave enumeration short shows all of the northeast lattice paths to the points (n,n) that don't pass below the line y=x where n ranges from 0 to 6. The number of such lattice paths is counted by the Catalan numbers. \n\nIf you like this video, consider subscribing to the channel or consider buying me a coffee: https://www.buymeacoffee.com/VisualProofs. Thanks!\n\nTo see a longer version of this that comes with a visualization of a proof for the recurrence relation, see https://youtu.be/__w_MDQeUYc .\n\nTo find out how to find a closed form for the Catalan numbers, check out https://youtu.be/kaInaIUABzY\n\nTo become more familiar with lattice path enumeration, check out https://youtu.be/9YUl0k2FYzc and https://youtu.be/KYpOoA2D63w. \n\n#catalannumbers #catalan\u200b #mathvideo\u200b #math\u200b #mtbos\u200b #manim\u200b #animation\u200b #theorem\u200b\u200b #visualproof\u200b #proof\u200b #iteachmath #mathematics #binomialcoefficients #latticepaths #discretemath #combinatorics #enumeration #synthwave #recurrence #quadraticrecurrence\n\nThis animation is based on the well-known interpretation of Catalan numbers as counting restricted lattice paths. You can find more in many textbooks such as Oscar Levin's Discrete Mathematics An Open Introduction (http://discrete.openmathbooks.org/dmoi2/sec_counting-binom.html). \n\nTo learn more about animating with manim, check out:\nhttps://manim.community", "lengthSeconds": 43, "width": 360, "height": 640}
{"video_id": "XXBHFOxp6pE", "transcript": [{"start": 0.0, "duration": 4.359, "text": "hello everyone in this video we will"}, {"start": 1.68, "duration": 5.52, "text": "prove Pythagoras Theorem"}, {"start": 4.359, "duration": 5.961, "text": "visually in this Square we draw a"}, {"start": 7.2, "duration": 7.479, "text": "rectangle with one side a and the other"}, {"start": 10.32, "duration": 8.039, "text": "side is B with its diagonal as C now we"}, {"start": 14.679, "duration": 6.44, "text": "copy paste this over here so this these"}, {"start": 18.359, "duration": 6.561, "text": "all lengths are a and these all lengths"}, {"start": 21.119, "duration": 7.201, "text": "are B now from here this unshaded area"}, {"start": 24.92, "duration": 5.759, "text": "is equal to a square while this unshaded"}, {"start": 28.32, "duration": 4.759, "text": "area is equal to b square"}, {"start": 30.679, "duration": 7.361, "text": "we have to prove that this unshaded area"}, {"start": 33.079, "duration": 9.32, "text": "is equal to c\u00b2 now let move the figure"}, {"start": 38.04, "duration": 8.6, "text": "like this one we move this triangle to"}, {"start": 42.399, "duration": 7.241, "text": "this region and this this unshaded"}, {"start": 46.64, "duration": 5.759, "text": "region is a square with its side length"}, {"start": 49.64, "duration": 7.12, "text": "C and thus this area is equal to C"}, {"start": 52.399, "duration": 9.12, "text": "Square by this figure we have a square +"}, {"start": 56.76, "duration": 4.759, "text": "B sare is = c\u00b2 and that is the answer"}], "title": "Visual Proof of Pythagoras Theorem", "description": "In this video, we prove through visual the existence of Pythagoras theorem\n#Shorts #shortsfeed #pythagoras_theorem #mathematics #geometry #area_problem", "lengthSeconds": 60, "width": 480, "height": 640}
{"video_id": "4UpBU7onP_0", "transcript": [{"start": 0.16, "duration": 3.68, "text": "James Garfield the 20th president of the"}, {"start": 2.08, "duration": 3.56, "text": "United States gave this proof of the"}, {"start": 3.84, "duration": 3.519, "text": "Pythagorean theorem start with a right"}, {"start": 5.64, "duration": 4.079, "text": "triangle with legs of length A and B and"}, {"start": 7.359, "duration": 4.32, "text": "hypotenuse of length C place a rotated"}, {"start": 9.719, "duration": 3.761, "text": "copy so that complimentary angles fit"}, {"start": 11.679, "duration": 4.44, "text": "together creating a right isoceles"}, {"start": 13.48, "duration": 5.2, "text": "triangle with leg length C the area of"}, {"start": 16.119, "duration": 4.0, "text": "the newly created triangle is 12 c^ 2"}, {"start": 18.68, "duration": 3.92, "text": "and the area of the original right"}, {"start": 20.119, "duration": 4.641, "text": "triangle is 12 a * B so the area"}, {"start": 22.6, "duration": 4.96, "text": "enclosed in this trapezoid diagram is"}, {"start": 24.76, "duration": 4.679, "text": "c^2 + a * B on the other hand using the"}, {"start": 27.56, "duration": 5.44, "text": "formula for the area of a trapezoid we"}, {"start": 29.439, "duration": 5.761, "text": "see that the is 12 a + b * A + B and"}, {"start": 33.0, "duration": 5.16, "text": "this simplifies to 1 12 * the quantity a"}, {"start": 35.2, "duration": 4.8, "text": "2 + 2 a + b^ 2 because these two"}, {"start": 38.16, "duration": 6.76, "text": "quantities measure the same area we"}, {"start": 40.0, "duration": 6.68, "text": "conclude that c^2 + a = A2 + 2 A + B ^2"}, {"start": 44.92, "duration": 4.72, "text": "applying some Elementary algebra shows"}, {"start": 46.68, "duration": 4.399, "text": "us that c^2 equal A2 + b^2 When A and B"}, {"start": 49.64, "duration": 3.399, "text": "are the legs of a right triangle with"}, {"start": 51.079, "duration": 3.841, "text": "hypotenuse C this is Garfield's"}, {"start": 53.039, "duration": 4.32, "text": "trapezoid proof of the right triangle"}, {"start": 54.92, "duration": 2.439, "text": "theorem"}], "title": "Garfield\u2019s Pythagorean Proof", "description": "This is a short, animated visual proof of the Pythagorean theorem (the right triangle theorem) using the trapezoid that is now attributed to President James Garfield. This theorem states the square of the hypotenuse of a right triangle is equal to the sum of squares of the two other side lengths.\n\nTo buy me a coffee, head over to https://www.buymeacoffee.com/VisualProofs \nThanks!\n\nThis animation is based on a proof due to James A. Garfield. For a static version of this proof, see Roger Nelsen's first \"Proof Without Words: Exercises in Visual Thinking\" compendium (page 7) : https://bookstore.ams.org/clrm-1. You can also check out Howard Eves' \"Great Moments in Mathematics (Before 1650)\" page 34-36.\n\nYou can also see the original on MAA Convergence: https://maa.org/press/periodicals/convergence/mathematical-treasure-james-a-garfields-proof-of-the-pythagorean-theorem\n\nFor other proofs of this same fact check out the following playlist:\n\nhttps://youtube.com/playlist?list=PLZh9gzIvXQUvlRlnYryejWGHYQquM92rB&si=o_im7-_NHp1zhkcD\n\n\n#mathshorts #mathvideo #math #pythagoreantheorem #pythagorean #triangle #manim #animation #theorem #pww #proofwithoutwords #visualproof #proof\n\nTo learn more about animating with manim, check out:\nhttps://manim.community", "lengthSeconds": 55, "width": 360, "height": 640}
{"video_id": "1AJqS6njvFw", "transcript": [{"start": 0.0, "duration": 3.12, "text": "thank you"}, {"start": 0.99, "duration": 2.13, "text": "[Music]"}, {"start": 7.45, "duration": 28.509, "text": "[Music]"}, {"start": 36.559, "duration": 3.0, "text": "thank you"}, {"start": 42.21, "duration": 9.269, "text": "[Music]"}, {"start": 55.02, "duration": 10.91, "text": "[Music]"}, {"start": 70.18, "duration": 3.319, "text": "[Music]"}, {"start": 76.79, "duration": 4.189, "text": "[Music]"}, {"start": 84.29, "duration": 4.159, "text": "[Music]"}, {"start": 92.55, "duration": 8.069, "text": "[Music]"}, {"start": 98.72, "duration": 4.74, "text": "thank you"}, {"start": 100.619, "duration": 2.841, "text": "foreign"}], "title": "Viviani's Theorem II (visual proof via reflection and translation)", "description": "This is a short, animated visual proof of Viviani's theorem, which states that the sum of the distances from any interior point to the sides of an equilateral triangle is equal to the length of the triangle's altitude. #math\u200b #geometry #mtbos\u200b #manim\u200b #animation\u200b #theorem\u200b #pww\u200b #proofwithoutwords\u200b #visualproof\u200b #proof\u200b #iteachmath #polygons #triangle #mathshorts\u200b #mathvideo\u200b \n\nThis animation is based on a visual proof by James Tanton (@JamesTantonMath ) from the October 2001 issue of Mathematics Magazine (https://www.jstor.org/stable/2691103 page 313).\n\nHere's an alternative proof using rotations:\nhttps://youtu.be/5r2jLt0_PLM\n\nHere is a cool interactive GeoGebra applet to play with this theorem yourself:\nhttps://www.geogebra.org/m/Jgv9N6gR#:~:text=Observe%20the%20sum%20of%20the,or%20on%20an%20equilateral%20triangle.\n\nTo learn more about animating with manim, check out:\nhttps://manim.community\n_____________________________\nMusic in this video:\nCologne 1983 by Josh Kirsch/Media Right Productions", "lengthSeconds": 105, "width": 640, "height": 360}
{"video_id": "xTANODwNHHM", "transcript": [{"start": 0.64, "duration": 3.52, "text": "let's find a closed formula for the sum"}, {"start": 2.56, "duration": 5.04, "text": "of triangular"}, {"start": 4.16, "duration": 5.519, "text": "numbers the M triangular number T subm"}, {"start": 7.6, "duration": 4.68, "text": "is the sum of the first M positive"}, {"start": 9.679, "duration": 4.601, "text": "integers this number can be represented"}, {"start": 12.28, "duration": 3.519, "text": "by a triangular stack of squares like"}, {"start": 14.28, "duration": 3.4, "text": "this one here where there's one square"}, {"start": 15.799, "duration": 3.961, "text": "in the first row all the way down to M"}, {"start": 17.68, "duration": 4.64, "text": "squares in the bottom"}, {"start": 19.76, "duration": 4.24, "text": "row let's start with a triangular stack"}, {"start": 22.32, "duration": 3.879, "text": "of squares corresponding to the"}, {"start": 24.0, "duration": 4.96, "text": "Triangular number T subn that means"}, {"start": 26.199, "duration": 4.641, "text": "there's n blocks in the bottom row in"}, {"start": 28.96, "duration": 4.4, "text": "the first row place one token so we'"}, {"start": 30.84, "duration": 4.039, "text": "placed T1 tokens in the second row Place"}, {"start": 33.36, "duration": 2.96, "text": "one token in the first square and two"}, {"start": 34.879, "duration": 3.881, "text": "tokens in the second Square so that"}, {"start": 36.32, "duration": 4.759, "text": "we've placed T2 tokens in the second row"}, {"start": 38.76, "duration": 5.24, "text": "then Place T3 tokens in the third row as"}, {"start": 41.079, "duration": 5.601, "text": "1 + 2 + 3 continue this pattern in each"}, {"start": 44.0, "duration": 4.96, "text": "row placing I tokens in the I Square in"}, {"start": 46.68, "duration": 4.48, "text": "the row then the number of tokens in"}, {"start": 48.96, "duration": 5.16, "text": "this triangular diagram is given by the"}, {"start": 51.16, "duration": 6.12, "text": "sum of the first n triangular numbers T1"}, {"start": 54.12, "duration": 6.04, "text": "plus T2 and so on up to TN to find this"}, {"start": 57.28, "duration": 5.2, "text": "sum we perform a trick we create two two"}, {"start": 60.16, "duration": 4.12, "text": "more triangular arrays like this in the"}, {"start": 62.48, "duration": 4.08, "text": "first one we place the same number of"}, {"start": 64.28, "duration": 4.36, "text": "tokens in each row but this time we"}, {"start": 66.56, "duration": 3.4, "text": "place the tokens in reverse order"}, {"start": 68.64, "duration": 4.28, "text": "starting with the largest number of"}, {"start": 69.96, "duration": 5.28, "text": "tokens in a row and going down to one in"}, {"start": 72.92, "duration": 4.12, "text": "the third triangular array we place the"}, {"start": 75.24, "duration": 3.68, "text": "same number of tokens but this time we"}, {"start": 77.04, "duration": 4.28, "text": "go column by column starting from the"}, {"start": 78.92, "duration": 4.28, "text": "end so the last column has one token the"}, {"start": 81.32, "duration": 3.96, "text": "second from the last column has T2"}, {"start": 83.2, "duration": 4.959, "text": "tokens and so on so that the First"}, {"start": 85.28, "duration": 4.64, "text": "Column has TN tokens so the total number"}, {"start": 88.159, "duration": 4.441, "text": "of tokens spread across these three"}, {"start": 89.92, "duration": 5.36, "text": "three triangular arrays is 3 * the sum"}, {"start": 92.6, "duration": 5.28, "text": "of the first n triangular numbers now we"}, {"start": 95.28, "duration": 4.36, "text": "perform the trick stack all three of"}, {"start": 97.88, "duration": 3.68, "text": "these triangular arrays together"}, {"start": 99.64, "duration": 4.2, "text": "aggregating the number of tokens at any"}, {"start": 101.56, "duration": 4.8, "text": "spot it turns out that in any single"}, {"start": 103.84, "duration": 5.879, "text": "spot we always have exactly n plus 2"}, {"start": 106.36, "duration": 5.32, "text": "tokens on that spot and therefore 3 *"}, {"start": 109.719, "duration": 5.121, "text": "the sum of the first n triangular"}, {"start": 111.68, "duration": 4.88, "text": "numbers must be equal to n + 2times the"}, {"start": 114.84, "duration": 4.0, "text": "number of squares in the Triangular"}, {"start": 116.56, "duration": 5.68, "text": "array which is T subn and that"}, {"start": 118.84, "duration": 5.48, "text": "simplifies to n * n + 1 over2 so we used"}, {"start": 122.24, "duration": 4.12, "text": "three stacks of triangular arrays with a"}, {"start": 124.32, "duration": 4.079, "text": "triangular number of tokens in each row"}, {"start": 126.36, "duration": 5.4, "text": "to see that the sum of the first n"}, {"start": 128.399, "duration": 6.401, "text": "triangular numbers is n * n + 1 * n + 2"}, {"start": 131.76, "duration": 5.16, "text": "all divided 6 in his first proof without"}, {"start": 134.8, "duration": 4.28, "text": "words compendium Roger Nelson attributes"}, {"start": 136.92, "duration": 3.8, "text": "this proof to Richard K guy can you see"}, {"start": 139.08, "duration": 3.92, "text": "other identities you can prove using"}, {"start": 140.72, "duration": 3.879, "text": "stacks of tokens on certain arrays check"}, {"start": 143.0, "duration": 3.04, "text": "the linked video where we show how to"}, {"start": 144.599, "duration": 4.761, "text": "use this technique to find the sum of"}, {"start": 146.04, "duration": 3.32, "text": "the first n squares"}], "title": "Tri-Nums Sums! (visual proof IV)", "description": "This is a short, animated visual proof showing how to find the sum of the first n triangular numbers (which themselves are sums of the first n integers) using stacks of tokens on three triangular stacks of squares.\n\nIf you like this video, consider subscribing to the channel or consider buying me a coffee: https://www.buymeacoffee.com/VisualProofs. Thanks! \n\nHere are some related videos with sums of triangular numbers: \nhttps://youtu.be/HrmOQJ3HBhI\nhttps://youtu.be/N0ETyJ5K6j0\nhttps://youtu.be/pucFDbdEyuE\nhttps://youtu.be/LBls3jRfxeY\n\nHere is a sum of squares visual proof using a similar technique:\nhttps://youtu.be/UqVmocdLFGc\n\nThis animation is based on a visual proof attributed to Richard K. Guy by Roger B. Nelsen in his first Proofs without Words compendium (I may receive a small commission at no cost to you for these affiliate links): \nhttps://amzn.to/48mjeX1\n\n#manim #math #mathvideo #mathshorts #sequences #triangles #animation #theorem #pww #proofwithoutwords #visualproof #proof #triangularnumbers #sums #pww\u200b #proofwithoutwords\u200b \u200b #proof\u200b #algebra #finitesums #mathematics\u200b #mathvideo\u200b #mtbos \n\nTo learn more about animating with manim, check out:\nhttps://manim.community", "lengthSeconds": 147, "width": 640, "height": 360}
{"video_id": "WKJQBTjtDr8", "transcript": [{"start": 6.14, "duration": 5.44, "text": "start with four positive real numbers a"}, {"start": 9.12, "duration": 5.34, "text": "b c and d and suppose that a times D is"}, {"start": 11.58, "duration": 5.959, "text": "greater than b times C then draw a right"}, {"start": 14.46, "duration": 5.88, "text": "triangle with legs of length a d and AC"}, {"start": 17.539, "duration": 5.32, "text": "rotate a copy of this triangle and scale"}, {"start": 20.34, "duration": 5.22, "text": "by the value B over a creating a right"}, {"start": 22.859, "duration": 5.041, "text": "triangle with legs of length b c and b d"}, {"start": 25.56, "duration": 5.039, "text": "place the two triangles so they meet at"}, {"start": 27.9, "duration": 4.199, "text": "complementary angles connect the apexes"}, {"start": 30.599, "duration": 3.781, "text": "of these two triangles with a straight"}, {"start": 32.099, "duration": 4.741, "text": "line to create a third triangle as shown"}, {"start": 34.38, "duration": 4.44, "text": "here we can use the Pythagorean theorem"}, {"start": 36.84, "duration": 3.719, "text": "on the original two triangles to find"}, {"start": 38.82, "duration": 4.38, "text": "two of the side lengths of this new"}, {"start": 40.559, "duration": 4.441, "text": "triangle the first is B times the square"}, {"start": 43.2, "duration": 3.72, "text": "root of c squared plus D Squared while"}, {"start": 45.0, "duration": 4.079, "text": "the second is a times the square root of"}, {"start": 46.92, "duration": 3.84, "text": "c squared plus D Squared because the"}, {"start": 49.079, "duration": 4.14, "text": "first two triangles fit together along"}, {"start": 50.76, "duration": 4.5, "text": "complementary angles we know that the"}, {"start": 53.219, "duration": 3.84, "text": "blue triangle is also a right angle"}, {"start": 55.26, "duration": 3.72, "text": "triangle so we can then use the"}, {"start": 57.059, "duration": 3.781, "text": "Pythagorean theorem to find the final"}, {"start": 58.98, "duration": 3.599, "text": "side length or the hypotenuse of the"}, {"start": 60.84, "duration": 3.6, "text": "blue triangle to be this square root of"}, {"start": 62.579, "duration": 3.481, "text": "a squared plus b squared times the"}, {"start": 64.44, "duration": 4.62, "text": "square root of c squared plus D Squared"}, {"start": 66.06, "duration": 4.98, "text": "the resulting diagram is a trapezoid we"}, {"start": 69.06, "duration": 3.66, "text": "can then draw another right triangle in"}, {"start": 71.04, "duration": 5.04, "text": "the upper right corner to complete this"}, {"start": 72.72, "duration": 5.579, "text": "trapezoid to a rectangle one side length"}, {"start": 76.08, "duration": 4.2, "text": "of this new right triangle is a D minus"}, {"start": 78.299, "duration": 4.32, "text": "BC because it's equal to the difference"}, {"start": 80.28, "duration": 4.8, "text": "of the two parallel side lengths of the"}, {"start": 82.619, "duration": 4.981, "text": "original two triangles the other side"}, {"start": 85.08, "duration": 5.34, "text": "length of the new right triangle is AC"}, {"start": 87.6, "duration": 5.339, "text": "plus BD because that length is equal to"}, {"start": 90.42, "duration": 4.98, "text": "the sum of the two adjacent side lengths"}, {"start": 92.939, "duration": 4.261, "text": "of the original two right triangles"}, {"start": 95.4, "duration": 3.539, "text": "but now we can apply the Pythagorean"}, {"start": 97.2, "duration": 4.08, "text": "theorem one last time to this new right"}, {"start": 98.939, "duration": 3.661, "text": "triangle from this theorem we see that"}, {"start": 101.28, "duration": 2.64, "text": "the square root of a squared plus b"}, {"start": 102.6, "duration": 3.6, "text": "squared times the square root of c"}, {"start": 103.92, "duration": 5.159, "text": "squared plus D Squared all squared must"}, {"start": 106.2, "duration": 5.94, "text": "equal the quantity AC plus b d squared"}, {"start": 109.079, "duration": 5.461, "text": "plus the quantity a D minus BC squared"}, {"start": 112.14, "duration": 4.38, "text": "if we Square the left side we get the"}, {"start": 114.54, "duration": 3.719, "text": "resulting formula that the quantity a"}, {"start": 116.52, "duration": 3.959, "text": "squared plus b squared times the"}, {"start": 118.259, "duration": 5.04, "text": "quantity c squared plus d squared equals"}, {"start": 120.479, "duration": 6.121, "text": "the quantity AC plus b d squared plus"}, {"start": 123.299, "duration": 5.46, "text": "the quantity a D minus BC squared this"}, {"start": 126.6, "duration": 4.499, "text": "theorem says that the product of two"}, {"start": 128.759, "duration": 4.681, "text": "sums of squares must be a sum of squares"}, {"start": 131.099, "duration": 4.801, "text": "itself the result is of particular"}, {"start": 133.44, "duration": 5.1, "text": "interest when all the numbers a b c and"}, {"start": 135.9, "duration": 4.919, "text": "d involved are actually integers and not"}, {"start": 138.54, "duration": 6.5, "text": "just real numbers in this case we call"}, {"start": 140.819, "duration": 4.221, "text": "this diaphanes's sum of squares identity"}], "title": "Diophantus's Sum of Squares Identity I (visual proof)", "description": "This is a short, animated visual proof of Diophantus's sum of squares identity (also sometimes called the Brahmagupta\u2013Fibonacci identity) using \"Garfield's trapezoid\" and scaling of right triangles. \n\nIf you like this video, consider subscribing to the channel or consider buying me a coffee: https://www.buymeacoffee.com/VisualProofs. Thanks!\n\nTo see a related proof of the Cauchy-Schwarz inequality, check out:\nhttps://youtu.be/OWR6QCPYjqI\n\nThis animation is based on a visual proof by Roger Nelsen from the April 2017 issue of Mathematics Magazine (https://doi.org/10.4169/math.mag.90.2.134 - page 134).\n\n#math\u200b #inequality \u200b #manim\u200b #animation\u200b #theorem\u200b #pww\u200b #proofwithoutwords\u200b #visualproof\u200b #proof\u200b #iteachmath #algebra #areas #mathematics #diophantus #algebraicidentity #mathshorts\u200b #mathvideo\u200b #mtbos #triangle #scaling #garfieldtrapezoid #trapezoid #identity\n\nTo learn more about animating with manim, check out:\nhttps://manim.community\n_____________________________\nMusic in this video:\nKiss the Sky by Aakash Gandhi", "lengthSeconds": 151, "width": 640, "height": 360}
{"video_id": "lJ5O22R9b7U", "transcript": [{"start": 0.16, "duration": 4.24, "text": "the square pyramidal numbers represent"}, {"start": 2.32, "duration": 3.999, "text": "the number of unit cubes in a pyramid of"}, {"start": 4.4, "duration": 4.64, "text": "stacked Square arrays with sides of"}, {"start": 6.319, "duration": 4.961, "text": "length one up to n for instance this"}, {"start": 9.04, "duration": 3.92, "text": "diagram shows the 24th square pyramidal"}, {"start": 11.28, "duration": 3.68, "text": "number because the bottom square has"}, {"start": 12.96, "duration": 4.68, "text": "side length 24 and the top square has"}, {"start": 14.96, "duration": 5.239, "text": "side length 1 it turns out that this"}, {"start": 17.64, "duration": 4.52, "text": "particular square pyramidal number 4900"}, {"start": 20.199, "duration": 4.761, "text": "is special other than the trivial"}, {"start": 22.16, "duration": 5.039, "text": "Solutions of 0 and 1 it is the only such"}, {"start": 24.96, "duration": 4.44, "text": "number that can also be arranged in one"}, {"start": 27.199, "duration": 4.201, "text": "large Square array we can lay down the"}, {"start": 29.4, "duration": 4.96, "text": "square array levels like this only"}, {"start": 31.4, "duration": 5.36, "text": "splitting up the 4x4 and the 6x6 squares"}, {"start": 34.36, "duration": 5.719, "text": "to fill gaps once we've done this we"}, {"start": 36.76, "duration": 5.479, "text": "have an entire 70x 70 square filled in"}, {"start": 40.079, "duration": 3.921, "text": "thus 4900 is the only non-trivial"}, {"start": 42.239, "duration": 3.361, "text": "solution to the so-called Cannonball"}, {"start": 44.0, "duration": 3.719, "text": "problem which asks for numbers that are"}, {"start": 45.6, "duration": 4.72, "text": "simultaneously square pyramidal and"}, {"start": 47.719, "duration": 2.601, "text": "square"}], "title": "Cannonball Problem!", "description": "In this short, we show the only nontrivial solution to the cannonball problem, which asks for numbers that are simultaneously square pyramidal and square. The problem gets its name because cannonballs stack nicely in square pyramidal arrays. \n\nCan you prove that 0,1, and 4900 are the only such solutions to the problem?\n\nIf you like this video, consider subscribing to the channel or consider buying me a coffee: https://www.buymeacoffee.com/VisualProofs. Thanks! \n\n#math\u200b \u200b#manim\u200b #animation\u200b #theorem\u200b #iteachmath #mathematics ##mathshorts\u200b #mathvideo\u200b #mtbos #mathematician #squarepyramidal #numbertheory #cannonballproblem #cannonballrun \n\nTo learn more about animating with manim, check out:\nhttps://manim.community", "lengthSeconds": 48, "width": 360, "height": 640}
{"video_id": "LYfKapE_g1Q", "transcript": [{"start": 0.64, "duration": 4.96, "text": "let's see two formulas for the sum of"}, {"start": 2.32, "duration": 6.12, "text": "the first n cubes the number I cubed can"}, {"start": 5.6, "duration": 4.999, "text": "be represented by an I by I by I cubical"}, {"start": 8.44, "duration": 4.72, "text": "stack of unit cubes so this stack"}, {"start": 10.599, "duration": 5.76, "text": "represents 1 cubed this one represents 2"}, {"start": 13.16, "duration": 5.879, "text": "cubed this one represents 3 cubed and so"}, {"start": 16.359, "duration": 5.321, "text": "on where we can end at n cubed in this"}, {"start": 19.039, "duration": 4.48, "text": "case we've pictured Nal 4 when we put"}, {"start": 21.68, "duration": 3.919, "text": "these Stacks together we see we have a"}, {"start": 23.519, "duration": 5.84, "text": "representation for the sum of the first"}, {"start": 25.599, "duration": 6.84, "text": "n positive cubes now if we lay out each"}, {"start": 29.359, "duration": 6.001, "text": "of the cubes flat each Cube forms an I"}, {"start": 32.439, "duration": 5.921, "text": "by I squ rectangle so the first Cube"}, {"start": 35.36, "duration": 5.679, "text": "forms a 1x1 rectangle the next one a 2x4"}, {"start": 38.36, "duration": 4.32, "text": "rectangle then a 3x9 rectangle all the"}, {"start": 41.039, "duration": 5.16, "text": "way down to the final one which forms an"}, {"start": 42.68, "duration": 6.039, "text": "N byn squ rectangle the first rectangle"}, {"start": 46.199, "duration": 4.481, "text": "represents the number one the second"}, {"start": 48.719, "duration": 4.921, "text": "rectangle represents the sum of the two"}, {"start": 50.68, "duration": 4.76, "text": "odd numbers 3 + 5 as shown here the"}, {"start": 53.64, "duration": 5.12, "text": "third rectangle represents the sum of"}, {"start": 55.44, "duration": 5.56, "text": "the odd numbers 7 + 9 + 11 so each of"}, {"start": 58.76, "duration": 5.399, "text": "these rectangles can be decomposed into"}, {"start": 61.0, "duration": 5.36, "text": "a sum of consecutive odd numbers the"}, {"start": 64.159, "duration": 6.28, "text": "last rectangle can be done so by"}, {"start": 66.36, "duration": 9.32, "text": "decomposing it into n * n -1 + 1 + n * n"}, {"start": 70.439, "duration": 8.241, "text": "-1 + 3 and so on up to n * n -1 + 2 N -"}, {"start": 75.68, "duration": 4.6, "text": "1 but this last number simplifies to n *"}, {"start": 78.68, "duration": 4.439, "text": "n + 1 -"}, {"start": 80.28, "duration": 5.36, "text": "1 from these rectangular diagrams then"}, {"start": 83.119, "duration": 5.161, "text": "we see that the sum of the first n"}, {"start": 85.64, "duration": 5.36, "text": "positive cubes is equal to the sum of"}, {"start": 88.28, "duration": 7.24, "text": "the odd integers starting from one and"}, {"start": 91.0, "duration": 6.36, "text": "ending at n * n + 1us 1 this formula is"}, {"start": 95.52, "duration": 4.48, "text": "amazing but we can use these rectangles"}, {"start": 97.36, "duration": 4.719, "text": "to say even more we can break each"}, {"start": 100.0, "duration": 4.04, "text": "rectangle into two separate rectangles"}, {"start": 102.079, "duration": 4.0, "text": "where one has dimensions and by the nth"}, {"start": 104.04, "duration": 4.719, "text": "triangular number we can take this"}, {"start": 106.079, "duration": 4.68, "text": "smaller rectangle rotate it and glue it"}, {"start": 108.759, "duration": 5.04, "text": "onto the other side of the rectangle"}, {"start": 110.759, "duration": 5.441, "text": "forming an L-shaped nomon in each Case"}, {"start": 113.799, "duration": 4.721, "text": "by construction these gons all fit"}, {"start": 116.2, "duration": 4.64, "text": "together perfectly to form a"}, {"start": 118.52, "duration": 4.959, "text": "square because because of the rectangle"}, {"start": 120.84, "duration": 5.599, "text": "Dimensions we see that this square has a"}, {"start": 123.479, "duration": 5.76, "text": "side length given by 1 + 2 up to n or"}, {"start": 126.439, "duration": 4.401, "text": "the sum of the first n positive integers"}, {"start": 129.239, "duration": 3.761, "text": "this means the total number of unit"}, {"start": 130.84, "duration": 5.479, "text": "cubes which was given by the sum of the"}, {"start": 133.0, "duration": 5.879, "text": "odd numbers from 1 to n * n + 1 - one"}, {"start": 136.319, "duration": 5.681, "text": "must also equal the square of the sum of"}, {"start": 138.879, "duration": 5.121, "text": "the first n positive integers but then"}, {"start": 142.0, "duration": 4.12, "text": "we can finish this off by connecting the"}, {"start": 144.0, "duration": 4.48, "text": "first and last formulas so that we see"}, {"start": 146.12, "duration": 5.36, "text": "that the sum of the first n positive"}, {"start": 148.48, "duration": 5.28, "text": "cubes must equ the square of the sum of"}, {"start": 151.48, "duration": 4.119, "text": "the first n positive integers this"}, {"start": 153.76, "duration": 4.839, "text": "formula is attributed to the ancient"}, {"start": 155.599, "duration": 5.36, "text": "Greek philosopher nomus the visual proof"}, {"start": 158.599, "duration": 4.121, "text": "we just presented is due to aleno Flores"}, {"start": 160.959, "duration": 4.081, "text": "but there are plenty of other visual"}, {"start": 162.72, "duration": 4.519, "text": "proofs of the same fact you can see"}, {"start": 165.04, "duration": 4.44, "text": "others on my Channel or you can find a"}, {"start": 167.239, "duration": 5.681, "text": "variety of them in Roger Nelson's books"}, {"start": 169.48, "duration": 5.32, "text": "proof without words volumes 1 2 and 3"}, {"start": 172.92, "duration": 5.84, "text": "have a look and let me know which visual"}, {"start": 174.8, "duration": 6.96, "text": "proof of nomiss theorem is your"}, {"start": 178.76, "duration": 3.0, "text": "favorite"}], "title": "Sum of Cubes as Sum of Odds and more (visual proof VI)", "description": "This is a short, animated visual proof demonstrating two formulas for the sum of the first n positive cubes, one in terms of consecutive odd numbers and the other in terms of the square of a sum of consecutive positive integers. We end with a bonus visualization of a direct version of Nicomachus's theorem. \n\nIf you like this video, consider subscribing to the channel or consider buying me a coffee: https://www.buymeacoffee.com/VisualProofs. Thanks!\n\nHere are five other visual proofs of sum of cubes formulas:\n\nhttps://youtu.be/jWpyrXYZNiI\nhttps://youtu.be/YQLicI8R4Gs\nhttps://youtu.be/Ye9OPNqV9FA\nhttps://youtu.be/NxOcT_VKQR0\nhttps://youtu.be/d1yM6Rq7Tfw\n\nThis animation is based on a visual proof by Alfinio Flores from the January 1998 issue of The College Mathematics Journal  (https://www.jstor.org/stable/2687639 page 61).\n\nTo learn more about animating with manim, check out:\nhttps://manim.community\n\n#mathshorts\u200b #mathvideo\u200b #math\u200b #calculus #mtbos\u200b #manim\u200b #animation\u200b #theorem\u200b #pww\u200b #proofwithoutwords\u200b #visualproof\u200b #proof\u200b #iteachmath #finitesums #discretemath #calculus #sum #induction", "lengthSeconds": 182, "width": 640, "height": 360}
{"video_id": "HiDe2Ngcq2c", "transcript": [{"start": 2.43, "duration": 7.929, "text": "[Music]"}, {"start": 13.679, "duration": 3.52, "text": "to make sense of this visual proof"}, {"start": 15.44, "duration": 3.679, "text": "we first need to see that the number n"}, {"start": 17.199, "duration": 3.041, "text": "squared which is typically represented"}, {"start": 19.119, "duration": 3.281, "text": "by a square array"}, {"start": 20.24, "duration": 4.4, "text": "can be decomposed into two triangular"}, {"start": 22.4, "duration": 4.16, "text": "arrays with bases differing by one"}, {"start": 24.64, "duration": 3.2, "text": "and these arrays can be morphed into"}, {"start": 26.56, "duration": 3.44, "text": "arrays of triangles"}, {"start": 27.84, "duration": 3.839, "text": "that fit together like this to form a"}, {"start": 30.0, "duration": 3.52, "text": "triangular tiling of an equilateral"}, {"start": 31.679, "duration": 3.761, "text": "triangle by equilateral triangles"}, {"start": 33.52, "duration": 4.16, "text": "so each square can be visualized this"}, {"start": 35.44, "duration": 2.24, "text": "way"}, {"start": 38.0, "duration": 4.32, "text": "starting with the square of a multiple"}, {"start": 39.84, "duration": 4.96, "text": "of 3 let's say 3k"}, {"start": 42.32, "duration": 3.52, "text": "we can consider the trapezoid tile in"}, {"start": 44.8, "duration": 5.279, "text": "the lower left"}, {"start": 45.84, "duration": 7.199, "text": "with bases 2k and k"}, {"start": 50.079, "duration": 5.201, "text": "three copies of this trapezoid tile can"}, {"start": 53.039, "duration": 5.68, "text": "be used to cover the entire triangle"}, {"start": 55.28, "duration": 5.599, "text": "in this way so that 3k squared"}, {"start": 58.719, "duration": 3.761, "text": "is three times the size of the trapezoid"}, {"start": 60.879, "duration": 3.761, "text": "but the trapezoid is a difference of two"}, {"start": 62.48, "duration": 2.639, "text": "triangular tilings so it contains two k"}, {"start": 64.64, "duration": 3.92, "text": "squared"}, {"start": 65.119, "duration": 5.04, "text": "minus k squared triangles next"}, {"start": 68.56, "duration": 3.04, "text": "start with a square of a number that is"}, {"start": 70.159, "duration": 4.241, "text": "one less than a multiple of three"}, {"start": 71.6, "duration": 4.879, "text": "say three k minus one again we consider"}, {"start": 74.4, "duration": 4.719, "text": "the trapezoid tile in the lower left"}, {"start": 76.479, "duration": 4.401, "text": "this time with bases two k minus one and"}, {"start": 79.119, "duration": 3.36, "text": "k minus one"}, {"start": 80.88, "duration": 3.2, "text": "using three copies of this tile this"}, {"start": 82.479, "duration": 4.081, "text": "time covers all but"}, {"start": 84.08, "duration": 3.039, "text": "one of the small triangles in the tiling"}, {"start": 86.56, "duration": 3.519, "text": "therefore"}, {"start": 87.119, "duration": 3.761, "text": "the quantity 3k minus 1 squared is equal"}, {"start": 90.079, "duration": 2.801, "text": "to 1"}, {"start": 90.88, "duration": 3.599, "text": "plus 3 times the quantity 2k minus 1"}, {"start": 92.88, "duration": 3.12, "text": "squared minus k minus 1 squared"}, {"start": 94.479, "duration": 4.0, "text": "as the trapezoid is again the difference"}, {"start": 96.0, "duration": 4.56, "text": "of two squares"}, {"start": 98.479, "duration": 3.92, "text": "finally if we start with a square of a"}, {"start": 100.56, "duration": 4.64, "text": "number one more than a multiple of three"}, {"start": 102.399, "duration": 4.881, "text": "say three k plus one then we can again"}, {"start": 105.2, "duration": 4.559, "text": "use three copies of a trapezoid tile"}, {"start": 107.28, "duration": 3.119, "text": "but this time the bases are two k plus"}, {"start": 109.759, "duration": 3.601, "text": "one"}, {"start": 110.399, "duration": 4.161, "text": "and k plus one and again these cover the"}, {"start": 113.36, "duration": 4.48, "text": "entire triangle"}, {"start": 114.56, "duration": 6.239, "text": "except for one middle triangle"}, {"start": 117.84, "duration": 3.76, "text": "therefore we conclude that three k plus"}, {"start": 120.799, "duration": 2.96, "text": "one squared"}, {"start": 121.6, "duration": 3.199, "text": "is one plus three times the quantity two"}, {"start": 123.759, "duration": 2.96, "text": "k plus one squared"}, {"start": 124.799, "duration": 3.201, "text": "minus k plus one squared as again the"}, {"start": 126.719, "duration": 3.68, "text": "trapezoid covers the difference of"}, {"start": 128.0, "duration": 2.399, "text": "squares"}, {"start": 130.8, "duration": 3.519, "text": "therefore we've seen that n squared is"}, {"start": 132.8, "duration": 3.76, "text": "congruent to 0 mod 3"}, {"start": 134.319, "duration": 4.241, "text": "if n is congruent to 0 mod 3 and n"}, {"start": 136.56, "duration": 4.88, "text": "squared is congruent to 1 mod 3"}, {"start": 138.56, "duration": 4.64, "text": "if n is not congruent to 0 mod 3. since"}, {"start": 141.44, "duration": 3.68, "text": "every number is either a multiple of 3"}, {"start": 143.2, "duration": 3.28, "text": "one more than a multiple of 3 or 1 less"}, {"start": 145.12, "duration": 2.64, "text": "than a multiple of 3"}, {"start": 146.48, "duration": 11.36, "text": "we've seen that every square is"}, {"start": 147.76, "duration": 10.08, "text": "congruent to 0 or 1 mod 3."}, {"start": 161.92, "duration": 2.08, "text": "you"}], "title": "Squares Modulo 3 (visual proof)", "description": "This is a short, animated visual proof demonstrating how to visualize the congruence classes of squares modulo 3. #mathshorts\u200b #mathvideo\u200b #math\u200b #numbertheory #mtbos\u200b #manim\u200b #animation\u200b #theorem\u200b #pww\u200b #proofwithoutwords\u200b #visualproof\u200b #proof\u200b #iteachmath #squares #modulararithmetic #numbertheory\n\nThis animation is based on a visual proof by Roger B. Nelsen from the September 2013 issue of The College Mathematics Journal (https://doi.org/10.4169/college.math.j.44.4.283 - page 283).\n\nTo learn more about animating with manim, check out:\nhttps://manim.community\n__________________________________________________________________\nBackground music, used according to CC BY-SA 3.0, is \"The Road\" from Esteban Orlando:\n\nEsteban Orlando:\nhttps://soundcloud.com/orlando-esteba...\nhttps://www.instagram.com/estebanorla...\n\nhttps://creativecommons.org/licenses/...\nMusic provided by RFM: https://youtu.be/b4KCM1bIq50", "lengthSeconds": 162, "width": 640, "height": 360}
{"video_id": "gRq6pX8WvVw", "transcript": [{"start": 0.91, "duration": 30.84, "text": "[Music]"}, {"start": 28.96, "duration": 32.8, "text": "oh e"}, {"start": 31.75, "duration": 30.01, "text": "[Music]"}, {"start": 63.76, "duration": 15.47, "text": "[Music]"}, {"start": 81.27, "duration": 27.48, "text": "[Music]"}], "title": "Proof of the Completing the Square Method", "description": "Completing the square might sound like an intimidating term, but it\u2019s actually a fascinating way to connect algebra and geometry. This method transforms a quadratic equation into a perfect square form, giving you a visual and intuitive understanding of the solutions. It\u2019s not just about finding the answer\u2014it\u2019s about appreciating the elegance of mathematics. While it\u2019s a classical method that dates back centuries, it still feels fresh and exciting when you see it in action. So, let\u2019s explore this timeless technique and uncover the joy of solving quadratic equations with a geometric twist!\n\nHere are some links for those who want to learn more about educational social media and support:\nhttps://sociabuzz.com/philosophizeus\n\nA website with a collection of free learning materials (all you need is dedication and an internet connection):\nhttps://philosophizeus.org/\n\nStay motivated,\nSuntar Jono, S.T., M.Si.\n\nNote:\nContent Creator: Suntar Jono\nAnimations and opening music created using CS 3 user\n\nFor feedback, please contact via email:\nSuntar@philosophizeus.org", "lengthSeconds": 107, "width": 360, "height": 360}
{"video_id": "MUBE4o0TymQ", "transcript": [{"start": 13.679, "duration": 5.321, "text": "I Got That good that you"}, {"start": 15.76, "duration": 6.4, "text": "want let me be your push push"}, {"start": 19.0, "duration": 7.68, "text": "push I Got That good ST that"}, {"start": 22.16, "duration": 8.599, "text": "you let me be push push Push"}, {"start": 26.68, "duration": 5.78, "text": "[M\u00fasica]"}, {"start": 30.759, "duration": 29.131, "text": "poio"}, {"start": 32.46, "duration": 27.43, "text": "[M\u00fasica]"}], "title": "Can you explain the pattern?", "description": "This enumeration shows all the tilings of a circular grid divided into n pieces from n = 1 to n = 8 using tiles that cover one or two cells. Can you tell the pattern? Can you prove why or how the Lucas numbers appear? If you are looking for more information about this, I recommend the wonderful textbook Proofs that Really Count: The Art of Combinatorial Proof from Arthur Benjamin and Jennifer Quinn: https://bookstore.ams.org/dol-27\n\nIf you like this video, consider subscribing to the channel or consider buying me a coffee: https://www.buymeacoffee.com/VisualProofs. Thanks!\n\nIf you enjoy this enumeration, check out the longer form: https://youtu.be/4S85bHkmM_Y\nor check out a Fibonacci tiling video : https://youtu.be/AFAcKDTmYXI\n\nIf you like this video, check out my others and consider subscribing. Thanks!\n\n#dominotilings #fibonacci #lucas #circulargrid ##countingtiling  #count #mathvideo\u200b #math\u200b #mtbos\u200b #manim\u200b #animation\u200b\u200b\u200b #iteachmath #mathematics #discretemath #combinatorics #enumeration #synthwave", "lengthSeconds": 58, "width": 360, "height": 640}
{"video_id": "XIC7SIdl7pI", "transcript": [{"start": 0.4, "duration": 18.799, "text": "[Music]"}, {"start": 15.28, "duration": 3.919, "text": "start with a semi-circle of radius one"}, {"start": 20.72, "duration": 4.479, "text": "then take the radius and rotate it a"}, {"start": 23.359, "duration": 4.721, "text": "fixed angle"}, {"start": 25.199, "duration": 5.84, "text": "the radius has length one and let's call"}, {"start": 28.08, "duration": 4.639, "text": "the angle created t drop a perpendicular"}, {"start": 31.039, "duration": 2.961, "text": "chord from the endpoint of the radius on"}, {"start": 32.719, "duration": 3.041, "text": "the circle down to the horizontal"}, {"start": 34.0, "duration": 3.28, "text": "diameter the length of this chord is the"}, {"start": 35.76, "duration": 3.04, "text": "sine of t"}, {"start": 37.28, "duration": 2.959, "text": "the third leg of the triangle formed by"}, {"start": 38.8, "duration": 3.759, "text": "the radius and this chord has length"}, {"start": 40.239, "duration": 4.881, "text": "cosine of t so that the remainder of the"}, {"start": 42.559, "duration": 4.761, "text": "horizontal radius has length one minus"}, {"start": 45.12, "duration": 4.16, "text": "the cosine of t"}, {"start": 47.32, "duration": 3.879, "text": "[Music]"}, {"start": 49.28, "duration": 3.36, "text": "the line connecting the other end of the"}, {"start": 51.199, "duration": 4.081, "text": "horizontal diameter to the endpoint of"}, {"start": 52.64, "duration": 4.16, "text": "the radius creates this triangle which"}, {"start": 55.28, "duration": 3.119, "text": "is isosceles because it has two"}, {"start": 56.8, "duration": 3.84, "text": "congruent sides"}, {"start": 58.399, "duration": 4.32, "text": "the supplementary angle to t is 180"}, {"start": 60.64, "duration": 4.08, "text": "minus t and because the other two angles"}, {"start": 62.719, "duration": 5.121, "text": "must be equal they must be"}, {"start": 64.72, "duration": 3.12, "text": "both t over two"}, {"start": 68.0, "duration": 4.4, "text": "but then"}, {"start": 69.68, "duration": 6.479, "text": "t over two is an angle inside"}, {"start": 72.4, "duration": 6.88, "text": "of this highlighted right triangle"}, {"start": 76.159, "duration": 4.32, "text": "so that the tangent of t over 2"}, {"start": 79.28, "duration": 3.519, "text": "is equal"}, {"start": 80.479, "duration": 5.761, "text": "to the sine of t"}, {"start": 82.799, "duration": 5.521, "text": "over 1 plus the cosine of t"}, {"start": 86.24, "duration": 4.0, "text": "furthermore if we create this line"}, {"start": 88.32, "duration": 3.52, "text": "segment we have built a large right"}, {"start": 90.24, "duration": 3.519, "text": "angle triangle by thaley's triangle"}, {"start": 91.84, "duration": 5.52, "text": "theorem and consequently the two"}, {"start": 93.759, "duration": 5.601, "text": "highlighted subtriangles must be similar"}, {"start": 97.36, "duration": 4.0, "text": "equating the leg ratios then tells us"}, {"start": 99.36, "duration": 3.36, "text": "that the sine of t over one plus the"}, {"start": 101.36, "duration": 3.92, "text": "cosine of t"}, {"start": 102.72, "duration": 5.2, "text": "is equal to one minus the cosine of t"}, {"start": 105.28, "duration": 5.04, "text": "over the sine of t"}, {"start": 107.92, "duration": 6.22, "text": "the semicircle has given us two formulas"}, {"start": 110.32, "duration": 8.85, "text": "for the tangent of half an angle"}, {"start": 114.14, "duration": 5.03, "text": "[Music]"}, {"start": 130.0, "duration": 2.08, "text": "you"}], "title": "Half Angle Tangent Formulas (visual proof; trigonometry)", "description": "This is a short, animated visual proof of the half angle formula for the tangent using Thales triangle theorem and similar triangles. This theorem gives two ways to compute the tangent of a half angle, like tan(t/2). #mathshorts #mathvideo #math #trigonometry #triangle #manim #animation #theorem #pww #proofwithoutwords #visualproof #proof #circle #thalestheorem #similartriangles #semicircle #identity #halfangle #trigidentity #trig\n\nPlease consider subscribing to the channel if you enjoy the content.\n\nThis animation is based on a proof due to R.J. Walker from the May 1942 issue of The American Mathematical Monthly (https://doi.org/10.2307/2303100) page 325.\n\nTo learn more about animating with manim, check out:\nhttps://manim.community", "lengthSeconds": 130, "width": 640, "height": 360}
{"video_id": "bEtdKgbomSo", "transcript": [{"start": 0.299, "duration": 4.921, "text": "the quantity x squared plus b times x"}, {"start": 2.879, "duration": 6.001, "text": "can be represented geometrically using"}, {"start": 5.22, "duration": 6.9, "text": "an X by x square as pictured here and a"}, {"start": 8.88, "duration": 5.12, "text": "b by X rectangle placed underneath then"}, {"start": 12.12, "duration": 4.919, "text": "x squared plus BX represents this area"}, {"start": 14.0, "duration": 6.4, "text": "but we can take that b by X rectangle"}, {"start": 17.039, "duration": 5.641, "text": "and cut it into two equal pieces so that"}, {"start": 20.4, "duration": 4.68, "text": "one rectangle is a b over 2 by X"}, {"start": 22.68, "duration": 5.099, "text": "rectangle and the other one is a b over"}, {"start": 25.08, "duration": 4.199, "text": "2 by X rectangle but we can place one of"}, {"start": 27.779, "duration": 4.5, "text": "them on the right side of the square"}, {"start": 29.279, "duration": 5.341, "text": "like this now the area represented is"}, {"start": 32.279, "duration": 4.741, "text": "almost an X plus b over 2 by X plus b"}, {"start": 34.62, "duration": 4.02, "text": "over 2 square the only thing missing is"}, {"start": 37.02, "duration": 4.559, "text": "this lower right square that has"}, {"start": 38.64, "duration": 5.04, "text": "Dimensions B over 2 by B over two so the"}, {"start": 41.579, "duration": 4.561, "text": "area we're interested in is X plus b"}, {"start": 43.68, "duration": 4.44, "text": "over 2 squared minus B over 2 squared"}, {"start": 46.14, "duration": 4.16, "text": "this process is known as completing the"}, {"start": 48.12, "duration": 2.18, "text": "square"}], "title": "Completing the Square Visually", "description": "This is a short, animated visual proof showing what is meant when we \"complete\" the square algebraically and justifying why the formula holds for positive real number\u200b #math\u200b #completingthesquare #mtbos\u200b #manim\u200b #animation\u200b #theorem\u200b #pww\u200b #proofwithoutwords\u200b #visualproof\u200b #proof\u200b #iteachmath #algebra s. #mathshorts\u200b #mathvideo\n\nSee the original video here: https://youtu.be/T0HyWIFbsHQ\n\nTo learn more about animating with manim, check out:\nhttps://manim.community\n\n______________________________________________\nMusic in this video:\nPrelude No. 9 by Chris Zabriskie is licensed under a Creative Commons Attribution 4.0 license. https://creativecommons.org/licenses/by/4.0/\n\nSource: http://chriszabriskie.com/preludes/\n\nArtist: http://chriszabriskie.com/", "lengthSeconds": 53, "width": 360, "height": 640}
{"video_id": "AuxqpiDTCyU", "transcript": [{"start": 1.14, "duration": 3.25, "text": "[Music]"}, {"start": 9.599, "duration": 4.241, "text": "suppose that a b c and d are all"}, {"start": 11.759, "duration": 5.041, "text": "positive real numbers and that a over b"}, {"start": 13.84, "duration": 5.599, "text": "is less than c over d the line"}, {"start": 16.8, "duration": 4.88, "text": "connecting the origin to the point b a"}, {"start": 19.439, "duration": 4.08, "text": "has a slope of a over b"}, {"start": 21.68, "duration": 5.599, "text": "the line connecting the origin and the"}, {"start": 23.519, "duration": 5.6, "text": "point dc has a slope c over d"}, {"start": 27.279, "duration": 5.921, "text": "therefore by assumption the line"}, {"start": 29.119, "duration": 5.6, "text": "connecting 0 0 to b a has a slope that's"}, {"start": 33.2, "duration": 5.44, "text": "less than the slope of the line"}, {"start": 34.719, "duration": 3.921, "text": "connecting 0 0 to d c"}, {"start": 39.04, "duration": 4.4, "text": "let's connect the points b over a and d"}, {"start": 41.28, "duration": 4.4, "text": "over c with a line and then highlight"}, {"start": 43.44, "duration": 4.4, "text": "the midpoint between these two points"}, {"start": 45.68, "duration": 4.16, "text": "the x coordinate of this midpoint is"}, {"start": 47.84, "duration": 3.6, "text": "halfway between b and d and the y"}, {"start": 49.84, "duration": 3.76, "text": "coordinate of this point is halfway"}, {"start": 51.44, "duration": 4.24, "text": "between a and c so the coordinates of"}, {"start": 53.6, "duration": 4.0, "text": "this point are given by b plus d over 2"}, {"start": 55.68, "duration": 3.76, "text": "a plus c over 2."}, {"start": 57.6, "duration": 4.32, "text": "the line connecting the origin to this"}, {"start": 59.44, "duration": 3.919, "text": "point lies between the first two lines"}, {"start": 61.92, "duration": 3.44, "text": "we started with"}, {"start": 63.359, "duration": 4.881, "text": "therefore the slope of the line"}, {"start": 65.36, "duration": 4.96, "text": "connecting 0 0 to b a is less than the"}, {"start": 68.24, "duration": 4.96, "text": "slope of the line connecting 0 0 to the"}, {"start": 70.32, "duration": 6.56, "text": "midpoint and that is less than the slope"}, {"start": 73.2, "duration": 5.52, "text": "of the line connecting 0 0 to dc"}, {"start": 76.88, "duration": 4.879, "text": "as we've already seen the slope of the"}, {"start": 78.72, "duration": 5.039, "text": "line connecting 0 0 to b a is a over b"}, {"start": 81.759, "duration": 5.04, "text": "the slope of the line connecting 0 0 to"}, {"start": 83.759, "duration": 4.72, "text": "d c is c over d and now we can check"}, {"start": 86.799, "duration": 3.841, "text": "that the slope of the line connecting 0"}, {"start": 88.479, "duration": 3.28, "text": "0 to the midpoint is a plus c over b"}, {"start": 90.64, "duration": 3.04, "text": "plus d"}, {"start": 91.759, "duration": 5.121, "text": "therefore the condition that a over b is"}, {"start": 93.68, "duration": 5.36, "text": "less than c over d implies that a over b"}, {"start": 96.88, "duration": 4.48, "text": "is less than a plus c over b plus d"}, {"start": 99.04, "duration": 4.719, "text": "which is less than c over d"}, {"start": 101.36, "duration": 4.719, "text": "the fraction a plus c over b plus d is"}, {"start": 103.759, "duration": 4.081, "text": "the median of a over b and c over d and"}, {"start": 106.079, "duration": 4.761, "text": "this inequality is known as the mediant"}, {"start": 107.84, "duration": 3.0, "text": "inequality"}, {"start": 123.04, "duration": 2.079, "text": "you"}], "title": "Mediant Inequality III (visual proof)", "description": "This is a short, animated visual proof demonstrating what one might call the mediant inequality. #math\u200b #inequality #numbertheory #mtbos\u200b #manim\u200b #animation\u200b #theorem\u200b #pww\u200b #proofwithoutwords\u200b #visualproof\u200b #proof\u200b #iteachmath #fractions #mediant  #mathematics #fareyfractions #farey #mathshorts\u200b #mathvideo\u200b \n\nIf you like this video, consider subscribing and checking out my other videos.\n\nFor an alternate visual proofs of this fact, see these video: https://youtu.be/RKeBuWL1B4A\nhttps://youtu.be/pODylgdI3p8\n\nThis animation is based on a visual proof by Li Changming from the January 1988 issue of Mathematics Teacher (https://www.jstor.org/stable/27965677 - page 63).\n\nTo learn more about animating with manim, check out:\nhttps://manim.community\n_____________________________\nMusic in this video:\nMelodic Interlude by Alexander Nakarada | https://www.serpentsoundstudios.com\nMusic promoted by https://www.free-stock-music.com\nAttribution 4.0 International (CC BY 4.0)\nhttps://creativecommons.org/licenses/by/4.0/", "lengthSeconds": 123, "width": 640, "height": 360}
{"video_id": "Mr5cpZyYHX0", "transcript": [{"start": 0.24, "duration": 3.6, "text": "let's consider the function f of x"}, {"start": 1.979, "duration": 4.32, "text": "equals e to the X and plot it on the"}, {"start": 3.84, "duration": 4.14, "text": "axis here we can draw the tangent line"}, {"start": 6.299, "duration": 3.781, "text": "to the curve which looks like the curve"}, {"start": 7.98, "duration": 4.199, "text": "locally and whose slope is the instant"}, {"start": 10.08, "duration": 4.019, "text": "rate of change of the Curve we can"}, {"start": 12.179, "duration": 4.561, "text": "compute the slope of this line and we"}, {"start": 14.099, "duration": 4.741, "text": "see that here it's .08 as we move the"}, {"start": 16.74, "duration": 4.74, "text": "point along the curve we see that the"}, {"start": 18.84, "duration": 4.859, "text": "slopes of the tangent lines get larger"}, {"start": 21.48, "duration": 4.559, "text": "in fact something even more interesting"}, {"start": 23.699, "duration": 4.441, "text": "happens in this example if we also plot"}, {"start": 26.039, "duration": 4.74, "text": "the y coordinate of the point we see"}, {"start": 28.14, "duration": 4.68, "text": "that as the point moves along the curve"}, {"start": 30.779, "duration": 3.96, "text": "the slope of the tangent line actually"}, {"start": 32.82, "duration": 4.68, "text": "equals the Y value"}, {"start": 34.739, "duration": 5.16, "text": "we can take this one step further where"}, {"start": 37.5, "duration": 5.04, "text": "we plot a new graph on a new set of axes"}, {"start": 39.899, "duration": 4.801, "text": "here the Y value will be the slope of"}, {"start": 42.54, "duration": 4.199, "text": "the tangent line from the first graph as"}, {"start": 44.7, "duration": 4.5, "text": "we move the point along the curve we map"}, {"start": 46.739, "duration": 4.681, "text": "out this graph the function f of x"}, {"start": 49.2, "duration": 4.199, "text": "equals e to the x is a special function"}, {"start": 51.42, "duration": 6.38, "text": "because its derivative or its slope"}, {"start": 53.399, "duration": 4.401, "text": "function f Prime of X is also e to the x"}], "title": "Exponential derivative visual", "description": "A visual of the derivative of f(x)=e^x. We show how to think about the derivative of a function visually. \n\n#manim #calculus #derivatives #derivative #tangentline #slope #parabola #mathvideo #mathshorts #math #visualmath #graph #exponential #linearapproximation", "lengthSeconds": 56, "width": 360, "height": 640}
{"video_id": "EklS3JyOBd0", "transcript": [{"start": 0.08, "duration": 3.079, "text": "why is the sum of a positive real number"}, {"start": 1.719, "duration": 4.08, "text": "and its reciprocal always greater than"}, {"start": 3.159, "duration": 4.001, "text": "or equal to 2 let X be a positive real"}, {"start": 5.799, "duration": 4.121, "text": "number and draw a rectangle with"}, {"start": 7.16, "duration": 5.04, "text": "Dimensions X by 1 /x and create four"}, {"start": 9.92, "duration": 4.919, "text": "rotated copies like this each of these"}, {"start": 12.2, "duration": 5.44, "text": "rectangles has an area of 1x * X which"}, {"start": 14.839, "duration": 5.44, "text": "is 1 so the four rectangles each enclose"}, {"start": 17.64, "duration": 5.68, "text": "an area of one these four rectangles lie"}, {"start": 20.279, "duration": 5.201, "text": "inside of an x + 1X by x + 1X s"}, {"start": 23.32, "duration": 5.16, "text": "enclosing an area of the quantity x +"}, {"start": 25.48, "duration": 5.0, "text": "1x^2 the four rectangles almost never"}, {"start": 28.48, "duration": 4.16, "text": "fill the entire square there's a missing"}, {"start": 30.48, "duration": 5.64, "text": "sub Square inside and that means that x"}, {"start": 32.64, "duration": 6.439, "text": "+ 1X 2 is greater than or equal to 1 + 1"}, {"start": 36.12, "duration": 4.0, "text": "+ 1 + 1 which is equal to 4 taking the"}, {"start": 39.079, "duration": 2.881, "text": "square root of both sides and"}, {"start": 40.12, "duration": 3.84, "text": "remembering that X is positive means"}, {"start": 41.96, "duration": 4.0, "text": "that x + 1X is greater than or equal to"}, {"start": 43.96, "duration": 3.52, "text": "2 can you see how to use the Center"}, {"start": 45.96, "duration": 6.0, "text": "Square to produce a condition to"}, {"start": 47.48, "duration": 4.48, "text": "guarantee x + 1x equal 2"}], "title": "Sum of a positive number and its reciprocal from four rectangles in a square", "description": "This is a short, animated visual proof demonstrating that sum of a positive real number and its reciprocal is always greater than or equal to 2. \n\nTo buy me a coffee, head over to https://www.buymeacoffee.com/VisualProofs \nThanks!\n\nFor a wider format and slightly longer version of this proof, see https://youtu.be/IghOHBl0Do8?si=CSvPRS3Dr3AGBWVO\n\nIt turns out that this theorem is equivalent to the Arithmetic Mean-Geometric Mean inequality. The equivalence is implied by the fact that his video resembles a similar visual proof for the AM-GM inequality found here: https://youtu.be/62G9fak1vyk\n\nThis animation is based on a proof by Roger B. Nelsen from the December 1994 issue of Mathematics Magazine, page 374 (https://doi.org/10.2307/2690999).\n\n#math\u200b #manim\u200b #visualproof\u200b #mathvideo\u200b  #geometry #mathshorts\u200b  #algebra #mtbos\u200b  #animation\u200b #theorem\u200b #pww\u200b #proofwithoutwords\u200b #proof\u200b #iteachmath #inequality #amgm #maths #mathematics #education \n\nTo learn more about animating with manim, check out:\nhttps://manim.community", "lengthSeconds": 50, "width": 360, "height": 640}
{"video_id": "R8RgR4wvzic", "transcript": [{"start": 0.0, "duration": 3.78, "text": "start with a cube with side length one"}, {"start": 1.86, "duration": 3.3, "text": "over square root 1. on top of that place"}, {"start": 3.78, "duration": 3.24, "text": "a cube with side length one over square"}, {"start": 5.16, "duration": 3.12, "text": "root of two on top of that place a cube"}, {"start": 7.02, "duration": 2.76, "text": "with the side length of 1 over the"}, {"start": 8.28, "duration": 3.3, "text": "square root of three and continue this"}, {"start": 9.78, "duration": 3.479, "text": "process indefinitely the resulting"}, {"start": 11.58, "duration": 3.96, "text": "structure sometimes called the infinite"}, {"start": 13.259, "duration": 4.201, "text": "GIF consists of an infinite stack of"}, {"start": 15.54, "duration": 4.02, "text": "Cubes each with a side length of 1 over"}, {"start": 17.46, "duration": 4.02, "text": "the square root of n each Cube has a"}, {"start": 19.56, "duration": 3.96, "text": "side with area given by the square root"}, {"start": 21.48, "duration": 3.959, "text": "of one over N squared so the total"}, {"start": 23.52, "duration": 3.72, "text": "surface area of this structure is"}, {"start": 25.439, "duration": 3.901, "text": "greater than or equal to four times the"}, {"start": 27.24, "duration": 5.28, "text": "infinite sum where n ranges from 1 to"}, {"start": 29.34, "duration": 5.58, "text": "Infinity of 1 over root N squared on the"}, {"start": 32.52, "duration": 4.14, "text": "other hand the volume of each cube is"}, {"start": 34.92, "duration": 3.84, "text": "given by the quantity one over the"}, {"start": 36.66, "duration": 4.079, "text": "square root of n cubed so the volume of"}, {"start": 38.76, "duration": 3.959, "text": "this structure is the infinite sum where"}, {"start": 40.739, "duration": 4.741, "text": "n ranges from 1 to Infinity of the"}, {"start": 42.719, "duration": 5.101, "text": "quantity one over root n cubed it turns"}, {"start": 45.48, "duration": 4.02, "text": "out that the area series diverges"}, {"start": 47.82, "duration": 3.78, "text": "because it's the harmonic series times"}, {"start": 49.5, "duration": 3.84, "text": "four and the volume infinite series"}, {"start": 51.6, "duration": 5.04, "text": "converges to a value that's"}, {"start": 53.34, "duration": 4.859, "text": "approximately 2.61 this structure is a"}, {"start": 56.64, "duration": 5.12, "text": "discrete version of Gabriel's horn and"}, {"start": 58.199, "duration": 3.561, "text": "can be used for the painter's paradox"}], "title": "Infinite gift and the painter\u2019s #paradox", "description": "In this animation, we show an approximation of the mathematical object colloquially referred to as the \"infinite gift.\" This gift is somewhat paradoxical in nature because it is infinitely long and requires an infinite amount of wrapping paper to cover, yet it only encloses a finite area. This object is a discrete analog of the more famous \"Gabriel's horn,\" which is an object studied in most integral calculus classes as it is an object enclosing finite volume but has an infinite surface area.\n\nOne major purpose of this video is to investigate the different, but related, infinite sums of 1/n and 1/sqrt(n cubed). These sums both have interpretations in terms of the infinite gift, and one of them diverges while the other converges. \n\nTo buy me a coffee, head over to https://www.buymeacoffee.com/VisualProofs \nThanks!\n\nHere is a slower, wide format, high definition alternate version of this video:\n\nhttps://youtu.be/YCCNWNXXSx4\n\nFor related information, I would begin with the Wikipedia article about Gabriel's horn and the painter's paradox: https://en.wikipedia.org/wiki/Gabriel%27s_horn\n\nHere is one of the first places I remember seeing this object (there are earlier ones, but I can't track them down): https://twitter.com/fermatslibrary/status/1166324490760065024?s=20&t=CLJ2N2LDs-yj0ASNTGzK-Q\n\n#manim #infinitegift #paintersparadox #gabrielshorn #harmonicseries #infiniteseries #convergence #divergence\n\nTo learn more about animating with manim, check out:\nhttps://manim.community", "lengthSeconds": 60, "width": 360, "height": 640}
{"video_id": "9BC1klE8jMk", "transcript": [{"start": 0.0, "duration": 7.82, "text": "[Music]"}, {"start": 12.679, "duration": 3.061, "text": "thank you"}, {"start": 17.9, "duration": 17.92, "text": "[Music]"}, {"start": 32.82, "duration": 3.0, "text": "foreign"}, {"start": 38.81, "duration": 20.08, "text": "[Music]"}, {"start": 62.69, "duration": 4.02, "text": "[Music]"}, {"start": 64.44, "duration": 15.07, "text": "foreign"}, {"start": 66.71, "duration": 12.8, "text": "[Music]"}, {"start": 86.69, "duration": 17.37, "text": "[Music]"}, {"start": 101.82, "duration": 2.24, "text": "foreign"}, {"start": 109.04, "duration": 8.89, "text": "[Music]"}], "title": "Halloween Visual Proof: Alternating Sum of Odds #math", "description": "This is a short, animated visual proof showing one method to prove that the alternating sum of the first n odd positive integers (starting with 2n-1 positive) is n. #manim #math #mathvideo #mathshorts #sequences #triangles #animation #theorem #pww #proofwithoutwords #visualproof #proof #squarenumbers #squares #oddnumbers #odds #sums #pww\u200b #proofwithoutwords\u200b \u200b #proof\u200b #algebra #finitesums #mathematics\u200b #mathvideo\u200b #mtbos \n\nIf you like this video, consider subscribing to the channel or consider buying me a coffee: https://www.buymeacoffee.com/VisualProofs. Thanks!\n\nThis animation is based on visual proof by Art Benjamin from the December 2005 issue of Mathematics Magazine, page 385 (https://www.jstor.org/stable/30044195)\n\nTo learn more about animating with manim, check out:\nhttps://manim.community\n_____________________________\nMusic in this video:\nHalloween Theme 1, written by Alexander Nakarada.\nCourtesy of https://freepd.com/horror.php", "lengthSeconds": 120, "width": 640, "height": 360}
{"video_id": "ltwnceNYWds", "transcript": [{"start": 0.28, "duration": 3.4, "text": "there are numerous ways that a function"}, {"start": 1.88, "duration": 3.879, "text": "might fail to be differentiable at a"}, {"start": 3.68, "duration": 4.24, "text": "point you might have a removable"}, {"start": 5.759, "duration": 4.88, "text": "discontinuity or a so-called hole at the"}, {"start": 7.92, "duration": 4.4, "text": "point similarly you might have a jump"}, {"start": 10.639, "duration": 3.2, "text": "discontinuity in this case it's"}, {"start": 12.32, "duration": 4.24, "text": "impossible to have a uniquely defined"}, {"start": 13.839, "duration": 3.561, "text": "tangent line in fact any discontinuity"}, {"start": 16.56, "duration": 3.04, "text": "leads to a point of"}, {"start": 17.4, "duration": 4.119, "text": "non-differentiability so vertical ASM"}, {"start": 19.6, "duration": 3.8, "text": "tootes also create problems such as the"}, {"start": 21.519, "duration": 4.441, "text": "one you might find in the function 1 /x"}, {"start": 23.4, "duration": 4.799, "text": "like this you might also have a point"}, {"start": 25.96, "duration": 4.2, "text": "where there's an Abrupt change in slope"}, {"start": 28.199, "duration": 3.32, "text": "this is often called a cusp or a corner"}, {"start": 30.16, "duration": 3.96, "text": "and it appears in functions like the"}, {"start": 31.519, "duration": 4.161, "text": "absolute value function another way is"}, {"start": 34.12, "duration": 3.119, "text": "that you might have a vertical tangent"}, {"start": 35.68, "duration": 3.68, "text": "line like the one appearing in the cube"}, {"start": 37.239, "duration": 4.8, "text": "root function this would correspond to"}, {"start": 39.36, "duration": 4.719, "text": "an infinite slope which is undefined so"}, {"start": 42.039, "duration": 4.241, "text": "we've seen that discontinuities places"}, {"start": 44.079, "duration": 3.8, "text": "with abrupt changes in slope and points"}, {"start": 46.28, "duration": 2.919, "text": "where there is a vertical tangent line"}, {"start": 47.879, "duration": 3.16, "text": "all create problems for"}, {"start": 49.199, "duration": 3.441, "text": "differentiability can you think of any"}, {"start": 51.039, "duration": 4.721, "text": "other situations where a function won't"}, {"start": 52.64, "duration": 3.12, "text": "be differentiable"}], "title": "Non-differentiability", "description": "In this short, we show five different graphs of functions with a point where the function is not differentiable.\n\nIf you like this video, consider subscribing to the channel or consider buying me a coffee: https://www.buymeacoffee.com/VisualProofs. Thanks! \n\n#math\u200b \u200b#manim\u200b #animation\u200b #theorem\u200b #iteachmath #mathematics ##mathshorts\u200b #mathvideo\u200b #mtbos #mathematician #calculus #differentiability  #functions #derivative #discontinuity #cusp #verticalasymptote #tangentline \n\nTo learn more about animating with manim, check out:\nhttps://manim.community", "lengthSeconds": 54, "width": 360, "height": 640}
{"video_id": "q28H5NY5UVU", "transcript": [{"start": 2.47, "duration": 14.41, "text": "[Music]"}, {"start": 14.88, "duration": 2.96, "text": "consider a right triangle with legs of"}, {"start": 16.88, "duration": 4.479, "text": "length a"}, {"start": 17.84, "duration": 6.0, "text": "and b and a hypotenuse of length c"}, {"start": 21.359, "duration": 4.16, "text": "first let's scale this triangle by the"}, {"start": 23.84, "duration": 4.16, "text": "value of c"}, {"start": 25.519, "duration": 3.361, "text": "this creates a right triangle and the"}, {"start": 28.0, "duration": 4.0, "text": "new legs"}, {"start": 28.88, "duration": 6.48, "text": "are length a times c and b times c"}, {"start": 32.0, "duration": 5.28, "text": "the hypotenuse is c squared"}, {"start": 35.36, "duration": 5.76, "text": "next we scale the triangle by the value"}, {"start": 37.28, "duration": 5.599, "text": "of b to obtain another right triangle"}, {"start": 41.12, "duration": 3.279, "text": "this right triangle has leg lengths a"}, {"start": 42.879, "duration": 3.761, "text": "times b and b"}, {"start": 44.399, "duration": 4.16, "text": "squared and a hypotenuse of length c"}, {"start": 46.64, "duration": 3.919, "text": "times b"}, {"start": 48.559, "duration": 4.32, "text": "finally we scale the original triangle"}, {"start": 50.559, "duration": 4.561, "text": "one more time by the value a"}, {"start": 52.879, "duration": 3.281, "text": "this produces a right triangle and in"}, {"start": 55.12, "duration": 3.84, "text": "this instance"}, {"start": 56.16, "duration": 5.12, "text": "the leg lengths are a squared b times a"}, {"start": 58.96, "duration": 4.079, "text": "and hypotenuse ca"}, {"start": 61.28, "duration": 3.44, "text": "notice that the second and third scaled"}, {"start": 63.039, "duration": 2.801, "text": "triangles fit together along their sides"}, {"start": 64.72, "duration": 3.759, "text": "of common length a b"}, {"start": 65.84, "duration": 4.08, "text": "equals b a glued together these"}, {"start": 68.479, "duration": 2.64, "text": "triangles create yet another right"}, {"start": 69.92, "duration": 3.04, "text": "triangle"}, {"start": 71.119, "duration": 5.121, "text": "this can be verified as the two acute"}, {"start": 72.96, "duration": 5.36, "text": "angles are complementary by construction"}, {"start": 76.24, "duration": 4.08, "text": "finally note that this new triangle is"}, {"start": 78.32, "duration": 3.6, "text": "congruent to the first scale triangle as"}, {"start": 80.32, "duration": 3.28, "text": "they have legs of the same lengths and"}, {"start": 81.92, "duration": 4.0, "text": "are both right triangles"}, {"start": 83.6, "duration": 3.44, "text": "therefore the hypotenuses must be equal"}, {"start": 85.92, "duration": 4.159, "text": "so that a squared"}, {"start": 87.04, "duration": 12.32, "text": "plus b squared equals c squared this is"}, {"start": 90.079, "duration": 9.281, "text": "the pythagorean theorem"}], "title": "Pythagorean Theorem I (visual proof)", "description": "This is a short, animated visual proof of the Pythagorean theorem (the right triangle theorem) using the hypotenuses of scaled triangles. This theorem states the square of the hypotenuse of a right triangle is equal to the sum of squares of the two other side lengths. #mathshorts #mathvideo #math #pythagoreantheorem #pythagorean #triangle #manim #animation #theorem #pww #proofwithoutwords #visualproof #proof\n\nThis animation is based on a visual proof by Frank Burk from the November 1996 issue of The College Mathematics Journal (https://doi.org/10.2307/2687335 page 409).\n\nTo learn more about animating with manim, check out:\nhttps://manim.community\n__________________________________________________________________\nBackground music, used according to CC BY-SA 3.0, is \"The Road\" from Esteban Orlando:\n\nEsteban Orlando:\nhttps://soundcloud.com/orlando-esteban-2\u200b\nhttps://www.instagram.com/estebanorla...\u200b\n\nhttps://creativecommons.org/licenses/...\u200b\nMusic provided by RFM: https://youtu.be/b4KCM1bIq50", "lengthSeconds": 98, "width": 640, "height": 360}
{"video_id": "G1JrE3875M8", "transcript": [{"start": 0.0, "duration": 3.0, "text": "foreign"}, {"start": 3.4, "duration": 7.67, "text": "[Music]"}, {"start": 19.54, "duration": 3.62, "text": "[Music]"}, {"start": 31.08, "duration": 3.42, "text": "foreign"}, {"start": 33.18, "duration": 8.399, "text": "foreign"}, {"start": 34.5, "duration": 7.079, "text": "[Music]"}], "title": "Summing odds using four stacks", "description": "This is a short, animated visual proof demonstrating how to find the sum of the first n positive odd integers. #math\u200b #numbertheory #mtbos\u200b #manim\u200b #animation\u200b #theorem\u200b #pww\u200b #proofwithoutwords\u200b #visualproof\u200b #proof\u200b #iteachmath #squares #oddintegers #numbertheory #sum #mathshorts\u200b #mathvideo\u200b \n\nHere is the original widescreen high def version: https://youtu.be/ZeEOgbLo0Rg\n\nThis animation is based on a visual proof by Roger B. Nelsen from his first compendium of visual proofs:\n\nProofs without Words I by Roger B. Nelsen (MAA, 1993) (https://bookstore.ams.org/clrm-1)\n\nTo learn more about animating with manim, check out:\nhttps://manim.community", "lengthSeconds": 44, "width": 360, "height": 640}
{"video_id": "1zKBUwNZA3g", "transcript": [{"start": 0.16, "duration": 2.88, "text": "this is one of the ways to prove the"}, {"start": 1.599, "duration": 3.281, "text": "pythagorean theorem of c"}, {"start": 3.04, "duration": 4.319, "text": "squared is equal to a squared plus b"}, {"start": 4.88, "duration": 3.52, "text": "squared using four similar triangles"}, {"start": 7.359, "duration": 3.841, "text": "with sides a b"}, {"start": 8.4, "duration": 3.92, "text": "and c we can arrange the four triangles"}, {"start": 11.2, "duration": 3.76, "text": "in such a way"}, {"start": 12.32, "duration": 4.08, "text": "that the hypotenuse side c would be"}, {"start": 14.96, "duration": 3.12, "text": "touching each other"}, {"start": 16.4, "duration": 4.879, "text": "and this would leave us with a big empty"}, {"start": 18.08, "duration": 5.279, "text": "space in between with a side c"}, {"start": 21.279, "duration": 3.441, "text": "we can compute the area of this empty"}, {"start": 23.359, "duration": 3.601, "text": "space as c times c"}, {"start": 24.72, "duration": 5.2, "text": "or c squared now let's rearrange the"}, {"start": 26.96, "duration": 4.239, "text": "same four right triangles but this time"}, {"start": 29.92, "duration": 4.319, "text": "the sides a and b"}, {"start": 31.199, "duration": 5.36, "text": "will be touching each other unlike the"}, {"start": 34.239, "duration": 2.961, "text": "first one which has one big empty space"}, {"start": 36.559, "duration": 3.281, "text": "left"}, {"start": 37.2, "duration": 4.8, "text": "this time we have two small spaces or"}, {"start": 39.84, "duration": 5.12, "text": "two small squares"}, {"start": 42.0, "duration": 3.84, "text": "one and two the area of one square is a"}, {"start": 44.96, "duration": 3.599, "text": "times a"}, {"start": 45.84, "duration": 3.84, "text": "or a squared and the other is b times v"}, {"start": 48.559, "duration": 2.881, "text": "or b squared"}, {"start": 49.68, "duration": 3.44, "text": "since we didn't change the areas of our"}, {"start": 51.44, "duration": 2.24, "text": "right triangles that means that the big"}, {"start": 53.12, "duration": 2.48, "text": "area c"}, {"start": 53.68, "duration": 3.92, "text": "squared is equal to the sum of b squared"}, {"start": 55.6, "duration": 4.24, "text": "plus a squared or the pythagorean"}, {"start": 57.6, "duration": 2.24, "text": "theorem"}], "title": "Visual Proof of the Pythagorean Theorem | Math Help and Tutorial #shorts", "description": "Have you learned this in high school? Here's a visual proof of the popular Pythagorean Theorem #Shorts\nThis channel is for anyone wanting for math help, algebra help, calculus help, engineering math help, and any tricks on math.\n\n\nFollow us on Tiktok: https://www.tiktok.com/@engineeringmath?lang=en", "lengthSeconds": 59, "width": 202, "height": 360}
{"video_id": "XJ-GaUrY-LE", "transcript": [{"start": 0.0, "duration": 3.84, "text": "here's an amazing diagram due to Rex woo"}, {"start": 2.04, "duration": 4.14, "text": "this is a 3x2 rectangle where we've"}, {"start": 3.84, "duration": 4.14, "text": "inscribed a triangle because these two"}, {"start": 6.18, "duration": 3.54, "text": "highlighted triangles are similar the"}, {"start": 7.98, "duration": 3.48, "text": "inscribed triangle is a right triangle"}, {"start": 9.72, "duration": 3.78, "text": "that happens to be isosceles so these"}, {"start": 11.46, "duration": 3.839, "text": "other two angles are pi over four we can"}, {"start": 13.5, "duration": 4.38, "text": "compute this Arc Length of pi over four"}, {"start": 15.299, "duration": 4.32, "text": "by using these two triangles here which"}, {"start": 17.88, "duration": 3.12, "text": "have angle measurements of arc tangent"}, {"start": 19.619, "duration": 3.42, "text": "one over two and arctangent one over"}, {"start": 21.0, "duration": 3.96, "text": "three using the rectangular diagram we"}, {"start": 23.039, "duration": 3.48, "text": "can compute the other pi over 4 angle as"}, {"start": 24.96, "duration": 3.12, "text": "the difference of the angles pictured in"}, {"start": 26.519, "duration": 3.061, "text": "these two triangles which are arc"}, {"start": 28.08, "duration": 3.42, "text": "tangent of three and arc tangent of one"}, {"start": 29.58, "duration": 3.659, "text": "half respectively using the diagram we"}, {"start": 31.5, "duration": 3.18, "text": "can compute that same pi over 4 angle"}, {"start": 33.239, "duration": 3.061, "text": "using the difference of the angles in"}, {"start": 34.68, "duration": 3.539, "text": "these two triangles which are arc"}, {"start": 36.3, "duration": 3.66, "text": "tangent of two and arctangent of 1 3"}, {"start": 38.219, "duration": 3.601, "text": "respectively we can also compute the"}, {"start": 39.96, "duration": 3.119, "text": "upper right pi over 2 angle as the sum"}, {"start": 41.82, "duration": 3.18, "text": "of the angles in these three picture"}, {"start": 43.079, "duration": 3.541, "text": "triangles so we get that pi over two is"}, {"start": 45.0, "duration": 3.66, "text": "arc tangent one plus arctangent a half"}, {"start": 46.62, "duration": 4.14, "text": "plus arctangent a third and finally we"}, {"start": 48.66, "duration": 3.84, "text": "can compute this Pi angle as the sum of"}, {"start": 50.76, "duration": 3.54, "text": "the angles in these three triangles so"}, {"start": 52.5, "duration": 3.36, "text": "we get that Pi is arctangent of one plus"}, {"start": 54.3, "duration": 3.599, "text": "arc tangent of two plus arctangent of"}, {"start": 55.86, "duration": 4.879, "text": "three this diagram has given us five"}, {"start": 57.899, "duration": 2.84, "text": "formulas for pi"}], "title": "A Five Pi Diagram (pi day short)", "description": "In this short, we give five different formulas for pi (or related constants) using only a single diagram consisting of a 3x2 rectangle and a triangle inside. Using the fact that the tangent of an angle is the ratio of the opposite and adjacent sides of an angle (and you can get the angle using the arctangent of the appropriate ratio), we decompose angles (of measure pi/4, pi/2, and pi) using certain triangles with visually computable arctangents using the diagram.\n\nThis visual proof is due to Rex Wu and comes from the March 2003 issue of The College Mathematics Journal (https://www.jstor.org/stable/3595783 ) see pages 115 and 138.\n\nFor more Pi-related videos, check out my playlist:\nhttps://youtube.com/playlist?list=PLZh9gzIvXQUsGRDvzvXc02lDIKHaVeFwq\n\n#manim #irrational #Pi #mathvideo\u200b #math\u200b #mtbos\u200b \u200b #animation\u200b\u200b\u200b #iteachmath #mathematics #piday #shorts #trigonometry #arctangent #tangent #triangle #rectangle #identities \n\nTo learn more about animating with manim, check out:\nhttps://manim.community", "lengthSeconds": 59, "width": 360, "height": 640}
{"video_id": "NxOcT_VKQR0", "transcript": [{"start": 2.43, "duration": 12.49, "text": "[Music]"}, {"start": 15.04, "duration": 3.96, "text": "let's build an expanding triangular"}, {"start": 16.84, "duration": 5.0, "text": "array of squares where the first row has"}, {"start": 19.0, "duration": 5.24, "text": "1 1X one square the second row has 2 2x"}, {"start": 21.84, "duration": 4.56, "text": "two squares the third row has 3 3x3"}, {"start": 24.24, "duration": 4.799, "text": "squares the fourth row has four 4x4"}, {"start": 26.4, "duration": 4.719, "text": "squares and so on until the last row row"}, {"start": 29.039, "duration": 4.321, "text": "N has n and by n"}, {"start": 31.119, "duration": 4.6, "text": "squares the side length of this array is"}, {"start": 33.36, "duration": 5.199, "text": "the sum of the first n positive integers"}, {"start": 35.719, "duration": 7.241, "text": "which is n * n + 1 /"}, {"start": 38.559, "duration": 7.0, "text": "2 the line of slope -2 starting from the"}, {"start": 42.96, "duration": 4.8, "text": "top left hits the upper left corner of"}, {"start": 45.559, "duration": 5.081, "text": "the last Square in every row and"}, {"start": 47.76, "duration": 4.84, "text": "produces a triangle this triangle has"}, {"start": 50.64, "duration": 5.12, "text": "base length n * n +"}, {"start": 52.6, "duration": 5.439, "text": "1 the small triangles excluded from the"}, {"start": 55.76, "duration": 4.639, "text": "large triangle by the line can be"}, {"start": 58.039, "duration": 5.401, "text": "shifted and rotated to fill in miss gaps"}, {"start": 60.399, "duration": 6.441, "text": "at the large triangle like this"}, {"start": 63.44, "duration": 8.999, "text": "therefore the area of the original array"}, {"start": 66.84, "duration": 8.2, "text": "is 12 * n * n + 1 / 2 * n * n + 1 which"}, {"start": 72.439, "duration": 4.441, "text": "is the area of the triangle from this we"}, {"start": 75.04, "duration": 4.04, "text": "can see that the sum of the first n"}, {"start": 76.88, "duration": 6.68, "text": "positive cubes is equal to the square of"}, {"start": 79.08, "duration": 4.48, "text": "n * n + 1 / 2"}, {"start": 84.08, "duration": 5.579, "text": "[Music]"}], "title": "Sum Of Cubes I (visual proof)", "description": "This is a short, animated visual proof giving a formula for the sum of the first n positive cubes. #mathshorts\u200b #mathvideo\u200b #math\u200b #numbertheory #mtbos\u200b #manim\u200b #animation\u200b #theorem\u200b #pww\u200b #proofwithoutwords\u200b #visualproof\u200b #proof\u200b #iteachmath #cubes #mathematics #sum\n\nThis animation is based on a visual proof by Georg Schrage from the June 1992 issue of Mathematics Magazine (https://doi.org/10.2307/2691330 - page 185).\n\nTo learn more about animating with manim, check out:\nhttps://manim.community\n__________________________________________________________________\nBackground music, used according to CC BY-SA 3.0, is \"The Road\" from Esteban Orlando:\n\nEsteban Orlando:\nhttps://soundcloud.com/orlando-esteba...\nhttps://www.instagram.com/estebanorla...\n\nhttps://creativecommons.org/licenses/...\nMusic provided by RFM: https://youtu.be/b4KCM1bIq50", "lengthSeconds": 98, "width": 640, "height": 360}
{"video_id": "I2m7YTHEZos", "transcript": [{"start": 0.44, "duration": 4.08, "text": "let's suppose that you have two finite"}, {"start": 2.399, "duration": 5.081, "text": "sets A and B and you want to find the"}, {"start": 4.52, "duration": 4.56, "text": "cardinality of their Union a union B you"}, {"start": 7.48, "duration": 3.96, "text": "might think that the cardinality of the"}, {"start": 9.08, "duration": 4.0, "text": "Union is simply the sum of the"}, {"start": 11.44, "duration": 3.68, "text": "cardinalities that is that the"}, {"start": 13.08, "duration": 4.359, "text": "cardinality of a union B is the"}, {"start": 15.12, "duration": 4.84, "text": "cardinality of a added to the"}, {"start": 17.439, "duration": 4.641, "text": "cardinality of B but note that elements"}, {"start": 19.96, "duration": 4.079, "text": "in the set a intersect B have been"}, {"start": 22.08, "duration": 4.88, "text": "counted twice in this computation once"}, {"start": 24.039, "duration": 5.24, "text": "in a and once in B so we must remove one"}, {"start": 26.96, "duration": 4.84, "text": "copy of the cardinality of a intersect B"}, {"start": 29.279, "duration": 4.12, "text": "so the cardinal it of a union B is equal"}, {"start": 31.8, "duration": 3.72, "text": "to the cardinality of a plus the"}, {"start": 33.399, "duration": 4.281, "text": "cardinality of B minus the cardinality"}, {"start": 35.52, "duration": 4.44, "text": "of the intersection of A and B an"}, {"start": 37.68, "duration": 4.12, "text": "alternative is to think of the Union in"}, {"start": 39.96, "duration": 3.759, "text": "three separate pieces so that the"}, {"start": 41.8, "duration": 4.439, "text": "cardinality of a union B is the"}, {"start": 43.719, "duration": 4.68, "text": "cardinality of a set minus B plus the"}, {"start": 46.239, "duration": 6.081, "text": "cardinality of a intersect B plus the"}, {"start": 48.399, "duration": 3.921, "text": "cardinality of B set minus a"}], "title": "General Sum Principle #venndiagram  #math", "description": "In this video, we show the general sum principle for sets (the formula for the cardinality of A union B). The key feature is that if the sets intersect, then we have to account for elements that may have been double counted.\n\nIf you like this video, consider subscribing to the channel or consider buying me a coffee: https://www.buymeacoffee.com/VisualProofs. Thanks!\n\nFor a more dramatic version of this video, check out: \nhttps://youtu.be/wXdoqPMlVRU (which includes the product principle too).\n\nIf you want to learn more about these principles, check out:\nhttps://youtu.be/sbIcWGYNA5I\n\nA related textbook forthis is the open-source textbook by Oscar Levin:\nhttp://discrete.openmathbooks.org/dmoi3.html\n\n#logic #settheory #union #propositions  #intersection #setdifference  #setminus #setconnectives #subsets #venndiagram #visualproof #math #manim #discretemathematics #sumprinciple #inclusionexclusion\n\nTo learn more about animating with manim, check out:\nhttps://manim.community", "lengthSeconds": 50, "width": 360, "height": 640}
{"video_id": "mCnjpnhLeDU", "transcript": [{"start": 3.18, "duration": 3.85, "text": "[Music]"}, {"start": 10.8, "duration": 3.04, "text": "let a"}, {"start": 12.48, "duration": 2.64, "text": "b"}, {"start": 13.84, "duration": 3.359, "text": "and c"}, {"start": 15.12, "duration": 4.64, "text": "all be sets"}, {"start": 17.199, "duration": 3.521, "text": "and suppose that f is a function from a"}, {"start": 19.76, "duration": 3.759, "text": "to b"}, {"start": 20.72, "duration": 5.12, "text": "and g is a function from b to c"}, {"start": 23.519, "duration": 5.041, "text": "then the function g composed f"}, {"start": 25.84, "duration": 4.8, "text": "is the function from a to c"}, {"start": 28.56, "duration": 4.88, "text": "defined by the rule that g composed f of"}, {"start": 30.64, "duration": 5.68, "text": "a is equal to g of f a"}, {"start": 33.44, "duration": 4.799, "text": "that is we send a into f"}, {"start": 36.32, "duration": 3.36, "text": "and send the resulting output from b"}, {"start": 38.239, "duration": 4.32, "text": "into g"}, {"start": 39.68, "duration": 5.359, "text": "the composition takes a directly to that"}, {"start": 42.559, "duration": 5.761, "text": "output so that g composed f of a is"}, {"start": 45.039, "duration": 3.281, "text": "equal to g of f of a"}, {"start": 49.92, "duration": 4.72, "text": "let's see this definition in action as"}, {"start": 52.239, "duration": 4.16, "text": "we prove that function composition is"}, {"start": 54.64, "duration": 3.84, "text": "associative"}, {"start": 56.399, "duration": 4.881, "text": "this means that if we have three"}, {"start": 58.48, "duration": 5.2, "text": "functions f g and h"}, {"start": 61.28, "duration": 5.44, "text": "then the composition of h with g"}, {"start": 63.68, "duration": 4.88, "text": "composed f is equal to h composed g"}, {"start": 66.72, "duration": 3.52, "text": "composed with f"}, {"start": 68.56, "duration": 3.36, "text": "in particular no matter what element"}, {"start": 70.24, "duration": 5.919, "text": "from the domain a we plug into these"}, {"start": 71.92, "duration": 4.239, "text": "functions the outputs will be equal"}, {"start": 76.64, "duration": 4.32, "text": "let's see this proof visually let a b"}, {"start": 80.24, "duration": 2.96, "text": "c"}, {"start": 80.96, "duration": 5.28, "text": "and d all be sets"}, {"start": 83.2, "duration": 6.239, "text": "and suppose that f is a function from a"}, {"start": 86.24, "duration": 4.96, "text": "to b g is a function from b to c and h"}, {"start": 89.439, "duration": 3.68, "text": "is a function from c to d"}, {"start": 91.2, "duration": 4.64, "text": "then we know that g composed f is a"}, {"start": 93.119, "duration": 6.921, "text": "function from a to c and h compose g is"}, {"start": 95.84, "duration": 4.2, "text": "a function from b to d"}, {"start": 101.28, "duration": 5.04, "text": "start with an arbitrary element x from"}, {"start": 103.68, "duration": 5.68, "text": "the domain a"}, {"start": 106.32, "duration": 6.0, "text": "we can send x through f"}, {"start": 109.36, "duration": 5.84, "text": "send the output through g"}, {"start": 112.32, "duration": 7.2, "text": "and send that output through h resulting"}, {"start": 115.2, "duration": 6.559, "text": "in the element h of g of f of x"}, {"start": 119.52, "duration": 5.44, "text": "now we end at the exact same element"}, {"start": 121.759, "duration": 5.68, "text": "when we send x through g composed f"}, {"start": 124.96, "duration": 5.439, "text": "and then the resulting output through h"}, {"start": 127.439, "duration": 6.081, "text": "so h composed with a g composed f of x"}, {"start": 130.399, "duration": 6.081, "text": "is equal to h of g composed f of x which"}, {"start": 133.52, "duration": 4.96, "text": "is equal to h of g of f of x"}, {"start": 136.48, "duration": 4.32, "text": "similarly we ended that exact same"}, {"start": 138.48, "duration": 4.56, "text": "element when we send x through f and the"}, {"start": 140.8, "duration": 5.04, "text": "resulting output through h composed g"}, {"start": 143.04, "duration": 5.199, "text": "so h composed g composed with f x is"}, {"start": 145.84, "duration": 5.039, "text": "equal to h composed g of f of x which is"}, {"start": 148.239, "duration": 5.521, "text": "equal to h of g of f x"}, {"start": 150.879, "duration": 5.601, "text": "therefore the two compositions map x to"}, {"start": 153.76, "duration": 4.8, "text": "the same element no matter what x is"}, {"start": 156.48, "duration": 4.42, "text": "so the two compositions are equal as"}, {"start": 158.56, "duration": 5.52, "text": "functions"}, {"start": 160.9, "duration": 3.18, "text": "[Music]"}, {"start": 173.92, "duration": 2.08, "text": "you"}], "title": "Function Composition is Associative  (visual proof)", "description": "This is a proof, with a visualization, of the classic set-theoretic proof that the operation of function composition is associative. The video includes a \"visual definition\" of the composition operation and then shows how to prove the theorem. Implicit in this explanation is the style of proof required when proving a universal statement (here that two functions are equal by showing they behave the same for every element in the domain). #manim #composition #associativeproperty #visualmath #math #mtbos #visualproof #settheory #introductiontoproofs #associative #function #functioncomposition #universalstatement #functiondiagram\n\nTo learn more about animating with manim, check out:\nhttps://manim.community", "lengthSeconds": 174, "width": 640, "height": 360}
{"video_id": "HlZTLvKNH7c", "transcript": [{"start": 0.04, "duration": 4.16, "text": "start with a right triangle with legs of"}, {"start": 1.839, "duration": 4.721, "text": "length A and B and a hypotenuse of"}, {"start": 4.2, "duration": 4.439, "text": "length C use the hypotenuse as the"}, {"start": 6.56, "duration": 4.6, "text": "radius of a circle centered at the left"}, {"start": 8.639, "duration": 4.08, "text": "point of the triangle with radius C the"}, {"start": 11.16, "duration": 4.32, "text": "horizontal distance from the right angle"}, {"start": 12.719, "duration": 4.161, "text": "to the circle is then C minus a when we"}, {"start": 15.48, "duration": 3.6, "text": "inscribe a triangle in the top"}, {"start": 16.88, "duration": 4.479, "text": "semicircle like this by the's triangle"}, {"start": 19.08, "duration": 3.88, "text": "theorem this triangle is right angled"}, {"start": 21.359, "duration": 3.121, "text": "one consequence of this fact is that"}, {"start": 22.96, "duration": 3.84, "text": "these two pictured right triangles here"}, {"start": 24.48, "duration": 4.6, "text": "must be similar as their top angles are"}, {"start": 26.8, "duration": 5.0, "text": "compliment because the two triangles are"}, {"start": 29.08, "duration": 6.639, "text": "similar their leg ratios must be equal"}, {"start": 31.8, "duration": 7.36, "text": "that means that C + a / B must equal B /"}, {"start": 35.719, "duration": 8.921, "text": "c - A but this equality means that C + a"}, {"start": 39.16, "duration": 8.44, "text": "* c - A = b^ 2 so that c^2 - A 2 = B ^2"}, {"start": 44.64, "duration": 4.68, "text": "from this we see that c^ 2al A2 + b^2"}, {"start": 47.6, "duration": 3.84, "text": "where A and B are the legs of a right"}, {"start": 49.32, "duration": 3.879, "text": "triangle with hypotenuse C we used a"}, {"start": 51.44, "duration": 4.32, "text": "semicircle to prove the Pythagorean"}, {"start": 53.199, "duration": 2.561, "text": "theorem"}], "title": "Pythagorean theorem from a (semi) circle!", "description": "This is a short, animated visual proof of the Pythagorean theorem (the right triangle theorem) using the semicircle and Thales triangle theorem. This theorem states the square of the hypotenuse of a right triangle is equal to the sum of squares of the two other side lengths. \n\nThis animation is based on a proof due to Michael Hardy from the November, 1986 issue of The College Math Journal (https://doi.org/10.2307/2686255).\n\nTo buy me a coffee, head over to https://www.buymeacoffee.com/VisualProofs \nThanks!\n\nFor other proofs of this same fact check out the following playlist:\n\nhttps://youtube.com/playlist?list=PLZh9gzIvXQUvlRlnYryejWGHYQquM92rB&si=o_im7-_NHp1zhkcD\n#mathshorts #mathvideo #math #pythagoreantheorem #pythagorean #triangle #manim #animation #theorem #pww #proofwithoutwords #visualproof #proof\n\nTo learn more about animating with manim, check out:\nhttps://manim.community", "lengthSeconds": 54, "width": 360, "height": 640}
{"video_id": "XbTPJ0eaezE", "transcript": [{"start": 0.199, "duration": 4.961, "text": "to find the area formula for a trapezoid"}, {"start": 2.56, "duration": 4.799, "text": "with bases of length B1 and B2 and a"}, {"start": 5.16, "duration": 4.399, "text": "height of length H we can perform the"}, {"start": 7.359, "duration": 4.601, "text": "following interesting process take one"}, {"start": 9.559, "duration": 4.761, "text": "of the side lengths and bict it cutting"}, {"start": 11.96, "duration": 4.76, "text": "it into two equal length pieces like"}, {"start": 14.32, "duration": 4.36, "text": "this now connect the midpoint to one of"}, {"start": 16.72, "duration": 4.2, "text": "the opposite Corners creating a small"}, {"start": 18.68, "duration": 4.28, "text": "triangle we can cut this triangle off"}, {"start": 20.92, "duration": 4.76, "text": "and rotate it around to create a larger"}, {"start": 22.96, "duration": 4.719, "text": "triangle with base of length B1 plus B2"}, {"start": 25.68, "duration": 6.04, "text": "and height of length H but then this"}, {"start": 27.679, "duration": 5.761, "text": "triangle has an area of/ 12 h * B1 + B2"}, {"start": 31.72, "duration": 3.679, "text": "as this triangle has the same area as"}, {"start": 33.44, "duration": 4.439, "text": "the original trapezoid we see that the"}, {"start": 35.399, "duration": 4.32, "text": "area formula for a trapezoid is 1/2 *"}, {"start": 37.879, "duration": 4.641, "text": "the height times the sum of the two"}, {"start": 39.719, "duration": 2.801, "text": "bases"}], "title": "Trapezoid Area Visual Proof", "description": "This is a short animation showing a standard fact about how to find the area of a trapezoid by finding a triangle with the same area. \n\nIf you like this video, consider subscribing to the channel or consider buying me a coffee: https://www.buymeacoffee.com/VisualProofs. Thanks!\n\nHere is the original video in wide-format, wordless, with dramatic music:\nhttps://youtu.be/4ZypJJmd5ZY\n\nHere is a related visual proof for the area of a parallelogram:\nhttps://youtu.be/QjWbBexfJvw\n\n#area #triangle #trapezoid #geometry #visualproof #math #manim\n\nThis video was inspired by a beautiful GeoGebra applet created by Tim Brzezinski:\n\nhttps://www.geogebra.org/m/ZeDN7ZTW\n\nTo learn more about animating with manim, check out:\nhttps://manim.community", "lengthSeconds": 41, "width": 360, "height": 640}
{"video_id": "A0ZvO_F-LZo", "transcript": [{"start": 11.44, "duration": 4.159, "text": "start with a 4x2 rectangle"}, {"start": 15.759, "duration": 4.081, "text": "and subdivide both sides"}, {"start": 20.72, "duration": 4.96, "text": "half of the width of the rectangle has"}, {"start": 23.439, "duration": 5.041, "text": "length two"}, {"start": 25.68, "duration": 4.48, "text": "we can swing a matching length around"}, {"start": 28.48, "duration": 3.119, "text": "until it intersects the line cutting"}, {"start": 30.16, "duration": 3.28, "text": "through the middle of the rectangle like"}, {"start": 31.599, "duration": 3.681, "text": "this"}, {"start": 33.44, "duration": 3.92, "text": "and we use that line"}, {"start": 35.28, "duration": 3.76, "text": "to create an equilateral triangle"}, {"start": 37.36, "duration": 4.32, "text": "sitting inside the left half of the"}, {"start": 39.04, "duration": 4.48, "text": "rectangle"}, {"start": 41.68, "duration": 3.76, "text": "because the triangle is equilateral and"}, {"start": 43.52, "duration": 4.48, "text": "has side length 2"}, {"start": 45.44, "duration": 4.77, "text": "the horizontal altitude of the triangle"}, {"start": 48.0, "duration": 5.92, "text": "must be the square root of 3."}, {"start": 50.21, "duration": 6.989, "text": "[Music]"}, {"start": 53.92, "duration": 6.08, "text": "connect the two edges labeled 2"}, {"start": 57.199, "duration": 4.641, "text": "and we create an isosceles triangle as"}, {"start": 60.0, "duration": 4.4, "text": "the two side lengths are equal"}, {"start": 61.84, "duration": 4.72, "text": "the interior obtuse angle is formed from"}, {"start": 64.4, "duration": 4.719, "text": "a 90 degree angle and a 60 degree angle"}, {"start": 66.56, "duration": 4.239, "text": "so it measures 150 degrees"}, {"start": 69.119, "duration": 3.841, "text": "consequently the other two angles in the"}, {"start": 70.799, "duration": 4.161, "text": "isoscele triangle must be equal and so"}, {"start": 72.96, "duration": 5.839, "text": "are 15 degrees and therefore the"}, {"start": 74.96, "duration": 5.68, "text": "complementary angle here is 75 degrees"}, {"start": 78.799, "duration": 4.721, "text": "using a similar argument we can create"}, {"start": 80.64, "duration": 5.119, "text": "another isosceles triangle seen here"}, {"start": 83.52, "duration": 4.16, "text": "one angle is supplementary to 150"}, {"start": 85.759, "duration": 3.441, "text": "degrees so it's 30 degrees this means"}, {"start": 87.68, "duration": 5.119, "text": "the other two angles which are equal"}, {"start": 89.2, "duration": 3.599, "text": "must both be 75 degrees"}, {"start": 93.36, "duration": 4.48, "text": "now we focus only on the lower half of"}, {"start": 95.2, "duration": 4.64, "text": "the rectangle with side lengths of one"}, {"start": 97.84, "duration": 4.4, "text": "using this highlighted right triangle we"}, {"start": 99.84, "duration": 4.639, "text": "determined that the tangent of 75"}, {"start": 102.24, "duration": 4.0, "text": "degrees is the opposite side length over"}, {"start": 104.479, "duration": 6.401, "text": "the adjacent side length of one so the"}, {"start": 106.24, "duration": 4.64, "text": "tangent of 75 is two plus root three"}, {"start": 111.2, "duration": 4.4, "text": "we can use this other highlighted"}, {"start": 113.2, "duration": 4.8, "text": "triangle with angle 15 degrees because"}, {"start": 115.6, "duration": 6.08, "text": "it's complementary to 75 to see that the"}, {"start": 118.0, "duration": 5.439, "text": "tangent of 15 is 2 minus root 3 all"}, {"start": 121.68, "duration": 3.6, "text": "divided by 1."}, {"start": 123.439, "duration": 3.761, "text": "we have used a rectangle and not the"}, {"start": 125.28, "duration": 4.56, "text": "half angle identity for tangent to"}, {"start": 127.2, "duration": 5.36, "text": "determine that the tangent of 75 is 2"}, {"start": 129.84, "duration": 4.84, "text": "plus root 3 and the tangent of 15 is 2"}, {"start": 132.56, "duration": 14.84, "text": "minus root 3."}, {"start": 134.68, "duration": 12.72, "text": "[Music]"}, {"start": 149.12, "duration": 2.08, "text": "you"}], "title": "Tangent 75 and tangent of 15 (visual proof; trigonometry)", "description": "This is a short, animated visual proof of the showing how to find the tangent of 75 and the tangent of 15 without using the half-angle identities for the tangent function. The idea is to embed some interesting triangles into a 2x4 rectangle. #mathshorts #mathvideo #math #trigonometry #triangle #manim #animation #theorem #pww #proofwithoutwords #visualproof #proof #circle #tangent #righttriangles #rectangle #identity #halfangle #trigidentity #trig\n\nPlease consider subscribing to the channel if you enjoy the content.\n\nAnother way to find these measurements is to use a half-angle tangent identity, such as the ones found in this video: https://youtu.be/XIC7SIdl7pI\n\nThis animation is based on a proof due to Garc\u00eda Capit\u00e1n Francisco Javier from the January 2017 issue of The College Mathematics Journal (https://doi.org/10.4169/college.math.j.48.1.35) page 35.\n\nTo learn more about animating with manim, check out:\nhttps://manim.community", "lengthSeconds": 149, "width": 640, "height": 360}
{"video_id": "XFHm0Ot3PP0", "transcript": [{"start": 5.899, "duration": 5.921, "text": "suppose you have a right triangle with"}, {"start": 8.519, "duration": 4.441, "text": "legs of length A and B and hypotenuse of"}, {"start": 11.82, "duration": 3.3, "text": "length C"}, {"start": 12.96, "duration": 5.04, "text": "you can draw squares off the legs"}, {"start": 15.12, "duration": 4.499, "text": "enclosing an area of a squared and of B"}, {"start": 18.0, "duration": 3.18, "text": "squared respectively"}, {"start": 19.619, "duration": 3.721, "text": "you can also draw a square off the"}, {"start": 21.18, "duration": 4.14, "text": "hypotenuse enclosing an area of c"}, {"start": 23.34, "duration": 4.259, "text": "squared let's see an interesting"}, {"start": 25.32, "duration": 5.1, "text": "dissection that'll give a formula for c"}, {"start": 27.599, "duration": 5.341, "text": "squared we can take these two lines off"}, {"start": 30.42, "duration": 4.139, "text": "the C by C square and shift them so that"}, {"start": 32.94, "duration": 3.959, "text": "they pass through the middle of the B by"}, {"start": 34.559, "duration": 4.321, "text": "b square notice we can shift the"}, {"start": 36.899, "duration": 3.66, "text": "original triangle so that it lies along"}, {"start": 38.88, "duration": 3.66, "text": "one of those lines and then we can"}, {"start": 40.559, "duration": 4.02, "text": "rotate the triangle about that line to"}, {"start": 42.54, "duration": 3.48, "text": "form a rectangle that covers part of the"}, {"start": 44.579, "duration": 4.14, "text": "B by b square"}, {"start": 46.02, "duration": 5.16, "text": "the two uncovered rectangles in the B by"}, {"start": 48.719, "duration": 4.801, "text": "b square have the same width by Symmetry"}, {"start": 51.18, "duration": 4.199, "text": "and this width can be computed then as"}, {"start": 53.52, "duration": 3.48, "text": "half the difference between the square"}, {"start": 55.379, "duration": 4.02, "text": "length and the leg length of the"}, {"start": 57.0, "duration": 4.26, "text": "triangle which is B minus a over two so"}, {"start": 59.399, "duration": 4.32, "text": "that the remaining side of the B by b"}, {"start": 61.26, "duration": 5.039, "text": "square has a length of B plus a over two"}, {"start": 63.719, "duration": 4.801, "text": "Again by symmetry this means we've cut"}, {"start": 66.299, "duration": 4.14, "text": "the B by b square into four congruent"}, {"start": 68.52, "duration": 4.86, "text": "trapezoids all with the same dimensions"}, {"start": 70.439, "duration": 5.22, "text": "we can shift these trapezoids as follows"}, {"start": 73.38, "duration": 3.96, "text": "over to the C by C Square"}, {"start": 75.659, "duration": 3.901, "text": "but notice there's a remaining Square"}, {"start": 77.34, "duration": 3.9, "text": "left on the inside the side length of"}, {"start": 79.56, "duration": 3.9, "text": "this Square can be computed as the"}, {"start": 81.24, "duration": 5.04, "text": "difference of B plus a over 2 and B"}, {"start": 83.46, "duration": 5.1, "text": "minus a over two but that's just a so we"}, {"start": 86.28, "duration": 4.199, "text": "can then shift the a by a square up and"}, {"start": 88.56, "duration": 4.86, "text": "now we've covered the entire C by C"}, {"start": 90.479, "duration": 5.581, "text": "Square therefore c squared must be equal"}, {"start": 93.42, "duration": 4.26, "text": "to B squared plus a squared"}, {"start": 96.06, "duration": 3.419, "text": "there is nothing special about the"}, {"start": 97.68, "duration": 3.96, "text": "particular right triangle that we chose"}, {"start": 99.479, "duration": 4.5, "text": "if a and b are the leg lengths of any"}, {"start": 101.64, "duration": 4.019, "text": "right triangle with hypotenuse C then a"}, {"start": 103.979, "duration": 5.161, "text": "squared plus b squared will always equal"}, {"start": 105.659, "duration": 5.341, "text": "c squared here we show a variety of"}, {"start": 109.14, "duration": 3.659, "text": "different right triangles morphing"}, {"start": 111.0, "duration": 4.56, "text": "through and each time we show the"}, {"start": 112.799, "duration": 5.041, "text": "dissection given by this proof it turns"}, {"start": 115.56, "duration": 4.14, "text": "out that the dissection of the B by B"}, {"start": 117.84, "duration": 4.68, "text": "squared and the a by a square will"}, {"start": 119.7, "duration": 4.98, "text": "always cover the C by C Square this is"}, {"start": 122.52, "duration": 5.18, "text": "the Pythagorean theorem As proved by"}, {"start": 124.68, "duration": 3.02, "text": "Henry dudenay"}], "title": "Pythagorean Theorem XI (Dudeney's dissection visual proof)", "description": "This is an animated visual proof of the Pythagorean theorem (the right triangle theorem) using a Henry Dudeney's dissection argument (congruent-by-addition proof).\n\nIf you like this video, consider subscribing to the channel or consider buying me a coffee: https://www.buymeacoffee.com/VisualProofs. Thanks! \n\nThis animation is based on a proof from Henry Dudeney and is discussed in Howard Eve's book \"Great Moments in Mathematics (before 1650).\" You can also find this in Roger Nelsen's first PWW compendium on page 6.\n\nHere is a shorts version of this proof:\nhttps://youtube.com/shorts/azzrmVw8O-8\n\nFor other proofs of this same fact check out: \nhttps://youtu.be/q28H5NY5UVU\nhttps://youtu.be/yfGtbNgcrQ8\nhttps://youtu.be/JRjPsuJ7S-g\nhttps://youtu.be/TsKTteGJpZU\nhttps://youtu.be/ZlGaQdNRdqA\nhttps://youtu.be/uqGjs_n-f7U\nhttps://youtu.be/wWEZ-rUlBDQ\nhttps://youtu.be/I75HYC8TRNY\nhttps://youtu.be/LWVog8hg74I?si=4GSzTLZP7XIWeaxJ\n\n#math #manim #pythagoreantheorem #pythagorean #triangle #animation #theorem #pww #proofwithoutwords #visualproof #proof  #mathshorts #mathvideo \n\nTo learn more about animating with manim, check out:\nhttps://manim.community", "lengthSeconds": 130, "width": 640, "height": 360}
{"video_id": "KFtlSDQmzYk", "transcript": [{"start": 0.06, "duration": 4.14, "text": "select 5000 positive integers uniformly"}, {"start": 2.639, "duration": 3.841, "text": "at random from the range one to five"}, {"start": 4.2, "duration": 4.5, "text": "million to plot points in the plane"}, {"start": 6.48, "duration": 4.079, "text": "if the pair is relatively prime plot the"}, {"start": 8.7, "duration": 3.42, "text": "point in red and otherwise plot the"}, {"start": 10.559, "duration": 3.901, "text": "point in blue"}, {"start": 12.12, "duration": 4.079, "text": "here's an animation of one example of"}, {"start": 14.46, "duration": 3.72, "text": "this random process"}, {"start": 16.199, "duration": 3.781, "text": "it turns out that the ratio of the"}, {"start": 18.18, "duration": 4.14, "text": "number of red dots to the total number"}, {"start": 19.98, "duration": 4.74, "text": "of dots gives a good approximation for"}, {"start": 22.32, "duration": 4.199, "text": "the fraction 6 over pi squared"}, {"start": 24.72, "duration": 3.66, "text": "therefore we can use this particular"}, {"start": 26.519, "duration": 7.58, "text": "simulation to get an approximation for"}, {"start": 28.38, "duration": 5.719, "text": "pi as 3.142 not great but not terrible"}], "title": "A random approximation for Pi (pi day short)", "description": "In this short, we randomly sample 5000 pairs of positive integers and use the pairs to plot points in the plane. If a pair is relatively prime, we shade the dot red; otherwise we shade the dot blue. The theory behind this process says that roughly 6/pi*pi of the dots will be red. So we can use this simulation to get an approximation for 6/pi*pi and in turn use that to get an approximation for Pi. We get 3.142, which is not that bad, but not really good either. \n\nFor more information about why this works, check out the wikipedia site:\n\nhttps://en.wikipedia.org/wiki/Coprime_integers\n\nFor more Pi-related videos, check out my playlist:\nhttps://youtube.com/playlist?list=PLZh9gzIvXQUsGRDvzvXc02lDIKHaVeFwq\n\n#manim #approximation  #relativelyprime #zetafunction #baselproblem #irrational #Pi #mathvideo\u200b #math\u200b #mtbos\u200b \u200b #animation\u200b\u200b\u200b #iteachmath #mathematics #piday \n\n\nTo learn more about animating with manim, check out:\nhttps://manim.community", "lengthSeconds": 32, "width": 360, "height": 640}
{"video_id": "if800mpoJaY", "transcript": [{"start": 0.0, "duration": 6.299, "text": "let A and B be two positive real numbers"}, {"start": 3.0, "duration": 6.12, "text": "and consider an a by B rectangle that"}, {"start": 6.299, "duration": 5.76, "text": "has area a times B"}, {"start": 9.12, "duration": 5.34, "text": "now create four copies of this rectangle"}, {"start": 12.059, "duration": 3.481, "text": "in closing an area of four times a times"}, {"start": 14.46, "duration": 3.3, "text": "B"}, {"start": 15.54, "duration": 5.819, "text": "these four rectangles can be organized"}, {"start": 17.76, "duration": 5.76, "text": "to fit inside of an A plus b by a plus b"}, {"start": 21.359, "duration": 4.981, "text": "square however there's a missing"}, {"start": 23.52, "duration": 4.44, "text": "interior square with a side length of B"}, {"start": 26.34, "duration": 4.5, "text": "minus a"}, {"start": 27.96, "duration": 6.119, "text": "from this diagram we see that the"}, {"start": 30.84, "duration": 6.6, "text": "quantity a plus b squared minus the"}, {"start": 34.079, "duration": 6.541, "text": "quantity B minus a squared must be equal"}, {"start": 37.44, "duration": 5.58, "text": "to the quantity four times a times B"}, {"start": 40.62, "duration": 4.619, "text": "if we remove the inner Square we see"}, {"start": 43.02, "duration": 4.02, "text": "that the quantity a plus b squared is"}, {"start": 45.239, "duration": 4.861, "text": "greater than or equal to 4 times a times"}, {"start": 47.04, "duration": 5.039, "text": "B this is true for all positive Pairs"}, {"start": 50.1, "duration": 4.32, "text": "and we see that we get equality between"}, {"start": 52.079, "duration": 4.5, "text": "a plus b squared and 4 a b if and only"}, {"start": 54.42, "duration": 3.959, "text": "if a equals B taking square roots and"}, {"start": 56.579, "duration": 4.941, "text": "dividing by 2 yields the arithmetic mean"}, {"start": 58.379, "duration": 3.141, "text": "geometric mean inequality"}], "title": "AM-GM inequality visually", "description": "This is a short, animated visual proof of the arithmetic mean-geometric mean inequality using areas. This theorem states that the average of two positive numbers is greater than or equal to the square root of the product of the same two numbers (with equality only when the two numbers are the same). #math #amgminequality #area #manim #animation #theorem #pww #proofwithoutwords #visualproof #proof #mathshorts #mathvideo \n\nIf you like this video, please like, subscribe and check out my others. \n\nThis animation is based on a visual proof by Doris Schattschneider from the February 1986 issue of Mathematics Magazine (https://doi.org/10.2307/2690011).\n\nHere are some other visual proof of the AMGM inequality (and others):\nhttps://youtu.be/62G9fak1vyk\nhttps://youtu.be/XJ3643WhKjY\nhttps://youtu.be/Pt8IX_Q5U1A\nhttps://youtu.be/02yM5LqMWUs\nhttps://youtu.be/jNx05gXSblE\nhttps://youtu.be/IJGwvvQdfgg\nhttps://youtu.be/SEPkdPWKdO0\n\nAnd here is a playlist with AMGM type videos: https://youtube.com/playlist?list=PLZh9gzIvXQUuoRR4kTd628cWdFQ1BDyS6\n\nTo learn more about animating with manim, check out:\nhttps://manim.community", "lengthSeconds": 60, "width": 360, "height": 640}
{"video_id": "i12p4Acb3L4", "transcript": [{"start": 6.17, "duration": 3.07, "text": "[Music]"}, {"start": 14.79, "duration": 8.43, "text": "[Music]"}, {"start": 25.45, "duration": 3.179, "text": "[Applause]"}, {"start": 31.39, "duration": 3.089, "text": "[Music]"}, {"start": 37.01, "duration": 13.99, "text": "[Applause]"}, {"start": 37.22, "duration": 13.78, "text": "[Music]"}, {"start": 53.53, "duration": 8.36, "text": "[Music]"}, {"start": 60.359, "duration": 21.251, "text": "a"}, {"start": 61.89, "duration": 19.72, "text": "[Music]"}, {"start": 87.45, "duration": 3.629, "text": "[Music]"}, {"start": 98.42, "duration": 9.51, "text": "[Music]"}, {"start": 110.14, "duration": 39.9, "text": "[Applause]"}, {"start": 115.08, "duration": 37.96, "text": "[Music]"}, {"start": 150.04, "duration": 3.0, "text": "a"}, {"start": 153.74, "duration": 12.429, "text": "[Music]"}], "title": "Sum of Cubes VII (visual proof without words)", "description": "This is a short, animated (wordless) visual proof demonstrating the sum of the first n positive cubes by rearranging two stacks of cubic arrays.\n\nIf you like this video, consider subscribing to the channel or consider buying me a coffee: https://www.buymeacoffee.com/VisualProofs. Thanks!\n\nHere are six other visual proofs of sum of cubes formulas:\n\nhttps://youtu.be/LYfKapE_g1Q\nhttps://youtu.be/jWpyrXYZNiI\nhttps://youtu.be/YQLicI8R4Gs\nhttps://youtu.be/Ye9OPNqV9FA\nhttps://youtu.be/NxOcT_VKQR0\nhttps://youtu.be/d1yM6Rq7Tfw\n\nThis animation is based on a visual proof by Tom Edgar from the September 2024 issue of Mathematics Magazine (https://doi.org/10.1080/0025570X.2024.2379213 page ??).\n\nTo learn more about animating with manim, check out:\nhttps://manim.community\n\n#mathshorts\u200b #mathvideo\u200b #math\u200b #calculus #mtbos\u200b #manim\u200b #animation\u200b #theorem\u200b #pww\u200b #proofwithoutwords\u200b #visualproof\u200b #proof\u200b #iteachmath #finitesums #discretemath #calculus #sum #induction #faulhaber #sumofcubes #cubes\n___________________________________________\nMusic in this video:\nEpic Cyberpunk | GLORY by Alex-Productions | https://onsound.eu/\nMusic promoted by https://www.free-stock-music.com\nCreative Commons / Attribution 3.0 Unported License (CC BY 3.0)\nhttps://creativecommons.org/licenses/by/3.0/deed.en_US", "lengthSeconds": 165, "width": 640, "height": 360}
{"video_id": "ngJFwvzzMuU", "transcript": [{"start": 2.47, "duration": 13.53, "text": "[Music]"}, {"start": 14.08, "duration": 2.959, "text": "start with a real number a between minus"}, {"start": 16.0, "duration": 2.56, "text": "one and one"}, {"start": 17.039, "duration": 3.441, "text": "to compute the integral of the square"}, {"start": 18.56, "duration": 3.44, "text": "root of one minus x squared between a"}, {"start": 20.48, "duration": 3.68, "text": "and one using the fundamental theorem of"}, {"start": 22.0, "duration": 5.439, "text": "calculus you need a trig substitution"}, {"start": 24.16, "duration": 4.72, "text": "and some trig identities however we can"}, {"start": 27.439, "duration": 2.801, "text": "also remember that this integral"}, {"start": 28.88, "duration": 3.04, "text": "represents the area under the square"}, {"start": 30.24, "duration": 2.4, "text": "root of one minus x squared between x"}, {"start": 31.92, "duration": 4.4, "text": "equals a"}, {"start": 32.64, "duration": 6.32, "text": "and x equals one like this to find this"}, {"start": 36.32, "duration": 4.559, "text": "area we can first find the area of the"}, {"start": 38.96, "duration": 4.4, "text": "full sector covering the region"}, {"start": 40.879, "duration": 4.241, "text": "if this sector has angle alpha then the"}, {"start": 43.36, "duration": 4.08, "text": "area is one half alpha because the"}, {"start": 45.12, "duration": 5.119, "text": "radius is one"}, {"start": 47.44, "duration": 5.04, "text": "the area we want is this area minus the"}, {"start": 50.239, "duration": 4.0, "text": "area of this right triangle"}, {"start": 52.48, "duration": 4.48, "text": "with base a and height square root of"}, {"start": 54.239, "duration": 4.96, "text": "one minus a squared"}, {"start": 56.96, "duration": 3.599, "text": "note that this right triangle also shows"}, {"start": 59.199, "duration": 4.401, "text": "that the cosine of alpha"}, {"start": 60.559, "duration": 5.681, "text": "is a so that alpha is the arc cosine"}, {"start": 63.6, "duration": 5.6, "text": "of a giving the formula for the area we"}, {"start": 66.24, "duration": 4.879, "text": "want in terms of a"}, {"start": 69.2, "duration": 3.04, "text": "but this diagram only works if a is"}, {"start": 71.119, "duration": 4.401, "text": "positive"}, {"start": 72.24, "duration": 6.72, "text": "if a is negative the region that we want"}, {"start": 75.52, "duration": 3.44, "text": "looks a little something like this"}, {"start": 81.92, "duration": 3.839, "text": "luckily this region is now the union of"}, {"start": 83.84, "duration": 4.639, "text": "a sector in a right triangle"}, {"start": 85.759, "duration": 3.68, "text": "again if this sector has angle alpha"}, {"start": 88.479, "duration": 4.161, "text": "then the area"}, {"start": 89.439, "duration": 3.201, "text": "is one-half alpha"}, {"start": 96.32, "duration": 3.2, "text": "this time we add the area of the"}, {"start": 98.72, "duration": 3.679, "text": "triangle"}, {"start": 99.52, "duration": 4.0, "text": "which now has base negative a and height"}, {"start": 102.399, "duration": 4.321, "text": "square root of 1 minus a"}, {"start": 103.52, "duration": 6.48, "text": "squared as before alpha"}, {"start": 106.72, "duration": 3.28, "text": "is the arc cosine of a"}, {"start": 110.32, "duration": 4.159, "text": "and we can think of adding the negative"}, {"start": 112.0, "duration": 6.399, "text": "as subtracting"}, {"start": 114.479, "duration": 3.92, "text": "this yields the same formula as before"}, {"start": 120.56, "duration": 4.08, "text": "so in either of these two cases the"}, {"start": 123.439, "duration": 4.32, "text": "integral that we want"}, {"start": 124.64, "duration": 5.44, "text": "is given by the arc cosine of a over two"}, {"start": 127.759, "duration": 15.2, "text": "minus one half a times the square root"}, {"start": 130.08, "duration": 14.96, "text": "of one minus a squared"}, {"start": 142.959, "duration": 2.081, "text": "you"}], "title": "An Integral without Trigonometric Substitution (visual proof)", "description": "This is a short, animated visual proof demonstrating how to find a certain integral that is challenging to do by the fundamental theorem of calculus by instead using an area dissection  . #mathshorts\u200b #mathvideo\u200b #math\u200b #calculus #mtbos\u200b #manim\u200b #animation\u200b #theorem\u200b #pww\u200b #proofwithoutwords\u200b #visualproof\u200b #proof\u200b #iteachmath #integral #substitution #trigsub\n\nThis animation is based on a visual proof by Marc Chamberland from the February 2001 issue of Mathematics Magazine (https://doi.org/10.2307/2691155).\n\nTo learn more about animating with manim, check out:\nhttps://manim.community\n__________________________________________________________________\nBackground music, used according to CC BY-SA 3.0, is \"The Road\" from Esteban Orlando:\n\nEsteban Orlando:\nhttps://soundcloud.com/orlando-esteba...\nhttps://www.instagram.com/estebanorla...\n\nhttps://creativecommons.org/licenses/...\nMusic provided by RFM: https://youtu.be/b4KCM1bIq50", "lengthSeconds": 143, "width": 640, "height": 360}
{"video_id": "Pt8IX_Q5U1A", "transcript": [{"start": 2.47, "duration": 15.01, "text": "[Music]"}, {"start": 15.0, "duration": 4.56, "text": "imagine two positive real numbers X and"}, {"start": 17.48, "duration": 4.08, "text": "Y represented by lengths of connected"}, {"start": 19.56, "duration": 4.0, "text": "line segments and then draw the"}, {"start": 21.56, "duration": 4.36, "text": "semicircle with diameter given by the"}, {"start": 23.56, "duration": 6.0, "text": "connected segments so that the radius of"}, {"start": 25.92, "duration": 5.08, "text": "the semicircle is x + y/ 2 then let Z be"}, {"start": 29.56, "duration": 3.4, "text": "the length of the half cord from the"}, {"start": 31.0, "duration": 3.92, "text": "point where X and Y meet to the"}, {"start": 32.96, "duration": 4.119, "text": "semicircle connecting the ends of the"}, {"start": 34.92, "duration": 4.76, "text": "diameter to the intersection of Z on the"}, {"start": 37.079, "duration": 4.401, "text": "circle yields a triangle and by the's"}, {"start": 39.68, "duration": 2.559, "text": "triangle theorem this triangle must be"}, {"start": 41.48, "duration": 3.72, "text": "right"}, {"start": 42.239, "duration": 6.32, "text": "angled consequently these two subt"}, {"start": 45.2, "duration": 7.44, "text": "triangles are similar so that the ratios"}, {"start": 48.559, "duration": 8.121, "text": "z overx and y over Z are equal we"}, {"start": 52.64, "duration": 8.919, "text": "conclude z^2 equals x * y so that Z is"}, {"start": 56.68, "duration": 4.879, "text": "the square root of x * y"}, {"start": 63.68, "duration": 4.56, "text": "no matter the lengths of X and Y the"}, {"start": 66.2, "duration": 4.8, "text": "chord length Z is always less than or"}, {"start": 68.24, "duration": 6.8, "text": "equal to the radius of the semicircle by"}, {"start": 71.0, "duration": 7.439, "text": "construction therefore the < TK of x * Y"}, {"start": 75.04, "duration": 6.68, "text": "is less than or equal to x + y / 2 with"}, {"start": 78.439, "duration": 5.201, "text": "equality if and only if xal y this is"}, {"start": 81.72, "duration": 4.92, "text": "the arithmetic mean geometric mean"}, {"start": 83.64, "duration": 3.0, "text": "inequality"}, {"start": 87.53, "duration": 5.569, "text": "[Music]"}], "title": "AM-GM Inequality II (visual proof)", "description": "This is a short, animated visual proof demonstrating the arithmetic mean geometric mean inequality using Thales theorem . #mathshorts\u200b #mathvideo\u200b #math\u200b #amgminequality #mtbos\u200b #manim\u200b #animation\u200b #theorem\u200b #pww\u200b #proofwithoutwords\u200b #visualproof\u200b #proof\u200b #iteachmath #calculus #inequality \n\nThis animation is based on a visual proof by Charles D. Gallant from the March 1977 issue of Mathematics Magazine (https://doi.org/10.2307/2689737).\n\nTo learn more about animating with manim, check out:\nhttps://manim.community\n__________________________________________________________________\nBackground music, used according to CC BY-SA 3.0, is \"The Road\" from Esteban Orlando:\n\nEsteban Orlando:\nhttps://soundcloud.com/orlando-esteba...\nhttps://www.instagram.com/estebanorla...\n\nhttps://creativecommons.org/licenses/...\nMusic provided by RFM: https://youtu.be/b4KCM1bIq50", "lengthSeconds": 99, "width": 640, "height": 360}
{"video_id": "-tJhH_k2LaM", "transcript": [{"start": 2.47, "duration": 13.53, "text": "[Music]"}, {"start": 14.48, "duration": 2.4, "text": "let's add up the first four perfect"}, {"start": 16.0, "duration": 5.68, "text": "squares"}, {"start": 16.88, "duration": 6.8, "text": "one plus four plus nine plus sixteen"}, {"start": 21.68, "duration": 3.519, "text": "by arranging the l-shaped nomens from"}, {"start": 23.68, "duration": 4.16, "text": "each of this square"}, {"start": 25.199, "duration": 3.521, "text": "we get this tower which consists of a"}, {"start": 27.84, "duration": 4.8, "text": "one by seven"}, {"start": 28.72, "duration": 6.48, "text": "a two by 5 a 3 by 3 and a 4 by 1 stack"}, {"start": 32.64, "duration": 2.56, "text": "of squares"}, {"start": 35.36, "duration": 3.199, "text": "we can then use the sum of squares twice"}, {"start": 37.92, "duration": 7.6, "text": "more"}, {"start": 38.559, "duration": 9.441, "text": "to build a rectangle"}, {"start": 45.52, "duration": 3.44, "text": "notice that this rectangle has width two"}, {"start": 48.0, "duration": 3.52, "text": "times four"}, {"start": 48.96, "duration": 2.56, "text": "plus one"}, {"start": 52.239, "duration": 3.441, "text": "and the length of this rectangle is one"}, {"start": 54.48, "duration": 4.96, "text": "plus two plus three"}, {"start": 55.68, "duration": 5.6, "text": "plus four therefore we see that 3 times"}, {"start": 59.44, "duration": 4.56, "text": "the sum of the first four squares"}, {"start": 61.28, "duration": 5.6, "text": "is 2 times 4 plus 1 multiplied by the"}, {"start": 64.0, "duration": 4.32, "text": "sum of the first four positive integers"}, {"start": 66.88, "duration": 3.599, "text": "there's nothing special about four"}, {"start": 68.32, "duration": 3.52, "text": "squares the analogous geometric"}, {"start": 70.479, "duration": 1.841, "text": "construction will work for the sum of"}, {"start": 71.84, "duration": 2.319, "text": "the first"}, {"start": 72.32, "duration": 3.2, "text": "n squares showing that three times the"}, {"start": 74.159, "duration": 3.521, "text": "sum of the first n squares"}, {"start": 75.52, "duration": 5.599, "text": "is two n plus one multiplied by the sum"}, {"start": 77.68, "duration": 5.68, "text": "of the first n positive integers"}, {"start": 81.119, "duration": 3.921, "text": "applying a known sum formula for the sum"}, {"start": 83.36, "duration": 4.96, "text": "of integers"}, {"start": 85.04, "duration": 4.48, "text": "and then dividing by 3 yields a formula"}, {"start": 88.32, "duration": 11.36, "text": "for the sum of the first"}, {"start": 89.52, "duration": 10.16, "text": "n squares"}], "title": "Sum of Squares I (visual proof)", "description": "This is a short, animated visual proof of the formula that computes that sum of the first n squares using 3 copies of the sum of squares to build a rectangle . #mathshorts\u200b #mathvideo\u200b #math\u200b #sumofsquares\u200b #mtbos\u200b #manim\u200b #animation\u200b #theorem\u200b #pww\u200b #proofwithoutwords\u200b #visualproof\u200b #proof\u200b #iteachmath\n\nThis animation is based on (independently discovered) visual proofs by Dan Kalman from the March 1991 issue of The College Mathematics Journal (https://www.jstor.org/stable/2686447) and Martin Gardner from the October 1973 Scientific American (https://www.jstor.org/stable/24923225). \n\nTo learn more about animating with manim, check out:\nhttps://manim.community\n__________________________________________________________________\nBackground music, used according to CC BY-SA 3.0, is \"The Road\" from Esteban Orlando:\n\nEsteban Orlando:\nhttps://soundcloud.com/orlando-esteba...\u200b\nhttps://www.instagram.com/estebanorla...\u200b\n\nhttps://creativecommons.org/licenses/...\u200b\nMusic provided by RFM: https://youtu.be/b4KCM1bIq50", "lengthSeconds": 98, "width": 640, "height": 360}
{"video_id": "UFZZW6r-IIU", "transcript": [{"start": 0.55, "duration": 3.809, "text": "[\u97f3\u697d]"}, {"start": 12.79, "duration": 2.1, "text": "[\u97f3\u697d]"}], "title": "(a+b) ^3 visual proof. #maths  #formula", "description": "", "lengthSeconds": 13, "width": 360, "height": 640}
{"video_id": "hvIusavrRf8", "transcript": [{"start": 2.35, "duration": 7.49, "text": "[Music]"}, {"start": 6.799, "duration": 5.38, "text": "let's let H sub K represent the harmonic"}, {"start": 9.84, "duration": 4.56, "text": "sum 1 plus a half plus a third up to one"}, {"start": 12.179, "duration": 5.041, "text": "over k"}, {"start": 14.4, "duration": 5.7, "text": "now consider a triangular array with n"}, {"start": 17.22, "duration": 5.28, "text": "minus one rows in the first row place"}, {"start": 20.1, "duration": 5.64, "text": "the harmonics on H1 in the second row"}, {"start": 22.5, "duration": 6.96, "text": "place the harmonic sum H2 and so on so"}, {"start": 25.74, "duration": 7.1, "text": "that the entry in row I and column J is"}, {"start": 29.46, "duration": 3.38, "text": "the fraction one over J"}, {"start": 32.88, "duration": 6.06, "text": "now consider a triangular array of boxes"}, {"start": 35.82, "duration": 5.759, "text": "with n rows take the number n and split"}, {"start": 38.94, "duration": 5.34, "text": "it into n copies of one then equally"}, {"start": 41.579, "duration": 6.0, "text": "distribute 1 into each column using"}, {"start": 44.28, "duration": 5.34, "text": "fractions so that column J contains all"}, {"start": 47.579, "duration": 3.48, "text": "entries of fractions one over J as"}, {"start": 49.62, "duration": 4.38, "text": "pictured"}, {"start": 51.059, "duration": 6.121, "text": "now for the trick we can combine these"}, {"start": 54.0, "duration": 4.379, "text": "two triangular arrays to make an N by n"}, {"start": 57.18, "duration": 4.1, "text": "Square"}, {"start": 58.379, "duration": 6.901, "text": "however if we now look at this Square"}, {"start": 61.28, "duration": 7.06, "text": "row by row we see that each row sums up"}, {"start": 65.28, "duration": 6.6, "text": "to the same value Row 1 is hn Row 2 is"}, {"start": 68.34, "duration": 5.099, "text": "hn and so on each row sums to the value"}, {"start": 71.88, "duration": 4.32, "text": "H sub n"}, {"start": 73.439, "duration": 4.921, "text": "but in this case there are n total rows"}, {"start": 76.2, "duration": 6.239, "text": "and therefore the sum of all the entries"}, {"start": 78.36, "duration": 5.939, "text": "is given by n times H sub n"}, {"start": 82.439, "duration": 4.381, "text": "but now we have an interesting identity"}, {"start": 84.299, "duration": 4.68, "text": "we see that if we sum up the harmonic"}, {"start": 86.82, "duration": 6.24, "text": "sums where K ranges from 1 to n minus 1"}, {"start": 88.979, "duration": 6.0, "text": "we have n times H sub n minus n"}, {"start": 93.06, "duration": 4.98, "text": "this is a formula for the sum of"}, {"start": 94.979, "duration": 5.46, "text": "harmonic sums but wait there's more that"}, {"start": 98.04, "duration": 4.439, "text": "we can do with this identity"}, {"start": 100.439, "duration": 4.021, "text": "we just proved visually that the sum of"}, {"start": 102.479, "duration": 4.381, "text": "harmonic sums where K ranges from 1 to n"}, {"start": 104.46, "duration": 3.659, "text": "minus 1 is equal to n times H sub n"}, {"start": 106.86, "duration": 3.36, "text": "minus n"}, {"start": 108.119, "duration": 5.341, "text": "we can get an equivalent identity by"}, {"start": 110.22, "duration": 5.82, "text": "adding H sub n to both sides"}, {"start": 113.46, "duration": 4.56, "text": "a theorem due to Ernesto Cesaro says"}, {"start": 116.04, "duration": 4.079, "text": "that if we have an infinite series that"}, {"start": 118.02, "duration": 4.08, "text": "converges with some s and we consider"}, {"start": 120.119, "duration": 4.32, "text": "the sequence of partial sums given by S"}, {"start": 122.1, "duration": 5.4, "text": "Sub n then the sequence of average"}, {"start": 124.439, "duration": 5.401, "text": "partial sums shown here also converges"}, {"start": 127.5, "duration": 4.92, "text": "to the same sum s"}, {"start": 129.84, "duration": 3.96, "text": "let's combine these two facts to prove"}, {"start": 132.42, "duration": 3.6, "text": "something interesting about the harmonic"}, {"start": 133.8, "duration": 4.26, "text": "series let's assume that the harmonic"}, {"start": 136.02, "duration": 3.66, "text": "series which is the sum where K ranges"}, {"start": 138.06, "duration": 3.96, "text": "from 1 to Infinity of the fractions 1"}, {"start": 139.68, "duration": 4.32, "text": "over K converges so that it has a sum s"}, {"start": 142.02, "duration": 3.78, "text": "this means that the limit as n"}, {"start": 144.0, "duration": 2.76, "text": "approaches Infinity of H sub n is equal"}, {"start": 145.8, "duration": 2.82, "text": "to s"}, {"start": 146.76, "duration": 3.54, "text": "then according to the theorem we have"}, {"start": 148.62, "duration": 2.94, "text": "that s is equal to the Limit as n"}, {"start": 150.3, "duration": 3.48, "text": "approaches Infinity of the average"}, {"start": 151.56, "duration": 4.8, "text": "partial sums but according to our"}, {"start": 153.78, "duration": 5.039, "text": "formula this limit can be replaced by"}, {"start": 156.36, "duration": 5.76, "text": "the limit as n approaches Infinity of H"}, {"start": 158.819, "duration": 5.64, "text": "sub n minus 1 plus h sub n over n and"}, {"start": 162.12, "duration": 5.28, "text": "this limit is equal to S minus 1 because"}, {"start": 164.459, "duration": 4.801, "text": "H sub n approaches s 1 approaches 1 and"}, {"start": 167.4, "duration": 4.5, "text": "N grows without bound"}, {"start": 169.26, "duration": 4.02, "text": "therefore we've proved that s equals s"}, {"start": 171.9, "duration": 3.72, "text": "minus 1."}, {"start": 173.28, "duration": 3.78, "text": "but this is a contradiction and that"}, {"start": 175.62, "duration": 3.72, "text": "means that our assumption was incorrect"}, {"start": 177.06, "duration": 5.06, "text": "so that in fact the harmonic series"}, {"start": 179.34, "duration": 2.78, "text": "diverges"}], "title": "Harmonic Sums and Divergence of Harmonic Series (visual proof)", "description": "This is a short, animated visual proof showing a formula for the sum of harmonic sums using triangular diagrams. As a bonus feature (watch until the end), we utilize this formula along with a theorem due to Ernesto Ces\u00e1ro to prove that the Harmonic Series diverges. #math #manim #harmonicseries #harmonicsum #divergent #diverges #triangle #animation #theorem #pww #proofwithoutwords #visualproof #proof\n\nIf you liked this video, please consider subscribing to the channel and checking out my other videos.\n\nFor a static version of the visual proof about Harmonic sums, see Roger Nelsen's \"Proof Without Words II: More Exercises in Visual Thinking\" compendium (page 116) : https://bookstore.ams.org/clrm-14/. To see the result about the divergence of the Harmonic series, check out this paper by Steven J. Kifowit https://stevekifowit.com/pubs/harm2.pdf (proof 39).\n\nTo learn more about animating with manim, check out:\nhttps://manim.community\n___________________________________\nMusic in this video:\nCold Blue by Astron", "lengthSeconds": 188, "width": 640, "height": 360}
{"video_id": "dOCk6SSpWLo", "transcript": [{"start": 0.0, "duration": 5.16, "text": "one way to picture the number n cubed is"}, {"start": 2.46, "duration": 5.16, "text": "to lay down n copies of an N by n Square"}, {"start": 5.16, "duration": 4.32, "text": "side by side flat if we use this"}, {"start": 7.62, "duration": 4.32, "text": "representation here we get a diagram"}, {"start": 9.48, "duration": 4.5, "text": "that represents the sum of the first n"}, {"start": 11.94, "duration": 4.38, "text": "cubes in this case we've shown n equals"}, {"start": 13.98, "duration": 4.44, "text": "four no matter what cubic stack we start"}, {"start": 16.32, "duration": 4.08, "text": "with we can now take four copies and"}, {"start": 18.42, "duration": 4.26, "text": "strategically shift and rotate them as"}, {"start": 20.4, "duration": 4.139, "text": "shown if we do that right we fill in a"}, {"start": 22.68, "duration": 4.919, "text": "perfect square with side length N"}, {"start": 24.539, "duration": 5.281, "text": "squared plus n therefore four copies of"}, {"start": 27.599, "duration": 4.381, "text": "the sum of the first n cubes must be"}, {"start": 29.82, "duration": 4.14, "text": "equal to the square of N squared plus n"}, {"start": 31.98, "duration": 3.9, "text": "this means that the sum of the first n"}, {"start": 33.96, "duration": 6.38, "text": "cubes must be equal to the quantity N"}, {"start": 35.88, "duration": 4.46, "text": "squared plus N squared divided by four"}], "title": "Summing cubes with lines of squares", "description": "This is a short, animated (wordless) visual proof demonstrating the sum of the first n positive cubes. \n\nIf you like this video, consider subscribing to the channel or consider buying me a coffee: https://www.buymeacoffee.com/VisualProofs. Thanks!\n\nTo see a longer, wide format, wordless (and perhaps more epic) version, see the original: https://youtu.be/Ye9OPNqV9FA\n\nThis animation is based on a visual proofs by Warren Lushbaugh (from an article by Solomon Golomb) in the May 1965 issue of the Mathematical Gazette (https://doi.org/10.2307/3612319) and, independently, Antonella Cupillari from the October 1989 issue of Mathematics Magazine page 259 (https://doi.org/10.2307/2689765).\n\n#mathshorts\u200b #mathvideo\u200b #math\u200b #calculus #mtbos\u200b #manim\u200b #animation\u200b #theorem\u200b #pww\u200b #proofwithoutwords\u200b #visualproof\u200b #proof\u200b #iteachmath #finitesums #discretemath #calculus #sum #induction \n\nTo learn more about animating with manim, check out:\nhttps://manim.community", "lengthSeconds": 40, "width": 360, "height": 640}
{"video_id": "F4XmmekehoE", "transcript": [{"start": 0.44, "duration": 4.8, "text": "hello everyone in this video we are"}, {"start": 2.399, "duration": 5.48, "text": "going to give the visual proof of a + b"}, {"start": 5.24, "duration": 5.399, "text": "square let we draw a square with side"}, {"start": 7.879, "duration": 5.76, "text": "length a definitely the area of this"}, {"start": 10.639, "duration": 6.361, "text": "square is equal to a square and having a"}, {"start": 13.639, "duration": 7.681, "text": "rectangle with the side length A and"}, {"start": 17.0, "duration": 8.24, "text": "B its area is equal to AB we draw"}, {"start": 21.32, "duration": 6.799, "text": "another square with side length as B"}, {"start": 25.24, "duration": 5.32, "text": "then area of this square is equal to b"}, {"start": 28.119, "duration": 5.28, "text": "square and definitely this is is a"}, {"start": 30.56, "duration": 6.28, "text": "rectangle with side length A and B and"}, {"start": 33.399, "duration": 6.48, "text": "having area is equal to a so this"}, {"start": 36.84, "duration": 7.64, "text": "complete figure is a square with side"}, {"start": 39.879, "duration": 8.881, "text": "length as a + b and its area is equal to"}, {"start": 44.48, "duration": 8.8, "text": "a + b s which is equal to sum of these"}, {"start": 48.76, "duration": 6.799, "text": "four areas which are these one and after"}, {"start": 53.28, "duration": 6.0, "text": "simplification we have a + b sare is"}, {"start": 55.559, "duration": 5.921, "text": "equal to a s + 2 a + b\u00b2 that is the"}, {"start": 59.28, "duration": 2.2, "text": "answer"}], "title": "Visual Proof of (a+b)\u00b2 \ud83d\udc4c\ud83d\udc4d #maths", "description": "#amsmath #geometry #visualproof #shortsfeed #shortsfeed2024 #education #a+bsquare #mathematicspuzzle", "lengthSeconds": 60, "width": 360, "height": 640}
{"video_id": "1bg7VpNW9bo", "transcript": [{"start": 11.76, "duration": 16.739, "text": "[\u0645\u0648\u0633\u064a\u0642\u0649]"}, {"start": 29.96, "duration": 4.78, "text": "\u200fy"}, {"start": 31.64, "duration": 3.1, "text": "[\u0645\u0648\u0633\u064a\u0642\u0649]"}], "title": "#trending #mathematics Visual Proof: Factors of Quadratic Equation Polynomial PART-4 x^2-2x-3", "description": "#trending #mathematics Visual Proof: Factors of Quadratic Equation Polynomial PART-4 x^2-2x-3\n\n\nOther Videos\nFibonacci Sequence, Golden Ratio, How to draw Golden Spiral:\nhttps://youtu.be/yHBa3EvDPco\nRadius and Tangent Vectors: \nhttps://youtube.com/shorts/KHSafRET3Sk\nDe Morgan's laws: Proof by Venn Diagrams:\nhttps://youtube.com/shorts/pR-REo4aBYk?feature=share\nVisual Proof of Roots of Quadratic Equation\nhttps://youtube.com/shorts/FdE7rHDZRd8\nVisual Proof: Factors of Quadratic Equation PART-1 x^2+3x+2\nhttps://youtube.com/shorts/lIqv5zKAKKE\nVisual Proof: Factors of Quadratic Polynomial PART-2 x^2-3x+2\nhttps://youtube.com/shorts/5pRYbKFWjUM\nVisual Proof: Factors of Quadratic Polynomial PART-2 x^2+2x-3\nhttps://youtube.com/shorts/AzAsz5itOfQ", "lengthSeconds": 33, "width": 360, "height": 640}
{"video_id": "Jq_I1ftJ9uc", "transcript": [{"start": 0.0, "duration": 4.74, "text": "here's a geometric way to compute the"}, {"start": 2.159, "duration": 5.341, "text": "infinite sum of the powers of 1 6 1 6"}, {"start": 4.74, "duration": 6.0, "text": "plus 1 6 squared plus 1 6 cubed and so"}, {"start": 7.5, "duration": 5.58, "text": "on take a circle of Area 1 and draw a"}, {"start": 10.74, "duration": 4.74, "text": "smaller circle inside with Area 1 6."}, {"start": 13.08, "duration": 4.44, "text": "then the annulus around that Circle can"}, {"start": 15.48, "duration": 4.62, "text": "be divided into five equal pieces all of"}, {"start": 17.52, "duration": 5.099, "text": "size 1 6. we can shade one of them to"}, {"start": 20.1, "duration": 4.74, "text": "represent the Area 1 6. now we can"}, {"start": 22.619, "duration": 5.641, "text": "repeat the process on the inner 1 6"}, {"start": 24.84, "duration": 6.12, "text": "circle taking 1 6 of 1 6 and decomposing"}, {"start": 28.26, "duration": 5.94, "text": "the annular region into five regions of"}, {"start": 30.96, "duration": 5.82, "text": "area 136 or 1 over 6 squared continuing"}, {"start": 34.2, "duration": 4.859, "text": "this process indefinitely produces a"}, {"start": 36.78, "duration": 5.099, "text": "region whose area represents the"}, {"start": 39.059, "duration": 5.761, "text": "infinite sum of the powers of one-sixth"}, {"start": 41.879, "duration": 5.281, "text": "but we also now see that the Shaded area"}, {"start": 44.82, "duration": 4.8, "text": "can be replicated four more times to"}, {"start": 47.16, "duration": 5.52, "text": "fill in this circle therefore the sum of"}, {"start": 49.62, "duration": 5.9, "text": "the powers of 1 6 represents one-fifth"}, {"start": 52.68, "duration": 2.84, "text": "of the full circle"}], "title": "Infinite sum of powers of sixths!", "description": "This is a short, animated visual proof showing the sum of the infinite geometric series of powers of 1/6 (starting with 1/6) is 1/5 using a general method of dissecting a circle into a small circle and annular sectors.\n\nFor a different version using polygons (and without words) check out : https://youtu.be/o0AyHmRV8RA\n\nThis animation is based on a proof by James Tanton from the March 2008 issue of The College Mathematics Journal page 106. (https://www.jstor.org/stable/27646594).\n\nTo learn more about animating with manim, check out:\nhttps://manim.community\n\n#manim #math\u200b\u200b #mathshorts\u200b #mathvideo\u200b  #mtbos\u200b #manim\u200b #animation\u200b #theorem\u200b #pww\u200b #proofwithoutwords\u200b #visualproof\u200b #proof\u200b #iteachmath #calculus #series #geometricseries #infiniteseries  #circle", "lengthSeconds": 54, "width": 360, "height": 640}
{"video_id": "FB-nBkjLixU", "transcript": [{"start": 0.719, "duration": 21.151, "text": "foreign"}, {"start": 0.99, "duration": 20.88, "text": "[Music]"}], "title": "Visual math", "description": "What is the sum of all numbers in a times table?\n\nFull video: https://youtu.be/bjjmuOd1fdc\n\nSubscribe: https://www.youtube.com/user/MindYourDecisions?sub_confirmation=1\n\nSend me suggestions by email (address at end of many videos). I may not reply but I do consider all ideas!\n\nIf you purchase through these links, I may be compensated for purchases made on Amazon. As an Amazon Associate I earn from qualifying purchases. This does not affect the price you pay. \n\nBook ratings are from January 2022.\n\nMy Books (worldwide links)\nhttps://mindyourdecisions.com/blog/my-books/#worldwide\n\nMy Books (US links)\nMind Your Decisions: Five Book Compilation\nhttps://amzn.to/2pbJ4wR\nA collection of 5 books:\n\"The Joy of Game Theory\" rated 4.2/5 stars on 224 reviews\nhttp://amzn.to/1uQvA20\n\"The Irrationality Illusion: How To Make Smart Decisions And Overcome Bias\" rated 4/5 stars on 24 reviews\nhttp://amzn.to/1o3FaAg\n\"40 Paradoxes in Logic, Probability, and Game Theory\" rated 4.1/5 stars on 38 reviews\nhttp://amzn.to/1LOCI4U\n\"The Best Mental Math Tricks\" rated 4.2/5 stars on 76 reviews\nhttp://amzn.to/18maAdo\n\"Multiply Numbers By Drawing Lines\" rated 4.3/5 stars on 30 reviews\nhttp://amzn.to/XRm7M4\n\nMind Your Puzzles: Collection Of Volumes 1 To 3\nhttps://amzn.to/2mMdrJr\nA collection of 3 books:\n\"Math Puzzles Volume 1\" rated 4.4/5 stars on 87 reviews\nhttp://amzn.to/1GhUUSH\n\"Math Puzzles Volume 2\" rated 4.1/5 stars on 24 reviews\nhttp://amzn.to/1NKbyCs\n\"Math Puzzles Volume 3\" rated 4.2/5 stars on 22 reviews\nhttp://amzn.to/1NKbGlp\n\n2017 Shorty Awards Nominee. Mind Your Decisions was nominated in the STEM category (Science, Technology, Engineering, and Math) along with eventual winner Bill Nye; finalists Adam Savage, Dr. Sandra Lee, Simone Giertz, Tim Peake, Unbox Therapy; and other nominees Elon Musk, Gizmoslip, Hope Jahren, Life Noggin, and Nerdwriter.\n\nMy Blog\nhttps://mindyourdecisions.com/blog/\n\nTwitter\nhttps://twitter.com/preshtalwalkar\n\nInstagram\nhttps://www.instagram.com/preshtalwalkar/\n\nMerch\nhttps://teespring.com/stores/mind-your-decisions\n\nPatreon\nhttps://www.patreon.com/mindyourdecisions\n\nPress\nhttps://mindyourdecisions.com/blog/press", "lengthSeconds": 20, "width": 360, "height": 640}
{"video_id": "Cmd6YE4dVhE", "transcript": [{"start": 0.04, "duration": 3.32, "text": "start with two positive real numbers a"}, {"start": 1.56, "duration": 3.92, "text": "and b and draw a rectangle with width"}, {"start": 3.36, "duration": 4.36, "text": "root B plus root B and length root a"}, {"start": 5.48, "duration": 4.48, "text": "plus root a inside this rectangular"}, {"start": 7.72, "duration": 4.52, "text": "diagram we can shade two sub rectangles"}, {"start": 9.96, "duration": 5.24, "text": "like this the unshaded area in this"}, {"start": 12.24, "duration": 4.92, "text": "rectangle is then given by two < of B by"}, {"start": 15.2, "duration": 5.36, "text": "of a rectangles for a total unshaded"}, {"start": 17.16, "duration": 5.48, "text": "area of 2 * the of a * B but now we can"}, {"start": 20.56, "duration": 4.16, "text": "cut those two shaded rectangles into two"}, {"start": 22.64, "duration": 4.08, "text": "triangles each and slide those triangles"}, {"start": 24.72, "duration": 3.52, "text": "around the rectangle like this the"}, {"start": 26.72, "duration": 3.319, "text": "resulting unshaded area is now a"}, {"start": 28.24, "duration": 3.439, "text": "parallelogram we can call this angle"}, {"start": 30.039, "duration": 3.84, "text": "Thea and the side lengths are both the"}, {"start": 31.679, "duration": 3.601, "text": "of a plus b by the Pythagorean theorem"}, {"start": 33.879, "duration": 3.161, "text": "using the formula for the area of a"}, {"start": 35.28, "duration": 5.119, "text": "parallelogram we see that the unshaded"}, {"start": 37.04, "duration": 5.8, "text": "area is theun of a plus b^ 2 * the S of"}, {"start": 40.399, "duration": 3.881, "text": "theta but now in this equality we see"}, {"start": 42.84, "duration": 3.879, "text": "that the S of theta is always less than"}, {"start": 44.28, "duration": 4.48, "text": "or equal to 1 so that 2 * the S of a * B"}, {"start": 46.719, "duration": 4.0, "text": "is less than or equal to a + b or"}, {"start": 48.76, "duration": 3.959, "text": "equivalently that the of a * B is less"}, {"start": 50.719, "duration": 3.64, "text": "than or equal to a + b / 2 we get"}, {"start": 52.719, "duration": 3.801, "text": "equality if and only if s of theta"}, {"start": 54.359, "duration": 3.921, "text": "equals 1 so that the angle is 90\u00b0 and a"}, {"start": 56.52, "duration": 5.08, "text": "equals B this is the arithmetic mean"}, {"start": 58.28, "duration": 3.32, "text": "geometric mean inequality"}], "title": "AM-GM Inequality via negative space", "description": "This is a short, animated visual proof demonstrating the arithmetic mean geometric mean inequality using algebraic areas and the Side-Angle-Side formula for a parallelogram.\n\nIf you like this video, consider subscribing to the channel or consider buying me a coffee: https://www.buymeacoffee.com/VisualProofs. Thanks! \n\nFor other visual proofs of this fact see:\n\nhttps://youtu.be/Pt8IX_Q5U1A\nhttps://youtu.be/62G9fak1vyk\nhttps://youtu.be/HFGSsPbYc-s\n\nFor a nearly identical visual proof of another inequality (Cauchy-Schwarz) see\nhttps://youtu.be/mKg_gVagHy8\n\nThis animation is based on a visual proof that appears in Icons of Mathematics by Claudi Alsina and Roger B. Nelsen (MAA, 2011): \n\n#mathshorts\u200b #mathvideo\u200b #math\u200b #amgminequality #mtbos\u200b #manim\u200b #animation\u200b #theorem\u200b #pww\u200b #proofwithoutwords\u200b #visualproof\u200b #proof\u200b #iteachmath #calculus #inequality \n\nTo learn more about animating with manim, check out:\nhttps://manim.community", "lengthSeconds": 60, "width": 360, "height": 640}
{"video_id": "g3C3hs8_nzU", "transcript": [{"start": 0.96, "duration": 4.28, "text": "let's see two proofs of the Pythagorean"}, {"start": 2.879, "duration": 4.641, "text": "theorem inspired by"}, {"start": 5.24, "duration": 5.68, "text": "uid imagine that we have a right"}, {"start": 7.52, "duration": 6.68, "text": "triangle with legs of length A and B and"}, {"start": 10.92, "duration": 6.119, "text": "a hypotenuse of length C we can draw a"}, {"start": 14.2, "duration": 6.319, "text": "square off the leg of length a and this"}, {"start": 17.039, "duration": 6.441, "text": "Square encloses an area of a * a or a"}, {"start": 20.519, "duration": 5.801, "text": "squ similarly we can draw a square off"}, {"start": 23.48, "duration": 6.08, "text": "the leg of length B enclosing an area of"}, {"start": 26.32, "duration": 4.32, "text": "b^ s we can also draw a square off the"}, {"start": 29.56, "duration": 3.92, "text": "hypoten"}, {"start": 30.64, "duration": 5.8, "text": "and this has an area of c^ 2 in the"}, {"start": 33.48, "duration": 7.2, "text": "picture diagram the Shaded area is a 2 +"}, {"start": 36.44, "duration": 7.2, "text": "b^ 2 ID realized that we can Shear"}, {"start": 40.68, "duration": 5.76, "text": "squares and we maintain the same area we"}, {"start": 43.64, "duration": 5.439, "text": "can also rotate shapes to maintain area"}, {"start": 46.44, "duration": 6.88, "text": "so if we Shear both squares along the"}, {"start": 49.079, "duration": 6.401, "text": "opposite leg of the triangle rotate 90\u00b0"}, {"start": 53.32, "duration": 4.6, "text": "and then Shear again down this vertical"}, {"start": 55.48, "duration": 5.12, "text": "line where they meet we end up filling"}, {"start": 57.92, "duration": 6.36, "text": "the C by C Square Off the hypotenuse"}, {"start": 60.6, "duration": 5.68, "text": "this means that A2 + B2 = c^2 where A"}, {"start": 64.28, "duration": 4.6, "text": "and B are the legs of a right triangle"}, {"start": 66.28, "duration": 5.159, "text": "with hypotenuse C this is essentially"}, {"start": 68.88, "duration": 5.36, "text": "ID's proof of the Pythagorean"}, {"start": 71.439, "duration": 4.921, "text": "theorem the shearing and rotation proof"}, {"start": 74.24, "duration": 4.6, "text": "also suggests a dissection proof of the"}, {"start": 76.36, "duration": 4.28, "text": "Pythagorean theorem we can cut the B by"}, {"start": 78.84, "duration": 3.84, "text": "b square like this so that the"}, {"start": 80.64, "duration": 4.56, "text": "Triangular region matches the original"}, {"start": 82.68, "duration": 4.92, "text": "right triangle we can rotate this shape"}, {"start": 85.2, "duration": 4.36, "text": "down and then use the triangle to cut"}, {"start": 87.6, "duration": 5.479, "text": "this region into two shapes dropping"}, {"start": 89.56, "duration": 6.4, "text": "down this top pictured triangle here we"}, {"start": 93.079, "duration": 4.921, "text": "can also cut the a by a square like this"}, {"start": 95.96, "duration": 4.28, "text": "dropping this blue region down into the"}, {"start": 98.0, "duration": 4.439, "text": "original triangle and using the line"}, {"start": 100.24, "duration": 3.8, "text": "shown here to cut another little piece"}, {"start": 102.439, "duration": 4.801, "text": "to fill the last little triangular"}, {"start": 104.04, "duration": 5.52, "text": "region here we rotate 90\u00b0 and use the"}, {"start": 107.24, "duration": 4.64, "text": "original triangle to create a new small"}, {"start": 109.56, "duration": 4.559, "text": "triangle that then drops down into place"}, {"start": 111.88, "duration": 4.839, "text": "in the C by C Square from this"}, {"start": 114.119, "duration": 5.0, "text": "dissection proof we see that a^2 + b^2"}, {"start": 116.719, "duration": 5.121, "text": "equal c^2 by cutting and moving pieces"}, {"start": 119.119, "duration": 2.721, "text": "around"}, {"start": 122.039, "duration": 3.801, "text": "let's see the original shearing and"}, {"start": 123.799, "duration": 4.24, "text": "rotating proof one more time where we"}, {"start": 125.84, "duration": 4.6, "text": "shear and rotate both rectangles"}, {"start": 128.039, "duration": 4.56, "text": "together you tell me do you like the"}, {"start": 130.44, "duration": 4.159, "text": "shearing and rotating proof better or do"}, {"start": 132.599, "duration": 4.401, "text": "you prefer the dissection that it"}, {"start": 134.599, "duration": 4.601, "text": "implies either way it's nice to have"}, {"start": 137.0, "duration": 4.76, "text": "multiple proofs of one fact and it's"}, {"start": 139.2, "duration": 5.56, "text": "amazing that ukids proof produces the"}, {"start": 141.76, "duration": 3.0, "text": "dissection proof as"}, {"start": 148.76, "duration": 3.0, "text": "well"}], "title": "Pythagorean Theorem XII (visual proof)", "description": "This is a short, animated visual proof of the Pythagorean theorem (the right triangle theorem) following essentially Euclid's proof. This theorem states the square of the hypotenuse of a right triangle is equal to the sum of squares of the two other side lengths.  As a bonus, we show how to use Euclid's shear and rotate proof to create a dissection proof of the Pythagorean theorem. Which of these proofs do you prefer? \n\nIf you like this video, consider subscribing to the channel or consider buying me a coffee: https://www.buymeacoffee.com/VisualProofs. Thanks! \n\nFor other proofs of this same fact check out: \nhttps://youtu.be/o2mbN452ywA (compilation)\nhttps://youtu.be/XFHm0Ot3PP0\nhttps://youtu.be/q28H5NY5UVU\nhttps://youtu.be/yfGtbNgcrQ8\nhttps://youtu.be/JRjPsuJ7S-g\nhttps://youtu.be/TsKTteGJpZU\nhttps://youtu.be/ZlGaQdNRdqA\nhttps://youtu.be/uqGjs_n-f7U\nhttps://youtu.be/wWEZ-rUlBDQ\nhttps://youtu.be/I75HYC8TRNY\nhttps://youtu.be/LWVog8hg74I?si=4GSzTLZP7XIWeaxJ\n\nor check out my Pythagorean theorem playlist:\n\nhttps://www.youtube.com/playlist?list=PLZh9gzIvXQUvlRlnYryejWGHYQquM92rB\n\nThis animation is based on a proof due to Euclid. For a static version of this proof, see Roger Nelsen's first \"Proof Without Words: Exercises in Visual Thinking\" compendium (page 5) : https://bookstore.ams.org/clrm-1. You can also check out Howard Eves' \"Great Moments in Mathematics (Before 1650)\" page 31-33.\n\n(I may receive a small commission at no cost to you for these affiliate links): \n\nhttps://amzn.to/48mjeX1\nhttps://amzn.to/3Tr0OjW\n\nThe dissection proof was suggested to me by Glen Whitney (https://studioinfinity.org/), and Glen then found a similar dissection in the compendium of Pythagorean theorem proofs from Elisha Loomis  (Proof 21 on p. 122 ).\n(I may receive a small commission at no cost to you for this affiliate link):  https://amzn.to/3RL7rfN\n\n#math #pythagoreantheorem #pythagorean #triangle #manim #animation #theorem #pww #proofwithoutwords #visualproof #proof  #mathshorts #mathvideo \n\nTo learn more about animating with manim, check out:\nhttps://manim.community", "lengthSeconds": 151, "width": 640, "height": 360}
{"video_id": "hJDjv7S0Efg", "transcript": [{"start": 0.0, "duration": 14.56, "text": "[Music]"}, {"start": 18.7, "duration": 8.01, "text": "[Music]"}, {"start": 28.96, "duration": 3.0, "text": "a"}, {"start": 35.13, "duration": 4.589, "text": "[Music]"}, {"start": 36.719, "duration": 3.0, "text": "a"}], "title": "Surface Area Of Sphere| Proof | Visual | Mathletes |Sunny Sir  #maths #algebra #foryou #satisfying", "description": "Surface Area Of Sphere| Proof | Visual | Mathletes | Sunny Sir  #maths #algebra #foryou #satisfying  #cbse #funnyshorts\n\nDo Like Share and Subscribe For more interesting videos\n\nJoin our telegram channel: https://t.me/Mathletes_29\n\nFollow us on Instagram: https://www.instagram.com/mathletes19?igsh=MWo2ZzdlY3cxdjg1MA==", "lengthSeconds": 38, "width": 460, "height": 640}
{"video_id": "tcBTI1_1Fuw", "transcript": [{"start": 0.199, "duration": 4.991, "text": "oh"}, {"start": 2.52, "duration": 5.769, "text": "[Applause]"}, {"start": 5.19, "duration": 3.099, "text": "[Music]"}, {"start": 14.11, "duration": 5.98, "text": "[Music]"}], "title": "Visual proof of complete the square or vertex form method \ufffc#maths #visualization #completethesquare", "description": "", "lengthSeconds": 25, "width": 360, "height": 640}
{"video_id": "xY7cO13K-eo", "transcript": [{"start": 1.34, "duration": 3.24, "text": "thank you"}, {"start": 5.67, "duration": 4.08, "text": "[Music]"}, {"start": 17.12, "duration": 3.489, "text": "[Music]"}], "title": "Math is easy | math formula visual proof", "description": "Follow @EduTechBytes \nmath is easy,\nmath is easy song,\nmath is easy meme,\nmath is easy subliminal,\nmath is easy channel,\nmath is easy shihab sir,\nmath is easy now,\nmath is easy with alaa,\nmath is easy edit,\nmath is easy differentiation,\nmath is easy or hard,\nmaths easy activity,\nmaths easy addition,\nmath is so easy garten of banban gh's animation,\nmath made easy by chetan setia,\nmath is very easy by nitin sir,\nmath ko easy banane ka tarika,\nmaths is easy by deepak pandey,\nwhich math is easy basic or standard,\nmath easy by surendra sir,\nmath made easy by studypug f3.0.0,\nmaths easy by rishi,\nmath easy book,\nmath made easy by anuradha singh,\nmath or biology which is easy,\nreel math is so easy garten of banban gh's animation,\nmath easy chapter for jee mains,\nmaths easy chapter class 12,\nmaths easy chapters for eamcet,\nmaths easy chapters for jee advanced,\nmaths easy concept,\nmaths is easy class 10,\nmath easy chart,\nmaths is easy class 9,\nmath easy chapter,\nmath easy chapter ssc gd,\nmath divide easy trick,\nmath sayaw easy dance steps,\nmaths easy division tricks,\nmaths easy division,\nmath easy drawing,\nmaths easy division tricks malayalam,\nmath division easy way,\nmath easy divide,\nmath design easy,\nmath dance easy steps,\ndifferentiation math is easy,\nmaths experiments easy,\nmaths easy exam tips,\nmath technique easy education,\nmath easy equations,\nmath easy formula,\nmaths easy front page,\nmath ka easy formula,\nwhy math is hard for me,\nf math,\nmath made easy grade 10,\nmaths easy ga nerchukovadam ela telugu,\nmath made easy grade 10 unit 3,\nmath made easy grade 11,\nmath made easy grade 10 unit 2,\nmath made easy grade 10 unit 4,\nmath made easy grade 10 unit 5,\nmath made easy grade 10 unit 3 exercise 3.2,\nmath made easy grade 11 unit 3,\nmath made easy grade 11 unit 2,\nmath easy hai ya biology,\nmath easy hai,\nmath ki easy,\nmaths easy homework,\nmath is easy math is helpful,\nmath easy home,\nmath hacks easy,\nis math easy or hard,\nis maths easy in class 11,\nmath is not hard it's a language,\nmaths easy in tamil,\nmath in easy,\nmaths easy japanese multiplication,\nmath easy japanese,\nmathematics with mr. j,\nmath with mr j standard algorithm,\nmath with mr j,\nmath ko easy kaise banaye,\nmaths easy kill topics for jee mains,\nmath ka easy trick,\nmath ko easy kaise banaen,\nmath ka easy model,\nmath easy kaise banaye,\nmath ka easy question,\nmath ke easy question,\nmath ko easy kaise karen,\nmaths easy kill topics,\nmaths easy learning tricks,\nmaths easy learning tricks in tamil,\nmaths easy learning,\nmaths easy learning tricks malayalam,\nmaths easy learning tips,\nmath is hard let's go shopping,\nmaths easy learning class 8,\nmath learning easy way,\nmaths easy learning tricks in telugu,\nmath hard level,\nlearn math love math,\nwhy math is easy,\nl math,\neveryone can learn math,\nmaths easy method,\nmath easy model,\nmaths easy method in tamil,\nmaths easy model for class 8,\nmaths easy model for class 5,\nmath easy multiplication,\nmath is hard meme,\nmaths easy model for class 10,\nmaths easy method in telugu,\nmath easy model class 7,\nmath is not hard,\nmath is not easy,\nbasic maths is easy or standard,\nbsc maths is easy or tough,\nmaths easy tricks multiplication,\nmath formula visual proof,\nmath formula proof,\nmath formula analysis,\nmath formula 12,\nmath a+b formulas,\nmaths formula a + b square,\na + b all formula,\nmath formula (a+b)3,\nmath formula (a+b)2,\nmath formula sin cos,\nformula of sin cos,\nhow to derive a formula in maths,\nformula proved,\ne^i theta formula,\ne^i theta proof,\ne^ipi+1=0 proof,\nhow to use math formulas,\nmath formula with example,\nproof of formulas,\nlog a/b formula,\nformula of optional math,\np(a/b) formula,\n(a+b)6 formula,\n(x+a)(x+b) formula proof,\n(x+a)(x+b) formula,\nmath formula x y,\na to z math formula,\n(a+b)3 formula proof,\nmath formula form 2,\nmath 3 formulas,\na+b)5 formula,\nmaths 9 formula,\n9 class maths formulas", "lengthSeconds": 23, "width": 360, "height": 640}
{"video_id": "jB0OZYA_EI0", "transcript": [{"start": 0.0, "duration": 4.62, "text": "given two positive real numbers X and Y"}, {"start": 2.22, "duration": 4.26, "text": "Place lines of length X and Y side by"}, {"start": 4.62, "duration": 4.92, "text": "side to form the diameter of a circle"}, {"start": 6.48, "duration": 4.68, "text": "whose radius is X Plus y over two"}, {"start": 9.54, "duration": 3.479, "text": "at the intersection point of the two"}, {"start": 11.16, "duration": 4.8, "text": "lines create a half chord to the circle"}, {"start": 13.019, "duration": 4.861, "text": "of length Z the triangle formed here"}, {"start": 15.96, "duration": 3.6, "text": "must be a right triangle by thales"}, {"start": 17.88, "duration": 3.659, "text": "triangle theorem and so these two"}, {"start": 19.56, "duration": 4.62, "text": "subtriangles must be similar this means"}, {"start": 21.539, "duration": 5.281, "text": "that Z over X has to be equal to Y over"}, {"start": 24.18, "duration": 4.8, "text": "Z and we can simplify this to see that Z"}, {"start": 26.82, "duration": 3.9, "text": "squared equals x times y so that the"}, {"start": 28.98, "duration": 3.419, "text": "length of Z is equal to the square root"}, {"start": 30.72, "duration": 3.839, "text": "of x times y"}, {"start": 32.399, "duration": 5.101, "text": "notice that no matter the lengths of X"}, {"start": 34.559, "duration": 5.641, "text": "and Y this half chord is always less"}, {"start": 37.5, "duration": 4.68, "text": "than or equal to the radius therefore we"}, {"start": 40.2, "duration": 3.48, "text": "see that the square root of x times Y is"}, {"start": 42.18, "duration": 4.02, "text": "less than or equal to the quantity X"}, {"start": 43.68, "duration": 4.74, "text": "Plus y over 2. the only way that these"}, {"start": 46.2, "duration": 3.12, "text": "two quantities are equal is if x equals"}, {"start": 48.42, "duration": 2.7, "text": "y"}, {"start": 49.32, "duration": 5.36, "text": "this is known as the arithmetic mean"}, {"start": 51.12, "duration": 3.56, "text": "geometric mean inequality"}], "title": "AM-GM Inequality II", "description": "This is a short, animated visual proof demonstrating the arithmetic mean geometric mean inequality using Thales theorem . #math\u200b #amgminequality #mtbos\u200b #manim\u200b #animation\u200b #theorem\u200b #pww\u200b #proofwithoutwords\u200b #visualproof\u200b #proof\u200b #iteachmath #calculus #inequality #mathshorts\u200b #mathvideo\u200b \n\nTo see my original, longer form, animation of this proof without words, check out: https://youtu.be/Pt8IX_Q5U1A\n\nThis animation is based on a visual proof by Charles D. Gallant from the March 1977 issue of Mathematics Magazine (https://doi.org/10.2307/2689737).\n\nTo learn more about animating with manim, check out:\nhttps://manim.community", "lengthSeconds": 54, "width": 360, "height": 640}
{"video_id": "O_5CnZh-G7Q", "transcript": [{"start": 0.76, "duration": 4.67, "text": "[Music]"}, {"start": 2.3, "duration": 3.13, "text": "[Applause]"}, {"start": 7.06, "duration": 5.68, "text": "[Music]"}, {"start": 20.41, "duration": 7.31, "text": "[Music]"}, {"start": 24.359, "duration": 3.361, "text": "hwood that"}], "title": "Summing cubes without words #phonk #phonkmath", "description": "This is a short, animated (wordless) visual proof demonstrating the sum of the first n positive cubes by rearranging cubes into a flat square. \n\nIf you like this video, consider subscribing to the channel or consider buying me a coffee: https://www.buymeacoffee.com/VisualProofs. Thanks!\n\nHere is a longer version of this proof : https://youtu.be/YQLicI8R4Gs\n\nHere are three other visual proofs of sum of cubes formulas:\n\nhttps://youtu.be/Ye9OPNqV9FA\nhttps://youtu.be/NxOcT_VKQR0\nhttps://youtu.be/d1yM6Rq7Tfw\n\nThis animation is based on independently discovered, separate visual proofs by J. Barry Love from the March 1977 issue of Mathematics Magazine (https://www.jstor.org/stable/2689727), page 74, and Alan L. Fry from the January 1985 issue of Mathematics Magazine (https://www.jstor.org/stable/2690228) , page 11.\n\nThis is my version of a classic visual proof. Here are other renditions:\nhttps://youtu.be/glearwgR1Ls by   @ThinkTwiceLtu  \nhttps://youtu.be/YG_gS5vZoyQ by   @Mathocube  \nhttps://youtu.be/WY5X_3q80WY by  @Mathologer\n\n#mathshorts\u200b #mathvideo\u200b #math\u200b #calculus #mtbos\u200b #manim\u200b #animation\u200b #theorem\u200b #pww\u200b #proofwithoutwords\u200b #visualproof\u200b #proof\u200b #iteachmath #finitesums #discretemath #calculus #sum #induction \n\nTo learn more about animating with manim, check out:\nhttps://manim.community", "lengthSeconds": 26, "width": 360, "height": 640}
{"video_id": "C39T8Ueoc5k", "transcript": [{"start": 0.46, "duration": 12.72, "text": "[\u0e40\u0e1e\u0e25\u0e07]"}, {"start": 21.73, "duration": 3.07, "text": "[\u0e40\u0e1e\u0e25\u0e07]"}, {"start": 26.359, "duration": 2.361, "text": " K "}], "title": "Visual proof pythagoras theorem #mathematics #maths Beauty of mathematics", "description": "", "lengthSeconds": 27, "width": 360, "height": 640}
{"video_id": "2V3Mt1XzhRc", "transcript": [{"start": 0.0, "duration": 5.1, "text": "here's a right triangle and let's label"}, {"start": 1.92, "duration": 5.339, "text": "the legs A and B in the hypotenuse C we"}, {"start": 5.1, "duration": 4.68, "text": "can draw and shade an a by a square off"}, {"start": 7.259, "duration": 5.101, "text": "the a leg and we can draw and shade a b"}, {"start": 9.78, "duration": 5.04, "text": "by b square off the B leg like this we"}, {"start": 12.36, "duration": 4.86, "text": "can also draw but leave unshaded a c by"}, {"start": 14.82, "duration": 4.62, "text": "C Square Off the hypotenuse so the"}, {"start": 17.22, "duration": 4.979, "text": "unshaded area in this diagram is given"}, {"start": 19.44, "duration": 4.98, "text": "by c squared but now we're going to"}, {"start": 22.199, "duration": 4.381, "text": "shift and rotate objects around this"}, {"start": 24.42, "duration": 4.14, "text": "shape to produce a different unshaded"}, {"start": 26.58, "duration": 3.84, "text": "area first we use the triangle to"}, {"start": 28.56, "duration": 4.26, "text": "decompose the B by b square into two"}, {"start": 30.42, "duration": 4.92, "text": "pieces like this then we rotate the"}, {"start": 32.82, "duration": 4.919, "text": "original triangle down 90 degrees shift"}, {"start": 35.34, "duration": 5.1, "text": "the original a by a square like this and"}, {"start": 37.739, "duration": 4.801, "text": "then we shift and rotate and then break"}, {"start": 40.44, "duration": 4.86, "text": "along the hinge the original B by b"}, {"start": 42.54, "duration": 4.859, "text": "square filling the diagram now we see"}, {"start": 45.3, "duration": 4.5, "text": "that the unshaded area is given by a"}, {"start": 47.399, "duration": 4.441, "text": "squared plus b squared this means that"}, {"start": 49.8, "duration": 4.439, "text": "in a right triangle with legs of length"}, {"start": 51.84, "duration": 4.5, "text": "A and B and hypotenuse c c squared"}, {"start": 54.239, "duration": 5.181, "text": "equals a squared plus b squared this is"}, {"start": 56.34, "duration": 3.08, "text": "the Pythagorean theorem"}], "title": "Simple Pythagorean Dissection Proof", "description": "This is a short, animated (wordless) visual proof of the Pythagorean theorem (the right triangle theorem) using a relatively elementary dissection proof. \n\nIf you like this video, consider subscribing to the channel or consider buying me a coffee: https://www.buymeacoffee.com/VisualProofs. Thanks! \n\nThis animation is based on a proof from Gr\u00e9goire Nicollier, Aljo\u0161a Peperko & Janez \u0160ter in the December 2018 issue of Mathematics Magazine (https://doi.org/10.1080/0025570X.2018.1473701 ) pages 382-383.\n\nFor other proofs of this same fact check out: \nhttps://youtu.be/q28H5NY5UVU\nhttps://youtu.be/yfGtbNgcrQ8\nhttps://youtu.be/JRjPsuJ7S-g\nhttps://youtu.be/TsKTteGJpZU\nhttps://youtu.be/ZlGaQdNRdqA\nhttps://youtu.be/uqGjs_n-f7U\nhttps://youtu.be/wWEZ-rUlBDQ\nhttps://youtu.be/I75HYC8TRNY\n\n#math #manim #pythagoreantheorem #pythagorean #triangle #animation #theorem #pww #proofwithoutwords #visualproof #proof  #mathshorts #mathvideo \n\nTo learn more about animating with manim, check out:\nhttps://manim.community", "lengthSeconds": 58, "width": 360, "height": 640}
{"video_id": "2Wv_toNTqxs", "transcript": [{"start": 0.0, "duration": 3.24, "text": "let's assume that the square root of 2"}, {"start": 1.62, "duration": 3.48, "text": "is rational so that the square root of 2"}, {"start": 3.24, "duration": 3.96, "text": "equals A over B with A and B integers"}, {"start": 5.1, "duration": 3.78, "text": "this means 2B squared equals a squared"}, {"start": 7.2, "duration": 3.18, "text": "or B squared plus b squared equals a"}, {"start": 8.88, "duration": 3.12, "text": "squared so that there exists an"}, {"start": 10.38, "duration": 3.96, "text": "isosceles right triangle with a side"}, {"start": 12.0, "duration": 3.599, "text": "length of b and a hypotenuse of a we can"}, {"start": 14.34, "duration": 2.939, "text": "swing the side length over to the"}, {"start": 15.599, "duration": 3.841, "text": "hypotenuse like this and create a"}, {"start": 17.279, "duration": 4.141, "text": "subtriangle using the construction shown"}, {"start": 19.44, "duration": 4.98, "text": "we can determine that this subtriangle"}, {"start": 21.42, "duration": 5.04, "text": "is also an isosceles right triangle the"}, {"start": 24.42, "duration": 3.599, "text": "side lengths of this new isosceles right"}, {"start": 26.46, "duration": 4.139, "text": "triangle are a minus B and the"}, {"start": 28.019, "duration": 5.461, "text": "hypotenuse is 2b minus a but now we can"}, {"start": 30.599, "duration": 5.401, "text": "use this new right isosceles triangle as"}, {"start": 33.48, "duration": 4.8, "text": "proof that two copies of a minus B"}, {"start": 36.0, "duration": 4.14, "text": "squared is equal to the quantity 2B"}, {"start": 38.28, "duration": 3.18, "text": "minus a squared but as we saw"}, {"start": 40.14, "duration": 3.66, "text": "geometrically and can verify"}, {"start": 41.46, "duration": 4.619, "text": "analytically two B minus a is a positive"}, {"start": 43.8, "duration": 3.96, "text": "integer less than a and a minus B is a"}, {"start": 46.079, "duration": 3.421, "text": "positive integer less than b so we have"}, {"start": 47.76, "duration": 3.6, "text": "found two smaller positive integers"}, {"start": 49.5, "duration": 3.78, "text": "satisfying the same equation As A and B"}, {"start": 51.36, "duration": 3.96, "text": "we can repeat this process indefinitely"}, {"start": 53.28, "duration": 3.36, "text": "producing a geometric infinite Descent"}, {"start": 55.32, "duration": 3.3, "text": "of positive integers which is a"}, {"start": 56.64, "duration": 4.22, "text": "contradiction so the square root of 2 is"}, {"start": 58.62, "duration": 2.24, "text": "irrational"}], "title": "Root Two is Irrational Visually", "description": "In this short, we use a similar triangle argument to prove that the square root of two is irrational by infinite descent.\n\nIf you like this video, consider subscribing to the channel or consider buying me a coffee: https://www.buymeacoffee.com/VisualProofs. Thanks!\n\nFor an alternate visual proof of this fact, see this video: https://youtu.be/-dbzvN4jnfY\n\n#irrationalnumbers #realnumbers\u200b #manim\u200b #math\u200b #mtbos\u200b \u200b #animation\u200b #theorem\u200b\u200b #visualproof\u200b #proof\u200b #iteachmath #mathematics #irrational #carpetstheorem #proofbycontradiction #root2 #algebra #infinitedescent\n\nThis animation is based on an argument due to Tom Apostol from issue 9 of the 2000 American Mathematical Monthly: https://doi.org/10.1080/00029890.2000.12005280\n\nTo learn more about animating with manim, check out:\nhttps://manim.community", "lengthSeconds": 59, "width": 360, "height": 640}
{"video_id": "CmWblKmxhYQ", "transcript": [{"start": 0.48, "duration": 10.84, "text": "[Music]"}, {"start": 8.32, "duration": 3.0, "text": "m"}], "title": "Visual Proof of Pythagorean Theorem", "description": "Visual Proof of Pythagorean Theorem", "lengthSeconds": 9, "width": 360, "height": 640}
{"video_id": "RXHtCIin2A8", "transcript": [{"start": 0.12, "duration": 33.64, "text": "ah"}, {"start": 2.2, "duration": 34.56, "text": "[M\u00fasica]"}, {"start": 33.76, "duration": 3.0, "text": "f"}, {"start": 36.9, "duration": 12.69, "text": "[M\u00fasica]"}, {"start": 56.25, "duration": 3.119, "text": "[M\u00fasica]"}], "title": "Geometric Series (Not my video, Mathematical Visual Proofs on Youtube) #facts #maths #geometry", "description": "", "lengthSeconds": 59, "width": 360, "height": 640}
{"video_id": "WNk6pvKyNLg", "transcript": [{"start": 0.0, "duration": 29.87, "text": "thank you"}, {"start": 0.29, "duration": 29.58, "text": "[Music]"}, {"start": 37.8, "duration": 3.0, "text": "foreign"}, {"start": 43.57, "duration": 8.309, "text": "[Music]"}], "title": "Viviani\u2019s theorem visual proof", "description": "", "lengthSeconds": 50, "width": 360, "height": 640}
{"video_id": "ocbI2R13Jxw", "transcript": [{"start": 0.04, "duration": 2.76, "text": "here's the sum of the first n positive"}, {"start": 1.56, "duration": 2.96, "text": "squares like you've never seen them"}, {"start": 2.8, "duration": 3.44, "text": "before as the levels get higher the"}, {"start": 4.52, "duration": 4.999, "text": "visual proofs get"}, {"start": 6.24, "duration": 6.599, "text": "Wilder turn it"}, {"start": 9.519, "duration": 6.641, "text": "up turn it"}, {"start": 12.839, "duration": 6.641, "text": "up turn it"}, {"start": 16.16, "duration": 3.32, "text": "up turn"}, {"start": 19.68, "duration": 6.28, "text": "it sometimes it's good to"}, {"start": 22.76, "duration": 7.759, "text": "whisper and keep your voice down"}, {"start": 25.96, "duration": 6.68, "text": "low but sometimes you got to risk it and"}, {"start": 30.519, "duration": 5.481, "text": "let the whole world"}, {"start": 32.64, "duration": 6.64, "text": "know don't want to be too"}, {"start": 36.0, "duration": 6.559, "text": "quiet I'd rather get real"}, {"start": 39.28, "duration": 7.84, "text": "loud we going to start a"}, {"start": 42.559, "duration": 4.561, "text": "riot come on show you"}, {"start": 47.52, "duration": 6.6, "text": "how let's get a little bit"}, {"start": 50.879, "duration": 6.601, "text": "louder let's get a little bit"}, {"start": 54.12, "duration": 7.439, "text": "louder got to get a little bit"}, {"start": 57.48, "duration": 6.28, "text": "louder let's get a little bit turn it up"}, {"start": 61.559, "duration": 4.641, "text": "let's just see how high we can go turn"}, {"start": 63.76, "duration": 5.48, "text": "it up no stopping till we lose our"}, {"start": 66.2, "duration": 5.48, "text": "control turn it up until we feel it down"}, {"start": 69.24, "duration": 2.44, "text": "in our"}, {"start": 72.4, "duration": 6.52, "text": "soul let's get your heartbeat"}, {"start": 75.799, "duration": 6.561, "text": "pumping this is the honest truth this is"}, {"start": 78.92, "duration": 6.76, "text": "the honest truth if we get these bodies"}, {"start": 82.36, "duration": 4.88, "text": "jumped we're going to shake it Loose"}, {"start": 85.68, "duration": 4.92, "text": "Shake It"}, {"start": 87.24, "duration": 6.64, "text": "Loose let's get a little bit like"}, {"start": 90.6, "duration": 6.64, "text": "Lou let's get a little bit"}, {"start": 93.88, "duration": 7.48, "text": "louder got to get a little bit"}, {"start": 97.24, "duration": 6.239, "text": "louder let's get a little bit turn it up"}, {"start": 101.36, "duration": 4.6, "text": "let's just see how high we can go turn"}, {"start": 103.479, "duration": 5.561, "text": "it up no stopping till we lose our"}, {"start": 105.96, "duration": 6.199, "text": "control turn it up until we feel it down"}, {"start": 109.04, "duration": 6.759, "text": "in our soul turn it"}, {"start": 112.159, "duration": 6.761, "text": "up turn it up let's just see how high we"}, {"start": 115.799, "duration": 6.24, "text": "can go turn it up no stop until we lose"}, {"start": 118.92, "duration": 6.76, "text": "our control turn it up until we feel it"}, {"start": 122.039, "duration": 3.641, "text": "down in our soul turn it"}, {"start": 127.08, "duration": 6.56, "text": "up let's get a little bit"}, {"start": 130.36, "duration": 6.64, "text": "louder let's get a little bit"}, {"start": 133.64, "duration": 6.64, "text": "louder got to get a little bit"}, {"start": 137.0, "duration": 6.599, "text": "louder let's get a little"}, {"start": 140.28, "duration": 6.12, "text": "bit let's get a little bit"}, {"start": 143.599, "duration": 5.72, "text": "louder let's get a little bit louder"}, {"start": 146.4, "duration": 4.0, "text": "let's go come on got get a little bit"}, {"start": 149.319, "duration": 4.161, "text": "louder"}, {"start": 150.4, "duration": 6.68, "text": "yeah let's get a little"}, {"start": 153.48, "duration": 6.56, "text": "bit turn it up let's just see how high"}, {"start": 157.08, "duration": 5.84, "text": "we can go turn it up don't stop until we"}, {"start": 160.04, "duration": 6.76, "text": "lose our control turn it up until we"}, {"start": 162.92, "duration": 7.52, "text": "feel it down in our soul turn it"}, {"start": 166.8, "duration": 6.719, "text": "up turn it up let's just see how high we"}, {"start": 170.44, "duration": 6.2, "text": "can go turn it up no stop until we lose"}, {"start": 173.519, "duration": 5.241, "text": "our control turn it up until we feel it"}, {"start": 176.64, "duration": 4.36, "text": "down in our soul turn it"}, {"start": 178.76, "duration": 4.92, "text": "up"}, {"start": 181.0, "duration": 5.56, "text": "turn it up let's just see how high we"}, {"start": 183.68, "duration": 6.04, "text": "can go turn it up don't stop until we"}, {"start": 186.56, "duration": 6.759, "text": "lose our control turn it until we feel"}, {"start": 189.72, "duration": 3.599, "text": "it down in our soul turn"}, {"start": 194.239, "duration": 4.801, "text": "it comment down below which level is"}, {"start": 196.799, "duration": 2.241, "text": "your"}, {"start": 200.68, "duration": 3.0, "text": "favorite"}], "title": "Sum of Squares Visual Proofs: From Level 1 to Level 5", "description": "This video is a compilation of five different visual proofs showing the sum formula for the sum of the first n positive squares that can be found on my channel. This is a new style of video for me inspired by \u00a0@ThatsAmazing\u00a0's trick shots at various levels and \u00a0@dudeperfect\u00a0's soundtracks. I hope you enjoy it. These visual proofs are likely too fast to follow completely in this format, so the originals are linked below.\n\nIf you like this video, consider subscribing to the channel or consider buying me a coffee: https://www.buymeacoffee.com/VisualProofs. Thanks! \n\nFor the original animations, see here:\nLevel 1, Induction Dissection: https://youtu.be/a8j3sBrXchg?si=zH0Wk9McXyJVxsKc\nLevel 2, Tokens on Triangular Array: https://youtu.be/UqVmocdLFGc?si=IzMGco38Zv-K7w3L\nLevel 3, Rectangular Dissection: https://youtu.be/-tJhH_k2LaM?si=EgLkd06b7vk9pt9u\nLevel 4, Pyramidal Stacks: https://youtu.be/WidzHiUFWNA?si=kbjqvkXtKaGoNXBO\nLevel 5, Stacking Cuboids: https://youtu.be/VYaEGvClg7Q?si=-2UC2ukk_UnAjQNp\n\nYou can see short versions of a levels 3 and 4 here:\nhttps://youtu.be/30TaNnzEMDo?si=CovsP9ZGUfJ7mnhz\nhttps://youtu.be/PVH5uxHFIGQ?si=_fqLjLTUZtua_YMa\n\nOriginal sources:\nLevel 1: This animation is based on a visual proof by Tom Edgar from the International Journal of Mathematical Education in Science and Technology (https://doi.org/10.1080/0020739X.2022.2131642 ). \nLevel 2: This animation is based on a visual proof by Sidney H. Kung from the May 1989 issue of The College Mathematics Journal  (https://doi.org/10.2307/2686767) page 205.\nLevel 3: This animation is based on (independently discovered) visual proofs by Dan Kalman from the March 1991 issue of The College Mathematics Journal (https://www.jstor.org/stable/2686447) and Martin Gardner from the October 1973 Scientific American (https://www.jstor.org/stable/24923225). \nLevel 4: This animation is based on a visual proof by Man-Keung Siu from the March 1984 issue of Mathematics Magazine (https://www.jstor.org/stable/2689589), page 92.\nLevel 5: This animation is based on a visual proof by Sasho Kalajdzievski from the June 2000 issue of The Mathematical Intelligencer (https://doi.org/10.1007/BF03025282 pages 47--49 ). \n\n#math #manim\u200b #mathvideo\u200b\u200b #sumofsquares\u200b #mtbos\u200b #manim\u200b #animation\u200b #theorem\u200b #pww\u200b #proofwithoutwords\u200b #visualproof\u200b #proof\u200b #iteachmath #finitesums #discretemath #induction\n\nTo learn more about animating with manim, check out:\nhttps://manim.community", "lengthSeconds": 202, "width": 640, "height": 360}
{"video_id": "hP9NMAuwED8", "transcript": [{"start": 0.29, "duration": 9.18, "text": "[Music]"}, {"start": 11.6, "duration": 9.559, "text": "[Music]"}, {"start": 22.68, "duration": 3.0, "text": "do"}], "title": "how surface of area of sphere is calculated (visual proof) || Fs bukhari Official #maths #mathlogic", "description": "how surface of area of sphere is calculated (visual proof) || Fs bukhari Official #maths #mathlogic", "lengthSeconds": 24, "width": 360, "height": 640}
{"video_id": "PBHf9Icpvsg", "transcript": [{"start": 0.0, "duration": 7.62, "text": "\u092e\u0948\u0925\u094d\u0938 \u0915\u093e \u092c\u0939\u0941\u0924 \u0939\u0940 \u092b\u0947\u092e\u0938 \u092b\u0949\u0930\u094d\u092e\u0942\u0932\u093e \u090f + b\u00b2 ="}, {"start": 3.899, "duration": 6.301, "text": "a\u00b2 + b\u00b2 + 2ab \u092f\u0939 \u0906\u0916\u093f\u0930 \u0915\u0939\u093e\u0902 \u0938\u0947 \u0921\u0947\u0930\u093f\u0935 \u0939\u0941\u0906"}, {"start": 7.62, "duration": 3.69, "text": "\u0938\u092e\u091d\u093f\u090f \u0907\u0938 \u0935\u0940\u0921\u093f\u092f\u094b \u0915\u0940 \u0939\u0947\u0932\u094d\u092a \u0938\u0947 \u0935\u0940\u0921\u093f\u092f\u094b \u0915\u094b"}, {"start": 10.2, "duration": 11.199, "text": "\u0932\u093e\u0938\u094d\u091f \u0924\u0915 \u091c\u0930\u0942\u0930 \u0926\u0947\u0916\u093f\u090f"}, {"start": 11.31, "duration": 10.089, "text": "[\u0938\u0902\u0917\u0940\u0924]"}, {"start": 24.86, "duration": 7.92, "text": "[\u0938\u0902\u0917\u0940\u0924]"}], "title": "Visual proof of Maths formula (a + b)2. #chalopadhaikare #shorts #maths", "description": "", "lengthSeconds": 33, "width": 360, "height": 640}
{"video_id": "u0L7rvNumjw", "transcript": [{"start": 0.0, "duration": 3.6, "text": "start with a regular heptagon and draw"}, {"start": 1.86, "duration": 4.08, "text": "another regular heptagon inside with"}, {"start": 3.6, "duration": 4.86, "text": "Area 1 8 of the original this decomposes"}, {"start": 5.94, "duration": 4.92, "text": "the heptagon into eight pieces of size 1"}, {"start": 8.46, "duration": 4.62, "text": "8. one of them a trapezoid at the bottom"}, {"start": 10.86, "duration": 4.26, "text": "represents 1 8. now let's continue this"}, {"start": 13.08, "duration": 5.039, "text": "again on the inner heptagon now we have"}, {"start": 15.12, "duration": 5.7, "text": "another shaded region of 1 8 times 1 8"}, {"start": 18.119, "duration": 5.761, "text": "or 1 8 squared we can continue this"}, {"start": 20.82, "duration": 5.1, "text": "process indefinitely each time we cut up"}, {"start": 23.88, "duration": 4.92, "text": "the heptagon into eight equal area"}, {"start": 25.92, "duration": 5.22, "text": "pieces and take one of them this means"}, {"start": 28.8, "duration": 4.02, "text": "that each piece we have is 1 8 of the"}, {"start": 31.14, "duration": 4.2, "text": "previous 1 8 of the area"}, {"start": 32.82, "duration": 4.5, "text": "if we do this forever we see that this"}, {"start": 35.34, "duration": 4.02, "text": "shaded region now has an area"}, {"start": 37.32, "duration": 5.04, "text": "represented by the sum of the powers of"}, {"start": 39.36, "duration": 5.64, "text": "1 8. but on the other hand we see that"}, {"start": 42.36, "duration": 5.34, "text": "in terms of the full picture this shaded"}, {"start": 45.0, "duration": 5.28, "text": "area is exactly 1 7 of the entire"}, {"start": 47.7, "duration": 6.74, "text": "heptagon therefore the sum of powers of"}, {"start": 50.28, "duration": 4.16, "text": "1 8 is equal to 1 7."}], "title": "Summing powers of 1/8 visually!", "description": "This is a short, animated visual proof showing the sum of the infinite geometric series of powers of 1/8 (starting with 1/8) is 1/7 using self similar trapezoids in a regular heptagon. #manim #math\u200b\u200b #mathshorts\u200b #mathvideo\u200b  #mtbos\u200b #manim\u200b #animation\u200b #theorem\u200b #pww\u200b #proofwithoutwords\u200b #visualproof\u200b #proof\u200b #iteachmath #calculus #series #geometricseries #infiniteseries  #heptagon #trapezoids\n\nHere is an alternate dissection proof for this series: https://youtu.be/wPgwgoj_aKA\n\nHere is a longer form video with lots of related infinite geometric sums: https://youtu.be/JteQEN1XPyc\n\nIf you like this video, consider subscribing to the channel or consider buying me a coffee: https://www.buymeacoffee.com/VisualProofs. Thanks!\n\nThis animation is based on a proof by James Tanton from the March 2008 issue of The College Mathematics Journal page 106. (https://www.jstor.org/stable/27646594).\n\nTo learn more about animating with manim, check out:\nhttps://manim.community", "lengthSeconds": 53, "width": 360, "height": 640}
{"video_id": "tjsAxN2oOGA", "transcript": [{"start": 0.02, "duration": 10.829, "text": "[Muzic\u0103]"}, {"start": 19.4, "duration": 3.06, "text": "[Muzic\u0103]"}, {"start": 28.96, "duration": 3.0, "text": "time"}, {"start": 43.39, "duration": 5.97, "text": "[Muzic\u0103]"}, {"start": 46.36, "duration": 3.0, "text": "temp"}], "title": "(a+b)^3 Visual Proof #visualproof #education #mathematics #algebra #identities #algebraicidentities", "description": "", "lengthSeconds": 47, "width": 360, "height": 640}
{"video_id": "SZsMVGqUiic", "transcript": [{"start": 0.0, "duration": 4.38, "text": "let's represent the sum of the first n"}, {"start": 2.04, "duration": 4.98, "text": "cubes using cubic stacks of Cubes like"}, {"start": 4.38, "duration": 4.98, "text": "this here we pictured 1 cubed plus two"}, {"start": 7.02, "duration": 3.84, "text": "cubed plus three cubed and so on in this"}, {"start": 9.36, "duration": 3.779, "text": "particular instance we stopped at n"}, {"start": 10.86, "duration": 4.92, "text": "equals 4 but we can imagine a similar"}, {"start": 13.139, "duration": 5.761, "text": "stack for any n now we decompose each"}, {"start": 15.78, "duration": 5.22, "text": "cubic stack into an L-shaped flat nomon"}, {"start": 18.9, "duration": 4.08, "text": "the levels of the odd cubes break up"}, {"start": 21.0, "duration": 4.74, "text": "perfectly to fit around the previous"}, {"start": 22.98, "duration": 4.92, "text": "gnomon but the even cubes don't work"}, {"start": 25.74, "duration": 4.56, "text": "quite as well instead we have to take"}, {"start": 27.9, "duration": 4.74, "text": "one level and split it into two separate"}, {"start": 30.3, "duration": 4.62, "text": "pieces as shown here when n is four once"}, {"start": 32.64, "duration": 4.5, "text": "we've broken up each Cube into a flat"}, {"start": 34.92, "duration": 4.02, "text": "nomon we see that all the small cubes"}, {"start": 37.14, "duration": 3.72, "text": "have been arranged in a perfect square"}, {"start": 38.94, "duration": 4.02, "text": "where the side length is given by the"}, {"start": 40.86, "duration": 4.08, "text": "sum of the first and positive integers"}, {"start": 42.96, "duration": 4.259, "text": "this means that the sum of the first n"}, {"start": 44.94, "duration": 6.26, "text": "positive cubes is equal to the square of"}, {"start": 47.219, "duration": 3.981, "text": "the sum of the first n positive integers"}], "title": "Visual Cube Sums! (from Nicomachus)", "description": "This is a short, animated (wordless) visual proof demonstrating the sum of the first n positive cubes by rearranging cubes into a flat square. \n\nIf you like this video, consider subscribing to the channel or consider buying me a coffee: https://www.buymeacoffee.com/VisualProofs. Thanks!\n\nHere is a longer version of this proof : https://youtu.be/YQLicI8R4Gs\n\nHere are three other visual proofs of sum of cubes formulas:\n\nhttps://youtu.be/Ye9OPNqV9FA\nhttps://youtu.be/NxOcT_VKQR0\nhttps://youtu.be/d1yM6Rq7Tfw\n\nThis animation is based on independently discovered, separate visual proofs by J. Barry Love from the March 1977 issue of Mathematics Magazine (https://www.jstor.org/stable/2689727), page 74, and Alan L. Fry from the January 1985 issue of Mathematics Magazine (https://www.jstor.org/stable/2690228) , page 11.\n\nThis is my version of a classic visual proof. Here are other renditions:\nhttps://youtu.be/glearwgR1Ls by   @ThinkTwiceLtu  \nhttps://youtu.be/YG_gS5vZoyQ by   @Mathocube  \nhttps://youtu.be/WY5X_3q80WY by  @Mathologer\n\n#mathshorts\u200b #mathvideo\u200b #math\u200b #calculus #mtbos\u200b #manim\u200b #animation\u200b #theorem\u200b #pww\u200b #proofwithoutwords\u200b #visualproof\u200b #proof\u200b #iteachmath #finitesums #discretemath #calculus #sum #induction \n\nTo learn more about animating with manim, check out:\nhttps://manim.community", "lengthSeconds": 49, "width": 360, "height": 640}
{"video_id": "bg7SxQ1x5rE", "transcript": [{"start": 0.0, "duration": 4.319, "text": "imagine two cubes one with a side length"}, {"start": 2.22, "duration": 4.32, "text": "a and one with a side length B the"}, {"start": 4.319, "duration": 4.441, "text": "number a cubed plus B Cubed is given by"}, {"start": 6.54, "duration": 5.22, "text": "the volume of these two cubes let's add"}, {"start": 8.76, "duration": 5.039, "text": "an A plus b by a by B rectangular prism"}, {"start": 11.76, "duration": 4.92, "text": "to this diagram which encloses a volume"}, {"start": 13.799, "duration": 5.281, "text": "of a plus b times a times B now we can"}, {"start": 16.68, "duration": 4.98, "text": "decompose this region into two different"}, {"start": 19.08, "duration": 4.74, "text": "cuboids one of them the shorter and"}, {"start": 21.66, "duration": 5.279, "text": "deeper green one shown can be rotated"}, {"start": 23.82, "duration": 5.76, "text": "and shifted as pictured notice that"}, {"start": 26.939, "duration": 5.581, "text": "these two cuboids sit inside an A plus b"}, {"start": 29.58, "duration": 5.94, "text": "outer shell and if we look at the top of"}, {"start": 32.52, "duration": 5.4, "text": "this diagram we see an a by a square and"}, {"start": 35.52, "duration": 4.68, "text": "a b by B squared so the area is a"}, {"start": 37.92, "duration": 4.74, "text": "squared plus b squared therefore the"}, {"start": 40.2, "duration": 5.28, "text": "volume of these two shapes is a plus b"}, {"start": 42.66, "duration": 5.04, "text": "times a squared plus b squared but now"}, {"start": 45.48, "duration": 3.78, "text": "we can reshift the cuboids together and"}, {"start": 47.7, "duration": 3.66, "text": "remove the region that we originally"}, {"start": 49.26, "duration": 4.5, "text": "added by subtracting that from both"}, {"start": 51.36, "duration": 4.8, "text": "sides of the equation therefore a cubed"}, {"start": 53.76, "duration": 6.38, "text": "plus B Cubed is equal to a plus b times"}, {"start": 56.16, "duration": 3.98, "text": "a squared plus b squared minus a times B"}], "title": "Factor a sum of cubes", "description": "This is a short, animated visual proof of showing how to factor a sum of two cubes using two actual cubes sitting inside a larger cube. #manim #math #mathvideo #factor #cubes #mathshorts #geometry  #animation #theorem #pww #proofwithoutwords #visualproof #proof #area #algebra #pww\u200b\u200b \u200b #proof\u200b #areas #mathematics\u200b\u200b #mtbos \nIf you like this video, please click \"like\" and consider subscribing and checking out my other videos.\n\nTo buy me a coffee, head over to https://www.buymeacoffee.com/VisualProofs  \nThanks!\n\nHere is the original wide form video without words (that moves a bit slower and with dramatic music): https://youtu.be/BQBnA9d-wAk\n\nHere are two related videos:\nhttps://youtu.be/2_vqruzQzkc\nhttps://youtu.be/IAsaNUUrfj4\n\nTo learn more about animating with manim, check out:\nhttps://manim.community", "lengthSeconds": 58, "width": 360, "height": 640}
{"video_id": "HFGSsPbYc-s", "transcript": [{"start": 2.35, "duration": 8.13, "text": "[Music]"}, {"start": 8.24, "duration": 3.84, "text": "let a and b both be real numbers with a"}, {"start": 10.48, "duration": 3.76, "text": "greater than or equal to b and b greater"}, {"start": 12.08, "duration": 4.4, "text": "than or equal to zero consider a line"}, {"start": 14.24, "duration": 3.68, "text": "segment with length a"}, {"start": 16.48, "duration": 4.4, "text": "and then overlay a line segment of"}, {"start": 17.92, "duration": 4.64, "text": "length b like this"}, {"start": 20.88, "duration": 3.84, "text": "use half of the uncovered length of the"}, {"start": 22.56, "duration": 2.959, "text": "first segment a length of a minus b over"}, {"start": 24.72, "duration": 2.96, "text": "two"}, {"start": 25.519, "duration": 4.571, "text": "to create a circle with a radius of a"}, {"start": 27.68, "duration": 5.52, "text": "minus b over two"}, {"start": 30.09, "duration": 5.27, "text": "[Music]"}, {"start": 33.2, "duration": 3.6, "text": "the yellow line segment created here"}, {"start": 35.36, "duration": 3.44, "text": "connecting the end of the line segment"}, {"start": 36.8, "duration": 4.88, "text": "to the center of the circle has length a"}, {"start": 38.8, "duration": 2.88, "text": "plus b over 2."}, {"start": 41.92, "duration": 3.76, "text": "draw a radius perpendicular to the line"}, {"start": 43.92, "duration": 3.76, "text": "segment and connect the endpoints of"}, {"start": 45.68, "duration": 3.52, "text": "this radius and the end of the extended"}, {"start": 47.68, "duration": 3.12, "text": "line segment"}, {"start": 49.2, "duration": 3.28, "text": "the resulting triangle is a right"}, {"start": 50.8, "duration": 3.68, "text": "triangle so we can use the pythagorean"}, {"start": 52.48, "duration": 3.759, "text": "theorem to determine that the hypotenuse"}, {"start": 54.48, "duration": 4.0, "text": "has a length of square root of a minus b"}, {"start": 56.239, "duration": 4.32, "text": "over 2 squared plus a plus b over two"}, {"start": 58.48, "duration": 3.759, "text": "squared which simplifies to the square"}, {"start": 60.559, "duration": 3.921, "text": "root of a squared plus b squared over"}, {"start": 62.239, "duration": 2.241, "text": "two"}, {"start": 64.799, "duration": 2.801, "text": "next draw the tangent line to the circle"}, {"start": 66.4, "duration": 3.28, "text": "that passes through the endpoint of the"}, {"start": 67.6, "duration": 3.68, "text": "extended line segment and draw a radius"}, {"start": 69.68, "duration": 4.32, "text": "to this point creating another right"}, {"start": 71.28, "duration": 4.56, "text": "triangle highlighted here"}, {"start": 74.0, "duration": 3.2, "text": "we again use the pythagorean theorem to"}, {"start": 75.84, "duration": 3.44, "text": "determine that the missing leg has a"}, {"start": 77.2, "duration": 4.4, "text": "length of square root of a plus b over 2"}, {"start": 79.28, "duration": 4.4, "text": "squared minus the quantity a minus b"}, {"start": 81.6, "duration": 5.68, "text": "over 2 squared which simplifies to the"}, {"start": 83.68, "duration": 5.52, "text": "square root of a times b"}, {"start": 87.28, "duration": 3.36, "text": "finally drop a perpendicular from the"}, {"start": 89.2, "duration": 3.279, "text": "tangent point on the circle to the"}, {"start": 90.64, "duration": 3.68, "text": "extended line segment"}, {"start": 92.479, "duration": 3.361, "text": "one of the newly created triangles in"}, {"start": 94.32, "duration": 4.32, "text": "the subdivision is similar to the"}, {"start": 95.84, "duration": 2.8, "text": "original triangle"}, {"start": 98.72, "duration": 3.6, "text": "we can see that they're similar because"}, {"start": 100.4, "duration": 4.399, "text": "they're both right triangles and they"}, {"start": 102.32, "duration": 4.079, "text": "share one of the other angles"}, {"start": 104.799, "duration": 3.68, "text": "using the similarity of these triangles"}, {"start": 106.399, "duration": 4.08, "text": "we can find that the leg labeled x"}, {"start": 108.479, "duration": 3.6, "text": "satisfies the equality x over the square"}, {"start": 110.479, "duration": 4.32, "text": "root of a times b equals the square root"}, {"start": 112.079, "duration": 5.36, "text": "of a times b over a plus b over two"}, {"start": 114.799, "duration": 5.511, "text": "we can solve for x to see that x is two"}, {"start": 117.439, "duration": 5.521, "text": "times a times b over a plus b"}, {"start": 120.31, "duration": 4.73, "text": "[Music]"}, {"start": 122.96, "duration": 4.32, "text": "working backward we see that two times a"}, {"start": 125.04, "duration": 3.999, "text": "times b divided by a plus b is the leg"}, {"start": 127.28, "duration": 4.319, "text": "of a right triangle with hypotenuse the"}, {"start": 129.039, "duration": 4.641, "text": "square root of a times b so two a b over"}, {"start": 131.599, "duration": 4.0, "text": "a plus b is less than or equal to the"}, {"start": 133.68, "duration": 3.52, "text": "square root of a times b but the square"}, {"start": 135.599, "duration": 3.28, "text": "root of a times b is the leg of a right"}, {"start": 137.2, "duration": 3.28, "text": "triangle with hypotenuse a plus b over"}, {"start": 138.879, "duration": 3.921, "text": "two so the square root of a times b is"}, {"start": 140.48, "duration": 4.32, "text": "less than or equal to a plus b over two"}, {"start": 142.8, "duration": 3.6, "text": "and then a plus b over two is a leg and"}, {"start": 144.8, "duration": 3.04, "text": "a right triangle with hypotenuse the"}, {"start": 146.4, "duration": 3.04, "text": "square root of a squared plus b squared"}, {"start": 147.84, "duration": 3.679, "text": "over two"}, {"start": 149.44, "duration": 3.2, "text": "so a plus b over two is less than or"}, {"start": 151.519, "duration": 2.881, "text": "equal to the square root of a squared"}, {"start": 152.64, "duration": 4.239, "text": "plus b squared over two"}, {"start": 154.4, "duration": 3.36, "text": "two a b over a plus b is the harmonic"}, {"start": 156.879, "duration": 2.561, "text": "mean"}, {"start": 157.76, "duration": 4.0, "text": "the square root of a times b is the"}, {"start": 159.44, "duration": 4.48, "text": "geometric mean and a plus b over 2 is"}, {"start": 161.76, "duration": 3.6, "text": "the arithmetic mean and the square root"}, {"start": 163.92, "duration": 3.599, "text": "of a squared plus b squared over 2 is"}, {"start": 165.36, "duration": 3.68, "text": "the root mean squared the resulting set"}, {"start": 167.519, "duration": 3.44, "text": "of inequalities is known as the mean"}, {"start": 169.04, "duration": 3.52, "text": "inequalities we note that we get"}, {"start": 170.959, "duration": 3.681, "text": "equality throughout if and only if a"}, {"start": 172.56, "duration": 4.08, "text": "equals b in this case the entire picture"}, {"start": 174.64, "duration": 4.9, "text": "collapses to a single line segment"}, {"start": 176.64, "duration": 6.9, "text": "representing all of the values"}, {"start": 179.54, "duration": 4.0, "text": "[Music]"}, {"start": 190.959, "duration": 2.081, "text": "you"}], "title": "Mean Inequalities (visual proof of AM-GM, HM-GM, and AM-QM inequalities)", "description": "This short animated proof demonstrates the two variable mean inequalities for the arithmetic mean, geometric mean, harmonic mean and root mean square (or quadratic mean). #math\u200b \u200b#manim\u200b #animation\u200b #theorem\u200b #pww\u200b #proofwithoutwords\u200b #visualproof\u200b #proof\u200b #inequality #iteachmath #mathematics #pythagoreantheorem #similartriangles #means #meaninequality #mathshorts\u200b #mathvideo\u200b #mtbos\n\nThis animation of the inequalities is based on a visual proof by Roger Nelsen from the June 1987 issue of Mathematics Magazine (https://www.jstor.org/stable/2689561) p. 158.\n\nIf you like visual proofs of inequalities, see these links:\n\nAM-GM inequality via rectangles/squares: https://youtu.be/62G9fak1vyk\n\nAM-GM inequality via circles: https://youtu.be/Pt8IX_Q5U1A\n\nAM-GM-HM-QM via Moments: https://youtu.be/XJ3643WhKjY\n\nTo learn more about animating with manim, check out:\nhttps://manim.community", "lengthSeconds": 191, "width": 640, "height": 360}
{"video_id": "enzxA-5MCGQ", "transcript": [{"start": 0.199, "duration": 4.521, "text": "these Stacks each represent sums of n"}, {"start": 2.639, "duration": 4.76, "text": "consecutive positive integers the first"}, {"start": 4.72, "duration": 4.52, "text": "one from 1 to n the second from 2 to n +"}, {"start": 7.399, "duration": 4.36, "text": "1 all the way up to the last one which"}, {"start": 9.24, "duration": 4.359, "text": "goes from n up to 2 N minus one when we"}, {"start": 11.759, "duration": 3.96, "text": "glue these Stacks together we get this"}, {"start": 13.599, "duration": 4.52, "text": "interesting shape here we can take a"}, {"start": 15.719, "duration": 5.281, "text": "second copy of this shape and rotate it"}, {"start": 18.119, "duration": 4.961, "text": "twice once the second copy is rotated"}, {"start": 21.0, "duration": 4.119, "text": "appropriately we see that the two copies"}, {"start": 23.08, "duration": 4.72, "text": "fit together perfectly to form a cuboid"}, {"start": 25.119, "duration": 4.16, "text": "with Dimensions n by n by 2 N that means"}, {"start": 27.8, "duration": 5.24, "text": "the number of cubes in the original"}, {"start": 29.279, "duration": 7.001, "text": "stack we started with is 12 * 2 n * n *"}, {"start": 33.04, "duration": 5.679, "text": "n this equals n cubed this means that"}, {"start": 36.28, "duration": 4.279, "text": "the sum of the consecutive sums of n"}, {"start": 38.719, "duration": 4.16, "text": "positive integers that we started with"}, {"start": 40.559, "duration": 4.241, "text": "equals n cubed so that we've uncovered"}, {"start": 42.879, "duration": 4.881, "text": "an interesting double sum formula for"}, {"start": 44.8, "duration": 2.96, "text": "cubic numbers"}], "title": "Double sum produces cubes", "description": "In this video, we demonstrate how each integer cube is a special double sum.  In particular each cube is a sum of n sums of n consecutive integers. \n\nIf you like this video, consider subscribing to the channel or consider buying me a coffee: https://www.buymeacoffee.com/VisualProofs. Thanks!\n\nTo see this animation a bit slower (and wordless), check out: https://youtu.be/_dltGrVlW3w\n\nHere are a few other related visual proofs:\nhttps://youtu.be/OECW5qqZN9I\nhttps://youtu.be/936eKowyvR4\nhttps://youtu.be/WidzHiUFWNA\nhttps://youtu.be/BBrtHbRcpXs\nhttps://youtu.be/YQLicI8R4Gs\n\nThis animation is based on a visual proof by Roger B. Nelsen from the March 2002 issue of the College Mathematics Journal  (https://www.jstor.org/stable/1559005 page 171 ). \n\nTo learn more about animating with manim, check out:\nhttps://manim.community\n\n#math #manim\u200b #mathvideo\u200b\u200b #cubes\u200b #mtbos\u200b #manim\u200b #animation\u200b #theorem\u200b #pww\u200b #proofwithoutwords\u200b #visualproof\u200b #proof\u200b #iteachmath #finitesums #discretemath #induction #doublesum", "lengthSeconds": 46, "width": 360, "height": 640}
{"video_id": "sZ5fR0gWjbQ", "transcript": [{"start": 0.08, "duration": 4.12, "text": "start with an equilateral triangle and"}, {"start": 2.04, "duration": 4.239, "text": "cut it into nine congruent equilateral"}, {"start": 4.2, "duration": 3.84, "text": "triangles and shade four of them thus"}, {"start": 6.279, "duration": 3.921, "text": "shading 49ths of the area of the"}, {"start": 8.04, "duration": 4.16, "text": "equilateral triangle now repeat this"}, {"start": 10.2, "duration": 4.08, "text": "process on the top four nths cutting it"}, {"start": 12.2, "duration": 3.919, "text": "into nine pieces and shading four of"}, {"start": 14.28, "duration": 4.4, "text": "them so that we've shaded four nths of"}, {"start": 16.119, "duration": 5.08, "text": "four kns of the area we can repeat this"}, {"start": 18.68, "duration": 4.48, "text": "process over and over again each time"}, {"start": 21.199, "duration": 3.84, "text": "taking the unshaded top four kns from"}, {"start": 23.16, "duration": 4.119, "text": "before cutting it into nine pieces and"}, {"start": 25.039, "duration": 3.961, "text": "shading four of them if we imagine"}, {"start": 27.279, "duration": 3.521, "text": "carrying this process out infinitely"}, {"start": 29.0, "duration": 4.48, "text": "often then we get a diagram that"}, {"start": 30.8, "duration": 4.72, "text": "represents the infinite sum the infinite"}, {"start": 33.48, "duration": 4.32, "text": "sum is a limit so in this limiting"}, {"start": 35.52, "duration": 5.08, "text": "diagram we've shaded the infinite sum of"}, {"start": 37.8, "duration": 5.12, "text": "the powers of 49ths on the other hand if"}, {"start": 40.6, "duration": 4.52, "text": "we look at each level along the way we"}, {"start": 42.92, "duration": 4.36, "text": "see that in each level we've shaded four"}, {"start": 45.12, "duration": 4.72, "text": "of five congruent triangles therefore"}, {"start": 47.28, "duration": 6.4, "text": "the infinite sum of the powers of 49ths"}, {"start": 49.84, "duration": 3.84, "text": "must equal 4 fths"}], "title": "Infinite sum of powers of 4/9", "description": "This is a short, animated visual proof demonstrating the sum of the infinite geometric series with ratio 4/9 using a decomposition of an equilateral triangle. \n\nIf you like this video, consider subscribing to the channel or consider buying me a coffee: https://www.buymeacoffee.com/VisualProofs. Thanks! \n\nThis animation is based on a visual proof by Tom Edgar from the June 2016 issue of Mathematics Magazine page 191 (https://doi.org/10.4169/math.mag.89.3.191).\n\n#mathshorts\u200b #mathvideo\u200b #math\u200b #calculus\n#mtbos\u200b #manim\u200b #animation\u200b #theorem\u200b #pww\u200b #proofwithoutwords\u200b #visualproof\u200b #proof\u200b #iteachmath #geometricseries #infiniteseries #fractals #sum #induction\n\nTo learn more about animating with manim, check out:\nhttps://manim.community", "lengthSeconds": 52, "width": 360, "height": 640}
{"video_id": "NUwRR-Fsxnw", "transcript": [{"start": 0.59, "duration": 10.39, "text": "[Music]"}, {"start": 26.15, "duration": 5.81, "text": "[Music]"}, {"start": 28.96, "duration": 3.0, "text": "he"}, {"start": 32.87, "duration": 3.02, "text": "[Music]"}], "title": "(a+b)\u00b2 Expanded Visual #Proof", "description": "This video was made with Clipchamp\n#clipchamp #proof #math", "lengthSeconds": 45, "width": 270, "height": 480}
{"video_id": "DBYXbjyw9sc", "transcript": [{"start": 0.0, "duration": 5.66, "text": "[Music]"}, {"start": 10.58, "duration": 8.999, "text": "thank you"}, {"start": 12.69, "duration": 6.889, "text": "[Music]"}, {"start": 22.87, "duration": 12.95, "text": "[Music]"}, {"start": 39.3, "duration": 8.28, "text": "foreign"}, {"start": 40.49, "duration": 7.09, "text": "[Music]"}, {"start": 50.5, "duration": 3.229, "text": "[Music]"}, {"start": 58.64, "duration": 3.06, "text": "[Music]"}], "title": "Sum of First n Even Integers | Visual Proof | Geometric Proof | Proofs Without WORDs", "description": "Thanks for watching!\n\n#shorts #viral #viralshorts #shortsvideo #mathtricks #math #mathematics #mathideas #mathanimation #animation #animated #visualization #geometric #proof #proofwithoutword #algebric #algebra #algebricequation #algebricexpression #visualproof #geometricproof #amazingmath #funfacts #mathfacts #abstractart #abstractmath #mathvisualization #mathart #geometry #shapes #equation #expression #square #proofing #proofof #proofofconcept #amazingahorts #amazingfacts #factshorts #intution #intutive #funmath #mathideas #mathworld #magicmath #magicmathtricks #mathmagic #mathzone", "lengthSeconds": 60, "width": 360, "height": 640}
{"video_id": "M1dQNwyBjg0", "transcript": [{"start": 0.12, "duration": 3.24, "text": "to prove that the set of integers is"}, {"start": 1.8, "duration": 3.06, "text": "accountable set we have to find a"}, {"start": 3.36, "duration": 3.3, "text": "one-to-one and onto function from the"}, {"start": 4.86, "duration": 3.42, "text": "natural numbers to the integers to do"}, {"start": 6.66, "duration": 3.66, "text": "that we imagine the integers form a"}, {"start": 8.28, "duration": 4.5, "text": "doubly infinite line of dots and we"}, {"start": 10.32, "duration": 4.319, "text": "start at zero and spiral out like this"}, {"start": 12.78, "duration": 3.78, "text": "we see that every integer will be hit"}, {"start": 14.639, "duration": 3.961, "text": "once and only once to make this more"}, {"start": 16.56, "duration": 5.16, "text": "formal we can Define the function f of x"}, {"start": 18.6, "duration": 5.46, "text": "equal to X plus one over two if x is odd"}, {"start": 21.72, "duration": 3.84, "text": "and negative X over 2 if x is even and"}, {"start": 24.06, "duration": 3.12, "text": "this gives the required one to one and"}, {"start": 25.56, "duration": 2.82, "text": "onto map from the natural numbers to the"}, {"start": 27.18, "duration": 3.179, "text": "integers"}, {"start": 28.38, "duration": 3.899, "text": "we can do something similar to prove"}, {"start": 30.359, "duration": 3.481, "text": "that Z Cross C is a countable set we"}, {"start": 32.279, "duration": 3.421, "text": "plot the integer coordinate points in"}, {"start": 33.84, "duration": 4.559, "text": "the real plane as shown beginning at"}, {"start": 35.7, "duration": 4.98, "text": "zero zero Des McHale showed this spiral"}, {"start": 38.399, "duration": 3.901, "text": "which will eventually visit every single"}, {"start": 40.68, "duration": 4.379, "text": "integer coordinate point in the real"}, {"start": 42.3, "duration": 5.099, "text": "plane therefore this spiral proves the"}, {"start": 45.059, "duration": 4.261, "text": "existence of a one to one and onto map"}, {"start": 47.399, "duration": 4.201, "text": "from the natural numbers to pairs of"}, {"start": 49.32, "duration": 4.02, "text": "integers can you find a formula for the"}, {"start": 51.6, "duration": 4.279, "text": "one to one and onto map guaranteed by"}, {"start": 53.34, "duration": 2.539, "text": "this spiral"}], "title": "Two Countable Sets", "description": "This is a short animation with two animations showing that the set of integers is a countable set (in bijection with the set of Natural numbers) and that the set of ordered pairs of integers is also a countable set. \n\nIf you like this video, consider subscribing to the channel or consider buying me a coffee: https://www.buymeacoffee.com/VisualProofs. Thanks!\n\nThis animation is based on a visual proof by Des MacHale from the February 2004 issue of Mathematics Magazine  (https://www.jstor.org/stable/3219231 page 55).\n\nHere is a related video proving that function composition is associative:\nhttps://youtu.be/mCnjpnhLeDU\n\n#shorts #visualproof #mathshorts\u200b #mathvideo\u200b #math\u200b #bijection #mtbos\u200b #manim\u200b #animation\u200b #theorem\u200b #pww\u200b #proofwithoutwords\u200b #visualproof\u200b #proof\u200b #iteachmath #function #injection #surjectivefunction #injectivefunction #surjective #injective  #naturalnumbers #integers #countableset #infinity \n\nTo learn more about animating with manim, check out:\nhttps://manim.community", "lengthSeconds": 55, "width": 360, "height": 640}
{"video_id": "e5wtveGt_tY", "transcript": [{"start": 0.16, "duration": 4.199, "text": "let's play the chaos game in a pentagon"}, {"start": 2.44, "duration": 3.68, "text": "first start with a DOT anywhere inside"}, {"start": 4.359, "duration": 4.561, "text": "the Pentagon then we're going to iterate"}, {"start": 6.12, "duration": 3.639, "text": "a procedure where at any given stage we"}, {"start": 8.92, "duration": 3.0, "text": "move"}, {"start": 9.759, "duration": 4.8, "text": "0.618 of the distance between the"}, {"start": 11.92, "duration": 4.52, "text": "current dot and a randomly chosen vertex"}, {"start": 14.559, "duration": 4.201, "text": "on the Pentagon think of rolling an"}, {"start": 16.44, "duration": 4.24, "text": "imaginary five-sided die and then moving"}, {"start": 18.76, "duration": 3.519, "text": "towards the corresponding vertex what"}, {"start": 20.68, "duration": 4.599, "text": "happens when we iterate this process"}, {"start": 22.279, "duration": 4.881, "text": "30,000 times in this picture we've"}, {"start": 25.279, "duration": 3.84, "text": "shaded the dots corresponding to the"}, {"start": 27.16, "duration": 5.04, "text": "vertex to which we move at any given"}, {"start": 29.119, "duration": 5.28, "text": "stage after 30,000 iterations if we"}, {"start": 32.2, "duration": 4.12, "text": "throw away the first few iterations we"}, {"start": 34.399, "duration": 4.041, "text": "see that the limiting shape will always"}, {"start": 36.32, "duration": 4.52, "text": "look like this diagram no matter where"}, {"start": 38.44, "duration": 4.68, "text": "we started this is an amazing fractal"}, {"start": 40.84, "duration": 4.84, "text": "generated from a simple process note"}, {"start": 43.12, "duration": 6.599, "text": "that the number 0.618 is special it's"}, {"start": 45.68, "duration": 4.039, "text": "approximately one over the golden ratio"}], "title": "Pentagonal Chaos Game", "description": "In this short, we show what happens when iterating the procedure of choosing a pentagon vertex at random and moving 0.168 (golden ratio reciprocal) the distance from the current dot to the chosen point. \n\nIf you like this video, consider subscribing to the channel or consider buying me a coffee: https://www.buymeacoffee.com/VisualProofs. Thanks!\n\nIf you want to know more about the Chaos game, see the following links:\nhttps://en.wikipedia.org/wiki/Chaos_game\nhttps://mathworld.wolfram.com/ChaosGa...\nhttps://math.bu.edu/DYSYS/chaos-game/...\nhttps://youtu.be/kbKtFN71Lfs\nhttps://youtu.be/IGlGvSXkRGI\nhttps://youtube.com/shorts/yDHkNDrHfIw\n\n#chaos #chaosgame #pentagon #mathvideo\u200b #math\u200b #mtbos\u200b #manim\u200b #animation\u200b #theorem\u200b\u200b\u200b #iteachmath #mathematics #dynamicalsystems  #iteratedfunctionsystem #dynamics #fractal #fractals \n\nTo learn more about animating with manim, check out:\nhttps://manim.community", "lengthSeconds": 48, "width": 360, "height": 640}
{"video_id": "8MyXwsDvHnc", "transcript": [{"start": 0.0, "duration": 3.72, "text": "here's a way to make geometric sense of"}, {"start": 2.22, "duration": 3.599, "text": "the infinite sum of the alternating"}, {"start": 3.72, "duration": 4.139, "text": "powers of one-third start with a one by"}, {"start": 5.819, "duration": 4.081, "text": "one square of area one cut it into three"}, {"start": 7.859, "duration": 3.78, "text": "pieces and remove one of those now cut"}, {"start": 9.9, "duration": 4.739, "text": "the diagram into three pieces vertically"}, {"start": 11.639, "duration": 5.341, "text": "and use that to re-add one third of the"}, {"start": 14.639, "duration": 4.921, "text": "missing area using these triangular"}, {"start": 16.98, "duration": 4.5, "text": "pieces now we repeat the process on the"}, {"start": 19.56, "duration": 3.9, "text": "Central Square created by the first two"}, {"start": 21.48, "duration": 4.2, "text": "divisions we cut that square into three"}, {"start": 23.46, "duration": 4.26, "text": "pieces remove one third and re-add"}, {"start": 25.68, "duration": 4.2, "text": "one-third with triangular diagrams we"}, {"start": 27.72, "duration": 3.6, "text": "continue this process indefinitely each"}, {"start": 29.88, "duration": 3.3, "text": "time decomposing the Central Square"}, {"start": 31.32, "duration": 4.98, "text": "which has one-ninth area of the previous"}, {"start": 33.18, "duration": 4.86, "text": "large Square using this process Shades"}, {"start": 36.3, "duration": 4.259, "text": "the alternating sum of the powers of"}, {"start": 38.04, "duration": 4.92, "text": "one-third on the other hand we see that"}, {"start": 40.559, "duration": 4.5, "text": "the Shaded area represents exactly three"}, {"start": 42.96, "duration": 3.96, "text": "quarters of the entire Square therefore"}, {"start": 45.059, "duration": 3.721, "text": "the infinite sum of the powers of"}, {"start": 46.92, "duration": 4.26, "text": "negative one-third is equal to"}, {"start": 48.78, "duration": 4.75, "text": "three-fourths"}, {"start": 51.18, "duration": 2.35, "text": "[Music]"}], "title": "Alternating infinite sum!", "description": "This is a short, animated visual proof demonstrating the sum of the infinite geometric series with ratio -1/3. #math\u200b #calculus #mtbos\u200b #manim\u200b #animation\u200b #theorem\u200b #pww\u200b #proofwithoutwords\u200b #visualproof\u200b #proof\u200b #iteachmath #geometricseries #infiniteseries #fractals #sum #induction #mathshorts\u200b #mathvideo\u200b \n\nIf you like this animation, please consider subscribing.\n\nHere is my original version (longer, wide format, with no words): https://youtu.be/MyCn7gIPxNg\n\nThis animation is based on a visual proof by Hasan Unal from the January 2009 issue of The College Math Journal page 39 (https://www.jstor.org/stable/27646717).\n \nHere is an alternating series dissection for powers of -1/2: https://youtu.be/NKFrHfBGDNE\n\nAfter I made this video, I found the proof animated very nicely on Think Twice's channel too: https://youtu.be/re3Xh74M8g0 \n\nTo learn more about animating with manim, check out:\nhttps://manim.community", "lengthSeconds": 52, "width": 360, "height": 640}
{"video_id": "7WKbDyWbxGk", "transcript": [{"start": 0.0, "duration": 3.0, "text": "foreign"}, {"start": 7.2, "duration": 3.3, "text": "start with a length one line segment and"}, {"start": 9.24, "duration": 2.64, "text": "use a straight edge Encompass to"}, {"start": 10.5, "duration": 3.24, "text": "construct a perpendicular length one"}, {"start": 11.88, "duration": 3.959, "text": "line segment connect the endpoints like"}, {"start": 13.74, "duration": 4.619, "text": "this and label the length of the"}, {"start": 15.839, "duration": 4.261, "text": "hypotenuse of this right triangle by a"}, {"start": 18.359, "duration": 3.721, "text": "because the triangle satisfies the"}, {"start": 20.1, "duration": 3.24, "text": "Pythagorean theorem we see that a equals"}, {"start": 22.08, "duration": 3.539, "text": "the square root of two"}, {"start": 23.34, "duration": 4.56, "text": "now use the Straight Edge Encompass to"}, {"start": 25.619, "duration": 3.9, "text": "draw a line segment of length one that"}, {"start": 27.9, "duration": 4.02, "text": "is perpendicular to the diagonal of"}, {"start": 29.519, "duration": 5.22, "text": "length root 2. connect the endpoints"}, {"start": 31.92, "duration": 4.74, "text": "again and now we have a new length a and"}, {"start": 34.739, "duration": 3.901, "text": "this time by the Pythagorean theorem we"}, {"start": 36.66, "duration": 3.96, "text": "can determine that a squared is 3 so"}, {"start": 38.64, "duration": 4.2, "text": "that a is the square root of 3. let's"}, {"start": 40.62, "duration": 3.599, "text": "repeat this process one more time use"}, {"start": 42.84, "duration": 3.059, "text": "the Straight Edge Encompass to create a"}, {"start": 44.219, "duration": 2.941, "text": "circle of radius 1 and then a line"}, {"start": 45.899, "duration": 3.0, "text": "segment of length one that is"}, {"start": 47.16, "duration": 3.78, "text": "perpendicular to the line segment of"}, {"start": 48.899, "duration": 3.601, "text": "length root 3. connect the endpoints"}, {"start": 50.94, "duration": 3.0, "text": "like this and use the Pythagorean"}, {"start": 52.5, "duration": 2.82, "text": "theorem to determine that the length of"}, {"start": 53.94, "duration": 2.959, "text": "the hypotenuse of that new right"}, {"start": 55.32, "duration": 4.259, "text": "triangle is the square root of four"}, {"start": 56.899, "duration": 4.541, "text": "using a straight edge Encompass we have"}, {"start": 59.579, "duration": 4.14, "text": "constructed line segments of length"}, {"start": 61.44, "duration": 4.44, "text": "square root of 2 square root of 3 and"}, {"start": 63.719, "duration": 5.58, "text": "square root of 4 but there's no reason"}, {"start": 65.88, "duration": 5.58, "text": "to stop there we can keep applying the"}, {"start": 69.299, "duration": 4.801, "text": "exact same process over and over again"}, {"start": 71.46, "duration": 4.56, "text": "to the new links that we create and we"}, {"start": 74.1, "duration": 4.019, "text": "see that we can create line segments of"}, {"start": 76.02, "duration": 5.66, "text": "length square root of 5 square root of 6"}, {"start": 78.119, "duration": 3.561, "text": "and square root of 7."}, {"start": 82.68, "duration": 5.52, "text": "how far can we take this process can we"}, {"start": 85.619, "duration": 5.04, "text": "just keep going forever the object shown"}, {"start": 88.2, "duration": 5.04, "text": "here is the first six stages of the"}, {"start": 90.659, "duration": 4.981, "text": "Spiral of theodorus"}, {"start": 93.24, "duration": 5.22, "text": "the spiral can be extended indefinitely"}, {"start": 95.64, "duration": 5.58, "text": "but just for our purposes let's take it"}, {"start": 98.46, "duration": 4.979, "text": "out to I don't know how about n equals"}, {"start": 101.22, "duration": 5.16, "text": "one thousand we'll see the first 1000"}, {"start": 103.439, "duration": 5.22, "text": "levels of the Spiral of theodorus and"}, {"start": 106.38, "duration": 5.34, "text": "along the way we're constructing the"}, {"start": 108.659, "duration": 6.661, "text": "square root of n for every single n"}, {"start": 111.72, "duration": 6.06, "text": "value between two and a thousand"}, {"start": 115.32, "duration": 5.04, "text": "once we get to the end of our spiral as"}, {"start": 117.78, "duration": 4.74, "text": "we've done right here we see that the"}, {"start": 120.36, "duration": 4.14, "text": "final line that we've drawn has a length"}, {"start": 122.52, "duration": 4.02, "text": "of size square root of one thousand"}, {"start": 124.5, "duration": 4.2, "text": "for appearances we can construct the"}, {"start": 126.54, "duration": 4.74, "text": "circle with a radius of square root 1000"}, {"start": 128.7, "duration": 5.16, "text": "and draw a radius in the same direction"}, {"start": 131.28, "duration": 4.44, "text": "as our original length one line segment"}, {"start": 133.86, "duration": 3.54, "text": "while this may not be the most efficient"}, {"start": 135.72, "duration": 4.2, "text": "way to get the length of square root of"}, {"start": 137.4, "duration": 4.32, "text": "1000 using this process we can construct"}, {"start": 139.92, "duration": 4.02, "text": "the square root of n for any positive"}, {"start": 141.72, "duration": 4.32, "text": "integer n this means that the spiral of"}, {"start": 143.94, "duration": 4.2, "text": "theodorus constructs square roots and"}, {"start": 146.04, "duration": 3.3, "text": "shows us how to prove that square roots"}, {"start": 148.14, "duration": 5.06, "text": "are constructable using Straight Edge"}, {"start": 149.34, "duration": 3.86, "text": "Encompass using proof by induction"}], "title": "Spiral of Theodorus Constructs Square Roots (visual proof; straightedge and compass)", "description": "This is a short, animated visual proof demonstrating how to construct square roots of any positive integer using the Spiral of Theodorus #manim #math\u200b\u200b #mathshorts\u200b #mathvideo\u200b  #trisect #construction #geometry #mtbos\u200b #manim\u200b #animation\u200b #theorem\u200b #pww\u200b #proofwithoutwords\u200b #visualproof\u200b #proof\u200b #iteachmath \n\n\nTo learn more about animating with manim, check out:\nhttps://manim.community\n\n___________________________________________________________\nMusic in this video:\nGlimpsing Infinity by Asher Fulero", "lengthSeconds": 153, "width": 640, "height": 360}
{"video_id": "lRDtdIyFhpE", "transcript": [{"start": 0.38, "duration": 2.68, "text": "[Music]"}, {"start": 0.78, "duration": 8.46, "text": "foreign"}, {"start": 3.06, "duration": 8.4, "text": "[Music]"}, {"start": 9.24, "duration": 5.88, "text": "start with a triangle with sides of"}, {"start": 11.46, "duration": 5.88, "text": "length a b and c"}, {"start": 15.12, "duration": 4.2, "text": "the angle bisectors of this triangle"}, {"start": 17.34, "duration": 4.199, "text": "meet at a point"}, {"start": 19.32, "duration": 4.619, "text": "this point is known as the in center of"}, {"start": 21.539, "duration": 4.5, "text": "the triangle and it's also the center of"}, {"start": 23.939, "duration": 4.021, "text": "a circle known as The Inn circle of the"}, {"start": 26.039, "duration": 3.9, "text": "triangle which is a circle that is"}, {"start": 27.96, "duration": 3.54, "text": "tangent to all three sides and the"}, {"start": 29.939, "duration": 4.441, "text": "largest Circle that fits inside the"}, {"start": 31.5, "duration": 5.04, "text": "triangle if we draw the radii that are"}, {"start": 34.38, "duration": 4.32, "text": "perpendicular to each of the sides we"}, {"start": 36.54, "duration": 4.019, "text": "get a shape that looks like this and we"}, {"start": 38.7, "duration": 4.74, "text": "can fill the six regions with different"}, {"start": 40.559, "duration": 5.221, "text": "colored triangles"}, {"start": 43.44, "duration": 4.92, "text": "now let's unfold these triangular"}, {"start": 45.78, "duration": 5.279, "text": "regions so that they all lay out in a"}, {"start": 48.36, "duration": 5.219, "text": "line of triangles like this with bases"}, {"start": 51.059, "duration": 4.98, "text": "that are all parallel"}, {"start": 53.579, "duration": 5.941, "text": "if we take a second copy of this shape"}, {"start": 56.039, "duration": 6.66, "text": "flip it upside down and then shift the"}, {"start": 59.52, "duration": 6.26, "text": "triangles as we see here we end up with"}, {"start": 62.699, "duration": 6.001, "text": "a nice rectangular shape"}, {"start": 65.78, "duration": 5.379, "text": "in fact we can find the dimensions of"}, {"start": 68.7, "duration": 4.02, "text": "this rectangle using the original labels"}, {"start": 71.159, "duration": 3.541, "text": "of the triangle"}, {"start": 72.72, "duration": 4.14, "text": "as shown in the picture"}, {"start": 74.7, "duration": 6.239, "text": "we see that the length of the rectangle"}, {"start": 76.86, "duration": 6.36, "text": "is a plus b plus C while the width of"}, {"start": 80.939, "duration": 5.521, "text": "the rectangle is the in radius length R"}, {"start": 83.22, "duration": 5.759, "text": "but now we see a nice fact 2 times the"}, {"start": 86.46, "duration": 5.22, "text": "area of the original triangle must be"}, {"start": 88.979, "duration": 5.46, "text": "equal to R the length of the in radius"}, {"start": 91.68, "duration": 4.74, "text": "times a plus b plus C the length of the"}, {"start": 94.439, "duration": 4.261, "text": "perimeter but this means that the area"}, {"start": 96.42, "duration": 5.339, "text": "of the triangle is equal to R times a"}, {"start": 98.7, "duration": 5.52, "text": "plus b plus C over 2 which is R times s"}, {"start": 101.759, "duration": 4.261, "text": "where s is known as the semiperimeter so"}, {"start": 104.22, "duration": 3.3, "text": "we've proved that the area of a triangle"}, {"start": 106.02, "duration": 4.22, "text": "is the product of the radius in the"}, {"start": 107.52, "duration": 2.72, "text": "semiperimeter"}], "title": "Triangle Area from Inradius and Semiperimeter (visual proof)", "description": "This is a short, animated visual proof showing that the area of a triangle is given by the product of the inradius and the semiperimeter of the triangle. #manim #math #mathvideo #mathshorts #geometry #circle #triangle  #animation #theorem #pww #proofwithoutwords #visualproof #proof #incircle #inradius #incircle #semiperimeter\n\nThis animation is included in the following sources:\n\nProofs without Words II by Roger B. Nelsen (MAA, 2000) (https://bookstore.ams.org/clrm-14/) page 14\n\nTo learn more about animating with manim, check out:\nhttps://manim.community\n____________________________________________________\nBackground music:\nLuminous Rain\u00a0by Kevin MacLeod is licensed under a Creative Commons Attribution 4.0 license.\u00a0https://creativecommons.org/licenses/by/4.0/\nSource:\u00a0http://incompetech.com/music/royalty-free/index.html?isrc=USUAN1100169\nArtist:\u00a0http://incompetech.com/", "lengthSeconds": 115, "width": 640, "height": 360}
{"video_id": "TT4-VWjyozY", "transcript": [{"start": 0.12, "duration": 3.759, "text": "here's an intriguing way to use a"}, {"start": 1.839, "duration": 4.321, "text": "rectangle to find the sum of various"}, {"start": 3.879, "duration": 6.081, "text": "geometric series with ratio"}, {"start": 6.16, "duration": 5.639, "text": "1/2 first let's start with a 1x1 Square"}, {"start": 9.96, "duration": 3.44, "text": "the diagonal length in this square is"}, {"start": 11.799, "duration": 4.04, "text": "the square otk of two by the Pythagorean"}, {"start": 13.4, "duration": 3.879, "text": "theorem so if we draw a line segment to"}, {"start": 15.839, "duration": 4.721, "text": "the midpoint of the square on that"}, {"start": 17.279, "duration": 4.881, "text": "diagonal it has a length ofun of 2 over2"}, {"start": 20.56, "duration": 4.2, "text": "now we can draw a horizontal line back"}, {"start": 22.16, "duration": 4.199, "text": "to the square that has a length of 1/2"}, {"start": 24.76, "duration": 3.48, "text": "but then we can repeat this exact"}, {"start": 26.359, "duration": 4.721, "text": "picture in the upper right corner we"}, {"start": 28.24, "duration": 5.04, "text": "take half of the diagonal line to be 2 4"}, {"start": 31.08, "duration": 4.12, "text": "half of the horizontal line to be 1/4"}, {"start": 33.28, "duration": 4.279, "text": "and we can repeat this process over and"}, {"start": 35.2, "duration": 3.92, "text": "over each time we take half of the"}, {"start": 37.559, "duration": 4.16, "text": "previous diagonal and half of the"}, {"start": 39.12, "duration": 4.32, "text": "previous horizontal along the way each"}, {"start": 41.719, "duration": 4.36, "text": "of these line segments has a length of"}, {"start": 43.44, "duration": 4.799, "text": "size 1/ < tk2 raised to the K where K is"}, {"start": 46.079, "duration": 4.561, "text": "some integer starting at one and going"}, {"start": 48.239, "duration": 4.12, "text": "on forever now the infinite sum"}, {"start": 50.64, "duration": 4.239, "text": "represents a limit so we need to think"}, {"start": 52.359, "duration": 4.36, "text": "of the limiting process in this diagram"}, {"start": 54.879, "duration": 4.32, "text": "in the limiting process we see that the"}, {"start": 56.719, "duration": 3.961, "text": "diagonal and horizontal lines fill in"}, {"start": 59.199, "duration": 3.32, "text": "all the way up to the top right corner"}, {"start": 60.68, "duration": 3.839, "text": "of this square and so the sum of the"}, {"start": 62.519, "duration": 4.721, "text": "lengths of this zigzag diagram represent"}, {"start": 64.519, "duration": 4.721, "text": "our infinite sum but now we perform our"}, {"start": 67.24, "duration": 3.919, "text": "trick we rotate the horizontal line"}, {"start": 69.24, "duration": 4.0, "text": "segments 90\u00b0 about their end point"}, {"start": 71.159, "duration": 3.801, "text": "filling in the right side of the square"}, {"start": 73.24, "duration": 3.519, "text": "and then we shift the diagonal line"}, {"start": 74.96, "duration": 4.519, "text": "segments over to the main diagonal of"}, {"start": 76.759, "duration": 4.481, "text": "the square filling that entirely as well"}, {"start": 79.479, "duration": 4.121, "text": "and this means that the sum of the"}, {"start": 81.24, "duration": 4.64, "text": "lengths of the zigzag diagram must be"}, {"start": 83.6, "duration": 4.6, "text": "equal to the sum of the side length of"}, {"start": 85.88, "duration": 4.599, "text": "the square plus the diagonal of the"}, {"start": 88.2, "duration": 5.76, "text": "square that means our infinite sum is"}, {"start": 90.479, "duration": 6.0, "text": "equal to 1 + the < TK 2 so from this"}, {"start": 93.96, "duration": 5.6, "text": "diagram we see that the sum where K"}, {"start": 96.479, "duration": 6.96, "text": "ranges from 1 to Infinity of 1K 2 raised"}, {"start": 99.56, "duration": 5.599, "text": "to the K power is equal to 1 + < TK of 2"}, {"start": 103.439, "duration": 4.96, "text": "as a bonus fact we note that the number"}, {"start": 105.159, "duration": 5.841, "text": "1 + the of two is known as the Silver"}, {"start": 108.399, "duration": 5.601, "text": "ratio let's see this process in action"}, {"start": 111.0, "duration": 4.88, "text": "again with a 1/2x one rectangle instead"}, {"start": 114.0, "duration": 4.28, "text": "the diagonal length in this rectangle is"}, {"start": 115.88, "duration": 5.12, "text": "the < TK of 5/2 by the Pythagorean"}, {"start": 118.28, "duration": 4.4, "text": "theorem if we take half of the diagonal"}, {"start": 121.0, "duration": 3.52, "text": "and then Traverse back down to the side"}, {"start": 122.68, "duration": 5.76, "text": "of the rectangle we end up with a length"}, {"start": 124.52, "duration": 6.12, "text": "of < TK 5 4 + 1/4 and now we repeat this"}, {"start": 128.44, "duration": 4.2, "text": "process on the lower one quarter by 1/2"}, {"start": 130.64, "duration": 4.319, "text": "rectangle we scale the diagonal length"}, {"start": 132.64, "duration": 4.239, "text": "by half and the vertical line by half"}, {"start": 134.959, "duration": 5.121, "text": "giving us two line segments that sum to"}, {"start": 136.879, "duration": 5.801, "text": "> of 5 8 + 1/8 and then we repeat this"}, {"start": 140.08, "duration": 5.72, "text": "process again and again each time taking"}, {"start": 142.68, "duration": 5.08, "text": "1/2 of the previous sum of two lengths"}, {"start": 145.8, "duration": 3.64, "text": "once again we are interested in the"}, {"start": 147.76, "duration": 3.68, "text": "infinite sum of the lengths of the"}, {"start": 149.44, "duration": 4.159, "text": "zigzag diagram shown here and that means"}, {"start": 151.44, "duration": 3.68, "text": "we're interested in a limiting process"}, {"start": 153.599, "duration": 3.64, "text": "we can think about this limiting process"}, {"start": 155.12, "duration": 4.0, "text": "geometrically as the zigzag diagram"}, {"start": 157.239, "duration": 5.321, "text": "converging in on the lower left corner"}, {"start": 159.12, "duration": 6.64, "text": "filling in entirely again we can rotate"}, {"start": 162.56, "duration": 5.679, "text": "the vertical pink segments 90\u00b0 and they"}, {"start": 165.76, "duration": 4.96, "text": "fill in half of the side length of one"}, {"start": 168.239, "duration": 4.321, "text": "we can also shift the blue diagonal line"}, {"start": 170.72, "duration": 4.36, "text": "segments up to the main diagonal of the"}, {"start": 172.56, "duration": 4.039, "text": "rectangle filling in the entire diagonal"}, {"start": 175.08, "duration": 3.6, "text": "and that means that the sum that we're"}, {"start": 176.599, "duration": 5.36, "text": "interested in the infinite sum is the"}, {"start": 178.68, "duration": 6.36, "text": "sum of these two lengths so it's 12 + <"}, {"start": 181.959, "duration": 7.041, "text": "TK 5/ 2 this is a geometric series with"}, {"start": 185.04, "duration": 5.44, "text": "first term < TK 5 4 + 1/4 and ratio 12"}, {"start": 189.0, "duration": 3.319, "text": "and we see from this diagram that the"}, {"start": 190.48, "duration": 5.08, "text": "infinite sum where K ranges from 0 to"}, {"start": 192.319, "duration": 7.081, "text": "Infinity of the < TK of 5 + 1 over 4 * 2"}, {"start": 195.56, "duration": 5.36, "text": "K is 12 + > 5/2 this latter number is"}, {"start": 199.4, "duration": 4.0, "text": "known as the golden"}, {"start": 200.92, "duration": 4.56, "text": "ratio now it's your turn why don't you"}, {"start": 203.4, "duration": 4.759, "text": "try this process on a rectangle with"}, {"start": 205.48, "duration": 4.479, "text": "generic side lengths A and B what"}, {"start": 208.159, "duration": 4.72, "text": "infinite sum do you find the sum of"}, {"start": 209.959, "duration": 4.961, "text": "geometrically using this particular"}, {"start": 212.879, "duration": 4.401, "text": "argument this visual proof is due to"}, {"start": 214.92, "duration": 4.84, "text": "anhel Plaza if you liked this video"}, {"start": 217.28, "duration": 4.239, "text": "check out my playlist on geometric sums"}, {"start": 219.76, "duration": 5.119, "text": "that show other ways to find the sum of"}, {"start": 221.519, "duration": 3.36, "text": "geometric series"}], "title": "Sum Geometric Series with Ratio 1/2 in Rectangles (visual proofs)", "description": "In this video, we show an animated version of a recent proof without words that finds the sum of the geometric series with first term 1/root(2)+1/2 and ratio 1/2 using the unit square. Then we show how to find other geometric series sums with ratio 1/2 using different sizes of rectangles. Careful, the final bonus question might require a different technique than rotating :)\n\nIf you like this video, consider subscribing to the channel or consider buying me a coffee: https://www.buymeacoffee.com/VisualProofs. Thanks! \n\nAlso, check out my playlist on geometric sums/series: https://youtube.com/playlist?list=PLZh9gzIvXQUsgw8W5TUVDtF0q4jEJ3iaw\n\nThis animation is based on a proof by \u00c1ngel Plaza in the June 2024 issue of Mathematics Magazine (https://doi.org/10.1080/0025570X.2024.2336443 page 328).\n\n#mathshorts\u200b #mathvideo\u200b #math\u200b #calculus #mtbos\u200b #manim\u200b #animation\u200b #theorem\u200b #pww\u200b #proofwithoutwords\u200b #visualproof\u200b #proof\u200b #iteachmath #geometricsums #series #infinitesums #infiniteseries #geometric #geometricseries #rectangle #silverratio #goldenratio  \n\nTo learn more about animating with manim, check out:\nhttps://manim.community", "lengthSeconds": 223, "width": 640, "height": 360}
{"video_id": "TGguzFLVMZU", "transcript": [{"start": 0.16, "duration": 3.559, "text": "suppose we have a right triangle with"}, {"start": 1.68, "duration": 4.48, "text": "legs of length A and B and a hypotenuse"}, {"start": 3.719, "duration": 4.681, "text": "of C we can draw an A by a square off"}, {"start": 6.16, "duration": 4.599, "text": "the side of length a we can draw a b by"}, {"start": 8.4, "duration": 4.0, "text": "b square off the side of length B and we"}, {"start": 10.759, "duration": 3.561, "text": "can draw a c by C Square Off the"}, {"start": 12.4, "duration": 4.639, "text": "hypotenuse though we leave the C by C"}, {"start": 14.32, "duration": 5.92, "text": "Square unshaded the total shaded area in"}, {"start": 17.039, "duration": 5.721, "text": "this diagram is a^2 + b^2 let's perform"}, {"start": 20.24, "duration": 4.56, "text": "a few area preserving Transformations"}, {"start": 22.76, "duration": 4.32, "text": "first we Shear each of the two squares"}, {"start": 24.8, "duration": 4.799, "text": "along the opposite side of the triangle"}, {"start": 27.08, "duration": 4.92, "text": "we rotate these sheared regions and then"}, {"start": 29.599, "duration": 4.921, "text": "we Shear them down again filling the C"}, {"start": 32.0, "duration": 4.92, "text": "by C Square since shearing and rotating"}, {"start": 34.52, "duration": 5.44, "text": "doesn't affect the area we see that a^2"}, {"start": 36.92, "duration": 4.52, "text": "+ b^2 must equal to c^2 so here we have"}, {"start": 39.96, "duration": 3.16, "text": "a visual proof of the Pythagorean"}, {"start": 41.44, "duration": 3.48, "text": "theorem which says that the sum of the"}, {"start": 43.12, "duration": 6.04, "text": "squares of the legs of a right triangle"}, {"start": 44.92, "duration": 4.24, "text": "must equal the square of the hypotenuse"}], "title": "Pythagorean theorem: Euclid\u2019s proof reimagined", "description": "This is a short, animated visual proof of the Pythagorean theorem (the right triangle theorem) following essentially Euclid's proof. This theorem states the square of the hypotenuse of a right triangle is equal to the sum of squares of the two other side lengths.\n\nIf you like this video, consider subscribing to the channel or consider buying me a coffee: https://www.buymeacoffee.com/VisualProofs. Thanks! \n\nFor a longer version of this video with a bonus dissection proof based on this proof, see\nhttps://youtu.be/g3C3hs8_nzU\n\nYou can also see other Pythagorean theorem proofs here:\nhttps://youtu.be/o2mbN452ywA (compilation)\nhttps://youtu.be/XFHm0Ot3PP0\nhttps://youtu.be/q28H5NY5UVU\nhttps://youtu.be/yfGtbNgcrQ8\nhttps://youtu.be/JRjPsuJ7S-g\nhttps://youtu.be/TsKTteGJpZU\nhttps://youtu.be/ZlGaQdNRdqA\nhttps://youtu.be/uqGjs_n-f7U\nhttps://youtu.be/wWEZ-rUlBDQ\nhttps://youtu.be/I75HYC8TRNY\nhttps://youtu.be/LWVog8hg74I?si=4GSzTLZP7XIWeaxJ\n\nor check out my Pythagorean theorem playlist:\n\nhttps://www.youtube.com/playlist?list=PLZh9gzIvXQUvlRlnYryejWGHYQquM92rB\n\nThis animation is based on a proof due to Euclid. For a static version of this proof, see Roger Nelsen's first \"Proof Without Words: Exercises in Visual Thinking\" compendium (page 5) : https://bookstore.ams.org/clrm-1. You can also check out Howard Eves' \"Great Moments in Mathematics (Before 1650)\" page 31-33.\n\n(I may receive a small commission at no cost to you for these affiliate links): \n\nhttps://amzn.to/48mjeX1\nhttps://amzn.to/3Tr0OjW\n\n#math #pythagoreantheorem #pythagorean #triangle #manim #animation #theorem #pww #proofwithoutwords #visualproof #proof  #mathshorts #mathvideo \n\nTo learn more about animating with manim, check out:\nhttps://manim.community", "lengthSeconds": 48, "width": 360, "height": 640}
{"video_id": "TjBV5UK8wIE", "transcript": [{"start": 20.64, "duration": 7.399, "text": "we are to prove Pythagoras Theorem c\u00b2 ="}, {"start": 25.199, "duration": 6.88, "text": "a sare +"}, {"start": 28.039, "duration": 7.961, "text": "b\u00b2 that is hypot square is equal to sum"}, {"start": 32.079, "duration": 7.841, "text": "of squares of other two"}, {"start": 36.0, "duration": 7.8, "text": "sides we will prove it by the Yan"}, {"start": 39.92, "duration": 5.319, "text": "property this property says that square"}, {"start": 43.8, "duration": 4.0, "text": "a"}, {"start": 45.239, "duration": 5.64, "text": "square will be changed to rectangle"}, {"start": 47.8, "duration": 5.32, "text": "whose area will be a square and square"}, {"start": 50.879, "duration": 7.0, "text": "with area b square will be changed to"}, {"start": 53.12, "duration": 4.759, "text": "rectangle whose area will be b s"}, {"start": 63.76, "duration": 4.84, "text": "now see the animation"}, {"start": 83.439, "duration": 3.401, "text": "process first"}, {"start": 88.64, "duration": 5.72, "text": "slider ch ches scares to"}, {"start": 96.68, "duration": 7.64, "text": "parallelogram and these have area same"}, {"start": 100.88, "duration": 6.08, "text": "because we have applied rule on the"}, {"start": 104.32, "duration": 5.04, "text": "right hand side given area and"}, {"start": 106.96, "duration": 4.4, "text": "parallelogram area of the both sides"}, {"start": 109.36, "duration": 5.48, "text": "figures are same because they have the"}, {"start": 111.36, "duration": 8.2, "text": "same base and same"}, {"start": 114.84, "duration": 4.72, "text": "height now see animation"}, {"start": 123.52, "duration": 7.24, "text": "these have three steps so off slider one"}, {"start": 127.159, "duration": 6.881, "text": "off slider two offsider"}, {"start": 130.76, "duration": 7.839, "text": "three you can repeat it first second"}, {"start": 134.04, "duration": 4.559, "text": "third third second 1 1 2"}, {"start": 138.64, "duration": 5.2, "text": "3 C animation again"}, {"start": 153.08, "duration": 4.08, "text": "step third second"}, {"start": 158.84, "duration": 13.0, "text": "one step first second"}, {"start": 163.36, "duration": 13.64, "text": "third third second 1 1 step two step"}, {"start": 171.84, "duration": 8.52, "text": "three therefore a square + b square"}, {"start": 177.0, "duration": 7.519, "text": "are merged in the big Square"}, {"start": 180.36, "duration": 9.92, "text": "c\u00b2 therefore c\u00b2 = to a sare + b square"}, {"start": 184.519, "duration": 5.761, "text": "Please Subscribe and share my channel"}], "title": "Pythagoroas Theorem ,Pythagoroas  Theorem Proof, Mathematical Visual proof ,Animation In Geogebra", "description": "Thanks for whatching\nkataria math animation,\nContact Me :-  rattanlalkataria@gmail.com  \nOn YouTube :- @KatariaMathAnimationOfficial\n Your Queries :-\nPythagoroas Theorem \n Pythagoroas Theorem Proof\n Kataria Math Animation\nAnimation In Geogebra\nGeogebra In Mathematics Teaching \nPythagorean Theorem Animation\nExterior Angles Theorem\nExterior Angles Properties\nProve Exterior Angles Of Triangle = Sum of  Opposite Interior Angles\nMaths Animation\nTeaching of  Maths\nLearning of Maths\nSum Of angles of triangles\n\nGeometry\n Algebra\nGeoGebra\nAnimation\nEasy Maths\nOn Line Teaching\nMath Teacher\nTriangles\nSquare\nQuadrilateral \nProperty of Triangles\nProperty of Circle\nArea\nVolume\nFind Nets of Solid Figure\nStudent's favorite method\nBefore classroom \nIn The Classroom \n#Geogebra\n#Angles\n#Sumofangles\n#favorite \n#Method\n#Maths\n#GeoGebra\n#viralmath\n#onlineteaching\n#Quiz For Kids Visual Addition\n#Quiz On Geogebra#\n# Visual Math\n# Animation\nPythagoroas Theoram \nEasy Math\n Mathematical Visual proof \n Only 2 Minutes\n#mathvisualproofs", "lengthSeconds": 204, "width": 640, "height": 360}
{"video_id": "TU0043SuGsM", "transcript": [{"start": 5.7, "duration": 7.38, "text": "imagine we have a triangle with sides of"}, {"start": 8.519, "duration": 6.361, "text": "length a b and c and the angles across"}, {"start": 13.08, "duration": 6.24, "text": "from the sides are labeled capital a"}, {"start": 14.88, "duration": 6.54, "text": "capital B and capital c respectively"}, {"start": 19.32, "duration": 4.5, "text": "drop in altitude from the vertex where"}, {"start": 21.42, "duration": 4.98, "text": "angle a is to the side of length a and"}, {"start": 23.82, "duration": 4.92, "text": "label it h this altitude cuts the"}, {"start": 26.4, "duration": 4.139, "text": "triangle into two right triangles from"}, {"start": 28.74, "duration": 5.4, "text": "the first one we see that the sine of"}, {"start": 30.539, "duration": 4.921, "text": "angle b equals H divided by c as H is"}, {"start": 34.14, "duration": 3.9, "text": "the opposite side and C is the"}, {"start": 35.46, "duration": 5.58, "text": "hypotenuse from the other right triangle"}, {"start": 38.04, "duration": 5.76, "text": "we see that the sine of C is equal to H"}, {"start": 41.04, "duration": 5.339, "text": "divided by B as H is the opposite side"}, {"start": 43.8, "duration": 4.739, "text": "and B is the hypotenuse of this triangle"}, {"start": 46.379, "duration": 5.221, "text": "as H is part of both of these equations"}, {"start": 48.539, "duration": 7.2, "text": "we can see that H equals B times sine of"}, {"start": 51.6, "duration": 6.119, "text": "C and H equals c times sine of B"}, {"start": 55.739, "duration": 4.5, "text": "from this we see that b times the sine"}, {"start": 57.719, "duration": 5.221, "text": "of angle C equals c times the sine of"}, {"start": 60.239, "duration": 5.1, "text": "angle B dividing both quantities by B"}, {"start": 62.94, "duration": 4.74, "text": "times C we see that sine of angle C"}, {"start": 65.339, "duration": 4.621, "text": "divided by C is equal to the sine of"}, {"start": 67.68, "duration": 4.2, "text": "angle B divided by B"}, {"start": 69.96, "duration": 4.62, "text": "now we can do a similar thing for the"}, {"start": 71.88, "duration": 4.5, "text": "angles A and B but this time we have to"}, {"start": 74.58, "duration": 3.539, "text": "be a little more careful this time when"}, {"start": 76.38, "duration": 4.32, "text": "we try to drop an altitude from the"}, {"start": 78.119, "duration": 4.441, "text": "vertex at angle C we see that we get a"}, {"start": 80.7, "duration": 3.72, "text": "new triangle created outside the"}, {"start": 82.56, "duration": 4.32, "text": "original like this we can label this"}, {"start": 84.42, "duration": 4.8, "text": "altitude by H and note that the newly"}, {"start": 86.88, "duration": 4.5, "text": "created outside triangle has an angle of"}, {"start": 89.22, "duration": 4.14, "text": "measure 180 degrees minus a the"}, {"start": 91.38, "duration": 3.96, "text": "supplementary angle to a"}, {"start": 93.36, "duration": 4.2, "text": "from this newly created right triangle"}, {"start": 95.34, "duration": 4.2, "text": "here we see that the sine of angle B"}, {"start": 97.56, "duration": 4.86, "text": "must be the opposite side length H"}, {"start": 99.54, "duration": 4.74, "text": "divided by the hypotenuse a on the other"}, {"start": 102.42, "duration": 4.739, "text": "hand from the small right triangle"}, {"start": 104.28, "duration": 5.1, "text": "created here we see that the sine of 180"}, {"start": 107.159, "duration": 5.1, "text": "degrees minus a must equal the opposite"}, {"start": 109.38, "duration": 4.919, "text": "side H divided by the hypotenuse B"}, {"start": 112.259, "duration": 4.68, "text": "once again both of these equalities"}, {"start": 114.299, "duration": 5.1, "text": "involve the quantity H so we see that H"}, {"start": 116.939, "duration": 5.881, "text": "is equal to B times the sine of 180"}, {"start": 119.399, "duration": 4.621, "text": "minus a and H equals a times the sine of"}, {"start": 122.82, "duration": 3.119, "text": "angle B"}, {"start": 124.02, "duration": 4.86, "text": "we can divide both of these quantities"}, {"start": 125.939, "duration": 6.481, "text": "by a times B so that sine of 180 degrees"}, {"start": 128.88, "duration": 6.66, "text": "minus a over a is equal to sine of angle"}, {"start": 132.42, "duration": 5.819, "text": "B divided by little B but now we use the"}, {"start": 135.54, "duration": 5.1, "text": "fact that sine of 180 degrees minus a is"}, {"start": 138.239, "duration": 4.321, "text": "equal to the sine of angle a this means"}, {"start": 140.64, "duration": 4.26, "text": "that the sine of angle a divided by side"}, {"start": 142.56, "duration": 4.86, "text": "length a is equal to the sine of angle B"}, {"start": 144.9, "duration": 3.96, "text": "divided by the side length little B when"}, {"start": 147.42, "duration": 3.36, "text": "we couple this result with our first"}, {"start": 148.86, "duration": 4.86, "text": "computation we see that the sine of"}, {"start": 150.78, "duration": 5.819, "text": "angle a divided by a equals the sine of"}, {"start": 153.72, "duration": 6.3, "text": "angle B divided by B and that equals the"}, {"start": 156.599, "duration": 6.181, "text": "sine of angle C divided by C where a B"}, {"start": 160.02, "duration": 5.46, "text": "and C are the side lengths of a triangle"}, {"start": 162.78, "duration": 5.64, "text": "with opposite angles capital a capital b"}, {"start": 165.48, "duration": 6.86, "text": "capital c respectively this is known as"}, {"start": 168.42, "duration": 3.92, "text": "The Law of Sines for triangles"}], "title": "Law of Sines Visual Proof", "description": "This is a short, animated visual proof of the Law of Sines using the definition of sine and by dropping appropriate altitudes of the triangle for given angles. The altitude may intersect the opposite side or an extended segment of the opposite side.\n\nIf you like this video, consider subscribing to the channel or consider buying me a coffee: https://www.buymeacoffee.com/VisualProofs. Thanks! \n\nThis animation is the classic visual proof the fact but most likely not the one Euclid used.\n\nFor visual proofs of the Law of Cosines, check out:\nhttps://youtu.be/NHxJ3Z_58Lw or\nhttps://youtu.be/vNCe7ciFwVc\n\n#mathshorts #mathvideo #math #trigonometry #lawofsines  #triangle #manim #animation #theorem #pww #proofwithoutwords #visualproof #proof #circle #obtuseangle #acuteangle #angle #trigidentities #identity #righttriangle\n\nTo learn more about animating with manim, check out:\nhttps://manim.community", "lengthSeconds": 178, "width": 640, "height": 360}
{"video_id": "oJvjbFicOtM", "transcript": [{"start": 0.04, "duration": 3.679, "text": "here's an amazing geometric structure"}, {"start": 1.56, "duration": 4.52, "text": "called the fanel plane start with seven"}, {"start": 3.719, "duration": 4.441, "text": "points laid out like this by a line we"}, {"start": 6.08, "duration": 4.72, "text": "meet a collection of three points so"}, {"start": 8.16, "duration": 4.639, "text": "draw these seven lines note that the"}, {"start": 10.8, "duration": 3.64, "text": "final one is actually going to look like"}, {"start": 12.799, "duration": 3.801, "text": "a circle even though it's just a"}, {"start": 14.44, "duration": 3.88, "text": "collection of three points we see that"}, {"start": 16.6, "duration": 4.72, "text": "there are three points on each line and"}, {"start": 18.32, "duration": 4.719, "text": "there are three lines through each point"}, {"start": 21.32, "duration": 4.6, "text": "we can realize this structure using the"}, {"start": 23.039, "duration": 4.761, "text": "three-dimensional Vector space over Z2"}, {"start": 25.92, "duration": 3.48, "text": "the points correspond to the non-zero"}, {"start": 27.8, "duration": 3.759, "text": "vectors in the vector space so they"}, {"start": 29.4, "duration": 4.319, "text": "correspond to the onedimensional linear"}, {"start": 31.559, "duration": 3.84, "text": "subspaces the lines correspond to the"}, {"start": 33.719, "duration": 3.641, "text": "two-dimensional linear subspaces and"}, {"start": 35.399, "duration": 4.601, "text": "that means each line consists of three"}, {"start": 37.36, "duration": 5.519, "text": "vectors that coordinate wise add up to"}, {"start": 40.0, "duration": 5.079, "text": "zero mod 2 you can check that the three"}, {"start": 42.879, "duration": 4.84, "text": "vectors on each line as highlighted here"}, {"start": 45.079, "duration": 4.8, "text": "do in fact add up to zero mod 2 when you"}, {"start": 47.719, "duration": 4.201, "text": "perform coordinate wise addition this is"}, {"start": 49.879, "duration": 3.601, "text": "the world's smallest projective geometry"}, {"start": 51.92, "duration": 4.4, "text": "and it shows up in a lot of places in"}, {"start": 53.48, "duration": 2.84, "text": "mathematics"}], "title": "Fano Plane: The World\u2019s Smallest Projective Geometry", "description": "This is short animation of the Fano Plane, which is the smallest projective geometry. We also show how to realize the geometry as the set of linear sub spaces of a three dimensional vector space over the field with two elements. \n\nTo buy me a coffee, head over to https://www.buymeacoffee.com/VisualProofs \nThanks!\n\n\n#mathshorts\u200b #mathvideo\u200b #math\u200b #geometry #mtbos\u200b #manim\u200b #animation\u200b #theorem\u200b #pww\u200b #proofwithoutwords\u200b #visualproof\u200b #proof\u200b #iteachmath #vectorspaces #projective #projectivegeometry #shorts #fanoplane #fano \n\nTo learn more about animating with manim, check out:\nhttps://manim.community", "lengthSeconds": 54, "width": 360, "height": 640}
{"video_id": "m8WVsRAntqk", "transcript": [{"start": 5.16, "duration": 4.64, "text": "here are two standard infinite series"}, {"start": 7.319, "duration": 4.161, "text": "that are related sure you can find the"}, {"start": 9.8, "duration": 4.2, "text": "sum of these series using classical"}, {"start": 11.48, "duration": 4.239, "text": "techniques but let's see a geometric way"}, {"start": 14.0, "duration": 4.84, "text": "that computes the sum of both of these"}, {"start": 15.719, "duration": 5.441, "text": "infinite series using regular polygons"}, {"start": 18.84, "duration": 4.759, "text": "start with an equilateral triangle the"}, {"start": 21.16, "duration": 5.16, "text": "interior angle of this triangle is pi"}, {"start": 23.599, "duration": 6.361, "text": "over 3 which can be Rewritten as 2 piun"}, {"start": 26.32, "duration": 5.24, "text": "3 * 2 now draw a square around that"}, {"start": 29.96, "duration": 4.08, "text": "equilateral triangle with side length"}, {"start": 31.56, "duration": 5.92, "text": "equal to the base the interior angle of"}, {"start": 34.04, "duration": 5.64, "text": "the square is 2 Pi / 4 and so the angle"}, {"start": 37.48, "duration": 5.88, "text": "between the square in the triangle is 2"}, {"start": 39.68, "duration": 6.719, "text": "piun 4 - < 3 which simplifies to 2 piun"}, {"start": 43.36, "duration": 4.719, "text": "/ 4 * 3 Let's extend this again by"}, {"start": 46.399, "duration": 3.761, "text": "drawing a pentagon with side length"}, {"start": 48.079, "duration": 4.521, "text": "equal to the square length again the"}, {"start": 50.16, "duration": 4.48, "text": "interior angle of the Pentagon is 3 pi"}, {"start": 52.6, "duration": 4.04, "text": "over 5 so the difference in angle"}, {"start": 54.64, "duration": 5.52, "text": "between the Pentagon and the square is 3"}, {"start": 56.64, "duration": 7.04, "text": "piun / 5 - 2 piun 4 and this simplifies"}, {"start": 60.16, "duration": 5.639, "text": "to 2 piun / 5 * 4 we can continue this"}, {"start": 63.68, "duration": 4.72, "text": "extension process by drawing larger and"}, {"start": 65.799, "duration": 4.281, "text": "larger polygons around so we start with"}, {"start": 68.4, "duration": 4.2, "text": "the hexagon and then we move to the"}, {"start": 70.08, "duration": 4.399, "text": "heptagon and so on using the interior"}, {"start": 72.6, "duration": 3.68, "text": "angle theorem we see that the angle"}, {"start": 74.479, "duration": 5.801, "text": "difference between the N gon and the"}, {"start": 76.28, "duration": 7.44, "text": "nus1 gone is n - 2 * piun / n minus the"}, {"start": 80.28, "duration": 7.28, "text": "quantity n- 3 * piun / nus1 and this"}, {"start": 83.72, "duration": 5.64, "text": "simplifies to 2 piun / n * nus1 so the"}, {"start": 87.56, "duration": 5.199, "text": "angle between the regular n gon and the"}, {"start": 89.36, "duration": 6.759, "text": "regular n -1 gone is always 2 piun / n *"}, {"start": 92.759, "duration": 5.36, "text": "nus1 as we extend this picture out but"}, {"start": 96.119, "duration": 4.561, "text": "now we can just keep extending this"}, {"start": 98.119, "duration": 4.841, "text": "picture drawing more and more regular"}, {"start": 100.68, "duration": 4.92, "text": "polygons where the number of sides"}, {"start": 102.96, "duration": 4.96, "text": "increases indefinitely in fact we can"}, {"start": 105.6, "duration": 4.96, "text": "consider a limiting shape of this"}, {"start": 107.92, "duration": 5.4, "text": "process the limiting outer shell of the"}, {"start": 110.56, "duration": 4.599, "text": "resulting shape is a circle by zooming"}, {"start": 113.32, "duration": 4.839, "text": "back into the original equilateral"}, {"start": 115.159, "duration": 4.88, "text": "triangle we get an amazing fact the"}, {"start": 118.159, "duration": 4.481, "text": "angles that we've determined can be"}, {"start": 120.039, "duration": 4.521, "text": "added together so we see that on one"}, {"start": 122.64, "duration": 6.28, "text": "hand the full angle measurement is 2"}, {"start": 124.56, "duration": 7.36, "text": "piun 3 * 2 + 2un 4 * 3 + 2 piun over 5 *"}, {"start": 128.92, "duration": 6.599, "text": "4 and so on each time adding up angles"}, {"start": 131.92, "duration": 5.76, "text": "of the form 2 pi/ n * nus1 on the other"}, {"start": 135.519, "duration": 3.761, "text": "hand all of these angles together form"}, {"start": 137.68, "duration": 3.44, "text": "the full angle measurement of a"}, {"start": 139.28, "duration": 3.92, "text": "horizontal line since this line is"}, {"start": 141.12, "duration": 4.8, "text": "tangent to the circle and therefore the"}, {"start": 143.2, "duration": 4.44, "text": "sum of the angles must equal Pi but now"}, {"start": 145.92, "duration": 3.48, "text": "we can cancel the pi and the numerators"}, {"start": 147.64, "duration": 4.959, "text": "of the fractions on the left with the pi"}, {"start": 149.4, "duration": 7.88, "text": "on the right and we see that 2 3 * 2 + 2"}, {"start": 152.599, "duration": 7.081, "text": "4 * 3 + 2 5 * 4 and so on must equal 1"}, {"start": 157.28, "duration": 4.44, "text": "if we then add 2 over 2 * 1 on the left"}, {"start": 159.68, "duration": 4.08, "text": "and 2 over 2 * 1 on the right which is"}, {"start": 161.72, "duration": 4.159, "text": "one then we get our first famous"}, {"start": 163.76, "duration": 5.199, "text": "infinite sum notice that each fraction"}, {"start": 165.879, "duration": 5.161, "text": "on the left is 1/ n choose 2 where n"}, {"start": 168.959, "duration": 4.241, "text": "choose 2 is a binomial coefficient or a"}, {"start": 171.04, "duration": 4.119, "text": "triangular number this means that the"}, {"start": 173.2, "duration": 5.64, "text": "infinite sum where n ranges from 2 to"}, {"start": 175.159, "duration": 5.281, "text": "Infinity of 1/ n choose 2 is equal to 2"}, {"start": 178.84, "duration": 3.759, "text": "we can also divide both sides of the"}, {"start": 180.44, "duration": 4.32, "text": "equation by two producing another famous"}, {"start": 182.599, "duration": 5.241, "text": "infinite sum the one where n ranges from"}, {"start": 184.76, "duration": 5.759, "text": "1 to Infinity of 1 n * n + 1 and this"}, {"start": 187.84, "duration": 4.479, "text": "sum equals 1 these infinite sums are"}, {"start": 190.519, "duration": 4.201, "text": "usually computed by recognizing the"}, {"start": 192.319, "duration": 3.801, "text": "infinite series as a telescoping series"}, {"start": 194.72, "duration": 3.68, "text": "but here we've shown how to find this"}, {"start": 196.12, "duration": 4.759, "text": "infinite sum using a limiting process on"}, {"start": 198.4, "duration": 5.24, "text": "an amazing diagram constructed by larger"}, {"start": 200.879, "duration": 4.44, "text": "and larger regular polygons if you like"}, {"start": 203.64, "duration": 4.159, "text": "this video check out my playlist on"}, {"start": 205.319, "duration": 5.041, "text": "infinite series there I've animated"}, {"start": 207.799, "duration": 7.36, "text": "classic visual proofs that Compu the sum"}, {"start": 210.36, "duration": 4.799, "text": "of infinite Series in interesting ways"}], "title": "Two Infinite Series Sums from Regular Polygons (visual proof)", "description": "This is a short, animated visual proof computing the sums of two series - one of reciprocals of triangular numbers (i.e., certain binomial coefficients) and the other a classic series that is used to demonstrate telescoping series.\n\nIf you like this video, consider subscribing to the channel or consider buying me a coffee: https://www.buymeacoffee.com/VisualProofs. Thanks! \n\nFor some related videos see these videos:\nhttps://youtu.be/moxeqJx2YsY\nhttps://youtu.be/I0juicS3SIc\n\nThis animation is based on a visual proof by Paul Stephenson from the December 2022 issue of Mathematics Magazine (https://doi.org/10.1080/0025570X.2022.2126173 page 572 ).\n\n\n#mathshorts\u200b #mathvideo\u200b #math\u200b #infiniteseries #mtbos\u200b #manim\u200b #animation\u200b #theorem\u200b #pww\u200b #proofwithoutwords\u200b #visualproof\u200b #proof\u200b #iteachmath #calculus #triangularnumbers \u200b\u200b #animation\u200b #theorem\u200b\u200b #proof\u200b #iteachmath #series #infinitesums #infiniteseries #pascaltriangle #binomialcoefficients #binomials #binomialtheorem #telescoping \n\nTo learn more about animating with manim, check out:\nhttps://manim.community", "lengthSeconds": 215, "width": 640, "height": 360}
{"video_id": "qiufkVTkLmg", "transcript": [{"start": 0.16, "duration": 28.57, "text": "aah"}, {"start": 2.1, "duration": 26.63, "text": "[Music]"}], "title": "Volume of cone by ingration visual proof #shorts", "description": "Discover how to calculate the volume of a cone with integration! Watch this short video to see the mathematical process and graphical proof that brings the formula to life. Perfect for math enthusiasts and students alike!\n#trending #shorts#viralshort #MathWeb", "lengthSeconds": 27, "width": 270, "height": 480}
{"video_id": "bpTZi01Mlqw", "transcript": [{"start": 0.0, "duration": 4.38, "text": "let's use integral calculus to find the"}, {"start": 2.159, "duration": 3.901, "text": "area under the curve Y equals X plus a"}, {"start": 4.38, "duration": 3.36, "text": "half"}, {"start": 6.06, "duration": 3.72, "text": "according to the fundamental theorem of"}, {"start": 7.74, "duration": 4.319, "text": "calculus the integral from 0 to n of X"}, {"start": 9.78, "duration": 4.62, "text": "plus a half DX is equal to the quantity"}, {"start": 12.059, "duration": 5.281, "text": "x squared over 2 plus X over 2 evaluated"}, {"start": 14.4, "duration": 5.1, "text": "at n and zero when we plug in n and 0"}, {"start": 17.34, "duration": 4.62, "text": "and subtract we get N squared over 2"}, {"start": 19.5, "duration": 4.859, "text": "plus n over 2 for the integral"}, {"start": 21.96, "duration": 4.62, "text": "on the other hand we can also compute"}, {"start": 24.359, "duration": 4.26, "text": "this integral by breaking up the region"}, {"start": 26.58, "duration": 4.859, "text": "into a bunch of different pieces like"}, {"start": 28.619, "duration": 5.28, "text": "this each of the pieces will then be the"}, {"start": 31.439, "duration": 5.161, "text": "definite integral from I minus 1 to I"}, {"start": 33.899, "duration": 4.261, "text": "under the function X plus a half so we"}, {"start": 36.6, "duration": 2.88, "text": "can add up all of these integrals"}, {"start": 38.16, "duration": 4.26, "text": "according to properties of integrals"}, {"start": 39.48, "duration": 4.919, "text": "once we do this we again get that each"}, {"start": 42.42, "duration": 4.38, "text": "integral is x squared over 2 plus X over"}, {"start": 44.399, "duration": 4.14, "text": "2 evaluated at the endpoints and thus we"}, {"start": 46.8, "duration": 4.62, "text": "get that the integral in question is"}, {"start": 48.539, "duration": 3.661, "text": "given by the sum from I equals 1 to n of"}, {"start": 51.42, "duration": 2.459, "text": "I"}, {"start": 52.2, "duration": 3.6, "text": "but now that we've computed the definite"}, {"start": 53.879, "duration": 3.36, "text": "integral in two separate ways we see"}, {"start": 55.8, "duration": 2.939, "text": "that the sum of the first n positive"}, {"start": 57.239, "duration": 4.46, "text": "integers is equal to N squared over two"}, {"start": 58.739, "duration": 2.96, "text": "plus n over 2."}], "title": "Summing integers using calculus!", "description": "This is a clip from a longer video showing a dozen ways to sum the first n integers. This proof uses integral calculus to get the famous formula. Full video is here : https://youtu.be/eHbtc50-qXo\n\nTo buy me a coffee, head over to https://www.buymeacoffee.com/VisualProofs \nThanks!\n\nThis proof is adapted from a static diagram due to Jaime Gaspar: Proof without words: using trapezoids to compute triangular numbers, Math. Mag. 91 (2018), no. 3, 206\u2013207. (https://www.jstor.org/stable/48665541)\n\n#manim #math #proof #visualproof #mathshorts #proofwithoutwords #sums #integers #sumformula \n\nTo learn more about animating with manim, check out:\nhttps://manim.community", "lengthSeconds": 60, "width": 360, "height": 640}
{"video_id": "v6n8XKlFY3k", "transcript": [{"start": 2.43, "duration": 7.96, "text": "[Music]"}, {"start": 13.99, "duration": 2.97, "text": "[Music]"}, {"start": 15.679, "duration": 3.6, "text": "in a previous video"}, {"start": 16.96, "duration": 3.76, "text": "we saw a visual proof of this formula"}, {"start": 19.279, "duration": 4.881, "text": "for the sum of the first"}, {"start": 20.72, "duration": 3.44, "text": "n positive squares"}, {"start": 26.56, "duration": 4.16, "text": "what happens if we instead consider the"}, {"start": 29.679, "duration": 5.681, "text": "alternating sum"}, {"start": 30.72, "duration": 4.64, "text": "of squares is there a nice formula now"}, {"start": 36.0, "duration": 2.719, "text": "let's take a look"}, {"start": 40.16, "duration": 7.04, "text": "a square array with side length n"}, {"start": 43.84, "duration": 5.6, "text": "represents the value n squared"}, {"start": 47.2, "duration": 3.039, "text": "removing the upper n minus 1 by n minus"}, {"start": 49.44, "duration": 2.88, "text": "1 square"}, {"start": 50.239, "duration": 3.281, "text": "corresponds to subtracting n minus 1"}, {"start": 52.32, "duration": 3.28, "text": "squared"}, {"start": 53.52, "duration": 3.76, "text": "we can add back an n minus 2 by n minus"}, {"start": 55.6, "duration": 4.08, "text": "2 square and then"}, {"start": 57.28, "duration": 4.64, "text": "remove the upper right n minus 3 by n"}, {"start": 59.68, "duration": 5.36, "text": "minus 3 square"}, {"start": 61.92, "duration": 4.64, "text": "we can continue this process alternating"}, {"start": 65.04, "duration": 4.56, "text": "between adding a square back"}, {"start": 66.56, "duration": 3.84, "text": "and removing the next square array until"}, {"start": 69.6, "duration": 5.839, "text": "eventually"}, {"start": 70.4, "duration": 5.039, "text": "we add or subtract the two by two square"}, {"start": 76.4, "duration": 6.0, "text": "and then subtract or add back the one by"}, {"start": 79.759, "duration": 2.641, "text": "one square"}, {"start": 82.479, "duration": 7.041, "text": "the resulting l shaped gnomons can then"}, {"start": 86.159, "duration": 6.0, "text": "be squished together"}, {"start": 89.52, "duration": 3.12, "text": "to produce a triangular array with base"}, {"start": 92.159, "duration": 3.6, "text": "n"}, {"start": 92.64, "duration": 6.08, "text": "which has t n squares where t"}, {"start": 95.759, "duration": 4.641, "text": "n is the nth triangular number therefore"}, {"start": 98.72, "duration": 17.12, "text": "the alternating sum of the first"}, {"start": 100.4, "duration": 15.44, "text": "n squares is the nth triangular number"}, {"start": 120.88, "duration": 2.08, "text": "you"}], "title": "Alternating Sum Of Squares (visual proof)", "description": "This is a short, animated visual proof demonstrating how to visualize the alternating sum of squares. #mathshorts\u200b #mathvideo\u200b #math\u200b #numbertheory #mtbos\u200b #manim\u200b #animation\u200b #theorem\u200b #pww\u200b #proofwithoutwords\u200b #visualproof\u200b #proof\u200b #iteachmath #squares #triangularnumbers\n\nThis animation is based on a visual proof by Steven L. Snover from the April 1992 issue of Mathematics Magazine (https://doi.org/10.2307/2690486 - page 90).\n\nCompare this formula to the regular sum of squares formula visualized here: https://youtu.be/-tJhH_k2LaM\n\nHere is an alternate visual proof of this same fact: https://youtu.be/stSftarO82s\n\nTo learn more about animating with manim, check out:\nhttps://manim.community\n__________________________________________________________________\nBackground music, used according to CC BY-SA 3.0, is \"The Road\" from Esteban Orlando:\n\nEsteban Orlando:\nhttps://soundcloud.com/orlando-esteba...\nhttps://www.instagram.com/estebanorla...\n\nhttps://creativecommons.org/licenses/...\nMusic provided by RFM: https://youtu.be/b4KCM1bIq50", "lengthSeconds": 121, "width": 640, "height": 360}
{"video_id": "owFwvyozcn8", "transcript": [{"start": 0.5, "duration": 4.12, "text": "[Music]"}, {"start": 1.86, "duration": 5.34, "text": "here's an isosceles right triangle with"}, {"start": 4.62, "duration": 4.559, "text": "legs of length 1 so that the hypotenuse"}, {"start": 7.2, "duration": 3.66, "text": "has a length square root of 2. if we"}, {"start": 9.179, "duration": 3.96, "text": "walk from the upper corner to the lower"}, {"start": 10.86, "duration": 4.319, "text": "corner going one unit East and then one"}, {"start": 13.139, "duration": 4.381, "text": "unit South we travel a total of two"}, {"start": 15.179, "duration": 4.801, "text": "units but we can keep poking in the"}, {"start": 17.52, "duration": 4.98, "text": "corners of these paths so that we walk"}, {"start": 19.98, "duration": 5.1, "text": "east south east south east south all the"}, {"start": 22.5, "duration": 5.519, "text": "way down the diagonal to the corner if"}, {"start": 25.08, "duration": 5.4, "text": "we do this process indefinitely have we"}, {"start": 28.019, "duration": 4.201, "text": "produced a proof that the square root of"}, {"start": 30.48, "duration": 4.919, "text": "2 which is the length of the hypotenuse"}, {"start": 32.22, "duration": 5.339, "text": "is exactly the same as 2 which is the"}, {"start": 35.399, "duration": 5.041, "text": "length of the path that we walk how"}, {"start": 37.559, "duration": 4.68, "text": "could this be let's take a zoom in and"}, {"start": 40.44, "duration": 4.799, "text": "see what's going on"}, {"start": 42.239, "duration": 3.0, "text": "foreign"}, {"start": 46.399, "duration": 4.961, "text": "so no this isn't a visual proof that"}, {"start": 48.96, "duration": 4.02, "text": "square root of 2 equals 2. but can you"}, {"start": 51.36, "duration": 3.719, "text": "explain what's going on"}, {"start": 52.98, "duration": 3.78, "text": "if you want to see some honest visual"}, {"start": 55.079, "duration": 4.221, "text": "proofs of some real facts then go check"}, {"start": 56.76, "duration": 2.54, "text": "out my channel"}], "title": "Is square root of 2 equal to 2? (Spoiler : no)", "description": "This is a short animation of an oft-claimed visual proof that the square root of 2 is equal to 2. Can you see why the two lengths aren't equal? Can you give a good, brief explanation? #math #manim #mathshorts #mathvideo #visualproof #proofwithoutwords #false #mathfail\n\nTo learn more about animating with manim, check out:\nhttps://manim.community\n_________________________\nMusic in this episode: \nPrelude No. 5\u00a0by Chris Zabriskie is licensed under a Creative Commons Attribution 4.0 license.\u00a0https://creativecommons.org/licenses/by/4.0/\nSource:\u00a0http://chriszabriskie.com/preludes/\nArtist:\u00a0http://chriszabriskie.com/", "lengthSeconds": 58, "width": 360, "height": 640}
{"video_id": "WrlngkU82Jc", "transcript": [{"start": 0.08, "duration": 3.92, "text": "have you ever wondered if e to the pi or"}, {"start": 1.92, "duration": 3.399, "text": "Pi the E is the larger number here's a"}, {"start": 4.0, "duration": 3.639, "text": "way to figure it out without a"}, {"start": 5.319, "duration": 4.881, "text": "calculator first plot the function y ="}, {"start": 7.639, "duration": 5.561, "text": "1X then plot the x coordinate e and the"}, {"start": 10.2, "duration": 4.639, "text": "x coordinate Pi notice e is less than Pi"}, {"start": 13.2, "duration": 3.32, "text": "we can find the area under this curve"}, {"start": 14.839, "duration": 4.321, "text": "between e and pi as the definite"}, {"start": 16.52, "duration": 3.999, "text": "integral where X ranges from E to Pi of"}, {"start": 19.16, "duration": 3.84, "text": "1x"}, {"start": 20.519, "duration": 4.161, "text": "DX we can also draw this rectangle"}, {"start": 23.0, "duration": 3.32, "text": "living over e to the pi and we see that"}, {"start": 24.68, "duration": 4.12, "text": "the area under the curve is less than"}, {"start": 26.32, "duration": 4.56, "text": "the area of the rectangle which is 1 E *"}, {"start": 28.8, "duration": 3.599, "text": "piun - e according to the fundamental"}, {"start": 30.88, "duration": 3.48, "text": "theorem of calculus the definite"}, {"start": 32.399, "duration": 3.281, "text": "integral is the natural log of Pi - 1"}, {"start": 34.36, "duration": 4.039, "text": "and we can simplify the area of the"}, {"start": 35.68, "duration": 4.399, "text": "rectangle to Pi eus 1 from this we see"}, {"start": 38.399, "duration": 4.441, "text": "that the natural log of Pi is less than"}, {"start": 40.079, "duration": 5.16, "text": "Pi e we can multiply both sides by E and"}, {"start": 42.84, "duration": 4.76, "text": "then apply two natural log rules to see"}, {"start": 45.239, "duration": 5.121, "text": "that Pi the E is less than e to the pi"}, {"start": 47.6, "duration": 4.92, "text": "so using the curve y = 1X we've proved"}, {"start": 50.36, "duration": 5.12, "text": "analytically that pi to the E is less"}, {"start": 52.52, "duration": 2.96, "text": "than e to the pi"}], "title": "e^Pi vs Pi^e : which is larger? (pi day battle)", "description": "This is a short, animated visual proof demonstrating which of the values e^pi or pi^e is larger (where pi is the circle constant and e is Euler's constant) . \n\nIf you like this video, consider subscribing to the channel or consider buying me a coffee: https://www.buymeacoffee.com/VisualProofs. Thanks! \n\nThis animation is based on a visual proof by Bikash Chakraborty from volume 41 issue no. 56 of the Mathematical Intelligencer in 2019 (https://doi.org/10.1007/s00283-018-9816-4)\n\nHere is wide format, high def, slightly longer version: https://youtu.be/I7wiS9rH2h0\n\nFor two visual proofs of a more general version of this inequality, see  \nhttps://youtu.be/JE_YeVlSqLY ;\nhttps://youtu.be/SaYs856QYlI ;\nand\nhttps://youtu.be/DOcsgckdvxU\n\n#math\u200b #inequality #manim\u200b #animation\u200b #theorem\u200b #pww\u200b #proofwithoutwords\u200b #visualproof\u200b #proof\u200b #iteachmath #calculus #mathshorts\u200b #mathvideo\u200b #mtbos\u200b #euler #pi #piday\n\nTo learn more about animating with manim, check out:\nhttps://manim.community", "lengthSeconds": 54, "width": 360, "height": 640}
{"video_id": "g1NFwURsn50", "transcript": [{"start": 0.979, "duration": 5.141, "text": "here's something interesting the area of"}, {"start": 4.02, "duration": 4.56, "text": "a regular octagon equals the product of"}, {"start": 6.12, "duration": 5.099, "text": "its longest and shortest diagonal here's"}, {"start": 8.58, "duration": 6.059, "text": "a visual proof"}, {"start": 11.219, "duration": 5.281, "text": "we can split the Octagon like this"}, {"start": 14.639, "duration": 4.481, "text": "and get these triangles"}, {"start": 16.5, "duration": 6.359, "text": "let's rearrange these parts"}, {"start": 19.12, "duration": 5.48, "text": "[Music]"}, {"start": 22.859, "duration": 4.281, "text": "all the parts combine to form this"}, {"start": 24.6, "duration": 2.54, "text": "rectangle"}, {"start": 27.92, "duration": 5.56, "text": "so the area of the Octagon is same as"}, {"start": 30.779, "duration": 4.141, "text": "the area of this rectangle"}, {"start": 33.48, "duration": 3.419, "text": "and we know that the area of the"}, {"start": 34.92, "duration": 3.84, "text": "rectangle equals the product of its"}, {"start": 36.899, "duration": 4.161, "text": "length and breadth"}, {"start": 38.76, "duration": 4.08, "text": "now let's go back to the octagon"}, {"start": 41.06, "duration": 3.7, "text": "[Music]"}, {"start": 42.84, "duration": 3.84, "text": "the length of the rectangle will be same"}, {"start": 44.76, "duration": 3.66, "text": "as the longest diagonal and the breadth"}, {"start": 46.68, "duration": 4.32, "text": "of the rectangle will be same as the"}, {"start": 48.42, "duration": 4.92, "text": "shortest diagonal"}, {"start": 51.0, "duration": 4.079, "text": "so the area of the Octagon is same as"}, {"start": 53.34, "duration": 3.31, "text": "the product of its longest and shortest"}, {"start": 55.079, "duration": 5.641, "text": "diagonals"}, {"start": 56.65, "duration": 4.07, "text": "[Music]"}], "title": "A cool way to find the area of an octagon", "description": "The area of a regular octagon equals the product of its shortest and longest diagonals.\n\n\"Learn more about the Cuemath Way of learning Math: \nVISIT our Website: https://s.cuemath.com/QwbIxn \n\nFall in love with math with more fun content on all social platforms\n- LIKE us on Facebook: https://www.facebook.com/Cuemath\n- FOLLOW us on Twitter: https://twitter.com/cuemath\n- FOLLOW us on Instagram: https://www.instagram.com/cuemath\"\n\n#math #geometry #animation #education #learning #proof #shorts #visual", "lengthSeconds": 59, "width": 360, "height": 640}
{"video_id": "OCIHS2vmdRA", "transcript": [{"start": 7.759, "duration": 6.161, "text": "start with two angles alpha and beta"}, {"start": 10.0, "duration": 6.64, "text": "both in the interval zero to pi over two"}, {"start": 13.92, "duration": 5.519, "text": "draw a triangle with horizontal base and"}, {"start": 16.64, "duration": 5.84, "text": "an altitude that cuts the triangle into"}, {"start": 19.439, "duration": 5.201, "text": "pieces with angle alpha and angle beta"}, {"start": 22.48, "duration": 5.92, "text": "let's suppose that the legs on this"}, {"start": 24.64, "duration": 3.76, "text": "triangle are called a and b"}, {"start": 29.359, "duration": 3.921, "text": "the area of this triangle can be"}, {"start": 31.199, "duration": 4.161, "text": "determined using the side angle side"}, {"start": 33.28, "duration": 4.32, "text": "formula for the area of the triangle and"}, {"start": 35.36, "duration": 4.4, "text": "is thus one half a times b times the"}, {"start": 37.6, "duration": 3.68, "text": "sine of alpha plus beta"}, {"start": 39.76, "duration": 3.68, "text": "suppose the altitude from the angle"}, {"start": 41.28, "duration": 5.04, "text": "alpha plus beta has length y"}, {"start": 43.44, "duration": 4.56, "text": "this altitude cuts the triangle into two"}, {"start": 46.32, "duration": 4.0, "text": "sub-triangles that are both right"}, {"start": 48.0, "duration": 4.0, "text": "triangles and both with a side length of"}, {"start": 50.32, "duration": 3.6, "text": "y"}, {"start": 52.0, "duration": 5.039, "text": "according to the left triangle y is"}, {"start": 53.92, "duration": 5.279, "text": "adjacent to alpha so y is equal to a"}, {"start": 57.039, "duration": 4.561, "text": "times the cosine of alpha and in the"}, {"start": 59.199, "duration": 5.761, "text": "right triangle y is adjacent to beta so"}, {"start": 61.6, "duration": 5.12, "text": "y is equal to b times the cosine of beta"}, {"start": 64.96, "duration": 4.96, "text": "but the value y is part of both"}, {"start": 66.72, "duration": 6.079, "text": "triangles so we can use opposite values"}, {"start": 69.92, "duration": 6.08, "text": "on opposite triangles like this"}, {"start": 72.799, "duration": 5.761, "text": "now the original triangle must have area"}, {"start": 76.0, "duration": 4.32, "text": "given by the sum of the areas of the two"}, {"start": 78.56, "duration": 4.239, "text": "sub triangles"}, {"start": 80.32, "duration": 4.64, "text": "but these two areas can be computed"}, {"start": 82.799, "duration": 4.801, "text": "using the side angle side formula for a"}, {"start": 84.96, "duration": 5.04, "text": "triangle again so we have that one"}, {"start": 87.6, "duration": 4.08, "text": "triangle is one half times a times b"}, {"start": 90.0, "duration": 4.0, "text": "times the cosine of beta times sine of"}, {"start": 91.68, "duration": 4.56, "text": "alpha and the other one is one half b"}, {"start": 94.0, "duration": 3.84, "text": "times a cosine alpha times the sine of"}, {"start": 96.24, "duration": 3.68, "text": "beta"}, {"start": 97.84, "duration": 4.16, "text": "we can cancel the one half a times b"}, {"start": 99.92, "duration": 3.839, "text": "from both sides of this equation and"}, {"start": 102.0, "duration": 3.68, "text": "that results in the formula sine of"}, {"start": 103.759, "duration": 3.68, "text": "alpha plus beta equals the sine of alpha"}, {"start": 105.68, "duration": 4.96, "text": "times the cosine of beta plus the cosine"}, {"start": 107.439, "duration": 7.881, "text": "of alpha times the sine of beta"}, {"start": 110.64, "duration": 4.68, "text": "this is the sine of a sum formula"}], "title": "Sine of a Sum I (visual proof; trigonometry)", "description": "This is a short, animated visual proof of the sum formula for the sine function using the Side-Angle-Side triangle area formula. This theorem relates the sine of a sum to the sum of products of sines and cosines. #mathshorts #mathvideo #math #trigonometry #sine #sinofsum  #triangle #manim #animation #theorem #pww #proofwithoutwords #visualproof #proof #circle #sideangleside #trianglearea \n\nThis animation is based on a proof due to Christopher Brueningsen from the April 1993 issue of Mathematics Magazine (https://doi.org/10.2307/2691126) page 135.\n\nTo learn more about animating with manim, check out:\nhttps://manim.community\n____________________________________________________\nBackground music:\nCinematic Ambient Beat | STEPS by Alex-Productions | https://www.youtube.com/channel/UCx0_M61F81Nfb-BRXE-SeVA\nMusic promoted by https://www.free-stock-music.com\nCreative Commons Attribution 3.0 Unported License\nhttps://creativecommons.org/licenses/by/3.0/deed.en_US", "lengthSeconds": 127, "width": 640, "height": 360}
{"video_id": "XjVTfw5E0bQ", "transcript": [{"start": 7.61, "duration": 3.04, "text": "[\u0645\u0648\u0633\u064a\u0642\u0649]"}, {"start": 20.39, "duration": 3.059, "text": "[\u0645\u0648\u0633\u064a\u0642\u0649]"}], "title": "quadratic formula#proof#visual#amazinn#reel#for you", "description": "", "lengthSeconds": 23, "width": 360, "height": 640}
{"video_id": "jMtoy-wEKx4", "transcript": [{"start": 0.12, "duration": 4.32, "text": "the sum of the first n positive integers"}, {"start": 2.46, "duration": 3.899, "text": "can be represented by a triangular array"}, {"start": 4.44, "duration": 4.56, "text": "of dots as pictured the first row has"}, {"start": 6.359, "duration": 4.801, "text": "one dot and the nthrow has n dots we can"}, {"start": 9.0, "duration": 4.32, "text": "also add the N plus first row like this"}, {"start": 11.16, "duration": 3.72, "text": "where there are n plus one dots now we"}, {"start": 13.32, "duration": 3.539, "text": "can build a one-to-one correspondence"}, {"start": 14.88, "duration": 3.54, "text": "between the dots in the Triangular array"}, {"start": 16.859, "duration": 3.541, "text": "representing the sum of the first n"}, {"start": 18.42, "duration": 4.38, "text": "positive integers and pairs of dots in"}, {"start": 20.4, "duration": 4.619, "text": "the N plus first row the dot in Row 1"}, {"start": 22.8, "duration": 4.979, "text": "maps to the pair of dots in row N plus"}, {"start": 25.019, "duration": 5.34, "text": "one that are n units apart the two dots"}, {"start": 27.779, "duration": 4.5, "text": "in row two map to the two pairs in row"}, {"start": 30.359, "duration": 4.2, "text": "one plus one that are n minus one units"}, {"start": 32.279, "duration": 4.881, "text": "apart this process continues all the way"}, {"start": 34.559, "duration": 5.52, "text": "down until we see the dots in row N"}, {"start": 37.16, "duration": 5.5, "text": "correspond to the pairs of dots in row N"}, {"start": 40.079, "duration": 4.381, "text": "plus one that are one unit apart sure"}, {"start": 42.66, "duration": 4.5, "text": "enough we've mapped to every possible"}, {"start": 44.46, "duration": 4.619, "text": "pair of two dots in row N plus one"}, {"start": 47.16, "duration": 3.84, "text": "therefore the sum of the first and"}, {"start": 49.079, "duration": 3.96, "text": "positive integers must be n plus one"}, {"start": 51.0, "duration": 4.76, "text": "choose two which is n plus one times n"}, {"start": 53.039, "duration": 2.721, "text": "divided by two"}], "title": "Bijective sum!", "description": "This is a short, we explore the famous formula for the sum of the first n positive integers via a bijective technique.\n\nIf you like this video, consider subscribing to the channel or consider buying me a coffee: https://www.buymeacoffee.com/VisualProofs. Thanks!\n\nThis proof comes from a compilation video showing 12 ways to prove this formula: https://youtu.be/eHbtc50-qXo\n\nThis animation is based a proof from Loren Larson's article in the last issue of the 1985 College Mathematics Journal : Loren C. Larson, A discrete look at 1 + 2 ++ n, The College Mathematics Journal 16 (1985), no. 5, 369\u2013382 (https://doi.org/10.2307/2686996)\n\nTo learn more about animating with manim, check out:\nhttps://manim.community\n\n#sumformula #sumintegers #integers\u200b #mathvideo\u200b #math\u200b #mtbos\u200b  #animation\u200b #theorem\u200b #pww\u200b #proofwithoutwords\u200b\u200b #proof\u200b #iteachmath #mathematics #combinatorialproof #combinatorics #bijection", "lengthSeconds": 55, "width": 360, "height": 640}
{"video_id": "T6b9xUcBXgI", "transcript": [{"start": 0.53, "duration": 12.96, "text": "[Music]"}, {"start": 18.04, "duration": 3.1, "text": "[Music]"}, {"start": 28.4, "duration": 2.48, "text": "the"}, {"start": 30.119, "duration": 4.761, "text": "going"}, {"start": 30.88, "duration": 4.0, "text": "to I"}, {"start": 37.12, "duration": 6.029, "text": "[Music]"}, {"start": 51.03, "duration": 3.089, "text": "[Music]"}], "title": "Geometric Series (Not my video, Mathematical Visual Proofs on Youtube) #facts #maths #geometry", "description": "", "lengthSeconds": 59, "width": 360, "height": 640}
{"video_id": "er9j4r3hkng", "transcript": [{"start": 0.0, "duration": 5.84, "text": "[Music]"}, {"start": 5.899, "duration": 4.541, "text": "in the cold"}, {"start": 8.22, "duration": 6.14, "text": "alone"}, {"start": 10.44, "duration": 3.92, "text": "there are times when"}, {"start": 14.42, "duration": 4.56, "text": "I'm almost letting it go"}, {"start": 19.38, "duration": 5.059, "text": "the cards were stacked against us both"}, {"start": 27.05, "duration": 8.55, "text": "[Music]"}, {"start": 33.78, "duration": 5.4, "text": "I know"}, {"start": 35.6, "duration": 4.42, "text": "[Music]"}, {"start": 39.18, "duration": 12.39, "text": "I know it's a member"}, {"start": 40.02, "duration": 11.55, "text": "[Music]"}], "title": "North/East Lattice Paths", "description": "This enumeration shows all of the northeast lattice paths on the 4x4 integer grid. The number of such lattice paths to a point (a,b) is counted by the binomial coefficient {a+b choose a}, as is discussed in this video: https://youtu.be/KYpOoA2D63w. So as you watch this video, Pascal's triangle (also known as Halayudha's triangle or just the arithmetic triangle) appears as the counts.\n\nHere is a link to a longer \"synthwave enumeration\" of the same idea (but up to the 5x5 grid and a bit slower): https://youtu.be/9YUl0k2FYzc\n\nIf you like this video, consider subscribing to the channel or consider buying me a coffee: https://www.buymeacoffee.com/VisualProofs. Thanks! \n\nThis animation is based on the well-known interpretation of binomial coefficients as counting lattice paths. You can find more in many textbooks such as Oscar Levin's Discrete Mathematics An Open Introduction (http://discrete.openmathbooks.org/dmoi2/sec_counting-binom.html). \n\n#pascaltriangle #pascalstriangle\u200b #mathvideo\u200b #math\u200b #mtbos\u200b #manim\u200b #animation\u200b #theorem\u200b\u200b #visualproof\u200b #proof\u200b #iteachmath #mathematics #binomialcoefficients #latticepaths #discretemath #combinatorics #enumeration #synthwave \n\nTo learn more about animating with manim, check out:\nhttps://manim.community", "lengthSeconds": 59, "width": 360, "height": 640}
{"video_id": "s4pdePg_m3w", "transcript": [{"start": 0.08, "duration": 4.48, "text": "suppose that the > of 2 is rational so"}, {"start": 2.159, "duration": 5.2, "text": "there exist positive integers A and B"}, {"start": 4.56, "duration": 4.84, "text": "with a^2 equal to 2 b^2 that means that"}, {"start": 7.359, "duration": 5.28, "text": "we have an a by a square that has the"}, {"start": 9.4, "duration": 5.159, "text": "same area as 2 B by B squares according"}, {"start": 12.639, "duration": 4.281, "text": "to the carpet's theorem if we overlay"}, {"start": 14.559, "duration": 4.72, "text": "the 2 B by B squares into the a by a"}, {"start": 16.92, "duration": 4.359, "text": "square since the areas are equal we must"}, {"start": 19.279, "duration": 5.24, "text": "have that the area of overlap which is a"}, {"start": 21.279, "duration": 5.561, "text": "2 b - A by 2 b - a square is equal to"}, {"start": 24.519, "duration": 5.68, "text": "the uncovered area in the a by a square"}, {"start": 26.84, "duration": 6.559, "text": "which is 2 Aus B by A- B squares but"}, {"start": 30.199, "duration": 7.241, "text": "this means that the quantity 2 b - A 2"}, {"start": 33.399, "duration": 5.601, "text": "is equal to 2 * the quantity a - b^ 2 so"}, {"start": 37.44, "duration": 3.4, "text": "we've produced a pair of smaller"}, {"start": 39.0, "duration": 5.0, "text": "positive integers satisfying the same"}, {"start": 40.84, "duration": 5.12, "text": "equation x^2 = 2 y^2 we can repeat this"}, {"start": 44.0, "duration": 3.64, "text": "process over and over again producing a"}, {"start": 45.96, "duration": 3.84, "text": "list of smaller and smaller positive"}, {"start": 47.64, "duration": 4.439, "text": "integers but such an infinite list of"}, {"start": 49.8, "duration": 4.079, "text": "decreasing positive integers can't exist"}, {"start": 52.079, "duration": 3.96, "text": "so we've produced a contradiction this"}, {"start": 53.879, "duration": 3.761, "text": "means that a squ can't equal 2 b^ s with"}, {"start": 56.039, "duration": 4.601, "text": "A and B integers so the square root of"}, {"start": 57.64, "duration": 3.0, "text": "two is irrational"}], "title": "Irrational Root 2!", "description": "In this video, we introduce and prove the \"Carpets Theorem\" and then utilize the theorem to prove that both the square root of two and the square root of three are irrational by a visual infinite descent argument.\n\nIf you like this video, consider subscribing to the channel or consider buying me a coffee: https://www.buymeacoffee.com/VisualProofs. Thanks!\n\nThis animation is based on an argument from Stanley Tennebaum. If you are interested, I recommend this article from John Conway and Joseph Shipman about the irrationality of square root of two (and others):\n\nhttp://dev.mccme.ru/~merzon/mirror/mathtabletalks/files/irrational-conway.pdf\n\n\n#irrationalnumbers #realnumbers\u200b #manim\u200b #math\u200b #mtbos\u200b \u200b #animation\u200b #theorem\u200b\u200b #visualproof\u200b #proof\u200b #iteachmath #mathematics #irrational #carpetstheorem #proofbycontradiction #root2 #algebra #infinitedescent\n\n\nTo learn more about animating with manim, check out:\nhttps://manim.community", "lengthSeconds": 59, "width": 360, "height": 640}
{"video_id": "Za-F8jqKxjA", "transcript": [{"start": 0.05, "duration": 2.51, "text": "[Music]"}, {"start": 8.559, "duration": 3.0, "text": "back"}], "title": "Difference Identity| Visual Proof | Mathletes |Sunny Sir  #maths #algebra #foryou #satisfying", "description": "Difference Identity| Visual Proof| Mathletes | Sunny Sir  #maths #algebra #foryou #satisfying  #cbse #funnyshorts\n\nDo Like Share and Subscribe For more interesting videos\n\nJoin our telegram channel: https://t.me/Mathletes_29\n\nFollow us on Instagram: https://www.instagram.com/mathletes19?igsh=MWo2ZzdlY3cxdjg1MA==", "lengthSeconds": 10, "width": 360, "height": 640}
{"video_id": "T31jS8GPSVw", "transcript": [{"start": 0.0, "duration": 6.06, "text": "the sum of the powers of one-fifth is"}, {"start": 2.879, "duration": 5.64, "text": "one-fourth but why is that well consider"}, {"start": 6.06, "duration": 5.28, "text": "a square whose set length is one and"}, {"start": 8.519, "duration": 5.401, "text": "connect each vertex to the midpoint of"}, {"start": 11.34, "duration": 4.68, "text": "the following Edge like this the Central"}, {"start": 13.92, "duration": 4.56, "text": "Square will have an area of one-fifth"}, {"start": 16.02, "duration": 5.099, "text": "because look I can build five identical"}, {"start": 18.48, "duration": 4.559, "text": "squares and therefore also this red"}, {"start": 21.119, "duration": 4.021, "text": "triangle we have an area of one-fifth"}, {"start": 23.039, "duration": 4.5, "text": "and I can repeat the process in the"}, {"start": 25.14, "duration": 5.52, "text": "Central Square like this and I'll get a"}, {"start": 27.539, "duration": 5.581, "text": "new red triangle whose area is one-fifth"}, {"start": 30.66, "duration": 4.739, "text": "times one-fifth that is one over five"}, {"start": 33.12, "duration": 5.22, "text": "squared and I can keep repeating the"}, {"start": 35.399, "duration": 6.18, "text": "process like this till I get this shape"}, {"start": 38.34, "duration": 6.2, "text": "that fits four times in the Square"}, {"start": 41.579, "duration": 2.961, "text": "and therefore"}], "title": "Genius Visual Proof. Sum of Powers.", "description": "Today's math video is about a genius visual proof that the sum of the powers \nof 1 fifth is 1 forth.\n\nThis animation is based on a proof by Rick Mabry from his website\nhttps://lsusmath.rickmabry.org/rmabry/fivesquares/fsq2.gif\nand featured in Roger B. Nelsen's third Proofs Without Words compendium\nhttps://bookstore.ams.org/clrm-52/\n\n\nNew Puzzles every Sunday and Thursday.\nI post math puzzles and their solutions with animations.\nSubscribe: https://www.youtube.com/@guzmat?sub_confirmation=1\n\n\nIf you prefer you can watch the videos in\n\ud83c\uddea\ud83c\uddf8 Spanish: https://www.youtube.com/@guzmat-espanol\n\n\ud83c\uddee\ud83c\uddf9 Italian: https://www.youtube.com/@guzmat-quesiti\n\nEnglish:  https://www.youtube.com/@guzmat\n\n\n             \ud83c\udf1fMy Links\ud83c\udf1f\nTwitter (english): https://twitter.com/GuzMatPuzzles\nFaceBook (english): https://www.facebook.com/GuzMat\nFaceBook (italian): https://www.facebook.com/GuzMatQuesiti\nFaceBook (profile): https://www.facebook.com/GuzMatBase\nInstagram (italian): https://www.instagram.com/guzmat_puzzles/\n\nPersonal Website: http://www.guzman.it\nPersonal Blog: https://guzman-tierno.blogspot.com/", "lengthSeconds": 43, "width": 360, "height": 640}
{"video_id": "wWEZ-rUlBDQ", "transcript": [{"start": 0.0, "duration": 2.179, "text": "foreign"}, {"start": 3.24, "duration": 9.42, "text": "[Music]"}, {"start": 9.719, "duration": 5.88, "text": "suppose we have a right triangle with"}, {"start": 12.66, "duration": 4.379, "text": "sides of length A and B and a hypotenuse"}, {"start": 15.599, "duration": 3.421, "text": "of length C"}, {"start": 17.039, "duration": 4.921, "text": "we can take a second copy of this"}, {"start": 19.02, "duration": 6.14, "text": "triangle and place it here which forms a"}, {"start": 21.96, "duration": 3.2, "text": "square shape like this"}, {"start": 25.38, "duration": 6.18, "text": "we can also fill out the two triangles"}, {"start": 28.56, "duration": 5.64, "text": "to rectangles in this manner here"}, {"start": 31.56, "duration": 5.159, "text": "all three of these new shapes the yellow"}, {"start": 34.2, "duration": 5.82, "text": "Square the blue rectangle and the green"}, {"start": 36.719, "duration": 5.581, "text": "rectangle all have a side of length C"}, {"start": 40.02, "duration": 4.98, "text": "we can use the altitude of the original"}, {"start": 42.3, "duration": 4.14, "text": "triangles to then create the final"}, {"start": 45.0, "duration": 3.899, "text": "missing square up in the upper right"}, {"start": 46.44, "duration": 4.32, "text": "corner here"}, {"start": 48.899, "duration": 4.381, "text": "the full diagram that we have just"}, {"start": 50.76, "duration": 4.68, "text": "created is a square with a side length"}, {"start": 53.28, "duration": 4.68, "text": "of C plus h"}, {"start": 55.44, "duration": 6.36, "text": "therefore the area contained in this"}, {"start": 57.96, "duration": 6.839, "text": "square is the quantity C plus h squared"}, {"start": 61.8, "duration": 5.58, "text": "but now move the four triangles used to"}, {"start": 64.799, "duration": 4.201, "text": "create the two rectangles in this"}, {"start": 67.38, "duration": 6.36, "text": "fashion here"}, {"start": 69.0, "duration": 7.619, "text": "the resulting shape is now two squares"}, {"start": 73.74, "duration": 3.96, "text": "the upper right square has a side length"}, {"start": 76.619, "duration": 3.54, "text": "of H"}, {"start": 77.7, "duration": 5.12, "text": "and the new square that is diagonally"}, {"start": 80.159, "duration": 5.82, "text": "tilted has a side length of a plus B"}, {"start": 82.82, "duration": 6.4, "text": "therefore the area contained is the"}, {"start": 85.979, "duration": 6.541, "text": "quantity a plus b squared added to the"}, {"start": 89.22, "duration": 5.64, "text": "quantity H squared"}, {"start": 92.52, "duration": 5.04, "text": "this formula implies the Pythagorean"}, {"start": 94.86, "duration": 4.259, "text": "theorem let's see how by revisiting the"}, {"start": 97.56, "duration": 5.16, "text": "original triangle"}, {"start": 99.119, "duration": 6.301, "text": "this triangle had sides of length A and"}, {"start": 102.72, "duration": 4.98, "text": "B and a hypotenuse of length C and an"}, {"start": 105.42, "duration": 4.68, "text": "altitude of length h"}, {"start": 107.7, "duration": 4.739, "text": "we can expand the formula algebraically"}, {"start": 110.1, "duration": 4.559, "text": "like this and then we compute the area"}, {"start": 112.439, "duration": 3.901, "text": "of the triangle in two ways one-half"}, {"start": 114.659, "duration": 5.041, "text": "times the base times the height to"}, {"start": 116.34, "duration": 5.34, "text": "realize that a times b equals c times h"}, {"start": 119.7, "duration": 4.08, "text": "therefore in the equation the 2 times c"}, {"start": 121.68, "duration": 4.32, "text": "times H cancels with the 2 times a times"}, {"start": 123.78, "duration": 4.319, "text": "B and the H squares cancel from both"}, {"start": 126.0, "duration": 4.2, "text": "sides resulting in the equation c"}, {"start": 128.099, "duration": 4.261, "text": "squared equals a squared plus b squared"}, {"start": 130.2, "duration": 4.2, "text": "this ladder formula is known as the"}, {"start": 132.36, "duration": 3.599, "text": "Pythagorean theorem it says that the"}, {"start": 134.4, "duration": 3.479, "text": "square of a hypotenuse of a right"}, {"start": 135.959, "duration": 5.121, "text": "triangle is equal to the sum of the"}, {"start": 137.879, "duration": 3.201, "text": "squares of the leg lengths"}, {"start": 141.66, "duration": 8.689, "text": "foreign"}, {"start": 143.86, "duration": 6.489, "text": "[Music]"}], "title": "Pythagorean Theorem VII (visual proof)", "description": "This is a short, animated visual proof of an extended version of the Pythagorean theorem (the right triangle theorem) that implies the Pythagorean theorem. This theorem states the square of the hypotenuse of a right triangle is equal to the sum of squares of the two other side lengths, and we prove the theorem using rigid translations of objects and the triangle area formula. #mathshorts #mathvideo #math #pythagoreantheorem #pythagorean #triangle #manim #animation #theorem #pww #proofwithoutwords #visualproof #proof #leonardodavinci #davinci\n\nIf you liked this video, please consider subscribing to the channel and checking out my other videos.\n\nThis animation is based on a recent proof by Roger B. Nelsen and Claudi Alsina from the January 2022 issue of The College Math Journal (https://doi.org/10.1080/07468342.2022.1997306). \n\nFor other visual proofs of this same fact check out: \nhttps://youtu.be/q28H5NY5UVU\nhttps://youtu.be/yfGtbNgcrQ8\nhttps://youtu.be/JRjPsuJ7S-g\nhttps://youtu.be/TsKTteGJpZU\nhttps://youtu.be/ZlGaQdNRdqA\nhttps://youtu.be/uqGjs_n-f7U\n\nTo learn more about animating with manim, check out:\nhttps://manim.community\n___________________________________\nMusic in this video:\nTranquil Fields (Tense) by Alexander Nakarada | https://www.serpentsoundstudios.com\nMusic promoted by https://www.free-stock-music.com\nAttribution 4.0 International (CC BY 4.0)\nhttps://creativecommons.org/licenses/by/4.0/", "lengthSeconds": 149, "width": 640, "height": 360}
{"video_id": "xeZU4oWO6p0", "transcript": [{"start": 0.34, "duration": 14.86, "text": "[Music]"}, {"start": 12.139, "duration": 3.061, "text": "thank you"}, {"start": 18.39, "duration": 14.199, "text": "[Music]"}], "title": "Why is \u03c0 = 3.14\u2026??? #pivalue #circle #circles #geometry  #maths  #mathematics", "description": "", "lengthSeconds": 31, "width": 360, "height": 640}
{"video_id": "nZlLCFSGWG8", "transcript": [{"start": 0.0, "duration": 4.62, "text": "let A and D be fixed real numbers in N"}, {"start": 2.52, "duration": 4.799, "text": "be a non-negative integer then consider"}, {"start": 4.62, "duration": 5.519, "text": "the sum a plus a plus d plus a plus 2"}, {"start": 7.319, "duration": 4.441, "text": "times D and so on up to a plus n times D"}, {"start": 10.139, "duration": 3.54, "text": "we can represent this sum of an"}, {"start": 11.76, "duration": 3.539, "text": "arithmetic sequence using a staircase"}, {"start": 13.679, "duration": 3.481, "text": "diagram consisting of a bunch of"}, {"start": 15.299, "duration": 4.021, "text": "rectangles the first level has a"}, {"start": 17.16, "duration": 4.5, "text": "rectangle with area a the next one has"}, {"start": 19.32, "duration": 4.619, "text": "two rectangles in closing area a plus d"}, {"start": 21.66, "duration": 5.1, "text": "and so on up to the last level which has"}, {"start": 23.939, "duration": 5.1, "text": "an area in closing a plus n times D but"}, {"start": 26.76, "duration": 4.2, "text": "then imagine a second staircase diagram"}, {"start": 29.039, "duration": 3.661, "text": "rotated and fit together with the"}, {"start": 30.96, "duration": 4.439, "text": "original to form a rectangle as shown"}, {"start": 32.7, "duration": 5.16, "text": "here this rectangle has a length given"}, {"start": 35.399, "duration": 4.801, "text": "by 2 times a plus n times d and a width"}, {"start": 37.86, "duration": 5.28, "text": "given by n plus one so the area enclosed"}, {"start": 40.2, "duration": 5.1, "text": "in this rectangle is n plus 1 times 2A"}, {"start": 43.14, "duration": 4.439, "text": "plus n times D and that must be equal to"}, {"start": 45.3, "duration": 4.02, "text": "2 times the original arithmetic sum this"}, {"start": 47.579, "duration": 4.741, "text": "means that the sum where I ranges from 0"}, {"start": 49.32, "duration": 7.46, "text": "to n of a plus I D must be n plus 1"}, {"start": 52.32, "duration": 4.46, "text": "times 2A plus ND all divided by 2."}], "title": "Visual Arithmetic Sums", "description": "This is a short, animated visual proof demonstrating how to visualize the sum of an arithmetic sequence, producing a closed formula for such a sum. \n\nIf you like this video, consider subscribing to the channel or consider buying me a coffee: https://www.buymeacoffee.com/VisualProofs. Thanks!\n\nFor a wide format wordless version (with epic music), check this link : https://youtu.be/uYaFE-1Lm5Y\n\nThis animation is based on a well known visual proof technique (and it can be found in Roger Nelsen\u2019s book Icons of Mathematics). Specializing to a = 0 and d = 1 gives the sum of the first n integers (triangular numbers) formula. Specializing to a = 1 and d = 2 gives the sum of the odd integers (which yields the squares). In general, we can get any polygonal number formula from this fact (in addition to many others).\n\nTo learn more about animating with manim, check out:\nhttps://manim.community\n\n#mathshorts \u200b #mathvideo\u200b #math\u200b #numbertheory #mtbos\u200b #manim\u200b #animation\u200b #theorem\u200b #pww\u200b #proofwithoutwords\u200b #visualproof\u200b #proof\u200b #iteachmath #squares #triangularnumbers #arithmeticsum #arithmeticsequence #finitesums", "lengthSeconds": 55, "width": 360, "height": 640}
{"video_id": "BojCVIc9hf4", "transcript": [{"start": 0.599, "duration": 5.7, "text": "let's learn a fascinating fact about an"}, {"start": 3.54, "duration": 6.18, "text": "amazing curve"}, {"start": 6.299, "duration": 5.281, "text": "imagine a circle lying on a flat line"}, {"start": 9.72, "duration": 4.14, "text": "let's label one point on the circle"}, {"start": 11.58, "duration": 4.44, "text": "that's tangent to the line and let's"}, {"start": 13.86, "duration": 4.94, "text": "start rolling this circle over and over"}, {"start": 16.02, "duration": 2.78, "text": "and over again"}, {"start": 19.619, "duration": 7.561, "text": "if we Trace out the path traversed by"}, {"start": 22.619, "duration": 7.021, "text": "the Red Dot we get this curve here"}, {"start": 27.18, "duration": 4.74, "text": "this curve is known as the cycloid and"}, {"start": 29.64, "duration": 5.7, "text": "it will continue forever as long as we"}, {"start": 31.92, "duration": 6.36, "text": "let the circle roll but we can see that"}, {"start": 35.34, "duration": 6.239, "text": "the cycloid is really one curve repeated"}, {"start": 38.28, "duration": 5.64, "text": "over and over and over again so let's"}, {"start": 41.579, "duration": 4.021, "text": "study this one section of the cycloid"}, {"start": 43.92, "duration": 6.139, "text": "and see if we can find something"}, {"start": 45.6, "duration": 4.459, "text": "interesting about that particular curve"}, {"start": 50.579, "duration": 5.581, "text": "so assume that we have a circle with a"}, {"start": 52.92, "duration": 4.92, "text": "radius of R and we label one point right"}, {"start": 56.16, "duration": 3.84, "text": "at the bottom of the circle tangent to"}, {"start": 57.84, "duration": 4.92, "text": "the surface on which we roll"}, {"start": 60.0, "duration": 5.159, "text": "we roll This Circle exactly one full"}, {"start": 62.76, "duration": 5.399, "text": "turn to create one portion of the"}, {"start": 65.159, "duration": 5.28, "text": "cycloid as shown here one question of"}, {"start": 68.159, "duration": 4.441, "text": "interest about this curve is can we find"}, {"start": 70.439, "duration": 4.441, "text": "the area contained underneath the"}, {"start": 72.6, "duration": 5.04, "text": "cycloid and above the line upon which we"}, {"start": 74.88, "duration": 4.8, "text": "rolled that is can we find this area in"}, {"start": 77.64, "duration": 3.839, "text": "terms of the radius of the circle that"}, {"start": 79.68, "duration": 3.78, "text": "we rolled we know that the distance"}, {"start": 81.479, "duration": 4.68, "text": "between the two endpoints of the cycloid"}, {"start": 83.46, "duration": 4.92, "text": "is 2 pi r because it corresponds to the"}, {"start": 86.159, "duration": 4.32, "text": "circumference of the circle we rolled"}, {"start": 88.38, "duration": 4.32, "text": "we can take this circle we rolled and"}, {"start": 90.479, "duration": 4.981, "text": "place it in the middle like this and the"}, {"start": 92.7, "duration": 5.16, "text": "circle covers some of the area it also"}, {"start": 95.46, "duration": 5.94, "text": "splits the rest of the cycloid into two"}, {"start": 97.86, "duration": 5.7, "text": "symmetric pieces on either side"}, {"start": 101.4, "duration": 4.44, "text": "but now we do something interesting we"}, {"start": 103.56, "duration": 3.12, "text": "apply the two-dimensional cavalieri's"}, {"start": 105.84, "duration": 3.84, "text": "principle"}, {"start": 106.68, "duration": 4.74, "text": "to do this we will go along each of the"}, {"start": 109.68, "duration": 4.02, "text": "two symmetric shapes on either side of"}, {"start": 111.42, "duration": 4.32, "text": "this circle and we will take the width"}, {"start": 113.7, "duration": 4.559, "text": "of the shape at a given point and shift"}, {"start": 115.74, "duration": 5.4, "text": "it over to the left as shown here"}, {"start": 118.259, "duration": 4.741, "text": "because we keep the widths the same at"}, {"start": 121.14, "duration": 4.799, "text": "each point and don't change the height"}, {"start": 123.0, "duration": 5.819, "text": "the resulting shape that arises out of"}, {"start": 125.939, "duration": 6.02, "text": "this process will have the same area as"}, {"start": 128.819, "duration": 3.14, "text": "the shape we started with"}, {"start": 132.72, "duration": 4.8, "text": "it might not seem any easier to find the"}, {"start": 135.48, "duration": 3.96, "text": "area of these resulting shapes but now"}, {"start": 137.52, "duration": 4.56, "text": "we can apply another property the"}, {"start": 139.44, "duration": 4.2, "text": "symmetry of the circle if we draw the"}, {"start": 142.08, "duration": 3.659, "text": "lines coming from the center of the"}, {"start": 143.64, "duration": 4.739, "text": "cycloid to the top of these new shapes"}, {"start": 145.739, "duration": 5.58, "text": "we see that because of the symmetry of"}, {"start": 148.379, "duration": 5.701, "text": "the circle this area above the line"}, {"start": 151.319, "duration": 4.92, "text": "matches the missing area below the line"}, {"start": 154.08, "duration": 3.72, "text": "and likewise this happens on the other"}, {"start": 156.239, "duration": 3.901, "text": "side as well"}, {"start": 157.8, "duration": 4.2, "text": "after applying cavalieri's principle and"}, {"start": 160.14, "duration": 3.78, "text": "the Symmetry argument we see that we've"}, {"start": 162.0, "duration": 4.319, "text": "decomposed the area under the cycloid"}, {"start": 163.92, "duration": 5.76, "text": "into a circle of radius R and two"}, {"start": 166.319, "duration": 5.221, "text": "triangles of height 2R and base pi times"}, {"start": 169.68, "duration": 5.22, "text": "r"}, {"start": 171.54, "duration": 5.82, "text": "the area of the circle is pi times r"}, {"start": 174.9, "duration": 5.04, "text": "squared the area of both of the"}, {"start": 177.36, "duration": 5.22, "text": "triangles is also pi times r squared so"}, {"start": 179.94, "duration": 5.46, "text": "the area under the cycloid is 3 times pi"}, {"start": 182.58, "duration": 5.76, "text": "times r squared where R is the radius of"}, {"start": 185.4, "duration": 5.52, "text": "the circle rolled to create the cycloid"}, {"start": 188.34, "duration": 5.34, "text": "this is a truly amazing fact the area"}, {"start": 190.92, "duration": 5.22, "text": "under the cycloid is three times the"}, {"start": 193.68, "duration": 5.0, "text": "area of the circle rolled to create the"}, {"start": 196.14, "duration": 2.54, "text": "cycloid"}], "title": "The Amazing Cycloid Area (Visual Proof via Cavalieri)", "description": "In this video, we show how to find the area under one arch of the cycloid, which is a curve generated by tracing the path of a single point on a circle as the circle rolls on a flat surface. The cycloid area has a surprisingly nice connection to the area of the rolled circle.\n\nIf you like this video, consider subscribing to the channel or consider buying me a coffee: https://www.buymeacoffee.com/VisualProofs. Thanks!\n\nThis animation is based on a visual proof by Richard M. Beekman from the February 1993 issue of Mathematics Magazine (https://www.jstor.org/stable/2690472 - page 39).\n\n\nThis animation is based can also be found in Roger Nelsen\u2019s book Proofs without Words (I may receive a commission from Amazon, at no cost to you, if you click the link below):\n\nhttps://amzn.to/456nvfg\n\n#cycloid #cavalieri  #cavalieriprinciple\u200b #manim\u200b #math\u200b #mtbos\u200b #animation\u200b #theorem\u200b\u200b #visualproof\u200b #proof\u200b #iteachmath #mathematics #area #circle #dissection #algebra #triangle\n\nTo learn more about animating with manim, check out:\nhttps://manim.community", "lengthSeconds": 205, "width": 640, "height": 360}
{"video_id": "ecgIbvq_XzE", "transcript": [{"start": 1.18, "duration": 58.88, "text": "[\u0645\u0648\u0633\u064a\u0642\u0649]"}], "title": "Visual proof of (A + B)^2 #shorts #maths #cbse #algebra  #academicdiary", "description": "think of a square when you see x^2. What about x^3?? Cubes :)", "lengthSeconds": 58, "width": 360, "height": 640}
{"video_id": "p3_7bOjiWm8", "transcript": [{"start": 1.26, "duration": 4.2, "text": "thank you"}, {"start": 2.33, "duration": 3.13, "text": "[Music]"}, {"start": 7.62, "duration": 5.34, "text": "suppose we have a right triangle and off"}, {"start": 10.62, "duration": 4.8, "text": "each of the legs we have a semicircle"}, {"start": 12.96, "duration": 3.6, "text": "with diameter equal to the leg of the"}, {"start": 15.42, "duration": 3.779, "text": "triangle"}, {"start": 16.56, "duration": 5.4, "text": "furthermore imagine a semi-circle that"}, {"start": 19.199, "duration": 4.861, "text": "has a diameter along the hypotenuse"}, {"start": 21.96, "duration": 3.84, "text": "according to the Pythagorean theorem the"}, {"start": 24.06, "duration": 4.2, "text": "area of the large semicircle on the"}, {"start": 25.8, "duration": 4.799, "text": "hypotenuse equals the sum of the areas"}, {"start": 28.26, "duration": 4.38, "text": "of the other two semicircles"}, {"start": 30.599, "duration": 4.861, "text": "so in this picture the Blue Area"}, {"start": 32.64, "duration": 4.86, "text": "counteracts the yellow area now when we"}, {"start": 35.46, "duration": 4.259, "text": "reflect the semicircle on the hypotenuse"}, {"start": 37.5, "duration": 4.62, "text": "about the hypotenuse and cancel the blue"}, {"start": 39.719, "duration": 4.261, "text": "and yellow areas of overlap we see that"}, {"start": 42.12, "duration": 4.14, "text": "the right triangle is filled and we've"}, {"start": 43.98, "duration": 4.62, "text": "created two loons off of the legs of the"}, {"start": 46.26, "duration": 4.44, "text": "right triangle like this"}, {"start": 48.6, "duration": 3.959, "text": "but from the Pythagorean theorem we know"}, {"start": 50.7, "duration": 4.74, "text": "that the blue shaded area and the yellow"}, {"start": 52.559, "duration": 6.48, "text": "shaded area are still equal"}, {"start": 55.44, "duration": 5.759, "text": "this means that the combined area of the"}, {"start": 59.039, "duration": 4.801, "text": "two loons constructed on the legs of a"}, {"start": 61.199, "duration": 5.1, "text": "given right triangle must be equal to"}, {"start": 63.84, "duration": 4.68, "text": "the area of that right triangle"}, {"start": 66.299, "duration": 2.851, "text": "this is a theorem due to Hippocrates of"}, {"start": 68.52, "duration": 5.179, "text": "chios"}, {"start": 69.15, "duration": 4.549, "text": "[Music]"}], "title": "Lunes off a Right Triangle (visual proof)", "description": "This is a short, animated visual proof showing that the area of a square inscribed in a circle is equal to the area of the four lunes created by drawing semicircles off of the sides of the square. #manim #math #mathvideo #mathshorts #geometry #animation #theorem #pww #proofwithoutwords #visualproof #proof #area #root3 #pww\u200b #proofwithoutwords\u200b \u200b #proof\u200b #algebra #areas #mathematics\u200b #mathvideo\u200b #mtbos \n\nCheck out these related videos:\nhttps://youtu.be/-IKyUjRcFDw\nhttps://youtu.be/5VaQVXECaVM\nhttps://youtu.be/6DnnJlzbCC4\nhttps://youtu.be/PNQxS6antBY\nhttps://youtu.be/vq853dCS1Fs\n\nIf you like this video, please click \"like\" and consider subscribing and checking out my other videos.\n\nThis animation is based on a proof by Eugene A. Margerum and Michael M. McDonnell from the December 1997  issues of Mathematics Magazine (https://doi.org/10.2307/2691177 ) page 380. The letter to the editor in the  June 1998 issue of the same journal pointed out that Hippocrates of Chios proved this in the 5th century BC E(https://www.jstor.org/stable/2691213 ) page 224. \n\nTo learn more about animating with manim, check out:\nhttps://manim.community\n_____________________________\nMusic in this video:\nCambodian Odyssey by Kevin MacLeod is licensed under a Creative Commons Attribution 4.0 license. https://creativecommons.org/licenses/by/4.0/\n\nSource: http://incompetech.com/music/royalty-free/index.html?isrc=USUAN1100585\n\nArtist: http://incompetech.com/", "lengthSeconds": 76, "width": 640, "height": 360}
{"video_id": "lIqv5zKAKKE", "transcript": [{"start": 11.76, "duration": 16.739, "text": "[\u0645\u0648\u0633\u064a\u0642\u0649]"}, {"start": 29.96, "duration": 4.78, "text": "\u200fy"}, {"start": 31.64, "duration": 3.1, "text": "[\u0645\u0648\u0633\u064a\u0642\u0649]"}], "title": "#trending #mathematics Visual Proof: Factors of Quadratic Equation Part 1 x^2+3x+2", "description": "#trending #mathematics Visual Proof: Factors of Quadratic Equation Part 1 x^2+3x+2\n\nOther Videos\nFibonacci Sequence, Golden Ratio, How to draw Golden Spiral:\nhttps://youtu.be/yHBa3EvDPco\nRadius and Tangent Vectors: \nhttps://youtube.com/shorts/KHSafRET3Sk\nDe Morgan's laws: Proof by Venn Diagrams:\nhttps://youtube.com/shorts/pR-REo4aBYk?feature=share\nVisual Proof of Roots of Quadratic Equation\nhttps://youtube.com/shorts/FdE7rHDZRd8", "lengthSeconds": 33, "width": 360, "height": 640}
{"video_id": "hUtAmgNu9dI", "transcript": [{"start": 0.0, "duration": 6.359, "text": "why is the infinite sum of the powers of"}, {"start": 3.06, "duration": 5.459, "text": "one-third called a geometric series how"}, {"start": 6.359, "duration": 3.721, "text": "can we make sense of the value of this"}, {"start": 8.519, "duration": 3.961, "text": "sum and what does it have to do with"}, {"start": 10.08, "duration": 4.5, "text": "geometry imagine a rectangle with area"}, {"start": 12.48, "duration": 4.08, "text": "one that can be subdivided into three"}, {"start": 14.58, "duration": 4.08, "text": "congruent rectangles each similar to the"}, {"start": 16.56, "duration": 4.26, "text": "original if we shade the top rectangle"}, {"start": 18.66, "duration": 4.92, "text": "we've shaded one third of the area now"}, {"start": 20.82, "duration": 5.039, "text": "we can repeat this process each time we"}, {"start": 23.58, "duration": 4.619, "text": "shade one of the rectangles and we take"}, {"start": 25.859, "duration": 5.221, "text": "the central rectangle and decompose it"}, {"start": 28.199, "duration": 5.641, "text": "into three congruent rectangles shading"}, {"start": 31.08, "duration": 5.94, "text": "one and continuing indefinitely if we do"}, {"start": 33.84, "duration": 6.12, "text": "this process forever or infinitely it"}, {"start": 37.02, "duration": 4.98, "text": "turns out that we have shaded exactly"}, {"start": 39.96, "duration": 4.14, "text": "the infinite sum one-third plus"}, {"start": 42.0, "duration": 4.44, "text": "one-third squared and so on of the"}, {"start": 44.1, "duration": 4.979, "text": "rectangle"}, {"start": 46.44, "duration": 5.459, "text": "but amazingly we see that this shaded"}, {"start": 49.079, "duration": 4.861, "text": "sum fills in exactly half of the"}, {"start": 51.899, "duration": 3.901, "text": "original rectangle since the original"}, {"start": 53.94, "duration": 5.84, "text": "rectangle had an area of one the"}, {"start": 55.8, "duration": 3.98, "text": "infinite sum has an area of one-half"}], "title": "Geometry behind the geometric sum of powers of 1/3!", "description": "This is a short, animated visual proof demonstrating the sum of the infinite series of the powers of 1/3. #shorts\u200b\u200b #math\u200b #calculus #mtbos\u200b #manim\u200b #animation\u200b #theorem\u200b #pww\u200b #proofwithoutwords\u200b #visualproof\u200b #proof\u200b #iteachmath #geometricseries #series #infiniteseries #fractal \n\nTo see the original wordless version of this video, check out: https://youtu.be/RmTZmNrkqss\n\nHere is a longer form video with lots of related infinite geometric sums: https://youtu.be/JteQEN1XPyc\n\nThis animation is based on a proof by Rick Mabry from the January 2001 issue of The College Mathematics Journal page 19 ( https://www.jstor.org/stable/2687214).\n\nFor an alternate series decomposition for the same series, check out \nhttps://youtu.be/vfEDDI3vfHU\n\nTo learn more about animating with manim, check out:\nhttps://manim.community", "lengthSeconds": 59, "width": 360, "height": 640}
{"video_id": "n9icb0lwrqM", "transcript": [{"start": 0.199, "duration": 3.64, "text": "let's see an approximation of a"}, {"start": 1.839, "duration": 4.44, "text": "fascinating curve called the virr"}, {"start": 3.839, "duration": 5.401, "text": "function this function is continuous"}, {"start": 6.279, "duration": 4.561, "text": "everywhere but differentiable nowhere in"}, {"start": 9.24, "duration": 3.76, "text": "particular that means that at every"}, {"start": 10.84, "duration": 4.2, "text": "point on the curve there is no uniquely"}, {"start": 13.0, "duration": 4.119, "text": "defined tangent line as the curve is"}, {"start": 15.04, "duration": 4.76, "text": "essentially an infinite collection of"}, {"start": 17.119, "duration": 4.761, "text": "spikes however there are also no breaks"}, {"start": 19.8, "duration": 5.08, "text": "in the curve it's one continuous"}, {"start": 21.88, "duration": 5.08, "text": "function if you zoom in you see a type"}, {"start": 24.88, "duration": 4.04, "text": "of self-similarity exhibited by this"}, {"start": 26.96, "duration": 3.719, "text": "function that allows it to have these"}, {"start": 28.92, "duration": 4.36, "text": "amazing properties"}, {"start": 30.679, "duration": 4.361, "text": "to get the full vros function replace"}, {"start": 33.28, "duration": 5.52, "text": "the finite sum shown here with an"}, {"start": 35.04, "duration": 3.76, "text": "infinite sum instead"}], "title": "Continuous everywhere but differentiable nowhere : Weierstrass Function Visualization!", "description": "This is a visualization of an approximation of the Weierstrass function, which is a function that is continuous everywhere but differentiable nowhere. The curve is defined as an infinite sum and there are several curves like this given different parameters.\n\nTo buy me a coffee, head over to https://www.buymeacoffee.com/VisualProofs \nThanks!\n\n\n#manim #math\u200b\u200b #mathshorts\u200b #mathvideo\u200b \n #construction #calculus #mtbos\u200b #manim\u200b #animation\u200b #theorem\u200b \u200b\u200b #iteachmath #spiral #weierstrass #continuouseverywhere #continuity #differentiation #differentiablenowhere #function \n\n\nTo learn more about animating with manim, check out:\nhttps://manim.community", "lengthSeconds": 37, "width": 360, "height": 640}
{"video_id": "cqS1HyNYGjM", "transcript": [{"start": 0.12, "duration": 4.44, "text": "here's an interesting fact about the 3 4"}, {"start": 2.28, "duration": 3.88, "text": "five right triangle find the in Center"}, {"start": 4.56, "duration": 3.999, "text": "as the intersection of the three angle"}, {"start": 6.16, "duration": 5.08, "text": "bis sectors draw the in radius to each"}, {"start": 8.559, "duration": 4.601, "text": "of the sides and draw the in circle do"}, {"start": 11.24, "duration": 4.599, "text": "you know the area of the in circle of"}, {"start": 13.16, "duration": 4.16, "text": "the 345 right triangle one way to figure"}, {"start": 15.839, "duration": 3.6, "text": "this out is to compute the area of the"}, {"start": 17.32, "duration": 5.0, "text": "triangle two ways first of all the area"}, {"start": 19.439, "duration": 4.641, "text": "is 1 12 * 3 * 4 but the area of the"}, {"start": 22.32, "duration": 3.879, "text": "triangle can also be found as the sum of"}, {"start": 24.08, "duration": 4.16, "text": "the areas of these three subt triangles"}, {"start": 26.199, "duration": 4.281, "text": "each with a height of R and with bases"}, {"start": 28.24, "duration": 6.319, "text": "of length three four and five so the"}, {"start": 30.48, "duration": 6.88, "text": "area is also 1 12 3 * r + 1 12 4 * r +"}, {"start": 34.559, "duration": 5.801, "text": "12 5 * R from these two computations"}, {"start": 37.36, "duration": 5.679, "text": "then we see that 6 = 6 * R but that"}, {"start": 40.36, "duration": 4.719, "text": "means the in radius R is of length 1 so"}, {"start": 43.039, "duration": 4.801, "text": "that the in circle has a radius of one"}, {"start": 45.079, "duration": 4.881, "text": "giving it an area of Pi it's amazing"}, {"start": 47.84, "duration": 4.68, "text": "that we get PI from the 345 right"}, {"start": 49.96, "duration": 2.56, "text": "triangle"}], "title": "Incircle area in 3-4-5 triangle?", "description": "This short demonstrates an intriguing fact about the area of the incircle of the 3-4-5 right triangle. Using two different methods of computing the area of the 3-4-5 rectangle, we find that the inradius has a length of one so that the area of the incircle is Pi!\n\nTo buy me a coffee, head over to https://www.buymeacoffee.com/VisualProofs \nThanks!\n\nTo see a related video, showing the general connection between the area of a triangle and the inradius, see https://youtu.be/lRDtdIyFhpE?si=QcluFxSoJzEZMcmR\n\n#manim #math #mathvideo #mathshorts #geometry #circle #triangle  #animation #theorem #pww #proofwithoutwords #visualproof #proof #incircle #inradius #incircle #semiperimeter #incenter #pi #righttriangle #area \n\nTo learn more about animating with manim, check out:\nhttps://manim.community", "lengthSeconds": 51, "width": 360, "height": 640}
{"video_id": "yfGtbNgcrQ8", "transcript": [{"start": 2.47, "duration": 14.25, "text": "[Music]"}, {"start": 15.12, "duration": 4.8, "text": "start with a right triangle"}, {"start": 16.72, "duration": 6.399, "text": "with legs of length a and b"}, {"start": 19.92, "duration": 7.84, "text": "and hypotenuse of length c"}, {"start": 23.119, "duration": 6.961, "text": "now draw an a plus b by a plus b square"}, {"start": 27.76, "duration": 3.2, "text": "we can fit four rotated copies of the"}, {"start": 30.08, "duration": 4.159, "text": "triangle"}, {"start": 30.96, "duration": 3.279, "text": "in the square like this"}, {"start": 35.76, "duration": 3.2, "text": "notice that the space in the square not"}, {"start": 37.52, "duration": 5.039, "text": "covered by the triangles"}, {"start": 38.96, "duration": 8.64, "text": "is a square of side length c therefore"}, {"start": 42.559, "duration": 5.041, "text": "this unshaded area is c squared"}, {"start": 50.399, "duration": 3.68, "text": "we can also move the same colored"}, {"start": 52.48, "duration": 4.64, "text": "triangles together"}, {"start": 54.079, "duration": 5.441, "text": "to form two different a by b rectangles"}, {"start": 57.12, "duration": 3.36, "text": "now the unshaded area is given by an a"}, {"start": 59.52, "duration": 3.6, "text": "by a square"}, {"start": 60.48, "duration": 3.12, "text": "and a b by b square so that the unshaded"}, {"start": 63.12, "duration": 3.84, "text": "area"}, {"start": 63.6, "duration": 5.28, "text": "is a squared plus b squared consequently"}, {"start": 66.96, "duration": 3.44, "text": "we have seen that c squared must be"}, {"start": 68.88, "duration": 17.919, "text": "equal to a squared plus b"}, {"start": 70.4, "duration": 16.399, "text": "squared this is the pythagorean theorem"}], "title": "Pythagorean Theorem II (visual proof)", "description": "This is a short, animated visual proof of the Pythagorean theorem (the right triangle theorem) using a dissection of a square in two different ways. This theorem states the square of the hypotenuse of a right triangle is equal to the sum of squares of the two other side lengths. #mathshorts #mathvideo #math #pythagoreantheorem #pythagorean #triangle #manim #animation #theorem #pww #proofwithoutwords #visualproof #proof\n\nThis animation is based on one of the oldest known visual proofs of the fact. According to page 1 of Roger Nelsen's first compendium (Proofs Without Words : https://www.amazon.com/Proofs-without-Words-Exercises-Classroom/dp/0883857006), this proof is adapted from the Chou pei suan ching.\n\nFor another proof of this same fact check out: https://youtu.be/q28H5NY5UVU\n\nTo learn more about animating with manim, check out:\nhttps://manim.community\n__________________________________________________________________\nBackground music, used according to CC BY-SA 3.0, is \"The Road\" from Esteban Orlando:\n\nEsteban Orlando:\nhttps://soundcloud.com/orlando-esteba...\nhttps://www.instagram.com/estebanorla...\n\nhttps://creativecommons.org/licenses/...\nMusic provided by RFM: https://youtu.be/b4KCM1bIq50", "lengthSeconds": 85, "width": 640, "height": 360}
{"video_id": "-1CqCz6hQ7I", "transcript": [{"start": 0.0, "duration": 4.02, "text": "here's an interesting infinite sum one"}, {"start": 2.22, "duration": 3.72, "text": "plus two times a half plus three times a"}, {"start": 4.02, "duration": 3.66, "text": "quarter plus four times an eighth and so"}, {"start": 5.94, "duration": 3.6, "text": "on do you know how to find the sum of"}, {"start": 7.68, "duration": 3.839, "text": "this infinite series what if I showed"}, {"start": 9.54, "duration": 3.48, "text": "you this geometric picture would this"}, {"start": 11.519, "duration": 12.181, "text": "help you figure it out"}, {"start": 13.02, "duration": 10.68, "text": "[Music]"}, {"start": 27.72, "duration": 4.639, "text": "can you figure out the sum now let me"}, {"start": 29.699, "duration": 2.66, "text": "know in the comments"}], "title": "What's this infinite sum?", "description": "This is a short, animated visual proof computing the sum of a differentiated geometric series with ratio given by r=1/2. The proof uses a dissection proof of a 2 by 2 square. \n\nIf you like this video, consider subscribing to the channel or consider buying me a coffee: https://www.buymeacoffee.com/VisualProofs. Thanks!\n\nTo see a bit longer/slower version with the final sum, check out this wordless video: https://youtu.be/2d4CTG1uJs8\n\nThis animation is based on a visual proof by Roger B. Nelsen from the December 1989 issue of Mathematics Magazine (https://www.jstor.org/stable/2689487 p. 332-333).\n\n#math\u200b #infiniteseries #mtbos\u200b #manim\u200b #animation\u200b #theorem\u200b #pww\u200b #proofwithoutwords\u200b #visualproof\u200b #proof\u200b #iteachmath #calculus #geometricseries #series #mathshorts\u200b #mathvideo\u200b \n\n\nTo learn more about animating with manim, check out:\nhttps://manim.community", "lengthSeconds": 31, "width": 360, "height": 640}
{"video_id": "Mi_Uo4eRWcE", "transcript": [{"start": 0.12, "duration": 4.159, "text": "let's see how to use geometry to find"}, {"start": 1.959, "duration": 5.281, "text": "the exact trig values of multiples of"}, {"start": 4.279, "duration": 5.921, "text": "18\u00b0 draw an isoceles triangle with base"}, {"start": 7.24, "duration": 5.88, "text": "angles given by 72\u00b0 vertex angle given"}, {"start": 10.2, "duration": 5.519, "text": "by 36\u00b0 base length of 1 and a side"}, {"start": 13.12, "duration": 4.52, "text": "length of x to construct this triangle"}, {"start": 15.719, "duration": 4.64, "text": "note that it appears naturally inside a"}, {"start": 17.64, "duration": 4.76, "text": "regular pentagon with side length one"}, {"start": 20.359, "duration": 4.881, "text": "use the angle bis sector of the lower"}, {"start": 22.4, "duration": 5.24, "text": "left 72\u00b0 angle to create two subt"}, {"start": 25.24, "duration": 6.08, "text": "triangles one of them the one pictured"}, {"start": 27.64, "duration": 6.439, "text": "in yellow has angles of 36\u00b0 in 72\u00b0 so"}, {"start": 31.32, "duration": 5.04, "text": "the other angle must also be 72\u00b0 that"}, {"start": 34.079, "duration": 5.401, "text": "means this yellow triangle is isoceles"}, {"start": 36.36, "duration": 5.08, "text": "so its other leg length is one but then"}, {"start": 39.48, "duration": 5.36, "text": "the other subt triangle the one shown in"}, {"start": 41.44, "duration": 5.56, "text": "purple here has two 36\u00b0 angles which"}, {"start": 44.84, "duration": 4.28, "text": "means that it's also isoceles and that"}, {"start": 47.0, "duration": 4.399, "text": "means that its other leg length is one"}, {"start": 49.12, "duration": 4.0, "text": "as well and this fact allows us to see"}, {"start": 51.399, "duration": 5.041, "text": "that the base length of the yellow"}, {"start": 53.12, "duration": 5.2, "text": "triangle must be x -1 notice that this"}, {"start": 56.44, "duration": 3.759, "text": "yellow triangle is similar to the"}, {"start": 58.32, "duration": 4.12, "text": "original triangle because they both have"}, {"start": 60.199, "duration": 5.6, "text": "the three same angle measurements but"}, {"start": 62.44, "duration": 7.96, "text": "that means that the ratio x / 1 is equal"}, {"start": 65.799, "duration": 8.161, "text": "to 1 / X - 1 this equation implies that"}, {"start": 70.4, "duration": 7.359, "text": "x^2 - x - 1 must be 0 and the positive"}, {"start": 73.96, "duration": 5.68, "text": "root of this equation is x = 1 + > 5/2"}, {"start": 77.759, "duration": 4.241, "text": "this number is the golden ratio and is"}, {"start": 79.64, "duration": 3.64, "text": "often denoted by the Greek letter Fe"}, {"start": 82.0, "duration": 3.36, "text": "that means the side length of the"}, {"start": 83.28, "duration": 3.32, "text": "original triangle is the golden ratio"}, {"start": 85.36, "duration": 2.92, "text": "and the base length of the yellow"}, {"start": 86.6, "duration": 3.72, "text": "triangle is one less than the golden"}, {"start": 88.28, "duration": 4.519, "text": "ratio here's how we can use use this"}, {"start": 90.32, "duration": 4.0, "text": "diagram to find interesting trig values"}, {"start": 92.799, "duration": 4.921, "text": "first notice that the supplementary"}, {"start": 94.32, "duration": 6.36, "text": "angle to the 72\u00b0 angle is 108\u00b0 as shown"}, {"start": 97.72, "duration": 4.759, "text": "here when we BCT the 108\u00b0 angle like"}, {"start": 100.68, "duration": 4.52, "text": "this cutting the other side in half we"}, {"start": 102.479, "duration": 5.401, "text": "create a right triangle with angles 54\u00b0"}, {"start": 105.2, "duration": 4.0, "text": "and 36\u00b0 and a side length that is half"}, {"start": 107.88, "duration": 4.36, "text": "the golden"}, {"start": 109.2, "duration": 6.12, "text": "ratio we can also bisect the lower left"}, {"start": 112.24, "duration": 6.6, "text": "36\u00b0 angle like this creating a right"}, {"start": 115.32, "duration": 6.88, "text": "triangle with angles 18\u00b0 and 72\u00b0 and a"}, {"start": 118.84, "duration": 5.08, "text": "leg length of the golden ratio - 1 / 2"}, {"start": 122.2, "duration": 3.559, "text": "notice that both these triangles"}, {"start": 123.92, "duration": 4.479, "text": "pictured in blue and purple have a"}, {"start": 125.759, "duration": 5.361, "text": "hypotenuse of length one using the blue"}, {"start": 128.399, "duration": 5.281, "text": "right triangle we see that the S of 54\u00b0"}, {"start": 131.12, "duration": 5.0, "text": "is equal to the cosine of 36\u00b0 and they"}, {"start": 133.68, "duration": 6.48, "text": "are both the golden ratio divided 2"}, {"start": 136.12, "duration": 5.839, "text": "which ends up being the < TK 5 + 1 / 4"}, {"start": 140.16, "duration": 4.0, "text": "from the purple right triangle we see"}, {"start": 141.959, "duration": 5.481, "text": "that the sign of 18\u00b0 which is equal to"}, {"start": 144.16, "duration": 5.76, "text": "the cosine of 72\u00b0 is equal to 1 less"}, {"start": 147.44, "duration": 5.92, "text": "than the golden ratio / 2 and this"}, {"start": 149.92, "duration": 5.319, "text": "simplifies to the < TK 5 - 1 4 we've"}, {"start": 153.36, "duration": 4.599, "text": "been able to use this interesting Golden"}, {"start": 155.239, "duration": 5.801, "text": "Triangle to find trig values for various"}, {"start": 157.959, "duration": 5.0, "text": "multiples of 18\u00b0 can you use the diagram"}, {"start": 161.04, "duration": 3.4, "text": "or perhaps other trig identities to find"}, {"start": 162.959, "duration": 4.0, "text": "the missing trig functions of the"}, {"start": 164.44, "duration": 4.12, "text": "multiples of 18\u00b0 in the first quadrant"}, {"start": 166.959, "duration": 5.721, "text": "for instance can you find the cosine of"}, {"start": 168.56, "duration": 5.24, "text": "54\u00b0 or cosine of 18\u00b0 or the S of 36\u00b0 and"}, {"start": 172.68, "duration": 3.279, "text": "the S of"}, {"start": 173.8, "duration": 3.96, "text": "72\u00b0 let me know in the comments and"}, {"start": 175.959, "duration": 5.601, "text": "check out my playlist on trigonometry if"}, {"start": 177.76, "duration": 3.8, "text": "you like this kind of video"}], "title": "Exact Trig Values for Multiples of 18 Degrees Geometrically (visual proof)", "description": "This is a short, animated visual proof demonstrating how to compute the various exact trigonometric values for multiples of 18 degrees using the geometry of the Golden Triangle. \n\nIf you like this video, consider subscribing to the channel or consider buying me a coffee: https://www.buymeacoffee.com/VisualProofs. Thanks! \n\nCheck out these related videos about trigonometry:\nhttps://youtu.be/88-uO2RFm78 (tangents of 22.5 degrees and 67.5 degrees)\nhttps://youtu.be/A0ZvO_F-LZo?si=9HmF52L34P3LzhEx (tangents of 15 and 75 degrees)\nhttps://youtu.be/XIC7SIdl7pI?si=vM3inxCJHEFqo70f (half angle tangent formula that you can use to get this result)\nhttps://youtu.be/XJ-GaUrY-LE?si=vIpTUyAjHs5wzxvK (arctangents)\n\nThis animation is based on a visual proof from Brian Bradie in the September 2002 issue of The College Mathematics Journal ( https://www.jstor.org/stable/1559057 page 318--319)\n\nTo learn more about animating with manim, check out:\nhttps://manim.community\n\n #mathshorts\u200b #mathvideo\u200b #math\u200b #trigonometry #mtbos\u200b #manim\u200b #animation\u200b #theorem\u200b #pww\u200b #proofwithoutwords\u200b #visualproof\u200b #proof\u200b #iteachmath #squares #pi #triangle #root5 #squareroot #18 #18degrees #36 #54 #72 #degrees #isoscelestriangle #sine  #cosine #sinefunction #cosinefunction #goldenratio #goldentriangle", "lengthSeconds": 180, "width": 640, "height": 360}
{"video_id": "f-kGtZlUpX4", "transcript": [{"start": 0.28, "duration": 4.84, "text": "let's see an elegant semicircle proof of"}, {"start": 2.56, "duration": 5.76, "text": "the double angle formulas for S and"}, {"start": 5.12, "duration": 5.519, "text": "cosine in the XY plane imagine the"}, {"start": 8.32, "duration": 5.439, "text": "origin as a point labeled"}, {"start": 10.639, "duration": 6.361, "text": "O then imagine a point labeled a that's"}, {"start": 13.759, "duration": 5.961, "text": "exactly distance one from o the line"}, {"start": 17.0, "duration": 4.76, "text": "segment O A has a length of one so we"}, {"start": 19.72, "duration": 4.44, "text": "can use it to create a semicircle of"}, {"start": 21.76, "duration": 5.64, "text": "radius one let's label the opposite"}, {"start": 24.16, "duration": 5.959, "text": "point of the semicircle as B we can take"}, {"start": 27.4, "duration": 5.319, "text": "the radius and rotate to some point C on"}, {"start": 30.119, "duration": 6.681, "text": "the semicircle let's imagine that this"}, {"start": 32.719, "duration": 5.801, "text": "Arc AC subtains an angle of 2 * T we can"}, {"start": 36.8, "duration": 3.599, "text": "create a right triangle by dropping a"}, {"start": 38.52, "duration": 3.8, "text": "perpendicular from point C down to the"}, {"start": 40.399, "duration": 5.761, "text": "x-axis and labeling the intersection"}, {"start": 42.32, "duration": 6.2, "text": "Point D in the right triangle OD d c the"}, {"start": 46.16, "duration": 5.559, "text": "leg opposite to the angle 2T has a"}, {"start": 48.52, "duration": 5.719, "text": "length of sine of 2T and the leg"}, {"start": 51.719, "duration": 3.84, "text": "adjacent to the angle of 2T has a length"}, {"start": 54.239, "duration": 3.681, "text": "of cosine"}, {"start": 55.559, "duration": 5.121, "text": "2T now we draw another triangle"}, {"start": 57.92, "duration": 5.639, "text": "connecting the points a c and B like"}, {"start": 60.68, "duration": 4.68, "text": "this by ph's triangle theorem this is a"}, {"start": 63.559, "duration": 5.161, "text": "right angled triangle and the right"}, {"start": 65.36, "duration": 7.84, "text": "angle occurs at Point C now consider the"}, {"start": 68.72, "duration": 6.68, "text": "triangle B pictured here the angle B is"}, {"start": 73.2, "duration": 5.0, "text": "supplementary to the angle 2T and"}, {"start": 75.4, "duration": 5.44, "text": "therefore has a measurement of 180 minus"}, {"start": 78.2, "duration": 4.959, "text": "2T because two of the side lengths of"}, {"start": 80.84, "duration": 3.84, "text": "this triangle are radi of the circle"}, {"start": 83.159, "duration": 4.32, "text": "they both have a length one and that"}, {"start": 84.68, "duration": 4.6, "text": "means that this triangle is isoceles so"}, {"start": 87.479, "duration": 3.441, "text": "the other two angles in this triangle"}, {"start": 89.28, "duration": 4.24, "text": "must be equal equal and therefore they"}, {"start": 90.92, "duration": 4.839, "text": "must both be T so that the full angle"}, {"start": 93.52, "duration": 5.279, "text": "measurement of the triangle is"}, {"start": 95.759, "duration": 5.32, "text": "180\u00b0 notice that the right triangle ACB"}, {"start": 98.799, "duration": 5.441, "text": "has a hypotenuse of length two since it"}, {"start": 101.079, "duration": 5.201, "text": "consists of two radi of the semicircle"}, {"start": 104.24, "duration": 4.36, "text": "the leg opposite to angle T in this"}, {"start": 106.28, "duration": 5.0, "text": "triangle therefore has a length 2 * the"}, {"start": 108.6, "duration": 6.0, "text": "S of T and the leg adjacent to the angle"}, {"start": 111.28, "duration": 5.72, "text": "T pictured has a length 2 * a cosine of"}, {"start": 114.6, "duration": 4.799, "text": "T now we have enough information to"}, {"start": 117.0, "duration": 4.56, "text": "prove our double angle identities this"}, {"start": 119.399, "duration": 4.64, "text": "yellow shaded triangle right here is a"}, {"start": 121.56, "duration": 4.4, "text": "right angled triangle with one angle T"}, {"start": 124.039, "duration": 4.401, "text": "and this red shaded triangle here is"}, {"start": 125.96, "duration": 4.279, "text": "also a right triangle with one angle T"}, {"start": 128.44, "duration": 3.36, "text": "so these two triangles have the same"}, {"start": 130.239, "duration": 4.481, "text": "angle measurements and that means that"}, {"start": 131.8, "duration": 5.04, "text": "they're similar right triangles using"}, {"start": 134.72, "duration": 3.76, "text": "similarity we know that the ratios of"}, {"start": 136.84, "duration": 4.399, "text": "the short leg to the hypotenuse in the"}, {"start": 138.48, "duration": 6.24, "text": "two triangles are equal from this we can"}, {"start": 141.239, "duration": 9.161, "text": "conclude that the S of 2T / 2 * the"}, {"start": 144.72, "duration": 8.36, "text": "cosine of T is equal to 2 * the S of T /"}, {"start": 150.4, "duration": 4.839, "text": "2 but now the twos cancel on the right"}, {"start": 153.08, "duration": 5.72, "text": "side and we can rearrange this equation"}, {"start": 155.239, "duration": 6.521, "text": "so that it reads s of 2T = 2 * the S of"}, {"start": 158.8, "duration": 5.159, "text": "T * the cosine of T we can also use"}, {"start": 161.76, "duration": 4.68, "text": "similarity to see that the ratios of the"}, {"start": 163.959, "duration": 4.161, "text": "long leg length to the hypotenuse are"}, {"start": 166.44, "duration": 4.56, "text": "equal in both the"}, {"start": 168.12, "duration": 6.72, "text": "triangles from this we see that 1 + the"}, {"start": 171.0, "duration": 6.64, "text": "cosine of 2T / 2 * the cosine of T must"}, {"start": 174.84, "duration": 4.479, "text": "equal 2 * the cosine of T divided Again"}, {"start": 177.64, "duration": 4.48, "text": "by the hypotenuse of the big triangle"}, {"start": 179.319, "duration": 4.48, "text": "which is 2 2 we can cancel the twos on"}, {"start": 182.12, "duration": 4.199, "text": "the right side again and clear"}, {"start": 183.799, "duration": 6.121, "text": "denominators to see that cosine of 2T ="}, {"start": 186.319, "duration": 6.761, "text": "2 * the cosine s of tus 1 the identity"}, {"start": 189.92, "duration": 5.16, "text": "on the left sin of 2T = 2 * the S of T *"}, {"start": 193.08, "duration": 4.239, "text": "the cosine of T is the double angle"}, {"start": 195.08, "duration": 4.92, "text": "identity for the sign function and the"}, {"start": 197.319, "duration": 5.64, "text": "identity on the right cosine of 2 T = 2"}, {"start": 200.0, "duration": 5.2, "text": "* cosine 2 of tus 1 is the double angle"}, {"start": 202.959, "duration": 4.481, "text": "identity for the cosine function"}, {"start": 205.2, "duration": 4.679, "text": "together these form the double angle"}, {"start": 207.44, "duration": 5.04, "text": "identities for S and cosine and we prove"}, {"start": 209.879, "duration": 4.881, "text": "them using a semicircle diagram this"}, {"start": 212.48, "duration": 3.839, "text": "proof is due to Roger B Nelson check the"}, {"start": 214.76, "duration": 3.479, "text": "description for more information about"}, {"start": 216.319, "duration": 4.12, "text": "this proof or for a link to another"}, {"start": 218.239, "duration": 4.2, "text": "proof of the same facts that uses the"}, {"start": 220.439, "duration": 5.0, "text": "law of signs and the law of cosines in a"}, {"start": 222.439, "duration": 3.0, "text": "triangle"}], "title": "Trig Double Angle Formulas from Semicircle (visual proof)", "description": "This is a short, animated visual proof of the Double angle identities for sine and cosine. To get the formulas we use a semicircle diagram and rely on similarity of two right triangles formed inside.\n\nIf you like this video, consider subscribing to the channel or consider buying me a coffee: https://www.buymeacoffee.com/VisualProofs. Thanks! \n\nFor another visual proof of this fact using the laws of sines and cosines, check out this animation:\nhttps://youtu.be/0ggGQ96eaiE\n\nThis animation is based on a visual proof by Roger B. Nelsen  from the January 1989 issue of The College Mathematics Journal (https://www.jstor.org/stable/2686819 - page 51).\n\n#mathshorts #mathvideo #math #trigonometry #lawofsines #lawofcosines  #triangle #manim #animation #theorem #pww #proofwithoutwords #visualproof #proof #circle #obtuseangle #acuteangle #angle #trigidentities #identity #righttriangle #isoscelestriangle #doubleangle \n\nTo learn more about animating with manim, check out:\nhttps://manim.community", "lengthSeconds": 224, "width": 640, "height": 360}
{"video_id": "-m_I5qVhlJA", "transcript": [{"start": 0.04, "duration": 3.399, "text": "start with a line segment of length a"}, {"start": 1.599, "duration": 5.121, "text": "and cut it into two segments of length a"}, {"start": 3.439, "duration": 5.001, "text": "plus b/2 and a minus b/2 use the Aus b"}, {"start": 6.72, "duration": 3.24, "text": "over2 line segment as the radius of a"}, {"start": 8.44, "duration": 3.199, "text": "circle connecting these two points"}, {"start": 9.96, "duration": 4.28, "text": "creates a right triangle with hypotenuse"}, {"start": 11.639, "duration": 4.4, "text": "of length OT of a s + b^ S ID 2 by the"}, {"start": 14.24, "duration": 3.56, "text": "Pythagorean theorem the hypotenuse is"}, {"start": 16.039, "duration": 3.761, "text": "greater than the leg length so a plus b/"}, {"start": 17.8, "duration": 4.76, "text": "2 is less than or equal to the < TK of a"}, {"start": 19.8, "duration": 4.319, "text": "+ b s / 2 now draw this radius so that"}, {"start": 22.56, "duration": 3.28, "text": "connecting these two points creates a"}, {"start": 24.119, "duration": 3.361, "text": "tangent to the circle Again by the"}, {"start": 25.84, "duration": 4.48, "text": "Pythagorean theorem the side length here"}, {"start": 27.48, "duration": 5.04, "text": "isun of a * B so theun of a * B is less"}, {"start": 30.32, "duration": 3.84, "text": "than or equal to a plus b/ 2 this yellow"}, {"start": 32.52, "duration": 3.44, "text": "shaded triangle is similar to this"}, {"start": 34.16, "duration": 3.239, "text": "larger green shaded triangle and by"}, {"start": 35.96, "duration": 4.16, "text": "similarity that means this side length"}, {"start": 37.399, "duration": 4.961, "text": "is 2 a b / a plus b and that side length"}, {"start": 40.12, "duration": 4.119, "text": "2ab ID a plus b is less than or equal to"}, {"start": 42.36, "duration": 3.44, "text": "the hypotenuse square root of a any"}, {"start": 44.239, "duration": 3.8, "text": "choice of non- negative real numbers A"}, {"start": 45.8, "duration": 4.12, "text": "and B leads to a similar diagram except"}, {"start": 48.039, "duration": 3.641, "text": "in the case when a equals B then all the"}, {"start": 49.92, "duration": 3.52, "text": "line segments collapse to the same so we"}, {"start": 51.68, "duration": 3.92, "text": "have equality throughout if and only if"}, {"start": 53.44, "duration": 3.599, "text": "a equals B we used one diagram to"}, {"start": 55.6, "duration": 3.2, "text": "produce an inequality relating the"}, {"start": 57.039, "duration": 4.801, "text": "harmonic geometric arithmetic and"}, {"start": 58.8, "duration": 3.04, "text": "quadratic means"}], "title": "Mean Inequalities", "description": "This short animated proof demonstrates the two variable mean inequalities for the arithmetic mean, geometric mean, harmonic mean and root mean square (or quadratic mean).\n\nIf you like this video, consider subscribing to the channel or consider buying me a coffee: https://www.buymeacoffee.com/VisualProofs. Thanks! \n\nThis animation of the inequalities is based on a visual proof by Roger Nelsen from the June 1987 issue of Mathematics Magazine (https://www.jstor.org/stable/2689561) p. 158.\n\nIf you like visual proofs of inequalities, see these links:\n\nAM-GM inequality via rectangles/squares: https://youtu.be/62G9fak1vyk\n\nAM-GM inequality via circles: https://youtu.be/Pt8IX_Q5U1A\n\nAM-GM-HM-QM via Moments: https://youtu.be/XJ3643WhKjY\n\n#math\u200b \u200b#manim\u200b #animation\u200b #theorem\u200b #pww\u200b #proofwithoutwords\u200b #visualproof\u200b #proof\u200b #inequality #iteachmath #mathematics #pythagoreantheorem #similartriangles #means #meaninequality #mathshorts\u200b #mathvideo\u200b #mtbos\n\nTo learn more about animating with manim, check out:\nhttps://manim.community", "lengthSeconds": 60, "width": 360, "height": 640}
{"video_id": "befcLpAer8A", "transcript": [{"start": 0.0, "duration": 3.84, "text": "the Pythagorean theorem says that the"}, {"start": 2.28, "duration": 3.539, "text": "area of the square should be equal to"}, {"start": 3.84, "duration": 3.66, "text": "the sum of these two squares but we can"}, {"start": 5.819, "duration": 3.961, "text": "really understand why it's true by"}, {"start": 7.5, "duration": 4.139, "text": "looking at these three triangles the"}, {"start": 9.78, "duration": 4.08, "text": "large green triangle is a reflection of"}, {"start": 11.639, "duration": 5.461, "text": "the original triangle the pink one is a"}, {"start": 13.86, "duration": 5.46, "text": "scaled down copy a similar triangle and"}, {"start": 17.1, "duration": 4.259, "text": "the blue one is similar again because"}, {"start": 19.32, "duration": 4.32, "text": "the triangles are similar these angles"}, {"start": 21.359, "duration": 5.041, "text": "must be equal so we can fit this pink"}, {"start": 23.64, "duration": 4.5, "text": "triangle nicely inside the big one at a"}, {"start": 26.4, "duration": 4.08, "text": "right angle to the hypotenuse"}, {"start": 28.14, "duration": 4.5, "text": "blue triangle works the same way it must"}, {"start": 30.48, "duration": 4.32, "text": "form a right angle with hypotenuse which"}, {"start": 32.64, "duration": 4.86, "text": "means that blue and pink areas must add"}, {"start": 34.8, "duration": 4.8, "text": "to give the original triangle cool right"}, {"start": 37.5, "duration": 4.62, "text": "but how does that prove that the sum of"}, {"start": 39.6, "duration": 4.32, "text": "the squares is equal well remember again"}, {"start": 42.12, "duration": 4.08, "text": "that the smaller triangles are just"}, {"start": 43.92, "duration": 4.74, "text": "scaled down copies of the big one so if"}, {"start": 46.2, "duration": 5.28, "text": "one square is K times the area of one"}, {"start": 48.66, "duration": 5.16, "text": "triangle that factor K must apply to all"}, {"start": 51.48, "duration": 4.86, "text": "squares and triangles and that is enough"}, {"start": 53.82, "duration": 5.66, "text": "to prove that a squared plus b squared"}, {"start": 56.34, "duration": 3.14, "text": "is equal to c squared"}], "title": "Understand WHY the Pythagorean Theorem is True - Nice Visual Proof", "description": "The Pythagorean Theorem is one of the most famous theorems in Geometry and all of Mathematics. This is a really nice proof using similar triangles.\nhttps://medium.com/nice-math-problems\n\nMusic:\nIsland by Luke Bergs | https://soundcloud.com/bergscloud/\nMusic promoted by https://www.chosic.com/free-music/all/\nCreative Commons CC BY-SA 3.0\nhttps://creativecommons.org/licenses/by-sa/3.0/", "lengthSeconds": 59, "width": 360, "height": 640}
{"video_id": "7O5cAcwEp8o", "transcript": [{"start": 0.29, "duration": 43.899, "text": "[\u0938\u0902\u0917\u0940\u0924]"}, {"start": 45.96, "duration": 3.0, "text": "a"}, {"start": 49.17, "duration": 145.91, "text": "[\u0938\u0902\u0917\u0940\u0924]"}], "title": "Factorial Product Formula (visual proof)", "description": "This is a short, animated visual proof showing an identity involving various factorials by taking the products of all the integers in a certain n x (n+1) grid in two different ways. #manim #math #mathvideo #mathshorts #sequences #triangles #animation #theorem #pww #proofwithoutwords #visualproof #proof #factorials  #product #pww\u200b #proofwithoutwords\u200b \u200b #proof\u200b #algebra #factorial #mathematics\u200b #mathvideo\u200b #mtbos \n\nIf you like this video, please click \"like\" and consider subscribing and checking out my other videos.\n\nThis animation is based on a visual proof by Edward T.H. Wang from the March 1989 issue of The College Mathematics Journal (https://www.jstor.org/stable/2686271- page 152).\n\nTo learn more about animating with manim, check out:\nhttps://manim.community\n\n_____________________________\nMusic in this video:\nReaching The Sky (2020) by Alexander Nakarada | https://www.serpentsoundstudios.com\nMusic promoted by https://www.free-stock-music.com\nAttribution 4.0 International (CC BY 4.0)\nhttps://creativecommons.org/licenses/by/4.0/", "lengthSeconds": 197, "width": 640, "height": 360}
{"video_id": "OP5GOKPAjSA", "transcript": [{"start": 0.0, "duration": 3.659, "text": "I've got a right angled triangle having"}, {"start": 1.68, "duration": 3.36, "text": "happiness 10 centimeters side B is 8"}, {"start": 3.659, "duration": 2.521, "text": "centimeters and side C is six"}, {"start": 5.04, "duration": 2.819, "text": "centimeters we are going to prove the"}, {"start": 6.18, "duration": 3.66, "text": "Pythagoras Theorem so in five seconds"}, {"start": 7.859, "duration": 3.841, "text": "theorem we are to prove that hypotenuse"}, {"start": 9.84, "duration": 3.419, "text": "10 centimeter square is equal to the sum"}, {"start": 11.7, "duration": 2.939, "text": "of 6 centimeters square and eight"}, {"start": 13.259, "duration": 3.181, "text": "centimeter Square so let's prove the"}, {"start": 14.639, "duration": 4.201, "text": "bicycle theorem I have got four copies"}, {"start": 16.44, "duration": 5.04, "text": "of this trendal ADC we are going to plot"}, {"start": 18.84, "duration": 4.56, "text": "these four triangles in this Square p q"}, {"start": 21.48, "duration": 3.6, "text": "r s Having side 14 centimeters so first"}, {"start": 23.4, "duration": 3.18, "text": "we are going to create the square on the"}, {"start": 25.08, "duration": 3.66, "text": "hypotenuse so you can see that by using"}, {"start": 26.58, "duration": 4.199, "text": "the four copies of the triangle ABC we"}, {"start": 28.74, "duration": 3.72, "text": "have created a white square inside 10"}, {"start": 30.779, "duration": 3.901, "text": "centimeters that is the hypotenuse of"}, {"start": 32.46, "duration": 3.36, "text": "the triangle ABC area of this y Square"}, {"start": 34.68, "duration": 2.34, "text": "hundred centimeter Square now let's"}, {"start": 35.82, "duration": 2.28, "text": "prove the another part of the particular"}, {"start": 37.02, "duration": 2.4, "text": "theorem so this is the second part of"}, {"start": 38.1, "duration": 2.88, "text": "the particular theorem here we have b"}, {"start": 39.42, "duration": 3.12, "text": "square is 6 into 6 is 36 centimeter"}, {"start": 40.98, "duration": 4.62, "text": "centimeter Square whereas the other"}, {"start": 42.54, "duration": 5.4, "text": "white part is 8 into 1864 64 square"}, {"start": 45.6, "duration": 4.56, "text": "centimeters we see that 64 plus 36 is"}, {"start": 47.94, "duration": 4.2, "text": "100 square centimeters so uh before we"}, {"start": 50.16, "duration": 4.2, "text": "have 100 square centimeters as C square"}, {"start": 52.14, "duration": 3.96, "text": "and now we have 64 plus 36 that is 100"}, {"start": 54.36, "duration": 3.12, "text": "square centimeters as b square plus C"}, {"start": 56.1, "duration": 2.82, "text": "Square so a square equals to b square"}, {"start": 57.48, "duration": 4.04, "text": "plus C square and hence Pythagoras"}, {"start": 58.92, "duration": 2.6, "text": "Theorem is proved"}], "title": "Visual Proof of Pythagoras Theorem #pythagorustheorem #pythagoreantheorem #mathshorts", "description": "The Visual Proof of the Pythagoras Theorem\na\u00b2=b\u00b2+c\u00b2\nHow to prove pythagoras theorem practically?\nEnjoy this video and learn something!\n\nComment your doubts and suggestions.\n\n*You may enjoy my other videos as follows:-*\nMy Maths Shorts playlist:-\nhttps://youtube.com/playlist?list=PLOPjB5QEzAe2z6q6Jmvl9g1HDp9XBXykS\n\nMy Maths Problem and Solution playlist:-\nhttps://youtube.com/playlist?list=PLOPjB5QEzAe3Nk93GXyHhBhI_uarZZL-v\n\nHashtags:-\n#pythagorustheorem \n#pythagorastheorem \n#youtubemath \n#pythagoras_theorem \n#proofofpythagorastheorem \n#pythagorastheoremproof \n#youtubemaths \n#geometry \n#rekhaganit \n#umeshkhetan \n#viralmathshorts \n#ytmathshorts\n\nProof of the pythagoras theorem,Pythagoras theorem,Pythagorean theorem,derive pythagoras theorem,how to prove pythagoras theorem,how to prove pythagorean theorem,pythagoras theorem class 10,pythagoras theorem proof,pythagorean theorem explained,pythagorean theorem proof,pythagorus theorem,what is pythagoras theorem,How to prove pythagoras theorem practically?,How to prove pythagorean theorem practically?,Pythagoras pramey ko siddh karen,\u092a\u093e\u0907\u0925\u093e\u0917\u094b\u0930\u0938 \u092a\u094d\u0930\u092e\u0947\u092f \u0915\u093e \u0938\u0942\u0924\u094d\u0930", "lengthSeconds": 60, "width": 360, "height": 640}
{"video_id": "PAJMMS-J2SY", "transcript": [{"start": 0.0, "duration": 3.84, "text": "let's see the geometry behind the"}, {"start": 1.74, "duration": 4.8, "text": "infinite sum of the powers of one-fifth"}, {"start": 3.84, "duration": 4.86, "text": "start with a one by one square if we"}, {"start": 6.54, "duration": 4.62, "text": "connect the vertices to the midpoints"}, {"start": 8.7, "duration": 5.34, "text": "like this it turns out that we divide"}, {"start": 11.16, "duration": 4.8, "text": "the square into five equal pieces the"}, {"start": 14.04, "duration": 3.78, "text": "four Triangles around the outside and"}, {"start": 15.96, "duration": 4.02, "text": "the inner Square all consist of"}, {"start": 17.82, "duration": 4.32, "text": "one-fifth of the area of the square so"}, {"start": 19.98, "duration": 3.84, "text": "this triangle is one-fifth but then we"}, {"start": 22.14, "duration": 4.08, "text": "can repeat the process on the inner"}, {"start": 23.82, "duration": 4.5, "text": "Square to get one-fifth of one-fifth and"}, {"start": 26.22, "duration": 4.68, "text": "so on we can continue this process"}, {"start": 28.32, "duration": 4.8, "text": "indefinitely each time taking one-fifth"}, {"start": 30.9, "duration": 4.08, "text": "of the inner Square which is one-fifth"}, {"start": 33.12, "duration": 3.3, "text": "of the previous Square we spiral this"}, {"start": 34.98, "duration": 3.9, "text": "down forever and it turns out that we've"}, {"start": 36.42, "duration": 4.92, "text": "shaded one-fifth plus one-fifth squared"}, {"start": 38.88, "duration": 5.699, "text": "and so on of the entire Square"}, {"start": 41.34, "duration": 6.42, "text": "but this shape can be rotated three more"}, {"start": 44.579, "duration": 5.16, "text": "times to fill in the entire Square"}, {"start": 47.76, "duration": 4.2, "text": "therefore the infinite sum of powers of"}, {"start": 49.739, "duration": 4.461, "text": "one-fifth must be one-fourth"}, {"start": 51.96, "duration": 2.24, "text": "foreign"}], "title": "Geometry of sums of powers of 1/5", "description": "This is a short, animated visual proof demonstrating the infinite sum of the powers of 1/5. \u200b \n\nHere is the longer form (wordless) version of this video: https://youtu.be/IguRXWNwrn8\n\nHere is an alternate dissection proof for this series: https://youtu.be/yp7afEXYeC4\n\nHere is a longer form video with lots of related infinite geometric sums: https://youtu.be/JteQEN1XPyc\n\n\nIf you like these videos, please consider subscribing.\n\nThis animation is based on a proof by Rick Mabry from his website https://lsusmath.rickmabry.org/rmabry/fivesquares/fsq2.gif and featured in Roger B. Nelsen's third Proofs Without Words compendium (https://bookstore.ams.org/clrm-52/).\n\n\nTo learn more about animating with manim, check out:\nhttps://manim.community\n\n#math\u200b #calculus\u200b #manim\u200b #animation\u200b #theorem\u200b #pww\u200b #proofwithoutwords\u200b #visualproof\u200b #proof\u200b #iteachmath #geometricseries #series #infinitesums #infiniteseries #mathshorts\u200b #mathvideo #mtbos", "lengthSeconds": 53, "width": 360, "height": 640}
{"video_id": "Z1B5a80Bqko", "transcript": [{"start": 0.0, "duration": 4.08, "text": "you've probably heard of the Pythagorean"}, {"start": 1.92, "duration": 4.14, "text": "theorem a squared plus b squared equals"}, {"start": 4.08, "duration": 4.26, "text": "c squared but what is it actually and"}, {"start": 6.06, "duration": 4.02, "text": "why is it true let's draw this out A and"}, {"start": 8.34, "duration": 3.66, "text": "B are these two sides and C is the"}, {"start": 10.08, "duration": 3.24, "text": "hypotenuse if you squared say the"}, {"start": 12.0, "duration": 3.42, "text": "hypotenuse then you would get the area"}, {"start": 13.32, "duration": 4.02, "text": "of a square with a side length C the"}, {"start": 15.42, "duration": 3.779, "text": "Pythagorean theorem states that for any"}, {"start": 17.34, "duration": 4.38, "text": "right angle triangle the square of the"}, {"start": 19.199, "duration": 4.441, "text": "hypotenuse C is equal to the sum of the"}, {"start": 21.72, "duration": 3.96, "text": "squared of the other two sides A and B"}, {"start": 23.64, "duration": 4.26, "text": "also called the feet but don't just take"}, {"start": 25.68, "duration": 4.08, "text": "my word for it let's prove it using a"}, {"start": 27.9, "duration": 3.84, "text": "visual proof this is a right triangle"}, {"start": 29.76, "duration": 3.479, "text": "these are four identical right triangles"}, {"start": 31.74, "duration": 3.72, "text": "you can arrange them in a way that they"}, {"start": 33.239, "duration": 3.721, "text": "form a square of side length a plus b as"}, {"start": 35.46, "duration": 3.36, "text": "you can see from the side length the"}, {"start": 36.96, "duration": 3.66, "text": "square in the top left has an area of a"}, {"start": 38.82, "duration": 3.72, "text": "squared and the one in the bottom left"}, {"start": 40.62, "duration": 3.779, "text": "has one of B squared however these"}, {"start": 42.54, "duration": 3.66, "text": "triangles can be rearranged to form an"}, {"start": 44.399, "duration": 3.781, "text": "identical square of side length a plus b"}, {"start": 46.2, "duration": 3.78, "text": "now however the middle square has formed"}, {"start": 48.18, "duration": 3.539, "text": "an area of c squared hence from this"}, {"start": 49.98, "duration": 3.18, "text": "proof a squared plus b squared equals c"}, {"start": 51.719, "duration": 3.301, "text": "squared now there are many other"}, {"start": 53.16, "duration": 4.14, "text": "numerical ways to prove this however I"}, {"start": 55.02, "duration": 3.6, "text": "find this visual proof so elegant hope"}, {"start": 57.3, "duration": 4.34, "text": "you've learned something stay curious"}, {"start": 58.62, "duration": 3.02, "text": "and I'll see you in the next one"}], "title": "An elegant visual proof of the Pythagorean Theorem #shorts", "description": "Welcome to DeSTEMber, a series where I answer your questions about math and science throughount the month of December. Do stay tuned, for more knowledge is on the way!\n\nIn this video, I'll be describing the Pythagorean Theorem and introducing a famous visual proof for it.", "lengthSeconds": 60, "width": 270, "height": 480}
{"video_id": "0jNuqakofMw", "transcript": [{"start": 0.0, "duration": 5.49, "text": "e o jogo por favor parte agora inteira"}, {"start": 2.399, "duration": 5.101, "text": "que \u00e9 que tu Square of here a esse"}, {"start": 5.49, "duration": 6.95, "text": "Guanambi Square Orange is the new Black"}, {"start": 7.5, "duration": 10.17, "text": "2 and White is a Long skirt and Lines of"}, {"start": 12.44, "duration": 9.55, "text": "Range of the Idiot widgets Will of Seeds"}, {"start": 17.67, "duration": 9.5, "text": "Where does cancros da pele da s4devs"}, {"start": 21.99, "duration": 5.18, "text": "Where e depois do Circular 1"}], "title": "Visual Proof of Pythagorean Theorem | Mathematics | JEE Mains - Advanced | NEET | KVPY | Olympiads", "description": "Pythagoras Theorem - Visual Proof\n\nA\u00b2 + B\u00b2 = C\u00b2\n\nComment your views\n\nTag a friend to show this\nTurn on post notifications\n\nFollow : @championsacademy.in\n\nTags :\n\n#Pythagoras #Theorem #Math #Triangle #Proof #PythagorasTheorem #Trigonometry #RightAngle #Hypotenuse #ProofOfPythagorasTheorem #VisualProof #StudyFromHome #MathAnimation #MathTutorial #MathematicalCreativity\n\n#ChampionsAcademy #AcademicSystem #Coaching #Learning #Online #JEE #NEET #Physics #Chemistry #Biology #CBSE #ICSE #EasyLearning #LearningMadeEasy #OnlineLearning", "lengthSeconds": 26, "width": 202, "height": 360}
{"video_id": "vi_IAh7bfEY", "transcript": [{"start": 1.319, "duration": 3.0, "text": "foreign"}], "title": "visual proof of (a+b)\u00b2 | algebra formula solution trick| #maths #algebra #jee #engineering", "description": "visual proof of (a+b)\u00b2 | algebra formula solution trick| #maths #algebra #jee #engineering\n\nvisual proof of (a+b)\u00b2 || #mathematics #proofing #formula1 #video #mathstricks #shorts\n\n@akaash.sain04 @PhysicsWallah \n#mathstricks  #viral #physics \n@ABHINAYMATHS\n\nmath formula class\na plus b ka whole 2wise", "lengthSeconds": 23, "width": 360, "height": 640}
{"video_id": "C14XbaqmuMc", "transcript": [{"start": 1.14, "duration": 2.78, "text": "foreign"}, {"start": 8.24, "duration": 3.06, "text": "[Music]"}, {"start": 14.41, "duration": 3.349, "text": "[Music]"}], "title": "Visual Proof of Maths Formula", "description": "#Hukabi #HukabiBD\nPuzzles and math tricks can have a number of benefits, including:\n1. Boosting brain power: Puzzles and math tricks can help to stimulate the brain and improve cognitive function, memory, and problem-solving skills.\n2. Improving concentration: When working on puzzles or math problems, individuals need to concentrate and focus, which can help to improve overall concentration and attention span.\n3. Enhancing creativity: Puzzles and math tricks can help to foster creativity by encouraging individuals to think outside the box and approach problems in new and innovative ways.\n4. Developing critical thinking skills: Puzzles and math tricks require individuals to think critically and logically, which can help to develop these important skills.\n5. Reducing stress: Engaging in puzzles and math problems can provide a welcome distraction from everyday stressors, helping individuals to relax and unwind.\n6. Building confidence: Successfully completing puzzles and math problems can help individuals feel a sense of accomplishment and build self-confidence.\n7. Improving math skills: Math tricks can help individuals to develop a deeper understanding of mathematical concepts and improve their overall math skills.\n8. Providing entertainment: Puzzles and math tricks can be a fun and engaging way to pass the time, providing hours of entertainment for individuals of all ages.\n\n#Math #Maths #Trick #Tricks #IQ #IQTest #Quiz #Riddle #MathChallenge #MathRiddle #Technic #MathTricks #Magic #MathMagic #MagicMath #MathMemes #MathSkills #MathStudent #ScienceMath #Mathematician #MathProblems #MathTeacher #Teacher #Motivation #ScienceMemes #Puzzle #MathPuzzle #Learning #Building #Education #Study #Studygram #Chemistry #Physics #Biology #Engineering #Art #BabyToy #LearningToy #EducationalToy #Reels #Shorts #ViralVideo #Hukabi\n\nLike, Follow & Subscribe!!!\nInstagram \u25b6 https://www.instagram.com/hukabi.bd\nFacebook \u25b6 https://www.facebook.com/hukabi.bd\nTweeter \u25b6 https://twitter.com/HukabiBD\nTikTok  \u25b6 https://www.tiktok.com/@hukabi.bd\nLinkedin \u25b6 https://www.linkedin.com/company/hukabi\n\n\ud83d\udc9c Helpful? Do Subscribe (#Hukabi)\n\n------------------------------------------------------------------------------\n\ud83d\udc49For business inquiries: hukabi.bd@gmail.com\n------------------------------------------------------------------------------", "lengthSeconds": 27, "width": 360, "height": 640}
{"video_id": "6S-yi2VbbVM", "transcript": [{"start": 6.74, "duration": 3.48, "text": "I know"}, {"start": 17.03, "duration": 5.009, "text": "[Music]"}, {"start": 19.039, "duration": 3.0, "text": "yeah"}, {"start": 22.49, "duration": 3.05, "text": "[Music]"}, {"start": 32.78, "duration": 3.939, "text": "I can do this every morning every"}, {"start": 35.16, "duration": 4.22, "text": "evening if you screaming straight back"}, {"start": 36.719, "duration": 2.661, "text": "to Sunrise"}, {"start": 40.079, "duration": 5.481, "text": "foreign"}, {"start": 42.51, "duration": 3.05, "text": "[Music]"}, {"start": 49.08, "duration": 3.12, "text": "[Music]"}], "title": "visual proof of sum of first n natural numbers #viral #youtubeshorts #maths #shorts #formula1", "description": "visual proof of sum of first n natural numbers #viral #youtubeshorts #maths #shorts #formula1", "lengthSeconds": 57, "width": 360, "height": 640}
{"video_id": "a8YMq_q4HRs", "transcript": [{"start": 0.04, "duration": 3.56, "text": "start with a square of side length one"}, {"start": 2.04, "duration": 3.44, "text": "cut the square into nine smaller"}, {"start": 3.6, "duration": 4.04, "text": "congruent squares and shade eight of"}, {"start": 5.48, "duration": 4.36, "text": "them like this shading a total of 8 9ths"}, {"start": 7.64, "duration": 4.8, "text": "of the square now repeat the process on"}, {"start": 9.84, "duration": 4.919, "text": "the unshaded Square shading 8 9th of"}, {"start": 12.44, "duration": 5.08, "text": "that 1 nth thus shading an additional 8"}, {"start": 14.759, "duration": 4.801, "text": "over 9 squar of the outer square if we"}, {"start": 17.52, "duration": 5.08, "text": "repeat this process indefinitely we"}, {"start": 19.56, "duration": 6.16, "text": "shade 8 over 9 cubed and then 8 over 9"}, {"start": 22.6, "duration": 5.8, "text": "4th and so on at each stage we shade 8"}, {"start": 25.72, "duration": 4.76, "text": "over 9 to the N if we then think of this"}, {"start": 28.4, "duration": 3.6, "text": "as an infinite process or a limiting"}, {"start": 30.48, "duration": 3.48, "text": "process we can then think of the"}, {"start": 32.0, "duration": 4.079, "text": "limiting diagram which is the entire"}, {"start": 33.96, "duration": 3.48, "text": "shaded square and therefore the infinite"}, {"start": 36.079, "duration": 3.361, "text": "sum that we're interested in which is"}, {"start": 37.44, "duration": 4.16, "text": "the Shaded area is equal to the Shaded"}, {"start": 39.44, "duration": 4.279, "text": "area of the full square or one one"}, {"start": 41.6, "duration": 5.0, "text": "interpretation of this is that 0.8"}, {"start": 43.719, "duration": 4.881, "text": "repeating in base 9 is 1 another"}, {"start": 46.6, "duration": 4.52, "text": "interpretation is that 8 * the infinite"}, {"start": 48.6, "duration": 4.2, "text": "sum of the positive powers of 1 9th is 1"}, {"start": 51.12, "duration": 5.68, "text": "or that the infinite sum of the positive"}, {"start": 52.8, "duration": 4.0, "text": "powers of 1 nth is 1/8"}], "title": "0.8888\u2026 = 1 (in base 9)", "description": "This is a short, animated visual proof showing the sum of the infinite geometric series with first term 8/9 and ratio 1/9, which in turn allows us to compute the sum of the series of powers of 1/9 and determine an interesting base 9 representation of 1. \n\nIf you like this video, consider subscribing to the channel or consider buying me a coffee: https://www.buymeacoffee.com/VisualProofs. Thanks! \n\nFor a longer, wordless version (more dramatic) of this animation see\nhttps://youtu.be/C4t_ps3VKvI\n\nAlso, check out my playlist on geometric sums/series: https://youtube.com/playlist?list=PLZh9gzIvXQUsgw8W5TUVDtF0q4jEJ3iaw\n\nThis animation is inspired by the a proof by Elizabeth M. Markham from the October 1993 issue of Mathematics Magazine page 242 (https://doi.org/10.2307/2690738).\n\n#mathshorts\u200b #mathvideo\u200b #math\u200b #calculus #mtbos\u200b #manim\u200b #animation\u200b #theorem\u200b #pww\u200b #proofwithoutwords\u200b #visualproof\u200b #proof\u200b #iteachmath #geometricsums #series #infinitesums #infiniteseries #geometric #geometricseries #square \n\nTo learn more about animating with manim, check out:\nhttps://manim.community", "lengthSeconds": 55, "width": 360, "height": 640}
{"video_id": "YAGjrfpHq-Q", "transcript": [{"start": 0.16, "duration": 4.679, "text": "suppose A and B are two positive real"}, {"start": 2.2, "duration": 4.679, "text": "numbers draw a right isoceles triangle"}, {"start": 4.839, "duration": 5.481, "text": "with side length sare otk of a so that"}, {"start": 6.879, "duration": 5.241, "text": "its area is/ 12 * a off the top vertex"}, {"start": 10.32, "duration": 3.68, "text": "draw a right isoceles triangle with s"}, {"start": 12.12, "duration": 4.64, "text": "side length square root B so that it has"}, {"start": 14.0, "duration": 4.6, "text": "an area of 1 12 B these two triangles"}, {"start": 16.76, "duration": 3.56, "text": "form the boundary of a rectangle that"}, {"start": 18.6, "duration": 4.16, "text": "has Dimensions squ < TK of a and squ <"}, {"start": 20.32, "duration": 5.279, "text": "TK of B so it has an area of the squ <"}, {"start": 22.76, "duration": 4.599, "text": "TK of a * B in this diagram imagine"}, {"start": 25.599, "duration": 3.561, "text": "folds along the top and left of the"}, {"start": 27.359, "duration": 4.281, "text": "rectangle so that when we fold in the"}, {"start": 29.16, "duration": 4.919, "text": "two isoc right triangles they meet along"}, {"start": 31.64, "duration": 4.84, "text": "their diagonal like this because the"}, {"start": 34.079, "duration": 4.401, "text": "areas of the two triangles cover the"}, {"start": 36.48, "duration": 4.16, "text": "area of the rectangle we see that the"}, {"start": 38.48, "duration": 3.88, "text": "square root of a * B which is the area"}, {"start": 40.64, "duration": 4.439, "text": "of the rectangle is less than or equal"}, {"start": 42.36, "duration": 4.359, "text": "to 12 a + 1 12 B this inequality is"}, {"start": 45.079, "duration": 3.401, "text": "known as the arithmetic mean geometric"}, {"start": 46.719, "duration": 3.921, "text": "mean inequality note that we get"}, {"start": 48.48, "duration": 6.84, "text": "equality if and only if the Shaded area"}, {"start": 50.64, "duration": 4.68, "text": "is a square so that A and B are equal"}], "title": "Fold the AM-GM inequality", "description": "This is a short, animated visual proof demonstrating the arithmetic mean geometric mean inequality using paper folding.\n\nTo buy me a coffee, head over to https://www.buymeacoffee.com/VisualProofs \nThanks!\n\nFor other visual proofs of this fact see:\n\nhttps://youtu.be/Pt8IX_Q5U1A\nhttps://youtu.be/62G9fak1vyk\nhttps://youtu.be/HFGSsPbYc-s\nhttps://youtu.be/SEPkdPWKdO0\nhttps://youtu.be/jNx05gXSblE\n\nThis animation is based on a visual proof by Yukio Kobayashi from the 2002 issue of the Mathematical Gazette (page 293).\n\n#manim #shorts #mathshorts\u200b #mathvideo\u200b #math\u200b #mathematics #amgminequality #mtbos\u200b\u200b #animation\u200b #theorem\u200b #pww\u200b #proofwithoutwords\u200b #visualproof\u200b #proof\u200b #iteachmath #calculus #inequality \n\nTo learn more about animating with manim, check out:\nhttps://manim.community", "lengthSeconds": 53, "width": 360, "height": 640}
{"video_id": "azzrmVw8O-8", "transcript": [{"start": 0.01, "duration": 4.86, "text": "start with a right triangle with sides"}, {"start": 2.35, "duration": 4.799, "text": "of length A and B and a hypotenuse of"}, {"start": 4.87, "duration": 4.439, "text": "length C now draw squares off the side"}, {"start": 7.149, "duration": 4.86, "text": "lengths enclosing an area of a squared"}, {"start": 9.309, "duration": 4.5, "text": "and B squared respectively then draw a"}, {"start": 12.009, "duration": 4.081, "text": "square off the hypotenuse in closing an"}, {"start": 13.809, "duration": 4.261, "text": "area of c squared here's a proof that"}, {"start": 16.09, "duration": 3.9, "text": "the area contained in the C by C square"}, {"start": 18.07, "duration": 4.02, "text": "is equal to the sum of the areas in the"}, {"start": 19.99, "duration": 4.02, "text": "other two squares imagine these two"}, {"start": 22.09, "duration": 3.84, "text": "lines of length C pass through the"}, {"start": 24.01, "duration": 3.599, "text": "center of the B by b square then they"}, {"start": 25.93, "duration": 4.019, "text": "cut the side length of the B by b square"}, {"start": 27.609, "duration": 4.621, "text": "into lengths of B plus a over 2 and B"}, {"start": 29.949, "duration": 4.321, "text": "minus a over 2. we can then take the"}, {"start": 32.23, "duration": 4.38, "text": "four resulting pieces and shift them to"}, {"start": 34.27, "duration": 4.92, "text": "the C by C square like this the leftover"}, {"start": 36.61, "duration": 4.5, "text": "middle piece is also a square and you"}, {"start": 39.19, "duration": 3.66, "text": "can check that the side length is a so"}, {"start": 41.11, "duration": 3.78, "text": "we can shift to a by a square inside"}, {"start": 42.85, "duration": 4.2, "text": "therefore we've seen that c squared"}, {"start": 44.89, "duration": 3.78, "text": "equals B squared plus a squared where A"}, {"start": 47.05, "duration": 3.599, "text": "and B are the leg lengths of a right"}, {"start": 48.67, "duration": 3.959, "text": "triangle with a hypotenuse of length C"}, {"start": 50.649, "duration": 5.121, "text": "this is the Pythagorean theorem As"}, {"start": 52.629, "duration": 3.141, "text": "proved by Henry dudney"}], "title": "Pythagorean Theorem XI (Dudeney\u2019s Dissection)", "description": "This is a short, animated visual proof of the Pythagorean theorem (the right triangle theorem) using a Henry Dudeney's dissection argument (congruent-by-addition proof).\n\nIf you like this video, consider subscribing to the channel or consider buying me a coffee: https://www.buymeacoffee.com/VisualProofs. Thanks! \n\nThis animation is based on a proof from Henry Dudeney and is discussed in Howard Eve's book \"Great Moments in Mathematics (before 1650).\" You can also find this in Roger Nelsen's first PWW compendium on page 6.\n\nFor other proofs of this same fact check out: \nhttps://youtu.be/q28H5NY5UVU\nhttps://youtu.be/yfGtbNgcrQ8\nhttps://youtu.be/JRjPsuJ7S-g\nhttps://youtu.be/TsKTteGJpZU\nhttps://youtu.be/ZlGaQdNRdqA\nhttps://youtu.be/uqGjs_n-f7U\nhttps://youtu.be/wWEZ-rUlBDQ\nhttps://youtu.be/I75HYC8TRNY\n\n\n#math #manim #pythagoreantheorem #pythagorean #triangle #animation #theorem #pww #proofwithoutwords #visualproof #proof  #mathshorts #mathvideo \n\nTo learn more about animating with manim, check out:\nhttps://manim.community", "lengthSeconds": 54, "width": 360, "height": 640}
{"video_id": "VJO6AeAtERU", "transcript": [{"start": 0.0, "duration": 4.02, "text": "here's a beautiful visualization for the"}, {"start": 2.1, "duration": 3.719, "text": "alternating sum of squares starting with"}, {"start": 4.02, "duration": 4.02, "text": "n squared and ending at 1 squared N"}, {"start": 5.819, "duration": 4.681, "text": "squared can be represented by an N by n"}, {"start": 8.04, "duration": 4.32, "text": "array of boxes then we can remove the"}, {"start": 10.5, "duration": 4.44, "text": "lower n minus 1 by n minus 1 array"}, {"start": 12.36, "duration": 5.46, "text": "re-add the lower n minus 2 by n minus 2"}, {"start": 14.94, "duration": 5.52, "text": "array and so on all the way down until"}, {"start": 17.82, "duration": 5.16, "text": "we either add or subtract the bottom one"}, {"start": 20.46, "duration": 4.26, "text": "by one square array this process has"}, {"start": 22.98, "duration": 3.719, "text": "resulted in a collection of L-shaped"}, {"start": 24.72, "duration": 3.96, "text": "gnomons which we can shift together to"}, {"start": 26.699, "duration": 4.261, "text": "form a triangular array with base length"}, {"start": 28.68, "duration": 4.92, "text": "n this means that there are t sub n"}, {"start": 30.96, "duration": 4.68, "text": "boxes in the resulting array where T sub"}, {"start": 33.6, "duration": 3.6, "text": "n is the nth triangular number therefore"}, {"start": 35.64, "duration": 3.66, "text": "the alternating sum of squares starting"}, {"start": 37.2, "duration": 4.5, "text": "with n squared and ending at 1 is given"}, {"start": 39.3, "duration": 5.599, "text": "by n times n plus 1 over 2 by the"}, {"start": 41.7, "duration": 3.199, "text": "formula for the Triangular numbers"}], "title": "Alternating sum of squares is triangular!", "description": "This is a short, animated visual proof demonstrating how to visualize the alternating sum of squares. \n\nIf you like this video, consider subscribing to the channel or consider buying me a coffee: https://www.buymeacoffee.com/VisualProofs. Thanks!\n\nThis animation is based on a visual proof by Steven L. Snover from the April 1992 issue of Mathematics Magazine (https://doi.org/10.2307/2690486 - page 90).\n\nCompare this formula to the regular sum of squares formula visualized here: https://youtu.be/-tJhH_k2LaM\n\nHere is an alternate visual proof of this same fact: https://youtu.be/5u34ifloAB4\n\n#mathshorts\u200b #mathvideo\u200b #math\u200b #numbertheory #mtbos\u200b #manim\u200b #animation\u200b #theorem\u200b #pww\u200b #proofwithoutwords\u200b #visualproof\u200b #proof\u200b #iteachmath #squares #triangularnumbers\n\nTo learn more about animating with manim, check out:\nhttps://manim.community", "lengthSeconds": 44, "width": 360, "height": 640}
{"video_id": "jq5AYCgkciE", "transcript": [{"start": 0.56, "duration": 4.32, "text": "here's an alternate interesting way to"}, {"start": 2.24, "duration": 5.639, "text": "find the sum of the first n positive odd"}, {"start": 4.88, "duration": 6.2, "text": "integers visually start with a square"}, {"start": 7.879, "duration": 5.321, "text": "array of cells as pictured here if we"}, {"start": 11.08, "duration": 5.359, "text": "imagine both dimensions to be n then we"}, {"start": 13.2, "duration": 5.999, "text": "have a total of n squ cells in this"}, {"start": 16.439, "duration": 5.161, "text": "array now let's go through and place two"}, {"start": 19.199, "duration": 5.4, "text": "tokens on every cell in the"}, {"start": 21.6, "duration": 6.04, "text": "array because there are n s cells and"}, {"start": 24.599, "duration": 6.52, "text": "two tokens per cell there are a total of"}, {"start": 27.64, "duration": 5.439, "text": "2 * n^ 2 tokens in this diagram"}, {"start": 31.119, "duration": 5.241, "text": "but now let's count the number of tokens"}, {"start": 33.079, "duration": 5.16, "text": "in a different way we can instead divide"}, {"start": 36.36, "duration": 4.12, "text": "the square array into triangular and"}, {"start": 38.239, "duration": 4.881, "text": "trapezoidal shapes each containing a"}, {"start": 40.48, "duration": 4.84, "text": "single diagonal line of tokens the first"}, {"start": 43.12, "duration": 4.52, "text": "one contains one token the next one"}, {"start": 45.32, "duration": 5.919, "text": "contains three tokens followed by five"}, {"start": 47.64, "duration": 5.52, "text": "tokens and so on each trapezoidal strip"}, {"start": 51.239, "duration": 4.241, "text": "contains exactly two more tokens than"}, {"start": 53.16, "duration": 4.8, "text": "the previous one so we get an ascending"}, {"start": 55.48, "duration": 5.559, "text": "list of consecutive odd integers ending"}, {"start": 57.96, "duration": 4.88, "text": "at the odd number 2 N minus one"}, {"start": 61.039, "duration": 3.921, "text": "but notice that this part of the diagram"}, {"start": 62.84, "duration": 3.84, "text": "can be repeated on the bottom half as"}, {"start": 64.96, "duration": 4.04, "text": "well and that means that the total"}, {"start": 66.68, "duration": 6.119, "text": "number of tokens is 2 * the sum of the"}, {"start": 69.0, "duration": 7.119, "text": "first n odd positive integers 1 + 3 + 5"}, {"start": 72.799, "duration": 5.401, "text": "and so on up to 2 nus 1 but now if we"}, {"start": 76.119, "duration": 4.32, "text": "cancel the two from both sides we get a"}, {"start": 78.2, "duration": 4.919, "text": "formula for the sum of the first n odd"}, {"start": 80.439, "duration": 5.68, "text": "positive integers in particular we see"}, {"start": 83.119, "duration": 6.36, "text": "that n^2 is equal to the sum where K"}, {"start": 86.119, "duration": 5.32, "text": "ranges from 1 to n of 2K minus 1 by"}, {"start": 89.479, "duration": 3.721, "text": "Counting tokens on a square array of"}, {"start": 91.439, "duration": 4.241, "text": "cells we've seen that the sum of the"}, {"start": 93.2, "duration": 5.279, "text": "first n odd positive integers is equal"}, {"start": 95.68, "duration": 4.16, "text": "to n^ S if you liked this visual proof"}, {"start": 98.479, "duration": 3.32, "text": "check out the description for other"}, {"start": 99.84, "duration": 3.319, "text": "visual proofs of the same fact and let"}, {"start": 101.799, "duration": 4.96, "text": "me know in the comments which one is"}, {"start": 103.159, "duration": 3.6, "text": "your favorite"}], "title": "Summing Odds using Tokens (visual proof III)", "description": "This is a short, animated visual proof showing one method to find the sum of the first n odd positive integers by stacking tokens in a square array of cells.\n\nIf you like this video, consider subscribing to the channel or consider buying me a coffee: https://www.buymeacoffee.com/VisualProofs. Thanks!\n\nTo see the other methods for finding the sum odd integers visually, check out the following videos: \nhttps://youtu.be/I4A2T55k0cg\nhttps://youtu.be/x3qfFBNRRDg\nhttps://youtu.be/ZeEOgbLo0Rg\nhttps://youtu.be/6H6nCpGW1nU\n\nFor other sums related to odd integers, check out these visuals:\nhttps://youtu.be/WKxoi2i0EaE (summing odd squares)\nhttps://studio.youtube.com/video/9BC1klE8jMk/edit (alternating odd sums)\nhttps://youtu.be/MmOTqrtbtFQ (odd sum ratios)\nhttps://youtu.be/fTBvVeURb3Q (odd sum ratios II)\n\nThis animation is based on a visual proof due to Timoth\u00e9e Duval that appeared in the Mars-Avril 2007 issue of Tangente no 115 (link.  page 10).\n\nYou can also find this diagram in Roger Nelsen's book on page 114: (I may receive a small commission at no cost to you for this affiliate link): https://amzn.to/47kfG7w\n\n#manim #math #mathvideo #mathshorts #sequences #triangles #animation #theorem #pww #proofwithoutwords #visualproof #proof #squarenumbers #squares #oddnumbers #odds #sums #pww\u200b #proofwithoutwords\u200b \u200b #proof\u200b #algebra #finitesums #mathematics\u200b #mathvideo\u200b #mtbos \n\nTo learn more about animating with manim, check out:\nhttps://manim.community", "lengthSeconds": 105, "width": 640, "height": 360}
{"video_id": "BUBAHmERkOw", "transcript": [{"start": 0.04, "duration": 4.279, "text": "graph the lines yal X and Y = RX + a"}, {"start": 2.76, "duration": 3.88, "text": "where R and A are positive real numbers"}, {"start": 4.319, "duration": 3.44, "text": "with r less than 1 these intersect at"}, {"start": 6.64, "duration": 3.44, "text": "the point where the X and the y"}, {"start": 7.759, "duration": 4.121, "text": "coordinate are both a over 1 - R the"}, {"start": 10.08, "duration": 3.519, "text": "vertical distance on the y- AIS between"}, {"start": 11.88, "duration": 3.719, "text": "the two curves is a and that means this"}, {"start": 13.599, "duration": 3.76, "text": "horizontal distance is also a because it"}, {"start": 15.599, "duration": 4.361, "text": "forms an isoceles triangle using the"}, {"start": 17.359, "duration": 4.361, "text": "line yal X as we continue to step up the"}, {"start": 19.96, "duration": 3.2, "text": "staircase diagram we can measure the"}, {"start": 21.72, "duration": 3.24, "text": "vertical distance as the difference"}, {"start": 23.16, "duration": 3.439, "text": "between two curves and the horizontal"}, {"start": 24.96, "duration": 3.44, "text": "distance will always be the same as that"}, {"start": 26.599, "duration": 4.16, "text": "one because it again forms an isoc"}, {"start": 28.4, "duration": 3.92, "text": "triangle with yal x the vertical and"}, {"start": 30.759, "duration": 3.8, "text": "horizontal lengths in the staircase"}, {"start": 32.32, "duration": 4.559, "text": "diagram are all of the form a * R to the"}, {"start": 34.559, "duration": 4.201, "text": "I as I increases and this means that the"}, {"start": 36.879, "duration": 3.921, "text": "widths decrease as the staircase"}, {"start": 38.76, "duration": 3.76, "text": "converges to the intersection point if"}, {"start": 40.8, "duration": 3.12, "text": "we focus only on the horizontal line"}, {"start": 42.52, "duration": 4.96, "text": "segments and start dropping them to the"}, {"start": 43.92, "duration": 5.68, "text": "x-axis we see that a plus a r plus a r 2"}, {"start": 47.48, "duration": 3.759, "text": "plus a R cubed and so on eventually"}, {"start": 49.6, "duration": 4.479, "text": "creates the entire line segment on the"}, {"start": 51.239, "duration": 4.561, "text": "x- axis from 0 to a over 1 - R therefore"}, {"start": 54.079, "duration": 3.921, "text": "the infinite sum of the geometric series"}, {"start": 55.8, "duration": 5.88, "text": "with first term a and common ratio R"}, {"start": 58.0, "duration": 3.68, "text": "must equal a over 1 - r"}], "title": "Infinite Geometric Series Sum", "description": "This is a short, animated visual proof demonstrating the infinite geometric series formula for any positive ratio r with r less than 1 and with positive first term a. This series is important for many results in calculus, discrete mathematics, and combinatorics.\n\nIf you like this content, you can support my work at https://www.buymeacoffee.com/VisualProofs  Thanks!\n\nAlso, check out my playlist on geometric sums/series: https://youtube.com/playlist?list=PLZh9gzIvXQUsgw8W5TUVDtF0q4jEJ3iaw\n\nThis animation is based on a proof by The Viewpoints 2000 group from the October 2001 issue of Mathematics Magazine page 320 (https://www.jstor.org/stable/2691106 ).\n\n#mathshorts\u200b #mathvideo\u200b #math\u200b #calculus #mtbos\u200b #manim\u200b #animation\u200b #theorem\u200b #pww\u200b #proofwithoutwords\u200b #visualproof\u200b #proof\u200b #iteachmath #geometricsums #series #infinitesums #infiniteseries #geometric #geometricseries\n\nTo learn more about animating with manim, check out:\nhttps://manim.community", "lengthSeconds": 60, "width": 360, "height": 640}
{"video_id": "HFu5-kUjKz0", "transcript": [{"start": 7.04, "duration": 14.13, "text": "par"}, {"start": 9.14, "duration": 12.03, "text": "[M\u00fasica]"}, {"start": 23.43, "duration": 4.48, "text": "[M\u00fasica]"}, {"start": 30.599, "duration": 3.0, "text": "non"}], "title": "Visual proof of the theorem Pythagoras! #maths #geometry #pythagoras_theorem #youtubeshorts", "description": "", "lengthSeconds": 34, "width": 360, "height": 640}
{"video_id": "FFCDmDqTyjI", "transcript": [{"start": 0.08, "duration": 3.879, "text": "here's an interesting way to approximate"}, {"start": 1.8, "duration": 5.119, "text": "Pi on the positive integer grid up to"}, {"start": 3.959, "duration": 4.72, "text": "101 shade the coordinate XY blue if the"}, {"start": 6.919, "duration": 4.281, "text": "greatest common divisor of X and Y is"}, {"start": 8.679, "duration": 5.12, "text": "one and shade the coordinate XY red"}, {"start": 11.2, "duration": 5.839, "text": "otherwise if you do this you get this"}, {"start": 13.799, "duration": 6.001, "text": "image shown here it's a famous result"}, {"start": 17.039, "duration": 5.0, "text": "that the quantity 6 over pi^ 2 is"}, {"start": 19.8, "duration": 4.68, "text": "approximately equal to the ratio of the"}, {"start": 22.039, "duration": 4.121, "text": "number of blue dots to Total dots it"}, {"start": 24.48, "duration": 5.959, "text": "turns out in this picture there are"}, {"start": 26.16, "duration": 6.2, "text": "6,287 blue dots and 10,21 total dots"}, {"start": 30.439, "duration": 3.8, "text": "this means that 6 over pi^ 2 is"}, {"start": 32.36, "duration": 4.519, "text": "approximately 0."}, {"start": 34.239, "duration": 3.961, "text": "61631 and from this we conclude that Pi"}, {"start": 36.879, "duration": 3.921, "text": "is approximately"}, {"start": 38.2, "duration": 6.12, "text": "3.12 not the best approximation but"}, {"start": 40.8, "duration": 3.52, "text": "definitely a fun and beautiful one"}], "title": "A rough Pi approximation from coprime integers", "description": "In this short, plot all the points in the positive integer grid up to 101. If a coordinate (x,y) has numbers x and y that are relatively prime, we shade the dot blue; otherwise we shade the dot red. A famous result states that roughly 6/pi*pi of the dots will be blue. So we can use this image to get an approximation for 6/pi*pi and in turn use that to get an approximation for Pi. We get 3.12, which is not really good, but it\u2019s kind of amazing that such a small grid gets us that close. Using a larger grid will give you a better approximation. \n\nIf you like this video, consider subscribing to the channel or consider buying me a coffee: https://www.buymeacoffee.com/VisualProofs. Thanks! \n\nFor more information about why this works, check out the wikipedia site:\n\nhttps://en.wikipedia.org/wiki/Coprime_integers\n\nFor more Pi-related videos, check out my playlist:\nhttps://youtube.com/playlist?list=PLZh9gzIvXQUsGRDvzvXc02lDIKHaVeFwq\n\n#manim #approximation  #relativelyprime #zetafunction #baselproblem #irrational #Pi #mathvideo\u200b #math\u200b #mtbos\u200b \u200b #animation\u200b\u200b\u200b #iteachmath #mathematics #piday \n\n\nTo learn more about animating with manim, check out:\nhttps://manim.community", "lengthSeconds": 42, "width": 360, "height": 640}
{"video_id": "k8uOEnqxrS8", "transcript": [{"start": 0.0, "duration": 5.1, "text": "in the 12th century Rabbi Abraham Barr"}, {"start": 3.0, "duration": 4.68, "text": "here discovered an amazing visual proof"}, {"start": 5.1, "duration": 4.56, "text": "of a circle's area so imagine a circle"}, {"start": 7.68, "duration": 5.579, "text": "filled with concentric circles whose"}, {"start": 9.66, "duration": 5.82, "text": "radiuses are and circumference is 2 pi r"}, {"start": 13.259, "duration": 4.141, "text": "then cut along one of the radius length"}, {"start": 15.48, "duration": 4.68, "text": "and open the circle into straight line"}, {"start": 17.4, "duration": 5.039, "text": "segments if repeated this process for"}, {"start": 20.16, "duration": 4.56, "text": "infinitely fine line segments the final"}, {"start": 22.439, "duration": 4.861, "text": "shape will be a triangle whose altitude"}, {"start": 24.72, "duration": 6.18, "text": "is r and bass is the circumference of"}, {"start": 27.3, "duration": 5.82, "text": "circle 2 pi r the triangles area is then"}, {"start": 30.9, "duration": 4.74, "text": "half times two pi r times R which"}, {"start": 33.12, "duration": 7.099, "text": "evaluates to be pi r squared therefore"}, {"start": 35.64, "duration": 4.579, "text": "the circle's area is pi times r squared"}], "title": "Unbelievable Visual Demonstration of Circle's Area", "description": "Visual Proof of Circle's Area | Area of Circle by Rabbi Abraham bar \u1e24iyya| Stay Tuned\n\nIn the Mid 12th century, Rabbi Abraham bar Hiyya Hansai discovered an amazing visual proof of a circle's area.\n\nKnow About Rabbi Abraham bar Hiyya Hanasi:\nhttps://en.m.wikipedia.org/wiki/Abraham_bar_Hiyya\n\nArea of a Circle by Rabbi Abraham bar Hiyya Hanasi:\nhttps://u.cs.biu.ac.il/~tsaban/Circles.html\n\nAbout the Video:\nWelcome to our mind-blowing showcase of jaw-dropping visuals demonstrating the awe-inspiring concept of a circle's area! In this captivating video, we unravel the secrets behind calculating the area of a circle through an extraordinary visual representation like never seen before. Prepare to be amazed as we unveil mesmerizing geometric patterns and unveil the mathematical wonders hidden within circles. Our experts walk you through each step of the process, providing clear explanations and enlightening insights. Whether you're a math enthusiast or just curious about the beauty of geometry, this video is a must-watch! Join us on this unique journey of exploration and expand your understanding of the remarkable properties possessed by circles.\n\nStay Tuned brings learning to life through its captivating educational videos.\n\n#AreaOfCircle #Mensuration #AreaOfTriangle #Math #Mathematics #CircleArea #GeometricPatterns #MathematicalWonders #MathEnthusiast #GeometryBeauty #VisualDemonstration #JawDroppingVisuals #CalculateArea #StayTuned\n\nRelated Keywords:\ncircle, area of a circle, geometric patterns, mathematical wonders, math enthusiast, beauty of geometry, visual representation, jaw-dropping visuals, calculating area, remarkable properties, mind-blowing showcase, extraordinary, mesmerizing, captivating video, math concepts, unique journey, proof of area of circle\n\nSchedule: New Video Every Fortnight\n\nTo stay updated, subscribe to our YouTube channel: https://youtube.com/channel/UCUHsuNo5zuz_FYu2rwVkIKA", "lengthSeconds": 39, "width": 360, "height": 640}
{"video_id": "uKkVazoO1KU", "transcript": [{"start": 0.719, "duration": 5.361, "text": "let's check out an interesting infinite"}, {"start": 3.92, "duration": 4.2, "text": "series you may be familiar with this"}, {"start": 6.08, "duration": 5.04, "text": "infinite series the sum where n ranges"}, {"start": 8.12, "duration": 4.88, "text": "from 0 to Infinity of 1 / n factorial"}, {"start": 11.12, "duration": 4.759, "text": "this infinite sum is famously equal to"}, {"start": 13.0, "duration": 4.92, "text": "the value e consequently this infinite"}, {"start": 15.879, "duration": 6.921, "text": "sum where n ranges from 1 to Infinity of"}, {"start": 17.92, "duration": 7.279, "text": "1 n + 1 factorial must equal eus 2 what"}, {"start": 22.8, "duration": 4.16, "text": "if we modify this series slightly and we"}, {"start": 25.199, "duration": 5.121, "text": "consider the sum where n ranges from 1"}, {"start": 26.96, "duration": 5.56, "text": "to Infinity of n / n +1 factorial this"}, {"start": 30.32, "duration": 4.559, "text": "is the infinite sum 1 over 2 factorial"}, {"start": 32.52, "duration": 4.879, "text": "plus 2 3 factorial plus 3 over 4"}, {"start": 34.879, "duration": 5.281, "text": "factorial and so on what's this infinite"}, {"start": 37.399, "duration": 4.48, "text": "sum this is a series of positive terms"}, {"start": 40.16, "duration": 3.48, "text": "and we can show using a geometric"}, {"start": 41.879, "duration": 4.281, "text": "dissection argument that the partial"}, {"start": 43.64, "duration": 4.2, "text": "sums are bounded above moreover the"}, {"start": 46.16, "duration": 3.2, "text": "dissection argument will show us the"}, {"start": 47.84, "duration": 4.44, "text": "actual sum of the"}, {"start": 49.36, "duration": 5.92, "text": "series let's consider a square whose"}, {"start": 52.28, "duration": 4.759, "text": "side lengths are both one we can cut"}, {"start": 55.28, "duration": 3.439, "text": "this Square in half and shade one of"}, {"start": 57.039, "duration": 3.961, "text": "those halves so that we've shaded one"}, {"start": 58.719, "duration": 4.761, "text": "copy of one2 then we can cut the"}, {"start": 61.0, "duration": 5.08, "text": "remaining half into three equal pieces"}, {"start": 63.48, "duration": 6.04, "text": "each of Area 1 six and shade two of them"}, {"start": 66.08, "duration": 6.2, "text": "so that we've shade 2 * 1 over6 or 2 * 1"}, {"start": 69.52, "duration": 5.279, "text": "3 factorial we can cut the remaining 1"}, {"start": 72.28, "duration": 5.64, "text": "over 3 factorial area into four equal"}, {"start": 74.799, "duration": 5.28, "text": "pieces each of Area 1 24 or 1 4"}, {"start": 77.92, "duration": 5.36, "text": "factorial when we shade three of them"}, {"start": 80.079, "duration": 5.161, "text": "we've shaded 3 * 1 4 factorial now cut"}, {"start": 83.28, "duration": 4.159, "text": "the remaining 1 over4 factorial into"}, {"start": 85.24, "duration": 5.44, "text": "five equal pieces and shade four of them"}, {"start": 87.439, "duration": 5.32, "text": "thus shading 4 * 1 over 5 factorial"}, {"start": 90.68, "duration": 3.92, "text": "repeat this process taking the 1 over 5"}, {"start": 92.759, "duration": 3.921, "text": "factorial that's left cut it into six"}, {"start": 94.6, "duration": 5.04, "text": "equal pieces and shade five of them"}, {"start": 96.68, "duration": 4.64, "text": "shading 5 * 1 over 6 factorial let's do"}, {"start": 99.64, "duration": 3.88, "text": "this again cutting the remaining 1 over"}, {"start": 101.32, "duration": 4.479, "text": "6 factorial into seven equal pieces and"}, {"start": 103.52, "duration": 5.239, "text": "shading six of them for a total of 6 * 1"}, {"start": 105.799, "duration": 5.241, "text": "over 7 factorial we can continue this"}, {"start": 108.759, "duration": 4.841, "text": "process indefinitely each time cutting"}, {"start": 111.04, "duration": 4.52, "text": "the remaining 1 n factorial into n + 1"}, {"start": 113.6, "duration": 4.519, "text": "equal pieces and shading n of them"}, {"start": 115.56, "duration": 5.04, "text": "shading a total area of n / n + 1"}, {"start": 118.119, "duration": 4.64, "text": "factorial at each stage"}, {"start": 120.6, "duration": 4.4, "text": "but now we see that the infinite sum of"}, {"start": 122.759, "duration": 5.441, "text": "interest is the limiting value of the"}, {"start": 125.0, "duration": 5.8, "text": "shaded areas and more importantly we see"}, {"start": 128.2, "duration": 4.84, "text": "that in the limit the entire square area"}, {"start": 130.8, "duration": 4.32, "text": "is shaded therefore the infinite sum"}, {"start": 133.04, "duration": 5.559, "text": "that we're interested in must equal the"}, {"start": 135.12, "duration": 5.759, "text": "area of the square which is 1 * 1 so we"}, {"start": 138.599, "duration": 3.961, "text": "have seen by a geometric dissection that"}, {"start": 140.879, "duration": 4.841, "text": "the sum where n ranges from 1 to"}, {"start": 142.56, "duration": 5.6, "text": "Infinity of n / by n + 1 factorial must"}, {"start": 145.72, "duration": 4.64, "text": "be 1 this is truly amazing that it's"}, {"start": 148.16, "duration": 4.0, "text": "such a simple value given that the two"}, {"start": 150.36, "duration": 4.32, "text": "infinite series that this infinite sum"}, {"start": 152.16, "duration": 5.76, "text": "was related to had values of e and e"}, {"start": 154.68, "duration": 3.24, "text": "minus 2 which are both"}, {"start": 162.12, "duration": 3.0, "text": "irrational"}], "title": "A Factorial Series Geometric Dissection (visual proof)", "description": "This is animated visual proof demonstrating the sum of the infinite series of of factorials, scaled by an appropriate value each time. This series is closed related to the famous infinite series for Euler's constant e.\n\nIf you like this video, consider subscribing to the channel or consider buying me a coffee: https://www.buymeacoffee.com/VisualProofs. Thanks!\n\nFor more infinite series dissections, check out this playlist:\nhttps://www.youtube.com/playlist?list=PLZh9gzIvXQUv6JqhlqQDlnqQw1NczW5Ro\n\nHere is a longer form video with lots of infinite geometric sums: https://youtu.be/JteQEN1XPyc\n\nThis animation is based on a visual proof by John Mason  from the November 1995 issue of The College Mathematics Journal (https://www.jstor.org/stable/2687383 - page 381).\n\n\u200b #math\u200b #calculus #mtbos\u200b #manim\u200b #animation\u200b #theorem\u200b #pww\u200b #proofwithoutwords\u200b #visualproof\u200b #proof\u200b #iteachmath #factorials #factorial #series #infiniteseries #fractal #infinitesum  #algebra #eulerconstant #e\n\n\nTo learn more about animating with manim, check out:\nhttps://manim.community", "lengthSeconds": 163, "width": 640, "height": 360}
{"video_id": "VWbLEaFDztU", "transcript": [{"start": 0.42, "duration": 4.2, "text": "have you ever wanted to trisect an angle"}, {"start": 3.0, "duration": 4.5, "text": "stick around and I'll show you one"}, {"start": 4.62, "duration": 4.86, "text": "technique to do this"}, {"start": 7.5, "duration": 4.44, "text": "the problem of trisecting an angle"}, {"start": 9.48, "duration": 5.64, "text": "requires us to take a given angle Alpha"}, {"start": 11.94, "duration": 5.82, "text": "and using classical tools find two lines"}, {"start": 15.12, "duration": 4.62, "text": "that trisect the angle into three pieces"}, {"start": 17.76, "duration": 4.38, "text": "all with an angle measure of alpha over"}, {"start": 19.74, "duration": 4.68, "text": "three it turns out that you cannot"}, {"start": 22.14, "duration": 4.44, "text": "trisect a general angle using a straight"}, {"start": 24.42, "duration": 5.22, "text": "edge Encompass but it might be possible"}, {"start": 26.58, "duration": 5.4, "text": "if we add an extra tool"}, {"start": 29.64, "duration": 4.56, "text": "polar curve R equals Theta is called the"}, {"start": 31.98, "duration": 4.86, "text": "Archimedes spiral let's take a look at"}, {"start": 34.2, "duration": 4.44, "text": "the graph of this spiral right here"}, {"start": 36.84, "duration": 3.6, "text": "you cannot construct the Archimedes"}, {"start": 38.64, "duration": 5.04, "text": "spiral using Straight Edge Encompass"}, {"start": 40.44, "duration": 5.76, "text": "however if we add the spiral as one of"}, {"start": 43.68, "duration": 4.14, "text": "our tools we may be able to use this"}, {"start": 46.2, "duration": 4.26, "text": "tool along with the straight edge"}, {"start": 47.82, "duration": 4.5, "text": "Encompass to trisect the angle we won't"}, {"start": 50.46, "duration": 3.96, "text": "actually need the entire Archimedes"}, {"start": 52.32, "duration": 5.3, "text": "spiral to do this just the bit that lies"}, {"start": 54.42, "duration": 3.2, "text": "between 0 and 2 pi"}, {"start": 59.879, "duration": 3.961, "text": "let's imagine that we have the unit"}, {"start": 61.86, "duration": 4.32, "text": "circle drawn like this"}, {"start": 63.84, "duration": 5.52, "text": "and that we have two points one along"}, {"start": 66.18, "duration": 5.759, "text": "the x-axis and the other one creating a"}, {"start": 69.36, "duration": 4.439, "text": "line with an angle of alpha the arc on"}, {"start": 71.939, "duration": 5.461, "text": "the unit circle subtending the angle"}, {"start": 73.799, "duration": 5.941, "text": "Alpha has an arc length of length Alpha"}, {"start": 77.4, "duration": 4.68, "text": "now let's draw the Archimedes spiral R"}, {"start": 79.74, "duration": 4.879, "text": "equals Theta this intersects our Top"}, {"start": 82.08, "duration": 5.52, "text": "Line at a point let's call the point P"}, {"start": 84.619, "duration": 5.381, "text": "because P lies on the line created by"}, {"start": 87.6, "duration": 4.14, "text": "angle Alpha it has an angle of Alpha and"}, {"start": 90.0, "duration": 4.259, "text": "because it lies on the Archimedes spiral"}, {"start": 91.74, "duration": 4.739, "text": "it must create a radius of length Alpha"}, {"start": 94.259, "duration": 4.441, "text": "from the origin we can then use our"}, {"start": 96.479, "duration": 4.261, "text": "straight edge Encompass to divide the"}, {"start": 98.7, "duration": 4.44, "text": "line from the origin to the point P into"}, {"start": 100.74, "duration": 3.96, "text": "three equal pieces like this we use our"}, {"start": 103.14, "duration": 3.36, "text": "Compass to create this Arc of radius"}, {"start": 104.7, "duration": 3.84, "text": "Alpha over 3. it intersects the"}, {"start": 106.5, "duration": 3.96, "text": "archimedian spiral at a point q and"}, {"start": 108.54, "duration": 4.02, "text": "because Q has a radius of alpha over 3"}, {"start": 110.46, "duration": 4.56, "text": "its angle must be Alpha over 3."}, {"start": 112.56, "duration": 4.5, "text": "similarly we construct this Arc of"}, {"start": 115.02, "duration": 3.66, "text": "radius 2 Alpha over 3 which intersects"}, {"start": 117.06, "duration": 4.14, "text": "the Archimedean spiral at a point R"}, {"start": 118.68, "duration": 4.979, "text": "because R has a radius of 2 Alpha over"}, {"start": 121.2, "duration": 3.36, "text": "three its angle is 2 Alpha over three as"}, {"start": 123.659, "duration": 3.121, "text": "well"}, {"start": 124.56, "duration": 4.32, "text": "and now we can finish off our problem we"}, {"start": 126.78, "duration": 5.06, "text": "connect the origin to point q and the"}, {"start": 128.88, "duration": 2.96, "text": "origin to point R"}, {"start": 132.599, "duration": 4.621, "text": "and sure enough we have divided the"}, {"start": 134.64, "duration": 4.92, "text": "angle Alpha into three separate angles"}, {"start": 137.22, "duration": 4.14, "text": "all of measure Alpha over three"}, {"start": 139.56, "duration": 4.259, "text": "this means that we've used the"}, {"start": 141.36, "duration": 4.879, "text": "Archimedes spiral to trisect any angle"}, {"start": 143.819, "duration": 2.42, "text": "Alpha"}], "title": "Trisect an Angle with Archimedean Spiral (visual proof)", "description": "This is a short, animated visual proof demonstrating how to trisect any angle using the Archimedean spiral. #manim #math\u200b\u200b #mathshorts\u200b #mathvideo\u200b  #trisect #trisectangle #impossible #geometry #chords #mtbos\u200b #manim\u200b #animation\u200b #theorem\u200b #pww\u200b #proofwithoutwords\u200b #visualproof\u200b #proof\u200b #iteachmath #inequality \n\nThis animation is based on a visual proof from the wonderful book Charming Proofs by Claudi Alsina and Roger Nelsen: (Affiliate link; I may get small commissions) https://amzn.to/3LbcpgG\n\nTo learn more about animating with manim, check out:\nhttps://manim.community", "lengthSeconds": 153, "width": 640, "height": 360}
{"video_id": "5pRYbKFWjUM", "transcript": [{"start": 11.76, "duration": 16.739, "text": "[\u0645\u0648\u0633\u064a\u0642\u0649]"}, {"start": 29.96, "duration": 6.799, "text": "\u200fn"}, {"start": 31.64, "duration": 5.119, "text": "[\u0645\u0648\u0633\u064a\u0642\u0649]"}], "title": "#trending #mathematics Visual Proof: Factors of Quadratic Equation Polynomial x^2-3x+2", "description": "#trending #mathematics Visual Proof: Factors of Quadratic Polynomial PART-2 x^2-3x+2\n\nOther Videos\nFibonacci Sequence, Golden Ratio, How to draw Golden Spiral:\nhttps://youtu.be/yHBa3EvDPco\nRadius and Tangent Vectors: \nhttps://youtube.com/shorts/KHSafRET3Sk\nDe Morgan's laws: Proof by Venn Diagrams:\nhttps://youtube.com/shorts/pR-REo4aBYk?feature=share\nVisual Proof of Roots of Quadratic Equation\nhttps://youtube.com/shorts/FdE7rHDZRd8\nVisual Proof: Factors of Quadratic Equation PART-1 x^2+3x+2\nhttps://youtube.com/shorts/lIqv5zKAKKE\nVisual Proof: Factors of Quadratic Polynomial PART-2 x^2-3x+2\nhttps://youtube.com/shorts/5pRYbKFWjUM", "lengthSeconds": 36, "width": 360, "height": 640}
{"video_id": "PQZonEZ8usM", "transcript": [{"start": 0.08, "duration": 5.679, "text": "satu motion and leonardo da vinci"}, {"start": 2.639, "duration": 5.361, "text": "devised an ingenious method to find the"}, {"start": 5.759, "duration": 5.281, "text": "area of a circle"}, {"start": 8.0, "duration": 6.08, "text": "partition the circle into slices"}, {"start": 11.04, "duration": 5.44, "text": "split the circle in half and rearrange"}, {"start": 14.08, "duration": 3.92, "text": "corresponding slices to form a"}, {"start": 16.48, "duration": 3.04, "text": "parallelogram-like"}, {"start": 18.0, "duration": 4.4, "text": "shape"}, {"start": 19.52, "duration": 5.28, "text": "if we increase the number of slices and"}, {"start": 22.4, "duration": 4.639, "text": "repeat the entire process"}, {"start": 24.8, "duration": 4.319, "text": "the rearranged shape will approach the"}, {"start": 27.039, "duration": 4.4, "text": "shape of a rectangle"}, {"start": 29.119, "duration": 5.361, "text": "whose width is equal to the radius of"}, {"start": 31.439, "duration": 7.041, "text": "the circle and height is equal to half"}, {"start": 34.48, "duration": 5.68, "text": "the circumference or pi times r"}, {"start": 38.48, "duration": 4.32, "text": "the circle's area"}, {"start": 40.16, "duration": 5.52, "text": "is equal to the rectangle's area"}, {"start": 42.8, "duration": 6.0, "text": "which is r times pi r"}, {"start": 45.68, "duration": 6.8, "text": "and that equals pi r squared"}, {"start": 48.8, "duration": 6.64, "text": "thus the circle's area is pi r squared"}, {"start": 52.48, "duration": 2.96, "text": "incredible"}], "title": "Do you know Leonardo da Vinci's circle area proof?", "description": "Happy Pi Day! Sato Moshun and Leonardo da Vinci independently devised an incredible visualization to find the formula for a circle's area. #Shorts\n\n(Apologies there was no early access on Patreon--I finished making the video just in time for Pi Day).\n\nWikipedia\nhttps://en.wikipedia.org/wiki/Area_of_a_circle#Rearrangement_proof\nCut The Knot area of a circle\nhttps://www.cut-the-knot.org/Curriculum/Geometry/AreaOfCircle.shtml\nArea of a circle, formula explained (mathematicsonline)\nhttps://www.youtube.com/watch?v=YokKp3pwVFc\nUnderstanding the Formula for Area of a Circle #shorts (mathematicsonline)\nhttps://youtu.be/0RGz_rvXVpk\n\nSubscribe: https://www.youtube.com/user/MindYourDecisions?sub_confirmation=1\n\nSend me suggestions by email (address at end of many videos). I may not reply but I do consider all ideas!\n\nIf you purchase through these links, I may be compensated for purchases made on Amazon. As an Amazon Associate I earn from qualifying purchases. This does not affect the price you pay. \n\nBook ratings are from January 2022.\n\nMy Books (worldwide links)\nhttps://mindyourdecisions.com/blog/my-books/#worldwide\n\nMy Books (US links)\nMind Your Decisions: Five Book Compilation\nhttps://amzn.to/2pbJ4wR\nA collection of 5 books:\n\"The Joy of Game Theory\" rated 4.2/5 stars on 224 reviews\nhttp://amzn.to/1uQvA20\n\"The Irrationality Illusion: How To Make Smart Decisions And Overcome Bias\" rated 4/5 stars on 24 reviews\nhttp://amzn.to/1o3FaAg\n\"40 Paradoxes in Logic, Probability, and Game Theory\" rated 4.1/5 stars on 38 reviews\nhttp://amzn.to/1LOCI4U\n\"The Best Mental Math Tricks\" rated 4.2/5 stars on 76 reviews\nhttp://amzn.to/18maAdo\n\"Multiply Numbers By Drawing Lines\" rated 4.3/5 stars on 30 reviews\nhttp://amzn.to/XRm7M4\n\nMind Your Puzzles: Collection Of Volumes 1 To 3\nhttps://amzn.to/2mMdrJr\nA collection of 3 books:\n\"Math Puzzles Volume 1\" rated 4.4/5 stars on 87 reviews\nhttp://amzn.to/1GhUUSH\n\"Math Puzzles Volume 2\" rated 4.1/5 stars on 24 reviews\nhttp://amzn.to/1NKbyCs\n\"Math Puzzles Volume 3\" rated 4.2/5 stars on 22 reviews\nhttp://amzn.to/1NKbGlp\n\n2017 Shorty Awards Nominee. Mind Your Decisions was nominated in the STEM category (Science, Technology, Engineering, and Math) along with eventual winner Bill Nye; finalists Adam Savage, Dr. Sandra Lee, Simone Giertz, Tim Peake, Unbox Therapy; and other nominees Elon Musk, Gizmoslip, Hope Jahren, Life Noggin, and Nerdwriter.\n\nMy Blog\nhttps://mindyourdecisions.com/blog/\n\nTwitter\nhttps://twitter.com/preshtalwalkar\n\nMerch\nhttps://teespring.com/stores/mind-your-decisions\n\nPatreon\nhttps://www.patreon.com/mindyourdecisions\n\nPress\nhttps://mindyourdecisions.com/blog/press #math #maths #mathematics", "lengthSeconds": 54, "width": 202, "height": 360}
{"video_id": "Ocz2tqt6zz4", "transcript": [{"start": 1.02, "duration": 7.14, "text": "foreign"}, {"start": 2.72, "duration": 7.6, "text": "triangle with sides of length a b and c"}, {"start": 8.16, "duration": 4.019, "text": "draw the inner circle of this triangle"}, {"start": 10.32, "duration": 3.96, "text": "and draw the in radii that are"}, {"start": 12.179, "duration": 4.44, "text": "perpendicular to the three sides call"}, {"start": 14.28, "duration": 4.86, "text": "the length of the in radius R the angle"}, {"start": 16.619, "duration": 4.801, "text": "bisectors meet at the in center of the"}, {"start": 19.14, "duration": 4.26, "text": "triangle and together they create these"}, {"start": 21.42, "duration": 3.96, "text": "three triangular regions which we can"}, {"start": 23.4, "duration": 4.02, "text": "unfold"}, {"start": 25.38, "duration": 5.219, "text": "each of these triangles has a height of"}, {"start": 27.42, "duration": 6.36, "text": "length R so the area of the triangle is"}, {"start": 30.599, "duration": 5.521, "text": "one half times a times R plus one half"}, {"start": 33.78, "duration": 4.799, "text": "times B times R plus one-half times c"}, {"start": 36.12, "duration": 4.86, "text": "times R factoring this out we see that"}, {"start": 38.579, "duration": 4.921, "text": "the area of the triangle is R times the"}, {"start": 40.98, "duration": 3.419, "text": "quantity a plus b plus C all divided by"}, {"start": 43.5, "duration": 3.0, "text": "two"}, {"start": 44.399, "duration": 3.301, "text": "so the area of a triangle is given by"}, {"start": 46.5, "duration": 4.579, "text": "the product of the length of the in"}, {"start": 47.7, "duration": 3.379, "text": "radius and the semi-perimeter"}], "title": "Triangle area formula", "description": "This short video shows how to prove that the area of the triangle is given by the product of the inradius and the semiperimeter. Here is a longer, alternate version: https://youtu.be/lRDtdIyFhpE\n\nThanks to Colin Wright for suggesting this particular visualization.\n_______________________________\nMusic in this video:\nPrelude No. 9 by Chris Zabriskie is licensed under a Creative Commons Attribution 4.0 license. https://creativecommons.org/licenses/by/4.0/\n\nSource: http://chriszabriskie.com/preludes/\n\nArtist: http://chriszabriskie.com/", "lengthSeconds": 50, "width": 360, "height": 640}
{"video_id": "HZEvtMMq1TA", "transcript": [{"start": 1.85, "duration": 14.939, "text": "[\u0645\u0648\u0633\u064a\u0642\u0649]"}], "title": "Visual proof of:Surface area of Sphere and Volume of sphere#maths #mathematics", "description": "", "lengthSeconds": 15, "width": 360, "height": 640}
{"video_id": "HQgzTbymElc", "transcript": [{"start": 0.359, "duration": 4.121, "text": "\u0915\u0908 \u092c\u093e\u0930 \u091a\u0940\u091c\u094b\u0902 \u0915\u093e \u090f\u0915 \u092c\u0939\u0941\u0924 \u0939\u0940 \u0938\u0941\u0902\u0926\u0930 \u0935\u093f\u091c\u0941\u0905\u0932"}, {"start": 2.36, "duration": 5.32, "text": "\u0938\u0949\u0932\u094d\u092f\u0942\u0936\u0928 \u092d\u0940 \u0939\u094b\u0924\u093e \u0939\u0948 \u091c\u0948\u0938\u0947 \u0907\u0938 \u0936\u0947\u092a \u0915\u094b \u0915\u0939\u0924\u0947"}, {"start": 4.48, "duration": 6.0, "text": "\u0939\u0948\u0902 \u090f\u0932\u0930 \u0930\u093f\u0902\u0917 \u0914\u0930 \u0907\u0938\u0915\u093e \u090f\u0930\u093f\u092f\u093e \u0907\u0938\u0915\u0947 \u0907\u0928\u0930 \u0938\u0930\u0915\u092e"}, {"start": 7.68, "duration": 4.959, "text": "\u092b\u0930\u0947\u0902\u0938 \u0915\u0947 \u0915\u0949\u0921 \u091f\u0947\u0902\u091c \u092a\u0930 \u0939\u0940 \u0928\u093f\u0930\u094d\u092d\u0930 \u0915\u0930\u0924\u093e \u0939\u0948"}, {"start": 10.48, "duration": 4.279, "text": "\u092f\u0939 \u091a\u0940\u091c \u0906\u092a \u092c\u0921\u093c\u0947 \u0938\u0930\u094d\u0915\u0932 \u0938\u0947 \u091b\u094b\u091f\u0947 \u0938\u0930\u094d\u0915\u0932 \u0915\u093e"}, {"start": 12.639, "duration": 4.281, "text": "\u090f\u0930\u093f\u092f\u093e \u0938\u092c\u0938\u094d\u091f\u0948\u0915 \u0915\u0930\u0915\u0947 \u092b\u093f\u0930 \u092a\u093e\u0907\u0925\u093e\u0917\u094b\u0930\u0938 \u0925\u094d\u092f\u094b\u0930\u092e"}, {"start": 14.759, "duration": 4.241, "text": "\u0932\u0917\u093e \u0915\u0947 \u092d\u0940 \u0928\u093f\u0915\u093e\u0932 \u0938\u0915\u0924\u0947 \u0939\u094b \u0905\u092c \u0905\u0917\u0930 \u0935\u093f\u091c\u0941\u0905\u0932"}, {"start": 16.92, "duration": 4.48, "text": "\u0905\u092a\u094d\u0930\u094b\u091a \u0926\u0947\u0916\u0947\u0902 \u0924\u094b \u092e\u0948\u0902 \u0905\u0917\u0930 \u0907\u0938 \u090f\u0928\u0932 \u0930\u093f\u0902\u0917 \u0915\u094b"}, {"start": 19.0, "duration": 4.24, "text": "\u091b\u094b\u091f\u0947-\u091b\u094b\u091f\u0947 \u0938\u094d\u0932\u093e\u0907\u0938\u0947\u091c \u092e\u0947\u0902 \u0921\u093f\u0935\u093e\u0907\u0921 \u0915\u0930\u0942\u0902 \u0924\u094b \u092f\u0939"}, {"start": 21.4, "duration": 4.44, "text": "\u0936\u0947\u092a \u0905\u092c \u0906\u092a\u0915\u094b \u090f\u0915 \u0915\u0948\u092e\u0930\u093e \u090f\u092a\u0930\u094d\u091a\u0930 \u0915\u0940 \u0924\u0930\u0939 \u0926\u093f\u0916"}, {"start": 23.24, "duration": 5.359, "text": "\u0930\u0939\u093e \u0939\u094b\u0917\u093e \u0914\u0930 \u0905\u0917\u0930 \u092e\u0948\u0902 \u0907\u0938\u0915\u094b \u0938\u094d\u0915\u094d\u0935\u093f\u091c \u0915\u0930\u0942\u0902 \u0924\u094b"}, {"start": 25.84, "duration": 4.759, "text": "\u092f\u0939 \u0915\u0948\u092e\u0930\u093e \u0905\u092a\u0930\u094d\u091f \u0915\u094d\u0932\u094b\u091c \u0939\u094b\u0915\u0947 \u0932\u0917\u092d\u0917 \u090f\u0915 \u0938\u0930\u094d\u0915\u0932"}, {"start": 28.599, "duration": 4.0, "text": "\u0915\u093e \u090f\u092a\u094d\u0930\u094b\u0915\u094d\u0938\u0940\u092e\u0947\u0936\u0928 \u092c\u0928 \u091c\u093e\u090f\u0917\u093e"}, {"start": 30.599, "duration": 4.201, "text": "\u0915\u094d\u092f\u094b\u0902\u0915\u093f \u092f \u0938\u094d\u0932\u093e\u0907\u0938 \u0915\u0940 \u0938\u093e\u0907\u091c \u092c\u0921\u093c\u0940 \u0925\u0940 \u0924\u094b \u092f\u0939"}, {"start": 32.599, "duration": 4.241, "text": "\u0938\u0930\u094d\u0915\u0932 \u0915\u093e \u0915\u0947\u0935\u0932 \u090f\u092a\u094d\u0930\u094b\u0915\u094d\u0938\u0940\u092e\u0947\u0936\u0928 \u0939\u0948 \u092a\u0930 \u0905\u0917\u0930"}, {"start": 34.8, "duration": 3.96, "text": "\u092e\u0948\u0902 \u0915\u093e\u092b\u0940 \u091b\u094b\u091f\u0947 \u091b\u094b\u091f\u0947 \u0938\u094d\u0932\u093e\u0907\u0938 \u092f\u0942\u091c \u0915\u0930\u0942\u0902 \u0924\u094b"}, {"start": 36.84, "duration": 5.199, "text": "\u0906\u092a\u0915\u094b \u090f\u0915 \u092a\u0930\u092b\u0947\u0915\u094d\u091f \u0938\u0930\u094d\u0915\u0932 \u092e\u093f\u0932\u0947\u0917\u093e \u091c\u093f\u0938\u0915\u093e \u090f\u0930\u093f\u092f\u093e"}, {"start": 38.76, "duration": 5.119, "text": "\u0939\u094b\u0917\u093e \u092a\u093e \u0938\u094d\u0915\u094d\u0935\u093e \u092f\u093e\u0928\u0940 \u0915\u093f \u090f\u0928\u0930 \u0930\u093f\u0902\u0917 \u0915\u093e \u090f\u0930\u093f\u092f\u093e"}, {"start": 42.039, "duration": 4.04, "text": "\u0915\u0947\u0935\u0932 \u0907\u0928\u0930 \u0915 \u091f\u0947\u0902\u091c \u0915\u0947 \u0938\u094d\u0915\u094d\u0935\u093e\u092f\u0930 \u092a\u0930 \u0921\u093f\u092a\u0947\u0902\u0921"}, {"start": 43.879, "duration": 2.2, "text": "\u0915\u0930\u0924\u093e"}], "title": "Visual Proofs", "description": "", "lengthSeconds": 44, "width": 360, "height": 640}
{"video_id": "B272ylT_cZ0", "transcript": [{"start": 0.089, "duration": 4.521, "text": "\u0926\u094b\u0938\u094d\u0924\u094b\u0902 \u092e\u0948\u091a \u0915\u0940 \u090f\u0915 \u092b\u0947\u092e\u0938 \u0907\u0928\u093f\u0915\u094d\u0935\u093e\u0932\u093f\u091f\u0940 \u0939\u0948"}, {"start": 4.67, "duration": 5.08, "text": "\u0924\u094b \u0909\u0928\u0915\u093e \u0905\u0930\u094d\u0925\u092e\u0948\u091f\u093f\u0915 \u092e\u093f\u0928 \u090f + \u092c\u0940 / 2 \u092f\u093e \u0924\u094b"}, {"start": 7.83, "duration": 4.38, "text": "\u092c\u0921\u093c\u093e \u0939\u094b\u0917\u093e \u092f\u093e \u092c\u0930\u093e\u092c\u0930 \u0939\u094b\u0917\u093e \u0909\u0928\u0915\u0947 \u091c\u094d\u092f\u093e\u092e\u093f\u0924\u0940\u092f"}, {"start": 9.75, "duration": 4.799, "text": "\u092e\u093f\u0928 \u0938\u0947 \u092e\u093e\u0928\u0947 \u0938\u094d\u0915\u094d\u0935\u093e\u092f\u0930 \u0930\u0942\u091f \u0905\u092c \u0938\u0947 \u0910\u0938\u093e \u0915\u094d\u092f\u094b\u0902"}, {"start": 12.21, "duration": 4.02, "text": "\u0939\u094b\u0924\u093e \u0939\u0948 \u0935\u093f\u091c\u0941\u0905\u0932 \u0938\u092e\u091d\u0924\u0947 \u0939\u0948\u0902 \u090f\u0915 \u092a\u0949\u0907\u0902\u091f \u092a\u0940 \u0915\u0947"}, {"start": 14.549, "duration": 4.801, "text": "\u0926\u094b\u0928\u094b\u0902 \u0924\u0930\u092b \u0921\u094d\u0930\u0949 \u0915\u0930\u0924\u0947 \u0939\u0948\u0902 \u0932\u093e\u0907\u0928 \u0938\u0947\u0917\u092e\u0947\u0902\u091f\u094d\u0938"}, {"start": 16.23, "duration": 5.52, "text": "\u0932\u0947\u0902\u0925 \u090f \u0914\u0930 \u092c\u0940\u0915\u0947 \u0905\u092c \u090f + \u092c\u0940 \u0915\u094b \u0921\u093e\u092f\u092e\u0940\u091f\u0930 \u092e\u093e\u0902"}, {"start": 19.35, "duration": 4.8, "text": "\u0915\u0930 \u0914\u0930 \u092a\u0949\u0907\u0902\u091f \u0938\u0940 \u0915\u094b \u0938\u0947\u0902\u091f\u0930 \u092e\u093e\u0902 \u0915\u0947 \u0921\u094d\u0930\u0949 \u0915\u0930\u0924\u0947"}, {"start": 21.75, "duration": 5.359, "text": "\u0939\u0948\u0902 \u090f\u0915 \u0938\u0930\u094d\u0915\u0932 \u0924\u094b \u092f\u0947 \u0938\u0930\u094d\u0915\u0932 \u0915\u093e \u0930\u0947\u0921\u093f\u092f\u0938 \u0939\u0948 \u0914\u0930"}, {"start": 24.15, "duration": 6.719, "text": "\u092d\u0940 \u0915\u093e \u0905\u0930\u094d\u0925\u092e\u0948\u091f\u093f\u0915 \u092e\u0948\u0902 \u092e\u093e\u0928\u0947 \u090f + \u092c\u0940 ="}, {"start": 27.109, "duration": 5.681, "text": "\u092f\u0947\u0932\u094b \u0932\u093e\u0907\u0928 \u0938\u0947 \u0905\u092c \u0905\u0917\u0930 \u092a\u0949\u0907\u0902\u091f \u092a\u0940 \u092a\u0947 \u0921\u093e\u092f\u092e\u0940\u091f\u0930"}, {"start": 30.869, "duration": 4.081, "text": "\u0915\u0947 \u092a\u0930\u092a\u0947\u0902\u0921\u093f\u0915\u0941\u0932\u0930 \u090f\u0915 \u0914\u0930 \u092c\u0928\u093e\u090f\u0902 \u0914\u0930 \u0915\u0949\u0930\u094d\u0921 \u0915\u0947"}, {"start": 32.79, "duration": 4.62, "text": "\u0906\u0927\u0947 \u0939\u093f\u0938\u094d\u0938\u0947 \u0915\u094b \u0928\u093e\u092e \u0926\u0947 \u0939 \u0924\u094b \u0938\u0930\u094d\u0915\u0932 \u0915\u0940 \u0915\u0949\u0932\u094d\u0921"}, {"start": 34.95, "duration": 5.64, "text": "\u0907\u0902\u091f\u0930\u0938\u0947\u0915\u094d\u0936\u0928 \u0925\u094d\u092f\u094b\u0930\u092e \u0938\u0947 \u092e\u093f\u0932\u0947\u0917\u093e \u0939 * \u0939 = \u0905\u092c"}, {"start": 37.41, "duration": 5.46, "text": "\u092f\u093e \u0939 = \u0938\u094d\u0915\u094d\u0935\u093e\u092f\u0930 \u0930\u0942\u091f \u0905\u092c \u092e\u093e\u0928\u0947 \u0939 \u092c\u0928\u093e \u090f \u0914\u0930"}, {"start": 40.59, "duration": 3.66, "text": "\u092c\u0940 \u0915\u093e \u091c\u094d\u092f\u093e\u092e\u093f\u0924\u0940\u092f \u092e\u0948\u0902 \u0924\u094b \u092b\u093e\u0907\u0928\u0932\u0940 \u0932\u0947\u092b\u094d\u091f \u0935\u093e\u0932\u0940"}, {"start": 42.87, "duration": 3.06, "text": "\u092f\u0947\u0932\u094b \u0932\u093e\u0907\u0928 \u0915\u0940 \u0932\u0947\u0902\u0925 \u0939\u0948 \u090f \u0914\u0930 \u092c\u0940 \u0915\u093e"}, {"start": 44.25, "duration": 4.8, "text": "\u091c\u094d\u092f\u093e\u092e\u093f\u0924\u0940\u092f \u092e\u0948\u0902 \u0914\u0930 \u0938\u0947\u0902\u091f\u0930 \u0935\u093e\u0932\u0940 \u092f\u0947\u0932\u094b \u0932\u093e\u0907\u0928 \u0915\u0940"}, {"start": 45.93, "duration": 4.74, "text": "\u0932\u0947\u0902\u0925 \u0939\u0948 \u0905\u0930\u094d\u0925\u092e\u0948\u091f\u093f\u0915 \u092e\u0948\u0902 \u0915\u094d\u0932\u093f\u092f\u0930 \u0939\u0948 \u0915\u0940 \u090f \u0914\u0930"}, {"start": 49.05, "duration": 3.54, "text": "\u092c\u0940 \u0915\u093f\u0938\u0940 \u092d\u0940 \u092a\u094d\u0930\u094b\u092a\u094b\u0930\u0947\u0936\u0928 \u092e\u0947\u0902 \u0939\u094b \u0938\u0947\u0902\u091f\u0930 \u0935\u093e\u0932\u0940"}, {"start": 50.67, "duration": 3.42, "text": "\u0932\u093e\u0907\u0928 \u0915\u0940 \u0932\u0947\u0902\u0925 \u0932\u0947\u092b\u094d\u091f \u0935\u093e\u0932\u0940 \u0932\u093e\u0907\u0928 \u0915\u0940 \u0932\u0947\u0902\u0925 \u0938\u0947"}, {"start": 52.59, "duration": 3.84, "text": "\u092c\u0921\u093c\u0940 \u0939\u094b\u0917\u0940 \u092f\u093e \u0909\u0938\u0915\u0947 \u092c\u0930\u093e\u092c\u0930 \u0939\u094b\u0917\u0940 \u092e\u0924\u0932\u092c"}, {"start": 54.09, "duration": 5.899, "text": "\u0905\u0930\u094d\u0925\u092e\u0948\u091f\u093f\u0915 \u092e\u0948\u0902 \u0939\u092e\u0947\u0936\u093e \u091c\u094b\u092e\u0947\u091f\u094d\u0930\u093f\u0915 \u092e\u0948\u0902 \u0915\u0947"}, {"start": 56.43, "duration": 3.559, "text": "\u092c\u0930\u093e\u092c\u0930 \u0939\u094b\u0917\u093e \u092f\u093e \u0909\u0938\u0938\u0947 \u092c\u0921\u093c\u093e \u0939\u094b\u0917\u093e"}], "title": "Visual Proof - AM GM Inequality", "description": "Visual Proof - AM GM Inequality", "lengthSeconds": 58, "width": 360, "height": 640}
{"video_id": "JiX9QI8zwwo", "transcript": [{"start": 0.399, "duration": 5.36, "text": "draw three mutually tangent unit circles"}, {"start": 3.04, "duration": 5.28, "text": "like this these three circles enclose an"}, {"start": 5.759, "duration": 4.84, "text": "area as shown here can you determine the"}, {"start": 8.32, "duration": 3.72, "text": "area of the enclosed shape pause the"}, {"start": 10.599, "duration": 3.641, "text": "video and think about it before I show"}, {"start": 12.04, "duration": 3.96, "text": "you an answer one way to answer this"}, {"start": 14.24, "duration": 4.0, "text": "problem is to realize that the centers"}, {"start": 16.0, "duration": 4.279, "text": "of the three circles can be connected to"}, {"start": 18.24, "duration": 4.199, "text": "form an equilateral triangle with a side"}, {"start": 20.279, "duration": 4.201, "text": "length of two this has an area ofare <"}, {"start": 22.439, "duration": 3.84, "text": "TK of three but then this equilateral"}, {"start": 24.48, "duration": 3.719, "text": "triangle can be decomposed into these"}, {"start": 26.279, "duration": 4.041, "text": "three wedge shaped pieces in the area we"}, {"start": 28.199, "duration": 5.561, "text": "want each of the wed shaped pieces has"}, {"start": 30.32, "duration": 6.8, "text": "an area ofk 6 so the area we want is the"}, {"start": 33.76, "duration": 6.04, "text": "< TK 3 - 3 * > 6 which simplifies to the"}, {"start": 37.12, "duration": 4.119, "text": "< of 3 - pi/ 2 let me know in the"}, {"start": 39.8, "duration": 3.84, "text": "comments if you solved it a different"}, {"start": 41.239, "duration": 2.401, "text": "way"}], "title": "What\u2019s the area?", "description": "This is a short, animated visual proof finding the area bounded between three mutually tangent unit circles.\n\nHave a different solution? Share it in the comments!\n\nIf you like this video, consider subscribing to the channel or consider buying me a coffee: https://www.buymeacoffee.com/VisualProofs. Thanks! \n\nCheck out these related videos:\nhttps://youtu.be/5VaQVXECaVM\nhttps://youtu.be/6DnnJlzbCC4\nhttps://youtu.be/PNQxS6antBY\n\nThis animation is based on an exercise from Roger Nelsen's and Claudi Alsina's wonderful book \"Icons of Mathematics\". You can find the book and other information about it at the links below\n\n(I may receive commissions from this paid link) : Icons of Mathematics from Amazon https://amzn.to/3L5U44S\n\nhttps://www.maa.org/press/maa-reviews/icons-of-mathematics-an-exploration-of-twenty-key-images\n\nhttps://bookstore.ams.org/dol-56/\n\n #manim #math #mathvideo #mathshorts #geometry #hexagon #animation #theorem #pww #proofwithoutwords #visualproof #proof #area #root3 #pww\u200b #proofwithoutwords\u200b \u200b #proof\u200b #algebra #areas #mathematics\u200b #mathvideo\u200b #mtbos \n\nTo learn more about animating with manim, check out:\nhttps://manim.community", "lengthSeconds": 42, "width": 360, "height": 640}
{"video_id": "_eH-wbN7WZw", "transcript": [{"start": 0.32, "duration": 4.959, "text": "visual proof of area of circle split the"}, {"start": 3.12, "duration": 4.0, "text": "circle into four parts and arrange them"}, {"start": 5.279, "duration": 4.24, "text": "like this do it again with eight"}, {"start": 7.12, "duration": 4.96, "text": "partitions do it again with 16"}, {"start": 9.519, "duration": 3.601, "text": "partitions repeat this pattern infinite"}, {"start": 12.08, "duration": 3.2, "text": "times"}, {"start": 13.12, "duration": 4.079, "text": "and you will see that entire circle fits"}, {"start": 15.28, "duration": 4.96, "text": "in a rectangle the dimensions of the"}, {"start": 17.199, "duration": 6.0, "text": "rectangle will be pi r and r"}, {"start": 20.24, "duration": 4.59, "text": "which gives us the area effectively pi r"}, {"start": 23.199, "duration": 5.481, "text": "square"}, {"start": 24.83, "duration": 3.85, "text": "[Music]"}], "title": "Visual proof that Area of Circle is equal to Area of Rectangle #VisualRepresentation #MathAnimation", "description": "Visual proof of Area of Circle.\n\nAs we know the area of a rectangle is equal to \u201cbase x height\u201d. \n\nThe height of our rectangle will be the same as the radius. \n\nTo find the base we will need the circumference of the circle. \n\nWhen we compared the base and the circumference we will get the base is equal to one-half the circumference. \n\nSince the circumference is equal to 2\u03c0r, the base has to be \u03c0r.\n\nNow \u201cbase x height\u201d becomes \u201c\u03c0r x r\u201d. \n\nThat gives us \u03c0r\u00b2 which is equal to the area of the rectangle which is equal to the area of the circle.\n\nFor more such educational reels follow @championsacademy.in \n\nTag your friend\nShare the knowledge!\n\n#ProofOfAreaOfCircle #AreaOfCircle #Radius #Circumference #Diameter #Circle #Algebra #Rectangle #Base #Height #Geometry #Mathematics #Diagrams #VisualRepresentation #StudyFromHome \n\n#ChampionsAcademy #AcademicSystem #Coaching #Learning #Online #JEE #NEET #Physics #Chemistry #Biology #CBSE #ICSE #EasyLearning #LearningMadeEasy #OnlineLearning", "lengthSeconds": 27, "width": 202, "height": 360}
{"video_id": "hSCGgi3O_rU", "transcript": [{"start": 0.0, "duration": 2.39, "text": "[\u0c38\u0c02\u0c17\u0c40\u0c24\u0c02]"}, {"start": 1.42, "duration": 4.229, "text": "[\u0c2a\u0c4d\u0c30\u0c36\u0c02\u0c38]"}, {"start": 2.39, "duration": 3.259, "text": "[\u0c38\u0c02\u0c17\u0c40\u0c24\u0c02]"}, {"start": 6.25, "duration": 4.27, "text": "[\u0c2a\u0c4d\u0c30\u0c36\u0c02\u0c38]"}, {"start": 6.36, "duration": 7.319, "text": "[\u0c38\u0c02\u0c17\u0c40\u0c24\u0c02]"}, {"start": 10.52, "duration": 6.159, "text": "[\u0c2a\u0c4d\u0c30\u0c36\u0c02\u0c38]"}, {"start": 13.679, "duration": 3.0, "text": "\u0c06"}, {"start": 18.01, "duration": 6.069, "text": "[\u0c38\u0c02\u0c17\u0c40\u0c24\u0c02]"}], "title": "Happy Tau Day!", "description": "In this short video, we show one way to visualize the constant tau, which is twice pi, as the area between the curves 1/(1+x^2) and its negation over the entire real line.\n\nHappy Tau Day if you celebrate!\n\nTo buy me a coffee, head over to https://www.buymeacoffee.com/VisualProofs \nThanks!\n\n\n#mathshorts #mathvideo #math #short #math #manim #radian #circle #radius #tauday #taurus \n#calculus \n\nTo learn more about animating with manim, check out:\nhttps://manim.community", "lengthSeconds": 27, "width": 360, "height": 640}
{"video_id": "yrB4Ek5OuF8", "transcript": [{"start": 0.55, "duration": 3.809, "text": "[\u97f3\u697d]"}, {"start": 7.28, "duration": 3.0, "text": "\u3042"}], "title": "Math visual proofs, geometry of binomial theorem, visual proof of binomial theorem, #viral #shorts", "description": "geometry of binomial theorem,\nMath visual proofs,proof without words, math, Mathematics visualization, mathematics, Mathematical visual proofs, maths, Easy math, maths for kids, Math basic concepts, basic maths for beginners, maths basic concepts,\n visual proof of binomial theorem\nbinomial theorem formula\nbinomial distribution formula\nbinomial theorem class 11\nbinomial theorem examples\nbinomial theorem proof\nbinomial theorem class 12\nbinomial theorem formula class 11\n#binomialtheorem \n#binomial \n#viral \n #shortsvideo\n #shortsviral \n#youtubeshorts \n#youtubeshort \n#shorts\n #short\n#trending \n #shortvideo\n #shortfeed", "lengthSeconds": 8, "width": 360, "height": 640}
{"video_id": "T0HyWIFbsHQ", "transcript": [{"start": 2.47, "duration": 13.93, "text": "[Music]"}, {"start": 15.679, "duration": 3.841, "text": "consider an"}, {"start": 16.4, "duration": 6.32, "text": "x by x square a joined"}, {"start": 19.52, "duration": 6.32, "text": "with an a by x rectangle"}, {"start": 22.72, "duration": 4.16, "text": "together these shapes enclose a total"}, {"start": 25.84, "duration": 4.32, "text": "area"}, {"start": 26.88, "duration": 8.4, "text": "of x squared plus a times"}, {"start": 30.16, "duration": 6.72, "text": "x now separate the a by x rectangle"}, {"start": 35.28, "duration": 4.16, "text": "cut it in half along the side with"}, {"start": 36.88, "duration": 4.8, "text": "length a and attach the two resulting"}, {"start": 39.44, "duration": 2.88, "text": "smaller rectangles to adjacent sides of"}, {"start": 41.68, "duration": 4.0, "text": "the square"}, {"start": 42.32, "duration": 5.52, "text": "like this these three rectangles now"}, {"start": 45.68, "duration": 3.199, "text": "almost fill a larger square with side"}, {"start": 47.84, "duration": 4.32, "text": "length x"}, {"start": 48.879, "duration": 6.0, "text": "plus a over two however"}, {"start": 52.16, "duration": 3.919, "text": "there is a smaller a over two by a over"}, {"start": 54.879, "duration": 4.081, "text": "two square"}, {"start": 56.079, "duration": 3.681, "text": "missing from the lower right corner"}, {"start": 58.96, "duration": 3.279, "text": "therefore"}, {"start": 59.76, "duration": 5.6, "text": "we see the original enclosed area is"}, {"start": 62.239, "duration": 6.801, "text": "quantity x plus a over two squared"}, {"start": 65.36, "duration": 6.72, "text": "minus the quantity a over two squared"}, {"start": 69.04, "duration": 5.6, "text": "converting x squared plus ax"}, {"start": 72.08, "duration": 4.24, "text": "to the form x plus a over 2 squared"}, {"start": 74.64, "duration": 12.88, "text": "minus a over 2 squared"}, {"start": 76.32, "duration": 11.2, "text": "is called completing the square"}], "title": "Completing the Square (visual proof)", "description": "This is a short, animated visual proof showing what is meant when we \"complete\" the square algebraically and justifying why the formula holds for positive real numbers. #mathshorts\u200b #mathvideo\u200b #math\u200b #completingthesquare #mtbos\u200b #manim\u200b #animation\u200b #theorem\u200b #pww\u200b #proofwithoutwords\u200b #visualproof\u200b #proof\u200b #iteachmath #algebra \n\n\n__________________________________________________________________\nBackground music, used according to CC BY-SA 3.0, is \"The Road\" from Esteban Orlando:\n\nEsteban Orlando:\nhttps://soundcloud.com/orlando-esteba...\nhttps://www.instagram.com/estebanorla...\n\nhttps://creativecommons.org/licenses/...\nMusic provided by RFM: https://youtu.be/b4KCM1bIq50", "lengthSeconds": 86, "width": 640, "height": 360}
{"video_id": "YJsb0tYenrI", "transcript": [{"start": 2.4, "duration": 7.04, "text": "[Music]"}, {"start": 10.8, "duration": 3.12, "text": "consider a right triangle with legs of"}, {"start": 12.719, "duration": 4.961, "text": "length a"}, {"start": 13.92, "duration": 5.92, "text": "b and hypotenuse of length c"}, {"start": 17.68, "duration": 4.48, "text": "the in circle of this triangle is the"}, {"start": 19.84, "duration": 5.12, "text": "largest circle contained in the triangle"}, {"start": 22.16, "duration": 5.359, "text": "it is tangent to each of the three sides"}, {"start": 24.96, "duration": 5.68, "text": "the radius r of this circle is called"}, {"start": 27.519, "duration": 5.361, "text": "the in radius of the triangle"}, {"start": 30.64, "duration": 4.72, "text": "the inradius can be used to decompose"}, {"start": 32.88, "duration": 5.44, "text": "the triangle into four triangles and a"}, {"start": 35.36, "duration": 6.16, "text": "square as demonstrated"}, {"start": 38.32, "duration": 4.96, "text": "if we glue a second rotated copy of this"}, {"start": 41.52, "duration": 3.92, "text": "triangle to the original along the"}, {"start": 43.28, "duration": 4.08, "text": "hypotenuse we end up with a rectangle"}, {"start": 45.44, "duration": 3.52, "text": "decomposed into eight triangles and two"}, {"start": 47.36, "duration": 3.679, "text": "squares"}, {"start": 48.96, "duration": 4.4, "text": "we can then arrange the pieces of this"}, {"start": 51.039, "duration": 3.701, "text": "rectangle dissection to form a different"}, {"start": 53.36, "duration": 19.759, "text": "rectangle"}, {"start": 54.74, "duration": 21.019, "text": "[Music]"}, {"start": 73.119, "duration": 6.081, "text": "from the decomposition we can see that"}, {"start": 75.759, "duration": 5.521, "text": "this new rectangle has length given by a"}, {"start": 79.2, "duration": 4.8, "text": "plus b"}, {"start": 81.28, "duration": 4.479, "text": "plus c"}, {"start": 84.0, "duration": 4.479, "text": "and the width of the rectangle is given"}, {"start": 85.759, "duration": 4.0, "text": "by the in radius r"}, {"start": 88.479, "duration": 3.441, "text": "because the areas of these two"}, {"start": 89.759, "duration": 5.04, "text": "rectangles must be equal"}, {"start": 91.92, "duration": 4.879, "text": "we can equate the areas to see that a"}, {"start": 94.799, "duration": 3.601, "text": "times b"}, {"start": 96.799, "duration": 4.721, "text": "must be equal"}, {"start": 98.4, "duration": 6.48, "text": "to r times"}, {"start": 101.52, "duration": 3.36, "text": "a plus b plus c"}, {"start": 105.84, "duration": 5.2, "text": "but then we can use this equality"}, {"start": 108.399, "duration": 3.841, "text": "to find that r"}, {"start": 111.04, "duration": 5.759, "text": "is equal"}, {"start": 112.24, "duration": 6.4, "text": "to a times b divided by a plus b plus c"}, {"start": 116.799, "duration": 4.241, "text": "giving us a formula for the n radius of"}, {"start": 118.64, "duration": 4.799, "text": "a right triangle in terms of its side"}, {"start": 121.04, "duration": 2.399, "text": "lengths"}, {"start": 124.0, "duration": 4.879, "text": "but that's not all"}, {"start": 126.479, "duration": 4.321, "text": "we can do more the four triangles from"}, {"start": 128.879, "duration": 4.241, "text": "the original decomposition"}, {"start": 130.8, "duration": 5.28, "text": "tell us extra information"}, {"start": 133.12, "duration": 5.92, "text": "the leg of one triangle is a minus r and"}, {"start": 136.08, "duration": 5.2, "text": "the leg of the other is b minus r"}, {"start": 139.04, "duration": 5.04, "text": "but these two legs side by side create"}, {"start": 141.28, "duration": 5.039, "text": "the hypotenuse of the triangle so that c"}, {"start": 144.08, "duration": 3.44, "text": "is equal to a minus r"}, {"start": 146.319, "duration": 4.161, "text": "plus"}, {"start": 147.52, "duration": 2.96, "text": "b minus r"}, {"start": 152.08, "duration": 2.879, "text": "we can again use this equation to solve"}, {"start": 153.92, "duration": 5.039, "text": "for r"}, {"start": 154.959, "duration": 7.92, "text": "and we'll see that r is equal to a plus"}, {"start": 158.959, "duration": 5.761, "text": "b minus c all divided by two"}, {"start": 162.879, "duration": 3.601, "text": "we have found two formulas for the n"}, {"start": 164.72, "duration": 3.98, "text": "radius of the right triangle in terms of"}, {"start": 166.48, "duration": 5.39, "text": "the side lengths"}, {"start": 168.7, "duration": 3.17, "text": "[Music]"}, {"start": 173.97, "duration": 5.349, "text": "[Music]"}, {"start": 187.84, "duration": 2.08, "text": "you"}], "title": "Inradius of a Right Triangle (visual proof)", "description": "This is a short, animated visual proof of two different formulas for the inradius of a right triangle in terms of its side lengths. #manim #math #mathvideo #mathshorts #geometry #circle #triangle  #animation #theorem #pww #proofwithoutwords #visualproof #proof #incircle #inradius #righttriangle \n\nThis animation is based on a proof due to Liu Hui. You can read more in the following sources:\n\nArticle by Philip D. Straffin Jr. in the June 1998 Mathematics Magazine (https://doi.org/10.2307/2691200) pages 163-181\n\nMathematical Morsels by Ross Honsberger (MAA, 1978)\n\nProofs without Words II by Roger B. Nelsen (MAA, 2000) (https://bookstore.ams.org/clrm-14/)\n\nTo learn more about animating with manim, check out:\nhttps://manim.community\n____________________________________________________\nBackground music:\nSpirit Of The Dream by Keys of Moon | https://soundcloud.com/keysofmoon\nMusic promoted by https://www.free-stock-music.com\nAttribution 4.0 International (CC BY 4.0)\nhttps://creativecommons.org/licenses/by/4.0/", "lengthSeconds": 188, "width": 640, "height": 360}
{"video_id": "OAs_GJR689I", "transcript": [{"start": 0.199, "duration": 6.08, "text": "start with a right triangle with lengths"}, {"start": 2.24, "duration": 6.72, "text": "a b and c create four congruent copies"}, {"start": 6.279, "duration": 5.161, "text": "of such triangle arrange the congruent"}, {"start": 8.96, "duration": 5.04, "text": "triangles to form a square the empty"}, {"start": 11.44, "duration": 4.839, "text": "yellow square has a side length of C"}, {"start": 14.0, "duration": 4.8, "text": "arrange the congruent triangles so that"}, {"start": 16.279, "duration": 4.561, "text": "they look exactly like the yellow Square"}, {"start": 18.8, "duration": 4.559, "text": "the small red square has a side length"}, {"start": 20.84, "duration": 6.24, "text": "of B minus a the area of the four"}, {"start": 23.359, "duration": 5.641, "text": "triangles is 12 a b * 4 plus the area of"}, {"start": 27.08, "duration": 3.4, "text": "the small red square which is the square"}, {"start": 29.0, "duration": 3.48, "text": "of B minus a"}, {"start": 30.48, "duration": 5.84, "text": "these are all equal to the area of the"}, {"start": 32.48, "duration": 6.399, "text": "yellow Square which is c\u00b2 12 of ab * 4"}, {"start": 36.32, "duration": 5.2, "text": "becomes 2ab plus the expansion of the"}, {"start": 38.879, "duration": 10.401, "text": "square of B minus a which is square of a"}, {"start": 41.52, "duration": 11.519, "text": "- 2 a + b\u00b2 = c\u00b2 2 a and -2ab is 0 so the"}, {"start": 49.28, "duration": 9.079, "text": "resulting equation is square of a + b\u00b2 ="}, {"start": 53.039, "duration": 5.32, "text": "c\u00b2 this is the Pythagorean theorem"}], "title": "Visual + Algebraic Proof of Pythagorean Theorem #mathviral #maths #algebra #mathtrick  #mathisfun", "description": "", "lengthSeconds": 60, "width": 360, "height": 640}
{"video_id": "LRriv5ujNwo", "transcript": [{"start": 1.18, "duration": 44.67, "text": "[\u0645\u0648\u0633\u064a\u0642\u0649]"}], "title": "Triangle angle sum property visual proof #academicdiary #maths #triangle", "description": "a simple visual proof of angle sum property of a triangle.", "lengthSeconds": 44, "width": 360, "height": 640}
{"video_id": "ZlGaQdNRdqA", "transcript": [{"start": 3.24, "duration": 8.519, "text": "[Music]"}, {"start": 9.76, "duration": 3.12, "text": "start with a right triangle with legs of"}, {"start": 11.759, "duration": 2.081, "text": "length a"}, {"start": 12.88, "duration": 4.159, "text": "and b"}, {"start": 13.84, "duration": 4.88, "text": "and a hypotenuse of length c"}, {"start": 17.039, "duration": 5.921, "text": "the square drawn using the side of"}, {"start": 18.72, "duration": 5.44, "text": "length a has an area of a squared"}, {"start": 22.96, "duration": 2.639, "text": "similarly"}, {"start": 24.16, "duration": 5.279, "text": "the square drawn using the side of"}, {"start": 25.599, "duration": 5.68, "text": "length b has an area of b squared"}, {"start": 29.439, "duration": 4.881, "text": "and finally the square drawn using the"}, {"start": 31.279, "duration": 4.161, "text": "hypotenuse of length c has an area of c"}, {"start": 34.32, "duration": 2.8, "text": "squared"}, {"start": 35.44, "duration": 3.279, "text": "[Music]"}, {"start": 37.12, "duration": 3.68, "text": "then add two more copies of the right"}, {"start": 38.719, "duration": 4.641, "text": "triangle one reflected about the angle"}, {"start": 40.8, "duration": 4.4, "text": "between the legs and one rotated and"}, {"start": 43.36, "duration": 4.719, "text": "shifted to the bottom of the square off"}, {"start": 45.2, "duration": 2.879, "text": "of the hypotenuse"}, {"start": 48.64, "duration": 4.8, "text": "next we draw two lines of symmetry"}, {"start": 51.199, "duration": 3.891, "text": "separating two hexagons inside of the"}, {"start": 53.44, "duration": 2.959, "text": "diagram like this"}, {"start": 55.09, "duration": 4.19, "text": "[Music]"}, {"start": 56.399, "duration": 5.041, "text": "we use the new regions as follows"}, {"start": 59.28, "duration": 3.599, "text": "this top quadrilateral is congruent to"}, {"start": 61.44, "duration": 2.48, "text": "its reflection along the first line of"}, {"start": 62.879, "duration": 3.121, "text": "symmetry"}, {"start": 63.92, "duration": 4.0, "text": "the resulting quadrilateral is congruent"}, {"start": 66.0, "duration": 4.159, "text": "to its rotation of 90 degrees clockwise"}, {"start": 67.92, "duration": 4.64, "text": "about the left triangle vertex and this"}, {"start": 70.159, "duration": 4.161, "text": "quadrilateral is congruent by 180 degree"}, {"start": 72.56, "duration": 3.44, "text": "rotation to the other half of the second"}, {"start": 74.32, "duration": 4.24, "text": "hexagon cut by the second line of"}, {"start": 76.0, "duration": 2.56, "text": "symmetry"}, {"start": 79.2, "duration": 4.959, "text": "therefore the two overlapping hexagons"}, {"start": 81.84, "duration": 5.2, "text": "in the diagram are both built from two"}, {"start": 84.159, "duration": 5.041, "text": "copies of the same quadrilateral"}, {"start": 87.04, "duration": 5.68, "text": "that means that these two hexagons"}, {"start": 89.2, "duration": 5.36, "text": "doesn't close the exact same area"}, {"start": 92.72, "duration": 3.52, "text": "but notice that both of these hexagons"}, {"start": 94.56, "duration": 2.8, "text": "use two copies of the original right"}, {"start": 96.24, "duration": 3.04, "text": "triangle"}, {"start": 97.36, "duration": 4.719, "text": "so removing those triangles from both"}, {"start": 99.28, "duration": 5.199, "text": "diagrams shows that the large square"}, {"start": 102.079, "duration": 4.72, "text": "encloses the exact same area as the two"}, {"start": 104.479, "duration": 5.92, "text": "smaller squares which means that c"}, {"start": 106.799, "duration": 5.171, "text": "squared is the same as a squared plus"}, {"start": 110.399, "duration": 2.961, "text": "b squared"}, {"start": 111.97, "duration": 3.39, "text": "[Music]"}, {"start": 113.36, "duration": 3.52, "text": "remembering that a and b"}, {"start": 115.36, "duration": 4.0, "text": "are the lengths of two legs of a right"}, {"start": 116.88, "duration": 3.919, "text": "triangle with hypotenuse of length c"}, {"start": 119.36, "duration": 4.719, "text": "we have shown that for such a right"}, {"start": 120.799, "duration": 4.241, "text": "triangle c squared is equal to a squared"}, {"start": 124.079, "duration": 2.64, "text": "plus"}, {"start": 125.04, "duration": 3.359, "text": "b squared"}, {"start": 126.719, "duration": 5.591, "text": "this is the pythagorean theorem"}, {"start": 128.399, "duration": 17.23, "text": "as proved by da vinci"}, {"start": 132.31, "duration": 13.319, "text": "[Music]"}, {"start": 146.08, "duration": 2.08, "text": "you"}], "title": "Pythagorean Theorem V (visual proof; Leonardo da Vinci)", "description": "This is a short, animated visual proof of the Pythagorean theorem (the right triangle theorem) using the a diagram that is now attributed to Leonardo da Vinci. The proof uses reflection and rotation symmetry arguments. This theorem states the square of the hypotenuse of a right triangle is equal to the sum of squares of the two other side lengths. #mathshorts #mathvideo #math #pythagoreantheorem #pythagorean #triangle #manim #animation #theorem #pww #proofwithoutwords #visualproof #proof #leonardodavinci #davinci\n\n\n01:30 I said something incorrect so made a small cut. I meant to say the two hexagons are made from congruent parts. \n\nIf you liked this video, please consider subscribing to the channel and checking out my other videos.\n\nThis animation is based on a proof due to Leonardo da Vinci. For a static version of this proof, see Roger Nelsen's first \"Proof Without Words II: More Exercises in Visual Thinking\" compendium (page 5) : https://bookstore.ams.org/clrm-14/. You can also check out Howard Eves' \"Great Moments in Mathematics (Before 1650)\" page 29--31 and Roger Nelsen's book \"Math Made Visual: Creating Images for Understanding Mathematics\" : https://bookstore.ams.org/clrm-28.\n\nFor other proofs of this same fact check out: \nhttps://youtu.be/q28H5NY5UVU\nhttps://youtu.be/yfGtbNgcrQ8\nhttps://youtu.be/JRjPsuJ7S-g\nhttps://youtu.be/TsKTteGJpZU\n\nTo learn more about animating with manim, check out:\nhttps://manim.community\n___________________________________\nMusic in this video:\nTranquil Fields (Tense) by Alexander Nakarada | https://www.serpentsoundstudios.com\nMusic promoted by https://www.free-stock-music.com\nAttribution 4.0 International (CC BY 4.0)\nhttps://creativecommons.org/licenses/by/4.0/", "lengthSeconds": 146, "width": 640, "height": 360}
{"video_id": "-CQk-QTKXsg", "transcript": [{"start": 1.08, "duration": 29.22, "text": "[Music]"}, {"start": 36.87, "duration": 3.09, "text": "[Music]"}, {"start": 57.82, "duration": 3.119, "text": "[Music]"}, {"start": 86.05, "duration": 4.88, "text": "[Music]"}, {"start": 94.88, "duration": 7.43, "text": "[Music]"}, {"start": 106.7, "duration": 7.95, "text": "[Music]"}, {"start": 126.079, "duration": 15.851, "text": "hmm"}, {"start": 127.68, "duration": 14.25, "text": "[Music]"}, {"start": 146.02, "duration": 11.95, "text": "[Music]"}, {"start": 164.89, "duration": 4.07, "text": "[Music]"}, {"start": 176.0, "duration": 5.82, "text": "so"}, {"start": 178.76, "duration": 3.06, "text": "[Music]"}, {"start": 190.77, "duration": 3.169, "text": "[Music]"}, {"start": 206.319, "duration": 3.0, "text": "hmm"}, {"start": 214.06, "duration": 16.399, "text": "[Music]"}, {"start": 233.439, "duration": 2.08, "text": "you"}], "title": "Pell Number Lattice Path Enumeration and visual recurrence formula (synthwave; visual proof)", "description": "This synthwave enumeration shows all of the lattice paths with steps (1,1), (-1,1) and (2,0) from the origin (0,0) to the line y = n-1. The number of such lattice paths is counted by the Pell numbers, and we use the visual enumeration to show how to produce the (linear) recurrence formula for the Pell numbers. To see other lattice path enumerations, check out https://youtu.be/9YUl0k2FYzc, https://youtu.be/KYpOoA2D63w, and https://youtu.be/__w_MDQeUYc. \n\nIf you like this video, check out my others and consider subscribing. Thanks!\n\n#pellnumbers #pell\u200b #mathvideo\u200b #math\u200b #mtbos\u200b #manim\u200b #animation\u200b #theorem\u200b\u200b #visualproof\u200b #proof\u200b #iteachmath #mathematics #linearrecurrence #linear #latticepaths #discretemath #combinatorics #enumeration #synthwave #recurrence \n\nTo learn more about animating with manim, check out:\nhttps://manim.community\n_________________________________________\nMusic in this video:\nVercetti by jiglr | https://soundcloud.com/jiglrmusic\nMusic promoted by https://www.free-stock-music.com\nCreative Commons Attribution 3.0 Unported License\nhttps://creativecommons.org/licenses/by/3.0/deed.en_US", "lengthSeconds": 234, "width": 640, "height": 360}
{"video_id": "VYaEGvClg7Q", "transcript": [{"start": 1.319, "duration": 3.921, "text": "here's an intriguing three-dimensional"}, {"start": 2.879, "duration": 6.201, "text": "way to sum"}, {"start": 5.24, "duration": 5.8, "text": "squares start with 6 1X 1 by 1 cuboids"}, {"start": 9.08, "duration": 4.88, "text": "as shown here each of these cuboids"}, {"start": 11.04, "duration": 5.28, "text": "contains a volume of 1^ 2 * 1 so the"}, {"start": 13.96, "duration": 4.6, "text": "total volume of these six is 6 * 1"}, {"start": 16.32, "duration": 5.0, "text": "squared we can glue these six together"}, {"start": 18.56, "duration": 6.32, "text": "to form this cuboid now we can enclose"}, {"start": 21.32, "duration": 5.68, "text": "this shape using six 2x 2x1 cuboids two"}, {"start": 24.88, "duration": 4.52, "text": "on top and two on bottom and one in the"}, {"start": 27.0, "duration": 5.52, "text": "front and the back each each of these"}, {"start": 29.4, "duration": 4.52, "text": "cuboids contains a volume of 2^ 2 * 1 so"}, {"start": 32.52, "duration": 5.08, "text": "when we glue them together to form a"}, {"start": 33.92, "duration": 5.959, "text": "shell we add a volume of 6 * 2^ 2 now"}, {"start": 37.6, "duration": 5.439, "text": "let's create a new shell for this shape"}, {"start": 39.879, "duration": 4.921, "text": "by using six 3x3 by1 cuboids two on the"}, {"start": 43.039, "duration": 3.721, "text": "left and the right each and one each in"}, {"start": 44.8, "duration": 5.32, "text": "the front and the back each of these"}, {"start": 46.76, "duration": 5.479, "text": "cuboids contains a volume of 3^ 2 * 1 so"}, {"start": 50.12, "duration": 5.239, "text": "the full shell created here encloses a"}, {"start": 52.239, "duration": 6.241, "text": "volume of 6 * 3^ 2 we can continue this"}, {"start": 55.359, "duration": 6.801, "text": "process as long as we want at each step"}, {"start": 58.48, "duration": 5.44, "text": "we add six copies of a k by K by1 cuboid"}, {"start": 62.16, "duration": 3.56, "text": "alternating between putting two on top"}, {"start": 63.92, "duration": 4.12, "text": "and two on bottom and two on the left"}, {"start": 65.72, "duration": 4.92, "text": "and two on the right when we finally"}, {"start": 68.04, "duration": 5.68, "text": "stop our last stage consists of adding"}, {"start": 70.64, "duration": 4.799, "text": "six copies of an N byn by one cuboid"}, {"start": 73.72, "duration": 4.439, "text": "each of these cuboids contains a volume"}, {"start": 75.439, "duration": 6.561, "text": "of n squ * 1 so the final shell shown"}, {"start": 78.159, "duration": 6.161, "text": "here encloses a volume of 6 * n^ 2 this"}, {"start": 82.0, "duration": 4.4, "text": "process of adding shells at each stage"}, {"start": 84.32, "duration": 4.28, "text": "has built a filled in cuboid whose"}, {"start": 86.4, "duration": 6.2, "text": "volume is given by this finite sum shown"}, {"start": 88.6, "duration": 7.4, "text": "here 6 * 1 2 + 6 * 2^ 2 added all the"}, {"start": 92.6, "duration": 6.36, "text": "way up to 6 * n^ 2 on the other hand"}, {"start": 96.0, "duration": 5.68, "text": "this cuboid has Dimensions n by n + 1 by"}, {"start": 98.96, "duration": 5.6, "text": "2 n + 1 because the final cuboids we"}, {"start": 101.68, "duration": 6.56, "text": "added were n by n by 1 but this means"}, {"start": 104.56, "duration": 8.04, "text": "that the volume of the cuboid is n * n +"}, {"start": 108.24, "duration": 6.839, "text": "1 * 2 n + 1 from this we see that the"}, {"start": 112.6, "duration": 6.799, "text": "finite sum where I ranges from 1 to n of"}, {"start": 115.079, "duration": 6.04, "text": "i^ 2 is n * n + 1 * 2 n + 1 all divided"}, {"start": 119.399, "duration": 4.481, "text": "6"}, {"start": 121.119, "duration": 5.561, "text": "to recap we've just proved the famous"}, {"start": 123.88, "duration": 5.28, "text": "sum of squares formula by stacking"}, {"start": 126.68, "duration": 5.04, "text": "cuboidal arrays inside of cuboidal"}, {"start": 129.16, "duration": 4.96, "text": "shells as shown here the resulting"}, {"start": 131.72, "duration": 6.36, "text": "formula says that the sum of the first n"}, {"start": 134.12, "duration": 6.839, "text": "squares is equal to n * n + 1 * 2 n + 1"}, {"start": 138.08, "duration": 2.879, "text": "all divided"}, {"start": 148.56, "duration": 3.0, "text": "6"}], "title": "An Amazing way to Sum Squares (visual proof V using cuboids)", "description": "This is a short, animated visual proof for the sum of squares formula using cuboids and cuboidal shells. This is the fifth in a series showing various ways to find the famous sum of squares formula.\n\nIf you like this video, consider subscribing to the channel or consider buying me a coffee: https://www.buymeacoffee.com/VisualProofs. Thanks!\n\nHere are other visual proofs about the sum of squares formulas:\n\nhttps://youtu.be/a8j3sBrXchg\nhttps://youtu.be/WidzHiUFWNA\nhttps://youtu.be/-tJhH_k2LaM\nhttps://youtu.be/UqVmocdLFGc\nhttps://youtu.be/Q-frL0Ot2m4\n\nThis animation is based on a visual proof by Sasho Kalajdzievski from the June 2000 issue of The Mathematical Intelligencer (https://doi.org/10.1007/BF03025282 pages 47--49 ). \n\nTo learn more about animating with manim, check out:\nhttps://manim.community\n\n#math #manim\u200b #mathvideo\u200b\u200b #sumofsquares\u200b #mtbos\u200b #manim\u200b #animation\u200b #theorem\u200b #pww\u200b #proofwithoutwords\u200b #visualproof\u200b #proof\u200b #iteachmath #finitesums #discretemath #induction", "lengthSeconds": 150, "width": 640, "height": 360}
{"video_id": "LvSwvZt5NNg", "transcript": [{"start": 0.48, "duration": 3.0, "text": "foreign"}, {"start": 6.91, "duration": 18.549, "text": "[Music]"}], "title": "Visual Proof of Pythagoras' Theorem", "description": "Made using Pivot Animator 5. As the triangles move, the grey area doesn't change, so we can equate the two. #shorts", "lengthSeconds": 24, "width": 360, "height": 480}
{"video_id": "KhfZK5IIK9E", "transcript": [{"start": 7.759, "duration": 3.84, "text": "using the classical tools of a straight"}, {"start": 9.519, "duration": 4.16, "text": "edge and compass for which n can we"}, {"start": 11.599, "duration": 4.24, "text": "divide a circle into n parts of equal"}, {"start": 13.679, "duration": 3.52, "text": "area"}, {"start": 15.839, "duration": 2.961, "text": "if we start playing around with this"}, {"start": 17.199, "duration": 3.521, "text": "problem at first it doesn't seem too"}, {"start": 18.8, "duration": 3.68, "text": "interesting we can construct two"}, {"start": 20.72, "duration": 3.52, "text": "antipodal points on a circle and that"}, {"start": 22.48, "duration": 3.28, "text": "divides the circle into two pieces of"}, {"start": 24.24, "duration": 3.84, "text": "equal area"}, {"start": 25.76, "duration": 4.48, "text": "we can construct an equilateral triangle"}, {"start": 28.08, "duration": 4.72, "text": "on the circle and that will allow us to"}, {"start": 30.24, "duration": 5.36, "text": "divide the circle into three pieces of"}, {"start": 32.8, "duration": 2.8, "text": "equal area"}, {"start": 35.92, "duration": 4.56, "text": "proceeding in a similar manner we can"}, {"start": 38.079, "duration": 4.241, "text": "construct the regular square"}, {"start": 40.48, "duration": 4.0, "text": "on the boundary of the circle dividing"}, {"start": 42.32, "duration": 4.48, "text": "the circle into four pieces of equal"}, {"start": 44.48, "duration": 2.32, "text": "area"}, {"start": 46.96, "duration": 3.439, "text": "and then we can do the same thing with"}, {"start": 48.399, "duration": 4.8, "text": "the regular pentagon and the regular"}, {"start": 50.399, "duration": 4.961, "text": "hexagon dividing the circle into five"}, {"start": 53.199, "duration": 5.36, "text": "pieces of equal area and six pieces of"}, {"start": 55.36, "duration": 4.719, "text": "equal area respectively"}, {"start": 58.559, "duration": 3.361, "text": "we might think we can continue this"}, {"start": 60.079, "duration": 4.241, "text": "pattern and just construct the regular"}, {"start": 61.92, "duration": 4.64, "text": "heptagon on the circle which then"}, {"start": 64.32, "duration": 4.08, "text": "divides the circle into seven equal"}, {"start": 66.56, "duration": 5.12, "text": "pieces"}, {"start": 68.4, "duration": 5.52, "text": "but there's a problem here it turns out"}, {"start": 71.68, "duration": 4.799, "text": "the regular heptagon is not"}, {"start": 73.92, "duration": 5.12, "text": "constructable using the classical tools"}, {"start": 76.479, "duration": 4.401, "text": "of a straight edge and compass"}, {"start": 79.04, "duration": 4.64, "text": "so that leaves us with a question can we"}, {"start": 80.88, "duration": 5.36, "text": "divide a circle into seven equal parts"}, {"start": 83.68, "duration": 5.2, "text": "using just a straight edge and compass"}, {"start": 86.24, "duration": 2.64, "text": "let's find out"}, {"start": 89.439, "duration": 3.521, "text": "start with a line segment that will"}, {"start": 91.119, "duration": 3.04, "text": "serve as the diameter of a semi-circle"}, {"start": 92.96, "duration": 3.199, "text": "like this"}, {"start": 94.159, "duration": 4.241, "text": "then divide the diameter of the"}, {"start": 96.159, "duration": 3.761, "text": "semicircle into seven congruent pieces"}, {"start": 98.4, "duration": 3.12, "text": "which can be done with a straight edging"}, {"start": 99.92, "duration": 3.519, "text": "compass let's call the length of these"}, {"start": 101.52, "duration": 4.48, "text": "pieces d"}, {"start": 103.439, "duration": 4.96, "text": "after bisecting segments when necessary"}, {"start": 106.0, "duration": 5.6, "text": "use the compass to construct these"}, {"start": 108.399, "duration": 6.68, "text": "nested semicircles for a total of seven"}, {"start": 111.6, "duration": 3.479, "text": "pictured semicircles"}, {"start": 116.96, "duration": 4.159, "text": "let's compute the area of the"}, {"start": 118.64, "duration": 4.72, "text": "semicircles first the first semi-circle"}, {"start": 121.119, "duration": 4.241, "text": "has area pi times d squared over eight"}, {"start": 123.36, "duration": 4.48, "text": "the second one is four pi d squared over"}, {"start": 125.36, "duration": 5.039, "text": "eight then nine pi d squared over 8 and"}, {"start": 127.84, "duration": 6.16, "text": "so on each time we have the nth square"}, {"start": 130.399, "duration": 6.401, "text": "times pi d squared over 8."}, {"start": 134.0, "duration": 4.319, "text": "if we call the area pi d squared over 8"}, {"start": 136.8, "duration": 4.079, "text": "the value x"}, {"start": 138.319, "duration": 5.201, "text": "then the areas of the circles are x 4x"}, {"start": 140.879, "duration": 4.321, "text": "9x 16x and so on each time we have the"}, {"start": 143.52, "duration": 3.68, "text": "next square times x"}, {"start": 145.2, "duration": 4.08, "text": "we can use these values to determine the"}, {"start": 147.2, "duration": 6.24, "text": "areas in the circle the first region has"}, {"start": 149.28, "duration": 6.4, "text": "an area of x then 3x then 5x 7x and so"}, {"start": 153.44, "duration": 4.0, "text": "on each time we're taking the difference"}, {"start": 155.68, "duration": 3.839, "text": "of successive squares to get the next"}, {"start": 157.44, "duration": 4.159, "text": "odd number"}, {"start": 159.519, "duration": 4.481, "text": "and now we apply our trick"}, {"start": 161.599, "duration": 4.881, "text": "this time we construct the exact same"}, {"start": 164.0, "duration": 4.72, "text": "regions in the bottom semicircle but"}, {"start": 166.48, "duration": 4.56, "text": "this time from right to left"}, {"start": 168.72, "duration": 4.4, "text": "once we adjoin the regions like this we"}, {"start": 171.04, "duration": 5.199, "text": "see that each of the seven created"}, {"start": 173.12, "duration": 5.039, "text": "regions has an area of 14 times x"}, {"start": 176.239, "duration": 5.441, "text": "that means that each of these regions"}, {"start": 178.159, "duration": 5.921, "text": "has exactly the same area"}, {"start": 181.68, "duration": 5.36, "text": "consequently this means that we have"}, {"start": 184.08, "duration": 5.12, "text": "divided the circle into seven pieces of"}, {"start": 187.04, "duration": 5.16, "text": "equal area using a straight edge and"}, {"start": 189.2, "duration": 3.0, "text": "compass"}, {"start": 203.04, "duration": 2.08, "text": "you"}], "title": "Dividing a Circle into Seven Equal Areas (visual proof)", "description": "This is a short, animated proof showing how to decompose a circle into seven pieces each with the same area. This process can be done using ruler and compass constructions, unlike the process of constructing the regular heptagon. #manim\u200b #mathshorts\u200b #mathvideo\u200b #math\u200b #geometry #mtbos\u200b #animation\u200b #theorem\u200b\u200b\u200b #visualproof\u200b #proof\u200b #iteachmath #circle #mathematics #rulerandcompass #construction #pizza \n\nPerhaps this should be titled \"The Best Way to Cut Pizza into 7 pieces\" based on the comments in this video :)\n\nThere were also a number of comments asking about how to divide a line into seven equal lengths, so I have created a short, wordless animation showing that procedure if you're interested: https://youtu.be/afZiAGaKEY4\n\nThis animation is based on a visual proof from Roger Nelsen's and Claudi Alsina's wonderful book \"Icons of Mathematics\" (Chapter 14.1 on page 76). You can find the book and other information about it at the links below\n\n(I may receive commissions from this paid link) : Icons of Mathematics from Amazon https://amzn.to/3L5U44S\n\nhttps://www.maa.org/press/maa-reviews/icons-of-mathematics-an-exploration-of-twenty-key-images\n\nhttps://bookstore.ams.org/dol-56/\n\nTo learn more about animating with manim, check out:\nhttps://manim.community\n____________________________________________\nMusic in this episode:\nDivider by Chris Zabriskie is licensed under a Creative Commons Attribution 4.0 license. https://creativecommons.org/licenses/by/4.0/\n\nSource: http://chriszabriskie.com/divider/\n\nArtist: http://chriszabriskie.com/", "lengthSeconds": 203, "width": 640, "height": 360}
{"video_id": "_e4Yn5uGznI", "transcript": [{"start": 0.19, "duration": 5.05, "text": "[Music]"}, {"start": 6.0, "duration": 4.08, "text": "you cannot square the circle using"}, {"start": 7.759, "duration": 3.92, "text": "classical tools but you can if you use"}, {"start": 10.08, "duration": 3.84, "text": "other techniques"}, {"start": 11.679, "duration": 5.121, "text": "if we let b be greater than zero then"}, {"start": 13.92, "duration": 5.04, "text": "the polar curve r equals b times theta"}, {"start": 16.8, "duration": 4.0, "text": "is called the archimedean spiral and it"}, {"start": 18.96, "duration": 3.92, "text": "looks like this"}, {"start": 20.8, "duration": 4.96, "text": "let's use the archimedean spiral to"}, {"start": 22.88, "duration": 2.88, "text": "square the circle"}, {"start": 25.84, "duration": 5.599, "text": "start with a radius a greater than zero"}, {"start": 30.64, "duration": 3.599, "text": "then"}, {"start": 31.439, "duration": 4.391, "text": "consider the circle of radius a centered"}, {"start": 34.239, "duration": 6.801, "text": "at the origin like this"}, {"start": 35.83, "duration": 6.97, "text": "[Music]"}, {"start": 41.04, "duration": 4.64, "text": "this circle"}, {"start": 42.8, "duration": 4.16, "text": "has an area given by pi times the radius"}, {"start": 45.68, "duration": 3.84, "text": "squared"}, {"start": 46.96, "duration": 5.599, "text": "so the area of the circle is equal to pi"}, {"start": 49.52, "duration": 3.039, "text": "times a squared"}, {"start": 53.36, "duration": 5.679, "text": "extend the radius to the other side"}, {"start": 55.92, "duration": 5.76, "text": "creating a line segment of length two a"}, {"start": 59.039, "duration": 5.2, "text": "and then draw the archimedean spiral a"}, {"start": 61.68, "duration": 5.6, "text": "times theta"}, {"start": 64.239, "duration": 5.92, "text": "this spiral intersects the y axis at the"}, {"start": 67.28, "duration": 4.96, "text": "y coordinate a pi over two"}, {"start": 70.159, "duration": 4.64, "text": "so we can draw a rectangle using these"}, {"start": 72.24, "duration": 7.199, "text": "two line segments and the rectangle has"}, {"start": 74.799, "duration": 6.801, "text": "lengths 2a and a pi over 2."}, {"start": 79.439, "duration": 3.441, "text": "therefore we see that the area of the"}, {"start": 81.6, "duration": 4.72, "text": "rectangle"}, {"start": 82.88, "duration": 6.32, "text": "must be a pi over 2 times 2a which is pi"}, {"start": 86.32, "duration": 5.119, "text": "a squared or the area of the circle"}, {"start": 89.2, "duration": 5.2, "text": "now let's square the rectangle"}, {"start": 91.439, "duration": 5.441, "text": "to do this we can draw the circle"}, {"start": 94.4, "duration": 4.56, "text": "with radius given by xy which is the"}, {"start": 96.88, "duration": 3.76, "text": "short edge of the rectangle and we can"}, {"start": 98.96, "duration": 4.72, "text": "extend"}, {"start": 100.64, "duration": 6.799, "text": "line segment xz out to this circle to"}, {"start": 103.68, "duration": 3.759, "text": "create a point called e"}, {"start": 108.159, "duration": 3.92, "text": "now use e and z"}, {"start": 110.24, "duration": 4.71, "text": "to find the midpoint"}, {"start": 112.079, "duration": 5.921, "text": "which we'll call m"}, {"start": 114.95, "duration": 4.09, "text": "[Music]"}, {"start": 118.0, "duration": 3.68, "text": "next"}, {"start": 119.04, "duration": 4.56, "text": "draw the circle centered at m"}, {"start": 121.68, "duration": 4.01, "text": "with radius m z"}, {"start": 123.6, "duration": 3.44, "text": "like this"}, {"start": 125.69, "duration": 2.87, "text": "[Music]"}, {"start": 127.04, "duration": 3.919, "text": "then extend"}, {"start": 128.56, "duration": 4.88, "text": "the line segment y x"}, {"start": 130.959, "duration": 6.241, "text": "with a half chord down to the circle"}, {"start": 133.44, "duration": 3.76, "text": "creating an intersection point s"}, {"start": 138.0, "duration": 5.74, "text": "we can use the line segment x s"}, {"start": 140.64, "duration": 7.22, "text": "to draw a square like this"}, {"start": 143.74, "duration": 4.12, "text": "[Music]"}, {"start": 148.4, "duration": 4.8, "text": "according to the intersecting chords"}, {"start": 150.239, "duration": 3.681, "text": "theorem we see that x s multiplied by x"}, {"start": 153.2, "duration": 3.28, "text": "s"}, {"start": 153.92, "duration": 3.679, "text": "is going to be equal to xz multiplied by"}, {"start": 156.48, "duration": 4.32, "text": "xe"}, {"start": 157.599, "duration": 6.241, "text": "but because xe equals xy"}, {"start": 160.8, "duration": 5.68, "text": "that means that the area of the square"}, {"start": 163.84, "duration": 4.88, "text": "and the area of the rectangle must be"}, {"start": 166.48, "duration": 4.399, "text": "equal"}, {"start": 168.72, "duration": 4.4, "text": "putting these three facts together"}, {"start": 170.879, "duration": 4.64, "text": "we see that the area of the square is"}, {"start": 173.12, "duration": 4.56, "text": "equal to the area of the rectangle which"}, {"start": 175.519, "duration": 4.64, "text": "in turn is equal to the area of the"}, {"start": 177.68, "duration": 4.0, "text": "original circle"}, {"start": 180.159, "duration": 4.481, "text": "but this means"}, {"start": 181.68, "duration": 7.4, "text": "that we have used the archimedean spiral"}, {"start": 184.64, "duration": 4.44, "text": "to square the original circle"}, {"start": 189.98, "duration": 22.58, "text": "[Music]"}, {"start": 210.48, "duration": 2.08, "text": "you"}], "title": "Squaring the Circle with the Archimedean Spiral (animated visual proof)", "description": "This is a short, animated visual proof that we can square the circle IF we use the Archimedean spiral. Unfortunately, this is not a solution to the squaring the circle problem from antiquity because that requires it to be done with only a straightedge and compass. #mathshorts\u200b #mathvideo\u200b #math\u200b ##geometry #mtbos\u200b #manim\u200b #animation\u200b #theorem\u200b #pww\u200b #proofwithoutwords\u200b #visualproof\u200b #proof\u200b #iteachmath #circle #mathematics #thales #squaringthecircle #squaringtherectangle\n\nThis animation is based on a visual proof from Roger Nelsen's and Claudi Alsina's wonderul book \"Charming Proofs: A Journey into Elegant Mathematics\" (Theorem 9.5 from page 145). You can find the book and other information about it at the links below\n\nhttps://bookstore.ams.org/dol-42/\n\nhttps://www.maa.org/press/maa-reviews/charming-proofs-a-journey-into-elegant-mathematics\n\nhttps://www.amazon.com/Charming-Proofs-Mathematics-Mathematical-Expositions/dp/0883853485\n\nHere is another video that squares the circle with a nonclassical technique: https://youtu.be/0D3KiCmum90\n\nTo learn more about animating with manim, check out:\nhttps://manim.community", "lengthSeconds": 211, "width": 640, "height": 360}
{"video_id": "6fIkJj0zMLw", "transcript": [{"start": 0.0, "duration": 9.32, "text": "\u0939\u0940 \u0924\u0915\u0926\u0940\u0930 \u0924\u094b \u0915\u094d\u092f\u093e \u091f\u0942\u091f\u0940 \u0936\u092e\u0936\u0940\u0930 \u0924\u094b \u0915\u094d\u092f\u093e \u091f\u0942\u091f\u0940"}, {"start": 6.08, "duration": 8.88, "text": "\u0936\u092e\u0936\u0940\u0930 \u0938\u0947 \u0939\u0940"}, {"start": 9.32, "duration": 5.64, "text": "\u0939\u094b \u0915\u0930 \u0939\u0930 \u092e\u0948\u0926\u093e\u0928 \u092a\u0924\u0947"}], "title": "Sum of odd no.(Visual proof)\ud83d\ude33\ud83d\udcdd#viral #trending #dsssb #math #shorts", "description": "Sum of odd no.(Visual proof)\ud83d\ude33\ud83d\udcdd#viral #trending #dsssb #math #shorts", "lengthSeconds": 13, "width": 360, "height": 640}
{"video_id": "XJtjU_Ipjno", "transcript": [{"start": 0.0, "duration": 3.959, "text": "here's a visual take on a math fact that"}, {"start": 2.159, "duration": 4.261, "text": "often creates controversy what is the"}, {"start": 3.959, "duration": 5.161, "text": "number 0.9 repeating start with a circle"}, {"start": 6.42, "duration": 4.38, "text": "of Area 1 and take a scaled copy that"}, {"start": 9.12, "duration": 3.54, "text": "has an area of 1 10. then the"}, {"start": 10.8, "duration": 3.839, "text": "surrounding annulus has an area of nine"}, {"start": 12.66, "duration": 3.78, "text": "tenths now we can repeat this process on"}, {"start": 14.639, "duration": 3.241, "text": "the inner circle this time we take one"}, {"start": 16.44, "duration": 3.12, "text": "tenth of one-tenth for the smaller"}, {"start": 17.88, "duration": 3.3, "text": "Circle which leaves us with nine tenths"}, {"start": 19.56, "duration": 3.54, "text": "of one-tenths for the annulus region"}, {"start": 21.18, "duration": 3.839, "text": "which is nine over a hundred imagine"}, {"start": 23.1, "duration": 4.019, "text": "we've repeated this process infinitely"}, {"start": 25.019, "duration": 4.621, "text": "so each time we have a small circle we"}, {"start": 27.119, "duration": 4.681, "text": "take a smaller sub circle of Area 1 10"}, {"start": 29.64, "duration": 4.68, "text": "and shade the annular region of nine"}, {"start": 31.8, "duration": 5.099, "text": "tenths of the previous small circle the"}, {"start": 34.32, "duration": 4.32, "text": "total area shaded is nine tens plus nine"}, {"start": 36.899, "duration": 4.32, "text": "one hundredths plus nine one thousandths"}, {"start": 38.64, "duration": 4.98, "text": "and so on but because the sequence of"}, {"start": 41.219, "duration": 4.381, "text": "radii of the inner circles converges to"}, {"start": 43.62, "duration": 4.02, "text": "zero this means every single point in"}, {"start": 45.6, "duration": 3.959, "text": "the circle except the center point is"}, {"start": 47.64, "duration": 3.66, "text": "contained in one of the annular regions"}, {"start": 49.559, "duration": 3.541, "text": "by the Archimedean principle so the"}, {"start": 51.3, "duration": 3.54, "text": "total shaded area is the area of the"}, {"start": 53.1, "duration": 4.5, "text": "circle which is one but this infinite"}, {"start": 54.84, "duration": 6.199, "text": "sum is precisely 0.9 repeating which"}, {"start": 57.6, "duration": 3.439, "text": "means 0.9 repeating is one"}], "title": "0.999999\u2026 = 1", "description": "In this short, we use a geometric representation and the Archimedean principle to show why 0.99999... equals 1. This fact tends to be a bit controversial in the mathematics classroom, but the only way to make sense of an infinite sum is to treat it as a limit. Thus, we can use the term equality when discussing infinite sums like 0.999... and there is no number this sum could equal besides 1.\n\nIf you like this video, please click \"like\" and consider subscribing and checking out my other videos.\n\nTo buy me a coffee, head over to https://www.buymeacoffee.com/VisualProofs  \nThanks!\n\nTo see a longer video with a similar fact (and perhaps more justification at the end), see\nhttps://youtu.be/BGsJjHbjcnA\n\nHere is a playlist with other geometric sums dissection proofs:\nhttps://youtube.com/playlist?list=PLZh9gzIvXQUsgw8W5TUVDtF0q4jEJ3iaw\n\n#infiniteseries #paradox #manim #math #mathshorts #visualproof #proofwithoutwords \n\nTo learn more about animating with manim, check out:\nhttps://manim.community", "lengthSeconds": 59, "width": 360, "height": 640}
{"video_id": "E6CgspdRLqo", "transcript": [{"start": 3.58, "duration": 19.249, "text": "[Music]"}, {"start": 19.75, "duration": 3.079, "text": "[Applause]"}, {"start": 24.71, "duration": 3.919, "text": "[Music]"}, {"start": 25.51, "duration": 3.119, "text": "[Applause]"}, {"start": 30.02, "duration": 4.61, "text": "[Music]"}, {"start": 38.79, "duration": 10.93, "text": "[Music]"}, {"start": 49.08, "duration": 3.37, "text": "[Applause]"}, {"start": 49.72, "duration": 5.519, "text": "[Music]"}, {"start": 52.45, "duration": 5.789, "text": "[Applause]"}, {"start": 55.239, "duration": 3.0, "text": "n"}], "title": "(a+b)^2 formula, visual proof|a plus b whole square formula |#square #maths #classviii", "description": "(a+b) ^2 formula visual proof\nstep by step explanation\nclass 8 formula proof\nsubscribe for watching more like this", "lengthSeconds": 56, "width": 360, "height": 640}
{"video_id": "mWhhFOTHPZY", "transcript": [{"start": 0.02, "duration": 3.84, "text": "approved Pythagoras Theorem really"}, {"start": 1.76, "duration": 4.019, "text": "quickly I've drawn a square here with"}, {"start": 3.86, "duration": 4.5, "text": "another Square inside it and this gives"}, {"start": 5.779, "duration": 4.86, "text": "me four congruent right angled triangles"}, {"start": 8.36, "duration": 4.02, "text": "which have these side lengths A and B"}, {"start": 10.639, "duration": 3.66, "text": "all equal to each other if we calculate"}, {"start": 12.38, "duration": 3.659, "text": "the area of the large Square we can do"}, {"start": 14.299, "duration": 3.48, "text": "that in two different ways I can just"}, {"start": 16.039, "duration": 4.441, "text": "think of it as a large square that has"}, {"start": 17.779, "duration": 4.861, "text": "side length a plus b so its area is a"}, {"start": 20.48, "duration": 4.44, "text": "plus b squared and if you multiply out a"}, {"start": 22.64, "duration": 4.5, "text": "plus b squared you get a squared plus b"}, {"start": 24.92, "duration": 3.96, "text": "squared plus 2 a b or I can think of it"}, {"start": 27.14, "duration": 3.6, "text": "as these four congruent right angle"}, {"start": 28.88, "duration": 4.74, "text": "triangles their areas are half base"}, {"start": 30.74, "duration": 5.22, "text": "times height so a half a times B so I"}, {"start": 33.62, "duration": 3.779, "text": "get a half a times B times four because"}, {"start": 35.96, "duration": 3.3, "text": "there are four of them and then I also"}, {"start": 37.399, "duration": 3.48, "text": "need to include this Square in the"}, {"start": 39.26, "duration": 3.9, "text": "middle which is a square of side length"}, {"start": 40.879, "duration": 4.561, "text": "C so it has area c squared so four times"}, {"start": 43.16, "duration": 3.719, "text": "a half is two that's two a B plus c"}, {"start": 45.44, "duration": 3.72, "text": "squared but these two expressions"}, {"start": 46.879, "duration": 3.961, "text": "represent the same area so they must be"}, {"start": 49.16, "duration": 4.14, "text": "equal to one another but they both"}, {"start": 50.84, "duration": 4.5, "text": "contain two a b so I can subtract that"}, {"start": 53.3, "duration": 3.599, "text": "from both sides of the equation so that"}, {"start": 55.34, "duration": 3.42, "text": "leaves us with Pythagoras theorem in the"}, {"start": 56.899, "duration": 4.881, "text": "familiar form a squared plus b squared"}, {"start": 58.76, "duration": 3.02, "text": "equals c squared and so that's how"}], "title": "Easiest Proof of Pythagoras' Theorem", "description": "Proof of Pythagoras' Theorem using just basic areas and algebra in less than a minute!\n______________________________________ \nFree online maths challenge courses:\nPrimary (age 9-11): https://bit.ly/primarymathschallenge\nJunior (age 10-13):  https://bit.ly/ukmtjuniormathschallenge\nIntermediate (age 13-16): https://bit.ly/intermediatemathschallenge\nSenior (age 15-18): https://bit.ly/seniormathschallenge\nAll online courses: https://courses.mathsaurus.com/\n______________________________________ \nMake your own online courses with Thinkific and turn your expertise into a thriving business: https://try.thinkific.com/cyfkxypcaasn\n______________________________________ \nMathsaurus website: https://mathsaurus.com/\nAll online courses: https://courses.mathsaurus.com/\nGet ready for A-level maths: https://bit.ly/getreadyforalevelmaths\nInstagram: https://www.instagram.com/mathsaurus/\nTwitter: https://twitter.com/mathsaurus  \nFacebook: https://www.facebook.com/Mathsaurus \nDiscord server: https://discord.gg/rc6VhmKv8y \nMailing list: http://newsletter.mathsaurus.com/\nTuition/contact: https://mathsaurus.com/contact/tuition/\nCalcualtor guide: https://bit.ly/whichcalculator\nSetup for online teaching/how I make my videos: https://bit.ly/teachingmathsonline", "lengthSeconds": 60, "width": 360, "height": 640}
{"video_id": "tqPqC2Waf3k", "transcript": [{"start": 1.599, "duration": 4.961, "text": "let's consider a right triangle with"}, {"start": 3.679, "duration": 5.521, "text": "perpendicular sides are a and b units"}, {"start": 6.56, "duration": 5.039, "text": "and the hypotenuse is c units"}, {"start": 9.2, "duration": 4.8, "text": "then according to pythagoras theorem a"}, {"start": 11.599, "duration": 3.361, "text": "square plus b square is equal to c"}, {"start": 14.0, "duration": 3.44, "text": "square"}, {"start": 14.96, "duration": 4.64, "text": "that is sum of squares of perpendicular"}, {"start": 17.44, "duration": 5.759, "text": "sides is equal to the square of"}, {"start": 19.6, "duration": 7.679, "text": "hypotenuse but my question is why"}, {"start": 23.199, "duration": 5.92, "text": "why is this true for any right triangle"}, {"start": 27.279, "duration": 4.4, "text": "for the detailed explanation see the"}, {"start": 29.119, "duration": 6.6, "text": "comment box and for more videos please"}, {"start": 31.679, "duration": 4.04, "text": "visit and subscribe the channel"}], "title": "Pythagoras theorem - proof with visual explanation | Hypotenuse of Right triangle | Mathematics", "description": "Link for explanation : https://youtu.be/0SorlZmPNOo\n\nAny queries regarding the subject or videos are invited.\nGmail ID : problemanalysis99@gmail.com\nInstagram ID : problem_analysis\n\n#shorts #shortvideos #maths #mathematics #pythagorastheorem #righttriangle #hypotenuse #geometry", "lengthSeconds": 35, "width": 360, "height": 360}
{"video_id": "FdE7rHDZRd8", "transcript": [{"start": 11.76, "duration": 16.739, "text": "[\u0645\u0648\u0633\u064a\u0642\u0649]"}, {"start": 30.08, "duration": 16.66, "text": "\u200fh"}, {"start": 31.64, "duration": 15.1, "text": "[\u0645\u0648\u0633\u064a\u0642\u0649]"}], "title": "#trending #mathematics Visual Proof of Roots of Quadratic Equation", "description": "#trending #mathematics Visual Proof of Roots of Quadratic Equation\n\nOther Videos\nFibonacci Sequence, Golden Ratio, How to draw Golden Spiral:\nhttps://youtu.be/yHBa3EvDPco\nRadius and Tangent Vectors: \nhttps://youtube.com/shorts/KHSafRET3Sk\nDe Morgan's laws: Proof by Venn Diagrams:\nhttps://youtube.com/shorts/pR-REo4aBYk?feature=share", "lengthSeconds": 45, "width": 360, "height": 640}
{"video_id": "S42uQzogM5o", "transcript": [{"start": 0.0, "duration": 4.799, "text": "start with a circle where the radius is"}, {"start": 2.46, "duration": 4.74, "text": "length R here's an interesting question"}, {"start": 4.799, "duration": 4.141, "text": "how many arcs of length R can we fit"}, {"start": 7.2, "duration": 4.2, "text": "around the circle"}, {"start": 8.94, "duration": 4.8, "text": "certainly one is not enough and we can"}, {"start": 11.4, "duration": 4.56, "text": "see pretty quickly that two Arc lengths"}, {"start": 13.74, "duration": 4.14, "text": "with a length of R is not enough how"}, {"start": 15.96, "duration": 4.44, "text": "about three that looks like it could"}, {"start": 17.88, "duration": 5.52, "text": "almost cover half"}, {"start": 20.4, "duration": 5.4, "text": "we can check that 4 won't do the job"}, {"start": 23.4, "duration": 4.26, "text": "and that 5 won't do the job either as"}, {"start": 25.8, "duration": 3.84, "text": "there's still plenty of circle left over"}, {"start": 27.66, "duration": 5.099, "text": "however when we try to cover this circle"}, {"start": 29.64, "duration": 4.68, "text": "with six arcs of length R That Just"}, {"start": 32.759, "duration": 4.14, "text": "About Does it and only leaves a little"}, {"start": 34.32, "duration": 4.32, "text": "bit if we try with seven different arcs"}, {"start": 36.899, "duration": 4.5, "text": "of radius R it covers the entire circle"}, {"start": 38.64, "duration": 5.88, "text": "and there's overlap so we can fit six"}, {"start": 41.399, "duration": 5.701, "text": "arcs of length R around the circle each"}, {"start": 44.52, "duration": 4.68, "text": "such Arc of length R subtends an angle"}, {"start": 47.1, "duration": 5.34, "text": "of one radian so this means that we can"}, {"start": 49.2, "duration": 5.22, "text": "fit more than six radians in a circle in"}, {"start": 52.44, "duration": 3.48, "text": "fact the total number two Pi of radians"}, {"start": 54.42, "duration": 4.5, "text": "it fits in a circle is approximately"}, {"start": 55.92, "duration": 3.0, "text": "6.28"}], "title": "How many radii fit around a circle? #tauday #tau", "description": "In this short, we investigate how many radii of a circle would fit if you lay them in arcs around a circle. The method of laying a single radius around the circle creates an angle subtended by that arc - that angle is known as 1 radian. So in this video we see that there are more than 6 radians and less than 7 radians in a circle. The full measurement is actually around 6.28 and is the constant 2*Pi.\n\n#short #math #manim #radian #circle #radius #tauday #tau", "lengthSeconds": 59, "width": 360, "height": 640}
{"video_id": "PwYmy-k3nRg", "transcript": [{"start": 0.0, "duration": 5.16, "text": "start with a right triangle with sides"}, {"start": 2.46, "duration": 3.78, "text": "of length A and B and a hypotenuse of"}, {"start": 5.16, "duration": 3.84, "text": "length C"}, {"start": 6.24, "duration": 6.0, "text": "then draw an A by a square and next to"}, {"start": 9.0, "duration": 5.4, "text": "that draw a b by b square we can glue"}, {"start": 12.24, "duration": 3.899, "text": "these two squares together to form this"}, {"start": 14.4, "duration": 4.32, "text": "shape right here"}, {"start": 16.139, "duration": 5.161, "text": "inside of this shape we can fit one copy"}, {"start": 18.72, "duration": 4.68, "text": "of the triangle vertically like this and"}, {"start": 21.3, "duration": 4.68, "text": "we can fit a second copy horizontally"}, {"start": 23.4, "duration": 5.219, "text": "like this now we can take the two"}, {"start": 25.98, "duration": 5.459, "text": "triangular regions and rotate them as"}, {"start": 28.619, "duration": 6.061, "text": "pictured here the resulting shape is a c"}, {"start": 31.439, "duration": 5.221, "text": "by C Square this means that if a and b"}, {"start": 34.68, "duration": 4.379, "text": "are the legs of a right triangle and C"}, {"start": 36.66, "duration": 4.62, "text": "is the hypotenuse we can take an a by a"}, {"start": 39.059, "duration": 4.741, "text": "square and a b by b square and decompose"}, {"start": 41.28, "duration": 4.86, "text": "it to fit in a c by c squared that means"}, {"start": 43.8, "duration": 4.5, "text": "a squared plus b squared equals c"}, {"start": 46.14, "duration": 4.2, "text": "squared When A and B are the legs of a"}, {"start": 48.3, "duration": 5.06, "text": "right triangle with hypotenuse C this is"}, {"start": 50.34, "duration": 3.02, "text": "the Pythagorean theorem"}], "title": "Pythagorean Theorem IX (hinged visual proof)", "description": "This is a short visual proof of the Pythagorean theorem using swinging hinges.\n\nIf you like this video, consider subscribing to the channel or consider buying me a coffee: https://www.buymeacoffee.com/VisualProofs. Thanks!\n\nHere is a playlist with all my videos that prove or feature the Pythagorean theorem: https://www.youtube.com/playlist?list=PLZh9gzIvXQUvlRlnYryejWGHYQquM92rB\n\nThis animation is based on Proof 2 in Bogomolny's site cut the knot: https://www.cut-the-knot.org/pythagoras/. Check that link for more about the history of this proof.\n\n#math #manim #shorts #pythagoreantheorem #proofwithoutwords #visualproof #theorem #proof", "lengthSeconds": 52, "width": 360, "height": 640}
{"video_id": "1gFB7dG_t00", "transcript": [{"start": 0.299, "duration": 6.181, "text": "start with a right triangle with legs of"}, {"start": 2.7, "duration": 6.78, "text": "length a b and a hypotenuse of length C"}, {"start": 6.48, "duration": 6.119, "text": "first take a copy of this triangle and"}, {"start": 9.48, "duration": 5.34, "text": "scale up by the value B"}, {"start": 12.599, "duration": 5.161, "text": "next take another copy of the triangle"}, {"start": 14.82, "duration": 5.459, "text": "and scale up by the value a"}, {"start": 17.76, "duration": 4.26, "text": "because the leg lengths a b and b a are"}, {"start": 20.279, "duration": 3.781, "text": "equal we see that the triangles fit"}, {"start": 22.02, "duration": 3.419, "text": "together this way and because the two"}, {"start": 24.06, "duration": 3.719, "text": "non-right angles in the original"}, {"start": 25.439, "duration": 4.08, "text": "triangle are complementary this larger"}, {"start": 27.779, "duration": 4.021, "text": "triangle we formed is a right triangle"}, {"start": 29.519, "duration": 5.581, "text": "now scale the original triangle up one"}, {"start": 31.8, "duration": 6.24, "text": "more time by the value C we now have two"}, {"start": 35.1, "duration": 4.74, "text": "right triangles with equal leg lengths"}, {"start": 38.04, "duration": 3.6, "text": "this means that the two triangles are"}, {"start": 39.84, "duration": 3.899, "text": "congruent and so their hypotenuse"}, {"start": 41.64, "duration": 5.64, "text": "lengths are equal"}, {"start": 43.739, "duration": 5.581, "text": "therefore a squared plus b squared must"}, {"start": 47.28, "duration": 3.66, "text": "be equal to c squared where A and B are"}, {"start": 49.32, "duration": 4.02, "text": "the legs of a right triangle and C is"}, {"start": 50.94, "duration": 4.7, "text": "the hypotenuse this is the Pythagorean"}, {"start": 53.34, "duration": 2.3, "text": "theorem"}], "title": "Pythagorean theorem proof by scaling", "description": "This is a short, animated visual proof of the Pythagorean theorem (the right triangle theorem) using the hypotenuses of scaled triangles. This theorem states the square of the hypotenuse of a right triangle is equal to the sum of squares of the two other side lengths. #mathshorts #mathvideo #math #pythagoreantheorem #pythagorean #triangle #manim #animation #theorem #pww #proofwithoutwords #visualproof #proof\n\nThis animation is based on a visual proof by Frank Burk from the November 1996 issue of The College Mathematics Journal (https://doi.org/10.2307/2687335 page 409).\n\nHere is my original animation of this proof: https://youtu.be/q28H5NY5UVU\n\nTo learn more about animating with manim, check out:\nhttps://manim.community", "lengthSeconds": 54, "width": 360, "height": 640}
{"video_id": "6KPSmajeseI", "transcript": [{"start": 0.38, "duration": 52.0, "text": "[Music]"}, {"start": 61.04, "duration": 54.88, "text": "[Music]"}, {"start": 113.84, "duration": 2.08, "text": "you"}], "title": "Heron's formula visual proof | mathocube |", "description": "Give us Suggestions about Course or Video you may like to watch\nhttps://forms.gle/5Uv4SMfsQ8yvPAL58\nin this video, we are going to learn about Heron's formula visual proof \nSo don't forget to watch till the end \ud83e\udd17\ud83e\udd17\n////here is a free gift for you\n-\u2795\u2797\u2716\ufe0f\u2796-A FREE COURSE ON HUMAN CALCULATOR-\u2795\u2797\u2716\ufe0f\u2796-\nhttps://www.youtube.com/playlist?list=PLMTt7lFOKaqpRfEMWHHTNzYeH9n1tm-70\nif you have any questions you can ask me in the comments.\nI hope you will watch this video till the end and subscribe to my channel for more upcoming videos.\nthanks for watching\nfollow us on Facebook \nhttps://www.facebook.com/mathocube", "lengthSeconds": 115, "width": 640, "height": 360}
{"video_id": "a9WnjJJy1QI", "transcript": [{"start": 0.0, "duration": 3.54, "text": "here's one way to make sense of the"}, {"start": 1.86, "duration": 4.38, "text": "alternating sums of the powers of"}, {"start": 3.54, "duration": 4.92, "text": "one-half start with this nomon shaped"}, {"start": 6.24, "duration": 4.56, "text": "object here and assume that it has area"}, {"start": 8.46, "duration": 3.9, "text": "one we can divide the gnomon into four"}, {"start": 10.8, "duration": 3.66, "text": "smaller gnomons and remove two of them"}, {"start": 12.36, "duration": 4.74, "text": "so we removed half and then we can add"}, {"start": 14.46, "duration": 5.1, "text": "one of those back adding one-fourth back"}, {"start": 17.1, "duration": 4.74, "text": "now we can repeat this process on the"}, {"start": 19.56, "duration": 4.32, "text": "added back nomon removing half and then"}, {"start": 21.84, "duration": 4.08, "text": "adding one half of that back we can"}, {"start": 23.88, "duration": 4.62, "text": "continue this process indefinitely each"}, {"start": 25.92, "duration": 5.04, "text": "time we re-add one of the gnomons we"}, {"start": 28.5, "duration": 4.62, "text": "divide it into four nomons remove two"}, {"start": 30.96, "duration": 4.86, "text": "and then add one back we can continue"}, {"start": 33.12, "duration": 4.08, "text": "this process forever or indefinitely and"}, {"start": 35.82, "duration": 3.239, "text": "it turns out that we've shaded the"}, {"start": 37.2, "duration": 3.96, "text": "infinite sum of the alternating powers"}, {"start": 39.059, "duration": 3.901, "text": "of one-half on the other hand if we look"}, {"start": 41.16, "duration": 4.559, "text": "at the picture we see that we've shaded"}, {"start": 42.96, "duration": 5.279, "text": "exactly two-thirds of the diagram so the"}, {"start": 45.719, "duration": 6.141, "text": "sum of the powers of negative one-half"}, {"start": 48.239, "duration": 3.621, "text": "must be equal to two-thirds"}], "title": "A visual alternating sum!", "description": "This is a short, animated visual proof demonstrating the sum of the infinite geometric series with ratio -1/2. \n\nFor a longer version of this animation (with dramatic music only), check out : https://youtu.be/wLPsEULfPnk\n\nThis animation is based on a visual proof by Roger B. Nelsen from the November 2012 issue of The College Math Journal page 370 (https://doi.org/10.4169/college.math.j.43.5.370).\n\n#mathshorts\u200b #mathvideo\u200b #math\u200b #calculus #mtbos\u200b #manim\u200b #animation\u200b #theorem\u200b #pww\u200b #proofwithoutwords\u200b #visualproof\u200b #proof\u200b #iteachmath #geometricseries #infiniteseries #fractals #sum #induction \n\nTo learn more about animating with manim, check out:\nhttps://manim.community", "lengthSeconds": 51, "width": 360, "height": 640}
{"video_id": "yDHkNDrHfIw", "transcript": [{"start": 0.04, "duration": 3.44, "text": "let's play the chaos game in a square"}, {"start": 1.959, "duration": 3.161, "text": "start with a point anywhere in the"}, {"start": 3.48, "duration": 4.12, "text": "interior of the square now we're going"}, {"start": 5.12, "duration": 4.76, "text": "to roll an eight-sided die each side"}, {"start": 7.6, "duration": 5.119, "text": "corresponds to a Vertex or a midpoint of"}, {"start": 9.88, "duration": 4.44, "text": "the square we then add a new DOT 2/3 of"}, {"start": 12.719, "duration": 3.881, "text": "the way from the current dot that we're"}, {"start": 14.32, "duration": 4.52, "text": "at and the corresponding Vertex or"}, {"start": 16.6, "duration": 4.72, "text": "midpoint from the dice roll let's"}, {"start": 18.84, "duration": 4.679, "text": "iterate this process 30,000 times to see"}, {"start": 21.32, "duration": 2.199, "text": "what"}, {"start": 24.4, "duration": 5.08, "text": "happens amazingly this is the diagram"}, {"start": 27.4, "duration": 3.199, "text": "that arises as the limiting shape of the"}, {"start": 29.48, "duration": 2.959, "text": "process"}, {"start": 30.599, "duration": 3.401, "text": "this particular object is known as the"}, {"start": 32.439, "duration": 4.561, "text": "serinsky"}, {"start": 34.0, "duration": 3.0, "text": "carpet"}], "title": "Chaos Game in a Square", "description": "In this short, we show what happens when iterating the procedure of choosing a square vertex or edge midpoint at random and moving two thirds the distance from the current dot to the chosen point. \n\nIf you like this video, consider subscribing to the channel or consider buying me a coffee: https://www.buymeacoffee.com/VisualProofs. Thanks!\n\n#chaos #chaosgame #square #sierpinski #sierpinskicarpet #mathvideo\u200b #math\u200b #mtbos\u200b #manim\u200b #animation\u200b #theorem\u200b\u200b\u200b #iteachmath #mathematics #dynamicalsystems  #iteratedfunctionsystem #dynamics #fractals \n\nIf you want to know more about the Chaos game, see the following links:\nhttps://en.wikipedia.org/wiki/Chaos_game\nhttps://mathworld.wolfram.com/ChaosGa...\nhttps://math.bu.edu/DYSYS/chaos-game/...\nhttps://youtu.be/kbKtFN71Lfs\nhttps://youtu.be/IGlGvSXkRGI\n\nTo learn more about animating with manim, check out:\nhttps://manim.community", "lengthSeconds": 35, "width": 360, "height": 640}
{"video_id": "7I9SAYnon70", "transcript": [{"start": 1.55, "duration": 12.53, "text": "[Music]"}, {"start": 11.519, "duration": 4.881, "text": "start with a circle of radius r"}, {"start": 14.08, "duration": 5.52, "text": "and then draw an inscribed regular two"}, {"start": 16.4, "duration": 3.2, "text": "end gone like this"}, {"start": 19.68, "duration": 3.839, "text": "let's figure out the area of this"}, {"start": 21.439, "duration": 5.441, "text": "regular two n gone"}, {"start": 23.519, "duration": 5.84, "text": "to do that we can decompose the polygon"}, {"start": 26.88, "duration": 4.88, "text": "into triangles"}, {"start": 29.359, "duration": 4.72, "text": "the radius of the circle is r and there"}, {"start": 31.76, "duration": 4.72, "text": "are two n copies of this triangle that"}, {"start": 34.079, "duration": 4.721, "text": "can be used to build the regular two end"}, {"start": 36.48, "duration": 2.32, "text": "gone"}, {"start": 40.16, "duration": 5.44, "text": "so the area of the regular two n gone is"}, {"start": 42.719, "duration": 5.201, "text": "2n times the area of one of these"}, {"start": 45.6, "duration": 4.959, "text": "triangles"}, {"start": 47.92, "duration": 4.56, "text": "now let's draw the inscribed regular"}, {"start": 50.559, "duration": 4.561, "text": "n-gon like this"}, {"start": 52.48, "duration": 4.559, "text": "and suppose that this n-gon has a side"}, {"start": 55.12, "duration": 3.52, "text": "length of s"}, {"start": 57.039, "duration": 5.04, "text": "then the triangle that we're interested"}, {"start": 58.64, "duration": 5.36, "text": "in has a height of length s over two and"}, {"start": 62.079, "duration": 4.161, "text": "a base of length r"}, {"start": 64.0, "duration": 4.64, "text": "therefore the area of the regular two n"}, {"start": 66.24, "duration": 5.12, "text": "gone is two n times one half times r"}, {"start": 68.64, "duration": 6.64, "text": "times s over two but this simplifies to"}, {"start": 71.36, "duration": 6.32, "text": "r over 2 times n s and n s is the"}, {"start": 75.28, "duration": 3.68, "text": "perimeter of the regular n-gon inscribed"}, {"start": 77.68, "duration": 3.04, "text": "in the circle"}, {"start": 78.96, "duration": 3.92, "text": "but this means that we have shown that"}, {"start": 80.72, "duration": 4.32, "text": "the area of the regular 2-n-gon"}, {"start": 82.88, "duration": 4.4, "text": "inscribed in a circle of radius r is"}, {"start": 85.04, "duration": 4.32, "text": "equal to r over 2 times the perimeter of"}, {"start": 87.28, "duration": 4.479, "text": "the regular n-gon inscribed in the same"}, {"start": 89.36, "duration": 2.399, "text": "circle"}, {"start": 92.24, "duration": 4.32, "text": "but that's not all we can say"}, {"start": 94.24, "duration": 5.919, "text": "because both the two n-gone and the"}, {"start": 96.56, "duration": 5.919, "text": "n-gon approach the circle as we let n go"}, {"start": 100.159, "duration": 5.28, "text": "to infinity we get the result that the"}, {"start": 102.479, "duration": 4.481, "text": "area of the circle is equal to r over 2"}, {"start": 105.439, "duration": 3.201, "text": "multiplied by the circumference of the"}, {"start": 106.96, "duration": 4.32, "text": "circle if we remember that the"}, {"start": 108.64, "duration": 4.479, "text": "circumference of the circle is 2 pi r"}, {"start": 111.28, "duration": 4.32, "text": "then we see that we've found a formula"}, {"start": 113.119, "duration": 6.521, "text": "for the area of a circle of radius r to"}, {"start": 115.6, "duration": 4.04, "text": "be pi times r squared"}], "title": "Areas and Perimeters of Regular Polygons (visual proof)", "description": "This is a short, animated visual proof demonstrating that the area of the regular 2n-gon inscribed in a circle of radius r is equal to one-half the product of r and the perimeter of the regular n-gon inscribed in the same circle. We finish with a bonus proof that the area of a circle is one-half the product of its radius and its circumference. #math\u200b #geometry #mtbos\u200b #manim\u200b #animation\u200b #theorem\u200b #pww\u200b #proofwithoutwords\u200b #visualproof\u200b #proof\u200b #iteachmath #polygons #regularpolygon #circlearea #area #perimeter #mathshorts\u200b #mathvideo\u200b \n\nThis animation is based on a visual proof by Roger B. Nelsen from his second compendium of visual proofs:\n\nProofs without Words I by Roger B. Nelsen (MAA, 2000) (https://bookstore.ams.org/clrm-14/)\n\nTo learn more about animating with manim, check out:\nhttps://manim.community\n_____________________________\nMusic in this video:\nOnly Memories Remain by Hayden Folker | https://soundcloud.com/hayden-folker\nMusic promoted by https://www.free-stock-music.com\nCreative Commons Attribution 3.0 Unported License\nhttps://creativecommons.org/licenses/by/3.0/deed.en_US", "lengthSeconds": 127, "width": 640, "height": 360}
{"video_id": "OSI00BWYeNY", "transcript": [{"start": 0.27, "duration": 3.1, "text": "[\u043c\u0443\u0437\u044b\u043a\u0430]"}, {"start": 7.79, "duration": 3.26, "text": "[\u043c\u0443\u0437\u044b\u043a\u0430]"}, {"start": 29.96, "duration": 3.0, "text": "A"}, {"start": 34.63, "duration": 2.729, "text": "[\u043c\u0443\u0437\u044b\u043a\u0430]"}], "title": "Visual Proof: Natural Numbers Formula#maths", "description": "Sum of first n natural number formula visually.#maths #mindsphere\n\nAlgebraIsAwesome #CalculusCrunch #GeometryGuru #ShapesOfMath #StatsSaturday #ProbabilityPuzzle #PrimePower #NumberNerd #ArithmeticAdventures #TrigTuesdays #MathMagic #MathematicsMatters #GeometricGenius #CalculusConcepts #AlgebraicExpressions #GeometryGoals #StatisticalInsights #ProbabilityPlays #PrimeNumbersOnly #NumberTheory\n#AlgebraIsAwesome #CalculusCrunch #GeometryGuru #ShapesOfMath #StatsSaturday #ProbabilityPuzzle #PrimePower #NumberNerd #ArithmeticAdventures #TrigTuesdays #MathMagic #MathematicsMatters #GeometricGenius #CalculusConcepts #AlgebraicExpressions #GeometryGoals #StatisticalInsights #ProbabilityPlays #PrimeNumbersOnly #NumberTheory #ArithmeticArt #TrigonometryTricks #MathematicalMind #MathMania #AlgebraAllDay #CalculusCraze #GeometryGenius #StatisticalStudies #ProbabilityPatterns #PrimeNumberPuzzles #NumberNinja #ArithmeticAddict #TrigonometryTales #MathematicalMaster #AlgebraicAdventures #CalculusChallenges #GeometryGreatness #StatisticalSolutions #ProbabilityProblems #PrimePowerhouse #NumberNerdiness #ArithmeticAficionado #TrigonometryTips #MathematicalMusings #AlgebraAcademy #CalculusConundrums #GeometryGalore #StatisticalAnalysis", "lengthSeconds": 36, "width": 360, "height": 640}
{"video_id": "dklWNdM9WSg", "transcript": [{"start": 0.04, "duration": 3.48, "text": "here's an interesting chaos game variant"}, {"start": 1.8, "duration": 3.12, "text": "start with these two matrices A and B"}, {"start": 3.52, "duration": 3.72, "text": "and consider the apine linear"}, {"start": 4.92, "duration": 4.56, "text": "Transformations ax and BX"}, {"start": 7.24, "duration": 4.16, "text": "-5.5 these two Transformations are"}, {"start": 9.48, "duration": 3.48, "text": "applied to the plane and shown here"}, {"start": 11.4, "duration": 4.119, "text": "start with a point in the plane and"}, {"start": 12.96, "duration": 4.44, "text": "apply both Transformations choose one of"}, {"start": 15.519, "duration": 3.441, "text": "the two output points at random Say by"}, {"start": 17.4, "duration": 3.68, "text": "flipping a coin and apply both"}, {"start": 18.96, "duration": 4.36, "text": "Transformations again then repeat this"}, {"start": 21.08, "duration": 4.039, "text": "process each time select one of the"}, {"start": 23.32, "duration": 4.32, "text": "output points at random and apply both"}, {"start": 25.119, "duration": 6.841, "text": "Transformations let's do this 25,000"}, {"start": 27.64, "duration": 4.32, "text": "times to see what the resulting shape is"}, {"start": 33.239, "duration": 3.48, "text": "no matter the starting point or the"}, {"start": 34.96, "duration": 4.32, "text": "outcome of the random coin flips along"}, {"start": 36.719, "duration": 5.121, "text": "the way the limiting shape will always"}, {"start": 39.28, "duration": 4.48, "text": "look like this the resulting fractal"}, {"start": 41.84, "duration": 4.28, "text": "shape is called the levy C curve or the"}, {"start": 43.76, "duration": 4.0, "text": "levy Dragon it can be generated in other"}, {"start": 46.12, "duration": 5.56, "text": "ways but this is how we generate it"}, {"start": 47.76, "duration": 3.92, "text": "using an iterated function system"}], "title": "Levy Curve from the Chaos Game", "description": "In this video, we show how to use a random process of iteratively applying two linear transformations in the real plane to generate fractal curve known as the L\u00e9vy Dragon. What happens if you do something similar with different affine linear transformations?\n\nIf you like this video, consider subscribing to the channel or consider buying me a coffee: https://www.buymeacoffee.com/VisualProofs. Thanks!\n\nCheck out these videos for related constructions:\nhttps://youtu.be/rLxx6b-oJGo\nhttps://youtu.be/NBI_5GdvqUo\n\nThis animation was inspired by an article written by Larry Riddle that appeared in the the November 2016 issue of Math Horizons:\nhttps://doi.org/10.4169/mathhorizons.24.2.18\n\nSee more here (https://larryriddle.agnesscott.org/symmetricfractals.pdf) and on his website: https://larryriddle.agnesscott.org/ifs/levy/levy.htm\n\n#chaos #chaosgame #levy #levydragon #mathvideo\u200b #math\u200b #mtbos\u200b #manim\u200b #animation\u200b #theorem\u200b\u200b\u200b #iteachmath #mathematics #dynamicalsystems  #iteratedfunctionsystem #dynamics #fractals \n\nTo learn more about animating with manim, check out:\nhttps://manim.community", "lengthSeconds": 50, "width": 360, "height": 640}
{"video_id": "x29K5VrtiLM", "transcript": [{"start": 0.0, "duration": 5.04, "text": "imagine this diagram with two squares 1"}, {"start": 2.639, "duration": 4.801, "text": "1 4 the size of the other encloses a"}, {"start": 5.04, "duration": 4.559, "text": "total area of one now cut both squares"}, {"start": 7.44, "duration": 4.26, "text": "into four equal pieces and remove one"}, {"start": 9.599, "duration": 4.261, "text": "this removes a total of one-fourth of"}, {"start": 11.7, "duration": 4.08, "text": "the area of the diagram on the removed"}, {"start": 13.86, "duration": 4.56, "text": "area cut that into four pieces and"}, {"start": 15.78, "duration": 4.32, "text": "re-add one piece thus adding one-fourth"}, {"start": 18.42, "duration": 4.8, "text": "of one-fourth back we can imagine"}, {"start": 20.1, "duration": 5.099, "text": "repeating this process indefinitely each"}, {"start": 23.22, "duration": 4.799, "text": "time removing one quarter of the area"}, {"start": 25.199, "duration": 5.281, "text": "and then re-adding one quarter of that"}, {"start": 28.019, "duration": 4.501, "text": "removed area back if we do this"}, {"start": 30.48, "duration": 4.739, "text": "infinitely then the resulting limiting"}, {"start": 32.52, "duration": 4.559, "text": "diagram has a shaded area given by the"}, {"start": 35.219, "duration": 4.321, "text": "infinite sum of the powers of negative"}, {"start": 37.079, "duration": 4.021, "text": "one-fourth and now here's the trick we"}, {"start": 39.54, "duration": 3.66, "text": "can shift the Shaded part of the small"}, {"start": 41.1, "duration": 4.26, "text": "square up to the larger Square thus"}, {"start": 43.2, "duration": 4.14, "text": "filling in all of the larger square but"}, {"start": 45.36, "duration": 3.78, "text": "by construction the larger Square was"}, {"start": 47.34, "duration": 3.96, "text": "four times bigger than the small square"}, {"start": 49.14, "duration": 4.259, "text": "and therefore we've shaded exactly"}, {"start": 51.3, "duration": 3.66, "text": "four-fifths of the original area this"}, {"start": 53.399, "duration": 4.881, "text": "means the sum of the powers of negative"}, {"start": 54.96, "duration": 3.32, "text": "one fourth is four-fifths"}], "title": "Alternating 1/4 Sum", "description": "This is a short, animated visual proof demonstrating the sum of the infinite geometric series with ratio -1/4. \n\nIf you like this video, consider subscribing to the channel or consider buying me a coffee: https://www.buymeacoffee.com/VisualProofs. Thanks!\n\nHere is a longer, wordless, high-definition version of this animation: https://youtu.be/SeKGZ2DLAwo\n\nHere is an alternating series dissection for powers of -1/2: https://youtu.be/NKFrHfBGDNE\n\nAnd here is an alternating series dissection for powers of -1/3:\nhttps://youtu.be/MyCn7gIPxNg\n\nThis proof is based on a visual proof by Ivica Martinjak & Ana Mimica from the May 2021 issue of the College Mathematics Journal (https://doi.org/10.1080/07468342.2021.1909963 page 204).\n\n#math\u200b #calculus #mtbos\u200b #manim\u200b #animation\u200b #theorem\u200b #pww\u200b #proofwithoutwords\u200b #visualproof\u200b #proof\u200b #iteachmath #geometricseries #infiniteseries #fractals #sum #induction #mathshorts\u200b #mathvideo\u200b \n\nTo learn more about animating with manim, check out:\nhttps://manim.community", "lengthSeconds": 57, "width": 360, "height": 640}
{"video_id": "enQdwZwoyyo", "transcript": [{"start": 0.12, "duration": 6.38, "text": "foreign"}, {"start": 1.52, "duration": 7.78, "text": "[Music]"}, {"start": 6.5, "duration": 4.96, "text": "start with four positive real numbers a"}, {"start": 9.3, "duration": 5.28, "text": "b x and y and assume a is less than or"}, {"start": 11.46, "duration": 5.22, "text": "equal to B and X is less than y create a"}, {"start": 14.58, "duration": 4.14, "text": "right triangle with legs of length A and"}, {"start": 16.68, "duration": 4.439, "text": "B and let's assume this angle in the"}, {"start": 18.72, "duration": 4.92, "text": "upper left is Alpha"}, {"start": 21.119, "duration": 4.981, "text": "first create a scaled copy of the right"}, {"start": 23.64, "duration": 5.639, "text": "triangle where the scale value is y"}, {"start": 26.1, "duration": 5.7, "text": "then create a rotated and scaled copy of"}, {"start": 29.279, "duration": 5.221, "text": "the triangle where the scale value is X"}, {"start": 31.8, "duration": 4.32, "text": "we see that the angle Alpha appears in"}, {"start": 34.5, "duration": 4.14, "text": "these two locations in the scaled"}, {"start": 36.12, "duration": 4.259, "text": "triangles connecting the other two"}, {"start": 38.64, "duration": 4.38, "text": "vertices of the triangle together like"}, {"start": 40.379, "duration": 4.261, "text": "this creates a right triangle due to the"}, {"start": 43.02, "duration": 2.94, "text": "Pythagorean theorem the legs of this"}, {"start": 44.64, "duration": 3.06, "text": "right triangle are x times the square"}, {"start": 45.96, "duration": 3.06, "text": "root of a squared plus b squared and Y"}, {"start": 47.7, "duration": 3.3, "text": "times the square root of a squared plus"}, {"start": 49.02, "duration": 4.92, "text": "b squared and let's label the angle in"}, {"start": 51.0, "duration": 4.5, "text": "the upper left beta finally we can"}, {"start": 53.94, "duration": 3.54, "text": "create a rectangular diagram from this"}, {"start": 55.5, "duration": 4.8, "text": "trapezoid by adding this right triangle"}, {"start": 57.48, "duration": 5.46, "text": "with legs of length b y minus ax and a y"}, {"start": 60.3, "duration": 4.679, "text": "plus b x the bottom angle in this"}, {"start": 62.94, "duration": 3.6, "text": "triangle must be Alpha plus beta by"}, {"start": 64.979, "duration": 3.241, "text": "construction because it's complementary"}, {"start": 66.54, "duration": 3.84, "text": "to the other acute angle which is"}, {"start": 68.22, "duration": 4.079, "text": "complementary to Alpha plus beta"}, {"start": 70.38, "duration": 3.779, "text": "now we can get an interesting formula by"}, {"start": 72.299, "duration": 5.161, "text": "Computing Alpha plus beta in two ways"}, {"start": 74.159, "duration": 6.421, "text": "first of all it's the arc tangent of a y"}, {"start": 77.46, "duration": 5.339, "text": "plus BX divided by b y minus ax"}, {"start": 80.58, "duration": 4.74, "text": "according to the upper right triangle on"}, {"start": 82.799, "duration": 5.221, "text": "the other hand Alpha is the arctangent"}, {"start": 85.32, "duration": 4.979, "text": "of a y divided by b y according to the"}, {"start": 88.02, "duration": 4.02, "text": "left triangle and beta is the arc"}, {"start": 90.299, "duration": 3.841, "text": "tangent of x times the square root of a"}, {"start": 92.04, "duration": 3.48, "text": "squared plus b squared divided by y"}, {"start": 94.14, "duration": 2.88, "text": "times the square root of a squared plus"}, {"start": 95.52, "duration": 3.419, "text": "b squared according to the inner"}, {"start": 97.02, "duration": 3.54, "text": "triangle notice that we can cancel the"}, {"start": 98.939, "duration": 3.481, "text": "Y's and cancel the square root of a"}, {"start": 100.56, "duration": 3.419, "text": "squared plus b squareds and we end up"}, {"start": 102.42, "duration": 3.72, "text": "with the formula that arc tangent of A"}, {"start": 103.979, "duration": 4.381, "text": "over B plus the arc tangent of X over Y"}, {"start": 106.14, "duration": 5.1, "text": "is equal to the arctangent of a y plus"}, {"start": 108.36, "duration": 4.98, "text": "BX divided by b y minus ax setting both"}, {"start": 111.24, "duration": 3.839, "text": "A over B and X over y to one-fifth in"}, {"start": 113.34, "duration": 3.239, "text": "this equation yields that 2 times the"}, {"start": 115.079, "duration": 4.021, "text": "arctangent of a fifth is equal to the"}, {"start": 116.579, "duration": 5.101, "text": "arctangent of 5 12. setting both A over"}, {"start": 119.1, "duration": 4.32, "text": "B and X over y to 5 12 reveals that 2"}, {"start": 121.68, "duration": 4.02, "text": "times the arctangent of 5 12 is equal to"}, {"start": 123.42, "duration": 3.96, "text": "the arc tangent of 120 over 119 and"}, {"start": 125.7, "duration": 3.539, "text": "finally setting A over B equal to one"}, {"start": 127.38, "duration": 4.56, "text": "over one and X over y equal to 1 over"}, {"start": 129.239, "duration": 4.981, "text": "239 we get the at pi over 4 plus the arc"}, {"start": 131.94, "duration": 5.4, "text": "tangent of 1 over 239 is equal to the"}, {"start": 134.22, "duration": 4.26, "text": "arc tangent of 120 divided by 119. now"}, {"start": 137.34, "duration": 3.18, "text": "let's string these three equations"}, {"start": 138.48, "duration": 4.68, "text": "together use the second equation to"}, {"start": 140.52, "duration": 4.56, "text": "replace arctangent 120 over 119 by 2"}, {"start": 143.16, "duration": 3.96, "text": "times arctangent 5 12. then replace"}, {"start": 145.08, "duration": 3.659, "text": "arctangent 5 12 by 2 times the"}, {"start": 147.12, "duration": 3.6, "text": "arctangent of one-fifth using the first"}, {"start": 148.739, "duration": 4.201, "text": "equation if we rearrange the resulting"}, {"start": 150.72, "duration": 4.32, "text": "formula we get that pi over 4 is equal"}, {"start": 152.94, "duration": 4.62, "text": "to 4 times the arctangent of one-fifth"}, {"start": 155.04, "duration": 4.199, "text": "minus the arc tangent of 1 over 239 this"}, {"start": 157.56, "duration": 3.3, "text": "is called the matron formula and is"}, {"start": 159.239, "duration": 4.321, "text": "attributed to John Machin who discovered"}, {"start": 160.86, "duration": 5.04, "text": "it in 1706 when he used it to compute pi"}, {"start": 163.56, "duration": 4.14, "text": "to 100 decimal places"}, {"start": 165.9, "duration": 3.839, "text": "here are the first 100 decimal places"}, {"start": 167.7, "duration": 4.92, "text": "that Machin would have found"}, {"start": 169.739, "duration": 5.761, "text": "in 1873 William Shanks used the same"}, {"start": 172.62, "duration": 5.82, "text": "formula but this time got 707 digits of"}, {"start": 175.5, "duration": 6.239, "text": "pi as listed here"}, {"start": 178.44, "duration": 5.7, "text": "he was so close but in 1946 an error was"}, {"start": 181.739, "duration": 3.541, "text": "found in his work in the 528th digit"}, {"start": 184.14, "duration": 2.58, "text": "shown here"}, {"start": 185.28, "duration": 3.179, "text": "how many digits of pi do you think"}, {"start": 186.72, "duration": 5.36, "text": "modern computers can compute using this"}, {"start": 188.459, "duration": 3.621, "text": "formula leave a comment let me know"}], "title": "Machin Formula Visualization (Pi day special)", "description": "This video describes a relatively new visual proof of the famous Machin formula, which is a formula that allows for relatively speedy computation of the digits of Pi.  As a Pi day bonus, we see Pi to 100 digits, as computed by Machin as well as to 707 digits as computed by Shanks. How many digits do you think it can be used to compute using modern technological tools? #math #pi #piday #geometry #inscribed #circumscribed  #triangle #manim #animation #theorem #pww #proofwithoutwords #visualproof #proof  #similartriangles #mathvideo #machinformula #machintheorem #digitsofpi #trapezoid \n\nIf you enjoyed this video, please consider liking and subscribing. Thanks for watching!\n\nThis animation is based on a proof due to Roger Nelsen from the November 2021 issue of The College Mathematics Journal  (https://doi.org/10.1080/07468342.2021.1969860 ) page 355.\n\nHere are two more Pi day gems:\n\nhttps://youtu.be/-LVpjnaqtwU\nhttps://youtu.be/JE_YeVlSqLY\n\nTo learn more about animating with manim, check out:\nhttps://manim.community\n____________________________________________________\nBackground music:\nAcoustic Meditation 2\u00a0by Audionautix is licensed under a Creative Commons Attribution 4.0 license.\u00a0https://creativecommons.org/licenses/by/4.0/\nArtist:\u00a0http://audionautix.com/", "lengthSeconds": 194, "width": 640, "height": 360}
{"video_id": "69TvfNA0Lxc", "transcript": [{"start": 0.179, "duration": 3.0, "text": "foreign"}, {"start": 6.74, "duration": 5.56, "text": "let's let T sub K represent the sum of"}, {"start": 9.72, "duration": 4.62, "text": "the first K positive integers the sum"}, {"start": 12.3, "duration": 3.899, "text": "can be represented by a triangular array"}, {"start": 14.34, "duration": 3.96, "text": "of Cubes like the one pictured here"}, {"start": 16.199, "duration": 5.281, "text": "where there's K cubes in the bottom row"}, {"start": 18.3, "duration": 5.76, "text": "and one cube in the top row"}, {"start": 21.48, "duration": 4.62, "text": "so let's consider T sub n plus 1 which"}, {"start": 24.06, "duration": 4.26, "text": "can be represented by a triangular array"}, {"start": 26.1, "duration": 3.06, "text": "of cubes with n plus 1 cubes on the"}, {"start": 28.32, "duration": 3.959, "text": "bottom"}, {"start": 29.16, "duration": 5.04, "text": "now construct n different copies of this"}, {"start": 32.279, "duration": 4.741, "text": "triangular array and place them side by"}, {"start": 34.2, "duration": 6.12, "text": "side like this this staircase diagram"}, {"start": 37.02, "duration": 5.1, "text": "contains a total of n times T sub n plus"}, {"start": 40.32, "duration": 4.739, "text": "1 with cubes"}, {"start": 42.12, "duration": 5.22, "text": "let's remove the N by n plus 1 rectangle"}, {"start": 45.059, "duration": 4.381, "text": "in the back of this staircase this"}, {"start": 47.34, "duration": 4.559, "text": "rectangle can be decomposed into two"}, {"start": 49.44, "duration": 4.98, "text": "triangular arrays each of these"}, {"start": 51.899, "duration": 5.34, "text": "triangular arrays has n cubes in the"}, {"start": 54.42, "duration": 5.22, "text": "bottom so represents the number T sub n"}, {"start": 57.239, "duration": 4.021, "text": "we can take these two arrays and place"}, {"start": 59.64, "duration": 4.079, "text": "them on either side of the remaining"}, {"start": 61.26, "duration": 5.34, "text": "staircase we see that the staircase"}, {"start": 63.719, "duration": 5.041, "text": "diagram now has a total of n plus two"}, {"start": 66.6, "duration": 3.78, "text": "copies of the Triangular array"}, {"start": 68.76, "duration": 4.02, "text": "representing the number T sub n"}, {"start": 70.38, "duration": 5.16, "text": "therefore the total number of Cubes is n"}, {"start": 72.78, "duration": 5.1, "text": "plus 2 times T sub n but we can use this"}, {"start": 75.54, "duration": 5.34, "text": "formula to see that t sub n plus 1 is"}, {"start": 77.88, "duration": 5.7, "text": "equal to n plus 2 over n multiplied by T"}, {"start": 80.88, "duration": 4.559, "text": "sub n the numbers given by T sub K are"}, {"start": 83.58, "duration": 5.359, "text": "called the Triangular numbers so this is"}, {"start": 85.439, "duration": 3.5, "text": "a recurrence for the Triangular numbers"}], "title": "A Multiplicative Triangular Number Recurrence (visual proof)", "description": "This is a short, animated visual proof demonstrating how to multiplicatively obtain the (n+1)st triangle number from the nth triangular number. #manim #math #mathvideo #mathshorts #triangularnumbers  #numbertheory  #triangle  #animation #theorem #pww #proofwithoutwords #visualproof #proof\n\nThis animation is included in the following sources:\n\nProofs without Words II by Roger B. Nelsen (MAA, 2000) (https://bookstore.ams.org/clrm-14/) page 14\n\nTo learn more about animating with manim, check out:\nhttps://manim.community\n____________________________________________________\nBackground music:\nTimeless by Lauren Duski", "lengthSeconds": 95, "width": 640, "height": 360}
{"video_id": "0D3KiCmum90", "transcript": [{"start": 10.639, "duration": 4.88, "text": "suppose we are given a circle with"}, {"start": 12.16, "duration": 5.92, "text": "radius r so that its area is a equals pi"}, {"start": 15.519, "duration": 2.561, "text": "r squared"}, {"start": 18.64, "duration": 4.719, "text": "let's paint half the circumference blue"}, {"start": 21.279, "duration": 3.84, "text": "and then roll the circle one half turn"}, {"start": 23.359, "duration": 3.92, "text": "so that we are left with a line of"}, {"start": 25.119, "duration": 3.761, "text": "length pi times r"}, {"start": 27.279, "duration": 4.481, "text": "extend the line by the length of the"}, {"start": 28.88, "duration": 5.839, "text": "radius r and we get a semicircle of"}, {"start": 31.76, "duration": 4.479, "text": "diameter pi r plus r"}, {"start": 34.719, "duration": 2.881, "text": "then create the half chord from the"}, {"start": 36.239, "duration": 4.48, "text": "bottom of the original circle to the"}, {"start": 37.6, "duration": 4.799, "text": "larger semicircle and call its length x"}, {"start": 40.719, "duration": 3.761, "text": "connecting the ends of the diameter to"}, {"start": 42.399, "duration": 3.921, "text": "the end of this half chord results in a"}, {"start": 44.48, "duration": 3.68, "text": "right triangle according to thaley's"}, {"start": 46.32, "duration": 4.0, "text": "triangle theorem"}, {"start": 48.16, "duration": 6.48, "text": "consequently the two subtriangles with"}, {"start": 50.32, "duration": 4.32, "text": "side length x are similar"}, {"start": 54.879, "duration": 6.961, "text": "this implies that pi r over x"}, {"start": 59.359, "duration": 2.481, "text": "equals"}, {"start": 62.079, "duration": 2.881, "text": "x"}, {"start": 62.879, "duration": 3.441, "text": "over r"}, {"start": 64.96, "duration": 5.04, "text": "therefore"}, {"start": 66.32, "duration": 5.68, "text": "x squared equals pi r squared"}, {"start": 70.0, "duration": 4.88, "text": "we can also use the half chord to create"}, {"start": 72.0, "duration": 3.84, "text": "a square with side length x so that its"}, {"start": 74.88, "duration": 3.52, "text": "area"}, {"start": 75.84, "duration": 5.76, "text": "is a equals x squared"}, {"start": 78.4, "duration": 5.759, "text": "but x squared equals pi r squared and"}, {"start": 81.6, "duration": 6.64, "text": "thus we have constructed a square with"}, {"start": 84.159, "duration": 5.92, "text": "the same area as the given circle"}, {"start": 88.24, "duration": 3.13, "text": "that is to say we have squared the"}, {"start": 90.079, "duration": 4.36, "text": "circle"}, {"start": 91.37, "duration": 3.069, "text": "[Music]"}, {"start": 104.479, "duration": 2.081, "text": "you"}], "title": "Squaring the Circle by Rolling (animated visual proof)", "description": "This is a short, animated visual proof that we can square the circle IF we allow the circle to roll. Unfortunately, this is not a solution to the squaring the circle problem from antiquity because that requires it to be done with only a straightedge and compass. #mathshorts\u200b #mathvideo\u200b #math\u200b ##geometry #mtbos\u200b #manim\u200b #animation\u200b #theorem\u200b #pww\u200b #proofwithoutwords\u200b #visualproof\u200b #proof\u200b #iteachmath #circle #mathematics #thales #squaringthecircle\n\nThis animation is based on a visual proof by Thomas Elsner from the May 1977 issue of Mathematics Magazine (https://doi.org/10.2307/2689507 - page 162).\n\nTo learn more about animating with manim, check out:\nhttps://manim.community", "lengthSeconds": 105, "width": 640, "height": 360}
{"video_id": "uWZsVx7wqeA", "transcript": [{"start": 0.16, "duration": 3.96, "text": "suppose the golden ratio were a rational"}, {"start": 2.36, "duration": 3.519, "text": "number that means we could write it as A"}, {"start": 4.12, "duration": 4.12, "text": "over B where A and B are positive"}, {"start": 5.879, "duration": 4.241, "text": "integers and this is in lowest terms the"}, {"start": 8.24, "duration": 3.8, "text": "existence of two such integers means"}, {"start": 10.12, "duration": 3.76, "text": "that we can draw this regular pentagon"}, {"start": 12.04, "duration": 4.12, "text": "with a diagonal of length a and a side"}, {"start": 13.88, "duration": 4.319, "text": "length of B the pictured green triangle"}, {"start": 16.16, "duration": 4.24, "text": "is known as a Golden Triangle when we"}, {"start": 18.199, "duration": 4.24, "text": "swing this side length down into the"}, {"start": 20.4, "duration": 4.28, "text": "Golden Triangle we create a second"}, {"start": 22.439, "duration": 4.121, "text": "triangle pictured in red that's also a"}, {"start": 24.68, "duration": 4.24, "text": "Golden Triangle with corresponding side"}, {"start": 26.56, "duration": 4.24, "text": "lengths B and A minus B but because"}, {"start": 28.92, "duration": 5.24, "text": "these two triangles are similar that"}, {"start": 30.8, "duration": 6.24, "text": "means that A over B is equal to B / a"}, {"start": 34.16, "duration": 5.399, "text": "minus B this means that the golden ratio"}, {"start": 37.04, "duration": 4.32, "text": "Fe is equal to B over a minus B but"}, {"start": 39.559, "duration": 4.241, "text": "that's a contradiction since B is less"}, {"start": 41.36, "duration": 4.6, "text": "than a and a minus B is less than b so"}, {"start": 43.8, "duration": 4.2, "text": "FAL A over B couldn't have been in"}, {"start": 45.96, "duration": 6.84, "text": "lowest terms this proof means that the"}, {"start": 48.0, "duration": 4.8, "text": "golden ratio fee must be irrational"}], "title": "Golden IrRATIOnal", "description": "In this video, we use the fact that that the golden ratio is the ratio of a diagonal in a regular pentagon to the side length to prove that the golden ratio is irrational, that is, that it cannot be written as the ratio of two integers.\n\nIf you like this video, consider subscribing to the channel or consider buying me a coffee: https://www.buymeacoffee.com/VisualProofs. Thanks!\n\nIf you want to see a longer version with justification about the similarity of pictured triangles, see \nhttps://youtu.be/lL9VN8ELH1M\n\nThis animation is based on an argument in Roger Nelsen\u2019s book Nuggets of Number Theory (I may receive a commission from Amazon, at no cost to you, if you click the link below):\n\nhttps://amzn.to/3RfbRf2\n\nHere are other videos related to the Golden ratio:\nhttps://www.youtube.com/watch?v=yeHDXdv5KH4\nhttps://www.youtube.com/watch?v=eMMErKcw8yM\nhttps://www.youtube.com/watch?v=1ctRFgWwg5E&t=4s\n\n#goldenratio #irrationalnumbers #realnumbers\u200b #manim\u200b #math\u200b #mtbos\u200b #animation\u200b #theorem\u200b\u200b #visualproof\u200b #proof\u200b #iteachmath #mathematics #irrational #pentagon #proofbycontradiction #algebra #infinitedescent #similartriangles #triangle #trianglesimilarity #similarity\n\nTo learn more about animating with manim, check out:\nhttps://manim.community", "lengthSeconds": 51, "width": 360, "height": 640}
{"video_id": "CTzBVi7ej_c", "transcript": [{"start": 0.04, "duration": 3.919, "text": "start with one Circle representing the"}, {"start": 1.76, "duration": 4.24, "text": "number one take seven copies of this"}, {"start": 3.959, "duration": 4.081, "text": "circle by placing one in the center and"}, {"start": 6.0, "duration": 4.36, "text": "six around the outer ring like this to"}, {"start": 8.04, "duration": 4.32, "text": "represent the number seven repeat this"}, {"start": 10.36, "duration": 3.84, "text": "process as often as you want taking"}, {"start": 12.36, "duration": 4.36, "text": "seven copies of the previous shape by"}, {"start": 14.2, "duration": 4.919, "text": "placing one in the center and six around"}, {"start": 16.72, "duration": 4.28, "text": "the outer ring this technique produces"}, {"start": 19.119, "duration": 3.441, "text": "visual representations for the powers of"}, {"start": 21.0, "duration": 3.76, "text": "seven to create the visual"}, {"start": 22.56, "duration": 4.6, "text": "representation for 7 to the N we can do"}, {"start": 24.76, "duration": 4.359, "text": "this one more time using seven copies of"}, {"start": 27.16, "duration": 4.0, "text": "the visual representation for 7 to the N"}, {"start": 29.119, "duration": 4.561, "text": "minus one but an alternate way to"}, {"start": 31.16, "duration": 5.12, "text": "produce this same diagram is to take the"}, {"start": 33.68, "duration": 5.12, "text": "sum of the powers of 7 from 1 up to 7"}, {"start": 36.28, "duration": 5.08, "text": "the n minus1 and create six different"}, {"start": 38.8, "duration": 4.759, "text": "copies living around the central circle"}, {"start": 41.36, "duration": 4.199, "text": "once we add the central circle we end up"}, {"start": 43.559, "duration": 4.561, "text": "with the same total number of circles in"}, {"start": 45.559, "duration": 5.201, "text": "the diagram as the one from 7 to the N"}, {"start": 48.12, "duration": 5.84, "text": "therefore 6 * the sum of the powers of 7"}, {"start": 50.76, "duration": 5.599, "text": "from 1 up to 7 the N minus 1 + 1 is"}, {"start": 53.96, "duration": 4.439, "text": "equal to 7 the N this gives us a formula"}, {"start": 56.359, "duration": 4.561, "text": "for the finite geometric sum with ratio"}, {"start": 58.399, "duration": 2.521, "text": "7"}], "title": "Geometric sum of powers of 7", "description": "This is a short, animated visual proof demonstrating the finite geometric sum formula for any integer n with n greater than 3 (explicitly showing the case n=7 with k=3). This series (and its infinite analog when x less than 1) is important for many results in calculus, discrete mathematics, and combinatorics. \n\nTo buy me a coffee, head over to https://www.buymeacoffee.com/VisualProofs \nThanks!\n\nFor a longer version of this video with other geometric sums (to see how this generalizes), check out: https://youtu.be/1wIdJxSfUz4\n\nAlso, check out my playlist on geometric sums/series: https://youtube.com/playlist?list=PLZh9gzIvXQUsgw8W5TUVDtF0q4jEJ3iaw\n\nThis animation is based on a proof by Mingjang Chen from the December 2004 issue of Mathematics Magazine page 373 (https://doi.org/10.2307/3219202).\n\n#manim\u200b #math #geometricsums #series\u200b #calculus #mtbos\u200b  #animation\u200b #theorem\u200b #pww\u200b #proofwithoutwords\u200b #visualproof\u200b #proof\u200b #iteachmath #finitesums #finiteseries #geometric  #mathshorts\u200b #mathvideo\u200b\n\nTo learn more about animating with manim, check out:\nhttps://manim.community", "lengthSeconds": 59, "width": 360, "height": 640}
{"video_id": "ZeEOgbLo0Rg", "transcript": [{"start": 0.95, "duration": 7.91, "text": "[Music]"}, {"start": 10.08, "duration": 3.84, "text": "let's build a centered stack of squares"}, {"start": 11.92, "duration": 4.4, "text": "consisting of one square in the top row"}, {"start": 13.92, "duration": 5.439, "text": "three squares in the second row five"}, {"start": 16.32, "duration": 5.119, "text": "squares in the third row and so on"}, {"start": 19.359, "duration": 5.201, "text": "adding the odd numbers in each"}, {"start": 21.439, "duration": 5.6, "text": "consecutive row"}, {"start": 24.56, "duration": 6.08, "text": "the resulting stack is a representation"}, {"start": 27.039, "duration": 6.081, "text": "of the sum of the first n odd numbers"}, {"start": 30.64, "duration": 5.68, "text": "but we can rotate this stack creating"}, {"start": 33.12, "duration": 8.2, "text": "four total copies and when we do that"}, {"start": 36.32, "duration": 5.0, "text": "the resulting shape is a perfect square"}, {"start": 42.84, "duration": 5.48, "text": "[Music]"}, {"start": 45.52, "duration": 5.92, "text": "the area of the resulting 2n by 2n"}, {"start": 48.32, "duration": 6.43, "text": "square is just 2n times 2n which"}, {"start": 51.44, "duration": 6.56, "text": "simplifies to 4 times n squared"}, {"start": 54.75, "duration": 4.289, "text": "[Music]"}, {"start": 58.0, "duration": 4.8, "text": "therefore"}, {"start": 59.039, "duration": 8.961, "text": "four times the sum of the first n odd"}, {"start": 62.8, "duration": 7.12, "text": "numbers is equal to four times n squared"}, {"start": 68.0, "duration": 4.479, "text": "if we divide both sides by four we see"}, {"start": 69.92, "duration": 6.36, "text": "that the sum of the first n odd integers"}, {"start": 72.479, "duration": 3.801, "text": "is equal to n squared"}, {"start": 86.4, "duration": 2.079, "text": "you"}], "title": "Sum of the first n odd numbers (visual proof)", "description": "This is a short, animated visual proof demonstrating how to find the sum of the first n positive odd integers. #math\u200b #numbertheory #mtbos\u200b #manim\u200b #animation\u200b #theorem\u200b #pww\u200b #proofwithoutwords\u200b #visualproof\u200b #proof\u200b #iteachmath #squares #oddintegers #numbertheory #sum #mathshorts\u200b #mathvideo\u200b \n\nThis animation is based on a visual proof by Roger B. Nelsen from his first compendium of visual proofs:\n\nProofs without Words I by Roger B. Nelsen (MAA, 1993) (https://bookstore.ams.org/clrm-1)\n\nTo learn more about animating with manim, check out:\nhttps://manim.community\n_____________________________\nMusic in this video:\nDramatic Interlude by Alexander Nakarada | https://www.serpentsoundstudios.com\nMusic promoted by https://www.free-stock-music.com\nAttribution 4.0 International (CC BY 4.0)\nhttps://creativecommons.org/licenses/by/4.0/", "lengthSeconds": 87, "width": 640, "height": 360}
{"video_id": "Evbsk8yuoSQ", "transcript": [{"start": 0.0, "duration": 5.64, "text": "let's find a formula for the sum a plus"}, {"start": 2.7, "duration": 5.819, "text": "ax plus ax squared up to ax to the K"}, {"start": 5.64, "duration": 4.8, "text": "minus 1 where X is greater than 1 a is a"}, {"start": 8.519, "duration": 4.141, "text": "positive number and K is a positive"}, {"start": 10.44, "duration": 4.199, "text": "integer we can make sense of this sum by"}, {"start": 12.66, "duration": 5.22, "text": "considering a line with lengths marked"}, {"start": 14.639, "duration": 4.741, "text": "at 1 x x squared x cubed and so on up to"}, {"start": 17.88, "duration": 3.659, "text": "x to the k then we can draw a"}, {"start": 19.38, "duration": 4.92, "text": "perpendicular line and start to build a"}, {"start": 21.539, "duration": 5.4, "text": "sequence of rectangles of width a over x"}, {"start": 24.3, "duration": 5.219, "text": "minus one the first rectangle has area a"}, {"start": 26.939, "duration": 5.34, "text": "the next rectangle has area ax the next"}, {"start": 29.519, "duration": 4.38, "text": "rectangle has an area of ax squared and"}, {"start": 32.279, "duration": 4.261, "text": "so on all the way until the last"}, {"start": 33.899, "duration": 5.101, "text": "rectangle which has an area of a x to"}, {"start": 36.54, "duration": 4.26, "text": "the K minus 1. so the total shaded area"}, {"start": 39.0, "duration": 3.84, "text": "in this rectangle is the finite sum"}, {"start": 40.8, "duration": 3.72, "text": "we're interested in but we can also just"}, {"start": 42.84, "duration": 3.18, "text": "compute the area of the rectangle in"}, {"start": 44.52, "duration": 3.24, "text": "terms of its endpoints we see the"}, {"start": 46.02, "duration": 4.26, "text": "rectangle has a length of x to the K"}, {"start": 47.76, "duration": 4.68, "text": "minus 1 so that the area enclosed is a"}, {"start": 50.28, "duration": 4.5, "text": "times x to the K minus a divided by x"}, {"start": 52.44, "duration": 4.139, "text": "minus 1. by Computing the area of this"}, {"start": 54.78, "duration": 3.18, "text": "rectangle in two separate ways we've"}, {"start": 56.579, "duration": 4.101, "text": "produced a formula for a finite"}, {"start": 57.96, "duration": 2.72, "text": "geometric sum"}], "title": "Finite Geometric Sum Formula", "description": "This is a short, animated visual proof demonstrating the finite geometric for any ratio x with x greater than 1. This series (and its infinite analog when x is less than 1) is important for many results in calculus, discrete mathematics, and combinatorics.\n\nIf you like this video, consider subscribing to the channel or consider buying me a coffee: https://www.buymeacoffee.com/VisualProofs. Thanks!\n\nFor a wide format, slower, wordless version of this video check out https://youtu.be/ROovDaRQVl0\n\nAlso, check out my playlist on geometric sums/series: https://youtube.com/playlist?list=PLZh9gzIvXQUsgw8W5TUVDtF0q4jEJ3iaw\n\nThis animation is based on a proof by both Carlos G. Spaht and Craig M. Johnson from the March 2001 issue of The College Math Journal page 109 (https://doi.org/10.2307/2687114).\n\n#mathshorts\u200b #mathvideo\u200b #math\u200b #calculus #mtbos\u200b #manim\u200b #animation\u200b #theorem\u200b #pww\u200b #proofwithoutwords\u200b #visualproof\u200b #proof\u200b #iteachmath #geometricsums #series #finitesums #finiteseries #geometric \n\nTo learn more about animating with manim, check out:\nhttps://manim.community", "lengthSeconds": 59, "width": 360, "height": 640}
{"video_id": "vbG_YBTiN38", "transcript": [{"start": 0.0, "duration": 2.0, "text": "\uc73c"}, {"start": 34.04, "duration": 4.039, "text": "\uc800\ud76c"}, {"start": 35.09, "duration": 2.989, "text": "\uc608 5"}], "title": "Pythagoras\u2019 theorem, an animated explanation!", "description": "(a^2)+(b^2) does indeed equal (c^2) !!!\nA visual proof!\n\nTechnical info:\nComputer Generated motion graphics, created in Adobe After effects.\n\nCredit:\nSound effects sourced from:\npacdv.com\nfreewavesamples.com\nsoundbible.com", "lengthSeconds": 40, "width": 640, "height": 360}
{"video_id": "YOA09clho1Q", "transcript": [{"start": 0.06, "duration": 34.76, "text": "[\u0645\u0648\u0633\u064a\u0642\u0649]"}], "title": "Visual Proof of the Pythagorean Theorem | 2", "description": "Here's a famous visual proof of the Pythagorean theorem.\n#maths #visualization #geometry #algebra #python #proof #visualproof \n\nVisualization Tool: Manim library, Python", "lengthSeconds": 33, "width": 360, "height": 640}
{"video_id": "0ggGQ96eaiE", "transcript": [{"start": 0.48, "duration": 3.68, "text": "let's see two trigonometric facts proved"}, {"start": 2.44, "duration": 4.56, "text": "visually in 2"}, {"start": 4.16, "duration": 4.72, "text": "minutes let's draw a right triangle"}, {"start": 7.0, "duration": 4.12, "text": "where one of the angles is labeled by"}, {"start": 8.88, "duration": 4.2, "text": "the Greek letter Theta assume the"}, {"start": 11.12, "duration": 3.84, "text": "hypotenuse has a length of one and the"}, {"start": 13.08, "duration": 4.68, "text": "opposite side then has a length of s of"}, {"start": 14.96, "duration": 5.96, "text": "theta the complimentary angle is Pi / 2"}, {"start": 17.76, "duration": 5.4, "text": "minus Theta we can take a second copy of"}, {"start": 20.92, "duration": 5.519, "text": "this triangle and reflect it creating a"}, {"start": 23.16, "duration": 5.48, "text": "larger isos triangle as shown here this"}, {"start": 26.439, "duration": 4.401, "text": "triangle has an angle 2 Theta and the"}, {"start": 28.64, "duration": 5.28, "text": "opposite side length is then 2 * the S"}, {"start": 30.84, "duration": 6.039, "text": "of theta we can use this triangle in two"}, {"start": 33.92, "duration": 7.72, "text": "different ways first of all from the law"}, {"start": 36.879, "duration": 9.32, "text": "of ss we see that s of 2 thet / 2 sin of"}, {"start": 41.64, "duration": 8.2, "text": "thet must equal the S ofun / 2 - thet /"}, {"start": 46.199, "duration": 7.121, "text": "1 this means that sin of 2 thet = 2 sin"}, {"start": 49.84, "duration": 6.28, "text": "of thet * the sin ofk 2us thet but the S"}, {"start": 53.32, "duration": 5.759, "text": "ofk 2 - thet is equal to the cosine of"}, {"start": 56.12, "duration": 5.959, "text": "thet therefore we see that sin of 2 thet"}, {"start": 59.079, "duration": 5.841, "text": "= 2 * * the S of thet * the cosine of"}, {"start": 62.079, "duration": 5.321, "text": "theta now let's apply the law of cosin"}, {"start": 64.92, "duration": 6.72, "text": "to the triangle pictured from the law of"}, {"start": 67.4, "duration": 9.84, "text": "cosin we see that 2 * the S of th^ 2 = 1"}, {"start": 71.64, "duration": 8.64, "text": "2 + 1^ 2 - 2 * 1 * 1 * the cosine of 2"}, {"start": 77.24, "duration": 5.32, "text": "thet dividing this equality by two and"}, {"start": 80.28, "duration": 7.32, "text": "isolating the cosine of 2 Theta we see"}, {"start": 82.56, "duration": 7.16, "text": "that cosine of 2 thet = 1 - 2 * sin^2 th"}, {"start": 87.6, "duration": 3.72, "text": "we have used one triangle diagram along"}, {"start": 89.72, "duration": 4.439, "text": "with with the law of ss and the law of"}, {"start": 91.32, "duration": 5.56, "text": "cosin to prove that sin of 2 thet = 2 *"}, {"start": 94.159, "duration": 6.28, "text": "the sin of thet * cosine thet and cosine"}, {"start": 96.88, "duration": 4.96, "text": "of 2 thet = 1 - 2 * the sin^2 of thet"}, {"start": 100.439, "duration": 4.601, "text": "these are known as the double angle"}, {"start": 101.84, "duration": 5.04, "text": "identities for S and cosine implicit in"}, {"start": 105.04, "duration": 3.88, "text": "this diagram is that the angle Theta is"}, {"start": 106.88, "duration": 4.96, "text": "acute do you think the identities hold"}, {"start": 108.92, "duration": 4.44, "text": "when Theta is obtuse can you prove it is"}, {"start": 111.84, "duration": 4.639, "text": "there a visual proof showing the double"}, {"start": 113.36, "duration": 3.119, "text": "angle identities in that"}, {"start": 118.84, "duration": 3.0, "text": "case"}], "title": "A proof to remember: Double Angle Formulas I (visual proof)", "description": "This is a short, animated visual proof of the Double angle identities for sine and cosine. To get the formulas we employ the Law of Sines and the Law of Cosines to an isosceles triangle created by from a reflected copy of a given triangle.\n\nIf you like this video, consider subscribing to the channel or consider buying me a coffee: https://www.buymeacoffee.com/VisualProofs. Thanks! \n\nThis animation is the classic visual proof the fact but most likely not the one Euclid used.\n\nFor visual proofs of the Law of Sines and Law of Cosines, check out:\nhttps://youtu.be/TU0043SuGsM\nhttps://youtu.be/NHxJ3Z_58Lw or\nhttps://youtu.be/vNCe7ciFwVc\n\nThis animation is based on a visual proof by Sidney H. Kung  from the March 1996 issue of The College Mathematics Journal (https://www.jstor.org/stable/2687407 - page 155).\n\n#mathshorts #mathvideo #math #trigonometry #lawofsines #lawofcosines  #triangle #manim #animation #theorem #pww #proofwithoutwords #visualproof #proof #circle #obtuseangle #acuteangle #angle #trigidentities #identity #righttriangle #isoscelestriangle #doubleangle \n\nTo learn more about animating with manim, check out:\nhttps://manim.community", "lengthSeconds": 120, "width": 640, "height": 360}
{"video_id": "vJTchOUHT7M", "transcript": [{"start": 11.76, "duration": 16.739, "text": "[\u0645\u0648\u0633\u064a\u0642\u0649]"}, {"start": 30.08, "duration": 14.62, "text": "\u200fh"}, {"start": 31.64, "duration": 13.06, "text": "[\u0645\u0648\u0633\u064a\u0642\u0649]"}], "title": "#trending #mathematics Visual Solve Algebraic Equation 2^x=8-x^2", "description": "#trending #mathematics Visual Proof: Factors of Quadratic Equation 2^x=8-x^2\n\nOther Videos\nFibonacci Sequence, Golden Ratio, How to draw Golden Spiral:\nhttps://youtu.be/yHBa3EvDPco\nRadius and Tangent Vectors: \nhttps://youtube.com/shorts/KHSafRET3Sk\nDe Morgan's laws: Proof by Venn Diagrams:\nhttps://youtube.com/shorts/pR-REo4aBYk?feature=share\nVisual Proof of Roots of Quadratic Equation\nhttps://youtube.com/shorts/FdE7rHDZRd8", "lengthSeconds": 43, "width": 360, "height": 640}
{"video_id": "8Q5HsVrcu6g", "transcript": [{"start": 0.0, "duration": 3.959, "text": "consider the function f of x equals x"}, {"start": 2.04, "duration": 3.719, "text": "squared where the output is obtained by"}, {"start": 3.959, "duration": 3.181, "text": "multiplying the input by itself the"}, {"start": 5.759, "duration": 3.781, "text": "graph of this function looks like this"}, {"start": 7.14, "duration": 3.78, "text": "if we plot a point on the curve we can"}, {"start": 9.54, "duration": 3.3, "text": "draw a line called the tangent line"}, {"start": 10.92, "duration": 3.66, "text": "which looks like the curve locally then"}, {"start": 12.84, "duration": 3.48, "text": "we can compute the slope of this line at"}, {"start": 14.58, "duration": 3.3, "text": "that point and now let's compute the"}, {"start": 16.32, "duration": 3.42, "text": "slopes of the tangent lines at every"}, {"start": 17.88, "duration": 3.659, "text": "point along the curve like this we see"}, {"start": 19.74, "duration": 3.299, "text": "that as we move along the curve the"}, {"start": 21.539, "duration": 4.08, "text": "slope of the associated tangent line"}, {"start": 23.039, "duration": 4.98, "text": "changes we can go one step further and"}, {"start": 25.619, "duration": 4.08, "text": "create a new set of axes on this set of"}, {"start": 28.019, "duration": 3.18, "text": "axes we're going to plot the y"}, {"start": 29.699, "duration": 3.36, "text": "coordinate as the slope for the"}, {"start": 31.199, "duration": 3.721, "text": "corresponding x coordinates in the curve"}, {"start": 33.059, "duration": 4.441, "text": "above when we do that we see a linear"}, {"start": 34.92, "duration": 4.799, "text": "function map out in particular the slope"}, {"start": 37.5, "duration": 5.46, "text": "function for f of x equals x squared is"}, {"start": 39.719, "duration": 5.641, "text": "the line y equals 2x we use the notation"}, {"start": 42.96, "duration": 4.259, "text": "F Prime of x equals 2x for this slope"}, {"start": 45.36, "duration": 4.26, "text": "function and say the derivative of x"}, {"start": 47.219, "duration": 4.32, "text": "squared is 2x because the derivative"}, {"start": 49.62, "duration": 3.599, "text": "function is the function that outputs"}, {"start": 51.539, "duration": 3.961, "text": "the slopes of the tangent lines to"}, {"start": 53.219, "duration": 3.721, "text": "verify this fact requires more work but"}, {"start": 55.5, "duration": 3.98, "text": "at least we can see that the formula"}, {"start": 56.94, "duration": 2.54, "text": "holds visually"}], "title": "Visual derivative of x squared", "description": "A visual of the derivative of f(x)=x squared. We show how to think about the derivative of a function visually. \n\n#manim #calculus #derivatives #derivative #tangentline #slope #parabola #mathvideo #mathshorts #math #visualmath #graph #parabola #linearapproximation", "lengthSeconds": 58, "width": 360, "height": 640}
{"video_id": "YVj-BgN2hd0", "transcript": [{"start": 13.28, "duration": 5.6, "text": "now it is a first"}, {"start": 15.68, "duration": 6.96, "text": "position where there are two squares"}, {"start": 18.88, "duration": 3.76, "text": "with area a square and B"}, {"start": 24.519, "duration": 6.76, "text": "Square now it is a second position"}, {"start": 32.96, "duration": 4.119, "text": "where two triangles are"}, {"start": 34.64, "duration": 5.16, "text": "selected whose sides are ab and"}, {"start": 37.079, "duration": 7.48, "text": "hypotenuse is"}, {"start": 39.8, "duration": 4.759, "text": "C now it is a third"}, {"start": 47.36, "duration": 8.039, "text": "position where triangles are"}, {"start": 50.68, "duration": 4.719, "text": "moved to the new position"}, {"start": 69.88, "duration": 6.44, "text": "check by sliders triangles are moving as"}, {"start": 74.0, "duration": 2.32, "text": "Arrow"}, {"start": 84.04, "duration": 6.719, "text": "shows it is a third"}, {"start": 87.28, "duration": 5.24, "text": "position where figure is"}, {"start": 90.759, "duration": 4.561, "text": "moved to one"}, {"start": 92.52, "duration": 5.32, "text": "square with area C"}, {"start": 95.32, "duration": 5.2, "text": "sare first there were two"}, {"start": 97.84, "duration": 3.88, "text": "squares now it is a single square with"}, {"start": 100.52, "duration": 3.959, "text": "area C"}, {"start": 101.72, "duration": 7.84, "text": "sare therefore theorem is"}, {"start": 104.479, "duration": 5.081, "text": "proved a square + B sare is = to"}, {"start": 109.88, "duration": 9.239, "text": "c\u00b2 if you like Please Subscribe it share"}, {"start": 114.799, "duration": 4.32, "text": "it like it"}, {"start": 121.92, "duration": 7.08, "text": "subscribe"}, {"start": 123.88, "duration": 5.12, "text": "it thanks for watching"}], "title": "Pythagoroas Theorem ,Pythagoroas  Theorem Proof, Mathematical Visual proof ,Animation In Geogebra", "description": "Thanks for whatching\nkataria math animation,\nContact Me :-  rattanlalkataria@gmail.com  \nOn YouTube :- @KatariaMathAnimationOfficial\n Your Queries :-\n Kataria Math Animation\nPythagoroas Theorem ,Pythagoroas  Theorem Proof, Mathematical Visual proof ,Animation In Geogebra\nExterior Angles Theorem\nExterior Angles Properties\nProve Exterior Angles Of Triangle = Sum of  Opposite Interior Angles\nMaths Animation\nTeaching of  Maths\nLearning of Maths\nSum Of angles of triangles\n#pythagorastheorem \nGeometry\n Algebra\nGeoGebra\nAnimation\nEasy Maths\nOn Line Teaching\nMath Teacher\nTriangles\nSquare\nQuadrilateral \nProperty of Triangles\nProperty of Circle\nArea\nVolume\nFind Nets of Solid Figure\nStudent's favorite method\nBefore classroom \nIn The Classroom \n#Geogebra\n#Angles\n#Sumofangles\n#favorite \n#Method\n#Maths\n#GeoGebra\n#viralmath\n#onlineteaching\n#Quiz For Kids Visual Addition\n#Quiz On Geogebra#\n# Visual Math\n# Animation\nPythagoroas Theoram \nEasy Math\n Mathematical Visual proof \n Only 2 Minutes\n#mathvisualproofs\n#upsc \n#nda \n#neet \n#ssc \n#ibps \n#maths \n#pstet \n#mathshorts \n#GeoGebra\n#viralmath\n#onlineteaching\n#Quiz For Kids Visual Addition\n#matholympiad \n#gate \n#iit \n#cat \n#Quiz On Geogebra#\n# Visual Math\n# Animation\nPythagoroas Theoram \nEasy Math\n Mathematical Visual proof \n Only 2 Minutes\n#mathvisualproofs\n#upsc \n#nda \n#neet \n#ssc \n#ibps \n#maths \n#pstet \n#mathshorts \n#GeoGebra\n#viralmath\n#onlineteaching\n#Quiz For Kids Visual Addition\n#matholympiad \n#gate \n#iit \n#cat \n#Quiz On Geogebra#\n# Visual Math\n# Animation\nPythagoroas Theoram \nEasy Math\n Mathematical Visual proof \n Only 2 Minutes\n#mathvisualproofs#upsc \n#nda \n#neet \n#ssc \n#ibps \n#maths \n#pstet \n#mathshorts \n#GeoGebra\n#viralmath\n#onlineteaching\n#Quiz For Kids Visual Addition\n#matholympiad \n#gate \n#iit \n#cat \n#Quiz On Geogebra#\n# Visual Math\n# Animation\nPythagoroas Theoram \nEasy Math\n Mathematical Visual proof \n Only 2 Minutes\n#mathvisualproofs\n#upsc \n#nda \n#neet \n#ssc \n#ibps \n#maths \n#pstet \n#mathshorts \n#GeoGebra\n#viralmath\n#onlineteaching\n#Quiz For Kids Visual Addition\n#matholympiad \n#gate \n#iit \n#cat \n#Quiz On Geogebra#\n# Visual Math\n# Animation\nPythagoroas Theoram \nEasy Math\n Mathematical Visual proof \n Only 2 Minutes\n#mathvisualproofs", "lengthSeconds": 148, "width": 640, "height": 360}
{"video_id": "m7YLVtlghfI", "transcript": [{"start": 0.08, "duration": 3.319, "text": "let X be a positive real number and"}, {"start": 1.88, "duration": 4.519, "text": "assume that X is greater than or equal"}, {"start": 3.399, "duration": 5.001, "text": "to 1x consider a right triangle with one"}, {"start": 6.399, "duration": 3.401, "text": "leg of length 2 and the other leg of"}, {"start": 8.4, "duration": 3.92, "text": "length x -"}, {"start": 9.8, "duration": 4.16, "text": "1X as this is a right triangle we can"}, {"start": 12.32, "duration": 3.6, "text": "use the Pythagorean theorem to find the"}, {"start": 13.96, "duration": 3.96, "text": "length of the hypotenuse in terms of the"}, {"start": 15.92, "duration": 4.16, "text": "legs in this case we see that the"}, {"start": 17.92, "duration": 4.84, "text": "hypotenuse length is the < TK of 2^2"}, {"start": 20.08, "duration": 6.119, "text": "plus the quantity x - 1x^2 and this"}, {"start": 22.76, "duration": 5.359, "text": "simplifies to x + 1 overx but in any"}, {"start": 26.199, "duration": 3.801, "text": "right triangle the hypotenuse is longer"}, {"start": 28.119, "duration": 3.881, "text": "than the leg lengths and so in this"}, {"start": 30.0, "duration": 4.719, "text": "situation we see that the quantity x +"}, {"start": 32.0, "duration": 4.76, "text": "1X is greater than or equal to 2 this"}, {"start": 34.719, "duration": 3.921, "text": "inequality is true for any positive real"}, {"start": 36.76, "duration": 4.4, "text": "number X because we can interchange X"}, {"start": 38.64, "duration": 5.919, "text": "and 1 /x can you see from the diagram"}, {"start": 41.16, "duration": 3.399, "text": "when equality holds"}], "title": "Sum of positive number and its reciprocal from a right triangle", "description": "This is a short, animated visual proof demonstrating that sum of a positive real number and its reciprocal is always greater than or equal to 2. \n\nTo buy me a coffee, head over to https://www.buymeacoffee.com/VisualProofs \nThanks!\n\nFor a wider format and slightly longer version of this proof, see https://youtu.be/zBUK8C6wSqs?si=dQIql-H09Q-G1Xt7\n\nIt turns out that this theorem is equivalent to the Arithmetic Mean-Geometric Mean inequality. The equivalence is implied by the fact that his video resembles a similar visual proof for the AM-GM inequality found here: https://youtu.be/62G9fak1vyk\n\nThis animation is based on a proof by Roger B. Nelsen from the December 1994 issue of Mathematics Magazine, page 374 (https://doi.org/10.2307/2690999).\n\n#math\u200b #manim\u200b #visualproof\u200b #mathvideo\u200b  #geometry #mathshorts\u200b  #algebra #mtbos\u200b  #animation\u200b #theorem\u200b #pww\u200b #proofwithoutwords\u200b #proof\u200b #iteachmath #inequality #amgm #maths #mathematics #education \n\nTo learn more about animating with manim, check out:\nhttps://manim.community", "lengthSeconds": 43, "width": 360, "height": 640}
{"video_id": "of-nEuChxCE", "transcript": [{"start": 0.12, "duration": 3.88, "text": "let me show you a geometric"}, {"start": 1.839, "duration": 6.321, "text": "visualization of the classic difference"}, {"start": 4.0, "duration": 9.12, "text": "of squares identity which says x^2 - y^2"}, {"start": 8.16, "duration": 10.24, "text": "always equals x + y * x - y here's an X"}, {"start": 13.12, "duration": 11.56, "text": "by x\u00b2 and if I cut a y by y\u00b2 out of the"}, {"start": 18.4, "duration": 9.24, "text": "corner the area left is X2 - y^ 2 these"}, {"start": 24.68, "duration": 7.92, "text": "little lengths are y with these longer"}, {"start": 27.64, "duration": 10.64, "text": "ones being x - y Y and if I cut this"}, {"start": 32.6, "duration": 11.0, "text": "rectangular strip off of the side making"}, {"start": 38.28, "duration": 9.88, "text": "this side now a whole X and that an x -"}, {"start": 43.6, "duration": 11.279, "text": "y this x - y fits perfectly with that"}, {"start": 48.16, "duration": 9.719, "text": "one to add together to an x - y * x + y"}, {"start": 54.879, "duration": 3.0, "text": "rectangle"}], "title": "A Visual Proof of the \"Difference of Squares\" Rule... #shorts", "description": "This short is a few months old. (It was previously on my other channel but most of you haven't seen it. Sorry if you already have)... Make sure you've also seen my latest main @ComboClass  episode about numerical opposites here: https://youtu.be/VNH_PaUsnJE?si=4s8eySoXyrLljU0J\n\nSpecial thanks to Evan Clark and to all of my Patreon supporters:\nMax, George Carozzi, Peter Offut, Tybie Fitzhugh, Henry Spencer, Mitch Harding, YbabFlow, Joseph Rissler, Plenty W, Quinn Moyer, Julius 420, Philip Rogers, Ilmori Fajt, Brandon, August Taub, Ira Sanborn, Matthew Chudleigh, Cornelis Van Der Bent, Craig Butz, Mark S, Thorbjorn M H, Mathias Ermatinger, Edward Clarke, Christopher Masto, Joshua S, Joost Doesberg, Adam, Chris Reisenbichler, Stan Seibert, Izeck, Beugul, OmegaRogue, Florian, William Hawkes, Michael Friemann, Claudio Fanelli, The Green Way, Julian Zassenhaus, Bailey Douglass, Jan Bosenberg, Brooks Boutwell, David Irvine, qe, George Sharabidze, Jack Dwyer, Fredrik, and Dave Brondsema!.. And if I'm missing any more recent names, thanks to you too, and I'll update this list soon :)\nTo join that list of people supporting this channel, and get some cool bonus content, check out the Combo Class Patreon at https://www.patreon.com/comboclass\n\nIf you want to send me anything (including old extra clocks/dice/etc.), here's a private mailbox address which I check every couple weeks for mail from viewers. If you're going to send anything, please watch this short video first: https://youtu.be/_-gSvbvO8W4\nDomotro\n1442 A Walnut Street, Box # 401 \nBerkeley, CA 94709\n\nLinks:\nMain channel: www.youtube.com/comboclass\nPatreon: https://www.patreon.com/comboclass\nDiscord: https://discord.gg/cHHvDcPPuc\nSubreddit: https://www.reddit.com/r/comboClass/\n\nIf you want to reach out to possibly collaborate or help with Combo Class somehow, my email is: combouniversity (at) gmail (dot) com\n\nCombo Class, taught by me (Domotro), is a crazy educational show where you can have fun learning rare things about math, science, language, and more! This is actually kind of my \"bonus\" channel for my livestreams, shorts, and extra content - make sure you're also subscribed to the main @ComboClass channel where my main episodes go!\n\nDISCLAIMER: any use of fire, tools, eating unexpected things, or other dangerous-seeming actions in this series are experiments that are conducted safely and professionally, and/or are accidents that I include in the videos for educational purposes. Do NOT try to copy any experiments (or accidents) that you see in these videos yourself.", "lengthSeconds": 56, "width": 270, "height": 480}
{"video_id": "OWR6QCPYjqI", "transcript": [{"start": 2.76, "duration": 6.78, "text": "[Music]"}, {"start": 5.819, "duration": 5.761, "text": "suppose that a b c and d are all real"}, {"start": 9.54, "duration": 4.38, "text": "numbers then we can build a right"}, {"start": 11.58, "duration": 4.08, "text": "triangle whose leg lengths are given by"}, {"start": 13.92, "duration": 3.6, "text": "the absolute value of B times the"}, {"start": 15.66, "duration": 4.08, "text": "absolute value of c and the absolute"}, {"start": 17.52, "duration": 5.22, "text": "value of a times the absolute value of C"}, {"start": 19.74, "duration": 4.92, "text": "if we assume that c is non-zero then we"}, {"start": 22.74, "duration": 3.779, "text": "can scale this triangle by the value"}, {"start": 24.66, "duration": 4.08, "text": "absolute value of D over the absolute"}, {"start": 26.519, "duration": 3.901, "text": "value of C producing this triangle which"}, {"start": 28.74, "duration": 3.539, "text": "has leg lengths absolute value of B"}, {"start": 30.42, "duration": 3.659, "text": "times absolute value of D and absolute"}, {"start": 32.279, "duration": 4.381, "text": "value of a times absolute value of D"}, {"start": 34.079, "duration": 4.32, "text": "because this new triangle is a scaled"}, {"start": 36.66, "duration": 3.899, "text": "and rotated copy of the original right"}, {"start": 38.399, "duration": 4.621, "text": "triangle when the two triangles are"}, {"start": 40.559, "duration": 4.68, "text": "placed side by side like this they"}, {"start": 43.02, "duration": 4.5, "text": "produce a right angle so that this"}, {"start": 45.239, "duration": 4.5, "text": "picture triangle is a right triangle as"}, {"start": 47.52, "duration": 4.5, "text": "well we can use the Pythagorean theorem"}, {"start": 49.739, "duration": 3.84, "text": "on the original two right triangles to"}, {"start": 52.02, "duration": 3.539, "text": "see that the leg lengths of this new"}, {"start": 53.579, "duration": 3.361, "text": "right triangle are the absolute value of"}, {"start": 55.559, "duration": 3.121, "text": "D times the square root of a squared"}, {"start": 56.94, "duration": 3.24, "text": "plus b squared and the absolute value of"}, {"start": 58.68, "duration": 3.719, "text": "c times the square root of a squared"}, {"start": 60.18, "duration": 3.78, "text": "plus b squared because this new triangle"}, {"start": 62.399, "duration": 3.001, "text": "is also right angled we can use the"}, {"start": 63.96, "duration": 3.3, "text": "Pythagorean theorem to find the length"}, {"start": 65.4, "duration": 3.719, "text": "of the high hotness to be the square"}, {"start": 67.26, "duration": 3.3, "text": "root of a squared plus b squared times"}, {"start": 69.119, "duration": 3.36, "text": "the square root of c squared plus D"}, {"start": 70.56, "duration": 4.32, "text": "Squared but now we know that the"}, {"start": 72.479, "duration": 4.441, "text": "absolute value of AC plus b d is less"}, {"start": 74.88, "duration": 4.26, "text": "than or equal to the absolute value of a"}, {"start": 76.92, "duration": 4.62, "text": "times C plus the absolute value of B"}, {"start": 79.14, "duration": 4.5, "text": "times D from the triangle inequality on"}, {"start": 81.54, "duration": 4.439, "text": "the other hand from this trapezoidal"}, {"start": 83.64, "duration": 4.38, "text": "shape we see that the base length which"}, {"start": 85.979, "duration": 4.081, "text": "is the absolute value of a times C plus"}, {"start": 88.02, "duration": 4.26, "text": "the absolute value of B times D must be"}, {"start": 90.06, "duration": 4.26, "text": "less than or equal to the length of the"}, {"start": 92.28, "duration": 3.42, "text": "hypotenuse of that top triangle which is"}, {"start": 94.32, "duration": 2.88, "text": "the square root of a squared plus b"}, {"start": 95.7, "duration": 3.599, "text": "squared times the square root of c"}, {"start": 97.2, "duration": 4.62, "text": "squared plus D Squared therefore the"}, {"start": 99.299, "duration": 4.14, "text": "absolute value of AC plus b d is less"}, {"start": 101.82, "duration": 3.36, "text": "than or equal to the square root of a"}, {"start": 103.439, "duration": 4.32, "text": "squared plus b squared times the square"}, {"start": 105.18, "duration": 5.759, "text": "root of c squared plus D Squared where a"}, {"start": 107.759, "duration": 5.581, "text": "b c and d are real numbers this is often"}, {"start": 110.939, "duration": 5.96, "text": "referred to as the two variable Koshi"}, {"start": 113.34, "duration": 3.559, "text": "bonyakowski Schwartz inequality"}, {"start": 118.38, "duration": 2.18, "text": "foreign"}], "title": "Cauchy-Bunyakovsky-Schwarz Inequality II (visual proof)", "description": "This is a short, animated visual proof of the two-dimensional Cauchy-Schwarz inequality (sometimes called Cauchy\u2013Bunyakovsky\u2013Schwarz inequality) using \"Garfield's trapezoid\" and scaling of right triangles. \n\nIf you like this video, consider subscribing to the channel or consider buying me a coffee: https://www.buymeacoffee.com/VisualProofs. Thanks!\n\nTo see an alternate proof of this fact, check out\nhttps://youtu.be/mKg_gVagHy8 (wordless)\nhttps://youtube.com/shorts/Xy97t7l6MYg (shorts version)\n\nThis animation is based on a visual proof by Claudi Alsina and Roger Nelsen from the April 2015 issue of Mathematics Magazine (https://www.jstor.org/stable/10.4169/math.mag.88.2.144 - page 144-145).\n\n#math\u200b #inequality \u200b #manim\u200b #animation\u200b #theorem\u200b #pww\u200b #proofwithoutwords\u200b #visualproof\u200b #proof\u200b #iteachmath #algebra #areas #mathematics #cauchyschwarz #algebraicidentity #mathshorts\u200b #mathvideo\u200b #mtbos \n\nTo learn more about animating with manim, check out:\nhttps://manim.community\n_____________________________\nMusic in this video:\nWhen We Found The Horizon by Late Night Feeler", "lengthSeconds": 124, "width": 640, "height": 360}
{"video_id": "gnGxF0XcGZ8", "transcript": [{"start": 0.52, "duration": 5.0, "text": "let's see four iconic proofs without"}, {"start": 2.56, "duration": 5.16, "text": "words in under 60 seconds by using two"}, {"start": 5.52, "duration": 4.999, "text": "triangular arrays of unit squares to"}, {"start": 7.72, "duration": 5.16, "text": "form an N by n +1 rectangle we see that"}, {"start": 10.519, "duration": 6.481, "text": "the sum of the first n positive integers"}, {"start": 12.88, "duration": 5.88, "text": "is equal to n * n + 1 / 2 next by using"}, {"start": 17.0, "duration": 4.16, "text": "four congruent right triangles to"}, {"start": 18.76, "duration": 4.4, "text": "partially fill a square diagram and then"}, {"start": 21.16, "duration": 4.6, "text": "measuring empty space in two different"}, {"start": 23.16, "duration": 6.16, "text": "ways by moving Triangles around we see"}, {"start": 25.76, "duration": 5.32, "text": "that c^2 = A2 + B2 where A and B are the"}, {"start": 29.32, "duration": 5.0, "text": "legs of a right triangle with hypotenuse"}, {"start": 31.08, "duration": 5.799, "text": "C third by rotating the bottom half of"}, {"start": 34.32, "duration": 4.88, "text": "odd Stacks around their central element"}, {"start": 36.879, "duration": 4.721, "text": "we obtain a square showing that the sum"}, {"start": 39.2, "duration": 6.12, "text": "of the first n positive odd integers is"}, {"start": 41.6, "duration": 5.92, "text": "n s and finally by taking a 1x one unit"}, {"start": 45.32, "duration": 4.48, "text": "square and repeatedly shading half of"}, {"start": 47.52, "duration": 4.039, "text": "the unshaded remaining area we get a"}, {"start": 49.8, "duration": 4.239, "text": "formula for the infinite sum of the"}, {"start": 51.559, "duration": 6.32, "text": "powers of 1/2 in particular this"}, {"start": 54.039, "duration": 3.84, "text": "infinite sum is one"}], "title": "Top four visual proofs?", "description": "In this short, we show animations of four of the most famous proofs without words: the formula for the sum of the first n integers; the pythagorean theorem using negative space/sliding rectangles/Chou pei suan ching; the formula for the sum of the first n odd integers; and the infinite geometric series of positive powers of 1/2 (using a rectangle/square dissection of a unit area square). We include brief justifications for these visual proofs.\n\nIf you like this video, consider subscribing to the channel or consider buying me a coffee: https://www.buymeacoffee.com/VisualProofs. Thanks!\n\nFor a slightly different version of this short without words, see:\nhttps://youtube.com/shorts/eG72HxNceEs\n\nFor longer videos with related animations, see\nSumming Integers: https://youtu.be/eHbtc50-qXo\nPythagorean Theorem:  https://youtu.be/o2mbN452ywA (and old version: https://youtu.be/yfGtbNgcrQ8)\nSumming Odds: https://youtu.be/I4A2T55k0cg\nGeometric Series: https://youtu.be/JteQEN1XPyc\n\nThe first proof was known to the ancient greeks (cited by Martin Gardner), the second proof is adapted from the Chou pei suan ching (around 200 BCE according to Roger Nelsen), the third is attributed to Nicomachus of Gerasa by Roger Nelsen, and the final one is attributed to Warren Page (from  the September 1981 issue of Mathematics Magazine, page 201 - https://www.jstor.org/stable/2689632 ). The first three can all be found in Roger Nelsen's first compendium, \"Proofs Without Words: Exercises in Visual Thinking: https://bookstore.ams.org/view?ProductCode=CLRM/1 .\n\n#math #manim #pythagoreantheorem #pythagorean #triangle #animation #theorem #pww #proofwithoutwords #visualproof #proof  #mathshorts #mathvideo #geometricseries #infiniteseries #finitesums #infinitesums #oddsumformula #sumsofintegers #sums #shorts \n\nTo learn more about animating with manim, check out:\nhttps://manim.community", "lengthSeconds": 56, "width": 360, "height": 640}
{"video_id": "-eltBKAANCk", "transcript": [{"start": 0.0, "duration": 3.72, "text": "we can make sense of the powers of three"}, {"start": 1.92, "duration": 3.899, "text": "by starting with one triangle taking"}, {"start": 3.72, "duration": 3.72, "text": "three copies to form this triangle with"}, {"start": 5.819, "duration": 3.36, "text": "three small triangles taking three"}, {"start": 7.44, "duration": 4.32, "text": "copies of that to represent three"}, {"start": 9.179, "duration": 4.141, "text": "squared and so on each time taking three"}, {"start": 11.76, "duration": 3.539, "text": "copies of the previous diagram to"}, {"start": 13.32, "duration": 3.9, "text": "produce the next power of three if we do"}, {"start": 15.299, "duration": 4.021, "text": "that here's a representation for three"}, {"start": 17.22, "duration": 3.78, "text": "to the N plus one but now let's see"}, {"start": 19.32, "duration": 3.78, "text": "something interesting we can take two"}, {"start": 21.0, "duration": 3.96, "text": "copies of three to the n and stack them"}, {"start": 23.1, "duration": 3.599, "text": "next to each other on top of that take"}, {"start": 24.96, "duration": 4.319, "text": "two copies of three to the N minus one"}, {"start": 26.699, "duration": 4.5, "text": "like this and keep going until we have"}, {"start": 29.279, "duration": 3.96, "text": "two copies of three squared on top of"}, {"start": 31.199, "duration": 4.681, "text": "that we place two copies of three and"}, {"start": 33.239, "duration": 5.16, "text": "finally we place two copies of one if we"}, {"start": 35.88, "duration": 4.859, "text": "then stack one last copy of one on top"}, {"start": 38.399, "duration": 4.381, "text": "it turns out that we produce the exact"}, {"start": 40.739, "duration": 3.721, "text": "same diagram as the diagram for three to"}, {"start": 42.78, "duration": 3.48, "text": "the N plus one so we get equality"}, {"start": 44.46, "duration": 4.38, "text": "between these two formulas from this"}, {"start": 46.26, "duration": 4.56, "text": "equality we see that the sum where I"}, {"start": 48.84, "duration": 3.96, "text": "ranges from zero to n of three to the I"}, {"start": 50.82, "duration": 4.259, "text": "must be three to the N plus one minus"}, {"start": 52.8, "duration": 5.96, "text": "one over two this is a formula for the"}, {"start": 55.079, "duration": 3.681, "text": "geometric sum of the powers of three"}], "title": "Summing Powers of Three", "description": "This is a short, animated visual proof demonstrating the sums of finite geometric series of powers of 3.\n\nIf you like this video, consider subscribing to the channel or consider buying me a coffee: https://www.buymeacoffee.com/VisualProofs. Thanks! \n\nHere is a longer, wide format version: Geometric series: sums of powers of 3 (visual proof)\nhttps://youtu.be/9IAm75UY2U8\n\n\nThis animation is based on a visual proof by David B. Sher from the Spring 1997 issue of Mathematics and Computer Education page 190.\n\n#mathshorts\u200b #mathvideo\u200b #math\u200b #calculus #mtbos\u200b #manim\u200b #animation\u200b #theorem\u200b #pww\u200b #proofwithoutwords\u200b #visualproof\u200b #proof\u200b #iteachmath #geometricsums #series #fractals #finitesums #induction \n\nTo learn more about animating with manim, check out:\nhttps://manim.community", "lengthSeconds": 57, "width": 360, "height": 640}
{"video_id": "xF1QNlmV7zw", "transcript": [{"start": 0.399, "duration": 3.761, "text": "have you ever wondered that how can you"}, {"start": 1.88, "duration": 5.76, "text": "replace sin x by simply X for smaller"}, {"start": 4.16, "duration": 4.88, "text": "values of X well those zoman expansion"}, {"start": 7.64, "duration": 3.76, "text": "might know that you can just ignore the"}, {"start": 9.04, "duration": 4.599, "text": "higher powers of X but here is a visual"}, {"start": 11.4, "duration": 5.28, "text": "proof for it if you plot the function"}, {"start": 13.639, "duration": 4.72, "text": "sinx and the function y = x then you can"}, {"start": 16.68, "duration": 3.519, "text": "see that in this region both of these"}, {"start": 18.359, "duration": 3.92, "text": "curves almost overlap each other that"}, {"start": 20.199, "duration": 4.041, "text": "means that no matter what input you give"}, {"start": 22.279, "duration": 4.041, "text": "in this region the output of both of"}, {"start": 24.24, "duration": 4.16, "text": "these functions is almost equal to each"}, {"start": 26.32, "duration": 4.359, "text": "other and you can do the same thing for"}, {"start": 28.4, "duration": 4.76, "text": "more functions like tangent X sin"}, {"start": 30.679, "duration": 4.521, "text": "inverse X tangent inverse X and even sin"}, {"start": 33.16, "duration": 3.719, "text": "hyperbolic X and tangent hyperbolic X"}, {"start": 35.2, "duration": 3.08, "text": "behaves the same if you're interested in"}, {"start": 36.879, "duration": 3.921, "text": "such science and math video subscribe"}, {"start": 38.28, "duration": 2.52, "text": "for more"}], "title": "Visual proof Of Sin(x) \u2248 x", "description": "Visualization of sin(x) = x for smaller angles proof using Maclaurin series expansion and graphical method to visualise it.", "lengthSeconds": 39, "width": 360, "height": 640}
{"video_id": "v75E_yfX3l4", "transcript": [{"start": 0.08, "duration": 3.119, "text": "start with this 2x2 matrix a and"}, {"start": 2.08, "duration": 4.36, "text": "consider the apine linear"}, {"start": 3.199, "duration": 4.881, "text": "Transformations ax and ax + 1 Z these"}, {"start": 6.44, "duration": 4.159, "text": "two Transformations are shown here"}, {"start": 8.08, "duration": 4.519, "text": "applied to the plane now start with any"}, {"start": 10.599, "duration": 4.08, "text": "point in the plane and apply both"}, {"start": 12.599, "duration": 4.041, "text": "Transformations then choose one point at"}, {"start": 14.679, "duration": 4.68, "text": "random and apply both Transformations"}, {"start": 16.64, "duration": 4.559, "text": "again repeat this process over and over"}, {"start": 19.359, "duration": 3.641, "text": "selecting one of the outputs at random"}, {"start": 21.199, "duration": 3.641, "text": "and applying both apine linear"}, {"start": 23.0, "duration": 5.439, "text": "Transformations let's repeat this"}, {"start": 24.84, "duration": 5.64, "text": "process 25,000 times to see what happens"}, {"start": 28.439, "duration": 3.24, "text": "each point is colorcoded yellow or blue"}, {"start": 30.48, "duration": 3.68, "text": "depending on the apine linear"}, {"start": 31.679, "duration": 4.72, "text": "transformation that outputs that point"}, {"start": 34.16, "duration": 4.64, "text": "as we iterate this procedure the points"}, {"start": 36.399, "duration": 4.401, "text": "stay within this enclosed region this"}, {"start": 38.8, "duration": 4.04, "text": "fractal shape is known as the Twin"}, {"start": 40.8, "duration": 4.239, "text": "Dragon and it's the limiting shape or"}, {"start": 42.84, "duration": 4.0, "text": "the attractor of this iterated function"}, {"start": 45.039, "duration": 5.121, "text": "system produced by these two apine"}, {"start": 46.84, "duration": 3.32, "text": "linear transformations"}], "title": "Twin Dragon from Chaos Game", "description": "In this video, we show how to use a random process of iteratively applying two (affine) linear transformations in the real plane to generate a 2-rep-tile known as the Twin Dragon as the attractor of the iterated function system. What happens if you do something similar with different affine linear transformations?\n\nHere is a bit longer version (without words): https://youtu.be/fFIyqMiycs4\n\nCheck out these videos for related constructions:\nhttps://youtube.com/shorts/dklWNdM9WSg\nhttps://youtu.be/rLxx6b-oJGo\nhttps://youtu.be/NBI_5GdvqUo\n\nIf you like this video, consider subscribing to the channel or consider buying me a coffee: https://www.buymeacoffee.com/VisualProofs. Thanks!\n\nThis animation was inspired by an article written by Lorelei Koss that appeared in the Proceedings of Bridges 2015: https://archive.bridgesmathart.org/2015/bridges2015-423.pdf . For more information, you should also check out the related article \"Fractal Tilings in the Plane\" by Richard Darst, Judith Palagallo and Thomas Price from the February 1998 Mathematics Magazine: https://www.jstor.org/stable/2691339\n\n#chaos #chaosgame #twindragon #twindragon #mathvideo\u200b #math\u200b #mtbos\u200b #manim\u200b #animation\u200b #theorem\u200b\u200b\u200b #iteachmath #mathematics #dynamicalsystems  #iteratedfunctionsystem #dynamics #fractals \n\nTo learn more about animating with manim, check out:\nhttps://manim.community", "lengthSeconds": 49, "width": 360, "height": 640}
{"video_id": "X-uesNEHv7U", "transcript": [{"start": 0.0, "duration": 4.5, "text": "many people draw this diagram when they"}, {"start": 2.46, "duration": 4.26, "text": "want to represent a four set Venn"}, {"start": 4.5, "duration": 4.08, "text": "diagram unfortunately this is not valid"}, {"start": 6.72, "duration": 2.82, "text": "can you figure out what's missing from"}, {"start": 8.58, "duration": 3.54, "text": "this diagram"}, {"start": 9.54, "duration": 5.4, "text": "but we can actually draw a valid four"}, {"start": 12.12, "duration": 5.1, "text": "set Venn diagram here's one possible way"}, {"start": 14.94, "duration": 4.02, "text": "to draw such a diagram we can check that"}, {"start": 17.22, "duration": 3.719, "text": "it's valid by finding all the regions"}, {"start": 18.96, "duration": 4.92, "text": "here's the regions for elements only in"}, {"start": 20.939, "duration": 5.281, "text": "a the ones only in B the ones only in C"}, {"start": 23.88, "duration": 4.08, "text": "and the ones only in D next are the"}, {"start": 26.22, "duration": 4.02, "text": "double intersection regions here's the"}, {"start": 27.96, "duration": 4.74, "text": "region for elements only in A and B this"}, {"start": 30.24, "duration": 4.68, "text": "one for those only in a and C this one"}, {"start": 32.7, "duration": 5.22, "text": "for those in A and D this one for those"}, {"start": 34.92, "duration": 5.1, "text": "in B and C only and in B and D only and"}, {"start": 37.92, "duration": 3.9, "text": "this one for those in C and D only next"}, {"start": 40.02, "duration": 5.34, "text": "we have the triple intersection regions"}, {"start": 41.82, "duration": 6.18, "text": "a b and c only a b and d only a c and d"}, {"start": 45.36, "duration": 4.679, "text": "only and b c and d only here's the"}, {"start": 48.0, "duration": 3.539, "text": "region for elements in all four sets and"}, {"start": 50.039, "duration": 3.781, "text": "here's the region for elements in none"}, {"start": 51.539, "duration": 4.981, "text": "of the four sets we've pictured all 16"}, {"start": 53.82, "duration": 4.879, "text": "possible regions so this is a valid Venn"}, {"start": 56.52, "duration": 2.179, "text": "diagram"}], "title": "Four Set Venn Diagram?", "description": "In this video, we show a typical non-example of a proposed four-set Venn diagram. We then show an example of a four-set Venn Diagram by highlighting all sixteen possible regions. \n\nIf you like this video, consider subscribing to the channel or consider buying me a coffee: https://www.buymeacoffee.com/VisualProofs. Thanks!\n\nHere is a longer, wordless video, showing the same thing along with a five set Venn diagram as well : https://youtu.be/WBot4kqc3ro\n\nThe image in this video was inspired by the wonderful article \"Venn Diagrams and Independent Families of Sets\" by Branko Gr\u00fcnbaum  (https://www.maa.org/sites/default/files/pdf/upload_library/22/Ford/BrankoGrunbaum.pdf)\n\n#logic #settheory #union #propositions  #intersection #setdifference  #setminus #setconnectives #subsets #venndiagram #visualproof #foursets #math #manim #discretemathematics #fivesets #complement\n\nTo learn more about animating with manim, check out:\nhttps://manim.community", "lengthSeconds": 57, "width": 360, "height": 640}
{"video_id": "i1JFx-0mHYM", "transcript": [{"start": 0.16, "duration": 3.519, "text": "here's a visualization of the"}, {"start": 1.68, "duration": 3.599, "text": "intersecting chords theorem"}, {"start": 3.679, "duration": 3.521, "text": "stick around and we'll see a formal"}, {"start": 5.279, "duration": 4.64, "text": "statement of the theorem along with a"}, {"start": 7.2, "duration": 2.719, "text": "visual proof"}, {"start": 10.32, "duration": 4.6, "text": "start with a circle"}, {"start": 12.24, "duration": 5.68, "text": "and draw two chords"}, {"start": 14.92, "duration": 4.84, "text": "inside assume that these two cords"}, {"start": 17.92, "duration": 6.4, "text": "intersect at a point"}, {"start": 19.76, "duration": 8.0, "text": "that then cuts each cord into two pieces"}, {"start": 24.32, "duration": 6.16, "text": "use the two pieces from each cord"}, {"start": 27.76, "duration": 5.759, "text": "as the side lengths of a rectangle"}, {"start": 30.48, "duration": 3.039, "text": "as pictured here"}, {"start": 34.399, "duration": 4.401, "text": "now let's compute the area of these"}, {"start": 36.559, "duration": 5.281, "text": "rectangles"}, {"start": 38.8, "duration": 6.24, "text": "it turns out that no matter how you draw"}, {"start": 41.84, "duration": 6.719, "text": "the chords as long as they intersect the"}, {"start": 45.04, "duration": 8.679, "text": "areas of these two rectangles will be"}, {"start": 48.559, "duration": 5.16, "text": "exactly the same as pictured here"}, {"start": 55.53, "duration": 3.089, "text": "[Music]"}, {"start": 60.0, "duration": 3.52, "text": "the intersecting chords theorem can be"}, {"start": 61.76, "duration": 3.84, "text": "stated as follows"}, {"start": 63.52, "duration": 3.84, "text": "suppose two chords in a circle intersect"}, {"start": 65.6, "duration": 3.76, "text": "at a point dividing one chord into"}, {"start": 67.36, "duration": 3.84, "text": "lengths of a and b and the other chord"}, {"start": 69.36, "duration": 4.079, "text": "into lengths of c and d"}, {"start": 71.2, "duration": 4.88, "text": "then the product a times b equals the"}, {"start": 73.439, "duration": 5.201, "text": "product c times d"}, {"start": 76.08, "duration": 4.32, "text": "let's check out a proof"}, {"start": 78.64, "duration": 4.24, "text": "first construct two triangles by"}, {"start": 80.4, "duration": 4.32, "text": "connecting the chords like this"}, {"start": 82.88, "duration": 5.36, "text": "now there are two angles one in each"}, {"start": 84.72, "duration": 6.56, "text": "triangle subtended by this arc"}, {"start": 88.24, "duration": 3.04, "text": "there's this angle"}, {"start": 91.52, "duration": 3.279, "text": "and this angle"}, {"start": 93.28, "duration": 3.6, "text": "because these angles are subtended by"}, {"start": 94.799, "duration": 3.36, "text": "the same arc they must be congruent"}, {"start": 96.88, "duration": 3.279, "text": "angles"}, {"start": 98.159, "duration": 4.481, "text": "likewise there's an angle in each"}, {"start": 100.159, "duration": 5.041, "text": "triangle subtended by this arc"}, {"start": 102.64, "duration": 4.88, "text": "this one here"}, {"start": 105.2, "duration": 4.4, "text": "and this one in the other triangle"}, {"start": 107.52, "duration": 3.599, "text": "once again because these two angles are"}, {"start": 109.6, "duration": 2.32, "text": "subtended by the same arc they're"}, {"start": 111.119, "duration": 2.561, "text": "congruent"}, {"start": 111.92, "duration": 3.6, "text": "[Music]"}, {"start": 113.68, "duration": 5.2, "text": "but this means that the two picture"}, {"start": 115.52, "duration": 5.279, "text": "triangles that we created are similar"}, {"start": 118.88, "duration": 4.64, "text": "now we can scale the two picture"}, {"start": 120.799, "duration": 4.96, "text": "triangles the top one we can scale by b"}, {"start": 123.52, "duration": 4.48, "text": "and the bottom one we can scale by c and"}, {"start": 125.759, "duration": 4.401, "text": "connect them like this this construction"}, {"start": 128.0, "duration": 4.72, "text": "creates a parallelogram and therefore"}, {"start": 130.16, "duration": 6.0, "text": "the opposite side lengths a times b and"}, {"start": 132.72, "duration": 7.48, "text": "c times d must be equal just as we need"}, {"start": 136.16, "duration": 4.04, "text": "for the intersecting chords theorem"}, {"start": 140.61, "duration": 6.769, "text": "[Music]"}, {"start": 148.879, "duration": 2.08, "text": "you"}], "title": "Intersecting Chords Theorem (visual proof)", "description": "This is a short, animated visual proof demonstrating and proving the intersecting chords theorem from geometry. #manim #math\u200b\u200b #mathshorts\u200b #mathvideo\u200b  #intersectingchords #geometry #chords #mtbos\u200b #manim\u200b #animation\u200b #theorem\u200b #pww\u200b #proofwithoutwords\u200b #visualproof\u200b #proof\u200b #iteachmath #inequality \n\nThis animation is based on a visual proof from Hubert Shutrick that appears on the beautiful math website Cut-the-knot (https://www.cut-the-knot.org/Curriculum/Geometry/GeoGebra/HubertIntersectingChords.shtml)\n\nTo learn more about animating with manim, check out:\nhttps://manim.community", "lengthSeconds": 149, "width": 640, "height": 360}
{"video_id": "vcO5pa7iZOU", "transcript": [{"start": 1.5, "duration": 21.98, "text": "[Music]"}, {"start": 21.14, "duration": 24.03, "text": "thank you"}, {"start": 23.48, "duration": 21.69, "text": "[Music]"}, {"start": 50.82, "duration": 13.93, "text": "[Music]"}, {"start": 67.28, "duration": 3.0, "text": "thank you"}, {"start": 73.14, "duration": 8.03, "text": "thank you"}, {"start": 75.22, "duration": 5.95, "text": "[Music]"}, {"start": 83.49, "duration": 3.28, "text": "[Music]"}, {"start": 89.05, "duration": 8.09, "text": "[Music]"}, {"start": 94.14, "duration": 3.0, "text": "foreign"}, {"start": 98.09, "duration": 9.02, "text": "[Music]"}], "title": "Geometric series: sums of powers of 8 (visual proof)", "description": "This is a short, animated (wordless) visual proof demonstrating the sums of finite geometric series of powers of 8 using cubes. #math #manim\u200b #mathvideo\u200b #calculus #mtbos\u200b\u200b #animation\u200b #theorem\u200b #pww\u200b #proofwithoutwords\u200b #visualproof\u200b #proof\u200b #iteachmath #geometricsums #series #squares #finitesums #induction #discretemath\n\nThis animation is based on a visual proof by Rajib Mukherjee from the December 2021 issue of The Mathematical Intelligencer (https://doi.org/10.1007/s00283-021-10087-5) pages 60-61.\n\nHere are two a related proofs for powers of 3, 4 and 9:\nhttps://youtu.be/9IAm75UY2U8\nhttps://youtu.be/yTpzDEDP090\nhttps://youtu.be/Ch7GFdsc9pQ\n\n____________________________________\nMusic in the video:\nMountain Trail by Vlad Gluschenko | https://soundcloud.com/vgl9\nMusic promoted by https://www.free-stock-music.com\nCreative Commons Attribution 3.0 Unported License\nhttps://creativecommons.org/licenses/by/3.0/deed.en_US", "lengthSeconds": 108, "width": 640, "height": 360}
{"video_id": "Il9w86d9mT8", "transcript": [{"start": 0.04, "duration": 3.359, "text": "let X be a positive real number and"}, {"start": 1.8, "duration": 3.559, "text": "first suppose that X is greater than or"}, {"start": 3.399, "duration": 4.4, "text": "equal to 1 start with a right triangle"}, {"start": 5.359, "duration": 5.001, "text": "with legs of length x and 1 take a"}, {"start": 7.799, "duration": 5.161, "text": "rotated copy and then scale that copy by"}, {"start": 10.36, "duration": 5.0, "text": "the value 1 /x creating a right triangle"}, {"start": 12.96, "duration": 4.0, "text": "with side lengths of 1 and 1 /x the"}, {"start": 15.36, "duration": 4.12, "text": "purple triangle that's pictured has an"}, {"start": 16.96, "duration": 4.96, "text": "area of 1/2 * X while the smaller red"}, {"start": 19.48, "duration": 5.119, "text": "triangle has an area of 1/ 12 *"}, {"start": 21.92, "duration": 5.679, "text": "1/x but now notice that the area of"}, {"start": 24.599, "duration": 5.641, "text": "these two triangles encloses the area of"}, {"start": 27.599, "duration": 5.8, "text": "the 1X one square as shown here and"}, {"start": 30.24, "duration": 4.8, "text": "therefore 1 12 * x + 1 12 * 1/x is"}, {"start": 33.399, "duration": 3.561, "text": "greater than or equal to 1 and"}, {"start": 35.04, "duration": 4.96, "text": "multiplying both sides by 2 we get that"}, {"start": 36.96, "duration": 5.279, "text": "x + 1X is greater than or equal to 2 by"}, {"start": 40.0, "duration": 4.16, "text": "interchanging X and 1 /x we see that"}, {"start": 42.239, "duration": 4.081, "text": "this inequality holds for all positive"}, {"start": 44.16, "duration": 6.28, "text": "real numbers X can you see from the"}, {"start": 46.32, "duration": 4.12, "text": "diagram when equality holds"}], "title": "Sum of positive number and its reciprocal from two triangles", "description": "This is a short, animated visual proof demonstrating that sum of a positive real number and its reciprocal is always greater than or equal to 2. \n\nTo buy me a coffee, head over to https://www.buymeacoffee.com/VisualProofs \nThanks!\n\nFor a wider format and slightly longer version of this proof, see https://youtu.be/BkQvnriVblY?si=cldTUiQAtPuZLvto\n\nIt turns out that this theorem is equivalent to the Arithmetic Mean-Geometric Mean inequality. The equivalence is implied by the fact that his video resembles a similar visual proof for the AM-GM inequality found here: https://youtu.be/62G9fak1vyk\n\nThis animation is based on a proof by Roger B. Nelsen from the December 1994 issue of Mathematics Magazine, page 374 (https://doi.org/10.2307/2690999).\n\n#math\u200b #manim\u200b #visualproof\u200b #mathvideo\u200b  #geometry #mathshorts\u200b  #algebra #mtbos\u200b  #animation\u200b #theorem\u200b #pww\u200b #proofwithoutwords\u200b #proof\u200b #iteachmath #inequality #amgm #maths #mathematics #education \n\nTo learn more about animating with manim, check out:\nhttps://manim.community", "lengthSeconds": 49, "width": 360, "height": 640}
{"video_id": "QK2SJo-AxT4", "transcript": [{"start": 0.56, "duration": 4.16, "text": "let's play the chaos game inside of an"}, {"start": 2.32, "duration": 4.88, "text": "equilateral triangle start with a dot"}, {"start": 4.72, "duration": 4.64, "text": "anywhere inside the triangle now"}, {"start": 7.2, "duration": 3.84, "text": "iteratively add a point one half of the"}, {"start": 9.36, "duration": 4.159, "text": "distance between the current dot's"}, {"start": 11.04, "duration": 3.599, "text": "location and a uniformly randomly chosen"}, {"start": 13.519, "duration": 2.881, "text": "vertex"}, {"start": 14.639, "duration": 4.4, "text": "here are the first few iterations of the"}, {"start": 16.4, "duration": 4.879, "text": "chaos game"}, {"start": 19.039, "duration": 4.881, "text": "but don't just do this a few times let's"}, {"start": 21.279, "duration": 3.881, "text": "do this 30 000 times and see exactly"}, {"start": 23.92, "duration": 6.09, "text": "what happens"}, {"start": 25.16, "duration": 4.85, "text": "[Music]"}, {"start": 34.48, "duration": 4.16, "text": "the limiting shape is the famous fractal"}, {"start": 36.64, "duration": 3.12, "text": "known as the sierpinski triangle"}, {"start": 38.64, "duration": 2.8, "text": "check out the linked video in the"}, {"start": 39.76, "duration": 5.319, "text": "description to find other constructions"}, {"start": 41.44, "duration": 3.639, "text": "of this awesome triangle"}], "title": "Chaos game - Sierpinski triangle", "description": "This is a recreation of a short clip from a long form video showing six different ways to construct the Sierpinski triangle: https://youtu.be/IZHiBJGcrqI\n\nIn this short, we play the chaos game using randomness to find order and create the Sierpinski triangle. Can you explain why this works?\n\n#math #manim #fractal #fractal #sierpinski #chaosgame #chaos #mathshort #visualmath \n\n_____________________\nMusic in this short (a 10 second cut):\nWhen The Lights Go On by Vlad Gluschenko | https://soundcloud.com/vgl9\nMusic promoted by https://www.free-stock-music.com\nCreative Commons Attribution 3.0 Unported License\nhttps://creativecommons.org/licenses/by/3.0/deed.en_US", "lengthSeconds": 44, "width": 360, "height": 640}
{"video_id": "8TDSID0BhwI", "transcript": [{"start": 0.12, "duration": 4.719, "text": "start with a right Isles triangle of"}, {"start": 2.2, "duration": 5.559, "text": "area one we can cut this triangle into"}, {"start": 4.839, "duration": 4.961, "text": "eight congruent right isoceles triangles"}, {"start": 7.759, "duration": 4.521, "text": "if we fill in exactly seven of those"}, {"start": 9.8, "duration": 5.719, "text": "triangles like this then we've shaded an"}, {"start": 12.28, "duration": 5.68, "text": "area of 7/8 now we can take that diagram"}, {"start": 15.519, "duration": 4.881, "text": "and scale it and put it in the last 1/8"}, {"start": 17.96, "duration": 5.04, "text": "therefore shading an extra 7/8 of 1/8 or"}, {"start": 20.4, "duration": 5.48, "text": "7 over 8^ squar if we do this again we"}, {"start": 23.0, "duration": 5.88, "text": "shade 7 over 8 cubed and then 7 over 8"}, {"start": 25.88, "duration": 5.479, "text": "4th and so on as we repeat this process"}, {"start": 28.88, "duration": 5.48, "text": "we keep shading 7/ 8 the N of the"}, {"start": 31.359, "duration": 4.681, "text": "triangle as n increases if we do this"}, {"start": 34.36, "duration": 3.68, "text": "indefinitely meaning we consider a"}, {"start": 36.04, "duration": 4.56, "text": "limiting process we eventually shade the"}, {"start": 38.04, "duration": 5.08, "text": "entire triangle therefore the Shaded"}, {"start": 40.6, "duration": 4.76, "text": "area is also one one interpretation of"}, {"start": 43.12, "duration": 4.48, "text": "this equality is that 7 repeating in"}, {"start": 45.36, "duration": 4.44, "text": "base 8 is equal to 1 another"}, {"start": 47.6, "duration": 4.599, "text": "interpretation is that 7 * the infinite"}, {"start": 49.8, "duration": 4.559, "text": "sum of the powers of 1/8 is 1 so that"}, {"start": 52.199, "duration": 5.16, "text": "the infinite sum of the powers of 1/8 is"}, {"start": 54.359, "duration": 3.0, "text": "17th"}], "title": "0.7777\u2026 = 1 (in base 8)", "description": "This is a short, animated visual proof showing the sum of the infinite geometric series with first term 7/8 and ratio 1/8, which in turn allows us to compute the sum of the series of powers of 1/8 and determine an interesting base 8 representation of 1. \n\nTo buy me a coffee, head over to https://www.buymeacoffee.com/VisualProofs \nThanks!\n\nFor a wide-format wordless version of this geometric series dissection, check out\n\nhttps://youtu.be/wPgwgoj_aKA?si=W12LT6bUYEhgaFNv\n\nThis animation is based on a visual proof from Roger B. Nelsen in the February 2006 issue of Mathematics Magazine (https://doi.org/10.2307/27642904 or https://www.jstor.org/stable/27642904), page 60.\n\n #manim #math\u200b\u200b #mathshorts\u200b #mathvideo\u200b  #mtbos\u200b #manim\u200b #animation\u200b #theorem\u200b #pww\u200b #proofwithoutwords\u200b #visualproof\u200b #proof\u200b #iteachmath #calculus #series #geometricseries #infiniteseries  \n\nTo learn more about animating with manim, check out:\nhttps://manim.community", "lengthSeconds": 56, "width": 360, "height": 640}
{"video_id": "Fq7-mkUafLI", "transcript": [{"start": 0.0, "duration": 3.24, "text": "created ternary branching tree by"}, {"start": 1.979, "duration": 3.78, "text": "starting with a line segment of unit"}, {"start": 3.24, "duration": 4.619, "text": "length scaling the line segment by a"}, {"start": 5.759, "duration": 4.38, "text": "half and rotating by the multiples of 2"}, {"start": 7.859, "duration": 4.021, "text": "pi over three repeat this scaling and"}, {"start": 10.139, "duration": 3.241, "text": "rotating process for each new set of"}, {"start": 11.88, "duration": 3.839, "text": "branches"}, {"start": 13.38, "duration": 4.92, "text": "if we continue this process for as long"}, {"start": 15.719, "duration": 4.621, "text": "as we can and eventually remove all but"}, {"start": 18.3, "duration": 4.62, "text": "the last level of the leaves you should"}, {"start": 20.34, "duration": 4.62, "text": "see an interesting shape appear"}, {"start": 22.92, "duration": 3.96, "text": "to see other amazing ways to construct"}, {"start": 24.96, "duration": 4.94, "text": "the sierpinski triangle check the video"}, {"start": 26.88, "duration": 3.02, "text": "that's Linked In the description"}], "title": "Sierpinski from ternary branching", "description": "This is a recreation of a short clip from a long form video showing six different ways to construct the Sierpinski triangle: https://youtu.be/IZHiBJGcrqI\n\nIn this short, we create the Sierpinski triangle using a ternary branching process. Can you explain why this works?\n\n#math #manim #fractal #fractal #sierpinski #pascaltriangle #pascal #mathshort #visualmath", "lengthSeconds": 30, "width": 360, "height": 640}
{"video_id": "D-x3eACMlCU", "transcript": [{"start": 1.39, "duration": 3.16, "text": "[Music]"}, {"start": 7.57, "duration": 3.16, "text": "[Music]"}, {"start": 14.1, "duration": 3.21, "text": "[Music]"}, {"start": 28.69, "duration": 17.11, "text": "[Music]"}, {"start": 42.8, "duration": 3.0, "text": "um"}, {"start": 47.24, "duration": 4.239, "text": "[Music]"}, {"start": 54.8, "duration": 3.0, "text": "so"}, {"start": 60.87, "duration": 3.2, "text": "[Music]"}, {"start": 66.69, "duration": 3.549, "text": "[Music]"}, {"start": 73.27, "duration": 12.12, "text": "[Music]"}, {"start": 88.4, "duration": 6.86, "text": "[Music]"}, {"start": 103.8, "duration": 6.659, "text": "[Music]"}, {"start": 112.6, "duration": 9.639, "text": "[Music]"}, {"start": 128.0, "duration": 2.08, "text": "you"}], "title": "The Beauty of Math Constants Visualized", "description": "A geometric visualization of 9 major mathematical constants showing the elegance of their derivation from first principles.\n\nSubmission for 3Blue1Brown's Summer of Math Exposition, #SoME1. Stay tuned for my submission this year for #SoME4, it'll be on Algebraic Numbers!\n\nMusic: \"Zero\" - Sebastian Kamae x Aylior", "lengthSeconds": 128, "width": 640, "height": 360}
{"video_id": "ITEvaacpDUU", "transcript": [{"start": 0.0, "duration": 5.66, "text": "[Music]"}, {"start": 10.58, "duration": 8.999, "text": "thank you"}, {"start": 12.69, "duration": 6.889, "text": "[Music]"}, {"start": 22.87, "duration": 7.709, "text": "[Music]"}, {"start": 31.52, "duration": 4.18, "text": "foreign foreign"}], "title": "Pythagorean Theorem | Visual Proof | Geometric Proof | Proof Without WORDs", "description": "Unveil the elegance of geometry in our illuminating YouTube Shorts video, where we present a mesmerizing visual proof of the Pythagorean theorem: c\u00b2 = a\u00b2 + b\u00b2. Through clever animations and intuitive illustrations, we unravel the mystery behind this timeless theorem, showcasing how the relationship between the sides of a right triangle can be proven with sheer geometric insight. Prepare to be captivated as right angles, squares, and triangles harmoniously come together to unveil the profound connection between areas and side lengths. Whether you're a math aficionado or simply curious about the secrets embedded within shapes, this concise video will leave you with a newfound appreciation for the artistry and logic embedded in the realm of mathematics.\n\nThanks for watching! If you found these tips helpful, please like and subscribe to our channel. Share your reviews in the comments below!\n\nLink to playlist \"Scientific WorkPlace Crash Course\":\nhttps://www.youtube.com/playlist?list=PLtxDg_npmNDCrZ4Q6wVPz4gk6c7LlYM6v\n\n#shorts #mathtricks #math #mathematics #mathideas #mathanimation #animation #animated #visualization #geometric #geometricpattern  #funfacts #mathfacts #abstractart #abstractmath #mathvisualization #mathart #geometry #shapes #amazingahorts #amazingfacts #factshorts #intution #intutive #funmath #mathideas #mathworld #magicmath #magicmathtricks #mathmagic #mathzone #spirograph #spirography #spirograhart #spirographdesign #pattren #art #creative #creativity #creativeart  #geometric #proof #proofwithoutword #algebric #algebra #algebricequation #algebricexpression #visualproof #geometricproof #amazingmath  #equation #expression #square #proofing #proofof #proofofconcept #amazingahorts #amazingfacts #factshorts #intution #intutive #funmath #mathideas #mathworld #magicmath #magicmathtricks #mathmagic #mathzone #pythagoras_theorem #pythagoreantheorem #theorem", "lengthSeconds": 35, "width": 360, "height": 640}
{"video_id": "kA-i429r4kE", "transcript": [{"start": 0.599, "duration": 13.681, "text": "\u0938\u0935\u0947\u0930 \u0915\u093e \u092e\u0947\u0930\u0947 \u0924\u0942 \u0938\u0942\u0930\u091c \u0932\u093e\u0917\u0947 \u0924\u0942 \u092e\u0947\u0930\u093e \u0915\u094b\u0908 \u0928\u093e"}, {"start": 8.24, "duration": 11.84, "text": "\u0939\u094b\u0915\u0947 \u092d\u0940 \u0915\u0941\u091b \u0932\u093e\u0917\u0947 \u0924\u0942 \u092e\u0947\u0930\u093e \u0915\u094b\u0908 \u0928\u093e \u0939\u094b\u0915\u0947 \u092d\u0940"}, {"start": 14.28, "duration": 5.8, "text": "\u0915\u0941\u091b \u0932\u093e\u0917\u0947 \u0924\u0942 \u092e\u0947\u0930\u093e \u0915\u094b\u0908 \u0928\u093e"}], "title": "visual proof of sum of first n positive integers\ud83d\ude31\ud83d\udcdd#viral #trending #dsssb #math #shorts", "description": "visual proof of sum of first n positive integers\ud83d\ude31\ud83d\udcdd#viral #trending #dsssb #math #shorts", "lengthSeconds": 18, "width": 360, "height": 640}
{"video_id": "eE4IGCAzmqA", "transcript": [{"start": 0.0, "duration": 4.68, "text": "suppose we have a function f and we draw"}, {"start": 2.28, "duration": 5.099, "text": "the graph y as a function of X like this"}, {"start": 4.68, "duration": 4.499, "text": "now imagine we have an input x equals a"}, {"start": 7.379, "duration": 3.84, "text": "and we want to know what the instant"}, {"start": 9.179, "duration": 4.321, "text": "rate of change of the function at x"}, {"start": 11.219, "duration": 4.321, "text": "equals a is how can we figure this out"}, {"start": 13.5, "duration": 4.38, "text": "one way is to take a separate Point"}, {"start": 15.54, "duration": 4.26, "text": "labeled X and draw the secant line"}, {"start": 17.88, "duration": 3.78, "text": "connecting the two points the slope of"}, {"start": 19.8, "duration": 4.2, "text": "this secant line is the average rate of"}, {"start": 21.66, "duration": 4.5, "text": "change of f between X and A and it's"}, {"start": 24.0, "duration": 4.56, "text": "given by the slope formula f of x minus"}, {"start": 26.16, "duration": 4.26, "text": "F of a over x minus a now here's the"}, {"start": 28.56, "duration": 4.26, "text": "trick we can compute this average rate"}, {"start": 30.42, "duration": 4.44, "text": "of change for any value X except for x"}, {"start": 32.82, "duration": 4.68, "text": "equals a because that would require us"}, {"start": 34.86, "duration": 5.1, "text": "to divide by zero so what we do is we"}, {"start": 37.5, "duration": 4.079, "text": "actually take a limiting value of the"}, {"start": 39.96, "duration": 3.48, "text": "slopes of the secant lines as X"}, {"start": 41.579, "duration": 3.961, "text": "approaches a and this gives us the"}, {"start": 43.44, "duration": 4.68, "text": "instant rate of change of F at x equals"}, {"start": 45.54, "duration": 4.92, "text": "a so we can Define f Prime of a to be"}, {"start": 48.12, "duration": 4.74, "text": "this limit as X approaches a of f of x"}, {"start": 50.46, "duration": 4.2, "text": "minus F of a over x minus a provided the"}, {"start": 52.86, "duration": 3.839, "text": "limit exists we call this the derivative"}, {"start": 54.66, "duration": 3.719, "text": "of F at a and it measures the slope of"}, {"start": 56.699, "duration": 4.52, "text": "the tangent line or the instant rate of"}, {"start": 58.379, "duration": 2.84, "text": "change of F at a"}], "title": "Visual Derivative Definition!", "description": "This video animates the idea behind the derivative of a function. We show how to think about the definition of the derivative of a function visually by using a limiting process of slopes of secant lines to obtain the slope of the tangent line (which measures the instant rate of change of a function at a point).\n\n#manim #calculus #derivatives #derivative #tangentline #slope #parabola #mathvideo #mathshorts #math #visualmath #graph #linearapproximation #secantline #instantrateofchange #averagerateofchange", "lengthSeconds": 59, "width": 360, "height": 640}
{"video_id": "K2zbTf4ysHw", "transcript": [{"start": 1.15, "duration": 95.77, "text": "[\u0645\u0648\u0633\u064a\u0642\u0649]"}], "title": "Olympiad Right Triangle Inequality (Canada 1969 - visual proof)", "description": "This is a short, animated proof without words demonstrating an inequality about the side lengths of a right triangle that was Problem 3 in the 1969 Canadian Mathematical Olympiad. #manim #math #mathvideo #righttriangle  #inequality #mathshorts #geometry  #animation #theorem #pww #proofwithoutwords #visualproof #proof #algebra\u200b\u200b \u200b #proof #mathematics\u200b\u200b #mtbos \n\nIf you like this video, please click \"like\" and consider subscribing and checking out my other videos.\n\nThis animation is based on a visual proof from Roger B. Nelsen's second proofs without words compendium (page 12).\n\nTo learn more about animating with manim, check out:\nhttps://manim.community\n_____________________________\nMusic in this video:\nSprite Star from Saidbysed", "lengthSeconds": 97, "width": 640, "height": 360}
{"video_id": "OD6WBF5lVwA", "transcript": [{"start": 0.0, "duration": 6.08, "text": "let's plot the function f of x equals"}, {"start": 2.22, "duration": 3.86, "text": "sine of X on this axis below"}, {"start": 6.54, "duration": 4.2, "text": "we notice that if we plot a point on"}, {"start": 8.76, "duration": 3.9, "text": "this curve we can draw the linear"}, {"start": 10.74, "duration": 3.899, "text": "approximation or the tangent line to the"}, {"start": 12.66, "duration": 3.6, "text": "curve and we can compute the slope of"}, {"start": 14.639, "duration": 3.661, "text": "this tangent line"}, {"start": 16.26, "duration": 3.9, "text": "we can then move the point along the"}, {"start": 18.3, "duration": 3.78, "text": "curve and we note that the slope of the"}, {"start": 20.16, "duration": 4.619, "text": "corresponding tangent line changes as we"}, {"start": 22.08, "duration": 4.439, "text": "move the point we can see that the slope"}, {"start": 24.779, "duration": 4.561, "text": "of the tangent line ranges between"}, {"start": 26.519, "duration": 5.34, "text": "negative one and one in this picture"}, {"start": 29.34, "duration": 4.02, "text": "so let's take this one step further and"}, {"start": 31.859, "duration": 4.441, "text": "let's plot the slopes of the tangent"}, {"start": 33.36, "duration": 5.219, "text": "lines on a separate axis system as we"}, {"start": 36.3, "duration": 4.68, "text": "move the point along the curve f of x"}, {"start": 38.579, "duration": 4.381, "text": "equals sine X we map out the slopes of"}, {"start": 40.98, "duration": 3.0, "text": "the tangent lines on the second axis"}, {"start": 42.96, "duration": 3.18, "text": "below"}, {"start": 43.98, "duration": 5.399, "text": "it turns out and you can see it visually"}, {"start": 46.14, "duration": 5.46, "text": "that the slope function of f of x equals"}, {"start": 49.379, "duration": 4.441, "text": "sine X is the function f Prime of x"}, {"start": 51.6, "duration": 3.779, "text": "equals cosine X to prove this requires"}, {"start": 53.82, "duration": 3.0, "text": "more work but this is a nice visual"}, {"start": 55.379, "duration": 4.7, "text": "representation of the fact that the"}, {"start": 56.82, "duration": 3.259, "text": "derivative of sine is cosine"}], "title": "Visualizing the derivative of sin(x)", "description": "A visual of the derivative of f(x)=sin(x). We show how to think about the derivative of a function visually. \n\n#manim #calculus #derivatives #derivative #tangentline #slope #parabola #mathvideo #mathshorts #math #visualmath #graph #sinefunction #trigonometry #linearapproximation", "lengthSeconds": 58, "width": 360, "height": 640}
{"video_id": "buT9V5_ro5c", "transcript": [{"start": 0.06, "duration": 4.8, "text": "play the chaos game in a regular hexagon"}, {"start": 2.46, "duration": 4.26, "text": "we apply an iterative algorithm with a"}, {"start": 4.86, "duration": 3.659, "text": "random component start with a DOT"}, {"start": 6.72, "duration": 3.839, "text": "anywhere in the interior of the polygon"}, {"start": 8.519, "duration": 3.961, "text": "and then iteratively add a new Point"}, {"start": 10.559, "duration": 3.721, "text": "two-thirds of the distance between the"}, {"start": 12.48, "duration": 4.02, "text": "current dot and a uniformly randomly"}, {"start": 14.28, "duration": 4.02, "text": "chosen vertex on the hexagon you can"}, {"start": 16.5, "duration": 3.539, "text": "think of rolling a fair six-sided die"}, {"start": 18.3, "duration": 3.66, "text": "and then moving two-thirds of the way"}, {"start": 20.039, "duration": 4.201, "text": "from the dot you're currently at to the"}, {"start": 21.96, "duration": 4.44, "text": "vertex corresponding to the die roll"}, {"start": 24.24, "duration": 4.799, "text": "here's the amazing shape that arises"}, {"start": 26.4, "duration": 5.279, "text": "after applying this process 35"}, {"start": 29.039, "duration": 4.86, "text": "000 times no matter the original Point"}, {"start": 31.679, "duration": 3.961, "text": "location or the random outcomes of the"}, {"start": 33.899, "duration": 3.721, "text": "dice rolls along the way the limiting"}, {"start": 35.64, "duration": 3.48, "text": "shape that you see here is always the"}, {"start": 37.62, "duration": 3.86, "text": "limiting shape that results from this"}, {"start": 39.12, "duration": 2.36, "text": "process"}], "title": "Chaos Game in Hexagon (bestagon?)", "description": "In this short, we show what happens when iterating the procedure of choosing a hexagon vertex at random and moving wo thirds the distance from the current dot to the chosen vertex. \n\nIf you like this video, consider subscribing to the channel or consider buying me a coffee: https://www.buymeacoffee.com/VisualProofs. Thanks!\n\n#chaos #chaosgame #hexagon #mathvideo\u200b #math\u200b #mtbos\u200b #manim\u200b #animation\u200b #theorem\u200b\u200b\u200b #iteachmath #mathematics #dynamicalsystems  #iteratedfunctionsystem #dynamics #fractals \n\nHere's a longer version of this video: https://youtu.be/NBI_5GdvqUo\n\n\nIf you want to know more about the Chaos game, see the following links:\nhttps://en.wikipedia.org/wiki/Chaos_game\nhttps://mathworld.wolfram.com/ChaosGa...\nhttps://math.bu.edu/DYSYS/chaos-game/...\nhttps://youtu.be/kbKtFN71Lfs\nhttps://youtu.be/IGlGvSXkRGI\n\nTo learn more about animating with manim, check out:\nhttps://manim.community", "lengthSeconds": 41, "width": 360, "height": 640}
{"video_id": "Pm6_9fCS0d8", "transcript": [{"start": 1.56, "duration": 3.0, "text": "foreign"}], "title": "(a+b)\u00b2 through Visual Proof. #visualproof #maths #realnumbers #iba #jobs #shorttrick #tricks #jobsea", "description": "", "lengthSeconds": 23, "width": 360, "height": 640}
{"video_id": "DpgS-GF1vI8", "transcript": [{"start": 0.42, "duration": 3.0, "text": "foreign"}, {"start": 8.3, "duration": 5.14, "text": "ER can be represented by a square array"}, {"start": 11.16, "duration": 3.479, "text": "of squares with a side length of 2K Plus"}, {"start": 13.44, "duration": 3.3, "text": "1."}, {"start": 14.639, "duration": 4.021, "text": "and the nth triangular number can be"}, {"start": 16.74, "duration": 5.699, "text": "represented by a triangular array of"}, {"start": 18.66, "duration": 5.22, "text": "dots where the bottom row has n dots"}, {"start": 22.439, "duration": 4.021, "text": "when we multiply these two numbers"}, {"start": 23.88, "duration": 5.219, "text": "together the product can be represented"}, {"start": 26.46, "duration": 4.979, "text": "by a triangular array of square arrays"}, {"start": 29.099, "duration": 6.661, "text": "as pictured here the number of small"}, {"start": 31.439, "duration": 7.441, "text": "squares in the base is 2K plus 1 times n"}, {"start": 35.76, "duration": 5.34, "text": "but now each Square array that's at the"}, {"start": 38.88, "duration": 4.32, "text": "end of one of the rows has a triangular"}, {"start": 41.1, "duration": 4.02, "text": "array with base length K in the upper"}, {"start": 43.2, "duration": 4.4, "text": "right corner which we can swing around"}, {"start": 45.12, "duration": 5.759, "text": "and drop down into place like this"}, {"start": 47.6, "duration": 6.1, "text": "resulting in an array with a base of 2K"}, {"start": 50.879, "duration": 5.401, "text": "plus 1 times n plus k"}, {"start": 53.7, "duration": 4.92, "text": "if we look carefully this array is"}, {"start": 56.28, "duration": 6.599, "text": "almost triangular but it's just missing"}, {"start": 58.62, "duration": 7.4, "text": "a base K triangular array from the top"}, {"start": 62.879, "duration": 5.641, "text": "therefore we see that 2K plus 1 squared"}, {"start": 66.02, "duration": 4.0, "text": "multiplied by the nth triangular number"}, {"start": 68.52, "duration": 5.88, "text": "T sub n"}, {"start": 70.02, "duration": 7.5, "text": "is given by the 2K plus 1 times n plus K"}, {"start": 74.4, "duration": 6.0, "text": "triangular number minus the K triangular"}, {"start": 77.52, "duration": 5.4, "text": "number T sub K therefore the product of"}, {"start": 80.4, "duration": 5.37, "text": "an odd square and a triangular number is"}, {"start": 82.92, "duration": 7.64, "text": "the difference of two triangular numbers"}, {"start": 85.77, "duration": 4.79, "text": "[Music]"}], "title": "Products of Odd Squares and Triangular Numbers (visual proof - updated version)", "description": "This is a short, animated visual proof demonstrating that the product of an odd square and a triangular number is the difference of two triangular numbers. #mathshorts\u200b #mathvideo\u200b #math\u200b #squares #mtbos\u200b #manim\u200b #animation\u200b #theorem\u200b #pww\u200b #proofwithoutwords\u200b #visualproof\u200b #proof\u200b #iteachmath #calculus #triangularnumbers \n\nThis animation is based on a visual proof by Brian Hopkins from the February 2018 issue of Mathematics Magazine (https://doi.org/10.1080/0025570X.2017.1409534).\n\nTo learn more about animating with manim, check out:\nhttps://manim.community\n__________________________________________________________________\nMusic in this video:\nReflections by MK2", "lengthSeconds": 92, "width": 640, "height": 360}
{"video_id": "4ri5pCirFsQ", "transcript": [{"start": 0.16, "duration": 5.199, "text": "in an equilateral triangle with area one"}, {"start": 3.159, "duration": 4.921, "text": "first start by tricing the three sides"}, {"start": 5.359, "duration": 4.641, "text": "of the triangle then bict the three"}, {"start": 8.08, "duration": 3.88, "text": "sides of the triangle as well connect"}, {"start": 10.0, "duration": 3.4, "text": "vertices to opposite Tri sectors like"}, {"start": 11.96, "duration": 3.639, "text": "this and connect the remaining Tri"}, {"start": 13.4, "duration": 5.2, "text": "sectors to the midpoints each of these"}, {"start": 15.599, "duration": 4.561, "text": "six shaded regions makes up exactly 17th"}, {"start": 18.6, "duration": 4.24, "text": "of the triangle as you can see by"}, {"start": 20.16, "duration": 4.959, "text": "rotating here therefore we've shaded 67"}, {"start": 22.84, "duration": 4.08, "text": "of the triangle the center triangle is"}, {"start": 25.119, "duration": 3.801, "text": "therefore 17th of the area of the"}, {"start": 26.92, "duration": 5.199, "text": "original we can repeat this process on"}, {"start": 28.92, "duration": 5.68, "text": "the inner 17th shading 67 of 17th so"}, {"start": 32.119, "duration": 4.561, "text": "shading 6 of 7^ squar repeat this"}, {"start": 34.6, "duration": 4.36, "text": "process over and over again each time"}, {"start": 36.68, "duration": 4.719, "text": "shading 67 of the remaining 1 over 7 to"}, {"start": 38.96, "duration": 3.72, "text": "the N if we consider the infinite sum"}, {"start": 41.399, "duration": 3.201, "text": "which corresponds to the limiting"}, {"start": 42.68, "duration": 3.559, "text": "process eventually the entire triangle"}, {"start": 44.6, "duration": 4.08, "text": "will be shaded and that means that the"}, {"start": 46.239, "duration": 4.361, "text": "infinite sum we have is equal to 1 one"}, {"start": 48.68, "duration": 4.559, "text": "interpretation is that 6 repeating in"}, {"start": 50.6, "duration": 4.52, "text": "base 7 is equal to 1 another one is that"}, {"start": 53.239, "duration": 4.12, "text": "6 * the infinite sum of the powers of"}, {"start": 55.12, "duration": 6.52, "text": "17th is 1 so that the infinite sum of"}, {"start": 57.359, "duration": 4.281, "text": "the powers of 17th is 1 16"}], "title": "0.66666\u2026 = 1 (in base 7)", "description": "This is a short, animated visual proof showing the sum of the infinite geometric series with first term 6/7 and ratio 1/7, which in turn allows us to compute the sum of the series of powers of 1/7 and determine an interesting base 7 representation of 1. \n\nIf you like this video, consider subscribing to the channel or consider buying me a coffee: https://www.buymeacoffee.com/VisualProofs. Thanks! \n\nFor a longer, wordless version of this animation with two other proofs, see https://youtu.be/L-pxcCk3Xi4?si=hCnLn61vVxSOp2b4\n\nAlso, check out my playlist on geometric sums/series: https://youtube.com/playlist?list=PLZh9gzIvXQUsgw8W5TUVDtF0q4jEJ3iaw\n\nThis animation is based on a proof by Stephan Berendonk (2020) from the November 2020 issue of The College Mathematics Journal, (https://doi.org/10.1080/07468342.2020.1830649 p. 385)\n\n#mathshorts\u200b #mathvideo\u200b #math\u200b #calculus #mtbos\u200b #manim\u200b #animation\u200b #theorem\u200b #pww\u200b #proofwithoutwords\u200b #visualproof\u200b #proof\u200b #iteachmath #geometricsums #series #infinitesums #infiniteseries #geometric #geometricseries #equilateraltriangle \n\nTo learn more about animating with manim, check out:\nhttps://manim.community", "lengthSeconds": 60, "width": 360, "height": 640}
{"video_id": "zhriP_4QpJQ", "transcript": [{"start": 2.45, "duration": 6.85, "text": "[Music]"}, {"start": 6.319, "duration": 5.981, "text": "consider a semi-circle and then along"}, {"start": 9.3, "duration": 6.12, "text": "the diameter of the semicircle draw two"}, {"start": 12.3, "duration": 5.04, "text": "smaller tangent semicircles the Shaded"}, {"start": 15.42, "duration": 3.72, "text": "region shown here lying inside the"}, {"start": 17.34, "duration": 4.019, "text": "larger semicircle and outside the"}, {"start": 19.14, "duration": 4.32, "text": "smaller semicircle is called an arbolos"}, {"start": 21.359, "duration": 4.741, "text": "the term arbolos literally means"}, {"start": 23.46, "duration": 5.159, "text": "shoemaker's knife by varying the sizes"}, {"start": 26.1, "duration": 4.8, "text": "of the inner two semicircles we get"}, {"start": 28.619, "duration": 4.44, "text": "various different arbolo shapes all"}, {"start": 30.9, "duration": 4.26, "text": "lying inside the same outer semicircle"}, {"start": 33.059, "duration": 4.561, "text": "here's an amazing theorem about the area"}, {"start": 35.16, "duration": 4.5, "text": "of the arbolos consider the semi chord"}, {"start": 37.62, "duration": 4.08, "text": "from the tangent point of the circles to"}, {"start": 39.66, "duration": 4.559, "text": "the larger semicircle and consider this"}, {"start": 41.7, "duration": 4.8, "text": "circle that has that semicord as its"}, {"start": 44.219, "duration": 5.16, "text": "diameter as we vary the sizes of the"}, {"start": 46.5, "duration": 5.28, "text": "inner semicircles we see that this new"}, {"start": 49.379, "duration": 4.86, "text": "circle's area changes just like the"}, {"start": 51.78, "duration": 4.799, "text": "arbolos area changes the amazing"}, {"start": 54.239, "duration": 5.581, "text": "principle here is that the area of the"}, {"start": 56.579, "duration": 5.64, "text": "arbolos is actually equal to the area of"}, {"start": 59.82, "duration": 4.559, "text": "that Circle that has the semi chord as"}, {"start": 62.219, "duration": 4.141, "text": "its diameter let's see a visual proof"}, {"start": 64.379, "duration": 4.861, "text": "justifying this fact"}, {"start": 66.36, "duration": 4.799, "text": "first let's complete the semicircle so"}, {"start": 69.24, "duration": 3.6, "text": "that it's now a full circle and then"}, {"start": 71.159, "duration": 3.841, "text": "rotate the semi chord down to this"}, {"start": 72.84, "duration": 4.56, "text": "bottom semicircle we can use this"}, {"start": 75.0, "duration": 4.32, "text": "semi-cord to create this triangle which"}, {"start": 77.4, "duration": 3.96, "text": "happens to be a right angled triangle by"}, {"start": 79.32, "duration": 4.32, "text": "thales triangle theorem next we can"}, {"start": 81.36, "duration": 5.399, "text": "decompose this right triangle into these"}, {"start": 83.64, "duration": 5.339, "text": "two smaller right triangles and split"}, {"start": 86.759, "duration": 5.22, "text": "the diagram into three separate pieces"}, {"start": 88.979, "duration": 5.221, "text": "one for each of the right triangles"}, {"start": 91.979, "duration": 4.68, "text": "for each one of these triangles we'll"}, {"start": 94.2, "duration": 4.32, "text": "draw semicircles off the three sides of"}, {"start": 96.659, "duration": 4.381, "text": "the triangle like this"}, {"start": 98.52, "duration": 4.919, "text": "according to the Pythagorean theorem the"}, {"start": 101.04, "duration": 4.439, "text": "sum of the areas of the semicircles off"}, {"start": 103.439, "duration": 4.081, "text": "the legs of the right triangle must be"}, {"start": 105.479, "duration": 4.561, "text": "equal to the area of the semicircle off"}, {"start": 107.52, "duration": 4.68, "text": "the hypotenuse of the right triangle so"}, {"start": 110.04, "duration": 4.98, "text": "in each of these diagrams the blue area"}, {"start": 112.2, "duration": 4.8, "text": "is equal to the pink area but now we can"}, {"start": 115.02, "duration": 4.139, "text": "glue these three diagrams back together"}, {"start": 117.0, "duration": 4.259, "text": "one at a time remembering that pink"}, {"start": 119.159, "duration": 4.381, "text": "areas in blue areas counteract one"}, {"start": 121.259, "duration": 4.5, "text": "another when we do that we see that the"}, {"start": 123.54, "duration": 3.98, "text": "only area surviving is the pink area of"}, {"start": 125.759, "duration": 3.78, "text": "the arbolos and the Blue Area"}, {"start": 127.52, "duration": 4.359, "text": "corresponding to the circle whose"}, {"start": 129.539, "duration": 5.041, "text": "diameter is the semicord but remember"}, {"start": 131.879, "duration": 4.86, "text": "that the sum of the blue areas was equal"}, {"start": 134.58, "duration": 4.32, "text": "to the sum of the pink areas and"}, {"start": 136.739, "duration": 4.5, "text": "therefore the area of the arbolos is in"}, {"start": 138.9, "duration": 4.62, "text": "fact equal to the area of the pictured"}, {"start": 141.239, "duration": 4.561, "text": "Circle no matter how we drew this"}, {"start": 143.52, "duration": 4.439, "text": "diagram thales triangle theorem would"}, {"start": 145.8, "duration": 3.96, "text": "hold for the semi chord in question and"}, {"start": 147.959, "duration": 6.381, "text": "so the area of the arbolos is always"}, {"start": 149.76, "duration": 4.58, "text": "equal to the area of the related Circle"}], "title": "Arbelos Area (visual proof)", "description": "This is a short, animated visual proof proving that the area of an arbelos is equal to the area of the circle with diameter given by the half circle chord obtained from the arbelos construction (from the tangent point of the two interior circles).\n\nIf you like this video, consider subscribing to the channel or consider buying me a coffee: https://www.buymeacoffee.com/VisualProofs. Thanks! \n\nFor other similar visual proofs check these out:\n\nhttps://youtu.be/p3_7bOjiWm8\nhttps://youtu.be/-IKyUjRcFDw\n\nThis animation is based on a visual proof from Roger B. Nelsen that appeared in the April 2002 issue of Mathematics Magazine (https://www.jstor.org/stable/3219152  p. 144)\n\nTo learn more about animating with manim, check out:\nhttps://manim.community\n\n#math\u200b #arbelos #geometry #pythagoreantheorem  #thalestheorem  #mtbos\u200b #manim\u200b #animation\u200b #theorem\u200b #pww\u200b #proofwithoutwords\u200b #visualproof\u200b #proof\u200b #iteachmath #area #circle #geometricconstruction \n_______________________________________\nMusic in this episode:\nDescent by Houses of Heaven", "lengthSeconds": 160, "width": 640, "height": 360}
{"video_id": "Ok5ilUTiN6Y", "transcript": [{"start": 0.21, "duration": 28.65, "text": "[\u97f3\u697d]"}, {"start": 28.96, "duration": 20.269, "text": "38"}, {"start": 32.509, "duration": 16.72, "text": "[\u97f3\u697d]"}], "title": "Sum of first N odd natural numbers  - Visual Proof #maths #visualproof #funlearning", "description": "Visual proof of showing that the sum of first n odd natural number is n^2.\n\n#maths #visualization #sum #series #learning #educational #funlearning", "lengthSeconds": 48, "width": 360, "height": 640}
{"video_id": "M90S9Nl50v0", "transcript": [{"start": 0.0, "duration": 3.36, "text": "one way to represent the sum of the"}, {"start": 1.5, "duration": 3.96, "text": "first n positive integers is with a"}, {"start": 3.36, "duration": 3.84, "text": "triangular array of squares where the"}, {"start": 5.46, "duration": 4.38, "text": "first row has one square and the bottom"}, {"start": 7.2, "duration": 4.56, "text": "row has n squares then we can cut this"}, {"start": 9.84, "duration": 4.98, "text": "triangular array with this diagonal"}, {"start": 11.76, "duration": 5.34, "text": "creating one large triangle and N small"}, {"start": 14.82, "duration": 4.32, "text": "one by one triangles therefore the sum"}, {"start": 17.1, "duration": 3.78, "text": "of the first n positive integers is"}, {"start": 19.14, "duration": 3.719, "text": "given by the area of all these triangles"}, {"start": 20.88, "duration": 4.319, "text": "together the large triangle has an area"}, {"start": 22.859, "duration": 4.381, "text": "of one half N squared and each of the"}, {"start": 25.199, "duration": 4.201, "text": "small triangles has an area of one-half"}, {"start": 27.24, "duration": 3.779, "text": "giving a total combined area of one-half"}, {"start": 29.4, "duration": 3.72, "text": "n therefore the sum of the first n"}, {"start": 31.019, "duration": 4.821, "text": "positive integers is one half N squared"}, {"start": 33.12, "duration": 2.72, "text": "plus one half n"}], "title": "Summing positive integers with triangles", "description": "This is a clip from a longer video showing a dozen ways to sum the first n integers. This proof uses triangular areas to get the famous sum formula. Full video is here : https://youtu.be/eHbtc50-qXo\n\nIf you like this video, consider subscribing to the channel or consider buying me a coffee: https://www.buymeacoffee.com/VisualProofs. Thanks!\n\nThis animation is based on the following source:\n\nIan Richards, Proof without Words: sum of integers, Mathematics Magazine (March 1984 page 104): https://www.maa.org/sites/default/files/Richards31975.pdf.\n\n#manim #math #proof #visualproof #mathshorts #proofwithoutwords #sums #integers #sumformula \n\nTo learn more about animating with manim, check out:\nhttps://manim.community", "lengthSeconds": 34, "width": 360, "height": 640}
{"video_id": "xLQFOBo71w0", "transcript": [{"start": 0.06, "duration": 6.18, "text": "we can represent three finite sets a b"}, {"start": 3.179, "duration": 4.861, "text": "and c by the Venn diagram shown here now"}, {"start": 6.24, "duration": 4.14, "text": "suppose we want to find the cardinality"}, {"start": 8.04, "duration": 5.88, "text": "or the number of elements in the union a"}, {"start": 10.38, "duration": 5.819, "text": "union B Union C to do this we start by"}, {"start": 13.92, "duration": 4.02, "text": "taking the number of elements in a and"}, {"start": 16.199, "duration": 3.781, "text": "adding that to the number of elements in"}, {"start": 17.94, "duration": 4.98, "text": "B and adding that to the number of"}, {"start": 19.98, "duration": 4.62, "text": "elements in C unfortunately this is too"}, {"start": 22.92, "duration": 4.32, "text": "much because we've over counted the"}, {"start": 24.6, "duration": 4.86, "text": "elements in A and B so we remove the"}, {"start": 27.24, "duration": 4.379, "text": "cardinality of a intersect B"}, {"start": 29.46, "duration": 4.56, "text": "likewise we need to remove the"}, {"start": 31.619, "duration": 4.801, "text": "cardinality of a intersect C since we've"}, {"start": 34.02, "duration": 4.379, "text": "counted those elements twice and we need"}, {"start": 36.42, "duration": 3.959, "text": "to subtract the cardinality of B"}, {"start": 38.399, "duration": 4.68, "text": "intersect C since we've counted those"}, {"start": 40.379, "duration": 4.2, "text": "elements twice but now notice that the"}, {"start": 43.079, "duration": 3.541, "text": "elements that are in the intersection of"}, {"start": 44.579, "duration": 4.14, "text": "all three sets have been added three"}, {"start": 46.62, "duration": 4.2, "text": "times and subtracted three times so we"}, {"start": 48.719, "duration": 4.68, "text": "need to re-add the cardinality of a"}, {"start": 50.82, "duration": 4.079, "text": "intersect B intersect C this formula is"}, {"start": 53.399, "duration": 4.64, "text": "known as the three set inclusion"}, {"start": 54.899, "duration": 3.14, "text": "exclusion formula"}], "title": "Visual Inclusion/Exclusion", "description": "In this video, we show how to begin to deal with the general Sum principle for Sets when you have more than two sets involved. Can you see how to keep extending this idea for more and more sets?\n\nIf you like this video, consider subscribing to the channel or consider buying me a coffee: https://www.buymeacoffee.com/VisualProofs. Thanks!\n\nTo see a longer (slower) version of this video, without words, check this out: https://youtu.be/vVZwe3TCJT8\n\nIf you want to see an alternate video, check this: https://youtu.be/wXdoqPMlVRU\n\nTo see more details about this, check this longer video: https://youtu.be/sbIcWGYNA5I\n\n#logic #settheory #inclusion/exclusion #intersection #setdifference  #setminus #setconnectives #subsets #venndiagram #visualproof #math #manim #discretemathematics #sumprinciple \n\nTo learn more about animating with manim, check out:\nhttps://manim.community", "lengthSeconds": 57, "width": 360, "height": 640}
{"video_id": "gY3rw6S6q5I", "transcript": [{"start": 0.27, "duration": 4.17, "text": "[Music]"}, {"start": 1.68, "duration": 5.04, "text": "let's see a geometric way to find this"}, {"start": 4.44, "duration": 4.26, "text": "infinite sum"}, {"start": 6.72, "duration": 4.499, "text": "start with a positive real number X"}, {"start": 8.7, "duration": 4.92, "text": "that's less than one then draw a square"}, {"start": 11.219, "duration": 5.281, "text": "with side length given by the infinite"}, {"start": 13.62, "duration": 4.62, "text": "sum of the powers of X in particular the"}, {"start": 16.5, "duration": 4.26, "text": "side length can be decomposed into a"}, {"start": 18.24, "duration": 4.799, "text": "bunch of line segments of length 1 x x"}, {"start": 20.76, "duration": 3.12, "text": "squared x cubed x to the fourth and so"}, {"start": 23.039, "duration": 3.121, "text": "on"}, {"start": 23.88, "duration": 4.44, "text": "applying the sum formula for an infinite"}, {"start": 26.16, "duration": 3.72, "text": "convergent geometric series we see that"}, {"start": 28.32, "duration": 3.539, "text": "the side length of this Square must be"}, {"start": 29.88, "duration": 4.26, "text": "one over one minus X"}, {"start": 31.859, "duration": 4.021, "text": "now we can use the labeled side lengths"}, {"start": 34.14, "duration": 3.72, "text": "to decompose the square into a bunch of"}, {"start": 35.88, "duration": 3.66, "text": "rectangles first in the lower left"}, {"start": 37.86, "duration": 4.859, "text": "corner we have a one by one square with"}, {"start": 39.54, "duration": 5.58, "text": "area one next to that we have two x by 1"}, {"start": 42.719, "duration": 5.34, "text": "rectangles both containing area X so"}, {"start": 45.12, "duration": 5.16, "text": "together they enclose an area of 2x"}, {"start": 48.059, "duration": 4.5, "text": "on top of those two rectangles we can"}, {"start": 50.28, "duration": 4.68, "text": "shade in a 1 by x squared rectangle and"}, {"start": 52.559, "duration": 3.601, "text": "X by X rectangle and an x squared by one"}, {"start": 54.96, "duration": 3.48, "text": "rectangle"}, {"start": 56.16, "duration": 5.219, "text": "together these three rectangles enclose"}, {"start": 58.44, "duration": 5.459, "text": "an area of 3x squared on top of those"}, {"start": 61.379, "duration": 4.501, "text": "rectangles we can place four rectangles"}, {"start": 63.899, "duration": 4.321, "text": "each with area X cubed and closing an"}, {"start": 65.88, "duration": 4.44, "text": "area of 4X Cubed on top of those"}, {"start": 68.22, "duration": 4.02, "text": "rectangles we have five rectangles each"}, {"start": 70.32, "duration": 4.74, "text": "enclosing an area of x to the fourth for"}, {"start": 72.24, "duration": 4.44, "text": "a total area of five x to the fourth and"}, {"start": 75.06, "duration": 3.96, "text": "then we have these six rectangles"}, {"start": 76.68, "duration": 3.84, "text": "enclosing a total area of 6X to the"}, {"start": 79.02, "duration": 3.0, "text": "fifth"}, {"start": 80.52, "duration": 4.2, "text": "we can continue this process"}, {"start": 82.02, "duration": 5.099, "text": "indefinitely each time we add n"}, {"start": 84.72, "duration": 4.56, "text": "rectangles each enclosing an area of x"}, {"start": 87.119, "duration": 3.841, "text": "to the N minus one but now we can"}, {"start": 89.28, "duration": 4.26, "text": "imagine that this process has been"}, {"start": 90.96, "duration": 5.1, "text": "carried out infinitely often that means"}, {"start": 93.54, "duration": 4.86, "text": "that we're looking at a limiting process"}, {"start": 96.06, "duration": 4.32, "text": "in this infinite limiting process the"}, {"start": 98.4, "duration": 4.859, "text": "entire Square will be shaded by these"}, {"start": 100.38, "duration": 5.16, "text": "rectangles that means that the infinite"}, {"start": 103.259, "duration": 5.461, "text": "sum that we're thinking about must equal"}, {"start": 105.54, "duration": 5.46, "text": "the area of the square but the area of"}, {"start": 108.72, "duration": 3.899, "text": "this Square can be computed just by"}, {"start": 111.0, "duration": 3.479, "text": "taking the product of the two side"}, {"start": 112.619, "duration": 3.78, "text": "lengths which are both one over one"}, {"start": 114.479, "duration": 4.68, "text": "minus X"}, {"start": 116.399, "duration": 6.601, "text": "therefore the infinite sum where n"}, {"start": 119.159, "duration": 7.081, "text": "ranges from 1 to Infinity of n times x"}, {"start": 123.0, "duration": 6.539, "text": "to the N minus 1 must equal the quantity"}, {"start": 126.24, "duration": 6.24, "text": "one over one minus x squared and this is"}, {"start": 129.539, "duration": 6.441, "text": "in fact the formula for the sum of the"}, {"start": 132.48, "duration": 3.5, "text": "differentiated geometric series"}], "title": "General Differentiated Geometric Series | visual proof", "description": "This is a short, animated visual proof computing the sum of a differentiated geometric series with ratio given by positive real number x less than 1. The proof uses a dissection proof of a 1/(1-x) by 1/(1-x) square using the formula for the infinite geometric series with ratio x.\n\nIf you like this video, consider subscribing to the channel or consider buying me a coffee: https://www.buymeacoffee.com/VisualProofs. Thanks!\n\nThis animation is based on a visual proof by Roger B. Nelsen from the September 2001 issue of The College Mathematics Journal (https://www.jstor.org/stable/2687557 p. 257).\n\nFor related videos, see:\nhttps://youtu.be/2d4CTG1uJs8\nhttps://youtu.be/2WuSNveg-1c\n\n#math\u200b #infiniteseries #mtbos\u200b #manim\u200b #animation\u200b #theorem\u200b #pww\u200b #proofwithoutwords\u200b #visualproof\u200b #proof\u200b #iteachmath #calculus #geometricseries #series #mathshorts\u200b #mathvideo\u200b #arithmeticogeometricseries #differentiatedseries\n\n\nTo learn more about animating with manim, check out:\nhttps://manim.community\n\n_______________\nMusic in this video:\nEvent Departure by Silent Partner", "lengthSeconds": 139, "width": 640, "height": 360}
{"video_id": "lL9VN8ELH1M", "transcript": [{"start": 7.16, "duration": 5.019, "text": "suppose you have a square that has a"}, {"start": 10.2, "duration": 4.5, "text": "side length of one"}, {"start": 12.179, "duration": 4.621, "text": "then the diagonal line that cuts through"}, {"start": 14.7, "duration": 3.6, "text": "the square has a length given by the"}, {"start": 16.8, "duration": 2.46, "text": "square root of 2 by the Pythagorean"}, {"start": 18.3, "duration": 3.18, "text": "theorem"}, {"start": 19.26, "duration": 4.019, "text": "here's a related question what if we"}, {"start": 21.48, "duration": 5.219, "text": "start with a regular pentagon with side"}, {"start": 23.279, "duration": 6.061, "text": "length one what's the length x of the"}, {"start": 26.699, "duration": 5.641, "text": "diagonal cutting through this shape"}, {"start": 29.34, "duration": 5.879, "text": "let's find out using a visual proof"}, {"start": 32.34, "duration": 5.219, "text": "first we draw a second diagonal line"}, {"start": 35.219, "duration": 4.86, "text": "like this which creates this shaded"}, {"start": 37.559, "duration": 4.921, "text": "isosceles triangle when we draw another"}, {"start": 40.079, "duration": 4.32, "text": "diagonal like this we create a second"}, {"start": 42.48, "duration": 4.86, "text": "triangle shaded in red that's a"}, {"start": 44.399, "duration": 5.221, "text": "subtriangle of the blue triangle"}, {"start": 47.34, "duration": 4.02, "text": "now if we consider the three pictured"}, {"start": 49.62, "duration": 3.779, "text": "angles where the top one is labeled"}, {"start": 51.36, "duration": 5.4, "text": "Alpha and we draw the circumscribed"}, {"start": 53.399, "duration": 6.061, "text": "circle we see that Alpha is subtended by"}, {"start": 56.76, "duration": 5.16, "text": "an arc and so are the other two pictured"}, {"start": 59.46, "duration": 5.22, "text": "angles with question marks but all three"}, {"start": 61.92, "duration": 5.699, "text": "of those arcs are exactly one-fifth of"}, {"start": 64.68, "duration": 5.88, "text": "the entire circle and therefore each of"}, {"start": 67.619, "duration": 5.281, "text": "the angles must be Alpha this also means"}, {"start": 70.56, "duration": 4.919, "text": "that the other angle that is in the blue"}, {"start": 72.9, "duration": 4.02, "text": "triangle and the red triangle must be 2"}, {"start": 75.479, "duration": 3.721, "text": "times Alpha"}, {"start": 76.92, "duration": 4.44, "text": "but now the four pictured angles give us"}, {"start": 79.2, "duration": 4.919, "text": "the full measure of the triangle so we"}, {"start": 81.36, "duration": 5.7, "text": "see that 5 times Alpha must be 180"}, {"start": 84.119, "duration": 5.581, "text": "degrees and therefore Alpha must be 36"}, {"start": 87.06, "duration": 4.379, "text": "degrees but this then means that both of"}, {"start": 89.7, "duration": 4.98, "text": "the pictured triangles the blue and the"}, {"start": 91.439, "duration": 5.22, "text": "red have 36 degree angles and 272 degree"}, {"start": 94.68, "duration": 4.38, "text": "angles and both of them are therefore"}, {"start": 96.659, "duration": 3.901, "text": "isosceles triangles but it turns out"}, {"start": 99.06, "duration": 3.78, "text": "that we also see we have a third"}, {"start": 100.56, "duration": 4.739, "text": "triangle shaded in green here this"}, {"start": 102.84, "duration": 4.739, "text": "triangle has two 36 degree angles and"}, {"start": 105.299, "duration": 4.14, "text": "therefore it is isosceles as well"}, {"start": 107.579, "duration": 4.201, "text": "using the fact that these three"}, {"start": 109.439, "duration": 4.441, "text": "triangles are isosceles we can see that"}, {"start": 111.78, "duration": 4.199, "text": "this side length here is one so that the"}, {"start": 113.88, "duration": 3.84, "text": "short side length on the red triangle is"}, {"start": 115.979, "duration": 4.5, "text": "x minus 1."}, {"start": 117.72, "duration": 5.64, "text": "but now we can use the fact that the two"}, {"start": 120.479, "duration": 4.74, "text": "triangles shown the red and the blue are"}, {"start": 123.36, "duration": 4.14, "text": "similar as they have exactly the same"}, {"start": 125.219, "duration": 4.26, "text": "angle measurements"}, {"start": 127.5, "duration": 3.959, "text": "the fact that the two triangles are"}, {"start": 129.479, "duration": 4.26, "text": "similar means that the ratio of their"}, {"start": 131.459, "duration": 6.301, "text": "side lengths are equal so that x divided"}, {"start": 133.739, "duration": 6.961, "text": "by one must equal 1 divided by x minus"}, {"start": 137.76, "duration": 5.04, "text": "one now when we clear denominators we"}, {"start": 140.7, "duration": 4.259, "text": "get that x times x minus 1 equals one"}, {"start": 142.8, "duration": 4.98, "text": "and this means that x squared minus x"}, {"start": 144.959, "duration": 5.341, "text": "minus 1 is equal to zero this equation"}, {"start": 147.78, "duration": 4.679, "text": "has two roots and only one of them is"}, {"start": 150.3, "duration": 3.84, "text": "positive which is x equals Phi which is"}, {"start": 152.459, "duration": 4.381, "text": "the golden ratio"}, {"start": 154.14, "duration": 5.04, "text": "therefore the length of the diagonal in"}, {"start": 156.84, "duration": 5.22, "text": "this regular pentagon with side length 1"}, {"start": 159.18, "duration": 4.919, "text": "must be Phi the golden ratio"}, {"start": 162.06, "duration": 4.5, "text": "here's a cool way to use this diagram"}, {"start": 164.099, "duration": 4.081, "text": "let's suppose that the golden ratio is"}, {"start": 166.56, "duration": 4.08, "text": "rational that means that it can be"}, {"start": 168.18, "duration": 4.98, "text": "written as a ratio of two integers say A"}, {"start": 170.64, "duration": 4.679, "text": "over B and let's assume that this ratio"}, {"start": 173.16, "duration": 5.04, "text": "is in lowest terms"}, {"start": 175.319, "duration": 4.5, "text": "the fact that Phi equals A over B means"}, {"start": 178.2, "duration": 3.72, "text": "that there is a regular pentagon like"}, {"start": 179.819, "duration": 3.981, "text": "this where the diagonal is length a and"}, {"start": 181.92, "duration": 4.74, "text": "the side length of the Pentagon is B"}, {"start": 183.8, "duration": 4.9, "text": "once again we can swing the side length"}, {"start": 186.66, "duration": 4.14, "text": "down like this to see that the small"}, {"start": 188.7, "duration": 4.44, "text": "subtriangle created inside the larger"}, {"start": 190.8, "duration": 4.74, "text": "triangle pictured here must have length"}, {"start": 193.14, "duration": 4.56, "text": "a minus B but once again these two"}, {"start": 195.54, "duration": 6.96, "text": "triangles are similar and therefore A"}, {"start": 197.7, "duration": 7.38, "text": "over B must be equal to B over a minus B"}, {"start": 202.5, "duration": 5.28, "text": "but Phi equals A over B so that means"}, {"start": 205.08, "duration": 4.86, "text": "Phi equals B over a minus B and this is"}, {"start": 207.78, "duration": 4.86, "text": "a contradiction because B is less than a"}, {"start": 209.94, "duration": 4.799, "text": "and a minus B is less than b so this"}, {"start": 212.64, "duration": 4.739, "text": "contradicts that we had Phi in lowest"}, {"start": 214.739, "duration": 4.14, "text": "terms when we wrote it as A over B but"}, {"start": 217.379, "duration": 3.541, "text": "that means our original assumption was"}, {"start": 218.879, "duration": 4.621, "text": "wrong so that the golden ratio fee"}, {"start": 220.92, "duration": 4.8, "text": "cannot be written as the ratio of two"}, {"start": 223.5, "duration": 4.94, "text": "integer values and so the golden ratio"}, {"start": 225.72, "duration": 2.72, "text": "is irrational"}], "title": "Golden Ratio is Irrational from a Regular Pentagon (visual proof)", "description": "In this video, we show how to prove that the golden ratio is the length of a diagonal in a regular pentagon with a side length of 1. We do this using similar triangles and angle analysis using the circumscribed circle. As a bonus, we use nearly the same diagram (just scaled appropriately if needed) to prove that the golden ratio is irrational, that is, that it cannot be written as the ratio of two integers.\n\nIf you like this video, consider subscribing to the channel or consider buying me a coffee: https://www.buymeacoffee.com/VisualProofs. Thanks!\n\nThis animation is based on an argument in Roger Nelsen\u2019s book Nuggets of Number Theory (I may receive a commission from Amazon, at no cost to you, if you click the link below):\n\nhttps://amzn.to/3RfbRf2\n\nHere are other videos related to the Golden ratio:\nhttps://www.youtube.com/watch?v=yeHDXdv5KH4\nhttps://www.youtube.com/watch?v=eMMErKcw8yM\nhttps://www.youtube.com/watch?v=1ctRFgWwg5E&t=4s\n\n#goldenratio #irrationalnumbers #realnumbers\u200b #manim\u200b #math\u200b #mtbos\u200b #animation\u200b #theorem\u200b\u200b #visualproof\u200b #proof\u200b #iteachmath #mathematics #irrational #pentagon #proofbycontradiction #algebra #infinitedescent #similartriangles #triangle #trianglesimilarity #similarity\n\nTo learn more about animating with manim, check out:\nhttps://manim.community", "lengthSeconds": 236, "width": 640, "height": 360}
{"video_id": "3Y4hOt_2-VM", "transcript": [{"start": 0.08, "duration": 4.52, "text": "Welcome to our Channel today we are"}, {"start": 2.639, "duration": 5.441, "text": "going to discuss important question from"}, {"start": 4.6, "duration": 6.84, "text": "CBC class 10 mathematics specifically"}, {"start": 8.08, "duration": 10.279, "text": "trometry let us consider a question if 3"}, {"start": 11.44, "duration": 10.64, "text": "c a = 4 check whether 1 - tan a / 1 + t\u00b2"}, {"start": 18.359, "duration": 6.84, "text": "a = cos\u00b2 a - sin"}, {"start": 22.08, "duration": 6.039, "text": "a to solve this let consider a right"}, {"start": 25.199, "duration": 9.561, "text": "angle triangle ABC with the right angle"}, {"start": 28.119, "duration": 12.321, "text": "at B give given that t a given that 3 C"}, {"start": 34.76, "duration": 8.72, "text": "= 4 Rite the equation c a = 4"}, {"start": 40.44, "duration": 8.0, "text": "by3 we know the"}, {"start": 43.48, "duration": 8.719, "text": "relation tan a = 4 by3 since t a equal"}, {"start": 48.44, "duration": 9.2, "text": "to opposite side ided by Adent side we"}, {"start": 52.199, "duration": 8.561, "text": "have tan a = to BC / by AB now let's now"}, {"start": 57.64, "duration": 7.159, "text": "let's substitute the value BC = to 3K"}, {"start": 60.76, "duration": 6.2, "text": "units and a = 4K units where K is a real"}, {"start": 64.799, "duration": 6.401, "text": "positive number we need to calculate"}, {"start": 66.96, "duration": 5.36, "text": "hypotenuse AC using Pythagorean theorem"}, {"start": 71.2, "duration": 6.48, "text": "AC"}, {"start": 72.32, "duration": 9.52, "text": "s a\u00b2 = to a\u00b2 + BC squ after substituting"}, {"start": 77.68, "duration": 7.68, "text": "the value AC = to 5K"}, {"start": 81.84, "duration": 8.56, "text": "unit now we have hypotenuse value so we"}, {"start": 85.36, "duration": 8.439, "text": "can calculate sin a and cos a sin AAL to"}, {"start": 90.4, "duration": 8.28, "text": "opposite side divid hypot which is equal"}, {"start": 93.799, "duration": 7.881, "text": "to BC / AC with 3x 4 similarly cos a"}, {"start": 98.68, "duration": 8.64, "text": "equal to Adent side/ hypotenuse which is"}, {"start": 101.68, "duration": 7.92, "text": "equal to a / AC with value 4 by 5 to"}, {"start": 107.32, "duration": 8.079, "text": "check the given equation let consider"}, {"start": 109.6, "duration": 7.28, "text": "the left hand side LHS 1 - tan S A / 1 +"}, {"start": 115.399, "duration": 4.601, "text": "t s"}, {"start": 116.88, "duration": 9.0, "text": "a substituting the value of T A we got"}, {"start": 120.0, "duration": 5.88, "text": "got 1 - 3x 4\u00b2 1 + 3x"}, {"start": 125.92, "duration": 8.319, "text": "4\u00b2 you simplified 7 by 25 now let us"}, {"start": 130.8, "duration": 6.56, "text": "consider the right hand side rhs cos\u00b2 A-"}, {"start": 134.239, "duration": 8.881, "text": "sin S A substituting the value of cos a"}, {"start": 137.36, "duration": 9.8, "text": "and sin a we got 4x 3\u00b2 - 3x b\u00b2 which is"}, {"start": 143.12, "duration": 7.52, "text": "Al which is simplified to 7 by"}, {"start": 147.16, "duration": 5.96, "text": "25 since LHS equal to rhs"}, {"start": 150.64, "duration": 4.92, "text": "we have proved that given equation is"}, {"start": 153.12, "duration": 6.16, "text": "correct this is the one of the easiest"}, {"start": 155.56, "duration": 3.72, "text": "way to solve this problem"}], "title": "If 3 cot A = 4, check whether (1 \u2013 tan2A)/(1 + tan2A) = cos2 A \u2013 sin2 A or not.", "description": "rigonometry Challenge: Prove the Identity!\n\nIn this video, we'll tackle an intriguing trigonometry problem:\n\nGiven: 3 cot A = 4\n\nProve: (1 - tan\u00b2A) / (1 + tan\u00b2A) = cos\u00b2A - sin\u00b2A\n\nWe'll break down the problem step-by-step, using fundamental trigonometric identities and algebraic manipulations. You'll learn how to simplify complex expressions and arrive at the desired result.\n\nWhy Watch This Video?\n\nClear and concise explanations\nStep-by-step solutions\nVisual aids for better understanding\nTips and tricks to solve similar problems\n#class10maths #trigonometry #cotA #tanA #cosA #sinA #trigonometricidentities #mathstutorial #education #studytips #exampreparation #cbse #ncert #trigonometryproofs #mathproblemsolving\n\nclass 10 maths, trigonometry, cot A, tan A, cos A, sin A, trigonometric identities, math tutorial, education, study tips, exam preparation, cbse, ncert, trigonometry proofs, math problem solving, prove trigonometric identity, class 10 trigonometry problems, trigonometry practice problems, how to solve trigonometry problems, trigonometry tricks, trigonometry formulas\nCBSE Class 10 Maths\nClass 10 Maths\n10th Maths\nMaths Tutorial\nMath Tricks\nCBSE Board\nNCERT Solutions\nExam Preparation\nStudy Tips\nSpecific Topic Keywords:\n\nTrigonometry\nAlgebra\nGeometry\nArithmetic\nStatistics\nProbability\nReal Numbers\nPolynomials\nPair of Linear Equations in Two Variables\nQuadratic Equations\nArithmetic Progressions\nCoordinate Geometry \u00a0 \nIntroduction to Trigonometry \u00a0 \nTrigonometric Identities\nHeights and Distances\nCircles\nConstructions\nAreas Related to Circles\nSurface Areas and Volumes\nStatistics\nProbability\nLong-Tail Keywords:\n\nCBSE Class 10 Maths Chapter-wise Solutions\nCBSE Class 10 Maths NCERT Solutions\nCBSE Class 10 Maths Important Questions\nCBSE Class 10 Maths Previous Year Papers\nCBSE Class 10 Maths Sample Papers\nHow to score good marks in CBSE Class 10 Maths\nBest tips for CBSE Class 10 Maths\nEasy way to learn CBSE Class 10 Maths\nFun way to learn CBSE Class 10 Maths", "lengthSeconds": 157, "width": 640, "height": 360}
{"video_id": "jNx05gXSblE", "transcript": [{"start": 9.44, "duration": 4.56, "text": "let a and b be two positive real numbers"}, {"start": 12.0, "duration": 3.44, "text": "and consider line segments of length a"}, {"start": 14.0, "duration": 3.199, "text": "and b"}, {"start": 15.44, "duration": 4.24, "text": "use these two line segments to create a"}, {"start": 17.199, "duration": 5.441, "text": "circle of diameter a"}, {"start": 19.68, "duration": 5.28, "text": "and a circle of diameter b"}, {"start": 22.64, "duration": 3.92, "text": "then move these two circles together so"}, {"start": 24.96, "duration": 3.12, "text": "that they are tangent"}, {"start": 26.56, "duration": 4.08, "text": "the lengths of the radii of the two"}, {"start": 28.08, "duration": 3.84, "text": "circles are a over two and b over two so"}, {"start": 30.64, "duration": 2.56, "text": "the length of the line segment"}, {"start": 31.92, "duration": 3.92, "text": "connecting the centers of the two"}, {"start": 33.2, "duration": 5.12, "text": "circles is given by the sum a plus b"}, {"start": 35.84, "duration": 4.08, "text": "over two the horizontal line coming from"}, {"start": 38.32, "duration": 3.28, "text": "the center of the smaller circle and"}, {"start": 39.92, "duration": 4.0, "text": "intersecting the diameter of the larger"}, {"start": 41.6, "duration": 4.639, "text": "circle creates a line segment of length"}, {"start": 43.92, "duration": 4.0, "text": "absolute value of a minus b over 2. to"}, {"start": 46.239, "duration": 3.521, "text": "find the length of the line segment that"}, {"start": 47.92, "duration": 3.52, "text": "we just drew that represents the"}, {"start": 49.76, "duration": 3.52, "text": "distance between the x-coordinates of"}, {"start": 51.44, "duration": 3.84, "text": "the centers of the circle we can use the"}, {"start": 53.28, "duration": 4.72, "text": "pythagorean theorem since it forms one"}, {"start": 55.28, "duration": 5.439, "text": "leg of a right triangle"}, {"start": 58.0, "duration": 4.399, "text": "the square of a plus b over two minus"}, {"start": 60.719, "duration": 4.001, "text": "the square of the absolute value of a"}, {"start": 62.399, "duration": 4.641, "text": "minus b over two is equal to the"}, {"start": 64.72, "duration": 5.52, "text": "quantity a times b so the desired length"}, {"start": 67.04, "duration": 3.2, "text": "is the square root of a times b"}, {"start": 70.72, "duration": 4.24, "text": "but then from this diagram we can see"}, {"start": 72.72, "duration": 4.48, "text": "that the value a plus b over 2 which is"}, {"start": 74.96, "duration": 4.0, "text": "the hypotenuse of a right triangle is"}, {"start": 77.2, "duration": 3.44, "text": "greater than or equal to the square root"}, {"start": 78.96, "duration": 3.92, "text": "of a times b which is the length of a"}, {"start": 80.64, "duration": 4.64, "text": "leg of the same right triangle no matter"}, {"start": 82.88, "duration": 4.48, "text": "how we change the sizes of a and b we"}, {"start": 85.28, "duration": 4.64, "text": "see that a plus b over 2 is always the"}, {"start": 87.36, "duration": 4.799, "text": "hypotenuse and square root of a b is"}, {"start": 89.92, "duration": 4.8, "text": "always the leg of a right triangle"}, {"start": 92.159, "duration": 5.28, "text": "they only become equal if a and b are"}, {"start": 94.72, "duration": 4.32, "text": "exactly the same so we see that a plus b"}, {"start": 97.439, "duration": 3.921, "text": "over 2 is equal to the square root of a"}, {"start": 99.04, "duration": 4.0, "text": "b if and only if a equals b the"}, {"start": 101.36, "duration": 3.28, "text": "inequality that we've just proved using"}, {"start": 103.04, "duration": 3.119, "text": "this circle diagram is called the"}, {"start": 104.64, "duration": 4.519, "text": "arithmetic mean geometric mean"}, {"start": 106.159, "duration": 3.0, "text": "inequality"}, {"start": 115.119, "duration": 2.081, "text": "you"}], "title": "AM-GM Inequality IV (visual proof)", "description": "This is a short, animated visual proof demonstrating the arithmetic mean geometric mean inequality using two tangent circles. #mathshorts\u200b #mathvideo\u200b #math\u200b #amgminequality #mtbos\u200b #manim\u200b #animation\u200b #theorem\u200b #pww\u200b #proofwithoutwords\u200b #visualproof\u200b #proof\u200b #iteachmath #calculus #inequality \n\nFor other visual proofs of this fact see:\n\nhttps://youtu.be/Pt8IX_Q5U1A\nhttps://youtu.be/62G9fak1vyk\nhttps://youtu.be/HFGSsPbYc-s\nhttps://youtu.be/SEPkdPWKdO0\n\nThis animation is based on a visual proof from Roland H. Eddy that appeared in the June 1985 issue of the College Mathematics Journal (https://www.jstor.org/stable/2686572)\n\nTo learn more about animating with manim, check out:\nhttps://manim.community\n_______________________________________\nMusic in this episode:\nProcession -- Geographer", "lengthSeconds": 115, "width": 640, "height": 360}
{"video_id": "JRjPsuJ7S-g", "transcript": [{"start": 1.71, "duration": 8.929, "text": "[Music]"}, {"start": 8.559, "duration": 5.2, "text": "start with a right triangle with legs of"}, {"start": 10.639, "duration": 4.241, "text": "length a and b and hypotenuse of length"}, {"start": 13.759, "duration": 2.881, "text": "c"}, {"start": 14.88, "duration": 3.76, "text": "draw the circle containing the triangle"}, {"start": 16.64, "duration": 3.44, "text": "with radius of length c and given by the"}, {"start": 18.64, "duration": 3.84, "text": "hypotenuse"}, {"start": 20.08, "duration": 4.56, "text": "extend the leg labeled a to a full"}, {"start": 22.48, "duration": 5.52, "text": "diameter by adding the radius of length"}, {"start": 24.64, "duration": 4.719, "text": "c and a segment of length c minus a"}, {"start": 28.0, "duration": 3.519, "text": "connecting the lines from the ends of"}, {"start": 29.359, "duration": 4.321, "text": "this diameter to the top of the triangle"}, {"start": 31.519, "duration": 5.841, "text": "like this yields a right angled triangle"}, {"start": 33.68, "duration": 3.68, "text": "according to thaley's triangle theorem"}, {"start": 37.68, "duration": 5.36, "text": "consequently these two triangles sharing"}, {"start": 40.16, "duration": 7.04, "text": "an edge of length b are similar"}, {"start": 43.04, "duration": 7.12, "text": "this implies that c plus a over b"}, {"start": 47.2, "duration": 4.48, "text": "is equal to the quantity b over c minus"}, {"start": 50.16, "duration": 2.879, "text": "a"}, {"start": 51.68, "duration": 4.0, "text": "in turn"}, {"start": 53.039, "duration": 5.52, "text": "this implies that c plus a"}, {"start": 55.68, "duration": 4.48, "text": "times c minus a is equal to b squared"}, {"start": 58.559, "duration": 3.361, "text": "and expanding the product we get c"}, {"start": 60.16, "duration": 4.16, "text": "squared minus a squared is equal to b"}, {"start": 61.92, "duration": 4.16, "text": "squared so that c squared is equal to a"}, {"start": 64.32, "duration": 5.479, "text": "squared plus b squared this is the"}, {"start": 66.08, "duration": 3.719, "text": "pythagorean theorem"}, {"start": 74.14, "duration": 5.28, "text": "[Music]"}, {"start": 80.32, "duration": 2.08, "text": "you"}], "title": "Pythagorean Theorem III (visual proof)", "description": "This is a short, animated visual proof of the Pythagorean theorem (the right triangle theorem) using the semicircle and Thales triangle theorem. This theorem states the square of the hypotenuse of a right triangle is equal to the sum of squares of the two other side lengths. #mathshorts #mathvideo #math #pythagoreantheorem #pythagorean #triangle #manim #animation #theorem #pww #proofwithoutwords #visualproof #proof\n\nThis animation is based on a proof due to Michael Hardy from the November, 1986 issue of The College Math Journal (https://doi.org/10.2307/2686255).\n\nTo learn more about animating with manim, check out:\nhttps://manim.community\n\nFor other proofs of this same fact check out: \nhttps://youtu.be/q28H5NY5UVU\nhttps://youtu.be/yfGtbNgcrQ8", "lengthSeconds": 81, "width": 640, "height": 360}
{"video_id": "I4EFHf0sXbM", "transcript": [{"start": 0.16, "duration": 2.96, "text": "consider the set of positive real"}, {"start": 1.68, "duration": 4.48, "text": "numbers and draw the graph of the"}, {"start": 3.12, "duration": 4.719, "text": "function f ofx = 1 /x on this domain the"}, {"start": 6.16, "duration": 5.92, "text": "derivative of this function is frime of"}, {"start": 7.839, "duration": 6.641, "text": "x = -1 /x^2 if we consider the input x ="}, {"start": 12.08, "duration": 6.039, "text": "1 we see that the point on the graph is"}, {"start": 14.48, "duration": 5.719, "text": "1 1 and the derivative frime of 1 is 1"}, {"start": 18.119, "duration": 5.281, "text": "this means that the tangent line to the"}, {"start": 20.199, "duration": 5.08, "text": "function 1 /x at x = 1 has a slope of -"}, {"start": 23.4, "duration": 5.84, "text": "1 and therefore the equation of the"}, {"start": 25.279, "duration": 7.241, "text": "tangent line is Y = 2 - x because FX ="}, {"start": 29.24, "duration": 5.319, "text": "1X concave up we see that every point on"}, {"start": 32.52, "duration": 4.4, "text": "the tangent line lies below the"}, {"start": 34.559, "duration": 4.921, "text": "corresponding point for the same x value"}, {"start": 36.92, "duration": 5.88, "text": "on the curve 1/x this means that in"}, {"start": 39.48, "duration": 6.239, "text": "general 2 - x is less than or equal to 1"}, {"start": 42.8, "duration": 5.12, "text": "/x by adding X to both sides we see that"}, {"start": 45.719, "duration": 4.881, "text": "for any positive real number x 2 is less"}, {"start": 47.92, "duration": 6.639, "text": "than or equal to 1x + x can you see from"}, {"start": 50.6, "duration": 3.959, "text": "the diagram when equality holds"}], "title": "Sum of a Positive Number and its Reciprocal from Calculus", "description": "This is a short, animated visual proof demonstrating that sum of a positive real number and its reciprocal is always greater than or equal to 2.  \n\nTo buy me a coffee, head over to https://www.buymeacoffee.com/VisualProofs \nThanks!\n\nIt turns out that this theorem is equivalent to the Arithmetic Mean-Geometric Mean inequality. The equivalence is implied by the following two proofs: https://youtu.be/62G9fak1vyk\nhttps://youtu.be/IghOHBl0Do8\n\nAnd here are two alternate ways to prove this fact via the Pythagorean Theorem and triangle areas:\nhttps://youtu.be/zBUK8C6wSqs\nhttps://youtu.be/BkQvnriVblY\n\nThis animation is based on a proof by Roger B. Nelsen from the December 1994 issue of Mathematics Magazine, page 374 (https://doi.org/10.2307/2690999). You can find this here too: https://www.maa.org/sites/default/files/Nelsen99959975.pdf\n\n#math\u200b #manim\u200b #visualproof\u200b #mathvideo\u200b  #geometry #mathshorts\u200b  #algebra #mtbos\u200b  #animation\u200b #theorem\u200b #pww\u200b #proofwithoutwords\u200b #proof\u200b #iteachmath #inequality #calculus #derivative #tangentline #slope #amgm\n\nTo learn more about animating with manim, check out:\nhttps://manim.community", "lengthSeconds": 53, "width": 360, "height": 640}
{"video_id": "72Ssn7ZVjtU", "transcript": [{"start": 0.93, "duration": 10.749, "text": "[Music]"}, {"start": 8.16, "duration": 7.04, "text": "start with a square of side length y"}, {"start": 11.679, "duration": 5.921, "text": "covering an area of y squared"}, {"start": 15.2, "duration": 4.88, "text": "then remove a square from the lower left"}, {"start": 17.6, "duration": 5.12, "text": "corner with side length z so that the"}, {"start": 20.08, "duration": 4.24, "text": "resulting area is y squared minus z"}, {"start": 22.72, "duration": 4.319, "text": "squared"}, {"start": 24.32, "duration": 4.64, "text": "we can cut the resulting nomen in half"}, {"start": 27.039, "duration": 3.921, "text": "along the diagonal creating two"}, {"start": 28.96, "duration": 4.48, "text": "trapezoids"}, {"start": 30.96, "duration": 3.759, "text": "move one of the trapezoids down"}, {"start": 33.44, "duration": 3.279, "text": "rotate it"}, {"start": 34.719, "duration": 4.881, "text": "and then flip it to connect it to the"}, {"start": 36.719, "duration": 5.761, "text": "other trapezoid resulting in a rectangle"}, {"start": 39.6, "duration": 6.32, "text": "the rectangle has side length y minus z"}, {"start": 42.48, "duration": 5.759, "text": "and the other side length is y plus z"}, {"start": 45.92, "duration": 4.639, "text": "therefore the original area of y squared"}, {"start": 48.239, "duration": 4.561, "text": "minus z squared is the same as the"}, {"start": 50.559, "duration": 4.16, "text": "product of y plus z"}, {"start": 52.8, "duration": 3.919, "text": "and y minus z"}, {"start": 54.719, "duration": 3.84, "text": "this formula is known as the difference"}, {"start": 56.719, "duration": 5.041, "text": "of squares formula"}, {"start": 58.559, "duration": 6.401, "text": "we can plug in x plus a over 2 for y and"}, {"start": 61.76, "duration": 6.399, "text": "a over 2 for z so that y plus z is x"}, {"start": 64.96, "duration": 4.0, "text": "plus a and y minus z is x"}, {"start": 68.159, "duration": 3.041, "text": "then"}, {"start": 68.96, "duration": 5.6, "text": "the resulting equation becomes x plus a"}, {"start": 71.2, "duration": 5.919, "text": "over 2 squared minus a over 2 squared is"}, {"start": 74.56, "duration": 3.44, "text": "equal to x plus a times"}, {"start": 77.119, "duration": 3.281, "text": "x"}, {"start": 78.0, "duration": 5.119, "text": "but the latter quantity is just x"}, {"start": 80.4, "duration": 4.88, "text": "squared plus a times x"}, {"start": 83.119, "duration": 5.04, "text": "therefore the quantity x plus a over 2"}, {"start": 85.28, "duration": 5.199, "text": "squared minus a over 2 squared is equal"}, {"start": 88.159, "duration": 4.081, "text": "to x squared plus ax"}, {"start": 90.479, "duration": 2.64, "text": "this equation is known as completing the"}, {"start": 92.24, "duration": 2.32, "text": "square"}, {"start": 93.119, "duration": 2.96, "text": "and therefore we have used the"}, {"start": 94.56, "duration": 3.27, "text": "difference of squares formula to"}, {"start": 96.079, "duration": 13.571, "text": "complete the square"}, {"start": 97.83, "duration": 11.82, "text": "[Music]"}, {"start": 111.84, "duration": 2.08, "text": "you"}], "title": "Completing The Square via the Difference of Squares (visual proof)", "description": "This is a short, animated visual proof showing how to derive the formula for \"completing the square.\" This proof first uncovers the formula for the difference of squares. The resulting algebraic formula can be used to write a polynomial of the form x^2 +ax as a difference of two squares. #mathshorts\u200b #mathvideo\u200b #math\u200b #completingthesquare #mtbos\u200b #manim\u200b #animation\u200b #theorem\u200b #pww\u200b #proofwithoutwords\u200b #visualproof\u200b #proof\u200b #iteachmath #algebra #completingthesquare\n\nThis animation is based on a visual proof by Munir Mahmood from the January 2014 issue of The College Mathematics Journal  (https://doi.org/10.4169/college.math.j.45.1.021) page 21.\n\nHere is another visual proof of the same fact:\nhttps://youtu.be/T0HyWIFbsHQ\n\nTo learn more about animating with manim, check out:\nhttps://manim.community", "lengthSeconds": 112, "width": 640, "height": 360}
{"video_id": "RKAtmeT-g9A", "transcript": [{"start": 0.199, "duration": 3.0, "text": "T"}, {"start": 25.3, "duration": 33.33, "text": "[\u97f3\u697d]"}], "title": "Area of Circle, visual proof", "description": "Watch as we visually demonstrate the fascinating formula for calculating the area of a circle! Using simple geometric shapes and intuitive explanations, this short video breaks down how \u03c0r\u00b2 comes together. Perfect for anyone curious about the magic behind circles and their areas. Like, subscribe, and share for more engaging math insights!\"\n#youtube #viral #math#areaofcircle", "lengthSeconds": 57, "width": 360, "height": 640}
{"video_id": "a41V8L5nNIU", "transcript": [{"start": 0.0, "duration": 3.899, "text": "have you ever wondered about the"}, {"start": 1.38, "duration": 4.74, "text": "infinite sum shown here 1 4 plus 1 4"}, {"start": 3.899, "duration": 4.261, "text": "squared plus one-fourth cubed and so on"}, {"start": 6.12, "duration": 3.72, "text": "one way to make sense of this sum is to"}, {"start": 8.16, "duration": 4.019, "text": "start with an equilateral triangle and"}, {"start": 9.84, "duration": 3.719, "text": "decompose it into four equal pieces then"}, {"start": 12.179, "duration": 3.481, "text": "shade the middle triangle which"}, {"start": 13.559, "duration": 4.14, "text": "represents one-fourth of the area now do"}, {"start": 15.66, "duration": 4.8, "text": "the same thing with the upper triangle"}, {"start": 17.699, "duration": 4.801, "text": "and repeat over and over again each time"}, {"start": 20.46, "duration": 3.96, "text": "shading one-fourth of the remaining"}, {"start": 22.5, "duration": 4.019, "text": "upper triangle so that we're shading"}, {"start": 24.42, "duration": 4.92, "text": "powers of one-fourth"}, {"start": 26.519, "duration": 4.741, "text": "notice that the total shaded area in"}, {"start": 29.34, "duration": 4.14, "text": "this equilateral triangle can be"}, {"start": 31.26, "duration": 4.799, "text": "represented by the infinite sum of the"}, {"start": 33.48, "duration": 4.98, "text": "powers of one-fourth"}, {"start": 36.059, "duration": 3.66, "text": "but now here's the amazing trick this"}, {"start": 38.46, "duration": 3.599, "text": "process has divided the original"}, {"start": 39.719, "duration": 5.701, "text": "equilateral triangle into three pieces"}, {"start": 42.059, "duration": 5.281, "text": "which are all stacks of triangles at"}, {"start": 45.42, "duration": 3.959, "text": "each level the triangles are the same"}, {"start": 47.34, "duration": 3.86, "text": "size so we've divided the equilateral"}, {"start": 49.379, "duration": 4.441, "text": "triangle into three equal pieces"}, {"start": 51.2, "duration": 6.3, "text": "therefore the sum of the powers of"}, {"start": 53.82, "duration": 3.68, "text": "one-fourth is equal to one-third"}], "title": "A visual infinite sum like you\u2019ve never seen!", "description": "This is a short, animated visual proof demonstrating the sum of the infinite series of the powers of 1/4. #shorts\u200b\u200b #math\u200b #calculus #mtbos\u200b #manim\u200b #animation\u200b #theorem\u200b #pww\u200b #proofwithoutwords\u200b #visualproof\u200b #proof\u200b #iteachmath #geometricseries #series #infiniteseries #fractal \n\nTo see the original wordless version of this video, check out: https://youtu.be/8i1xj5ORwUw\n\nHere is a longer form video with lots of related infinite geometric sums: https://youtu.be/JteQEN1XPyc\n\nThis animation is based on a visual proof by Rick Mabry from the February 1999 issue of Mathematics Magazine (https://www.jstor.org/stable/2691318 - page 63).\n\nTo learn more about animating with manim, check out:\nhttps://manim.community\n\n_______________________________\nMusic in this:\nPrelude No. 6 by Chris Zabriskie is licensed under a Creative Commons Attribution 4.0 license. https://creativecommons.org/licenses/by/4.0/\n\nSource: http://chriszabriskie.com/preludes/\n\nArtist: http://chriszabriskie.com/", "lengthSeconds": 57, "width": 360, "height": 640}
{"video_id": "aMRDYqPPJf4", "transcript": [{"start": 11.76, "duration": 16.739, "text": "[Music]"}, {"start": 29.16, "duration": 5.32, "text": "a"}, {"start": 31.48, "duration": 3.0, "text": "oh"}, {"start": 35.72, "duration": 19.209, "text": "[Music]"}, {"start": 60.23, "duration": 6.28, "text": "[Music]"}, {"start": 71.5, "duration": 3.17, "text": "[Music]"}, {"start": 79.75, "duration": 11.45, "text": "[Music]"}, {"start": 93.8, "duration": 23.109, "text": "[Music]"}], "title": "#trending #mathematics Visual Proof: Factors of Quadratic Equation Polynomials", "description": "#trendingshorts  #mathematics Visual Proof: Factors of Quadratic Equation Polynomials\n\n\nOther Videos\nFibonacci Sequence, Golden Ratio, How to draw Golden Spiral:\nhttps://youtu.be/yHBa3EvDPco\nRadius and Tangent Vectors: \nhttps://youtube.com/shorts/KHSafRET3Sk\nDe Morgan's laws: Proof by Venn Diagrams:\nhttps://youtube.com/shorts/pR-REo4aBYk?feature=share\nVisual Proof of Roots of Quadratic Equation\nhttps://youtube.com/shorts/FdE7rHDZRd8\nVisual Proof: Factors of Quadratic Equation PART-1 x^2+3x+2\nhttps://youtube.com/shorts/lIqv5zKAKKE\nVisual Proof: Factors of Quadratic Polynomial PART-2 x^2-3x+2\nhttps://youtube.com/shorts/5pRYbKFWjUM\nVisual Proof: Factors of Quadratic Polynomial PART-2 x^2+2x-3\nhttps://youtube.com/shorts/AzAsz5itOfQ", "lengthSeconds": 116, "width": 640, "height": 360}
{"video_id": "qk1IPvvWfr0", "transcript": [{"start": 0.04, "duration": 4.36, "text": "here's a classic visual infinite sum"}, {"start": 2.28, "duration": 3.92, "text": "start with a unit square cut it into"}, {"start": 4.4, "duration": 4.319, "text": "four pieces and shade three of them"}, {"start": 6.2, "duration": 4.92, "text": "shading an area of 3/4s cut the"}, {"start": 8.719, "duration": 4.0, "text": "remaining 1/4 into four pieces and shade"}, {"start": 11.12, "duration": 4.2, "text": "three of them so that we've now shaded"}, {"start": 12.719, "duration": 4.4, "text": "an additional 3 over4 squar continue"}, {"start": 15.32, "duration": 3.84, "text": "repeating this process cutting the top"}, {"start": 17.119, "duration": 4.16, "text": "one quarter into four equal pieces and"}, {"start": 19.16, "duration": 4.16, "text": "shading three of them at each stage this"}, {"start": 21.279, "duration": 4.16, "text": "means we've shaded 3 over 4 to the end"}, {"start": 23.32, "duration": 4.199, "text": "of the total area of the square if we"}, {"start": 25.439, "duration": 4.201, "text": "continue this indefinitely Or imagine"}, {"start": 27.519, "duration": 4.281, "text": "the limiting process we see that in the"}, {"start": 29.64, "duration": 3.8, "text": "limit the entire Square diagram will be"}, {"start": 31.8, "duration": 4.36, "text": "shaded and therefore we've shaded an"}, {"start": 33.44, "duration": 5.639, "text": "area of one this means that 3 * the"}, {"start": 36.16, "duration": 5.44, "text": "infinite sum of the powers of 1/4 is one"}, {"start": 39.079, "duration": 4.561, "text": "one interpretation is that 0.3 repeating"}, {"start": 41.6, "duration": 3.84, "text": "in base 4 is 1 and another"}, {"start": 43.64, "duration": 4.439, "text": "interpretation is that the sum of the"}, {"start": 45.44, "duration": 5.639, "text": "infinite poers of 1/4 is"}, {"start": 48.079, "duration": 3.0, "text": "1/3"}], "title": "0.3333\u2026 = 1 (in base 4)", "description": "This is a short, animated visual proof demonstrating the infinite sum of the powers of 1/4.\n\nTo buy me a coffee, head over to https://www.buymeacoffee.com/VisualProofs \nThanks!\n\nTo see two other versions of this sum, check out\nhttps://youtu.be/8i1xj5ORwUw?si=5SlPwm_0ym2NyGWl\nhttps://youtu.be/UePC62RciUc?si=gdtUgL7gPSgiImYC\n\nTo see the related base 10 fact, check out:\nhttps://youtube.com/shorts/XJtjU_Ipjno?feature=share\n\nAnd to see the related binary fact, check out:\nhttps://youtube.com/shorts/RoovN01q83Y?feature=share\n\nTo see related base-3 fact, check out:\nhttps://youtu.be/vfEDDI3vfHU\nor\nhttps://youtube.com/shorts/GwIVlNRV-M0\n\nTo see a related base-5 fact, check out\nhttps://youtube.com/shorts/ltKZJ8wumUI\n\nThis animation is based on a proof by Sunday A. Ajose from the June 1994 issue of Mathematics Magazine (https://www.jstor.org/stable/2690616 page 230).\n\n#mathshorts\u200b #mathvideo\u200b #math\u200b #calculus #mtbos\u200b #manim\u200b #animation\u200b #theorem\u200b #pww\u200b #proofwithoutwords\u200b #visualproof\u200b #proof\u200b #iteachmath #geometricseries #series #infinitesums #infiniteseries #shorts\n\nTo learn more about animating with manim, check out:\nhttps://manim.community", "lengthSeconds": 49, "width": 360, "height": 640}
{"video_id": "F-wmWXC4xfw", "transcript": [{"start": 2.95, "duration": 7.729, "text": "[Music]"}, {"start": 13.04, "duration": 3.92, "text": "GL loneliness"}], "title": "Why Area of Circle = \u03c0r^2 | Visual Proof | Math fun | Chetan Setia | Best Coaching Institute", "description": "\u0a2e\u0a3e\u0a38\u0a1f\u0a30 \u0a15\u0a48\u0a21\u0a30 \u0a26\u0a40 \u0a24\u0a3f\u0a06\u0a30\u0a40 | \u0a1a\u0a47\u0a24\u0a28 \u0a38\u0a47\u0a24\u0a3f\u0a06\nCTET Best Coaching Online | Memes | Aspirants | Student | Math fun | Chetan Setia \nHappy to help- Call us:  86997-02667 & 73473-02667\nJoin VIP Community:-  https://chat.whatsapp.com/Lx6fvc7Q1Iw...\nFor Free Materials click here: http://wa.me/918699702667 \nLinks to download MME App\nFor Android:- https://play.google.com/store/apps/de...\nFor iOS:- https://apps.apple.com/us/app/classpl... ORG code:- mme \n\nCTET | Memes | Aspirants | Student | Math fun | Chetan Setia | Best Coaching Online\n\nAbout Channel: It is an elite educational channel that provides online coaching in:\n- Master Cadre Math\n- Master Cadre Science\n- Lecture Cadre Math\n- IIT-JEE\n- NDA\n- B.Sc. Hons. Math (all the universities)\nWith the help of world-class animation and visualization. This channel has an unmatchable record in giving maximum selections in all competitive exams. It has given about 700+ selections in Master Cadre Maths and Master Cadre Science in the last 2 years.\n\nCTET | Memes | Aspirants | Student | Math fun | Chetan Setia | Best Coaching Online\n \nAbout Chetan Setia: He is the founder and Director of Chetan Setia Education.\nSome wonders about him:  \n- 24 years experience: he has coached M.Sc. students while pursuing his Graduation\n- Shortcut tricks: He has developed shortcut tricks for all the topics of math\n- He is among the best Vedic math teacher in north India\n- He relates all the mathematics topics to real life\n\nOur playlists:\nMaster Cadre Math 2022 Achievers  https://www.youtube.com/watch?v=kFy7KMoWRO0&list=PL7C15KYeZyKSkUg6wx-T-mLsZMHMmy0oy&pp=iAQB \nMaster Cadre Maths Preparation 2023\nhttps://www.youtube.com/watch?v=tb4vwOvI288&list=PL7C15KYeZyKSg1kMcW9yuZRK7G8ndXoOn&pp=iAQB \nMaster Cadre Maths 2023\nhttps://www.youtube.com/watch?v=COeZkmZbjMs&list=PL7C15KYeZyKTX7dkEpJwjjhHxcfC2dM3j&pp=iAQB \nIntroduction to Class 11th Maths with Fun\ud83e\udd73 & Games\nhttps://www.youtube.com/watch?v=t0B_lPGHRKE&list=PL7C15KYeZyKSrS8f0XmxH94Y5_MqMQbH5&pp=iAQB \n\nSocial Media links:\nInstagram handle: https://instagram.com/chetan.setia_mme?\nFacebook: https://www.facebook.com/chetan.setia \nTelegram Group: http://chetan.us/govtjobtelegram\n\nCTET | Memes | Aspirants | Student | Math fun | Chetan Setia | Best Coaching Onliner \n\nGoogle Form:\nTgt/Pgt Interest Form: http://chetan.us/mclcinterest\nUltimate Success Formula(USF):  http://chetan.us/usfinterest\n\n#chetansetia  #mastercadremaths2023  #mastercadresmath  #mastercadrenewupdate  #mastercadremathshortcuttricks   #mastercadremaths2023  #mastercadremathpraciticesets  #punjabmastercadremath  #mastercadremathshortcuts  #mastercadremathpraciticetests  #mastercadremathmocktest #mastercadremathpreparation #mastercadremathpreviouspapersolved  #mathematics #aspirantlife #Examresult #entranceexam #Student #Questions #mathstricks #maths #mathfun #memes #mathsmasti #ctet #aspirantlife \n#areaofcircle #visualeffects #visualization", "lengthSeconds": 15, "width": 360, "height": 640}
{"video_id": "AHotEMrXTX8", "transcript": [{"start": 0.0, "duration": 4.38, "text": "here's the graph of the function f"}, {"start": 1.68, "duration": 5.76, "text": "defined by f of x equals 1 over X along"}, {"start": 4.38, "duration": 5.58, "text": "with one label at the input x equals one"}, {"start": 7.44, "duration": 4.68, "text": "now consider another input labeled X"}, {"start": 9.96, "duration": 4.62, "text": "that can vary and draw a line from the"}, {"start": 12.12, "duration": 4.8, "text": "x-axis up to the point on the curve as"}, {"start": 14.58, "duration": 4.32, "text": "we move x to the right we can measure"}, {"start": 16.92, "duration": 4.74, "text": "the area under the curve and above the"}, {"start": 18.9, "duration": 4.68, "text": "x-axis between 1 and the input X the"}, {"start": 21.66, "duration": 4.199, "text": "area is 0 when x equals one and"}, {"start": 23.58, "duration": 4.5, "text": "increases as X increases we can do more"}, {"start": 25.859, "duration": 4.561, "text": "let's actually plot the area on a"}, {"start": 28.08, "duration": 4.26, "text": "separate set of axes like this as X"}, {"start": 30.42, "duration": 4.2, "text": "varies we end up with the curve of a new"}, {"start": 32.34, "duration": 4.26, "text": "function we denote this new function by"}, {"start": 34.62, "duration": 4.439, "text": "capital f of x equals the integral from"}, {"start": 36.6, "duration": 4.26, "text": "1 to X of 1 over T DT and we call this"}, {"start": 39.059, "duration": 3.66, "text": "an area accumulation function of the"}, {"start": 40.86, "duration": 3.66, "text": "function f this particular area"}, {"start": 42.719, "duration": 3.781, "text": "accumulation function is actually one"}, {"start": 44.52, "duration": 3.84, "text": "way to define the natural log notice"}, {"start": 46.5, "duration": 3.899, "text": "that we can let the x value vary less"}, {"start": 48.36, "duration": 4.08, "text": "than one as well resulting in a negative"}, {"start": 50.399, "duration": 4.261, "text": "orientation for the area so this defines"}, {"start": 52.44, "duration": 4.7, "text": "the natural log of x for any X greater"}, {"start": 54.66, "duration": 2.48, "text": "than zero"}], "title": "Defining the natural logarithm", "description": "The definition of the natural log as an area accumulation function. #shorts #math #manim #logarithms #calculus #integral", "lengthSeconds": 55, "width": 360, "height": 640}
{"video_id": "Be2neKFhli8", "transcript": [{"start": 0.48, "duration": 4.08, "text": "here's a non-standard way to find the"}, {"start": 2.12, "duration": 5.88, "text": "sum of the first n odd positive integers"}, {"start": 4.56, "duration": 6.44, "text": "using areas the sum of the first n odd"}, {"start": 8.0, "duration": 5.639, "text": "positive integers 1 + 3 + 5 and so on up"}, {"start": 11.0, "duration": 5.32, "text": "to 2 N minus 1 can be represented by an"}, {"start": 13.639, "duration": 4.72, "text": "array of 1x one squares like this where"}, {"start": 16.32, "duration": 4.44, "text": "the first row has one square the second"}, {"start": 18.359, "duration": 4.641, "text": "row has three squares and so on all the"}, {"start": 20.76, "duration": 5.48, "text": "way down until the very last row which"}, {"start": 23.0, "duration": 5.64, "text": "has 2 N - one squares this array has a"}, {"start": 26.24, "duration": 4.6, "text": "height of size n and a base of size 2 N"}, {"start": 28.64, "duration": 4.28, "text": "minus one like this"}, {"start": 30.84, "duration": 4.2, "text": "to find the area of this array we can"}, {"start": 32.92, "duration": 4.68, "text": "cut it into pieces using this diagonal"}, {"start": 35.04, "duration": 5.199, "text": "line of slope - one2 starting at the"}, {"start": 37.6, "duration": 5.439, "text": "bottom right corner of the top"}, {"start": 40.239, "duration": 5.16, "text": "square we can use that line to break the"}, {"start": 43.039, "duration": 5.721, "text": "diagram up into a variety of rectangular"}, {"start": 45.399, "duration": 7.561, "text": "and triangular pieces as shown here"}, {"start": 48.76, "duration": 6.439, "text": "first on the left side we see a 1xn"}, {"start": 52.96, "duration": 5.68, "text": "rectangle the triangle in the middle is"}, {"start": 55.199, "duration": 6.721, "text": "2 N - 2 by nus1 and the nus1 triangles"}, {"start": 58.64, "duration": 6.4, "text": "on the right are all 2 2x1 triangles the"}, {"start": 61.92, "duration": 6.16, "text": "rectangle has an area of n * 1 or n the"}, {"start": 65.04, "duration": 7.399, "text": "large triangle has an area of 12 N - 1 *"}, {"start": 68.08, "duration": 7.96, "text": "2 N - 2 which is n -1 * n -1 or n^2 - 2"}, {"start": 72.439, "duration": 5.921, "text": "n + 1 and finally the N -1 triangles on"}, {"start": 76.04, "duration": 5.88, "text": "the right all have an area of 1/ 12 * 1"}, {"start": 78.36, "duration": 5.719, "text": "* 2 or 1 combining these areas together"}, {"start": 81.92, "duration": 6.04, "text": "we get that the sum of the first n odd"}, {"start": 84.079, "duration": 8.0, "text": "positive integers is n + n^ 2 - 2 n + 1"}, {"start": 87.96, "duration": 6.04, "text": "+ N - 1 but this simplifies to n^2 and"}, {"start": 92.079, "duration": 4.04, "text": "therefore the sum of the first n odd"}, {"start": 94.0, "duration": 5.119, "text": "positive integers is"}, {"start": 96.119, "duration": 5.601, "text": "n^2 to write this more compactly we have"}, {"start": 99.119, "duration": 6.6, "text": "that the sum where K ranges from 1 to n"}, {"start": 101.72, "duration": 6.12, "text": "of 2 K minus 1 must equal n^ 2 this is a"}, {"start": 105.719, "duration": 5.201, "text": "classic formula but we proved it using"}, {"start": 107.84, "duration": 3.08, "text": "areas of rectangles and"}, {"start": 115.719, "duration": 3.0, "text": "triangles"}], "title": "Adding Odds using Areas (Sum of odds visual proof IV)", "description": "This is a short, animated visual proof showing one method to find the sum of the first n odd positive integers decomposing an array of unit squares into rectangular and triangular pieces and using area formulas for those pieces.\n\nIf you like this video, consider subscribing to the channel or consider buying me a coffee: https://www.buymeacoffee.com/VisualProofs. Thanks!\n\nTo see the other methods for finding the sum odd integers visually, check out the following videos: \nhttps://youtu.be/jq5AYCgkciE?si=U6bxLBmWjyBD_HQZ\nhttps://youtu.be/I4A2T55k0cg\nhttps://youtu.be/x3qfFBNRRDg\nhttps://youtu.be/ZeEOgbLo0Rg\nhttps://youtu.be/6H6nCpGW1nU\n\nFor other sums related to odd integers, check out these visuals:\nhttps://youtu.be/WKxoi2i0EaE (summing odd squares)\nhttps://studio.youtube.com/video/9BC1klE8jMk/edit (alternating odd sums)\nhttps://youtu.be/MmOTqrtbtFQ (odd sum ratios)\nhttps://youtu.be/fTBvVeURb3Q (odd sum ratios II)\n\nThis animation is based on a visual proof due to Samuel G. Moreno that appeared in the October 2017 issue of Mathematics Magazine (https://doi.org/10.4169/math.mag.90.4.298 page 298).\n\nYou can find other diagrams in Roger Nelsen's book: (I may receive a small commission at no cost to you for this affiliate link): https://amzn.to/47kfG7w\n\n#manim #math #mathvideo #mathshorts #sequences #triangles #animation #theorem #pww #proofwithoutwords #visualproof #proof #squarenumbers #squares #oddnumbers #odds #sums #pww\u200b #proofwithoutwords\u200b \u200b #proof\u200b #algebra #finitesums #mathematics\u200b #mathvideo\u200b #mtbos \n\nTo learn more about animating with manim, check out:\nhttps://manim.community", "lengthSeconds": 117, "width": 640, "height": 360}
{"video_id": "RKeBuWL1B4A", "transcript": [{"start": 2.43, "duration": 14.45, "text": "[Music]"}, {"start": 15.12, "duration": 3.12, "text": "suppose the fraction a over b is less"}, {"start": 16.88, "duration": 4.08, "text": "than or equal to the fraction c"}, {"start": 18.24, "duration": 4.799, "text": "over d both of these fractions can be"}, {"start": 20.96, "duration": 4.399, "text": "represented by the slopes of hypotenuses"}, {"start": 23.039, "duration": 4.0, "text": "of right triangles and the inequality"}, {"start": 25.359, "duration": 3.441, "text": "means that the slope of the hypotenuse"}, {"start": 27.039, "duration": 3.121, "text": "of the triangle with legs a and b"}, {"start": 28.8, "duration": 3.04, "text": "is less than or equal to the slope of"}, {"start": 30.16, "duration": 3.28, "text": "the hypotenuse of the triangle with legs"}, {"start": 31.84, "duration": 3.44, "text": "c and d"}, {"start": 33.44, "duration": 4.0, "text": "aligning the two triangles this way"}, {"start": 35.28, "duration": 3.599, "text": "allows us to draw them inside of a third"}, {"start": 37.44, "duration": 3.279, "text": "right triangle"}, {"start": 38.879, "duration": 4.721, "text": "this new right triangle has legs of"}, {"start": 40.719, "duration": 4.16, "text": "length b plus d and a plus c"}, {"start": 43.6, "duration": 3.2, "text": "now comparing the slopes of the"}, {"start": 44.879, "duration": 4.641, "text": "hypotenuses of these three triangles"}, {"start": 46.8, "duration": 4.88, "text": "reveals that a over b is less than or"}, {"start": 49.52, "duration": 5.359, "text": "equal to the slope of the big triangle"}, {"start": 51.68, "duration": 4.48, "text": "which is a plus c over b plus d and that"}, {"start": 54.879, "duration": 3.36, "text": "in turn is less than or equal to the"}, {"start": 56.16, "duration": 5.36, "text": "slope of the top right triangle"}, {"start": 58.239, "duration": 6.081, "text": "which is c over d the middle quantity"}, {"start": 61.52, "duration": 3.359, "text": "a plus c over b plus d is known as the"}, {"start": 64.32, "duration": 2.96, "text": "median"}, {"start": 64.879, "duration": 3.441, "text": "of the two fractions a over b and c over"}, {"start": 67.28, "duration": 3.56, "text": "d"}, {"start": 68.32, "duration": 5.52, "text": "therefore this is known as the mediant"}, {"start": 70.84, "duration": 3.0, "text": "inequality"}, {"start": 74.84, "duration": 5.539, "text": "[Music]"}, {"start": 88.159, "duration": 2.081, "text": "you"}], "title": "The Mediant Inequality I (visual proof)", "description": "This is a short, animated visual proof demonstrating what one might called the mediant inequality. #mathshorts\u200b #mathvideo\u200b #math\u200b #numbertheory #mtbos\u200b #manim\u200b #animation\u200b #theorem\u200b #pww\u200b #proofwithoutwords\u200b #visualproof\u200b #proof\u200b #iteachmath #fractions #mediant #inequality #mathematics #fareyfractions #farey\n\nThis animation is based on a visual proof by Richard A. Gibbs from the June 1990 issue of Mathematics Magazine (https://doi.org/10.2307/2691137  - page 172).\n\nTo learn more about animating with manim, check out:\nhttps://manim.community\n__________________________________________________________________\nBackground music, used according to CC BY-SA 3.0, is \"The Road\" from Esteban Orlando:\n\nEsteban Orlando:\nhttps://soundcloud.com/orlando-esteba...\nhttps://www.instagram.com/estebanorla...\n\nhttps://creativecommons.org/licenses/...\nMusic provided by RFM: https://youtu.be/b4KCM1bIq50", "lengthSeconds": 88, "width": 640, "height": 360}
{"video_id": "bPAm1nRUJWw", "transcript": [{"start": 11.76, "duration": 16.739, "text": "[Music]"}, {"start": 29.16, "duration": 5.32, "text": "a"}, {"start": 31.48, "duration": 3.0, "text": "oh"}, {"start": 35.72, "duration": 19.019, "text": "[Music]"}], "title": "#trending #mathematics #maths #calculus Visual Proof: Trigonometric Identity sin^2 (x) +cos^2 (x)=1", "description": "#trending #mathematics #maths #calculus Visual Proof: Trigonometric Identity sin^2 (x) +cos^2 (x)=1\n\nOther Videos\nFibonacci Sequence, Golden Ratio, How to draw Golden Spiral:\nhttps://youtu.be/yHBa3EvDPco\nRadius and Tangent Vectors: \nhttps://youtube.com/shorts/KHSafRET3Sk\nDe Morgan's laws: Proof by Venn Diagrams:\nhttps://youtube.com/shorts/pR-REo4aBYk?feature=share\nVisual Proof of Roots of Quadratic Equation\nhttps://youtube.com/shorts/FdE7rHDZRd8\nVisual Proof: Factors of Quadratic Equation PART-1 x^2+3x+2\nhttps://youtube.com/shorts/lIqv5zKAKKE\nVisual Proof: Factors of Quadratic Polynomial PART-2 x^2-3x+2\nhttps://youtube.com/shorts/5pRYbKFWjUM\nVisual Proof: Factors of Quadratic Polynomial PART-2 x^2+2x-3\nhttps://youtube.com/shorts/AzAsz5itOfQ\nVisual Proof: Factors of Quadratic Polynomial PART-2 x^2+2x-3\nhttps://youtube.com/shorts/AzAsz5itOfQ\nVisual Proof: Trigonometric Identity sin^2 (x) +cos^2 (x)=1\nhttps://youtube.com/shorts/bPAm1nRUJWw", "lengthSeconds": 53, "width": 360, "height": 640}
{"video_id": "LBls3jRfxeY", "transcript": [{"start": 0.52, "duration": 3.199, "text": "here's how to find the total number of"}, {"start": 2.04, "duration": 4.44, "text": "gifts in the 12 Days of Christmas using"}, {"start": 3.719, "duration": 4.761, "text": "a geometric argument on the first day of"}, {"start": 6.48, "duration": 4.8, "text": "Christmas you get one partridge in a"}, {"start": 8.48, "duration": 4.96, "text": "pear tree on the second day you get the"}, {"start": 11.28, "duration": 4.559, "text": "Partridge in the pear tree and the two"}, {"start": 13.44, "duration": 4.88, "text": "turtle doves on day three you get three"}, {"start": 15.839, "duration": 4.761, "text": "French hens and the gifts from day two"}, {"start": 18.32, "duration": 4.879, "text": "on day four you get four calling Birds"}, {"start": 20.6, "duration": 4.64, "text": "and the gifts from day three on day five"}, {"start": 23.199, "duration": 4.801, "text": "you get five golden rings and the gifts"}, {"start": 25.24, "duration": 5.4, "text": "from day four on day six you get six"}, {"start": 28.0, "duration": 5.12, "text": "geese and all the gifts from day five on"}, {"start": 30.64, "duration": 4.96, "text": "day seven you get seven swans and all"}, {"start": 33.12, "duration": 4.2, "text": "the gifts from day six on day eight you"}, {"start": 35.6, "duration": 4.959, "text": "get eight maids and all of the previous"}, {"start": 37.32, "duration": 5.44, "text": "gifts on day nine you get nine ladies"}, {"start": 40.559, "duration": 4.241, "text": "dancing in all the previous gifts on day"}, {"start": 42.76, "duration": 5.08, "text": "10 you get 10 Lords and all of the"}, {"start": 44.8, "duration": 4.919, "text": "previous gifts on day 11 you get 11"}, {"start": 47.84, "duration": 4.68, "text": "Pipers piping and all of the previous"}, {"start": 49.719, "duration": 4.961, "text": "gifts and finally on day 12 you get 12"}, {"start": 52.52, "duration": 3.96, "text": "drummers and the previous gifts so how"}, {"start": 54.68, "duration": 4.199, "text": "many total gifts have you obtained over"}, {"start": 56.48, "duration": 5.12, "text": "the 12 days to figure this out we employ"}, {"start": 58.879, "duration": 4.601, "text": "a trick we imagine three people getting"}, {"start": 61.6, "duration": 4.28, "text": "all of these gifts so that we actually"}, {"start": 63.48, "duration": 4.16, "text": "have three copies of the gifts we take"}, {"start": 65.88, "duration": 3.68, "text": "the gifts from the second person and"}, {"start": 67.64, "duration": 4.76, "text": "glue them together like this with the"}, {"start": 69.56, "duration": 5.64, "text": "gifts from the first person we then form"}, {"start": 72.4, "duration": 5.2, "text": "this stack finally we can use the stack"}, {"start": 75.2, "duration": 5.4, "text": "of gifts from the third person to lay"}, {"start": 77.6, "duration": 5.479, "text": "them down in order like this as we fill"}, {"start": 80.6, "duration": 5.479, "text": "in this diagram we see that the final"}, {"start": 83.079, "duration": 4.521, "text": "shape will be a perfect rectangle by"}, {"start": 86.079, "duration": 3.161, "text": "construction the length of this"}, {"start": 87.6, "duration": 4.92, "text": "rectangle is two more than the total"}, {"start": 89.24, "duration": 5.72, "text": "number of days so it's 12 + 2 which is"}, {"start": 92.52, "duration": 4.639, "text": "14 on the other hand the width of this"}, {"start": 94.96, "duration": 4.24, "text": "rectangle is equal to the total number"}, {"start": 97.159, "duration": 3.841, "text": "of new gifts each day so it's equal to"}, {"start": 99.2, "duration": 3.84, "text": "the sum of the first 12 positive"}, {"start": 101.0, "duration": 3.799, "text": "integers but this sum is the total"}, {"start": 103.04, "duration": 4.119, "text": "number of gifts obtained on day 12 which"}, {"start": 104.799, "duration": 4.721, "text": "is half of this 13x 12 rectangle"}, {"start": 107.159, "duration": 5.481, "text": "pictured here so the sum of the first 12"}, {"start": 109.52, "duration": 4.639, "text": "positive integers is 13 * 12 / 2"}, {"start": 112.64, "duration": 3.36, "text": "therefore this construction shows us"}, {"start": 114.159, "duration": 3.401, "text": "that 3 * the total number of gifts"}, {"start": 116.0, "duration": 5.28, "text": "received is equal to the area of this"}, {"start": 117.56, "duration": 5.36, "text": "rectangle which is 14 * 13 * 12 12 ID 2"}, {"start": 121.28, "duration": 5.439, "text": "but this means the total number of gifts"}, {"start": 122.92, "duration": 6.88, "text": "is 14 * 13 * 12 / 6 which is"}, {"start": 126.719, "duration": 6.721, "text": "364 in the song The 12 Days of Christmas"}, {"start": 129.8, "duration": 3.64, "text": "they've given 364"}, {"start": 139.0, "duration": 3.0, "text": "gifts"}], "title": "Geometry of the Twelve Days of X-mas (visual proof)", "description": "You might know that in the song \"The Twelve Days of Christmas\" there are a total of 364 gifts given over the 12-day span. You might even know how to prove this mathematically (there are great videos about this - see links below). But did you know that you can demonstrate this fact geometrically? In this video, we show one way to think about the sum of the first 12 positive integers. Not that this technique also can work in general; see this earlier (more dramatic, and wordless) video showing the general formula and general visual proof:\n\nhttps://youtu.be/N0ETyJ5K6j0\n\nIf you like this video, consider subscribing to the channel or consider buying me a coffee: https://www.buymeacoffee.com/VisualProofs. Thanks! \n\nThis animation is based on a visual proof by Monte J. Zerger from the December 1990 issue of Mathematics Magazine (https://www.jstor.org/stable/2690899- page 314).\n\nTo see other amazing videos discussing the math behind the Twelve Days of Christmas, check out:\n\nhttps://youtu.be/fC8W4s6N9HQ?si=5r2gRH-r1o9V6LIq by @TippingPointMath \nhttps://youtu.be/cNl7GypEH_A?si=v4ZDSYrmwxDapMpo by @MindYourDecisions \nhttps://youtu.be/7adXOcUweaY?si=t-5QxwlypzdQtu5W by @polymathematic \n\n #manim #math #mathvideo #mathshorts #sequences #triangles #animation #theorem #pww #proofwithoutwords #visualproof #proof #triangularnumbers #sums #pww\u200b #proofwithoutwords\u200b \u200b #proof\u200b #algebra #finitesums #mathematics\u200b #mathvideo\u200b #mtbos \n\nTo learn more about animating with manim, check out:\nhttps://manim.community\n\nThumbnail for video includes part of an image from Xavier Romero-Frias, entitled English: The Twelve Days of Christmas song poster. This image was used according to Creative Commons Attribution-Share Alike 3.0 Unported (https://creativecommons.org/licenses/by-sa/3.0/deed.en ).", "lengthSeconds": 140, "width": 640, "height": 360}
{"video_id": "00xLLlfowJ0", "transcript": [{"start": 0.52, "duration": 34.3, "text": "[\u0645\u0648\u0633\u064a\u0642\u0649]"}], "title": "Visual Proof of the Pythagorean Theorem | 1", "description": "In this short the most famous and intuitive visual proof of the Pythagorean theorem. #maths #visualization #geometry #algebra #python #proof #visualproof", "lengthSeconds": 33, "width": 360, "height": 640}
{"video_id": "j-xvHyolrGM", "transcript": [{"start": 0.29, "duration": 9.18, "text": "[Music]"}, {"start": 11.6, "duration": 18.999, "text": "[Music]"}, {"start": 27.599, "duration": 3.0, "text": "do"}], "title": "Visual proof of difference of squares of two numbers || visualize math || FS bukhari official #maths", "description": "Visual proof of difference of squares of two numbers || visualize math || FS bukhari official #maths #mathematics_facts #mathlogic #mathematics", "lengthSeconds": 29, "width": 360, "height": 640}
{"video_id": "IAsaNUUrfj4", "transcript": [{"start": 0.179, "duration": 36.54, "text": "foreign"}, {"start": 2.1, "duration": 35.41, "text": "[Music]"}, {"start": 36.719, "duration": 25.69, "text": "foreign"}, {"start": 37.51, "duration": 24.899, "text": "[Music]"}, {"start": 64.619, "duration": 5.411, "text": "foreign"}, {"start": 66.91, "duration": 3.12, "text": "[Music]"}, {"start": 72.12, "duration": 23.74, "text": "[Music]"}, {"start": 96.6, "duration": 3.0, "text": "foreign"}], "title": "Factor a Sum of two Squares (visual proof)", "description": "This is a short, animated visual proof of showing how to factor a sum of two squares. #manim #math #mathvideo #arithmeticprogression #squares #mathshorts #geometry  #animation #theorem #pww #proofwithoutwords #visualproof #proof #area #algebra #pww\u200b\u200b \u200b #proof\u200b #areas #mathematics\u200b\u200b #mtbos \n\nIf you like this video, please click \"like\" and consider subscribing and checking out my other videos.\n\nThis animation is based on a visual proof from Roger B. Nelsen's second proofs without words compendium.\n\nTo learn more about animating with manim, check out:\nhttps://manim.community\n_____________________________\nMusic in this video:\nFriends Forever by FSM TEAM\nMusic by https://www.free-stock-music.com", "lengthSeconds": 101, "width": 640, "height": 360}
{"video_id": "kaInaIUABzY", "transcript": [{"start": 0.9, "duration": 5.999, "text": "foreign"}, {"start": 2.09, "duration": 7.87, "text": "[Music]"}, {"start": 6.899, "duration": 6.481, "text": "suppose we want to walk from the point 0"}, {"start": 9.96, "duration": 5.7, "text": "0 to the point n comma n Only traveling"}, {"start": 13.38, "duration": 5.34, "text": "north and east and not passing below"}, {"start": 15.66, "duration": 5.1, "text": "this diagonal Y equals X"}, {"start": 18.72, "duration": 4.26, "text": "this problem consists of counting what"}, {"start": 20.76, "duration": 4.74, "text": "are called the Northeast integer lattice"}, {"start": 22.98, "duration": 5.58, "text": "paths and they look like this"}, {"start": 25.5, "duration": 5.279, "text": "in a previous video I enumerated lots of"}, {"start": 28.56, "duration": 4.199, "text": "these and showed how to find a recursive"}, {"start": 30.779, "duration": 3.96, "text": "formula for counting the number of such"}, {"start": 32.759, "duration": 3.96, "text": "lattice paths"}, {"start": 34.739, "duration": 6.0, "text": "right here we see that the first view"}, {"start": 36.719, "duration": 8.101, "text": "values of this sequence are one one two"}, {"start": 40.739, "duration": 7.921, "text": "five Fourteen and here we see the 42"}, {"start": 44.82, "duration": 7.02, "text": "lattice paths traveling from 0 0 to 5 5."}, {"start": 48.66, "duration": 5.76, "text": "as I said we know already how to find a"}, {"start": 51.84, "duration": 4.26, "text": "recursion for this sequence but if we"}, {"start": 54.42, "duration": 4.619, "text": "let C sub n be the number of such"}, {"start": 56.1, "duration": 5.16, "text": "lattice pass to n comma n can we now"}, {"start": 59.039, "duration": 3.98, "text": "find a closed formula for this integer"}, {"start": 61.26, "duration": 4.2, "text": "sequence"}, {"start": 63.019, "duration": 4.121, "text": "here's one way to provide a closed"}, {"start": 65.46, "duration": 4.26, "text": "formula for the number of such paths"}, {"start": 67.14, "duration": 4.92, "text": "here we let N equal 5 represent the"}, {"start": 69.72, "duration": 4.259, "text": "general situation instead of counting"}, {"start": 72.06, "duration": 3.9, "text": "the restricted lattice paths we can"}, {"start": 73.979, "duration": 4.081, "text": "first enumerate all the lattice paths"}, {"start": 75.96, "duration": 4.74, "text": "traveling Northeast they go from 0 0 to"}, {"start": 78.06, "duration": 3.78, "text": "n n in a previous video I showed that"}, {"start": 80.7, "duration": 3.36, "text": "this is counted by a binomial"}, {"start": 81.84, "duration": 5.279, "text": "coefficient but we want to avoid the"}, {"start": 84.06, "duration": 4.739, "text": "paths that pass below Y equals X those"}, {"start": 87.119, "duration": 3.901, "text": "must be the ones that intersect this"}, {"start": 88.799, "duration": 5.161, "text": "line y equals x minus 1."}, {"start": 91.02, "duration": 5.52, "text": "for each such path that intersects the"}, {"start": 93.96, "duration": 4.26, "text": "red line we can reflect the remaining"}, {"start": 96.54, "duration": 4.8, "text": "part of the path after the first"}, {"start": 98.22, "duration": 5.939, "text": "intersection across that line"}, {"start": 101.34, "duration": 4.739, "text": "here we do it with all of the lines that"}, {"start": 104.159, "duration": 2.931, "text": "we don't want to include in our final"}, {"start": 106.079, "duration": 4.261, "text": "count"}, {"start": 107.09, "duration": 6.73, "text": "[Music]"}, {"start": 110.34, "duration": 6.18, "text": "as we reflect and remove the bad lattice"}, {"start": 113.82, "duration": 5.339, "text": "paths we notice something special about"}, {"start": 116.52, "duration": 4.62, "text": "all of these Reflections look at what"}, {"start": 119.159, "duration": 4.441, "text": "happens to the end point of these"}, {"start": 121.14, "duration": 4.799, "text": "lattice paths they all seem to be"}, {"start": 123.6, "duration": 4.08, "text": "hitting on the same point so what we're"}, {"start": 125.939, "duration": 3.901, "text": "getting from this collection of"}, {"start": 127.68, "duration": 4.139, "text": "reflected lattice paths seems to be a"}, {"start": 129.84, "duration": 4.8, "text": "collection of lattice paths that still"}, {"start": 131.819, "duration": 6.741, "text": "starts at the origin zero zero but this"}, {"start": 134.64, "duration": 3.92, "text": "time ends at a different point"}, {"start": 139.98, "duration": 4.8, "text": "in fact all of the reflected lattice"}, {"start": 142.5, "duration": 4.2, "text": "paths end at the point n plus 1 comma n"}, {"start": 144.78, "duration": 4.74, "text": "minus 1 which is the reflection of the"}, {"start": 146.7, "duration": 4.56, "text": "point n n over the red dotted line but"}, {"start": 149.52, "duration": 3.96, "text": "now if we enumerate all the Northeast"}, {"start": 151.26, "duration": 4.74, "text": "lattice paths from 0 0 to the point n"}, {"start": 153.48, "duration": 4.619, "text": "plus 1 comma n minus one we see that"}, {"start": 156.0, "duration": 4.44, "text": "each of them crosses the red dotted line"}, {"start": 158.099, "duration": 4.92, "text": "and therefore must have been one of the"}, {"start": 160.44, "duration": 5.159, "text": "reflected paths because the reflection"}, {"start": 163.019, "duration": 4.561, "text": "process is invertible we see there's a"}, {"start": 165.599, "duration": 5.22, "text": "one-to-one correspondence between the"}, {"start": 167.58, "duration": 5.28, "text": "bad paths traveling to n comma n and all"}, {"start": 170.819, "duration": 4.56, "text": "of the paths traveling to n plus one"}, {"start": 172.86, "duration": 4.98, "text": "comma n minus one but now we have our"}, {"start": 175.379, "duration": 5.341, "text": "strategy to count the good lattice paths"}, {"start": 177.84, "duration": 4.56, "text": "the ones from 0 0 to NN traveling"}, {"start": 180.72, "duration": 4.86, "text": "Northeast and not passing below the"}, {"start": 182.4, "duration": 5.52, "text": "diagonal in order to do this we must"}, {"start": 185.58, "duration": 5.579, "text": "first have all the paths that travel"}, {"start": 187.92, "duration": 4.92, "text": "Northeast from 0 0 to NN as I mentioned"}, {"start": 191.159, "duration": 4.44, "text": "before this is counted by the binomial"}, {"start": 192.84, "duration": 5.94, "text": "coefficient 2N choose n then we're going"}, {"start": 195.599, "duration": 5.28, "text": "to subtract all of the bad paths but we"}, {"start": 198.78, "duration": 3.959, "text": "know that those correspond to the"}, {"start": 200.879, "duration": 4.621, "text": "Northeast lattice paths traveling from 0"}, {"start": 202.739, "duration": 4.441, "text": "0 to n plus one comma n minus 1 which is"}, {"start": 205.5, "duration": 4.62, "text": "also counted by the binomial coefficient"}, {"start": 207.18, "duration": 4.74, "text": "2N choose n plus one but then we can use"}, {"start": 210.12, "duration": 4.02, "text": "binomial coefficient identities and"}, {"start": 211.92, "duration": 4.8, "text": "formulas to see that c sub n is equal to"}, {"start": 214.14, "duration": 4.56, "text": "1 divided by n plus 1 times 2N choose n"}, {"start": 216.72, "duration": 4.14, "text": "this is a closed formula for the"}, {"start": 218.7, "duration": 3.5, "text": "sequence c n which are known as the"}, {"start": 220.86, "duration": 4.019, "text": "Catalan numbers"}, {"start": 222.2, "duration": 6.42, "text": "you can find this result in any standard"}, {"start": 224.879, "duration": 3.741, "text": "text describing the Catalan numbers"}], "title": "Walking city streets: Catalan Closed Form (visual proof from lattice paths)", "description": "In this video, we show how to provide a closed form for the number of northeast lattice paths to the point (n,n) that don't pass below the line y=x. The number of such lattice paths is counted by the famous Catalan numbers. \n\nFor other videos discussing lattice paths (and those mentioned in this video), check out \n1) https://youtu.be/9YUl0k2FYzc\n2) https://youtu.be/KYpOoA2D63w \n3) https://youtu.be/__w_MDQeUYc \n\nIf you like this video, check out my others and consider subscribing. Thanks!\n\n#catalannumbers #catalan\u200b #manim\u200b #math\u200b #mtbos\u200b \u200b #animation\u200b #theorem\u200b\u200b #visualproof\u200b #proof\u200b #iteachmath #mathematics #binomialcoefficients #latticepaths #discretemath #combinatorics #enumeration #closedform\n\nThis animation is based on the well-known interpretation of Catalan numbers as counting restricted lattice paths. I recommend the Wikipedia site or the books Enumerative Combinatorics I and II by Richard Stanley for someone wanting to know more about Catalan numbers.\n\nTo learn more about animating with manim, check out:\nhttps://manim.community\n_________________________________________\nMusic in this video:\nCylinder Two by Chris Zabriskie is licensed under a Creative Commons Attribution 4.0 license. https://creativecommons.org/licenses/by/4.0/\n\nSource: http://chriszabriskie.com/cylinders/\n\nArtist: http://chriszabriskie.com/", "lengthSeconds": 228, "width": 640, "height": 360}
{"video_id": "BBrtHbRcpXs", "transcript": [{"start": 0.42, "duration": 6.56, "text": "foreign"}, {"start": 1.45, "duration": 8.45, "text": "[Music]"}, {"start": 6.98, "duration": 4.48, "text": "here's an intriguing number pattern one"}, {"start": 9.9, "duration": 3.54, "text": "plus two is three"}, {"start": 11.46, "duration": 3.059, "text": "four plus five plus six is seven plus"}, {"start": 13.44, "duration": 3.54, "text": "eight"}, {"start": 14.519, "duration": 5.401, "text": "nine plus ten plus eleven plus twelve"}, {"start": 16.98, "duration": 5.7, "text": "is 13 plus 14 plus 15 and 16 plus 17"}, {"start": 19.92, "duration": 6.359, "text": "plus 18 plus 19 plus 20 is 21 plus 22"}, {"start": 22.68, "duration": 6.54, "text": "plus 23 plus 24 and so on this pattern"}, {"start": 26.279, "duration": 5.281, "text": "continues as long as we take it out the"}, {"start": 29.22, "duration": 4.38, "text": "question I have for you is can you"}, {"start": 31.56, "duration": 4.92, "text": "determine the general pattern and the"}, {"start": 33.6, "duration": 4.74, "text": "general formula and if so can you"}, {"start": 36.48, "duration": 4.87, "text": "provide a proof that this formula always"}, {"start": 38.34, "duration": 6.159, "text": "holds pause the video and give it a try"}, {"start": 41.35, "duration": 3.149, "text": "[Music]"}, {"start": 45.44, "duration": 4.959, "text": "if you need a hint the first term and"}, {"start": 48.18, "duration": 5.28, "text": "every sum on the left is of the form N"}, {"start": 50.399, "duration": 5.401, "text": "squared for an integer n"}, {"start": 53.46, "duration": 5.579, "text": "in the first term in every sum on the"}, {"start": 55.8, "duration": 6.0, "text": "right is of the form N squared plus n"}, {"start": 59.039, "duration": 4.621, "text": "plus 1 for the same value of n let's"}, {"start": 61.8, "duration": 3.66, "text": "investigate A visual representation of"}, {"start": 63.66, "duration": 5.34, "text": "these sums in order to uncover the"}, {"start": 65.46, "duration": 5.82, "text": "formula Take N squared times n plus 1"}, {"start": 69.0, "duration": 7.56, "text": "cubes and place them in a cuboid with"}, {"start": 71.28, "duration": 7.86, "text": "Dimensions n by n plus 1 by n on top of"}, {"start": 76.56, "duration": 6.18, "text": "this cuboid place a triangular array of"}, {"start": 79.14, "duration": 6.36, "text": "cubes with a base of length n this adds"}, {"start": 82.74, "duration": 7.46, "text": "an extra n times n plus 1 divided by two"}, {"start": 85.5, "duration": 4.7, "text": "cubes on top of the cuboid foreign"}, {"start": 90.26, "duration": 3.34, "text": "[Music]"}, {"start": 91.74, "duration": 3.839, "text": "utilizing this structure in two"}, {"start": 93.6, "duration": 5.22, "text": "different ways will reveal the formula"}, {"start": 95.579, "duration": 5.521, "text": "that we want if we first separate the"}, {"start": 98.82, "duration": 4.56, "text": "structure into horizontal arrays like"}, {"start": 101.1, "duration": 4.92, "text": "this we see that all of the green"}, {"start": 103.38, "duration": 5.279, "text": "rectangular arrays are n by n"}, {"start": 106.02, "duration": 5.34, "text": "so there is a total number of N squared"}, {"start": 108.659, "duration": 5.521, "text": "plus N squared plus 1 up to N squared"}, {"start": 111.36, "duration": 5.1, "text": "plus n cubes on the left on the other"}, {"start": 114.18, "duration": 4.68, "text": "hand if we separate the structure"}, {"start": 116.46, "duration": 5.22, "text": "vertically we end up with rectangular"}, {"start": 118.86, "duration": 4.799, "text": "arrays of the size n plus 1 by n with"}, {"start": 121.68, "duration": 4.32, "text": "cubes on top so there's a total of N"}, {"start": 123.659, "duration": 4.8, "text": "squared plus n plus one up to N squared"}, {"start": 126.0, "duration": 3.84, "text": "plus n plus n total cubes on the right"}, {"start": 128.459, "duration": 3.181, "text": "side"}, {"start": 129.84, "duration": 4.02, "text": "therefore the formula that we're looking"}, {"start": 131.64, "duration": 4.56, "text": "for says that the sum of the numbers"}, {"start": 133.86, "duration": 4.08, "text": "from N squared to N squared plus n is"}, {"start": 136.2, "duration": 4.02, "text": "equal to the sum of the numbers from N"}, {"start": 137.94, "duration": 4.76, "text": "squared plus n plus 1 up to N squared"}, {"start": 140.22, "duration": 2.48, "text": "plus 2N"}], "title": "Unbelievable Equal Consecutive Sums (3D visual proof)", "description": "This is a short, animated visual proof showing how two related sums of consecutive integers are equal using a 3D cuboid with an extra triangular array on top. #manim #math #mathvideo #mathshorts #triangularnumbers  #numbertheory  #triangle  #animation #theorem #pww #proofwithoutwords #visualproof #proof\n\nThis animation is is based on a proof without words by Roger B. Nelsen from the February 1990 issue of Mathematics Magazine (https://doi.org/10.2307/2691505) page 25.\n\nHere is a recent related video turned out by @Mathologer : https://youtu.be/XR2u7izwZ04\n\nTo learn more about animating with manim, check out:\nhttps://manim.community\n____________________________________________________\nBackground music: \nRun Until Your Wings Grow\nLate Night Feeler", "lengthSeconds": 148, "width": 640, "height": 360}
{"video_id": "I5IrNug26S8", "transcript": [{"start": 0.12, "duration": 4.04, "text": "here's four proofs that the area of the"}, {"start": 1.92, "duration": 4.12, "text": "regular do deagon in a unit circle is"}, {"start": 4.16, "duration": 4.08, "text": "three the higher the level the better"}, {"start": 6.04, "duration": 2.2, "text": "the"}, {"start": 8.31, "duration": 9.65, "text": "[Music]"}, {"start": 16.359, "duration": 5.76, "text": "proof"}, {"start": 17.96, "duration": 6.6, "text": "yeah yeah sing in the street the sun is"}, {"start": 22.119, "duration": 6.041, "text": "in the East"}, {"start": 24.56, "duration": 6.519, "text": "yeah yeah calling out the Beast the"}, {"start": 28.16, "duration": 6.559, "text": "tiger underneath"}, {"start": 31.079, "duration": 7.441, "text": "yeah yeah I'm starting to believe the"}, {"start": 34.719, "duration": 6.881, "text": "world is at my feet"}, {"start": 38.52, "duration": 7.08, "text": "yeah I'm ready for the feast it's all"}, {"start": 41.6, "duration": 10.0, "text": "within my reach it's all within my"}, {"start": 45.6, "duration": 9.799, "text": "reach I'm going all the way I'm going"}, {"start": 51.6, "duration": 10.32, "text": "yeah my blood is Rebel I'm going all the"}, {"start": 55.399, "duration": 6.521, "text": "way I'm going to another level"}, {"start": 62.24, "duration": 3.159, "text": "[Applause]"}, {"start": 68.83, "duration": 8.329, "text": "[Applause]"}, {"start": 72.68, "duration": 7.92, "text": "if I'm on the loose running after you"}, {"start": 77.159, "duration": 6.561, "text": "yeah yeah there's nothing you can do"}, {"start": 80.6, "duration": 7.6, "text": "I'll catch you when it's through"}, {"start": 83.72, "duration": 9.2, "text": "yeah yeah hle for the juice yeah 120"}, {"start": 88.2, "duration": 7.84, "text": "poof yeah yeah yeah yeah I'm blowing up"}, {"start": 92.92, "duration": 5.28, "text": "the roof blowing up the roof got nothing"}, {"start": 96.04, "duration": 8.359, "text": "left to"}, {"start": 98.2, "duration": 11.519, "text": "Pro I'm going all the way I'm going yeah"}, {"start": 104.399, "duration": 9.241, "text": "my blood is reel going all the way I'm"}, {"start": 109.719, "duration": 3.921, "text": "going to another"}, {"start": 114.85, "duration": 3.99, "text": "[Applause]"}, {"start": 116.119, "duration": 4.32, "text": "level let"}, {"start": 118.84, "duration": 4.639, "text": "go"}, {"start": 120.439, "duration": 3.04, "text": "got to get my hands"}, {"start": 124.2, "duration": 9.24, "text": "dirty chain breaker Earth Shaker rain"}, {"start": 129.52, "duration": 6.6, "text": "maker yeah another level chain breaker"}, {"start": 133.44, "duration": 6.6, "text": "Earth Shaker rain"}, {"start": 136.12, "duration": 6.56, "text": "maker yeah another level chain breaker"}, {"start": 140.04, "duration": 6.559, "text": "Earth Shaker rain"}, {"start": 142.68, "duration": 8.44, "text": "maker yeah another level chain breaker"}, {"start": 146.599, "duration": 10.241, "text": "Earth Shaker rain maker oh to another"}, {"start": 151.12, "duration": 9.52, "text": "level I'm going all the way I'm going"}, {"start": 156.84, "duration": 9.479, "text": "yeah my blood is re I'm going all the"}, {"start": 160.64, "duration": 5.679, "text": "way I'm going to another"}, {"start": 167.45, "duration": 4.67, "text": "[Applause]"}, {"start": 169.28, "duration": 6.239, "text": "level another"}, {"start": 172.12, "duration": 6.72, "text": "Lev let's go let's go let's"}, {"start": 175.519, "duration": 6.321, "text": "go to another"}, {"start": 178.84, "duration": 3.0, "text": "level"}, {"start": 185.28, "duration": 3.4, "text": "you let's get"}, {"start": 191.76, "duration": 6.039, "text": "it let me know in the comments which of"}, {"start": 194.04, "duration": 3.759, "text": "the four is your favorite thanks for"}, {"start": 198.76, "duration": 3.0, "text": "watching"}], "title": "Area of a Regular Dodecagon | Proofs without Words Levels 1 to 4", "description": "This is a compilation of four animated visual proofs demonstrating that the area of a regular dodecagon inscribed in the unit circle has an area of exactly 3.  \n\nIf you like this video, consider subscribing to the channel or consider buying me a coffee: https://www.buymeacoffee.com/VisualProofs. Thanks! \n\nLevel 1 is a standard construction using trigonometry that you can find in many sources.\n\nLevel 2 is based on a dissection from the book The Penguin Book of Curious and Interesting Puzzles by David Wells. https://amzn.to/3O4CnUF : I may receive commissions from this link\n\nThanks to Matthew Scroggs for pointing me to the source. \n\nLevel 3 is based on a proof by Roger B. Nelsen from the January 2015 issue of The College Mathematics Journal page 10 (https://doi.org/10.4169/college.math.j.46.1.10 ).\n\nLevel 4 is based on a dissection by J. K\u00fcrsch\u00e1k that appears in the following sources:\n\nMathematical Morsels by Ross Honsberger (MAA, 1978)\n\nProofs without Words II by Roger B. Nelsen (MAA, 2000) (https://bookstore.ams.org/clrm-14/)\n\nSee these videos for standalone versions of these dissections:\nhttps://youtu.be/LbCghvu6G5w\nhttps://youtu.be/aSdo2ug4EyE\nhttps://youtu.be/7lCJDdD_fHs\nhttps://youtube.com/shorts/uxL7K2IZBf8\nhttps://youtube.com/shorts/f_y-V75O93c\nhttps://youtube.com/shorts/AgdgwQbWFvM\n\n#math\u200b #manim\u200b #visualproof\u200b #mathvideo\u200b  #geometry #mathshorts\u200b  #geometry #mtbos\u200b  #animation\u200b #theorem\u200b #pww\u200b #proofwithoutwords\u200b #proof\u200b #iteachmath #dodecagon #area #dissection \n\nTo learn more about animating with manim, check out:\nhttps://manim.community", "lengthSeconds": 200, "width": 640, "height": 360}
{"video_id": "2WuSNveg-1c", "transcript": [{"start": 0.04, "duration": 8.92, "text": "[Music]"}, {"start": 7.2, "duration": 3.84, "text": "start with the real number r between 0"}, {"start": 8.96, "duration": 4.16, "text": "and 1."}, {"start": 11.04, "duration": 3.84, "text": "draw an infinite vertical line connected"}, {"start": 13.12, "duration": 3.999, "text": "to a horizontal line of length given by"}, {"start": 14.88, "duration": 4.0, "text": "the sum of the positive powers of r"}, {"start": 17.119, "duration": 4.16, "text": "which is a geometric series and thus"}, {"start": 18.88, "duration": 4.159, "text": "converges to a finite value"}, {"start": 21.279, "duration": 3.681, "text": "from the end of the horizontal line"}, {"start": 23.039, "duration": 4.0, "text": "build a staircase type diagram where the"}, {"start": 24.96, "duration": 4.639, "text": "stair heights are all one and the stair"}, {"start": 27.039, "duration": 4.881, "text": "depths are the positive powers of r so"}, {"start": 29.599, "duration": 4.001, "text": "that each stair platform gets narrower"}, {"start": 31.92, "duration": 3.76, "text": "this object has been called gabriel's"}, {"start": 33.6, "duration": 4.4, "text": "staircase it has an infinite perimeter"}, {"start": 35.68, "duration": 4.32, "text": "but encloses a finite area"}, {"start": 38.0, "duration": 3.2, "text": "let's compute the enclosed area in two"}, {"start": 40.0, "duration": 2.879, "text": "ways"}, {"start": 41.2, "duration": 3.76, "text": "first we decompose the staircase into"}, {"start": 42.879, "duration": 3.84, "text": "columns of rectangles the first one has"}, {"start": 44.96, "duration": 4.4, "text": "height 1 and width r so it contains an"}, {"start": 46.719, "duration": 4.961, "text": "area of 1 times r"}, {"start": 49.36, "duration": 4.16, "text": "the next one has height 2 and width r"}, {"start": 51.68, "duration": 4.719, "text": "squared so it contains an area of 2"}, {"start": 53.52, "duration": 2.879, "text": "times r squared"}, {"start": 56.96, "duration": 5.2, "text": "the third one has height three"}, {"start": 60.0, "duration": 4.72, "text": "and width r cubed so it encloses an area"}, {"start": 62.16, "duration": 4.8, "text": "of three times r cubed and the next one"}, {"start": 64.72, "duration": 4.719, "text": "encloses an area of four times r to the"}, {"start": 66.96, "duration": 4.56, "text": "fourth we continue this process forever"}, {"start": 69.439, "duration": 5.761, "text": "with narrower and narrower rectangles"}, {"start": 71.52, "duration": 3.68, "text": "giving an infinite sum for the area"}, {"start": 79.04, "duration": 4.0, "text": "we can also decompose the staircase into"}, {"start": 81.28, "duration": 4.159, "text": "rows of rectangles"}, {"start": 83.04, "duration": 3.439, "text": "the first such rectangle has height 1"}, {"start": 85.439, "duration": 3.121, "text": "and the width is the sum of the"}, {"start": 86.479, "duration": 4.481, "text": "geometric series with ratio r and first"}, {"start": 88.56, "duration": 5.36, "text": "term r which converges to r over 1 minus"}, {"start": 90.96, "duration": 5.119, "text": "r so the area enclosed is r over 1 minus"}, {"start": 93.92, "duration": 2.159, "text": "r"}, {"start": 96.159, "duration": 3.92, "text": "the second rectangle also has height 1"}, {"start": 98.56, "duration": 3.519, "text": "and this time the width is the geometric"}, {"start": 100.079, "duration": 3.841, "text": "series with ratio r and first term r"}, {"start": 102.079, "duration": 4.08, "text": "squared which converges to r squared"}, {"start": 103.92, "duration": 4.559, "text": "over 1 minus r so that the area is r"}, {"start": 106.159, "duration": 4.081, "text": "squared over 1 minus r"}, {"start": 108.479, "duration": 4.0, "text": "the next two rectangles are also of"}, {"start": 110.24, "duration": 5.12, "text": "height 1 and widths of geometric series"}, {"start": 112.479, "duration": 4.721, "text": "and thus they enclose areas of r cubed"}, {"start": 115.36, "duration": 6.96, "text": "over one minus r"}, {"start": 117.2, "duration": 5.12, "text": "and r to the fourth over one minus r"}, {"start": 122.96, "duration": 4.08, "text": "we again continue this process forever"}, {"start": 125.2, "duration": 3.6, "text": "with smaller and smaller rectangles"}, {"start": 127.04, "duration": 4.079, "text": "giving a different infinite sum for the"}, {"start": 128.8, "duration": 2.319, "text": "area"}, {"start": 131.52, "duration": 4.079, "text": "these two infinite sums can be shown to"}, {"start": 133.68, "duration": 4.4, "text": "both converge to a finite value because"}, {"start": 135.599, "duration": 4.961, "text": "r is between 0 and 1. the sums enclose"}, {"start": 138.08, "duration": 3.84, "text": "the same area so they must be equal thus"}, {"start": 140.56, "duration": 3.6, "text": "we have that the sum from one to"}, {"start": 141.92, "duration": 4.24, "text": "infinity of k r to the k"}, {"start": 144.16, "duration": 4.24, "text": "is equal to the sum from one to infinity"}, {"start": 146.16, "duration": 4.159, "text": "of r to the k over one minus r"}, {"start": 148.4, "duration": 3.839, "text": "but we can factor one minus r out of the"}, {"start": 150.319, "duration": 3.681, "text": "ladder sum leaving us with the geometric"}, {"start": 152.239, "duration": 4.161, "text": "series with ratio r and first term r"}, {"start": 154.0, "duration": 4.72, "text": "again which we know converges to r over"}, {"start": 156.4, "duration": 3.919, "text": "one minus r thus the sum from one to"}, {"start": 158.72, "duration": 3.84, "text": "infinity of k"}, {"start": 160.319, "duration": 4.961, "text": "times r to the k is equal to r over the"}, {"start": 162.56, "duration": 5.88, "text": "quantity one minus r squared provided r"}, {"start": 165.28, "duration": 6.46, "text": "is between zero and one"}, {"start": 168.44, "duration": 3.3, "text": "[Music]"}, {"start": 180.28, "duration": 5.16, "text": "[Music]"}, {"start": 183.36, "duration": 2.08, "text": "you"}], "title": "Arithmetico-Geometric Series Visualized (visual proof using Gabriel's Staircase)", "description": "This is a short, animated visual proof computing the sum of the differentiated geometric series with terms of the form k times r^k where r is between 0 and 1. The proof uses a fascinating object called Gabriel's staircase, which has infinite perimeter but encloses a finite area. #math\u200b #infiniteseries #mtbos\u200b #manim\u200b #animation\u200b #theorem\u200b #pww\u200b #proofwithoutwords\u200b #visualproof\u200b #proof\u200b #iteachmath #calculus #geometricseries #series #mathshorts\u200b #mathvideo\u200b \n\nIf you like this video, consider subscribing to the channel. Thanks!\n\nThis animation is based on a visual proof by Stuart G. Swain from the June 1994 issue of Mathematics Magazine (https://doi.org/10.2307/2690612 p. 209).\n\nTo learn more about animating with manim, check out:\nhttps://manim.community\n__________________________________________________________________\nMusic in this video:\nAutumn Day by Kevin MacLeod is licensed under a Creative Commons Attribution 4.0 license. https://creativecommons.org/licenses/by/4.0/\n\nSource: http://incompetech.com/music/royalty-free/index.html?isrc=USUAN1100765\n\nArtist: http://incompetech.com/", "lengthSeconds": 184, "width": 640, "height": 360}
{"video_id": "y7D6U7vnrUw", "transcript": [{"start": 0.12, "duration": 3.88, "text": "here's a problem from a 1976 Journal of"}, {"start": 2.48, "duration": 4.16, "text": "recreational mathematics issue that"}, {"start": 4.0, "duration": 5.88, "text": "Roger Nelson calls the Christmas tree"}, {"start": 6.64, "duration": 6.28, "text": "problem start with a circle and inside"}, {"start": 9.88, "duration": 6.2, "text": "the circle inscribe an isoceles triangle"}, {"start": 12.92, "duration": 5.519, "text": "along the diameter like this inside of"}, {"start": 16.08, "duration": 5.84, "text": "the isoceles triangle let's inscribe a"}, {"start": 18.439, "duration": 6.321, "text": "circle the bottom radius shown bcts the"}, {"start": 21.92, "duration": 5.64, "text": "diameter we can use the two quadrants"}, {"start": 24.76, "duration": 4.04, "text": "created to inscribe two more circles"}, {"start": 27.56, "duration": 3.84, "text": "like"}, {"start": 28.8, "duration": 4.48, "text": "this the the problem in the journal asks"}, {"start": 31.4, "duration": 4.159, "text": "you to prove that these three small"}, {"start": 33.28, "duration": 3.76, "text": "inscribed circles are all congruent"}, {"start": 35.559, "duration": 5.0, "text": "pause the video and think about this"}, {"start": 37.04, "duration": 6.519, "text": "before I reveal one possible"}, {"start": 40.559, "duration": 5.121, "text": "answer are you ready to see a solution"}, {"start": 43.559, "duration": 5.041, "text": "it turns out that we can prove this fact"}, {"start": 45.68, "duration": 4.84, "text": "visually by changing our point of view"}, {"start": 48.6, "duration": 4.56, "text": "we take a second copy of the Outer"}, {"start": 50.52, "duration": 6.039, "text": "Circle and shift it up by a distance"}, {"start": 53.16, "duration": 6.12, "text": "equal to the radius then rotate 45\u00b0 or <"}, {"start": 56.559, "duration": 4.52, "text": "over 4 radians from this perspective we"}, {"start": 59.28, "duration": 4.36, "text": "see that the top circle inscribed in the"}, {"start": 61.079, "duration": 4.44, "text": "triangle is actually one of the pair of"}, {"start": 63.64, "duration": 4.519, "text": "the two circles inscribed in the two"}, {"start": 65.519, "duration": 4.92, "text": "quadrants of this new Circle therefore"}, {"start": 68.159, "duration": 4.121, "text": "these two circles are congruent to the"}, {"start": 70.439, "duration": 4.081, "text": "original two in the bottom quadrant of"}, {"start": 72.28, "duration": 4.24, "text": "the original Circle and this means that"}, {"start": 74.52, "duration": 3.919, "text": "all three of the original inscribed"}, {"start": 76.52, "duration": 4.32, "text": "small circles are congruent to one"}, {"start": 78.439, "duration": 4.081, "text": "another thanks for watching if you like"}, {"start": 80.84, "duration": 3.599, "text": "this video and others on my channel you"}, {"start": 82.52, "duration": 3.36, "text": "might enjoy Roger Nelson's books proofs"}, {"start": 84.439, "duration": 3.521, "text": "without words there are three different"}, {"start": 85.88, "duration": 5.08, "text": "compendia Linked In the"}, {"start": 87.96, "duration": 3.0, "text": "description"}], "title": "Recreational Math Christmas Tree Problem (with visual proof)", "description": "Like this video? You might like Roger Nelsen's Proofs without Words books (I may receive a small commission at no cost to you for these affiliate links): \nhttps://amzn.to/41Cci6j\nhttps://amzn.to/41BtR6n\nhttps://amzn.to/48mjeX1\n\nIn this short video, we pose a fun problem from the 1976 Journal of Recreational Mathematics and give a visual proof that is featured in Roger Nelsen's Third Proof without Words Compendium. \n\nIf you like this video, consider subscribing to the channel or consider buying me a coffee: https://www.buymeacoffee.com/VisualProofs. Thanks! \n\nFor other visual geometric proofs like this check out: \n\nFor a static version of this proof, see Roger Nelsen's third \"Proof Without Words: Exercises in Visual Thinking\" compendium (page 39) : \n(I may receive a small commission at no cost to you for these affiliate links): \nhttps://amzn.to/41Cci6j\n \n#math #circle #pythagorean #triangle #manim #animation #theorem #pww #proofwithoutwords #visualproof #proof  #mathshorts #mathvideo #incsribed #recreationalmath\n\nTo learn more about animating with manim, check out:\nhttps://manim.community", "lengthSeconds": 89, "width": 640, "height": 360}
{"video_id": "351rFH40uwk", "transcript": [{"start": 0.0, "duration": 13.14, "text": "e o peitinho primeiro \u00e9"}, {"start": 4.35, "duration": 8.79, "text": "[M\u00fasica]"}], "title": "Visual Proof for 1/3 + 1/9 + 1/27+ 1/81 + ... = 1/2 #eduwow #reels #shorts", "description": "Visual Proof for 1/3 + 1/9 + 1/27 + 1/81 + ... = 1/2\n\n#shorts #math", "lengthSeconds": 13, "width": 202, "height": 360}
{"video_id": "MEzgGGq2l5E", "transcript": [{"start": 0.359, "duration": 3.881, "text": "here's a fascinating way to approximate"}, {"start": 2.24, "duration": 3.119, "text": "the serinsky triangle with a single"}, {"start": 4.24, "duration": 3.8, "text": "curve in the"}, {"start": 5.359, "duration": 5.121, "text": "plane let's consider a relatively simple"}, {"start": 8.04, "duration": 4.92, "text": "Linden Meer system we start with the"}, {"start": 10.48, "duration": 6.84, "text": "string a and the production rules tell"}, {"start": 12.96, "duration": 10.04, "text": "us that at each stage we replace a by B-"}, {"start": 17.32, "duration": 7.92, "text": "a-b and we replace B With A + B + a and"}, {"start": 23.0, "duration": 5.0, "text": "we will let dashes and pluses stay the"}, {"start": 25.24, "duration": 4.72, "text": "same at each stage let's see the first"}, {"start": 28.0, "duration": 3.96, "text": "few stages of this process start"}, {"start": 29.96, "duration": 3.56, "text": "starting with a the first stage results"}, {"start": 31.96, "duration": 4.439, "text": "in B-"}, {"start": 33.52, "duration": 11.68, "text": "a-b then the second stage results in the"}, {"start": 36.399, "duration": 12.84, "text": "string a plus B+ a dash B- A- B Das a +"}, {"start": 45.2, "duration": 7.08, "text": "B+ a we can continue this process over"}, {"start": 49.239, "duration": 4.48, "text": "and over as long as we like each time"}, {"start": 52.28, "duration": 4.24, "text": "applying the production"}, {"start": 53.719, "duration": 4.64, "text": "rules eventually we can stop at a"}, {"start": 56.52, "duration": 4.24, "text": "particular string and we can use the"}, {"start": 58.359, "duration": 5.481, "text": "string to walk on the plane"}, {"start": 60.76, "duration": 5.0, "text": "if we see a character of A or B we will"}, {"start": 63.84, "duration": 4.8, "text": "step one unit forward in the direction"}, {"start": 65.76, "duration": 5.56, "text": "we're facing if we see a plus sign we"}, {"start": 68.64, "duration": 6.64, "text": "will rotate 60\u00b0 to the left and if we"}, {"start": 71.32, "duration": 6.68, "text": "see a dash we will rotate 60\u00b0 to the"}, {"start": 75.28, "duration": 5.0, "text": "right let's see what shape emerges from"}, {"start": 78.0, "duration": 4.24, "text": "this Turtle Graphics process when we"}, {"start": 80.28, "duration": 3.6, "text": "investigate the string that results"}, {"start": 82.24, "duration": 3.96, "text": "after the 10th iteration of the"}, {"start": 83.88, "duration": 5.0, "text": "production rules described by this L"}, {"start": 86.2, "duration": 5.12, "text": "system here are the first few characters"}, {"start": 88.88, "duration": 6.08, "text": "of the process let's use this string to"}, {"start": 91.32, "duration": 3.64, "text": "walk on the plane to some dramatic"}, {"start": 96.59, "duration": 3.39, "text": "[Music]"}, {"start": 110.61, "duration": 3.07, "text": "[Music]"}, {"start": 118.84, "duration": 3.0, "text": "music"}, {"start": 136.03, "duration": 3.169, "text": "[Music]"}, {"start": 154.16, "duration": 3.879, "text": "[Music]"}, {"start": 180.43, "duration": 3.129, "text": "[Music]"}, {"start": 189.64, "duration": 3.16, "text": "[Music]"}, {"start": 195.48, "duration": 3.21, "text": "[Music]"}, {"start": 208.68, "duration": 3.0, "text": "n"}, {"start": 227.17, "duration": 3.07, "text": "[Music]"}, {"start": 231.799, "duration": 3.16, "text": "if you liked this video check out the"}, {"start": 233.36, "duration": 3.64, "text": "links in the description for other"}, {"start": 234.959, "duration": 5.601, "text": "videos drawing other shapes using Turtle"}, {"start": 237.0, "duration": 3.56, "text": "graphics and L systems"}], "title": "Sierpinski Arrowhead Curve L-system (118098 steps!)", "description": "In this video, we describe two how to draw a single amazing curve - the Sierpinski Arrowhead curve - using the simple processes of Turtle graphics and L-systems. We draw 118098 line segments, which is double 3 to the 10th. It is amazing that such a simple process can draw these complex curves.\n\nIf you liked this video, consider liking or subscribing. Or you can support me by buying me a coffee at https://www.buymeacoffee.com/VisualProofs \nThanks!\n\nIf you want to see two other constructions of like this, see\nhttps://youtu.be/8Q_zuRKKh2g\n\nIf you want to see other ways to construct this curve or other ways to build the Sierpinski triangle, check out this video:\nhttps://youtu.be/IZHiBJGcrqI\n\nIf you want to know more about L-systems or turtle graphics, check out these wikipedia sites:\n\nhttps://en.wikipedia.org/wiki/L-system\nhttps://en.wikipedia.org/wiki/Turtle_graphics\n\nTo learn more about animating with manim, check out:\nhttps://manim.community\n\n#turtlegraphics #fractal #math #manim #mathvideo #Lsystem #sierpinski #sierpinskitriangle #triangle #arrowheadcurve \n\n___________________________________________\nMusic in this video:\nMajestic by Alex-Productions | https://onsound.eu/\nMusic promoted by https://www.free-stock-music.com\nCreative Commons / Attribution 3.0 Unported License (CC BY 3.0)\nhttps://creativecommons.org/licenses/by/3.0/deed.en_US", "lengthSeconds": 239, "width": 640, "height": 360}
{"video_id": "rtzghwwfVoc", "transcript": [{"start": 0.0, "duration": 3.48, "text": "I bet you've seen an infinite chocolate"}, {"start": 1.62, "duration": 3.3, "text": "trick before but have you seen one that"}, {"start": 3.48, "duration": 3.899, "text": "allows you to share the chocolate with a"}, {"start": 4.92, "duration": 5.099, "text": "friend start with this chocolate bar and"}, {"start": 7.379, "duration": 5.041, "text": "split it into these pieces as we've done"}, {"start": 10.019, "duration": 4.381, "text": "here there are actually two squares of"}, {"start": 12.42, "duration": 4.08, "text": "chocolate left inside you can split"}, {"start": 14.4, "duration": 4.139, "text": "those apart eat one yourself and share"}, {"start": 16.5, "duration": 3.72, "text": "the other with your favorite friend but"}, {"start": 18.539, "duration": 3.24, "text": "now to make sure that nobody knows that"}, {"start": 20.22, "duration": 3.54, "text": "you've done this you can take the"}, {"start": 21.779, "duration": 4.021, "text": "remaining pieces of the chocolate and"}, {"start": 23.76, "duration": 4.56, "text": "reorganize the triangular pieces like"}, {"start": 25.8, "duration": 4.799, "text": "this on either side and then take both"}, {"start": 28.32, "duration": 4.8, "text": "of the L-shaped pieces and drop them"}, {"start": 30.599, "duration": 4.261, "text": "down into place this way once you've"}, {"start": 33.12, "duration": 3.779, "text": "done this the full original chocolate"}, {"start": 34.86, "duration": 4.379, "text": "bar has been restored and no chocolate"}, {"start": 36.899, "duration": 4.201, "text": "is missing this triangular diagram is"}, {"start": 39.239, "duration": 3.66, "text": "based on the curry triangle which is a"}, {"start": 41.1, "duration": 4.5, "text": "dissection fallacy that allows you to"}, {"start": 42.899, "duration": 5.241, "text": "quote unquote prove that 60 59 and 58"}, {"start": 45.6, "duration": 2.54, "text": "are all equal"}], "title": "The Infinite Chocolate Trick with a Friendly Twist #shorts", "description": "In this short, we show a dissection fallacy that allows you to show visually (though incorrectly) that 58=59=60. This is based on the Curry triangle. \n\nIf you like this video, consider subscribing to the channel or consider buying me a coffee: https://www.buymeacoffee.com/VisualProofs. Thanks!\n\nFor more on the Curry triangle shown here, check out this longer video that exposes the dissection fallacy: https://youtu.be/iMgFDhpa000\n\nThis visual was created by L. Vosburgh Lyons based on work by Paul Curry. See the MathWorld page for more information:\nhttps://mathworld.wolfram.com/CurryTriangle.html\n\n\u200b\u2060@JamesTantonMath did a real life cake version of this! https://youtu.be/o0_ysMQ9ne0\n\n#math\u200b #paradox #mtbos\u200b #manim\u200b #animation\u200b #theorem\u200b #pww\u200b #proofwithoutwords\u200b #visualproof\u200b #proof\u200b #iteachmath #calculus #dissection #area #currytriangle #missingsquare #mathshorts\u200b #mathvideo\u200b #infinitechocolate #chocolate #sharing #fallacy \n\nHere are some other resources with infinite chocolate games:\nhttps://en.wikipedia.org/wiki/Missing_square_puzzle\nhttps://youtube.com/shorts/wGYrdpKDrb4?feature=share (maxdesignpro)\nhttps://youtube.com/shorts/6aDB9VLnyZQ?feature=share (mindyourdecisions)\nhttps://youtube.com/shorts/rBA0ujypLgI?feature=share (mindyourdecisions)\nhttps://youtube.com/shorts/j8bok-ce-iU?feature=share (kylestook)\nhttps://youtube.com/shorts/SkipjSkqifE?feature=share (_vector_)\n\nTo learn more about animating with manim, check out:\nhttps://manim.community", "lengthSeconds": 47, "width": 360, "height": 640}
{"video_id": "1ycDCxmmOx8", "transcript": [{"start": 0.12, "duration": 3.559, "text": "suppose we have a right triangle with"}, {"start": 1.64, "duration": 4.48, "text": "legs of length A and B and a hypotenuse"}, {"start": 3.679, "duration": 4.681, "text": "of C we can draw an A by a square off"}, {"start": 6.12, "duration": 4.599, "text": "the side of length a we can draw a b by"}, {"start": 8.36, "duration": 4.0, "text": "b square off the side of length B and we"}, {"start": 10.719, "duration": 3.561, "text": "can draw a c by C Square Off the"}, {"start": 12.36, "duration": 4.64, "text": "hypotenuse though we leave the C by C"}, {"start": 14.28, "duration": 2.72, "text": "Square unshaped"}], "title": "Pythagorean theorem \ud83d\udcd0| Pythagorean theorem visual proof | #fyp\u30b7 #fyp #shorts  #pythagoras_theorem", "description": "Pythagorean theorem \ud83d\udcd0| Pythagorean theorem visual proof | #fyp\u30b7 #fyp #shorts  #pythagoras_theorem #ytshorts #pythagoreantheorem #pythagorean #maths #mathstrick #mathematics", "lengthSeconds": 15, "width": 360, "height": 640}
{"video_id": "02yM5LqMWUs", "transcript": [{"start": 0.0, "duration": 3.72, "text": "[Music]"}, {"start": 1.22, "duration": 4.18, "text": "what do these diagrams have to do with"}, {"start": 3.72, "duration": 5.78, "text": "the arithmetic mean geometric mean"}, {"start": 5.4, "duration": 4.1, "text": "inequality let's find out"}, {"start": 9.88, "duration": 4.879, "text": "[Music]"}, {"start": 12.36, "duration": 5.52, "text": "start with a right triangle with legs of"}, {"start": 14.759, "duration": 6.421, "text": "length A and B"}, {"start": 17.88, "duration": 6.479, "text": "use four copies of the triangle rotated"}, {"start": 21.18, "duration": 7.259, "text": "so that we create this shape which is a"}, {"start": 24.359, "duration": 7.141, "text": "square with a side length of a plus b"}, {"start": 28.439, "duration": 5.041, "text": "inside there's a missing Square reflect"}, {"start": 31.5, "duration": 4.86, "text": "each triangle about its hypotenuse"}, {"start": 33.48, "duration": 5.64, "text": "creating this diagram using eight"}, {"start": 36.36, "duration": 3.6, "text": "triangles notice there's still a missing"}, {"start": 39.12, "duration": 3.0, "text": "Square inside"}, {"start": 39.96, "duration": 4.619, "text": "[Music]"}, {"start": 42.12, "duration": 5.16, "text": "to verify that this missing inside shape"}, {"start": 44.579, "duration": 5.341, "text": "is a square notice that the side length"}, {"start": 47.28, "duration": 5.459, "text": "can be computed as the difference of the"}, {"start": 49.92, "duration": 5.279, "text": "leg lengths of the original triangle so"}, {"start": 52.739, "duration": 5.401, "text": "we see that the square has a side of"}, {"start": 55.199, "duration": 5.581, "text": "length a minus B"}, {"start": 58.14, "duration": 5.879, "text": "now we can find the area of this shape"}, {"start": 60.78, "duration": 5.46, "text": "in two different ways first we note that"}, {"start": 64.019, "duration": 5.1, "text": "it is an A plus b by a plus b rectangle"}, {"start": 66.24, "duration": 4.919, "text": "so the area is a plus b quantity squared"}, {"start": 69.119, "duration": 4.381, "text": "on the other hand we've used eight"}, {"start": 71.159, "duration": 4.261, "text": "triangles plus the inner Square so we"}, {"start": 73.5, "duration": 4.92, "text": "have an area of eight times a half a B"}, {"start": 75.42, "duration": 4.92, "text": "plus the quantity a minus B squared we"}, {"start": 78.42, "duration": 4.559, "text": "can simplify the eight triangles to have"}, {"start": 80.34, "duration": 4.56, "text": "the area four times a times B now we"}, {"start": 82.979, "duration": 4.261, "text": "note that the quantity a minus B squared"}, {"start": 84.9, "duration": 4.98, "text": "is always non-negative so if we leave it"}, {"start": 87.24, "duration": 4.199, "text": "off we obtain the inequality a plus b"}, {"start": 89.88, "duration": 3.0, "text": "squared is greater than or equal to 4"}, {"start": 91.439, "duration": 4.261, "text": "times a times B"}, {"start": 92.88, "duration": 4.62, "text": "as we vary A and B we see that the inner"}, {"start": 95.7, "duration": 4.62, "text": "Square gets larger and smaller depending"}, {"start": 97.5, "duration": 5.159, "text": "on the sizes of A and B but we also note"}, {"start": 100.32, "duration": 4.32, "text": "that the square is always there except"}, {"start": 102.659, "duration": 3.6, "text": "in the one instance when A and B are"}, {"start": 104.64, "duration": 3.9, "text": "exactly the same so that that square"}, {"start": 106.259, "duration": 4.081, "text": "shrinks to zero area"}, {"start": 108.54, "duration": 4.619, "text": "from this we conclude that we have"}, {"start": 110.34, "duration": 5.4, "text": "equality amongst the inequality if and"}, {"start": 113.159, "duration": 3.78, "text": "only if a minus B is zero so that a"}, {"start": 115.74, "duration": 3.239, "text": "equals B"}, {"start": 116.939, "duration": 4.201, "text": "upon taking square roots and dividing"}, {"start": 118.979, "duration": 4.14, "text": "both sides by 2 we obtain the inequality"}, {"start": 121.14, "duration": 4.019, "text": "a plus b over 2 is greater than or equal"}, {"start": 123.119, "duration": 3.78, "text": "to the square root of a times B this is"}, {"start": 125.159, "duration": 4.401, "text": "the arithmetic mean geometric mean"}, {"start": 126.899, "duration": 2.661, "text": "inequality"}], "title": "AM-GM Inequality VI (visual proof)", "description": "This is a short, animated visual proof demonstrating the arithmetic mean geometric mean inequality using 8 triangles arranged in a famous shape for visual proofs. #mathshorts\u200b #mathvideo\u200b #math\u200b #amgminequality #mtbos\u200b #manim\u200b #animation\u200b #theorem\u200b #pww\u200b #proofwithoutwords\u200b #visualproof\u200b #proof\u200b #iteachmath #inequality #arithmeticmean #geometricmean \n\nFor other visual proofs of this fact see:\n\nhttps://youtu.be/Pt8IX_Q5U1A\nhttps://youtu.be/62G9fak1vyk\nhttps://youtu.be/HFGSsPbYc-s\nhttps://youtu.be/SEPkdPWKdO0\nhttps://youtu.be/jNx05gXSblE\nhttps://youtu.be/IJGwvvQdfgg\n\nThis animation is based on a visual proof from Ayoub B. Ayoub that appeared in the Spring 1997 issue of Mathematics and Computer Education (vol 31, no 2 p191).\n\nTo learn more about animating with manim, check out:\nhttps://manim.community", "lengthSeconds": 135, "width": 640, "height": 360}
{"video_id": "0QNWU-He7uc", "transcript": [{"start": 0.46, "duration": 12.72, "text": "[Music]"}, {"start": 21.73, "duration": 3.07, "text": "[Music]"}, {"start": 26.48, "duration": 3.0, "text": "keep"}, {"start": 38.84, "duration": 3.68, "text": "going to Le but"}], "title": "Trapezoid Area Formula: Visual Proof with Maths Animation #ssc #maths #ytshorts", "description": "Trapezoid Area Formula: Visual Proof with Maths Animation #ssc #maths #ytshorts. In this video, we simplify the Trapezoid Area Formula using engaging visual proof and maths animation. Perfect for SSC exam aspirants and anyone looking to understand geometry concepts more clearly, this short tutorial breaks down the formula step by step. Whether you're struggling with this topic or just want to reinforce your knowledge, our clear and concise explanation will help you grasp the concept quickly. Make sure to watch till the end for a full understanding and share with your friends who might find this useful. Don't forget to like, subscribe, and hit the bell icon for more educational content!\nalfaproton\nalphaproton\nalpha proton\n\n\nHashtags\n\n#TrapezoidArea\n#Maths\n#YtShorts\n#ssc #upsccsat\n#mathsanimation #alfaproton\n#Geometry\n#MathsTutorial\n#ExamPreparation\n#EasyMaths\n#LearningMadeEasy\nTags\ntrapezoid area formula, visual proof, maths animation, ssc, maths, ytshorts, geometry, trapezoid, easy maths, maths tutorial, exam preparation, trapezoid formula, visual maths, maths tricks, visual geometry, trapezoid area, maths short, learning maths, geometry animation, trapezoid explanation, ssc maths, quick maths, maths guide, area of trapezoid, visual learning, maths hacks, ssc preparation, maths concepts, geometry formula, learning made easy, maths video, educational content, trapezoid area explained, quick geometry, study maths, trapezoid breakdown, maths revision", "lengthSeconds": 41, "width": 360, "height": 640}
{"video_id": "KYpOoA2D63w", "transcript": [{"start": 1.72, "duration": 6.409, "text": "[Music]"}, {"start": 9.44, "duration": 3.84, "text": "recall that the binomial coefficient n"}, {"start": 11.44, "duration": 3.92, "text": "choose a represents the number of ways"}, {"start": 13.28, "duration": 4.64, "text": "of selecting a unordered objects from a"}, {"start": 15.36, "duration": 4.32, "text": "set of size n"}, {"start": 17.92, "duration": 3.199, "text": "consider the lattice of points in the"}, {"start": 19.68, "duration": 3.679, "text": "plane with non-negative integer"}, {"start": 21.119, "duration": 4.0, "text": "coordinates let's label the origin and"}, {"start": 23.359, "duration": 3.92, "text": "the point a b"}, {"start": 25.119, "duration": 4.961, "text": "how many northeast lattice paths are"}, {"start": 27.279, "duration": 4.721, "text": "there from 0 0 to a b where a northeast"}, {"start": 30.08, "duration": 4.24, "text": "path is one that only allows steps"}, {"start": 32.0, "duration": 4.719, "text": "directly right or up"}, {"start": 34.32, "duration": 5.36, "text": "let's enumerate them for a equals 2 and"}, {"start": 36.719, "duration": 2.961, "text": "b equals 3."}, {"start": 41.36, "duration": 4.56, "text": "as we build all the paths we realize"}, {"start": 44.0, "duration": 5.039, "text": "that each path has the same number of"}, {"start": 45.92, "duration": 5.36, "text": "steps namely a plus b"}, {"start": 49.039, "duration": 5.281, "text": "moreover we see that each path has"}, {"start": 51.28, "duration": 3.759, "text": "exactly a steps to the right and b steps"}, {"start": 54.32, "duration": 3.2, "text": "up"}, {"start": 55.039, "duration": 5.461, "text": "so we only need to keep track of where"}, {"start": 57.52, "duration": 5.039, "text": "and when the right steps occur"}, {"start": 60.5, "duration": 4.54, "text": "[Music]"}, {"start": 62.559, "duration": 5.041, "text": "we can keep track of these steps by"}, {"start": 65.04, "duration": 3.92, "text": "listing off the order of the steps"}, {"start": 67.6, "duration": 4.72, "text": "for instance"}, {"start": 68.96, "duration": 5.68, "text": "the first path is r r u u u because the"}, {"start": 72.32, "duration": 5.76, "text": "right steps happen first and the last"}, {"start": 74.64, "duration": 6.799, "text": "path is u u u r r because the two right"}, {"start": 78.08, "duration": 3.359, "text": "steps happen at the end"}, {"start": 81.6, "duration": 4.559, "text": "a consequence of this encoding is that"}, {"start": 83.6, "duration": 5.04, "text": "the north east lattice paths to av"}, {"start": 86.159, "duration": 6.561, "text": "correspond to strings of r's and u's"}, {"start": 88.64, "duration": 4.08, "text": "with ars and bu's"}, {"start": 92.88, "duration": 5.12, "text": "thus such a lattice path corresponds to"}, {"start": 95.759, "duration": 4.801, "text": "a selection of a unordered positions"}, {"start": 98.0, "duration": 5.28, "text": "from a plus b spots to place the rs and"}, {"start": 100.56, "duration": 6.16, "text": "the u's go in the other spots"}, {"start": 103.28, "duration": 5.28, "text": "hence northeast lattice paths to a b are"}, {"start": 106.72, "duration": 4.88, "text": "counted by the binomial coefficient a"}, {"start": 108.56, "duration": 3.04, "text": "plus b choose a"}, {"start": 112.479, "duration": 5.441, "text": "returning to our lattice paths"}, {"start": 115.28, "duration": 3.92, "text": "we see that we can group them by their"}, {"start": 117.92, "duration": 3.519, "text": "final step"}, {"start": 119.2, "duration": 5.04, "text": "so in our example"}, {"start": 121.439, "duration": 5.521, "text": "four end with a right step and six end"}, {"start": 124.24, "duration": 5.2, "text": "with an up step"}, {"start": 126.96, "duration": 4.32, "text": "notice that all the paths ending in r"}, {"start": 129.44, "duration": 5.04, "text": "must first pass through the point a"}, {"start": 131.28, "duration": 3.2, "text": "minus one comma b"}, {"start": 135.03, "duration": 6.33, "text": "[Music]"}, {"start": 138.8, "duration": 4.96, "text": "similarly all the paths ending in an up"}, {"start": 141.36, "duration": 5.5, "text": "step u must first pass through the point"}, {"start": 143.76, "duration": 8.0, "text": "a comma b minus one"}, {"start": 146.86, "duration": 7.54, "text": "[Music]"}, {"start": 151.76, "duration": 5.36, "text": "these two collections of paths are"}, {"start": 154.4, "duration": 5.44, "text": "clearly disjoint and make up all of the"}, {"start": 157.12, "duration": 4.88, "text": "possible northeast lattice paths to the"}, {"start": 159.84, "duration": 3.759, "text": "point a b"}, {"start": 162.0, "duration": 4.56, "text": "thus"}, {"start": 163.599, "duration": 5.92, "text": "a plus b choose a"}, {"start": 166.56, "duration": 5.12, "text": "is equal to a minus 1 plus b choose a"}, {"start": 169.519, "duration": 4.481, "text": "minus 1 which is the number of lattice"}, {"start": 171.68, "duration": 3.52, "text": "paths to a minus 1 comma b"}, {"start": 174.0, "duration": 3.44, "text": "plus"}, {"start": 175.2, "duration": 4.08, "text": "the binomial coefficient a plus b minus"}, {"start": 177.44, "duration": 5.6, "text": "1 choose a which is the number of"}, {"start": 179.28, "duration": 6.239, "text": "lattice paths to a comma b minus 1."}, {"start": 183.04, "duration": 5.199, "text": "replacing a plus b by n"}, {"start": 185.519, "duration": 5.761, "text": "results in n choose a equals n minus 1"}, {"start": 188.239, "duration": 5.891, "text": "choose a minus 1 plus n minus 1 choose a"}, {"start": 191.28, "duration": 13.489, "text": "this is the binomial recurrence"}, {"start": 194.13, "duration": 10.639, "text": "[Music]"}, {"start": 206.0, "duration": 2.08, "text": "you"}], "title": "The Binomial Recurrence from Lattice Paths (visual proof)", "description": "This short animated proof demonstrates one combinatorial proof for the recurrence satisfied by the binomial coefficients. #mathshorts\u200b #mathvideo\u200b #math\u200b #mtbos\u200b #manim\u200b #animation\u200b #theorem\u200b\u200b #visualproof\u200b #proof\u200b #iteachmath #mathematics #binomialcoefficients #latticepaths #discretemath #combinatorics #combinatorialproof \n\nThis animation is based on the well-known interpretation of binomial coefficients as counting lattice paths. You can find more in many textbooks such as Oscar Levin's Discrete Mathematics An Open Introduction (http://discrete.openmathbooks.org/dmoi2/sec_counting-binom.html). \n\nThis visual proof was inspired by the work of Steven Klee with his undergraduates:\n\nhttps://cs.uwaterloo.ca/journals/JIS/VOL21/Klee/klee2.pdf\n\nand\n\nhttps://cs.uwaterloo.ca/journals/JIS/VOL23/Klee/klee4.pdf .\n\nTo learn more about animating with manim, check out:\nhttps://manim.community", "lengthSeconds": 206, "width": 640, "height": 360}
{"video_id": "hP5ExUA5P8A", "transcript": [{"start": 2.43, "duration": 7.96, "text": "[Music]"}, {"start": 13.99, "duration": 3.77, "text": "[Music]"}, {"start": 15.92, "duration": 4.0, "text": "an odd square number can be visualized"}, {"start": 17.76, "duration": 4.72, "text": "as a square array with side length two n"}, {"start": 19.92, "duration": 2.56, "text": "plus one"}, {"start": 25.199, "duration": 6.0, "text": "this array has a triangular array"}, {"start": 28.96, "duration": 3.759, "text": "with base n sitting in the upper left"}, {"start": 31.199, "duration": 3.36, "text": "corner"}, {"start": 32.719, "duration": 3.921, "text": "if we swing the triangular array out and"}, {"start": 34.559, "duration": 3.281, "text": "down it forms the beginning of a larger"}, {"start": 36.64, "duration": 4.4, "text": "triangular array"}, {"start": 37.84, "duration": 4.48, "text": "with side length 3n plus 1. to complete"}, {"start": 41.04, "duration": 3.6, "text": "this larger array"}, {"start": 42.32, "duration": 3.36, "text": "we need a base n triangular array at the"}, {"start": 44.64, "duration": 3.52, "text": "top"}, {"start": 45.68, "duration": 3.84, "text": "therefore two n plus one squared is"}, {"start": 48.16, "duration": 2.8, "text": "equal to the three n plus first"}, {"start": 49.52, "duration": 5.199, "text": "triangular number"}, {"start": 50.96, "duration": 5.919, "text": "minus the nth triangular number that is"}, {"start": 54.719, "duration": 5.121, "text": "each odd square is a difference of two"}, {"start": 56.879, "duration": 2.961, "text": "triangular numbers"}, {"start": 61.76, "duration": 5.569, "text": "[Music]"}, {"start": 75.439, "duration": 2.081, "text": "you"}], "title": "Odd Squares as Difference of Triangular Numbers (visual proof)", "description": "This is a short, animated visual proof demonstrating how to visualize odd squares as the difference of two triangular numbers. #mathshorts\u200b #mathvideo\u200b #math\u200b #numbertheory #mtbos\u200b #manim\u200b #animation\u200b #theorem\u200b #pww\u200b #proofwithoutwords\u200b #visualproof\u200b #proof\u200b #iteachmath #squares #triangular\n\nThis animation is based on a visual proof by Roger B. Nelsen from the March 1996 issue of The College Mathematics Journal (https://www.jstor.org/stable/2687399 - page 118).\n\nNote that this technique is also utilized in the related result proved in one of my previous videos : https://youtu.be/sDbPEjOtp50\n\nTo learn more about animating with manim, check out:\nhttps://manim.community\n__________________________________________________________________\nBackground music, used according to CC BY-SA 3.0, is \"The Road\" from Esteban Orlando:\n\nEsteban Orlando:\nhttps://soundcloud.com/orlando-esteba...\nhttps://www.instagram.com/estebanorla...\n\nhttps://creativecommons.org/licenses/...\nMusic provided by RFM: https://youtu.be/b4KCM1bIq50", "lengthSeconds": 76, "width": 640, "height": 360}
{"video_id": "mKg_gVagHy8", "transcript": [{"start": 0.0, "duration": 2.03, "text": "\u0e02\u0e2d"}, {"start": 4.08, "duration": 3.089, "text": "[\u0e40\u0e1e\u0e25\u0e07]"}, {"start": 9.7, "duration": 15.17, "text": "[\u0e40\u0e1e\u0e25\u0e07]"}, {"start": 22.189, "duration": 26.18, "text": "\u0e14\u0e39\u0e2b\u0e19\u0e31\u0e07\u0e42\u0e1b\u0e4a\u0e2b\u0e19\u0e48\u0e2d\u0e22"}, {"start": 24.87, "duration": 23.499, "text": "[\u0e40\u0e1e\u0e25\u0e07]"}, {"start": 58.43, "duration": 4.58, "text": "[\u0e40\u0e1e\u0e25\u0e07]"}, {"start": 60.01, "duration": 3.0, "text": "\u0e21.\u0e04"}, {"start": 64.04, "duration": 39.719, "text": "[\u0e40\u0e1e\u0e25\u0e07]"}], "title": "Cauchy Bunyakovsky Schwarz Inequality I (visual proof)", "description": "This is a short, animated visual proof of the two-dimensional Cauchy-Schwarz inequality (sometimes called Cauchy\u2013Bunyakovsky\u2013Schwarz inequality) using the Side-angle-side formula for the area of a parallelogram. #math\u200b #inequality \u200b #manim\u200b #animation\u200b #theorem\u200b #pww\u200b #proofwithoutwords\u200b #visualproof\u200b #proof\u200b #iteachmath #algebra #areas #mathematics #cauchyschwarz #algebraicidentity #mathshorts\u200b #mathvideo\u200b #mtbos \n\nIf you like this video, please click \"like\" and consider subscribing and checking out my other videos.\n\nThis animation is based on a visual proof by Sidney H. Kung from the February 2008 issue of Mathematics Magazine (https://www.jstor.org/stable/27643084 - page 69).\n\nTo learn more about animating with manim, check out:\nhttps://manim.community\n_____________________________\nMusic in this video:\nCyberpunk Gaming Electro | STARDUST by Alex-Productions | https://www.youtube.com/channel/UCx0_M61F81Nfb-BRXE-SeVA\nMusic promoted by https://www.free-stock-music.com\nCreative Commons Attribution 3.0 Unported License\nhttps://creativecommons.org/licenses/by/3.0/deed.en_US", "lengthSeconds": 106, "width": 640, "height": 360}
{"video_id": "TsKTteGJpZU", "transcript": [{"start": 8.24, "duration": 3.359, "text": "start with a right triangle that has"}, {"start": 10.08, "duration": 2.479, "text": "legs of length a"}, {"start": 11.599, "duration": 4.08, "text": "and b"}, {"start": 12.559, "duration": 5.761, "text": "and a hypotenuse of length c"}, {"start": 15.679, "duration": 5.36, "text": "rotate and shift a copy of this triangle"}, {"start": 18.32, "duration": 4.719, "text": "so that they align like this and notice"}, {"start": 21.039, "duration": 5.441, "text": "that the two connected angles are"}, {"start": 23.039, "duration": 3.441, "text": "complementary by construction"}, {"start": 27.519, "duration": 5.761, "text": "we can add this line to make a trapezoid"}, {"start": 30.8, "duration": 5.2, "text": "and the interior triangle here is right"}, {"start": 33.28, "duration": 4.4, "text": "angled by construction"}, {"start": 36.0, "duration": 3.92, "text": "let's compute the area of the resulting"}, {"start": 37.68, "duration": 5.44, "text": "trapezoid called garfield's trapezoid in"}, {"start": 39.92, "duration": 3.2, "text": "two different ways"}, {"start": 43.28, "duration": 4.32, "text": "first of all the area of the trapezoid"}, {"start": 46.0, "duration": 4.399, "text": "is the product of the average of the two"}, {"start": 47.6, "duration": 5.279, "text": "bases with the height so the area is a"}, {"start": 50.399, "duration": 4.8, "text": "plus b over two times a plus b which"}, {"start": 52.879, "duration": 6.36, "text": "expands to a squared plus two times a b"}, {"start": 55.199, "duration": 4.04, "text": "plus b squared over two"}, {"start": 59.28, "duration": 2.88, "text": "alternatively"}, {"start": 60.64, "duration": 4.4, "text": "the trapezoid is built from three"}, {"start": 62.16, "duration": 7.319, "text": "triangles so its area is the sum of the"}, {"start": 65.04, "duration": 4.439, "text": "areas of those three triangles"}, {"start": 71.97, "duration": 3.18, "text": "[Music]"}, {"start": 77.2, "duration": 5.919, "text": "the first two triangles in this sum are"}, {"start": 80.4, "duration": 5.28, "text": "congruent and they have the same area of"}, {"start": 83.119, "duration": 6.32, "text": "one-half times a times b"}, {"start": 85.68, "duration": 3.759, "text": "or a times b over two"}, {"start": 91.52, "duration": 4.8, "text": "the last triangle has area one half"}, {"start": 94.0, "duration": 4.72, "text": "times c squared so the area of the"}, {"start": 96.32, "duration": 5.04, "text": "trapezoid is two times a times b plus c"}, {"start": 98.72, "duration": 4.64, "text": "squared over two"}, {"start": 101.36, "duration": 5.28, "text": "but this means that a squared plus two a"}, {"start": 103.36, "duration": 6.48, "text": "b plus b squared over two"}, {"start": 106.64, "duration": 6.08, "text": "is equal to two times a times b"}, {"start": 109.84, "duration": 4.8, "text": "plus c squared over two"}, {"start": 112.72, "duration": 4.48, "text": "multiplying this equality by 2 and"}, {"start": 114.64, "duration": 4.96, "text": "subtracting 2 times a times b results in"}, {"start": 117.2, "duration": 6.31, "text": "a squared plus b squared is equal to c"}, {"start": 119.6, "duration": 11.879, "text": "squared this is the pythagorean theorem"}, {"start": 123.51, "duration": 7.969, "text": "[Music]"}, {"start": 138.4, "duration": 2.08, "text": "you"}], "title": "Pythagorean Theorem IV (visual proof; Garfield's trapezoid)", "description": "This is a short, animated visual proof of the Pythagorean theorem (the right triangle theorem) using the trapezoid that is now attributed to President James Garfield. This theorem states the square of the hypotenuse of a right triangle is equal to the sum of squares of the two other side lengths. #mathshorts #mathvideo #math #pythagoreantheorem #pythagorean #triangle #manim #animation #theorem #pww #proofwithoutwords #visualproof #proof\n\nThis animation is based on a proof due to James A. Garfield. For a static version of this proof, see Roger Nelsen's first \"Proof Without Words: Exercises in Visual Thinking\" compendium (page 7) : https://bookstore.ams.org/clrm-1. You can also check out Howard Eves' \"Great Moments in Mathematics (Before 1650)\" page 34-36.\n\nFor other proofs of this same fact check out: \nhttps://youtu.be/q28H5NY5UVU\nhttps://youtu.be/yfGtbNgcrQ8\nhttps://youtu.be/JRjPsuJ7S-g\n\nTo learn more about animating with manim, check out:\nhttps://manim.community\n________________________________________\nMusic in this video:\nHarmony by Vlad Gluschenko | https://soundcloud.com/vgl9\nMusic promoted by https://www.free-stock-music.com\nCreative Commons Attribution 3.0 Unported License\nhttps://creativecommons.org/licenses/by/3.0/deed.en_US", "lengthSeconds": 139, "width": 640, "height": 360}
{"video_id": "GGY-EpA8PJo", "transcript": [{"start": 0.0, "duration": 4.38, "text": "which is larger e to the pi or pi to the"}, {"start": 2.399, "duration": 3.781, "text": "E to figure this out graph the function"}, {"start": 4.38, "duration": 3.96, "text": "y equals the natural log of x over X"}, {"start": 6.18, "duration": 3.839, "text": "this function has a derivative equal to"}, {"start": 8.34, "duration": 4.2, "text": "1 minus the natural log of x over x"}, {"start": 10.019, "duration": 5.58, "text": "squared and this derivative is zero if"}, {"start": 12.54, "duration": 5.1, "text": "and only if x is equal to E"}, {"start": 15.599, "duration": 4.44, "text": "this means that the function attains a"}, {"start": 17.64, "duration": 4.26, "text": "maximum value at x equals e and in"}, {"start": 20.039, "duration": 2.82, "text": "particular that maximum value is 1 over"}, {"start": 21.9, "duration": 3.84, "text": "e"}, {"start": 22.859, "duration": 4.981, "text": "now because Pi is not equal to E when we"}, {"start": 25.74, "duration": 4.859, "text": "plug Pi into the function we get the"}, {"start": 27.84, "duration": 4.2, "text": "value natural log of pi over pi and we"}, {"start": 30.599, "duration": 2.701, "text": "see that this value is less than one"}, {"start": 32.04, "duration": 2.82, "text": "over e"}, {"start": 33.3, "duration": 3.54, "text": "therefore we conclude that the natural"}, {"start": 34.86, "duration": 4.98, "text": "log of pi over Pi is less than one over"}, {"start": 36.84, "duration": 5.58, "text": "e multiply both sides by E and Pi then"}, {"start": 39.84, "duration": 4.98, "text": "apply one natural log rule followed by"}, {"start": 42.42, "duration": 4.5, "text": "another natural log rule at this point"}, {"start": 44.82, "duration": 3.719, "text": "we found our winner calculus has proved"}, {"start": 46.92, "duration": 1.95, "text": "to us that pi to the E is less than e to"}, {"start": 48.539, "duration": 2.94, "text": "the pi"}, {"start": 48.87, "duration": 2.609, "text": "[Music]"}], "title": "Pi to e versus e to pi: which is larger? (Pi day short)", "description": "This is a short, animated visual proof demonstrating which of the values e^pi or pi^e is larger (where pi is the circle constant and e is Euler's constant) . #math\u200b #inequality #manim\u200b #animation\u200b #theorem\u200b #pww\u200b #proofwithoutwords\u200b #visualproof\u200b #proof\u200b #iteachmath #calculus #mathshorts\u200b #mathvideo\u200b #mtbos\u200b #euler #pi #piday\n\nThis animation is based on a visual proof by Fouad Nakhli from the June 1987 issue of Mathematics Magazine (https://www.jstor.org/stable/2689563). \n\nTo see the longer, wordless video with this result, check out https://youtu.be/JE_YeVlSqLY\n\nFor two visual proofs of a more general version of this inequality, see:\nhttps://youtu.be/SaYs856QYlI\nand\nhttps://youtu.be/DOcsgckdvxU\n\nTo learn more about animating with manim, check out:\nhttps://manim.community", "lengthSeconds": 50, "width": 360, "height": 640}
{"video_id": "Vt61xtJilHA", "transcript": [{"start": 0.0, "duration": 5.68, "text": "what's one over one plus one over two"}, {"start": 2.399, "duration": 5.761, "text": "plus one over three so on forever you"}, {"start": 5.68, "duration": 4.48, "text": "keep adding positive values so maybe"}, {"start": 8.16, "duration": 4.32, "text": "it's infinity but the numbers are"}, {"start": 10.16, "duration": 4.479, "text": "getting smaller and smaller so maybe"}, {"start": 12.48, "duration": 4.96, "text": "this sum is finite"}, {"start": 14.639, "duration": 4.881, "text": "here's the one represented as an area"}, {"start": 17.44, "duration": 4.88, "text": "here's the one half as an area the"}, {"start": 19.52, "duration": 5.519, "text": "one-third and so on forever"}, {"start": 22.32, "duration": 5.199, "text": "and here's the function one over x"}, {"start": 25.039, "duration": 5.201, "text": "graphed from one to infinity"}, {"start": 27.519, "duration": 5.04, "text": "notice how the rectangles line up nicely"}, {"start": 30.24, "duration": 4.96, "text": "and more importantly notice that the"}, {"start": 32.559, "duration": 5.761, "text": "area of the rectangles is larger than"}, {"start": 35.2, "duration": 5.76, "text": "the area under the curve of the function"}, {"start": 38.32, "duration": 4.72, "text": "you can use technology or calculus to"}, {"start": 40.96, "duration": 4.88, "text": "find the area under this curve and"}, {"start": 43.04, "duration": 5.359, "text": "realize it's infinite"}, {"start": 45.84, "duration": 4.0, "text": "yet the area of the rectangles is bigger"}, {"start": 48.399, "duration": 3.121, "text": "than that"}, {"start": 49.84, "duration": 3.28, "text": "since the area of the rectangles"}, {"start": 51.52, "duration": 3.12, "text": "represent our sum"}, {"start": 53.12, "duration": 5.84, "text": "we have our answer"}, {"start": 54.64, "duration": 4.32, "text": "so now what if i ask you this question"}], "title": "Every Student Should See This", "description": "This has always been one of my absolute favorite visual proofs!\n\nThis sum 1 + 1/2 +1/3 +... is call the harmonic series. This is a great math proof showing why the harmonic series diverges! (We basically use the integral test for series)\n\n\u25baBECOME A CHANNEL MEMBER\nhttps://www.youtube.com/channel/UChVUSXFzV8QCOKNWGfE56YQ/join\n\n\u25baWEBSITE \nhttps://www.brithemathguy.com\n\n\u25baMY COURSE\nProve It Like A Mathematician! (Intro To Math Proofs)\nhttps://www.udemy.com/course/prove-it-like-a-mathematician/?referralCode=D4A14680C629BCC9D84C\n\nDisclaimer: This video is for entertainment purposes only and should not be considered academic. Though all information is provided in good faith, no warranty of any kind, expressed or implied, is made with regards to the accuracy, validity, reliability, consistency, adequacy, or completeness of this information.\n\n\n#math #brithemathguy #shorts", "lengthSeconds": 57, "width": 360, "height": 640}
{"video_id": "GmDiE6Gnl10", "transcript": [{"start": 0.13, "duration": 4.01, "text": "[Music]"}, {"start": 1.14, "duration": 3.0, "text": "foreign"}, {"start": 6.5, "duration": 3.82, "text": "with any triangle such as the one"}, {"start": 9.059, "duration": 4.261, "text": "pictured here"}, {"start": 10.32, "duration": 4.859, "text": "then place a square on each side of the"}, {"start": 13.32, "duration": 3.059, "text": "triangle so that we have three squares"}, {"start": 15.179, "duration": 3.18, "text": "as pictured"}, {"start": 16.379, "duration": 4.441, "text": "these three squares allow us to create"}, {"start": 18.359, "duration": 5.281, "text": "three new triangles connecting the"}, {"start": 20.82, "duration": 4.74, "text": "adjacent vertices of adjacent squares as"}, {"start": 23.64, "duration": 3.6, "text": "shown here's an interesting question"}, {"start": 25.56, "duration": 4.44, "text": "about the four triangles that we've"}, {"start": 27.24, "duration": 5.22, "text": "created which of them has the greatest"}, {"start": 30.0, "duration": 4.5, "text": "area remember that I'm asking this"}, {"start": 32.46, "duration": 4.259, "text": "question in general not for the specific"}, {"start": 34.5, "duration": 4.559, "text": "triangle that I drew so I want to know"}, {"start": 36.719, "duration": 4.981, "text": "the answer no matter which triangle we"}, {"start": 39.059, "duration": 4.201, "text": "start with is it possible that the"}, {"start": 41.7, "duration": 3.96, "text": "answer doesn't depend on the starting"}, {"start": 43.26, "duration": 4.799, "text": "triangle or will the answer change as"}, {"start": 45.66, "duration": 4.379, "text": "the initial triangle changes"}, {"start": 48.059, "duration": 3.011, "text": "pause the video and see if you can"}, {"start": 50.039, "duration": 2.761, "text": "figure this out on your own"}, {"start": 51.07, "duration": 3.829, "text": "[Music]"}, {"start": 52.8, "duration": 4.2, "text": "it turns out that this is a bit of a"}, {"start": 54.899, "duration": 3.84, "text": "trick question if we erase the three"}, {"start": 57.0, "duration": 4.32, "text": "squares and rotate each of the three"}, {"start": 58.739, "duration": 4.741, "text": "created triangles 90 degrees we get this"}, {"start": 61.32, "duration": 4.32, "text": "shape pictured here in this shape each"}, {"start": 63.48, "duration": 4.319, "text": "of the three created triangles has a"}, {"start": 65.64, "duration": 3.9, "text": "side length that extends one of the side"}, {"start": 67.799, "duration": 4.201, "text": "lengths of the original triangle and"}, {"start": 69.54, "duration": 5.16, "text": "matches that length now when we draw"}, {"start": 72.0, "duration": 5.1, "text": "three altitudes in the original triangle"}, {"start": 74.7, "duration": 4.44, "text": "one perpendicular to each side we see"}, {"start": 77.1, "duration": 4.44, "text": "something interesting not only do the"}, {"start": 79.14, "duration": 4.619, "text": "newly created triangles share a base"}, {"start": 81.54, "duration": 4.259, "text": "with the original triangle they also"}, {"start": 83.759, "duration": 4.321, "text": "share a perpendicular height with the"}, {"start": 85.799, "duration": 4.621, "text": "original triangle this means that the"}, {"start": 88.08, "duration": 5.579, "text": "areas of each of the three created"}, {"start": 90.42, "duration": 5.22, "text": "triangles is equal to the area of the"}, {"start": 93.659, "duration": 4.621, "text": "original triangle but that means that"}, {"start": 95.64, "duration": 3.9, "text": "all four triangles have exactly the same"}, {"start": 98.28, "duration": 3.659, "text": "area"}, {"start": 99.54, "duration": 4.5, "text": "this argument holds No Matter What"}, {"start": 101.939, "duration": 4.201, "text": "triangle we started with it turns out"}, {"start": 104.04, "duration": 4.68, "text": "that whatever triangle you start with"}, {"start": 106.14, "duration": 4.86, "text": "you create three new triangles and when"}, {"start": 108.72, "duration": 4.92, "text": "you rotate them 90 degrees you will"}, {"start": 111.0, "duration": 4.5, "text": "always have one side length that matches"}, {"start": 113.64, "duration": 3.839, "text": "the side length of the original triangle"}, {"start": 115.5, "duration": 4.2, "text": "and you will always get this nicely"}, {"start": 117.479, "duration": 4.5, "text": "glued together three triangle shape so"}, {"start": 119.7, "duration": 4.86, "text": "that when you draw altitudes for the"}, {"start": 121.979, "duration": 5.221, "text": "original triangle those altitudes become"}, {"start": 124.56, "duration": 5.239, "text": "altitudes of one of the three created"}, {"start": 127.2, "duration": 2.599, "text": "triangles"}], "title": "Four Triangles: Which is Largest? (visual proof)", "description": "This is a short, animated visual proof demonstrating a surprising relationship between three triangles created in a special way from a given fourth triangle.\n\nIf you like this video, consider subscribing to the channel or consider buying me a coffee: https://www.buymeacoffee.com/VisualProofs. Thanks! \n\nFor similar visual proofs see:\n\nhttps://youtu.be/zhriP_4QpJQ\nhttps://youtu.be/_emvEr4IB_Y\nhttps://youtu.be/PNQxS6antBY\nhttps://youtube.com/shorts/NA8UsiZ3mMI\n\nThis animation is based on a visual proof from Steven Snover that appeared in Roger B. Nelsen's second Proofs without Words Compendium (page 15).\n\nTo learn more about animating with manim, check out:\nhttps://manim.community\n\n#mathshorts\u200b #mathvideo\u200b #math\u200b #triangle #area #mtbos\u200b #manim\u200b #animation\u200b #theorem\u200b #pww\u200b #proofwithoutwords\u200b #visualproof\u200b #proof\u200b #iteachmath #geometry #algebra #inequality \n\n_______________________________\nMusic in this video\nTil Death Parts Us by Aakash Gandhi", "lengthSeconds": 135, "width": 640, "height": 360}
{"video_id": "PQTHxiHeohY", "transcript": [{"start": 0.08, "duration": 4.679, "text": "start with a unit square with an area of"}, {"start": 2.04, "duration": 4.92, "text": "one connect each vertex to an opposite"}, {"start": 4.759, "duration": 4.321, "text": "Edge midpoint like this the region"}, {"start": 6.96, "duration": 4.44, "text": "bounded between the four Drawn Lines"}, {"start": 9.08, "duration": 4.519, "text": "forms a square itself can you find the"}, {"start": 11.4, "duration": 4.08, "text": "area of this Square pause the video and"}, {"start": 13.599, "duration": 4.68, "text": "think about it before I reveal an"}, {"start": 15.48, "duration": 5.48, "text": "answer we can rotate the four small"}, {"start": 18.279, "duration": 4.721, "text": "triangles pictured here to form four"}, {"start": 20.96, "duration": 4.239, "text": "squares congruent to the original shaded"}, {"start": 23.0, "duration": 3.119, "text": "square but that means the Shaded area"}, {"start": 25.199, "duration": 3.92, "text": "must be"}, {"start": 26.119, "duration": 3.0, "text": "1/5"}], "title": "A SquAREA within a Square?", "description": "This is a short, animated visual proof of a classic proof about the area of the square obtained by connecting the vertices of a square to the midpoints of the opposite sides of the square. \n\nIf you like this video, consider subscribing to the channel or consider buying me a coffee: https://www.buymeacoffee.com/VisualProofs. Thanks!\n\nThis animation is based on a proof that appears in the following sources:\n\nMathematical Morsels by Ross Honsberger (MAA, 1978)\n\nProofs without Words II by Roger B. Nelsen (MAA, 2000) (https://bookstore.ams.org/clrm-14/)\n\nAnd can be traced to an article by an author known as Nev R. Mind.\n\nTo learn more about animating with manim, check out:\nhttps://manim.community\n\n#manim #math #mathvideo #mathshorts #geometry #square #animation #theorem #pww #proofwithoutwords #visualproof #proof #area #onefifth", "lengthSeconds": 27, "width": 360, "height": 640}
{"video_id": "2lNSnzNbAm0", "transcript": [{"start": 0.12, "duration": 4.199, "text": "so you need to divide a circular Pizza"}, {"start": 2.159, "duration": 4.14, "text": "seven equal ways and all you have is a"}, {"start": 4.319, "duration": 4.261, "text": "straight edge Encompass no worries I've"}, {"start": 6.299, "duration": 4.081, "text": "got you covered first take the diameter"}, {"start": 8.58, "duration": 3.479, "text": "of the pizza and cut it into seven equal"}, {"start": 10.38, "duration": 4.679, "text": "pieces which you can do with a straight"}, {"start": 12.059, "duration": 6.001, "text": "edge Encompass now use your compass to"}, {"start": 15.059, "duration": 5.88, "text": "draw these seven semi-circles you should"}, {"start": 18.06, "duration": 4.92, "text": "have one semi-circle of diameter ND for"}, {"start": 20.939, "duration": 3.241, "text": "each value of n from one up to seven"}, {"start": 22.98, "duration": 3.66, "text": "like this"}, {"start": 24.18, "duration": 4.2, "text": "now here's the trick you want to do the"}, {"start": 26.64, "duration": 3.84, "text": "exact same construction on the bottom"}, {"start": 28.38, "duration": 3.719, "text": "half of the pizza except this time from"}, {"start": 30.48, "duration": 3.66, "text": "right to left"}, {"start": 32.099, "duration": 4.201, "text": "these are the resulting seven pieces"}, {"start": 34.14, "duration": 4.14, "text": "that you have and they're all equal in"}, {"start": 36.3, "duration": 3.779, "text": "area if you want to know more about why"}, {"start": 38.28, "duration": 3.0, "text": "this works check out the link in the"}, {"start": 40.079, "duration": 4.761, "text": "description to the longer video"}, {"start": 41.28, "duration": 3.56, "text": "describing all the details"}], "title": "Divide pizza into 7 using only straightedge and compass", "description": "In this short, we show how to decompose a circle into seven pieces each with the same area. This process can be done using ruler and compass constructions, unlike the process of constructing the regular heptagon. #manim\u200b #mathshorts\u200b #mathvideo\u200b #math\u200b #geometry #mtbos\u200b #animation\u200b #theorem\u200b\u200b\u200b #visualproof\u200b #proof\u200b #iteachmath #circle #mathematics #rulerandcompass #construction #pizza \n\nHere is a longer video filling the details that the seven regions are actually equal in area: https://youtu.be/KhfZK5IIK9E\n\nHere is a video showing how to divide a line into seven equal lengths: https://youtu.be/afZiAGaKEY4\n\nThis animation is based on a visual proof from Roger Nelsen's and Claudi Alsina's wonderful book \"Icons of Mathematics\" (Chapter 14.1 on page 76). You can find the book and other information about it at the links below\n\n(I may receive commissions from this paid link) : Icons of Mathematics from Amazon https://amzn.to/3L5U44S\n\nhttps://www.maa.org/press/maa-reviews/icons-of-mathematics-an-exploration-of-twenty-key-images\n\nhttps://bookstore.ams.org/dol-56/\n\nTo learn more about animating with manim, check out:\nhttps://manim.community", "lengthSeconds": 44, "width": 360, "height": 640}
{"video_id": "SqLKCxBLMXo", "transcript": [{"start": 11.76, "duration": 16.739, "text": "[Music]"}, {"start": 29.16, "duration": 5.32, "text": "a"}, {"start": 31.48, "duration": 3.0, "text": "oh"}], "title": "#mathematics #calculus Indeterminant Form (0xinfinity): L'Hospital rule", "description": "#mathematics #calculus Indeterminant Form (0xinfinity)\n\nMathematics Shorts: https://www.youtube.com/playlist?list=PLN1Cp7TpZOL7LH2zGFtSejZ18P4swqnDA \n\nOther Videos\nFibonacci Sequence, Golden Ratio, How to draw Golden Spiral:\nhttps://youtu.be/yHBa3EvDPco\nRadius and Tangent Vectors: \nhttps://youtube.com/shorts/KHSafRET3Sk\nDe Morgan's laws: Proof by Venn Diagrams:\nhttps://youtube.com/shorts/pR-REo4aBYk?feature=share\nVisual Proof of Roots of Quadratic Equation\nhttps://youtube.com/shorts/FdE7rHDZRd8\nVisual Proof: Factors of Quadratic Equation PART-1 x^2+3x+2\nhttps://youtube.com/shorts/lIqv5zKAKKE\nVisual Proof: Factors of Quadratic Polynomial PART-2 x^2-3x+2\nhttps://youtube.com/shorts/5pRYbKFWjUM\nVisual Proof: Factors of Quadratic Polynomial PART-2 x^2+2x-3\nhttps://youtube.com/shorts/AzAsz5itOfQ\nVisual Proof: Factors of Quadratic Polynomial PART-2 x^2+2x-3\nhttps://youtube.com/shorts/AzAsz5itOfQ\nVisual Proof: Trigonometric Identity sin^2 (x) +cos^2 (x)=1\nhttps://youtube.com/shorts/bPAm1nRUJWw", "lengthSeconds": 36, "width": 360, "height": 640}
{"video_id": "9Qyba_swKTI", "transcript": [{"start": 0.89, "duration": 10.15, "text": "[Music]"}, {"start": 6.919, "duration": 6.34, "text": "let A and B be two positive real numbers"}, {"start": 11.04, "duration": 4.62, "text": "first draw a square with side length a"}, {"start": 13.259, "duration": 5.52, "text": "and adjacent to that draw a square with"}, {"start": 15.66, "duration": 5.1, "text": "side length B in each Square we can draw"}, {"start": 18.779, "duration": 4.141, "text": "the diagonals and use the intersection"}, {"start": 20.76, "duration": 3.9, "text": "to draw a line from the center of the"}, {"start": 22.92, "duration": 3.779, "text": "square to the midpoint of the base of"}, {"start": 24.66, "duration": 4.74, "text": "the square this process has created"}, {"start": 26.699, "duration": 4.92, "text": "isosceles right triangles in either"}, {"start": 29.4, "duration": 3.839, "text": "square one of them with a side length of"}, {"start": 31.619, "duration": 3.6, "text": "a over two and one of them with a side"}, {"start": 33.239, "duration": 3.901, "text": "length of B over 2. now we can connect"}, {"start": 35.219, "duration": 4.02, "text": "the center points of the two squares to"}, {"start": 37.14, "duration": 4.32, "text": "create a right triangle pictured here"}, {"start": 39.239, "duration": 3.601, "text": "the legs of this right triangle are a"}, {"start": 41.46, "duration": 2.939, "text": "divided by the square root of 2 and B"}, {"start": 42.84, "duration": 3.48, "text": "divided by the square root of 2 since"}, {"start": 44.399, "duration": 4.081, "text": "both of those are hypotenuses of the"}, {"start": 46.32, "duration": 3.78, "text": "previously described right triangles but"}, {"start": 48.48, "duration": 3.059, "text": "then we can use the Pythagorean theorem"}, {"start": 50.1, "duration": 3.119, "text": "to find that the hypotenuse of this"}, {"start": 51.539, "duration": 4.261, "text": "triangle is the square root of a squared"}, {"start": 53.219, "duration": 4.86, "text": "over 2 plus b squared over two the two"}, {"start": 55.8, "duration": 4.739, "text": "isosceles right triangles and the Shaded"}, {"start": 58.079, "duration": 4.5, "text": "purple triangle create a trapezoid from"}, {"start": 60.539, "duration": 4.141, "text": "this trapezoid we see that the one side"}, {"start": 62.579, "duration": 3.781, "text": "length given by a over 2 plus b over 2"}, {"start": 64.68, "duration": 3.24, "text": "is less than or equal to the longer side"}, {"start": 66.36, "duration": 3.299, "text": "length which is given by the square root"}, {"start": 67.92, "duration": 4.86, "text": "of a squared over 2 plus b squared over"}, {"start": 69.659, "duration": 5.221, "text": "two no matter this sizes of A and B we"}, {"start": 72.78, "duration": 4.32, "text": "see that we can always create this"}, {"start": 74.88, "duration": 4.739, "text": "diagram with an a by a square and a b by"}, {"start": 77.1, "duration": 5.159, "text": "b square and a triangle connecting their"}, {"start": 79.619, "duration": 5.461, "text": "Center points this triangle is part of a"}, {"start": 82.259, "duration": 4.741, "text": "trapezoid and the trapezoid has one long"}, {"start": 85.08, "duration": 4.56, "text": "side and one short side so the"}, {"start": 87.0, "duration": 4.92, "text": "inequality is always strict except in"}, {"start": 89.64, "duration": 5.159, "text": "the one situation where the side length"}, {"start": 91.92, "duration": 5.04, "text": "a equals the side length B therefore a"}, {"start": 94.799, "duration": 3.96, "text": "over 2 plus b over 2 is less than or"}, {"start": 96.96, "duration": 3.6, "text": "equal to the square root of a squared"}, {"start": 98.759, "duration": 4.021, "text": "over 2 plus b squared over 2 and"}, {"start": 100.56, "duration": 4.68, "text": "equality occurs if and only if a equals"}, {"start": 102.78, "duration": 4.199, "text": "B A over two plus b over 2 is the"}, {"start": 105.24, "duration": 3.419, "text": "arithmetic mean of A and B and the"}, {"start": 106.979, "duration": 3.541, "text": "square root of a squared over 2 plus b"}, {"start": 108.659, "duration": 4.021, "text": "squared over 2 is the root mean square"}, {"start": 110.52, "duration": 3.599, "text": "of A and B so this is known as the"}, {"start": 112.68, "duration": 4.04, "text": "arithmetic mean root mean Square"}, {"start": 114.119, "duration": 2.601, "text": "inequality"}], "title": "Arithmetic Mean-Root Mean Square Inequality (visual proof)", "description": "This is a short, animated visual proof demonstrating the arithmetic mean - root mean square inequality using two adjacent squares.\n\nIf you like this video, consider subscribing to the channel or consider buying me a coffee: https://www.buymeacoffee.com/VisualProofs. Thanks! \n\nFor other visual proofs of inequalities like this see:\n\nhttps://youtu.be/jNx05gXSblE\nhttps://youtu.be/Pt8IX_Q5U1A\nhttps://youtu.be/62G9fak1vyk\nhttps://youtu.be/HFGSsPbYc-s\nhttps://youtu.be/SEPkdPWKdO0\n\nor this playlist: https://www.youtube.com/playlist?list=PLZh9gzIvXQUuKcFHdtjFkwAVELDxu9nH4\n\nThis animation is based on a visual proof from Juan-Bosco Romero M\u00e1rquez that appeared in the March 2001 issue of the College Mathematics Journal (https://www.jstor.org/stable/2687116  p. 118)\n\nTo learn more about animating with manim, check out:\nhttps://manim.community\n\n#mathshorts\u200b #mathvideo\u200b #math\u200b #amrmsinequality #arithmeticmean #rootmeansquare #rootmean #mtbos\u200b #manim\u200b #animation\u200b #theorem\u200b #pww\u200b #proofwithoutwords\u200b #visualproof\u200b #proof\u200b #iteachmath #calculus #inequality\n_______________________________________\nMusic in this episode:\nProcession -- Geographer", "lengthSeconds": 121, "width": 640, "height": 360}
{"video_id": "RoovN01q83Y", "transcript": [{"start": 0.0, "duration": 3.36, "text": "here's one way to make sense of the"}, {"start": 1.68, "duration": 4.079, "text": "infinite sum of the powers of one-half"}, {"start": 3.36, "duration": 4.68, "text": "start with an isosceles triangle of Area"}, {"start": 5.759, "duration": 4.441, "text": "1 and cut it into two congruent"}, {"start": 8.04, "duration": 3.84, "text": "isosceles triangles shading one of them"}, {"start": 10.2, "duration": 4.2, "text": "so that we've shaded an area of a half"}, {"start": 11.88, "duration": 4.56, "text": "then repeat on the unshaded isosceles"}, {"start": 14.4, "duration": 4.32, "text": "triangle cutting it in half and shading"}, {"start": 16.44, "duration": 4.08, "text": "one so that we've shaded a quarter we"}, {"start": 18.72, "duration": 4.8, "text": "can continue this process indefinitely"}, {"start": 20.52, "duration": 5.22, "text": "each time we take the unshaded isosceles"}, {"start": 23.52, "duration": 4.56, "text": "triangle cut it in half and shade one"}, {"start": 25.74, "duration": 4.32, "text": "piece the area of this piece will be one"}, {"start": 28.08, "duration": 4.44, "text": "half the area of the previously shaded"}, {"start": 30.06, "duration": 4.8, "text": "triangle but as we do this infinitely we"}, {"start": 32.52, "duration": 4.86, "text": "end up with the Shaded region being the"}, {"start": 34.86, "duration": 4.859, "text": "sum of the powers of one-half on the"}, {"start": 37.38, "duration": 4.8, "text": "other hand as we do this forever we end"}, {"start": 39.719, "duration": 5.041, "text": "up filling the entire isosceles triangle"}, {"start": 42.18, "duration": 5.64, "text": "which has area one therefore the sum of"}, {"start": 44.76, "duration": 6.02, "text": "the powers of one-half is equal to one"}, {"start": 47.82, "duration": 2.96, "text": "foreign"}], "title": "Adding powers of 1/2", "description": "This is a short, animated visual proof demonstrating the infinite sum of the powers of 1/2 using an isosceles triangle. #mathshorts\u200b #mathvideo\u200b #math\u200b #calculus #mtbos\u200b #manim\u200b #animation\u200b #theorem\u200b #pww\u200b #proofwithoutwords\u200b #visualproof\u200b #proof\u200b #iteachmath #geometricseries #series #infinitesums #infiniteseries \n\nHere is a bit longer and slower version (with dramatic music and no words): https://youtu.be/saJJGNlsfn8\n\nThis animation is based on a proof by Roger B. Nelsen from the February 2006 issue of Mathematics Magazine page 60. (https://doi.org/10.2307/27642904).\n\nTo learn more about animating with manim, check out:\nhttps://manim.community", "lengthSeconds": 51, "width": 360, "height": 640}
{"video_id": "xobfnvLuE08", "transcript": [{"start": 11.76, "duration": 16.739, "text": "[Music]"}, {"start": 29.16, "duration": 5.32, "text": "a"}, {"start": 31.48, "duration": 3.0, "text": "oh"}, {"start": 35.72, "duration": 6.02, "text": "[Music]"}], "title": "##mathematics #calculus Visual : Generalized  Parabola y=a(x-h)^2+k", "description": "##mathematics #calculus Visual : Generalized  Parabola y=a(x-h)^2+k\n\nOther Videos\nFibonacci Sequence, Golden Ratio, How to draw Golden Spiral:\nhttps://youtu.be/yHBa3EvDPco\nRadius and Tangent Vectors: \nhttps://youtube.com/shorts/KHSafRET3Sk\nDe Morgan's laws: Proof by Venn Diagrams:\nhttps://youtube.com/shorts/pR-REo4aBYk?feature=share\nVisual Proof of Roots of Quadratic Equation\nhttps://youtube.com/shorts/FdE7rHDZRd8\nVisual Proof: Factors of Quadratic Equation PART-1 x^2+3x+2\nhttps://youtube.com/shorts/lIqv5zKAKKE\nVisual Proof: Factors of Quadratic Polynomial PART-2 x^2-3x+2\nhttps://youtube.com/shorts/5pRYbKFWjUM\nVisual Proof: Factors of Quadratic Polynomial PART-2 x^2+2x-3\nhttps://youtube.com/shorts/AzAsz5itOfQ\nVisual Proof: Factors of Quadratic Polynomial PART-2 x^2+2x-3\nhttps://youtube.com/shorts/AzAsz5itOfQ\nVisual Proof: Trigonometric Identity sin^2 (x) +cos^2 (x)=1\nhttps://youtube.com/shorts/bPAm1nRUJWw", "lengthSeconds": 40, "width": 360, "height": 640}
{"video_id": "-dwSzCLaso4", "transcript": [{"start": 1.38, "duration": 5.42, "text": "[Music]"}, {"start": 6.859, "duration": 4.66, "text": "a positive integer is said to be perfect"}, {"start": 9.54, "duration": 3.179, "text": "if it is equal to the sum of its proper"}, {"start": 11.519, "duration": 3.841, "text": "divisors"}, {"start": 12.719, "duration": 4.32, "text": "for example here's the number six with"}, {"start": 15.36, "duration": 4.2, "text": "its proper divisors two three and one"}, {"start": 17.039, "duration": 6.121, "text": "and here's the number 28 with its proper"}, {"start": 19.56, "duration": 6.959, "text": "divisors 4 14 2 7 and 1. we can see that"}, {"start": 23.16, "duration": 4.98, "text": "6 is equal to 1 plus 2 plus 3 and 28 is"}, {"start": 26.519, "duration": 4.621, "text": "equal to one plus two plus four plus"}, {"start": 28.14, "duration": 5.84, "text": "seven plus fourteen therefore both 6 and"}, {"start": 31.14, "duration": 6.419, "text": "28 are perfect numbers"}, {"start": 33.98, "duration": 5.56, "text": "separated by about 2000 years Euclid and"}, {"start": 37.559, "duration": 4.141, "text": "Euler proved the two halves of the"}, {"start": 39.54, "duration": 4.74, "text": "theorem that states that n is an even"}, {"start": 41.7, "duration": 4.5, "text": "perfect number if and only if n can be"}, {"start": 44.28, "duration": 4.68, "text": "written as the form 2 raised to the"}, {"start": 46.2, "duration": 5.879, "text": "quantity P minus 1 multiplied by 2 to"}, {"start": 48.96, "duration": 6.06, "text": "the P minus 1 where 2 to the P minus 1"}, {"start": 52.079, "duration": 5.041, "text": "is prime according to this theorem if n"}, {"start": 55.02, "duration": 5.1, "text": "is an even perfect number we can arrange"}, {"start": 57.12, "duration": 5.22, "text": "n dots in a rectangle with Dimensions 2"}, {"start": 60.12, "duration": 5.1, "text": "raised to the P minus 1 and 2 to the P"}, {"start": 62.34, "duration": 5.52, "text": "minus one but then the side of length 2"}, {"start": 65.22, "duration": 5.04, "text": "to the P minus 1 can be split into two"}, {"start": 67.86, "duration": 4.98, "text": "pieces one of length two raised to the P"}, {"start": 70.26, "duration": 5.16, "text": "minus 1 and the other with a length of 2"}, {"start": 72.84, "duration": 4.8, "text": "raised to the P minus 1 minus one now"}, {"start": 75.42, "duration": 3.96, "text": "consider the Triangular array with base"}, {"start": 77.64, "duration": 3.78, "text": "of length 2 raised to the P minus 1"}, {"start": 79.38, "duration": 4.559, "text": "minus 1 sitting in the upper right of"}, {"start": 81.42, "duration": 5.519, "text": "this rectangle we can swing this array"}, {"start": 83.939, "duration": 5.281, "text": "down forming a larger triangular array"}, {"start": 86.939, "duration": 4.68, "text": "in this case the base of the new"}, {"start": 89.22, "duration": 4.74, "text": "triangular array is 2 to the P minus 1"}, {"start": 91.619, "duration": 4.86, "text": "plus 2 to the P minus 1 minus one which"}, {"start": 93.96, "duration": 4.56, "text": "simplifies to 2 to the P minus 1 but"}, {"start": 96.479, "duration": 5.46, "text": "this means that any even perfect number"}, {"start": 98.52, "duration": 5.76, "text": "can fit into a triangular array with a"}, {"start": 101.939, "duration": 4.441, "text": "base of length 2 to the P minus 1. so"}, {"start": 104.28, "duration": 4.74, "text": "that n is an even perfect number if and"}, {"start": 106.38, "duration": 4.68, "text": "only if n is the 2 to the P minus 1"}, {"start": 109.02, "duration": 4.02, "text": "triangular number where 2 to the P minus"}, {"start": 111.06, "duration": 3.599, "text": "1 is prime it turns out that there"}, {"start": 113.04, "duration": 3.899, "text": "aren't many primes of the form 2 to the"}, {"start": 114.659, "duration": 4.081, "text": "P minus 1 that we know these primes are"}, {"start": 116.939, "duration": 3.601, "text": "called mercen primes and it's an open"}, {"start": 118.74, "duration": 4.08, "text": "problem to figure out if there are"}, {"start": 120.54, "duration": 4.38, "text": "infinitely many such primes equivalently"}, {"start": 122.82, "duration": 3.54, "text": "it's an open problem to figure out if"}, {"start": 124.92, "duration": 3.78, "text": "there are infinitely many even perfect"}, {"start": 126.36, "duration": 5.34, "text": "numbers"}, {"start": 128.7, "duration": 3.0, "text": "foreign"}], "title": "Perfect Numbers (even ones) are Triangular (visual proof) -- with bonus open problem!", "description": "This is a short, animated visual proof showing that all even perfect numbers are triangular numbers (and thus all currently known perfect numbers are triangular numbers). #math\u200b #euler #euclid\u200b #mtbos\u200b #manim\u200b #animation\u200b #theorem\u200b #pww\u200b #proofwithoutwords\u200b #visualproof\u200b #proof\u200b #iteachmath #mersenne #openproblem #unsolvedproblem\n\nThis animation is based on a visual proof by Roger Nelsen from the May 2016 issue of The College Mathematics Journal (https://doi.org/10.4169/college.math.j.47.3.171 ).\n\nTo learn more about animating with manim, check out:\nhttps://manim.community\n__________________________________________________________________\nMusic in this video:\nTil Death Parts Us by Aakash Gandhi", "lengthSeconds": 135, "width": 640, "height": 360}
{"video_id": "e52w9H1OWEA", "transcript": [{"start": 0.08, "duration": 4.96, "text": "let's say you have to multiply two"}, {"start": 1.439, "duration": 5.32, "text": "binomials for example x + 3 and x - 2"}, {"start": 5.04, "duration": 4.679, "text": "you multiply the first terms of each"}, {"start": 6.759, "duration": 4.88, "text": "binomial the outer terms the inner terms"}, {"start": 9.719, "duration": 3.8, "text": "and then the last terms you lastly"}, {"start": 11.639, "duration": 4.0, "text": "combine any like terms to get the final"}, {"start": 13.519, "duration": 4.001, "text": "result this is known as the foil method"}, {"start": 15.639, "duration": 4.241, "text": "and here's a visualization showing how"}, {"start": 17.52, "duration": 4.48, "text": "it works say we have a rectangle where"}, {"start": 19.88, "duration": 3.8, "text": "its area is equal to its width time its"}, {"start": 22.0, "duration": 4.48, "text": "height if the width and height are each"}, {"start": 23.68, "duration": 4.8, "text": "broken into two variables A and B and C"}, {"start": 26.48, "duration": 3.48, "text": "and D we would have to multiply these"}, {"start": 28.48, "duration": 3.199, "text": "binomials to find the area of the re"}, {"start": 29.96, "duration": 3.759, "text": "rectangle an alternative approach would"}, {"start": 31.679, "duration": 3.72, "text": "be to break the rectangle down into four"}, {"start": 33.719, "duration": 3.601, "text": "smaller rectangles it's easier to"}, {"start": 35.399, "duration": 3.84, "text": "visualize their area and we can find the"}, {"start": 37.32, "duration": 4.16, "text": "area of the total rectangle by adding up"}, {"start": 39.239, "duration": 4.561, "text": "all the smaller areas by doing so we"}, {"start": 41.48, "duration": 5.32, "text": "find the total area is equal to AC plus"}, {"start": 43.8, "duration": 4.48, "text": "a d plus BC plus BD which yields the"}, {"start": 46.8, "duration": 3.439, "text": "exact same result as if you were to"}, {"start": 48.28, "duration": 3.48, "text": "follow the foil method the same approach"}, {"start": 50.239, "duration": 3.201, "text": "can be used to visualize multiplying"}, {"start": 51.76, "duration": 3.04, "text": "larger polinomial the order in which you"}, {"start": 53.44, "duration": 2.72, "text": "add the terms doesn't matter as long as"}, {"start": 54.8, "duration": 2.559, "text": "you're multiplying every possible pair"}, {"start": 56.16, "duration": 2.359, "text": "between left and right comment your"}, {"start": 57.359, "duration": 4.16, "text": "thoughts below and subscribe for more"}, {"start": 58.519, "duration": 3.0, "text": "stem content"}], "title": "FOIL method visual proof #maths #animation #education", "description": "Here\u2019s a potentially useful visualization of binomial multiplication. Comment below for what content you\u2019d like to see form this channel!\nMusic: Inhale - Recall", "lengthSeconds": 60, "width": 360, "height": 640}
{"video_id": "Hb4bAn6MmU0", "transcript": [{"start": 0.399, "duration": 3.761, "text": "the halleyuda pascal triangle is a"}, {"start": 2.399, "duration": 2.721, "text": "triangular array of numbers built as"}, {"start": 4.16, "duration": 2.4, "text": "follows"}, {"start": 5.12, "duration": 2.639, "text": "every row begins with a one and ends"}, {"start": 6.56, "duration": 2.959, "text": "with a one"}, {"start": 7.759, "duration": 4.161, "text": "and every other entry in the triangle is"}, {"start": 9.519, "duration": 4.481, "text": "obtained by adding the two entries"}, {"start": 11.92, "duration": 3.599, "text": "directly above the entry in question we"}, {"start": 14.0, "duration": 3.119, "text": "can fill in the rest of the triangle"}, {"start": 15.519, "duration": 4.401, "text": "using this pattern"}, {"start": 17.119, "duration": 6.24, "text": "here we've only pictured up to row seven"}, {"start": 19.92, "duration": 5.279, "text": "in the halle utah pascal triangle"}, {"start": 23.359, "duration": 5.521, "text": "now let's go through this triangle and"}, {"start": 25.199, "duration": 3.681, "text": "shade only the odd entries"}, {"start": 29.199, "duration": 5.681, "text": "we get a picture that looks like this"}, {"start": 31.519, "duration": 6.081, "text": "but if we zoom out to show 256 rows of"}, {"start": 34.88, "duration": 5.12, "text": "pascal's triangle and shade only the odd"}, {"start": 37.6, "duration": 6.44, "text": "entries something amazing happens we"}, {"start": 40.0, "duration": 4.04, "text": "produce the sierpinski triangle"}, {"start": 44.16, "duration": 2.079, "text": "you"}], "title": "Sierpinski from Pascal", "description": "This is a recreation of a short clip from a long form video showing six different ways to construct the Sierpinski triangle: https://youtu.be/IZHiBJGcrqI\n\nIn this short, we shade odd entries of the Halayuda/Pascal triangle to obtain the Sierpinski triangle. Can you explain why this works?\n\n#math #manim #fractal #fractal #sierpinski #pascaltriangle #pascal #mathshort #visualmath", "lengthSeconds": 44, "width": 360, "height": 640}
{"video_id": "WLh6r31VXp8", "transcript": [{"start": 11.76, "duration": 16.739, "text": "[Music]"}, {"start": 29.16, "duration": 5.32, "text": "a"}, {"start": 31.48, "duration": 3.0, "text": "oh"}, {"start": 35.72, "duration": 8.02, "text": "[Music]"}], "title": "#trending #mathematics Visual Proof: Algebraic Identity (a+b)^3=a^3+3a^2b+3ab^2+b^3", "description": "#trending #mathematics Visual Proof: Factors of Quadratic Equation Polynomial PART-4 x^2-2x-3\n\n\nOther Videos\nFibonacci Sequence, Golden Ratio, How to draw Golden Spiral:\nhttps://youtu.be/yHBa3EvDPco\nRadius and Tangent Vectors: \nhttps://youtube.com/shorts/KHSafRET3Sk\nDe Morgan's laws: Proof by Venn Diagrams:\nhttps://youtube.com/shorts/pR-REo4aBYk?feature=share\nVisual Proof of Roots of Quadratic Equation\nhttps://youtube.com/shorts/FdE7rHDZRd8\nVisual Proof: Factors of Quadratic Equation PART-1 x^2+3x+2\nhttps://youtube.com/shorts/lIqv5zKAKKE\nVisual Proof: Factors of Quadratic Polynomial PART-2 x^2-3x+2\nhttps://youtube.com/shorts/5pRYbKFWjUM\nVisual Proof: Factors of Quadratic Polynomial PART-2 x^2+2x-3\nhttps://youtube.com/shorts/AzAsz5itOfQ", "lengthSeconds": 42, "width": 360, "height": 640}
{"video_id": "wiaUC271di4", "transcript": [{"start": 0.37, "duration": 7.38, "text": "[Music]"}, {"start": 9.28, "duration": 3.0, "text": "sh"}], "title": "Visual Proof of PT \ud83d\ude07 #shorts #youtube #trending #viral #youtubeshorts #maths #mathstricks #pt #fun", "description": "", "lengthSeconds": 10, "width": 360, "height": 640}
{"video_id": "o0AyHmRV8RA", "transcript": [{"start": 2.37, "duration": 6.41, "text": "[Music]"}, {"start": 13.759, "duration": 3.061, "text": "thank you"}, {"start": 19.77, "duration": 7.429, "text": "[Music]"}, {"start": 41.04, "duration": 2.24, "text": "foreign"}, {"start": 44.5, "duration": 6.67, "text": "[Music]"}, {"start": 54.36, "duration": 19.209, "text": "[Music]"}, {"start": 74.939, "duration": 14.38, "text": "foreign"}, {"start": 76.75, "duration": 12.569, "text": "[Music]"}], "title": "Geometric Series: sum of powers of 1/6 (visual proof)", "description": "This is a short, animated visual proof showing the sum of the infinite geometric series of powers of 1/6 (starting with 1/6) is 1/5 using self similar trapezoids in a regular pentagon. #manim #math\u200b\u200b #mathshorts\u200b #mathvideo\u200b  #mtbos\u200b #manim\u200b #animation\u200b #theorem\u200b #pww\u200b #proofwithoutwords\u200b #visualproof\u200b #proof\u200b #iteachmath #calculus #series #geometricseries #infiniteseries  #pentagon #trapezoids\n\nThis animation is based on a proof by James Tanton from the March 2008 issue of The College Mathematics Journal page 106. (https://www.jstor.org/stable/27646594).\n\nTo learn more about animating with manim, check out:\nhttps://manim.community\n___________________________________\nMusic: \nTale Forest by Vlad Gluschenko | https://soundcloud.com/vgl9\nMusic promoted by https://www.free-stock-music.com\nCreative Commons Attribution 3.0 Unported License\nhttps://creativecommons.org/licenses/by/3.0/deed.en_US", "lengthSeconds": 91, "width": 640, "height": 360}
{"video_id": "_dltGrVlW3w", "transcript": [{"start": 1.079, "duration": 3.0, "text": "foreign"}, {"start": 7.15, "duration": 15.569, "text": "[Music]"}, {"start": 30.599, "duration": 3.0, "text": "foreign"}, {"start": 45.08, "duration": 9.379, "text": "[Music]"}, {"start": 56.71, "duration": 6.25, "text": "[Music]"}, {"start": 65.72, "duration": 3.0, "text": "thank you"}, {"start": 76.55, "duration": 9.139, "text": "[Music]"}, {"start": 88.1, "duration": 3.0, "text": "thank you"}, {"start": 94.06, "duration": 5.579, "text": "[Music]"}], "title": "Cubes from double sums (visual proof without words)", "description": "In this (wordless) video, we demonstrate how each integer cube is a special double sum.\n\nIf you like this video, consider subscribing to the channel or consider buying me a coffee: https://www.buymeacoffee.com/VisualProofs. Thanks!\n\nHere are a few other related visual proofs:\nhttps://youtu.be/OECW5qqZN9I\nhttps://youtu.be/936eKowyvR4\nhttps://youtu.be/WidzHiUFWNA\nhttps://youtu.be/BBrtHbRcpXs\nhttps://youtu.be/YQLicI8R4Gs\n\nThis animation is based on a visual proof by Roger B. Nelsen from the March 2002 issue of the College Mathematics Journal  (https://www.jstor.org/stable/1559005 page 171 ). \n\nTo learn more about animating with manim, check out:\nhttps://manim.community\n\n#math #manim\u200b #mathvideo\u200b\u200b #cubes\u200b #mtbos\u200b #manim\u200b #animation\u200b #theorem\u200b #pww\u200b #proofwithoutwords\u200b #visualproof\u200b #proof\u200b #iteachmath #finitesums #discretemath #induction #doublesum\n\n___________________________________________\nMusic in this video:\nDance Party | JUMP by Alex-Productions | https://onsound.eu/\nMusic promoted by https://www.free-stock-music.com\nCreative Commons / Attribution 3.0 Unported License (CC BY 3.0)\nhttps://creativecommons.org/licenses/by/3.0/deed.en_US", "lengthSeconds": 99, "width": 640, "height": 360}
{"video_id": "77vlRweCtGE", "transcript": [{"start": 0.04, "duration": 4.08, "text": "draw the circle of radius 1 at the"}, {"start": 1.64, "duration": 4.4, "text": "origin rotate an angle Theta the point"}, {"start": 4.12, "duration": 3.36, "text": "on the unit circle for Theta has an x"}, {"start": 6.04, "duration": 3.599, "text": "coordinate of cosine Theta and a"}, {"start": 7.48, "duration": 3.6, "text": "y-coordinate of sin Theta if we draw the"}, {"start": 9.639, "duration": 3.92, "text": "tangent line to the circle at the point"}, {"start": 11.08, "duration": 4.719, "text": "and connect to the x-axis by triangle"}, {"start": 13.559, "duration": 5.681, "text": "similarity the distance labeled D here"}, {"start": 15.799, "duration": 6.001, "text": "satisfies d/ 1 is equal to sin thet over"}, {"start": 19.24, "duration": 4.16, "text": "cosine thet which is the tangent of thet"}, {"start": 21.8, "duration": 3.319, "text": "also by triangle similarity the"}, {"start": 23.4, "duration": 3.199, "text": "hypotenuse of the picture triangle is 1"}, {"start": 25.119, "duration": 3.801, "text": "over the cosine of theta so it's the"}, {"start": 26.599, "duration": 4.68, "text": "secant of theta if we extend the tangent"}, {"start": 28.92, "duration": 4.2, "text": "line to the Y AIS like this then the two"}, {"start": 31.279, "duration": 4.841, "text": "picture triangles are similar so that"}, {"start": 33.12, "duration": 4.84, "text": "the labeled distance D satisfies D 1al"}, {"start": 36.12, "duration": 4.0, "text": "cosine Theta over sin Theta which is the"}, {"start": 37.96, "duration": 4.68, "text": "cent of theta and the hypotenuse of the"}, {"start": 40.12, "duration": 4.599, "text": "green triangle is 1/ sin Theta which is"}, {"start": 42.64, "duration": 3.48, "text": "cosecant Theta these definitions of the"}, {"start": 44.719, "duration": 3.201, "text": "six standard trig functions can be"}, {"start": 46.12, "duration": 3.2, "text": "extended to the other quadrants as well"}, {"start": 47.92, "duration": 3.68, "text": "and this animation demonstrates their"}, {"start": 49.32, "duration": 3.759, "text": "range restrictions the prefix Co stands"}, {"start": 51.6, "duration": 3.04, "text": "for complimentary because the two"}, {"start": 53.079, "duration": 3.8, "text": "picture triangles have complimentary"}, {"start": 54.64, "duration": 3.64, "text": "angles Theta and pi/ 2 minus Theta so"}, {"start": 56.879, "duration": 4.36, "text": "the cosine is the sign of the"}, {"start": 58.28, "duration": 2.959, "text": "complimentary angle"}], "title": "Six Trigs in 60 Seconds! #math #trigonometry", "description": "In this video, we show a single diagram consisting of various triangles that connects the six primary trig functions (sine, cosine, tangent, secant, cosecant, and cotangent) to lengths of line segments created from the unit circle (circle of radius 1).  We also briefly discuss the fact that the \"co\" on three of the functions refer to the \"complementary angle.\"\n\nIf you like this video, consider subscribing to the channel or consider buying me a coffee: https://www.buymeacoffee.com/VisualProofs. Thanks!\n\nFor a longer version of this video with more details and information, see\nhttps://youtu.be/dUkCgTOOpQ0\n\nThis animation is based on a classic diagram for describing the six main trig functions. If you want to know more trig identities, check out my playlist:\nhttps://youtube.com/playlist?list=PLZh9gzIvXQUvc6SufrIpkVrVLdhcGthSy&si=2IXs0Jx5j30_-spb\n\nor check out this Wikipedia article:\nhttps://en.wikipedia.org/wiki/List_of_trigonometric_identities\n\nHere is an interactive version from Tien Chih using Desmos:\n\nhttps://www.desmos.com/calculator/26ftp9is44\n\nHere are alternate videos with the same diagram:\n\nhttps://youtu.be/Dsf6ADwJ66E?si=1N8GEyBe3abyoan1\n\nhttps://youtu.be/68Pi4RJ1Rnw?si=aTCeeGa73-R3eApe\n\n#math\u200b #mathvideo #math #trigonometry #trigidentities #identity  #triangle #manim #animation #theorem #pww #proofwithoutwords #visualproof #proof #circle #pythagoreantheorem #obtuseangle #acuteangle #angle #sine #cosine #tangent #secant #cosecant #cotangent #learnmaths \n\nTo learn more about animating with manim, check out:\nhttps://manim.community", "lengthSeconds": 59, "width": 360, "height": 640}
{"video_id": "lk310mk01OU", "transcript": [{"start": 0.04, "duration": 4.64, "text": "the Fibonacci sequence starts with fub1"}, {"start": 1.8, "duration": 6.68, "text": "equal to 1 so f1^ squar is the area of a"}, {"start": 4.68, "duration": 6.36, "text": "1x1 Square FS2 is also equal to 1 so FS2"}, {"start": 8.48, "duration": 4.96, "text": "^2 is the area of a 1x one square as"}, {"start": 11.04, "duration": 4.08, "text": "well when n is greater than two the nth"}, {"start": 13.44, "duration": 3.88, "text": "Fibonacci number is the sum of the"}, {"start": 15.12, "duration": 3.76, "text": "previous two Fibonacci numbers notice we"}, {"start": 17.32, "duration": 3.44, "text": "can keep attaching squares to this"}, {"start": 18.88, "duration": 3.72, "text": "diagram in such a way that the side"}, {"start": 20.76, "duration": 3.439, "text": "length is the sum of the previous two"}, {"start": 22.6, "duration": 3.96, "text": "Fibonacci numbers and is therefore for"}, {"start": 24.199, "duration": 4.521, "text": "the next Fibonacci number so as we add"}, {"start": 26.56, "duration": 3.76, "text": "squares to this diagram the I Square"}, {"start": 28.72, "duration": 3.519, "text": "will have an area given by this square"}, {"start": 30.32, "duration": 4.28, "text": "of the I Fibonacci number we can"}, {"start": 32.239, "duration": 4.48, "text": "continue this process as long as we like"}, {"start": 34.6, "duration": 4.279, "text": "if we eventually stop say at the number"}, {"start": 36.719, "duration": 4.641, "text": "F subn then we have a rectangular"}, {"start": 38.879, "duration": 4.401, "text": "diagram enclosing an area given by the"}, {"start": 41.36, "duration": 4.199, "text": "sum of the squares of the first n"}, {"start": 43.28, "duration": 3.919, "text": "Fibonacci numbers on the other hand the"}, {"start": 45.559, "duration": 3.48, "text": "dimensions of this rectangular diagram"}, {"start": 47.199, "duration": 4.161, "text": "are the N Fibonacci number by the N plus"}, {"start": 49.039, "duration": 4.281, "text": "1 Fibonacci number so the enclosed area"}, {"start": 51.36, "duration": 3.719, "text": "is the product of these two numbers this"}, {"start": 53.32, "duration": 4.039, "text": "means that the sum of the squares of the"}, {"start": 55.079, "duration": 6.48, "text": "first n Fibonacci numbers is the product"}, {"start": 57.359, "duration": 4.2, "text": "of two consecutive Fibonacci numbers"}], "title": "Sums of Fibonacci Squares", "description": "This is a short, animated visual proof demonstrating the sum of the squares of the first n consecutive Fibonacci numbers. \n\nTo buy me a coffee, head over to https://www.buymeacoffee.com/VisualProofs \nThanks!\n\nTo see a slower, wider high definition version of this visual identity (without words), check out this video: https://youtu.be/2EsoiBdnIAM?si=3bW8M_zxxx-ueAhC\n\nAlso, check out my playlist on other finite sums:\nhttps://youtube.com/playlist?list=PLZh9gzIvXQUubr38YfIlul9j7_54hXZy_\n\nThis animation is based on a proof by Alfred Brouusseau from \"A Primer for the Fibonacci Numbers by M. Bicknell and V.E. Hoggatt Jr (eds.). You can also find this proof in Roger Nelsen's first Proof Without Words compendium (page 83).\n\nGet Nelsen's Amazing Book (Affiliate link so I may receive a commission): https://amzn.to/3QF6Zvz\n\n#fibonacci #math\u200b #manim\u200b #math\u200bvideo #calculus #mtbos\u200b #mathshorts\u200b #animation\u200b #theorem\u200b #pww\u200b #proofwithoutwords\u200b #visualproof\u200b #proof\u200b #iteachmath #series #finitesums #finiteseries #geometric #squares\n\nTo learn more about animating with manim, check out:\nhttps://manim.community", "lengthSeconds": 60, "width": 360, "height": 640}
{"video_id": "iVLE8gEJfPA", "transcript": [{"start": 0.04, "duration": 4.56, "text": "you can use a 1x one square to make"}, {"start": 1.92, "duration": 4.999, "text": "sense of the infinite sum 1/2 plus 1/4 +"}, {"start": 4.6, "duration": 4.68, "text": "1/8 and so on first cut the square into"}, {"start": 6.919, "duration": 4.481, "text": "two parts and shade 1/2 then shade 1/2"}, {"start": 9.28, "duration": 4.239, "text": "of the remaining half shading a quarter"}, {"start": 11.4, "duration": 4.639, "text": "then shade 1/2 of the remaining 1/2"}, {"start": 13.519, "duration": 4.721, "text": "shading an eighth continue this process"}, {"start": 16.039, "duration": 5.441, "text": "over and over again each time shading"}, {"start": 18.24, "duration": 5.359, "text": "1/2 of the remaining 1 over 2 to the N"}, {"start": 21.48, "duration": 4.36, "text": "if we continue shading indefinitely that"}, {"start": 23.599, "duration": 4.321, "text": "is if we consider the limiting process"}, {"start": 25.84, "duration": 3.679, "text": "eventually in the limit the entire"}, {"start": 27.92, "duration": 3.92, "text": "Square will be shaded by the sum of"}, {"start": 29.519, "duration": 4.641, "text": "powers of 12 but the square has an area"}, {"start": 31.84, "duration": 4.84, "text": "of 1 so that the infinite sum where K"}, {"start": 34.16, "duration": 5.36, "text": "ranges from 1 to Infinity of 1 2 the K"}, {"start": 36.68, "duration": 5.44, "text": "is equal to 1 another interpretation of"}, {"start": 39.52, "duration": 6.8, "text": "this infinite sum is that 0.1 repeating"}, {"start": 42.12, "duration": 4.2, "text": "in base 2 is equal to 1"}], "title": "0.1111\u2026 = 1 (in base 2)", "description": "This is a short, animated visual proof showing the sum of the infinite geometric series with first term 1/2 and ratio 1/2 using a 1 by 1 square. This infinite sum allows us to determine an interesting base 2 (or binary) representation of 1. \n\nTo buy me a coffee, head over to https://www.buymeacoffee.com/VisualProofs \nThanks!\n\nFor an alternate wide-format wordless geometric series dissection of the same fact, check out\n\nhttps://youtu.be/saJJGNlsfn8?si=7eCh6og1dkB9GrWB\n\nThis animation is based on a visual proof by Warren Page from the September 1981 issue of Mathematics Magazine, (page 201 - https://www.jstor.org/stable/2689632 ).\n\n #manim #math\u200b\u200b #mathshorts\u200b #mathvideo\u200b  #mtbos\u200b #manim\u200b #animation\u200b #theorem\u200b #pww\u200b #proofwithoutwords\u200b #visualproof\u200b #proof\u200b #iteachmath #calculus #series #geometricseries #infiniteseries  #base2 #binary #zeno #paradox \n\nTo learn more about animating with manim, check out:\nhttps://manim.community", "lengthSeconds": 44, "width": 360, "height": 640}
{"video_id": "Qb9EYqzCVsQ", "transcript": [{"start": 0.78, "duration": 4.38, "text": "[Music]"}, {"start": 8.119, "duration": 4.07, "text": "[Music]"}, {"start": 15.0, "duration": 13.42, "text": "[Music]"}, {"start": 28.96, "duration": 7.229, "text": "yeah"}, {"start": 31.39, "duration": 4.799, "text": "[Music]"}, {"start": 39.08, "duration": 4.23, "text": "[Music]"}], "title": "#VISUAL #PROOF #OF #DIFFERENCE #OF #SQUARES #x\u00b2 \u2014 y\u00b2", "description": "Dear viewers,\nWelcome to    MATH BRIDGES  FUN WITH MATH \nTo enhance the learning process  and  make the math concepts alive  and fun loving through online medium for all..", "lengthSeconds": 42, "width": 360, "height": 640}
{"video_id": "iygc5S7XWMI", "transcript": [{"start": 0.0, "duration": 5.66, "text": "[Music]"}, {"start": 10.58, "duration": 8.999, "text": "thank you"}, {"start": 12.69, "duration": 6.889, "text": "[Music]"}, {"start": 22.87, "duration": 12.95, "text": "[Music]"}, {"start": 40.49, "duration": 9.069, "text": "[Music]"}, {"start": 46.559, "duration": 3.0, "text": "foreign"}, {"start": 50.5, "duration": 3.229, "text": "[Music]"}, {"start": 56.19, "duration": 3.3, "text": "[Music]"}], "title": "Sum of First n Odd Integers | Visual Proof | Geometric Proof | Proof Without WORDs", "description": "Thanks for watching! If you found these tips helpful, please like and subscribe to our channel. Share your reviews in the comments below!\n\n#shorts #mathtricks #math #mathematics #mathideas #mathanimation #animation #animated #visualization #geometric #proof #proofwithoutword #algebric #algebra #algebricequation #algebricexpression #visualproof #geometricproof #amazingmath #funfacts #mathfacts #abstractart #abstractmath #mathvisualization #mathart #geometry #shapes #equation #expression #square #proofing #proofof #proofofconcept #amazingahorts #amazingfacts #factshorts #intution #intutive #funmath #mathideas #mathworld #magicmath #magicmathtricks #mathmagic #mathzone #abstractview #proofs", "lengthSeconds": 58, "width": 360, "height": 640}
{"video_id": "1zHfp20jn4c", "transcript": [{"start": 0.04, "duration": 4.4, "text": "here are two methods to find the area of"}, {"start": 1.92, "duration": 4.76, "text": "a circle using the method of exhaustion"}, {"start": 4.44, "duration": 4.04, "text": "in the top image we start peeling off"}, {"start": 6.68, "duration": 3.879, "text": "circumferences one at a time and lay"}, {"start": 8.48, "duration": 4.199, "text": "them flat as lines because the"}, {"start": 10.559, "duration": 3.841, "text": "circumference varies linearly with"}, {"start": 12.679, "duration": 4.161, "text": "respect to the radius we see that the"}, {"start": 14.4, "duration": 5.0, "text": "shape will eventually form a triangle in"}, {"start": 16.84, "duration": 4.4, "text": "the bottom diagram we split the circle"}, {"start": 19.4, "duration": 4.879, "text": "into an even number of wedge-shaped"}, {"start": 21.24, "duration": 5.32, "text": "pieces as we create smaller and smaller"}, {"start": 24.279, "duration": 4.281, "text": "wedges we can split the circle apart"}, {"start": 26.56, "duration": 3.879, "text": "into those wedge shaped pieces and glue"}, {"start": 28.56, "duration": 4.039, "text": "the wedges together to form shapes that"}, {"start": 30.439, "duration": 4.401, "text": "look more and more like a rectangle it"}, {"start": 32.599, "duration": 4.321, "text": "turns out that in the top shape we end"}, {"start": 34.84, "duration": 5.16, "text": "up with a triangle that has a height of"}, {"start": 36.92, "duration": 5.119, "text": "R and a base of length 2 pi r and in the"}, {"start": 40.0, "duration": 3.48, "text": "bottom shape we end up with a rectangle"}, {"start": 42.039, "duration": 3.921, "text": "where the length is given by half the"}, {"start": 43.48, "duration": 4.68, "text": "circumference which is pi * R and a"}, {"start": 45.96, "duration": 4.56, "text": "height given by the radius of R so in"}, {"start": 48.16, "duration": 5.32, "text": "the top diagram we get an area of 1 12 *"}, {"start": 50.52, "duration": 5.16, "text": "2i R * R which is p piun r^ 2 and in the"}, {"start": 53.48, "duration": 4.399, "text": "bottom diagram we get an area of piun *"}, {"start": 55.68, "duration": 6.0, "text": "r^ 2 let me know in the comments which"}, {"start": 57.879, "duration": 3.801, "text": "visualization you like better"}], "title": "Circle area by exhaustion (two visual area techniques)", "description": "In this short, we show two fascinating methods of determining the area of a circle using the \"method of exhaustion.\" Which one do you like better?\n\nIf you like this video, consider subscribing to the channel or consider buying me a coffee: https://www.buymeacoffee.com/VisualProofs. Thanks! \n\nThe top animation unrolls successive circumferences of nested circular shells. In this manner, the circle area gets transformed into the area of a triangle with base 2pi times r and height r.\n\nHere are other versions from me:\nhttps://youtu.be/GjmtmCT5N8Q\nhttps://youtube.com/shorts/3L_GorNnk4Q\n\nTo see an alternate (and quite viral) video showing this in stop motion, check out this one from  @MinutePhysics  : https://youtu.be/whYqhpc6S6g\n\nFor more information about this construction, see\nhttps://personal.math.ubc.ca/~cass/courses/m446-03/exhaustion.pdf\n\nor check out this nice survey article by David Richeson from the May 2015 issue of The College Math Journal: https://doi.org/10.4169/college.math.j.46.3.162 .\n\nThis bottom animation of a classic visual proof showing how to find the area of a circle by using more and more wedges and arranging them in a rectangle.\n\nThis proof can be traced to both Sat\u014d Moshun and Leonardo da Vinci (see Smith, David Eugene; Mikami, Yoshio (1914), A history of Japanese mathematics, https://archive.org/details/historyofjapanes00smituoft, page 130-132 and Beckmann, Petr (1976), A History of Pi, St. Martin's Griffin, page 19). \n\nHere are other versions from me:\nhttps://youtu.be/-UBLutigNMM\nhttps://youtube.com/shorts/X-kTZgFWUFQ\n\nYou can also read more about this in a great NYT article by Steven Strogatz: https://archive.nytimes.com/opinionator.blogs.nytimes.com/2010/04/04/take-it-to-the-limit/\n\n#math #manim #visualproof #proofwithoutwords #circle #circlearea #archimedes #radius #area #areaofcircle #pi #piday #shorts #circle #archimedes #infinite #methodofexhaustion\n\nTo learn more about animating with manim, check out:\nhttps://manim.community", "lengthSeconds": 60, "width": 360, "height": 640}
{"video_id": "UqVmocdLFGc", "transcript": [{"start": 8.96, "duration": 4.719, "text": "start with a triangular array of"}, {"start": 10.559, "duration": 5.28, "text": "triangles with n triangles on the base"}, {"start": 13.679, "duration": 4.241, "text": "imagine placing one token in the first"}, {"start": 15.839, "duration": 4.081, "text": "triangle for one squared tokens"}, {"start": 17.92, "duration": 4.0, "text": "then place two tokens in each of the"}, {"start": 19.92, "duration": 3.92, "text": "triangles in the second diagonal adding"}, {"start": 21.92, "duration": 3.76, "text": "two squared tokens"}, {"start": 23.84, "duration": 4.0, "text": "place three tokens on each of the three"}, {"start": 25.68, "duration": 4.56, "text": "triangles in the third diagonal adding"}, {"start": 27.84, "duration": 3.519, "text": "three squared total tokens and continue"}, {"start": 30.24, "duration": 3.6, "text": "this process"}, {"start": 31.359, "duration": 4.561, "text": "so that the n minus 1 tokens in each of"}, {"start": 33.84, "duration": 5.12, "text": "the n minus 1 triangles in the next to"}, {"start": 35.92, "duration": 4.08, "text": "last diagonal add n minus 1 squared"}, {"start": 38.96, "duration": 3.04, "text": "tokens"}, {"start": 40.0, "duration": 4.48, "text": "and the n tokens in each of the n"}, {"start": 42.0, "duration": 4.399, "text": "triangles in the last diagonal add n"}, {"start": 44.48, "duration": 4.239, "text": "squared tokens"}, {"start": 46.399, "duration": 5.761, "text": "so the total number of tokens is the sum"}, {"start": 48.719, "duration": 3.441, "text": "of the first n squares"}, {"start": 53.6, "duration": 6.88, "text": "now create a copy of the diagram"}, {"start": 56.64, "duration": 3.84, "text": "and rotate it 120 degrees"}, {"start": 60.81, "duration": 2.229, "text": "[Music]"}, {"start": 62.079, "duration": 3.44, "text": "then"}, {"start": 63.039, "duration": 4.961, "text": "create a copy from this one and rotate"}, {"start": 65.519, "duration": 5.6, "text": "it another 120 degrees resulting in"}, {"start": 68.0, "duration": 3.119, "text": "three total copies"}, {"start": 71.439, "duration": 4.641, "text": "notice that at any position if you add"}, {"start": 74.479, "duration": 3.921, "text": "the number of tokens at that position"}, {"start": 76.08, "duration": 3.679, "text": "over the three copies you always get two"}, {"start": 78.4, "duration": 3.2, "text": "n plus one"}, {"start": 79.759, "duration": 4.241, "text": "thus when we stack all three copies"}, {"start": 81.6, "duration": 4.08, "text": "together we have a triangular array of"}, {"start": 84.0, "duration": 4.64, "text": "stacks of tokens with two n plus one"}, {"start": 85.68, "duration": 5.04, "text": "tokens so the total number of tokens is"}, {"start": 88.64, "duration": 3.119, "text": "two n plus one"}, {"start": 90.72, "duration": 4.16, "text": "times"}, {"start": 91.759, "duration": 5.441, "text": "the nth triangular number"}, {"start": 94.88, "duration": 6.64, "text": "but this is equal to two n plus one"}, {"start": 97.2, "duration": 6.16, "text": "times n times n plus one over two"}, {"start": 101.52, "duration": 3.12, "text": "consequently"}, {"start": 103.36, "duration": 3.759, "text": "three"}, {"start": 104.64, "duration": 4.56, "text": "multiplied by the sum of the first n"}, {"start": 107.119, "duration": 5.601, "text": "squares"}, {"start": 109.2, "duration": 6.239, "text": "is equal to two n plus one times n"}, {"start": 112.72, "duration": 5.28, "text": "times n plus one over two"}, {"start": 115.439, "duration": 4.0, "text": "dividing both sides by three"}, {"start": 118.0, "duration": 4.24, "text": "gives the formula for the sum of the"}, {"start": 119.439, "duration": 5.68, "text": "first n squares to be equal to two n"}, {"start": 122.24, "duration": 4.79, "text": "plus one times n times n plus one over"}, {"start": 125.119, "duration": 18.801, "text": "six"}, {"start": 127.03, "duration": 18.97, "text": "[Music]"}, {"start": 143.92, "duration": 2.08, "text": "you"}], "title": "Sum of Squares II (visual proof)", "description": "This is a short, animated visual proof of the formula that computes that sum of the first n squares using 3 copies of a triangular array containing stacks of tokens equaling the sum of the first n squares . #mathshorts\u200b #mathvideo\u200b #math\u200b #sumofsquares\u200b #mtbos\u200b #manim\u200b #animation\u200b #theorem\u200b #pww\u200b #proofwithoutwords\u200b #visualproof\u200b #proof\u200b #iteachmath #finitesums #discretemath\n\nThis animation is based on a visual proof by Sidney H. Kung from the May 1989 issue of The College Mathematics Journal  (https://doi.org/10.2307/2686767) page 205.\n\nHere is another visual proof of the same fact:\nhttps://youtu.be/-tJhH_k2LaM\n\nTo learn more about animating with manim, check out:\nhttps://manim.community", "lengthSeconds": 144, "width": 640, "height": 360}
{"video_id": "lEcyafR51tM", "transcript": [{"start": 9.92, "duration": 5.759, "text": "let d and n both be positive integers"}, {"start": 13.28, "duration": 3.28, "text": "this means that we can arrange d dots in"}, {"start": 15.679, "duration": 5.041, "text": "a line"}, {"start": 16.56, "duration": 4.16, "text": "and we can arrange n dots in a line"}, {"start": 21.279, "duration": 3.92, "text": "the theorem we're interested in says"}, {"start": 22.8, "duration": 4.08, "text": "there exists unique integers q and r"}, {"start": 25.199, "duration": 4.48, "text": "with 0 less than or equal to r less than"}, {"start": 26.88, "duration": 4.799, "text": "d satisfying the equality n equals q"}, {"start": 29.679, "duration": 5.361, "text": "times d plus r"}, {"start": 31.679, "duration": 6.88, "text": "to prove the existence of q and r we"}, {"start": 35.04, "duration": 5.519, "text": "start subtracting copies of d from n"}, {"start": 38.559, "duration": 4.32, "text": "as we do this we create a list of"}, {"start": 40.559, "duration": 4.641, "text": "decreasing positive integers"}, {"start": 42.879, "duration": 4.321, "text": "by the least integer principle"}, {"start": 45.2, "duration": 4.24, "text": "of positive integers we know that this"}, {"start": 47.2, "duration": 4.48, "text": "set must have a least element as we see"}, {"start": 49.44, "duration": 3.279, "text": "here where we can no longer subtract off"}, {"start": 51.68, "duration": 4.32, "text": "d"}, {"start": 52.719, "duration": 4.881, "text": "this is going to be the value of r"}, {"start": 56.0, "duration": 4.719, "text": "as we see in the diagram we have"}, {"start": 57.6, "duration": 5.52, "text": "arranged the n dots into an a d by q"}, {"start": 60.719, "duration": 3.44, "text": "array with r dots left over and a line"}, {"start": 63.12, "duration": 3.679, "text": "on top"}, {"start": 64.159, "duration": 3.521, "text": "this means that n equals"}, {"start": 66.799, "duration": 4.281, "text": "q"}, {"start": 67.68, "duration": 3.4, "text": "times d"}, {"start": 73.04, "duration": 5.36, "text": "plus a remainder of r"}, {"start": 75.92, "duration": 4.08, "text": "to prove uniqueness of the pair q and r"}, {"start": 78.4, "duration": 3.44, "text": "let's imagine that we have another pair"}, {"start": 80.0, "duration": 5.119, "text": "q prime and r prime satisfying the same"}, {"start": 81.84, "duration": 5.279, "text": "conditions because these two are equal"}, {"start": 85.119, "duration": 4.32, "text": "we can subtract to see that q prime"}, {"start": 87.119, "duration": 4.081, "text": "times d minus q times d is equal to r"}, {"start": 89.439, "duration": 4.0, "text": "minus r prime"}, {"start": 91.2, "duration": 4.16, "text": "but now because r and r prime are both"}, {"start": 93.439, "duration": 4.561, "text": "between 0 and d their difference is"}, {"start": 95.36, "duration": 4.88, "text": "between negative d and d"}, {"start": 98.0, "duration": 4.56, "text": "this means that q prime times d minus q"}, {"start": 100.24, "duration": 4.48, "text": "times d is a multiple of d that lives"}, {"start": 102.56, "duration": 4.08, "text": "between minus d and d"}, {"start": 104.72, "duration": 4.399, "text": "from this we see that q prime times d"}, {"start": 106.64, "duration": 4.96, "text": "minus q times d must be zero and"}, {"start": 109.119, "duration": 3.761, "text": "consequently that means r minus r prime"}, {"start": 111.6, "duration": 3.28, "text": "must be zero"}, {"start": 112.88, "duration": 4.239, "text": "dividing the second equality by d which"}, {"start": 114.88, "duration": 5.279, "text": "is non-zero shows us that q prime minus"}, {"start": 117.119, "duration": 5.68, "text": "q is zero and r minus r prime is zero"}, {"start": 120.159, "duration": 4.96, "text": "therefore r must be equal to r prime and"}, {"start": 122.799, "duration": 4.881, "text": "q must be equal to q prime this means"}, {"start": 125.119, "duration": 4.401, "text": "that r and q are unique"}, {"start": 127.68, "duration": 3.359, "text": "this concludes our proof of the theorem"}, {"start": 129.52, "duration": 4.09, "text": "known as the division theorem or the"}, {"start": 131.039, "duration": 20.52, "text": "division algorithm"}, {"start": 133.61, "duration": 17.949, "text": "[Music]"}, {"start": 151.92, "duration": 2.08, "text": "you"}], "title": "Division Theorem/Division Algorithm", "description": "This is a proof, with a visualization, of the classic number-theoretic proof of the division theorem/division algorithm. The video includes a \"visual demonstration\" of the existence part of the theorem (where we also implicitly show how it is an algorithm). We also investigate the well-ordering principle (or least integer principle) and discuss existence and uniqueness style proofs. #math #proof #manim #numbertheory #visualmath #mtbos #visualproof #division #introductiontoproofs #divisiontheorem #divisionalgorithm #existence #uniqueness #subtraction\n\nIf you liked this video, please consider liking and subscribing and checking out a few of my other videos.\n\nTo learn more about animating with manim, check out:\nhttps://manim.community\n\n_________________________________\nMusic in this video:\n\u65e5\u672c\u5ead\u5712 (Japanese Garden) by Alex-Productions | https://www.youtube.com/channel/UCx0_M61F81Nfb-BRXE-SeVA\nMusic promoted by https://www.free-stock-music.com\nCreative Commons Attribution 3.0 Unported License\nhttps://creativecommons.org/licenses/by/3.0/deed.en_US", "lengthSeconds": 152, "width": 640, "height": 360}
{"video_id": "mZujswzs7Qg", "transcript": [{"start": 2.52, "duration": 5.769, "text": "[Applause]"}, {"start": 5.19, "duration": 3.099, "text": "[Music]"}, {"start": 14.11, "duration": 5.97, "text": "[Music]"}, {"start": 17.08, "duration": 3.0, "text": "oh"}], "title": "Best sum visual proof?", "description": "In this video, we show an animation of perhaps the best and most iconic visual proof of the formula for the sum of the first n positive integers.\n\nIf you like this video, consider subscribing to the channel or consider buying me a coffee: https://www.buymeacoffee.com/VisualProofs. Thanks!\n\nTo see more about this proof and a generalization, check out this video:\nhttps://youtu.be/eHbtc50-qXo?si=ztvGFf1G-dLzr-m3\n\nThis proof is attributed to the \"ancient Greeks\" by Roger Nelsen (referencing Martin Gardner) and can be found in Roger Nelsen's first compendium, \"Proofs Without Words: Exercises in Visual Thinking: \n\n(I may receive a small commission at no cost to you for this affiliate link): https://amzn.to/40cQL3k\n\n#math #manim #sumformula #naturalnumbers  #triangle #animation #theorem #pww #proofwithoutwords #visualproof #proof  #mathshorts #mathvideo #finitesums  #sums #shorts #triangularnumbers \n\nTo learn more about animating with manim, check out:\nhttps://manim.community", "lengthSeconds": 18, "width": 360, "height": 640}
{"video_id": "30TaNnzEMDo", "transcript": [{"start": 0.0, "duration": 4.92, "text": "here's my favorite diagram for finding"}, {"start": 2.1, "duration": 4.92, "text": "the sum of the first n squares imagine a"}, {"start": 4.92, "duration": 3.719, "text": "stack of squares is pictured here we see"}, {"start": 7.02, "duration": 3.72, "text": "one squared plus 2 squared plus 3"}, {"start": 8.639, "duration": 4.141, "text": "squared and so on up to N squared where"}, {"start": 10.74, "duration": 4.38, "text": "in this case we've pictured n equals 4."}, {"start": 12.78, "duration": 5.04, "text": "now we create a tower using the L-shaped"}, {"start": 15.12, "duration": 4.919, "text": "gnomons we have one row with two n minus"}, {"start": 17.82, "duration": 4.26, "text": "one squares two rows with two n minus"}, {"start": 20.039, "duration": 4.5, "text": "three squares and so on all the way down"}, {"start": 22.08, "duration": 4.74, "text": "until we have n rows of one square then"}, {"start": 24.539, "duration": 4.5, "text": "we can use one stack of the sum of the"}, {"start": 26.82, "duration": 4.44, "text": "first n squares on the left side like"}, {"start": 29.039, "duration": 4.2, "text": "this and we can use another stack of the"}, {"start": 31.26, "duration": 4.08, "text": "sum of the first n squares on the right"}, {"start": 33.239, "duration": 4.201, "text": "side of the tower like this to create a"}, {"start": 35.34, "duration": 3.719, "text": "rectangular diagram we've used three"}, {"start": 37.44, "duration": 3.54, "text": "copies of the sum of the squares to"}, {"start": 39.059, "duration": 3.84, "text": "create this rectangular diagram which"}, {"start": 40.98, "duration": 3.36, "text": "has a width of two n plus one and a"}, {"start": 42.899, "duration": 3.541, "text": "length of the sum of the first n"}, {"start": 44.34, "duration": 4.02, "text": "positive integers so three times the sum"}, {"start": 46.44, "duration": 3.54, "text": "of the first n positive squares is equal"}, {"start": 48.36, "duration": 3.66, "text": "to the sum of the first n positive"}, {"start": 49.98, "duration": 3.54, "text": "integers times two n plus one we apply"}, {"start": 52.02, "duration": 3.179, "text": "the formula for the sum of the first n"}, {"start": 53.52, "duration": 3.9, "text": "positive integers to see that the sum of"}, {"start": 55.199, "duration": 6.02, "text": "the first n squares is n times n plus 1"}, {"start": 57.42, "duration": 3.799, "text": "times 2N plus one divided by six"}], "title": "Summing Squares in a Rectangle", "description": "This is a short, animated visual proof of the formula that computes that sum of the first n squares using 3 copies of the sum of squares to build a rectangle . \n\nIf you like this video, consider subscribing to the channel or consider buying me a coffee: https://www.buymeacoffee.com/VisualProofs. Thanks!\n\nHere is a slower version of this video: https://youtu.be/-tJhH_k2LaM\n\nHere are three other visual proofs about the sum of squares formulas:\n\nhttps://youtu.be/WidzHiUFWNA (or https://youtube.com/shorts/PVH5uxHFIGQ)\nhttps://youtu.be/UqVmocdLFGc\nhttps://youtu.be/Q-frL0Ot2m4\n\nThis animation is based on (independently discovered) visual proofs by Dan Kalman from the March 1991 issue of The College Mathematics Journal (https://www.jstor.org/stable/2686447) and Martin Gardner from the October 1973 Scientific American (https://www.jstor.org/stable/24923225). \n\nTo learn more about animating with manim, check out:\nhttps://manim.community\n\n#mathshorts\u200b #mathvideo\u200b #math\u200b #sumofsquares\u200b #mtbos\u200b #manim\u200b #animation\u200b #theorem\u200b #pww\u200b #proofwithoutwords\u200b #visualproof\u200b #proof\u200b #iteachmath", "lengthSeconds": 59, "width": 360, "height": 640}
{"video_id": "0GcctoG_eTk", "transcript": [{"start": 0.48, "duration": 4.44, "text": "here's a quick proof of the Pythagorean"}, {"start": 2.08, "duration": 5.56, "text": "theorem first take four copies of your"}, {"start": 4.92, "duration": 5.4, "text": "right triangle flip them over as such"}, {"start": 7.64, "duration": 4.76, "text": "then arrange them to form a big Square"}, {"start": 10.32, "duration": 4.88, "text": "that's blue and green notice that these"}, {"start": 12.4, "duration": 6.12, "text": "angles are complimentary so the red"}, {"start": 15.2, "duration": 5.319, "text": "lines form a square Now find the area of"}, {"start": 18.52, "duration": 2.96, "text": "the big blue and green square by taking"}, {"start": 20.519, "duration": 4.161, "text": "the side"}, {"start": 21.48, "duration": 5.199, "text": "lengths next find the area alternatively"}, {"start": 24.68, "duration": 5.519, "text": "by multiplying by four times the"}, {"start": 26.679, "duration": 5.361, "text": "triangles and 1 * the Red Square now now"}, {"start": 30.199, "duration": 4.921, "text": "we have the following equation for the"}, {"start": 32.04, "duration": 5.48, "text": "area which we simplify and finally we"}, {"start": 35.12, "duration": 5.84, "text": "get Pythagoras's beautiful identity"}, {"start": 37.52, "duration": 3.44, "text": "thank you for watching"}], "title": "A Visual Proof of the Pythagorean Theorem", "description": "Manim animation of Pythagoras", "lengthSeconds": 40, "width": 360, "height": 640}
{"video_id": "LPAH4mzWgjc", "transcript": [{"start": 0.0, "duration": 11.339, "text": "foreign"}, {"start": 0.82, "duration": 10.519, "text": "[Music]"}, {"start": 14.38, "duration": 5.679, "text": "[Music]"}], "title": "Pythagoras visual proof \ud83e\uddd0 #studyhub #interestingfacts #factshorts #pythagoras_theorem", "description": "pythagoras \npythagoras theorem\ntheory proof visual\n2d proof\nkuk\nuniversity \nvisual proof\ngeometry \npractical \nexperiment \nscience\nsquare\ntriangle\nclass 10\nclass 11\nclass 12\nbsc\nmsc\n\n\n\n\n\n\ncomedy \nbachon ki video\ncricket video\nmsd\ndhoni\nvirat kohli \nrohit sharma\nkhwaja\nt20\ncricket live\nsports car\nsports\nlive video\njadeja\nashwin\nbb ki vines\n\nneet short video\n\n\nnew full movie\n\n\nnomadic indian\n\n\n\noscar dhruv rathee\n\nosman season 4 episode 118 trailer\n\n\n\npathan whatsapp status\n\n\n\nranveer allahbadia\n\n\nshot put\n\n\n\nsnap x baarishein\n\nsong\n\ntakeshi castle\n\ntarak mehta ka bhoot wala \n\nupsc motivational video\n\ncooking\n\n\ndj remix song\n\n\neat chalk\n\n\n\na little\n\n\n\ngolden boy\n\n\nhame to loot liya milke husn walon ne\n\nhot sex\n\n\n\nmourning\n\n\njiraiya stop war between sand and leaf\n\n\nkajal comedy\n\n\n\nkrishna radha\n\n\nkumkum bhagya full episode today old\n\n\n\nmc stan ki video\n\n\nmehandi\n\nGroup theory \nM.sc topic\nAlgebric number theory\nGroups and rings\nMathematics \nMaths learning\nMaths theorem\nB.sc maths\nFinite group \nSimple group\nSymmetry \nEarlier 1950\nLegends \nWriting proof\nTheorem \nProofs\nFacts\nInteresting things to know\nShorts\nExperiment \nMathsvlog\nstories\nOnly legends can understand\n\nonly stories\npalak dance\npetta song\nscience\nscience experiments\nscience project\nscience project for class 2\nsourav joshi vlogs\nstories kannada\nTark matka oolta chasma\ntum tum\nzero in the moonlight ep 1 eng sub\nWayne\nAND\nthe second measure\nasma ul husna\nattitude song status\nbl series\nc drama romantic scenes\nchildish girl mature boy whatsapp status\nwere\nf4 thailand\nfather daughter emotional moments\nfine by taeyeon\nflash mob dance\nfood vlogs\nSimplification Trick Simplification short trick | math short tricks #shorts #shortvideo #maths\n\nsimplification tricks simplification tricks for bank exams\n\nsimplification tricks in telugu simplification tricks in tamil simplification tricks adda247 simplification tricks in tamil tnpsc group 4\n\nsimplification tricks for tnpsc group 2\n\nsimplification tricks by ashish arora simplification tricks tnpsc tamil simplification tricks in tamil tnpsc simplification short tricks simplificationshort tricks for bank exam\n\nsimplification short tricks for bank exams in telugu simplification short tricks tnpsc simplification short tricks for ssc simplification short tricks for tnpsc group\n\n\n\nsimplification short tricks by dear sir simplification short tricks by imran sir vedic maths\n\nvedic maths short tricks", "lengthSeconds": 18, "width": 360, "height": 640}
{"video_id": "nQb11RIEXkI", "transcript": [{"start": 0.0, "duration": 3.0, "text": "foreign"}, {"start": 22.48, "duration": 9.7, "text": "[Music]"}, {"start": 29.4, "duration": 2.78, "text": "thank you"}, {"start": 33.08, "duration": 9.01, "text": "[Music]"}, {"start": 42.54, "duration": 27.569, "text": "foreign"}, {"start": 44.62, "duration": 25.489, "text": "[Music]"}, {"start": 74.28, "duration": 3.0, "text": "foreign"}, {"start": 77.87, "duration": 18.089, "text": "[Music]"}, {"start": 92.88, "duration": 3.079, "text": "[Applause]"}], "title": "Geometric Series: sum of powers of 1/9 (visual proof)", "description": "This is a short, animated visual proof showing the sum of the infinite geometric series of powers of 1/9 (starting with 1/9) is 1/8 using self-similar trapezoidal shapes in arrays of squares. #manim #math\u200b\u200b #mathshorts\u200b #mathvideo\u200b  #mtbos\u200b #manim\u200b #animation\u200b #theorem\u200b #pww\u200b #proofwithoutwords\u200b #visualproof\u200b #proof\u200b #iteachmath #calculus #series #geometricseries #infiniteseries  #pentagon #trapezoids\n\nTo learn more about animating with manim, check out:\nhttps://manim.community\n___________________________________\nMusic: \nWhat Words Can\u2019t Describe by Vlad Gluschenko | https://soundcloud.com/vgl9\nMusic promoted by https://www.free-stock-music.com\nCreative Commons Attribution 3.0 Unported License\nhttps://creativecommons.org/licenses/by/3.0/deed.en_US", "lengthSeconds": 101, "width": 640, "height": 360}
{"video_id": "cI92ZID95g4", "transcript": [{"start": 0.04, "duration": 3.72, "text": "start with rational numbers A over B"}, {"start": 1.719, "duration": 4.12, "text": "less than or equal to C over D we can"}, {"start": 3.76, "duration": 3.599, "text": "represent the rational number A over B"}, {"start": 5.839, "duration": 3.561, "text": "as the slope of the hypotenuse of a"}, {"start": 7.359, "duration": 4.121, "text": "right triangle with side lengths A and B"}, {"start": 9.4, "duration": 3.52, "text": "and we can represent C over d as the"}, {"start": 11.48, "duration": 3.96, "text": "slope of the hypotenuse of a right"}, {"start": 12.92, "duration": 4.08, "text": "triangle with legs of length C and D by"}, {"start": 15.44, "duration": 3.759, "text": "assumption the latter triangle has a"}, {"start": 17.0, "duration": 3.92, "text": "steeper hypotenuse if we shift the"}, {"start": 19.199, "duration": 3.561, "text": "triangle so their vertices connect like"}, {"start": 20.92, "duration": 3.64, "text": "this we can fit them inside of a larger"}, {"start": 22.76, "duration": 4.759, "text": "right triangle with sides of length B"}, {"start": 24.56, "duration": 4.84, "text": "plus d and a plus c by investigating the"}, {"start": 27.519, "duration": 4.121, "text": "slopes of the hypotenuses in this"}, {"start": 29.4, "duration": 4.72, "text": "diagram we see that A over B is less"}, {"start": 31.64, "duration": 4.52, "text": "than or equal to the quantity a + C / B"}, {"start": 34.12, "duration": 4.119, "text": "+ D and that's less than or equal to C"}, {"start": 36.16, "duration": 4.919, "text": "over D the fraction in the middle of"}, {"start": 38.239, "duration": 4.081, "text": "this inequality a plus c over B + D is"}, {"start": 41.079, "duration": 3.48, "text": "known as the mediant of the two"}, {"start": 42.32, "duration": 4.399, "text": "fractions so we can define an operation"}, {"start": 44.559, "duration": 4.84, "text": "where we combine A over B and C over D"}, {"start": 46.719, "duration": 4.761, "text": "to produce a plus c over B plus d and"}, {"start": 49.399, "duration": 4.041, "text": "this is called the median operation due"}, {"start": 51.48, "duration": 3.719, "text": "to this terminology the inequality we"}, {"start": 53.44, "duration": 4.56, "text": "proved is sometimes called the mediant"}, {"start": 55.199, "duration": 2.801, "text": "inequality"}], "title": "Mediant Inequality I", "description": "This is a short, animated visual proof demonstrating what one might called the mediant inequality.\n\nIf you like this video, consider subscribing to the channel or consider buying me a coffee: https://www.buymeacoffee.com/VisualProofs. Thanks!\n\nThis animation is based on a visual proof by Richard A. Gibbs from the June 1990 issue of Mathematics Magazine (https://doi.org/10.2307/2691137  - page 172).\n\n#mathshorts\u200b #mathvideo\u200b #math\u200b #numbertheory #mtbos\u200b #manim\u200b #animation\u200b #theorem\u200b #pww\u200b #proofwithoutwords\u200b #visualproof\u200b #proof\u200b #iteachmath #fractions #mediant #inequality #mathematics #fareyfractions #farey\n\nTo learn more about animating with manim, check out:\nhttps://manim.community", "lengthSeconds": 56, "width": 360, "height": 640}
{"video_id": "vNCe7ciFwVc", "transcript": [{"start": 1.99, "duration": 3.21, "text": "[Music]"}, {"start": 8.88, "duration": 3.44, "text": "consider a triangle with sides of length"}, {"start": 11.28, "duration": 2.0, "text": "a"}, {"start": 12.32, "duration": 2.08, "text": "b"}, {"start": 13.28, "duration": 4.079, "text": "and c"}, {"start": 14.4, "duration": 5.12, "text": "and angle theta opposite to c"}, {"start": 17.359, "duration": 5.601, "text": "draw the circle containing the triangle"}, {"start": 19.52, "duration": 6.16, "text": "using the side length of a as the radius"}, {"start": 22.96, "duration": 4.8, "text": "extend the radius to a full diameter"}, {"start": 25.68, "duration": 3.599, "text": "and extend the side length c to a"}, {"start": 27.76, "duration": 5.12, "text": "diameter by connecting a radius of"}, {"start": 29.279, "duration": 5.761, "text": "length a and a segment of length a minus"}, {"start": 32.88, "duration": 2.16, "text": "c"}, {"start": 36.0, "duration": 3.6, "text": "extending the side length b to the"}, {"start": 38.559, "duration": 2.481, "text": "circle"}, {"start": 39.6, "duration": 3.44, "text": "and connecting that line to the other"}, {"start": 41.04, "duration": 4.48, "text": "end of the horizontal diameter"}, {"start": 43.04, "duration": 5.76, "text": "results in a right triangle according to"}, {"start": 45.52, "duration": 3.28, "text": "thaley's triangle theorem"}, {"start": 55.28, "duration": 5.599, "text": "this right triangle has a hypotenuse of"}, {"start": 58.239, "duration": 5.041, "text": "length 2 times a"}, {"start": 60.879, "duration": 5.201, "text": "this means that the leg adjacent to the"}, {"start": 63.28, "duration": 4.24, "text": "angle theta has length 2a cosine of"}, {"start": 66.08, "duration": 3.76, "text": "theta"}, {"start": 67.52, "duration": 3.32, "text": "therefore the line segment extending the"}, {"start": 69.84, "duration": 5.2, "text": "length"}, {"start": 70.84, "duration": 7.56, "text": "b to the circle has length two a cosine"}, {"start": 75.04, "duration": 3.36, "text": "theta minus b"}, {"start": 79.2, "duration": 4.9, "text": "now we have two intersecting chords in"}, {"start": 81.84, "duration": 4.56, "text": "the circle"}, {"start": 84.1, "duration": 5.9, "text": "[Music]"}, {"start": 86.4, "duration": 6.56, "text": "so by the intersecting chords theorem"}, {"start": 90.0, "duration": 4.72, "text": "we get that the product of two a cosine"}, {"start": 92.96, "duration": 4.24, "text": "theta minus b"}, {"start": 94.72, "duration": 3.68, "text": "times b"}, {"start": 97.2, "duration": 4.32, "text": "equals"}, {"start": 98.4, "duration": 4.48, "text": "the product a minus c"}, {"start": 101.52, "duration": 3.84, "text": "times"}, {"start": 102.88, "duration": 4.72, "text": "a plus c"}, {"start": 105.36, "duration": 3.68, "text": "expanding both products produces the"}, {"start": 107.6, "duration": 5.44, "text": "equality"}, {"start": 109.04, "duration": 8.079, "text": "2 a times b cosine theta minus b squared"}, {"start": 113.04, "duration": 6.64, "text": "equals a squared minus c squared"}, {"start": 117.119, "duration": 5.441, "text": "adding b squared plus c squared to both"}, {"start": 119.68, "duration": 5.36, "text": "sides results in two a times b cosine"}, {"start": 122.56, "duration": 3.919, "text": "theta plus c squared equals a squared"}, {"start": 125.04, "duration": 4.8, "text": "plus b squared"}, {"start": 126.479, "duration": 3.361, "text": "this is the law of cosines"}, {"start": 130.59, "duration": 19.049, "text": "[Music]"}, {"start": 150.56, "duration": 2.08, "text": "you"}], "title": "Law Of Cosines I (visual proof)", "description": "This is a short, animated visual proof of the Law of Cosines using Thales triangle theorem and the intersecting chord theorem. This theorem relates the side of a triangle with the other two sides and the angle between those two sides. #mathshorts #mathvideo #math #trigonometry #lawofcosines  #triangle #manim #animation #theorem #pww #proofwithoutwords #visualproof #proof #circle #thalestheorem #intersecting chords\n\nThis animation is based on a proof due to Sidney H. Kung from the December 1990 issue of Mathematics Magazine (https://doi.org/10.2307/2690911) page 342.\n\nTo learn more about animating with manim, check out:\nhttps://manim.community\n____________________________________________________\nBackground music:\nThe Temperature of the Air on the Bow of the Kaleetan by Chris Zabriskie is licensed under a Creative Commons Attribution 4.0 license. https://creativecommons.org/licenses/by/4.0/\n\nSource: http://chriszabriskie.com/uvp/\n\nArtist: http://chriszabriskie.com/", "lengthSeconds": 151, "width": 640, "height": 360}
{"video_id": "Sl81AOB2Bb8", "transcript": [{"start": 7.24, "duration": 4.84, "text": "suppose we start with a regular pentagon"}, {"start": 9.4, "duration": 4.56, "text": "and let's assume the side length is one"}, {"start": 12.08, "duration": 3.639, "text": "the diagonal length that cuts across the"}, {"start": 13.96, "duration": 4.28, "text": "Pentagon here is well known to have a"}, {"start": 15.719, "duration": 4.841, "text": "length given by the Golden Ratio"}, {"start": 18.24, "duration": 4.24, "text": "fee another well-known fact about the"}, {"start": 20.56, "duration": 4.0, "text": "regular pentagon is that this diagonal"}, {"start": 22.48, "duration": 4.36, "text": "shown here along with the original"}, {"start": 24.56, "duration": 4.119, "text": "diagonal drawn creates an isoceles"}, {"start": 26.84, "duration": 4.04, "text": "triangle as pictured here which has"}, {"start": 28.679, "duration": 4.761, "text": "sides of length one and a base of length"}, {"start": 30.88, "duration": 4.359, "text": "1 divided by the golden ratio to see a"}, {"start": 33.44, "duration": 3.84, "text": "proof of this fact check the video"}, {"start": 35.239, "duration": 4.241, "text": "Linked In the description the purple"}, {"start": 37.28, "duration": 5.08, "text": "shaded triangle shown in this diagram is"}, {"start": 39.48, "duration": 5.239, "text": "known as the Golden Triangle the outer"}, {"start": 42.36, "duration": 3.879, "text": "containing triangle is often referred to"}, {"start": 44.719, "duration": 4.441, "text": "as the golden"}, {"start": 46.239, "duration": 5.441, "text": "nomon now we scale the Golden Triangle"}, {"start": 49.16, "duration": 4.44, "text": "by the value one over the Golden Ratio"}, {"start": 51.68, "duration": 4.08, "text": "this allows us to fit a new triangle"}, {"start": 53.6, "duration": 4.32, "text": "right next to the original with sides of"}, {"start": 55.76, "duration": 4.56, "text": "length 1 over the golden ratio and base"}, {"start": 57.92, "duration": 4.56, "text": "given by 1 over the golden ratio squar"}, {"start": 60.32, "duration": 4.559, "text": "this is actually just another Golden"}, {"start": 62.48, "duration": 4.12, "text": "Triangle as it's similar to the original"}, {"start": 64.879, "duration": 3.441, "text": "but now we can repeat this process"}, {"start": 66.6, "duration": 3.92, "text": "scaling the new triangle by one over the"}, {"start": 68.32, "duration": 4.4, "text": "golden ratio and rotating it and fitting"}, {"start": 70.52, "duration": 3.88, "text": "it in the empty space and continuing"}, {"start": 72.72, "duration": 3.32, "text": "again scaling by one over the golden"}, {"start": 74.4, "duration": 4.24, "text": "ratio and fitting the new triangle in"}, {"start": 76.04, "duration": 3.719, "text": "the empty space we can continue this"}, {"start": 78.64, "duration": 3.76, "text": "process"}, {"start": 79.759, "duration": 4.72, "text": "indefinitely each stage we take the"}, {"start": 82.4, "duration": 5.0, "text": "previously scaled copy of the Golden"}, {"start": 84.479, "duration": 5.841, "text": "Triangle and scale Again by the value 1"}, {"start": 87.4, "duration": 6.2, "text": "divid by the Golden Ratio rotate and fit"}, {"start": 90.32, "duration": 5.68, "text": "it in the empty space if we imagine that"}, {"start": 93.6, "duration": 5.799, "text": "we let this process go on forever or"}, {"start": 96.0, "duration": 6.24, "text": "infinitely often the limiting shape is"}, {"start": 99.399, "duration": 4.76, "text": "the entire golden Noman and that means"}, {"start": 102.24, "duration": 5.559, "text": "that we've filled in the entire golden"}, {"start": 104.159, "duration": 5.28, "text": "Noman with these scaled golden triangles"}, {"start": 107.799, "duration": 3.64, "text": "but then we can compute the length of"}, {"start": 109.439, "duration": 5.161, "text": "the long side of the golden Noman"}, {"start": 111.439, "duration": 5.081, "text": "triangle two different ways we know that"}, {"start": 114.6, "duration": 4.439, "text": "on the one hand it's given by the golden"}, {"start": 116.52, "duration": 5.0, "text": "ratio on the other hand it's given by"}, {"start": 119.039, "duration": 5.281, "text": "the sum of the the lengths of the purple"}, {"start": 121.52, "duration": 5.199, "text": "Golden Triangle sitting inside but each"}, {"start": 124.32, "duration": 4.6, "text": "of those purple triangles has a length"}, {"start": 126.719, "duration": 4.801, "text": "given by 1 over an even power of the"}, {"start": 128.92, "duration": 5.959, "text": "golden ratio this means that the golden"}, {"start": 131.52, "duration": 5.64, "text": "ratio is equal to the infinite sum of"}, {"start": 134.879, "duration": 5.08, "text": "the reciprocals of the even powers of"}, {"start": 137.16, "duration": 4.88, "text": "the golden ratio as shown here but we"}, {"start": 139.959, "duration": 4.0, "text": "can also get a related fact by looking"}, {"start": 142.04, "duration": 4.8, "text": "at the pink shaded Golden Triangle"}, {"start": 143.959, "duration": 5.28, "text": "sitting inside this golden nomon on the"}, {"start": 146.84, "duration": 5.16, "text": "one hand the side of the golden nomon"}, {"start": 149.239, "duration": 4.961, "text": "has a length one on the other hand it's"}, {"start": 152.0, "duration": 5.2, "text": "also given by the sum of the lengths of"}, {"start": 154.2, "duration": 4.759, "text": "the pink shaded triangles sitting inside"}, {"start": 157.2, "duration": 3.8, "text": "but each of the pink shaded triangles"}, {"start": 158.959, "duration": 4.761, "text": "has a length given by one over an odd"}, {"start": 161.0, "duration": 4.64, "text": "power of the golden ratio therefore one"}, {"start": 163.72, "duration": 3.64, "text": "is equal to the infinite sum of the"}, {"start": 165.64, "duration": 4.8, "text": "reciprocal of the odd powers of the"}, {"start": 167.36, "duration": 5.0, "text": "golden ratio notice that these two facts"}, {"start": 170.44, "duration": 4.2, "text": "are actually one and the same as we can"}, {"start": 172.36, "duration": 4.36, "text": "obtain the first fact from the second"}, {"start": 174.64, "duration": 5.239, "text": "fact simply by multiplying the entire"}, {"start": 176.72, "duration": 4.96, "text": "equality by the Golden Ratio Fe either"}, {"start": 179.879, "duration": 3.72, "text": "of these two infinite series could be"}, {"start": 181.68, "duration": 3.4, "text": "called the Golden series as they deal"}, {"start": 183.599, "duration": 3.481, "text": "with the infinite powers of the"}, {"start": 185.08, "duration": 4.079, "text": "reciprocal of the golden ratio can you"}, {"start": 187.08, "duration": 4.2, "text": "see what happens if you try to add up"}, {"start": 189.159, "duration": 5.281, "text": "all of the powers of the reciprocal of"}, {"start": 191.28, "duration": 3.16, "text": "the golden ratio"}], "title": "A Golden Ratio Infinite Series Dissection (visual proof)", "description": "This is animated visual proof demonstrating the sum of the infinite series of the reciprocals of even powers of the golden ratio is the golden ratio. We also get a related fact showing the sum of the reciprocals of the reciprocals of the odd powers of the golden ratio. This dissection proof utilizes the so-called \"golden gnomon\" and the \"golden triangle\". \n\nIf you like this video, consider subscribing to the channel or consider buying me a coffee: https://www.buymeacoffee.com/VisualProofs. Thanks!\n\nTo see the proof that the golden triangle has the dimensions stated, see this video: https://youtu.be/lL9VN8ELH1M\n\nFor more infinite series dissections, check out this playlist:\nhttps://www.youtube.com/playlist?list=PLZh9gzIvXQUv6JqhlqQDlnqQw1NczW5Ro\n\nHere is a longer form video with lots of infinite geometric sums: https://youtu.be/JteQEN1XPyc\n\nThis animation is based on a visual proof by Steven Edwards from the March 2014 issue of The College Mathematics Journal (https://doi.org/10.4169/college.math.j.45.2.120 - page 120).\n\n#shorts\u200b\u200b #math\u200b #calculus #mtbos\u200b #manim\u200b #animation\u200b #theorem\u200b #pww\u200b #proofwithoutwords\u200b #visualproof\u200b #proof\u200b #iteachmath #geometricseries #series #infiniteseries #fractal #infinitesum  #goldenratio #pentagon #algebra #similartriangles #triangle #trianglesimilarity #similarity #goldentriangle #goldengnomon\n\n\nTo learn more about animating with manim, check out:\nhttps://manim.community", "lengthSeconds": 193, "width": 640, "height": 360}
{"video_id": "u8X9ymLUC38", "transcript": [{"start": 0.12, "duration": 5.54, "text": "foreign"}, {"start": 2.67, "duration": 5.849, "text": "[Music]"}, {"start": 5.66, "duration": 5.14, "text": "the sum of the first K positive integers"}, {"start": 8.519, "duration": 4.381, "text": "is called the kth Triangular number is"}, {"start": 10.8, "duration": 4.14, "text": "denoted T sub K and can be visually"}, {"start": 12.9, "duration": 3.959, "text": "represented by a triangular array of"}, {"start": 14.94, "duration": 4.32, "text": "dots where there's K dots in the last"}, {"start": 16.859, "duration": 4.5, "text": "row and one dot in the first row in this"}, {"start": 19.26, "duration": 5.099, "text": "video we'll investigate divisibility of"}, {"start": 21.359, "duration": 5.041, "text": "triangular numbers by three"}, {"start": 24.359, "duration": 3.901, "text": "because we're dividing by three there"}, {"start": 26.4, "duration": 3.42, "text": "are three cases to consider suppose"}, {"start": 28.26, "duration": 3.72, "text": "first that we're looking at a triangular"}, {"start": 29.82, "duration": 3.899, "text": "array with 3K plus one dots in the base"}, {"start": 31.98, "duration": 3.66, "text": "so that we're looking at the 3K plus"}, {"start": 33.719, "duration": 3.781, "text": "first triangular a number then we can"}, {"start": 35.64, "duration": 4.62, "text": "focus on a trapezoidal array at the"}, {"start": 37.5, "duration": 5.1, "text": "bottom that has 2K plus 1 Dots and a"}, {"start": 40.26, "duration": 4.44, "text": "height of length K we can then take"}, {"start": 42.6, "duration": 4.5, "text": "three copies of this trapezoidal array"}, {"start": 44.7, "duration": 4.379, "text": "rotate it and move it around the diagram"}, {"start": 47.1, "duration": 4.26, "text": "so that it almost fills up the entire"}, {"start": 49.079, "duration": 4.561, "text": "diagram there's only one dot left over"}, {"start": 51.36, "duration": 4.379, "text": "this means that the 3K plus first"}, {"start": 53.64, "duration": 4.14, "text": "triangular number must be equal to one"}, {"start": 55.739, "duration": 3.541, "text": "plus three copies of the number of dots"}, {"start": 57.78, "duration": 2.939, "text": "in the trapezoidal array but the"}, {"start": 59.28, "duration": 2.88, "text": "trapezoidal array dots can be"}, {"start": 60.719, "duration": 3.66, "text": "represented as the 2K plus first"}, {"start": 62.16, "duration": 3.959, "text": "triangular number minus the K plus first"}, {"start": 64.379, "duration": 3.841, "text": "triangular number as it's really the"}, {"start": 66.119, "duration": 3.54, "text": "difference of two triangular arrays so"}, {"start": 68.22, "duration": 3.6, "text": "in this case the number of dots is"}, {"start": 69.659, "duration": 3.661, "text": "congruent to one mod 3. for our next"}, {"start": 71.82, "duration": 3.299, "text": "case let's suppose that the number of"}, {"start": 73.32, "duration": 3.6, "text": "dots in the base of the Triangular array"}, {"start": 75.119, "duration": 3.721, "text": "is divisible by three so that it's three"}, {"start": 76.92, "duration": 3.42, "text": "times k for some integer k then we're"}, {"start": 78.84, "duration": 4.2, "text": "looking at the array representing the"}, {"start": 80.34, "duration": 4.5, "text": "Triangular number T sub 3K this time we"}, {"start": 83.04, "duration": 3.66, "text": "can form a trapezoidal array in the"}, {"start": 84.84, "duration": 4.2, "text": "bottom left with two K dots in the"}, {"start": 86.7, "duration": 4.38, "text": "bottom and the height of size k then we"}, {"start": 89.04, "duration": 4.38, "text": "can rotate and shift three copies of"}, {"start": 91.08, "duration": 5.16, "text": "this trapezoidal array to perfectly fill"}, {"start": 93.42, "duration": 5.519, "text": "in the entire diagram this time this"}, {"start": 96.24, "duration": 4.559, "text": "means that t sub 3K is equal to 3 times"}, {"start": 98.939, "duration": 3.301, "text": "the number of dots in the trapezoidal"}, {"start": 100.799, "duration": 3.601, "text": "array which is given by the difference"}, {"start": 102.24, "duration": 4.62, "text": "of the 2K triangular number and the kth"}, {"start": 104.4, "duration": 5.999, "text": "Triangular number so in this case we see"}, {"start": 106.86, "duration": 6.0, "text": "that t sub 3K is congruent to 0 mod 3."}, {"start": 110.399, "duration": 5.04, "text": "for our final case we imagine that our"}, {"start": 112.86, "duration": 4.619, "text": "triangular array has 3K plus two dots in"}, {"start": 115.439, "duration": 4.381, "text": "the base so that we're looking at T sub"}, {"start": 117.479, "duration": 3.721, "text": "three K plus 2. once again we form a"}, {"start": 119.82, "duration": 4.14, "text": "trapezoidal array in the bottom left"}, {"start": 121.2, "duration": 5.16, "text": "with a base consisting of two k plus 1"}, {"start": 123.96, "duration": 4.799, "text": "Dots and a height consisting of K plus 1"}, {"start": 126.36, "duration": 4.619, "text": "dots once again we can use three rotated"}, {"start": 128.759, "duration": 4.441, "text": "copies of this trapezoidal array of dots"}, {"start": 130.979, "duration": 4.441, "text": "to fill in the entire triangular array"}, {"start": 133.2, "duration": 4.8, "text": "that we're interested in this means that"}, {"start": 135.42, "duration": 4.26, "text": "t sub 3 K plus 2 is equal to 3 times the"}, {"start": 138.0, "duration": 3.239, "text": "number of dots in the trapezoidal array"}, {"start": 139.68, "duration": 3.48, "text": "which is given by the difference of the"}, {"start": 141.239, "duration": 4.08, "text": "2K plus first triangular number and the"}, {"start": 143.16, "duration": 4.32, "text": "kth Triangular number as shown in this"}, {"start": 145.319, "duration": 5.101, "text": "case we see that t sub 3 K plus 2 is"}, {"start": 147.48, "duration": 5.399, "text": "congruent to 0 mod 3. using these three"}, {"start": 150.42, "duration": 5.16, "text": "cases we see that t sub n is congruent"}, {"start": 152.879, "duration": 5.041, "text": "to 1 or 0 mod 3 and it's congruent to 1"}, {"start": 155.58, "duration": 4.62, "text": "if and only if n is congruent to 1 mod"}, {"start": 157.92, "duration": 6.26, "text": "3. check out the linked video for the"}, {"start": 160.2, "duration": 3.98, "text": "analogous result for square numbers"}], "title": "Triangular Numbers Modulo 3 (visual proof)", "description": "We present an animation of a visual proof demonstrating how to visualize the the congruence classes of triangular numbers modulo 3. \n\nIf you like this video, consider subscribing to the channel or consider buying me a coffee: https://www.buymeacoffee.com/VisualProofs. Thanks! \n\nIf you want to see a related video (about congruence classes of squares), check out : https://youtu.be/HiDe2Ngcq2c\n\nThis animation is based on a visual proof by Roger B. Nelsen from his first Proofs Without Words Compendium: https://bookstore.ams.org/view?ProductCode=CLRM/1\n\nTo learn more about animating with manim, check out:\nhttps://manim.community\n\n#mathshorts\u200b #mathvideo\u200b #math\u200b #numbertheory #mtbos\u200b #manim\u200b #animation\u200b #theorem\u200b #pww\u200b #proofwithoutwords\u200b #visualproof\u200b #proof\u200b #iteachmath #squares #modulararithmetic #numbertheory \n__________________________________________\nMusic in this video:\nTwinkle in the Night by Aakash Gandhi", "lengthSeconds": 166, "width": 640, "height": 360}
{"video_id": "tcxLMMncWZ0", "transcript": [{"start": 0.28, "duration": 4.36, "text": "let's say we have a line comprised of"}, {"start": 2.12, "duration": 5.08, "text": "segments A and B the total length of"}, {"start": 4.64, "duration": 5.4, "text": "this line would be a plus b how would we"}, {"start": 7.2, "duration": 6.0, "text": "find a square of this let's visualize by"}, {"start": 10.04, "duration": 5.679, "text": "rotating the line by 90\u00b0 this would form"}, {"start": 13.2, "duration": 4.96, "text": "a square made out of one red square one"}, {"start": 15.719, "duration": 5.441, "text": "blue square and two purple rectangles"}, {"start": 18.16, "duration": 6.64, "text": "the area of the red square is a * a or a"}, {"start": 21.16, "duration": 5.92, "text": "squ so we add a squ to the equation next"}, {"start": 24.8, "duration": 5.84, "text": "we have a purple rectangle of lengths A"}, {"start": 27.08, "duration": 5.519, "text": "and B its area would be AB same goes for"}, {"start": 30.64, "duration": 5.36, "text": "the other purple rectangle this"}, {"start": 32.599, "duration": 5.841, "text": "simplifies into positive 2ab lastly the"}, {"start": 36.0, "duration": 5.559, "text": "area of the blue squared is a b * B"}, {"start": 38.44, "duration": 8.2, "text": "simply b^ 2 therefore we have proven"}, {"start": 41.559, "duration": 8.251, "text": "that a + b 2 is equal to a 2 + 2 a + b s"}, {"start": 46.64, "duration": 9.509, "text": "subscribe to Mika Pine 91"}, {"start": 49.81, "duration": 6.339, "text": "[Music]"}], "title": "Visual proof of (a+b)^2", "description": "Thank you @MikaPie91 for helping out! \n\nPS. first math yt video. please give feedback!\n.\n.\n.\n.\n.\n.\n.\n.\n.\n.\n.\n#maths  #mathshorts #visualproof #algebra #matheducation #mathtutorial \n#mathexplained #mathconcepts #learnmaths #educationalshorts\n#mathisfun #mathematics #mathtricks #geometry #algebratricks \n#mathlessons #stem #quickmath #mathvideo \n#youtubeshorts #mathproblems", "lengthSeconds": 54, "width": 360, "height": 360}
{"video_id": "NHxJ3Z_58Lw", "transcript": [{"start": 0.06, "duration": 6.14, "text": "foreign"}, {"start": 1.88, "duration": 7.36, "text": "[Music]"}, {"start": 6.2, "duration": 6.58, "text": "suppose we have a triangle with sides of"}, {"start": 9.24, "duration": 6.24, "text": "length a b and c and suppose that the"}, {"start": 12.78, "duration": 4.86, "text": "angle opposite of C is called Theta the"}, {"start": 15.48, "duration": 5.28, "text": "law of cosines provides a relationship"}, {"start": 17.64, "duration": 5.04, "text": "between these four values to derive the"}, {"start": 20.76, "duration": 3.72, "text": "law of cosines we drop a perpendicular"}, {"start": 22.68, "duration": 4.38, "text": "from the top of the triangle to the base"}, {"start": 24.48, "duration": 4.619, "text": "and that creates two sub right triangles"}, {"start": 27.06, "duration": 3.66, "text": "the hypotenuse of the yellow triangle is"}, {"start": 29.099, "duration": 3.361, "text": "B which means that the height of that"}, {"start": 30.72, "duration": 3.54, "text": "yellow triangle is B times the sine of"}, {"start": 32.46, "duration": 3.96, "text": "theta while the base of that triangle is"}, {"start": 34.26, "duration": 4.38, "text": "B times the cosine of theta this means"}, {"start": 36.42, "duration": 4.5, "text": "that the base of the blue right triangle"}, {"start": 38.64, "duration": 4.32, "text": "is a minus B times the cosine of theta"}, {"start": 40.92, "duration": 4.44, "text": "but now that we have a right triangle"}, {"start": 42.96, "duration": 4.8, "text": "pictured we can draw squares off the"}, {"start": 45.36, "duration": 4.56, "text": "legs of the right triangle and a square"}, {"start": 47.76, "duration": 4.799, "text": "off the hypotenuse of the right triangle"}, {"start": 49.92, "duration": 5.04, "text": "like this these squares have area B"}, {"start": 52.559, "duration": 5.041, "text": "times the sine of theta squared"}, {"start": 54.96, "duration": 5.7, "text": "quantity a minus B cosine Theta squared"}, {"start": 57.6, "duration": 6.24, "text": "and c squared by the Pythagorean theorem"}, {"start": 60.66, "duration": 5.459, "text": "we know that c squared equals the"}, {"start": 63.84, "duration": 4.44, "text": "quantity a minus B times cosine Theta"}, {"start": 66.119, "duration": 4.561, "text": "squared plus the quantity B times sine"}, {"start": 68.28, "duration": 4.26, "text": "of theta squared but now we can apply"}, {"start": 70.68, "duration": 3.9, "text": "algebraic rules along with the"}, {"start": 72.54, "duration": 3.96, "text": "Pythagorean trigonometric identity to"}, {"start": 74.58, "duration": 4.5, "text": "see that c squared equals a squared plus"}, {"start": 76.5, "duration": 4.92, "text": "b squared minus 2 times a times B times"}, {"start": 79.08, "duration": 3.78, "text": "cosine Theta this is the law of cosines"}, {"start": 81.42, "duration": 3.54, "text": "and this means that we've used the"}, {"start": 82.86, "duration": 4.92, "text": "Pythagorean theorem to prove the law of"}, {"start": 84.96, "duration": 4.799, "text": "cosines we can also plug theta equals pi"}, {"start": 87.78, "duration": 3.78, "text": "over 2 into the law of cosines to prove"}, {"start": 89.759, "duration": 4.561, "text": "the Pythagorean theorem and that means"}, {"start": 91.56, "duration": 4.5, "text": "that the two theorems are equivalent"}, {"start": 94.32, "duration": 3.72, "text": "you may be wondering if the diagram"}, {"start": 96.06, "duration": 3.96, "text": "works when the angle Theta is obtuse"}, {"start": 98.04, "duration": 4.2, "text": "instead here is such a triangular"}, {"start": 100.02, "duration": 5.099, "text": "diagram with sides of length a b and c"}, {"start": 102.24, "duration": 5.22, "text": "and an obtuse angle Theta across from C"}, {"start": 105.119, "duration": 4.921, "text": "in this instance we can drop an altitude"}, {"start": 107.46, "duration": 5.339, "text": "and then draw a horizontal line to build"}, {"start": 110.04, "duration": 4.68, "text": "this right triangle inside this blue"}, {"start": 112.799, "duration": 4.5, "text": "right triangle we see a yellow right"}, {"start": 114.72, "duration": 4.5, "text": "triangle with an angle Pi minus Theta so"}, {"start": 117.299, "duration": 3.78, "text": "its height is B times the sine of Pi"}, {"start": 119.22, "duration": 4.98, "text": "minus Theta while its base is B times"}, {"start": 121.079, "duration": 5.521, "text": "the cosine of pi minus Theta however we"}, {"start": 124.2, "duration": 4.38, "text": "know that b times the sine of Pi minus"}, {"start": 126.6, "duration": 4.139, "text": "Theta is actually equal to B times the"}, {"start": 128.58, "duration": 4.14, "text": "sine of theta and B times the cosine of"}, {"start": 130.739, "duration": 4.021, "text": "pi minus Theta is negative B times the"}, {"start": 132.72, "duration": 4.08, "text": "cosine of theta this means that the"}, {"start": 134.76, "duration": 4.14, "text": "pictured blue right triangle has a"}, {"start": 136.8, "duration": 4.38, "text": "height of B times sine of theta and a"}, {"start": 138.9, "duration": 4.32, "text": "base of length a minus B times a cosine"}, {"start": 141.18, "duration": 3.72, "text": "of theta so we can apply the Pythagorean"}, {"start": 143.22, "duration": 3.72, "text": "theorem again to get this exact same"}, {"start": 144.9, "duration": 4.08, "text": "identity that c squared equals the"}, {"start": 146.94, "duration": 4.439, "text": "quantity a minus B cosine Theta squared"}, {"start": 148.98, "duration": 3.839, "text": "plus b times sine Theta squared once"}, {"start": 151.379, "duration": 4.821, "text": "again algebraic manipulation will"}, {"start": 152.819, "duration": 3.381, "text": "provide the law of cosines in this case"}], "title": "Law Of Cosines II (visual proof)", "description": "This is a short, animated visual proof of the Law of Cosines using the Pythagorean theorem. This theorem relates the side of a triangle with the other two sides and the angle between those two sides. \n\nIf you like this video, consider subscribing to the channel or consider buying me a coffee: https://www.buymeacoffee.com/VisualProofs. Thanks! \n\nThis animation is based on a proof from Timothy Sipka in the October 1988 issue of Mathematics Magazine (https://www.jstor.org/stable/2689363 ) page 259.\n\nFor another visual proof of this same fact check out: \nhttps://youtu.be/vNCe7ciFwVc\n\n#mathshorts #mathvideo #math #trigonometry #lawofcosines  #triangle #manim #animation #theorem #pww #proofwithoutwords #visualproof #proof #circle #pythagoreantheorem #obtuseangle #acuteangle #angle \n\nTo learn more about animating with manim, check out:\nhttps://manim.community\n_________________________________\nMusic in this video:\nElegy by Asher Fulero", "lengthSeconds": 162, "width": 640, "height": 360}
{"video_id": "_0RLAnO53Ao", "transcript": [{"start": 0.12, "duration": 4.04, "text": "here's an interesting Pi conundrum start"}, {"start": 2.2, "duration": 4.119, "text": "with a semicircle with radius one so"}, {"start": 4.16, "duration": 4.6, "text": "that it has a circumference of Pi now we"}, {"start": 6.319, "duration": 3.921, "text": "can draw two semicircles inside each has"}, {"start": 8.76, "duration": 3.959, "text": "a radius of a half so has a"}, {"start": 10.24, "duration": 4.88, "text": "circumference of pi/ 2 for a combined"}, {"start": 12.719, "duration": 4.761, "text": "length of Pi again now draw two"}, {"start": 15.12, "duration": 4.48, "text": "semicircles in each of those semicircles"}, {"start": 17.48, "duration": 3.76, "text": "so that each has a radius of 1/4 and a"}, {"start": 19.6, "duration": 3.96, "text": "circumference of pi over 4 for a"}, {"start": 21.24, "duration": 4.4, "text": "combined total length of Pi again we can"}, {"start": 23.56, "duration": 4.639, "text": "continue this process over and over"}, {"start": 25.64, "duration": 5.399, "text": "again each time we double the number of"}, {"start": 28.199, "duration": 4.88, "text": "semicircles but we have the radius this"}, {"start": 31.039, "duration": 4.241, "text": "means that the total combined lengths of"}, {"start": 33.079, "duration": 5.561, "text": "their circumferences will always remain"}, {"start": 35.28, "duration": 5.959, "text": "pi as we keep doing this we see that the"}, {"start": 38.64, "duration": 4.239, "text": "semicircles squash down and flattened to"}, {"start": 41.239, "duration": 4.201, "text": "almost match the diameter of the"}, {"start": 42.879, "duration": 3.961, "text": "original semicircle does that mean that"}, {"start": 45.44, "duration": 4.08, "text": "the length of the diameter of the"}, {"start": 46.84, "duration": 4.92, "text": "original semicircle is actually equal to"}, {"start": 49.52, "duration": 5.92, "text": "Pi when we consider the limiting process"}, {"start": 51.76, "duration": 5.279, "text": "is pi equal to 2 of course Pi isn't"}, {"start": 55.44, "duration": 4.68, "text": "equal to two but can you see where this"}, {"start": 57.039, "duration": 3.081, "text": "argument fails"}], "title": "Does Pi Equal 2?? (Spoiler: no)", "description": "In this short, we show a thought experiment that arises by doubling the number of semicircles drawn in a semicircle but halving their radii. The process produces an infinite collection of smaller and smaller chains of more and more semicircles that always have combined circumference length equal to pi. If we think about the limiting process, it seems like maybe this technique shows that pi = 2. But Pi can definitely be shown to be greater than 3 - so where does this argument fail?\n\nIf you like this video, consider subscribing to the channel or consider buying me a coffee: https://www.buymeacoffee.com/VisualProofs. Thanks! \n\nHere is a related false proof that shows root 2 is equal to 2:\nhttps://youtube.com/shorts/owFwvyozcn8\n\nThis false argument was suggested to me by Jeff Stuart as a nice alternative to the more classic Pi = 4 argument.\n\n#math #manim #visualproof #proofwithoutwords #circle #circumference  #radius #pi #piday #shorts #infinite #falseproof\n\nTo learn more about animating with manim, check out:\nhttps://manim.community", "lengthSeconds": 58, "width": 360, "height": 640}
{"video_id": "hZDXA1R8dkA", "transcript": [{"start": 0.96, "duration": 4.2, "text": "let's see an intriguing diagram that"}, {"start": 2.639, "duration": 4.601, "text": "gives us five formulas for"}, {"start": 5.16, "duration": 5.2, "text": "pi start with a rectangle with"}, {"start": 7.24, "duration": 6.12, "text": "Dimensions 2 by3 inside the rectangle"}, {"start": 10.36, "duration": 5.319, "text": "draw this triangle as shown here the two"}, {"start": 13.36, "duration": 5.079, "text": "highlighted pink triangles are both 1X"}, {"start": 15.679, "duration": 3.921, "text": "two right triangles so they're congruent"}, {"start": 18.439, "duration": 2.92, "text": "because they fit together at"}, {"start": 19.6, "duration": 4.439, "text": "complimentary angles that means that the"}, {"start": 21.359, "duration": 4.481, "text": "blue shaded triangle is a right triangle"}, {"start": 24.039, "duration": 3.761, "text": "the hypotenuses on the pink triangles"}, {"start": 25.84, "duration": 4.279, "text": "are equal and that means that the blue"}, {"start": 27.8, "duration": 3.92, "text": "triangle has two sides of equal length"}, {"start": 30.119, "duration": 3.481, "text": "therefore the blue triangle is a right"}, {"start": 31.72, "duration": 4.839, "text": "isoceles triangle so the other two"}, {"start": 33.6, "duration": 4.88, "text": "angles are both Pi over4 the pi over4"}, {"start": 36.559, "duration": 3.881, "text": "angle on the left of this diagram can be"}, {"start": 38.48, "duration": 4.2, "text": "obtained as the sum of two angles from"}, {"start": 40.44, "duration": 4.52, "text": "these pictured triangles one of them has"}, {"start": 42.68, "duration": 4.96, "text": "an angle of arctangent 1/3 and the other"}, {"start": 44.96, "duration": 5.52, "text": "one has an angle of arc tangent"}, {"start": 47.64, "duration": 4.88, "text": "1/2 the upper right Pi over4 angle can"}, {"start": 50.48, "duration": 3.96, "text": "be obtained as the difference of two"}, {"start": 52.52, "duration": 3.6, "text": "angles one in this triangle which has an"}, {"start": 54.44, "duration": 3.119, "text": "angle measurement of arctangent 3 and"}, {"start": 56.12, "duration": 4.56, "text": "one in this triangle which has an angle"}, {"start": 57.559, "duration": 4.721, "text": "measurement of AR tangent of a half"}, {"start": 60.68, "duration": 3.479, "text": "another way to get pi over 4 in this"}, {"start": 62.28, "duration": 3.839, "text": "case is as the difference of two angles"}, {"start": 64.159, "duration": 3.721, "text": "in these triangles one with an angle"}, {"start": 66.119, "duration": 3.121, "text": "measurement of arct tangent of two and"}, {"start": 67.88, "duration": 4.2, "text": "the other one with an angle measurement"}, {"start": 69.24, "duration": 4.519, "text": "of the arct tangent of 1/3 the angle in"}, {"start": 72.08, "duration": 4.0, "text": "the upper right corner of the rectangle"}, {"start": 73.759, "duration": 4.321, "text": "is pi/ 2 and this can be obtained as the"}, {"start": 76.08, "duration": 3.2, "text": "sum of three angles where one of them is"}, {"start": 78.08, "duration": 3.16, "text": "the angle measurement of the blue"}, {"start": 79.28, "duration": 3.799, "text": "triangle which is arct tangent one the"}, {"start": 81.24, "duration": 3.559, "text": "next is the arct tangent of 1/2 in this"}, {"start": 83.079, "duration": 3.921, "text": "yellow triangle and the last is the arc"}, {"start": 84.799, "duration": 4.36, "text": "tangent of 1/3 in this yellow"}, {"start": 87.0, "duration": 4.159, "text": "triangle finally the angle on the left"}, {"start": 89.159, "duration": 3.64, "text": "side of the diagram pictured here is pi"}, {"start": 91.159, "duration": 3.441, "text": "and it's the sum of three angles in"}, {"start": 92.799, "duration": 3.761, "text": "three triangles the blue triangle gives"}, {"start": 94.6, "duration": 3.519, "text": "us an angle measurement of arc tangent 1"}, {"start": 96.56, "duration": 3.44, "text": "the bottom yellow triangle here gives us"}, {"start": 98.119, "duration": 3.32, "text": "an angle measurement of arct tangent 2"}, {"start": 100.0, "duration": 3.719, "text": "and the top yellow triangle here gives"}, {"start": 101.439, "duration": 4.921, "text": "us an angle measurement arct tangent"}, {"start": 103.719, "duration": 4.801, "text": "3 this amazing diagram has given us five"}, {"start": 106.36, "duration": 3.52, "text": "formulas for pi let me know in the"}, {"start": 108.52, "duration": 3.44, "text": "comments which of these five you like"}, {"start": 109.88, "duration": 4.0, "text": "the best and check out my Pi playlist"}, {"start": 111.96, "duration": 4.119, "text": "for other videos and animations related"}, {"start": 113.88, "duration": 5.199, "text": "to the circle"}, {"start": 116.079, "duration": 3.0, "text": "constant"}], "title": "Five Pi Formulas in \"Pi/2 minutes\" (Pi Day visual proofs)", "description": "In this short video, we give five different formulas for pi (or related constants) using only a single diagram consisting of a 2x3 rectangle and a triangle inside. Using the fact that the tangent of an angle is the ratio of the opposite and adjacent sides of an angle (and you can get the angle using the arctangent of the appropriate ratio), we decompose angles (of measure pi/4, pi/2, and pi) using certain triangles with visually computable arctangents using the diagram.\n\nIf you like this video, consider subscribing to the channel or consider buying me a coffee: https://www.buymeacoffee.com/VisualProofs. Thanks! \n\nThis visual proof is due to Rex Wu and comes from the March 2003 issue of The College Mathematics Journal (https://www.jstor.org/stable/3595783 ) see pages 115 and 138.\n\nFor more Pi-related videos, check out my playlist:\nhttps://youtube.com/playlist?list=PLZh9gzIvXQUsGRDvzvXc02lDIKHaVeFwq\n\n#manim #irrational #Pi #mathvideo\u200b #math\u200b #mtbos\u200b \u200b #animation\u200b\u200b\u200b #iteachmath #mathematics #piday #shorts #trigonometry #arctangent #tangent #triangle #rectangle #identities \n\nTo learn more about animating with manim, check out:\nhttps://manim.community", "lengthSeconds": 117, "width": 640, "height": 360}
{"video_id": "Noc2zM3iRAI", "transcript": [{"start": 4.08, "duration": 3.16, "text": "in"}, {"start": 5.78, "duration": 5.1, "text": "[Music]"}, {"start": 7.24, "duration": 5.88, "text": "this them c i your energy"}, {"start": 10.88, "duration": 6.16, "text": "baby look at you crazy got to cut the"}, {"start": 13.12, "duration": 3.92, "text": "lights ons cut"}, {"start": 17.22, "duration": 10.02, "text": "[Music]"}, {"start": 25.4, "duration": 6.0, "text": "the"}, {"start": 27.24, "duration": 7.2, "text": "got the birds the I don't want to see I"}, {"start": 31.4, "duration": 6.4, "text": "want to see when the be drop theop got"}, {"start": 34.44, "duration": 3.36, "text": "to shake the Sheck"}], "title": "3\u00b2+4\u00b2=5\u00b2 Visual Proof, Desmos 3D #Desmos3D #DesmosArt #MathArt #pythagorastheorem", "description": "3\u00b2+4\u00b2=5\u00b2 Visual Proof, Desmos 3D #Desmos3D #DesmosArt #MathArt #pythagorastheorem\n\nThis video was created by:\n\nhttps://www.twitter.com/tiago_hands\n\nFor more mathematics content, check out:\n\nhttps://www.threads.net/@mathematics.proofs\n\nhttps://www.instagram.com/mathematics.proofs\n\nThanks for watching this video and make sure you have subscribed if you want to see more videos like this!", "lengthSeconds": 37, "width": 360, "height": 640}
{"video_id": "sujdwst2qh8", "transcript": [{"start": 0.77, "duration": 3.129, "text": "[\u0e40\u0e1e\u0e25\u0e07]"}, {"start": 5.95, "duration": 3.139, "text": "[\u0e40\u0e1e\u0e25\u0e07]"}, {"start": 12.66, "duration": 4.81, "text": "[\u0e40\u0e1e\u0e25\u0e07]"}, {"start": 19.84, "duration": 12.88, "text": "[\u0e40\u0e1e\u0e25\u0e07]"}, {"start": 28.96, "duration": 3.76, "text": " H  y "}], "title": "Visual Proof of (a+b)^2 #math", "description": "", "lengthSeconds": 31, "width": 360, "height": 640}
{"video_id": "AtHKh6b_bPo", "transcript": [{"start": 3.7, "duration": 6.62, "text": "[Music]"}, {"start": 22.62, "duration": 3.169, "text": "[Music]"}, {"start": 28.96, "duration": 4.809, "text": "yeah"}, {"start": 30.62, "duration": 3.149, "text": "[Music]"}, {"start": 46.68, "duration": 5.68, "text": "[Music]"}, {"start": 58.92, "duration": 3.0, "text": "he"}, {"start": 63.09, "duration": 3.099, "text": "[Music]"}, {"start": 71.07, "duration": 7.13, "text": "[Music]"}, {"start": 82.61, "duration": 9.23, "text": "[Music]"}, {"start": 88.84, "duration": 3.0, "text": "oh"}, {"start": 94.59, "duration": 3.539, "text": "[Music]"}, {"start": 103.07, "duration": 3.12, "text": "[Music]"}], "title": "Infinite Sum of Powers of -1/7 in a Square (visual proof without words)", "description": "This is a short, animated (wordless) visual proof demonstrating the sum of the infinite geometric series with ratio -1/7. \n\nIf you like this video, consider subscribing to the channel or consider buying me a coffee: https://www.buymeacoffee.com/VisualProofs. Thanks! \n\nHere are two alternating series dissections for powers of -1/2: https://youtu.be/NKFrHfBGDNE\nhttps://youtu.be/wLPsEULfPnk\n\nAnd here is an alternating series dissection for powers of -1/3:\nhttps://youtu.be/MyCn7gIPxNg\n\nAnd here is an alternating series dissection for powers of -1/4:\nhttps://youtu.be/SeKGZ2DLAwo\n\n#math\u200b #calculus #mtbos\u200b #manim\u200b #animation\u200b #theorem\u200b #pww\u200b #proofwithoutwords\u200b #visualproof\u200b #proof\u200b #iteachmath #geometricseries #infiniteseries #fractals #sum #induction #mathshorts\u200b #mathvideo\u200b \n\nTo learn more about animating with manim, check out:\nhttps://manim.community\n_______________________________\nMusic in this video:\nDoing My Best by Alex-Productions | https://onsound.eu/\nMusic promoted by https://www.free-stock-music.com\nCreative Commons / Attribution 3.0 Unported License (CC BY 3.0)\nhttps://creativecommons.org/licenses/by/3.0/deed.en_US", "lengthSeconds": 108, "width": 640, "height": 360}
{"video_id": "_ssQqXs7l-c", "transcript": [{"start": 0.26, "duration": 2.71, "text": "[Music]"}, {"start": 9.2, "duration": 5.599, "text": "start with two real numbers theta less"}, {"start": 11.599, "duration": 5.841, "text": "than phi both less than pi over two"}, {"start": 14.799, "duration": 4.881, "text": "on the domain from 0 to pi over 2"}, {"start": 17.44, "duration": 4.8, "text": "graph the functions y equals the tangent"}, {"start": 19.68, "duration": 5.04, "text": "of x which is concave up and increasing"}, {"start": 22.24, "duration": 4.32, "text": "and the function y equals sine of x"}, {"start": 24.72, "duration": 4.92, "text": "which is concave down and increasing on"}, {"start": 26.56, "duration": 3.08, "text": "the domain"}, {"start": 30.08, "duration": 5.36, "text": "secant line connecting the origin to the"}, {"start": 32.719, "duration": 5.121, "text": "point theta sine of theta"}, {"start": 35.44, "duration": 4.959, "text": "has the equation y equals the sine of"}, {"start": 37.84, "duration": 4.879, "text": "theta over theta times x and this secant"}, {"start": 40.399, "duration": 5.361, "text": "line lies above sine of x for x larger"}, {"start": 42.719, "duration": 4.721, "text": "than theta similarly the secant line"}, {"start": 45.76, "duration": 3.84, "text": "connecting the origin to theta tangent"}, {"start": 47.44, "duration": 4.48, "text": "of theta has the equation y equals the"}, {"start": 49.6, "duration": 4.08, "text": "tangent of theta over theta times x and"}, {"start": 51.92, "duration": 3.52, "text": "this secant line lies below the tangent"}, {"start": 53.68, "duration": 4.8, "text": "function"}, {"start": 55.44, "duration": 5.759, "text": "if we plug phi into the sine function we"}, {"start": 58.48, "duration": 4.559, "text": "see that we have sine of phi"}, {"start": 61.199, "duration": 4.641, "text": "and this is less than what happens when"}, {"start": 63.039, "duration": 5.201, "text": "we plug phi into the secant line"}, {"start": 65.84, "duration": 5.2, "text": "which is the sine of theta over theta"}, {"start": 68.24, "duration": 5.36, "text": "times phi"}, {"start": 71.04, "duration": 5.04, "text": "likewise if we plug fee into the secant"}, {"start": 73.6, "duration": 4.879, "text": "line of the tangent function"}, {"start": 76.08, "duration": 4.16, "text": "we get a value of tangent theta over"}, {"start": 78.479, "duration": 3.841, "text": "theta times phi"}, {"start": 80.24, "duration": 3.6, "text": "and that must be less than what happens"}, {"start": 82.32, "duration": 2.72, "text": "when we plug phi into the tangent"}, {"start": 83.84, "duration": 3.7, "text": "function"}, {"start": 85.04, "duration": 4.96, "text": "which gives us tangent of phi"}, {"start": 87.54, "duration": 4.3, "text": "[Music]"}, {"start": 90.0, "duration": 3.439, "text": "we can divide the first inequality by"}, {"start": 91.84, "duration": 3.36, "text": "the positive value sine of theta"}, {"start": 93.439, "duration": 4.161, "text": "resulting in sine of phi over sine of"}, {"start": 95.2, "duration": 4.16, "text": "theta is less than v over theta and we"}, {"start": 97.6, "duration": 3.28, "text": "can divide the second inequality by"}, {"start": 99.36, "duration": 3.52, "text": "tangent of theta"}, {"start": 100.88, "duration": 5.519, "text": "resulting in phi over theta is less than"}, {"start": 102.88, "duration": 5.04, "text": "tangent of phi over tangent of theta"}, {"start": 106.399, "duration": 3.601, "text": "but then we can string these two"}, {"start": 107.92, "duration": 4.4, "text": "inequalities together"}, {"start": 110.0, "duration": 4.479, "text": "and we see that sine of phi over sine of"}, {"start": 112.32, "duration": 4.159, "text": "theta is less than phi over theta which"}, {"start": 114.479, "duration": 3.201, "text": "is less than tangent to phi over tangent"}, {"start": 116.479, "duration": 3.201, "text": "of theta"}, {"start": 117.68, "duration": 4.01, "text": "this is known as aristarchus's"}, {"start": 119.68, "duration": 13.729, "text": "inequalities"}, {"start": 121.69, "duration": 11.719, "text": "[Music]"}], "title": "Aristarchus's Inequality (visual proof)", "description": "This is a short, animated visual proofs of the Aristarchus's inequality using the concavity of the sine and tangent functions. #mathshorts #mathvideo #math #aristarchusinequality #aristarchus #inequality #logarithm #tangentfunction #sinefunction #manim #animation #theorem #pww #proofwithoutwords #visualproof #proof\n\nThis animation is based on a proof due to Roger B. Nelsen from the February, 1993 issue of Mathematics Magazine page 65 (https://doi.org/10.2307/2690480).\n\nTo learn more about animating with manim, check out:\nhttps://manim.community\n________________________________________________\nMusic in this video:\nThe Voyage by Audionautix is licensed under a Creative Commons Attribution 4.0 license. https://creativecommons.org/licenses/by/4.0/\n\nArtist: http://audionautix.com/", "lengthSeconds": 135, "width": 640, "height": 360}
{"video_id": "YokKp3pwVFc", "transcript": [{"start": 1.06, "duration": 2.72, "text": "\u03c0r^2 is de formule voor de oppervlakte van een cirkel"}, {"start": 3.78, "duration": 3.64, "text": "Maar vanwaar komt deze formule eigenlijk?"}, {"start": 7.42, "duration": 1.19, "text": "Ten eerste"}, {"start": 8.61, "duration": 3.42, "text": "Tekenen we een cirkel, en kleuren we de oppervlakte groen."}, {"start": 12.03, "duration": 5.36, "text": "Daarna verdelen we de cirkel in grote, gelijke delen"}, {"start": 17.39, "duration": 0.929, "text": "En we schuiven we ze in elkaar"}, {"start": 18.319, "duration": 3.331, "text": "Zoals je kan zien benadert dit nog niet echt een rechthoek"}, {"start": 21.65, "duration": 4.799, "text": "Dus daarom gaan we de cirkel in kleinere deeltjes verdelen"}, {"start": 26.449, "duration": 3.23, "text": "En op dezelfde manier schuiven we deze in elkaar"}, {"start": 29.679, "duration": 4.821, "text": "Dit lijkt er al meer op!"}, {"start": 34.5, "duration": 5.1, "text": "Als we de cirkel steeds in kleinere deeltjes blijven verdelen"}, {"start": 39.6, "duration": 7.0, "text": "benaderen de in elkaar geschoven deeltjes steeds beter een rechthoek"}, {"start": 48.03, "duration": 2.97, "text": "Maar hoe klein moeten die deeltjes nu precies zijn"}, {"start": 51.0, "duration": 7.0, "text": "Om een perfecte rechthoek te vormen?"}, {"start": 61.78, "duration": 5.22, "text": "Wel, we blijven de cirkel gewoon verdelen in steeds kleinere en kleinere stukjes"}, {"start": 67.0, "duration": 3.96, "text": "Tot het kleinste stukje dat je kan maken"}, {"start": 70.96, "duration": 4.85, "text": "Eigenlijk zouden we oneindig moeten blijven verdelen"}, {"start": 75.81, "duration": 1.93, "text": "Totdat we de lijntjes niet meer van elkaar kunnen onderscheiden"}, {"start": 77.74, "duration": 1.99, "text": "En pas dan"}, {"start": 79.73, "duration": 7.0, "text": "bekomen we een perfecte rechthoek"}, {"start": 88.74, "duration": 4.55, "text": "Dus de oppervlakte van de cirkel is gelijk aan de oppervlakte van de rechthoek en deze is gelijk aan"}, {"start": 93.29, "duration": 3.43, "text": "basis maal hoogte"}, {"start": 96.72, "duration": 1.96, "text": "De hoogte van de rechthoek"}, {"start": 98.68, "duration": 3.06, "text": "is zoals je kunt zien even lang als de straal van de cirkel"}, {"start": 101.74, "duration": 3.69, "text": "Dus de lengte van de hoogte is gelijk aan de straal"}, {"start": 105.43, "duration": 2.51, "text": "Om nu de basis te vinden"}, {"start": 107.94, "duration": 3.99, "text": "Moeten we eens kijken naar de omtrek van de cirkel"}, {"start": 111.93, "duration": 4.47, "text": "Wanneer we de basis hebben vergeleken met de omtrek van de cirkel, zien we dat de basis"}, {"start": 116.4, "duration": 2.109, "text": "gelijk is aan"}, {"start": 118.509, "duration": 5.161, "text": "de helft van de omtrek van de cirkel"}, {"start": 123.67, "duration": 4.789, "text": "Herinner je dat de omtrek van een cirkel gelijk is aan 2\u03c0r"}, {"start": 128.459, "duration": 2.56, "text": "En wanneer we dit combineren met \u00e9\u00e9n tweede"}, {"start": 131.019, "duration": 2.62, "text": "dan kunnen we de tweetjes schrappen"}, {"start": 133.639, "duration": 4.8, "text": "en de basis is dus gelijk aan \u03c0r"}, {"start": 138.439, "duration": 5.19, "text": "Dus de formule basis maal hoogte wordt dan \u03c0r  maal r"}, {"start": 143.629, "duration": 2.32, "text": "Als we de r's bij elkaar zetten, krijgen we"}, {"start": 145.949, "duration": 1.78, "text": "\u03c0r^2"}, {"start": 147.729, "duration": 3.11, "text": "Wat gelijk is aan de oppervlakte van de rechthoek"}, {"start": 150.839, "duration": 5.13, "text": "en dit is dan weer gelijk aan de oppervlakte van de cirkel"}, {"start": 155.969, "duration": 5.02, "text": "Dus de formule voor de oppervlakte van elke mogelijke cirkel is \u03c0r^2 of \u03c0rr"}], "title": "Area of a circle, formula explained", "description": "https://mathematicsonline.etsy.com\n\nEnjoyed the video? Show your love for math by checking out our exclusive math merch! Click the link above to grab your favorite items and support our channel. Your contribution helps us keep creating content you enjoy. Thank you for being a part of our community!\n\n\u03c0r(squared) gives you the area of a circle, but where does it come from? Here is a simple explanation. The area of a circle is the space occupied by the circle in a two-dimensional plane. The space occupied within the boundary/circumference of a circle is called the area of the circle. The formula for the area of a circle is A = \u03c0r2, where r is the radius of the circle.", "lengthSeconds": 167, "width": 640, "height": 360}
{"video_id": "F4zYDx-EfZI", "transcript": [{"start": 8.64, "duration": 4.959, "text": "[Music]"}, {"start": 11.2, "duration": 5.68, "text": "an integer partition of an integer n is"}, {"start": 13.599, "duration": 5.121, "text": "a list of integers l1 l2 up to lk"}, {"start": 16.88, "duration": 4.319, "text": "satisfying the properties that n is the"}, {"start": 18.72, "duration": 3.68, "text": "sum of the integers in the list and the"}, {"start": 21.199, "duration": 2.561, "text": "integers in the list are weakly"}, {"start": 22.4, "duration": 3.52, "text": "decreasing and greater than or equal to"}, {"start": 23.76, "duration": 3.2, "text": "1. the elements in the list are called"}, {"start": 25.92, "duration": 3.92, "text": "parts"}, {"start": 26.96, "duration": 4.719, "text": "for example the list 43211 is an integer"}, {"start": 29.84, "duration": 3.68, "text": "partition of eleven because eleven is"}, {"start": 31.679, "duration": 3.841, "text": "equal to four plus three plus two plus"}, {"start": 33.52, "duration": 4.08, "text": "one plus one this is a partition of"}, {"start": 35.52, "duration": 4.48, "text": "eleven into five parts"}, {"start": 37.6, "duration": 5.52, "text": "the young diagram of a partition is an"}, {"start": 40.0, "duration": 4.96, "text": "array of boxes with li boxes in row i"}, {"start": 43.12, "duration": 4.959, "text": "so for example the young diagram of the"}, {"start": 44.96, "duration": 4.96, "text": "partition 43211 is pictured here because"}, {"start": 48.079, "duration": 4.16, "text": "there are four boxes in the first row"}, {"start": 49.92, "duration": 3.44, "text": "then three then two then one and then"}, {"start": 52.239, "duration": 2.561, "text": "one"}, {"start": 53.36, "duration": 4.16, "text": "let's let p of n be the number of"}, {"start": 54.8, "duration": 4.88, "text": "integer partitions of n and p sub k of n"}, {"start": 57.52, "duration": 4.08, "text": "be the number of integer partitions of n"}, {"start": 59.68, "duration": 4.4, "text": "with k parts"}, {"start": 61.6, "duration": 5.519, "text": "if we're trying to compute p of n we can"}, {"start": 64.08, "duration": 5.12, "text": "list out all of the young diagrams for n"}, {"start": 67.119, "duration": 4.961, "text": "in this case we have n is nine and here"}, {"start": 69.2, "duration": 5.44, "text": "are all the partitions"}, {"start": 72.08, "duration": 4.8, "text": "natural way to organize the partitions"}, {"start": 74.64, "duration": 4.32, "text": "is to group them together with other"}, {"start": 76.88, "duration": 5.07, "text": "partitions with the same number of parts"}, {"start": 78.96, "duration": 10.0, "text": "as we've done here"}, {"start": 81.95, "duration": 8.77, "text": "[Music]"}, {"start": 88.96, "duration": 3.92, "text": "the smallest number of parts a partition"}, {"start": 90.72, "duration": 4.56, "text": "can have is one and the largest number"}, {"start": 92.88, "duration": 4.879, "text": "of parts it can have is n where each"}, {"start": 95.28, "duration": 4.24, "text": "part has size one this visual shows us"}, {"start": 97.759, "duration": 3.441, "text": "that the number of partitions of n is"}, {"start": 99.52, "duration": 3.84, "text": "equal to the sum of the number of"}, {"start": 101.2, "duration": 4.64, "text": "partitions of n into k parts where k"}, {"start": 103.36, "duration": 4.24, "text": "ranges from 1 to n"}, {"start": 105.84, "duration": 3.68, "text": "now we can present our primary result"}, {"start": 107.6, "duration": 3.28, "text": "about the number of partitions of n into"}, {"start": 109.52, "duration": 3.279, "text": "k parts"}, {"start": 110.88, "duration": 4.72, "text": "if we have all the partitions of n into"}, {"start": 112.799, "duration": 5.441, "text": "k parts we can divide these into two"}, {"start": 115.6, "duration": 4.96, "text": "disjoint sets the first set are the"}, {"start": 118.24, "duration": 3.76, "text": "partitions that have a part of size one"}, {"start": 120.56, "duration": 3.519, "text": "and the second set consists of the"}, {"start": 122.0, "duration": 4.399, "text": "partitions with no part of size one the"}, {"start": 124.079, "duration": 4.16, "text": "partitions that have a part of size one"}, {"start": 126.399, "duration": 4.161, "text": "are in one-to-one correspondence with"}, {"start": 128.239, "duration": 5.441, "text": "those partitions where we remove one of"}, {"start": 130.56, "duration": 5.039, "text": "the parts of size one"}, {"start": 133.68, "duration": 3.919, "text": "but these partitions are just the"}, {"start": 135.599, "duration": 4.161, "text": "partitions of n minus one into k minus"}, {"start": 137.599, "duration": 4.72, "text": "one parts so we can count them using p"}, {"start": 139.76, "duration": 4.4, "text": "sub k minus one of n minus one for each"}, {"start": 142.319, "duration": 4.481, "text": "of the partitions that have no part of"}, {"start": 144.16, "duration": 5.36, "text": "size one we can remove one from each"}, {"start": 146.8, "duration": 5.519, "text": "part resulting in a partition of n minus"}, {"start": 149.52, "duration": 5.04, "text": "k into k parts these can be counted by p"}, {"start": 152.319, "duration": 4.161, "text": "sub k of n minus k both of these"}, {"start": 154.56, "duration": 3.92, "text": "processes are reversible and because the"}, {"start": 156.48, "duration": 4.32, "text": "two sets have no elements in common we"}, {"start": 158.48, "duration": 5.28, "text": "see that p sub k of n is equal to p sub"}, {"start": 160.8, "duration": 4.4, "text": "k minus one of n minus one plus p sub k"}, {"start": 163.76, "duration": 3.92, "text": "of n minus k"}, {"start": 165.2, "duration": 4.88, "text": "this is a recursive formula for p sub k"}, {"start": 167.68, "duration": 5.12, "text": "of n the number of partitions of n into"}, {"start": 170.08, "duration": 4.799, "text": "k parts we now see that we do not need"}, {"start": 172.8, "duration": 3.84, "text": "to enumerate all of the partitions to"}, {"start": 174.879, "duration": 4.64, "text": "know how many there are"}, {"start": 176.64, "duration": 5.12, "text": "instead we can use the two formulas that"}, {"start": 179.519, "duration": 4.241, "text": "we just encountered to find the number"}, {"start": 181.76, "duration": 4.479, "text": "of partitions of size n"}, {"start": 183.76, "duration": 5.839, "text": "namely we can realize that the number of"}, {"start": 186.239, "duration": 5.201, "text": "partitions of n is equal to the sum of"}, {"start": 189.599, "duration": 4.081, "text": "the number of partitions of n into k"}, {"start": 191.44, "duration": 4.32, "text": "parts where k ranges from one to n and"}, {"start": 193.68, "duration": 4.479, "text": "that the number of partitions of n into"}, {"start": 195.76, "duration": 4.64, "text": "k parts satisfies the recursion given"}, {"start": 198.159, "duration": 3.761, "text": "here"}, {"start": 200.4, "duration": 2.8, "text": "of course this is only one such"}, {"start": 201.92, "duration": 3.28, "text": "technique to count the number of"}, {"start": 203.2, "duration": 5.8, "text": "partitions of n can you think of a"}, {"start": 205.2, "duration": 3.8, "text": "different way to count them"}, {"start": 218.879, "duration": 2.08, "text": "you"}], "title": "Recurrence for partitions into k parts (visual proof)", "description": "In this video, we define partitions and show how to think visually about integer partitions. We then visually prove a recurrence formula for the number of integer partitions of n with exactly k parts.\n\nIf you like integer partitions, check out these videos:\nhttps://youtu.be/p0FKjVA_sRE (synthwave enumeration) \nhttps://youtu.be/M4TmnYxS4gk from @behackl \n\nIf you like this video,  please check out my others and consider subscribing. Thanks!\n\nHere is a nice resource if you want to know more:\n\nhttps://www.whitman.edu/mathematics/cgt_online/book/section03.03.html\n\n#integerpartitions #integers #partition #youngdiagram #counting #count #mathvideo\u200b #math\u200b #mtbos\u200b #manim\u200b #animation\u200b #theorem\u200b\u200b #visualproof\u200b #proof\u200b #iteachmath #mathematics #discretemath #combinatorics #enumeration #synthwave #algorithm\n\nTo learn more about animating with manim, check out:\nhttps://manim.community\n_________________________________________\nMusic in this video:\nElysium by Alexander Nakarada | https://www.serpentsoundstudios.com\nMusic promoted by https://www.free-stock-music.com\nAttribution 4.0 International (CC BY 4.0)\nhttps://creativecommons.org/licenses/by/4.0/", "lengthSeconds": 219, "width": 640, "height": 360}
{"video_id": "feg4RqF-xCA", "transcript": [{"start": 0.21, "duration": 10.32, "text": "ok guys for this problem we're going to\nwant to generate this compound starting"}, {"start": 10.53, "duration": 3.6, "text": "from benzene using any any other\ncompounds you want. so as a hint we're probably"}, {"start": 14.13, "duration": 3.76, "text": "gonna be using some electrophilic\naromatic substitution so go ahead and"}, {"start": 17.89, "duration": 1.48, "text": "see if you can build this molecule"}, {"start": 41.54, "duration": 5.959, "text": "ok so let's take a look at the two substituents that are on the target\nmolecule so first we have an alkyl group"}, {"start": 47.499, "duration": 4.891, "text": "we know that we can attach an alkyl\ngroup onto a benzene ring by Friedel"}, {"start": 52.39, "duration": 21.06, "text": "Crafts alkylation and then we also have\nnitration. but what order are we"}, {"start": 73.45, "duration": 3.269, "text": "gonna do them in? in order to answer this\nquestion we have to understand something"}, {"start": 76.719, "duration": 4.091, "text": "about the directing abilities of these\ngroups. now we know that an alkyl group"}, {"start": 80.81, "duration": 6.28, "text": "is an ortho/para director and we know\nthat a nitro group is a meta director"}, {"start": 87.09, "duration": 5.919, "text": "and we also understand that this spatial\nrelationship between them on the"}, {"start": 93.009, "duration": 4.79, "text": "molecule is ortho, so this is gonna tell us\nwhat order we have to do the reactions"}, {"start": 97.799, "duration": 5.53, "text": "in because given that they have an ortho\nrelationship to one another it must be"}, {"start": 103.329, "duration": 5.631, "text": "the ortho/para director that directed\nthe other group to its position and so"}, {"start": 108.96, "duration": 3.509, "text": "what we're saying is that we must\nstarting with benzene have done the"}, {"start": 112.469, "duration": 6.14, "text": "Friedel Crafts alkylation here with this alkyl chloride to get this"}, {"start": 118.609, "duration": 6.481, "text": "intermediate because if we have this\nintermediate then upon nitration we will"}, {"start": 125.09, "duration": 2.5, "text": "get a mixture of ortho and para"}, {"start": 127.59, "duration": 5.45, "text": "nitrated products whereas if we were to\ndo the nitration first if we did the"}, {"start": 133.04, "duration": 5.71, "text": "nitration first and let's say this was\nour nitrated product then if we went"}, {"start": 138.75, "duration": 9.7, "text": "and did the alkylation we would get meta\ndirection from the nitro group and so"}, {"start": 148.45, "duration": 5.56, "text": "the nitro group cannot have been the\ncompound that was added first because that"}, {"start": 154.01, "duration": 3.699, "text": "will not give us a structure that\nmatches our target molecule, instead"}, {"start": 157.709, "duration": 5.681, "text": "we're recognizing that alkyl is ortho/para, nitro group is meta, so it must"}, {"start": 163.39, "duration": 5.209, "text": "have been the alkyl group that directed\nthe nitro group to the ortho position, we"}, {"start": 168.599, "duration": 4.48, "text": "can get this mixture, separate them out\nand there is going to be our target molecule"}, {"start": 196.44, "duration": 4.32, "text": "thanks for watching guys, subscribe to my channel for more tutorials, and as always feel free to email me"}], "title": "Practice Problem: Electrophilic Aromatic Substitution Retrosynthesis", "description": "You gotta know your EAS reactions for this one! Make sure you do them in the right order, too.\n\nTry all of the organic chemistry practice problems: http://bit.ly/ProfDaveOCPP\n\nOrganic Chemistry Tutorials: http://bit.ly/ProfDaveOrgChem\n\nEMAIL\u25ba ProfessorDaveExplains@gmail.com\nPATREON\u25ba http://patreon.com/ProfessorDaveExplains\n\nCheck out \"Is This Wi-Fi Organic?\", my book on disarming pseudoscience!\nAmazon: https://amzn.to/2HtNpVH\nBookshop: https://bit.ly/39cKADM\n\rBarnes and Noble: https://bit.ly/3pUjmrn \nBook Depository: http://bit.ly/3aOVDlT", "lengthSeconds": 204, "width": 640, "height": 360}
{"video_id": "Fy4Gvlnc1aA", "transcript": [{"start": 0.08, "duration": 5.36, "text": "number five asks draw the correct form"}, {"start": 3.08, "duration": 4.56, "text": "of the resonance hybrid given the"}, {"start": 5.44, "duration": 4.279, "text": "following resonance structures so let's"}, {"start": 7.64, "duration": 4.56, "text": "take a look at what we've got here we"}, {"start": 9.719, "duration": 4.8, "text": "have an oxyanion right there so we have"}, {"start": 12.2, "duration": 3.839, "text": "an O minus right there and then how are"}, {"start": 14.519, "duration": 4.081, "text": "we getting to each of these resonance"}, {"start": 16.039, "duration": 6.4, "text": "structures we can take this negative"}, {"start": 18.6, "duration": 6.999, "text": "charge and form a CO Pi Bond right there"}, {"start": 22.439, "duration": 5.641, "text": "if we then take this Pi Bond and push"}, {"start": 25.599, "duration": 5.0, "text": "that up into the carbon atom to get that"}, {"start": 28.08, "duration": 4.12, "text": "next annion right there so as we can see"}, {"start": 30.599, "duration": 4.681, "text": "we've got the pie Bond there and then we"}, {"start": 32.2, "duration": 5.359, "text": "have the new location for the anion so"}, {"start": 35.28, "duration": 5.56, "text": "we've shifted that o minus over to a c"}, {"start": 37.559, "duration": 4.881, "text": "minus now let's take that and let's put"}, {"start": 40.84, "duration": 4.0, "text": "that right there we can make another"}, {"start": 42.44, "duration": 5.08, "text": "piie Bond there if we take this piie"}, {"start": 44.84, "duration": 5.6, "text": "Bond and we push it up onto that carbon"}, {"start": 47.52, "duration": 5.92, "text": "so now we have a pi Bond here and we"}, {"start": 50.44, "duration": 4.799, "text": "have a new location for the carbanion"}, {"start": 53.44, "duration": 4.84, "text": "then to get to that fourth resonance"}, {"start": 55.239, "duration": 5.28, "text": "structure we are going to put that right"}, {"start": 58.28, "duration": 2.239, "text": "here"}], "title": "Structure and Bonding OCHEM Exam Question 5", "description": "This is an abbreviated version of this question for YouTube Shorts. Check out the full version in the linked video.", "lengthSeconds": 59, "width": 360, "height": 640}
{"video_id": "o7UHgK7j8qA", "transcript": [{"start": 1.269, "duration": 1.071, "text": "[Music] button"}, {"start": 2.96, "duration": 11.86, "text": "[Music] Professor Dave and Chegg here we've\u00a0\nlearned a lot about proton NMR Spectra and this\u00a0\u00a0"}, {"start": 14.82, "duration": 5.52, "text": "is indeed the single most important technique in\u00a0\norganic chemistry for the characterization of any\u00a0\u00a0"}, {"start": 20.34, "duration": 5.4, "text": "molecule however there are other related tools\u00a0\nthat can also be quite useful and one of them\u00a0\u00a0"}, {"start": 25.74, "duration": 7.08, "text": "is carbon 13 NMR spectroscopy let's learn how\u00a0\nthis works as well as another form of nuclear\u00a0\u00a0"}, {"start": 32.82, "duration": 7.14, "text": "magnetic resonance spectroscopy this operates by\u00a0\nprecisely the same principles as proton NMR it is\u00a0\u00a0"}, {"start": 39.96, "duration": 5.94, "text": "simply focused on a different nuclide instead\u00a0\nof protium an isotope of hydrogen data will be\u00a0\u00a0"}, {"start": 45.9, "duration": 6.36, "text": "collected regarding carbon 13 nuclides now it\u00a0\nis true that only about one percent of carbon\u00a0\u00a0"}, {"start": 52.26, "duration": 5.58, "text": "atoms will be carbon 13 Isotopes so most of\u00a0\nthe individual molecules in a sample will\u00a0\u00a0"}, {"start": 57.84, "duration": 6.0, "text": "not contain any carbon 13. but this is okay\u00a0\nbecause given a macroscopic sample trillions\u00a0\u00a0"}, {"start": 63.84, "duration": 6.24, "text": "upon trillions of molecules within that sample\u00a0\nwill contain a carbon 13 atom at a particular\u00a0\u00a0"}, {"start": 70.08, "duration": 5.88, "text": "location in the molecule thus generating data\u00a0\nfor us to examine regarding that position on the\u00a0\u00a0"}, {"start": 75.96, "duration": 5.76, "text": "molecule and the same goes for every other carbon\u00a0\nin the molecule so we can therefore gather data\u00a0\u00a0"}, {"start": 81.72, "duration": 6.12, "text": "regarding the chemical environment experienced\u00a0\nby every carbon atom in any structure this\u00a0\u00a0"}, {"start": 87.84, "duration": 7.38, "text": "approach can be described as signal averaging\u00a0\nas performed by Fourier transform NMR or ft NMR\u00a0\u00a0"}, {"start": 95.76, "duration": 5.46, "text": "the tricky thing about signal averaging is that\u00a0\nthere is a lot of noise to deal with the signals\u00a0\u00a0"}, {"start": 101.22, "duration": 6.0, "text": "from the 13c nuclides are weak in comparison\u00a0\nto background electronic noise but we can run\u00a0\u00a0"}, {"start": 107.22, "duration": 6.12, "text": "hundreds or even thousands of Trials and average\u00a0\nthem together hence the term signal averaging and\u00a0\u00a0"}, {"start": 113.34, "duration": 7.26, "text": "digitally lock on to any data that is non-random\u00a0\nand therefore due to 13c nuclides the random noise\u00a0\u00a0"}, {"start": 120.6, "duration": 5.76, "text": "can then be minimized so that the important data\u00a0\ncan stand out the problem is that taking so many\u00a0\u00a0"}, {"start": 126.36, "duration": 6.12, "text": "measurements is extremely time consuming that's\u00a0\nwhere the Ft NMR comes into play a short pulse\u00a0\u00a0"}, {"start": 132.48, "duration": 7.5, "text": "of RF energy irradiates the sample all 1H and\u00a0\n13c nuclides resonate at once and the resulting\u00a0\u00a0"}, {"start": 139.98, "duration": 5.46, "text": "complex signal is manipulated using Fourier\u00a0\ntransforms allowing for data to be collected\u00a0\u00a0"}, {"start": 145.44, "duration": 6.3, "text": "in mere seconds we will examine carbon 13 Spectra\u00a0\nlater but for now there is one interesting fact\u00a0\u00a0"}, {"start": 151.74, "duration": 6.42, "text": "to be aware of we will not see splitting patterns\u00a0\nthe way we do with proton NMR the reason for this\u00a0\u00a0"}, {"start": 158.16, "duration": 5.58, "text": "has to do with the low abundance of this isotope\u00a0\nit is unlikely that carbon 13 nuclides will be\u00a0\u00a0"}, {"start": 163.74, "duration": 5.22, "text": "adjacent in any particular molecule so there is\u00a0\nno opportunity for them to split each other's\u00a0\u00a0"}, {"start": 168.96, "duration": 6.06, "text": "signals the way protons do coupling with protons\u00a0\nthemselves also does not occur because these\u00a0\u00a0"}, {"start": 175.02, "duration": 6.24, "text": "Spectra are recorded with Broadband decoupling a\u00a0\nsecond irradiation makes hydrogen's spin flip so\u00a0\u00a0"}, {"start": 181.26, "duration": 5.58, "text": "rapidly that their local magnetic fields average\u00a0\nto zero and no coupling with carbon will occur"}, {"start": 189.06, "duration": 4.92, "text": "so with that we understand roughly\u00a0\nhow carbon 13 NMR Works through\u00a0\u00a0"}, {"start": 193.98, "duration": 4.86, "text": "signal averaging and Fourier transforms\u00a0\nProfessor day for Chegg see you next time"}], "title": "What is 13C-NMR Spectroscopy? Ft. Professor Dave", "description": "1-H NMR spectroscopy is the most important technique in organic chemistry for the characterization of any molecule. But there are other useful related tools, and one of them is 13C-NMR spectroscopy. @ProfessorDaveExplains explores the topic of 13C-NMR spectroscopy in this video, delving into how it works, and explaining the process of signal averaging as performed through Fourier transform NMR.\n\nGet more homework help from Chegg at https://che.gg/3HbtG8Y\nWatch the whole Organic Chemistry playlist: https://www.youtube.com/playlist?list=PLllCkartRRdMPnATf1X8qVdo5RzM8gA_M\n\nAbout Chegg:\nChegg is a leading education technology company that provides students with affordable and accessible study resources. Chegg's mission is to help students save time, save money and get smarter.  \n\nSubscribe: https://www.youtube.com/@chegg\nTikTok: https://www.tiktok.com/@chegg\nInstagram: https://www.instagram.com/chegg/\nTwitter: https://twitter.com/chegg\n\n00:00 Intro\n00:30 1-H NMR spectroscopy\n00:42 13C-NMR spectroscopy\n01:27 Signal averaging / Fourier transform NMR\n02:26 13C-NMR spectra features", "lengthSeconds": 209, "width": 640, "height": 360}
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\u0627\u0644\u062f\u0642\u064a\u0642 \u0644\u0644\u062d\u0645\u0636 \u0623\u0648 \u0627\u0644\u0642\u0627\u0639\u062f\u0629 \u0627\u0644\u062a\u064a \u0633\u062a\u062a\u0641\u0627\u0639\u0644 \u0645\u0639\u0647 \u0628\u0634\u0643\u0644 \u062a\u0627\u0645"}, {"start": 21.68, "duration": 2.6, "text": "\u0625\u0646\u0647 \u0645\u062b\u0644 \u0646\u0648\u0639 \u0645\u0646 \u0627\u0644\u062d\u0633\u0627\u0628\u0627\u062a \u0627\u0644\u0643\u064a\u0645\u064a\u0627\u0626\u064a\u0629"}, {"start": 24.6, "duration": 3.18, "text": "\u0639\u0644\u0649 \u0645\u0646\u062d\u0646\u0649 \u0645\u0639\u0627\u064a\u0631\u0629 \u0644\u0645\u062d\u0644\u0648\u0644 \u062d\u0645\u0636\u064a"}, {"start": 27.94, "duration": 4.54, "text": "\u064a\u062a\u0645 \u062a\u062e\u0637\u064a\u0637 \u0627\u0644\u0631\u0642\u0645 \u0627\u0644\u0647\u064a\u062f\u0631\u0648\u062c\u064a\u0646\u064a (pH) \u0645\u0642\u0627\u0628\u0644 \u062d\u062c\u0645 \u0627\u0644\u0642\u0627\u0639\u062f\u0629 \u0627\u0644\u062a\u064a \u062a\u062a\u0645 \u0625\u0636\u0627\u0641\u062a\u0647\u0627 \u062a\u062f\u0631\u064a\u062c\u064a\u0627\u064b"}, {"start": 32.88, "duration": 2.82, "text": "\u0639\u0646\u062f \u0627\u0636\u0627\u0641\u0629 \u0642\u0627\u0639\u062f\u0629 \u064a\u0631\u062a\u0641\u0639PH \u0628\u0628\u0637\u0621"}, {"start": 36.02, "duration": 2.94, "text": "\u062b\u0645 \u064a\u0631\u062a\u0641\u0639 \u0628\u0634\u0643\u0644 \u062d\u0627\u062f \u0646\u062d\u0648 \u0646\u0642\u0637\u0629 \u0627\u0644\u062a\u0643\u0627\u0641\u0624"}, {"start": 39.64, "duration": 3.68, "text": "\u0647\u0630\u0647 \u0647\u064a \u0627\u0644\u0646\u0642\u0637\u0629 \u0627\u0644\u062a\u064a \u062a\u0645\u062a \u0641\u064a\u0647\u0627 \u0625\u0636\u0627\u0641\u0629 \u0642\u0627\u0639\u062f\u0629 \u0628\u0643\u0645\u064a\u0629 \u0643\u0627\u0641\u064a\u0629"}, {"start": 43.32, "duration": 3.66, "text": "\u0644\u062a\u062d\u064a\u064a\u062f \u0643\u0645\u064a\u0629 \u0627\u0644\u062d\u0645\u0636 \u0627\u0644\u0641\u0639\u0644\u064a\u0629 \u0641\u064a \u0627\u0644\u0645\u062d\u0644\u0648\u0644"}, {"start": 47.3, "duration": 5.84, "text": "\u0639\u0646\u062f \u0625\u062c\u0631\u0627\u0621 \u0627\u0644\u0645\u0639\u0627\u064a\u0631\u0629\u060c \u0646\u062a\u0639\u0631\u0641 \u0639\u0644\u0649 \u0646\u0642\u0637\u0629 \u0627\u0644\u062a\u0643\u0627\u0641\u0624 \u0628\u0627\u0633\u062a\u062e\u062f\u0627\u0645 \u0634\u064a\u0621 \u064a\u0633\u0645\u0649 \u0627\u0644\u0645\u0624\u0634\u0631."}, {"start": 53.62, "duration": 5.0, "text": "\u0627\u0644\u0645\u0624\u0634\u0631 \u0647\u0648 \u0645\u0627\u062f\u0629 \u0633\u062a\u062a\u062d\u0648\u0644 \u0625\u0644\u0649 \u0644\u0648\u0646 \u0648\u0627\u0636\u062d \u0628\u0645\u062c\u0631\u062f \u0628\u0644\u0648\u063a \u0646\u0642\u0637\u0629 \u0627\u0644\u062a\u0643\u0627\u0641\u0624"}, {"start": 59.6, "duration": 5.46, "text": "\u0639\u0646\u062f\u0645\u0627 \u0646\u062d\u064a\u062f \u062d\u0645\u0636\u064b\u0627 \u0642\u0648\u064a\u064b\u0627 \u0628\u0642\u0627\u0639\u062f\u0629 \u0642\u0648\u064a\u0629\u060c \u062a\u0643\u0648\u0646 \u0646\u0642\u0637\u0629 \u0627\u0644\u062a\u0643\u0627\u0641\u0624 7"}, {"start": 65.7, "duration": 3.88, "text": "\u0625\u0630\u0627 \u0643\u0627\u0646 \u0623\u064a \u0645\u0646 \u0627\u0644\u0646\u0648\u0639\u064a\u0646 \u0623\u0648 \u0643\u0644\u064a\u0647\u0645\u0627 \u0636\u0639\u064a\u0641\u064b\u0627 \u060c \u0641\u0633\u064a\u062e\u062a\u0644\u0641 \u0647\u0630\u0627 \u0627\u0644\u0639\u062f\u062f"}, {"start": 70.34, "duration": 5.8, "text": "\u0639\u0644\u0649 \u0633\u0628\u064a\u0644 \u0627\u0644\u0645\u062b\u0627\u0644\u060c \u0644\u0646\u0641\u062a\u0631\u0636 \u0623\u0646 \u0647\u0646\u0627\u0643 \u0645\u062d\u0644\u0648\u0644 \u0647\u064a\u062f\u0631\u0648\u0643\u0633\u064a\u062f \u0627\u0644\u0635\u0648\u062f\u064a\u0648\u0645 \u0648\u0646\u0631\u064a\u062f \u0623\u0646 \u0646\u0639\u0631\u0641 \u062a\u0631\u0643\u064a\u0632\u0647"}, {"start": 76.78, "duration": 5.68, "text": "\u0627\u0641\u062a\u0631\u0636 \u0625\u0646\u0646\u0627 \u0646\u0641\u0627\u0639\u0644 \u0645\u0639 \u062d\u0645\u0636 \u0627\u0644\u0643\u0628\u0631\u064a\u062a\u064a\u0643 3.0 \u0645 \u0628\u062d\u062c\u0645 25 \u0645\u0644  \u0645\u0646 \u0627\u0644\u0642\u0627\u0639\u062f\u0629"}, {"start": 82.74, "duration": 5.52, "text": "\u0648\u0646\u062c\u062f \u0623\u0646\u0647 \u064a\u0633\u062a\u063a\u0631\u0642 11.6 \u0645\u0644 \u0645\u0646 \u0627\u0644\u062d\u0645\u0636 \u0644\u0644\u0648\u0635\u0648\u0644 \u0625\u0644\u0649 \u0646\u0642\u0637\u0629 \u0627\u0644\u062a\u0643\u0627\u0641\u0624."}, {"start": 88.64, "duration": 5.8, "text": "\u0625\u0630\u0627 \u062d\u0648\u0644\u0646\u0627 \u0625\u0644\u0649 \u0644\u062a\u0631\u0627\u062a \u062b\u0645 \u0644\u062f\u064a\u0646\u0627 \u0639\u0648\u0627\u0645\u0644 \u062a\u062d\u0648\u064a\u0644 \u0644\u062a\u0631\u0643\u064a\u0632 \u0627\u0644\u062d\u0645\u0636"}, {"start": 94.64, "duration": 5.6, "text": "\u062b\u0645 \u0646\u0633\u0628\u0629 \u0627\u0644\u0642\u064a\u0627\u0633 \u0627\u0644\u0643\u064a\u0645\u064a\u0627\u0626\u064a \u060c \u064a\u0645\u0643\u0646\u0646\u0627 \u0627\u0644\u0648\u0635\u0648\u0644 \u0625\u0644\u0649 \u062a\u0631\u0643\u064a\u0632 \u0627\u0644\u0645\u0648\u0627\u062f \u0627\u0644\u0645\u062a\u0641\u0627\u0639\u0644\u0629"}, {"start": 100.46, "duration": 2.54, "text": "\u0644\u0646\u062a\u062d\u0642\u0642 \u0645\u0646 \u0645\u062f\u0649 \u0627\u0644\u0625\u0633\u062a\u064a\u0639\u0627\u0628"}, {"start": 132.36, "duration": 3.42, "text": "\u0634\u0643\u0631\u064b\u0627 \u0639\u0644\u0649 \u0627\u0644\u0645\u0634\u0627\u0647\u062f\u0629 \u0627\u0634\u062a\u0631\u0643 \u0641\u064a \u0642\u0646\u0627\u062a\u064a \u0644\u0645\u0632\u064a\u062f \u0645\u0646 \u0627\u0644\u062f\u0631\u0648\u0633"}, {"start": 135.78, "duration": 2.02, "text": "\u0648\u0643\u0645\u0627 \u0647\u0648 \u0627\u0644\u062d\u0627\u0644 \u062f\u0627\u0626\u0645\u064b\u0627 \u060c \u0644\u0627 \u062a\u062a\u0631\u062f\u062f \u0641\u064a \u0645\u0631\u0627\u0633\u0644\u062a\u064a \u0639\u0628\u0631 \u0627\u0644\u0628\u0631\u064a\u062f \u0627\u0644\u0625\u0644\u0643\u062a\u0631\u0648\u0646\u064a"}], "title": "Acid-Base Titration", "description": "Any introductory chemistry class will include titrations, and to do these, you have to do math. But you get to see pretty colors, too! Here's a quick run through of the stoichiometry of it all.\n\nWatch the whole General Chemistry playlist: http://bit.ly/ProfDaveGenChem\n\nStudy for the AP Chemistry exam with me: https://bit.ly/ProfDaveAPChem\n\nOrganic Chemistry Tutorials: http://bit.ly/ProfDaveOrgChem\nBiochemistry Tutorials: http://bit.ly/ProfDaveBiochem\nBiology Tutorials: http://bit.ly/ProfDaveBio\nClassical Physics Tutorials: http://bit.ly/ProfDavePhysics1\nModern Physics Tutorials: http://bit.ly/ProfDavePhysics2\nMathematics Tutorials: http://bit.ly/ProfDaveMaths\n\nEMAIL\u25ba ProfessorDaveExplains@gmail.com\nPATREON\u25ba http://patreon.com/ProfessorDaveExplains\n\nCheck out \"Is This Wi-Fi Organic?\", my book on disarming pseudoscience!\nAmazon: https://amzn.to/2HtNpVH\nBookshop: https://bit.ly/39cKADM\n\rBarnes and Noble: https://bit.ly/3pUjmrn \nBook Depository: http://bit.ly/3aOVDlT", "lengthSeconds": 160, "width": 640, "height": 360}
{"video_id": "H9GLj_j6K10", "transcript": [{"start": 0.08, "duration": 5.48, "text": "five asks how many stereoisomers are"}, {"start": 2.879, "duration": 4.92, "text": "possible for the following compound we"}, {"start": 5.56, "duration": 4.36, "text": "need to understand that this depends on"}, {"start": 7.799, "duration": 4.441, "text": "how many stereo centers there are on the"}, {"start": 9.92, "duration": 4.719, "text": "compound for example if you have one"}, {"start": 12.24, "duration": 4.4, "text": "chyro Center that chyro Center can be r"}, {"start": 14.639, "duration": 3.441, "text": "or it can be S and then if you have two"}, {"start": 16.64, "duration": 4.12, "text": "chyro centers now we have more"}, {"start": 18.08, "duration": 6.199, "text": "possibilities so what the formula is is"}, {"start": 20.76, "duration": 5.679, "text": "2 to the N the number of chyro centers n"}, {"start": 24.279, "duration": 4.281, "text": "you take two and raise it to that power"}, {"start": 26.439, "duration": 4.92, "text": "and that's how many stereo isomers are"}, {"start": 28.56, "duration": 4.6, "text": "possible how many KY Centers do we have"}, {"start": 31.359, "duration": 3.761, "text": "well we have one right here there's that"}, {"start": 33.16, "duration": 4.12, "text": "bromine the implied hydrogen and then"}, {"start": 35.12, "duration": 4.52, "text": "the two directions of the Ring we have"}, {"start": 37.28, "duration": 3.92, "text": "one right here as well because we have"}, {"start": 39.64, "duration": 3.12, "text": "this substituent and then the two"}, {"start": 41.2, "duration": 4.44, "text": "directions of the ring as well as the"}, {"start": 42.76, "duration": 5.0, "text": "implied hydrogen and then this also is a"}, {"start": 45.64, "duration": 3.8, "text": "chyro center because that chlorine can"}, {"start": 47.76, "duration": 2.88, "text": "be on the dash or the wedge and that"}, {"start": 49.44, "duration": 3.24, "text": "will be different because we've got"}, {"start": 50.64, "duration": 4.96, "text": "chlorine hydrogen methyl and then the"}, {"start": 52.68, "duration": 5.519, "text": "rest of the molecule so if we have three"}, {"start": 55.6, "duration": 4.919, "text": "chyal centers then we're saying that's"}, {"start": 58.199, "duration": 2.32, "text": "two"}], "title": "Determining All Possible Stereoisomers", "description": "This is an abbreviated version of this question for YouTube Shorts. Check out the full version in the linked video.", "lengthSeconds": 59, "width": 360, "height": 640}
{"video_id": "VjI4LtotC2o", "transcript": [{"start": 0.46, "duration": 3.44, "text": "Professor Dave here, let\u2019s prove the Pythagorean\ntheorem."}, {"start": 10.56, "duration": 5.2, "text": "As I mentioned before when we initially learned\nthe Pythagorean theorem, there are very elegant"}, {"start": 15.769, "duration": 5.702, "text": "ways to prove that the theorem is true, and\nnow that we know how to calculate area, we"}, {"start": 21.471, "duration": 3.839, "text": "are ready to perform one such proof ourselves."}, {"start": 25.31, "duration": 6.509, "text": "Let\u2019s say we draw a right triangle like\nthis one, and label the sides A, B, and C,"}, {"start": 31.819, "duration": 1.771, "text": "like we are used to."}, {"start": 33.59, "duration": 4.6, "text": "Now let\u2019s make three more copies of this\ntriangle and arrange them like this, so that"}, {"start": 38.19, "duration": 2.23, "text": "we get two squares."}, {"start": 40.42, "duration": 5.72, "text": "The larger square has a side length of A plus\nB, while the smaller inner square has a side"}, {"start": 46.14, "duration": 1.46, "text": "length of C."}, {"start": 47.6, "duration": 5.52, "text": "We can calculate the areas of all of these\nshapes."}, {"start": 53.12, "duration": 5.119, "text": "We know that the area of one of the triangles\nwill be one half base times height, or one"}, {"start": 58.239, "duration": 7.761, "text": "half A times B. The area of the large square\nwill be the side length squared, or A plus"}, {"start": 66.0, "duration": 2.149, "text": "B quantity squared."}, {"start": 68.149, "duration": 4.281, "text": "And the small square will have an area of\nC squared."}, {"start": 72.43, "duration": 3.82, "text": "Now let\u2019s set up a relationship between\nthese areas."}, {"start": 76.25, "duration": 6.07, "text": "The area of the large square is equal to A\nplus B quantity squared as we just said, but"}, {"start": 82.32, "duration": 6.17, "text": "it is also equal to the area of the small\nsquare plus the area of all the triangles,"}, {"start": 88.49, "duration": 4.03, "text": "since these shapes comprise the larger square."}, {"start": 92.52, "duration": 3.15, "text": "So we can make an equation that states the\nfollowing."}, {"start": 95.67, "duration": 6.54, "text": "A plus B quantity squared, the area of the\nbig square, is equal to four times one half"}, {"start": 102.21, "duration": 7.449, "text": "AB, which is four times the area of one triangle,\nsince there are four triangles, plus C squared,"}, {"start": 109.659, "duration": 2.591, "text": "the area of the small square."}, {"start": 112.25, "duration": 4.859, "text": "We know how to square a binomial by using\nthe FOIL method, so let\u2019s take away the"}, {"start": 117.109, "duration": 3.821, "text": "exponent and just write the binomial out twice."}, {"start": 120.93, "duration": 9.82, "text": "First gives us A squared, outer gives us AB,\ninner gives us AB, and last gives us B squared."}, {"start": 130.75, "duration": 6.7, "text": "Combining like terms, that\u2019s A squared plus\ntwo AB plus B squared."}, {"start": 137.45, "duration": 7.39, "text": "On the right, we can multiply four and one\nhalf, and we get two AB plus C squared."}, {"start": 144.84, "duration": 6.491, "text": "As it happens, we have two AB on both sides,\nso those will cancel out, and wouldn\u2019t you"}, {"start": 151.331, "duration": 3.329, "text": "know it, take a look at what we are left with."}, {"start": 154.66, "duration": 4.31, "text": "The Pythagorean theorem."}, {"start": 158.97, "duration": 5.34, "text": "I hope this result makes you feel warm and\nfuzzy inside, because proofs like this are"}, {"start": 164.31, "duration": 2.44, "text": "what math is really all about."}, {"start": 166.75, "duration": 5.62, "text": "There are many other proofs of this theorem\nthat involve similar calculations with triangles"}, {"start": 172.37, "duration": 5.01, "text": "and squares and other shapes, so if you\u2019re\njust dying to know them all, it\u2019s time to"}, {"start": 177.38, "duration": 1.0, "text": "hit the internet."}, {"start": 178.38, "duration": 4.02, "text": "Otherwise, let\u2019s move forward and wrap things\nup with geometry."}], "title": "Proving the Pythagorean Theorem", "description": "We learned about the Pythagorean Theorem, but where did it come from? How do we know it's definitely true? What if old Pythag just made it up off the top of his mystical skull? Lucky for us, in math we can proof that things are definitely true, and there are tons of ways to prove that the Pythagorean theorem is true. Here's one of the simpler ones for you to memorize, so you can bust it out at a party and be the coolest kid in the room!\n\nWatch the whole Mathematics playlist: http://bit.ly/ProfDaveMath\n\nClassical Physics Tutorials: http://bit.ly/ProfDavePhysics1\nModern Physics Tutorials: http://bit.ly/ProfDavePhysics2\nGeneral Chemistry Tutorials: http://bit.ly/ProfDaveGenChem\nOrganic Chemistry Tutorials: http://bit.ly/ProfDaveOrgChem\nBiochemistry Tutorials: http://bit.ly/ProfDaveBiochem\nBiology Tutorials: http://bit.ly/ProfDaveBio\n\nEMAIL\u25ba ProfessorDaveExplains@gmail.com\nPATREON\u25ba http://patreon.com/ProfessorDaveExplains\n\nCheck out \"Is This Wi-Fi Organic?\", my book on disarming pseudoscience!\nAmazon: https://amzn.to/2HtNpVH\nBookshop: https://bit.ly/39cKADM\n\rBarnes and Noble: https://bit.ly/3pUjmrn \nBook Depository: http://bit.ly/3aOVDlT", "lengthSeconds": 214, "width": 640, "height": 360}
{"video_id": "RI7xCYBmDqQ", "transcript": [{"start": 6.14, "duration": 4.68, "text": "okay let's do some practice regarding\nenthalpy of combustion so the question"}, {"start": 10.83, "duration": 6.72, "text": "asks how much heat is released upon the\ncombustion of 1.00 litres of isooctane"}, {"start": 17.55, "duration": 7.139, "text": "and that has the formula C8H18. now in\norder to answer this question we will"}, {"start": 24.689, "duration": 4.83, "text": "need to know that the density of\nisooctane is zero point six nine two"}, {"start": 29.519, "duration": 5.191, "text": "grams per milliliter and we also need to\nknow that the Delta H of combustion for"}, {"start": 34.71, "duration": 7.259, "text": "iso-octane is negative 5461 kilojoules\nper mole so this is all the data you"}, {"start": 41.969, "duration": 1.951, "text": "need go ahead and give this a try."}, {"start": 61.84, "duration": 5.06, "text": "so let's start out with our one point\nzero zero liters of isooctane what are"}, {"start": 66.9, "duration": 3.899, "text": "we gonna do first well we know we're\ngonna do some kind of stoichiometry so"}, {"start": 70.799, "duration": 3.811, "text": "we're gonna want to try to get two moles\nso how do we do this well we've got"}, {"start": 74.61, "duration": 4.29, "text": "liters and our density was in grams per\nmilliliter so we're gonna need in"}, {"start": 78.9, "duration": 4.92, "text": "milliliters so let's just convert this\ninto milliliters first we've got one"}, {"start": 83.82, "duration": 4.74, "text": "point zero zero liters times a thousand\nand since we can only have three sig"}, {"start": 88.56, "duration": 6.51, "text": "figs that is going to be one point zero\nzero times 10 to the 3 milliliters. now"}, {"start": 95.07, "duration": 5.549, "text": "as we said we have that density so we\ncan convert to grams one point zero zero"}, {"start": 100.619, "duration": 6.14, "text": "times 10 to the 3 milliliters times zero\npoint six nine two grams per milliliter"}, {"start": 106.759, "duration": 8.711, "text": "milliliters will cancel and we'll get\n692 grams so now that we have a mass of"}, {"start": 115.47, "duration": 5.64, "text": "isooctane we want to use that mass to\nget moles because that way we can use"}, {"start": 121.11, "duration": 6.81, "text": "the Delta H of combustion that we were\ngiven so 692 grams times 1 mol / 114"}, {"start": 127.92, "duration": 3.6, "text": "grams that's what we get just by adding\nup all the carbon and hydrogen atoms"}, {"start": 131.52, "duration": 5.53, "text": "that is going to give us six\npoint zero seven moles so we have six"}, {"start": 137.05, "duration": 5.97, "text": "point zero seven moles of isooctane in\nthat one liter of isooctane and then"}, {"start": 143.02, "duration": 4.92, "text": "lastly we can use our thermochemical\ndata we can do six point zero seven"}, {"start": 147.94, "duration": 6.24, "text": "moles times negative five thousand 461\nkilojoules that are released for every"}, {"start": 154.18, "duration": 5.88, "text": "mole of iso-octane and we get negative\nthree point three one times ten to the"}, {"start": 160.06, "duration": 8.88, "text": "four kilojoules that is how much energy is associated with the combustion of one liter of isooctane."}], "title": "Practice Problem: Enthalpy of Combustion", "description": "Stoichiometry for combustion? You bet! If we know about moles, we can calculate energy changes. Give it a try!\n\nTry all of the general chemistry practice problems: http://bit.ly/ProfDaveGCPP\n\nGeneral Chemistry Tutorials: http://bit.ly/ProfDaveGenChem\n\nEMAIL\u25ba ProfessorDaveExplains@gmail.com\nPATREON\u25ba http://patreon.com/ProfessorDaveExplains\n\nCheck out \"Is This Wi-Fi Organic?\", my book on disarming pseudoscience!\nAmazon: https://amzn.to/2HtNpVH\nBookshop: https://bit.ly/39cKADM\n\rBarnes and Noble: https://bit.ly/3pUjmrn \nBook Depository: http://bit.ly/3aOVDlT", "lengthSeconds": 200, "width": 640, "height": 360}
{"video_id": "n9fgcm0Pwgs", "transcript": [{"start": 0.16, "duration": 2.68, "text": "Hey it\u2019s Professor Dave; let\u2019s work with\npercentages."}, {"start": 9.72, "duration": 5.14, "text": "We are all aware of percentages, as they are\nubiquitous in society."}, {"start": 14.86, "duration": 2.61, "text": "There is a sale today, thirty percent off."}, {"start": 17.47, "duration": 2.88, "text": "Leave a twenty percent tip after the meal."}, {"start": 20.35, "duration": 3.3, "text": "Today there is a fifty percent chance of rain."}, {"start": 23.65, "duration": 1.65, "text": "So what are these things?"}, {"start": 25.3, "duration": 5.89, "text": "Well in Latin, centum means one hundred, so\npercent literally means out of a hundred."}, {"start": 31.19, "duration": 6.25, "text": "It\u2019s just a way of conveying magnitude,\nby modifying some total into a hundred, and"}, {"start": 37.44, "duration": 5.11, "text": "then reporting a value of interest as some\nfraction of that hundred."}, {"start": 42.55, "duration": 6.98, "text": "If you get a ninety percent on your math quiz,\nyou could have gotten nine points out of ten,"}, {"start": 49.53, "duration": 5.189, "text": "eighteen points out of twenty, four hundred\nfifty out of five hundred, or any other number"}, {"start": 54.719, "duration": 3.91, "text": "that obeys this ratio of ninety out of a hundred."}, {"start": 58.629, "duration": 4.971, "text": "Whatever we have to do to the larger number\nto make it one hundred, we do that same thing"}, {"start": 63.6, "duration": 2.0, "text": "to the other number."}, {"start": 65.6, "duration": 6.98, "text": "Ten must be multiplied by ten to get a hundred,\nso the nine is also multiplied by ten."}, {"start": 72.58, "duration": 6.52, "text": "In this way, percentages are ratios, they\nare just very specific ratios that compare"}, {"start": 79.1, "duration": 5.809, "text": "numbers to a hundred, because although this\nnumber is totally arbitrary, it is an ideal"}, {"start": 84.909, "duration": 2.45, "text": "size for comparison."}, {"start": 87.359, "duration": 6.401, "text": "One hundred is small enough that our brains\ncan easily comprehend its magnitude, but also"}, {"start": 93.76, "duration": 6.88, "text": "large enough that there are sufficient gradations\nbetween zero and a hundred so as to make meaningful"}, {"start": 100.64, "duration": 1.7, "text": "comparisons."}, {"start": 102.34, "duration": 5.209, "text": "In this way, we can look at percentages as\nsimple fractions."}, {"start": 107.549, "duration": 3.301, "text": "Eighty percent is eighty one-hundredths."}, {"start": 110.85, "duration": 5.089, "text": "We learned how to simplify fractions, so let\u2019s\ndivide both of these numbers by twenty."}, {"start": 115.939, "duration": 2.591, "text": "We should get four fifths."}, {"start": 118.53, "duration": 6.479, "text": "This means that eighty percent and four fifths\nare equivalent; they mean the same thing."}, {"start": 125.009, "duration": 5.78, "text": "Moving to another example, fifty percent is\nfifty one hundredths, and dividing both numbers"}, {"start": 130.789, "duration": 3.191, "text": "by fifty, we get one half."}, {"start": 133.98, "duration": 3.619, "text": "This makes sense, as fifty is half of a hundred."}, {"start": 137.599, "duration": 5.14, "text": "If you have something like three percent,\nthat will be three one hundredths, and that"}, {"start": 142.739, "duration": 3.191, "text": "can\u2019t be simplified any further."}, {"start": 145.93, "duration": 6.389, "text": "We should be able to convert between percentages,\nfractions, and even decimals in this way with"}, {"start": 152.32, "duration": 2.7, "text": "relative ease, so let\u2019s check comprehension."}], "title": "Working With Percentages", "description": "Oh my! There's a big sale at your favorite store! 30% off, in fact! But wait, what does that mean exactly? Percentages are convenient ways of conveying magnitude by comparing some value to one hundred. Why a hundred? Because it's not too big and not too small. Let's learn how to work with these babies.\n\nWatch the whole Mathematics playlist: http://bit.ly/ProfDaveMath\n\nClassical Physics Tutorials: http://bit.ly/ProfDavePhysics1\nModern Physics Tutorials: http://bit.ly/ProfDavePhysics2\nGeneral Chemistry Tutorials: http://bit.ly/ProfDaveGenChem\nOrganic Chemistry Tutorials: http://bit.ly/ProfDaveOrgChem\nBiochemistry Tutorials: http://bit.ly/ProfDaveBiochem\nBiology Tutorials: http://bit.ly/ProfDaveBio\n\nEMAIL\u25ba ProfessorDaveExplains@gmail.com\nPATREON\u25ba http://patreon.com/ProfessorDaveExplains\n\nCheck out \"Is This Wi-Fi Organic?\", my book on disarming pseudoscience!\nAmazon: https://amzn.to/2HtNpVH\nBookshop: https://bit.ly/39cKADM\n\rBarnes and Noble: https://bit.ly/3pUjmrn \nBook Depository: http://bit.ly/3aOVDlT", "lengthSeconds": 216, "width": 640, "height": 360}
{"video_id": "5OVjgKea-a8", "transcript": [{"start": 0.0, "duration": 9.38, "text": "[Music]"}, {"start": 9.8, "duration": 4.759, "text": "Professor Dave and CH here we know about"}, {"start": 12.16, "duration": 4.36, "text": "markovnikov's rule which dictates that"}, {"start": 14.559, "duration": 4.761, "text": "in a reaction like electrophilic"}, {"start": 16.52, "duration": 5.4, "text": "addition when a carbocation intermediate"}, {"start": 19.32, "duration": 5.16, "text": "is formed it is specifically the more"}, {"start": 21.92, "duration": 5.24, "text": "substituted carbocation of the two that"}, {"start": 24.48, "duration": 5.52, "text": "are possible which will form but why is"}, {"start": 27.16, "duration": 5.72, "text": "this let's look at carbocation stability"}, {"start": 30.0, "duration": 5.8, "text": "to understand how this works first a few"}, {"start": 32.88, "duration": 5.519, "text": "details about carbocations these are"}, {"start": 35.8, "duration": 4.4, "text": "carbons with a formal positive charge"}, {"start": 38.399, "duration": 4.561, "text": "because they have lost an electron"}, {"start": 40.2, "duration": 5.08, "text": "domain because of this they are SP2"}, {"start": 42.96, "duration": 4.96, "text": "hybridized and trigonal planer so they"}, {"start": 45.28, "duration": 6.0, "text": "are totally flat about the carbocation"}, {"start": 47.92, "duration": 5.799, "text": "with 120\u00b0 Bond angles this means that"}, {"start": 51.28, "duration": 4.4, "text": "there is one remaining unhybridized p"}, {"start": 53.719, "duration": 4.121, "text": "orbital and this will be vacant"}, {"start": 55.68, "duration": 4.399, "text": "extending on loes perpendicular to the"}, {"start": 57.84, "duration": 4.039, "text": "plane of the molecule now let's let's"}, {"start": 60.079, "duration": 4.48, "text": "revisit the mechanism for an addition"}, {"start": 61.879, "duration": 5.321, "text": "reaction to contextualize carbocation"}, {"start": 64.559, "duration": 5.801, "text": "stability recall that in the first step"}, {"start": 67.2, "duration": 5.72, "text": "of a hydrohalogenation a new CH Bond was"}, {"start": 70.36, "duration": 4.079, "text": "produced leaving us with a carbocation"}, {"start": 72.92, "duration": 3.839, "text": "but it was the case that the new"}, {"start": 74.439, "duration": 5.04, "text": "hydrogen ended up exclusively on the"}, {"start": 76.759, "duration": 5.36, "text": "less substituted carbon leaving us with"}, {"start": 79.479, "duration": 5.241, "text": "this specific carbocation which we can"}, {"start": 82.119, "duration": 4.36, "text": "call a tertiary carbocation since the"}, {"start": 84.72, "duration": 4.399, "text": "carbon that bears the formal positive"}, {"start": 86.479, "duration": 5.761, "text": "charge is bonded to three other carbons"}, {"start": 89.119, "duration": 5.36, "text": "in this case three methyl groups we did"}, {"start": 92.24, "duration": 3.839, "text": "not see the reverse situation where the"}, {"start": 94.479, "duration": 3.801, "text": "hydrogen ended up on the more"}, {"start": 96.079, "duration": 4.241, "text": "substituted carbon which would leave us"}, {"start": 98.28, "duration": 4.479, "text": "with this carbocation which we would"}, {"start": 100.32, "duration": 4.399, "text": "call a primary carbocation since the"}, {"start": 102.759, "duration": 4.68, "text": "carbon would be bonded to only one other"}, {"start": 104.719, "duration": 4.72, "text": "carbon the rest of the molecule only the"}, {"start": 107.439, "duration": 4.72, "text": "first pathway occurs and never the"}, {"start": 109.439, "duration": 5.441, "text": "second the reasoning behind this is that"}, {"start": 112.159, "duration": 5.041, "text": "carbocations become more stable as they"}, {"start": 114.88, "duration": 5.4, "text": "are more substituted meaning when they"}, {"start": 117.2, "duration": 5.16, "text": "have more adjacent alkal groups this has"}, {"start": 120.28, "duration": 4.64, "text": "to do with a phenomenon called hyper"}, {"start": 122.36, "duration": 5.439, "text": "conjugation where electrons in adjacent"}, {"start": 124.92, "duration": 5.36, "text": "CH bonds which are roughly parallel to"}, {"start": 127.799, "duration": 5.08, "text": "the lobes of the vacant unhybridized p"}, {"start": 130.28, "duration": 5.52, "text": "orbital lend some electron density to"}, {"start": 132.879, "duration": 5.561, "text": "the carbocation thereby stabilizing it"}, {"start": 135.8, "duration": 4.88, "text": "the more alkal groups there are adjacent"}, {"start": 138.44, "duration": 4.079, "text": "the greater the effect so the methyl"}, {"start": 140.68, "duration": 4.4, "text": "cation would be the least stable"}, {"start": 142.519, "duration": 5.481, "text": "carbocation possible followed by a"}, {"start": 145.08, "duration": 5.28, "text": "primary carbocation with one alkal group"}, {"start": 148.0, "duration": 4.879, "text": "then a secondary carbocation with two"}, {"start": 150.36, "duration": 4.84, "text": "alkal groups and the most stable would"}, {"start": 152.879, "duration": 5.201, "text": "be a tertiary carbocation with three"}, {"start": 155.2, "duration": 5.6, "text": "alkal groups this same phenomenon is"}, {"start": 158.08, "duration": 5.2, "text": "also behind the stabilization of alkenes"}, {"start": 160.8, "duration": 4.56, "text": "with greater substitution another reason"}, {"start": 163.28, "duration": 4.52, "text": "for this stabilization has to do with"}, {"start": 165.36, "duration": 4.28, "text": "inductive effects this refers to"}, {"start": 167.8, "duration": 4.6, "text": "electronic effects that are occurring"}, {"start": 169.64, "duration": 4.959, "text": "through Sigma bonds the electron density"}, {"start": 172.4, "duration": 4.28, "text": "surrounding alkal groups is more"}, {"start": 174.599, "duration": 4.681, "text": "polarizable through the sigma Bond"}, {"start": 176.68, "duration": 4.96, "text": "system than that around a hydrogen atom"}, {"start": 179.28, "duration": 5.12, "text": "so more alal results in a greater"}, {"start": 181.64, "duration": 5.2, "text": "inductive effect which is"}, {"start": 184.4, "duration": 5.52, "text": "stabilizing with that we understand how"}, {"start": 186.84, "duration": 4.959, "text": "hyper conjugation and inductive effects"}, {"start": 189.92, "duration": 4.0, "text": "both contribute to the carbocation"}, {"start": 191.799, "duration": 4.681, "text": "stability Trend which says that more"}, {"start": 193.92, "duration": 4.599, "text": "highly substituted carbocations meaning"}, {"start": 196.48, "duration": 4.52, "text": "those with more alkal groups will be"}, {"start": 198.519, "duration": 4.881, "text": "more stable Professor Dave for che see"}, {"start": 201.0, "duration": 2.4, "text": "you next"}, {"start": 208.68, "duration": 5.32, "text": "time"}, {"start": 211.4, "duration": 2.6, "text": "go"}], "title": "Understanding Carbocation Structure and Stability Ft. Professor Dave", "description": "Now that we've covered Markovnikov's rule, let's delve further into the concept of carbocation. We'll take a look at carbocation properties, we'll revisit the mechanism for a specific addition reaction to contextualize carbocation stability, and we'll finally talk about hyperconjugation and inductive effects in the context of carbocation stability.\n\nGet more homework help from Chegg at https://che.gg/3HbtG8Y\nWatch the whole Organic Chemistry playlist: https://www.youtube.com/playlist?list=PLllCkartRRdMPnATf1X8qVdo5RzM8gA_M\n\nAbout Chegg:\nChegg is a leading education technology company that provides students with affordable and accessible study resources. Chegg's mission is to help students save time, save money and get smarter.  \n\nSubscribe: https://www.youtube.com/@chegg\nTikTok: https://www.tiktok.com/@chegg\nInstagram: https://www.instagram.com/chegg/\nTwitter: https://twitter.com/chegg\n\n00:00 Intro\n00:31 Carbocation properties\n00:59 Addition mechanism\n01:50 Hyperconjugation\n02:42 Inductive effects", "lengthSeconds": 212, "width": 640, "height": 360}
{"video_id": "m0IH3Hgy3oI", "transcript": [{"start": 1.85, "duration": 4.48, "text": "when you study chemistry there's a lot"}, {"start": 4.259, "duration": 4.321, "text": "of things that come up when you're"}, {"start": 6.33, "duration": 4.139, "text": "making measurements and will refer to it"}, {"start": 8.58, "duration": 4.679, "text": "being an intensive or an extensive"}, {"start": 10.469, "duration": 6.39, "text": "property or variable and what we mean by"}, {"start": 13.259, "duration": 5.341, "text": "these is we really mean it's a property"}, {"start": 16.859, "duration": 3.481, "text": "that changes with amount or it's a"}, {"start": 18.6, "duration": 3.75, "text": "property that's more intensive which"}, {"start": 20.34, "duration": 3.66, "text": "doesn't change with them out and right"}, {"start": 22.35, "duration": 4.05, "text": "there is really what your definitions"}, {"start": 24.0, "duration": 4.65, "text": "are any property that you can measure"}, {"start": 26.4, "duration": 3.959, "text": "that as you change the amount of"}, {"start": 28.65, "duration": 3.96, "text": "substance but that property doesn't"}, {"start": 30.359, "duration": 5.011, "text": "change with the amount is called"}, {"start": 32.61, "duration": 5.25, "text": "intensive a really great example of this"}, {"start": 35.37, "duration": 5.43, "text": "is temperature it's one of our premier"}, {"start": 37.86, "duration": 5.24, "text": "intensive variables if you will a great"}, {"start": 40.8, "duration": 5.97, "text": "example if you had a glass of water"}, {"start": 43.1, "duration": 7.06, "text": "that's 25 degrees C and you pour half"}, {"start": 46.77, "duration": 4.8, "text": "the water into another glass and ask"}, {"start": 50.16, "duration": 3.87, "text": "yourself now what's the temperature"}, {"start": 51.57, "duration": 5.009, "text": "temperature still 25 degrees because"}, {"start": 54.03, "duration": 4.98, "text": "it's an intensive property it absolutely"}, {"start": 56.579, "duration": 6.48, "text": "does not matter about the amount that"}, {"start": 59.01, "duration": 6.359, "text": "you have it's intensive now let's"}, {"start": 63.059, "duration": 3.721, "text": "compare that to the mass of the water if"}, {"start": 65.369, "duration": 4.651, "text": "I have a glass of water"}, {"start": 66.78, "duration": 6.14, "text": "I know it's mass is say 500 grams and I"}, {"start": 70.02, "duration": 6.75, "text": "pour half of it out now the mass is not"}, {"start": 72.92, "duration": 6.82, "text": "500 grams it's 250 grams that's because"}, {"start": 76.77, "duration": 6.15, "text": "that's an extensive variable extensive"}, {"start": 79.74, "duration": 6.18, "text": "properties vary with the amount and we"}, {"start": 82.92, "duration": 6.18, "text": "love to quantify amounts we quantify in"}, {"start": 85.92, "duration": 7.46, "text": "mass and grams we quantify ammount"}, {"start": 89.1, "duration": 8.4, "text": "itself in moles of substance we quantify"}, {"start": 93.38, "duration": 6.669, "text": "length how long is something how tall is"}, {"start": 97.5, "duration": 4.439, "text": "something volume is a great example of"}, {"start": 100.049, "duration": 4.201, "text": "an extensive property that actually"}, {"start": 101.939, "duration": 5.07, "text": "brings in length three times height"}, {"start": 104.25, "duration": 4.59, "text": "depth and width you multiply all three"}, {"start": 107.009, "duration": 4.741, "text": "of them you get yourself a volume which"}, {"start": 108.84, "duration": 6.3, "text": "is very extensive you get more amount"}, {"start": 111.75, "duration": 5.64, "text": "you get a bigger volume an interesting"}, {"start": 115.14, "duration": 5.4, "text": "thing though about extensive properties"}, {"start": 117.39, "duration": 5.85, "text": "if you ratio one extensive property"}, {"start": 120.54, "duration": 5.009, "text": "against another extensive property you"}, {"start": 123.24, "duration": 5.28, "text": "get an intensive property a great"}, {"start": 125.549, "duration": 6.151, "text": "example of this is density density is"}, {"start": 128.52, "duration": 4.54, "text": "mass per unit volume the mass is"}, {"start": 131.7, "duration": 3.46, "text": "extensive the more you"}, {"start": 133.06, "duration": 4.14, "text": "have the more mass you have volume is"}, {"start": 135.16, "duration": 5.25, "text": "extensive the more volume you have the"}, {"start": 137.2, "duration": 6.21, "text": "more the number goes up if you Rea show"}, {"start": 140.41, "duration": 5.939, "text": "them mass divided by volume you get"}, {"start": 143.41, "duration": 5.46, "text": "density and that's intensive the water"}, {"start": 146.349, "duration": 4.621, "text": "example I gave earlier when I pour it"}, {"start": 148.87, "duration": 4.71, "text": "doesn't matter when I split it the"}, {"start": 150.97, "duration": 5.04, "text": "density is still approximately one gram"}, {"start": 153.58, "duration": 4.98, "text": "per milliliter even in the part I pour"}, {"start": 156.01, "duration": 4.29, "text": "it off it's one gram per milliliter so"}, {"start": 158.56, "duration": 4.2, "text": "this is an important concept you will"}, {"start": 160.3, "duration": 5.28, "text": "see over and over when you're answering"}, {"start": 162.76, "duration": 5.49, "text": "problems numerically in a chemistry"}, {"start": 165.58, "duration": 6.65, "text": "class is the answer an intensive or is"}, {"start": 168.25, "duration": 7.5, "text": "it an extensive property one last thing"}, {"start": 172.23, "duration": 5.979, "text": "extensive properties by themselves tend"}, {"start": 175.75, "duration": 5.849, "text": "not to reveal much about what the"}, {"start": 178.209, "duration": 5.431, "text": "substance actually is for example let's"}, {"start": 181.599, "duration": 5.161, "text": "say I have a substance and I tell you"}, {"start": 183.64, "duration": 5.22, "text": "it's 500 grams that really tells you"}, {"start": 186.76, "duration": 5.37, "text": "nothing about the substance other than"}, {"start": 188.86, "duration": 5.87, "text": "the amount I have but if I tell you I"}, {"start": 192.13, "duration": 6.57, "text": "have a substance that has a density of"}, {"start": 194.73, "duration": 5.68, "text": "one gram per milliliter it really limits"}, {"start": 198.7, "duration": 4.319, "text": "you on your choices of what that could"}, {"start": 200.41, "duration": 4.44, "text": "be and so what we tend to do in the"}, {"start": 203.019, "duration": 5.071, "text": "sciences and what we do in chemistry is"}, {"start": 204.85, "duration": 5.639, "text": "we list intensive properties and if you"}, {"start": 208.09, "duration": 5.07, "text": "get a good listing of them it truly"}, {"start": 210.489, "duration": 5.22, "text": "identify that substance as being a"}, {"start": 213.16, "duration": 5.49, "text": "unique version of those intensive"}, {"start": 215.709, "duration": 7.11, "text": "properties meaning we like to"}, {"start": 218.65, "duration": 6.78, "text": "characterize matter by listing intensive"}, {"start": 222.819, "duration": 4.2, "text": "properties and that's really what the"}, {"start": 225.43, "duration": 5.33, "text": "importance of both of these things are"}, {"start": 227.019, "duration": 3.741, "text": "and you need to keep track of them"}], "title": "Intensive Extensive Properites", "description": "Intensive and Extensive properties.", "lengthSeconds": 230, "width": 640, "height": 360}
{"video_id": "fDHuEwhS3zU", "transcript": [{"start": 0.08, "duration": 4.64, "text": "number four what is the iup pack name"}, {"start": 2.399, "duration": 4.96, "text": "for the following compound so now we"}, {"start": 4.72, "duration": 4.919, "text": "have a cyclohexene derivative so we have"}, {"start": 7.359, "duration": 5.081, "text": "a double bond inside the cyclohexane"}, {"start": 9.639, "duration": 5.08, "text": "ring which makes it cyclohexene and now"}, {"start": 12.44, "duration": 4.72, "text": "it's not arbitrary the way we number we"}, {"start": 14.719, "duration": 5.32, "text": "need for the two carbons participating"}, {"start": 17.16, "duration": 5.359, "text": "in the double bond to be carbons one and"}, {"start": 20.039, "duration": 4.441, "text": "two but there are two options possible"}, {"start": 22.519, "duration": 4.041, "text": "we could either have this be number one"}, {"start": 24.48, "duration": 3.719, "text": "and this be number two or we could have"}, {"start": 26.56, "duration": 4.639, "text": "this be number one and this be number"}, {"start": 28.199, "duration": 5.04, "text": "two and we need to choose the option"}, {"start": 31.199, "duration": 4.921, "text": "that will give any substituents on the"}, {"start": 33.239, "duration": 8.761, "text": "molecule occurring soonest what we end"}, {"start": 36.12, "duration": 10.279, "text": "up realizing is that we have to go 1 2 3"}, {"start": 42.0, "duration": 7.36, "text": "4 5 6 like that there's a few common"}, {"start": 46.399, "duration": 4.8, "text": "errors here some will try to label the"}, {"start": 49.36, "duration": 4.44, "text": "carbon with the chlorine as number one"}, {"start": 51.199, "duration": 4.68, "text": "that doesn't work again it's cyclohexene"}, {"start": 53.8, "duration": 3.72, "text": "which means the carbons participating in"}, {"start": 55.879, "duration": 4.64, "text": "the double bond needs to be carbons one"}, {"start": 57.52, "duration": 2.999, "text": "and two some"}], "title": "Naming a Compound With Multiple Functional Groups", "description": "This is an abbreviated version of this question for YouTube Shorts. Check out the full version in the linked video.", "lengthSeconds": 59, "width": 360, "height": 640}
{"video_id": "rK5IQwmykiI", "transcript": [{"start": 0.04, "duration": 4.6, "text": "four asks the following pair of"}, {"start": 2.0, "duration": 5.72, "text": "structures are identical enantiomers"}, {"start": 4.64, "duration": 5.039, "text": "diamer and constitutional isomers so we"}, {"start": 7.72, "duration": 4.44, "text": "have some straight chain alkanes with"}, {"start": 9.679, "duration": 4.761, "text": "bromo groups on them and why don't we go"}, {"start": 12.16, "duration": 4.08, "text": "ahead and number these parent chains so"}, {"start": 14.44, "duration": 3.48, "text": "on the left which direction would we"}, {"start": 16.24, "duration": 9.359, "text": "number this we would start from the"}, {"start": 17.92, "duration": 9.679, "text": "right and we would get 1 2 3 4 5 6 7 8"}, {"start": 25.599, "duration": 3.76, "text": "so that is an octane and then on the"}, {"start": 27.599, "duration": 2.881, "text": "right this one we would number from left"}, {"start": 29.359, "duration": 8.161, "text": "to right because that will give the"}, {"start": 30.48, "duration": 10.64, "text": "bromine occurring soonest 1 2 3 4 5 6 7"}, {"start": 37.52, "duration": 6.4, "text": "8 so where are the bromo groups over"}, {"start": 41.12, "duration": 4.2, "text": "here they're on carbons three and five"}, {"start": 43.92, "duration": 4.119, "text": "and then on the other structure they're"}, {"start": 45.32, "duration": 4.36, "text": "on carbons 2 and four so these are"}, {"start": 48.039, "duration": 4.561, "text": "actually completely different molecules"}, {"start": 49.68, "duration": 5.519, "text": "these are not stereoisomers these are"}, {"start": 52.6, "duration": 4.72, "text": "constitutional isomers they have"}, {"start": 55.199, "duration": 5.32, "text": "different connectivity remember that"}, {"start": 57.32, "duration": 3.199, "text": "stereoisomer have"}], "title": "Determining the Type of Isomer", "description": "This is an abbreviated version of this question for YouTube Shorts. Check out the full version in the linked video.", "lengthSeconds": 59, "width": 360, "height": 640}
{"video_id": "aDXQBE4r65Y", "transcript": [{"start": 0.0, "duration": 3.38, "text": "Hey it's professor Dave, let's define internal energy."}, {"start": 10.3, "duration": 0.919, "text": "We now understand"}, {"start": 11.219, "duration": 1.531, "text": "temperature as a measure of the"}, {"start": 12.75, "duration": 2.339, "text": "molecular kinetic energy present in the"}, {"start": 15.089, "duration": 2.971, "text": "particles of a substance. As we said, this"}, {"start": 18.06, "duration": 2.01, "text": "kinetic energy is distributed amongst"}, {"start": 20.07, "duration": 2.699, "text": "translational motion, rotational motion"}, {"start": 22.769, "duration": 2.551, "text": "and vibrational motion. There is also"}, {"start": 25.32, "duration": 2.52, "text": "molecular potential energy by virtue of"}, {"start": 27.84, "duration": 2.55, "text": "the electromagnetic force acting amongst"}, {"start": 30.39, "duration": 2.099, "text": "the atoms of an individual molecule and"}, {"start": 32.489, "duration": 3.451, "text": "between each separate molecule. The sum"}, {"start": 35.94, "duration": 2.43, "text": "of all of these types of energy that are"}, {"start": 38.37, "duration": 1.98, "text": "exhibited by the particles of a"}, {"start": 40.35, "duration": 3.24, "text": "substance is called the internal energy"}, {"start": 43.59, "duration": 2.699, "text": "of that substance, represented by an"}, {"start": 46.289, "duration": 3.721, "text": "upper case U. This is the energy"}, {"start": 50.01, "duration": 2.88, "text": "associated with atomic motion and it is"}, {"start": 52.89, "duration": 2.489, "text": "a quantity that is directly proportional"}, {"start": 55.379, "duration": 2.311, "text": "to the temperature of a sample. Higher"}, {"start": 57.69, "duration": 2.7, "text": "temperatures mean more internal energy,"}, {"start": 60.39, "duration": 2.759, "text": "lower temperatures mean less internal"}, {"start": 63.149, "duration": 2.79, "text": "energy. In this way, we must understand"}, {"start": 65.939, "duration": 2.731, "text": "that a substance does not contain heat,"}, {"start": 68.67, "duration": 3.42, "text": "it contains internal energy, like the"}, {"start": 72.09, "duration": 1.92, "text": "kinetic energy of all the particles, and"}, {"start": 74.01, "duration": 2.7, "text": "it is the transfer of this energy from"}, {"start": 76.71, "duration": 2.159, "text": "areas of high temperature to areas of"}, {"start": 78.869, "duration": 2.371, "text": "low temperature that we can label as"}, {"start": 81.24, "duration": 3.57, "text": "heat. Internal energy is an important"}, {"start": 84.81, "duration": 2.34, "text": "concept in the study of thermodynamics"}, {"start": 87.15, "duration": 2.1, "text": "which is the study of heat and"}, {"start": 89.25, "duration": 2.7, "text": "temperature and their relation to energy"}, {"start": 91.95, "duration": 2.82, "text": "and work. The internal energy of a system"}, {"start": 94.77, "duration": 2.97, "text": "can be increased due to heat transfer"}, {"start": 97.74, "duration": 2.519, "text": "but also due to things like friction or"}, {"start": 100.259, "duration": 2.97, "text": "structural deformation when bending a"}, {"start": 103.229, "duration": 1.771, "text": "piece of metal or stretching a rubber"}, {"start": 105.0, "duration": 2.85, "text": "band. But as we learned earlier, for a"}, {"start": 107.85, "duration": 2.1, "text": "particular system there will always be"}, {"start": 109.95, "duration": 4.11, "text": "conservation of energy. We already know"}, {"start": 114.06, "duration": 2.19, "text": "about the kinetic energy and potential"}, {"start": 116.25, "duration": 2.189, "text": "energy of an object moving through a"}, {"start": 118.439, "duration": 2.731, "text": "gravitational field so let's add the"}, {"start": 121.17, "duration": 2.549, "text": "internal energy of the particles in the"}, {"start": 123.719, "duration": 2.79, "text": "object to this list, and restate"}, {"start": 126.509, "duration": 2.161, "text": "conservation of energy with the"}, {"start": 128.67, "duration": 2.97, "text": "following equation: change in potential"}, {"start": 131.64, "duration": 1.26, "text": "energy plus"}, {"start": 132.9, "duration": 2.64, "text": "change in kinetic energy plus change in"}, {"start": 135.54, "duration": 3.18, "text": "internal energy will always be zero. That"}, {"start": 138.72, "duration": 2.129, "text": "means that whenever there is a change in"}, {"start": 140.849, "duration": 1.95, "text": "any one of these quantities the"}, {"start": 142.799, "duration": 3.151, "text": "difference in energy must transform into"}, {"start": 145.95, "duration": 3.659, "text": "or be provided by one of the other forms."}, {"start": 149.609, "duration": 3.091, "text": "So when two objects participate in an"}, {"start": 152.7, "duration": 2.58, "text": "inelastic collision, some of the kinetic"}, {"start": 155.28, "duration": 2.549, "text": "energy is transferred into internal"}, {"start": 157.829, "duration": 2.22, "text": "energy, which will be absorbed by the"}, {"start": 160.049, "duration": 3.33, "text": "object. Internal energy can in turn be"}, {"start": 163.379, "duration": 2.851, "text": "used to do work through heat transfer,"}, {"start": 166.23, "duration": 2.759, "text": "pressure-volume work by an expanding gas"}, {"start": 168.989, "duration": 2.791, "text": "or some other process. We should also"}, {"start": 171.78, "duration": 2.37, "text": "note that internal energy is a state"}, {"start": 174.15, "duration": 2.1, "text": "function, which means that the internal"}, {"start": 176.25, "duration": 2.7, "text": "energy of a system depends only on the"}, {"start": 178.95, "duration": 2.939, "text": "state of the system and not how it got"}, {"start": 181.889, "duration": 2.43, "text": "to that state. We will learn more about"}, {"start": 184.319, "duration": 3.12, "text": "state functions later. These kinds of"}, {"start": 187.439, "duration": 2.311, "text": "concepts are precisely what we will be"}, {"start": 189.75, "duration": 2.069, "text": "examining when we discuss the laws of"}, {"start": 191.819, "duration": 2.611, "text": "thermodynamics, so let's move on to this"}, {"start": 194.43, "duration": 1.73, "text": "incredibly important subject."}, {"start": 197.22, "duration": 1.68, "text": "Thanks for watching, guys. Subscribe to my channel"}, {"start": 198.9, "duration": 1.949, "text": "for more tutorials, support me on patreon"}, {"start": 200.849, "duration": 2.311, "text": "so I can keep making content, and as"}, {"start": 203.16, "duration": 1.48, "text": "always, feel free to email me:"}], "title": "Internal Energy", "description": "We know about kinetic energy and potential energy, which can interchange when an object moves through a gravitational field, so let's add to that list the internal energy of the object. This gives us a new and comprehensive way to describe conservation of energy. Check it out!\n\nWatch the whole Classical Physics playlist: http://bit.ly/ProfDavePhysics1\n\nModern Physics Tutorials: http://bit.ly/ProfDavePhysics2\nMathematics Tutorials: http://bit.ly/ProfDaveMaths\nGeneral Chemistry Tutorials: http://bit.ly/ProfDaveGenChem\nOrganic Chemistry Tutorials: http://bit.ly/ProfDaveOrgChem\nBiochemistry Tutorials: http://bit.ly/ProfDaveBiochem\nBiology Tutorials: http://bit.ly/ProfDaveBio\n\nEMAIL\u25ba ProfessorDaveExplains@gmail.com\nPATREON\u25ba http://patreon.com/ProfessorDaveExplains\n\nCheck out \"Is This Wi-Fi Organic?\", my book on disarming pseudoscience!\nAmazon: https://amzn.to/2HtNpVH\nBookshop: https://bit.ly/39cKADM\n\rBarnes and Noble: https://bit.ly/3pUjmrn \nBook Depository: http://bit.ly/3aOVDlT", "lengthSeconds": 227, "width": 640, "height": 360}
{"video_id": "EJqOUcEq_SA", "transcript": [{"start": 0.12, "duration": 4.159, "text": "number five what is the iup pack name"}, {"start": 2.32, "duration": 5.76, "text": "for the following compound here we have"}, {"start": 4.279, "duration": 6.28, "text": "an Aline and we need to number the"}, {"start": 8.08, "duration": 5.12, "text": "parent chain so as to give the triple"}, {"start": 10.559, "duration": 10.56, "text": "bond occurring soonest so that is going"}, {"start": 13.2, "duration": 9.919, "text": "to go like this 1 2 3 4 5 6 7 8 9 now"}, {"start": 21.119, "duration": 3.92, "text": "there are a couple common errors here it"}, {"start": 23.119, "duration": 4.041, "text": "does look funny sometimes with the"}, {"start": 25.039, "duration": 5.4, "text": "triple bond there there are students"}, {"start": 27.16, "duration": 5.439, "text": "that don't realize this is a carbon and"}, {"start": 30.439, "duration": 4.561, "text": "this is a carbon some students will have"}, {"start": 32.599, "duration": 5.241, "text": "a tendency to see this just as two"}, {"start": 35.0, "duration": 4.879, "text": "carbons with a very long triple bond but"}, {"start": 37.84, "duration": 4.399, "text": "remember that carbons participating in"}, {"start": 39.879, "duration": 4.52, "text": "triple bonds are SP hybridized and"}, {"start": 42.239, "duration": 4.521, "text": "therefore exhibit linear molecular"}, {"start": 44.399, "duration": 5.68, "text": "geometry so we do have carbons 2 and"}, {"start": 46.76, "duration": 5.4, "text": "four totally 180\u00b0 apart as well as three"}, {"start": 50.079, "duration": 4.16, "text": "and five so this geometry the way that"}, {"start": 52.16, "duration": 3.68, "text": "we draw this is very deliberate that is"}, {"start": 54.239, "duration": 4.081, "text": "accurately depicting the geometry of the"}, {"start": 55.84, "duration": 4.679, "text": "molecule so make sure we understand that"}, {"start": 58.32, "duration": 2.199, "text": "and"}], "title": "IUPAC Nomenclature of Alkynes", "description": "This is an abbreviated version of this question for YouTube Shorts. Check out the full version in the linked video.", "lengthSeconds": 59, "width": 360, "height": 640}
{"video_id": "nQBC00jIiHE", "transcript": [{"start": 6.3, "duration": 6.68, "text": "okay let's practice using solubility\nproduct constant fluorite or CaF2 is"}, {"start": 12.99, "duration": 5.16, "text": "slightly water soluble and dissolves\naccording to the following equation so"}, {"start": 18.15, "duration": 5.43, "text": "we have fluoride solid in equilibrium\nwith the calcium 2 plus ion and 2"}, {"start": 23.58, "duration": 6.959, "text": "fluoride ions now a saturated fluoride\nsolution is measured as having a calcium"}, {"start": 30.539, "duration": 7.351, "text": "ion concentration of 2.1 times 10 to the\nnegative 4 molar so what is the"}, {"start": 37.89, "duration": 5.669, "text": "solubility product for fluorite so\nthat's all the information we need go"}, {"start": 43.56, "duration": 1.34, "text": "ahead and give this a try."}, {"start": 62.98, "duration": 3.58, "text": "so let's put\nour equilibrium up top and once again we"}, {"start": 66.57, "duration": 6.54, "text": "know that the calcium ion concentration\nis 2.1 times 10 to the negative 4 molar"}, {"start": 73.11, "duration": 5.009, "text": "so what can we do with this information\nwell we know that fluorite is going to"}, {"start": 78.119, "duration": 6.421, "text": "dissolve to give a calcium ion and to\nfluoride ions so if enough fluoride has"}, {"start": 84.54, "duration": 6.509, "text": "dissolved so as to give a calcium ion\nconcentration of that magnitude then we"}, {"start": 91.049, "duration": 4.921, "text": "know that the fluoride ion concentration\nmust be double that because there are"}, {"start": 95.97, "duration": 5.189, "text": "two fluorides that form for every\ncalcium ion so that means the fluoride"}, {"start": 101.159, "duration": 5.221, "text": "ion concentration must be 4.2 times 10\nto the negative 4 molar right that has"}, {"start": 106.38, "duration": 3.66, "text": "exactly double the calcium ion\nconcentration so we figured that out"}, {"start": 110.04, "duration": 5.939, "text": "just by stoichiometry now let's write\nour Ksp expression so this is just like"}, {"start": 115.979, "duration": 4.801, "text": "any other equilibrium expression it\ninvolves only aqueous or gaseous species"}, {"start": 120.78, "duration": 4.979, "text": "and so that means we're just going to\nhave the calcium ion concentration times"}, {"start": 125.759, "duration": 4.261, "text": "the fluoride ion concentration squared\nand then we don't have to put that over"}, {"start": 130.02, "duration": 3.689, "text": "any reactants because the reactant is a solid"}, {"start": 133.709, "duration": 3.9, "text": "now we have all the information we need\nwe have the calcium ion concentration"}, {"start": 137.609, "duration": 3.96, "text": "and we have the fluoride ion\nconcentration so we just plug those"}, {"start": 141.569, "duration": 5.34, "text": "numbers in and then we do the arithmetic\nand we get a Ksp value of three point"}, {"start": 146.909, "duration": 4.67, "text": "seven times ten to the negative eleven."}], "title": "Practice Problem: Solubility Product Constant Calculations", "description": "We've learned that there are ionic solids that are insoluble in water, and while for the most part that's accurate, we are bending the truth a little bit. Even compounds that are totally water insoluble will still dissolve a teeny tiny bit. We can communicate just how much using solubility product constants, which are another type of equilibrium constant. Let's learn how to calculate these now!\n\nTry all of the general chemistry practice problems: http://bit.ly/ProfDaveGCPP\n\nGeneral Chemistry Tutorials: http://bit.ly/ProfDaveGenChem\n\nEMAIL\u25ba ProfessorDaveExplains@gmail.com\nPATREON\u25ba http://patreon.com/ProfessorDaveExplains\n\nCheck out \"Is This Wi-Fi Organic?\", my book on disarming pseudoscience!\nAmazon: https://amzn.to/2HtNpVH\nBookshop: https://bit.ly/39cKADM\n\rBarnes and Noble: https://bit.ly/3pUjmrn \nBook Depository: http://bit.ly/3aOVDlT", "lengthSeconds": 182, "width": 640, "height": 360}
{"video_id": "EklmJ9Wvrh0", "transcript": [{"start": 0.12, "duration": 4.239, "text": "question number one says in terms of"}, {"start": 2.2, "duration": 4.48, "text": "intermolecular forces which of the"}, {"start": 4.359, "duration": 4.441, "text": "following compounds is only capable of"}, {"start": 6.68, "duration": 4.039, "text": "London dispersion so we have different"}, {"start": 8.8, "duration": 3.759, "text": "kinds of intermolecular forces possible"}, {"start": 10.719, "duration": 3.84, "text": "there could be molecules that do dipole"}, {"start": 12.559, "duration": 3.921, "text": "dipole interactions for example but one"}, {"start": 14.559, "duration": 4.041, "text": "of these is only capable of London"}, {"start": 16.48, "duration": 3.84, "text": "dispersion in order to check out what's"}, {"start": 18.6, "duration": 3.56, "text": "going on we want to draw these molecules"}, {"start": 20.32, "duration": 4.84, "text": "out so we can identify what kinds of"}, {"start": 22.16, "duration": 6.56, "text": "bonds are present ch3"}, {"start": 25.16, "duration": 7.119, "text": "ch2 we've got that o group there"}, {"start": 28.72, "duration": 6.6, "text": "ch3f we got that Central carbon atom and"}, {"start": 32.279, "duration": 5.001, "text": "then we've got the F and then we've got"}, {"start": 35.32, "duration": 6.44, "text": "some hydrogens so that's got that"}, {"start": 37.28, "duration": 6.599, "text": "tetrahedral geometry ch2 ch2 that means"}, {"start": 41.76, "duration": 4.319, "text": "that we have a double bond because every"}, {"start": 43.879, "duration": 4.081, "text": "carbon is going to have four bonds there"}, {"start": 46.079, "duration": 3.521, "text": "and then we have hbr so what do we"}, {"start": 47.96, "duration": 3.599, "text": "notice about the types of bonds that are"}, {"start": 49.6, "duration": 4.4, "text": "present in order for something to do"}, {"start": 51.559, "duration": 5.0, "text": "dipole dipole interactions we have to"}, {"start": 54.0, "duration": 5.28, "text": "have some Bond dipoles we need to have"}, {"start": 56.559, "duration": 3.96, "text": "some polar bonds there can we find some"}, {"start": 59.28, "duration": 3.16, "text": "polar bonds"}, {"start": 60.519, "duration": 4.801, "text": "most of these do have polar bonds so for"}, {"start": 62.44, "duration": 5.76, "text": "example on ethanol there we've got an O"}, {"start": 65.32, "duration": 6.479, "text": "bond this is partial minus this is"}, {"start": 68.2, "duration": 5.32, "text": "partial plus so that is a polar bond and"}, {"start": 71.799, "duration": 3.841, "text": "that Bond dipole is going to be able to"}, {"start": 73.52, "duration": 3.8, "text": "interact with other bond dipoles to make"}, {"start": 75.64, "duration": 3.479, "text": "dipole dipole interactions in this case"}, {"start": 77.32, "duration": 4.72, "text": "they're specifically hydrogen bonds and"}, {"start": 79.119, "duration": 4.96, "text": "then over here we have a polar CF Bond"}, {"start": 82.04, "duration": 3.399, "text": "right this is also polarized towards the"}, {"start": 84.079, "duration": 3.0, "text": "Florine the carbon is going to be"}, {"start": 85.439, "duration": 3.32, "text": "partially positive the Florine is"}, {"start": 87.079, "duration": 4.521, "text": "partially negative the same thing with"}, {"start": 88.759, "duration": 5.481, "text": "hbr we've got a partially positive H and"}, {"start": 91.6, "duration": 5.559, "text": "we've got a partially minus bromine and"}, {"start": 94.24, "duration": 4.159, "text": "so those are examples of polar bonds"}, {"start": 97.159, "duration": 3.28, "text": "that are going to be able to make"}, {"start": 98.399, "duration": 4.72, "text": "electrostatic interactions with other"}, {"start": 100.439, "duration": 5.441, "text": "polar bonds that will be dipole dipole"}, {"start": 103.119, "duration": 4.6, "text": "interactions or hydrogen bonds that are"}, {"start": 105.88, "duration": 4.04, "text": "stronger interactions than London"}, {"start": 107.719, "duration": 5.201, "text": "dispersion however over here we don't"}, {"start": 109.92, "duration": 5.44, "text": "have any of those with ch2 ch2 these are"}, {"start": 112.92, "duration": 4.36, "text": "all non-polar bonds so carbon carbon"}, {"start": 115.36, "duration": 4.119, "text": "bonds are obviously non-polar completely"}, {"start": 117.28, "duration": 3.92, "text": "non-polar and CH bonds there's not"}, {"start": 119.479, "duration": 3.521, "text": "enough an electro negativity difference"}, {"start": 121.2, "duration": 2.959, "text": "there to consider that a polar bond so"}, {"start": 123.0, "duration": 3.719, "text": "we would say that these are all"}, {"start": 124.159, "duration": 5.16, "text": "non-polar bonds and therefore it is not"}, {"start": 126.719, "duration": 4.16, "text": "able to do any dipo dipole interactions"}, {"start": 129.319, "duration": 3.881, "text": "or anything of that magnitude we're"}, {"start": 130.879, "duration": 5.041, "text": "going to say C right here that is ch2"}, {"start": 133.2, "duration": 6.679, "text": "ch2 that is the only molecule that is"}, {"start": 135.92, "duration": 3.959, "text": "only capable of London dispersion"}], "title": "Professor Dave Explains Intermolecular Forces", "description": "\ud83d\udcafCheck out Professor Dave's FREE Assessment Exam and more OCHEM RESOURCES here: https://chemmunity.info/dave \ud83d\ude4c", "lengthSeconds": 138, "width": 360, "height": 640}
{"video_id": "mq_2WN0s6L0", "transcript": [{"start": 0.12, "duration": 4.92, "text": "welcome to MooMoo Math and Science in"}, {"start": 2.7, "duration": 6.0, "text": "this video let's talk about Solutions"}, {"start": 5.04, "duration": 6.179, "text": "suspensions and colloids a solution is a"}, {"start": 8.7, "duration": 5.64, "text": "mixture of two or more substances the"}, {"start": 11.219, "duration": 5.461, "text": "particles in a solution cannot be seen"}, {"start": 14.34, "duration": 4.32, "text": "light is not visible when it passes"}, {"start": 16.68, "duration": 4.439, "text": "through a solution"}, {"start": 18.66, "duration": 3.24, "text": "and it cannot be separated using a"}, {"start": 21.119, "duration": 4.561, "text": "filter"}, {"start": 21.9, "duration": 6.84, "text": "an example would be sugar and water"}, {"start": 25.68, "duration": 6.48, "text": "sugar is the solute and water is the"}, {"start": 28.74, "duration": 6.24, "text": "solvent the solute is the item being"}, {"start": 32.16, "duration": 5.34, "text": "dissolved and the solvent is what the"}, {"start": 34.98, "duration": 4.739, "text": "item is being dissolved into"}, {"start": 37.5, "duration": 4.68, "text": "this may help you remember"}, {"start": 39.719, "duration": 6.84, "text": "the difference between a solute and a"}, {"start": 42.18, "duration": 8.039, "text": "solvent think of placing the loot into a"}, {"start": 46.559, "duration": 5.52, "text": "vent to hide it so the solute goes into"}, {"start": 50.219, "duration": 4.141, "text": "the solvent"}, {"start": 52.079, "duration": 4.741, "text": "a suspension is a mixture of a fluid"}, {"start": 54.36, "duration": 4.5, "text": "with undissolved particles in it the"}, {"start": 56.82, "duration": 4.44, "text": "solid particles may be visible and will"}, {"start": 58.86, "duration": 4.8, "text": "settle over time in addition the"}, {"start": 61.26, "duration": 4.56, "text": "particles can be filtered out"}, {"start": 63.66, "duration": 3.6, "text": "light is visible when it passes through"}, {"start": 65.82, "duration": 3.6, "text": "a suspension"}, {"start": 67.26, "duration": 4.859, "text": "the solid particles are called the"}, {"start": 69.42, "duration": 4.68, "text": "internal phase and the fluid is called"}, {"start": 72.119, "duration": 6.301, "text": "the external phase"}, {"start": 74.1, "duration": 6.72, "text": "an example would be muddy water"}, {"start": 78.42, "duration": 4.44, "text": "and sand and water"}, {"start": 80.82, "duration": 2.88, "text": "the sand does not dissolve into the"}, {"start": 82.86, "duration": 3.899, "text": "water"}, {"start": 83.7, "duration": 5.58, "text": "but it's mixed into the water and over"}, {"start": 86.759, "duration": 4.86, "text": "time it will settle"}, {"start": 89.28, "duration": 5.76, "text": "a colloid is a mixture consisting of"}, {"start": 91.619, "duration": 5.04, "text": "very small insoluble particles dispersed"}, {"start": 95.04, "duration": 3.78, "text": "into a fluid"}, {"start": 96.659, "duration": 3.721, "text": "light is visible when it passes through"}, {"start": 98.82, "duration": 3.96, "text": "the colloid"}, {"start": 100.38, "duration": 5.279, "text": "you cannot use a filter to filter out"}, {"start": 102.78, "duration": 4.979, "text": "what is dispersed and the particles do"}, {"start": 105.659, "duration": 4.92, "text": "not settle over time"}, {"start": 107.759, "duration": 5.281, "text": "examples include glue"}, {"start": 110.579, "duration": 5.881, "text": "milk and fog"}, {"start": 113.04, "duration": 5.82, "text": "a colloid has a dispersed phase which"}, {"start": 116.46, "duration": 4.799, "text": "are the suspended particles and a"}, {"start": 118.86, "duration": 4.86, "text": "continuous phase which is what the"}, {"start": 121.259, "duration": 4.801, "text": "particles are suspended in"}, {"start": 123.72, "duration": 4.379, "text": "The Continuous phase can be a solid"}, {"start": 126.06, "duration": 3.78, "text": "liquid or gas"}, {"start": 128.099, "duration": 3.481, "text": "I hope that was helpful thanks for"}, {"start": 129.84, "duration": 4.2, "text": "watching and remember kindness"}, {"start": 131.58, "duration": 5.18, "text": "multiplies kindness be kind to someone"}, {"start": 134.04, "duration": 2.72, "text": "today"}], "title": "Solution Suspension Colloid", "description": "Learn the difference between a solution,suspension, and a colloid.\nThis video will help with the following Science  standard\nS8P1. Obtain, evaluate, and communicate information about the structure and properties\nof matter.\na. Develop and use a model to compare and contrast pure substances (elements and\ncompounds) and mixtures.\n(Clarification statement: Include heterogeneous and homogeneous mixtures\n\nIn this video I compare these three types of mixtures. A solution and a colloid are homogenous mixtures and the suspension is a heterogenous mixture.", "lengthSeconds": 136, "width": 640, "height": 360}
{"video_id": "Tzl2BJeJ45Q", "transcript": [{"start": 0.179, "duration": 4.201, "text": "hi my name is Paul from physics hype"}, {"start": 2.34, "duration": 7.039, "text": "today I want to give you a very quick"}, {"start": 4.38, "duration": 4.999, "text": "overview of the concept of lenses law"}, {"start": 11.099, "duration": 3.54, "text": "please like share and subscribe and"}, {"start": 12.78, "duration": 3.54, "text": "maybe buy me a coffee the link is in the"}, {"start": 14.639, "duration": 3.541, "text": "description below if you want a more"}, {"start": 16.32, "duration": 3.539, "text": "fulsome discussion of lenses law please"}, {"start": 18.18, "duration": 3.24, "text": "check out the link there or the link in"}, {"start": 19.859, "duration": 4.381, "text": "the description below where I give a"}, {"start": 21.42, "duration": 5.4, "text": "more detailed analysis today is just a"}, {"start": 24.24, "duration": 4.799, "text": "short version of it what is lenses law"}, {"start": 26.82, "duration": 4.5, "text": "well lenses law is related to the"}, {"start": 29.039, "duration": 4.141, "text": "concept of Faraday's law and let's write"}, {"start": 31.32, "duration": 4.919, "text": "the Faraday's law down which basically"}, {"start": 33.18, "duration": 5.399, "text": "says that if you generate an EMF because"}, {"start": 36.239, "duration": 5.16, "text": "so we have a change of flux with respect"}, {"start": 38.579, "duration": 5.041, "text": "to time that a metal is experiencing"}, {"start": 41.399, "duration": 5.101, "text": "then there's a negative sign that refers"}, {"start": 43.62, "duration": 5.64, "text": "to the concept of lenses law and lenses"}, {"start": 46.5, "duration": 5.52, "text": "law says that whatever the EMF that is"}, {"start": 49.26, "duration": 5.22, "text": "generated will always produce a polarity"}, {"start": 52.02, "duration": 5.28, "text": "that opposes the one that generates it"}, {"start": 54.48, "duration": 5.88, "text": "and so one way we can describe this is"}, {"start": 57.3, "duration": 4.68, "text": "using a coil so let's say I have a coil"}, {"start": 60.36, "duration": 4.439, "text": "like this"}, {"start": 61.98, "duration": 5.819, "text": "and I'm going to connect this to a"}, {"start": 64.799, "duration": 6.061, "text": "voltmeter and I've insert a North Pole"}, {"start": 67.799, "duration": 5.101, "text": "and I push that into the coil we have a"}, {"start": 70.86, "duration": 3.84, "text": "magnetic field around my bar magnet and"}, {"start": 72.9, "duration": 3.899, "text": "it's going to enter the coil which"}, {"start": 74.7, "duration": 4.14, "text": "results in an EMF Genie being generated"}, {"start": 76.799, "duration": 4.801, "text": "and as a result in this case a current"}, {"start": 78.84, "duration": 4.74, "text": "is produced now the card that is"}, {"start": 81.6, "duration": 3.9, "text": "produced will always produce a magnetic"}, {"start": 83.58, "duration": 6.42, "text": "field that opposes it so as I push this"}, {"start": 85.5, "duration": 6.659, "text": "in I am going to produce a North Pole in"}, {"start": 90.0, "duration": 4.14, "text": "this end it's going to not want to go in"}, {"start": 92.159, "duration": 4.201, "text": "it's going to experience a force of"}, {"start": 94.14, "duration": 4.5, "text": "repulsion now because of the fact that"}, {"start": 96.36, "duration": 4.02, "text": "there's a North Pole over here that"}, {"start": 98.64, "duration": 4.38, "text": "means I will generate a South bolt over"}, {"start": 100.38, "duration": 5.46, "text": "here and So based on my diagram my code"}, {"start": 103.02, "duration": 5.279, "text": "is going to go from there to there so if"}, {"start": 105.84, "duration": 4.98, "text": "I pull that magnet out the north is now"}, {"start": 108.299, "duration": 4.86, "text": "moving away that will produce a polarity"}, {"start": 110.82, "duration": 4.619, "text": "here that is South to oppose it it's"}, {"start": 113.159, "duration": 3.96, "text": "moving away it will create a force that"}, {"start": 115.439, "duration": 3.96, "text": "wants to pull it back in which will"}, {"start": 117.119, "duration": 4.381, "text": "therefore oppose the change that causes"}, {"start": 119.399, "duration": 5.04, "text": "it and so therefore the kind therefore"}, {"start": 121.5, "duration": 4.56, "text": "that is in used is determined by what is"}, {"start": 124.439, "duration": 4.8, "text": "happening over here and it'll always"}, {"start": 126.06, "duration": 5.7, "text": "oppose and as a result that is going to"}, {"start": 129.239, "duration": 4.681, "text": "be consistent with the conservation of"}, {"start": 131.76, "duration": 4.08, "text": "energy because as you pull it in you"}, {"start": 133.92, "duration": 4.8, "text": "require energy to push it down and so"}, {"start": 135.84, "duration": 5.759, "text": "that requires energy to be produced over"}, {"start": 138.72, "duration": 4.379, "text": "here so that is lenses law please like"}, {"start": 141.599, "duration": 4.201, "text": "share and subscribe put a comment down"}, {"start": 143.099, "duration": 6.921, "text": "below and consider supporting me by"}, {"start": 145.8, "duration": 4.22, "text": "buying me a coffee take care bye for now"}], "title": "A quick \"cheat sheet \" Lenz's Law", "description": "This video review the nature of Lenz's as a subset of Faraday's Law\n\nFor a further elaboration, see my longer videos on the topic...\n\nEddy currents and Lenz's Law - https://youtu.be/yU_wwWC-K9c\nLenz's Law in Action - https://youtu.be/0QS3blZFzUg\nelectromagnetic Braking Explained  -  https://youtu.be/BIsKthqeKSo\n\n\nLike what I do? Support by buying me a coffee - www.buymeacoffee.com/physicshigh\n\nSubscribe - www.youtube.com/c/physicshigh\n\nFor on going support, support me at Patreon: www.patreon.com/physicshigh\n\nLIKE and SHARE with your peers. And please add a COMMENT to let me know I have helped you.\n\nPhysics High is committed to producing content that teaches physics concepts at a level a high schooler can understand. \nSee www.physicshigh.com for all my videos and other resources.\n\nAs well as this I produce a podcast series called Deep Impact  - interviews with science communicators as to what they do, what drives them to communicate their craft\nYou will find these on the channel and also on podcast sites such as Spotify, Apple and Google podcasts\n\n \ud83d\udc65 Social\n---------------------------------------------------------\nFollow me on \nfacebook: @physicshigh\ntwitter: @physicshigh\nInstagram: @physicshigh", "lengthSeconds": 148, "width": 604, "height": 360}
{"video_id": "Oa-yZGz2AdY", "transcript": [{"start": 1.88, "duration": 3.72, "text": "hey everyone Professor Davis here again"}, {"start": 3.399, "duration": 4.601, "text": "to talk to you a little bit about hyper"}, {"start": 5.6, "duration": 4.48, "text": "conjugation specifically how that"}, {"start": 8.0, "duration": 5.28, "text": "applies to carbocation"}, {"start": 10.08, "duration": 4.92, "text": "stability so carbocations are most"}, {"start": 13.28, "duration": 3.72, "text": "stable when they're more substituted we"}, {"start": 15.0, "duration": 4.279, "text": "know this from our lecture"}, {"start": 17.0, "duration": 4.56, "text": "course and we also know that alkal"}, {"start": 19.279, "duration": 3.801, "text": "groups are weekly electron donating and"}, {"start": 21.56, "duration": 3.12, "text": "that's the mechanism by which they"}, {"start": 23.08, "duration": 3.599, "text": "stabilize"}, {"start": 24.68, "duration": 4.8, "text": "carbocations because reducing or"}, {"start": 26.679, "duration": 6.321, "text": "delocalizing charges in small Organics"}, {"start": 29.48, "duration": 6.12, "text": "gives those ions greater stability so we"}, {"start": 33.0, "duration": 4.079, "text": "expect methyl carbocations to be very"}, {"start": 35.6, "duration": 4.639, "text": "rare because they're so"}, {"start": 37.079, "duration": 6.32, "text": "unstable primary is not much better but"}, {"start": 40.239, "duration": 5.921, "text": "when we get on to secondary and tertiary"}, {"start": 43.399, "duration": 5.201, "text": "carbocations we see these constantly in"}, {"start": 46.16, "duration": 4.919, "text": "organic chemistry these are Staples of"}, {"start": 48.6, "duration": 4.56, "text": "first order substitution and elimination"}, {"start": 51.079, "duration": 3.681, "text": "reactions and the reason is that those"}, {"start": 53.16, "duration": 3.719, "text": "alkal groups are increasing the"}, {"start": 54.76, "duration": 4.16, "text": "stability of the"}, {"start": 56.879, "duration": 4.16, "text": "carbocation so let's take a look at what"}, {"start": 58.92, "duration": 4.639, "text": "it is about these R groups that allows"}, {"start": 61.039, "duration": 7.041, "text": "them to donate electrons or to put"}, {"start": 63.559, "duration": 4.521, "text": "electrons close to that carbocation"}, {"start": 69.68, "duration": 5.52, "text": "Center so here I've drawn for you the"}, {"start": 71.88, "duration": 5.48, "text": "methyl cation and the ethyl cation and"}, {"start": 75.2, "duration": 4.44, "text": "these two carbocations you're not really"}, {"start": 77.36, "duration": 3.92, "text": "going to see in a genuine Bonafide"}, {"start": 79.64, "duration": 4.0, "text": "reaction because they're just not stable"}, {"start": 81.28, "duration": 3.879, "text": "enough but they're the simplest examples"}, {"start": 83.64, "duration": 3.799, "text": "that will allow me to demonstrate how"}, {"start": 85.159, "duration": 3.721, "text": "hyper conjugation works so let's set"}, {"start": 87.439, "duration": 4.04, "text": "aside the fact that these are"}, {"start": 88.88, "duration": 4.4, "text": "unreasonably stable and just take a look"}, {"start": 91.479, "duration": 3.96, "text": "at them right now"}, {"start": 93.28, "duration": 4.28, "text": "anyway so I'm going to turn them into"}, {"start": 95.439, "duration": 3.121, "text": "three-dimensional representations and"}, {"start": 97.56, "duration": 3.04, "text": "then I'm going to turn these"}, {"start": 98.56, "duration": 4.64, "text": "representations on their"}, {"start": 100.6, "duration": 5.4, "text": "side and when I do this I can place the"}, {"start": 103.2, "duration": 5.8, "text": "unoccupied p orbital of each carbocation"}, {"start": 106.0, "duration": 5.64, "text": "Center vertically in the plane of our"}, {"start": 109.0, "duration": 5.439, "text": "page now what's different between the"}, {"start": 111.64, "duration": 6.36, "text": "ethyl and methyl cations is that the"}, {"start": 114.439, "duration": 6.121, "text": "ethyl cation has a neighboring CH Sigma"}, {"start": 118.0, "duration": 5.119, "text": "bonding molecular orbital and this"}, {"start": 120.56, "duration": 4.32, "text": "bonding molecular orbital is actually"}, {"start": 123.119, "duration": 3.441, "text": "filled with electrons that's how it's"}, {"start": 124.88, "duration": 3.76, "text": "holding on to the"}, {"start": 126.56, "duration": 4.0, "text": "hydrogen but we also know that these"}, {"start": 128.64, "duration": 4.0, "text": "electrons though I've drawn them here in"}, {"start": 130.56, "duration": 4.08, "text": "between the two bonded atoms we know"}, {"start": 132.64, "duration": 4.44, "text": "that they actually have free run of that"}, {"start": 134.64, "duration": 3.52, "text": "entire volume that is defined by the"}, {"start": 137.08, "duration": 3.439, "text": "molecular"}, {"start": 138.16, "duration": 6.159, "text": "orbital so they can run all around up"}, {"start": 140.519, "duration": 7.241, "text": "and down and as you notice occasionally"}, {"start": 144.319, "duration": 6.241, "text": "they find their way over to the empty p"}, {"start": 147.76, "duration": 4.68, "text": "orbital on the carbocation"}, {"start": 150.56, "duration": 4.84, "text": "and when this happens they slightly"}, {"start": 152.44, "duration": 5.28, "text": "attenuate the positive charge thereby"}, {"start": 155.4, "duration": 5.04, "text": "decreasing it and remember what we said"}, {"start": 157.72, "duration": 5.32, "text": "previously anything that decreases or"}, {"start": 160.44, "duration": 5.879, "text": "delocalizes charge in small organic"}, {"start": 163.04, "duration": 5.88, "text": "molecules or ions is going to stabilize"}, {"start": 166.319, "duration": 5.481, "text": "them so this is why having additional"}, {"start": 168.92, "duration": 5.239, "text": "alkal groups around a carbocation center"}, {"start": 171.8, "duration": 3.799, "text": "makes them more stable and therefore"}, {"start": 174.159, "duration": 3.521, "text": "more likely to"}, {"start": 175.599, "duration": 4.801, "text": "form so next time we'll talk a little"}, {"start": 177.68, "duration": 4.72, "text": "bit about how this hyper conjugate ation"}, {"start": 180.4, "duration": 4.68, "text": "and the stabilizing effect of alkal"}, {"start": 182.4, "duration": 5.44, "text": "groups on carbocations leads to things"}, {"start": 185.08, "duration": 6.12, "text": "like hydride and methyl shifts and I'll"}, {"start": 187.84, "duration": 3.36, "text": "see you then"}], "title": "Hyperconjugation Explained", "description": "More substituted carbocations tend to be more stable.  Here, Professor Davis explains the role of hyperconjugation in the process of stabilizing this critical class of organic ions.  Find more videos like this one at www.youtube.com/chemsurvival", "lengthSeconds": 200, "width": 480, "height": 360}
{"video_id": "JzOouZHde2Q", "transcript": [{"start": 0.36, "duration": 2.64, "text": "It\u2019s Professor Dave, let\u2019s check out Legionnaires\u2019 Disease."}, {"start": 9.82, "duration": 6.5, "text": "Legionnaires\u2019 disease, also called Legionellosis,\nis a severe, often lethal form of pneumonia,"}, {"start": 16.33, "duration": 3.03, "text": "which just means lung inflammation."}, {"start": 19.36, "duration": 5.29, "text": "Symptoms may include cough, muscle aches,\ndiarrhea, fever, and chills."}, {"start": 24.65, "duration": 4.941, "text": "In the U.S., somewhere between 10,000 and\n18,000 people get infected each year, some"}, {"start": 29.591, "duration": 3.069, "text": "of which end up in intensive care units."}, {"start": 32.66, "duration": 6.1, "text": "We first learned about this disease in 1976,\nwhen a mysterious outbreak of pneumonia occurred"}, {"start": 38.76, "duration": 4.54, "text": "at an American Legion convention in Philadelphia,\nhence the name."}, {"start": 43.3, "duration": 4.349, "text": "Because of the fatal nature of some of the\ncases, health officials launched an intensive"}, {"start": 47.649, "duration": 6.241, "text": "investigation that resulted in the definition\nof a new family of pathogenic bacteria, the"}, {"start": 53.89, "duration": 1.74, "text": "Legionellaceae."}, {"start": 55.63, "duration": 7.41, "text": "The family Legionellaceae consists of a single\ngenus, Legionella, with 48 species and 70"}, {"start": 63.04, "duration": 1.0, "text": "serogroups."}, {"start": 64.04, "duration": 5.71, "text": "These bacteria are slender gram-negative rods\nthat are pleomorphic, which means they can"}, {"start": 69.75, "duration": 5.15, "text": "change shape in response to their environment."}, {"start": 74.9, "duration": 6.08, "text": "Infection with Legionella pneumophila has\nthree possible outcomes: One is no symptoms at all."}, {"start": 81.56, "duration": 4.88, "text": "Two, an influenza-like illness called Pontiac\nfever."}, {"start": 86.44, "duration": 5.41, "text": "And three, legionnaires\u2019 disease, a severe\nform of pneumonia."}, {"start": 91.85, "duration": 5.39, "text": "Legionella is naturally found in freshwater\nlakes and streams, but can be particularly"}, {"start": 97.24, "duration": 6.11, "text": "dangerous to humans when it spreads in man-made\nbuilding water systems like cooling towers,"}, {"start": 103.35, "duration": 4.24, "text": "plumbing systems, hot water heaters, or decorative\nfountains."}, {"start": 107.59, "duration": 5.65, "text": "In particular, these bacteria thrive in the\nmist created by air-conditioning ducts, and"}, {"start": 113.24, "duration": 6.949, "text": "can infect an entire building of people, like\nthat fateful summer of 1976 in Philadelphia."}, {"start": 120.189, "duration": 6.22, "text": "A person becomes infected by breathing in\nsmall droplets of water that contain Legionella,"}, {"start": 126.409, "duration": 3.1, "text": "and typically can\u2019t spread the disease to\nother people."}, {"start": 129.509, "duration": 5.581, "text": "Legionnaires\u2019 disease most often develops\nin people with an already increased likelihood"}, {"start": 135.09, "duration": 4.77, "text": "of developing respiratory sickness, such as\nthose who smoke cigarettes or those with a"}, {"start": 139.86, "duration": 2.9, "text": "compromised immune system."}, {"start": 142.76, "duration": 5.36, "text": "The Legionella, if able to establish an infection,\nreplicate within cells of the immune system."}, {"start": 148.12, "duration": 6.92, "text": "After an incubation period of two to ten days,\nan infection of L. pneumophila can cause acute"}, {"start": 155.04, "duration": 6.91, "text": "illness, which can escalate to multi-organ\ndisease of the gastrointestinal tract, liver,"}, {"start": 161.95, "duration": 3.48, "text": "central nervous system, or the kidneys."}, {"start": 165.43, "duration": 5.66, "text": "To diagnose a case of Legionnaires\u2019 disease,\nclinicians typically take a chest x-ray, and"}, {"start": 171.09, "duration": 5.44, "text": "might conduct a urine test or saliva test\nto confirm the presence of the bacteria."}, {"start": 176.53, "duration": 5.51, "text": "Typically, antibiotics can clear an infection\nof legionnaires\u2019 disease, but approximately"}, {"start": 182.04, "duration": 5.53, "text": "10% of people who get sick with the disease\nwill die due to complications."}, {"start": 187.57, "duration": 5.59, "text": "The best way to prevent Legionnaires\u2019 disease\nis to properly maintain building water systems,"}, {"start": 193.16, "duration": 1.69, "text": "hot tubs, and fountains."}, {"start": 194.85, "duration": 5.57, "text": "So if you have many hot tubs and fountains\nin your mansion, be sure to take precautions now."}], "title": "Legionnaires\u2019 Disease: Legionella pneumophila", "description": "One day in 1976, there was a terrible outbreak of an unknown disease at an American Legion convention in Philadelphia. What was the pathogen responsible for this so-called Legionnaires' disease? Let's find out!\n\nScript by Kellie Vinal\n\nWatch the whole Microbiology playlist: http://bit.ly/ProfDaveMicrobio\n\nGeneral Chemistry Tutorials: http://bit.ly/ProfDaveGenChem\nOrganic Chemistry Tutorials: http://bit.ly/ProfDaveOrgChem\nBiochemistry Tutorials: http://bit.ly/ProfDaveBiochem\nBiology/Genetics Tutorials: http://bit.ly/ProfDaveBio\nAnatomy & Physiology Tutorials: http://bit.ly/ProfDaveAnatPhys\nBiopsychology Tutorials: http://bit.ly/ProfDaveBiopsych\nImmunology Tutorials: http://bit.ly/ProfDaveImmuno\nHistory of Drugs Videos: http://bit.ly/ProfDaveHistoryDrugs\n\nEMAIL\u25ba ProfessorDaveExplains@gmail.com\nPATREON\u25ba http://patreon.com/ProfessorDaveExplains\n\nCheck out \"Is This Wi-Fi Organic?\", my book on disarming pseudoscience!\nAmazon: https://amzn.to/2HtNpVH\nBookshop: https://bit.ly/39cKADM\n\rBarnes and Noble: https://bit.ly/3pUjmrn \nBook Depository: http://bit.ly/3aOVDlT", "lengthSeconds": 231, "width": 640, "height": 360}
{"video_id": "ZGmGapKNDQ8", "transcript": [{"start": 0.0, "duration": 4.34, "text": "[Music]"}, {"start": 1.8, "duration": 2.54, "text": "button"}, {"start": 4.75, "duration": 6.95, "text": "[Music]"}, {"start": 9.32, "duration": 4.359, "text": "Professor Dave and Chegg here an"}, {"start": 11.7, "duration": 4.019, "text": "interesting aspect of organic molecules"}, {"start": 13.679, "duration": 4.801, "text": "is that they can join together to form"}, {"start": 15.719, "duration": 5.521, "text": "polymers these are long chain-like"}, {"start": 18.48, "duration": 5.52, "text": "molecules consisting of many repeats of"}, {"start": 21.24, "duration": 5.22, "text": "a small subunit called a monomer let's"}, {"start": 24.0, "duration": 4.859, "text": "learn about these now the best way to"}, {"start": 26.46, "duration": 4.44, "text": "learn about polymers is by examining the"}, {"start": 28.859, "duration": 3.961, "text": "different types of polymers so that we"}, {"start": 30.9, "duration": 4.8, "text": "can see the monomers they are comprised"}, {"start": 32.82, "duration": 6.0, "text": "of and how they link together ethene can"}, {"start": 35.7, "duration": 5.879, "text": "polymerize to form polyethylene this is"}, {"start": 38.82, "duration": 5.22, "text": "just thousands of fully saturated ch2"}, {"start": 41.579, "duration": 5.521, "text": "units perpetuating like a never-ending"}, {"start": 44.04, "duration": 4.62, "text": "straight chain alkane this substance is"}, {"start": 47.1, "duration": 4.32, "text": "a flexible plastic with many"}, {"start": 48.66, "duration": 5.399, "text": "applications if we replace the hydrogen"}, {"start": 51.42, "duration": 4.979, "text": "atoms on ethene with fluorine atoms and"}, {"start": 54.059, "duration": 5.16, "text": "then allow this to polymerize we get a"}, {"start": 56.399, "duration": 5.16, "text": "material called Teflon this is what"}, {"start": 59.219, "duration": 4.741, "text": "gives certain containers their non-stick"}, {"start": 61.559, "duration": 3.42, "text": "coating because the CF bonds are quite"}, {"start": 63.96, "duration": 3.36, "text": "inert"}, {"start": 64.979, "duration": 4.201, "text": "they are almost entirely unreactive"}, {"start": 67.32, "duration": 4.619, "text": "making the material tough and"}, {"start": 69.18, "duration": 4.979, "text": "non-flammable here is a table depicting"}, {"start": 71.939, "duration": 4.021, "text": "many other synthetic polymers according"}, {"start": 74.159, "duration": 4.32, "text": "to the monomers they are comprised of"}, {"start": 75.96, "duration": 4.26, "text": "some of them may sound familiar and"}, {"start": 78.479, "duration": 3.541, "text": "their applications will certainly be"}, {"start": 80.22, "duration": 3.079, "text": "familiar as these materials are"}, {"start": 82.02, "duration": 3.959, "text": "ubiquitous"}, {"start": 83.299, "duration": 5.561, "text": "polymerization can occur by a variety of"}, {"start": 85.979, "duration": 4.741, "text": "mechanisms sometimes they go by addition"}, {"start": 88.86, "duration": 4.02, "text": "polymerization which is a radical"}, {"start": 90.72, "duration": 4.079, "text": "mechanism meaning that it employs"}, {"start": 92.88, "duration": 4.919, "text": "radical intermediates which have"}, {"start": 94.799, "duration": 5.82, "text": "unpaired electrons another mechanism is"}, {"start": 97.799, "duration": 5.101, "text": "condensation polymerization where each"}, {"start": 100.619, "duration": 6.0, "text": "time two monomers are joint a water"}, {"start": 102.9, "duration": 5.52, "text": "molecule is lost a homopolymer is one in"}, {"start": 106.619, "duration": 4.86, "text": "which there is only one repeating"}, {"start": 108.42, "duration": 4.86, "text": "monomeric Unit A copolymer is one in"}, {"start": 111.479, "duration": 4.68, "text": "which there are two different monomers"}, {"start": 113.28, "duration": 6.619, "text": "that must combine to form the chain"}, {"start": 116.159, "duration": 5.701, "text": "nylon is an example of a copolymer"}, {"start": 119.899, "duration": 4.061, "text": "polyethylenes are some of the most"}, {"start": 121.86, "duration": 6.0, "text": "important polymers and there are two"}, {"start": 123.96, "duration": 7.44, "text": "forms low density polyethylene or ldpe"}, {"start": 127.86, "duration": 6.36, "text": "and high density polyethylene or HDPE"}, {"start": 131.4, "duration": 5.16, "text": "the low density variety is as such"}, {"start": 134.22, "duration": 4.32, "text": "because of the extensive branching which"}, {"start": 136.56, "duration": 4.56, "text": "causes the molecule to not be able to"}, {"start": 138.54, "duration": 4.68, "text": "pack together as tightly as for the high"}, {"start": 141.12, "duration": 3.54, "text": "density variety which is comprised"}, {"start": 143.22, "duration": 4.379, "text": "predominantly of straight chain"}, {"start": 144.66, "duration": 5.34, "text": "molecules with these compounds if a"}, {"start": 147.599, "duration": 5.161, "text": "substituent is present like this methyl"}, {"start": 150.0, "duration": 5.28, "text": "group this can result in an isotactic"}, {"start": 152.76, "duration": 4.92, "text": "chain where the methyl groups are all on"}, {"start": 155.28, "duration": 4.98, "text": "the same side of the molecule or an"}, {"start": 157.68, "duration": 4.74, "text": "atactic chain where they alternate from"}, {"start": 160.26, "duration": 4.979, "text": "one side to the other as is depicted"}, {"start": 162.42, "duration": 5.28, "text": "with these Dash and wedge bonds if the"}, {"start": 165.239, "duration": 5.701, "text": "substituent is a chlorine atom we get"}, {"start": 167.7, "duration": 5.34, "text": "polyvinyl chloride or PVC if the"}, {"start": 170.94, "duration": 5.04, "text": "substituent is a phenyl group meaning a"}, {"start": 173.04, "duration": 5.279, "text": "Benzene ring we get polystyrene if"}, {"start": 175.98, "duration": 4.86, "text": "styrene monomers are combined with some"}, {"start": 178.319, "duration": 4.681, "text": "other monomer to make a copolymer we can"}, {"start": 180.84, "duration": 5.66, "text": "get a variety of other useful polymers"}, {"start": 183.0, "duration": 3.5, "text": "with various applications"}, {"start": 187.56, "duration": 4.5, "text": "with that we have taken a glimpse into"}, {"start": 189.72, "duration": 4.14, "text": "the Incredible World of polymers which"}, {"start": 192.06, "duration": 4.5, "text": "have transformed the way we make"}, {"start": 193.86, "duration": 4.58, "text": "industrial materials Professor Dave for"}, {"start": 196.56, "duration": 4.679, "text": "Chegg see you next time"}, {"start": 198.44, "duration": 4.98, "text": "all right"}, {"start": 201.239, "duration": 2.181, "text": "foreign"}], "title": "Introduction to Polymers | Professor Dave & Chegg Explain", "description": "In this video, @ProfessorDaveExplains provides a comprehensive introduction to polymers. Polymers are made up of monomers linked together in chains. He explains the different types of polymers, the mechanisms behind polymerization, and provides examples of polyethylenes such as low-density and high-density polymers.\n\nGet more homework help from Chegg at https://che.gg/3HbtG8Y\nWatch the whole General Chemistry playlist: https://www.youtube.com/playlist?list=PLllCkartRRdPQdEE-oyt5Di31wIwJc5S6\n\nAbout Chegg:\nChegg is a leading education technology company that provides students with affordable and accessible study resources. Chegg's mission is to help students save time, save money and get smarter.  \n\nSubscribe: https://www.youtube.com/@chegg\nTikTok: https://www.tiktok.com/@chegg\nInstagram: https://www.instagram.com/chegg/\nTwitter: https://twitter.com/chegg\n\n00:00 Intro\n00:27 Types of polymers\n01:28 Polymerization mechanisms\n02:00 Polyethylenes", "lengthSeconds": 208, "width": 640, "height": 360}
{"video_id": "bwzzD6UGPr4", "transcript": [{"start": 6.46, "duration": 3.94, "text": "ok so for today's problem we've got this\ncompound here with a carbonyl and a"}, {"start": 10.41, "duration": 4.059, "text": "methyl group and our target molecule has\na methyl and a bromo group and what we"}, {"start": 14.469, "duration": 4.98, "text": "want to be able to do is figure out how\nto get from here to here in two steps, give it a shot."}, {"start": 34.94, "duration": 2.58, "text": "as we're analyzing this, let's take a look at the molecule"}, {"start": 37.53, "duration": 3.02, "text": "we're realizing that we have to lose oxygen, we\nhave to figure out a way to get that"}, {"start": 40.55, "duration": 3.71, "text": "oxygen atom off of there, and we also have to\nfigure out a way to get a bromine atom"}, {"start": 44.26, "duration": 3.299, "text": "on there, and so we have two steps to\nplay with, either one is gonna get rid of"}, {"start": 47.559, "duration": 3.761, "text": "the oxygen one is gonna add the bromine\nor somehow both can happen at once but"}, {"start": 51.32, "duration": 3.48, "text": "what we definitely want to do is get rid of\nthat oxygen, so we don't really know of a"}, {"start": 54.8, "duration": 5.349, "text": "way to remove a carbonyl oxygen so why\ndon't we reduce it. let's take sodium"}, {"start": 60.149, "duration": 4.25, "text": "borohydride, just a nice soft reducing\nagent, and let's as step 1 reduce the"}, {"start": 64.399, "duration": 4.591, "text": "carbonyl. we're gonna get an alcohol. now\nthat we have an alcohol we have some"}, {"start": 68.99, "duration": 3.82, "text": "different possibilities to think about.\nnow one possibility that we might wanna"}, {"start": 72.81, "duration": 5.29, "text": "use is some strong acid to\nprotonate that hydroxyl group, get it"}, {"start": 78.1, "duration": 4.18, "text": "to leave as a water molecule, but then what we want to"}, {"start": 82.28, "duration": 4.769, "text": "do is choose a strong acid selectively that\nmight perform the other transformation"}, {"start": 87.049, "duration": 5.221, "text": "that we also want to occur, so why don't we\nuse hydrobromic acid and see what happens."}, {"start": 92.27, "duration": 5.69, "text": "so once again the first step was that\nwe had some hydride and that went ahead"}, {"start": 97.96, "duration": 5.35, "text": "and reduced it so here is our alcohol,\nnow if we use hydrobromic acid"}, {"start": 103.31, "duration": 8.04, "text": "we're gonna get that right there, so the\nhydroxyl has been protonated, now water"}, {"start": 111.35, "duration": 3.67, "text": "is a good leaving group, we are in acidic\nconditions, so carbocations are going to be"}, {"start": 115.02, "duration": 5.099, "text": "fine, so that can go ahead and leave. now\nthe interesting thing here is if we have"}, {"start": 120.119, "duration": 3.851, "text": "a secondary carbocation, and remember\nthat a secondary carbocation that is"}, {"start": 123.97, "duration": 4.539, "text": "adjacent to a tertiary or quaternary\ncarbon, you're probably gonna get some"}, {"start": 128.509, "duration": 5.561, "text": "kind of hydride or or methanide shift.\nand so what's happening here is we've"}, {"start": 134.07, "duration": 7.74, "text": "got this implied hydrogen and what we\ncan do is it'll spontaneously just do a"}, {"start": 141.81, "duration": 4.45, "text": "hydride shift and so if this hydride\nmoves over there that's going to"}, {"start": 146.26, "duration": 4.03, "text": "neutralize that carbocation, it's gonna leave a\ncarbocation at the tertiary position. once"}, {"start": 150.29, "duration": 5.69, "text": "again this happens because the tertiary carbocation is\nmore stable than a secondary carbocation, so that"}, {"start": 155.98, "duration": 4.05, "text": "is going to happen spontaneously and\npretty much instantly. and now since we"}, {"start": 160.03, "duration": 4.76, "text": "used hydrobromic acid, when this\ndeprotonated, its gonna leave a lot of"}, {"start": 164.79, "duration": 3.509, "text": "bromide swimming around in solution and\nso that's going to kill two birds with"}, {"start": 168.299, "duration": 4.571, "text": "one stone because if there's a lot of\nbromide and one finds this carbocation"}, {"start": 172.87, "duration": 5.26, "text": "it is absolutely going to react and\ngenerate the intended product. so because"}, {"start": 178.13, "duration": 4.23, "text": "we were clever in the selection of acid,\nwe've been able to induce this in two steps."}, {"start": 182.36, "duration": 5.0, "text": "the first thing we did was reduce that\nketone to the alcohol and then because"}, {"start": 187.36, "duration": 5.56, "text": "we used HBr we are able to get that\nwater to leave, allow this carbocation"}, {"start": 192.92, "duration": 4.27, "text": "rearrangement to occur spontaneously and\nthen the free bromide ion swimming"}, {"start": 197.19, "duration": 4.09, "text": "around in solution finds the carbocation and there we go, we get our product"}, {"start": 223.46, "duration": 3.419, "text": "thanks for watching guys, subscribe to my\nchannel for more tutorials and as always"}, {"start": 226.879, "duration": 1.12, "text": "feel free to email me"}], "title": "Practice Problem: Synthetic Strategy", "description": "We've got starting material and we've got a target molecule, and we've gotta figure out how to make the transformation in just two steps. Sift through that bag of synthetic tricks!\n\nTry all of the organic chemistry practice problems: http://bit.ly/ProfDaveOCPP\n\nOrganic Chemistry Tutorials: http://bit.ly/ProfDaveOrgChem\n\nEMAIL\u25ba ProfessorDaveExplains@gmail.com\nPATREON\u25ba http://patreon.com/ProfessorDaveExplains\n\nCheck out \"Is This Wi-Fi Organic?\", my book on disarming pseudoscience!\nAmazon: https://amzn.to/2HtNpVH\nBookshop: https://bit.ly/39cKADM\n\rBarnes and Noble: https://bit.ly/3pUjmrn \nBook Depository: http://bit.ly/3aOVDlT", "lengthSeconds": 231, "width": 640, "height": 360}
{"video_id": "xweYA-IJTqs", "transcript": [{"start": 13.309, "duration": 5.351, "text": "the sodium potassium pump is an active"}, {"start": 16.53, "duration": 4.77, "text": "transport mechanism that is driven by"}, {"start": 18.66, "duration": 5.04, "text": "the breakdown of ATP and works through a"}, {"start": 21.3, "duration": 6.39, "text": "series of conformational changes in a"}, {"start": 23.7, "duration": 6.27, "text": "transmembrane protein 3 sodium ions bind"}, {"start": 27.69, "duration": 4.499, "text": "to the cytoplasmic side of the protein"}, {"start": 29.97, "duration": 4.53, "text": "causing the protein to change its"}, {"start": 32.189, "duration": 4.651, "text": "conformation in its new conformation the"}, {"start": 34.5, "duration": 4.739, "text": "molecule becomes phosphorylated at the"}, {"start": 36.84, "duration": 4.469, "text": "expense of a molecule of ATP the"}, {"start": 39.239, "duration": 4.5, "text": "phosphorylation induces a second"}, {"start": 41.309, "duration": 4.291, "text": "conformational change the translocates"}, {"start": 43.739, "duration": 4.17, "text": "the three sodium ions across the"}, {"start": 45.6, "duration": 4.68, "text": "membrane in this new conformation the"}, {"start": 47.909, "duration": 4.801, "text": "protein has a low affinity for sodium"}, {"start": 50.28, "duration": 4.89, "text": "ions and the three bound sodium ions"}, {"start": 52.71, "duration": 4.5, "text": "dissociate from the protein and diffuse"}, {"start": 55.17, "duration": 3.959, "text": "into the extracellular fluid the new"}, {"start": 57.21, "duration": 4.649, "text": "conformation has a high affinity for"}, {"start": 59.129, "duration": 4.951, "text": "potassium ions two of which bind to the"}, {"start": 61.859, "duration": 4.8, "text": "extracellular side of the protein the"}, {"start": 64.08, "duration": 4.01, "text": "bound phosphate now dissociates and the"}, {"start": 66.659, "duration": 3.991, "text": "protein reverts to its original"}, {"start": 68.09, "duration": 5.559, "text": "conformation exposing the two potassium"}, {"start": 70.65, "duration": 5.069, "text": "ions to the cytoplasm on the inside of"}, {"start": 73.649, "duration": 4.711, "text": "the cell this conformation has a low"}, {"start": 75.719, "duration": 5.25, "text": "affinity for potassium ions so the two"}, {"start": 78.36, "duration": 5.039, "text": "bound potassium ions dissociate from the"}, {"start": 80.969, "duration": 5.031, "text": "protein and diffuse into the interior of"}, {"start": 83.399, "duration": 2.601, "text": "the cell"}], "title": "Sodium potassium pump animation", "description": "\ud83d\udccc \ud835\udc05\ud835\udc28\ud835\udc25\ud835\udc25\ud835\udc28\ud835\udc30 \ud835\udc28\ud835\udc27 \ud835\udc08\ud835\udc27\ud835\udc2c\ud835\udc2d\ud835\udc1a\ud835\udc20\ud835\udc2b\ud835\udc1a\ud835\udc26:- https://www.instagram.com/drgbhanuprakash\n\ud83d\udccc\ud835\udddd\ud835\uddfc\ud835\uddf6\ud835\uddfb \ud835\udde2\ud835\ude02\ud835\uddff \ud835\udde7\ud835\uddf2\ud835\uddf9\ud835\uddf2\ud835\uddf4\ud835\uddff\ud835\uddee\ud835\uddfa \ud835\uddd6\ud835\uddf5\ud835\uddee\ud835\uddfb\ud835\uddfb\ud835\uddf2\ud835\uddf9 \ud835\udddb\ud835\uddf2\ud835\uddff\ud835\uddf2:- https://t.me/bhanuprakashdr\n\ud83d\udccc\ud835\udde6\ud835\ude02\ud835\uddef\ud835\ude00\ud835\uddf0\ud835\uddff\ud835\uddf6\ud835\uddef\ud835\uddf2 \ud835\udde7\ud835\uddfc \ud835\udde0\ud835\ude06 \ud835\udde0\ud835\uddee\ud835\uddf6\ud835\uddf9\ud835\uddf6\ud835\uddfb\ud835\uddf4 \ud835\udddf\ud835\uddf6\ud835\ude00\ud835\ude01:- https://linktr.ee/DrGBhanuprakash\n\nSODIUM-POTASSIUM PUMP ANIMATED LECTURE\n\nThe #Sodium-PotassiumPump. The process of moving sodium and potassium ions across the cell membrance is an active transport process involving the hydrolysis of ATP to provide the necessary energy. It involves an enzyme referred to as Na+/K+-ATPase.\n \nNa+-K+-ATPase (Sodium potassium ATPase)\n------------------------------------------------------------------------\nA transmembrane protein pump that actively transports three Na+ ions out of the cell for every two K+ ions that it moves into the cell. Uses ATP for energy. The ATP binding site is on the cytosolic side. Na+-K+-ATPase maintains the concentration gradients of Na+ and K+ and is thus responsible for the resting membrane potential of all cells. The pump is phosporylated when it transports sodium and dephosphorylated when it transports potassium.\n\n#sodiumpotassiumpump #sodiumpotassiumpumpanimation #sodiumpotassiumpumpphysiology #activetransport #usmlevideos #membranephysiology #usmlestep1videos #neetpg #mbbs #fmge #drgbhanuprakash #physiologyvideos #animatedmedicalvideos", "lengthSeconds": 99, "width": 640, "height": 360}
{"video_id": "ANQXarFkWyE", "transcript": [{"start": 0.0, "duration": 9.38, "text": "[Music]"}, {"start": 9.88, "duration": 4.959, "text": "Professor Dave and CH here we know that"}, {"start": 12.04, "duration": 4.6, "text": "alkanes are fully saturated hydrocarbons"}, {"start": 14.839, "duration": 4.041, "text": "meaning they possess the maximum"}, {"start": 16.64, "duration": 4.96, "text": "hydrogen content possible so every"}, {"start": 18.88, "duration": 5.12, "text": "carbon has as many bonds to hydrogen as"}, {"start": 21.6, "duration": 4.4, "text": "possible but some hydrocarbons are"}, {"start": 24.0, "duration": 4.48, "text": "unsaturated so let's learn about the"}, {"start": 26.0, "duration": 5.96, "text": "various degrees of unsaturation that are"}, {"start": 28.48, "duration": 5.72, "text": "possible if sat saturation means maximum"}, {"start": 31.96, "duration": 5.08, "text": "possible hydrogen content then"}, {"start": 34.2, "duration": 5.48, "text": "unsaturation means less than the maximum"}, {"start": 37.04, "duration": 4.72, "text": "possible hydrogen content in other words"}, {"start": 39.68, "duration": 4.719, "text": "carbon atoms are participating in"}, {"start": 41.76, "duration": 4.959, "text": "additional carboncarbon bonds or bonds"}, {"start": 44.399, "duration": 6.081, "text": "to other elements instead of bonding to"}, {"start": 46.719, "duration": 6.241, "text": "hydrogen so ethane c2h6 is fully"}, {"start": 50.48, "duration": 4.559, "text": "saturated because the two carbon atoms"}, {"start": 52.96, "duration": 6.559, "text": "can't have any more bonds to hydrogen"}, {"start": 55.039, "duration": 7.081, "text": "atoms ethine or ethylene c2h4 has one"}, {"start": 59.519, "duration": 5.241, "text": "degree of unsaturation due to the carbon"}, {"start": 62.12, "duration": 4.999, "text": "carbon Pi bond this Pi bond is"}, {"start": 64.76, "duration": 4.679, "text": "preventing two additional hydrogen atoms"}, {"start": 67.119, "duration": 4.401, "text": "from coordinating we can say the same"}, {"start": 69.439, "duration": 5.921, "text": "thing about straight chain butane or"}, {"start": 71.52, "duration": 6.8, "text": "c4h10 versus cyob butane or"}, {"start": 75.36, "duration": 5.16, "text": "c4h8 because the molecule closes in on"}, {"start": 78.32, "duration": 4.36, "text": "itself to form a ring the two end"}, {"start": 80.52, "duration": 4.36, "text": "carbons must have a bond between them"}, {"start": 82.68, "duration": 5.36, "text": "and this additional Bond results in a"}, {"start": 84.88, "duration": 6.36, "text": "degree of unsaturation in general every"}, {"start": 88.04, "duration": 5.96, "text": "Pi Bond and ring in a molecule adds"}, {"start": 91.24, "duration": 5.64, "text": "another degree of unsaturation which"}, {"start": 94.0, "duration": 8.04, "text": "will result in two hydrogens lost from"}, {"start": 96.88, "duration": 8.0, "text": "the maximum possible given by cnh h2n +2"}, {"start": 102.04, "duration": 4.88, "text": "which describes alkanes knowing the"}, {"start": 104.88, "duration": 4.239, "text": "degree of unsaturation can help us"}, {"start": 106.92, "duration": 3.799, "text": "predict molecular structure let's say"}, {"start": 109.119, "duration": 4.161, "text": "that we know the molecular mass of a"}, {"start": 110.719, "duration": 4.76, "text": "hydrocarbon as being 82 atomic mass"}, {"start": 113.28, "duration": 6.0, "text": "units which corresponds with a formula"}, {"start": 115.479, "duration": 5.68, "text": "of C6 h10 if a six carbon molecule were"}, {"start": 119.28, "duration": 5.68, "text": "fully Satur saturated it would have the"}, {"start": 121.159, "duration": 7.24, "text": "formula C6 h14 since there are only 10"}, {"start": 124.96, "duration": 5.519, "text": "hydrogens this molecule must have 2\u00b0 of"}, {"start": 128.399, "duration": 4.361, "text": "unsaturation this does not tell us"}, {"start": 130.479, "duration": 4.361, "text": "precisely what the molecule is but we"}, {"start": 132.76, "duration": 5.839, "text": "know that the molecule must either have"}, {"start": 134.84, "duration": 6.56, "text": "two piie bonds two rings or one Pi Bond"}, {"start": 138.599, "duration": 5.64, "text": "and one ring so it could be"}, {"start": 141.4, "duration": 5.16, "text": "cyclohexene it could be hexine or many"}, {"start": 144.239, "duration": 4.601, "text": "other possibilities degrees of"}, {"start": 146.56, "duration": 4.679, "text": "unsaturation can extend to compounds"}, {"start": 148.84, "duration": 4.56, "text": "with hetero atoms as well for more"}, {"start": 151.239, "duration": 4.241, "text": "involved examples we will use this"}, {"start": 153.4, "duration": 4.839, "text": "equation instead where the number of"}, {"start": 155.48, "duration": 4.8, "text": "degrees of unsaturation will be equal to"}, {"start": 158.239, "duration": 5.321, "text": "twice the number of carbon atoms in the"}, {"start": 160.28, "duration": 6.08, "text": "molecule plus two plus the number of"}, {"start": 163.56, "duration": 5.759, "text": "nitrogen atoms minus the number of"}, {"start": 166.36, "duration": 5.92, "text": "halogens minus the number of hydrogens"}, {"start": 169.319, "duration": 7.361, "text": "all over two so for example take the"}, {"start": 172.28, "duration": 7.2, "text": "formula C6 h10 NCL how many degrees of"}, {"start": 176.68, "duration": 5.44, "text": "unsaturation must this molecule have"}, {"start": 179.48, "duration": 6.24, "text": "well twice the number of carbons is 12"}, {"start": 182.12, "duration": 7.24, "text": "plus 2 plus the one nitrogen minus the"}, {"start": 185.72, "duration": 6.84, "text": "one hogen minus 10 hydrogens all over"}, {"start": 189.36, "duration": 5.799, "text": "two so that's 4 over 2 which is equal to"}, {"start": 192.56, "duration": 4.679, "text": "two and in fact this could be a molecule"}, {"start": 195.159, "duration": 4.401, "text": "like this and with the one ring and the"}, {"start": 197.239, "duration": 5.401, "text": "one Pi Bond we do indeed see that there"}, {"start": 199.56, "duration": 5.84, "text": "are two degrees of"}, {"start": 202.64, "duration": 5.08, "text": "unsaturation this concept of degrees of"}, {"start": 205.4, "duration": 5.039, "text": "unsaturation will be useful for certain"}, {"start": 207.72, "duration": 4.92, "text": "applications especially spect R roscopy"}, {"start": 210.439, "duration": 4.681, "text": "which we will examine later for now we"}, {"start": 212.64, "duration": 4.84, "text": "must simply know what this term means"}, {"start": 215.12, "duration": 5.039, "text": "and be able to determine molecules which"}, {"start": 217.48, "duration": 4.92, "text": "exhibit a specific number of degrees of"}, {"start": 220.159, "duration": 4.8, "text": "unsaturation Professor Dave for check"}, {"start": 222.4, "duration": 2.559, "text": "see you next"}, {"start": 232.48, "duration": 3.0, "text": "time"}], "title": "How to Calculate Degrees of Unsaturation Ft. Professor Dave", "description": "We know that alkanes are fully saturated hydrocarbons. But not all hydrocarbons are fully saturated. With @ProfessorDaveExplains, let's take a look at the concept of degrees of unsaturation, including how to calculate the degree of unsaturation of a hydrocarbon, using a molecular formula.\n\nGet more homework help from Chegg at https://che.gg/3HbtG8Y\nWatch the whole Organic Chemistry playlist: https://www.youtube.com/playlist?list=PLllCkartRRdMPnATf1X8qVdo5RzM8gA_M\n\nAbout Chegg:\nChegg is a leading education technology company that provides students with affordable and accessible study resources. Chegg's mission is to help students save time, save money and get smarter.  \n\nSubscribe: https://www.youtube.com/@chegg\nTikTok: https://www.tiktok.com/@chegg\nInstagram: https://www.instagram.com/chegg/\nTwitter: https://twitter.com/chegg\n\n00:00 Intro\n00:29 Degrees of unsaturation", "lengthSeconds": 234, "width": 640, "height": 360}
{"video_id": "K53EOjjMt4E", "transcript": [{"start": 0.199, "duration": 4.881, "text": "in this video we'll look at how the Lac"}, {"start": 2.52, "duration": 6.28, "text": "operon functions in"}, {"start": 5.08, "duration": 7.04, "text": "eoli this is the Lac operon here you can"}, {"start": 8.8, "duration": 6.919, "text": "see the three structural genes Lac Z Lac"}, {"start": 12.12, "duration": 4.64, "text": "Y and Lac a Lac Z encodes the protein"}, {"start": 15.719, "duration": 4.001, "text": "beta"}, {"start": 16.76, "duration": 7.24, "text": "galactosidase Lac y encodes the enzyme"}, {"start": 19.72, "duration": 6.16, "text": "permease and Lac a encodes the protein"}, {"start": 24.0, "duration": 3.84, "text": "transacetylase all three of these"}, {"start": 25.88, "duration": 4.159, "text": "proteins work together to help the cell"}, {"start": 27.84, "duration": 4.52, "text": "bring in and break down lactose when"}, {"start": 30.039, "duration": 4.561, "text": "it's available in the environment"}, {"start": 32.36, "duration": 4.12, "text": "lactose is a sugar that the cell can use"}, {"start": 34.6, "duration": 4.799, "text": "for energy and"}, {"start": 36.48, "duration": 5.96, "text": "growth so these genes all share a"}, {"start": 39.399, "duration": 5.441, "text": "promoter and an operator further"}, {"start": 42.44, "duration": 5.08, "text": "Upstream of the Lac operon you see the"}, {"start": 44.84, "duration": 4.76, "text": "regulatory Gene Lac ey which encodes an"}, {"start": 47.52, "duration": 4.24, "text": "active"}, {"start": 49.6, "duration": 5.56, "text": "repressor the cell wants to be able to"}, {"start": 51.76, "duration": 5.4, "text": "break lactose down when it's present so"}, {"start": 55.16, "duration": 3.52, "text": "expression of these genes is necessary"}, {"start": 57.16, "duration": 4.32, "text": "when lactose is present in the"}, {"start": 58.68, "duration": 4.559, "text": "environment when lactose is not present"}, {"start": 61.48, "duration": 3.88, "text": "it would be a waste of the cell's energy"}, {"start": 63.239, "duration": 4.801, "text": "and resources to continue to make these"}, {"start": 65.36, "duration": 4.72, "text": "proteins so transcription of these"}, {"start": 68.04, "duration": 4.439, "text": "structural genes will be"}, {"start": 70.08, "duration": 5.24, "text": "limited so how does the cell alter"}, {"start": 72.479, "duration": 5.081, "text": "structural gene expression when lactose"}, {"start": 75.32, "duration": 5.159, "text": "is absent from the environment the Lac"}, {"start": 77.56, "duration": 5.04, "text": "eye repressor is active it binds to the"}, {"start": 80.479, "duration": 4.921, "text": "operator and prevents RNA polymerase"}, {"start": 82.6, "duration": 5.44, "text": "from binding to the promoter this limits"}, {"start": 85.4, "duration": 5.64, "text": "transcription of the lack genes in this"}, {"start": 88.04, "duration": 5.56, "text": "state the operon is said to be"}, {"start": 91.04, "duration": 4.6, "text": "repressed when lactose is present in the"}, {"start": 93.6, "duration": 3.879, "text": "cell the low concentration of"}, {"start": 95.64, "duration": 4.36, "text": "metabolizing proteins that are always"}, {"start": 97.479, "duration": 6.361, "text": "available start to convert lactose into"}, {"start": 100.0, "duration": 7.32, "text": "Alo lactose this Alo lactose acts as an"}, {"start": 103.84, "duration": 5.599, "text": "inducer binding to the Lac ey repressor"}, {"start": 107.32, "duration": 4.88, "text": "making it"}, {"start": 109.439, "duration": 4.561, "text": "inactive the repressor becomes inactive"}, {"start": 112.2, "duration": 3.919, "text": "because The Binding of alol lactose"}, {"start": 114.0, "duration": 4.64, "text": "causes its DNA binding domain to change"}, {"start": 116.119, "duration": 5.721, "text": "shape so it's unable to attach to the"}, {"start": 118.64, "duration": 3.2, "text": "DNA operator"}, {"start": 122.799, "duration": 4.24, "text": "in this state the repressor cannot block"}, {"start": 125.2, "duration": 4.559, "text": "The Binding of RNA polymerase to the"}, {"start": 127.039, "duration": 5.681, "text": "promoter and therefore transcription of"}, {"start": 129.759, "duration": 4.76, "text": "the structural genes can occur this"}, {"start": 132.72, "duration": 4.2, "text": "means that the lack proteins are"}, {"start": 134.519, "duration": 5.561, "text": "produced and the cell is able to break"}, {"start": 136.92, "duration": 5.599, "text": "down lactose at a higher rate in this"}, {"start": 140.08, "duration": 4.4, "text": "state the operon is said to be induced"}, {"start": 142.519, "duration": 4.761, "text": "by the presence of"}, {"start": 144.48, "duration": 5.44, "text": "lactose the Lac operon is said to be"}, {"start": 147.28, "duration": 4.239, "text": "inducible because lactose must be"}, {"start": 149.92, "duration": 3.319, "text": "present for the structural genes of the"}, {"start": 151.519, "duration": 4.72, "text": "operon to be"}, {"start": 153.239, "duration": 5.08, "text": "expressed that's the Lac operon when"}, {"start": 156.239, "duration": 4.241, "text": "lactose is present the repressor is"}, {"start": 158.319, "duration": 4.441, "text": "inactive and the genes that encode the"}, {"start": 160.48, "duration": 3.52, "text": "proteins that help break down lactose"}, {"start": 162.76, "duration": 4.44, "text": "are"}, {"start": 164.0, "duration": 5.959, "text": "expressed when lactose is absent the"}, {"start": 167.2, "duration": 4.8, "text": "repressor is active and the cell limits"}, {"start": 169.959, "duration": 5.92, "text": "production of those lactose metabolizing"}, {"start": 172.0, "duration": 6.12, "text": "proteins as a way to save energy and"}, {"start": 175.879, "duration": 4.28, "text": "resources if you'd like to delve deeper"}, {"start": 178.12, "duration": 4.8, "text": "into the Lac operon check out out my"}, {"start": 180.159, "duration": 4.881, "text": "Advanced La operon video if you want to"}, {"start": 182.92, "duration": 5.84, "text": "learn about another operon check out my"}, {"start": 185.04, "duration": 3.72, "text": "video on the trip operon"}], "title": "The lac Operon Explained", "description": "Basics of the lac operon, including how lactose presences alters repressor shape to modify lac gene expression.", "lengthSeconds": 187, "width": 480, "height": 360}
{"video_id": "8v-j2kUeEbM", "transcript": [{"start": 0.0, "duration": 4.34, "text": "[Music]"}, {"start": 1.8, "duration": 2.54, "text": "button"}, {"start": 4.75, "duration": 7.07, "text": "[Music]"}, {"start": 9.26, "duration": 4.959, "text": "Professor Dave and Chegg here the main"}, {"start": 11.82, "duration": 4.62, "text": "factor that influences reaction rates is"}, {"start": 14.219, "duration": 4.081, "text": "the concentration of each reactant we"}, {"start": 16.44, "duration": 3.9, "text": "can describe the relationship between a"}, {"start": 18.3, "duration": 4.139, "text": "reaction rate and the concentration of"}, {"start": 20.34, "duration": 4.38, "text": "its reactants by using something called"}, {"start": 22.439, "duration": 4.381, "text": "a rate law it will be important for us"}, {"start": 24.72, "duration": 4.26, "text": "to be able to derive these for any given"}, {"start": 26.82, "duration": 4.199, "text": "system but before we get to that let's"}, {"start": 28.98, "duration": 4.98, "text": "first make sure we understand precisely"}, {"start": 31.019, "duration": 4.801, "text": "what these are a rate law will take this"}, {"start": 33.96, "duration": 4.8, "text": "general form where the rate will be"}, {"start": 35.82, "duration": 4.919, "text": "equal to a rate constant K times the"}, {"start": 38.76, "duration": 4.38, "text": "product of the molar concentrations of"}, {"start": 40.739, "duration": 5.16, "text": "the reactants each raised to a"}, {"start": 43.14, "duration": 5.28, "text": "particular exponent these exponents are"}, {"start": 45.899, "duration": 4.5, "text": "almost always positive integers though"}, {"start": 48.42, "duration": 4.139, "text": "it is technically possible for them to"}, {"start": 50.399, "duration": 4.32, "text": "be fractions or negative numbers and"}, {"start": 52.559, "duration": 4.861, "text": "they are not evident from looking at the"}, {"start": 54.719, "duration": 5.16, "text": "Stoichiometry of the reaction let us"}, {"start": 57.42, "duration": 5.22, "text": "state that one more time for emphasis as"}, {"start": 59.879, "duration": 5.341, "text": "this is an extremely common error among"}, {"start": 62.64, "duration": 5.64, "text": "students the X exponents that we see"}, {"start": 65.22, "duration": 6.3, "text": "here in the rate law are not related to"}, {"start": 68.28, "duration": 5.46, "text": "and not necessarily equivalent to the"}, {"start": 71.52, "duration": 4.62, "text": "stoichiometric coefficients from a"}, {"start": 73.74, "duration": 4.14, "text": "balanced equation instead these"}, {"start": 76.14, "duration": 4.56, "text": "exponents must be determined"}, {"start": 77.88, "duration": 5.22, "text": "experimentally the rate constant is"}, {"start": 80.7, "duration": 4.62, "text": "independent of these concentrations but"}, {"start": 83.1, "duration": 4.68, "text": "it does depend on other parameters like"}, {"start": 85.32, "duration": 4.2, "text": "temperature and surface area the"}, {"start": 87.78, "duration": 4.32, "text": "exponents in the rate law tell us"}, {"start": 89.52, "duration": 4.62, "text": "something called the reaction order for"}, {"start": 92.1, "duration": 5.04, "text": "a reaction with two reactants the rate"}, {"start": 94.14, "duration": 4.86, "text": "law would look like this if m is 1 we"}, {"start": 97.14, "duration": 5.04, "text": "would say the reaction is first order"}, {"start": 99.0, "duration": 6.24, "text": "with respect to a if n is one it is"}, {"start": 102.18, "duration": 5.579, "text": "first order with respect to B if n is 2"}, {"start": 105.24, "duration": 5.4, "text": "it would be second order in B if it was"}, {"start": 107.759, "duration": 5.82, "text": "0 we would say it was Zero order in B"}, {"start": 110.64, "duration": 5.7, "text": "the overall reaction order is simply the"}, {"start": 113.579, "duration": 6.241, "text": "sum of the individual reaction orders so"}, {"start": 116.34, "duration": 6.12, "text": "if M and N are both one the overall"}, {"start": 119.82, "duration": 3.78, "text": "reaction order is two since one plus one"}, {"start": 122.46, "duration": 3.78, "text": "equals two"}, {"start": 123.6, "duration": 5.04, "text": "here are some examples of rate loss with"}, {"start": 126.24, "duration": 4.26, "text": "the reaction orders listed let's make"}, {"start": 128.64, "duration": 4.62, "text": "sure we can write these as they will be"}, {"start": 130.5, "duration": 4.56, "text": "important in our study of kinetics say"}, {"start": 133.26, "duration": 4.979, "text": "we are examining the reaction of"}, {"start": 135.06, "duration": 5.52, "text": "nitrogen dioxide and carbon monoxide to"}, {"start": 138.239, "duration": 5.761, "text": "form nitrogen monoxide and carbon"}, {"start": 140.58, "duration": 5.4, "text": "dioxide if we acquire kinetic data and"}, {"start": 144.0, "duration": 5.819, "text": "find that the reaction is second order"}, {"start": 145.98, "duration": 6.36, "text": "in NO2 and zero order in Co what will be"}, {"start": 149.819, "duration": 4.681, "text": "the rate law for this reaction well in"}, {"start": 152.34, "duration": 4.5, "text": "the rate law each concentration will be"}, {"start": 154.5, "duration": 4.56, "text": "raised to a certain power equal to the"}, {"start": 156.84, "duration": 4.02, "text": "reaction order for that reactant so"}, {"start": 159.06, "duration": 4.319, "text": "given the data we collected the"}, {"start": 160.86, "duration": 4.56, "text": "concentration of NO2 will be raised to"}, {"start": 163.379, "duration": 4.801, "text": "the second power and the concentration"}, {"start": 165.42, "duration": 5.459, "text": "of Co will be raised to the zero power"}, {"start": 168.18, "duration": 4.68, "text": "which makes it equal to one it can"}, {"start": 170.879, "duration": 3.541, "text": "therefore drop out of the rate law and"}, {"start": 172.86, "duration": 4.2, "text": "we will see that the rate law for this"}, {"start": 174.42, "duration": 5.64, "text": "reaction will simply involve the NO2"}, {"start": 177.06, "duration": 5.58, "text": "concentration squared let's try one more"}, {"start": 180.06, "duration": 5.039, "text": "for this reaction the rate law is as"}, {"start": 182.64, "duration": 4.98, "text": "follows what is the reaction order with"}, {"start": 185.099, "duration": 5.161, "text": "respect to each reactant and what is the"}, {"start": 187.62, "duration": 4.619, "text": "overall reaction order we can see from"}, {"start": 190.26, "duration": 4.32, "text": "the exponents that this reaction is"}, {"start": 192.239, "duration": 5.461, "text": "second order in no because of the two"}, {"start": 194.58, "duration": 5.579, "text": "and first order in H2 because if no"}, {"start": 197.7, "duration": 4.679, "text": "exponent is listed it's the same as an"}, {"start": 200.159, "duration": 4.8, "text": "exponent of one and the reaction is"}, {"start": 202.379, "duration": 5.101, "text": "therefore third order overall since two"}, {"start": 204.959, "duration": 5.221, "text": "plus one equals three"}, {"start": 207.48, "duration": 4.74, "text": "this may all seem pretty confusing but"}, {"start": 210.18, "duration": 4.02, "text": "for now we simply need to focus on these"}, {"start": 212.22, "duration": 4.62, "text": "definitions as we will soon put them to"}, {"start": 214.2, "duration": 4.86, "text": "good use when we derive reaction orders"}, {"start": 216.84, "duration": 4.44, "text": "experimentally which will make this"}, {"start": 219.06, "duration": 5.84, "text": "whole concept much more clear Professor"}, {"start": 221.28, "duration": 3.62, "text": "Dave for check see you next time"}, {"start": 225.9, "duration": 3.0, "text": "foreign"}], "title": "Introduction to Rate Laws | Professor Dave & Chegg Explain", "description": "In this video, @ProfessorDaveExplains explains rate laws and their importance in determining reaction rates. The concentration of each reactant is a key factor that affects the reaction rate. By understanding the definition of rate laws and examining some examples, we can progress to experimental derivation of reaction orders.\n\nGet more homework help from Chegg at https://che.gg/3HbtG8Y\nWatch the whole General Chemistry playlist: https://www.youtube.com/playlist?list=PLllCkartRRdPQdEE-oyt5Di31wIwJc5S6\n\nAbout Chegg:\nChegg is a leading education technology company that provides students with affordable and accessible study resources. Chegg's mission is to help students save time, save money and get smarter.  \n\nSubscribe: https://www.youtube.com/@chegg\nTikTok: https://www.tiktok.com/@chegg\nInstagram: https://www.instagram.com/chegg/\nTwitter: https://twitter.com/chegg\n\n00:00 Intro\n00:22 What are rate laws?\n01:27 Reaction order\n02:16 Example", "lengthSeconds": 233, "width": 640, "height": 360}
{"video_id": "ulb_HuYdLzM", "transcript": [{"start": 6.32, "duration": 5.04, "text": "okay let's do some calculations\nregarding the molar mass of a gas. if we"}, {"start": 11.37, "duration": 5.07, "text": "have a one point three one gram sample\nof a hydrocarbon we vaporize it and it"}, {"start": 16.44, "duration": 6.39, "text": "is found to occupy two point zero liters\nat 25 degrees Celsius and one point zero"}, {"start": 22.83, "duration": 6.0, "text": "atmospheres what is the gas we want to\nidentify this gas now we're gonna"}, {"start": 28.83, "duration": 4.35, "text": "probably need to use some information\nregarding ideal gases to answer this"}, {"start": 33.18, "duration": 4.08, "text": "question so if you need to review ideal\ngases go ahead and check out that"}, {"start": 37.26, "duration": 2.72, "text": "tutorial and when you're ready give this a try."}, {"start": 58.76, "duration": 1.599, "text": "so we have our given information one"}, {"start": 60.359, "duration": 5.881, "text": "point three one grams 2.0 liters 25\ndegrees Celsius and 1 atmosphere so"}, {"start": 66.24, "duration": 4.08, "text": "we're definitely going to need to use\nthe ideal gas law in order to answer"}, {"start": 70.32, "duration": 4.17, "text": "this question\nso that's PV equals nRT and if we want"}, {"start": 74.49, "duration": 4.83, "text": "to identify that gas we're probably\ngoing to need the molar mass and molar"}, {"start": 79.32, "duration": 5.58, "text": "mass is grams per mole we have grams but\nwe don't have moles so let's solve for"}, {"start": 84.9, "duration": 5.039, "text": "it let's solve for n in the ideal gas\nlaw so n is going to be equal to PV over"}, {"start": 89.939, "duration": 5.881, "text": "RT so now let's just plug in what we\nknow n or the number of moles in this"}, {"start": 95.82, "duration": 5.25, "text": "sample of gas are going to be equal to\none point zero zero atmospheres times"}, {"start": 101.07, "duration": 5.61, "text": "two point zero liters that's the volume\nof the sample divided by the gas"}, {"start": 106.68, "duration": 4.64, "text": "constant R which in this case we're\ngoing to use the version that is 0.0821"}, {"start": 111.32, "duration": 3.939, "text": "liters atmospheres per mole Kelvin\nbecause we're going to want to cancel"}, {"start": 115.259, "duration": 4.741, "text": "out those other units and then the\ntemperature is 25 degrees Celsius but"}, {"start": 120.0, "duration": 3.93, "text": "remember we always want to express that\nin Kelvin when we're doing calculations"}, {"start": 123.93, "duration": 5.43, "text": "so that is going to be 298 Kelvin and\nthen if we plug all that into the"}, {"start": 129.36, "duration": 7.03, "text": "calculator we get 0.08 to\nso that is how many moles are in this"}, {"start": 136.39, "duration": 4.74, "text": "sample so now if we know the mass of\nthis sample of gas and we know how many"}, {"start": 141.13, "duration": 5.58, "text": "moles it represents all we have to do is\ntake the value in grams and divide by"}, {"start": 146.71, "duration": 7.08, "text": "the value in moles and we'll get grams\nper mole so 1.31 grams over 0.08 two"}, {"start": 153.79, "duration": 6.36, "text": "moles is approximately 16 grams per mole\nand now while this could be a few"}, {"start": 160.15, "duration": 4.92, "text": "different things we know it is a\nhydrocarbon and so the only hydrocarbon"}, {"start": 165.07, "duration": 3.48, "text": "that is gonna have this molar mass is\ngonna be methane because we need at"}, {"start": 168.55, "duration": 3.87, "text": "least one carbon atom that's 12 and then\nwe have no more room for a second one"}, {"start": 172.42, "duration": 4.53, "text": "and just add four hydrogen atoms on\nthere and we've gotten methane so this"}, {"start": 176.95, "duration": 3.09, "text": "gas must have been methane."}], "title": "Practice Problem: Molar Mass of a Gas", "description": "We know about the ideal gas law, but what are some clever things we can do with it? Well, how about identifying an unknown gas by solving for its molar mass? Sure, why not!\n\nTry all of the general chemistry practice problems: http://bit.ly/ProfDaveGCPP\n\nGeneral Chemistry Tutorials: http://bit.ly/ProfDaveGenChem\n\nEMAIL\u25ba ProfessorDaveExplains@gmail.com\nPATREON\u25ba http://patreon.com/ProfessorDaveExplains\n\nCheck out \"Is This Wi-Fi Organic?\", my book on disarming pseudoscience!\nAmazon: https://amzn.to/2HtNpVH\nBookshop: https://bit.ly/39cKADM\n\rBarnes and Noble: https://bit.ly/3pUjmrn \nBook Depository: http://bit.ly/3aOVDlT", "lengthSeconds": 211, "width": 640, "height": 360}
{"video_id": "ojMemmYifxM", "transcript": [{"start": 0.1, "duration": 2.52, "text": "Professor Dave again, let\u2019s talk about midsegments."}, {"start": 9.74, "duration": 5.36, "text": "We just looked at all kinds of bisectors and\nother lines that pertain to triangles, so"}, {"start": 15.11, "duration": 2.3, "text": "let\u2019s learn just one more type."}, {"start": 17.41, "duration": 5.83, "text": "A midsegment of a triangle is a line segment\nthat joins the midpoints of two sides of the"}, {"start": 23.24, "duration": 1.18, "text": "triangle."}, {"start": 24.42, "duration": 6.81, "text": "Every triangle will therefore have precisely\nthree midsegments, and these three line segments"}, {"start": 31.23, "duration": 4.27, "text": "form another triangle, the midsegment triangle."}, {"start": 35.5, "duration": 5.819, "text": "Since the vertices of the midsegment triangle\nare the midpoints of these sides, we can know"}, {"start": 41.319, "duration": 4.731, "text": "that the segments on either side of a midpoint\nare equal to one another."}, {"start": 46.05, "duration": 5.339, "text": "This is signified by these tick marks, one\nline here and one line here means these are"}, {"start": 51.389, "duration": 1.361, "text": "the same."}, {"start": 52.75, "duration": 5.739, "text": "Two lines here and two lines here means these\nare the same as each other, but not necessarily"}, {"start": 58.489, "duration": 3.87, "text": "the same as the segments with one tick mark."}, {"start": 62.359, "duration": 2.37, "text": "Three tick marks, same idea."}, {"start": 64.729, "duration": 6.191, "text": "Furthermore, the triangle midsegment theorem\nstates that any midsegment of a triangle is"}, {"start": 70.92, "duration": 6.12, "text": "parallel to one of the sides of the triangle\nand is also precisely half the length of that"}, {"start": 77.04, "duration": 1.0, "text": "side."}, {"start": 78.04, "duration": 5.7, "text": "So here we can see one midsegment and the\nside of the triangle that it is parallel to."}, {"start": 83.74, "duration": 6.1, "text": "The midsegment is half the length of the side,\nand since the side is split up into two equal"}, {"start": 89.84, "duration": 5.47, "text": "sections, the midsegment is equal to both\nof these two sections."}, {"start": 95.31, "duration": 5.32, "text": "We can say the same thing for the other two\nmidsegments and their corresponding parallel"}, {"start": 100.63, "duration": 1.01, "text": "sides."}, {"start": 101.64, "duration": 5.689, "text": "This allows us to draw all kinds of figures\nlike this one, where some lengths are given"}, {"start": 107.329, "duration": 2.771, "text": "and others can be determined."}, {"start": 110.1, "duration": 5.24, "text": "If we know the length of this midsegment,\nwe know this side length."}, {"start": 115.34, "duration": 5.309, "text": "If we know this other side length, then we\nknow the length of this other midsegment."}, {"start": 120.649, "duration": 5.091, "text": "There\u2019s plenty of algebra we can do with\nthis theorem, so let\u2019s check comprehension."}], "title": "The Triangle Midsegment Theorem", "description": "We just went over a bunch of different theorems, but we need to do one more! It's kind of neat. I don't have anything else to say, just watch this, it's really short.\n\nWatch the whole Mathematics playlist: http://bit.ly/ProfDaveMath\n\nClassical Physics Tutorials: http://bit.ly/ProfDavePhysics1\nModern Physics Tutorials: http://bit.ly/ProfDavePhysics2\nGeneral Chemistry Tutorials: http://bit.ly/ProfDaveGenChem\nOrganic Chemistry Tutorials: http://bit.ly/ProfDaveOrgChem\nBiochemistry Tutorials: http://bit.ly/ProfDaveBiochem\nBiology Tutorials: http://bit.ly/ProfDaveBio\n\nEMAIL\u25ba ProfessorDaveExplains@gmail.com\nPATREON\u25ba http://patreon.com/ProfessorDaveExplains\n\nCheck out \"Is This Wi-Fi Organic?\", my book on disarming pseudoscience!\nAmazon: https://amzn.to/2HtNpVH\nBookshop: https://bit.ly/39cKADM\n\rBarnes and Noble: https://bit.ly/3pUjmrn \nBook Depository: http://bit.ly/3aOVDlT", "lengthSeconds": 187, "width": 640, "height": 360}
{"video_id": "UKbrwPL3wXE", "transcript": [{"start": 0.57, "duration": 5.459, "text": "in a document if we suspect we've"}, {"start": 3.209, "duration": 4.921, "text": "misspelled a word we can use the fine"}, {"start": 6.029, "duration": 5.94, "text": "function to highlight the error and"}, {"start": 8.13, "duration": 6.69, "text": "correct it or delete it within our DNA"}, {"start": 11.969, "duration": 7.171, "text": "that function is taken on by a system"}, {"start": 14.82, "duration": 6.12, "text": "called CRISPR casts nine CRISPR is short"}, {"start": 19.14, "duration": 5.73, "text": "for clustered regularly interspaced"}, {"start": 20.94, "duration": 6.6, "text": "short palindromic repeats CRISPR"}, {"start": 24.87, "duration": 6.24, "text": "consists of two components the caste"}, {"start": 27.54, "duration": 6.33, "text": "nine protein that can cut DNA and a"}, {"start": 31.11, "duration": 5.91, "text": "guide RNA that can recognize the"}, {"start": 33.87, "duration": 5.67, "text": "sequence of DNA to be edited to use"}, {"start": 37.02, "duration": 4.47, "text": "CRISPR casts nine scientists first"}, {"start": 39.54, "duration": 4.41, "text": "identify the sequence of the human"}, {"start": 41.49, "duration": 6.45, "text": "genome that's causing a health problem"}, {"start": 43.95, "duration": 6.66, "text": "then they create a specific guide RNA to"}, {"start": 47.94, "duration": 7.59, "text": "recognize that particular stretch of a's"}, {"start": 50.61, "duration": 7.62, "text": "t's g's and c's in the DNA the guide RNA"}, {"start": 55.53, "duration": 6.03, "text": "is attached to the DNA cutting enzyme"}, {"start": 58.23, "duration": 5.84, "text": "cast 9 and then this complex is"}, {"start": 61.56, "duration": 5.37, "text": "introduced to the target cells it"}, {"start": 64.07, "duration": 6.82, "text": "locates the target letter sequence and"}, {"start": 66.93, "duration": 6.81, "text": "cuts the DNA at that point scientists"}, {"start": 70.89, "duration": 6.18, "text": "can then edit the existing genome by"}, {"start": 73.74, "duration": 5.88, "text": "either modifying deleting or inserting"}, {"start": 77.07, "duration": 6.06, "text": "new sequences it effectively makes"}, {"start": 79.62, "duration": 6.72, "text": "CRISPR cast 9 a cut-and-paste tool for"}, {"start": 83.13, "duration": 6.27, "text": "DNA editing in the future scientists"}, {"start": 86.34, "duration": 5.959, "text": "hope to use CRISPR cast 9 to develop"}, {"start": 89.4, "duration": 8.48, "text": "critical advances in patient care or"}, {"start": 92.299, "duration": 5.581, "text": "even cure lifelong inherited diseases"}], "title": "CRISPR Explained", "description": "This video is an explanation of CRISPR-Cas 9.\n____________________________________________\nFOR THE PUBLIC: More health and medical news on the Mayo Clinic News Network. https://newsnetwork.mayoclinic.org/\n\nFOR THE MEDIA ONLY: Register at https://newsnetwork.mayoclinic.org/re... to access clean and nat sound versions of this video on the Mayo Clinic News Network. https://newsnetwork.mayoclinic.org/\n\nMayo Clinic https://mayocl.in/3tNMAdF Follow Mayo Clinic on Instagram: https://www.instagram.com/mayoclinic/ Like Mayo Clinic on Facebook: https://www.facebook.com/ Follow Mayo Clinic on Twitter: https://twitter.com/MayoClinic", "lengthSeconds": 98, "width": 640, "height": 360}
{"video_id": "8Ji3g4yp4VE", "transcript": [{"start": 0.03, "duration": 4.499, "text": "hormones are chemical messengers"}, {"start": 2.01, "duration": 5.49, "text": "synthesized at one location that deliver"}, {"start": 4.529, "duration": 5.19, "text": "messages to a different location five"}, {"start": 7.5, "duration": 5.849, "text": "main types of hormones influence plant"}, {"start": 9.719, "duration": 7.65, "text": "growth oxen gibberellin cytokine in"}, {"start": 13.349, "duration": 5.731, "text": "ethylene and abscisic acid auxin is the"}, {"start": 17.369, "duration": 3.691, "text": "hormone that allows plants to bend"}, {"start": 19.08, "duration": 5.459, "text": "towards the light this is called"}, {"start": 21.06, "duration": 6.26, "text": "phototropism tropism is the turning of"}, {"start": 24.539, "duration": 5.91, "text": "an organism towards an external stimulus"}, {"start": 27.32, "duration": 6.13, "text": "auxin is synthesized at stem tips and"}, {"start": 30.449, "duration": 5.281, "text": "travels downward that's why if you"}, {"start": 33.45, "duration": 4.32, "text": "remove the stem tip the plant will lose"}, {"start": 35.73, "duration": 4.47, "text": "the ability to grow towards the light"}, {"start": 37.77, "duration": 4.44, "text": "aux-in makes the plant bend towards the"}, {"start": 40.2, "duration": 5.01, "text": "light by moving to the shaded part of"}, {"start": 42.21, "duration": 5.61, "text": "the stem gibberellins are important at"}, {"start": 45.21, "duration": 5.07, "text": "several stages of plant development for"}, {"start": 47.82, "duration": 4.739, "text": "instance they are required in order for"}, {"start": 50.28, "duration": 5.43, "text": "a plant to germinate and our hands up"}, {"start": 52.559, "duration": 4.801, "text": "regulated at that time they also promote"}, {"start": 55.71, "duration": 5.04, "text": "elongation of internodes"}, {"start": 57.36, "duration": 5.21, "text": "aka the stem between nodes which are"}, {"start": 60.75, "duration": 4.86, "text": "despot from which the leaves grow"}, {"start": 62.57, "duration": 5.92, "text": "cytokinin counter senescence in plants"}, {"start": 65.61, "duration": 6.96, "text": "it is also important in making new plant"}, {"start": 68.49, "duration": 6.449, "text": "organs aka roots versus shoots this"}, {"start": 72.57, "duration": 5.1, "text": "hormone is produced in the root apical"}, {"start": 74.939, "duration": 5.07, "text": "meristem in other words the tips of"}, {"start": 77.67, "duration": 4.89, "text": "roots and is passively carried up with"}, {"start": 80.009, "duration": 4.531, "text": "water through the plants xylem the plant"}, {"start": 82.56, "duration": 4.349, "text": "uses the ratio of aux-in versus"}, {"start": 84.54, "duration": 5.579, "text": "cytokinin to determine if it should form"}, {"start": 86.909, "duration": 5.46, "text": "roots or shoots more auxin promotes the"}, {"start": 90.119, "duration": 4.221, "text": "formation of roots while more cytokinin"}, {"start": 92.369, "duration": 4.5, "text": "promotes the formation of shoots"}, {"start": 94.34, "duration": 5.319, "text": "cytokinin also works with the next"}, {"start": 96.869, "duration": 5.911, "text": "hormone will discuss ethylene to cost"}, {"start": 99.659, "duration": 7.231, "text": "abscission or shedding of leaves flowers"}, {"start": 102.78, "duration": 7.199, "text": "and fruits ethylene the magical ripening"}, {"start": 106.89, "duration": 5.67, "text": "hormone if you put a ripe banana in a"}, {"start": 109.979, "duration": 4.971, "text": "bag with unripe bananas the unripe"}, {"start": 112.56, "duration": 5.25, "text": "bananas will ripen much more quickly"}, {"start": 114.95, "duration": 5.68, "text": "ethylene is gaseous with low solubility"}, {"start": 117.81, "duration": 4.86, "text": "in water it diffuses out of cells and"}, {"start": 120.63, "duration": 4.5, "text": "more of it gets produced by rapidly"}, {"start": 122.67, "duration": 4.979, "text": "growing and dividing cells this is"}, {"start": 125.13, "duration": 3.989, "text": "important for germinating plants since"}, {"start": 127.649, "duration": 3.631, "text": "the high production of this hormone"}, {"start": 129.119, "duration": 4.321, "text": "inhibits the expansion of leaves and"}, {"start": 131.28, "duration": 2.61, "text": "allows the chute to dig upwards through"}, {"start": 133.44, "duration": 3.78, "text": "the soil"}, {"start": 133.89, "duration": 6.24, "text": "until it reaches light abscisic acid is"}, {"start": 137.22, "duration": 4.739, "text": "a hormone that signals dehydration it is"}, {"start": 140.13, "duration": 4.2, "text": "synthesized in the chloroplasts of"}, {"start": 141.959, "duration": 4.621, "text": "leaves when a plant is water stressed"}, {"start": 144.33, "duration": 4.17, "text": "this hormone travels through its xylem"}, {"start": 146.58, "duration": 3.75, "text": "and phloem and causes the leaves to"}, {"start": 148.5, "duration": 4.5, "text": "close their stomata to reduce water loss"}, {"start": 150.33, "duration": 4.08, "text": "through photo transpiration these are"}, {"start": 153.0, "duration": 3.93, "text": "just five types of hormones that"}, {"start": 154.41, "duration": 5.07, "text": "influence plant growth however there are"}, {"start": 156.93, "duration": 5.16, "text": "many more if you liked this video please"}, {"start": 159.48, "duration": 4.74, "text": "like and subscribe it would help me make"}, {"start": 162.09, "duration": 3.96, "text": "more videos and make sure to comment"}, {"start": 164.22, "duration": 4.92, "text": "with any topics you'd like me to cover"}, {"start": 166.05, "duration": 4.89, "text": "in future videos also it would be really"}, {"start": 169.14, "duration": 4.879, "text": "nice if you could support me on patreon"}, {"start": 170.94, "duration": 3.079, "text": "thank you"}], "title": "PLANT HORMONES - Auxin Gibberellin Cytokinin Ethylene Abscisic Acid", "description": "", "lengthSeconds": 172, "width": 640, "height": 360}
{"video_id": "ZuWa827qAao", "transcript": [{"start": 0.199, "duration": 2.921, "text": "\u064a\u0627 \u0634\u0628\u0627\u0628 \u0623\u0646\u0627 \u0623\u0633\u062a\u0627\u0630 \u062f\u064a\u0641 \u0648\u0623\u0631\u064a\u062f\n\u0644\u0623\u062e\u0628\u0631\u0643 \u0639\u0646 \u0627\u0644\u0643\u064a\u0645\u064a\u0627\u0621"}, {"start": 9.62, "duration": 2.019, "text": "\u0645\u0627 \u0647\u064a \u0627\u0644\u0643\u064a\u0645\u064a\u0627\u0621 \u0644\u0645\u0627\u0630\u0627 \u064a\u062c\u0628 \u0639\u0644\u064a\u0646\u0627\n\u0627\u0639\u0631\u0641 \u0639\u0646\u0647\u0627"}, {"start": 11.639, "duration": 3.691, "text": "\u0627\u0644\u0643\u064a\u0645\u064a\u0627\u0621 \u0647\u064a \u062f\u0631\u0627\u0633\u0629 \u0645\u0627 \u064a\u0647\u0645\n\u0645\u0635\u0646\u0648\u0639\u0629 \u0645\u0646 \u0648\u0643\u064a\u0641 \u064a\u062a\u063a\u064a\u0631"}, {"start": 15.33, "duration": 3.0, "text": "\u0645\u0646 \u0627\u0644\u0645\u0647\u0645 \u0623\u0646 \u062a\u0639\u0631\u0641 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Yes, you there. Normal Jack or Jill. Do you want to learn science? What's that? Oh, you don't know anything about science? Yes, salt is salty. That's one thing, I guess. It doesn't matter. If you want to start from scratch, you've come to the right place. Learn chemistry first! There's no easier way than with Professor Dave.\n\nWatch the whole General Chemistry playlist: http://bit.ly/ProfDaveGenChem\n\nStudy for the AP Chemistry exam with me: https://bit.ly/ProfDaveAPChem\n\nOrganic Chemistry Tutorials: http://bit.ly/ProfDaveOrgChem\nBiochemistry Tutorials: http://bit.ly/ProfDaveBiochem\nBiology Tutorials: http://bit.ly/ProfDaveBio\nClassical Physics Tutorials: http://bit.ly/ProfDavePhysics1\nModern Physics Tutorials: http://bit.ly/ProfDavePhysics2\nMathematics Tutorials: http://bit.ly/ProfDaveMaths\n\nEMAIL\u25ba ProfessorDaveExplains@gmail.com\nPATREON\u25ba http://patreon.com/ProfessorDaveExplains\n\nCheck out \"Is This Wi-Fi Organic?\", my book on disarming pseudoscience!\nAmazon: https://amzn.to/2HtNpVH\nBookshop: https://bit.ly/39cKADM\n\rBarnes and Noble: https://bit.ly/3pUjmrn \nBook Depository: http://bit.ly/3aOVDlT", "lengthSeconds": 141, "width": 640, "height": 360}
{"video_id": "1XIsgaOIxOw", "transcript": [{"start": 0.0, "duration": 12.0, "text": "If 3A minus B equals 23, and 4B plus 3C equals 22.\u00a0\nWhat is the average of A, B and C? The average of\u00a0\u00a0"}, {"start": 12.0, "duration": 7.2, "text": "three items is the sum of the items divided by\u00a0\nthree. To find the sum, add the given equations.\u00a0\u00a0"}, {"start": 20.04, "duration": 8.46, "text": "So you have 3A plus 3B plus 3C equals\u00a0\nforty-five. Now dividing both sides of\u00a0\u00a0"}, {"start": 28.5, "duration": 6.36, "text": "the equation by 3, will give us the sum\u00a0\nequals 15. Therefore, the average is 5."}], "title": "Solve an Average of 3 Items from two give equations | SAT Math", "description": "This videos explains how to solve an average of three items from two give equations in an easy way.\n\n#sat \n#satmath \n#satmathtestprep \n\nsat math, satprep, sat math practice, sat math tips, sat math prep, sat math walkthrough, sat math formulas to know, sat math tricks, sat math section, sat math no calculator, sat math no calculator walkthrough, sat math tips tricks and strategies\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\nThe organic chemistry tutor , Professor Dave Explains , TabletClass Math ,  Mr H Tutoring , AnilKumar , Math and Science , tecmath , Cognito , Brian McLogan ,  Math with Mr J , UltimateAlgebra ,  Math Teacher Gon , Math meeting , Eddie Woo ,  GreeneMath.com , NancyPi ,  The Math Sorcerer , Justice Shepard , PreMath , Scalar Learning ,", "lengthSeconds": 36, "width": 360, "height": 640}
{"video_id": "2sNyzST_NuI", "transcript": [{"start": 0.08, "duration": 6.799, "text": "number one asks a bronzee lowy acid"}, {"start": 3.399, "duration": 5.081, "text": "blank while a Lewis acid blank we have"}, {"start": 6.879, "duration": 3.401, "text": "these two different paradigms the Bron"}, {"start": 8.48, "duration": 3.56, "text": "stead Lowry definition and the Lewis"}, {"start": 10.28, "duration": 3.399, "text": "definition and we need to know that for"}, {"start": 12.04, "duration": 4.079, "text": "Bron stead Lowry we're talking about"}, {"start": 13.679, "duration": 4.6, "text": "transferring protons while with Lewis"}, {"start": 16.119, "duration": 6.481, "text": "we're talking about electrons a bronzee"}, {"start": 18.279, "duration": 7.08, "text": "Lowry acid is one that donates protons"}, {"start": 22.6, "duration": 5.12, "text": "and a Lewis acid is one that accepts"}, {"start": 25.359, "duration": 5.601, "text": "electrons so this one is going to be"}, {"start": 27.72, "duration": 6.839, "text": "option b remember a bronze said y acid"}, {"start": 30.96, "duration": 6.36, "text": "donates protons alouis acid accepts"}, {"start": 34.559, "duration": 2.761, "text": "electrons"}], "title": "Important Acids and Bases Definitions #organicchemistry", "description": "\ud83d\udcfd\ufe0fThis is a clip from Professor Dave's Acids and Bases Organic Chemistry Practice Exam which is only available on Chemmunity.com \n\n\ud83d\udcafCheck out Professor Dave's Practice Exams and more OCHEM RESOURCES here: https://chemmunity.info/dave \ud83d\ude4c", "lengthSeconds": 35, "width": 360, "height": 640}
{"video_id": "ULiLt2rtpAg", "transcript": [{"start": 0.48, "duration": 2.38, "text": "hey it's professor Dave, let's talk about\ncarbon"}, {"start": 9.12, "duration": 1.58, "text": "you may have heard that carbon is the"}, {"start": 10.71, "duration": 3.639, "text": "element of life, but what does that\nmean? what determines whether an atom is"}, {"start": 14.349, "duration": 1.321, "text": "of carbon in the first place?"}, {"start": 15.67, "duration": 3.59, "text": "an atom consists of positively charged protons"}, {"start": 19.26, "duration": 3.8, "text": "neutrons which have no charge, and\nnegatively charged electrons"}, {"start": 23.06, "duration": 3.86, "text": "an element is defined by the number of\nprotons in the nucleus"}, {"start": 26.92, "duration": 3.09, "text": "one proton is hydrogen, two is helium"}, {"start": 30.01, "duration": 3.11, "text": "3 lithium, 4  beryllium, 5"}, {"start": 33.12, "duration": 4.8, "text": "boron and six is carbon. the number of\nneutrons and electrons can change but"}, {"start": 37.92, "duration": 3.7, "text": "by definition any atom with six protons\nis a carbon atom"}, {"start": 41.62, "duration": 4.15, "text": "how do they form? hydrogen the simplest and\nmost abundant of all"}, {"start": 45.77, "duration": 3.699, "text": "elements accumulates into clumps due to\ngravity which become stars"}, {"start": 49.469, "duration": 4.101, "text": "the tremendous inward pressure triggers\nnuclear fusion which is when protons and"}, {"start": 53.57, "duration": 1.94, "text": "neutrons smash together and fuse"}, {"start": 55.51, "duration": 3.389, "text": "many elements including carbon and\nlarger ones like iron"}, {"start": 58.899, "duration": 4.0, "text": "are made in stars. much larger ones need\nthe immense energy a supernova"}, {"start": 62.899, "duration": 0.741, "text": "explosion"}, {"start": 63.64, "duration": 3.94, "text": "or particle accelerators here on earth.\nso what's so special about carbon?"}, {"start": 67.58, "duration": 4.09, "text": "let's look at the electrons. carbon atoms\nhave six. the first two are in the inner"}, {"start": 71.67, "duration": 0.5, "text": "shell"}, {"start": 72.17, "duration": 3.339, "text": "and the other four are the so-called\nvalence electrons in the outer shell"}, {"start": 75.509, "duration": 3.43, "text": "these are the ones available for bonding\nwith other atoms which happens when two"}, {"start": 78.939, "duration": 1.82, "text": "atoms share two electrons"}, {"start": 80.759, "duration": 3.481, "text": "because elements in this part of the\nperiodic table want to have eight"}, {"start": 84.24, "duration": 1.62, "text": "electrons in the outermost shell"}, {"start": 85.86, "duration": 3.6, "text": "carbon has a tendency to make four bonds.\nthese can be single"}, {"start": 89.46, "duration": 3.71, "text": "double or triple bonds and with many\ndifferent elements. in addition while"}, {"start": 93.17, "duration": 4.46, "text": "an atom that's bound to two or three other\natoms will adopt a linear or flat shape"}, {"start": 97.63, "duration": 3.57, "text": "a carbon atom that's bound to four other\natoms will take on a three-dimensional"}, {"start": 101.2, "duration": 0.93, "text": "shape"}, {"start": 102.13, "duration": 3.379, "text": "because that's how the electron clouds\non the four atoms can be as far away"}, {"start": 105.509, "duration": 1.36, "text": "from each other as possible"}, {"start": 106.869, "duration": 3.86, "text": "so it's the ability of carbon to bond in\na wide variety of shapes with so many"}, {"start": 110.729, "duration": 0.651, "text": "elements"}, {"start": 111.38, "duration": 3.15, "text": "including itself that makes it so\nspecial. also"}, {"start": 114.53, "duration": 3.619, "text": "carbon-carbon bonds are strong enough to\nbe stable but not so strong that they"}, {"start": 118.149, "duration": 1.421, "text": "can't break and rearrange"}, {"start": 119.57, "duration": 3.13, "text": "which makes them excellent building\nblocks. when by itself"}, {"start": 122.7, "duration": 3.32, "text": "carbon has many allotropes, or ways an\nelement can be arranged"}, {"start": 126.02, "duration": 3.17, "text": "there's the graphite in your pencil\nwhich consists of slippery sheets of"}, {"start": 129.19, "duration": 1.42, "text": "carbon that rub off easily"}, {"start": 130.61, "duration": 2.34, "text": "there's diamond, the hardest substance known"}, {"start": 132.95, "duration": 3.3, "text": "and we can even make interesting\nstructures like nanotubes and fullerenes"}, {"start": 136.81, "duration": 3.7, "text": "that's just carbon by itself. when\ncombined with hydrogen"}, {"start": 140.51, "duration": 3.369, "text": "oxygen, and nitrogen and a few other\nelements it makes all the diverse"}, {"start": 143.879, "duration": 1.061, "text": "structures in your body"}, {"start": 144.94, "duration": 4.85, "text": "including carbohydrates, proteins, and DNA.\nwe can take nature a step further and"}, {"start": 149.79, "duration": 2.94, "text": "synthesize novel materials with an array\nof purposes"}, {"start": 152.73, "duration": 3.68, "text": "putting carbons in the empty spots in an iron\nlattice makes it stronger"}, {"start": 156.41, "duration": 3.78, "text": "we call this steel which is used in most\nlarge structures we know today"}, {"start": 160.19, "duration": 4.22, "text": "teflon, a carbon-fluorine polymer is a\nnon-stick substance that is highly"}, {"start": 164.41, "duration": 1.57, "text": "resistant to chemical degradation"}, {"start": 165.98, "duration": 3.68, "text": "and carbon based synthetic drugs like\ninhibitors that can silence a faulty"}, {"start": 169.66, "duration": 0.65, "text": "enzyme"}, {"start": 170.31, "duration": 3.73, "text": "hold the prospect of curing so many\ndiseases. that's a lotta reasons to love"}, {"start": 174.04, "duration": 0.52, "text": "carbon"}, {"start": 174.56, "duration": 2.61, "text": "it really is the element of life. see you\nnext time"}], "title": "Carbon: The Element of Life", "description": "You may have heard that carbon is the element of life. What does that mean? Let's find out!\n\nGeneral Chemistry Tutorials: http://bit.ly/ProfDaveGenChem\nOrganic Chemistry Tutorials: http://bit.ly/ProfDaveOrgChem\nBiochemistry Tutorials: http://bit.ly/ProfDaveBiochem\nBiology Tutorials: http://bit.ly/ProfDaveBio\nClassical Physics Tutorials: http://bit.ly/ProfDavePhysics1\nModern Physics Tutorials: http://bit.ly/ProfDavePhysics2\nMathematics Tutorials: http://bit.ly/ProfDaveMaths\n\nEMAIL\u25ba ProfessorDaveExplains@gmail.com\nPATREON\u25ba http://patreon.com/ProfessorDaveExplains\n\nCheck out \"Is This Wi-Fi Organic?\", my book on disarming pseudoscience!\nAmazon: https://amzn.to/2HtNpVH\nBookshop: https://bit.ly/39cKADM\n\rBarnes and Noble: https://bit.ly/3pUjmrn \nBook Depository: http://bit.ly/3aOVDlT", "lengthSeconds": 177, "width": 640, "height": 360}
{"video_id": "wioqhkDpI_I", "transcript": [{"start": 6.12, "duration": 1.88, "text": "we hear the words\nnatural and synthetic"}, {"start": 8.0, "duration": 3.53, "text": "all the time when talking about food,\nmedicine, clothes"}, {"start": 11.53, "duration": 4.029, "text": "and other things. but what do they mean?\nif you took a poll"}, {"start": 15.559, "duration": 3.921, "text": "most people would tell you that natural\nhas to do with their juice cleanse"}, {"start": 19.48, "duration": 3.049, "text": "organic groceries, yoga and other healthy activities"}, {"start": 22.529, "duration": 3.561, "text": "but for synthetic they'll warn you of toxins"}, {"start": 26.09, "duration": 3.939, "text": "scary chemicals and mean confusing\nchemistry teachers"}, {"start": 30.029, "duration": 4.521, "text": "with all these charged emotions it's not\nsurprising that a lot of people equate"}, {"start": 34.55, "duration": 4.62, "text": "these words with good and bad. but this discounts"}, {"start": 39.17, "duration": 3.87, "text": "everything chemistry has to offer\nsociety, so we need to change this view"}, {"start": 43.04, "duration": 4.48, "text": "natural simply means made by nature, the\nsame benevolent force that brings us"}, {"start": 47.52, "duration": 2.17, "text": "deadly viruses and toxic blowfish"}, {"start": 49.69, "duration": 3.33, "text": "which looks strangely similar. synthetic means"}, {"start": 53.02, "duration": 3.39, "text": "made by humans who save lives with\nantibiotics"}, {"start": 56.41, "duration": 3.93, "text": "and synthetic microbes that eat up the\noil from a tanker spill. now that we've"}, {"start": 60.34, "duration": 2.289, "text": "decoupled these words from good and bad"}, {"start": 62.629, "duration": 3.401, "text": "let's look at something both nature and humans"}, {"start": 66.03, "duration": 3.949, "text": "make. a vitamin is something we must\nconsume in small amounts"}, {"start": 69.979, "duration": 3.741, "text": "in order for our cells to keep normal\nfunction and metabolism"}, {"start": 73.72, "duration": 4.34, "text": "most plants and animals make these on\ntheir own but somewhere along biological"}, {"start": 78.06, "duration": 0.78, "text": "evolution"}, {"start": 78.84, "duration": 3.06, "text": "a few mammals lost the ability so we\nhave to eat them"}, {"start": 81.9, "duration": 3.93, "text": "take vitamin C. for thousands of years\nwe've gotten this from fruit"}, {"start": 85.83, "duration": 4.2, "text": "but with modern science we can identify\nthe structure of this and other"}, {"start": 90.03, "duration": 0.86, "text": "compounds"}, {"start": 90.89, "duration": 3.04, "text": "and make them ourselves. the question is"}, {"start": 93.93, "duration": 3.0, "text": "are they the same? to answer that"}, {"start": 96.93, "duration": 3.78, "text": "let's look at how a vitamin works. a\nvitamin has just the right shape to fit"}, {"start": 100.71, "duration": 0.9, "text": "into an"}, {"start": 101.61, "duration": 4.829, "text": "active site of an enzyme and make\ninteractions that turn on the enzyme"}, {"start": 106.439, "duration": 4.311, "text": "like a lock and key. and just as it\ndoesn't matter who makes your key"}, {"start": 110.75, "duration": 4.11, "text": "as long as it's the right shape for the\nlock, same goes for the vitamin"}, {"start": 114.86, "duration": 3.5, "text": "that's why any molecule that these\nspecific atoms"}, {"start": 118.36, "duration": 3.18, "text": "arranged in this way is vitamin C"}, {"start": 121.54, "duration": 4.039, "text": "nature makes this just fine, always has\nbut when nature does chemistry"}, {"start": 125.579, "duration": 4.581, "text": "it is limited to the normal pH and\ntemperature inside a living organism"}, {"start": 130.16, "duration": 2.93, "text": "so it has to use big bulky molecules"}, {"start": 133.09, "duration": 3.29, "text": "to do blind chemistry. when we do\nchemistry"}, {"start": 136.38, "duration": 5.18, "text": "we can use any exotic compounds and\nconditions we want to build"}, {"start": 141.56, "duration": 3.55, "text": "any molecule we want, and even though the pathways"}, {"start": 145.11, "duration": 4.31, "text": "are different the end product is the\nexact same"}, {"start": 149.42, "duration": 4.71, "text": "molecule. so the bioactivity of a\nmolecule can not depend on how it came"}, {"start": 154.13, "duration": 0.69, "text": "to be"}, {"start": 154.82, "duration": 3.77, "text": "an enzyme simply can't tell the\ndifference. furthermore"}, {"start": 158.59, "duration": 4.72, "text": "it is often cheaper and easier to\nsynthesize these in mass quantities"}, {"start": 163.31, "duration": 3.69, "text": "than to have to extract and purify from\nnatural sources"}, {"start": 167.0, "duration": 3.6, "text": "so if we want to keep the growing global population"}, {"start": 170.6, "duration": 3.82, "text": "healthy and well-fed, we have to get over\nthe idea"}, {"start": 174.42, "duration": 3.51, "text": "that science looks like this, because\nscience"}, {"start": 177.93, "duration": 1.17, "text": "looks like this"}], "title": "Will Synthetic Vitamins Make Me Explode?", "description": "This was my final submission for the American Chemical Society's ChemChamps contest in 2015.\n\nThe words natural and synthetic are thrown around a lot these days, when talking about foods, medicine, and other things. But what do they really mean, and how are they different? Is one necessarily better than the other? Can the bioactivity of a molecule depend on its pathway? Let's answer these questions and more.\n\nGeneral Chemistry Tutorials: http://bit.ly/ProfDaveGenChem\nOrganic Chemistry Tutorials: http://bit.ly/ProfDaveOrgChem\nBiochemistry Tutorials: http://bit.ly/ProfDaveBiochem\nBiology Tutorials: http://bit.ly/ProfDaveBio\nClassical Physics Tutorials: http://bit.ly/ProfDavePhysics1\nModern Physics Tutorials: http://bit.ly/ProfDavePhysics2\nMathematics Tutorials: http://bit.ly/ProfDaveMaths\n\nEMAIL\u25ba ProfessorDaveExplains@gmail.com\nPATREON\u25ba http://patreon.com/ProfessorDaveExplains\n\nCheck out \"Is This Wi-Fi Organic?\", my book on disarming pseudoscience!\nAmazon: https://amzn.to/2HtNpVH\nBookshop: https://bit.ly/39cKADM\n\rBarnes and Noble: https://bit.ly/3pUjmrn \nBook Depository: http://bit.ly/3aOVDlT", "lengthSeconds": 180, "width": 640, "height": 360}
{"video_id": "LC_R2Zh66fU", "transcript": [{"start": 0.5, "duration": 3.32, "text": "It\u2019s Professor Dave; let\u2019s introduce the\ndistributive property."}, {"start": 10.58, "duration": 4.88, "text": "As we learn math, we will see that there are\ncertain properties of numbers that are very"}, {"start": 15.47, "duration": 1.26, "text": "important."}, {"start": 16.73, "duration": 5.79, "text": "We already learned about the commutative property\nand the associative property, and we saw how"}, {"start": 22.52, "duration": 5.99, "text": "addition and multiplication abide by these\nproperties, while subtraction and division"}, {"start": 28.51, "duration": 1.07, "text": "do not."}, {"start": 29.58, "duration": 4.15, "text": "Now let\u2019s learn another property, the distributive\nproperty."}, {"start": 33.73, "duration": 3.96, "text": "To see how this works, let\u2019s look at some\nmore apples."}, {"start": 37.69, "duration": 3.689, "text": "Say we have five piles of seven apples."}, {"start": 41.379, "duration": 5.381, "text": "Five piles times seven apples per pile gives\nus thirty-five apples."}, {"start": 46.76, "duration": 4.539, "text": "What if we split each pile up into a group\nof three and a group of four?"}, {"start": 51.299, "duration": 2.591, "text": "There are a few ways we could represent this."}, {"start": 53.89, "duration": 5.269, "text": "Let\u2019s take the seven and split it up into\nthree plus four, in parentheses."}, {"start": 59.159, "duration": 5.351, "text": "This is clearly the same value we once had,\nbecause three plus four is seven, which makes"}, {"start": 64.51, "duration": 5.93, "text": "sense because rearranging the apples shouldn\u2019t\nchange the number of them that are present."}, {"start": 70.44, "duration": 6.34, "text": "But we could also say that we have five piles\nof three and five piles of four."}, {"start": 76.78, "duration": 3.65, "text": "That means five times three plus five times\nfour."}, {"start": 80.43, "duration": 5.93, "text": "Visually, we are just describing what we see,\nbut mathematically, we have just derived the"}, {"start": 86.36, "duration": 2.06, "text": "distributive property."}, {"start": 88.42, "duration": 5.69, "text": "When we have a number that is multiplied by\na sum or difference, we can distribute that"}, {"start": 94.11, "duration": 4.47, "text": "number across each value within the sum or\ndifference."}, {"start": 98.58, "duration": 4.97, "text": "In this case, that means five times three\nplus five times four."}, {"start": 103.55, "duration": 5.72, "text": "That gives us fifteen plus twenty, which is\nindeed still thirty-five."}, {"start": 109.27, "duration": 5.08, "text": "So whether we add the numbers in the parentheses\nfirst and then multiply by the other number,"}, {"start": 114.35, "duration": 6.14, "text": "or distribute the number across the sum, we\nwill get the same result, and that is the"}, {"start": 120.49, "duration": 6.25, "text": "essence of the distributive property, which\nwill be very important once we get to algebra."}, {"start": 126.74, "duration": 2.559, "text": "We can even use this property in reverse."}, {"start": 129.299, "duration": 3.5, "text": "Let\u2019s say we have five times seventeen."}, {"start": 132.799, "duration": 6.22, "text": "If we don\u2019t want to write down a calculation,\nwe can easily do this in our heads if we split"}, {"start": 139.019, "duration": 3.53, "text": "seventeen up into ten plus five plus two."}, {"start": 142.549, "duration": 6.681, "text": "Now, we can distribute the five across the\nsum, and rewrite this as five times ten plus"}, {"start": 149.23, "duration": 3.679, "text": "five times five plus five times two."}, {"start": 152.909, "duration": 5.541, "text": "These sums can be done easily in your head,\nand we get fifty plus twenty-five plus ten,"}, {"start": 158.45, "duration": 1.989, "text": "and that\u2019s eighty-five."}, {"start": 160.439, "duration": 4.81, "text": "This sort of mental trickery is very useful\nfor multiplying large numbers, so let\u2019s"}, {"start": 165.249, "duration": 2.291, "text": "move on to that topic next."}], "title": "The Distributive Property for Arithmetic", "description": "We know about the commutative property and the associative property, but there are more to learn! This next one, the distributive property, is very important in algebra, so let's introduce it now so that when we get there we already have this covered.\n\nWatch the whole Mathematics playlist: http://bit.ly/ProfDaveMath\n\nClassical Physics Tutorials: http://bit.ly/ProfDavePhysics1\nModern Physics Tutorials: http://bit.ly/ProfDavePhysics2\nGeneral Chemistry Tutorials: http://bit.ly/ProfDaveGenChem\nOrganic Chemistry Tutorials: http://bit.ly/ProfDaveOrgChem\nBiochemistry Tutorials: http://bit.ly/ProfDaveBiochem\nBiology Tutorials: http://bit.ly/ProfDaveBio\n\nEMAIL\u25ba ProfessorDaveExplains@gmail.com\nPATREON\u25ba http://patreon.com/ProfessorDaveExplains\n\nCheck out \"Is This Wi-Fi Organic?\", my book on disarming pseudoscience!\nAmazon: https://amzn.to/2HtNpVH\nBookshop: https://bit.ly/39cKADM\n\rBarnes and Noble: https://bit.ly/3pUjmrn \nBook Depository: http://bit.ly/3aOVDlT", "lengthSeconds": 199, "width": 640, "height": 360}
{"video_id": "sdEw2ztuAsY", "transcript": [{"start": 0.0, "duration": 3.23, "text": "[Music]"}, {"start": 4.48, "duration": 3.08, "text": "the Pythagorean theorem is one of the"}, {"start": 6.04, "duration": 4.16, "text": "most well-known theorems in the history"}, {"start": 7.56, "duration": 6.959, "text": "of mankind it states that for any right"}, {"start": 10.2, "duration": 6.4, "text": "triangle a s + B2 is equal to c^2 most"}, {"start": 14.519, "duration": 3.881, "text": "people have seen this theorem but not"}, {"start": 16.6, "duration": 5.08, "text": "that many actually know the reason why"}, {"start": 18.4, "duration": 5.879, "text": "it's true today we're showing just that"}, {"start": 21.68, "duration": 4.88, "text": "to do this we take four copies of any"}, {"start": 24.279, "duration": 6.961, "text": "right triangle as shown here and then we"}, {"start": 26.56, "duration": 7.64, "text": "flip them to create the following image"}, {"start": 31.24, "duration": 4.999, "text": "then we want to arrange them in a way"}, {"start": 34.2, "duration": 5.359, "text": "like so to form a"}, {"start": 36.239, "duration": 5.441, "text": "square notice that the angles shown here"}, {"start": 39.559, "duration": 4.921, "text": "are complimentary so that the"}, {"start": 41.68, "duration": 6.48, "text": "intersection of the red lines actually"}, {"start": 44.48, "duration": 7.879, "text": "form 90\u00b0 angles as shown"}, {"start": 48.16, "duration": 7.199, "text": "below now we have a red square basically"}, {"start": 52.359, "duration": 6.081, "text": "inscribed within a larger blue and green"}, {"start": 55.359, "duration": 5.281, "text": "square let's go further by asking what's"}, {"start": 58.44, "duration": 4.32, "text": "the area of the blue and green green"}, {"start": 60.64, "duration": 5.479, "text": "square the area of the blue and green"}, {"start": 62.76, "duration": 5.719, "text": "square can be given by its side lengths"}, {"start": 66.119, "duration": 6.561, "text": "so the side lengths are a plus b so the"}, {"start": 68.479, "duration": 6.361, "text": "area is a + b^ 2 which is equal to a 2 +"}, {"start": 72.68, "duration": 5.439, "text": "2 a + b"}, {"start": 74.84, "duration": 6.16, "text": "2 but alternatively we could have also"}, {"start": 78.119, "duration": 5.96, "text": "found the area from our figure by"}, {"start": 81.0, "duration": 5.68, "text": "noticing that the area is just four"}, {"start": 84.079, "duration": 6.481, "text": "copies of the area of each right"}, {"start": 86.68, "duration": 6.719, "text": "triangle plus the area of the Red Square"}, {"start": 90.56, "duration": 5.68, "text": "Square so now we have two alternative"}, {"start": 93.399, "duration": 5.481, "text": "definitions for the area of the large"}, {"start": 96.24, "duration": 5.48, "text": "blue and green square and we write it"}, {"start": 98.88, "duration": 5.48, "text": "below as shown"}, {"start": 101.72, "duration": 5.28, "text": "here now it just remains for us to"}, {"start": 104.36, "duration": 5.6, "text": "algebraically simplify this in the"}, {"start": 107.0, "duration": 4.84, "text": "following fashion and then to notice"}, {"start": 109.96, "duration": 4.96, "text": "that if we just subtract"}, {"start": 111.84, "duration": 7.239, "text": "2ab from both sides and do a little bit"}, {"start": 114.92, "duration": 6.879, "text": "of algebra we have exactly that a 2 + B2"}, {"start": 119.079, "duration": 5.281, "text": "is equal to C squ and this holds for any"}, {"start": 121.799, "duration": 4.801, "text": "generic right triangle I hope that's"}, {"start": 124.36, "duration": 4.679, "text": "shown some of the beauty behind the"}, {"start": 126.6, "duration": 4.159, "text": "mathematical proof of Pythagoras's"}, {"start": 129.039, "duration": 4.161, "text": "Theorem and actually showing you some of"}, {"start": 130.759, "duration": 4.081, "text": "the intuition and reasoning behind this"}, {"start": 133.2, "duration": 3.64, "text": "beautiful and very well-known"}, {"start": 134.84, "duration": 5.0, "text": "mathematical theorem thank you for"}, {"start": 136.84, "duration": 3.0, "text": "watching"}, {"start": 141.13, "duration": 18.039, "text": "[Music]"}], "title": "Proving the Pythagorean Theorem", "description": "Visualizing the Pythagorean theorem using Manim. Hope you enjoy, feedback is appreciated!\n\n\nMusic Credits:\nRoyalty Free Music: Bensound.com/royalty-free-music\nLicense code: HEQ5SRCKWR9JYMGK\n\nChapters:\n\n0:00 Introduction\n0:20 Proving the Pythagorean Theorem\n1:55 Ending", "lengthSeconds": 158, "width": 640, "height": 360}
{"video_id": "uvvP0oaH5-A", "transcript": [{"start": 0.0, "duration": 9.38, "text": "[Music]"}, {"start": 9.96, "duration": 3.879, "text": "Professor Dave and CH here a very"}, {"start": 12.04, "duration": 3.759, "text": "important thing to be able to do in"}, {"start": 13.839, "duration": 4.52, "text": "organic chemistry is to be able to"}, {"start": 15.799, "duration": 5.4, "text": "assign the formal charge of any atom in"}, {"start": 18.359, "duration": 5.041, "text": "a particular molecule atoms can be of"}, {"start": 21.199, "duration": 4.281, "text": "neutral charge or they can have a"}, {"start": 23.4, "duration": 4.0, "text": "positive or negative charge and"}, {"start": 25.48, "duration": 4.0, "text": "identifying atoms that have these"}, {"start": 27.4, "duration": 4.48, "text": "charges will help us predict what"}, {"start": 29.48, "duration": 5.04, "text": "molecule will do so let's make sure we"}, {"start": 31.88, "duration": 4.28, "text": "know how to do this the main thing to"}, {"start": 34.52, "duration": 3.92, "text": "understand is that an atom will be of"}, {"start": 36.16, "duration": 4.8, "text": "neutral charge when it is contributing a"}, {"start": 38.44, "duration": 5.84, "text": "number of electrons to a leis structure"}, {"start": 40.96, "duration": 5.88, "text": "that is equal to its veence for example"}, {"start": 44.28, "duration": 4.959, "text": "carbon has four veence electrons so when"}, {"start": 46.84, "duration": 5.12, "text": "it makes four bonds it contributes one"}, {"start": 49.239, "duration": 5.201, "text": "electron to each and is thus neutral"}, {"start": 51.96, "duration": 4.36, "text": "nitrogen has five veence electrons so"}, {"start": 54.44, "duration": 4.799, "text": "when it makes three bonds and has one"}, {"start": 56.32, "duration": 5.039, "text": "lone pair it is neutral If instead it"}, {"start": 59.239, "duration": 4.441, "text": "makes four bonds it contributes only"}, {"start": 61.359, "duration": 4.921, "text": "four electrons which is one less than"}, {"start": 63.68, "duration": 5.88, "text": "its veence so it has a formal positive"}, {"start": 66.28, "duration": 5.32, "text": "charge oxygen has six veence electrons"}, {"start": 69.56, "duration": 4.64, "text": "so when it makes two bonds and has two"}, {"start": 71.6, "duration": 5.04, "text": "lone pairs it is neutral but if it makes"}, {"start": 74.2, "duration": 5.04, "text": "only one Bond and has three loone pairs"}, {"start": 76.64, "duration": 4.92, "text": "it contributes seven electrons which is"}, {"start": 79.24, "duration": 5.239, "text": "one more than its veence so it will have"}, {"start": 81.56, "duration": 5.199, "text": "a formal negative charge this is the"}, {"start": 84.479, "duration": 5.041, "text": "easiest way to tell the formal charge of"}, {"start": 86.759, "duration": 5.36, "text": "any atom within a molecule here are the"}, {"start": 89.52, "duration": 5.08, "text": "most common situations we will be seeing"}, {"start": 92.119, "duration": 5.241, "text": "over and over again in organic chemistry"}, {"start": 94.6, "duration": 5.199, "text": "starting with carbon species here is a"}, {"start": 97.36, "duration": 5.079, "text": "carbon radical with three bonds and one"}, {"start": 99.799, "duration": 4.521, "text": "unpaired electron a carbo cation with"}, {"start": 102.439, "duration": 4.96, "text": "three bonds and an empty fourth"}, {"start": 104.32, "duration": 6.119, "text": "coordination site and a carban ion with"}, {"start": 107.399, "duration": 5.161, "text": "three bonds and one loone pair these are"}, {"start": 110.439, "duration": 4.64, "text": "neutral positively charged and"}, {"start": 112.56, "duration": 4.32, "text": "negatively charged respectively carbon"}, {"start": 115.079, "duration": 3.561, "text": "has four veence electrons but the"}, {"start": 116.88, "duration": 4.48, "text": "electrons being contributed to these"}, {"start": 118.64, "duration": 4.92, "text": "structures are four 3 and five"}, {"start": 121.36, "duration": 4.119, "text": "respectively so a carbon radical is"}, {"start": 123.56, "duration": 4.0, "text": "neutral just like a carbon that is"}, {"start": 125.479, "duration": 4.76, "text": "participating in four bonds a"}, {"start": 127.56, "duration": 5.16, "text": "carbocation has a formal positive charge"}, {"start": 130.239, "duration": 5.08, "text": "and a carban ion has a formal negative"}, {"start": 132.72, "duration": 5.48, "text": "charge then for nitrogen we have three"}, {"start": 135.319, "duration": 5.361, "text": "bonds and one loone pair four bonds and"}, {"start": 138.2, "duration": 5.24, "text": "no loone pairs or two bonds and two"}, {"start": 140.68, "duration": 5.12, "text": "loone pairs nitrogen has five veence"}, {"start": 143.44, "duration": 5.64, "text": "electrons and here we see contributions"}, {"start": 145.8, "duration": 4.799, "text": "of five 4 and six that means we have"}, {"start": 149.08, "duration": 3.84, "text": "neutral nitrogen"}, {"start": 150.599, "duration": 5.481, "text": "positively charged nitrogen and"}, {"start": 152.92, "duration": 5.48, "text": "negatively charged nitrogen moving on to"}, {"start": 156.08, "duration": 5.0, "text": "oxygen again we have two bonds and two"}, {"start": 158.4, "duration": 5.119, "text": "loone pairs three bonds and one loone"}, {"start": 161.08, "duration": 5.6, "text": "pair and then one Bond and three loone"}, {"start": 163.519, "duration": 5.8, "text": "pairs oxygen has six veence electrons"}, {"start": 166.68, "duration": 5.44, "text": "but these contribute six 5 and seven"}, {"start": 169.319, "duration": 5.241, "text": "respectively so we have neutral oxygen"}, {"start": 172.12, "duration": 5.52, "text": "positively charged oxygen and negatively"}, {"start": 174.56, "duration": 5.599, "text": "charged oxygen or an oxyanion which we"}, {"start": 177.64, "duration": 4.959, "text": "will see frequently then sulfur will"}, {"start": 180.159, "duration": 5.241, "text": "behave Just Like Oxygen since it also"}, {"start": 182.599, "duration": 5.401, "text": "has six veence electrons so by replacing"}, {"start": 185.4, "duration": 4.919, "text": "these oxygen atoms with sulfur atoms all"}, {"start": 188.0, "duration": 4.959, "text": "the same charges will apply and then"}, {"start": 190.319, "duration": 4.64, "text": "phosphorus is an analog of nitrogen and"}, {"start": 192.959, "duration": 4.56, "text": "we will sometimes see a phosphorus"}, {"start": 194.959, "duration": 3.481, "text": "cation when it is making four bonds to"}, {"start": 197.519, "duration": 3.321, "text": "other"}, {"start": 198.44, "duration": 4.56, "text": "atoms it is very important that this"}, {"start": 200.84, "duration": 5.08, "text": "ability of assigning formal charge to"}, {"start": 203.0, "duration": 4.36, "text": "any atom in a molecule becomes intuitive"}, {"start": 205.92, "duration": 4.039, "text": "because we need to know where the"}, {"start": 207.36, "duration": 5.079, "text": "charges are on a molecule to understand"}, {"start": 209.959, "duration": 6.241, "text": "how it reacts with other molecules"}, {"start": 212.439, "duration": 3.761, "text": "Professor Dave for che see you next"}, {"start": 223.76, "duration": 3.0, "text": "time"}], "title": "What Are Formal Charges? Ft. Professor Dave", "description": "In organic chemistry, it's important to be able to assign the formal charge of any atom in a particular molecule, in order to determine how these atoms may behave. Atoms can have a positive, negative, or neutral charge\u2014let's learn how exactly to determine an atom's formal charge, and how the concepts of bonds and valences fit in with this determination.\n\nGet more homework help from Chegg at https://che.gg/3HbtG8Y\nWatch the whole Organic Chemistry playlist: https://www.youtube.com/playlist?list=PLllCkartRRdMPnATf1X8qVdo5RzM8gA_M\n\nAbout Chegg:\nChegg is a leading education technology company that provides students with affordable and accessible study resources. Chegg's mission is to help students save time, save money and get smarter.  \n\nSubscribe: https://www.youtube.com/@chegg\nTikTok: https://www.tiktok.com/@chegg\nInstagram: https://www.instagram.com/chegg/\nTwitter: https://twitter.com/chegg\n\n00:00 Intro\n00:33 Valence electrons\n01:28 Carbon species\n02:13 Nitrogen species\n02:35 Oxygen/sulfur species", "lengthSeconds": 225, "width": 640, "height": 360}
{"video_id": "H0MDRWDB1Go", "transcript": [{"start": 0.0, "duration": 3.52, "text": "hey professor dave let's check out the"}, {"start": 1.12, "duration": 2.4, "text": "night sky"}, {"start": 4.05, "duration": 3.39, "text": "[Music]"}, {"start": 5.6, "duration": 3.36, "text": "it's taken almost 10 billion years but"}, {"start": 7.44, "duration": 2.48, "text": "we finally made it we started with a big"}, {"start": 8.96, "duration": 2.88, "text": "bang"}, {"start": 9.92, "duration": 3.36, "text": "we watched adam's form"}, {"start": 11.84, "duration": 3.28, "text": "we saw these collect forms stars and"}, {"start": 13.28, "duration": 3.04, "text": "galaxies we saw those stars explode to"}, {"start": 15.12, "duration": 2.72, "text": "disperse the materials that would become"}, {"start": 16.32, "duration": 3.2, "text": "all the planets and we watched our own"}, {"start": 17.84, "duration": 3.599, "text": "solar system form including the planet"}, {"start": 19.52, "duration": 3.04, "text": "earth thus begins a new chapter of the"}, {"start": 21.439, "duration": 3.68, "text": "story"}, {"start": 22.56, "duration": 4.4, "text": "upon this earth biomolecules arose"}, {"start": 25.119, "duration": 4.0, "text": "polymerized and became contained in cell"}, {"start": 26.96, "duration": 3.84, "text": "membranes life was born which slowly"}, {"start": 29.119, "duration": 3.12, "text": "evolved and eventually brought about"}, {"start": 30.8, "duration": 2.8, "text": "human beings like you and me this"}, {"start": 32.239, "duration": 2.561, "text": "fascinating chapter of the story is told"}, {"start": 33.6, "duration": 2.88, "text": "by the field of general chemistry"}, {"start": 34.8, "duration": 3.52, "text": "organic chemistry biochemistry and"}, {"start": 36.48, "duration": 3.28, "text": "biology check out my tutorials on those"}, {"start": 38.32, "duration": 2.72, "text": "subjects if you want to know more about"}, {"start": 39.76, "duration": 3.08, "text": "that whole show since it's more than a"}, {"start": 41.04, "duration": 3.6, "text": "little involved as one might"}, {"start": 42.84, "duration": 3.32, "text": "imagine i just thought it'd be fun to"}, {"start": 44.64, "duration": 3.599, "text": "have a little persona professor dave"}, {"start": 46.16, "duration": 2.879, "text": "it's just a little name um you know i"}, {"start": 48.239, "duration": 2.241, "text": "don't know"}, {"start": 49.039, "duration": 2.721, "text": "anything but i you know did uh work in"}, {"start": 50.48, "duration": 2.96, "text": "um you know i took courses without"}, {"start": 51.76, "duration": 2.799, "text": "components uh in as an undergrad and"}, {"start": 53.44, "duration": 2.799, "text": "then i did most of the masters in"}, {"start": 54.559, "duration": 5.281, "text": "synthetic but i didn't really last long"}, {"start": 56.239, "duration": 3.601, "text": "i dropped out of my semester"}, {"start": 60.48, "duration": 3.44, "text": "again for music opportunities and i"}, {"start": 62.48, "duration": 2.719, "text": "eventually got this teaching job uh for"}, {"start": 63.92, "duration": 2.559, "text": "organic chemistry at this little trade"}, {"start": 65.199, "duration": 2.401, "text": "university it was pretty great because"}, {"start": 66.479, "duration": 2.481, "text": "the bachelor's degree i was supposed to"}, {"start": 67.6, "duration": 2.559, "text": "teach the laboratory portion but on the"}, {"start": 68.96, "duration": 4.0, "text": "first day the lecture instructor had"}, {"start": 70.159, "duration": 2.801, "text": "this freakout we just took"}, {"start": 78.33, "duration": 3.07, "text": "[Music]"}, {"start": 85.36, "duration": 2.48, "text": "you're not enjoying chemistry but"}, {"start": 86.56, "duration": 3.28, "text": "through a bunch of bureaucracy quite"}, {"start": 87.84, "duration": 3.84, "text": "frankly i lost that job withdrawal yet"}, {"start": 89.84, "duration": 3.68, "text": "again from from the master program a lot"}, {"start": 91.68, "duration": 3.36, "text": "of you asked very infectiously are you a"}, {"start": 93.52, "duration": 3.44, "text": "real professor i sit literally in this"}, {"start": 95.04, "duration": 3.52, "text": "room uh like 60 to 70 hours a week and"}, {"start": 96.96, "duration": 3.04, "text": "uh i chose the name really without much"}, {"start": 98.56, "duration": 2.559, "text": "thought uh it was kind of telling you to"}, {"start": 100.0, "duration": 2.479, "text": "be honest right um i just thought it'd"}, {"start": 101.119, "duration": 3.841, "text": "be fun to have a little personal"}, {"start": 102.479, "duration": 3.68, "text": "professor it's just a little name"}, {"start": 104.96, "duration": 2.72, "text": "through a bunch of bureaucracy and i"}, {"start": 106.159, "duration": 3.521, "text": "lost that job i chose the name it was"}, {"start": 107.68, "duration": 3.04, "text": "kind of pungent to be honest so and even"}, {"start": 109.68, "duration": 2.16, "text": "at that time i had no interest in being"}, {"start": 110.72, "duration": 4.23, "text": "a chemist i do not enjoy the industries"}, {"start": 111.84, "duration": 24.5, "text": "i really really hate it"}, {"start": 114.95, "duration": 21.39, "text": "[Music]"}, {"start": 136.8, "duration": 2.079, "text": "you"}], "title": "Hey! it\u2019s Professor Dave \ud83d\ude01", "description": "we saw and watched it all... \ud83d\ude02\n\nit\u2019s not so much about WHAT he says as it\u2019s BS anyway... my point is: this guy who isn\u2018t even a real Professor has 2 Million followers!!! (how many bots... idk)\n\nso by that you can see how brainwashed and indoctrinated people are... I\u2018m sure he get\u2019s paid from the US tax money \ud83e\udd37\ud83c\udffb\u200d\u2640\ufe0f\n\nand this stupid intro... c\u2019mon!!! I wouldn't even put my children through that... SMH\n\n#indoctrination #brainwashedsociety \n#useyourbrain \n#washyourhands \ud83d\ude06\ud83d\ude02\ud83d\ude44\n#stayreal \ud83d\ude4f\ud83d\udd25", "lengthSeconds": 137, "width": 640, "height": 360}
{"video_id": "knKngmf_qh8", "transcript": [{"start": 0.0, "duration": 4.08, "text": "here's the homework problem as you know"}, {"start": 2.48, "duration": 5.68, "text": "last week"}, {"start": 4.08, "duration": 7.04, "text": "we caught a mistake with sal khan from"}, {"start": 8.16, "duration": 3.76, "text": "khan academy this week we found another"}, {"start": 11.12, "duration": 4.0, "text": "mistake from"}, {"start": 11.92, "duration": 6.48, "text": "another famous physics professor"}, {"start": 15.12, "duration": 5.52, "text": "his channel named professor dave"}, {"start": 18.4, "duration": 5.92, "text": "explains let's take a look he knows a"}, {"start": 20.64, "duration": 3.68, "text": "lot about the science stuff"}, {"start": 24.64, "duration": 6.24, "text": "if you compress the spring and put a"}, {"start": 27.199, "duration": 5.761, "text": "little 0.01 kilogram army guy on top"}, {"start": 30.88, "duration": 3.199, "text": "how high will it be launched when you"}, {"start": 32.96, "duration": 3.52, "text": "release the spring"}, {"start": 34.079, "duration": 3.041, "text": "and this is the answer as it happens we"}, {"start": 36.48, "duration": 3.599, "text": "could use"}, {"start": 37.12, "duration": 3.599, "text": "this same equation again and we could"}, {"start": 40.079, "duration": 4.081, "text": "plug in"}, {"start": 40.719, "duration": 4.16, "text": "our initial velocity as well as a final"}, {"start": 44.16, "duration": 3.919, "text": "velocity"}, {"start": 44.879, "duration": 5.761, "text": "of zero since velocity will be zero"}, {"start": 48.079, "duration": 3.201, "text": "at the moment that the object reaches"}, {"start": 50.64, "duration": 4.8, "text": "the peak"}, {"start": 51.28, "duration": 9.439, "text": "of the trajectory and we get 1.35 meters"}, {"start": 55.44, "duration": 8.16, "text": "this is wrong so i want you to"}, {"start": 60.719, "duration": 2.881, "text": "get the right answer"}], "title": "Lunching Things with Spring : Fix the mistake Professor Dave made", "description": "Learn Math & Science @ https://brilliant.org/BariScienceLab", "lengthSeconds": 62, "width": 640, "height": 360}
{"video_id": "s-aFj5GVkwA", "transcript": [{"start": 0.12, "duration": 5.199, "text": "okay number four asks which of the"}, {"start": 2.32, "duration": 4.8, "text": "following is the strongest base that"}, {"start": 5.319, "duration": 5.481, "text": "means that we're looking at the ability"}, {"start": 7.12, "duration": 6.599, "text": "of these compounds to accept a proton in"}, {"start": 10.8, "duration": 6.479, "text": "all of these cases we're looking at some"}, {"start": 13.719, "duration": 6.161, "text": "lone pairs on oxygen and nitrogen so"}, {"start": 17.279, "duration": 5.24, "text": "this oxygen has two lone pairs this"}, {"start": 19.88, "duration": 4.719, "text": "oxygen has two lone pairs this nitrogen"}, {"start": 22.519, "duration": 5.281, "text": "has one lone pair and this nitrogen has"}, {"start": 24.599, "duration": 5.641, "text": "one lone pair in each case an acid base"}, {"start": 27.8, "duration": 5.04, "text": "reaction is going to occur whereby"}, {"start": 30.24, "duration": 5.6, "text": "either oxygen or nitrogen is accepting a"}, {"start": 32.84, "duration": 5.879, "text": "proton with a lone pair and so which is"}, {"start": 35.84, "duration": 5.16, "text": "able to do it best now what we have to"}, {"start": 38.719, "duration": 5.121, "text": "understand is that these lone pairs are"}, {"start": 41.0, "duration": 5.28, "text": "not all equally available for acid base"}, {"start": 43.84, "duration": 4.84, "text": "reactions some of them are tied up in"}, {"start": 46.28, "duration": 5.64, "text": "resonance so for example over here on"}, {"start": 48.68, "duration": 5.719, "text": "this phol we've got a lone pair that can"}, {"start": 51.92, "duration": 4.4, "text": "get pushed down here and then all of"}, {"start": 54.399, "duration": 3.84, "text": "this Pi electron density can be"}, {"start": 56.32, "duration": 4.199, "text": "delocalized we have a lot of different"}, {"start": 58.239, "duration": 2.28, "text": "res"}], "title": "Which is the Strongest Base?", "description": "This is an abbreviated version of this question for YouTube Shorts. Check out the full version in the linked video.", "lengthSeconds": 59, "width": 360, "height": 640}
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No wait, that's what gets reduced, or is it the reducing agent? Ahh! Stupid binary concepts! OK, settle down. Watch this clip and learn everything you need to know about electron transfer.\n\nWatch the whole General Chemistry playlist: http://bit.ly/ProfDaveGenChem\n\nStudy for the AP Chemistry exam with me: https://bit.ly/ProfDaveAPChem\n\nOrganic Chemistry Tutorials: http://bit.ly/ProfDaveOrgChem\nBiochemistry Tutorials: http://bit.ly/ProfDaveBiochem\nBiology Tutorials: http://bit.ly/ProfDaveBio\nClassical Physics Tutorials: http://bit.ly/ProfDavePhysics1\nModern Physics Tutorials: http://bit.ly/ProfDavePhysics2\nMathematics Tutorials: http://bit.ly/ProfDaveMaths\n\nEMAIL\u25ba ProfessorDaveExplains@gmail.com\nPATREON\u25ba http://patreon.com/ProfessorDaveExplains\n\nCheck out \"Is This Wi-Fi Organic?\", my book on disarming pseudoscience!\nAmazon: https://amzn.to/2HtNpVH\nBookshop: https://bit.ly/39cKADM\n\rBarnes and Noble: https://bit.ly/3pUjmrn \nBook Depository: http://bit.ly/3aOVDlT", "lengthSeconds": 231, "width": 640, "height": 360}
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{"video_id": "sAPT6HTRcEs", "transcript": [{"start": 6.14, "duration": 4.2, "text": "so let's practice calculating\nequilibrium concentrations"}, {"start": 10.34, "duration": 6.82, "text": "consider the following equilibrium with\nN2 + 3H2 in equilibrium with 2NH3"}, {"start": 17.16, "duration": 7.289, "text": "now at equilibrium the nitrogen\nconcentration is 4 point 2 6 molar the"}, {"start": 24.449, "duration": 5.521, "text": "hydrogen concentration is 2 point zero 9\nmolar and the equilibrium constant for"}, {"start": 29.97, "duration": 6.06, "text": "this system is zero point zero six zero\nzero what is the equilibrium"}, {"start": 36.03, "duration": 6.02, "text": "concentration of NH3? so if this seems\nconfusing check out my tutorial on"}, {"start": 42.05, "duration": 2.35, "text": "equilibria and then give this a try."}, {"start": 62.82, "duration": 2.58, "text": "so this is a very straightforward situation"}, {"start": 65.4, "duration": 4.23, "text": "here is our equilibrium now the first\nthing we want to do is generate our"}, {"start": 69.63, "duration": 5.37, "text": "equilibrium constant expression now\nrecall that will involve placing the"}, {"start": 75.0, "duration": 5.159, "text": "concentrations of the products over the\nconcentrations of the reactants and then"}, {"start": 80.159, "duration": 4.371, "text": "the concentrations are going to be\nraised to the power of the relevant"}, {"start": 84.53, "duration": 5.08, "text": "stoichiometric coefficient so whatever\nthe coefficient is on a substance the"}, {"start": 89.61, "duration": 5.549, "text": "concentration of that substance will be\nraised to that power so in this case we"}, {"start": 95.159, "duration": 5.07, "text": "have our equilibrium constant being\nequal to the concentration of ammonia"}, {"start": 100.229, "duration": 4.71, "text": "squared because ammonia has a\ncoefficient of 2 over the concentration"}, {"start": 104.939, "duration": 5.371, "text": "of nitrogen times the concentration of\nhydrogen raised to the 3rd power because"}, {"start": 110.31, "duration": 6.629, "text": "hydrogen has a coefficient of 3 so that\nis our expression and now we just plug"}, {"start": 116.939, "duration": 5.04, "text": "in what we know and as it turns out we\nknow everything except one thing so we"}, {"start": 121.979, "duration": 5.1, "text": "have that constant it is zero point zero\nsix zero zero we have our nitrogen"}, {"start": 127.079, "duration": 4.111, "text": "concentration it's four point two six\nand we have our hydrogen concentration"}, {"start": 131.19, "duration": 4.169, "text": "it's two point zero nine\nand that will be raised to the third"}, {"start": 135.359, "duration": 4.86, "text": "power so here all we have to do is a\nlittle bit of arithmetic and we get the"}, {"start": 140.219, "duration": 5.011, "text": "ammonia concentration squared as being\nequal to two point three three and then"}, {"start": 145.23, "duration": 5.929, "text": "we take the square root and our answer\nwill be one point five three molar."}], "title": "Practice Problem: Calculating Equilibrium Concentrations", "description": "We know how to write equilibrium constant expressions for any equilibrium, and we know that they relate the equilibrium constant to the equilibrium concentrations for all of the substances in the system. Let's see if we can calculate a missing equilibrium concentration when other data is given!\n\nTry all of the general chemistry practice problems: http://bit.ly/ProfDaveGCPP\n\nGeneral Chemistry Tutorials: http://bit.ly/ProfDaveGenChem\n\nEMAIL\u25ba ProfessorDaveExplains@gmail.com\nPATREON\u25ba http://patreon.com/ProfessorDaveExplains\n\nCheck out \"Is This Wi-Fi Organic?\", my book on disarming pseudoscience!\nAmazon: https://amzn.to/2HtNpVH\nBookshop: https://bit.ly/39cKADM\n\rBarnes and Noble: https://bit.ly/3pUjmrn \nBook Depository: http://bit.ly/3aOVDlT", "lengthSeconds": 182, "width": 640, "height": 360}
{"video_id": "msTcsxCxgmY", "transcript": [{"start": 6.32, "duration": 4.26, "text": "okay let's try a question\nrelating to spontaneity consider the"}, {"start": 10.59, "duration": 6.72, "text": "conversion of limestone to lime so we\nhave calcium carbonate caco3 yielding"}, {"start": 17.31, "duration": 8.61, "text": "calcium oxide CA o and co2 gas the\nvalues of Delta H and Delta S are 179"}, {"start": 25.92, "duration": 6.63, "text": "point 1 kilojoules per mole and 160\npoint 2 joules per Kelvin respectively"}, {"start": 32.55, "duration": 5.939, "text": "at 298 K and 1 bar assuming that these\nvalues do not change with temperature"}, {"start": 38.489, "duration": 6.361, "text": "what is the temperature above which\nconversion of limestone to lime will be"}, {"start": 44.85, "duration": 6.18, "text": "spontaneous so we need to know some\nthings about Gibbs free energy enthalpy"}, {"start": 51.03, "duration": 4.59, "text": "entropy we need to know how those values\ncorrelate we need to know something"}, {"start": 55.62, "duration": 4.95, "text": "about the conditions within which a\nreaction will be spontaneous so go ahead"}, {"start": 60.57, "duration": 4.65, "text": "and check out my tutorial on Gibbs free\nenergy if you need some review and when"}, {"start": 65.22, "duration": 1.3, "text": "you're ready give it a try"}, {"start": 81.98, "duration": 2.28, "text": "so the first\nthing we have to remember is that"}, {"start": 84.27, "duration": 5.88, "text": "reactions are spontaneous when Delta G\nis less than zero so we need Delta G to"}, {"start": 90.15, "duration": 5.67, "text": "be negative in order for a reaction to\nbe spontaneous so if we have Delta G"}, {"start": 95.82, "duration": 5.7, "text": "equals Delta H minus T Delta s and we\nneed Delta G to be less than zero then"}, {"start": 101.52, "duration": 4.709, "text": "we also need Delta H minus T Delta s to\nbe less than zero because those are"}, {"start": 106.229, "duration": 5.64, "text": "equivalent now because we're trying to\nfigure out the temperature above which"}, {"start": 111.869, "duration": 3.96, "text": "conversion of limestone to lime will be\nspontaneous that means we're going to be"}, {"start": 115.829, "duration": 4.741, "text": "solving for a temperature so let's\nrearrange this inequality to solve for"}, {"start": 120.57, "duration": 5.579, "text": "temperature all we do is add T Delta s\nto both sides and then divide by Delta s"}, {"start": 126.149, "duration": 7.17, "text": "and we get T is greater than Delta H\nover Delta S so we can just go ahead and"}, {"start": 133.319, "duration": 3.601, "text": "plug in the\nuse that we know now we have 179.1"}, {"start": 136.92, "duration": 5.58, "text": "kilojoules per mole for Delta H but our\nentropy value has joules so these are"}, {"start": 142.5, "duration": 4.08, "text": "gonna need to agree let's just go ahead\nand multiply this one by a thousand that"}, {"start": 146.58, "duration": 3.09, "text": "way we'll have it in joules so one\nhundred and seventy-nine thousand one"}, {"start": 149.67, "duration": 4.679, "text": "hundred joules per mole and then we can\ndivide that by a hundred and sixty point"}, {"start": 154.349, "duration": 6.531, "text": "two joules per Kelvin that way the\njoules will cancel and we end up with"}, {"start": 160.88, "duration": 7.18, "text": "1118 Kelvin so that means that despite\nthe enthalpic unfavorability we see that"}, {"start": 168.06, "duration": 3.87, "text": "this is endothermic right one hundred\nseventy nine point one kilojoules per"}, {"start": 171.93, "duration": 4.71, "text": "mole this is an endothermic reaction\nwhich we typically expect to not be"}, {"start": 176.64, "duration": 4.89, "text": "spontaneous however it is entropically\nfavorable we're producing more"}, {"start": 181.53, "duration": 4.08, "text": "substances we're producing a gas\ntherefore if we raise the temperature"}, {"start": 185.61, "duration": 5.219, "text": "high enough that T Delta s term will\noutweigh the Delta H term and it happens"}, {"start": 190.829, "duration": 7.911, "text": "to be this value 1118 Kelvin above that\ntemperature this will be a spontaneous process."}], "title": "IIT/JEE Chemistry Practice #30: Spontaneity", "description": "Practice REAL problems from actual past IIT/JEE exams with Professor Dave!\n\nTry all of the IIT/JEE chemistry practice problems: http://bit.ly/ProfDaveIITJEEChem\n\nGeneral Chemistry Tutorials: http://bit.ly/ProfDaveGenChem\nOrganic Chemistry Tutorials: http://bit.ly/ProfDaveOrgChem\n\nEMAIL\u25ba ProfessorDaveExplains@gmail.com\nPATREON\u25ba http://patreon.com/ProfessorDaveExplains\n\nCheck out \"Is This Wi-Fi Organic?\", my book on disarming pseudoscience!\nAmazon: https://amzn.to/2HtNpVH\nBookshop: https://bit.ly/39cKADM\n\rBarnes and Noble: https://bit.ly/3pUjmrn \nBook Depository: http://bit.ly/3aOVDlT", "lengthSeconds": 231, "width": 640, "height": 360}
{"video_id": "24QIDWndxqQ", "transcript": [{"start": 0.12, "duration": 3.76, "text": "okay number eight asks which is the"}, {"start": 2.0, "duration": 3.68, "text": "lowest energy confirmation for"}, {"start": 3.88, "duration": 3.919, "text": "cyclohexene so this one's pretty"}, {"start": 5.68, "duration": 4.44, "text": "straightforward option A is chair let's"}, {"start": 7.799, "duration": 3.96, "text": "just draw the chair really quick and so"}, {"start": 10.12, "duration": 4.639, "text": "what we need to understand about the"}, {"start": 11.759, "duration": 5.36, "text": "chair is it does a couple of things"}, {"start": 14.759, "duration": 5.52, "text": "first it allows for all of the carbons"}, {"start": 17.119, "duration": 5.641, "text": "to obey the perfect tetrahedral geometry"}, {"start": 20.279, "duration": 4.601, "text": "that they enjoy for example we've got"}, {"start": 22.76, "duration": 4.519, "text": "two hydrogens right here we can"}, {"start": 24.88, "duration": 4.84, "text": "essentially imagine this as being a"}, {"start": 27.279, "duration": 4.48, "text": "wedge Bond and this as being a dash Bond"}, {"start": 29.72, "duration": 4.24, "text": "down to those other carbons but however"}, {"start": 31.759, "duration": 5.401, "text": "you want to visualize it it has that"}, {"start": 33.96, "duration": 5.16, "text": "perfect tetrahedral geometry with 109.5"}, {"start": 37.16, "duration": 4.36, "text": "degree Bond angles and then the other"}, {"start": 39.12, "duration": 4.759, "text": "thing it does is that all of the Newman"}, {"start": 41.52, "duration": 5.559, "text": "projections for all of these bonds would"}, {"start": 43.879, "duration": 4.961, "text": "put the different alkal groups as anti"}, {"start": 47.079, "duration": 3.601, "text": "to one another so for example this"}, {"start": 48.84, "duration": 3.519, "text": "carbon carbon bond is anti to this"}, {"start": 50.68, "duration": 3.96, "text": "carbon carbon Bond and you can do that"}, {"start": 52.359, "duration": 4.481, "text": "for all the carboncarbon bonds so number"}, {"start": 54.64, "duration": 5.879, "text": "one there are no eclipsing interactions"}, {"start": 56.84, "duration": 3.679, "text": "the alal are not eclipsing"}], "title": "Conformations of Cyclohexane", "description": "This is an abbreviated version of this question for YouTube Shorts. Check out the full version in the linked video.", "lengthSeconds": 59, "width": 360, "height": 640}
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\u0648\u0644\u064a\u0633\u062a \u0634\u064a\u0621 \u0641\u0648\u0642\u0647\u060c \u0628\u064a\u0646\u0645\u0627 \u0627\u0644\u0642\u0627\u0631\u0627\u062a"}, {"start": 137.81, "duration": 1.439, "text": "\u0647\u064a \u0627\u0644\u0623\u0645\u062a\u0639\u0629.\n \n\u062a\u0631\u062c\u0645\u0629 \u0641\u0631\u064a\u0642 @autrjim"}], "title": "Plate Tectonics Explained", "description": "How do plate tectonics REALLY work?\n\nPlease support us on Patreon at: http://www.patreon.com/minuteearth\n\nAnd subscribe! - http://www.youtube.com/user/minuteearth?sub_confirmation=1\n\nThanks to our Patreon patrons:\n   - @AntoineCoeur\n   - salar tel\n   - @TodayIFoundOut1\n   - @82abhilash\n\n________________________\nCreated by Henry Reich\nProduction and Writing Team: Alex Reich, Peter Reich, Emily Elert, Ever Salazar, Kate Yoshida, and Henry Reich\nMusic by Nathaniel Schroeder: http://www.soundcloud.com/drschroeder\n\n________________________\nFree iTunes podcasts of MinuteEarth! - https://goo.gl/sfwS6n\nFacebook - http://facebook.com/minuteearth\nTwitter - http://twitter.com/MinuteEarth\n\nMinuteEarth provides an energetic and entertaining view of trends in earth's environment -- in just a few minutes!\n\n________________________\nReferences\n\nAnderson, D. L. (2001). Top-down tectonics?. Science, 293(5537), 2016-2018. http://www.sciencemag.org/content/293/5537/2016.short\n\nConrad, C. P., & Lithgow-Bertelloni, C. (2002). How mantle slabs drive plate tectonics. Science, 298(5591), 207-209. ftp://128.171.151.230/engels/Stanley/Textbook_update/Science_298/Conrad-02.pdf\n\nConrad, C.P., 2014. Personal Communication\n\nDietz, R. S. (1961). Continent and ocean basin evolution by spreading of the sea floor. Nature, 190(4779), 854-857. http://eps.mcgill.ca/~courses/c350/lecturestuff/jan28/Dietz_1961.pdf\n\nForsyth, D., & Uyeda, S. (1975). On the relative importance of the driving forces of plate motion. Geophysical Journal International, 43(1), 163-200. http://gji.oxfordjournals.org/content/43/1/163.short\n\nHecht, J. 2014, Aug 29. Earth's tectonic plates have doubled their speed. New Scientist. http://www.newscientist.com/article/mg22329843.000-earths-tectonic-plates-have-doubled-their-speed.html#.VBDAJ7ywJ_Q\n\nSmil, V. 2007. Energy in Nature and Society: General Energetics of Complex Systems. MIT Press. http://mitpress.mit.edu/books/energy-nature-and-society\n\nDriving forces of plate tectonics (website):\nhttp://www.columbia.edu/~vjd1/driving_forces_basic.htm", "lengthSeconds": 156, "width": 640, "height": 360}
{"video_id": "kuGmsnzjbpE", "transcript": [{"start": 0.0, "duration": 2.1, "text": "Professor Dave here, let's talk about the"}, {"start": 2.1, "duration": 1.78, "text": "third law of thermodynamics."}, {"start": 10.86, "duration": 2.06, "text": "We've gone through three of the laws of"}, {"start": 12.929, "duration": 2.761, "text": "thermodynamics, which leaves just one to"}, {"start": 15.69, "duration": 2.04, "text": "go, and the third law of thermodynamics"}, {"start": 17.73, "duration": 3.15, "text": "deals with another strange concept: the"}, {"start": 20.88, "duration": 2.79, "text": "idea of a temperature of absolute zero."}, {"start": 23.67, "duration": 3.0, "text": "We know that temperatures can get very"}, {"start": 26.67, "duration": 2.97, "text": "hot and very cold, but not everyone is"}, {"start": 29.64, "duration": 2.219, "text": "familiar with the idea of the coldest"}, {"start": 31.859, "duration": 2.731, "text": "possible temperature. However, with our"}, {"start": 34.59, "duration": 1.68, "text": "new understanding of heat and"}, {"start": 36.27, "duration": 2.49, "text": "temperature, it should make perfect sense."}, {"start": 38.76, "duration": 3.33, "text": "We know that temperature is a measure of"}, {"start": 42.09, "duration": 3.059, "text": "available heat energy, which is just a"}, {"start": 45.149, "duration": 2.401, "text": "collection of kinetic energies of motion."}, {"start": 47.55, "duration": 2.669, "text": "We know that as the temperature"}, {"start": 50.219, "duration": 2.61, "text": "decreases, the internal energy of a"}, {"start": 52.829, "duration": 2.761, "text": "system decreases, as does the kinetic"}, {"start": 55.59, "duration": 3.239, "text": "energy of each particle. As the"}, {"start": 58.829, "duration": 2.521, "text": "temperature becomes very cold, particles"}, {"start": 61.35, "duration": 2.699, "text": "slow to a complete stop, and there is no"}, {"start": 64.049, "duration": 2.791, "text": "more kinetic energy. No more kinetic"}, {"start": 66.84, "duration": 2.01, "text": "energy of motion means no more"}, {"start": 68.85, "duration": 2.04, "text": "temperature by any real definition of"}, {"start": 70.89, "duration": 2.7, "text": "the term, so we will call this absolute"}, {"start": 73.59, "duration": 3.12, "text": "zero. At absolute zero, every substance"}, {"start": 76.71, "duration": 1.769, "text": "would be a solid, even things like"}, {"start": 78.479, "duration": 2.551, "text": "hydrogen and helium, and one way of"}, {"start": 81.03, "duration": 1.71, "text": "summarizing the third law of"}, {"start": 82.74, "duration": 2.4, "text": "thermodynamics is by saying that the"}, {"start": 85.14, "duration": 2.58, "text": "entropy of any crystalline substance at"}, {"start": 87.72, "duration": 4.71, "text": "absolute zero will be equal to zero. This"}, {"start": 92.43, "duration": 1.71, "text": "is a necessary consequence of"}, {"start": 94.14, "duration": 2.88, "text": "Boltzmann's equation, S equals k times"}, {"start": 97.02, "duration": 2.7, "text": "the natural log of W. If there is no"}, {"start": 99.72, "duration": 2.82, "text": "kinetic energy, nothing is moving, so"}, {"start": 102.54, "duration": 2.25, "text": "there is only one possible microstate"}, {"start": 104.79, "duration": 2.43, "text": "that the particles can inhabit. With a"}, {"start": 107.22, "duration": 3.329, "text": "value for W equal to one, the natural log"}, {"start": 110.549, "duration": 3.871, "text": "of 1 is 0, so at absolute zero the"}, {"start": 114.42, "duration": 2.729, "text": "entropy of any substance must also be"}, {"start": 117.149, "duration": 3.271, "text": "zero. However, absolute zero can never"}, {"start": 120.42, "duration": 2.46, "text": "truly be reached, which is communicated"}, {"start": 122.88, "duration": 2.4, "text": "in a rephrasing of the third law which"}, {"start": 125.28, "duration": 2.369, "text": "states that it is impossible to lower"}, {"start": 127.649, "duration": 1.95, "text": "the temperature of any system to"}, {"start": 129.6, "duration": 2.92, "text": "absolute zero in a finite number of steps."}, {"start": 133.54, "duration": 2.14, "text": "Satisfactory analysis of this law will"}, {"start": 135.68, "duration": 1.71, "text": "require the quantum mechanical"}, {"start": 137.39, "duration": 2.55, "text": "zero-point energy, and will thus have to"}, {"start": 139.94, "duration": 1.86, "text": "wait until the modern physics course."}, {"start": 141.8, "duration": 3.51, "text": "This concludes our analysis of the laws"}, {"start": 145.31, "duration": 2.37, "text": "of thermodynamics, a near mythical"}, {"start": 147.68, "duration": 1.98, "text": "accomplishment in the history of science."}, {"start": 149.66, "duration": 2.73, "text": "It also means we have covered some of"}, {"start": 152.39, "duration": 2.28, "text": "the harder stuff in physics, so now we"}, {"start": 154.67, "duration": 1.92, "text": "get to move on to fun things like waves"}, {"start": 156.59, "duration": 3.13, "text": "and magnets. But first, let's check comprehension."}, {"start": 189.98, "duration": 2.1, "text": "Thanks for watching, guys. Subscribe to my channel for"}, {"start": 192.08, "duration": 2.46, "text": "more tutorials, support me on patreon so I can"}, {"start": 194.54, "duration": 2.16, "text": "keep making content, and as always feel"}, {"start": 196.7, "duration": 0.78, "text": "free to email me:"}], "title": "The Third Law of Thermodynamics: Absolute Zero", "description": "Brr, it's so cold today! Could it get any colder? Is there a coldest possible temperature? Yes, there is! That seems strange, but now we know that temperature is just a measure of kinetic energy, so zero kinetic energy must mean zero temperature! Let's learn more about this winter wonderland of physics as we wrap up our adventures with the laws of thermodynamics.\n\nWatch the whole Classical Physics playlist: http://bit.ly/ProfDavePhysics1\n\nModern Physics Tutorials: http://bit.ly/ProfDavePhysics2\nMathematics Tutorials: http://bit.ly/ProfDaveMaths\nGeneral Chemistry Tutorials: http://bit.ly/ProfDaveGenChem\nOrganic Chemistry Tutorials: http://bit.ly/ProfDaveOrgChem\nBiochemistry Tutorials: http://bit.ly/ProfDaveBiochem\nBiology Tutorials: http://bit.ly/ProfDaveBio\n\nEMAIL\u25ba ProfessorDaveExplains@gmail.com\nPATREON\u25ba http://patreon.com/ProfessorDaveExplains\n\nCheck out \"Is This Wi-Fi Organic?\", my book on disarming pseudoscience!\nAmazon: https://amzn.to/2HtNpVH\nBookshop: https://bit.ly/39cKADM\n\rBarnes and Noble: https://bit.ly/3pUjmrn \nBook Depository: http://bit.ly/3aOVDlT", "lengthSeconds": 220, "width": 640, "height": 360}
{"video_id": "t-apvn2jvmU", "transcript": [{"start": 0.08, "duration": 4.199, "text": "three what is the iup pack name for the"}, {"start": 2.6, "duration": 4.839, "text": "following compound we have another one"}, {"start": 4.279, "duration": 5.161, "text": "here we have a cyclohexane so the way"}, {"start": 7.439, "duration": 4.12, "text": "that we name cyclohexanes is a little"}, {"start": 9.44, "duration": 4.48, "text": "bit different from linear molecules what"}, {"start": 11.559, "duration": 6.121, "text": "we need to do is we need to find the"}, {"start": 13.92, "duration": 6.16, "text": "lowest set of locant possible so we have"}, {"start": 17.68, "duration": 4.599, "text": "three substituents and we need to number"}, {"start": 20.08, "duration": 4.92, "text": "the ring so as to give them the lowest"}, {"start": 22.279, "duration": 4.521, "text": "set of locant possible one could be any"}, {"start": 25.0, "duration": 4.08, "text": "of these carbons and then we go either"}, {"start": 26.8, "duration": 4.12, "text": "clockwise or counterclockwise but it"}, {"start": 29.08, "duration": 4.999, "text": "happens to be the case that this is the"}, {"start": 30.92, "duration": 6.959, "text": "lowest set of locant that we can get is"}, {"start": 34.079, "duration": 6.64, "text": "if we do 1 2 4 so if we started with the"}, {"start": 37.879, "duration": 5.641, "text": "ethyl we could get 134 if we started"}, {"start": 40.719, "duration": 5.241, "text": "with the bromo we could get 125 right"}, {"start": 43.52, "duration": 4.08, "text": "none of these are the lowest possible"}, {"start": 45.96, "duration": 4.2, "text": "the one that I've drawn right here"}, {"start": 47.6, "duration": 4.4, "text": "that's the lowest set of loans possible"}, {"start": 50.16, "duration": 4.28, "text": "we have to do it that way because these"}, {"start": 52.0, "duration": 5.8, "text": "alkal and hallogen substituents are all"}, {"start": 54.44, "duration": 6.079, "text": "of equal priority so we just go 1 2 3 4"}, {"start": 57.8, "duration": 2.719, "text": "5 6 like that"}], "title": "IUPAC Nomenclature of a Cycloalkane", "description": "This is an abbreviated version of this question for YouTube Shorts. Check out the full version in the linked video.", "lengthSeconds": 59, "width": 360, "height": 640}
{"video_id": "Y_w07A7chnk", "transcript": [{"start": 5.0, "duration": 4.0, "text": "Before we take a look at how osmosis and tonicity affect"}, {"start": 9.0, "duration": 4.0, "text": "a cell, let\u2019s review what each of these terms means."}, {"start": 13.0, "duration": 4.0, "text": "Osmosis represents the diffusion of water across a semi-permeable"}, {"start": 17.0, "duration": 1.933, "text": "membrane."}, {"start": 18.933, "duration": 4.0, "text": "The term tonicity refers to the relative solute concentration"}, {"start": 22.933, "duration": 4.0, "text": "of two environments separated by a semipermeable membrane."}, {"start": 26.933, "duration": 4.0, "text": "In other words, by comparing the tonicity of the solution, "}, {"start": 30.933, "duration": 4.0, "text": "you can determine the direction in which osmosis will occur."}, {"start": 34.933, "duration": 1.3, "text": " "}, {"start": 36.233, "duration": 4.0, "text": "To demonstrate how tonicity affects a cell, "}, {"start": 40.233, "duration": 4.0, "text": "let\u2019s place some red blood cells into a beaker containing pure water."}, {"start": 44.233, "duration": 4.0, "text": "In this case, the solute concentration is greater inside"}, {"start": 48.233, "duration": 4.0, "text": "the cells, than in the surrounding water.  In other words, "}, {"start": 52.233, "duration": 4.0, "text": "the contents of the cells are hypertonic in relation to the hypotonic"}, {"start": 56.233, "duration": 4.0, "text": "contents of the beaker.  Because of this, osmotic"}, {"start": 60.233, "duration": 4.0, "text": "pressure results in the diffusion of water across the membrane and into the"}, {"start": 64.233, "duration": 4.0, "text": "cells.  Over time, if enough water enters the cells, "}, {"start": 68.233, "duration": 4.0, "text": "the cell membranes may burst.  This is called lysis."}, {"start": 72.233, "duration": 0.967, "text": " "}, {"start": 73.2, "duration": 4.0, "text": "Now lets place the red blood cells in a beaker containing a solution "}, {"start": 77.2, "duration": 4.0, "text": "of salt, such as sodium chloride.  Since the contents of the beaker"}, {"start": 81.2, "duration": 4.0, "text": "beaker are hypertonic in relation to the interior of the cell, "}, {"start": 85.2, "duration": 4.0, "text": "the water within the cell will diffuse across the membrane and into the"}, {"start": 89.2, "duration": 4.0, "text": "contents of the beaker.  This causes crenation or shrinking of"}, {"start": 93.2, "duration": 2.5, "text": "the cells."}, {"start": 95.7, "duration": 4.0, "text": "If the red blood cells are placed in a beaker whose contents"}, {"start": 99.7, "duration": 4.0, "text": "match the tonicity within the cells, then there is no net gain"}, {"start": 103.7, "duration": 4.0, "text": "or loss of water.  The environments within the beaker"}, {"start": 107.7, "duration": 4.0, "text": "and inside the cells are said to be isotonic, or the same."}, {"start": 111.7, "duration": 0.166, "text": " "}, {"start": 111.866, "duration": 4.0, "text": "An important thing to remember, is that osmotic pressure"}, {"start": 115.866, "duration": 4.0, "text": "always causes water to move from a hypotonic environment toward"}, {"start": 119.866, "duration": 4.0, "text": "a hypertonic environment.  In other words, water moves toward"}, {"start": 123.866, "duration": 4.0, "text": "areas of high salt or sugar concentrations.  This simple"}, {"start": 127.866, "duration": 4.0, "text": "process is used to drive the operation of our kidneys and explains"}, {"start": 131.866, "duration": 4.0, "text": "some of the physiological consequences of diseases such as diabetes."}, {"start": 135.866, "duration": 3.0, "text": " "}], "title": "Osmosis and Tonicity", "description": "This 2 minute animation describes the relationship beteween osmosis and tonicity.  Find more free tutorials, videos and readings for the science classroom at ricochetscience.com", "lengthSeconds": 144, "width": 640, "height": 360}
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\u0639\u064a\u0646\u0629 0.342 \u062c\u0631\u0627\u0645"}, {"start": 85.14, "duration": 3.67, "text": "\u0648\u0646\u062c\u0645\u0639 1.156 \u062c\u0631\u0627\u0645\u064b\u0627 \u0645\u0646 \u062b\u0627\u0646\u064a \u0623\u0643\u0633\u064a\u062f \u0627\u0644\u0643\u0631\u0628\u0648\u0646"}, {"start": 88.81, "duration": 4.14, "text": "\u0647\u0646\u0627, \u0644\u0623\u0646 \u062b\u0627\u0646\u064a \u0623\u0643\u0633\u064a\u062f \u0627\u0644\u0643\u0631\u0628\u0648\u0646 \u064a\u0632\u0646 44 \u0648\u062d\u062f\u0629 \u0643\u062a\u0644\u0629 \u0630\u0631\u064a\u0629"}, {"start": 92.95, "duration": 3.63, "text": "\u0648\u062a\u0632\u0646 \u0630\u0631\u0629 \u0627\u0644\u0643\u0631\u0628\u0648\u0646 12 \u060c \u0643\u0644 \u062c\u0632\u064a\u0621 \u0645\u0646 \u062b\u0627\u0646\u064a \u0623\u0643\u0633\u064a\u062f \u0627\u0644\u0643\u0631\u0628\u0648\u0646"}, {"start": 96.58, "duration": 4.67, "text": "27.3\u066a \u0643\u0631\u0628\u0648\u0646 \u0628\u0627\u0644\u0643\u062a\u0644\u0629"}, {"start": 101.25, "duration": 3.7, "text": "\u0647\u0630\u0627 \u064a\u0639\u0646\u064a \u0623\u0646\u0647 \u064a\u0648\u062c\u062f \u0641\u064a \u062b\u0627\u0646\u064a \u0623\u0643\u0633\u064a\u062f \u0627\u0644\u0643\u0631\u0628\u0648\u0646 0.316 \u062c\u0631\u0627\u0645\u064b\u0627"}, {"start": 104.95, "duration": 3.43, "text": "\u0645\u0646 \u0627\u0644\u0643\u0631\u0628\u0648\u0646, \u0625\u0630\u0627\u064b \u0645\u0646\u0644\u0639\u064a\u0646\u0629 \u0627\u0644\u0647\u064a\u062f\u0631\u0648\u0643\u0631\u0628\u0648\u0646 \u0627\u0644\u0623\u0635\u0644\u064a\u0629"}, {"start": 108.38, "duration": 3.41, "text": "\u0628\u0647\u0630\u0627 \u0627\u0644\u0642\u062f\u0631 \u0645\u0646\u0647\u0627 \u064a\u062c\u0628 \u0623\u0646 \u064a\u0643\u0648\u0646 \u0645\u0639\u0638\u0645\u0647\u0627 \u0645\u0646 \u0627\u0644\u0643\u0631\u0628\u0648\u0646 \u0648\u0627\u0644\u0628\u0627\u0642\u064a"}, {"start": 111.79, "duration": 2.64, "text": "\u064a\u062c\u0628 \u0623\u0646 \u064a\u0643\u0648\u0646 \u0627\u0644\u0647\u064a\u062f\u0631\u0648\u062c\u064a\u0646"}, {"start": 114.43, "duration": 3.43, "text": "\u0647\u0630\u0627 \u064a\u062c\u0639\u0644 \u0627\u0644\u0647\u064a\u062f\u0631\u0648\u0643\u0631\u0628\u0648\u0646 92.4\u066a \u0643\u0631\u0628\u0648\u0646"}, {"start": 117.86, "duration": 4.149, "text": "\u0648 7.6\u066a \u0647\u064a\u062f\u0631\u0648\u062c\u064a\u0646 \u0628\u0627\u0644\u0643\u062a\u0644\u0629"}, {"start": 122.009, "duration": 4.25, "text": "\u0646\u0633\u0645\u064a\u0647\u0627 \u0646\u0633\u0628\u0629 \u062a\u0643\u0648\u064a\u0646 \u0627\u0644\u0628\u0646\u0632\u064a\u0646, \u0644\u0646\u0630\u0647\u0628 \u062e\u0637\u0648\u0629 \u0623\u0628\u0639\u062f \u0648\u0646\u062c\u062f"}, {"start": 126.259, "duration": 2.441, "text": "\u0627\u0644\u0635\u064a\u063a\u0629 \u0627\u0644\u062a\u062c\u0631\u064a\u0628\u064a\u0629"}, {"start": 128.7, "duration": 2.7, "text": "\u0625\u0630\u0627 \u0627\u0639\u062a\u0628\u0631\u0646\u0627 \u0623\u0646 \u0644\u062f\u064a\u0646\u0627 100 \u062c\u0631\u0627\u0645 \u0645\u0646 \u0627\u0644\u0647\u064a\u062f\u0631\u0648\u0643\u0631\u0628\u0648\u0646"}, {"start": 131.4, "duration": 4.49, "text": "92.4 \u062c\u0631\u0627\u0645\u064b\u0627 \u0645\u0646 \u0627\u0644\u0643\u0631\u0628\u0648\u0646 \u0648 7.6 \u062c\u0631\u0627\u0645\u064b\u0627 \u0645\u0646 \u0627\u0644\u0647\u064a\u062f\u0631\u0648\u062c\u064a\u0646"}, {"start": 135.89, "duration": 5.21, "text": "\u0628\u0627\u0633\u062a\u062e\u062f\u0627\u0645 \u0643\u062a\u0644\u0647\u0627 \u0627\u0644\u0630\u0631\u064a\u0629 \u0644\u0644\u062a\u062d\u0648\u0644 \u0625\u0644\u0649 \u0627\u0644\u0645\u0648\u0644\u0627\u062a \u0646\u062d\u0635\u0644 \u0639\u0644\u0649 7.6 \u0645\u0648\u0644\u0627\u062a \u062a\u0642\u0631\u064a\u0628\u064b\u0627 \u0644\u0643\u0644 \u0648\u0627\u062d\u062f"}, {"start": 141.12, "duration": 3.86, "text": "\u0645\u0645\u0627 \u064a\u0634\u064a\u0631 \u0625\u0644\u0649 \u0646\u0633\u0628\u0629 \u0648\u0627\u062d\u062f \u0625\u0644\u0649 \u0648\u0627\u062d\u062f. \u0648\u0628\u0627\u0644\u062a\u0627\u0644\u064a"}, {"start": 144.98, "duration": 4.65, "text": "\u0627\u0644\u0635\u064a\u063a\u0629 \u0627\u0644\u062a\u062c\u0631\u064a\u0628\u064a\u0629 \u0644\u0644\u0628\u0646\u0632\u064a\u0646 \u0647\u064a CH. \u0647\u0630\u0627 \u064a\u0639\u0646\u064a \u0623\u0646 \u0647\u0646\u0627\u0643 \u0646\u0641\u0633 \u0627\u0644\u0639\u062f\u062f \u0645\u0646"}, {"start": 149.63, "duration": 0.78, "text": "\u0630\u0631\u0627\u062a \u0627\u0644\u0643\u0631\u0628\u0648\u0646"}, {"start": 150.41, "duration": 2.62, "text": "\u0645\u062b\u0644 \u0630\u0631\u0627\u062a \u0627\u0644\u0647\u064a\u062f\u0631\u0648\u062c\u064a\u0646 \u0641\u064a \u0627\u0644\u062c\u0632\u064a\u0621"}, {"start": 153.03, "duration": 3.37, "text": "\u0625\u0630\u0627 \u0623\u0631\u062f\u0646\u0627 \u0625\u064a\u062c\u0627\u062f \u0627\u0644\u0635\u064a\u063a\u0629 \u0627\u0644\u062c\u0632\u064a\u0626\u064a\u0629 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78 \u0645\u0642\u0633\u0648\u0645\u0629 \u0639\u0644\u0649 13 \u0647\u064a \u0633\u062a\u0629, \u0648\u0647\u0630\u0627 \u064a\u0639\u0646\u064a \u0623\u0646\u0646\u0627 \u0628\u062d\u0627\u062c\u0629 \u0625\u0644\u0649 \u0633\u062a \u0648\u062d\u062f\u0627\u062a"}, {"start": 172.21, "duration": 4.88, "text": "\u0627\u0644\u0635\u064a\u063a\u0629 \u0627\u0644\u062a\u062c\u0631\u064a\u0628\u064a\u0629 \u0644\u0644\u062d\u0635\u0648\u0644 \u0639\u0644\u0649 \u0627\u0644\u0635\u064a\u063a\u0629 \u0627\u0644\u062c\u0632\u064a\u0626\u064a\u0629 \u0623\u0648 C6H6"}, {"start": 177.52, "duration": 1.0, "text": "\u062f\u0639\u0648\u0646\u0627 \u0646\u062a\u062d\u0642\u0642 \u0645\u0646 \u0645\u062f\u0649 \u0627\u0644\u0627\u0633\u062a\u064a\u0639\u0627\u0628"}, {"start": 208.84, "duration": 2.92, "text": "\u0634\u0643\u0631\u0627 \u0644\u0645\u0634\u0627\u0647\u062f\u062a\u0643\u0645 \u0627\u0634\u062a\u0631\u0643\u0648\u0627 \u0641\u064a \u0642\u0646\u0627\u062a\u064a \u0644\u0644\u0645\u0632\u064a\u062f \u0645\u0646 \u0627\u0644\u062f\u0631\u0648\u0633"}, {"start": 211.76, "duration": 1.98, "text": "\u0648\u0643\u0627\u0644\u0639\u0627\u062f\u0629 \u0644\u0627 \u062a\u062a\u0631\u062f\u062f \u0641\u064a \u0645\u0631\u0627\u0633\u0644\u062a\u064a \u0639\u0628\u0631 \u0627\u0644\u0628\u0631\u064a\u062f \u0627\u0644\u0625\u0644\u0643\u062a\u0631\u0648\u0646\u064a"}], "title": "Elemental Analysis: Empirical and Molecular Formulas", "description": "There's a thing with carbon and hydrogen in it. But how many of each?! That's the kind of thing a chemist should know. So let's do some elemental analysis!\n\nWatch the whole General Chemistry playlist: http://bit.ly/ProfDaveGenChem\n\nStudy for the AP Chemistry exam with me: https://bit.ly/ProfDaveAPChem\n\nOrganic Chemistry Tutorials: http://bit.ly/ProfDaveOrgChem\nBiochemistry Tutorials: http://bit.ly/ProfDaveBiochem\nBiology Tutorials: http://bit.ly/ProfDaveBio\nClassical Physics Tutorials: http://bit.ly/ProfDavePhysics1\nModern Physics Tutorials: http://bit.ly/ProfDavePhysics2\nMathematics Tutorials: http://bit.ly/ProfDaveMaths\n\nEMAIL\u25ba ProfessorDaveExplains@gmail.com\nPATREON\u25ba http://patreon.com/ProfessorDaveExplains\n\nCheck out \"Is This Wi-Fi Organic?\", my book on disarming pseudoscience!\nAmazon: https://amzn.to/2HtNpVH\nBookshop: https://bit.ly/39cKADM\n\rBarnes and Noble: https://bit.ly/3pUjmrn \nBook Depository: http://bit.ly/3aOVDlT", "lengthSeconds": 236, "width": 640, "height": 360}
{"video_id": "8CWLG0HuGtA", "transcript": [{"start": 0.0, "duration": 4.34, "text": "[Music]"}, {"start": 1.8, "duration": 2.54, "text": "button"}, {"start": 4.75, "duration": 7.13, "text": "[Music]"}, {"start": 9.32, "duration": 4.899, "text": "Professor Dave and Chegg here we've been"}, {"start": 11.88, "duration": 4.56, "text": "discussing galvanic cells from a more"}, {"start": 14.219, "duration": 4.501, "text": "practical standpoint but it will also be"}, {"start": 16.44, "duration": 4.499, "text": "useful to talk about the thermodynamics"}, {"start": 18.72, "duration": 4.2, "text": "behind them so that we can get a more"}, {"start": 20.939, "duration": 4.561, "text": "fundamental understanding of this"}, {"start": 22.92, "duration": 4.98, "text": "phenomenon so let's do that now we know"}, {"start": 25.5, "duration": 4.98, "text": "that a galvanic cell involves a"}, {"start": 27.9, "duration": 4.26, "text": "spontaneous redox reaction generating an"}, {"start": 30.48, "duration": 3.72, "text": "electric current that can be used for"}, {"start": 32.16, "duration": 4.2, "text": "some practical purpose this is because"}, {"start": 34.2, "duration": 4.859, "text": "the flow of electrons which is what an"}, {"start": 36.36, "duration": 4.74, "text": "electric current is can do work just"}, {"start": 39.059, "duration": 4.02, "text": "like water spontaneously flowing"}, {"start": 41.1, "duration": 4.08, "text": "Downstream in a river can do work and"}, {"start": 43.079, "duration": 4.921, "text": "push a water wheel which can then drive"}, {"start": 45.18, "duration": 5.16, "text": "some mechanical function so too can a"}, {"start": 48.0, "duration": 5.52, "text": "spontaneous flow of electrons do work"}, {"start": 50.34, "duration": 5.039, "text": "this electromotive force is defined in"}, {"start": 53.52, "duration": 4.26, "text": "terms of a potential difference between"}, {"start": 55.379, "duration": 4.801, "text": "two points in a circuit measured in"}, {"start": 57.78, "duration": 4.439, "text": "volts kind of like a gravitational"}, {"start": 60.18, "duration": 4.5, "text": "potential difference from a higher"}, {"start": 62.219, "duration": 4.26, "text": "elevation where a river starts and some"}, {"start": 64.68, "duration": 5.1, "text": "point of lower elevation Where the River"}, {"start": 66.479, "duration": 5.701, "text": "Flows to recall that a volt represents a"}, {"start": 69.78, "duration": 6.839, "text": "jewel of work per coulomb of charge"}, {"start": 72.18, "duration": 6.6, "text": "transferred so V equals J over C in the"}, {"start": 76.619, "duration": 3.901, "text": "context of an electrochemical cell we"}, {"start": 78.78, "duration": 4.019, "text": "consider work to be negative when"}, {"start": 80.52, "duration": 4.62, "text": "flowing out of the system such as when a"}, {"start": 82.799, "duration": 4.561, "text": "galvanic cell produces a current cell"}, {"start": 85.14, "duration": 4.56, "text": "potential is related to work and charge"}, {"start": 87.36, "duration": 4.5, "text": "by this equation so we can see that a"}, {"start": 89.7, "duration": 4.2, "text": "negative value for work results in a"}, {"start": 91.86, "duration": 4.86, "text": "positive cell potential which is also"}, {"start": 93.9, "duration": 4.38, "text": "characteristic of a galvanic cell as we"}, {"start": 96.72, "duration": 3.359, "text": "know from learning about the second law"}, {"start": 98.28, "duration": 4.5, "text": "of Thermodynamics some of the work"}, {"start": 100.079, "duration": 4.5, "text": "produced by galvanic cell will always be"}, {"start": 102.78, "duration": 4.08, "text": "lost to the environment rather than"}, {"start": 104.579, "duration": 3.9, "text": "going toward the cell potential the wire"}, {"start": 106.86, "duration": 3.899, "text": "that allows for current flow will"}, {"start": 108.479, "duration": 4.14, "text": "produce heat such that the maximum work"}, {"start": 110.759, "duration": 4.201, "text": "possible is not physically attained"}, {"start": 112.619, "duration": 4.081, "text": "again this is because the entropy of the"}, {"start": 114.96, "duration": 4.14, "text": "universe must increase for any"}, {"start": 116.7, "duration": 4.5, "text": "spontaneous process and this dispersal"}, {"start": 119.1, "duration": 4.26, "text": "of energy in the form of heat is what"}, {"start": 121.2, "duration": 3.12, "text": "produces this increase in the entropy of"}, {"start": 123.36, "duration": 2.7, "text": "the universe"}, {"start": 124.32, "duration": 4.32, "text": "despite the fact that we can never"}, {"start": 126.06, "duration": 4.32, "text": "harvest the maximum work possible from a"}, {"start": 128.64, "duration": 3.48, "text": "galvanic cell we can still make"}, {"start": 130.38, "duration": 4.14, "text": "approximations with the following"}, {"start": 132.12, "duration": 4.5, "text": "equation where work equals the opposite"}, {"start": 134.52, "duration": 4.56, "text": "of charge transferred which is the moles"}, {"start": 136.62, "duration": 4.68, "text": "of electrons transferred times Faraday's"}, {"start": 139.08, "duration": 3.18, "text": "constant times the actual potential"}, {"start": 141.3, "duration": 3.299, "text": "difference"}, {"start": 142.26, "duration": 4.44, "text": "this can also be expressed in terms of"}, {"start": 144.599, "duration": 3.661, "text": "the change in free energy and if we want"}, {"start": 146.7, "duration": 3.6, "text": "to know about the standard free energy"}, {"start": 148.26, "duration": 4.199, "text": "change we simply change the cell"}, {"start": 150.3, "duration": 4.62, "text": "potential to the standard cell potential"}, {"start": 152.459, "duration": 4.561, "text": "with the appropriate superscript this"}, {"start": 154.92, "duration": 3.899, "text": "means that the maximum cell potential is"}, {"start": 157.02, "duration": 4.079, "text": "directly related to the free energy"}, {"start": 158.819, "duration": 4.92, "text": "difference between reactants and"}, {"start": 161.099, "duration": 4.86, "text": "products in the cell for example given"}, {"start": 163.739, "duration": 4.021, "text": "this galvanic cell with copper and iron"}, {"start": 165.959, "duration": 4.021, "text": "and their Associated reduction"}, {"start": 167.76, "duration": 4.74, "text": "potentials what is the standard change"}, {"start": 169.98, "duration": 4.44, "text": "in free energy for the cell well first"}, {"start": 172.5, "duration": 5.28, "text": "we calculate the cell potential which"}, {"start": 174.42, "duration": 5.399, "text": "will be 0.78 volts that goes in the"}, {"start": 177.78, "duration": 4.38, "text": "equation along with Faraday's constant"}, {"start": 179.819, "duration": 4.021, "text": "and also two for moles of electrons"}, {"start": 182.16, "duration": 4.62, "text": "because two electrons are being"}, {"start": 183.84, "duration": 5.759, "text": "transferred for every copper ion doing"}, {"start": 186.78, "duration": 5.72, "text": "the math we get Negative 1.5 times 10 to"}, {"start": 189.599, "duration": 2.901, "text": "the 5 joules"}, {"start": 192.9, "duration": 4.44, "text": "and now we have a better understanding"}, {"start": 194.879, "duration": 4.14, "text": "of how cell potential relates to"}, {"start": 197.34, "duration": 4.08, "text": "electrical work and free energy"}, {"start": 199.019, "duration": 4.58, "text": "Professor Dave for check see you next"}, {"start": 201.42, "duration": 2.179, "text": "time"}, {"start": 205.04, "duration": 3.059, "text": "thank you"}], "title": "Cell Potential, Electrical Work, and Free Energy | Professor Dave & Chegg Explain", "description": "Professor Dave explains the basics of cell potential, electrical work and free energy. He defines galvanic cells, explains how to calculate cell potential, and covers the formulas of work and cell potential.\n\nGet more homework help from Chegg at https://che.gg/3HbtG8Y\nWatch the whole General Chemistry playlist: https://www.youtube.com/playlist?list=PLllCkartRRdPQdEE-oyt5Di31wIwJc5S6\n\nAbout Chegg:\nChegg is a leading education technology company that provides students with affordable and accessible study resources. Chegg's mission is to help students save time, save money and get smarter.  \n\nSubscribe: https://www.youtube.com/@chegg\nTikTok: https://www.tiktok.com/@chegg\nInstagram: https://www.instagram.com/chegg/\nTwitter: https://twitter.com/chegg\n\n00:00 Intro\n00:11 Galvanic cells\n00:50 Cell potential\n01:37 Second law of thermodynamics\n02:08 Work and cell potential", "lengthSeconds": 212, "width": 640, "height": 360}
{"video_id": "evT3wAL5ZBg", "transcript": [{"start": 0.0, "duration": 9.38, "text": "[Music]"}, {"start": 9.88, "duration": 3.799, "text": "Professor D Dave and CH here we should"}, {"start": 11.719, "duration": 4.4, "text": "remember quite a bit about acids and"}, {"start": 13.679, "duration": 4.76, "text": "bases especially the Bron stead Lowry"}, {"start": 16.119, "duration": 4.92, "text": "model which is the model of acidity and"}, {"start": 18.439, "duration": 4.961, "text": "basicity we will most frequently use"}, {"start": 21.039, "duration": 4.761, "text": "describing acidbase reactions in the"}, {"start": 23.4, "duration": 4.199, "text": "context of proton transfer but there is"}, {"start": 25.8, "duration": 4.479, "text": "another model that we will sometimes be"}, {"start": 27.599, "duration": 5.561, "text": "using in organic chemistry and that is"}, {"start": 30.279, "duration": 4.96, "text": "the Lewis model let's review this model"}, {"start": 33.16, "duration": 4.28, "text": "as well since it will be relevant to a"}, {"start": 35.239, "duration": 4.84, "text": "number of important mechanisms we see"}, {"start": 37.44, "duration": 5.32, "text": "later rather than focusing on proton"}, {"start": 40.079, "duration": 5.361, "text": "transfer this model looks at electrons"}, {"start": 42.76, "duration": 5.2, "text": "alouis acid is a substance which accepts"}, {"start": 45.44, "duration": 5.2, "text": "a pair of electrons while a Lewis base"}, {"start": 47.96, "duration": 4.279, "text": "is one which donates a pair of electrons"}, {"start": 50.64, "duration": 3.439, "text": "in truth there isn't much difference"}, {"start": 52.239, "duration": 4.441, "text": "between a Lewis base and a bron stad"}, {"start": 54.079, "duration": 5.32, "text": "Lowry base as even when proton transfer"}, {"start": 56.68, "duration": 4.839, "text": "is involved it requires a lone pair from"}, {"start": 59.399, "duration": 5.08, "text": "some base which can coordinate to the"}, {"start": 61.519, "duration": 4.521, "text": "proton hence an electron pair donor but"}, {"start": 64.479, "duration": 3.801, "text": "the difference is quite significant when"}, {"start": 66.04, "duration": 4.36, "text": "looking at aets because the leis model"}, {"start": 68.28, "duration": 4.08, "text": "will often involve a base coordinating"}, {"start": 70.4, "duration": 4.64, "text": "to some atom rather than accepting a"}, {"start": 72.36, "duration": 5.88, "text": "proton for example let's look at Boron"}, {"start": 75.04, "duration": 5.32, "text": "Tri fluide BF3 this Boron atom has a"}, {"start": 78.24, "duration": 4.0, "text": "substantial partial positive charge"}, {"start": 80.36, "duration": 4.439, "text": "because of the polarity of these Boron"}, {"start": 82.24, "duration": 4.64, "text": "Florine bonds because of this electron"}, {"start": 84.799, "duration": 4.36, "text": "deficiency it is prone to attack by"}, {"start": 86.88, "duration": 4.199, "text": "electrons since positive charges and"}, {"start": 89.159, "duration": 3.761, "text": "negative charges attract which is"}, {"start": 91.079, "duration": 4.841, "text": "essentially the central theme of the"}, {"start": 92.92, "duration": 5.28, "text": "entire field of chemistry so a lwi base"}, {"start": 95.92, "duration": 5.08, "text": "like this fluoride ion can attack the"}, {"start": 98.2, "duration": 5.959, "text": "Boron atom to form a new Bond leaving us"}, {"start": 101.0, "duration": 5.719, "text": "with BF4 minus we will call this product"}, {"start": 104.159, "duration": 4.401, "text": "an acid base adduct so we won't get"}, {"start": 106.719, "duration": 4.521, "text": "conjugate species like with the Bron"}, {"start": 108.56, "duration": 4.879, "text": "stead Lowry model one of the reasons"}, {"start": 111.24, "duration": 4.64, "text": "this is relevant in organic chemistry is"}, {"start": 113.439, "duration": 5.241, "text": "that we will frequently see metal cat I"}, {"start": 115.88, "duration": 4.8, "text": "acting as Lewis acids take the Magnesium"}, {"start": 118.68, "duration": 4.0, "text": "2+ ion"}, {"start": 120.68, "duration": 4.52, "text": "this is electron deficient due to the"}, {"start": 122.68, "duration": 4.84, "text": "formal positive charge so electrons can"}, {"start": 125.2, "duration": 5.0, "text": "coordinate to it this means that we can"}, {"start": 127.52, "duration": 4.719, "text": "often find metal cations like this"}, {"start": 130.2, "duration": 3.96, "text": "coordinated to Regions with a formal"}, {"start": 132.239, "duration": 4.0, "text": "negative charge on certain kinds of"}, {"start": 134.16, "duration": 4.799, "text": "molecules especially"}, {"start": 136.239, "duration": 5.64, "text": "oxyanion like here with this phosphate"}, {"start": 138.959, "duration": 6.401, "text": "ion transition metals also frequently"}, {"start": 141.879, "duration": 5.161, "text": "form Lewis acids compounds like"}, {"start": 145.36, "duration": 3.2, "text": "ti4"}, {"start": 147.04, "duration": 5.72, "text": "fecl3"}, {"start": 148.56, "duration": 6.44, "text": "zncl2 and sncl4 are all Lewis acids due"}, {"start": 152.76, "duration": 4.64, "text": "to having a central electron deficient"}, {"start": 155.0, "duration": 5.599, "text": "atom which is susceptible to attack from"}, {"start": 157.4, "duration": 5.0, "text": "a LS base again with bases things are"}, {"start": 160.599, "duration": 4.201, "text": "not much different conceptually from a"}, {"start": 162.4, "duration": 4.479, "text": "bron dead Lowry base the difference is"}, {"start": 164.8, "duration": 4.32, "text": "that a Lewis base doesn't specifically"}, {"start": 166.879, "duration": 4.041, "text": "have to coordinate to a proton it can"}, {"start": 169.12, "duration": 5.24, "text": "coordinate to something else and will"}, {"start": 170.92, "duration": 5.8, "text": "still qualify as a LS base many oxygen"}, {"start": 174.36, "duration": 4.879, "text": "and nitrogen containing compounds can"}, {"start": 176.72, "duration": 6.079, "text": "act as Lewis bases due to the lone pair"}, {"start": 179.239, "duration": 6.241, "text": "of electrons on these atoms as well as"}, {"start": 182.799, "duration": 4.72, "text": "sulfur with that we should have a basic"}, {"start": 185.48, "duration": 4.72, "text": "understanding of the Lewis model for"}, {"start": 187.519, "duration": 5.44, "text": "acidity and basicity as well as some"}, {"start": 190.2, "duration": 4.679, "text": "important examples of Lewis acids that"}, {"start": 192.959, "duration": 4.481, "text": "will come up as we study organic"}, {"start": 194.879, "duration": 5.761, "text": "chemistry professor Dave for che see you"}, {"start": 197.44, "duration": 3.2, "text": "next time"}], "title": "What Are Lewis Acids and Bases? Ft. Professor Dave", "description": "Now that we've covered the Br\u00f8nsted-Lowry model of acidity and basicity, let's take a look at a different model: the Lewis model. With @ProfessorDaveExplains, we'll learn about how the Lewis model looks at electrons instead of proton transfer, and then we'll cover the concept of an acid-base adduct with the example of boron trifluoride. Finally we'll discuss how metal cations can act as Lewis acids.\n\nGet more homework help from Chegg at https://che.gg/3HbtG8Y\nWatch the whole Organic Chemistry playlist: https://www.youtube.com/playlist?list=PLllCkartRRdMPnATf1X8qVdo5RzM8gA_M\n\nAbout Chegg:\nChegg is a leading education technology company that provides students with affordable and accessible study resources. Chegg's mission is to help students save time, save money and get smarter.  \n\nSubscribe: https://www.youtube.com/@chegg\nTikTok: https://www.tiktok.com/@chegg\nInstagram: https://www.instagram.com/chegg/\nTwitter: https://twitter.com/chegg\n\n00:00 Intro\n00:38 Lewis model\n01:13 Examples\n01:50 Metal cations as acids", "lengthSeconds": 208, "width": 640, "height": 360}
{"video_id": "tm5QKQ39G7o", "transcript": [{"start": 6.6, "duration": 4.1, "text": "hey everybody, for today's problem we have a\nreaction here that I've drawn, this is"}, {"start": 10.7, "duration": 5.609, "text": "this compound reacting with sulfuric\nacid to generate this product, and what"}, {"start": 16.309, "duration": 2.611, "text": "we're gonna do is we want to show the\nmechanism, that means every single"}, {"start": 18.92, "duration": 3.6, "text": "intermediate and every single electron\npushing arrow, so other than the fact that"}, {"start": 22.52, "duration": 3.14, "text": "we're reacting with with a strong acid i don't\nwant to give too much away"}, {"start": 25.7, "duration": 1.42, "text": "go ahead and give it a shot"}, {"start": 42.42, "duration": 2.46, "text": "ok so this is a pretty interesting\nmechanism let's take a look at what"}, {"start": 44.88, "duration": 4.56, "text": "happens, now we know that this is a\nstrong acid, it has the capacity to"}, {"start": 49.44, "duration": 3.78, "text": "protonate something. the only thing we're\ngonna be able to protonate is this hydroxyl"}, {"start": 53.22, "duration": 3.82, "text": "group so anytime you see an acid-base\nreaction, we've got a strong acid, we know"}, {"start": 57.04, "duration": 3.8, "text": "that strong acid is gonna donate a proton so\nwe want to look at the substrate and say"}, {"start": 60.84, "duration": 3.26, "text": "where's the proton gonna go? well once\nagain the only thing here that's gonna"}, {"start": 64.1, "duration": 2.869, "text": "work, we have a lone pair on oxygen,\nthat's the only thing that can"}, {"start": 66.969, "duration": 4.1, "text": "accommodate a proton, so that is gonna\nget protonated and now we have basically"}, {"start": 71.069, "duration": 4.521, "text": "a water molecule right here. now whereas\na hydroxyl group is a is a terrible"}, {"start": 75.59, "duration": 3.639, "text": "leaving group, a water molecule is a\ngreat leaving group and we are in acidic"}, {"start": 79.229, "duration": 7.09, "text": "condition so positive charges are ok, so\nthis water can leave so that if"}, {"start": 86.319, "duration": 3.97, "text": "water leaves, now we're left with a carbocation right here. now this is the"}, {"start": 90.289, "duration": 4.67, "text": "key step, this is a carbocation\nrearrangement step. if we remember from"}, {"start": 94.959, "duration": 3.62, "text": "the other lectures we can have a hydride\nshift or we can have a methanide"}, {"start": 98.579, "duration": 4.35, "text": "shift where actual alkyl groups are\nmoving, but the interesting thing here is"}, {"start": 102.929, "duration": 5.19, "text": "that this is gonna be able to rearrange\nto turn a secondary carbocation into a"}, {"start": 108.119, "duration": 3.48, "text": "tertiary carbocation and the way\nit's going to be able to do it is with"}, {"start": 111.599, "duration": 5.38, "text": "this methanide shift right there. or\nit's not really a methyl but an alkyl"}, {"start": 116.979, "duration": 5.68, "text": "shift where this bond is going to\ndetach from that carbon and reattach"}, {"start": 122.659, "duration": 4.361, "text": "to that carbon so this is what we're\ngonna get. this is favorable for two"}, {"start": 127.02, "duration": 0.79, "text": "reasons"}, {"start": 127.81, "duration": 2.74, "text": "number one: we turn a secondary carbocation"}, {"start": 130.55, "duration": 4.16, "text": "into a tertiary carbocation so that's a driving force"}, {"start": 134.71, "duration": 2.98, "text": "but then the other thing we're doing is\nturning a five-membered ring into a"}, {"start": 137.69, "duration": 3.96, "text": "six-membered ring, so there's a lot going\non here that makes this a favorable"}, {"start": 141.65, "duration": 3.54, "text": "rearrangement. and so now from the\nsecondary carbocation we have this"}, {"start": 145.19, "duration": 5.25, "text": "tertiary carbocation, and we are in aqueous acidic\nsolution so we can always eliminate and"}, {"start": 150.44, "duration": 6.49, "text": "what's going to happen is this can grab\nthis proton here and form the pi bond"}, {"start": 156.93, "duration": 3.0, "text": "there, and there's our product."}, {"start": 181.46, "duration": 3.499, "text": "thanks for watching guys, subscribe to my\nchannel for more tutorials and as always"}, {"start": 184.959, "duration": 0.98, "text": "feel free to email me"}], "title": "Practice Problem: Mechanism - Acid Catalysis", "description": "For this one we need to know what happens to alcohols under acidic conditions. Beware of tricky carbocations!\n\nTry all of the organic chemistry practice problems: http://bit.ly/ProfDaveOCPP\n\nOrganic Chemistry Tutorials: http://bit.ly/ProfDaveOrgChem\n\nEMAIL\u25ba ProfessorDaveExplains@gmail.com\nPATREON\u25ba http://patreon.com/ProfessorDaveExplains\n\nCheck out \"Is This Wi-Fi Organic?\", my book on disarming pseudoscience!\nAmazon: https://amzn.to/2HtNpVH\nBookshop: https://bit.ly/39cKADM\n\rBarnes and Noble: https://bit.ly/3pUjmrn \nBook Depository: http://bit.ly/3aOVDlT", "lengthSeconds": 189, "width": 640, "height": 360}
{"video_id": "zFZ0yLRupfU", "transcript": [{"start": 6.32, "duration": 5.08, "text": "let's try a practice problem regarding\nenthalpy of vaporization. so the question"}, {"start": 11.4, "duration": 7.889, "text": "is how much heat is required to vaporize\n1.500 liters of water at 37 degrees"}, {"start": 19.289, "duration": 5.341, "text": "Celsius so in order to answer this we\nneed to know that the density of water"}, {"start": 24.63, "duration": 5.37, "text": "at 37 degrees Celsius is actually just a\nhair under one gram per milliliter which"}, {"start": 30.0, "duration": 4.26, "text": "is what we're used to we're going to use\npoint nine nine three four grams per"}, {"start": 34.26, "duration": 6.51, "text": "milliliter to be very specific and then\nthe heat of vaporization of water at 37"}, {"start": 40.77, "duration": 6.39, "text": "degrees Celsius is forty three point\nfour six kilojoules per mole so this is"}, {"start": 47.16, "duration": 3.12, "text": "all the data you need to answer this\nquestion go ahead and give it a try."}, {"start": 68.96, "duration": 4.6, "text": "let's take our one point five zero zero\nliters of water and we're gonna want to"}, {"start": 73.56, "duration": 4.59, "text": "do some stoichiometric calculations so\nwe've got to get two moles our goal is"}, {"start": 78.15, "duration": 4.289, "text": "always going to be to get two moles\nsomehow with these things so let's take"}, {"start": 82.439, "duration": 4.47, "text": "our one point five zero zero liters and\nwe want milliliters because our density"}, {"start": 86.909, "duration": 4.71, "text": "was grams per milliliter so let's\nmultiply by a thousand milliliters over"}, {"start": 91.619, "duration": 5.82, "text": "one liter and we're gonna get 1,500\nmilliliters now that we have milliliters"}, {"start": 97.439, "duration": 5.25, "text": "we can use our density 1500 milliliters\ntimes zero point nine nine three four"}, {"start": 102.689, "duration": 5.551, "text": "grams per milliliter and that is going\nto give us a mass of fourteen ninety"}, {"start": 108.24, "duration": 5.089, "text": "grams so that is the mass of water\ncontained in that volume of water"}, {"start": 113.329, "duration": 3.881, "text": "specifically at this temperature 37\ndegrees Celsius right if we had a"}, {"start": 117.21, "duration": 3.869, "text": "different temperature things might be\nslightly different then we take our"}, {"start": 121.079, "duration": 4.801, "text": "fourteen ninety grams of water and we're\ngonna multiply by one mol / eighteen"}, {"start": 125.88, "duration": 4.56, "text": "point oh two grams that's the molar mass\nof water and we know we therefore have"}, {"start": 130.44, "duration": 5.44, "text": "eighty two point six nine moles of water\nthat's how many moles of water are in"}, {"start": 135.88, "duration": 5.49, "text": "this volume of water at this temperature\nthen lastly we can just use the given"}, {"start": 141.37, "duration": 5.07, "text": "thermo chemical data we know that eighty\ntwo point six nine moles times forty"}, {"start": 146.44, "duration": 5.37, "text": "three point four six kilojoules of\nenergy required per mole for"}, {"start": 151.81, "duration": 4.2, "text": "vaporization to occur that's going to\ngive us a total of three thousand five"}, {"start": 156.01, "duration": 7.01, "text": "hundred ninety four kilojoules of energy required to vaporize this volume of water at this temperature."}], "title": "Practice Problem: Enthalpy of Vaporization", "description": "Can we do stoichiometry regarding phase changes? Sure! If we know how many moles of a substance we have, and the energy associated with each mole of that substance undergoing a particular phase change, we can get the energy associated with a specific sample undergoing that phase change. It's easy, try it!\n\nTry all of the general chemistry practice problems: http://bit.ly/ProfDaveGCPP\n\nGeneral Chemistry Tutorials: http://bit.ly/ProfDaveGenChem\n\nEMAIL\u25ba ProfessorDaveExplains@gmail.com\nPATREON\u25ba http://patreon.com/ProfessorDaveExplains\n\nCheck out \"Is This Wi-Fi Organic?\", my book on disarming pseudoscience!\nAmazon: https://amzn.to/2HtNpVH\nBookshop: https://bit.ly/39cKADM\n\rBarnes and Noble: https://bit.ly/3pUjmrn \nBook Depository: http://bit.ly/3aOVDlT", "lengthSeconds": 194, "width": 640, "height": 360}
{"video_id": "YIGBqkSOxtM", "transcript": [{"start": 9.357, "duration": 0.483, "text": "[Music]"}, {"start": 9.84, "duration": 4.8, "text": "Professor Dave and CH here we just learned\u00a0\nabout enantiomers so we know that they are\u00a0\u00a0"}, {"start": 14.64, "duration": 6.08, "text": "pairs of molecules which are non-identical mirror\u00a0\nimages of one another but what does this have to\u00a0\u00a0"}, {"start": 20.72, "duration": 7.8, "text": "do with how molecules behave do entiers have any\u00a0\ndifferent properties let's investigate given that\u00a0\u00a0"}, {"start": 28.52, "duration": 5.76, "text": "enantiomers are precise I mirror images they\u00a0\nhave nearly identical physical properties they\u00a0\u00a0"}, {"start": 34.28, "duration": 6.24, "text": "have exactly the same density polarity melting\u00a0\npoint and boiling point one way of distinguishing\u00a0\u00a0"}, {"start": 40.52, "duration": 5.68, "text": "between a pair of an antier has to do with the\u00a0\ndirection that they rotate plain polarized light\u00a0\u00a0"}, {"start": 46.2, "duration": 5.4, "text": "when a beam of ordinary light passes through\u00a0\nsomething called a polarizer only the light\u00a0\u00a0"}, {"start": 51.6, "duration": 6.04, "text": "oscillating in a particular plane will pass\u00a0\nthrough and this is said to be plain polarized\u00a0\u00a0"}, {"start": 57.64, "duration": 7.04, "text": "as light oscillating along any other plane is\u00a0\nblocked out any chyal compound will be optically\u00a0\u00a0"}, {"start": 64.68, "duration": 6.04, "text": "active meaning that it will rotate plain polarized\u00a0\nlight by some angle depending on the structure of\u00a0\u00a0"}, {"start": 70.72, "duration": 5.04, "text": "the molecule as measured by and analyzed on\u00a0\nthe other side of the molecule with respect\u00a0\u00a0"}, {"start": 75.76, "duration": 5.04, "text": "to the light source when looking at a pair of an\u00a0\nanti immerse they will rotate this light by the\u00a0\u00a0"}, {"start": 80.8, "duration": 7.68, "text": "same angle but in opposite directions one will be\u00a0\ndextrorotatory signified with a plus sign meaning\u00a0\u00a0"}, {"start": 88.48, "duration": 6.96, "text": "the rotation is clockwise and the other will be\u00a0\nLev rotatory signified with a minus sign meaning\u00a0\u00a0"}, {"start": 95.44, "duration": 6.32, "text": "the rotation is counterclockwise we can measure\u00a0\nthe optical activity of many different compounds\u00a0\u00a0"}, {"start": 101.76, "duration": 5.44, "text": "and tabulate them and testing the optical\u00a0\nactivity of a sample is a way of determining\u00a0\u00a0"}, {"start": 107.2, "duration": 6.4, "text": "the enantiomeric purity of the sample if both\u00a0\nenantiomers are present in equal amounts the\u00a0\u00a0"}, {"start": 113.6, "duration": 6.64, "text": "dexr rotatory and Lev rotatory activity will\u00a0\ncancel out and the sample will not be optic\u00a0\u00a0"}, {"start": 120.24, "duration": 5.24, "text": "active overall if one enantiomer is pure\u00a0\nit will rotate the light according to the\u00a0\u00a0"}, {"start": 125.48, "duration": 6.12, "text": "tabulated value for that enantiomer and then if\u00a0\nthe optical activity is some fraction of that\u00a0\u00a0"}, {"start": 131.6, "duration": 5.36, "text": "of the pure en antier we can do math to find\u00a0\nout the ratio of one enantiomer to the other\u00a0\u00a0"}, {"start": 136.96, "duration": 7.28, "text": "in the sample and so we now have a little more\u00a0\ncontext surrounding the concept of an antier\u00a0\u00a0"}, {"start": 144.24, "duration": 5.76, "text": "in the way of a specific application namely a\u00a0\nphysical property that differs from one in antier\u00a0\u00a0"}, {"start": 150.0, "duration": 4.64, "text": "to the other we can think of optically\u00a0\nactive as more or less a synonym for\u00a0\u00a0"}, {"start": 154.64, "duration": 16.16, "text": "Kyu but we also now know exactly what the term\u00a0\nmeans Professor Dave for che see you next time"}], "title": "Enantiomers and Optical Activity Ft. Professor Dave", "description": "Now that we've learned about enantiomers\u2014pairs of molecules that are non-identical mirror images of each other\u2014let's learn what enantiomers can tell us about how molecules behave. With @ProfessorDaveExplains, we'll learn how enantiomers are distinguished by their optical activity: the ability to rotate plane-polarized light in different directions. We'll also learn how we can measure the particular degree of rotation of an optically active compound to determine its enantiomeric purity.\n\nGet more homework help from Chegg at https://che.gg/3HbtG8Y\nWatch the whole Organic Chemistry playlist: https://www.youtube.com/playlist?list=PLllCkartRRdMPnATf1X8qVdo5RzM8gA_M\n\nAbout Chegg:\nChegg is a leading education technology company that provides students with affordable and accessible study resources. Chegg's mission is to help students save time, save money and get smarter.  \n\nSubscribe: https://www.youtube.com/@chegg\nTikTok: https://www.tiktok.com/@chegg\nInstagram: https://www.instagram.com/chegg/\nTwitter: https://twitter.com/chegg\n\n00:00 Intro\n00:39 Plane-polarized light\n01:01 Optical activity", "lengthSeconds": 171, "width": 640, "height": 360}
{"video_id": "sZ1OR_0yQJY", "transcript": [{"start": 3.14, "duration": 4.74, "text": "Some of the important reactions in Organic Chemistry involve generating new carbon-carbon bonds."}, {"start": 8.14, "duration": 7.274, "text": "Especially ones in which we add new alkyl groups to a benzene ring or some other aromatic structure."}, {"start": 15.42, "duration": 4.2, "text": "So one of these reactions is the Friedel-Crafts Alkylation."}, {"start": 19.62, "duration": 7.46, "text": "This is where we are able to add some alkyl fragment to a benzene ring or some other aromatic structure."}, {"start": 27.08, "duration": 5.6, "text": "So what is happening here is we have a benzene ring and we have an alkyl chloride \u2013 could be any alkyl chloride."}, {"start": 32.76, "duration": 2.5, "text": "I've drawn this one arbitrarily."}, {"start": 35.26, "duration": 11.44, "text": "And then in the presence of aluminum trichloride, lewis acid catalyst, we're going to get this benzene ring with the alkyl substituent and then HCl byproduct."}, {"start": 46.7, "duration": 5.04, "text": "So this actually operates in a very similar manner to the halogenation we just saw."}, {"start": 51.84, "duration": 6.16, "text": "So once again the first step is one in which we're generating a catalytic intermediate."}, {"start": 58.06, "duration": 12.78, "text": "So a lone pair on the chlorine atom is going to go and interact with the aluminum atom in the aluminum trichloride very similarly to the way we were seeing the iron compound behaving before."}, {"start": 70.84, "duration": 10.66, "text": "And so we're going to generate this compound here.  Once again this chlorine is going to have a formal positive charge, because it is contributing 6 electrons to this lewis dot structure."}, {"start": 81.5, "duration": 7.48, "text": "And then the aluminum atom will have a formal negative charge, because it is now contributing 4 electrons to this lewis dot structure. "}, {"start": 89.1, "duration": 4.82, "text": "Now the only difference here from the halogenation is that there is now a further step."}, {"start": 93.96, "duration": 3.32, "text": "The benzene ring is not going to directly interact with this intermediate."}, {"start": 97.34, "duration": 14.72, "text": "  Instead what is going to happen is this carbon-chlorine bond is going to cleave and it's going to do so because it will then stabilize this chlorine atom and neutralize it and then we'll have an alkyl carbocation."}, {"start": 112.06, "duration": 11.3, "text": "We'll have this alkyl cation intermediate instead. So now we have our isopropyl cation and then we have the remainder of the catalytic complex. "}, {"start": 123.5, "duration": 8.78, "text": "The aluminum will still have a formal negative charge there.  Aluminum has 3 valence electrons, here it is showing 4.  That means it has a formal negative charge."}, {"start": 132.36, "duration": 20.26, "text": "So now this is a big difference because whereas a benzene ring would have a very difficult time interacting with this neutral chloroalkane, there's going to be a much lower activation energy associated with benzene interacting with this highly unstable carbocation."}, {"start": 152.691, "duration": 7.0, "text": "The instability dictates it's going to be much more reactive and therefore able to react even with benzene."}, {"start": 159.72, "duration": 3.58, "text": "So this pi bond is going to interact with the carbocation."}, {"start": 163.3, "duration": 3.74, "text": "Now that alkyl fragment is attached to the ring just as we've seen before."}, {"start": 167.04, "duration": 4.4, "text": "Now we have a cation on the ring and then the remainder of this structure"}, {"start": 171.44, "duration": 6.42, "text": "In order to neutralize this aluminum atom, the chlorine atom leaves with the electrons in this bond"}, {"start": 177.86, "duration": 4.76, "text": "Goes extracts the proton, the electrons left behind form the new pi bond"}, {"start": 182.62, "duration": 5.36, "text": "And we've restored aromaticity.  So now we have that alkyl fragment present on the benzene ring."}, {"start": 187.98, "duration": 4.0, "text": "And then we have HCl and then we've regenerated the catalyst."}, {"start": 192.06, "duration": 3.42, "text": "So this is extremely similar to the halogenation mechanism we just saw."}, {"start": 195.48, "duration": 5.52, "text": "The only difference is that the benzene ring doesn't immediately react with this catalytic complex."}, {"start": 201.0, "duration": 11.12, "text": "There is yet a second step where this carbon-chlorine bond cleaves generating a carbocation intermediate and it is the carbocation intermediate that interacts initially with the benzene ring."}, {"start": 213.04, "duration": 5.4, "text": "Thanks for watching, guys.  Subscribe to my channel for more tutorials, and as always, feel free to email me with questions"}], "title": "Friedel-Crafts Alkylation", "description": "How do you attach alkyl groups to benzene rings? The Friedel-Crafts reaction is one way. This is one of the Electrophilic Aromatic substitutions we will examine.\n\nWatch the whole Organic Chemistry playlist: http://bit.ly/ProfDaveOrgChem\n\nGeneral Chemistry Tutorials: http://bit.ly/ProfDaveGenChem\nBiochemistry Tutorials: http://bit.ly/ProfDaveBiochem\nBiology Tutorials: http://bit.ly/ProfDaveBio\nClassical Physics Tutorials: http://bit.ly/ProfDavePhysics1\nModern Physics Tutorials: http://bit.ly/ProfDavePhysics2\nMathematics Tutorials: http://bit.ly/ProfDaveMath\n\nEMAIL\u25ba ProfessorDaveExplains@gmail.com\nPATREON\u25ba http://patreon.com/ProfessorDaveExplains\n\nCheck out \"Is This Wi-Fi Organic?\", my book on disarming pseudoscience!\nAmazon: https://amzn.to/2HtNpVH\nBookshop: https://bit.ly/39cKADM\n\rBarnes and Noble: https://bit.ly/3pUjmrn \nBook Depository: http://bit.ly/3aOVDlT", "lengthSeconds": 236, "width": 640, "height": 360}
{"video_id": "JaoaJ62FnYQ", "transcript": [{"start": 2.77, "duration": 5.67, "text": "in eukaryotes a number of proteins"}, {"start": 5.77, "duration": 5.609, "text": "called transcription factors must bind"}, {"start": 8.44, "duration": 6.21, "text": "to a promoter before RNA polymerase 2"}, {"start": 11.379, "duration": 5.371, "text": "can bind and initiate transcription the"}, {"start": 14.65, "duration": 6.75, "text": "first transcription factor to bind has"}, {"start": 16.75, "duration": 6.3, "text": "been named tf2d tf2d binds to a region"}, {"start": 21.4, "duration": 3.84, "text": "of the promoter called the tata box"}, {"start": 23.05, "duration": 7.41, "text": "which contains the sequence"}, {"start": 25.24, "duration": 6.9, "text": "ta ta T following tf2d binding a"}, {"start": 30.46, "duration": 4.55, "text": "succession of other transcription"}, {"start": 32.14, "duration": 6.51, "text": "factors and RNA polymerase to bind"}, {"start": 35.01, "duration": 5.83, "text": "forming an initiation complex although"}, {"start": 38.65, "duration": 4.53, "text": "this complex is often capable of"}, {"start": 40.84, "duration": 4.68, "text": "initiating transcription for many genes"}, {"start": 43.18, "duration": 5.04, "text": "other proteins are required for a high"}, {"start": 45.52, "duration": 4.89, "text": "rate of transcription in the promoter"}, {"start": 48.22, "duration": 5.25, "text": "there may be a region of DNA that can"}, {"start": 50.41, "duration": 5.46, "text": "bind to regulator proteins a regulator"}, {"start": 53.47, "duration": 3.81, "text": "protein binds to the regulator region"}, {"start": 55.87, "duration": 5.76, "text": "and to the transcription initiation"}, {"start": 57.28, "duration": 6.45, "text": "complex thereby activating it enhancer"}, {"start": 61.63, "duration": 4.44, "text": "regions are yet other transcriptionally"}, {"start": 63.73, "duration": 4.11, "text": "important regions on the DNA and they"}, {"start": 66.07, "duration": 3.93, "text": "are frequently located some distance"}, {"start": 67.84, "duration": 5.16, "text": "from the promoter often tens of"}, {"start": 70.0, "duration": 5.25, "text": "thousands of base pairs away activator"}, {"start": 73.0, "duration": 3.869, "text": "proteins bind to enhancers and then"}, {"start": 75.25, "duration": 6.17, "text": "attach to the transcription initiation"}, {"start": 76.869, "duration": 4.551, "text": "complex stimulating it strongly"}, {"start": 82.74, "duration": 3.96, "text": "if a cell has all of the appropriate DNA"}, {"start": 84.899, "duration": 4.11, "text": "sequences and if all the necessary"}, {"start": 86.7, "duration": 4.19, "text": "regulator proteins and transcription"}, {"start": 89.009, "duration": 4.321, "text": "factors are present and in place"}, {"start": 90.89, "duration": 5.5, "text": "transcription begins and the gene is"}, {"start": 93.33, "duration": 4.74, "text": "expressed otherwise no transcription or"}, {"start": 96.39, "duration": 5.39, "text": "very little transcription takes place"}, {"start": 98.07, "duration": 3.71, "text": "and the gene is silent"}], "title": "Initation of eukaryote Transcription", "description": "", "lengthSeconds": 107, "width": 480, "height": 360}
{"video_id": "Vd0mBXPslTM", "transcript": [{"start": 0.0, "duration": 14.48, "text": "[Music]"}, {"start": 11.44, "duration": 5.12, "text": "hi everyone it's dr a and in this video"}, {"start": 14.48, "duration": 4.559, "text": "we're going to discuss a process"}, {"start": 16.56, "duration": 5.84, "text": "of bone growth and development called"}, {"start": 19.039, "duration": 4.641, "text": "intramembranous ossification"}, {"start": 22.4, "duration": 3.76, "text": "the first thing that i'd like to share"}, {"start": 23.68, "duration": 4.32, "text": "with you is that bone is a type of"}, {"start": 26.16, "duration": 4.32, "text": "replacement tissue"}, {"start": 28.0, "duration": 4.96, "text": "in other words bone tissue develops to"}, {"start": 30.48, "duration": 4.8, "text": "replace an existing tissue"}, {"start": 32.96, "duration": 4.32, "text": "and that existing tissue is known as"}, {"start": 35.28, "duration": 4.32, "text": "cartilage so we can think about"}, {"start": 37.28, "duration": 5.119, "text": "cartilage as a blueprint for the"}, {"start": 39.6, "duration": 6.08, "text": "building of bone tissue"}, {"start": 42.399, "duration": 6.16, "text": "so in this process we start with unique"}, {"start": 45.68, "duration": 3.92, "text": "specialized cells called mesenchymal"}, {"start": 48.559, "duration": 3.361, "text": "cells"}, {"start": 49.6, "duration": 4.56, "text": "and this type of cell is a precursor"}, {"start": 51.92, "duration": 4.799, "text": "cell for connective tissue"}, {"start": 54.16, "duration": 5.199, "text": "in other words it's this cell that leads"}, {"start": 56.719, "duration": 4.401, "text": "to the development of a variety of"}, {"start": 59.359, "duration": 4.081, "text": "connective tissues"}, {"start": 61.12, "duration": 4.72, "text": "and for the purposes of this video we're"}, {"start": 63.44, "duration": 5.999, "text": "going to utilize a green circle"}, {"start": 65.84, "duration": 5.279, "text": "to represent the mesenchymal cell"}, {"start": 69.439, "duration": 3.04, "text": "and it's through intramembranous"}, {"start": 71.119, "duration": 3.441, "text": "ossification"}, {"start": 72.479, "duration": 4.801, "text": "that we have the development of flat"}, {"start": 74.56, "duration": 6.4, "text": "bones like those of the skull"}, {"start": 77.28, "duration": 6.72, "text": "the ribs and the clavicle"}, {"start": 80.96, "duration": 6.32, "text": "so ultimately this process begins with"}, {"start": 84.0, "duration": 5.68, "text": "the embryonic skeleton and remember that"}, {"start": 87.28, "duration": 3.44, "text": "here we're starting with the cartilage"}, {"start": 89.68, "duration": 3.36, "text": "model"}, {"start": 90.72, "duration": 6.56, "text": "and in the cartilage we begin with"}, {"start": 93.04, "duration": 6.64, "text": "mesenchymal cells which gather together"}, {"start": 97.28, "duration": 5.119, "text": "these mesenchymal cells then"}, {"start": 99.68, "duration": 3.92, "text": "differentiate into specific connective"}, {"start": 102.399, "duration": 3.121, "text": "tissues"}, {"start": 103.6, "duration": 5.36, "text": "some of these differentiate into"}, {"start": 105.52, "duration": 5.12, "text": "capillaries and others become osteogenic"}, {"start": 108.96, "duration": 4.32, "text": "cells"}, {"start": 110.64, "duration": 5.68, "text": "and for note-taking purposes osteogenic"}, {"start": 113.28, "duration": 7.28, "text": "cells are the precursor cells meaning"}, {"start": 116.32, "duration": 7.439, "text": "that they ultimately become bone cells"}, {"start": 120.56, "duration": 6.32, "text": "now these osteogenic cells differentiate"}, {"start": 123.759, "duration": 5.761, "text": "and become osteoblast"}, {"start": 126.88, "duration": 4.4, "text": "and these osteoblasts appear as a"}, {"start": 129.52, "duration": 4.56, "text": "cluster of cells"}, {"start": 131.28, "duration": 7.28, "text": "and this cluster of cells is called or"}, {"start": 134.08, "duration": 7.64, "text": "referred to as an ossification center"}, {"start": 138.56, "duration": 7.2, "text": "these osteoblasts then secrete an"}, {"start": 141.72, "duration": 6.44, "text": "unmineralized material called osteoid"}, {"start": 145.76, "duration": 4.96, "text": "and although not pictured once this"}, {"start": 148.16, "duration": 5.439, "text": "osteoid is secreted it allows for the"}, {"start": 150.72, "duration": 4.32, "text": "deposit of collagen fibers into the"}, {"start": 153.599, "duration": 2.881, "text": "osteoid"}, {"start": 155.04, "duration": 4.0, "text": "and following this"}, {"start": 156.48, "duration": 5.36, "text": "we'll soon have the deposit of minerals"}, {"start": 159.04, "duration": 5.36, "text": "such as calcium and phosphorus which"}, {"start": 161.84, "duration": 5.28, "text": "allow the tissue to begin heartening"}, {"start": 164.4, "duration": 5.44, "text": "over the course of a few days"}, {"start": 167.12, "duration": 5.44, "text": "and through this process the"}, {"start": 169.84, "duration": 4.08, "text": "become entrapped in the surrounding"}, {"start": 172.56, "duration": 3.44, "text": "tissue"}, {"start": 173.92, "duration": 4.399, "text": "now what's unique to this process is"}, {"start": 176.0, "duration": 5.12, "text": "that in the areas where osteoid is"}, {"start": 178.319, "duration": 6.241, "text": "secreted around the capillaries it"}, {"start": 181.12, "duration": 6.399, "text": "becomes what we call trabecular bone"}, {"start": 184.56, "duration": 5.92, "text": "and as this process continues we'll end"}, {"start": 187.519, "duration": 5.841, "text": "up having two distinct portions"}, {"start": 190.48, "duration": 6.479, "text": "the outer portion of this bony tissue is"}, {"start": 193.36, "duration": 6.08, "text": "called the periosteum or compact bone"}, {"start": 196.959, "duration": 5.041, "text": "and again the inner portion which is"}, {"start": 199.44, "duration": 5.519, "text": "largely surrounded by capillaries is"}, {"start": 202.0, "duration": 4.48, "text": "referred to as trabecular bone"}, {"start": 204.959, "duration": 4.081, "text": "and this is the process of"}, {"start": 206.48, "duration": 3.839, "text": "intramembranous ossification in a"}, {"start": 209.04, "duration": 3.44, "text": "nutshell"}, {"start": 210.319, "duration": 4.081, "text": "well thank you for watching this video i"}, {"start": 212.48, "duration": 4.72, "text": "hope it's been helpful and if you indeed"}, {"start": 214.4, "duration": 5.04, "text": "found value in it please like comment"}, {"start": 217.2, "duration": 6.599, "text": "share and subscribe i'll look forward to"}, {"start": 219.44, "duration": 4.359, "text": "seeing you in the next video"}, {"start": 233.76, "duration": 2.08, "text": "you"}], "title": "Intramembranous Ossification", "description": "Intramembranous Ossification! What is Intramembranous Ossification? How does understanding this process of ossificaiton help me to better understand the body, and the skeleton? Let's make intramembranous ossification an easy process to figure out.", "lengthSeconds": 234, "width": 640, "height": 360}
{"video_id": "ZO5mcBor-vo", "transcript": [{"start": 0.08, "duration": 6.239, "text": "number four asks which of the marked"}, {"start": 2.56, "duration": 6.44, "text": "loan pairs on the following compound is"}, {"start": 6.319, "duration": 5.081, "text": "delocalized so what this is asking is is"}, {"start": 9.0, "duration": 3.719, "text": "there any loone pair on here such that"}, {"start": 11.4, "duration": 3.319, "text": "we could draw another resonance"}, {"start": 12.719, "duration": 4.48, "text": "structure by turning that loone pair"}, {"start": 14.719, "duration": 4.241, "text": "into a calent bond pushing some other P"}, {"start": 17.199, "duration": 3.601, "text": "electron density onto some other portion"}, {"start": 18.96, "duration": 4.639, "text": "of the molecule which one of these lone"}, {"start": 20.8, "duration": 4.84, "text": "pairs can be moved around essentially so"}, {"start": 23.599, "duration": 4.68, "text": "let's look at number one first we have"}, {"start": 25.64, "duration": 4.559, "text": "an oxygen we have two lone pairs there"}, {"start": 28.279, "duration": 3.96, "text": "is there anywhere that these can go can"}, {"start": 30.199, "duration": 5.2, "text": "we for example take this lone pair and"}, {"start": 32.239, "duration": 5.401, "text": "put it right there well we can't here"}, {"start": 35.399, "duration": 4.521, "text": "because this carbon already has four"}, {"start": 37.64, "duration": 4.079, "text": "bonds it's got a bond to oxygen two"}, {"start": 39.92, "duration": 3.959, "text": "bonds to these two carbon atoms there's"}, {"start": 41.719, "duration": 3.84, "text": "an implied hydrogen it is not going to"}, {"start": 43.879, "duration": 4.081, "text": "be able to accommodate any other bonds"}, {"start": 45.559, "duration": 4.0, "text": "we need access to other Pi electrons"}, {"start": 47.96, "duration": 3.84, "text": "that can Shuffle around in some other"}, {"start": 49.559, "duration": 3.84, "text": "way we have a pi Bond over here but"}, {"start": 51.8, "duration": 4.239, "text": "that's just too far away it's not going"}, {"start": 53.399, "duration": 4.68, "text": "to be able to interact with this one so"}, {"start": 56.039, "duration": 4.68, "text": "that's going to be no let's look at"}, {"start": 58.079, "duration": 5.201, "text": "number two we have a lone pair here is"}, {"start": 60.719, "duration": 4.601, "text": "it the case that this can go down here"}, {"start": 63.28, "duration": 4.92, "text": "well in this case it actually can"}, {"start": 65.32, "duration": 5.92, "text": "because this is adjacent to this Pi bond"}, {"start": 68.2, "duration": 5.88, "text": "this CN Sigma bond is adjacent to this"}, {"start": 71.24, "duration": 5.04, "text": "CC double bond which has a sigma and a"}, {"start": 74.08, "duration": 4.76, "text": "pi Bond so if we push this loan paired"}, {"start": 76.28, "duration": 4.64, "text": "down like that we can also take this Pi"}, {"start": 78.84, "duration": 4.599, "text": "Bond and push it right there and we"}, {"start": 80.92, "duration": 5.28, "text": "could put the anion on that carbon"}, {"start": 83.439, "duration": 6.64, "text": "instead we would end up with n plus and"}, {"start": 86.2, "duration": 5.68, "text": "C minus this is the case that this loone"}, {"start": 90.079, "duration": 3.641, "text": "pair is delocalized that is going to be"}, {"start": 91.88, "duration": 4.279, "text": "number two let's just look at the other"}, {"start": 93.72, "duration": 4.759, "text": "options with the Florine atom we've got"}, {"start": 96.159, "duration": 4.28, "text": "three lone pairs here is it the case"}, {"start": 98.479, "duration": 4.241, "text": "that we can take a lone pair and push it"}, {"start": 100.439, "duration": 4.28, "text": "there well no we can't because again"}, {"start": 102.72, "duration": 3.92, "text": "we've got a carbon atom here that's got"}, {"start": 104.719, "duration": 3.881, "text": "four bonds already it's not going to be"}, {"start": 106.64, "duration": 4.28, "text": "able to accommodate another Bond because"}, {"start": 108.6, "duration": 4.479, "text": "it cannot lose a bond the way that this"}, {"start": 110.92, "duration": 4.0, "text": "carbon atom did right this carbon atom"}, {"start": 113.079, "duration": 3.521, "text": "was able to accommodate a new Bond"}, {"start": 114.92, "duration": 4.0, "text": "because it was able to lose a bond as"}, {"start": 116.6, "duration": 4.479, "text": "well and remain coordinated with four"}, {"start": 118.92, "duration": 4.0, "text": "bonds there so this one can't do it"}, {"start": 121.079, "duration": 4.04, "text": "that's no good and then the same thing"}, {"start": 122.92, "duration": 4.32, "text": "here with oxygen this just isn't going"}, {"start": 125.119, "duration": 3.881, "text": "to work this oxygen could take one of"}, {"start": 127.24, "duration": 4.56, "text": "these piie bonds but we're not going to"}, {"start": 129.0, "duration": 5.64, "text": "form yet another Bond here because this"}, {"start": 131.8, "duration": 5.36, "text": "carbon has no bond that it can lose so"}, {"start": 134.64, "duration": 4.319, "text": "this also is not going to work again in"}, {"start": 137.16, "duration": 3.6, "text": "order for the lone pair to be"}, {"start": 138.959, "duration": 3.92, "text": "delocalized we have to be able to draw"}, {"start": 140.76, "duration": 4.28, "text": "another resonance structure like we did"}, {"start": 142.879, "duration": 4.521, "text": "here where this loone pair goes down we"}, {"start": 145.04, "duration": 4.36, "text": "form a pi Bond there and then we lose"}, {"start": 147.4, "duration": 4.24, "text": "that piie bond that flies up and gives"}, {"start": 149.4, "duration": 4.199, "text": "you the Ion on the carbon atom so that"}, {"start": 151.64, "duration": 4.959, "text": "one is going to be B lone pair number"}, {"start": 153.599, "duration": 3.0, "text": "two"}], "title": "Professor Dave Explains Delocalized Electrons #organicchemistry", "description": "Take Professor Dave's Structure and Bonding Organic Chemistry Exam here \ud83d\udc49 https://chemmunity.info/structureandbonding\n\n\ud83d\udcafCheck out Professor Dave's Practice Exams and more OCHEM RESOURCES here: https://chemmunity.info/dave \ud83d\ude4c", "lengthSeconds": 155, "width": 270, "height": 480}
{"video_id": "OY_lPQVXENI", "transcript": [{"start": 6.04, "duration": 4.78, "text": "so let's look at a problem\nregarding isotopic abundance and atomic"}, {"start": 10.83, "duration": 6.0, "text": "mass. the question says the abundances of\nthe following silicon isotopes are as"}, {"start": 16.83, "duration": 8.31, "text": "follows: silicon 28 is ninety two point\ntwo three percent, silicon 29 is four"}, {"start": 25.14, "duration": 6.959, "text": "point six seven percent, and silicon 30\nis 3.10% so that means out of all the"}, {"start": 32.099, "duration": 4.591, "text": "silicon atoms in the universe ninety two\npoint two three percent of them weigh"}, {"start": 36.69, "duration": 3.93, "text": "twenty-eight atomic mass units, four\npoint six seven percent of them weigh"}, {"start": 40.62, "duration": 5.369, "text": "twenty nine atomic mass units, and three\npoint one zero percent of them weigh"}, {"start": 45.989, "duration": 6.421, "text": "thirty atomic mass units. so the question\nis what is the average atomic mass of"}, {"start": 52.41, "duration": 6.3, "text": "silicon. so if this is confusing go and\ncheck out my tutorial on nuclides"}, {"start": 58.71, "duration": 5.97, "text": "symbols, we want to understand atomic\nmass we want to understand atomic number"}, {"start": 64.68, "duration": 5.46, "text": "and mass number and we want to\nunderstand what isotopes are, so if you"}, {"start": 70.14, "duration": 5.2, "text": "are unclear go ahead and check out that\ntutorial first, if everything makes sense give this a shot."}, {"start": 93.46, "duration": 2.56, "text": "so here are the\nrelevant values, now what are we going to"}, {"start": 96.03, "duration": 3.93, "text": "do with them? as it turns out to get the\naverage atomic mass all we need to do is"}, {"start": 99.96, "duration": 6.689, "text": "multiply the mass number of each isotope\nby its abundance, so let's do that now."}, {"start": 106.649, "duration": 6.33, "text": "first we have silicon 28 so let's put 28\ntimes zero point nine two two three"}, {"start": 112.979, "duration": 6.96, "text": "that is 92 point two three percent of\none, or a whole. to that we will add 29"}, {"start": 119.939, "duration": 5.761, "text": "for the mass number of silicon 29 and\nmultiply that by zero point zero four"}, {"start": 125.7, "duration": 5.67, "text": "six seven, that is four point six seven\npercent of one or a whole, which"}, {"start": 131.37, "duration": 5.89, "text": "represents the atomic mass\nof silicon. and lastly we have 30 for the"}, {"start": 137.26, "duration": 5.19, "text": "mass number of silicon 30 and we will\nmultiply that by zero point zero three"}, {"start": 142.45, "duration": 5.7, "text": "one zero, which is three point one zero\npercent of one. and now all we need to do"}, {"start": 148.15, "duration": 6.03, "text": "is add these up. let's also be clear that\nour answer should have three sig figs"}, {"start": 154.18, "duration": 5.85, "text": "because these values in the parentheses\nhave four sig figs, three sig figs, and"}, {"start": 160.03, "duration": 5.06, "text": "three sig figs respectively, so that will\nlimit our answer to three sig figs. and"}, {"start": 165.09, "duration": 6.01, "text": "if we plug all of that into a calculator\nwe do get 28 point one and that is the"}, {"start": 171.1, "duration": 5.4, "text": "average atomic mass of silicon that we\nwould find on the periodic table. so that"}, {"start": 176.5, "duration": 5.01, "text": "means if we took a sample of silicon\natoms and added all of their masses"}, {"start": 181.51, "duration": 4.17, "text": "together and then divided by the number\nof atoms there are, we would get twenty"}, {"start": 185.68, "duration": 5.56, "text": "eight point one atomic mass units as the\naverage mass of all of those silicon atoms."}], "title": "Practice Problem: Isotopic Abundance and Atomic Mass", "description": "In the general chemistry series we learned about nuclide symbols, which all imply a specific atomic number and mass number. Associated with this were isotopes, and average atomic mass. Given the relative abundances of all the isotopes of an element, how can we find the average atomic mass of that element? Let's practice that calculation here!\n\nTry all of the general chemistry practice problems: http://bit.ly/ProfDaveGCPP\n\nGeneral Chemistry Tutorials: http://bit.ly/ProfDaveGenChem\n\nEMAIL\u25ba ProfessorDaveExplains@gmail.com\nPATREON\u25ba http://patreon.com/ProfessorDaveExplains\n\nCheck out \"Is This Wi-Fi Organic?\", my book on disarming pseudoscience!\nAmazon: https://amzn.to/2HtNpVH\nBookshop: https://bit.ly/39cKADM\n\rBarnes and Noble: https://bit.ly/3pUjmrn \nBook Depository: http://bit.ly/3aOVDlT", "lengthSeconds": 223, "width": 640, "height": 360}
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"duration": 4.049, "text": "\u062e\u0627\u0631\u062c \u0627\u0644\u0639\u0635\u0628\u0648\u0646. \u0627\u0644\u0627\u062e\u062a\u0644\u0627\u0641\n\u0641\u064a \u0627\u0644\u0634\u062d\u0646\u0629 \u0627\u0644\u0643\u0647\u0631\u0628\u0627\u0626\u064a\u0629 \u064a\u062a\u0637\u0648\u0631 \u0628\u0633\u0628\u0628 \u0627\u0644\u062a\u062c\u0645\u0639"}, {"start": 25.609, "duration": 4.791, "text": "\u0645\u0646 \u0623\u064a\u0648\u0646\u0627\u062a \u062f\u0627\u062e\u0644 \u0648\u062e\u0627\u0631\u062c \u0627\u0644\u063a\u0634\u0627\u0621.\n\u0627\u0644\u0623\u064a\u0648\u0646\u0627\u062a \u0647\u064a \u0630\u0631\u0627\u062a \u0625\u0645\u0627 \u0641\u0642\u062f\u062a \u0623\u0648 \u0627\u0643\u062a\u0633\u0628\u062a"}, {"start": 30.4, "duration": 3.87, "text": "\u0648\u0628\u0627\u0644\u062a\u0627\u0644\u064a \u062a\u0643\u0648\u0646 \u0644\u0644\u0625\u0644\u0643\u062a\u0631\u0648\u0646\u0627\u062a \u0645\u0648\u062c\u0628\u0629 \u0623\u0648 \u0633\u0627\u0644\u0628\u0629\n\u0627\u0644\u0634\u062d\u0646\u0629."}, {"start": 34.27, "duration": 3.87, "text": "\u0647\u0646\u0627\u0643 \u0627\u0644\u0639\u062f\u064a\u062f \u0645\u0646 \u0627\u0644\u0623\u064a\u0648\u0646\u0627\u062a \u0627\u0644\u062a\u064a \u062a\u0644\u0639\u0628 \u062f\u0648\u0631\u064b\u0627 \u0645\u0647\u0645\u064b\u0627\n\u062f\u0648\u0631 \u0641\u064a \u0625\u0645\u0643\u0627\u0646\u0627\u062a \u0627\u0644\u063a\u0634\u0627\u0621 \u0644\u0644\u062e\u0644\u0627\u064a\u0627 \u0627\u0644\u0639\u0635\u0628\u064a\u0629."}, {"start": 38.14, "duration": 4.3, "text": "\u0647\u0646\u0627\u0643 \u0623\u064a\u0648\u0646\u0627\u062a \u0635\u0648\u062f\u064a\u0648\u0645 \u0630\u0627\u062a \u0634\u062d\u0646\u0629 \u0645\u0648\u062c\u0628\u0629 \u060c\n\u0645\u0645\u062b\u0644\u0629 \u0628\u0647\u0630\u0647 \u0627\u0644\u062f\u0648\u0627\u0626\u0631 \u0627\u0644\u0632\u0631\u0642\u0627\u0621 \u0648\u0633\u0644\u0628\u064a\u0629"}, {"start": 42.44, "duration": 4.79, 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(\u0623\u0646\u064a\u0648\u0646"}, {"start": 61.769, "duration": 5.001, "text": "\u0628\u0628\u0633\u0627\u0637\u0629 \u0645\u0635\u0637\u0644\u062d \u0623\u064a\u0648\u0646 \u0633\u0627\u0644\u0628\u0629 \u0627\u0644\u0634\u062d\u0646\u0629).\n\u0639\u0646\u062f\u0645\u0627 \u064a\u0643\u0648\u0646 \u0627\u0644\u0639\u0635\u0628\u0648\u0646 \u0641\u064a \u062d\u0627\u0644\u0629 \u0627\u0633\u062a\u0631\u0627\u062d\u0629 \u060c \u0641\u0625\u0646 \u0623\u064a\u0648\u0646\u0627\u062a \u0627\u0644\u0628\u0648\u062a\u0627\u0633\u064a\u0648\u0645"}, {"start": 66.77, "duration": 4.25, "text": "\u0648\u0627\u0644\u0623\u0646\u064a\u0648\u0646\u0627\u062a \u0627\u0644\u0639\u0636\u0648\u064a\u0629 \u0623\u0643\u062b\u0631 \u0627\u0646\u062a\u0634\u0627\u0631\u064b\u0627 \u0641\u064a \u0627\u0644\u062f\u0627\u062e\u0644\n\u0627\u0644\u062e\u0644\u064a\u0629. \u0641\u064a \u0628\u0642\u064a\u0629 \u0627\u0644\u062e\u0644\u0627\u064a\u0627 \u0627\u0644\u0639\u0635\u0628\u064a\u0629"}, {"start": 71.02, "duration": 3.61, "text": "\u0623\u0643\u062b\u0631 \u0633\u0644\u0628\u064a\u0629 \u0645\u0646 \u0627\u0644\u062e\u0627\u0631\u062c \u060c\n\u0645\u0645\u0627 \u064a\u0633\u0628\u0628 \u0627\u062d\u062a\u0645\u0627\u0644\u064a\u0629 \u063a\u0634\u0627\u0621 \u0627\u0644\u0631\u0627\u062d\u0629"}, {"start": 74.63, "duration": 3.78, "text": "\u064a\u0628\u0644\u063a \u0645\u062a\u0648\u0633\u0637 \u200b\u200b\u0627\u0644\u062e\u0644\u0627\u064a\u0627 \u0627\u0644\u0639\u0635\u0628\u064a\u0629 \u062d\u0648\u0627\u0644\u064a -70 \u0645\u0644\u064a \u0641\u0648\u0644\u062a."}, {"start": 78.41, "duration": 3.7, "text": "\u0625\u062d\u062f\u0649 \u0627\u0644\u0637\u0631\u0642 \u0644\u0644\u062d\u0641\u0627\u0638 \u0639\u0644\u0649 \u0647\u0630\u0647 \u0627\u0644\u0625\u0645\u0643\u0627\u0646\u0627\u062a \u0647\u064a \u0645\u0646 \u062e\u0644\u0627\u0644\n\u0622\u0644\u064a\u0629 \u062a\u0639\u0631\u0641 \u0628\u0627\u0633\u0645 \u0627\u0644\u0635\u0648\u062f\u064a\u0648\u0645 \u0648\u0627\u0644\u0628\u0648\u062a\u0627\u0633\u064a\u0648\u0645"}, {"start": 82.11, "duration": 4.55, "text": "\u0645\u0636\u062e\u0629. \u0647\u0630\u0627 \u0628\u0631\u0648\u062a\u064a\u0646 \u0646\u0642\u0644 \u064a\u0633\u062a\u062e\u062f\u0645\n\u0627\u0644\u0637\u0627\u0642\u0629 \u0644\u0636\u062e \u0628\u0627\u0633\u062a\u0645\u0631\u0627\u0631 \u062b\u0644\u0627\u062b\u0629 \u0623\u064a\u0648\u0646\u0627\u062a \u0627\u0644\u0635\u0648\u062f\u064a\u0648\u0645"}, {"start": 86.66, "duration": 4.98, "text": "\u0644\u0644\u062e\u0631\u0648\u062c \u0645\u0646 \u0627\u0644\u062e\u0644\u064a\u0629 \u0623\u062b\u0646\u0627\u0621 \u0627\u0644\u0636\u062e \u0641\u064a \u0646\u0641\u0633 \u0627\u0644\u0648\u0642\u062a\n\u0627\u062b\u0646\u064a\u0646 \u0645\u0646 \u0623\u064a\u0648\u0646\u0627\u062a \u0627\u0644\u0628\u0648\u062a\u0627\u0633\u064a\u0648\u0645 \u0641\u064a \u0627\u0644\u062e\u0644\u064a\u0629. \u0644\u0627\u0646"}, {"start": 91.64, "duration": 3.86, "text": "\u064a\u062a\u0645 \u0636\u062e \u0623\u064a\u0648\u0646\u0627\u062a \u0623\u0643\u062b\u0631 \u0625\u064a\u062c\u0627\u0628\u064a\u0629\n\u064a\u0633\u0627\u0639\u062f \u0639\u0644\u0649 \u0627\u0644\u062d\u0641\u0627\u0638 \u0639\u0644\u0649 \u0627\u0644\u063a\u0634\u0627\u0621"}, {"start": 95.5, "duration": 1.95, "text": "\u0627\u0644\u0633\u0644\u0628\u064a\u0629 \u0627\u0644\u0645\u062d\u062a\u0645\u0644\u0629."}, {"start": 97.45, "duration": 4.43, "text": "\u0639\u0644\u0649 \u0639\u0643\u0633 \u0627\u0644\u0623\u064a\u0648\u0646\u0627\u062a \u0627\u0644\u0623\u062e\u0631\u0649 \u060c \u064a\u0645\u064a\u0644 \u0627\u0644\u0628\u0648\u062a\u0627\u0633\u064a\u0648\u0645 \u0625\u0644\u0649 \u0627\u0644\u062a\u062d\u0631\u0643\n\u0628\u0633\u0647\u0648\u0644\u0629 \u0625\u0644\u0649 \u062d\u062f \u0645\u0627 \u0639\u0628\u0631 \u063a\u0634\u0627\u0621 \u0627\u0644\u062e\u0644\u064a\u0629 \u0645\u0646 \u062e\u0644\u0627\u0644"}, {"start": 101.88, "duration": 4.92, "text": "\u0627\u0644\u0642\u0646\u0648\u0627\u062a \u0627\u0644\u0623\u064a\u0648\u0646\u064a\u0629 \u060c \u0648\u0627\u0644\u062a\u064a \u062a\u0645\u062a\u062f \u0639\u0628\u0631 \u0627\u0644\u063a\u0634\u0627\u0621\n\u0627\u0644\u0628\u0631\u0648\u062a\u064a\u0646\u0627\u062a \u0627\u0644\u062a\u064a \u062a\u0633\u0645\u062d \u0644\u0644\u0623\u064a\u0648\u0646\u0627\u062a \u0628\u0627\u0644\u0645\u0631\u0648\u0631."}, {"start": 106.8, "duration": 3.63, "text": "\u0627\u0644\u0628\u0648\u062a\u0627\u0633\u064a\u0648\u0645 \u0633\u064a\u062e\u0631\u062c \u0645\u0646 \u0627\u0644\u062e\u0644\u0627\u064a\u0627 \u0627\u0644\u0639\u0635\u0628\u064a\u0629 \u062d\u062a\u0649\n\u064a\u0635\u0644 \u0625\u0644\u0649 \u0627\u0644\u0646\u0642\u0637\u0629 \u0627\u0644\u062a\u064a \u064a\u0643\u0648\u0646 \u0641\u064a\u0647\u0627 \u0641\u064a \u062d\u0627\u0644\u0629 \u062a\u0648\u0627\u0632\u0646"}, {"start": 110.43, "duration": 4.18, "text": "- \u0639\u0646\u062f\u0645\u0627 \u0644\u0627 \u062a\u062f\u0641\u0639 \u0642\u0648\u0649 \u0645\u062b\u0644 \u0627\u0644\u0627\u0646\u062a\u0634\u0627\u0631\n\u0641\u064a \u0627\u062a\u062c\u0627\u0647 \u0623\u0648 \u0622\u062e\u0631. \u0641\u064a \u0647\u0630\u0647"}, {"start": 114.61, "duration": 3.6, "text": "\u0627\u0644\u0646\u0642\u0637\u0629 \u060c \u0625\u0645\u0643\u0627\u0646\u0627\u062a \u0627\u0644\u063a\u0634\u0627\u0621 \u0644\u0644\u062e\u0644\u0627\u064a\u0627 \u0627\u0644\u0639\u0635\u0628\u064a\u0629\n\u062d\u0648\u0644 -65 \u0625\u0644\u0649 -70 mV \u060c \u0648\u0627\u0644\u062a\u064a \u062a\u0639\u0631\u0641 \u0628\u0627\u0633\u0645"}, {"start": 118.21, "duration": 1.5, "text": "\u0625\u0645\u0643\u0627\u0646\u0627\u062a \u0627\u0644\u063a\u0634\u0627\u0621 \u0627\u0644\u0628\u0627\u0642\u064a."}], "title": "2-Minute Neuroscience: Membrane Potential", "description": "In my 2-Minute Neuroscience videos I explain neuroscience topics in 2 minutes or less. In this video, I discuss membrane potential. Membrane potential refers to the difference in charge between the inside and outside of a neuron, which is created due to the unequal distribution of ions on both sides of the cell membrane. Membrane potential is maintained through mechanisms like the sodium-potassium pump, and an understanding of membrane potential is crucial to understanding what happens during an action potential.\n\nFor more neuroscience articles, videos, and a complete neuroscience glossary, check out my website at www.neuroscientificallychallenged.com !\n\nTRANSCRIPT:\n\nWelcome to 2 minute neuro, where I simplistically explain neuroscience topics in 2 minutes or less. In this installment I will discuss membrane potential.\n\nMembrane potential refers to the difference in electrical charge between the inside and outside of a neuron. This is the plasma or cell membrane of the neuron. It separates the inside of the cell from the outside environment; we\u2019ll say this is the inside and this is the outside of the neuron. The difference in electrical charge develops due to the grouping of ions on the inside and outside of the membrane. Ions are atoms that have either lost or gained electrons and thus have a positive or negative charge.\n\nThere are a variety of ions found in the human body; several play an important role in the membrane potential of neurons. There are positively charged sodium ions, represented by these blue circles and negatively charged chloride ions, represented by these green circles. When a neuron is at rest, the sodium ions and chloride ions are more prevalent outside of the cell. There are also positively charged potassium ions, represented by these yellow circles and various negatively charged ions, often referred to as organic anions represented by these grey circles (anion is simply a term for a negatively charged ion). When a neuron is at rest, the potassium ions and organic anions are more prevalent inside the cell. At rest, the inside of the neuron is more negatively charged than the outside, causing the resting membrane potential of an average neuron to be around -70 mV. \n\nOne way this potential is maintained is through a mechanism known as the sodium-potassium pump. This is a transport protein that uses energy to constantly pump three sodium ions out of the cell while at the same time pumping two potassium ions into the cell. Because there are more positive ions being pumped out than negative, it helps to keep the membrane potential negative.\n\nUnlike other ions, potassium tends to move fairly easily across the cell membrane through ion channels, which are membrane spanning proteins that allow ions to pass through. Potassium will pass out of the neuron until it reaches the point where it is at an equilibrium---when forces like diffusion aren\u2019t pushing it in one direction or the other. At this point, the membrane potential of the neuron is around -65 to -70 mV, which is known as the resting membrane potential.\n\nREFERENCE:\n\nPurves D, Augustine GJ, Fitzpatrick D, Hall WC, Lamantia AS, McNamara JO, White LE. Neuroscience. 4th ed. Sunderland, MA. Sinauer Associates; 2008.", "lengthSeconds": 121, "width": 640, "height": 360}
{"video_id": "uaog3UM7DT8", "transcript": [{"start": 0.57, "duration": 3.74, "text": "Hey guys, it\u2019s Dave, if you\u2019re here on\nmy Patreon page, it means that you\u2019re considering"}, {"start": 4.31, "duration": 2.97, "text": "supporting me in my quest to educate the world."}, {"start": 7.28, "duration": 3.82, "text": "I want to raise the bar of science literacy\nin our society, and I sure could use your help."}, {"start": 11.12, "duration": 3.12, "text": "Unlike some of the bigger educational channels,\nI don\u2019t have a staff, I don\u2019t have any"}, {"start": 14.24, "duration": 3.23, "text": "grants or funding, this is a low-budget one-man show."}, {"start": 17.47, "duration": 5.209, "text": "I write, edit, and animate all the content\non my channel, while managing all other aspects"}, {"start": 22.679, "duration": 1.0, "text": "of being an entrepreneur."}, {"start": 23.679, "duration": 4.281, "text": "It\u2019s a lot of work, and I\u2019m usually at\nit about sixty or seventy hours a week, so"}, {"start": 27.96, "duration": 2.729, "text": "calling it my passion is an understatement."}, {"start": 30.689, "duration": 4.46, "text": "The problem is that the ad revenue I earn\nfrom my YouTube channel is nowhere near commensurate"}, {"start": 35.149, "duration": 1.471, "text": "with the work I put in."}, {"start": 36.62, "duration": 3.759, "text": "Even with tens of thousands of views per day,\nit\u2019s just not a living wage."}, {"start": 40.379, "duration": 4.18, "text": "Beyond that, I have to pay people to check\nmy scripts for errors, operate camera and"}, {"start": 44.559, "duration": 3.8, "text": "audio when I shoot the content, and all kinds\nof other things, so I sure could use some"}, {"start": 48.359, "duration": 1.821, "text": "help covering those costs."}, {"start": 50.18, "duration": 4.429, "text": "If I can get enough support from you guys,\nthe viewers who benefit from all this hard"}, {"start": 54.609, "duration": 3.961, "text": "work, I can even hire people to help with\nsome of the writing and animating, which means"}, {"start": 58.57, "duration": 4.649, "text": "I could move much faster, and put out twice\nas much content every week."}, {"start": 63.219, "duration": 3.481, "text": "Even though I\u2019ve already covered a lot of\ntopics, from chemistry to biology to physics"}, {"start": 66.7, "duration": 4.44, "text": "and tons more, there are still so many other\nsubjects left to do, and I\u2019m not going to"}, {"start": 71.14, "duration": 1.75, "text": "stop until I do them all."}, {"start": 72.89, "duration": 2.53, "text": "So please check out the rewards on this page."}, {"start": 75.42, "duration": 4.809, "text": "For just a few dollars a month you can get\naccess to exclusive material, live chatting"}, {"start": 80.229, "duration": 4.871, "text": "with me, you can vote on what subjects I cover\nnext, or even get one-on-one tutoring."}, {"start": 85.1, "duration": 3.979, "text": "These are just some of the ways I want to\nthank you guys for taking the initiative to"}, {"start": 89.079, "duration": 4.271, "text": "help when you didn\u2019t even have to, because\nthat\u2019s how we are going to create the world"}, {"start": 93.35, "duration": 1.32, "text": "we all want to live in."}, {"start": 94.67, "duration": 0.989, "text": "I\u2019ll see you on YouTube."}], "title": "Professor Dave Patreon Trailer", "description": "This is the trailer for my Patreon page. If you're watching it on YouTube, go to Patreon instead! And while you're there, please consider supporting me!\n\nhttps://www.patreon.com/professordaveexplains", "lengthSeconds": 98, "width": 640, "height": 360}
{"video_id": "h_zkgd_P3Nk", "transcript": [{"start": 0.2, "duration": 10.679, "text": "ok so for this problem we've got a\nsubstrate here which has a ketone"}, {"start": 10.879, "duration": 4.21, "text": "functional group and a carboxylic acid\nfunctional group and we're gonna do two"}, {"start": 15.089, "duration": 4.111, "text": "reactions with two different reducing\nagents, we've got sodium borohydride and"}, {"start": 19.2, "duration": 3.5, "text": "we've got lithium aluminum hydride so if\nyou don't remember what these do go ahead and"}, {"start": 22.7, "duration": 3.44, "text": "check out my lecture on oxidation\nreduction in the organic chemistry"}, {"start": 26.14, "duration": 4.83, "text": "playlist and we we these are going to\ngenerate two different products based on"}, {"start": 30.97, "duration": 4.67, "text": "reducing capacity of these two reducing\nagents so go and see if you can draw the"}, {"start": 35.64, "duration": 2.4, "text": "correct product for these two\ntransformations"}, {"start": 52.94, "duration": 5.0, "text": "we've got our two different reducing agents, our sodium\nborohydride and a lithium"}, {"start": 57.95, "duration": 5.46, "text": "aluminum hydride, we know that those are\nboth sources of hydride, H- which is"}, {"start": 63.41, "duration": 4.209, "text": "going to act as a nucleophile and\nattack a carbonyl and reduce some carbonyl"}, {"start": 67.619, "duration": 3.29, "text": "containing compound but the key\ndifference here is that sodium"}, {"start": 70.909, "duration": 4.52, "text": "borohydride is a softer reducing agent, a\nbit weaker whereas lithium aluminum"}, {"start": 75.429, "duration": 3.55, "text": "hydride is a hard reducing agent, it's a little\nbit stronger, so there is a"}, {"start": 78.979, "duration": 4.25, "text": "discrepancy in what kind of carbonyl\ncontaining compounds they are able to"}, {"start": 83.229, "duration": 5.151, "text": "reduce. as it happens sodium borohydride is\nonly able to reduce ketones and"}, {"start": 88.38, "duration": 6.029, "text": "aldehydes, this is because the ketone\nan aldehyde functional group, the partial"}, {"start": 94.409, "duration": 4.46, "text": "positivity on this carbonyl carbon\ndiffers from that of say a carboxylic"}, {"start": 98.869, "duration": 5.54, "text": "acid because this carbon there, remember\nthat this adjacent oxygen atom is able to"}, {"start": 104.409, "duration": 5.181, "text": "put electrons onto that carbon via\nresonance and so that is the reason that"}, {"start": 109.59, "duration": 5.43, "text": "this is not as partially positive and\ntherefore not as electrophilic as a ketone"}, {"start": 115.02, "duration": 4.309, "text": "or aldehyde carbon so that means that\njust the discrepancy in electrophilicity"}, {"start": 119.329, "duration": 4.74, "text": "between this and this means that sodium\nborohydride is able to reduce this"}, {"start": 124.069, "duration": 4.84, "text": "ketone functional group but not the\ncarboxylic acid functional group so as a"}, {"start": 128.909, "duration": 1.541, "text": "result on the product"}, {"start": 130.45, "duration": 4.14, "text": "reducing with sodium borohydride we can\nsee that the ketone has been reduced to"}, {"start": 134.59, "duration": 4.24, "text": "the secondary alcohol whereas the\ncarboxylic acid has been untouched"}, {"start": 138.83, "duration": 3.95, "text": "it does not have the ability to reduce the\ncarboxylic acid so that won't change"}, {"start": 142.78, "duration": 5.56, "text": "however lithium aluminum hydride is able\nto reduce ketones, aldehydes, carboxylic"}, {"start": 148.34, "duration": 5.02, "text": "acids, esters, it is a stronger reducing\nagent because it's stronger it is"}, {"start": 153.36, "duration": 5.05, "text": "able to reduce not just this but also\nthis and so as a result we can see in"}, {"start": 158.41, "duration": 4.439, "text": "the product that the ketone has been\nreduced to the secondary alcohol just as"}, {"start": 162.849, "duration": 4.411, "text": "it has over here but in addition the\ncarboxylic acid has been reduced to the"}, {"start": 167.26, "duration": 4.02, "text": "primary alcohol. so the key here is\nnoticing the difference in reducing"}, {"start": 171.28, "duration": 5.22, "text": "strength of these two reducing agents and\nusing that information to draw the to"}, {"start": 176.5, "duration": 0.86, "text": "correct products"}, {"start": 199.2, "duration": 3.38, "text": "thanks for watching guys, subscribe to my\nchannel for more tutorials and as always"}, {"start": 202.59, "duration": 1.16, "text": "feel free to email me"}], "title": "Practice Problem: Reducing Agents", "description": "For this one we need to know the ability of different reducing agents to react with various functional groups.\n\nTry all of the organic chemistry practice problems: http://bit.ly/ProfDaveOCPP\n\nOrganic Chemistry Tutorials: http://bit.ly/ProfDaveOrgChem\n\nEMAIL\u25ba ProfessorDaveExplains@gmail.com\nPATREON\u25ba http://patreon.com/ProfessorDaveExplains\n\nCheck out \"Is This Wi-Fi Organic?\", my book on disarming pseudoscience!\nAmazon: https://amzn.to/2HtNpVH\nBookshop: https://bit.ly/39cKADM\n\rBarnes and Noble: https://bit.ly/3pUjmrn \nBook Depository: http://bit.ly/3aOVDlT", "lengthSeconds": 207, "width": 640, "height": 360}
{"video_id": "rZQVAxuT3pg", "transcript": [{"start": 0.08, "duration": 5.96, "text": "number four asks a catalyst lowers the"}, {"start": 3.24, "duration": 5.0, "text": "blank for a reaction is it activation"}, {"start": 6.04, "duration": 5.32, "text": "energy change in entropy change in"}, {"start": 8.24, "duration": 5.479, "text": "enthalpy change in free energy if we are"}, {"start": 11.36, "duration": 5.52, "text": "looking at an energy diagram we have"}, {"start": 13.719, "duration": 6.081, "text": "reactants we have products this is going"}, {"start": 16.88, "duration": 5.159, "text": "to be the activation energy that's the"}, {"start": 19.8, "duration": 6.36, "text": "activation energy right there what a"}, {"start": 22.039, "duration": 5.681, "text": "catalyst does it is going to lower that"}, {"start": 26.16, "duration": 3.92, "text": "value it is going to lower the"}, {"start": 27.72, "duration": 4.399, "text": "activation energy so here's the new"}, {"start": 30.08, "duration": 4.12, "text": "activation energy there so this is going"}, {"start": 32.119, "duration": 4.6, "text": "to be a what we need to understand about"}, {"start": 34.2, "duration": 4.44, "text": "the catalyst is that it does not alter"}, {"start": 36.719, "duration": 5.081, "text": "any of these other parameters it does"}, {"start": 38.64, "duration": 5.239, "text": "not change the position of reactants or"}, {"start": 41.8, "duration": 4.16, "text": "products on the energy diagram for"}, {"start": 43.879, "duration": 4.121, "text": "example change in enthalpy or change in"}, {"start": 45.96, "duration": 4.079, "text": "free energy that is determined from"}, {"start": 48.0, "duration": 4.079, "text": "where the reactants sit to where the"}, {"start": 50.039, "duration": 4.481, "text": "products sit so if those don't change"}, {"start": 52.079, "duration": 5.241, "text": "their location none of those values can"}, {"start": 54.52, "duration": 5.999, "text": "change either all that we're changing is"}, {"start": 57.32, "duration": 3.199, "text": "the energy of the transition"}], "title": "What Does a Catalyst Do?", "description": "This is an abbreviated version of this question for YouTube Shorts. Check out the full version in the linked video.\n\nGet $5 off your first month of Chemmunity with promo code: DAVE", "lengthSeconds": 59, "width": 360, "height": 640}
{"video_id": "4AlUprQGvY4", "transcript": [{"start": 0.38, "duration": 3.76, "text": "hey all, it'sPprofessor Dave, I want to\ncontinue talking about elimination reactions"}, {"start": 4.14, "duration": 1.78, "text": "specifically the E1 reaction."}, {"start": 12.23, "duration": 5.73, "text": "so the E1 reaction is to the E2 reaction\nas the SN1 reaction is to the SN2 reaction"}, {"start": 17.96, "duration": 3.819, "text": "so basically as you'd expect, the E1\nreaction is going to be"}, {"start": 21.779, "duration": 3.391, "text": "two steps with a unimolecular transition state"}, {"start": 25.17, "duration": 4.58, "text": "just like with the SN1"}, {"start": 29.75, "duration": 3.649, "text": "the first step in the E1 reaction is\nthat the leaving group leaves"}, {"start": 33.399, "duration": 3.731, "text": "so let's take a look at this substrate\nhere the bromine is gonna leave"}, {"start": 37.13, "duration": 5.17, "text": "that leaves a tertiary carbocation,\nnow it's very very important to understand"}, {"start": 42.3, "duration": 3.31, "text": "that while this substrate, this carbon is"}, {"start": 45.61, "duration": 4.05, "text": "sp3 hybridized and therefore has\ntetrahedral geometry"}, {"start": 49.66, "duration": 3.27, "text": "anytime a carbon that is sp3 hybridized"}, {"start": 52.93, "duration": 3.87, "text": "loses an electron domain, because\nthis bromine is leaving"}, {"start": 56.8, "duration": 4.71, "text": "with the electrons in that bond now this\ncarbon atom has only"}, {"start": 61.51, "duration": 4.74, "text": "three electron domains and anything surrounded by three electron domains will"}, {"start": 66.25, "duration": 5.21, "text": "exhibit trigonal planar geometry so it\nis the case that this methyl group is now"}, {"start": 71.46, "duration": 3.769, "text": "flat in the board, this is now a flat\nmolecule with respect to the carbons"}, {"start": 75.229, "duration": 3.261, "text": "so there is a formal positive charge now"}, {"start": 78.49, "duration": 3.9, "text": "and so that means even a weak nucleophile like water can go ahead do an"}, {"start": 82.39, "duration": 1.33, "text": "acid-base reaction"}, {"start": 83.72, "duration": 3.18, "text": "and grab that proton, so here is oxygen"}, {"start": 86.9, "duration": 3.179, "text": "interacting with this proton and then"}, {"start": 90.079, "duration": 4.551, "text": "the electrons in this carbon-hydrogen\nbond are the ones that go to form"}, {"start": 94.63, "duration": 3.199, "text": "the carbon-carbon pi bond in the product"}, {"start": 98.8, "duration": 3.28, "text": "thanks for watching, guys. subscribe to my\nchannel for more tutorials"}, {"start": 102.08, "duration": 2.12, "text": "and as always feel free to email me with questions"}], "title": "E1 Reaction", "description": "Here we examine the mechanism for the E1 reaction. Don't worry, it's just these four for now!\n\nWatch the whole Organic Chemistry playlist: http://bit.ly/ProfDaveOrgChem\n\nGeneral Chemistry Tutorials: http://bit.ly/ProfDaveGenChem\nBiochemistry Tutorials: http://bit.ly/ProfDaveBiochem\nBiology Tutorials: http://bit.ly/ProfDaveBio\nClassical Physics Tutorials: http://bit.ly/ProfDavePhysics1\nModern Physics Tutorials: http://bit.ly/ProfDavePhysics2\nMathematics Tutorials: http://bit.ly/ProfDaveMath\n\nEMAIL\u25ba ProfessorDaveExplains@gmail.com\nPATREON\u25ba http://patreon.com/ProfessorDaveExplains\n\nCheck out \"Is This Wi-Fi Organic?\", my book on disarming pseudoscience!\nAmazon: https://amzn.to/2HtNpVH\nBookshop: https://bit.ly/39cKADM\n\rBarnes and Noble: https://bit.ly/3pUjmrn \nBook Depository: http://bit.ly/3aOVDlT", "lengthSeconds": 122, "width": 640, "height": 360}
{"video_id": "AZSah-Ny5Is", "transcript": [{"start": 5.75, "duration": 3.96, "text": "today we're gonna do some titration calculations"}, {"start": 9.71, "duration": 4.78, "text": "consider the following acid-base\nreaction we have hydrochloric acid and"}, {"start": 14.49, "duration": 6.449, "text": "sodium hydroxide yielding sodium\nchloride and water if 50 milliliters of"}, {"start": 20.939, "duration": 5.551, "text": "an HCl solution with unknown\nconcentration are titrated with zero"}, {"start": 26.49, "duration": 6.12, "text": "point two five zero molar sodium\nhydroxide and then if it takes 35 point"}, {"start": 32.61, "duration": 5.699, "text": "two three milliliters of base to reach\nthe equivalence point what was the"}, {"start": 38.309, "duration": 5.101, "text": "concentration of the HCL solution so if\nyou're not sure how to approach this go"}, {"start": 43.41, "duration": 4.13, "text": "ahead and check out my tutorial on\ntitration and when you're ready give it a try."}, {"start": 65.44, "duration": 5.06, "text": "so let's put our reaction up top and\nthen what we need to remember about the"}, {"start": 70.5, "duration": 5.31, "text": "equivalence point is that it is the\npoint where the moles of acid equals"}, {"start": 75.81, "duration": 6.09, "text": "moles of base so it is the point where\nprecisely all of the acid has been"}, {"start": 81.9, "duration": 5.37, "text": "neutralized by the base so it is telling\nus something about stoichiometry when"}, {"start": 87.27, "duration": 4.559, "text": "the equivalence point is reached so\nsince we want to find out the moles of"}, {"start": 91.829, "duration": 5.881, "text": "acid let's find the moles of base so\nlet's take our volume of base we have"}, {"start": 97.71, "duration": 4.29, "text": "thirty five point two four milliliters\nof base that were required to reach the"}, {"start": 102.0, "duration": 4.829, "text": "equivalence point and let's first\nconvert that into leaders so dividing by"}, {"start": 106.829, "duration": 5.311, "text": "a thousand milliliters we have 0.03 five\nto four liters of Base to reach the"}, {"start": 112.14, "duration": 6.03, "text": "equivalence point now since we have the\nconcentration of that base we can"}, {"start": 118.17, "duration": 5.22, "text": "convert liters into moles so let's take\nour liters of base and multiply by 0.25"}, {"start": 123.39, "duration": 7.02, "text": "zero multiplied not means that was zero\npoint zero zero eight eight one moles of"}, {"start": 130.41, "duration": 6.0, "text": "base that were required to reach the\npoint now we have to look at the"}, {"start": 136.41, "duration": 5.49, "text": "stoichiometric ratio the acid in the\nbase are in a one-to-one ratio so that"}, {"start": 141.9, "duration": 5.28, "text": "means if that many moles of base were\nrequired to reach the equivalence point"}, {"start": 147.18, "duration": 4.95, "text": "then we multiply by one mole of acid\nover one mole of base and we get that"}, {"start": 152.13, "duration": 6.12, "text": "same number of moles back for the acid\nso when the equivalence point was"}, {"start": 158.25, "duration": 6.39, "text": "reached that many moles of base had\nprecisely neutralized that many moles of"}, {"start": 164.64, "duration": 5.34, "text": "acid and that is the number of moles of\nacid that were in the solution initially"}, {"start": 169.98, "duration": 5.01, "text": "for the HCl solution so now all we have\nto do is convert that many moles of acid"}, {"start": 174.99, "duration": 4.47, "text": "into a concentration and so\nconcentration is moles per liter so we"}, {"start": 179.46, "duration": 6.42, "text": "take our moles of acid and we divide by\n0.0500 liters because we know we had"}, {"start": 185.88, "duration": 4.14, "text": "fifty point zero milliliters of acid we\nknew the volume we just didn't know the"}, {"start": 190.02, "duration": 6.69, "text": "concentration so that many moles over\nthat many liters gives us 0.176 moles"}, {"start": 196.71, "duration": 8.15, "text": "per liter or point 176 molar HCl which\nwas the concentration of our unknown HCl solution."}], "title": "Practice Problem: Titration Calculations", "description": "Titration is a way to do stoichiometry with acids and bases. The equivalence point tells us something about the moles of acid and base that are present in solution at that moment. And if we know one of those values, we can know the other. Let's calculate some unknown concentrations!\n\nTry all of the general chemistry practice problems: http://bit.ly/ProfDaveGCPP\n\nGeneral Chemistry Tutorials: http://bit.ly/ProfDaveGenChem\n\nEMAIL\u25ba ProfessorDaveExplains@gmail.com\nPATREON\u25ba http://patreon.com/ProfessorDaveExplains\n\nCheck out \"Is This Wi-Fi Organic?\", my book on disarming pseudoscience!\nAmazon: https://amzn.to/2HtNpVH\nBookshop: https://bit.ly/39cKADM\n\rBarnes and Noble: https://bit.ly/3pUjmrn \nBook Depository: http://bit.ly/3aOVDlT", "lengthSeconds": 236, "width": 640, "height": 360}
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\u0623\u0646 \u064a\u062d\u062f\u062b \u0645\u0631\u0629 \u0623\u062e\u0631\u0649\u061f\n \u0644\u0647\u0630\u0627 \u0627\u0644\u0633\u0628\u0628 \u0637\u0648\u0631\u0646\u0627"}, {"start": 19.42, "duration": 3.75, "text": "\u0648\u062d\u062f\u0627\u062a \u0627\u0644\u0642\u064a\u0627\u0633\n \u064a\u0631\u062a\u0628\u0637 \u0628\u0639\u0636\u0647\u0627 \u0628\u0627\u0644\u0638\u0648\u0627\u0647\u0631 \u0627\u0644\u0637\u0628\u064a\u0639\u064a\u0629"}, {"start": 23.17, "duration": 3.41, "text": "\u0645\u062b\u0644 \u0627\u0644\u064a\u0648\u0645 \u0623\u0648 \u0627\u0644\u0633\u0646\u0629\n\u0645\u0628\u0646\u064a\u0629 \u0639\u0644\u0649 \u062d\u0631\u0643\u0629 \u0627\u0644\u0623\u0631\u0636"}, {"start": 26.58, "duration": 3.73, "text": "\u0644\u0643\u0646 \u0627\u0644\u0643\u062b\u064a\u0631 \u0645\u0646\u0647\u0627 \u062a\u0639\u0633\u0641\u064a \u0645\u062b\u0644 \u0645\u0639\u0638\u0645 \u0648\u062d\u062f\u0627\u062a \u0627\u0644\u0637\u0648\u0644 \u0648\u0627\u0644\u0643\u062a\u0644\u0629"}, {"start": 30.31, "duration": 4.26, "text": "\u0639\u0646\u062f\u0645\u0627 \u0646\u0634\u0627\u0631\u0643 \u0641\u064a \u0627\u0644\u0642\u064a\u0627\u0633 \u064a\u062c\u0628 \u0623\u0646 \u0646\u0643\u0648\u0646 \u0639\u0644\u0649 \u062f\u0631\u0627\u064a\u0629 \u0628\u0628\u0639\u0636 \u0627\u0644\u0634\u0631\u0648\u0637"}, {"start": 34.57, "duration": 4.21, "text": "\u0639\u0644\u0649 \u0633\u0628\u064a\u0644 \u0627\u0644\u0645\u062b\u0627\u0644 \u060c \u0643\u0645 \u0639\u062f\u062f \u0627\u0644\u0623\u0634\u062e\u0627\u0635 \u0641\u064a \u0647\u0630\u0627 \u0627\u0644\u062d\u0634\u062f\u061f\n\u064a\u0645\u0643\u0646 \u0623\u0646 \u0646\u0642\u0648\u0644"}, {"start": 38.78, "duration": 5.43, "text": "10,000\u061f\n \u0645\u0627\u0630\u0627 \u0639\u0646 10,400 \u0623\u0648 10,472"}, {"start": 44.21, "duration": 3.97, "text": "\u064a\u062c\u0628 \u0623\u0646 \u0646\u0643\u0648\u0646 \u062d\u0631\u064a\u0635\u064a\u0646 \u0639\u0644\u0649 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3.84, "text": "\u0647\u0646\u0627 \u0627\u0644\u0642\u0648\u0627\u0639\u062f \u0644\u062a\u062d\u062f\u064a\u062f \u0639\u062f\u062f \u0627\u0644\u0627\u0631\u0642\u0627\u0645 \u0627\u0644\u0645\u0639\u0646\u064a\u0629 \u0641\u064a \u0627\u0644\u0642\u064a\u0645\u0629"}, {"start": 80.44, "duration": 3.2, "text": "\u0648\u0627\u062d\u062f: \u0623\u064a \u0631\u0642\u0645 \u0644\u064a\u0633 \u0635\u0641\u0631\u064b\u0627"}, {"start": 83.64, "duration": 1.979, "text": "\u064a\u0639\u062a\u0628\u0631 \u0631\u0642\u0645 \u0645\u0639\u0646\u064a"}, {"start": 85.619, "duration": 3.421, "text": "\u0627\u062b\u0646\u0627\u0646: \u0627\u0644\u0623\u0635\u0641\u0627\u0631 \u0628\u064a\u0646 \u0627\u0644\u0623\u0631\u0642\u0627\u0645 \u0627\u0644\u0623\u062e\u0631\u0649 \u063a\u064a\u0631 \u0627\u0644\u0635\u0641\u0631\u064a\u0629"}, {"start": 89.04, "duration": 2.06, "text": "\u062a\u0639\u062a\u0628\u0631 \u0627\u0631\u0642\u0627\u0645 \u0645\u0639\u0646\u064a\u0629"}, {"start": 91.1, 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measurement or make a calculation, how many digits do we use? There's rules, friend! You must obey the sig figs. Don't worry, learn all about them in this clip, and you'll be the coolest kid on the block.\n\nWatch the whole General Chemistry playlist: http://bit.ly/ProfDaveGenChem\n\nStudy for the AP Chemistry exam with me: https://bit.ly/ProfDaveAPChem\n\nOrganic Chemistry Tutorials: http://bit.ly/ProfDaveOrgChem\nBiochemistry Tutorials: http://bit.ly/ProfDaveBiochem\nBiology Tutorials: http://bit.ly/ProfDaveBio\nClassical Physics Tutorials: http://bit.ly/ProfDavePhysics1\nModern Physics Tutorials: http://bit.ly/ProfDavePhysics2\nMathematics Tutorials: http://bit.ly/ProfDaveMaths\n\nEMAIL\u25ba ProfessorDaveExplains@gmail.com\nPATREON\u25ba http://patreon.com/ProfessorDaveExplains\n\nCheck out \"Is This Wi-Fi Organic?\", my book on disarming pseudoscience!\nAmazon: https://amzn.to/2HtNpVH\nBookshop: https://bit.ly/39cKADM\n\rBarnes and Noble: https://bit.ly/3pUjmrn \nBook Depository: http://bit.ly/3aOVDlT", "lengthSeconds": 218, "width": 640, "height": 360}
{"video_id": "euu3i7ZeY5A", "transcript": [{"start": 0.08, "duration": 4.16, "text": "number one asks a reaction with a"}, {"start": 2.32, "duration": 5.32, "text": "negative value for the change in free"}, {"start": 4.24, "duration": 6.479, "text": "energy is referred to as exothermic"}, {"start": 7.64, "duration": 4.72, "text": "endothermic exergonic endergonic so what"}, {"start": 10.719, "duration": 4.721, "text": "are we talking about we're talking about"}, {"start": 12.36, "duration": 5.16, "text": "free energy so Gibbs free energy is"}, {"start": 15.44, "duration": 3.839, "text": "associated with Delta G and we're"}, {"start": 17.52, "duration": 5.24, "text": "looking for a negative value for this"}, {"start": 19.279, "duration": 7.361, "text": "change exothermic and endothermic these"}, {"start": 22.76, "duration": 6.8, "text": "relate to enp exothermic means Delta H"}, {"start": 26.64, "duration": 5.52, "text": "is negative and endothermic means Delta"}, {"start": 29.56, "duration": 4.56, "text": "H is positive so those are not the ones"}, {"start": 32.16, "duration": 4.559, "text": "that we're looking for exergonic and"}, {"start": 34.12, "duration": 5.32, "text": "endergonic are words that relate instead"}, {"start": 36.719, "duration": 6.041, "text": "to Delta G to Gibs free energy a"}, {"start": 39.44, "duration": 7.2, "text": "negative Delta G is referred to as an"}, {"start": 42.76, "duration": 6.92, "text": "exergonic reaction endergonic would be"}, {"start": 46.64, "duration": 5.36, "text": "Delta G is positive a negative value for"}, {"start": 49.68, "duration": 4.519, "text": "the change in Gibbs free energy that's"}, {"start": 52.0, "duration": 5.359, "text": "Delta g equals negative that is going to"}, {"start": 54.199, "duration": 3.16, "text": "be exonic"}], "title": "Enthalpy and Free Energy", "description": "This is an abbreviated version of this question for YouTube Shorts. Check out the full version in the linked video.\n\nGet $5 off your first month of Chemmunity with promo code: DAVE", "lengthSeconds": 55, "width": 360, "height": 640}
{"video_id": "vzLO-1efXrQ", "transcript": [{"start": 0.08, "duration": 5.36, "text": "number three asks which is the strongest"}, {"start": 2.96, "duration": 4.64, "text": "acid when we're talking about acid"}, {"start": 5.44, "duration": 4.36, "text": "strength what we need to understand is"}, {"start": 7.6, "duration": 4.32, "text": "that the strength of an acid is"}, {"start": 9.8, "duration": 4.48, "text": "proportional to the stability of the"}, {"start": 11.92, "duration": 4.68, "text": "conjugate base that sounds like a lot"}, {"start": 14.28, "duration": 5.32, "text": "but what we really mean here is that the"}, {"start": 16.6, "duration": 5.519, "text": "more stable the thing is that you get"}, {"start": 19.6, "duration": 4.759, "text": "the more readily the acid wants to"}, {"start": 22.119, "duration": 7.4, "text": "donate the proton meaning the stronger"}, {"start": 24.359, "duration": 7.601, "text": "the acid is so if we have HF HCL hbr hi"}, {"start": 29.519, "duration": 5.241, "text": "what are the conjugate bases of these"}, {"start": 31.96, "duration": 7.48, "text": "things what are we going to get if these"}, {"start": 34.76, "duration": 8.88, "text": "lose protons HF becomes F minus HCL"}, {"start": 39.44, "duration": 6.759, "text": "becomes CL minus hbr becomes BR minus"}, {"start": 43.64, "duration": 5.88, "text": "and hi becomes I minus so we get the"}, {"start": 46.199, "duration": 5.801, "text": "halide ions so now the question becomes"}, {"start": 49.52, "duration": 5.28, "text": "what is the relative stability of the"}, {"start": 52.0, "duration": 4.6, "text": "halide ions that are being produced now"}, {"start": 54.8, "duration": 5.719, "text": "what is the trend we want to look at"}, {"start": 56.6, "duration": 3.919, "text": "here what we understand is that the"}], "title": "Which is the Strongest Acid?", "description": "This is an abbreviated version of this question for YouTube Shorts. Check out the full version in the linked video.", "lengthSeconds": 59, "width": 360, "height": 640}
{"video_id": "U0_MroEJBh0", "transcript": [{"start": 0.08, "duration": 4.8, "text": "number three asks which of the following"}, {"start": 2.28, "duration": 4.479, "text": "is the best electrophile so whereas a"}, {"start": 4.88, "duration": 4.96, "text": "nucleophile is something that has a lot"}, {"start": 6.759, "duration": 4.84, "text": "of electron excess an electrophile is"}, {"start": 9.84, "duration": 4.32, "text": "something that has an electron"}, {"start": 11.599, "duration": 5.161, "text": "deficiency so it seeks electron density"}, {"start": 14.16, "duration": 5.16, "text": "to coordinate to it electrophile file"}, {"start": 16.76, "duration": 5.16, "text": "meaning love loves electrons wants"}, {"start": 19.32, "duration": 5.279, "text": "electrons to come coordinate to it first"}, {"start": 21.92, "duration": 4.24, "text": "we have butane well there's no"}, {"start": 24.599, "duration": 4.121, "text": "electrophilicity there there is no"}, {"start": 26.16, "duration": 4.32, "text": "region of electron deficiency there's no"}, {"start": 28.72, "duration": 4.12, "text": "partial positive charge there's no"}, {"start": 30.48, "duration": 5.84, "text": "formal positive charge so that doesn't"}, {"start": 32.84, "duration": 6.0, "text": "work B ammonia we do have a positive"}, {"start": 36.32, "duration": 5.759, "text": "charge on that nitrogen because it is"}, {"start": 38.84, "duration": 4.64, "text": "coordinated to four hydrogens but it's"}, {"start": 42.079, "duration": 2.681, "text": "not really going to work as an"}, {"start": 43.48, "duration": 3.399, "text": "electrophile because it cannot"}, {"start": 44.76, "duration": 3.799, "text": "accommodate any more bonds right a"}, {"start": 46.879, "duration": 3.721, "text": "nucleophile cannot coordinate to the"}, {"start": 48.559, "duration": 4.0, "text": "nitrogen it's already coordinatively"}, {"start": 50.6, "duration": 4.279, "text": "saturated it cannot accommodate any more"}, {"start": 52.559, "duration": 5.64, "text": "bonds it's got a full octet so that's"}, {"start": 54.879, "duration": 5.64, "text": "not going to work water of course is not"}, {"start": 58.199, "duration": 2.32, "text": "really"}], "title": "Which is the Best Electrophile?", "description": "This is an abbreviated version of this question for YouTube Shorts. Check out the full version in the linked video.\n\nGet $5 off your first month of Chemmunity with promo code: DAVE", "lengthSeconds": 59, "width": 360, "height": 640}
{"video_id": "5gEWOh630b8", "transcript": [{"start": 0.44, "duration": 4.72, "text": "this video explains how the three types"}, {"start": 2.32, "duration": 4.999, "text": "of chemical bonding work first ionic"}, {"start": 5.16, "duration": 5.599, "text": "bonding between metal and non-metal"}, {"start": 7.319, "duration": 5.561, "text": "atoms for example sodium and chlorine"}, {"start": 10.759, "duration": 3.84, "text": "all metal atoms generally have between"}, {"start": 12.88, "duration": 4.36, "text": "one and three electrons in their outer"}, {"start": 14.599, "duration": 5.561, "text": "shell for example sodium has one in its"}, {"start": 17.24, "duration": 4.879, "text": "outer shell shown as a cross non-metal"}, {"start": 20.16, "duration": 4.08, "text": "atoms generally have between four and"}, {"start": 22.119, "duration": 4.56, "text": "eight electrons in their outer shell"}, {"start": 24.24, "duration": 5.4, "text": "chlorine for example has seven shown as"}, {"start": 26.679, "duration": 5.72, "text": "dots but all atoms want a full outer"}, {"start": 29.64, "duration": 5.28, "text": "shell so the metal and non-metal atoms"}, {"start": 32.399, "duration": 4.521, "text": "do a deal the metal sodium atom gives"}, {"start": 34.92, "duration": 5.119, "text": "its electron away and the non-metal"}, {"start": 36.92, "duration": 5.44, "text": "chlorine atom receives it now they both"}, {"start": 40.039, "duration": 5.281, "text": "have full outer shells but where does"}, {"start": 42.36, "duration": 5.4, "text": "the ionic bond come from both atoms were"}, {"start": 45.32, "duration": 5.12, "text": "neutral but after trading the electron"}, {"start": 47.76, "duration": 5.08, "text": "the sodium is now a positive ion and the"}, {"start": 50.44, "duration": 4.88, "text": "chlorine is now a negative ion the"}, {"start": 52.84, "duration": 5.359, "text": "oppositely charged ions attract causing"}, {"start": 55.32, "duration": 5.12, "text": "the ionic bond many ions attract to form"}, {"start": 58.199, "duration": 4.401, "text": "a giant lattice structure ionic"}, {"start": 60.44, "duration": 4.039, "text": "compounds have interesting properties"}, {"start": 62.6, "duration": 3.8, "text": "because ionic bonds are strong and act"}, {"start": 64.479, "duration": 4.161, "text": "in all directions in the lattice it"}, {"start": 66.4, "duration": 4.359, "text": "takes a lot of energy to break the ionic"}, {"start": 68.64, "duration": 3.119, "text": "bonds giving ionic compounds high"}, {"start": 70.759, "duration": 3.761, "text": "melting"}, {"start": 71.759, "duration": 5.161, "text": "points second Cove valent bonding"}, {"start": 74.52, "duration": 5.559, "text": "between non-metal atoms let's look at"}, {"start": 76.92, "duration": 5.559, "text": "water again all atoms want a full outer"}, {"start": 80.079, "duration": 5.241, "text": "shell oxygen has six outer shell"}, {"start": 82.479, "duration": 4.761, "text": "electrons but needs eight hydrogen has"}, {"start": 85.32, "duration": 4.799, "text": "one electron in its outer shell but"}, {"start": 87.24, "duration": 5.72, "text": "needs two but rather than swap electrons"}, {"start": 90.119, "duration": 5.241, "text": "like ionic bonding coent compounds share"}, {"start": 92.96, "duration": 4.119, "text": "electrons the oxygen atom shares one"}, {"start": 95.36, "duration": 4.799, "text": "pair of electrons with each hydrogen"}, {"start": 97.079, "duration": 5.561, "text": "atom each hydrogen now has two electrons"}, {"start": 100.159, "duration": 5.32, "text": "and the oxygen has eight all of them are"}, {"start": 102.64, "duration": 4.56, "text": "happy so where does the bond come from"}, {"start": 105.479, "duration": 4.441, "text": "the shared pairs of electrons are"}, {"start": 107.2, "duration": 5.44, "text": "negative and the nuclei of all atoms are"}, {"start": 109.92, "duration": 4.879, "text": "positive the attraction between them is"}, {"start": 112.64, "duration": 5.159, "text": "the Cove valent Bond and like ionic"}, {"start": 114.799, "duration": 5.521, "text": "bonds they're very strong water is one"}, {"start": 117.799, "duration": 5.32, "text": "of three kinds of calent compound called"}, {"start": 120.32, "duration": 4.92, "text": "a simple calent compound although calent"}, {"start": 123.119, "duration": 4.36, "text": "bonds are strong the forces between the"}, {"start": 125.24, "duration": 3.879, "text": "small molecules are weak so it takes"}, {"start": 127.479, "duration": 4.12, "text": "little energy to break these weak"}, {"start": 129.119, "duration": 4.721, "text": "intermolecular forces meaning that water"}, {"start": 131.599, "duration": 4.28, "text": "and other simple calent compounds have"}, {"start": 133.84, "duration": 4.52, "text": "very low boiling points however giant"}, {"start": 135.879, "duration": 4.801, "text": "calent compounds like diamond have very"}, {"start": 138.36, "duration": 5.08, "text": "high melting points due to the calent"}, {"start": 140.68, "duration": 6.12, "text": "bonds and finally we have metallic"}, {"start": 143.44, "duration": 5.799, "text": "bonding between metal atoms such as iron"}, {"start": 146.8, "duration": 4.88, "text": "ion has two electrons on its outer shell"}, {"start": 149.239, "duration": 4.241, "text": "But Metal at work as a group and share"}, {"start": 151.68, "duration": 4.12, "text": "their electrons between them the"}, {"start": 153.48, "duration": 4.2, "text": "electrons form a sea of delocalized"}, {"start": 155.8, "duration": 3.96, "text": "electrons which are free to roam"}, {"start": 157.68, "duration": 4.32, "text": "throughout the giant lattice where does"}, {"start": 159.76, "duration": 4.96, "text": "the bonding come from well the positive"}, {"start": 162.0, "duration": 5.04, "text": "nuclei are all held together by the ca"}, {"start": 164.72, "duration": 5.04, "text": "of delocalized electrons like a strong"}, {"start": 167.04, "duration": 4.32, "text": "glue the bonding is very strong meaning"}, {"start": 169.76, "duration": 2.6, "text": "that metal generally has very high"}, {"start": 171.36, "duration": 3.519, "text": "melting"}, {"start": 172.36, "duration": 4.76, "text": "points so these are the three types of"}, {"start": 174.879, "duration": 4.28, "text": "bonding for a closer look at calent"}, {"start": 177.12, "duration": 4.08, "text": "compounds with some examples see the"}, {"start": 179.159, "duration": 5.401, "text": "following video and don't forget to like"}, {"start": 181.2, "duration": 3.36, "text": "And subscribe"}], "title": "Chemical Bonding Explained | Ionic, Covalent and Metallic | GCSE Chemistry", "description": "Chemical bonding allows atoms to combine into more complex molecules. Learn how the 3 types of chemical bonding work in this video.", "lengthSeconds": 183, "width": 640, "height": 360}
{"video_id": "YT_r8Fi2spQ", "transcript": [{"start": 0.08, "duration": 4.0, "text": "number two what is the iup pack name for"}, {"start": 2.52, "duration": 4.279, "text": "the following compound we have another"}, {"start": 4.08, "duration": 5.24, "text": "one here we need to find the longest"}, {"start": 6.799, "duration": 4.241, "text": "carbon chain this is not always going to"}, {"start": 9.32, "duration": 3.96, "text": "be just the chain that is given to you"}, {"start": 11.04, "duration": 4.599, "text": "from left to right we need to make sure"}, {"start": 13.28, "duration": 4.48, "text": "we get the longest one possible and so"}, {"start": 15.639, "duration": 4.48, "text": "this one snakes up over here that is"}, {"start": 17.76, "duration": 4.4, "text": "going to be the longest carbon chain now"}, {"start": 20.119, "duration": 4.08, "text": "we have to number left to right or right"}, {"start": 22.16, "duration": 4.6, "text": "to left and we want the first"}, {"start": 24.199, "duration": 5.281, "text": "substituent occurring soonest so we will"}, {"start": 26.76, "duration": 7.6, "text": "in this case number right to left 1 two"}, {"start": 29.48, "duration": 7.239, "text": "3 four five 6 7 eight because we have a"}, {"start": 34.36, "duration": 4.12, "text": "substituent on three if we numbered left"}, {"start": 36.719, "duration": 4.241, "text": "to right we wouldn't have a substituent"}, {"start": 38.48, "duration": 4.559, "text": "until carbon 4 then we see what"}, {"start": 40.96, "duration": 5.4, "text": "substituents we have we have a methyl"}, {"start": 43.039, "duration": 6.321, "text": "here we have a methyl here and we have"}, {"start": 46.36, "duration": 5.6, "text": "an ethyl there if we want to number"}, {"start": 49.36, "duration": 6.359, "text": "these alphabetically we will do the"}, {"start": 51.96, "duration": 5.64, "text": "ethyl first so we need five ethyl then"}, {"start": 55.719, "duration": 4.8, "text": "we would name the methyls at the same"}, {"start": 57.6, "duration": 2.919, "text": "time because they're the"}], "title": "IUPAC Nomenclature of Alkanes", "description": "This is an abbreviated version of this question for YouTube Shorts. Check out the full version in the linked video.", "lengthSeconds": 59, "width": 360, "height": 640}
{"video_id": "5VjgMPcmURg", "transcript": [{"start": 6.7, "duration": 4.71, "text": "One day in physics class, you\u2019re learning\nabout gravity, and your friend tells you that"}, {"start": 11.41, "duration": 6.39, "text": "they exert a stronger gravitational force\non you than the nearest star beyond our sun,"}, {"start": 17.8, "duration": 1.52, "text": "proxima centauri."}, {"start": 19.32, "duration": 6.93, "text": "You have a mass of sixty kilograms, as does\nyour friend, who is sitting two meters away."}, {"start": 26.25, "duration": 7.48, "text": "Proxima centauri has a mass of 2.45 times\nten to the twenty-nine kilograms, and it is"}, {"start": 33.73, "duration": 4.39, "text": "4.01 times ten to the sixteen meters away."}, {"start": 38.12, "duration": 1.9, "text": "So is your friend correct?"}, {"start": 40.02, "duration": 3.5, "text": "Which exerts more gravity on you, your friend,\nor the star?"}, {"start": 43.52, "duration": 5.53, "text": "To answer this, we will need to use Newton\u2019s\nlaw of universal gravitation, so if you missed"}, {"start": 49.05, "duration": 2.67, "text": "that tutorial, check it out right now."}, {"start": 51.72, "duration": 2.9, "text": "If you know what to do, give this calculation\na try."}, {"start": 77.34, "duration": 3.36, "text": "This is easy to solve if we know this equation."}, {"start": 80.719, "duration": 6.521, "text": "All we do is plug in the masses and distances\ninvolved, as well as the gravitational constant."}, {"start": 87.24, "duration": 5.979, "text": "First, for your friend, we plug in sixty and\nsixty as the masses, and two meters as the"}, {"start": 93.219, "duration": 1.76, "text": "distance between you."}, {"start": 94.979, "duration": 6.67, "text": "We square that and evaluate, and we should\nget six times ten to the negative eight Newtons."}, {"start": 101.649, "duration": 5.64, "text": "This is an extremely tiny force, which explains\nwhy people don\u2019t slowly drift towards each"}, {"start": 107.289, "duration": 4.03, "text": "other when sitting in a room."}, {"start": 111.319, "duration": 2.21, "text": "Now let\u2019s do one for the star."}, {"start": 113.529, "duration": 5.461, "text": "We replace the mass of your friend with the\nmass of the star, but we also change this"}, {"start": 118.99, "duration": 6.97, "text": "value to the distance to the star, so we have\na huge number on top, but we have a pretty"}, {"start": 125.96, "duration": 6.14, "text": "big number on the bottom, too, and we have\nto square it, so when we evaluate this, we"}, {"start": 132.1, "duration": 5.54, "text": "get 6.1 times ten to the negative thirteen\nNewtons."}, {"start": 137.64, "duration": 6.56, "text": "As small as the first number was, this one\nis a hundred thousand times smaller, so it"}, {"start": 144.2, "duration": 5.95, "text": "looks like your friend was right, none of\nthe stars in the universe apart from the sun"}, {"start": 150.15, "duration": 4.35, "text": "can have any significant gravitational influence\non the earth."}, {"start": 154.5, "duration": 1.66, "text": "Take that, astrology!"}], "title": "Practice Problem: Comparing Gravitational Forces", "description": "Gravity is mysterious, isn't it? If all matter exerts gravity, why don't we all just drift towards each other all the time? Well, as you know, it depends on the masses of the two objects and the distance between them. Your friend says that she exerts more gravity on you than a nearby star. Is that possible? Stars are so massive! But they're also so far away. Let's do some calculations and see what we can come up with.\n\nTry all of the Classical Physics Practice Problems: http://bit.ly/ProfDaveCPPP\n\nClassical Physics Tutorials: http://bit.ly/ProfDavePhysics1\nModern Physics Tutorials: http://bit.ly/ProfDavePhysics2\nMathematics Tutorials: http://bit.ly/ProfDaveMaths\nGeneral Chemistry Tutorials: http://bit.ly/ProfDaveGenChem\nOrganic Chemistry Tutorials: http://bit.ly/ProfDaveOrgChem\nBiochemistry Tutorials: http://bit.ly/ProfDaveBiochem\nBiology Tutorials: http://bit.ly/ProfDaveBio\n\nEMAIL\u25ba ProfessorDaveExplains@gmail.com\nPATREON\u25ba http://patreon.com/ProfessorDaveExplains\n\nCheck out \"Is This Wi-Fi Organic?\", my book on disarming pseudoscience!\nAmazon: https://amzn.to/2HtNpVH\nBookshop: https://bit.ly/39cKADM\n\rBarnes and Noble: https://bit.ly/3pUjmrn \nBook Depository: http://bit.ly/3aOVDlT", "lengthSeconds": 187, "width": 640, "height": 360}
{"video_id": "hSDv2G_I3tc", "transcript": [{"start": 0.08, "duration": 4.92, "text": "number three asks which of the following"}, {"start": 2.24, "duration": 5.199, "text": "functional groups does not contain a"}, {"start": 5.0, "duration": 6.44, "text": "carbonal so again what is a carbonal a"}, {"start": 7.439, "duration": 5.921, "text": "carbonal is cble Bond o and then there's"}, {"start": 11.44, "duration": 3.64, "text": "stuff on either side and different"}, {"start": 13.36, "duration": 4.32, "text": "functional groups will contain a"}, {"start": 15.08, "duration": 4.76, "text": "carbonal depending on what that stuff on"}, {"start": 17.68, "duration": 3.88, "text": "either side is we will call them"}, {"start": 19.84, "duration": 6.72, "text": "different functional groups so what is"}, {"start": 21.56, "duration": 7.639, "text": "an alahh an alahh is C double bond o"}, {"start": 26.56, "duration": 4.6, "text": "where we have alkal on one side and a"}, {"start": 29.199, "duration": 4.641, "text": "hydrogen on the the other side so the"}, {"start": 31.16, "duration": 5.719, "text": "thing that makes it an alahh is that one"}, {"start": 33.84, "duration": 5.32, "text": "side of the carbonal has hydrogen it is"}, {"start": 36.879, "duration": 4.761, "text": "basically the end of a carbon chain what"}, {"start": 39.16, "duration": 6.84, "text": "about a carboxilic acid a carboxilic"}, {"start": 41.64, "duration": 6.32, "text": "acid is cbond O and then o so instead of"}, {"start": 46.0, "duration": 4.0, "text": "just an H we have o there and the"}, {"start": 47.96, "duration": 4.88, "text": "presence of that other oxygen atom is"}, {"start": 50.0, "duration": 6.399, "text": "what makes this a carboxilic acid then"}, {"start": 52.84, "duration": 7.679, "text": "we have ether so an ether is an oxygen"}, {"start": 56.399, "duration": 4.12, "text": "atom with alkal on either side"}], "title": "Structure and Bonding OCHEM Exam Question 3", "description": "This is an abbreviated version of this question for YouTube Shorts. Check out the full version in the linked video.", "lengthSeconds": 59, "width": 360, "height": 640}
{"video_id": "hLJw6O4p2Lk", "transcript": [{"start": 6.52, "duration": 4.22, "text": "okay let's try a problem\nregarding enthalpy of combustion this"}, {"start": 10.74, "duration": 5.13, "text": "one asks the standard enthalpies of\nformation for carbon dioxide liquid"}, {"start": 15.87, "duration": 5.669, "text": "water and solid glucose at 25 degrees\nCelsius are negative 400 negative 300"}, {"start": 21.539, "duration": 4.83, "text": "and negative 1,300 kilojoules per mole\nrespectively what is the standard"}, {"start": 26.369, "duration": 7.141, "text": "enthalpy of combustion per gram of\nglucose at 25 degrees Celsius so if you"}, {"start": 33.51, "duration": 3.9, "text": "need to check out some of my thermo\nchemistry tutorials to help you figure"}, {"start": 37.41, "duration": 2.29, "text": "this out and when you're ready give it a try"}, {"start": 55.34, "duration": 2.32, "text": "so first things first we're talking"}, {"start": 57.66, "duration": 5.34, "text": "about the standard enthalpy of\ncombustion and so this must be a"}, {"start": 63.0, "duration": 4.729, "text": "combustion reaction so let's write out\nthe combustion reaction we're performing"}, {"start": 67.729, "duration": 5.411, "text": "combustion on glucose so let's write\nthis out we have glucose which we know"}, {"start": 73.14, "duration": 5.22, "text": "is c6h12o6 and that's going to be a\nsolid and then combustion means we're"}, {"start": 78.36, "duration": 5.579, "text": "reacting with oxygen and so we're going\nto need o2 gas and then in any"}, {"start": 83.939, "duration": 4.351, "text": "combustion we're always producing carbon\ndioxide and water that is always going"}, {"start": 88.29, "duration": 4.71, "text": "to be the case and we need this to be\nbalanced so we have six six and six for"}, {"start": 93.0, "duration": 5.1, "text": "oxygen carbon dioxide and water so this\nis our combustion reaction and now we"}, {"start": 98.1, "duration": 7.049, "text": "can do thermo chemical analysis on it as\nwe recall the standard enthalpy change"}, {"start": 105.149, "duration": 5.101, "text": "for any reaction is going to be equal to\nthe sum of the standard enthalpies of"}, {"start": 110.25, "duration": 4.68, "text": "formation of the products minus the sum\nof the standard enthalpies of formation"}, {"start": 114.93, "duration": 4.38, "text": "of the reactants so let's go ahead and\nplug in what we know"}, {"start": 119.31, "duration": 5.73, "text": "so first starting with the products\nwe're looking at carbon dioxide and we"}, {"start": 125.04, "duration": 3.63, "text": "know that that is negative 400 and we're\ngoing to have to multiply that by six"}, {"start": 128.67, "duration": 4.57, "text": "because we have a coefficient of six in\nthe balanced equation"}, {"start": 133.24, "duration": 3.96, "text": "then we're adding to that because this\nis a sum we're adding to that negative"}, {"start": 137.2, "duration": 6.57, "text": "300 for water again times six because we\nhave a coefficient of six then from all"}, {"start": 143.77, "duration": 4.53, "text": "of that we're going to subtract the\nenthalpies of formation of the reactants"}, {"start": 148.3, "duration": 6.18, "text": "and so we have negative 1300 for glucose\nand then we're going to have zero for"}, {"start": 154.48, "duration": 4.56, "text": "oxygen because as we know for any\nelement this is going to be zero we"}, {"start": 159.04, "duration": 5.79, "text": "don't have heats of formation for\nelements only for compounds so we're"}, {"start": 164.83, "duration": 4.86, "text": "going to do the arithmetic and we find\nthat the enthalpy of combustion for"}, {"start": 169.69, "duration": 6.96, "text": "glucose is negative 2900 kilojoules per\nmole so this is per mole of glucose but"}, {"start": 176.65, "duration": 4.65, "text": "the question asked per gram of glucose\nso we have one more calculation to do"}, {"start": 181.3, "duration": 5.04, "text": "it's very simple we just have to take\nnegative 2900 kilojoules per mole and"}, {"start": 186.34, "duration": 5.73, "text": "then divide that by the molar mass of\nglucose which is 180 grams per mole that"}, {"start": 192.07, "duration": 5.94, "text": "way the moles cancel and we're going to\nend up with kilojoules per gram and we"}, {"start": 198.01, "duration": 7.57, "text": "do the arithmetic and we get negative\nsixteen point one one kilojoules per gram of glucose."}], "title": "IIT/JEE Chemistry Practice #26: Enthalpy of Combustion", "description": "Practice REAL problems from actual past IIT/JEE exams with Professor Dave!\n\nTry all of the IIT/JEE chemistry practice problems: http://bit.ly/ProfDaveIITJEEChem\n\nGeneral Chemistry Tutorials: http://bit.ly/ProfDaveGenChem\nOrganic Chemistry Tutorials: http://bit.ly/ProfDaveOrgChem\n\nEMAIL\u25ba ProfessorDaveExplains@gmail.com\nPATREON\u25ba http://patreon.com/ProfessorDaveExplains\n\nCheck out \"Is This Wi-Fi Organic?\", my book on disarming pseudoscience!\nAmazon: https://amzn.to/2HtNpVH\nBookshop: https://bit.ly/39cKADM\n\rBarnes and Noble: https://bit.ly/3pUjmrn \nBook Depository: http://bit.ly/3aOVDlT", "lengthSeconds": 237, "width": 640, "height": 360}
{"video_id": "2hU_SXmPZhg", "transcript": [{"start": 0.08, "duration": 3.639, "text": "number two the following pair of"}, {"start": 1.76, "duration": 5.24, "text": "structures are and again we have these"}, {"start": 3.719, "duration": 5.481, "text": "options identical enantiomers derse and"}, {"start": 7.0, "duration": 4.08, "text": "constitutional isomers we want to"}, {"start": 9.2, "duration": 3.92, "text": "compare these structures in the same way"}, {"start": 11.08, "duration": 4.0, "text": "we have something in line notation and"}, {"start": 13.12, "duration": 3.76, "text": "then we have a Newman projection you can"}, {"start": 15.08, "duration": 3.92, "text": "go either way we could convert the line"}, {"start": 16.88, "duration": 3.92, "text": "notation to a Newan projection but I"}, {"start": 19.0, "duration": 3.4, "text": "think we're used to looking at molecules"}, {"start": 20.8, "duration": 4.399, "text": "in line notation so why don't we go"}, {"start": 22.4, "duration": 5.24, "text": "ahead and convert the Newan projection"}, {"start": 25.199, "duration": 5.681, "text": "to line notation what we want to do here"}, {"start": 27.64, "duration": 6.439, "text": "is clearly we are looking at a be"}, {"start": 30.88, "duration": 5.28, "text": "we have this methyl down and this methyl"}, {"start": 34.079, "duration": 4.441, "text": "up in the back that's these two right"}, {"start": 36.16, "duration": 4.52, "text": "here now what else do we have in order"}, {"start": 38.52, "duration": 4.199, "text": "to look at this Newman projection what"}, {"start": 40.68, "duration": 4.68, "text": "we're really doing is we're looking down"}, {"start": 42.719, "duration": 5.241, "text": "this Bond right here if we were to do"}, {"start": 45.36, "duration": 5.519, "text": "that what would we see well if we see"}, {"start": 47.96, "duration": 5.2, "text": "the bromine going up and to the left"}, {"start": 50.879, "duration": 5.081, "text": "that means that that must be on the dash"}, {"start": 53.16, "duration": 7.359, "text": "Bond here because the dash bond is going"}, {"start": 55.96, "duration": 4.559, "text": "to the left it's going past the screen"}], "title": "Identifying the Type of Isomer", "description": "This is an abbreviated version of this question for YouTube Shorts. Check out the full version in the linked video.", "lengthSeconds": 59, "width": 360, "height": 640}
{"video_id": "oa7JP77GLxw", "transcript": [{"start": 0.08, "duration": 5.44, "text": "s asks what is the relationship between"}, {"start": 2.6, "duration": 6.16, "text": "the compound in line notation and the"}, {"start": 5.52, "duration": 5.32, "text": "adjacent Newman projection very clearly"}, {"start": 8.76, "duration": 4.68, "text": "looking at this Newman projection we've"}, {"start": 10.84, "duration": 5.56, "text": "got an O on the front and a CL on the"}, {"start": 13.44, "duration": 6.16, "text": "back so we're definitely looking at this"}, {"start": 16.4, "duration": 5.879, "text": "like this we are looking down this"}, {"start": 19.6, "duration": 5.4, "text": "carboncarbon Bond right here so that is"}, {"start": 22.279, "duration": 4.601, "text": "the carboncarbon bond that we cannot see"}, {"start": 25.0, "duration": 3.96, "text": "it is obscured by the carbon atom that"}, {"start": 26.88, "duration": 4.719, "text": "we are looking at and so what do we"}, {"start": 28.96, "duration": 4.48, "text": "notice if we are looking down that Bond"}, {"start": 31.599, "duration": 4.441, "text": "we're going to have a methyl group"}, {"start": 33.44, "duration": 5.88, "text": "projecting directly upwards so that's"}, {"start": 36.04, "duration": 5.359, "text": "right there we will have o down and to"}, {"start": 39.32, "duration": 4.919, "text": "our right because if we are in the plane"}, {"start": 41.399, "duration": 5.0, "text": "of the screen and that o is projecting"}, {"start": 44.239, "duration": 4.64, "text": "forwards from the screen it is going to"}, {"start": 46.399, "duration": 4.361, "text": "be down and to our right and then the"}, {"start": 48.879, "duration": 4.16, "text": "implied hydrogen will be down and to the"}, {"start": 50.76, "duration": 5.16, "text": "left so that's fine then looking at the"}, {"start": 53.039, "duration": 5.0, "text": "back carbon we have a chlorine atom that"}, {"start": 55.92, "duration": 4.599, "text": "is going to be up and to our right"}, {"start": 58.039, "duration": 2.48, "text": "because again"}], "title": "Newman Projections and Isomerism", "description": "This is an abbreviated version of this question for YouTube Shorts. Check out the full version in the linked video.", "lengthSeconds": 59, "width": 360, "height": 640}
{"video_id": "5KcQ751tXTU", "transcript": [{"start": 0.84, "duration": 10.75, "text": "n"}, {"start": 2.62, "duration": 8.97, "text": "[Music]"}], "title": "anomer organic compound|anomers biochemistry|anomers maltose|anomer formation #science #chemistry", "description": "anomers biochemistry\nepimers and anomers biochemistry\nanomers mutarotation biochemistry\nepimers and anomers in carbohydrates biochemistry\nanomers of carbohydrates biochemistry\nbiochemistry 10\nbiochem professor dave\nd and l carbohydrate\nd and l monosaccharides\ncarbohydrates and nucleic acids\nprofessor dave explains macromolecules\nd-glyceraldehyde\nformation of anomer\nfe biochemistry\nalpha beta carbohydrates\nbio chem lecture\nglucose anomers\nanomers and epimers of carbohydrates\nhb biochemistry\nwhat is anomers in biochemistry\nju biochemistru\nj chemistry carbohydrates\nkcat biochemistry\nkeq biochemistry\nanomeric carbon in carbohydrates\ncarbohydrates biochemistry andrey k\nalpha l glucose structure\nl and d enantiomers amino acids\nna biochemistry\nnaming sugars biochemistry\nnmr biochemistry\nnaming sugars organic chemistry\nprof dave biochemistry\nprobe biochemistry\nprofessor dave biochemistry\nprofessor dave biochem\nalpha and beta anomers of carbohydrates\nwhat are anomers and epimers\nwhat is an anomeric carbon\nanomers of carbohydrates\nwhat are anomers\nanalysis of biomolecules\nanomeric carbon\nbiochemistry molecules\nbiochemistry lecture on carbohydrates\nbiochemistry chapter 1 carbohydrates\nunit first biochemistry\nunit 5 biochemistry\nanomers vs epimers\nprofessor dave explains biochem\nalpha helix biochemistry\nanomeric carbon mcat\nprofessor dave explains amino acids\nprofessor dave monosaccharides\n2 3 bpg biochemistry\nas level biology carbohydrates\nalpha and beta anomers of glucose\nbioorganic chemistry carbohydrates\ncarbohydrates biochemistry a level\n6 classes of enzymes\nanomers in carbohydrates\nbiochemistry 8\na level biology biomolecules\nbiochemistry unit 1 biomolecules\nbiochemistry 9th grade\nbiochemistry 9\n@science+ #science+", "lengthSeconds": 10, "width": 360, "height": 640}
{"video_id": "wzQstigxbuo", "transcript": [{"start": 0.18, "duration": 3.08, "text": "It\u2019s Professor Dave, let\u2019s learn the distance\nformula."}, {"start": 10.22, "duration": 5.22, "text": "As we navigate the coordinate plane in our\nalgebraic adventures, one thing we might want"}, {"start": 15.44, "duration": 3.17, "text": "to do is find the distance between two points."}, {"start": 18.61, "duration": 5.92, "text": "Luckily, we are well equipped to do this,\nsince we know the Pythagorean Theorem."}, {"start": 24.53, "duration": 5.31, "text": "If you need to review it, check out the tutorials\non how to use the theorem, as well as how"}, {"start": 29.84, "duration": 4.719, "text": "the theorem was derived, in the geometry portion\nof this playlist."}, {"start": 34.56, "duration": 7.32, "text": "Otherwise, let\u2019s just select two points\nin the coordinate plane, and draw a line between them."}, {"start": 42.0, "duration": 4.409, "text": "The length of this line is what we are trying\nto figure out, and in order to get it, we"}, {"start": 46.409, "duration": 2.29, "text": "can make a right triangle."}, {"start": 48.699, "duration": 5.741, "text": "This line will be the hypotenuse, and the\nlegs will be the horizontal and vertical line"}, {"start": 54.44, "duration": 3.189, "text": "segments we need to construct the triangle."}, {"start": 57.629, "duration": 4.941, "text": "Now let\u2019s label the hypotenuse as D, the\ndistance we want to know."}, {"start": 62.57, "duration": 6.15, "text": "The legs can be expressed as the difference\nbetween the respective points that mark their ends."}, {"start": 68.8, "duration": 5.52, "text": "This horizontal line segment has a length\nof X two minus X one, since it only moves"}, {"start": 74.33, "duration": 1.62, "text": "in the X direction."}, {"start": 75.95, "duration": 5.06, "text": "The vertical line segment has a length of\nY two minus Y one, since it only moves in"}, {"start": 81.01, "duration": 1.23, "text": "the Y direction."}, {"start": 82.24, "duration": 3.36, "text": "Now, we just use the Pythagorean Theorem."}, {"start": 85.6, "duration": 7.34, "text": "The square of this leg plus the square of\nthe other leg equals the hypotenuse squared,"}, {"start": 92.94, "duration": 1.3, "text": "or D squared."}, {"start": 94.24, "duration": 5.06, "text": "Then, all we do is take the square root of\nboth sides to solve for D."}, {"start": 99.3, "duration": 5.55, "text": "We will only take the positive value here,\nas negative length doesn\u2019t make any sense."}, {"start": 104.85, "duration": 3.23, "text": "And there you have it, we have derived the\ndistance formula."}, {"start": 108.08, "duration": 1.85, "text": "Let\u2019s give it a shot."}, {"start": 109.93, "duration": 4.09, "text": "Take the points negative two, two, and two,\nfive."}, {"start": 114.02, "duration": 4.12, "text": "What is the distance covered by the line segment\nthat connects these two points?"}, {"start": 118.14, "duration": 7.34, "text": "Well, let\u2019s take our X values and our Y\nvalues, and plug them into the formula."}, {"start": 125.48, "duration": 6.45, "text": "That gives us four, which squared is sixteen,\nand three, which squared is nine."}, {"start": 131.93, "duration": 6.13, "text": "Add them up and we get twenty-five, take the\nsquare root, and that gives us five for the"}, {"start": 138.06, "duration": 2.11, "text": "distance."}, {"start": 140.17, "duration": 5.46, "text": "This is the same value we would get if we\ncontructed a triangle and performed the Pythagorean"}, {"start": 145.63, "duration": 1.0, "text": "Theorem."}, {"start": 146.63, "duration": 6.189, "text": "So the distance formula is a generalized way\nto simply plug in the coordinates of two points,"}, {"start": 152.82, "duration": 2.08, "text": "and find the distance between them."}, {"start": 154.98, "duration": 1.18, "text": "Let's check comprehension."}], "title": "The Distance Formula: Finding the Distance Between Two Points", "description": "One thing we may want to do in algebra is compute the distance between two points on the coordinate plane. Luckily, there is a handy dandy equation for doing just that! Let's derive it first, and then practice using it.\n\nWatch the whole Mathematics playlist: http://bit.ly/ProfDaveMath\n\nClassical Physics Tutorials: http://bit.ly/ProfDavePhysics1\nModern Physics Tutorials: http://bit.ly/ProfDavePhysics2\nGeneral Chemistry Tutorials: http://bit.ly/ProfDaveGenChem\nOrganic Chemistry Tutorials: http://bit.ly/ProfDaveOrgChem\nBiochemistry Tutorials: http://bit.ly/ProfDaveBiochem\nBiology Tutorials: http://bit.ly/ProfDaveBio\n\nEMAIL\u25ba ProfessorDaveExplains@gmail.com\nPATREON\u25ba http://patreon.com/ProfessorDaveExplains\n\nCheck out \"Is This Wi-Fi Organic?\", my book on disarming pseudoscience!\nAmazon: https://amzn.to/2HtNpVH\nBookshop: https://bit.ly/39cKADM\n\rBarnes and Noble: https://bit.ly/3pUjmrn \nBook Depository: http://bit.ly/3aOVDlT", "lengthSeconds": 217, "width": 640, "height": 360}
{"video_id": "JG8EnqCt6mI", "transcript": [{"start": 0.0, "duration": 9.38, "text": "[Music]"}, {"start": 9.96, "duration": 4.2, "text": "Professor Dave and CH here an extremely"}, {"start": 12.24, "duration": 4.48, "text": "common type of reaction in organic"}, {"start": 14.16, "duration": 4.879, "text": "chemistry involves some nucleophile"}, {"start": 16.72, "duration": 5.24, "text": "attacking a carbonal carbon often in the"}, {"start": 19.039, "duration": 4.961, "text": "form of an alahh or Ketone what are"}, {"start": 21.96, "duration": 4.159, "text": "these nucleophiles and what kinds of"}, {"start": 24.0, "duration": 4.96, "text": "compounds can be produced let's check"}, {"start": 26.119, "duration": 5.761, "text": "out a few examples when a nucleophile"}, {"start": 28.96, "duration": 6.2, "text": "attacks a carbon carbon it will do so at"}, {"start": 31.88, "duration": 6.519, "text": "a specific angle called the Bergie dunit"}, {"start": 35.16, "duration": 5.919, "text": "angle which is around 75\u00b0 from the"}, {"start": 38.399, "duration": 5.48, "text": "horizontal this is so that it can access"}, {"start": 41.079, "duration": 5.441, "text": "the lowest unoccupied molecular orbital"}, {"start": 43.879, "duration": 6.0, "text": "and bond to the carbon thus Shifting the"}, {"start": 46.52, "duration": 6.12, "text": "carbonal carbon from SP2 to sp3"}, {"start": 49.879, "duration": 5.36, "text": "hybridized this pushes the pi Bond up"}, {"start": 52.64, "duration": 4.759, "text": "onto the oxygen to generate the"}, {"start": 55.239, "duration": 4.921, "text": "oxyanion generally this is then"}, {"start": 57.399, "duration": 4.96, "text": "protonated to give the alcohol what"}, {"start": 60.16, "duration": 4.8, "text": "kinds of nucleophiles are capable of"}, {"start": 62.359, "duration": 5.481, "text": "this transformation many of them are"}, {"start": 64.96, "duration": 7.799, "text": "negatively charged such as hydroxide"}, {"start": 67.84, "duration": 7.16, "text": "hydde carbanion alkoxide or cyanide but"}, {"start": 72.759, "duration": 5.68, "text": "sometimes even neutral nucleophiles can"}, {"start": 75.0, "duration": 6.32, "text": "attack the carbonal these include water"}, {"start": 78.439, "duration": 5.081, "text": "alcohol ammonia or other"}, {"start": 81.32, "duration": 4.439, "text": "amines sometimes depending on the"}, {"start": 83.52, "duration": 5.16, "text": "identity of the nucleophile we don't"}, {"start": 85.759, "duration": 4.921, "text": "stop at the alcohol instead the alcohol"}, {"start": 88.68, "duration": 4.6, "text": "can be further proteinated and a second"}, {"start": 90.68, "duration": 4.799, "text": "Bond can form from the nucleophile to"}, {"start": 93.28, "duration": 4.4, "text": "what was originally the carbonal carbon"}, {"start": 95.479, "duration": 4.64, "text": "resulting in a pi bond this will"}, {"start": 97.68, "duration": 4.759, "text": "frequently happen with amines which will"}, {"start": 100.119, "duration": 4.761, "text": "attack the carbonal but then proton"}, {"start": 102.439, "duration": 5.0, "text": "transfer will occur such that the same"}, {"start": 104.88, "duration": 4.8, "text": "nitrogen can attack again kicking water"}, {"start": 107.439, "duration": 5.801, "text": "off and producing what is called an"}, {"start": 109.68, "duration": 6.24, "text": "imine in terms of reactivity alahh are"}, {"start": 113.24, "duration": 5.28, "text": "generally more reactive than ketones in"}, {"start": 115.92, "duration": 4.92, "text": "the context of nucleophilic addition"}, {"start": 118.52, "duration": 5.16, "text": "both for steric and and electronic"}, {"start": 120.84, "duration": 5.48, "text": "reasons sterically there is one less"}, {"start": 123.68, "duration": 4.68, "text": "alkal group projecting from the carbonal"}, {"start": 126.32, "duration": 5.279, "text": "so a nucleophile can more easily"}, {"start": 128.36, "duration": 5.879, "text": "approach but also an alahh carbonal is"}, {"start": 131.599, "duration": 4.64, "text": "slightly more polarized than a ketone"}, {"start": 134.239, "duration": 4.561, "text": "because alkal groups are slightly"}, {"start": 136.239, "duration": 4.561, "text": "electron donating through induction so"}, {"start": 138.8, "duration": 4.799, "text": "the carbonal carbon is not quite as"}, {"start": 140.8, "duration": 6.04, "text": "partially positive in a ketone as it is"}, {"start": 143.599, "duration": 5.92, "text": "for an alahh to extend this to aromatic"}, {"start": 146.84, "duration": 6.32, "text": "systems something like benzal deide will"}, {"start": 149.519, "duration": 5.121, "text": "be less reactive than an aliphatic alahh"}, {"start": 153.16, "duration": 3.76, "text": "because of the resonance from the"}, {"start": 154.64, "duration": 4.679, "text": "Benzene ring which makes the carbonal"}, {"start": 156.92, "duration": 4.48, "text": "carbon less"}, {"start": 159.319, "duration": 4.441, "text": "electrophilic there is much more we can"}, {"start": 161.4, "duration": 4.88, "text": "do with alahh and ketones in the way of"}, {"start": 163.76, "duration": 5.199, "text": "nucleophilic substitution but for now"}, {"start": 166.28, "duration": 7.52, "text": "that serves as a reasonable introduction"}, {"start": 168.959, "duration": 4.841, "text": "Professor Dave for che see you next time"}, {"start": 174.98, "duration": 3.11, "text": "[Music]"}, {"start": 180.64, "duration": 2.599, "text": "go"}], "title": "Nucleophilic Addition Reactions of Aldehydes and Ketones Ft. Professor Dave", "description": "An extremely common type of reaction in organic chemistry involves some nucleophile attacking a carbonyl carbon, often an aldehyde or ketones. Let's learn what these nucleophiles are, and what compounds are produced from these attacks. We'll also cover the concept of the Burgi-Dunitz angle\u2014the angle at which nucleophilic attacks on carbonyls occur\u2014as well as the reactivity of carbonyls.\n\nGet more homework help from Chegg at https://che.gg/3HbtG8Y\nWatch the whole Organic Chemistry playlist: https://www.youtube.com/playlist?list=PLllCkartRRdMPnATf1X8qVdo5RzM8gA_M\n\nAbout Chegg:\nChegg is a leading education technology company that provides students with affordable and accessible study resources. Chegg's mission is to help students save time, save money and get smarter.  \n\nSubscribe: https://www.youtube.com/@chegg\nTikTok: https://www.tiktok.com/@chegg\nInstagram: https://www.instagram.com/chegg/\nTwitter: https://twitter.com/chegg\n\n00:00 Intro\n00:27 Burgi-Dunitz angle\n01:04 Survey of nucleophiles\n01:51 Comparing reactivity", "lengthSeconds": 181, "width": 640, "height": 360}
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problems\n\n#ytshorts\n#shortsindia\n#short\n#shorts\n#ethers\n#iupac\n#how\n#what\n#trivial\n#ethoxy\n#methoxy\n#NEET2022\n#NEET2023\n#JEE2022\n\nethers organic chemistry,\netherscan api,\netherscan withdraw,\netherscan to metamask,\nethereal,\nethers and epoxides,\nethers and epoxides thiols and sulfides,\nethers and esters,\nethers and epoxides bsc 2nd year,\nethers aldehydes and ketones,\nethers ambient,\nethers and thiols,\nethers and epoxides organic chemistry bsc 2nd year,\nethers and alcohols,\nethers and epoxides notes,\nalcohol phenol and ethers class 12,\nalcohol phenol and ethers class 12 one shot,\nalcohol phenol and ethers class 12 physics wallah,\nalcohol phenol and ethers class 12 vt sir,\nalcohol phenol and ethers class 12 mht cet,\nalcohol phenol and ethers class 12 unacademy,\nalcohol phenol and ethers class 12 apni kaksha,\n\nbest for neet alcohols phenols and ethers,\nbasics of alcohol phenol and ethers class 12,\nbiomentors alcohol phenol and ethers,\nboiling point of ethers,\nbreaking the ethers,\nwhere to buy ethers in pokemon sword,\nalcohol phenol and ethers class 12 by pankaj sir,\nalcohol phenol and ethers class 12 by vt sir,\nalcohol phenol and ethers class 12 by physics wallah,\nalcohol phenol and ethers class 12 by apni kaksha,\nethers class 12,\nethers class 12 one shot,\nethers chemical reactions,\nethers chemistry,\nethers class 12 one shot vt sir,\nethers class 12 neet,\nethers class 12 unacademy,\nethers class 12 tamil,\nethers class 12 physics wallah,\nethers class 12 chemistry,\ncrown ethers,\nclass 12 alcohols phenols and ethers one shot,\nchemistry alcohols phenols and ethers notes for class 12,\nch alcohol phenol and ethers class 12,\ncrown ethers and cryptands,\nconversion of alcohol phenol and ethers class 12,\ncleavage of c-o bond in ethers class 12,\ncleavage of ethers,\nclass 12 chemistry alcohols phenols and ethers,\ncleavage of ethers with hydrogen iodide,\nethereum difficulty bomb,\nethereum documentary,\nethereum dev call,\nethereum development,\nethereum drop,\nethereum deflationary,\nethereum down,\nethereum delay,\nethereum dip,\nethereum dropping today,\ndehydration of alcohols to ethers,\ndrawing ethers,\ndo alcohols phenols and ethers,\nd chemist hydroxy compounds and ethers,\nalkoxymercuration-demercuration of ether,\nnaming and drawing ethers,\npolybrominated diphenyl ethers,\nprofessor dave explains ethers,\nalcohol phenol and ethers class 12\nalcohol phenol and ethers class 12 \n\n\nTrick to solve numericals based upon molality and density || #molality\nhttps://youtube.com/shorts/pFGyqeZkCaQ?feature=share\n\n\n\n\nethers epoxides and sulfides,\nethers epoxides and thioethers,\nethers events,\nethers epoxides,\nethers edge,\nethers ending,\nethers eip712,\nethers ecrecover,\nethers ethereum,\nethers estimate gas gas limit,\nesters and ethers,\nexplain chapter alcohol phenol and ethers class 12,\nexplain alcohol phenol and ethers class 12,\nelectrophilic substitution in aromatic ethers,\nepoxides and ethers,\neduport alcohol phenol and ethers class 12,\nnaming esters and ethers,\nnaming epoxides and ethers,\nalcohol phenol and ethers class 12 exercise solutions,\nalcohol phenol and ethers class 12 new indian era,\nethers functional group,\nethers formatether,\nesther's follies,\nethers fromwei,\nethers fixed number,\nethers from mnemonic,\nethers flutter,\nethers fractional component exceeds decimals,\nethers from private key,\nethers float to bignumber,\nfour ethers,\nformation of ethers,\nfragmentation of ether,\nfragmentation pattern of ethers,\nformation of glycosides ethers and esters in hindi,\nfathers vision alcohols phenols and ethers,\nformation of ethers and epoxides,\nserpentwithfeet four ethers live,\niupac naming for ethers,\nnotes for alcohols phenols and ethers,\nethers get block timestamp,\nethers get signer,\nethers get nonce,\nethers get latest block,\nethers get wallet balance,\nethers get signer from private key,\nethers get private key,\nethers gethistory,\nethers generate private key,\nethers get nfts,\ngethers songs,\ngethers,\ngethers funeral home,\nglycerol ethers\nformation of glycosides ethers and esters,\nhow to get ethers in pokemon bdsp,\nalcohol phenol and ethers class 12 gravity circle,\nalcohol phenol and ethers class 12\nsilyl ethers as protecting groups,\nalcohol phenol and ethers class 12 in gujarati,\nethers \nethers \nethers have angular v-shaped geometry like,\nethers hash,\nethers htrick to find", "lengthSeconds": 55, "width": 360, "height": 610}
{"video_id": "rYjo774UpHI", "transcript": [{"start": 0.12, "duration": 4.56, "text": "Coulomb's law"}, {"start": 2.52, "duration": 5.279, "text": "if two objects have electrical charges"}, {"start": 4.68, "duration": 5.52, "text": "q1 and Q2 and if they're separated by a"}, {"start": 7.799, "duration": 4.081, "text": "distance R then Coulomb's law says that"}, {"start": 10.2, "duration": 5.16, "text": "the electrical or coulombic force"}, {"start": 11.88, "duration": 4.56, "text": "between these objects is kq1 Q2 over r"}, {"start": 15.36, "duration": 4.2, "text": "squared"}, {"start": 16.44, "duration": 5.58, "text": "K which you write here as K sub e e for"}, {"start": 19.56, "duration": 4.979, "text": "electrical is a constant approximately"}, {"start": 22.02, "duration": 5.339, "text": "equal to 9 times 10 to the 9 newton"}, {"start": 24.539, "duration": 5.641, "text": "meters squared per coulomb squared"}, {"start": 27.359, "duration": 5.521, "text": "now matter could be electrically charged"}, {"start": 30.18, "duration": 5.219, "text": "charges can be positive negative or"}, {"start": 32.88, "duration": 4.32, "text": "neutral if an object is neutral it has"}, {"start": 35.399, "duration": 3.601, "text": "zero electrical charge"}, {"start": 37.2, "duration": 3.48, "text": "here's the representation we'll use for"}, {"start": 39.0, "duration": 6.059, "text": "a positive charge"}, {"start": 40.68, "duration": 6.539, "text": "a negative charge and a neutral charge"}, {"start": 45.059, "duration": 4.741, "text": "Coulomb's law says that like charges"}, {"start": 47.219, "duration": 4.081, "text": "that is charges with the same sign repel"}, {"start": 49.8, "duration": 4.079, "text": "each other while opposite charges"}, {"start": 51.3, "duration": 4.62, "text": "attract if we put two positive charges"}, {"start": 53.879, "duration": 4.621, "text": "close together they'll repel each other"}, {"start": 55.92, "duration": 6.799, "text": "since they have the same sign"}, {"start": 58.5, "duration": 4.219, "text": "the same goes for two negative charges"}, {"start": 63.059, "duration": 5.401, "text": "if a positive and a negative charge are"}, {"start": 64.92, "duration": 5.4, "text": "near each other they'll attract"}, {"start": 68.46, "duration": 5.28, "text": "and neutral charges won't interact with"}, {"start": 70.32, "duration": 6.36, "text": "electrically charged objects"}, {"start": 73.74, "duration": 4.739, "text": "now the closer two charges are the"}, {"start": 76.68, "duration": 3.96, "text": "stronger the force between them will be"}, {"start": 78.479, "duration": 4.381, "text": "so let's return to our positive and"}, {"start": 80.64, "duration": 4.019, "text": "negative charges when they're far apart"}, {"start": 82.86, "duration": 3.899, "text": "like we show them here the force between"}, {"start": 84.659, "duration": 4.201, "text": "them is relatively weak as indicated by"}, {"start": 86.759, "duration": 4.381, "text": "the short arrows as we move the charges"}, {"start": 88.86, "duration": 3.899, "text": "closer together the forces get stronger"}, {"start": 91.14, "duration": 3.839, "text": "which we show by the arrows getting"}, {"start": 92.759, "duration": 4.86, "text": "longer"}, {"start": 94.979, "duration": 4.381, "text": "Coulomb's law is also quantitative it"}, {"start": 97.619, "duration": 3.481, "text": "relates the force between two objects to"}, {"start": 99.36, "duration": 3.719, "text": "their electrical charges here's"}, {"start": 101.1, "duration": 5.159, "text": "Coulomb's law again the force is equal"}, {"start": 103.079, "duration": 5.701, "text": "to K times q1 times Q2 over r squared"}, {"start": 106.259, "duration": 4.441, "text": "where q1 and Q2 are the charges and R is"}, {"start": 108.78, "duration": 3.6, "text": "the distance between them one of the"}, {"start": 110.7, "duration": 3.54, "text": "keys to Coulomb's law is that the"}, {"start": 112.38, "duration": 3.66, "text": "charges are multiplied together which"}, {"start": 114.24, "duration": 4.08, "text": "means they're signs whether positive"}, {"start": 116.04, "duration": 5.039, "text": "negative or neutral also get multiplied"}, {"start": 118.32, "duration": 5.579, "text": "let's say q1 is positive and let's say"}, {"start": 121.079, "duration": 4.921, "text": "Q2 is positive as well multiplying two"}, {"start": 123.899, "duration": 4.381, "text": "positive charges results in a positive"}, {"start": 126.0, "duration": 4.319, "text": "force and when a force is positive here"}, {"start": 128.28, "duration": 4.44, "text": "it means repulsive pushing the charges"}, {"start": 130.319, "duration": 4.741, "text": "away from each other"}, {"start": 132.72, "duration": 4.8, "text": "now let's suppose q1 is negative and"}, {"start": 135.06, "duration": 4.319, "text": "that Q2 is also negative multiplying two"}, {"start": 137.52, "duration": 6.6, "text": "negatives again results in a positive"}, {"start": 139.379, "duration": 6.72, "text": "force so again q1 and Q2 will repel"}, {"start": 144.12, "duration": 3.72, "text": "if one of the charges is positive while"}, {"start": 146.099, "duration": 3.421, "text": "the other is negative then their product"}, {"start": 147.84, "duration": 4.68, "text": "will also be negative which means the"}, {"start": 149.52, "duration": 4.859, "text": "force is attractive"}, {"start": 152.52, "duration": 3.719, "text": "and finally if at least one of the"}, {"start": 154.379, "duration": 3.72, "text": "charges is neutral meaning it has zero"}, {"start": 156.239, "duration": 3.36, "text": "electrical charge this zero carries"}, {"start": 158.099, "duration": 4.881, "text": "through in the multiplication and the"}, {"start": 159.599, "duration": 3.381, "text": "force between them is zero"}, {"start": 163.319, "duration": 3.721, "text": "so multiplying the charges is one key to"}, {"start": 165.48, "duration": 3.539, "text": "Coulomb's law"}, {"start": 167.04, "duration": 3.96, "text": "the other key is the r squared in the"}, {"start": 169.019, "duration": 4.021, "text": "denominator suppose we doubled the"}, {"start": 171.0, "duration": 3.239, "text": "distance between two charges how would"}, {"start": 173.04, "duration": 2.58, "text": "the force change"}, {"start": 174.239, "duration": 3.661, "text": "doubling the distance means we're"}, {"start": 175.62, "duration": 5.16, "text": "changing the r to 2R"}, {"start": 177.9, "duration": 4.38, "text": "squaring 2R gives 4r squared now we can"}, {"start": 180.78, "duration": 3.959, "text": "move the 4 in front of the fraction"}, {"start": 182.28, "duration": 4.319, "text": "keeping it in the denominator"}, {"start": 184.739, "duration": 3.78, "text": "so now we have one quarter of our"}, {"start": 186.599, "duration": 3.121, "text": "original Force back when the distance"}, {"start": 188.519, "duration": 2.94, "text": "was r"}, {"start": 189.72, "duration": 4.08, "text": "doubling the distance results in a"}, {"start": 191.459, "duration": 4.081, "text": "quarter of the force similarly having"}, {"start": 193.8, "duration": 4.26, "text": "the distance would have quadrupled the"}, {"start": 195.54, "duration": 4.08, "text": "force in other words Coulomb's law is"}, {"start": 198.06, "duration": 3.66, "text": "what's known as an inverse Square law"}, {"start": 199.62, "duration": 5.24, "text": "like the law of gravity and so distances"}, {"start": 201.72, "duration": 3.14, "text": "are very important"}, {"start": 204.959, "duration": 3.841, "text": "finally just a little housekeeping but"}, {"start": 207.18, "duration": 4.74, "text": "not critical to understanding Coulomb's"}, {"start": 208.8, "duration": 5.04, "text": "law we've thrown around this number K 9"}, {"start": 211.92, "duration": 3.899, "text": "times 10 to the 9 newton meters squared"}, {"start": 213.84, "duration": 5.22, "text": "per coulomb squared another way to write"}, {"start": 215.819, "duration": 6.0, "text": "K is 1 over 4 Pi Epsilon naught where Pi"}, {"start": 219.06, "duration": 4.56, "text": "is the numerical constant 3.14159"}, {"start": 221.819, "duration": 4.081, "text": "Etc and Epsilon naught is what's known"}, {"start": 223.62, "duration": 4.38, "text": "as the vacuum permittivity also known as"}, {"start": 225.9, "duration": 4.559, "text": "the permittivity of free space and it's"}, {"start": 228.0, "duration": 4.44, "text": "approximately equal to 8.85 times 10 to"}, {"start": 230.459, "duration": 4.7, "text": "the minus 12 coulomb squared per Newton"}, {"start": 232.44, "duration": 2.719, "text": "meter squared"}], "title": "Coulomb's law", "description": "An explanation of Coulomb's law. For more content visit schoolyourself.org.", "lengthSeconds": 234, "width": 640, "height": 360}
{"video_id": "Gw-XAXqxppI", "transcript": [{"start": 0.0, "duration": 4.76, "text": "Hey everyone, Professor Dave here, I wanna talk a little more about nucleophilic substitutions"}, {"start": 4.76, "duration": 2.64, "text": "specifically, the SN1 reaction"}, {"start": 13.92, "duration": 2.88, "text": "so the SN1 is another nucleophilic substitution reaction"}, {"start": 16.8, "duration": 3.56, "text": "so it's similar to the SN2, but there's a couple key mechanistic differences."}, {"start": 20.36, "duration": 2.476, "text": "so let's take a look at what's going on"}, {"start": 22.836, "duration": 3.423, "text": "the first difference here, is that it is a two-step reaction"}, {"start": 26.26, "duration": 3.26, "text": "so i know that's confusing, SN1 being two steps and SN2 being 1"}, {"start": 29.52, "duration": 3.84, "text": "but once again it's about the transition state of the reaction. so let's see here"}, {"start": 33.46, "duration": 3.78, "text": "so the first step is that the leaving group leaves"}, {"start": 37.24, "duration": 6.6, "text": "so this is prior to any contact with a nucleophile, this carbon is able to lose a bond to this chlorine"}, {"start": 43.84, "duration": 9.52, "text": "chloride is very stable by itself, we have a carbocation intermediate here. so one key thing to understand"}, {"start": 53.84, "duration": 12.7, "text": "is that we're going from an sp3 hybridized tetrahedral carbon, to an sp2 hybridized trigonal planar situation"}, {"start": 66.54, "duration": 3.4, "text": "because this carbon is losing an electron domain"}, {"start": 69.94, "duration": 5.54, "text": "so now that this carbon with a formal positive charge has only three electron domains is gonna have"}, {"start": 75.48, "duration": 5.74, "text": "trigonal planar geometry, so basically this molecule is a plane perpendicular to the board"}, {"start": 81.22, "duration": 6.8, "text": "we are looking at it edge on like this. now that we have the carbocation intermediate"}, {"start": 88.02, "duration": 3.36, "text": "the second step is for the nucleophile to attack"}, {"start": 91.38, "duration": 4.1, "text": "here let's say it's just a molecule of water solvent"}, {"start": 95.48, "duration": 5.06, "text": "it is gonna go ahead and attack here, and we'll have a water molecule attached"}, {"start": 100.54, "duration": 7.16, "text": "and then another molecule of water will go ahead, snag that proton, deprotonate, and we'll have just the alcohol"}, {"start": 108.2, "duration": 7.2, "text": "one very important part about the SN1 mechanism is that because a carbocation intermediate"}, {"start": 115.4, "duration": 4.8, "text": "is trigonal planar, that means that a nucleophile can attack from either side"}, {"start": 120.2, "duration": 3.88, "text": "remember that any time we have a reaction occurring"}, {"start": 124.08, "duration": 6.3, "text": "something is dumping electrons from its highest occupied molecular orbital"}, {"start": 130.38, "duration": 6.1, "text": "and putting them into the lowest unoccupied molecular orbital, and in this case, if we have a trigonal planar"}, {"start": 136.48, "duration": 6.28, "text": "situation, the lowest unoccupied molecular orbital is the unhybridized remaining p orbital"}, {"start": 142.76, "duration": 7.44, "text": "so these are sp2, and then the unhybridized third p orbital is each lobe extending in this direction"}, {"start": 150.2, "duration": 5.38, "text": "perpendicular to the plane of the molecule. so that means that water can attack from this side"}, {"start": 155.58, "duration": 6.26, "text": "or water can attack from that side. and if you are generating a chiral center you are gonna get something"}, {"start": 161.84, "duration": 2.32, "text": "called a racemic mixture"}, {"start": 169.98, "duration": 5.4, "text": "of enantiomers, or of stereoisomers at that location"}, {"start": 175.38, "duration": 7.06, "text": "which means in 50% of the resulting molecules you'll have an R stereocenter and in 50% you'll have S"}, {"start": 182.44, "duration": 2.779, "text": "so that's something to be mindful of."}, {"start": 185.219, "duration": 3.86, "text": "thanks for watching, guys. subscribe to my channel for more tutorials"}, {"start": 189.08, "duration": 2.0, "text": "and as always feel free to email me with questions"}], "title": "SN1 Reaction", "description": "Why is SN2 a one-step reaction and SN1 a two-step reaction! Argh! Just listen to Dave, he'll tell you about the mechanism, intermediate, and lack of stereospecificity for the SN1 reaction. Look out for racemic mixtures!\n\nWatch the whole Organic Chemistry playlist: http://bit.ly/ProfDaveOrgChem\n\nGeneral Chemistry Tutorials: http://bit.ly/ProfDaveGenChem\nBiochemistry Tutorials: http://bit.ly/ProfDaveBiochem\nBiology Tutorials: http://bit.ly/ProfDaveBio\nClassical Physics Tutorials: http://bit.ly/ProfDavePhysics1\nModern Physics Tutorials: http://bit.ly/ProfDavePhysics2\nMathematics Tutorials: http://bit.ly/ProfDaveMath\n\nEMAIL\u25ba ProfessorDaveExplains@gmail.com\nPATREON\u25ba http://patreon.com/ProfessorDaveExplains\n\nCheck out \"Is This Wi-Fi Organic?\", my book on disarming pseudoscience!\nAmazon: https://amzn.to/2HtNpVH\nBookshop: https://bit.ly/39cKADM\n\rBarnes and Noble: https://bit.ly/3pUjmrn \nBook Depository: http://bit.ly/3aOVDlT", "lengthSeconds": 209, "width": 640, "height": 360}
{"video_id": "WdCRrcfan44", "transcript": [{"start": 0.0, "duration": 1.0, "text": "The skeleton has"}, {"start": 1.0, "duration": 1.0, "text": "206"}, {"start": 2.0, "duration": 2.2, "text": "bones in an adult"}, {"start": 4.295, "duration": 1.405, "text": "Babies have 270"}, {"start": 5.8, "duration": 2.14, "text": "which fuse for this result"}, {"start": 7.94, "duration": 2.0, "text": "The skeleton is divided"}, {"start": 10.02, "duration": 1.76, "text": "into two main zones"}, {"start": 11.78, "duration": 3.92, "text": "which are called the axial and appendicular bones"}, {"start": 15.86, "duration": 4.04, "text": "The axial skeleton is topped by the skull"}, {"start": 20.08, "duration": 3.66, "text": "Your brain is in the cranium which acts as a hull"}, {"start": 23.96, "duration": 3.6, "text": "The mandible or jawbone is able to expand"}, {"start": 27.84, "duration": 4.1, "text": "which is useful for talking, as you can understand"}, {"start": 32.1, "duration": 3.54, "text": "The ossicles are small bones found inside your ear"}, {"start": 35.96, "duration": 3.64, "text": "The maleus, incus, stapes, which all help you hear"}, {"start": 39.98, "duration": 3.94, "text": "Sound waves are amplified by these three bones"}, {"start": 44.06, "duration": 3.82, "text": "And then move to the cochlea which discerns the tones"}, {"start": 48.08, "duration": 3.74, "text": "The ribcage helps to protect the lungs and the heart"}, {"start": 52.02, "duration": 3.94, "text": "With 12 ribs on each side, the sternum in the front"}, {"start": 56.28, "duration": 3.46, "text": "Your vertebral column, made up of vertebrae"}, {"start": 59.82, "duration": 4.2, "text": "Are stacked up very neatly in a vertical array"}, {"start": 64.12, "duration": 3.7, "text": "The cervical vertebrae make up your neck"}, {"start": 68.04, "duration": 3.78, "text": "The spinal cord keeps your reflexes on deck"}, {"start": 71.98, "duration": 3.98, "text": "The ribs are supported by thoracic vertebrae"}, {"start": 76.14, "duration": 3.72, "text": "Lumbar ones are at the end of the spinal cord pathway"}, {"start": 80.04, "duration": 4.02, "text": "Five sacral vertebrae are fused to form your sacrum"}, {"start": 84.08, "duration": 3.92, "text": "Which holds your upper body and is found in your bum"}, {"start": 88.1, "duration": 3.4, "text": "Below it is the coccyx or the tailbone"}, {"start": 91.82, "duration": 4.02, "text": "In humans the coccyx has not really grown"}, {"start": 96.04, "duration": 4.26, "text": "The appendicular skeleton, 126 bones"}, {"start": 100.52, "duration": 3.44, "text": "In shoulders, arms and legs, as we have here shown"}, {"start": 104.12, "duration": 3.8, "text": "Your scapula, also known as your shoulderblade"}, {"start": 108.1, "duration": 3.84, "text": "Next to the clavicle or collarbone is laid"}, {"start": 112.0, "duration": 3.94, "text": "You have three arm bones, one above and two below"}, {"start": 116.16, "duration": 3.74, "text": "Here they come together to meet at the elbow"}, {"start": 119.9, "duration": 4.04, "text": "The bones of the forearm are the ulna and the radius"}, {"start": 124.1, "duration": 3.76, "text": "The humerus in the upper arm so don't you forget this"}, {"start": 128.04, "duration": 3.8, "text": "You have 54 bones in one hand and one wrist"}, {"start": 132.06, "duration": 3.74, "text": "Of carpals, metacarpals and phalanges these consist"}, {"start": 136.0, "duration": 3.88, "text": "The proximal, intermediate and distal phalanges"}, {"start": 140.08, "duration": 3.78, "text": "Make up your fingers which you can move with ease"}, {"start": 144.1, "duration": 3.74, "text": "Remember that your hands are where you find the carpals"}, {"start": 148.04, "duration": 3.7, "text": "And the feet are where you will find all your tarsals"}, {"start": 151.92, "duration": 3.84, "text": "The hip bone can be divided into three"}, {"start": 155.92, "duration": 3.72, "text": "The ilium, ischium and pubis as you can see"}, {"start": 159.88, "duration": 3.62, "text": "The bones in your leg, well there are four"}, {"start": 163.82, "duration": 4.06, "text": "The patella is your kneecap, and then there are three more"}, {"start": 168.16, "duration": 3.6, "text": "The tibia and fibula are found below the knee"}, {"start": 172.06, "duration": 3.64, "text": "And the femur's found above it, as we can all agree"}, {"start": 176.04, "duration": 3.48, "text": "There are 52 bones in your ankles and your feet"}, {"start": 179.92, "duration": 3.5, "text": "Once you know the hand bones, these will be a treat"}, {"start": 184.12, "duration": 3.84, "text": "There are the tarsals, metatarsals and phalanges"}, {"start": 188.1, "duration": 3.7, "text": "Remember that they are proximal, intermediate and distal"}, {"start": 193.12, "duration": 2.955, "text": "If you enjoyed this video, like and subscribe!"}, {"start": 196.075, "duration": 3.06, "text": "You can also support me by following the link to my Patreon"}, {"start": 199.14, "duration": 3.54, "text": "If you have any topics you'd like me to cover, please leave a comment :)"}], "title": "SKELETON BONES SONG - LEARN IN 3 MINUTES!!!", "description": "HAPPY HALLOWEEN! Here's a song for you to memorize the bones in 3 minutes!\n\nThe skeleton has 2-0-6 bones in an adult,\nBabies have 2-70, which fuse for this result.\nThe skeleton is divided into two main zones,\nWhich are called the axial and appendicular bones.\n \nThe axial skeleton is topped by the skull,\nYour brain is in the cranium, which acts as a hull.\nThe mandible or jawbone is able to expand,\nWhich is useful for talking, as you can understand.\n \nThe ossicles are small bones, found inside your ear,\nThe maleus, incus, stapes, which all help you hear.\nSound waves are amplified by these three bones,\nAnd then move to the cochlea, which discerns the tones!\n\nThe rib cage helps to protect the lungs and the heart,\nWith 12 ribs on each side, the sternum in the front.\nYour vertebral column, made up of vertebrae,\nAre stacked up very neatly, in a vertical array.\n \nThe cervical vertebrae make up your neck,\nThe spinal chord keeps your reflexes on deck!\nThe ribs are supported by thoracic vertebrae, \nLumbar ones are at the end of the spinal cord pathway!\n \nFive sacral vertebrae are fused to form your sacrum,\nWhich holds your upper body, and is found in your bum.\nBelow it is the coccyx, or the tailbone,\nIn humans the coccyx has not really grown.\n \nThe appendicular skeleton  - a  126  bones,\nIn shoulders, arms, and legs, as we have here shown,\nYour scapula, also known, as your shoulder blade,\nNext to the clavicle, or collarbone, is laid.\n \nYou have three arm bones, one above and two below,\nHere they come together, to meet at the elbow.\nThe bones of the forearm are the ulna and the radius,\nThe humerus in the upper arm, so don\u2019t you forget this.\n \nYou have 54 bones in one hand and one wrist,\nOf carpals, metacarpals, and phalanges these consist.\nThe proximal, intermediate, and distal phalanges,\nMake up your fingers, which you can move with ease.\n \nRemember that your hands are where you find the carpals,\nAnd the feet are where you will find all your tarsals.\nThe hip bone can be divided into three,\nThe ilium, ischium, and pubis, as you can see!\n \nThe bones in your leg? Well, there are four,\nThe patella is your kneecap, and then there are three more,\nThe tibia and fibula are found below the knee,\nAnd the femur\u2019s found above it, as we can all agree!\n \nThere are 52 bones, in your ankles and your feet,\nOnce you know the hand bones, these will be a treat!\nThere are the tarsals, metatarsals, and phalanges,\nRemember that they are proximal, intermediate, and distal!\n\nMusic/Lyrics/Visual \u00a9 Neural Academy\n\n3D models:\nhttps://videohive.net/item/skeleton-street-dance-/24358994\nhttps://videohive.net/item/human-skeleton-transforming-2/19767274\nhttps://videohive.net/item/running-human-skeleton-2pack/22684573\nhttps://videohive.net/item/ear-anatomy-2k/23878870\nhttps://videohive.net/item/skeleton-sexy-dance/18743642\nhttps://videohive.net/item/skeleton-dubstep-dance/18594801", "lengthSeconds": 204, "width": 640, "height": 360}
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\u0633\u0627\u0644\u0628 \u0627\u0644\u0634\u062d\u0646\u0629, \u0627\u0644\u062c\u0633\u064a\u0645\u0627\u062a \u0630\u0627\u062a \u0627\u0644\u0634\u062d\u0646\u0629 \u0627\u0644\u0645\u0639\u0627\u0643\u0633\u0629 \u0644\u0647\u0627"}, {"start": 44.67, "duration": 4.319, "text": "\u062c\u0627\u0630\u0628\u064a\u0629 \u0643\u0647\u0631\u0648\u0633\u062a\u0627\u062a\u064a\u0643\u064a\u0629 \u0628\u062d\u064a\u062b \u064a\u062a\u062c\u0648\u0644 \u0647\u0630\u064a\u0646 \u0627\u0644\u0623\u064a\u0648\u0646\u064a\u0646 \u0645\u0639\u064b\u0627 \u0645\u0646 \u0645\u0634\u0643\u0644\u064a\u0646"}, {"start": 48.989, "duration": 4.361, "text": "\u0631\u0627\u0628\u0637\u0629 \u0623\u064a\u0648\u0646\u064a\u0629, \u0647\u0630\u0647  \u0646\u0642\u0637\u0629 \u062c\u0630\u0628 \u0642\u0648\u064a\u0629 \u0644\u0623\u0646\u0647 \u064a\u062d\u062f\u062b"}, {"start": 53.35, "duration": 7.709, "text": "\u0628\u064a\u0646 \u0627\u0644\u0623\u064a\u0648\u0646\u0627\u062a \u0627\u0644\u062a\u064a 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"\u062a\u0634\u062a\u0631\u0643 \u0627\u0644\u0630\u0631\u0627\u062a \u0641\u064a \u0627\u0644\u0625\u0644\u0643\u062a\u0631\u0648\u0646\u064a\u0646 \u0644\u062a\u0634\u0643\u064a\u0644 \u0631\u0627\u0628\u0637\u0629 \u062a\u0633\u0627\u0647\u0645\u064a\u0629, \u064a\u0645\u0643\u0646 \u0623\u0646 \u062a\u0643\u0648\u0646 \u0627\u0644\u0631\u0648\u0627\u0628\u0637 \u0627\u0644\u062a\u0633\u0627\u0647\u0645\u064a\u0629"}, {"start": 86.33, "duration": 3.899, "text": "\u0642\u0637\u0628\u064a\u0629 \u0623\u0648 \u063a\u064a\u0631 \u0642\u0637\u0628\u064a\u0629, \u0625\u0630\u0627 \u0643\u0627\u0646 \u0641\u0631\u0642 \u0627\u0644\u0633\u0627\u0644\u0628\u064a\u0629 \u0627\u0644\u0643\u0647\u0631\u0628\u0627\u0626\u064a\u0629 \u0628\u064a\u0646"}, {"start": 90.229, "duration": 4.451, "text": "\u0627\u0644\u0630\u0631\u0627\u062a \u0644\u064a\u0633\u062a \u0639\u0627\u0644\u064a\u0629 \u0628\u0645\u0627 \u064a\u0643\u0641\u064a \u0644\u0644\u062a\u0633\u0628\u0628 \u0641\u064a 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4.809, "text": "\u0627\u0644\u0646\u0627\u062a\u062c\u0629,  \u0643\u0644\u0645\u0627 \u0632\u0627\u062f\u062a \u0630\u0631\u0629 \u0643\u0647\u0631\u0633\u0644\u0628\u064a\u0629 \u0641\u064a \u0631\u0627\u0628\u0637\u0629 \u062a\u0633\u0627\u0647\u0645\u064a\u0629 \u0642\u0637\u0628\u064a\u0629 \u062a\u0633\u062d\u0628"}, {"start": 108.799, "duration": 5.14, "text": "\u0643\u062b\u0627\u0641\u0629 \u0627\u0644\u0625\u0644\u0643\u062a\u0631\u0648\u0646\u0627\u062a \u062a\u062c\u0627\u0647 \u0646\u0641\u0633\u0647\u0627 \u0648\u0646\u062a\u064a\u062c\u0629 \u0644\u0632\u064a\u0627\u062f\u0629 \u0627\u0644\u0625\u0644\u0643\u062a\u0631\u0648\u0646 \u0627\u0644\u0637\u0641\u064a\u0641\u0629"}, {"start": 113.939, "duration": 4.89, "text": "\u0646\u0633\u0645\u064a \u0647\u0630\u0647 \u0627\u0644\u0630\u0631\u0629 \u0633\u0627\u0644\u0628\u0629 \u062c\u0632\u0626\u064a\u064b\u0627, \u064a\u0631\u0645\u0632 \u0625\u0644\u064a\u0647\u0627 \u0628\u0627\u0644\u0631\u0645\u0632: 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{"start": 132.76, "duration": 5.339, "text": "\u0641\u064a \u0627\u0644\u0631\u0627\u0628\u0637\u0629 \u0627\u0644\u062a\u0633\u0627\u0647\u0645\u064a\u0629 \u063a\u064a\u0631 \u0627\u0644\u0642\u0637\u0628\u064a\u0629, \u064a\u062a\u0645 \u062a\u0642\u0627\u0633\u0645 \u0627\u0644\u0625\u0644\u0643\u062a\u0631\u0648\u0646\u0627\u062a \u0628\u0634\u0643\u0644 \u0645\u062a\u0633\u0627\u0648\u064d \u062a\u0642\u0631\u064a\u0628\u064b\u0627"}, {"start": 138.099, "duration": 5.47, "text": "\u0623\u0648 \u0641\u064a \u062d\u0627\u0644\u0629 \u0630\u0631\u062a\u064a\u0646 \u0645\u0646 \u0646\u0641\u0633 \u0627\u0644\u0639\u0646\u0635\u0631, \u0645\u062a\u0633\u0627\u0648\u064a\u064a\u0646 \u0628\u0627\u0644\u062a\u062d\u062f\u064a\u062f"}, {"start": 143.569, "duration": 4.041, "text": "\u0644\u0646 \u064a\u0643\u0648\u0646 \u0644\u0644\u0630\u0631\u0627\u062a \u0634\u062d\u0646\u0627\u062a \u062c\u0632\u0626\u064a\u0629 \u0639\u0644\u0649 \u0627\u0644\u0625\u0637\u0644\u0627\u0642. \u0628\u062e\u0644\u0627\u0641 \u0627\u0644\u0638\u0648\u0627\u0647\u0631 \u0627\u0644\u063a\u0631\u064a\u0628\u0629 \u0645\u062b\u0644"}, {"start": 147.61, "duration": 3.68, "text": "\u0627\u0644\u0631\u0627\u0628\u0637\u0629 \u0627\u0644\u0641\u0644\u0632\u064a\u0629, \u0647\u0630\u0647 \u0647\u064a \u0627\u0644\u0637\u0631\u0642 \u0627\u0644\u0631\u0626\u064a\u0633\u064a\u0629 \u0644\u0644\u0630\u0631\u0627\u062a \u0644\u0643\u064a\n\u062a\u0631\u062a\u0628\u0637 \u0628\u0628\u0639\u0636\u0647\u0627 \u0627\u0644\u0628\u0639\u0636"}, {"start": 151.29, "duration": 3.4, "text": "\u062a\u0630\u0643\u0631 \u0623\u0646\u0647 \u064a\u0645\u0643\u0646\u0643 \u0627\u0644\u062a\u0646\u0628\u0624 \u0628\u0646\u0648\u0639 \u0627\u0644\u0631\u0627\u0628\u0637\u0629 \u0627\u0644\u062a\u064a \u0633\u062a\u062a\u0634\u0643\u0644 \u0628\u064a\u0646 \u0639\u0646\u0635\u0631\u064a\u064a\u0646"}, {"start": 154.69, "duration": 4.82, "text": "\u0628\u0645\u0642\u0627\u0631\u0646\u0629 \u0646\u0634\u0627\u0637\u0647\u0645\u0627 \u0627\u0644\u0625\u0644\u0643\u062a\u0631\u0648\u0646\u064a, \u0625\u0630\u0627 \u0643\u0627\u0646 \u0627\u0644\u0641\u0631\u0642 \u0645\u0646\u062e\u0641\u0636 \u0644\u0644\u063a\u0627\u064a\u0629"}, {"start": 159.51, "duration": 5.67, "text": "\u0623\u0642\u0644 \u0645\u0646 0.5 \u0623\u0648 \u0646\u062d\u0648 \u0630\u0644\u0643, \u0633\u062a\u0643\u0648\u0646 \u0631\u0627\u0628\u0637\u0629 \u062a\u0633\u0627\u0647\u0645\u064a\u0629 \u063a\u064a\u0631 \u0642\u0637\u0628\u064a\u0629 \u062d\u064a\u062b"}, {"start": 165.18, "duration": 5.88, "text": "\u064a\u062a\u0645 \u062a\u0642\u0627\u0633\u0645 \u0627\u0644\u0625\u0644\u0643\u062a\u0631\u0648\u0646\u0627\u062a \u0628\u0627\u0644\u062a\u0633\u0627\u0648\u064a, \u0645\u0627 \u0628\u064a\u0646 \u062d\u0648\u0627\u0644\u064a 0.5 \u0648 1.7 \u0633\u062a\u0643\u0648\u0646 \u0627\u0644\u0631\u0627\u0628\u0637\u0629 \u0642\u0637\u0628\u064a\u0629"}, {"start": 171.06, "duration": 4.03, "text": "\u062a\u0633\u0627\u0647\u0645\u064a\u0629 \u0645\u0639 \u0627\u0644\u0625\u0644\u0643\u062a\u0631\u0648\u0646\u0627\u062a \u0645\u0634\u062a\u0631\u0643\u0629 \u0648\u0644\u0643\u0646 \u0630\u0631\u0629 \u0648\u0627\u062d\u062f\u0629 \u062a\u062d\u0645\u0644\u0647\u0627 \u062a\u0645\u0633\u0643\u0647\u0627 \u0628\u0627\u062d\u0643\u0627\u0645"}, {"start": 175.09, "duration": 5.399, "text": "\u0625\u0630\u0627 \u0641\u0648\u0642 2 \u0623\u0648 \u0646\u062d\u0648 \u0630\u0644\u0643 \u0633\u062a\u0642\u0648\u0645 \u0627\u0644\u0630\u0631\u0629 \u0628\u0633\u0644\u0628 \u0627\u0644\u0643\u062a\u0631\u0648\u0646 \u0648\u0627\u062d\u062f \u0645\u0646 \u0630\u0631\u0629"}, {"start": 180.489, "duration": 4.87, "text": "\u0648\u0633\u062a\u062d\u0635\u0644 \u0639\u0644\u0649 \u0623\u064a\u0648\u0646\u0627\u062a \u0645\u0631\u062a\u0628\u0637\u0629 \u0628\u0631\u0627\u0628\u0637\u0629 \u0623\u064a\u0648\u0646\u064a"}, {"start": 185.36, "duration": 3.22, "text": "\u0634\u0643\u0631\u0627 \u0644\u0643\u0645 \u0639\u0644\u0649 \u0627\u0644\u0645\u0634\u0627\u0647\u062f\u0629 \u0627\u0634\u062a\u0631\u0627\u0643\u0648 \u0641\u064a \u0642\u0646\u0627\u062a\u064a \u0644\u0645\u0632\u064a\u062f \u0645\u0646 \u0627\u0644\u062f\u0631\u0648\u0633"}, {"start": 188.58, "duration": 1.9, "text": "\u0643\u0645\u0627 \u0647\u0648 \u0627\u0644\u062d\u0627\u0644 \u062f\u0627\u0626\u0645\u0627 \u0644\u0627 \u062a\u062a\u0631\u062f\u062f \u0641\u064a \u0645\u0631\u0627\u0633\u0644\u062a\u064a \u0639\u0644\u0649 \u0627\u0644\u0627\u064a\u0645\u064a\u0644 \u0627\u0644\u062a\u0627\u0644\u064a"}], "title": "The Chemical Bond: Covalent vs. Ionic and Polar vs. Nonpolar", "description": "Ionic Bond, Covalent Bond, James Bond, so many bonds! What dictates which kind of bond will form? Electronegativity values, of course. Let's go through each type and what they're all about.\n\nWatch the whole General Chemistry playlist: http://bit.ly/ProfDaveGenChem\n\nStudy for the AP Chemistry exam with me: https://bit.ly/ProfDaveAPChem\n\nOrganic Chemistry Tutorials: http://bit.ly/ProfDaveOrgChem\nBiochemistry Tutorials: http://bit.ly/ProfDaveBiochem\nBiology Tutorials: http://bit.ly/ProfDaveBio\nClassical Physics Tutorials: http://bit.ly/ProfDavePhysics1\nModern Physics Tutorials: http://bit.ly/ProfDavePhysics2\nMathematics Tutorials: http://bit.ly/ProfDaveMaths\n\nEMAIL\u25ba ProfessorDaveExplains@gmail.com\nPATREON\u25ba http://patreon.com/ProfessorDaveExplains\n\nCheck out \"Is This Wi-Fi Organic?\", my book on disarming pseudoscience!\nAmazon: https://amzn.to/2HtNpVH\nBookshop: https://bit.ly/39cKADM\n\rBarnes and Noble: https://bit.ly/3pUjmrn \nBook Depository: http://bit.ly/3aOVDlT", "lengthSeconds": 213, "width": 640, "height": 360}
{"video_id": "LiSWU1p_zIk", "transcript": [{"start": 0.04, "duration": 3.74, "text": "[Music]"}, {"start": 0.78, "duration": 3.0, "text": "foreign"}, {"start": 5.04, "duration": 4.86, "text": "Professor Dave and Chegg here when using"}, {"start": 7.379, "duration": 4.8, "text": "line notation to depict molecules we"}, {"start": 9.9, "duration": 4.92, "text": "sometimes disregard the true geometry of"}, {"start": 12.179, "duration": 4.981, "text": "the molecule by flattening it out but if"}, {"start": 14.82, "duration": 5.4, "text": "we are depicting molecules with their"}, {"start": 17.16, "duration": 5.94, "text": "true geometry substituents must be on"}, {"start": 20.22, "duration": 5.639, "text": "Wedge or Dash Bonds in order to obey the"}, {"start": 23.1, "duration": 5.4, "text": "tetrahedral geometry that these sp3"}, {"start": 25.859, "duration": 5.281, "text": "centers demand and with that we have set"}, {"start": 28.5, "duration": 4.04, "text": "the stage for an important concept the"}, {"start": 31.14, "duration": 4.62, "text": "notion of"}, {"start": 32.54, "duration": 6.1, "text": "stereoisomerism and chirality we already"}, {"start": 35.76, "duration": 4.38, "text": "know what structural isomers are these"}, {"start": 38.64, "duration": 3.239, "text": "are molecules that have the same"}, {"start": 40.14, "duration": 4.02, "text": "molecular formula but differing"}, {"start": 41.879, "duration": 4.5, "text": "connectivity this would be like a"}, {"start": 44.16, "duration": 4.079, "text": "straight Chain alkane versus a branched"}, {"start": 46.379, "duration": 4.02, "text": "alkane with the same number of carbons"}, {"start": 48.239, "duration": 4.261, "text": "or even something like dimethyl ether"}, {"start": 50.399, "duration": 3.961, "text": "versus ethanol which produces different"}, {"start": 52.5, "duration": 3.6, "text": "functional groups but even when"}, {"start": 54.36, "duration": 4.46, "text": "molecules have precisely the same"}, {"start": 56.1, "duration": 4.459, "text": "connectivity we can still have"}, {"start": 58.82, "duration": 3.94, "text": "stereoisomerism"}, {"start": 60.559, "duration": 4.96, "text": "stereoisomers are molecules that have"}, {"start": 62.76, "duration": 5.039, "text": "the same molecular formula and the same"}, {"start": 65.519, "duration": 4.561, "text": "connectivity but differ in the way the"}, {"start": 67.799, "duration": 5.221, "text": "atoms are distributed in space"}, {"start": 70.08, "duration": 4.92, "text": "take this molecule to bromobutane this"}, {"start": 73.02, "duration": 4.2, "text": "bromo group must be on either a dash"}, {"start": 75.0, "duration": 4.02, "text": "Bond or a wedge Bond as we know but if"}, {"start": 77.22, "duration": 3.78, "text": "we draw both options we must understand"}, {"start": 79.02, "duration": 4.68, "text": "that these are not the same molecule"}, {"start": 81.0, "duration": 4.56, "text": "they are mirror images of one another if"}, {"start": 83.7, "duration": 3.959, "text": "we take this one on the wedge Bond and"}, {"start": 85.56, "duration": 4.14, "text": "reflect it across this mirror plane we"}, {"start": 87.659, "duration": 4.201, "text": "get this mirror image and if we then"}, {"start": 89.7, "duration": 4.559, "text": "flip it over to the other side to try to"}, {"start": 91.86, "duration": 4.799, "text": "line up the main chain this bromo group"}, {"start": 94.259, "duration": 4.5, "text": "pointing towards us will end up pointing"}, {"start": 96.659, "duration": 4.261, "text": "away from us and therefore on a dash"}, {"start": 98.759, "duration": 5.04, "text": "Bond and will not line up with this"}, {"start": 100.92, "duration": 4.98, "text": "first bromo group these molecules are"}, {"start": 103.799, "duration": 4.5, "text": "mirror images just like your hands are"}, {"start": 105.9, "duration": 5.1, "text": "mirror images your hands are not"}, {"start": 108.299, "duration": 4.801, "text": "superposable that is to say if one hand"}, {"start": 111.0, "duration": 4.38, "text": "is placed over the other with the Palms"}, {"start": 113.1, "duration": 4.979, "text": "facing the same direction we can't get"}, {"start": 115.38, "duration": 4.739, "text": "the fingers to line up properly when a"}, {"start": 118.079, "duration": 4.141, "text": "molecule is not superposable on its"}, {"start": 120.119, "duration": 5.401, "text": "Mirror Image in precisely this manner"}, {"start": 122.22, "duration": 5.879, "text": "this molecule is said to be chiral and"}, {"start": 125.52, "duration": 3.84, "text": "specifically this carbon is said to be a"}, {"start": 128.099, "duration": 3.421, "text": "chiral Center"}, {"start": 129.36, "duration": 4.68, "text": "there are a lot of synonyms for this"}, {"start": 131.52, "duration": 4.5, "text": "term we could say stereocenter or"}, {"start": 134.04, "duration": 4.26, "text": "stereogenic Center and we'd be talking"}, {"start": 136.02, "duration": 4.2, "text": "about the same thing we would generally"}, {"start": 138.3, "duration": 3.96, "text": "be talking about a carbon atom that is"}, {"start": 140.22, "duration": 5.159, "text": "connected to four different groups in"}, {"start": 142.26, "duration": 5.339, "text": "this case methyl ethyl bromo hydrogen"}, {"start": 145.379, "duration": 4.981, "text": "this is the reason the mirror image is"}, {"start": 147.599, "duration": 4.741, "text": "different if we changed this ethyl to a"}, {"start": 150.36, "duration": 3.36, "text": "methyl when we get the mirror image we"}, {"start": 152.34, "duration": 3.899, "text": "would find that it is absolutely"}, {"start": 153.72, "duration": 4.86, "text": "identical this is due to the internal"}, {"start": 156.239, "duration": 4.681, "text": "symmetry of the molecule because the"}, {"start": 158.58, "duration": 5.22, "text": "mirror image is identical this molecule"}, {"start": 160.92, "duration": 3.959, "text": "is not chiral which we can also call a"}, {"start": 163.8, "duration": 3.24, "text": "chiral"}, {"start": 164.879, "duration": 4.86, "text": "it will be very important to be able to"}, {"start": 167.04, "duration": 5.04, "text": "identify chiral centers take a look at"}, {"start": 169.739, "duration": 4.561, "text": "this molecule take a moment and see if"}, {"start": 172.08, "duration": 6.239, "text": "you can identify all the chiral centers"}, {"start": 174.3, "duration": 6.6, "text": "bear in mind that SP and SP2 hybridized"}, {"start": 178.319, "duration": 5.821, "text": "carbons will not qualify we are almost"}, {"start": 180.9, "duration": 5.58, "text": "always talking about sp3 carbons when"}, {"start": 184.14, "duration": 4.379, "text": "referring to chiral centers also"}, {"start": 186.48, "duration": 4.74, "text": "remember that hydrogen atoms bound to"}, {"start": 188.519, "duration": 4.681, "text": "carbon are implied and not shown so we"}, {"start": 191.22, "duration": 4.26, "text": "must keep these in mind as well when"}, {"start": 193.2, "duration": 3.36, "text": "considering the four groups bound to any"}, {"start": 195.48, "duration": 3.899, "text": "carbon"}, {"start": 196.56, "duration": 5.039, "text": "are you ready here are all the chiral"}, {"start": 199.379, "duration": 4.86, "text": "centers in this molecule marked with an"}, {"start": 201.599, "duration": 4.86, "text": "asterisk each of the four groups bound"}, {"start": 204.239, "duration": 4.321, "text": "to these carbons is different either"}, {"start": 206.459, "duration": 4.081, "text": "different elements or different kinds of"}, {"start": 208.56, "duration": 4.5, "text": "carbon chains that are not precisely"}, {"start": 210.54, "duration": 5.1, "text": "identical we must be able to do this"}, {"start": 213.06, "duration": 4.62, "text": "with any organic molecule"}, {"start": 215.64, "duration": 4.379, "text": "so we should Now understand what is"}, {"start": 217.68, "duration": 4.86, "text": "meant by the term chiral as well as the"}, {"start": 220.019, "duration": 5.401, "text": "chiral Center on a molecule and be able"}, {"start": 222.54, "duration": 5.4, "text": "to identify such chiral centers on any"}, {"start": 225.42, "duration": 4.94, "text": "molecule Professor Dave for Chegg see"}, {"start": 227.94, "duration": 2.42, "text": "you next time"}], "title": "Understanding Chirality in Molecules Ft. Professor Dave", "description": "Professor Dave delves into the notions of stereoisomerism and chirality in this video. We'll see how stereo isomers differ from structural isomers, learn the definition of a chiral molecule, and start learning how to label chiral centers and work with chirality.\n\nGet more homework help from Chegg at https://che.gg/3HbtG8Y\nWatch the whole Organic Chemistry playlist: https://www.youtube.com/playlist?list=PLllCkartRRdMPnATf1X8qVdo5RzM8gA_M\n\nAbout Chegg:\nChegg is a leading education technology company that provides students with affordable and accessible study resources. Chegg's mission is to help students save time, save money and get smarter.  \n\nSubscribe: https://www.youtube.com/@chegg\nTikTok: https://www.tiktok.com/@chegg\nInstagram: https://www.instagram.com/chegg/\nTwitter: https://twitter.com/chegg\n\n00:00 Intro\n00:37 Structural isomers\n00:59 Stereo isomers\n02:01 Chirality\n02:49 Identifying chiral centers", "lengthSeconds": 239, "width": 640, "height": 360}
{"video_id": "KNLUzqW8IuA", "transcript": [{"start": 0.08, "duration": 3.76, "text": "welcome to moomoomath and science in the"}, {"start": 2.159, "duration": 4.001, "text": "carbon cycle"}, {"start": 3.84, "duration": 4.96, "text": "carbon is an element that is vital to"}, {"start": 6.16, "duration": 4.559, "text": "most living organisms on earth"}, {"start": 8.8, "duration": 3.52, "text": "it is also a key component of our"}, {"start": 10.719, "duration": 3.601, "text": "atmosphere"}, {"start": 12.32, "duration": 4.0, "text": "carbon has the ability to cycle through"}, {"start": 14.32, "duration": 5.039, "text": "our earth the ocean"}, {"start": 16.32, "duration": 4.4, "text": "living factors abiotic factors in the"}, {"start": 19.359, "duration": 3.92, "text": "air"}, {"start": 20.72, "duration": 3.76, "text": "the carbon recycles among five major"}, {"start": 23.279, "duration": 4.641, "text": "locations"}, {"start": 24.48, "duration": 6.559, "text": "these items include the atmosphere"}, {"start": 27.92, "duration": 6.56, "text": "the terrestrial biosphere or land"}, {"start": 31.039, "duration": 6.001, "text": "the earth's interior the ocean"}, {"start": 34.48, "duration": 5.12, "text": "and human influence let's take a look at"}, {"start": 37.04, "duration": 4.32, "text": "what happens at each of these areas"}, {"start": 39.6, "duration": 3.92, "text": "carbon is found in the atmosphere in at"}, {"start": 41.36, "duration": 3.519, "text": "least two forms carbon dioxide and"}, {"start": 43.52, "duration": 3.28, "text": "methane"}, {"start": 44.879, "duration": 4.321, "text": "this atmospheric carbon can be absorbed"}, {"start": 46.8, "duration": 4.96, "text": "by autotrophs like plants and plankton"}, {"start": 49.2, "duration": 4.56, "text": "to be used for photosynthesis it can"}, {"start": 51.76, "duration": 5.119, "text": "also be absorbed by bodies of water"}, {"start": 53.76, "duration": 4.88, "text": "and the ocean when carbon is absorbed by"}, {"start": 56.879, "duration": 4.16, "text": "the ocean it reacts with the water and"}, {"start": 58.64, "duration": 4.719, "text": "creates carbonic acid"}, {"start": 61.039, "duration": 4.48, "text": "the terrestrial biosphere which is"}, {"start": 63.359, "duration": 5.12, "text": "another term for the earth's land"}, {"start": 65.519, "duration": 4.001, "text": "has several paths of carbon that it can"}, {"start": 68.479, "duration": 2.721, "text": "take"}, {"start": 69.52, "duration": 3.04, "text": "first there is an exchange between"}, {"start": 71.2, "duration": 3.04, "text": "plants and animals"}, {"start": 72.56, "duration": 4.239, "text": "plants absorb carbon in the form of"}, {"start": 74.24, "duration": 4.96, "text": "carbon dioxide for photosynthesis"}, {"start": 76.799, "duration": 5.36, "text": "and animals release carbon dioxide"}, {"start": 79.2, "duration": 5.2, "text": "during cellular respiration"}, {"start": 82.159, "duration": 4.0, "text": "also heterotrophs eat plants that"}, {"start": 84.4, "duration": 3.92, "text": "contain carbon"}, {"start": 86.159, "duration": 3.6, "text": "animals also release carbon as methane"}, {"start": 88.32, "duration": 3.6, "text": "during digestion"}, {"start": 89.759, "duration": 4.081, "text": "and the soil contains decomposers that"}, {"start": 91.92, "duration": 3.44, "text": "release carbon into the atmosphere and"}, {"start": 93.84, "duration": 3.599, "text": "the soil"}, {"start": 95.36, "duration": 3.36, "text": "carbon is also stored in the earth's"}, {"start": 97.439, "duration": 3.441, "text": "interior"}, {"start": 98.72, "duration": 4.8, "text": "carbon in the lithosphere includes"}, {"start": 100.88, "duration": 6.32, "text": "fossil fuels like coal"}, {"start": 103.52, "duration": 5.599, "text": "oil and natural gas along with deposits"}, {"start": 107.2, "duration": 4.48, "text": "like limestone"}, {"start": 109.119, "duration": 3.521, "text": "and volcanoes may release some of this"}, {"start": 111.68, "duration": 4.719, "text": "carbon stored"}, {"start": 112.64, "duration": 6.64, "text": "in the lithosphere when they erupt"}, {"start": 116.399, "duration": 3.521, "text": "up next the ocean the ocean has the"}, {"start": 119.28, "duration": 3.36, "text": "greatest"}, {"start": 119.92, "duration": 5.04, "text": "exchange quantity of cycled carbon and"}, {"start": 122.64, "duration": 4.159, "text": "stores a large amount of carbon"}, {"start": 124.96, "duration": 3.519, "text": "the ocean absorbs carbon in the form of"}, {"start": 126.799, "duration": 3.921, "text": "carbon dioxide"}, {"start": 128.479, "duration": 3.361, "text": "the ocean also has a large amount of"}, {"start": 130.72, "duration": 3.68, "text": "plankton"}, {"start": 131.84, "duration": 4.64, "text": "that absorbs carbon dioxide that it uses"}, {"start": 134.4, "duration": 4.479, "text": "for photosynthesis"}, {"start": 136.48, "duration": 3.52, "text": "humans also influence the movement of"}, {"start": 138.879, "duration": 3.281, "text": "carbon"}, {"start": 140.0, "duration": 4.959, "text": "humans burn fossil fuel which releases"}, {"start": 142.16, "duration": 5.28, "text": "carbon dioxide into the atmosphere"}, {"start": 144.959, "duration": 4.561, "text": "also the production of clinker which is"}, {"start": 147.44, "duration": 5.04, "text": "used for cement from limestone"}, {"start": 149.52, "duration": 5.439, "text": "also releases carbon in addition"}, {"start": 152.48, "duration": 4.399, "text": "deforestation from humans can cause"}, {"start": 154.959, "duration": 3.761, "text": "the amount of carbon in the atmosphere"}, {"start": 156.879, "duration": 4.401, "text": "to increase"}, {"start": 158.72, "duration": 4.56, "text": "so there you go an overview of the path"}, {"start": 161.28, "duration": 4.48, "text": "of an atom of carbon"}, {"start": 163.28, "duration": 3.84, "text": "that may take place during the carbon"}, {"start": 165.76, "duration": 3.28, "text": "cycle"}, {"start": 167.12, "duration": 3.119, "text": "thanks for watching and moon math"}, {"start": 169.04, "duration": 3.279, "text": "uploads a new map"}, {"start": 170.239, "duration": 5.041, "text": "and science video every day please"}, {"start": 172.319, "duration": 2.961, "text": "subscribe and"}, {"start": 175.879, "duration": 3.0, "text": "share"}], "title": "The Carbon Cycle Process", "description": "What is the Carbon cycle? The carbon cycle is one of several biogeochemical cycles found on Earth. Carbon is found in almost all living organisms and makes up around .04 percent of our atmosphere.\nWhat are the steps of the carbon cycle and how does the Carbon cycle work? \nCarbon may travel several paths including,\nThe atmosphere, the terrestrial biosphere, the earth\u2019s interior which includes the lithosphere, the ocean, and human influence like burning fossil fuels and other objects that store carbon.\nIn the video, I go through a carbon cycle diagram and explain each step.\nCarbon can be found in several forms including carbon dioxide and methane.\nThe carbon cycle is essential for many ecosystems on Earth.\n\nThe Carbon Cycle\nhttps://moomoomath.com/explain-the-carbon-cycle/\n\nBiogeochemical Cycles - How the Earth Recycles\nhttps://moomoomath.com/how-the-earth-recycles-elements-biogeochemical-cycles/", "lengthSeconds": 177, "width": 640, "height": 360}
{"video_id": "pYVgB2lnztY", "transcript": [{"start": 0.459, "duration": 1.801, "text": "Welcome to Professor Dave Explains."}, {"start": 2.26, "duration": 2.57, "text": "I\u2019m Dave, and this channel is for learning."}, {"start": 4.83, "duration": 2.429, "text": "There are a lot of YouTube channels for learning,\nbut this one is different."}, {"start": 7.259, "duration": 4.55, "text": "I\u2019m not going to dazzle you with science-flavored\nedutainment that\u2019s fun to watch but falls"}, {"start": 11.809, "duration": 1.701, "text": "short at delivering real knowledge."}, {"start": 13.51, "duration": 4.07, "text": "I\u2019m also not going to just stand here and\nlecture at a blackboard for two hours at a time."}, {"start": 17.62, "duration": 1.919, "text": "I want to give you something in between."}, {"start": 19.539, "duration": 5.61, "text": "The shortest, easiest route there has ever\nbeen to a working understanding of everything"}, {"start": 25.149, "duration": 5.18, "text": "around you, from the infinitesimally small\nto the incomprehensibly large."}, {"start": 30.329, "duration": 4.011, "text": "I want to teach you chemistry, so you can\nunderstand atoms and molecules."}, {"start": 34.34, "duration": 5.28, "text": "I want to teach you biochemistry, so you can\nunderstand big molecules, like proteins and DNA."}, {"start": 39.72, "duration": 5.359, "text": "I want to teach you biology, so you can understand\nthe components that make up living organisms."}, {"start": 45.08, "duration": 6.1, "text": "I want to teach you physics, so you can understand the way the universe works on the most fundamental level."}, {"start": 51.24, "duration": 4.02, "text": "I want to teach you astronomy, so you can\nunderstand all the fascinating objects we"}, {"start": 55.269, "duration": 2.121, "text": "can see in this great big universe."}, {"start": 57.39, "duration": 4.21, "text": "I want to teach you mathematics, so you can\nunderstand the important calculations that"}, {"start": 61.6, "duration": 3.37, "text": "scientists have to perform, and reproduce them yourself."}, {"start": 64.97, "duration": 3.63, "text": "And that\u2019s just the tip of the iceberg."}, {"start": 68.6, "duration": 1.97, "text": "But why bother learning all this stuff?"}, {"start": 70.57, "duration": 3.979, "text": "Well for one thing, when you read about science-related\nnews, regarding nutrition, or medicine, or"}, {"start": 74.549, "duration": 2.36, "text": "technology, wouldn\u2019t you like to know what\u2019s\ngoing on?"}, {"start": 76.909, "duration": 4.051, "text": "To make informed decisions for yourself, rather\nthan just repeating what others are saying?"}, {"start": 80.96, "duration": 4.33, "text": "The only way to get there is to learn the\nfundamentals, and this channel is the fastest"}, {"start": 85.29, "duration": 1.02, "text": "way to do it."}, {"start": 86.31, "duration": 4.53, "text": "I cover typical high school and undergraduate\ncurricula, so if you\u2019re a student, stick around."}, {"start": 90.84, "duration": 4.4, "text": "I will explain exactly what you are learning\nin class in ways that will finally make sense."}, {"start": 95.24, "duration": 5.26, "text": "My tutorials are succinct, with animations\ngeared towards maximizing comprehension rather"}, {"start": 100.5, "duration": 4.46, "text": "than amusement, and they stress the most important\ndetails, while skipping all the fluff."}, {"start": 104.96, "duration": 4.75, "text": "It\u2019s like I took your textbook and I\u2019m\ngiving you the cliffs notes version, in easy-to-watch"}, {"start": 109.71, "duration": 1.0, "text": "segments."}, {"start": 110.71, "duration": 4.35, "text": "And the best news is that because I focus\non the main concepts, my content is also great"}, {"start": 115.06, "duration": 4.199, "text": "for anyone who just wants to learn science\nwithout having to go to school or read a lot"}, {"start": 119.259, "duration": 2.761, "text": "of books, and no prior knowledge is assumed."}, {"start": 122.02, "duration": 4.11, "text": "I\u2019ll start you at the very beginning with\ngeneral chemistry, and you can climb the ladder"}, {"start": 126.13, "duration": 1.07, "text": "from there."}, {"start": 127.2, "duration": 4.32, "text": "You may not find any explosions or songs or\nflashy gimmicks along the way, but we don\u2019t"}, {"start": 131.52, "duration": 4.6, "text": "have time for that, we are here to understand\nthe universe, which is pretty fun in its own right."}, {"start": 136.18, "duration": 2.48, "text": "And it doesn\u2019t matter who you are, either."}, {"start": 138.66, "duration": 4.44, "text": "Education is for everyone, and you should\nwant it, because knowledge is cool."}, {"start": 143.1, "duration": 3.94, "text": "With knowledge comes wisdom, and if everyone\ncan get wise enough, we can mold the world"}, {"start": 147.04, "duration": 1.51, "text": "as we see fit."}, {"start": 148.55, "duration": 4.24, "text": "So subscribe to my channel for updates about\nnew content, support me on Patreon so I can"}, {"start": 152.79, "duration": 3.27, "text": "make lots of it as fast as possible, and let\u2019s\nlearn some science."}], "title": "Welcome to Professor Dave Explains!", "description": "Welcome to my YouTube channel! My goal is to provide the best resource for self-education in existence. I've already covered a lot of subjects, but I plan to cover literally every single academic subject in the coming decade or two. My videos have just enough detail for high school and undergraduate students, but are visually engaging and general enough for the common viewer that just wants to learn a few things. Get started today by looking through my playlists!\n\nGeneral Chemistry Tutorials: http://bit.ly/ProfDaveGenChem\nOrganic Chemistry Tutorials: http://bit.ly/ProfDaveOrgChem\nBiochemistry Tutorials: http://bit.ly/ProfDaveBiochem\nBiology Tutorials: http://bit.ly/ProfDaveBio\nClassical Physics Tutorials: http://bit.ly/ProfDavePhysics1\nModern Physics Tutorials: http://bit.ly/ProfDavePhysics2\nMathematics Tutorials: http://bit.ly/ProfDaveMath\nAmerican History Tutorials: http://bit.ly/ProfDaveAmericanHistory\n\nEMAIL\u25ba ProfessorDaveExplains@gmail.com\nPATREON\u25ba http://patreon.com/ProfessorDaveExplains\n\nCheck out \"Is This Wi-Fi Organic?\", my book on disarming pseudoscience!\nAmazon: https://amzn.to/2HtNpVH\nBookshop: https://bit.ly/39cKADM\n\rBarnes and Noble: https://bit.ly/3pUjmrn \nBook Depository: http://bit.ly/3aOVDlT", "lengthSeconds": 156, "width": 640, "height": 360}
{"video_id": "Hiq2ZT7H9tc", "transcript": [{"start": 0.44, "duration": 5.2, "text": "hey it's Professor Dave let's talk about"}, {"start": 3.6, "duration": 4.68, "text": "carbon"}, {"start": 5.64, "duration": 4.879, "text": "science"}, {"start": 8.28, "duration": 4.16, "text": "Dave you may have heard that carbon is"}, {"start": 10.519, "duration": 3.841, "text": "the element of life but what does that"}, {"start": 12.44, "duration": 4.28, "text": "mean what determines whether an atom is"}, {"start": 14.36, "duration": 5.16, "text": "of carbon in the first place an atom"}, {"start": 16.72, "duration": 4.559, "text": "consists of positively charged protons"}, {"start": 19.52, "duration": 4.32, "text": "neutrons which have no charge and"}, {"start": 21.279, "duration": 4.92, "text": "negatively charged electrons an element"}, {"start": 23.84, "duration": 5.359, "text": "is defined by the number of protons in"}, {"start": 26.199, "duration": 6.88, "text": "the nucleus one proton is hydrogen two"}, {"start": 29.199, "duration": 6.88, "text": "is helium three lithium four burum five"}, {"start": 33.079, "duration": 5.201, "text": "Boron and six is carbon the number of"}, {"start": 36.079, "duration": 4.64, "text": "neutrons and electrons can change but by"}, {"start": 38.28, "duration": 5.759, "text": "definition any atom with six protons is"}, {"start": 40.719, "duration": 4.961, "text": "a carbon atom how do they form hydrogen"}, {"start": 44.039, "duration": 3.721, "text": "the simplest and most abundant of all"}, {"start": 45.68, "duration": 4.039, "text": "elements accumulates into clumps due to"}, {"start": 47.76, "duration": 3.6, "text": "gravity which becomes Stars the"}, {"start": 49.719, "duration": 3.801, "text": "tremendous inward pressure triggers"}, {"start": 51.36, "duration": 4.719, "text": "nuclear fusion which is when protons and"}, {"start": 53.52, "duration": 4.4, "text": "neutrons smash together in fuse many"}, {"start": 56.079, "duration": 4.401, "text": "elements including carbon and larger"}, {"start": 57.92, "duration": 4.479, "text": "ones like iron are made in Stars much"}, {"start": 60.48, "duration": 3.88, "text": "larger ones need the immense energy of a"}, {"start": 62.399, "duration": 4.08, "text": "supernova explosion or particle"}, {"start": 64.36, "duration": 3.96, "text": "accelerators here on Earth so what's so"}, {"start": 66.479, "duration": 4.28, "text": "special about carbon let's look at the"}, {"start": 68.32, "duration": 4.119, "text": "electrons carbon atoms have six the"}, {"start": 70.759, "duration": 3.36, "text": "first two are in the inner shell and the"}, {"start": 72.439, "duration": 3.441, "text": "other four are the so-called veence"}, {"start": 74.119, "duration": 3.201, "text": "electrons in the outer shell these are"}, {"start": 75.88, "duration": 3.559, "text": "the ones available for bonding with"}, {"start": 77.32, "duration": 4.88, "text": "other atoms which happens when two atoms"}, {"start": 79.439, "duration": 4.32, "text": "share two electrons because elements in"}, {"start": 82.2, "duration": 3.239, "text": "this part of the periodic table want to"}, {"start": 83.759, "duration": 4.04, "text": "have eight electrons in their outermost"}, {"start": 85.439, "duration": 4.561, "text": "shell carbon has a tendency to make four"}, {"start": 87.799, "duration": 3.841, "text": "bonds these can be single double or or"}, {"start": 90.0, "duration": 3.6, "text": "triple bonds and with many different"}, {"start": 91.64, "duration": 3.88, "text": "elements in addition while an atom"}, {"start": 93.6, "duration": 4.479, "text": "that's bound to two or three other atoms"}, {"start": 95.52, "duration": 4.16, "text": "will adopt a linear or flat shape a"}, {"start": 98.079, "duration": 3.601, "text": "carbon atom that's bound to four other"}, {"start": 99.68, "duration": 3.88, "text": "atoms will take on a three-dimensional"}, {"start": 101.68, "duration": 3.759, "text": "shape because that's how the electron"}, {"start": 103.56, "duration": 4.199, "text": "clouds on the four atoms can be as far"}, {"start": 105.439, "duration": 4.0, "text": "away from each other as possible so it's"}, {"start": 107.759, "duration": 3.601, "text": "the ability of carbon to bond in a wide"}, {"start": 109.439, "duration": 3.72, "text": "variety of shapes with so many elements"}, {"start": 111.36, "duration": 4.359, "text": "including itself that makes it so"}, {"start": 113.159, "duration": 4.56, "text": "special also carbon carbon bonds are"}, {"start": 115.719, "duration": 3.121, "text": "strong enough to be stable but not so"}, {"start": 117.719, "duration": 2.881, "text": "strong that they can't break and"}, {"start": 118.84, "duration": 4.16, "text": "rearrange which makes them excellent"}, {"start": 120.6, "duration": 4.519, "text": "building blocks when by itself carbon"}, {"start": 123.0, "duration": 4.0, "text": "has many allotropes or ways an element"}, {"start": 125.119, "duration": 3.681, "text": "can be arranged there's the graphite in"}, {"start": 127.0, "duration": 3.759, "text": "your pencil which consists of slippery"}, {"start": 128.8, "duration": 3.56, "text": "sheets of carbon that rub off easily"}, {"start": 130.759, "duration": 3.441, "text": "there's Diamond the hardest substance"}, {"start": 132.36, "duration": 4.12, "text": "known and we can even make interesting"}, {"start": 134.2, "duration": 5.2, "text": "structures like nanot tubes and"}, {"start": 136.48, "duration": 5.119, "text": "ferin that's just carbon by itself when"}, {"start": 139.4, "duration": 3.96, "text": "combined with hydrogen oxygen nitrogen"}, {"start": 141.599, "duration": 3.441, "text": "and a few other elements it makes all"}, {"start": 143.36, "duration": 4.68, "text": "the diverse structures in your body"}, {"start": 145.04, "duration": 4.96, "text": "including carbohydrates proteins and DNA"}, {"start": 148.04, "duration": 3.88, "text": "we can take nature a step further and"}, {"start": 150.0, "duration": 4.319, "text": "synthesize novel materials with an array"}, {"start": 151.92, "duration": 4.0, "text": "of purposes putting carbons in the empty"}, {"start": 154.319, "duration": 3.761, "text": "spots of an iron lattice makes it"}, {"start": 155.92, "duration": 3.64, "text": "stronger we call this steel which is"}, {"start": 158.08, "duration": 3.96, "text": "used in most of the large structures we"}, {"start": 159.56, "duration": 4.599, "text": "know today Teflon a carbon Florine"}, {"start": 162.04, "duration": 4.119, "text": "polymer is a non-stick substance that's"}, {"start": 164.159, "duration": 3.761, "text": "highly resistant to chemical degradation"}, {"start": 166.159, "duration": 3.601, "text": "and carbon-based synthetic drugs like"}, {"start": 167.92, "duration": 3.88, "text": "Inhibitors that can silence a faulty"}, {"start": 169.76, "duration": 4.199, "text": "enzyme hold the prospect of curing so"}, {"start": 171.8, "duration": 4.24, "text": "many diseases that's a lot of reasons to"}, {"start": 173.959, "duration": 5.321, "text": "love carbon it really is the element of"}, {"start": 176.04, "duration": 3.24, "text": "Life see you next time"}], "title": "Professor Dave Explains: Carbon \u2013 Dave Farina", "description": "It's been said that carbon is the element of life! Let's take a look at why. Semifinalist for ChemChamps 2015\n\nSubscribe! http://bit.ly/AmerChemSOc \nFacebook! https://www.facebook.com/AmericanChem... \nTwitter! https://twitter.com/ACSpressroom\n\nFor more information on Prized Science, please visit the website: \nhttps://www.acs.org/content/acs/en/funding-and-awards/awards/national/prizedscience.html\n\nYou might also like: \n\nACS Catalysis Lectureship Award Video Series:\nhttps://www.youtube.com/watch?v=TQapgoOJ80M&list=PLLG7h7fPoH8LSS6DsWOzWF0lEPxkou-J6 \n\nHeroes of Chemistry:\nhttps://www.youtube.com/watch?v=dUoRI79AN0Y&list=PLLG7h7fPoH8Jym-hI_gXIheoIos2pKonw \n\nChemistry for Innovation:\nhttps://www.youtube.com/watch?v=unk9d9fqyXk&list=PLLG7h7fPoH8IeB8dlRGTrNBFSBSV0lBUP \n\nMusic: From Audioblocks \n\nProduced by the American Chemical Society, the world\u2019s largest scientific society. ACS is a global leader in providing access to chemistry-related information and research through its multiple databases, peer-reviewed journals and scientific conferences. \n\nJoin the American Chemical Society! https://bit.ly/Join_ACS", "lengthSeconds": 177, "width": 640, "height": 360}
{"video_id": "cSj82GG6MX4", "transcript": [{"start": 0.0, "duration": 6.18, "text": "the rank of a matrix is the number of"}, {"start": 2.94, "duration": 5.579, "text": "linearly independent rows that it has or"}, {"start": 6.18, "duration": 5.16, "text": "equivalently the number of linearly"}, {"start": 8.519, "duration": 5.461, "text": "independent columns the rank of a matrix"}, {"start": 11.34, "duration": 5.76, "text": "is always positive except in the case of"}, {"start": 13.98, "duration": 5.16, "text": "the zero Matrix which has rank 0. let's"}, {"start": 17.1, "duration": 3.78, "text": "go through five examples of calculating"}, {"start": 19.14, "duration": 3.36, "text": "the rank of a matrix we'll do the first"}, {"start": 20.88, "duration": 3.42, "text": "four real quick and we'll be a bit more"}, {"start": 22.5, "duration": 4.68, "text": "thorough with the last one here's a"}, {"start": 24.3, "duration": 5.219, "text": "matrix a it's not the zero Matrix so its"}, {"start": 27.18, "duration": 4.2, "text": "rank is positive generally to find the"}, {"start": 29.519, "duration": 4.921, "text": "rank of a matrix we'll want to put it"}, {"start": 31.38, "duration": 5.76, "text": "into row Echelon form doing this to a"}, {"start": 34.44, "duration": 5.16, "text": "gives us this Matrix and this tells us"}, {"start": 37.14, "duration": 5.36, "text": "the number of linearly independent rows"}, {"start": 39.6, "duration": 6.84, "text": "that a has it's simply the number of"}, {"start": 42.5, "duration": 7.12, "text": "non-zero rows in the row Echelon form in"}, {"start": 46.44, "duration": 6.54, "text": "this case we see that's three so the"}, {"start": 49.62, "duration": 6.06, "text": "rank of a is equal to three similarly"}, {"start": 52.98, "duration": 5.52, "text": "for this Matrix B if we put it into row"}, {"start": 55.68, "duration": 5.28, "text": "Echelon form this is what we get the"}, {"start": 58.5, "duration": 6.36, "text": "number of linearly independent it rows"}, {"start": 60.96, "duration": 6.839, "text": "in B is the number of non-zero rows in"}, {"start": 64.86, "duration": 6.0, "text": "the row Echelon form in this case there"}, {"start": 67.799, "duration": 5.461, "text": "are two non-zero rows so the rank of B"}, {"start": 70.86, "duration": 4.02, "text": "is two some matrices are simple enough"}, {"start": 73.26, "duration": 4.26, "text": "that we don't need to put them in row"}, {"start": 74.88, "duration": 4.8, "text": "Echelon form this Matrix C is not the"}, {"start": 77.52, "duration": 5.279, "text": "zero Matrix so it does have a positive"}, {"start": 79.68, "duration": 6.0, "text": "rank but I notice the two rows are not"}, {"start": 82.799, "duration": 5.761, "text": "independent the second row is simply"}, {"start": 85.68, "duration": 6.9, "text": "equal to negative two times the first"}, {"start": 88.56, "duration": 6.36, "text": "row so C has only one independent row"}, {"start": 92.58, "duration": 5.88, "text": "and its rank which we could also denote"}, {"start": 94.92, "duration": 6.78, "text": "like this is equal to one this Matrix D"}, {"start": 98.46, "duration": 5.82, "text": "has fewer columns than it has rows so it"}, {"start": 101.7, "duration": 4.8, "text": "may be simpler to look at the columns we"}, {"start": 104.28, "duration": 4.68, "text": "notice the first two columns are"}, {"start": 106.5, "duration": 5.1, "text": "linearly independent there's no way to"}, {"start": 108.96, "duration": 6.06, "text": "multiply column one to get column two"}, {"start": 111.6, "duration": 6.479, "text": "however column three is a linear"}, {"start": 115.02, "duration": 5.279, "text": "combination of the first two columns in"}, {"start": 118.079, "duration": 5.58, "text": "fact if we just add column one in in"}, {"start": 120.299, "duration": 6.541, "text": "column 2 that gives us column three so"}, {"start": 123.659, "duration": 5.281, "text": "the Matrix D has only two linearly"}, {"start": 126.84, "duration": 5.28, "text": "independent columns and we would say"}, {"start": 128.94, "duration": 4.92, "text": "that the rank of D is equal to two if"}, {"start": 132.12, "duration": 3.9, "text": "all else fails we can always put the"}, {"start": 133.86, "duration": 4.5, "text": "Matrix into row Echelon form and count"}, {"start": 136.02, "duration": 5.16, "text": "the non-zero rows let's do one more"}, {"start": 138.36, "duration": 5.16, "text": "example of that with this Matrix e we"}, {"start": 141.18, "duration": 5.58, "text": "want to make the entries below each"}, {"start": 143.52, "duration": 6.24, "text": "leading entry equal to zero so below"}, {"start": 146.76, "duration": 6.96, "text": "this two we want only zeros to get that"}, {"start": 149.76, "duration": 6.24, "text": "we'll set row two equal to row two minus"}, {"start": 153.72, "duration": 4.68, "text": "row four so those ones will cancel out"}, {"start": 156.0, "duration": 5.28, "text": "to get rid of this one we will subtract"}, {"start": 158.4, "duration": 5.46, "text": "row two from row four to again cancel"}, {"start": 161.28, "duration": 5.22, "text": "out the ones that gets us here we have"}, {"start": 163.86, "duration": 4.8, "text": "zeros below the two like we want now we"}, {"start": 166.5, "duration": 4.8, "text": "want to get zeros below the next leading"}, {"start": 168.66, "duration": 6.18, "text": "entry of negative one to make this two"}, {"start": 171.3, "duration": 6.36, "text": "equal to zero We'll add two copies of"}, {"start": 174.84, "duration": 5.7, "text": "row two into this third row and that"}, {"start": 177.66, "duration": 4.98, "text": "gets us here now we also want to get rid"}, {"start": 180.54, "duration": 4.32, "text": "of this one that's below the leading"}, {"start": 182.64, "duration": 5.28, "text": "entry of negative one and in order to do"}, {"start": 184.86, "duration": 5.7, "text": "that we'll simply add Row 2 to Row 4"}, {"start": 187.92, "duration": 5.459, "text": "which will eliminate Row 4 completely so"}, {"start": 190.56, "duration": 5.34, "text": "Row 4 is all zeros and we can see we are"}, {"start": 193.379, "duration": 6.36, "text": "now in row Echelon form and there are"}, {"start": 195.9, "duration": 6.3, "text": "three non-zero rows thus the rank of E"}, {"start": 199.739, "duration": 5.301, "text": "equals three thanks for watching let me"}, {"start": 202.2, "duration": 2.84, "text": "know if you have any questions"}], "title": "How to Find the Rank of a Matrix (with echelon form) | Linear Algebra", "description": "The rank of a matrix is the number of linearly independent rows or the number of linearly independent columns the matrix has. These definitions are equivalent. To find this number, we can reduce a matrix to row echelon form and count the nonzero rows, whose leading entries are called pivot numbers. We'll solve five rank of a matrix examples in this lesson. #linearalgebra  \n\nJoin Wrath of Math to get exclusive videos, music, and more:\nhttps://www.youtube.com/channel/UCyEKvaxi8mt9FMc62MHcliw/join\n\nFind Rank of a Matrix by Inspection: https://youtu.be/A-RZ2oOoAwk\nFind Rank of a 3x3 Matrix: https://www.youtube.com/watch?v=bcnZm2paFAM\nFind Rank of a 4x4 Matrix: https://youtu.be/-o8QOexmEWA\nFinding Basis for the Row Space of a Matrix: https://youtu.be/F74TXjoruwg\n\nLinear Algebra course: https://www.youtube.com/playlist?list=PLztBpqftvzxWT5z53AxSqkSaWDhAeToDG\nLinear Algebra exercises: https://www.youtube.com/playlist?list=PLztBpqftvzxVmiiFW7KtPwBpnHNkTVeJc\n\n\u25c9Textbooks I Like\u25c9\nGraph Theory: https://amzn.to/3JHQtZj\nReal Analysis: https://amzn.to/3CMdgjI\nProofs and Set Theory: https://amzn.to/367VBXP (available for free online)\nStatistics: https://amzn.to/3tsaEER\nAbstract Algebra: https://amzn.to/3IjoZaO\nDiscrete Math: https://amzn.to/3qfhoUn\nNumber Theory: https://amzn.to/3JqpOQd\n\n\u2605DONATE\u2605\n\u25c6 Support Wrath of Math on Patreon for early access to new videos and other exclusive benefits: https://www.patreon.com/join/wrathofmathlessons\n\u25c6 Donate on PayPal: https://www.paypal.me/wrathofmath\n\nThanks to Petar, dric, Rolf Waefler,  Robert Rennie, Barbara Sharrock, Joshua Gray, Karl Kristiansen, Katy, Mohamad Nossier, and Shadow Master for their generous support on Patreon!\n\nThanks to Crayon Angel, my favorite musician in the world, who upon my request gave me permission to use his music in my math lessons: https://crayonangel.bandcamp.com/\n\nFollow Wrath of Math on...\n\u25cf Instagram: https://www.instagram.com/wrathofmathedu\n\u25cf Facebook: https://www.facebook.com/WrathofMath\n\u25cf Twitter:  https://twitter.com/wrathofmathedu\n\nMy Math Rap channel: https://www.youtube.com/channel/UCQ2UBhg5nwWCL2aPC7_IpDQ/featured", "lengthSeconds": 204, "width": 640, "height": 360}
{"video_id": "B3S1r2kqPAQ", "transcript": [{"start": 0.16, "duration": 5.52, "text": "okay number two asks convert the"}, {"start": 2.36, "duration": 5.84, "text": "following compound in condensed notation"}, {"start": 5.68, "duration": 3.76, "text": "into the appropriate line notation so"}, {"start": 8.2, "duration": 3.479, "text": "now we're going the other way we're"}, {"start": 9.44, "duration": 4.88, "text": "taking our condensed notation and we're"}, {"start": 11.679, "duration": 5.561, "text": "expanding that into line notation we"}, {"start": 14.32, "duration": 5.68, "text": "start with ch3 so we have a methyl group"}, {"start": 17.24, "duration": 5.48, "text": "and then that is going to connect to co"}, {"start": 20.0, "duration": 5.16, "text": "So Co is our carbonal group so here is"}, {"start": 22.72, "duration": 6.36, "text": "our methyl and then we have our carbonal"}, {"start": 25.16, "duration": 5.959, "text": "next is going to be ch ch looking ahead"}, {"start": 29.08, "duration": 3.639, "text": "a little bit we know that we have a"}, {"start": 31.119, "duration": 4.241, "text": "carbon and then that is going to be"}, {"start": 32.719, "duration": 5.241, "text": "bound to just one hydrogen and then"}, {"start": 35.36, "duration": 5.28, "text": "another carbon that is also bound to"}, {"start": 37.96, "duration": 5.079, "text": "just one hydrogen but if both of these"}, {"start": 40.64, "duration": 3.72, "text": "carbons are just bound to one hydrogen"}, {"start": 43.039, "duration": 2.921, "text": "that means that there must be a double"}, {"start": 44.36, "duration": 4.16, "text": "bond between them because all these"}, {"start": 45.96, "duration": 6.16, "text": "carbons have to have four bonds and so"}, {"start": 48.52, "duration": 5.96, "text": "the way we get ch ch is by making a pi"}, {"start": 52.12, "duration": 5.48, "text": "Bond there between those carbons after"}, {"start": 54.48, "duration": 6.039, "text": "chch we have ch2 so that's just a"}, {"start": 57.6, "duration": 2.919, "text": "regular ch2"}], "title": "Condensed Notation to Line Notation: CH3COCHCHCH2CCH #organicchemistry", "description": "This is a clip from Professor Dave's Structure and Bonding Organic Chemistry Exam. \n\n\ud83d\udcafCheck out Professor Dave's Practice Exams and more OCHEM RESOURCES here: https://chemmunity.info/dave \ud83d\ude4c", "lengthSeconds": 59, "width": 360, "height": 640}
{"video_id": "S2hsaTO0bO8", "transcript": [{"start": 0.0, "duration": 2.61, "text": "Professor Dave here, let's discuss the"}, {"start": 2.61, "duration": 2.03, "text": "zeroth law of thermodynamics."}, {"start": 11.4, "duration": 2.18, "text": "Thermodynamics is a tricky subject that"}, {"start": 13.59, "duration": 2.37, "text": "must be studied in chemistry, physics,"}, {"start": 15.96, "duration": 2.55, "text": "engineering, and many other courses of"}, {"start": 18.51, "duration": 1.95, "text": "academic study, because it has an"}, {"start": 20.46, "duration": 2.46, "text": "incredible impact on the behavior of"}, {"start": 22.92, "duration": 3.54, "text": "matter and energy. At first, thermodynamic"}, {"start": 26.46, "duration": 2.52, "text": "concepts may seem impossible to"}, {"start": 28.98, "duration": 2.49, "text": "understand, but if you go over the basics"}, {"start": 31.47, "duration": 2.249, "text": "enough times they start to make sense"}, {"start": 33.719, "duration": 2.671, "text": "and you can continue to learn more about"}, {"start": 36.39, "duration": 3.06, "text": "the subject as needed for your field of"}, {"start": 39.45, "duration": 2.45, "text": "study or level of desired knowledge."}, {"start": 42.8, "duration": 2.079, "text": "We will begin our analysis with the"}, {"start": 44.879, "duration": 3.301, "text": "infamous laws of thermodynamics. Just"}, {"start": 48.18, "duration": 1.949, "text": "like Newton's laws of motion there is a"}, {"start": 50.129, "duration": 2.881, "text": "first, second, and third law, but later in"}, {"start": 53.01, "duration": 1.95, "text": "the development of the field a more"}, {"start": 54.96, "duration": 2.88, "text": "fundamental law was added and dubbed the"}, {"start": 57.84, "duration": 2.34, "text": "zeroth law, so let's start there."}, {"start": 61.17, "duration": 2.79, "text": "The zeroth law of thermodynamics deals with"}, {"start": 63.96, "duration": 2.61, "text": "thermal equilibrium, which as we recall"}, {"start": 66.57, "duration": 2.67, "text": "describes a situation in which there is"}, {"start": 69.24, "duration": 2.66, "text": "no net heat flow between two systems."}, {"start": 72.44, "duration": 2.38, "text": "The zeroth law states that if two systems"}, {"start": 74.82, "duration": 2.369, "text": "are in thermal equilibrium with a third"}, {"start": 77.189, "duration": 2.761, "text": "system, then those two systems are also"}, {"start": 79.95, "duration": 2.37, "text": "in thermal equilibrium with each other."}, {"start": 82.32, "duration": 2.549, "text": "In other words, if A is in thermal"}, {"start": 84.869, "duration": 2.461, "text": "equilibrium with B which is in thermal"}, {"start": 87.33, "duration": 3.99, "text": "equilibrium with C, A and C are also in"}, {"start": 91.32, "duration": 2.61, "text": "equilibrium. This may seem like a trivial"}, {"start": 93.93, "duration": 2.579, "text": "statement but it is an important one"}, {"start": 96.509, "duration": 2.581, "text": "because it means that two objects at the"}, {"start": 99.09, "duration": 2.73, "text": "same temperature will not exhibit heat"}, {"start": 101.82, "duration": 2.04, "text": "flow when they come into contact with"}, {"start": 103.86, "duration": 3.45, "text": "one another. This marks temperature as"}, {"start": 107.31, "duration": 2.94, "text": "the indicator of thermal equilibrium."}, {"start": 110.25, "duration": 2.46, "text": "If two objects are in thermal equilibrium"}, {"start": 112.71, "duration": 3.39, "text": "with thermometers that read the same"}, {"start": 116.1, "duration": 2.729, "text": "temperature, the objects will be in"}, {"start": 118.829, "duration": 2.191, "text": "thermal equilibrium with each other"}, {"start": 121.02, "duration": 2.76, "text": "which therefore allows us to gather all"}, {"start": 123.78, "duration": 2.699, "text": "kinds of thermodynamic data in our"}, {"start": 126.479, "duration": 3.42, "text": "laboratories. In studying thermodynamics"}, {"start": 129.899, "duration": 3.17, "text": "we must also learn certain definitions."}, {"start": 133.069, "duration": 1.831, "text": "We will want to be able to define a"}, {"start": 134.9, "duration": 2.55, "text": "system as the collection of objects we"}, {"start": 137.45, "duration": 2.22, "text": "are examining while the surroundings"}, {"start": 139.67, "duration": 3.06, "text": "represent the environment surrounding"}, {"start": 142.73, "duration": 2.459, "text": "the system, which is technically the rest"}, {"start": 145.189, "duration": 3.33, "text": "of the universe. Some systems allow heat"}, {"start": 148.519, "duration": 2.97, "text": "to flow in and out. These systems have"}, {"start": 151.489, "duration": 3.661, "text": "diathermal walls. Some systems do not"}, {"start": 155.15, "duration": 2.369, "text": "allow heat to flow in and out. These"}, {"start": 157.519, "duration": 3.661, "text": "systems have adiabatic walls. No system"}, {"start": 161.18, "duration": 2.669, "text": "can be truly adiabatic, but it is a"}, {"start": 163.849, "duration": 2.22, "text": "useful approximation when we do"}, {"start": 166.069, "duration": 2.64, "text": "calculations involving systems that"}, {"start": 168.709, "duration": 2.7, "text": "permit only negligible heat transfer."}, {"start": 171.409, "duration": 3.211, "text": "With these definitions out of the way, we"}, {"start": 174.62, "duration": 2.76, "text": "can move on to the first law of thermodynamics."}, {"start": 178.34, "duration": 2.18, "text": "Thanks for watching, guys. Subscribe to my channel for"}, {"start": 180.53, "duration": 2.429, "text": "more tutorials, support me on patreon so I can"}, {"start": 182.959, "duration": 2.191, "text": "keep making content, and as always feel"}, {"start": 185.15, "duration": 0.85, "text": "free to email me:"}], "title": "The Zeroth Law of Thermodynamics: Thermal Equilibrium", "description": "You've heard of the laws of thermodynamics, but did you know there are actually four of them? It's true, and since they already had laws one, two, and three, but then wanted to formulate another more fundamental one, the number right before one is zero so that's what they called it! Also, \"zeroth\" is fun to say. Let's say it together.\n\nWatch the whole Classical Physics playlist: http://bit.ly/ProfDavePhysics1\n\nModern Physics Tutorials: http://bit.ly/ProfDavePhysics2\nMathematics Tutorials: http://bit.ly/ProfDaveMaths\nGeneral Chemistry Tutorials: http://bit.ly/ProfDaveGenChem\nOrganic Chemistry Tutorials: http://bit.ly/ProfDaveOrgChem\nBiochemistry Tutorials: http://bit.ly/ProfDaveBiochem\nBiology Tutorials: http://bit.ly/ProfDaveBio\n\nEMAIL\u25ba ProfessorDaveExplains@gmail.com\nPATREON\u25ba http://patreon.com/ProfessorDaveExplains\n\nCheck out \"Is This Wi-Fi Organic?\", my book on disarming pseudoscience!\nAmazon: https://amzn.to/2HtNpVH\nBookshop: https://bit.ly/39cKADM\n\rBarnes and Noble: https://bit.ly/3pUjmrn \nBook Depository: http://bit.ly/3aOVDlT", "lengthSeconds": 209, "width": 640, "height": 360}
{"video_id": "w_ec9zYiEBU", "transcript": [{"start": 0.0, "duration": 9.38, "text": "[Music]"}, {"start": 9.92, "duration": 5.0, "text": "Professor Dave and CH here now that we"}, {"start": 12.0, "duration": 5.72, "text": "know what infrared spectroscopy is and"}, {"start": 14.92, "duration": 5.24, "text": "what simple IR Spectra depict it's time"}, {"start": 17.72, "duration": 4.479, "text": "to get more thorough and see how a wide"}, {"start": 20.16, "duration": 5.64, "text": "variety of functional groups will show"}, {"start": 22.199, "duration": 5.84, "text": "up on an IR spectrum first let's recall"}, {"start": 25.8, "duration": 4.639, "text": "what we are looking at the horizontal"}, {"start": 28.039, "duration": 4.921, "text": "axis tells us the wave number which is a"}, {"start": 30.439, "duration": 4.44, "text": "way of expressing frequency in inverse"}, {"start": 32.96, "duration": 3.759, "text": "centimeters so all the different"}, {"start": 34.879, "duration": 3.761, "text": "wavelengths of infrared light are"}, {"start": 36.719, "duration": 4.761, "text": "represented at different places on the"}, {"start": 38.64, "duration": 5.12, "text": "horizontal axis and the vertical axis"}, {"start": 41.48, "duration": 4.8, "text": "tells us the present transmittance at"}, {"start": 43.76, "duration": 4.799, "text": "that wave number if the transmittance is"}, {"start": 46.28, "duration": 4.16, "text": "very high almost all of that light got"}, {"start": 48.559, "duration": 3.921, "text": "through so almost none of it was"}, {"start": 50.44, "duration": 4.24, "text": "absorbed by the sample if the"}, {"start": 52.48, "duration": 4.399, "text": "transmittance is very low then very"}, {"start": 54.68, "duration": 4.6, "text": "little of it got through so most of it"}, {"start": 56.879, "duration": 4.721, "text": "was absorbed by the sample that is what"}, {"start": 59.28, "duration": 4.959, "text": "we see with these Peaks that dip down in"}, {"start": 61.6, "duration": 4.16, "text": "a pronounced way wherever we see these"}, {"start": 64.239, "duration": 3.321, "text": "Peaks we can be certain that the"}, {"start": 65.76, "duration": 4.52, "text": "corresponding functional group is"}, {"start": 67.56, "duration": 5.08, "text": "present in the molecule there are Peaks"}, {"start": 70.28, "duration": 4.8, "text": "that we will see over and over again so"}, {"start": 72.64, "duration": 4.839, "text": "let's simply list what they are give the"}, {"start": 75.08, "duration": 5.12, "text": "associated wave numbers and discuss"}, {"start": 77.479, "duration": 4.96, "text": "other characteristics when necessary for"}, {"start": 80.2, "duration": 3.919, "text": "alkanes an IR spectrum isn't too"}, {"start": 82.439, "duration": 5.36, "text": "interesting since all we have is the"}, {"start": 84.119, "duration": 6.401, "text": "saturated CH stretch and the CC stretch"}, {"start": 87.799, "duration": 5.041, "text": "most organic compounds have these bonds"}, {"start": 90.52, "duration": 5.32, "text": "so these Peaks will show up almost all"}, {"start": 92.84, "duration": 5.08, "text": "the time for alkenes we see stretches"}, {"start": 95.84, "duration": 5.0, "text": "for the vinyl proton and the"}, {"start": 97.92, "duration": 5.879, "text": "carboncarbon double bond there is also"}, {"start": 100.84, "duration": 4.919, "text": "outof plane bending for the vinyl proton"}, {"start": 103.799, "duration": 4.881, "text": "which helps determine the substitution"}, {"start": 105.759, "duration": 5.121, "text": "pattern with alkin we get a stretch for"}, {"start": 108.68, "duration": 5.039, "text": "the triple bond and the intensity tells"}, {"start": 110.88, "duration": 5.08, "text": "us whether this is terminal or internal"}, {"start": 113.719, "duration": 5.121, "text": "then for aromatic compounds we see"}, {"start": 115.96, "duration": 5.0, "text": "aromatic CH stretching there is also"}, {"start": 118.84, "duration": 2.919, "text": "data for the carbon carbon bonds in the"}, {"start": 120.96, "duration": 3.4, "text": "ring"}, {"start": 121.759, "duration": 5.561, "text": "itself moving on to functional groups"}, {"start": 124.36, "duration": 5.959, "text": "with hetero atoms there is the hydroxy"}, {"start": 127.32, "duration": 5.6, "text": "stretch this is perhaps the easiest to"}, {"start": 130.319, "duration": 4.801, "text": "spot of any functional group as the peak"}, {"start": 132.92, "duration": 5.12, "text": "will be very Broad and rounded which is"}, {"start": 135.12, "duration": 5.28, "text": "due to hydrogen bonding if you see this"}, {"start": 138.04, "duration": 5.68, "text": "and you can't miss it there is a hydroxy"}, {"start": 140.4, "duration": 5.68, "text": "group present amines have an NH stretch"}, {"start": 143.72, "duration": 4.44, "text": "that is easy to spot as well as it is"}, {"start": 146.08, "duration": 4.72, "text": "much sharper and less intense than the"}, {"start": 148.16, "duration": 4.799, "text": "hydroxy stretch car carbonal containing"}, {"start": 150.8, "duration": 4.36, "text": "compounds are also easy to spot as the"}, {"start": 152.959, "duration": 4.601, "text": "carbonal stretch is unmistakably long"}, {"start": 155.16, "duration": 4.88, "text": "and narrow the precise position of this"}, {"start": 157.56, "duration": 5.72, "text": "stretch will help us determine what kind"}, {"start": 160.04, "duration": 7.6, "text": "of carbonal it is whether aldah Ketone"}, {"start": 163.28, "duration": 6.84, "text": "carboxilic acid Esther amid or"}, {"start": 167.64, "duration": 4.36, "text": "otherwise once again every functional"}, {"start": 170.12, "duration": 4.119, "text": "group will have its own characteristic"}, {"start": 172.0, "duration": 4.879, "text": "Peak and we can refer to a table like"}, {"start": 174.239, "duration": 5.08, "text": "this to identify them we will never have"}, {"start": 176.879, "duration": 4.521, "text": "to memorize this information but the"}, {"start": 179.319, "duration": 4.84, "text": "characteris istic shapes of some of the"}, {"start": 181.4, "duration": 4.479, "text": "Peaks will be easy to remember all of"}, {"start": 184.159, "duration": 4.08, "text": "this allows us to figure out which"}, {"start": 185.879, "duration": 4.681, "text": "functional groups are on the molecule as"}, {"start": 188.239, "duration": 4.681, "text": "well as which functional groups are not"}, {"start": 190.56, "duration": 5.0, "text": "on the molecule if their peaks are not"}, {"start": 192.92, "duration": 5.64, "text": "present Professor Dave for see you next"}, {"start": 195.56, "duration": 3.0, "text": "time"}], "title": "Infrared Spectra of Common Functional Groups Ft. Professor Dave", "description": "Now that we've covered infrared spectroscopy, let's take a look at the IR spectra of some common functional groups. We'll revisit how to interpret IR spectra, including wavenumbers and transmittance, and then we'll take a look at the IR spectra of functional groups like alkanes, alkenes, alkynes, aromatics, amines, and more.\n\nGet more homework help from Chegg at https://che.gg/3HbtG8Y\nWatch the whole Organic Chemistry playlist: https://www.youtube.com/playlist?list=PLllCkartRRdMPnATf1X8qVdo5RzM8gA_M\n\nAbout Chegg:\nChegg is a leading education technology company that provides students with affordable and accessible study resources. Chegg's mission is to help students save time, save money and get smarter.  \n\nSubscribe: https://www.youtube.com/@chegg\nTikTok: https://www.tiktok.com/@chegg\nInstagram: https://www.instagram.com/chegg/\nTwitter: https://twitter.com/chegg\n\n00:00 Intro\n00:24 Defining the axes\n01:34 Alkyl FGs\n02:22 Amines", "lengthSeconds": 206, "width": 640, "height": 360}
{"video_id": "tURzBx_i7og", "transcript": [{"start": 0.04, "duration": 3.74, "text": "[Music]"}, {"start": 0.78, "duration": 3.0, "text": "foreign"}, {"start": 5.1, "duration": 4.26, "text": "Professor Dave and Chegg here we know"}, {"start": 7.2, "duration": 4.26, "text": "that spectroscopy is the study of the"}, {"start": 9.36, "duration": 3.96, "text": "interaction of light and matter we can"}, {"start": 11.46, "duration": 4.62, "text": "use this interaction to gather"}, {"start": 13.32, "duration": 5.719, "text": "information regarding the structure of a"}, {"start": 16.08, "duration": 5.58, "text": "molecule irradiating substances with"}, {"start": 19.039, "duration": 4.9, "text": "electromagnetic radiation of specific"}, {"start": 21.66, "duration": 5.879, "text": "wavelengths and observing the"}, {"start": 23.939, "duration": 5.58, "text": "interaction IR spectroscopy is one such"}, {"start": 27.539, "duration": 2.881, "text": "technique so let's learn how this works"}, {"start": 29.519, "duration": 4.38, "text": "now"}, {"start": 30.42, "duration": 6.06, "text": "the IR in IR spectroscopy stands for"}, {"start": 33.899, "duration": 4.741, "text": "infrared so as one might guess this"}, {"start": 36.48, "duration": 4.739, "text": "technique involves irradiating a sample"}, {"start": 38.64, "duration": 4.439, "text": "with infrared radiation this is the"}, {"start": 41.219, "duration": 4.381, "text": "region of the electromagnetic spectrum"}, {"start": 43.079, "duration": 5.041, "text": "shown here the next section lower in"}, {"start": 45.6, "duration": 4.92, "text": "energy after visible light the reason we"}, {"start": 48.12, "duration": 4.98, "text": "do this is that infrared light is able"}, {"start": 50.52, "duration": 4.859, "text": "to induce specific types of motion in"}, {"start": 53.1, "duration": 4.26, "text": "the molecules in the sample we already"}, {"start": 55.379, "duration": 4.101, "text": "know that molecules are involved in"}, {"start": 57.36, "duration": 4.859, "text": "translational and rotational motion"}, {"start": 59.48, "duration": 4.6, "text": "translational motion allows molecules to"}, {"start": 62.219, "duration": 4.141, "text": "move around in space with respect to"}, {"start": 64.08, "duration": 4.44, "text": "each other and rotational motion allows"}, {"start": 66.36, "duration": 4.619, "text": "for all the sigma Bonds in the molecule"}, {"start": 68.52, "duration": 4.919, "text": "to be rotating all the time producing"}, {"start": 70.979, "duration": 4.921, "text": "the various conformations but chemical"}, {"start": 73.439, "duration": 4.761, "text": "bonds are also doing other things as"}, {"start": 75.9, "duration": 5.219, "text": "they can exhibit vibrational motion"}, {"start": 78.2, "duration": 5.739, "text": "bonds can stretch which means they can"}, {"start": 81.119, "duration": 5.581, "text": "expand and contract slightly this can be"}, {"start": 83.939, "duration": 5.461, "text": "done in a symmetric way with respect to"}, {"start": 86.7, "duration": 5.52, "text": "a central atom which looks like this or"}, {"start": 89.4, "duration": 5.579, "text": "in asymmetric fashion we call this"}, {"start": 92.22, "duration": 5.52, "text": "symmetric stretch and asymmetric stretch"}, {"start": 94.979, "duration": 5.341, "text": "we can also get bending where atoms will"}, {"start": 97.74, "duration": 5.22, "text": "twist a bit to expand and contract the"}, {"start": 100.32, "duration": 4.92, "text": "bond angles again this can be done in"}, {"start": 102.96, "duration": 4.92, "text": "symmetric or asymmetric fashion to"}, {"start": 105.24, "duration": 5.339, "text": "produce what we call symmetric Bend and"}, {"start": 107.88, "duration": 5.699, "text": "asymmetric Bend all of these vibrational"}, {"start": 110.579, "duration": 4.801, "text": "modes are induced by infrared light and"}, {"start": 113.579, "duration": 3.961, "text": "the specific frequency that will cause"}, {"start": 115.38, "duration": 4.919, "text": "these behaviors will depend on the"}, {"start": 117.54, "duration": 5.96, "text": "precise identity of the bond so carbon"}, {"start": 120.299, "duration": 6.241, "text": "carbon carbon oxygen carbon hydrogen"}, {"start": 123.5, "duration": 5.44, "text": "oxygen hydrogen these will all absorb"}, {"start": 126.54, "duration": 5.279, "text": "infrared light of differing wavelengths"}, {"start": 128.94, "duration": 4.56, "text": "so when light of varying wavelengths"}, {"start": 131.819, "duration": 3.661, "text": "passes through the sample the"}, {"start": 133.5, "duration": 4.02, "text": "wavelengths that don't make it to the"}, {"start": 135.48, "duration": 4.32, "text": "detector must have been absorbed by the"}, {"start": 137.52, "duration": 4.02, "text": "sample and this tells us what kinds of"}, {"start": 139.8, "duration": 3.96, "text": "functional groups are present in the"}, {"start": 141.54, "duration": 5.16, "text": "molecule the way we will be reporting"}, {"start": 143.76, "duration": 5.16, "text": "the wavelengths of ir radiation is using"}, {"start": 146.7, "duration": 5.34, "text": "their wave numbers wave number"}, {"start": 148.92, "duration": 4.92, "text": "represented by new with a bar over it is"}, {"start": 152.04, "duration": 4.38, "text": "the reciprocal of wavelength in"}, {"start": 153.84, "duration": 4.14, "text": "centimeters so it is expressed in"}, {"start": 156.42, "duration": 4.38, "text": "inverse centimeters"}, {"start": 157.98, "duration": 5.7, "text": "the useful region in an IR spectrum is"}, {"start": 160.8, "duration": 6.12, "text": "from 4000 to 400 inverse centimeters"}, {"start": 163.68, "duration": 5.22, "text": "meaning that 400 to 4000 Cycles fit in"}, {"start": 166.92, "duration": 4.5, "text": "one centimeter"}, {"start": 168.9, "duration": 5.339, "text": "the main thing to understand is that"}, {"start": 171.42, "duration": 5.88, "text": "with ir spectroscopy we are irradiating"}, {"start": 174.239, "duration": 4.801, "text": "a sample with infrared radiation seeing"}, {"start": 177.3, "duration": 3.96, "text": "which wavelengths pass through the"}, {"start": 179.04, "duration": 4.32, "text": "sample to the detector and which do not"}, {"start": 181.26, "duration": 4.86, "text": "and taking those that are not"}, {"start": 183.36, "duration": 4.56, "text": "transmitted as inducing some kind of"}, {"start": 186.12, "duration": 3.42, "text": "molecular motion which will be"}, {"start": 187.92, "duration": 4.26, "text": "characteristic of the missing"}, {"start": 189.54, "duration": 4.919, "text": "frequencies this tells us about the"}, {"start": 192.18, "duration": 4.559, "text": "functional groups present on a molecule"}, {"start": 194.459, "duration": 4.461, "text": "Professor Dave for check see you next"}, {"start": 196.739, "duration": 2.181, "text": "time"}, {"start": 200.22, "duration": 2.84, "text": "foreign"}], "title": "Introduction to Infrared Spectroscopy Ft. Professor Dave", "description": "Professor Dave walks us through the interpretation of infrared spectra. We'll learn how to use spectroscopy \u2014 the study of the interaction of light and matter \u2014 to gather information about the structure of a molecule. Specifically, we'll look at the technique of infrared (IR) spectroscopy, how it relates to vibrational motion, and how IR spectroscopy can reveal wavenumbers that will tell us more information about a molecule.\n\nGet more homework help from Chegg at https://che.gg/3HbtG8Y\nWatch the whole Organic Chemistry playlist: https://www.youtube.com/playlist?list=PLllCkartRRdMPnATf1X8qVdo5RzM8gA_M\n\nAbout Chegg:\nChegg is a leading education technology company that provides students with affordable and accessible study resources. Chegg's mission is to help students save time, save money and get smarter.  \n\nSubscribe: https://www.youtube.com/@chegg\nTikTok: https://www.tiktok.com/@chegg\nInstagram: https://www.instagram.com/chegg/\nTwitter: https://twitter.com/chegg\n\n00:00 Intro\n00:09 Spectroscopy\n00:30 Infrared light and chemical bonds\n01:12 Types of vibrational motion\n02:22 Wavenumbers", "lengthSeconds": 207, "width": 640, "height": 360}
{"video_id": "3MdaUCGiOHE", "transcript": [{"start": 0.5, "duration": 3.62, "text": "hey everyone, it's professor Dave, I want to tell you some things about ozonolysis."}, {"start": 11.06, "duration": 3.96, "text": "ozonolysis is a reaction that utilizes ozone as a reagent"}, {"start": 15.02, "duration": 5.52, "text": "ozone is a triatomic allotrope of oxygen that exists as a zwitterion"}, {"start": 20.54, "duration": 3.08, "text": "that's ozone, that's the stuff up in the atmosphere"}, {"start": 25.84, "duration": 7.34, "text": "ozonolysis utilizes an alkene substrate and what happens is that this double bond"}, {"start": 33.18, "duration": 6.26, "text": "gets cleaved and pulled apart, and each of the carbons that was participating in the double bond"}, {"start": 40.56, "duration": 2.04, "text": "is now a carbonyl."}, {"start": 43.48, "duration": 5.42, "text": "so this is useful for transforming alkenes into ketones and aldehydes."}, {"start": 49.512, "duration": 6.4, "text": "we won't look at the mechanism here, but that is available on the internet for you to take a look at"}, {"start": 55.912, "duration": 2.171, "text": "it's a little bit interesting to look at"}, {"start": 59.6, "duration": 5.84, "text": "the reason this is a very useful reaction is that we can do some things like transforming"}, {"start": 65.64, "duration": 6.0, "text": "terminal alkenes into aldehydes which is something that can be very useful for synthesis"}, {"start": 71.64, "duration": 5.32, "text": "so basically any time you are looking at an ozonolysis reaction, remember that we are cleaving"}, {"start": 77.22, "duration": 5.08, "text": "that double bond and pulling apart the fragments and on each of the carbons that was participating"}, {"start": 82.3, "duration": 5.06, "text": "in the double bond, we will place an oxygen atom, and it will be in the form of a carbonyl"}, {"start": 87.36, "duration": 3.54, "text": "here we are generating this aldehyde and formaldehyde"}, {"start": 90.9, "duration": 5.36, "text": "which can easily be separated, we've changed the functionality of a part of this molecule"}, {"start": 96.26, "duration": 2.4, "text": "which may be useful for the next step in a synthesis"}, {"start": 100.58, "duration": 5.04, "text": "the other thing that's interesting about ozonolysis is if you perform ozonolysis on a cyclic alkene"}, {"start": 105.62, "duration": 6.66, "text": "you won't then get two molecules, you will simply pull apart that ring and get one linear product"}, {"start": 113.86, "duration": 5.04, "text": "this is useful to number sometimes when we have cyclic structures like this"}, {"start": 120.52, "duration": 2.32, "text": "we want to take these seven carbons"}, {"start": 125.771, "duration": 5.042, "text": "it's very easy to draw an incorrect product if we don't do something like numbering the carbons"}, {"start": 131.58, "duration": 6.2, "text": "what this is going to do for us is show that if we cleave this double bond and pull that molecule apart "}, {"start": 137.78, "duration": 7.893, "text": "we know that the resulting carbonyls when we draw the linear product must be on carbons 2 and 7"}, {"start": 145.673, "duration": 4.133, "text": "because those were the carbons that were participating in the double bond"}, {"start": 149.806, "duration": 3.2, "text": "that's just a trick to help you draw the correct ozonolysis product."}, {"start": 153.76, "duration": 2.918, "text": "thanks for watching, guys. subscribe to my channel for more tutorials"}, {"start": 157.2, "duration": 2.14, "text": "and as always feel free to email me with questions "}], "title": "Ozonolysis", "description": "Ozonolysis? Ozone is in the sky, right? That thing that shields us from UV light? That's right friend, but ozone is also used in the lab to cleave carbon-carbon double bonds. Check it out if you don't believe me. \n\nWatch the whole Organic Chemistry playlist: http://bit.ly/ProfDaveOrgChem\n\nGeneral Chemistry Tutorials: http://bit.ly/ProfDaveGenChem\nBiochemistry Tutorials: http://bit.ly/ProfDaveBiochem\nBiology Tutorials: http://bit.ly/ProfDaveBio\nClassical Physics Tutorials: http://bit.ly/ProfDavePhysics1\nModern Physics Tutorials: http://bit.ly/ProfDavePhysics2\nMathematics Tutorials: http://bit.ly/ProfDaveMath\n\nEMAIL\u25ba ProfessorDaveExplains@gmail.com\nPATREON\u25ba http://patreon.com/ProfessorDaveExplains\n\nCheck out \"Is This Wi-Fi Organic?\", my book on disarming pseudoscience!\nAmazon: https://amzn.to/2HtNpVH\nBookshop: https://bit.ly/39cKADM\n\rBarnes and Noble: https://bit.ly/3pUjmrn \nBook Depository: http://bit.ly/3aOVDlT", "lengthSeconds": 177, "width": 640, "height": 360}
{"video_id": "3OKIZlGZZkg", "transcript": [{"start": 0.0, "duration": 4.08, "text": "my girl is meant in equation is"}, {"start": 2.159, "duration": 3.961, "text": "mathematical model that describes the"}, {"start": 4.08, "duration": 4.019, "text": "rate at which an enzyme catalyzes a"}, {"start": 6.12, "duration": 4.2, "text": "chemical reaction it is based on the"}, {"start": 8.099, "duration": 4.381, "text": "idea that enzymes work by binding to"}, {"start": 10.32, "duration": 4.439, "text": "their substrate the chemical that they"}, {"start": 12.48, "duration": 4.559, "text": "are acting on and then converting it"}, {"start": 14.759, "duration": 4.381, "text": "into a product as the amount of"}, {"start": 17.039, "duration": 4.381, "text": "substrate decreases with time the amount"}, {"start": 19.14, "duration": 6.36, "text": "of product increases with time here you"}, {"start": 21.42, "duration": 6.359, "text": "can see the equation where V is the rate"}, {"start": 25.5, "duration": 4.5, "text": "of the reaction as measured by the"}, {"start": 27.779, "duration": 4.141, "text": "formation of product V Max is the"}, {"start": 30.0, "duration": 3.66, "text": "maximum velocity of the reaction which"}, {"start": 31.92, "duration": 4.86, "text": "is the rate of the reaction when the"}, {"start": 33.66, "duration": 6.239, "text": "enzyme is fully saturated with substrate"}, {"start": 36.78, "duration": 5.939, "text": "s is the concentration of substrate and"}, {"start": 39.899, "duration": 4.561, "text": "km is the Michael Ellis constant which"}, {"start": 42.719, "duration": 4.621, "text": "is a measure of the Affinity of the"}, {"start": 44.46, "duration": 4.8, "text": "enzyme for the substrate it is equal to"}, {"start": 47.34, "duration": 4.32, "text": "the substrate concentration at which the"}, {"start": 49.26, "duration": 3.6, "text": "enzyme is working at half of its maximum"}, {"start": 51.66, "duration": 3.419, "text": "velocity"}, {"start": 52.86, "duration": 5.519, "text": "the equation describes how the rate of"}, {"start": 55.079, "duration": 7.441, "text": "Direction B changes as the substrate"}, {"start": 58.379, "duration": 6.541, "text": "concentration s changes at low substrate"}, {"start": 62.52, "duration": 4.2, "text": "concentrations the rate of direction is"}, {"start": 64.92, "duration": 3.72, "text": "relatively slow because there are not"}, {"start": 66.72, "duration": 4.38, "text": "enough substrate molecules present for"}, {"start": 68.64, "duration": 4.56, "text": "the enzyme to work on as the substrate"}, {"start": 71.1, "duration": 5.04, "text": "concentration increases the rate of the"}, {"start": 73.2, "duration": 5.58, "text": "reaction increases until it reaches a"}, {"start": 76.14, "duration": 4.56, "text": "maximum V Max Beyond this point"}, {"start": 78.78, "duration": 4.26, "text": "increasing the substrate concentration"}, {"start": 80.7, "duration": 4.02, "text": "does not increase the rate of the"}, {"start": 83.04, "duration": 4.439, "text": "reaction because the enzyme is already"}, {"start": 84.72, "duration": 5.46, "text": "working at its maximum capacity the"}, {"start": 87.479, "duration": 4.561, "text": "Michael is constant km represents the"}, {"start": 90.18, "duration": 4.259, "text": "substrate concentration at which the"}, {"start": 92.04, "duration": 5.16, "text": "enzyme is working at half of its maximum"}, {"start": 94.439, "duration": 4.621, "text": "velocity if km is large it means that"}, {"start": 97.2, "duration": 3.779, "text": "the enzyme has a low affinity for the"}, {"start": 99.06, "duration": 3.9, "text": "substrate and requires a higher"}, {"start": 100.979, "duration": 4.801, "text": "substrate concentration to reach its"}, {"start": 102.96, "duration": 5.159, "text": "maximum velocity if km is small it means"}, {"start": 105.78, "duration": 4.32, "text": "that the enzyme has a high affinity for"}, {"start": 108.119, "duration": 3.721, "text": "the substrate and can reach its maximum"}, {"start": 110.1, "duration": 3.9, "text": "velocity at a lower substrate"}, {"start": 111.84, "duration": 4.559, "text": "concentration the the michaelisment"}, {"start": 114.0, "duration": 4.259, "text": "equation is useful for understanding how"}, {"start": 116.399, "duration": 3.841, "text": "enzymes work and how they can be"}, {"start": 118.259, "duration": 4.32, "text": "affected by changes in substrate"}, {"start": 120.24, "duration": 3.72, "text": "concentration and other factors it is"}, {"start": 122.579, "duration": 3.54, "text": "often used in Biochemistry and"}, {"start": 123.96, "duration": 5.78, "text": "pharmacology to study the kinetics of"}, {"start": 126.119, "duration": 3.621, "text": "enzyme catalyzed reactions"}], "title": "Michaelis Menten Equation Explained For Beginners", "description": "The Michaelis-Menten equation is a mathematical model that describes the rate at which an enzyme catalyzes a chemical reaction. It is based on the idea that enzymes work by binding to their substrate (the chemical that they are acting on) and then converting it into a product. As the amount of substrate decreases with time, the amount of product increases with time.\n\nThe equation is written as:\n\nv = Vmax[S]/(Km + [S])\n\nWhere:\n\n- v is the rate of the reaction (as measured by the formation of product)\n- Vmax is the maximum velocity of the reaction, which is the rate of the reaction when the enzyme is fully saturated with substrate\n- [S] is the concentration of substrate\n- Km is the Michaelis constant, which is a measure of the affinity of the enzyme for the substrate. It is equal to the substrate concentration at which the enzyme is working at half of its maximum velocity.\n\nThe equation describes how the rate of the reaction (v) changes as the substrate concentration ([S]) changes. At low substrate concentrations, the rate of the reaction is relatively slow because there are not enough substrate molecules present for the enzyme to work on. As the substrate concentration increases, the rate of the reaction increases until it reaches a maximum (Vmax). Beyond this point, increasing the substrate concentration does not increase the rate of the reaction because the enzyme is already working at its maximum capacity.\n\nThe Michaelis constant (Km) represents the substrate concentration at which the enzyme is working at half of its maximum velocity. If Km is large, it means that the enzyme has a low affinity for the substrate and requires a higher substrate concentration to reach its maximum velocity. If Km is small, it means that the enzyme has a high affinity for the substrate and can reach its maximum velocity at a lower substrate concentration.\n\nThe Michaelis-Menten equation is useful for understanding how enzymes work and how they can be affected by changes in substrate concentration and other factors. It is often used in biochemistry and pharmacology to study the kinetics of enzyme-catalyzed reactions.", "lengthSeconds": 131, "width": 640, "height": 360}
{"video_id": "M_YKG6atlGw", "transcript": [{"start": 6.8, "duration": 4.1, "text": "ok guys for today's problem we've got\nthis epoxide and we are going to react"}, {"start": 10.9, "duration": 4.46, "text": "with methoxide in methanol, we're going\nto get this product so what we need to"}, {"start": 15.36, "duration": 3.89, "text": "do is show the mechanism. if this\nconfusing you will want to take a look at"}, {"start": 19.25, "duration": 3.0, "text": "my lecture on epoxides and then give\nthis a shot."}, {"start": 36.82, "duration": 3.4, "text": "ok guys so to draw this mechanism we\nhave to recall a couple things about"}, {"start": 40.23, "duration": 4.099, "text": "epoxides. number one: eooxides are very\nsusceptible to nucleophilic attack"}, {"start": 44.329, "duration": 4.371, "text": "because there's a lot of ring stream\nfrom that small three-membered ring, that"}, {"start": 48.7, "duration": 3.43, "text": "means that either of these carbons could\npotentially be attacked by a nucleophile"}, {"start": 52.13, "duration": 3.79, "text": "so we do have a nucleophile, we've got\nmethoxide, that's gonna be a good"}, {"start": 55.92, "duration": 3.95, "text": "nucleophile, so we want to ask ourselves:\nwhich carbon is it gonna attack in the"}, {"start": 59.87, "duration": 4.419, "text": "epoxide? now sterics are going to\ndictate that it is going to go over here"}, {"start": 64.289, "duration": 5.341, "text": "it is the less sterically hindered side so\nit is more available for nucleophilic"}, {"start": 69.63, "duration": 4.779, "text": "attack, so if it attacks right there, it's\ngonna pop open the epoxide, we're"}, {"start": 74.409, "duration": 4.82, "text": "gonna get this here, so this is the\nmethoxide that attacked, this used to be"}, {"start": 79.229, "duration": 4.131, "text": "the epoxide here but now that popped open\nto leave the oxyanion"}, {"start": 83.36, "duration": 4.45, "text": "there and now the second step here is\nkinda like the second step of Williamson"}, {"start": 87.81, "duration": 4.75, "text": "ether synthesis, except that it's\nintramolecular because this oxyanion"}, {"start": 92.56, "duration": 5.76, "text": "can do an SN2 right here to kick off\nthis bromide and that'll lead us to the"}, {"start": 98.32, "duration": 5.29, "text": "product. and so what we can do just to\nmake sure we're able to draw that"}, {"start": 103.61, "duration": 5.01, "text": "properly, we can see that this is gonna\ncreate a five-membered ring because this"}, {"start": 108.62, "duration": 4.41, "text": "oxyanion is going to attack this\ncarbon so that generates this"}, {"start": 113.03, "duration": 7.61, "text": "five-membered ring, and then this group\nover here will still be there so there"}, {"start": 120.64, "duration": 6.41, "text": "we go we have an opening of the epoxide\nfollowed by intramolecular ring closure."}, {"start": 149.44, "duration": 4.44, "text": "thanks for watching, guys, subscribe to my channel for more tutorials and as always feel free to email me"}], "title": "Practice Problem: Mechanism - Reaction of an Epoxide", "description": "For this one we will have to know what epoxides do! It's a pretty neat little mechanism.\n\nTry all of the organic chemistry practice problems: http://bit.ly/ProfDaveOCPP\n\nOrganic Chemistry Tutorials: http://bit.ly/ProfDaveOrgChem\n\nEMAIL\u25ba ProfessorDaveExplains@gmail.com\nPATREON\u25ba http://patreon.com/ProfessorDaveExplains\n\nCheck out \"Is This Wi-Fi Organic?\", my book on disarming pseudoscience!\nAmazon: https://amzn.to/2HtNpVH\nBookshop: https://bit.ly/39cKADM\n\rBarnes and Noble: https://bit.ly/3pUjmrn \nBook Depository: http://bit.ly/3aOVDlT", "lengthSeconds": 157, "width": 640, "height": 360}
{"video_id": "T_wdWyCog98", "transcript": [{"start": 0.3, "duration": 2.96, "text": "It\u2019s Professor Dave, I want to tell you\nabout superlatives."}, {"start": 10.26, "duration": 5.11, "text": "There are certain things that the Italian\nlanguage does that are fun and unique, like"}, {"start": 15.37, "duration": 2.799, "text": "the altered words we learned before."}, {"start": 18.169, "duration": 4.26, "text": "Another such construct is the way Italian\nuses superlatives."}, {"start": 22.429, "duration": 4.26, "text": "It often does this by adding a suffix to a\nparticular adjective."}, {"start": 26.689, "duration": 4.541, "text": "But before we see these, let\u2019s distinguish\nbetween two types of superlatives."}, {"start": 31.23, "duration": 4.64, "text": "There is the absolute superlative, and the\nrelative superlative."}, {"start": 35.87, "duration": 3.95, "text": "Let\u2019s see an example of each."}, {"start": 52.52, "duration": 3.28, "text": "The first of these is an absolute superlative."}, {"start": 55.8, "duration": 2.66, "text": "It says that Fabio is very good."}, {"start": 58.46, "duration": 3.61, "text": "There is no specific limit to how good he is."}, {"start": 62.07, "duration": 3.67, "text": "Notice the suffix, issimo."}, {"start": 65.74, "duration": 3.269, "text": "The second example is a relative superlative."}, {"start": 69.009, "duration": 3.4, "text": "This is because his talent is limited to his school."}, {"start": 72.409, "duration": 8.051, "text": "The relative superlative is formed like the\ncomparative, but with an article in front."}, {"start": 101.229, "duration": 2.75, "text": "So let\u2019s go back to the absolute superlative."}, {"start": 103.979, "duration": 3.161, "text": "Let\u2019s practice forming these."}, {"start": 107.14, "duration": 4.179, "text": "If something is sweet, we would say that it is dolce."}, {"start": 111.319, "duration": 6.59, "text": "If we want to say that it is very sweet, we\ncould say molto dolce, but if we want to use"}, {"start": 117.909, "duration": 4.861, "text": "another form of superlative, we could say\nthat it is dolcissimo."}, {"start": 122.77, "duration": 3.97, "text": "Let\u2019s try a few more."}, {"start": 173.74, "duration": 3.76, "text": "That\u2019s how you make superlatives in Italian,\nso let\u2019s check comprehension."}], "title": "Superlatives in Italian: Relative and Absolute", "description": "One of the most delightful features of the Italian language is the way we can add certain suffixes to modify words in specific ways, the way we learned for our altered words. Superlatives are another way this can happen, which you may know if you've ever shouted: Bravissimo! Let's learn the ways that we can form superlatives in Italian.\n\nScript by Patrizia Farina, Professor of Italian at Western Connecticut State University and Purchase College.\n\nWatch the whole Italian playlist: http://bit.ly/ProfDaveItalian\n\nAmerican History Tutorials: http://bit.ly/ProfDaveAmericanHistory\nClassical Physics Tutorials: http://bit.ly/ProfDavePhysics1\nModern Physics Tutorials: http://bit.ly/ProfDavePhysics2\nGeneral Chemistry Tutorials: http://bit.ly/ProfDaveGenChem\nOrganic Chemistry Tutorials: http://bit.ly/ProfDaveOrgChem\nBiochemistry Tutorials: http://bit.ly/ProfDaveBiochem\nBiology Tutorials: http://bit.ly/ProfDaveBio\n\nEMAIL\u25ba ProfessorDaveExplains@gmail.com\nPATREON\u25ba http://patreon.com/ProfessorDaveExplains\n\nCheck out \"Is This Wi-Fi Organic?\", my book on disarming pseudoscience!\nAmazon: https://amzn.to/2HtNpVH\nBookshop: https://bit.ly/39cKADM\n\rBarnes and Noble: https://bit.ly/3pUjmrn \nBook Depository: http://bit.ly/3aOVDlT", "lengthSeconds": 239, "width": 640, "height": 360}
{"video_id": "G4NLidjwbEA", "transcript": [{"start": 0.03, "duration": 4.11, "text": "now I want you to observe something"}, {"start": 1.949, "duration": 3.961, "text": "about all of these different sets that"}, {"start": 4.14, "duration": 3.33, "text": "I've been talking about so you're just"}, {"start": 5.91, "duration": 5.13, "text": "one such example that we've been looking"}, {"start": 7.47, "duration": 7.14, "text": "at this is the even integers notice"}, {"start": 11.04, "duration": 7.26, "text": "however that the inside of my definition"}, {"start": 14.61, "duration": 6.54, "text": "of the even integers began n is an"}, {"start": 18.3, "duration": 6.45, "text": "element of the integers such that some"}, {"start": 21.15, "duration": 6.03, "text": "property is true so this a in its"}, {"start": 24.75, "duration": 6.599, "text": "definition is sort of living within the"}, {"start": 27.18, "duration": 6.559, "text": "larger set of this integers and this was"}, {"start": 31.349, "duration": 5.071, "text": "going to introduce this to the idea of a"}, {"start": 33.739, "duration": 3.971, "text": "universal set so imagine I just have"}, {"start": 36.42, "duration": 5.22, "text": "some sets floating around I got my a"}, {"start": 37.71, "duration": 5.7, "text": "I've got my B but what domain what"}, {"start": 41.64, "duration": 4.23, "text": "contents do we have are we talking about"}, {"start": 43.41, "duration": 4.86, "text": "integers are we talking about people or"}, {"start": 45.87, "duration": 4.65, "text": "we're talking about functions where are"}, {"start": 48.27, "duration": 6.15, "text": "we broadly living what's our larger"}, {"start": 50.52, "duration": 5.879, "text": "context so what we have is the notion of"}, {"start": 54.42, "duration": 4.709, "text": "a universe and we usually denote it by"}, {"start": 56.399, "duration": 6.241, "text": "this a big square box on the outside we"}, {"start": 59.129, "duration": 5.041, "text": "call it you for universe here and all of"}, {"start": 62.64, "duration": 4.17, "text": "the sets that we're talking about are"}, {"start": 64.17, "duration": 4.89, "text": "all contained within this sort of larger"}, {"start": 66.81, "duration": 5.7, "text": "contextual set in other words we're"}, {"start": 69.06, "duration": 6.599, "text": "going to say that the universe U is the"}, {"start": 72.51, "duration": 5.609, "text": "larger context for whatever we are"}, {"start": 75.659, "duration": 5.631, "text": "talking about it's just some big set and"}, {"start": 78.119, "duration": 6.0, "text": "all of the other sets are subsets of it"}, {"start": 81.29, "duration": 4.81, "text": "and then we can do an operation if we"}, {"start": 84.119, "duration": 4.561, "text": "know what the universal set is going to"}, {"start": 86.1, "duration": 4.769, "text": "be so for example let me just take one"}, {"start": 88.68, "duration": 4.649, "text": "set a and it's living inside of the"}, {"start": 90.869, "duration": 4.71, "text": "universal set I can define something"}, {"start": 93.329, "duration": 5.491, "text": "where I set a notice I put a little C up"}, {"start": 95.579, "duration": 7.29, "text": "here this stands for complement I'm"}, {"start": 98.82, "duration": 5.369, "text": "claiming that a complement is not a it's"}, {"start": 102.869, "duration": 4.471, "text": "all this stuff over here"}, {"start": 104.189, "duration": 6.901, "text": "all of the stuff which is around my set"}, {"start": 107.34, "duration": 6.95, "text": "a and all of the things that are not in"}, {"start": 111.09, "duration": 7.169, "text": "a but are still broadly in my universe"}, {"start": 114.29, "duration": 7.689, "text": "so for example what is the complement of"}, {"start": 118.259, "duration": 6.36, "text": "the even integers well it does depend on"}, {"start": 121.979, "duration": 5.041, "text": "what my universe is as the complement of"}, {"start": 124.619, "duration": 4.801, "text": "the even integers like within the all of"}, {"start": 127.02, "duration": 4.53, "text": "the integers is going to be the odd"}, {"start": 129.42, "duration": 4.47, "text": "integers right every integer that is not"}, {"start": 131.55, "duration": 4.44, "text": "even as odd so if my universe is"}, {"start": 133.89, "duration": 5.429, "text": "all of the integers the complement of"}, {"start": 135.99, "duration": 5.25, "text": "the evens is the odd but if my broader"}, {"start": 139.319, "duration": 4.681, "text": "universe that I'm considering is all"}, {"start": 141.24, "duration": 6.24, "text": "real integers things like pies and"}, {"start": 144.0, "duration": 6.33, "text": "things like 1/2 then the complement of"}, {"start": 147.48, "duration": 4.71, "text": "the even integers include the odd"}, {"start": 150.33, "duration": 4.17, "text": "integers but includes all of those other"}, {"start": 152.19, "duration": 4.98, "text": "rational and irrational numbers as well"}, {"start": 154.5, "duration": 4.65, "text": "that are not even integers so what"}, {"start": 157.17, "duration": 4.05, "text": "universe you're talking into is usually"}, {"start": 159.15, "duration": 4.68, "text": "contextually understood or perhaps"}, {"start": 161.22, "duration": 4.65, "text": "explicitly written down and if you know"}, {"start": 163.83, "duration": 4.23, "text": "what the universe is you can take the"}, {"start": 165.87, "duration": 4.88, "text": "complement of a set within a particular"}, {"start": 168.06, "duration": 2.69, "text": "universe"}], "title": "Universes and Complements in Set Theory", "description": "Learning Objectives: Given a set living in a Universe, compute the compliment of that set, relative to that universe. \n\n\u25baFULL DISCRETE MATH PLAYLIST: https://www.youtube.com/playlist?list=PLHXZ9OQGMqxersk8fUxiUMSIx0DBqsKZS\n\nOTHER COURSE PLAYLISTS:\n\u25baCALCULUS I: https://www.youtube.com/playlist?list=PLHXZ9OQGMqxfT9RMcReZ4WcoVILP4k6-m\n\u25ba CALCULUS II: https://www.youtube.com/playlist?list=PLHXZ9OQGMqxc4ySKTIW19TLrT91Ik9M4n\n\u25baMULTIVARIABLE CALCULUS (Calc III): https://www.youtube.com/playlist?list=PLHXZ9OQGMqxc_CvEy7xBKRQr6I214QJcd\n\u25baVECTOR CALCULUS (Calc IV): https://www.youtube.com/playlist?list=PLHXZ9OQGMqxfW0GMqeUE1bLKaYor6kbHa\n\u25baDIFFERENTIAL EQUATIONS: https://www.youtube.com/watch?v=xeeM3TT4Zgg&list=PLHXZ9OQGMqxcJXnLr08cyNaup4RDsbAl1\n\u25baLINEAR ALGEBRA: https://www.youtube.com/playlist?list=PLHXZ9OQGMqxfUl0tcqPNTJsb7R6BqSLo6\n\nOTHER PLAYLISTS:\n\u25ba Learning Math Series\nhttps://www.youtube.com/watch?v=LPH2lqis3D0&list=PLHXZ9OQGMqxfSkRtlL5KPq6JqMNTh_MBw\n\u25baCool Math Series:\nhttps://www.youtube.com/playlist?list=PLHXZ9OQGMqxelE_9RzwJ-cqfUtaFBpiho\n\nBECOME A MEMBER:\n\u25baJoin: https://www.youtube.com/channel/UC9rTsvTxJnx1DNrDA3Rqa6A/join\n\nMATH BOOKS & MERCH I LOVE:\n\u25ba My Amazon Affiliate Shop: https://www.amazon.com/shop/treforbazett\n\nSOCIALS:\n\u25baTwitter (math based): http://twitter.com/treforbazett\n\u25baInstagram (photography based): http://instagram.com/treforphotography", "lengthSeconds": 170, "width": 640, "height": 360}
{"video_id": "pHWaxuyfK2s", "transcript": [{"start": 0.17, "duration": 4.99, "text": "the next argument form on our list is"}, {"start": 3.12, "duration": 5.25, "text": "something referred to as generalization"}, {"start": 5.16, "duration": 6.66, "text": "and it is basically the idea that if I"}, {"start": 8.37, "duration": 5.55, "text": "have some statement I can always take"}, {"start": 11.82, "duration": 5.01, "text": "that statement over another statement if"}, {"start": 13.92, "duration": 4.41, "text": "the original is true then that statement"}, {"start": 16.83, "duration": 2.91, "text": "or something else is also going to be"}, {"start": 18.33, "duration": 3.81, "text": "true because an or statement only needs"}, {"start": 19.74, "duration": 5.16, "text": "one so if I have any statement P I can"}, {"start": 22.14, "duration": 5.7, "text": "generalize it to this larger statement P"}, {"start": 24.9, "duration": 5.25, "text": "or Q so for example I said I'm a"}, {"start": 27.84, "duration": 4.469, "text": "Canadian that's a statement P and I can"}, {"start": 30.15, "duration": 5.73, "text": "therefore conclude that I am a Canadian"}, {"start": 32.309, "duration": 6.991, "text": "or I am a unicorn one of those two"}, {"start": 35.88, "duration": 6.06, "text": "things must be true and indeed because"}, {"start": 39.3, "duration": 4.71, "text": "I'm a Canadian the I am a Canadian"}, {"start": 41.94, "duration": 4.2, "text": "portion of the I am Canadian or I am a"}, {"start": 44.01, "duration": 4.08, "text": "unicorn part is true and so this this"}, {"start": 46.14, "duration": 4.23, "text": "whole disjunction this whole or"}, {"start": 48.09, "duration": 4.59, "text": "statement is going to be true and this"}, {"start": 50.37, "duration": 4.11, "text": "says nothing about whether I am or am"}, {"start": 52.68, "duration": 3.93, "text": "not a unicorn it doesn't matter I only"}, {"start": 54.48, "duration": 5.309, "text": "need one of the two things I've got the"}, {"start": 56.61, "duration": 5.6, "text": "Canadian part that's all I need sort of"}, {"start": 59.789, "duration": 5.431, "text": "the opposite of generalization is"}, {"start": 62.21, "duration": 6.25, "text": "specialization if I have one statement"}, {"start": 65.22, "duration": 4.53, "text": "and some other statement so in other"}, {"start": 68.46, "duration": 2.76, "text": "words I have I have both of the"}, {"start": 69.75, "duration": 2.939, "text": "statements true at the same time I might"}, {"start": 71.22, "duration": 2.79, "text": "not care about both of them equally I"}, {"start": 72.689, "duration": 3.091, "text": "have both of them but I don't need both"}, {"start": 74.01, "duration": 3.24, "text": "of them I only care about one of them I"}, {"start": 75.78, "duration": 3.57, "text": "can just get rid of the one I don't care"}, {"start": 77.25, "duration": 5.34, "text": "about and I'm left with just therefore"}, {"start": 79.35, "duration": 6.629, "text": "say P so for example if I say I'm a"}, {"start": 82.59, "duration": 5.73, "text": "Canadian and I have a PhD but for my"}, {"start": 85.979, "duration": 3.57, "text": "argument about being the president"}, {"start": 88.32, "duration": 2.549, "text": "United States it didn't matter whether I"}, {"start": 89.549, "duration": 3.871, "text": "had a PhD or not that is nothing to do"}, {"start": 90.869, "duration": 4.351, "text": "with it so so well I have this statement"}, {"start": 93.42, "duration": 3.659, "text": "I am Canadian and I have a PhD I was"}, {"start": 95.22, "duration": 4.859, "text": "gonna toss away the PhD portion of it"}, {"start": 97.079, "duration": 4.61, "text": "and just specialize down to the"}, {"start": 100.079, "duration": 3.75, "text": "statement I'm Canadian which is needed"}, {"start": 101.689, "duration": 4.241, "text": "do both generalization and"}, {"start": 103.829, "duration": 4.86, "text": "specialization are both sort of quite"}, {"start": 105.93, "duration": 5.34, "text": "trivial and easy things and in fact in"}, {"start": 108.689, "duration": 4.561, "text": "everyday language we use this all the"}, {"start": 111.27, "duration": 4.11, "text": "time but we don't we don't use it in the"}, {"start": 113.25, "duration": 4.079, "text": "explicit structure that I'm talking"}, {"start": 115.38, "duration": 4.379, "text": "about here however when we're trying to"}, {"start": 117.329, "duration": 4.261, "text": "translate complicated long arguments to"}, {"start": 119.759, "duration": 4.621, "text": "mathematics we want to make sure that"}, {"start": 121.59, "duration": 6.18, "text": "that every single step is truly valid"}, {"start": 124.38, "duration": 5.379, "text": "and so even for relatively simple steps"}, {"start": 127.77, "duration": 4.57, "text": "like the generalization are this"}, {"start": 129.759, "duration": 4.501, "text": "chelation we want to make sure that we"}, {"start": 132.34, "duration": 3.209, "text": "know that this is true we could write"}, {"start": 134.26, "duration": 3.36, "text": "down the truth tables for it if we"}, {"start": 135.549, "duration": 4.201, "text": "really wanted to and therefore be"}, {"start": 137.62, "duration": 6.06, "text": "completely confident that I'm not making"}, {"start": 139.75, "duration": 5.76, "text": "any sort of mistake a final one a very"}, {"start": 143.68, "duration": 4.07, "text": "very important one in mathematics one"}, {"start": 145.51, "duration": 6.66, "text": "we're going to be using a whole bunch is"}, {"start": 147.75, "duration": 6.36, "text": "arguing by contradiction and the idea is"}, {"start": 152.17, "duration": 5.64, "text": "this I'm trying to prove statement P"}, {"start": 154.11, "duration": 7.57, "text": "what are you do is I assume I take as my"}, {"start": 157.81, "duration": 5.549, "text": "conditional if not P I assume that that"}, {"start": 161.68, "duration": 4.8, "text": "my statement is actually false"}, {"start": 163.359, "duration": 4.591, "text": "and I go along and I derive some sort of"}, {"start": 166.48, "duration": 2.759, "text": "contradiction or for example a"}, {"start": 167.95, "duration": 4.289, "text": "contradiction might be something like"}, {"start": 169.239, "duration": 4.801, "text": "zero is equal to one or I'm a mammal and"}, {"start": 172.239, "duration": 4.59, "text": "I'm a reptile something like this that"}, {"start": 174.04, "duration": 5.669, "text": "just makes no sense at all so if"}, {"start": 176.829, "duration": 4.981, "text": "assuming my statement was false leads me"}, {"start": 179.709, "duration": 4.61, "text": "to this nonsensical contradiction the"}, {"start": 181.81, "duration": 6.329, "text": "contradiction is always false then my"}, {"start": 184.319, "duration": 5.471, "text": "initial statement must indeed be true so"}, {"start": 188.139, "duration": 4.44, "text": "this is the logical form of a"}, {"start": 189.79, "duration": 6.15, "text": "contradiction not P implies a"}, {"start": 192.579, "duration": 5.97, "text": "contradiction then my initial claim must"}, {"start": 195.94, "duration": 5.31, "text": "be true and I'm actually going to delay"}, {"start": 198.549, "duration": 5.15, "text": "showing you an example of this until a"}, {"start": 201.25, "duration": 2.449, "text": "later video"}], "title": "Logical Argument Forms: Generalizations, Specialization, Contradiction", "description": "Learning Objectives:\n1) State the argument forms for generalization, specialization and contradiction\n2) Analyze a written argument as being of these forms\n\nTogether with Modus Tollens and Modus Ponens introduced in the previous video, these logical forms provide a collection of ways to construct valid arguments. In this video we see each of the logical forms symbolically and use them to interpret sentences. \n\n\u25baFull DISCRETE MATH Course Playlist: https://www.youtube.com/playlist?list=PLHXZ9OQGMqxersk8fUxiUMSIx0DBqsKZS\n\nOther Course Playlists:\n\n\u25baCALCULUS I: https://www.youtube.com/playlist?list=PLHXZ9OQGMqxfT9RMcReZ4WcoVILP4k6-m\n\n\u25baCALCULUS II: https://www.youtube.com/playlist?list=PLHXZ9OQGMqxc4ySKTIW19TLrT91Ik9M4n\n\n\u25baCALCULUS III: https://www.youtube.com/playlist?list=PLHXZ9OQGMqxc_CvEy7xBKRQr6I214QJcd\n\n\u25baLINEAR ALGEBRA: https://www.youtube.com/playlist?list=PLHXZ9OQGMqxfUl0tcqPNTJsb7R6BqSLo6\n\n\u25ba Want to learn math effectively? Check out my \"Learning Math\" Series: https://www.youtube.com/watch?v=LPH2lqis3D0&list=PLHXZ9OQGMqxfSkRtlL5KPq6JqMNTh_MBw\n\n\u25baWant some cool math? Check out my \"Cool Math\" Series: https://www.youtube.com/playlist?list=PLHXZ9OQGMqxelE_9RzwJ-cqfUtaFBpiho\n\n*****************************************************\nYOUR TURN! Learning math requires more than just watching math videos, so make sure you reflect, ask questions, and do lots of practice problems!\n\r****************************************************\n\n\u25baFollow me on Twitter: http://twitter.com/treforbazett\n\nBECOME A MEMBER:\n\u25baJoin: https://www.youtube.com/channel/UC9rTsvTxJnx1DNrDA3Rqa6A/join\n\nMATH BOOKS & MERCH I LOVE:\n\u25ba My Amazon Affiliate Shop: https://www.amazon.com/shop/treforbazett", "lengthSeconds": 202, "width": 640, "height": 360}
{"video_id": "ON7yAw6W9VY", "transcript": [{"start": 0.48, "duration": 5.24, "text": "here I have the example if I'm skilled"}, {"start": 3.04, "duration": 3.359, "text": "at poker then I'm going to win a bunch"}, {"start": 5.72, "duration": 4.04, "text": "of"}, {"start": 6.399, "duration": 6.721, "text": "money so go to the casino I won a bunch"}, {"start": 9.76, "duration": 5.44, "text": "of money playing poker and I'm tempted"}, {"start": 13.12, "duration": 2.8, "text": "to conclude that therefore I'm skilled"}, {"start": 15.2, "duration": 3.76, "text": "at"}, {"start": 15.92, "duration": 4.72, "text": "poker okay so lots of people might"}, {"start": 18.96, "duration": 3.92, "text": "actually believe this this would be a"}, {"start": 20.64, "duration": 3.36, "text": "very reasonable fallacy that that a lot"}, {"start": 22.88, "duration": 4.52, "text": "of people would make let's try to"}, {"start": 24.0, "duration": 5.4, "text": "analyze it and I claim this is not true"}, {"start": 27.4, "duration": 3.76, "text": "and the loose idea is that just cuz I"}, {"start": 29.4, "duration": 2.96, "text": "went in one a bunch of money I might"}, {"start": 31.16, "duration": 3.919, "text": "have just gotten lucky I might be"}, {"start": 32.36, "duration": 6.64, "text": "terrible at poker but I got really lucky"}, {"start": 35.079, "duration": 5.601, "text": "so I cannot claim that I am skilled at"}, {"start": 39.0, "duration": 4.0, "text": "poker from this"}, {"start": 40.68, "duration": 4.6, "text": "argument so let's try to analyze the"}, {"start": 43.0, "duration": 4.399, "text": "structure of this argument first I have"}, {"start": 45.28, "duration": 5.959, "text": "if I'm skilled at poker I'm going to"}, {"start": 47.399, "duration": 5.601, "text": "give that a p then I will win money I'm"}, {"start": 51.239, "duration": 5.12, "text": "going to give that as a"}, {"start": 53.0, "duration": 7.48, "text": "q so that was my first premise if P then"}, {"start": 56.359, "duration": 6.921, "text": "Q so I'll write this as P implies Q q"}, {"start": 60.48, "duration": 9.04, "text": "that is my first premise next up it say"}, {"start": 63.28, "duration": 9.04, "text": "I won money playing poker so this is AER"}, {"start": 69.52, "duration": 4.48, "text": "q and then what we're saying is p"}, {"start": 72.32, "duration": 3.32, "text": "implies q and I'm going to I going to"}, {"start": 74.0, "duration": 6.52, "text": "copy it over here so we have it all in"}, {"start": 75.64, "duration": 7.439, "text": "place P implies Q then you assume the"}, {"start": 80.52, "duration": 5.959, "text": "conclusion and you're going to get I"}, {"start": 83.079, "duration": 5.321, "text": "skill that I'm skilled at poker so the L"}, {"start": 86.479, "duration": 5.28, "text": "The Logical form that you have here is p"}, {"start": 88.4, "duration": 6.039, "text": "implies Q then then q and then you get"}, {"start": 91.759, "duration": 5.121, "text": "out P but that wasn't modus pents that"}, {"start": 94.439, "duration": 3.881, "text": "wasn't modus tones it was none of these"}, {"start": 96.88, "duration": 4.96, "text": "and the problem here is that you're kind"}, {"start": 98.32, "duration": 6.839, "text": "of arguing by the converse it it's it's"}, {"start": 101.84, "duration": 5.639, "text": "Q implies P would be the converse of the"}, {"start": 105.159, "duration": 4.92, "text": "statement not P implies Q if you had Q"}, {"start": 107.479, "duration": 5.201, "text": "implies P here then you could take q and"}, {"start": 110.079, "duration": 5.881, "text": "get P but not the other way around so"}, {"start": 112.68, "duration": 4.52, "text": "this is trying to use the converse of"}, {"start": 115.96, "duration": 4.079, "text": "the statement and we know that a"}, {"start": 117.2, "duration": 5.4, "text": "Converse is not logically equivalent to"}, {"start": 120.039, "duration": 6.161, "text": "the original statement so this argument"}, {"start": 122.6, "duration": 6.4, "text": "does not work so this argument does not"}, {"start": 126.2, "duration": 6.0, "text": "work and we are going to say that it is"}, {"start": 129.0, "duration": 5.879, "text": "invalid I disagree with this logical"}, {"start": 132.2, "duration": 2.679, "text": "structure"}], "title": "Analyzing an argument for validity", "description": "Learning Objectives: \n1) Analyze an argument to determine it's logical structure\n2) Decide if the logical form of an argument is valid or invalid\n\nThe main argument forms are Modus Ponens, Modus Tollens, Generalization, Specialization, and Contradiction. To see more on each of these, check out the following playlist:\n\n\u25baFull DISCRETE MATH Course Playlist: https://www.youtube.com/playlist?list=PLHXZ9OQGMqxersk8fUxiUMSIx0DBqsKZS\n\nOther Course Playlists:\n\n\u25baCALCULUS I: https://www.youtube.com/playlist?list=PLHXZ9OQGMqxfT9RMcReZ4WcoVILP4k6-m\n\n\u25baCALCULUS II: https://www.youtube.com/playlist?list=PLHXZ9OQGMqxc4ySKTIW19TLrT91Ik9M4n\n\n\u25baCALCULUS III: https://www.youtube.com/playlist?list=PLHXZ9OQGMqxc_CvEy7xBKRQr6I214QJcd\n\n\u25baLINEAR ALGEBRA: https://www.youtube.com/playlist?list=PLHXZ9OQGMqxfUl0tcqPNTJsb7R6BqSLo6\n\n\u25ba Want to learn math effectively? Check out my \"Learning Math\" Series: https://www.youtube.com/watch?v=LPH2lqis3D0&list=PLHXZ9OQGMqxfSkRtlL5KPq6JqMNTh_MBw\n\n\u25baWant some cool math? Check out my \"Cool Math\" Series: https://www.youtube.com/playlist?list=PLHXZ9OQGMqxelE_9RzwJ-cqfUtaFBpiho\n\n*****************************************************\nYOUR TURN! Learning math requires more than just watching math videos, so make sure you reflect, ask questions, and do lots of practice problems!\n\r****************************************************\n\n\u25baFollow me on Twitter: http://twitter.com/treforbazett\n\nBECOME A MEMBER:\n\u25baJoin: https://www.youtube.com/channel/UC9rTsvTxJnx1DNrDA3Rqa6A/join\n\nMATH BOOKS & MERCH I LOVE:\n\u25ba My Amazon Affiliate Shop: https://www.amazon.com/shop/treforbazett", "lengthSeconds": 133, "width": 640, "height": 360}
{"video_id": "XDcy_wqWQVs", "transcript": [{"start": 4.2, "duration": 7.24, "text": "let me do a really simple example next I"}, {"start": 6.839, "duration": 6.84, "text": "want to do the limit of a constant as X"}, {"start": 11.44, "duration": 3.439, "text": "goes to a value of a so for example I'm"}, {"start": 13.679, "duration": 2.961, "text": "going to bring up a graph here this is"}, {"start": 14.879, "duration": 5.601, "text": "the choice where my constant was one"}, {"start": 16.64, "duration": 5.0, "text": "what is the limit of one as X goes to a"}, {"start": 20.48, "duration": 2.4, "text": "well it doesn't matter where I am on"}, {"start": 21.64, "duration": 3.16, "text": "here if I if I want to talk about this"}, {"start": 22.88, "duration": 3.92, "text": "point right here the limit as you get"}, {"start": 24.8, "duration": 3.399, "text": "close to this point is one limit as you"}, {"start": 26.8, "duration": 3.479, "text": "get close to this point is one this"}, {"start": 28.199, "duration": 5.561, "text": "point every point you could get close"}, {"start": 30.279, "duration": 5.521, "text": "two the height of it's always one so no"}, {"start": 33.76, "duration": 4.36, "text": "matter what it is it's the value of one"}, {"start": 35.8, "duration": 4.68, "text": "or in the generic case when arbitrary C"}, {"start": 38.12, "duration": 4.84, "text": "here I'm going to say the limit is equal"}, {"start": 40.48, "duration": 4.36, "text": "to C okay here's sort of the next most"}, {"start": 42.96, "duration": 5.04, "text": "complicated function not a constant this"}, {"start": 44.84, "duration": 4.6, "text": "time it's the variable X so this is the"}, {"start": 48.0, "duration": 4.399, "text": "graph of a straight line it looks a"}, {"start": 49.44, "duration": 5.439, "text": "little bit like this and notice if I was"}, {"start": 52.399, "duration": 6.48, "text": "to do the limit as we get close to.5"}, {"start": 54.879, "duration": 6.081, "text": "here 0.5 is just equal to well 0. five"}, {"start": 58.879, "duration": 3.84, "text": "if I got close to the value of one up"}, {"start": 60.96, "duration": 3.519, "text": "here well that's going to have this"}, {"start": 62.719, "duration": 3.841, "text": "height of one over here if I go up to"}, {"start": 64.479, "duration": 5.801, "text": "two it's going to have this height of"}, {"start": 66.56, "duration": 5.84, "text": "two so as we get close to any point a"}, {"start": 70.28, "duration": 3.6, "text": "because it's just this straight line"}, {"start": 72.4, "duration": 4.28, "text": "that the height of it is going to be a"}, {"start": 73.88, "duration": 5.239, "text": "as and so the limit of the function X as"}, {"start": 76.68, "duration": 5.72, "text": "the variable X goes to a is just going"}, {"start": 79.119, "duration": 6.081, "text": "to be equal to a okay what about all"}, {"start": 82.4, "duration": 5.16, "text": "this nonsense we've got a whole polom"}, {"start": 85.2, "duration": 4.72, "text": "inside of here what we get to do is we"}, {"start": 87.56, "duration": 4.839, "text": "get to use these different rules these"}, {"start": 89.92, "duration": 4.12, "text": "limit laws to very quickly figure this"}, {"start": 92.399, "duration": 3.281, "text": "out and I'm going to go slowly this"}, {"start": 94.04, "duration": 3.0, "text": "first time but in a moment we're just"}, {"start": 95.68, "duration": 4.2, "text": "going to answer this"}, {"start": 97.04, "duration": 4.68, "text": "immediately the idea is first see was a"}, {"start": 99.88, "duration": 3.32, "text": "sum of three different things and we saw"}, {"start": 101.72, "duration": 4.399, "text": "that the limit of a sum was the sum of"}, {"start": 103.2, "duration": 4.44, "text": "the limits so I can break that up as the"}, {"start": 106.119, "duration": 4.841, "text": "sum of three different things the sum of"}, {"start": 107.64, "duration": 5.0, "text": "this first 3x cubed the sum of the X and"}, {"start": 110.96, "duration": 4.119, "text": "the sum of the minus one it's the it's"}, {"start": 112.64, "duration": 3.88, "text": "three different limits and we figured"}, {"start": 115.079, "duration": 4.121, "text": "out what the limit of a constant was and"}, {"start": 116.52, "duration": 4.199, "text": "the limit of x was we know both of those"}, {"start": 119.2, "duration": 4.559, "text": "and I'm also going to try to break this"}, {"start": 120.719, "duration": 7.641, "text": "one up notice we got a scalar and we got"}, {"start": 123.759, "duration": 6.401, "text": "a product X cubed is like x * x * X so"}, {"start": 128.36, "duration": 4.92, "text": "then when I Advance this what we're"}, {"start": 130.16, "duration": 5.48, "text": "going to have is first of all the the"}, {"start": 133.28, "duration": 5.0, "text": "triple product x * x * X is three"}, {"start": 135.64, "duration": 3.92, "text": "different limits here the scaler gets bu"}, {"start": 138.28, "duration": 3.84, "text": "built out of the front that was one of"}, {"start": 139.56, "duration": 4.0, "text": "our rules and then for the limit of x"}, {"start": 142.12, "duration": 3.52, "text": "and limit of the minus one I just plug"}, {"start": 143.56, "duration": 3.08, "text": "in those values we just saw the A and"}, {"start": 145.64, "duration": 3.56, "text": "the minus"}, {"start": 146.64, "duration": 4.4, "text": "one and now I've taken the X Cub down to"}, {"start": 149.2, "duration": 3.52, "text": "a limit of x X we just seen what that"}, {"start": 151.04, "duration": 6.279, "text": "was you plug in a and so what we're"}, {"start": 152.72, "duration": 7.0, "text": "going to get is 3 a cub + the Aus 1 in"}, {"start": 157.319, "duration": 3.961, "text": "other words what we've done here is we"}, {"start": 159.72, "duration": 4.519, "text": "just took this a that was there and we"}, {"start": 161.28, "duration": 6.48, "text": "just just plugged it in 3x Cub + x - 1"}, {"start": 164.239, "duration": 5.161, "text": "became 3 a cub + Aus one so polinomial"}, {"start": 167.76, "duration": 3.24, "text": "are really easy there's nothing to do"}, {"start": 169.4, "duration": 5.04, "text": "with a polinomial not have to graph it"}, {"start": 171.0, "duration": 4.8, "text": "you just plug the value in and indeed"}, {"start": 174.44, "duration": 3.4, "text": "it's true even slightly more General"}, {"start": 175.8, "duration": 4.359, "text": "it's true for something called rational"}, {"start": 177.84, "duration": 5.28, "text": "function so this is a rational function"}, {"start": 180.159, "duration": 5.72, "text": "f ofx it's a polinomial on the top and a"}, {"start": 183.12, "duration": 4.679, "text": "polom on the bottom and as long as you"}, {"start": 185.879, "duration": 3.92, "text": "don't have the polinomial at this point"}, {"start": 187.799, "duration": 3.961, "text": "a being equal to zero because you'd have"}, {"start": 189.799, "duration": 4.241, "text": "some sort of division by zero thing the"}, {"start": 191.76, "duration": 4.8, "text": "limit is super easy you just plug the"}, {"start": 194.04, "duration": 4.52, "text": "value in whatever rational function you"}, {"start": 196.56, "duration": 3.8, "text": "might have its limit is you just take"}, {"start": 198.56, "duration": 3.759, "text": "that a that you're taking the limit to"}, {"start": 200.36, "duration": 5.76, "text": "and you plug it in the same way this"}, {"start": 202.319, "duration": 5.761, "text": "polinomial we just plugged it in so for"}, {"start": 206.12, "duration": 3.24, "text": "a ton of these values as long as you"}, {"start": 208.08, "duration": 3.159, "text": "don't have a zero on the bottom as long"}, {"start": 209.36, "duration": 4.04, "text": "as you don't have that problem spot that"}, {"start": 211.239, "duration": 5.801, "text": "for these rational functions limits are"}, {"start": 213.4, "duration": 3.64, "text": "really easy you just plug them in"}], "title": "Building up to computing limits of rational functions", "description": "Watch first: Limit Laws: https://www.youtube.com/watch?v=dY5T7BcQ2Nc\n\nWe start with simple functions ( f(x)= c and f(x) = x ), and apply the Limit Laws of the previous video to build up to compute the limit of a class of more complicated functions: rational functions with nonzero denominators. \n\nLearning Objectives:\n1) Use graphs to compute the limits of \"simple\" functions\n2) Apply Limit Laws in success to compute the limit of more complicated functions. \n3) Apply the theorem about rational functions to quickly compute the limit of a rational function at points where the denominator is nonzero. \n\n\n\n\nNow it's your turn: \n1) Summarize the big idea of this video in your own words\n2) Write down anything you are unsure about to think about later\n3) What questions for the future do you have? Where are we going with this content?\n4) Can you come up with your own sample test problem on this material? Solve it!\n\nLearning mathematics is best done by actually DOING mathematics. A video like this can only ever be a starting point. I might show you the basic ideas, definitions, formulas, and examples, but to truly master calculus means that you have to spend time - a lot of time! - sitting down and trying problems yourself, asking questions, and thinking about mathematics. So before you go on to the next video, pause and go THINK. \n\nThis video is part of a Calculus course taught by Dr. Trefor Bazett at the University of Cincinnati.\n\nBECOME A MEMBER:\n\u25baJoin: https://www.youtube.com/channel/UC9rTsvTxJnx1DNrDA3Rqa6A/join\n\nMATH BOOKS & MERCH I LOVE:\n\u25ba My Amazon Affiliate Shop: https://www.amazon.com/shop/treforbazett", "lengthSeconds": 215, "width": 640, "height": 360}
{"video_id": "1gkUKbrIssY", "transcript": [{"start": 0.0, "duration": 6.48, "text": "let's look at an example to understand"}, {"start": 2.1, "duration": 6.6, "text": "what the composition of relations is in"}, {"start": 6.48, "duration": 6.119, "text": "this diagram we have two relations the"}, {"start": 8.7, "duration": 7.139, "text": "relation R from a set a to a set b and a"}, {"start": 12.599, "duration": 5.52, "text": "relation s from the set B to the set C"}, {"start": 15.839, "duration": 5.52, "text": "here are those two relations written out"}, {"start": 18.119, "duration": 6.301, "text": "precisely as sets for example we can see"}, {"start": 21.359, "duration": 5.521, "text": "under the relation r one is related to T"}, {"start": 24.42, "duration": 5.22, "text": "given this arrow that shouldn't be"}, {"start": 26.88, "duration": 5.82, "text": "surprising in the relation s for example"}, {"start": 29.64, "duration": 6.96, "text": "we see that s relates to Beta which is"}, {"start": 32.7, "duration": 6.6, "text": "indicated by this Arrow here s is also"}, {"start": 36.6, "duration": 5.639, "text": "related to Alpha and in the relation R"}, {"start": 39.3, "duration": 5.939, "text": "we can see that one is also related to"}, {"start": 42.239, "duration": 5.581, "text": "lowercase R so hopefully the diagram and"}, {"start": 45.239, "duration": 4.921, "text": "the set notation are both clear what"}, {"start": 47.82, "duration": 4.68, "text": "would happen if we compose these"}, {"start": 50.16, "duration": 5.579, "text": "relations what would it mean to compose"}, {"start": 52.5, "duration": 5.399, "text": "the relation r with the relation s it"}, {"start": 55.739, "duration": 5.16, "text": "works out probably just how you expect"}, {"start": 57.899, "duration": 6.66, "text": "under the composite relation for example"}, {"start": 60.899, "duration": 7.381, "text": "three relates to gamma because the"}, {"start": 64.559, "duration": 7.081, "text": "relation R relates 3 to q and then s"}, {"start": 68.28, "duration": 7.019, "text": "relates Q to gamma so in total under the"}, {"start": 71.64, "duration": 6.42, "text": "composition 3 is related to gamma in"}, {"start": 75.299, "duration": 5.64, "text": "plain language the elements relate from"}, {"start": 78.06, "duration": 6.059, "text": "a to c if there's a connection between"}, {"start": 80.939, "duration": 5.581, "text": "them given these two relations here is"}, {"start": 84.119, "duration": 4.561, "text": "the entire composite relation written"}, {"start": 86.52, "duration": 4.44, "text": "out you can see how I've notated the"}, {"start": 88.68, "duration": 4.74, "text": "composition we'll talk about notation in"}, {"start": 90.96, "duration": 4.979, "text": "a second but just for another example"}, {"start": 93.42, "duration": 5.519, "text": "you can see that 4 is related to Alpha"}, {"start": 95.939, "duration": 5.941, "text": "and 4 is related to Beta because 4 is"}, {"start": 98.939, "duration": 6.72, "text": "related to S but then s is related to"}, {"start": 101.88, "duration": 7.86, "text": "both Alpha and to beta one is related to"}, {"start": 105.659, "duration": 6.78, "text": "R but R isn't related to anything in C"}, {"start": 109.74, "duration": 6.54, "text": "so one only appears in the composition"}, {"start": 112.439, "duration": 6.18, "text": "once given that one relates to T and"}, {"start": 116.28, "duration": 4.619, "text": "then T relates to gamma so that's the"}, {"start": 118.619, "duration": 4.381, "text": "only time that 1 a p here is in the"}, {"start": 120.899, "duration": 4.68, "text": "composite relation now let's take a"}, {"start": 123.0, "duration": 5.459, "text": "quick look at the general definition of"}, {"start": 125.579, "duration": 6.24, "text": "a composition of relations you can see"}, {"start": 128.459, "duration": 5.881, "text": "how an ordered pair x z is in the"}, {"start": 131.819, "duration": 6.241, "text": "composition of relations if there's some"}, {"start": 134.34, "duration": 5.7, "text": "middle element connecting x and z you"}, {"start": 138.06, "duration": 4.319, "text": "probably notice the notation I'm using"}, {"start": 140.04, "duration": 4.919, "text": "here is the same as what's used for"}, {"start": 142.379, "duration": 5.281, "text": "function composition because of that"}, {"start": 144.959, "duration": 5.341, "text": "sometimes a different notation is used"}, {"start": 147.66, "duration": 6.6, "text": "since this is a composition of relations"}, {"start": 150.3, "duration": 6.96, "text": "not functions this semicolon notation is"}, {"start": 154.26, "duration": 5.58, "text": "one such option you may also recall that"}, {"start": 157.26, "duration": 5.64, "text": "with function composition the second"}, {"start": 159.84, "duration": 5.34, "text": "function is typically applied first the"}, {"start": 162.9, "duration": 4.199, "text": "order is from right to left there"}, {"start": 165.18, "duration": 4.08, "text": "doesn't seem to be an agreed-upon"}, {"start": 167.099, "duration": 4.881, "text": "standard for the composition of"}, {"start": 169.26, "duration": 5.16, "text": "relations in this case I'm going"}, {"start": 171.98, "duration": 4.66, "text": "opposite the way that it's done for"}, {"start": 174.42, "duration": 6.06, "text": "function composition I'm going left to"}, {"start": 176.64, "duration": 7.5, "text": "right the relation R relates X to y I"}, {"start": 180.48, "duration": 6.839, "text": "but then s relates y to Z and so in"}, {"start": 184.14, "duration": 5.94, "text": "total the composition relates X to Z"}, {"start": 187.319, "duration": 4.861, "text": "this is a left to right direction just"}, {"start": 190.08, "duration": 4.56, "text": "be wary you might also see a composition"}, {"start": 192.18, "duration": 5.04, "text": "of relations following the typical order"}, {"start": 194.64, "duration": 4.8, "text": "for function composition which is right"}, {"start": 197.22, "duration": 4.379, "text": "to left and that's a quick overview of"}, {"start": 199.44, "duration": 3.36, "text": "what a composition of relations Is let"}, {"start": 201.599, "duration": 2.761, "text": "me know in the comments if you have any"}, {"start": 202.8, "duration": 2.03, "text": "questions and thank you very much for"}, {"start": 204.36, "duration": 1.739, "text": "watching"}, {"start": 204.83, "duration": 4.269, "text": "[Music]"}, {"start": 206.099, "duration": 3.0, "text": "existence"}, {"start": 222.46, "duration": 3.109, "text": "[Music]"}], "title": "Composition of Relations | Discrete Math", "description": "We go over how to compose relations. The composition of relations is similar to the composition of functions, but in our example we'll see some differences in the notation for relation composition and all that good stuff. #discretemathematics \n\nIntro to Relations: https://youtu.be/HXFHVRS1ZW8\n\nDiscrete Math playlist: https://www.youtube.com/playlist?list=PLztBpqftvzxXgpx57CEhZScNk2lWs1LdU\nDiscrete Math exercises: https://www.youtube.com/playlist?list=PLztBpqftvzxW14CkPnOTIuGf2hUeX8jX0\n\n\u25c9Textbooks I Like\u25c9\nGraph Theory: https://amzn.to/3JHQtZj\nReal Analysis: https://amzn.to/3CMdgjI\nAbstract Algebra: https://amzn.to/3IjoZaO\nLinear Algebra: https://amzn.to/43xAWEz\nCalculus: https://amzn.to/3PieD1M\nProofs and Set Theory: https://amzn.to/367VBXP (available for free online)\nStatistics: https://amzn.to/3tsaEER\nDiscrete Math: https://amzn.to/3qfhoUn\nNumber Theory: https://amzn.to/3JqpOQd\n\n\u2605DONATE\u2605\n\u25c6 Support Wrath of Math on Patreon for early access to new videos and other exclusive benefits: https://www.patreon.com/join/wrathofmathlessons\n\u25c6 Donate on PayPal: https://www.paypal.me/wrathofmath\n\nThanks to Loke Tan, Ra\u00fcl Beienheimer, Matt Venia, Micheline, Doug Walker, Odd Hultberg, Marc, Shlome Ashkenazi, Barbora Sharrock, Mohamad Nossier, Rolf Waefler, Shadow Master, and James Mead for their generous support on Patreon!\n\nOutro music is mine. You cannot find it anywhere, for now.\n\nFollow Wrath of Math on...\n\u25cf Instagram: https://www.instagram.com/wrathofmathedu\n\u25cf Facebook: https://www.facebook.com/WrathofMath\n\u25cf Twitter:  https://twitter.com/wrathofmathedu\n\nMy Math Rap channel: https://www.youtube.com/channel/UCQ2UBhg5nwWCL2aPC7_IpDQ/featured", "lengthSeconds": 224, "width": 640, "height": 360}
{"video_id": "U2zPxuj8mnI", "transcript": [{"start": 0.84, "duration": 5.32, "text": "the next concept we want to introduce is"}, {"start": 2.879, "duration": 4.121, "text": "that of logical equivalence between two"}, {"start": 6.16, "duration": 4.399, "text": "different"}, {"start": 7.0, "duration": 5.12, "text": "statements now the idea here is that you"}, {"start": 10.559, "duration": 4.841, "text": "could have two statements and both of"}, {"start": 12.12, "duration": 5.4, "text": "them could be really long messy compound"}, {"start": 15.4, "duration": 3.56, "text": "statements of got lots of ores and ands"}, {"start": 17.52, "duration": 4.8, "text": "and Knots and all sorts of different"}, {"start": 18.96, "duration": 5.559, "text": "inputs and it's not clear whether those"}, {"start": 22.32, "duration": 4.44, "text": "two statements are the same thing but"}, {"start": 24.519, "duration": 5.16, "text": "just presented differently or whether"}, {"start": 26.76, "duration": 4.519, "text": "they truly are different things and the"}, {"start": 29.679, "duration": 3.441, "text": "way we're going to claim that two"}, {"start": 31.279, "duration": 4.28, "text": "statements that maybe superficially do"}, {"start": 33.12, "duration": 5.16, "text": "not look to be the same are in fact"}, {"start": 35.559, "duration": 4.48, "text": "logically equivalent if you have one you"}, {"start": 38.28, "duration": 3.36, "text": "get the other we're going to say they're"}, {"start": 40.039, "duration": 4.441, "text": "logically equivalent if their truth"}, {"start": 41.64, "duration": 4.439, "text": "tables are identical which is to say"}, {"start": 44.48, "duration": 3.399, "text": "we're saying something is logically"}, {"start": 46.079, "duration": 5.0, "text": "independent or logically equivalent"}, {"start": 47.879, "duration": 5.52, "text": "rather if it's the case that for every"}, {"start": 51.079, "duration": 4.761, "text": "pairing of their inputs all true false"}, {"start": 53.399, "duration": 5.521, "text": "pairings of their PS and their Q's that"}, {"start": 55.84, "duration": 5.519, "text": "their output is going to have the exact"}, {"start": 58.92, "duration": 4.599, "text": "same truth and falseness for every"}, {"start": 61.359, "duration": 5.001, "text": "different possible input so that is our"}, {"start": 63.519, "duration": 4.92, "text": "idea is having the same truth table so"}, {"start": 66.36, "duration": 6.96, "text": "here's a simple example I want you to"}, {"start": 68.439, "duration": 7.04, "text": "look at the statement not not p and and"}, {"start": 73.32, "duration": 3.76, "text": "we know that this idea of a double"}, {"start": 75.479, "duration": 3.921, "text": "negative is that a double negative in"}, {"start": 77.08, "duration": 5.039, "text": "English sort of cancels right if I say"}, {"start": 79.4, "duration": 5.16, "text": "that I am not not hungry then that's me"}, {"start": 82.119, "duration": 3.921, "text": "saying that I'm hungry that if I put a a"}, {"start": 84.56, "duration": 5.8, "text": "double negative in you can sort of"}, {"start": 86.04, "duration": 6.88, "text": "imagine it cancel it now let's just try"}, {"start": 90.36, "duration": 4.399, "text": "to verify that that this actually makes"}, {"start": 92.92, "duration": 3.839, "text": "logical sense according to what I've"}, {"start": 94.759, "duration": 3.36, "text": "told you which is the truth tables so"}, {"start": 96.759, "duration": 3.04, "text": "first of all we're going to go in and"}, {"start": 98.119, "duration": 3.96, "text": "we're going to try to fill in What's"}, {"start": 99.799, "duration": 6.561, "text": "Happening Here for p my two different"}, {"start": 102.079, "duration": 6.64, "text": "inputs are true and false so I always do"}, {"start": 106.36, "duration": 5.079, "text": "that first I I put all of the possible"}, {"start": 108.719, "duration": 5.44, "text": "input values that I could have my truth"}, {"start": 111.439, "duration": 4.64, "text": "or falseness of that input P so then if"}, {"start": 114.159, "duration": 5.761, "text": "I look at not P which is just sort of an"}, {"start": 116.079, "duration": 6.881, "text": "intermediate step here if p is true then"}, {"start": 119.92, "duration": 5.879, "text": "not p is going to be false and if p is"}, {"start": 122.96, "duration": 4.159, "text": "false then not p is going to be true so"}, {"start": 125.799, "duration": 4.44, "text": "I was just doing that as a temporary"}, {"start": 127.119, "duration": 6.76, "text": "thing now for the third column now I'm"}, {"start": 130.239, "duration": 6.0, "text": "looking at not not P so I'm going to not"}, {"start": 133.879, "duration": 6.201, "text": "p and taking the opposite of that so if"}, {"start": 136.239, "duration": 6.201, "text": "if not p is false then then the negation"}, {"start": 140.08, "duration": 5.08, "text": "of being false is true and so I'm going"}, {"start": 142.44, "duration": 6.56, "text": "to put a true over here and likewise if"}, {"start": 145.16, "duration": 5.84, "text": "my not p is true then the negation of a"}, {"start": 149.0, "duration": 5.04, "text": "true is false so I'm going to put a"}, {"start": 151.0, "duration": 6.36, "text": "false down here and then what I've"}, {"start": 154.04, "duration": 5.32, "text": "noticed here is that these two things"}, {"start": 157.36, "duration": 4.239, "text": "are going to be the same thing that the"}, {"start": 159.36, "duration": 4.36, "text": "the truth table for p which is a true"}, {"start": 161.599, "duration": 5.441, "text": "and a false is the same thing as the"}, {"start": 163.72, "duration": 6.239, "text": "truth table for not not P again a true"}, {"start": 167.04, "duration": 8.479, "text": "or false so this recovers for us the"}, {"start": 169.959, "duration": 8.441, "text": "idea that P is the same thing as not not"}, {"start": 175.519, "duration": 6.121, "text": "p and the way that I like to write this"}, {"start": 178.4, "duration": 4.199, "text": "I have my statement P that's my my one"}, {"start": 181.64, "duration": 3.56, "text": "and I'm trying to compare it to"}, {"start": 182.599, "duration": 5.801, "text": "something else and I use if it's logical"}, {"start": 185.2, "duration": 5.44, "text": "equivalence I use an equality sign but"}, {"start": 188.4, "duration": 3.919, "text": "one that's got like three horizontal"}, {"start": 190.64, "duration": 2.879, "text": "lines to it that's what I mean for"}, {"start": 192.319, "duration": 3.761, "text": "logical equivalence that's going to be"}, {"start": 193.519, "duration": 6.64, "text": "my short hand for it this is logically"}, {"start": 196.08, "duration": 6.0, "text": "equivalent to not not"}, {"start": 200.159, "duration": 5.241, "text": "p and it's going to turn out that"}, {"start": 202.08, "duration": 5.159, "text": "there's all sorts of different logical"}, {"start": 205.4, "duration": 4.039, "text": "forms that may appear to be very"}, {"start": 207.239, "duration": 4.761, "text": "different like not not P superficially"}, {"start": 209.439, "duration": 4.481, "text": "is in a different way than P but even"}, {"start": 212.0, "duration": 3.92, "text": "though it's superficially different they"}, {"start": 213.92, "duration": 4.0, "text": "truly are the same thing and our method"}, {"start": 215.92, "duration": 3.599, "text": "is just write out the truth tables for"}, {"start": 217.92, "duration": 5.36, "text": "both of them are they the same truth"}, {"start": 219.519, "duration": 3.761, "text": "table or are they different"}], "title": "Logical Equivalence of Two Statements", "description": "Two statements are Logically Equivalent if they have the same truth table. For instance, p and ~(~p) are logically equivalent. \n\n****************************************************\nYOUR TURN! Learning math requires more than just watching videos, so make sure you reflect, ask questions (leave them in the comments!), and do lots of practice problems.\n****************************************************\n\n\u25baFull Course Playlist: DISCRETE MATH: https://www.youtube.com/playlist?list=PLHXZ9OQGMqxersk8fUxiUMSIx0DBqsKZS\n\n****************************************************\nOther Course Playlists:\n\n\u25baCALCULUS I: https://www.youtube.com/playlist?list=PLHXZ9OQGMqxfT9RMcReZ4WcoVILP4k6-m\n\n\u25baCALCULUS II: https://www.youtube.com/playlist?list=PLHXZ9OQGMqxc4ySKTIW19TLrT91Ik9M4n\n\n\u25baLINEAR ALGEBRA: https://www.youtube.com/playlist?list=PLHXZ9OQGMqxfUl0tcqPNTJsb7R6BqSLo6\n\n***************************************************\n\n\u25ba Want to learn math effectively? Check out my \"Learning Math\" Series:\nhttps://www.youtube.com/watch?v=LPH2lqis3D0&list=PLHXZ9OQGMqxfSkRtlL5KPq6JqMNTh_MBw\n\n\n\u25baWant some cool math? Check out my \"Cool Math\" Series:\nhttps://www.youtube.com/playlist?list=PLHXZ9OQGMqxelE_9RzwJ-cqfUtaFBpiho\n\n*****************************************************\n\n\u25baCheck out my 2nd Channel for lower production quality \"live\" math videos: https://www.youtube.com/channel/UCFNL8G1CZ4dH0p-Jf7UXqdQ\n\n*****************************************************\n\n\u25baFollow me on Twitter: http://twitter.com/treforbazett\n\n*****************************************************\n\nThis video was created by Dr. Trefor Bazett", "lengthSeconds": 221, "width": 640, "height": 360}
{"video_id": "VgurXuuIBbQ", "transcript": [{"start": 3.78, "duration": 3.22, "text": "we've had a similar goal for quite some"}, {"start": 6.73, "duration": 2.52, "text": "time"}, {"start": 7.0, "duration": 4.049, "text": "it was compute the derivative other as"}, {"start": 9.25, "duration": 4.53, "text": "many functions as possible and we have"}, {"start": 11.049, "duration": 4.771, "text": "done a lot we figured out polynomials"}, {"start": 13.78, "duration": 4.71, "text": "and trigonometric terms we figured out"}, {"start": 15.82, "duration": 4.14, "text": "how to do things like the product we"}, {"start": 18.49, "duration": 3.51, "text": "figured out how to do things like a"}, {"start": 19.96, "duration": 3.72, "text": "composition we know how to do quotients"}, {"start": 22.0, "duration": 4.74, "text": "we know how to do sums we know an"}, {"start": 23.68, "duration": 4.82, "text": "enormous amount but there's one class"}, {"start": 26.74, "duration": 4.799, "text": "that we don't know yet and that's"}, {"start": 28.5, "duration": 6.64, "text": "Exponential's for example we do not know"}, {"start": 31.539, "duration": 6.48, "text": "how to take the derivative yet of X to"}, {"start": 35.14, "duration": 4.98, "text": "the power of sine of X so how can we"}, {"start": 38.019, "duration": 4.5, "text": "solve something like this what I want to"}, {"start": 40.12, "duration": 4.649, "text": "use what I'm going to exploit is a nice"}, {"start": 42.519, "duration": 4.621, "text": "little fact about logarithms and the"}, {"start": 44.769, "duration": 4.621, "text": "fact is this if I'm going to take the"}, {"start": 47.14, "duration": 5.099, "text": "natural log of a e to the power of B"}, {"start": 49.39, "duration": 4.739, "text": "then one of the rules of logarithms is"}, {"start": 52.239, "duration": 4.081, "text": "that this exponent that I have it can"}, {"start": 54.129, "duration": 5.25, "text": "come brought down at the front B times a"}, {"start": 56.32, "duration": 4.489, "text": "natural log of a now I want to suppose"}, {"start": 59.379, "duration": 4.591, "text": "you're taking the derivative of this"}, {"start": 60.809, "duration": 6.131, "text": "if exponents are hard things like X to"}, {"start": 63.97, "duration": 5.13, "text": "the sine X if those are hard well we do"}, {"start": 66.94, "duration": 3.45, "text": "know what the derivative of a product is"}, {"start": 69.1, "duration": 4.129, "text": "we've got the product rule for that"}, {"start": 70.39, "duration": 6.06, "text": "so because the natural log sort of"}, {"start": 73.229, "duration": 5.081, "text": "transforms exponents into products it's"}, {"start": 76.45, "duration": 3.39, "text": "going to be able to transform something"}, {"start": 78.31, "duration": 4.44, "text": "that's challenging into something as"}, {"start": 79.84, "duration": 3.87, "text": "effectively just a product rule so let's"}, {"start": 82.75, "duration": 4.71, "text": "see how this actually works in an"}, {"start": 83.71, "duration": 6.57, "text": "example I take my y equal to X the power"}, {"start": 87.46, "duration": 4.68, "text": "of sine of X and before I just try to"}, {"start": 90.28, "duration": 3.96, "text": "compute dy DX I'm going to go and take"}, {"start": 92.14, "duration": 3.899, "text": "the natural log of both sides first that"}, {"start": 94.24, "duration": 5.159, "text": "is I'm going to compute the natural log"}, {"start": 96.039, "duration": 6.811, "text": "of Y and the natural log of the X to the"}, {"start": 99.399, "duration": 5.191, "text": "power of sine of X and then our goal is"}, {"start": 102.85, "duration": 3.509, "text": "going to be if I have it written in this"}, {"start": 104.59, "duration": 4.8, "text": "form I'm gonna take the derivative of"}, {"start": 106.359, "duration": 5.55, "text": "both sides here now why have I helped"}, {"start": 109.39, "duration": 4.38, "text": "myself the key idea is this I've got the"}, {"start": 111.909, "duration": 4.081, "text": "X to the sine of X this is kind of like"}, {"start": 113.77, "duration": 4.32, "text": "an a to the B and I can apply my log"}, {"start": 115.99, "duration": 5.07, "text": "rule I can take this sine of X and I can"}, {"start": 118.09, "duration": 5.01, "text": "bring it down so what this is gonna have"}, {"start": 121.06, "duration": 4.949, "text": "on the right hand side is the derivative"}, {"start": 123.1, "duration": 5.34, "text": "of the sine of X times the natural log"}, {"start": 126.009, "duration": 5.551, "text": "of X what happened over here on the left"}, {"start": 128.44, "duration": 4.769, "text": "well this is a chain rule an outside"}, {"start": 131.56, "duration": 5.13, "text": "function the natural logarithm the"}, {"start": 133.209, "duration": 4.021, "text": "inside function y of X so what's its"}, {"start": 136.69, "duration": 2.46, "text": "derivative"}, {"start": 137.23, "duration": 4.229, "text": "it's the drew the upside which is one"}, {"start": 139.15, "duration": 5.19, "text": "over whatever the inside is one over Y"}, {"start": 141.459, "duration": 5.221, "text": "times the derivative the inside dy/dx"}, {"start": 144.34, "duration": 4.44, "text": "that's me apply my chain rule so apply"}, {"start": 146.68, "duration": 5.13, "text": "my chain rule over here I applied a log"}, {"start": 148.78, "duration": 4.62, "text": "rule over there I can take the"}, {"start": 151.81, "duration": 3.57, "text": "derivative of product another take the"}, {"start": 153.4, "duration": 3.809, "text": "derivative of sine times the natural log"}, {"start": 155.38, "duration": 3.0, "text": "of X I just use the product rule so what"}, {"start": 157.209, "duration": 3.78, "text": "do I have I'm going to leave the left"}, {"start": 158.38, "duration": 4.77, "text": "hand side sine goes to coast plus the"}, {"start": 160.989, "duration": 4.231, "text": "natural log plus sine times the dru the"}, {"start": 163.15, "duration": 5.94, "text": "natural log which is 1 over X so I've"}, {"start": 165.22, "duration": 5.28, "text": "applied my product rule here ok so I'm"}, {"start": 169.09, "duration": 2.88, "text": "happy with this but I want to do a"}, {"start": 170.5, "duration": 3.66, "text": "little bit more work let's get rid of"}, {"start": 171.97, "duration": 4.32, "text": "all this other nonsense what I want to"}, {"start": 174.16, "duration": 2.49, "text": "do is I don't want this Y here any"}, {"start": 176.29, "duration": 2.37, "text": "longer"}, {"start": 176.65, "duration": 4.44, "text": "I want my derivative to be some function"}, {"start": 178.66, "duration": 4.62, "text": "of X so I want to manipulate this but I"}, {"start": 181.09, "duration": 6.06, "text": "can both move the Y up to the other side"}, {"start": 183.28, "duration": 6.39, "text": "and I know that Y is X to the power of"}, {"start": 187.15, "duration": 4.619, "text": "sine of X so I can isolate it and take"}, {"start": 189.67, "duration": 4.47, "text": "this Y here move it over and what I get"}, {"start": 191.769, "duration": 5.101, "text": "is the derivative ektu X is there's the"}, {"start": 194.14, "duration": 5.49, "text": "Y exit sine of x times all of this other"}, {"start": 196.87, "duration": 5.07, "text": "stuff so that is my final answer for the"}, {"start": 199.63, "duration": 4.949, "text": "derivative with respect to X of this"}, {"start": 201.94, "duration": 4.829, "text": "function X to the sine of X and you can"}, {"start": 204.579, "duration": 3.75, "text": "see that it is kind of a long and messy"}, {"start": 206.769, "duration": 3.511, "text": "expression here but this is just how"}, {"start": 208.329, "duration": 4.56, "text": "this logarithmic differentiation works"}, {"start": 210.28, "duration": 4.17, "text": "it makes it possible for us to compute a"}, {"start": 212.889, "duration": 4.401, "text": "derivative that we didn't know how to do"}, {"start": 214.45, "duration": 2.84, "text": "before"}], "title": "Logarithmic Differentiation |  Example: x^sinx", "description": "Description: \nOne of the few major classes of functions we can't yet differentiate are functions like x^sin(x). That is, one function to the power of another. By taking ln(x) first we can use the power of log rules to convert this into a product, and then apply product rule. But we have to be careful to convert back to the original function at the end!\n\nLearning Objectives:\n1) Compute the derivative of one function to the power of another. \n\n\n\nNow it's your turn: \n1) Summarize the big idea of this video in your own words\n2) Write down anything you are unsure about to think about later\n3) What questions for the future do you have? Where are we going with this content?\n4) Can you come up with your own sample test problem on this material? Solve it!\n\nLearning mathematics is best done by actually DOING mathematics. A video like this can only ever be a starting point. I might show you the basic ideas, definitions, formulas, and examples, but to truly master calculus means that you have to spend time - a lot of time! - sitting down and trying problems yourself, asking questions, and thinking about mathematics. So before you go on to the next video, pause and go THINK. \n\nThis video is part of a Calculus course taught by Dr. Trefor Bazett at the University of Cincinnati.\n\nBECOME A MEMBER:\n\u25baJoin: https://www.youtube.com/channel/UC9rTsvTxJnx1DNrDA3Rqa6A/join\n\nMATH BOOKS & MERCH I LOVE:\n\u25ba My Amazon Affiliate Shop: https://www.amazon.com/shop/treforbazett", "lengthSeconds": 216, "width": 640, "height": 360}
{"video_id": "0YqZIHFmVzg", "transcript": [{"start": 0.32, "duration": 5.2, "text": "if you recall when we talked about a"}, {"start": 2.44, "duration": 5.8, "text": "normal implication P implies Q we've"}, {"start": 5.52, "duration": 6.64, "text": "seen before that P implies Q is"}, {"start": 8.24, "duration": 7.08, "text": "logically equivalent to not Q implies"}, {"start": 12.16, "duration": 5.24, "text": "not P so I'm going to make use of that"}, {"start": 15.32, "duration": 4.959, "text": "logical equivalency when I try to go out"}, {"start": 17.4, "duration": 4.48, "text": "and prove things in particular my first"}, {"start": 20.279, "duration": 2.601, "text": "goal is going to be to try to prove an"}, {"start": 21.88, "duration": 2.52, "text": "implication that's what I'm trying to do"}, {"start": 22.88, "duration": 4.159, "text": "I'm trying to show prove an assumption"}, {"start": 24.4, "duration": 5.0, "text": "leads to a conclusion but what I'm going"}, {"start": 27.039, "duration": 4.16, "text": "to do is treat this as the other way"}, {"start": 29.4, "duration": 4.839, "text": "around I'm know that P implies Q is the"}, {"start": 31.199, "duration": 5.801, "text": "same thing as not Q implies not P that"}, {"start": 34.239, "duration": 4.16, "text": "is I'm instead going to go and try to"}, {"start": 37.0, "duration": 4.92, "text": "prove the Contra positive I'm going to"}, {"start": 38.399, "duration": 5.241, "text": "try to prove not Q implies not p and"}, {"start": 41.92, "duration": 4.24, "text": "that's the way I'm going to demonstrate"}, {"start": 43.64, "duration": 5.68, "text": "that P implies Q so let's see an"}, {"start": 46.16, "duration": 7.28, "text": "example in this theorem I'm saying if"}, {"start": 49.32, "duration": 6.12, "text": "the square is even then n itself is also"}, {"start": 53.44, "duration": 4.56, "text": "going to be even now I want to not"}, {"start": 55.44, "duration": 3.919, "text": "carefully this is different from the"}, {"start": 58.0, "duration": 3.039, "text": "theorem that we've seen in a previous"}, {"start": 59.359, "duration": 3.801, "text": "video this is not not saying if n is"}, {"start": 61.039, "duration": 5.44, "text": "even then n s is even it's the other way"}, {"start": 63.16, "duration": 5.44, "text": "around it's saying if n squar is even"}, {"start": 66.479, "duration": 4.921, "text": "then n is going to be even I'm going to"}, {"start": 68.6, "duration": 5.6, "text": "just try to use a direct proof and show"}, {"start": 71.4, "duration": 6.8, "text": "you why it might be challenging so my"}, {"start": 74.2, "duration": 5.84, "text": "assumption might be that N2 is even"}, {"start": 78.2, "duration": 4.919, "text": "which is going to be that N2 is going to"}, {"start": 80.04, "duration": 4.68, "text": "be twice some k for some K I'm not going"}, {"start": 83.119, "duration": 3.481, "text": "to write all my symbols down because"}, {"start": 84.72, "duration": 3.399, "text": "this is my plain around step I don't"}, {"start": 86.6, "duration": 3.44, "text": "have to be all that"}, {"start": 88.119, "duration": 4.241, "text": "precise all right and I want to say"}, {"start": 90.04, "duration": 4.32, "text": "something about n I've got this Square"}, {"start": 92.36, "duration": 5.52, "text": "here so maybe what I should be doing is"}, {"start": 94.36, "duration": 5.039, "text": "taking a square root of both sides that"}, {"start": 97.88, "duration": 4.08, "text": "might give me my n if I took the"}, {"start": 99.399, "duration": 4.281, "text": "positive square root for instance but"}, {"start": 101.96, "duration": 5.72, "text": "then what I have is it being like equal"}, {"start": 103.68, "duration": 6.52, "text": "to < tk2 * < TK K which doesn't really"}, {"start": 107.68, "duration": 4.2, "text": "look anything at all like being even so"}, {"start": 110.2, "duration": 3.12, "text": "assuming that my n squar was even and"}, {"start": 111.88, "duration": 3.04, "text": "taking square roots just didn't leave"}, {"start": 113.32, "duration": 3.839, "text": "anywhere it's got all this messy rot2"}, {"start": 114.92, "duration": 4.64, "text": "stuff floating around so proving it"}, {"start": 117.159, "duration": 4.32, "text": "directly seems a little bit challenging"}, {"start": 119.56, "duration": 5.44, "text": "so let's let's get rid of"}, {"start": 121.479, "duration": 5.96, "text": "that since proving this directly wasn't"}, {"start": 125.0, "duration": 4.119, "text": "all that useful let's try to prove it by"}, {"start": 127.439, "duration": 4.681, "text": "the contrapositive we're we're going to"}, {"start": 129.119, "duration": 5.641, "text": "do it the other way around with"}, {"start": 132.12, "duration": 4.759, "text": "negations so when I write it as proof by"}, {"start": 134.76, "duration": 5.16, "text": "contradiction by Contra positive rather"}, {"start": 136.879, "duration": 5.08, "text": "note what happens first of all the the N"}, {"start": 139.92, "duration": 4.84, "text": "squar is even and the N is even they"}, {"start": 141.959, "duration": 4.36, "text": "flip orders but they also get two knots"}, {"start": 144.76, "duration": 4.32, "text": "in that's what happens when I do Contra"}, {"start": 146.319, "duration": 5.161, "text": "positive so in other words I'm assuming"}, {"start": 149.08, "duration": 6.84, "text": "that n is not even and I'm trying to"}, {"start": 151.48, "duration": 6.399, "text": "deduce that n^2 is not even by the way"}, {"start": 155.92, "duration": 5.319, "text": "the Assumption n is not even is the same"}, {"start": 157.879, "duration": 4.841, "text": "thing as n is odd and the conclusion n"}, {"start": 161.239, "duration": 5.64, "text": "squ is not even is the same as the"}, {"start": 162.72, "duration": 6.64, "text": "conclusion n s is odd but I've actually"}, {"start": 166.879, "duration": 6.921, "text": "seen that particular proof before I've"}, {"start": 169.36, "duration": 6.44, "text": "seen a proof that n is odd implies N"}, {"start": 173.8, "duration": 4.84, "text": "squared is odd and so we can sort of"}, {"start": 175.8, "duration": 4.92, "text": "bring that one in and this is the proof"}, {"start": 178.64, "duration": 5.56, "text": "that we've seen before"}, {"start": 180.72, "duration": 6.32, "text": "indeed this claim n^2 even implies n is"}, {"start": 184.2, "duration": 6.08, "text": "even is the same claim it's logically"}, {"start": 187.04, "duration": 5.479, "text": "equivalent to n is odd implies n squ is"}, {"start": 190.28, "duration": 4.28, "text": "odd it's just that when it's written"}, {"start": 192.519, "duration": 4.0, "text": "this way in its contrapositive form it's"}, {"start": 194.56, "duration": 3.8, "text": "quite tractable and we were able to do"}, {"start": 196.519, "duration": 4.241, "text": "its proof almost directly it sort of"}, {"start": 198.36, "duration": 4.2, "text": "just fell out by our assumptions in a"}, {"start": 200.76, "duration": 4.32, "text": "way that just wasn't true when I tried"}, {"start": 202.56, "duration": 5.48, "text": "to directly prove the original so it's a"}, {"start": 205.08, "duration": 5.4, "text": "wonderful illustration of how a a proof"}, {"start": 208.04, "duration": 4.399, "text": "that's challenging and hard if you try"}, {"start": 210.48, "duration": 3.399, "text": "to go and prove its condra positive the"}, {"start": 212.439, "duration": 7.241, "text": "condra positive while logically"}, {"start": 213.879, "duration": 5.801, "text": "equivalent might be much easier to prove"}], "title": "Proof by Contrapositive | Method & First Example", "description": "Learning objective: prove an implication by showing the contrapositive is true. \n\nThis video is part of a Discrete Math course taught at the University of Cincinnati.\n\nBECOME A MEMBER:\n\u25baJoin: https://www.youtube.com/channel/UC9rTsvTxJnx1DNrDA3Rqa6A/join\n\nMATH BOOKS & MERCH I LOVE:\n\u25ba My Amazon Affiliate Shop: https://www.amazon.com/shop/treforbazett", "lengthSeconds": 218, "width": 640, "height": 360}
{"video_id": "BJLi8uaruF8", "transcript": [{"start": 0.0, "duration": 7.349, "text": "[Music]"}, {"start": 12.24, "duration": 5.84, "text": "so let's look at an example of"}, {"start": 14.32, "duration": 6.0, "text": "computing the mass using line integrals"}, {"start": 18.08, "duration": 5.039, "text": "so we have the following example which"}, {"start": 20.32, "duration": 3.92, "text": "is a wire is in the shape of a circle of"}, {"start": 23.119, "duration": 3.761, "text": "radius 2"}, {"start": 24.24, "duration": 4.0, "text": "centered in the origin let's say and has"}, {"start": 26.88, "duration": 4.559, "text": "density function"}, {"start": 28.24, "duration": 6.24, "text": "rho which is 2 plus 1 half x"}, {"start": 31.439, "duration": 5.201, "text": "we want to find the mass of the wire now"}, {"start": 34.48, "duration": 3.919, "text": "of course the mass is given by the line"}, {"start": 36.64, "duration": 4.32, "text": "integral over the curve c"}, {"start": 38.399, "duration": 4.801, "text": "of the density function and so we first"}, {"start": 40.96, "duration": 4.4, "text": "have to parameterize the wire"}, {"start": 43.2, "duration": 3.6, "text": "and we'll do that by using the function"}, {"start": 45.36, "duration": 4.96, "text": "alpha t which is"}, {"start": 46.8, "duration": 5.68, "text": "2 sine t in the x component and 2"}, {"start": 50.32, "duration": 3.2, "text": "cosine t in the y component and we'll"}, {"start": 52.48, "duration": 4.079, "text": "vary t"}, {"start": 53.52, "duration": 4.48, "text": "from minus pi to pi now with respect to"}, {"start": 56.559, "duration": 5.361, "text": "this parametrization we see"}, {"start": 58.0, "duration": 6.0, "text": "that rho is just two plus sine t"}, {"start": 61.92, "duration": 3.92, "text": "in particular the line integral over the"}, {"start": 64.0, "duration": 4.96, "text": "curve c of rho of x"}, {"start": 65.84, "duration": 3.84, "text": "y d s is just the integral from minus pi"}, {"start": 68.96, "duration": 3.519, "text": "to pi"}, {"start": 69.68, "duration": 4.479, "text": "of two plus sine t that's just rho with"}, {"start": 72.479, "duration": 4.64, "text": "respect to the parametrization"}, {"start": 74.159, "duration": 3.921, "text": "and then the arc length term is just the"}, {"start": 77.119, "duration": 4.801, "text": "square root of 4"}, {"start": 78.08, "duration": 6.8, "text": "cosine squared t plus 4 sine squared t"}, {"start": 81.92, "duration": 4.08, "text": "dt the pythagorean identity of course"}, {"start": 84.88, "duration": 3.44, "text": "tells us that cosine"}, {"start": 86.0, "duration": 4.32, "text": "squared plus sine squared t is one so"}, {"start": 88.32, "duration": 2.799, "text": "we're left with the integral from minus"}, {"start": 90.32, "duration": 4.24, "text": "pi to pi"}, {"start": 91.119, "duration": 5.921, "text": "of two times two plus sine t dt"}, {"start": 94.56, "duration": 3.28, "text": "which is just two times two t minus"}, {"start": 97.04, "duration": 3.92, "text": "cosine t"}, {"start": 97.84, "duration": 6.959, "text": "from minus pi to pi and this expands to"}, {"start": 100.96, "duration": 7.519, "text": "two times two pi minus cosine of pi"}, {"start": 104.799, "duration": 4.801, "text": "minus two times minus pi plus cosine of"}, {"start": 108.479, "duration": 3.441, "text": "minus pi"}, {"start": 109.6, "duration": 4.72, "text": "of course since cosine of pi is minus"}, {"start": 111.92, "duration": 5.12, "text": "one and cosine of minus pi is also"}, {"start": 114.32, "duration": 3.04, "text": "minus 1. what we're left with is 2 times"}, {"start": 117.04, "duration": 3.84, "text": "2"}, {"start": 117.36, "duration": 12.48, "text": "pi plus 1 plus 2 pi minus 1"}, {"start": 120.88, "duration": 8.96, "text": "and this is just 8 pi"}, {"start": 135.599, "duration": 2.081, "text": "you"}], "title": "Line Integrals, Mass, and Density | Vector Calculus | Calculus 3", "description": "Vector Calculus and Differential Equations Playlist -- https://www.youtube.com/watch?v=UhuxxkOocdY&list=PL912tg7wFUfXIwSKeTyxU5LTqdRZ8Urza&index=1\n\nMore videos on geometric algebra and the exterior algebra:\nWhat is area? -- https://www.youtube.com/watch?v=3sGNR36eUR0\nWhy you don't understand the fundamental theorem of calculus -- https://www.youtube.com/watch?v=JV0vgVa4bCg\nWhy you don't understand Green's theorem -- https://www.youtube.com/watch?v=XEDtm09ol5o\nComputing gradient vector fields -- https://www.youtube.com/watch?v=oNKFhQTn-1s\nLine Integral Example 1: https://www.youtube.com/watch?v=fmnYFM0C9W8&ab_channel=KyleBroder\nLine Integral Example 2: https://youtu.be/dW7SdW5sRuY\nLine Integral Example 3: https://youtu.be/FXHYaFig-OE\nOlder video on Line Integrals, Mass, and Density: https://youtu.be/sYmSGrcA10k\nStokes' theorem and Path Independence: https://youtu.be/2FGKxFG5lSI\nConservative Vector Fields: https://youtu.be/eQhj4m8pt9E\nComputing Arc Length: https://youtu.be/UhuxxkOocdY\nIntroduction to Vector Calculus: https://youtu.be/fV0d-FE03EM\nHow conservative vector fields relate to Euler's number: https://youtu.be/oU5elvZL0uU\n\n\n\ud83d\udc4d To support the channel, hit the like button and subscribe. \n\n\ud83d\udc38 To learn how I make these animations using Grant Sanderson's (3Blue1Brown) program Manim, check out my quick intro to Manim on Skillshare: https://www.skillshare.com/r/profile/Kyle-B/344454857\n\n\ud83c\udfb8 Music in this video is from Epidemic Sound: https://www.epidemicsound.com/referral/yi0wzw/\nWays to support the channel: \n\ud83c\udfc6 Support the channel over on Patreon https://www.patreon.com/KyleBroder\n\ud83d\udc38 Support the channel by signing up to a free trial of Skillshare using the link https://www.skillshare.com/r/profile/Kyle-B/344454857 \n\nFYI, these are affiliate links \ud83d\udcaa\ud83d\ude4f\n\nThese videos are separate from my research and teaching roles at the Australian National University, University of Sydney, and Beijing University. \n\nHi, my name is Kyle and I'm currently doing my doctoral mathematics degree in complex differential geometry under the supervision of Professor Gang Tian and Professor Ben Andrews.", "lengthSeconds": 136, "width": 640, "height": 360}
{"video_id": "aFi9WTRkmpY", "transcript": [{"start": 0.359, "duration": 5.4, "text": "the spookiest math curve is the infamous"}, {"start": 2.879, "duration": 4.801, "text": "Witch of anessi originally named by"}, {"start": 5.759, "duration": 3.88, "text": "Maria anessi the first woman appointed"}, {"start": 7.68, "duration": 4.839, "text": "to the math professor it was then later"}, {"start": 9.639, "duration": 6.361, "text": "completely mistranslated to woman who is"}, {"start": 12.519, "duration": 5.481, "text": "against God and then to which now we can"}, {"start": 16.0, "duration": 4.519, "text": "construct it geometrically by starting"}, {"start": 18.0, "duration": 4.56, "text": "with the circle of radius a then taking"}, {"start": 20.519, "duration": 3.6, "text": "a point on the circle and drawing the"}, {"start": 22.56, "duration": 3.76, "text": "segment from the origin out to the point"}, {"start": 24.119, "duration": 4.041, "text": "P if you put a horizontal tangent at the"}, {"start": 26.32, "duration": 3.68, "text": "top of the circle and extend our segment"}, {"start": 28.16, "duration": 4.36, "text": "out until it intersects with that"}, {"start": 30.0, "duration": 4.64, "text": "horizontal finally drop it de down until"}, {"start": 32.52, "duration": 5.08, "text": "it matches the height of the point P"}, {"start": 34.64, "duration": 4.96, "text": "this gives us a new Point as we move"}, {"start": 37.6, "duration": 4.92, "text": "around the circle you sketch out a curve"}, {"start": 39.6, "duration": 6.439, "text": "this is the witch of anessi analytically"}, {"start": 42.52, "duration": 6.08, "text": "this curve is given by 8 a cub / x^2 + 4"}, {"start": 46.039, "duration": 4.641, "text": "a^ 2 where a is the radius of the circle"}, {"start": 48.6, "duration": 5.119, "text": "and I'm a particular fan of the special"}, {"start": 50.68, "duration": 7.199, "text": "case where a is equal to 12 and in this"}, {"start": 53.719, "duration": 7.561, "text": "case it simplifies to just be 1x^2 + 1"}, {"start": 57.879, "duration": 6.92, "text": "now any good mathemagician can make Pi"}, {"start": 61.28, "duration": 5.36, "text": "appear so let's integrate from 0 to 1"}, {"start": 64.799, "duration": 3.96, "text": "and the calculus students among you will"}, {"start": 66.64, "duration": 5.479, "text": "recognize that this is the derivative of"}, {"start": 68.759, "duration": 6.441, "text": "arctan and so evaluating at 0 and 1"}, {"start": 72.119, "duration": 6.961, "text": "gives the value of pi over 4 but if"}, {"start": 75.2, "duration": 6.959, "text": "instead we had looked at the inside and"}, {"start": 79.08, "duration": 8.12, "text": "expanded using geometric series to give"}, {"start": 82.159, "duration": 8.041, "text": "1 - x^2 + X4 - x 6 and so on then"}, {"start": 87.2, "duration": 5.599, "text": "integrate term by term evaluated at the"}, {"start": 90.2, "duration": 6.68, "text": "zero and the one and you get the series"}, {"start": 92.799, "duration": 9.081, "text": "1 - 1/3 plus 5th - 17th and so forth"}, {"start": 96.88, "duration": 5.0, "text": "this is a series expansion for pi"}], "title": "Spookiest Math Curve: Witch of Agnesi", "description": "Spooky!!\nBECOME A MEMBER:\n\u25baJoin: https://www.youtube.com/channel/UC9rTsvTxJnx1DNrDA3Rqa6A/join\n\nMATH BOOKS I LOVE (affilliate link):\n\u25ba https://www.amazon.com/shop/treforbazett\n\nCOURSE PLAYLISTS:\n\u25baDISCRETE MATH: https://www.youtube.com/playlist?list=PLHXZ9OQGMqxersk8fUxiUMSIx0DBqsKZS\n\u25baLINEAR ALGEBRA: https://www.youtube.com/playlist?list=PLHXZ9OQGMqxfUl0tcqPNTJsb7R6BqSLo6\n\u25baCALCULUS I: https://www.youtube.com/playlist?list=PLHXZ9OQGMqxfT9RMcReZ4WcoVILP4k6-m\n\u25ba CALCULUS II: https://www.youtube.com/playlist?list=PLHXZ9OQGMqxc4ySKTIW19TLrT91Ik9M4n\n\u25baMULTIVARIABLE CALCULUS (Calc III): https://www.youtube.com/playlist?list=PLHXZ9OQGMqxc_CvEy7xBKRQr6I214QJcd\n\u25baVECTOR CALCULUS (Calc IV) https://www.youtube.com/playlist?list=PLHXZ9OQGMqxfW0GMqeUE1bLKaYor6kbHa\n\u25baDIFFERENTIAL EQUATIONS: https://www.youtube.com/playlist?list=PLHXZ9OQGMqxde-SlgmWlCmNHroIWtujBw\n\u25baLAPLACE TRANSFORM: https://www.youtube.com/playlist?list=PLHXZ9OQGMqxcJXnLr08cyNaup4RDsbAl1\n\u25baGAME THEORY:  https://www.youtube.com/playlist?list=PLHXZ9OQGMqxdzD8KpTHz6_gsw9pPxRFlX\n\nOTHER PLAYLISTS:\n\u25ba Learning Math Series\nhttps://www.youtube.com/watch?v=LPH2lqis3D0&list=PLHXZ9OQGMqxfSkRtlL5KPq6JqMNTh_MBw\n\u25baCool Math Series:\nhttps://www.youtube.com/playlist?list=PLHXZ9OQGMqxelE_9RzwJ-cqfUtaFBpiho\n\nSOCIALS:\n\u25baX/Twitter: http://X.com/treforbazett\n\u25baTikTok: http://tiktok.com/@drtrefor\n\u25baInstagram (photography based): http://instagram.com/treforphotography", "lengthSeconds": 100, "width": 270, "height": 480}
{"video_id": "DG9NUPaPVbE", "transcript": [{"start": 0.08, "duration": 4.88, "text": "the final example we're going to do is a"}, {"start": 2.48, "duration": 3.359, "text": "incredibly important example of a vector"}, {"start": 4.96, "duration": 3.2, "text": "field one"}, {"start": 5.839, "duration": 3.201, "text": "that comes from just a normal old scalar"}, {"start": 8.16, "duration": 3.92, "text": "function f"}, {"start": 9.04, "duration": 6.32, "text": "of x y and z a a function from r"}, {"start": 12.08, "duration": 5.199, "text": "three in this case just to r well for"}, {"start": 15.36, "duration": 4.4, "text": "such a scalar function"}, {"start": 17.279, "duration": 5.281, "text": "you could take what we called previously"}, {"start": 19.76, "duration": 4.32, "text": "in multi-variable calculus the gradient"}, {"start": 22.56, "duration": 3.92, "text": "the i-hat component is the partial"}, {"start": 24.08, "duration": 3.439, "text": "respect to x the y-hat is the partial"}, {"start": 26.48, "duration": 3.039, "text": "with respect to y"}, {"start": 27.519, "duration": 3.121, "text": "and the k-hat is the partial with"}, {"start": 29.519, "duration": 3.281, "text": "respect to z"}, {"start": 30.64, "duration": 3.84, "text": "that is if you give me a scalar function"}, {"start": 32.8, "duration": 5.04, "text": "f from either r2"}, {"start": 34.48, "duration": 5.28, "text": "or r3 to r then i can give you a vector"}, {"start": 37.84, "duration": 2.399, "text": "field a corresponding vector field given"}, {"start": 39.76, "duration": 2.72, "text": "by the"}, {"start": 40.239, "duration": 3.201, "text": "gradient of f the gradient of f is a"}, {"start": 42.48, "duration": 3.599, "text": "vector field"}, {"start": 43.44, "duration": 3.2, "text": "and it also has a very nice geometric"}, {"start": 46.079, "duration": 2.721, "text": "meaning"}, {"start": 46.64, "duration": 3.759, "text": "so for example suppose that my function"}, {"start": 48.8, "duration": 4.239, "text": "was the function f of x y"}, {"start": 50.399, "duration": 4.081, "text": "equal to x squared plus y squared then"}, {"start": 53.039, "duration": 3.04, "text": "we could compute out what its gradient"}, {"start": 54.48, "duration": 3.599, "text": "is not too bad well the partial respect"}, {"start": 56.079, "duration": 2.64, "text": "to x is just 2x and the partial respect"}, {"start": 58.079, "duration": 4.081, "text": "to y"}, {"start": 58.719, "duration": 5.601, "text": "is just 2y so the gradient is just that"}, {"start": 62.16, "duration": 3.76, "text": "now we've seen this vector field except"}, {"start": 64.32, "duration": 3.2, "text": "for the fact that there's a 2 in both"}, {"start": 65.92, "duration": 2.64, "text": "places which just stretches every vector"}, {"start": 67.52, "duration": 3.279, "text": "by a multiple of 2."}, {"start": 68.56, "duration": 3.44, "text": "this is precisely the vector field we"}, {"start": 70.799, "duration": 3.041, "text": "saw earlier"}, {"start": 72.0, "duration": 3.36, "text": "and so now i can put everything together"}, {"start": 73.84, "duration": 5.04, "text": "in one picture so this is my"}, {"start": 75.36, "duration": 5.04, "text": "picture for this function what we see is"}, {"start": 78.88, "duration": 3.36, "text": "the graph of the function that's this"}, {"start": 80.4, "duration": 4.8, "text": "blue paraboloid"}, {"start": 82.24, "duration": 3.28, "text": "and then in the domain the domain is the"}, {"start": 85.2, "duration": 3.04, "text": "x"}, {"start": 85.52, "duration": 3.44, "text": "y plane we get a vector field in that"}, {"start": 88.24, "duration": 3.519, "text": "domain"}, {"start": 88.96, "duration": 4.72, "text": "and so i plotted the gradient vector"}, {"start": 91.759, "duration": 4.881, "text": "field in the domain"}, {"start": 93.68, "duration": 3.28, "text": "and then this function above the domain"}, {"start": 96.64, "duration": 2.64, "text": "and"}, {"start": 96.96, "duration": 3.44, "text": "as you might recall from multi-variable"}, {"start": 99.28, "duration": 2.879, "text": "calculus"}, {"start": 100.4, "duration": 3.999, "text": "the gradient vector points in the"}, {"start": 102.159, "duration": 4.801, "text": "direction of steepest"}, {"start": 104.399, "duration": 4.72, "text": "increase of your function so if you're"}, {"start": 106.96, "duration": 4.24, "text": "at some particular x and y-coordinate"}, {"start": 109.119, "duration": 4.241, "text": "and you're up there on the function"}, {"start": 111.2, "duration": 2.48, "text": "value at that x-y coordinate you want to"}, {"start": 113.36, "duration": 2.48, "text": "know"}, {"start": 113.68, "duration": 4.16, "text": "how do i climb this mountain represented"}, {"start": 115.84, "duration": 3.52, "text": "by this function as fast as possible"}, {"start": 117.84, "duration": 3.279, "text": "where is this function steepest what"}, {"start": 119.36, "duration": 3.6, "text": "direction should i go you should go in"}, {"start": 121.119, "duration": 3.521, "text": "the direction of the gradient"}, {"start": 122.96, "duration": 3.28, "text": "i hope you enjoyed this video and if you"}, {"start": 124.64, "duration": 3.119, "text": "did please give it a like for the"}, {"start": 126.24, "duration": 3.28, "text": "youtube algorithm because we're all"}, {"start": 127.759, "duration": 3.681, "text": "mathematicians here and we appreciate"}, {"start": 129.52, "duration": 4.48, "text": "algorithm just as much as youtube does"}, {"start": 131.44, "duration": 3.84, "text": "if you have a question leave it down in"}, {"start": 134.0, "duration": 3.36, "text": "the comments and"}, {"start": 135.28, "duration": 4.48, "text": "we'll do some more math as always in the"}, {"start": 137.36, "duration": 2.4, "text": "next video"}], "title": "The Gradient Vector Field", "description": "One prominent example of a vector field is the Gradient Vector Field. Given any scalar, multivariable function f: R^n\\to R, we can get a corresponding vector field that has a precise geometrical meaning: the vectors point in the direction of maximal increase of the function. \n\nMY VECTOR CALCULUS PLAYLIST: \n\u25baVECTOR CALCULUS (Calc IV) https://www.youtube.com/playlist?list=PLHXZ9OQGMqxfW0GMqeUE1bLKaYor6kbHa\n\nOTHER COURSE PLAYLISTS:\n\u25baDISCRETE MATH: https://www.youtube.com/playlist?list=PLHXZ9OQGMqxersk8fUxiUMSIx0DBqsKZS\n\u25baLINEAR ALGEBRA: https://www.youtube.com/playlist?list=PLHXZ9OQGMqxfUl0tcqPNTJsb7R6BqSLo6\n\u25baCALCULUS I: https://www.youtube.com/playlist?list=PLHXZ9OQGMqxfT9RMcReZ4WcoVILP4k6-m\n\u25ba CALCULUS II: https://www.youtube.com/playlist?list=PLHXZ9OQGMqxc4ySKTIW19TLrT91Ik9M4n\n\u25baMULTIVARIABLE CALCULUS (Calc III): https://www.youtube.com/playlist?list=PLHXZ9OQGMqxc_CvEy7xBKRQr6I214QJcd\n\u25baDIFFERENTIAL EQUATIONS: https://www.youtube.com/watch?v=xeeM3TT4Zgg&list=PLHXZ9OQGMqxcJXnLr08cyNaup4RDsbAl1\n\nOTHER PLAYLISTS:\n\u25ba Learning Math Series\nhttps://www.youtube.com/watch?v=LPH2lqis3D0&list=PLHXZ9OQGMqxfSkRtlL5KPq6JqMNTh_MBw\n\u25baCool Math Series:\nhttps://www.youtube.com/playlist?list=PLHXZ9OQGMqxelE_9RzwJ-cqfUtaFBpiho\n\nBECOME A MEMBER:\n\u25baJoin: https://www.youtube.com/channel/UC9rTsvTxJnx1DNrDA3Rqa6A/join\n\nMATH BOOKS & MERCH I LOVE:\n\u25ba My Amazon Affiliate Shop: https://www.amazon.com/shop/treforbazett\n\nSOCIALS:\n\u25baTwitter (math based): http://twitter.com/treforbazett\n\u25baInstagram (photography based): http://instagram.com/treforphotography", "lengthSeconds": 138, "width": 640, "height": 360}
{"video_id": "oHijygNdx_8", "transcript": [{"start": 0.03, "duration": 5.729, "text": "well it's true that generally and"}, {"start": 2.629, "duration": 4.451, "text": "implication and it's converse are not"}, {"start": 5.759, "duration": 3.661, "text": "the same thing they don't have the same"}, {"start": 7.08, "duration": 5.82, "text": "truth value what you can have is a"}, {"start": 9.42, "duration": 6.21, "text": "scenario where both directions this the"}, {"start": 12.9, "duration": 5.639, "text": "original conditional and it's converse"}, {"start": 15.63, "duration": 4.47, "text": "where those are both true and when both"}, {"start": 18.539, "duration": 3.601, "text": "of those are true what we can have is"}, {"start": 20.1, "duration": 3.24, "text": "something called the biconditional"}, {"start": 22.14, "duration": 3.27, "text": "and you'll notice here that in the"}, {"start": 23.34, "duration": 3.87, "text": "biconditional it's got this sort of"}, {"start": 25.41, "duration": 4.619, "text": "double sided arrow because it's saying"}, {"start": 27.21, "duration": 6.779, "text": "that P implies Q and it's the case that"}, {"start": 30.029, "duration": 5.701, "text": "Q implies P this is the idea where there"}, {"start": 33.989, "duration": 3.511, "text": "P and Q are sort of like equivalent"}, {"start": 35.73, "duration": 4.739, "text": "property if you have one you get the"}, {"start": 37.5, "duration": 5.489, "text": "other and if you have the other you get"}, {"start": 40.469, "duration": 3.901, "text": "the initial one so let's try to figure"}, {"start": 42.989, "duration": 5.701, "text": "out what this works in terms of some"}, {"start": 44.37, "duration": 7.02, "text": "statements if I study hard then I will"}, {"start": 48.69, "duration": 8.939, "text": "pass so that was the statement P implies"}, {"start": 51.39, "duration": 9.21, "text": "Q but I can also go the other way around"}, {"start": 57.629, "duration": 6.36, "text": "this probably applies to most of us"}, {"start": 60.6, "duration": 6.629, "text": "I could say if I pass then I studied"}, {"start": 63.989, "duration": 6.391, "text": "hard so this is the statement Q implies"}, {"start": 67.229, "duration": 5.161, "text": "P and you'll notice that what I'm having"}, {"start": 70.38, "duration": 3.54, "text": "here is if I combine these together and"}, {"start": 72.39, "duration": 3.51, "text": "I think of them as one statement"}, {"start": 73.92, "duration": 4.44, "text": "it's a conjunctive statement staying on"}, {"start": 75.9, "duration": 4.17, "text": "that and in it it got one implication P"}, {"start": 78.36, "duration": 6.149, "text": "implies Q and it got a second"}, {"start": 80.07, "duration": 6.75, "text": "implication Q implies P and the way you"}, {"start": 84.509, "duration": 4.261, "text": "should think about this is that if we"}, {"start": 86.82, "duration": 3.24, "text": "exclude people like me or people who"}, {"start": 88.77, "duration": 2.79, "text": "have a lot of amelie arity with"}, {"start": 90.06, "duration": 2.82, "text": "materials they can sort of sit down and"}, {"start": 91.56, "duration": 2.01, "text": "not bother studying at all and just pass"}, {"start": 92.88, "duration": 2.489, "text": "the test already"}, {"start": 93.57, "duration": 3.74, "text": "although truly for myself that's just"}, {"start": 95.369, "duration": 4.5, "text": "good I studied a long time in the past"}, {"start": 97.31, "duration": 4.9, "text": "if we exclude those kinds of cases this"}, {"start": 99.869, "duration": 5.131, "text": "is probably true you need to study hard"}, {"start": 102.21, "duration": 5.189, "text": "in order to pass and if you pass you did"}, {"start": 105.0, "duration": 3.75, "text": "indeed study hard so both of these"}, {"start": 107.399, "duration": 3.481, "text": "different implications are going to be"}, {"start": 108.75, "duration": 4.38, "text": "true and I can combine them in a kind of"}, {"start": 110.88, "duration": 5.43, "text": "nice way look at my free zoology here I"}, {"start": 113.13, "duration": 7.58, "text": "say I will pass I use this funny word"}, {"start": 116.31, "duration": 6.87, "text": "funny phrase if and only if I study hard"}, {"start": 120.71, "duration": 4.6, "text": "so whenever we write that whenever we"}, {"start": 123.18, "duration": 6.27, "text": "say they sort of guess and only a thing"}, {"start": 125.31, "duration": 5.74, "text": "here this is a way of saying that both"}, {"start": 129.45, "duration": 5.59, "text": "directions are going to be"}, {"start": 131.05, "duration": 7.83, "text": "it's a way of saying if I study hard"}, {"start": 135.04, "duration": 6.45, "text": "then I will pass and if I passed then I"}, {"start": 138.88, "duration": 5.28, "text": "study hard that's what I have here but"}, {"start": 141.49, "duration": 4.53, "text": "the first of these ifs is referring to"}, {"start": 144.16, "duration": 4.14, "text": "the stop statement and the second of the"}, {"start": 146.02, "duration": 4.86, "text": "one that said only it is referring to"}, {"start": 148.3, "duration": 5.4, "text": "the other way around if I pass the neck"}, {"start": 150.88, "duration": 5.16, "text": "started hard so mathematician is really"}, {"start": 153.7, "duration": 4.65, "text": "like this phrase if and only if as a"}, {"start": 156.04, "duration": 5.16, "text": "sort of a shorthand for talking about by"}, {"start": 158.35, "duration": 5.16, "text": "conditionals as a shorthand for having a"}, {"start": 161.2, "duration": 4.29, "text": "conjunctive statement with two different"}, {"start": 163.51, "duration": 4.68, "text": "conditionals written down they just put"}, {"start": 165.49, "duration": 4.23, "text": "if one thing this is the passing the"}, {"start": 168.19, "duration": 6.23, "text": "thing on what it has that's going to be"}, {"start": 169.72, "duration": 4.7, "text": "true if and only if I study hard"}], "title": "Biconditional Statements | \"if and only if\"", "description": "A biconditional statement is one of the form \"if and only if\", sometimes written as \"iff\". The statement \"p if and only if q\" means \"p implies q\" AND \"q implies p\". That is, it is a conjunction of two individual conditional statements. In this video we will see how to write and interpret sentences using if and only if notation. \n\n\u25baFull DISCRETE MATH Course Playlist: https://www.youtube.com/playlist?list=PLHXZ9OQGMqxersk8fUxiUMSIx0DBqsKZS\n\nOther Course Playlists:\n\n\u25baCALCULUS I: https://www.youtube.com/playlist?list=PLHXZ9OQGMqxfT9RMcReZ4WcoVILP4k6-m\n\n\u25baCALCULUS II: https://www.youtube.com/playlist?list=PLHXZ9OQGMqxc4ySKTIW19TLrT91Ik9M4n\n\n\u25baCALCULUS III: https://www.youtube.com/playlist?list=PLHXZ9OQGMqxc_CvEy7xBKRQr6I214QJcd\n\n\u25baLINEAR ALGEBRA: https://www.youtube.com/playlist?list=PLHXZ9OQGMqxfUl0tcqPNTJsb7R6BqSLo6\n\n\u25ba Want to learn math effectively? Check out my \"Learning Math\" Series: https://www.youtube.com/watch?v=LPH2lqis3D0&list=PLHXZ9OQGMqxfSkRtlL5KPq6JqMNTh_MBw\n\n\u25baWant some cool math? Check out my \"Cool Math\" Series: https://www.youtube.com/playlist?list=PLHXZ9OQGMqxelE_9RzwJ-cqfUtaFBpiho\n\n*****************************************************\nYOUR TURN! Learning math requires more than just watching math videos, so make sure you reflect, ask questions, and do lots of practice problems!\n\r****************************************************\n\n\u25baFollow me on Twitter: http://twitter.com/treforbazett", "lengthSeconds": 173, "width": 640, "height": 360}
{"video_id": "AQ0f4rsbsrQ", "transcript": [{"start": 0.24, "duration": 5.96, "text": "now I want to look at an example where"}, {"start": 2.72, "duration": 6.32, "text": "this vacuously true part applies"}, {"start": 6.2, "duration": 5.2, "text": "vacuously true meant that when the"}, {"start": 9.04, "duration": 4.519, "text": "hypothesis was false so it didn't even"}, {"start": 11.4, "duration": 3.96, "text": "get off the ground then we were going to"}, {"start": 13.559, "duration": 3.72, "text": "still say that the true the statement is"}, {"start": 15.36, "duration": 4.4, "text": "true but but true in a sort of"}, {"start": 17.279, "duration": 6.441, "text": "unimportant or or uninteresting or"}, {"start": 19.76, "duration": 7.16, "text": "vacuous sense so consider this statement"}, {"start": 23.72, "duration": 6.08, "text": "if Trevor is a unicorn then everyone"}, {"start": 26.92, "duration": 6.319, "text": "gets an A in the class so it is set up"}, {"start": 29.8, "duration": 6.04, "text": "this this is an if P then Q I can say"}, {"start": 33.239, "duration": 5.801, "text": "this is an if"}, {"start": 35.84, "duration": 6.48, "text": "P then Q statement or alternatively I"}, {"start": 39.04, "duration": 5.92, "text": "can say it is a p implies Q"}, {"start": 42.32, "duration": 5.2, "text": "statement now what we know is that"}, {"start": 44.96, "duration": 4.279, "text": "because I can turn a conditional like"}, {"start": 47.52, "duration": 4.8, "text": "this into a disjunctive into an or"}, {"start": 49.239, "duration": 6.761, "text": "statement of the form not P or Q let me"}, {"start": 52.32, "duration": 6.28, "text": "try to rewrite this in that form so in"}, {"start": 56.0, "duration": 5.039, "text": "this statement what I've said is either"}, {"start": 58.6, "duration": 4.24, "text": "Trevor is not unicorn so this is the"}, {"start": 61.039, "duration": 4.641, "text": "statement not P"}, {"start": 62.84, "duration": 4.36, "text": "now or everyone gets an a that's the"}, {"start": 65.68, "duration": 2.88, "text": "same statement I had before that is"}, {"start": 67.2, "duration": 4.0, "text": "again"}, {"start": 68.56, "duration": 4.199, "text": "Q when it's written in this form however"}, {"start": 71.2, "duration": 3.599, "text": "when it's written in its disjunctive"}, {"start": 72.759, "duration": 4.4, "text": "form I think it's more obvious that we"}, {"start": 74.799, "duration": 4.761, "text": "should say that that is true either"}, {"start": 77.159, "duration": 4.92, "text": "Trevor is not a unicorn or everyone gets"}, {"start": 79.56, "duration": 5.199, "text": "an a one of those two things is true it"}, {"start": 82.079, "duration": 5.961, "text": "turns out to be unfortunately the uh"}, {"start": 84.759, "duration": 5.441, "text": "Trevor is not a unicorn one however we"}, {"start": 88.04, "duration": 4.119, "text": "still have this logical form that is is"}, {"start": 90.2, "duration": 3.919, "text": "equivalent to the first one that's"}, {"start": 92.159, "duration": 4.24, "text": "written in its conditional or"}, {"start": 94.119, "duration": 5.36, "text": "implication form and it's because of"}, {"start": 96.399, "duration": 5.601, "text": "examples like this that we say that it's"}, {"start": 99.479, "duration": 5.121, "text": "vacuously true yes this statement down"}, {"start": 102.0, "duration": 4.159, "text": "here is true uh one of these two things"}, {"start": 104.6, "duration": 4.32, "text": "in myor statement is true the Trevor is"}, {"start": 106.159, "duration": 4.681, "text": "not a unicorn but it's not interesting"}, {"start": 108.92, "duration": 3.08, "text": "that it's not an important conclusion"}, {"start": 110.84, "duration": 2.72, "text": "the statement doesn't really mean"}, {"start": 112.0, "duration": 3.64, "text": "anything I can say anything I want here"}, {"start": 113.56, "duration": 4.479, "text": "if pigs fly then you get a million bucks"}, {"start": 115.64, "duration": 6.68, "text": "uh all of these statements are sort of"}, {"start": 118.039, "duration": 4.281, "text": "true but true in a vacuous sets"}], "title": "Vacuously True Statements", "description": "Learning Objectives: Determine when a conditional statement is vacuously true\n\nA conditional statement of the form \"if P then Q\" is called vacuously true whenever P is false. This might seem counterintuitive, but consider the reverse. The conditional \"if P then Q\" is false whenever the assumption P is true and the conclusion Q is false. So when the assumption is false it doesn't matter what the conclusion is, and we call the conditional true, but in a vacuous sense. \n\n\r\n****************************************************\n\u25baFull Course Playlist: DISCRETE MATH: https://www.youtube.com/playlist?list=PLHXZ9OQGMqxersk8fUxiUMSIx0DBqsKZS\r\n\n****************************************************\r\nOther Playlists:\r\n\n\u25baCALCULUS I: https://www.youtube.com/playlist?list=PLHXZ9OQGMqxfT9RMcReZ4WcoVILP4k6-m\r\n\n\u25baCALCULUS II: https://www.youtube.com/playlist?list=PLHXZ9OQGMqxc4ySKTIW19TLrT91Ik9M4n\r\n\n\u25baLINEAR ALGEBRA: https://www.youtube.com/playlist?list=PLHXZ9OQGMqxfUl0tcqPNTJsb7R6BqSLo6\r\n\n\u25ba Want to learn math effectively? Check out my \"Learning Math\" Series: https://www.youtube.com/watch?v=LPH2lqis3D0&list=PLHXZ9OQGMqxfSkRtlL5KPq6JqMNTh_MBw\r\n\n\u25baWant some cool math? Check out my \"Cool Math\" Series: https://www.youtube.com/playlist?list=PLHXZ9OQGMqxelE_9RzwJ-cqfUtaFBpiho\r\n\n*****************************************************\r\nYOUR TURN! Learning math requires more than just watching videos, so make sure you reflect, ask questions, and do lots of practice problems!\r\n****************************************************\r\n\n\u25baFollow me on Twitter: http://twitter.com/treforbazett\r\n\nThis video was created by Dr. Trefor Bazett", "lengthSeconds": 120, "width": 640, "height": 360}
{"video_id": "CJPZsu1mfDw", "transcript": [{"start": 0.35, "duration": 5.44, "text": "so now that we understand what a"}, {"start": 2.85, "duration": 5.7, "text": "proposition is let's go ahead and"}, {"start": 5.79, "duration": 4.11, "text": "introduce some notation in particular"}, {"start": 8.55, "duration": 3.18, "text": "because we're writing out these"}, {"start": 9.9, "duration": 4.71, "text": "propositions and the full form like this"}, {"start": 11.73, "duration": 9.69, "text": "can get really tedious so we use some"}, {"start": 14.61, "duration": 20.99, "text": "notation so that notation is referred to"}, {"start": 21.42, "duration": 17.389, "text": "as propositional variables so we use"}, {"start": 35.6, "duration": 15.04, "text": "propositional variables these are also"}, {"start": 38.809, "duration": 17.761, "text": "known as so aka sentential variables and"}, {"start": 50.64, "duration": 5.93, "text": "we use these things to represent"}, {"start": 60.14, "duration": 15.13, "text": "propositions and we typically use the"}, {"start": 63.12, "duration": 15.63, "text": "symbols like p q r s etc to represent"}, {"start": 75.27, "duration": 6.8, "text": "these different propositions and these"}, {"start": 78.75, "duration": 3.32, "text": "variables right here"}, {"start": 90.25, "duration": 3.82, "text": "can I have to take the value of true"}, {"start": 92.63, "duration": 3.659, "text": "because remember a proposition can"}, {"start": 94.07, "duration": 4.95, "text": "either be true or false and so we"}, {"start": 96.289, "duration": 11.551, "text": "usually represent true using a tea right"}, {"start": 99.02, "duration": 10.139, "text": "here or false F so for example instead"}, {"start": 107.84, "duration": 3.059, "text": "of having to write down one plus one"}, {"start": 109.159, "duration": 3.75, "text": "equals two every time I want to refer to"}, {"start": 110.899, "duration": 4.17, "text": "that proposition right there I could"}, {"start": 112.909, "duration": 5.49, "text": "just say well I'm going to define that"}, {"start": 115.069, "duration": 7.381, "text": "to be the propositional variable P and"}, {"start": 118.399, "duration": 6.481, "text": "then this proposition right here I can"}, {"start": 122.45, "duration": 5.789, "text": "define that to be the propositional"}, {"start": 124.88, "duration": 5.46, "text": "variable Q and this longer proposition"}, {"start": 128.239, "duration": 4.621, "text": "right here I can very compactly"}, {"start": 130.34, "duration": 4.979, "text": "represent that simply by using the"}, {"start": 132.86, "duration": 6.36, "text": "propositional variable or sentential"}, {"start": 135.319, "duration": 5.791, "text": "variable R right here and as this"}, {"start": 139.22, "duration": 4.799, "text": "notation right here indicates these"}, {"start": 141.11, "duration": 6.81, "text": "things only take on the value of true or"}, {"start": 144.019, "duration": 10.831, "text": "false so for example if we were to look"}, {"start": 147.92, "duration": 15.3, "text": "at our different variables and also look"}, {"start": 154.85, "duration": 11.37, "text": "at their truth value then in this"}, {"start": 163.22, "duration": 6.06, "text": "particular example right here our prop"}, {"start": 166.22, "duration": 5.25, "text": "up proposition P turns out to be true"}, {"start": 169.28, "duration": 5.82, "text": "because yes indeed one plus one equals"}, {"start": 171.47, "duration": 7.769, "text": "two whereas Q over here that proposition"}, {"start": 175.1, "duration": 10.469, "text": "it takes on the value false right there"}, {"start": 179.239, "duration": 8.941, "text": "and our also happens to be true so these"}, {"start": 185.569, "duration": 5.431, "text": "are propositional variables that have"}, {"start": 188.18, "duration": 6.119, "text": "particular truth values this just gives"}, {"start": 191.0, "duration": 5.94, "text": "us a really compact way to refer to what"}, {"start": 194.299, "duration": 6.391, "text": "could be a very long declarative"}, {"start": 196.94, "duration": 9.199, "text": "sentence right here by the way this"}, {"start": 200.69, "duration": 5.449, "text": "thing right here is referred to as a"}, {"start": 206.68, "duration": 4.83, "text": "truth table and it's going to show up"}, {"start": 209.51, "duration": 5.3, "text": "quite a bit as we talk about"}, {"start": 211.51, "duration": 3.3, "text": "propositional logic"}], "title": "propositional variables", "description": "", "lengthSeconds": 214, "width": 640, "height": 360}
{"video_id": "LoTaJE16uLk", "transcript": [{"start": 0.08, "duration": 5.82, "text": "Hey! Can you cover complex analysis? [Chatters\nabout people requesting complex analysis over the years]"}, {"start": 5.9, "duration": 6.33, "text": "Alright let\u2019s do this. But why\nshould anybody care about complex analysis?"}, {"start": 12.23, "duration": 5.209, "text": "Well, just like any other field of maths,\nit is beautiful, which is enough for pure"}, {"start": 17.439, "duration": 5.911, "text": "mathematicians, but for applied mathematicians,\nphysicists, engineers and the like, there"}, {"start": 23.35, "duration": 7.07, "text": "are quite surprisingly lots of profound applications.\nIt is very useful in solving differential"}, {"start": 30.42, "duration": 6.13, "text": "equations, because the real and imaginary\nparts of a complex function satisfy a really"}, {"start": 36.55, "duration": 7.74, "text": "important differential equation. This equation\narises in many areas of physics, like electrostatics,"}, {"start": 44.29, "duration": 5.94, "text": "fluid dynamics, and heat conduction, but even\nif you are not solving this particular equation,"}, {"start": 50.23, "duration": 6.28, "text": "for any other DEs, a common trick is to use\nFourier or Laplace transforms, which requires"}, {"start": 56.51, "duration": 3.21, "text": "tools in complex analysis."}, {"start": 59.72, "duration": 5.22, "text": "If that\u2019s not tantalising enough, it is\nconnected to some popular math topics like"}, {"start": 64.94, "duration": 5.73, "text": "the fundamental theorem of algebra, and the\nRiemann hypothesis. Well, if even that is"}, {"start": 70.67, "duration": 5.17, "text": "not attractive enough, it allows you to show\noff! No, seriously, how would you evaluate"}, {"start": 75.84, "duration": 6.42, "text": "this integral? Using partial fractions would\nbe a nightmare, but if we consider its complex"}, {"start": 82.26, "duration": 6.64, "text": "version, integrating on a complex plane, it\nbecomes massively easier. And actually, these"}, {"start": 88.9, "duration": 5.07, "text": "are only applications that hopefully *most*\nof you would find interesting, but obviously"}, {"start": 93.97, "duration": 6.439, "text": "there are more! All I\u2019m trying to say is\nthat complex analysis is very powerful!"}, {"start": 100.409, "duration": 6.212, "text": "And so I am going to make a video series about\ncomplex analysis, and this is my plan. The"}, {"start": 106.621, "duration": 6.039, "text": "first one is this video, a preview of what\u2019s\ncoming next. Then a crash course on complex"}, {"start": 112.66, "duration": 6.95, "text": "numbers, so that we can all be on the same\npage, and then how to visualise complex functions,"}, {"start": 119.61, "duration": 6.52, "text": "and the messy stuff about complex functions.\nAnd then a particular kind of complex function,"}, {"start": 126.13, "duration": 6.46, "text": "the M\u00f6bius map, and the slick way of visualising\nit using the Riemann sphere. We can\u2019t not"}, {"start": 132.59, "duration": 6.35, "text": "talk about complex differentiation, but it\nturns out that if we want to visualise integration,"}, {"start": 138.94, "duration": 5.6, "text": "we also have to know a bit of vector calculus\nfirst. But after all these, we are going to"}, {"start": 144.54, "duration": 5.419, "text": "see *some* of the applications mentioned,\npossibly in several videos."}, {"start": 149.959, "duration": 6.321, "text": "However, I want to emphasise that it isn\u2019t\nlike this hasn\u2019t been seen anywhere on YouTube."}, {"start": 156.28, "duration": 5.28, "text": "Welch Labs made a series on complex numbers\nand functions, M\u00f6bius transformations were"}, {"start": 161.56, "duration": 6.19, "text": "illustrated more than 10 years ago in this\nvideo, Zach Star made a video on complex derivatives,"}, {"start": 167.75, "duration": 5.09, "text": "loads of people have covered vector calculus\nbefore, and actually, this whole series will"}, {"start": 172.84, "duration": 6.59, "text": "be very much from this amazing book called\n\u201cVisual Complex Analysis\u201d by Tristan Needham."}, {"start": 179.43, "duration": 5.82, "text": "So think of this series as a more convenient\ncompilation and *my* take and animation on"}, {"start": 185.25, "duration": 6.35, "text": "this subject. However, a little note that\nmight be disappointing to the purists. This"}, {"start": 191.6, "duration": 6.27, "text": "video focuses much more on visual insights,\nand the rigour is not prioritised in the arguments."}, {"start": 197.87, "duration": 5.13, "text": "By all means, make the arguments in the videos\nmore rigorous! But in any case, I hope you"}, {"start": 203.0, "duration": 2.53, "text": "will enjoy the series ahead!"}, {"start": 205.53, "duration": 5.5, "text": "Thanks to these patrons, and as always, like\nand subscribe with notifications on, comment"}, {"start": 211.03, "duration": 19.299, "text": "and share the video as well! See you next\ntime!"}], "title": "Why care about complex analysis? | Essence of complex analysis #1", "description": "Complex analysis is an incredibly powerful tool used in many applications, specifically in solving differential equations (Laplace's and others via inverse Fourier / Laplace transforms), and of course, fundamental theorem of algebra, Riemann hypothesis, as well as solving complicated integrals to show off!\n\nThis is the start of a series on complex analysis, which focuses on the visual insights rather than the more traditional rigorous approach, so it is not very likely that we will touch upon more advanced, yet very remarkable theorems in complex analysis, like Riemann mapping theorem, or the Picard's theorems. This is more likely of more use to the applied mathematicians, physicists and engineers, at least in the first few videos, simply because the interesting theorems are most likely pushed towards the end, and grouped as \"applications of complex analysis\", like FTA and RH.\n\nThis is only currently my plan, and things are likely to change, but hopefully this is the backbone of what's coming next. I might not be uploading like weekly, but hopefully as frequently as I hope. \n\nVideos mentioned:\nhttps://www.youtube.com/playlist?list=PLiaHhY2iBX9g6KIvZ_703G3KJXapKkNaF (Welch Labs complex numbers playlist)\nhttps://www.youtube.com/watch?v=0z1fIsUNhO4 (M\u00f6bius transformations video)\nhttps://www.youtube.com/watch?v=b8_3PFjiJvY (Zach Star complex derivatives)\nhttps://www.youtube.com/playlist?list=PLSQl0a2vh4HC5feHa6Rc5c0wbRTx56nF7 (Khan academy playlist on vector calculus)\nhttps://www.youtube.com/playlist?list=PLHXZ9OQGMqxfW0GMqeUE1bLKaYor6kbHa (Dr Trefor Bazett\u2019s playlist on vector calculus)\n\nSome (sort of) unexpected applications / connections to complex analysis:\nhttps://en.wikipedia.org/wiki/Joukowsky_transform (historical aerofoil application)\nhttp://math.colorado.edu/~kstange/papers/notes-Spin.pdf (Relationship between M\u00f6bius map and special relativity)\n\nOther than commenting on the video, you are very welcome to fill in a Google form linked below, which helps me make better videos by catering for your math levels:\nhttps://forms.gle/QJ29hocF9uQAyZyH6\n\nIf you want to know more interesting Mathematics, stay tuned for the next video!\n\nSUBSCRIBE and see you in the next video!\n\nIf you are wondering how I made all these videos, even though it is stylistically similar to 3Blue1Brown, I don't use his animation engine Manim, but I will probably reveal how I did it in a potential subscriber milestone, so do subscribe!\n\nSocial media:\n\nFacebook: https://www.facebook.com/mathemaniacyt\nInstagram: https://www.instagram.com/_mathemaniac_/\nTwitter: https://twitter.com/mathemaniacyt\nPatreon: https://www.patreon.com/mathemaniac (support if you want to and can afford to!)\n\nFor my contact email, check my About page on a PC.\n\nSee you next time!", "lengthSeconds": 235, "width": 640, "height": 360}
{"video_id": "xQeH2vrQ-5A", "transcript": [{"start": 4.0, "duration": 5.74, "text": "associated to every matrix there are two"}, {"start": 6.74, "duration": 5.729, "text": "canonical subspaces but a null space of"}, {"start": 9.74, "duration": 4.41, "text": "a column space and this video I want to"}, {"start": 12.469, "duration": 3.601, "text": "figure out what is the dimension of the"}, {"start": 14.15, "duration": 4.73, "text": "null space and what is the dimension of"}, {"start": 16.07, "duration": 5.849, "text": "the column space now we know that"}, {"start": 18.88, "duration": 4.87, "text": "dimension is just the number of basis"}, {"start": 21.919, "duration": 4.32, "text": "vectors and in fact in previous videos"}, {"start": 23.75, "duration": 4.619, "text": "we've actually seen how to figure out a"}, {"start": 26.239, "duration": 4.531, "text": "basis for the null space and a basis for"}, {"start": 28.369, "duration": 4.05, "text": "the column space so you just figure out"}, {"start": 30.77, "duration": 3.51, "text": "those bases and that's gonna be the"}, {"start": 32.419, "duration": 3.781, "text": "dimension so in this video I'm gonna"}, {"start": 34.28, "duration": 3.84, "text": "quickly go through that process and in"}, {"start": 36.2, "duration": 3.87, "text": "the next video we're going to see some"}, {"start": 38.12, "duration": 3.09, "text": "kind of crazy which is how the dimension"}, {"start": 40.07, "duration": 3.21, "text": "of the null space of how the dimension"}, {"start": 41.21, "duration": 3.98, "text": "of the column space actually relate to"}, {"start": 43.28, "duration": 4.11, "text": "each other and are interconnected so"}, {"start": 45.19, "duration": 5.29, "text": "first off dimension of the null space"}, {"start": 47.39, "duration": 5.4, "text": "I'll remind you that the null space is"}, {"start": 50.48, "duration": 4.23, "text": "all the vectors that are killed off by"}, {"start": 52.79, "duration": 4.14, "text": "applying particular matrix a they're all"}, {"start": 54.71, "duration": 5.52, "text": "the vectors where ax is equal to 0 and"}, {"start": 56.93, "duration": 5.67, "text": "my steps to find a basis for this are"}, {"start": 60.23, "duration": 4.8, "text": "number one I want to go and reduce the"}, {"start": 62.6, "duration": 3.93, "text": "matrix so but ax equal to zero I wanna"}, {"start": 65.03, "duration": 4.05, "text": "see maybe like some example like this"}, {"start": 66.53, "duration": 4.62, "text": "where I've got my Augmented majors with"}, {"start": 69.08, "duration": 4.56, "text": "the zero column I'm gonna put it"}, {"start": 71.15, "duration": 4.41, "text": "actually into our our EF form for you"}, {"start": 73.64, "duration": 3.839, "text": "then what I'm going to do as I say that"}, {"start": 75.56, "duration": 3.72, "text": "there's these two free columns are going"}, {"start": 77.479, "duration": 4.471, "text": "to give me variables for that I read off"}, {"start": 79.28, "duration": 4.56, "text": "of the rows they're gonna tell me what"}, {"start": 81.95, "duration": 4.89, "text": "the rows are going to do and I can put"}, {"start": 83.84, "duration": 4.62, "text": "it then into vector format now if you're"}, {"start": 86.84, "duration": 3.75, "text": "not really speedy with this pause the"}, {"start": 88.46, "duration": 4.98, "text": "video and do it yourself carefully but"}, {"start": 90.59, "duration": 4.8, "text": "the point I want to make is this when I"}, {"start": 93.44, "duration": 3.539, "text": "go through the standard process of"}, {"start": 95.39, "duration": 2.94, "text": "Gaussian elimination and back"}, {"start": 96.979, "duration": 3.561, "text": "substitution to put the vectors in"}, {"start": 98.33, "duration": 5.28, "text": "vector form what I get out of it is a"}, {"start": 100.54, "duration": 4.68, "text": "basis for the null space these two"}, {"start": 103.61, "duration": 4.38, "text": "vectors over here they're linearly"}, {"start": 105.22, "duration": 4.66, "text": "independent and indeed they're by"}, {"start": 107.99, "duration": 3.059, "text": "definition going to be spanning all the"}, {"start": 109.88, "duration": 3.18, "text": "vectors since every vector can be"}, {"start": 111.049, "duration": 3.841, "text": "written in this way so that's the basis"}, {"start": 113.06, "duration": 3.3, "text": "so in this case I've got a big sis it's"}, {"start": 114.89, "duration": 4.53, "text": "got two different vectors in it and"}, {"start": 116.36, "duration": 5.46, "text": "since the dimension is going to simply"}, {"start": 119.42, "duration": 3.57, "text": "be the number of basis vectors the"}, {"start": 121.82, "duration": 4.32, "text": "dimension of the null space is going to"}, {"start": 122.99, "duration": 5.279, "text": "be 2 here so my steps are number one to"}, {"start": 126.14, "duration": 4.409, "text": "reduce the matrix as we just saw number"}, {"start": 128.269, "duration": 4.351, "text": "two to note that the number of free"}, {"start": 130.549, "duration": 4.95, "text": "variables is the same thing as the"}, {"start": 132.62, "duration": 3.28, "text": "number of basis vectors and then we can"}, {"start": 135.499, "duration": 1.6, "text": "say that"}, {"start": 135.9, "duration": 2.789, "text": "dimension of the null space is just"}, {"start": 137.099, "duration": 3.661, "text": "going to be whatever the number of free"}, {"start": 138.689, "duration": 3.781, "text": "variables is going to be so that's"}, {"start": 140.76, "duration": 4.11, "text": "dimension of the null space what about"}, {"start": 142.47, "duration": 4.32, "text": "dimension of the column space now well"}, {"start": 144.87, "duration": 5.149, "text": "what's the column space remember it is"}, {"start": 146.79, "duration": 6.33, "text": "the span of the cause of matrix a and"}, {"start": 150.019, "duration": 5.711, "text": "indeed because the basis has to both"}, {"start": 153.12, "duration": 3.959, "text": "span and be literally independent the"}, {"start": 155.73, "duration": 3.149, "text": "only thing to check in the column space"}, {"start": 157.079, "duration": 4.231, "text": "is the move your independent part cuz it"}, {"start": 158.879, "duration": 4.2, "text": "gets a span part for free so if I take"}, {"start": 161.31, "duration": 3.149, "text": "my matrix like here's the same a I want"}, {"start": 163.079, "duration": 3.69, "text": "to reduce it in the exact same way"}, {"start": 164.459, "duration": 4.801, "text": "because here's the trick if I look at"}, {"start": 166.769, "duration": 4.14, "text": "the first and second column here well"}, {"start": 169.26, "duration": 3.78, "text": "there are multiples of each other"}, {"start": 170.909, "duration": 5.041, "text": "the second column is just twice the"}, {"start": 173.04, "duration": 4.77, "text": "first column so this if I look at all"}, {"start": 175.95, "duration": 4.98, "text": "four vectors they're not linearly"}, {"start": 177.81, "duration": 6.629, "text": "independent however if I focus just on"}, {"start": 180.93, "duration": 5.339, "text": "the vectors with leading ones that still"}, {"start": 184.439, "duration": 3.571, "text": "spans it and that are going to be"}, {"start": 186.269, "duration": 4.08, "text": "linearly independent so the color"}, {"start": 188.01, "duration": 4.619, "text": "leading ones are going to be my basis"}, {"start": 190.349, "duration": 3.48, "text": "for the column space and therefore the"}, {"start": 192.629, "duration": 3.411, "text": "dimension of the column space is just"}, {"start": 193.829, "duration": 4.44, "text": "going to be the number of leading what's"}, {"start": 196.04, "duration": 4.12, "text": "now in the next video what we're gonna"}, {"start": 198.269, "duration": 3.36, "text": "see is we've got the dimension of the"}, {"start": 200.16, "duration": 3.9, "text": "column space we've got the dimension of"}, {"start": 201.629, "duration": 5.241, "text": "the null space but in fact they are"}, {"start": 204.06, "duration": 2.81, "text": "intimately related"}], "title": "Computing Dimension of Null Space & Column Space", "description": "The dimension of a subspace is the number of basis vectors. For the two canonical subspaces associated to any matrix - the Null Space and the Column Space - we repeat quickly the computation of basis vectors for them and thus are able to compute their dimensions.\n\n**************************************************\n\u25baFull Playlist: https://www.youtube.com/playlist?list=PLHXZ9OQGMqxfUl0tcqPNTJsb7R6BqSLo6\n**************************************************\n\n**Now it's your turn**\n1) Summarize the big idea of this video in your own words\n2) Write down anything you are unsure about to think about later\n3) What questions for the future do you have? Where are we going with this content?\n4) Can you come up with your own sample test problem on this material? Solve it!\n\nLearning mathematics is best done by actually DOING mathematics. A video like this can only ever be a starting point. I might show you the basic ideas, definitions, formulas, and examples, but to truly master math means that you have to spend time - a lot of time! - sitting down and trying problems yourself, asking questions, and thinking about mathematics. So before you go on to the next video, pause and go THINK. \n\n***************************************************\nWant more ideas for learning math effectively?\n\u25baHow to Watch Math Videos:\nhttps://www.youtube.com/watch?v=Ljq5y3ksk8o\n\n\u25ba5 Tips to Make Math Practice Problems Effective: https://www.youtube.com/watch?v=LPH2lqis3D0\n\n****************************************************\n\u25baWant some cool math? Check out my \"Cool Math\" Series:\nhttps://www.youtube.com/playlist?list=PLHXZ9OQGMqxelE_9RzwJ-cqfUtaFBpiho\n\n****************************************************\nCourse Playlists:\n\u25baCalculus I: https://www.youtube.com/playlist?list=PLHXZ9OQGMqxfT9RMcReZ4WcoVILP4k6-m\n\n\u25baCalculus II: https://www.youtube.com/playlist?list=PLHXZ9OQGMqxdQCBjYswqbn7LxL1pW4cW4\n\n\u25baDiscrete Math: https://www.youtube.com/playlist?list=PLHXZ9OQGMqxersk8fUxiUMSIx0DBqsKZS\n\n\u25baLinear Algebra: https://www.youtube.com/playlist?list=PLHXZ9OQGMqxfUl0tcqPNTJsb7R6BqSLo6\n\n*****************************************************\n\u25baFollow me on Twitter: http://twitter.com/treforbazett\n*****************************************************\n\nThis video was created by Dr. Trefor Bazett, an Assistant Professor, Educator at the University of Cincinnati.\n\nBECOME A MEMBER:\n\u25baJoin: https://www.youtube.com/channel/UC9rTsvTxJnx1DNrDA3Rqa6A/join\n\nMATH BOOKS & MERCH I LOVE:\n\u25ba My Amazon Affiliate Shop: https://www.amazon.com/shop/treforbazett", "lengthSeconds": 206, "width": 640, "height": 360}
{"video_id": "NBy1j5PZfjk", "transcript": [{"start": 0.03, "duration": 4.94, "text": "in this example we want to figure out"}, {"start": 1.949, "duration": 7.381, "text": "the probability that if I randomly"}, {"start": 4.97, "duration": 6.1, "text": "choose a number between 3 and 14 where"}, {"start": 9.33, "duration": 3.929, "text": "I'm including the end I'm cluding 3 and"}, {"start": 11.07, "duration": 5.64, "text": "I'm including 14 what is the probability"}, {"start": 13.259, "duration": 5.581, "text": "that an even integer now what I want to"}, {"start": 16.71, "duration": 3.979, "text": "partially illustrate here is that this"}, {"start": 18.84, "duration": 4.019, "text": "questionable probability is"}, {"start": 20.689, "duration": 3.941, "text": "intrinsically tied to the idea of"}, {"start": 22.859, "duration": 3.541, "text": "counting in that that in many ways"}, {"start": 24.63, "duration": 4.409, "text": "answering your probability problem is is"}, {"start": 26.4, "duration": 4.65, "text": "answering a question about counting so"}, {"start": 29.039, "duration": 4.981, "text": "let's pull up the particular integers"}, {"start": 31.05, "duration": 4.919, "text": "that were interested in so I've got 3 4"}, {"start": 34.02, "duration": 3.92, "text": "all the way down to 14 and I've"}, {"start": 35.969, "duration": 5.671, "text": "highlighted in yellow the even integers"}, {"start": 37.94, "duration": 8.349, "text": "so what our formula told us is that I"}, {"start": 41.64, "duration": 7.079, "text": "want to investigate the number of things"}, {"start": 46.289, "duration": 5.131, "text": "in our event so this case the number of"}, {"start": 48.719, "duration": 5.73, "text": "evens happens to be denoted by an E as"}, {"start": 51.42, "duration": 5.04, "text": "well but it stands for event divided out"}, {"start": 54.449, "duration": 4.651, "text": "by the number in our sample space and"}, {"start": 56.46, "duration": 6.869, "text": "that this is going to be the probability"}, {"start": 59.1, "duration": 6.48, "text": "of our event so we have to do these two"}, {"start": 63.329, "duration": 7.801, "text": "counties so let's count the even ones"}, {"start": 65.58, "duration": 7.77, "text": "first I've got 4 6 8 10 12 and 14 so"}, {"start": 71.13, "duration": 6.14, "text": "that looks like 6 even are appearing"}, {"start": 73.35, "duration": 7.23, "text": "here so this is going to give me 6 and"}, {"start": 77.27, "duration": 8.919, "text": "then the total number of things between"}, {"start": 80.58, "duration": 8.73, "text": "3 and 14 is 12 so divided up by 12 here"}, {"start": 86.189, "duration": 8.281, "text": "in other words this is going to be 0.5"}, {"start": 89.31, "duration": 6.68, "text": "or 50% as well but notice that if I"}, {"start": 94.47, "duration": 4.859, "text": "change this question a little bit"}, {"start": 95.99, "duration": 6.97, "text": "suppose I then went and said from all"}, {"start": 99.329, "duration": 5.191, "text": "the way from 3 up to 15 well how does"}, {"start": 102.96, "duration": 3.78, "text": "that change 15 is not an even integer"}, {"start": 104.52, "duration": 5.849, "text": "but it would increase the number down"}, {"start": 106.74, "duration": 6.059, "text": "here from 12 to 13 and so your"}, {"start": 110.369, "duration": 4.671, "text": "probability would change because you've"}, {"start": 112.799, "duration": 4.71, "text": "increased the size of your sample space"}, {"start": 115.04, "duration": 4.179, "text": "even though you haven't changed the"}, {"start": 117.509, "duration": 5.36, "text": "number of even if you haven't changed"}, {"start": 119.219, "duration": 3.65, "text": "the number in your event"}], "title": "Example: Computing Probabilities using P(E)=N(E)/N(S)", "description": "This video computes the probability of an event using the formula P(E)=N(E)/N(S). That is the probability of an Event occurring counts the number of times that event occurs an divides by the total number in the sample space. \n\n\u25baFULL DISCRETE MATH PLAYLIST: https://www.youtube.com/playlist?list=PLHXZ9OQGMqxersk8fUxiUMSIx0DBqsKZS\n\nOTHER COURSE PLAYLISTS:\n\u25baCALCULUS I: https://www.youtube.com/playlist?list=PLHXZ9OQGMqxfT9RMcReZ4WcoVILP4k6-m\n\u25ba CALCULUS II: https://www.youtube.com/playlist?list=PLHXZ9OQGMqxc4ySKTIW19TLrT91Ik9M4n\n\u25baMULTIVARIABLE CALCULUS (Calc III): https://www.youtube.com/playlist?list=PLHXZ9OQGMqxc_CvEy7xBKRQr6I214QJcd\n\u25baVECTOR CALCULUS (Calc IV): https://www.youtube.com/playlist?list=PLHXZ9OQGMqxfW0GMqeUE1bLKaYor6kbHa\n\u25baDIFFERENTIAL EQUATIONS: https://www.youtube.com/watch?v=xeeM3TT4Zgg&list=PLHXZ9OQGMqxcJXnLr08cyNaup4RDsbAl1\n\u25baLINEAR ALGEBRA: https://www.youtube.com/playlist?list=PLHXZ9OQGMqxfUl0tcqPNTJsb7R6BqSLo6\n\nOTHER PLAYLISTS:\n\u25ba Learning Math Series\nhttps://www.youtube.com/watch?v=LPH2lqis3D0&list=PLHXZ9OQGMqxfSkRtlL5KPq6JqMNTh_MBw\n\u25baCool Math Series:\nhttps://www.youtube.com/playlist?list=PLHXZ9OQGMqxelE_9RzwJ-cqfUtaFBpiho\n\nBECOME A MEMBER:\n\u25baJoin: https://www.youtube.com/channel/UC9rTsvTxJnx1DNrDA3Rqa6A/join\n\nMATH BOOKS & MERCH I LOVE:\n\u25ba My Amazon Affiliate Shop: https://www.amazon.com/shop/treforbazett\n\nSOCIALS:\n\u25baTwitter (math based): http://twitter.com/treforbazett\n\u25baInstagram (photography based): http://instagram.com/treforphotography", "lengthSeconds": 122, "width": 640, "height": 360}
{"video_id": "4XrfeNIGUKM", "transcript": [{"start": 0.0, "duration": 5.52, "text": "recall that if I had two different"}, {"start": 2.399, "duration": 5.941, "text": "logical statements P and Q then I could"}, {"start": 5.52, "duration": 7.71, "text": "have the conditional between them that"}, {"start": 8.34, "duration": 6.9, "text": "said if P then Q and I can do the same"}, {"start": 13.23, "duration": 3.98, "text": "idea not just between two different"}, {"start": 15.24, "duration": 4.879, "text": "statements but between two different"}, {"start": 17.21, "duration": 6.19, "text": "predicates so that's what the idea of a"}, {"start": 20.119, "duration": 6.191, "text": "universal conditional is so we use this"}, {"start": 23.4, "duration": 4.86, "text": "notation P of X and notice that it's a a"}, {"start": 26.31, "duration": 5.039, "text": "double arrow opposed to a single arrow"}, {"start": 28.26, "duration": 5.76, "text": "here a P of X which is a predicate is"}, {"start": 31.349, "duration": 4.611, "text": "going to imply a different predicate Q"}, {"start": 34.02, "duration": 5.28, "text": "of X and what I mean by that is that"}, {"start": 35.96, "duration": 6.07, "text": "every time I put it in an input then the"}, {"start": 39.3, "duration": 4.62, "text": "predicate becomes a statement so what"}, {"start": 42.03, "duration": 4.2, "text": "I'm saying is for every possible input"}, {"start": 43.92, "duration": 6.119, "text": "in my domain or for every X inside of my"}, {"start": 46.23, "duration": 5.52, "text": "domain then the statement P of X where I"}, {"start": 50.039, "duration": 4.531, "text": "plugged in a specific value of X implies"}, {"start": 51.75, "duration": 6.39, "text": "the statement Q of X for that same value"}, {"start": 54.57, "duration": 6.57, "text": "of x so in other words a universal"}, {"start": 58.14, "duration": 4.8, "text": "conditional is going to be just like a"}, {"start": 61.14, "duration": 3.809, "text": "normal conditional but where your"}, {"start": 62.94, "duration": 4.77, "text": "statements are replaced by predicates"}, {"start": 64.949, "duration": 6.24, "text": "and then the condition will work for"}, {"start": 67.71, "duration": 6.479, "text": "every single input value so let's take a"}, {"start": 71.189, "duration": 5.79, "text": "look at this mathematical sentence this"}, {"start": 74.189, "duration": 6.631, "text": "has two different predicates inside of"}, {"start": 76.979, "duration": 5.341, "text": "it first of all I have if X is the POTUS"}, {"start": 80.82, "duration": 1.979, "text": "if X is the president of the United"}, {"start": 82.32, "duration": 3.089, "text": "States"}, {"start": 82.799, "duration": 5.941, "text": "that's a predicate and I will call it P"}, {"start": 85.409, "duration": 5.761, "text": "of X and then it says that if this is"}, {"start": 88.74, "duration": 5.1, "text": "going to be true then X is a US citizen"}, {"start": 91.17, "duration": 6.57, "text": "and I'll write that as a second"}, {"start": 93.84, "duration": 8.04, "text": "predicate so depending on what your X is"}, {"start": 97.74, "duration": 7.35, "text": "some X's if X has sort of lives in the"}, {"start": 101.88, "duration": 7.37, "text": "universe of people for instance if I set"}, {"start": 105.09, "duration": 6.86, "text": "my domain to be equal to just humans"}, {"start": 109.25, "duration": 6.49, "text": "some of those people are US citizens"}, {"start": 111.95, "duration": 7.0, "text": "some of those people are US presidents"}, {"start": 115.74, "duration": 6.3, "text": "and what this claim is is that all the"}, {"start": 118.95, "duration": 5.309, "text": "ones that are US presidents that set of"}, {"start": 122.04, "duration": 6.809, "text": "people throughout history all of those"}, {"start": 124.259, "duration": 6.791, "text": "people are US citizens note that when I"}, {"start": 128.849, "duration": 5.651, "text": "phrase it here as an if"}, {"start": 131.05, "duration": 7.59, "text": "as a conditional that the universal part"}, {"start": 134.5, "duration": 7.44, "text": "they for all X in the people part is"}, {"start": 138.64, "duration": 5.4, "text": "just sort of not explicitly written but"}, {"start": 141.94, "duration": 4.86, "text": "it still is they're implicitly when I"}, {"start": 144.04, "duration": 5.85, "text": "say if X is going to be the president"}, {"start": 146.8, "duration": 5.25, "text": "United States I am saying for all people"}, {"start": 149.89, "duration": 4.89, "text": "that happen to be the president of the"}, {"start": 152.05, "duration": 4.26, "text": "United States so this is sort of the"}, {"start": 154.78, "duration": 4.44, "text": "equivalence between these two ideas"}, {"start": 156.31, "duration": 5.19, "text": "saying if some predicate is going to be"}, {"start": 159.22, "duration": 3.54, "text": "true in other words if X has the"}, {"start": 161.5, "duration": 4.02, "text": "property of being the president noted"}, {"start": 162.76, "duration": 6.36, "text": "states is the same thing as saying for"}, {"start": 165.52, "duration": 5.55, "text": "all people or every person who is the"}, {"start": 169.12, "duration": 3.66, "text": "president United States they are those"}, {"start": 171.07, "duration": 5.1, "text": "people are then going to become US"}, {"start": 172.78, "duration": 5.67, "text": "citizens another way to phrase this is"}, {"start": 176.17, "duration": 5.25, "text": "sort of natural English that might make"}, {"start": 178.45, "duration": 5.52, "text": "the for all or the every more explicit"}, {"start": 181.42, "duration": 4.95, "text": "is something like this every president"}, {"start": 183.97, "duration": 5.07, "text": "of the United States is a US citizen I"}, {"start": 186.37, "duration": 5.28, "text": "haven't put it in my formal if-then"}, {"start": 189.04, "duration": 5.49, "text": "conditional form but the every that I"}, {"start": 191.65, "duration": 4.53, "text": "have here is the same thing as this for"}, {"start": 194.53, "duration": 3.72, "text": "all right every is kind of like a for"}, {"start": 196.18, "duration": 4.11, "text": "all so this is saying every president of"}, {"start": 198.25, "duration": 4.44, "text": "the United States is the same thing as"}, {"start": 200.29, "duration": 5.01, "text": "if X is the president of States is a US"}, {"start": 202.69, "duration": 4.64, "text": "citizen same thing as that X is a US"}, {"start": 205.3, "duration": 2.03, "text": "citizen"}], "title": "Universal Conditionals P(x) implies Q(x)", "description": "We describe the notation for universal conditionals of the form P(x) implies Q(x). This is short hand for saying that FOR EVERY specific value x, the logical implication P(x) implies  Q(x) is true. That is, this is a claim about two PREDICATES saying that the implication is true for all STATEMENTS where x is an element in the domain. \n\n\u25baFull DISCRETE MATH Course Playlist: https://www.youtube.com/playlist?list=PLHXZ9OQGMqxersk8fUxiUMSIx0DBqsKZS\n\nOther Course Playlists:\n\n\u25baCALCULUS I: https://www.youtube.com/playlist?list=PLHXZ9OQGMqxfT9RMcReZ4WcoVILP4k6-m\n\n\u25baCALCULUS II: https://www.youtube.com/playlist?list=PLHXZ9OQGMqxc4ySKTIW19TLrT91Ik9M4n\n\n\u25baCALCULUS III: https://www.youtube.com/playlist?list=PLHXZ9OQGMqxc_CvEy7xBKRQr6I214QJcd\n\n\u25baLINEAR ALGEBRA: https://www.youtube.com/playlist?list=PLHXZ9OQGMqxfUl0tcqPNTJsb7R6BqSLo6\n\n\u25ba Want to learn math effectively? Check out my \"Learning Math\" Series: https://www.youtube.com/watch?v=LPH2lqis3D0&list=PLHXZ9OQGMqxfSkRtlL5KPq6JqMNTh_MBw\n\n\u25baWant some cool math? Check out my \"Cool Math\" Series: https://www.youtube.com/playlist?list=PLHXZ9OQGMqxelE_9RzwJ-cqfUtaFBpiho\n\n*****************************************************\nYOUR TURN! Learning math requires more than just watching math videos, so make sure you reflect, ask questions, and do lots of practice problems!\n\r****************************************************\n\n\u25baFollow me on Twitter: http://twitter.com/treforbazett\n\nBECOME A MEMBER:\n\u25baJoin: https://www.youtube.com/channel/UC9rTsvTxJnx1DNrDA3Rqa6A/join\n\nMATH BOOKS & MERCH I LOVE:\n\u25ba My Amazon Affiliate Shop: https://www.amazon.com/shop/treforbazett", "lengthSeconds": 206, "width": 640, "height": 360}
{"video_id": "nzYxGpnllHA", "transcript": [{"start": 3.62, "duration": 4.67, "text": "in this video I want to investigate"}, {"start": 6.62, "duration": 5.36, "text": "whether I can make this particular"}, {"start": 8.29, "duration": 5.74, "text": "piecewise function continuous everywhere"}, {"start": 11.98, "duration": 4.51, "text": "well I have a little bit of freedom"}, {"start": 14.03, "duration": 3.72, "text": "notice that there's a see inside of"}, {"start": 16.49, "duration": 3.629, "text": "there so the hope is that by"}, {"start": 17.75, "duration": 4.619, "text": "manipulating the C I can make it"}, {"start": 20.119, "duration": 4.231, "text": "continuous everywhere first I want to"}, {"start": 22.369, "duration": 4.561, "text": "note this but look at this function here"}, {"start": 24.35, "duration": 4.62, "text": "or this function individually like if"}, {"start": 26.93, "duration": 4.05, "text": "this applied for all values of X either"}, {"start": 28.97, "duration": 4.62, "text": "of those are just polynomials one's a"}, {"start": 30.98, "duration": 4.62, "text": "quadratic one's a linear and polynomials"}, {"start": 33.59, "duration": 3.96, "text": "are continuous absolutely everywhere"}, {"start": 35.6, "duration": 4.229, "text": "there's no division by zero there's no"}, {"start": 37.55, "duration": 4.4, "text": "infinite spikes there's no oscillatory"}, {"start": 39.829, "duration": 5.011, "text": "behavior these are continuous everywhere"}, {"start": 41.95, "duration": 4.449, "text": "so the only possibility the only spot"}, {"start": 44.84, "duration": 4.89, "text": "where there could be a problem with this"}, {"start": 46.399, "duration": 4.711, "text": "value of x equal to one that's where it"}, {"start": 49.73, "duration": 4.44, "text": "might be that the limit from the left"}, {"start": 51.11, "duration": 6.03, "text": "and I'll limit from the right and the"}, {"start": 54.17, "duration": 4.86, "text": "function value might be different okay"}, {"start": 57.14, "duration": 5.399, "text": "so let's match them our goal is this I"}, {"start": 59.03, "duration": 5.669, "text": "want to say that the limit as X goes to"}, {"start": 62.539, "duration": 7.341, "text": "the interesting spot the 1 as X goes to"}, {"start": 64.699, "duration": 7.95, "text": "1 from let's do the left first of f of X"}, {"start": 69.88, "duration": 5.29, "text": "what I'm hoping is that this is going to"}, {"start": 72.649, "duration": 6.0, "text": "be the same thing as the limit as X goes"}, {"start": 75.17, "duration": 6.359, "text": "to 1 now from the right of my function f"}, {"start": 78.649, "duration": 4.231, "text": "of X so I need that to be the case if I"}, {"start": 81.529, "duration": 3.48, "text": "have any hope for it to be continuous"}, {"start": 82.88, "duration": 3.599, "text": "and an indeed my third condition is that"}, {"start": 85.009, "duration": 2.82, "text": "I want it to be equal to F of 1 that"}, {"start": 86.479, "duration": 5.611, "text": "one's going to come more or less for"}, {"start": 87.829, "duration": 6.33, "text": "free ok well now let's do the one from"}, {"start": 92.09, "duration": 5.639, "text": "the left now from the left is this top"}, {"start": 94.159, "duration": 5.46, "text": "one up here X values less than one so in"}, {"start": 97.729, "duration": 5.16, "text": "other words this is the same thing as"}, {"start": 99.619, "duration": 8.16, "text": "the limit as X goes to 1 from the left"}, {"start": 102.889, "duration": 6.6, "text": "of see x squared plus 1 and and this one"}, {"start": 107.779, "duration": 6.36, "text": "over here is just going to be equal to"}, {"start": 109.489, "duration": 10.141, "text": "the limit as X goes to one now from the"}, {"start": 114.139, "duration": 6.78, "text": "right of 2 X minus C and the F of 1 is"}, {"start": 119.63, "duration": 6.449, "text": "just going to be plugging in here this"}, {"start": 120.919, "duration": 6.24, "text": "is going to be equal to 2 minus C so"}, {"start": 126.079, "duration": 3.96, "text": "these are the three different things"}, {"start": 127.159, "duration": 4.41, "text": "that I need to be equal ok well how do i"}, {"start": 130.039, "duration": 4.081, "text": "do these limits both of these limits"}, {"start": 131.569, "duration": 4.981, "text": "over here are easy in the sense that"}, {"start": 134.12, "duration": 5.04, "text": "these are polynomials so that"}, {"start": 136.55, "duration": 5.04, "text": "ulis so the computer limit I just plug"}, {"start": 139.16, "duration": 4.08, "text": "it in so in this case if I plug in one"}, {"start": 141.59, "duration": 4.17, "text": "for the value of X so this is equal to"}, {"start": 143.24, "duration": 3.69, "text": "two minus C the same value as the"}, {"start": 145.76, "duration": 3.93, "text": "function value as we would have expected"}, {"start": 146.93, "duration": 6.39, "text": "and then over here I'm gonna plug in one"}, {"start": 149.69, "duration": 5.79, "text": "and I'm gonna get c plus one so this"}, {"start": 153.32, "duration": 5.1, "text": "gives me an equation it tells me that I"}, {"start": 155.48, "duration": 6.9, "text": "want C plus one to equal to minus C so"}, {"start": 158.42, "duration": 8.13, "text": "let's do that we're gonna see plus 1 to"}, {"start": 162.38, "duration": 5.7, "text": "be equal to two minus C and then if I do"}, {"start": 166.55, "duration": 3.66, "text": "a little bit of manipulation on this"}, {"start": 168.08, "duration": 3.81, "text": "okay I'm gonna move this C over and this"}, {"start": 170.21, "duration": 5.28, "text": "one I'm gonna subtract off so I'm gonna"}, {"start": 171.89, "duration": 5.64, "text": "have a two C is equal to one and so"}, {"start": 175.49, "duration": 3.32, "text": "finally I have C equal to the value of"}, {"start": 177.53, "duration": 4.74, "text": "one half"}, {"start": 178.81, "duration": 6.1, "text": "and indeed when I do this this value of"}, {"start": 182.27, "duration": 4.92, "text": "C would have hat it fixes the potential"}, {"start": 184.91, "duration": 3.93, "text": "discontinuity at one the letter from the"}, {"start": 187.19, "duration": 3.84, "text": "left is equal to limit from the right so"}, {"start": 188.84, "duration": 5.01, "text": "yes indeed this is continuous at one and"}, {"start": 191.03, "duration": 5.01, "text": "because of continuous everywhere else as"}, {"start": 193.85, "duration": 4.4, "text": "they're polynomials it's continuous"}, {"start": 196.04, "duration": 2.21, "text": "everywhere"}], "title": "Example: When is a Piecewise Function Continuous?", "description": "In this example, we look at a piecewise function that has a parameter. The question is, can we find a value of the parameter that makes the function continuous everywhere? Indeed, if we check that the limit from the left, the limit from the right, and the function value are all equal at the interesting point in the piece wise function, then we will get equations in the parameter we can solve. \n\n**************************************************\nNow it's your turn: \n1) Summarize the big idea of this video in your own words\n2) Write down anything you are unsure about to think about later\n3) What questions for the future do you have? Where are we going with this content?\n4) Can you come up with your own sample test problem on this material? Solve it!\n\nLearning mathematics is best done by actually DOING mathematics. A video like this can only ever be a starting point. I might show you the basic ideas, definitions, formulas, and examples, but to truly master math means that you have to spend time - a lot of time! - sitting down and trying problems yourself, asking questions, and thinking about mathematics. So before you go on to the next video, pause and go THINK. \n\n***************************************************\nWant more ideas for learning math effectively?\n\u25baHow to Watch Math Videos:\nhttps://www.youtube.com/watch?v=Ljq5y3ksk8o\n\n\u25ba5 Tips to Make Math Practice Problems Effective: https://www.youtube.com/watch?v=LPH2lqis3D0\n\n****************************************************\n\u25baWant some cool math? Check out my \"Cool Math\" Series:\nhttps://www.youtube.com/playlist?list=PLHXZ9OQGMqxelE_9RzwJ-cqfUtaFBpiho\n\n****************************************************\nCourse Playlists:\n\u25baCalculus I: https://www.youtube.com/playlist?list=PLHXZ9OQGMqxfT9RMcReZ4WcoVILP4k6-m\n\n\u25baCalculus II: https://www.youtube.com/playlist?list=PLHXZ9OQGMqxdQCBjYswqbn7LxL1pW4cW4\n\n\u25baDiscrete Math: https://www.youtube.com/playlist?list=PLHXZ9OQGMqxersk8fUxiUMSIx0DBqsKZS\n\n\u25baLinear Algebra: https://www.youtube.com/playlist?list=PLHXZ9OQGMqxfUl0tcqPNTJsb7R6BqSLo6\n\n*****************************************************\n\u25baFollow me on Twitter: http://twitter.com/treforbazett\n*****************************************************\n\nThis video was created by Dr. Trefor Bazett, an Assistant Professor, Educator at the University of Cincinnati.\n\nBECOME A MEMBER:\n\u25baJoin: https://www.youtube.com/channel/UC9rTsvTxJnx1DNrDA3Rqa6A/join\n\nMATH BOOKS & MERCH I LOVE:\n\u25ba My Amazon Affiliate Shop: https://www.amazon.com/shop/treforbazett", "lengthSeconds": 198, "width": 640, "height": 360}
{"video_id": "jZ-eL71AyzU", "transcript": [{"start": 2.72, "duration": 5.22, "text": "in this video I need to show you an"}, {"start": 5.029, "duration": 5.28, "text": "effective method to be able to take one"}, {"start": 7.94, "duration": 4.17, "text": "vector and write it in terms of some"}, {"start": 10.309, "duration": 3.961, "text": "other basis to write it in some other"}, {"start": 12.11, "duration": 3.72, "text": "coordinate system and while I show you"}, {"start": 14.27, "duration": 3.06, "text": "the best way to do this kind of"}, {"start": 15.83, "duration": 3.09, "text": "computation I'm also going to introduce"}, {"start": 17.33, "duration": 3.96, "text": "a little bit of notation it's going to"}, {"start": 18.92, "duration": 4.77, "text": "help us going forward in the course so"}, {"start": 21.29, "duration": 4.05, "text": "the problem is this I have some"}, {"start": 23.69, "duration": 4.29, "text": "particular basis specified in this case"}, {"start": 25.34, "duration": 5.73, "text": "looks like the base is 1 1 and minus 2 1"}, {"start": 27.98, "duration": 5.19, "text": "then I have some vector for example the"}, {"start": 31.07, "duration": 4.41, "text": "vector 4 1 and what I want to know how"}, {"start": 33.17, "duration": 7.319, "text": "to do is how do I write that vector in"}, {"start": 35.48, "duration": 7.32, "text": "the B basis that is if I take my 4 1"}, {"start": 40.489, "duration": 4.291, "text": "what are the coefficients the C 1 and"}, {"start": 42.8, "duration": 5.039, "text": "the C 2 so that it can be written here"}, {"start": 44.78, "duration": 5.669, "text": "as a linear combination of the b one of"}, {"start": 47.839, "duration": 4.261, "text": "the B 2 now whatever you see a linear"}, {"start": 50.449, "duration": 3.69, "text": "combination I want to have a little bit"}, {"start": 52.1, "duration": 4.649, "text": "alarm bell go off in your mind because"}, {"start": 54.139, "duration": 5.281, "text": "linear combinations and multiply by a"}, {"start": 56.749, "duration": 4.5, "text": "matrix are the same thing indeed running"}, {"start": 59.42, "duration": 3.059, "text": "this linear combination is just the same"}, {"start": 61.249, "duration": 3.961, "text": "thing as saying that this is the"}, {"start": 62.479, "duration": 5.551, "text": "multiplication of the C 1 and C 2 by"}, {"start": 65.21, "duration": 5.55, "text": "these particular matrix the 1 1 minus 2"}, {"start": 68.03, "duration": 5.7, "text": "1 well what do I have I have a linear"}, {"start": 70.76, "duration": 4.35, "text": "system matrix times a vector is equal to"}, {"start": 73.73, "duration": 3.33, "text": "some other vector and I want to solve"}, {"start": 75.11, "duration": 3.81, "text": "that system and I could use some row"}, {"start": 77.06, "duration": 3.69, "text": "reductions I could use my normal trick"}, {"start": 78.92, "duration": 3.81, "text": "to solve a linear system of equations"}, {"start": 80.75, "duration": 5.37, "text": "but I want to talk about it in terms of"}, {"start": 82.73, "duration": 5.73, "text": "invertible matrices so the matrix that I"}, {"start": 86.12, "duration": 5.76, "text": "have here I can give it a name this"}, {"start": 88.46, "duration": 6.66, "text": "matrix is called P sub B this is a"}, {"start": 91.88, "duration": 5.55, "text": "matrix associated to every single basis"}, {"start": 95.12, "duration": 3.93, "text": "and you just take the vectors of the"}, {"start": 97.43, "duration": 3.33, "text": "basis and put them in as the columns of"}, {"start": 99.05, "duration": 4.62, "text": "the matrix and then the other thing to"}, {"start": 100.76, "duration": 6.63, "text": "note is that the C 1 C 2 we previously"}, {"start": 103.67, "duration": 5.67, "text": "define that to be the vector X written"}, {"start": 107.39, "duration": 4.29, "text": "in the B basis so this is just going to"}, {"start": 109.34, "duration": 5.37, "text": "be my vector X written in the B basis at"}, {"start": 111.68, "duration": 5.88, "text": "4 1 is my X and the C 1 C 2 is my X"}, {"start": 114.71, "duration": 4.59, "text": "written in the B basis now what I'm"}, {"start": 117.56, "duration": 4.23, "text": "talking about all of our two or all of"}, {"start": 119.3, "duration": 4.74, "text": "our three but the P matrix I'd get this"}, {"start": 121.79, "duration": 4.29, "text": "is going to be a square matrix we're"}, {"start": 124.04, "duration": 3.69, "text": "talking about different bases of our two"}, {"start": 126.08, "duration": 3.0, "text": "so they're all going to have two"}, {"start": 127.73, "duration": 3.99, "text": "components and there's going to be two"}, {"start": 129.08, "duration": 3.41, "text": "of them and in fact because the basis"}, {"start": 131.72, "duration": 3.48, "text": "where the vectors are linearly"}, {"start": 132.49, "duration": 3.1, "text": "independent if your row reduced is gonna"}, {"start": 135.2, "duration": 2.789, "text": "have a lead"}, {"start": 135.59, "duration": 4.47, "text": "one in every single column and so that"}, {"start": 137.989, "duration": 4.351, "text": "this matrix is going to be invertible so"}, {"start": 140.06, "duration": 4.35, "text": "I can go and reverse this process I've"}, {"start": 142.34, "duration": 4.56, "text": "got right now X is equal to a matrix"}, {"start": 144.41, "duration": 4.59, "text": "times the vector expert in the B basis I"}, {"start": 146.9, "duration": 4.26, "text": "can invert it and what I'm gonna get is"}, {"start": 149.0, "duration": 5.97, "text": "this I'm gonna get that the C 1 C 2 is"}, {"start": 151.16, "duration": 6.57, "text": "the inverse of that matrix times the 4 1"}, {"start": 154.97, "duration": 5.22, "text": "times the X and I know how to take the"}, {"start": 157.73, "duration": 4.259, "text": "inverse of a 2x2 matrix I swap the a and"}, {"start": 160.19, "duration": 4.68, "text": "the D put a minus sign in front of the B"}, {"start": 161.989, "duration": 5.671, "text": "and the C you divide out by ad minus BC"}, {"start": 164.87, "duration": 5.76, "text": "so I inverted this particular matrix and"}, {"start": 167.66, "duration": 5.73, "text": "now I can just multiply it out and what"}, {"start": 170.63, "duration": 4.59, "text": "do I get I get the value 2 minus 1"}, {"start": 173.39, "duration": 3.45, "text": "in fact we always have these two"}, {"start": 175.22, "duration": 5.25, "text": "different ways of solve systems you can"}, {"start": 176.84, "duration": 5.94, "text": "do the row reduction methodology or if"}, {"start": 180.47, "duration": 3.96, "text": "it's a square matrix you can just invert"}, {"start": 182.78, "duration": 3.48, "text": "it but the nice part about being will"}, {"start": 184.43, "duration": 3.54, "text": "invert it is now you can do it for every"}, {"start": 186.26, "duration": 3.3, "text": "single vector at once you just always"}, {"start": 187.97, "duration": 3.299, "text": "have to just do this multiplication you"}, {"start": 189.56, "duration": 5.22, "text": "don't have to repeat that row reduction"}, {"start": 191.269, "duration": 4.891, "text": "step every single time so if you can use"}, {"start": 194.78, "duration": 3.66, "text": "invertible matrices I usually suggest"}, {"start": 196.16, "duration": 4.979, "text": "that's the superior way to do it in"}, {"start": 198.44, "duration": 5.25, "text": "generality regardless I have now figured"}, {"start": 201.139, "duration": 5.85, "text": "it out my C 1 C 2 is equal to the 2"}, {"start": 203.69, "duration": 5.01, "text": "minus 1 and in doing so I actually can"}, {"start": 206.989, "duration": 4.741, "text": "come up with two sort of general"}, {"start": 208.7, "duration": 5.789, "text": "formulas the first is going to be that X"}, {"start": 211.73, "duration": 6.0, "text": "can always be written as this P matrix"}, {"start": 214.489, "duration": 5.101, "text": "times X written in the B basis or to"}, {"start": 217.73, "duration": 4.229, "text": "flip it around the other way X written"}, {"start": 219.59, "duration": 5.009, "text": "in the B basis can be the inverse of"}, {"start": 221.959, "duration": 5.581, "text": "this P matrix multiplied by n where the"}, {"start": 224.599, "duration": 5.371, "text": "where the PB matrix just has the first"}, {"start": 227.54, "duration": 5.15, "text": "and second basis vector as the first and"}, {"start": 229.97, "duration": 2.72, "text": "second call"}], "title": "Writing Vectors in a New Coordinate System   **Example**", "description": "The best method to algebraically express a vector in some given coordinate system (ie in terms of a given basis) is by introducing a matrix whose columns are the basis vectors, and then inverting that. We pick up some new notation along the way. \n\n**************************************************\nNow it's your turn: \n1) Summarize the big idea of this video in your own words\n2) Write down anything you are unsure about to think about later\n3) What questions for the future do you have? Where are we going with this content?\n4) Can you come up with your own sample test problem on this material? Solve it!\n\nLearning mathematics is best done by actually DOING mathematics. A video like this can only ever be a starting point. I might show you the basic ideas, definitions, formulas, and examples, but to truly master math means that you have to spend time - a lot of time! - sitting down and trying problems yourself, asking questions, and thinking about mathematics. So before you go on to the next video, pause and go THINK. \n\n***************************************************\nWant more ideas for learning math effectively?\n\u25baHow to Watch Math Videos:\nhttps://www.youtube.com/watch?v=Ljq5y3ksk8o\n\n\u25ba5 Tips to Make Math Practice Problems Effective: https://www.youtube.com/watch?v=LPH2lqis3D0\n\n****************************************************\n\u25baWant some cool math? Check out my \"Cool Math\" Series:\nhttps://www.youtube.com/playlist?list=PLHXZ9OQGMqxelE_9RzwJ-cqfUtaFBpiho\n\n****************************************************\nCourse Playlists:\n\u25baCalculus I: https://www.youtube.com/playlist?list=PLHXZ9OQGMqxfT9RMcReZ4WcoVILP4k6-m\n\n\u25baCalculus II: https://www.youtube.com/playlist?list=PLHXZ9OQGMqxdQCBjYswqbn7LxL1pW4cW4\n\n\u25baDiscrete Math: https://www.youtube.com/playlist?list=PLHXZ9OQGMqxersk8fUxiUMSIx0DBqsKZS\n\n\u25baLinear Algebra: https://www.youtube.com/playlist?list=PLHXZ9OQGMqxfUl0tcqPNTJsb7R6BqSLo6\n\n*****************************************************\n\u25baFollow me on Twitter: http://twitter.com/treforbazett\n*****************************************************\n\nThis video was created by Dr. Trefor Bazett, an Assistant Professor, Educator at the University of Cincinnati.\n\nBECOME A MEMBER:\n\u25baJoin: https://www.youtube.com/channel/UC9rTsvTxJnx1DNrDA3Rqa6A/join\n\nMATH BOOKS & MERCH I LOVE:\n\u25ba My Amazon Affiliate Shop: https://www.amazon.com/shop/treforbazett", "lengthSeconds": 231, "width": 640, "height": 360}
{"video_id": "rZcCPrqr1CU", "transcript": [{"start": 0.03, "duration": 5.039, "text": "okay so let's look at one more example"}, {"start": 2.22, "duration": 5.76, "text": "here where I have a series and my series"}, {"start": 5.069, "duration": 4.921, "text": "is like really just kind of complicated"}, {"start": 7.98, "duration": 3.659, "text": "if I try to write out for example what"}, {"start": 9.99, "duration": 3.75, "text": "SN was it would just like be this"}, {"start": 11.639, "duration": 3.241, "text": "horrible mess and we don't really know"}, {"start": 13.74, "duration": 3.18, "text": "how to do anything with it"}, {"start": 14.88, "duration": 4.29, "text": "indeed this pretty much often the case"}, {"start": 16.92, "duration": 4.019, "text": "in the telescoping series we could write"}, {"start": 19.17, "duration": 3.66, "text": "out the SN and everything canceled it"}, {"start": 20.939, "duration": 3.721, "text": "was really neat but in the vast majority"}, {"start": 22.83, "duration": 3.57, "text": "of cases if we try to write out what SN"}, {"start": 24.66, "duration": 5.64, "text": "was it's just gonna be this horrifying"}, {"start": 26.4, "duration": 6.839, "text": "mess so how do we deal with this and if"}, {"start": 30.3, "duration": 4.5, "text": "I look at this I'm like very tempted to"}, {"start": 33.239, "duration": 3.931, "text": "do an integration because I see an e to"}, {"start": 34.8, "duration": 4.829, "text": "the minus N squared and then multiplied"}, {"start": 37.17, "duration": 5.49, "text": "by a power of n on the on the bottom it"}, {"start": 39.629, "duration": 3.871, "text": "kind of like screams you subs to me but"}, {"start": 42.66, "duration": 3.149, "text": "that's something we did back in"}, {"start": 43.5, "duration": 4.86, "text": "integration so let's just compare this"}, {"start": 45.809, "duration": 5.101, "text": "to the improper interval indeed if I"}, {"start": 48.36, "duration": 6.539, "text": "consider the improper integral 1 to"}, {"start": 50.91, "duration": 5.579, "text": "infinity of 4 times X e to the minus x"}, {"start": 54.899, "duration": 3.21, "text": "squared so I'm replacing my n with X"}, {"start": 56.489, "duration": 4.14, "text": "here DX"}, {"start": 58.109, "duration": 4.441, "text": "well I'd first know that this is the"}, {"start": 60.629, "duration": 4.591, "text": "same thing as the limit as T goes to"}, {"start": 62.55, "duration": 5.88, "text": "infinity of the integral from 1 up to T"}, {"start": 65.22, "duration": 5.7, "text": "of 4 X e to the minus x squared DX"}, {"start": 68.43, "duration": 3.6, "text": "that's how I deal with improper under"}, {"start": 70.92, "duration": 3.33, "text": "both I write this as the sort of this"}, {"start": 72.03, "duration": 3.9, "text": "limit of these quote-unquote partial"}, {"start": 74.25, "duration": 4.2, "text": "integrals are sort of analogous to"}, {"start": 75.93, "duration": 4.32, "text": "partial sums in series so if I do a"}, {"start": 78.45, "duration": 4.89, "text": "little bit of the u sub maybe I'm going"}, {"start": 80.25, "duration": 7.2, "text": "to say that U is equal to x squared that"}, {"start": 83.34, "duration": 8.25, "text": "means that D U is equal to 2x DX"}, {"start": 87.45, "duration": 6.86, "text": "if X is equal to 1 that implies that U"}, {"start": 91.59, "duration": 6.15, "text": "is equal to 1 and if X is equal to T"}, {"start": 94.31, "duration": 5.44, "text": "that implies that U is equal to T"}, {"start": 97.74, "duration": 4.68, "text": "squared so I can replace this entire"}, {"start": 99.75, "duration": 7.619, "text": "thing this is just the limit as T goes"}, {"start": 102.42, "duration": 11.159, "text": "to infinity of the integral from 1 up to"}, {"start": 107.369, "duration": 11.221, "text": "T squared I have a 4x so this is a 2 do"}, {"start": 113.579, "duration": 9.301, "text": "you E - the - you do u and this is just"}, {"start": 118.59, "duration": 6.48, "text": "the limit as T goes to infinity of a"}, {"start": 122.88, "duration": 6.93, "text": "minus sign because the e to the minus u"}, {"start": 125.07, "duration": 8.19, "text": "T e to the minus u evaluated between 1"}, {"start": 129.81, "duration": 4.98, "text": "and T squared which is this"}, {"start": 133.26, "duration": 7.17, "text": "same thing as limit as T goes to"}, {"start": 134.79, "duration": 8.88, "text": "infinity of minus 2 e to the minus T"}, {"start": 140.43, "duration": 4.83, "text": "squared all subtracted because I've got"}, {"start": 143.67, "duration": 7.62, "text": "two minus signs I go put a plus sign all"}, {"start": 145.26, "duration": 8.819, "text": "plus 2 e to the minus 1 and then if I"}, {"start": 151.29, "duration": 5.19, "text": "evaluate this as T goes to infinity this"}, {"start": 154.079, "duration": 5.28, "text": "this whole first term is going to zero"}, {"start": 156.48, "duration": 6.75, "text": "so just left with this 2 over e and"}, {"start": 159.359, "duration": 5.25, "text": "therefore that's my answer so that's my"}, {"start": 163.23, "duration": 5.16, "text": "answer for the integral the integral"}, {"start": 164.609, "duration": 7.081, "text": "converges to 2 over e and therefore my"}, {"start": 168.39, "duration": 6.769, "text": "original series from 1 to infinity of 4"}, {"start": 171.69, "duration": 6.659, "text": "and E the minus N squared converges and"}, {"start": 175.159, "duration": 4.931, "text": "again note that I'm not saying it"}, {"start": 178.349, "duration": 4.26, "text": "converges to 2 over e it almost"}, {"start": 180.09, "duration": 5.269, "text": "certainly doesn't I'm merely saying that"}, {"start": 182.609, "duration": 2.75, "text": "it converges"}], "title": "Integral Test - Bonus Example", "description": "Learning Objectives: \n1) Understand the concept of a remainder\n2) Follow the geometric derivation of estimating the remainder by integrals\n3) State the Remainder Estimate\n4) Apply the estimate to determine how many terms are needed to achieve a specific level of uncertainty. \n\nThis video is part of a Calculus II course taught at the University of Cincinnati.", "lengthSeconds": 184, "width": 640, "height": 360}
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"[\u0938\u0902\u0917\u0940\u0924]"}], "title": "Introduction to Nested Quantifiers", "description": "Discrete Mathematics: The Introduction to Nested Quantifiers\nTopics discussed:\n1) The definition of nested quantifiers.\n2) The example of nested quantifiers.\n3) Different combinations of nested quantifiers.\n\nFollow Neso Academy on Instagram: @nesoacademy(https://bit.ly/2XP63OE)\nFollow me on Instagram: @jaspreetedu(https://bit.ly/2YX26E5)\n\nContribute: http://www.nesoacademy.org/donate\n\nMemberships: https://bit.ly/2U7YSPI\n\nBooks: http://www.nesoacademy.org/recommended-books\n\nWebsite \u25ba http://www.nesoacademy.org/\nForum \u25ba http://forum.nesoacademy.org/\nFacebook \u25ba https://goo.gl/Nt0PmB\nTwitter      \u25ba https://twitter.com/nesoacademy\n\nMusic:\nAxol x Alex Skrindo - You [NCS Release]\n\n#DiscreteMathematicsByNeso #DiscreteMaths #NestedQuantifiers", "lengthSeconds": 197, "width": 640, "height": 360}
{"video_id": "dlKcfGu-WpI", "transcript": [{"start": 3.14, "duration": 5.77, "text": "the even and all of integers are"}, {"start": 5.88, "duration": 4.71, "text": "familiar concepts to us however in this"}, {"start": 8.91, "duration": 3.99, "text": "particular video what we're going to do"}, {"start": 10.59, "duration": 5.339, "text": "is we're going to try to define exactly"}, {"start": 12.9, "duration": 4.469, "text": "what we mean by the even and odd"}, {"start": 15.929, "duration": 4.08, "text": "integers we're going to try to have the"}, {"start": 17.369, "duration": 4.681, "text": "most precise and formal and rigorous"}, {"start": 20.009, "duration": 4.921, "text": "definition of these concepts that we can"}, {"start": 22.05, "duration": 4.83, "text": "have and the idea is this when we're"}, {"start": 24.93, "duration": 4.259, "text": "trying to make claims about mathematics"}, {"start": 26.88, "duration": 4.32, "text": "and when we're trying to prove claims in"}, {"start": 29.189, "duration": 4.8, "text": "mathematics we need to have a high"}, {"start": 31.2, "duration": 4.859, "text": "degree of precision and simply saying we"}, {"start": 33.989, "duration": 3.69, "text": "understand what even numbers are their"}, {"start": 36.059, "duration": 3.691, "text": "numbers like two four six eight and"}, {"start": 37.679, "duration": 3.961, "text": "seeing that we understand what odd"}, {"start": 39.75, "duration": 4.289, "text": "numbers like three five seven and nine"}, {"start": 41.64, "duration": 5.34, "text": "are is insufficient we need to use"}, {"start": 44.039, "duration": 5.851, "text": "incredibly precise ideas for what these"}, {"start": 46.98, "duration": 5.34, "text": "concepts actually are now when I look at"}, {"start": 49.89, "duration": 5.699, "text": "this in formal definition there's a sort"}, {"start": 52.32, "duration": 5.94, "text": "of existential nature to it when we say"}, {"start": 55.589, "duration": 5.34, "text": "that an integer say n can be written as"}, {"start": 58.26, "duration": 6.119, "text": "twice something really what we're saying"}, {"start": 60.929, "duration": 6.891, "text": "is that there exists this existential"}, {"start": 64.379, "duration": 6.211, "text": "claim there exists some other integer K"}, {"start": 67.82, "duration": 5.11, "text": "where the N can be written as twice at"}, {"start": 70.59, "duration": 4.559, "text": "so if I want to take this informal"}, {"start": 72.93, "duration": 4.02, "text": "definition and make it into a formal one"}, {"start": 75.149, "duration": 6.691, "text": "I might write something like this I"}, {"start": 76.95, "duration": 8.58, "text": "would say that n is an integer if there"}, {"start": 81.84, "duration": 6.3, "text": "exists another integer K so so there"}, {"start": 85.53, "duration": 7.019, "text": "exists a K any integers but has the"}, {"start": 88.14, "duration": 6.45, "text": "property that the N is twice K now why"}, {"start": 92.549, "duration": 4.231, "text": "do I like this formal definition so much"}, {"start": 94.59, "duration": 5.849, "text": "it's because now there's actually an"}, {"start": 96.78, "duration": 5.25, "text": "equation here N equals to K and I know"}, {"start": 100.439, "duration": 2.07, "text": "something about what K is K is an"}, {"start": 102.03, "duration": 2.82, "text": "integer"}, {"start": 102.509, "duration": 5.07, "text": "so if I want to make a proof that"}, {"start": 104.85, "duration": 5.04, "text": "involves the even integers now I can"}, {"start": 107.579, "duration": 4.411, "text": "actually substitute it in I can write"}, {"start": 109.89, "duration": 4.11, "text": "equation I have a precise way of writing"}, {"start": 111.99, "duration": 4.44, "text": "what I mean when I talk about an even"}, {"start": 114.0, "duration": 5.21, "text": "integer ok same story but for the"}, {"start": 116.43, "duration": 5.16, "text": "auditors well okay what's an odd integer"}, {"start": 119.21, "duration": 7.119, "text": "informally odd integers are the integers"}, {"start": 121.59, "duration": 8.49, "text": "that are just not even as in if two four"}, {"start": 126.329, "duration": 5.791, "text": "six eight are even integers then the"}, {"start": 130.08, "duration": 2.73, "text": "other ones though the one three five"}, {"start": 132.12, "duration": 4.24, "text": "seven"}, {"start": 132.81, "duration": 6.31, "text": "those are the auditors"}, {"start": 136.36, "duration": 6.09, "text": "you are allowed to define something as"}, {"start": 139.12, "duration": 4.74, "text": "being not some other thing okay but"}, {"start": 142.45, "duration": 4.29, "text": "let's try to be a little bit more"}, {"start": 143.86, "duration": 5.58, "text": "precise about this if I take say six we"}, {"start": 146.74, "duration": 5.28, "text": "said that was even because it was twice"}, {"start": 149.44, "duration": 5.4, "text": "an integer but what about seven so seven"}, {"start": 152.02, "duration": 5.18, "text": "is odd but what I notice about seven is"}, {"start": 154.84, "duration": 6.3, "text": "that it's twice an integer twice three"}, {"start": 157.2, "duration": 6.66, "text": "plus one so all of the odd integers can"}, {"start": 161.14, "duration": 5.61, "text": "really be written as twice an integer"}, {"start": 163.86, "duration": 4.96, "text": "plus one so I maybe use that as my"}, {"start": 166.75, "duration": 3.6, "text": "formal definition again it's an"}, {"start": 168.82, "duration": 4.71, "text": "existential claim I'm saying there"}, {"start": 170.35, "duration": 5.57, "text": "exists a cave in the integers such that"}, {"start": 173.53, "duration": 6.63, "text": "your audience a Duran can be written as"}, {"start": 175.92, "duration": 6.67, "text": "twice K plus one and now again I have a"}, {"start": 180.16, "duration": 4.47, "text": "precise formal definition of what I mean"}, {"start": 182.59, "duration": 3.93, "text": "by all integers there's an equation here"}, {"start": 184.63, "duration": 3.42, "text": "I can send it to that equation into"}, {"start": 186.52, "duration": 4.17, "text": "other equations that I can use these"}, {"start": 188.05, "duration": 4.23, "text": "definitions to make proofs about the"}, {"start": 190.69, "duration": 3.57, "text": "even and odd integers and indeed in the"}, {"start": 192.28, "duration": 3.84, "text": "next video that's exactly what I'm going"}, {"start": 194.26, "duration": 3.66, "text": "to do I'm going to do our first proof"}, {"start": 196.12, "duration": 4.619, "text": "and I'm going to use these definitions"}, {"start": 197.92, "duration": 5.569, "text": "of even and odd in the proof that I'm"}, {"start": 200.739, "duration": 2.75, "text": "going to do in the next video"}], "title": "Formal Definitions in Math | Ex: Even & Odd Integers", "description": "We've all seen even and odd integers before. But how - exactly - are they defined? How would you use them in a proof about the even, and odd integers. In this example we properly define the even and odd integers. In the next video, we will use them in a proof about even and odd integers. \n\n\u25baFull DISCRETE MATH Course Playlist: https://www.youtube.com/playlist?list=PLHXZ9OQGMqxersk8fUxiUMSIx0DBqsKZS\n\nOther Course Playlists:\n\n\u25baCALCULUS I: https://www.youtube.com/playlist?list=PLHXZ9OQGMqxfT9RMcReZ4WcoVILP4k6-m\n\n\u25baCALCULUS II: https://www.youtube.com/playlist?list=PLHXZ9OQGMqxc4ySKTIW19TLrT91Ik9M4n\n\n\u25baCALCULUS III: https://www.youtube.com/playlist?list=PLHXZ9OQGMqxc_CvEy7xBKRQr6I214QJcd\n\n\u25baLINEAR ALGEBRA: https://www.youtube.com/playlist?list=PLHXZ9OQGMqxfUl0tcqPNTJsb7R6BqSLo6\n\n\u25ba Want to learn math effectively? Check out my \"Learning Math\" Series: https://www.youtube.com/watch?v=LPH2lqis3D0&list=PLHXZ9OQGMqxfSkRtlL5KPq6JqMNTh_MBw\n\n\u25baWant some cool math? Check out my \"Cool Math\" Series: https://www.youtube.com/playlist?list=PLHXZ9OQGMqxelE_9RzwJ-cqfUtaFBpiho\n\n*****************************************************\nYOUR TURN! Learning math requires more than just watching math videos, so make sure you reflect, ask questions, and do lots of practice problems!\n\r****************************************************\n\n\u25baFollow me on Twitter: http://twitter.com/treforbazett\n\nBECOME A MEMBER:\n\u25baJoin: https://www.youtube.com/channel/UC9rTsvTxJnx1DNrDA3Rqa6A/join\n\nMATH BOOKS & MERCH I LOVE:\n\u25ba My Amazon Affiliate Shop: https://www.amazon.com/shop/treforbazett", "lengthSeconds": 203, "width": 640, "height": 360}
{"video_id": "6RC70C9FNXI", "transcript": [{"start": 3.48, "duration": 3.72, "text": "before you learn about Rings you should"}, {"start": 5.2, "duration": 4.08, "text": "really learn about groups so if you"}, {"start": 7.2, "duration": 5.16, "text": "haven't already watched our group video"}, {"start": 9.28, "duration": 7.96, "text": "then hit the pause button and check it"}, {"start": 12.36, "duration": 4.88, "text": "out all wait go"}, {"start": 17.68, "duration": 4.4, "text": "on roughly speaking a ring is a set of"}, {"start": 20.279, "duration": 3.881, "text": "elements where you can add subtract and"}, {"start": 22.08, "duration": 4.24, "text": "multiply but you may not be able to"}, {"start": 24.16, "duration": 4.4, "text": "divide and multiplication may not be"}, {"start": 26.32, "duration": 4.48, "text": "commutative for example the 2 x two"}, {"start": 28.56, "duration": 4.24, "text": "matrices with real numbers form a ring"}, {"start": 30.8, "duration": 3.96, "text": "and matrix multiplication is not"}, {"start": 32.8, "duration": 4.16, "text": "commutative by not requiring a"}, {"start": 34.76, "duration": 3.92, "text": "multiplication to be commutative Rings"}, {"start": 36.96, "duration": 4.079, "text": "generalize things like matrices which"}, {"start": 38.68, "duration": 3.96, "text": "are pretty important by the way did you"}, {"start": 41.039, "duration": 4.321, "text": "notice I said elements instead of"}, {"start": 42.64, "duration": 5.239, "text": "numbers H that's because some rings"}, {"start": 45.36, "duration": 5.519, "text": "consist of numbers While others are made"}, {"start": 47.879, "duration": 5.441, "text": "of more exotic objects like pols or"}, {"start": 50.879, "duration": 4.84, "text": "matrices before we give a more precise"}, {"start": 53.32, "duration": 5.48, "text": "definition let's see some"}, {"start": 55.719, "duration": 4.961, "text": "examples the integers are a ring you can"}, {"start": 58.8, "duration": 4.279, "text": "add subtract and multiply LLY any two"}, {"start": 60.68, "duration": 5.36, "text": "integers and you'll get another integer"}, {"start": 63.079, "duration": 5.36, "text": "but can you divide two integers H well"}, {"start": 66.04, "duration": 4.399, "text": "yes but you may not get an integer if"}, {"start": 68.439, "duration": 4.801, "text": "you divide 3 by five you get a fraction"}, {"start": 70.439, "duration": 4.761, "text": "not an integer that's key if you add"}, {"start": 73.24, "duration": 4.4, "text": "subtract or multiply two elements in the"}, {"start": 75.2, "duration": 4.52, "text": "ring you get another element in the ring"}, {"start": 77.64, "duration": 4.24, "text": "we say the ring is closed under addition"}, {"start": 79.72, "duration": 5.039, "text": "and multiplication what happens in the"}, {"start": 81.88, "duration": 5.44, "text": "ring stays in the"}, {"start": 84.759, "duration": 4.161, "text": "ring another example of a ring is the"}, {"start": 87.32, "duration": 4.04, "text": "set of polinomial with integer"}, {"start": 88.92, "duration": 4.479, "text": "coefficients you can freely add subtract"}, {"start": 91.36, "duration": 4.439, "text": "and multiply two such polinomial and you"}, {"start": 93.399, "duration": 5.161, "text": "will get a third polinomial of this type"}, {"start": 95.799, "duration": 6.841, "text": "but division no I don't think so if you"}, {"start": 98.56, "duration": 9.28, "text": "divide x +1 by x -1 you get x +1 over x"}, {"start": 102.64, "duration": 7.36, "text": "-1 hey it's a thing but it's not a"}, {"start": 107.84, "duration": 4.319, "text": "polinomial so now for the official"}, {"start": 110.0, "duration": 4.24, "text": "definition a ring is a set of elements"}, {"start": 112.159, "duration": 4.121, "text": "with two operations addition and"}, {"start": 114.24, "duration": 4.0, "text": "multiplication if you add two elements"}, {"start": 116.28, "duration": 4.519, "text": "in a ring you get another element in the"}, {"start": 118.24, "duration": 5.68, "text": "ring but earlier I said you can add and"}, {"start": 120.799, "duration": 4.761, "text": "subtract in a ring what gives well"}, {"start": 123.92, "duration": 3.92, "text": "subtraction is just the same as adding"}, {"start": 125.56, "duration": 4.08, "text": "inverses so when I said you can add and"}, {"start": 127.84, "duration": 4.2, "text": "subtract I was essentially saying that"}, {"start": 129.64, "duration": 4.319, "text": "rings have negative elements and if you"}, {"start": 132.04, "duration": 4.76, "text": "add an element to its negative you get"}, {"start": 133.959, "duration": 5.441, "text": "zero so in a ring the elements are a"}, {"start": 136.8, "duration": 5.48, "text": "group under addition better yet addition"}, {"start": 139.4, "duration": 5.16, "text": "is commutative in"}, {"start": 142.28, "duration": 4.52, "text": "rings now here's where things get"}, {"start": 144.56, "duration": 4.039, "text": "serious if you multiply two elements in"}, {"start": 146.8, "duration": 4.76, "text": "a ring you get another element in the"}, {"start": 148.599, "duration": 5.28, "text": "ring it's close under multiplication no"}, {"start": 151.56, "duration": 4.679, "text": "surprise there but elements may not have"}, {"start": 153.879, "duration": 4.841, "text": "inverses under multiplication we may not"}, {"start": 156.239, "duration": 4.761, "text": "be able to divide so while the ring is a"}, {"start": 158.72, "duration": 4.599, "text": "nice aelan group under addition things"}, {"start": 161.0, "duration": 6.239, "text": "may not be so nice with multiplication"}, {"start": 163.319, "duration": 6.241, "text": "but hey at least multiplication is"}, {"start": 167.239, "duration": 3.841, "text": "associative and to wrap things up"}, {"start": 169.56, "duration": 4.24, "text": "there's a rule which connects addition"}, {"start": 171.08, "duration": 3.92, "text": "and multiplication the distributive rule"}, {"start": 173.8, "duration": 3.439, "text": "all the numbers you studied in"}, {"start": 175.0, "duration": 4.44, "text": "arithmetic obey the distributive rule"}, {"start": 177.239, "duration": 3.161, "text": "the expressions in algebra do as well so"}, {"start": 179.44, "duration": 3.04, "text": "we're good"}, {"start": 180.4, "duration": 5.72, "text": "this isn't something new ladies and"}, {"start": 182.48, "duration": 3.64, "text": "gentlemen I give you the"}, {"start": 191.519, "duration": 3.0, "text": "rank"}], "title": "Abstract Algebra: The definition of a Ring", "description": "Learn the definition of a ring, one of the central objects in abstract algebra.  We give several examples to illustrate this concept including matrices and polynomials.\n\nBe sure to subscribe so you don't miss new lessons from Socratica:  \nhttp://bit.ly/1ixuu9W \n\n\u2666\u2666\u2666\u2666\u2666\u2666\u2666\u2666\u2666\u2666\n\nWe recommend the following textbooks:\nDummit & Foote, Abstract Algebra 3rd Edition\nhttp://amzn.to/2oOBd5S\n\nMilne, Algebra Course Notes (available free online)\nhttp://www.jmilne.org/math/CourseNotes/index.html \n\n\u2666\u2666\u2666\u2666\u2666\u2666\u2666\u2666\u2666\u2666\n\nWays to support our channel:\n\n\u25ba  Join our Patreon : https://www.patreon.com/socratica\n\u25ba  Make a one-time PayPal donation: https://www.paypal.me/socratica\n\u25ba  We also accept Bitcoin @ 1EttYyGwJmpy9bLY2UcmEqMJuBfaZ1HdG9\n\nThank you!\n \n\u2666\u2666\u2666\u2666\u2666\u2666\u2666\u2666\u2666\u2666\n\nConnect with us!\n\nFacebook: https://www.facebook.com/SocraticaStudios/\nInstagram: https://www.instagram.com/SocraticaStudios/\nTwitter: https://twitter.com/Socratica\n\n\u2666\u2666\u2666\u2666\u2666\u2666\u2666\u2666\u2666\u2666\n\nTeaching\u200b \u200bAssistant:\u200b \u200b\u200b \u200bLiliana\u200b \u200bde\u200b \u200bCastro \nWritten\u200b \u200b&\u200b \u200bDirected\u200b \u200bby\u200b \u200bMichael\u200b \u200bHarrison \nProduced\u200b \u200bby\u200b \u200bKimberly\u200b \u200bHatch\u200b \u200bHarrison\n\n\u2666\u2666\u2666\u2666\u2666\u2666\u2666\u2666\u2666\u2666", "lengthSeconds": 193, "width": 640, "height": 360}
{"video_id": "Ji2vr-9duPI", "transcript": [{"start": 0.52, "duration": 6.799, "text": "one kind of funny statement is a toogy a"}, {"start": 4.279, "duration": 5.841, "text": "toogy and we we usually use T for that"}, {"start": 7.319, "duration": 5.32, "text": "opposed to P or Q or R we use the symbol"}, {"start": 10.12, "duration": 4.679, "text": "t for a topology this is the statement"}, {"start": 12.639, "duration": 4.72, "text": "that is always true and can never be"}, {"start": 14.799, "duration": 6.201, "text": "false so for instance consider the"}, {"start": 17.359, "duration": 5.321, "text": "statement that dog is a mammal well that"}, {"start": 21.0, "duration": 4.24, "text": "statement is always true doesn't matter"}, {"start": 22.68, "duration": 5.919, "text": "which dog you're referring to in the"}, {"start": 25.24, "duration": 5.08, "text": "concept of what a dog is implicitly is"}, {"start": 28.599, "duration": 3.401, "text": "the concept that it is going to be a"}, {"start": 30.32, "duration": 4.399, "text": "mammal that is one of the properties of"}, {"start": 32.0, "duration": 6.239, "text": "being a dog so if I if I have the"}, {"start": 34.719, "duration": 5.721, "text": "statement that dog is a mammal it does"}, {"start": 38.239, "duration": 4.921, "text": "not matter which dog I'm referring to so"}, {"start": 40.44, "duration": 6.32, "text": "this statement is always true"}, {"start": 43.16, "duration": 5.64, "text": "irregardless of my inputs so this is"}, {"start": 46.76, "duration": 3.56, "text": "sort of play some funny games when it"}, {"start": 48.8, "duration": 4.88, "text": "comes to looking at what the truth"}, {"start": 50.32, "duration": 5.239, "text": "tables are so I'm going to imagine that"}, {"start": 53.68, "duration": 5.719, "text": "I've got the statement I want to look at"}, {"start": 55.559, "duration": 5.68, "text": "what T or p is where I've got a toy T"}, {"start": 59.399, "duration": 5.4, "text": "and A ology it's truth table is very"}, {"start": 61.239, "duration": 5.401, "text": "simple it's just always true right the"}, {"start": 64.799, "duration": 4.441, "text": "the topology can never be false so it's"}, {"start": 66.64, "duration": 4.119, "text": "just true true true true everywhere and"}, {"start": 69.24, "duration": 3.84, "text": "then I also have this other statement p"}, {"start": 70.759, "duration": 4.201, "text": "p is just a normal old statement here so"}, {"start": 73.08, "duration": 4.28, "text": "it can be true or false so maybe its"}, {"start": 74.96, "duration": 4.56, "text": "first possibility will be true and a"}, {"start": 77.36, "duration": 5.6, "text": "second possibility will be"}, {"start": 79.52, "duration": 6.52, "text": "false and then if I want to take a"}, {"start": 82.96, "duration": 5.32, "text": "topology or some statement remember or"}, {"start": 86.04, "duration": 5.68, "text": "is true if if one of the two are going"}, {"start": 88.28, "duration": 5.0, "text": "to be true but the ology is always true"}, {"start": 91.72, "duration": 3.32, "text": "so it's always the case that one of them"}, {"start": 93.28, "duration": 3.519, "text": "is going to be true here so there's two"}, {"start": 95.04, "duration": 4.2, "text": "truths there which is good enough to"}, {"start": 96.799, "duration": 3.96, "text": "give me a true here there's only one"}, {"start": 99.24, "duration": 3.199, "text": "true here but that's fine it's an or"}, {"start": 100.759, "duration": 4.281, "text": "statement I only need one of them to be"}, {"start": 102.439, "duration": 5.481, "text": "true so that's going to be true there as"}, {"start": 105.04, "duration": 6.039, "text": "well I can also have contradictions"}, {"start": 107.92, "duration": 4.92, "text": "contradictions are labeled by a c now"}, {"start": 111.079, "duration": 4.08, "text": "for contradiction and this is a"}, {"start": 112.84, "duration": 5.919, "text": "statement that is always false so this"}, {"start": 115.159, "duration": 6.041, "text": "is an example that dog is a reptile"}, {"start": 118.759, "duration": 4.481, "text": "again in the notion of a dog it is"}, {"start": 121.2, "duration": 5.0, "text": "implicit that dogs are mammals and not"}, {"start": 123.24, "duration": 6.439, "text": "reptiles so it can never be true that"}, {"start": 126.2, "duration": 7.039, "text": "that dog is a reptile so this means that"}, {"start": 129.679, "duration": 6.2, "text": "in our truth table whenever I have a c"}, {"start": 133.239, "duration": 5.241, "text": "it is going to be a column of all"}, {"start": 135.879, "duration": 5.801, "text": "falsehoods so here I'm going to have a"}, {"start": 138.48, "duration": 4.64, "text": "false and here I'm going to have a false"}, {"start": 141.68, "duration": 4.32, "text": "and then I've also done a simple example"}, {"start": 143.12, "duration": 4.52, "text": "I'm trying to look at C and P where p is"}, {"start": 146.0, "duration": 3.36, "text": "just some other statements so let's fill"}, {"start": 147.64, "duration": 5.319, "text": "in a couple different possibilities for"}, {"start": 149.36, "duration": 5.44, "text": "p could be true and it could be false"}, {"start": 152.959, "duration": 5.401, "text": "and then if I want to look at my final"}, {"start": 154.8, "duration": 7.24, "text": "conclusion which was C and P remember"}, {"start": 158.36, "duration": 5.72, "text": "and requires both of them to be true but"}, {"start": 162.04, "duration": 3.559, "text": "because we have my contradiction always"}, {"start": 164.08, "duration": 3.6, "text": "been false it will never be the case"}, {"start": 165.599, "duration": 3.36, "text": "that both of them are true and so since"}, {"start": 167.68, "duration": 2.919, "text": "there's one false there there's going to"}, {"start": 168.959, "duration": 4.28, "text": "be a false there since there's a false"}, {"start": 170.599, "duration": 5.801, "text": "there there's going to be false there as"}, {"start": 173.239, "duration": 7.161, "text": "well so when I had an or statement with"}, {"start": 176.4, "duration": 5.44, "text": "a a a topology it forced that to be toy"}, {"start": 180.4, "duration": 3.919, "text": "it made it always going to be true if"}, {"start": 181.84, "duration": 3.759, "text": "you if you took or a toy you could just"}, {"start": 184.319, "duration": 3.64, "text": "take the value of the toy would be"}, {"start": 185.599, "duration": 4.121, "text": "always true if you take and a"}, {"start": 187.959, "duration": 4.161, "text": "contradiction then it's just going to"}, {"start": 189.72, "duration": 6.239, "text": "make it into its own contradiction which"}, {"start": 192.12, "duration": 9.88, "text": "is going to be all falses in other words"}, {"start": 195.959, "duration": 6.041, "text": "I can say that that c and P is a"}, {"start": 203.08, "duration": 5.799, "text": "contradiction it will always be false as"}, {"start": 205.84, "duration": 8.039, "text": "this table I mentioned and as we saw in"}, {"start": 208.879, "duration": 7.961, "text": "the previous slide if I take t or P that"}, {"start": 213.879, "duration": 5.961, "text": "is always true and so it is always a"}, {"start": 216.84, "duration": 3.0, "text": "topology"}], "title": "Tautologies and Contradictions", "description": "Learning Objectives: \n1) Tautologies: examples and truth table\n2) Contradictions: examples and truth table\n3) Use tautologies and contradictions in compound statements. \n\n****************************************************\r\n\u25baFull Course Playlist: DISCRETE MATH: https://www.youtube.com/playlist?list=PLHXZ9OQGMqxersk8fUxiUMSIx0DBqsKZS\n****************************************************\r\nOther Course Playlists:\r\n\u25baCALCULUS I: https://www.youtube.com/playlist?list=PLHXZ9OQGMqxfT9RMcReZ4WcoVILP4k6-m\r\n\n\u25baCALCULUS II: https://www.youtube.com/playlist?list=PLHXZ9OQGMqxc4ySKTIW19TLrT91Ik9M4n\r\n\n\u25baLINEAR ALGEBRA: https://www.youtube.com/playlist?list=PLHXZ9OQGMqxfUl0tcqPNTJsb7R6BqSLo6\n***************************************************\n\u25ba Want to learn math effectively? Check out my \"Learning Math\" Series: https://www.youtube.com/watch?v=LPH2lqis3D0&list=PLHXZ9OQGMqxfSkRtlL5KPq6JqMNTh_MBw\r\n\n\u25baWant some cool math? Check out my \"Cool Math\" Series: https://www.youtube.com/playlist?list=PLHXZ9OQGMqxelE_9RzwJ-cqfUtaFBpiho\r\n\n*****************************************************\nYOUR TURN! Learning math requires more than just watching videos, so make sure you reflect, ask questions, and do lots of practice problems!\n****************************************************\n\u25baFollow me on Twitter: http://twitter.com/treforbazett\n*****************************************************\r\n\nThis video was created by Dr. Trefor Bazett", "lengthSeconds": 221, "width": 640, "height": 360}
{"video_id": "6HIlT6oSvXc", "transcript": [{"start": 2.919, "duration": 6.6, "text": "\u0939\u093e\u0909 \u0935\u093f\u0932 \u092f\u0942 \u092a\u094d\u0932\u0949\u091f \u0926 \u0915\u0949\u092e\u094d\u0932\u0947\u0915\u094d\u0938 \u0928\u0902\u092c\u0930 a + \u092c"}, {"start": 5.6, "duration": 6.8, "text": "\u0911\u0928 \u0926 \u0906\u0930\u094d\u0917\u0928 \u092a\u094d\u0932\u0947\u0928 \u0905\u091c\u0942\u092e \u090f \u090f\u0902\u0921 \u092c\u0940 \u0906\u0930 \u092c\u094b\u0925"}, {"start": 9.519, "duration": 5.641, "text": "\u092a\u0949\u091c\u093f\u091f\u093f\u0935 \u0935\u0940 \u0935\u093f\u0932 \u092e\u0942\u0935 \u090f \u092f\u0942\u0928\u093f\u091f\u094d\u0938 \u0930\u093e\u0907\u091f \u0911\u0928 \u0926"}, {"start": 12.4, "duration": 4.4, "text": "\u0930\u093f\u092f\u0932 \u090f\u0915\u094d\u0938\u093f\u0938 \u090f\u0902\u0921 \u092c\u0940 \u092f\u0942\u0928\u093f\u091f\u094d\u0938 \u0905\u092a \u091f\u0942 \u0917\u0947\u091f \u091f\u0942"}, {"start": 15.16, "duration": 5.08, "text": "\u0926 \u092a\u0949\u0907\u0902\u091f a"}, {"start": 16.8, "duration": 7.36, "text": "\u092c\u0940 \u0935\u0940 \u0915\u0948\u0928 \u0915\u0949\u0932 \u0907\u091f \u092a\u0949\u0907\u0902\u091f p \u090f\u0902\u0921 \u0932\u0947\u091f\u094d\u0938 \u0938\u0947 \u0926"}, {"start": 20.24, "duration": 6.68, "text": "\u0913\u0930\u093f\u091c\u093f\u0928 \u0907\u091c \u092a\u0949\u0907\u0902\u091f \u0913 \u0928\u093e\u0909 \u091c\u0938\u094d\u091f \u092b\u094b\u0915\u0938 \u0911\u0928 \u092a\u0949\u0907\u0902\u091f"}, {"start": 24.16, "duration": 6.4, "text": "p \u0907\u092b \u092f\u0942 \u0935\u093e\u0902\u091f \u091f\u0942 \u0906\u0938\u094d\u0915 \u092f\u094b\u0930 \u092b\u094d\u0930\u0947\u0902\u0921 \u091f\u0942 \u092a\u094d\u0932\u0949\u091f"}, {"start": 26.92, "duration": 5.839, "text": "\u092a\u0949\u0907\u0902\u091f p \u0939\u093e\u0909 \u0935\u0921 \u092f\u0942 \u0921\u0942 \u0907\u091f \u092f\u094b\u0930 \u092b\u094d\u0930 \u091c\u0938\u094d\u091f \u0939\u0948\u091c"}, {"start": 30.56, "duration": 5.28, "text": "\u090f\u0928 \u090f\u092e\u091f\u0940 \u0917\u094d\u0930\u093e\u092b \u0907\u0928 \u092b\u094d\u0930\u0902\u091f \u0911\u092b \u0939\u0930 \u090f\u0902\u0921 \u092f\u0942 \u0935\u093e\u0902\u091f"}, {"start": 32.759, "duration": 5.201, "text": "\u091f\u0942 \u0906\u0938\u094d\u0915 \u0939\u0930 \u091f\u0942 \u092a\u094d\u0932\u0949\u091f \u092a\u0949\u0907\u0902\u091f p \u0932\u0947\u091f\u094d\u0938 \u092e\u0942\u0935"}, {"start": 35.84, "duration": 4.76, "text": "\u0926\u093f\u0938 \u0917\u094d\u0930\u093e\u092b \u091f\u0942 \u0926 \u0932\u0947\u092b\u094d\u091f \u090f\u0902\u0921 \u0939\u0948\u0935 \u090f\u0928 \u090f\u092e\u091f\u0940 \u0935\u0928"}, {"start": 37.96, "duration": 5.64, "text": "\u0911\u0928 \u0926 \u0930\u093e\u0907\u091f \u0938\u094b \u0935\u0928 \u0925\u093f\u0902\u0917 \u092f\u0942 \u0915\u0921 \u091f\u0947\u0932 \u092f\u094b\u0930"}, {"start": 40.6, "duration": 6.599, "text": "\u092b\u094d\u0930\u0947\u0902\u0921 \u0907\u091c \u092e\u0942\u0935 a \u092f\u0942\u0928\u093f\u091f\u094d\u0938 \u091f\u0942 \u0926 \u0930\u093e\u0907\u091f \u090f\u0902\u0921 b"}, {"start": 43.6, "duration": 5.52, "text": "\u092f\u0942\u0928\u093f\u091f\u094d\u0938 \u0905\u092a\u0935\u0930\u094d\u0921\u094d\u0938 \u090f\u0902\u0921 \u092f\u0942 \u0935\u093f\u0932 \u0917\u0947\u091f \u092a\u0949\u0907\u0902\u091f p"}, {"start": 47.199, "duration": 4.36, "text": "\u0907\u091c \u0926\u0947\u092f\u0930 \u090f\u0928\u0940 \u0905\u0926\u0930 \u0935\u0947 \u0907\u0928 \u0935\u094d\u0939\u093f\u091a \u092f\u0942 \u0915\u0948\u0928 \u0906\u0938\u094d\u0915"}, {"start": 49.12, "duration": 6.0, "text": "\u0939\u0930 \u091f\u0942 \u092a\u094d\u0932\u0949\u091f \u0926\u093f\u0938 \u092a\u0949\u0907\u0902\u091f \u0932\u0947\u091f \u092e\u0940 \u0917\u093f\u0935 \u092f\u0942 \u0905"}, {"start": 51.559, "duration": 8.32, "text": "\u0939\u093f\u0902\u091f \u0906\u0908 \u091c\u0949\u0907\u0928 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"\u0915\u0949\u092e\u094d\u092a\u094d\u0932\u0947\u0915\u094d\u0936\u0928 \u0935\u093f \u091c\u0938\u094d\u091f \u092f\u0942\u091c\u093f\u0902\u0917 \u0921\u093f\u092b\u0930\u0947\u0902\u091f"}, {"start": 116.64, "duration": 7.119, "text": "\u092a\u0948\u0930\u093e\u091f\u094d\u0938 \u091f\u0942 \u0930\u093f\u092a\u094d\u0930\u0947\u091c\u0947\u0902\u091f \u092a\u0949\u0907\u0902\u091f p \u0935\u0940\u0935"}, {"start": 120.439, "duration": 3.32, "text": "\u0935\u093f \u0926\u093f concept.com"}, {"start": 124.1, "duration": 6.61, "text": "[\u0938\u0902\u0917\u0940\u0924]"}], "title": "Complex Numbers - Polar Form Part 1 | Don't Memorise", "description": "\ud83c\udfafNEET 2024 Paper Solutions with NEET Answer Key: https://www.youtube.com/watch?v=fwXYZUBp4m0&list=PLmdFyQYShrjc4OSwBsTiCoyPgl0TJTgon&index=1\n\ud83d\udcc5\ud83c\udd93NEET Rank & College Predictor 2024: https://infinitylearn.com/neet-rank-predictor?utm_source=YouTube&utm_medium=NEETYT&utm_campaign=LIVE How do we understand the Polar representation of a Complex Number? Watch this video to know more\n\nTo access all videos related to Complex Numbers, enroll in our full course now:  https://infinitylearn.com/microcourses?utm_source=youtube&utm_medium=Soical&utm_campaign=DM&utm_content=6HIlT6oSvXc&utm_term=%7Bkeyword%7D\n\nIn this video, we will learn: \n0:00 Plotting complex numbers on Argand plane\n0:27 polar representation of a complex number\n\nWatch Complex Numbers - Polar Form Part 2 here - https://www.youtube.com/watch?v=nWJhQnzPZiM\n\nTo watch more Complex Numbers videos, click here: https://bit.ly/ComplexNumbers_DMYT\n\nDon\u2019t Memorise brings learning to life through its captivating educational videos. To Know More, visit https://InfinityLearn.com\n\nNew videos every week. To stay updated, subscribe to our YouTube channel : http://bit.ly/DontMemoriseYouTube\n\nRegister on our website to gain access to all videos and quizzes:\nhttps://infinitylearn.com/microcourses?utm_source=youtube&utm_medium=Soical&utm_campaign=DM&utm_content=6HIlT6oSvXc&utm_term=%7Bkeyword%7D\n\n\n#ComplexNumbers #ImaginaryNumbers #PolarForm #neet2024 #infinityLearnNEET #neetsyllabus #neet2025", "lengthSeconds": 129, "width": 640, "height": 360}
{"video_id": "k6AovDCSY8A", "transcript": [{"start": 2.739, "duration": 5.32, "text": "consider the graphs of two different"}, {"start": 5.09, "duration": 5.58, "text": "sequences first I've graphed a random a"}, {"start": 8.059, "duration": 5.041, "text": "n that I made up and then I went and"}, {"start": 10.67, "duration": 5.31, "text": "graphed the absolute value of that same"}, {"start": 13.1, "duration": 4.32, "text": "sequence and then the top sequence a"}, {"start": 15.98, "duration": 3.57, "text": "there were some positive and negative"}, {"start": 17.42, "duration": 4.109, "text": "terms when I take the absolute value it"}, {"start": 19.55, "duration": 4.799, "text": "takes all of those negatives and makes"}, {"start": 21.529, "duration": 4.89, "text": "them all positives now this is a"}, {"start": 24.349, "duration": 3.451, "text": "sequence and in both of these sequences"}, {"start": 26.419, "duration": 3.57, "text": "at least visually it appears that the"}, {"start": 27.8, "duration": 4.319, "text": "limits going to be 0 but just because"}, {"start": 29.989, "duration": 4.59, "text": "the limit of the terms goes to 0 doesn't"}, {"start": 32.119, "duration": 4.951, "text": "say anything about what the series does"}, {"start": 34.579, "duration": 5.07, "text": "so let's investigate that let's compare"}, {"start": 37.07, "duration": 4.98, "text": "the series that adds up the a n to the"}, {"start": 39.649, "duration": 5.46, "text": "series that adds up the absolute value"}, {"start": 42.05, "duration": 5.039, "text": "of the a n now because that program is"}, {"start": 45.109, "duration": 4.29, "text": "saying I happen to know what this series"}, {"start": 47.089, "duration": 4.891, "text": "is this is the series minus 1 to the n"}, {"start": 49.399, "duration": 5.82, "text": "minus 1 all divided by square root of n"}, {"start": 51.98, "duration": 5.19, "text": "so the absolute value the effect it has"}, {"start": 55.219, "duration": 3.691, "text": "is just takes all that minus 1 stuff and"}, {"start": 57.17, "duration": 4.739, "text": "gets rid of it it just all turns it into"}, {"start": 58.91, "duration": 5.19, "text": "plus 1 now these two sequences are ones"}, {"start": 61.909, "duration": 4.501, "text": "that we can do by previous methods we've"}, {"start": 64.1, "duration": 4.26, "text": "seen the top one it's an alternating"}, {"start": 66.41, "duration": 4.47, "text": "thing with the minus 1 to the n minus 1"}, {"start": 68.36, "duration": 5.09, "text": "and then it multiplies it by 1 over"}, {"start": 70.88, "duration": 5.61, "text": "spurred n which is a positive decreasing"}, {"start": 73.45, "duration": 4.15, "text": "sequence with limit 0 it satisfies all"}, {"start": 76.49, "duration": 4.2, "text": "conditions of the alternating series"}, {"start": 77.6, "duration": 4.98, "text": "test so that top one indeed converges by"}, {"start": 80.69, "duration": 5.16, "text": "the alternating series test as for the"}, {"start": 82.58, "duration": 6.21, "text": "bottom one this is AP test it's AP with"}, {"start": 85.85, "duration": 5.82, "text": "N equal to 1/2 and so this is a p-series"}, {"start": 88.79, "duration": 4.83, "text": "that diverges so the lesson here is that"}, {"start": 91.67, "duration": 4.23, "text": "taking absolute values of the terms"}, {"start": 93.62, "duration": 4.7, "text": "makes a big deal in this example it"}, {"start": 95.9, "duration": 5.16, "text": "converts it from a convergent to a"}, {"start": 98.32, "duration": 5.23, "text": "divergent series no why might that be"}, {"start": 101.06, "duration": 4.08, "text": "the case if you think about the one with"}, {"start": 103.55, "duration": 2.85, "text": "positives and negatives when you take a"}, {"start": 105.14, "duration": 4.17, "text": "positive term and you add it to a"}, {"start": 106.4, "duration": 5.16, "text": "negative term it cancels maybe not"}, {"start": 109.31, "duration": 4.83, "text": "entirely all the way but the combination"}, {"start": 111.56, "duration": 5.13, "text": "of positives and negatives they don't"}, {"start": 114.14, "duration": 4.92, "text": "add up as much as only adding positives"}, {"start": 116.69, "duration": 4.35, "text": "or alternatively only adding negatives"}, {"start": 119.06, "duration": 4.8, "text": "together so when you take the absolute"}, {"start": 121.04, "duration": 4.68, "text": "value you get rid of any form of"}, {"start": 123.86, "duration": 3.75, "text": "internal cancellation that you have so"}, {"start": 125.72, "duration": 4.11, "text": "it would make sense that the absolute"}, {"start": 127.61, "duration": 3.87, "text": "value one would be the larger that it"}, {"start": 129.83, "duration": 4.14, "text": "might diverge even though the one with"}, {"start": 131.48, "duration": 4.02, "text": "the minus signs that converged and"}, {"start": 133.97, "duration": 2.61, "text": "indeed this is generally the case"}, {"start": 135.5, "duration": 3.75, "text": "so if here"}, {"start": 136.58, "duration": 5.49, "text": "that we have is that if the sum of the"}, {"start": 139.25, "duration": 5.79, "text": "absolute value of the ANS converges then"}, {"start": 142.07, "duration": 5.91, "text": "the original series with just the end"}, {"start": 145.04, "duration": 4.68, "text": "that converges as well it says that you"}, {"start": 147.98, "duration": 3.78, "text": "put this absolute value around it you"}, {"start": 149.72, "duration": 3.96, "text": "have a stronger condition so when you"}, {"start": 151.76, "duration": 4.35, "text": "know it converges with the absolutes"}, {"start": 153.68, "duration": 4.56, "text": "there then it converges without them"}, {"start": 156.11, "duration": 4.65, "text": "deep a little bit of terminology here"}, {"start": 158.24, "duration": 4.59, "text": "we'll say that absolute convergence that"}, {"start": 160.76, "duration": 3.87, "text": "expression is what we mean when we take"}, {"start": 162.83, "duration": 4.59, "text": "the sum of the absolute value of a ends"}, {"start": 164.63, "duration": 5.94, "text": "so another way to say this is that if it"}, {"start": 167.42, "duration": 6.09, "text": "converges absolutely then it converges"}, {"start": 170.57, "duration": 5.43, "text": "in the normal sense but the other"}, {"start": 173.51, "duration": 4.68, "text": "direction is not true if something"}, {"start": 176.0, "duration": 3.63, "text": "converges like in the example we saw"}, {"start": 178.19, "duration": 3.99, "text": "before that converged by the alternating"}, {"start": 179.63, "duration": 5.19, "text": "series test it does not necessarily"}, {"start": 182.18, "duration": 5.55, "text": "converge absolutely it might but it"}, {"start": 184.82, "duration": 4.74, "text": "might not and in that scenario where it"}, {"start": 187.73, "duration": 3.69, "text": "doesn't we call this conditional"}, {"start": 189.56, "duration": 5.07, "text": "convergence so conditional convergence"}, {"start": 191.42, "duration": 6.45, "text": "is where the original series converges"}, {"start": 194.63, "duration": 4.98, "text": "but does not converge absolutely it"}, {"start": 197.87, "duration": 4.26, "text": "doesn't converge when you wrap it in"}, {"start": 199.61, "duration": 3.75, "text": "absolute value signs indeed this example"}, {"start": 202.13, "duration": 3.27, "text": "that we started the video with this"}, {"start": 203.36, "duration": 4.05, "text": "minus 1 to the n minus 1 over root n"}, {"start": 205.4, "duration": 5.37, "text": "that was a conditionally convergent"}, {"start": 207.41, "duration": 6.21, "text": "series it converged but it didn't"}, {"start": 210.77, "duration": 4.23, "text": "converge absolutely so these are just a"}, {"start": 213.62, "duration": 3.27, "text": "couple of pieces of terminology that are"}, {"start": 215.0, "duration": 6.41, "text": "going to be very important for our next"}, {"start": 216.89, "duration": 4.52, "text": "test the ratio test and the root test"}], "title": "Absolute Convergence vs Conditional Convergence vs Convergence", "description": "We've seen regular convergence of a series before, but now we consider two special cases. Absolute convergence is when we take the series of the absolute value of the terms, which gets rid of any possible cancellation like what happened a lot in the Alternating Series Test. Absolute convergence implies regular convergence. Conditional Convergence is when it converges but does not converge absolutely. \n\n****************************************************\nYOUR TURN! Learning math requires more than just watching videos, so make sure you reflect, ask questions, and do lots of practice problems!\n****************************************************\n\n\u25baFull Course Playlist: CALCULUS II: https://www.youtube.com/playlist?list=PLHXZ9OQGMqxc4ySKTIW19TLrT91Ik9M4n\n\n****************************************************\nOther Course Playlists:\n\n\u25baCALCULUS I: https://www.youtube.com/playlist?list=PLHXZ9OQGMqxfT9RMcReZ4WcoVILP4k6-m\n\n\u25baDISCRETE MATH: https://www.youtube.com/playlist?list=PLHXZ9OQGMqxersk8fUxiUMSIx0DBqsKZS\n\n\u25baLINEAR ALGEBRA: https://www.youtube.com/playlist?list=PLHXZ9OQGMqxfUl0tcqPNTJsb7R6BqSLo6\n\n***************************************************\n\n\u25ba Want to learn math effectively? Check out my \"Learning Math\" Series:\nhttps://www.youtube.com/watch?v=LPH2lqis3D0&list=PLHXZ9OQGMqxfSkRtlL5KPq6JqMNTh_MBw\n\n\n\u25baWant some cool math? Check out my \"Cool Math\" Series:\nhttps://www.youtube.com/playlist?list=PLHXZ9OQGMqxelE_9RzwJ-cqfUtaFBpiho\n\n*****************************************************\n\n\u25baCheck out my 2nd Channel for lower production quality \"live\" math videos: https://www.youtube.com/channel/UCFNL8G1CZ4dH0p-Jf7UXqdQ\n\n*****************************************************\n\n\u25baFollow me on Twitter: http://twitter.com/treforbazett\n\n*****************************************************\n\nThis video was created by Dr. Trefor Bazett, an Assistant Professor, Educator at the University of Cincinnati.\n\nBECOME A MEMBER:\n\u25baJoin: https://www.youtube.com/channel/UC9rTsvTxJnx1DNrDA3Rqa6A/join\n\nMATH BOOKS & MERCH I LOVE:\n\u25ba My Amazon Affiliate Shop: https://www.amazon.com/shop/treforbazett", "lengthSeconds": 220, "width": 640, "height": 360}
{"video_id": "3kezO88rEvE", "transcript": [{"start": 0.03, "duration": 3.03, "text": "let's investigate this sequence we were"}, {"start": 1.68, "duration": 3.42, "text": "just looking at a little bit more"}, {"start": 3.06, "duration": 5.129, "text": "carefully because I want to give a"}, {"start": 5.1, "duration": 5.51, "text": "different view on what a sequence really"}, {"start": 8.189, "duration": 4.53, "text": "is now currently I have it written here"}, {"start": 10.61, "duration": 3.76, "text": "horizontally but I could also go and"}, {"start": 12.719, "duration": 2.941, "text": "switch it around and write it vertically"}, {"start": 14.37, "duration": 2.97, "text": "as well that shouldn't make any"}, {"start": 15.66, "duration": 4.77, "text": "difference it's still an infinite"}, {"start": 17.34, "duration": 4.859, "text": "ordered list and I also have another one"}, {"start": 20.43, "duration": 3.48, "text": "that we've seen before this is the one"}, {"start": 22.199, "duration": 4.371, "text": "two three four five six seven eight and"}, {"start": 23.91, "duration": 6.51, "text": "so on so we can think of these as two"}, {"start": 26.57, "duration": 6.91, "text": "different sequences but because I sort"}, {"start": 30.42, "duration": 4.799, "text": "of lined them up in this nice way we we"}, {"start": 33.48, "duration": 5.009, "text": "should have this association that like"}, {"start": 35.219, "duration": 4.651, "text": "the one over here is associated to the"}, {"start": 38.489, "duration": 3.421, "text": "three in the sense that they're both the"}, {"start": 39.87, "duration": 3.75, "text": "first terms and the two here and the"}, {"start": 41.91, "duration": 4.559, "text": "minus six are associated senses that are"}, {"start": 43.62, "duration": 4.74, "text": "both the second terms and I can think of"}, {"start": 46.469, "duration": 5.43, "text": "this as a sort of a dynamic way I can"}, {"start": 48.36, "duration": 6.12, "text": "think of it as the one goes to the 3 and"}, {"start": 51.899, "duration": 5.851, "text": "the 2 goes to the minus 6 and the 3 goes"}, {"start": 54.48, "duration": 4.8, "text": "to the not indeed the association"}, {"start": 57.75, "duration": 4.14, "text": "between here is precisely what that"}, {"start": 59.28, "duration": 6.659, "text": "explicit formula tells me if I choose a"}, {"start": 61.89, "duration": 8.22, "text": "different K it goes to the minus 1 to"}, {"start": 65.939, "duration": 7.201, "text": "the K times 3 K so what do I really have"}, {"start": 70.11, "duration": 6.81, "text": "here well if I think about my sequences"}, {"start": 73.14, "duration": 6.39, "text": "as being my arrow diagram when I've got"}, {"start": 76.92, "duration": 5.64, "text": "a domain here and I've got a co domain"}, {"start": 79.53, "duration": 5.97, "text": "and then the elements inside of my"}, {"start": 82.56, "duration": 6.68, "text": "domain are all Matt to things in my co"}, {"start": 85.5, "duration": 6.27, "text": "domain what I'm describing is a function"}, {"start": 89.24, "duration": 7.059, "text": "every single thing in my domain goes"}, {"start": 91.77, "duration": 6.419, "text": "somewhere and more importantly it obeys"}, {"start": 96.299, "duration": 4.471, "text": "the vertical line test there's there's"}, {"start": 98.189, "duration": 4.64, "text": "no element in my domain here that that"}, {"start": 100.77, "duration": 5.129, "text": "splits into multiple different things"}, {"start": 102.829, "duration": 6.281, "text": "because my association is to take one"}, {"start": 105.899, "duration": 6.08, "text": "element say the case 1 to the 8th one"}, {"start": 109.11, "duration": 6.0, "text": "over here not to two different ones so"}, {"start": 111.979, "duration": 6.011, "text": "really what a sequence is going to be"}, {"start": 115.11, "duration": 6.75, "text": "then is this function from this"}, {"start": 117.99, "duration": 5.85, "text": "particular set to some other set or in"}, {"start": 121.86, "duration": 4.35, "text": "other words I'm going to say this is my"}, {"start": 123.84, "duration": 4.139, "text": "formal mathematically precise definition"}, {"start": 126.21, "duration": 4.859, "text": "a sequence is a function where I'm"}, {"start": 127.979, "duration": 5.291, "text": "taking things from the positive integers"}, {"start": 131.069, "duration": 4.751, "text": "that was the 1 2 3 4"}, {"start": 133.27, "duration": 4.5, "text": "and I'm going to some codomain that we"}, {"start": 135.82, "duration": 3.39, "text": "haven't specified and the other example"}, {"start": 137.77, "duration": 5.85, "text": "look like it was integers as well but"}, {"start": 139.21, "duration": 6.18, "text": "it's going to be some codomain so as an"}, {"start": 143.62, "duration": 5.1, "text": "example let's take the one we just"}, {"start": 145.39, "duration": 5.94, "text": "looked at if I wanted to prescribe it in"}, {"start": 148.72, "duration": 5.13, "text": "this language the language of a function"}, {"start": 151.33, "duration": 4.08, "text": "with a domain and with a co domain what"}, {"start": 153.85, "duration": 5.4, "text": "I'm going to say is that I have a"}, {"start": 155.41, "duration": 5.61, "text": "function ass and it inputs things inside"}, {"start": 159.25, "duration": 5.61, "text": "of my Co domain so it's going to input"}, {"start": 161.02, "duration": 5.43, "text": "some K and then I had to tell you what"}, {"start": 164.86, "duration": 5.07, "text": "it does to the K well it does precisely"}, {"start": 166.45, "duration": 7.07, "text": "what we just saw I got that minus 1 to"}, {"start": 169.93, "duration": 5.67, "text": "the K times 3 K this could be an example"}, {"start": 173.52, "duration": 6.3, "text": "in other words if I think of the"}, {"start": 175.6, "duration": 10.95, "text": "function it inputs the K and it sits out"}, {"start": 179.82, "duration": 9.43, "text": "this a sub K now this formal definition"}, {"start": 186.55, "duration": 5.07, "text": "of a function I want us to note it and"}, {"start": 189.25, "duration": 4.29, "text": "to be there and for us to think that"}, {"start": 191.62, "duration": 4.62, "text": "sequences really aren't anything new"}, {"start": 193.54, "duration": 5.49, "text": "they're just a special type of function"}, {"start": 196.24, "duration": 4.65, "text": "however it's really intuitively useful"}, {"start": 199.03, "duration": 5.19, "text": "to us if we if we actually think of"}, {"start": 200.89, "duration": 5.1, "text": "sequences more in the informal sense but"}, {"start": 204.22, "duration": 4.32, "text": "there's this ordered list of numbers in"}, {"start": 205.99, "duration": 4.92, "text": "other words I typically don't use the f"}, {"start": 208.54, "duration": 5.67, "text": "of K notation at all when I'm just"}, {"start": 210.91, "duration": 5.58, "text": "noting a sequence I will say a sub K the"}, {"start": 214.21, "duration": 4.8, "text": "K term of my sequence and I don't give"}, {"start": 216.49, "duration": 4.29, "text": "the formula that depends on K I don't"}, {"start": 219.01, "duration": 3.81, "text": "make explicit that I'm talking about a"}, {"start": 220.78, "duration": 4.77, "text": "function I don't make explicit a"}, {"start": 222.82, "duration": 7.7, "text": "particular s however that is what is"}, {"start": 225.55, "duration": 4.97, "text": "going on behind the idea of a sequence"}], "title": "The formal definition of a sequence.", "description": "We have an intuitive picture of sequences (infinite ordered lists). But there is a formal definition of sequences based out of the idea of a specific function between sets, specifically from the positive integers to the real numbers. \n\n\u25baFull DISCRETE MATH Course Playlist: https://www.youtube.com/playlist?list=PLHXZ9OQGMqxersk8fUxiUMSIx0DBqsKZS\n\n*********************************************************\nOther Course Playlists:\n\n\u25baCALCULUS I: https://www.youtube.com/playlist?list=PLHXZ9OQGMqxfT9RMcReZ4WcoVILP4k6-m\n\n\u25baCALCULUS II: https://www.youtube.com/playlist?list=PLHXZ9OQGMqxc4ySKTIW19TLrT91Ik9M4n\n\n\u25baCALCULUS III (multivariable): https://www.youtube.com/playlist?list=PLHXZ9OQGMqxc_CvEy7xBKRQr6I214QJcd\n\n\u25baDIFFERENTIAL EQUATIONS: https://www.youtube.com/playlist?list=PLHXZ9OQGMqxcJXnLr08cyNaup4RDsbAl1\n\n\u25baLINEAR ALGEBRA: https://www.youtube.com/playlist?list=PLHXZ9OQGMqxfUl0tcqPNTJsb7R6BqSLo6\n\n\u25ba Want to learn math effectively? Check out my \"Learning Math\" Series: https://www.youtube.com/watch?v=LPH2lqis3D0&list=PLHXZ9OQGMqxfSkRtlL5KPq6JqMNTh_MBw\n\n\u25baWant some cool math? Check out my \"Cool Math\" Series: https://www.youtube.com/playlist?list=PLHXZ9OQGMqxelE_9RzwJ-cqfUtaFBpiho\n\n*****************************************************\nYOUR TURN! Learning math requires more than just watching math videos, so make sure you reflect, ask questions, and do lots of practice problems!\n\r****************************************************\n\n\u25baFollow me on Twitter: http://twitter.com/treforbazett\n\nBECOME A MEMBER:\n\u25baJoin: https://www.youtube.com/channel/UC9rTsvTxJnx1DNrDA3Rqa6A/join\n\nMATH BOOKS & MERCH I LOVE:\n\u25ba My Amazon Affiliate Shop: https://www.amazon.com/shop/treforbazett", "lengthSeconds": 229, "width": 640, "height": 360}
{"video_id": "LQUy83xFL5E", "transcript": [{"start": 1.24, "duration": 5.679, "text": "okay here we're going to find"}, {"start": 3.0, "duration": 6.84, "text": "25% of the number 76 whenever you're"}, {"start": 6.919, "duration": 7.521, "text": "finding a percentage of a number you can"}, {"start": 9.84, "duration": 4.6, "text": "read of basically just replace it with"}, {"start": 14.719, "duration": 5.081, "text": "multiplication and the only thing we"}, {"start": 16.44, "duration": 4.759, "text": "need to do is we need to turn 25% either"}, {"start": 19.8, "duration": 5.04, "text": "into a"}, {"start": 21.199, "duration": 6.281, "text": "decimal or a fraction before we"}, {"start": 24.84, "duration": 7.84, "text": "multiply so that's what I'm going to do"}, {"start": 27.48, "duration": 7.96, "text": "well 25% uh we can write that 25 over"}, {"start": 32.68, "duration": 5.96, "text": "100 of is going to be our"}, {"start": 35.44, "duration": 6.959, "text": "multiplication of 76 I'm even going to"}, {"start": 38.64, "duration": 7.48, "text": "write that as 76 over 1 and now we can"}, {"start": 42.399, "duration": 6.761, "text": "just start simplifying so 25 over 100"}, {"start": 46.12, "duration": 6.4, "text": "that would actually reduce to 1 over4 if"}, {"start": 49.16, "duration": 7.399, "text": "we divide top and bottom by"}, {"start": 52.52, "duration": 7.08, "text": "25 so multiplied by 76 over"}, {"start": 56.559, "duration": 4.761, "text": "1 well that's just going to give us 76"}, {"start": 59.6, "duration": 6.72, "text": "over"}, {"start": 61.32, "duration": 7.96, "text": "four and four will go into 76"}, {"start": 66.32, "duration": 5.36, "text": "evenly four will go into seven once 1 *"}, {"start": 69.28, "duration": 5.08, "text": "4 is"}, {"start": 71.68, "duration": 4.6, "text": "four drop down subtract we'll get our"}, {"start": 74.36, "duration": 7.64, "text": "three and drop down our six and then"}, {"start": 76.28, "duration": 8.08, "text": "four will go into 36 19 times so 76 / 4"}, {"start": 82.0, "duration": 4.68, "text": "is 19"}, {"start": 84.36, "duration": 5.719, "text": "so if you had"}, {"start": 86.68, "duration": 7.68, "text": "25% of say $76"}, {"start": 90.079, "duration": 4.281, "text": "you would have exactly $19"}], "title": "Finding a Percent of a Number ( 25% of 76 in this video!)", "description": "Thanks to all of you who support me on Patreon. You da real mvps!   $1 per month helps!! :)  https://www.patreon.com/patrickjmt !! Finding a Percent of a Number ( 25% of 76 in this video!).  Here we look at an example of finding the percent of a number.", "lengthSeconds": 93, "width": 480, "height": 360}
{"video_id": "LNSfM86I8is", "transcript": [{"start": 0.68, "duration": 3.639, "text": "so to give you a little bit of a flavor"}, {"start": 2.399, "duration": 3.88, "text": "for how these things can can get a"}, {"start": 4.319, "duration": 5.36, "text": "little bit more complicated look at this"}, {"start": 6.279, "duration": 6.44, "text": "statement here so we're going to have"}, {"start": 9.679, "duration": 4.481, "text": "not P or not q and I want to try to"}, {"start": 12.719, "duration": 2.8, "text": "figure out what does this mean it's"}, {"start": 14.16, "duration": 2.56, "text": "already getting a little bit complicated"}, {"start": 15.519, "duration": 3.721, "text": "of course I can keep on adding I could"}, {"start": 16.72, "duration": 4.92, "text": "add an r i could add more variables and"}, {"start": 19.24, "duration": 5.359, "text": "keep on combining these negations and"}, {"start": 21.64, "duration": 4.719, "text": "conjunctions and ores as much as I wish"}, {"start": 24.599, "duration": 3.0, "text": "so I want to have this systematic way of"}, {"start": 26.359, "duration": 3.201, "text": "doing it and we're going to use the"}, {"start": 27.599, "duration": 2.881, "text": "truth table here so let's pull up our"}, {"start": 29.56, "duration": 2.119, "text": "template"}, {"start": 30.48, "duration": 3.0, "text": "and you're notice that it's going to be"}, {"start": 31.679, "duration": 5.56, "text": "a little bit bigger than it was last"}, {"start": 33.48, "duration": 5.239, "text": "time so if I go way off to the left"}, {"start": 37.239, "duration": 4.081, "text": "again we're going to have the same sort"}, {"start": 38.719, "duration": 5.721, "text": "of beginning Parts here so I want to"}, {"start": 41.32, "duration": 6.8, "text": "have an initial input variable"}, {"start": 44.44, "duration": 5.72, "text": "p and another initial input Q so those"}, {"start": 48.12, "duration": 5.2, "text": "are my same two inputs that we've had"}, {"start": 50.16, "duration": 5.199, "text": "before and I can go down and fill out"}, {"start": 53.32, "duration": 4.0, "text": "the different possibilities that we have"}, {"start": 55.359, "duration": 5.16, "text": "exactly we've done before I could have"}, {"start": 57.32, "duration": 7.119, "text": "true and true I could have true and"}, {"start": 60.519, "duration": 6.96, "text": "false I could have false and true and I"}, {"start": 64.439, "duration": 5.521, "text": "could have false and false so the same"}, {"start": 67.479, "duration": 4.881, "text": "sort of beginning moves here now I don't"}, {"start": 69.96, "duration": 3.64, "text": "want to jump right to the final column"}, {"start": 72.36, "duration": 3.92, "text": "over here I don't want to jump right to"}, {"start": 73.6, "duration": 4.32, "text": "this final answer I want to give some"}, {"start": 76.28, "duration": 3.96, "text": "intermediate ones that are going to help"}, {"start": 77.92, "duration": 5.36, "text": "me along so what I'm going to do in"}, {"start": 80.24, "duration": 6.4, "text": "these ones is I'm going to put not P"}, {"start": 83.28, "duration": 5.04, "text": "here and I'm going to put not Q over"}, {"start": 86.64, "duration": 3.76, "text": "there and I'm going to try to figure out"}, {"start": 88.32, "duration": 3.92, "text": "what those are and then find finally"}, {"start": 90.4, "duration": 3.96, "text": "after I'm done with everything I can"}, {"start": 92.24, "duration": 4.12, "text": "come down here and I can come and put"}, {"start": 94.36, "duration": 6.16, "text": "this inside of"}, {"start": 96.36, "duration": 6.64, "text": "there all right so not p is just exactly"}, {"start": 100.52, "duration": 4.8, "text": "the opposite of whatever p is if if p is"}, {"start": 103.0, "duration": 6.36, "text": "going to be true then not p is going to"}, {"start": 105.32, "duration": 8.0, "text": "be false p is true not p is false p is"}, {"start": 109.36, "duration": 6.039, "text": "false not p is true and P is false not p"}, {"start": 113.32, "duration": 5.04, "text": "is true all right so I'm able to fill"}, {"start": 115.399, "duration": 5.521, "text": "that in pretty easily same with not Q"}, {"start": 118.36, "duration": 4.84, "text": "not Q I just look at Q q and then I take"}, {"start": 120.92, "duration": 5.519, "text": "the negation of every one so if Q is"}, {"start": 123.2, "duration": 5.0, "text": "true then not Q is going to be false"}, {"start": 126.439, "duration": 3.561, "text": "false is going to turn into a true true"}, {"start": 128.2, "duration": 3.96, "text": "is going to turn into a false and false"}, {"start": 130.0, "duration": 4.959, "text": "is going to turn into a"}, {"start": 132.16, "duration": 5.68, "text": "true so now comes the tricky part I want"}, {"start": 134.959, "duration": 7.36, "text": "to put these two things together because"}, {"start": 137.84, "duration": 7.679, "text": "I have that it's going to be not P or"}, {"start": 142.319, "duration": 6.401, "text": "not q and so that tells me if one of the"}, {"start": 145.519, "duration": 6.561, "text": "two of these columns the not p and the"}, {"start": 148.72, "duration": 5.239, "text": "not Q columns is true then my output is"}, {"start": 152.08, "duration": 5.12, "text": "going to be true as well so what do I"}, {"start": 153.959, "duration": 4.56, "text": "have here I have a false and a false so"}, {"start": 157.2, "duration": 2.92, "text": "neither of them is true it's not the"}, {"start": 158.519, "duration": 4.36, "text": "case that one or the other one is going"}, {"start": 160.12, "duration": 4.399, "text": "to be so neither of them is true and so"}, {"start": 162.879, "duration": 5.121, "text": "therefore it is going to be"}, {"start": 164.519, "duration": 5.921, "text": "false down here one of the two is true"}, {"start": 168.0, "duration": 4.08, "text": "this one's true so one or the other is"}, {"start": 170.44, "duration": 2.439, "text": "true yeah that works that's going to be"}, {"start": 172.08, "duration": 3.36, "text": "a"}, {"start": 172.879, "duration": 4.72, "text": "true true and false one or the other"}, {"start": 175.44, "duration": 3.719, "text": "that's going to be another true and true"}, {"start": 177.599, "duration": 3.601, "text": "and true one of them is going to be true"}, {"start": 179.159, "duration": 4.881, "text": "in fact both of them true and so that's"}, {"start": 181.2, "duration": 7.399, "text": "going to be a final true so what do I"}, {"start": 184.04, "duration": 7.04, "text": "get false true true and true so these"}, {"start": 188.599, "duration": 6.081, "text": "truth tables are very useful at taking"}, {"start": 191.08, "duration": 6.28, "text": "these complicated compounded logical"}, {"start": 194.68, "duration": 4.639, "text": "statements and just trying to interpret"}, {"start": 197.36, "duration": 4.48, "text": "when are they false and when are they"}, {"start": 199.319, "duration": 6.441, "text": "true depending on the initial parameters"}, {"start": 201.84, "duration": 3.92, "text": "the P's and the q's"}], "title": "Truth Table Example: ~p V ~q", "description": "Learning Objective: We investigate the truth table for the more complicated logical form ~p V ~q\n\n****************************************************\nYOUR TURN! Learning math requires more than just watching videos, so make sure you reflect, ask questions (leave them in the comments!), and do lots of practice problems.\n****************************************************\n\n\u25baFull Course Playlist: DISCRETE MATH: https://www.youtube.com/playlist?list=PLHXZ9OQGMqxersk8fUxiUMSIx0DBqsKZS\n\n****************************************************\nOther Course Playlists:\n\n\u25baCALCULUS I: https://www.youtube.com/playlist?list=PLHXZ9OQGMqxfT9RMcReZ4WcoVILP4k6-m\n\n\u25baCALCULUS II: https://www.youtube.com/playlist?list=PLHXZ9OQGMqxc4ySKTIW19TLrT91Ik9M4n\n\n\u25baLINEAR ALGEBRA: https://www.youtube.com/playlist?list=PLHXZ9OQGMqxfUl0tcqPNTJsb7R6BqSLo6\n\n***************************************************\n\n\u25ba Want to learn math effectively? Check out my \"Learning Math\" Series:\nhttps://www.youtube.com/watch?v=LPH2lqis3D0&list=PLHXZ9OQGMqxfSkRtlL5KPq6JqMNTh_MBw\n\n\n\u25baWant some cool math? Check out my \"Cool Math\" Series:\nhttps://www.youtube.com/playlist?list=PLHXZ9OQGMqxelE_9RzwJ-cqfUtaFBpiho\n\n*****************************************************\n\n\u25baCheck out my 2nd Channel for lower production quality \"live\" math videos: https://www.youtube.com/channel/UCFNL8G1CZ4dH0p-Jf7UXqdQ\n\n*****************************************************\n\n\u25baFollow me on Twitter: http://twitter.com/treforbazett\n\n*****************************************************\n\nThis video was created by Dr. Trefor Bazett", "lengthSeconds": 204, "width": 640, "height": 360}
{"video_id": "oSV4qwO_kNY", "transcript": [{"start": 0.03, "duration": 4.8, "text": "because sets in the set operations like"}, {"start": 3.0, "duration": 5.22, "text": "Union and intersection and complement"}, {"start": 4.83, "duration": 7.11, "text": "are all defined in terms of some logic"}, {"start": 8.22, "duration": 6.869, "text": "condition like or or and or not in it"}, {"start": 11.94, "duration": 5.52, "text": "means that all of those rules of logic"}, {"start": 15.089, "duration": 5.581, "text": "that we got all those logical"}, {"start": 17.46, "duration": 5.04, "text": "equivalences can be applied to the"}, {"start": 20.67, "duration": 5.699, "text": "language of set and therefore we get a"}, {"start": 22.5, "duration": 5.73, "text": "lot of equalities of sets so for example"}, {"start": 26.369, "duration": 6.631, "text": "consider this claim I'm claiming that a"}, {"start": 28.23, "duration": 7.56, "text": "union B equals B Union a so how could we"}, {"start": 33.0, "duration": 4.44, "text": "prove a claim like this again I want you"}, {"start": 35.79, "duration": 4.89, "text": "to notice inequality so I want to show"}, {"start": 37.44, "duration": 6.63, "text": "if it's in the Union a union B then it's"}, {"start": 40.68, "duration": 5.969, "text": "in the B Union a and vice versa"}, {"start": 44.07, "duration": 4.71, "text": "if it's in B Union a then it's an a"}, {"start": 46.649, "duration": 4.051, "text": "union B I want to show both sides of"}, {"start": 48.78, "duration": 4.049, "text": "that so I want to say show that what my"}, {"start": 50.7, "duration": 3.24, "text": "proof is going forward everything is in"}, {"start": 52.829, "duration": 3.48, "text": "biconditional"}, {"start": 53.94, "duration": 4.29, "text": "and if and only if so what I'm going to"}, {"start": 56.309, "duration": 5.25, "text": "use the element method mainly I'm going"}, {"start": 58.23, "duration": 6.959, "text": "to start with an element X which is"}, {"start": 61.559, "duration": 9.541, "text": "inside of the a and the B inside of a"}, {"start": 65.189, "duration": 8.04, "text": "union B but the definition of a union B"}, {"start": 71.1, "duration": 4.589, "text": "and by the way all definitions are"}, {"start": 73.229, "duration": 3.691, "text": "inherently if and only if you don't have"}, {"start": 75.689, "duration": 4.531, "text": "to worry about that if you're applying a"}, {"start": 76.92, "duration": 6.449, "text": "definition it is always if anomia so X"}, {"start": 80.22, "duration": 7.02, "text": "is an a union B is the same thing as X"}, {"start": 83.369, "duration": 9.261, "text": "is in a or I'm going to put this symbol"}, {"start": 87.24, "duration": 8.58, "text": "for my disjunctive or X is inside of B"}, {"start": 92.63, "duration": 6.099, "text": "but now we can use a property that we"}, {"start": 95.82, "duration": 4.74, "text": "had for disjunctive we know that two"}, {"start": 98.729, "duration": 4.801, "text": "properties that are connected with an or"}, {"start": 100.56, "duration": 5.19, "text": "that I can reverse that order that I can"}, {"start": 103.53, "duration": 7.47, "text": "say that this is the same thing as X"}, {"start": 105.75, "duration": 7.56, "text": "inside of B or X is inside of a indeed"}, {"start": 111.0, "duration": 5.189, "text": "back in the day we would have verified"}, {"start": 113.31, "duration": 5.129, "text": "this logical property by looking at it"}, {"start": 116.189, "duration": 5.011, "text": "at level of truth tables this is P or Q"}, {"start": 118.439, "duration": 5.191, "text": "and this is Q or P and you can verify"}, {"start": 121.2, "duration": 5.43, "text": "that they have the same truth table so"}, {"start": 123.63, "duration": 6.96, "text": "this is a logical equivalence and then I"}, {"start": 126.63, "duration": 5.85, "text": "can go backwards X and B or X and a this"}, {"start": 130.59, "duration": 7.319, "text": "is precisely the same thing"}, {"start": 132.48, "duration": 8.399, "text": "saying that X is in B Union a and so I"}, {"start": 137.909, "duration": 5.821, "text": "have a chain of if and only if and I get"}, {"start": 140.879, "duration": 5.341, "text": "if I start in a union B I end up in B"}, {"start": 143.73, "duration": 6.42, "text": "Union a and the key thing to note here"}, {"start": 146.22, "duration": 6.599, "text": "is that the important property is"}, {"start": 150.15, "duration": 4.86, "text": "something about logic the the reversing"}, {"start": 152.819, "duration": 4.92, "text": "the order came down here is a logical"}, {"start": 155.01, "duration": 5.13, "text": "property and then because the Union was"}, {"start": 157.739, "duration": 4.771, "text": "defined in terms of this or statement"}, {"start": 160.14, "duration": 4.77, "text": "you could sort of bring it out of the"}, {"start": 162.51, "duration": 5.64, "text": "logical property into this statement"}, {"start": 164.91, "duration": 5.669, "text": "about unions but the same is true in a"}, {"start": 168.15, "duration": 4.649, "text": "much larger domain for instance we have"}, {"start": 170.579, "duration": 6.331, "text": "a de Morgan's law for logic we have a de"}, {"start": 172.799, "duration": 5.881, "text": "Morgan's law for sets and that idea is"}, {"start": 176.91, "duration": 5.54, "text": "going to continue for all of the"}, {"start": 178.68, "duration": 3.77, "text": "different logical equivalences"}], "title": "Proving equalities of sets using the element method", "description": "In this video we prove that we can commute unions, that is change the order of a union of two sets. This is just a stand in for a larger class of equalities between sets. To prove this we use the element method. We need to show an element in the left hand side is in the right hand side, AND vice versa. \n\n\u25baFULL DISCRETE MATH PLAYLIST: https://www.youtube.com/playlist?list=PLHXZ9OQGMqxersk8fUxiUMSIx0DBqsKZS\n\nOTHER COURSE PLAYLISTS:\n\u25baCALCULUS I: https://www.youtube.com/playlist?list=PLHXZ9OQGMqxfT9RMcReZ4WcoVILP4k6-m\n\u25ba CALCULUS II: https://www.youtube.com/playlist?list=PLHXZ9OQGMqxc4ySKTIW19TLrT91Ik9M4n\n\u25baMULTIVARIABLE CALCULUS (Calc III): https://www.youtube.com/playlist?list=PLHXZ9OQGMqxc_CvEy7xBKRQr6I214QJcd\n\u25baVECTOR CALCULUS (Calc IV): https://www.youtube.com/playlist?list=PLHXZ9OQGMqxfW0GMqeUE1bLKaYor6kbHa\n\u25baDIFFERENTIAL EQUATIONS: https://www.youtube.com/watch?v=xeeM3TT4Zgg&list=PLHXZ9OQGMqxcJXnLr08cyNaup4RDsbAl1\n\u25baLINEAR ALGEBRA: https://www.youtube.com/playlist?list=PLHXZ9OQGMqxfUl0tcqPNTJsb7R6BqSLo6\n\nOTHER PLAYLISTS:\n\u25ba Learning Math Series\nhttps://www.youtube.com/watch?v=LPH2lqis3D0&list=PLHXZ9OQGMqxfSkRtlL5KPq6JqMNTh_MBw\n\u25baCool Math Series:\nhttps://www.youtube.com/playlist?list=PLHXZ9OQGMqxelE_9RzwJ-cqfUtaFBpiho\n\nBECOME A MEMBER:\n\u25baJoin: https://www.youtube.com/channel/UC9rTsvTxJnx1DNrDA3Rqa6A/join\n\nMATH BOOKS & MERCH I LOVE:\n\u25ba My Amazon Affiliate Shop: https://www.amazon.com/shop/treforbazett\n\nSOCIALS:\n\u25baTwitter (math based): http://twitter.com/treforbazett\n\u25baInstagram (photography based): http://instagram.com/treforphotography", "lengthSeconds": 181, "width": 640, "height": 360}
{"video_id": "RdeTrQKz3fk", "transcript": [{"start": 3.7, "duration": 6.54, "text": "all right let's do one more example what"}, {"start": 7.51, "duration": 6.15, "text": "we have here is now the integral of e to"}, {"start": 10.24, "duration": 6.39, "text": "the 2x on the top all divided by 1 plus"}, {"start": 13.66, "duration": 5.51, "text": "e to the 2x on the bottom so they figure"}, {"start": 16.63, "duration": 4.71, "text": "out what's my f here and what's my you"}, {"start": 19.17, "duration": 5.02, "text": "now one of the things that I can sort of"}, {"start": 21.34, "duration": 5.279, "text": "know right off the top is that when I"}, {"start": 24.19, "duration": 6.179, "text": "take the derivative of 1 plus e to the"}, {"start": 26.619, "duration": 5.371, "text": "2x I'm going to get twice eetu 2x and"}, {"start": 30.369, "duration": 3.271, "text": "there's this factor of 2 there that's a"}, {"start": 31.99, "duration": 4.38, "text": "bit confusing but but it sort of looks"}, {"start": 33.64, "duration": 5.01, "text": "like the top here is related off by a"}, {"start": 36.37, "duration": 5.07, "text": "factor of 2 but it's related to the"}, {"start": 38.65, "duration": 5.369, "text": "derivative of the bottom so let me"}, {"start": 41.44, "duration": 5.399, "text": "experiment with setting u to be that"}, {"start": 44.019, "duration": 4.651, "text": "denominator ok so how does that work"}, {"start": 46.839, "duration": 4.201, "text": "U is the 1 plus e to the 2x it's"}, {"start": 48.67, "duration": 4.61, "text": "derivative us and we said was that twice"}, {"start": 51.04, "duration": 5.009, "text": "either 2x DX"}, {"start": 53.28, "duration": 5.47, "text": "well if that's my you what's my app look"}, {"start": 56.049, "duration": 5.97, "text": "this looks like the function 1 divided"}, {"start": 58.75, "duration": 5.58, "text": "by u times D U but we're off by this"}, {"start": 62.019, "duration": 5.64, "text": "factor of 2 all right so that's the U"}, {"start": 64.33, "duration": 6.179, "text": "and the D U but what is going to be in"}, {"start": 67.659, "duration": 5.13, "text": "that case the F now if I think about"}, {"start": 70.509, "duration": 4.5, "text": "this expression the place where you"}, {"start": 72.789, "duration": 5.22, "text": "occurs is on the denominator so we think"}, {"start": 75.009, "duration": 5.97, "text": "my F looks like 1 divide it out by U and"}, {"start": 78.009, "duration": 4.771, "text": "if they set my F to be 1 over u that"}, {"start": 80.979, "duration": 4.111, "text": "means that they capital F the"}, {"start": 82.78, "duration": 3.96, "text": "antiderivative of the lowercase F is"}, {"start": 85.09, "duration": 5.339, "text": "just gonna be the natural logarithm of"}, {"start": 86.74, "duration": 5.669, "text": "the absolute value of U now why is this"}, {"start": 90.429, "duration": 4.891, "text": "the case well if we take the derivative"}, {"start": 92.409, "duration": 5.49, "text": "of the natural logarithm of the absolute"}, {"start": 95.32, "duration": 7.38, "text": "value of U we're going to get 1 divided"}, {"start": 97.899, "duration": 7.83, "text": "by u so indeed capital F is an"}, {"start": 102.7, "duration": 4.769, "text": "antiderivative of a lower case of F ok"}, {"start": 105.729, "duration": 3.691, "text": "so now if I take what my expression is"}, {"start": 107.469, "duration": 4.231, "text": "notice how I'm going to relate it so we"}, {"start": 109.42, "duration": 4.86, "text": "have that syndra we want to compute but"}, {"start": 111.7, "duration": 6.449, "text": "what I've done here is I've divided out"}, {"start": 114.28, "duration": 5.729, "text": "by a 2 and I've multiplied by a 2 and I"}, {"start": 118.149, "duration": 4.801, "text": "can multiply and divide by 2 that's just"}, {"start": 120.009, "duration": 5.701, "text": "x 1 that doesn't doesn't do anything but"}, {"start": 122.95, "duration": 4.32, "text": "the reason why I do it is because now"}, {"start": 125.71, "duration": 3.659, "text": "when I've got this 2 on the top even"}, {"start": 127.27, "duration": 5.639, "text": "though is cancelled down here to e to"}, {"start": 129.369, "duration": 7.411, "text": "the 2x DX 2 e to the 2x DX that is just"}, {"start": 132.909, "duration": 4.861, "text": "D u that is what I have done is managed"}, {"start": 136.78, "duration": 6.27, "text": "to write this as"}, {"start": 137.77, "duration": 7.92, "text": "1/2 the integral of 1 over u du u so we"}, {"start": 143.05, "duration": 5.94, "text": "get the 1/2 that just comes along for"}, {"start": 145.69, "duration": 5.25, "text": "the ride the integral of 1 over u"}, {"start": 148.99, "duration": 4.17, "text": "becomes the natural logarithm and I"}, {"start": 150.94, "duration": 4.95, "text": "evaluate it at U where u is this 1 plus"}, {"start": 153.16, "duration": 5.19, "text": "e to the 2x and finally I add in the"}, {"start": 155.89, "duration": 4.95, "text": "plus C so that is going to be my general"}, {"start": 158.35, "duration": 5.94, "text": "antiderivative to this funky expression"}, {"start": 160.84, "duration": 4.98, "text": "and it's worth noting that integration"}, {"start": 164.29, "duration": 3.81, "text": "in general is actually quite a bit"}, {"start": 165.82, "duration": 3.48, "text": "harder than differentiation most of"}, {"start": 168.1, "duration": 3.72, "text": "these functions that you can combine"}, {"start": 169.3, "duration": 5.37, "text": "with quotients and products and"}, {"start": 171.82, "duration": 4.98, "text": "compositions and exponents of the"}, {"start": 174.67, "duration": 3.69, "text": "elementary functions we've been talking"}, {"start": 176.8, "duration": 4.05, "text": "about e to the ax and sine and"}, {"start": 178.36, "duration": 4.17, "text": "polynomials we can just apply our rules"}, {"start": 180.85, "duration": 3.96, "text": "and take the derivatives of all of them"}, {"start": 182.53, "duration": 5.49, "text": "but the other way around is not quite so"}, {"start": 184.81, "duration": 5.4, "text": "simple some integrands when you take the"}, {"start": 188.02, "duration": 4.2, "text": "integral of some functions it's able to"}, {"start": 190.21, "duration": 3.93, "text": "be done because it can be manipulated to"}, {"start": 192.22, "duration": 3.69, "text": "look into one of these nice forms like"}, {"start": 194.14, "duration": 4.23, "text": "we have here with the substitution rule"}, {"start": 195.91, "duration": 3.54, "text": "but that is in general not the case and"}, {"start": 198.37, "duration": 2.97, "text": "there's all sorts of integrals where we"}, {"start": 199.45, "duration": 4.37, "text": "don't have a nice rule like this to be"}, {"start": 201.34, "duration": 2.48, "text": "able to apply"}], "title": "Adjusting the Constant in Integration by Substitution", "description": "In this example, the derivative of the inside doesn't exactly line up in the integrand, we have to adjust the constant. \n\nNow it's your turn: \n1) Summarize the big idea of this video in your own words\n2) Write down anything you are unsure about to think about later\n3) What questions for the future do you have? Where are we going with this content?\n4) Can you come up with your own sample test problem on this material? Solve it!\n\nLearning mathematics is best done by actually DOING mathematics. A video like this can only ever be a starting point. I might show you the basic ideas, definitions, formulas, and examples, but to truly master calculus means that you have to spend time - a lot of time! - sitting down and trying problems yourself, asking questions, and thinking about mathematics. So before you go on to the next video, pause and go THINK. \n\nThis video is part of a Calculus course taught by Dr. Trefor Bazett at the University of Cincinnati.\n\nBECOME A MEMBER:\n\u25baJoin: https://www.youtube.com/channel/UC9rTsvTxJnx1DNrDA3Rqa6A/join\n\nMATH BOOKS & MERCH I LOVE:\n\u25ba My Amazon Affiliate Shop: https://www.amazon.com/shop/treforbazett", "lengthSeconds": 203, "width": 640, "height": 360}
{"video_id": "9VPN1qlkFzY", "transcript": [{"start": 1.16, "duration": 5.159, "text": "welcome back to logic 101 I'm William"}, {"start": 3.6, "duration": 4.68, "text": "Spaniel and this lecture is on the rule"}, {"start": 6.319, "duration": 4.2, "text": "of inference known as the constructive"}, {"start": 8.28, "duration": 4.239, "text": "dilemma this one is a little bit more"}, {"start": 10.519, "duration": 4.721, "text": "complicated than previous rules of"}, {"start": 12.519, "duration": 5.92, "text": "inference we've seen because it involves"}, {"start": 15.24, "duration": 5.48, "text": "three three premises we have two"}, {"start": 18.439, "duration": 5.641, "text": "implications to start off with P implies"}, {"start": 20.72, "duration": 5.719, "text": "q and R implies s notice that we have"}, {"start": 24.08, "duration": 4.32, "text": "four simple sentences in total there and"}, {"start": 26.439, "duration": 4.521, "text": "they're split evenly across those two"}, {"start": 28.4, "duration": 4.72, "text": "premises so we have just p and Q up top"}, {"start": 30.96, "duration": 4.599, "text": "and just R and S down bottom there is no"}, {"start": 33.12, "duration": 4.88, "text": "overlap between those two the third"}, {"start": 35.559, "duration": 5.041, "text": "premise is the disjunction between the"}, {"start": 38.0, "duration": 5.719, "text": "antecedence of those two implications so"}, {"start": 40.6, "duration": 5.24, "text": "we have P or R and if we have all of"}, {"start": 43.719, "duration": 4.36, "text": "that then we can conclude that the"}, {"start": 45.84, "duration": 5.16, "text": "disjunction between the second half of"}, {"start": 48.079, "duration": 5.48, "text": "those two implications QRS is true as"}, {"start": 51.0, "duration": 4.92, "text": "well essentially what's going on here is"}, {"start": 53.559, "duration": 4.8, "text": "that we're applying modus ponin across"}, {"start": 55.92, "duration": 5.439, "text": "the various different cases that part"}, {"start": 58.359, "duration": 6.48, "text": "three gives us so part three premise 3"}, {"start": 61.359, "duration": 6.76, "text": "tells us that P is true or R is true or"}, {"start": 64.839, "duration": 5.801, "text": "both P and R are true well in the first"}, {"start": 68.119, "duration": 5.281, "text": "case suppose p is true then we have q"}, {"start": 70.64, "duration": 5.519, "text": "being true through line one and modus"}, {"start": 73.4, "duration": 4.92, "text": "ponents if we have the second case where"}, {"start": 76.159, "duration": 5.401, "text": "just R is true then through modus"}, {"start": 78.32, "duration": 5.24, "text": "ponents in line two we have S being true"}, {"start": 81.56, "duration": 4.04, "text": "and if both of them are true then"}, {"start": 83.56, "duration": 4.68, "text": "through modus ponents and lines one and"}, {"start": 85.6, "duration": 4.32, "text": "two we have both q and s being true and"}, {"start": 88.24, "duration": 3.36, "text": "that's exactly what we see with the"}, {"start": 89.92, "duration": 4.199, "text": "conclusion there we see Q or S being"}, {"start": 91.6, "duration": 4.159, "text": "true which means Q could be true or S"}, {"start": 94.119, "duration": 4.241, "text": "could be true or both of those things"}, {"start": 95.759, "duration": 5.201, "text": "could be true to see this in action with"}, {"start": 98.36, "duration": 5.0, "text": "an example suppose that premise one said"}, {"start": 100.96, "duration": 5.439, "text": "if I am running I am happy and premise"}, {"start": 103.36, "duration": 5.079, "text": "two said if I am sleeping I am dreaming"}, {"start": 106.399, "duration": 4.36, "text": "and finally premise three said I'm"}, {"start": 108.439, "duration": 4.96, "text": "running or I'm sleeping well if I'm"}, {"start": 110.759, "duration": 4.801, "text": "running then I'm happy if I'm sleeping"}, {"start": 113.399, "duration": 4.08, "text": "then I'm dreaming and if I'm running and"}, {"start": 115.56, "duration": 3.599, "text": "sleeping perhaps I'm running in my sleep"}, {"start": 117.479, "duration": 3.721, "text": "then I'm both happy and I'm Dreaming"}, {"start": 119.159, "duration": 3.721, "text": "which is exactly again what the"}, {"start": 121.2, "duration": 4.839, "text": "conclusion says I'm happy or I'm"}, {"start": 122.88, "duration": 4.72, "text": "Dreaming or both now normally what I've"}, {"start": 126.039, "duration": 3.56, "text": "been doing for these rules of inference"}, {"start": 127.6, "duration": 4.32, "text": "is then showing you that this is true on"}, {"start": 129.599, "duration": 3.921, "text": "a truth table but as I mentioned at the"}, {"start": 131.92, "duration": 3.599, "text": "top we have four different simple"}, {"start": 133.52, "duration": 4.12, "text": "sentences here and in the past we've"}, {"start": 135.519, "duration": 4.961, "text": "explored what it looks like when you do"}, {"start": 137.64, "duration": 5.319, "text": "a truth table with four simple sentences"}, {"start": 140.48, "duration": 4.36, "text": "and of course we have 1 2 3 four complex"}, {"start": 142.959, "duration": 3.041, "text": "or compound sentences here which are"}, {"start": 144.84, "duration": 2.88, "text": "only going to make things more"}, {"start": 146.0, "duration": 3.64, "text": "complicated so I'm actually going to"}, {"start": 147.72, "duration": 3.72, "text": "hold off on doing that truth table"}, {"start": 149.64, "duration": 2.959, "text": "because again it's pretty complicated"}, {"start": 151.44, "duration": 3.6, "text": "something that you might want to do on"}, {"start": 152.599, "duration": 5.481, "text": "your own and instead of showing you how"}, {"start": 155.04, "duration": 4.64, "text": "to prove it using a truth table instead"}, {"start": 158.08, "duration": 3.879, "text": "later on in this course when we've"}, {"start": 159.68, "duration": 4.68, "text": "learned how to do proof by contradiction"}, {"start": 161.959, "duration": 4.761, "text": "I will use proof by contradiction to"}, {"start": 164.36, "duration": 3.879, "text": "prove the constructive dilemma through"}, {"start": 166.72, "duration": 4.0, "text": "what actually is going to turn out to be"}, {"start": 168.239, "duration": 3.801, "text": "a much simpler process than a truth"}, {"start": 170.72, "duration": 3.32, "text": "table so of course you don't know how to"}, {"start": 172.04, "duration": 3.52, "text": "do this quite yet but by the end of this"}, {"start": 174.04, "duration": 3.119, "text": "course when we start getting into proofs"}, {"start": 175.56, "duration": 3.2, "text": "you will learn how to do this and I will"}, {"start": 177.159, "duration": 3.401, "text": "take you through this step by step so"}, {"start": 178.76, "duration": 4.08, "text": "you can see that constructive dilemma is"}, {"start": 180.56, "duration": 5.08, "text": "true not using of course constructive"}, {"start": 182.84, "duration": 4.44, "text": "Dilemma to arrive at that conclusion all"}, {"start": 185.64, "duration": 3.679, "text": "right I hope you enjoyed this and I hope"}, {"start": 187.28, "duration": 4.319, "text": "to see you here next time when we talk"}, {"start": 189.319, "duration": 4.081, "text": "about the modus tolan's version of the"}, {"start": 191.599, "duration": 4.081, "text": "constructive dilemma which is known as"}, {"start": 193.4, "duration": 3.479, "text": "the destructive dilemma hope you enjoy"}, {"start": 195.68, "duration": 3.68, "text": "this again and I hope to see you next"}, {"start": 196.879, "duration": 2.481, "text": "time take"}, {"start": 199.64, "duration": 3.0, "text": "care"}], "title": "Logic 101 (#31): Constructive Dilemma", "description": "http://gametheory101.com/courses/logic-101/\n\nConstructive dilemma is a logical rule of inference that says if P implies Q, R implies S, and P or R is true, then Q or S is true as well. Essentially, the constructive dilemma passes the disjunction through two conditional statements. For example, if the statements\n\n\"If I am running, I am happy.\"\n\nand\n\n\"If I am sleeping, I am dreaming.\"\n\nand\n\n\"I am running or sleeping.\"\n\nare true, then it must also be true that I am happy or I am dreaming.", "lengthSeconds": 201, "width": 640, "height": 360}
{"video_id": "cQiyDUiIcOQ", "transcript": [{"start": 0.03, "duration": 5.22, "text": "in this video we're going to take a"}, {"start": 2.7, "duration": 3.869, "text": "closer look at the idea of relations"}, {"start": 5.25, "duration": 4.8, "text": "which are something that we've"}, {"start": 6.569, "duration": 6.99, "text": "previously seen for now you'll recall"}, {"start": 10.05, "duration": 5.7, "text": "that a relation is to let it involve two"}, {"start": 13.559, "duration": 4.081, "text": "different set you had a set a and you"}, {"start": 15.75, "duration": 3.48, "text": "had a set B and I'm put a bunch of"}, {"start": 17.64, "duration": 4.32, "text": "elements inside of a and I put a bunch"}, {"start": 19.23, "duration": 6.03, "text": "of elements inside of B then you could"}, {"start": 21.96, "duration": 5.909, "text": "visualize sort of an arrow diagram for a"}, {"start": 25.26, "duration": 6.179, "text": "relation which is connections from the"}, {"start": 27.869, "duration": 5.161, "text": "set a to the set B so for example I can"}, {"start": 31.439, "duration": 3.96, "text": "have these connections where the the a"}, {"start": 33.03, "duration": 3.99, "text": "one is going to go down to the B 2 and"}, {"start": 35.399, "duration": 4.91, "text": "the a five is goes to the B 2 and the a"}, {"start": 37.02, "duration": 7.02, "text": "3 is going to be 1 and the B 4 and so on"}, {"start": 40.309, "duration": 7.141, "text": "and then you would be able to sometimes"}, {"start": 44.04, "duration": 6.089, "text": "say that a relation was a function if"}, {"start": 47.45, "duration": 4.929, "text": "additionally it had the property of"}, {"start": 50.129, "duration": 4.02, "text": "passing the vertical line test and that"}, {"start": 52.379, "duration": 4.59, "text": "everything in the domain went somewhere"}, {"start": 54.149, "duration": 4.8, "text": "we've previously discussed that and then"}, {"start": 56.969, "duration": 4.471, "text": "our formal definition for what a"}, {"start": 58.949, "duration": 3.0, "text": "relation was is going to be the"}, {"start": 61.44, "duration": 5.27, "text": "following"}, {"start": 61.949, "duration": 8.131, "text": "it is a set of the Cartesian product"}, {"start": 66.71, "duration": 4.72, "text": "that is I look at all pairs something in"}, {"start": 70.08, "duration": 3.0, "text": "a something in B that's what the"}, {"start": 71.43, "duration": 4.29, "text": "Cartesian project pairs we choose a"}, {"start": 73.08, "duration": 6.03, "text": "first thing in a and a second thing in B"}, {"start": 75.72, "duration": 6.63, "text": "and then we say that some pair like say"}, {"start": 79.11, "duration": 6.03, "text": "how about a 1 B 2 that is an element of"}, {"start": 82.35, "duration": 5.82, "text": "your relation if when you look at this"}, {"start": 85.14, "duration": 6.06, "text": "arrow diagram a 1 really is related to B"}, {"start": 88.17, "duration": 5.04, "text": "2 and then the nomenclature we had is"}, {"start": 91.2, "duration": 6.45, "text": "we're going to say that an element in R"}, {"start": 93.21, "duration": 7.86, "text": "is written as a relates to be precisely"}, {"start": 97.65, "duration": 5.249, "text": "lame little a comma little B was an"}, {"start": 101.07, "duration": 4.17, "text": "element of your Cartesian product or"}, {"start": 102.899, "duration": 6.18, "text": "there was an arrow up here in your arrow"}, {"start": 105.24, "duration": 6.72, "text": "diagram now if I have a relation I can"}, {"start": 109.079, "duration": 6.93, "text": "also define something referred to as the"}, {"start": 111.96, "duration": 7.049, "text": "inverse relation and the idea of an"}, {"start": 116.009, "duration": 5.311, "text": "inverse relation is it's exactly doing"}, {"start": 119.009, "duration": 5.22, "text": "what the relation from A to B did but it"}, {"start": 121.32, "duration": 5.6, "text": "flips it around so for example let me"}, {"start": 124.229, "duration": 5.971, "text": "put up my relation R that I saw before"}, {"start": 126.92, "duration": 4.959, "text": "for the inverse relation we're going to"}, {"start": 130.2, "duration": 3.48, "text": "try to take the a and the B and we're"}, {"start": 131.879, "duration": 3.0, "text": "going to try to swap them so I"}, {"start": 133.68, "duration": 3.27, "text": "put them up I'm not going to fill in the"}, {"start": 134.879, "duration": 4.53, "text": "arrow just for the moment but for the in"}, {"start": 136.95, "duration": 4.98, "text": "Reverse relation first of all the the"}, {"start": 139.409, "duration": 4.651, "text": "starting set your domain if you will is"}, {"start": 141.93, "duration": 4.05, "text": "the be your codomain is the age so we"}, {"start": 144.06, "duration": 4.23, "text": "flip that order and then if I think"}, {"start": 145.98, "duration": 5.67, "text": "about how I should be connecting these"}, {"start": 148.29, "duration": 7.11, "text": "well for instance if I go over here the"}, {"start": 151.65, "duration": 5.819, "text": "B 1 is connected down to the a3 the the"}, {"start": 155.4, "duration": 6.149, "text": "B 2 is connected up to the a1 and the a5"}, {"start": 157.469, "duration": 7.291, "text": "and so on so I can fill in my line like"}, {"start": 161.549, "duration": 5.401, "text": "this where these connections are exactly"}, {"start": 164.76, "duration": 5.33, "text": "this picture but it's the mirror image"}, {"start": 166.95, "duration": 3.14, "text": "of that picture"}], "title": "Relations and their Inverses", "description": "We review the idea of a relation, and show how to express the inverse of a relation. A relation from A to B i a subset of the cartesian product A x B and we write an element in the relation as aRb to mean that a relates to b. \n\n\u25baFULL DISCRETE MATH PLAYLIST: https://www.youtube.com/playlist?list=PLHXZ9OQGMqxersk8fUxiUMSIx0DBqsKZS\n\nOTHER COURSE PLAYLISTS:\n\u25baCALCULUS I: https://www.youtube.com/playlist?list=PLHXZ9OQGMqxfT9RMcReZ4WcoVILP4k6-m\n\u25ba CALCULUS II: https://www.youtube.com/playlist?list=PLHXZ9OQGMqxc4ySKTIW19TLrT91Ik9M4n\n\u25baMULTIVARIABLE CALCULUS (Calc III): https://www.youtube.com/playlist?list=PLHXZ9OQGMqxc_CvEy7xBKRQr6I214QJcd\n\u25baVECTOR CALCULUS (Calc IV): https://www.youtube.com/playlist?list=PLHXZ9OQGMqxfW0GMqeUE1bLKaYor6kbHa\n\u25baDIFFERENTIAL EQUATIONS: https://www.youtube.com/watch?v=xeeM3TT4Zgg&list=PLHXZ9OQGMqxcJXnLr08cyNaup4RDsbAl1\n\u25baLINEAR ALGEBRA: https://www.youtube.com/playlist?list=PLHXZ9OQGMqxfUl0tcqPNTJsb7R6BqSLo6\n\nOTHER PLAYLISTS:\n\u25ba Learning Math Series\nhttps://www.youtube.com/watch?v=LPH2lqis3D0&list=PLHXZ9OQGMqxfSkRtlL5KPq6JqMNTh_MBw\n\u25baCool Math Series:\nhttps://www.youtube.com/playlist?list=PLHXZ9OQGMqxelE_9RzwJ-cqfUtaFBpiho\n\nBECOME A MEMBER:\n\u25baJoin: https://www.youtube.com/channel/UC9rTsvTxJnx1DNrDA3Rqa6A/join\n\nMATH BOOKS & MERCH I LOVE:\n\u25ba My Amazon Affiliate Shop: https://www.amazon.com/shop/treforbazett\n\nSOCIALS:\n\u25baTwitter (math based): http://twitter.com/treforbazett\n\u25baInstagram (photography based): http://instagram.com/treforphotography", "lengthSeconds": 169, "width": 640, "height": 360}
{"video_id": "j7qVcc6X8g4", "transcript": [{"start": 0.03, "duration": 5.19, "text": "now let's investigate what it means to"}, {"start": 2.34, "duration": 5.759, "text": "negate a conditional so in this case I"}, {"start": 5.22, "duration": 5.16, "text": "have this conditional P implies Q and"}, {"start": 8.099, "duration": 3.96, "text": "then the question is what happens if I'm"}, {"start": 10.38, "duration": 3.78, "text": "going to negate that particular"}, {"start": 12.059, "duration": 4.14, "text": "conditional now what I'm going to do is"}, {"start": 14.16, "duration": 3.51, "text": "I'm going to use just a bunch of logical"}, {"start": 16.199, "duration": 2.611, "text": "equivalences here I'm going to get some"}, {"start": 17.67, "duration": 3.51, "text": "result and then we're going to see"}, {"start": 18.81, "duration": 4.11, "text": "whether that result makes sense to us so"}, {"start": 21.18, "duration": 4.8, "text": "the first thing that I can say is that"}, {"start": 22.92, "duration": 5.429, "text": "we know that that I can take this P"}, {"start": 25.98, "duration": 4.139, "text": "implies Q and we've just shown that this"}, {"start": 28.349, "duration": 3.811, "text": "is logically equivalent to I'm not"}, {"start": 30.119, "duration": 4.321, "text": "changing the knot here at all but we"}, {"start": 32.16, "duration": 9.57, "text": "know that the inside the conditional is"}, {"start": 34.44, "duration": 9.799, "text": "logically equivalent to not P or Q so"}, {"start": 41.73, "duration": 4.919, "text": "that was one step and now I have a"}, {"start": 44.239, "duration": 4.691, "text": "disjunct of an or statement and Center"}, {"start": 46.649, "duration": 5.91, "text": "than a knot in front of it and we know"}, {"start": 48.93, "duration": 5.1, "text": "that one of de Morgan's laws applies for"}, {"start": 52.559, "duration": 4.051, "text": "a scenario like this that I can take"}, {"start": 54.03, "duration": 5.279, "text": "them not and I can distribute it through"}, {"start": 56.61, "duration": 6.21, "text": "but it does two things it takes the"}, {"start": 59.309, "duration": 5.06, "text": "order to an and and it takes a knot in"}, {"start": 62.82, "duration": 3.57, "text": "front of both of the two different sides"}, {"start": 64.369, "duration": 4.18, "text": "so in other words I'm going to claim"}, {"start": 66.39, "duration": 3.57, "text": "that this is logically equivalent to the"}, {"start": 68.549, "duration": 4.561, "text": "first thing is I'm going to bring the"}, {"start": 69.96, "duration": 7.35, "text": "knot to the inside so goes naught naught"}, {"start": 73.11, "duration": 7.23, "text": "P and then it flips the order to an and"}, {"start": 77.31, "duration": 4.26, "text": "so I go to an and and then I take the"}, {"start": 80.34, "duration": 4.279, "text": "knowledge and I put it in front of the"}, {"start": 81.57, "duration": 12.39, "text": "queue as well and so I go naught Q and"}, {"start": 84.619, "duration": 12.04, "text": "this was de Morgan and then I also note"}, {"start": 93.96, "duration": 4.619, "text": "that here I have this this double"}, {"start": 96.659, "duration": 3.691, "text": "negative and we've previously seen that"}, {"start": 98.579, "duration": 4.561, "text": "I can manipulate the double negatives"}, {"start": 100.35, "duration": 5.37, "text": "not not P is the same thing logically as"}, {"start": 103.14, "duration": 8.61, "text": "just P so I can say that this is the"}, {"start": 105.72, "duration": 8.429, "text": "first thing and not Q now this should"}, {"start": 111.75, "duration": 5.189, "text": "make some sense to us if we think about"}, {"start": 114.149, "duration": 5.97, "text": "what an implication was there was only"}, {"start": 116.939, "duration": 6.651, "text": "going to be one way that the implication"}, {"start": 120.119, "duration": 7.14, "text": "was not going to be true when the"}, {"start": 123.59, "duration": 5.71, "text": "Assumption the hypothesis was true was"}, {"start": 127.259, "duration": 2.551, "text": "that the conclusion was going to be"}, {"start": 129.3, "duration": 2.26, "text": "false"}, {"start": 129.81, "duration": 4.24, "text": "that was the only one in her"}, {"start": 131.56, "duration": 4.35, "text": "truth table that gave a value of false"}, {"start": 134.05, "duration": 4.11, "text": "for the application and that really has"}, {"start": 135.91, "duration": 4.65, "text": "here the first one has to be true the"}, {"start": 138.16, "duration": 4.5, "text": "assumption and it has to be the case"}, {"start": 140.56, "duration": 4.2, "text": "that the conclusion needs to be false"}, {"start": 142.66, "duration": 4.17, "text": "that was the one scenario the other"}, {"start": 144.76, "duration": 2.699, "text": "scenarios like for example when P was"}, {"start": 146.83, "duration": 3.15, "text": "false"}, {"start": 147.459, "duration": 4.711, "text": "those were bumped into the vacuously"}, {"start": 149.98, "duration": 4.5, "text": "true scenario and of course when P was"}, {"start": 152.17, "duration": 4.44, "text": "true and Q is true that was also going"}, {"start": 154.48, "duration": 4.35, "text": "to be true for the implication as well"}, {"start": 156.61, "duration": 4.29, "text": "so what we managed to prove here by"}, {"start": 158.83, "duration": 4.53, "text": "showing a bunch of logical equivalences"}, {"start": 160.9, "duration": 4.68, "text": "is the result is the exact same result"}, {"start": 163.36, "duration": 4.8, "text": "we would have gotten if we just gone to"}, {"start": 165.58, "duration": 4.5, "text": "the truth table and negated the column"}, {"start": 168.16, "duration": 4.909, "text": "for P implies Q so you've got the same"}, {"start": 170.08, "duration": 2.989, "text": "result which is good"}], "title": "Negating a Conditional Statement", "description": "Learning Objectives: Take the negation of a conditional not by using truth tables, but by using known logical equivalences.\n\u25baFull Course Playlist: DISCRETE MATH: https://www.youtube.com/playlist?list=PLHXZ9OQGMqxersk8fUxiUMSIx0DBqsKZS\r\n\nOther Playlists:\r\n\r\n\u25baCALCULUS I: https://www.youtube.com/playlist?list=PLHXZ9OQGMqxfT9RMcReZ4WcoVILP4k6-m\r\n\r\n\u25baCALCULUS II: https://www.youtube.com/playlist?list=PLHXZ9OQGMqxc4ySKTIW19TLrT91Ik9M4n\r\n\r\n\u25baLINEAR ALGEBRA: https://www.youtube.com/playlist?list=PLHXZ9OQGMqxfUl0tcqPNTJsb7R6BqSLo6\r\n\r\n\u25ba Want to learn math effectively? Check out my \"Learning Math\" Series: https://www.youtube.com/watch?v=LPH2lqis3D0&list=PLHXZ9OQGMqxfSkRtlL5KPq6JqMNTh_MBw\r\n\r\n\u25baWant some cool math? Check out my \"Cool Math\" Series: https://www.youtube.com/playlist?list=PLHXZ9OQGMqxelE_9RzwJ-cqfUtaFBpiho\r\n\r\n*****************************************************\r\nYOUR TURN! Learning math requires more than just watching videos, so make sure you reflect, ask questions, and do lots of practice problems!\r\n\r\n****************************************************\r\n\u25baFollow me on Twitter: http://twitter.com/treforbazett\r\n*****************************************************\r\n\r\nThis video was created by Dr. Trefor Bazett", "lengthSeconds": 172, "width": 640, "height": 360}
{"video_id": "JV0vgVa4bCg", "transcript": [{"start": 0.08, "duration": 3.92, "text": "let's recall the fundamental theorem of"}, {"start": 1.599, "duration": 3.76, "text": "calculus that we learned in kindergarten"}, {"start": 4.0, "duration": 3.44, "text": "so the formulation i'm going to take"}, {"start": 5.359, "duration": 5.2, "text": "here is that the integral"}, {"start": 7.44, "duration": 6.56, "text": "of a to b of f prime of x dx"}, {"start": 10.559, "duration": 5.441, "text": "is equal to f of b minus f of a"}, {"start": 14.0, "duration": 4.16, "text": "so if we look at the graph of f prime of"}, {"start": 16.0, "duration": 4.0, "text": "x the fundamental theorem of calculus"}, {"start": 18.16, "duration": 4.959, "text": "tells us that if we take two points a"}, {"start": 20.0, "duration": 6.48, "text": "and b then the area underneath"}, {"start": 23.119, "duration": 4.721, "text": "this graph is determined by the value of"}, {"start": 26.48, "duration": 4.4, "text": "another function namely"}, {"start": 27.84, "duration": 5.28, "text": "f at these boundary points alone"}, {"start": 30.88, "duration": 3.6, "text": "now a priori we need to know every point"}, {"start": 33.12, "duration": 2.959, "text": "between a and b"}, {"start": 34.48, "duration": 3.52, "text": "to determine the area but the"}, {"start": 36.079, "duration": 2.961, "text": "fundamental theorem of calculus tells us"}, {"start": 38.0, "duration": 2.879, "text": "that it's actually"}, {"start": 39.04, "duration": 3.44, "text": "only determined by the values of a"}, {"start": 40.879, "duration": 2.721, "text": "separate function namely its"}, {"start": 42.48, "duration": 3.52, "text": "antiderivative"}, {"start": 43.6, "duration": 4.08, "text": "at the endpoints now to get further"}, {"start": 46.0, "duration": 2.48, "text": "insight on the fundamental theorem of"}, {"start": 47.68, "duration": 2.8, "text": "calculus"}, {"start": 48.48, "duration": 4.399, "text": "what we'll do is we'll rewrite the"}, {"start": 50.48, "duration": 5.12, "text": "integral from a to b of f prime of x"}, {"start": 52.879, "duration": 3.281, "text": "dx as the integral over the closed"}, {"start": 55.6, "duration": 3.599, "text": "interval"}, {"start": 56.16, "duration": 3.919, "text": "a b f prime of x dx for us this is going"}, {"start": 59.199, "duration": 3.68, "text": "to be the same"}, {"start": 60.079, "duration": 3.761, "text": "and we'll also write f prime of x as df"}, {"start": 62.879, "duration": 4.801, "text": "dx"}, {"start": 63.84, "duration": 7.12, "text": "also write a b as the boundary"}, {"start": 67.68, "duration": 4.32, "text": "of the closed interval a b to denote the"}, {"start": 70.96, "duration": 4.159, "text": "boundary of a set"}, {"start": 72.0, "duration": 5.119, "text": "we use this partial symbol so the set"}, {"start": 75.119, "duration": 4.481, "text": "containing only the elements a b"}, {"start": 77.119, "duration": 3.521, "text": "is denoted by partial of the closed"}, {"start": 79.6, "duration": 2.48, "text": "interval a b"}, {"start": 80.64, "duration": 3.36, "text": "so now what the fundamental theorem of"}, {"start": 82.08, "duration": 4.32, "text": "calculus says is that the"}, {"start": 84.0, "duration": 3.68, "text": "integral over the closed interval a b of"}, {"start": 86.4, "duration": 3.12, "text": "df dx"}, {"start": 87.68, "duration": 3.6, "text": "is equal to the integral over the"}, {"start": 89.52, "duration": 6.0, "text": "boundary of the closed interval"}, {"start": 91.28, "duration": 6.72, "text": "of a b of f dx now just to make this"}, {"start": 95.52, "duration": 4.08, "text": "subsequent remark more transparent what"}, {"start": 98.0, "duration": 5.68, "text": "we can do is it can actually replace"}, {"start": 99.6, "duration": 5.6, "text": "df with partial f so from what we've"}, {"start": 103.68, "duration": 3.28, "text": "done here the way we've written the"}, {"start": 105.2, "duration": 3.76, "text": "fundamental theorem of calculus what we"}, {"start": 106.96, "duration": 2.24, "text": "see is that it's not that derivatives"}, {"start": 108.96, "duration": 2.0, "text": "and"}, {"start": 109.2, "duration": 4.16, "text": "integrals are the opposites of each"}, {"start": 110.96, "duration": 5.759, "text": "other it's that derivatives"}, {"start": 113.36, "duration": 5.52, "text": "are dual to the boundaries"}, {"start": 116.719, "duration": 13.121, "text": "of regions of integration and the"}, {"start": 118.88, "duration": 10.96, "text": "integral is the dual pairing here"}, {"start": 129.92, "duration": 2.08, "text": "you"}], "title": "Why you don't understand the Fundamental Theorem of Calculus | Geometric Algebra", "description": "In this video, we discuss how the derivative is not the \"opposite\" of the integral. We offer a perspective on the fundamental theorem of calculus from the point of de Rham cohomology. This is part of an earlier video that I made concerning Green's theorem: https://www.youtube.com/watch?v=XEDtm09ol5o The audio quality was not the best, so I fixed the audio (I hope), and focused only on the part of the video which treats the fundamental theorem of calculus. This perspective has been growing in interest, and commonly goes under the umbrella of Geometric Algebra in the YouTube community. For a swift introduction to geometric algebra, see the following video: https://youtu.be/60z_hpEAtD8\n\n\ud83d\udcaa\ud83d\ude4f Support the channel by signing up to a free trial of Skillshare using the affiliate link https://www.skillshare.com/r/profile/Kyle-B/344454857 \n\nIf you would like free access to the manim course without signing up to Skillshare, send me an email and I'll send you a free link to the course :)\n\nThese videos are separate from my research and teaching roles at the Australian National University, University of Sydney, and Beijing University. \n\nHi, my name is Kyle and I'm currently doing my doctoral mathematics degree in complex differential geometry under the supervision of Professor Gang Tian and Professor Ben Andrews.", "lengthSeconds": 130, "width": 640, "height": 360}
{"video_id": "ZMP6xIdim24", "transcript": [{"start": 3.639, "duration": 4.92, "text": "the function e to the X is a very"}, {"start": 6.729, "duration": 4.441, "text": "important very famous function within"}, {"start": 8.559, "duration": 4.681, "text": "mathematics and indeed it has this"}, {"start": 11.17, "duration": 5.07, "text": "interesting property which is that the"}, {"start": 13.24, "duration": 6.69, "text": "derivative of e to the X is just the"}, {"start": 16.24, "duration": 5.43, "text": "same function that is if I take a point"}, {"start": 19.93, "duration": 5.16, "text": "like how about 1 here all right"}, {"start": 21.67, "duration": 6.75, "text": "so 1 e to the 1 is going to have the"}, {"start": 25.09, "duration": 6.029, "text": "value just e to the 1 is e and so if I"}, {"start": 28.42, "duration": 5.849, "text": "figure out what the slope is that the"}, {"start": 31.119, "duration": 5.94, "text": "height of this which looks like a little"}, {"start": 34.269, "duration": 5.911, "text": "bit over 2 that's basically the value of"}, {"start": 37.059, "duration": 5.011, "text": "e here is the same thing as the slope of"}, {"start": 40.18, "duration": 5.1, "text": "this particular function they have the"}, {"start": 42.07, "duration": 5.1, "text": "same function value and slope and indeed"}, {"start": 45.28, "duration": 3.269, "text": "if I take in my slope down here or my"}, {"start": 47.17, "duration": 3.09, "text": "slope up here it doesn't really matter"}, {"start": 48.549, "duration": 4.591, "text": "it's always gonna have the same slope as"}, {"start": 50.26, "duration": 5.039, "text": "a function value indeed you could ask a"}, {"start": 53.14, "duration": 3.839, "text": "broader category like there's all these"}, {"start": 55.299, "duration": 3.631, "text": "different exponents like 2 to the X and"}, {"start": 56.979, "duration": 6.72, "text": "3 to the X and PI to the X and 100 to"}, {"start": 58.93, "duration": 7.259, "text": "the X e to the X is defined to be or one"}, {"start": 63.699, "duration": 5.791, "text": "way you can define it to be is defined"}, {"start": 66.189, "duration": 5.611, "text": "to be the number such that the"}, {"start": 69.49, "duration": 5.52, "text": "derivative of e to the X is itself we"}, {"start": 71.8, "duration": 5.28, "text": "define e to be the number where this"}, {"start": 75.01, "duration": 3.81, "text": "property is true so we really are gonna"}, {"start": 77.08, "duration": 2.88, "text": "like Exponential's as we go forward in"}, {"start": 78.82, "duration": 4.02, "text": "our course and they have this really"}, {"start": 79.96, "duration": 5.13, "text": "nice property indeed what you can do is"}, {"start": 82.84, "duration": 5.28, "text": "take the derivative of the derivative of"}, {"start": 85.09, "duration": 5.37, "text": "e to the X first wrote it takes UVs"}, {"start": 88.12, "duration": 4.26, "text": "second river takes you to the X you take"}, {"start": 90.46, "duration": 4.77, "text": "a hundred rooms in a row you get back to"}, {"start": 92.38, "duration": 5.31, "text": "e to the X and so forth so where are we"}, {"start": 95.23, "duration": 4.02, "text": "at we now know how to take the"}, {"start": 97.69, "duration": 3.48, "text": "derivatives of two different types of"}, {"start": 99.25, "duration": 4.05, "text": "functions the derivatives of an"}, {"start": 101.17, "duration": 6.39, "text": "exponential like e to the X and the"}, {"start": 103.3, "duration": 5.25, "text": "derivative of a power like X to the n so"}, {"start": 107.56, "duration": 3.03, "text": "we've got those sort of two like"}, {"start": 108.55, "duration": 3.39, "text": "baseline kind of functions and then the"}, {"start": 110.59, "duration": 3.12, "text": "additive 'ti in the scalar"}, {"start": 111.94, "duration": 3.81, "text": "multiplication those are ways that I can"}, {"start": 113.71, "duration": 3.75, "text": "combine them so for example I could have"}, {"start": 115.75, "duration": 4.35, "text": "a polynomial which is a bunch of power"}, {"start": 117.46, "duration": 4.68, "text": "rules or I can have like e to the X plus"}, {"start": 120.1, "duration": 4.41, "text": "x squared and I would be able to combine"}, {"start": 122.14, "duration": 3.63, "text": "them by these different rules so I"}, {"start": 124.51, "duration": 3.6, "text": "haven't got all functions by any means"}, {"start": 125.77, "duration": 4.519, "text": "but we're developing so far a decent"}, {"start": 128.11, "duration": 2.179, "text": "list"}], "title": "The derivative of e^x.", "description": "Description:\nThe derivative of e^x is....itself! e^x. \n\nLearning Objectives\n1) Compute the derivative of e^x\n\nNow it's your turn: \n1) Summarize the big idea of this video in your own words\n2) Write down anything you are unsure about to think about later\n3) What questions for the future do you have? Where are we going with this content?\n4) Can you come up with your own sample test problem on this material? Solve it!\n\nLearning mathematics is best done by actually DOING mathematics. A video like this can only ever be a starting point. I might show you the basic ideas, definitions, formulas, and examples, but to truly master calculus means that you have to spend time - a lot of time! - sitting down and trying problems yourself, asking questions, and thinking about mathematics. So before you go on to the next video, pause and go THINK. \n\nThis video is part of a Calculus course taught by Dr. Trefor Bazett at the University of Cincinnati.\n\nBECOME A MEMBER:\n\u25baJoin: https://www.youtube.com/channel/UC9rTsvTxJnx1DNrDA3Rqa6A/join\n\nMATH BOOKS & MERCH I LOVE:\n\u25ba My Amazon Affiliate Shop: https://www.amazon.com/shop/treforbazett", "lengthSeconds": 129, "width": 640, "height": 360}
{"video_id": "rbJlTAO_Oms", "transcript": [{"start": 0.08, "duration": 3.4, "text": "continuing in this fashion always"}, {"start": 1.76, "duration": 3.44, "text": "cutting up the rightmost piece into"}, {"start": 3.48, "duration": 6.199, "text": "those same proportions you'll find that"}, {"start": 5.2, "duration": 6.64, "text": "1 - p + P * 1 - P plus p ^ 2 * 1 - p on"}, {"start": 9.679, "duration": 5.721, "text": "and on always adding P to the next power"}, {"start": 11.84, "duration": 6.199, "text": "* 1 - P equal 1 dividing both sides by 1"}, {"start": 15.4, "duration": 5.36, "text": "- P we get this nice"}, {"start": 18.039, "duration": 5.0, "text": "formula in this formula the universe has"}, {"start": 20.76, "duration": 3.759, "text": "offered a weird form of nonsense even"}, {"start": 23.039, "duration": 3.281, "text": "though the way that you discovered it"}, {"start": 24.519, "duration": 4.481, "text": "only makes sense for values of P between"}, {"start": 26.32, "duration": 4.48, "text": "0 and 1 the right hand side still makes"}, {"start": 29.0, "duration": 4.92, "text": "sense when you replace ke with any other"}, {"start": 30.8, "duration": 7.72, "text": "number except maybe for one for instance"}, {"start": 33.92, "duration": 8.08, "text": "plugging in -1 the equation reads 1 - 1"}, {"start": 38.52, "duration": 7.6, "text": "+ 1us one on and on forever alternating"}, {"start": 42.0, "duration": 6.32, "text": "between the two equals 12 which feels"}, {"start": 46.12, "duration": 4.16, "text": "both pretty silly and kind of like the"}, {"start": 48.32, "duration": 6.919, "text": "only thing that it could be plugging in"}, {"start": 50.28, "duration": 10.32, "text": "two the equation reads 1 + 2 + 4 + 8 on"}, {"start": 55.239, "duration": 5.361, "text": "and on to Infinity = -1"}], "title": "this is MF DOOM (ft. 3blue1brown)", "description": "original video: https://youtu.be/XFDM1ip5HdU?si=dUjEakmKfR3wJI7x\n(just cut it up a teeny bit to make it fit more :3)", "lengthSeconds": 59, "width": 640, "height": 640}
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\u673a\uff0c"}, {"start": 195.698, "duration": 3.182, "text": "\u53ea\u6709\u5f53\u4e8b\u4ef6\u4e0d\u662f\u72ec\u7acb\u7684\u65f6\u624d\u5177\u6709\u5b9e\u8d28\u6027\u3002"}, {"start": 198.88, "duration": 3.035, "text": "\u8d1d\u53f6\u65af\u5b9a\u7406\u5c31\u662f\u4e00\u4e2a\u5b8c\u7f8e\u7684\u4f8b\u5b50\uff0c\u5b83\u51c6\u786e\u5730 "}, {"start": 201.915, "duration": 2.885, "text": "\u6d4b\u91cf\u4e86\u4e00\u4e2a\u53d8\u91cf\u5bf9\u53e6\u4e00\u4e2a\u53d8\u91cf\u7684\u4f9d\u8d56\u7a0b\u5ea6\u3002"}], "title": "The quick proof of Bayes' theorem", "description": "Including some added words on independence.\nMain video: https://youtu.be/HZGCoVF3YvM\nHelp fund future projects: https://www.patreon.com/3blue1brown\nAn equally valuable form of support is to simply share some of the videos.\nSpecial thanks to these supporters: http://3b1b.co/bayes-thanks\r\n\r\n------------------\r\n\r\nThese animations are largely made using manim, a scrappy open-source python library:  https://github.com/3b1b/manim\r\n\r\nIf you want to check it out, I feel compelled to warn you that it's not the most well-documented tool, and it has many other quirks you might expect in a library someone wrote with only their own use in mind.\r\n\r\nMusic by Vincent Rubinetti.\r\nDownload the music on Bandcamp:\r\nhttps://vincerubinetti.bandcamp.com/album/the-music-of-3blue1brown\r\n\r\nStream the music on Spotify:\r\nhttps://open.spotify.com/album/1dVyjwS8FBqXhRunaG5W5u\r\n\r\nIf you want to contribute translated subtitles or to help review those that have already been made by others and need approval, you can click the gear icon in the video and go to subtitles/cc, then \"add subtitles/cc\".  I really appreciate those who do this, as it helps make the lessons accessible to more people.\r\n\r\n------------------\r\n\r\n3blue1brown is a channel about animating math, in all senses of the word animate.  And you know the drill with YouTube, if you want to stay posted on new videos, subscribe: http://3b1b.co/subscribe\r\n\r\nVarious social media stuffs:\r\nWebsite: https://www.3blue1brown.com\r\nTwitter: https://twitter.com/3blue1brown\r\nReddit: https://www.reddit.com/r/3blue1brown\r\nInstagram: https://www.instagram.com/3blue1brown_animations/\r\nPatreon: https://patreon.com/3blue1brown\r\nFacebook: https://www.facebook.com/3blue1brown", "lengthSeconds": 227, "width": 640, "height": 360}
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{"video_id": "VFbyGEZLMZw", "transcript": [{"start": 0.08, "duration": 3.76, "text": "draw two points on a circle and connect"}, {"start": 1.92, "duration": 4.04, "text": "them obviously this divides the circle"}, {"start": 3.84, "duration": 3.72, "text": "into two different regions if you put a"}, {"start": 5.96, "duration": 3.04, "text": "third point on the circle and then draw"}, {"start": 7.56, "duration": 3.24, "text": "lines connecting it to the previous"}, {"start": 9.0, "duration": 4.2, "text": "points now the circle's been divided"}, {"start": 10.8, "duration": 4.2, "text": "into four separate regions at a fourth"}, {"start": 13.2, "duration": 3.88, "text": "Point add lines connecting it to all the"}, {"start": 15.0, "duration": 3.6, "text": "previous ones and again count up how"}, {"start": 17.08, "duration": 4.08, "text": "many regions the circle's been divided"}, {"start": 18.6, "duration": 4.4, "text": "into this time we get eight if you start"}, {"start": 21.16, "duration": 3.199, "text": "to see the pattern let's do one more at"}, {"start": 23.0, "duration": 3.439, "text": "a fifth Point connect it to all the"}, {"start": 24.359, "duration": 4.32, "text": "previous ones count up the regions and"}, {"start": 26.439, "duration": 3.68, "text": "this time we get 16 again maybe you can"}, {"start": 28.679, "duration": 4.241, "text": "guess what's coming next but would You"}, {"start": 30.119, "duration": 4.841, "text": "Bet Your Life on it well let's try I'll"}, {"start": 32.92, "duration": 4.12, "text": "add a sixth Point connect it to all of"}, {"start": 34.96, "duration": 3.64, "text": "the previous ones and this time if I go"}, {"start": 37.04, "duration": 3.359, "text": "through and meticulously count all the"}, {"start": 38.6, "duration": 3.68, "text": "regions you can pause and do it yourself"}, {"start": 40.399, "duration": 4.041, "text": "I'm not pulling any trickery here you"}, {"start": 42.28, "duration": 4.52, "text": "actually get 31 be careful of"}, {"start": 44.44, "duration": 3.959, "text": "extrapolating patterns without proof in"}, {"start": 46.8, "duration": 3.399, "text": "this case there's a very good reason why"}, {"start": 48.399, "duration": 4.041, "text": "it looks like powers of two until"}, {"start": 50.199, "duration": 3.921, "text": "suddenly breaking at the sixth iteration"}, {"start": 52.44, "duration": 3.4, "text": "it's a little too long for a short form"}, {"start": 54.12, "duration": 3.4, "text": "explanation but before watching the full"}, {"start": 55.84, "duration": 6.039, "text": "video on YouTube I highly encourage you"}, {"start": 57.52, "duration": 4.359, "text": "to try thinking it through yourself"}], "title": "Don't let it fool you! (Link above explains what's happening)", "description": "A link to the full video is at the bottom of the screen.\nOr, for reference: https://youtu.be/YtkIWDE36qU\n\nThanks to Dawid Ko\u0142odziej for editing together this short", "lengthSeconds": 60, "width": 360, "height": 640}
{"video_id": "GOSezO0CHss", "transcript": [{"start": 0.16, "duration": 3.559, "text": "this is a gton"}, {"start": 1.52, "duration": 4.239, "text": "board maybe you've seen one before it's"}, {"start": 3.719, "duration": 4.481, "text": "a popular demonstration of how even when"}, {"start": 5.759, "duration": 4.76, "text": "a single event is chaotic and random"}, {"start": 8.2, "duration": 3.84, "text": "with an effectively unknowable outcome"}, {"start": 10.519, "duration": 3.28, "text": "it's still possible to make precise"}, {"start": 12.04, "duration": 3.679, "text": "statements about a large number of"}, {"start": 13.799, "duration": 3.681, "text": "events namely how the relative"}, {"start": 15.719, "duration": 3.961, "text": "proportions for many different outcomes"}, {"start": 17.48, "duration": 4.4, "text": "are"}, {"start": 19.68, "duration": 3.64, "text": "distributed more specifically the gon"}, {"start": 21.88, "duration": 2.96, "text": "board illustrates one of the most"}, {"start": 23.32, "duration": 3.48, "text": "prominent distributions in all of"}, {"start": 24.84, "duration": 4.12, "text": "probability known as the normal"}, {"start": 26.8, "duration": 5.31, "text": "distribution more colloquially known as"}, {"start": 28.96, "duration": 11.909, "text": "a bell curve and also called a gaussian"}, {"start": 32.11, "duration": 8.759, "text": "[Music]"}, {"start": 43.89, "duration": 3.17, "text": "[Music]"}, {"start": 47.92, "duration": 3.0, "text": "distribution"}], "title": "Order from chaos", "description": "A link to the full video on the Central Limit Theorem is at the bottom of the screen.\nOr, for reference: https://youtu.be/zeJD6dqJ5lo\n\nThanks to Dawid Ko\u0142odziej from long-to-short editing", "lengthSeconds": 49, "width": 360, "height": 640}
{"video_id": "XPoU4xfXDhg", "transcript": [{"start": 0.28, "duration": 5.64, "text": "hello think about a forward gain"}, {"start": 2.36, "duration": 7.6, "text": "transfer function represented as 2 l/ s"}, {"start": 5.92, "duration": 6.799, "text": "+ 2 * s + 3 integrated to a system"}, {"start": 9.96, "duration": 5.48, "text": "featuring a Unity gain feedback"}, {"start": 12.719, "duration": 6.48, "text": "path the transer function for the closed"}, {"start": 15.44, "duration": 9.12, "text": "loop system can be written as GS over 1"}, {"start": 19.199, "duration": 5.361, "text": "plus GS and can be further simplified as"}, {"start": 25.279, "duration": 6.4, "text": "follows when we use a step input as our"}, {"start": 28.4, "duration": 6.52, "text": "reference signal the output YT will have"}, {"start": 31.679, "duration": 6.281, "text": "the shape of this form to achieve better"}, {"start": 34.92, "duration": 6.04, "text": "control of the output we can introduce a"}, {"start": 37.96, "duration": 5.599, "text": "controller to the system let's consider"}, {"start": 40.96, "duration": 6.0, "text": "adding a controller for our system"}, {"start": 43.559, "duration": 6.361, "text": "represented as s+ 2/ s+"}, {"start": 46.96, "duration": 7.399, "text": "10 now the close loop transfer function"}, {"start": 49.92, "duration": 7.319, "text": "for our system is gscs over 1 plus"}, {"start": 54.359, "duration": 5.88, "text": "gscs and can be further simplified as"}, {"start": 57.239, "duration": 3.0, "text": "follows"}], "title": "Animated Guide to Discrete Control Systems #3blue1brown#animation #shorts #manim#controlsystems", "description": "@BuddhiWijenayake2000\n#youtubeshorts #discretemathematics #manim #maths #electricalandelectronics #electronic #engineering", "lengthSeconds": 59, "width": 360, "height": 640}
{"video_id": "PE_0n9Ph_PQ", "transcript": [{"start": 0.63, "duration": 20.62, "text": "[M\u00fasica]"}, {"start": 29.32, "duration": 5.049, "text": "SA"}, {"start": 31.34, "duration": 3.029, "text": "[M\u00fasica]"}, {"start": 36.719, "duration": 11.011, "text": "at"}, {"start": 39.46, "duration": 8.27, "text": "[M\u00fasica]"}], "title": "Taylor & Maclaurin Series Explained with an Animated Example ! #3blue1brown #mathshorts #animation", "description": "", "lengthSeconds": 47, "width": 360, "height": 640}
{"video_id": "nAT-MCsJeik", "transcript": [{"start": 0.04, "duration": 3.639, "text": "I posed a certain puzzle in a recent"}, {"start": 1.76, "duration": 3.559, "text": "video and based on a couple comments I"}, {"start": 3.679, "duration": 4.08, "text": "realized that I could have been clearer"}, {"start": 5.319, "duration": 4.001, "text": "about what makes this puzzle tricky the"}, {"start": 7.759, "duration": 3.88, "text": "question starts by supposing that you"}, {"start": 9.32, "duration": 4.399, "text": "have a disc with radius one and you"}, {"start": 11.639, "duration": 3.92, "text": "cover it with a whole bunch of strips"}, {"start": 13.719, "duration": 4.281, "text": "when I say strip here I mean a region"}, {"start": 15.559, "duration": 4.521, "text": "that's Bound by two parallel lines and"}, {"start": 18.0, "duration": 4.24, "text": "the quantity that you'll care about is"}, {"start": 20.08, "duration": 4.88, "text": "the width of that strip the distance"}, {"start": 22.24, "duration": 4.359, "text": "between those lines the question is"}, {"start": 24.96, "duration": 3.96, "text": "supposing that your strips completely"}, {"start": 26.599, "duration": 4.92, "text": "cover the disc what is the smallest"}, {"start": 28.92, "duration": 5.2, "text": "possible value for the sum of all of"}, {"start": 31.519, "duration": 5.04, "text": "those widths for example if you just"}, {"start": 34.12, "duration": 4.2, "text": "used a bunch of parallel strips then the"}, {"start": 36.559, "duration": 4.52, "text": "sum of all of the widths would be the"}, {"start": 38.32, "duration": 5.0, "text": "diameter of the circle which is two Now"}, {"start": 41.079, "duration": 4.0, "text": "spoiler alert it turns out that two is"}, {"start": 43.32, "duration": 3.559, "text": "the smallest that you can get and a"}, {"start": 45.079, "duration": 3.441, "text": "couple commenters claimed that they"}, {"start": 46.879, "duration": 3.641, "text": "don't see the point this is obvious"}, {"start": 48.52, "duration": 3.359, "text": "parallel strips should be optimal CU"}, {"start": 50.52, "duration": 4.76, "text": "otherwise you would have some kind of"}, {"start": 51.879, "duration": 5.601, "text": "overlap and overlap is clearly wasteful"}, {"start": 55.28, "duration": 4.239, "text": "that would be true if we were trying to"}, {"start": 57.48, "duration": 4.32, "text": "minimize the total area that these"}, {"start": 59.519, "duration": 4.521, "text": "strips have in that case overlap is"}, {"start": 61.8, "duration": 4.16, "text": "necessarily a waste of area but that's"}, {"start": 64.04, "duration": 4.04, "text": "not the challenge the tricky part here"}, {"start": 65.96, "duration": 4.44, "text": "is that the width of a strip is not"}, {"start": 68.08, "duration": 4.44, "text": "proportional to the area it's perfectly"}, {"start": 70.4, "duration": 3.88, "text": "possible to have a very fat strip near"}, {"start": 72.52, "duration": 4.4, "text": "the edge of the circle with a bigger"}, {"start": 74.28, "duration": 4.839, "text": "width but a smaller area than a thinner"}, {"start": 76.92, "duration": 3.92, "text": "strip near the center of the circle so"}, {"start": 79.119, "duration": 3.721, "text": "for all you know there might be a clever"}, {"start": 80.84, "duration": 3.599, "text": "covering that has a little overlap here"}, {"start": 82.84, "duration": 3.919, "text": "and there which trades off an"}, {"start": 84.439, "duration": 4.521, "text": "inefficient use of area for a more"}, {"start": 86.759, "duration": 4.241, "text": "efficient use of the total width the"}, {"start": 88.96, "duration": 4.519, "text": "challenge is to find find a rigorous way"}, {"start": 91.0, "duration": 4.04, "text": "to prove that this is not possible that"}, {"start": 93.479, "duration": 3.521, "text": "the total width of the strips really"}, {"start": 95.04, "duration": 4.48, "text": "cannot be lower than the diameter of the"}, {"start": 97.0, "duration": 4.56, "text": "circle it's really not an obvious fact"}, {"start": 99.52, "duration": 4.0, "text": "and if you appreciate this it makes the"}, {"start": 101.56, "duration": 5.32, "text": "solution that we cover in the main video"}, {"start": 103.52, "duration": 3.36, "text": "all the more beautiful"}], "title": "This puzzle is trickier than it seems", "description": "From this full video: https://youtu.be/piJkuavhV50", "lengthSeconds": 105, "width": 360, "height": 640}
{"video_id": "SfYY8SSTyFU", "transcript": [{"start": 0.08, "duration": 3.72, "text": "when a large language model generates"}, {"start": 1.8, "duration": 3.8, "text": "text there's a parameter that can be set"}, {"start": 3.8, "duration": 4.68, "text": "called temperature which in a manner of"}, {"start": 5.6, "duration": 4.36, "text": "speaking controls how creative it is to"}, {"start": 8.48, "duration": 2.8, "text": "understand temperature you first have to"}, {"start": 9.96, "duration": 3.28, "text": "understand that large language models"}, {"start": 11.28, "duration": 4.359, "text": "are originally trained to take in a"}, {"start": 13.24, "duration": 4.4, "text": "passage of text and predict what word or"}, {"start": 15.639, "duration": 3.761, "text": "other common character combination comes"}, {"start": 17.64, "duration": 4.16, "text": "next this prediction looks like some"}, {"start": 19.4, "duration": 4.639, "text": "kind of probability distribution when"}, {"start": 21.8, "duration": 4.28, "text": "these models generate new text like an"}, {"start": 24.039, "duration": 4.48, "text": "answer to a user question with a chatbot"}, {"start": 26.08, "duration": 4.84, "text": "what's going on is that a full context"}, {"start": 28.519, "duration": 4.321, "text": "describing the interaction is input into"}, {"start": 30.92, "duration": 4.6, "text": "the model the model predicts what word"}, {"start": 32.84, "duration": 4.559, "text": "comes next it takes a random sample from"}, {"start": 35.52, "duration": 3.559, "text": "the distribution that it generates it"}, {"start": 37.399, "duration": 3.721, "text": "appends that random choice to the full"}, {"start": 39.079, "duration": 3.841, "text": "context and then it runs it all again on"}, {"start": 41.12, "duration": 3.599, "text": "this extended text to make another"}, {"start": 42.92, "duration": 3.959, "text": "prediction and take another sample and"}, {"start": 44.719, "duration": 4.041, "text": "so on and so on over and over the"}, {"start": 46.879, "duration": 3.36, "text": "temperature is a way to modify the"}, {"start": 48.76, "duration": 3.88, "text": "probability distributions that it"}, {"start": 50.239, "duration": 4.441, "text": "generates if you set a high temperature"}, {"start": 52.64, "duration": 3.84, "text": "it gives more weight to the less likely"}, {"start": 54.68, "duration": 3.92, "text": "Words which essentially gives the model"}, {"start": 56.48, "duration": 4.12, "text": "a better chance of selecting unusual"}, {"start": 58.6, "duration": 3.52, "text": "phrases and ideas whereas a low"}, {"start": 60.6, "duration": 3.36, "text": "temperature makes it more likely to"}, {"start": 62.12, "duration": 3.56, "text": "choose the most predictable words"}, {"start": 63.96, "duration": 5.839, "text": "there's more to say but as always the"}, {"start": 65.68, "duration": 4.119, "text": "full video goes into the full details"}], "title": "Check out @3blue1brown for more", "description": "", "lengthSeconds": 68, "width": 270, "height": 480}
{"video_id": "fThuRkf3af4", "transcript": [{"start": 0.0, "duration": 3.179, "text": "option e to the t is to ask what"}, {"start": 2.04, "duration": 2.58, "text": "properties does it have the most"}, {"start": 3.179, "duration": 2.821, "text": "important one and from some binding"}, {"start": 4.62, "duration": 3.239, "text": "property is that it is its own"}, {"start": 6.0, "duration": 3.36, "text": "derivative together with the added"}, {"start": 7.859, "duration": 3.121, "text": "condition that inputting zero returns"}, {"start": 9.36, "duration": 3.18, "text": "one it's actually the only function with"}, {"start": 10.98, "duration": 3.3, "text": "this property and you can illustrate"}, {"start": 12.54, "duration": 3.54, "text": "what this means with a physical model if"}, {"start": 14.28, "duration": 3.6, "text": "e to the T describes your position on a"}, {"start": 16.08, "duration": 3.539, "text": "number line as a function of time then"}, {"start": 17.88, "duration": 3.12, "text": "you start at the number one and what"}, {"start": 19.619, "duration": 3.601, "text": "this equation is saying is that your"}, {"start": 21.0, "duration": 4.439, "text": "velocity the derivative of position is"}, {"start": 23.22, "duration": 3.66, "text": "always equal to that position the"}, {"start": 25.439, "duration": 3.361, "text": "farther away from zero you are the"}, {"start": 26.88, "duration": 3.899, "text": "faster you move so even before knowing"}, {"start": 28.8, "duration": 3.36, "text": "how to compute e to the T exactly going"}, {"start": 30.779, "duration": 3.361, "text": "from a specific time to a specific"}, {"start": 32.16, "duration": 4.02, "text": "position this ability to associate each"}, {"start": 34.14, "duration": 3.36, "text": "position with a velocity paints a very"}, {"start": 36.18, "duration": 3.059, "text": "strong intuitive picture of how the"}, {"start": 37.5, "duration": 3.54, "text": "function must grow you know that you'll"}, {"start": 39.239, "duration": 3.301, "text": "be accelerating and at an accelerating"}, {"start": 41.04, "duration": 4.64, "text": "rate with an all-around feeling of"}, {"start": 42.54, "duration": 3.14, "text": "things getting out of hand quickly"}, {"start": 46.8, "duration": 4.079, "text": "and if you add a constant to that"}, {"start": 48.36, "duration": 3.84, "text": "exponent like e to the 2 times T the"}, {"start": 50.879, "duration": 3.601, "text": "chain rule tells us that the derivative"}, {"start": 52.2, "duration": 3.6, "text": "is now two times itself so at every"}, {"start": 54.48, "duration": 2.94, "text": "point on the number line rather than"}, {"start": 55.8, "duration": 6.02, "text": "attaching a vector corresponding to the"}, {"start": 57.42, "duration": 4.4, "text": "number itself first double the magnitude"}], "title": "\u202a@3blue1brown\u202c e^t #maths #viral", "description": "", "lengthSeconds": 60, "width": 360, "height": 640}
{"video_id": "BsleKGwiuc8", "transcript": [{"start": 0.46, "duration": 12.72, "text": "[Music]"}, {"start": 21.73, "duration": 3.07, "text": "[Music]"}, {"start": 26.48, "duration": 3.0, "text": "keep"}, {"start": 40.079, "duration": 5.511, "text": "to iimp"}, {"start": 42.54, "duration": 3.05, "text": "[Music]"}], "title": "Pythagore Demonstration Using Geometry ! #3blue1brown #mathshorts #animation #demonstration #maths", "description": "", "lengthSeconds": 45, "width": 360, "height": 640}
{"video_id": "IBzmAtVe9OQ", "transcript": [{"start": 0.46, "duration": 12.72, "text": "[Music]"}, {"start": 21.73, "duration": 3.07, "text": "[Music]"}, {"start": 26.48, "duration": 3.0, "text": "keep"}, {"start": 42.54, "duration": 3.05, "text": "[Music]"}, {"start": 46.36, "duration": 5.6, "text": "gang SL get that Tempo just up keep"}, {"start": 48.719, "duration": 3.241, "text": " tight make f"}], "title": "What is a derivative ? #3blue1brown #animation #mathshorts #derivative #explained #definition", "description": "", "lengthSeconds": 50, "width": 360, "height": 640}
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{"video_id": "EPAD0OnSslE", "transcript": [{"start": 0.0, "duration": 5.279, "text": "the surface area of a sphere is 4 pi"}, {"start": 2.639, "duration": 4.561, "text": "times its radius squared exactly four"}, {"start": 5.279, "duration": 4.801, "text": "times the area of a circle with the same"}, {"start": 7.2, "duration": 4.439, "text": "radius but why Archimedes had a very"}, {"start": 10.08, "duration": 3.84, "text": "beautiful proof that this surface area"}, {"start": 11.639, "duration": 4.2, "text": "is the same as the area of a cylinder"}, {"start": 13.92, "duration": 3.6, "text": "that encloses that sphere if you"}, {"start": 15.839, "duration": 3.601, "text": "disregard the circular caps of that"}, {"start": 17.52, "duration": 4.62, "text": "cylinder the idea is that if you shine"}, {"start": 19.44, "duration": 4.8, "text": "some light from the z-axis perpendicular"}, {"start": 22.14, "duration": 4.139, "text": "to that axis then if you compare the"}, {"start": 24.24, "duration": 4.799, "text": "area of a small rectangle drawn on that"}, {"start": 26.279, "duration": 4.801, "text": "sphere to the area of the shadow cast on"}, {"start": 29.039, "duration": 4.741, "text": "that cylinder those two areas turn out"}, {"start": 31.08, "duration": 4.319, "text": "to be the same this isn't obvious but"}, {"start": 33.78, "duration": 3.36, "text": "when you work it out that shadow is a"}, {"start": 35.399, "duration": 4.141, "text": "copy of the original Little rectangle"}, {"start": 37.14, "duration": 4.079, "text": "but squished down in One Direction and"}, {"start": 39.54, "duration": 3.42, "text": "stretched out in another when you"}, {"start": 41.219, "duration": 3.481, "text": "analyze the relevant geometry the two"}, {"start": 42.96, "duration": 4.02, "text": "effects actually cancel out perfectly"}, {"start": 44.7, "duration": 4.26, "text": "treating the sphere as a sum of all of"}, {"start": 46.98, "duration": 3.84, "text": "these little patches and the cylinder as"}, {"start": 48.96, "duration": 3.48, "text": "a sum of all of those Shadows we can"}, {"start": 50.82, "duration": 3.6, "text": "infer that the surface area of the"}, {"start": 52.44, "duration": 3.72, "text": "sphere is the same as the area of that"}, {"start": 54.42, "duration": 3.959, "text": "cylinder now the cylinder is something"}, {"start": 56.16, "duration": 3.6, "text": "you can unwrap into a rectangle where"}, {"start": 58.379, "duration": 3.5, "text": "one side length corresponds to the"}, {"start": 59.76, "duration": 2.119, "text": "circum"}], "title": "\u202a@3blue1brown\u202c #mathematics projections", "description": "Content @3blue1brown refer for full video.\nArea of sphere\n#mathematics \n#concept \n#geometry", "lengthSeconds": 60, "width": 360, "height": 640}
{"video_id": "JNVZK4ofW-Y", "transcript": [{"start": 0.0, "duration": 3.1, "text": "Has it ever bothered you how with exponents, logs, and roots, "}, {"start": 3.1, "duration": 3.8, "text": "we have three completely different types of notation to write the same fact?"}, {"start": 7.22, "duration": 2.601, "text": "If you write 2 cubed equals 8, it's the relative "}, {"start": 9.821, "duration": 3.079, "text": "position of the 3 over the 2 that indicates the operation."}, {"start": 13.4, "duration": 2.226, "text": "But when you write the cube root of 8 equals 2, "}, {"start": 15.626, "duration": 3.294, "text": "which is the same fact, you introduce this new squiggly radical symbol."}, {"start": 19.16, "duration": 3.591, "text": "And then writing log base 2 of 8 equals 3, again the same fact, "}, {"start": 22.751, "duration": 2.189, "text": "you write out a word for the operation."}, {"start": 24.94, "duration": 3.748, "text": "This weird discrepancy in notation isn't just counterintuitive, "}, {"start": 28.688, "duration": 1.992, "text": "I think it's counter-mathematical."}, {"start": 30.98, "duration": 4.421, "text": "Rather than making seemingly different facts look the same, which is what math should do, "}, {"start": 35.401, "duration": 4.079, "text": "it makes three facts that should obviously be the same look artificially different."}, {"start": 39.74, "duration": 2.374, "text": "I came across a really nice suggestion on math "}, {"start": 42.114, "duration": 2.526, "text": "stack exchange for a more symmetric notation here."}, {"start": 44.86, "duration": 3.663, "text": "In our example, the way this would work is you write a triangle with "}, {"start": 48.523, "duration": 3.557, "text": "a 2 in the lower left, a 3 on the top, and an 8 on the lower right."}, {"start": 52.08, "duration": 3.877, "text": "To express the operation 2 cubed, you remove that bottom right corner, "}, {"start": 55.957, "duration": 4.643, "text": "and the symbol as a whole represents the value that should go in that missing corner."}, {"start": 60.94, "duration": 4.228, "text": "To express log base 2 of 8, which is asking the question 2 to the what equals 8, "}, {"start": 65.168, "duration": 1.932, "text": "what you do is remove the top number."}, {"start": 67.52, "duration": 4.18, "text": "Again, the symbol as a whole represents the value that should go in that missing corner."}, {"start": 72.1, "duration": 1.08, "text": "You can guess the third case."}, {"start": 73.54, "duration": 3.74, "text": "If you want to express the cube root of 8, you remove the lower left corner."}, {"start": 77.8, "duration": 1.981, "text": "Once again, the symbol as a whole represents the "}, {"start": 79.781, "duration": 1.739, "text": "value that should go in the missing corner."}, {"start": 81.52, "duration": 3.9, "text": "This much more clearly expresses the relationship of all three operations."}, {"start": 85.82, "duration": 4.07, "text": "The definition alone is mildly pleasing, but where it becomes useful is in "}, {"start": 89.89, "duration": 4.45, "text": "seeing how the rules for exponentiation logs and radicals are all really the same."}, {"start": 94.8, "duration": 4.461, "text": "The most extreme example might be how with our current notation, there are six, "}, {"start": 99.261, "duration": 3.402, "text": "six distinct ways to express the various inverse operations, "}, {"start": 102.663, "duration": 1.897, "text": "and it looks like a complete mess."}, {"start": 104.98, "duration": 2.896, "text": "If you use the triangle of power, all of these "}, {"start": 107.876, "duration": 3.144, "text": "operations follow the same basic aesthetic pattern."}, {"start": 111.02, "duration": 4.02, "text": "Admittedly, it looks a little bit weird when your brain has already been trained with "}, {"start": 115.04, "duration": 3.88, "text": "the traditional notation, but our brains are really good at picking up on patterns."}, {"start": 119.4, "duration": 2.612, "text": "And in this case, once you get used to it a little, "}, {"start": 122.012, "duration": 3.768, "text": "you only need to remember one pattern which unlocks all six of these cases."}, {"start": 126.52, "duration": 3.755, "text": "Essentially, any rule that's associated with exponents logs and radicals "}, {"start": 130.275, "duration": 3.345, "text": "becomes essentially three times faster to learn and to recognize."}, {"start": 134.26, "duration": 2.923, "text": "If you have a pencil and paper, I highly encourage you to just take a "}, {"start": 137.183, "duration": 3.217, "text": "moment and mess around with what some of these rules for exponents look like."}, {"start": 140.4, "duration": 3.072, "text": "It's weird at first when you're trained with the traditional notation, "}, {"start": 143.472, "duration": 1.948, "text": "but it's super satisfying once it all clicks."}], "title": "The Triangle Of Power", "description": "The referenced stack exchange post by usename 2'5 9'2,  http://math.stackexchange.com/questions/30046/alternative-notation-for-exponents-logs-and-roots", "lengthSeconds": 146, "width": 360, "height": 640}
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{"video_id": "eJq0nzORwzU", "transcript": [{"start": 0.04, "duration": 3.32, "text": "take a look at this little grid of"}, {"start": 1.48, "duration": 3.839, "text": "values which involve some positive"}, {"start": 3.36, "duration": 3.68, "text": "numbers on the left and some negative"}, {"start": 5.319, "duration": 3.36, "text": "numbers on the right which I'll color"}, {"start": 7.04, "duration": 3.92, "text": "with blue and red"}, {"start": 8.679, "duration": 4.0, "text": "respectively all in all this little"}, {"start": 10.96, "duration": 3.919, "text": "process basically detects wherever"}, {"start": 12.679, "duration": 4.001, "text": "there's variation in the pixel value As"}, {"start": 14.879, "duration": 3.201, "text": "you move from left to right and so it"}, {"start": 16.68, "duration": 4.84, "text": "gives you a kind of way to pick up on"}, {"start": 18.08, "duration": 3.44, "text": "all the vertical edges from your"}, {"start": 21.76, "duration": 4.32, "text": "image and similarly if we rotated that"}, {"start": 24.359, "duration": 3.521, "text": "grid around so that it varies As you"}, {"start": 26.08, "duration": 3.48, "text": "move from the top to the bottom this"}, {"start": 27.88, "duration": 2.84, "text": "will be picking up on all the horizontal"}, {"start": 29.56, "duration": 2.879, "text": "edges"}, {"start": 30.72, "duration": 3.44, "text": "which in the case of our little pie"}, {"start": 32.439, "duration": 3.081, "text": "creature image does result in some"}, {"start": 34.16, "duration": 3.8, "text": "pretty demonic"}, {"start": 35.52, "duration": 4.4, "text": "eyes this smaller grid by the way is"}, {"start": 37.96, "duration": 3.56, "text": "often called a kernel and the beauty"}, {"start": 39.92, "duration": 3.08, "text": "here is how just by choosing a different"}, {"start": 41.52, "duration": 3.08, "text": "kernel you can get different image"}, {"start": 43.0, "duration": 3.32, "text": "processing effects not just blurring"}, {"start": 44.6, "duration": 3.52, "text": "your Edge detection but also things like"}, {"start": 46.32, "duration": 3.68, "text": "sharpening for those of you who have"}, {"start": 48.12, "duration": 3.72, "text": "heard of a convolutional neural network"}, {"start": 50.0, "duration": 3.64, "text": "the idea there is to use data to figure"}, {"start": 51.84, "duration": 3.8, "text": "out what the kernels should be in the"}, {"start": 53.64, "duration": 6.399, "text": "first place as determined by whatever"}, {"start": 55.64, "duration": 4.399, "text": "the neural network wants to detect"}], "title": "Deep learning math: Convolution #deeplearning #machinelerning #ai Credit:@3Blue1Brown", "description": "", "lengthSeconds": 58, "width": 360, "height": 640}
{"video_id": "teYL0Er4c3g", "transcript": [{"start": 0.08, "duration": 4.8, "text": "the international math Olympiad consists"}, {"start": 2.159, "duration": 4.68, "text": "of six problems broken up over 2 days"}, {"start": 4.88, "duration": 3.88, "text": "the three problems given on each day are"}, {"start": 6.839, "duration": 4.641, "text": "supposed to get progressively harder but"}, {"start": 8.76, "duration": 4.2, "text": "on the test from 2011 problem number two"}, {"start": 11.48, "duration": 3.6, "text": "was much more challenging than the"}, {"start": 12.96, "duration": 4.92, "text": "organizers intended for it to be here's"}, {"start": 15.08, "duration": 4.959, "text": "what it states let s be a finite set of"}, {"start": 17.88, "duration": 4.52, "text": "at least two points in the plane assume"}, {"start": 20.039, "duration": 4.601, "text": "that no three points of s are collinear"}, {"start": 22.4, "duration": 4.799, "text": "a windmill is a process that starts with"}, {"start": 24.64, "duration": 5.52, "text": "a line L going through a single point P"}, {"start": 27.199, "duration": 5.04, "text": "and S the line rotates clockwise about"}, {"start": 30.16, "duration": 3.64, "text": "the pivot P until the first time that"}, {"start": 32.239, "duration": 4.361, "text": "that line meets some other point"}, {"start": 33.8, "duration": 4.279, "text": "belonging to S this point Q takes over"}, {"start": 36.6, "duration": 3.76, "text": "as the new pivot and the line now"}, {"start": 38.079, "duration": 4.201, "text": "rotates clockwise around Q until the"}, {"start": 40.36, "duration": 4.48, "text": "next time it meets a point of s this"}, {"start": 42.28, "duration": 5.56, "text": "process continues indefinitely show that"}, {"start": 44.84, "duration": 5.16, "text": "you can choose a point p in s and a line"}, {"start": 47.84, "duration": 5.12, "text": "L going through P such that the"}, {"start": 50.0, "duration": 5.559, "text": "resulting windmill uses each point of s"}, {"start": 52.96, "duration": 4.36, "text": "as a pivot infinitely many times it's a"}, {"start": 55.559, "duration": 3.52, "text": "really beautiful problem and in the full"}, {"start": 57.32, "duration": 4.52, "text": "video we cover a very clever way that"}, {"start": 59.079, "duration": 2.761, "text": "you can solve it"}], "title": "A beautiful international math olympiad problem", "description": "The link to the full video is at the bottom of the screen. For reference, here it is: https://youtu.be/M64HUIJFTZM", "lengthSeconds": 60, "width": 360, "height": 640}
{"video_id": "rYUXDwUD3J4", "transcript": [{"start": 0.28, "duration": 3.48, "text": "inventing math is no joke and there is a"}, {"start": 2.399, "duration": 3.281, "text": "difference between being told why"}, {"start": 3.76, "duration": 3.96, "text": "something's true and actually generating"}, {"start": 5.68, "duration": 3.6, "text": "it from scratch but at all points I want"}, {"start": 7.72, "duration": 3.32, "text": "you to think to yourself if you were an"}, {"start": 9.28, "duration": 3.96, "text": "early mathematician pondering these"}, {"start": 11.04, "duration": 3.519, "text": "ideas and drawing out the right diagrams"}, {"start": 13.24, "duration": 2.959, "text": "does it feel reasonable that you could"}, {"start": 14.559, "duration": 4.281, "text": "have stumbled across these truths"}, {"start": 16.199, "duration": 4.08, "text": "yourself in this initial video I want to"}, {"start": 18.84, "duration": 3.0, "text": "show how you might stumble into the core"}, {"start": 20.279, "duration": 3.281, "text": "ideas of calculus by thinking very"}, {"start": 21.84, "duration": 4.519, "text": "deeply about one specific bit of"}, {"start": 23.56, "duration": 5.16, "text": "geometry the area of a circle maybe you"}, {"start": 26.359, "duration": 4.041, "text": "know that this is pi * radius squ but"}, {"start": 28.72, "duration": 4.08, "text": "why is there a nice way to think about"}, {"start": 30.4, "duration": 3.92, "text": "where this formula comes from well"}, {"start": 32.8, "duration": 2.84, "text": "contemplating this problem and leaving"}, {"start": 34.32, "duration": 2.96, "text": "yourself open to exploring the"}, {"start": 35.64, "duration": 3.12, "text": "interesting thoughts that come about can"}, {"start": 37.28, "duration": 3.68, "text": "actually lead you to a glimpse of three"}, {"start": 38.76, "duration": 5.56, "text": "big ideas in calculus integrals"}, {"start": 40.96, "duration": 3.36, "text": "derivatives and the fact that they're"}, {"start": 45.28, "duration": 3.0, "text": "opposites"}], "title": "Essence of Calculus by 3Blue1Brown #math #calculus #nerd #computerscience", "description": "", "lengthSeconds": 46, "width": 360, "height": 640}
{"video_id": "fwIKaZP8jMg", "transcript": [{"start": 0.24, "duration": 4.599, "text": "imagine you're at a restaurant you order"}, {"start": 2.639, "duration": 3.921, "text": "your favorite dish and in return you're"}, {"start": 4.839, "duration": 3.84, "text": "given a buzzer that will ring when your"}, {"start": 6.56, "duration": 4.119, "text": "order is ready this buzzer is like a"}, {"start": 8.679, "duration": 4.281, "text": "JavaScript promise it represents a"}, {"start": 10.679, "duration": 4.441, "text": "future value in this case your delicious"}, {"start": 12.96, "duration": 3.84, "text": "dish just like the buzzer can either"}, {"start": 15.12, "duration": 3.6, "text": "alert you that your meal is ready or"}, {"start": 16.8, "duration": 4.04, "text": "something went wrong maybe they're out"}, {"start": 18.72, "duration": 4.24, "text": "of your favorite dish a promise in"}, {"start": 20.84, "duration": 5.12, "text": "JavaScript can either be fulfilled or"}, {"start": 22.96, "duration": 4.68, "text": "rejected what's a promise A promise is"}, {"start": 25.96, "duration": 3.719, "text": "an object representing the eventual"}, {"start": 27.64, "duration": 4.599, "text": "completion or failure of an asynchronous"}, {"start": 29.679, "duration": 5.161, "text": "oper operation it's a way to handle"}, {"start": 32.239, "duration": 4.281, "text": "asynchronous operations in JavaScript"}, {"start": 34.84, "duration": 5.28, "text": "allowing you to write cleaner more"}, {"start": 36.52, "duration": 6.719, "text": "readable code a promise has three states"}, {"start": 40.12, "duration": 6.32, "text": "pending the initial State the operation"}, {"start": 43.239, "duration": 5.281, "text": "is still ongoing fulfilled the operation"}, {"start": 46.44, "duration": 5.52, "text": "completed successfully and the promise"}, {"start": 48.52, "duration": 5.76, "text": "now has a resolved value rejected the"}, {"start": 51.96, "duration": 4.8, "text": "operation failed and the promise has a"}, {"start": 54.28, "duration": 4.279, "text": "reason for the failure if you found this"}, {"start": 56.76, "duration": 5.119, "text": "content helpful don't forget to"}, {"start": 58.559, "duration": 3.32, "text": "subscribe to my channel"}], "title": "Promises in JavaScript #3blue1brown #javascript #learningcoding #animation #asynchronousjavascript", "description": "Dive into the world of JavaScript Promises through an engaging and relatable comparison to a restaurant buzzer system. This short video takes you on a journey from the suspense of waiting for your favorite dish, to understanding how Promises work in programming. Whether you're a coding enthusiast eager to tidy up your asynchronous operations, or just curious about the magic behind code execution, this analogy will illuminate the concept of JavaScript Promises in a fun and accessible way. Discover the states of a Promise, why they're a game-changer compared to traditional callback methods, and how to create your own Promise to manage asynchronous tasks more efficiently. \n\n#javascript  #programmingtips  #techtutorial  #coders  #async  #promises  #codingforbeginners  #webdevelopment  #viral #technology", "lengthSeconds": 60, "width": 360, "height": 640}
{"video_id": "tjIOqIr80ns", "transcript": [{"start": 0.08, "duration": 4.08, "text": "I'd like to show you a simple argument"}, {"start": 1.719, "duration": 4.241, "text": "for the fact that Pi is equal to 4 we"}, {"start": 4.16, "duration": 4.08, "text": "start off with a circle say with radius"}, {"start": 5.96, "duration": 4.28, "text": "1 and we ask how can we figure out its"}, {"start": 8.24, "duration": 3.439, "text": "circumference after all Pi is by"}, {"start": 10.24, "duration": 3.0, "text": "definition the ratio of this"}, {"start": 11.679, "duration": 3.601, "text": "circumference to the diameter of the"}, {"start": 13.24, "duration": 3.6, "text": "circle we start off by drawing the"}, {"start": 15.28, "duration": 3.88, "text": "square whose side lengths are all"}, {"start": 16.84, "duration": 4.04, "text": "tangent to that Circle it's not too hard"}, {"start": 19.16, "duration": 4.56, "text": "to see that the perimeter of this square"}, {"start": 20.88, "duration": 5.0, "text": "is eight then the argument proceeds by"}, {"start": 23.72, "duration": 4.319, "text": "producing a sequence of Curves all of"}, {"start": 25.88, "duration": 3.6, "text": "whom also have this perimeter of eight"}, {"start": 28.039, "duration": 3.601, "text": "but which more and more closely"}, {"start": 29.48, "duration": 4.16, "text": "approximate the circle and there are all"}, {"start": 31.64, "duration": 3.68, "text": "sorts of examples throughout calculus"}, {"start": 33.64, "duration": 3.759, "text": "when we talk about approximating one"}, {"start": 35.32, "duration": 4.04, "text": "thing we want to know as a limit of a"}, {"start": 37.399, "duration": 4.201, "text": "bunch of other things that are easier to"}, {"start": 39.36, "duration": 5.12, "text": "understand so the question at the heart"}, {"start": 41.6, "duration": 5.439, "text": "here is why exactly is it not okay to do"}, {"start": 44.48, "duration": 2.559, "text": "that in this"}, {"start": 49.239, "duration": 3.0, "text": "example"}], "title": "The limit of limiting arguments", "description": "A link to the full video is at the bottom of the screen.\nOr, for reference: https://youtu.be/VYQVlVoWoPY\n\nThat video gives multiple examples of lying with visual proofs\n\nEditing from the original video into this short by Dawid Ko\u0142odziej", "lengthSeconds": 50, "width": 360, "height": 640}
{"video_id": "EPDZTLavmcg", "transcript": [{"start": 0.0, "duration": 3.813, "text": "The surface area of a sphere is 4\u03c0 times its radius squared, "}, {"start": 3.813, "duration": 3.627, "text": "exactly 4 times the area of a circle with the same radius."}, {"start": 8.0, "duration": 0.42, "text": "But why?"}, {"start": 8.92, "duration": 4.35, "text": "Archimedes had a very beautiful proof that this surface area is the same as the area of "}, {"start": 13.27, "duration": 4.45, "text": "a cylinder that encloses that sphere, if you disregard the circular caps of that cylinder."}, {"start": 18.0, "duration": 4.711, "text": "The idea is that if you shine some light from the z-axis perpendicular to that axis, "}, {"start": 22.711, "duration": 4.6, "text": "then if you compare the area of a small rectangle drawn on that sphere to the area "}, {"start": 27.311, "duration": 4.269, "text": "of the shadow cast on that cylinder, those two areas turn out to be the same."}, {"start": 32.259, "duration": 4.13, "text": "This isn't obvious, but when you work it out, that shadow is a copy of the original "}, {"start": 36.389, "duration": 4.031, "text": "little rectangle, but squished down in one direction and stretched out in another."}, {"start": 41.12, "duration": 3.9, "text": "When you analyze the relevant geometry, the two effects actually cancel out perfectly."}, {"start": 45.76, "duration": 2.947, "text": "Treating the sphere as a sum of all of these little patches, "}, {"start": 48.707, "duration": 2.464, "text": "and the cylinder as a sum of all of those shadows, "}, {"start": 51.171, "duration": 4.349, "text": "we can infer that the surface area of the sphere is the same as the area of that cylinder."}, {"start": 56.0, "duration": 2.952, "text": "Now the cylinder is something you can unwrap into a rectangle, "}, {"start": 58.952, "duration": 2.999, "text": "where one side length corresponds to the circumference, 2 pi r, "}, {"start": 61.951, "duration": 3.469, "text": "and the other corresponds to the height of the sphere, 2 times its radius."}, {"start": 65.8, "duration": 2.58, "text": "This gives us the area we want, 4 pi r squared."}, {"start": 68.74, "duration": 3.627, "text": "And if you're also familiar with the trick of showing a circle's area, "}, {"start": 72.367, "duration": 4.496, "text": "by unwrapping it into a triangle, one whose height is r and whose base is 2 pi times r, "}, {"start": 76.863, "duration": 4.497, "text": "you can see how four of those unwrapped circles fit perfectly into this unwrapped shape."}], "title": "Sphere surface area proof sketch", "description": "Full video: https://youtu.be/GNcFjFmqEc8", "lengthSeconds": 85, "width": 360, "height": 640}
{"video_id": "IeiJ3mupoBc", "transcript": [{"start": 0.659, "duration": 7.501, "text": "\u0939\u0948 \u0906\u0908 \u090f\u092e \u0930\u0941\u0926\u094d\u0930\u093e\u0902\u0936 \u0907\u0928 \u091f\u0941\u0921\u0947\u0938 \u0935\u0940\u0921\u093f\u092f\u094b \u0935\u0947 \u092c\u093f\u0932"}, {"start": 5.22, "duration": 4.4, "text": "\u0932\u0941\u0915 \u090f\u091f \u090f\u0928 \u0917\u094d\u0930\u0947\u091f \u0905\u092a\u0930\u094b\u0915\u094d\u0937 \u0914\u0930 \u092e\u0920 \u0911\u092b"}, {"start": 8.16, "duration": 6.0, "text": "\u0925\u0930\u094d\u092e\u094b\u0921\u093e\u092f\u0928\u0947\u092e\u093f\u0915\u094d\u0938"}, {"start": 9.62, "duration": 4.54, "text": "\u0932\u0947\u091f \u0938\u094d\u091f\u093e\u0930\u094d\u091f \u0915\u0940 \u090f \u092b\u0948\u0915\u094d\u091f\u094b\u0930\u093f\u092f\u0932"}, {"start": 19.58, "duration": 13.84, "text": "[\u0938\u0902\u0917\u0940\u0924]"}, {"start": 31.759, "duration": 4.081, "text": "\u0925\u0930\u094d\u092e\u094b\u0921\u093e\u092f\u0928\u0947\u092e\u093f\u0915\u094d\u0938 \u0914\u0930 \u0938\u094d\u091f\u0948\u091f\u093f\u0938\u091f\u093f\u0915\u094d\u0938"}, {"start": 33.42, "duration": 2.42, "text": "\u092e\u0948\u0915\u0947\u0928\u093f\u0915\u094d\u0938"}, {"start": 36.059, "duration": 8.241, "text": "\u092c\u091f \u0925\u0947 \u092b\u0948\u0915\u094d\u091f\u094b\u0930\u093f\u092f\u0932 \u0915\u093e 70 \u0907\u0938 \u0907\u0928 \u0921\u0940 \u0911\u0930\u094d\u0921\u0930 \u0911\u092b"}, {"start": 39.3, "duration": 5.0, "text": "\u0917\u0942\u0917\u0932 \u0906\u0908\u091f\u0940 \u0907\u0938\u0947 \u0935\u0947\u0930\u0940 \u0921\u093f\u092b\u093f\u0915\u0932\u094d\u091f \u0924\u0942 \u0915\u0902\u092a\u094d\u092f\u0942\u091f\u0930"}, {"start": 51.71, "duration": 7.019, "text": "[\u0938\u0902\u0917\u0940\u0924]"}, {"start": 64.61, "duration": 8.829, "text": "[\u0938\u0902\u0917\u0940\u0924]"}, {"start": 75.86, "duration": 9.34, "text": 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\u0905\u092a\u0930\u094b\u0915\u094d\u0937 \u0907\u0938"}, {"start": 184.14, "duration": 5.28, "text": "\u0917\u0947\u091f\u093f\u0902\u0917 \u092e\u094b\u0930 \u090f\u0915\u094d\u092f\u0942\u0930\u0947\u091f"}, {"start": 186.54, "duration": 5.52, "text": "\u0906\u0908\u090f\u092b \u092f\u0942 \u0906\u0930 \u0907\u0902\u091f\u0930\u0947\u0938\u094d\u091f \u0930\u0947\u091f \u0907\u0928 \u0926\u093f\u0938 \u091f\u0949\u092a\u093f\u0915 \u092f\u0942"}, {"start": 189.42, "duration": 4.8, "text": "\u0915\u0948\u0928 \u0905\u0932\u0938\u094b \u0935\u0949\u091a \u0938\u092e \u0935\u0940\u0921\u093f\u092f\u094b \u0921\u091f\u0947\u0938\u094d\u091f\u0947\u0921 \u0914\u0930"}, {"start": 192.06, "duration": 5.12, "text": "\u0921\u093f\u0938\u094d\u0915\u094d\u0930\u093f\u092a\u094d\u0936\u0928 \u0914\u0930 \u0930\u0940\u0921 \u090f\u0928 \u0906\u0930\u094d\u091f\u093f\u0915\u0932 \u092b\u094d\u0930\u0949\u092e"}, {"start": 194.22, "duration": 2.96, "text": "\u0935\u0941\u0932\u094d\u092b \u092b\u094d\u0930\u0949\u092e \u092e\u0920 \u0935\u0930\u094d\u0932\u094d\u0921"}], "title": "Factorials and Physics : Revving up Thermodynamics with Stirling's Approximation @3blue1brown", "description": "In this video , you will learn about Stirling's relation and math of thermodynamics. suggest me some more topics and improvement in future.\nMusic by Vince Rubinetti: \nDr. Trefor Bazett\n Flammable math \nMichel van Biezen \nblackpenredpen \n\nWolfram Mathworld \n@DrTrefor\n @papaflammy \n@MichelvanBiezen\n @blackpenredpen\n @3blue1brown \n\nmath\nphysics\nmath and physics\nthermodynamics\nstatistical mechanics\nvideo\npermutation\ncombination\ncsir net\nprmo\nmaths\nfactorials\n\nCopyright Disclaimer Under Section 107 of the Copyright Act 1976, allowance is made for \"fair use\" for purposes such as criticism, commenting, news reporting, teaching, scholarship, and research. Fair use is a use permitted by copyright statute that might otherwise be infringing. Non-profit, educational or personal use tips the balance in favor of fair use.", "lengthSeconds": 196, "width": 638, "height": 360}
{"video_id": "AyzBLbayARo", "transcript": [{"start": 0.24, "duration": 4.96, "text": "\u062f\u0631 \u0645\u0631\u0628\u0639 \u0628 \u0632\u0644 \u06f4 \u062f\u0648 \u06a9\u0645\u0627\u0646 \u06cc\u06a9\u06cc \u0628\u0647 \u0645\u0631\u06a9\u0632 \u0631\u0633\u06cc"}, {"start": 2.76, "duration": 5.24, "text": "\u0627\u0632 \u0645\u0631\u0628\u0639 \u0648 \u062f\u06cc\u06af\u0631\u06cc \u0628\u0647 \u0645\u0631\u06a9\u0632 \u0648\u0633\u0637 \u06cc\u06a9 \u0632\u0644 \u0645\u0631\u0628\u0639"}, {"start": 5.2, "duration": 5.6, "text": "\u0631\u0633 \u0634\u062f\u0647\u200c\u0627\u0646\u062f \u0633\u0637\u062d \u0645\u062d\u0635\u0648\u0644 \u0627\u0633 \u06a9\u062f\u0627\u0645 \u0627\u0633\u062a \u0627\u0632"}, {"start": 8.0, "duration": 5.24, "text": "\u0645\u0631\u06a9\u0632\u0647\u0627\u06cc \u0646\u0642\u0637\u0647 \u0628\u0631\u062e\u0648\u0631\u062f \u062f\u0648 \u06a9\u0645\u0627\u0646 \u067e\u0627\u0631 \u062e\u0637\u200c\u0647\u0627\u06cc\u06cc"}, {"start": 10.8, "duration": 4.56, "text": "\u0631\u0633\u0645 \u0645\u06cc\u200c\u06a9\u0646\u06cc\u0645 \u0627\u06cc\u0646 \u067e\u0627\u0631 \u062e\u0637\u200c\u0647\u0627 \u0634\u0639\u0627\u0639\u200c\u0647\u0627\u06cc \u0645\u0631\u0628\u0648\u0637"}, {"start": 13.24, "duration": 4.84, "text": "\u0628\u0647 \u06a9\u0645\u0627\u0646\u0647\u0627\u06cc \u0631\u0633\u0645 \u0634\u062f\u0647\u200c\u0627\u0646\u062f \u067e\u0633 \u0637\u0648\u0644 \u06cc\u06a9\u06cc \u0627\u0632"}, {"start": 15.36, "duration": 4.679, "text": "\u0622\u0646\u0647\u0627 \u0686\u0627\u0631 \u0648 \u0637\u0648\u0644 \u062f\u06cc\u06af\u0631\u06cc \u0628\u0631\u0627\u0628\u0631 \u062f\u0648 \u0627\u0633\u062a \u062d\u0627\u0644\u0627"}, {"start": 18.08, "duration": 4.52, "text": "\u0627\u06af\u0631 \u062f\u0648 \u0645\u0631\u06a9\u0632 \u0631\u0627 \u0628\u0647 \u0647\u0645 \u0648\u0635\u0644 \u06a9\u0646\u06cc\u0645 \u062f\u0648 \u0645\u062b\u0644\u062b"}, {"start": 20.039, "duration": 5.681, "text": "\u0642\u0627\u0626\u0645 \u0648 \u0632\u0627\u0648\u06cc \u0647\u0645 \u0646\u0647\u0634\u062a \u062d\u0627\u0635\u0644 \u0645\u06cc\u200c\u0634\u0648\u062f \u062f\u0631 \u0634\u06a9\u0644"}, {"start": 22.6, "duration": 6.439, "text": "\u0631\u0648\u0628\u0647\u0631\u0648 \u0646\u0627\u062d\u06cc\u0647 s \u0641\u0635\u0644 \u0645\u0634\u062a\u0631\u06a9 \u062f\u0648 \u0642\u0637\u0627\u0639 \u0631\u0633 \u0634\u062f\u0647"}, {"start": 25.72, "duration": 6.48, "text": "\u0645\u06cc\u200c\u0628\u0627\u0634\u062f \u067e\u0633 \u0645\u0633\u0627\u062d\u062a \u0645\u062b\u0644\u062b\u200c\u0647\u0627 \u0628\u0627 \u0645\u062c\u0645\u0648\u0639 \u0645\u0633\u0627\u062d\u062a"}, {"start": 29.039, "duration": 5.641, "text": "\u0647\u0627\u06cc \u062f\u0648 \u0642\u0637\u0627\u0639 \u0645\u0646\u0647\u0627\u06cc \u0645\u0633\u0627\u062d\u062a \u0646\u0627\u062d\u06cc\u0647 \u0645\u062d\u0635\u0648\u0644 s"}, {"start": 32.2, "duration": 5.64, "text": "\u0628\u0631\u0627\u0628\u0631 \u0627\u0633\u062a \u0627\u06af\u0631 \u0632\u0627\u0648\u06cc\u0647 \u06a9\u0648\u0686\u06cc\u06a9\u200c\u062a\u0631 \u0645\u062b\u0644\u062b \u0631\u0627"}, {"start": 34.68, "duration": 6.199, "text": "\u0622\u0644\u0641\u0627 \u062f\u0631 \u0646\u0638\u0631 \u0628\u06af\u06cc\u0631\u06cc\u0645 \u0627\u06cc\u0646 \u0632\u0627\u0648\u06cc\u0647 \u06f2 \u0622\u0644\u0641\u0627 \u0648"}, {"start": 37.84, "duration": 6.039, "text": "\u0627\u06cc\u0646 \u0632\u0627\u0648\u06cc\u0647 \u067e\u06cc \u0645\u0646\u0647\u0627\u06cc \u062f\u0648 \u0622\u0644\u0641\u0627 \u0627\u0633\u062a \u0645\u0633\u0627\u062d\u062a"}, {"start": 40.879, "duration": 6.081, "text": "\u0645\u062b\u0644\u062b\u200c\u0647\u0627 \u0648 \u0642\u0637\u0639\u0647\u0627 \u0628\u0627 \u062a\u0648\u062c\u0647 \u0645\u0639\u0644\u0648\u0645 \u0628\u0648\u062f\u0646 \u0627\u062c\u0632\u0627\u06cc"}, {"start": 43.879, "duration": 3.081, "text": "\u0622\u0646\u200c\u0647\u0627 \u0642\u0627\u0628\u0644 \u0645\u062d\u0627\u0633\u0628\u0647"}, {"start": 48.8, "duration": 6.68, "text": "\u0627\u0646\u062f \u0628\u0627 \u062a\u0648\u062c\u0647 \u0628\u0647 \u0627\u06cc\u0646\u06a9\u0647 \u0622\u0644\u0641\u0627 \u0647\u0645\u0627\u0646 \u062a\u0627\u0646\u0698\u0627\u0646\u062a"}, {"start": 51.96, "duration": 7.96, "text": "\u0627\u06cc\u0646\u0648\u0631\u0633 \u06f1 \u062f\u0648\u0645 \u0627\u0633\u062a \u0627\u0633 \u0628\u0647 \u062f\u0633\u062a \u0622\u0645\u062f\u0647 \u0648 \u0628\u0647 \u0637\u0648\u0631"}, {"start": 55.48, "duration": 4.44, "text": "\u062a\u0642\u0631\u06cc\u0628\u06cc \u0628\u0631\u0627\u0628\u0631 \u0627\u06cc\u0646 \u0639\u062f\u062f \u062e\u0648\u0627\u0647\u062f \u0634\u062f"}], "title": "AreaBetweenTwoArcs #maths #rectangle #3blue1brown#olympiad", "description": "", "lengthSeconds": 58, "width": 360, "height": 640}
{"video_id": "VS9yOhwtVDY", "transcript": [{"start": 0.32, "duration": 3.72, "text": "sometimes it feels like the universe is"}, {"start": 1.92, "duration": 4.479, "text": "just messing with you I have up on"}, {"start": 4.04, "duration": 3.839, "text": "screen here a sequence of computations"}, {"start": 6.399, "duration": 3.721, "text": "and what I want you to notice is how the"}, {"start": 7.879, "duration": 4.601, "text": "sequence follows a very predictable if"}, {"start": 10.12, "duration": 4.84, "text": "random seeming pattern and how each"}, {"start": 12.48, "duration": 3.879, "text": "computation happens to equal Pi you"}, {"start": 14.96, "duration": 4.12, "text": "might think that this was a pattern that"}, {"start": 16.359, "duration": 4.561, "text": "would go on forever but it doesn't at"}, {"start": 19.08, "duration": 4.119, "text": "some point it stops and instead of"}, {"start": 20.92, "duration": 6.04, "text": "equaling Pi you get a value which is"}, {"start": 23.199, "duration": 5.961, "text": "just barely barely less than"}, {"start": 26.96, "duration": 4.159, "text": "pi and before you go thinking this is"}, {"start": 29.16, "duration": 2.88, "text": "the result of some numerical error maybe"}, {"start": 31.119, "duration": 2.801, "text": "because we're doing something with"}, {"start": 32.04, "duration": 3.88, "text": "floating Point arithmetic if you work"}, {"start": 33.92, "duration": 4.36, "text": "this out more precisely here is the"}, {"start": 35.92, "duration": 4.6, "text": "exact value of that last integral which"}, {"start": 38.28, "duration": 4.439, "text": "is a certain fraction of Pi where the"}, {"start": 40.52, "duration": 6.64, "text": "numerator and the denominator are absurd"}, {"start": 42.719, "duration": 4.441, "text": "they're both around 400 billion billion"}, {"start": 49.6, "duration": 3.0, "text": "billion"}], "title": "These integrals all equal \u03c0, until...", "description": "A link to the full video is at the bottom of the screen.\nOr, for reference: https://youtu.be/851U557j6HE\nThese are known as Borwein integrals\n\nEditing from long-form to short by Dawid Ko\u0142odziej", "lengthSeconds": 51, "width": 360, "height": 640}
{"video_id": "xbR3CiXgniw", "transcript": [{"start": 0.199, "duration": 5.241, "text": "first we need to figure out where ihat"}, {"start": 2.48, "duration": 5.56, "text": "goes after applying M1 the new"}, {"start": 5.44, "duration": 5.44, "text": "coordinates of ihat by definition are"}, {"start": 8.04, "duration": 6.0, "text": "given by that First Column of M1 namely"}, {"start": 10.88, "duration": 6.92, "text": "1 one to see what happens after applying"}, {"start": 14.04, "duration": 5.92, "text": "M2 multiply The Matrix for M2 by that"}, {"start": 17.8, "duration": 4.44, "text": "Vector 1"}, {"start": 19.96, "duration": 4.2, "text": "one working it out the way that I"}, {"start": 22.24, "duration": 2.959, "text": "described last video you'll get the"}, {"start": 24.16, "duration": 3.439, "text": "vector"}, {"start": 25.199, "duration": 3.761, "text": "21 this will be the first column of the"}, {"start": 27.599, "duration": 4.201, "text": "composition"}, {"start": 28.96, "duration": 5.08, "text": "Matrix likewise to follow J hat the"}, {"start": 31.8, "duration": 5.32, "text": "second column of M1 tells us that it"}, {"start": 34.04, "duration": 3.08, "text": "first lands on"}, {"start": 37.44, "duration": 7.24, "text": "-20 then when we apply M2 to that"}, {"start": 42.44, "duration": 6.16, "text": "Vector you can work out the Matrix"}, {"start": 44.68, "duration": 5.48, "text": "Vector product to get 0 -2 which becomes"}, {"start": 48.6, "duration": 4.56, "text": "the second column of our composition"}, {"start": 50.16, "duration": 3.0, "text": "Matrix"}], "title": "Linear Algebra for #Machinelearning #mathematicsformachinelearning Credit:@3Blue1Brown", "description": "", "lengthSeconds": 55, "width": 360, "height": 640}
{"video_id": "llgB0J98cuU", "transcript": [{"start": 0.12, "duration": 4.5, "text": "given a convex polygon how do you find"}, {"start": 2.639, "duration": 5.46, "text": "the area of it"}, {"start": 4.62, "duration": 5.519, "text": "let be zero be some fixed point we can"}, {"start": 8.099, "duration": 5.281, "text": "now divide the polygon into triangles"}, {"start": 10.139, "duration": 5.101, "text": "the area of each triangle can be found"}, {"start": 13.38, "duration": 3.84, "text": "by the length of the cross product of"}, {"start": 15.24, "duration": 4.26, "text": "these two vectors"}, {"start": 17.22, "duration": 4.74, "text": "you just sum up the area for all of the"}, {"start": 19.5, "duration": 4.38, "text": "triangles this algorithm Works in all"}, {"start": 21.96, "duration": 4.22, "text": "fan where n is the number of points of"}, {"start": 23.88, "duration": 2.3, "text": "the poly"}], "title": "Area of a Convex Polygon #3blue1brown #animations", "description": "", "lengthSeconds": 24, "width": 360, "height": 640}
{"video_id": "0Gu8KdoMIDs", "transcript": [{"start": 0.78, "duration": 3.96, "text": "good day everyone you're welcome to my"}, {"start": 2.58, "duration": 4.799, "text": "YouTube channel today we are giving that"}, {"start": 4.74, "duration": 4.98, "text": "X Plus Y is equal to 4 and X Y is equal"}, {"start": 7.379, "duration": 5.041, "text": "to one all over two then we're going to"}, {"start": 9.72, "duration": 5.4, "text": "evaluate X raised power 3 plus y raised"}, {"start": 12.42, "duration": 5.64, "text": "to power 3. so without finding the value"}, {"start": 15.12, "duration": 5.64, "text": "of X and Y so how do we do this we're"}, {"start": 18.06, "duration": 4.08, "text": "going to use the identity that a raised"}, {"start": 20.76, "duration": 5.099, "text": "to square 3"}, {"start": 22.14, "duration": 8.18, "text": "plus b square 3 is equal to"}, {"start": 25.859, "duration": 9.001, "text": "a plus b raised to square 3 minus 3 a b"}, {"start": 30.32, "duration": 6.04, "text": "Open brackets a plus b so now using this"}, {"start": 34.86, "duration": 3.48, "text": "identity here"}, {"start": 36.36, "duration": 6.3, "text": "we have"}, {"start": 38.34, "duration": 5.399, "text": "X respiratory plus y raise power 3 equal"}, {"start": 42.66, "duration": 7.86, "text": "to"}, {"start": 43.739, "duration": 8.66, "text": "X Plus y raised by 3 minus 3 times x y"}, {"start": 50.52, "duration": 6.48, "text": "then times"}, {"start": 52.399, "duration": 7.361, "text": "X Plus y so this is equal to X Plus y"}, {"start": 57.0, "duration": 7.32, "text": "from the given equation is four so for"}, {"start": 59.76, "duration": 9.6, "text": "raise power 3 minus 3 times x y is one"}, {"start": 64.32, "duration": 9.0, "text": "all over 2 times X Plus Y is what four"}, {"start": 69.36, "duration": 5.64, "text": "so here two one you're here two so we"}, {"start": 73.32, "duration": 7.5, "text": "have this to be"}, {"start": 75.0, "duration": 10.32, "text": "4X battery is 64 minus 3 times 2 and"}, {"start": 80.82, "duration": 5.9, "text": "this is equal to 64 minus 6 and this is"}, {"start": 85.32, "duration": 4.5, "text": "giving us what"}, {"start": 86.72, "duration": 5.259, "text": "58 and this is the right answer to the"}, {"start": 89.82, "duration": 3.659, "text": "question thank you for viewing and make"}, {"start": 91.979, "duration": 4.941, "text": "sure you subscribe to my YouTube channel"}, {"start": 93.479, "duration": 3.441, "text": "for more interesting videos"}], "title": "Algebra.", "description": "", "lengthSeconds": 95, "width": 636, "height": 360}
{"video_id": "yT4dxToCJpo", "transcript": [{"start": 0.32, "duration": 4.2, "text": "hello good viers welcome back once again"}, {"start": 2.28, "duration": 4.479, "text": "we're going to solve for the value of x"}, {"start": 4.52, "duration": 6.999, "text": "in the following nice looking"}, {"start": 6.759, "duration": 8.08, "text": "exponential equation 9^ x + 15^ x this"}, {"start": 11.519, "duration": 6.081, "text": "is equal to 25 to ^"}, {"start": 14.839, "duration": 6.161, "text": "x now let's go ahead and divide both"}, {"start": 17.6, "duration": 5.64, "text": "sides each St from both sides by 9 to^"}, {"start": 21.0, "duration": 2.24, "text": "of"}, {"start": 26.92, "duration": 9.159, "text": "X and again we record this property a a^"}, {"start": 30.64, "duration": 10.04, "text": "m/ B of M this is same as a / B all the^"}, {"start": 36.079, "duration": 9.521, "text": "of M but for here we have just one plus"}, {"start": 40.68, "duration": 10.399, "text": "here we have 15 / 9^ X this is will be"}, {"start": 45.6, "duration": 8.16, "text": "equal to 25 ra the power sorry 25 9 then"}, {"start": 51.079, "duration": 2.681, "text": "to the power"}, {"start": 54.0, "duration": 6.28, "text": "x so from here we have 1 plus if you"}, {"start": 58.039, "duration": 7.12, "text": "divide if you reduce this to its slope"}, {"start": 60.28, "duration": 10.0, "text": "term we just have 5 over 3 to the^ X"}, {"start": 65.159, "duration": 10.32, "text": "this is equal to 25 / 9 is same as 5 /"}, {"start": 70.28, "duration": 8.4, "text": "3 2 to the^ of X let us recall that M"}, {"start": 75.479, "duration": 8.921, "text": "power of n z we can interchange the"}, {"start": 78.68, "duration": 12.24, "text": "powers as M of Z of n so here we have 1"}, {"start": 84.4, "duration": 11.679, "text": "+ 5 / 3 to the^ X this is equal to"}, {"start": 90.92, "duration": 9.92, "text": "5/ 3 to^ X2 so make a substition we let"}, {"start": 96.079, "duration": 8.481, "text": "5/ 3 to the power of X = Y from the"}, {"start": 100.84, "duration": 9.919, "text": "right hand side you have y^ 2 minus here"}, {"start": 104.56, "duration": 10.519, "text": "y - 1 this is equal to zero now for real"}, {"start": 110.759, "duration": 6.801, "text": "value of x here y needs to be greater"}, {"start": 115.079, "duration": 4.881, "text": "than zero right so which actually mean"}, {"start": 117.56, "duration": 5.239, "text": "that we're not interested in the"}, {"start": 119.96, "duration": 7.479, "text": "negative value of y here but the only"}, {"start": 122.799, "duration": 7.401, "text": "Val positive value of y is 1 +un 5 all/"}, {"start": 127.439, "duration": 5.921, "text": "2 using the general quadratic formula"}, {"start": 130.2, "duration": 5.88, "text": "right if Y is negative here x will have"}, {"start": 133.36, "duration": 5.28, "text": "a complex solution if Y is equal to 0"}, {"start": 136.08, "duration": 5.64, "text": "then there will be no solution but when"}, {"start": 138.64, "duration": 8.319, "text": "Y is greater than Z we will have a r"}, {"start": 141.72, "duration": 8.32, "text": "solution so now from there 5 3^ x is = y"}, {"start": 146.959, "duration": 6.56, "text": "y is 1 + root"}, {"start": 150.04, "duration": 10.479, "text": "5/ 2 let us recall that the"}, {"start": 153.519, "duration": 12.08, "text": "logm of m to the base of M isal to 1"}, {"start": 160.519, "duration": 7.921, "text": "right and again the logarithm of M of B"}, {"start": 165.599, "duration": 6.601, "text": "to the power sorry to the base m is"}, {"start": 168.44, "duration": 6.6, "text": "equal to B so from here we can log both"}, {"start": 172.2, "duration": 5.28, "text": "sides but the logi must be to the base"}, {"start": 175.04, "duration": 6.76, "text": "5/ 3 so here we have the"}, {"start": 177.48, "duration": 7.64, "text": "logarithm of 5/ 3 to the power of x to"}, {"start": 181.8, "duration": 8.719, "text": "the base of 5 / 3 this is equal to the"}, {"start": 185.12, "duration": 9.72, "text": "logarithm of 1 +un 5 / 2 to the base of"}, {"start": 190.519, "duration": 8.761, "text": "5 / 3 so using this property we have"}, {"start": 194.84, "duration": 9.64, "text": "your X here this is equal to the"}, {"start": 199.28, "duration": 8.16, "text": "logarithm of 1 +un 5 / 2 to the base of"}, {"start": 204.48, "duration": 7.28, "text": "5 / 3 and this is the solution which is"}, {"start": 207.44, "duration": 4.32, "text": "approximately equal to"}, {"start": 213.08, "duration": 5.799, "text": "0.94 2"}, {"start": 216.4, "duration": 4.08, "text": "027"}, {"start": 218.879, "duration": 4.64, "text": "57"}, {"start": 220.48, "duration": 4.8, "text": "785 and this is the solution thank you"}, {"start": 223.519, "duration": 4.121, "text": "for watching please kindly subscribe to"}, {"start": 225.28, "duration": 5.84, "text": "my YouTube channel also like comment and"}, {"start": 227.64, "duration": 3.48, "text": "share bye-bye"}], "title": "Olympiad Math | Interesting exponential equation | #maths #olympiad", "description": "", "lengthSeconds": 229, "width": 640, "height": 360}
{"video_id": "PKG1BElbiZw", "transcript": [{"start": 0.76, "duration": 4.92, "text": "let's solve the"}, {"start": 2.159, "duration": 7.041, "text": "following radical equation we have theun"}, {"start": 5.68, "duration": 5.079, "text": "of X+ the otk of X this is equal to X so"}, {"start": 9.2, "duration": 4.76, "text": "here we're going to be solving for the"}, {"start": 10.759, "duration": 4.641, "text": "values of X so let's get started so from"}, {"start": 13.96, "duration": 6.52, "text": "the left hand side we have theare root"}, {"start": 15.4, "duration": 9.199, "text": "of x plus the < TK of X this is equal to"}, {"start": 20.48, "duration": 6.28, "text": "X so we know the otk of X+ the root of x"}, {"start": 24.599, "duration": 6.561, "text": "from"}, {"start": 26.76, "duration": 7.479, "text": "here this is equal to 2 * the squ of X"}, {"start": 31.16, "duration": 5.12, "text": "then this is equal to X now the next"}, {"start": 34.239, "duration": 5.561, "text": "thing to do let's Square both sides of"}, {"start": 36.28, "duration": 8.16, "text": "the equation right so we have 2un"}, {"start": 39.8, "duration": 6.8, "text": "x^ 2 this will be equal to x^ 2 now"}, {"start": 44.44, "duration": 9.799, "text": "let's use this identity we know that a *"}, {"start": 46.6, "duration": 13.08, "text": "m the^ b is = a the^ of B * m to power"}, {"start": 54.239, "duration": 11.561, "text": "B so from here this will be 2^ 2 * < TK"}, {"start": 59.68, "duration": 6.12, "text": "of x x to^ 2 then this is equal to x^"}, {"start": 66.28, "duration": 4.479, "text": "2 now we can change the sides of this"}, {"start": 69.24, "duration": 4.84, "text": "equation so let's make the right hand"}, {"start": 70.759, "duration": 7.081, "text": "side the left hand side so we get x2 is"}, {"start": 74.08, "duration": 5.64, "text": "equal to here 2 2 is 4 multip by square"}, {"start": 77.84, "duration": 6.12, "text": "we cancel square root here and we have"}, {"start": 79.72, "duration": 6.96, "text": "just X now there's a mistake students"}, {"start": 83.96, "duration": 5.519, "text": "always make when they arrive at this"}, {"start": 86.68, "duration": 6.36, "text": "point most students always divide both"}, {"start": 89.479, "duration": 6.881, "text": "sides by X thereby getting X is just"}, {"start": 93.04, "duration": 6.079, "text": "equal to 4 right but if you do so you"}, {"start": 96.36, "duration": 4.64, "text": "will miss a solution so the best way to"}, {"start": 99.119, "duration": 4.64, "text": "solve this problem from here is to"}, {"start": 101.0, "duration": 6.64, "text": "subtract 4X from both"}, {"start": 103.759, "duration": 7.32, "text": "sides so subtract 4X from both sides so"}, {"start": 107.64, "duration": 6.759, "text": "from the left hand side we have x^2 - 4x"}, {"start": 111.079, "duration": 5.0, "text": "this is equal to 4 x - 4x is 0 so we can"}, {"start": 114.399, "duration": 3.64, "text": "see that from the left hand side now we"}, {"start": 116.079, "duration": 4.881, "text": "have something common which is X so"}, {"start": 118.039, "duration": 5.561, "text": "let's Factor it out so we have x - 4"}, {"start": 120.96, "duration": 5.68, "text": "left then this is equal to Z so this"}, {"start": 123.6, "duration": 8.12, "text": "implies from here that X is equal to 0"}, {"start": 126.64, "duration": 8.52, "text": "or x - 4 is = to0 so from here we have X"}, {"start": 131.72, "duration": 6.2, "text": "is = 4 so you can see that we have this"}, {"start": 135.16, "duration": 5.64, "text": "solution of x is equal to0 which also"}, {"start": 137.92, "duration": 4.76, "text": "satisfy the equation thank you for"}, {"start": 140.8, "duration": 4.76, "text": "watching please kindly subscribe to my"}, {"start": 142.68, "duration": 5.88, "text": "channel also like comment and share"}, {"start": 145.56, "duration": 3.0, "text": "bye-bye"}], "title": "Interesting radical equation | Can you solve ?", "description": "", "lengthSeconds": 147, "width": 428, "height": 240}
{"video_id": "1Chsc1_AIjE", "transcript": [{"start": 0.96, "duration": 4.0, "text": "hello welcome back once again today"}, {"start": 3.0, "duration": 6.84, "text": "we're going to solve this interesting"}, {"start": 4.96, "duration": 7.639, "text": "exponential equation with cube root of x"}, {"start": 9.84, "duration": 8.599, "text": "as power we have 2 to the^ cube root of"}, {"start": 12.599, "duration": 9.241, "text": "x this equals 2 ^ of x / 2 - 12 and here"}, {"start": 18.439, "duration": 5.561, "text": "we're going to solve for x values right"}, {"start": 21.84, "duration": 5.08, "text": "so let us start with a substitution"}, {"start": 24.0, "duration": 7.0, "text": "first so we're going to"}, {"start": 26.92, "duration": 5.159, "text": "let the cube root of x"}, {"start": 31.0, "duration": 7.719, "text": "="}, {"start": 32.079, "duration": 9.121, "text": "M and x / 2 = N"}, {"start": 38.719, "duration": 5.601, "text": "right so from the equation we're going"}, {"start": 41.2, "duration": 7.839, "text": "to have 2 to the power of M will be"}, {"start": 44.32, "duration": 8.239, "text": "equal to 2 ^ of N -"}, {"start": 49.039, "duration": 10.281, "text": "12 so from here we're going to write"}, {"start": 52.559, "duration": 8.281, "text": "this as 2 ^ nus 2 the ^ M this is equal"}, {"start": 59.32, "duration": 5.64, "text": "to"}, {"start": 60.84, "duration": 7.56, "text": "12 so from here this 12 right over here"}, {"start": 64.96, "duration": 6.68, "text": "we're going to write it as this frence"}, {"start": 68.4, "duration": 7.68, "text": "of two numbers right so here we have 2^"}, {"start": 71.64, "duration": 9.96, "text": "of n minus 2^ M this will be equal to 12"}, {"start": 76.08, "duration": 7.96, "text": "is the same as 16 - 4 right okay I know"}, {"start": 81.6, "duration": 4.8, "text": "that this method may not be"}, {"start": 84.04, "duration": 5.28, "text": "mathematically always true right but I"}, {"start": 86.4, "duration": 4.759, "text": "think this is something really nice and"}, {"start": 89.32, "duration": 4.759, "text": "something we can can easily memorize"}, {"start": 91.159, "duration": 5.481, "text": "right so if we can have the difference"}, {"start": 94.079, "duration": 7.36, "text": "of two numbers in this form and here we"}, {"start": 96.64, "duration": 8.119, "text": "have basis of two then we can either try"}, {"start": 101.439, "duration": 9.04, "text": "relating this with some powers of two as"}, {"start": 104.759, "duration": 9.281, "text": "follow here we have 2^ of N - 2^ n and"}, {"start": 110.479, "duration": 8.28, "text": "this is equal to 2 to the^ of 4 right"}, {"start": 114.04, "duration": 7.16, "text": "then - 2 to the^ 2 right so if you have"}, {"start": 118.759, "duration": 4.921, "text": "something like this it is awesome right"}, {"start": 121.2, "duration": 6.88, "text": "because we can easily State now that n"}, {"start": 123.68, "duration": 8.199, "text": "is = 4 and the m is equal"}, {"start": 128.08, "duration": 8.439, "text": "2 remember we're solving for natural"}, {"start": 131.879, "duration": 7.841, "text": "numbers right for x m and n right so"}, {"start": 136.519, "duration": 6.0, "text": "here we found that n is equal 4 and M is"}, {"start": 139.72, "duration": 4.36, "text": "equal to 2 now let us recall again that"}, {"start": 142.519, "duration": 10.841, "text": "the cube root of"}, {"start": 144.08, "duration": 12.32, "text": "x is equal to M and the x / 2 is = n"}, {"start": 153.36, "duration": 4.36, "text": "now when m is equal to 2 so we get the"}, {"start": 156.4, "duration": 7.04, "text": "cube root of"}, {"start": 157.72, "duration": 10.0, "text": "x is = 2 when n is equal 4 we get x / 2"}, {"start": 163.44, "duration": 8.64, "text": "is = 4 now here CU both sides we get x ="}, {"start": 167.72, "duration": 8.08, "text": "2 cubed which is 8 and from here cross"}, {"start": 172.08, "duration": 5.6, "text": "multiply we also get that xal 8 and"}, {"start": 175.8, "duration": 7.359, "text": "therefore in the above solution we can"}, {"start": 177.68, "duration": 7.919, "text": "see that X is = to 8 is the only integer"}, {"start": 183.159, "duration": 4.761, "text": "solution thank you for watching please"}, {"start": 185.599, "duration": 6.321, "text": "kindly subscribe to my channel also like"}, {"start": 187.92, "duration": 4.0, "text": "comment and share bye-bye"}], "title": "Interesting exponential equation | Can you solve ?", "description": "", "lengthSeconds": 190, "width": 640, "height": 360}
{"video_id": "Tm65b3rxqnQ", "transcript": [{"start": 0.719, "duration": 6.241, "text": "hello good day everyone welcome back"}, {"start": 3.84, "duration": 5.96, "text": "once again to my YouTube Channel please"}, {"start": 6.96, "duration": 4.719, "text": "kindly support this channel by"}, {"start": 9.8, "duration": 5.4, "text": "subscribing to this"}, {"start": 11.679, "duration": 7.76, "text": "Channel please subscribe to this Channel"}, {"start": 15.2, "duration": 9.04, "text": "and share my videos today we have this"}, {"start": 19.439, "duration": 9.68, "text": "continued fraction here Z is = to 1 + 2"}, {"start": 24.24, "duration": 7.839, "text": "all over 1 + 2 all over 1 + 2 and so on"}, {"start": 29.119, "duration": 6.361, "text": "we are asked to find the exact value of"}, {"start": 32.079, "duration": 10.681, "text": "this continued fraction right okay let's"}, {"start": 35.48, "duration": 12.44, "text": "get started so here we have Z = 1 + 2 1"}, {"start": 42.76, "duration": 7.6, "text": "+ 2 over 1 + 2 and so on so if we"}, {"start": 47.92, "duration": 5.479, "text": "carefully look at this continued"}, {"start": 50.36, "duration": 6.56, "text": "fraction here you can see here we have Z"}, {"start": 53.399, "duration": 7.881, "text": "= 1 + 2 over and right over here we can"}, {"start": 56.92, "duration": 5.92, "text": "see here 1 + 2 over 1 + two so we can"}, {"start": 61.28, "duration": 3.76, "text": "actually"}, {"start": 62.84, "duration": 6.36, "text": "call"}, {"start": 65.04, "duration": 7.2, "text": "this Z right because it is still"}, {"start": 69.2, "duration": 6.44, "text": "repeating and be going we can see Z = 1"}, {"start": 72.24, "duration": 7.559, "text": "+ 2 all over right over here we have Z ="}, {"start": 75.64, "duration": 7.96, "text": "1 + 2 then we have another Z here which"}, {"start": 79.799, "duration": 4.841, "text": "goes on forever right so now we derive"}, {"start": 83.6, "duration": 7.76, "text": "this"}, {"start": 84.64, "duration": 9.72, "text": "equation Z is = to 1 + 2 all over Z"}, {"start": 91.36, "duration": 5.88, "text": "right so now the next idea here is to"}, {"start": 94.36, "duration": 6.92, "text": "multiply both sides of this equation by"}, {"start": 97.24, "duration": 7.08, "text": "Z right so here we have to multiply by z"}, {"start": 101.28, "duration": 7.6, "text": "z * and here we have to multiply by Z"}, {"start": 104.32, "duration": 10.399, "text": "Open Bracket 1 + 2 all over Z right so"}, {"start": 108.88, "duration": 9.0, "text": "here Z * Z is Z 2 is equal to Z * 1 is Z"}, {"start": 114.719, "duration": 7.201, "text": "Plus Z * 2 all over Z you know Z will"}, {"start": 117.88, "duration": 8.04, "text": "cancel Z we're left to it two so now let"}, {"start": 121.92, "duration": 5.44, "text": "us subtract Z + 2 from both sides so"}, {"start": 125.92, "duration": 9.399, "text": "here we get z^"}, {"start": 127.36, "duration": 10.12, "text": "2 - Z - 2 is equal to 0 right so this"}, {"start": 135.319, "duration": 5.64, "text": "quadratic expression at the left hand"}, {"start": 137.48, "duration": 6.0, "text": "side of this equation is factorizable so"}, {"start": 140.959, "duration": 8.081, "text": "if you factorize this this is going to"}, {"start": 143.48, "duration": 8.36, "text": "give us Z + 1 and what Z - 2 this is"}, {"start": 149.04, "duration": 2.8, "text": "equal Z"}, {"start": 153.08, "duration": 11.56, "text": "so then by solving here we have Z + 1 ="}, {"start": 157.519, "duration": 11.201, "text": "0 or Z - 2 = 0 you know here Z is = -1"}, {"start": 164.64, "duration": 6.959, "text": "or Z is equal to 2 but we need to know"}, {"start": 168.72, "duration": 5.519, "text": "that this continued fraction yelds a"}, {"start": 171.599, "duration": 5.64, "text": "positive value and not a negative value"}, {"start": 174.239, "duration": 6.561, "text": "so this is disregarded right so the"}, {"start": 177.239, "duration": 5.881, "text": "right answer is z equal to 2 so we can"}, {"start": 180.8, "duration": 5.76, "text": "actually say that this continued"}, {"start": 183.12, "duration": 5.64, "text": "fraction is exactly equal to two and"}, {"start": 186.56, "duration": 3.64, "text": "this is the right answer thank you for"}, {"start": 188.76, "duration": 5.88, "text": "watching do wait to subscribe to my"}, {"start": 190.2, "duration": 4.44, "text": "YouTube channel and share my videos"}], "title": "Continued Fraction Problem. From Chidex Hosea.  Very interesting problem", "description": "", "lengthSeconds": 193, "width": 636, "height": 360}
{"video_id": "uS7VDYFWuBw", "transcript": [{"start": 0.68, "duration": 3.88, "text": "hello good viewers welcome back once"}, {"start": 2.36, "duration": 6.08, "text": "again let's find solution to the"}, {"start": 4.56, "duration": 8.52, "text": "following given exponential equation 3^"}, {"start": 8.44, "duration": 9.12, "text": "of root of x this is equal to 9 ^"}, {"start": 13.08, "duration": 8.84, "text": "x we have here 3^ of root X this is"}, {"start": 17.56, "duration": 8.0, "text": "equal to 9 is the same as 3 S right then"}, {"start": 21.92, "duration": 6.759, "text": "ra of X now let us record this"}, {"start": 25.56, "duration": 8.4, "text": "property a to the power of B then in"}, {"start": 28.679, "duration": 9.641, "text": "bracket of C this is same as a^ of B * C"}, {"start": 33.96, "duration": 10.279, "text": "so from here we have 3^ ofun X this is"}, {"start": 38.32, "duration": 8.239, "text": "equal to 3 ^ of 2 x right now here we"}, {"start": 44.239, "duration": 5.241, "text": "have equal bases from both sides so we"}, {"start": 46.559, "duration": 5.881, "text": "can actually equate the powers right so"}, {"start": 49.48, "duration": 6.48, "text": "here we have root of x which is equal to"}, {"start": 52.44, "duration": 6.32, "text": "2 x now let's go ahead and square both"}, {"start": 55.96, "duration": 9.279, "text": "sides so we have root of"}, {"start": 58.76, "duration": 9.44, "text": "X2 this is equal to 2x^ 2 so from here"}, {"start": 65.239, "duration": 7.161, "text": "this root and the square we cancel out"}, {"start": 68.2, "duration": 5.959, "text": "we have here x is = to 2x2 will give us"}, {"start": 72.4, "duration": 5.92, "text": "4"}, {"start": 74.159, "duration": 7.481, "text": "x^2 now so many students makes mistake"}, {"start": 78.32, "duration": 5.72, "text": "dividing both side by X if you divide"}, {"start": 81.64, "duration": 6.64, "text": "both side by X from here you're going to"}, {"start": 84.04, "duration": 5.88, "text": "lose a solution which is X = 0 so"}, {"start": 88.28, "duration": 4.479, "text": "whenever you encounter this kind of"}, {"start": 89.92, "duration": 9.04, "text": "equation all you have to do is to"}, {"start": 92.759, "duration": 8.841, "text": "rewrite it as 4x^2 - x then = 0 from"}, {"start": 98.96, "duration": 5.119, "text": "here from here we can factor out X from"}, {"start": 101.6, "duration": 7.72, "text": "the left hand side we have X into"}, {"start": 104.079, "duration": 8.801, "text": "bracket 4x - 1 right then this is equal"}, {"start": 109.32, "duration": 6.399, "text": "to Z so from here we have X is equal to"}, {"start": 112.88, "duration": 7.32, "text": "0 or then in the bracket there which is"}, {"start": 115.719, "duration": 8.68, "text": "4x - 1 is = 0 so from here add 1 to both"}, {"start": 120.2, "duration": 9.96, "text": "side we get 4X is = 1 divide both side"}, {"start": 124.399, "duration": 8.761, "text": "by 4 we get X is = 1 / 4 so from here we"}, {"start": 130.16, "duration": 6.6, "text": "have the following solution X is equal"}, {"start": 133.16, "duration": 8.32, "text": "to0 and also we have another solution X"}, {"start": 136.76, "duration": 7.119, "text": "is = 1 / 4 thank you for watching please"}, {"start": 141.48, "duration": 4.64, "text": "kindly subscribe to my channel turn on"}, {"start": 143.879, "duration": 4.72, "text": "the notification Bell so that you be"}, {"start": 146.12, "duration": 5.479, "text": "notified anytime I upload new videos"}, {"start": 148.599, "duration": 3.0, "text": "bye-bye"}], "title": "Interesting exponential equation | step by step solution | #maths #mathematics", "description": "", "lengthSeconds": 150, "width": 640, "height": 360}
{"video_id": "VX1d38CM4u8", "transcript": [{"start": 0.799, "duration": 4.6, "text": "hello welcome back once again today"}, {"start": 3.04, "duration": 4.64, "text": "we're going to solve this exponential"}, {"start": 5.399, "duration": 5.12, "text": "equation from the left here in the"}, {"start": 7.68, "duration": 6.44, "text": "bracket we have root3 which is same as 3"}, {"start": 10.519, "duration": 8.361, "text": "to the^ 1 / 2"}, {"start": 14.12, "duration": 8.239, "text": "then of X this is equal to here 27 is"}, {"start": 18.88, "duration": 6.239, "text": "the same as 3 to the^ 3 then here we"}, {"start": 22.359, "duration": 4.76, "text": "have power ofare root of"}, {"start": 25.119, "duration": 5.64, "text": "x now"}, {"start": 27.119, "duration": 8.481, "text": "recall a to the power of B in bra"}, {"start": 30.759, "duration": 8.041, "text": "C this is equal to a to the power B * C"}, {"start": 35.6, "duration": 8.2, "text": "so from the left here we get 3 power of"}, {"start": 38.8, "duration": 8.64, "text": "1 / 2 * X the same as x / 2 then this is"}, {"start": 43.8, "duration": 6.84, "text": "equal to here 3 power of 3 * root X"}, {"start": 47.44, "duration": 5.959, "text": "which is same as 3un"}, {"start": 50.64, "duration": 4.919, "text": "X from here we have the same basis right"}, {"start": 53.399, "duration": 8.921, "text": "so we're going to equate the powers so"}, {"start": 55.559, "duration": 10.6, "text": "this implies that x / 2 = + 3un"}, {"start": 62.32, "duration": 8.76, "text": "x so from here this is over one so let's"}, {"start": 66.159, "duration": 8.721, "text": "go ahead and cross multiply so here x *"}, {"start": 71.08, "duration": 9.24, "text": "1 here will give us X this is equal to"}, {"start": 74.88, "duration": 8.4, "text": "here 2 * 3un X will give us 6 < TK"}, {"start": 80.32, "duration": 6.159, "text": "X okay so from here let's Square both"}, {"start": 83.28, "duration": 5.879, "text": "sides to get rid of this root sign so"}, {"start": 86.479, "duration": 6.401, "text": "here this is squared and from the right"}, {"start": 89.159, "duration": 7.721, "text": "here the whole of this is squ so X2 is"}, {"start": 92.88, "duration": 7.76, "text": "X2 right then this is equal to here we"}, {"start": 96.88, "duration": 9.4, "text": "have 6 * x 2 first of all 6 2 will give"}, {"start": 100.64, "duration": 7.96, "text": "us 36 and here root X oh root X2 I mean"}, {"start": 106.28, "duration": 3.839, "text": "Square we cancel the roots so we have"}, {"start": 108.6, "duration": 4.799, "text": "just"}, {"start": 110.119, "duration": 8.801, "text": "X now let's subtract 36 from both sides"}, {"start": 113.399, "duration": 9.921, "text": "that is x\u00b2 - 36 x this is equal to Z"}, {"start": 118.92, "duration": 8.64, "text": "factor out X so X into bracket here x\u00b2 /"}, {"start": 123.32, "duration": 8.8, "text": "X will give us xus 36 x / X will give us"}, {"start": 127.56, "duration": 9.959, "text": "36 and this is equal to Z so from here"}, {"start": 132.12, "duration": 11.24, "text": "this implies that X is equal to Z or x -"}, {"start": 137.519, "duration": 9.401, "text": "36 = 0 of which we solve to get x = 36"}, {"start": 143.36, "duration": 6.8, "text": "and here we have the solution and also"}, {"start": 146.92, "duration": 8.039, "text": "here we have a solution so we only have"}, {"start": 150.16, "duration": 8.04, "text": "two R Solutions x = 0 and x ="}, {"start": 154.959, "duration": 5.36, "text": "36 thank you for watching please kindly"}, {"start": 158.2, "duration": 6.319, "text": "subscribe to my channel also like"}, {"start": 160.319, "duration": 4.2, "text": "comment and share bye-bye"}], "title": "Exponential equation | Can you solve ?", "description": "", "lengthSeconds": 163, "width": 640, "height": 360}
{"video_id": "JLqCb4Zcglg", "transcript": [{"start": 0.56, "duration": 5.4, "text": "hello welcome back once again today let"}, {"start": 2.919, "duration": 5.961, "text": "us solve this equation and this is"}, {"start": 5.96, "duration": 7.92, "text": "indeed a very nice equation we have the"}, {"start": 8.88, "duration": 7.92, "text": "root of 2^ x / 2 = the cube root of 4^"}, {"start": 13.88, "duration": 5.28, "text": "of X and here we're going to solve for x"}, {"start": 16.8, "duration": 4.96, "text": "values okay now let's get"}, {"start": 19.16, "duration": 5.279, "text": "started from the left right over here"}, {"start": 21.76, "duration": 9.359, "text": "let us take note of this that the end"}, {"start": 24.439, "duration": 9.161, "text": "root of M is same as M to the^ of 1 / n"}, {"start": 31.119, "duration": 7.521, "text": "so from the left right over here that"}, {"start": 33.6, "duration": 7.119, "text": "will become 2 to the^ of x / 2 then with"}, {"start": 38.64, "duration": 5.599, "text": "that square root that will be to the^ 1"}, {"start": 40.719, "duration": 7.0, "text": "/ 2 and this is equal to from here that"}, {"start": 44.239, "duration": 8.32, "text": "will be 4^ X then the cube root will"}, {"start": 47.719, "duration": 6.761, "text": "become to the^ of 1 / 3 let us take note"}, {"start": 52.559, "duration": 5.401, "text": "of this when we have a to the power of B"}, {"start": 54.48, "duration": 6.52, "text": "to the power of C this is same as a to"}, {"start": 57.96, "duration": 6.519, "text": "the power of B * C right"}, {"start": 61.0, "duration": 8.08, "text": "so from here we're going to have 2 ^ x /"}, {"start": 64.479, "duration": 9.201, "text": "2 * 1 / 2 will give us x 4 then this"}, {"start": 69.08, "duration": 7.76, "text": "will be equal to here 4^ of x * 1 / 3"}, {"start": 73.68, "duration": 6.04, "text": "will give us x /"}, {"start": 76.84, "duration": 5.48, "text": "3 okay now remember that this four right"}, {"start": 79.72, "duration": 6.2, "text": "over here is the same as 2 s right so"}, {"start": 82.32, "duration": 6.799, "text": "here we have 2 to the power of x / 4"}, {"start": 85.92, "duration": 9.0, "text": "this is equal to 2^ 2 then to the power"}, {"start": 89.119, "duration": 10.081, "text": "of x ID 3 so from here we get 2^ x / 4"}, {"start": 94.92, "duration": 8.76, "text": "this is equal to 2^ of 2 * x / 3 will"}, {"start": 99.2, "duration": 6.76, "text": "give us 2x / 3 so from here we can see"}, {"start": 103.68, "duration": 4.119, "text": "that we have the same bases right so"}, {"start": 105.96, "duration": 6.519, "text": "we're going to equate the"}, {"start": 107.799, "duration": 8.68, "text": "powers and that will imply"}, {"start": 112.479, "duration": 8.521, "text": "that x / 4 that is by equating the"}, {"start": 116.479, "duration": 7.24, "text": "powers will be equal to 2X"}, {"start": 121.0, "duration": 2.719, "text": "divided"}, {"start": 123.88, "duration": 5.8, "text": "by 3"}, {"start": 126.68, "duration": 3.0, "text": "right"}, {"start": 130.76, "duration": 6.559, "text": "okay so from here let us cross multiply"}, {"start": 135.0, "duration": 5.36, "text": "so 3 * X will give us"}, {"start": 137.319, "duration": 8.28, "text": "3x and this is equal to 4"}, {"start": 140.36, "duration": 7.8, "text": "* 2x will give us 8 x right okay so from"}, {"start": 145.599, "duration": 5.881, "text": "here many may think that there's no"}, {"start": 148.16, "duration": 5.76, "text": "solution but obviously there solution do"}, {"start": 151.48, "duration": 4.96, "text": "not make the mistake of canceling X from"}, {"start": 153.92, "duration": 5.36, "text": "both sides because that will result to"}, {"start": 156.44, "duration": 7.159, "text": "no solution to the question but the best"}, {"start": 159.28, "duration": 9.08, "text": "thing we can do is either we say 8 x -"}, {"start": 163.599, "duration": 8.0, "text": "3x is = 0 right so that we get here"}, {"start": 168.36, "duration": 7.4, "text": "5x is equal to Z then dividing both side"}, {"start": 171.599, "duration": 9.36, "text": "by 5 we arrive at X is equal to Z and"}, {"start": 175.76, "duration": 7.839, "text": "obviously Z is the only solution to this"}, {"start": 180.959, "duration": 4.481, "text": "exponential equation thank you for"}, {"start": 183.599, "duration": 4.0, "text": "watching please kindly subscribe to my"}, {"start": 185.44, "duration": 5.159, "text": "channel also like comment and share"}, {"start": 187.599, "duration": 3.0, "text": "bye-bye"}], "title": "Interesting exponential equation | Can you solve ?", "description": "", "lengthSeconds": 189, "width": 640, "height": 360}
{"video_id": "CESwZXZkUmM", "transcript": [{"start": 0.919, "duration": 4.84, "text": "hello welcome back once again today we"}, {"start": 3.6, "duration": 4.68, "text": "have another interesting equation you"}, {"start": 5.759, "duration": 4.561, "text": "know why this equation is interesting"}, {"start": 8.28, "duration": 4.96, "text": "we're going to solve for both real and"}, {"start": 10.32, "duration": 5.04, "text": "complex values right so let's get"}, {"start": 13.24, "duration": 7.16, "text": "started so from the left hand side we"}, {"start": 15.36, "duration": 7.2, "text": "have x * root X and this is equal to 8"}, {"start": 20.4, "duration": 7.799, "text": "now let's Square both sides so we get x"}, {"start": 22.56, "duration": 8.4, "text": "* root X squ this will be equal to 8 squ"}, {"start": 28.199, "duration": 8.601, "text": "now recall when we have 8 is B all to"}, {"start": 30.96, "duration": 10.96, "text": "the^ n this is same as a^ NTI b^ n so"}, {"start": 36.8, "duration": 10.72, "text": "here we have X to the^ 2 * otk x^ 2"}, {"start": 41.92, "duration": 10.84, "text": "which is X then this is equal to 82 is"}, {"start": 47.52, "duration": 6.6, "text": "64 so from here this is X cubed - 64"}, {"start": 52.76, "duration": 6.88, "text": "this is equal to"}, {"start": 54.12, "duration": 8.279, "text": "0 from here x cubed - 64 is 4 Cub this"}, {"start": 59.64, "duration": 5.519, "text": "is = to 0 Let's make use of the"}, {"start": 62.399, "duration": 8.801, "text": "difference of two cubes formula remember"}, {"start": 65.159, "duration": 10.96, "text": "a cubus b Cub this is equal to a - b"}, {"start": 71.2, "duration": 9.08, "text": "into braet a 2 + a b + b"}, {"start": 76.119, "duration": 6.64, "text": "2 so from here this is going to be x -"}, {"start": 80.28, "duration": 8.839, "text": "4 into braet"}, {"start": 82.759, "duration": 10.241, "text": "X2 + 2 I mean that is 4 * X yes so 4 x"}, {"start": 89.119, "duration": 8.96, "text": "this is a here here then plus 4 2 which"}, {"start": 93.0, "duration": 7.6, "text": "is 16 then this is equal to 0 right okay"}, {"start": 98.079, "duration": 4.841, "text": "so from here we have these two products"}, {"start": 100.6, "duration": 4.559, "text": "which is equal to Z so we have either"}, {"start": 102.92, "duration": 7.199, "text": "this is equal to Z or this is equal to Z"}, {"start": 105.159, "duration": 8.401, "text": "so this implies from here x - 4 is = 0"}, {"start": 110.119, "duration": 9.801, "text": "of which X is = 4 so this is the only"}, {"start": 113.56, "duration": 9.159, "text": "real solution and here we have X2 + 4x +"}, {"start": 119.92, "duration": 4.799, "text": "16 this is equal to 0 we're going to"}, {"start": 122.719, "duration": 4.801, "text": "solve this using the general Quadra"}, {"start": 124.719, "duration": 7.6, "text": "quadratic formula so we get xal to will"}, {"start": 127.52, "duration": 10.799, "text": "be 4 plus or minus otk of b 2 is 4 2"}, {"start": 132.319, "duration": 9.56, "text": "which is 16 - 4 * a * C 4 * 16 is 64"}, {"start": 138.319, "duration": 8.721, "text": "then the whole of this / 2 * a which is"}, {"start": 141.879, "duration": 9.281, "text": "2 so from here we get X is = -4 +"}, {"start": 147.04, "duration": 8.72, "text": "orus < TK of -48"}, {"start": 151.16, "duration": 9.6, "text": "then / 2 from here we get X is ="}, {"start": 155.76, "duration": 9.96, "text": "to -4+ orus of-1 is"}, {"start": 160.76, "duration": 8.0, "text": "Iota then we know that 48 here is 16 * 3"}, {"start": 165.72, "duration": 6.36, "text": "so of 16 is 4 then we have"}, {"start": 168.76, "duration": 6.52, "text": "root3 the all of this divid two which"}, {"start": 172.08, "duration": 9.439, "text": "when you split this denominator 4/ 2 is"}, {"start": 175.28, "duration": 9.0, "text": "-2 plus or minus 4 I3 / 2 is 2 I <"}, {"start": 181.519, "duration": 5.241, "text": "tk3 so here we come up with the solution"}, {"start": 184.28, "duration": 7.64, "text": "so we have the only real solution at X"}, {"start": 186.76, "duration": 12.6, "text": "is equal to 4 and we have two complex"}, {"start": 191.92, "duration": 9.2, "text": "Solutions -2 plus or minus 2 I root of 3"}, {"start": 199.36, "duration": 4.159, "text": "thank you for watching please kindly"}, {"start": 201.12, "duration": 6.96, "text": "subscribe to my channel also like"}, {"start": 203.519, "duration": 4.561, "text": "comment and share bye-bye"}], "title": "Interesting radical equation | Can you solve ?", "description": "", "lengthSeconds": 207, "width": 552, "height": 360}
{"video_id": "1N377t3Of_Q", "transcript": [{"start": 0.919, "duration": 6.4, "text": "hello good viewers welcome back once"}, {"start": 3.159, "duration": 9.64, "text": "again let's find a solution to this"}, {"start": 7.319, "duration": 7.041, "text": "given equation we have here 4 xot of 27"}, {"start": 12.799, "duration": 4.681, "text": "this is equal to"}, {"start": 14.36, "duration": 5.759, "text": "4X now the left hand side of this"}, {"start": 17.48, "duration": 7.639, "text": "equation we will write it make use of"}, {"start": 20.119, "duration": 10.521, "text": "this property we know the end root of M"}, {"start": 25.119, "duration": 11.28, "text": "this is equal to m^ of 1 / n right now"}, {"start": 30.64, "duration": 9.32, "text": "from the left we have 27 to the^ of 1 /"}, {"start": 36.399, "duration": 5.601, "text": "4X then from the right hand side this is"}, {"start": 39.96, "duration": 6.16, "text": "equal to"}, {"start": 42.0, "duration": 6.559, "text": "4X now to get rid of this power here 1"}, {"start": 46.12, "duration": 6.0, "text": "over 4X we shall raise both sides of"}, {"start": 48.559, "duration": 9.081, "text": "this equation to the power of 4X right"}, {"start": 52.12, "duration": 8.959, "text": "so from here we have 27 ^ of 1 / 4X in"}, {"start": 57.64, "duration": 5.28, "text": "bracket rais power of 4X"}, {"start": 61.079, "duration": 5.801, "text": "this is equal to from the right hand"}, {"start": 62.92, "duration": 6.4, "text": "side 4X to the power of"}, {"start": 66.88, "duration": 5.08, "text": "4X now from the left hand side let us"}, {"start": 69.32, "duration": 7.0, "text": "recall another property here when we"}, {"start": 71.96, "duration": 9.0, "text": "have a to the power of B in bracket of C"}, {"start": 76.32, "duration": 6.88, "text": "this is same as a^ B * C now from here"}, {"start": 80.96, "duration": 5.4, "text": "when we multiply these Powers together 1"}, {"start": 83.2, "duration": 6.879, "text": "/ 4X * 4X that will give us 1 and here"}, {"start": 86.36, "duration": 7.2, "text": "we left with 27 right to the^ of 1 and"}, {"start": 90.079, "duration": 7.921, "text": "from the right hand side we have 4X ra"}, {"start": 93.56, "duration": 8.04, "text": "power 4 x now let us recall 27 is the"}, {"start": 98.0, "duration": 4.88, "text": "same as 3 to the^ of 3 then this is"}, {"start": 101.6, "duration": 5.72, "text": "equal to"}, {"start": 102.88, "duration": 9.919, "text": "4X power of 4X now from here this"}, {"start": 107.32, "duration": 8.399, "text": "implies that 4X is equal to 3 right from"}, {"start": 112.799, "duration": 5.121, "text": "the base we have 4X and from the base"}, {"start": 115.719, "duration": 4.161, "text": "the left hand side we have three three"}, {"start": 117.92, "duration": 4.0, "text": "right and the power here corres"}, {"start": 119.88, "duration": 5.279, "text": "responding to what we have as the basis"}, {"start": 121.92, "duration": 5.559, "text": "4X which is equal to 3 right so from"}, {"start": 125.159, "duration": 4.32, "text": "here soling for X we divide both sides"}, {"start": 127.479, "duration": 4.521, "text": "by"}, {"start": 129.479, "duration": 7.361, "text": "four and from here we come up with the"}, {"start": 132.0, "duration": 7.04, "text": "solution that X is = to 3/ 4 and this is"}, {"start": 136.84, "duration": 4.24, "text": "the right answer thank you for watching"}, {"start": 139.04, "duration": 6.68, "text": "please kindly subscribe to my channel"}, {"start": 141.08, "duration": 4.64, "text": "also like comment and share bye-bye"}], "title": "Japanese | Interesting exponential equation | #maths #mathematics", "description": "", "lengthSeconds": 144, "width": 520, "height": 360}
{"video_id": "NKfplLCJ0Oc", "transcript": [{"start": 0.76, "duration": 6.079, "text": "hello welcome back once again today"}, {"start": 2.879, "duration": 8.081, "text": "we're going to evaluate 18 /un x given"}, {"start": 6.839, "duration": 6.361, "text": "that 9^ ofun X is equal to root3 now"}, {"start": 10.96, "duration": 4.639, "text": "let's get started from the left hand"}, {"start": 13.2, "duration": 7.239, "text": "side this base right over here which is"}, {"start": 15.599, "duration": 9.561, "text": "9 we can write it as 3 2 so here we have"}, {"start": 20.439, "duration": 7.561, "text": "3 S then ra to the power of root X then"}, {"start": 25.16, "duration": 6.0, "text": "this is equal to now theare root of 3 is"}, {"start": 28.0, "duration": 6.879, "text": "the same as 3 to the^ of 1 all over"}, {"start": 31.16, "duration": 6.16, "text": "2 okay now from here let us take note of"}, {"start": 34.879, "duration": 5.601, "text": "this when we have a to the^ B then to"}, {"start": 37.32, "duration": 9.96, "text": "the power C this is equal to A to the^ B"}, {"start": 40.48, "duration": 10.399, "text": "* C so from here we have 3 to the^ 2 *"}, {"start": 47.28, "duration": 8.759, "text": "root X which will give us 2un X then"}, {"start": 50.879, "duration": 8.441, "text": "this is equal to 3 to the^ of 1 /"}, {"start": 56.039, "duration": 5.68, "text": "2 okay now from here we have equal bases"}, {"start": 59.32, "duration": 8.64, "text": "right so we can actually equate the"}, {"start": 61.719, "duration": 7.881, "text": "powers so here we get 2un X is = 1 / 2"}, {"start": 67.96, "duration": 2.4, "text": "now let's go ahead and divide both side"}, {"start": 69.6, "duration": 3.72, "text": "by"}, {"start": 70.36, "duration": 4.439, "text": "two and by two the reason for this is to"}, {"start": 73.32, "duration": 4.159, "text": "solve for root"}, {"start": 74.799, "duration": 5.64, "text": "X after getting the value for root X so"}, {"start": 77.479, "duration": 5.881, "text": "we can easily divide 18 by the value for"}, {"start": 80.439, "duration": 5.761, "text": "root X and there we have our answer"}, {"start": 83.36, "duration": 7.48, "text": "right so here this cancels this so here"}, {"start": 86.2, "duration": 8.48, "text": "we have root X is equal to 1 / 2 2 / 2"}, {"start": 90.84, "duration": 7.12, "text": "will give us 1 over 4 here we have the"}, {"start": 94.68, "duration": 8.56, "text": "value for root X and therefore we can go"}, {"start": 97.96, "duration": 11.159, "text": "ahead and evaluate this expression 18 /"}, {"start": 103.24, "duration": 11.68, "text": "otk X which should be equal to 18 / 1/ 4"}, {"start": 109.119, "duration": 8.881, "text": "which we can write as 18 / 1 / 1 4 which"}, {"start": 114.92, "duration": 5.119, "text": "is equal to 18 / 1 if we change this"}, {"start": 118.0, "duration": 5.56, "text": "multiplication then we take the"}, {"start": 120.039, "duration": 8.84, "text": "reciprocal of 1/ 4 which is 4/ 1 and"}, {"start": 123.56, "duration": 7.92, "text": "from here we get 18 * 4 which is 72 and"}, {"start": 128.879, "duration": 4.481, "text": "this is the right answer thank you for"}, {"start": 131.48, "duration": 4.24, "text": "watching please kindly subscribe to my"}, {"start": 133.36, "duration": 5.36, "text": "channel also like comment and share"}, {"start": 135.72, "duration": 3.0, "text": "bye-bye"}], "title": "Interesting algebra problem", "description": "", "lengthSeconds": 137, "width": 640, "height": 360}
{"video_id": "g0kooe-c_9U", "transcript": [{"start": 0.64, "duration": 5.119, "text": "hello welcome back once again today we"}, {"start": 3.04, "duration": 4.48, "text": "have this Olympiad mathematics problem"}, {"start": 5.759, "duration": 5.721, "text": "we're given the following Quantic"}, {"start": 7.52, "duration": 7.96, "text": "equation that x ^ 4 - 8 is equal to0 so"}, {"start": 11.48, "duration": 9.16, "text": "here we're going to evaluate the C power"}, {"start": 15.48, "duration": 7.76, "text": "of X that is x ^ 6 right okay from the"}, {"start": 20.64, "duration": 9.44, "text": "given equation we have x ^"}, {"start": 23.24, "duration": 11.92, "text": "4 - 8 which is equal to 0 right so here"}, {"start": 30.08, "duration": 7.319, "text": "we have X to the^ 2^ 2 right X to the^ 4"}, {"start": 35.16, "duration": 6.879, "text": "can be re written in this form because"}, {"start": 37.399, "duration": 7.241, "text": "we know 2 * 2 here is equal 4 then minus"}, {"start": 42.039, "duration": 5.0, "text": "so let's look for a number that when we"}, {"start": 44.64, "duration": 7.2, "text": "Square it it will give us 8 so I'm"}, {"start": 47.039, "duration": 9.881, "text": "thinking of 2 < tk2 but how did I get"}, {"start": 51.84, "duration": 10.08, "text": "this we can say that 8 is equal to < TK"}, {"start": 56.92, "duration": 11.76, "text": "8 squ right so which is equal"}, {"start": 61.92, "duration": 10.92, "text": "to < TK 4 * 2 all 2 which is equal to of"}, {"start": 68.68, "duration": 8.84, "text": "4 is 2 so we get here 2 < tk2 2 so you"}, {"start": 72.84, "duration": 8.68, "text": "can see so 2 < tk2 2 will give us 8 so"}, {"start": 77.52, "duration": 4.84, "text": "here we write as 2un of 2^ 2 which is"}, {"start": 81.52, "duration": 3.08, "text": "equal"}, {"start": 82.36, "duration": 3.759, "text": "to0 okay so from here we're going to"}, {"start": 84.6, "duration": 7.559, "text": "make use of the difference of two"}, {"start": 86.119, "duration": 10.201, "text": "squares formula a s - b 2 is = to a - b"}, {"start": 92.159, "duration": 10.561, "text": "* A + B so make use of this difference"}, {"start": 96.32, "duration": 12.159, "text": "of two squares so we get x^2 - 2 < tk2 *"}, {"start": 102.72, "duration": 7.84, "text": "X2 + 2 < tk2 and this is equal to Z so"}, {"start": 108.479, "duration": 6.64, "text": "these two products are equal to Z so"}, {"start": 110.56, "duration": 9.879, "text": "this implies from here that either X2 -"}, {"start": 115.119, "duration": 11.081, "text": "2 < tk2 is = 0 or X2 +"}, {"start": 120.439, "duration": 8.16, "text": "2 < tk2 is = 0 so from here we get x2 is"}, {"start": 126.2, "duration": 4.28, "text": "equal to add 2k2 to both side so this"}, {"start": 128.599, "duration": 4.961, "text": "will become pos2 <"}, {"start": 130.48, "duration": 6.0, "text": "tk2 so we leave this here so that is"}, {"start": 133.56, "duration": 6.319, "text": "going to help us in this video so X2"}, {"start": 136.48, "duration": 5.56, "text": "here will be 0 - 2 < tk2 will give us -2"}, {"start": 139.879, "duration": 2.161, "text": "<"}, {"start": 142.959, "duration": 5.801, "text": "tk2 remember we are solving for x to"}, {"start": 145.879, "duration": 5.961, "text": "the^ of 6 right okay so from the"}, {"start": 148.76, "duration": 7.64, "text": "original equation we have x ^ 4 - 8"}, {"start": 151.84, "duration": 6.759, "text": "which is = 0 so we have x ^ 4 is = 0 + 8"}, {"start": 156.4, "duration": 7.36, "text": "which is 8 now let's multiply both side"}, {"start": 158.599, "duration": 8.961, "text": "by x^ 2 since we are looking for x^ 6 so"}, {"start": 163.76, "duration": 4.8, "text": "we can see from here x to the of record"}, {"start": 167.56, "duration": 4.399, "text": "this"}, {"start": 168.56, "duration": 7.399, "text": "property oh that is an initial law right"}, {"start": 171.959, "duration": 5.56, "text": "a to the^ M MTI a the^ n since they have"}, {"start": 175.959, "duration": 5.0, "text": "the same base so we need to add the"}, {"start": 177.519, "duration": 6.601, "text": "powers right so this will be n+ + n so"}, {"start": 180.959, "duration": 5.56, "text": "here 2 + 4 will give us 6 so we can see"}, {"start": 184.12, "duration": 5.039, "text": "the x^ 6 we are looking for is already"}, {"start": 186.519, "duration": 7.72, "text": "right over here so it will be X it will"}, {"start": 189.159, "duration": 9.321, "text": "be 8 * X2 so this will be equal to 8 *"}, {"start": 194.239, "duration": 6.201, "text": "X2 is pos2 < tk2 and -2 < tk2 so when"}, {"start": 198.48, "duration": 6.44, "text": "you combine them so they will be written"}, {"start": 200.44, "duration": 7.079, "text": "in this form plus or minus 2 < tk2 right"}, {"start": 204.92, "duration": 4.959, "text": "so from here we get our final answer x^"}, {"start": 207.519, "duration": 6.161, "text": "6 will be equal to if we multiply this"}, {"start": 209.879, "duration": 9.44, "text": "will give us plus or minus 8 * 2 is 16"}, {"start": 213.68, "duration": 8.16, "text": "then < tk2 so this here is the right"}, {"start": 219.319, "duration": 5.241, "text": "answer thank you for watching please"}, {"start": 221.84, "duration": 7.24, "text": "kindly subscribe to my channel also like"}, {"start": 224.56, "duration": 4.52, "text": "comment and share bye-bye"}], "title": "Olympiad Math problem | Can you solve ?", "description": "", "lengthSeconds": 228, "width": 640, "height": 360}
{"video_id": "9RgzuMgN5Xg", "transcript": [{"start": 0.68, "duration": 5.04, "text": "hello everyone welcome back once again"}, {"start": 3.8, "duration": 5.32, "text": "please if today is your first day of"}, {"start": 5.72, "duration": 6.48, "text": "visiting this Channel please subscribe"}, {"start": 9.12, "duration": 5.719, "text": "to this Channel Please Subscribe and"}, {"start": 12.2, "duration": 4.4, "text": "share our videos thank you today we're"}, {"start": 14.839, "duration": 8.641, "text": "going to evaluate this expression of"}, {"start": 16.6, "duration": 9.2, "text": "number here 2^ 10 - 1 all/ 2^ 5 + 1"}, {"start": 23.48, "duration": 4.6, "text": "before we proceed I'm going to recall"}, {"start": 25.8, "duration": 3.92, "text": "the difference of two squares formula"}, {"start": 28.08, "duration": 10.56, "text": "here we are going to make use of this"}, {"start": 29.72, "duration": 11.519, "text": "idea identity a 2 - b 2 = A + B * Aus B"}, {"start": 38.64, "duration": 6.439, "text": "right okay now let us try to rewrite"}, {"start": 41.239, "duration": 9.361, "text": "this numerator here so this is equal to"}, {"start": 45.079, "duration": 9.8, "text": "you know this identity that a of B * C"}, {"start": 50.6, "duration": 6.24, "text": "is the same as a of B in bracket C right"}, {"start": 54.879, "duration": 9.28, "text": "so this is going to be"}, {"start": 56.84, "duration": 12.56, "text": "2 of 5 * 2 - 1 all over"}, {"start": 64.159, "duration": 7.761, "text": "2^ of 5 + 1 right so this is equal to"}, {"start": 69.4, "duration": 6.88, "text": "make use make use of this identity here"}, {"start": 71.92, "duration": 8.04, "text": "so we have 2 power of 5 in bracket then"}, {"start": 76.28, "duration": 10.799, "text": "squ minus you know 1 is the same as 1"}, {"start": 79.96, "duration": 10.72, "text": "squ then all/ 2 5 + one right so now"}, {"start": 87.079, "duration": 7.481, "text": "we're going to make use of this identity"}, {"start": 90.68, "duration": 9.799, "text": "right over here right so here we have 2"}, {"start": 94.56, "duration": 16.559, "text": "of 5 the base then plus the second base"}, {"start": 100.479, "duration": 13.441, "text": "+ 1 * by 2^ 5 - 1 all over 2^ 5 + 1"}, {"start": 111.119, "duration": 8.761, "text": "right so we're going to cancel out 2^ 5"}, {"start": 113.92, "duration": 9.879, "text": "+ 1 and 2 5 + 1 so this is equal to 2 5"}, {"start": 119.88, "duration": 8.84, "text": "- 1 and this is equal to you know 2^ of"}, {"start": 123.799, "duration": 7.361, "text": "5 is 32 - 1 and this is equal to 31 and"}, {"start": 128.72, "duration": 4.96, "text": "this is the right answer please"}, {"start": 131.16, "duration": 6.24, "text": "subscribe to this Channel and share our"}, {"start": 133.68, "duration": 3.72, "text": "videos thank you"}], "title": "Simplification of numbers. How faster can we do this?", "description": "", "lengthSeconds": 135, "width": 636, "height": 360}
{"video_id": "AT4oolz4uOQ", "transcript": [{"start": 1.079, "duration": 5.2, "text": "good day viewers welcome back once again"}, {"start": 3.12, "duration": 5.12, "text": "today we have an interesting exponent of"}, {"start": 6.279, "duration": 6.201, "text": "irrational numbers"}, {"start": 8.24, "duration": 8.44, "text": "right this is amazing right we have here"}, {"start": 12.48, "duration": 7.4, "text": "cube root of 6 to the^ of Cub root of 6"}, {"start": 16.68, "duration": 7.12, "text": "rais power of three and we are going to"}, {"start": 19.88, "duration": 6.479, "text": "evaluate this without calculators okay"}, {"start": 23.8, "duration": 5.479, "text": "now let's get into the video here we"}, {"start": 26.359, "duration": 5.521, "text": "have to rewrite the cube roots of"}, {"start": 29.279, "duration": 5.521, "text": "something as follows if we"}, {"start": 31.88, "duration": 8.56, "text": "recall this property when we have the"}, {"start": 34.8, "duration": 11.439, "text": "end root of a this is equal to a ^ 1 / n"}, {"start": 40.44, "duration": 11.2, "text": "right so from here we have -6 ^ 1 / 3"}, {"start": 46.239, "duration": 8.881, "text": "right ra the^ of -6 to^ 1 / 3 then ra"}, {"start": 51.64, "duration": 9.12, "text": "power of 3 now from here let us straet"}, {"start": 55.12, "duration": 10.119, "text": "this power first we have a negative"}, {"start": 60.76, "duration": 7.76, "text": "let's put it -1 * 6^ 1 / 3 of course"}, {"start": 65.239, "duration": 7.721, "text": "then of 3 this is equal to let us recall"}, {"start": 68.52, "duration": 11.76, "text": "another property a * b^ C this is equal"}, {"start": 72.96, "duration": 12.56, "text": "to a^ C * b^ C so from here we have -1 3"}, {"start": 80.28, "duration": 8.519, "text": "* 6^ 1 / 3^ 3 from here we all know"}, {"start": 85.52, "duration": 7.76, "text": "three is odd -1 to the power of an odd"}, {"start": 88.799, "duration": 6.36, "text": "number is 1 so we have1 times here we"}, {"start": 93.28, "duration": 6.119, "text": "need to multiply these Powers together"}, {"start": 95.159, "duration": 7.801, "text": "right we know this property a b c this"}, {"start": 99.399, "duration": 5.04, "text": "is equal to a b * C right so if we"}, {"start": 102.96, "duration": 3.24, "text": "multiply these Powers together this"}, {"start": 104.439, "duration": 6.601, "text": "three will cancel this three and here we"}, {"start": 106.2, "duration": 9.76, "text": "have six from here we have -6 amazing"}, {"start": 111.04, "duration": 10.8, "text": "right now remember that this this base"}, {"start": 115.96, "duration": 5.88, "text": "here is -6 to the^ of 1 / 3"}, {"start": 122.32, "duration": 7.32, "text": "1 3 3 and here we evaluated this power"}, {"start": 126.2, "duration": 9.24, "text": "right over here to be 6 so here we have"}, {"start": 129.64, "duration": 12.52, "text": "ra^ -6 from here we write as -1 * 6^ 1 /"}, {"start": 135.44, "duration": 10.879, "text": "3^ of -6 now we distribute -1 to^ of -6"}, {"start": 142.16, "duration": 7.76, "text": "* by 6 1 / 3 to^ of"}, {"start": 146.319, "duration": 8.321, "text": "-6 from here we need to recall another"}, {"start": 149.92, "duration": 10.399, "text": "property once again when we have a^ of n"}, {"start": 154.64, "duration": 7.12, "text": "this is equal to 1 a^ of n so from here"}, {"start": 160.319, "duration": 7.64, "text": "we"}, {"start": 161.76, "duration": 14.68, "text": "have -1 to^ of -6 * 6^ 1 /"}, {"start": 167.959, "duration": 13.2, "text": "3-6 this is the same as 1 -1 ^ 6 MTI 1"}, {"start": 176.44, "duration": 7.92, "text": "6^ 1 / 3 to^ of 6"}, {"start": 181.159, "duration": 7.0, "text": "and from here 6 is even so -1 to the"}, {"start": 184.36, "duration": 6.439, "text": "power of an even number is 1 1 over 1"}, {"start": 188.159, "duration": 5.921, "text": "times here we multiply these Powers"}, {"start": 190.799, "duration": 8.201, "text": "together so we have 1/"}, {"start": 194.08, "duration": 10.879, "text": "6 power of 1/ 3 * 6 this is equal to 1/"}, {"start": 199.0, "duration": 10.2, "text": "1 * 3 here 1 and here 2 1 all over 6 2"}, {"start": 204.959, "duration": 6.321, "text": "and this is equal to 1 over 36 and this"}, {"start": 209.2, "duration": 3.88, "text": "is the right answer thank you for"}, {"start": 211.28, "duration": 4.12, "text": "watching please kindly subscribe to my"}, {"start": 213.08, "duration": 5.96, "text": "YouTube channel also like comment and"}, {"start": 215.4, "duration": 3.64, "text": "share bye-bye"}], "title": "Interesting exponential problem involving irrational numbers | #maths #olympiad", "description": "", "lengthSeconds": 217, "width": 640, "height": 360}
{"video_id": "xigjP1U35YY", "transcript": [{"start": 0.12, "duration": 3.559, "text": "hello welcome back once again my lovely"}, {"start": 2.159, "duration": 5.521, "text": "people today we're going to find the"}, {"start": 3.679, "duration": 9.681, "text": "value of x + y + z given that x\u00b2 + y s +"}, {"start": 7.68, "duration": 10.08, "text": "z s = 1 x * y + y * Z + x * Z ="}, {"start": 13.36, "duration": 9.96, "text": "1 the solution to this that is the value"}, {"start": 17.76, "duration": 7.8, "text": "for x + y + z must be two right I not"}, {"start": 23.32, "duration": 7.24, "text": "mean that it is equal to two I mean we"}, {"start": 25.56, "duration": 7.56, "text": "should have two values for x + y + z you"}, {"start": 30.56, "duration": 5.519, "text": "know why take a look at this let's"}, {"start": 33.12, "duration": 10.599, "text": "assume that what what was given is X2 +"}, {"start": 36.079, "duration": 10.8, "text": "Y2 + Z2 = 3 and XY + y z + x z = 3 as"}, {"start": 43.719, "duration": 9.441, "text": "well so we can see we have TR solution X"}, {"start": 46.879, "duration": 11.0, "text": "= Y this = Y and this = Z meaning x = 1"}, {"start": 53.16, "duration": 8.079, "text": "y = 1 Z = 1 take a look 1 s + 1 + 1 is 3"}, {"start": 57.879, "duration": 5.481, "text": "1 * 1 + 1 * 1 + 1 * 1 is three right and"}, {"start": 61.239, "duration": 6.761, "text": "you can also see that X can"}, {"start": 63.36, "duration": 7.04, "text": "be1 y can be 1 and Z can be1 so with"}, {"start": 68.0, "duration": 6.6, "text": "this we can say that x + y +"}, {"start": 70.4, "duration": 7.24, "text": "z is going to be3 and positive 3 in this"}, {"start": 74.6, "duration": 6.839, "text": "case right so here we're expecting two"}, {"start": 77.64, "duration": 9.64, "text": "values for x + y + z so let's call x + y"}, {"start": 81.439, "duration": 10.521, "text": "+ z m right so squaring both sides M2"}, {"start": 87.28, "duration": 7.44, "text": "this is going to be x + y + z Z 2 so"}, {"start": 91.96, "duration": 8.439, "text": "from here M2 is going to be for expand"}, {"start": 94.72, "duration": 7.2, "text": "this we get x2 + y^ 2+ z^ 2 plus I can"}, {"start": 100.399, "duration": 6.841, "text": "factor out"}, {"start": 101.92, "duration": 8.239, "text": "to then y z + x z so here make"}, {"start": 107.24, "duration": 6.4, "text": "substitutions so M2 equals here the"}, {"start": 110.159, "duration": 7.081, "text": "value is given us 1 + 2 MTI the value"}, {"start": 113.64, "duration": 6.759, "text": "for this is also 1 so from here we get"}, {"start": 117.24, "duration": 5.76, "text": "M2 = 1 + 2 which is"}, {"start": 120.399, "duration": 5.841, "text": "take root of both side we get M = plus"}, {"start": 123.0, "duration": 7.08, "text": "orus theun 3 and here we come up with"}, {"start": 126.24, "duration": 7.359, "text": "the solution and this is a very quick"}, {"start": 130.08, "duration": 7.92, "text": "solution plus or minus root3 thanks to"}, {"start": 133.599, "duration": 5.961, "text": "God who invited mathematics right okay"}, {"start": 138.0, "duration": 3.08, "text": "mathematics is awesome and that's my"}, {"start": 139.56, "duration": 3.56, "text": "best subject thank you for watching"}, {"start": 141.08, "duration": 5.96, "text": "please kindly subscribe subscribe like"}, {"start": 143.12, "duration": 3.92, "text": "comment and share bye-bye"}], "title": "Interesting Olympiad Math", "description": "", "lengthSeconds": 146, "width": 548, "height": 360}
{"video_id": "uhVfNgm7wAo", "transcript": [{"start": 0.64, "duration": 5.279, "text": "hello welcome back once again today"}, {"start": 2.679, "duration": 5.721, "text": "we're going to solve this exponential"}, {"start": 5.919, "duration": 3.72, "text": "equation so from here we have the cube"}, {"start": 8.4, "duration": 5.48, "text": "root of"}, {"start": 9.639, "duration": 8.121, "text": "x of the cube root of x of the cube root"}, {"start": 13.88, "duration": 7.399, "text": "of x this will be equal to now 16 is the"}, {"start": 17.76, "duration": 6.24, "text": "same as 2 ra^"}, {"start": 21.279, "duration": 7.16, "text": "4 so from the left we have the cube root"}, {"start": 24.0, "duration": 7.16, "text": "of XIs by the cube root of XIs by the"}, {"start": 28.439, "duration": 7.001, "text": "cube root of x this will be equal to"}, {"start": 31.16, "duration": 7.44, "text": "here two now take note that this four"}, {"start": 35.44, "duration": 7.959, "text": "right over here is same as 2 to the^ of"}, {"start": 38.6, "duration": 6.479, "text": "two right so this will be 2 power of 2"}, {"start": 43.399, "duration": 4.48, "text": "okay now from here we're going to"}, {"start": 45.079, "duration": 6.201, "text": "compare this as follow here we have"}, {"start": 47.879, "duration": 6.48, "text": "something of the same thing of the same"}, {"start": 51.28, "duration": 7.04, "text": "thing equals another thing of the same"}, {"start": 54.359, "duration": 7.441, "text": "thing of the same thing right so either"}, {"start": 58.32, "duration": 6.159, "text": "we say that the cube of x = 2 or we say"}, {"start": 61.8, "duration": 5.959, "text": "the cube root of x = 2 or from here as"}, {"start": 64.479, "duration": 9.041, "text": "well the cube root of x = 2 right so"}, {"start": 67.759, "duration": 7.961, "text": "this implies that the cube root of x = 2"}, {"start": 73.52, "duration": 4.76, "text": "now from here Cube both"}, {"start": 75.72, "duration": 5.96, "text": "sides we know the cube and the cube root"}, {"start": 78.28, "duration": 6.64, "text": "will cancel and here we're left with X"}, {"start": 81.68, "duration": 6.24, "text": "which is equal to 2 ^ of 3 which is 8"}, {"start": 84.92, "duration": 5.36, "text": "and here is the right answer thank you"}, {"start": 87.92, "duration": 5.08, "text": "for watching please kindly Subs subbe to"}, {"start": 90.28, "duration": 5.72, "text": "my channel also like comment and share"}, {"start": 93.0, "duration": 3.0, "text": "bye-bye"}], "title": "Interesting exponential equation with cube root truncated powers.", "description": "", "lengthSeconds": 94, "width": 640, "height": 360}
{"video_id": "kXv2_1zQW6Q", "transcript": [{"start": 0.599, "duration": 5.121, "text": "hello good day viewers today we have"}, {"start": 2.56, "duration": 8.88, "text": "another Olympia ma problem here we given"}, {"start": 5.72, "duration": 9.8, "text": "that Aus B is equal to 2 a * B is equal"}, {"start": 11.44, "duration": 6.48, "text": "to 1 you we ask to find the sum of A and"}, {"start": 15.52, "duration": 4.56, "text": "B please if you have not subscribed to"}, {"start": 17.92, "duration": 4.92, "text": "this YouTube Channel please kindly"}, {"start": 20.08, "duration": 5.48, "text": "support this channel by subscribing and"}, {"start": 22.84, "duration": 6.199, "text": "also do not forget to share our"}, {"start": 25.56, "duration": 6.28, "text": "videos here I will be presenting a very"}, {"start": 29.039, "duration": 7.2, "text": "nice algebraic identity which I'm going"}, {"start": 31.84, "duration": 12.879, "text": "to use to solving this problem here we"}, {"start": 36.239, "duration": 15.0, "text": "have Aus b s is equal to a + b"}, {"start": 44.719, "duration": 9.52, "text": "s minus 4 a b right so this identity is"}, {"start": 51.239, "duration": 6.441, "text": "going to help us so much here here"}, {"start": 54.239, "duration": 5.8, "text": "recalling that a minus B is equal 2 so"}, {"start": 57.68, "duration": 6.439, "text": "here we're going to replace a minus B"}, {"start": 60.039, "duration": 7.76, "text": "with 2 so here we have 2 squ right CU"}, {"start": 64.119, "duration": 7.121, "text": "Aus B is s this is equal to remember we"}, {"start": 67.799, "duration": 5.241, "text": "are looking for a + b right so here we"}, {"start": 71.24, "duration": 7.8, "text": "have a +"}, {"start": 73.04, "duration": 9.6, "text": "b 2 so here we have minus 4 * a remember"}, {"start": 79.04, "duration": 7.2, "text": "a is equal to 1 so here we"}, {"start": 82.64, "duration": 6.479, "text": "have 4 * a is going to be just four"}, {"start": 86.24, "duration": 7.239, "text": "right so now the idea here is to solve"}, {"start": 89.119, "duration": 7.28, "text": "for a plus B right so now we can"}, {"start": 93.479, "duration": 12.401, "text": "exchange the sides of this equation so"}, {"start": 96.399, "duration": 17.08, "text": "this will become a + b 2 - 4 is = 2 of 2"}, {"start": 105.88, "duration": 11.159, "text": "right so here we have a + b 2 - 4 is = 2"}, {"start": 113.479, "duration": 7.081, "text": "2 is equal to 4 so you know this4"}, {"start": 117.039, "duration": 4.841, "text": "crossing over will BEC positive right"}, {"start": 120.56, "duration": 6.96, "text": "so here we"}, {"start": 121.88, "duration": 9.84, "text": "have a + b s is equal to here we have 4"}, {"start": 127.52, "duration": 8.16, "text": "+ 4 and that's equal 8 so here we take"}, {"start": 131.72, "duration": 8.44, "text": "root of both sides so we get a + b is"}, {"start": 135.68, "duration": 9.639, "text": "equal to plus or minus < TK 8 and this"}, {"start": 140.16, "duration": 7.96, "text": "is exactly equal to plus orus < TK 4 * 2"}, {"start": 145.319, "duration": 5.2, "text": "you know of 4 is 2 so this is exactly"}, {"start": 148.12, "duration": 7.08, "text": "equal to plus or minus"}, {"start": 150.519, "duration": 6.921, "text": "S 4 that is 2 then < tk2 and this is the"}, {"start": 155.2, "duration": 3.959, "text": "final answer thank you for watching"}, {"start": 157.44, "duration": 7.2, "text": "please do well to subscribe to our"}, {"start": 159.159, "duration": 5.481, "text": "YouTube channel and share our videos"}], "title": "Olympiad Math. Algebra Equations. #maths #olympics", "description": "", "lengthSeconds": 163, "width": 636, "height": 360}
{"video_id": "ATDuxAzyLcE", "transcript": [{"start": 0.28, "duration": 5.44, "text": "hello good day viewers in this video we"}, {"start": 3.439, "duration": 6.561, "text": "have a very nice problem here we're"}, {"start": 5.72, "duration": 8.56, "text": "given that 8 of M is equal to 27 here we"}, {"start": 10.0, "duration": 7.439, "text": "asked to evaluate 4 to the^ of M without"}, {"start": 14.28, "duration": 4.64, "text": "first finding the value of M right"}, {"start": 17.439, "duration": 3.401, "text": "please if you have not subscribed to my"}, {"start": 18.92, "duration": 5.0, "text": "Channel please kindly support this"}, {"start": 20.84, "duration": 6.519, "text": "channel by subscribing please do way to"}, {"start": 23.92, "duration": 8.639, "text": "like comment and share here we're given"}, {"start": 27.359, "duration": 9.761, "text": "that 8 R of M is equal to 27 right so"}, {"start": 32.559, "duration": 7.32, "text": "from here 8 is the same as 2^ of 3 then"}, {"start": 37.12, "duration": 7.04, "text": "here we have a power of M here and this"}, {"start": 39.879, "duration": 8.761, "text": "is equal to 27 right okay now let us"}, {"start": 44.16, "duration": 9.719, "text": "recall this law that a of B into bracket"}, {"start": 48.64, "duration": 9.439, "text": "of C is the same as a of C then of B"}, {"start": 53.879, "duration": 6.561, "text": "right so from here we're going to have 2"}, {"start": 58.079, "duration": 5.961, "text": "of M then R of 3"}, {"start": 60.44, "duration": 5.2, "text": "and this is equal to 27 right from here"}, {"start": 64.04, "duration": 4.48, "text": "we're going to take the cube root of"}, {"start": 65.64, "duration": 7.36, "text": "both sides of this equation so from here"}, {"start": 68.52, "duration": 7.04, "text": "we get 2 of M because from here cube"}, {"start": 73.0, "duration": 4.92, "text": "root of this here we have cube root of"}, {"start": 75.56, "duration": 6.72, "text": "27 you know the cube root and Cube"}, {"start": 77.92, "duration": 7.92, "text": "cancel out so we left 2 of M is equal to"}, {"start": 82.28, "duration": 6.799, "text": "the cube root of 27 that is equal to 3"}, {"start": 85.84, "duration": 7.52, "text": "so from here we find out that 2 of M is"}, {"start": 89.079, "duration": 7.521, "text": "equal to 3 okay now let us consider the"}, {"start": 93.36, "duration": 7.039, "text": "expression we are about to evaluate from"}, {"start": 96.6, "duration": 8.839, "text": "here so here we have 4^ of M you know"}, {"start": 100.399, "duration": 8.08, "text": "this is same as 2^ of 2 then power of M"}, {"start": 105.439, "duration": 6.921, "text": "right so this is equal to using this law"}, {"start": 108.479, "duration": 8.081, "text": "here so this going to be 2^ of M then of"}, {"start": 112.36, "duration": 6.679, "text": "2 and from here we have 2 of m = 3 so"}, {"start": 116.56, "duration": 5.839, "text": "this is going to be 3^ of 2 and that is"}, {"start": 119.039, "duration": 5.0, "text": "equal to 9 and this is the final answer"}, {"start": 122.399, "duration": 4.201, "text": "thank you for watching please kindly"}, {"start": 124.039, "duration": 6.241, "text": "subscribe to my channel like comment and"}, {"start": 126.6, "duration": 3.68, "text": "share bye-bye"}], "title": "Interesting exponential problem.", "description": "", "lengthSeconds": 129, "width": 640, "height": 360}
{"video_id": "itDmmIEaZZc", "transcript": [{"start": 0.919, "duration": 4.321, "text": "hello my wonderful viewers here we're"}, {"start": 2.879, "duration": 4.68, "text": "going to solve this cubic equation we"}, {"start": 5.24, "duration": 5.8, "text": "have X Cub ="}, {"start": 7.559, "duration": 6.521, "text": "16x now let's get started we have a x"}, {"start": 11.04, "duration": 6.36, "text": "cubed then we're going to subtract 16 x"}, {"start": 14.08, "duration": 4.959, "text": "right and this is equal to Z so from"}, {"start": 17.4, "duration": 4.16, "text": "here we have a common factor right and"}, {"start": 19.039, "duration": 7.48, "text": "that is X so we factor out X and we're"}, {"start": 21.56, "duration": 8.68, "text": "left with X2 - 16 this is equal to zero"}, {"start": 26.519, "duration": 7.72, "text": "now from here this implies that here x x"}, {"start": 30.24, "duration": 8.88, "text": "is = 0 right or from the bracket we have"}, {"start": 34.239, "duration": 8.081, "text": "x^2 - 16 = 0 now let's go ahead and"}, {"start": 39.12, "duration": 5.599, "text": "solve for x in this equation here we"}, {"start": 42.32, "duration": 4.759, "text": "already found a solution for X right and"}, {"start": 44.719, "duration": 5.641, "text": "that is xal 0 so from here we're going"}, {"start": 47.079, "duration": 6.881, "text": "to write this equation as x^2 - 16 is"}, {"start": 50.36, "duration": 5.039, "text": "the same as 4 2 and this is equal to0"}, {"start": 53.96, "duration": 4.72, "text": "now we're going to make use of the"}, {"start": 55.399, "duration": 8.8, "text": "difference of two squares formula here"}, {"start": 58.68, "duration": 11.2, "text": "right so we have a 2 - b 2 this is equal"}, {"start": 64.199, "duration": 10.96, "text": "to a + b * a - b right so make use of"}, {"start": 69.88, "duration": 8.64, "text": "that here we have x + 4 right then MTI x"}, {"start": 75.159, "duration": 8.081, "text": "- 4 this is equal to Z so from here we"}, {"start": 78.52, "duration": 8.559, "text": "have that x + 4 is = 0 the other bracket"}, {"start": 83.24, "duration": 7.519, "text": "here x - 4 = 0 so we have two linear"}, {"start": 87.079, "duration": 7.121, "text": "equations here and from here x = to4 4"}, {"start": 90.759, "duration": 8.081, "text": "from the other side here x = 4 and here"}, {"start": 94.2, "duration": 8.48, "text": "we found the solutions that x is = 0 we"}, {"start": 98.84, "duration": 8.48, "text": "also have that X is = to -4 and we also"}, {"start": 102.68, "duration": 7.2, "text": "have that X is = to 4 and this are the"}, {"start": 107.32, "duration": 5.119, "text": "solutions thank you for watching please"}, {"start": 109.88, "duration": 6.76, "text": "kindly subscribe to my channel also like"}, {"start": 112.439, "duration": 4.201, "text": "comment and share bye-bye"}], "title": "Nice cubic equation | #maths #mathematics", "description": "", "lengthSeconds": 115, "width": 576, "height": 360}
{"video_id": "kmdrsUwp78A", "transcript": [{"start": 0.68, "duration": 5.28, "text": "hello welcome back once again to my"}, {"start": 3.08, "duration": 4.28, "text": "YouTube Channel please kindly support"}, {"start": 5.96, "duration": 4.52, "text": "this YouTube"}, {"start": 7.36, "duration": 5.72, "text": "channel by subscribing to this YouTube"}, {"start": 10.48, "duration": 6.16, "text": "Channel Please Subscribe and share this"}, {"start": 13.08, "duration": 7.24, "text": "video here we have this limit problem"}, {"start": 16.64, "duration": 9.6, "text": "limit as x ts to Infinity of this"}, {"start": 20.32, "duration": 8.279, "text": "function here 1 / x^ of 1 all X right"}, {"start": 26.24, "duration": 4.279, "text": "okay now before we proceed let us take"}, {"start": 28.599, "duration": 3.12, "text": "note of this follow"}, {"start": 30.519, "duration": 5.36, "text": "algebraic"}, {"start": 31.719, "duration": 11.761, "text": "identity when we have a"}, {"start": 35.879, "duration": 11.441, "text": "m of n so this is same as a of n over M"}, {"start": 43.48, "duration": 6.32, "text": "of n right okay right over here we"}, {"start": 47.32, "duration": 9.239, "text": "have 1"}, {"start": 49.8, "duration": 11.88, "text": "x root of 1 x so this is same as 1^ 1 /"}, {"start": 56.559, "duration": 11.121, "text": "X all over x^ of 1 all X you know this"}, {"start": 61.68, "duration": 9.439, "text": "is equal to 1 1 / X is 1 then all X of"}, {"start": 67.68, "duration": 8.04, "text": "1x right okay let us take note of this"}, {"start": 71.119, "duration": 9.64, "text": "property again a power of 1 / n is the"}, {"start": 75.72, "duration": 10.359, "text": "same as the nend root of a right so this"}, {"start": 80.759, "duration": 10.601, "text": "actually mean that 1 over x^ 1 / X is"}, {"start": 86.079, "duration": 10.68, "text": "going to be 1 all over the X root of X"}, {"start": 91.36, "duration": 8.96, "text": "right so now our limit that is limit as"}, {"start": 96.759, "duration": 6.601, "text": "X is approaching Infinity of 1 x 1"}, {"start": 100.32, "duration": 6.72, "text": "= limit as X is approaching Infinity of"}, {"start": 103.36, "duration": 7.96, "text": "1 xot x you know we can say rewrite this"}, {"start": 107.04, "duration": 7.84, "text": "as follows 1 all over the"}, {"start": 111.32, "duration": 8.079, "text": "limit as X is approaching Infinity of"}, {"start": 114.88, "duration": 5.839, "text": "the X root of x so if you check our"}, {"start": 119.399, "duration": 4.801, "text": "recent"}, {"start": 120.719, "duration": 6.76, "text": "videos you will see where we find this"}, {"start": 124.2, "duration": 7.28, "text": "limit limit as X is approaching Infinity"}, {"start": 127.479, "duration": 7.241, "text": "of xot of X check our previous videos"}, {"start": 131.48, "duration": 7.119, "text": "this limit is equal to one right so this"}, {"start": 134.72, "duration": 7.76, "text": "is going to be 1 over 1 and this is"}, {"start": 138.599, "duration": 5.681, "text": "equal to 1 and this is the right answer"}, {"start": 142.48, "duration": 3.6, "text": "thank you for watching do wa to"}, {"start": 144.28, "duration": 5.64, "text": "subscribe to this YouTube channel and"}, {"start": 146.08, "duration": 3.84, "text": "share our videos"}], "title": "An interesting limit. How to deal?", "description": "", "lengthSeconds": 148, "width": 636, "height": 360}
{"video_id": "FXq4yG-nGn0", "transcript": [{"start": 0.919, "duration": 4.521, "text": "hello good day viewers welcome back once"}, {"start": 3.08, "duration": 5.4, "text": "again please kindly support my channel"}, {"start": 5.44, "duration": 6.359, "text": "by subscribing also like comment and"}, {"start": 8.48, "duration": 6.48, "text": "share today we are going to solve for x"}, {"start": 11.799, "duration": 5.601, "text": "in the following exponential equation"}, {"start": 14.96, "duration": 5.6, "text": "here we have x to the power of x to the^"}, {"start": 17.4, "duration": 6.6, "text": "of 4 this is equal to 64 here we're"}, {"start": 20.56, "duration": 10.199, "text": "going to find the value of x right so"}, {"start": 24.0, "duration": 8.719, "text": "here we have X x^ 4 this is equal to 64"}, {"start": 30.759, "duration": 5.001, "text": "now let us raise both sides of this"}, {"start": 32.719, "duration": 7.0, "text": "equation to the power of 4 and here we"}, {"start": 35.76, "duration": 6.76, "text": "raise this to the power of 4 right so"}, {"start": 39.719, "duration": 5.961, "text": "from here we're going to have you know"}, {"start": 42.52, "duration": 8.76, "text": "this property a to the^ of"}, {"start": 45.68, "duration": 8.399, "text": "b^ of C this is equal to a of B * C"}, {"start": 51.28, "duration": 6.04, "text": "right so from here the power of X here"}, {"start": 54.079, "duration": 6.601, "text": "is x ^ of 4 so we're going to multiply"}, {"start": 57.32, "duration": 5.6, "text": "the powers together that is x^ 4 * 4"}, {"start": 60.68, "duration": 9.759, "text": "right so here we"}, {"start": 62.92, "duration": 10.36, "text": "have X of x^ of 4 multip by 4 right so"}, {"start": 70.439, "duration": 7.081, "text": "this is equal to you know in the bracket"}, {"start": 73.28, "duration": 9.12, "text": "here 64 is the same as 8 to the^ of 2"}, {"start": 77.52, "duration": 7.559, "text": "and here ra power of 4 right okay now"}, {"start": 82.4, "duration": 7.0, "text": "let us recall this property again when"}, {"start": 85.079, "duration": 5.561, "text": "we have a to the power of B * C this is"}, {"start": 89.4, "duration": 6.12, "text": "same as"}, {"start": 90.64, "duration": 9.159, "text": "a power of C ra power of B right so from"}, {"start": 95.52, "duration": 10.199, "text": "here we're going to have X power of this"}, {"start": 99.799, "duration": 8.28, "text": "four then power of x to the^ 4 right so"}, {"start": 105.719, "duration": 4.04, "text": "this is equal to you know from here"}, {"start": 108.079, "duration": 7.601, "text": "we're going to multiply these Powers"}, {"start": 109.759, "duration": 10.521, "text": "together so we have 8 power of 2 * 4"}, {"start": 115.68, "duration": 6.16, "text": "right so from here we have"}, {"start": 120.28, "duration": 7.36, "text": "x^"}, {"start": 121.84, "duration": 8.039, "text": "4^ x^ 4 this is equal to 8 power of 8"}, {"start": 127.64, "duration": 5.92, "text": "right so you know we can put this in"}, {"start": 129.879, "duration": 8.841, "text": "bracket now we can see what is going on"}, {"start": 133.56, "duration": 9.36, "text": "here we have here x^ of 4 ra of x^ 4"}, {"start": 138.72, "duration": 10.12, "text": "this is equal to 8 to^ of 8 so from here"}, {"start": 142.92, "duration": 10.2, "text": "this implies that x^ 4 is equal to 8"}, {"start": 148.84, "duration": 6.28, "text": "right so from here we have x^ 4 this is"}, {"start": 153.12, "duration": 3.759, "text": "equal to 8 you know from here we're"}, {"start": 155.12, "duration": 5.199, "text": "going to take the fourth root of both"}, {"start": 156.879, "duration": 7.64, "text": "sides so here we get X is equal to the 4"}, {"start": 160.319, "duration": 5.761, "text": "root of 8 and this is the exact answer"}, {"start": 164.519, "duration": 4.761, "text": "but when we use calculator to"}, {"start": 166.08, "duration": 4.879, "text": "approximate it we actually have"}, {"start": 169.28, "duration": 9.92, "text": "1."}, {"start": 170.959, "duration": 10.28, "text": "63 sorry 681 I mean 6 81 7 99 3 and this"}, {"start": 179.2, "duration": 4.039, "text": "is the right answer thank you for"}, {"start": 181.239, "duration": 4.56, "text": "watching please kindly subscribe to my"}, {"start": 183.239, "duration": 5.56, "text": "channel also like comment and share"}, {"start": 185.799, "duration": 3.0, "text": "bye-bye"}], "title": "Olympiad Math | Another interesting exponential equation. #education #math #mathematics", "description": "", "lengthSeconds": 187, "width": 640, "height": 360}
{"video_id": "_mjY6-jJjTI", "transcript": [{"start": 0.539, "duration": 3.36, "text": "good viewers welcome back to my Channel"}, {"start": 2.94, "duration": 2.94, "text": "today"}, {"start": 3.899, "duration": 4.561, "text": "we have the following"}, {"start": 5.88, "duration": 5.639, "text": "functional equation"}, {"start": 8.46, "duration": 4.799, "text": "so we have F of seven all over 2 raised"}, {"start": 11.519, "duration": 6.241, "text": "to power x minus one"}, {"start": 13.259, "duration": 9.001, "text": "equals 10 over x squared plus one then"}, {"start": 17.76, "duration": 6.3, "text": "we're going to evaluate F of one okay"}, {"start": 22.26, "duration": 3.359, "text": "now"}, {"start": 24.06, "duration": 4.979, "text": "remember that"}, {"start": 25.619, "duration": 7.861, "text": "we are going to look for a real value of"}, {"start": 29.039, "duration": 5.52, "text": "x such that when you substitute into X"}, {"start": 33.48, "duration": 3.599, "text": "here"}, {"start": 34.559, "duration": 6.961, "text": "this inside this bracket here becomes"}, {"start": 37.079, "duration": 5.82, "text": "one so then the responding value here at"}, {"start": 41.52, "duration": 4.8, "text": "the right hand side should be the value"}, {"start": 42.899, "duration": 6.5, "text": "of f of one okay so in this case we'll"}, {"start": 46.32, "duration": 7.259, "text": "be equating this to be equal to one okay"}, {"start": 49.399, "duration": 7.721, "text": "so we're going to say let's 7 all over"}, {"start": 53.579, "duration": 5.101, "text": "2 raised to power x minus 1 equals one"}, {"start": 57.12, "duration": 5.279, "text": "okay"}, {"start": 58.68, "duration": 6.78, "text": "now put this over one then from here we"}, {"start": 62.399, "duration": 5.821, "text": "cross multiply so here we have two"}, {"start": 65.46, "duration": 6.979, "text": "raised power x minus one times one"}, {"start": 68.22, "duration": 8.219, "text": "so this gives us 2 raised to x minus one"}, {"start": 72.439, "duration": 6.521, "text": "equals seven times one so that's seven"}, {"start": 76.439, "duration": 4.621, "text": "so you know this negative one crossing"}, {"start": 78.96, "duration": 3.299, "text": "over so we have two raised power x"}, {"start": 81.06, "duration": 5.04, "text": "equals"}, {"start": 82.259, "duration": 6.781, "text": "eight okay now we have here two raised"}, {"start": 86.1, "duration": 6.18, "text": "by x equals eight is the same as two"}, {"start": 89.04, "duration": 6.539, "text": "over two two respiratory so from here we"}, {"start": 92.28, "duration": 5.82, "text": "get x equals three okay"}, {"start": 95.579, "duration": 8.301, "text": "so we're going to substitute into X here"}, {"start": 98.1, "duration": 5.78, "text": "as three so now we have F of"}, {"start": 105.92, "duration": 8.86, "text": "minus one is equal to 10"}, {"start": 110.1, "duration": 6.36, "text": "over three respirator plus one"}, {"start": 114.78, "duration": 3.92, "text": "okay"}, {"start": 116.46, "duration": 6.9, "text": "so from here we have F of"}, {"start": 118.7, "duration": 5.919, "text": "7 over 8 minus one close bracket is"}, {"start": 123.36, "duration": 4.259, "text": "equal to"}, {"start": 124.619, "duration": 4.86, "text": "10 all over you know here"}, {"start": 127.619, "duration": 7.141, "text": "nine"}, {"start": 129.479, "duration": 8.101, "text": "plus one so here we have F of"}, {"start": 134.76, "duration": 4.74, "text": "you know here we have H minus one as the"}, {"start": 137.58, "duration": 3.12, "text": "denominator here so that is seven and"}, {"start": 139.5, "duration": 4.14, "text": "seven seven"}, {"start": 140.7, "duration": 6.36, "text": "by seven is one so we have F of one is"}, {"start": 143.64, "duration": 5.16, "text": "equal to ten all over ten and this is"}, {"start": 147.06, "duration": 4.44, "text": "equal to one and this is the right"}, {"start": 148.8, "duration": 5.939, "text": "answer to the question please if you"}, {"start": 151.5, "duration": 5.58, "text": "enjoyed this video please do well to"}, {"start": 154.739, "duration": 4.261, "text": "subscribe please"}, {"start": 157.08, "duration": 5.36, "text": "subscribe to our channel for more"}, {"start": 159.0, "duration": 3.44, "text": "interesting videos thank you"}], "title": "A nice functional equation. f(1) = ?  Here is a unique way to solve this problem", "description": "", "lengthSeconds": 161, "width": 636, "height": 360}
{"video_id": "0T9fmJ5o6SY", "transcript": [{"start": 0.599, "duration": 4.601, "text": "hello welcome back once again today we"}, {"start": 3.159, "duration": 7.36, "text": "have this interesting exponential"}, {"start": 5.2, "duration": 7.559, "text": "equation we have 3 ^ x + 3^ x + 3^ x"}, {"start": 10.519, "duration": 5.281, "text": "this is equal to 1 and here we'll be"}, {"start": 12.759, "duration": 5.721, "text": "solving for the value of x now let's get"}, {"start": 15.8, "duration": 7.16, "text": "started from the left hand side we have"}, {"start": 18.48, "duration": 10.039, "text": "3 to the power of x + 3 to the power of"}, {"start": 22.96, "duration": 8.239, "text": "x + 3 to the^ of X this is equal to 1"}, {"start": 28.519, "duration": 6.641, "text": "okay now 3 to^ x is common here so let's"}, {"start": 31.199, "duration": 12.761, "text": "factor that out so we have 3 ^ of X into"}, {"start": 35.16, "duration": 15.559, "text": "bracket 3 ^ x / 3^ X is 1 + 3^ x / 3^ X"}, {"start": 43.96, "duration": 9.52, "text": "is 1 + 3^ x / 3^ X is 1 right and here"}, {"start": 50.719, "duration": 6.561, "text": "we have this is equal to 1 okay so from"}, {"start": 53.48, "duration": 6.96, "text": "here we have 3 to^ of X multiply by here"}, {"start": 57.28, "duration": 5.64, "text": "the sum of this one one and 1 is"}, {"start": 60.44, "duration": 4.48, "text": "right and here we have this is equal to"}, {"start": 62.92, "duration": 6.8, "text": "one now let's go ahead and divide both"}, {"start": 64.92, "duration": 9.44, "text": "sides by three and here by 3 so this"}, {"start": 69.72, "duration": 7.079, "text": "cancel this and here we have 3^ X is ="}, {"start": 74.36, "duration": 4.36, "text": "to now from the right we have 1 all over"}, {"start": 76.799, "duration": 5.68, "text": "3 remember the power of this three here"}, {"start": 78.72, "duration": 9.759, "text": "is one right now let's take note of this"}, {"start": 82.479, "duration": 10.6, "text": "1 / a to the power of n is = to a of n"}, {"start": 88.479, "duration": 7.28, "text": "so here this will be 3 of1 now we can"}, {"start": 93.079, "duration": 5.121, "text": "see that we have equal bases right so we"}, {"start": 95.759, "duration": 7.881, "text": "can equate the power and from here this"}, {"start": 98.2, "duration": 8.04, "text": "imply that X is = to -1 and this is the"}, {"start": 103.64, "duration": 5.2, "text": "correct answer thank you for watching"}, {"start": 106.24, "duration": 7.28, "text": "please kindly subscribe to my channel"}, {"start": 108.84, "duration": 4.68, "text": "also like comment and share bye-bye"}], "title": "Interesting exponential equation", "description": "", "lengthSeconds": 112, "width": 640, "height": 360}
{"video_id": "Hx9jeICU0s8", "transcript": [{"start": 0.32, "duration": 6.959, "text": "hello good day welcome back to another"}, {"start": 2.2, "duration": 8.8, "text": "video here we're given that x + 6 * 9 -"}, {"start": 7.279, "duration": 9.401, "text": "x is = to 5 and right over here we're"}, {"start": 11.0, "duration": 7.56, "text": "going to evaluate x + 6 ^ 2 + 9 - x ^ 2"}, {"start": 16.68, "duration": 4.2, "text": "please kindly support my channel by"}, {"start": 18.56, "duration": 4.84, "text": "subscribing to this Channel please also"}, {"start": 20.88, "duration": 5.639, "text": "like comment and share here we're going"}, {"start": 23.4, "duration": 7.279, "text": "to use this identity that a"}, {"start": 26.519, "duration": 11.001, "text": "2 + b 2 is equal to"}, {"start": 30.679, "duration": 14.481, "text": "a + b 2 - 2 a b right so right"}, {"start": 37.52, "duration": 9.64, "text": "here x + 6 the^ of 2 + 9 - x to the^ of"}, {"start": 45.16, "duration": 8.559, "text": "2 is going to be equal"}, {"start": 47.16, "duration": 10.28, "text": "to the sum of x + 6 and 9 - x x + 6 then"}, {"start": 53.719, "duration": 6.721, "text": "plus 9 - x right"}, {"start": 57.44, "duration": 5.599, "text": "squar then - 2"}, {"start": 60.44, "duration": 8.719, "text": "then the product of x + 6 and 9 - x that"}, {"start": 63.039, "duration": 8.44, "text": "is x + 6 and 9 - x right that is making"}, {"start": 69.159, "duration": 4.561, "text": "use of this identity here so this going"}, {"start": 71.479, "duration": 5.64, "text": "to be equal to you know here x and this"}, {"start": 73.72, "duration": 4.56, "text": "negative X cancel out here we have 6 + 9"}, {"start": 77.119, "duration": 2.68, "text": "and that is"}, {"start": 78.28, "duration": 8.08, "text": "15"}, {"start": 79.799, "duration": 9.481, "text": "s - 2 * the produ of x + 69 - x is 5 as"}, {"start": 86.36, "duration": 6.6, "text": "given so here we have five so this is"}, {"start": 89.28, "duration": 7.68, "text": "going to be equal to 15 squ from here so"}, {"start": 92.96, "duration": 7.36, "text": "that is going to give us"}, {"start": 96.96, "duration": 7.759, "text": "225 so here we have 2"}, {"start": 100.32, "duration": 7.159, "text": "25 then - 2 * 5 that is 10 and from here"}, {"start": 104.719, "duration": 6.281, "text": "we get"}, {"start": 107.479, "duration": 5.041, "text": "215 and this is the right answer thank"}, {"start": 111.0, "duration": 3.28, "text": "you for watching please do wor to"}, {"start": 112.52, "duration": 6.12, "text": "subscribe to my Channel please like"}, {"start": 114.28, "duration": 4.36, "text": "comment and share bye-bye"}], "title": "If and only if you know the trick.", "description": "", "lengthSeconds": 117, "width": 640, "height": 360}
{"video_id": "geiAEn0MdTs", "transcript": [{"start": 0.799, "duration": 4.761, "text": "hello good day viewers welcome back once"}, {"start": 3.08, "duration": 5.24, "text": "again to another video please subscribe"}, {"start": 5.56, "duration": 4.84, "text": "to this channel also like comment and"}, {"start": 8.32, "duration": 5.04, "text": "here here we're given the following"}, {"start": 10.4, "duration": 6.959, "text": "Olympia M problem here we're given that"}, {"start": 13.36, "duration": 6.72, "text": "x + 1 /x is = 2 right over here we ask"}, {"start": 17.359, "duration": 10.08, "text": "to evaluate x ^ of"}, {"start": 20.08, "duration": 9.359, "text": "2024 - 1 / x ^ of 2 and 24 right okay"}, {"start": 27.439, "duration": 5.041, "text": "carefully looking at this equation one"}, {"start": 29.439, "duration": 8.681, "text": "can easily say that the value of x is 1"}, {"start": 32.48, "duration": 10.52, "text": "right because here 1 + 1 / 1 is = 2"}, {"start": 38.12, "duration": 7.48, "text": "right so another way is to manipulate"}, {"start": 43.0, "duration": 5.68, "text": "this equation so by multiplying both"}, {"start": 45.6, "duration": 9.36, "text": "sides by X we"}, {"start": 48.68, "duration": 11.679, "text": "get x2 + 1 is = 2X and from here we get"}, {"start": 54.96, "duration": 7.399, "text": "x^2 - 2x + 1 this is equal to Z and here"}, {"start": 60.359, "duration": 5.241, "text": "this is a perfect square that is the"}, {"start": 62.359, "duration": 6.721, "text": "square of x -1 which is equal to Z from"}, {"start": 65.6, "duration": 6.24, "text": "here we we we can see again that X is"}, {"start": 69.08, "duration": 6.32, "text": "equal to 1 right so we can easily get"}, {"start": 71.84, "duration": 10.04, "text": "the value of this which is x^ of"}, {"start": 75.4, "duration": 9.84, "text": "2024 - 1 / x^ 2024 which is equal to 1 -"}, {"start": 81.88, "duration": 5.44, "text": "1 / 1 and this is equal to zero and this"}, {"start": 85.24, "duration": 4.199, "text": "is the right answer thank you for"}, {"start": 87.32, "duration": 4.479, "text": "watching please do wa to subscribe to"}, {"start": 89.439, "duration": 5.36, "text": "this channel also like comment and share"}, {"start": 91.799, "duration": 3.0, "text": "bye-bye"}], "title": "Interesting and easy problem.", "description": "", "lengthSeconds": 93, "width": 640, "height": 360}
{"video_id": "X9GdVeXx8FM", "transcript": [{"start": 0.719, "duration": 4.481, "text": "hello welcome back once again today"}, {"start": 3.04, "duration": 5.559, "text": "we're going to solve this interesting"}, {"start": 5.2, "duration": 10.519, "text": "exponential equation so from the left we"}, {"start": 8.599, "duration": 10.44, "text": "have 10 to the ^ x + 10 to^ x + 10 to"}, {"start": 15.719, "duration": 5.441, "text": "the^ of X this is equal to"}, {"start": 19.039, "duration": 4.601, "text": "600 so we can see from the left hand"}, {"start": 21.16, "duration": 5.959, "text": "side 10^ x is common so we're going to"}, {"start": 23.64, "duration": 7.039, "text": "factor it out so we have 10^ X into"}, {"start": 27.119, "duration": 7.801, "text": "bracket this / this is 1"}, {"start": 30.679, "duration": 8.2, "text": "plus this ID by this is 1 plus this ID"}, {"start": 34.92, "duration": 8.639, "text": "by this is 1 then this is equal to"}, {"start": 38.879, "duration": 8.481, "text": "600 from here we get 10^ x * 1 + 1 + 1"}, {"start": 43.559, "duration": 6.801, "text": "is 3 then this is equal to"}, {"start": 47.36, "duration": 5.16, "text": "600 so in our next step let's divide"}, {"start": 50.36, "duration": 5.8, "text": "both side by"}, {"start": 52.52, "duration": 6.039, "text": "three yes so this cancels this so from"}, {"start": 56.16, "duration": 5.399, "text": "the left hand side we left with 10^ X so"}, {"start": 58.559, "duration": 5.68, "text": "this will be equal to 600 ID 3 will give"}, {"start": 61.559, "duration": 5.361, "text": "us"}, {"start": 64.239, "duration": 4.761, "text": "200 so from here we're going to log both"}, {"start": 66.92, "duration": 6.64, "text": "sides of this equation so from the left"}, {"start": 69.0, "duration": 9.4, "text": "hand side we have log 10 to the^ of X so"}, {"start": 73.56, "duration": 6.44, "text": "this will be equal to log of 200 so with"}, {"start": 78.4, "duration": 6.759, "text": "the property of log reading we know that"}, {"start": 80.0, "duration": 9.759, "text": "the log of a the^ b is equal to B MTI"}, {"start": 85.159, "duration": 8.32, "text": "log of a right so this will become X MTI"}, {"start": 89.759, "duration": 8.881, "text": "log log of 10 this will be equal to log"}, {"start": 93.479, "duration": 8.32, "text": "of we know 200 is the same as 100 *"}, {"start": 98.64, "duration": 7.799, "text": "2 let us apply this law at the right"}, {"start": 101.799, "duration": 10.68, "text": "hand side we know that log of a MTI B is"}, {"start": 106.439, "duration": 9.04, "text": "equal to log of a + log of B so here"}, {"start": 112.479, "duration": 7.6, "text": "this will be x * the log of 10 this will"}, {"start": 115.479, "duration": 6.56, "text": "be equal to log of 100+"}, {"start": 120.079, "duration": 6.04, "text": "log of"}, {"start": 122.039, "duration": 9.001, "text": "2 so here we have x * the log of 10 this"}, {"start": 126.119, "duration": 7.721, "text": "is equal to log of 100 is same as 10"}, {"start": 131.04, "duration": 7.08, "text": "2+ log of"}, {"start": 133.84, "duration": 6.96, "text": "2 so from here x m the log of 10 this is"}, {"start": 138.12, "duration": 7.199, "text": "equal to you know this property here so"}, {"start": 140.8, "duration": 7.079, "text": "this will be 2 MTI the log of 10 plus"}, {"start": 145.319, "duration": 5.961, "text": "log of 2 so divide both by the"}, {"start": 147.879, "duration": 6.761, "text": "coefficient of x which is log of 10"}, {"start": 151.28, "duration": 6.64, "text": "here by the log of 10 so this will"}, {"start": 154.64, "duration": 6.2, "text": "cancel this and from here we get X isal"}, {"start": 157.92, "duration": 8.28, "text": "to splitting this denominator we get 2"}, {"start": 160.84, "duration": 11.92, "text": "log of 10 / log of 10"}, {"start": 166.2, "duration": 9.28, "text": "plus log of 2 / log of 10 so let us"}, {"start": 172.76, "duration": 6.119, "text": "record this property again we have log"}, {"start": 175.48, "duration": 5.96, "text": "of a / log of B if you want to write it"}, {"start": 178.879, "duration": 6.521, "text": "as a ling as a single logarithm I mean"}, {"start": 181.44, "duration": 6.879, "text": "so this will be log of a to the base B"}, {"start": 185.4, "duration": 4.96, "text": "so we're going to use it here so here"}, {"start": 188.319, "duration": 7.64, "text": "log 10 we cancel log 10 and here we have"}, {"start": 190.36, "duration": 9.84, "text": "our solution X is = to 2 plus use this"}, {"start": 195.959, "duration": 8.0, "text": "here so this will become log 2 to the"}, {"start": 200.2, "duration": 5.959, "text": "base 10 so here is our"}, {"start": 203.959, "duration": 6.041, "text": "solution thank you for watching please"}, {"start": 206.159, "duration": 5.36, "text": "do not forget to subscribe"}, {"start": 210.0, "duration": 4.519, "text": "please"}, {"start": 211.519, "duration": 7.28, "text": "like"}, {"start": 214.519, "duration": 7.561, "text": "comment and please share our videos"}, {"start": 218.799, "duration": 3.281, "text": "thank you"}], "title": "Olympiad Math problem | Can you solve ?", "description": "", "lengthSeconds": 223, "width": 640, "height": 360}
{"video_id": "DQa65MVlDFg", "transcript": [{"start": 1.16, "duration": 4.719, "text": "hello good viewers welcome back once"}, {"start": 3.08, "duration": 7.08, "text": "again in this video we'll be solving for"}, {"start": 5.879, "duration": 9.72, "text": "the value of a * B given that A + B is ="}, {"start": 10.16, "duration": 8.039, "text": "4 and a minus B is = 1 right in the"}, {"start": 15.599, "duration": 6.321, "text": "video I'm going to present two different"}, {"start": 18.199, "duration": 6.561, "text": "methods let's start with the first"}, {"start": 21.92, "duration": 6.32, "text": "method but you know what the first"}, {"start": 24.76, "duration": 7.6, "text": "method is awesome why the second one is"}, {"start": 28.24, "duration": 7.08, "text": "amazing right so here we have a + b this"}, {"start": 32.36, "duration": 5.519, "text": "is equal to 4 actually from here if we"}, {"start": 35.32, "duration": 7.399, "text": "try solving for B in terms of a we get B"}, {"start": 37.879, "duration": 7.121, "text": "is = 4 - A let's keep this in mind then"}, {"start": 42.719, "duration": 5.641, "text": "we have here A minus B this is equal to"}, {"start": 45.0, "duration": 6.12, "text": "1 let's go ahead and solve this equation"}, {"start": 48.36, "duration": 5.519, "text": "simultaneously if we add both equations"}, {"start": 51.12, "duration": 6.88, "text": "B will be eliminated right we have here"}, {"start": 53.879, "duration": 7.801, "text": "a + a which is 2 a this is equal to 4 +"}, {"start": 58.0, "duration": 10.68, "text": "1 that is 5 from here this is telling us"}, {"start": 61.68, "duration": 10.68, "text": "or implying that a is = 5 2 for B we"}, {"start": 68.68, "duration": 7.04, "text": "already have that 4 - A will give us B"}, {"start": 72.36, "duration": 6.56, "text": "which is 4 - 5 over 2 and this will give"}, {"start": 75.72, "duration": 9.359, "text": "us 3 all 2 we can come up with the"}, {"start": 78.92, "duration": 9.76, "text": "solution a * B will be 5 2 * 3 2 and"}, {"start": 85.079, "duration": 6.961, "text": "this will give us 15 all over"}, {"start": 88.68, "duration": 5.04, "text": "4 now let's see how the second method"}, {"start": 92.04, "duration": 4.6, "text": "would"}, {"start": 93.72, "duration": 5.52, "text": "be for the second method I'll be making"}, {"start": 96.64, "duration": 5.519, "text": "use of algebraic identity you guys know"}, {"start": 99.24, "duration": 4.72, "text": "me very well I'm a fan of that we have"}, {"start": 102.159, "duration": 9.681, "text": "to make use of this"}, {"start": 103.96, "duration": 13.24, "text": "property a * B is a + b 2 - a - b 2 all"}, {"start": 111.84, "duration": 8.76, "text": "/ 4 right so from here we have a * B"}, {"start": 117.2, "duration": 5.36, "text": "this is equal to a + b from the original"}, {"start": 120.6, "duration": 6.76, "text": "problem you know that is 4 then we have"}, {"start": 122.56, "duration": 10.36, "text": "4 2 minus a minus B is 1 we have 1 2"}, {"start": 127.36, "duration": 8.28, "text": "then all over 4 we know 4 2 is 16 - 1 2"}, {"start": 132.92, "duration": 6.52, "text": "which is 1 / 4 and here we come up with"}, {"start": 135.64, "duration": 5.72, "text": "the solution again 15 all over 4 and"}, {"start": 139.44, "duration": 3.64, "text": "this is the answer thank you for"}, {"start": 141.36, "duration": 3.92, "text": "watching please kindly subscribe to my"}, {"start": 143.08, "duration": 5.76, "text": "YouTube channel also like comment and"}, {"start": 145.28, "duration": 3.56, "text": "share bye-bye"}], "title": "A junior Olympiad Math | Two different methods | #maths #olympiad", "description": "", "lengthSeconds": 147, "width": 640, "height": 360}
{"video_id": "UZrxRjvkraw", "transcript": [{"start": 1.16, "duration": 4.639, "text": "hello welcome back once again today we"}, {"start": 3.08, "duration": 6.68, "text": "have a form problem on modular"}, {"start": 5.799, "duration": 8.681, "text": "arithmetic here 2 to the^ of 16 is"}, {"start": 9.76, "duration": 8.4, "text": "Congress to what in mod"}, {"start": 14.48, "duration": 6.76, "text": "257 and what does this mean the better"}, {"start": 18.16, "duration": 6.64, "text": "way to interpret this is what is the"}, {"start": 21.24, "duration": 6.32, "text": "remainder when 2 to the^ of 16 is"}, {"start": 24.8, "duration": 2.76, "text": "divided by"}, {"start": 28.599, "duration": 4.681, "text": "257 Okay so"}, {"start": 30.8, "duration": 5.24, "text": "let's take note that"}, {"start": 33.28, "duration": 5.119, "text": "2 to the power"}, {"start": 36.04, "duration": 6.359, "text": "of yes that is"}, {"start": 38.399, "duration": 4.0, "text": "8 is equal to"}, {"start": 42.96, "duration": 10.32, "text": "256 right and 256 is congr"}, {"start": 47.84, "duration": 5.44, "text": "to -1 mod"}, {"start": 53.96, "duration": 8.84, "text": "257 now from this equation here we found"}, {"start": 57.039, "duration": 7.4, "text": "out that 2 to the^ of 8 is Congress to"}, {"start": 62.8, "duration": 4.44, "text": "-1"}, {"start": 64.439, "duration": 6.121, "text": "mod"}, {"start": 67.24, "duration": 8.28, "text": "257 now let's raise both side to the"}, {"start": 70.56, "duration": 12.919, "text": "second power that will be 2 ^ of 8 SAR"}, {"start": 75.52, "duration": 7.959, "text": "this will be Congress to -1 SAR mod"}, {"start": 83.88, "duration": 8.84, "text": "257 Now using this power rule a to the"}, {"start": 87.92, "duration": 9.08, "text": "power of B of C this this = a ^ B * C"}, {"start": 92.72, "duration": 6.32, "text": "here we become 2 ^ 8 * 2 which is 16"}, {"start": 97.0, "duration": 4.159, "text": "this will be Congress to remember this"}, {"start": 99.04, "duration": 7.119, "text": "power here is given so this will result"}, {"start": 101.159, "duration": 8.081, "text": "to positive one which is in mode"}, {"start": 106.159, "duration": 6.881, "text": "257 therefore the remainder when 2 to"}, {"start": 109.24, "duration": 8.28, "text": "the^ of 16 is ID"}, {"start": 113.04, "duration": 6.039, "text": "257 is one and this is the right answer"}, {"start": 117.52, "duration": 3.44, "text": "thank you for watching please kindly"}, {"start": 119.079, "duration": 6.281, "text": "subscribe to my channel Channel also"}, {"start": 120.96, "duration": 4.4, "text": "like comment and share bye-bye"}], "title": "Number Theory | Modular Arithmetic |", "description": "", "lengthSeconds": 125, "width": 554, "height": 360}
{"video_id": "XPwpcwF4jOo", "transcript": [{"start": 0.719, "duration": 3.521, "text": "hello good viewers welcome back once"}, {"start": 2.36, "duration": 4.8, "text": "again let's solve this exponential"}, {"start": 4.24, "duration": 7.519, "text": "equation very quickly so here we have"}, {"start": 7.16, "duration": 9.48, "text": "square root of 3 < TK X this is equal to"}, {"start": 11.759, "duration": 10.201, "text": "3 of 1 /un x now let us recall that the"}, {"start": 16.64, "duration": 9.639, "text": "square root of M is same as m^ of 1 /"}, {"start": 21.96, "duration": 5.76, "text": "2 right so here the root of 3^ ofun x"}, {"start": 26.279, "duration": 4.641, "text": "will be same"}, {"start": 27.72, "duration": 8.92, "text": "as 3 of root"}, {"start": 30.92, "duration": 8.4, "text": "x 1 / 2 then this is equal to 3 of 1 /un"}, {"start": 36.64, "duration": 4.599, "text": "x again we have this"}, {"start": 39.32, "duration": 6.399, "text": "property that when we have something"}, {"start": 41.239, "duration": 8.041, "text": "like this we rewrite it as a b * C so"}, {"start": 45.719, "duration": 9.0, "text": "here this will be same as 3un x * 1 2"}, {"start": 49.28, "duration": 9.239, "text": "will beunk x 2 this is equal to 3 of 1"}, {"start": 54.719, "duration": 6.561, "text": "/un x right so now we have the same"}, {"start": 58.519, "duration": 7.441, "text": "basis right so we can equate the powers"}, {"start": 61.28, "duration": 8.56, "text": "so this implies that root of x / 2 is ="}, {"start": 65.96, "duration": 7.159, "text": "1 allun of x from here let's go ahead"}, {"start": 69.84, "duration": 9.279, "text": "and cross multiply root x * root X will"}, {"start": 73.119, "duration": 6.0, "text": "beunk x^2 right then this is equal"}, {"start": 79.36, "duration": 6.04, "text": "to so this is equal to oh sorry"}, {"start": 83.96, "duration": 2.76, "text": "shouldn't that should not be written"}, {"start": 85.4, "duration": 4.24, "text": "like this right but that would be"}, {"start": 86.72, "duration": 3.759, "text": "absolute value so root x * root X from"}, {"start": 89.64, "duration": 2.88, "text": "here"}, {"start": 90.479, "duration": 6.481, "text": "that will give"}, {"start": 92.52, "duration": 8.4, "text": "us root X like this squared right then"}, {"start": 96.96, "duration": 7.08, "text": "this is equal to 2 * 1 that will give us"}, {"start": 100.92, "duration": 6.559, "text": "2 so from"}, {"start": 104.04, "duration": 6.439, "text": "here the square and the root sign cancel"}, {"start": 107.479, "duration": 6.441, "text": "out right so here we're left with X isal"}, {"start": 110.479, "duration": 5.24, "text": "to 2 and this is the right answer thank"}, {"start": 113.92, "duration": 4.199, "text": "you for watching please kindly subscribe"}, {"start": 115.719, "duration": 5.921, "text": "to my channel also like comment and"}, {"start": 118.119, "duration": 3.521, "text": "share bye-bye"}], "title": "Interesting equation with radical powers | #maths #mathematics", "description": "", "lengthSeconds": 120, "width": 640, "height": 360}
{"video_id": "-_N006WhLsw", "transcript": [{"start": 0.96, "duration": 3.24, "text": "hello good viewers welcome back once"}, {"start": 2.639, "duration": 7.08, "text": "again today we're going to solve the"}, {"start": 4.2, "duration": 10.2, "text": "given equation x^ 1X = < tk2 here we are"}, {"start": 9.719, "duration": 10.32, "text": "going to solve for positive integers X"}, {"start": 14.4, "duration": 10.76, "text": "right okay here we have x^ 1"}, {"start": 20.039, "duration": 9.16, "text": "/x this is equal to < TK 2 we have here"}, {"start": 25.16, "duration": 9.04, "text": "x^ of 1 /x this is equal to < tk2 is the"}, {"start": 29.199, "duration": 9.68, "text": "same as 2 to^ of 1 / 2 now we can see"}, {"start": 34.2, "duration": 6.12, "text": "from here that obviously X is = 2 is a"}, {"start": 38.879, "duration": 5.121, "text": "solution"}, {"start": 40.32, "duration": 7.12, "text": "right okay now let's try manipulating"}, {"start": 44.0, "duration": 7.399, "text": "this equation to find out if we have"}, {"start": 47.44, "duration": 6.72, "text": "more positive integers of X that will"}, {"start": 51.399, "duration": 5.401, "text": "satisfy this equation so from here I'm"}, {"start": 54.16, "duration": 7.559, "text": "going to raise both sides to the power"}, {"start": 56.8, "duration": 9.4, "text": "of the LCM of X and 2 and that will be"}, {"start": 61.719, "duration": 9.521, "text": "2x right and from the right we have a 2"}, {"start": 66.2, "duration": 9.12, "text": "power of 1 / 2 to the power of 2"}, {"start": 71.24, "duration": 6.96, "text": "x now from here with this property a to"}, {"start": 75.32, "duration": 7.839, "text": "the^ of B to the power C this is equal"}, {"start": 78.2, "duration": 8.48, "text": "to a^ B * C now from here 1 x * 2X x"}, {"start": 83.159, "duration": 6.361, "text": "will cancel X and we have X the^ of 2"}, {"start": 86.68, "duration": 7.84, "text": "and this will be equal to 1 2 * 2 X2 we"}, {"start": 89.52, "duration": 8.559, "text": "cancel 2 and we have 2 to the^ of X okay"}, {"start": 94.52, "duration": 7.559, "text": "now from here we can actually make use"}, {"start": 98.079, "duration": 7.08, "text": "of the lambat W function but we have"}, {"start": 102.079, "duration": 8.161, "text": "some obvious Solutions here we all know"}, {"start": 105.159, "duration": 8.121, "text": "that 2^ 2 = 2^ 2 which we have found"}, {"start": 110.24, "duration": 8.919, "text": "earlier that x = 2 and again we know"}, {"start": 113.28, "duration": 9.479, "text": "that 4^ 2 = 2^ 4 which is an obvious"}, {"start": 119.159, "duration": 7.6, "text": "solution from here here that x ="}, {"start": 122.759, "duration": 6.36, "text": "4 from here we have the two positive"}, {"start": 126.759, "duration": 8.521, "text": "solutions that is positive integers for"}, {"start": 129.119, "duration": 10.2, "text": "X so we have x = 2 and we have x = 4 and"}, {"start": 135.28, "duration": 6.08, "text": "these are the two positive integers of X"}, {"start": 139.319, "duration": 5.361, "text": "that will satisfy the given"}, {"start": 141.36, "duration": 6.8, "text": "equation I know most of you guys may ask"}, {"start": 144.68, "duration": 6.04, "text": "if there are more positive integers for"}, {"start": 148.16, "duration": 4.799, "text": "X but when I cross"}, {"start": 150.72, "duration": 4.599, "text": "this equation I found out that the"}, {"start": 152.959, "duration": 6.161, "text": "equation has only two intersection"}, {"start": 155.319, "duration": 6.961, "text": "points which is at two and four and"}, {"start": 159.12, "duration": 5.399, "text": "these are the only two positive integers"}, {"start": 162.28, "duration": 4.12, "text": "that satisfy the equation thank you for"}, {"start": 164.519, "duration": 4.36, "text": "watching please kindly subscribe to my"}, {"start": 166.4, "duration": 5.479, "text": "channel also like comment and share"}, {"start": 168.879, "duration": 3.0, "text": "bye-bye"}], "title": "Interesting exponential equation", "description": "", "lengthSeconds": 170, "width": 568, "height": 360}
{"video_id": "uwUeReG7aEc", "transcript": [{"start": 1.319, "duration": 6.04, "text": "hello welcome back once again today we"}, {"start": 3.639, "duration": 8.641, "text": "have this interesting optic equation x^"}, {"start": 7.359, "duration": 8.44, "text": "8 + x^ 4 is equal to 2 and here we're"}, {"start": 12.28, "duration": 6.36, "text": "going to solve for the real value of x"}, {"start": 15.799, "duration": 8.881, "text": "right so let's get started from the left"}, {"start": 18.64, "duration": 8.399, "text": "hand side we have x to^ 8 + x^ 4 from"}, {"start": 24.68, "duration": 8.84, "text": "the right hand side this is equal to"}, {"start": 27.039, "duration": 9.52, "text": "2 now we have X to the^ 8 + x ^ 4 this"}, {"start": 33.52, "duration": 7.76, "text": "is equal to 2 is same as 1 +"}, {"start": 36.559, "duration": 9.921, "text": "1 so here we write x^ 8 - 1 the first"}, {"start": 41.28, "duration": 9.64, "text": "one here then plus x^ 4us the other one"}, {"start": 46.48, "duration": 9.36, "text": "this is equal to Z Now from here let us"}, {"start": 50.92, "duration": 9.92, "text": "rewrite x^ 8 will be x^ 4 2ar we all"}, {"start": 55.84, "duration": 11.319, "text": "know 4 * 2 is 8 minus here 1 2"}, {"start": 60.84, "duration": 8.68, "text": "then plus x^ 4 - 1 this is equal 0 From"}, {"start": 67.159, "duration": 6.681, "text": "Here let us apply the difference of two"}, {"start": 69.52, "duration": 11.72, "text": "squares formula here remember that a 2 -"}, {"start": 73.84, "duration": 11.639, "text": "b 2 is equal to a + b * Aus"}, {"start": 81.24, "duration": 9.6, "text": "B so from here that will become x ^ 4 +"}, {"start": 85.479, "duration": 10.32, "text": "1 * x^ 4 - 1 then Plus this let's put it"}, {"start": 90.84, "duration": 8.12, "text": "in bracket x^ 4 - 1 this is equal to 0"}, {"start": 95.799, "duration": 8.401, "text": "we can see x ^ 4 - 1 is common so Factor"}, {"start": 98.96, "duration": 8.4, "text": "it out we get x^ 4 - 1 then into bracket"}, {"start": 104.2, "duration": 6.48, "text": "this will be x ^ 4 + 1 left and here +"}, {"start": 107.36, "duration": 5.2, "text": "one left this is equal to Z so we have"}, {"start": 110.68, "duration": 5.6, "text": "these two products which is equal to Z"}, {"start": 112.56, "duration": 10.72, "text": "so this implies from here that either x^"}, {"start": 116.28, "duration": 10.68, "text": "4 - 1 is = 0 or X x^ 4 + 1 + 1 is 2 is"}, {"start": 123.28, "duration": 9.0, "text": "equal 0 so obviously there is no real"}, {"start": 126.96, "duration": 8.24, "text": "solution for X here right yes so no real"}, {"start": 132.28, "duration": 5.8, "text": "solution for X right over here so this"}, {"start": 135.2, "duration": 6.119, "text": "equation here we provide the real"}, {"start": 138.08, "duration": 7.519, "text": "solutions for X so let's solve this"}, {"start": 141.319, "duration": 8.64, "text": "equation so we have x^ 4 - 1 which is"}, {"start": 145.599, "duration": 10.121, "text": "equal 0 so we have this is same as x^ 2"}, {"start": 149.959, "duration": 7.241, "text": "s - 1 same as 1 2 this is equal to Z so"}, {"start": 155.72, "duration": 6.76, "text": "applying the difference of two squares"}, {"start": 157.2, "duration": 7.92, "text": "formula this will be X2 + 1 * x^2 - 1"}, {"start": 162.48, "duration": 4.759, "text": "this is equal to0 so we have these two"}, {"start": 165.12, "duration": 4.839, "text": "products which is equal to 0 so this"}, {"start": 167.239, "duration": 7.881, "text": "also implies from here that either X2 +"}, {"start": 169.959, "duration": 8.161, "text": "1 is = 0 or X2 - 1 is equal to Z and"}, {"start": 175.12, "duration": 6.479, "text": "again no real solution for this equation"}, {"start": 178.12, "duration": 5.92, "text": "here this equation only has imaginary"}, {"start": 181.599, "duration": 5.881, "text": "Solutions so here take one to the other"}, {"start": 184.04, "duration": 6.0, "text": "side we get x\u00b2 is = to 1 take the square"}, {"start": 187.48, "duration": 5.0, "text": "root of both side we get X is equal to"}, {"start": 190.04, "duration": 6.72, "text": "plus or"}, {"start": 192.48, "duration": 6.92, "text": "minus < TK 1 so here we have the only"}, {"start": 196.76, "duration": 7.399, "text": "solution is that only solution only"}, {"start": 199.4, "duration": 7.759, "text": "solutions so plus or minus one right so"}, {"start": 204.159, "duration": 7.401, "text": "therefore the only two real solutions"}, {"start": 207.159, "duration": 7.28, "text": "are X is = to Plus minus"}, {"start": 211.56, "duration": 5.92, "text": "one thank you for watching please kindly"}, {"start": 214.439, "duration": 7.041, "text": "subscribe to my channel also like"}, {"start": 217.48, "duration": 4.0, "text": "comment and share"}], "title": "Harvard University \ud83c\udf93 Entrance Exam | Can you solve ?", "description": "", "lengthSeconds": 224, "width": 428, "height": 240}
{"video_id": "dlRRxRY8VCE", "transcript": [{"start": 0.32, "duration": 4.32, "text": "hello good viewers welcome back once"}, {"start": 2.08, "duration": 5.48, "text": "again to this YouTube Channel please"}, {"start": 4.64, "duration": 5.32, "text": "kindly subscribe to this channel here we"}, {"start": 7.56, "duration": 8.68, "text": "have another interesting exponential"}, {"start": 9.96, "duration": 7.96, "text": "equation 1 / 8 ^ x + 2 is = 32 ^ of X"}, {"start": 16.24, "duration": 6.0, "text": "here we are asked to solve for the value"}, {"start": 17.92, "duration": 7.24, "text": "of x right okay now just like we did in"}, {"start": 22.24, "duration": 5.08, "text": "the previous videos all we need to do is"}, {"start": 25.16, "duration": 5.4, "text": "to make sure we have equal bases from"}, {"start": 27.32, "duration": 6.399, "text": "both sides of this equation so that can"}, {"start": 30.56, "duration": 5.76, "text": "equate the powers right okay having it"}, {"start": 33.719, "duration": 7.401, "text": "known that 1 / 8 is the same as 8 to"}, {"start": 36.32, "duration": 9.68, "text": "the^ of-1 right that is 1 / 8 this is"}, {"start": 41.12, "duration": 8.52, "text": "equal to 8 to the^ of -1 and again 8 is"}, {"start": 46.0, "duration": 5.64, "text": "the same as 2 to the^ of 3 then ra of1"}, {"start": 49.64, "duration": 4.599, "text": "so you know this law we are going to"}, {"start": 51.64, "duration": 7.599, "text": "multiply this three And1 together so we"}, {"start": 54.239, "duration": 6.201, "text": "have 1 / 8 is the same as 2^ of -3 right"}, {"start": 59.239, "duration": 6.121, "text": "and from the right"}, {"start": 60.44, "duration": 8.48, "text": "32 is the same as 2^ 5 okay now the"}, {"start": 65.36, "duration": 6.48, "text": "equation is totally transformed"}, {"start": 68.92, "duration": 11.519, "text": "to 2 ^"}, {"start": 71.84, "duration": 11.08, "text": "-3 power x + 2 this is equal 2^ 5 x so"}, {"start": 80.439, "duration": 7.761, "text": "we can multiply the powers together from"}, {"start": 82.92, "duration": 9.32, "text": "here so we have 2^ -3 into braet x + 2"}, {"start": 88.2, "duration": 6.04, "text": "this is equal to 2 5 * X right so we"}, {"start": 92.24, "duration": 5.28, "text": "have equal bases so we can equate the"}, {"start": 94.24, "duration": 7.96, "text": "powers right from here so we have -3"}, {"start": 97.52, "duration": 8.48, "text": "into braet x + 2 this is equal to 5x so"}, {"start": 102.2, "duration": 8.279, "text": "let us open this bracket we have a -3x -"}, {"start": 106.0, "duration": 6.92, "text": "6 this is equal to 5x right so 5x is"}, {"start": 110.479, "duration": 6.64, "text": "crossing over and -6 is crossing over to"}, {"start": 112.92, "duration": 8.839, "text": "the right hand side so here we get -3x -"}, {"start": 117.119, "duration": 7.0, "text": "5x this is equal to 6 right so from here"}, {"start": 121.759, "duration": 6.48, "text": "we get"}, {"start": 124.119, "duration": 10.161, "text": "-8x this is equal to 6 so divide both"}, {"start": 128.239, "duration": 9.041, "text": "side by 8 and here by 8 so here we get X"}, {"start": 134.28, "duration": 7.039, "text": "is equal to you know this is same as -6"}, {"start": 137.28, "duration": 8.08, "text": "all 8 and here simplifying we"}, {"start": 141.319, "duration": 5.881, "text": "get3 / 4 and this is the right answer"}, {"start": 145.36, "duration": 3.92, "text": "thank you for watching please kindly"}, {"start": 147.2, "duration": 6.119, "text": "subscribe to this channel also like"}, {"start": 149.28, "duration": 4.039, "text": "comment and share bye-bye"}], "title": "Interesting exponential equation.#maths #education", "description": "", "lengthSeconds": 152, "width": 640, "height": 360}
{"video_id": "EesQqAZP7ls", "transcript": [{"start": 0.76, "duration": 4.32, "text": "hello good day viewers welcome back once"}, {"start": 2.8, "duration": 4.44, "text": "again in this video we're going to solve"}, {"start": 5.08, "duration": 6.639, "text": "for the values of X in the following"}, {"start": 7.24, "duration": 8.16, "text": "quatic equation we have x ^ 4 + X2 this"}, {"start": 11.719, "duration": 8.081, "text": "is equal to 20 here let us solve for the"}, {"start": 15.4, "duration": 6.44, "text": "values of X for X to ^ of 4 we are going"}, {"start": 19.8, "duration": 7.84, "text": "to rewrite it"}, {"start": 21.84, "duration": 9.64, "text": "as x^ of 2 power of 2 right because 2 *"}, {"start": 27.64, "duration": 9.239, "text": "2 here is 4 then plus"}, {"start": 31.48, "duration": 7.16, "text": "X2 then - 20 this is equal Z Now from"}, {"start": 36.879, "duration": 7.641, "text": "here let us make the following"}, {"start": 38.64, "duration": 12.8, "text": "substitution we're going to let x^ 2 = y"}, {"start": 44.52, "duration": 9.839, "text": "from here we have y^ 2 + y - 20 this is"}, {"start": 51.44, "duration": 5.439, "text": "equal to 0 we are very lucky that from"}, {"start": 54.359, "duration": 7.561, "text": "here this quadratic equation is"}, {"start": 56.879, "duration": 11.92, "text": "factorizable we have y^2 plus y here we"}, {"start": 61.92, "duration": 10.08, "text": "can write it as 5 y - 4 y here - 20 this"}, {"start": 68.799, "duration": 7.401, "text": "is equal to Zer but my question here is"}, {"start": 72.0, "duration": 10.96, "text": "where is 5 Y and 4 y coming from now the"}, {"start": 76.2, "duration": 10.599, "text": "product of y^2 and -20 is -20 y^ 2 right"}, {"start": 82.96, "duration": 5.799, "text": "now we think about two terms that when"}, {"start": 86.799, "duration": 6.241, "text": "we multiply"}, {"start": 88.759, "duration": 7.081, "text": "them the will give us -2 y^2 and when we"}, {"start": 93.04, "duration": 9.96, "text": "add add them they will give us positive"}, {"start": 95.84, "duration": 13.76, "text": "y these two terms are 5 Y and -4"}, {"start": 103.0, "duration": 9.68, "text": "y so that is where 5 Y and this 4 Y come"}, {"start": 109.6, "duration": 7.839, "text": "from so from here we have a common"}, {"start": 112.68, "duration": 7.439, "text": "factor here Y into braet y + 5 from here"}, {"start": 117.439, "duration": 6.72, "text": "we have a common factor of -4 then we"}, {"start": 120.119, "duration": 9.881, "text": "have y + 5 this is equal to0 so from"}, {"start": 124.159, "duration": 8.041, "text": "here this implies that y - 4 * y + 5"}, {"start": 130.0, "duration": 9.519, "text": "this is equal to 0 from here we have"}, {"start": 132.2, "duration": 13.72, "text": "either y - 4 is = 0 or y + 5 is = 0 so"}, {"start": 139.519, "duration": 11.481, "text": "from here we have y is = 4 or Y is ="}, {"start": 145.92, "duration": 7.679, "text": "to5 now recall again X2 is = y"}, {"start": 151.0, "duration": 9.319, "text": "now when"}, {"start": 153.599, "duration": 9.601, "text": "Y is equal to 4 we get x\u00b2 is = 4 if you"}, {"start": 160.319, "duration": 6.481, "text": "take root of both sides we get X is = to"}, {"start": 163.2, "duration": 6.0, "text": "plus or minus the RO TK of 4 and this we"}, {"start": 166.8, "duration": 4.92, "text": "have plus or minus"}, {"start": 169.2, "duration": 4.679, "text": "2 now when"}, {"start": 171.72, "duration": 8.32, "text": "Y is equal"}, {"start": 173.879, "duration": 8.681, "text": "to5 we get x2 is = -5 if we take root of"}, {"start": 180.04, "duration": 4.119, "text": "both side this implies that X = to Plus"}, {"start": 182.56, "duration": 6.8, "text": "or - <"}, {"start": 184.159, "duration": 7.041, "text": "tk-5 which is equal to plus orus IUN 5"}, {"start": 189.36, "duration": 5.72, "text": "from here we come up with the"}, {"start": 191.2, "duration": 7.84, "text": "solutions we have X is = to plus or"}, {"start": 195.08, "duration": 5.64, "text": "minus 2 so we have two right we have X"}, {"start": 199.04, "duration": 4.88, "text": "is = to"}, {"start": 200.72, "duration": 9.76, "text": "-2 we have X is equal"}, {"start": 203.92, "duration": 10.8, "text": "to positive IUN 5 and we have X isal to"}, {"start": 210.48, "duration": 6.119, "text": "IUN 5 and these are the solutions thank"}, {"start": 214.72, "duration": 4.04, "text": "you for watching please kindly subscribe"}, {"start": 216.599, "duration": 5.761, "text": "to my channel also like comment and"}, {"start": 218.76, "duration": 3.6, "text": "share bye-bye"}], "title": "A Homemade Quartic Equation | Interesting problem | #maths #olympiad", "description": "", "lengthSeconds": 221, "width": 640, "height": 360}
{"video_id": "qIxIr_xvxRo", "transcript": [{"start": 0.64, "duration": 4.36, "text": "hello good viewers welcome back once"}, {"start": 2.24, "duration": 7.08, "text": "again today we're going to solve this"}, {"start": 5.0, "duration": 7.559, "text": "equation this equation looks very simple"}, {"start": 9.32, "duration": 6.279, "text": "actually it is very simple but do not be"}, {"start": 12.559, "duration": 5.521, "text": "surprised that this equation has two"}, {"start": 15.599, "duration": 7.961, "text": "solutions but not just one because of"}, {"start": 18.08, "duration": 9.959, "text": "how simple it looks we have here x 4"}, {"start": 23.56, "duration": 7.28, "text": "Plus 8 all X and this is equal to 3 we"}, {"start": 28.039, "duration": 6.161, "text": "can put all over 1 now we're to multiply"}, {"start": 30.84, "duration": 5.44, "text": "each of the terms from both sides by"}, {"start": 34.2, "duration": 7.76, "text": "something and that something will be the"}, {"start": 36.28, "duration": 8.56, "text": "LM of the whole denominators for x and y"}, {"start": 41.96, "duration": 9.759, "text": "and fortunately the"}, {"start": 44.84, "duration": 15.64, "text": "L of 4 x and 1 is = to 4 x right so here"}, {"start": 51.719, "duration": 14.961, "text": "we have 4X multi by x 4 plus here 4X * 8"}, {"start": 60.48, "duration": 9.04, "text": "X this is = to 4X * 3 / 1 so this 4 and"}, {"start": 66.68, "duration": 7.439, "text": "this 4 x x will give us"}, {"start": 69.52, "duration": 7.32, "text": "X2 then plus this X and this x 4 * 4"}, {"start": 74.119, "duration": 8.201, "text": "here will give us 32 and this is equal"}, {"start": 76.84, "duration": 9.16, "text": "to for x * 3 will give us a 12"}, {"start": 82.32, "duration": 4.759, "text": "x so from here we have x\u00b2 we subtract"}, {"start": 86.0, "duration": 6.479, "text": "this"}, {"start": 87.079, "duration": 8.241, "text": "12x then + 32 this is equal to zero okay"}, {"start": 92.479, "duration": 5.201, "text": "now this is a quadratic equation of"}, {"start": 95.32, "duration": 5.479, "text": "course this quadratic equation is"}, {"start": 97.68, "duration": 4.56, "text": "factorizable we look for two terms that"}, {"start": 100.799, "duration": 5.081, "text": "when we multiply that they will give it"}, {"start": 102.24, "duration": 6.72, "text": "2x2 and when we add them they give -2X"}, {"start": 105.88, "duration": 8.72, "text": "such terms should be both negative right"}, {"start": 108.96, "duration": 6.96, "text": "so we have -8x and -4x so the -2X will"}, {"start": 114.6, "duration": 6.799, "text": "write it in terms"}, {"start": 115.92, "duration": 9.72, "text": "of 8 s and oh in terms of 8 X and -4x"}, {"start": 121.399, "duration": 7.36, "text": "right okay so here this is equal to"}, {"start": 125.64, "duration": 6.0, "text": "Z now we're going to factor out the"}, {"start": 128.759, "duration": 5.041, "text": "common factor here which is X and we"}, {"start": 131.64, "duration": 4.879, "text": "have this and here minus 4 should be"}, {"start": 133.8, "duration": 4.2, "text": "factored out we have x - 8 and this is"}, {"start": 136.519, "duration": 6.8, "text": "equal to"}, {"start": 138.0, "duration": 8.92, "text": "Z so from here we have x - 4 into braet"}, {"start": 143.319, "duration": 7.521, "text": "x - 8 this is equal to 0 so from here"}, {"start": 146.92, "duration": 8.0, "text": "this implies that x - 4 is = 0 or from"}, {"start": 150.84, "duration": 9.16, "text": "here we have x - 8 is = 0 so solving for"}, {"start": 154.92, "duration": 8.12, "text": "x we have x = 4 and we also have this"}, {"start": 160.0, "duration": 5.319, "text": "solution that X isal to 8 and these are"}, {"start": 163.04, "duration": 4.16, "text": "the solutions thank you for watching"}, {"start": 165.319, "duration": 6.401, "text": "please kindly subscribe to my YouTube"}, {"start": 167.2, "duration": 7.52, "text": "channel also like comment and share"}, {"start": 171.72, "duration": 3.0, "text": "bye-bye"}], "title": "Interesting fractional equation that later turned into a polynomial of order 2 |  #mathematics", "description": "", "lengthSeconds": 173, "width": 640, "height": 360}
{"video_id": "VcxgCYAVLWQ", "transcript": [{"start": 0.68, "duration": 4.599, "text": "hello welcome back once again today we"}, {"start": 2.96, "duration": 7.48, "text": "have this interesting exponential"}, {"start": 5.279, "duration": 7.56, "text": "equation with irrational bases otk 2^ X"}, {"start": 10.44, "duration": 5.079, "text": "is = root 3^"}, {"start": 12.839, "duration": 7.641, "text": "X let's divide both sides of this"}, {"start": 15.519, "duration": 4.961, "text": "equation by root of 3 to the power of"}, {"start": 23.16, "duration": 6.32, "text": "X this will cancel out here and we left"}, {"start": 26.0, "duration": 7.04, "text": "it one right now remember we have the"}, {"start": 29.48, "duration": 10.28, "text": "same Powers here let us apply this law"}, {"start": 33.04, "duration": 10.16, "text": "we know a^ m / b^ m is equal to a / B"}, {"start": 39.76, "duration": 8.04, "text": "all to the^ n so therefore this will"}, {"start": 43.2, "duration": 8.359, "text": "become otk of 2 / Ro of 3 all to the"}, {"start": 47.8, "duration": 7.599, "text": "power of X and this is equal to 1 from"}, {"start": 51.559, "duration": 7.84, "text": "here we know this one right over here we"}, {"start": 55.399, "duration": 7.361, "text": "can write it in index power of root of 2"}, {"start": 59.399, "duration": 6.001, "text": "/ 3 remember we need to get the same"}, {"start": 62.76, "duration": 6.8, "text": "base from both sides so here we have"}, {"start": 65.4, "duration": 6.84, "text": "root of 2 / root 3 to the power of X so"}, {"start": 69.56, "duration": 7.64, "text": "this will be equal to we know rooot of 2"}, {"start": 72.24, "duration": 6.8, "text": "/un 3 to the^ of 0 will be equal to 1 so"}, {"start": 77.2, "duration": 4.68, "text": "from here we have the same base so we're"}, {"start": 79.04, "duration": 6.28, "text": "going to equate the power so from here"}, {"start": 81.88, "duration": 5.199, "text": "this implies that X is equal to zero and"}, {"start": 85.32, "duration": 4.24, "text": "this is the right"}, {"start": 87.079, "duration": 4.36, "text": "answer thank you for watching please"}, {"start": 89.56, "duration": 7.4, "text": "kindly please subscribe to my channel"}, {"start": 91.439, "duration": 5.521, "text": "also like comment and share bye-bye"}], "title": "Interesting exponential equation with irrational bases | Can you solve  ?", "description": "", "lengthSeconds": 96, "width": 640, "height": 360}
{"video_id": "B_-_vxd3KGQ", "transcript": [{"start": 1.04, "duration": 4.6, "text": "hello good viewers welcome back once"}, {"start": 2.919, "duration": 6.561, "text": "again today we're going to evaluate this"}, {"start": 5.64, "duration": 8.959, "text": "expression here x^ of"}, {"start": 9.48, "duration": 8.68, "text": "999 given that x^2 - x + 1 is ="}, {"start": 14.599, "duration": 6.321, "text": "0 now let's get into the video before we"}, {"start": 18.16, "duration": 8.64, "text": "proceed let us recall the sum of two"}, {"start": 20.92, "duration": 10.64, "text": "cubes formula we have a cubed + B Cubed"}, {"start": 26.8, "duration": 4.76, "text": "this is equal to a"}, {"start": 32.52, "duration": 18.48, "text": "that is A + B into braet a 2 - A B + b"}, {"start": 42.239, "duration": 13.8, "text": "2 okay now if we let a = x and we let b"}, {"start": 51.0, "duration": 8.32, "text": "= 1 then we obtain a cubed will be X"}, {"start": 56.039, "duration": 6.441, "text": "cubed plus B Cubed will be 1 cubed which"}, {"start": 59.32, "duration": 8.64, "text": "is 1 this is equal"}, {"start": 62.48, "duration": 10.6, "text": "to a + b which is x + 1 into bracket A 2"}, {"start": 67.96, "duration": 8.199, "text": "will be X2 - A which is x * 1 which is x"}, {"start": 73.08, "duration": 6.079, "text": "+ b 2 which is 1 2 which is"}, {"start": 76.159, "duration": 5.92, "text": "1 so we can see from the right hand side"}, {"start": 79.159, "duration": 6.041, "text": "of this equation from here we have X2 -"}, {"start": 82.079, "duration": 6.4, "text": "x + 1 and from the original equation"}, {"start": 85.2, "duration": 6.919, "text": "that is exactly the left hand"}, {"start": 88.479, "duration": 5.96, "text": "side so what is this trying to let us"}, {"start": 92.119, "duration": 7.121, "text": "know this is telling us that when we"}, {"start": 94.439, "duration": 8.64, "text": "multiply x^2 - x + 1 by x +1 this will"}, {"start": 99.24, "duration": 6.479, "text": "automatically turn to X Cub + 1 now"}, {"start": 103.079, "duration": 4.841, "text": "let's go ahead and do that here we have"}, {"start": 105.719, "duration": 8.521, "text": "from the equation"}, {"start": 107.92, "duration": 10.159, "text": "X2 - x + 1 this is equal to0 now let's"}, {"start": 114.24, "duration": 10.68, "text": "multiply x + 1 to both sides of this"}, {"start": 118.079, "duration": 12.641, "text": "equation so here we have x^2 - x + 1 * x"}, {"start": 124.92, "duration": 8.24, "text": "+ 1 this will be equal to 0 MTI x + 1"}, {"start": 130.72, "duration": 7.68, "text": "from the left hand side we all know that"}, {"start": 133.16, "duration": 10.28, "text": "will be X Cub + 1 and this is equal 0"}, {"start": 138.4, "duration": 9.28, "text": "from here this implies that X Cub is = 0"}, {"start": 143.44, "duration": 4.24, "text": "- 1 which is1"}, {"start": 150.319, "duration": 6.121, "text": "what about thetion we are trying to"}, {"start": 153.64, "duration": 9.16, "text": "evaluate luckily"}, {"start": 156.44, "duration": 8.879, "text": "999 is equal to 3 * 3 3 3 so from this"}, {"start": 162.8, "duration": 5.76, "text": "equation here if we raise both sides to"}, {"start": 165.319, "duration": 6.161, "text": "the power of 3 3 3 we will obtain X to"}, {"start": 168.56, "duration": 5.12, "text": "the^ of 9 999 from the left hand side"}, {"start": 171.48, "duration": 7.759, "text": "now let's go ahead and do that we have a"}, {"start": 173.68, "duration": 8.52, "text": "x cubed rais to the power of 3 3 3 this"}, {"start": 179.239, "duration": 8.321, "text": "will corres corly be equal"}, {"start": 182.2, "duration": 8.759, "text": "to1 3 3 3 now let's record this property"}, {"start": 187.56, "duration": 6.959, "text": "a to the^ of B to the^ of C this is"}, {"start": 190.959, "duration": 8.92, "text": "equal to a of B *"}, {"start": 194.519, "duration": 9.401, "text": "C so from here 3 * 3 3 3 will give us x^"}, {"start": 199.879, "duration": 7.72, "text": "of 9 9 9 this is equal to you know the"}, {"start": 203.92, "duration": 6.879, "text": "power here is odd -1 ra the power of odd"}, {"start": 207.599, "duration": 5.28, "text": "will give us -1 and and this is the"}, {"start": 210.799, "duration": 3.761, "text": "right answer thank you for watching"}, {"start": 212.879, "duration": 4.28, "text": "please kindly subscribe to my YouTube"}, {"start": 214.56, "duration": 5.12, "text": "channel also like comment and share"}, {"start": 217.159, "duration": 4.681, "text": "please support my channel by joining my"}, {"start": 219.68, "duration": 4.36, "text": "membership or you can send me a super"}, {"start": 221.84, "duration": 4.959, "text": "tank if you find this video really"}, {"start": 224.04, "duration": 6.559, "text": "interesting and helpful bye-bye see you"}, {"start": 226.799, "duration": 3.8, "text": "in the next video"}], "title": "Olympiad Math | Interesting problem | #maths #olympiad #mathcompetition", "description": "", "lengthSeconds": 230, "width": 640, "height": 360}
{"video_id": "OH51W2povX4", "transcript": [{"start": 0.48, "duration": 7.08, "text": "hello you are welcome here we're going"}, {"start": 2.8, "duration": 7.04, "text": "to find the value of root of X2 + Y2 + 1"}, {"start": 7.56, "duration": 9.879, "text": "given the following two systems of"}, {"start": 9.84, "duration": 10.48, "text": "equations x^2 = 19 x + y y^2 = 19 y + x"}, {"start": 17.439, "duration": 5.16, "text": "let's call this our equation 1 and let's"}, {"start": 20.32, "duration": 5.119, "text": "call this our equation"}, {"start": 22.599, "duration": 6.121, "text": "2 so let's start like this we're going"}, {"start": 25.439, "duration": 6.721, "text": "to subtract equation two from equation"}, {"start": 28.72, "duration": 6.48, "text": "one right so so we have to subtract oh"}, {"start": 32.16, "duration": 7.079, "text": "sorry that is equation 1 minus equation"}, {"start": 35.2, "duration": 7.359, "text": "2 so equation one the left hand side is"}, {"start": 39.239, "duration": 4.441, "text": "X2 minus equation 2 the left hand side"}, {"start": 42.559, "duration": 3.64, "text": "is y^"}, {"start": 43.68, "duration": 9.039, "text": "2 then this will be equal to for the"}, {"start": 46.199, "duration": 8.88, "text": "right hand side so we have 19 x +"}, {"start": 52.719, "duration": 6.401, "text": "Yus the equation 2 the right answer"}, {"start": 55.079, "duration": 5.64, "text": "which is 19 y + x so from here we have"}, {"start": 59.12, "duration": 6.92, "text": "X2"}, {"start": 60.719, "duration": 6.72, "text": "- y^ 2 this equals here 19 x - x will"}, {"start": 66.04, "duration": 6.28, "text": "give us 18"}, {"start": 67.439, "duration": 9.161, "text": "x then here 9 that is y - 19 y that will"}, {"start": 72.32, "duration": 6.159, "text": "give us -8 y so here remember this is"}, {"start": 76.6, "duration": 7.36, "text": "the difference of two squares so we"}, {"start": 78.479, "duration": 7.96, "text": "write it as x - y * x + y and from here"}, {"start": 83.96, "duration": 6.159, "text": "18 is common so we factor that out and"}, {"start": 86.439, "duration": 6.32, "text": "here we have x - y so remember that X is"}, {"start": 90.119, "duration": 6.28, "text": "not equal to Y right so here we cancel"}, {"start": 92.759, "duration": 7.881, "text": "this from both sides so here we get x +"}, {"start": 96.399, "duration": 8.201, "text": "y = 18"}, {"start": 100.64, "duration": 6.36, "text": "amazing okay now let us add the two"}, {"start": 104.6, "duration": 4.4, "text": "equations so we can derive something"}, {"start": 107.0, "duration": 5.24, "text": "very beautiful one more time so here we"}, {"start": 109.0, "duration": 5.96, "text": "have equation 1 plus equation 2 this"}, {"start": 112.24, "duration": 9.28, "text": "will imply from the left that would be"}, {"start": 114.96, "duration": 10.199, "text": "X2 + Y 2 that will be equal to 19 X + y"}, {"start": 121.52, "duration": 10.2, "text": "+ 19 y +"}, {"start": 125.159, "duration": 11.321, "text": "x okay so from here we have X2 + y^2 ="}, {"start": 131.72, "duration": 10.48, "text": "here 19 x + x that will give us 20 x"}, {"start": 136.48, "duration": 11.8, "text": "then plus y + 19 y will give us 20 y so"}, {"start": 142.2, "duration": 10.6, "text": "from here we have X2 + y^2 = here factor"}, {"start": 148.28, "duration": 10.2, "text": "out 20 and here we have x + y so from"}, {"start": 152.8, "duration": 8.0, "text": "here x + y is 18 so we get 20 * 18 right"}, {"start": 158.48, "duration": 5.64, "text": "and that will give us"}, {"start": 160.8, "duration": 4.92, "text": "360 so from here we have something"}, {"start": 164.12, "duration": 5.0, "text": "similar to the expression we asked to"}, {"start": 165.72, "duration": 8.08, "text": "evaluate which is < TK of"}, {"start": 169.12, "duration": 7.88, "text": "X2 + y^2 + 1 and this will be equal to"}, {"start": 173.8, "duration": 7.28, "text": "the square root of x + y s is 360 right"}, {"start": 177.0, "duration": 5.68, "text": "so we have 360 then we have + 1 1 this"}, {"start": 181.08, "duration": 5.48, "text": "equals to"}, {"start": 182.68, "duration": 6.4, "text": "361 right which square root is 19 and"}, {"start": 186.56, "duration": 4.599, "text": "here is the right answer thank you for"}, {"start": 189.08, "duration": 4.56, "text": "watching please kindly subscribe to my"}, {"start": 191.159, "duration": 5.481, "text": "channel also like comment and share"}, {"start": 193.64, "duration": 3.0, "text": "bye-bye"}], "title": "Interesting Olympiad Math", "description": "", "lengthSeconds": 195, "width": 640, "height": 360}
{"video_id": "kiQz7zSAETs", "transcript": [{"start": 0.88, "duration": 5.56, "text": "hello you're welcome today we're going"}, {"start": 3.159, "duration": 4.841, "text": "to solve this exponential equation from"}, {"start": 6.44, "duration": 6.239, "text": "here we have"}, {"start": 8.0, "duration": 7.08, "text": "256 root x = 64 to^ of"}, {"start": 12.679, "duration": 5.721, "text": "X now there's something really nice we"}, {"start": 15.08, "duration": 6.24, "text": "need to take note of here we have 256"}, {"start": 18.4, "duration": 4.879, "text": "and remember this is exactly equals 2 to"}, {"start": 21.32, "duration": 4.76, "text": "the^ of"}, {"start": 23.279, "duration": 5.441, "text": "8 and then from the other side the base"}, {"start": 26.08, "duration": 4.84, "text": "there which is 64 is the same as 2 to"}, {"start": 28.72, "duration": 4.2, "text": "the power of 6"}, {"start": 30.92, "duration": 5.56, "text": "so we're going to write the bases in"}, {"start": 32.92, "duration": 7.28, "text": "terms of the powers of two right so from"}, {"start": 36.48, "duration": 7.96, "text": "the left instead of 256 we write it as 2"}, {"start": 40.2, "duration": 8.039, "text": "to the^ of 8 then ra the power of root X"}, {"start": 44.44, "duration": 7.2, "text": "this equals 64 we write it as 2 to the"}, {"start": 48.239, "duration": 5.12, "text": "power of 6 then to the power of X now"}, {"start": 51.64, "duration": 4.399, "text": "let's take note of this a to the power"}, {"start": 53.359, "duration": 6.801, "text": "of B to the power of C this is same as a"}, {"start": 56.039, "duration": 9.16, "text": "to the power B * C so from here we have"}, {"start": 60.16, "duration": 8.639, "text": "2^ 8 * otk X will give us 8un X then"}, {"start": 65.199, "duration": 7.361, "text": "this is equal to here 2^ 6 * X will give"}, {"start": 68.799, "duration": 6.32, "text": "us 6 x so we can see that we have"}, {"start": 72.56, "duration": 5.68, "text": "obtained equal bases from both sides of"}, {"start": 75.119, "duration": 6.081, "text": "this equation so from here this implies"}, {"start": 78.24, "duration": 7.28, "text": "that 8un"}, {"start": 81.2, "duration": 8.12, "text": "x = 6 x that is equating the powers"}, {"start": 85.52, "duration": 6.4, "text": "right okay so from here let us Square"}, {"start": 89.32, "duration": 7.36, "text": "both sides to get rid of this root sign"}, {"start": 91.92, "duration": 10.519, "text": "so here we have 8un x s this will be"}, {"start": 96.68, "duration": 10.16, "text": "equal to 6 x^ 2 now take note of this a"}, {"start": 102.439, "duration": 8.28, "text": "* B to the^ of M this is same as a to"}, {"start": 106.84, "duration": 7.12, "text": "the power of M * B to the power of"}, {"start": 110.719, "duration": 6.281, "text": "M we can see from here 8 2 will give us"}, {"start": 113.96, "duration": 6.04, "text": "64 and the root x s Square will cancel"}, {"start": 117.0, "duration": 7.52, "text": "the root sign we have just X then this"}, {"start": 120.0, "duration": 7.479, "text": "be equal to 62 will give us 36 then"}, {"start": 124.52, "duration": 5.519, "text": "X2 now let's change the sides of this"}, {"start": 127.479, "duration": 7.12, "text": "equation so we can write as 36"}, {"start": 130.039, "duration": 7.681, "text": "x^2 = 64x but this does not affect the"}, {"start": 134.599, "duration": 3.121, "text": "original equation"}, {"start": 138.599, "duration": 7.081, "text": "right okay so from"}, {"start": 141.72, "duration": 9.36, "text": "here we have to subtract 64x from both"}, {"start": 145.68, "duration": 9.96, "text": "sides so we get 36 x^ 2 - 64x"}, {"start": 151.08, "duration": 8.92, "text": "then this equals z we have XX is common"}, {"start": 155.64, "duration": 7.84, "text": "so we factor that out so we have 36 x -"}, {"start": 160.0, "duration": 12.4, "text": "64 then this equal 0 so from here this"}, {"start": 163.48, "duration": 14.2, "text": "implies that x = 0 or 36 x - 64 = 0 add"}, {"start": 172.4, "duration": 8.72, "text": "64 to both side we get 6 36 x = 64 then"}, {"start": 177.68, "duration": 8.24, "text": "from here we divide both sides by 36"}, {"start": 181.12, "duration": 9.36, "text": "6 here by 36 36 cancel 36 and here we"}, {"start": 185.92, "duration": 9.239, "text": "left with X = here let us divide by 4 64"}, {"start": 190.48, "duration": 7.36, "text": "/ 4 will give us 16 then all over 36 / 4"}, {"start": 195.159, "duration": 10.0, "text": "will give us 9 so here we come up with"}, {"start": 197.84, "duration": 11.319, "text": "the solutions that x = 0 and also x = 16"}, {"start": 205.159, "duration": 5.401, "text": "/ 9 and these are the only two real"}, {"start": 209.159, "duration": 3.36, "text": "solutions"}, {"start": 210.56, "duration": 4.2, "text": "thank you for watching please kindly"}, {"start": 212.519, "duration": 6.321, "text": "subscribe to my channel also like"}, {"start": 214.76, "duration": 4.08, "text": "comment and share bye-bye"}], "title": "Interesting exponential equation", "description": "", "lengthSeconds": 217, "width": 640, "height": 360}
{"video_id": "bfJmXlMWGM4", "transcript": [{"start": 0.719, "duration": 3.361, "text": "good evening viewers welcome back to my"}, {"start": 2.76, "duration": 3.24, "text": "channel"}, {"start": 4.08, "duration": 3.239, "text": "today we have the following interesting"}, {"start": 6.0, "duration": 3.72, "text": "problem"}, {"start": 7.319, "duration": 5.521, "text": "if two raised for X all over three plus"}, {"start": 9.72, "duration": 5.52, "text": "two raised by S is equal to one over two"}, {"start": 12.84, "duration": 5.4, "text": "then eight raised per X all over three"}, {"start": 15.24, "duration": 4.32, "text": "plus eight raised by x c was Dutch"}, {"start": 18.24, "duration": 3.66, "text": "okay"}, {"start": 19.56, "duration": 5.16, "text": "wow please if you have not subscribed to"}, {"start": 21.9, "duration": 4.98, "text": "this my Channel please subscribe turn on"}, {"start": 24.72, "duration": 5.399, "text": "the notification Bell so that you'll be"}, {"start": 26.88, "duration": 4.979, "text": "notified anytime we upload new video"}, {"start": 30.119, "duration": 3.78, "text": "okay let's see how we're going to solve"}, {"start": 31.859, "duration": 4.081, "text": "this problem"}, {"start": 33.899, "duration": 3.361, "text": "this expression I'm going to rewrite it"}, {"start": 35.94, "duration": 4.38, "text": "as follows"}, {"start": 37.26, "duration": 6.42, "text": "so I will be writing this expression as"}, {"start": 40.32, "duration": 6.84, "text": "we have here H raised by X"}, {"start": 43.68, "duration": 5.1, "text": "all over three loss eight raised by X"}, {"start": 47.16, "duration": 2.879, "text": "okay"}, {"start": 48.78, "duration": 3.36, "text": "so"}, {"start": 50.039, "duration": 4.381, "text": "we all know that this age can be"}, {"start": 52.14, "duration": 4.559, "text": "expressed as two raised by three then"}, {"start": 54.42, "duration": 4.02, "text": "raised by X"}, {"start": 56.699, "duration": 4.981, "text": "all over"}, {"start": 58.44, "duration": 6.179, "text": "3 plus you know two raised battery"}, {"start": 61.68, "duration": 5.82, "text": "raised by X now we're going to use the"}, {"start": 64.619, "duration": 5.701, "text": "following in this is law"}, {"start": 67.5, "duration": 5.58, "text": "that's when we have a to the power B"}, {"start": 70.32, "duration": 6.839, "text": "in bracket raised by m"}, {"start": 73.08, "duration": 7.079, "text": "we can rewrite as follows a raise for M"}, {"start": 77.159, "duration": 6.721, "text": "and bracketing risk YB okay"}, {"start": 80.159, "duration": 8.041, "text": "so here we have"}, {"start": 83.88, "duration": 6.72, "text": "here to raise power x respiratory"}, {"start": 88.2, "duration": 4.14, "text": "all over"}, {"start": 90.6, "duration": 3.6, "text": "3 plus"}, {"start": 92.34, "duration": 4.2, "text": "2 raised by X"}, {"start": 94.2, "duration": 3.66, "text": "raised by three okay"}, {"start": 96.54, "duration": 3.66, "text": "wow"}, {"start": 97.86, "duration": 4.74, "text": "now I'm going to make a substitution"}, {"start": 100.2, "duration": 7.559, "text": "here so let me say"}, {"start": 102.6, "duration": 8.22, "text": "let's to raise power x equals k"}, {"start": 107.759, "duration": 6.481, "text": "now we have change"}, {"start": 110.82, "duration": 5.64, "text": "the Outlook of this question so"}, {"start": 114.24, "duration": 4.199, "text": "what we're having here now our new"}, {"start": 116.46, "duration": 4.439, "text": "equation here"}, {"start": 118.439, "duration": 4.021, "text": "now is going to be"}, {"start": 120.899, "duration": 7.981, "text": "okay"}, {"start": 122.46, "duration": 9.72, "text": "all over 3 plus k equals 1 over 2 of"}, {"start": 128.88, "duration": 5.88, "text": "which will be evaluating"}, {"start": 132.18, "duration": 4.98, "text": "of which we'll be evaluating"}, {"start": 134.76, "duration": 8.52, "text": "K Cube"}, {"start": 137.16, "duration": 9.48, "text": "all over 3 plus K Cube and this is"}, {"start": 143.28, "duration": 5.459, "text": "exactly equals to the value of the"}, {"start": 146.64, "duration": 5.7, "text": "expression we're about to evaluate a to"}, {"start": 148.739, "duration": 4.741, "text": "raise power x all over 3 plus 8 raised"}, {"start": 152.34, "duration": 4.979, "text": "by X okay"}, {"start": 153.48, "duration": 5.7, "text": "wow that's nice okay now cross multiply"}, {"start": 157.319, "duration": 5.341, "text": "here so if you cross apply we have two"}, {"start": 159.18, "duration": 5.94, "text": "times K so we have 2K is equal to you"}, {"start": 162.66, "duration": 4.859, "text": "know 3 plus K plus times this one here"}, {"start": 165.12, "duration": 4.8, "text": "so we have three plus okay you know"}, {"start": 167.519, "duration": 6.36, "text": "subtracting this K from both sides so we"}, {"start": 169.92, "duration": 6.179, "text": "get K is equal to three so finally we"}, {"start": 173.879, "duration": 4.5, "text": "get K is equal to three so we can easily"}, {"start": 176.099, "duration": 4.5, "text": "evaluate this expression which will also"}, {"start": 178.379, "duration": 4.86, "text": "give us the right value of this"}, {"start": 180.599, "duration": 6.86, "text": "expression here so since K is equal to 3"}, {"start": 183.239, "duration": 4.22, "text": "so this expression here is going to be"}, {"start": 188.34, "duration": 3.0, "text": "327"}, {"start": 195.54, "duration": 9.479, "text": "all over you know here 3 plus 27 and"}, {"start": 200.04, "duration": 7.26, "text": "here we have what 27 all over 30 and"}, {"start": 205.019, "duration": 4.621, "text": "simplifying this so simplifying the"}, {"start": 207.3, "duration": 5.4, "text": "using these three to divide both sides"}, {"start": 209.64, "duration": 5.519, "text": "to divide up and down I mean so we get 9"}, {"start": 212.7, "duration": 4.8, "text": "all over ten and this is the right"}, {"start": 215.159, "duration": 4.5, "text": "answer to the question thank you for"}, {"start": 217.5, "duration": 4.5, "text": "viewing and please make sure you"}, {"start": 219.659, "duration": 4.201, "text": "subscribe to our YouTube Channel please"}, {"start": 222.0, "duration": 5.58, "text": "subscribe"}, {"start": 223.86, "duration": 6.799, "text": "Please Subscribe share our video and"}, {"start": 227.58, "duration": 3.079, "text": "comment thank you"}], "title": "An Olympiad Math Problem", "description": "", "lengthSeconds": 229, "width": 636, "height": 360}
{"video_id": "vnPkk8fL4HE", "transcript": [{"start": 0.93, "duration": 18.699, "text": "[\u0938\u0902\u0917\u0940\u0924]"}, {"start": 22.519, "duration": 36.17, "text": "\u0938\u094d\u091f"}, {"start": 25.13, "duration": 33.559, "text": "[\u0938\u0902\u0917\u0940\u0924]"}, {"start": 60.91, "duration": 10.009, "text": "[\u0938\u0902\u0917\u0940\u0924]"}, {"start": 73.02, "duration": 23.31, "text": "[\u0938\u0902\u0917\u0940\u0924]"}, {"start": 97.439, "duration": 2.521, "text": "\u0938\u094d"}, {"start": 100.51, "duration": 20.4, "text": "[\u0938\u0902\u0917\u0940\u0924]"}, {"start": 118.84, "duration": 11.149, "text": "\u0938\u094d \u092c"}, {"start": 120.91, "duration": 9.079, "text": "[\u0938\u0902\u0917\u0940\u0924]"}, {"start": 132.91, "duration": 3.789, "text": "[\u0938\u0902\u0917\u0940\u0924]"}], "title": "Interesting surdic equation.                    #olympiad #maths #education", "description": "", "lengthSeconds": 135, "width": 640, "height": 360}
{"video_id": "019FmifBYKc", "transcript": [{"start": 0.0, "duration": 3.0, "text": "and we're going to find the last digit"}, {"start": 1.8, "duration": 4.2, "text": "of this"}, {"start": 3.0, "duration": 4.2, "text": "of this number here so we're going to"}, {"start": 6.0, "duration": 4.019, "text": "find the last digit using the"}, {"start": 7.2, "duration": 6.66, "text": "exponential method so we have two raised"}, {"start": 10.019, "duration": 8.641, "text": "for 2008. now 2008"}, {"start": 13.86, "duration": 6.96, "text": "is the same as four times five zero two"}, {"start": 18.66, "duration": 4.619, "text": "so now when we are looking for the last"}, {"start": 20.82, "duration": 5.219, "text": "digit of any number is equally the same"}, {"start": 23.279, "duration": 5.221, "text": "as divided the number by 10 so we have"}, {"start": 26.039, "duration": 5.281, "text": "two raised to the powerful this is just"}, {"start": 28.5, "duration": 5.399, "text": "application of common indices law now 2"}, {"start": 31.32, "duration": 5.94, "text": "raise power 4 is equal to 16 so we have"}, {"start": 33.899, "duration": 5.101, "text": "in Brackets 16 raise power 502 in the"}, {"start": 37.26, "duration": 3.72, "text": "bracket here the last digit here is six"}, {"start": 39.0, "duration": 4.68, "text": "so take it out we have six raised to"}, {"start": 40.98, "duration": 4.62, "text": "power five zero two remember six raised"}, {"start": 43.68, "duration": 4.74, "text": "for Any number greater than zero we"}, {"start": 45.6, "duration": 5.52, "text": "always have the last digit to be six and"}, {"start": 48.42, "duration": 5.659, "text": "this is division by 10 that is a model"}, {"start": 51.12, "duration": 6.72, "text": "ten so the last digit of two principle"}, {"start": 54.079, "duration": 5.14, "text": "2008 is equal to what six thank you for"}, {"start": 57.84, "duration": 2.94, "text": "viewing and make sure you subscribe to"}, {"start": 59.219, "duration": 3.741, "text": "my YouTube channel for more interesting"}, {"start": 60.78, "duration": 2.18, "text": "videos"}], "title": "The fastest method", "description": "", "lengthSeconds": 62, "width": 362, "height": 640}
{"video_id": "U5i36YNFf2E", "transcript": [{"start": 0.24, "duration": 3.96, "text": "hello welcome back my lovely people"}, {"start": 2.919, "duration": 5.201, "text": "today we're going to solve this"}, {"start": 4.2, "duration": 6.12, "text": "exponential equation we're given 6^ X2 ="}, {"start": 8.12, "duration": 3.76, "text": "15 so we're going to solve this problem"}, {"start": 10.32, "duration": 3.88, "text": "by using the natural logarithm so we're"}, {"start": 11.88, "duration": 7.8, "text": "going to Ln both sides of this equation"}, {"start": 14.2, "duration": 9.04, "text": "so Ln of 6^ of X2 this equals the Ln of"}, {"start": 19.68, "duration": 6.12, "text": "15 now with the property of the natural"}, {"start": 23.24, "duration": 4.68, "text": "log so this power here is going to cross"}, {"start": 25.8, "duration": 7.36, "text": "over to this place and that will become"}, {"start": 27.92, "duration": 7.84, "text": "x\u00b2 MTI the Ln of 6 this equals the Ln of"}, {"start": 33.16, "duration": 4.28, "text": "15 right so from here remember we're"}, {"start": 35.76, "duration": 4.08, "text": "suming for X right so we're going to"}, {"start": 37.44, "duration": 6.439, "text": "divide both side from here by the Ln of"}, {"start": 39.84, "duration": 7.8, "text": "6 here by the Ln of 6 so this will"}, {"start": 43.879, "duration": 8.68, "text": "cancel this and here we get x2 equal the"}, {"start": 47.64, "duration": 6.399, "text": "Ln of 15 / the Ln of 6 from here we're"}, {"start": 52.559, "duration": 3.401, "text": "going to take root of both sides and"}, {"start": 54.039, "duration": 5.52, "text": "here we come up with the solution X ="}, {"start": 55.96, "duration": 4.8, "text": "plus or minus the square root of Ln of"}, {"start": 59.559, "duration": 4.521, "text": "15"}, {"start": 60.76, "duration": 5.48, "text": "/ the Ln of six and this is the right"}, {"start": 64.08, "duration": 4.48, "text": "answer thank you for watching please"}, {"start": 66.24, "duration": 6.32, "text": "kindly subscribe to my channel also like"}, {"start": 68.56, "duration": 4.0, "text": "comment and share bye-bye"}], "title": "Interesting exponential equation", "description": "", "lengthSeconds": 71, "width": 538, "height": 360}
{"video_id": "URp7i0OayW8", "transcript": [{"start": 0.599, "duration": 5.041, "text": "hello good day viewers welcome back once"}, {"start": 2.72, "duration": 8.24, "text": "again today we're giv the following"}, {"start": 5.64, "duration": 10.36, "text": "equation x + 1 / x = 1 and here we're"}, {"start": 10.96, "duration": 8.319, "text": "going to evaluate x^2 - 1 /"}, {"start": 16.0, "duration": 5.24, "text": "x^2 before we proceed let us recall the"}, {"start": 19.279, "duration": 7.721, "text": "difference of two squares"}, {"start": 21.24, "duration": 12.56, "text": "formula we have a s - b 2 this is equal"}, {"start": 27.0, "duration": 9.96, "text": "to a + b MTI Aus B now this expression"}, {"start": 33.8, "duration": 6.8, "text": "here we are asked to evaluate we can"}, {"start": 36.96, "duration": 7.24, "text": "Factor it"}, {"start": 40.6, "duration": 11.84, "text": "as x + 1 /"}, {"start": 44.2, "duration": 10.64, "text": "X MTI by X - 1 /x right okay now from"}, {"start": 52.44, "duration": 6.04, "text": "the given equation we already have the"}, {"start": 54.84, "duration": 6.64, "text": "value for x + 1 /x so our major goal"}, {"start": 58.48, "duration": 7.88, "text": "here is to find the value of of x - 1 /"}, {"start": 61.48, "duration": 8.56, "text": "X from the given equation we have x + 1"}, {"start": 66.36, "duration": 6.24, "text": "/x this is equal to one now let's Square"}, {"start": 70.04, "duration": 4.8, "text": "both sides of this"}, {"start": 72.6, "duration": 5.32, "text": "equation from the left hand side we'll"}, {"start": 74.84, "duration": 10.239, "text": "be making use of the following identity"}, {"start": 77.92, "duration": 12.32, "text": "a + b 2 this is equal to a - b"}, {"start": 85.079, "duration": 7.441, "text": "s + 4 a b now make use of this this"}, {"start": 90.24, "duration": 8.6, "text": "identity the left hand side of this"}, {"start": 92.52, "duration": 11.0, "text": "equation will be X - 1 /x"}, {"start": 98.84, "duration": 9.44, "text": "2+ 4 a here will be 4 * x * 1 x which is"}, {"start": 103.52, "duration": 10.559, "text": "4 this is equal to 1 so from here we"}, {"start": 108.28, "duration": 11.08, "text": "have x -1 / x^ 2 this is = to 1 - 4"}, {"start": 114.079, "duration": 10.281, "text": "which is -3 from here we have X - 1 /x"}, {"start": 119.36, "duration": 8.959, "text": "this is = to plus or minus root of -3"}, {"start": 124.36, "duration": 9.08, "text": "from here we have x -1 / X this is equal"}, {"start": 128.319, "duration": 7.481, "text": "to plus or minus you know root of1 is I"}, {"start": 133.44, "duration": 4.92, "text": "then root of"}, {"start": 135.8, "duration": 8.159, "text": "3 therefore the expression we asked to"}, {"start": 138.36, "duration": 12.44, "text": "evaluate x^2 - 1 /x2 which we factored"}, {"start": 143.959, "duration": 10.721, "text": "as x + 1 /x * X - 1 /x this will be"}, {"start": 150.8, "duration": 7.4, "text": "equal to x + 1 / X is given us 1 then"}, {"start": 154.68, "duration": 7.72, "text": "multip X - 1 / X which is equal to plus"}, {"start": 158.2, "duration": 9.84, "text": "or minus IUN of 3 and from here we get"}, {"start": 162.4, "duration": 8.28, "text": "the solution as plus or minus I < tk3"}, {"start": 168.04, "duration": 4.64, "text": "and this is the right answer thank you"}, {"start": 170.68, "duration": 4.68, "text": "for watching please kindly subscribe to"}, {"start": 172.68, "duration": 5.68, "text": "my channel also like comment and share"}, {"start": 175.36, "duration": 3.0, "text": "bye-bye"}], "title": "Very tricky homemade math problem | #maths #mathematics", "description": "", "lengthSeconds": 177, "width": 454, "height": 360}
{"video_id": "SqDTx_muKZU", "transcript": [{"start": 0.76, "duration": 3.519, "text": "hello good day viewers welcome back once"}, {"start": 2.72, "duration": 5.199, "text": "again today we're going to solve for the"}, {"start": 4.279, "duration": 7.201, "text": "value of x in the following exponential"}, {"start": 7.919, "duration": 6.441, "text": "equation involving root signs right okay"}, {"start": 11.48, "duration": 2.88, "text": "let's get to the"}, {"start": 14.48, "duration": 5.44, "text": "video here we're going to square both"}, {"start": 16.76, "duration": 5.88, "text": "sides of this equation to get rid of the"}, {"start": 19.92, "duration": 2.72, "text": "first root"}, {"start": 23.68, "duration": 7.4, "text": "sign squar this will be 9 squ so from"}, {"start": 29.08, "duration": 4.12, "text": "here this one and the first one here we"}, {"start": 31.08, "duration": 7.24, "text": "cancel out and we're left with square <"}, {"start": 33.2, "duration": 9.48, "text": "TK of 1/ 9 to^ x this is equal to um"}, {"start": 38.32, "duration": 6.68, "text": "that is 9 squ right which is 8"}, {"start": 42.68, "duration": 4.399, "text": "to1 and let's go ahead and square both"}, {"start": 45.0, "duration": 4.48, "text": "sides"}, {"start": 47.079, "duration": 6.12, "text": "again this one and this one we cancel"}, {"start": 49.48, "duration": 6.919, "text": "out and we left with 1/ 9 to the^ of X"}, {"start": 53.199, "duration": 7.961, "text": "this is equal to um that is"}, {"start": 56.399, "duration": 8.12, "text": "81 2 so 81 is the same as 3 power of 4"}, {"start": 61.16, "duration": 4.959, "text": "then of two so when we multiply this"}, {"start": 64.519, "duration": 4.121, "text": "four and two together you know this"}, {"start": 66.119, "duration": 6.961, "text": "property m"}, {"start": 68.64, "duration": 8.36, "text": "n z this isal m n * Z so this will give"}, {"start": 73.08, "duration": 11.48, "text": "us 3 4 * 2 which is"}, {"start": 77.0, "duration": 14.6, "text": "8 now 1/ 9 let us recall again 1 m n = m"}, {"start": 84.56, "duration": 10.919, "text": "n so 1 / 9 is 1 / 3 2 x this is = to 3 3"}, {"start": 91.6, "duration": 10.36, "text": "8 now make use of this property here we"}, {"start": 95.479, "duration": 8.161, "text": "have in the bracket 3 -2 x this = to 3 8"}, {"start": 101.96, "duration": 5.68, "text": "so from here we multiply these Powers"}, {"start": 103.64, "duration": 8.479, "text": "together we have 3 -2 x this is equal to"}, {"start": 107.64, "duration": 7.2, "text": "3 of 8 we have equal bases so actually"}, {"start": 112.119, "duration": 7.161, "text": "we can equate the powers right so we"}, {"start": 114.84, "duration": 8.12, "text": "have -2X this is equal to 8 now let's go"}, {"start": 119.28, "duration": 3.68, "text": "ahead and divide both side by"}, {"start": 123.079, "duration": 9.961, "text": "-2 so here we have S is = to -8 / 2 that"}, {"start": 128.879, "duration": 5.64, "text": "will give us -4 and here is the solution"}, {"start": 133.04, "duration": 3.64, "text": "thank you for watching please kindly"}, {"start": 134.519, "duration": 6.36, "text": "subscribe to my YouTube channel also"}, {"start": 136.68, "duration": 4.199, "text": "like comment and share bye-bye"}], "title": "Interesting exponential equation | #maths #olympiad", "description": "", "lengthSeconds": 139, "width": 640, "height": 360}
{"video_id": "_8sq8NvYYJg", "transcript": [{"start": 0.24, "duration": 2.959, "text": "hello good viewers welcome back once"}, {"start": 1.8, "duration": 3.959, "text": "again today we're going to solve the"}, {"start": 3.199, "duration": 5.201, "text": "Bangladesh Olympia math we're given the"}, {"start": 5.759, "duration": 6.0, "text": "equation I think that's quadratic right"}, {"start": 8.4, "duration": 8.2, "text": "X2 - 3x + 1 = 0 here we're going to"}, {"start": 11.759, "duration": 7.44, "text": "evaluate this expression x^ 6 + 1 x^ 6"}, {"start": 16.6, "duration": 5.64, "text": "from this given equation we can divide"}, {"start": 19.199, "duration": 12.08, "text": "each time from both sides by X and see"}, {"start": 22.24, "duration": 12.4, "text": "what happens x\u00b2 / x - 3x / x + 1 / X"}, {"start": 31.279, "duration": 7.921, "text": "this is equal to 0 /"}, {"start": 34.64, "duration": 12.72, "text": "X from here x\u00b2 / X that will reduce to x"}, {"start": 39.2, "duration": 10.4, "text": "- 3x / X that is 3 + 1 /x this = 0 now"}, {"start": 47.36, "duration": 8.32, "text": "let's move this -3 to the right so we"}, {"start": 49.6, "duration": 8.799, "text": "have here x + 1 /x ="}, {"start": 55.68, "duration": 6.24, "text": "3 now from here we're going to CU both"}, {"start": 58.399, "duration": 3.521, "text": "sides of this equation"}, {"start": 63.039, "duration": 9.481, "text": "from the left let us make use of this"}, {"start": 65.6, "duration": 15.08, "text": "property A + B Cubed this equals a cubed"}, {"start": 72.52, "duration": 10.919, "text": "+ B Cubed + 3 * a * B into bracket A + B"}, {"start": 80.68, "duration": 4.72, "text": "and mind you when we Cube this second"}, {"start": 83.439, "duration": 7.0, "text": "expression here it will look like this"}, {"start": 85.4, "duration": 7.079, "text": "1X Cub this will be equal to 1/ X cubed"}, {"start": 90.439, "duration": 4.481, "text": "because I may not explain this in this"}, {"start": 92.479, "duration": 5.801, "text": "video okay now from the left we have the"}, {"start": 94.92, "duration": 6.159, "text": "first term cubed which is X cubed plus"}, {"start": 98.28, "duration": 6.879, "text": "the second term cubed which is 1 / X"}, {"start": 101.079, "duration": 7.241, "text": "cubed then plus 3 * the first term * the"}, {"start": 105.159, "duration": 5.721, "text": "second term first term which is x * 1 /"}, {"start": 108.32, "duration": 5.56, "text": "X will give us 1 so we left with three"}, {"start": 110.88, "duration": 5.72, "text": "into bracket the sum of X and the"}, {"start": 113.88, "duration": 6.839, "text": "reciprocal of X then this will be equal"}, {"start": 116.6, "duration": 7.28, "text": "to 3 Cub which is 27"}, {"start": 120.719, "duration": 8.52, "text": "from here remember that x + 1 x = 3 so"}, {"start": 123.88, "duration": 8.32, "text": "we have here x Cub + 1 / X Cub plus here"}, {"start": 129.239, "duration": 4.041, "text": "will be 3 * 3 right then this is equal"}, {"start": 132.2, "duration": 8.24, "text": "to"}, {"start": 133.28, "duration": 10.36, "text": "27 from here we have X Cub + 1 / X Cub +"}, {"start": 140.44, "duration": 6.4, "text": "3 * 3 which is 9 this ="}, {"start": 143.64, "duration": 7.879, "text": "27 subtract 9 from both side we arrived"}, {"start": 146.84, "duration": 7.479, "text": "at X Cub + 1 / X cub"}, {"start": 151.519, "duration": 7.281, "text": "= 18"}, {"start": 154.319, "duration": 7.401, "text": "right from here let us Square both"}, {"start": 158.8, "duration": 8.84, "text": "sides from the left we're going to make"}, {"start": 161.72, "duration": 9.84, "text": "use of this a + b 2 this equal a 2 + b 2"}, {"start": 167.64, "duration": 10.04, "text": "+ 2 * a * B so from here the first term"}, {"start": 171.56, "duration": 9.92, "text": "squar will be X Cub 2 plus this one will"}, {"start": 177.68, "duration": 8.839, "text": "be same as 1 / X cub Cub s right because"}, {"start": 181.48, "duration": 9.319, "text": "1 2 is 1 then plus 2 * the first time x"}, {"start": 186.519, "duration": 8.041, "text": "Cub * the second term 1 / X cub and this"}, {"start": 190.799, "duration": 3.761, "text": "equals 18 s which is"}, {"start": 195.92, "duration": 6.0, "text": "324 from here 3 * 2 here is 6 so we have"}, {"start": 199.68, "duration": 7.32, "text": "x ^ 6"}, {"start": 201.92, "duration": 7.519, "text": "+ 1 / x^ 6 and we can see the expression"}, {"start": 207.0, "duration": 4.959, "text": "we asked to evaluate has already popped"}, {"start": 209.439, "duration": 5.0, "text": "up right right so we have here plus this"}, {"start": 211.959, "duration": 3.961, "text": "we cancel this and we have two and this"}, {"start": 214.439, "duration": 4.201, "text": "is equal to"}, {"start": 215.92, "duration": 9.8, "text": "324 subtract two from both sides and we"}, {"start": 218.64, "duration": 10.319, "text": "have the solution x 6 + 1 / x 6 ="}, {"start": 225.72, "duration": 4.92, "text": "322 and this is the right answer thank"}, {"start": 228.959, "duration": 3.721, "text": "you for watching please kindly subscribe"}, {"start": 230.64, "duration": 5.56, "text": "to my channel also like comment and"}, {"start": 232.68, "duration": 3.52, "text": "share bye-bye"}], "title": "Bangladesh Olympiad Math | Interesting Algebra problem |", "description": "", "lengthSeconds": 234, "width": 564, "height": 360}
{"video_id": "dPVAHkU6gx0", "transcript": [{"start": 0.359, "duration": 4.681, "text": "hello good viewers welcome back once"}, {"start": 2.44, "duration": 5.88, "text": "again please kindly subscribe to this"}, {"start": 5.04, "duration": 5.08, "text": "channel also like comment and share here"}, {"start": 8.32, "duration": 6.64, "text": "we're given the following interesting"}, {"start": 10.12, "duration": 9.319, "text": "limit limit as x ts to 0 of < TK 2 - <"}, {"start": 14.96, "duration": 8.079, "text": "TK 2 - x all all over 2 here we asked to"}, {"start": 19.439, "duration": 5.84, "text": "evaluate this limit right if we try to"}, {"start": 23.039, "duration": 4.921, "text": "make direct substition here that is"}, {"start": 25.279, "duration": 7.16, "text": "we're going to have theare < TK of 2"}, {"start": 27.96, "duration": 6.84, "text": "minus the < TK of 2 - 0 right then all"}, {"start": 32.439, "duration": 6.041, "text": "over 0 and you know that division by 0"}, {"start": 34.8, "duration": 6.12, "text": "is undefined even though we simplify the"}, {"start": 38.48, "duration": 6.599, "text": "upstairs here we're going to have > of 2"}, {"start": 40.92, "duration": 7.08, "text": "- < TK of 2 / 0 and this equal 0/0 so"}, {"start": 45.079, "duration": 5.921, "text": "this is in in determinant form right so"}, {"start": 48.0, "duration": 5.239, "text": "let us go ahead and manipulate this"}, {"start": 51.0, "duration": 5.16, "text": "which we have in this bracket right over"}, {"start": 53.239, "duration": 5.721, "text": "here okay so what we're going to do here"}, {"start": 56.16, "duration": 5.28, "text": "is we are going to multiply the upstairs"}, {"start": 58.96, "duration": 7.24, "text": "and downstairs by the conjugate form of"}, {"start": 61.44, "duration": 8.84, "text": "this upstairs here so that is"}, {"start": 66.2, "duration": 7.48, "text": "limit as X is approaching zero so here"}, {"start": 70.28, "duration": 5.0, "text": "we have the < TK of 2 minus the < TK of"}, {"start": 73.68, "duration": 5.84, "text": "2 -"}, {"start": 75.28, "duration": 7.92, "text": "x then we multiply by the conjugate form"}, {"start": 79.52, "duration": 7.52, "text": "of this which is < TK of 2 plus the < TK"}, {"start": 83.2, "duration": 7.64, "text": "of 2 - x the same thing happens"}, {"start": 87.04, "duration": 10.84, "text": "downstairs so here we have X multip by"}, {"start": 90.84, "duration": 9.68, "text": "the < TK 2 + the < TK 2 - x right so"}, {"start": 97.88, "duration": 6.68, "text": "here this is equal to"}, {"start": 100.52, "duration": 5.72, "text": "limit as X is approaching zero so you"}, {"start": 104.56, "duration": 4.199, "text": "know here we're going to make use of the"}, {"start": 106.24, "duration": 7.479, "text": "difference of 2 squ formula and what is"}, {"start": 108.759, "duration": 10.081, "text": "this formula when we have a minus B * A"}, {"start": 113.719, "duration": 11.161, "text": "+ B this is going to give us a 2 - b 2"}, {"start": 118.84, "duration": 11.559, "text": "right so up here we have the < TK of 2"}, {"start": 124.88, "duration": 11.12, "text": "2us the < TK 2 - x squared right then"}, {"start": 130.399, "duration": 8.361, "text": "all / X into braet < TK 2 + the < TK of"}, {"start": 136.0, "duration": 6.36, "text": "2 -"}, {"start": 138.76, "duration": 5.559, "text": "x okay so here you know this square and"}, {"start": 142.36, "duration": 4.36, "text": "this square root will cancel out square"}, {"start": 144.319, "duration": 4.321, "text": "and square root will cancel out so you"}, {"start": 146.72, "duration": 5.879, "text": "know this is equal to"}, {"start": 148.64, "duration": 7.8, "text": "limit as X is approaching zero so upset"}, {"start": 152.599, "duration": 7.841, "text": "now we have 2 minus then into bracket"}, {"start": 156.44, "duration": 9.519, "text": "what we have there 2- X right then all"}, {"start": 160.44, "duration": 10.159, "text": "over X Open Bracket < TK 2 plus the sare"}, {"start": 165.959, "duration": 5.841, "text": "otk of 2 - x right so you know this is"}, {"start": 170.599, "duration": 5.201, "text": "equal to"}, {"start": 171.8, "duration": 8.159, "text": "limit as X is approaching zero so here"}, {"start": 175.8, "duration": 5.68, "text": "you know 2 - 2 here then plus X that is"}, {"start": 179.959, "duration": 5.681, "text": "what we have because of this negative"}, {"start": 181.48, "duration": 9.2, "text": "sign here so 2 - 2 is 0 and 0 + x is X"}, {"start": 185.64, "duration": 11.2, "text": "so here we have X then all over X into"}, {"start": 190.68, "duration": 8.919, "text": "braet < TK 2 + < TK 2 - x right you know"}, {"start": 196.84, "duration": 5.92, "text": "this will cancel out and from here we're"}, {"start": 199.599, "duration": 6.521, "text": "going to take our limit now so this is"}, {"start": 202.76, "duration": 8.36, "text": "going to be 1 all over you know here we"}, {"start": 206.12, "duration": 9.039, "text": "have < TK 2 + S < TK of 2 -"}, {"start": 211.12, "duration": 8.16, "text": "0 so this is equal to 1 all over the <"}, {"start": 215.159, "duration": 9.08, "text": "TK 2 + the < TK of 2 and this is equal"}, {"start": 219.28, "duration": 6.959, "text": "to 1 all over 2 < tk2 and this is the"}, {"start": 224.239, "duration": 3.72, "text": "right answer thank you for watching"}, {"start": 226.239, "duration": 4.161, "text": "please do well to subscribe to my"}, {"start": 227.959, "duration": 5.441, "text": "channel also like comment and share"}, {"start": 230.4, "duration": 3.0, "text": "bye-bye"}], "title": "Interesting limit. How to evaluate without L'hospital's rule ?", "description": "", "lengthSeconds": 232, "width": 640, "height": 360}
{"video_id": "drYfbRS-xwc", "transcript": [{"start": 0.44, "duration": 3.32, "text": "hello welcome back once again today"}, {"start": 2.12, "duration": 4.56, "text": "we're going to solve this simple"}, {"start": 3.76, "duration": 5.68, "text": "exponential equation from the left hand"}, {"start": 6.68, "duration": 6.72, "text": "side eight can be Express an index power"}, {"start": 9.44, "duration": 7.679, "text": "of 2 as 2 the^ 3 so here we have a"}, {"start": 13.4, "duration": 6.16, "text": "bracket 2^ 3 then ra to the power x this"}, {"start": 17.119, "duration": 6.4, "text": "will be equal to we know square root"}, {"start": 19.56, "duration": 8.719, "text": "means the^ of 1 / 2 so theare root of 2"}, {"start": 23.519, "duration": 8.521, "text": "is as 2^ of 1 / 2 from here with this"}, {"start": 28.279, "duration": 9.001, "text": "property a to the^ b^ C this is equal to"}, {"start": 32.04, "duration": 9.999, "text": "a^ B * C so this will become 2^ 3x which"}, {"start": 37.28, "duration": 7.32, "text": "is equal to here 2^ 1 / 2 we can see"}, {"start": 42.039, "duration": 4.641, "text": "from both sides we have the same base so"}, {"start": 44.6, "duration": 7.32, "text": "we're going to equate the power so this"}, {"start": 46.68, "duration": 10.64, "text": "implies from here that 3x is equal to 1"}, {"start": 51.92, "duration": 8.799, "text": "/ 2 right so here cross multiply so this"}, {"start": 57.32, "duration": 5.36, "text": "here we multiply this and this here we"}, {"start": 60.719, "duration": 3.12, "text": "multiply this that is the meaning of"}, {"start": 62.68, "duration": 4.759, "text": "cross"}, {"start": 63.839, "duration": 7.241, "text": "multiplication so here 3x * 2 here will"}, {"start": 67.439, "duration": 5.961, "text": "give us 6 x this will be equal to 1 * 1"}, {"start": 71.08, "duration": 5.56, "text": "here will give us 1 from here let's"}, {"start": 73.4, "duration": 6.16, "text": "divide both side by the coefficient of x"}, {"start": 76.64, "duration": 7.36, "text": "6 cancel 6 from here so here we have our"}, {"start": 79.56, "duration": 6.72, "text": "solution X is = 1 / 6 and this is the"}, {"start": 84.0, "duration": 4.68, "text": "right answer thank you for watching"}, {"start": 86.28, "duration": 7.92, "text": "please kindly subscribe to my channel"}, {"start": 88.68, "duration": 5.52, "text": "also like comment and share bye-bye"}], "title": "Simple exponential equation |", "description": "", "lengthSeconds": 93, "width": 640, "height": 360}
{"video_id": "MqhRcNnJx3E", "transcript": [{"start": 0.88, "duration": 5.16, "text": "hello good day viewers welcome back once"}, {"start": 3.08, "duration": 6.12, "text": "again today we have an improper integral"}, {"start": 6.04, "duration": 6.12, "text": "here and we're going to evaluate this we"}, {"start": 9.2, "duration": 8.919, "text": "have the integral from 0 to Infinity of"}, {"start": 12.16, "duration": 8.359, "text": "e^ of -5x * X to ^ of two in this video"}, {"start": 18.119, "duration": 6.16, "text": "I'll be making use of this special"}, {"start": 20.519, "duration": 6.641, "text": "function is a very beautiful one gamma"}, {"start": 24.279, "duration": 8.001, "text": "function and it has the integral"}, {"start": 27.16, "duration": 7.919, "text": "representation as the integral from 0 to"}, {"start": 32.28, "duration": 6.36, "text": "Infinity of"}, {"start": 35.079, "duration": 7.881, "text": "e^x * x^ n"}, {"start": 38.64, "duration": 5.8, "text": "-1 DX so our goal here is to transform"}, {"start": 42.96, "duration": 5.119, "text": "this integral to"}, {"start": 44.44, "duration": 6.04, "text": "look exactly like this in this form I"}, {"start": 48.079, "duration": 4.96, "text": "mean and then we can make use of the"}, {"start": 50.48, "duration": 6.719, "text": "gamma function remember that the gamma"}, {"start": 53.039, "duration": 6.2, "text": "of n = n -1"}, {"start": 57.199, "duration": 4.801, "text": "factorial we can just say that this"}, {"start": 59.239, "duration": 5.681, "text": "gamma function is chose the extension of"}, {"start": 62.0, "duration": 5.72, "text": "the factorial it's an amazing function"}, {"start": 64.92, "duration": 6.6, "text": "now let's get into this video from here"}, {"start": 67.72, "duration": 8.0, "text": "the power of E here is just --x and from"}, {"start": 71.52, "duration": 6.84, "text": "here the power of E here is -5 * X so"}, {"start": 75.72, "duration": 8.24, "text": "we're going to make a substitution by"}, {"start": 78.36, "duration": 10.6, "text": "letting 5x = y from here divide B by 5"}, {"start": 83.96, "duration": 9.6, "text": "we get X = Y / 5 then differentiate both"}, {"start": 88.96, "duration": 6.4, "text": "side we get DX = the Y / 5 then our"}, {"start": 93.56, "duration": 5.599, "text": "integral"}, {"start": 95.36, "duration": 7.56, "text": "becomes the integral from 0 to Infinity"}, {"start": 99.159, "duration": 8.361, "text": "yes 5 * 0 is 0 5 * infinity is infinity"}, {"start": 102.92, "duration": 10.96, "text": "then e to the^ of Y from here then * x 2"}, {"start": 107.52, "duration": 10.44, "text": "will be y / 5 2 which is y^ 2 all 25 *"}, {"start": 113.88, "duration": 9.04, "text": "the X which is the y/ 5 then from here"}, {"start": 117.96, "duration": 10.48, "text": "25 * 5 is 125 so we put it out that is 1"}, {"start": 122.92, "duration": 9.24, "text": "125 the integral from 0 to Infinity e y"}, {"start": 128.44, "duration": 6.36, "text": "multip by here y^ 2 we can write as y^ 3"}, {"start": 132.16, "duration": 7.0, "text": "- 1 then d"}, {"start": 134.8, "duration": 7.799, "text": "y now if you compare this with this here"}, {"start": 139.16, "duration": 7.04, "text": "we can see that here we have the gamma"}, {"start": 142.599, "duration": 5.761, "text": "of three which is equal to 3 - 1"}, {"start": 146.2, "duration": 6.16, "text": "factorial"}, {"start": 148.36, "duration": 8.04, "text": "right and that we give us 2 factorial"}, {"start": 152.36, "duration": 7.2, "text": "which is equal to 2 * 1 and that's equal"}, {"start": 156.4, "duration": 9.04, "text": "2 so from here our right answer becomes"}, {"start": 159.56, "duration": 7.64, "text": "1 all 125 * 2 / 1 which is equal to 2"}, {"start": 165.44, "duration": 5.28, "text": "all over"}, {"start": 167.2, "duration": 5.44, "text": "125 and this is the right answer thank"}, {"start": 170.72, "duration": 4.239, "text": "you for watching please kindly subscribe"}, {"start": 172.64, "duration": 5.879, "text": "to my channel also like comment and"}, {"start": 174.959, "duration": 3.56, "text": "share bye-bye"}], "title": "The Gamma Function", "description": "", "lengthSeconds": 177, "width": 548, "height": 360}
{"video_id": "mArHU0jLHSA", "transcript": [{"start": 0.32, "duration": 4.72, "text": "good day viewers welcome back once again"}, {"start": 2.56, "duration": 5.279, "text": "to my YouTube Channel please if you have"}, {"start": 5.04, "duration": 5.88, "text": "not subscribed to our Channel please"}, {"start": 7.839, "duration": 6.441, "text": "kindly support us by subscribing to our"}, {"start": 10.92, "duration": 4.92, "text": "Channel thank you here we have this"}, {"start": 14.28, "duration": 5.2, "text": "infinite"}, {"start": 15.84, "duration": 6.96, "text": "next radical right we have x equal to"}, {"start": 19.48, "duration": 6.719, "text": "theare < TK of 5+ the < TK of 5+ the <"}, {"start": 22.8, "duration": 5.44, "text": "TK of 5+ and so on and this goes on"}, {"start": 26.199, "duration": 6.281, "text": "forever right so we call this an"}, {"start": 28.24, "duration": 7.76, "text": "infinite nested radical right okay now"}, {"start": 32.48, "duration": 6.8, "text": "we asked to find the value of this x"}, {"start": 36.0, "duration": 8.6, "text": "right so let us get started here we have"}, {"start": 39.28, "duration": 9.439, "text": "X is = to the square < TK of 5 + the"}, {"start": 44.6, "duration": 9.16, "text": "square < TK of 5 + the < TK of 5 plus"}, {"start": 48.719, "duration": 8.68, "text": "and so on right okay so this is equal"}, {"start": 53.76, "duration": 10.0, "text": "to x = square root of you know here we"}, {"start": 57.399, "duration": 9.601, "text": "have 5+ so here root of 5 + and so on is"}, {"start": 63.76, "duration": 5.399, "text": "just X right because X is equal to this"}, {"start": 67.0, "duration": 4.64, "text": "so here we have squ root of 5 plus and"}, {"start": 69.159, "duration": 4.6, "text": "here we have from here we have exactly"}, {"start": 71.64, "duration": 5.76, "text": "what is here which is X so we replace"}, {"start": 73.759, "duration": 9.281, "text": "that with X so right over"}, {"start": 77.4, "duration": 8.52, "text": "here this is what we have now < TK 5 + x"}, {"start": 83.04, "duration": 5.439, "text": "is = X right so we're going to square"}, {"start": 85.92, "duration": 5.239, "text": "both sides to just eliminate this Square"}, {"start": 88.479, "duration": 5.721, "text": "sign so let us Square Square both"}, {"start": 91.159, "duration": 8.881, "text": "sides so this will cancel this right so"}, {"start": 94.2, "duration": 9.08, "text": "here we have X2 is just equal to 5 + x"}, {"start": 100.04, "duration": 9.92, "text": "so subtracting 5 + x from both sides so"}, {"start": 103.28, "duration": 12.08, "text": "we just get x^2 - 5 - x = 0 so let us"}, {"start": 109.96, "duration": 8.199, "text": "put this equation well x^2 - x - 5 is"}, {"start": 115.36, "duration": 5.24, "text": "equal to Z so right over here I'm going"}, {"start": 118.159, "duration": 6.161, "text": "to make use of the"}, {"start": 120.6, "duration": 7.359, "text": "braga's quadratic formula right so we"}, {"start": 124.32, "duration": 9.999, "text": "have that X is just equal to B plus or"}, {"start": 127.959, "duration": 9.481, "text": "minus < TK b ^ 2 - 4 a c all/ 2 a so"}, {"start": 134.319, "duration": 7.361, "text": "right here the coefficient of X2 is a"}, {"start": 137.44, "duration": 8.64, "text": "right and that is what 1 here B is equal"}, {"start": 141.68, "duration": 7.639, "text": "to coent of X here -1 and C is equal to"}, {"start": 146.08, "duration": 4.96, "text": "-5 which is the constant right so this"}, {"start": 149.319, "duration": 2.64, "text": "is equal"}, {"start": 151.04, "duration": 6.16, "text": "B"}, {"start": 151.959, "duration": 10.161, "text": "is1 which is + 1 then plus or minus"}, {"start": 157.2, "duration": 8.959, "text": "square otk of b 2 is1 2 which is 1 - 4 a"}, {"start": 162.12, "duration": 8.36, "text": "c means 4 * 1 * -5 which is"}, {"start": 166.159, "duration": 8.08, "text": "-20 all 2 a 2 * 1 which is 2 so this is"}, {"start": 170.48, "duration": 5.679, "text": "equal to 1+ or minus Square Ro TK of you"}, {"start": 174.239, "duration": 5.761, "text": "know this opening this bracket we have"}, {"start": 176.159, "duration": 8.36, "text": "positive sign and 1 + 20 is just 21 or"}, {"start": 180.0, "duration": 6.239, "text": "by two right so now take note"}, {"start": 184.519, "duration": 5.601, "text": "that"}, {"start": 186.239, "duration": 7.0, "text": "this this infinite next"}, {"start": 190.12, "duration": 6.399, "text": "radical has a result which is positive"}, {"start": 193.239, "duration": 5.881, "text": "and not negative right so which means"}, {"start": 196.519, "duration": 7.64, "text": "that the negative value for x will be"}, {"start": 199.12, "duration": 12.24, "text": "ignored right so here we have X = to 1 +"}, {"start": 204.159, "duration": 9.921, "text": "the < TK of 21 / 2 or 1us the RO of 21 /"}, {"start": 211.36, "duration": 5.84, "text": "2 and this value is negative because"}, {"start": 214.08, "duration": 4.96, "text": "sare root of 21 is bigger than one right"}, {"start": 217.2, "duration": 4.759, "text": "so to subtract this from one will give"}, {"start": 219.04, "duration": 7.0, "text": "us a negative result so this is"}, {"start": 221.959, "duration": 5.881, "text": "cancelled and this is the right answer"}, {"start": 226.04, "duration": 3.6, "text": "thank you for watching do wa to"}, {"start": 227.84, "duration": 5.399, "text": "subscribe to our YouTube channel and"}, {"start": 229.64, "duration": 3.599, "text": "share our videos"}], "title": "An infinite nested radical", "description": "", "lengthSeconds": 231, "width": 636, "height": 360}
{"video_id": "WJ5a95d0HhU", "transcript": [{"start": 0.84, "duration": 4.8, "text": "hello good viewers welcome back once"}, {"start": 2.84, "duration": 4.88, "text": "again today we have this interesting"}, {"start": 5.64, "duration": 3.76, "text": "problem here and here we're going to"}, {"start": 7.72, "duration": 4.879, "text": "solve this"}, {"start": 9.4, "duration": 4.68, "text": "problem we have here that 98% people"}, {"start": 12.599, "duration": 5.561, "text": "fail this"}, {"start": 14.08, "duration": 6.119, "text": "problem obviously this is true most"}, {"start": 18.16, "duration": 4.16, "text": "people may think that the only solution"}, {"start": 20.199, "duration": 5.121, "text": "here is X isal to"}, {"start": 22.32, "duration": 6.08, "text": "3 but is something very important we"}, {"start": 25.32, "duration": 5.08, "text": "need to consider here let us recall"}, {"start": 28.4, "duration": 6.319, "text": "about the absolute value forun function"}, {"start": 30.4, "duration": 7.96, "text": "of X remember that the absolute value I"}, {"start": 34.719, "duration": 6.801, "text": "mean the sare root of x\u00b2 I mean is equal"}, {"start": 38.36, "duration": 5.359, "text": "to the absolute value of"}, {"start": 41.52, "duration": 5.76, "text": "x and"}, {"start": 43.719, "duration": 6.401, "text": "again we have another interesting"}, {"start": 47.28, "duration": 4.72, "text": "property of this absolute value function"}, {"start": 50.12, "duration": 5.52, "text": "the absolute value of"}, {"start": 52.0, "duration": 3.64, "text": "x is equal"}, {"start": 56.44, "duration": 8.799, "text": "tox if and only if x is less than zero"}, {"start": 61.359, "duration": 9.001, "text": "right and again the absolute value of x"}, {"start": 65.239, "duration": 7.121, "text": "is equal to X if an only if x is greater"}, {"start": 70.36, "duration": 4.64, "text": "than or equal to"}, {"start": 72.36, "duration": 5.96, "text": "zero now let's take a look at this"}, {"start": 75.0, "duration": 4.96, "text": "problem here we have square root of"}, {"start": 78.32, "duration": 3.759, "text": "x\u00b2"}, {"start": 79.96, "duration": 7.44, "text": "remember square"}, {"start": 82.079, "duration": 8.521, "text": "root of x\u00b2 is not equal"}, {"start": 87.4, "duration": 5.96, "text": "to < TK X"}, {"start": 90.6, "duration": 5.4, "text": "squared these are not equal because this"}, {"start": 93.36, "duration": 5.719, "text": "is where many students get"}, {"start": 96.0, "duration": 5.24, "text": "confused right now here we can see we"}, {"start": 99.079, "duration": 4.4, "text": "have a different tax and that is"}, {"start": 101.24, "duration": 5.199, "text": "something like this so the left hand"}, {"start": 103.479, "duration": 7.68, "text": "side here is just the absolute value of"}, {"start": 106.439, "duration": 7.601, "text": "x right and this is equal to three now"}, {"start": 111.159, "duration": 4.161, "text": "let's take note of this if the absolute"}, {"start": 114.04, "duration": 6.719, "text": "value of"}, {"start": 115.32, "duration": 8.839, "text": "x is equal to a then therefore"}, {"start": 120.759, "duration": 6.64, "text": "X is = plus or minus"}, {"start": 124.159, "duration": 9.08, "text": "a so from here we can see the solutions"}, {"start": 127.399, "duration": 10.041, "text": "that X is equal to+ or - 3 and here we"}, {"start": 133.239, "duration": 8.761, "text": "have the solution to the problem x = 3"}, {"start": 137.44, "duration": 7.32, "text": "and x = -3 thank you for watching please"}, {"start": 142.0, "duration": 6.92, "text": "kindly subscribe to my channel also like"}, {"start": 144.76, "duration": 4.16, "text": "comment and share bye-bye"}], "title": "Interesting problem | Can you find all solutions ?", "description": "", "lengthSeconds": 147, "width": 640, "height": 360}
{"video_id": "LxdaQoncMJo", "transcript": [{"start": 0.599, "duration": 4.2, "text": "hello good viewers welcome back once"}, {"start": 2.76, "duration": 4.72, "text": "again let's find the solution to the"}, {"start": 4.799, "duration": 5.241, "text": "following given radical"}, {"start": 7.48, "duration": 5.72, "text": "equation there are different ways of"}, {"start": 10.04, "duration": 5.319, "text": "solving this problem but in this video I"}, {"start": 13.2, "duration": 5.28, "text": "will start by making a"}, {"start": 15.359, "duration": 7.961, "text": "substitution first of all I will let"}, {"start": 18.48, "duration": 8.84, "text": "root of x = Y when you square both sides"}, {"start": 23.32, "duration": 7.439, "text": "we get X is = Y"}, {"start": 27.32, "duration": 6.079, "text": "2 now from here we have root X and that"}, {"start": 30.759, "duration": 6.441, "text": "is y so we write"}, {"start": 33.399, "duration": 7.361, "text": "y+ X here will be y^"}, {"start": 37.2, "duration": 5.519, "text": "2 and this is equal to 2 * root X will"}, {"start": 40.76, "duration": 5.2, "text": "be 2 *"}, {"start": 42.719, "duration": 5.32, "text": "y now from here let's try"}, {"start": 45.96, "duration": 9.079, "text": "simplifying from the left hand side we"}, {"start": 48.039, "duration": 11.0, "text": "have y^ 2 then + y from the right hand"}, {"start": 55.039, "duration": 6.48, "text": "side will be Min - 2 y right and this is"}, {"start": 59.039, "duration": 6.841, "text": "equal to zer"}, {"start": 61.519, "duration": 8.6, "text": "now from here we get y^ 2 now y - 2 y"}, {"start": 65.88, "duration": 6.64, "text": "will give us - Y and this is equal to Z"}, {"start": 70.119, "duration": 6.481, "text": "so let's go ahead and factor out y we"}, {"start": 72.52, "duration": 8.52, "text": "have Y into braet y - 1 this is equal to"}, {"start": 76.6, "duration": 9.519, "text": "Z and from here this implies that Y is"}, {"start": 81.04, "duration": 8.24, "text": "equal to0 or in the bracket y -1 is = 0"}, {"start": 86.119, "duration": 6.881, "text": "so from here we get solutions for y y is"}, {"start": 89.28, "duration": 7.24, "text": "= 0 or Y is ="}, {"start": 93.0, "duration": 7.119, "text": "1 now let's recall that X is actually"}, {"start": 96.52, "duration": 10.84, "text": "equal to Y2 right so"}, {"start": 100.119, "duration": 9.441, "text": "when Y is = 0 we get x is = 0 2 and that"}, {"start": 107.36, "duration": 5.52, "text": "is equal to"}, {"start": 109.56, "duration": 3.32, "text": "Z and"}, {"start": 113.119, "duration": 11.32, "text": "again when Y is = 1 X is = 1 2 and that"}, {"start": 120.439, "duration": 8.96, "text": "is one so from here we have Solutions we"}, {"start": 124.439, "duration": 8.681, "text": "have X is equal to Z and we have that"}, {"start": 129.399, "duration": 5.881, "text": "also X is = to 1 and these are the"}, {"start": 133.12, "duration": 4.64, "text": "solutions thank you for watching please"}, {"start": 135.28, "duration": 6.56, "text": "kindly subscribe to my channel also like"}, {"start": 137.76, "duration": 4.08, "text": "comment and share bye-bye"}], "title": "Indians | Interesting surdic equation | #maths #mathematics", "description": "", "lengthSeconds": 140, "width": 522, "height": 360}
{"video_id": "FvVqATrgEQ0", "transcript": [{"start": 0.32, "duration": 5.64, "text": "hello good viewers in this video I'll be"}, {"start": 3.0, "duration": 5.88, "text": "providing a solution to this nice"}, {"start": 5.96, "duration": 6.32, "text": "Olympia mat problem here here we given"}, {"start": 8.88, "duration": 7.479, "text": "that the < TK of 7 to the^ of < TK of 24"}, {"start": 12.28, "duration": 7.159, "text": "- < TK X is = 49 here we asked to find"}, {"start": 16.359, "duration": 4.76, "text": "the value of values of X please if you"}, {"start": 19.439, "duration": 3.401, "text": "have not subscribed to my Channel please"}, {"start": 21.119, "duration": 3.92, "text": "kindly support this channel by"}, {"start": 22.84, "duration": 4.88, "text": "subscribing please like comment and"}, {"start": 25.039, "duration": 8.04, "text": "share here we're giving < TK of 7 to"}, {"start": 27.72, "duration": 8.2, "text": "the^ ofun of 24 - < TK of 8 X is = to 49"}, {"start": 33.079, "duration": 7.841, "text": "here we can put this in bracket otk of 7"}, {"start": 35.92, "duration": 10.36, "text": "is the same as 7^ of 1 all/ 2 right then"}, {"start": 40.92, "duration": 9.24, "text": "ra the power of 24 - < TK X and this is"}, {"start": 46.28, "duration": 6.56, "text": "equal to 49 right so according to the"}, {"start": 50.16, "duration": 4.6, "text": "law of indices you know we can multiply"}, {"start": 52.84, "duration": 3.12, "text": "these Powers here together so we're"}, {"start": 54.76, "duration": 7.599, "text": "going to have"}, {"start": 55.96, "duration": 10.159, "text": "7 of square root of 24 - < TK of X all"}, {"start": 62.359, "duration": 7.001, "text": "all over 2 this is equal to 49 is the"}, {"start": 66.119, "duration": 5.841, "text": "same as 7^ of 2 so we can see from both"}, {"start": 69.36, "duration": 5.56, "text": "sides we have equal bases so we can"}, {"start": 71.96, "duration": 7.76, "text": "easily equate the powers right so from"}, {"start": 74.92, "duration": 8.12, "text": "here we have square root of 24 - < TK x"}, {"start": 79.72, "duration": 7.16, "text": "all/ 2 this is equal to 2 so cross"}, {"start": 83.04, "duration": 9.0, "text": "multiply so here we get sare < TK of 24"}, {"start": 86.88, "duration": 7.279, "text": "- < TK X is = 2 * two here that is four"}, {"start": 92.04, "duration": 4.48, "text": "right so from here we're going to square"}, {"start": 94.159, "duration": 5.561, "text": "both sides of this equation here we"}, {"start": 96.52, "duration": 4.919, "text": "Square here and here we Square here"}, {"start": 99.72, "duration": 3.439, "text": "right so we can see that this square and"}, {"start": 101.439, "duration": 4.96, "text": "square root cancel out so we're left"}, {"start": 103.159, "duration": 6.521, "text": "with 24 minus theare root of x this is"}, {"start": 106.399, "duration": 6.521, "text": "equal to 4 S that is 16 so from here"}, {"start": 109.68, "duration": 5.16, "text": "we're going to subtract 16 sorry from"}, {"start": 112.92, "duration": 4.44, "text": "here we're going to collect L ter so"}, {"start": 114.84, "duration": 5.52, "text": "here we're going to have < TK of X is"}, {"start": 117.36, "duration": 8.24, "text": "also equal to 24 -"}, {"start": 120.36, "duration": 10.56, "text": "16 right so from here we getun X is = 24"}, {"start": 125.6, "duration": 7.359, "text": "- 16 that is equal 8 okay so from here"}, {"start": 130.92, "duration": 6.16, "text": "now we have the square root of x is"}, {"start": 132.959, "duration": 8.36, "text": "equal to 8 so we're going to square both"}, {"start": 137.08, "duration": 7.08, "text": "sides here we have of X2 this is equal"}, {"start": 141.319, "duration": 6.601, "text": "to 8 2 so square and square root cancel"}, {"start": 144.16, "duration": 5.799, "text": "out so here we get x = 82 and that is 64"}, {"start": 147.92, "duration": 4.0, "text": "and this is the right answer thank you"}, {"start": 149.959, "duration": 5.0, "text": "for watching please kindly support to my"}, {"start": 151.92, "duration": 6.039, "text": "channel subscribe like comment and share"}, {"start": 154.959, "duration": 3.0, "text": "bye-bye"}], "title": "Interesting problem for Olympiad.", "description": "", "lengthSeconds": 156, "width": 640, "height": 360}
{"video_id": "CjEsxI-UO5k", "transcript": [{"start": 0.56, "duration": 4.0, "text": "hello good day everyone welcome back"}, {"start": 2.2, "duration": 5.52, "text": "once again today we have this nice and"}, {"start": 4.56, "duration": 8.64, "text": "interesting exponential equation we have"}, {"start": 7.72, "duration": 7.959, "text": "2^ root X is = 16 ^ x / 4 and here we'll"}, {"start": 13.2, "duration": 4.96, "text": "be solving for the values of X that we"}, {"start": 15.679, "duration": 5.641, "text": "satisfy this awesome looking"}, {"start": 18.16, "duration": 5.84, "text": "equation now let's get into this video"}, {"start": 21.32, "duration": 5.52, "text": "from the left hand side we have 2 to the"}, {"start": 24.0, "duration": 6.039, "text": "power of root of x then this will be"}, {"start": 26.84, "duration": 7.12, "text": "equal to remember 16 can be expressed in"}, {"start": 30.039, "duration": 8.161, "text": "index power of 2 so therefore 16 is 2 to"}, {"start": 33.96, "duration": 7.96, "text": "the^ 4 then remember we have 4 or I mean"}, {"start": 38.2, "duration": 6.879, "text": "that is X right yes so x / 4 as the"}, {"start": 41.92, "duration": 7.2, "text": "power here now let's take note of this"}, {"start": 45.079, "duration": 8.081, "text": "property a the^ B to the^ C this is"}, {"start": 49.12, "duration": 7.52, "text": "equal to a^ B * C so from here we have"}, {"start": 53.16, "duration": 6.32, "text": "from the left hand side 2 the^ < TK X"}, {"start": 56.64, "duration": 8.239, "text": "then this will be equal to 2 to the^ of"}, {"start": 59.48, "duration": 5.399, "text": "4 multi by x /"}, {"start": 65.88, "duration": 5.76, "text": "4 okay so from here we have this nice"}, {"start": 69.6, "duration": 4.519, "text": "cancellation remember this is 4 all over"}, {"start": 71.64, "duration": 6.36, "text": "one so here this four we cancel this"}, {"start": 74.119, "duration": 7.921, "text": "four here so we have 2 to the^ of otk X"}, {"start": 78.0, "duration": 7.6, "text": "this will be equal to 2 to the^ of"}, {"start": 82.04, "duration": 5.2, "text": "X so from here we have equal basis so"}, {"start": 85.6, "duration": 4.32, "text": "from here we're going to equate the"}, {"start": 87.24, "duration": 6.559, "text": "powers so this implies from here that"}, {"start": 89.92, "duration": 6.8, "text": "that the square root of x is equal to"}, {"start": 93.799, "duration": 4.841, "text": "X from here to get rid of this root sign"}, {"start": 96.72, "duration": 4.56, "text": "let's go ahead and square both sides of"}, {"start": 98.64, "duration": 5.799, "text": "this equation so we know that the square"}, {"start": 101.28, "duration": 4.72, "text": "and the root signs are en right so the"}, {"start": 104.439, "duration": 3.201, "text": "square and the root sign they are en so"}, {"start": 106.0, "duration": 5.479, "text": "they will cancel out from here so we"}, {"start": 107.64, "duration": 6.92, "text": "have here x is ="}, {"start": 111.479, "duration": 5.28, "text": "x\u00b2 now let's switch the sides of this"}, {"start": 114.56, "duration": 5.64, "text": "equation so from the right hand side let"}, {"start": 116.759, "duration": 6.96, "text": "us make it the left hand side so x\u00b2 is"}, {"start": 120.2, "duration": 7.68, "text": "equal to X subtracting X from both side"}, {"start": 123.719, "duration": 6.081, "text": "we get x^2 - x this is equal 0 so we can"}, {"start": 127.88, "duration": 5.639, "text": "see from here x is common so we Factor"}, {"start": 129.8, "duration": 7.88, "text": "it out and we have x -1 which is equal"}, {"start": 133.519, "duration": 8.281, "text": "to0 okay so from"}, {"start": 137.68, "duration": 10.279, "text": "here we have x * x - y = 0 so we have"}, {"start": 141.8, "duration": 9.24, "text": "either x is = 0 or X - 1 is = 0 and from"}, {"start": 147.959, "duration": 6.081, "text": "here x is = 1 so therefore we have only"}, {"start": 151.04, "duration": 7.4, "text": "two real solutions that satisfy the"}, {"start": 154.04, "duration": 8.08, "text": "given equation so we have xal 0 or X is"}, {"start": 158.44, "duration": 6.48, "text": "equal to 1 thank you for watching please"}, {"start": 162.12, "duration": 7.32, "text": "kindly subscribe to my channel also like"}, {"start": 164.92, "duration": 4.52, "text": "comment and share bye-bye"}], "title": "Interesting exponential equation | can you solve ?", "description": "", "lengthSeconds": 168, "width": 640, "height": 360}
{"video_id": "tkWXjJDDrK0", "transcript": [{"start": 0.799, "duration": 4.441, "text": "hello good day viewers welcome back once"}, {"start": 2.52, "duration": 5.64, "text": "again today let us solve this simple"}, {"start": 5.24, "duration": 6.04, "text": "exponential equation here we be solving"}, {"start": 8.16, "duration": 9.52, "text": "for the value of x so from here we have"}, {"start": 11.28, "duration": 9.759, "text": "8^ of x + 8 to ^ of X which is equal to"}, {"start": 17.68, "duration": 6.32, "text": "1 right so from the left hand side h8"}, {"start": 21.039, "duration": 8.56, "text": "to^ of X is common so let's factor out"}, {"start": 24.0, "duration": 11.96, "text": "8^ X so that will become 8 ^ of X into"}, {"start": 29.599, "duration": 12.041, "text": "bracket 8 a^ x / a^ X we left with 1"}, {"start": 35.96, "duration": 9.4, "text": "plus a^ x / 8^ X that would be 1 right"}, {"start": 41.64, "duration": 9.04, "text": "then this is equal to 1 so from here we"}, {"start": 45.36, "duration": 10.16, "text": "have 8 to the^ X MTI by 1 + 1 from here"}, {"start": 50.68, "duration": 7.879, "text": "is 2 then this is equal to 1 so from"}, {"start": 55.52, "duration": 9.12, "text": "here let's divide both sides by two so"}, {"start": 58.559, "duration": 11.361, "text": "we have 8 to^ x x * 2 / 2 this will be"}, {"start": 64.64, "duration": 9.64, "text": "equal to 1 / 2 so from here 2 we cancel"}, {"start": 69.92, "duration": 9.04, "text": "out 2 so we have here 8 to^ x which is"}, {"start": 74.28, "duration": 9.56, "text": "equal to 1 / 2 so let's take note of"}, {"start": 78.96, "duration": 8.519, "text": "this property we know 1 all a to the^ n"}, {"start": 83.84, "duration": 13.44, "text": "this is equal to a to the^"}, {"start": 87.479, "duration": 12.68, "text": "n right so here 1 / 2 is = to 1 / 2^ 1"}, {"start": 97.28, "duration": 7.839, "text": "which is equal to using this algebraic"}, {"start": 100.159, "duration": 4.96, "text": "identity here so this will be 2"}, {"start": 106.6, "duration": 8.439, "text": "of1 so from here we have 8 to the^ of X"}, {"start": 111.56, "duration": 6.519, "text": "which is equal to 2 ra power"}, {"start": 115.039, "duration": 6.641, "text": "of1 from here we know this base here can"}, {"start": 118.079, "duration": 6.601, "text": "be expressed in index power of 2 so that"}, {"start": 121.68, "duration": 6.28, "text": "is 2 to the^ 3 which is equal to this 8"}, {"start": 124.68, "duration": 5.32, "text": "then we have a power of X this is equal"}, {"start": 127.96, "duration": 5.2, "text": "to"}, {"start": 130.0, "duration": 7.56, "text": "2^1 from here let us consider another"}, {"start": 133.16, "duration": 9.52, "text": "algebraic identity We Know a^ M in"}, {"start": 137.56, "duration": 10.72, "text": "bracket n is = to a^ M *"}, {"start": 142.68, "duration": 10.32, "text": "n so from here this is equal to 2^ 3 * X"}, {"start": 148.28, "duration": 7.72, "text": "which is 3x then this is equal to 2"}, {"start": 153.0, "duration": 6.92, "text": "of1 we can see from both sides of this"}, {"start": 156.0, "duration": 5.72, "text": "equation we have the same base right so"}, {"start": 159.92, "duration": 5.52, "text": "we're going to equate the power so this"}, {"start": 161.72, "duration": 6.519, "text": "implies from here the power of two here"}, {"start": 165.44, "duration": 5.96, "text": "3x is equal to the power of two from the"}, {"start": 168.239, "duration": 5.201, "text": "right hand side -1 since we're solving"}, {"start": 171.4, "duration": 7.36, "text": "for x let's divide both side by the"}, {"start": 173.44, "duration": 8.2, "text": "coefficient of x soide by 3/ 3 so this"}, {"start": 178.76, "duration": 8.72, "text": "will cancel this and here we have our"}, {"start": 181.64, "duration": 8.599, "text": "solution X is = -1 / 3 and this is the"}, {"start": 187.48, "duration": 5.28, "text": "right answer thank you for watching"}, {"start": 190.239, "duration": 8.08, "text": "please kindly subscribe to my channel"}, {"start": 192.76, "duration": 5.559, "text": "also like comment and share bye-bye"}], "title": "Tricky exponential equation", "description": "", "lengthSeconds": 197, "width": 640, "height": 360}
{"video_id": "UdfwSuWJlmQ", "transcript": [{"start": 0.64, "duration": 4.96, "text": "hello good viewers welcome back once"}, {"start": 2.56, "duration": 5.8, "text": "again today we have an indefinite"}, {"start": 5.6, "duration": 6.68, "text": "integral here we're going to evaluate"}, {"start": 8.36, "duration": 6.279, "text": "this using integration by parts"}, {"start": 12.28, "duration": 6.28, "text": "formula I know we're very familiar with"}, {"start": 14.639, "duration": 6.601, "text": "this property here the integral of U DV"}, {"start": 18.56, "duration": 6.4, "text": "this is equal to u v minus the integral"}, {"start": 21.24, "duration": 6.199, "text": "of V duu the right over here we have"}, {"start": 24.96, "duration": 7.799, "text": "this indefinite integral here integral"}, {"start": 27.439, "duration": 7.761, "text": "of x * e^ X DX X now let us make the"}, {"start": 32.759, "duration": 7.48, "text": "following substitutions in terms of u"}, {"start": 35.2, "duration": 10.4, "text": "and v here we're going to let u"}, {"start": 40.239, "duration": 8.361, "text": "= x right so if you differentiate this"}, {"start": 45.6, "duration": 5.76, "text": "we have du this is equal to the"}, {"start": 48.6, "duration": 5.799, "text": "derivative of x with respect to X is 1"}, {"start": 51.36, "duration": 6.519, "text": "and here we have"}, {"start": 54.399, "duration": 8.401, "text": "DX okay the next substitutions in terms"}, {"start": 57.879, "duration": 7.801, "text": "of V we have V is equal to now the"}, {"start": 62.8, "duration": 8.44, "text": "integral of e to the power of"}, {"start": 65.68, "duration": 7.72, "text": "X the integral of e to the power of X DX"}, {"start": 71.24, "duration": 5.64, "text": "so if we differentiate this we actually"}, {"start": 73.4, "duration": 7.079, "text": "have DV is equal to the integral of e"}, {"start": 76.88, "duration": 6.599, "text": "that is e sorry that is d v if you"}, {"start": 80.479, "duration": 6.441, "text": "differentiate the integral of e to^ x"}, {"start": 83.479, "duration": 4.64, "text": "actually here we have V is = e to the"}, {"start": 86.92, "duration": 3.92, "text": "power of"}, {"start": 88.119, "duration": 8.96, "text": "X so here if if you differentiate both"}, {"start": 90.84, "duration": 9.599, "text": "side we get DV this is equal to e to^ x"}, {"start": 97.079, "duration": 8.161, "text": "DX Now using this property here we have"}, {"start": 100.439, "duration": 10.121, "text": "u v and here this is U and this is V so"}, {"start": 105.24, "duration": 9.4, "text": "here we have x * e to the power of X"}, {"start": 110.56, "duration": 8.32, "text": "then minus the integral of v and what is"}, {"start": 114.64, "duration": 7.2, "text": "v v is e to the power of X and we have"}, {"start": 118.88, "duration": 2.96, "text": "here"}, {"start": 123.399, "duration": 7.64, "text": "DX okay because du here is equal to DX"}, {"start": 127.759, "duration": 8.2, "text": "as you can see here so this from here we"}, {"start": 131.039, "duration": 9.001, "text": "have x * e^ of x minus the integral of"}, {"start": 135.959, "duration": 8.241, "text": "e^ x remains e^ of X right and here we"}, {"start": 140.04, "duration": 5.72, "text": "have plus C and this is the right answer"}, {"start": 144.2, "duration": 3.6, "text": "thank you for watching please kindly"}, {"start": 145.76, "duration": 6.32, "text": "subscribe to my channel also like"}, {"start": 147.8, "duration": 4.28, "text": "comment and share bye-bye but"}], "title": "Integration by parts | #maths #mathematics", "description": "", "lengthSeconds": 150, "width": 560, "height": 360}
{"video_id": "gj11pxf7N8E", "transcript": [{"start": 0.64, "duration": 5.04, "text": "hello welcome back once again in this"}, {"start": 2.56, "duration": 6.2, "text": "video let us solve for the value of x"}, {"start": 5.68, "duration": 6.36, "text": "that will satisfy this equation from the"}, {"start": 8.76, "duration": 7.079, "text": "left hand side we have the < TK of X"}, {"start": 12.04, "duration": 5.12, "text": "plus the < TK of X and this is equal to"}, {"start": 15.839, "duration": 4.041, "text": "the < TK of"}, {"start": 17.16, "duration": 6.4, "text": "8 from the left hand side we know < TK"}, {"start": 19.88, "duration": 6.799, "text": "of x + < TK X will give us 2 * the < TK"}, {"start": 23.56, "duration": 4.119, "text": "X then this is equal to the square root"}, {"start": 26.679, "duration": 6.161, "text": "of"}, {"start": 27.679, "duration": 8.321, "text": "8 now we know that squ < TK of 8 we can"}, {"start": 32.84, "duration": 5.8, "text": "actually rewrite it right from the left"}, {"start": 36.0, "duration": 4.559, "text": "hand side here we have 2 * root of x so"}, {"start": 38.64, "duration": 6.12, "text": "this will be equal to the square Ro TK"}, {"start": 40.559, "duration": 9.121, "text": "of 8 here is as 4 *"}, {"start": 44.76, "duration": 9.0, "text": "2 then we have here 2un x is = to so"}, {"start": 49.68, "duration": 8.96, "text": "we're going to write this as < TK 4 MTI"}, {"start": 53.76, "duration": 8.36, "text": "the < TK of 2 so from here we have 2un X"}, {"start": 58.64, "duration": 7.28, "text": "this is equal of 4 is 2 then multiply by"}, {"start": 62.12, "duration": 7.52, "text": "< tk2 right so from here two we cancel"}, {"start": 65.92, "duration": 7.16, "text": "two from both sides we have your < TK X"}, {"start": 69.64, "duration": 6.119, "text": "is = to < TK 2 to solve for x simply"}, {"start": 73.08, "duration": 5.399, "text": "Square both"}, {"start": 75.759, "duration": 4.68, "text": "sides Square cancel square root square"}, {"start": 78.479, "duration": 5.241, "text": "cancel square root and here we get our"}, {"start": 80.439, "duration": 5.68, "text": "right answer that X is = to 2 and this"}, {"start": 83.72, "duration": 4.16, "text": "is the correct answer thank you for"}, {"start": 86.119, "duration": 5.121, "text": "watching please kindly subscribe to my"}, {"start": 87.88, "duration": 6.36, "text": "channel also like comment and share"}, {"start": 91.24, "duration": 3.0, "text": "bye-bye"}], "title": "Simple radical equation | Can you solve ?", "description": "", "lengthSeconds": 93, "width": 428, "height": 240}
{"video_id": "1_Hh-w5ZBfw", "transcript": [{"start": 0.68, "duration": 4.04, "text": "hello good viewers welcome back once"}, {"start": 2.48, "duration": 4.919, "text": "again today we're going to solve for the"}, {"start": 4.72, "duration": 6.799, "text": "all real values of X that will satisfy"}, {"start": 7.399, "duration": 8.8, "text": "this given exponential equation we have"}, {"start": 11.519, "duration": 7.281, "text": "e^ x\u00b2 this is equal to e^ of X now the"}, {"start": 16.199, "duration": 5.561, "text": "first thing to do is to Ln both sides of"}, {"start": 18.8, "duration": 6.96, "text": "this equation if we have the idea that"}, {"start": 21.76, "duration": 7.439, "text": "the Ln of e to any power is equal to"}, {"start": 25.76, "duration": 7.2, "text": "that power so from here we have the Ln"}, {"start": 29.199, "duration": 7.921, "text": "of e to the of x\u00b2 this is equal to the"}, {"start": 32.96, "duration": 6.24, "text": "Ln of e to power of X so making use of"}, {"start": 37.12, "duration": 3.8, "text": "this property from here we have the"}, {"start": 39.2, "duration": 4.64, "text": "power of E which is"}, {"start": 40.92, "duration": 6.56, "text": "x\u00b2 then this is equal to the power of E"}, {"start": 43.84, "duration": 6.559, "text": "here which is X from here do not make"}, {"start": 47.48, "duration": 6.16, "text": "mistake of dividing both side by X if"}, {"start": 50.399, "duration": 6.441, "text": "you do that you will lose a solution so"}, {"start": 53.64, "duration": 6.719, "text": "the best thing to do here is to subtract"}, {"start": 56.84, "duration": 6.239, "text": "X from both sides and here equals zero"}, {"start": 60.359, "duration": 7.281, "text": "from here we factor out X and we're left"}, {"start": 63.079, "duration": 7.08, "text": "with x\u00b2 / x with X then x / X"}, {"start": 67.64, "duration": 6.839, "text": "is1 and this is equal"}, {"start": 70.159, "duration": 8.561, "text": "to0 so from here we have X is equal to 0"}, {"start": 74.479, "duration": 7.481, "text": "or the one in bracket X - 1 = 0 from"}, {"start": 78.72, "duration": 6.399, "text": "here we have X is = 1 so from here we"}, {"start": 81.96, "duration": 8.56, "text": "have the following solution that x is ="}, {"start": 85.119, "duration": 7.841, "text": "0 and x is = 1 and this are the real"}, {"start": 90.52, "duration": 4.959, "text": "solutions thank you for watching please"}, {"start": 92.96, "duration": 6.56, "text": "kindly subscribe to my channel also like"}, {"start": 95.479, "duration": 4.041, "text": "comment and share bye-bye"}], "title": "Germany | Nice exponential equation | #maths #mathematics", "description": "", "lengthSeconds": 98, "width": 640, "height": 360}
{"video_id": "4YinRyiLXvg", "transcript": [{"start": 0.799, "duration": 3.841, "text": "hello welcome back once again today"}, {"start": 2.879, "duration": 4.121, "text": "we're going to evaluate this with"}, {"start": 4.64, "duration": 8.56, "text": "calculator and here is how we're going"}, {"start": 7.0, "duration": 11.64, "text": "to do this we have 6^ 12 / 12 power 6"}, {"start": 13.2, "duration": 10.68, "text": "which we can write as 6^ 12 ided by here"}, {"start": 18.64, "duration": 8.399, "text": "12 is the same as 2 * 6 then power of 6"}, {"start": 23.88, "duration": 7.68, "text": "now with this basic property in algebra"}, {"start": 27.039, "duration": 9.881, "text": "remember when we have a * B all of n we"}, {"start": 31.56, "duration": 9.4, "text": "can express this as a to the^ of n * B"}, {"start": 36.92, "duration": 9.04, "text": "to the power of n right so from here"}, {"start": 40.96, "duration": 11.04, "text": "this is going to be equal to 6^ 12 /"}, {"start": 45.96, "duration": 11.0, "text": "here we get 2 ^ 6 * 6^"}, {"start": 52.0, "duration": 8.719, "text": "6 from here we can express this as 1 all"}, {"start": 56.96, "duration": 10.479, "text": "over here 2^ 6"}, {"start": 60.719, "duration": 10.281, "text": "multiply by here 6^ 12 / 6^"}, {"start": 67.439, "duration": 7.281, "text": "6 now let us take note of this again"}, {"start": 71.0, "duration": 5.08, "text": "when we have a to the power of n / B to"}, {"start": 74.72, "duration": 4.759, "text": "the power of"}, {"start": 76.08, "duration": 6.679, "text": "n i mean a to the^ of n so sorry for"}, {"start": 79.479, "duration": 5.481, "text": "that so we have a the power n ided by"}, {"start": 82.759, "duration": 5.601, "text": "the same base here a to the power of"}, {"start": 84.96, "duration": 8.24, "text": "another power we can actually subtract"}, {"start": 88.36, "duration": 10.52, "text": "the power that will be N - n right so"}, {"start": 93.2, "duration": 8.599, "text": "from here we have this equal 1 all 2^ 6"}, {"start": 98.88, "duration": 10.239, "text": "then multip by here this is going to be"}, {"start": 101.799, "duration": 11.481, "text": "6^ 12 - 6 so from here we get 1 2^ 6"}, {"start": 109.119, "duration": 8.04, "text": "multip by here this will become 6 ^ 6"}, {"start": 113.28, "duration": 9.439, "text": "then from here put this over one 6^ 6 *"}, {"start": 117.159, "duration": 7.801, "text": "1 will give us 6^ 6 / 2^ 6 * 1 will give"}, {"start": 122.719, "duration": 4.801, "text": "us 2^"}, {"start": 124.96, "duration": 4.719, "text": "6 there's another nice property we're"}, {"start": 127.52, "duration": 8.16, "text": "going to take note of"}, {"start": 129.679, "duration": 8.481, "text": "again when we have a^ of m / B the^ of M"}, {"start": 135.68, "duration": 6.16, "text": "since we have the same Powers we can"}, {"start": 138.16, "duration": 8.52, "text": "rewrite it as a / B all to the power of"}, {"start": 141.84, "duration": 10.16, "text": "M right so from here that will"}, {"start": 146.68, "duration": 9.8, "text": "be 6 / 2 all power of 6 and from here 6"}, {"start": 152.0, "duration": 7.84, "text": "2 is 3 we get 3 to the^ of 6 which is"}, {"start": 156.48, "duration": 6.679, "text": "equal to"}, {"start": 159.84, "duration": 5.16, "text": "729 and this is the right answer thank"}, {"start": 163.159, "duration": 4.121, "text": "you for watching please kindly subscribe"}, {"start": 165.0, "duration": 5.76, "text": "to my channel also like comment and"}, {"start": 167.28, "duration": 3.48, "text": "share bye-bye"}], "title": "Can you simplify without calculators ?", "description": "", "lengthSeconds": 169, "width": 566, "height": 360}
{"video_id": "bY4t288USiU", "transcript": [{"start": 0.08, "duration": 3.359, "text": "hello good day everyone welcome back"}, {"start": 1.88, "duration": 4.36, "text": "once again today we're going to solve"}, {"start": 3.439, "duration": 6.12, "text": "for the value of x in the following"}, {"start": 6.24, "duration": 5.24, "text": "Olympiad M problem yes this is an"}, {"start": 9.559, "duration": 4.681, "text": "interesting equation we are going to"}, {"start": 11.48, "duration": 5.6, "text": "find that number which when four is"}, {"start": 14.24, "duration": 4.6, "text": "raised to the power of that number then"}, {"start": 17.08, "duration": 4.08, "text": "that will give us the reciprocal of that"}, {"start": 18.84, "duration": 4.84, "text": "number and what is this number let's go"}, {"start": 21.16, "duration": 5.8, "text": "ahead and figure it out okay this"}, {"start": 23.68, "duration": 5.679, "text": "question is very simple right so let's"}, {"start": 26.96, "duration": 4.36, "text": "get into the video from the right hand"}, {"start": 29.359, "duration": 7.081, "text": "side I want to re write the equation as"}, {"start": 31.32, "duration": 7.12, "text": "1 /x this is = to 4 of X now the next"}, {"start": 36.44, "duration": 5.2, "text": "thing to do here is"}, {"start": 38.44, "duration": 6.24, "text": "to is to get rid of this power of four"}, {"start": 41.64, "duration": 5.32, "text": "here which is which is X so for us to"}, {"start": 44.68, "duration": 5.48, "text": "get rid of this x we need to raise both"}, {"start": 46.96, "duration": 6.119, "text": "sides of this equation to the reciprocal"}, {"start": 50.16, "duration": 8.32, "text": "of X which is 1 /x"}, {"start": 53.079, "duration": 8.0, "text": "right and here by 1 /x so from here we"}, {"start": 58.48, "duration": 7.72, "text": "get um that is 1"}, {"start": 61.079, "duration": 7.72, "text": "/x of 1 /x this will give us from here"}, {"start": 66.2, "duration": 9.32, "text": "we know we can multiply This Together"}, {"start": 68.799, "duration": 8.601, "text": "recall that M of b n isal m b * n so we"}, {"start": 75.52, "duration": 6.12, "text": "multiply This Together X time the"}, {"start": 77.4, "duration": 4.24, "text": "reciprocal is 1 so here we have"}, {"start": 83.28, "duration": 7.32, "text": "four so from here we have 1 /x of 1 /x"}, {"start": 88.52, "duration": 6.04, "text": "this is equal to 4 is the SU"}, {"start": 90.6, "duration": 6.04, "text": "2 2 so from here this implies that"}, {"start": 94.56, "duration": 5.76, "text": "either we equate the bases or equate the"}, {"start": 96.64, "duration": 6.92, "text": "powers so because we have um a pattern"}, {"start": 100.32, "duration": 6.479, "text": "of exponents here the base 1 / x isal 2"}, {"start": 103.56, "duration": 8.559, "text": "and also the power 1/ x is = 2 so we"}, {"start": 106.799, "duration": 9.201, "text": "have 1 /x this is equal to 2 okay now"}, {"start": 112.119, "duration": 7.161, "text": "let's um cross multiply x * 2 we give us"}, {"start": 116.0, "duration": 6.84, "text": "2x this is equal to 1 * 1 that is 1"}, {"start": 119.28, "duration": 7.32, "text": "divide both by 2 and here we get x = 1 /"}, {"start": 122.84, "duration": 5.479, "text": "2 and this is the solution thank you for"}, {"start": 126.6, "duration": 3.719, "text": "watching please kindly subscribe to my"}, {"start": 128.319, "duration": 5.92, "text": "YouTube channel also like comment and"}, {"start": 130.319, "duration": 3.92, "text": "share bye-bye"}], "title": "Olympiad Math | Interesting exponential equation | #maths #olympiad", "description": "", "lengthSeconds": 133, "width": 640, "height": 360}
{"video_id": "PSmRrIdSNFs", "transcript": [{"start": 0.359, "duration": 3.001, "text": "good everyone viewers here welcome back"}, {"start": 2.04, "duration": 4.5, "text": "to my Channel today we have the"}, {"start": 3.36, "duration": 5.76, "text": "following equation 3.8 times 2 raised by"}, {"start": 6.54, "duration": 5.46, "text": "x squared is equal to one where X to"}, {"start": 9.12, "duration": 4.559, "text": "find the value of values of X okay"}, {"start": 12.0, "duration": 4.26, "text": "now I'm going to start by logging both"}, {"start": 13.679, "duration": 5.041, "text": "sides of this equation remember that the"}, {"start": 16.26, "duration": 8.34, "text": "logarithm of a times B"}, {"start": 18.72, "duration": 9.84, "text": "is the same as log a plus log B okay now"}, {"start": 24.6, "duration": 6.06, "text": "I have log of 3 raised by S times 2"}, {"start": 28.56, "duration": 4.679, "text": "raised to power x squared is equal to"}, {"start": 30.66, "duration": 4.32, "text": "log of 1."}, {"start": 33.239, "duration": 6.901, "text": "now I have"}, {"start": 34.98, "duration": 7.259, "text": "log of 3 raised for X here plus log of 2"}, {"start": 40.14, "duration": 5.099, "text": "raised power x squared is equal to log"}, {"start": 42.239, "duration": 5.521, "text": "of 1 is 0. now using the property of"}, {"start": 45.239, "duration": 4.561, "text": "logarithm so this x we multiply the"}, {"start": 47.76, "duration": 3.72, "text": "entire logarithm here and X we multiply"}, {"start": 49.8, "duration": 7.62, "text": "data logarithm here"}, {"start": 51.48, "duration": 9.239, "text": "so I have a x Times log of 3 plus x"}, {"start": 57.42, "duration": 6.72, "text": "squared times log of 2 is equal to 0."}, {"start": 60.719, "duration": 6.601, "text": "now I'm going to factor out X here so I"}, {"start": 64.14, "duration": 6.36, "text": "have X to be factored out X log 3"}, {"start": 67.32, "duration": 6.06, "text": "divided by S we are left with log 3 plus"}, {"start": 70.5, "duration": 6.42, "text": "x square log 2 divided by X we are left"}, {"start": 73.38, "duration": 6.48, "text": "with X log 2 close bracket is equal to"}, {"start": 76.92, "duration": 9.059, "text": "zero so from here we have either x equal"}, {"start": 79.86, "duration": 9.72, "text": "to 0 or you know log 3 plus X log 2"}, {"start": 85.979, "duration": 4.621, "text": "equals zero and here we have S equals 0"}, {"start": 89.58, "duration": 3.24, "text": "or"}, {"start": 90.6, "duration": 5.4, "text": "you know X log"}, {"start": 92.82, "duration": 4.86, "text": "2 is equal to you know if this log 3"}, {"start": 96.0, "duration": 3.84, "text": "should cross over this side so it will"}, {"start": 97.68, "duration": 5.28, "text": "change to a negative"}, {"start": 99.84, "duration": 6.48, "text": "log three and here we have X is equal to"}, {"start": 102.96, "duration": 5.4, "text": "zero or divided by the question of s"}, {"start": 106.32, "duration": 5.939, "text": "this is over log 2 here"}, {"start": 108.36, "duration": 7.14, "text": "and here over log 2 log 2 cancel log 2"}, {"start": 112.259, "duration": 7.081, "text": "so half s is equal to negative log three"}, {"start": 115.5, "duration": 6.18, "text": "all over log two so we have come to the"}, {"start": 119.34, "duration": 4.62, "text": "end of today's video that"}, {"start": 121.68, "duration": 4.619, "text": "in the above equation"}, {"start": 123.96, "duration": 3.72, "text": "the values of s will have S is equal to"}, {"start": 126.299, "duration": 5.701, "text": "zero or"}, {"start": 127.68, "duration": 7.559, "text": "negative log three all over"}, {"start": 132.0, "duration": 4.5, "text": "log two thank you for viewing and please"}, {"start": 135.239, "duration": 4.941, "text": "make sure you subscribe to my YouTube"}, {"start": 136.5, "duration": 3.68, "text": "channel for more interesting videos"}], "title": "An interesting exponential equation. Feel free to solve !", "description": "", "lengthSeconds": 139, "width": 636, "height": 360}
{"video_id": "3qnQMArWHkI", "transcript": [{"start": 0.84, "duration": 3.56, "text": "hello good viewers welcome back once"}, {"start": 2.56, "duration": 5.52, "text": "again today we have another nice"}, {"start": 4.4, "duration": 8.359, "text": "algebraic problem here we have x\u00b2 + y s"}, {"start": 8.08, "duration": 6.679, "text": "= 58 we have the square of the sum = 10"}, {"start": 12.759, "duration": 4.44, "text": "right over here we're going to evaluate"}, {"start": 14.759, "duration": 7.44, "text": "x -1 all"}, {"start": 17.199, "duration": 10.281, "text": "squar from the equation one we have X2 +"}, {"start": 22.199, "duration": 8.961, "text": "y^2 = 58 now with the one of the"}, {"start": 27.48, "duration": 5.919, "text": "properties of algebra the left hand side"}, {"start": 31.16, "duration": 10.96, "text": "we can write it as"}, {"start": 33.399, "duration": 11.68, "text": "X + Y 2 then - 2x Y and this is equal to"}, {"start": 42.12, "duration": 6.32, "text": "58 right so from equation 2 we already"}, {"start": 45.079, "duration": 5.64, "text": "have that x + y^ 2 is 10 so here this"}, {"start": 48.44, "duration": 6.279, "text": "will be replaced with 10 and here we"}, {"start": 50.719, "duration": 10.081, "text": "have - 2x y this is equal to"}, {"start": 54.719, "duration": 7.441, "text": "58 so from here we have 2X y = 10 - 58"}, {"start": 60.8, "duration": 5.28, "text": "that will give us"}, {"start": 62.16, "duration": 6.36, "text": "-48 now let us multiply both by two I"}, {"start": 66.08, "duration": 8.6, "text": "have reason for this and we're going to"}, {"start": 68.52, "duration": 10.2, "text": "know very soon so here we have 4X y this"}, {"start": 74.68, "duration": 6.84, "text": "will be equal to 96 now the reason for"}, {"start": 78.72, "duration": 4.92, "text": "doing this will come up soon so just let"}, {"start": 81.52, "duration": 7.959, "text": "us pay attention and listening so here"}, {"start": 83.64, "duration": 12.519, "text": "we have to evaluate x - Y 2 which we can"}, {"start": 89.479, "duration": 8.241, "text": "use this identity x + y^ 2 - 4 x y so"}, {"start": 96.159, "duration": 3.521, "text": "the reason why I did this here by"}, {"start": 97.72, "duration": 5.6, "text": "multiply both by two that is finding the"}, {"start": 99.68, "duration": 6.039, "text": "value for 4X y because I know that 4X y"}, {"start": 103.32, "duration": 4.36, "text": "must come up somewhere in the evaluation"}, {"start": 105.719, "duration": 5.241, "text": "of this and we can see it right over"}, {"start": 107.68, "duration": 5.399, "text": "here right okay so from here we can"}, {"start": 110.96, "duration": 7.28, "text": "easily evaluate this remember that x +"}, {"start": 113.079, "duration": 9.521, "text": "Y2 is 10 then minus 4X y"}, {"start": 118.24, "duration": 7.36, "text": "is96 then this will turn out to be 96"}, {"start": 122.6, "duration": 5.359, "text": "positive so here we have to add this up"}, {"start": 125.6, "duration": 5.68, "text": "and from here we get"}, {"start": 127.959, "duration": 4.92, "text": "1006 and this is the right answer thank"}, {"start": 131.28, "duration": 3.4, "text": "you for watching please kindly subscribe"}, {"start": 132.879, "duration": 4.401, "text": "to my Channel please turn on the"}, {"start": 134.68, "duration": 5.04, "text": "notification Bell so that so that you'll"}, {"start": 137.28, "duration": 7.2, "text": "be notified anytime we upload new videos"}, {"start": 139.72, "duration": 4.76, "text": "byebye I'm going to cut it short"}], "title": "Interesting Algebra problem | #maths #mathematics", "description": "", "lengthSeconds": 143, "width": 572, "height": 360}
{"video_id": "_4CyDcZ6Rds", "transcript": [{"start": 0.68, "duration": 3.639, "text": "good day viewers welcome back to my"}, {"start": 2.28, "duration": 7.16, "text": "Channel today we have the following"}, {"start": 4.319, "duration": 8.001, "text": "interesting s mathematical problem"}, {"start": 9.44, "duration": 6.56, "text": "okay now today we're having a first"}, {"start": 12.32, "duration": 7.0, "text": "class on the S problem and anyone who"}, {"start": 16.0, "duration": 5.48, "text": "want to apply or register for this set"}, {"start": 19.32, "duration": 5.32, "text": "you can follow up my channel by"}, {"start": 21.48, "duration": 4.6, "text": "subscribing please subscribe so that the"}, {"start": 24.64, "duration": 4.799, "text": "whole"}, {"start": 26.08, "duration": 4.88, "text": "classes will reach you okay now today we"}, {"start": 29.439, "duration": 5.081, "text": "have the following problem this happens"}, {"start": 30.96, "duration": 6.599, "text": "to be their problem number number 10"}, {"start": 34.52, "duration": 8.519, "text": "okay here we have the function G of X is"}, {"start": 37.559, "duration": 10.241, "text": "equal to ax\u00b2 + 24 now for the function G"}, {"start": 43.039, "duration": 10.401, "text": "defined above a is a constant and G of 4"}, {"start": 47.8, "duration": 7.759, "text": "= 8 what is the value of G of -4 many PE"}, {"start": 53.44, "duration": 4.079, "text": "many people may initially think that the"}, {"start": 55.559, "duration": 5.32, "text": "answer to this question"}, {"start": 57.519, "duration": 5.56, "text": "is8 because here we have the 4 and"}, {"start": 60.879, "duration": 4.841, "text": "because we have pos4 maybe the"}, {"start": 63.079, "duration": 5.36, "text": "corresponding values will change signs"}, {"start": 65.72, "duration": 5.36, "text": "but that is very wrong okay now let's"}, {"start": 68.439, "duration": 6.481, "text": "get started now the first thing we need"}, {"start": 71.08, "duration": 5.8, "text": "to do we can just substitute into x-4 in"}, {"start": 74.92, "duration": 5.44, "text": "order to find the value of this"}, {"start": 76.88, "duration": 6.12, "text": "expression here G of4 okay we need to"}, {"start": 80.36, "duration": 4.92, "text": "First find this unknown number here"}, {"start": 83.0, "duration": 8.079, "text": "which is a constant here okay how do we"}, {"start": 85.28, "duration": 8.28, "text": "do that now we have our function G of x"}, {"start": 91.079, "duration": 9.881, "text": "is = a"}, {"start": 93.56, "duration": 11.32, "text": "X2 + 24 okay now we are given that g of"}, {"start": 100.96, "duration": 6.96, "text": "4 is equal to 8 okay now let us"}, {"start": 104.88, "duration": 5.4, "text": "substitute into X here for and see what"}, {"start": 107.92, "duration": 5.199, "text": "we're going to have as our corresponding"}, {"start": 110.28, "duration": 3.56, "text": "right hand side value so here we have G"}, {"start": 113.119, "duration": 7.881, "text": "of"}, {"start": 113.84, "duration": 11.36, "text": "four to be equal to you know a 4 2 then"}, {"start": 121.0, "duration": 8.16, "text": "+ 24 and this happens to be equals 8"}, {"start": 125.2, "duration": 6.399, "text": "okay so we need to solve this simple"}, {"start": 129.16, "duration": 6.079, "text": "equation here and find the value of this"}, {"start": 131.599, "duration": 9.041, "text": "constant a here okay here we have here 4"}, {"start": 135.239, "duration": 10.041, "text": "S is equal to 16 so we have here 16 a +"}, {"start": 140.64, "duration": 7.52, "text": "24 here is equal to 8 okay now 6 24 here"}, {"start": 145.28, "duration": 5.0, "text": "is crossing over so we have 16 a is"}, {"start": 148.16, "duration": 4.48, "text": "equal to"}, {"start": 150.28, "duration": 6.84, "text": "16 now dividable side by the coefficient"}, {"start": 152.64, "duration": 8.959, "text": "of a and that is over 16 and here over"}, {"start": 157.12, "duration": 7.96, "text": "16 16 cancel 16 so here we have a = to"}, {"start": 161.599, "duration": 9.56, "text": "-1 okay now our original function here"}, {"start": 165.08, "duration": 12.32, "text": "is G of x = to here a is = to1 so we"}, {"start": 171.159, "duration": 10.121, "text": "have - x 2 + 24 okay now from here we"}, {"start": 177.4, "duration": 7.0, "text": "have gotten what we first need finding"}, {"start": 181.28, "duration": 5.879, "text": "that constant a and that happens to be1"}, {"start": 184.4, "duration": 8.64, "text": "now we can easily find the value of G"}, {"start": 187.159, "duration": 10.36, "text": "of-4 and here we have G of -4 isal"}, {"start": 193.04, "duration": 7.559, "text": "to -4 2 +"}, {"start": 197.519, "duration": 5.44, "text": "24 here we have negative remember that"}, {"start": 200.599, "duration": 4.92, "text": "in bracket -4 s is"}, {"start": 202.959, "duration": 8.761, "text": "16 +"}, {"start": 205.519, "duration": 11.241, "text": "24 here we can switch it as 24- - 16 and"}, {"start": 211.72, "duration": 8.519, "text": "this is equal to 8 and G of -4 is equal"}, {"start": 216.76, "duration": 5.16, "text": "to 8 so option A here is correct thank"}, {"start": 220.239, "duration": 3.801, "text": "you for viewing and please make sure you"}, {"start": 221.92, "duration": 5.48, "text": "subscribe to my YouTube channel for more"}, {"start": 224.04, "duration": 3.36, "text": "interesting videos"}], "title": "SAT math problem. Number (10). In this video, l showed how to prepare for SAT exam.", "description": "", "lengthSeconds": 226, "width": 636, "height": 360}
{"video_id": "CNAB2TWC3kA", "transcript": [{"start": 0.28, "duration": 7.2, "text": "hello welcome here we have an"}, {"start": 3.799, "duration": 7.92, "text": "interesting question here so we have"}, {"start": 7.48, "duration": 7.359, "text": "here the W of the natural logarithm of"}, {"start": 11.719, "duration": 4.441, "text": "four but the question here is what is"}, {"start": 14.839, "duration": 4.52, "text": "this"}, {"start": 16.16, "duration": 7.24, "text": "W so like is this a"}, {"start": 19.359, "duration": 7.32, "text": "function or this is just a written"}, {"start": 23.4, "duration": 7.6, "text": "expression so but take note that this W"}, {"start": 26.679, "duration": 7.04, "text": "Here is known as the lamber W function"}, {"start": 31.0, "duration": 6.399, "text": "so we can still call it the product"}, {"start": 33.719, "duration": 6.761, "text": "logarithm and here is how the function"}, {"start": 37.399, "duration": 6.68, "text": "works so when we"}, {"start": 40.48, "duration": 10.399, "text": "have the lambar W function of let's say"}, {"start": 44.079, "duration": 12.681, "text": "we have X time a here a is a number then"}, {"start": 50.879, "duration": 8.281, "text": "power of X so this simplifies to X right"}, {"start": 56.76, "duration": 7.399, "text": "so mind you that whatever that is here"}, {"start": 59.16, "duration": 8.44, "text": "must be equal to the power of this e so"}, {"start": 64.159, "duration": 6.521, "text": "now how does this relate to this problem"}, {"start": 67.6, "duration": 4.76, "text": "okay let's see how to solve this problem"}, {"start": 70.68, "duration": 5.759, "text": "so now take"}, {"start": 72.36, "duration": 8.399, "text": "note the natural logarithm of"}, {"start": 76.439, "duration": 6.161, "text": "four is equal to the Natural logi of you"}, {"start": 80.759, "duration": 3.561, "text": "know four is the same as"}, {"start": 82.6, "duration": 5.96, "text": "2 of"}, {"start": 84.32, "duration": 6.119, "text": "2 right so you know the one of the"}, {"start": 88.56, "duration": 5.48, "text": "properties of logarithm right"}, {"start": 90.439, "duration": 8.32, "text": "right so here the power is going to come"}, {"start": 94.04, "duration": 6.64, "text": "over to this place so like this right so"}, {"start": 98.759, "duration": 7.36, "text": "here we're going to have this to be"}, {"start": 100.68, "duration": 11.24, "text": "equal to 2 * the natural logarithm of 2"}, {"start": 106.119, "duration": 6.921, "text": "right okay now this expression here is"}, {"start": 111.92, "duration": 2.76, "text": "going to"}, {"start": 113.04, "duration": 7.6, "text": "be"}, {"start": 114.68, "duration": 6.96, "text": "lambat W function of 2 * the natural"}, {"start": 120.64, "duration": 4.92, "text": "of"}, {"start": 121.64, "duration": 6.119, "text": "two okay but"}, {"start": 125.56, "duration": 5.199, "text": "here we don't have"}, {"start": 127.759, "duration": 4.761, "text": "something looking similar to this right"}, {"start": 130.759, "duration": 5.84, "text": "so here we have e to the power of"}, {"start": 132.52, "duration": 7.52, "text": "something but here we don't have e even"}, {"start": 136.599, "duration": 5.081, "text": "e at all in this bracket so now let us"}, {"start": 140.04, "duration": 3.279, "text": "recall this"}, {"start": 141.68, "duration": 8.279, "text": "property"}, {"start": 143.319, "duration": 9.28, "text": "that e of natural log of x is equal to X"}, {"start": 149.959, "duration": 5.64, "text": "so we're going to use this property here"}, {"start": 152.599, "duration": 6.121, "text": "so we're going to write this as the"}, {"start": 155.599, "duration": 6.28, "text": "natural the lambar W function of you"}, {"start": 158.72, "duration": 6.0, "text": "know two we write two in this form so"}, {"start": 161.879, "duration": 6.0, "text": "we're going to have e of the natural log"}, {"start": 164.72, "duration": 5.4, "text": "of two mind you that the E the natural"}, {"start": 167.879, "duration": 5.841, "text": "log of two will give all this two here"}, {"start": 170.12, "duration": 6.56, "text": "so here we Times by the natural log of"}, {"start": 173.72, "duration": 6.0, "text": "two so we can see that we have"}, {"start": 176.68, "duration": 5.6, "text": "similar expression like what we have at"}, {"start": 179.72, "duration": 4.96, "text": "the the left hand side of this equation"}, {"start": 182.28, "duration": 6.0, "text": "so you know we can still rewrite this as"}, {"start": 184.68, "duration": 10.76, "text": "the lamb W function of the natural log"}, {"start": 188.28, "duration": 9.319, "text": "of 2 right time eal log of 2 so here we"}, {"start": 195.44, "duration": 4.84, "text": "can see that we what we have here is"}, {"start": 197.599, "duration": 6.321, "text": "exactly equal to the power of E so we"}, {"start": 200.28, "duration": 6.28, "text": "can now see that this simplifies to the"}, {"start": 203.92, "duration": 5.48, "text": "natural log of two and this is the right"}, {"start": 206.56, "duration": 7.64, "text": "answer thank you for watching please do"}, {"start": 209.4, "duration": 4.8, "text": "well to like comment and share"}], "title": "The Lambert W Function. ProductLog. how to deal without calculators. #maths #olympics", "description": "", "lengthSeconds": 212, "width": 640, "height": 360}
{"video_id": "0uL18W66JFA", "transcript": [{"start": 0.68, "duration": 5.119, "text": "hello good viewers in this video we're"}, {"start": 2.639, "duration": 7.561, "text": "going to solve for the value of a * b +"}, {"start": 5.799, "duration": 10.0, "text": "B * C + a * C given that A + B + C is ="}, {"start": 10.2, "duration": 7.04, "text": "1 also a s + B sare + c\u00b2 is = 1 please"}, {"start": 15.799, "duration": 4.081, "text": "if you have not subscribed to my Channel"}, {"start": 17.24, "duration": 3.92, "text": "please kindly subscribe and also like"}, {"start": 19.88, "duration": 4.28, "text": "comment and"}, {"start": 21.16, "duration": 9.16, "text": "share now let's get started from"}, {"start": 24.16, "duration": 8.919, "text": "equation one we have a + b + C this is ="}, {"start": 30.32, "duration": 3.8, "text": "to 1 I would like to square both sides"}, {"start": 33.079, "duration": 3.281, "text": "of this"}, {"start": 34.12, "duration": 4.919, "text": "equation now from the left when you"}, {"start": 36.36, "duration": 5.0, "text": "expand this we have a"}, {"start": 39.039, "duration": 13.281, "text": "2 + b"}, {"start": 41.36, "duration": 13.4, "text": "2 + C 2 + 2 into braet a b + B C + a c"}, {"start": 52.32, "duration": 3.64, "text": "and this is equal to 1 2 that will give"}, {"start": 54.76, "duration": 3.36, "text": "us"}, {"start": 55.96, "duration": 4.279, "text": "one from here we're going to make a"}, {"start": 58.12, "duration": 4.72, "text": "substitution"}, {"start": 60.239, "duration": 6.401, "text": "we already have that aare + B sare + c"}, {"start": 62.84, "duration": 4.76, "text": "sare = 1 so from here we can replace"}, {"start": 66.64, "duration": 5.56, "text": "this"}, {"start": 67.6, "duration": 11.559, "text": "as 1 then we have here + 2 into braet a"}, {"start": 72.2, "duration": 9.16, "text": "b + B C + a c then this is equal to one"}, {"start": 79.159, "duration": 4.481, "text": "so from here this one is crossing over"}, {"start": 81.36, "duration": 5.16, "text": "and that will be negative so we're going"}, {"start": 83.64, "duration": 8.759, "text": "to subract one from both sides so by so"}, {"start": 86.52, "duration": 9.48, "text": "doing we have 2 into bracket a b + B C +"}, {"start": 92.399, "duration": 6.441, "text": "a c then this is equal to 1 - 1 that"}, {"start": 96.0, "duration": 5.68, "text": "will give us Z so from here let's go"}, {"start": 98.84, "duration": 5.08, "text": "ahead and divide both sides by two since"}, {"start": 101.68, "duration": 5.399, "text": "we're interested in finding the value of"}, {"start": 103.92, "duration": 5.799, "text": "a plus b c plus a c so here this will"}, {"start": 107.079, "duration": 7.08, "text": "cancel out"}, {"start": 109.719, "duration": 7.281, "text": "right okay let's finish it up"}, {"start": 114.159, "duration": 7.721, "text": "here so from here two cancel two and"}, {"start": 117.0, "duration": 10.079, "text": "here we have a + B"}, {"start": 121.88, "duration": 8.32, "text": "C+ a c this is equal to 0 / 2 and that"}, {"start": 127.079, "duration": 4.8, "text": "is 0 and this is the right answer thank"}, {"start": 130.2, "duration": 3.96, "text": "you for watching please kindly subscribe"}, {"start": 131.879, "duration": 5.761, "text": "to my channel also like comment and"}, {"start": 134.16, "duration": 3.48, "text": "share bye-bye"}], "title": "Japanese | A nice tricky Olympiad Math | #maths #mathematics", "description": "", "lengthSeconds": 136, "width": 554, "height": 360}
{"video_id": "fuOB-_VUpnQ", "transcript": [{"start": 1.0, "duration": 3.88, "text": "hello good day viewers welcome back once"}, {"start": 2.96, "duration": 3.839, "text": "again to another video today we're going"}, {"start": 4.88, "duration": 7.0, "text": "to find the square root of a sodic"}, {"start": 6.799, "duration": 7.161, "text": "expression the > of 10 + 2un of 21"}, {"start": 11.88, "duration": 6.12, "text": "right"}, {"start": 13.96, "duration": 5.12, "text": "okay we have theare root of 10 + 2 of"}, {"start": 18.0, "duration": 4.32, "text": "root of"}, {"start": 19.08, "duration": 8.959, "text": "21 we can write this as square root of"}, {"start": 22.32, "duration": 11.96, "text": "10 is the same as 7 + 3 right then + 2"}, {"start": 28.039, "duration": 9.641, "text": "root of 21 is same as 7 * 3 so from here"}, {"start": 34.28, "duration": 7.64, "text": "this is equal to the square root of 7"}, {"start": 37.68, "duration": 8.44, "text": "here is same as sare < TK of 7 s then"}, {"start": 41.92, "duration": 6.479, "text": "plus 3 here is same as the < TK of 3"}, {"start": 46.12, "duration": 6.56, "text": "S then"}, {"start": 48.399, "duration": 7.121, "text": "plus 2 multi by you know otk 7 * 3 let"}, {"start": 52.68, "duration": 6.76, "text": "us recall that the root of M * n same as"}, {"start": 55.52, "duration": 9.08, "text": "root of M * root of n so here we have"}, {"start": 59.44, "duration": 7.999, "text": "root of 7 * the root of 3 now from here"}, {"start": 64.6, "duration": 10.76, "text": "let us recall the expansion of this"}, {"start": 67.439, "duration": 10.481, "text": "binomial a + b 2 is = a 2 + b 2 + 2 a b"}, {"start": 75.36, "duration": 4.799, "text": "right and if you compare this with what"}, {"start": 77.92, "duration": 5.519, "text": "we have under this root sign we will"}, {"start": 80.159, "duration": 5.32, "text": "find out that a is equal to root of 7"}, {"start": 83.439, "duration": 4.04, "text": "and B is equal to root of 3 if you"}, {"start": 85.479, "duration": 4.121, "text": "follow the pattern a squ will have"}, {"start": 87.479, "duration": 3.68, "text": "something squ plus b squ you have"}, {"start": 89.6, "duration": 4.72, "text": "another something which is different"}, {"start": 91.159, "duration": 5.081, "text": "from the first something squ plus 2 *"}, {"start": 94.32, "duration": 5.04, "text": "something * the different something and"}, {"start": 96.24, "duration": 7.04, "text": "here we have 2 * a * B right so this"}, {"start": 99.36, "duration": 9.32, "text": "implies that we have the square root of"}, {"start": 103.28, "duration": 8.199, "text": "< TK 7 + < tk3 S and from here this is"}, {"start": 108.68, "duration": 9.719, "text": "same as the absolute value of Ro < TK 7"}, {"start": 111.479, "duration": 9.28, "text": "+ < tk3 which is = to < TK 7 + < tk3 and"}, {"start": 118.399, "duration": 4.161, "text": "this is the right answer thank you for"}, {"start": 120.759, "duration": 4.0, "text": "watching please kindly subscribe to my"}, {"start": 122.56, "duration": 5.68, "text": "YouTube channel also like comment and"}, {"start": 124.759, "duration": 3.481, "text": "share bye-bye"}], "title": "Junior Olympiad Math | Can you find the square root of a surdic expression? | #maths", "description": "", "lengthSeconds": 126, "width": 640, "height": 360}
{"video_id": "KHTcVHKCOoU", "transcript": [{"start": 0.919, "duration": 3.601, "text": "hello welcome back once again today"}, {"start": 2.84, "duration": 4.16, "text": "we're going to solve this awesome"}, {"start": 4.52, "duration": 5.68, "text": "looking equation and we're going to"}, {"start": 7.0, "duration": 7.04, "text": "solve for all real values of X that will"}, {"start": 10.2, "duration": 6.159, "text": "satisfy this equation right so let's get"}, {"start": 14.04, "duration": 10.079, "text": "started from the left hand side we have"}, {"start": 16.359, "duration": 9.881, "text": "x to^ x to^ 4 this is equal to 64 right"}, {"start": 24.119, "duration": 4.841, "text": "now in our next step let us raise both"}, {"start": 26.24, "duration": 7.28, "text": "sides of this equation to the power of"}, {"start": 28.96, "duration": 7.72, "text": "four so here raise it to the^ 4 and from"}, {"start": 33.52, "duration": 6.44, "text": "the right hand side raise it to the^"}, {"start": 36.68, "duration": 6.48, "text": "4 okay now we're going to make use of"}, {"start": 39.96, "duration": 6.48, "text": "this exponential rule we know when we"}, {"start": 43.16, "duration": 7.719, "text": "have a to the^ B in bracket to the^ C"}, {"start": 46.44, "duration": 6.24, "text": "this is same as a to the^ B * C we're"}, {"start": 50.879, "duration": 4.081, "text": "going to apply this rule at the left"}, {"start": 52.68, "duration": 5.519, "text": "hand side so that will become we know"}, {"start": 54.96, "duration": 9.919, "text": "the power of X here is x^ 4 so that will"}, {"start": 58.199, "duration": 10.521, "text": "become x x x^ 4 * 4 this will be equal"}, {"start": 64.879, "duration": 4.56, "text": "to we know 64 is = 8 2 right so we have"}, {"start": 68.72, "duration": 5.039, "text": "here"}, {"start": 69.439, "duration": 7.881, "text": "\u00b2 then ra the^ 4 now this exponential"}, {"start": 73.759, "duration": 9.121, "text": "rule here let us apply it here okay so"}, {"start": 77.32, "duration": 10.96, "text": "we have here x^ x^ 4 * 4 this will be"}, {"start": 82.88, "duration": 7.279, "text": "equal to 8^ 2 * 4 which is equal to 8"}, {"start": 88.28, "duration": 5.32, "text": "this exponential rule when we have have"}, {"start": 90.159, "duration": 6.681, "text": "this product we can take either a either"}, {"start": 93.6, "duration": 4.879, "text": "B or C inside bracket and a and the"}, {"start": 96.84, "duration": 3.919, "text": "other one outside don't know what is"}, {"start": 98.479, "duration": 4.121, "text": "wrong with me today okay so here we're"}, {"start": 100.759, "duration": 4.921, "text": "going to write this here you know we"}, {"start": 102.6, "duration": 7.119, "text": "have this product between this power"}, {"start": 105.68, "duration": 10.119, "text": "here we have x^ 4 * 4 so we can actually"}, {"start": 109.719, "duration": 10.561, "text": "write as x ^ 4 then ra x^ 4 then this is"}, {"start": 115.799, "duration": 8.201, "text": "equal to 8 to the^ 8 okay now you can"}, {"start": 120.28, "duration": 7.959, "text": "see we have the same pattern with bases"}, {"start": 124.0, "duration": 8.0, "text": "and exponents we have x ^ 4 is = to 8"}, {"start": 128.239, "duration": 5.881, "text": "from the exponents we have x^ 4 isal 8"}, {"start": 132.0, "duration": 5.48, "text": "so we're going to equate either the"}, {"start": 134.12, "duration": 7.28, "text": "bases or the powers so this implies that"}, {"start": 137.48, "duration": 8.96, "text": "x to^ 4 is equal"}, {"start": 141.4, "duration": 9.0, "text": "to 8 right okay now from here we're"}, {"start": 146.44, "duration": 6.76, "text": "going to solve for x but we"}, {"start": 150.4, "duration": 5.44, "text": "interested in solving for the real value"}, {"start": 153.2, "duration": 7.0, "text": "for x so we write the left hand side as"}, {"start": 155.84, "duration": 7.6, "text": "X2 to the^ 2 then this is equal"}, {"start": 160.2, "duration": 6.6, "text": "to 8 now take the square root of both"}, {"start": 163.44, "duration": 7.4, "text": "sides we get x2 is equal to plus or"}, {"start": 166.8, "duration": 9.359, "text": "minus the square root of"}, {"start": 170.84, "duration": 9.0, "text": "8 right okay so here we have the square"}, {"start": 176.159, "duration": 5.44, "text": "root of 8 now remember we're solving for"}, {"start": 179.84, "duration": 6.64, "text": "for the ray value so we're going to"}, {"start": 181.599, "duration": 4.881, "text": "discard the negative value so for r"}, {"start": 187.799, "duration": 9.201, "text": "x we get x2 will be equal to the square"}, {"start": 192.599, "duration": 6.161, "text": "root of 8 right okay now we're going to"}, {"start": 197.0, "duration": 5.319, "text": "take the square root of both side one"}, {"start": 198.76, "duration": 6.759, "text": "more time so we get X will be equal to"}, {"start": 202.319, "duration": 5.881, "text": "plus or minus this square root of square"}, {"start": 205.519, "duration": 5.72, "text": "otk of 8 we know square root of square"}, {"start": 208.2, "duration": 6.679, "text": "root of something is that to the of 1 4"}, {"start": 211.239, "duration": 7.72, "text": "which is equal to plus or minus the 4"}, {"start": 214.879, "duration": 5.801, "text": "root of 8 so therefore we have our two"}, {"start": 218.959, "duration": 6.48, "text": "real solutions that satisfy this"}, {"start": 220.68, "duration": 6.04, "text": "equation x is equal to plus or minus the"}, {"start": 225.439, "duration": 5.72, "text": "four"}, {"start": 226.72, "duration": 5.92, "text": "root of 8 and this is the correct answer"}, {"start": 231.159, "duration": 4.0, "text": "thank you for watching please kindly"}, {"start": 232.64, "duration": 6.36, "text": "subscribe to my channel also like"}, {"start": 235.159, "duration": 3.841, "text": "comment and share"}], "title": "Oxford University \ud83c\udf93 Entrance Exam | Can you solve ?", "description": "", "lengthSeconds": 238, "width": 428, "height": 240}
{"video_id": "k-5Ypsfd_gc", "transcript": [{"start": 0.56, "duration": 4.799, "text": "hello welcome back once again today we"}, {"start": 2.44, "duration": 5.88, "text": "have this nice and interesting"}, {"start": 5.359, "duration": 6.041, "text": "exponential equation and here we'll be"}, {"start": 8.32, "duration": 5.68, "text": "solving for the values of X okay now"}, {"start": 11.4, "duration": 5.68, "text": "let's get started first of all let us"}, {"start": 14.0, "duration": 8.8, "text": "take note of this the M"}, {"start": 17.08, "duration": 9.519, "text": "root of a is equal to a to the^ 1 / M"}, {"start": 22.8, "duration": 8.479, "text": "right so from here remember we have the"}, {"start": 26.599, "duration": 8.601, "text": "X root of 5 and everything is RA to the^"}, {"start": 31.279, "duration": 6.481, "text": "1 /x this is equal to"}, {"start": 35.2, "duration": 6.76, "text": "25 so from here make use of this"}, {"start": 37.76, "duration": 11.4, "text": "property so in the bracket we have 5^ 1"}, {"start": 41.96, "duration": 10.079, "text": "/x then 1 / X then this is equal to"}, {"start": 49.16, "duration": 6.879, "text": "25 okay in our next step let us record"}, {"start": 52.039, "duration": 8.961, "text": "this property a to the^ of B in bracket"}, {"start": 56.039, "duration": 10.52, "text": "C this is equal to a^ B * C so from here"}, {"start": 61.0, "duration": 5.559, "text": "that will be 5^ 1X * 1"}, {"start": 67.04, "duration": 11.2, "text": "/x you know that would be 1 / x^ 2 so we"}, {"start": 72.84, "duration": 9.44, "text": "have 5 ^ 1 / x^ 2 this will be equal to"}, {"start": 78.24, "duration": 6.28, "text": "25 is the same as 5 to the^ 2 so we can"}, {"start": 82.28, "duration": 5.04, "text": "see we have the same basis from both"}, {"start": 84.52, "duration": 6.36, "text": "sides so let us equate the power so this"}, {"start": 87.32, "duration": 8.28, "text": "implies from here the 1 /x x"}, {"start": 90.88, "duration": 9.4, "text": "2 is = to 2 so put this over one and"}, {"start": 95.6, "duration": 9.08, "text": "cross multiply so x * 2 will give us 2"}, {"start": 100.28, "duration": 6.839, "text": "X2 this is equal to 1 * 1 which is 1 so"}, {"start": 104.68, "duration": 6.64, "text": "divide both sides by"}, {"start": 107.119, "duration": 9.241, "text": "2 so 2 we cancel two so here we have"}, {"start": 111.32, "duration": 6.56, "text": "X2 is equal to 1 / 2 from here let us"}, {"start": 116.36, "duration": 3.799, "text": "take root of both sides since we're"}, {"start": 117.88, "duration": 6.72, "text": "solving for x so X will be equal to plus"}, {"start": 120.159, "duration": 6.96, "text": "or minus the sare < TK of 1 / 2 which is"}, {"start": 124.6, "duration": 4.84, "text": "equal to plus or minus we have the"}, {"start": 127.119, "duration": 4.961, "text": "square root of the numerator / the"}, {"start": 129.44, "duration": 5.159, "text": "square root of the denominator so from"}, {"start": 132.08, "duration": 8.12, "text": "here this is equal to plus or minus otk"}, {"start": 134.599, "duration": 7.761, "text": "of 1 is 1 / root of 2 and this is the"}, {"start": 140.2, "duration": 4.72, "text": "right answer thank you for watching"}, {"start": 142.36, "duration": 7.959, "text": "please kindly subscribe to my channel"}, {"start": 144.92, "duration": 5.399, "text": "also like comment and share bye-bye"}], "title": "Interesting exponential equation | Can you solve ?", "description": "", "lengthSeconds": 149, "width": 640, "height": 360}
{"video_id": "moXeY8Lp6C4", "transcript": [{"start": 0.56, "duration": 4.56, "text": "hello good day viewers welcome back once"}, {"start": 2.52, "duration": 4.839, "text": "again please kindly support this channel"}, {"start": 5.12, "duration": 5.24, "text": "by subscribing also like comment and"}, {"start": 7.359, "duration": 5.681, "text": "share today we have an interesting"}, {"start": 10.36, "duration": 7.72, "text": "exponential equation problem here we're"}, {"start": 13.04, "duration": 8.239, "text": "given that 2^ of 2^ X is equal to 3^ 3^"}, {"start": 18.08, "duration": 7.92, "text": "X and right over here we are solving for"}, {"start": 21.279, "duration": 6.641, "text": "the value of x right okay so the first"}, {"start": 26.0, "duration": 4.439, "text": "thing we're going to do here is to elen"}, {"start": 27.92, "duration": 7.68, "text": "both sides of this equation right so"}, {"start": 30.439, "duration": 8.161, "text": "here we have the Ln of 2^ of 2 to the^ X"}, {"start": 35.6, "duration": 7.68, "text": "you know this is equal to the Ln of 3"}, {"start": 38.6, "duration": 6.0, "text": "to^ of 3 to^ of X right okay now recall"}, {"start": 43.28, "duration": 4.64, "text": "this"}, {"start": 44.6, "duration": 6.88, "text": "property the Ln of a to the power of B"}, {"start": 47.92, "duration": 6.439, "text": "is equal to B * the Ln of a right here"}, {"start": 51.48, "duration": 7.96, "text": "the power of 2 is 2 the^ of X"}, {"start": 54.359, "duration": 11.44, "text": "so we we have from here 2^ x * the Ln of"}, {"start": 59.44, "duration": 9.24, "text": "2 this is equal to 3^ of x * the Ln of 3"}, {"start": 65.799, "duration": 6.161, "text": "right okay so from here we divide both"}, {"start": 68.68, "duration": 6.799, "text": "side by the Ln of two so here we have"}, {"start": 71.96, "duration": 6.36, "text": "over Ln of two and from here over the Ln"}, {"start": 75.479, "duration": 9.32, "text": "of two so this will cancel this here we"}, {"start": 78.32, "duration": 9.439, "text": "have 2^ X is equal to 3^ x * by the Ln"}, {"start": 84.799, "duration": 6.601, "text": "of 3 over the Ln of"}, {"start": 87.759, "duration": 5.04, "text": "2 okay so what what do we do from here"}, {"start": 91.4, "duration": 7.44, "text": "from here we're going to divide both"}, {"start": 92.799, "duration": 9.96, "text": "sides Again by 3 ^ of X so here 3 ^ x"}, {"start": 98.84, "duration": 5.879, "text": "and right over here 3 ^ of X so this"}, {"start": 102.759, "duration": 4.32, "text": "will cancel this and from here we're"}, {"start": 104.719, "duration": 6.281, "text": "going to make use of this property when"}, {"start": 107.079, "duration": 6.36, "text": "we have a to the power of B over another"}, {"start": 111.0, "duration": 5.799, "text": "base with the same power C to the power"}, {"start": 113.439, "duration": 6.121, "text": "B this is same as a/ C all to the power"}, {"start": 116.799, "duration": 6.401, "text": "of B right so from here we're going to"}, {"start": 119.56, "duration": 8.08, "text": "have 2 / 3 to the power of X this is"}, {"start": 123.2, "duration": 7.08, "text": "equal to the Ln of 3 over the Ln of 2"}, {"start": 127.64, "duration": 6.0, "text": "okay so what do we do from here from"}, {"start": 130.28, "duration": 7.2, "text": "here I'm going to Len both sides again"}, {"start": 133.64, "duration": 7.879, "text": "so here we have the Ln of 2/ 3 to the^"}, {"start": 137.48, "duration": 7.759, "text": "of X this is equal to the Ln of the Ln"}, {"start": 141.519, "duration": 7.0, "text": "of 3 over the Ln of 2 so from here this"}, {"start": 145.239, "duration": 5.321, "text": "we cross over here so we have x * the Ln"}, {"start": 148.519, "duration": 5.521, "text": "of 2/ 3"}, {"start": 150.56, "duration": 6.319, "text": "this is equal to the Ln of the Ln of 3"}, {"start": 154.04, "duration": 6.479, "text": "over the Ln of 2 so we're solving for x"}, {"start": 156.879, "duration": 8.161, "text": "so here we divide both by the Ln of 2/ 3"}, {"start": 160.519, "duration": 7.241, "text": "and here Ln of 2/ 3 so this cancel this"}, {"start": 165.04, "duration": 7.44, "text": "and here we get the value of x which is"}, {"start": 167.76, "duration": 9.32, "text": "the Ln of Ln of 3 over the Ln of"}, {"start": 172.48, "duration": 6.96, "text": "2ide by the Ln of 2 over 3 and this is"}, {"start": 177.08, "duration": 4.84, "text": "the right answer thank you for watching"}, {"start": 179.44, "duration": 7.079, "text": "please kindly subscribe to my channel"}, {"start": 181.92, "duration": 4.599, "text": "also like comment and share bye-bye"}], "title": "Interesting exponential equation. How to solve for real value of  x", "description": "", "lengthSeconds": 185, "width": 640, "height": 360}
{"video_id": "Paj1SenERVI", "transcript": [{"start": 0.88, "duration": 5.999, "text": "from the left hand side cube root of x"}, {"start": 3.0, "duration": 5.4, "text": "is the same as x^ 1 / 3 this is equal to"}, {"start": 6.879, "duration": 6.481, "text": "from the right hand side sare root of x"}, {"start": 8.4, "duration": 7.6, "text": "is same as x^ 1 / 2 we know the LCM of 2"}, {"start": 13.36, "duration": 5.2, "text": "and 3 is equal to 6 raise both side to"}, {"start": 16.0, "duration": 5.199, "text": "the power of 6 so here raise this to"}, {"start": 18.56, "duration": 4.799, "text": "the^ of 6 and here raise this to the"}, {"start": 21.199, "duration": 6.681, "text": "power 6 from the left hand side we have"}, {"start": 23.359, "duration": 9.88, "text": "X to the^ 1 / 3 * 6 is 2 then this is"}, {"start": 27.88, "duration": 10.48, "text": "equal to x^ 1 2 * 3 * 6 I mean is 3 from"}, {"start": 33.239, "duration": 7.441, "text": "here we have X Cub - X2 is = 0 so we"}, {"start": 38.36, "duration": 4.84, "text": "have X2 which is common factor it out we"}, {"start": 40.68, "duration": 7.0, "text": "have X - 1 which is equal to 0 so we"}, {"start": 43.2, "duration": 6.199, "text": "have either x\u00b2 is = to 0 or X - 1 is = 0"}, {"start": 47.68, "duration": 5.92, "text": "from here take root of both side we get"}, {"start": 49.399, "duration": 7.281, "text": "x = 0 and this is indeed a solution and"}, {"start": 53.6, "duration": 7.0, "text": "here we have x = 1 which is a solution"}, {"start": 56.68, "duration": 3.92, "text": "subscribe please"}], "title": "Interesting radical equation | Can you solve ?  #maths #mathematics #exam", "description": "", "lengthSeconds": 60, "width": 360, "height": 640}
{"video_id": "71p0rciZ7DU", "transcript": [{"start": 1.12, "duration": 4.759, "text": "hello welcome back once again today we"}, {"start": 3.719, "duration": 5.561, "text": "have this interesting"}, {"start": 5.879, "duration": 7.521, "text": "problem we're going to simplify the"}, {"start": 9.28, "duration": 5.72, "text": "following given division of irrational"}, {"start": 13.4, "duration": 3.52, "text": "numbers"}, {"start": 15.0, "duration": 4.039, "text": "right so let's see how we're going to"}, {"start": 16.92, "duration": 6.04, "text": "solve this"}, {"start": 19.039, "duration": 6.801, "text": "problem now let us split the denominator"}, {"start": 22.96, "duration": 10.119, "text": "so we're going to have the square < TK"}, {"start": 25.84, "duration": 11.559, "text": "of 5 / 8 all over theare < TK of 10 then"}, {"start": 33.079, "duration": 8.041, "text": "plus theare < TK of 8 all over"}, {"start": 37.399, "duration": 6.68, "text": "5id by the sare root of"}, {"start": 41.12, "duration": 6.599, "text": "10 okay so from here we're going to make"}, {"start": 44.079, "duration": 5.201, "text": "use of the following rule when we have"}, {"start": 47.719, "duration": 4.601, "text": "the square root of"}, {"start": 49.28, "duration": 9.64, "text": "m / the square root of"}, {"start": 52.32, "duration": 8.84, "text": "n this isal to theun of m /"}, {"start": 58.92, "duration": 4.959, "text": "n so we're going to make use of this"}, {"start": 61.16, "duration": 6.4, "text": "rule here so this will"}, {"start": 63.879, "duration": 11.361, "text": "become the square root of so we have"}, {"start": 67.56, "duration": 12.64, "text": "here 5 all 8 then ID 10 then"}, {"start": 75.24, "duration": 12.16, "text": "plus the square root of we have here 8"}, {"start": 80.2, "duration": 10.08, "text": "all 5 / 10 right okay so now let's talk"}, {"start": 87.4, "duration": 5.48, "text": "about this fraction"}, {"start": 90.28, "duration": 2.6, "text": "we know"}, {"start": 93.52, "duration": 11.44, "text": "that 5 8 / 10 put it like this over one"}, {"start": 100.52, "duration": 7.04, "text": "this is same as 5 all over 8 * then we"}, {"start": 104.96, "duration": 7.04, "text": "take the reciprocal of this fraction"}, {"start": 107.56, "duration": 8.919, "text": "here which is 1 all over 10 so from here"}, {"start": 112.0, "duration": 8.64, "text": "we can see that 5 here is 1 5 here is 2"}, {"start": 116.479, "duration": 7.881, "text": "so we get 1 all over 16"}, {"start": 120.64, "duration": 9.759, "text": "and for the other part here we have 8 /"}, {"start": 124.36, "duration": 9.959, "text": "5 then / 10 / 1 so this is = to 8 / 5 *"}, {"start": 130.399, "duration": 6.401, "text": "then the reciprocal 1 / 10 when you use"}, {"start": 134.319, "duration": 7.28, "text": "two here we get four we used to here we"}, {"start": 136.8, "duration": 9.04, "text": "get five we can see we have 4 all over"}, {"start": 141.599, "duration": 8.881, "text": "25 so this will become the square root"}, {"start": 145.84, "duration": 8.28, "text": "of so the first one we have 1 / 16"}, {"start": 150.48, "duration": 6.72, "text": "then plus the square root of here we get"}, {"start": 154.12, "duration": 8.119, "text": "4 all over"}, {"start": 157.2, "duration": 8.0, "text": "25 so from here remember this rule here"}, {"start": 162.239, "duration": 7.041, "text": "this particular rule here so we're going"}, {"start": 165.2, "duration": 7.52, "text": "to take the square root of the numerator"}, {"start": 169.28, "duration": 6.84, "text": "ID by the square root of the denominator"}, {"start": 172.72, "duration": 6.519, "text": "plus the square root of the numerator ID"}, {"start": 176.12, "duration": 4.32, "text": "by the square root of the denominator so"}, {"start": 179.239, "duration": 8.28, "text": "this will be"}, {"start": 180.44, "duration": 12.079, "text": "of 1 is 1 /un of 16 is 4 plus 4 is"}, {"start": 187.519, "duration": 7.64, "text": "2id of 25 is 5 so here we're going to"}, {"start": 192.519, "duration": 7.44, "text": "simplify this fraction so we know the"}, {"start": 195.159, "duration": 9.961, "text": "LCM of 4 and 5 is 20 so we have 20 20 /"}, {"start": 199.959, "duration": 10.2, "text": "4 is 5 then this one * 5 will give us"}, {"start": 205.12, "duration": 5.96, "text": "5+ 20 / 5 is 4 then this 2 * 4 we give"}, {"start": 210.159, "duration": 5.921, "text": "us"}, {"start": 211.08, "duration": 9.92, "text": "8 right so here we get our solution 13"}, {"start": 216.08, "duration": 6.48, "text": "over 20 and this is the correct answer"}, {"start": 221.0, "duration": 4.08, "text": "thank you for watching please kindly"}, {"start": 222.56, "duration": 5.28, "text": "subscribe to my channel also like"}, {"start": 225.08, "duration": 5.1, "text": "comment and share bye-bye"}, {"start": 227.84, "duration": 2.34, "text": "[Music]"}], "title": "Germany | Nice Olympiad Math | Simplification of radicals", "description": "", "lengthSeconds": 228, "width": 428, "height": 240}
{"video_id": "Fi5Gja08dEo", "transcript": [{"start": 0.48, "duration": 5.0, "text": "hello welcome back once again let's find"}, {"start": 2.72, "duration": 5.919, "text": "the solution to this problem we have X2"}, {"start": 5.48, "duration": 6.68, "text": "+ x + 1 this is equal to 0 so from here"}, {"start": 8.639, "duration": 6.481, "text": "we're going to evaluate x^ of"}, {"start": 12.16, "duration": 6.68, "text": "2025 before we proceed let us take note"}, {"start": 15.12, "duration": 8.8, "text": "of the following difference of two cubes"}, {"start": 18.84, "duration": 11.88, "text": "formula we have a cubed minus B Cubed"}, {"start": 23.92, "duration": 9.56, "text": "this is equal to a - b then * by a\u00b2 + a"}, {"start": 30.72, "duration": 8.04, "text": "b + b"}, {"start": 33.48, "duration": 15.04, "text": "2 so from this from this given"}, {"start": 38.76, "duration": 14.479, "text": "identity let a = x and let b = 1 so we"}, {"start": 48.52, "duration": 7.039, "text": "have X Cub - y Cub which is 1 this is"}, {"start": 53.239, "duration": 8.361, "text": "equal to x"}, {"start": 55.559, "duration": 10.441, "text": "-1 then MTI X2 + a will be x * 1 which"}, {"start": 61.6, "duration": 7.68, "text": "is x + b\u00b2 will be 1 2 which is 1 and"}, {"start": 66.0, "duration": 8.04, "text": "here from the given equation X2 + x + 1"}, {"start": 69.28, "duration": 13.68, "text": "is equal 0 so here this is equal to0 so"}, {"start": 74.04, "duration": 13.119, "text": "we have X Cub -1 this is equal to x -1 *"}, {"start": 82.96, "duration": 9.119, "text": "0 so from here we have X Cub - 1 this is"}, {"start": 87.159, "duration": 8.881, "text": "equal to0 so from here x X Cub is = 0 +"}, {"start": 92.079, "duration": 6.641, "text": "1 which is 1 now remember we're asked to"}, {"start": 96.04, "duration": 7.719, "text": "evaluate x^ of"}, {"start": 98.72, "duration": 9.439, "text": "2025 right but luckily 2025 is a"}, {"start": 103.759, "duration": 6.561, "text": "multiple of three right remember 2025 is"}, {"start": 108.159, "duration": 5.161, "text": "equal to 3"}, {"start": 110.32, "duration": 3.0, "text": "MTI"}, {"start": 113.6, "duration": 5.4, "text": "675 now what happens if we raise both"}, {"start": 116.719, "duration": 4.36, "text": "sides of this equation to the power of"}, {"start": 119.0, "duration": 3.399, "text": "67"}, {"start": 121.079, "duration": 4.281, "text": "you will see that from the left hand"}, {"start": 122.399, "duration": 6.121, "text": "side we will have exactly x^ of"}, {"start": 125.36, "duration": 5.28, "text": "2025 so let's go ahead and do that so"}, {"start": 128.52, "duration": 4.079, "text": "here we have X cubed then we have to"}, {"start": 130.64, "duration": 6.76, "text": "raise to the power of"}, {"start": 132.599, "duration": 7.801, "text": "675 this is equal to 1 to the power of"}, {"start": 137.4, "duration": 4.72, "text": "675 so from the left here we have x to"}, {"start": 140.4, "duration": 5.72, "text": "the power of"}, {"start": 142.12, "duration": 6.08, "text": "225 and here comes the answer one and"}, {"start": 146.12, "duration": 3.839, "text": "that's the right answer thank you for"}, {"start": 148.2, "duration": 4.2, "text": "watching please kindly subscribe to my"}, {"start": 149.959, "duration": 5.441, "text": "Channel also like comment and share"}, {"start": 152.4, "duration": 3.0, "text": "bye-bye"}], "title": "Japanese | Nice Olympiad Math | #maths #mathematics", "description": "", "lengthSeconds": 154, "width": 640, "height": 360}
{"video_id": "ucWGtYhQ02w", "transcript": [{"start": 0.24, "duration": 3.04, "text": "good day viewers welcome back to our"}, {"start": 1.68, "duration": 4.079, "text": "Channel today we have the following"}, {"start": 3.28, "duration": 8.0, "text": "definite integral problem integral from"}, {"start": 5.759, "duration": 6.84, "text": "0 to 4 16 all 16 + x\u00b2 DX touch let's see"}, {"start": 11.28, "duration": 9.48, "text": "how we're going to solve this problem"}, {"start": 12.599, "duration": 14.481, "text": "take a note integral of 1 over X2 + a"}, {"start": 20.76, "duration": 9.839, "text": "2 the x is = 1 all a * the tan inverse"}, {"start": 27.08, "duration": 6.68, "text": "of x a + C"}, {"start": 30.599, "duration": 5.441, "text": "so a here is a real number and which is"}, {"start": 33.76, "duration": 5.28, "text": "greater than zero okay so we're going to"}, {"start": 36.04, "duration": 6.64, "text": "use this here so I can write this limit"}, {"start": 39.04, "duration": 6.24, "text": "as follows 16 * integral from 0 to"}, {"start": 42.68, "duration": 9.399, "text": "4 1 all"}, {"start": 45.28, "duration": 9.079, "text": "over X2 plus of 16 here so 4 2 DX so we"}, {"start": 52.079, "duration": 5.721, "text": "can see that this is applicable here now"}, {"start": 54.359, "duration": 6.121, "text": "so we have here 16 all over you know"}, {"start": 57.8, "duration": 6.88, "text": "4 the tan inverse"}, {"start": 60.48, "duration": 7.56, "text": "of X All 4 close bracket with our limit"}, {"start": 64.68, "duration": 8.56, "text": "0 to 4 so now putting our upper limit so"}, {"start": 68.04, "duration": 7.04, "text": "we get 16 / 4 is 4 the tan inverse of 4"}, {"start": 73.24, "duration": 7.6, "text": "all over 4 is"}, {"start": 75.08, "duration": 10.56, "text": "1us 4 the tan inverse of 0 All 4 is 0"}, {"start": 80.84, "duration": 7.2, "text": "and this is equal to 4 tan inverse of 1"}, {"start": 85.64, "duration": 4.76, "text": "so this is equal to 4 * the tan inverse"}, {"start": 88.04, "duration": 5.84, "text": "of 1 is pi 4"}, {"start": 90.4, "duration": 6.84, "text": "P four canel four here so we finally get"}, {"start": 93.88, "duration": 4.84, "text": "pie wow thank you for viewing and please"}, {"start": 97.24, "duration": 6.04, "text": "make sure you subscribe to my YouTube"}, {"start": 98.72, "duration": 4.56, "text": "channel for more exciting videos"}], "title": "Definite integral. The result is so so good.", "description": "", "lengthSeconds": 101, "width": 636, "height": 360}
{"video_id": "_7Y3y6iqSPc", "transcript": [{"start": 0.42, "duration": 4.68, "text": "good if you was welcome back to my"}, {"start": 3.179, "duration": 3.66, "text": "channel if you have not subscribed to"}, {"start": 5.1, "duration": 4.98, "text": "this Channel please"}, {"start": 6.839, "duration": 5.521, "text": "subscribe share our videos and comment"}, {"start": 10.08, "duration": 4.979, "text": "thank you today we have the following"}, {"start": 12.36, "duration": 5.04, "text": "limits we have a solution here we're not"}, {"start": 15.059, "duration": 4.321, "text": "going to use the so-called L hobby to"}, {"start": 17.4, "duration": 4.82, "text": "scroll that is differentiating up and"}, {"start": 19.38, "duration": 5.399, "text": "down so we'll be approaching this"}, {"start": 22.22, "duration": 4.66, "text": "through algebra okay"}, {"start": 24.779, "duration": 4.26, "text": "so how do we solve this problem remember"}, {"start": 26.88, "duration": 3.84, "text": "if we make direct optician here we have"}, {"start": 29.039, "duration": 3.421, "text": "four minus four"}, {"start": 30.72, "duration": 3.839, "text": "so if you make that resolution we get"}, {"start": 32.46, "duration": 3.9, "text": "four minus one zero all over you know"}, {"start": 34.559, "duration": 4.081, "text": "square root of 4 here is two then we"}, {"start": 36.36, "duration": 4.379, "text": "have 2 minus two this is equal to zero"}, {"start": 38.64, "duration": 3.96, "text": "all over zero which is not permitted in"}, {"start": 40.739, "duration": 5.281, "text": "mathematics this is an indeterminate"}, {"start": 42.6, "duration": 5.88, "text": "form okay now since we have we can e we"}, {"start": 46.02, "duration": 6.96, "text": "can solve this question in different"}, {"start": 48.48, "duration": 6.719, "text": "ways okay so we can rationalize or make"}, {"start": 52.98, "duration": 5.04, "text": "substitution but I will go for making"}, {"start": 55.199, "duration": 8.241, "text": "substitution because that will be easier"}, {"start": 58.02, "duration": 5.42, "text": "okay so solution let's get started"}, {"start": 65.159, "duration": 4.381, "text": "now I'm going to approach this problem"}, {"start": 67.2, "duration": 6.66, "text": "by first making substitution let's"}, {"start": 69.54, "duration": 8.7, "text": "square root of x equals y okay so you"}, {"start": 73.86, "duration": 7.439, "text": "know X is going to be y resp Square two"}, {"start": 78.24, "duration": 6.48, "text": "so we have here x is equal to Y raised"}, {"start": 81.299, "duration": 6.081, "text": "by 2 okay so we have our limit now as"}, {"start": 84.72, "duration": 2.66, "text": "limit"}, {"start": 87.42, "duration": 6.42, "text": "as Y is approaching so let's see what"}, {"start": 91.5, "duration": 5.939, "text": "why we approach as X is approaching 4"}, {"start": 93.84, "duration": 6.66, "text": "okay so here we have square root of x if"}, {"start": 97.439, "duration": 5.04, "text": "you substitute into x 4 we have square"}, {"start": 100.5, "duration": 7.2, "text": "root of 4 which is 2. so Y is"}, {"start": 102.479, "duration": 8.701, "text": "approaching 2. so up here we have"}, {"start": 107.7, "duration": 6.419, "text": "instead of X we have y squared minus one"}, {"start": 111.18, "duration": 8.34, "text": "and down here we have e sub square root"}, {"start": 114.119, "duration": 7.561, "text": "of x we get y minus two okay so we'll be"}, {"start": 119.52, "duration": 5.279, "text": "using the difference of two"}, {"start": 121.68, "duration": 6.24, "text": "squares formula we have a raised to"}, {"start": 124.799, "duration": 5.94, "text": "power 2 minus b e raised part 2 equal is"}, {"start": 127.92, "duration": 7.679, "text": "equal to a minus B"}, {"start": 130.739, "duration": 7.621, "text": "times a plus b okay so we have a squared"}, {"start": 135.599, "duration": 7.981, "text": "minus B squared equal to a minus B times"}, {"start": 138.36, "duration": 9.06, "text": "a plus b okay so y squared minus 4 we"}, {"start": 143.58, "duration": 5.82, "text": "can write as follows y squared minus you"}, {"start": 147.42, "duration": 6.539, "text": "know"}, {"start": 149.4, "duration": 9.12, "text": "to sweat okay so using this property"}, {"start": 153.959, "duration": 10.021, "text": "above so we get we get y minus two"}, {"start": 158.52, "duration": 7.74, "text": "then times y plus two okay so here we"}, {"start": 163.98, "duration": 6.899, "text": "have our limit"}, {"start": 166.26, "duration": 8.04, "text": "as Y is approaching two so up there we"}, {"start": 170.879, "duration": 6.961, "text": "have here y minus 2"}, {"start": 174.3, "duration": 7.62, "text": "we have here y plus two"}, {"start": 177.84, "duration": 6.72, "text": "we have here y minus 2."}, {"start": 181.92, "duration": 6.12, "text": "then close brackets so we can do"}, {"start": 184.56, "duration": 5.759, "text": "cancellation here so y minus 2 and Y"}, {"start": 188.04, "duration": 4.86, "text": "minus 2 cancel out so we have this equal"}, {"start": 190.319, "duration": 5.761, "text": "to limit"}, {"start": 192.9, "duration": 4.08, "text": "so we have limits as Y is approaching"}, {"start": 196.08, "duration": 5.879, "text": "two"}, {"start": 196.98, "duration": 6.96, "text": "of Y Plus 2. so here we have as Y is"}, {"start": 201.959, "duration": 5.7, "text": "Operation so we'll get to that"}, {"start": 203.94, "duration": 6.659, "text": "substitute into y two so we get two plus"}, {"start": 207.659, "duration": 5.101, "text": "two and this is equal to four so this is"}, {"start": 210.599, "duration": 3.78, "text": "the right answer to the problem thank"}, {"start": 212.76, "duration": 3.839, "text": "you for viewing and please make sure you"}, {"start": 214.379, "duration": 4.14, "text": "subscribe to our YouTube channel for"}, {"start": 216.599, "duration": 4.401, "text": "more interesting videos see you in our"}, {"start": 218.519, "duration": 2.481, "text": "next video"}], "title": "Limit. Avoid L'Hospital rule. Here we showed the algebraic approach.", "description": "", "lengthSeconds": 219, "width": 636, "height": 360}
{"video_id": "ptv9JuPcgQM", "transcript": [{"start": 0.399, "duration": 8.721, "text": "hello welcome back today we have to find"}, {"start": 4.24, "duration": 10.359, "text": "this limit here so this is read as limit"}, {"start": 9.12, "duration": 11.36, "text": "as X is approaching one of this function"}, {"start": 14.599, "duration": 8.76, "text": "here x\u00b2 -1 all over x"}, {"start": 20.48, "duration": 7.559, "text": "-1 so before we proceed let us try to"}, {"start": 23.359, "duration": 7.361, "text": "make a direct substitution here so if we"}, {"start": 28.039, "duration": 4.161, "text": "make direct substitution here so we're"}, {"start": 30.72, "duration": 6.6, "text": "going to"}, {"start": 32.2, "duration": 9.76, "text": "have X to be equal to 1 so we have 1 s"}, {"start": 37.32, "duration": 9.879, "text": "right then - 1 all over so down here we"}, {"start": 41.96, "duration": 11.16, "text": "have 1 - 1 and 1 s is 1 1 - 1 we have"}, {"start": 47.199, "duration": 9.081, "text": "here 0 over 1 - 1 here this is 0 so 0/ 0"}, {"start": 53.12, "duration": 7.04, "text": "is an indeterminate form and this is not"}, {"start": 56.28, "duration": 6.279, "text": "allowed in mathematics right so but the"}, {"start": 60.16, "duration": 5.68, "text": "most interesting part of this solution"}, {"start": 62.559, "duration": 8.88, "text": "is that I'm not going to make use of the"}, {"start": 65.84, "duration": 8.24, "text": "L hope to so let us see a very nice way"}, {"start": 71.439, "duration": 6.201, "text": "to solve this problem so we're going to"}, {"start": 74.08, "duration": 7.76, "text": "recall the difference of two squares"}, {"start": 77.64, "duration": 7.439, "text": "formula so recall we have this a"}, {"start": 81.84, "duration": 10.8, "text": "2us b 2 is ="}, {"start": 85.079, "duration": 11.921, "text": "to a + b * a - B right so now let us"}, {"start": 92.64, "duration": 7.159, "text": "start to factorize this numerator here"}, {"start": 97.0, "duration": 6.6, "text": "so here we're going to have x^2 minus"}, {"start": 99.799, "duration": 7.241, "text": "you know 1 is the same as 1 2 so going"}, {"start": 103.6, "duration": 10.479, "text": "to rewrite this in this form so this is"}, {"start": 107.04, "duration": 11.48, "text": "going to be equal to x + 1 all over X -"}, {"start": 114.079, "duration": 6.161, "text": "1 right so our limit now is going to be"}, {"start": 118.52, "duration": 4.44, "text": "limit"}, {"start": 120.24, "duration": 9.2, "text": "as X is approaching"}, {"start": 122.96, "duration": 12.999, "text": "one of up here we have x + 1 * by x"}, {"start": 129.44, "duration": 10.72, "text": "-1 all over x -1 right so we can see"}, {"start": 135.959, "duration": 6.041, "text": "that x -1 and x -1 will cancel out so"}, {"start": 140.16, "duration": 8.439, "text": "this becomes"}, {"start": 142.0, "duration": 11.04, "text": "limit as X is approaching one of x + 1"}, {"start": 148.599, "duration": 10.28, "text": "right so this is going to be equal to"}, {"start": 153.04, "duration": 8.96, "text": "you know put x = 1 so here we have 1 + 1"}, {"start": 158.879, "duration": 6.201, "text": "and this is equal to two and that's the"}, {"start": 162.0, "duration": 5.84, "text": "right answer to this problem thank you"}, {"start": 165.08, "duration": 6.92, "text": "for watching please do wa to like"}, {"start": 167.84, "duration": 4.16, "text": "comment and share"}], "title": "Interesting Limit", "description": "Interesting limit.", "lengthSeconds": 171, "width": 640, "height": 360}
{"video_id": "lfRgfPA1FOs", "transcript": [{"start": 0.68, "duration": 4.78, "text": "in this place we are given the following"}, {"start": 3.3, "duration": 4.2, "text": "equation 2 raised by x minus 2 is equal"}, {"start": 5.46, "duration": 3.9, "text": "to 100 then we're asked to find the"}, {"start": 7.5, "duration": 3.78, "text": "value of 2 raised to power x"}, {"start": 9.36, "duration": 3.96, "text": "if you want to solve this equation to"}, {"start": 11.28, "duration": 4.68, "text": "solve for the value of x here we're"}, {"start": 13.32, "duration": 4.86, "text": "going to use the natural logarithm but"}, {"start": 15.96, "duration": 4.8, "text": "we just use some algebraic manipulation"}, {"start": 18.18, "duration": 5.64, "text": "approach here we have 2 raised for S"}, {"start": 20.76, "duration": 5.16, "text": "minus 2 equals 100 2 raised by S"}, {"start": 23.82, "duration": 5.119, "text": "according to indices law over two raised"}, {"start": 25.92, "duration": 3.019, "text": "by 2 is equal to 100"}, {"start": 30.859, "duration": 6.7, "text": "is 4 times 100 therefore 2 raised to S"}, {"start": 34.32, "duration": 5.46, "text": "is equal to 400 please subscribe to this"}, {"start": 37.559, "duration": 7.221, "text": "Channel and turn on the navigation Bell"}, {"start": 39.78, "duration": 5.0, "text": "if you find our videos interesting bye"}], "title": "Algebra is beautiful", "description": "", "lengthSeconds": 44, "width": 360, "height": 640}
{"video_id": "_Jm4I04gq10", "transcript": [{"start": 0.799, "duration": 4.04, "text": "hello welcome back once again in our"}, {"start": 3.08, "duration": 4.24, "text": "today's video we're going to solve the"}, {"start": 4.839, "duration": 5.001, "text": "following given Olympia mathematics"}, {"start": 7.32, "duration": 6.199, "text": "problem so from the left hand side we"}, {"start": 9.84, "duration": 7.439, "text": "have 27 to the^ X so this is equal to"}, {"start": 13.519, "duration": 6.281, "text": "from the right hand side we have 1 / X"}, {"start": 17.279, "duration": 4.801, "text": "now for us to get rid of this power here"}, {"start": 19.8, "duration": 5.12, "text": "we're going to raise both sides of this"}, {"start": 22.08, "duration": 7.08, "text": "equation to the power of the reciprocal"}, {"start": 24.92, "duration": 7.0, "text": "of X right so here raise this to the"}, {"start": 29.16, "duration": 8.28, "text": "power of the recip of X which is 1 / X"}, {"start": 31.92, "duration": 10.68, "text": "and here 2 1 / X so we can see x * 1 / X"}, {"start": 37.44, "duration": 11.72, "text": "is 1 so from there we have 27 to^ 1 this"}, {"start": 42.6, "duration": 8.52, "text": "is = to 1 /x ra^ 1/ x right okay so from"}, {"start": 49.16, "duration": 3.32, "text": "here let's make the left hand side the"}, {"start": 51.12, "duration": 6.52, "text": "right hand side and the right hand side"}, {"start": 52.48, "duration": 8.28, "text": "the left hand side so get 1 /x to^ 1 /x"}, {"start": 57.64, "duration": 8.2, "text": "this is equal to 27"}, {"start": 60.76, "duration": 9.999, "text": "okay so from here we get 1 /x to^ 1 /x"}, {"start": 65.84, "duration": 7.48, "text": "this isal 7 same as 3 to the^ 3 right so"}, {"start": 70.759, "duration": 6.68, "text": "we have this power 1 /x = 3 from the"}, {"start": 73.32, "duration": 8.56, "text": "base 1 / x = 3 so this implies from here"}, {"start": 77.439, "duration": 7.72, "text": "that 1 /x is = to 3 right so if the"}, {"start": 81.88, "duration": 4.599, "text": "reciprocal of X is equal to 3 then this"}, {"start": 85.159, "duration": 6.841, "text": "implies"}, {"start": 86.479, "duration": 7.6, "text": "that X is = 1 / 3 right or if you want"}, {"start": 92.0, "duration": 5.2, "text": "to find out you can put this over one"}, {"start": 94.079, "duration": 7.201, "text": "and cross multiply x * 3 is"}, {"start": 97.2, "duration": 5.599, "text": "3x this is = to 1 * 1 is 1 divide by the"}, {"start": 101.28, "duration": 4.519, "text": "coefficient of"}, {"start": 102.799, "duration": 7.0, "text": "x this will cancel this and here we get"}, {"start": 105.799, "duration": 6.081, "text": "X is = 1 / 3 and this is the correct"}, {"start": 109.799, "duration": 4.28, "text": "answer thank you for watching please"}, {"start": 111.88, "duration": 7.239, "text": "kindly subscribe to my channel alsoo"}, {"start": 114.079, "duration": 5.04, "text": "like comment and share bye-bye"}], "title": "Olympiad Math | Can you solve ?", "description": "", "lengthSeconds": 118, "width": 428, "height": 240}
{"video_id": "7fCRbe4MoJ0", "transcript": [{"start": 0.719, "duration": 3.481, "text": "hello good viewers welcome back once"}, {"start": 2.44, "duration": 4.96, "text": "again today we're going to solve for the"}, {"start": 4.2, "duration": 7.0, "text": "value of x in the following given"}, {"start": 7.4, "duration": 7.6, "text": "equation we have 2^ of 3 to ^ of 4 to ^"}, {"start": 11.2, "duration": 3.8, "text": "of X this is equal to"}, {"start": 15.16, "duration": 6.08, "text": "512 if you notice this you will know"}, {"start": 17.68, "duration": 7.0, "text": "that 512 can be expressed in powers of"}, {"start": 21.24, "duration": 7.039, "text": "two right and from here we have the base"}, {"start": 24.68, "duration": 6.32, "text": "here to so let's go ahead and express"}, {"start": 28.279, "duration": 7.321, "text": "512 as power of"}, {"start": 31.0, "duration": 9.039, "text": "2 recall 512 is exactly equal to 2 the"}, {"start": 35.6, "duration": 10.16, "text": "Power of 9 right so here we have 2^ of"}, {"start": 40.039, "duration": 8.961, "text": "3^ 4 x this will be equal to 2^ of"}, {"start": 45.76, "duration": 6.4, "text": "9 now carefully looking at this equation"}, {"start": 49.0, "duration": 6.359, "text": "we can see that we have equal bases so"}, {"start": 52.16, "duration": 8.199, "text": "we can equate the powers so equating the"}, {"start": 55.359, "duration": 9.041, "text": "powers this implies that 3^ 4^ X this is"}, {"start": 60.359, "duration": 8.041, "text": "equal to 9 again we can express 9 in"}, {"start": 64.4, "duration": 11.079, "text": "index powers of three right so here we"}, {"start": 68.4, "duration": 9.48, "text": "have 3 ^ 4^ x this is equal to 3 2 from"}, {"start": 75.479, "duration": 5.28, "text": "here once again we can see that we have"}, {"start": 77.88, "duration": 5.919, "text": "equal bases so we can equate the powers"}, {"start": 80.759, "duration": 6.68, "text": "from here this implies that 4 to^ of X"}, {"start": 83.799, "duration": 6.32, "text": "is equal to 2 do not forget that 4 is"}, {"start": 87.439, "duration": 5.561, "text": "equal 2 2 right then with this power"}, {"start": 90.119, "duration": 7.32, "text": "here remember we have 2 2 then to the^"}, {"start": 93.0, "duration": 8.24, "text": "of X this is equal to 2 you know 2 * X"}, {"start": 97.439, "duration": 7.881, "text": "is 2 X so we have 2 ^ 2x this is equal"}, {"start": 101.24, "duration": 6.36, "text": "to 2 ^ of 1 again we have equal bases so"}, {"start": 105.32, "duration": 6.24, "text": "we can equate the power so we have here"}, {"start": 107.6, "duration": 7.199, "text": "2x is = 1 dividing both side by 2 we get"}, {"start": 111.56, "duration": 6.4, "text": "the solution that X is = 1 over two"}, {"start": 114.799, "duration": 4.841, "text": "amazing and this is a right answer thank"}, {"start": 117.96, "duration": 4.159, "text": "you for watching please kindly subscribe"}, {"start": 119.64, "duration": 6.04, "text": "to my my channel also like comment and"}, {"start": 122.119, "duration": 3.561, "text": "share bye-bye"}], "title": "China | Interesting exponential equation | #maths #mathematics", "description": "", "lengthSeconds": 125, "width": 552, "height": 360}
{"video_id": "1MffctB6NT0", "transcript": [{"start": 0.919, "duration": 5.801, "text": "hello welcome back once again today we"}, {"start": 3.76, "duration": 6.2, "text": "need to discuss something very"}, {"start": 6.72, "duration": 6.76, "text": "important a lot of students fails to"}, {"start": 9.96, "duration": 7.399, "text": "provide all solutions to this"}, {"start": 13.48, "duration": 5.639, "text": "problem many would think like"}, {"start": 17.359, "duration": 8.441, "text": "this we"}, {"start": 19.119, "duration": 11.64, "text": "have x - 9 to the^ 2 which is equal to"}, {"start": 25.8, "duration": 6.84, "text": "81 many we go ahead and expand this"}, {"start": 30.759, "duration": 3.081, "text": "then transforming everything to a"}, {"start": 32.64, "duration": 4.36, "text": "quadratic"}, {"start": 33.84, "duration": 6.36, "text": "equation but that is really a long way"}, {"start": 37.0, "duration": 5.039, "text": "to go and many would think that the best"}, {"start": 40.2, "duration": 4.8, "text": "way to solve is just to take root of"}, {"start": 42.039, "duration": 5.321, "text": "both sides in this form here take root"}, {"start": 45.0, "duration": 7.239, "text": "of the left hand side that would be x -"}, {"start": 47.36, "duration": 10.08, "text": "9 which will be equal to < TK 81 then we"}, {"start": 52.239, "duration": 9.721, "text": "get x - 9 is equal to > of 81 is 9 add 9"}, {"start": 57.44, "duration": 7.32, "text": "to both sides so we get X is = to9 9 + 9"}, {"start": 61.96, "duration": 5.76, "text": "which is equal to"}, {"start": 64.76, "duration": 8.2, "text": "18 yes of course this is a"}, {"start": 67.72, "duration": 7.6, "text": "solution but this only provide one"}, {"start": 72.96, "duration": 3.519, "text": "solution so this is not the best way to"}, {"start": 75.32, "duration": 3.88, "text": "solve this"}, {"start": 76.479, "duration": 4.081, "text": "problem there's something the person"}, {"start": 79.2, "duration": 4.2, "text": "really"}, {"start": 80.56, "duration": 5.16, "text": "forget when we're taking roots of both"}, {"start": 83.4, "duration": 5.8, "text": "sides the right hand side should carry"}, {"start": 85.72, "duration": 6.039, "text": "plus or minus so the best to the best"}, {"start": 89.2, "duration": 9.08, "text": "solution to this problem should be we"}, {"start": 91.759, "duration": 9.32, "text": "have x - 9 2 which is equal to 81 now"}, {"start": 98.28, "duration": 6.96, "text": "taking root of both sides so we get x -"}, {"start": 101.079, "duration": 5.72, "text": "9 this should be plus or minus square"}, {"start": 105.24, "duration": 5.879, "text": "root of"}, {"start": 106.799, "duration": 5.881, "text": "81 yes this is 100%"}, {"start": 111.119, "duration": 6.64, "text": "true"}, {"start": 112.68, "duration": 9.399, "text": "okay so from here we get x - 9 is equal"}, {"start": 117.759, "duration": 7.761, "text": "to plus or minus squ of 81 is"}, {"start": 122.079, "duration": 9.641, "text": "9 so now we have two linear equations to"}, {"start": 125.52, "duration": 9.6, "text": "solve the first one is x - 9 is = POS 9"}, {"start": 131.72, "duration": 5.159, "text": "and the second one is x - 9 is"}, {"start": 135.12, "duration": 4.88, "text": "="}, {"start": 136.879, "duration": 6.72, "text": "9 so from here add 9 to both sides so we"}, {"start": 140.0, "duration": 8.8, "text": "get X is = 9 + 9 which is"}, {"start": 143.599, "duration": 9.28, "text": "18 yes this is indeed a solution now add"}, {"start": 148.8, "duration": 8.4, "text": "n to both Sid from here we get X is = -9"}, {"start": 152.879, "duration": 8.041, "text": "+ 9 here we get another solution x is ="}, {"start": 157.2, "duration": 8.16, "text": "0 we can see now with this approach we"}, {"start": 160.92, "duration": 5.92, "text": "found all solutions x equal to 0 and x"}, {"start": 165.36, "duration": 4.44, "text": "equal to"}, {"start": 166.84, "duration": 5.319, "text": "18 thank you for watching please kindly"}, {"start": 169.8, "duration": 6.88, "text": "subscribe to my channel also like"}, {"start": 172.159, "duration": 4.521, "text": "comment and share bye-bye"}], "title": "It's easy but we need to be careful | Interesting algebraic problem", "description": "", "lengthSeconds": 175, "width": 640, "height": 360}
{"video_id": "4RYivbz95Gw", "transcript": [{"start": 0.32, "duration": 5.88, "text": "hello good day everyone welcome back"}, {"start": 3.32, "duration": 4.2, "text": "once again to my YouTube Channel please"}, {"start": 6.2, "duration": 6.12, "text": "if you have not subscribed to this"}, {"start": 7.52, "duration": 4.8, "text": "Channel please kindly subscribe to this"}, {"start": 13.12, "duration": 5.4, "text": "channel"}, {"start": 15.08, "duration": 6.52, "text": "subscribe to this YouTube channel and"}, {"start": 18.52, "duration": 9.079, "text": "also share this video here we giv the"}, {"start": 21.6, "duration": 10.24, "text": "following problem that A + B + C is = 1"}, {"start": 27.599, "duration": 7.601, "text": "a ^ of 2 + B to the ^ of 2 + c^ of 2 is"}, {"start": 31.84, "duration": 8.64, "text": "= 3 and right over here we asked to find"}, {"start": 35.2, "duration": 7.839, "text": "the value of a * b + B * C + a *"}, {"start": 40.48, "duration": 5.64, "text": "C okay now let's see how to solve this"}, {"start": 43.039, "duration": 4.52, "text": "problem let us consider expanding the"}, {"start": 46.12, "duration": 3.84, "text": "following"}, {"start": 47.559, "duration": 7.16, "text": "binomial A +"}, {"start": 49.96, "duration": 6.48, "text": "B + C ra^ of 2 if we expand this"}, {"start": 54.719, "duration": 3.64, "text": "binomial we're going to"}, {"start": 56.44, "duration": 4.16, "text": "have"}, {"start": 58.359, "duration": 10.881, "text": "a\u00b2 Plus"}, {"start": 60.6, "duration": 16.199, "text": "b 2 + C 2 + 2 a b +"}, {"start": 69.24, "duration": 16.96, "text": "2 b c then + 2 a c right so here we have"}, {"start": 76.799, "duration": 11.96, "text": "a + b + C ra^ of 2 is = to a 2 + b 2 + C"}, {"start": 86.2, "duration": 4.52, "text": "2 now the whole three terms here we can"}, {"start": 88.759, "duration": 4.601, "text": "see that two is Comm on here so we're"}, {"start": 90.72, "duration": 10.0, "text": "going to factor out two so here we have"}, {"start": 93.36, "duration": 10.439, "text": "+ 2 Open Bracket a * b + B * C + a * C"}, {"start": 100.72, "duration": 5.719, "text": "right so we can see that we can make"}, {"start": 103.799, "duration": 5.521, "text": "substitutions from here we given that A"}, {"start": 106.439, "duration": 6.521, "text": "+ B + C is equal 1 and right over here"}, {"start": 109.32, "duration": 7.6, "text": "we have a + b + C ra to the power of 2"}, {"start": 112.96, "duration": 8.24, "text": "so this going to become 1^ of 2 is equal"}, {"start": 116.92, "duration": 7.0, "text": "to a 2 + b 2 + c\u00b2 the value is given in"}, {"start": 121.2, "duration": 5.959, "text": "the problem as three so here we're going"}, {"start": 123.92, "duration": 9.319, "text": "to have this to be replaced with three"}, {"start": 127.159, "duration": 11.201, "text": "then plus 2 Open brackets a * b + B *"}, {"start": 133.239, "duration": 7.921, "text": "C+ a c close bracket right so now we can"}, {"start": 138.36, "duration": 4.28, "text": "make the right hand side to be the left"}, {"start": 141.16, "duration": 3.799, "text": "hand side and the left hand side to be"}, {"start": 142.64, "duration": 7.08, "text": "the right hand side since we are solving"}, {"start": 144.959, "duration": 9.36, "text": "for a * b + B * C + a * C so we're going"}, {"start": 149.72, "duration": 12.36, "text": "to have here two Open brackets a * b + B"}, {"start": 154.319, "duration": 9.721, "text": "* C + a * C then + 3 is = 1 power of 2"}, {"start": 162.08, "duration": 4.2, "text": "so we can see that this does not affect"}, {"start": 164.04, "duration": 8.4, "text": "the question right so here we're going"}, {"start": 166.28, "duration": 12.16, "text": "to have two Open Bracket a * b + B * C +"}, {"start": 172.44, "duration": 8.04, "text": "a * C then + 3 is = to 1 2 is 1 so we're"}, {"start": 178.44, "duration": 4.76, "text": "going to subtract three from both sides"}, {"start": 180.48, "duration": 9.759, "text": "right so here we have two Open brackets"}, {"start": 183.2, "duration": 10.08, "text": "a * b + B * C + a * C is equal to you"}, {"start": 190.239, "duration": 7.0, "text": "know 1 - 3 that"}, {"start": 193.28, "duration": 7.16, "text": "is 2 right so from here since we are"}, {"start": 197.239, "duration": 5.72, "text": "looking for the value of a * b + B * C +"}, {"start": 200.44, "duration": 5.6, "text": "a * C so we can divide both sides of the"}, {"start": 202.959, "duration": 7.48, "text": "equation by the coefficient of a * b + B"}, {"start": 206.04, "duration": 6.88, "text": "* C + a * C so here we divide by 2 and"}, {"start": 210.439, "duration": 7.601, "text": "here we divide by two so this cancel out"}, {"start": 212.92, "duration": 10.12, "text": "so we get the value for a * B plus b * C"}, {"start": 218.04, "duration": 7.96, "text": "+ a * C to be equal to -2 2 and that is"}, {"start": 223.04, "duration": 5.88, "text": "-1 and this is the right answer thank"}, {"start": 226.0, "duration": 6.64, "text": "you for watching please do where to like"}, {"start": 228.92, "duration": 6.72, "text": "please comment and also share this video"}, {"start": 232.64, "duration": 3.0, "text": "bye-bye"}], "title": "Interesting Olympiad Math. Involving three variables with only two equations.", "description": "", "lengthSeconds": 234, "width": 568, "height": 320}
{"video_id": "DPrciTnZPcc", "transcript": [{"start": 0.76, "duration": 4.72, "text": "hello welcome back once again today we"}, {"start": 3.04, "duration": 5.72, "text": "have this interesting equation here with"}, {"start": 5.48, "duration": 7.239, "text": "a negative base we have the negative pi^"}, {"start": 8.76, "duration": 7.56, "text": "x = a and we have two interesting"}, {"start": 12.719, "duration": 9.841, "text": "constants here we have pi and the"}, {"start": 16.32, "duration": 8.76, "text": "constant e how can pk^ X be equal to a"}, {"start": 22.56, "duration": 4.92, "text": "and carefully looking at this equation"}, {"start": 25.08, "duration": 5.039, "text": "one can easily say there's no real"}, {"start": 27.48, "duration": 4.72, "text": "solution right yes I strongly believe"}, {"start": 30.119, "duration": 4.921, "text": "that the only solution for X must be"}, {"start": 32.2, "duration": 4.56, "text": "complex Solutions right now let's get"}, {"start": 35.04, "duration": 3.519, "text": "into this video in this video we're"}, {"start": 36.76, "duration": 4.799, "text": "going to solve for just a complex"}, {"start": 38.559, "duration": 5.801, "text": "solution we're not here to solve for"}, {"start": 41.559, "duration": 5.441, "text": "general form or the general solution"}, {"start": 44.36, "duration": 4.679, "text": "right okay now this go into this video"}, {"start": 47.0, "duration": 4.96, "text": "and we're going to start by natural"}, {"start": 49.039, "duration": 7.121, "text": "loging both sides of this equation so"}, {"start": 51.96, "duration": 7.119, "text": "here we have the natural log of piun to"}, {"start": 56.16, "duration": 4.6, "text": "the^ of X this is equal to the natural"}, {"start": 59.079, "duration": 3.201, "text": "log of a"}, {"start": 60.76, "duration": 3.52, "text": "now with the property of the natural"}, {"start": 62.28, "duration": 5.479, "text": "logarithm remember this power here is"}, {"start": 64.28, "duration": 5.24, "text": "crossing over here we have X MTI by the"}, {"start": 67.759, "duration": 8.321, "text": "natural log"}, {"start": 69.52, "duration": 9.16, "text": "ofk is same as -1 * pi right and this is"}, {"start": 76.08, "duration": 4.52, "text": "equal to the natural log of e is one"}, {"start": 78.68, "duration": 3.64, "text": "right because the base of the natural"}, {"start": 80.6, "duration": 4.6, "text": "logm is"}, {"start": 82.32, "duration": 4.68, "text": "e okay now from here let us take note of"}, {"start": 85.2, "duration": 4.84, "text": "this property of the natural logarithm"}, {"start": 87.0, "duration": 5.96, "text": "when we have the natural log of a * B"}, {"start": 90.04, "duration": 5.24, "text": "this is same as the natural log of a"}, {"start": 92.96, "duration": 5.6, "text": "plus the natural log of"}, {"start": 95.28, "duration": 6.28, "text": "B so from here this will be X into"}, {"start": 98.56, "duration": 6.64, "text": "bracket so this will be natural log of"}, {"start": 101.56, "duration": 7.599, "text": "-1 plus the natural log of"}, {"start": 105.2, "duration": 5.8, "text": "Pi and this is equal to one so from here"}, {"start": 109.159, "duration": 4.841, "text": "let's take note of something really very"}, {"start": 111.0, "duration": 9.159, "text": "important and amazing"}, {"start": 114.0, "duration": 8.52, "text": "from -1 is equal to e to power of I pi"}, {"start": 120.159, "duration": 5.121, "text": "right now let's natural log both sides"}, {"start": 122.52, "duration": 5.12, "text": "so we have natural log of1 this will be"}, {"start": 125.28, "duration": 4.72, "text": "equal to natural log of e to the^ of I"}, {"start": 127.64, "duration": 4.52, "text": "pi and with the property of the natural"}, {"start": 130.0, "duration": 4.48, "text": "logarithm so this power here is coming"}, {"start": 132.16, "duration": 6.439, "text": "down here so here we have the natural"}, {"start": 134.48, "duration": 6.64, "text": "log of1 to be equal to I Pi MTI the"}, {"start": 138.599, "duration": 5.561, "text": "natural of e and we all know that the"}, {"start": 141.12, "duration": 7.24, "text": "natural log of e is equal to 1 so we get"}, {"start": 144.16, "duration": 7.48, "text": "this is equal to I Pi * 1 which is I pi"}, {"start": 148.36, "duration": 6.8, "text": "and therefore we have X into bracket so"}, {"start": 151.64, "duration": 6.959, "text": "instead of natural logarithm of1 so we"}, {"start": 155.16, "duration": 6.799, "text": "replace that with I Pi then plus the"}, {"start": 158.599, "duration": 6.481, "text": "natural log of Pi then this is equal to"}, {"start": 161.959, "duration": 5.881, "text": "1 since we are solving for x let's go"}, {"start": 165.08, "duration": 6.4, "text": "ahead and divide both sides by this"}, {"start": 167.84, "duration": 4.75, "text": "coefficient of x which is I pi plus the"}, {"start": 171.48, "duration": 5.52, "text": "natural log of"}, {"start": 172.59, "duration": 7.569, "text": "[Music]"}, {"start": 177.0, "duration": 5.8, "text": "Pi and from here this will cancel this"}, {"start": 180.159, "duration": 9.201, "text": "right and here we get the solution X is"}, {"start": 182.8, "duration": 10.24, "text": "= 1 all over I Pi + the natural log of"}, {"start": 189.36, "duration": 5.64, "text": "Pi and this is the complex solution"}, {"start": 193.04, "duration": 4.119, "text": "thank you for watching please kindly"}, {"start": 195.0, "duration": 6.28, "text": "subscribe to my channel also like"}, {"start": 197.159, "duration": 4.121, "text": "comment and share bye-bye"}], "title": "Interesting equation involving the two most interesting constants.", "description": "", "lengthSeconds": 200, "width": 640, "height": 360}
{"video_id": "9KFzY1_16C0", "transcript": [{"start": 1.12, "duration": 5.36, "text": "hello welcome back once again today we"}, {"start": 3.6, "duration": 7.36, "text": "have this nice question here we have"}, {"start": 6.48, "duration": 8.879, "text": "this cubic equation x Cub + X2 = 11 and"}, {"start": 10.96, "duration": 11.159, "text": "here we're going to evaluate this hexic"}, {"start": 15.359, "duration": 10.361, "text": "expression x ^ 6 + 2 x ^ 5 + x ^ 4 - 2 X"}, {"start": 22.119, "duration": 6.48, "text": "Cub - 2 x^ 2 +"}, {"start": 25.72, "duration": 6.96, "text": "15 how are we going to do"}, {"start": 28.599, "duration": 8.561, "text": "this okay now let us see from the given"}, {"start": 32.68, "duration": 8.0, "text": "equation we have X cubed +"}, {"start": 37.16, "duration": 7.84, "text": "X2 which is equal to"}, {"start": 40.68, "duration": 6.44, "text": "11 right okay now from here let's go"}, {"start": 45.0, "duration": 8.12, "text": "ahead and subtract one from both sides"}, {"start": 47.12, "duration": 9.2, "text": "so here we have X cubed + x^2 - 1 this"}, {"start": 53.12, "duration": 9.88, "text": "will be equal to 11 -"}, {"start": 56.32, "duration": 9.799, "text": "1 so from here we have X Cub + + x^2 - 1"}, {"start": 63.0, "duration": 6.72, "text": "this will be = 11 - 1 which is 10 let's"}, {"start": 66.119, "duration": 7.921, "text": "Square both sides of this equation from"}, {"start": 69.72, "duration": 14.079, "text": "here so from the left that is same as X"}, {"start": 74.04, "duration": 14.079, "text": "Cub + X2 - 1 * X Cub + X2 - 1 this is ="}, {"start": 83.799, "duration": 6.64, "text": "to 100 so from here x Cub * X Cub will"}, {"start": 88.119, "duration": 8.841, "text": "give us X to the^ 6"}, {"start": 90.439, "duration": 9.521, "text": "+ x * X2 will give us x ^ 5 - x * -1"}, {"start": 96.96, "duration": 6.4, "text": "that ISX cubed so continue with the"}, {"start": 99.96, "duration": 9.96, "text": "second term here we get x\u00b2 * X Cub which"}, {"start": 103.36, "duration": 10.16, "text": "is x^ 5 plus x * X2 X ^ 4 - x 2 * 1 here"}, {"start": 109.92, "duration": 8.239, "text": "will give us X2 then -1 we multiply"}, {"start": 113.52, "duration": 8.68, "text": "everything right so we getx Cub - x 2"}, {"start": 118.159, "duration": 7.441, "text": "then + a 1 then this equals"}, {"start": 122.2, "duration": 5.8, "text": "100 so from here let us simplify we have"}, {"start": 125.6, "duration": 7.16, "text": "here x to the^"}, {"start": 128.0, "duration": 9.68, "text": "6 Plus from here x ^ 5 + x ^ 5 will give"}, {"start": 132.76, "duration": 9.88, "text": "us 2 x ^ 5 right and from here we have +"}, {"start": 137.68, "duration": 10.88, "text": "x ^ 4 here x Cub - x Cub will give us"}, {"start": 142.64, "duration": 7.92, "text": "-2X Cub here x^2 - X2 will give us -2X 2"}, {"start": 148.56, "duration": 5.28, "text": "and here we have + 1"}, {"start": 150.56, "duration": 4.6, "text": "then this equals 100 okay now let's go"}, {"start": 153.84, "duration": 3.88, "text": "back to the expression we asked to"}, {"start": 155.16, "duration": 5.2, "text": "evaluate so if you look at this this is"}, {"start": 157.72, "duration": 6.48, "text": "almost the same with the Hees expression"}, {"start": 160.36, "duration": 6.28, "text": "we asked to evaluate the only difference"}, {"start": 164.2, "duration": 5.319, "text": "here is that we have plus one here and"}, {"start": 166.64, "duration": 6.28, "text": "here we have + 15 so what happens when"}, {"start": 169.519, "duration": 4.881, "text": "we add 14 to both sides of this equation"}, {"start": 172.92, "duration": 4.319, "text": "you will find out that the left hand"}, {"start": 174.4, "duration": 4.8, "text": "side will automatically turn to the heic"}, {"start": 177.239, "duration": 4.761, "text": "expression right so let's go ahead and"}, {"start": 179.2, "duration": 10.48, "text": "do that so let us add 14 to both sides"}, {"start": 182.0, "duration": 10.76, "text": "so here we get X cubed + 2 x ^ 5 + x ^ 4"}, {"start": 189.68, "duration": 3.08, "text": "- 2 X"}, {"start": 193.159, "duration": 9.561, "text": "Cub - 2 x^ 2 plus here we're adding 14"}, {"start": 198.599, "duration": 8.121, "text": "so is that 14 yes so 1 + 14 will give us"}, {"start": 202.72, "duration": 5.28, "text": "15 right and here this equals 100 + 14"}, {"start": 206.72, "duration": 5.079, "text": "which is"}, {"start": 208.0, "duration": 5.92, "text": "114 and here is the right answer thank"}, {"start": 211.799, "duration": 4.36, "text": "you for watching please kindly subscribe"}, {"start": 213.92, "duration": 5.879, "text": "to my channel also like comment and"}, {"start": 216.159, "duration": 3.64, "text": "share bye-bye"}], "title": "Interesting Olympiad Math", "description": "", "lengthSeconds": 218, "width": 640, "height": 360}
{"video_id": "qCJp927zzG8", "transcript": [{"start": 0.96, "duration": 5.04, "text": "hello you're welcome back once again"}, {"start": 3.6, "duration": 3.4, "text": "today we're going to solve these systems"}, {"start": 6.0, "duration": 4.08, "text": "of"}, {"start": 7.0, "duration": 7.32, "text": "equation so we have from equation 1 we"}, {"start": 10.08, "duration": 9.119, "text": "have that 8 + B is = 1 we have from"}, {"start": 14.32, "duration": 9.52, "text": "equation 2 that b + C is = 2 and we have"}, {"start": 19.199, "duration": 7.121, "text": "from equation 3 that a + C is ="}, {"start": 23.84, "duration": 6.96, "text": "3 okay now let's get"}, {"start": 26.32, "duration": 9.0, "text": "started let us start by subtracting"}, {"start": 30.8, "duration": 7.079, "text": "equation 3 from equation 1 so here we"}, {"start": 35.32, "duration": 8.64, "text": "have equation 1"}, {"start": 37.879, "duration": 12.441, "text": "A + B is = to 1 and we have the equation"}, {"start": 43.96, "duration": 7.16, "text": "3 A+ C is equal 3 so let's subtract"}, {"start": 50.32, "duration": 3.8, "text": "these"}, {"start": 51.12, "duration": 3.0, "text": "equations"}, {"start": 54.68, "duration": 8.8, "text": "okay so now here we have a - A that will"}, {"start": 59.64, "duration": 7.24, "text": "give us 0 then we have B minus C from"}, {"start": 63.48, "duration": 8.6, "text": "the left hand side so we have B minus C"}, {"start": 66.88, "duration": 11.32, "text": "this is equal to 1 - 3 that will give us"}, {"start": 72.08, "duration": 12.44, "text": "-2 so now we have our equation 4 as b -"}, {"start": 78.2, "duration": 9.2, "text": "c is = to -2 so this is our equation 4"}, {"start": 84.52, "duration": 6.0, "text": "now let us go back to equation 2 we can"}, {"start": 87.4, "duration": 6.64, "text": "see from equation 2 we have B and C at"}, {"start": 90.52, "duration": 7.08, "text": "the left hand side and this equation 4"}, {"start": 94.04, "duration": 7.2, "text": "also contain B and C so let's make use"}, {"start": 97.6, "duration": 8.08, "text": "of equation 2 and equation 4 so this is"}, {"start": 101.24, "duration": 9.36, "text": "equation 4 equation 2 I mean b + C I"}, {"start": 105.68, "duration": 7.0, "text": "mean is equal to 2 right so now what are"}, {"start": 110.6, "duration": 4.36, "text": "we going to do are we going to subtract"}, {"start": 112.68, "duration": 3.32, "text": "or are we going to add obviously we're"}, {"start": 114.96, "duration": 6.519, "text": "going to"}, {"start": 116.0, "duration": 8.92, "text": "add so now b + B will give us 2 b"}, {"start": 121.479, "duration": 7.12, "text": "then C + C will give us 0 and this is"}, {"start": 124.92, "duration": 5.839, "text": "equal to -2 + 2 will give us 0 from here"}, {"start": 128.599, "duration": 3.521, "text": "let's solve for b divide by the"}, {"start": 130.759, "duration": 4.881, "text": "coefficient of"}, {"start": 132.12, "duration": 8.72, "text": "B this will cancel this from here this"}, {"start": 135.64, "duration": 5.2, "text": "implies that b is equal to"}, {"start": 141.08, "duration": 8.0, "text": "Z okay now from this equation two we can"}, {"start": 145.519, "duration": 4.921, "text": "easily solve for C right we have that b"}, {"start": 149.08, "duration": 5.6, "text": "+ C"}, {"start": 150.44, "duration": 10.4, "text": "is = 2 now substitute the value for B"}, {"start": 154.68, "duration": 8.839, "text": "then we have 0 + C is equal to 2"}, {"start": 160.84, "duration": 6.72, "text": "subtract 0 from both sides so we get C"}, {"start": 163.519, "duration": 9.0, "text": "is = to 2 - 0 which is 2 so we have"}, {"start": 167.56, "duration": 6.72, "text": "found a solution for c c is equal to 2"}, {"start": 172.519, "duration": 4.281, "text": "then we can use any of the three"}, {"start": 174.28, "duration": 4.52, "text": "equations to solve for a so let's make"}, {"start": 176.8, "duration": 4.32, "text": "use of equation one we know that"}, {"start": 178.8, "duration": 6.079, "text": "equation one we have A +"}, {"start": 181.12, "duration": 6.039, "text": "B which is equal to 1 remember that b is"}, {"start": 184.879, "duration": 4.121, "text": "equal to Z so we substitute the value of"}, {"start": 187.159, "duration": 5.561, "text": "B in terms"}, {"start": 189.0, "duration": 6.28, "text": "of B oh is that in terms of no exactly"}, {"start": 192.72, "duration": 6.28, "text": "equal to Zer right so B is equal to Z so"}, {"start": 195.28, "duration": 6.72, "text": "we have a + 0 is = 1 so when you"}, {"start": 199.0, "duration": 6.44, "text": "subtract 0 from both sides so we get a"}, {"start": 202.0, "duration": 6.28, "text": "is = 1 - 0 which is"}, {"start": 205.44, "duration": 4.84, "text": "one so we have our"}, {"start": 208.28, "duration": 4.0, "text": "solution a"}, {"start": 210.28, "duration": 10.44, "text": "is = to"}, {"start": 212.28, "duration": 10.64, "text": "1 B is equal to Z and C is equal to 2"}, {"start": 220.72, "duration": 5.12, "text": "right thank you for watching please"}, {"start": 222.92, "duration": 7.519, "text": "kindly subscribe to my channel also like"}, {"start": 225.84, "duration": 4.599, "text": "comment and share bye-bye"}], "title": "Interesting systems of equations | Can you solve ?", "description": "", "lengthSeconds": 229, "width": 428, "height": 240}
{"video_id": "7weE2y6EWqw", "transcript": [{"start": 0.76, "duration": 4.36, "text": "hello good viewers welcome back once"}, {"start": 2.76, "duration": 5.64, "text": "again please kindly subscribe to my"}, {"start": 5.12, "duration": 5.16, "text": "channel also like comment and share in"}, {"start": 8.4, "duration": 5.359, "text": "this video we're going to solve for x in"}, {"start": 10.28, "duration": 6.88, "text": "the following exponential equation right"}, {"start": 13.759, "duration": 4.241, "text": "now let's get started here we have 2 ^"}, {"start": 17.16, "duration": 3.8, "text": "of"}, {"start": 18.0, "duration": 5.199, "text": "X+ here we're going to write this second"}, {"start": 20.96, "duration": 5.36, "text": "term at the left hand side using the"}, {"start": 23.199, "duration": 7.4, "text": "property a to the power of b + C this is"}, {"start": 26.32, "duration": 9.399, "text": "equal to a power B * a^ of C"}, {"start": 30.599, "duration": 6.521, "text": "so this will be 2^ x * 2^ 3 this is"}, {"start": 35.719, "duration": 3.041, "text": "equal to"}, {"start": 37.12, "duration": 7.16, "text": "144"}, {"start": 38.76, "duration": 9.72, "text": "right so from here we have 2 ^ of x + 2"}, {"start": 44.28, "duration": 5.72, "text": "^ x * 2 cubed which is 8 this is equal"}, {"start": 48.48, "duration": 5.84, "text": "to"}, {"start": 50.0, "duration": 8.16, "text": "144 so let us factor out 2 the^ of X and"}, {"start": 54.32, "duration": 6.64, "text": "here we are left with 1 + 8 right this"}, {"start": 58.16, "duration": 7.56, "text": "is equal to 144"}, {"start": 60.96, "duration": 7.799, "text": "so from here this is equal to 2 x * 9"}, {"start": 65.72, "duration": 5.0, "text": "this is equal to 144 so from here let's"}, {"start": 68.759, "duration": 6.201, "text": "divide both sides by"}, {"start": 70.72, "duration": 11.28, "text": "9 so this cancel this from here we have"}, {"start": 74.96, "duration": 9.36, "text": "2 x this is equal to you know 144 / 9 we"}, {"start": 82.0, "duration": 7.479, "text": "are going to"}, {"start": 84.32, "duration": 9.439, "text": "have 16 right so from here we have 2^ of"}, {"start": 89.479, "duration": 6.401, "text": "x is equal to 16 is the same as 2 ^ 4 so"}, {"start": 93.759, "duration": 4.961, "text": "here we have equal bases we can equate"}, {"start": 95.88, "duration": 6.16, "text": "the powers and from here we get X isal"}, {"start": 98.72, "duration": 4.679, "text": "to 4 and that's the right answer thank"}, {"start": 102.04, "duration": 3.359, "text": "you for watching please don't foret to"}, {"start": 103.399, "duration": 6.04, "text": "subscribe to my channel also like"}, {"start": 105.399, "duration": 4.04, "text": "comment and share bye-bye"}], "title": "Very interesting exponential equation #maths #education", "description": "", "lengthSeconds": 108, "width": 640, "height": 360}
{"video_id": "x6T4--4kkkw", "transcript": [{"start": 0.68, "duration": 4.2, "text": "hello good day viewers welcome back once"}, {"start": 2.52, "duration": 5.12, "text": "again to another video please kindly"}, {"start": 4.88, "duration": 5.12, "text": "support my channel by subscribing also"}, {"start": 7.64, "duration": 4.84, "text": "like comment and share today we're going"}, {"start": 10.0, "duration": 8.279, "text": "to find the following limits limit as X"}, {"start": 12.48, "duration": 7.28, "text": "is approaching -2 of 3x + 6 all X Cub +"}, {"start": 18.279, "duration": 5.0, "text": "8"}, {"start": 19.76, "duration": 6.32, "text": "right here we're going to make use of"}, {"start": 23.279, "duration": 6.76, "text": "algebraic manipulations we are not going"}, {"start": 26.08, "duration": 5.999, "text": "to use the lhop to scroll right okay now"}, {"start": 30.039, "duration": 4.761, "text": "we're going to manipulate this function"}, {"start": 32.079, "duration": 5.121, "text": "here and I think we're going to make"}, {"start": 34.8, "duration": 4.88, "text": "some nice cancellations because from"}, {"start": 37.2, "duration": 6.44, "text": "here if you if we do direct substitution"}, {"start": 39.68, "duration": 6.76, "text": "here we are going to have 0/ 0 and this"}, {"start": 43.64, "duration": 5.28, "text": "is in indeterminate form right so let's"}, {"start": 46.44, "duration": 4.799, "text": "see what we're going to do so now let's"}, {"start": 48.92, "duration": 6.88, "text": "talk about this function here and see"}, {"start": 51.239, "duration": 9.721, "text": "the behavior we have here 3x + 6 all"}, {"start": 55.8, "duration": 8.439, "text": "over X Cub + 8 you know is that from the"}, {"start": 60.96, "duration": 7.04, "text": "numerator we can factor out three and we"}, {"start": 64.239, "duration": 7.441, "text": "have x + 2 right and this denominator"}, {"start": 68.0, "duration": 6.96, "text": "here we can write as follows X Cub plus"}, {"start": 71.68, "duration": 4.64, "text": "you know 8 is the same as 2 cubed okay"}, {"start": 74.96, "duration": 3.64, "text": "now from the denominator here we're"}, {"start": 76.32, "duration": 4.96, "text": "going to make use of the sum of two"}, {"start": 78.6, "duration": 7.159, "text": "cubes right and we have a formula for"}, {"start": 81.28, "duration": 7.24, "text": "that when we have a cubed + B Cubed you"}, {"start": 85.759, "duration": 8.0, "text": "know this is equal to a +"}, {"start": 88.52, "duration": 8.52, "text": "b multip by a 2 sorry"}, {"start": 93.759, "duration": 8.881, "text": "sorry this is minus from here"}, {"start": 97.04, "duration": 12.52, "text": "right oh so this is equal to a + b"}, {"start": 102.64, "duration": 11.04, "text": "* a 2 - A B then + b^ 2 right so from"}, {"start": 109.56, "duration": 6.479, "text": "the denominator here we have X Cub + 2"}, {"start": 113.68, "duration": 6.68, "text": "Cub you know this is going to be equal"}, {"start": 116.039, "duration": 9.601, "text": "to x + 2 * X2"}, {"start": 120.36, "duration": 10.239, "text": "- 2 * X which is 2 X then + 2^ 2 and"}, {"start": 125.64, "duration": 9.959, "text": "that's 4 right so from here we have 3x +"}, {"start": 130.599, "duration": 9.881, "text": "6 all X Cub + 8 you know this is equal"}, {"start": 135.599, "duration": 12.241, "text": "to here 3 into braet x + 2 then all"}, {"start": 140.48, "duration": 10.32, "text": "over x + 2 * X2 - 2x + 4 right so from"}, {"start": 147.84, "duration": 7.16, "text": "here we cancel out x + 2"}, {"start": 150.8, "duration": 9.159, "text": "and here we have this is = to 3 all over"}, {"start": 155.0, "duration": 9.28, "text": "x^2 - 2x + 4 now we can take the limit"}, {"start": 159.959, "duration": 10.841, "text": "so we have the limit as X is approaching"}, {"start": 164.28, "duration": 10.12, "text": "-2 of 3 x + 6 all X Cub + 8 you know"}, {"start": 170.8, "duration": 4.92, "text": "this is equal to limit as X is"}, {"start": 174.4, "duration": 8.559, "text": "approaching"}, {"start": 175.72, "duration": 9.68, "text": "-2 of 3 all over x^2 - - 2x + 4 and here"}, {"start": 182.959, "duration": 11.2, "text": "we make direct sub from here we have 3"}, {"start": 185.4, "duration": 12.16, "text": "all over here -2 2 - 2 * -2 + 4 right so"}, {"start": 194.159, "duration": 8.16, "text": "let us simplify this properly so here we"}, {"start": 197.56, "duration": 6.52, "text": "have this equals to 3 all over you -2"}, {"start": 202.319, "duration": 4.161, "text": "here is RA to the power of an even"}, {"start": 204.08, "duration": 5.079, "text": "number so we're going to have a positive"}, {"start": 206.48, "duration": 7.0, "text": "result so here we have -2 s which will"}, {"start": 209.159, "duration": 6.28, "text": "be four and here minus * minus is plus"}, {"start": 213.48, "duration": 5.52, "text": "here 2 * 2 which is 4 and here we have"}, {"start": 215.439, "duration": 8.201, "text": "four as well so this simplifies to 3 all"}, {"start": 219.0, "duration": 6.879, "text": "over 12 which is equal to 1/ 4 and this"}, {"start": 223.64, "duration": 3.879, "text": "is the right answer thank you for"}, {"start": 225.879, "duration": 4.08, "text": "watching please do wa to subscribe to my"}, {"start": 227.519, "duration": 5.44, "text": "channel also like comment and share"}, {"start": 229.959, "duration": 3.0, "text": "bye-bye"}], "title": "Interesting limit. Can you do this without L'hospital's rule ?  #maths #education", "description": "", "lengthSeconds": 231, "width": 640, "height": 360}
{"video_id": "CXoY7a0YWU8", "transcript": [{"start": 0.68, "duration": 5.639, "text": "hello welcome back once again welcome"}, {"start": 3.36, "duration": 4.88, "text": "back to shid online math class please if"}, {"start": 6.319, "duration": 4.921, "text": "today is the first day of visiting this"}, {"start": 8.24, "duration": 5.559, "text": "Channel please kindly support us by"}, {"start": 11.24, "duration": 4.519, "text": "subscribing to our channel so we can see"}, {"start": 13.799, "duration": 5.121, "text": "it over here subscribe please support"}, {"start": 15.759, "duration": 6.001, "text": "this our Channel thank you now we're"}, {"start": 18.92, "duration": 7.08, "text": "going to evaluate this the fourth root"}, {"start": 21.76, "duration": 9.04, "text": "of two all to the power of8 right so let"}, {"start": 26.0, "duration": 10.04, "text": "us see how to do this let us recall that"}, {"start": 30.8, "duration": 8.52, "text": "the M root of n is equal to n of 1 all n"}, {"start": 36.04, "duration": 9.6, "text": "right so this expression will"}, {"start": 39.32, "duration": 11.44, "text": "become in bracket 2 of 1 over 4 then"}, {"start": 45.64, "duration": 9.88, "text": "close bracket R of 8 right so let us"}, {"start": 50.76, "duration": 8.799, "text": "recall a r of M in bracket R power of n"}, {"start": 55.52, "duration": 6.44, "text": "is equal to a power of M * n right so"}, {"start": 59.559, "duration": 9.56, "text": "here this is equal to"}, {"start": 61.96, "duration": 8.88, "text": "2 of 1 / 4 * 8 right okay this is equal"}, {"start": 69.119, "duration": 5.561, "text": "to"}, {"start": 70.84, "duration": 9.68, "text": "2 of you know this is 8 all over one so"}, {"start": 74.68, "duration": 11.2, "text": "here 4 1 and here 2 2 of 2 and this is"}, {"start": 80.52, "duration": 11.32, "text": "equal to 2 * 2 and this is equal to 4"}, {"start": 85.88, "duration": 5.96, "text": "and this is the final answer 10 you"}], "title": "4th root radical Expression", "description": "", "lengthSeconds": 90, "width": 636, "height": 360}
{"video_id": "TZ62pgbPSH0", "transcript": [{"start": 0.799, "duration": 2.84, "text": "hello good viers welcome back once again"}, {"start": 2.639, "duration": 2.921, "text": "today we're going to solve for the"}, {"start": 3.639, "duration": 7.521, "text": "values of X in the following given"}, {"start": 5.56, "duration": 8.56, "text": "equation x^ 4 is equal to x^ of X yes of"}, {"start": 11.16, "duration": 6.24, "text": "course we one can easily say that X is"}, {"start": 14.12, "duration": 6.76, "text": "equal to 1 right but there's a way to"}, {"start": 17.4, "duration": 6.68, "text": "solve for the values of X course I found"}, {"start": 20.88, "duration": 5.36, "text": "out that X has two real values okay"}, {"start": 24.08, "duration": 4.08, "text": "let's go ahead and figure them out we"}, {"start": 26.24, "duration": 6.4, "text": "have to take the natural log of both"}, {"start": 28.16, "duration": 8.48, "text": "sides so we have here the L L of x^ of 4"}, {"start": 32.64, "duration": 7.439, "text": "this is equal to the Ln of X power of"}, {"start": 36.64, "duration": 7.0, "text": "X now recall the property of logarithm"}, {"start": 40.079, "duration": 8.16, "text": "which STS that the Ln of a to the power"}, {"start": 43.64, "duration": 9.88, "text": "of B this is equal to B MTI the Ln of a"}, {"start": 48.239, "duration": 10.561, "text": "right so here this is 4 * the Ln of X"}, {"start": 53.52, "duration": 7.359, "text": "this isal to x * the Ln of X now let's"}, {"start": 58.8, "duration": 5.719, "text": "subtract this Ln"}, {"start": 60.879, "duration": 10.441, "text": "X Ln of x from both sides so here we"}, {"start": 64.519, "duration": 9.6, "text": "have 4 * the Ln of x - x * the Ln of X"}, {"start": 71.32, "duration": 5.96, "text": "this is equal to Z Now from here we can"}, {"start": 74.119, "duration": 7.64, "text": "factor out the Ln of X and here we have"}, {"start": 77.28, "duration": 7.32, "text": "4 - x then this is equal to Z so from"}, {"start": 81.759, "duration": 7.561, "text": "here we have that Ln of X is = to Z that"}, {"start": 84.6, "duration": 9.559, "text": "is this implies okay or here we have 4 -"}, {"start": 89.32, "duration": 10.079, "text": "X8 is equal to0 so you know from here x"}, {"start": 94.159, "duration": 9.6, "text": "is = to e to the^ of 0 or from here x ="}, {"start": 99.399, "duration": 9.441, "text": "4 - 0 so from here we get the value x ="}, {"start": 103.759, "duration": 7.4, "text": "1 or here we have X is equal to 4 and"}, {"start": 108.84, "duration": 4.0, "text": "these are the real solutions thank you"}, {"start": 111.159, "duration": 3.92, "text": "for watching please kindly subscribe to"}, {"start": 112.84, "duration": 5.68, "text": "my YouTube channel also like comment and"}, {"start": 115.079, "duration": 3.441, "text": "share bye-bye"}], "title": "An interesting exponential equation | #maths #olympiad #mathcompetition", "description": "", "lengthSeconds": 117, "width": 640, "height": 360}
{"video_id": "9AAQlegsvY8", "transcript": [{"start": 0.52, "duration": 5.6, "text": "hello welcome back here we have an"}, {"start": 3.839, "duration": 6.401, "text": "interesting summation"}, {"start": 6.12, "duration": 10.84, "text": "problem here we're given the sum as n"}, {"start": 10.24, "duration": 10.44, "text": "goes from 0 to Infinity of1 of n all n +"}, {"start": 16.96, "duration": 8.52, "text": "one so here I'm going to make use"}, {"start": 20.68, "duration": 9.4, "text": "of integral calculus here so so how do"}, {"start": 25.48, "duration": 6.64, "text": "we find this sum how can we find the"}, {"start": 30.08, "duration": 7.68, "text": "exact value of this"}, {"start": 32.12, "duration": 8.8, "text": "sum okay now let us recall that the"}, {"start": 37.76, "duration": 11.88, "text": "integral of x of"}, {"start": 40.92, "duration": 14.6, "text": "n DX is equal to x n + 1 all over n + 1"}, {"start": 49.64, "duration": 9.919, "text": "+ C right but in the definite"}, {"start": 55.52, "duration": 4.96, "text": "form so we have integral from 0 to 1 of"}, {"start": 59.559, "duration": 7.401, "text": "x"}, {"start": 60.48, "duration": 11.6, "text": "n d x so this is going to be x n + 1 / n"}, {"start": 66.96, "duration": 10.199, "text": "+ 1 from 0 to 1 so if inside the limit"}, {"start": 72.08, "duration": 9.359, "text": "we're going to have 1 over n + 1 right"}, {"start": 77.159, "duration": 10.521, "text": "so here this sum is the same as sum as n"}, {"start": 81.439, "duration": 8.801, "text": "goes from 0 to Infinity of NE 1 n * 1 n"}, {"start": 87.68, "duration": 5.96, "text": "+ 1"}, {"start": 90.24, "duration": 6.72, "text": "so right over here this is going to be"}, {"start": 93.64, "duration": 8.479, "text": "sum as n goes from 0 to"}, {"start": 96.96, "duration": 8.479, "text": "Infinity 1 n so 1 / n + 1 is equal to"}, {"start": 102.119, "duration": 5.96, "text": "this definite integral here right so"}, {"start": 105.439, "duration": 6.401, "text": "here we're going to"}, {"start": 108.079, "duration": 6.4, "text": "have integral from 0 to 1 of x to the^"}, {"start": 111.84, "duration": 6.08, "text": "of n"}, {"start": 114.479, "duration": 6.201, "text": "DX okay so from here this is going to be"}, {"start": 117.92, "duration": 6.6, "text": "you know we can write out this"}, {"start": 120.68, "duration": 10.199, "text": "integral then here we have some as n"}, {"start": 124.52, "duration": 7.64, "text": "goes from 0 to Infinity of -1 n * x n"}, {"start": 130.879, "duration": 5.521, "text": "then"}, {"start": 132.16, "duration": 8.56, "text": "DX so we all know that this is just an"}, {"start": 136.4, "duration": 7.119, "text": "infinite geometric series here right so"}, {"start": 140.72, "duration": 6.2, "text": "and right over here this is exactly"}, {"start": 143.519, "duration": 6.401, "text": "equal to 1/ x +"}, {"start": 146.92, "duration": 8.959, "text": "1 so this is going to be equal to"}, {"start": 149.92, "duration": 7.76, "text": "integral from 0 to 1 1 /x + 1 DX so you"}, {"start": 155.879, "duration": 5.161, "text": "know the integral of this is just the"}, {"start": 157.68, "duration": 7.44, "text": "natural log of x + 1 and here we have"}, {"start": 161.04, "duration": 5.64, "text": "the limit 0 to 1 so here if you insert"}, {"start": 165.12, "duration": 3.44, "text": "the upper limit first we're going to"}, {"start": 166.68, "duration": 5.559, "text": "have the natural log of"}, {"start": 168.56, "duration": 5.16, "text": "two right then minus if inside the lower"}, {"start": 172.239, "duration": 4.321, "text": "limit we're going to have the natural"}, {"start": 173.72, "duration": 5.32, "text": "log of one we all know that the natural"}, {"start": 176.56, "duration": 5.72, "text": "log of one is zero so this is going to"}, {"start": 179.04, "duration": 7.119, "text": "be the natural log of 2 - 0 and that is"}, {"start": 182.28, "duration": 7.48, "text": "the natural log of 2 therefore the Su is"}, {"start": 186.159, "duration": 6.08, "text": "exactly equal to the natural log of two"}, {"start": 189.76, "duration": 5.8, "text": "thank you for watching please do well to"}, {"start": 192.239, "duration": 6.521, "text": "subscribe to this channel like comment"}, {"start": 195.56, "duration": 3.2, "text": "and share"}], "title": "Interesting Sum", "description": "", "lengthSeconds": 197, "width": 640, "height": 360}
{"video_id": "Bn0oxmJZKec", "transcript": [{"start": 0.719, "duration": 4.08, "text": "hello welcome back once again today we"}, {"start": 2.8, "duration": 5.52, "text": "have this nice Olympia mathematics"}, {"start": 4.799, "duration": 8.361, "text": "problem we're given that x + y is equal"}, {"start": 8.32, "duration": 9.36, "text": "3 X2 + y^2 is = 5 and right over here"}, {"start": 13.16, "duration": 9.0, "text": "we'll be evaluating x - y okay now let's"}, {"start": 17.68, "duration": 8.64, "text": "get started from equation one we have x"}, {"start": 22.16, "duration": 6.0, "text": "+ y this is equal to 3 now let's go"}, {"start": 26.32, "duration": 4.24, "text": "ahead and square both sides of this"}, {"start": 28.16, "duration": 5.239, "text": "equation one"}, {"start": 30.56, "duration": 5.2, "text": "now you know how to expand this binomial"}, {"start": 33.399, "duration": 5.201, "text": "so we have"}, {"start": 35.76, "duration": 7.6, "text": "X2 +"}, {"start": 38.6, "duration": 9.0, "text": "y\u00b2 + 2 * x * y then this will be equal"}, {"start": 43.36, "duration": 8.8, "text": "to 3 2 which is 9 now take note of this"}, {"start": 47.6, "duration": 8.639, "text": "X2 + y^2 has a value of 5 as you can see"}, {"start": 52.16, "duration": 7.44, "text": "in equation 2 so here we replace this"}, {"start": 56.239, "duration": 4.28, "text": "with five so we have here x\u00b2 + y s which"}, {"start": 59.6, "duration": 7.199, "text": "is 5"}, {"start": 60.519, "duration": 9.24, "text": "then plus 2X y this will be equal to 9"}, {"start": 66.799, "duration": 4.721, "text": "okay so from here this five is positive"}, {"start": 69.759, "duration": 6.4, "text": "so if it's crossing over it will become"}, {"start": 71.52, "duration": 10.279, "text": "negative so we have here 2x Y is = to 9"}, {"start": 76.159, "duration": 9.521, "text": "- 5 so from here we have 2X Y is = to 9"}, {"start": 81.799, "duration": 7.481, "text": "- 5 which is 4 from here divide both"}, {"start": 85.68, "duration": 7.6, "text": "side by the coefficient of x y so here"}, {"start": 89.28, "duration": 6.4, "text": "we get theu of X and Y which is equal to"}, {"start": 93.28, "duration": 5.28, "text": "2"}, {"start": 95.68, "duration": 5.759, "text": "amazing okay now let's go back to the"}, {"start": 98.56, "duration": 7.4, "text": "original problem remember we solving for"}, {"start": 101.439, "duration": 6.121, "text": "x - y now let us recall this nice"}, {"start": 105.96, "duration": 4.119, "text": "algebraic"}, {"start": 107.56, "duration": 12.879, "text": "identity that"}, {"start": 110.079, "duration": 15.481, "text": "a squar + b s is equal to a - b s"}, {"start": 120.439, "duration": 7.28, "text": "then plus 2 * a * B okay so this is a"}, {"start": 125.56, "duration": 5.16, "text": "very powerful identity that will be"}, {"start": 127.719, "duration": 6.24, "text": "making use of here so if you expand this"}, {"start": 130.72, "duration": 5.64, "text": "from here okay if you try to prove this"}, {"start": 133.959, "duration": 8.161, "text": "so when you expand a minus b s it will"}, {"start": 136.36, "duration": 9.56, "text": "give you a 2 + b 2 - 2 a b and when you"}, {"start": 142.12, "duration": 7.08, "text": "add this to it that will be + 2 a so we"}, {"start": 145.92, "duration": 5.84, "text": "can see that -2 a + 2 a is zero so"}, {"start": 149.2, "duration": 7.08, "text": "therefore we have a + b\u00b2 which is equal"}, {"start": 151.76, "duration": 10.36, "text": "to a\u00b2 + b\u00b2 so this identity holds okay"}, {"start": 156.28, "duration": 8.8, "text": "so now from equation 2 we have X2 + y^ 2"}, {"start": 162.12, "duration": 6.88, "text": "which is equal to 5 now making use of"}, {"start": 165.08, "duration": 10.68, "text": "this identity from here we get here x -"}, {"start": 169.0, "duration": 9.8, "text": "y^ 2 then + 2 x y this will Beal to 5 so"}, {"start": 175.76, "duration": 5.399, "text": "here we have the value for 2x y right"}, {"start": 178.8, "duration": 9.159, "text": "which is 2 so so here we replace this"}, {"start": 181.159, "duration": 10.08, "text": "with 2 so we get here x - y^ 2 + 2 * 2"}, {"start": 187.959, "duration": 6.041, "text": "which is 4 this is equal to 5 so"}, {"start": 191.239, "duration": 8.321, "text": "subtract four from both side we get x -"}, {"start": 194.0, "duration": 8.56, "text": "y^ 2 this is = to 5 - 4 which is 1 now"}, {"start": 199.56, "duration": 8.8, "text": "taking root of both sides so we get x -"}, {"start": 202.56, "duration": 10.12, "text": "Y is = plus orus < TK 1 which is = to+"}, {"start": 208.36, "duration": 11.48, "text": "or - 1 so therefore we have our solution"}, {"start": 212.68, "duration": 9.479, "text": "that x - Y is = to plus orus one thank"}, {"start": 219.84, "duration": 5.16, "text": "you for watching please kindly subscribe"}, {"start": 222.159, "duration": 6.481, "text": "to my channel also like comment and"}, {"start": 225.0, "duration": 3.64, "text": "share bye-bye"}], "title": "Interesting Olympiad Math problem", "description": "", "lengthSeconds": 227, "width": 640, "height": 360}
{"video_id": "gFecSAg153g", "transcript": [{"start": 0.599, "duration": 4.921, "text": "hello welcome back once again today"}, {"start": 2.96, "duration": 6.2, "text": "we're going to solve this exponential"}, {"start": 5.52, "duration": 5.999, "text": "equation with fractional basis right"}, {"start": 9.16, "duration": 6.76, "text": "here we have 1 /"}, {"start": 11.519, "duration": 7.321, "text": "2^x which is equal to 1 all 4 now let's"}, {"start": 15.92, "duration": 7.92, "text": "get started let us take note of this"}, {"start": 18.84, "duration": 9.32, "text": "property we know 1 / a to^ n is equal to"}, {"start": 23.84, "duration": 8.08, "text": "a to the power of n right so let us"}, {"start": 28.16, "duration": 7.28, "text": "rewrite our equation as 1 over 2 here"}, {"start": 31.92, "duration": 6.92, "text": "has a power of 1 then raise the power"}, {"start": 35.44, "duration": 8.76, "text": "ofx then this will be equal to 1 all"}, {"start": 38.84, "duration": 7.719, "text": "over 4 is same as 2 to the^ 2 okay so"}, {"start": 44.2, "duration": 5.16, "text": "from here in the bracket right over here"}, {"start": 46.559, "duration": 3.881, "text": "make use of this property so we have 2"}, {"start": 49.36, "duration": 3.76, "text": "to^"}, {"start": 50.44, "duration": 5.48, "text": "of-1 then"}, {"start": 53.12, "duration": 6.439, "text": "ofx this will be equal to here this will"}, {"start": 55.92, "duration": 6.319, "text": "be 2 to the^ of -2 there's another"}, {"start": 59.559, "duration": 9.641, "text": "interesting algebraic property remember"}, {"start": 62.239, "duration": 10.641, "text": "a b ra C is equal to a^ B * C so from"}, {"start": 69.2, "duration": 8.12, "text": "here this will be 2^ of-1 * - x will"}, {"start": 72.88, "duration": 8.64, "text": "give us POS X then this is equal to 2 to"}, {"start": 77.32, "duration": 6.96, "text": "the^ of -2 we can see we have the same"}, {"start": 81.52, "duration": 5.12, "text": "base from both sides of this equation so"}, {"start": 84.28, "duration": 6.879, "text": "we can kindly equate the powers right so"}, {"start": 86.64, "duration": 9.479, "text": "this implies from here that X is equal"}, {"start": 91.159, "duration": 7.361, "text": "to -2 and this is the right answer thank"}, {"start": 96.119, "duration": 5.36, "text": "you for watching please kindly subscribe"}, {"start": 98.52, "duration": 6.559, "text": "to my channel also like comment and"}, {"start": 101.479, "duration": 3.6, "text": "share bye-bye"}], "title": "Simple exponential equation | tricky one with fractional bases", "description": "", "lengthSeconds": 104, "width": 640, "height": 360}
{"video_id": "PkPEyCQRd9M", "transcript": [{"start": 0.52, "duration": 4.239, "text": "hello welcome back once again today we"}, {"start": 2.44, "duration": 7.239, "text": "have an interesting problem here we're"}, {"start": 4.759, "duration": 7.121, "text": "going to evaluate I to the^ of 5,000 -"}, {"start": 9.679, "duration": 5.401, "text": "one for those of us that are not"}, {"start": 11.88, "duration": 8.479, "text": "familiar with this I it is also known as"}, {"start": 15.08, "duration": 5.279, "text": "the Iota right which is the imaginary"}, {"start": 20.96, "duration": 9.319, "text": "unit so this is the imaginary unit so I"}, {"start": 25.72, "duration": 8.159, "text": "here is equal to according to Leonard a"}, {"start": 30.279, "duration": 4.96, "text": "1 right so if you square both sides we"}, {"start": 33.879, "duration": 4.68, "text": "found out that"}, {"start": 35.239, "duration": 8.081, "text": "i\u00b2 will be equal"}, {"start": 38.559, "duration": 8.961, "text": "to1 okay so from here"}, {"start": 43.32, "duration": 6.559, "text": "5,000 is a multiple of two right so here"}, {"start": 47.52, "duration": 4.76, "text": "I to the^"}, {"start": 49.879, "duration": 8.68, "text": "5,000 -"}, {"start": 52.28, "duration": 9.0, "text": "1 we can write as I to the^ 2 multip by"}, {"start": 58.559, "duration": 7.611, "text": "5,000 / 2 give us"}, {"start": 61.28, "duration": 6.36, "text": "2,500 then we have here minus1"}, {"start": 66.17, "duration": 4.83, "text": "[Music]"}, {"start": 67.64, "duration": 7.28, "text": "right okay so from here let us take note"}, {"start": 71.0, "duration": 7.24, "text": "of this property you know a to^ b * C"}, {"start": 74.92, "duration": 6.44, "text": "this is same as a to the^ B in bracket"}, {"start": 78.24, "duration": 8.48, "text": "then ra power C so from here this will"}, {"start": 81.36, "duration": 8.799, "text": "be equal to I squar in bracket then"}, {"start": 86.72, "duration": 3.439, "text": "power of 2,500"}, {"start": 90.24, "duration": 9.12, "text": "then - 1 so from here this will be equal"}, {"start": 94.6, "duration": 6.6, "text": "to i\u00b2 remember that is -1 so have1 to"}, {"start": 99.36, "duration": 6.52, "text": "the^ of"}, {"start": 101.2, "duration": 8.04, "text": "2,500 - 1 so we know that -1 to the"}, {"start": 105.88, "duration": 5.199, "text": "power of an even number is one right so"}, {"start": 109.24, "duration": 5.239, "text": "this power here is even so therefore"}, {"start": 111.079, "duration": 6.881, "text": "this will be one so we have 1 - 1 which"}, {"start": 114.479, "duration": 6.121, "text": "is equal to0 so therefore our right"}, {"start": 117.96, "duration": 4.6, "text": "answer is0"}, {"start": 120.6, "duration": 4.479, "text": "thank you for watching please kindly"}, {"start": 122.56, "duration": 7.04, "text": "subscribe to my channel also like"}, {"start": 125.079, "duration": 4.521, "text": "comment and share bye-bye"}], "title": "Can you evaluate ?  i^5000 - 1 = ?", "description": "", "lengthSeconds": 128, "width": 640, "height": 360}
{"video_id": "44GNNco7pl4", "transcript": [{"start": 0.48, "duration": 5.08, "text": "hello welcome back once again today let"}, {"start": 2.76, "duration": 5.32, "text": "us solve for the integral value of x in"}, {"start": 5.56, "duration": 5.92, "text": "the following given nice exponential"}, {"start": 8.08, "duration": 6.759, "text": "equation from the left we have X to the^"}, {"start": 11.48, "duration": 11.2, "text": "X then this equals here let us take note"}, {"start": 14.839, "duration": 13.84, "text": "of this a to^ m + n is = a^ M * a^ n so"}, {"start": 22.68, "duration": 8.32, "text": "here this will be 5^ x * 5 to the^ 25"}, {"start": 28.679, "duration": 6.921, "text": "from here let us divide both sides by 5"}, {"start": 31.0, "duration": 7.719, "text": "to^ of X and here by 5 to the^ X we can"}, {"start": 35.6, "duration": 6.04, "text": "see this we cancel this from here and"}, {"start": 38.719, "duration": 6.721, "text": "from the left here remember we have the"}, {"start": 41.64, "duration": 9.04, "text": "same Powers right so let us take note of"}, {"start": 45.44, "duration": 10.04, "text": "this once again a^ m / b^ M this will be"}, {"start": 50.68, "duration": 8.679, "text": "equal to a / B all to the power M so"}, {"start": 55.48, "duration": 8.639, "text": "here this will be same as x / 5 all to"}, {"start": 59.359, "duration": 8.561, "text": "the^ X X then this equals 5^"}, {"start": 64.119, "duration": 6.0, "text": "25 now here let us manipulate the right"}, {"start": 67.92, "duration": 7.72, "text": "hand side of this equation as"}, {"start": 70.119, "duration": 9.561, "text": "follow here 5 is the same as 25 / 5 then"}, {"start": 75.64, "duration": 6.64, "text": "r^ of 25 now from here let us compare"}, {"start": 79.68, "duration": 6.36, "text": "here we have five and we have five here"}, {"start": 82.28, "duration": 8.04, "text": "x = 25 from the power x is also equal to"}, {"start": 86.04, "duration": 6.759, "text": "25 right so whatever position X is 25 5"}, {"start": 90.32, "duration": 6.439, "text": "is right over there right so from here"}, {"start": 92.799, "duration": 6.96, "text": "this implies that X is equal to 25 and"}, {"start": 96.759, "duration": 4.761, "text": "this is the right answer thank you for"}, {"start": 99.759, "duration": 4.121, "text": "watching please kindly subscribe to my"}, {"start": 101.52, "duration": 5.36, "text": "channel also like comment and share"}, {"start": 103.88, "duration": 3.0, "text": "bye-bye"}], "title": "Olympiad Math | Nice exponential equation", "description": "", "lengthSeconds": 105, "width": 640, "height": 360}
{"video_id": "Hmcbu8Ws8Oc", "transcript": [{"start": 0.96, "duration": 4.04, "text": "hello good viewers welcome back once"}, {"start": 2.72, "duration": 4.88, "text": "again today we have the following simple"}, {"start": 5.0, "duration": 6.519, "text": "exponential equation and here we'll be"}, {"start": 7.6, "duration": 11.599, "text": "solving for the value of x for the left"}, {"start": 11.519, "duration": 10.361, "text": "hand side we have 6^ x + 6^ x + 6^ x and"}, {"start": 19.199, "duration": 5.24, "text": "this is equal to 36 from the left hand"}, {"start": 21.88, "duration": 5.52, "text": "side we can see that 6^ X is common so"}, {"start": 24.439, "duration": 7.401, "text": "we Factor it out and that become 6^ X"}, {"start": 27.4, "duration": 9.4, "text": "into bracket this ID by this is 1 plus"}, {"start": 31.84, "duration": 7.36, "text": "this/ this is 1 plus this/ this is 1"}, {"start": 36.8, "duration": 6.759, "text": "then this is equal to"}, {"start": 39.2, "duration": 7.359, "text": "36 from here this is 6^ X mtip by this"}, {"start": 43.559, "duration": 6.201, "text": "plus this plus this is 3 then this is"}, {"start": 46.559, "duration": 6.401, "text": "equal to 36 so from here let's go ahead"}, {"start": 49.76, "duration": 8.08, "text": "and divide both side by three this will"}, {"start": 52.96, "duration": 10.2, "text": "cancel this and here we have 6^ X is ="}, {"start": 57.84, "duration": 8.12, "text": "36 / 3 is 12 right okay now let us make"}, {"start": 63.16, "duration": 5.88, "text": "use of the log rthm so here we're going"}, {"start": 65.96, "duration": 6.6, "text": "to simplify the answer in terms of log"}, {"start": 69.04, "duration": 5.2, "text": "RM but it will be well simplified so"}, {"start": 72.56, "duration": 6.12, "text": "we're going to take the log on both"}, {"start": 74.24, "duration": 8.4, "text": "sides so we have the log of 6^ x this is"}, {"start": 78.68, "duration": 5.92, "text": "equal to the log of 12 now with the"}, {"start": 82.64, "duration": 7.08, "text": "following properties of logarithm we"}, {"start": 84.6, "duration": 6.68, "text": "have the log of a the^ b isal to B MTI"}, {"start": 89.72, "duration": 8.64, "text": "the log of"}, {"start": 91.28, "duration": 12.4, "text": "a and we also have that the log of a * B"}, {"start": 98.36, "duration": 8.439, "text": "is equal to the log of a plus the log of"}, {"start": 103.68, "duration": 6.2, "text": "B so we're going to make use of these"}, {"start": 106.799, "duration": 5.841, "text": "amazing properties of logarithm so here"}, {"start": 109.88, "duration": 6.72, "text": "this will come over here and that will"}, {"start": 112.64, "duration": 7.439, "text": "become X multi log of 6 then this will"}, {"start": 116.6, "duration": 6.479, "text": "be equal to the log of instead of 12 we"}, {"start": 120.079, "duration": 6.96, "text": "write it as 2 *"}, {"start": 123.079, "duration": 10.401, "text": "6 okay so from here we have x * the log"}, {"start": 127.039, "duration": 8.961, "text": "of 6 this is equal to the log of 2 + the"}, {"start": 133.48, "duration": 5.399, "text": "log of"}, {"start": 136.0, "duration": 4.36, "text": "6 okay now from here we're solving for x"}, {"start": 138.879, "duration": 4.201, "text": "let's go ahead and divide both side by"}, {"start": 140.36, "duration": 6.599, "text": "the coefficient of x which is log of 6"}, {"start": 143.08, "duration": 7.2, "text": "and here we divide by the log of 6 this"}, {"start": 146.959, "duration": 6.521, "text": "will cancel this from here so we have X"}, {"start": 150.28, "duration": 7.56, "text": "is = to we split that denominator that"}, {"start": 153.48, "duration": 10.0, "text": "we become log of 2 / log of"}, {"start": 157.84, "duration": 9.2, "text": "6+ log of 6 / log of"}, {"start": 163.48, "duration": 6.64, "text": "6 now from here this will cancel this"}, {"start": 167.04, "duration": 4.32, "text": "and here we have one again let us recall"}, {"start": 170.12, "duration": 6.479, "text": "another interesting property of the"}, {"start": 171.36, "duration": 8.64, "text": "logarithm the log of a / the log of B is"}, {"start": 176.599, "duration": 6.321, "text": "equal to this logm we can write it in in"}, {"start": 180.0, "duration": 8.0, "text": "a single logarithm so that will become"}, {"start": 182.92, "duration": 7.039, "text": "the log of a to the base B right so from"}, {"start": 188.0, "duration": 4.68, "text": "here we can see we have our solution for"}, {"start": 189.959, "duration": 4.92, "text": "X that X is equal to let's start from"}, {"start": 192.68, "duration": 6.0, "text": "here remember this divid by this is one"}, {"start": 194.879, "duration": 6.521, "text": "so we have one plus so here we have that"}, {"start": 198.68, "duration": 7.559, "text": "to rewrite it in a single logarithm as"}, {"start": 201.4, "duration": 7.36, "text": "the log of 2 to the base 6 and here we"}, {"start": 206.239, "duration": 4.161, "text": "have the solution to the problem thank"}, {"start": 208.76, "duration": 4.36, "text": "you for watching please"}, {"start": 210.4, "duration": 6.759, "text": "kindly subscribe to my channel also like"}, {"start": 213.12, "duration": 4.039, "text": "comment and share bye-bye"}], "title": "Watch this special method | Interesting exponential equation.", "description": "", "lengthSeconds": 216, "width": 640, "height": 360}
{"video_id": "xAwmT02bYIc", "transcript": [{"start": 0.64, "duration": 4.119, "text": "hello you're welcome today we're going"}, {"start": 2.52, "duration": 5.48, "text": "to solve for the R value of x in the"}, {"start": 4.759, "duration": 5.001, "text": "following given radical equation so we"}, {"start": 8.0, "duration": 5.08, "text": "can see from the left hand side we have"}, {"start": 9.76, "duration": 9.56, "text": "the square root of x MTI theare root of"}, {"start": 13.08, "duration": 8.72, "text": "x and this is equal to 64 right okay so"}, {"start": 19.32, "duration": 4.4, "text": "let us go ahead and square both sides we"}, {"start": 21.8, "duration": 3.92, "text": "can see we have root sign at the left"}, {"start": 23.72, "duration": 4.719, "text": "hand side and we know when we Square"}, {"start": 25.72, "duration": 6.719, "text": "both Sid this root sign will get cancel"}, {"start": 28.439, "duration": 4.0, "text": "out right so Square both"}, {"start": 32.84, "duration": 6.12, "text": "sides so now we know the square always"}, {"start": 35.96, "duration": 7.56, "text": "cancel the square root right so we have"}, {"start": 38.96, "duration": 8.68, "text": "here x * the root of x and this is equal"}, {"start": 43.52, "duration": 6.559, "text": "to 64 to the^ 2 now let's go ahead and"}, {"start": 47.64, "duration": 3.599, "text": "square both sides one more time so"}, {"start": 50.079, "duration": 3.881, "text": "Square"}, {"start": 51.239, "duration": 5.12, "text": "here and square the right hand"}, {"start": 53.96, "duration": 4.52, "text": "side so at the left hand side we're"}, {"start": 56.359, "duration": 5.561, "text": "going to make use of this property we"}, {"start": 58.48, "duration": 8.24, "text": "know that 8 * n m to the power of B for"}, {"start": 61.92, "duration": 10.12, "text": "R values A and M this is equal to a to"}, {"start": 66.72, "duration": 9.64, "text": "the power B MTI M to the^ of B right and"}, {"start": 72.04, "duration": 8.719, "text": "again we know that A to the^ B to the^ C"}, {"start": 76.36, "duration": 8.36, "text": "is equal to A to the^ B * C so the left"}, {"start": 80.759, "duration": 3.961, "text": "hand side this is going to be this"}, {"start": 85.799, "duration": 7.721, "text": "x\u00b2 multip by < TK of X2 Square will"}, {"start": 90.4, "duration": 8.44, "text": "cancel the square root so we have just X"}, {"start": 93.52, "duration": 8.36, "text": "so this is equal to 64 to the^ of 2 * 2"}, {"start": 98.84, "duration": 5.68, "text": "which is equal to 4 so from the left"}, {"start": 101.88, "duration": 4.12, "text": "hand side you know the power here is one"}, {"start": 104.52, "duration": 5.199, "text": "since we are multiplying we're going to"}, {"start": 106.0, "duration": 6.68, "text": "add the powers so we get x to^ 3 this"}, {"start": 109.719, "duration": 9.241, "text": "will be equal to 64 as the base here is"}, {"start": 112.68, "duration": 9.16, "text": "4 to^ 3 then ra the power of 4 right yes"}, {"start": 118.96, "duration": 7.64, "text": "okay now we can interchange these Powers"}, {"start": 121.84, "duration": 7.52, "text": "here so this is going to be x^ 3 isal to"}, {"start": 126.6, "duration": 5.08, "text": "so that can be written as 4 to the^ 4"}, {"start": 129.36, "duration": 5.48, "text": "then to the^ of 3 so you can see this"}, {"start": 131.68, "duration": 6.04, "text": "one goes out and this one comes in right"}, {"start": 134.84, "duration": 6.28, "text": "okay so now from here we have something"}, {"start": 137.72, "duration": 7.04, "text": "cubed and we also have something cubed"}, {"start": 141.12, "duration": 8.199, "text": "so this implies from here that the base"}, {"start": 144.76, "duration": 8.839, "text": "X is = 4 to ^ 4 so from here we get our"}, {"start": 149.319, "duration": 8.441, "text": "real solution that X is equal to"}, {"start": 153.599, "duration": 7.881, "text": "256 and this is the correct answer right"}, {"start": 157.76, "duration": 6.52, "text": "okay now we equated the bases here"}, {"start": 161.48, "duration": 6.28, "text": "because we are looking for the real"}, {"start": 164.28, "duration": 5.64, "text": "solution if the equation ask to solve in"}, {"start": 167.76, "duration": 5.399, "text": "total both the real and the complex"}, {"start": 169.92, "duration": 5.959, "text": "solution no need for only equating the"}, {"start": 173.159, "duration": 6.08, "text": "bases we still going to do some algebric"}, {"start": 175.879, "duration": 6.201, "text": "manipulations to get the two remaining"}, {"start": 179.239, "duration": 6.28, "text": "complex Solutions right so anytime you"}, {"start": 182.08, "duration": 5.719, "text": "arrive at this point unless the question"}, {"start": 185.519, "duration": 4.561, "text": "asks for only R solution then you can"}, {"start": 187.799, "duration": 4.961, "text": "equate the bases but if the question"}, {"start": 190.08, "duration": 6.36, "text": "asks you to solve in total you are going"}, {"start": 192.76, "duration": 5.96, "text": "to solve the cubic equation thank you"}, {"start": 196.44, "duration": 6.0, "text": "for watching this is the correct answer"}, {"start": 198.72, "duration": 3.72, "text": "x equal to 256"}], "title": "Tricky Olympiad Math | Can you solve ?", "description": "", "lengthSeconds": 203, "width": 428, "height": 240}
{"video_id": "CItbZ6RulFg", "transcript": [{"start": 1.0, "duration": 5.24, "text": "hello good viewers welcome back once"}, {"start": 3.28, "duration": 6.96, "text": "again let's find a solution to the"}, {"start": 6.24, "duration": 9.479, "text": "following given exponential equation we"}, {"start": 10.24, "duration": 7.4, "text": "have x^ x^ 5 this is equal to 100 now"}, {"start": 15.719, "duration": 4.601, "text": "let's get"}, {"start": 17.64, "duration": 6.16, "text": "started we're going to raise both sides"}, {"start": 20.32, "duration": 6.24, "text": "of this equation to the power of five so"}, {"start": 23.8, "duration": 7.6, "text": "from the left hand side we have x to the"}, {"start": 26.56, "duration": 8.56, "text": "power of x^ 5 then rais power of 5 from"}, {"start": 31.4, "duration": 8.08, "text": "the right hand side this is equal to 100"}, {"start": 35.12, "duration": 7.48, "text": "100 is the same as 10 power of two right"}, {"start": 39.48, "duration": 5.16, "text": "then we have a power of"}, {"start": 42.6, "duration": 5.92, "text": "five"}, {"start": 44.64, "duration": 6.079, "text": "okay now from here we'll be making use"}, {"start": 48.52, "duration": 6.719, "text": "of this"}, {"start": 50.719, "duration": 7.961, "text": "property when we have a r of B and"}, {"start": 55.239, "duration": 7.28, "text": "bracket of C we can interchange the"}, {"start": 58.68, "duration": 4.879, "text": "powers as a a to the power of C then ra"}, {"start": 62.519, "duration": 4.92, "text": "power of"}, {"start": 63.559, "duration": 6.281, "text": "B and similarly we can still rewrite it"}, {"start": 67.439, "duration": 5.121, "text": "as a to the power of B *"}, {"start": 69.84, "duration": 6.959, "text": "C now from the left hand side the power"}, {"start": 72.56, "duration": 6.84, "text": "of X here is X to the^ of 5 and we have"}, {"start": 76.799, "duration": 4.32, "text": "this power outside this bracket here now"}, {"start": 79.4, "duration": 6.88, "text": "make use of this property we can"}, {"start": 81.119, "duration": 8.881, "text": "interchange that as x to the power of 5"}, {"start": 86.28, "duration": 6.72, "text": "then ra power of x to the^ of 5 right"}, {"start": 90.0, "duration": 6.24, "text": "right then this is equal to from here we"}, {"start": 93.0, "duration": 7.28, "text": "make use of this property here so this"}, {"start": 96.24, "duration": 7.6, "text": "will become 10 to the^ of 2 * 5 which is"}, {"start": 100.28, "duration": 10.64, "text": "10 right so from here we"}, {"start": 103.84, "duration": 11.919, "text": "have x^ of 5 then of x^ of 5 this is"}, {"start": 110.92, "duration": 7.96, "text": "equal to 10 the power of 10 from here we"}, {"start": 115.759, "duration": 6.521, "text": "can see we have the same pattern of the"}, {"start": 118.88, "duration": 7.64, "text": "basis with the X exponents x^"}, {"start": 122.28, "duration": 9.08, "text": "5^ x^ 5 this is equal to 10^ 10 so this"}, {"start": 126.52, "duration": 7.96, "text": "implies that X to the^ of 5 is equal to"}, {"start": 131.36, "duration": 5.08, "text": "10 right so from here we take the fifth"}, {"start": 134.48, "duration": 4.96, "text": "root of both sides since we're"}, {"start": 136.44, "duration": 6.68, "text": "interested in the real solution we get X"}, {"start": 139.44, "duration": 6.32, "text": "is equal to the fifth root of 10 and"}, {"start": 143.12, "duration": 4.36, "text": "this is the right answer thank you for"}, {"start": 145.76, "duration": 4.32, "text": "watching please kindly subscribe to my"}, {"start": 147.48, "duration": 5.6, "text": "channel also like comment and share"}, {"start": 150.08, "duration": 3.0, "text": "bye-bye"}], "title": "China| Nice Olympiad Math | Interesting exponential equation | #maths #mathematics", "description": "", "lengthSeconds": 151, "width": 500, "height": 360}
{"video_id": "D-hLoizKuGo", "transcript": [{"start": 0.799, "duration": 4.841, "text": "hello good viewers welcome back once"}, {"start": 2.8, "duration": 6.96, "text": "again today we're going to solve for the"}, {"start": 5.64, "duration": 9.199, "text": "value of a s + b s given that a cub - B"}, {"start": 9.76, "duration": 8.279, "text": "Cub is = 56 Aus B is ="}, {"start": 14.839, "duration": 8.081, "text": "2 now we're going to make use of this"}, {"start": 18.039, "duration": 7.64, "text": "property a cubed - B Cubed this is equal"}, {"start": 22.92, "duration": 9.56, "text": "to Aus B"}, {"start": 25.679, "duration": 6.801, "text": "Cubed + 3 a b into braet a minus"}, {"start": 32.76, "duration": 6.28, "text": "B this tells us that we're going to"}, {"start": 35.92, "duration": 3.92, "text": "solve for a * B from here we're going to"}, {"start": 39.04, "duration": 5.08, "text": "make"}, {"start": 39.84, "duration": 8.039, "text": "substitutions a cubus b Cub is 56 this"}, {"start": 44.12, "duration": 8.88, "text": "is equal to A- B that would be 2 then"}, {"start": 47.879, "duration": 8.281, "text": "cubed plus 3 a b * Aus b"}, {"start": 53.0, "duration": 7.199, "text": "2 from the right hand side 2 cubed is 8"}, {"start": 56.16, "duration": 7.48, "text": "we have 8 plus 3 * 2 * a b that will"}, {"start": 60.199, "duration": 6.24, "text": "give us 6 a b this is equal to"}, {"start": 63.64, "duration": 6.6, "text": "56 let us subtract eight from both sides"}, {"start": 66.439, "duration": 6.801, "text": "we get 6 a b is = to 56 - 8 that will"}, {"start": 70.24, "duration": 4.84, "text": "give us 48 from here we divide both"}, {"start": 73.24, "duration": 6.28, "text": "sides by"}, {"start": 75.08, "duration": 8.84, "text": "6 and from here we get a * B is = to 48"}, {"start": 79.52, "duration": 8.52, "text": "/ 6 that will give us 8"}, {"start": 83.92, "duration": 8.96, "text": "right okay now a 2 + b"}, {"start": 88.04, "duration": 9.56, "text": "2 can be written in terms of a minus b s"}, {"start": 92.88, "duration": 9.76, "text": "then + 2 A B from here A minus B is 2 we"}, {"start": 97.6, "duration": 9.92, "text": "have 2 2 + 2 * a b which is 2 * 8 this"}, {"start": 102.64, "duration": 8.88, "text": "is equal to 4 + 2 * 8 is 16 right and"}, {"start": 107.52, "duration": 6.12, "text": "from here we get this is equal to 20 and"}, {"start": 111.52, "duration": 3.919, "text": "this is right answer thank you for"}, {"start": 113.64, "duration": 4.04, "text": "watching please kindly subscribe to my"}, {"start": 115.439, "duration": 6.04, "text": "YouTube channel also like comment and"}, {"start": 117.68, "duration": 3.799, "text": "share bye-bye"}], "title": "Junior Olympiad Math | Interesting algebraic problem | #maths #olympiad", "description": "", "lengthSeconds": 120, "width": 640, "height": 360}
{"video_id": "uHV8PXb9ZhQ", "transcript": [{"start": 0.76, "duration": 4.48, "text": "hello welcome back once again today we"}, {"start": 3.32, "duration": 4.16, "text": "have this interesting Olympiad"}, {"start": 5.24, "duration": 6.319, "text": "mathematics problem and here we're going"}, {"start": 7.48, "duration": 5.279, "text": "to be solving for the value of x right"}, {"start": 11.559, "duration": 4.24, "text": "now let's get"}, {"start": 12.759, "duration": 5.841, "text": "started now let us manipulate the right"}, {"start": 15.799, "duration": 4.281, "text": "hand s side of this equation we know"}, {"start": 18.6, "duration": 6.599, "text": "that 3 <"}, {"start": 20.08, "duration": 7.92, "text": "tk3 is equal to 3 here isun"}, {"start": 25.199, "duration": 5.4, "text": "9 then multiply by this"}, {"start": 28.0, "duration": 5.12, "text": "root3 so since we have the products of"}, {"start": 30.599, "duration": 6.881, "text": "two numbers under root sign so we can"}, {"start": 33.12, "duration": 9.08, "text": "write as root of 9 * 3 so which gives"}, {"start": 37.48, "duration": 7.28, "text": "root of 27 so root of 3 < tk3 I mean is"}, {"start": 42.2, "duration": 7.08, "text": "equal to root of"}, {"start": 44.76, "duration": 7.08, "text": "27 right okay so"}, {"start": 49.28, "duration": 8.279, "text": "again root of"}, {"start": 51.84, "duration": 9.64, "text": "27 is equal to otk of 3 to the^ 3"}, {"start": 57.559, "duration": 6.521, "text": "because 3 to the^ 3 is 27 right now let"}, {"start": 61.48, "duration": 5.16, "text": "us make use of this identity we know"}, {"start": 64.08, "duration": 6.12, "text": "that the sare root of a to the power of"}, {"start": 66.64, "duration": 3.56, "text": "B is same"}, {"start": 70.24, "duration": 9.64, "text": "as < TK of a in bracket ra the power of"}, {"start": 75.799, "duration": 9.68, "text": "B so from here this is going to be equal"}, {"start": 79.88, "duration": 9.76, "text": "to squ < TK 3 in bracket ra^"}, {"start": 85.479, "duration": 8.121, "text": "3 and do not forget that this we can as"}, {"start": 89.64, "duration": 8.28, "text": "well rewrite as this three as Square <"}, {"start": 93.6, "duration": 7.04, "text": "TK 3 in braet squared because Square"}, {"start": 97.92, "duration": 5.0, "text": "always cancel the square root so now"}, {"start": 100.64, "duration": 6.64, "text": "let's go back to the original equation"}, {"start": 102.92, "duration": 7.0, "text": "our original equation is x^ x^ 2 this is"}, {"start": 107.28, "duration": 5.72, "text": "equal to 3 <"}, {"start": 109.92, "duration": 6.76, "text": "tk3 so using this idea here we can write"}, {"start": 113.0, "duration": 7.079, "text": "the right hand side we have X x^ 2 this"}, {"start": 116.68, "duration": 9.039, "text": "will be equal to < tk3"}, {"start": 120.079, "duration": 8.481, "text": "R of < tk3 then squar now let's compare"}, {"start": 125.719, "duration": 4.641, "text": "we have X here we have root3 we have X"}, {"start": 128.56, "duration": 4.88, "text": "we have root3 we have square here we"}, {"start": 130.36, "duration": 7.44, "text": "have square so this implies from here"}, {"start": 133.44, "duration": 7.2, "text": "that X is equal to square root of 3 and"}, {"start": 137.8, "duration": 4.64, "text": "this is the right answer thank you for"}, {"start": 140.64, "duration": 5.319, "text": "watching please kindly subscribe to my"}, {"start": 142.44, "duration": 6.519, "text": "channel also like comment and share"}, {"start": 145.959, "duration": 3.0, "text": "bye-bye"}], "title": "Interesting Olympiad Math | Can you solve ?", "description": "", "lengthSeconds": 148, "width": 376, "height": 240}
{"video_id": "eHo-Sq5Rnu4", "transcript": [{"start": 0.56, "duration": 4.16, "text": "hello welcome back once again today we"}, {"start": 2.8, "duration": 4.36, "text": "have interesting cubic equation for"}, {"start": 4.72, "duration": 5.48, "text": "Olympiad and here we'll be solving for"}, {"start": 7.16, "duration": 9.08, "text": "the values of X in the given equation so"}, {"start": 10.2, "duration": 11.68, "text": "here we have x^ 3 + x this is equal to"}, {"start": 16.24, "duration": 13.08, "text": "10 is same as 8 + 2 okay so here we have"}, {"start": 21.88, "duration": 12.12, "text": "X to the^ 3 then - 8 + x here - 2 this"}, {"start": 29.32, "duration": 10.6, "text": "is equal 0 so from here we have x ^ 3 -"}, {"start": 34.0, "duration": 8.76, "text": "8 is as 2^ 3 then here we have + x - 2"}, {"start": 39.92, "duration": 4.76, "text": "this is equal to Z so from here we're"}, {"start": 42.76, "duration": 4.639, "text": "going to make use of the difference of"}, {"start": 44.68, "duration": 6.16, "text": "two cubes formula remember when we have"}, {"start": 47.399, "duration": 11.241, "text": "a cubus B Cubed then this can be"}, {"start": 50.84, "duration": 9.84, "text": "factored as Aus B multip by a 2 + a b +"}, {"start": 58.64, "duration": 4.2, "text": "b 2"}, {"start": 60.68, "duration": 3.04, "text": "so make use of this property from here"}, {"start": 62.84, "duration": 7.4, "text": "we"}, {"start": 63.72, "duration": 11.8, "text": "have x - 2 into bracket x 2 + 2 * X"}, {"start": 70.24, "duration": 9.76, "text": "which is 2 x + 2 2 which is 4 then plus"}, {"start": 75.52, "duration": 7.08, "text": "here we have x - 2 this is equal to0 so"}, {"start": 80.0, "duration": 5.4, "text": "from the left hand side x - 2 is common"}, {"start": 82.6, "duration": 5.479, "text": "so let's Factor it out and we are left"}, {"start": 85.4, "duration": 5.16, "text": "with X2 +"}, {"start": 88.079, "duration": 5.281, "text": "2x + 4"}, {"start": 90.56, "duration": 6.559, "text": "then plus here we left with one then"}, {"start": 93.36, "duration": 12.2, "text": "this is equal to Z so from here we have"}, {"start": 97.119, "duration": 12.241, "text": "x - 2 * X2 + 2x + 5 then this is equal"}, {"start": 105.56, "duration": 6.199, "text": "to 0 so from here we have these two"}, {"start": 109.36, "duration": 6.439, "text": "products which is equal to Z so we this"}, {"start": 111.759, "duration": 5.04, "text": "implies from here that either x - 2 is ="}, {"start": 115.799, "duration": 8.881, "text": "0"}, {"start": 116.799, "duration": 9.881, "text": "or X2 + 2 x + 5 is = 0 so from here add"}, {"start": 124.68, "duration": 6.279, "text": "two to both sides so we get our real"}, {"start": 126.68, "duration": 6.199, "text": "solution that X is = 2 so from here"}, {"start": 130.959, "duration": 5.601, "text": "let's make use of the general quadratic"}, {"start": 132.879, "duration": 9.601, "text": "formula so we have X = to B will be"}, {"start": 136.56, "duration": 12.0, "text": "2+ or minus sare otk of b 2 is 4 - 4 * a"}, {"start": 142.48, "duration": 9.52, "text": "* C is 20 then we are dividing by 2 so"}, {"start": 148.56, "duration": 6.92, "text": "here x will be equal = to -2 plus or"}, {"start": 152.0, "duration": 6.76, "text": "minus this right over here is"}, {"start": 155.48, "duration": 5.56, "text": "-16 so the root of -6 first of all"}, {"start": 158.76, "duration": 6.72, "text": "square of this the netive one there is"}, {"start": 161.04, "duration": 7.52, "text": "Iota and the square root of 16 is 4 and"}, {"start": 165.48, "duration": 5.36, "text": "we are dividing by 2 so if you split"}, {"start": 168.56, "duration": 7.8, "text": "this denominator here we get our complex"}, {"start": 170.84, "duration": 9.679, "text": "solution x = -2 / 2 will give us -1 plus"}, {"start": 176.36, "duration": 8.84, "text": "or minus 4 I / 2 will give us 2"}, {"start": 180.519, "duration": 6.561, "text": "I and here we have the complex Solutions"}, {"start": 185.2, "duration": 4.28, "text": "thank you for watching please kindly"}, {"start": 187.08, "duration": 6.84, "text": "subscribe to my channel also like"}, {"start": 189.48, "duration": 4.44, "text": "comment and share bye-bye"}], "title": "Oxford University \ud83c\udf93 Entrance Exam | Can you solve ?", "description": "", "lengthSeconds": 193, "width": 640, "height": 360}
{"video_id": "pVup3wuYN-M", "transcript": [{"start": 0.56, "duration": 4.6, "text": "hello good day viewers welcome back once"}, {"start": 2.76, "duration": 4.919, "text": "again to another video please subscribe"}, {"start": 5.16, "duration": 5.2, "text": "to this channel also like comment and"}, {"start": 7.679, "duration": 4.601, "text": "share here we are asked to evaluate this"}, {"start": 10.36, "duration": 5.999, "text": "interesting limit limit as n is"}, {"start": 12.28, "duration": 10.44, "text": "approaching Infinity of n ^ 4 - 2 n 2 +"}, {"start": 16.359, "duration": 9.401, "text": "1 all 3 n ^ 4 - n 2 + 5 so mind you that"}, {"start": 22.72, "duration": 6.44, "text": "this is an easy problem to"}, {"start": 25.76, "duration": 6.959, "text": "solve so what do we do what we're going"}, {"start": 29.16, "duration": 6.6, "text": "to do here is to divide both upstairs"}, {"start": 32.719, "duration": 6.721, "text": "and downstairs by the first term here"}, {"start": 35.76, "duration": 6.08, "text": "which has the highest power right so"}, {"start": 39.44, "duration": 6.2, "text": "this limit is the same as"}, {"start": 41.84, "duration": 5.48, "text": "limit as n is oppressing infinity so"}, {"start": 45.64, "duration": 9.64, "text": "here we have to"}, {"start": 47.32, "duration": 14.32, "text": "divide n ^ 4 - 2 n 2 + 1 then over n^ 4"}, {"start": 55.28, "duration": 11.68, "text": "then everything over 3 n^ 4 - n 2 + 5"}, {"start": 61.64, "duration": 8.44, "text": "then Over N the^ of 4 right so from here"}, {"start": 66.96, "duration": 6.08, "text": "we need to split so here we have limit"}, {"start": 70.08, "duration": 11.8, "text": "as n is approaching Infinity so here we"}, {"start": 73.04, "duration": 15.88, "text": "have n ^ 4/ n ^ 4 - 2 n 2 / n ^ 4 + 1 /"}, {"start": 81.88, "duration": 16.04, "text": "n ^ 4 then we have all over 3 n^ 4 / n ^"}, {"start": 88.92, "duration": 10.68, "text": "4 - n 2 / n ^ 4 + 5 / n ^ of 4 so from"}, {"start": 97.92, "duration": 4.76, "text": "here we're going to simplify where"}, {"start": 99.6, "duration": 6.68, "text": "necessary so here this is equal to"}, {"start": 102.68, "duration": 5.439, "text": "limit as n is approaching Infinity so"}, {"start": 106.28, "duration": 6.6, "text": "you know this will cancel out so we have"}, {"start": 108.119, "duration": 10.841, "text": "1 minus you know this reduces to 2 / n^"}, {"start": 112.88, "duration": 9.32, "text": "of 2 then + 1 / n ^ 4 then all over this"}, {"start": 118.96, "duration": 6.56, "text": "will cancel out with left to 3 minus you"}, {"start": 122.2, "duration": 8.8, "text": "know this reduces to 1 n 2 and here we"}, {"start": 125.52, "duration": 9.04, "text": "have + 5/ n^ of 4 right so this is equal"}, {"start": 131.0, "duration": 6.56, "text": "to 1us you know 2/ n\u00b2 as n is"}, {"start": 134.56, "duration": 5.6, "text": "approaching Infinity is zero right"}, {"start": 137.56, "duration": 5.679, "text": "because the button will keep getting"}, {"start": 140.16, "duration": 6.159, "text": "larger larger larger larger and larger"}, {"start": 143.239, "duration": 6.28, "text": "right so and that will T to zero right"}, {"start": 146.319, "duration": 5.521, "text": "so here we have plus here the bottom is"}, {"start": 149.519, "duration": 4.961, "text": "larger right so we're going to see have"}, {"start": 151.84, "duration": 4.72, "text": "zero as n is approaching Infinity right"}, {"start": 154.48, "duration": 6.039, "text": "that will turn to zero and here we have"}, {"start": 156.56, "duration": 6.92, "text": "3 minus here we have 0 then + 0 and here"}, {"start": 160.519, "duration": 5.201, "text": "we have 1 / 3 and this is the right"}, {"start": 163.48, "duration": 4.6, "text": "answer thank you for watching please do"}, {"start": 165.72, "duration": 6.64, "text": "well to subscribe to this channel also"}, {"start": 168.08, "duration": 4.28, "text": "like comment and share bye-bye"}], "title": "Advanced limit. Interesting limit.", "description": "", "lengthSeconds": 172, "width": 640, "height": 360}
{"video_id": "qlPpXu1Ud6I", "transcript": [{"start": 0.52, "duration": 4.92, "text": "hello welcome back once again to my"}, {"start": 2.8, "duration": 6.0, "text": "YouTube Channel please if today is your"}, {"start": 5.44, "duration": 8.52, "text": "first day of visiting my Channel please"}, {"start": 8.8, "duration": 8.28, "text": "kindly support us by subscribing to our"}, {"start": 13.96, "duration": 6.96, "text": "Channel please and please subscribe to"}, {"start": 17.08, "duration": 7.32, "text": "this Channel and share our video today"}, {"start": 20.92, "duration": 5.24, "text": "we are asked to evaluate the double"}, {"start": 24.4, "duration": 5.279, "text": "factorial of"}, {"start": 26.16, "duration": 7.719, "text": "five we are going to evaluate this the"}, {"start": 29.679, "duration": 7.161, "text": "double factorial of five by definition"}, {"start": 33.879, "duration": 7.241, "text": "let us know what is double factorial is"}, {"start": 36.84, "duration": 8.8, "text": "all about so here we have by definition"}, {"start": 41.12, "duration": 10.72, "text": "if n is even the double factorial of n"}, {"start": 45.64, "duration": 9.079, "text": "is = to n * N - 2 * N - 4 all the way"}, {"start": 51.84, "duration": 6.719, "text": "till 4 *"}, {"start": 54.719, "duration": 9.281, "text": "2 then if n is OD the double factoral of"}, {"start": 58.559, "duration": 11.361, "text": "n is equal to n N - 2 * N - 4 * all the"}, {"start": 64.0, "duration": 8.4, "text": "way till * 3 * 1 and we have seen this"}, {"start": 69.92, "duration": 6.239, "text": "here that the double factor of n is not"}, {"start": 72.4, "duration": 7.079, "text": "equal to the N factor in bracket then"}, {"start": 76.159, "duration": 5.761, "text": "another factorial right okay now here we"}, {"start": 79.479, "duration": 5.841, "text": "have to evaluate the"}, {"start": 81.92, "duration": 6.36, "text": "five double factorial right here we can"}, {"start": 85.32, "duration": 6.159, "text": "see that five is an odd number right so"}, {"start": 88.28, "duration": 5.04, "text": "we're going to make use of this prop to"}, {"start": 91.479, "duration": 7.92, "text": "here so this is going to"}, {"start": 93.32, "duration": 6.079, "text": "be 5 * 5 -"}, {"start": 99.479, "duration": 7.121, "text": "2 right which is 3 * 5 -"}, {"start": 107.28, "duration": 11.36, "text": "4 and that is one right so this is going"}, {"start": 110.799, "duration": 10.121, "text": "to be 5 * 5 - 2 that is 3 * 5 - 4 that"}, {"start": 118.64, "duration": 4.519, "text": "is 1 so we can see this property holds"}, {"start": 120.92, "duration": 5.44, "text": "here this is odd number and this is"}, {"start": 123.159, "duration": 5.281, "text": "equal to 15 and this is the right answer"}, {"start": 126.36, "duration": 5.64, "text": "please can you support us by subscribing"}, {"start": 128.44, "duration": 3.56, "text": "to our Channel"}], "title": "Double Factorial.", "description": "", "lengthSeconds": 130, "width": 636, "height": 360}
{"video_id": "cIc-pmfeUHo", "transcript": [{"start": 0.28, "duration": 5.8, "text": "hello welcome back once again today we"}, {"start": 2.44, "duration": 7.8, "text": "have this nice Olympiad mathematics"}, {"start": 6.08, "duration": 7.88, "text": "problem we're given a to the^ x + 1 ^ x"}, {"start": 10.24, "duration": 8.24, "text": "= 9 and here at this ugly expression"}, {"start": 13.96, "duration": 5.8, "text": "here we asked to evaluate it 2^ 9x + 1 /"}, {"start": 18.48, "duration": 5.36, "text": "2^"}, {"start": 19.76, "duration": 7.56, "text": "9x now let's get started now the base"}, {"start": 23.84, "duration": 9.52, "text": "here is 8 which we can express in index"}, {"start": 27.32, "duration": 10.559, "text": "power of two right so 8 is as 2^ of 3"}, {"start": 33.36, "duration": 10.32, "text": "right then with this power of X here 8"}, {"start": 37.879, "duration": 8.801, "text": "power x will be equal to 2^ 3x right so"}, {"start": 43.68, "duration": 11.48, "text": "here we have 2 to^"}, {"start": 46.68, "duration": 11.44, "text": "3x + 1 / 2^ 3x this is = to 9 right and"}, {"start": 55.16, "duration": 6.559, "text": "this expression here we can actually"}, {"start": 58.12, "duration": 7.16, "text": "rewrite it as 2 3x then to the power of"}, {"start": 61.719, "duration": 6.601, "text": "something xse right so we have to take"}, {"start": 65.28, "duration": 6.96, "text": "note of this that 9x is = to"}, {"start": 68.32, "duration": 3.92, "text": "3x *"}, {"start": 72.799, "duration": 15.081, "text": "3 right so that expression will"}, {"start": 78.24, "duration": 11.919, "text": "become 2 ^ 3x in bracket 3 + 1 / 2 ^ 3x"}, {"start": 87.88, "duration": 5.519, "text": "in bracket 3 and this is what what we're"}, {"start": 90.159, "duration": 5.721, "text": "going to evaluate so to clear the"}, {"start": 93.399, "duration": 4.241, "text": "complicated stuff here we need to make"}, {"start": 95.88, "duration": 7.559, "text": "the following substitution we're going"}, {"start": 97.64, "duration": 11.56, "text": "to let 2^ 3x = M therefore the original"}, {"start": 103.439, "duration": 7.72, "text": "equation will become m + 1 / m is = 9"}, {"start": 109.2, "duration": 5.36, "text": "and therefore what are we going to"}, {"start": 111.159, "duration": 8.28, "text": "evaluate we're going to evaluate M Cub +"}, {"start": 114.56, "duration": 8.479, "text": "1 / M Cub hope you get it right so here"}, {"start": 119.439, "duration": 4.561, "text": "let us take note of this famous identity"}, {"start": 123.039, "duration": 2.961, "text": "in"}, {"start": 124.0, "duration": 11.48, "text": "algebra"}, {"start": 126.0, "duration": 15.239, "text": "that X cubed + 1 / X Cub is = x + 1 / X"}, {"start": 135.48, "duration": 9.24, "text": "Cub - 3 into x + 1 /x with the help of"}, {"start": 141.239, "duration": 8.441, "text": "this identity I think we are almost done"}, {"start": 144.72, "duration": 13.08, "text": "right so therefore MB + 1 / M Cub will"}, {"start": 149.68, "duration": 13.279, "text": "be = n + 1 / M Cub - 3 into n + 1 / M"}, {"start": 157.8, "duration": 11.04, "text": "and from here this is equal to 9 Cub - 3"}, {"start": 162.959, "duration": 11.64, "text": "* 9 so this is equal to 9 cubed which is"}, {"start": 168.84, "duration": 9.479, "text": "729 - 3 * 9 which is 27 from here we get"}, {"start": 174.599, "duration": 6.961, "text": "70 2 right so in our conclusion now"}, {"start": 178.319, "duration": 10.801, "text": "we're going to say that"}, {"start": 181.56, "duration": 10.599, "text": "2^ 9 x + 1 / 2^ 9x is ="}, {"start": 189.12, "duration": 5.52, "text": "72 and this is the right answer thank"}, {"start": 192.159, "duration": 4.961, "text": "you for watching please kindly subscribe"}, {"start": 194.64, "duration": 6.36, "text": "to my channel also like comment and"}, {"start": 197.12, "duration": 3.88, "text": "share bye-bye"}], "title": "Interesting Olympiad Math | Here is the trick !", "description": "", "lengthSeconds": 201, "width": 640, "height": 360}
{"video_id": "gt2hRxO_c5E", "transcript": [{"start": 0.64, "duration": 5.199, "text": "hello everyone welcome back once"}, {"start": 3.56, "duration": 5.88, "text": "again please if you have not subscribed"}, {"start": 5.839, "duration": 6.241, "text": "to our Channel please do not forget to"}, {"start": 9.44, "duration": 4.96, "text": "subscribe and share our videos today we"}, {"start": 12.08, "duration": 6.959, "text": "have this beautiful exponential"}, {"start": 14.4, "duration": 7.56, "text": "problem e r of 2x is equal to 16 we are"}, {"start": 19.039, "duration": 5.48, "text": "asked to find the value of"}, {"start": 21.96, "duration": 7.0, "text": "x"}, {"start": 24.519, "duration": 8.56, "text": "solution now let us recall the natural"}, {"start": 28.96, "duration": 8.439, "text": "logarithm of X is equal to the logarithm"}, {"start": 33.079, "duration": 8.32, "text": "of x to the base e right and again the"}, {"start": 37.399, "duration": 8.0, "text": "natural logarithm of e is just equal to"}, {"start": 41.399, "duration": 6.68, "text": "one right because the base is what e we"}, {"start": 45.399, "duration": 5.961, "text": "can see it here the natural L of e is"}, {"start": 48.079, "duration": 6.28, "text": "equal to the log of e to the base of e"}, {"start": 51.36, "duration": 5.039, "text": "which is equal to 1 right okay now we"}, {"start": 54.359, "duration": 4.081, "text": "have e of"}, {"start": 56.399, "duration": 4.441, "text": "2x ="}, {"start": 58.44, "duration": 4.88, "text": "16 we're going to to take the natural"}, {"start": 60.84, "duration": 6.04, "text": "log of both sides right so here we have"}, {"start": 63.32, "duration": 8.76, "text": "the natural log of e of"}, {"start": 66.88, "duration": 7.04, "text": "2x is equal to the natural log of 16 so"}, {"start": 72.08, "duration": 3.76, "text": "now let us use"}, {"start": 73.92, "duration": 3.76, "text": "this"}, {"start": 75.84, "duration": 4.0, "text": "logarithmic"}, {"start": 77.68, "duration": 5.88, "text": "identity so we have here that the"}, {"start": 79.84, "duration": 6.12, "text": "natural log of K power of let's say B is"}, {"start": 83.56, "duration": 4.96, "text": "equal to what this B crossing over to"}, {"start": 85.96, "duration": 4.44, "text": "multiply the entire log so we have B *"}, {"start": 88.52, "duration": 4.48, "text": "the natural log of K"}, {"start": 90.4, "duration": 6.64, "text": "right so here this is coming over here"}, {"start": 93.0, "duration": 7.6, "text": "so we have 2X * the natural log of e"}, {"start": 97.04, "duration": 6.039, "text": "equal to the natural log of 16 so right"}, {"start": 100.6, "duration": 7.36, "text": "over here we know this is equal to 1 and"}, {"start": 103.079, "duration": 7.72, "text": "2x * 1 is = 2x then this is equal to the"}, {"start": 107.96, "duration": 5.04, "text": "natural log of"}, {"start": 110.799, "duration": 4.481, "text": "16 so we can divide both side by the"}, {"start": 113.0, "duration": 5.88, "text": "coefficient of x to get the value of x"}, {"start": 115.28, "duration": 5.72, "text": "so here over two and here over two so"}, {"start": 118.88, "duration": 6.199, "text": "this cancel this well with X is just"}, {"start": 121.0, "duration": 5.64, "text": "equal to the natural log of 16 / 2 but"}, {"start": 125.079, "duration": 3.841, "text": "we can simplify"}, {"start": 126.64, "duration": 5.92, "text": "further knowing that the natural log of"}, {"start": 128.92, "duration": 6.88, "text": "16 is the natural log of 4 of 2 then"}, {"start": 132.56, "duration": 5.28, "text": "over two so you know this property here"}, {"start": 135.8, "duration": 5.719, "text": "so this coming over so we have 2 * the"}, {"start": 137.84, "duration": 5.64, "text": "natural log of 4 / 2 so this two cancel"}, {"start": 141.519, "duration": 5.281, "text": "this two and we are left"}, {"start": 143.48, "duration": 5.92, "text": "with the natural log of four and this is"}, {"start": 146.8, "duration": 4.6, "text": "the right answer thank you for watching"}, {"start": 149.4, "duration": 5.88, "text": "do to subscribe to this Channel and"}, {"start": 151.4, "duration": 3.88, "text": "share our videos"}], "title": "An exponential equation. e^(2x) = 16, x = ?", "description": "", "lengthSeconds": 153, "width": 636, "height": 360}
{"video_id": "1uYuutzLjqI", "transcript": [{"start": 0.93, "duration": 18.699, "text": "[\u0938\u0902\u0917\u0940\u0924]"}, {"start": 22.519, "duration": 36.39, "text": "\u0938\u094d\u091f \u0938\u094d"}, {"start": 24.39, "duration": 34.519, "text": "[\u0938\u0902\u0917\u0940\u0924]"}, {"start": 60.91, "duration": 9.09, "text": "[\u0938\u0902\u0917\u0940\u0924]"}, {"start": 72.07, "duration": 24.26, "text": "[\u0938\u0902\u0917\u0940\u0924]"}], "title": "Implicit differentiation. #olympiad #education #math", "description": "", "lengthSeconds": 95, "width": 640, "height": 360}
{"video_id": "gXtybFwVufg", "transcript": [{"start": 0.88, "duration": 5.479, "text": "hello welcome back once again today"}, {"start": 4.319, "duration": 4.641, "text": "we're going to evaluate this infinite"}, {"start": 6.359, "duration": 7.681, "text": "series we have this Su as n goes from 0"}, {"start": 8.96, "duration": 5.879, "text": "to Infinity of n / 5^ n and if you check"}, {"start": 14.04, "duration": 4.48, "text": "my"}, {"start": 14.839, "duration": 6.841, "text": "thumbnail I said there that I did this"}, {"start": 18.52, "duration": 6.44, "text": "within seconds and that is"}, {"start": 21.68, "duration": 4.839, "text": "true now here is how we're going to"}, {"start": 24.96, "duration": 7.36, "text": "evaluate"}, {"start": 26.519, "duration": 8.121, "text": "this let us remember that 1 iD 1 - x"}, {"start": 32.32, "duration": 6.96, "text": "which is an infinite geometric series"}, {"start": 34.64, "duration": 9.68, "text": "equals sum as n goes from 0 to Infinity"}, {"start": 39.28, "duration": 7.24, "text": "of x to the^ of n right okay now we're"}, {"start": 44.32, "duration": 4.559, "text": "going to differentiate both sides so"}, {"start": 46.52, "duration": 8.519, "text": "here we have D"}, {"start": 48.879, "duration": 9.921, "text": "DX of 1 / 1 - x this will be equal to"}, {"start": 55.039, "duration": 7.68, "text": "the partial derivative with respect to X"}, {"start": 58.8, "duration": 6.04, "text": "of some as n goes from 0 to Infinity of"}, {"start": 62.719, "duration": 5.08, "text": "X the^ of"}, {"start": 64.84, "duration": 7.4, "text": "n now here we're going to make use of"}, {"start": 67.799, "duration": 9.441, "text": "the quent rule right remember that the D"}, {"start": 72.24, "duration": 10.36, "text": "DX of these two functions of x g of x /"}, {"start": 77.24, "duration": 12.8, "text": "the H of X this equals here G Prime of x"}, {"start": 82.6, "duration": 10.519, "text": "m the H of xus G of x * the H Prime of x"}, {"start": 90.04, "duration": 6.84, "text": "then divided by h of"}, {"start": 93.119, "duration": 6.64, "text": "X2 right okay now from the left when you"}, {"start": 96.88, "duration": 6.48, "text": "differentiate this or when you when you"}, {"start": 99.759, "duration": 7.921, "text": "find the derivative of 1 / 1 - x we're"}, {"start": 103.36, "duration": 5.84, "text": "going to have 1 / 1 -"}, {"start": 107.68, "duration": 4.399, "text": "x"}, {"start": 109.2, "duration": 5.08, "text": "s and this equals here the partial"}, {"start": 112.079, "duration": 5.241, "text": "derivative with respect to X that's"}, {"start": 114.28, "duration": 9.119, "text": "going to be Sol from 0 to Infinity the"}, {"start": 117.32, "duration": 9.24, "text": "power here which is n m x 12 the^ minus1"}, {"start": 123.399, "duration": 7.241, "text": "right okay now let's just take note of"}, {"start": 126.56, "duration": 10.52, "text": "something like this when we have a nus n"}, {"start": 130.64, "duration": 12.2, "text": "this is equal to a m / a of n so here we"}, {"start": 137.08, "duration": 10.08, "text": "have 1 / 1 - x^ 2 this equals Su as n"}, {"start": 142.84, "duration": 8.08, "text": "goes from 0 to Infinity of n MTI x^ n"}, {"start": 147.16, "duration": 6.92, "text": "and here / x^ 1 which is X"}, {"start": 150.92, "duration": 3.16, "text": "multiply both side by"}, {"start": 154.56, "duration": 8.319, "text": "X from the left here we're going to have"}, {"start": 157.599, "duration": 9.961, "text": "x / 1 - x^ 2 and this going to be here"}, {"start": 162.879, "duration": 9.72, "text": "this we cancel this Su from 0 to"}, {"start": 167.56, "duration": 9.2, "text": "Infinity of n x^ of n now let us recall"}, {"start": 172.599, "duration": 4.161, "text": "that the sum the given sum equals"}, {"start": 176.84, "duration": 8.88, "text": "this right so here we're going to let X"}, {"start": 181.0, "duration": 8.159, "text": "exactly equal 1 5 right so when we let X"}, {"start": 185.72, "duration": 7.72, "text": "= 1 5 here we get from the right sum"}, {"start": 189.159, "duration": 7.921, "text": "from 0 to Infinity of n if s = 1 5 that"}, {"start": 193.44, "duration": 10.64, "text": "is going to be / 5^ n this will be equal"}, {"start": 197.08, "duration": 11.239, "text": "to 1 5 / 1 - 1 5 2 which is equal to 1/"}, {"start": 204.08, "duration": 8.879, "text": "5 / 1 - 1 5 is 4 5 and when you square"}, {"start": 208.319, "duration": 7.84, "text": "that that is 16 / 2 25 thisal 1 5 like"}, {"start": 212.959, "duration": 7.081, "text": "this ID 16 all"}, {"start": 216.159, "duration": 8.041, "text": "25 and from there this equals 1/ 5 *"}, {"start": 220.04, "duration": 9.8, "text": "then the reciprocal 25 all over"}, {"start": 224.2, "duration": 8.239, "text": "16 right and from here this equal 5/ 16"}, {"start": 229.84, "duration": 4.8, "text": "and this is the right answer thank you"}, {"start": 232.439, "duration": 4.961, "text": "for watching please kindly subscribe to"}, {"start": 234.64, "duration": 5.76, "text": "my channel also like comment and share"}, {"start": 237.4, "duration": 3.0, "text": "bye-bye"}], "title": "Interesting advanced infinite series", "description": "", "lengthSeconds": 239, "width": 640, "height": 360}
{"video_id": "6TpaSsihu2E", "transcript": [{"start": 0.719, "duration": 4.12, "text": "hello good viewers in this equation"}, {"start": 2.52, "duration": 7.039, "text": "we're going to solve for the R value of"}, {"start": 4.839, "duration": 5.96, "text": "values of X we're given a 2^ 2 x + 2^ x"}, {"start": 9.559, "duration": 3.681, "text": "this is equal to"}, {"start": 10.799, "duration": 5.521, "text": "20 now let's get"}, {"start": 13.24, "duration": 6.24, "text": "started but before we proceed let us"}, {"start": 16.32, "duration": 7.24, "text": "recall that a to the power of b c this"}, {"start": 19.48, "duration": 6.92, "text": "can be written as a to the power of B"}, {"start": 23.56, "duration": 8.119, "text": "then ra power of C we can actually write"}, {"start": 26.4, "duration": 10.56, "text": "it as a to the power of C ra power of B"}, {"start": 31.679, "duration": 12.001, "text": "so for 2^ 2 x we can write it as 2^"}, {"start": 36.96, "duration": 9.599, "text": "x^ 2 then + 2 power of X right then this"}, {"start": 43.68, "duration": 4.92, "text": "is equal to 20 so from here we're going"}, {"start": 46.559, "duration": 7.921, "text": "to make a substitution we're going to"}, {"start": 48.6, "duration": 9.84, "text": "let 2 X = Y now for a real solution for"}, {"start": 54.48, "duration": 6.12, "text": "X here 2 xal to Y we're expecting y to"}, {"start": 58.44, "duration": 4.16, "text": "be greater than zero right"}, {"start": 60.6, "duration": 7.08, "text": "so in the equation we derive now in"}, {"start": 62.6, "duration": 8.04, "text": "terms of Y y^ 2 + y is = 20 we"}, {"start": 67.68, "duration": 7.96, "text": "interested in solving the positive value"}, {"start": 70.64, "duration": 8.44, "text": "of y so let's proceed we have y^2 + y -"}, {"start": 75.64, "duration": 9.04, "text": "20 this equal to0 so y here I can write"}, {"start": 79.08, "duration": 8.48, "text": "it as y^ 2 + 5 y - 4 y - 20 this is"}, {"start": 84.68, "duration": 5.92, "text": "equal to Zer so we have a common factor"}, {"start": 87.56, "duration": 5.64, "text": "of Y we Factor it out we are left to y +"}, {"start": 90.6, "duration": 6.839, "text": "5 in the bracket and here -4 should be"}, {"start": 93.2, "duration": 8.72, "text": "factored out we left with y + 5 and from"}, {"start": 97.439, "duration": 9.04, "text": "here we get y - 4 * y + 5 this is equal"}, {"start": 101.92, "duration": 10.44, "text": "0 so from here y - 4 is = 0 or we have"}, {"start": 106.479, "duration": 9.481, "text": "here y + 5 is = 0 so from here Y is = 4"}, {"start": 112.36, "duration": 6.24, "text": "from here we have y isal to -5 for this"}, {"start": 115.96, "duration": 4.799, "text": "we don't have a real solution for X"}, {"start": 118.6, "duration": 6.36, "text": "right so we're interested in the"}, {"start": 120.759, "duration": 9.401, "text": "positive value of y so let us"}, {"start": 124.96, "duration": 10.48, "text": "recall that 2 X isal Y since Y is = 4 we"}, {"start": 130.16, "duration": 10.6, "text": "have that 2 x = 4 so here we have 2 x ="}, {"start": 135.44, "duration": 8.2, "text": "4 is 2^ s so since we have obtain equal"}, {"start": 140.76, "duration": 5.839, "text": "bases from both sides so this implies"}, {"start": 143.64, "duration": 5.8, "text": "that we can equate the powers and that's"}, {"start": 146.599, "duration": 4.441, "text": "we have the solution X is equal to 2"}, {"start": 149.44, "duration": 3.64, "text": "thank you for watching please kindly"}, {"start": 151.04, "duration": 6.24, "text": "subscribe to my YouTube channel also"}, {"start": 153.08, "duration": 4.2, "text": "like comment and share bye-bye"}], "title": "Exponential equation | #maths #olympiad #mathcompetition", "description": "", "lengthSeconds": 156, "width": 640, "height": 360}
{"video_id": "-eF8-dmBXUw", "transcript": [{"start": 0.96, "duration": 6.96, "text": "hello welcome back once again today we"}, {"start": 3.399, "duration": 8.12, "text": "have this interesting equation x^ X is ="}, {"start": 7.92, "duration": 6.0, "text": "12 and here we're going to solve for the"}, {"start": 11.519, "duration": 5.401, "text": "value of x that will satisfy this"}, {"start": 13.92, "duration": 6.68, "text": "equation so let's get started from the"}, {"start": 16.92, "duration": 7.199, "text": "left hand side we have X to the^ X then"}, {"start": 20.6, "duration": 6.2, "text": "this is equal to 12 so here we're going"}, {"start": 24.119, "duration": 5.0, "text": "to log both sides of this equation so"}, {"start": 26.8, "duration": 5.759, "text": "let's take the natural log we have the"}, {"start": 29.119, "duration": 7.241, "text": "natural log of x x to^ x this will be"}, {"start": 32.559, "duration": 5.721, "text": "equal to the natural log of 12 right we"}, {"start": 36.36, "duration": 5.64, "text": "know this property of logarithm this"}, {"start": 38.28, "duration": 6.52, "text": "power here we come down here right so"}, {"start": 42.0, "duration": 6.28, "text": "this will become X MTI the natural log"}, {"start": 44.8, "duration": 7.2, "text": "of x this is equal to the natural log of"}, {"start": 48.28, "duration": 7.919, "text": "12 right now this x here we can actually"}, {"start": 52.0, "duration": 7.079, "text": "rewrite it using this property that a"}, {"start": 56.199, "duration": 6.601, "text": "can be written as e to the power of Ln"}, {"start": 59.079, "duration": 7.161, "text": "of a we know that the the base of the"}, {"start": 62.8, "duration": 7.679, "text": "natural logarithm is e so e to the^ of L"}, {"start": 66.24, "duration": 6.48, "text": "of anything will be that right so here"}, {"start": 70.479, "duration": 5.721, "text": "this x here let's rewrite it so that"}, {"start": 72.72, "duration": 6.439, "text": "will become e^ of Ln of X then multiply"}, {"start": 76.2, "duration": 6.599, "text": "by that Ln of X at the side this will be"}, {"start": 79.159, "duration": 6.161, "text": "equal to the Ln of 12 now we're going to"}, {"start": 82.799, "duration": 5.441, "text": "make use of the Lamb function but you"}, {"start": 85.32, "duration": 5.88, "text": "need to recall that lamb function of x *"}, {"start": 88.24, "duration": 5.879, "text": "e^ X where the power of E and the"}, {"start": 91.2, "duration": 6.12, "text": "coefficients are exactly equal that is"}, {"start": 94.119, "duration": 6.64, "text": "coefficient right so this simplifies to"}, {"start": 97.32, "duration": 5.92, "text": "X so now we can see that the power of a"}, {"start": 100.759, "duration": 5.72, "text": "here natural log of x and coefficient"}, {"start": 103.24, "duration": 5.44, "text": "natural log of x they exactly equal so"}, {"start": 106.479, "duration": 4.92, "text": "this allows us to use the lambar W"}, {"start": 108.68, "duration": 7.359, "text": "function so we have the W of the left"}, {"start": 111.399, "duration": 6.561, "text": "hand side e^ Ln of X time the Ln of X"}, {"start": 116.039, "duration": 5.921, "text": "this will be equal to W of the right"}, {"start": 117.96, "duration": 7.28, "text": "hand side the Ln of 12 so the left hand"}, {"start": 121.96, "duration": 8.919, "text": "side simplifies to the Ln of X this is"}, {"start": 125.24, "duration": 10.079, "text": "equal to the W of the Ln of 12 so from"}, {"start": 130.879, "duration": 9.401, "text": "here so from here recall that if the Ln"}, {"start": 135.319, "duration": 10.841, "text": "of X is equal to a then this implies"}, {"start": 140.28, "duration": 8.52, "text": "that X isal to e of a right so from here"}, {"start": 146.16, "duration": 5.6, "text": "L of X is equal to this so that implies"}, {"start": 148.8, "duration": 8.36, "text": "that X is equal to e to the power of the"}, {"start": 151.76, "duration": 8.199, "text": "Lamb function of Ln of 12 and we find a"}, {"start": 157.16, "duration": 5.92, "text": "very nice approximation for x x is"}, {"start": 159.959, "duration": 3.121, "text": "approximately equal to"}, {"start": 163.319, "duration": 3.0, "text": "2.60"}, {"start": 167.84, "duration": 4.96, "text": "2955 and this is the correct answer"}, {"start": 171.08, "duration": 6.159, "text": "thank you for watching please kindly"}, {"start": 172.8, "duration": 4.439, "text": "like comment and share"}], "title": "Oxford University \ud83c\udf93 Entrance Exam | Can you solve ?", "description": "", "lengthSeconds": 176, "width": 428, "height": 240}
{"video_id": "Igi1zFBPwhg", "transcript": [{"start": 0.68, "duration": 4.52, "text": "hello good viewers welcome back once"}, {"start": 2.84, "duration": 5.6, "text": "again please do wa to subscribe to my"}, {"start": 5.2, "duration": 6.12, "text": "channel also like comment and share"}, {"start": 8.44, "duration": 8.56, "text": "today we have this math Olympiad problem"}, {"start": 11.32, "duration": 10.92, "text": "x + y is = 3 X2 + Y2 is = 7 right over"}, {"start": 17.0, "duration": 8.84, "text": "here we are going to evaluate x^ 4 + y ^"}, {"start": 22.24, "duration": 8.64, "text": "4 right okay let's get started from"}, {"start": 25.84, "duration": 7.719, "text": "equation one we have x + y and this is"}, {"start": 30.88, "duration": 5.519, "text": "equal to three now we're going to square"}, {"start": 33.559, "duration": 5.32, "text": "both sides of this equation and from the"}, {"start": 36.399, "duration": 9.041, "text": "left we're going to make use of this a +"}, {"start": 38.879, "duration": 8.441, "text": "b 2 this is = to a 2 + b 2 + 2 * a * B"}, {"start": 45.44, "duration": 9.48, "text": "from the left we have"}, {"start": 47.32, "duration": 11.0, "text": "X2 + Y 2 + 2 x y this is equal to 3\u00b2"}, {"start": 54.92, "duration": 7.119, "text": "which is 9"}, {"start": 58.32, "duration": 10.6, "text": "right so from here you know X2 + y^2 is"}, {"start": 62.039, "duration": 9.521, "text": "= 7 right so here we have 7 + 2x y this"}, {"start": 68.92, "duration": 7.48, "text": "is equal to 9 so we're going to subtract"}, {"start": 71.56, "duration": 8.28, "text": "7 from both sides and we get 2x Y is = 9"}, {"start": 76.4, "duration": 10.92, "text": "- 7 which is 2 and this implies from"}, {"start": 79.84, "duration": 7.48, "text": "here that x * Y is = 2 / 2 which is one"}, {"start": 87.6, "duration": 5.479, "text": "right okay"}, {"start": 90.2, "duration": 7.959, "text": "now we're going to find the value of x"}, {"start": 93.079, "duration": 8.961, "text": "the power of 4 + B to the power of 4 +"}, {"start": 98.159, "duration": 9.441, "text": "y^ 4 I mean right so from equation 2 we"}, {"start": 102.04, "duration": 8.88, "text": "have X2 + Y 2 and that is equal to 7 so"}, {"start": 107.6, "duration": 6.4, "text": "let's go ahead and square both"}, {"start": 110.92, "duration": 6.6, "text": "sides so from the left we make use of"}, {"start": 114.0, "duration": 6.439, "text": "this property as well so we have here X2"}, {"start": 117.52, "duration": 8.879, "text": "^ 2 which is x ^ 4"}, {"start": 120.439, "duration": 12.321, "text": "+ y^ 2^ 2 which is y ^ 4 then here + 2"}, {"start": 126.399, "duration": 10.92, "text": "X2 y\u00b2 and this is equal to 49 right so"}, {"start": 132.76, "duration": 9.68, "text": "here we write as follows x ^ 4 + y ^ 4"}, {"start": 137.319, "duration": 9.321, "text": "plus here 2 into braet x y^ 2 and this"}, {"start": 142.44, "duration": 10.6, "text": "is equal to 49 so remember that x * Y is"}, {"start": 146.64, "duration": 13.319, "text": "= 1 right so from here we get x^ 4 + y^"}, {"start": 153.04, "duration": 13.96, "text": "4 + 2 * 1 2 this is = to 49 so and from"}, {"start": 159.959, "duration": 9.601, "text": "there x^ 4 + y^ 4 then here + 2 is = to"}, {"start": 167.0, "duration": 4.4, "text": "49 so let's go ahead and subtract two"}, {"start": 169.56, "duration": 5.959, "text": "from both side and here we find the"}, {"start": 171.4, "duration": 8.759, "text": "value for x^ 4 + y^ 4 which is equal to"}, {"start": 175.519, "duration": 6.921, "text": "49 - 2 and that is equal to 47 and and"}, {"start": 180.159, "duration": 4.121, "text": "this is the right answer thank you for"}, {"start": 182.44, "duration": 4.24, "text": "watching please kindly subscribe to my"}, {"start": 184.28, "duration": 5.4, "text": "channel also like comment and share"}, {"start": 186.68, "duration": 3.0, "text": "bye-bye"}], "title": "Olympiad Math. I think algebraic manipulations are going to play roles here. #maths", "description": "", "lengthSeconds": 188, "width": 640, "height": 360}
{"video_id": "QZoiEXLnWUw", "transcript": [{"start": 0.19, "duration": 2.729, "text": "[Music]"}, {"start": 0.68, "duration": 4.8, "text": "hello good viewers welcome back once"}, {"start": 2.919, "duration": 5.521, "text": "again please kindly support my channel"}, {"start": 5.48, "duration": 6.0, "text": "by subscribing also like comment and"}, {"start": 8.44, "duration": 6.199, "text": "share in this video we are going to find"}, {"start": 11.48, "duration": 6.52, "text": "a solution to this problem here we have"}, {"start": 14.639, "duration": 5.921, "text": "the root of three mtip by the root of 3"}, {"start": 18.0, "duration": 6.56, "text": "multip by the root of 3 multi by the"}, {"start": 20.56, "duration": 7.639, "text": "root of 3 and so on till Infinity we"}, {"start": 24.56, "duration": 8.2, "text": "asked to find the exact value"}, {"start": 28.199, "duration": 8.841, "text": "right okay now we're going to"}, {"start": 32.76, "duration": 6.84, "text": "let we're going to let x"}, {"start": 37.04, "duration": 7.88, "text": "equals"}, {"start": 39.6, "duration": 8.479, "text": "this root of three multip by root of 3"}, {"start": 44.92, "duration": 10.2, "text": "multip by root of"}, {"start": 48.079, "duration": 10.521, "text": "3 multi by and so on right so from here"}, {"start": 55.12, "duration": 5.56, "text": "we're going to square both sides so here"}, {"start": 58.6, "duration": 5.16, "text": "we have"}, {"start": 60.68, "duration": 7.84, "text": "X2 this will be exactly the whole of"}, {"start": 63.76, "duration": 10.039, "text": "this squared right so from here we have"}, {"start": 68.52, "duration": 8.2, "text": "x\u00b2 is equal to you know this square and"}, {"start": 73.799, "duration": 6.921, "text": "this root here we get cancelled out"}, {"start": 76.72, "duration": 8.52, "text": "right and here we have 3 multi by the"}, {"start": 80.72, "duration": 9.399, "text": "root of 3 MTI by the root of 3 multi by"}, {"start": 85.24, "duration": 7.28, "text": "the root of 3 multi by and so on right"}, {"start": 90.119, "duration": 5.561, "text": "now carefully looking at this equation"}, {"start": 92.52, "duration": 8.36, "text": "from the right you will observe that"}, {"start": 95.68, "duration": 8.079, "text": "here we have three multiply by X right"}, {"start": 100.88, "duration": 8.96, "text": "because this is exactly what we started"}, {"start": 103.759, "duration": 8.32, "text": "with so here we have X2 = 3 * X so from"}, {"start": 109.84, "duration": 4.68, "text": "here we subtract 3 X from both sides we"}, {"start": 112.079, "duration": 5.96, "text": "get x^2 -"}, {"start": 114.52, "duration": 8.4, "text": "3x this is equal to Zer so from the left"}, {"start": 118.039, "duration": 9.601, "text": "we factorize X into 2 bracket x - 3 this"}, {"start": 122.92, "duration": 12.64, "text": "is equal to0 so from here we get X is ="}, {"start": 127.64, "duration": 14.04, "text": "to 0 or x - 3 is = 0 that is we have x ="}, {"start": 135.56, "duration": 10.12, "text": "0 or X is = to 3 but we all know that"}, {"start": 141.68, "duration": 7.44, "text": "the value of this is going to be greater"}, {"start": 145.68, "duration": 4.68, "text": "than zero right which actually mean that"}, {"start": 149.12, "duration": 4.68, "text": "the whole of that"}, {"start": 150.36, "duration": 7.72, "text": "expression cannot be equal to zero right"}, {"start": 153.8, "duration": 8.4, "text": "so this value is an extraneous value so"}, {"start": 158.08, "duration": 6.159, "text": "from here we go with this value here so"}, {"start": 162.2, "duration": 5.6, "text": "from here we can see"}, {"start": 164.239, "duration": 7.521, "text": "that the root of 3 MTI the root of 3 m"}, {"start": 167.8, "duration": 6.439, "text": "the root of 3 and so on is equal to 3"}, {"start": 171.76, "duration": 4.28, "text": "and this is the right answer thank you"}, {"start": 174.239, "duration": 4.64, "text": "for watching please kindly subscribe to"}, {"start": 176.04, "duration": 5.8, "text": "my channel also like comment and share"}, {"start": 178.879, "duration": 2.961, "text": "bye-bye"}], "title": "An infinite nested radical | Interesting problem #education #mathematics #math", "description": "", "lengthSeconds": 180, "width": 640, "height": 360}
{"video_id": "f4QEiuF68ug", "transcript": [{"start": 0.719, "duration": 4.6, "text": "hello good day viewers welcome back once"}, {"start": 2.76, "duration": 5.04, "text": "again today let's find the solution to"}, {"start": 5.319, "duration": 7.24, "text": "the following given nice functional"}, {"start": 7.8, "duration": 7.919, "text": "equation F of 2x -1 + f of 3 this is ="}, {"start": 12.559, "duration": 7.161, "text": "to 4x - 1 from here we're going to find"}, {"start": 15.719, "duration": 7.48, "text": "F of 2 from the given equation we're"}, {"start": 19.72, "duration": 9.04, "text": "going to let x ="}, {"start": 23.199, "duration": 10.921, "text": "2 so from here 2 * 2 here is 4 4 - 1 is"}, {"start": 28.76, "duration": 9.88, "text": "3 right so we have F of"}, {"start": 34.12, "duration": 9.599, "text": "3+ F of 3 this is equal to remember X is"}, {"start": 38.64, "duration": 7.36, "text": "= 2 so we have 4 * to 8 8 - 1 that is 7"}, {"start": 43.719, "duration": 6.241, "text": "so from here F of 3 + f of 3 this will"}, {"start": 46.0, "duration": 9.28, "text": "give us 2 F of 3 right this is equal to"}, {"start": 49.96, "duration": 9.36, "text": "7 so from here divide both sides by"}, {"start": 55.28, "duration": 9.36, "text": "two 2 cancel two so this implies that F"}, {"start": 59.32, "duration": 8.159, "text": "of 3 is = to 7 / 2 so from the original"}, {"start": 64.64, "duration": 7.72, "text": "equation we can then rewrite it as F of"}, {"start": 67.479, "duration": 10.481, "text": "2x - 1 plus now instead of f of 3 we use"}, {"start": 72.36, "duration": 8.24, "text": "7 over 2 then this is equal to 4X -1 now"}, {"start": 77.96, "duration": 6.799, "text": "let's think about that value of x which"}, {"start": 80.6, "duration": 6.879, "text": "will make 2x - 1 to be 2 right so for us"}, {"start": 84.759, "duration": 9.0, "text": "to get this value of x we should make 2x"}, {"start": 87.479, "duration": 11.081, "text": "- 1 = 2 so we need to let 2x - 1 = 2 now"}, {"start": 93.759, "duration": 6.801, "text": "when you sol for X here x is = to X is ="}, {"start": 98.56, "duration": 5.48, "text": "3"}, {"start": 100.56, "duration": 12.68, "text": "2 so from here we're going to let x = 3"}, {"start": 104.04, "duration": 16.32, "text": "/ 2 so we get F of 2 * 3 / 2 - 1 + 7 2"}, {"start": 113.24, "duration": 11.559, "text": "this is = to 4 * 3 2 - 1"}, {"start": 120.36, "duration": 8.28, "text": "so from here 2 / 2 here is 1 so we left"}, {"start": 124.799, "duration": 9.001, "text": "with f of 3 * 1 is 3 then 3 - 1 will"}, {"start": 128.64, "duration": 9.28, "text": "give us 2 then plus 7 2 this is equal to"}, {"start": 133.8, "duration": 8.64, "text": "from here 4 / 2 here that will give us 2"}, {"start": 137.92, "duration": 6.92, "text": "then 2 * 3 is 6 6 - 1 will give us 5 so"}, {"start": 142.44, "duration": 7.96, "text": "from here let's solve for f of 2 so we"}, {"start": 144.84, "duration": 12.0, "text": "have F of 2 is = 5 / 1 - 7"}, {"start": 150.4, "duration": 10.08, "text": "/ 2 so 5 1 is same as 10 / 2 - 7 / 2 so"}, {"start": 156.84, "duration": 7.52, "text": "from the numerators 10 - 7 is 3 then"}, {"start": 160.48, "duration": 5.28, "text": "over 2 and this is the right answer"}, {"start": 164.36, "duration": 3.36, "text": "thank you for watching please kindly"}, {"start": 165.76, "duration": 5.92, "text": "subscribe to my channel also like"}, {"start": 167.72, "duration": 3.96, "text": "comment and share bye-bye"}], "title": "Interesting functional equation | #maths #mathematics", "description": "", "lengthSeconds": 170, "width": 508, "height": 360}
{"video_id": "msf2FoTdUlU", "transcript": [{"start": 1.0, "duration": 3.92, "text": "hello good viewers welcome back once"}, {"start": 2.84, "duration": 8.6, "text": "again today we're going to find the"}, {"start": 4.92, "duration": 9.639, "text": "square root of 641 squ + 641 +"}, {"start": 11.44, "duration": 6.48, "text": "642 and we're going to do this without"}, {"start": 14.559, "duration": 4.841, "text": "calculators now I will start like this"}, {"start": 17.92, "duration": 3.0, "text": "I'm going to"}, {"start": 19.4, "duration": 5.959, "text": "let"}, {"start": 20.92, "duration": 7.08, "text": "641 = X right so if we add one to both"}, {"start": 25.359, "duration": 5.641, "text": "sides from the left we have 641 + 1"}, {"start": 28.0, "duration": 6.68, "text": "which is 642 and from the right we have"}, {"start": 31.0, "duration": 7.84, "text": "x + 1 therefore this expression in terms"}, {"start": 34.68, "duration": 9.8, "text": "of X is going to be 641 squ is the same"}, {"start": 38.84, "duration": 7.52, "text": "as X squ right plus 641 which is"}, {"start": 44.48, "duration": 7.64, "text": "X+"}, {"start": 46.36, "duration": 8.08, "text": "642 which is x + 1 right so from here"}, {"start": 52.12, "duration": 4.04, "text": "this is equal to the square root"}, {"start": 54.44, "duration": 8.639, "text": "of"}, {"start": 56.16, "duration": 9.12, "text": "X2 + x + x that will give us 2X + 1 and"}, {"start": 63.079, "duration": 4.641, "text": "I'm very sure we're very familiar with"}, {"start": 65.28, "duration": 4.96, "text": "this quadratic expression here that"}, {"start": 67.72, "duration": 5.52, "text": "quadratic expression is exactly the"}, {"start": 70.24, "duration": 7.84, "text": "square of x + 1 right so this is equal"}, {"start": 73.24, "duration": 7.239, "text": "to the square root of the square of x +"}, {"start": 78.08, "duration": 4.84, "text": "1 and do not forget when we have"}, {"start": 80.479, "duration": 6.241, "text": "something like this this is the absolute"}, {"start": 82.92, "duration": 5.76, "text": "value of x +1 remember that X is a ray"}, {"start": 86.72, "duration": 8.8, "text": "value right so the absolute value of x"}, {"start": 88.68, "duration": 7.799, "text": "+1 is = x + 1 right so therefore square"}, {"start": 95.52, "duration": 2.599, "text": "root"}, {"start": 96.479, "duration": 4.64, "text": "of"}, {"start": 98.119, "duration": 7.761, "text": "x\u00b2 +"}, {"start": 101.119, "duration": 13.881, "text": "x + x + 1 which is equal to the square"}, {"start": 105.88, "duration": 12.919, "text": "root of X2 + 2x + 1 is = x + 1 now let"}, {"start": 115.0, "duration": 7.96, "text": "us recall that let us recall that X is"}, {"start": 118.799, "duration": 9.32, "text": "equal to 6 41 so when we substitute here"}, {"start": 122.96, "duration": 7.88, "text": "we get that the square root of"}, {"start": 128.119, "duration": 5.601, "text": "641 SAR"}, {"start": 130.84, "duration": 4.479, "text": "plus here"}, {"start": 133.72, "duration": 3.4, "text": "641"}, {"start": 135.319, "duration": 7.481, "text": "+"}, {"start": 137.12, "duration": 8.08, "text": "642 right that this is equal to 641 + 1"}, {"start": 142.8, "duration": 5.6, "text": "which is equal to"}, {"start": 145.2, "duration": 5.039, "text": "642 and this is the right answer thank"}, {"start": 148.4, "duration": 5.24, "text": "you for watching please kindly subscribe"}, {"start": 150.239, "duration": 6.841, "text": "to my channel also like comment and"}, {"start": 153.64, "duration": 3.44, "text": "share bye-bye"}], "title": "China| Can you evaluate this ? | No calculators please | #maths #mathematics", "description": "", "lengthSeconds": 155, "width": 576, "height": 360}
{"video_id": "pYdaRb61F18", "transcript": [{"start": 0.799, "duration": 3.96, "text": "hello welcome back once again today"}, {"start": 2.72, "duration": 6.64, "text": "we're going to solve the following given"}, {"start": 4.759, "duration": 9.401, "text": "exponential equation we have here 2^ 4"}, {"start": 9.36, "duration": 9.44, "text": "to^ 3^ x this is equal to here 32 is the"}, {"start": 14.16, "duration": 9.24, "text": "same as 2^ 5 right now from the left the"}, {"start": 18.8, "duration": 7.2, "text": "base here is 2 y 4^ 3^ x is the Power of"}, {"start": 23.4, "duration": 5.799, "text": "Two And from the right here the base is"}, {"start": 26.0, "duration": 5.279, "text": "two why just five is the power right so"}, {"start": 29.199, "duration": 4.52, "text": "we're going to to the power since we"}, {"start": 31.279, "duration": 9.8, "text": "have obtained equal bases right so from"}, {"start": 33.719, "duration": 9.84, "text": "here this implies that 4 to^ 3 to^ x = 5"}, {"start": 41.079, "duration": 5.48, "text": "now from here we're going to make use of"}, {"start": 43.559, "duration": 7.761, "text": "log reading let us log both sides so"}, {"start": 46.559, "duration": 9.68, "text": "here we have the log of 4^ 3^ x this"}, {"start": 51.32, "duration": 8.199, "text": "equals the logm of 5 now let us take"}, {"start": 56.239, "duration": 6.32, "text": "note of this the log of a to the power"}, {"start": 59.519, "duration": 7.721, "text": "of P same as the power here crossing"}, {"start": 62.559, "duration": 7.761, "text": "over becoming B the log of a right so"}, {"start": 67.24, "duration": 4.72, "text": "here the Bas is 4 the power is 3 to^ x"}, {"start": 70.32, "duration": 6.0, "text": "so this power will cross over so that"}, {"start": 71.96, "duration": 8.799, "text": "will become 3 to the^ X multi by the log"}, {"start": 76.32, "duration": 6.04, "text": "of four this is equal to the log of five"}, {"start": 80.759, "duration": 4.161, "text": "now from here let's go ahead and divide"}, {"start": 82.36, "duration": 6.16, "text": "both side by the log of four so divide"}, {"start": 84.92, "duration": 6.44, "text": "here by the log of four also divide here"}, {"start": 88.52, "duration": 4.919, "text": "by the log of four"}, {"start": 91.36, "duration": 5.039, "text": "now from here this will cancel this we"}, {"start": 93.439, "duration": 4.201, "text": "have 3 to the power of X this will be"}, {"start": 96.399, "duration": 7.521, "text": "equal to the"}, {"start": 97.64, "duration": 9.92, "text": "log of 5 / the log of four okay now from"}, {"start": 103.92, "duration": 6.199, "text": "here let us log both sides again let us"}, {"start": 107.56, "duration": 6.72, "text": "log both sides again so here we have the"}, {"start": 110.119, "duration": 10.6, "text": "log of 3 to the^ X this is equal to the"}, {"start": 114.28, "duration": 9.32, "text": "log of log of 5 / log of four"}, {"start": 120.719, "duration": 4.68, "text": "now with the same property of logi this"}, {"start": 123.6, "duration": 4.84, "text": "power of three here which is X will"}, {"start": 125.399, "duration": 5.361, "text": "cross over that will become X log of"}, {"start": 128.44, "duration": 8.56, "text": "three this is equal to the"}, {"start": 130.76, "duration": 8.64, "text": "log of log of 5 / the log of 4 now from"}, {"start": 137.0, "duration": 5.04, "text": "here we're solving for x so divide both"}, {"start": 139.4, "duration": 6.6, "text": "side by log of 3 that is the coefficient"}, {"start": 142.04, "duration": 6.96, "text": "of x right so here by log of three this"}, {"start": 146.0, "duration": 5.4, "text": "will cancel this so from here we get the"}, {"start": 149.0, "duration": 8.2, "text": "exact solution which is the"}, {"start": 151.4, "duration": 10.4, "text": "log of log of 5/ the log of four then"}, {"start": 157.2, "duration": 7.679, "text": "the all of this ided by the log of 3"}, {"start": 161.8, "duration": 5.48, "text": "right so which is approximately equal to"}, {"start": 164.879, "duration": 5.401, "text": "0."}, {"start": 167.28, "duration": 4.679, "text": "135"}, {"start": 170.28, "duration": 5.679, "text": "853"}, {"start": 171.959, "duration": 6.041, "text": "87 and here is the right answer thank"}, {"start": 175.959, "duration": 4.2, "text": "you for watching please kindly subscribe"}, {"start": 178.0, "duration": 5.64, "text": "to my channel also like comment and"}, {"start": 180.159, "duration": 3.481, "text": "share bye-bye"}], "title": "Interesting exponential equation |", "description": "#maths #physics #exam #chemistry", "lengthSeconds": 182, "width": 540, "height": 360}
{"video_id": "QAgzA7NMyz8", "transcript": [{"start": 0.24, "duration": 5.119, "text": "hello welcome back once again here we"}, {"start": 2.56, "duration": 7.119, "text": "have another question here we're given"}, {"start": 5.359, "duration": 7.121, "text": "that 2 ^ of X is equal to 3 we're asked"}, {"start": 9.679, "duration": 5.04, "text": "to evaluate 16 to the power"}, {"start": 12.48, "duration": 4.04, "text": "ofx please if you have not subscribed to"}, {"start": 14.719, "duration": 5.56, "text": "this Channel please kindly subscribe to"}, {"start": 16.52, "duration": 5.64, "text": "this Channel and also share this video"}, {"start": 20.279, "duration": 4.16, "text": "many may think that the best way to"}, {"start": 22.16, "duration": 5.56, "text": "evaluate this is to find the value of x"}, {"start": 24.439, "duration": 5.84, "text": "in this equation but mind you that"}, {"start": 27.72, "duration": 5.359, "text": "before you find the value of x here we"}, {"start": 30.279, "duration": 5.8, "text": "need to make use of logarithm but to"}, {"start": 33.079, "duration": 6.64, "text": "avoid this I'm going to show you an"}, {"start": 36.079, "duration": 5.64, "text": "easier way to get this problem solved so"}, {"start": 39.719, "duration": 4.401, "text": "here we're given"}, {"start": 41.719, "duration": 6.281, "text": "16 ra power"}, {"start": 44.12, "duration": 7.919, "text": "ofx so you know this is equal to 16 is"}, {"start": 48.0, "duration": 6.359, "text": "the same as 2 power of 4 then power"}, {"start": 52.039, "duration": 4.401, "text": "ofx so these powers are going to"}, {"start": 54.359, "duration": 5.88, "text": "multiply together so here we're going to"}, {"start": 56.44, "duration": 6.599, "text": "have 2 of -4x"}, {"start": 60.239, "duration": 8.401, "text": "which we can also rewrite as follows 2"}, {"start": 63.039, "duration": 9.241, "text": "power of X then ra power of -4 so mind"}, {"start": 68.64, "duration": 6.76, "text": "you that 2 power of X is already equal"}, {"start": 72.28, "duration": 6.199, "text": "to 3 right so right over here we're"}, {"start": 75.4, "duration": 8.88, "text": "going to replace this with 3 so we have"}, {"start": 78.479, "duration": 9.841, "text": "16 ofx is going to be equal to 3^"}, {"start": 84.28, "duration": 9.72, "text": "of4 so you know with this negative sign"}, {"start": 88.32, "duration": 11.52, "text": "up here we're going to have 1/ 3^ 4 so"}, {"start": 94.0, "duration": 9.72, "text": "this is = to 1 / 3 ^ 4 is 3 * 3 * 3 * 3"}, {"start": 99.84, "duration": 5.879, "text": "and that is 81 this is the final answer"}, {"start": 103.72, "duration": 6.359, "text": "thank you for watching please do well to"}, {"start": 105.719, "duration": 4.36, "text": "like comments and share"}], "title": "Interesting exponential problem. #maths #olympics", "description": "", "lengthSeconds": 108, "width": 640, "height": 360}
{"video_id": "WtQQbSlpCJk", "transcript": [{"start": 0.76, "duration": 5.84, "text": "hello good viewers today we have this"}, {"start": 3.159, "duration": 6.4, "text": "nice interesting exponential"}, {"start": 6.6, "duration": 4.999, "text": "equation okay now the equation looks a"}, {"start": 9.559, "duration": 5.721, "text": "little bit"}, {"start": 11.599, "duration": 9.801, "text": "but believe me is a very simple question"}, {"start": 15.28, "duration": 9.96, "text": "23 to the^ 27 x in bracket 27 x again"}, {"start": 21.4, "duration": 6.08, "text": "this is equal to 23 ^ 9^ 3 before we"}, {"start": 25.24, "duration": 5.76, "text": "proceed let us make a substitution let"}, {"start": 27.48, "duration": 7.96, "text": "us let this 27 of x equals something"}, {"start": 31.0, "duration": 11.039, "text": "it's a y and here we have 23 to^"}, {"start": 35.44, "duration": 11.92, "text": "y y this is equal 23 9 3 let us record"}, {"start": 42.039, "duration": 9.04, "text": "this property m n z this isal to m n * Z"}, {"start": 47.36, "duration": 9.56, "text": "so here we multiply Y and Y we have 23"}, {"start": 51.079, "duration": 7.841, "text": "y^ 2 this is equal to 23 9 3 we have"}, {"start": 56.92, "duration": 4.08, "text": "equal bases right you know what is going"}, {"start": 58.92, "duration": 5.12, "text": "to happen we're going to to equate the"}, {"start": 61.0, "duration": 8.84, "text": "powers so here we have y^ 2 this is"}, {"start": 64.04, "duration": 10.719, "text": "equal to here we have 9^ of 3 right so"}, {"start": 69.84, "duration": 9.239, "text": "but 9 here that is 9 3 so we have y^ 2 ="}, {"start": 74.759, "duration": 9.441, "text": "9 is the same as 3 S then of"}, {"start": 79.079, "duration": 9.201, "text": "3 so here we have Y 2 is equal"}, {"start": 84.2, "duration": 6.279, "text": "to 3 S of 3 here making use of this"}, {"start": 88.28, "duration": 3.479, "text": "property this becomes 3"}, {"start": 90.479, "duration": 5.361, "text": "of 2"}, {"start": 91.759, "duration": 8.081, "text": "* 3 which is 6 so from here we take the"}, {"start": 95.84, "duration": 8.2, "text": "root of both side we have y is equal to"}, {"start": 99.84, "duration": 9.44, "text": "the square otk of 3^"}, {"start": 104.04, "duration": 8.28, "text": "6 so recall that the N root of M is = m"}, {"start": 109.28, "duration": 9.24, "text": "1 / n so here Y is"}, {"start": 112.32, "duration": 9.32, "text": "= 3 I mean 1 / two and from here Y is"}, {"start": 118.52, "duration": 5.32, "text": "equal to you know 6 * 1/ 2 is 3 so we"}, {"start": 121.64, "duration": 5.479, "text": "get 3 power of"}, {"start": 123.84, "duration": 10.2, "text": "3 okay now let us recall"}, {"start": 127.119, "duration": 11.441, "text": "that 27 x was = to Y 27 here is 3 3 so"}, {"start": 134.04, "duration": 8.16, "text": "we have 3 3 right then R of X this is"}, {"start": 138.56, "duration": 6.399, "text": "equal to Y which is 3 to the^ of"}, {"start": 142.2, "duration": 6.679, "text": "3 so from here we multiply these Powers"}, {"start": 144.959, "duration": 7.521, "text": "together here we have 3 3x this is equal"}, {"start": 148.879, "duration": 5.521, "text": "to 3 3 we have equal bases right so this"}, {"start": 152.48, "duration": 7.32, "text": "implies that the powers we can equate"}, {"start": 154.4, "duration": 8.119, "text": "them 3x is = 3 divide both sides by"}, {"start": 159.8, "duration": 7.0, "text": "3 and here we come up with the solution"}, {"start": 162.519, "duration": 6.36, "text": "X is = 3 / 3 which is 1 and this is the"}, {"start": 166.8, "duration": 4.24, "text": "solution thank you for watching please"}, {"start": 168.879, "duration": 6.841, "text": "kindly subscribe to my YouTube channel"}, {"start": 171.04, "duration": 4.68, "text": "also like comment and share bye-bye"}], "title": "Can you solve this complicated looking exponential equation? | #maths #olympiad #mathcompetition", "description": "", "lengthSeconds": 174, "width": 640, "height": 360}
{"video_id": "eO7Z2MTqlDU", "transcript": [{"start": 0.74, "duration": 2.3, "text": "[Music]"}, {"start": 1.16, "duration": 5.239, "text": "hello good day viewers welcome back once"}, {"start": 3.04, "duration": 7.04, "text": "again today we have an interesting"}, {"start": 6.399, "duration": 6.721, "text": "problem and this is an Olympiad mat we"}, {"start": 10.08, "duration": 6.44, "text": "are given the following equation 36 x ^"}, {"start": 13.12, "duration": 8.399, "text": "4 + 1 that this is equal to 0 right over"}, {"start": 16.52, "duration": 7.16, "text": "here we're going to evaluate 3x + 1 / 2x"}, {"start": 21.519, "duration": 2.961, "text": "from the given equation we can re write"}, {"start": 23.68, "duration": 9.6, "text": "it"}, {"start": 24.48, "duration": 13.0, "text": "as 9 x^ 2 * 4 x 2 right CU 9 * 4 is 36"}, {"start": 33.28, "duration": 8.2, "text": "x\u00b2 * X will give us x^ 4 from here then"}, {"start": 37.48, "duration": 8.239, "text": "+ 1 this is equal to zero now from both"}, {"start": 41.48, "duration": 6.079, "text": "sides we're going to divide each term by"}, {"start": 45.719, "duration": 11.601, "text": "4"}, {"start": 47.559, "duration": 15.521, "text": "X2 so here 9 X2 * 4 X2 all over 4 x 2 +"}, {"start": 57.32, "duration": 9.44, "text": "1 over 4 X2 this is equal to 0 all over"}, {"start": 63.08, "duration": 6.44, "text": "4x2 so from here this and this will"}, {"start": 66.76, "duration": 5.96, "text": "cancel out and here we have 9"}, {"start": 69.52, "duration": 7.16, "text": "x2+ here 1 /"}, {"start": 72.72, "duration": 10.6, "text": "4x2 this is equal to 0 so from here this"}, {"start": 76.68, "duration": 9.92, "text": "is same as 3x^ 2 then plus here 1 / 2x^"}, {"start": 83.32, "duration": 8.04, "text": "2 this is equal to zero now let us"}, {"start": 86.6, "duration": 12.559, "text": "recall this property of algebra a a 2 +"}, {"start": 91.36, "duration": 10.88, "text": "b 2 this is same as a + b 2 - 2 * a * B"}, {"start": 99.159, "duration": 6.0, "text": "right so using this property from here"}, {"start": 102.24, "duration": 5.44, "text": "we have 3x right which is the first time"}, {"start": 105.159, "duration": 8.161, "text": "here squ plus the second term squ here"}, {"start": 107.68, "duration": 9.84, "text": "which is 1 / 2x then s - 2 * by the"}, {"start": 113.32, "duration": 9.52, "text": "first term 3x multi by the second term 1"}, {"start": 117.52, "duration": 7.72, "text": "/ 2x 2 1/ 2x I mean sorry for that so"}, {"start": 122.84, "duration": 6.24, "text": "that is 1/ 2x that is the second term"}, {"start": 125.24, "duration": 10.999, "text": "here and this is equal to"}, {"start": 129.08, "duration": 10.96, "text": "0 so from here we have 3x + 1 / 2x^ 2"}, {"start": 136.239, "duration": 5.921, "text": "then minus here 2 and two we cancel out"}, {"start": 140.04, "duration": 4.96, "text": "x and x will cancel out and we're left"}, {"start": 142.16, "duration": 9.079, "text": "with -3 this is ="}, {"start": 145.0, "duration": 9.92, "text": "Z from here we have 3x + 1 / 2X x 2 this"}, {"start": 151.239, "duration": 7.481, "text": "is equal 0 + 3 that will give us 3 so"}, {"start": 154.92, "duration": 10.36, "text": "here we arrive at the solution that 3x +"}, {"start": 158.72, "duration": 8.76, "text": "1 / 2x is = to + orus otk of 3 and this"}, {"start": 165.28, "duration": 3.879, "text": "is the right answer thank you for"}, {"start": 167.48, "duration": 4.16, "text": "watching please kindly subscribe to my"}, {"start": 169.159, "duration": 5.481, "text": "channel also like comment and share"}, {"start": 171.64, "duration": 3.0, "text": "bye-bye"}], "title": "Japanese | Can you solve ? | Nice Olympiad Math | #maths #mathematics", "description": "", "lengthSeconds": 173, "width": 640, "height": 360}
{"video_id": "EDOTzO9MecM", "transcript": [{"start": 0.359, "duration": 4.161, "text": "hello welcome back once again today"}, {"start": 2.24, "duration": 6.8, "text": "we're going to solve this exponential"}, {"start": 4.52, "duration": 7.48, "text": "equation we have 5^ X2 = 150 and here"}, {"start": 9.04, "duration": 7.68, "text": "we'll be solving for the values of X"}, {"start": 12.0, "duration": 8.039, "text": "right okay now let's begin by Ling both"}, {"start": 16.72, "duration": 6.08, "text": "sides of this equation okay so here we"}, {"start": 20.039, "duration": 7.281, "text": "have the Ln of 5 to^"}, {"start": 22.8, "duration": 6.52, "text": "X2 this will be equal to the Ln of 150"}, {"start": 27.32, "duration": 5.32, "text": "right so now with the property of"}, {"start": 29.32, "duration": 8.16, "text": "logarithm this power will come over here"}, {"start": 32.64, "duration": 10.04, "text": "so we have x\u00b2 * the Ln of 5 this will be"}, {"start": 37.48, "duration": 10.559, "text": "equal to from here Ln of 150 is same as"}, {"start": 42.68, "duration": 8.559, "text": "Ln of 5 * 30 right and with the property"}, {"start": 48.039, "duration": 6.601, "text": "of logarithm this is same as Ln of 5"}, {"start": 51.239, "duration": 7.081, "text": "plus the Ln of"}, {"start": 54.64, "duration": 7.439, "text": "30 okay so from here we have the Ln of 5"}, {"start": 58.32, "duration": 6.239, "text": "plus the Ln of 30 so in our next step"}, {"start": 62.079, "duration": 6.881, "text": "let's divide both side by the Ln of 5 so"}, {"start": 64.559, "duration": 5.961, "text": "we have x\u00b2 MTI the Ln of 5 this is equal"}, {"start": 68.96, "duration": 7.92, "text": "to Ln of"}, {"start": 70.52, "duration": 6.36, "text": "5+ Ln of 30 so the by the Ln of"}, {"start": 78.56, "duration": 7.04, "text": "5 so here this will cancel this we have"}, {"start": 81.64, "duration": 7.2, "text": "here x\u00b2 is equal to here we split this"}, {"start": 85.6, "duration": 7.799, "text": "denominator so we have Ln of 5 / Ln 5"}, {"start": 88.84, "duration": 11.08, "text": "plus Ln 30 ID Ln of 5 so from here Ln of"}, {"start": 93.399, "duration": 10.641, "text": "5 Ln of 5 is 1 then plus Ln of 30 / the"}, {"start": 99.92, "duration": 9.4, "text": "Ln of 5 right okay so from here once"}, {"start": 104.04, "duration": 10.16, "text": "again Ln of 30 is some as Ln of 5 * 6"}, {"start": 109.32, "duration": 10.32, "text": "which is same as Ln of 5 + Ln of"}, {"start": 114.2, "duration": 11.0, "text": "6 so from here we have X2 is = to 1+"}, {"start": 119.64, "duration": 12.16, "text": "that will be Ln of 5 plus Ln of 6 then /"}, {"start": 125.2, "duration": 10.52, "text": "by the Ln of 5 so we have here X2 is = 1"}, {"start": 131.8, "duration": 8.799, "text": "+ Ln 5/ Ln 5 is"}, {"start": 135.72, "duration": 8.04, "text": "1+ Ln of 6 / the Ln of"}, {"start": 140.599, "duration": 6.041, "text": "5 let us take note of this the Ln of a"}, {"start": 143.76, "duration": 6.6, "text": "divid by the Ln of B is equal to the"}, {"start": 146.64, "duration": 6.28, "text": "logarithm of a to the base B right so"}, {"start": 150.36, "duration": 7.84, "text": "we're going to apply that law here so we"}, {"start": 152.92, "duration": 9.679, "text": "have here X2 is = 1 + 1 which is 2 then"}, {"start": 158.2, "duration": 7.36, "text": "plus so this will be the log of 6 to the"}, {"start": 162.599, "duration": 5.441, "text": "base of 5 so remember we're solving for"}, {"start": 165.56, "duration": 6.28, "text": "x so let's take the root of both sides"}, {"start": 168.04, "duration": 6.44, "text": "we have X is equal to plus or minus the"}, {"start": 171.84, "duration": 7.72, "text": "square root of"}, {"start": 174.48, "duration": 7.2, "text": "2+ the log of 6 to the base 5 and here"}, {"start": 179.56, "duration": 4.36, "text": "we have have the right answer thank you"}, {"start": 181.68, "duration": 5.559, "text": "for watching please kindly subscribe to"}, {"start": 183.92, "duration": 6.319, "text": "my channel also like comment and share"}, {"start": 187.239, "duration": 3.0, "text": "bye-bye"}], "title": "Nice exponential equation | Can you solve ?", "description": "", "lengthSeconds": 189, "width": 640, "height": 360}
{"video_id": "cNcjRFFCpe4", "transcript": [{"start": 0.56, "duration": 5.36, "text": "hello good day everyone welcome back"}, {"start": 3.04, "duration": 5.72, "text": "once again please if today is your first"}, {"start": 5.92, "duration": 6.599, "text": "day of visiting this Channel please"}, {"start": 8.76, "duration": 7.439, "text": "kindly support us by subscribing to our"}, {"start": 12.519, "duration": 7.001, "text": "YouTube channel and sharing our videos"}, {"start": 16.199, "duration": 6.441, "text": "thank you today we have this interesting"}, {"start": 19.52, "duration": 5.519, "text": "limits right over here we have limit as"}, {"start": 22.64, "duration": 6.639, "text": "n is approaching Infinity of this"}, {"start": 25.039, "duration": 8.441, "text": "function right over here n\u00b2 + 2 all/ 4"}, {"start": 29.279, "duration": 6.44, "text": "n\u00b2 - 3 if we try to make a direct"}, {"start": 33.48, "duration": 6.36, "text": "substitution right over here we're going"}, {"start": 35.719, "duration": 9.081, "text": "to get Infinity S which is infinity + 2"}, {"start": 39.84, "duration": 8.359, "text": "is infinity IDE 4 * Infinity 2 is"}, {"start": 44.8, "duration": 6.16, "text": "infinity - 3 which is infinity and this"}, {"start": 48.199, "duration": 7.121, "text": "is not allowed in mathematics"}, {"start": 50.96, "duration": 8.599, "text": "right okay now let us try to see what we"}, {"start": 55.32, "duration": 8.479, "text": "can do here let us try to divide both"}, {"start": 59.559, "duration": 7.92, "text": "the numerator and denominator by n^ of"}, {"start": 63.799, "duration": 7.32, "text": "2 so this is equal to"}, {"start": 67.479, "duration": 9.121, "text": "limit as n is approaching"}, {"start": 71.119, "duration": 14.561, "text": "Infinity n 2 + 2 all/"}, {"start": 76.6, "duration": 10.36, "text": "n\u00b2 / 4 n 2 - 3 all/ n 2 right so this is"}, {"start": 85.68, "duration": 4.88, "text": "equal to"}, {"start": 86.96, "duration": 6.88, "text": "limit as n is approaching Infinity"}, {"start": 90.56, "duration": 12.28, "text": "so let us try to split the denominator"}, {"start": 93.84, "duration": 15.599, "text": "we have N2 / n 2 + 2 / n 2 all/ 4 n 2 /"}, {"start": 102.84, "duration": 9.16, "text": "n 2 - 3 / n\u00b2 right now let us make a"}, {"start": 109.439, "duration": 7.881, "text": "little bit cancellation"}, {"start": 112.0, "duration": 9.399, "text": "here so we have here n 2 canel n 2 and"}, {"start": 117.32, "duration": 5.28, "text": "here we have n\u00b2 canel n 2 so we are left"}, {"start": 121.399, "duration": 4.4, "text": "with"}, {"start": 122.6, "duration": 10.639, "text": "limit as n is approaching"}, {"start": 125.799, "duration": 14.201, "text": "Infinity here we are left it 1 + 2 n 2"}, {"start": 133.239, "duration": 10.64, "text": "over here we leftt with 4 - 3 all n of 2"}, {"start": 140.0, "duration": 8.0, "text": "so right over here we have now if n is"}, {"start": 143.879, "duration": 6.681, "text": "approaching Infinity 2 all n 2 as n is"}, {"start": 148.0, "duration": 4.959, "text": "getting larger larger"}, {"start": 150.56, "duration": 6.92, "text": "larger and larger is approaching zero"}, {"start": 152.959, "duration": 8.841, "text": "right so here we have 1 + 0 all over"}, {"start": 157.48, "duration": 6.96, "text": "here we have 4 minus you know 3 n 2 as n"}, {"start": 161.8, "duration": 5.64, "text": "is approaching Infinity meaning the"}, {"start": 164.44, "duration": 5.799, "text": "bottom is getting larger larger larger"}, {"start": 167.44, "duration": 6.24, "text": "and larger so that way approach zero"}, {"start": 170.239, "duration": 6.841, "text": "right so we are left with just one all"}, {"start": 173.68, "duration": 6.16, "text": "over four and this is the right answer"}, {"start": 177.08, "duration": 5.519, "text": "thank you for watching please do well to"}, {"start": 179.84, "duration": 6.88, "text": "to subscribe to this Channel and share"}, {"start": 182.599, "duration": 4.121, "text": "our videos bye-bye"}], "title": "Limit of a function. interesting limit", "description": "", "lengthSeconds": 185, "width": 636, "height": 360}
{"video_id": "t1vQKlWnqRY", "transcript": [{"start": 1.0, "duration": 5.2, "text": "hello welcome back once again today we"}, {"start": 3.0, "duration": 4.759, "text": "have this amazing exponential equation"}, {"start": 6.2, "duration": 7.479, "text": "and here we'll be solving for the value"}, {"start": 7.759, "duration": 8.761, "text": "of x we have here 2 to^ of x^ of X this"}, {"start": 13.679, "duration": 5.841, "text": "is equal to 3 we need to take note of"}, {"start": 16.52, "duration": 6.36, "text": "something really very important here the"}, {"start": 19.52, "duration": 4.56, "text": "base here is 2 and X the^ of X is the"}, {"start": 22.88, "duration": 4.08, "text": "power of"}, {"start": 24.08, "duration": 4.92, "text": "two now we're going to natural log both"}, {"start": 26.96, "duration": 6.48, "text": "sides of this equation so here we have"}, {"start": 29.0, "duration": 7.76, "text": "the natural log of 2^ x^ X then this"}, {"start": 33.44, "duration": 6.04, "text": "will be equal to the natural log of 3"}, {"start": 36.76, "duration": 5.88, "text": "now since we know that x the^ x is the"}, {"start": 39.48, "duration": 6.68, "text": "power of two then this power right over"}, {"start": 42.64, "duration": 7.32, "text": "here we can move it to the front right"}, {"start": 46.16, "duration": 6.32, "text": "and that will become X to^ of X MTI the"}, {"start": 49.96, "duration": 5.96, "text": "natural log of 2 and this is equal to"}, {"start": 52.48, "duration": 5.32, "text": "the natural log of 3 at this point we"}, {"start": 55.92, "duration": 4.68, "text": "need to divide both side by the natural"}, {"start": 57.8, "duration": 5.919, "text": "log of two and here"}, {"start": 60.6, "duration": 5.64, "text": "log of two this will cancel this and"}, {"start": 63.719, "duration": 5.201, "text": "from the left hand side we have x to the"}, {"start": 66.24, "duration": 6.239, "text": "power of X this is equal to the natural"}, {"start": 68.92, "duration": 7.239, "text": "log of 3 / the natural log of"}, {"start": 72.479, "duration": 6.601, "text": "2 okay now again let us natural log both"}, {"start": 76.159, "duration": 6.201, "text": "sides of this equation one more time so"}, {"start": 79.08, "duration": 6.039, "text": "here we have the natural log of x^ X"}, {"start": 82.36, "duration": 5.36, "text": "this is equal to the natural log of the"}, {"start": 85.119, "duration": 3.281, "text": "natural log of 3 ID by the natural log"}, {"start": 87.72, "duration": 4.52, "text": "of"}, {"start": 88.4, "duration": 6.84, "text": "two now with power rule so this we cross"}, {"start": 92.24, "duration": 5.4, "text": "over and here we have X MTI the natural"}, {"start": 95.24, "duration": 6.0, "text": "log of x this is equal to the natural"}, {"start": 97.64, "duration": 4.32, "text": "log of the natural log of 3/ natural log"}, {"start": 101.24, "duration": 5.12, "text": "of"}, {"start": 101.96, "duration": 8.6, "text": "2 okay now from here this x here we can"}, {"start": 106.36, "duration": 6.799, "text": "rewrite it as e to the^ of Ln of X we"}, {"start": 110.56, "duration": 4.48, "text": "all know that this e to the^ of Ln"}, {"start": 113.159, "duration": 4.481, "text": "cancel out to be one and here we're left"}, {"start": 115.04, "duration": 5.28, "text": "with just X right and here we're"}, {"start": 117.64, "duration": 5.439, "text": "multiplying by the natural of x then"}, {"start": 120.32, "duration": 6.24, "text": "this is equal to the natural log of the"}, {"start": 123.079, "duration": 6.481, "text": "natural log of 3 / the natural log of"}, {"start": 126.56, "duration": 7.52, "text": "2 okay now from here we're going to make"}, {"start": 129.56, "duration": 7.52, "text": "use of the lumber W"}, {"start": 134.08, "duration": 6.159, "text": "function okay but we need to understand"}, {"start": 137.08, "duration": 5.08, "text": "this function first the lamb function"}, {"start": 140.239, "duration": 6.441, "text": "with the following argument e to the"}, {"start": 142.16, "duration": 7.24, "text": "power of x m x is equal to"}, {"start": 146.68, "duration": 5.839, "text": "X so we can see that the power of E here"}, {"start": 149.4, "duration": 5.4, "text": "Isa equal to the coefficient so here"}, {"start": 152.519, "duration": 4.201, "text": "let's take the lamb function on both"}, {"start": 154.8, "duration": 5.799, "text": "sides so here we have the"}, {"start": 156.72, "duration": 9.0, "text": "lambat function of e to power of Ln X"}, {"start": 160.599, "duration": 10.601, "text": "MTI the Ln X this is equal to the W of"}, {"start": 165.72, "duration": 8.439, "text": "Ln of 3 / the Ln of 2 from here remember"}, {"start": 171.2, "duration": 5.039, "text": "the power of E here is equal to the"}, {"start": 174.159, "duration": 6.921, "text": "coefficient so this simplifies to just"}, {"start": 176.239, "duration": 7.321, "text": "the of X and this is equal to the W oh I"}, {"start": 181.08, "duration": 4.799, "text": "think I missing something here yes so"}, {"start": 183.56, "duration": 6.039, "text": "this is the natural log of the all of"}, {"start": 185.879, "duration": 6.881, "text": "this right so sorry for that and here we"}, {"start": 189.599, "duration": 7.28, "text": "have the W of the natural log of the"}, {"start": 192.76, "duration": 8.44, "text": "natural log of 3 / the natural log of 2"}, {"start": 196.879, "duration": 8.161, "text": "so from here x is equal to e to the"}, {"start": 201.2, "duration": 6.92, "text": "power of the W of the natural log of the"}, {"start": 205.04, "duration": 7.279, "text": "natural log of 3 / the natural log of"}, {"start": 208.12, "duration": 8.44, "text": "two and using my calculator this is"}, {"start": 212.319, "duration": 4.241, "text": "approximately equal to"}, {"start": 217.72, "duration": 8.439, "text": "1392 1"}, {"start": 219.68, "duration": 8.52, "text": "26 612 3 1 and this is the right answer"}, {"start": 226.159, "duration": 4.321, "text": "thank you for watching please kindly"}, {"start": 228.2, "duration": 6.44, "text": "subscribe to my channel also like"}, {"start": 230.48, "duration": 4.16, "text": "comment and share bye-bye"}], "title": "Interesting exponential equation | The Lambert W Function is a good way, trust me.", "description": "", "lengthSeconds": 233, "width": 640, "height": 360}
{"video_id": "-45pyQvik8E", "transcript": [{"start": 0.24, "duration": 3.92, "text": "hello dear friends welcome back once"}, {"start": 2.08, "duration": 4.279, "text": "again today we're going to solve this"}, {"start": 4.16, "duration": 5.519, "text": "cubic equation and we will solve for"}, {"start": 6.359, "duration": 6.881, "text": "both real and complex Solutions right"}, {"start": 9.679, "duration": 8.121, "text": "okay let's start we have here"}, {"start": 13.24, "duration": 8.44, "text": "solution so we have here x to the^ 3"}, {"start": 17.8, "duration": 7.44, "text": "minus 1 here is as 1 to^ 3 and this is"}, {"start": 21.68, "duration": 5.08, "text": "equal to 0 so from the left hand side"}, {"start": 25.24, "duration": 4.4, "text": "we're going to make use of the"}, {"start": 26.76, "duration": 13.2, "text": "difference of two cubes formula remember"}, {"start": 29.64, "duration": 11.599, "text": "a Cub - B Cub is = to a - b * a 2 + a b"}, {"start": 39.96, "duration": 4.32, "text": "+"}, {"start": 41.239, "duration": 9.121, "text": "b\u00b2 okay so make use of this property the"}, {"start": 44.28, "duration": 10.4, "text": "left hand side will become X - 1 * x 2 +"}, {"start": 50.36, "duration": 5.719, "text": "x * 1 which is x + 1 2 which is 1 and"}, {"start": 54.68, "duration": 4.879, "text": "this is equal"}, {"start": 56.079, "duration": 6.041, "text": "to0 so if these two products is equal to"}, {"start": 59.559, "duration": 9.761, "text": "0 so we have either x"}, {"start": 62.12, "duration": 11.28, "text": "-1 is = 0 or from this we have X2 + x +"}, {"start": 69.32, "duration": 8.68, "text": "1 is = 0 so from here we have the r"}, {"start": 73.4, "duration": 6.24, "text": "solution that X is = 1 right okay so"}, {"start": 78.0, "duration": 4.04, "text": "from here let us solve this quadratic"}, {"start": 79.64, "duration": 4.92, "text": "equation we're going to make use of the"}, {"start": 82.04, "duration": 6.96, "text": "general quadratic formula so X will be"}, {"start": 84.56, "duration": 12.599, "text": "equal to B plus or minus Square < TK of"}, {"start": 89.0, "duration": 10.64, "text": "b 2 - 4 a c then / by 2 * a right so if"}, {"start": 97.159, "duration": 4.28, "text": "you if you carefully look at this"}, {"start": 99.64, "duration": 4.119, "text": "expression on the left hand side of this"}, {"start": 101.439, "duration": 4.32, "text": "equation so we realize that a is equal"}, {"start": 103.759, "duration": 4.561, "text": "to the leading"}, {"start": 105.759, "duration": 5.281, "text": "coefficient the leading time the"}, {"start": 108.32, "duration": 5.2, "text": "coefficient which is one and B here is"}, {"start": 111.04, "duration": 5.999, "text": "the coefficient of x which is one and C"}, {"start": 113.52, "duration": 6.919, "text": "the constant is also one okay so here we"}, {"start": 117.039, "duration": 6.68, "text": "have X is = to B will be-1"}, {"start": 120.439, "duration": 8.8, "text": "plus or minus otk of b 2 will be 1 2"}, {"start": 123.719, "duration": 9.481, "text": "which is 1 - 4 * a * C which is 4 then /"}, {"start": 129.239, "duration": 7.681, "text": "2 * 1 which is 2 so here we have the"}, {"start": 133.2, "duration": 9.88, "text": "complex solution -1 plus or minus < TK"}, {"start": 136.92, "duration": 11.48, "text": "of -3 / 2 okay so from here we have X is"}, {"start": 143.08, "duration": 7.879, "text": "= to1 plus or minus first of all of1 is"}, {"start": 148.4, "duration": 6.559, "text": "Iota right then we have root of three"}, {"start": 150.959, "duration": 6.721, "text": "right over there then / by two so these"}, {"start": 154.959, "duration": 4.961, "text": "are the two complex Solutions thank you"}, {"start": 157.68, "duration": 5.639, "text": "for watching please kindly subscribe to"}, {"start": 159.92, "duration": 6.399, "text": "my channel also like comment and share"}, {"start": 163.319, "duration": 3.0, "text": "bye-bye"}], "title": "Interesting cubic equation | can you find all solutions ?", "description": "", "lengthSeconds": 165, "width": 640, "height": 360}
{"video_id": "a7O4YFuPtaY", "transcript": [{"start": 0.24, "duration": 3.84, "text": "good day everyone welcome back once"}, {"start": 2.399, "duration": 3.561, "text": "again today we're going to solve for the"}, {"start": 4.08, "duration": 5.559, "text": "value of x in the"}, {"start": 5.96, "duration": 6.92, "text": "following um fractional"}, {"start": 9.639, "duration": 5.641, "text": "exponential equation right oh if my"}, {"start": 12.88, "duration": 8.2, "text": "English is too bad sorry for that so"}, {"start": 15.28, "duration": 11.079, "text": "here we have 6^ x - 2 x / 2 x this is ="}, {"start": 21.08, "duration": 14.4, "text": "to 8 right okay 6 of X the base we can"}, {"start": 26.359, "duration": 11.961, "text": "write as 2 * 3 then of x - 2 x / 2^ X"}, {"start": 35.48, "duration": 8.68, "text": "this is equal to 8 now let's recall this"}, {"start": 38.32, "duration": 10.28, "text": "property M * n power of Z the same as M"}, {"start": 44.16, "duration": 13.199, "text": "of Z * n power of"}, {"start": 48.6, "duration": 13.32, "text": "Z so here we have 2^ of x * 3^ x - 2^ x"}, {"start": 57.359, "duration": 6.36, "text": "/ 2^ X this is is equal to 8 so for the"}, {"start": 61.92, "duration": 7.32, "text": "numerator here we have a common factor"}, {"start": 63.719, "duration": 11.121, "text": "of 2 x so we have 2 x into bracket 3 X -"}, {"start": 69.24, "duration": 8.84, "text": "1 / 2 x this is equal to 8 you know from"}, {"start": 74.84, "duration": 7.4, "text": "here this two get cancel out and we left"}, {"start": 78.08, "duration": 9.039, "text": "with 3 X - 1 this is equal to 8 add one"}, {"start": 82.24, "duration": 8.68, "text": "to both side we get 3 x is equal to"}, {"start": 87.119, "duration": 7.04, "text": "9 from here we have three of X is equal"}, {"start": 90.92, "duration": 6.28, "text": "to 9 is the same as 3 S since we have"}, {"start": 94.159, "duration": 6.521, "text": "obtained equal bases so we can equate"}, {"start": 97.2, "duration": 5.04, "text": "the powers and X becomes two thank you"}, {"start": 100.68, "duration": 4.2, "text": "for watching please kindly subscribe to"}, {"start": 102.24, "duration": 6.159, "text": "my YouTube channel also like comment and"}, {"start": 104.88, "duration": 3.519, "text": "share bye-bye"}], "title": "A very tricky exponential equation | #maths #olympiad", "description": "", "lengthSeconds": 107, "width": 640, "height": 360}
{"video_id": "pMZX3vKjGJo", "transcript": [{"start": 0.799, "duration": 4.321, "text": "hello welcome back once again today"}, {"start": 3.199, "duration": 5.121, "text": "we're going to solve for the value of x"}, {"start": 5.12, "duration": 5.0, "text": "in the following given equation so here"}, {"start": 8.32, "duration": 7.12, "text": "we have root of"}, {"start": 10.12, "duration": 7.48, "text": "2x + x this is equal to 1 now from here"}, {"start": 15.44, "duration": 7.32, "text": "let's subtract X from both sides so we"}, {"start": 17.6, "duration": 8.16, "text": "get < TK of 2x this is equal to 1 - x"}, {"start": 22.76, "duration": 4.96, "text": "right okay now to get rid of this root"}, {"start": 25.76, "duration": 6.16, "text": "sign here we're going to square both"}, {"start": 27.72, "duration": 6.48, "text": "sides of this equation so we Square here"}, {"start": 31.92, "duration": 5.36, "text": "and from here Square we cancel this root"}, {"start": 34.2, "duration": 8.64, "text": "sign we have 2 x left then this is equal"}, {"start": 37.28, "duration": 11.84, "text": "to Here let us take note of this a - b 2"}, {"start": 42.84, "duration": 12.32, "text": "this is equal to a 2 + b 2 - 2 * a * B"}, {"start": 49.12, "duration": 10.0, "text": "right okay so here 1 2 is 1 + x"}, {"start": 55.16, "duration": 5.76, "text": "2 - 2 * a * B in this case will be 2 * 1"}, {"start": 59.12, "duration": 4.119, "text": "* X which is"}, {"start": 60.92, "duration": 6.16, "text": "2x okay now let's make the right hand"}, {"start": 63.239, "duration": 7.2, "text": "side the left hand side so we have 1 +"}, {"start": 67.08, "duration": 7.2, "text": "X2 - 2x then this is equal to from the"}, {"start": 70.439, "duration": 7.841, "text": "left 2x so subtract 2 X from both Sid"}, {"start": 74.28, "duration": 6.56, "text": "let's start with this we have x\u00b2 - 2x -"}, {"start": 78.28, "duration": 7.159, "text": "2x"}, {"start": 80.84, "duration": 10.08, "text": "right then we have + 1 this is = z so"}, {"start": 85.439, "duration": 8.081, "text": "from here we have x^2 - 4x + 1 1 this is"}, {"start": 90.92, "duration": 5.04, "text": "equal to Z at this point this is a"}, {"start": 93.52, "duration": 4.639, "text": "quadratic equation which we're going to"}, {"start": 95.96, "duration": 5.839, "text": "solve using the general quadratic"}, {"start": 98.159, "duration": 5.561, "text": "formula so we have xal to B will be 4"}, {"start": 101.799, "duration": 5.841, "text": "because B is negative so it's negative"}, {"start": 103.72, "duration": 9.48, "text": "will be positive then plus orus otk of b"}, {"start": 107.64, "duration": 10.119, "text": "s b s is -4 which is 16 - 4 * a * B 4 *"}, {"start": 113.2, "duration": 7.879, "text": "1 * 1 which is 4 then / 2 so this is"}, {"start": 117.759, "duration": 8.521, "text": "equal to 4 plus or minus square root of"}, {"start": 121.079, "duration": 9.641, "text": "12 right then ID by two so here we have"}, {"start": 126.28, "duration": 8.2, "text": "X is = to 4 plus or minus remember this"}, {"start": 130.72, "duration": 7.44, "text": "12 right over here is 4 * 3 so theun of"}, {"start": 134.48, "duration": 6.44, "text": "12 will beun of 4 * 3un of 4 is 2 and we"}, {"start": 138.16, "duration": 5.84, "text": "have root3 then we are dividing this by"}, {"start": 140.92, "duration": 9.44, "text": "two so from here let's split the"}, {"start": 144.0, "duration": 10.16, "text": "denominator 4 / 2+ or minus 2 < tk3 / 2"}, {"start": 150.36, "duration": 8.519, "text": "of which we can see that 4 3 is 2 plus"}, {"start": 154.16, "duration": 5.76, "text": "or minus 2K3 / 2 2 cancel two we left"}, {"start": 158.879, "duration": 5.761, "text": "with"}, {"start": 159.92, "duration": 7.88, "text": "root3 okay so now when I checked the"}, {"start": 164.64, "duration": 8.72, "text": "values which one satisfy this equation I"}, {"start": 167.8, "duration": 8.24, "text": "found out that X is = 2 + < tk3 is an"}, {"start": 173.36, "duration": 2.68, "text": "extrus"}, {"start": 178.519, "duration": 4.72, "text": "root it is is called an estrous root"}, {"start": 181.519, "duration": 4.481, "text": "because it does not satisfy the given"}, {"start": 183.239, "duration": 6.56, "text": "equation and therefore the only"}, {"start": 186.0, "duration": 6.64, "text": "satisfying root here is 2 minus < tk3"}, {"start": 189.799, "duration": 5.0, "text": "and this is the right answer thank you"}, {"start": 192.64, "duration": 5.04, "text": "for watching please kindly subscribe to"}, {"start": 194.799, "duration": 5.881, "text": "my channel also like comment and share"}, {"start": 197.68, "duration": 3.0, "text": "bye-bye"}], "title": "Interesting radical equation.", "description": "", "lengthSeconds": 200, "width": 640, "height": 360}
{"video_id": "4tqn0qVr9n0", "transcript": [{"start": 0.76, "duration": 5.4, "text": "hello welcome back once again today we"}, {"start": 3.04, "duration": 5.04, "text": "have this simple exponential equation"}, {"start": 6.16, "duration": 6.16, "text": "and here we're going to solve for the"}, {"start": 8.08, "duration": 9.44, "text": "value of x from the left we're given 4"}, {"start": 12.32, "duration": 7.68, "text": "to the ^ of x + 4 to the^ X then this"}, {"start": 17.52, "duration": 6.2, "text": "equals"}, {"start": 20.0, "duration": 5.32, "text": "16 okay now take note of this when we"}, {"start": 23.72, "duration": 7.2, "text": "have"}, {"start": 25.32, "duration": 9.28, "text": "A+ a you know this is equal to 2 okay"}, {"start": 30.92, "duration": 8.319, "text": "right so from the left remember 4^ X is"}, {"start": 34.6, "duration": 9.4, "text": "the same as 4^ X so to add this together"}, {"start": 39.239, "duration": 9.081, "text": "it will be same as 2 * 4 to the power of"}, {"start": 44.0, "duration": 7.64, "text": "X right then this equals"}, {"start": 48.32, "duration": 7.239, "text": "16 from here let us divide both sides by"}, {"start": 51.64, "duration": 7.919, "text": "two divide this by two and divide here"}, {"start": 55.559, "duration": 8.281, "text": "by two this will cancel this then we are"}, {"start": 59.559, "duration": 8.681, "text": "left with 4 to the^ X this equals 2 now"}, {"start": 63.84, "duration": 4.4, "text": "16 / 2 that will give us"}, {"start": 68.479, "duration": 8.28, "text": "8 now in our next step we're going to"}, {"start": 71.84, "duration": 8.36, "text": "rewrite four as 2^ two right because 4 ="}, {"start": 76.759, "duration": 7.481, "text": "2 to the^ 2 and from the right 8 which"}, {"start": 80.2, "duration": 6.2, "text": "is equal to 2 to the^ 3 so here this"}, {"start": 84.24, "duration": 7.199, "text": "will be 2 to the^"}, {"start": 86.4, "duration": 7.679, "text": "2 x then this equals 2 to the power of 3"}, {"start": 91.439, "duration": 4.881, "text": "now take note of this when we have a to"}, {"start": 94.079, "duration": 6.521, "text": "the power of B then to the power of C"}, {"start": 96.32, "duration": 7.2, "text": "this is same as a to the power of B *"}, {"start": 100.6, "duration": 6.159, "text": "C so here we're going to apply this Rule"}, {"start": 103.52, "duration": 8.52, "text": "and here we get 2 to the^ of 2 * X which"}, {"start": 106.759, "duration": 7.72, "text": "is 2x this equal 2 to the^ 3 from here"}, {"start": 112.04, "duration": 5.079, "text": "we have the same basis right so let's go"}, {"start": 114.479, "duration": 5.881, "text": "ahead and equate the powers so from here"}, {"start": 117.119, "duration": 5.96, "text": "this implies that 2X"}, {"start": 120.36, "duration": 5.439, "text": "= 3 remember we're solving for x so"}, {"start": 123.079, "duration": 5.441, "text": "let's divide both sides by two so divide"}, {"start": 125.799, "duration": 5.641, "text": "this by two and divide here by two this"}, {"start": 128.52, "duration": 6.16, "text": "cancel this and here we get the solution"}, {"start": 131.44, "duration": 5.96, "text": "that x = 3 all"}, {"start": 134.68, "duration": 4.839, "text": "2 thank you for watching please kindly"}, {"start": 137.4, "duration": 6.199, "text": "subscribe to my channel also like"}, {"start": 139.519, "duration": 4.08, "text": "comment and share bye-bye"}], "title": "Exponential equation |", "description": "", "lengthSeconds": 142, "width": 640, "height": 360}
{"video_id": "ONbqDfJrqWY", "transcript": [{"start": 0.52, "duration": 4.4, "text": "hello good day viewers welcome back once"}, {"start": 2.32, "duration": 5.36, "text": "again today we're going to provide"}, {"start": 4.92, "duration": 7.2, "text": "solution to the following given"}, {"start": 7.68, "duration": 10.48, "text": "exponential equation we have 8^ x + 8^ x"}, {"start": 12.12, "duration": 9.079, "text": "+ 8^ x / 4 x + 4 x this is equal to"}, {"start": 18.16, "duration": 5.32, "text": "12 now let us recall something very"}, {"start": 21.199, "duration": 4.84, "text": "important and very nice you know when we"}, {"start": 23.48, "duration": 4.799, "text": "suming three variables that is same"}, {"start": 26.039, "duration": 6.321, "text": "variables this will be same as three"}, {"start": 28.279, "duration": 8.561, "text": "multip by that Vari right or when we"}, {"start": 32.36, "duration": 9.719, "text": "have b + B this will be same as 2 * B so"}, {"start": 36.84, "duration": 10.52, "text": "from here we have 8 x + 8 x + 8 x that"}, {"start": 42.079, "duration": 8.721, "text": "would be same as 3 * 8^ of X then"}, {"start": 47.36, "duration": 7.879, "text": "divided by here that will be 2 * 4 to"}, {"start": 50.8, "duration": 7.599, "text": "the^ X then this is equal to 12 from"}, {"start": 55.239, "duration": 5.201, "text": "here we have 3 multi by 8 is same as 2"}, {"start": 58.399, "duration": 9.121, "text": "cubed right"}, {"start": 60.44, "duration": 7.84, "text": "that is 2 cubed then of x / 2 * 4 is as"}, {"start": 67.52, "duration": 4.08, "text": "2"}, {"start": 68.28, "duration": 6.36, "text": "s x then this is equal to"}, {"start": 71.6, "duration": 7.96, "text": "12 you know from here with this property"}, {"start": 74.64, "duration": 8.479, "text": "a^ b c this is same as a b * C so this"}, {"start": 79.56, "duration": 11.36, "text": "will become 3 * 2^ 3"}, {"start": 83.119, "duration": 10.561, "text": "x / 2 * 2^ 2x this is equal to 12"}, {"start": 90.92, "duration": 7.32, "text": "now take a note the left hand side we"}, {"start": 93.68, "duration": 7.0, "text": "can write it as 3 all 2 then multiply by"}, {"start": 98.24, "duration": 7.479, "text": "now let us recall another interesting"}, {"start": 100.68, "duration": 8.56, "text": "property when we have a^ B / a^ C this"}, {"start": 105.719, "duration": 6.561, "text": "is same as a the^ of B minus C since we"}, {"start": 109.24, "duration": 7.76, "text": "have the same bases right so here this"}, {"start": 112.28, "duration": 8.28, "text": "will be same as 2 to the^ of 3x -"}, {"start": 117.0, "duration": 10.119, "text": "2x then this is equal to 12"}, {"start": 120.56, "duration": 10.0, "text": "from here we have 3 2 then multi 2^ of X"}, {"start": 127.119, "duration": 5.681, "text": "this is equal to 12 right so from here"}, {"start": 130.56, "duration": 6.44, "text": "let's go ahead and multiply both sides"}, {"start": 132.8, "duration": 8.12, "text": "by two so here multiply by two and here"}, {"start": 137.0, "duration": 7.599, "text": "MTI two this two and this two so here we"}, {"start": 140.92, "duration": 6.959, "text": "have 3 * 2 to the^ X this will be equal"}, {"start": 144.599, "duration": 7.241, "text": "to that will give us 24 right so from"}, {"start": 147.879, "duration": 3.961, "text": "here divide both side by three"}, {"start": 152.92, "duration": 6.84, "text": "so from here we get 2 to^ x this is"}, {"start": 155.44, "duration": 6.6, "text": "equal to 24 3 will give us 8 right so"}, {"start": 159.76, "duration": 5.92, "text": "from here we have 2 to the^ X this is"}, {"start": 162.04, "duration": 5.919, "text": "equal to 8 is the same as 2^ 3 since we"}, {"start": 165.68, "duration": 5.04, "text": "have equal bases here this implies that"}, {"start": 167.959, "duration": 5.28, "text": "we can equate the power here we have X"}, {"start": 170.72, "duration": 4.599, "text": "is equal to 3 and this is the right"}, {"start": 173.239, "duration": 4.36, "text": "answer thank you for watching please"}, {"start": 175.319, "duration": 6.241, "text": "kindly subscribe to my channel also like"}, {"start": 177.599, "duration": 3.961, "text": "comment and share bye-bye"}], "title": "Exponential equation.", "description": "", "lengthSeconds": 180, "width": 640, "height": 360}
{"video_id": "VY3RPCv16NU", "transcript": [{"start": 0.399, "duration": 5.521, "text": "hello welcome back once again to my"}, {"start": 2.56, "duration": 5.68, "text": "YouTube Channel please kindly support us"}, {"start": 5.92, "duration": 5.48, "text": "by subscribing to our"}, {"start": 8.24, "duration": 6.68, "text": "Channel please and please subscribe to"}, {"start": 11.4, "duration": 7.28, "text": "this Channel and share our videos today"}, {"start": 14.92, "duration": 8.439, "text": "we have this exponential equation"}, {"start": 18.68, "duration": 7.72, "text": "problem here we have 6 ^ of x = 13 then"}, {"start": 23.359, "duration": 8.281, "text": "we are asked to find the value of x"}, {"start": 26.4, "duration": 6.36, "text": "right so here we have 6 of x"}, {"start": 31.64, "duration": 4.4, "text": "is equal to"}, {"start": 32.76, "duration": 5.6, "text": "13 so now we're going to make use of the"}, {"start": 36.04, "duration": 5.88, "text": "natural logarithm right so I'm going to"}, {"start": 38.36, "duration": 10.08, "text": "Ln both sides of this equation so right"}, {"start": 41.92, "duration": 11.2, "text": "over here I have the Ln of 6 of X is"}, {"start": 48.44, "duration": 7.48, "text": "equal to the Ln of 13 right so we know"}, {"start": 53.12, "duration": 5.32, "text": "this logarithmic property the power of"}, {"start": 55.92, "duration": 4.56, "text": "this C is going to come over to this"}, {"start": 58.44, "duration": 5.359, "text": "place and multiply the entire"}, {"start": 60.48, "duration": 10.12, "text": "right so here we have x"}, {"start": 63.799, "duration": 8.521, "text": "* the Ln of 6 is equal to the Ln of 13"}, {"start": 70.6, "duration": 2.8, "text": "so here we're solving for x so we're"}, {"start": 72.32, "duration": 3.32, "text": "going to divide both side by the"}, {"start": 73.4, "duration": 5.48, "text": "coefficient of x and the coefficient of"}, {"start": 75.64, "duration": 5.56, "text": "x here is Ln of 6 right so here we have"}, {"start": 78.88, "duration": 6.96, "text": "to divide by Ln of"}, {"start": 81.2, "duration": 7.72, "text": "6 and right over here by Ln of six so"}, {"start": 85.84, "duration": 9.44, "text": "this cancel this so we can see now that"}, {"start": 88.92, "duration": 10.479, "text": "X is equal to the Ln of 13 all/ by the"}, {"start": 95.28, "duration": 7.68, "text": "Ln of 6 so this is approximately equal"}, {"start": 99.399, "duration": 6.32, "text": "to so this is approximately equal"}, {"start": 102.96, "duration": 5.839, "text": "to"}, {"start": 105.719, "duration": 10.241, "text": "1.4 3"}, {"start": 108.799, "duration": 10.561, "text": "1 5 3 right okay now let us cross check"}, {"start": 115.96, "duration": 4.439, "text": "and see if this can satisfy the above"}, {"start": 119.36, "duration": 5.84, "text": "equation"}, {"start": 120.399, "duration": 4.801, "text": "so the equation here is so let us CR"}, {"start": 125.399, "duration": 6.721, "text": "check so the equation here is 6 of x"}, {"start": 129.16, "duration": 5.159, "text": "equal to 13 so now let us insert the"}, {"start": 132.12, "duration": 7.24, "text": "value of x here the approximate value of"}, {"start": 134.319, "duration": 7.881, "text": "x so here we have 6 rais power of 1."}, {"start": 139.36, "duration": 6.92, "text": "4315 so we can actually approximate it"}, {"start": 142.2, "duration": 8.28, "text": "to four decimal places so this is"}, {"start": 146.28, "duration": 8.319, "text": "approximately equal to 12"}, {"start": 150.48, "duration": 7.6, "text": "9 9 9 right so we can see that this is"}, {"start": 154.599, "duration": 6.881, "text": "approximately equal to what 13 thank you"}, {"start": 158.08, "duration": 8.12, "text": "for watching please do well to subscribe"}, {"start": 161.48, "duration": 7.72, "text": "to my YouTube channel like share and"}, {"start": 166.2, "duration": 3.0, "text": "comment"}], "title": "An exponential equation problem", "description": "", "lengthSeconds": 167, "width": 636, "height": 360}
{"video_id": "A7jawjW7pqI", "transcript": [{"start": 0.799, "duration": 5.6, "text": "hello welcome back once again today we"}, {"start": 2.84, "duration": 5.16, "text": "have this interesting equation to solve"}, {"start": 6.399, "duration": 3.761, "text": "and we're going to solve for the real"}, {"start": 8.0, "duration": 4.48, "text": "values of X that will satisfy this"}, {"start": 10.16, "duration": 4.399, "text": "equation so let's get"}, {"start": 12.48, "duration": 4.36, "text": "started we're going to start with the"}, {"start": 14.559, "duration": 6.201, "text": "substitution we're going to let the"}, {"start": 16.84, "duration": 9.279, "text": "square < TK of x = y and when you square"}, {"start": 20.76, "duration": 9.519, "text": "both side we get X is = y^ 2 right so"}, {"start": 26.119, "duration": 7.681, "text": "here we have X which is equal to y^ 2"}, {"start": 30.279, "duration": 6.841, "text": "multi by we have root of x which isal to"}, {"start": 33.8, "duration": 6.4, "text": "Y this will be equal to here we have X"}, {"start": 37.12, "duration": 6.8, "text": "which is equal to Y plus here we have"}, {"start": 40.2, "duration": 6.76, "text": "this I mean X here is equal to y^2 so"}, {"start": 43.92, "duration": 6.0, "text": "sorry for that right yes okay so here we"}, {"start": 46.96, "duration": 6.919, "text": "have plusun X ="}, {"start": 49.92, "duration": 6.639, "text": "Y so from here Y 2 * y will give us y"}, {"start": 53.879, "duration": 6.32, "text": "Cub then - y^"}, {"start": 56.559, "duration": 6.48, "text": "2 then - Y and the whole of this is"}, {"start": 60.199, "duration": 6.321, "text": "equal 0 so we can see from the left hand"}, {"start": 63.039, "duration": 10.4, "text": "side Y is common so Factor it out we get"}, {"start": 66.52, "duration": 11.32, "text": "y^ 2 - y - 1 this is equal Z so from"}, {"start": 73.439, "duration": 11.761, "text": "here this implies that either Y is = 0"}, {"start": 77.84, "duration": 9.0, "text": "or y^ 2 - y - 1 is = 0 so this is a"}, {"start": 85.2, "duration": 4.08, "text": "quadratic equation we're going to solve"}, {"start": 86.84, "duration": 5.48, "text": "it using the general quadratic formula"}, {"start": 89.28, "duration": 6.36, "text": "and when we use that b here will be 1"}, {"start": 92.32, "duration": 10.0, "text": "plus or minus the root of B 2 is 1 which"}, {"start": 95.64, "duration": 9.159, "text": "is 1 - 4 a 4 A is4 1 - - 4 is 1 + 5 1 +"}, {"start": 102.32, "duration": 5.399, "text": "5 I mean and that is 5 and we're very"}, {"start": 104.799, "duration": 4.841, "text": "familiar with the solution to quadratic"}, {"start": 107.719, "duration": 6.4, "text": "equations right so when you solve this"}, {"start": 109.64, "duration": 6.32, "text": "you're going to get 1 + orus otk 5 / 2"}, {"start": 114.119, "duration": 6.32, "text": "now"}, {"start": 115.96, "duration": 9.72, "text": "recall our x value is equal to y^2 right"}, {"start": 120.439, "duration": 10.641, "text": "so here we have when Y is = 0 x will be"}, {"start": 125.68, "duration": 9.68, "text": "= to 0 2 which is equal Z so when Y is ="}, {"start": 131.08, "duration": 11.08, "text": "to the positive 1 +un 5 oh the coding"}, {"start": 135.36, "duration": 10.08, "text": "ratio X will be equal to 1 +un 5 / 2 2"}, {"start": 142.16, "duration": 10.56, "text": "and when you square this you get 3 + <"}, {"start": 145.44, "duration": 15.68, "text": "TK 5 / 2 again we have when y = 1 - < TK"}, {"start": 152.72, "duration": 11.4, "text": "5 / 2 x will be = to 1 - root5 / 2 2 and"}, {"start": 161.12, "duration": 7.08, "text": "when you square this you get 3 - < TK 5"}, {"start": 164.12, "duration": 7.399, "text": "/ 2 but unfortunately this root here is"}, {"start": 168.2, "duration": 5.92, "text": "an exous root because it does not"}, {"start": 171.519, "duration": 5.08, "text": "satisfy the equation so therefore we"}, {"start": 174.12, "duration": 6.759, "text": "have only two real solutions for X we"}, {"start": 176.599, "duration": 10.72, "text": "have xal to0 and we also have X is = to"}, {"start": 180.879, "duration": 8.761, "text": "3 + < TK 5 / 2 thank you for watching"}, {"start": 187.319, "duration": 7.64, "text": "please kindly subscribe to my channel"}, {"start": 189.64, "duration": 5.319, "text": "also like comment and share bye-bye"}], "title": "Interesting radical equation | Can you solve ?", "description": "", "lengthSeconds": 193, "width": 428, "height": 240}
{"video_id": "UFWgGFT1uVo", "transcript": [{"start": 0.96, "duration": 3.919, "text": "hello welcome back once again today"}, {"start": 3.6, "duration": 5.079, "text": "we're going to"}, {"start": 4.879, "duration": 5.88, "text": "evaluate the natural log of a negative"}, {"start": 8.679, "duration": 4.96, "text": "number"}, {"start": 10.759, "duration": 6.44, "text": "right if I could recall vividly when I"}, {"start": 13.639, "duration": 7.8, "text": "was in secondary school that I think in"}, {"start": 17.199, "duration": 7.521, "text": "grade 8 and nine right so we're thinking"}, {"start": 21.439, "duration": 6.08, "text": "that this is impossible right well this"}, {"start": 24.72, "duration": 6.12, "text": "is impossible in the real world but very"}, {"start": 27.519, "duration": 6.161, "text": "possible in the complex word right"}, {"start": 30.84, "duration": 5.28, "text": "okay we're going to evaluate the natural"}, {"start": 33.68, "duration": 6.399, "text": "log of"}, {"start": 36.12, "duration": 7.16, "text": "-1 and how are we going to do"}, {"start": 40.079, "duration": 7.201, "text": "this first of all we need to recall this"}, {"start": 43.28, "duration": 9.32, "text": "wonderful Yow identity we"}, {"start": 47.28, "duration": 5.32, "text": "know that E rais power of I"}, {"start": 53.399, "duration": 6.84, "text": "Pi +"}, {"start": 55.559, "duration": 6.041, "text": "1 is equal to Z right if you subtract"}, {"start": 60.239, "duration": 3.96, "text": "one from both"}, {"start": 61.6, "duration": 6.24, "text": "sides we'll find out"}, {"start": 64.199, "duration": 7.361, "text": "that e of I"}, {"start": 67.84, "duration": 7.12, "text": "Pi is equal to"}, {"start": 71.56, "duration": 6.32, "text": "-1 right"}, {"start": 74.96, "duration": 8.4, "text": "okay so the natural log"}, {"start": 77.88, "duration": 9.68, "text": "of1 which we have here the natural log"}, {"start": 83.36, "duration": 8.16, "text": "of1 is going to be equal to the natural"}, {"start": 87.56, "duration": 8.28, "text": "log of e of I"}, {"start": 91.52, "duration": 8.0, "text": "Pi since we know that E power of I Pi is"}, {"start": 95.84, "duration": 6.559, "text": "equal to1 from here right so instead of"}, {"start": 99.52, "duration": 6.8, "text": "writing this netive 1 so we write e to"}, {"start": 102.399, "duration": 6.08, "text": "the power of I Pi now let us recall this"}, {"start": 106.32, "duration": 6.2, "text": "property of"}, {"start": 108.479, "duration": 4.041, "text": "logarithm so we know"}, {"start": 112.56, "duration": 7.839, "text": "that the natural log of a to the power"}, {"start": 116.56, "duration": 8.68, "text": "of B is equal to"}, {"start": 120.399, "duration": 7.2, "text": "B multip by the natural log of a right"}, {"start": 125.24, "duration": 6.88, "text": "so with this property"}, {"start": 127.599, "duration": 6.801, "text": "here so we're going to have from"}, {"start": 132.12, "duration": 5.28, "text": "here I"}, {"start": 134.4, "duration": 7.52, "text": "Pi multiply by the natural"}, {"start": 137.4, "duration": 8.04, "text": "log of e and we know the natural log of"}, {"start": 141.92, "duration": 5.16, "text": "e is equal to 1 right so this will be"}, {"start": 145.44, "duration": 4.92, "text": "equal to I"}, {"start": 147.08, "duration": 7.519, "text": "Pi multi by 1"}, {"start": 150.36, "duration": 7.72, "text": "so which is equal to I pi and this is"}, {"start": 154.599, "duration": 3.481, "text": "the right answer"}], "title": "Is that possible \ud83d\ude46  ???", "description": "", "lengthSeconds": 161, "width": 640, "height": 358}
{"video_id": "jYNpjigNdKM", "transcript": [{"start": 0.48, "duration": 2.8, "text": "good day viewers welcome back to my"}, {"start": 1.88, "duration": 6.44, "text": "Channel today we have the following"}, {"start": 3.28, "duration": 14.519, "text": "addressing problem A + B = 4 a 2 + b\u00b2 is"}, {"start": 8.32, "duration": 9.479, "text": "= 10 then a 4 + B 4 = dash wow okay now"}, {"start": 18.88, "duration": 11.68, "text": "solution let us recall that a 2 + b 2 is"}, {"start": 23.359, "duration": 9.001, "text": "equal to a + b 2 - 2 a Okay so"}, {"start": 30.56, "duration": 6.319, "text": "here I'm going to rewrite this as"}, {"start": 32.36, "duration": 12.16, "text": "follows a 2 + b 2 so to be written as a"}, {"start": 36.879, "duration": 10.52, "text": "+ b^ 2 - 2 a b = 10 my major priority"}, {"start": 44.52, "duration": 7.199, "text": "here is to find the value of a * B first"}, {"start": 47.399, "duration": 10.361, "text": "okay so I'm going to replace a + b with"}, {"start": 51.719, "duration": 13.041, "text": "four here so I have 4 S - 2 a b = 10 so"}, {"start": 57.76, "duration": 10.28, "text": "4 squ is 16 - 2 a = 10 you know 16 - 10"}, {"start": 64.76, "duration": 6.88, "text": "and 16 is crossing over so we get 2 a is"}, {"start": 68.04, "duration": 5.079, "text": "equal to 6 so from here divide both side"}, {"start": 71.64, "duration": 8.64, "text": "by"}, {"start": 73.119, "duration": 9.96, "text": "two so a * b equal 3 okay now we can"}, {"start": 80.28, "duration": 10.28, "text": "easily get the value of a power"}, {"start": 83.079, "duration": 12.801, "text": "4 then plus b 4 so we writing as a 2 2 +"}, {"start": 90.56, "duration": 15.839, "text": "b 2 squ okay so this is equal to using"}, {"start": 95.88, "duration": 15.239, "text": "this so a 2 + b 2 2 - 2 a 2 b 2 so this"}, {"start": 106.399, "duration": 9.72, "text": "is equal to you know a 2 + b 2 is equal"}, {"start": 111.119, "duration": 9.401, "text": "to 10 so we have here 10^ 2 - 2 remember"}, {"start": 116.119, "duration": 10.68, "text": "a 2 + b 2 is the same as a a * B bracket"}, {"start": 120.52, "duration": 9.719, "text": "squ so this is equal to 10 2 - 2 Open"}, {"start": 126.799, "duration": 5.121, "text": "Bracket you know 8 * B is 3 S and is"}, {"start": 130.239, "duration": 8.801, "text": "equal"}, {"start": 131.92, "duration": 12.44, "text": "to 100 - 2 * 3 S is 9 so this is equal"}, {"start": 139.04, "duration": 7.919, "text": "to this is equal to 100 - 2 * 9"}, {"start": 144.36, "duration": 4.4, "text": "18 and this is equal"}, {"start": 146.959, "duration": 3.36, "text": "to"}, {"start": 148.76, "duration": 3.199, "text": "82"}, {"start": 150.319, "duration": 3.321, "text": "thank you for viewing and please make"}, {"start": 151.959, "duration": 5.241, "text": "sure you subscribe to my YouTube channel"}, {"start": 153.64, "duration": 3.56, "text": "for more interesting videos"}], "title": "Can you do this without finding the values of a and b ? Watch out. Algebraic Manipulation is best.", "description": "", "lengthSeconds": 155, "width": 636, "height": 360}
{"video_id": "5rDv5H3qczc", "transcript": [{"start": 1.24, "duration": 4.519, "text": "hello welcome back once again today"}, {"start": 3.639, "duration": 6.681, "text": "we're going to find the remainder when"}, {"start": 5.759, "duration": 8.76, "text": "29 race report of 999"}, {"start": 10.32, "duration": 7.16, "text": "999999 is divided by 30 then in modular"}, {"start": 14.519, "duration": 5.201, "text": "form that is a modular arithmetic form"}, {"start": 17.48, "duration": 4.799, "text": "this is how it should be"}, {"start": 19.72, "duration": 5.559, "text": "written now let's get started from the"}, {"start": 22.279, "duration": 3.0, "text": "page here"}, {"start": 25.32, "duration": 10.16, "text": "29 is Congress to -1 mode 30 right you"}, {"start": 32.559, "duration": 10.881, "text": "know why when you add the modul we get"}, {"start": 35.48, "duration": 13.44, "text": "-1 + 30 then mod 30 from here1 + 30 will"}, {"start": 43.44, "duration": 9.16, "text": "give us 29 mod 30 right so which"}, {"start": 48.92, "duration": 7.439, "text": "actually mean that 29 is similar to -1"}, {"start": 52.6, "duration": 6.119, "text": "mod 30 therefore the power of this 29"}, {"start": 56.359, "duration": 9.481, "text": "which is 999"}, {"start": 58.719, "duration": 8.641, "text": "999 9 999 will be similar or congr to -1"}, {"start": 65.84, "duration": 4.8, "text": "999"}, {"start": 67.36, "duration": 8.119, "text": "999 999"}, {"start": 70.64, "duration": 8.519, "text": "mode 30 right okay so from here this"}, {"start": 75.479, "duration": 7.401, "text": "power here must be odd right you know"}, {"start": 79.159, "duration": 7.121, "text": "why the last digit here is odd therefore"}, {"start": 82.88, "duration": 6.52, "text": "this power must be odd And1 to the power"}, {"start": 86.28, "duration": 6.72, "text": "of an odd number will give US1 then we"}, {"start": 89.4, "duration": 5.48, "text": "have here mode 30 from here you know"}, {"start": 93.0, "duration": 5.64, "text": "this simplifies so let's put this"}, {"start": 94.88, "duration": 7.0, "text": "similar so this simplifies to what 29"}, {"start": 98.64, "duration": 3.24, "text": "right so"}, {"start": 104.759, "duration": 10.32, "text": "therefore when 29 to the^ of 999 999"}, {"start": 111.439, "duration": 3.64, "text": "999 is"}, {"start": 115.24, "duration": 7.72, "text": "divided by"}, {"start": 118.28, "duration": 6.92, "text": "30 = 29 amazing and that's the right"}, {"start": 122.96, "duration": 4.799, "text": "answer thank you for watching please"}, {"start": 125.2, "duration": 6.72, "text": "kindly subscribe to my channel also like"}, {"start": 127.759, "duration": 4.161, "text": "comment and share bye-bye"}], "title": "Application of Modular Arithmetic.", "description": "#maths #mathematics #olympics", "lengthSeconds": 130, "width": 572, "height": 360}
{"video_id": "p_t6uNvK10Y", "transcript": [{"start": 0.52, "duration": 4.92, "text": "hello welcome back once again today"}, {"start": 3.399, "duration": 4.641, "text": "we're going to solve this radical"}, {"start": 5.44, "duration": 6.56, "text": "equation here and we'll be solving for"}, {"start": 8.04, "duration": 8.44, "text": "the integral values of X now let's get"}, {"start": 12.0, "duration": 8.519, "text": "started so here we have square root of"}, {"start": 16.48, "duration": 6.92, "text": "rootx this is equal to X remember"}, {"start": 20.519, "duration": 5.6, "text": "anytime root sign is involved in"}, {"start": 23.4, "duration": 6.16, "text": "equation what usually come to the mind"}, {"start": 26.119, "duration": 5.48, "text": "is to square the equation right so here"}, {"start": 29.56, "duration": 6.48, "text": "let's go ahead and square both sides so"}, {"start": 31.599, "duration": 9.64, "text": "here we have the square root ofun X"}, {"start": 36.04, "duration": 7.8, "text": "squ so this will be equal to x 2 now"}, {"start": 41.239, "duration": 5.441, "text": "here this Square here we cancel the"}, {"start": 43.84, "duration": 8.76, "text": "first root sign here so here we have"}, {"start": 46.68, "duration": 8.08, "text": "root X is = to X2 right okay in our next"}, {"start": 52.6, "duration": 5.84, "text": "step let's just go ahead and square both"}, {"start": 54.76, "duration": 9.52, "text": "sides one more time so we have root of x"}, {"start": 58.44, "duration": 8.64, "text": "s so this is = to X2 2 so from here"}, {"start": 64.28, "duration": 5.04, "text": "Square cancels the square root sign so"}, {"start": 67.08, "duration": 6.32, "text": "from the right hand side so here we have"}, {"start": 69.32, "duration": 8.64, "text": "just X here then this is equal to x^ 2 *"}, {"start": 73.4, "duration": 5.719, "text": "2 here is 4 now let us make the right"}, {"start": 77.96, "duration": 2.519, "text": "hand side left hand side and left hand"}, {"start": 79.119, "duration": 3.401, "text": "side the right hand side and that does"}, {"start": 80.479, "duration": 7.121, "text": "not affect the equation right so here we"}, {"start": 82.52, "duration": 8.4, "text": "have x^ 4 is = to X now here we get x^ 4"}, {"start": 87.6, "duration": 6.04, "text": "- x this is equal to 0"}, {"start": 90.92, "duration": 4.96, "text": "now X is common here so factor out X so"}, {"start": 93.64, "duration": 5.6, "text": "we get X Cub"}, {"start": 95.88, "duration": 5.559, "text": "-1 this is equal to 0 so if you have"}, {"start": 99.24, "duration": 7.44, "text": "this product to be equal to zero either"}, {"start": 101.439, "duration": 7.32, "text": "this is equal to Z or this is equal to 0"}, {"start": 106.68, "duration": 4.36, "text": "so here we have our first Solution"}, {"start": 108.759, "duration": 8.0, "text": "that's X is = to"}, {"start": 111.04, "duration": 9.039, "text": "0 and here x cubed - 1 is = to 0 we have"}, {"start": 116.759, "duration": 5.761, "text": "X Cub minus let's write this as 1 cub"}, {"start": 120.079, "duration": 4.4, "text": "this is equal to Z making use of the"}, {"start": 122.52, "duration": 5.44, "text": "difference of two cubes"}, {"start": 124.479, "duration": 9.961, "text": "formula a cubus b Cub is same as a minus"}, {"start": 127.96, "duration": 12.04, "text": "B into bracket a 2 + a b + b 2 so here"}, {"start": 134.44, "duration": 9.6, "text": "we have X - 1 into braet x 2 + 1 * X is"}, {"start": 140.0, "duration": 6.0, "text": "x + 1 2 which is 1 this is equal to 0"}, {"start": 144.04, "duration": 5.559, "text": "here we have another product so either"}, {"start": 146.0, "duration": 6.84, "text": "this is equal to0 or this is equal to0"}, {"start": 149.599, "duration": 5.72, "text": "so we have X - 1 = 0 add one to both"}, {"start": 152.84, "duration": 5.44, "text": "side we arrive at another solution that"}, {"start": 155.319, "duration": 5.161, "text": "X is = 1 from here we have this"}, {"start": 158.28, "duration": 5.72, "text": "quadratic expression and we're equating"}, {"start": 160.48, "duration": 5.08, "text": "it to zero right so here checking the"}, {"start": 164.0, "duration": 3.84, "text": "discriminant we find out that"}, {"start": 165.56, "duration": 5.92, "text": "discriminant is less than zero and"}, {"start": 167.84, "duration": 7.8, "text": "therefore no real solution exist here"}, {"start": 171.48, "duration": 7.24, "text": "check it out X here would be B which is"}, {"start": 175.64, "duration": 7.84, "text": "1 plus or minus Ro of b\u00b2 which is 1 - 4"}, {"start": 178.72, "duration": 6.4, "text": "a c which is3 that is Iota root of 3 / 2"}, {"start": 183.48, "duration": 4.0, "text": "so we can see that this is a complex"}, {"start": 185.12, "duration": 5.399, "text": "solution therefore we don't need the"}, {"start": 187.48, "duration": 8.44, "text": "complex value for x so here we have only"}, {"start": 190.519, "duration": 8.481, "text": "two integral solution X = to 1 and X is"}, {"start": 195.92, "duration": 5.76, "text": "equal to Z thank you for watching please"}, {"start": 199.0, "duration": 6.92, "text": "kindly subscribe to my channel also like"}, {"start": 201.68, "duration": 4.24, "text": "comment and share bye-bye"}], "title": "Interesting radical equation | Can you solve ?", "description": "", "lengthSeconds": 204, "width": 640, "height": 360}
{"video_id": "WhKokHM_lks", "transcript": [{"start": 0.84, "duration": 4.439, "text": "hello good viewers welcome back once"}, {"start": 2.52, "duration": 7.76, "text": "again today we have this exponential"}, {"start": 5.279, "duration": 8.001, "text": "equation problem here we have 64 m + 64"}, {"start": 10.28, "duration": 6.36, "text": "M this is equal to 1 here we'll be"}, {"start": 13.28, "duration": 9.68, "text": "solving for the value of M now let's get"}, {"start": 16.64, "duration": 9.44, "text": "started we have 64 m + 64 ^ M this is"}, {"start": 22.96, "duration": 5.559, "text": "equal to 1 from the left hand side 64"}, {"start": 26.08, "duration": 5.96, "text": "the^ m is common so let's Factor it out"}, {"start": 28.519, "duration": 8.281, "text": "so here we have 64 to of M into bracket"}, {"start": 32.04, "duration": 7.839, "text": "this this is 1 plus this ID this is 1"}, {"start": 36.8, "duration": 8.84, "text": "then this is equal to 1 so from here we"}, {"start": 39.879, "duration": 8.68, "text": "have 64 to the^ M * 1 + 1 is 2 then this"}, {"start": 45.64, "duration": 4.64, "text": "isal to 1 now let's canly divide both"}, {"start": 48.559, "duration": 4.761, "text": "side by"}, {"start": 50.28, "duration": 7.919, "text": "two this will cancel this and here we"}, {"start": 53.32, "duration": 7.079, "text": "have 64 the^ M this is equal to 1 / 2"}, {"start": 58.199, "duration": 6.081, "text": "remember the power of this two here is 1"}, {"start": 60.399, "duration": 6.841, "text": "right now let's take note of this 1 / a^"}, {"start": 64.28, "duration": 5.92, "text": "n is same as a^"}, {"start": 67.24, "duration": 5.48, "text": "n so the the right hand side we're going"}, {"start": 70.2, "duration": 6.16, "text": "to write it as 2 to the^"}, {"start": 72.72, "duration": 7.12, "text": "of-1 and again do not forget that 64 can"}, {"start": 76.36, "duration": 10.56, "text": "be expressed in index form with base 2"}, {"start": 79.84, "duration": 8.84, "text": "so here 64 is 2 to^ 6 right so 2^ 6^ M"}, {"start": 86.92, "duration": 5.92, "text": "this is equal to"}, {"start": 88.68, "duration": 6.96, "text": "2^1 so so here we get 2^ 6 * m here is 6"}, {"start": 92.84, "duration": 5.959, "text": "M this is ="}, {"start": 95.64, "duration": 5.56, "text": "2^1 remember we have equal Bas right so"}, {"start": 98.799, "duration": 5.561, "text": "let us equate the powers so from here"}, {"start": 101.2, "duration": 8.68, "text": "this implies that 6 m is ="}, {"start": 104.36, "duration": 10.439, "text": "to1 divide by 6 6 cancel 6 and here we"}, {"start": 109.88, "duration": 7.44, "text": "get m is = -1 / 6 and this is the right"}, {"start": 114.799, "duration": 5.24, "text": "answer thank you for watching please"}, {"start": 117.32, "duration": 6.759, "text": "kindly subscribe to my channel also like"}, {"start": 120.039, "duration": 4.04, "text": "comment and share bye-bye"}], "title": "How to solve simple exponential equation", "description": "", "lengthSeconds": 123, "width": 640, "height": 360}
{"video_id": "9fgnSeF19dU", "transcript": [{"start": 0.88, "duration": 6.32, "text": "hello welcome back once again today we"}, {"start": 3.36, "duration": 6.84, "text": "have this exponential equation and we're"}, {"start": 7.2, "duration": 5.92, "text": "going to solve for the value of x that"}, {"start": 10.2, "duration": 6.159, "text": "will satisfy this equation so from the"}, {"start": 13.12, "duration": 6.8, "text": "left side we have 64 to the power of X"}, {"start": 16.359, "duration": 8.68, "text": "we know 64 can be expressed in index"}, {"start": 19.92, "duration": 9.199, "text": "power of 8 as 8 to the^ of 2 right so"}, {"start": 25.039, "duration": 7.04, "text": "left hand side will become 8 to the^ 2"}, {"start": 29.119, "duration": 6.201, "text": "then ra to the power of X then this is"}, {"start": 32.079, "duration": 8.281, "text": "equal to we know square root"}, {"start": 35.32, "duration": 11.04, "text": "of 8 is equal to 8 to the^ of 1 / 2"}, {"start": 40.36, "duration": 8.16, "text": "right then the^ of 12 okay now from here"}, {"start": 46.36, "duration": 5.879, "text": "let's take note of this property we have"}, {"start": 48.52, "duration": 8.0, "text": "a the^ b a braet the^ c this is equal to"}, {"start": 52.239, "duration": 11.12, "text": "a^ B * C so from here this will become"}, {"start": 56.52, "duration": 9.36, "text": "8^ 2 * X and this is equal to 8^ of we"}, {"start": 63.359, "duration": 6.561, "text": "know 1 2 *"}, {"start": 65.88, "duration": 9.239, "text": "12 put this over one this will be 12 / 2"}, {"start": 69.92, "duration": 7.519, "text": "which is equal to 6 so here 1 2 * 12 is"}, {"start": 75.119, "duration": 4.281, "text": "6 so we can see from here we have the"}, {"start": 77.439, "duration": 4.881, "text": "same base from both sides so we're going"}, {"start": 79.4, "duration": 5.399, "text": "to equate the powers so this implies"}, {"start": 82.32, "duration": 6.72, "text": "from here that"}, {"start": 84.799, "duration": 6.801, "text": "2x is equal to 6 from here we're solving"}, {"start": 89.04, "duration": 5.439, "text": "for x let's divide by the coefficient of"}, {"start": 91.6, "duration": 5.92, "text": "x so divide this by two and divide here"}, {"start": 94.479, "duration": 4.68, "text": "by two so this two we cancel this two we"}, {"start": 97.52, "duration": 5.2, "text": "can see from the left hand side we are"}, {"start": 99.159, "duration": 6.441, "text": "left with X which is equal to 6 / 2"}, {"start": 102.72, "duration": 5.399, "text": "which is 3 and this is the right answer"}, {"start": 105.6, "duration": 5.04, "text": "to the problem thank you for watching"}, {"start": 108.119, "duration": 8.121, "text": "please kindly subscribe to my channel"}, {"start": 110.64, "duration": 5.6, "text": "also like comment and share bye-bye"}], "title": "Tricky exponential equation | can you solve ?", "description": "", "lengthSeconds": 115, "width": 640, "height": 360}
{"video_id": "KTRF8GcaEss", "transcript": [{"start": 0.199, "duration": 6.6, "text": "good day viewers today we having the"}, {"start": 2.959, "duration": 8.001, "text": "following Nigerian oad mathematical"}, {"start": 6.799, "duration": 9.601, "text": "problem X Cub + y Cub ="}, {"start": 10.96, "duration": 8.2, "text": "8,100 x + y = 30 then x\u00b2 + Y2 = that"}, {"start": 16.4, "duration": 6.28, "text": "okay now to solve out the values of X"}, {"start": 19.16, "duration": 5.72, "text": "and Y before Computing this expression"}, {"start": 22.68, "duration": 4.359, "text": "the value of this expression should not"}, {"start": 24.88, "duration": 5.639, "text": "be the right decision to be made here"}, {"start": 27.039, "duration": 6.241, "text": "okay so let us try to make some"}, {"start": 30.519, "duration": 5.72, "text": "manipulations or introduce some albic"}, {"start": 33.28, "duration": 4.32, "text": "manipulations approaches okay okay now"}, {"start": 36.239, "duration": 4.561, "text": "let us see how we're going to solve this"}, {"start": 37.6, "duration": 11.04, "text": "problem so I'm using the following ideas"}, {"start": 40.8, "duration": 16.279, "text": "that a cub + B Cub is equal to a + b Cub"}, {"start": 48.64, "duration": 13.879, "text": "- 3 A B A + B and again a 2 + b 2 is = a"}, {"start": 57.079, "duration": 7.72, "text": "+ b 2 - 2 a okay so you may ask the"}, {"start": 62.519, "duration": 4.361, "text": "reason why I introduced the following"}, {"start": 64.799, "duration": 4.201, "text": "properties I did this in order to"}, {"start": 66.88, "duration": 4.68, "text": "compute the value of a * B because I've"}, {"start": 69.0, "duration": 5.799, "text": "seen that what I'm about to evaluate the"}, {"start": 71.56, "duration": 5.8, "text": "value of a is needed okay so we only"}, {"start": 74.799, "duration": 5.041, "text": "have the value of x + y so we don't have"}, {"start": 77.36, "duration": 6.04, "text": "x * y so we need to compute the value of"}, {"start": 79.84, "duration": 6.52, "text": "x * y so let us try to write the left"}, {"start": 83.4, "duration": 6.6, "text": "hand side of this first equation here so"}, {"start": 86.36, "duration": 10.68, "text": "writing it we write as follows x + y"}, {"start": 90.0, "duration": 12.84, "text": "Cube - 3 * X Y Open Bracket x + y is ="}, {"start": 97.04, "duration": 12.039, "text": "to 8, 100 okay so now the value of x + y"}, {"start": 102.84, "duration": 12.279, "text": "is 30 right so we have 30 Cub - 3 you"}, {"start": 109.079, "duration": 8.68, "text": "know 3x y * you know x + y here is 30"}, {"start": 115.119, "duration": 6.481, "text": "okay so this is equal to"}, {"start": 117.759, "duration": 9.201, "text": "8,100 so here 30 cu cube is"}, {"start": 121.6, "duration": 8.8, "text": "27,000 so minus 3 * 90 3 * 30 here is 90"}, {"start": 126.96, "duration": 9.559, "text": "so we get X Y = to"}, {"start": 130.4, "duration": 9.52, "text": "8,100 so remember we have 90 x y is = to"}, {"start": 136.519, "duration": 7.041, "text": "27,000 -"}, {"start": 139.92, "duration": 8.48, "text": "8,100 okay so now subtracting this so we"}, {"start": 143.56, "duration": 7.16, "text": "get 90 x y is equal to you know"}, {"start": 148.4, "duration": 5.88, "text": "27,000 -"}, {"start": 150.72, "duration": 6.32, "text": "8,100 so this is giving us"}, {"start": 154.28, "duration": 5.8, "text": "18,900 okay so we'll be dividing both"}, {"start": 157.04, "duration": 11.279, "text": "side by 90 to get the value of x * y so"}, {"start": 160.08, "duration": 13.36, "text": "we get x * y = 1890 0 all over 90 so 0 0"}, {"start": 168.319, "duration": 9.801, "text": "so this is giving us 2 1 0 okay so wow"}, {"start": 173.44, "duration": 7.159, "text": "so we can get the value of X2 + y^ 2"}, {"start": 178.12, "duration": 3.64, "text": "which is the same as X"}, {"start": 180.599, "duration": 7.481, "text": "+"}, {"start": 181.76, "duration": 15.36, "text": "Y 2 - 2 * x * Y and this = to X + Y 2 is"}, {"start": 188.08, "duration": 13.439, "text": "30 2 - 2 * you know XY is 2 1 0 okay so"}, {"start": 197.12, "duration": 7.56, "text": "this is given us so we have here 30 s so"}, {"start": 201.519, "duration": 7.0, "text": "that is what so we get"}, {"start": 204.68, "duration": 7.32, "text": "900 minus here we have"}, {"start": 208.519, "duration": 5.161, "text": "420 okay okay so and this is exactly"}, {"start": 212.0, "duration": 4.56, "text": "equal to"}, {"start": 213.68, "duration": 4.559, "text": "480 and this is the right answer thank"}, {"start": 216.56, "duration": 3.599, "text": "you for viewing and please make sure you"}, {"start": 218.239, "duration": 5.241, "text": "subscribe to my YouTube channel for more"}, {"start": 220.159, "duration": 3.321, "text": "interesting videos"}], "title": "Nice Olympiad math problem. Do not solve for the values of x and y.", "description": "", "lengthSeconds": 222, "width": 636, "height": 360}
{"video_id": "NcHmWXdHRfg", "transcript": [{"start": 0.799, "duration": 5.121, "text": "hello welcome back once again today"}, {"start": 3.24, "duration": 5.24, "text": "we're going to solve this beautiful"}, {"start": 5.92, "duration": 4.839, "text": "equation with fractional"}, {"start": 8.48, "duration": 5.48, "text": "powers and here we'll be solving for"}, {"start": 10.759, "duration": 4.6, "text": "both real and complex values for X now"}, {"start": 13.96, "duration": 3.64, "text": "let's get"}, {"start": 15.359, "duration": 7.041, "text": "started okay now from the left hand side"}, {"start": 17.6, "duration": 7.56, "text": "we have X to the^ 3 / 2 this is equal to"}, {"start": 22.4, "duration": 5.84, "text": "5 the^ 3 /"}, {"start": 25.16, "duration": 5.76, "text": "2 when we say that x equal to 5 here"}, {"start": 28.24, "duration": 4.839, "text": "obviously this is a solution"}, {"start": 30.92, "duration": 5.92, "text": "but the question here is is that the"}, {"start": 33.079, "duration": 7.48, "text": "only solution the answer is no but this"}, {"start": 36.84, "duration": 6.879, "text": "happens to be the only real solution and"}, {"start": 40.559, "duration": 5.561, "text": "we have two more complex"}, {"start": 43.719, "duration": 5.121, "text": "Solutions okay now let's get into this"}, {"start": 46.12, "duration": 5.599, "text": "video now let's go ahead to let's go"}, {"start": 48.84, "duration": 7.0, "text": "ahead and square both sides so that we"}, {"start": 51.719, "duration": 9.32, "text": "can eliminate this two right over here"}, {"start": 55.84, "duration": 10.84, "text": "so we have here x^ 3 2 2"}, {"start": 61.039, "duration": 9.561, "text": "this will be equal to 5 3 2"}, {"start": 66.68, "duration": 7.72, "text": "2 so we can see from here 3 2 * 2 2"}, {"start": 70.6, "duration": 7.519, "text": "cancels 2 here 2 cancels 2 so we have X"}, {"start": 74.4, "duration": 7.32, "text": "the^ of 3 this is = to 5^"}, {"start": 78.119, "duration": 9.601, "text": "3 so subtract 5^ 3 from both side we get"}, {"start": 81.72, "duration": 7.2, "text": "x^ 3us 5^ 3 this is equal to Z from here"}, {"start": 87.72, "duration": 5.32, "text": "we're going to make use of the"}, {"start": 88.92, "duration": 9.559, "text": "difference of two cubes formula a cub -"}, {"start": 93.04, "duration": 9.2, "text": "B Cub this is equal to Aus B multip by a"}, {"start": 98.479, "duration": 8.721, "text": "2 + a b + b"}, {"start": 102.24, "duration": 11.12, "text": "2 so from here this will become x - 5"}, {"start": 107.2, "duration": 7.4, "text": "into braet X2 + 5 * X then + 5 2 which"}, {"start": 113.36, "duration": 4.0, "text": "is"}, {"start": 114.6, "duration": 5.36, "text": "25 and this will be equal to"}, {"start": 117.36, "duration": 4.52, "text": "zero so from here if this time this"}, {"start": 119.96, "duration": 5.36, "text": "equal to 0 so we have either this is"}, {"start": 121.88, "duration": 7.599, "text": "equal to 0 or this is equal to 0 so here"}, {"start": 125.32, "duration": 6.559, "text": "x - 5 is = 0 here we found the solution"}, {"start": 129.479, "duration": 9.161, "text": "that x = 5 which is the only real"}, {"start": 131.879, "duration": 9.241, "text": "solution and here we have X2 + 5x + 25"}, {"start": 138.64, "duration": 6.12, "text": "this is equal 0 now using the general"}, {"start": 141.12, "duration": 7.36, "text": "quadratic formula we get x = b will be5"}, {"start": 144.76, "duration": 9.28, "text": "plus or minus otk of b 2 is 5 S which is"}, {"start": 148.48, "duration": 8.92, "text": "25 - 4 * a * C 4 * 1 * 25 will give us"}, {"start": 154.04, "duration": 8.32, "text": "100 then /"}, {"start": 157.4, "duration": 12.16, "text": "2 so from here we get X is = -5 plus or"}, {"start": 162.36, "duration": 11.76, "text": "minus sare < TK of -75 right then / by 2"}, {"start": 169.56, "duration": 8.56, "text": "so from here x = to -5 plus or minus"}, {"start": 174.12, "duration": 7.32, "text": "already square root of1 is iot right and"}, {"start": 178.12, "duration": 6.36, "text": "do not forget that 75 is the same thing"}, {"start": 181.44, "duration": 11.0, "text": "as 25 *"}, {"start": 184.48, "duration": 13.119, "text": "3 then / 2 so here we get X = -5 + or"}, {"start": 192.44, "duration": 9.159, "text": "minus of 25 is 5 so we get 5 I root of 3"}, {"start": 197.599, "duration": 7.0, "text": "then / 2 and here we have the other two"}, {"start": 201.599, "duration": 5.2, "text": "compx solutions for X thank you for"}, {"start": 204.599, "duration": 4.761, "text": "watching please kindly subscribe to my"}, {"start": 206.799, "duration": 5.961, "text": "channel also like comment and share"}, {"start": 209.36, "duration": 3.4, "text": "bye-bye I"}], "title": "Interesting Olympiad Math | Can you find all solutions ?", "description": "", "lengthSeconds": 211, "width": 640, "height": 360}
{"video_id": "-M-kUCxdiA8", "transcript": [{"start": 1.04, "duration": 3.88, "text": "hello welcome back once"}, {"start": 2.76, "duration": 5.24, "text": "again today we're going to solve the"}, {"start": 4.92, "duration": 5.2, "text": "following given exponential"}, {"start": 8.0, "duration": 4.719, "text": "equation and here we be solving for the"}, {"start": 10.12, "duration": 5.04, "text": "value of x that will satisfy this"}, {"start": 12.719, "duration": 10.48, "text": "equation from the left hand side we have"}, {"start": 15.16, "duration": 11.439, "text": "5 to the power of x * 5^ x * 5^ X we"}, {"start": 23.199, "duration": 6.56, "text": "know this is equal to 10 now let us"}, {"start": 26.599, "duration": 9.8, "text": "recall this law of algebra we know a to"}, {"start": 29.759, "duration": 9.48, "text": "the of B * a power of C * a power of D"}, {"start": 36.399, "duration": 8.521, "text": "this will be equal to a the power of b +"}, {"start": 39.239, "duration": 7.681, "text": "C + D since we have the same basis right"}, {"start": 44.92, "duration": 4.6, "text": "so we can see from the left hand side we"}, {"start": 46.92, "duration": 9.279, "text": "can apply this law and that will become"}, {"start": 49.52, "duration": 11.48, "text": "5^ x + x + x and then this is equal to"}, {"start": 56.199, "duration": 8.04, "text": "10 from here we have 5 to the ^ 3x X"}, {"start": 61.0, "duration": 6.08, "text": "this is equal to 10 now we're going to"}, {"start": 64.239, "duration": 5.481, "text": "apply logarithm on both sides of this"}, {"start": 67.08, "duration": 7.76, "text": "equation so let's log both sides so here"}, {"start": 69.72, "duration": 9.52, "text": "we have log of 5^ 3x this will be equal"}, {"start": 74.84, "duration": 8.44, "text": "to log of 10 we know this property of"}, {"start": 79.24, "duration": 8.72, "text": "logarithm log of a to the^ B is equal to"}, {"start": 83.28, "duration": 7.12, "text": "B MTI the log of a so this power of 5"}, {"start": 87.96, "duration": 7.68, "text": "here which is 3x will come down here so"}, {"start": 90.4, "duration": 9.96, "text": "we get 3x * the log of 5 this will be"}, {"start": 95.64, "duration": 9.96, "text": "equal to log of we know 10 is same as 5"}, {"start": 100.36, "duration": 12.48, "text": "* 2 we know this Lo this logarithmic law"}, {"start": 105.6, "duration": 10.04, "text": "that log of a * B is equal to log of a +"}, {"start": 112.84, "duration": 4.639, "text": "log of B so we can see that this law"}, {"start": 115.64, "duration": 4.519, "text": "here we can apply it at the right hand"}, {"start": 117.479, "duration": 6.64, "text": "side so remember the left hand side is"}, {"start": 120.159, "duration": 11.08, "text": "still 3x * log of 5 then this will be"}, {"start": 124.119, "duration": 8.921, "text": "equal to log of 5 plus log of 2 since"}, {"start": 131.239, "duration": 4.561, "text": "you're soling for X let's divide both"}, {"start": 133.04, "duration": 5.4, "text": "side by the log of"}, {"start": 135.8, "duration": 5.92, "text": "5 by the log of"}, {"start": 138.44, "duration": 7.0, "text": "5 so this will cancel this and we're"}, {"start": 141.72, "duration": 10.56, "text": "left 3x isal to splitting this"}, {"start": 145.44, "duration": 9.96, "text": "denominator we get log of 5 / log of 5"}, {"start": 152.28, "duration": 5.36, "text": "+ log of 2"}, {"start": 155.4, "duration": 5.759, "text": "/ log of"}, {"start": 157.64, "duration": 7.08, "text": "5 we know this here we cancel to one"}, {"start": 161.159, "duration": 12.641, "text": "right because log of 5 log of 5 is 1 so"}, {"start": 164.72, "duration": 11.12, "text": "we have 3x is = 1 + log of 2 / log of 5"}, {"start": 173.8, "duration": 5.159, "text": "so we know this another"}, {"start": 175.84, "duration": 4.92, "text": "important logarithmic law we have that"}, {"start": 178.959, "duration": 6.481, "text": "the log of a"}, {"start": 180.76, "duration": 8.199, "text": "/ the log of B is equal to the log of a"}, {"start": 185.44, "duration": 5.96, "text": "to the base B so this law we're going to"}, {"start": 188.959, "duration": 5.241, "text": "apply it right over here so this will"}, {"start": 191.4, "duration": 6.72, "text": "become 3x is ="}, {"start": 194.2, "duration": 6.119, "text": "1+ so that will be log of 2 to the base"}, {"start": 198.12, "duration": 3.92, "text": "5 so let's divide both by the"}, {"start": 200.319, "duration": 4.801, "text": "coefficient of"}, {"start": 202.04, "duration": 6.279, "text": "x so this will cancel this so from here"}, {"start": 205.12, "duration": 8.0, "text": "we have our solution that X is equal to"}, {"start": 208.319, "duration": 12.241, "text": "split this denominator that is 1 / 3"}, {"start": 213.12, "duration": 9.839, "text": "plus log of 2 to the base 5 then / 3 and"}, {"start": 220.56, "duration": 4.56, "text": "this is our solution thank you for"}, {"start": 222.959, "duration": 5.56, "text": "watching please kindly subscribe to my"}, {"start": 225.12, "duration": 6.399, "text": "channel also like comment and share"}, {"start": 228.519, "duration": 3.0, "text": "bye-bye"}], "title": "Germany | Olympiad Math | Can you solve ?", "description": "", "lengthSeconds": 230, "width": 640, "height": 360}
{"video_id": "oDUUUItU6yo", "transcript": [{"start": 0.84, "duration": 4.439, "text": "hello good viewers welcome back once"}, {"start": 2.76, "duration": 5.4, "text": "again today we have this interesting"}, {"start": 5.279, "duration": 4.001, "text": "Olympiad mathematics problem and here"}, {"start": 8.16, "duration": 7.12, "text": "we'll be"}, {"start": 9.28, "duration": 7.28, "text": "evaluating 5x + xun x given that x +un x"}, {"start": 15.28, "duration": 4.759, "text": "is ="}, {"start": 16.56, "duration": 8.92, "text": "4 now let's get started from the given"}, {"start": 20.039, "duration": 7.441, "text": "equation we have X Plus root of x this"}, {"start": 25.48, "duration": 4.959, "text": "is equal to"}, {"start": 27.48, "duration": 6.32, "text": "4 now let's go ahead and multiply both"}, {"start": 30.439, "duration": 6.761, "text": "sides by root of x so here we have root"}, {"start": 33.8, "duration": 6.64, "text": "of x into braet"}, {"start": 37.2, "duration": 8.08, "text": "X+ root of"}, {"start": 40.44, "duration": 8.84, "text": "x then this is equal to root of x *"}, {"start": 45.28, "duration": 9.2, "text": "4 okay so from here root of x * X will"}, {"start": 49.28, "duration": 7.51, "text": "give us X root X then we have plus root"}, {"start": 54.48, "duration": 4.0, "text": "of x * root of"}, {"start": 56.79, "duration": 3.89, "text": "[Music]"}, {"start": 58.48, "duration": 3.52, "text": "x we give"}, {"start": 60.68, "duration": 3.72, "text": "root of"}, {"start": 62.0, "duration": 5.72, "text": "X2 of which this Square will cancel the"}, {"start": 64.4, "duration": 7.399, "text": "square root and we have just X so right"}, {"start": 67.72, "duration": 10.48, "text": "over here we have X and this is equal to"}, {"start": 71.799, "duration": 9.0, "text": "4ot of X so let's call this our equation"}, {"start": 78.2, "duration": 9.12, "text": "one okay now let's go back to the"}, {"start": 80.799, "duration": 9.32, "text": "original equation we have here x + otk x"}, {"start": 87.32, "duration": 5.6, "text": "this is equal to 4 now let's go ah head"}, {"start": 90.119, "duration": 11.36, "text": "and multiply both sides by 4 so we have"}, {"start": 92.92, "duration": 12.199, "text": "4 into braet x +un x this is = to 4 * 4"}, {"start": 101.479, "duration": 6.401, "text": "so from here we have 4 * X is 4 x then"}, {"start": 105.119, "duration": 6.32, "text": "we have plus 4 * root of x will give us"}, {"start": 107.88, "duration": 8.04, "text": "4un X then this will be equal to"}, {"start": 111.439, "duration": 8.0, "text": "16 now let's call this our equation 2"}, {"start": 115.92, "duration": 7.799, "text": "okay now from equation one we have 4"}, {"start": 119.439, "duration": 6.441, "text": "root X to be equal to xun x + x so right"}, {"start": 123.719, "duration": 6.76, "text": "over here we're going to replace this"}, {"start": 125.88, "duration": 9.88, "text": "with xun x + x so here we get 4X plus so"}, {"start": 130.479, "duration": 9.4, "text": "instead of this we we write as xun x + x"}, {"start": 135.76, "duration": 6.92, "text": "right then this is equal 16 so from here"}, {"start": 139.879, "duration": 7.401, "text": "4x + x will give us"}, {"start": 142.68, "duration": 7.36, "text": "5x then we have plus xun"}, {"start": 147.28, "duration": 4.8, "text": "X okay so this is the left hand side now"}, {"start": 150.04, "duration": 4.0, "text": "let's go back to the original problem"}, {"start": 152.08, "duration": 4.56, "text": "let us know what we are asked to"}, {"start": 154.04, "duration": 6.0, "text": "evaluate so we can see that we have this"}, {"start": 156.64, "duration": 7.44, "text": "expression here to evaluate 5un x + xun"}, {"start": 160.04, "duration": 7.72, "text": "x and right from here we already have"}, {"start": 164.08, "duration": 7.2, "text": "the expression here and this is equal to"}, {"start": 167.76, "duration": 5.559, "text": "16 which is the correct answer thank you"}, {"start": 171.28, "duration": 4.92, "text": "for watching please kindly subscribe to"}, {"start": 173.319, "duration": 5.881, "text": "my channel also like comment and share"}, {"start": 176.2, "duration": 3.0, "text": "bye-bye"}], "title": "Tricky Olympiad Math problem | Can you solve ?", "description": "", "lengthSeconds": 178, "width": 640, "height": 360}
{"video_id": "TtMIjj7ouGo", "transcript": [{"start": 0.95, "duration": 18.69, "text": "[Music]"}, {"start": 22.56, "duration": 7.639, "text": "slit spe"}, {"start": 25.14, "duration": 8.059, "text": "[Music]"}, {"start": 30.199, "duration": 3.0, "text": "oh"}, {"start": 34.15, "duration": 24.559, "text": "[Music]"}, {"start": 60.239, "duration": 3.0, "text": "a"}, {"start": 63.97, "duration": 32.379, "text": "[Music]"}, {"start": 97.52, "duration": 33.239, "text": "slit space"}, {"start": 100.53, "duration": 30.229, "text": "[Music]"}, {"start": 132.93, "duration": 9.089, "text": "[Music]"}, {"start": 144.87, "duration": 23.479, "text": "[Music]"}, {"start": 168.8, "duration": 11.159, "text": "slit Beats"}, {"start": 171.54, "duration": 11.419, "text": "[Music]"}, {"start": 179.959, "duration": 3.0, "text": "la"}, {"start": 183.54, "duration": 12.529, "text": "[Music]"}, {"start": 198.82, "duration": 4.09, "text": "[Music]"}, {"start": 206.09, "duration": 5.94, "text": "[Music]"}, {"start": 210.599, "duration": 6.07, "text": "oh"}, {"start": 212.03, "duration": 4.639, "text": "[Music]"}], "title": "Junior Olympiad Math. |Simplification of powers of numbers #education #maths", "description": "", "lengthSeconds": 215, "width": 640, "height": 360}
{"video_id": "tx5onTp2sd8", "transcript": [{"start": 1.79, "duration": 29.19, "text": "[\u0938\u0902\u0917\u0940\u0924]"}, {"start": 28.96, "duration": 29.899, "text": "\u0938\u094d\u0932\u0940\u091f \u0914\u0930"}, {"start": 30.98, "duration": 27.879, "text": "[\u0938\u0902\u0917\u0940\u0924]"}, {"start": 61.77, "duration": 9.09, "text": "[\u0938\u0902\u0917\u0940\u0924]"}, {"start": 73.9, "duration": 21.569, "text": "[\u0938\u0902\u0917\u0940\u0924]"}, {"start": 98.36, "duration": 33.149, "text": "\u0938\u094d\u0932\u093f\u091f \u0938\u094d \u092a\u0940\u091f"}, {"start": 101.37, "duration": 30.139, "text": "[\u0938\u0902\u0917\u0940\u0924]"}, {"start": 133.77, "duration": 9.739, "text": "[\u0938\u0902\u0917\u0940\u0924]"}, {"start": 145.81, "duration": 6.03, "text": "[\u0938\u0902\u0917\u0940\u0924]"}, {"start": 149.879, "duration": 10.04, "text": "\u092e\u0948\u0902"}, {"start": 151.84, "duration": 8.079, "text": "[\u0938\u0902\u0917\u0940\u0924]"}], "title": "Junior Olympiad Math. | Interesting problem. #education", "description": "", "lengthSeconds": 158, "width": 640, "height": 360}
{"video_id": "4MeKJYj3cBk", "transcript": [{"start": 0.799, "duration": 5.841, "text": "hello welcome back once again today we"}, {"start": 3.159, "duration": 5.681, "text": "have this awesome looking problem here"}, {"start": 6.64, "duration": 5.44, "text": "this is an Olympian mathematics problem"}, {"start": 8.84, "duration": 6.96, "text": "yes indeed this problem is very"}, {"start": 12.08, "duration": 6.68, "text": "tricky imagine we have just an equation"}, {"start": 15.8, "duration": 7.2, "text": "here involving two different variables"}, {"start": 18.76, "duration": 5.64, "text": "and we asked to evaluate a / B 2 plus b"}, {"start": 23.0, "duration": 3.96, "text": "/ a"}, {"start": 24.4, "duration": 5.719, "text": "s now let's go ahead and solve this"}, {"start": 26.96, "duration": 7.0, "text": "problem from the given equation let us"}, {"start": 30.119, "duration": 9.481, "text": "expand the left hand side we have here a"}, {"start": 33.96, "duration": 11.2, "text": "- b 2 which is equal to 6 a b expanding"}, {"start": 39.6, "duration": 10.4, "text": "this this will give us a 2 + b^ 2 - 2 a"}, {"start": 45.16, "duration": 10.12, "text": "b and this is equal to 6 a add 2 a to"}, {"start": 50.0, "duration": 10.039, "text": "both sides we get a 2 + b 2 is equal to"}, {"start": 55.28, "duration": 8.959, "text": "6 a + 2 a we give us 8 a"}, {"start": 60.039, "duration": 8.44, "text": "this point let us divide both sides by a"}, {"start": 64.239, "duration": 7.761, "text": "* B so divide this by a b and divide"}, {"start": 68.479, "duration": 10.841, "text": "here by a split this denominator from"}, {"start": 72.0, "duration": 10.4, "text": "here we get a 2 / a plus b 2 / a this"}, {"start": 79.32, "duration": 3.799, "text": "will be equal to a will cancel a we left"}, {"start": 82.4, "duration": 4.8, "text": "with"}, {"start": 83.119, "duration": 8.081, "text": "8 a s / a will give us just a remember"}, {"start": 87.2, "duration": 6.84, "text": "we have over B plus b Sid B we are left"}, {"start": 91.2, "duration": 7.72, "text": "with b and we have over a and this is"}, {"start": 94.04, "duration": 6.96, "text": "equal to 8 wonderful right now let us"}, {"start": 98.92, "duration": 9.12, "text": "recall the expression we asked to"}, {"start": 101.0, "duration": 8.84, "text": "evaluate which is a / B 2 plus b/ a s so"}, {"start": 108.04, "duration": 4.359, "text": "let us manipulate this expression but"}, {"start": 109.84, "duration": 12.319, "text": "before that take note of this property"}, {"start": 112.399, "duration": 11.0, "text": "in algebra a 2 + b 2 is = a + b 2 - 2 *"}, {"start": 122.159, "duration": 5.721, "text": "a *"}, {"start": 123.399, "duration": 6.121, "text": "B so with this identity here remember"}, {"start": 127.88, "duration": 4.12, "text": "something is squared here and something"}, {"start": 129.52, "duration": 5.4, "text": "is squared here so it's going to be into"}, {"start": 132.0, "duration": 6.519, "text": "bracket that's first something a / B"}, {"start": 134.92, "duration": 6.319, "text": "plus the second something B / a then"}, {"start": 138.519, "duration": 5.681, "text": "squared according to this property here"}, {"start": 141.239, "duration": 5.801, "text": "minus 2 * the first something here which"}, {"start": 144.2, "duration": 5.8, "text": "is a / B multip the second something"}, {"start": 147.04, "duration": 4.68, "text": "which is B / a"}, {"start": 150.0, "duration": 7.319, "text": "so from"}, {"start": 151.72, "duration": 9.64, "text": "here this is a / B + B /"}, {"start": 157.319, "duration": 6.881, "text": "a\u00b2 minus just two because these two"}, {"start": 161.36, "duration": 7.48, "text": "cancel out and from here we have the"}, {"start": 164.2, "duration": 11.64, "text": "solution remember a/ b + B / a b is 8 so"}, {"start": 168.84, "duration": 11.72, "text": "we have 8 2 - 2 8 2 is 64 then - 2 which"}, {"start": 175.84, "duration": 13.28, "text": "is equal to 62 so therefore"}, {"start": 180.56, "duration": 10.959, "text": "a / B 2 + B / a 2 is = 62 and this is"}, {"start": 189.12, "duration": 4.88, "text": "the right answer thank you for watching"}, {"start": 191.519, "duration": 8.041, "text": "please kindly subscribe to my channel"}, {"start": 194.0, "duration": 5.56, "text": "also like comment and share bye-bye"}], "title": "Interesting Olympiad Math problem | Can you solve ?", "description": "", "lengthSeconds": 198, "width": 640, "height": 360}
{"video_id": "u_q_c1wdGjc", "transcript": [{"start": 0.44, "duration": 5.919, "text": "welcome back my"}, {"start": 3.2, "duration": 6.68, "text": "fans here we have this mind"}, {"start": 6.359, "duration": 7.44, "text": "calling problem so we are asked to at"}, {"start": 9.88, "duration": 6.8, "text": "least find all the values of X"}, {"start": 13.799, "duration": 5.601, "text": "right okay now let's see how to solve"}, {"start": 16.68, "duration": 5.48, "text": "this problem but before we proceed"}, {"start": 19.4, "duration": 4.879, "text": "please kindly support this channel by"}, {"start": 22.16, "duration": 5.119, "text": "subscribing to this YouTube"}, {"start": 24.279, "duration": 9.561, "text": "Channel please subscribe to the channel"}, {"start": 27.279, "duration": 14.161, "text": "and share this video here we have x^ 4"}, {"start": 33.84, "duration": 10.12, "text": "is = to 1 we have x^ 4 - 1 is equal 0"}, {"start": 41.44, "duration": 7.88, "text": "you know this x 4 we can write it as"}, {"start": 43.96, "duration": 9.32, "text": "follows X of 2 squar minus you know 1 is"}, {"start": 49.32, "duration": 7.28, "text": "the same as 1 2 and this is equal to Z"}, {"start": 53.28, "duration": 6.759, "text": "right so right from here we can make use"}, {"start": 56.6, "duration": 14.24, "text": "of the difference of two squares mainly"}, {"start": 60.039, "duration": 14.881, "text": "a 2 - b 2 is equal to a + b * a minus B"}, {"start": 70.84, "duration": 6.12, "text": "right okay here we're going to have"}, {"start": 74.92, "duration": 4.48, "text": "X2 +"}, {"start": 76.96, "duration": 9.6, "text": "1 *"}, {"start": 79.4, "duration": 12.16, "text": "X2 - 1 is = 0 so from here now we can"}, {"start": 86.56, "duration": 9.919, "text": "say X2 + 1 is equal to 0"}, {"start": 91.56, "duration": 9.559, "text": "or x^2 - 1 is = 0"}, {"start": 96.479, "duration": 9.721, "text": "right okay now let us solve this"}, {"start": 101.119, "duration": 8.561, "text": "first we have X2 - 1 is ="}, {"start": 106.2, "duration": 3.48, "text": "0 so we"}, {"start": 109.799, "duration": 9.721, "text": "have X2 is = 1 that's when this1 crosses"}, {"start": 115.2, "duration": 8.44, "text": "over it will have opposite sign right so"}, {"start": 119.52, "duration": 7.44, "text": "we can go ahead and take the root of"}, {"start": 123.64, "duration": 9.679, "text": "both sides so we have xal to plus or"}, {"start": 126.96, "duration": 10.84, "text": "minus < TK 1 so we get X = to+ or"}, {"start": 133.319, "duration": 6.481, "text": "minus 1 right so you can see we have at"}, {"start": 137.8, "duration": 4.799, "text": "least found"}, {"start": 139.8, "duration": 5.56, "text": "two values of X and these are the real"}, {"start": 142.599, "duration": 6.761, "text": "values right here we have + one"}, {"start": 145.36, "duration": 5.519, "text": "or1 now let us go ahead and solve this"}, {"start": 149.36, "duration": 5.8, "text": "problem"}, {"start": 150.879, "duration": 10.601, "text": "here we have x^2 + 1 is ="}, {"start": 155.16, "duration": 8.999, "text": "0 so from here we have X2 is = to -1"}, {"start": 161.48, "duration": 5.92, "text": "since pos1 is crossing over so it will"}, {"start": 164.159, "duration": 6.72, "text": "have opposite sign so we have here x ="}, {"start": 167.4, "duration": 8.4, "text": "to plus or minus root"}, {"start": 170.879, "duration": 7.801, "text": "of1 and we all know that theun of1 is"}, {"start": 175.8, "duration": 5.519, "text": "equal to I that is the imaginary unit"}, {"start": 178.68, "duration": 7.76, "text": "right so this this is going to give us"}, {"start": 181.319, "duration": 6.64, "text": "plus or minus I right so we can go"}, {"start": 186.44, "duration": 6.48, "text": "ahead"}, {"start": 187.959, "duration": 8.441, "text": "and write down the possible values of X"}, {"start": 192.92, "duration": 6.08, "text": "in that equation so we have"}, {"start": 196.4, "duration": 5.88, "text": "1 we have -"}, {"start": 199.0, "duration": 8.48, "text": "1 we have positive"}, {"start": 202.28, "duration": 7.679, "text": "I and then we have I and this are the"}, {"start": 207.48, "duration": 5.959, "text": "right answers thank you for watching"}, {"start": 209.959, "duration": 6.401, "text": "please kindly support this channel by"}, {"start": 213.439, "duration": 7.761, "text": "subscribing to the channel subscribe"}, {"start": 216.36, "duration": 4.84, "text": "please and share this video bye-bye"}], "title": "A very nice equation. x\u2074 = 1.", "description": "", "lengthSeconds": 219, "width": 636, "height": 360}
{"video_id": "pENDRibJMQQ", "transcript": [{"start": 0.12, "duration": 4.36, "text": "hello wonderful viewers here we have"}, {"start": 2.08, "duration": 5.319, "text": "another interesting question please"}, {"start": 4.48, "duration": 6.039, "text": "subscribe to this channel also like"}, {"start": 7.399, "duration": 9.96, "text": "comment and share here we're given that"}, {"start": 10.519, "duration": 9.481, "text": "x^ of 1 1999 - 1 / x ^ of 1 1999 is"}, {"start": 17.359, "duration": 9.041, "text": "equal to 11 and here we asked to"}, {"start": 20.0, "duration": 8.24, "text": "evaluate x ^ of 1 1999 + 1 /x ^ of 1"}, {"start": 26.4, "duration": 4.52, "text": "1999"}, {"start": 28.24, "duration": 5.159, "text": "right okay well this question looks"}, {"start": 30.92, "duration": 5.4, "text": "crazy but an easy"}, {"start": 33.399, "duration": 6.16, "text": "problem let's see how we can find the"}, {"start": 36.32, "duration": 9.16, "text": "value of this expression"}, {"start": 39.559, "duration": 7.52, "text": "here okay now from the equation given x"}, {"start": 45.48, "duration": 6.599, "text": "1"}, {"start": 47.079, "duration": 9.0, "text": "1999 - 1 / x 1"}, {"start": 52.079, "duration": 7.041, "text": "1999 is equal to 11 let's Square both"}, {"start": 56.079, "duration": 5.841, "text": "sides of this equation so if you square"}, {"start": 59.12, "duration": 5.64, "text": "both sides"}, {"start": 61.92, "duration": 5.04, "text": "here we have 11 squ from the left we're"}, {"start": 64.76, "duration": 12.039, "text": "going to make use of the"}, {"start": 66.96, "duration": 11.519, "text": "property a - b s this is equal to a + b"}, {"start": 76.799, "duration": 9.161, "text": "s"}, {"start": 78.479, "duration": 9.801, "text": "minus 4 a b right okay cool so if we"}, {"start": 85.96, "duration": 5.119, "text": "make use of this property at the left"}, {"start": 88.28, "duration": 5.28, "text": "hand side we going to have"}, {"start": 91.079, "duration": 8.561, "text": "x 1"}, {"start": 93.56, "duration": 6.08, "text": "1999 + 1 x 1"}, {"start": 99.72, "duration": 7.12, "text": "1999 then here"}, {"start": 103.04, "duration": 10.16, "text": "squared right then minus here we have 4"}, {"start": 106.84, "duration": 9.959, "text": "* x 1 1999 * 1 / x 1 1999 you know this"}, {"start": 113.2, "duration": 7.64, "text": "we cancel out we are left with four and"}, {"start": 116.799, "duration": 8.24, "text": "here this is equal to 11 s which is 12 1"}, {"start": 120.84, "duration": 8.279, "text": "right so from here we have x^"}, {"start": 125.039, "duration": 6.92, "text": "1999 + 1 / X for"}, {"start": 129.119, "duration": 6.321, "text": "1999 squ this is equal to you know if"}, {"start": 131.959, "duration": 5.961, "text": "you add four to both side we get"}, {"start": 135.44, "duration": 5.12, "text": "125 so from here we're going to take"}, {"start": 137.92, "duration": 5.16, "text": "root of both sides and there we find the"}, {"start": 140.56, "duration": 5.039, "text": "value of this expression and the value"}, {"start": 143.08, "duration": 7.519, "text": "is going to be X for"}, {"start": 145.599, "duration": 8.961, "text": "199 + 1 / X is for 1 1999"}, {"start": 150.599, "duration": 6.681, "text": "this is equal to plus or minus < TK 125"}, {"start": 154.56, "duration": 7.959, "text": "so simplifying this s we get plus or"}, {"start": 157.28, "duration": 7.4, "text": "minus 5 < TK 5 and this is the final"}, {"start": 162.519, "duration": 3.601, "text": "answer to the problem thank you for"}, {"start": 164.68, "duration": 3.88, "text": "watching please do wait to subscribe to"}, {"start": 166.12, "duration": 5.44, "text": "my channel also like comment and share"}, {"start": 168.56, "duration": 3.0, "text": "bye-bye"}], "title": "The crazy looking Olympiad Math. How to evaluate x^1999 + \u00b9/x^1999.", "description": "", "lengthSeconds": 170, "width": 640, "height": 360}
{"video_id": "SyJ5TBWqpBM", "transcript": [{"start": 0.64, "duration": 5.119, "text": "hello welcome back once again today we"}, {"start": 2.84, "duration": 5.679, "text": "have another interesting problem here"}, {"start": 5.759, "duration": 6.92, "text": "we're going to give a step by step"}, {"start": 8.519, "duration": 7.2, "text": "solution to this radical equation so"}, {"start": 12.679, "duration": 7.44, "text": "here we have square root of square root"}, {"start": 15.719, "duration": 7.72, "text": "of 2x + 1 which is equal to 3 so here"}, {"start": 20.119, "duration": 5.92, "text": "we'll be solving for the value of x"}, {"start": 23.439, "duration": 5.16, "text": "right so let's start the first thing"}, {"start": 26.039, "duration": 5.681, "text": "we'll do here is to square both sides of"}, {"start": 28.599, "duration": 6.12, "text": "this equation right right so we going to"}, {"start": 31.72, "duration": 7.4, "text": "square this and equally Square the right"}, {"start": 34.719, "duration": 5.84, "text": "hand side as we know for example Square"}, {"start": 39.12, "duration": 6.36, "text": "< TK of"}, {"start": 40.559, "duration": 7.761, "text": "a^ of 2 is equal to a because Square we"}, {"start": 45.48, "duration": 5.52, "text": "cancel the square root right but here we"}, {"start": 48.32, "duration": 5.759, "text": "have two square root signs so this"}, {"start": 51.0, "duration": 8.64, "text": "Square will cancel the first square root"}, {"start": 54.079, "duration": 7.201, "text": "so here we are left with square OT of 2x"}, {"start": 59.64, "duration": 4.239, "text": "+"}, {"start": 61.28, "duration": 7.199, "text": "1 right which is equal to from the right"}, {"start": 63.879, "duration": 6.801, "text": "hand side 3 S is 9 okay again we have"}, {"start": 68.479, "duration": 5.761, "text": "another square root so let's go ahead"}, {"start": 70.68, "duration": 5.759, "text": "and square both sides okay so here"}, {"start": 74.24, "duration": 4.96, "text": "Square this"}, {"start": 76.439, "duration": 6.201, "text": "side and equally Square the right hand"}, {"start": 79.2, "duration": 7.84, "text": "side we know this Square will cancel the"}, {"start": 82.64, "duration": 8.88, "text": "square roots and here we have 2X then +"}, {"start": 87.04, "duration": 8.119, "text": "1 which is equal to 9^ 2 is"}, {"start": 91.52, "duration": 7.16, "text": "81 okay so look at this left hand side"}, {"start": 95.159, "duration": 5.481, "text": "of this equation we have positive one so"}, {"start": 98.68, "duration": 4.399, "text": "what happens when we subtract one from"}, {"start": 100.64, "duration": 5.4, "text": "both sides we see that the positive one"}, {"start": 103.079, "duration": 6.121, "text": "here and the negative one will vanish"}, {"start": 106.04, "duration": 5.28, "text": "and that will be left with 2x right so"}, {"start": 109.2, "duration": 8.64, "text": "let's do that so here we have"}, {"start": 111.32, "duration": 10.96, "text": "2X then + 1 - 1 this will be equal to 81"}, {"start": 117.84, "duration": 7.08, "text": "- 1 so from here we can see that + 1 - 1"}, {"start": 122.28, "duration": 6.8, "text": "this is zero so here we have"}, {"start": 124.92, "duration": 8.08, "text": "2X which should be equal to 81 - 1 will"}, {"start": 129.08, "duration": 3.92, "text": "give us 80"}, {"start": 133.28, "duration": 5.76, "text": "right okay so from here the next thing"}, {"start": 136.44, "duration": 5.28, "text": "to do is to divide both sides by the"}, {"start": 139.04, "duration": 7.44, "text": "coefficient of x and the coefficient of"}, {"start": 141.72, "duration": 9.32, "text": "x in this case is two so divide this by"}, {"start": 146.48, "duration": 6.8, "text": "two and divide here by two so this two"}, {"start": 151.04, "duration": 6.24, "text": "we cancel this two and here we left with"}, {"start": 153.28, "duration": 8.2, "text": "X so our answer now we get X is equal to"}, {"start": 157.28, "duration": 8.599, "text": "80 divid by 2 which should give us 40 so"}, {"start": 161.48, "duration": 6.399, "text": "the right answer to this problem is 40"}, {"start": 165.879, "duration": 4.521, "text": "thank you for watching please kindly"}, {"start": 167.879, "duration": 7.161, "text": "subscribe to my channel also like"}, {"start": 170.4, "duration": 4.64, "text": "comment and share bye-bye"}], "title": "Very nice radical equation | Step by step solution", "description": "", "lengthSeconds": 176, "width": 640, "height": 332}
{"video_id": "zZgm97YI6Zg", "transcript": [{"start": 0.88, "duration": 5.08, "text": "hello good day viewers welcome back once"}, {"start": 3.0, "duration": 6.24, "text": "again please cly support my channel by"}, {"start": 5.96, "duration": 5.4, "text": "subscribing also like comment and share"}, {"start": 9.24, "duration": 5.359, "text": "in this video we are going to solve for"}, {"start": 11.36, "duration": 6.679, "text": "x in the following Olympia math"}, {"start": 14.599, "duration": 6.281, "text": "problem here we are given that x ^ of X"}, {"start": 18.039, "duration": 6.32, "text": "is = to 2^ of"}, {"start": 20.88, "duration": 6.719, "text": "248 in this video we are going to find"}, {"start": 24.359, "duration": 6.561, "text": "the solution for X"}, {"start": 27.599, "duration": 8.601, "text": "right let's get started we have here x"}, {"start": 30.92, "duration": 10.159, "text": "to the^ of X this is equal to 2^ of"}, {"start": 36.2, "duration": 9.199, "text": "248 right so from here we have X to the^"}, {"start": 41.079, "duration": 6.081, "text": "of X this is equal to 2 power of you"}, {"start": 45.399, "duration": 9.48, "text": "know"}, {"start": 47.16, "duration": 10.12, "text": "2048 is the same as 2 multi by 1 24"}, {"start": 54.879, "duration": 8.32, "text": "right now let's record this"}, {"start": 57.28, "duration": 10.68, "text": "property a of B * C this is same as a"}, {"start": 63.199, "duration": 7.681, "text": "power of B then in bracket power of C so"}, {"start": 67.96, "duration": 3.88, "text": "from here we have X the^ of X this is"}, {"start": 70.88, "duration": 4.44, "text": "equal"}, {"start": 71.84, "duration": 9.08, "text": "to you know in bracket 2"}, {"start": 75.32, "duration": 10.52, "text": "2 then ra power of 1 to 4 and from here"}, {"start": 80.92, "duration": 8.64, "text": "this is equal to 4 ra power of 1 to"}, {"start": 85.84, "duration": 7.8, "text": "4 okay now from here we have x to the"}, {"start": 89.56, "duration": 8.04, "text": "power of X this is equal to 4 power of"}, {"start": 93.64, "duration": 5.72, "text": "1024 is the same as 2"}, {"start": 97.6, "duration": 5.64, "text": "*"}, {"start": 99.36, "duration": 7.399, "text": "512 right so we make use of this"}, {"start": 103.24, "duration": 8.159, "text": "property once again from here we get x^"}, {"start": 106.759, "duration": 5.801, "text": "of X this is equal to 4^ 2 then ra power"}, {"start": 111.399, "duration": 4.161, "text": "of"}, {"start": 112.56, "duration": 8.72, "text": "512 and from here the bracket here we"}, {"start": 115.56, "duration": 12.64, "text": "have 16 to^ 51 2 now from here we have"}, {"start": 121.28, "duration": 9.92, "text": "x^ X this is equal to 16 power of 2 *"}, {"start": 128.2, "duration": 8.399, "text": "512 is 2 *"}, {"start": 131.2, "duration": 7.16, "text": "256 and from here we have 16 2 ra the^"}, {"start": 136.599, "duration": 6.0, "text": "of"}, {"start": 138.36, "duration": 7.2, "text": "256 and from here we have x ^ of X this"}, {"start": 142.599, "duration": 5.561, "text": "is equal to 16 s which is"}, {"start": 145.56, "duration": 5.36, "text": "256 to the power of"}, {"start": 148.16, "duration": 7.76, "text": "256 isn't this"}, {"start": 150.92, "duration": 6.319, "text": "so from here this implies that X is"}, {"start": 155.92, "duration": 4.64, "text": "equal to"}, {"start": 157.239, "duration": 5.161, "text": "256 and this is the solution thank you"}, {"start": 160.56, "duration": 4.56, "text": "for watching please kindly subscribe to"}, {"start": 162.4, "duration": 5.72, "text": "my channel also like comment and share"}, {"start": 165.12, "duration": 3.0, "text": "bye-bye"}], "title": "Japanese | Nice Olympiad exponential equation | #maths #mathematics", "description": "", "lengthSeconds": 167, "width": 640, "height": 360}
{"video_id": "tN9o2FAz4ZM", "transcript": [{"start": 0.719, "duration": 5.281, "text": "good if he was today shidex is showing"}, {"start": 3.419, "duration": 5.1, "text": "his concern on this problem"}, {"start": 6.0, "duration": 5.7, "text": "this has been a trending problem online"}, {"start": 8.519, "duration": 4.2, "text": "huh the negative 3 squared equals Dutch"}, {"start": 11.7, "duration": 4.679, "text": "okay"}, {"start": 12.719, "duration": 4.921, "text": "now before we go into this problem"}, {"start": 16.379, "duration": 3.361, "text": "please"}, {"start": 17.64, "duration": 5.46, "text": "subscribe to my channel for more"}, {"start": 19.74, "duration": 4.98, "text": "interesting videos please subscribe you"}, {"start": 23.1, "duration": 4.14, "text": "can share"}, {"start": 24.72, "duration": 4.26, "text": "you can comment"}, {"start": 27.24, "duration": 4.439, "text": "and so"}, {"start": 28.98, "duration": 4.86, "text": "you can like now"}, {"start": 31.679, "duration": 4.201, "text": "let's say we have"}, {"start": 33.84, "duration": 5.399, "text": "Negative X"}, {"start": 35.88, "duration": 7.019, "text": "part 2 like this in Brackets okay so"}, {"start": 39.239, "duration": 5.34, "text": "here you know any variable or number in"}, {"start": 42.899, "duration": 4.32, "text": "Brackets where it means the number"}, {"start": 44.579, "duration": 6.601, "text": "multiply itself twice so here we have"}, {"start": 47.219, "duration": 8.941, "text": "negative x times negative X okay so we"}, {"start": 51.18, "duration": 7.5, "text": "get this equals you know minus here"}, {"start": 56.16, "duration": 6.0, "text": "so here we have minus times minus is"}, {"start": 58.68, "duration": 4.76, "text": "plus okay here we have x times x so that"}, {"start": 62.16, "duration": 5.639, "text": "is X2"}, {"start": 63.44, "duration": 7.539, "text": "so this is equal to X raised part two"}, {"start": 67.799, "duration": 5.64, "text": "okay now"}, {"start": 70.979, "duration": 5.221, "text": "let's say we have Negative X"}, {"start": 73.439, "duration": 4.081, "text": "you raise poetry so this is equal to"}, {"start": 76.2, "duration": 5.7, "text": "Negative X"}, {"start": 77.52, "duration": 6.66, "text": "times negative x times negative X so"}, {"start": 81.9, "duration": 5.16, "text": "this is giving us what so you know here"}, {"start": 84.18, "duration": 6.06, "text": "minus times minus here so that is plus"}, {"start": 87.06, "duration": 6.78, "text": "then times this minus here so we get"}, {"start": 90.24, "duration": 7.32, "text": "minus so we get minus x times x times x"}, {"start": 93.84, "duration": 5.4, "text": "that is SQ so we have a negative sign so"}, {"start": 97.56, "duration": 4.64, "text": "this means that whenever we have"}, {"start": 99.24, "duration": 2.96, "text": "Negative X"}, {"start": 102.659, "duration": 4.741, "text": "even numbers"}, {"start": 104.88, "duration": 4.62, "text": "so this will give us answer in positive"}, {"start": 107.4, "duration": 5.46, "text": "form so"}, {"start": 109.5, "duration": 5.159, "text": "in positive form and also when we have"}, {"start": 112.86, "duration": 4.86, "text": "Negative X"}, {"start": 114.659, "duration": 5.581, "text": "raised by odd numbers"}, {"start": 117.72, "duration": 4.679, "text": "so this will give us answer in negative"}, {"start": 120.24, "duration": 5.339, "text": "form okay"}, {"start": 122.399, "duration": 6.301, "text": "so this is what we have now"}, {"start": 125.579, "duration": 5.821, "text": "let's say we have Negative X raised part"}, {"start": 128.7, "duration": 7.679, "text": "two in this one so this is equal to"}, {"start": 131.4, "duration": 6.24, "text": "negative one times x raised power 2. so"}, {"start": 136.379, "duration": 4.021, "text": "most of us"}, {"start": 137.64, "duration": 6.54, "text": "has been assuming that negative x"}, {"start": 140.4, "duration": 6.3, "text": "squared is the same as Negative X raised"}, {"start": 144.18, "duration": 4.68, "text": "by two like this then answer becoming"}, {"start": 146.7, "duration": 6.06, "text": "positive but this is absolutely wrong"}, {"start": 148.86, "duration": 6.239, "text": "okay so here we have negative 1 times x"}, {"start": 152.76, "duration": 5.24, "text": "squared so"}, {"start": 155.099, "duration": 5.821, "text": "in this case we have negative three"}, {"start": 158.0, "duration": 4.3, "text": "respirator so this is equal to negative"}, {"start": 160.92, "duration": 5.52, "text": "one"}, {"start": 162.3, "duration": 7.799, "text": "times negative sorry times three"}, {"start": 166.44, "duration": 7.26, "text": "squared okay so here we have minus one"}, {"start": 170.099, "duration": 5.941, "text": "then times three squared is nine so we"}, {"start": 173.7, "duration": 4.38, "text": "get our answer as negative nine and this"}, {"start": 176.04, "duration": 3.96, "text": "is the right answer to the problem thank"}, {"start": 178.08, "duration": 3.299, "text": "you for viewing and please make sure you"}, {"start": 180.0, "duration": 4.099, "text": "subscribe to our channel for more"}, {"start": 181.379, "duration": 2.72, "text": "interesting videos"}], "title": "My take on this.", "description": "", "lengthSeconds": 183, "width": 636, "height": 360}
{"video_id": "VqlM4E73nnE", "transcript": [{"start": 0.399, "duration": 3.641, "text": "hello good viewers in this video we're"}, {"start": 2.04, "duration": 4.56, "text": "going to solve for the value of x in the"}, {"start": 4.04, "duration": 6.28, "text": "following exponential"}, {"start": 6.6, "duration": 6.36, "text": "equation we have here 2 to ^ of x + 1"}, {"start": 10.32, "duration": 6.039, "text": "you know we can move this to the right"}, {"start": 12.96, "duration": 7.52, "text": "hand side that will give us 3 to ^ of x"}, {"start": 16.359, "duration": 10.201, "text": "+ 2 now let us record this property M of"}, {"start": 20.48, "duration": 11.24, "text": "n + P this is same as M of n * m power"}, {"start": 26.56, "duration": 10.679, "text": "of P so from here we have 2^ X Multi 2 1"}, {"start": 31.72, "duration": 6.64, "text": "which is 2 this is equal to 3 x * 3 2"}, {"start": 37.239, "duration": 3.8, "text": "that will give us"}, {"start": 38.36, "duration": 4.879, "text": "9 so from here let's go ahead and divide"}, {"start": 41.039, "duration": 5.0, "text": "both sides by"}, {"start": 43.239, "duration": 9.681, "text": "two this will cancel this we have here"}, {"start": 46.039, "duration": 6.881, "text": "2^ x this is = 3 x * 9 /"}, {"start": 52.96, "duration": 7.8, "text": "2 from here we can go ahead and divide"}, {"start": 55.76, "duration": 5.0, "text": "both sides by 3 to the^ of x"}, {"start": 60.8, "duration": 5.72, "text": "this should cancel this and here we have"}, {"start": 63.559, "duration": 7.041, "text": "you know we have the same power so we"}, {"start": 66.52, "duration": 7.76, "text": "can write it as 2 over 3 all X this will"}, {"start": 70.6, "duration": 6.519, "text": "give us 9 all 2 now let us recall that"}, {"start": 74.28, "duration": 6.799, "text": "the logarithm of M to the power of B to"}, {"start": 77.119, "duration": 5.64, "text": "the base of M is exactly equal to B so"}, {"start": 81.079, "duration": 4.201, "text": "from here we're going to log both sides"}, {"start": 82.759, "duration": 4.36, "text": "but the logarithm must be to the base 2/"}, {"start": 85.28, "duration": 5.799, "text": "3 so we have the"}, {"start": 87.119, "duration": 7.04, "text": "logarithm of 2/ 3 of x x to the base of"}, {"start": 91.079, "duration": 7.201, "text": "2 / 3 this is equal to the"}, {"start": 94.159, "duration": 6.92, "text": "logarithm of 9 / 2 to the base of 2 over"}, {"start": 98.28, "duration": 4.04, "text": "3 so we are left with just X here"}, {"start": 101.079, "duration": 5.121, "text": "because the whole of this will cancel"}, {"start": 102.32, "duration": 9.479, "text": "out we have X is equal to the"}, {"start": 106.2, "duration": 9.16, "text": "logarithm of 9 / 2 to the base of 2/ 3"}, {"start": 111.799, "duration": 9.201, "text": "which is approximately equal"}, {"start": 115.36, "duration": 9.24, "text": "to -3 point this is three"}, {"start": 121.0, "duration": 7.08, "text": "-3 709"}, {"start": 124.6, "duration": 5.799, "text": "513 okay 113 I mean thank you for"}, {"start": 128.08, "duration": 4.4, "text": "watching please kindly subscribe to my"}, {"start": 130.399, "duration": 5.081, "text": "channel also like comment and share"}, {"start": 132.48, "duration": 3.0, "text": "bye-bye"}], "title": "Exponential equation | #maths #mathematics", "description": "", "lengthSeconds": 134, "width": 640, "height": 360}
{"video_id": "fMnZZjj01fA", "transcript": [{"start": 0.84, "duration": 4.519, "text": "hello good day viewers let's find a"}, {"start": 3.12, "duration": 3.6, "text": "solution to the following given"}, {"start": 5.359, "duration": 5.761, "text": "exponential"}, {"start": 6.72, "duration": 7.6, "text": "equation we have 4^ of otk X this is"}, {"start": 11.12, "duration": 5.319, "text": "equal to 9 now we're going to make use"}, {"start": 14.32, "duration": 4.76, "text": "of the natural"}, {"start": 16.439, "duration": 4.521, "text": "logarithm so from the both side we'll be"}, {"start": 19.08, "duration": 6.4, "text": "taking the natural logarithm so here we"}, {"start": 20.96, "duration": 6.639, "text": "have the Ln of 4^ ofun X this is equal"}, {"start": 25.48, "duration": 6.0, "text": "to the Ln of"}, {"start": 27.599, "duration": 7.681, "text": "9 so from here you know the property of"}, {"start": 31.48, "duration": 8.36, "text": "the logarithm this power we cross over"}, {"start": 35.28, "duration": 8.52, "text": "so here we have the root x * the Ln of 4"}, {"start": 39.84, "duration": 6.52, "text": "this is equal to the Ln of 9 so from"}, {"start": 43.8, "duration": 4.72, "text": "here remember we're solving for x so let"}, {"start": 46.36, "duration": 7.8, "text": "us go ahead and divide both side by the"}, {"start": 48.52, "duration": 9.08, "text": "Ln of 4 so we have otk x * the Ln of 4 /"}, {"start": 54.16, "duration": 8.6, "text": "by the Ln of 4 this is equal to the Ln"}, {"start": 57.6, "duration": 8.04, "text": "of 9 / the Ln of four so from here this"}, {"start": 62.76, "duration": 7.92, "text": "cancel this so we have root X this is"}, {"start": 65.64, "duration": 9.32, "text": "equal to the Ln of 9 is the same as 3 2"}, {"start": 70.68, "duration": 9.079, "text": "/ 4 is the same as 2"}, {"start": 74.96, "duration": 7.4, "text": "2 so from here we have rootx this is"}, {"start": 79.759, "duration": 4.881, "text": "equal to with the property of logarithm"}, {"start": 82.36, "duration": 5.92, "text": "this power is crossing over so we have 2"}, {"start": 84.64, "duration": 6.36, "text": "MTI the Ln of 3 divided by the same"}, {"start": 88.28, "duration": 5.36, "text": "thing happens here so we have two M the"}, {"start": 91.0, "duration": 6.32, "text": "Ln of 2 so this two and this two we"}, {"start": 93.64, "duration": 8.759, "text": "divide out so here we have root X this"}, {"start": 97.32, "duration": 7.96, "text": "is equal to the Ln of 3 / the Ln of"}, {"start": 102.399, "duration": 4.801, "text": "2 now we're solving for x so we're going"}, {"start": 105.28, "duration": 5.439, "text": "to square both side to get rid of this"}, {"start": 107.2, "duration": 6.48, "text": "root sign so here we Square this and"}, {"start": 110.719, "duration": 4.801, "text": "from here we Square this so the square"}, {"start": 113.68, "duration": 7.84, "text": "and the root sign we cancel out and here"}, {"start": 115.52, "duration": 10.68, "text": "we have X is equal to the Ln of 3 / the"}, {"start": 121.52, "duration": 8.96, "text": "Ln of 2 s and this is approximately"}, {"start": 126.2, "duration": 6.08, "text": "equal to 2"}, {"start": 130.48, "duration": 5.08, "text": "51210"}, {"start": 132.28, "duration": 5.0, "text": "613 and this is the right answer thank"}, {"start": 135.56, "duration": 3.959, "text": "you for watching please kindly subscribe"}, {"start": 137.28, "duration": 5.72, "text": "to my channel also like comment and"}, {"start": 139.519, "duration": 3.481, "text": "share bye-bye"}], "title": "Interesting exponential equation | #maths #mathematics", "description": "", "lengthSeconds": 141, "width": 640, "height": 360}
{"video_id": "FUa5ees7QW0", "transcript": [{"start": 0.16, "duration": 4.92, "text": "hello welcome back once again let us"}, {"start": 2.44, "duration": 6.119, "text": "solve this exponential equation from the"}, {"start": 5.08, "duration": 7.679, "text": "left we have 3 of root of x this will be"}, {"start": 8.559, "duration": 7.881, "text": "equal to here 9 is same as 3 S right"}, {"start": 12.759, "duration": 6.36, "text": "then power of the CU root of x now take"}, {"start": 16.44, "duration": 6.96, "text": "note of this when we have a^ B to the"}, {"start": 19.119, "duration": 7.961, "text": "power C this is same as a^ B * C so from"}, {"start": 23.4, "duration": 7.8, "text": "here we get 3^ ofun X this equals here"}, {"start": 27.08, "duration": 7.12, "text": "will be 3^ 2 MTI the cube root of"}, {"start": 31.2, "duration": 5.039, "text": "x okay now from here we have the same"}, {"start": 34.2, "duration": 8.24, "text": "bases right so we're going to equate the"}, {"start": 36.239, "duration": 9.761, "text": "powers so we get root of x this equal 2"}, {"start": 42.44, "duration": 6.48, "text": "* the cube root of"}, {"start": 46.0, "duration": 3.93, "text": "x okay now how do we solve this equation"}, {"start": 48.92, "duration": 4.24, "text": "from"}, {"start": 49.93, "duration": 5.75, "text": "[Music]"}, {"start": 53.16, "duration": 5.879, "text": "here so how do we solve this equation"}, {"start": 55.68, "duration": 4.839, "text": "from here so for my solution let us"}, {"start": 59.039, "duration": 3.36, "text": "first reite the"}, {"start": 60.519, "duration": 5.721, "text": "square root and the cube root sign so"}, {"start": 62.399, "duration": 12.04, "text": "this is same as x^ 1 / 2 then this is"}, {"start": 66.24, "duration": 12.36, "text": "equal to 2 * X to the^ 1 / 3 now let x"}, {"start": 74.439, "duration": 8.401, "text": "equals remember the LCM of 2 and 3 is 6"}, {"start": 78.6, "duration": 7.0, "text": "so we're going to let it equals y^ of 6"}, {"start": 82.84, "duration": 5.4, "text": "right so from here we can see that when"}, {"start": 85.6, "duration": 7.799, "text": "we raise both sides to the^ of 1 / 2 we"}, {"start": 88.24, "duration": 8.76, "text": "get that x^ 1 / 2 2 will be y to the^ of"}, {"start": 93.399, "duration": 6.72, "text": "3 right because 6 * 1 2 is 3 and raise"}, {"start": 97.0, "duration": 7.36, "text": "both sides to the power of 1/3 we get X"}, {"start": 100.119, "duration": 8.761, "text": "of 1/3 = y^"}, {"start": 104.36, "duration": 11.32, "text": "2 okay so from here otk x which is x^ 1"}, {"start": 108.88, "duration": 10.0, "text": "/ 2 will become y^ 3 = 2 * by x^ 1 / 3"}, {"start": 115.68, "duration": 8.96, "text": "which is y^ 2 move this term to the left"}, {"start": 118.88, "duration": 7.64, "text": "so we get y Cub - 2 y^ 2 this = 0 so we"}, {"start": 124.64, "duration": 5.039, "text": "have something common here and that is"}, {"start": 126.52, "duration": 9.799, "text": "y^ 2 we factor that out and we left with"}, {"start": 129.679, "duration": 11.28, "text": "y - 2 then the whole of this equals z"}, {"start": 136.319, "duration": 10.761, "text": "right okay now from here we have that"}, {"start": 140.959, "duration": 10.681, "text": "y^2 = 0 we have from the bracket y - 2 ="}, {"start": 147.08, "duration": 9.0, "text": "0 so from here you know this is l y = 0"}, {"start": 151.64, "duration": 8.4, "text": "and from here Y = 2 now let us recall"}, {"start": 156.08, "duration": 6.48, "text": "again that x = y^ 6 so we have the first"}, {"start": 160.04, "duration": 6.199, "text": "value of x let's call it X1 let's call"}, {"start": 162.56, "duration": 7.24, "text": "This y1 And here Y 2 right so for my"}, {"start": 166.239, "duration": 6.561, "text": "first solution for x x = 0 and is a"}, {"start": 169.8, "duration": 6.0, "text": "valid solution right and for the second"}, {"start": 172.8, "duration": 7.92, "text": "solution which is X2 this"}, {"start": 175.8, "duration": 8.76, "text": "equals y ^ 6 which is 2^ 6 and 2 6 will"}, {"start": 180.72, "duration": 6.32, "text": "give us 64 and here we found the"}, {"start": 184.56, "duration": 4.959, "text": "solutions thank you for watching please"}, {"start": 187.04, "duration": 6.6, "text": "kindly subscribe to my channel also like"}, {"start": 189.519, "duration": 4.121, "text": "comment and share bye-bye"}], "title": "Interesting exponential equation", "description": "", "lengthSeconds": 192, "width": 640, "height": 360}
{"video_id": "wjade-yR3uY", "transcript": [{"start": 0.48, "duration": 4.6, "text": "hello welcome back once again today we"}, {"start": 2.44, "duration": 7.199, "text": "have this nice radical equation we have"}, {"start": 5.08, "duration": 7.36, "text": "root of root of x = 1 / root of x and we"}, {"start": 9.639, "duration": 5.2, "text": "are given that X is not equal to Z"}, {"start": 12.44, "duration": 4.96, "text": "without even letting us know we all know"}, {"start": 14.839, "duration": 5.241, "text": "that obviously X must not be equal to"}, {"start": 17.4, "duration": 5.6, "text": "zero right so now let's get into this"}, {"start": 20.08, "duration": 10.48, "text": "video so we have here the square root of"}, {"start": 23.0, "duration": 9.759, "text": "root of x this is equal to 1 / root of x"}, {"start": 30.56, "duration": 5.88, "text": "right so now let's go ahead and square"}, {"start": 32.759, "duration": 6.001, "text": "both sides because this is involving the"}, {"start": 36.44, "duration": 5.2, "text": "root sign we all know that the square"}, {"start": 38.76, "duration": 5.52, "text": "and the root signs are enemies right"}, {"start": 41.64, "duration": 3.96, "text": "they always get cancel out so here let's"}, {"start": 44.28, "duration": 4.52, "text": "Square both"}, {"start": 45.6, "duration": 6.279, "text": "sides so we Square both"}, {"start": 48.8, "duration": 5.399, "text": "sides we Square both sides and for the"}, {"start": 51.879, "duration": 5.121, "text": "right hand side here let us take note of"}, {"start": 54.199, "duration": 7.961, "text": "this a /"}, {"start": 57.0, "duration": 10.119, "text": "B to the power of n this is equal to a^"}, {"start": 62.16, "duration": 6.999, "text": "n / p the^ n so but from the left hand"}, {"start": 67.119, "duration": 3.481, "text": "side we know this square and this root"}, {"start": 69.159, "duration": 5.121, "text": "sign they are enemies so they will"}, {"start": 70.6, "duration": 8.12, "text": "cancel out and we are left with one root"}, {"start": 74.28, "duration": 7.4, "text": "sign under this X or X is under 8 I mean"}, {"start": 78.72, "duration": 7.56, "text": "so here this will be equal to 1 s which"}, {"start": 81.68, "duration": 8.759, "text": "is 1 / root of x\u00b2 remember the square"}, {"start": 86.28, "duration": 5.879, "text": "will cancel this root so we have 1 / X"}, {"start": 90.439, "duration": 3.68, "text": "so in our next step let's go ahead and"}, {"start": 92.159, "duration": 5.721, "text": "square both sides again because of this"}, {"start": 94.119, "duration": 7.121, "text": "root sign here so Square both sides and"}, {"start": 97.88, "duration": 8.64, "text": "here Square both sides so this cancel"}, {"start": 101.24, "duration": 10.879, "text": "this so we have here x is = to 1 2 is 1"}, {"start": 106.52, "duration": 10.559, "text": "/ X2 which is x^ 2 so in our next step"}, {"start": 112.119, "duration": 10.761, "text": "Let's cross multiply so this we multiply"}, {"start": 117.079, "duration": 9.521, "text": "this and this we multiply this right so"}, {"start": 122.88, "duration": 7.28, "text": "here x\u00b2 * X will give us x^ 3 this will"}, {"start": 126.6, "duration": 6.12, "text": "be equal to 1 * 1 which is 1 so let's"}, {"start": 130.16, "duration": 7.52, "text": "subtract one from both side we get x^ 3"}, {"start": 132.72, "duration": 8.36, "text": "- 1 this is equal to 0 so from here x"}, {"start": 137.68, "duration": 6.279, "text": "Cub - 1 is same as 1 Cub this is equal"}, {"start": 141.08, "duration": 5.159, "text": "to Z so from here let's go ahead and"}, {"start": 143.959, "duration": 7.721, "text": "make use of the difference of two cubes"}, {"start": 146.239, "duration": 12.64, "text": "formula a cub minus B Cub this is = to a"}, {"start": 151.68, "duration": 10.639, "text": "- b into braet a 2 + a b + b 2 so make"}, {"start": 158.879, "duration": 8.08, "text": "use of that we have here x -1 into braet"}, {"start": 162.319, "duration": 6.84, "text": "X2 + 1 * X which is x + 1 s which is one"}, {"start": 166.959, "duration": 3.961, "text": "then this is equal to Z so whenever we"}, {"start": 169.159, "duration": 4.921, "text": "have these two products which is equal"}, {"start": 170.92, "duration": 5.92, "text": "to Zer so we all think about this that"}, {"start": 174.08, "duration": 6.96, "text": "either this is equal to Z or this right"}, {"start": 176.84, "duration": 9.24, "text": "over here is equal to Z so when X - 1 is"}, {"start": 181.04, "duration": 8.24, "text": "= 0 we found our real solution that X is"}, {"start": 186.08, "duration": 4.92, "text": "= 1 and this is the only real solution"}, {"start": 189.28, "duration": 6.0, "text": "and for this quadratic expression we're"}, {"start": 191.0, "duration": 6.68, "text": "going to equate it equal zero right so"}, {"start": 195.28, "duration": 3.679, "text": "here we form this quadratic equation"}, {"start": 197.68, "duration": 3.839, "text": "which we're going to solve with the"}, {"start": 198.959, "duration": 6.881, "text": "general quadratic formulative B here"}, {"start": 201.519, "duration": 8.64, "text": "will be1 plus or minus b s b s is 1 - 4"}, {"start": 205.84, "duration": 8.679, "text": "a which is -3 of -3 is Iota root of"}, {"start": 210.159, "duration": 6.841, "text": "three then divid by two here we have the"}, {"start": 214.519, "duration": 4.92, "text": "two complex"}, {"start": 217.0, "duration": 5.4, "text": "Solutions thank you for watching please"}, {"start": 219.439, "duration": 7.641, "text": "kindly subscribe to my channel also like"}, {"start": 222.4, "duration": 4.68, "text": "comment and share bye-bye"}], "title": "Interesting radical equation | Can you solve ?", "description": "", "lengthSeconds": 226, "width": 640, "height": 360}
{"video_id": "7TQ4_0X2eLA", "transcript": [{"start": 0.84, "duration": 4.8, "text": "hello good day viewers welcome back once"}, {"start": 2.56, "duration": 5.199, "text": "again today we have this nice cubic"}, {"start": 5.64, "duration": 5.999, "text": "equation and here we'll be solving for"}, {"start": 7.759, "duration": 8.801, "text": "the values of X now let's get started"}, {"start": 11.639, "duration": 13.601, "text": "from here we have X cubed plus"}, {"start": 16.56, "duration": 8.68, "text": "x\u00b2 + x this is equal to Z okay now from"}, {"start": 26.24, "duration": 4.199, "text": "here many student may think that the"}, {"start": 28.519, "duration": 6.241, "text": "best way to approach this problem from"}, {"start": 30.439, "duration": 7.521, "text": "here is to divide through by X but if"}, {"start": 34.76, "duration": 6.959, "text": "you do that you will forget a solution"}, {"start": 37.96, "duration": 5.8, "text": "that X is equal to Z and this solution"}, {"start": 41.719, "duration": 5.081, "text": "you're definitely not going to have this"}, {"start": 43.76, "duration": 4.92, "text": "solution again so what do we do from"}, {"start": 46.8, "duration": 6.32, "text": "here look at the left hand side of this"}, {"start": 48.68, "duration": 8.8, "text": "equation x is common so we Factor it out"}, {"start": 53.12, "duration": 8.239, "text": "so here x Cub / X will give us"}, {"start": 57.48, "duration": 9.08, "text": "x2+ x\u00b2 / X will give us"}, {"start": 61.359, "duration": 8.841, "text": "X+ x / X that is 1 then this is equal to"}, {"start": 66.56, "duration": 6.36, "text": "Z okay now here we have this product to"}, {"start": 70.2, "duration": 5.52, "text": "be equal to Z then we have either X is"}, {"start": 72.92, "duration": 10.0, "text": "equal to0 so we can see we have this"}, {"start": 75.72, "duration": 10.2, "text": "solution here or the bracket x\u00b2 + x + 1"}, {"start": 82.92, "duration": 5.199, "text": "is = 0 so from here we're going to make"}, {"start": 85.92, "duration": 4.839, "text": "use of the general quadratic formula so"}, {"start": 88.119, "duration": 6.841, "text": "here we get X isal to B in this case"}, {"start": 90.759, "duration": 8.801, "text": "will be -1 plus or minus Square Ro TK of"}, {"start": 94.96, "duration": 9.88, "text": "b s will be 1 s which is 1us 4 * a * c"}, {"start": 99.56, "duration": 8.199, "text": "will give us 4 then ID 2 so from here"}, {"start": 104.84, "duration": 8.8, "text": "this is equal to1 + or"}, {"start": 107.759, "duration": 12.241, "text": "minus < TK of -3 / 2 now simplifying we"}, {"start": 113.64, "duration": 11.799, "text": "get X is = -1 + or minus now here s of1"}, {"start": 120.0, "duration": 6.479, "text": "is Iota and we have root of 3 then / 2"}, {"start": 125.439, "duration": 5.641, "text": "therefore we have the following"}, {"start": 126.479, "duration": 9.241, "text": "solutions for X so we have x is = 0 and"}, {"start": 131.08, "duration": 10.159, "text": "x is also = to -1 + or minus Iota root"}, {"start": 135.72, "duration": 8.32, "text": "of 3 / by 2 thank you for watching"}, {"start": 141.239, "duration": 7.601, "text": "please kindly subscribe to my channel"}, {"start": 144.04, "duration": 4.8, "text": "also like comment and share bye-bye"}], "title": "Interesting cubic equation", "description": "", "lengthSeconds": 147, "width": 640, "height": 360}
{"video_id": "Gl_9xzzpJog", "transcript": [{"start": 0.88, "duration": 4.16, "text": "hello good viewers welcome back once"}, {"start": 2.679, "duration": 5.08, "text": "again today we have this interesting"}, {"start": 5.04, "duration": 7.479, "text": "Olympiad mathematics problem involving"}, {"start": 7.759, "duration": 8.921, "text": "the sub factor of 4 here we have 6^ of <"}, {"start": 12.519, "duration": 7.201, "text": "TK of x^ X this is equal to 4 * the subf"}, {"start": 16.68, "duration": 5.32, "text": "factoral of four here we'll be solving"}, {"start": 19.72, "duration": 6.36, "text": "for the R value of"}, {"start": 22.0, "duration": 6.8, "text": "x what is this subf factor of four if"}, {"start": 26.08, "duration": 5.679, "text": "you check my videos I've made video"}, {"start": 28.8, "duration": 5.2, "text": "about this particular topic the sub"}, {"start": 31.759, "duration": 5.401, "text": "factoral but I'm going to talk less"}, {"start": 34.0, "duration": 6.76, "text": "about this here we"}, {"start": 37.16, "duration": 6.96, "text": "have the property here we have the sub"}, {"start": 40.76, "duration": 8.119, "text": "factoral of n or N Sub factoral is equal"}, {"start": 44.12, "duration": 10.68, "text": "to n factorial multip by this sum as K"}, {"start": 48.879, "duration": 8.281, "text": "goes from 0 to n of -1 to the power of k"}, {"start": 54.8, "duration": 4.48, "text": "/ K"}, {"start": 57.16, "duration": 5.399, "text": "factorial we also have the rough"}, {"start": 59.28, "duration": 6.0, "text": "approximation for the sub factorial of n"}, {"start": 62.559, "duration": 5.121, "text": "which is approximately equal to n"}, {"start": 65.28, "duration": 5.96, "text": "factorial /"}, {"start": 67.68, "duration": 6.0, "text": "a now the sub factoral of four using"}, {"start": 71.24, "duration": 4.0, "text": "this s of approximation here is"}, {"start": 73.68, "duration": 4.68, "text": "approximately equal to the four"}, {"start": 75.24, "duration": 6.72, "text": "factorial ID a and this is approximately"}, {"start": 78.36, "duration": 7.799, "text": "equal to 9 but when we use this we get"}, {"start": 81.96, "duration": 8.44, "text": "exactly 9 so therefore this right over"}, {"start": 86.159, "duration": 6.64, "text": "here is equal to 9 therefore 4 * 9 will"}, {"start": 90.4, "duration": 3.64, "text": "give us 36 therefore the left hand side"}, {"start": 92.799, "duration": 3.881, "text": "is equal to"}, {"start": 94.04, "duration": 5.96, "text": "36 now let's go ahead and solve the"}, {"start": 96.68, "duration": 8.32, "text": "problem so here we have 6^ of theare <"}, {"start": 100.0, "duration": 8.439, "text": "TK of x x this is equal to"}, {"start": 105.0, "duration": 7.479, "text": "36 now from here we have 6^ of the < TK"}, {"start": 108.439, "duration": 5.64, "text": "of x^ X this is equal to 36 is the same"}, {"start": 112.479, "duration": 5.521, "text": "as 6"}, {"start": 114.079, "duration": 8.841, "text": "s now take note of this when we have a^"}, {"start": 118.0, "duration": 9.0, "text": "B = a^ C since they have equal base we"}, {"start": 122.92, "duration": 5.959, "text": "can easily say that b is equal to C so"}, {"start": 127.0, "duration": 3.879, "text": "therefore we have equal bases here so we"}, {"start": 128.879, "duration": 7.64, "text": "can equate the power so here we have the"}, {"start": 130.879, "duration": 7.881, "text": "square root of x of X this is equal to 2"}, {"start": 136.519, "duration": 4.72, "text": "to get rid of this root sign we need to"}, {"start": 138.76, "duration": 4.759, "text": "square both sides of this"}, {"start": 141.239, "duration": 6.121, "text": "equation so this Square will cancel the"}, {"start": 143.519, "duration": 6.921, "text": "square roots so we have here x of X this"}, {"start": 147.36, "duration": 5.44, "text": "is equal to 2 to the^ 2"}, {"start": 150.44, "duration": 6.92, "text": "so from here we can see that this imply"}, {"start": 152.8, "duration": 7.64, "text": "that X is = 2 and therefore this is the"}, {"start": 157.36, "duration": 5.159, "text": "solution we have X as the base from the"}, {"start": 160.44, "duration": 4.2, "text": "right hand side we have two at the base"}, {"start": 162.519, "duration": 3.561, "text": "the same X which is the power here the"}, {"start": 164.64, "duration": 3.679, "text": "right hand side we have the same two"}, {"start": 166.08, "duration": 5.799, "text": "which is the power here therefore this"}, {"start": 168.319, "duration": 5.761, "text": "simply implies that xal 2 and here is"}, {"start": 171.879, "duration": 4.681, "text": "the solution thank you for watching"}, {"start": 174.08, "duration": 7.159, "text": "please kindly subscribe to my channel"}, {"start": 176.56, "duration": 4.679, "text": "also like comment and share bye-bye"}], "title": "Interesting exponential equation involving subfactorial.", "description": "", "lengthSeconds": 180, "width": 640, "height": 360}
{"video_id": "doOhKIOKqkg", "transcript": [{"start": 0.359, "duration": 3.721, "text": "hello welcome back once again today"}, {"start": 2.24, "duration": 4.28, "text": "we'll be finding the following given"}, {"start": 4.08, "duration": 8.08, "text": "limits we have limits as X is"}, {"start": 6.52, "duration": 8.119, "text": "approaching for of 4 - x / 2 - < TK of"}, {"start": 12.16, "duration": 8.0, "text": "X when we try to make direct"}, {"start": 14.639, "duration": 9.241, "text": "substitution here we get 4 - 4 / 2 - <"}, {"start": 20.16, "duration": 5.84, "text": "TK4 and here we get this 0 all over 0"}, {"start": 23.88, "duration": 4.92, "text": "which is in indetermined"}, {"start": 26.0, "duration": 5.039, "text": "form in this video the most interesting"}, {"start": 28.8, "duration": 5.88, "text": "part is that when we are going to make"}, {"start": 31.039, "duration": 5.04, "text": "use of the litus rule so therefore we're"}, {"start": 34.68, "duration": 3.84, "text": "going to solve this"}, {"start": 36.079, "duration": 4.041, "text": "algebraically so let us start by making"}, {"start": 38.52, "duration": 2.68, "text": "the following substitution so we're"}, {"start": 40.12, "duration": 4.119, "text": "going to"}, {"start": 41.2, "duration": 7.16, "text": "let this square root of x at the bottom"}, {"start": 44.239, "duration": 8.721, "text": "here equals y now Square both sides we"}, {"start": 48.36, "duration": 9.64, "text": "get X will be equal to Y 2 so therefore"}, {"start": 52.96, "duration": 7.88, "text": "our limit will be limit now y will be"}, {"start": 58.0, "duration": 6.199, "text": "approaching remember if X is approaching"}, {"start": 60.84, "duration": 7.279, "text": "4 y will be approaching theare root of 4"}, {"start": 64.199, "duration": 9.721, "text": "which is 2 so here we have 4us instead"}, {"start": 68.119, "duration": 9.601, "text": "of X so we have y^ 2 then / 2us inste of"}, {"start": 73.92, "duration": 7.04, "text": "root of x we replace with Y so from here"}, {"start": 77.72, "duration": 8.359, "text": "this will be equal to limit as Y is"}, {"start": 80.96, "duration": 10.96, "text": "approaching 2 so here 4 is as 2 2 then -"}, {"start": 86.079, "duration": 8.801, "text": "y^ 2 / 2 - y"}, {"start": 91.92, "duration": 4.6, "text": "y so at this point here let us record"}, {"start": 94.88, "duration": 8.64, "text": "the difference of two squares formula"}, {"start": 96.52, "duration": 11.919, "text": "remember a 2us b 2 is equal to a + b *"}, {"start": 103.52, "duration": 10.48, "text": "Aus B so from here we have limit as Y is"}, {"start": 108.439, "duration": 10.161, "text": "approaching 2 so up here we have 2 + y"}, {"start": 114.0, "duration": 7.24, "text": "MTI 2 - Y and down below there we have 2"}, {"start": 118.6, "duration": 5.479, "text": "- y so we can see we have this nice"}, {"start": 121.24, "duration": 8.719, "text": "cancellation here so this is equal to"}, {"start": 124.079, "duration": 7.761, "text": "limit as Y is approaching to of 2 + y so"}, {"start": 129.959, "duration": 6.481, "text": "here we can make our direct substitution"}, {"start": 131.84, "duration": 7.84, "text": "now to get 2 + y here which is 2 which"}, {"start": 136.44, "duration": 5.879, "text": "is equal to 4 and this is the right"}, {"start": 139.68, "duration": 5.52, "text": "answer thank you for watching please"}, {"start": 142.319, "duration": 7.441, "text": "kindly subscribe to my channel also like"}, {"start": 145.2, "duration": 4.56, "text": "comment and share bye-bye"}], "title": "Very nice Limit | No L'hospital's rule !", "description": "", "lengthSeconds": 148, "width": 640, "height": 360}
{"video_id": "FzwddVpN7Hg", "transcript": [{"start": 0.64, "duration": 6.04, "text": "hello good day viewers today I'll be"}, {"start": 2.76, "duration": 8.6, "text": "solving this nice Junior Olympiad matth"}, {"start": 6.68, "duration": 10.04, "text": "so we have here 2 power of 5 ra power of"}, {"start": 11.36, "duration": 9.52, "text": "< TK of 2 power of 0 is equal to D we"}, {"start": 16.72, "duration": 5.559, "text": "asked to evaluate this right please if"}, {"start": 20.88, "duration": 3.96, "text": "you have not subscribed to my YouTube"}, {"start": 22.279, "duration": 5.521, "text": "Channel please kindly support this"}, {"start": 24.84, "duration": 7.439, "text": "channel by subscribing and also share my"}, {"start": 27.8, "duration": 7.32, "text": "videos right how do we evaluate this so"}, {"start": 32.279, "duration": 6.361, "text": "I'm going to start with square < TK of"}, {"start": 35.12, "duration": 8.36, "text": "2^ of 0 right over here right so let us"}, {"start": 38.64, "duration": 9.559, "text": "recall that X power of 0 is always 1"}, {"start": 43.48, "duration": 9.8, "text": "provided that X is not equal to 0 right"}, {"start": 48.199, "duration": 7.36, "text": "and here we have sare < TK 2^ of 0 here"}, {"start": 53.28, "duration": 4.959, "text": "we see that square root of 2 is not"}, {"start": 55.559, "duration": 7.32, "text": "equal to zero right so this is going to"}, {"start": 58.239, "duration": 10.321, "text": "be equal to one right so the expression"}, {"start": 62.879, "duration": 9.841, "text": "2 5 otk of 2 0 is going to be the same"}, {"start": 68.56, "duration": 7.44, "text": "thing as 2^ of 5 then power of"}, {"start": 72.72, "duration": 5.759, "text": "1 so this is going to be equal to now"}, {"start": 76.0, "duration": 6.159, "text": "take this power from here we only know"}, {"start": 78.479, "duration": 5.401, "text": "that 5 power of 1 is equal to 5 right so"}, {"start": 82.159, "duration": 9.28, "text": "this going to be equal to"}, {"start": 83.88, "duration": 10.96, "text": "2 5 and 2 5 is the same as 2 * 2 * 2 2 *"}, {"start": 91.439, "duration": 6.68, "text": "2 * 2 that is in five places and this"}, {"start": 94.84, "duration": 6.2, "text": "resulted to 32 and this is the right"}, {"start": 98.119, "duration": 5.401, "text": "answer thank you for watching please do"}, {"start": 101.04, "duration": 6.8, "text": "well to subscribe to my YouTube channel"}, {"start": 103.52, "duration": 4.32, "text": "and also share my videos"}], "title": "Junior Olympiad Math. How to deal? #maths #olympics", "description": "", "lengthSeconds": 106, "width": 636, "height": 360}
{"video_id": "29Eyh24KYmI", "transcript": [{"start": 1.0, "duration": 4.48, "text": "hello welcome back once again today"}, {"start": 3.199, "duration": 6.561, "text": "we're going to solve this interesting"}, {"start": 5.48, "duration": 8.079, "text": "equation so we have X - 1 to^ 2 all 3"}, {"start": 9.76, "duration": 5.6, "text": "this is equal to 25 so in our video"}, {"start": 13.559, "duration": 5.8, "text": "we're going to solve for the value of"}, {"start": 15.36, "duration": 5.96, "text": "values of X so let's get started from"}, {"start": 19.359, "duration": 6.601, "text": "the left hand side of this equation we"}, {"start": 21.32, "duration": 10.359, "text": "can rewrite it as follow we have X - 1"}, {"start": 25.96, "duration": 8.04, "text": "then 2 over 3 is same as 2 * 1/ 3 then"}, {"start": 31.679, "duration": 3.88, "text": "from the right hand side this is equal"}, {"start": 34.0, "duration": 5.0, "text": "to"}, {"start": 35.559, "duration": 8.041, "text": "25 now let us make use of this property"}, {"start": 39.0, "duration": 8.44, "text": "we know that a to the power B * C this"}, {"start": 43.6, "duration": 7.08, "text": "is equal to a^ B in bracket to the power"}, {"start": 47.44, "duration": 7.24, "text": "of C this is also equal to a to the"}, {"start": 50.68, "duration": 5.679, "text": "power C in bracket to the power of B so"}, {"start": 54.68, "duration": 5.44, "text": "using this property the left hand side"}, {"start": 56.359, "duration": 11.321, "text": "of this equation we can write it as X X"}, {"start": 60.12, "duration": 10.4, "text": "- 1 to the^ 1 / 3 in bracket to the^ 2"}, {"start": 67.68, "duration": 5.32, "text": "this is equal to"}, {"start": 70.52, "duration": 4.279, "text": "25 now in our next step let's take the"}, {"start": 73.0, "duration": 8.0, "text": "square root of both sides so the left"}, {"start": 74.799, "duration": 9.081, "text": "hand side will become x -1 to^ 1 / 3"}, {"start": 81.0, "duration": 5.159, "text": "this is equal to plus or minus the"}, {"start": 83.88, "duration": 9.239, "text": "square root of"}, {"start": 86.159, "duration": 11.041, "text": "25 from here we have x - 1 to the^ 1 / 3"}, {"start": 93.119, "duration": 8.36, "text": "this is equal to plus or minus >"}, {"start": 97.2, "duration": 5.239, "text": "25 is five right so from here we have"}, {"start": 101.479, "duration": 6.721, "text": "two"}, {"start": 102.439, "duration": 10.401, "text": "equations we have the first one x -1 to^"}, {"start": 108.2, "duration": 7.76, "text": "1 / 3 is = to POS 5 and we have the"}, {"start": 112.84, "duration": 8.559, "text": "second one here x -"}, {"start": 115.96, "duration": 8.32, "text": "1^ 1 / 3 is = to"}, {"start": 121.399, "duration": 4.921, "text": "5 now in our next step let's CU both"}, {"start": 124.28, "duration": 9.12, "text": "sides so we"}, {"start": 126.32, "duration": 11.24, "text": "have X - 1 to the^ 1 / 3 cubed this is"}, {"start": 133.4, "duration": 11.0, "text": "equal to 5 cubed and from"}, {"start": 137.56, "duration": 7.8, "text": "here x -1 to^ of 1/ 3 cubed this is"}, {"start": 144.4, "duration": 2.839, "text": "equal"}, {"start": 145.36, "duration": 7.4, "text": "to"}, {"start": 147.239, "duration": 11.0, "text": "-5 cubed so we know 1 3 * 3 is 1 right"}, {"start": 152.76, "duration": 8.36, "text": "so 3 will cancel 3 so we have X - 1 is ="}, {"start": 158.239, "duration": 5.321, "text": "5 cubed is"}, {"start": 161.12, "duration": 7.0, "text": "125 from here we know three we cancel"}, {"start": 163.56, "duration": 7.72, "text": "three here so we have X - 1 is equal to"}, {"start": 168.12, "duration": 3.16, "text": "-5 cubed will give"}, {"start": 171.36, "duration": 5.48, "text": "us15 because the power is"}, {"start": 174.04, "duration": 6.96, "text": "odd then from here add one to both sides"}, {"start": 176.84, "duration": 6.52, "text": "so we get X is = to 125 + 1 which is"}, {"start": 181.0, "duration": 5.159, "text": "equal to"}, {"start": 183.36, "duration": 5.28, "text": "126 so this is a"}, {"start": 186.159, "duration": 7.401, "text": "solution then from here add one to both"}, {"start": 188.64, "duration": 7.08, "text": "sides so we get X is = to -125 + 1 we"}, {"start": 193.56, "duration": 2.16, "text": "give"}, {"start": 195.76, "duration": 6.0, "text": "us14 and this is our second solution"}, {"start": 200.239, "duration": 4.041, "text": "thank you for watching please kindly"}, {"start": 201.76, "duration": 7.119, "text": "subscribe to my channel also like"}, {"start": 204.28, "duration": 4.599, "text": "comment and share bye-bye"}], "title": "Interesting equation with fractional power | Can you solve ?", "description": "", "lengthSeconds": 207, "width": 428, "height": 240}
{"video_id": "ejpu3WtYPh0", "transcript": [{"start": 0.799, "duration": 7.201, "text": "hello welcome back once again today we"}, {"start": 3.56, "duration": 6.159, "text": "have to solve this problem right when"}, {"start": 8.0, "duration": 3.24, "text": "students always make mistake when"}, {"start": 9.719, "duration": 3.601, "text": "solving this"}, {"start": 11.24, "duration": 3.72, "text": "problem most students think that the"}, {"start": 13.32, "duration": 6.2, "text": "best way to solve this problem is to"}, {"start": 14.96, "duration": 6.399, "text": "First div B by x\u00b2 right but if you do so"}, {"start": 19.52, "duration": 3.96, "text": "you miss some"}, {"start": 21.359, "duration": 3.281, "text": "solutions so this is how we're going to"}, {"start": 23.48, "duration": 6.24, "text": "solve this"}, {"start": 24.64, "duration": 9.48, "text": "problem we have x to^ 6 which is equal"}, {"start": 29.72, "duration": 8.96, "text": "to 2 x^ 2 this time let's take it to the"}, {"start": 34.12, "duration": 8.48, "text": "left hand side we have x ^ 6 - 2"}, {"start": 38.68, "duration": 6.16, "text": "x\u00b2 then this is equal to Z from the left"}, {"start": 42.6, "duration": 8.279, "text": "hand side X2 is common so let's Factor"}, {"start": 44.84, "duration": 9.84, "text": "it out so we have x 2 into braet x ^ 4 -"}, {"start": 50.879, "duration": 6.52, "text": "2 this is equal to 0 from here we have"}, {"start": 54.68, "duration": 6.679, "text": "two products which is equal to zero so"}, {"start": 57.399, "duration": 11.561, "text": "this implies from here that either x 2"}, {"start": 61.359, "duration": 12.08, "text": "is = 0 or x^ 4 - 2 is = 0 from here take"}, {"start": 68.96, "duration": 8.4, "text": "root of both side we get X isal to 0"}, {"start": 73.439, "duration": 7.201, "text": "right so this is one of the solution we"}, {"start": 77.36, "duration": 8.32, "text": "would have missed if we started by"}, {"start": 80.64, "duration": 7.68, "text": "dividing both sides by x\u00b2 right so here"}, {"start": 85.68, "duration": 6.64, "text": "we're going to manipulate the left hand"}, {"start": 88.32, "duration": 7.159, "text": "side so let us reite this as x^2 to the^"}, {"start": 92.32, "duration": 6.159, "text": "2 we know 2 * 2 is 4 then minus rewrite"}, {"start": 95.479, "duration": 5.361, "text": "this one as < tk2 2 we know Square"}, {"start": 98.479, "duration": 5.401, "text": "cancel square root then this is equal to"}, {"start": 100.84, "duration": 9.879, "text": "Zer you know why now we can make use of"}, {"start": 103.88, "duration": 8.96, "text": "this formula a s - b 2 = A + B * Aus B"}, {"start": 110.719, "duration": 4.841, "text": "called the difference of two squares"}, {"start": 112.84, "duration": 4.919, "text": "right so without formula we get"}, {"start": 115.56, "duration": 4.96, "text": "x2 + <"}, {"start": 117.759, "duration": 7.241, "text": "tk2 * X"}, {"start": 120.52, "duration": 9.0, "text": "- < tk2 then this is equal Z again this"}, {"start": 125.0, "duration": 12.28, "text": "implies from here that either X2 + < tk2"}, {"start": 129.52, "duration": 11.079, "text": "is = 0 or X2 - < tk2 is = 0 now subtract"}, {"start": 137.28, "duration": 6.08, "text": "root2 from both side here we get X = <"}, {"start": 140.599, "duration": 5.081, "text": "tk2 and here add root2 to both side we"}, {"start": 143.36, "duration": 4.56, "text": "get x2 isal"}, {"start": 145.68, "duration": 7.52, "text": "tok2 take theare root of both side we"}, {"start": 147.92, "duration": 7.8, "text": "get xal plus or minus theun of < tk2"}, {"start": 153.2, "duration": 7.119, "text": "again here take the of both get X Plus"}, {"start": 155.72, "duration": 8.159, "text": "or theun ofun of two from here"}, {"start": 160.319, "duration": 6.92, "text": "simplifying we get xal plus or minus of1"}, {"start": 163.879, "duration": 6.36, "text": "is aota and of root2 is same as the 4"}, {"start": 167.239, "duration": 6.36, "text": "root of 2 please do do not forget and"}, {"start": 170.239, "duration": 9.241, "text": "here we get xal plus or minus the 4th"}, {"start": 173.599, "duration": 7.56, "text": "root of 2 remember 0 is a repeated"}, {"start": 179.48, "duration": 7.56, "text": "solution right so here we have the"}, {"start": 181.159, "duration": 11.761, "text": "following solutions for x x = 0 x = 4un"}, {"start": 187.04, "duration": 13.08, "text": "2 x = 4ot of 2 then imaginary Solutions"}, {"start": 192.92, "duration": 10.879, "text": "X = iot posi 4ot of 2 and X = Iota 4"}, {"start": 200.12, "duration": 5.08, "text": "root of 2 and these are the solutions"}, {"start": 203.799, "duration": 3.841, "text": "thank you for watching please kindly"}, {"start": 205.2, "duration": 7.679, "text": "subscribe to my channel also like"}, {"start": 207.64, "duration": 5.239, "text": "comment and share bye-bye he"}], "title": "Interesting hexic equation | You will probably miss a solution !!!", "description": "", "lengthSeconds": 211, "width": 428, "height": 240}
{"video_id": "jZ7jN7Ec3yo", "transcript": [{"start": 0.52, "duration": 5.52, "text": "let us find a solution to this problem"}, {"start": 4.56, "duration": 4.36, "text": "please if today is your first day of"}, {"start": 6.04, "duration": 5.24, "text": "visiting this channel kindly support us"}, {"start": 8.92, "duration": 3.879, "text": "by subscribing to this YouTube Channel"}, {"start": 11.28, "duration": 4.239, "text": "please subscribe to this Channel and"}, {"start": 12.799, "duration": 4.48, "text": "share this video here we are giving this"}, {"start": 15.519, "duration": 8.08, "text": "nice exponential"}, {"start": 17.279, "duration": 8.281, "text": "problem 5^ 2 n^ 2 - 1 is ="}, {"start": 23.599, "duration": 5.361, "text": "to"}, {"start": 25.56, "duration": 7.8, "text": "125 okay here we know that"}, {"start": 28.96, "duration": 7.68, "text": "125 is the same as 5^ 3 so let us go"}, {"start": 33.36, "duration": 10.48, "text": "ahead and replace that with 5^ 3 so here"}, {"start": 36.64, "duration": 9.96, "text": "we have 5^ of 2 n^ 2 - 1 = 5^ 3 so we"}, {"start": 43.84, "duration": 6.16, "text": "have equal bases right from both Sid so"}, {"start": 46.6, "duration": 8.08, "text": "we can easily equate the powers right so"}, {"start": 50.0, "duration": 8.559, "text": "here we have 2 n^ 2 - 1 is = 3 add one"}, {"start": 54.68, "duration": 6.559, "text": "to both side we get 2 n\u00b2 is = to 4"}, {"start": 58.559, "duration": 5.0, "text": "divide both side by 2"}, {"start": 61.239, "duration": 5.081, "text": "here over two and here over two we get"}, {"start": 63.559, "duration": 6.401, "text": "this canceled out we left with n^2 is ="}, {"start": 66.32, "duration": 8.2, "text": "4 2 is 2 so let us take the root of both"}, {"start": 69.96, "duration": 7.36, "text": "sides so here we have n = plus or minus"}, {"start": 74.52, "duration": 4.639, "text": "the sare RO TK of 2 right and this is"}, {"start": 77.32, "duration": 6.72, "text": "the right"}, {"start": 79.159, "duration": 4.881, "text": "answer thank you for watching do well to"}, {"start": 84.24, "duration": 5.32, "text": "subscribe and share the video thank you"}], "title": "An exponential equation.", "description": "", "lengthSeconds": 88, "width": 636, "height": 360}
{"video_id": "jbBl8htBf1M", "transcript": [{"start": 0.64, "duration": 4.44, "text": "hello good viewers welcome back once"}, {"start": 2.639, "duration": 3.72, "text": "again let's find a solution to this"}, {"start": 5.08, "duration": 4.8, "text": "sodic"}, {"start": 6.359, "duration": 7.601, "text": "equation root of x - root of 6 this is"}, {"start": 9.88, "duration": 7.6, "text": "equal to root of 24 now we can easily"}, {"start": 13.96, "duration": 8.04, "text": "rewrite root of 24 in terms of root of 6"}, {"start": 17.48, "duration": 7.76, "text": "if we recall that 24 is same as 4 * 6"}, {"start": 22.0, "duration": 8.119, "text": "therefore root of 24 will be same as <"}, {"start": 25.24, "duration": 8.72, "text": "TK 4 * 6 which is same as otk of 4"}, {"start": 30.119, "duration": 7.6, "text": "multip by root of 6 from here we get 2 <"}, {"start": 33.96, "duration": 10.32, "text": "TK 6 so from here our equation will"}, {"start": 37.719, "duration": 9.201, "text": "become < TK x - otk 6 this is = 2un 6"}, {"start": 44.28, "duration": 10.2, "text": "now we add root of 6 to both sides we"}, {"start": 46.92, "duration": 12.56, "text": "get otk X is = 2un 6 +un 6 from here we"}, {"start": 54.48, "duration": 7.36, "text": "get root of x is = 3 < TK 6 from here we"}, {"start": 59.48, "duration": 4.56, "text": "go and and square both side to get rid"}, {"start": 61.84, "duration": 4.8, "text": "of this root"}, {"start": 64.04, "duration": 7.28, "text": "sign the square and the root will cancel"}, {"start": 66.64, "duration": 12.2, "text": "out we get X is equal to here 3un 6 2 is"}, {"start": 71.32, "duration": 11.92, "text": "same as 3un 6 * 3un 6 now simplifying 3"}, {"start": 78.84, "duration": 8.0, "text": "* 3 here is 9 then * root of 6 * root of"}, {"start": 83.24, "duration": 6.879, "text": "6 will give us root of 36 from here this"}, {"start": 86.84, "duration": 5.36, "text": "is 9 * root of 36 which is 6 and and"}, {"start": 90.119, "duration": 6.96, "text": "from here oh let me use"}, {"start": 92.2, "duration": 7.72, "text": "this so from here 9 * 6 is 54 right so"}, {"start": 97.079, "duration": 6.801, "text": "here we come up with the solution X isal"}, {"start": 99.92, "duration": 5.64, "text": "to 54 and this is the right answer thank"}, {"start": 103.88, "duration": 3.839, "text": "you for watching please kindly subscribe"}, {"start": 105.56, "duration": 5.8, "text": "to my channel also like comment and"}, {"start": 107.719, "duration": 3.641, "text": "share bye-bye"}], "title": "Easy Surdic Equation | #maths #mathematics", "description": "", "lengthSeconds": 111, "width": 600, "height": 360}
{"video_id": "NeKeU0sm160", "transcript": [{"start": 1.0, "duration": 3.56, "text": "hello good viewers welcome back once"}, {"start": 2.8, "duration": 6.52, "text": "again today we're given the following"}, {"start": 4.56, "duration": 8.119, "text": "exponential equation x^ x^ 4 = 4 here"}, {"start": 9.32, "duration": 8.56, "text": "we're going to evaluate this expression"}, {"start": 12.679, "duration": 8.281, "text": "x^ x^ 8 + x^ X the power of"}, {"start": 17.88, "duration": 5.68, "text": "2 now there's a way I want to solve this"}, {"start": 20.96, "duration": 7.12, "text": "equation and I think that would be very"}, {"start": 23.56, "duration": 12.159, "text": "cool right this equals 4 I'm going to"}, {"start": 28.08, "duration": 12.28, "text": "let X to the^ 4 = Y and obviously X = Y"}, {"start": 35.719, "duration": 7.801, "text": "to^ 1 / 4 so from here remember this"}, {"start": 40.36, "duration": 8.039, "text": "base here has the power of x^ 4 so X"}, {"start": 43.52, "duration": 8.839, "text": "here will be y^ 1/ 4 then all the^ of x^"}, {"start": 48.399, "duration": 8.64, "text": "4 which is y then this equals"}, {"start": 52.359, "duration": 9.68, "text": "4 from here with this property a^ b^ c ="}, {"start": 57.039, "duration": 8.561, "text": "a^ b * C now we have y^ 1 4 * Y is the"}, {"start": 62.039, "duration": 5.72, "text": "same as y / 4 then this is equal"}, {"start": 65.6, "duration": 4.92, "text": "to"}, {"start": 67.759, "duration": 5.521, "text": "4 from here let's raise both side to the"}, {"start": 70.52, "duration": 6.76, "text": "power of four I have reason for"}, {"start": 73.28, "duration": 6.96, "text": "that whatever I do I have good raising"}, {"start": 77.28, "duration": 4.72, "text": "so ra 4 and from here we can multiply"}, {"start": 80.24, "duration": 7.48, "text": "this and four we cancel for we left it"}, {"start": 82.0, "duration": 9.439, "text": "y^ y = 4^ 4 nice reason so from here"}, {"start": 87.72, "duration": 8.48, "text": "this implies that y = 4 and that implies"}, {"start": 91.439, "duration": 9.04, "text": "also that x = y^ 1 4 which is 4^ 1 4"}, {"start": 96.2, "duration": 5.8, "text": "remember 4 is as 2 2 then to^ 1 / 4"}, {"start": 100.479, "duration": 5.881, "text": "multiplying these Powers together we get"}, {"start": 102.0, "duration": 7.759, "text": "2^ 2 / 4 which is equal to 2^ 1 / 2"}, {"start": 106.36, "duration": 6.52, "text": "which is equal to the < TK of 2 amazing"}, {"start": 109.759, "duration": 6.0, "text": "so in the given equation here x = the <"}, {"start": 112.88, "duration": 5.559, "text": "TK of 2 now let's go back to the"}, {"start": 115.759, "duration": 5.481, "text": "expression we ask to"}, {"start": 118.439, "duration": 8.801, "text": "evaluate right so this will be equal to"}, {"start": 121.24, "duration": 6.879, "text": "the < TK of 2 of < TK of 2 of 8 plus the"}, {"start": 127.24, "duration": 5.68, "text": "< TK of"}, {"start": 128.119, "duration": 8.4, "text": "2 < TK 2 of 8 of two I mean sorry for"}, {"start": 132.92, "duration": 7.399, "text": "that okay so from"}, {"start": 136.519, "duration": 8.841, "text": "here first of all we know that the < TK"}, {"start": 140.319, "duration": 8.401, "text": "of 2 2 = 2 remember this8 we can Factor"}, {"start": 145.36, "duration": 5.04, "text": "it as 2 * 4 then the root of 2 of the"}, {"start": 148.72, "duration": 4.32, "text": "two will give us two"}, {"start": 150.4, "duration": 5.8, "text": "which implies that of 2^ 8 is the same"}, {"start": 153.04, "duration": 8.839, "text": "as 2^ of"}, {"start": 156.2, "duration": 9.759, "text": "4 so here we have the < TK of 2 2^ 4"}, {"start": 161.879, "duration": 8.841, "text": "then plus the < TK of 2un of 2^ 2 here"}, {"start": 165.959, "duration": 10.241, "text": "will be two so from here again 2^ 4 is"}, {"start": 170.72, "duration": 11.36, "text": "16 right so let's put it well 2 16 +un"}, {"start": 176.2, "duration": 9.319, "text": "2^ 2 from here remember of 2 2 is 2 16"}, {"start": 182.08, "duration": 5.719, "text": "is same as 2 * 8 so this root of 2 of"}, {"start": 185.519, "duration": 5.961, "text": "the 2 will give us two so this will same"}, {"start": 187.799, "duration": 7.241, "text": "as 2 to the^ of that 8 then plus here"}, {"start": 191.48, "duration": 7.96, "text": "this is two right so this equals"}, {"start": 195.04, "duration": 7.759, "text": "256 + 2 which is equal to"}, {"start": 199.44, "duration": 5.04, "text": "258 and this is the right answer thank"}, {"start": 202.799, "duration": 3.72, "text": "you for watching please kindly subscribe"}, {"start": 204.48, "duration": 5.56, "text": "to my channel also like comment and"}, {"start": 206.519, "duration": 3.521, "text": "share bye-bye"}], "title": "Oxford University Entrance Exam", "description": "", "lengthSeconds": 208, "width": 562, "height": 360}
{"video_id": "d1yU_2Peiyo", "transcript": [{"start": 0.76, "duration": 4.88, "text": "hello good viewers welcome back once"}, {"start": 2.56, "duration": 8.56, "text": "again let's solve this exponential"}, {"start": 5.64, "duration": 8.84, "text": "equation very quickly we have 2^ of 2^"}, {"start": 11.12, "duration": 5.8, "text": "of X this is equal to 3 now what we're"}, {"start": 14.48, "duration": 5.52, "text": "going to do is to make use of the"}, {"start": 16.92, "duration": 5.92, "text": "natural logarithm so we're going to Ln"}, {"start": 20.0, "duration": 8.64, "text": "both sides of this equation so we have"}, {"start": 22.84, "duration": 10.719, "text": "here the Ln of 2 of 2 x this is equal to"}, {"start": 28.64, "duration": 9.239, "text": "the Ln of 3 now we call this property of"}, {"start": 33.559, "duration": 8.84, "text": "logarithm that the Ln of a to the^ of B"}, {"start": 37.879, "duration": 7.481, "text": "this is equal to B * the Ln of a right"}, {"start": 42.399, "duration": 6.241, "text": "so here the power of 2 here is 2 to the^"}, {"start": 45.36, "duration": 7.359, "text": "of X so this power here is crossing over"}, {"start": 48.64, "duration": 8.64, "text": "so here we have 2 to the^ of X MTI the"}, {"start": 52.719, "duration": 7.32, "text": "Ln of 2 this is equal to the Ln of"}, {"start": 57.28, "duration": 6.68, "text": "3 so from here we're going to divide"}, {"start": 60.039, "duration": 6.44, "text": "both side by the Ln of two so here we"}, {"start": 63.96, "duration": 5.159, "text": "divide by the Ln of two and from this"}, {"start": 66.479, "duration": 5.881, "text": "side we divide by the Ln of 2 so we can"}, {"start": 69.119, "duration": 7.441, "text": "see that this Ln of 2 and this Ln of two"}, {"start": 72.36, "duration": 9.56, "text": "divide out so here we have 2 ^ of X this"}, {"start": 76.56, "duration": 8.16, "text": "is equal to the Ln of 3 / the Ln of 2"}, {"start": 81.92, "duration": 6.0, "text": "now let us recall another property of"}, {"start": 84.72, "duration": 5.719, "text": "this logarithm again recall that when we"}, {"start": 87.92, "duration": 6.0, "text": "have the log"}, {"start": 90.439, "duration": 9.0, "text": "of a to the power of B to the base a"}, {"start": 93.92, "duration": 7.92, "text": "this simplifies to B right so now from"}, {"start": 99.439, "duration": 6.521, "text": "both sides we're going to take the"}, {"start": 101.84, "duration": 8.12, "text": "logarithm and this log must be in base 2"}, {"start": 105.96, "duration": 7.28, "text": "which is the base of this x here so here"}, {"start": 109.96, "duration": 7.04, "text": "we have to log both side to the base"}, {"start": 113.24, "duration": 7.559, "text": "two and from the right we have the log"}, {"start": 117.0, "duration": 5.119, "text": "of the Ln of 3 / the Ln of 2"}, {"start": 120.799, "duration": 5.521, "text": "to the base"}, {"start": 122.119, "duration": 9.76, "text": "2 so from here with this property here"}, {"start": 126.32, "duration": 10.96, "text": "so this simplifies to X so X is equal to"}, {"start": 131.879, "duration": 8.321, "text": "the log of the Ln of 3 over the L of 2"}, {"start": 137.28, "duration": 6.4, "text": "to the base 2 and this is approximately"}, {"start": 140.2, "duration": 3.48, "text": "equal to"}, {"start": 144.0, "duration": 9.56, "text": "0.66 4 4 4 right then 8 7"}, {"start": 150.56, "duration": 4.8, "text": "one and this is the right answer thank"}, {"start": 153.56, "duration": 4.039, "text": "you for watching please kindly subscribe"}, {"start": 155.36, "duration": 5.76, "text": "to my channel also like comment and"}, {"start": 157.599, "duration": 3.521, "text": "share bye-bye"}], "title": "China | Interesting exponential equation | can you solve ? | #maths #mathematics", "description": "", "lengthSeconds": 159, "width": 640, "height": 360}
{"video_id": "IaBQQ779-R0", "transcript": [{"start": 0.719, "duration": 4.321, "text": "hello good day everyone welcome back"}, {"start": 2.919, "duration": 4.48, "text": "once again today we're going to solve"}, {"start": 5.04, "duration": 8.0, "text": "this equation here we have the fourth"}, {"start": 7.399, "duration": 7.841, "text": "root of x = 12 / 7 - 4 root of x from"}, {"start": 13.04, "duration": 5.319, "text": "here we're going to make a substitution"}, {"start": 15.24, "duration": 5.52, "text": "we're going to let the fourth root of x"}, {"start": 18.359, "duration": 5.201, "text": "equals another variable let's call it y"}, {"start": 20.76, "duration": 8.519, "text": "right so from here if we raise both"}, {"start": 23.56, "duration": 8.199, "text": "sides to the^ of 4 we get X is = y ^ 4"}, {"start": 29.279, "duration": 5.12, "text": "right so so from here we're going to"}, {"start": 31.759, "duration": 5.8, "text": "write this equation in terms of Y so"}, {"start": 34.399, "duration": 9.241, "text": "here 4 root of x will be Y which is"}, {"start": 37.559, "duration": 8.281, "text": "equal to 12 / 7 - 4 root of x which is y"}, {"start": 43.64, "duration": 4.599, "text": "now from here we're going to cross"}, {"start": 45.84, "duration": 3.52, "text": "multiply when we cross multiply we have"}, {"start": 48.239, "duration": 7.361, "text": "y"}, {"start": 49.36, "duration": 8.8, "text": "* 7 - y this will be equal to 1 * 12"}, {"start": 55.6, "duration": 6.759, "text": "which is 12 so from here let's open this"}, {"start": 58.16, "duration": 8.48, "text": "bracket y * 7 will give us 7 y y * y"}, {"start": 62.359, "duration": 7.201, "text": "will give us y^ 2 then we can subtract"}, {"start": 66.64, "duration": 7.32, "text": "12 from here and this will be equal to"}, {"start": 69.56, "duration": 11.96, "text": "Zer From Here let us multiply both sides"}, {"start": 73.96, "duration": 11.519, "text": "by-1 so here1 into bracket 7 y - y^ 2 -"}, {"start": 81.52, "duration": 8.919, "text": "12 this is equal to 0 -"}, {"start": 85.479, "duration": 11.881, "text": "-1 from here we have -1 * 7 y y will"}, {"start": 90.439, "duration": 11.841, "text": "give us -7 y -1 * Y 2 will give us + y^"}, {"start": 97.36, "duration": 7.24, "text": "2 and here1 * -12 will give us + 12 and"}, {"start": 102.28, "duration": 8.159, "text": "this will give us 0 from here let's"}, {"start": 104.6, "duration": 8.479, "text": "rewrite this equation as y^ 2 - 7 y + 12"}, {"start": 110.439, "duration": 6.561, "text": "this will be equal to0 now here we have"}, {"start": 113.079, "duration": 6.281, "text": "a quadratic equation which we can"}, {"start": 117.0, "duration": 5.439, "text": "factorize right so this quadratic"}, {"start": 119.36, "duration": 7.119, "text": "equation is factorizable right so we're"}, {"start": 122.439, "duration": 8.001, "text": "going to look for two terms which when"}, {"start": 126.479, "duration": 7.84, "text": "we multiply them we get 2 y^2 and when"}, {"start": 130.44, "duration": 11.56, "text": "we add them we get -7 Y and these two"}, {"start": 134.319, "duration": 9.841, "text": "terms are -4 Y and -3 y so -7 y we can"}, {"start": 142.0, "duration": 8.319, "text": "write it in terms of these two terms and"}, {"start": 144.16, "duration": 11.48, "text": "that will give us y^ 2 - 4 y - 3 y + +"}, {"start": 150.319, "duration": 9.321, "text": "12 = 0 so from here we factor out y we"}, {"start": 155.64, "duration": 7.64, "text": "have y - 4 and here we factor out -3 and"}, {"start": 159.64, "duration": 6.84, "text": "we get y - 4 and the whole of this equal"}, {"start": 163.28, "duration": 8.64, "text": "0 so from here we get y -"}, {"start": 166.48, "duration": 9.88, "text": "4 * y - 3 which is equal 0 and this"}, {"start": 171.92, "duration": 12.12, "text": "implies that y - 4 is = 0 and from here"}, {"start": 176.36, "duration": 10.799, "text": "Y is = 4 also y - 3 3 = 0 so from here Y"}, {"start": 184.04, "duration": 7.88, "text": "is = 3 now let's recall"}, {"start": 187.159, "duration": 8.44, "text": "that 4 root of x = y of which when we"}, {"start": 191.92, "duration": 6.84, "text": "raise both s to the^ of 4 we get X is ="}, {"start": 195.599, "duration": 7.56, "text": "y to the^ 4 now from here we can get the"}, {"start": 198.76, "duration": 8.24, "text": "two real values of X X1 which is 4 ^ of"}, {"start": 203.159, "duration": 7.041, "text": "4 and this will give us"}, {"start": 207.0, "duration": 6.72, "text": "256 and the second value of x which will"}, {"start": 210.2, "duration": 4.92, "text": "be y to ^ 4 which is 3 to ^ 4 and this"}, {"start": 213.72, "duration": 6.439, "text": "will give"}, {"start": 215.12, "duration": 6.839, "text": "us 81 so these are the solutions thank"}, {"start": 220.159, "duration": 4.041, "text": "you for watching please kindly subscribe"}, {"start": 221.959, "duration": 5.761, "text": "to my channel also like comment and"}, {"start": 224.2, "duration": 3.52, "text": "share bye-bye"}], "title": "Interesting Olympiad Math.", "description": "", "lengthSeconds": 226, "width": 524, "height": 360}
{"video_id": "6EspkWRrJrM", "transcript": [{"start": 0.56, "duration": 5.16, "text": "hello welcome back once again today we"}, {"start": 3.04, "duration": 6.96, "text": "have this interesting exponential"}, {"start": 5.72, "duration": 6.64, "text": "equation and involving a fractional"}, {"start": 10.0, "duration": 6.0, "text": "base and we're going to solve for the"}, {"start": 12.36, "duration": 5.839, "text": "value of x right now let's get started"}, {"start": 16.0, "duration": 4.92, "text": "from the left hand side here we have"}, {"start": 18.199, "duration": 9.0, "text": "25 /"}, {"start": 20.92, "duration": 11.159, "text": "16 ra power of X then this is = to 4 / 5"}, {"start": 27.199, "duration": 9.321, "text": "now take note of this 25 here is 5 S and"}, {"start": 32.079, "duration": 10.96, "text": "16 here is 4 S so this will be same as 5"}, {"start": 36.52, "duration": 13.96, "text": "/ 4 all squ then remember we have of X"}, {"start": 43.039, "duration": 10.241, "text": "and this is equal to 4 / 5 okay with one"}, {"start": 50.48, "duration": 4.84, "text": "of the laws of indices this power here"}, {"start": 53.28, "duration": 4.16, "text": "we multiply together right so we have"}, {"start": 55.32, "duration": 6.68, "text": "here 5/"}, {"start": 57.44, "duration": 8.16, "text": "4 2 * X which is 2 X then this is equal"}, {"start": 62.0, "duration": 8.119, "text": "to 4 /"}, {"start": 65.6, "duration": 8.76, "text": "5 okay now from here let us take note of"}, {"start": 70.119, "duration": 9.121, "text": "this when we have a/"}, {"start": 74.36, "duration": 7.6, "text": "B of n if we want to replace a and b"}, {"start": 79.24, "duration": 8.32, "text": "then we have a negative power that means"}, {"start": 81.96, "duration": 8.0, "text": "that will become B / a ra of n right"}, {"start": 87.56, "duration": 5.839, "text": "then remember this 4 5 here they have a"}, {"start": 89.96, "duration": 5.72, "text": "power of 1 so if you try to replace four"}, {"start": 93.399, "duration": 8.121, "text": "and 5 then we have a negative power so"}, {"start": 95.68, "duration": 12.28, "text": "here we have 5 / 4 to the^ 2x this is be"}, {"start": 101.52, "duration": 9.04, "text": "equal to 5 / 4 of1 so we can see here we"}, {"start": 107.96, "duration": 5.24, "text": "have the same base right so we can"}, {"start": 110.56, "duration": 8.199, "text": "equate the power so here this implies"}, {"start": 113.2, "duration": 7.36, "text": "that 2x is = to -1 since we're solving"}, {"start": 118.759, "duration": 5.161, "text": "for x let's divide both by the"}, {"start": 120.56, "duration": 5.68, "text": "coefficient of x so 2 cancel two and"}, {"start": 123.92, "duration": 6.399, "text": "from here we get the right answer that x"}, {"start": 126.24, "duration": 6.88, "text": "= -1 over two thank you for watching"}, {"start": 130.319, "duration": 7.601, "text": "please kindly subscribe to my channel"}, {"start": 133.12, "duration": 4.8, "text": "also like comment and share bye-bye"}], "title": "Interesting exponential equation with fractional bases", "description": "", "lengthSeconds": 136, "width": 640, "height": 360}
{"video_id": "Ph2eu_6KgTc", "transcript": [{"start": 0.68, "duration": 4.52, "text": "hello good day viewers welcome back"}, {"start": 2.56, "duration": 4.799, "text": "today to another video please subscribe"}, {"start": 5.2, "duration": 4.24, "text": "to my channel if you have not subscribed"}, {"start": 7.359, "duration": 4.24, "text": "to this Channel Please Subscribe please"}, {"start": 9.44, "duration": 5.88, "text": "like comment and share here we're going"}, {"start": 11.599, "duration": 10.961, "text": "to evaluate this expression here right"}, {"start": 15.32, "duration": 13.64, "text": "here we have 7^ of < TK 7 / 7 < TK 3 all"}, {"start": 22.56, "duration": 9.679, "text": "ra the power of < TK 7 + < TK 3 right"}, {"start": 28.96, "duration": 5.64, "text": "okay now take a look at this place we"}, {"start": 32.239, "duration": 5.081, "text": "have equal bases here but we have"}, {"start": 34.6, "duration": 5.92, "text": "different powers so there's a law of"}, {"start": 37.32, "duration": 9.36, "text": "indices we're going to make use here we"}, {"start": 40.52, "duration": 10.4, "text": "have a m over a n so we have a basis so"}, {"start": 46.68, "duration": 6.8, "text": "this is same as a power of m- n so this"}, {"start": 50.92, "duration": 6.479, "text": "is the division law right so in the"}, {"start": 53.48, "duration": 5.12, "text": "bracket we're going to have 7^ of < TK"}, {"start": 57.399, "duration": 8.321, "text": "of"}, {"start": 58.6, "duration": 13.159, "text": "7- < TK 3 then in bracket < TK 7"}, {"start": 65.72, "duration": 10.48, "text": "+un 3 okay now there's another law that"}, {"start": 71.759, "duration": 6.961, "text": "when we have a of B of C we can multiply"}, {"start": 76.2, "duration": 5.959, "text": "the powers together so that becomes a b"}, {"start": 78.72, "duration": 6.92, "text": "* C right okay now here we're going to"}, {"start": 82.159, "duration": 12.481, "text": "have this to be equal to 7 ra the power"}, {"start": 85.64, "duration": 11.4, "text": "of < TK 7 - < TK 3 m by < TK 7 + < TK 3"}, {"start": 94.64, "duration": 7.479, "text": "so from here you know this is difference"}, {"start": 97.04, "duration": 10.039, "text": "of two squares right so recall A + B *"}, {"start": 102.119, "duration": 6.521, "text": "Aus B this is equal to a 2 - b 2 right"}, {"start": 107.079, "duration": 6.521, "text": "so from here this is going to be equal"}, {"start": 108.64, "duration": 4.96, "text": "to 7^ of < TK 7"}, {"start": 114.04, "duration": 7.359, "text": "2us < TK 3 2 so this is equal to 7^ of"}, {"start": 119.92, "duration": 5.239, "text": "you know square and square root cancel"}, {"start": 121.399, "duration": 7.72, "text": "out so we have 7 minus square and square"}, {"start": 125.159, "duration": 8.16, "text": "root cancel out we have 3 right so from"}, {"start": 129.119, "duration": 11.2, "text": "here this is equal to 7^ 4 and this is ="}, {"start": 133.319, "duration": 10.041, "text": "to 7 * 7 * 7 * 7 and this equal"}, {"start": 140.319, "duration": 4.761, "text": "241 and this is the right answer thank"}, {"start": 143.36, "duration": 3.64, "text": "you for watching please do wa to"}, {"start": 145.08, "duration": 5.96, "text": "subscribe to my Channel please like"}, {"start": 147.0, "duration": 4.04, "text": "comment and share bye-bye"}], "title": "Interesting problem for everyone.", "description": "", "lengthSeconds": 149, "width": 640, "height": 360}
{"video_id": "e9S2AF_ePBo", "transcript": [{"start": 0.799, "duration": 5.361, "text": "hello welcome back once again in this"}, {"start": 3.439, "duration": 7.841, "text": "video let us find the derivative of this"}, {"start": 6.16, "duration": 9.24, "text": "function x / by the natural logarithm of"}, {"start": 11.28, "duration": 7.2, "text": "X we have here a numerator and a"}, {"start": 15.4, "duration": 6.28, "text": "denominator which is a rational function"}, {"start": 18.48, "duration": 5.76, "text": "here who knows whether it is rational or"}, {"start": 21.68, "duration": 5.16, "text": "irrational I think this function is"}, {"start": 24.24, "duration": 6.0, "text": "irrational right because of this natural"}, {"start": 26.84, "duration": 4.84, "text": "logarithm of x"}, {"start": 30.24, "duration": 3.92, "text": "so here we're going to find the"}, {"start": 31.68, "duration": 5.48, "text": "derivative let us recall the quotient"}, {"start": 34.16, "duration": 7.0, "text": "rule because that would be the rule to"}, {"start": 37.16, "duration": 8.2, "text": "be applied here we know the derivative"}, {"start": 41.16, "duration": 10.64, "text": "with respect to X of this function of x"}, {"start": 45.36, "duration": 9.08, "text": "f of x / G of X will be equal to G of x"}, {"start": 51.8, "duration": 8.8, "text": "* frime of"}, {"start": 54.44, "duration": 13.16, "text": "xus f ofx * G Prime of x then the whole"}, {"start": 60.6, "duration": 7.0, "text": "of this / by g^ 2 of X or G of"}, {"start": 67.68, "duration": 7.16, "text": "X2 so therefore we have here y isal x /"}, {"start": 72.479, "duration": 5.32, "text": "the natural logarithm of X take note of"}, {"start": 74.84, "duration": 8.959, "text": "this the derivative of the natural log"}, {"start": 77.799, "duration": 9.32, "text": "of x is = to 1 /x well known so here"}, {"start": 83.799, "duration": 6.401, "text": "differentiating we have Dy DX is equal"}, {"start": 87.119, "duration": 5.601, "text": "to with this property here we first have"}, {"start": 90.2, "duration": 7.48, "text": "the natural log of x MTI by the"}, {"start": 92.72, "duration": 7.84, "text": "derivative of the numerator X then minus"}, {"start": 97.68, "duration": 7.719, "text": "the numerator multip by the derivative"}, {"start": 100.56, "duration": 7.44, "text": "of the denominator right then the all of"}, {"start": 105.399, "duration": 4.201, "text": "this / by the square of"}, {"start": 108.0, "duration": 6.159, "text": "the"}, {"start": 109.6, "duration": 7.879, "text": "denominator so here we have Dy DX isal"}, {"start": 114.159, "duration": 6.201, "text": "to here derivative of x is 1 1 * natural"}, {"start": 117.479, "duration": 5.92, "text": "log of x is natural log of x"}, {"start": 120.36, "duration": 8.679, "text": "minus here the derivative of natural log"}, {"start": 123.399, "duration": 9.601, "text": "of x is 1 / x x * 1 x is 1 then divided"}, {"start": 129.039, "duration": 6.441, "text": "by the square of the natural log of x"}, {"start": 133.0, "duration": 4.76, "text": "and this is the right answer thank you"}, {"start": 135.48, "duration": 5.68, "text": "for watching please kindly subscribe to"}, {"start": 137.76, "duration": 6.4, "text": "my channel also like comment and share"}, {"start": 141.16, "duration": 3.0, "text": "bye-bye"}], "title": "The Quotient Rule | Differentiation", "description": "", "lengthSeconds": 143, "width": 640, "height": 360}
{"video_id": "kgJMoSZIArE", "transcript": [{"start": 0.799, "duration": 4.48, "text": "hello welcome back once again today"}, {"start": 2.639, "duration": 5.96, "text": "we're going to solve the following given"}, {"start": 5.279, "duration": 8.801, "text": "exponential equation we have squ root of"}, {"start": 8.599, "duration": 7.2, "text": "xun x = x here clearly X cannot be equal"}, {"start": 14.08, "duration": 4.88, "text": "to Z right because from the left we're"}, {"start": 15.799, "duration": 6.56, "text": "going to have 0 to the^ of 0 which is"}, {"start": 18.96, "duration": 6.2, "text": "undefined so what do we do now for my"}, {"start": 22.359, "duration": 8.561, "text": "solution we're going to natural log both"}, {"start": 25.16, "duration": 9.6, "text": "sides so here we have the Ln ofun X X"}, {"start": 30.92, "duration": 5.56, "text": "this will be equal to the Ln of X now"}, {"start": 34.76, "duration": 3.799, "text": "with the property of logarithm you know"}, {"start": 36.48, "duration": 5.079, "text": "this power here will come over here so"}, {"start": 38.559, "duration": 8.16, "text": "we're going to have the root of x * the"}, {"start": 41.559, "duration": 8.081, "text": "root of x this equals the root of"}, {"start": 46.719, "duration": 5.441, "text": "x what did I call that that is natural L"}, {"start": 49.64, "duration": 4.599, "text": "of root of x by the way okay now from"}, {"start": 52.16, "duration": 5.239, "text": "here we have the squ root of x multiply"}, {"start": 54.239, "duration": 6.601, "text": "by the natural log of you know of X same"}, {"start": 57.399, "duration": 6.84, "text": "as x^ of 1 / 2 then this equals the"}, {"start": 60.84, "duration": 7.319, "text": "natural of X so from here this power"}, {"start": 64.239, "duration": 7.041, "text": "here is coming over so we have of x * 1/"}, {"start": 68.159, "duration": 5.921, "text": "2 * the natural log of x then let's"}, {"start": 71.28, "duration": 6.159, "text": "subtract this minus natural log of x"}, {"start": 74.08, "duration": 5.64, "text": "this equal 0 so from here we have to"}, {"start": 77.439, "duration": 6.04, "text": "factor out the natural L of X and here"}, {"start": 79.72, "duration": 6.16, "text": "we have the otk of x / 2 right that is"}, {"start": 83.479, "duration": 5.841, "text": "what is left here then minus here will"}, {"start": 85.88, "duration": 6.8, "text": "be one then this equals z so from here"}, {"start": 89.32, "duration": 10.2, "text": "this implies is that the natural of x is"}, {"start": 92.68, "duration": 10.439, "text": "= 0 and from here x 2 - 1 = 0 so from"}, {"start": 99.52, "duration": 6.12, "text": "here we have to raise both"}, {"start": 103.119, "duration": 5.761, "text": "sides um to the power of E here we have"}, {"start": 105.64, "duration": 5.759, "text": "to the power of0 right so here this will"}, {"start": 108.88, "duration": 5.599, "text": "cancel this and we're left with x = e"}, {"start": 111.399, "duration": 5.601, "text": "the^ 0 which is 1 and here this is a"}, {"start": 114.479, "duration": 6.24, "text": "solution then from here let's add one to"}, {"start": 117.0, "duration": 7.2, "text": "both side we get of x / 2 = 1"}, {"start": 120.719, "duration": 7.04, "text": "so cross multiply we have the root of x"}, {"start": 124.2, "duration": 6.32, "text": "= 2 now to get rid of this root sign so"}, {"start": 127.759, "duration": 5.321, "text": "let's go ahead and square both"}, {"start": 130.52, "duration": 5.64, "text": "sides so the square we canc square roots"}, {"start": 133.08, "duration": 6.56, "text": "and here we find another solution x = 2"}, {"start": 136.16, "duration": 6.439, "text": "2 which is four and here we have the two"}, {"start": 139.64, "duration": 5.16, "text": "real solutions thank you for watching"}, {"start": 142.599, "duration": 6.761, "text": "please kindly subscribe to my channel"}, {"start": 144.8, "duration": 4.56, "text": "also like comment and share bye-bye"}], "title": "Interesting exponential radical equation", "description": "", "lengthSeconds": 147, "width": 530, "height": 360}
{"video_id": "X7pglXrw5Ps", "transcript": [{"start": 0.799, "duration": 5.52, "text": "hello welcome back once again today we"}, {"start": 2.96, "duration": 6.599, "text": "have this Olympian ma problem we have x"}, {"start": 6.319, "duration": 5.921, "text": "-1 to^ x ="}, {"start": 9.559, "duration": 6.56, "text": "1024 here we're going to find the"}, {"start": 12.24, "duration": 4.92, "text": "integral value of x for my solution"}, {"start": 16.119, "duration": 4.121, "text": "we're going to start with the"}, {"start": 17.16, "duration": 7.32, "text": "substitution we're going to let the base"}, {"start": 20.24, "duration": 6.56, "text": "which is x -1 = M from here trying to"}, {"start": 24.48, "duration": 6.4, "text": "solve for x in terms of M that would be"}, {"start": 26.8, "duration": 7.759, "text": "x = m + 1 right because to add one to"}, {"start": 30.88, "duration": 6.6, "text": "both sides okay so the"}, {"start": 34.559, "duration": 6.281, "text": "equation will be written in terms of M"}, {"start": 37.48, "duration": 7.2, "text": "as the base which is M the power the"}, {"start": 40.84, "duration": 6.6, "text": "power which is X and that is m + 1 then"}, {"start": 44.68, "duration": 7.6, "text": "this equals 1 to"}, {"start": 47.44, "duration": 7.08, "text": "4 from here we have m^ n + 1 this is"}, {"start": 52.28, "duration": 6.239, "text": "equal to"}, {"start": 54.52, "duration": 9.199, "text": "1024 is same as 4 to the^ of"}, {"start": 58.519, "duration": 8.881, "text": "5 so from here we have m^ n + 1 this is"}, {"start": 63.719, "duration": 8.241, "text": "equal to 4 now this 5 here is same as 4"}, {"start": 67.4, "duration": 8.88, "text": "+ 1 so from here we can obviously say"}, {"start": 71.96, "duration": 9.479, "text": "that m is equal to 4 right here we have"}, {"start": 76.28, "duration": 9.12, "text": "m n + 1 thisal 4^ 4 + 1 so we can say"}, {"start": 81.439, "duration": 7.72, "text": "that m is equal 4 right okay now let us"}, {"start": 85.4, "duration": 9.24, "text": "recall that x = n +1 and therefore we"}, {"start": 89.159, "duration": 7.96, "text": "have our solution x = 4 + 1 which is = 5"}, {"start": 94.64, "duration": 4.479, "text": "and here is the right answer thank you"}, {"start": 97.119, "duration": 4.801, "text": "for watching please kindly subscribe to"}, {"start": 99.119, "duration": 5.801, "text": "my channel also like comment and share"}, {"start": 101.92, "duration": 3.0, "text": "bye-bye"}], "title": "Interesting Olympiad Math | Nice exponential equation", "description": "", "lengthSeconds": 104, "width": 640, "height": 360}
{"video_id": "OF1GFFQq2II", "transcript": [{"start": 0.32, "duration": 4.4, "text": "good day viewers welcome back to my"}, {"start": 2.24, "duration": 4.199, "text": "channel once again today we have the"}, {"start": 4.72, "duration": 6.56, "text": "following"}, {"start": 6.439, "duration": 10.081, "text": "interesting problem if Y is = the S of"}, {"start": 11.28, "duration": 7.079, "text": "3x + 5 then d y DX equal that okay"}, {"start": 16.52, "duration": 5.32, "text": "please if you have not subscribed to my"}, {"start": 18.359, "duration": 3.481, "text": "Channel Please"}, {"start": 22.599, "duration": 6.201, "text": "Subscribe like"}, {"start": 25.119, "duration": 7.841, "text": "share my videos thank you"}, {"start": 28.8, "duration": 5.759, "text": "now with we have different methods of"}, {"start": 32.96, "duration": 4.68, "text": "differentiating"}, {"start": 34.559, "duration": 6.441, "text": "functions here we have a trigonometric"}, {"start": 37.64, "duration": 6.8, "text": "function okay so I'll be making a"}, {"start": 41.0, "duration": 8.52, "text": "substitution here so in this case I'll"}, {"start": 44.44, "duration": 9.88, "text": "be using the Shan rule okay so let's get"}, {"start": 49.52, "duration": 7.12, "text": "started here we have y is equal to"}, {"start": 54.32, "duration": 7.28, "text": "S of"}, {"start": 56.64, "duration": 7.479, "text": "3x + 5 okay so this is what we have I'm"}, {"start": 61.6, "duration": 4.8, "text": "going to make a substitution here I will"}, {"start": 64.119, "duration": 6.961, "text": "say let"}, {"start": 66.4, "duration": 4.68, "text": "U let U"}, {"start": 71.4, "duration": 10.52, "text": "equal"}, {"start": 73.2, "duration": 11.12, "text": "3x + 5 okay U is = 3x + 5 so now our y"}, {"start": 81.92, "duration": 7.559, "text": "will be"}, {"start": 84.32, "duration": 6.04, "text": "S of youu okay wow okay so here I'll be"}, {"start": 89.479, "duration": 3.481, "text": "different"}, {"start": 90.36, "duration": 5.119, "text": "both sides"}, {"start": 92.96, "duration": 5.439, "text": "so from here I have"}, {"start": 95.479, "duration": 5.68, "text": "du is equal to you know to differentiate"}, {"start": 98.399, "duration": 6.68, "text": "this with respect to X so we have"}, {"start": 101.159, "duration": 6.721, "text": "three DX okay and right"}, {"start": 105.079, "duration": 9.601, "text": "here I be differentiate both sides so I"}, {"start": 107.88, "duration": 12.44, "text": "have d y is equal to you know cine of U"}, {"start": 114.68, "duration": 10.28, "text": "du okay now from here if I divide both"}, {"start": 120.32, "duration": 8.24, "text": "sides by DX so I will get"}, {"start": 124.96, "duration": 6.76, "text": "du/ DX is equal to"}, {"start": 128.56, "duration": 6.0, "text": "three and to this place if I divide both"}, {"start": 131.72, "duration": 9.12, "text": "by du I will get"}, {"start": 134.56, "duration": 12.64, "text": "dy over du is equal to the cine of U"}, {"start": 140.84, "duration": 8.28, "text": "okay wow now how do we get the Y the x"}, {"start": 147.2, "duration": 3.16, "text": "value okay so let's see what we're going"}, {"start": 149.12, "duration": 2.96, "text": "to do"}, {"start": 150.36, "duration": 6.04, "text": "d y"}, {"start": 152.08, "duration": 8.4, "text": "DX is equal to if you multiply du"}, {"start": 156.4, "duration": 6.919, "text": "u i mean d y/ du"}, {"start": 160.48, "duration": 7.08, "text": "U and"}, {"start": 163.319, "duration": 6.161, "text": "du over DX together so you know this and"}, {"start": 167.56, "duration": 7.24, "text": "this will get cancel out so we left with"}, {"start": 169.48, "duration": 10.36, "text": "this Dy DX and this is what 3 * the"}, {"start": 174.8, "duration": 8.64, "text": "cosine of U so this is equal to 3 cosine"}, {"start": 179.84, "duration": 7.24, "text": "of youu okay so what is our U our U is"}, {"start": 183.44, "duration": 10.28, "text": "3x + 5 so we get d"}, {"start": 187.08, "duration": 11.4, "text": "y DX is = to 3 the cine of"}, {"start": 193.72, "duration": 6.599, "text": "3x + five and that's the answer thank"}, {"start": 198.48, "duration": 3.72, "text": "you for viewing and please make sure you"}, {"start": 200.319, "duration": 5.081, "text": "subscribe to my YouTube channel for more"}, {"start": 202.2, "duration": 3.2, "text": "interesting videos"}], "title": "Differentiation. Chain rule. Differentiation of trigonometric function", "description": "", "lengthSeconds": 203, "width": 636, "height": 360}
{"video_id": "DpwXPvCdvdM", "transcript": [{"start": 0.96, "duration": 4.6, "text": "hello welcome back once again today"}, {"start": 2.76, "duration": 5.839, "text": "we're going to solve the following given"}, {"start": 5.56, "duration": 5.079, "text": "exponential equation for x so let's get"}, {"start": 8.599, "duration": 5.841, "text": "started from the left hand side we have"}, {"start": 10.639, "duration": 7.121, "text": "4 to^ of X and we know this is equal to"}, {"start": 14.44, "duration": 6.079, "text": "48 now we're going to make use of the"}, {"start": 17.76, "duration": 6.359, "text": "natural logarithm so let's natural log"}, {"start": 20.519, "duration": 7.6, "text": "both sides so we have the Ln of 4^ X"}, {"start": 24.119, "duration": 6.0, "text": "this is equal to the Ln of 48 so we know"}, {"start": 28.119, "duration": 4.361, "text": "the property of the logarithm so this"}, {"start": 30.119, "duration": 5.28, "text": "will come down here right and that will"}, {"start": 32.48, "duration": 5.84, "text": "become x * the natural log of 4 so this"}, {"start": 35.399, "duration": 9.401, "text": "is equal to the natural log of 48 is"}, {"start": 38.32, "duration": 8.879, "text": "same as 16 * 3 right so now this is x *"}, {"start": 44.8, "duration": 4.52, "text": "the natural log of 4 so this is equal to"}, {"start": 47.199, "duration": 4.84, "text": "you know the property of logarithm the"}, {"start": 49.32, "duration": 6.2, "text": "natural of a * B is same as natural log"}, {"start": 52.039, "duration": 5.761, "text": "of a plus the natural log of B that is"}, {"start": 55.52, "duration": 5.8, "text": "the addition rule right so this will be"}, {"start": 57.8, "duration": 7.16, "text": "the natural log of 16 plus the natural"}, {"start": 61.32, "duration": 6.24, "text": "log of 3 so here we have x * the natural"}, {"start": 64.96, "duration": 6.32, "text": "log of 4 this is equal to the natural"}, {"start": 67.56, "duration": 4.76, "text": "log of 4 S right then plus the natural"}, {"start": 71.28, "duration": 4.199, "text": "log of"}, {"start": 72.32, "duration": 5.439, "text": "3 so from here we have x * the natural"}, {"start": 75.479, "duration": 4.561, "text": "log of 4 this is equal to the property"}, {"start": 77.759, "duration": 4.921, "text": "of logarithm this will come down here so"}, {"start": 80.04, "duration": 5.16, "text": "it will become 2 * the natural log of 4"}, {"start": 82.68, "duration": 5.799, "text": "plus the natural log of"}, {"start": 85.2, "duration": 5.68, "text": "three okay so from"}, {"start": 88.479, "duration": 4.521, "text": "here let's divide div both side by the"}, {"start": 90.88, "duration": 5.12, "text": "coefficient of x which is the natural"}, {"start": 93.0, "duration": 5.56, "text": "log of four so divide both side by the"}, {"start": 96.0, "duration": 5.079, "text": "coefficient of x the natural L of 4 this"}, {"start": 98.56, "duration": 5.599, "text": "will cancel this from here so have X is"}, {"start": 101.079, "duration": 5.36, "text": "equal to split the denominator here 2 *"}, {"start": 104.159, "duration": 6.361, "text": "the natural log of 4 / the natural log"}, {"start": 106.439, "duration": 5.161, "text": "of 4 plus natural log of 3 / the natural"}, {"start": 110.52, "duration": 3.84, "text": "log of"}, {"start": 111.6, "duration": 5.92, "text": "4 so from here this will cancel this and"}, {"start": 114.36, "duration": 7.439, "text": "we have just two left so we have X is ="}, {"start": 117.52, "duration": 7.199, "text": "2 plus now this right over here we can"}, {"start": 121.799, "duration": 5.24, "text": "actually rewrite it so remember that the"}, {"start": 124.719, "duration": 7.441, "text": "natural log of a / by the natural log of"}, {"start": 127.039, "duration": 8.84, "text": "B is same as the log of a to the base B"}, {"start": 132.16, "duration": 7.4, "text": "so this right over here will be log of 3"}, {"start": 135.879, "duration": 5.961, "text": "to the base 4 and here we have the"}, {"start": 139.56, "duration": 5.0, "text": "solution thank you for watching please"}, {"start": 141.84, "duration": 7.16, "text": "kindly subscribe to my channel also like"}, {"start": 144.56, "duration": 4.44, "text": "comment and share bye-bye"}], "title": "Interesting exponential equation | can you solve ?", "description": "", "lengthSeconds": 148, "width": 404, "height": 240}
{"video_id": "y7uQXzoGjwU", "transcript": [{"start": 0.799, "duration": 4.401, "text": "hello good viewers welcome back once"}, {"start": 2.399, "duration": 8.761, "text": "again today we have the following"}, {"start": 5.2, "duration": 7.479, "text": "interesting problem x^ n + 1 x^ n = 4"}, {"start": 11.16, "duration": 7.32, "text": "right over here we're going to evaluate"}, {"start": 12.679, "duration": 6.921, "text": "x^ N - 1 / x^ n provided that n is not"}, {"start": 18.48, "duration": 3.32, "text": "equal to"}, {"start": 19.6, "duration": 6.439, "text": "zero now we'll be making use of this"}, {"start": 21.8, "duration": 9.0, "text": "property remember that a + b s this"}, {"start": 26.039, "duration": 9.321, "text": "equals a - b s"}, {"start": 30.8, "duration": 11.12, "text": "+ 4 * a * B now from the given equation"}, {"start": 35.36, "duration": 8.6, "text": "x^ of n + 1 / x^ n = 4 let's go ahead"}, {"start": 41.92, "duration": 4.68, "text": "and square both sides of this"}, {"start": 43.96, "duration": 4.56, "text": "equation from the left hand side will'll"}, {"start": 46.6, "duration": 4.479, "text": "be making use of this property so we"}, {"start": 48.52, "duration": 9.96, "text": "have here x^"}, {"start": 51.079, "duration": 12.16, "text": "nus 1 / x^ n then s + 4 MTI by the first"}, {"start": 58.48, "duration": 9.679, "text": "time here x n MTI the second term here 1"}, {"start": 63.239, "duration": 13.841, "text": "/ x^ n this isal 4 which is"}, {"start": 68.159, "duration": 8.921, "text": "16 so from here we have x^ N - 1 / x^ n"}, {"start": 77.119, "duration": 6.96, "text": "2 then"}, {"start": 80.2, "duration": 6.84, "text": "plus here we have just four because this"}, {"start": 84.079, "duration": 9.601, "text": "will cancel this then this is equal to"}, {"start": 87.04, "duration": 10.88, "text": "16 so from here we have x n N - 1 / x n"}, {"start": 93.68, "duration": 5.84, "text": "2 this is = 16 - 4 that will give us 12"}, {"start": 97.92, "duration": 6.4, "text": "from here let's take root of both we get"}, {"start": 99.52, "duration": 8.12, "text": "x n - 1 / x n this is equal plus or"}, {"start": 104.32, "duration": 7.6, "text": "minus < TK 12 which simplifies to plus"}, {"start": 107.64, "duration": 6.4, "text": "or minus 2 < tk3 and this is the right"}, {"start": 111.92, "duration": 4.44, "text": "answer thank you for watching please"}, {"start": 114.04, "duration": 6.32, "text": "kindly subscribe to my channel also like"}, {"start": 116.36, "duration": 4.0, "text": "comment and share bye-bye"}], "title": "Can you solve ?  Nice algebra problem.", "description": "", "lengthSeconds": 118, "width": 546, "height": 360}
{"video_id": "vpUogFmfo4E", "transcript": [{"start": 0.28, "duration": 5.479, "text": "hello you are welcome here we're going"}, {"start": 2.52, "duration": 7.24, "text": "to solve this exponential equation we"}, {"start": 5.759, "duration": 8.04, "text": "have here M to the^ of 2 m this is equal"}, {"start": 9.76, "duration": 6.439, "text": "to 1 all over 2 now for us to get rid of"}, {"start": 13.799, "duration": 6.001, "text": "this two right over here we're going to"}, {"start": 16.199, "duration": 6.32, "text": "raise both sides of this equation to the"}, {"start": 19.8, "duration": 6.44, "text": "power of the reciprocal of two and we"}, {"start": 22.519, "duration": 7.321, "text": "know the reciprocal of 2 is 1 all/ 2 so"}, {"start": 26.24, "duration": 7.68, "text": "here we have M to the^ of 2 m ra power"}, {"start": 29.84, "duration": 9.96, "text": "of 1 / 2 this is = to 1 all"}, {"start": 33.92, "duration": 8.479, "text": "2^ of 1 / 2 from here we can multiply"}, {"start": 39.8, "duration": 5.0, "text": "these Powers together with the help of"}, {"start": 42.399, "duration": 6.881, "text": "one of the basic laws of IND thises"}, {"start": 44.8, "duration": 7.239, "text": "right that this is same as a^ B * C so 2"}, {"start": 49.28, "duration": 6.72, "text": "m * 1 / 2 2 we cancel 2 and we're left"}, {"start": 52.039, "duration": 10.241, "text": "with M so we have m m this will be equal"}, {"start": 56.0, "duration": 8.76, "text": "to 1 / 2 of 1 / 2 from here obviously m"}, {"start": 62.28, "duration": 5.76, "text": "= 1 /"}, {"start": 64.76, "duration": 4.359, "text": "2 but that's not all we still have"}, {"start": 68.04, "duration": 4.759, "text": "another"}, {"start": 69.119, "duration": 5.241, "text": "solution now we have M of M which is"}, {"start": 72.799, "duration": 5.161, "text": "equal to from"}, {"start": 74.36, "duration": 6.079, "text": "here 1 / 2 is the same as the square <"}, {"start": 77.96, "duration": 6.4, "text": "TK of 1 / 4"}, {"start": 80.439, "duration": 7.761, "text": "right then we have power of 1 / 2 right"}, {"start": 84.36, "duration": 9.56, "text": "over here now take note the square root"}, {"start": 88.2, "duration": 10.08, "text": "of a is same as a^ of 1 / 2 so from here"}, {"start": 93.92, "duration": 8.64, "text": "we get M to the^ of M this is equal to"}, {"start": 98.28, "duration": 8.76, "text": "here we have 1 / 4 ra power of 1 / 2"}, {"start": 102.56, "duration": 7.159, "text": "then ra power of 1 / 2 from here we get"}, {"start": 107.04, "duration": 5.88, "text": "m to the power of M this is equal to"}, {"start": 109.719, "duration": 6.121, "text": "from here we have 1/ 4 then we can"}, {"start": 112.92, "duration": 6.159, "text": "multiply this power together 1 / 2 * 1 /"}, {"start": 115.84, "duration": 4.559, "text": "2 which is 1 / 4 from here through this"}, {"start": 119.079, "duration": 5.96, "text": "manipulation"}, {"start": 120.399, "duration": 7.92, "text": "we still find out that m = 1 4 and this"}, {"start": 125.039, "duration": 5.36, "text": "is the other real solution thank you for"}, {"start": 128.319, "duration": 4.801, "text": "watching please kindly subscribe to my"}, {"start": 130.399, "duration": 5.721, "text": "channel also like comment and share"}, {"start": 133.12, "duration": 3.0, "text": "bye-bye"}], "title": "Tricky Algebra problem | Nice exponential equation", "description": "", "lengthSeconds": 135, "width": 548, "height": 360}
{"video_id": "5vFIfZOGfw4", "transcript": [{"start": 0.68, "duration": 4.119, "text": "hello good viewers welcome back once"}, {"start": 2.44, "duration": 5.399, "text": "again let's solve the following given"}, {"start": 4.799, "duration": 7.681, "text": "interesting radical equation we have"}, {"start": 7.839, "duration": 7.88, "text": "root x + root ofx this is equal to < tk2"}, {"start": 12.48, "duration": 5.559, "text": "now let's get started from here we can"}, {"start": 15.719, "duration": 9.601, "text": "write this as otk"}, {"start": 18.039, "duration": 8.881, "text": "ofx this is equal to < tk2 - < TK of X"}, {"start": 25.32, "duration": 3.92, "text": "now from here we're going to square both"}, {"start": 26.92, "duration": 6.319, "text": "sides but before we proceed we need to"}, {"start": 29.24, "duration": 12.2, "text": "record this property a - b 2 this is"}, {"start": 33.239, "duration": 11.32, "text": "equal to a 2 + b 2 - 2 * a * B right so"}, {"start": 41.44, "duration": 9.68, "text": "from here we have to square both side"}, {"start": 44.559, "duration": 8.64, "text": "root ofx 2 this is equal to < TK 2 - otk"}, {"start": 51.12, "duration": 5.16, "text": "x^"}, {"start": 53.199, "duration": 6.921, "text": "2 and from here they know the square and"}, {"start": 56.28, "duration": 6.4, "text": "this root cancel out we have ax"}, {"start": 60.12, "duration": 6.08, "text": "is equal to now the first time squ will"}, {"start": 62.68, "duration": 8.04, "text": "give us 2 plus the second time squ will"}, {"start": 66.2, "duration": 10.04, "text": "give us X then - 2 * 2 2 * < tk2 * < TK"}, {"start": 70.72, "duration": 7.92, "text": "X which is same as 2 < tk2 * < TK"}, {"start": 76.24, "duration": 5.239, "text": "X now from here remember we're"}, {"start": 78.64, "duration": 4.96, "text": "subtracting this term from this term so"}, {"start": 81.479, "duration": 4.521, "text": "if you if you wish to color L term from"}, {"start": 83.6, "duration": 8.08, "text": "here 2 <"}, {"start": 86.0, "duration": 8.32, "text": "tk2 * rootx will be also equal = to 2 +"}, {"start": 91.68, "duration": 7.119, "text": "x"}, {"start": 94.32, "duration": 7.52, "text": "--x from the left from here 2 < tk2 we"}, {"start": 98.799, "duration": 6.92, "text": "all know this is same Asun 8 right then"}, {"start": 101.84, "duration": 8.279, "text": "multi by root X this is equal to from"}, {"start": 105.719, "duration": 5.961, "text": "here 2 then x - - x this minus and minus"}, {"start": 110.119, "duration": 7.241, "text": "here will be plus as they are"}, {"start": 111.68, "duration": 6.96, "text": "multiplying so we have here 2 + 2 x from"}, {"start": 117.36, "duration": 4.2, "text": "here let's go ahead and square both"}, {"start": 118.64, "duration": 4.439, "text": "sides again so here we have to square"}, {"start": 121.56, "duration": 4.72, "text": "both"}, {"start": 123.079, "duration": 6.641, "text": "sides so from here we have root 8 squ"}, {"start": 126.28, "duration": 6.319, "text": "will give us 8 multip root x^2 will give"}, {"start": 129.72, "duration": 6.159, "text": "us X right then this is equal to from"}, {"start": 132.599, "duration": 10.041, "text": "here there's another binomial expansion"}, {"start": 135.879, "duration": 10.561, "text": "we need to recall a + b 2 is = a 2 + b 2"}, {"start": 142.64, "duration": 6.84, "text": "+ 2 * a * B so from here the first time"}, {"start": 146.44, "duration": 7.32, "text": "squ which is 4 plus the Second Time"}, {"start": 149.48, "duration": 7.119, "text": "Square give us 4 x 2 + 2 * the first"}, {"start": 153.76, "duration": 7.92, "text": "term * the second term that will give us"}, {"start": 156.599, "duration": 6.601, "text": "8 x right so here we have 2 * 2 4 * 2 x"}, {"start": 161.68, "duration": 4.12, "text": "which is 8"}, {"start": 163.2, "duration": 10.88, "text": "x so from here from the right hand side"}, {"start": 165.8, "duration": 11.56, "text": "we have 4 x^2 + 8 x + 4 this is = to 8"}, {"start": 174.08, "duration": 8.28, "text": "x now we're going to collect like terms"}, {"start": 177.36, "duration": 7.959, "text": "so we have here 4 x2+ plus from here 8 x"}, {"start": 182.36, "duration": 5.92, "text": "- 8 x that is 0 and here we are left"}, {"start": 185.319, "duration": 5.361, "text": "with 4 then this is equal to 0 Let's go"}, {"start": 188.28, "duration": 8.48, "text": "ahead and divide both sides of this"}, {"start": 190.68, "duration": 12.32, "text": "equation by 4 so here we have 4 X2 / 4 +"}, {"start": 196.76, "duration": 8.479, "text": "4 / 4 this is equal 0 / 4 so from here"}, {"start": 203.0, "duration": 6.04, "text": "this will divide out we have"}, {"start": 205.239, "duration": 5.241, "text": "x2+ here 4 / 4 which is 1 this is equal"}, {"start": 209.04, "duration": 6.68, "text": "to0"}, {"start": 210.48, "duration": 7.479, "text": "so from here we have X2 is = -1 from"}, {"start": 215.72, "duration": 6.239, "text": "here take root of both side we get X is"}, {"start": 217.959, "duration": 7.521, "text": "= to plus orus root of1 which is equal"}, {"start": 221.959, "duration": 7.081, "text": "to plus or minus I and here we have the"}, {"start": 225.48, "duration": 7.08, "text": "solution that X is equal to plus or"}, {"start": 229.04, "duration": 6.16, "text": "minus I amazing thank you for watching"}, {"start": 232.56, "duration": 7.319, "text": "please kindly subscribe to my channel"}, {"start": 235.2, "duration": 4.679, "text": "also like comment and share bye-bye"}], "title": "Japanese | Nice Olympiad Math | #maths #mathematics", "description": "", "lengthSeconds": 239, "width": 522, "height": 360}
{"video_id": "bJlmSSYr-3E", "transcript": [{"start": 0.599, "duration": 4.841, "text": "good day viewers today we have the"}, {"start": 1.959, "duration": 9.68, "text": "following interesting problem a 2 - b 2"}, {"start": 5.44, "duration": 8.0, "text": "= 56 Aus b = 2 then a + b = that please"}, {"start": 11.639, "duration": 3.2, "text": "if you have not subscribed to my Channel"}, {"start": 13.44, "duration": 5.16, "text": "Please"}, {"start": 14.839, "duration": 9.68, "text": "Subscribe like comment and share thank"}, {"start": 18.6, "duration": 9.4, "text": "you now we have a^ 2us b^ 2 ="}, {"start": 24.519, "duration": 7.04, "text": "56 remember we have the difference of"}, {"start": 28.0, "duration": 7.68, "text": "two squares formula here so that is A +"}, {"start": 31.559, "duration": 12.561, "text": "B * a - b is = to"}, {"start": 35.68, "duration": 12.96, "text": "56 here a - b = 2 so here we have a + b"}, {"start": 44.12, "duration": 6.959, "text": "here * 2 is equal to 56 here we'll be"}, {"start": 48.64, "duration": 6.0, "text": "dividing both side by two so this is"}, {"start": 51.079, "duration": 10.16, "text": "over two and here over two so 2 cancel"}, {"start": 54.64, "duration": 9.479, "text": "two here we get a + b = 56 divide by two"}, {"start": 61.239, "duration": 5.041, "text": "so that is equal to 28 and this is the"}, {"start": 64.119, "duration": 3.481, "text": "right answer thank you for viewing and"}, {"start": 66.28, "duration": 2.879, "text": "please make sure you subscribe to my"}, {"start": 67.6, "duration": 4.559, "text": "YouTube channel for more interesting"}, {"start": 69.159, "duration": 3.0, "text": "videos"}], "title": "A simple Algebra trick.", "description": "", "lengthSeconds": 70, "width": 636, "height": 360}
{"video_id": "zzQgzpAo5qE", "transcript": [{"start": 0.599, "duration": 5.68, "text": "hello welcome back once again let's find"}, {"start": 3.399, "duration": 10.16, "text": "this limit so we have limit as X is"}, {"start": 6.279, "duration": 12.121, "text": "approaching two of x^ 4 - 16 / x -"}, {"start": 13.559, "duration": 8.921, "text": "2 we have limit as X is approaching 2 so"}, {"start": 18.4, "duration": 7.719, "text": "here x^ 4 we going to re write it as x^"}, {"start": 22.48, "duration": 8.92, "text": "2 2 right you know 2 * 2 here will give"}, {"start": 26.119, "duration": 9.481, "text": "us 4 then - 16 is 4 to the^ 2 2 then the"}, {"start": 31.4, "duration": 5.12, "text": "whole of this / x - 2 okay now from here"}, {"start": 35.6, "duration": 4.68, "text": "we're going to make use of the"}, {"start": 36.52, "duration": 11.359, "text": "difference of two squares remember a 2 -"}, {"start": 40.28, "duration": 9.56, "text": "b 2 is same as a + b * a minus B so make"}, {"start": 47.879, "duration": 4.84, "text": "use of that to get"}, {"start": 49.84, "duration": 13.12, "text": "limit as X is approaching"}, {"start": 52.719, "duration": 14.841, "text": "2 so we have here X2 - 4 * X2 + 4 then"}, {"start": 62.96, "duration": 8.08, "text": "ided by x -"}, {"start": 67.56, "duration": 8.0, "text": "2 okay now from here we get"}, {"start": 71.04, "duration": 6.399, "text": "limit as X is approaching 2 so now in"}, {"start": 75.56, "duration": 3.919, "text": "the first bracket here we're still going"}, {"start": 77.439, "duration": 5.0, "text": "to make use of difference of two squares"}, {"start": 79.479, "duration": 9.121, "text": "right so we get x2 minus 4 which is same"}, {"start": 82.439, "duration": 11.36, "text": "as 2^ 2 then here we have X2 + 4 then"}, {"start": 88.6, "duration": 6.92, "text": "ided by X x - 2 so in that first bracket"}, {"start": 93.799, "duration": 4.441, "text": "let us make use of the difference of two"}, {"start": 95.52, "duration": 8.68, "text": "squares formula right so there we're"}, {"start": 98.24, "duration": 9.479, "text": "going to have x - 2 * x + 2 right then"}, {"start": 104.2, "duration": 9.4, "text": "in the next bracket we have x 2 + 4 the"}, {"start": 107.719, "duration": 10.801, "text": "whole of this ided by x - 2 so from here"}, {"start": 113.6, "duration": 8.119, "text": "this we cancel this right then we get"}, {"start": 118.52, "duration": 11.879, "text": "limit as X is approaching"}, {"start": 121.719, "duration": 10.481, "text": "2 of x + 2 * x 2 + 4 right so here we"}, {"start": 130.399, "duration": 9.281, "text": "can make our direct substitution and"}, {"start": 132.2, "duration": 13.48, "text": "here we get 2 + 2 * 2 2 + 4 which is"}, {"start": 139.68, "duration": 9.839, "text": "equal to here 2 + 2 is 4 * by here 2 2 2"}, {"start": 145.68, "duration": 6.199, "text": "2 is 4 4 + 4 which is 8 and from here we"}, {"start": 149.519, "duration": 5.201, "text": "get the right answer which is"}, {"start": 151.879, "duration": 4.961, "text": "32 thank you for watching please kindly"}, {"start": 154.72, "duration": 6.159, "text": "subscribe to my channel also like"}, {"start": 156.84, "duration": 4.039, "text": "comment and share bye-bye"}], "title": "Interesting limit.", "description": "", "lengthSeconds": 159, "width": 640, "height": 360}
{"video_id": "-YjwIdqffkU", "transcript": [{"start": 0.56, "duration": 4.36, "text": "hello welcome back once again today"}, {"start": 2.72, "duration": 6.999, "text": "we're going to solve the following given"}, {"start": 4.92, "duration": 7.2, "text": "exponential equation we have x + < tk3"}, {"start": 9.719, "duration": 5.481, "text": "of x + < tk3 = <"}, {"start": 12.12, "duration": 7.8, "text": "tk3 here we're going to solve for the"}, {"start": 15.2, "duration": 8.04, "text": "real value of x first of all let us"}, {"start": 19.92, "duration": 6.88, "text": "natural log both sides of this equation"}, {"start": 23.24, "duration": 7.48, "text": "so here we have the natural log"}, {"start": 26.8, "duration": 6.599, "text": "of x + < tk3"}, {"start": 30.72, "duration": 5.12, "text": "of x + <"}, {"start": 33.399, "duration": 6.121, "text": "tk3 this will be equal to the natural"}, {"start": 35.84, "duration": 6.399, "text": "log of < tk3 now with the property of"}, {"start": 39.52, "duration": 5.24, "text": "natural logarithm you know this power"}, {"start": 42.239, "duration": 6.281, "text": "here is coming over here right that is"}, {"start": 44.76, "duration": 10.04, "text": "from here to here and that will"}, {"start": 48.52, "duration": 10.64, "text": "become x + < tk3 * by the natural log of"}, {"start": 54.8, "duration": 6.919, "text": "x + < tk3 and from the right this equals"}, {"start": 59.16, "duration": 7.76, "text": "the natural log of"}, {"start": 61.719, "duration": 7.321, "text": "ot3 now from here remember a can be"}, {"start": 66.92, "duration": 6.199, "text": "written as e to the power of natural log"}, {"start": 69.04, "duration": 7.439, "text": "of a right so from here x +3 we can"}, {"start": 73.119, "duration": 7.481, "text": "actually write it as e of the natural"}, {"start": 76.479, "duration": 8.28, "text": "log of x + < tk3 then multiply by the"}, {"start": 80.6, "duration": 8.08, "text": "natural log of x + < tk3 from the right"}, {"start": 84.759, "duration": 5.281, "text": "this equals the natural log of root3"}, {"start": 88.68, "duration": 4.28, "text": "from here we're going to make use of of"}, {"start": 90.04, "duration": 8.92, "text": "the lambar W function recall that the"}, {"start": 92.96, "duration": 8.88, "text": "lamb W function of x * e^ of x = x since"}, {"start": 98.96, "duration": 6.119, "text": "the power of E here is exactly equal to"}, {"start": 101.84, "duration": 6.68, "text": "the coefficient and from here we have"}, {"start": 105.079, "duration": 5.32, "text": "the same argument here right so now"}, {"start": 108.52, "duration": 5.76, "text": "making use of the Lamb W function on"}, {"start": 110.399, "duration": 9.601, "text": "both sides natural log of x + <"}, {"start": 114.28, "duration": 9.759, "text": "tk3 e the natural log of x + < tk3"}, {"start": 120.0, "duration": 5.56, "text": "will be equal to W of the natural log of"}, {"start": 124.039, "duration": 4.401, "text": "<"}, {"start": 125.56, "duration": 5.44, "text": "tk3 so from here this left side"}, {"start": 128.44, "duration": 7.519, "text": "simplifies to just the natural log of x"}, {"start": 131.0, "duration": 6.68, "text": "+ < tk3 and this equals the W of the"}, {"start": 135.959, "duration": 8.28, "text": "natural log of Ro <"}, {"start": 137.68, "duration": 9.32, "text": "tk3 from here we have x + < tk3 you know"}, {"start": 144.239, "duration": 7.08, "text": "the power rule here will be e to the"}, {"start": 147.0, "duration": 6.959, "text": "power of the W of the natural log of ot3"}, {"start": 151.319, "duration": 5.56, "text": "subtract ro3 from both sides and here"}, {"start": 153.959, "duration": 4.801, "text": "comes the value of x which is e to the"}, {"start": 156.879, "duration": 6.401, "text": "power of the Lamb function of the"}, {"start": 158.76, "duration": 8.7, "text": "natural log of < tk3 minus <"}, {"start": 163.28, "duration": 6.0, "text": "tk3 and this is approximately equal"}, {"start": 167.46, "duration": 4.82, "text": "[Music]"}, {"start": 169.28, "duration": 3.0, "text": "to0"}, {"start": 172.599, "duration": 7.64, "text": "274"}, {"start": 175.44, "duration": 7.079, "text": "381 6 4 7 and this is a right right"}, {"start": 180.239, "duration": 4.881, "text": "answer thank you for watching please"}, {"start": 182.519, "duration": 6.8, "text": "kindly subscribe to my channel also like"}, {"start": 185.12, "duration": 4.199, "text": "comment and share bye-bye"}], "title": "Interesting exponential equation.", "description": "", "lengthSeconds": 188, "width": 566, "height": 360}
{"video_id": "hJUwQlERbcI", "transcript": [{"start": 0.359, "duration": 3.401, "text": "hello good viewers welcome back once"}, {"start": 2.04, "duration": 4.64, "text": "again let's find a solution to this"}, {"start": 3.76, "duration": 5.48, "text": "Olympiad Mass problem we given that X"}, {"start": 6.68, "duration": 6.56, "text": "the first Power Plus y the 1 power is ="}, {"start": 9.24, "duration": 7.48, "text": "to 1 x the power of 2 that is the second"}, {"start": 13.24, "duration": 7.64, "text": "power + y the power of 2 is = 2 then"}, {"start": 16.72, "duration": 6.559, "text": "we're going to evaluate x^ 4 + y^ 4 many"}, {"start": 20.88, "duration": 4.96, "text": "may assume that four is the right answer"}, {"start": 23.279, "duration": 5.84, "text": "but unfortunately four is not the right"}, {"start": 25.84, "duration": 7.32, "text": "answer what do we do the first idea is"}, {"start": 29.119, "duration": 7.081, "text": "to solve for or x * y cuz that would be"}, {"start": 33.16, "duration": 6.48, "text": "very helpful here so from equation 1 we"}, {"start": 36.2, "duration": 5.839, "text": "have x + y this is equal to 1 Let's"}, {"start": 39.64, "duration": 9.16, "text": "Square both sides from the left we have"}, {"start": 42.039, "duration": 9.36, "text": "X2 + Y 2 + 2 x y this will give us 1 X2"}, {"start": 48.8, "duration": 7.16, "text": "+ y^2 is already 2 so we replace we have"}, {"start": 51.399, "duration": 6.561, "text": "2 + 2 x y this is equal to 1 when we"}, {"start": 55.96, "duration": 7.439, "text": "subtract two from both side and divide"}, {"start": 57.96, "duration": 7.64, "text": "by two we find out that X Y = 1 - 2 / 2"}, {"start": 63.399, "duration": 8.08, "text": "which is a -1 /"}, {"start": 65.6, "duration": 8.559, "text": "2 okay now from equation 2 X2 + y^2 is ="}, {"start": 71.479, "duration": 8.121, "text": "2 let's Square both"}, {"start": 74.159, "duration": 10.121, "text": "sides from the left here we have x\u00b2 2"}, {"start": 79.6, "duration": 7.6, "text": "which is x ^ 4 + y ^ 4 then + 2 into"}, {"start": 84.28, "duration": 4.68, "text": "braet x y^ 2 if you expand this this is"}, {"start": 87.2, "duration": 5.04, "text": "exactly what you're going to have and"}, {"start": 88.96, "duration": 5.159, "text": "this is equal to 2 which is 4 remember"}, {"start": 92.24, "duration": 5.919, "text": "XY is ="}, {"start": 94.119, "duration": 7.161, "text": "to1 / 2 and it is squ think about it"}, {"start": 98.159, "duration": 8.56, "text": "when we Square this we find out that x"}, {"start": 101.28, "duration": 7.519, "text": "y^ 2 is = to 1 1/ 2 2 will give us 1 4"}, {"start": 106.719, "duration": 7.241, "text": "and we multiply by"}, {"start": 108.799, "duration": 9.201, "text": "2 we find out that 2 * x y^ 2 will be 1"}, {"start": 113.96, "duration": 6.32, "text": "/ 2 so from here this is 1 /2 and we're"}, {"start": 118.0, "duration": 10.679, "text": "subtracting 1/2 from both sides so we"}, {"start": 120.28, "duration": 12.679, "text": "get x^ 4 + y ^ 4 will be = to 4 - 1 2"}, {"start": 128.679, "duration": 6.64, "text": "right okay 4 is the same as 8 / 2 then"}, {"start": 132.959, "duration": 6.801, "text": "we are subtracting 1 / two from here we"}, {"start": 135.319, "duration": 6.401, "text": "get um 7 / 2 and this is the right"}, {"start": 139.76, "duration": 3.839, "text": "answer thank you for watching please"}, {"start": 141.72, "duration": 6.4, "text": "kindly subscribe to my YouTube channel"}, {"start": 143.599, "duration": 4.521, "text": "also like comment and share bye-bye"}], "title": "Olympiad Math problem | Interesting algebraic problem | #maths #mathematics", "description": "", "lengthSeconds": 146, "width": 640, "height": 360}
{"video_id": "-Ket7WbCoDQ", "transcript": [{"start": 1.12, "duration": 4.44, "text": "hello welcome back once again today"}, {"start": 3.399, "duration": 5.961, "text": "we're going to solve for the value of x"}, {"start": 5.56, "duration": 7.36, "text": "in the following given exponential"}, {"start": 9.36, "duration": 10.44, "text": "equation we have the cube"}, {"start": 12.92, "duration": 9.48, "text": "root of 12 to the^ of X this is equal to"}, {"start": 19.8, "duration": 5.36, "text": "2 from here we're going to cube both"}, {"start": 22.4, "duration": 7.959, "text": "sides of this equation to get rid of"}, {"start": 25.16, "duration": 8.8, "text": "this cube root sign so here we Cube this"}, {"start": 30.359, "duration": 5.801, "text": "and from the right we Cube as well so we"}, {"start": 33.96, "duration": 5.32, "text": "know the cube and the cube root will"}, {"start": 36.16, "duration": 7.239, "text": "cancel out we are left with 12 to the^"}, {"start": 39.28, "duration": 7.64, "text": "of X this is equal to 2 cubed which is"}, {"start": 43.399, "duration": 7.401, "text": "8 from here we're going to make use of"}, {"start": 46.92, "duration": 8.2, "text": "log reading we going to take the log of"}, {"start": 50.8, "duration": 9.439, "text": "both sides but the log must be in Bas 12"}, {"start": 55.12, "duration": 9.359, "text": "so here we have the log of 12 to^ of x x"}, {"start": 60.239, "duration": 8.321, "text": "to the base 12 this is equal to the log"}, {"start": 64.479, "duration": 8.28, "text": "of 8 to the base 12 but what happens to"}, {"start": 68.56, "duration": 9.12, "text": "the left hand side let us record this"}, {"start": 72.759, "duration": 9.04, "text": "property the log of a ra of B to the"}, {"start": 77.68, "duration": 8.92, "text": "base a this simplifies to be the power"}, {"start": 81.799, "duration": 9.481, "text": "of a here so and here we can see we have"}, {"start": 86.6, "duration": 8.36, "text": "similar TX here we have log a to the"}, {"start": 91.28, "duration": 5.56, "text": "base a log of a b to the base a this"}, {"start": 94.96, "duration": 5.439, "text": "simplifies to the power of a right over"}, {"start": 96.84, "duration": 6.08, "text": "here so here we have the log of 12 x the"}, {"start": 100.399, "duration": 6.241, "text": "base 12 so this will cancel out and we"}, {"start": 102.92, "duration": 8.839, "text": "are left with X so from here we get X is"}, {"start": 106.64, "duration": 10.04, "text": "equal to the log of 8 to the base 12 and"}, {"start": 111.759, "duration": 4.921, "text": "this is approximately equal to"}, {"start": 117.92, "duration": 6.559, "text": "0.836 8"}, {"start": 120.36, "duration": 5.679, "text": "to 9 and this is the correct answer"}, {"start": 124.479, "duration": 3.721, "text": "thank you for watching please kindly"}, {"start": 126.039, "duration": 6.401, "text": "subscribe to my YouTube channel also"}, {"start": 128.2, "duration": 4.24, "text": "like comment and share bye-bye"}], "title": "Exponential equation 2 | #maths #mathematics", "description": "", "lengthSeconds": 131, "width": 544, "height": 360}
{"video_id": "YRgognGCJRs", "transcript": [{"start": 0.599, "duration": 4.321, "text": "hello good day everyone welcome back"}, {"start": 2.76, "duration": 5.68, "text": "once again today we're going to solve"}, {"start": 4.92, "duration": 4.719, "text": "this exponential equation involving root"}, {"start": 8.44, "duration": 5.079, "text": "of"}, {"start": 9.639, "duration": 7.761, "text": "x so from the left we have X to the^ of"}, {"start": 13.519, "duration": 6.321, "text": "root of x this equals now this base"}, {"start": 17.4, "duration": 7.24, "text": "right over here which is root of x we"}, {"start": 19.84, "duration": 10.279, "text": "can write it as x^ 1 / 2 then we have to"}, {"start": 24.64, "duration": 8.36, "text": "the^ of X so here we have X of root of x"}, {"start": 30.119, "duration": 4.921, "text": "then this will be equal to X remember we"}, {"start": 33.0, "duration": 5.92, "text": "can multiply this power together right"}, {"start": 35.04, "duration": 7.679, "text": "so we have 1 / 2 * X that will give us X"}, {"start": 38.92, "duration": 7.08, "text": "all over two okay now for my solution"}, {"start": 42.719, "duration": 6.401, "text": "here I'm going to natural log both sides"}, {"start": 46.0, "duration": 7.44, "text": "of this equation remember X cannot be"}, {"start": 49.12, "duration": 6.32, "text": "equal to Z right okay so here if you"}, {"start": 53.44, "duration": 5.24, "text": "natural log both side we get natural log"}, {"start": 55.44, "duration": 5.759, "text": "of x to the^ root X this equals the"}, {"start": 58.68, "duration": 5.559, "text": "natural log of x"}, {"start": 61.199, "duration": 5.321, "text": "x / 2 now with the property of natural"}, {"start": 64.239, "duration": 5.401, "text": "logarithm remember this is crossing over"}, {"start": 66.52, "duration": 6.32, "text": "here and this is crossing over here so"}, {"start": 69.64, "duration": 9.799, "text": "that will become root X the natural log"}, {"start": 72.84, "duration": 10.88, "text": "of x = here x 2 m natural log of"}, {"start": 79.439, "duration": 9.401, "text": "x okay from here we have root x m natur"}, {"start": 83.72, "duration": 7.719, "text": "log of x - x 2 m natural X the whole of"}, {"start": 88.84, "duration": 5.76, "text": "this equal 0 so you know natural log of"}, {"start": 91.439, "duration": 4.801, "text": "x is common so we factor that out and we"}, {"start": 94.6, "duration": 7.199, "text": "left with root of"}, {"start": 96.24, "duration": 8.28, "text": "xus x 2 then this equals 0 okay so from"}, {"start": 101.799, "duration": 6.96, "text": "here this implies that the natural log"}, {"start": 104.52, "duration": 11.519, "text": "of x = 0 or from the bracket we have <"}, {"start": 108.759, "duration": 12.0, "text": "TK x - x 2 = 0 and from here we have x ="}, {"start": 116.039, "duration": 7.04, "text": "e^ 0 which is 1 right so here we find a"}, {"start": 120.759, "duration": 5.72, "text": "solution and from here take this time to"}, {"start": 123.079, "duration": 7.641, "text": "the right hand side we get root x = 0 +"}, {"start": 126.479, "duration": 8.241, "text": "x / 2 is X /2 this is over one so we"}, {"start": 130.72, "duration": 7.36, "text": "cross multiply from there"}, {"start": 134.72, "duration": 9.08, "text": "right so when you cross multiply we get"}, {"start": 138.08, "duration": 8.64, "text": "2 * otk X is 2un X then this equal x * 1"}, {"start": 143.8, "duration": 4.799, "text": "is X now for us to get rid of this root"}, {"start": 146.72, "duration": 3.799, "text": "sign we need to square both sides of"}, {"start": 148.599, "duration": 6.36, "text": "this equation"}, {"start": 150.519, "duration": 6.401, "text": "so here first of all 2 2 is 4 root x 2"}, {"start": 154.959, "duration": 5.28, "text": "remember Square we cancel the root sign"}, {"start": 156.92, "duration": 7.319, "text": "we get just X then this equals"}, {"start": 160.239, "duration": 8.041, "text": "x\u00b2 from the right hand side we can write"}, {"start": 164.239, "duration": 6.761, "text": "this as X2 then subtracting for X then"}, {"start": 168.28, "duration": 7.599, "text": "this equal 0 factor out X and here we"}, {"start": 171.0, "duration": 8.36, "text": "have x - 4 = 0 so from here we get x is"}, {"start": 175.879, "duration": 7.561, "text": "= 0 or from the bracket x - 4 = 0"}, {"start": 179.36, "duration": 6.28, "text": "implies that x = 4 but X is not equal to"}, {"start": 183.44, "duration": 5.84, "text": "Z that is not a solution here we find"}, {"start": 185.64, "duration": 7.04, "text": "another solution that x = 4 and"}, {"start": 189.28, "duration": 7.039, "text": "therefore the two real values of X that"}, {"start": 192.68, "duration": 6.8, "text": "satisfy the above equation is only one"}, {"start": 196.319, "duration": 4.92, "text": "and four and these are the right answers"}, {"start": 199.48, "duration": 3.88, "text": "thank you for watching please kindly"}, {"start": 201.239, "duration": 6.201, "text": "subscribe to my channel also like"}, {"start": 203.36, "duration": 4.08, "text": "comment and share bye-bye"}], "title": "Interesting exponential equation | can you solve ?", "description": "", "lengthSeconds": 206, "width": 640, "height": 360}
{"video_id": "Dfw7W5u-quo", "transcript": [{"start": 0.48, "duration": 5.24, "text": "okay here we have theun of x + 1 is ="}, {"start": 3.52, "duration": 5.4, "text": "root3 so let's go ahead and square both"}, {"start": 5.72, "duration": 6.72, "text": "sides since we can see we have root sign"}, {"start": 8.92, "duration": 4.759, "text": "everywhere right so Square both sides"}, {"start": 12.44, "duration": 3.359, "text": "then we can see that this Square will"}, {"start": 13.679, "duration": 4.44, "text": "cancel the square root and we can see"}, {"start": 15.799, "duration": 4.081, "text": "here Square we cancel the square root so"}, {"start": 18.119, "duration": 6.681, "text": "we have here just"}, {"start": 19.88, "duration": 6.88, "text": "X2 + 1 which is equal to 3 subtract one"}, {"start": 24.8, "duration": 5.92, "text": "from both sides so we get"}, {"start": 26.76, "duration": 5.839, "text": "x2 is equal to 3 - 1 which is 2 now you"}, {"start": 30.72, "duration": 4.76, "text": "can take the square root of both sides"}, {"start": 32.599, "duration": 7.241, "text": "so here we get X is equal to plus or"}, {"start": 35.48, "duration": 6.84, "text": "minus theare < TK of 2 and here we have"}, {"start": 39.84, "duration": 5.48, "text": "the solution please subscribe to my"}, {"start": 42.32, "duration": 3.0, "text": "channel"}], "title": "Interesting solution  |  #maths #exam #mathematics", "description": "", "lengthSeconds": 49, "width": 360, "height": 640}
{"video_id": "2w1gOBvfa_s", "transcript": [{"start": 0.48, "duration": 5.319, "text": "hello guys welcome back once again today"}, {"start": 3.08, "duration": 6.68, "text": "is circle theor right so we're going to"}, {"start": 5.799, "duration": 6.561, "text": "find the value of x here Y and Z as well"}, {"start": 9.76, "duration": 6.24, "text": "so let's see how to solve this problem"}, {"start": 12.36, "duration": 6.52, "text": "right so now let us recall this theorem"}, {"start": 16.0, "duration": 4.519, "text": "that states that angles in opposite"}, {"start": 18.88, "duration": 4.159, "text": "segments are"}, {"start": 20.519, "duration": 4.68, "text": "supplementary what I mean by two angles"}, {"start": 23.039, "duration": 5.961, "text": "are supplementary so let's assume we"}, {"start": 25.199, "duration": 6.0, "text": "have angle a and we have angle B so if"}, {"start": 29.0, "duration": 6.76, "text": "these two angles are supplementary so"}, {"start": 31.199, "duration": 8.561, "text": "this means that a + b is equal to"}, {"start": 35.76, "duration": 7.119, "text": "180\u00b0 right so now here we have the"}, {"start": 39.76, "duration": 6.479, "text": "opposite angles in this segment here"}, {"start": 42.879, "duration": 6.68, "text": "here we have X and we have 102\u00b0 so this"}, {"start": 46.239, "duration": 5.881, "text": "actually means that X and 102 are"}, {"start": 49.559, "duration": 6.16, "text": "supplementary angles right so we have"}, {"start": 52.12, "duration": 6.599, "text": "that X Plus"}, {"start": 55.719, "duration": 4.721, "text": "102\u00b0 is equal to"}, {"start": 58.719, "duration": 4.48, "text": "180\u00b0 right"}, {"start": 60.44, "duration": 5.44, "text": "so according to this theorem right so"}, {"start": 63.199, "duration": 5.201, "text": "right here we going to find the value of"}, {"start": 65.88, "duration": 6.0, "text": "x here so X here is equal to what"}, {"start": 68.4, "duration": 5.12, "text": "remember 102\u00b0 is positive here so if"}, {"start": 71.88, "duration": 4.16, "text": "it's crossing over it will change to"}, {"start": 73.52, "duration": 5.68, "text": "negative so here we have"}, {"start": 76.04, "duration": 6.36, "text": "180\u00b0 minus"}, {"start": 79.2, "duration": 4.2, "text": "102\u00b0 right and this gives the value of x"}, {"start": 82.4, "duration": 6.079, "text": "as"}, {"start": 83.4, "duration": 8.759, "text": "78 degrees wow wonderful okay now let us"}, {"start": 88.479, "duration": 5.6, "text": "find the value of Y here so if you take"}, {"start": 92.159, "duration": 4.681, "text": "a very good look at this place you can"}, {"start": 94.079, "duration": 6.601, "text": "see that we have a straight line here so"}, {"start": 96.84, "duration": 9.599, "text": "here we have this angle 81\u00b0 here we have"}, {"start": 100.68, "duration": 8.6, "text": "y so this actually mean that this y+ 81\u00b0"}, {"start": 106.439, "duration": 5.521, "text": "should be 180\u00b0 because this is sum of"}, {"start": 109.28, "duration": 4.24, "text": "angles on a straight line right so here"}, {"start": 111.96, "duration": 4.519, "text": "we have"}, {"start": 113.52, "duration": 7.4, "text": "y plus"}, {"start": 116.479, "duration": 6.92, "text": "81\u00b0 is equal to what 180\u00b0 that is sum of"}, {"start": 120.92, "duration": 6.44, "text": "angles on a straight line so here we"}, {"start": 123.399, "duration": 9.681, "text": "have y is equal to what"}, {"start": 127.36, "duration": 11.48, "text": "180\u00b0 minus 81\u00b0 right and this gives"}, {"start": 133.08, "duration": 8.64, "text": "99\u00b0 for y right okay now let us find the"}, {"start": 138.84, "duration": 5.44, "text": "value of Z here right so let us record"}, {"start": 141.72, "duration": 5.44, "text": "this theorem the same theorem we apply"}, {"start": 144.28, "duration": 4.959, "text": "in the first one that angles in opposite"}, {"start": 147.16, "duration": 5.359, "text": "segments are equal so the angles in"}, {"start": 149.239, "duration": 5.28, "text": "opposite M here is z and y right so"}, {"start": 152.519, "duration": 5.641, "text": "since they are supplementary so this"}, {"start": 154.519, "duration": 8.601, "text": "means that y + z isal 180\u00b0 so here we"}, {"start": 158.16, "duration": 9.04, "text": "have Z Plus the value of y here is 99\u00b0"}, {"start": 163.12, "duration": 8.64, "text": "right so 99\u00b0 is equal to"}, {"start": 167.2, "duration": 5.84, "text": "180\u00b0 right so here we have Z is equal to"}, {"start": 171.76, "duration": 4.119, "text": "what"}, {"start": 173.04, "duration": 7.32, "text": "180\u00b0 -"}, {"start": 175.879, "duration": 7.601, "text": "99\u00b0 and this is giving us what 81"}, {"start": 180.36, "duration": 5.879, "text": "\u00b0 and that's the answer so here we have"}, {"start": 183.48, "duration": 7.0, "text": "X is equal to what"}, {"start": 186.239, "duration": 5.28, "text": "78\u00b0 here Y is equal to what Y is equal"}, {"start": 190.48, "duration": 4.52, "text": "to"}, {"start": 191.519, "duration": 6.961, "text": "99\u00b0 and Z is equal to what"}, {"start": 195.0, "duration": 5.92, "text": "81\u00b0 and these are the answers thank you"}, {"start": 198.48, "duration": 5.759, "text": "for watching please do well to like"}, {"start": 200.92, "duration": 6.319, "text": "comment and share my videos thank you"}, {"start": 204.239, "duration": 3.0, "text": "bye-bye"}], "title": "Circle Theorem \ud83d\ude0d\ud83e\udd29", "description": "", "lengthSeconds": 205, "width": 636, "height": 360}
{"video_id": "35QnbAOsmus", "transcript": [{"start": 0.48, "duration": 5.2, "text": "hello you're welcome today we're going"}, {"start": 2.52, "duration": 7.48, "text": "to solve this nice interesting equation"}, {"start": 5.68, "duration": 7.2, "text": "here we have the fourth root of x plus"}, {"start": 10.0, "duration": 4.599, "text": "the root of x and this is equal to 1 so"}, {"start": 12.88, "duration": 4.159, "text": "let's go ahead and solve for the value"}, {"start": 14.599, "duration": 5.721, "text": "of x that will satisfy this beautiful"}, {"start": 17.039, "duration": 5.16, "text": "looking equation now let's get started"}, {"start": 20.32, "duration": 7.4, "text": "let us take note of this that the end"}, {"start": 22.199, "duration": 10.24, "text": "root of a is equal to a^ 1 / n so the"}, {"start": 27.72, "duration": 8.999, "text": "fourth root of x will be X to 1 / 4 plus"}, {"start": 32.439, "duration": 9.881, "text": "theun of X will be x^ 1 / 2 and this is"}, {"start": 36.719, "duration": 10.721, "text": "equal to 1 so from here we have x^ 1 / 4"}, {"start": 42.32, "duration": 9.28, "text": "plus so here we can rewrite this as x^ 1"}, {"start": 47.44, "duration": 10.959, "text": "/ 4 in bracket power of 2 we know this"}, {"start": 51.6, "duration": 10.0, "text": "property that a^ m n is = a^ M * n we"}, {"start": 58.399, "duration": 6.321, "text": "know 1 4 * 2 we give + 2 4 which is 1 /"}, {"start": 61.6, "duration": 5.519, "text": "2 and that is the power of X here then"}, {"start": 64.72, "duration": 4.6, "text": "this is equal to 1 so it is very clear"}, {"start": 67.119, "duration": 8.241, "text": "to make the following substitution"}, {"start": 69.32, "duration": 8.439, "text": "letting 1 x^ 1 4 = K so if you if you"}, {"start": 75.36, "duration": 6.799, "text": "raise both sides to the^ of 4 we get"}, {"start": 77.759, "duration": 7.841, "text": "that X will be equal to K to the^ 4 all"}, {"start": 82.159, "duration": 8.481, "text": "right so here we have here started from"}, {"start": 85.6, "duration": 7.12, "text": "here we have k 2 then plus k subtracting"}, {"start": 90.64, "duration": 6.439, "text": "one from both sides then this will be"}, {"start": 92.72, "duration": 6.88, "text": "equal to zero right okay so from here"}, {"start": 97.079, "duration": 4.441, "text": "let us solve this quadratic equation"}, {"start": 99.6, "duration": 8.199, "text": "using the general quadratic formula so"}, {"start": 101.52, "duration": 9.12, "text": "this will be K isal to -1 that is B plus"}, {"start": 107.799, "duration": 7.841, "text": "or minus so this will be square root of"}, {"start": 110.64, "duration": 9.04, "text": "B2 is 1 - 4 a c which is -4 1 - - 4 is 5"}, {"start": 115.64, "duration": 14.88, "text": "so we have here 5 / 2 so we can see here"}, {"start": 119.68, "duration": 19.68, "text": "that K1 is = -1 + < TK 5 / 2 and K2 is ="}, {"start": 130.52, "duration": 12.439, "text": "to -1 - < TK 5 / 2 okay so here we're"}, {"start": 139.36, "duration": 9.239, "text": "going to solve for x and x is K to^ 4 so"}, {"start": 142.959, "duration": 10.441, "text": "here we have our X1 is = to K1 ^ 4 which"}, {"start": 148.599, "duration": 8.241, "text": "is -1 + otk 5 /"}, {"start": 153.4, "duration": 6.08, "text": "2^ of 4 so I'm not going to show you"}, {"start": 156.84, "duration": 4.96, "text": "guys how to evaluate this here if you"}, {"start": 159.48, "duration": 9.119, "text": "watch my previous video on how to"}, {"start": 161.8, "duration": 10.68, "text": "evaluate -1 /2 +un 5 / 2 to the^ 9 you"}, {"start": 168.599, "duration": 8.241, "text": "will find how the fourth power of this"}, {"start": 172.48, "duration": 12.039, "text": "is evaluated right so by doing it here"}, {"start": 176.84, "duration": 10.8, "text": "we have here 7 / 2 - 3 < TK 5 all 2 and"}, {"start": 184.519, "duration": 5.521, "text": "when you plug this in the given equation"}, {"start": 187.64, "duration": 6.44, "text": "this satisfies the"}, {"start": 190.04, "duration": 7.08, "text": "equation so here our X2 will be K2 2"}, {"start": 194.08, "duration": 8.4, "text": "which is -1"}, {"start": 197.12, "duration": 7.44, "text": "sorry so that would be -1 - < TK 5 / 2^"}, {"start": 202.48, "duration": 7.88, "text": "4 which should be the conjugate of this"}, {"start": 204.56, "duration": 8.759, "text": "which is 7 / 2 + 3un 5 / 2 but this root"}, {"start": 210.36, "duration": 4.879, "text": "here is an extranos root so this root"}, {"start": 213.319, "duration": 7.441, "text": "does not satisfy the equation and"}, {"start": 215.239, "duration": 9.121, "text": "therefore X is = 7 /2 - 3un 5 / 2 which"}, {"start": 220.76, "duration": 5.759, "text": "is the only satisfying root of the"}, {"start": 224.36, "duration": 4.799, "text": "equation thank you for watching please"}, {"start": 226.519, "duration": 7.241, "text": "kindly subscribe to my channel also like"}, {"start": 229.159, "duration": 4.601, "text": "comment and share bye-bye"}], "title": "Olympiad Math problem | Can you solve ?", "description": "", "lengthSeconds": 232, "width": 640, "height": 360}
{"video_id": "Fk8M_ATKyKI", "transcript": [{"start": 0.719, "duration": 5.401, "text": "hello welcome back once again today we"}, {"start": 2.6, "duration": 6.28, "text": "have this nice olymp mathematics problem"}, {"start": 6.12, "duration": 5.679, "text": "we're given that X to the^ X is equal to"}, {"start": 8.88, "duration": 5.32, "text": "9 so in this video we're going to solve"}, {"start": 11.799, "duration": 6.56, "text": "for x so let's get"}, {"start": 14.2, "duration": 8.6, "text": "started we know that 2 to the^ of 2 is"}, {"start": 18.359, "duration": 8.641, "text": "equal 4 and obviously 4 is less than 9"}, {"start": 22.8, "duration": 8.0, "text": "and again 3 to the^ 3 is equal to 27 and"}, {"start": 27.0, "duration": 7.76, "text": "obviously 27 is greater than 9 so we can"}, {"start": 30.8, "duration": 8.88, "text": "see that the value of x lies between 2"}, {"start": 34.76, "duration": 7.68, "text": "and 3 right yes okay so now let's get"}, {"start": 39.68, "duration": 7.039, "text": "into the main video from the left hand"}, {"start": 42.44, "duration": 8.2, "text": "side we have X to the^ X this is equal"}, {"start": 46.719, "duration": 5.721, "text": "to 9 so we're going to Len both sides of"}, {"start": 50.64, "duration": 5.439, "text": "this equation so here we have the"}, {"start": 52.44, "duration": 7.52, "text": "natural log of x to the^ X this will be"}, {"start": 56.079, "duration": 6.401, "text": "equal to the natural log of 9 right so"}, {"start": 59.96, "duration": 6.44, "text": "we know this property of logm the log of"}, {"start": 62.48, "duration": 7.279, "text": "a to the power B is equal to B the log"}, {"start": 66.4, "duration": 7.52, "text": "of a so here this power here will come"}, {"start": 69.759, "duration": 7.121, "text": "down here right so this will become x *"}, {"start": 73.92, "duration": 3.96, "text": "the log of x so this is equal to the log"}, {"start": 76.88, "duration": 5.48, "text": "of"}, {"start": 77.88, "duration": 8.52, "text": "9 now X here we can actually write it"}, {"start": 82.36, "duration": 6.2, "text": "recall that X is same as e to the^ Ln of"}, {"start": 86.4, "duration": 5.16, "text": "X we know that the base of the natural"}, {"start": 88.56, "duration": 6.16, "text": "logarithm is e so with that base here"}, {"start": 91.56, "duration": 6.519, "text": "this will cancel out leaving us with X"}, {"start": 94.72, "duration": 7.32, "text": "so here x is same as e to the power of"}, {"start": 98.079, "duration": 8.36, "text": "Ln of X so we have multip by this Ln of"}, {"start": 102.04, "duration": 8.0, "text": "X then this is equal to the Ln of 9"}, {"start": 106.439, "duration": 6.64, "text": "right okay now let us recall the lambar"}, {"start": 110.04, "duration": 7.16, "text": "W function we know the lambar W function"}, {"start": 113.079, "duration": 8.881, "text": "with the following argument or E input x"}, {"start": 117.2, "duration": 7.599, "text": "* e to X is = to X right since the power"}, {"start": 121.96, "duration": 5.36, "text": "of E and the coefficient are exactly"}, {"start": 124.799, "duration": 5.32, "text": "equal so here we can see the power of E"}, {"start": 127.32, "duration": 5.24, "text": "is the is exactly with the coefficient"}, {"start": 130.119, "duration": 4.801, "text": "natural log of x and natural log of x so"}, {"start": 132.56, "duration": 4.319, "text": "we're going to take the lamb W on both"}, {"start": 134.92, "duration": 8.36, "text": "sides so here we get the"}, {"start": 136.879, "duration": 11.241, "text": "W of e to^ Ln X the Ln X this is equal"}, {"start": 143.28, "duration": 7.239, "text": "to the W of Ln of 9 so the left hand"}, {"start": 148.12, "duration": 4.6, "text": "side with this property we simplify to"}, {"start": 150.519, "duration": 5.36, "text": "something very nice so that would be the"}, {"start": 152.72, "duration": 8.72, "text": "natural log of x which is equal to here"}, {"start": 155.879, "duration": 8.921, "text": "the W of the natural log of 9 right so"}, {"start": 161.44, "duration": 7.56, "text": "from here we're going to make a simple"}, {"start": 164.8, "duration": 7.04, "text": "base of e right we know that e to the^"}, {"start": 169.0, "duration": 6.0, "text": "of Ln of anything let's use a is just"}, {"start": 171.84, "duration": 5.64, "text": "equal to a so if you make this base of e"}, {"start": 175.0, "duration": 5.28, "text": "so the left hand side we have e to the^"}, {"start": 177.48, "duration": 5.479, "text": "of the Ln of X so this will be equal the"}, {"start": 180.28, "duration": 5.879, "text": "right hand side e to the power of the W"}, {"start": 182.959, "duration": 6.0, "text": "of the natural log of 9 so we can see"}, {"start": 186.159, "duration": 6.08, "text": "this here simplifies to X which we are"}, {"start": 188.959, "duration": 7.481, "text": "looking for and this is equal to e to"}, {"start": 192.239, "duration": 7.481, "text": "the power of the W of the Ln of 9 and"}, {"start": 196.44, "duration": 7.92, "text": "with the help of my calculator this is"}, {"start": 199.72, "duration": 4.64, "text": "finally approximated as"}, {"start": 205.2, "duration": 6.56, "text": "2.45"}, {"start": 208.159, "duration": 3.601, "text": "953 9"}, {"start": 212.0, "duration": 4.959, "text": "2802 and this is the correct answer"}, {"start": 215.48, "duration": 3.88, "text": "thank you for watching please kindly"}, {"start": 216.959, "duration": 5.151, "text": "subscribe to my channel also like"}, {"start": 219.36, "duration": 5.369, "text": "comment and share bye-bye"}, {"start": 222.11, "duration": 2.619, "text": "[Music]"}], "title": "Oxford University Entrance Exam | Can you solve ?", "description": "", "lengthSeconds": 223, "width": 428, "height": 240}
{"video_id": "iFAbWSicEcU", "transcript": [{"start": 0.52, "duration": 5.72, "text": "hello welcome back once again please"}, {"start": 3.76, "duration": 3.959, "text": "kindly support this channel by"}, {"start": 6.24, "duration": 5.0, "text": "subscribing to this"}, {"start": 7.719, "duration": 6.681, "text": "Channel please subscribe to this Channel"}, {"start": 11.24, "duration": 6.36, "text": "and share our videos please support this"}, {"start": 14.4, "duration": 6.76, "text": "channel by subscribing to the channel"}, {"start": 17.6, "duration": 9.599, "text": "thank you here we have this function of"}, {"start": 21.16, "duration": 9.32, "text": "X here we have f of x is = X all over x"}, {"start": 27.199, "duration": 7.36, "text": "+ 1 so we are asked to find the first"}, {"start": 30.48, "duration": 8.48, "text": "derivative of this function at x = 1"}, {"start": 34.559, "duration": 7.52, "text": "right okay now let's see how to do this"}, {"start": 38.96, "duration": 8.52, "text": "okay now here we have this function of X"}, {"start": 42.079, "duration": 8.281, "text": "here f of x = x 1 + x so the first thing"}, {"start": 47.48, "duration": 5.719, "text": "to do is to find the derivative of this"}, {"start": 50.36, "duration": 5.839, "text": "function with respect to X so we can see"}, {"start": 53.199, "duration": 7.32, "text": "that this is a rational function right x"}, {"start": 56.199, "duration": 7.641, "text": "1 + x so we call the quotient rule"}, {"start": 60.519, "duration": 7.6, "text": "so we have these functions of X U of X"}, {"start": 63.84, "duration": 7.24, "text": "and V of X so if we are given y equal to"}, {"start": 68.119, "duration": 6.68, "text": "U of X over V of X so these are both"}, {"start": 71.08, "duration": 8.359, "text": "functions of X right then the derivative"}, {"start": 74.799, "duration": 6.081, "text": "is going to be the bottom V of x * the"}, {"start": 79.439, "duration": 4.72, "text": "derivative of the"}, {"start": 80.88, "duration": 6.32, "text": "numerator minus the numerator time the"}, {"start": 84.159, "duration": 5.481, "text": "derivative of the denominator all divide"}, {"start": 87.2, "duration": 3.36, "text": "by the denominator rais to the^ of two"}, {"start": 89.64, "duration": 3.6, "text": "right"}, {"start": 90.56, "duration": 6.64, "text": "so here we have the derivative of this"}, {"start": 93.24, "duration": 7.68, "text": "function FR Prime of X is going to be"}, {"start": 97.2, "duration": 6.32, "text": "making use of this property so the"}, {"start": 100.92, "duration": 7.519, "text": "denominator 1 +"}, {"start": 103.52, "duration": 7.12, "text": "x time the derivative of the numerator"}, {"start": 108.439, "duration": 7.121, "text": "so the numerator here is X so the"}, {"start": 110.64, "duration": 7.839, "text": "derivative of x is just one right so"}, {"start": 115.56, "duration": 5.199, "text": "minus here we can see the numerator that"}, {"start": 118.479, "duration": 6.081, "text": "is X"}, {"start": 120.759, "duration": 6.521, "text": "time the derivative of the denominator"}, {"start": 124.56, "duration": 7.48, "text": "so we have it 1 + x so the derivative of"}, {"start": 127.28, "duration": 7.44, "text": "1 + x with respect to X is just one so"}, {"start": 132.04, "duration": 8.0, "text": "all divide by the square of the"}, {"start": 134.72, "duration": 7.76, "text": "denominator so 1 + x ra^ of 2"}, {"start": 140.04, "duration": 6.0, "text": "right so we have"}, {"start": 142.48, "duration": 6.0, "text": "here F Prime of X that is to simplify"}, {"start": 146.04, "duration": 9.919, "text": "this well so here we open up this"}, {"start": 148.48, "duration": 14.679, "text": "brackets 1 * 1 is 1 + 1 * X is x - 1 * X"}, {"start": 155.959, "duration": 10.36, "text": "is X all over 1 + x of 2 so this is"}, {"start": 163.159, "duration": 5.8, "text": "simplified to you know this X and this x"}, {"start": 166.319, "duration": 7.401, "text": "cancel out so we left with just 1 all"}, {"start": 168.959, "duration": 7.761, "text": "over 1 + x of 2 so this is the"}, {"start": 173.72, "duration": 6.32, "text": "derivative of this function of X here so"}, {"start": 176.72, "duration": 6.799, "text": "now we are here to evaluate the frime of"}, {"start": 180.04, "duration": 6.04, "text": "1 so here we have fime of X so we're"}, {"start": 183.519, "duration": 9.321, "text": "going to let X ="}, {"start": 186.08, "duration": 12.439, "text": "1 then we get fime of 1 is equal to 1"}, {"start": 192.84, "duration": 10.319, "text": "all over 1 + 1^ of 2 so this is equal to"}, {"start": 198.519, "duration": 8.121, "text": "1 all over you know 1 + 1 is 2 then of 2"}, {"start": 203.159, "duration": 5.961, "text": "and this is exactly equal to 1 over 4"}, {"start": 206.64, "duration": 4.48, "text": "and this is the right answer thank you"}, {"start": 209.12, "duration": 4.759, "text": "for watching way to subscribe to this"}, {"start": 211.12, "duration": 5.759, "text": "YouTube channel and share our"}, {"start": 213.879, "duration": 3.0, "text": "videos"}], "title": "A nice function of x. f(x) = x/(1+x), f'(1)=?", "description": "", "lengthSeconds": 215, "width": 636, "height": 360}
{"video_id": "TRbbc87mGSo", "transcript": [{"start": 0.44, "duration": 4.76, "text": "good day everyone welcome back once"}, {"start": 2.84, "duration": 6.0, "text": "again please if you have not subscribed"}, {"start": 5.2, "duration": 6.08, "text": "to my Channel please and please support"}, {"start": 8.84, "duration": 5.4, "text": "this my channel by"}, {"start": 11.28, "duration": 5.999, "text": "subscribing thank you today we have this"}, {"start": 14.24, "duration": 4.56, "text": "limit problem here the limit as X is"}, {"start": 17.279, "duration": 5.521, "text": "approaching"}, {"start": 18.8, "duration": 6.96, "text": "Infinity of this function over here 2x -"}, {"start": 22.8, "duration": 6.16, "text": "8 all divide by 6 x -"}, {"start": 25.76, "duration": 5.24, "text": "4 right if you try to make direct"}, {"start": 28.96, "duration": 4.64, "text": "substitution here we have here 2 *"}, {"start": 31.0, "duration": 7.68, "text": "Infinity which is infinity and infinity"}, {"start": 33.6, "duration": 8.32, "text": "minus 8 is infinity over 6 * infinity is"}, {"start": 38.68, "duration": 6.44, "text": "infinity infinity minus 4 is infinity so"}, {"start": 41.92, "duration": 6.119, "text": "we will have Infinity over infinity"}, {"start": 45.12, "duration": 6.8, "text": "right and this is in indeterminate form"}, {"start": 48.039, "duration": 6.281, "text": "this is not allowed in mathematics right"}, {"start": 51.92, "duration": 5.159, "text": "but this function here as X is"}, {"start": 54.32, "duration": 5.879, "text": "approaching Infinity that is as X is"}, {"start": 57.079, "duration": 5.8, "text": "getting larger larger larger and larg"}, {"start": 60.199, "duration": 6.24, "text": "larger must be"}, {"start": 62.879, "duration": 6.801, "text": "approaching a particular value"}, {"start": 66.439, "duration": 6.121, "text": "right okay now let's see what to do the"}, {"start": 69.68, "duration": 6.2, "text": "idea here is to divide the numerator"}, {"start": 72.56, "duration": 6.879, "text": "throughout by X and also the denominator"}, {"start": 75.88, "duration": 7.52, "text": "throughout by X okay let us do that and"}, {"start": 79.439, "duration": 8.72, "text": "see here this is equal to"}, {"start": 83.4, "duration": 4.759, "text": "limit as X is approaching"}, {"start": 88.52, "duration": 4.879, "text": "Infinity here we have"}, {"start": 90.84, "duration": 10.0, "text": "2x -"}, {"start": 93.399, "duration": 14.241, "text": "8 / X then again the denominator 6 x -"}, {"start": 100.84, "duration": 6.8, "text": "4 / X right so this is equal"}, {"start": 109.159, "duration": 8.121, "text": "to limit as X is approaching"}, {"start": 114.6, "duration": 4.24, "text": "Infinity you know we're going to split"}, {"start": 117.28, "duration": 8.24, "text": "here we have"}, {"start": 118.84, "duration": 13.64, "text": "2x/ x - 88 / X and"}, {"start": 125.52, "duration": 11.12, "text": "again 6 x / x - 4 /"}, {"start": 132.48, "duration": 8.2, "text": "X right so let us try to simplify here"}, {"start": 136.64, "duration": 7.36, "text": "we have X cancel X and here x cancel X"}, {"start": 140.68, "duration": 6.68, "text": "so we have this is equal to"}, {"start": 144.0, "duration": 8.44, "text": "limit as X is approaching"}, {"start": 147.36, "duration": 8.84, "text": "Infinity of 2us 8 all"}, {"start": 152.44, "duration": 7.2, "text": "x/ 6 - 4"}, {"start": 156.2, "duration": 6.2, "text": "/x so this is equal to now let's see now"}, {"start": 159.64, "duration": 6.04, "text": "8 x as X is getting larger is"}, {"start": 162.4, "duration": 6.119, "text": "approaching zero right so we can deduce"}, {"start": 165.68, "duration": 5.24, "text": "that the limit of 8 x as X is"}, {"start": 168.519, "duration": 9.041, "text": "approaching Infinity is z right so here"}, {"start": 170.92, "duration": 8.959, "text": "we have 2 - 0 over 6 here 4X as X is"}, {"start": 177.56, "duration": 7.44, "text": "approaching Infinity is approaching zero"}, {"start": 179.879, "duration": 8.961, "text": "so 6 - 0 and this equal to 2 over 6 this"}, {"start": 185.0, "duration": 6.44, "text": "simplifies finally to 1 over three and"}, {"start": 188.84, "duration": 5.039, "text": "this is the final answer thank you for"}, {"start": 191.44, "duration": 8.079, "text": "watching do where to subscribe to this"}, {"start": 193.879, "duration": 5.64, "text": "channel share our videos bye-bye"}], "title": "Fascinating limit", "description": "", "lengthSeconds": 198, "width": 636, "height": 360}
{"video_id": "ilC_P5NU6sI", "transcript": [{"start": 0.76, "duration": 5.92, "text": "hello good day ladies and gentlemen"}, {"start": 3.919, "duration": 4.521, "text": "welcome back once again today we are"}, {"start": 6.68, "duration": 3.999, "text": "going to solve for the values of X in"}, {"start": 8.44, "duration": 6.64, "text": "the following cubic"}, {"start": 10.679, "duration": 8.001, "text": "equation we have X Cub - 1 this is equal"}, {"start": 15.08, "duration": 6.56, "text": "to 0 now let's get"}, {"start": 18.68, "duration": 5.04, "text": "started for the left hand side we are"}, {"start": 21.64, "duration": 7.6, "text": "going to write it as"}, {"start": 23.72, "duration": 8.76, "text": "follows X cubed - 1 is the same as 1"}, {"start": 29.24, "duration": 5.96, "text": "cubed this is equal to"}, {"start": 32.48, "duration": 5.599, "text": "zero now from the left we are going to"}, {"start": 35.2, "duration": 5.8, "text": "make use of the difference of two cubes"}, {"start": 38.079, "duration": 11.241, "text": "formula we have a"}, {"start": 41.0, "duration": 14.52, "text": "cubus B Cubed this is equal to a - b * a"}, {"start": 49.32, "duration": 9.32, "text": "2 + a b + b"}, {"start": 55.52, "duration": 7.76, "text": "2 so from the left make use of this"}, {"start": 58.64, "duration": 7.92, "text": "formula here we have x"}, {"start": 63.28, "duration": 11.479, "text": "-1 into braet"}, {"start": 66.56, "duration": 10.72, "text": "X2 + x * 1 which is x + 1 2 which is 1"}, {"start": 74.759, "duration": 9.241, "text": "and this is equal"}, {"start": 77.28, "duration": 9.72, "text": "0 so from here we have that X - 1 is = 0"}, {"start": 84.0, "duration": 6.799, "text": "also we have"}, {"start": 87.0, "duration": 5.36, "text": "x\u00b2 + x + 1"}, {"start": 90.799, "duration": 4.881, "text": "this is equal to"}, {"start": 92.36, "duration": 6.0, "text": "Z for the left hand side here we have"}, {"start": 95.68, "duration": 5.16, "text": "just a linear equation and right hand"}, {"start": 98.36, "duration": 6.2, "text": "side we have a quadratic"}, {"start": 100.84, "duration": 8.0, "text": "equation so for this equation here x is"}, {"start": 104.56, "duration": 6.599, "text": "= to 1 and this is a"}, {"start": 108.84, "duration": 4.84, "text": "solution now for the equation here we"}, {"start": 111.159, "duration": 6.041, "text": "are going to make use of the general"}, {"start": 113.68, "duration": 10.92, "text": "quadratic formula we have"}, {"start": 117.2, "duration": 11.16, "text": "here x is equal to B and here a is equal"}, {"start": 124.6, "duration": 7.279, "text": "to the coefficient of x\u00b2 which is 1 B is"}, {"start": 128.36, "duration": 6.64, "text": "equal to the coent of X which is 1 and C"}, {"start": 131.879, "duration": 5.801, "text": "which is the constant is also one right"}, {"start": 135.0, "duration": 6.68, "text": "and the formula we are going to make use"}, {"start": 137.68, "duration": 10.4, "text": "of it's B plus or"}, {"start": 141.68, "duration": 10.16, "text": "minus the square root of B 2 - 4 a"}, {"start": 148.08, "duration": 3.76, "text": "c all/ 2"}, {"start": 151.92, "duration": 10.599, "text": "a so from here we have X is = to B which"}, {"start": 157.44, "duration": 11.159, "text": "is 1 plus or minus the root of B 2 is 1"}, {"start": 162.519, "duration": 10.44, "text": "2 which is 1 - 4 a c 4 * a * C is the"}, {"start": 168.599, "duration": 8.0, "text": "same as 4 * 1 * 1 and that is 4 then"}, {"start": 172.959, "duration": 9.241, "text": "over 2 a which is 2 * 1 which is"}, {"start": 176.599, "duration": 11.841, "text": "2 so from here we have X is is = to -1"}, {"start": 182.2, "duration": 11.48, "text": "plus or minus theare < TK of -3 then 2"}, {"start": 188.44, "duration": 11.799, "text": "and from here we have -1 plus or minus"}, {"start": 193.68, "duration": 7.96, "text": "of -3 is the same as I < tk3 then all 2"}, {"start": 200.239, "duration": 5.161, "text": "and from here we come up with the"}, {"start": 201.64, "duration": 7.239, "text": "solutions we have one real solution X is"}, {"start": 205.4, "duration": 5.759, "text": "= to 1 we also have two complex"}, {"start": 208.879, "duration": 10.601, "text": "Solutions x"}, {"start": 211.159, "duration": 16.44, "text": "= -1 + I < tk3 / 2 and the other complex"}, {"start": 219.48, "duration": 10.479, "text": "solution X is = -1 - I < tk3 over two"}, {"start": 227.599, "duration": 4.081, "text": "and these are the solutions thank you"}, {"start": 229.959, "duration": 4.48, "text": "for watching please kindly subscribe to"}, {"start": 231.68, "duration": 5.759, "text": "my channel also like comment and share"}, {"start": 234.439, "duration": 3.0, "text": "bye-bye"}], "title": "Another well-known cubic equation | Interesting problem | #maths #olympiad", "description": "", "lengthSeconds": 236, "width": 640, "height": 360}
{"video_id": "rfpl_gO1Wdg", "transcript": [{"start": 0.0, "duration": 3.959, "text": "hello welcome today we're going to solve"}, {"start": 2.04, "duration": 4.2, "text": "this simultaneous equation"}, {"start": 3.959, "duration": 4.801, "text": "within some seconds how do we solve it"}, {"start": 6.24, "duration": 5.52, "text": "remember X and Y belongs to real numbers"}, {"start": 8.76, "duration": 5.759, "text": "okay now let us add the equations"}, {"start": 11.76, "duration": 4.859, "text": "together so we have X Plus X here so we"}, {"start": 14.519, "duration": 4.68, "text": "get 2X"}, {"start": 16.619, "duration": 5.641, "text": "and here we have minus y plus Y is zero"}, {"start": 19.199, "duration": 6.361, "text": "so is equal to 2 plus 4 here we get six"}, {"start": 22.26, "duration": 6.24, "text": "so here x is equal to 6 all over 2 and"}, {"start": 25.56, "duration": 4.86, "text": "that is equal to 3 okay how do we solve"}, {"start": 28.5, "duration": 5.1, "text": "for y let's subtract the equations so we"}, {"start": 30.42, "duration": 7.139, "text": "have x minus X is zero minus one minus y"}, {"start": 33.6, "duration": 7.32, "text": "is minus 2y is equal to you know 2 minus"}, {"start": 37.559, "duration": 6.541, "text": "4 here so we get what negative two here"}, {"start": 40.92, "duration": 6.659, "text": "Y is equal to negative 2 over negative 2"}, {"start": 44.1, "duration": 5.7, "text": "and Y is equal to one therefore X is"}, {"start": 47.579, "duration": 5.401, "text": "equal to 3 and"}, {"start": 49.8, "duration": 6.32, "text": "Y is equal to one thank you for viewing"}, {"start": 52.98, "duration": 3.14, "text": "please subscribe to this channel"}], "title": "Best way to solve simultaneous equation within seconds", "description": "", "lengthSeconds": 60, "width": 360, "height": 640}
{"video_id": "1Mavoxs1DOk", "transcript": [{"start": 0.799, "duration": 3.601, "text": "hello good viewers welcome back once"}, {"start": 2.48, "duration": 6.28, "text": "again today we have an interesting"}, {"start": 4.4, "duration": 8.6, "text": "problem we're given that 72 ra^ of 1 - 2"}, {"start": 8.76, "duration": 7.96, "text": "x = 8 and here we're going to evaluate 3"}, {"start": 13.0, "duration": 5.8, "text": "^ of X let's get started now from the"}, {"start": 16.72, "duration": 4.92, "text": "left hand side of this equation we shall"}, {"start": 18.8, "duration": 5.36, "text": "make use of the following algebraic"}, {"start": 21.64, "duration": 6.92, "text": "property remember when we have a to the"}, {"start": 24.16, "duration": 10.239, "text": "power of m- n this is same as a to the"}, {"start": 28.56, "duration": 10.36, "text": "power of M / a to the^ of n right so"}, {"start": 34.399, "duration": 6.201, "text": "here we have 1 - 2 X so we can write"}, {"start": 38.92, "duration": 9.04, "text": "this as"}, {"start": 40.6, "duration": 10.799, "text": "72 of 1 / 72 of 2 all X then from the"}, {"start": 47.96, "duration": 6.759, "text": "right hand side this is equal to 8 now"}, {"start": 51.399, "duration": 7.361, "text": "from here let us cross multiply remember"}, {"start": 54.719, "duration": 6.601, "text": "this is 8 over 1 so here 72 1 is 72"}, {"start": 58.76, "duration": 5.28, "text": "right then then if you multiply by 1 you"}, {"start": 61.32, "duration": 9.56, "text": "will have 72 and from here"}, {"start": 64.04, "duration": 10.6, "text": "72 2 x * 8 so that will give us 8 *"}, {"start": 70.88, "duration": 7.32, "text": "72 2 all X then from the right this is"}, {"start": 74.64, "duration": 6.479, "text": "equal to 72 now from here let's go ahead"}, {"start": 78.2, "duration": 7.4, "text": "and divide both side by 8 so divide here"}, {"start": 81.119, "duration": 8.96, "text": "by 8 divide here by 8 here8 cancel 8"}, {"start": 85.6, "duration": 8.199, "text": "here then from the left we have here 72"}, {"start": 90.079, "duration": 7.281, "text": "ra the^ of 2 all over X then this is"}, {"start": 93.799, "duration": 8.081, "text": "equal to 72 / 8 that will give us a 9"}, {"start": 97.36, "duration": 7.96, "text": "now from here let us raise both sides to"}, {"start": 101.88, "duration": 7.279, "text": "the power of X right so here to the"}, {"start": 105.32, "duration": 6.52, "text": "power of X and here to the power of"}, {"start": 109.159, "duration": 5.161, "text": "X now notice that from the left hand"}, {"start": 111.84, "duration": 5.319, "text": "side these Powers here can actually"}, {"start": 114.32, "duration": 6.159, "text": "multiply right that is if we recall that"}, {"start": 117.159, "duration": 6.761, "text": "a to the power of B in bracket of"}, {"start": 120.479, "duration": 6.32, "text": "this is same as a^ B * C so when we"}, {"start": 123.92, "duration": 7.36, "text": "multiply 2 x * X you know X will cancel"}, {"start": 126.799, "duration": 9.281, "text": "X then we left with what 72 then ra^ of"}, {"start": 131.28, "duration": 7.959, "text": "2 which is equal to 9 to the^ of X now"}, {"start": 136.08, "duration": 6.239, "text": "from here let us change the sides of"}, {"start": 139.239, "duration": 4.321, "text": "this equation so we have 9^ X which is"}, {"start": 142.319, "duration": 5.64, "text": "equal to"}, {"start": 143.56, "duration": 7.28, "text": "72 of 2 now from here we shall take"}, {"start": 147.959, "duration": 5.961, "text": "square root of both sides so here theare"}, {"start": 150.84, "duration": 9.039, "text": "< TK of 9 ^ x this is equal to the"}, {"start": 153.92, "duration": 9.399, "text": "square < TK of 72 2 right okay now from"}, {"start": 159.879, "duration": 6.961, "text": "the left recall that 9 is the same as 3"}, {"start": 163.319, "duration": 7.361, "text": "S then with this x that will become 2x"}, {"start": 166.84, "duration": 7.2, "text": "right then square root the same as 1 2"}, {"start": 170.68, "duration": 6.08, "text": "that is the power of 1 2 then from here"}, {"start": 174.04, "duration": 6.08, "text": "remember that when we have the square"}, {"start": 176.76, "duration": 6.479, "text": "root of x\u00b2 in this form this is actually"}, {"start": 180.12, "duration": 5.24, "text": "the absolute value of x right so from"}, {"start": 183.239, "duration": 5.401, "text": "here this is same as the absolute value"}, {"start": 185.36, "duration": 8.28, "text": "of 72 right and that will give us"}, {"start": 188.64, "duration": 7.8, "text": "72 so here from the right we have here"}, {"start": 193.64, "duration": 4.76, "text": "72 now from here notice that this power"}, {"start": 196.44, "duration": 5.079, "text": "can multiply together right so we have 2"}, {"start": 198.4, "duration": 6.119, "text": "x * 1 / 2 2 will cancel 2 and we left"}, {"start": 201.519, "duration": 7.92, "text": "with 3 to the power of X which should be"}, {"start": 204.519, "duration": 6.961, "text": "equal to 72 and this is the right answer"}, {"start": 209.439, "duration": 4.16, "text": "thank you for watching please kindly"}, {"start": 211.48, "duration": 6.24, "text": "subscribe to my channel also like"}, {"start": 213.599, "duration": 4.121, "text": "comment and share bye-bye"}], "title": "Very interesting homemade algebra problem | #maths #mathematics", "description": "", "lengthSeconds": 217, "width": 608, "height": 360}
{"video_id": "Q5bvZ1PqcY4", "transcript": [{"start": 0.52, "duration": 4.56, "text": "hello good day viewers once again"}, {"start": 2.32, "duration": 5.24, "text": "welcome back again in today's video"}, {"start": 5.08, "duration": 9.16, "text": "we're going to solve this exponential"}, {"start": 7.56, "duration": 10.719, "text": "equation right 4^ x * 16 3 X - 1 is = to"}, {"start": 14.24, "duration": 8.199, "text": "8 here we ask to solve for the value of"}, {"start": 18.279, "duration": 4.881, "text": "x right okay please kindly subscribe to"}, {"start": 22.439, "duration": 4.521, "text": "my"}, {"start": 23.16, "duration": 6.439, "text": "channel also like comment and share okay"}, {"start": 26.96, "duration": 5.04, "text": "what do we need to do here first the"}, {"start": 29.599, "duration": 4.96, "text": "first thing we need to do here is to"}, {"start": 32.0, "duration": 4.32, "text": "make sure that from both sides the whole"}, {"start": 34.559, "duration": 5.721, "text": "bases are"}, {"start": 36.32, "duration": 7.079, "text": "equal right and then we can equate the"}, {"start": 40.28, "duration": 7.16, "text": "powers and here we have four 4 is the"}, {"start": 43.399, "duration": 7.64, "text": "same as 2^ 2 we have 16 here 16 is the"}, {"start": 47.44, "duration": 7.439, "text": "same as 2 ^ 4 here we have 8 8 is the"}, {"start": 51.039, "duration": 7.04, "text": "same as 2 to^ of 3 and do not forget"}, {"start": 54.879, "duration": 6.36, "text": "that a to the^ of B in bracket of C is"}, {"start": 58.079, "duration": 5.721, "text": "the same as a^ of B * C"}, {"start": 61.239, "duration": 6.88, "text": "okay now let's see how this will help in"}, {"start": 63.8, "duration": 12.84, "text": "this solution here we have to express as"}, {"start": 68.119, "duration": 13.881, "text": "2 ^ 2 then X multip by here 2^ 4^ 3 X -"}, {"start": 76.64, "duration": 7.92, "text": "1 this is equal to 2^ of 3 right okay so"}, {"start": 82.0, "duration": 4.36, "text": "now we make use of this law here that"}, {"start": 84.56, "duration": 8.68, "text": "will give us 2^"}, {"start": 86.36, "duration": 11.6, "text": "2x * here 2^ 4 into braet 3x - 1 this is"}, {"start": 93.24, "duration": 7.6, "text": "equal to 2^ of 3 so you know from here"}, {"start": 97.96, "duration": 6.4, "text": "let us recall this property once again A"}, {"start": 100.84, "duration": 7.4, "text": "to the^ B * a the^ c this is same as"}, {"start": 104.36, "duration": 6.0, "text": "since we have equal bases a^ b + C so"}, {"start": 108.24, "duration": 8.0, "text": "from here we have 2^"}, {"start": 110.36, "duration": 9.799, "text": "2X + 4 into bracket 3x - 1 this is equal"}, {"start": 116.24, "duration": 5.879, "text": "to 2^ 3 right okay from here here we"}, {"start": 120.159, "duration": 5.081, "text": "have equal bases from both sides so we"}, {"start": 122.119, "duration": 8.84, "text": "can equate the powers so here we have 2X"}, {"start": 125.24, "duration": 8.6, "text": "+ 4 into braet 3x - 1 this is equal to 3"}, {"start": 130.959, "duration": 6.841, "text": "so here we have 2X + open this bracket"}, {"start": 133.84, "duration": 5.08, "text": "with four we have here 12x - 4 this is"}, {"start": 137.8, "duration": 4.96, "text": "equal to"}, {"start": 138.92, "duration": 8.599, "text": "3 so from here we have 2X + 12 x that is"}, {"start": 142.76, "duration": 7.8, "text": "14 x - 4 this is equal to 3 so from here"}, {"start": 147.519, "duration": 6.121, "text": "we add four to both sides we get 14 X is"}, {"start": 150.56, "duration": 4.759, "text": "= to 3 + 4 which is 7 so from here we"}, {"start": 153.64, "duration": 4.28, "text": "divide both side by the coefficient of x"}, {"start": 155.319, "duration": 6.081, "text": "which is 14 here"}, {"start": 157.92, "duration": 7.8, "text": "14 so from here we get X is equal to you"}, {"start": 161.4, "duration": 7.119, "text": "know 7 over 14 is the same as 1 over 2"}, {"start": 165.72, "duration": 5.72, "text": "because 7 here 1 and 7 here two and here"}, {"start": 168.519, "duration": 6.08, "text": "we get 1 / 2"}, {"start": 171.44, "duration": 5.4, "text": "or5 and this is the solution thank you"}, {"start": 174.599, "duration": 4.401, "text": "for watching please kindly subscribe to"}, {"start": 176.84, "duration": 5.88, "text": "this channel also do where to like"}, {"start": 179.0, "duration": 3.72, "text": "comment and share bye-bye bye"}], "title": "Interesting exponential equation. #olympiadmathematics #maths #education", "description": "", "lengthSeconds": 181, "width": 640, "height": 360}
{"video_id": "d39bY5YYJb4", "transcript": [{"start": 0.48, "duration": 4.44, "text": "hello good viewers welcome back once"}, {"start": 2.399, "duration": 6.28, "text": "again let us find a solution to the"}, {"start": 4.92, "duration": 8.16, "text": "following radical equation we have here"}, {"start": 8.679, "duration": 10.44, "text": "root of x * root 3 this is equal to"}, {"start": 13.08, "duration": 10.76, "text": "2 we have root of x * root of 3 this is"}, {"start": 19.119, "duration": 4.721, "text": "equal to 2 now let's Square both"}, {"start": 24.279, "duration": 5.92, "text": "sides now when you square both sides"}, {"start": 27.199, "duration": 6.441, "text": "here the square and this root sign we"}, {"start": 30.199, "duration": 5.04, "text": "cancel out and we're left with x * root"}, {"start": 33.64, "duration": 6.52, "text": "of 3"}, {"start": 35.239, "duration": 8.881, "text": "right so x * root of 3 then this is"}, {"start": 40.16, "duration": 5.32, "text": "equal to 2 2 which is 4 remember we"}, {"start": 44.12, "duration": 5.04, "text": "solving"}, {"start": 45.48, "duration": 5.759, "text": "for x right so from here we divide both"}, {"start": 49.16, "duration": 5.84, "text": "side by the coefficient of x which is"}, {"start": 51.239, "duration": 7.241, "text": "root3 right so divided by root3 from the"}, {"start": 55.0, "duration": 7.359, "text": "right hand side 4 /"}, {"start": 58.48, "duration": 7.039, "text": "root3 we Sol that is"}, {"start": 62.359, "duration": 5.041, "text": "root3 okay now from here root3 we cancel"}, {"start": 65.519, "duration": 8.64, "text": "root3 and here we come up with the"}, {"start": 67.4, "duration": 9.759, "text": "solution X is = 4 all over < tk3 or we"}, {"start": 74.159, "duration": 6.481, "text": "can rationalize this by multiply byun3"}, {"start": 77.159, "duration": 5.96, "text": "over3 that is 4/ <"}, {"start": 80.64, "duration": 5.2, "text": "tk3 multi"}, {"start": 83.119, "duration": 6.36, "text": "byun3 over <"}, {"start": 85.84, "duration": 8.84, "text": "tk3 now from here we get 4 * < tk3 which"}, {"start": 89.479, "duration": 8.68, "text": "is 4 < tk3 over < tk3 * tk3 isun 9 and"}, {"start": 94.68, "duration": 6.52, "text": "that will give usun of 9 which is 3 so"}, {"start": 98.159, "duration": 7.32, "text": "here we come up with a solution X is ="}, {"start": 101.2, "duration": 5.959, "text": "43 / 3 and that's the right answer thank"}, {"start": 105.479, "duration": 3.881, "text": "you for watching please kindly subscribe"}, {"start": 107.159, "duration": 5.721, "text": "to my channel also like comment and"}, {"start": 109.36, "duration": 3.52, "text": "share bye-bye"}], "title": "Radical equation | We need to be careful with the root sign. #maths #mathematics", "description": "", "lengthSeconds": 111, "width": 534, "height": 360}
{"video_id": "b46PGD_Smsc", "transcript": [{"start": 0.84, "duration": 4.519, "text": "hello good day everyone welcome back"}, {"start": 3.32, "duration": 5.279, "text": "once again today we have this"}, {"start": 5.359, "duration": 5.001, "text": "interesting Olympiad mathematics problem"}, {"start": 8.599, "duration": 8.481, "text": "we're given the following quadratic"}, {"start": 10.36, "duration": 10.56, "text": "equation X2 + 7 x + 49 is = 0 and here"}, {"start": 17.08, "duration": 6.08, "text": "we'll be evaluating x to the thir"}, {"start": 20.92, "duration": 5.48, "text": "power now let's get"}, {"start": 23.16, "duration": 5.359, "text": "started from the original equation we"}, {"start": 26.4, "duration": 5.199, "text": "have that"}, {"start": 28.519, "duration": 7.841, "text": "x\u00b2 Plus 7"}, {"start": 31.599, "duration": 8.161, "text": "x + 49 is ="}, {"start": 36.36, "duration": 6.76, "text": "to0 now from here let's move this second"}, {"start": 39.76, "duration": 6.52, "text": "term and the constant here to the right"}, {"start": 43.12, "duration": 5.68, "text": "remember both are positive here so if"}, {"start": 46.28, "duration": 5.72, "text": "they are crossing over they will become"}, {"start": 48.8, "duration": 7.8, "text": "negative so here we have"}, {"start": 52.0, "duration": 8.64, "text": "x\u00b2 is equal to this will become"}, {"start": 56.6, "duration": 8.76, "text": "-7x - 49"}, {"start": 60.64, "duration": 6.68, "text": "right okay so here we have that X2 is"}, {"start": 65.36, "duration": 7.64, "text": "equal"}, {"start": 67.32, "duration": 9.4, "text": "to7 x - 49 this will be very"}, {"start": 73.0, "duration": 8.6, "text": "important now let's proceed so far we"}, {"start": 76.72, "duration": 9.56, "text": "have gotten x\u00b2 to be equal to"}, {"start": 81.6, "duration": 8.519, "text": "-7x - 49 remember the question asks for"}, {"start": 86.28, "duration": 6.119, "text": "X to^ of 3 so for us to get X to to the"}, {"start": 90.119, "duration": 4.561, "text": "power of 3 at the left hand side we need"}, {"start": 92.399, "duration": 3.441, "text": "to multiply both sides of this equation"}, {"start": 94.68, "duration": 5.36, "text": "by x to"}, {"start": 95.84, "duration": 11.4, "text": "the to the 4 power right so here we have"}, {"start": 100.04, "duration": 11.399, "text": "x^ 1 * X2 this will be equal to x ^ 1"}, {"start": 107.24, "duration": 9.159, "text": "into braet --x -"}, {"start": 111.439, "duration": 8.96, "text": "49 so now let us recall this property a"}, {"start": 116.399, "duration": 8.0, "text": "m a n remember they have the same base"}, {"start": 120.399, "duration": 6.68, "text": "so this will be a^ n + n"}, {"start": 124.399, "duration": 4.92, "text": "right so here we have the same base we"}, {"start": 127.079, "duration": 5.641, "text": "need to add the power so here we get X"}, {"start": 129.319, "duration": 7.64, "text": "to^ of 3 this will be equal to remember"}, {"start": 132.72, "duration": 10.2, "text": "x^ of 1 is just X right so here we have"}, {"start": 136.959, "duration": 8.28, "text": "x * -7 x will give us -7 x 2 then minus"}, {"start": 142.92, "duration": 5.36, "text": "x * 49 will give us"}, {"start": 145.239, "duration": 6.401, "text": "49x so we can see what is going on here"}, {"start": 148.28, "duration": 5.44, "text": "so here we have X cu op this will be"}, {"start": 151.64, "duration": 6.239, "text": "equal to remember this is"}, {"start": 153.72, "duration": 8.799, "text": "-77 right then this x\u00b2 here check back"}, {"start": 157.879, "duration": 7.0, "text": "this is the value in terms of x -7x - 49"}, {"start": 162.519, "duration": 9.321, "text": "so we replace that"}, {"start": 164.879, "duration": 11.72, "text": "with 7 x - 49 then we have -"}, {"start": 171.84, "duration": 11.64, "text": "49x so from here we get X cubed is = to"}, {"start": 176.599, "duration": 6.881, "text": "-7 * -7 x is that 7 X oh sorry this"}, {"start": 185.72, "duration": 15.36, "text": "is -7 oh okay so -7 * -7 x will give us"}, {"start": 191.92, "duration": 14.52, "text": "POS 49x right then -7 * -49 we give us+"}, {"start": 201.08, "duration": 9.6, "text": "343 then minus remember we have 49x here"}, {"start": 206.44, "duration": 6.68, "text": "so obviously 49x - 49x is Z and here we"}, {"start": 210.68, "duration": 5.0, "text": "have our value for x cubed which is"}, {"start": 213.12, "duration": 5.56, "text": "equal to"}, {"start": 215.68, "duration": 5.479, "text": "343 amazing thank you for watching"}, {"start": 218.68, "duration": 7.279, "text": "please kindly subscribe to my channel"}, {"start": 221.159, "duration": 4.8, "text": "also like comment and share bye-bye"}], "title": "Olympiad Math problem | Can you solve ?", "description": "", "lengthSeconds": 225, "width": 640, "height": 360}
{"video_id": "RcFyFT8Fc2w", "transcript": [{"start": 0.799, "duration": 4.56, "text": "hello good viewers welcome back once"}, {"start": 2.879, "duration": 6.001, "text": "again let's find solution to the"}, {"start": 5.359, "duration": 7.561, "text": "following Olympiad ma problem we have"}, {"start": 8.88, "duration": 7.56, "text": "here Ro otk X +un Y this is equal to 1"}, {"start": 12.92, "duration": 5.64, "text": "from here we get otk x * y this isal to"}, {"start": 16.44, "duration": 7.44, "text": "1 and right over here we're going to"}, {"start": 18.56, "duration": 7.719, "text": "evaluate this expression here X2 + y^ 2"}, {"start": 23.88, "duration": 6.2, "text": "now from this equation two"}, {"start": 26.279, "duration": 8.041, "text": "here when you square both sides we have"}, {"start": 30.08, "duration": 8.08, "text": "that x * y will also be equal to 1"}, {"start": 34.32, "duration": 7.96, "text": "right okay now let's see how to solve"}, {"start": 38.16, "duration": 9.64, "text": "for X2 + Y 2 from equation 1 we have otk"}, {"start": 42.28, "duration": 8.32, "text": "x + otk y this is = one now let's go"}, {"start": 47.8, "duration": 4.36, "text": "ahead and square both sides from the"}, {"start": 50.6, "duration": 2.639, "text": "left hand side we'll be making use of"}, {"start": 52.16, "duration": 7.96, "text": "this"}, {"start": 53.239, "duration": 13.841, "text": "property a + b 2 this is equal to a 2 +"}, {"start": 60.12, "duration": 8.96, "text": "+ b 2 + 2 * a * B now make use of this"}, {"start": 67.08, "duration": 6.64, "text": "property we have the first 10 squ which"}, {"start": 69.08, "duration": 7.039, "text": "is otk X2 which will give us X plus the"}, {"start": 73.72, "duration": 6.28, "text": "second 10 square which is root y^2 that"}, {"start": 76.119, "duration": 7.921, "text": "will give us y plus 2 * the first term"}, {"start": 80.0, "duration": 7.799, "text": "otk X MTI by the second term here root y"}, {"start": 84.04, "duration": 9.92, "text": "then this is equal to"}, {"start": 87.799, "duration": 9.401, "text": "1 so from here we have have x + y + 2"}, {"start": 93.96, "duration": 8.08, "text": "now otk x * otk Y is the same as root of"}, {"start": 97.2, "duration": 7.32, "text": "x * Y and this is equal to 1 now from"}, {"start": 102.04, "duration": 5.759, "text": "the equation two we have the value for"}, {"start": 104.52, "duration": 8.32, "text": "root X Y which is 1 so here we"}, {"start": 107.799, "duration": 9.64, "text": "substitute we have x + y + 2 * 1 which"}, {"start": 112.84, "duration": 7.36, "text": "is 2 this is equal to 1 so from here we"}, {"start": 117.439, "duration": 9.081, "text": "subtract two from both sides and we get"}, {"start": 120.2, "duration": 8.919, "text": "x + y this is = to 1 - 2 which is a"}, {"start": 126.52, "duration": 4.76, "text": "-1 now from here let's go ahead and"}, {"start": 129.119, "duration": 6.041, "text": "square both sides once again because"}, {"start": 131.28, "duration": 6.679, "text": "we're solving for X2 + Y 2 now when you"}, {"start": 135.16, "duration": 2.799, "text": "square both"}, {"start": 138.319, "duration": 10.2, "text": "sides from the left you know this"}, {"start": 140.599, "duration": 11.481, "text": "property here we we have x\u00b2 + y^ 2 + 2 *"}, {"start": 148.519, "duration": 7.281, "text": "x * y this is equal to now -1 to the"}, {"start": 152.08, "duration": 7.92, "text": "power of an even number will give us 1"}, {"start": 155.8, "duration": 7.04, "text": "now do not forget from here x * Y is = 1"}, {"start": 160.0, "duration": 11.0, "text": "so let's go ahead and substitute we get"}, {"start": 162.84, "duration": 11.72, "text": "here X2 + Y 2 + 2 * 1 right x * 1 x * Y"}, {"start": 171.0, "duration": 6.36, "text": "is 1 so 2 * 1 will give us 2 and this is"}, {"start": 174.56, "duration": 6.84, "text": "equal to one now subtract two from both"}, {"start": 177.36, "duration": 10.12, "text": "sides we arrive at the solution that x^2"}, {"start": 181.4, "duration": 8.32, "text": "+ y^2 is = 1 - 2 which is -1 and this is"}, {"start": 187.48, "duration": 4.36, "text": "the right answer thank you for watching"}, {"start": 189.72, "duration": 6.84, "text": "please kindly subscribe to my channel"}, {"start": 191.84, "duration": 4.72, "text": "also like comment and share bye-bye"}], "title": "Chinese | Interesting Olympiad Math | #maths #mathematics", "description": "", "lengthSeconds": 195, "width": 548, "height": 360}
{"video_id": "UdLqt_aUcXA", "transcript": [{"start": 0.399, "duration": 4.481, "text": "wow hello welcome back once again please"}, {"start": 3.439, "duration": 4.321, "text": "subscribe to my channel if you have not"}, {"start": 4.88, "duration": 6.52, "text": "done that please like comment and"}, {"start": 7.76, "duration": 8.24, "text": "share okay so here we have the ramanan"}, {"start": 11.4, "duration": 7.48, "text": "problem here we given that x - y = 5 and"}, {"start": 16.0, "duration": 5.119, "text": "also the root of x +un of Y is equal to"}, {"start": 18.88, "duration": 6.08, "text": "5 here we're going to find the value of"}, {"start": 21.119, "duration": 5.961, "text": "values of X and Y right okay I'm going"}, {"start": 24.96, "duration": 3.76, "text": "to solve this problem like this so I'm"}, {"start": 27.08, "duration": 6.999, "text": "going to"}, {"start": 28.72, "duration": 9.12, "text": "let this squ < TK of x = a so here squ <"}, {"start": 34.079, "duration": 7.0, "text": "TK of y = b so Square both side we get X"}, {"start": 37.84, "duration": 9.76, "text": "is = a 2 and here Square both side Y is"}, {"start": 41.079, "duration": 11.401, "text": "= b 2 right okay so from equation 1 x -"}, {"start": 47.6, "duration": 6.24, "text": "y this is going to be a 2 - b 2 and this"}, {"start": 52.48, "duration": 5.079, "text": "is equal"}, {"start": 53.84, "duration": 7.44, "text": "to 5 and from here we"}, {"start": 57.559, "duration": 8.441, "text": "have of X which is a + of Y which is B"}, {"start": 61.28, "duration": 8.76, "text": "this is also equal to 5 right so this is"}, {"start": 66.0, "duration": 7.0, "text": "our equation one in terms of A and B and"}, {"start": 70.04, "duration": 4.719, "text": "here equation two okay so from the"}, {"start": 73.0, "duration": 3.72, "text": "equation one from the left hand side we"}, {"start": 74.759, "duration": 6.0, "text": "have difference of two squares right so"}, {"start": 76.72, "duration": 8.2, "text": "that is going to be a + b MTI by a minus"}, {"start": 80.759, "duration": 6.4, "text": "B this is equal to 5 right so but in"}, {"start": 84.92, "duration": 4.519, "text": "equation two we have the value for a + b"}, {"start": 87.159, "duration": 6.32, "text": "which is 5 so right over here we replace"}, {"start": 89.439, "duration": 7.801, "text": "it with 5 then multi a minus B and this"}, {"start": 93.479, "duration": 5.28, "text": "is equal to 5 so divide both sides by 5"}, {"start": 97.24, "duration": 5.12, "text": "and here by"}, {"start": 98.759, "duration": 6.121, "text": "5 okay so here we get a minus B which is"}, {"start": 102.36, "duration": 6.079, "text": "equal to 1"}, {"start": 104.88, "duration": 5.8, "text": "right okay so we can call this equation"}, {"start": 108.439, "duration": 6.761, "text": "three right so we're going to find the"}, {"start": 110.68, "duration": 7.84, "text": "value of a and b using using the"}, {"start": 115.2, "duration": 5.4, "text": "equation two and equation three right"}, {"start": 118.52, "duration": 3.559, "text": "and we're going to solve the two"}, {"start": 120.6, "duration": 4.799, "text": "equations"}, {"start": 122.079, "duration": 7.201, "text": "simultaneously so here we have a + b is"}, {"start": 125.399, "duration": 6.761, "text": "= to 2 is equal to 5 right so this is"}, {"start": 129.28, "duration": 7.36, "text": "equation one equation two I mean and"}, {"start": 132.16, "duration": 7.52, "text": "here A minus B is equal to 1 and this is"}, {"start": 136.64, "duration": 5.28, "text": "equation three okay so we're going to"}, {"start": 139.68, "duration": 5.44, "text": "use the elimination method so let us add"}, {"start": 141.92, "duration": 7.08, "text": "both equations together to eliminate B"}, {"start": 145.12, "duration": 7.72, "text": "Because B has opposite signs right so"}, {"start": 149.0, "duration": 7.239, "text": "here A+ a which is 2 a this is equal to"}, {"start": 152.84, "duration": 4.96, "text": "5 + 1 is 6 so from here we divide both"}, {"start": 156.239, "duration": 5.28, "text": "side by"}, {"start": 157.8, "duration": 7.12, "text": "2 and here we get a is equal to"}, {"start": 161.519, "duration": 6.521, "text": "3 right so we can solve for the value of"}, {"start": 164.92, "duration": 4.88, "text": "B from any of the"}, {"start": 168.04, "duration": 6.119, "text": "equations"}, {"start": 169.8, "duration": 6.84, "text": "so we can make use of equation two or"}, {"start": 174.159, "duration": 6.08, "text": "equation three so let us make use of"}, {"start": 176.64, "duration": 6.56, "text": "equation two a plus b is given us five"}, {"start": 180.239, "duration": 6.801, "text": "here we have a is = 3 so here we have 3"}, {"start": 183.2, "duration": 7.759, "text": "+ B is = 5 so subtract from both side we"}, {"start": 187.04, "duration": 10.16, "text": "get B isal 5 - 3 which is 2 right so let"}, {"start": 190.959, "duration": 10.441, "text": "us recall that X is = a 2 and also B is"}, {"start": 197.2, "duration": 9.56, "text": "X I mean Y is equal to B2 right so here"}, {"start": 201.4, "duration": 8.52, "text": "x is = to a 2 which is 3 S and here Y is"}, {"start": 206.76, "duration": 7.759, "text": "= b 2 which is 2^ 2 so from here we get"}, {"start": 209.92, "duration": 8.8, "text": "X = to 3 S which is 9 and also here Y is"}, {"start": 214.519, "duration": 6.241, "text": "= 4 and this are the solution to the"}, {"start": 218.72, "duration": 4.48, "text": "problem thank you for watching please"}, {"start": 220.76, "duration": 5.16, "text": "kindly subscribe to my YouTube Channel"}, {"start": 223.2, "duration": 6.319, "text": "please like my videos comment and also"}, {"start": 225.92, "duration": 3.599, "text": "share bye-bye"}], "title": "Sir Srinivasa Ramanujan'a like problem", "description": "", "lengthSeconds": 228, "width": 640, "height": 360}
{"video_id": "tqSdYvrWeUg", "transcript": [{"start": 1.16, "duration": 4.679, "text": "hello welcome back once again today we"}, {"start": 3.28, "duration": 5.279, "text": "have this interesting exponential"}, {"start": 5.839, "duration": 5.481, "text": "equation and this is an Olympia"}, {"start": 8.559, "duration": 5.521, "text": "mathematics problem so we're given from"}, {"start": 11.32, "duration": 4.52, "text": "the left hand side that the square root"}, {"start": 14.08, "duration": 4.08, "text": "of"}, {"start": 15.84, "duration": 8.24, "text": "x root of"}, {"start": 18.16, "duration": 10.039, "text": "x is equal to X right so in this video I"}, {"start": 24.08, "duration": 6.0, "text": "would like to use log rithm right so"}, {"start": 28.199, "duration": 5.121, "text": "we're going to take the log of both"}, {"start": 30.08, "duration": 5.479, "text": "sides so here we have the natural log"}, {"start": 33.32, "duration": 6.52, "text": "ofun"}, {"start": 35.559, "duration": 8.601, "text": "xun X this will be equal to the natural"}, {"start": 39.84, "duration": 7.64, "text": "log of x so we know this property of"}, {"start": 44.16, "duration": 7.96, "text": "log the natural log of a to the power of"}, {"start": 47.48, "duration": 8.36, "text": "B is equal to B MTI the natural log of a"}, {"start": 52.12, "duration": 7.16, "text": "right so we apply this property here so"}, {"start": 55.84, "duration": 7.039, "text": "this will become the root of x * the"}, {"start": 59.28, "duration": 6.04, "text": "natural log of root of x then this is"}, {"start": 62.879, "duration": 4.92, "text": "equal to the natural log of"}, {"start": 65.32, "duration": 6.56, "text": "x okay we know that the root of x is"}, {"start": 67.799, "duration": 9.521, "text": "same as x^ 1 / 2 right so we have here"}, {"start": 71.88, "duration": 9.8, "text": "root of x * the natural log of x^ 1 / 2"}, {"start": 77.32, "duration": 7.68, "text": "square root means to the^ of 1 / 2 then"}, {"start": 81.68, "duration": 6.399, "text": "we have here minus natural log of x this"}, {"start": 85.0, "duration": 6.24, "text": "one here then this is equal to"}, {"start": 88.079, "duration": 6.241, "text": "zero okay so from"}, {"start": 91.24, "duration": 7.919, "text": "here we use this property again so this"}, {"start": 94.32, "duration": 8.2, "text": "will become 1 / 2 * X Asun x /"}, {"start": 99.159, "duration": 6.361, "text": "2 then times the natural log of x and"}, {"start": 102.52, "duration": 4.919, "text": "here minus the natural log of x this"}, {"start": 105.52, "duration": 4.68, "text": "isal"}, {"start": 107.439, "duration": 4.96, "text": "to0 so from here we can see the natural"}, {"start": 110.2, "duration": 5.68, "text": "log of x is common so let's Factor it"}, {"start": 112.399, "duration": 5.281, "text": "out so we have the natural log of x into"}, {"start": 115.88, "duration": 5.8, "text": "bracket so here we are left with square"}, {"start": 117.68, "duration": 8.32, "text": "root of x / 2 then minus here we're left"}, {"start": 121.68, "duration": 6.999, "text": "with one then this is equal to zero so"}, {"start": 126.0, "duration": 6.28, "text": "from here we have these two product s is"}, {"start": 128.679, "duration": 6.241, "text": "equal to Z so which implies from here"}, {"start": 132.28, "duration": 11.56, "text": "that either the natural of X is equal"}, {"start": 134.92, "duration": 13.76, "text": "to0 or the < TK x / 2 - 1 is = 0 from"}, {"start": 143.84, "duration": 7.399, "text": "here we take a common base of a right so"}, {"start": 148.68, "duration": 5.559, "text": "here we have X will be equal to e to"}, {"start": 151.239, "duration": 8.521, "text": "the^ of 0 so here we get our solution"}, {"start": 154.239, "duration": 7.761, "text": "that X is = 1 right okay and from here"}, {"start": 159.76, "duration": 7.479, "text": "take1 to the right hand side to become"}, {"start": 162.0, "duration": 10.159, "text": "positive we getun x / 2 isal to 1 then"}, {"start": 167.239, "duration": 9.801, "text": "cross multiply this is over 1 x * 1 isun"}, {"start": 172.159, "duration": 7.72, "text": "x so this isal to 2 * 1 is 2 so from"}, {"start": 177.04, "duration": 5.119, "text": "here for us to get rid of the root sign"}, {"start": 179.879, "duration": 4.601, "text": "we are going to square both sides of the"}, {"start": 182.159, "duration": 4.281, "text": "equation so this Square will cancel the"}, {"start": 184.48, "duration": 7.08, "text": "square root and here we have our second"}, {"start": 186.44, "duration": 8.4, "text": "solution X is = to 2^ 2 which is 4 so"}, {"start": 191.56, "duration": 5.399, "text": "here we have found the two R solutions"}, {"start": 194.84, "duration": 4.319, "text": "that satisfy the given"}, {"start": 196.959, "duration": 5.041, "text": "equation thank you for watching please"}, {"start": 199.159, "duration": 7.481, "text": "kindly subscribe to my channel also like"}, {"start": 202.0, "duration": 4.64, "text": "comment and share bye-bye"}], "title": "Interesting Olympiad Math | Can you solve this equation ?", "description": "", "lengthSeconds": 205, "width": 428, "height": 240}
{"video_id": "UChu1RZE0po", "transcript": [{"start": 0.08, "duration": 4.679, "text": "hello welcome back to another video here"}, {"start": 2.12, "duration": 4.799, "text": "we have this interesting Olympia matth"}, {"start": 4.759, "duration": 4.84, "text": "problem please if you not subscribed to"}, {"start": 6.919, "duration": 4.88, "text": "my channel kindly support my channel by"}, {"start": 9.599, "duration": 4.841, "text": "subscribing please like comment and"}, {"start": 11.799, "duration": 7.161, "text": "share here we're given that A + B is"}, {"start": 14.44, "duration": 7.8, "text": "equal to 2 a\u00b2 + B2 is = 5 and right over"}, {"start": 18.96, "duration": 6.239, "text": "here we asked to evaluate a cubed + B"}, {"start": 22.24, "duration": 7.4, "text": "Cubed right okay we're going to make use"}, {"start": 25.199, "duration": 11.961, "text": "of this identity in this video a cubed +"}, {"start": 29.64, "duration": 9.079, "text": "b Cub is equal to a + b Cub - 3 * a * B"}, {"start": 37.16, "duration": 6.84, "text": "*"}, {"start": 38.719, "duration": 7.881, "text": "A+ B right so we have a problem here and"}, {"start": 44.0, "duration": 4.719, "text": "the only problem we have here is the"}, {"start": 46.6, "duration": 4.08, "text": "value of a * B which is not given in the"}, {"start": 48.719, "duration": 4.84, "text": "original problem but right over here A"}, {"start": 50.68, "duration": 5.32, "text": "plus b and A+ B the value is already"}, {"start": 53.559, "duration": 7.361, "text": "here as two right so how do we figure"}, {"start": 56.0, "duration": 8.799, "text": "out the value of a times B right"}, {"start": 60.92, "duration": 9.559, "text": "okay we're going to do that by also"}, {"start": 64.799, "duration": 10.241, "text": "recording this identity a\u00b2 + b\u00b2 = a + b^"}, {"start": 70.479, "duration": 7.161, "text": "2 - 2 a b right so from here we're going"}, {"start": 75.04, "duration": 6.16, "text": "to make substitutions here a s + b s is"}, {"start": 77.64, "duration": 9.799, "text": "5 so here we have 5 this is equal to a +"}, {"start": 81.2, "duration": 9.16, "text": "b 2 that is 2 2 and here we have - 2 a b"}, {"start": 87.439, "duration": 6.64, "text": "so here we have 5 is = to 2 Square which"}, {"start": 90.36, "duration": 7.719, "text": "is 4 - 2 a b so here col terms we're"}, {"start": 94.079, "duration": 6.671, "text": "going to have 2 a b is = 4 - 5 which"}, {"start": 98.079, "duration": 4.121, "text": "is1 so divide both sides by"}, {"start": 100.75, "duration": 5.53, "text": "[Music]"}, {"start": 102.2, "duration": 9.519, "text": "2 so from here we have a * B is equal to"}, {"start": 106.28, "duration": 8.64, "text": "-1 all over 2 cool so we're going to"}, {"start": 111.719, "duration": 7.44, "text": "find the value of a cub + B Cubed from"}, {"start": 114.92, "duration": 10.119, "text": "here right so here we have a cubed + B"}, {"start": 119.159, "duration": 9.361, "text": "Cubed which is = to a + b Cub - 3 * a *"}, {"start": 125.039, "duration": 7.961, "text": "B A + B so this is going to be equal to"}, {"start": 128.52, "duration": 10.999, "text": "a + b that is 2 so we have here 2 cubus"}, {"start": 133.0, "duration": 9.68, "text": "3 * a * B is1 / 2 then times a + b so"}, {"start": 139.519, "duration": 7.921, "text": "that is 2 right so this is going to be"}, {"start": 142.68, "duration": 9.839, "text": "equal to 2 cubed is 8 and here 2 cancel"}, {"start": 147.44, "duration": 7.0, "text": "2 so here we have -3 * 1 which is pos3"}, {"start": 152.519, "duration": 5.121, "text": "and right over here we get this to be"}, {"start": 154.44, "duration": 5.64, "text": "equal to 11 therefore the value of a cub"}, {"start": 157.64, "duration": 4.8, "text": "plus b Cub is 11 thank you for watching"}, {"start": 160.08, "duration": 5.239, "text": "please if you find my video interesting"}, {"start": 162.44, "duration": 5.4, "text": "kindly support my channel by subscribing"}, {"start": 165.319, "duration": 5.521, "text": "also please like comment and share"}, {"start": 167.84, "duration": 3.0, "text": "bye-bye"}], "title": "Interesting Olympiad Math.", "description": "", "lengthSeconds": 169, "width": 640, "height": 360}
{"video_id": "_p-a5Z6PcK0", "transcript": [{"start": 0.96, "duration": 5.439, "text": "hello welcome back once again today we"}, {"start": 3.36, "duration": 4.68, "text": "have this interesting quadratic equation"}, {"start": 6.399, "duration": 6.921, "text": "so we're going to solve for the values"}, {"start": 8.04, "duration": 8.28, "text": "of X so we have from the left we have X"}, {"start": 13.32, "duration": 5.0, "text": "to the^ 4 this is equal to from the"}, {"start": 16.32, "duration": 5.719, "text": "right we have"}, {"start": 18.32, "duration": 9.879, "text": "81 right so on the left hand side we can"}, {"start": 22.039, "duration": 8.961, "text": "rewrite it as X to the^ 2 then squared"}, {"start": 28.199, "duration": 6.761, "text": "since we know that 2 * 2 is 4 4 then"}, {"start": 31.0, "duration": 7.559, "text": "this will be equal to 81 is same as 9"}, {"start": 34.96, "duration": 13.4, "text": "s now subtract 9 squ from both sides so"}, {"start": 38.559, "duration": 11.361, "text": "we get x ^ 2 2 - 9^ 2 this is equal to Z"}, {"start": 48.36, "duration": 3.6, "text": "from here we're going to make use of the"}, {"start": 49.92, "duration": 3.959, "text": "difference of two squares formula"}, {"start": 51.96, "duration": 10.759, "text": "remember that a"}, {"start": 53.879, "duration": 13.28, "text": "s - b 2 is equal to a + b B *"}, {"start": 62.719, "duration": 6.44, "text": "Aus B so from here this is the"}, {"start": 67.159, "duration": 4.721, "text": "difference of 2 squares formula so we"}, {"start": 69.159, "duration": 6.161, "text": "going to apply this rule here so the"}, {"start": 71.88, "duration": 7.64, "text": "base here is X2 so we're going to have"}, {"start": 75.32, "duration": 7.64, "text": "X2 plus the base here is 9 we have 9"}, {"start": 79.52, "duration": 5.84, "text": "then multiply by the subtraction that is"}, {"start": 82.96, "duration": 3.36, "text": "the difference of the two bases that"}, {"start": 85.36, "duration": 6.64, "text": "would be"}, {"start": 86.32, "duration": 8.52, "text": "X2 - 9 then this is equal to0 so from"}, {"start": 92.0, "duration": 6.24, "text": "here we have these two products which is"}, {"start": 94.84, "duration": 14.08, "text": "equal to Z so this implies from here"}, {"start": 98.24, "duration": 12.199, "text": "that either X2 + 9 is = 0 or x\u00b2 - 9 is ="}, {"start": 108.92, "duration": 5.36, "text": "to"}, {"start": 110.439, "duration": 5.64, "text": "0 so from here this 9 is positive if it"}, {"start": 114.28, "duration": 6.4, "text": "is crossing over it will become negative"}, {"start": 116.079, "duration": 7.441, "text": "so we get x\u00b2 is = to"}, {"start": 120.68, "duration": 6.24, "text": "9 take the square root of both sides you"}, {"start": 123.52, "duration": 5.28, "text": "get X will be equal to the plus or minus"}, {"start": 126.92, "duration": 8.12, "text": "the root"}, {"start": 128.8, "duration": 10.719, "text": "of9 so here we get X is equal to of1 is"}, {"start": 135.04, "duration": 7.72, "text": "Iota so we get plus or minus aota then"}, {"start": 139.519, "duration": 4.961, "text": "sare root of 9 is 3 so we can see this"}, {"start": 142.76, "duration": 5.8, "text": "solution is"}, {"start": 144.48, "duration": 7.6, "text": "imaginary then over to this here we have"}, {"start": 148.56, "duration": 6.399, "text": "x\u00b2 is equal to this is negative crossing"}, {"start": 152.08, "duration": 6.239, "text": "over it becomes POS 9 take the square"}, {"start": 154.959, "duration": 8.0, "text": "root of both sides we get X is equal to+"}, {"start": 158.319, "duration": 11.601, "text": "or minus the < TK of 9 then from here we"}, {"start": 162.959, "duration": 10.321, "text": "get X is = + or - 3 so here we have the"}, {"start": 169.92, "duration": 6.56, "text": "solutions X is = to plus or minus 3 is"}, {"start": 173.28, "duration": 7.599, "text": "the only two R Solutions and X is equal"}, {"start": 176.48, "duration": 8.679, "text": "to plus or minus Iota root of nine which"}, {"start": 180.879, "duration": 6.64, "text": "is Iota 3 this is the two imaginary"}, {"start": 185.159, "duration": 3.881, "text": "Solutions and the two real solutions"}, {"start": 187.519, "duration": 3.841, "text": "thank you for watching please kindly"}, {"start": 189.04, "duration": 6.88, "text": "subscribe to my channel also like"}, {"start": 191.36, "duration": 4.56, "text": "comment and share bye-bye"}], "title": "Interesting quartic equation | Can you solve ?", "description": "", "lengthSeconds": 194, "width": 428, "height": 240}
{"video_id": "aPQ5QEza8Us", "transcript": [{"start": 0.539, "duration": 4.26, "text": "good evening viewers welcome back to my"}, {"start": 2.639, "duration": 5.401, "text": "Channel today we have this simple"}, {"start": 4.799, "duration": 6.121, "text": "problem if a is equal to negative 1 and"}, {"start": 8.04, "duration": 5.34, "text": "B is equal to negative 2 then a to the"}, {"start": 10.92, "duration": 5.04, "text": "power 3 minus B to the power 3 over a"}, {"start": 13.38, "duration": 4.26, "text": "minus b equals Dutch"}, {"start": 15.96, "duration": 3.6, "text": "please if today is the first day of"}, {"start": 17.64, "duration": 3.6, "text": "visiting this Channel please subscribe"}, {"start": 19.56, "duration": 3.6, "text": "to this Channel and turn on the"}, {"start": 21.24, "duration": 4.74, "text": "notification Bell so that you'll be"}, {"start": 23.16, "duration": 4.56, "text": "notified anytime we upload new videos"}, {"start": 25.98, "duration": 3.24, "text": "okay now let's see how we're going to"}, {"start": 27.72, "duration": 3.24, "text": "solve this problem now we're going to"}, {"start": 29.22, "duration": 4.62, "text": "use the following identity"}, {"start": 30.96, "duration": 6.119, "text": "that a raised power 3 minus B raise"}, {"start": 33.84, "duration": 5.58, "text": "power 3 is the same as a minus B"}, {"start": 37.079, "duration": 4.561, "text": "times a squared"}, {"start": 39.42, "duration": 6.299, "text": "plus b squared"}, {"start": 41.64, "duration": 6.78, "text": "plus a b okay now"}, {"start": 45.719, "duration": 6.121, "text": "we have a raise power 3 minus B raise"}, {"start": 48.42, "duration": 4.38, "text": "power 3 all over a minus B to be equal"}, {"start": 51.84, "duration": 5.28, "text": "to"}, {"start": 52.8, "duration": 9.06, "text": "a minus B times a squared plus b squared"}, {"start": 57.12, "duration": 7.8, "text": "plus a b close bracket then all over"}, {"start": 61.86, "duration": 6.299, "text": "a minus B okay"}, {"start": 64.92, "duration": 5.879, "text": "now we're going to cancel out a minus B"}, {"start": 68.159, "duration": 4.981, "text": "from the numerator and from the"}, {"start": 70.799, "duration": 5.82, "text": "denominator here so we are left with"}, {"start": 73.14, "duration": 8.159, "text": "what a squared plus b squared"}, {"start": 76.619, "duration": 6.661, "text": "plus a b okay now recall that a is equal"}, {"start": 81.299, "duration": 4.621, "text": "to negative 1 and B is equal to negative"}, {"start": 83.28, "duration": 6.18, "text": "two so we have a raised to power 3 minus"}, {"start": 85.92, "duration": 7.8, "text": "B raise power 3 all over a minus b equal"}, {"start": 89.46, "duration": 6.979, "text": "to you know a squared plus b squared"}, {"start": 93.72, "duration": 6.899, "text": "plus a b and this is equal to what"}, {"start": 96.439, "duration": 5.5, "text": "negative 1 squared plus negative 2"}, {"start": 100.619, "duration": 4.86, "text": "squared"}, {"start": 101.939, "duration": 5.82, "text": "plus negative 1 times negative two okay"}, {"start": 105.479, "duration": 4.561, "text": "we're going to evaluate it from here so"}, {"start": 107.759, "duration": 6.72, "text": "we have this you know negative 1 squared"}, {"start": 110.04, "duration": 6.719, "text": "2 is even so this is one plus four then"}, {"start": 114.479, "duration": 4.441, "text": "plus you know negative times negative is"}, {"start": 116.759, "duration": 4.801, "text": "positive then one times two is what two"}, {"start": 118.92, "duration": 4.92, "text": "and this is equal to seven and this is"}, {"start": 121.56, "duration": 3.78, "text": "the right answer to the problem thank"}, {"start": 123.84, "duration": 3.059, "text": "you for viewing and please make sure you"}, {"start": 125.34, "duration": 4.279, "text": "subscribe to my YouTube channel for more"}, {"start": 126.899, "duration": 2.72, "text": "interesting videos"}], "title": "Algebra trick", "description": "", "lengthSeconds": 128, "width": 636, "height": 360}
{"video_id": "5n_YuJBFVXI", "transcript": [{"start": 0.719, "duration": 4.441, "text": "hello good viewers welcome back once"}, {"start": 2.48, "duration": 5.52, "text": "again a lot of student find this very"}, {"start": 5.16, "duration": 8.24, "text": "difficult to evaluate some say that the"}, {"start": 8.0, "duration": 8.32, "text": "answer is one thinking that 2^ 2 to^ 2^"}, {"start": 13.4, "duration": 6.039, "text": "2 is the base here and therefore this ra"}, {"start": 16.32, "duration": 6.24, "text": "to the^ of 0 is one beter will tell you"}, {"start": 19.439, "duration": 6.481, "text": "the truth the answer to this is not"}, {"start": 22.56, "duration": 6.719, "text": "one now let's get into this"}, {"start": 25.92, "duration": 8.639, "text": "video solution here we're going to start"}, {"start": 29.279, "duration": 6.521, "text": "like this we have 2^ 2^ 2^ 2^ 0 this"}, {"start": 34.559, "duration": 5.52, "text": "will be equal"}, {"start": 35.8, "duration": 9.68, "text": "to here we have to take this Force 2"}, {"start": 40.079, "duration": 10.401, "text": "the^ of 0 so that will become 2^ 2^ of 2"}, {"start": 45.48, "duration": 9.919, "text": "now 2^ of 0 here is 1 right okay next we"}, {"start": 50.48, "duration": 10.16, "text": "take this 2 the^ of 1 so here we get 2^"}, {"start": 55.399, "duration": 9.84, "text": "of 2 then 2^ of 1 is 2 right"}, {"start": 60.64, "duration": 6.96, "text": "then next we take this 2 to the^ of 2"}, {"start": 65.239, "duration": 7.401, "text": "and therefore we get two ra the power of"}, {"start": 67.6, "duration": 10.32, "text": "you know 2^ 2 is 4 and therefore 2^ 4 is"}, {"start": 72.64, "duration": 8.32, "text": "2 * 2 * 2 * 2 four places right and that"}, {"start": 77.92, "duration": 6.519, "text": "will give us 16 and therefore the right"}, {"start": 80.96, "duration": 5.64, "text": "answer to this problem is 16 thank you"}, {"start": 84.439, "duration": 4.921, "text": "for watching please kindly subscribe to"}, {"start": 86.6, "duration": 6.44, "text": "my channel also like comment and share"}, {"start": 89.36, "duration": 3.68, "text": "bye-bye I"}], "title": "An easy way of evaluating tower of powers of numbers", "description": "", "lengthSeconds": 91, "width": 640, "height": 360}
{"video_id": "2huo2L_yAdk", "transcript": [{"start": 0.56, "duration": 4.4, "text": "hello everyone welcome back once again"}, {"start": 3.52, "duration": 4.48, "text": "please and please if you have not"}, {"start": 4.96, "duration": 5.559, "text": "subscribed to our Channel please kindly"}, {"start": 8.0, "duration": 5.12, "text": "subscribe subscribe to this Channel and"}, {"start": 10.519, "duration": 5.0, "text": "share our videos today we're going to"}, {"start": 13.12, "duration": 9.04, "text": "evaluate or simplify this expression"}, {"start": 15.519, "duration": 10.321, "text": "here 49 of 2 - 1 all over 50 so let us"}, {"start": 22.16, "duration": 9.279, "text": "try to rewrite the numerator over here"}, {"start": 25.84, "duration": 8.0, "text": "so here we have this is equal to 49 2 -"}, {"start": 31.439, "duration": 6.96, "text": "1M as 1"}, {"start": 33.84, "duration": 7.0, "text": "2ide by what 50 right okay now we can"}, {"start": 38.399, "duration": 3.881, "text": "see that we can make use of difference"}, {"start": 40.84, "duration": 7.28, "text": "of two squares"}, {"start": 42.28, "duration": 12.759, "text": "formula namely a s minus b 2 is equal to"}, {"start": 48.12, "duration": 11.04, "text": "a + b * Aus B right so now making use of"}, {"start": 55.039, "duration": 12.08, "text": "this property here so we have up here 49"}, {"start": 59.16, "duration": 10.88, "text": "that is the base here + 1 then 49 - 1"}, {"start": 67.119, "duration": 8.841, "text": "all IDE"}, {"start": 70.04, "duration": 7.36, "text": "50 so this is equal to 49 + 1 is 50 * 49"}, {"start": 75.96, "duration": 6.72, "text": "- 1 is"}, {"start": 77.4, "duration": 8.56, "text": "48 oride by 50 see 50 cancel 50 here and"}, {"start": 82.68, "duration": 5.6, "text": "we are left with 48 and this is the"}, {"start": 85.96, "duration": 4.32, "text": "right answer please if you're enjoying"}, {"start": 88.28, "duration": 4.08, "text": "this exciting videos please please"}, {"start": 90.28, "duration": 4.36, "text": "kindly subscribe to our Channel and"}, {"start": 92.36, "duration": 5.28, "text": "share our videos"}, {"start": 94.64, "duration": 3.0, "text": "bye-bye"}], "title": "Difference of two squares is the best way out. (49\u00b2-1)/50 = ?", "description": "", "lengthSeconds": 96, "width": 636, "height": 360}
{"video_id": "nIObj2FCPt0", "transcript": [{"start": 0.24, "duration": 4.36, "text": "hello good viewers here we have an"}, {"start": 2.48, "duration": 4.879, "text": "interesting exponential problem here to"}, {"start": 4.6, "duration": 6.4, "text": "solve we are given that the fifth root"}, {"start": 7.359, "duration": 7.681, "text": "of three of nus 5 is = to"}, {"start": 11.0, "duration": 5.4, "text": "729 here we ask to find the value of n"}, {"start": 15.04, "duration": 4.44, "text": "please if you have not subscribe to my"}, {"start": 16.4, "duration": 6.959, "text": "channel kindly subscribe to this channel"}, {"start": 19.48, "duration": 5.32, "text": "like comment and share here we have the"}, {"start": 23.359, "duration": 8.601, "text": "fifth"}, {"start": 24.8, "duration": 7.16, "text": "route of 3 of N - 5 is equal to"}, {"start": 32.12, "duration": 8.48, "text": "729 let us recall this property when we"}, {"start": 35.399, "duration": 10.8, "text": "have the end root of a of B this is same"}, {"start": 40.6, "duration": 10.279, "text": "as a of B all over n right so from here"}, {"start": 46.199, "duration": 8.961, "text": "this is going to be equal to 3 of N - 5"}, {"start": 50.879, "duration": 8.441, "text": "all over 5 this is equal to 729 is the"}, {"start": 55.16, "duration": 6.559, "text": "same as 3 ra to the^ of 6 so from here"}, {"start": 59.32, "duration": 4.68, "text": "we can see that we have equal bases"}, {"start": 61.719, "duration": 5.241, "text": "right so we're going to equate the"}, {"start": 64.0, "duration": 9.04, "text": "powers or the exponents so here we get N"}, {"start": 66.96, "duration": 9.88, "text": "- 5 all over 5 is equal to 6 so cross"}, {"start": 73.04, "duration": 7.759, "text": "multiply from here we get N - 5 is equal"}, {"start": 76.84, "duration": 7.52, "text": "to 5 * 6 that is 30 add 5 to both sides"}, {"start": 80.799, "duration": 6.881, "text": "we get n is equal to"}, {"start": 84.36, "duration": 5.68, "text": "35 and this is the right answer thank"}, {"start": 87.68, "duration": 3.96, "text": "you for watching please do where to like"}, {"start": 90.04, "duration": 5.759, "text": "comment and share and don't forget to"}, {"start": 91.64, "duration": 4.159, "text": "subscribe to my channel bye-bye"}], "title": "lnteresting exponential equation.", "description": "", "lengthSeconds": 94, "width": 640, "height": 360}
{"video_id": "YRIaxYeLWd8", "transcript": [{"start": 0.64, "duration": 3.719, "text": "hello good viewers welcome back once"}, {"start": 2.28, "duration": 4.639, "text": "again today we have a very interesting"}, {"start": 4.359, "duration": 8.44, "text": "equation here and this is an infinite"}, {"start": 6.919, "duration": 8.84, "text": "Tower right we have x^ x^ x^ x^ x^ X and"}, {"start": 12.799, "duration": 6.881, "text": "so on that this is equal to 2 then we're"}, {"start": 15.759, "duration": 6.081, "text": "going to figure out the value of x but"}, {"start": 19.68, "duration": 5.439, "text": "before we solve this equation there's"}, {"start": 21.84, "duration": 10.84, "text": "something we need to take note of when"}, {"start": 25.119, "duration": 10.561, "text": "we have x x x x X and so on let's assume"}, {"start": 32.68, "duration": 5.719, "text": "that this is equal to n now there's a"}, {"start": 35.68, "duration": 6.039, "text": "condition that this infinite Tower here"}, {"start": 38.399, "duration": 5.921, "text": "will converge exactly to the right hand"}, {"start": 41.719, "duration": 10.041, "text": "side of the equation and the condition"}, {"start": 44.32, "duration": 11.44, "text": "is that 1 a is less than n and n is less"}, {"start": 51.76, "duration": 7.56, "text": "than a right so if n exceeds the value"}, {"start": 55.76, "duration": 4.88, "text": "of e then this infinite Tower cannot"}, {"start": 59.32, "duration": 3.12, "text": "converge"}, {"start": 60.64, "duration": 3.239, "text": "now let's talk about the right hand side"}, {"start": 62.44, "duration": 4.48, "text": "of that equation let's bring the"}, {"start": 63.879, "duration": 6.721, "text": "equation down right over here we have x^"}, {"start": 66.92, "duration": 7.0, "text": "x^ x to the power x and so on that this"}, {"start": 70.6, "duration": 5.839, "text": "is equal to 2 now we all know that e is"}, {"start": 73.92, "duration": 4.04, "text": "approximately"}, {"start": 76.439, "duration": 4.68, "text": "2.718"}, {"start": 77.96, "duration": 6.6, "text": "28 and we can say that e is greater than"}, {"start": 81.119, "duration": 5.0, "text": "two right or we can say 2 is less than a"}, {"start": 84.56, "duration": 4.96, "text": "so which actually mean that this"}, {"start": 86.119, "duration": 5.481, "text": "infinite Tower will converge now we can"}, {"start": 89.52, "duration": 7.72, "text": "go ahead and and solve our"}, {"start": 91.6, "duration": 8.559, "text": "question so here we have x^ x^ x^ x^ X"}, {"start": 97.24, "duration": 6.64, "text": "and so on that this is equal to two so"}, {"start": 100.159, "duration": 7.56, "text": "now we can write this as X the power of"}, {"start": 103.88, "duration": 6.16, "text": "then we can put this in this form then"}, {"start": 107.719, "duration": 4.4, "text": "we can put the powers x to the power of"}, {"start": 110.04, "duration": 6.32, "text": "x to the power of x to the power of X"}, {"start": 112.119, "duration": 6.881, "text": "and so on then this is equal to two now"}, {"start": 116.36, "duration": 4.719, "text": "think about this power here this power"}, {"start": 119.0, "duration": 3.68, "text": "here is exactly"}, {"start": 121.079, "duration": 3.921, "text": "what is at the left hand side of this"}, {"start": 122.68, "duration": 5.639, "text": "equation which is exactly equal to two"}, {"start": 125.0, "duration": 7.44, "text": "so meaning we can replace this with two"}, {"start": 128.319, "duration": 6.721, "text": "right therefore we get x^ of 2 then this"}, {"start": 132.44, "duration": 5.72, "text": "is equal to 2 now we take root of both"}, {"start": 135.04, "duration": 6.4, "text": "sides then we get X is equal to the sare"}, {"start": 138.16, "duration": 5.6, "text": "< TK of 2 and this is the right answer"}, {"start": 141.44, "duration": 5.799, "text": "think about it check your calculator and"}, {"start": 143.76, "duration": 9.0, "text": "continue the infinite Tower < TK of 2 <"}, {"start": 147.239, "duration": 6.601, "text": "TK 2 < TK 2 to the^ two to and so on"}, {"start": 152.76, "duration": 4.36, "text": "this"}, {"start": 153.84, "duration": 5.08, "text": "will converge to two and this is the"}, {"start": 157.12, "duration": 3.6, "text": "right answer thank you for watching"}, {"start": 158.92, "duration": 3.76, "text": "please cly subscribe to my YouTube"}, {"start": 160.72, "duration": 4.96, "text": "channel also like comment and share"}, {"start": 162.68, "duration": 3.0, "text": "bye-bye"}], "title": "An infinite Tower | Exponentiating Exponents \ud83d\ude0d", "description": "", "lengthSeconds": 164, "width": 640, "height": 360}
{"video_id": "HJmpyQxVGxM", "transcript": [{"start": 0.24, "duration": 4.24, "text": "good day welcome back once again today"}, {"start": 2.0, "duration": 5.64, "text": "we have these nice systems of equations"}, {"start": 4.48, "duration": 7.6, "text": "here we have a cub minus B Cub which"}, {"start": 7.64, "duration": 8.159, "text": "isal to 3 we have Aus b = 1 and here"}, {"start": 12.08, "duration": 6.24, "text": "we'll be solving for a * B right okay"}, {"start": 15.799, "duration": 7.72, "text": "now from the second equation here we"}, {"start": 18.32, "duration": 8.359, "text": "have a - b which is equal to 1 so let's"}, {"start": 23.519, "duration": 5.801, "text": "go ahead and CU both sides of this"}, {"start": 26.679, "duration": 11.641, "text": "equation now from the left hand side let"}, {"start": 29.32, "duration": 12.36, "text": "us right as a - b * a - b * a -"}, {"start": 38.32, "duration": 5.64, "text": "b and this is equal to 1 Cub which is 1"}, {"start": 41.68, "duration": 7.0, "text": "so from here let us take this product"}, {"start": 43.96, "duration": 8.04, "text": "first right so here we have a - b and"}, {"start": 48.68, "duration": 4.96, "text": "into bracket a - b * Aus B that will"}, {"start": 52.0, "duration": 8.6, "text": "give us a"}, {"start": 53.64, "duration": 10.199, "text": "s + b 2 - 2 * a * B right"}, {"start": 60.6, "duration": 7.0, "text": "then this is equal to"}, {"start": 63.839, "duration": 7.681, "text": "one okay now from here let us take this"}, {"start": 67.6, "duration": 8.92, "text": "product here so from here a * A 2 will"}, {"start": 71.52, "duration": 6.76, "text": "give us a cubed then plus a * b 2 will"}, {"start": 76.52, "duration": 6.16, "text": "give us a b"}, {"start": 78.28, "duration": 9.32, "text": "2 then minus a * 2 a here will give us 2"}, {"start": 82.68, "duration": 9.16, "text": "a 2 B okay now we have here B * a s will"}, {"start": 87.6, "duration": 10.32, "text": "give us a\u00b2 B"}, {"start": 91.84, "duration": 10.599, "text": "then B * POS b 2 will give us B Cub B *"}, {"start": 97.92, "duration": 7.72, "text": "2 a will give us positive 2 a b 2 then"}, {"start": 102.439, "duration": 4.36, "text": "this is equal to 1 so from here we have"}, {"start": 105.64, "duration": 5.159, "text": "a"}, {"start": 106.799, "duration": 6.36, "text": "cubed we have here minus B"}, {"start": 110.799, "duration": 7.761, "text": "Cubed then we have"}, {"start": 113.159, "duration": 9.441, "text": "here we have here 2 a\u00b2 B minus sorry"}, {"start": 118.56, "duration": 10.64, "text": "that is -2 a s b - a b a s b that will"}, {"start": 122.6, "duration": 10.4, "text": "give us -3 a\u00b2 B then plus a b 2 + 2 a b"}, {"start": 129.2, "duration": 7.44, "text": "2 will give us 3 a b 2 then this is"}, {"start": 133.0, "duration": 9.36, "text": "equal to 1 so from here we have a cued"}, {"start": 136.64, "duration": 8.959, "text": "minus B Cubed then here we have minus 3"}, {"start": 142.36, "duration": 7.08, "text": "a which is common here so we're going to"}, {"start": 145.599, "duration": 5.401, "text": "have here a then minus B then this is"}, {"start": 149.44, "duration": 4.28, "text": "equal to"}, {"start": 151.0, "duration": 7.56, "text": "1 okay now let's go back to the original"}, {"start": 153.72, "duration": 7.92, "text": "problem remember a cubus b Cub is = 3"}, {"start": 158.56, "duration": 6.319, "text": "and we have a minus b = 1 of which we"}, {"start": 161.64, "duration": 8.4, "text": "are solving for a * B right okay so from"}, {"start": 164.879, "duration": 9.761, "text": "here a CU - B Cub is 3 so we have 3 - 3"}, {"start": 170.04, "duration": 9.24, "text": "a b multi by a - b which is 1 then this"}, {"start": 174.64, "duration": 5.8, "text": "is equal to 1 so from here we have 3 - 3"}, {"start": 179.28, "duration": 4.2, "text": "a b"}, {"start": 180.44, "duration": 8.159, "text": "which is equal to 1 okay now from here"}, {"start": 183.48, "duration": 7.039, "text": "we have 3 a b is = to 3 - 1 which is 2"}, {"start": 188.599, "duration": 4.0, "text": "so remember we're solving for a so let's"}, {"start": 190.519, "duration": 5.481, "text": "divide both by coefficient of a which is"}, {"start": 192.599, "duration": 8.161, "text": "three so cancel this here then we have"}, {"start": 196.0, "duration": 6.84, "text": "here a * B is = 2/ 3 and this is the"}, {"start": 200.76, "duration": 4.44, "text": "right answer thank you for watching"}, {"start": 202.84, "duration": 7.28, "text": "please kindly subscribe to my channel"}, {"start": 205.2, "duration": 4.92, "text": "also like comment and share bye-bye"}], "title": "Interesting algebraic problem | Can you solve ?", "description": "", "lengthSeconds": 209, "width": 640, "height": 360}
{"video_id": "yQIxlYAEMVU", "transcript": [{"start": 0.799, "duration": 4.0, "text": "hello good day viewers welcome back once"}, {"start": 2.44, "duration": 6.44, "text": "again today we have a very nice Olympiad"}, {"start": 4.799, "duration": 6.241, "text": "math problem we're given x + 1 /x = 7"}, {"start": 8.88, "duration": 7.159, "text": "and here we're going to evaluate this"}, {"start": 11.04, "duration": 8.68, "text": "expression x\u00b2 ID by this quatic"}, {"start": 16.039, "duration": 7.041, "text": "expression here x^ 4 + X2 +"}, {"start": 19.72, "duration": 11.559, "text": "1 now let's start with this expression"}, {"start": 23.08, "duration": 10.44, "text": "here x^ 2 / x ^ 4 + X2 + 1 1 now let's"}, {"start": 31.279, "duration": 9.001, "text": "divide the numerator and the denominator"}, {"start": 33.52, "duration": 12.519, "text": "by x ^ 2 so here we have x^ 2 / x^ 2"}, {"start": 40.28, "duration": 10.599, "text": "then the whole of this x^ 4 + X2 + 1 all"}, {"start": 46.039, "duration": 7.68, "text": "/ x^ 2 from the numerator this will"}, {"start": 50.879, "duration": 5.68, "text": "cancel this and we left with one then"}, {"start": 53.719, "duration": 10.52, "text": "divided by let's split this denominator"}, {"start": 56.559, "duration": 10.801, "text": "here that will be x^ 4 / X X2 + X2 / X2"}, {"start": 64.239, "duration": 9.121, "text": "+ 1 /"}, {"start": 67.36, "duration": 8.119, "text": "X2 so from here we have 1 / here x^ 4 /"}, {"start": 73.36, "duration": 5.88, "text": "X2 will give us"}, {"start": 75.479, "duration": 8.521, "text": "X2 plus here this will cancel this we"}, {"start": 79.24, "duration": 11.0, "text": "have 1 and here plus 1 / x\u00b2 of which we"}, {"start": 84.0, "duration": 10.84, "text": "can actually rewrite as 1 / x ^ 2 + 1/"}, {"start": 90.24, "duration": 10.48, "text": "x^ 2 then + 1 right now let's go back to"}, {"start": 94.84, "duration": 9.559, "text": "the original equation we have x + 1 /x ="}, {"start": 100.72, "duration": 6.16, "text": "7 now let's try squaring both sides of"}, {"start": 104.399, "duration": 6.08, "text": "this equation and from the left we shall"}, {"start": 106.88, "duration": 10.04, "text": "make use of this formula remember this"}, {"start": 110.479, "duration": 9.56, "text": "equal a 2 + b 2 + 2 * a * B for this"}, {"start": 116.92, "duration": 6.4, "text": "here our a is going to be X and our B is"}, {"start": 120.039, "duration": 12.801, "text": "going to be 1 / X and with this we have"}, {"start": 123.32, "duration": 13.4, "text": "x 2 plus we have here 1 /x 2 + 2 * x * 1"}, {"start": 132.84, "duration": 8.52, "text": "/x and this equal"}, {"start": 136.72, "duration": 7.159, "text": "49 so from here we have x^ 2 plus here"}, {"start": 141.36, "duration": 10.08, "text": "we're going to make use of this a over"}, {"start": 143.879, "duration": 12.241, "text": "b^ n thisal a^ n / b^ n so from here 1 2"}, {"start": 151.44, "duration": 8.28, "text": "is 1 / x 2 right and here plus this"}, {"start": 156.12, "duration": 6.479, "text": "cancel this we have 2 here this equals"}, {"start": 159.72, "duration": 8.159, "text": "49 subtract two from both sides we have"}, {"start": 162.599, "duration": 9.601, "text": "X2 + 1 / X2 ="}, {"start": 167.879, "duration": 7.241, "text": "47 now remember this is the expression"}, {"start": 172.2, "duration": 5.6, "text": "we are evaluating and if you take very"}, {"start": 175.12, "duration": 6.119, "text": "good look at the bottom here we actually"}, {"start": 177.8, "duration": 5.24, "text": "have x\u00b2 + 1 / x\u00b2 right over here so let"}, {"start": 181.239, "duration": 4.601, "text": "us add one to both sides of this"}, {"start": 183.04, "duration": 5.36, "text": "equation to get the exact value of this"}, {"start": 185.84, "duration": 7.24, "text": "denominator right so add one to both"}, {"start": 188.4, "duration": 8.28, "text": "side from here we get x2 + 1 X2 + 1 this"}, {"start": 193.08, "duration": 5.56, "text": "will be equal to 47 + 1 which is 48 so"}, {"start": 196.68, "duration": 7.6, "text": "therefore the expression we asked to"}, {"start": 198.64, "duration": 8.44, "text": "evaluate equals 1 all over 48 and here"}, {"start": 204.28, "duration": 4.56, "text": "is the right answer thank you for"}, {"start": 207.08, "duration": 4.159, "text": "watching please kindly subscribe to my"}, {"start": 208.84, "duration": 5.399, "text": "channel also like comment and share"}, {"start": 211.239, "duration": 3.0, "text": "bye-bye"}], "title": "Interesting Olympiad Math | Here is the trick", "description": "", "lengthSeconds": 213, "width": 556, "height": 360}
{"video_id": "tfIeYhKirp8", "transcript": [{"start": 1.0, "duration": 3.56, "text": "hello good viewers welcome back once"}, {"start": 2.84, "duration": 4.479, "text": "again in our today's video we're going"}, {"start": 4.56, "duration": 5.84, "text": "to solve the following given simple"}, {"start": 7.319, "duration": 5.521, "text": "exponential equation I made this video"}, {"start": 10.4, "duration": 5.04, "text": "to let some people who still find this"}, {"start": 12.84, "duration": 5.88, "text": "difficult to understand how we can make"}, {"start": 15.44, "duration": 5.32, "text": "use of the logarithm when solving this"}, {"start": 18.72, "duration": 5.16, "text": "kind of exponential"}, {"start": 20.76, "duration": 6.72, "text": "equation so from here we have 5 to^ of"}, {"start": 23.88, "duration": 6.399, "text": "xal 10 so let let us use the logarithm"}, {"start": 27.48, "duration": 7.48, "text": "on both sides so here we have the log"}, {"start": 30.279, "duration": 8.161, "text": "log of 5^ of X this will be equal to the"}, {"start": 34.96, "duration": 4.64, "text": "log of 10 now let us record this"}, {"start": 38.44, "duration": 3.76, "text": "property of the"}, {"start": 39.6, "duration": 6.24, "text": "logm which states that the log of a to"}, {"start": 42.2, "duration": 7.879, "text": "the power of B is same as B MTI the log"}, {"start": 45.84, "duration": 7.16, "text": "of a where this is in base 10 and this"}, {"start": 50.079, "duration": 5.401, "text": "is also in base 10 or they can be in any"}, {"start": 53.0, "duration": 6.12, "text": "base so"}, {"start": 55.48, "duration": 7.399, "text": "here moving this x to the back here so"}, {"start": 59.12, "duration": 7.56, "text": "that will become X X MTI the log of 5"}, {"start": 62.879, "duration": 5.801, "text": "then this is equal to the log of 10 now"}, {"start": 66.68, "duration": 4.68, "text": "since we're solving for x let's divide"}, {"start": 68.68, "duration": 4.72, "text": "both side by the coefficient of x the"}, {"start": 71.36, "duration": 4.64, "text": "coefficient of x is a number here which"}, {"start": 73.4, "duration": 6.039, "text": "is the log of 5 so divide here by the"}, {"start": 76.0, "duration": 6.32, "text": "log of five and equally divide this side"}, {"start": 79.439, "duration": 4.72, "text": "by the log of five so this will cancel"}, {"start": 82.32, "duration": 8.68, "text": "this from here and we can see that our X"}, {"start": 84.159, "duration": 11.161, "text": "is fre X = log of 10 / the log"}, {"start": 91.0, "duration": 7.32, "text": "of five which is approximately equal"}, {"start": 95.32, "duration": 3.0, "text": "to"}, {"start": 98.84, "duration": 3.2, "text": "1.43"}, {"start": 100.41, "duration": 4.63, "text": "[Music]"}, {"start": 102.04, "duration": 3.0, "text": "676"}, {"start": 105.759, "duration": 5.441, "text": "55807 and this is the right answer thank"}, {"start": 109.159, "duration": 4.041, "text": "you for watching please kindly subscribe"}, {"start": 111.2, "duration": 5.48, "text": "to my channel also like comment and"}, {"start": 113.2, "duration": 3.48, "text": "share bye-bye"}], "title": "How to solve simple exponential equation using logarithm", "description": "", "lengthSeconds": 115, "width": 640, "height": 360}
{"video_id": "WvSR6QdXnuw", "transcript": [{"start": 0.799, "duration": 3.681, "text": "hello good viewers welcome back once"}, {"start": 2.6, "duration": 4.159, "text": "again today we're going to solve for the"}, {"start": 4.48, "duration": 7.079, "text": "values of X in the following given"}, {"start": 6.759, "duration": 8.161, "text": "exponential equation we have e^ x = 3X"}, {"start": 11.559, "duration": 9.601, "text": "now let's get started we have here e to^"}, {"start": 14.92, "duration": 8.08, "text": "x is = to 3 x right okay now from here"}, {"start": 21.16, "duration": 5.359, "text": "let's go ahead and divide both sides of"}, {"start": 23.0, "duration": 8.8, "text": "this equation by 3 so here we have a^ x"}, {"start": 26.519, "duration": 8.161, "text": "/ 3 this is equal to 3x divided by"}, {"start": 31.8, "duration": 4.56, "text": "3 okay from the right this three we"}, {"start": 34.68, "duration": 6.559, "text": "cancel this three and from the right we"}, {"start": 36.36, "duration": 8.359, "text": "have X = from the left here we have e^ X"}, {"start": 41.239, "duration": 5.881, "text": "multip by we can write it as 1 /"}, {"start": 44.719, "duration": 8.52, "text": "3 now from here let's go ahead and"}, {"start": 47.12, "duration": 14.8, "text": "divide both by e^ of X so here x / e^ X"}, {"start": 53.239, "duration": 8.681, "text": "this equals e^ x * 1 / 3 / e^ X"}, {"start": 62.68, "duration": 9.479, "text": "so from here this will cancel this and"}, {"start": 65.08, "duration": 12.079, "text": "here we have x * 1 over e^ X this = 1 /"}, {"start": 72.159, "duration": 13.361, "text": "3 now let us record this property 1 a^ n"}, {"start": 77.159, "duration": 13.0, "text": "= a n so from here this is x * e^ of --x"}, {"start": 85.52, "duration": 7.84, "text": "this = 1 / 3 now we'll be making use of"}, {"start": 90.159, "duration": 5.041, "text": "the Lamb W function remember that the"}, {"start": 93.36, "duration": 5.719, "text": "lamb W function with the following"}, {"start": 95.2, "duration": 7.0, "text": "argument x * e^ x ="}, {"start": 99.079, "duration": 8.08, "text": "x this means that the power of E here"}, {"start": 102.2, "duration": 7.4, "text": "must be exactly the coefficient right so"}, {"start": 107.159, "duration": 5.081, "text": "the power needs to be equal with the"}, {"start": 109.6, "duration": 5.76, "text": "coefficient and here the power of E here"}, {"start": 112.24, "duration": 5.839, "text": "ISX but the coefficient is POS X so"}, {"start": 115.36, "duration": 6.64, "text": "let's go ahead and multiply both by1 and"}, {"start": 118.079, "duration": 8.801, "text": "by so doing we get X x *"}, {"start": 122.0, "duration": 7.2, "text": "e^x this will Beal to -1 / 3 now we can"}, {"start": 126.88, "duration": 4.56, "text": "make use of the Lamb function so let's"}, {"start": 129.2, "duration": 5.52, "text": "lze both sides so we"}, {"start": 131.44, "duration": 6.64, "text": "getx *"}, {"start": 134.72, "duration": 7.519, "text": "e^x then this is equal to the lambat"}, {"start": 138.08, "duration": 8.28, "text": "function of-1 / 3 so from here this will"}, {"start": 142.239, "duration": 8.521, "text": "evaluate tox then this is equal to W of"}, {"start": 146.36, "duration": 11.4, "text": "-1 / 3 from here multiply by1 we get X"}, {"start": 150.76, "duration": 9.8, "text": "is = to W with a subscr n then -1 / 3"}, {"start": 157.76, "duration": 7.28, "text": "okay now this is the most interesting"}, {"start": 160.56, "duration": 8.16, "text": "part of this the d w function has some"}, {"start": 165.04, "duration": 6.04, "text": "Branch points and some points usually"}, {"start": 168.72, "duration": 6.64, "text": "give us the real solutions to the"}, {"start": 171.08, "duration": 7.72, "text": "equation right so and that is at n = 0"}, {"start": 175.36, "duration": 9.84, "text": "and at n = -1 so here we have the first"}, {"start": 178.8, "duration": 9.159, "text": "solution X1 = W with Subs 0 of -1 all"}, {"start": 185.2, "duration": 5.56, "text": "over 3 and this is approximately equal"}, {"start": 187.959, "duration": 2.801, "text": "to"}, {"start": 191.08, "duration": 3.0, "text": "0.619"}, {"start": 194.12, "duration": 10.039, "text": "0613 right and we have the other Branch"}, {"start": 197.84, "duration": 10.2, "text": "C which is W which sub -1 then -1 / 3"}, {"start": 204.159, "duration": 10.0, "text": "which is approximately equal to 1."}, {"start": 208.04, "duration": 9.64, "text": "512 13 4 5 5 and these are the real"}, {"start": 214.159, "duration": 5.201, "text": "values of X in the given equation thank"}, {"start": 217.68, "duration": 3.72, "text": "you for watching please kindly subscribe"}, {"start": 219.36, "duration": 5.599, "text": "to my channel also like comment and"}, {"start": 221.4, "duration": 3.559, "text": "share bye-bye"}], "title": "Oxford University Entrance Exam | Can you find the two real roots ? | Lambert W Function application", "description": "", "lengthSeconds": 224, "width": 560, "height": 360}
{"video_id": "nFAZBEGLNlU", "transcript": [{"start": 1.04, "duration": 5.64, "text": "hello welcome back once again today we"}, {"start": 3.48, "duration": 5.4, "text": "have this interesting equation and we're"}, {"start": 6.68, "duration": 4.999, "text": "going to solve for the value of x that"}, {"start": 8.88, "duration": 5.56, "text": "will satisfy this equation it's"}, {"start": 11.679, "duration": 7.321, "text": "obviously one can clearly say that one"}, {"start": 14.44, "duration": 8.12, "text": "is a solution yes indeed because 1^ 1 is"}, {"start": 19.0, "duration": 8.0, "text": "= to 1 so therefore we can see that X is"}, {"start": 22.56, "duration": 8.959, "text": "equal to 1 now what about X is equal"}, {"start": 27.0, "duration": 10.16, "text": "to0 can that be a solution no because 0"}, {"start": 31.519, "duration": 8.56, "text": "^ of 0 is not equal to 1 rather 0^ 0 is"}, {"start": 37.16, "duration": 5.239, "text": "undefined but how can we prove that X is"}, {"start": 40.079, "duration": 5.441, "text": "equal to 1 is the only"}, {"start": 42.399, "duration": 5.601, "text": "solution now let's see now let's start"}, {"start": 45.52, "duration": 6.519, "text": "by introducing the natural log on both"}, {"start": 48.0, "duration": 8.76, "text": "sides so remember our equation is X to"}, {"start": 52.039, "duration": 8.081, "text": "the^ X is equal to 1 now let do natural"}, {"start": 56.76, "duration": 7.719, "text": "log both sides so we have the Ln of x to"}, {"start": 60.12, "duration": 6.6, "text": "the^ of X this is equal to the Ln of 1"}, {"start": 64.479, "duration": 4.561, "text": "now with the with the property of"}, {"start": 66.72, "duration": 5.719, "text": "logarithm this power of X will come down"}, {"start": 69.04, "duration": 6.119, "text": "here so that will become X MTI the Ln of"}, {"start": 72.439, "duration": 6.281, "text": "X this is equal to the natural log of 1"}, {"start": 75.159, "duration": 6.121, "text": "is z okay now we can see we have this"}, {"start": 78.72, "duration": 5.8, "text": "product x * natural log of x which is"}, {"start": 81.28, "duration": 8.68, "text": "equal to Z so this implies from here"}, {"start": 84.52, "duration": 9.4, "text": "that either X is equal to0 or the"}, {"start": 89.96, "duration": 5.56, "text": "natural log of x is equal to0 and we"}, {"start": 93.92, "duration": 5.8, "text": "know for the natural log of x to be"}, {"start": 95.52, "duration": 6.68, "text": "equal to0 x has to be equal to 1 right"}, {"start": 99.72, "duration": 7.759, "text": "so from here we have X will be equal to"}, {"start": 102.2, "duration": 9.0, "text": "e to^ of 0 right and here we get X is ="}, {"start": 107.479, "duration": 6.0, "text": "1 but 0 is not a solution right"}, {"start": 111.2, "duration": 5.519, "text": "therefore the only solution here is X"}, {"start": 113.479, "duration": 6.041, "text": "isal to 1 thank you for watching please"}, {"start": 116.719, "duration": 7.44, "text": "kindly subscribe to my channel also like"}, {"start": 119.52, "duration": 4.639, "text": "comment and share bye-bye"}], "title": "Oxford University \ud83c\udf93 Entrance Exam | Can you solve ?", "description": "", "lengthSeconds": 123, "width": 428, "height": 240}
{"video_id": "EhuS9KggGG8", "transcript": [{"start": 0.88, "duration": 3.919, "text": "hello good viewers welcome back once"}, {"start": 2.6, "duration": 5.319, "text": "again today we have an interesting"}, {"start": 4.799, "duration": 6.8, "text": "exponential equation problem we are"}, {"start": 7.919, "duration": 7.281, "text": "giving that 2^ of otk X this is equal to"}, {"start": 11.599, "duration": 9.241, "text": "8^ of X now let's go ahead and solve for"}, {"start": 15.2, "duration": 8.52, "text": "the values of X we have here 2 of root X"}, {"start": 20.84, "duration": 8.679, "text": "this is equal to in bracket it8 is can"}, {"start": 23.72, "duration": 7.359, "text": "be expressed as 2^ 3 then power of X now"}, {"start": 29.519, "duration": 6.921, "text": "let's call this"}, {"start": 31.079, "duration": 9.041, "text": "property M power of n ra the power of K"}, {"start": 36.44, "duration": 7.88, "text": "this is equal to m^ of n *"}, {"start": 40.12, "duration": 6.4, "text": "K so from here we have 2^ ofun X this"}, {"start": 44.32, "duration": 6.879, "text": "will be equal to"}, {"start": 46.52, "duration": 9.16, "text": "2 of 3 * X which is 3 x so from here let"}, {"start": 51.199, "duration": 7.801, "text": "us recall this when we have b r power of"}, {"start": 55.68, "duration": 7.6, "text": "let's say m and this is equal to let's"}, {"start": 59.0, "duration": 7.04, "text": "say we have B ra the power of um let's"}, {"start": 63.28, "duration": 5.8, "text": "say we have one or we have K so let's"}, {"start": 66.04, "duration": 5.68, "text": "let's use that so since we have equal"}, {"start": 69.08, "duration": 4.92, "text": "bases from both sides we can easily"}, {"start": 71.72, "duration": 6.52, "text": "equate the powers that M will be equal"}, {"start": 74.0, "duration": 7.4, "text": "to K right okay so from here we can see"}, {"start": 78.24, "duration": 5.879, "text": "that we have equal bases right so we can"}, {"start": 81.4, "duration": 6.64, "text": "equate the powers that is this square"}, {"start": 84.119, "duration": 6.481, "text": "root of x this will be equal to 3 x now"}, {"start": 88.04, "duration": 5.48, "text": "let's Square both sides to get r of this"}, {"start": 90.6, "duration": 6.879, "text": "root sign so we have root"}, {"start": 93.52, "duration": 6.959, "text": "x s this will be equal to"}, {"start": 97.479, "duration": 3.0, "text": "3x"}, {"start": 100.6, "duration": 5.96, "text": "squared now from the left hand side here"}, {"start": 104.479, "duration": 5.201, "text": "the square and the root will cancel out"}, {"start": 106.56, "duration": 4.879, "text": "we have X will be equal to Here let us"}, {"start": 109.68, "duration": 5.32, "text": "record this"}, {"start": 111.439, "duration": 11.32, "text": "MN the power of B this will be same as"}, {"start": 115.0, "duration": 13.36, "text": "m^ of B * n^ of B so here we have 3"}, {"start": 122.759, "duration": 11.601, "text": "2 * x 2 so from the right we have 32"}, {"start": 128.36, "duration": 9.239, "text": "which is 9 * X2 this will be equal to"}, {"start": 134.36, "duration": 8.519, "text": "X now let's subract X from both sides we"}, {"start": 137.599, "duration": 9.961, "text": "obtain 9 x^ 2 - x this will be equal 0"}, {"start": 142.879, "duration": 7.0, "text": "factor out X we have X into braet 9x - 1"}, {"start": 147.56, "duration": 8.679, "text": "this is equal Z so from here this"}, {"start": 149.879, "duration": 9.681, "text": "implies that x is = 0 or 9 x - 1 this is"}, {"start": 156.239, "duration": 7.841, "text": "equal 0 so from here we get X is equal"}, {"start": 159.56, "duration": 8.12, "text": "to0 or we have here x will be equal to 1"}, {"start": 164.08, "duration": 6.28, "text": "all over 9 right so here we have come to"}, {"start": 167.68, "duration": 8.12, "text": "the end of today's video that in the"}, {"start": 170.36, "duration": 11.2, "text": "above exponential equation x is = to"}, {"start": 175.8, "duration": 7.84, "text": "0 and x is = to 1 over9 and these are"}, {"start": 181.56, "duration": 3.759, "text": "the solutions thank you for watching"}, {"start": 183.64, "duration": 3.879, "text": "please kindly subscribe to my YouTube"}, {"start": 185.319, "duration": 5.2, "text": "channel also like comment and share"}, {"start": 187.519, "duration": 3.0, "text": "bye-bye"}], "title": "Interesting exponential equation | Not just one value of x ! | #maths #olympiad", "description": "", "lengthSeconds": 189, "width": 640, "height": 360}
{"video_id": "kCMBB21ddFw", "transcript": [{"start": 0.48, "duration": 5.6, "text": "hello welcome back once again let us"}, {"start": 3.12, "duration": 8.32, "text": "solve for the value of x in the given"}, {"start": 6.08, "duration": 10.36, "text": "radical equation here we have otk of 2x"}, {"start": 11.44, "duration": 7.12, "text": "this is equal to < TK 2 * < tkx now in"}, {"start": 16.44, "duration": 4.839, "text": "our first step here we're going to"}, {"start": 18.56, "duration": 6.0, "text": "square both sides of this equation to"}, {"start": 21.279, "duration": 4.881, "text": "get rid of this root signs so here we"}, {"start": 24.56, "duration": 6.039, "text": "have root of"}, {"start": 26.16, "duration": 10.0, "text": "2x 2 then this will be equal to"}, {"start": 30.599, "duration": 7.64, "text": "root of 2 * root X all 2 as we know"}, {"start": 36.16, "duration": 5.879, "text": "Square will cancel the square root so"}, {"start": 38.239, "duration": 7.84, "text": "here we have 2X is equal to here Square"}, {"start": 42.039, "duration": 6.801, "text": "we cancel the square root we have 2ot of"}, {"start": 46.079, "duration": 6.761, "text": "X okay now from here let us go ahead and"}, {"start": 48.84, "duration": 9.12, "text": "divide both sides by two so we have 2X /"}, {"start": 52.84, "duration": 8.64, "text": "2 this will be equal to 2un x / 2 so we"}, {"start": 57.96, "duration": 8.4, "text": "know 2 cancel two here 2 can 2 here and"}, {"start": 61.48, "duration": 6.76, "text": "we have X is = to otk X in our next step"}, {"start": 66.36, "duration": 4.32, "text": "let us go ahead and square both sides"}, {"start": 68.24, "duration": 4.28, "text": "again to get rid of this root sign so"}, {"start": 70.68, "duration": 5.119, "text": "here we have"}, {"start": 72.52, "duration": 8.4, "text": "X2 this will be equal to root of"}, {"start": 75.799, "duration": 6.96, "text": "x^2 so here we have X2 is equal to"}, {"start": 80.92, "duration": 4.6, "text": "square cancels the square root so we"}, {"start": 82.759, "duration": 7.201, "text": "have X now subtract X from both side we"}, {"start": 85.52, "duration": 7.599, "text": "get x^2 - x this is equal to0 now X is"}, {"start": 89.96, "duration": 7.04, "text": "is common we factor out X so we have x"}, {"start": 93.119, "duration": 6.36, "text": "-1 then this is equal to Z from here"}, {"start": 97.0, "duration": 8.88, "text": "either X is equal"}, {"start": 99.479, "duration": 8.161, "text": "0 and this is a solution or x -1 is = 0"}, {"start": 105.88, "duration": 4.08, "text": "which when we add one to both side we"}, {"start": 107.64, "duration": 5.519, "text": "get another solution that X is equal to"}, {"start": 109.96, "duration": 7.32, "text": "1 so therefore in the above equation we"}, {"start": 113.159, "duration": 7.56, "text": "have two possible solutions X = to 0 and"}, {"start": 117.28, "duration": 6.119, "text": "x = to 1 thank you for watching please"}, {"start": 120.719, "duration": 6.881, "text": "kindly subscribe to my channel also like"}, {"start": 123.399, "duration": 4.201, "text": "comment and share bye-bye"}], "title": "Easy and interesting radical equation.", "description": "", "lengthSeconds": 126, "width": 640, "height": 360}
{"video_id": "WKcMfuuXFpg", "transcript": [{"start": 0.719, "duration": 4.161, "text": "hello welcome back once again today we"}, {"start": 2.8, "duration": 5.32, "text": "have this interesting Olympiad"}, {"start": 4.88, "duration": 6.24, "text": "mathematics problem so in this equation"}, {"start": 8.12, "duration": 7.28, "text": "we're going to solve for M M is an"}, {"start": 11.12, "duration": 5.919, "text": "integer we have 3^ m + 2^ m this is"}, {"start": 15.4, "duration": 5.48, "text": "equal to"}, {"start": 17.039, "duration": 6.24, "text": "35 okay now before we proceed let us"}, {"start": 20.88, "duration": 5.04, "text": "take note of the following"}, {"start": 23.279, "duration": 4.721, "text": "functions now let us Define this"}, {"start": 25.92, "duration": 4.56, "text": "function f of"}, {"start": 28.0, "duration": 9.04, "text": "M is equal to"}, {"start": 30.48, "duration": 9.68, "text": "3 to the^ m + 2 to the^ of M now the"}, {"start": 37.04, "duration": 5.4, "text": "graph of this function looks exactly"}, {"start": 40.16, "duration": 2.28, "text": "like"}, {"start": 43.079, "duration": 6.721, "text": "this so the graph of this function looks"}, {"start": 46.44, "duration": 3.36, "text": "exactly like"}, {"start": 50.719, "duration": 5.32, "text": "this if I'm not mistaken so it is an"}, {"start": 54.559, "duration": 5.0, "text": "increasing"}, {"start": 56.039, "duration": 5.721, "text": "function if I'm not mistaken at all so"}, {"start": 59.559, "duration": 5.041, "text": "the gra have somehow look like this"}, {"start": 61.76, "duration": 7.2, "text": "which we can see obviously that this is"}, {"start": 64.6, "duration": 8.24, "text": "an increasing function why the bigger"}, {"start": 68.96, "duration": 7.479, "text": "the value for M the bigger the result"}, {"start": 72.84, "duration": 9.8, "text": "right so if m is equal to 1 if m is"}, {"start": 76.439, "duration": 9.68, "text": "equal to 1 we get our outcome 3^ 1 + 2^"}, {"start": 82.64, "duration": 6.32, "text": "1 which is 5 then if m is equal to 2 you"}, {"start": 86.119, "duration": 5.401, "text": "can see it is an increasing function"}, {"start": 88.96, "duration": 5.04, "text": "right okay okay now if you look at the"}, {"start": 91.52, "duration": 5.559, "text": "right hand side we have just"}, {"start": 94.0, "duration": 6.52, "text": "35 which we know that that is just a"}, {"start": 97.079, "duration": 7.881, "text": "straight line so if you check along this"}, {"start": 100.52, "duration": 7.4, "text": "Y axis here this vertical Y axis so we"}, {"start": 104.96, "duration": 5.92, "text": "going to have 35 at some points here"}, {"start": 107.92, "duration": 5.159, "text": "right so the graph for this one will"}, {"start": 110.88, "duration": 5.96, "text": "pass through like this which shows that"}, {"start": 113.079, "duration": 6.561, "text": "there's only one intersection point so"}, {"start": 116.84, "duration": 4.879, "text": "therefore for integer M there will only"}, {"start": 119.64, "duration": 4.079, "text": "be one real"}, {"start": 121.719, "duration": 7.001, "text": "solution now let's go back to the"}, {"start": 123.719, "duration": 11.24, "text": "original problem so we have 3 to the^ m"}, {"start": 128.72, "duration": 12.28, "text": "+ 2 to the^ M this is equal to"}, {"start": 134.959, "duration": 10.121, "text": "35 so here we have 3 to the^ m + 2 to"}, {"start": 141.0, "duration": 4.959, "text": "the^ M this is equal to now 35 is same"}, {"start": 145.08, "duration": 5.28, "text": "as"}, {"start": 145.959, "duration": 9.64, "text": "27 + 8"}, {"start": 150.36, "duration": 10.44, "text": "now we have here 3 to the^ m + 2 to the^"}, {"start": 155.599, "duration": 9.761, "text": "M this is equal to 277 3 to the^ 3 then"}, {"start": 160.8, "duration": 6.64, "text": "Plus 8 as 2 to the^ 3 so I think from"}, {"start": 165.36, "duration": 4.799, "text": "here we can compare and from here this"}, {"start": 167.44, "duration": 5.879, "text": "implies that our solution is out and we"}, {"start": 170.159, "duration": 5.121, "text": "have this only real I mean is that not"}, {"start": 173.319, "duration": 5.2, "text": "integer yes of course so the only"}, {"start": 175.28, "duration": 5.48, "text": "integer solution for m is three and this"}, {"start": 178.519, "duration": 3.681, "text": "is the correct answer"}, {"start": 180.76, "duration": 3.839, "text": "thank you for watching please kindly"}, {"start": 182.2, "duration": 6.92, "text": "subscribe to my channel also like"}, {"start": 184.599, "duration": 4.521, "text": "comment and share bye-bye"}], "title": "India | Nice Olympiad Math problem | Can you solve ?", "description": "", "lengthSeconds": 188, "width": 428, "height": 240}
{"video_id": "pEZObHwAXec", "transcript": [{"start": 0.52, "duration": 3.88, "text": "good day viewers welcome back to my"}, {"start": 2.08, "duration": 5.319, "text": "channel again today we have our second"}, {"start": 4.4, "duration": 6.48, "text": "class on this problem so this is problem"}, {"start": 7.399, "duration": 7.041, "text": "number eight the S mathematical problem"}, {"start": 10.88, "duration": 7.2, "text": "here we have if a all Bal 2 What is the"}, {"start": 14.44, "duration": 6.88, "text": "value of 4 B all over a options are here"}, {"start": 18.08, "duration": 3.24, "text": "okay let's get started"}, {"start": 21.92, "duration": 8.48, "text": "solution now let us bring down this"}, {"start": 25.599, "duration": 10.081, "text": "expression 4 b a so you remember this"}, {"start": 30.4, "duration": 7.92, "text": "expression is the same as 4 * B all a"}, {"start": 35.68, "duration": 5.6, "text": "okay so this is what will be"}, {"start": 38.32, "duration": 7.64, "text": "evaluating okay now remember the"}, {"start": 41.28, "duration": 6.88, "text": "equation we have here is a/ b = 2 okay"}, {"start": 45.96, "duration": 10.16, "text": "now I'm going to take the reciprocal of"}, {"start": 48.16, "duration": 7.96, "text": "both sides so here I have a/ B close"}, {"start": 56.32, "duration": 9.6, "text": "bracket1 is = to 2"}, {"start": 61.239, "duration": 7.321, "text": "are1 so now remember if I want to"}, {"start": 65.92, "duration": 5.64, "text": "rewrite this according to this negative"}, {"start": 68.56, "duration": 5.8, "text": "power the power will become positive"}, {"start": 71.56, "duration": 5.52, "text": "okay so reciprocal of numbers for"}, {"start": 74.36, "duration": 6.36, "text": "example the reciprocal of 1 over 2 is"}, {"start": 77.08, "duration": 7.52, "text": "equal to 2 the reciprocal of 3 is equal"}, {"start": 80.72, "duration": 9.68, "text": "to 1 over 3 and so on so here I'm going"}, {"start": 84.6, "duration": 10.08, "text": "to have B all over a in bracket"}, {"start": 90.4, "duration": 8.84, "text": "to the positive power is equal to 1 all"}, {"start": 94.68, "duration": 7.119, "text": "2 okay now remember B all a to the first"}, {"start": 99.24, "duration": 7.68, "text": "power is still the same so here we have"}, {"start": 101.799, "duration": 8.041, "text": "B all a is = 1 2 now I'm going to"}, {"start": 106.92, "duration": 5.44, "text": "multiply both sides by four in order to"}, {"start": 109.84, "duration": 6.0, "text": "obtain this expression which we are"}, {"start": 112.36, "duration": 10.84, "text": "asked to evaluate okay so here I have 4"}, {"start": 115.84, "duration": 10.199, "text": "* B all a is = to 4 4 * 1 all over 2 so"}, {"start": 123.2, "duration": 6.72, "text": "from here this is over one so 4 * B is 4"}, {"start": 126.039, "duration": 6.601, "text": "B all over a * 1 here is a and here we"}, {"start": 129.92, "duration": 6.2, "text": "have this over one two here one and two"}, {"start": 132.64, "duration": 5.92, "text": "here is two so we get 4 B all a is equal"}, {"start": 136.12, "duration": 4.119, "text": "to two and this is the right answer so"}, {"start": 138.56, "duration": 5.52, "text": "we go to the"}, {"start": 140.239, "duration": 7.0, "text": "option and we see that the option"}, {"start": 144.08, "duration": 4.56, "text": "C is correct thank you for viewing and"}, {"start": 147.239, "duration": 3.08, "text": "please make sure you subscribe to my"}, {"start": 148.64, "duration": 4.52, "text": "YouTube channel for more interesting"}, {"start": 150.319, "duration": 2.841, "text": "videos"}], "title": "SAT math Problem 8.", "description": "", "lengthSeconds": 151, "width": 636, "height": 360}
{"video_id": "lSoxRX5KBx4", "transcript": [{"start": 0.88, "duration": 4.479, "text": "hello good viewers welcome back once"}, {"start": 2.72, "duration": 6.4, "text": "again let's find a solution to this"}, {"start": 5.359, "duration": 7.36, "text": "Olympiad exponential equation problem"}, {"start": 9.12, "duration": 6.12, "text": "we're given that 27 ^ ofun x this is"}, {"start": 12.719, "duration": 5.241, "text": "equal to 1 allun"}, {"start": 15.24, "duration": 5.16, "text": "x before we proceed let us make the"}, {"start": 17.96, "duration": 8.8, "text": "following substitution we're going to"}, {"start": 20.4, "duration": 11.679, "text": "let rootx = Y when you square both sides"}, {"start": 26.76, "duration": 8.639, "text": "we say that X is = to Y 2 right so now"}, {"start": 32.079, "duration": 10.841, "text": "this equation in terms of Y is going to"}, {"start": 35.399, "duration": 11.281, "text": "be 27 ra^ of Y this is equal to 1 / y"}, {"start": 42.92, "duration": 7.319, "text": "now the next thing is to get rid of this"}, {"start": 46.68, "duration": 5.64, "text": "power right over here so what do we do"}, {"start": 50.239, "duration": 4.8, "text": "now what we're going to do is to raise"}, {"start": 52.32, "duration": 6.399, "text": "both sides to the power of the"}, {"start": 55.039, "duration": 6.801, "text": "reciprocal of Y right now from here we"}, {"start": 58.719, "duration": 9.841, "text": "have 27 to^ of"}, {"start": 61.84, "duration": 11.88, "text": "y^ of 1 / y this is equal to 1 / y power"}, {"start": 68.56, "duration": 6.72, "text": "of 1 / y right okay now from the left"}, {"start": 73.72, "duration": 5.92, "text": "hand side we're going to make use of"}, {"start": 75.28, "duration": 8.56, "text": "this property when we have a^ of b c"}, {"start": 79.64, "duration": 5.799, "text": "this is same as a^ of B * C so here we"}, {"start": 83.84, "duration": 9.84, "text": "have"}, {"start": 85.439, "duration": 13.081, "text": "27 of y * 1 / y this is equal to 1 / y^"}, {"start": 93.68, "duration": 8.28, "text": "1 / y now we can see here that Y and Y"}, {"start": 98.52, "duration": 7.68, "text": "we cancel out and here we have 27 ^ 1"}, {"start": 101.96, "duration": 6.0, "text": "which is 27 this is equal 1 / y to the^"}, {"start": 106.2, "duration": 6.559, "text": "of 1 /"}, {"start": 107.96, "duration": 8.799, "text": "y okay now let us recall that 27 is"}, {"start": 112.759, "duration": 5.921, "text": "actually equal to 3 to the power of 3 so"}, {"start": 116.759, "duration": 4.04, "text": "the left hand side we rewrite it in"}, {"start": 118.68, "duration": 8.399, "text": "terms of the powers of 3 and here we"}, {"start": 120.799, "duration": 10.08, "text": "have 3^ of 3 this is equal to 1 / y^ of"}, {"start": 127.079, "duration": 9.081, "text": "1 / y so from here we can see this"}, {"start": 130.879, "duration": 9.041, "text": "pattern here this implies that 1 / Y is"}, {"start": 136.16, "duration": 7.04, "text": "equal to 3 we have the bases here 1 / y"}, {"start": 139.92, "duration": 6.44, "text": "= 3 and from here we have the powers 1/"}, {"start": 143.2, "duration": 7.16, "text": "y = 3 so with this pattern this implies"}, {"start": 146.36, "duration": 7.879, "text": "that 1 / Y is = 3 and when we cross"}, {"start": 150.36, "duration": 6.8, "text": "multiply we get 3 y this is equal to 1"}, {"start": 154.239, "duration": 7.121, "text": "right that is when you cross multiply"}, {"start": 157.16, "duration": 6.999, "text": "that is y * 3 3 y this is equal to 1 * 1"}, {"start": 161.36, "duration": 5.04, "text": "which is 1 so when we divide both sides"}, {"start": 164.159, "duration": 7.36, "text": "by"}, {"start": 166.4, "duration": 10.68, "text": "3 3 cancel 3 and from here we get Y is ="}, {"start": 171.519, "duration": 11.44, "text": "1 / 3 okay now let us recall"}, {"start": 177.08, "duration": 9.76, "text": "again recall that X is = y^ 2 and from"}, {"start": 182.959, "duration": 10.041, "text": "here remember Y is = 1/ 3 so here x is ="}, {"start": 186.84, "duration": 9.16, "text": "to 1/ 3 S and this will give us 1 / 9"}, {"start": 193.0, "duration": 4.72, "text": "and this is the solution thank you for"}, {"start": 196.0, "duration": 4.12, "text": "watching please kindly subscribe to my"}, {"start": 197.72, "duration": 5.4, "text": "channel also like comment and share"}, {"start": 200.12, "duration": 3.0, "text": "bye-bye"}], "title": "Indians | Interesting Olympiad Math | #maths #mathematics", "description": "", "lengthSeconds": 201, "width": 548, "height": 360}
{"video_id": "0AuZ2dXsCSQ", "transcript": [{"start": 0.84, "duration": 4.72, "text": "hello good viewers welcome back once"}, {"start": 3.359, "duration": 4.401, "text": "again today we're going to find the"}, {"start": 5.56, "duration": 6.959, "text": "square root of a sodic"}, {"start": 7.76, "duration": 4.759, "text": "expression we have theare otk of 6 + <"}, {"start": 18.439, "duration": 11.201, "text": "tk32 now this is equal to the square"}, {"start": 22.279, "duration": 10.561, "text": "root of 6 + otk of 16 * 2 right with the"}, {"start": 29.64, "duration": 4.439, "text": "following property root a * B this is"}, {"start": 32.84, "duration": 5.0, "text": "equal to root"}, {"start": 34.079, "duration": 6.601, "text": "ATI root B right so this is going to be"}, {"start": 37.84, "duration": 6.0, "text": "equal to the square root of first of all"}, {"start": 40.68, "duration": 7.24, "text": "we are going to rewrite six as the sum"}, {"start": 43.84, "duration": 9.12, "text": "of two numbers that will be 4 and"}, {"start": 47.92, "duration": 7.92, "text": "2 then plus the root of 16 which is 4"}, {"start": 52.96, "duration": 6.239, "text": "then multiply by the square < TK of 2"}, {"start": 55.84, "duration": 7.8, "text": "right so from here this is equal to the"}, {"start": 59.199, "duration": 8.041, "text": "square root of 4 is the same as 2 2 plus"}, {"start": 63.64, "duration": 6.2, "text": "here two we can rewrite it rewrite two"}, {"start": 67.24, "duration": 5.84, "text": "as the square of a number and that will"}, {"start": 69.84, "duration": 8.04, "text": "be the square of <"}, {"start": 73.08, "duration": 7.96, "text": "tk2 then plus 4 is the same as 2 multi"}, {"start": 77.88, "duration": 6.76, "text": "by 2 then * <"}, {"start": 81.04, "duration": 5.64, "text": "tk2 okay now let's recall the following"}, {"start": 84.64, "duration": 5.839, "text": "algebraic"}, {"start": 86.68, "duration": 8.84, "text": "property that has to do with a B"}, {"start": 90.479, "duration": 12.441, "text": "expansion right so we have here A + B 2"}, {"start": 95.52, "duration": 10.639, "text": "this is equal to a 2 + b 2 + 2 * a * B"}, {"start": 102.92, "duration": 5.4, "text": "so now let's us compare the right hand"}, {"start": 106.159, "duration": 5.081, "text": "side of this equation from here with"}, {"start": 108.32, "duration": 5.36, "text": "what we have under this root sign so if"}, {"start": 111.24, "duration": 5.6, "text": "we compare this together we have a s and"}, {"start": 113.68, "duration": 5.759, "text": "here we have something squar plus b s we"}, {"start": 116.84, "duration": 7.319, "text": "have another something squared plus 2"}, {"start": 119.439, "duration": 7.32, "text": "MTI by a * B and here we have + 2 MTI by"}, {"start": 124.159, "duration": 5.201, "text": "something time another something right"}, {"start": 126.759, "duration": 8.401, "text": "so if we compare this we find out that a"}, {"start": 129.36, "duration": 9.04, "text": "is equal to 2 y b isal to < tk2 right so"}, {"start": 135.16, "duration": 4.359, "text": "from here this is equal to the square"}, {"start": 138.4, "duration": 7.64, "text": "root"}, {"start": 139.519, "duration": 6.521, "text": "of 2 + < tk2 to the^ of"}, {"start": 146.08, "duration": 10.239, "text": "two and from here this is equal to the"}, {"start": 150.44, "duration": 10.4, "text": "absolute value of 2 + < tk2 right which"}, {"start": 156.319, "duration": 6.601, "text": "is exactly equal to 2 + < tk2 and this"}, {"start": 160.84, "duration": 3.88, "text": "is the right answer thank you for"}, {"start": 162.92, "duration": 4.28, "text": "watching please kindly subscribe to my"}, {"start": 164.72, "duration": 5.48, "text": "channel also like comment and share"}, {"start": 167.2, "duration": 3.0, "text": "bye-bye"}], "title": "Chinese junior Olympiad Math | The square root surdic expression. #olympiad #exam", "description": "", "lengthSeconds": 168, "width": 640, "height": 360}
{"video_id": "Z4sukkgkPgw", "transcript": [{"start": 1.12, "duration": 5.199, "text": "hello welcome back once again today we"}, {"start": 3.159, "duration": 5.52, "text": "have this interesting radical equation"}, {"start": 6.319, "duration": 4.28, "text": "and here we'll be solving for the values"}, {"start": 8.679, "duration": 5.441, "text": "of"}, {"start": 10.599, "duration": 5.481, "text": "X so let's get started from the left"}, {"start": 14.12, "duration": 4.6, "text": "hand side we"}, {"start": 16.08, "duration": 4.72, "text": "have the square root of"}, {"start": 18.72, "duration": 8.44, "text": "x then"}, {"start": 20.8, "duration": 9.319, "text": "plus the < TK of x - 9 this is equal to"}, {"start": 27.16, "duration": 6.0, "text": "9 now"}, {"start": 30.119, "duration": 5.921, "text": "this term right over here let's move it"}, {"start": 33.16, "duration": 7.76, "text": "to the right hand"}, {"start": 36.04, "duration": 10.8, "text": "side so we are left with square < TK of"}, {"start": 40.92, "duration": 8.159, "text": "x - 9 then this will be equal to 9 - the"}, {"start": 46.84, "duration": 5.0, "text": "sare root of"}, {"start": 49.079, "duration": 5.32, "text": "x in our next step let's Square both"}, {"start": 51.84, "duration": 8.92, "text": "sides so here we"}, {"start": 54.399, "duration": 10.841, "text": "have the S < TK of x - 9 squared this is"}, {"start": 60.76, "duration": 6.2, "text": "equal to 9 - < TK"}, {"start": 65.24, "duration": 6.08, "text": "x"}, {"start": 66.96, "duration": 6.479, "text": "2 okay so from the right hand side let"}, {"start": 71.32, "duration": 6.119, "text": "us make use of this identity we know"}, {"start": 73.439, "duration": 7.081, "text": "that a minus b"}, {"start": 77.439, "duration": 10.04, "text": "s this is equal to a"}, {"start": 80.52, "duration": 8.48, "text": "2 + b 2 - 2 * a * B so we're going to"}, {"start": 87.479, "duration": 3.721, "text": "make use of this identity at the right"}, {"start": 89.0, "duration": 4.079, "text": "hand side but here we know Square we"}, {"start": 91.2, "duration": 5.64, "text": "cancel square"}, {"start": 93.079, "duration": 6.32, "text": "roots and we have x - 9 this will be"}, {"start": 96.84, "duration": 6.48, "text": "equal to we have 9"}, {"start": 99.399, "duration": 9.481, "text": "2 + otk x"}, {"start": 103.32, "duration": 10.36, "text": "2 - 2 * 9 * otk of"}, {"start": 108.88, "duration": 8.599, "text": "X so from here we have x - 9 this is = 9"}, {"start": 113.68, "duration": 10.359, "text": "s is 81 Square cancel square root we get"}, {"start": 117.479, "duration": 10.0, "text": "+ x - 18 root of"}, {"start": 124.039, "duration": 6.84, "text": "x we can see that X is positive at the"}, {"start": 127.479, "duration": 5.84, "text": "left hand side and X is also positive at"}, {"start": 130.879, "duration": 6.481, "text": "the right hand side so we need to cancel"}, {"start": 133.319, "duration": 7.481, "text": "them out we all know that if we subtract"}, {"start": 137.36, "duration": 6.0, "text": "X from both sides X will be eliminated"}, {"start": 140.8, "duration": 2.56, "text": "from both"}, {"start": 143.519, "duration": 7.44, "text": "sides so what do we have next so we have"}, {"start": 146.56, "duration": 9.72, "text": "from the left hand side -9 this is = to"}, {"start": 150.959, "duration": 9.28, "text": "81 - 18 otk of"}, {"start": 156.28, "duration": 5.52, "text": "X so from here let us make the right"}, {"start": 160.239, "duration": 3.64, "text": "hand side the left hand side and the"}, {"start": 161.8, "duration": 5.359, "text": "left hand side the right hand side so"}, {"start": 163.879, "duration": 5.761, "text": "move this term to the left and take this"}, {"start": 167.159, "duration": 5.521, "text": "term to the"}, {"start": 169.64, "duration": 8.04, "text": "right so we're going to have 81 don't"}, {"start": 172.68, "duration": 6.72, "text": "change any sign - 18K X this is equal"}, {"start": 177.68, "duration": 3.24, "text": "to9"}, {"start": 179.4, "duration": 4.16, "text": "[Music]"}, {"start": 180.92, "duration": 4.959, "text": "okay so we have this minus this to give"}, {"start": 183.56, "duration": 4.8, "text": "us this so This actually mean that this"}, {"start": 185.879, "duration": 9.321, "text": "is also equal to this minus this so we"}, {"start": 188.36, "duration": 11.879, "text": "have 18 < tkx is = 81 -"}, {"start": 195.2, "duration": 9.44, "text": "9 from here we have 18 tkx is equal to"}, {"start": 200.239, "duration": 7.481, "text": "that will give us 90 divide by 18 so"}, {"start": 204.64, "duration": 7.799, "text": "divide here by 18 and divide here by 18"}, {"start": 207.72, "duration": 10.439, "text": "18 will cancel 18 so we have otk X is ="}, {"start": 212.439, "duration": 5.72, "text": "90 18 is 5 Square both"}, {"start": 218.72, "duration": 7.439, "text": "sides so Square we cancel square root"}, {"start": 221.439, "duration": 7.041, "text": "and here we get our solution X is = 25"}, {"start": 226.159, "duration": 5.041, "text": "and this is the only real solution that"}, {"start": 228.48, "duration": 4.399, "text": "satisfy the equation thank you for"}, {"start": 231.2, "duration": 5.0, "text": "watching please kindly subscribe to my"}, {"start": 232.879, "duration": 6.321, "text": "channel also like comment and share"}, {"start": 236.2, "duration": 3.0, "text": "bye-bye"}], "title": "Germany | Nice Olympiad Math | Interesting radical equation | #maths #education #puzzle", "description": "#maths #education #mathematics #puzzle #learning", "lengthSeconds": 238, "width": 428, "height": 240}
{"video_id": "8ZRGcj1t1p8", "transcript": [{"start": 1.12, "duration": 4.44, "text": "hello good day viewers welcome back once"}, {"start": 2.879, "duration": 5.521, "text": "again today we're going to solve for the"}, {"start": 5.56, "duration": 5.959, "text": "value of x in the following given"}, {"start": 8.4, "duration": 10.0, "text": "exponential equation we have square Ro"}, {"start": 11.519, "duration": 10.161, "text": "TK of 8^ of X this is equal to"}, {"start": 18.4, "duration": 5.68, "text": "16 okay now we're going to get rid of"}, {"start": 21.68, "duration": 5.08, "text": "this root sign here and how do we do"}, {"start": 24.08, "duration": 5.84, "text": "this we're going to square both sides of"}, {"start": 26.76, "duration": 3.16, "text": "the equation"}, {"start": 34.76, "duration": 6.16, "text": "from the left we know the square and the"}, {"start": 37.32, "duration": 7.079, "text": "root sign we cancel out and we have 8 to"}, {"start": 40.92, "duration": 6.6, "text": "the^ of X this is equal to in the"}, {"start": 44.399, "duration": 8.201, "text": "bracket there 16 is the same as 2 to"}, {"start": 47.52, "duration": 7.559, "text": "the^ of 4 then rest power of 2 now from"}, {"start": 52.6, "duration": 6.439, "text": "the left again we need to at least"}, {"start": 55.079, "duration": 5.921, "text": "obtain the same basis from both sides of"}, {"start": 59.039, "duration": 6.841, "text": "this equation right"}, {"start": 61.0, "duration": 12.36, "text": "so here we have 8 to be written as 2^ 3"}, {"start": 65.88, "duration": 9.279, "text": "then X this is equal to 2^ 4 squar now"}, {"start": 73.36, "duration": 7.2, "text": "we're going to make use of this"}, {"start": 75.159, "duration": 11.0, "text": "property a of B of C this is equal to a"}, {"start": 80.56, "duration": 11.239, "text": "of B * C so from here this equals 2^ 3 *"}, {"start": 86.159, "duration": 6.721, "text": "X then this is equal to 2 of 4 * 2 which"}, {"start": 91.799, "duration": 4.521, "text": "is"}, {"start": 92.88, "duration": 7.239, "text": "8 so since we have equal base from both"}, {"start": 96.32, "duration": 6.799, "text": "sides we can equate the power so this"}, {"start": 100.119, "duration": 7.801, "text": "implies that"}, {"start": 103.119, "duration": 7.881, "text": "3x is equal to 8 from here if you divide"}, {"start": 107.92, "duration": 3.08, "text": "both sides by"}, {"start": 111.159, "duration": 5.96, "text": "three then we come up with a solution"}, {"start": 114.32, "duration": 7.24, "text": "that X is ="}, {"start": 117.119, "duration": 6.201, "text": "8/ 3 and this is right answer thank you"}, {"start": 121.56, "duration": 3.96, "text": "for watching please kindly subscribe to"}, {"start": 123.32, "duration": 5.719, "text": "my YouTube channel also like comment and"}, {"start": 125.52, "duration": 3.519, "text": "share bye-bye"}], "title": "Exponential equation | #maths #mathematics", "description": "", "lengthSeconds": 127, "width": 640, "height": 360}
{"video_id": "YSTICWJn57A", "transcript": [{"start": 0.44, "duration": 4.76, "text": "hello good viewers welcome back once"}, {"start": 2.08, "duration": 4.92, "text": "again we're given that a = -8 and here"}, {"start": 5.2, "duration": 5.04, "text": "we're going to evaluate the square root"}, {"start": 7.0, "duration": 6.799, "text": "of a squ I have a reason for saying that"}, {"start": 10.24, "duration": 6.16, "text": "99% fails this please forgive me if my"}, {"start": 13.799, "duration": 4.56, "text": "English is wrong because my main"}, {"start": 16.4, "duration": 5.6, "text": "language is not"}, {"start": 18.359, "duration": 8.281, "text": "English okay now let's get started we"}, {"start": 22.0, "duration": 9.48, "text": "have here a = 8 if we try evaluating"}, {"start": 26.64, "duration": 10.32, "text": "this like this square root of a s = the"}, {"start": 31.48, "duration": 7.36, "text": "square < TK of -8 s and from here this"}, {"start": 36.96, "duration": 6.56, "text": "square and this square root will cancel"}, {"start": 38.84, "duration": 6.199, "text": "out and we still arrive at -8 obviously"}, {"start": 43.52, "duration": 5.84, "text": "this is very"}, {"start": 45.039, "duration": 8.761, "text": "wrong so what do we do let us take note"}, {"start": 49.36, "duration": 8.519, "text": "that square < TK of x to the^ of 2 like"}, {"start": 53.8, "duration": 7.12, "text": "this is not equal to the square root of"}, {"start": 57.879, "duration": 6.32, "text": "x all"}, {"start": 60.92, "duration": 7.839, "text": "squar for instance assuming the equation"}, {"start": 64.199, "duration": 6.841, "text": "was the square root of a all squared"}, {"start": 68.759, "duration": 7.081, "text": "then that will be a different case this"}, {"start": 71.04, "duration": 8.56, "text": "will be -8 I agree but here we are"}, {"start": 75.84, "duration": 6.12, "text": "giving a s under this root sign so let's"}, {"start": 79.6, "duration": 4.159, "text": "take note that the square root of x to"}, {"start": 81.96, "duration": 6.799, "text": "the^ of two in this"}, {"start": 83.759, "duration": 8.281, "text": "form signifies the absolute value of"}, {"start": 88.759, "duration": 8.441, "text": "x and and we know the absolute value of"}, {"start": 92.04, "duration": 9.759, "text": "x equal --x when X is less"}, {"start": 97.2, "duration": 7.519, "text": "than 0 right and the absolute value of x"}, {"start": 101.799, "duration": 5.521, "text": "equal positive X when X is greater than"}, {"start": 104.719, "duration": 5.68, "text": "or equals 0"}, {"start": 107.32, "duration": 6.839, "text": "amazing so with from the given problem"}, {"start": 110.399, "duration": 6.76, "text": "the square root of a squar"}, {"start": 114.159, "duration": 7.6, "text": "equals the absolute value of a and"}, {"start": 117.159, "duration": 6.92, "text": "remember a = -8 right so we have that"}, {"start": 121.759, "duration": 5.32, "text": "this will be equal to the absolute value"}, {"start": 124.079, "duration": 5.8, "text": "of 8 and we know the absolute value of"}, {"start": 127.079, "duration": 6.041, "text": "negative number result to the positive"}, {"start": 129.879, "duration": 5.72, "text": "of that number so the right solution is"}, {"start": 133.12, "duration": 5.119, "text": "eight thank you for watching please"}, {"start": 135.599, "duration": 6.801, "text": "kindly subscribe to my channel also like"}, {"start": 138.239, "duration": 4.161, "text": "comment and share bye-bye"}], "title": "99% keeps failing this !", "description": "", "lengthSeconds": 141, "width": 564, "height": 360}
{"video_id": "f3N305-T2lc", "transcript": [{"start": 0.919, "duration": 5.44, "text": "hello welcome back once again let's"}, {"start": 3.199, "duration": 7.48, "text": "solve for the real value of x in the"}, {"start": 6.359, "duration": 8.36, "text": "following given equation we have X Cub +"}, {"start": 10.679, "duration": 6.481, "text": "x Cub = 162 here we'll be solving for"}, {"start": 14.719, "duration": 4.761, "text": "only real value of"}, {"start": 17.16, "duration": 4.48, "text": "x now from the left hand side of this"}, {"start": 19.48, "duration": 5.16, "text": "equation we can see that this term here"}, {"start": 21.64, "duration": 5.52, "text": "and this term here are exactly the"}, {"start": 24.64, "duration": 7.24, "text": "same let us take note of this when we"}, {"start": 27.16, "duration": 9.36, "text": "have a + a this is same as 2 * a which"}, {"start": 31.88, "duration": 8.839, "text": "is 2 a right and here we have X Cub + x"}, {"start": 36.52, "duration": 6.199, "text": "Cub which should be same as 2 * X cubed"}, {"start": 40.719, "duration": 3.52, "text": "right then this"}, {"start": 42.719, "duration": 5.441, "text": "equals"}, {"start": 44.239, "duration": 7.521, "text": "162 from here let's divide both sides by"}, {"start": 48.16, "duration": 6.96, "text": "two this we cancel this we have a x"}, {"start": 51.76, "duration": 6.04, "text": "cubed equals here if you divide this by"}, {"start": 55.12, "duration": 5.32, "text": "two that will give us"}, {"start": 57.8, "duration": 4.88, "text": "81 now from here remember we solving for"}, {"start": 60.44, "duration": 4.32, "text": "the real value of x therefore we're"}, {"start": 62.68, "duration": 6.68, "text": "going to take the cube root on both"}, {"start": 64.76, "duration": 6.76, "text": "sides so from here we get X = the cube"}, {"start": 69.36, "duration": 6.32, "text": "root of"}, {"start": 71.52, "duration": 8.639, "text": "81 which obviously we can simplify right"}, {"start": 75.68, "duration": 8.439, "text": "remember 81 is the same as 27 *"}, {"start": 80.159, "duration": 5.361, "text": "3 so from here we can see that when we"}, {"start": 84.119, "duration": 6.441, "text": "have the N"}, {"start": 85.52, "duration": 5.8, "text": "root of M * K this is same as the n root"}, {"start": 90.56, "duration": 5.32, "text": "of"}, {"start": 91.32, "duration": 8.439, "text": "M the N root of K so therefore here will"}, {"start": 95.88, "duration": 8.12, "text": "be same as the cube root of 27 MTI the"}, {"start": 99.759, "duration": 7.081, "text": "cube root of 3 from here we get X = the"}, {"start": 104.0, "duration": 6.36, "text": "cube root of 27 is 3 then multiply by"}, {"start": 106.84, "duration": 5.559, "text": "the cube root of three and this is the"}, {"start": 110.36, "duration": 4.039, "text": "right answer thank you for watching"}, {"start": 112.399, "duration": 6.72, "text": "please kindly subscribe to my channel"}, {"start": 114.399, "duration": 4.72, "text": "also like comment and share bye-bye"}], "title": "Junior Olympiad Math", "description": "", "lengthSeconds": 117, "width": 640, "height": 360}
{"video_id": "-F7WcWWdd-0", "transcript": [{"start": 0.599, "duration": 3.801, "text": "hello good viewers welcome back once"}, {"start": 2.32, "duration": 8.04, "text": "again today we have this tricky math"}, {"start": 4.4, "duration": 7.88, "text": "problem we have x^ 60 + 1 x^ 60 = 47 and"}, {"start": 10.36, "duration": 7.16, "text": "right over here we're going to evaluate"}, {"start": 12.28, "duration": 7.919, "text": "x ^ 30 + 1 / x ^ 30 from the given"}, {"start": 17.52, "duration": 9.759, "text": "equation we can actually write it as x"}, {"start": 20.199, "duration": 11.56, "text": "to the power of 30 * 2 + 1 all X to the^"}, {"start": 27.279, "duration": 7.6, "text": "of 30 * 2 and this is equal to"}, {"start": 31.759, "duration": 6.201, "text": "47 from the left hand side here let us"}, {"start": 34.879, "duration": 5.921, "text": "take note of this when we have a^ of B *"}, {"start": 37.96, "duration": 6.36, "text": "C we can actually Express this as a to"}, {"start": 40.8, "duration": 9.759, "text": "the power of B then ra power of C so"}, {"start": 44.32, "duration": 11.68, "text": "from here we get X the power of 30 2 + 1"}, {"start": 50.559, "duration": 8.121, "text": "x ^ 30 2 this is equal to"}, {"start": 56.0, "duration": 9.32, "text": "47 now from the left side let us take"}, {"start": 58.68, "duration": 13.759, "text": "note of this a 2 + b s can be written as"}, {"start": 65.32, "duration": 9.839, "text": "a + b all squ then - 2 * a * B so from"}, {"start": 72.439, "duration": 11.201, "text": "the left making use of this so we have"}, {"start": 75.159, "duration": 10.721, "text": "x^ 30 right then plus 1 / x ^ of 30 you"}, {"start": 83.64, "duration": 3.839, "text": "know why remember we have everything"}, {"start": 85.88, "duration": 3.64, "text": "here squared but here we have one/"}, {"start": 87.479, "duration": 5.241, "text": "something squared what happens to one"}, {"start": 89.52, "duration": 9.08, "text": "here actually this we can express this"}, {"start": 92.72, "duration": 13.52, "text": "as 1 X all to the^ of 30 right so here"}, {"start": 98.6, "duration": 12.559, "text": "we have this 2 then - 2 * x ^ 30 * 1 / X"}, {"start": 106.24, "duration": 7.96, "text": "the^ of 30 then this will be equal to"}, {"start": 111.159, "duration": 11.96, "text": "47 from here this cancel this right so"}, {"start": 114.2, "duration": 11.12, "text": "we have here x ^ 30 + 1 / X 30 or 2 - 2"}, {"start": 123.119, "duration": 8.361, "text": "this equal"}, {"start": 125.32, "duration": 11.199, "text": "47 add two to both side we get x^ 30 + 1"}, {"start": 131.48, "duration": 8.24, "text": "/ x^ 30 2 this equal"}, {"start": 136.519, "duration": 6.161, "text": "49 take root of both sides we get the"}, {"start": 139.72, "duration": 9.28, "text": "right answer to the question that x^ 30"}, {"start": 142.68, "duration": 8.919, "text": "+ 1 x^ 30 = plus orus < 49 which is 7"}, {"start": 149.0, "duration": 4.36, "text": "and this is the right answer thank you"}, {"start": 151.599, "duration": 4.441, "text": "for watching please kindly subscribe to"}, {"start": 153.36, "duration": 5.68, "text": "my channel also like comment and share"}, {"start": 156.04, "duration": 3.0, "text": "bye-bye"}], "title": "Tricky Algebra Math", "description": "", "lengthSeconds": 157, "width": 560, "height": 360}
{"video_id": "gGQ1kKd_UDY", "transcript": [{"start": 1.35, "duration": 18.709, "text": "[\u0938\u0902\u0917\u0940\u0924]"}, {"start": 22.88, "duration": 36.239, "text": "\u0938\u094d\u091f \u092a\u0940"}, {"start": 25.55, "duration": 33.569, "text": "[\u0938\u0902\u0917\u0940\u0924]"}, {"start": 61.33, "duration": 9.09, "text": "[\u0938\u0902\u0917\u0940\u0924]"}, {"start": 73.37, "duration": 23.38, "text": "[\u0938\u0902\u0917\u0940\u0924]"}, {"start": 97.96, "duration": 33.15, "text": "\u0938\u094d\u0932\u093f\u091f \u092a\u0940\u091f\u0938"}, {"start": 100.94, "duration": 30.17, "text": "[\u0938\u0902\u0917\u0940\u0924]"}, {"start": 133.33, "duration": 18.07, "text": "[\u0938\u0902\u0917\u0940\u0924]"}, {"start": 149.879, "duration": 10.341, "text": "\u092c"}, {"start": 151.4, "duration": 8.82, "text": "[\u0938\u0902\u0917\u0940\u0924]"}], "title": "Can you solve for x without logarithms ?   #maths #education", "description": "", "lengthSeconds": 158, "width": 640, "height": 360}
{"video_id": "mE80TTLTXB4", "transcript": [{"start": 0.88, "duration": 5.2, "text": "hello welcome back once again today we"}, {"start": 3.32, "duration": 5.48, "text": "have this interesting equation and here"}, {"start": 6.08, "duration": 6.439, "text": "we'll be solving for the values of x"}, {"start": 8.8, "duration": 6.56, "text": "from the left hand side we have x to^ 2"}, {"start": 12.519, "duration": 5.121, "text": "to^ 3 this is equal to from the right"}, {"start": 15.36, "duration": 7.2, "text": "hand side x^"}, {"start": 17.64, "duration": 8.2, "text": "2^ 2 right now we're going to simplify"}, {"start": 22.56, "duration": 7.52, "text": "each each exponents from both sides so"}, {"start": 25.84, "duration": 9.96, "text": "we start like this 2 to^ 3 from here"}, {"start": 30.08, "duration": 9.76, "text": "is equal to 8 right and here 2^ 2"}, {"start": 35.8, "duration": 8.4, "text": "here is equal to 4 so from the left hand"}, {"start": 39.84, "duration": 8.0, "text": "side we have x to^ 8 this is equal to x"}, {"start": 44.2, "duration": 9.76, "text": "to^ 4 now move this term to the left we"}, {"start": 47.84, "duration": 8.64, "text": "get x^ 8 - x^ 4 this is equal to 0 we"}, {"start": 53.96, "duration": 8.04, "text": "can see x^ 4 is common so let's Factor"}, {"start": 56.48, "duration": 10.6, "text": "it out so get x^ 4 into bracket X 8 x^ 4"}, {"start": 62.0, "duration": 8.6, "text": "will give us x^ 4 - x^ 4 x^ 4 will give"}, {"start": 67.08, "duration": 7.359, "text": "us 1 and this is equal to0 so this"}, {"start": 70.6, "duration": 9.8, "text": "implies from here that either x^ 4 is"}, {"start": 74.439, "duration": 7.72, "text": "equal 0 or x ^ 4 - 1 is = 0 from here"}, {"start": 80.4, "duration": 4.0, "text": "take the square take the fourth root of"}, {"start": 82.159, "duration": 5.92, "text": "both sides so we get X = fourth root of"}, {"start": 84.4, "duration": 6.48, "text": "0 is still zero right so this is our"}, {"start": 88.079, "duration": 4.161, "text": "first solution and here we're going to"}, {"start": 90.88, "duration": 2.96, "text": "make use of the difference of two"}, {"start": 92.24, "duration": 6.12, "text": "squares but we need to rewrite the"}, {"start": 93.84, "duration": 9.559, "text": "equation as follow x 2 2 we know 2 * 2"}, {"start": 98.36, "duration": 7.359, "text": "is 4 - 1^ 2 this is equal to 0 so making"}, {"start": 103.399, "duration": 7.04, "text": "use of the difference of 2"}, {"start": 105.719, "duration": 13.241, "text": "squar we know this is equal to a + b *"}, {"start": 110.439, "duration": 13.36, "text": "Aus B so that will become X2 + 1 * X2 -"}, {"start": 118.96, "duration": 7.04, "text": "1 this is equal to Z so from here we"}, {"start": 123.799, "duration": 6.721, "text": "have two products which is equal to Z so"}, {"start": 126.0, "duration": 12.0, "text": "this implies that either x\u00b2 + 1 is = 0"}, {"start": 130.52, "duration": 7.48, "text": "or X2 - 1 is = 0 so from"}, {"start": 138.8, "duration": 6.96, "text": "here subtract one from both sides so we"}, {"start": 141.72, "duration": 7.12, "text": "get x2 is = to -1 and you add one to"}, {"start": 145.76, "duration": 6.08, "text": "both side we get x2 is = 1 taking the"}, {"start": 148.84, "duration": 5.36, "text": "square root of both sides so we get X ="}, {"start": 151.84, "duration": 7.88, "text": "Plus orus theun"}, {"start": 154.2, "duration": 9.72, "text": "of1 and here X = plus orus the root of 1"}, {"start": 159.72, "duration": 6.439, "text": "so we know according TOA theot of1 is I"}, {"start": 163.92, "duration": 5.319, "text": "which is Iota so this is equal plus or"}, {"start": 166.159, "duration": 6.881, "text": "minus I and here is equal to plus or"}, {"start": 169.239, "duration": 7.881, "text": "minus 1 so in the above equation we have"}, {"start": 173.04, "duration": 9.24, "text": "the following solutions for X so x is ="}, {"start": 177.12, "duration": 9.72, "text": "0 x is = + or minus one and for"}, {"start": 182.28, "duration": 6.64, "text": "Imaginary Solutions X = plus or minus"}, {"start": 186.84, "duration": 4.959, "text": "Iota thank you for watching please"}, {"start": 188.92, "duration": 7.36, "text": "kindly subscribe to my channel also like"}, {"start": 191.799, "duration": 4.481, "text": "comment and share bye-bye"}], "title": "Harvard University \ud83c\udf93 Entrance Exam | Can you solve ?", "description": "", "lengthSeconds": 195, "width": 428, "height": 240}
{"video_id": "Fd9D4E_20_8", "transcript": [{"start": 0.96, "duration": 4.319, "text": "hello welcome back once"}, {"start": 2.56, "duration": 5.44, "text": "again in the following equation we're"}, {"start": 5.279, "duration": 5.681, "text": "going to solve for the value of x and"}, {"start": 8.0, "duration": 6.08, "text": "here we're given that 0 is less than x"}, {"start": 10.96, "duration": 5.2, "text": "and x is less than 1 so going to solve"}, {"start": 14.08, "duration": 6.88, "text": "for the value of x that we satisfy this"}, {"start": 16.16, "duration": 7.279, "text": "equation within this range 0 and one so"}, {"start": 20.96, "duration": 3.68, "text": "let's get started from the left hand"}, {"start": 23.439, "duration": 5.16, "text": "side you have"}, {"start": 24.64, "duration": 6.2, "text": "256 to the^ of X this is equal to from"}, {"start": 28.599, "duration": 5.28, "text": "the right 1"}, {"start": 30.84, "duration": 5.399, "text": "X now let's raise both sides of this"}, {"start": 33.879, "duration": 4.0, "text": "equation to the power of the reciprocal"}, {"start": 36.239, "duration": 5.241, "text": "of the power of"}, {"start": 37.879, "duration": 6.721, "text": "256 so we know the power of 256 is X so"}, {"start": 41.48, "duration": 7.079, "text": "its reciprocal is 1 / X so raise both"}, {"start": 44.6, "duration": 8.2, "text": "sides to the power of 1 / X and raas"}, {"start": 48.559, "duration": 9.121, "text": "this side to the^ of 1 / X we know x * 1"}, {"start": 52.8, "duration": 8.36, "text": "/ X is 1 right so here we have just"}, {"start": 57.68, "duration": 7.119, "text": "256 and this is equal to 1 /"}, {"start": 61.16, "duration": 5.959, "text": "X ra the^ of 1"}, {"start": 64.799, "duration": 5.521, "text": "/x fortunately"}, {"start": 67.119, "duration": 8.04, "text": "256 can be expressing index power of 4"}, {"start": 70.32, "duration": 7.24, "text": "as 4^ 4 so from the left hand side from"}, {"start": 75.159, "duration": 6.64, "text": "the right I mean we have 1"}, {"start": 77.56, "duration": 8.84, "text": "/x we have to the ^ 1 /x this will be"}, {"start": 81.799, "duration": 6.921, "text": "equal to 4 to^ 4 so now we can compy as"}, {"start": 86.4, "duration": 4.48, "text": "follow so we have the base is equal to"}, {"start": 88.72, "duration": 6.28, "text": "the base or the power is equal to the"}, {"start": 90.88, "duration": 10.36, "text": "power so from here this implies that 1"}, {"start": 95.0, "duration": 6.24, "text": "/x is = 4 put this over one and cross"}, {"start": 101.399, "duration": 5.76, "text": "multiply so here we"}, {"start": 103.759, "duration": 7.601, "text": "have x * 4 we give us"}, {"start": 107.159, "duration": 7.361, "text": "4X this is equal to 1 * 1 will give us 1"}, {"start": 111.36, "duration": 6.96, "text": "divide by the coefficient of x which is"}, {"start": 114.52, "duration": 8.36, "text": "four four we cancel four so from here we"}, {"start": 118.32, "duration": 7.56, "text": "have the solution X is is equal to 1 / 4"}, {"start": 122.88, "duration": 3.0, "text": "or"}, {"start": 126.24, "duration": 5.12, "text": "0.25 and here we have the"}, {"start": 129.239, "duration": 5.08, "text": "solution thank you for watching please"}, {"start": 131.36, "duration": 7.519, "text": "kindly subscribe to my channel also like"}, {"start": 134.319, "duration": 4.56, "text": "comment and share bye-bye"}], "title": "India | Nice Olympiad Math | Can you solve ?", "description": "", "lengthSeconds": 137, "width": 536, "height": 360}
{"video_id": "0ahfz39KY08", "transcript": [{"start": 0.84, "duration": 4.8, "text": "hello welcome back once again today"}, {"start": 2.52, "duration": 6.119, "text": "we're going to solve this equation we"}, {"start": 5.64, "duration": 6.2, "text": "have from the left hand side x * x * X"}, {"start": 8.639, "duration": 8.001, "text": "which is X cubed right so this is equal"}, {"start": 11.84, "duration": 8.4, "text": "to x + x + x which is 3 x so move this"}, {"start": 16.64, "duration": 7.16, "text": "time to the left so we get X Cub -"}, {"start": 20.24, "duration": 6.64, "text": "3x this is equal to Z we can see from"}, {"start": 23.8, "duration": 6.719, "text": "here x is common so we have X into"}, {"start": 26.88, "duration": 6.8, "text": "bracket X2 - 3"}, {"start": 30.519, "duration": 6.56, "text": "then this is equal to Z from here this"}, {"start": 33.68, "duration": 9.24, "text": "implies that either X is equal to Z and"}, {"start": 37.079, "duration": 10.48, "text": "indeed this is a solution or x^2 - 3 is"}, {"start": 42.92, "duration": 7.6, "text": "= to 0 add 3 to both side we get x\u00b2 is ="}, {"start": 47.559, "duration": 7.64, "text": "3 take root of both sides so here we get"}, {"start": 50.52, "duration": 8.12, "text": "X is = to plus or minus < TK of 3 so"}, {"start": 55.199, "duration": 8.16, "text": "here we have the solutions so three real"}, {"start": 58.64, "duration": 8.2, "text": "solutions exist for equation x = 0 and x"}, {"start": 63.359, "duration": 5.281, "text": "= plus orus otk of 3 thank you for"}, {"start": 66.84, "duration": 4.88, "text": "watching please kindly subscribe to my"}, {"start": 68.64, "duration": 6.08, "text": "channel also like comment and share"}, {"start": 71.72, "duration": 3.0, "text": "bye-bye"}], "title": "Interesting Olympiad Math | Can you solve ?", "description": "", "lengthSeconds": 74, "width": 548, "height": 360}
{"video_id": "eKk86NQA0jo", "transcript": [{"start": 0.599, "duration": 4.841, "text": "hello welcome back once again today we"}, {"start": 3.0, "duration": 5.88, "text": "have another interesting systems of"}, {"start": 5.44, "duration": 6.0, "text": "equations we have a + b equal to 1 we"}, {"start": 8.88, "duration": 7.68, "text": "have the product of A and B is equal to"}, {"start": 11.44, "duration": 8.8, "text": "2 and here we'll be evaluating a 2 - b 2"}, {"start": 16.56, "duration": 9.039, "text": "interesting right now let's get"}, {"start": 20.24, "duration": 7.6, "text": "started from equation one we have a + b"}, {"start": 25.599, "duration": 6.361, "text": "this is equal to 1 now let's go ahead"}, {"start": 27.84, "duration": 4.12, "text": "and square both sides of this equation"}, {"start": 32.16, "duration": 2.919, "text": "okay now from the left hand side let us"}, {"start": 33.76, "duration": 6.24, "text": "make use of this"}, {"start": 35.079, "duration": 9.64, "text": "property that a + b s this is equal to"}, {"start": 40.0, "duration": 9.76, "text": "in terms of aus B that will be a - b 2 +"}, {"start": 44.719, "duration": 7.16, "text": "4 * a * B amazing property right so at"}, {"start": 49.76, "duration": 6.72, "text": "the left hand side we be making use of"}, {"start": 51.879, "duration": 10.68, "text": "this property and that will become a - b"}, {"start": 56.48, "duration": 7.959, "text": "2 + 4 * a * B this is equal to 1 2 is 1"}, {"start": 62.559, "duration": 5.24, "text": "okay from the original problem remember"}, {"start": 64.439, "duration": 9.121, "text": "a * B is equal 2 so we replace here with"}, {"start": 67.799, "duration": 9.201, "text": "2 that will become a - b 2 + 4 * 2 which"}, {"start": 73.56, "duration": 7.36, "text": "is 8 this is equal to 1 subtract it8"}, {"start": 77.0, "duration": 7.84, "text": "from both sides we get a - b 2 this is ="}, {"start": 80.92, "duration": 6.92, "text": "to 1 - 8 is -7 from here take root of"}, {"start": 84.84, "duration": 6.959, "text": "both side we get a minus B is equal to"}, {"start": 87.84, "duration": 8.639, "text": "plus or minus < TK of 7 which is equal"}, {"start": 91.799, "duration": 8.96, "text": "to plus or minus remember of1 is I then"}, {"start": 96.479, "duration": 7.201, "text": "we have otk 7 okay okay now the we ask"}, {"start": 100.759, "duration": 5.801, "text": "evaluate a s minus b 2 is simply the"}, {"start": 103.68, "duration": 7.119, "text": "difference of two squares right which is"}, {"start": 106.56, "duration": 6.599, "text": "factorized as a + b * a minus B so you"}, {"start": 110.799, "duration": 5.161, "text": "can see the reason why we solve for a"}, {"start": 113.159, "duration": 6.0, "text": "minus B already from the problem we"}, {"start": 115.96, "duration": 5.799, "text": "given that A + B is equal to 1 and from"}, {"start": 119.159, "duration": 6.081, "text": "here we have found Aus B to be equal to"}, {"start": 121.759, "duration": 5.881, "text": "plus orus IUN 7 so we can easily"}, {"start": 125.24, "duration": 4.239, "text": "evaluate that expression Now by"}, {"start": 127.64, "duration": 4.319, "text": "multiplying the value for this and the"}, {"start": 129.479, "duration": 6.76, "text": "value for this together so we have here"}, {"start": 131.959, "duration": 9.441, "text": "a 2us b 2 will be equal to a + b that is"}, {"start": 136.239, "duration": 7.561, "text": "1 * a minus B that is plus or minus IUN"}, {"start": 141.4, "duration": 7.839, "text": "7 and from here we get the right answer"}, {"start": 143.8, "duration": 7.68, "text": "that this equals plus orus I root of 7"}, {"start": 149.239, "duration": 4.401, "text": "thank you for watching please kindly"}, {"start": 151.48, "duration": 6.2, "text": "subscribe to my channel also like"}, {"start": 153.64, "duration": 4.04, "text": "comment and share bye-bye"}], "title": "Interesting Olympiad Math | Algebra problem", "description": "", "lengthSeconds": 156, "width": 640, "height": 360}
{"video_id": "0W9MvqVzpXQ", "transcript": [{"start": 0.399, "duration": 5.721, "text": "hello welcome back once again today I"}, {"start": 3.04, "duration": 5.479, "text": "have cooked another nice problem here"}, {"start": 6.12, "duration": 5.92, "text": "and today we're going to find the sum of"}, {"start": 8.519, "duration": 8.281, "text": "the six powers of A and B we give you"}, {"start": 12.04, "duration": 9.399, "text": "that a + b = 2 a 2 + b\u00b2 = 3 and here"}, {"start": 16.8, "duration": 6.96, "text": "we're going to evaluate a to ^ 6 + b^ of"}, {"start": 21.439, "duration": 4.881, "text": "6 okay the first thing to do here is to"}, {"start": 23.76, "duration": 5.2, "text": "find the product of A and B I think that"}, {"start": 26.32, "duration": 8.44, "text": "will be very helpful right so from"}, {"start": 28.96, "duration": 8.16, "text": "equation one we have a + b = 2 so we cut"}, {"start": 34.76, "duration": 9.6, "text": "to square both sides then from the left"}, {"start": 37.12, "duration": 10.279, "text": "this = a 2 + b 2 + 2 * a * B this = 2 2"}, {"start": 44.36, "duration": 9.24, "text": "which is 4 from the given equation a s +"}, {"start": 47.399, "duration": 9.201, "text": "b\u00b2 = 3 so we get 3 + 2 a b this = 4"}, {"start": 53.6, "duration": 9.36, "text": "subract three from both side we get 2 a"}, {"start": 56.6, "duration": 9.08, "text": "= 1 and from here we get A8 * b = 1 / 2"}, {"start": 62.96, "duration": 8.56, "text": "amazing so what do we do"}, {"start": 65.68, "duration": 10.479, "text": "next I will adice us to First find the"}, {"start": 71.52, "duration": 9.559, "text": "sum of the thir power of A and B right"}, {"start": 76.159, "duration": 8.241, "text": "that is a cub + B Cub so here a cubed +"}, {"start": 81.079, "duration": 6.601, "text": "B Cubed is given by the following"}, {"start": 84.4, "duration": 8.079, "text": "property or formula that this = A + B"}, {"start": 87.68, "duration": 8.32, "text": "Cub - 3 * a * B into bracket A + B so"}, {"start": 92.479, "duration": 5.441, "text": "you can see the reason why a * B's value"}, {"start": 96.0, "duration": 5.119, "text": "is very helpful right so you can see it"}, {"start": 97.92, "duration": 6.04, "text": "has come up here then this equals from"}, {"start": 101.119, "duration": 8.201, "text": "the given equation A + B is 2 so 2"}, {"start": 103.96, "duration": 7.72, "text": "cubus 3 * 1 over 2 then times a + b"}, {"start": 109.32, "duration": 4.799, "text": "which is 2 from here we get 2 cubed"}, {"start": 111.68, "duration": 6.32, "text": "which is 8 this cancel this and we left"}, {"start": 114.119, "duration": 9.721, "text": "with minus 3 this equals 5"}, {"start": 118.0, "duration": 10.52, "text": "beautiful then what of a ^ of 6 + b ^ 6"}, {"start": 123.84, "duration": 9.759, "text": "remember we can write this as a cub"}, {"start": 128.52, "duration": 12.32, "text": "2+ B Cubed 2 but before we proceed take"}, {"start": 133.599, "duration": 12.241, "text": "note a 2 + b 2 = A + B is a wonderful"}, {"start": 140.84, "duration": 7.2, "text": "Property 2 - 2 * a * B with this"}, {"start": 145.84, "duration": 7.2, "text": "property"}, {"start": 148.04, "duration": 7.52, "text": "here we call a Cub a we call B Cub B and"}, {"start": 153.04, "duration": 8.44, "text": "from there we derive that this equal a"}, {"start": 155.56, "duration": 10.679, "text": "cub + B Cub then squar - 2 a cub B Cub"}, {"start": 161.48, "duration": 8.32, "text": "the same as 2 I mean a * B all"}, {"start": 166.239, "duration": 7.841, "text": "cubed so from here this equals a cub +"}, {"start": 169.8, "duration": 8.0, "text": "BB is 5 right so we have 5^ 2 - 2 into"}, {"start": 174.08, "duration": 7.76, "text": "bracket here A = 1 /2 then we Cube it"}, {"start": 177.8, "duration": 10.68, "text": "right so from here this = 5^ 2 which is"}, {"start": 181.84, "duration": 9.039, "text": "25 - 2 * this will be 1 / 8 right this"}, {"start": 188.48, "duration": 5.039, "text": "equal 25"}, {"start": 190.879, "duration": 7.601, "text": "1us that would be 1/"}, {"start": 193.519, "duration": 8.761, "text": "4 remember 25 is the same as 100 / 4"}, {"start": 198.48, "duration": 5.319, "text": "we're trying to make equal denominator"}, {"start": 202.28, "duration": 6.519, "text": "and from here we can subtract Den"}, {"start": 203.799, "duration": 6.481, "text": "numerator 100 - 1 is 99 / 4 and here is"}, {"start": 208.799, "duration": 3.561, "text": "the right answer"}, {"start": 210.28, "duration": 4.12, "text": "amazing thank you for watching please if"}, {"start": 212.36, "duration": 4.04, "text": "you really enjoy this content please"}, {"start": 214.4, "duration": 3.96, "text": "kindly subscribe turn on the"}, {"start": 216.4, "duration": 4.52, "text": "notification Bell so that you be"}, {"start": 218.36, "duration": 6.799, "text": "notified whenever I upload the next"}, {"start": 220.92, "duration": 4.239, "text": "video thank you bye-bye"}], "title": "Can you solve this Oxford University \ud83c\udf93 Math problem ?", "description": "", "lengthSeconds": 224, "width": 552, "height": 360}
{"video_id": "9ix7-y50RaM", "transcript": [{"start": 0.799, "duration": 4.0, "text": "hello good viewers welcome back once"}, {"start": 2.639, "duration": 4.561, "text": "again please kindly support my channel"}, {"start": 4.799, "duration": 4.84, "text": "by subscribing also like comment and"}, {"start": 7.2, "duration": 6.0, "text": "share today we're going to simplify this"}, {"start": 9.639, "duration": 7.161, "text": "sodic fraction here we have ro3 + 1 over"}, {"start": 13.2, "duration": 6.68, "text": "the conjugate plus ro3 - 1/ conjugates"}, {"start": 16.8, "duration": 5.8, "text": "so we asked to simplify this right so"}, {"start": 19.88, "duration": 5.52, "text": "let's get started now the first step"}, {"start": 22.6, "duration": 5.079, "text": "we're going to find the L let me let me"}, {"start": 25.4, "duration": 4.359, "text": "put it that way of this and this and"}, {"start": 27.679, "duration": 4.281, "text": "that is actually multiplying them"}, {"start": 29.759, "duration": 2.201, "text": "together"}, {"start": 32.44, "duration": 5.4, "text": "right and then the next step the product"}, {"start": 35.92, "duration": 4.28, "text": "of this if you divide it by this"}, {"start": 37.84, "duration": 5.719, "text": "denominator here we're going to have <"}, {"start": 40.2, "duration": 5.92, "text": "tk3 + 1 CU this and this will cancel out"}, {"start": 43.559, "duration": 5.881, "text": "and then we multiply this numerator"}, {"start": 46.12, "duration": 7.84, "text": "byun3 + 1 which is actually the square"}, {"start": 49.44, "duration": 6.84, "text": "of < tk3 + 1 right here we have a plus"}, {"start": 53.96, "duration": 5.559, "text": "sign you know if you divide this by this"}, {"start": 56.28, "duration": 5.04, "text": "we left with3 -1 and if you multiply by"}, {"start": 59.519, "duration": 8.361, "text": "this that is actually the square of <"}, {"start": 61.32, "duration": 9.56, "text": "tk3 - 1 right so from here we have < tk3"}, {"start": 67.88, "duration": 8.879, "text": "+ 1 if you square this we are going to"}, {"start": 70.88, "duration": 9.76, "text": "have 3 + 1 + 2 < tk3 then plus if you"}, {"start": 76.759, "duration": 8.441, "text": "square this we're going to have 3 + 1 -"}, {"start": 80.64, "duration": 6.839, "text": "2 < tk3 then all over do not forget that"}, {"start": 85.2, "duration": 5.04, "text": "down here we have to make use of the"}, {"start": 87.479, "duration": 7.6, "text": "difference of two squares right so here"}, {"start": 90.24, "duration": 6.48, "text": "this is going to be the < tk3 2 - 1 2"}, {"start": 95.079, "duration": 6.32, "text": "I'm very sure we're very familiar with"}, {"start": 96.72, "duration": 8.759, "text": "this when we have a + b * a minus B this"}, {"start": 101.399, "duration": 6.161, "text": "is same as a s - b 2 right so from here"}, {"start": 105.479, "duration": 5.521, "text": "we have this is equal to you know this"}, {"start": 107.56, "duration": 6.48, "text": "and this will cancel out and here 4 + 4"}, {"start": 111.0, "duration": 6.039, "text": "which is 8 then all over you know this"}, {"start": 114.04, "duration": 7.32, "text": "and this will cancel out we have 3 -"}, {"start": 117.039, "duration": 7.161, "text": "1 so from here we have 8 all over two"}, {"start": 121.36, "duration": 4.92, "text": "and that is equal to 4 and this is the"}, {"start": 124.2, "duration": 4.16, "text": "right answer thank you for watching"}, {"start": 126.28, "duration": 6.759, "text": "please do wa to subscribe to my channel"}, {"start": 128.36, "duration": 4.679, "text": "also like comment and share bye-bye"}], "title": "simplification of surds. |Rationalization #maths #education #mathematics", "description": "", "lengthSeconds": 131, "width": 640, "height": 360}
{"video_id": "P_go8b9dua4", "transcript": [{"start": 0.799, "duration": 3.641, "text": "hello good viewers welcome back once"}, {"start": 2.52, "duration": 5.039, "text": "again today we're going to solve for the"}, {"start": 4.44, "duration": 7.72, "text": "values of X in the following given"}, {"start": 7.559, "duration": 6.761, "text": "exponential equation we have x^ of X"}, {"start": 12.16, "duration": 5.959, "text": "this is = to 4 -"}, {"start": 14.32, "duration": 7.119, "text": "2x^ of X many may think that the best"}, {"start": 18.119, "duration": 6.561, "text": "way to solve this problem is to equate"}, {"start": 21.439, "duration": 9.08, "text": "the bases since we have equal"}, {"start": 24.68, "duration": 9.12, "text": "powers that is letting x = 4 - 2x"}, {"start": 30.519, "duration": 6.681, "text": "yes of course luckily a"}, {"start": 33.8, "duration": 6.2, "text": "solution will come up that is 4 all over"}, {"start": 37.2, "duration": 5.039, "text": "3 and that is IR rational"}, {"start": 40.0, "duration": 5.64, "text": "solution what if there are other"}, {"start": 42.239, "duration": 6.281, "text": "Solutions how do we figure them out now"}, {"start": 45.64, "duration": 5.239, "text": "when I try plotting the graph of the two"}, {"start": 48.52, "duration": 9.519, "text": "different functions of"}, {"start": 50.879, "duration": 9.961, "text": "X and G of x equal 4 - 2x^ X I found out"}, {"start": 58.039, "duration": 7.321, "text": "that these two graphs"}, {"start": 60.84, "duration": 7.12, "text": "have two intersection points and one is"}, {"start": 65.36, "duration": 6.48, "text": "an integral solution the other is a"}, {"start": 67.96, "duration": 7.76, "text": "rational solution just as we found here"}, {"start": 71.84, "duration": 7.72, "text": "how do we solve for all real values of X"}, {"start": 75.72, "duration": 7.68, "text": "in this given equation now let's get"}, {"start": 79.56, "duration": 6.4, "text": "started here we have X the^ of X this is"}, {"start": 83.4, "duration": 5.96, "text": "equal to 4 -"}, {"start": 85.96, "duration": 7.6, "text": "2x of X let go ahead and divide both"}, {"start": 89.36, "duration": 8.039, "text": "side by X X power of X and here by x^ of"}, {"start": 93.56, "duration": 7.239, "text": "x from the left hand side this way"}, {"start": 97.399, "duration": 6.921, "text": "cancel out to be one we have here now"}, {"start": 100.799, "duration": 8.881, "text": "let's use this property a^ B over M the"}, {"start": 104.32, "duration": 8.52, "text": "power B this is = a/ m to the power of B"}, {"start": 109.68, "duration": 7.28, "text": "so for here we have 4 -"}, {"start": 112.84, "duration": 7.4, "text": "2x over X then to the power of"}, {"start": 116.96, "duration": 6.24, "text": "X so here we have 1 is equal so when you"}, {"start": 120.24, "duration": 9.999, "text": "split this denominator here we have 4"}, {"start": 123.2, "duration": 10.08, "text": "/x - 2 then power of X okay now we have"}, {"start": 130.239, "duration": 11.441, "text": "some important things to discuss"}, {"start": 133.28, "duration": 12.2, "text": "here if x is even then 4 /x - 2 can be"}, {"start": 141.68, "duration": 6.88, "text": "equal to 1 right so this is our case"}, {"start": 145.48, "duration": 7.039, "text": "one because 1 to the power of an even"}, {"start": 148.56, "duration": 6.2, "text": "number is equal to one then the second"}, {"start": 152.519, "duration": 3.841, "text": "case to be considered"}, {"start": 154.76, "duration": 6.52, "text": "here"}, {"start": 156.36, "duration": 7.08, "text": "when the base here is odd that is if the"}, {"start": 161.28, "duration": 8.28, "text": "base here is"}, {"start": 163.44, "duration": 10.6, "text": "-1 that is 4 /x - 2 to be equal to -1"}, {"start": 169.56, "duration": 6.56, "text": "then X can also be even right because -1"}, {"start": 174.04, "duration": 5.96, "text": "to the power of an even number will give"}, {"start": 176.12, "duration": 7.96, "text": "us one right so if x is even the the"}, {"start": 180.0, "duration": 7.12, "text": "base here can be equal to 1 or"}, {"start": 184.08, "duration": 9.2, "text": "-1 so from here when you add two to both"}, {"start": 187.12, "duration": 9.199, "text": "side we get 4 /x is = to1 sorry 1 + 2"}, {"start": 193.28, "duration": 7.519, "text": "that is 3 right and from here cross"}, {"start": 196.319, "duration": 7.401, "text": "multiply we get 4 is = to 3x and from"}, {"start": 200.799, "duration": 6.64, "text": "here when you divide both by 3 we get X"}, {"start": 203.72, "duration": 6.519, "text": "is = to 4 / 3 which is the rational"}, {"start": 207.439, "duration": 6.961, "text": "solution and from here when you add 2 to"}, {"start": 210.239, "duration": 6.801, "text": "both sides we get 4 /x is = to1 + 2"}, {"start": 214.4, "duration": 5.839, "text": "which is 1 when you cross multiply we"}, {"start": 217.04, "duration": 8.96, "text": "get X is = 4 and here we find the"}, {"start": 220.239, "duration": 9.08, "text": "solutions X is = to 4 and also X is = to"}, {"start": 226.0, "duration": 4.879, "text": "4 over 3 and these are the solutions"}, {"start": 229.319, "duration": 3.56, "text": "thank you for watching please kindly"}, {"start": 230.879, "duration": 6.041, "text": "subscribe to my channel also like"}, {"start": 232.879, "duration": 4.041, "text": "comment and share bye-bye"}], "title": "Germany | Olympiad Math | Interesting exponential equation | #maths #mathematics", "description": "", "lengthSeconds": 235, "width": 554, "height": 360}
{"video_id": "G3FTtBCW0DI", "transcript": [{"start": 0.399, "duration": 6.641, "text": "hello good day everyone welcome back"}, {"start": 3.28, "duration": 8.92, "text": "once again to my YouTube Channel please"}, {"start": 7.04, "duration": 7.92, "text": "kindly support my channel by"}, {"start": 12.2, "duration": 5.76, "text": "subscribing please subscribe to this"}, {"start": 14.96, "duration": 4.84, "text": "YouTube channel and share my videos"}, {"start": 17.96, "duration": 6.399, "text": "subscribe to this Channel please and"}, {"start": 19.8, "duration": 7.559, "text": "share my videos here we are giving this"}, {"start": 24.359, "duration": 6.201, "text": "radical equation right we have the"}, {"start": 27.359, "duration": 6.321, "text": "square root of otk of X +qu root of X is"}, {"start": 30.56, "duration": 7.08, "text": "equal to otk of X right so we are asked"}, {"start": 33.68, "duration": 7.68, "text": "to find the value or values of X so here"}, {"start": 37.64, "duration": 5.36, "text": "we have the equation the sare root of <"}, {"start": 41.36, "duration": 7.679, "text": "TK x"}, {"start": 43.0, "duration": 9.48, "text": "+ < TK X is = to < TK of X right so"}, {"start": 49.039, "duration": 6.881, "text": "obviously we can see our tral solution"}, {"start": 52.48, "duration": 6.84, "text": "that X is equal to 0 because Zer satisfy"}, {"start": 55.92, "duration": 6.319, "text": "this equation right but let us go ahead"}, {"start": 59.32, "duration": 5.36, "text": "and work it out okay now the first thing"}, {"start": 62.239, "duration": 7.441, "text": "I'm going to do here I'm going to"}, {"start": 64.68, "duration": 11.479, "text": "let the sare root of x to be equals K"}, {"start": 69.68, "duration": 9.479, "text": "right so right here I will have X = kqu"}, {"start": 76.159, "duration": 6.521, "text": "of two if you square both sides right so"}, {"start": 79.159, "duration": 9.96, "text": "here I have the square root of you know"}, {"start": 82.68, "duration": 10.32, "text": "K plus K is equal to K right so here we"}, {"start": 89.119, "duration": 7.721, "text": "have the square root root of 2 K is"}, {"start": 93.0, "duration": 6.96, "text": "equal to K so I'm going to go ahead and"}, {"start": 96.84, "duration": 7.239, "text": "square both sides of this equation so"}, {"start": 99.96, "duration": 6.64, "text": "here I have this squ is equal to this"}, {"start": 104.079, "duration": 6.921, "text": "squ so we have to eliminate this root"}, {"start": 106.6, "duration": 9.4, "text": "sign and this way so here we have 2 K"}, {"start": 111.0, "duration": 6.92, "text": "isal to K of 2 so you know we can make"}, {"start": 116.0, "duration": 4.84, "text": "the right hand side to be the left hand"}, {"start": 117.92, "duration": 5.799, "text": "side and left hand side to be the right"}, {"start": 120.84, "duration": 6.639, "text": "hand side so here we have K2 is exactly"}, {"start": 123.719, "duration": 8.081, "text": "equal 2 K so now we're going to subtract"}, {"start": 127.479, "duration": 8.521, "text": "2 K from both sides so here we have k^ 2"}, {"start": 131.8, "duration": 7.4, "text": "- 2 K is equal 0 so the common factor"}, {"start": 136.0, "duration": 10.0, "text": "here is K so we Factor it out K Open"}, {"start": 139.2, "duration": 6.8, "text": "Bracket K - 2 is equal to 0 right"}, {"start": 146.319, "duration": 15.761, "text": "so here we have k isal 0 or k - 2 = 0 so"}, {"start": 154.92, "duration": 9.399, "text": "we have here K = 0 or k isal 2 right so"}, {"start": 162.08, "duration": 6.72, "text": "from here we can go ahead and find the"}, {"start": 164.319, "duration": 11.84, "text": "values of X so here we have when"}, {"start": 168.8, "duration": 10.799, "text": "K is equal to0 x = to k 2 which is 0 S"}, {"start": 176.159, "duration": 4.201, "text": "and that is 0 so this is our false"}, {"start": 179.599, "duration": 3.0, "text": "solution"}, {"start": 180.36, "duration": 10.4, "text": "solution so"}, {"start": 182.599, "duration": 10.801, "text": "when k = 2 x = k 2 which is equal to 2 2"}, {"start": 190.76, "duration": 4.64, "text": "and this is equal to 4 and this is our"}, {"start": 193.4, "duration": 6.119, "text": "second solution so we can see that the"}, {"start": 195.4, "duration": 6.24, "text": "equation has two real solutions Z and"}, {"start": 199.519, "duration": 6.28, "text": "four thank you for watching do wa to"}, {"start": 201.64, "duration": 4.159, "text": "subscribe and share our videos"}], "title": "An interesting radical equation.", "description": "", "lengthSeconds": 204, "width": 636, "height": 360}
{"video_id": "tTPVxhBmDoc", "transcript": [{"start": 1.48, "duration": 4.64, "text": "hello welcome back once again today"}, {"start": 3.159, "duration": 5.721, "text": "we're going to solve this nice Olympiad"}, {"start": 6.12, "duration": 7.559, "text": "problem we have this exponential"}, {"start": 8.88, "duration": 9.319, "text": "equation 3 to^ < TK of X this is equal"}, {"start": 13.679, "duration": 6.921, "text": "to 27 to the of"}, {"start": 18.199, "duration": 5.24, "text": "X we know that from the right hand side"}, {"start": 20.6, "duration": 6.519, "text": "27 can be expressed in index power of"}, {"start": 23.439, "duration": 6.68, "text": "three we know this is equal to 3 * 3 * 3"}, {"start": 27.119, "duration": 6.241, "text": "in three places which is equal to 3 to"}, {"start": 30.119, "duration": 7.081, "text": "the^ 3 so from the right from the left"}, {"start": 33.36, "duration": 9.199, "text": "hand side we have 3 to the^ < TK X this"}, {"start": 37.2, "duration": 9.039, "text": "is equal to so 27 is 3 cubed then ra of"}, {"start": 42.559, "duration": 7.881, "text": "X so from here we have 3 to the^ sare ro"}, {"start": 46.239, "duration": 9.48, "text": "of X this is equal to we know that 3 * X"}, {"start": 50.44, "duration": 8.0, "text": "is 3 x so we get here 3^ 3 x we can see"}, {"start": 55.719, "duration": 4.601, "text": "we have the same denominators from both"}, {"start": 58.44, "duration": 4.88, "text": "sides so we're going to equate the"}, {"start": 60.32, "duration": 6.96, "text": "powers so this implies from here that"}, {"start": 63.32, "duration": 8.24, "text": "theun of x isal to"}, {"start": 67.28, "duration": 8.32, "text": "3x so Square both sides so we get S of X"}, {"start": 71.56, "duration": 6.52, "text": "is equal to 3x and we're going to square"}, {"start": 75.6, "duration": 4.32, "text": "both sides so from the left hand side"}, {"start": 78.08, "duration": 6.359, "text": "Square will cancel square roots we have"}, {"start": 79.92, "duration": 7.239, "text": "X is = to 3x2 is 9"}, {"start": 84.439, "duration": 5.801, "text": "x\u00b2 so let's change the sides of this"}, {"start": 87.159, "duration": 6.761, "text": "equation so we have this the same as 9 x"}, {"start": 90.24, "duration": 5.32, "text": "is = to X so here subtract X from both"}, {"start": 93.92, "duration": 6.28, "text": "sides so we get 9"}, {"start": 95.56, "duration": 7.64, "text": "X2 - x this is equal to Z we can see X"}, {"start": 100.2, "duration": 9.559, "text": "is common here so Factor it out and we"}, {"start": 103.2, "duration": 13.36, "text": "have 9x - 1 is equal Z so from here this"}, {"start": 109.759, "duration": 10.481, "text": "implies that X is equal 0 or 9x - 1 is ="}, {"start": 116.56, "duration": 7.28, "text": "0 add one to both sides we get 9x is ="}, {"start": 120.24, "duration": 8.6, "text": "to 1 and divide by the coefficient of"}, {"start": 123.84, "duration": 8.88, "text": "x so 9 cancel 9 and here we have X is ="}, {"start": 128.84, "duration": 10.88, "text": "1 / 9 so here we have our"}, {"start": 132.72, "duration": 9.64, "text": "solution x is = 0 and x is equal to 1 /"}, {"start": 139.72, "duration": 5.12, "text": "9 thank you for watching please kindly"}, {"start": 142.36, "duration": 7.0, "text": "subscribe to my channel also like"}, {"start": 144.84, "duration": 4.52, "text": "comment and share bye-bye"}], "title": "Interesting exponential equation | can you solve ?", "description": "#maths #mathematics #education #puzzle #learning", "lengthSeconds": 148, "width": 428, "height": 240}
{"video_id": "EOPQljyvUQk", "transcript": [{"start": 0.919, "duration": 4.88, "text": "hello good viewers welcome back once"}, {"start": 3.4, "duration": 4.439, "text": "again today we're going to solve for the"}, {"start": 5.799, "duration": 3.561, "text": "real value of x in the following"}, {"start": 7.839, "duration": 3.8, "text": "exponential"}, {"start": 9.36, "duration": 6.159, "text": "equation notice that this equation is an"}, {"start": 11.639, "duration": 8.64, "text": "Olympian math problem we have here x^ of"}, {"start": 15.519, "duration": 8.84, "text": "X the^ of 3 this is equal to 36 here let"}, {"start": 20.279, "duration": 11.241, "text": "us solve for the value of x we have here"}, {"start": 24.359, "duration": 11.001, "text": "x power of x^ 3 this is equal to 3 six"}, {"start": 31.52, "duration": 6.359, "text": "right now the idea from here is to raise"}, {"start": 35.36, "duration": 7.719, "text": "both sides of this equation to the power"}, {"start": 37.879, "duration": 9.041, "text": "of three right now a very smart question"}, {"start": 43.079, "duration": 6.361, "text": "here we come in this form why did I"}, {"start": 46.92, "duration": 7.319, "text": "raise both sides of this equation to the"}, {"start": 49.44, "duration": 7.279, "text": "power of three or why three not any"}, {"start": 54.239, "duration": 6.201, "text": "other number"}, {"start": 56.719, "duration": 5.68, "text": "right now anytime you encounter this"}, {"start": 60.44, "duration": 4.399, "text": "type of"}, {"start": 62.399, "duration": 5.241, "text": "question this is how you will start"}, {"start": 64.839, "duration": 4.361, "text": "solving it and to raise both sides to"}, {"start": 67.64, "duration": 5.64, "text": "the power of any"}, {"start": 69.2, "duration": 8.16, "text": "number we are considering the power of X"}, {"start": 73.28, "duration": 7.479, "text": "right over here for example if we have X"}, {"start": 77.36, "duration": 5.799, "text": "power of X let's say power of n let's"}, {"start": 80.759, "duration": 5.561, "text": "say this is for general and this is"}, {"start": 83.159, "duration": 5.161, "text": "equal to let's say capital letter N we"}, {"start": 86.32, "duration": 4.72, "text": "are going to raise both side to the"}, {"start": 88.32, "duration": 5.64, "text": "power of M that is is the power of this"}, {"start": 91.04, "duration": 2.92, "text": "x right over"}, {"start": 94.439, "duration": 7.121, "text": "here okay now from here we all know this"}, {"start": 98.96, "duration": 7.28, "text": "property is a very nice property by the"}, {"start": 101.56, "duration": 9.96, "text": "way a to the^ of B of C this is equal to"}, {"start": 106.24, "duration": 8.12, "text": "a of B * C we can still interchange the"}, {"start": 111.52, "duration": 6.36, "text": "powers as a to the power of C then power"}, {"start": 114.36, "duration": 6.039, "text": "of B now from the left hand side here we"}, {"start": 117.88, "duration": 6.12, "text": "have X power of"}, {"start": 120.399, "duration": 6.36, "text": "x^ 3 *"}, {"start": 124.0, "duration": 9.28, "text": "3 and this is equal to in the bracket"}, {"start": 126.759, "duration": 10.041, "text": "here 36 is 6 2 then power of 3 from here"}, {"start": 133.28, "duration": 6.28, "text": "we have here x Cub * 3 as the power of X"}, {"start": 136.8, "duration": 8.04, "text": "right over here now we can interchange"}, {"start": 139.56, "duration": 11.2, "text": "the power we have here x power of 3 then"}, {"start": 144.84, "duration": 7.16, "text": "x^ of 3 this is equal to 6 of 2 * 3"}, {"start": 150.76, "duration": 4.28, "text": "which is"}, {"start": 152.0, "duration": 6.959, "text": "6 so from here this"}, {"start": 155.04, "duration": 7.16, "text": "implies that X Cub is equal to 6 we can"}, {"start": 158.959, "duration": 5.36, "text": "see the base and we can see the powers"}, {"start": 162.2, "duration": 5.72, "text": "the bases and the powers corresponds"}, {"start": 164.319, "duration": 6.961, "text": "right X cubed as the base here six as"}, {"start": 167.92, "duration": 6.36, "text": "the base here x Cub as the power here"}, {"start": 171.28, "duration": 7.84, "text": "and six as the power here so here we"}, {"start": 174.28, "duration": 7.44, "text": "have x 3 this is equal to 6 so if you"}, {"start": 179.12, "duration": 6.039, "text": "take the CU cube root of both sides we"}, {"start": 181.72, "duration": 6.599, "text": "get here x is equal to the cube root of"}, {"start": 185.159, "duration": 4.961, "text": "6 and this is the right answer thank you"}, {"start": 188.319, "duration": 4.441, "text": "for watching please kindly subscribe to"}, {"start": 190.12, "duration": 5.64, "text": "my channel also like comment and share"}, {"start": 192.76, "duration": 3.0, "text": "bye-bye"}], "title": "Koreans | Olympiad Math | Interesting exponential equation | #maths #mathematics", "description": "", "lengthSeconds": 195, "width": 640, "height": 360}
{"video_id": "JDUvRciejgs", "transcript": [{"start": 1.0, "duration": 4.48, "text": "hello good day everyone welcome back"}, {"start": 3.24, "duration": 5.319, "text": "once again today we're going to solve"}, {"start": 5.48, "duration": 7.64, "text": "the following given Quantic equation we"}, {"start": 8.559, "duration": 8.64, "text": "have m^ 4 = 5 here we're going to find"}, {"start": 13.12, "duration": 8.44, "text": "both the real and complex values of M"}, {"start": 17.199, "duration": 8.08, "text": "what if m is imaginary we can also find"}, {"start": 21.56, "duration": 6.639, "text": "that now let's get started for the left"}, {"start": 25.279, "duration": 8.321, "text": "hand side we can write that as M to the^"}, {"start": 28.199, "duration": 7.761, "text": "of 2 then of two you know why this 2 *"}, {"start": 33.6, "duration": 3.959, "text": "this 2 will give us four and let's"}, {"start": 35.96, "duration": 5.04, "text": "subtract"}, {"start": 37.559, "duration": 8.401, "text": "5 and that will be equal to zero now"}, {"start": 41.0, "duration": 7.96, "text": "take note of this we can write a as the"}, {"start": 45.96, "duration": 5.84, "text": "square root of a in this form then"}, {"start": 48.96, "duration": 5.239, "text": "squared so five here we can rewrite it"}, {"start": 51.8, "duration": 3.52, "text": "as square root of 5 squ right so here we"}, {"start": 54.199, "duration": 7.401, "text": "get"}, {"start": 55.32, "duration": 9.519, "text": "M2 squ then minus here root of 5 then"}, {"start": 61.6, "duration": 5.92, "text": "squar and this is equal to zero the"}, {"start": 64.839, "duration": 6.121, "text": "reason why I'm doing this is so that we"}, {"start": 67.52, "duration": 6.959, "text": "can factorize this equation if we recall"}, {"start": 70.96, "duration": 10.04, "text": "that this is just the difference of two"}, {"start": 74.479, "duration": 9.921, "text": "squares which is equal to a + b * Aus"}, {"start": 81.0, "duration": 5.4, "text": "B with this idea we can factorize this"}, {"start": 84.4, "duration": 4.52, "text": "expression at the left hand side of this"}, {"start": 86.4, "duration": 11.48, "text": "equation which is going to be"}, {"start": 88.92, "duration": 12.519, "text": "m\u00b2 + < TK 5 * m 2 - < TK 5 and the whole"}, {"start": 97.88, "duration": 6.04, "text": "of this equals z so from here this"}, {"start": 101.439, "duration": 11.441, "text": "implies that we have"}, {"start": 103.92, "duration": 13.159, "text": "M2 + < tk5 = 0 and we have m^2 - < TK 5"}, {"start": 112.88, "duration": 8.36, "text": "= 0 from here take positive root 5 to"}, {"start": 117.079, "duration": 8.121, "text": "the other side we get m^2 ="}, {"start": 121.24, "duration": 7.28, "text": "root of 5 and from here take root 5 to"}, {"start": 125.2, "duration": 5.48, "text": "the other side that will become posun 5"}, {"start": 128.52, "duration": 5.359, "text": "right and take"}, {"start": 130.68, "duration": 7.16, "text": "note when we are looking for the square"}, {"start": 133.879, "duration": 7.401, "text": "root of a square root of a number a this"}, {"start": 137.84, "duration": 7.039, "text": "is same as the fourth root of"}, {"start": 141.28, "duration": 7.319, "text": "a right so from here let's take root of"}, {"start": 144.879, "duration": 5.561, "text": "both side we get M = plus or minus"}, {"start": 148.599, "duration": 7.241, "text": "square root of NE"}, {"start": 150.44, "duration": 7.4, "text": "5 and from here M = plus or minus < TK"}, {"start": 155.84, "duration": 5.84, "text": "of square < TK of"}, {"start": 157.84, "duration": 9.8, "text": "5 from here we can actually rewrite this"}, {"start": 161.68, "duration": 9.12, "text": "as plus or minus < TK of -1 MTI root of"}, {"start": 167.64, "duration": 6.959, "text": "5 from here we already have plus or"}, {"start": 170.8, "duration": 5.88, "text": "minus the fourth root of 5 which is a"}, {"start": 174.599, "duration": 6.201, "text": "real answer"}, {"start": 176.68, "duration": 10.839, "text": "right so from here we get M = plus minor"}, {"start": 180.8, "duration": 9.28, "text": "otk of -1 is I and ofun 5 is 4 root of 5"}, {"start": 187.519, "duration": 6.241, "text": "so here we come up with the solutions we"}, {"start": 190.08, "duration": 7.68, "text": "have the imaginary Solutions Plus I 4ot"}, {"start": 193.76, "duration": 6.479, "text": "of 5 and I 4 root of 5 and here the real"}, {"start": 197.76, "duration": 5.199, "text": "solutions here plus or minus the four"}, {"start": 200.239, "duration": 4.92, "text": "root of 5 and these are the right"}, {"start": 202.959, "duration": 4.801, "text": "answers thank you for watching please"}, {"start": 205.159, "duration": 6.601, "text": "kindly subscribe to my channel also like"}, {"start": 207.76, "duration": 4.0, "text": "comment and share bye-bye"}], "title": "Nice quartic equation", "description": "", "lengthSeconds": 210, "width": 544, "height": 360}
{"video_id": "IKIJkAMf-uw", "transcript": [{"start": 0.96, "duration": 4.319, "text": "good day"}, {"start": 2.28, "duration": 6.92, "text": "everyone welcome back to my Facebook"}, {"start": 5.279, "duration": 6.201, "text": "page this is chid online M class today"}, {"start": 9.2, "duration": 5.359, "text": "we are giv the following sodic"}, {"start": 11.48, "duration": 8.76, "text": "expression we are asked to evaluate"}, {"start": 14.559, "duration": 5.681, "text": "this okay let's see how to solve this"}, {"start": 20.279, "duration": 4.84, "text": "problem"}, {"start": 22.279, "duration": 8.24, "text": "now this is in bracket we have the first"}, {"start": 25.119, "duration": 7.521, "text": "one square < TK of 448 + < TK of 2024"}, {"start": 30.519, "duration": 5.401, "text": "ultip by < TK of"}, {"start": 32.64, "duration": 8.919, "text": "448 - < TK of"}, {"start": 35.92, "duration": 7.68, "text": "2024 so if we try to expand this like"}, {"start": 41.559, "duration": 3.0, "text": "multiplying whatever that is in the"}, {"start": 43.6, "duration": 5.68, "text": "second"}, {"start": 44.559, "duration": 8.8, "text": "bracket by this plus multiplying Again"}, {"start": 49.28, "duration": 7.72, "text": "by this that will take a little bit time"}, {"start": 53.359, "duration": 7.601, "text": "right okay now let's see the tricky way"}, {"start": 57.0, "duration": 6.519, "text": "to solve this problem now we will be"}, {"start": 60.96, "duration": 6.36, "text": "making use of the difference of two"}, {"start": 63.519, "duration": 10.361, "text": "squares formula right so here we"}, {"start": 67.32, "duration": 14.439, "text": "go so we have that A + B * Aus B is = to"}, {"start": 73.88, "duration": 9.879, "text": "a 2 - b 2 right okay now if we carefully"}, {"start": 81.759, "duration": 5.32, "text": "look at the left hand side of this"}, {"start": 83.759, "duration": 5.961, "text": "equation so we can see that it look very"}, {"start": 87.079, "duration": 3.36, "text": "similar to what we have to evaluate"}, {"start": 89.72, "duration": 3.48, "text": "right"}, {"start": 90.439, "duration": 6.36, "text": "so here we have a + b and here we have"}, {"start": 93.2, "duration": 6.959, "text": "otk of 448 + < TK of"}, {"start": 96.799, "duration": 4.801, "text": "2024 here we have a minus B here we have"}, {"start": 100.159, "duration": 4.161, "text": "square < TK of"}, {"start": 101.6, "duration": 7.559, "text": "448 minus < TK of"}, {"start": 104.32, "duration": 9.56, "text": "2024 so which actually mean that this"}, {"start": 109.159, "duration": 6.841, "text": "expression is the same as here a s so we"}, {"start": 113.88, "duration": 6.48, "text": "have square root"}, {"start": 116.0, "duration": 10.719, "text": "of 4 0 48"}, {"start": 120.36, "duration": 9.84, "text": "squ right here minus b 2 here we"}, {"start": 126.719, "duration": 8.001, "text": "have square otk of"}, {"start": 130.2, "duration": 7.56, "text": "2024 squared right okay now this is"}, {"start": 134.72, "duration": 5.12, "text": "equal to here we all know that square"}, {"start": 137.76, "duration": 7.52, "text": "cancel out square root so we are left"}, {"start": 139.84, "duration": 8.0, "text": "with 4 0 48 minus here we have square"}, {"start": 145.28, "duration": 7.039, "text": "cancel square root so we are left to two"}, {"start": 147.84, "duration": 6.36, "text": "0 2 4 so if we subtract this we will"}, {"start": 152.319, "duration": 5.681, "text": "have"}, {"start": 154.2, "duration": 6.039, "text": "2024 and this is the right answer thank"}, {"start": 158.0, "duration": 5.12, "text": "you for watching please do wa to like"}, {"start": 160.239, "duration": 6.841, "text": "comment and share bye-bye seeing you in"}, {"start": 163.12, "duration": 3.96, "text": "our next video"}], "title": "surds.   Evaluation of surdic expression", "description": "", "lengthSeconds": 165, "width": 636, "height": 360}
{"video_id": "CdBhJdu3JAY", "transcript": [{"start": 0.68, "duration": 4.76, "text": "hello good day viewers welcome back once"}, {"start": 3.319, "duration": 4.48, "text": "again here I'm going to solve this"}, {"start": 5.44, "duration": 5.96, "text": "problem again using a different method"}, {"start": 7.799, "duration": 6.441, "text": "from the previous video I made"}, {"start": 11.4, "duration": 5.639, "text": "right in this video I'm going to make"}, {"start": 14.24, "duration": 6.48, "text": "use of the demov"}, {"start": 17.039, "duration": 6.281, "text": "theorem the problem here is x + 1 / X ="}, {"start": 20.72, "duration": 8.0, "text": "tok3 we are going to compute the value"}, {"start": 23.32, "duration": 9.079, "text": "of x power of 2025 + 1 /x to^ of 2025"}, {"start": 28.72, "duration": 5.8, "text": "right now now here we have x + 1 /x"}, {"start": 32.399, "duration": 5.721, "text": "right so we have a positive sign in"}, {"start": 34.52, "duration": 4.96, "text": "between these two terms right so which"}, {"start": 38.12, "duration": 4.119, "text": "actually mean that we're going to make"}, {"start": 39.48, "duration": 11.599, "text": "use of the following demov theorem the"}, {"start": 42.239, "duration": 14.081, "text": "one of x^ of n + 1 / x^ of n is = to 2 *"}, {"start": 51.079, "duration": 9.241, "text": "the cosine of n Theta right so remember"}, {"start": 56.32, "duration": 6.0, "text": "that the power of X here is one right so"}, {"start": 60.32, "duration": 11.68, "text": "here we're going to let"}, {"start": 62.32, "duration": 14.24, "text": "N = 1 and we get x + 1 /x is = to 2 *"}, {"start": 72.0, "duration": 8.079, "text": "the cosine of theta right now remember"}, {"start": 76.56, "duration": 6.879, "text": "from this equation here x + 1 / X ="}, {"start": 80.079, "duration": 6.72, "text": "root3 so here we replace this with root3"}, {"start": 83.439, "duration": 6.081, "text": "and that is equal to 2 * the cosine of"}, {"start": 86.799, "duration": 8.64, "text": "theta right so divide both side by two"}, {"start": 89.52, "duration": 10.72, "text": "we get cosine of theta = < tk3 / 2 and"}, {"start": 95.439, "duration": 10.72, "text": "from here we get Theta isal the are COS"}, {"start": 100.24, "duration": 8.64, "text": "of < tk3 / 2 and this is equal to < / 6"}, {"start": 106.159, "duration": 6.841, "text": "right so now the expression here we"}, {"start": 108.88, "duration": 10.199, "text": "asked to evaluate x^ of"}, {"start": 113.0, "duration": 7.64, "text": "2025 + 1 /x the^ of 2025 is going to be"}, {"start": 119.079, "duration": 4.881, "text": "equal to"}, {"start": 120.64, "duration": 8.2, "text": "you know here we already have Theta as"}, {"start": 123.96, "duration": 7.68, "text": "Pi all 6 right which means the general"}, {"start": 128.84, "duration": 10.16, "text": "formula for this problem here we have"}, {"start": 131.64, "duration": 8.28, "text": "gotten as x^ of n + 1 / x^ n is = 2 *"}, {"start": 139.0, "duration": 6.519, "text": "the"}, {"start": 139.92, "duration": 7.36, "text": "cine of n * < all 6 right so here we're"}, {"start": 145.519, "duration": 7.44, "text": "going to let"}, {"start": 147.28, "duration": 10.039, "text": "N = 2020 5 and from here we now"}, {"start": 152.959, "duration": 10.401, "text": "get 2 * the cosine of"}, {"start": 157.319, "duration": 8.56, "text": "2025 * < 6 and this is equal to Z and"}, {"start": 163.36, "duration": 4.72, "text": "that's the answer thank you for watching"}, {"start": 165.879, "duration": 7.041, "text": "please kindly subscribe to my channel"}, {"start": 168.08, "duration": 4.84, "text": "also like comment and share bye-bye"}], "title": "Application of Demoivre's Theorem.   #maths #education", "description": "", "lengthSeconds": 171, "width": 640, "height": 360}
{"video_id": "Y_i6G_qRgo4", "transcript": [{"start": 0.48, "duration": 3.32, "text": "hello good viewers today we're going to"}, {"start": 1.92, "duration": 9.0, "text": "find the square root of this expression"}, {"start": 3.8, "duration": 8.959, "text": "involving X and Y 6.25 x^2 + 5x y + y^ 2"}, {"start": 10.92, "duration": 4.36, "text": "but before we proceed you know I'm a fan"}, {"start": 12.759, "duration": 5.321, "text": "of algebraic identities there's an"}, {"start": 15.28, "duration": 5.8, "text": "identity we're going to make sure we use"}, {"start": 18.08, "duration": 7.279, "text": "here a + b"}, {"start": 21.08, "duration": 4.279, "text": "s this is equal to a"}, {"start": 26.16, "duration": 9.68, "text": "2+ 2 a b + b s so this is actually"}, {"start": 33.2, "duration": 4.679, "text": "telling us that we need to at least get"}, {"start": 35.84, "duration": 3.32, "text": "an expression that will look exactly"}, {"start": 37.879, "duration": 3.121, "text": "like the expression at the right hand"}, {"start": 39.16, "duration": 4.2, "text": "side of this equation so I will know"}, {"start": 41.0, "duration": 4.039, "text": "that the square root of that will be the"}, {"start": 43.36, "duration": 5.719, "text": "sum of A and B"}, {"start": 45.039, "duration": 7.0, "text": "squ corresponding to X and Y with their"}, {"start": 49.079, "duration": 5.441, "text": "coefficients right okay now this is what"}, {"start": 52.039, "duration": 3.761, "text": "we're going to do first of all observe"}, {"start": 54.52, "duration": 3.96, "text": "that"}, {"start": 55.8, "duration": 7.88, "text": "6.25 is equal to"}, {"start": 58.48, "duration": 10.44, "text": "625 all 100 right which reduces"}, {"start": 63.68, "duration": 8.799, "text": "to if you use 25 that is 25 all over 100"}, {"start": 68.92, "duration": 6.72, "text": "25 that is 4 and this is equal to 5 all"}, {"start": 72.479, "duration": 6.32, "text": "2 the whole of this s so for that we're"}, {"start": 75.64, "duration": 8.479, "text": "going to write we have the square root"}, {"start": 78.799, "duration": 8.761, "text": "of here we have 5 / 2 we can put we can"}, {"start": 84.119, "duration": 6.68, "text": "attach X and then we Square"}, {"start": 87.56, "duration": 6.72, "text": "everything then Plus"}, {"start": 90.799, "duration": 7.121, "text": "here we have to at least have the"}, {"start": 94.28, "duration": 5.92, "text": "coefficient of x and y as 5 2 but before"}, {"start": 97.92, "duration": 5.68, "text": "doing that we need to multiply by two"}, {"start": 100.2, "duration": 8.559, "text": "then we put the five as 5/ 2 then we"}, {"start": 103.6, "duration": 7.159, "text": "have X Y then for here we have y^ 2"}, {"start": 108.759, "duration": 6.4, "text": "right okay the reason for putting this"}, {"start": 110.759, "duration": 6.161, "text": "two * 5 over 2 we need to get something"}, {"start": 115.159, "duration": 4.481, "text": "very similar to what we have at the"}, {"start": 116.92, "duration": 5.0, "text": "right hand side and if you carefully"}, {"start": 119.64, "duration": 4.28, "text": "look at this which we have right over"}, {"start": 121.92, "duration": 4.199, "text": "here we have something squared and here"}, {"start": 123.92, "duration": 4.28, "text": "we have something squared plus 2 *"}, {"start": 126.119, "duration": 4.0, "text": "something and here we have plus 2 time"}, {"start": 128.2, "duration": 4.759, "text": "something then plus something squ and"}, {"start": 130.119, "duration": 4.76, "text": "here we have something squar so this is"}, {"start": 132.959, "duration": 6.761, "text": "actually telling us that this will be"}, {"start": 134.879, "duration": 9.881, "text": "the square root of now we pick the one"}, {"start": 139.72, "duration": 9.159, "text": "the first term here that is 5 over 2 x"}, {"start": 144.76, "duration": 6.8, "text": "then plus we pick the last 10 which is"}, {"start": 148.879, "duration": 5.241, "text": "squared which is Y and the whole of this"}, {"start": 151.56, "duration": 4.759, "text": "is squared we know what is going to"}, {"start": 154.12, "duration": 4.88, "text": "happen here so this is the absolute"}, {"start": 156.319, "duration": 4.401, "text": "value of this or we can easily just"}, {"start": 159.0, "duration": 3.44, "text": "cancel this since we have expression"}, {"start": 160.72, "duration": 6.879, "text": "right over there so this is going to"}, {"start": 162.44, "duration": 7.36, "text": "give us 5 all over 2 x + y and this is"}, {"start": 167.599, "duration": 3.961, "text": "the right answer thank you for watching"}, {"start": 169.8, "duration": 3.92, "text": "please kindly subscribe to my YouTube"}, {"start": 171.56, "duration": 5.16, "text": "channel also like comment and share"}, {"start": 173.72, "duration": 3.0, "text": "bye-bye"}], "title": "A square root problem | Can you find the square of this expression? | #maths #olympiad", "description": "", "lengthSeconds": 175, "width": 640, "height": 360}
{"video_id": "kX1AyTD-8oM", "transcript": [{"start": 0.68, "duration": 3.36, "text": "hello good viewers welcome back once"}, {"start": 2.32, "duration": 4.64, "text": "again today we're going to solve for the"}, {"start": 4.04, "duration": 5.519, "text": "values of X in the following given"}, {"start": 6.96, "duration": 4.839, "text": "exponential equation when we see this"}, {"start": 9.559, "duration": 6.04, "text": "type of equation all we need to think"}, {"start": 11.799, "duration": 5.961, "text": "about is to natural log both sides right"}, {"start": 15.599, "duration": 5.201, "text": "so let's go ahead and do that so from"}, {"start": 17.76, "duration": 5.88, "text": "the left we have the natural log of 5 ^"}, {"start": 20.8, "duration": 6.08, "text": "of X2 this will be equal to the natural"}, {"start": 23.64, "duration": 4.879, "text": "log of 7 to the^ of X now with the"}, {"start": 26.88, "duration": 4.719, "text": "property of logarithm you know the"}, {"start": 28.519, "duration": 6.481, "text": "natural log of a to the^ of B will be b"}, {"start": 31.599, "duration": 5.721, "text": "m the natural log of a right so here the"}, {"start": 35.0, "duration": 5.879, "text": "power of 5 here is X2 and that will"}, {"start": 37.32, "duration": 6.719, "text": "cross over so here we have X2 * the"}, {"start": 40.879, "duration": 6.36, "text": "natural log of 5 this will be equal to x"}, {"start": 44.039, "duration": 4.441, "text": "* the natural log of 7 now the term at"}, {"start": 47.239, "duration": 3.561, "text": "the right hand side let's move it to the"}, {"start": 48.48, "duration": 3.68, "text": "left hand side so we have x^2 * the"}, {"start": 50.8, "duration": 4.84, "text": "natural log of"}, {"start": 52.16, "duration": 7.039, "text": "5us x * the natural log of 7 this is"}, {"start": 55.64, "duration": 5.68, "text": "equal to Z so from the left we have a"}, {"start": 59.199, "duration": 4.121, "text": "common factor and that is X then we"}, {"start": 61.32, "duration": 3.28, "text": "factor that out we left at x * the"}, {"start": 63.32, "duration": 4.439, "text": "natural log of"}, {"start": 64.6, "duration": 6.159, "text": "5us the natural log of 7 and this is"}, {"start": 67.759, "duration": 7.04, "text": "equal to 0 so from here this implies"}, {"start": 70.759, "duration": 6.68, "text": "that X isal to 0 or from the bracket x *"}, {"start": 74.799, "duration": 6.64, "text": "the natural of 5 minus the natural log"}, {"start": 77.439, "duration": 6.921, "text": "of 7 is also equal to Z so add natural"}, {"start": 81.439, "duration": 5.081, "text": "log of 7 to the both sides so we get x *"}, {"start": 84.36, "duration": 5.2, "text": "the natural log of 5 this will be equal"}, {"start": 86.52, "duration": 4.639, "text": "to the natural log of 7 so remember"}, {"start": 89.56, "duration": 4.199, "text": "we're soling for X so we need to divide"}, {"start": 91.159, "duration": 5.361, "text": "both side by the coefficient of x which"}, {"start": 93.759, "duration": 5.441, "text": "is the natural log of five right by the"}, {"start": 96.52, "duration": 4.8, "text": "natural L of five so here this cancel"}, {"start": 99.2, "duration": 4.68, "text": "this and here we're left with xal the"}, {"start": 101.32, "duration": 5.56, "text": "natural log of 7 all over the natural L"}, {"start": 103.88, "duration": 5.48, "text": "of 5 and this is a solution but there's"}, {"start": 106.88, "duration": 4.519, "text": "something we need to recall recall that"}, {"start": 109.36, "duration": 5.119, "text": "when we have the natural log of a all"}, {"start": 111.399, "duration": 6.68, "text": "over the natural log of B this can be"}, {"start": 114.479, "duration": 7.841, "text": "written in in logarithmic form as the"}, {"start": 118.079, "duration": 6.121, "text": "log of a to the base B right so this"}, {"start": 122.32, "duration": 4.6, "text": "actually mean that our answer can be"}, {"start": 124.2, "duration": 8.399, "text": "written as the log of 7 to the base 5"}, {"start": 126.92, "duration": 11.36, "text": "also right so we have solution x = 0 and"}, {"start": 132.599, "duration": 7.961, "text": "x = log 7 to the base 5 thank you for"}, {"start": 138.28, "duration": 4.56, "text": "watching please kindly subscribe to my"}, {"start": 140.56, "duration": 5.679, "text": "channel thank you once again thank you"}, {"start": 142.84, "duration": 3.399, "text": "thank you bye-bye"}], "title": "Olympiad Math | Interesting exponential equation | #mathematics #maths", "description": "", "lengthSeconds": 145, "width": 580, "height": 360}
{"video_id": "Mo5fOf4xd2c", "transcript": [{"start": 0.599, "duration": 5.961, "text": "hello welcome back once again to my"}, {"start": 3.12, "duration": 6.24, "text": "YouTube Channel please and please"}, {"start": 6.56, "duration": 5.72, "text": "support this channel by subscribing to"}, {"start": 9.36, "duration": 6.16, "text": "this Channel please subscribe to my"}, {"start": 12.28, "duration": 6.44, "text": "YouTube channel and share my"}, {"start": 15.52, "duration": 9.28, "text": "videos right here we are given that A +"}, {"start": 18.72, "duration": 10.959, "text": "B = < tk2 A minus B = < tk3 we asked to"}, {"start": 24.8, "duration": 7.24, "text": "find the value of a * B in this solution"}, {"start": 29.679, "duration": 4.681, "text": "I'll making use of this"}, {"start": 32.04, "duration": 7.92, "text": "property we have this property in"}, {"start": 34.36, "duration": 7.32, "text": "algebra that a b is equal to the square"}, {"start": 39.96, "duration": 7.88, "text": "of A +"}, {"start": 41.68, "duration": 10.92, "text": "B minus the square of a minus B all"}, {"start": 47.84, "duration": 6.559, "text": "divide by 4 so if we Square this and"}, {"start": 52.6, "duration": 5.16, "text": "square this when we subtract them and"}, {"start": 54.399, "duration": 6.32, "text": "divide by 4 we are going to have a * B"}, {"start": 57.76, "duration": 5.599, "text": "right so we are going to make of this"}, {"start": 60.719, "duration": 7.641, "text": "identity right over here so in our"}, {"start": 63.359, "duration": 7.881, "text": "solution we have that a b is equal to"}, {"start": 68.36, "duration": 5.799, "text": "the square of a + b here we have a + b"}, {"start": 71.24, "duration": 7.919, "text": "is = < tk2 so here we have <"}, {"start": 74.159, "duration": 8.121, "text": "tk2 2 minus the square of aus B here we"}, {"start": 79.159, "duration": 8.681, "text": "have Aus B is = < tk3 so we have here <"}, {"start": 82.28, "duration": 8.08, "text": "tk3 2 all divide by four so this is"}, {"start": 87.84, "duration": 4.76, "text": "going to be no the square root and the"}, {"start": 90.36, "duration": 4.52, "text": "square cancel out we are left with 2"}, {"start": 92.6, "duration": 5.68, "text": "minus the square root and the square"}, {"start": 94.88, "duration": 8.76, "text": "cancel out we are left with 3 all over 4"}, {"start": 98.28, "duration": 8.08, "text": "you know 2 - 3 is = -1 all over 4 and"}, {"start": 103.64, "duration": 5.36, "text": "this is the right answer thank you for"}, {"start": 106.36, "duration": 6.079, "text": "watching please do well to subscribe to"}, {"start": 109.0, "duration": 6.439, "text": "my YouTube channel and share my"}, {"start": 112.439, "duration": 3.0, "text": "videos"}], "title": "An interesting problem. Watch out the way I solved this problem", "description": "", "lengthSeconds": 114, "width": 636, "height": 360}
{"video_id": "bfjlIZPNgng", "transcript": [{"start": 0.84, "duration": 5.6, "text": "hello good viewers welcome back once"}, {"start": 2.84, "duration": 7.4, "text": "again today we're going to simplify this"}, {"start": 6.44, "duration": 8.319, "text": "expression of number here we have 9 with"}, {"start": 10.24, "duration": 9.16, "text": "power of 49 and 48 right okay let's get"}, {"start": 14.759, "duration": 7.401, "text": "started we given 9 ^"}, {"start": 19.4, "duration": 7.119, "text": "49- 9 ^"}, {"start": 22.16, "duration": 8.599, "text": "48 / by 9^ of"}, {"start": 26.519, "duration": 9.521, "text": "48 okay now from here 9 to the^ of 49 we"}, {"start": 30.759, "duration": 11.761, "text": "can write it as 9 ^ 48 + 1 right then -"}, {"start": 36.04, "duration": 12.159, "text": "9 ^ 48 then divided by 9^ of"}, {"start": 42.52, "duration": 8.519, "text": "48 okay now 9^ of 48 + 1 we can actually"}, {"start": 48.199, "duration": 6.2, "text": "rewrite it right then we'll be making"}, {"start": 51.039, "duration": 7.601, "text": "use of this basic algebraic identity"}, {"start": 54.399, "duration": 6.12, "text": "when we have a^ of m + n this is same as"}, {"start": 58.64, "duration": 8.6, "text": "a^ of M"}, {"start": 60.519, "duration": 17.001, "text": "* ^ n right so 9^ 48 + 1 we can rewrite"}, {"start": 67.24, "duration": 14.16, "text": "it as 9^ 48 * 9^ 1 then- 9^ 48 divided"}, {"start": 77.52, "duration": 5.84, "text": "by 9^ 48 so we can see that from this"}, {"start": 81.4, "duration": 5.44, "text": "numerator here we have a common factor"}, {"start": 83.36, "duration": 6.079, "text": "of 9 to the^ of 48 then when we Factor"}, {"start": 86.84, "duration": 3.639, "text": "it out we are left with 9 ^ 1 here which"}, {"start": 89.439, "duration": 6.64, "text": "is 9"}, {"start": 90.479, "duration": 9.801, "text": "minus 9 48 9^ 8 we left with 1 here then"}, {"start": 96.079, "duration": 7.841, "text": "the whole of this ID 9 to the^ of 48 so"}, {"start": 100.28, "duration": 6.08, "text": "from here this and this we cancel right"}, {"start": 103.92, "duration": 4.879, "text": "and we are left"}, {"start": 106.36, "duration": 7.52, "text": "with and we're left"}, {"start": 108.799, "duration": 7.36, "text": "with 9 - 1 and that is equal to 8 and"}, {"start": 113.88, "duration": 4.159, "text": "this is the right answer thank you for"}, {"start": 116.159, "duration": 4.401, "text": "watching please kindly subscribe to my"}, {"start": 118.039, "duration": 5.521, "text": "channel also like comment and share"}, {"start": 120.56, "duration": 3.0, "text": "bye-bye"}], "title": "Can you find out the trick ? No calculators please !", "description": "#maths #learnhowtosolvetheexponentialequation #learning #mathematics #automobile", "lengthSeconds": 122, "width": 576, "height": 360}
{"video_id": "A-VLP8iZN24", "transcript": [{"start": 0.84, "duration": 5.959, "text": "hello welcome back once again today"}, {"start": 2.8, "duration": 8.28, "text": "we're given this crazy looking equation"}, {"start": 6.799, "duration": 7.121, "text": "x^ of aota is equal to Iota we know that"}, {"start": 11.08, "duration": 6.199, "text": "Iota according to Ila is equal to the"}, {"start": 13.92, "duration": 5.6, "text": "square root of1 right so here we're"}, {"start": 17.279, "duration": 4.121, "text": "going to solve for the value of x that"}, {"start": 19.52, "duration": 5.24, "text": "we satisfy this"}, {"start": 21.4, "duration": 7.08, "text": "equation Iota is the imaginary unit"}, {"start": 24.76, "duration": 5.839, "text": "right if you're new to it it is equal to"}, {"start": 28.48, "duration": 6.919, "text": "theare root of -1"}, {"start": 30.599, "duration": 8.761, "text": "but do not be surprised X is a real"}, {"start": 35.399, "duration": 7.32, "text": "number is that possible so let's see now"}, {"start": 39.36, "duration": 6.96, "text": "we need to take note of the following we"}, {"start": 42.719, "duration": 9.761, "text": "know Iota in exponential form is equal"}, {"start": 46.32, "duration": 9.52, "text": "to e to the power of Iota Pi / 2 so is a"}, {"start": 52.48, "duration": 8.12, "text": "very nice identity and again we're going"}, {"start": 55.84, "duration": 7.12, "text": "to use this logarithmic rule a to the"}, {"start": 60.6, "duration": 5.4, "text": "power of B in exponential form we can"}, {"start": 62.96, "duration": 8.159, "text": "write it as e to the power of the power"}, {"start": 66.0, "duration": 7.4, "text": "B MTI the Ln of the base a right so the"}, {"start": 71.119, "duration": 5.64, "text": "left hand side of the equation which is"}, {"start": 73.4, "duration": 6.56, "text": "X to the power of aota we can rewrite it"}, {"start": 76.759, "duration": 6.921, "text": "as e to the power of Iota time the"}, {"start": 79.96, "duration": 7.799, "text": "natural log of x and the right hand side"}, {"start": 83.68, "duration": 9.68, "text": "iot we're going to rewrite it as e to"}, {"start": 87.759, "duration": 11.201, "text": "the power of I pi ID 2 so the equation"}, {"start": 93.36, "duration": 10.52, "text": "x^ of I is = I we"}, {"start": 98.96, "duration": 10.92, "text": "become e to^ i * the natural log of x"}, {"start": 103.88, "duration": 8.199, "text": "this is equal to e to the of I Pi / 2 so"}, {"start": 109.88, "duration": 4.559, "text": "we can see we have obtained equal bases"}, {"start": 112.079, "duration": 5.64, "text": "from both sides so we're going to equate"}, {"start": 114.439, "duration": 7.401, "text": "the powers so this implies from here"}, {"start": 117.719, "duration": 5.08, "text": "that I * the natural log of x X is = to"}, {"start": 121.84, "duration": 5.559, "text": "I"}, {"start": 122.799, "duration": 7.841, "text": "Pi / 2 so we can see we can cancel I"}, {"start": 127.399, "duration": 9.48, "text": "from both sides and then we have the"}, {"start": 130.64, "duration": 10.08, "text": "natural of X is = to / 2 so make a"}, {"start": 136.879, "duration": 9.041, "text": "simple base of e we can see that X is"}, {"start": 140.72, "duration": 8.799, "text": "equal to e to the power of Pi / two wow"}, {"start": 145.92, "duration": 5.8, "text": "amazing so X is finally real and that's"}, {"start": 149.519, "duration": 7.481, "text": "is the correct answer so in the above"}, {"start": 151.72, "duration": 7.799, "text": "equation x = e to the power of PK / 2"}, {"start": 157.0, "duration": 4.28, "text": "and this is the correct answer thank you"}, {"start": 159.519, "duration": 5.161, "text": "for watching please kindly subscribe to"}, {"start": 161.28, "duration": 6.4, "text": "my channel also like comment and share"}, {"start": 164.68, "duration": 3.0, "text": "bye-bye"}], "title": "Oxford University \ud83c\udf93 Entrance Exam | Could you believe that x is real  ?", "description": "", "lengthSeconds": 166, "width": 428, "height": 240}
{"video_id": "-16tvb2fQSg", "transcript": [{"start": 0.44, "duration": 3.12, "text": "hello welcome back once again today"}, {"start": 2.08, "duration": 4.44, "text": "we're going to solve the following"}, {"start": 3.56, "duration": 6.119, "text": "radical equation we have here square"}, {"start": 6.52, "duration": 7.36, "text": "root of 3x which we can rewrite as 3x"}, {"start": 9.679, "duration": 8.321, "text": "all the^ 1 / 2 then this equals cube"}, {"start": 13.88, "duration": 7.44, "text": "root of 4X which we can rewrite as 4x^ 1"}, {"start": 18.0, "duration": 6.48, "text": "/ 3 now we're going to raise both sides"}, {"start": 21.32, "duration": 6.08, "text": "to the power of a particular number and"}, {"start": 24.48, "duration": 6.879, "text": "how can we figure out what the number is"}, {"start": 27.4, "duration": 7.08, "text": "the LM of 2 and 3 is 6 right so let's"}, {"start": 31.359, "duration": 9.761, "text": "raise both sides to the power of 6 so"}, {"start": 34.48, "duration": 8.399, "text": "3x^ 1 / 2 all ra to^ of 6 this equal 4X"}, {"start": 41.12, "duration": 7.32, "text": "^ 1 /"}, {"start": 42.879, "duration": 12.121, "text": "3 all power of 6 with this property a^ b"}, {"start": 48.44, "duration": 9.68, "text": "to^ c = a^ b * C we can multiply 1/2"}, {"start": 55.0, "duration": 6.32, "text": "with 6 and from there we get 3x in"}, {"start": 58.12, "duration": 7.28, "text": "bracket then 1/ 2 * 6 will give us 3"}, {"start": 61.32, "duration": 7.64, "text": "right then this equals here 4X in braet"}, {"start": 65.4, "duration": 9.079, "text": "1 3 * 6 that will give us"}, {"start": 68.96, "duration": 9.519, "text": "2 so from here 3x cubed will give us 27"}, {"start": 74.479, "duration": 6.64, "text": "x cubed right then this equals 4x2 will"}, {"start": 78.479, "duration": 5.28, "text": "give us 16"}, {"start": 81.119, "duration": 5.721, "text": "x^2 now let's move this term from the"}, {"start": 83.759, "duration": 8.561, "text": "right to the left so we have here 27 x"}, {"start": 86.84, "duration": 8.4, "text": "Cub - 16 x^2 this equals z from here we"}, {"start": 92.32, "duration": 3.92, "text": "have a common factor of x\u00b2 right so we"}, {"start": 95.24, "duration": 2.919, "text": "have here"}, {"start": 96.24, "duration": 6.72, "text": "27x -"}, {"start": 98.159, "duration": 11.841, "text": "16 this equal 0 so from here we have X2"}, {"start": 102.96, "duration": 9.32, "text": "= 0 and from here we have 27 x - 16 = 0"}, {"start": 110.0, "duration": 5.759, "text": "from here we take Ro of both side x is"}, {"start": 112.28, "duration": 7.28, "text": "exactly equal to 0 this is a solution"}, {"start": 115.759, "duration": 5.921, "text": "and from here move -6 to the right hand"}, {"start": 119.56, "duration": 8.039, "text": "side will become positive and divide by"}, {"start": 121.68, "duration": 8.079, "text": "27 we get x = 16 / 27 and these are the"}, {"start": 127.599, "duration": 4.561, "text": "solutions thank you for watching please"}, {"start": 129.759, "duration": 6.48, "text": "kindly subscribe to my channel also like"}, {"start": 132.16, "duration": 4.079, "text": "comment and share bye-bye"}], "title": "Interesting radical equation | Algebra problem", "description": "", "lengthSeconds": 134, "width": 556, "height": 360}
{"video_id": "Jm68zTIMwz0", "transcript": [{"start": 0.56, "duration": 5.64, "text": "hello welcome back to another video"}, {"start": 3.12, "duration": 6.519, "text": "today we're going to evaluate the root"}, {"start": 6.2, "duration": 7.0, "text": "of some irrational numbers right so we"}, {"start": 9.639, "duration": 6.361, "text": "have rooot of 2 + < tk3 + root of 2 - <"}, {"start": 13.2, "duration": 5.96, "text": "tk3 and here we're going to evaluate"}, {"start": 16.0, "duration": 6.92, "text": "this without calculators right now let's"}, {"start": 19.16, "duration": 3.76, "text": "start with this we're going to"}, {"start": 23.16, "duration": 8.68, "text": "let"}, {"start": 24.76, "duration": 12.68, "text": "t equal the expression > 2 + < tk3"}, {"start": 31.84, "duration": 7.8, "text": "+ < TK 2 - < tk3 right okay we need to"}, {"start": 37.44, "duration": 4.4, "text": "be very careful because we're expecting"}, {"start": 39.64, "duration": 5.2, "text": "T to be greater than zero"}, {"start": 41.84, "duration": 6.12, "text": "right okay now from here let's go ahead"}, {"start": 44.84, "duration": 8.64, "text": "and square both sides so we get t^ 2"}, {"start": 47.96, "duration": 11.079, "text": "will be equal to < TK 2 + < tk3"}, {"start": 53.48, "duration": 10.24, "text": "+un 2 - < tk3 all 2 and from the right"}, {"start": 59.039, "duration": 12.241, "text": "hand side let us take note of this a + b"}, {"start": 63.72, "duration": 10.88, "text": "2 this is equal to a 2 + b 2 + 2 * a * B"}, {"start": 71.28, "duration": 6.4, "text": "right so from here we get t^ 2 will be"}, {"start": 74.6, "duration": 7.0, "text": "equal to the first term here squared"}, {"start": 77.68, "duration": 6.68, "text": "that is otk of 2 + < tk3"}, {"start": 81.6, "duration": 7.08, "text": "squar then plus the second term here"}, {"start": 84.36, "duration": 8.759, "text": "squared which is otk of 2 - < tk3 2 then"}, {"start": 88.68, "duration": 6.88, "text": "we have + 2 2 * the first term < TK 2 +"}, {"start": 93.119, "duration": 6.36, "text": "< tk3 multip by the second term in the"}, {"start": 95.56, "duration": 7.199, "text": "bracket < TK 2 -"}, {"start": 99.479, "duration": 5.401, "text": "root3 okay so from here we have t^2 is"}, {"start": 102.759, "duration": 4.121, "text": "equal to remember this Square we cancel"}, {"start": 104.88, "duration": 6.4, "text": "this Roots this Square will cancel this"}, {"start": 106.88, "duration": 10.159, "text": "root sign so we have 2 + < tk3 plus here"}, {"start": 111.28, "duration": 9.4, "text": "2 - < tk3 then plus 2 multiply by"}, {"start": 117.039, "duration": 5.68, "text": "remember this are under the same r signs"}, {"start": 120.68, "duration": 6.32, "text": "right so we can actually write that as"}, {"start": 122.719, "duration": 8.921, "text": "root of 2 +3 * 2 - < tk3 remember this"}, {"start": 127.0, "duration": 9.64, "text": "is difference of 2 squares right recall"}, {"start": 131.64, "duration": 8.879, "text": "that A + B * a minus B this is equal to"}, {"start": 136.64, "duration": 6.08, "text": "a 2us b 2 right and we can see we have"}, {"start": 140.519, "duration": 4.681, "text": "the only difference here this plus sign"}, {"start": 142.72, "duration": 5.12, "text": "and this negative sign so to multiply"}, {"start": 145.2, "duration": 5.2, "text": "this and this together that would be 2^"}, {"start": 147.84, "duration": 5.679, "text": "2us < tk3"}, {"start": 150.4, "duration": 6.199, "text": "squ okay so from here we have t^2 is"}, {"start": 153.519, "duration": 6.521, "text": "equal to this and this become Z right"}, {"start": 156.599, "duration": 8.36, "text": "root3 - root3 is 0 then we have 2 + 2"}, {"start": 160.04, "duration": 7.72, "text": "which is 4 then + 2 now from here this"}, {"start": 164.959, "duration": 6.961, "text": "is 4 - 3 because this Square will cancel"}, {"start": 167.76, "duration": 8.64, "text": "this root sign and 2 2 is 4 4 - 3 is 1"}, {"start": 171.92, "duration": 7.56, "text": "and root of 1 is 1 2 * 1 is 2 so we can"}, {"start": 176.4, "duration": 6.08, "text": "see here we have 2 2 = 4 + 2 which is ="}, {"start": 179.48, "duration": 6.319, "text": "6 now take root of both side remember T"}, {"start": 182.48, "duration": 6.36, "text": "is greater than Z so no need for plus or"}, {"start": 185.799, "duration": 6.241, "text": "minus we only take the positive square"}, {"start": 188.84, "duration": 7.399, "text": "root so we have t is equal to square otk"}, {"start": 192.04, "duration": 8.4, "text": "of 6 and this is the right answer so we"}, {"start": 196.239, "duration": 11.72, "text": "cannot say that the square < TK of 2 + <"}, {"start": 200.44, "duration": 10.079, "text": "tk3 + S < TK of 2 - < tk3 is = to < TK 6"}, {"start": 207.959, "duration": 4.56, "text": "and this is the right answer thank you"}, {"start": 210.519, "duration": 4.72, "text": "for watching please kindly subscribe to"}, {"start": 212.519, "duration": 5.72, "text": "my channel also like comment and share"}, {"start": 215.239, "duration": 3.0, "text": "bye-bye"}], "title": "Interesting radical expression | Can you Denest ?", "description": "", "lengthSeconds": 217, "width": 640, "height": 360}
{"video_id": "oGlK6o1FmwM", "transcript": [{"start": 0.64, "duration": 4.36, "text": "hello good viewers welcome back once"}, {"start": 2.679, "duration": 5.001, "text": "again please kindly subscribe to my"}, {"start": 5.0, "duration": 4.88, "text": "channel also like comment and share"}, {"start": 7.68, "duration": 6.039, "text": "today we are given the following nested"}, {"start": 9.88, "duration": 6.92, "text": "radical equation x is = the < TK 12 +"}, {"start": 13.719, "duration": 6.161, "text": "the < TK of 12 + the < TK of 12+ and so"}, {"start": 16.8, "duration": 5.559, "text": "on here we're going to compute the exact"}, {"start": 19.88, "duration": 6.28, "text": "value of x"}, {"start": 22.359, "duration": 7.68, "text": "right okay let's get started we have X"}, {"start": 26.16, "duration": 8.28, "text": "is = to the square < TK of 12 + this"}, {"start": 30.039, "duration": 9.801, "text": "sare < TK of 12 + the < TK of 12"}, {"start": 34.44, "duration": 8.0, "text": "+ and so on right so here we have X is"}, {"start": 39.84, "duration": 3.84, "text": "equal to the square root of you know"}, {"start": 42.44, "duration": 5.2, "text": "this is"}, {"start": 43.68, "duration": 7.28, "text": "12+ now carefully look at this this is"}, {"start": 47.64, "duration": 6.12, "text": "exactly equal to X since X is equal otk"}, {"start": 50.96, "duration": 5.599, "text": "of 12 +un of 12 + the < TK of 12 and so"}, {"start": 53.76, "duration": 6.24, "text": "on and from here we have the < TK of 12"}, {"start": 56.559, "duration": 7.121, "text": "+ the < TK of 12 plus and so on so we"}, {"start": 60.0, "duration": 5.799, "text": "replace that with X now to get rid of"}, {"start": 63.68, "duration": 5.439, "text": "this root sign we are going to square"}, {"start": 65.799, "duration": 4.841, "text": "both sides of this equation right so we"}, {"start": 69.119, "duration": 3.601, "text": "know from the right the root and the"}, {"start": 70.64, "duration": 8.64, "text": "square will cancel out so here we have"}, {"start": 72.72, "duration": 9.759, "text": "X2 is = 12 + x so here we're going to"}, {"start": 79.28, "duration": 6.6, "text": "move the whole term from the right to"}, {"start": 82.479, "duration": 8.041, "text": "the left so we're going to have X2 here"}, {"start": 85.88, "duration": 7.44, "text": "- x - 12 this is equal to 0"}, {"start": 90.52, "duration": 4.48, "text": "so we need to bear this in mind that X"}, {"start": 93.32, "duration": 5.2, "text": "must be greater than zero because we're"}, {"start": 95.0, "duration": 6.079, "text": "dealing with Ned radical that is root"}, {"start": 98.52, "duration": 4.32, "text": "sign right so X is going to be greater"}, {"start": 101.079, "duration": 4.04, "text": "than zero"}, {"start": 102.84, "duration": 4.959, "text": "right so from"}, {"start": 105.119, "duration": 4.921, "text": "here we are going to use the general"}, {"start": 107.799, "duration": 4.481, "text": "quadratic formula when we have the"}, {"start": 110.04, "duration": 4.0, "text": "general form of any quadratic equation a"}, {"start": 112.28, "duration": 7.32, "text": "X2 +"}, {"start": 114.04, "duration": 10.28, "text": "BX + C is equal to Z then X is given by"}, {"start": 119.6, "duration": 9.079, "text": "B plus or minus theun of b^ 2 - 4 a c"}, {"start": 124.32, "duration": 8.44, "text": "all 2 a right"}, {"start": 128.679, "duration": 7.601, "text": "okay so from here a is equal to coent of"}, {"start": 132.76, "duration": 6.24, "text": "x\u00b2 which is 1 B is equal to coent of X"}, {"start": 136.28, "duration": 6.08, "text": "which is -1 and C is equal to the"}, {"start": 139.0, "duration": 6.08, "text": "constant which is -12 so X is going to"}, {"start": 142.36, "duration": 2.72, "text": "be"}, {"start": 145.319, "duration": 11.081, "text": "B-1 plus or minus the root of B S1 s"}, {"start": 150.64, "duration": 10.239, "text": "- 4 * a which is 1 * C which is -12"}, {"start": 156.4, "duration": 7.559, "text": "right then over 2 * a which is 2 * 1 so"}, {"start": 160.879, "duration": 7.28, "text": "this is equal to here 1 plus or minus"}, {"start": 163.959, "duration": 8.721, "text": "theare root of 1 1 sare which is 1 here-"}, {"start": 168.159, "duration": 7.401, "text": "* minus we have plus 4 * 1 4 * 12 which"}, {"start": 172.68, "duration": 7.36, "text": "is 48 10 /"}, {"start": 175.56, "duration": 8.599, "text": "2 so from here we get X is = 1 + so"}, {"start": 180.04, "duration": 7.16, "text": "minus you know this is from here this is"}, {"start": 184.159, "duration": 7.281, "text": "equal to the root of 49 which is 7 right"}, {"start": 187.2, "duration": 10.319, "text": "so we get 7 / 2 so this is equal to 1 +"}, {"start": 191.44, "duration": 10.68, "text": "7 / 2 or 1 - 7 / 2 so this is equal to"}, {"start": 197.519, "duration": 9.161, "text": "you know 1 + 7 is 8 / 2 which is 4 or 1"}, {"start": 202.12, "duration": 7.28, "text": "- 7 -6 / 2 which is -3 we all know that"}, {"start": 206.68, "duration": 5.52, "text": "this value is not acceptable right since"}, {"start": 209.4, "duration": 6.119, "text": "X is is greater than Z here we go for"}, {"start": 212.2, "duration": 6.84, "text": "this value so here we can deduce that"}, {"start": 215.519, "duration": 7.681, "text": "the < TK of 12 + the < TK of 12 + the <"}, {"start": 219.04, "duration": 7.919, "text": "TK of 12 plus and so on which was equal"}, {"start": 223.2, "duration": 5.959, "text": "to X is exactly equal to 4 and this is"}, {"start": 226.959, "duration": 4.521, "text": "the right answer thank you for watching"}, {"start": 229.159, "duration": 7.08, "text": "please do wa to subscribe to my channel"}, {"start": 231.48, "duration": 4.759, "text": "also like comment and share bye-bye"}], "title": "An infinite nested radical equation. #maths #education", "description": "", "lengthSeconds": 234, "width": 640, "height": 360}
{"video_id": "KGFzvJCMDPM", "transcript": [{"start": 1.079, "duration": 4.081, "text": "hello good day viewers welcome back once"}, {"start": 2.8, "duration": 5.68, "text": "again let's solve the following given"}, {"start": 5.16, "duration": 8.08, "text": "quartic equation very quickly so we're"}, {"start": 8.48, "duration": 11.6, "text": "given that x ^ 4 - 9 is = 0 now we're"}, {"start": 13.24, "duration": 10.84, "text": "going to write x ^ 4 as x ^ 2 then s"}, {"start": 20.08, "duration": 6.4, "text": "then - 9 is the same as 3 2 and this is"}, {"start": 24.08, "duration": 4.84, "text": "equal to Z Now from the left hand side"}, {"start": 26.48, "duration": 5.639, "text": "of this equation we're going to make use"}, {"start": 28.92, "duration": 11.04, "text": "of the difference of two squ formula a 2"}, {"start": 32.119, "duration": 10.241, "text": "- b 2 is = A + B * a - b right so from"}, {"start": 39.96, "duration": 6.96, "text": "the left we have here"}, {"start": 42.36, "duration": 9.56, "text": "X2 + 3 then multiply by"}, {"start": 46.92, "duration": 11.479, "text": "X2 - 3 and this is equal to Z so from"}, {"start": 51.92, "duration": 8.76, "text": "here this implies that x\u00b2 + 3 is = 0 or"}, {"start": 58.399, "duration": 7.201, "text": "we have from here x\u00b2"}, {"start": 60.68, "duration": 9.68, "text": "- 3 is = 0 now from here subtract from"}, {"start": 65.6, "duration": 9.8, "text": "both sides we get x2 is = 0 - 3 will"}, {"start": 70.36, "duration": 7.0, "text": "give us -3 and from this side X2 is = 0"}, {"start": 75.4, "duration": 4.56, "text": "+ 3 which is"}, {"start": 77.36, "duration": 5.92, "text": "three from here let's take root of both"}, {"start": 79.96, "duration": 6.72, "text": "side we get X is equal to plus or minus"}, {"start": 83.28, "duration": 5.96, "text": "square otk of -3 and from here take root"}, {"start": 86.68, "duration": 6.28, "text": "of both side we get X isal to plus or"}, {"start": 89.24, "duration": 6.76, "text": "minus Square t of"}, {"start": 92.96, "duration": 7.04, "text": "3 now do not forget that the square root"}, {"start": 96.0, "duration": 7.079, "text": "of1 is I that is the imaginary unit"}, {"start": 100.0, "duration": 7.84, "text": "right so we have here plus or minus I"}, {"start": 103.079, "duration": 10.241, "text": "root of 3 so here we have Solutions we"}, {"start": 107.84, "duration": 7.8, "text": "have X1 equals here let's take the the"}, {"start": 113.32, "duration": 7.96, "text": "real roots for so we have <"}, {"start": 115.64, "duration": 8.759, "text": "tk3 we have X2 = root three these are"}, {"start": 121.28, "duration": 5.6, "text": "the real Roots now let's go to complex"}, {"start": 124.399, "duration": 8.281, "text": "Solutions X1"}, {"start": 126.88, "duration": 8.68, "text": "equals for complex Roots X1 = I root of"}, {"start": 132.68, "duration": 4.44, "text": "3 then X2"}, {"start": 135.56, "duration": 5.84, "text": "="}, {"start": 137.12, "duration": 6.28, "text": "IUN of 3 and these are the solutions"}, {"start": 141.4, "duration": 4.28, "text": "thank you for watching please kindly"}, {"start": 143.4, "duration": 6.4, "text": "subscribe to my channel also like"}, {"start": 145.68, "duration": 4.12, "text": "comment and share bye-bye"}], "title": "A nice quartic equation.", "description": "#maths #mathematics #simplification #exponents", "lengthSeconds": 148, "width": 568, "height": 360}
{"video_id": "oNCFE4ZkOQA", "transcript": [{"start": 0.599, "duration": 4.361, "text": "hello good viewers welcome back once"}, {"start": 2.48, "duration": 5.24, "text": "again to another video please kindly"}, {"start": 4.96, "duration": 5.24, "text": "support my channel by subscribing also"}, {"start": 7.72, "duration": 5.999, "text": "like comment and share today we have"}, {"start": 10.2, "duration": 9.159, "text": "this nice algebraic problem here we're"}, {"start": 13.719, "duration": 7.121, "text": "given X2 + Y2 is = 34 x - Y is = 2 right"}, {"start": 19.359, "duration": 5.16, "text": "over here we're going to compute the"}, {"start": 20.84, "duration": 7.24, "text": "value of the product of X and Y"}, {"start": 24.519, "duration": 5.84, "text": "right okay in this video we're going to"}, {"start": 28.08, "duration": 4.96, "text": "make use of some nice subject break"}, {"start": 30.359, "duration": 4.481, "text": "identities right I love these identities"}, {"start": 33.04, "duration": 5.76, "text": "because they are really cool right they"}, {"start": 34.84, "duration": 7.96, "text": "helps us in solving all these"}, {"start": 38.8, "duration": 10.919, "text": "complicated problems right here is the"}, {"start": 42.8, "duration": 14.48, "text": "identity X2 + y s can be written as x -"}, {"start": 49.719, "duration": 11.121, "text": "Y 2 right + 2 * x * Y is that not cool"}, {"start": 57.28, "duration": 5.2, "text": "yes of course this is really nice"}, {"start": 60.84, "duration": 6.04, "text": "okay this actually mean that this"}, {"start": 62.48, "duration": 6.4, "text": "equation x^2 y^2 + Y 2 I mean which is"}, {"start": 66.88, "duration": 10.12, "text": "equal to 34 we can write the left hand"}, {"start": 68.88, "duration": 11.64, "text": "side as x - y^ 2 + 2x Y and this is"}, {"start": 77.0, "duration": 5.52, "text": "equal to 34 now there's a nice question"}, {"start": 80.52, "duration": 5.8, "text": "that should be asked here why did I make"}, {"start": 82.52, "duration": 6.36, "text": "use of x - y the reason is because I"}, {"start": 86.32, "duration": 6.6, "text": "have seen that in the second equation"}, {"start": 88.88, "duration": 6.48, "text": "the value of x - + y is given right so"}, {"start": 92.92, "duration": 6.479, "text": "we can still rewrite this in terms of x"}, {"start": 95.36, "duration": 7.759, "text": "+ y but the reason here is clear that x"}, {"start": 99.399, "duration": 6.68, "text": "- Y is given us two right so from here"}, {"start": 103.119, "duration": 6.241, "text": "we can make the substitution from here"}, {"start": 106.079, "duration": 8.561, "text": "so x - Y is 2 so here we have 2^ 2 and"}, {"start": 109.36, "duration": 9.6, "text": "here + 2 x y this is equal to 34 right"}, {"start": 114.64, "duration": 9.32, "text": "so you know 2 2 is 4 so here we have 4 +"}, {"start": 118.96, "duration": 7.04, "text": "2 x y this is equal to 34 so from here"}, {"start": 123.96, "duration": 5.919, "text": "we subtract four from both sides and"}, {"start": 126.0, "duration": 8.0, "text": "here we get 2 * X and Y you know this is"}, {"start": 129.879, "duration": 8.72, "text": "equal to 30 so from here we're going to"}, {"start": 134.0, "duration": 6.879, "text": "divide both sides by two right so here"}, {"start": 138.599, "duration": 5.64, "text": "over two and here over two this two"}, {"start": 140.879, "duration": 7.521, "text": "cancel this two and here we get x * Y is"}, {"start": 144.239, "duration": 6.241, "text": "= 30 / 2 which is 15 and this is the"}, {"start": 148.4, "duration": 4.0, "text": "right answer take thank you for watching"}, {"start": 150.48, "duration": 6.52, "text": "please do wa to subscribe to my channel"}, {"start": 152.4, "duration": 4.6, "text": "also like comment and share bye-bye"}], "title": "Interesting algebraic problem. Can you do this without finding the value of x and y ? #maths", "description": "", "lengthSeconds": 155, "width": 640, "height": 360}
{"video_id": "jv0jfygSszg", "transcript": [{"start": 0.88, "duration": 5.04, "text": "hello welcome back once again today we"}, {"start": 3.48, "duration": 5.56, "text": "have this quatic equation and we're"}, {"start": 5.92, "duration": 4.16, "text": "going to solve for all values of X now"}, {"start": 9.04, "duration": 3.16, "text": "let's get"}, {"start": 10.08, "duration": 4.4, "text": "started now in this video we're not"}, {"start": 12.2, "duration": 5.239, "text": "going to make any substitution so we're"}, {"start": 14.48, "duration": 5.44, "text": "going to solve this problem"}, {"start": 17.439, "duration": 8.281, "text": "straight so let us get started from the"}, {"start": 19.92, "duration": 9.56, "text": "left hand side we have x to^ 4 plus x\u00b2"}, {"start": 25.72, "duration": 11.76, "text": "then this is equal to 20 so here we have"}, {"start": 29.48, "duration": 11.36, "text": "x to the 4 + X2 this is = 20 as 16 +"}, {"start": 37.48, "duration": 5.44, "text": "4 so 16 we're going to take it to the"}, {"start": 40.84, "duration": 4.239, "text": "left and also four take it to the left"}, {"start": 42.92, "duration": 9.36, "text": "so they're going to have opposite signs"}, {"start": 45.079, "duration": 9.361, "text": "so we have X to the^ 4 - 16 + X2 - 4"}, {"start": 52.28, "duration": 8.72, "text": "this is equal to"}, {"start": 54.44, "duration": 10.079, "text": "0 so from here we have X to the^ 2 in"}, {"start": 61.0, "duration": 10.52, "text": "bracket to the^ 2 CU 2 * 2 will give us"}, {"start": 64.519, "duration": 9.161, "text": "4 - 16 is same as 4 2 then plus x^2 - 4"}, {"start": 71.52, "duration": 4.8, "text": "this is equal to"}, {"start": 73.68, "duration": 4.96, "text": "0 From Here let us make use of the"}, {"start": 76.32, "duration": 9.72, "text": "difference of two squares formula we"}, {"start": 78.64, "duration": 11.64, "text": "have a 2 - b 2 is = A + B * Aus"}, {"start": 86.04, "duration": 5.48, "text": "B okay now sorry for the noise so here"}, {"start": 90.28, "duration": 4.08, "text": "we're going to make use of this"}, {"start": 91.52, "duration": 10.08, "text": "difference of two squares formula here"}, {"start": 94.36, "duration": 9.24, "text": "so that is going to be X2 + 4 * x^2 - 4"}, {"start": 101.6, "duration": 6.32, "text": "then here we have plus let's put in"}, {"start": 103.6, "duration": 6.559, "text": "bracket x^2 - 4 then this is equal to Z"}, {"start": 107.92, "duration": 4.239, "text": "so we can see that X2 - 4 is Common from"}, {"start": 110.159, "duration": 6.401, "text": "the left so let's Factor it out so we"}, {"start": 112.159, "duration": 8.24, "text": "have x^ - 4 then into bracket here we"}, {"start": 116.56, "duration": 5.4, "text": "have X2 + 4 left then Plus because in"}, {"start": 120.399, "duration": 5.36, "text": "this brackets here when you factor out"}, {"start": 121.96, "duration": 6.6, "text": "this x - 4 we have just one then this is"}, {"start": 125.759, "duration": 5.641, "text": "equal to Z so we have these two products"}, {"start": 128.56, "duration": 8.399, "text": "which is equal to Z so we have either so"}, {"start": 131.4, "duration": 11.88, "text": "this implies that either X2 - 4 is = 0"}, {"start": 136.959, "duration": 8.961, "text": "or X2 + 4 + 1 is 5 is = to 0 from here"}, {"start": 143.28, "duration": 9.64, "text": "take4 to the right so this will become"}, {"start": 145.92, "duration": 11.24, "text": "X2 is = to pos4 here x\u00b2 is = -5 take"}, {"start": 152.92, "duration": 10.56, "text": "root of both sides so we get X is = plus"}, {"start": 157.16, "duration": 11.64, "text": "or- 4 and here x is = to+ orus < TK"}, {"start": 163.48, "duration": 9.16, "text": "-5 so from here we get X is = to+ or - 2"}, {"start": 168.8, "duration": 7.359, "text": "so we have only two real solutions and"}, {"start": 172.64, "duration": 7.72, "text": "here x is = to otk of1 is Iota so we"}, {"start": 176.159, "duration": 7.681, "text": "have plus or minus Iota root of 5"}, {"start": 180.36, "duration": 5.68, "text": "and these are the only two imaginary"}, {"start": 183.84, "duration": 5.08, "text": "Solutions thank you for watching please"}, {"start": 186.04, "duration": 7.4, "text": "kindly subscribe to my channel also like"}, {"start": 188.92, "duration": 4.52, "text": "comment and share bye-bye"}], "title": "Stanford University \ud83c\udf93 Entrance Exam | Can you solve ?", "description": "", "lengthSeconds": 192, "width": 370, "height": 240}
{"video_id": "Xv8U7U5fbHQ", "transcript": [{"start": 1.12, "duration": 3.84, "text": "hello good viewers welcome back once"}, {"start": 2.96, "duration": 4.799, "text": "again today we're going to find the"}, {"start": 4.96, "duration": 8.48, "text": "following limit limit as X is"}, {"start": 7.759, "duration": 8.481, "text": "approaching to X CU - 8 all x - 2 when"}, {"start": 13.44, "duration": 8.4, "text": "we make the direct substitution here we"}, {"start": 16.24, "duration": 9.08, "text": "get 2 Cub - 8 2 - 2 and this is equal to"}, {"start": 21.84, "duration": 7.32, "text": "0 0 you know this is an indeterminate"}, {"start": 25.32, "duration": 7.6, "text": "form right so we need to manipulate this"}, {"start": 29.16, "duration": 7.96, "text": "function here I think the function can"}, {"start": 32.92, "duration": 6.68, "text": "further be simplified right now let us"}, {"start": 37.12, "duration": 7.759, "text": "talk about this function right over here"}, {"start": 39.6, "duration": 8.88, "text": "we have X Cub - 8 all x - 2 this is"}, {"start": 44.879, "duration": 8.641, "text": "equal to X Cub - 8 is the same as 2"}, {"start": 48.48, "duration": 8.759, "text": "cubed then all x - 2 now let us recall"}, {"start": 53.52, "duration": 7.039, "text": "the difference of 2 Cub formula a cubus"}, {"start": 57.239, "duration": 9.081, "text": "b Cub this is equal to a minus B into"}, {"start": 60.559, "duration": 12.721, "text": "braet a 2 + a b + b 2 from here this"}, {"start": 66.32, "duration": 12.439, "text": "will be x - 2 into braet X2 + 2 x right"}, {"start": 73.28, "duration": 8.839, "text": "then + 2 2 which is 4 then all over x -"}, {"start": 78.759, "duration": 9.281, "text": "2 so from here we cancel out and here"}, {"start": 82.119, "duration": 11.081, "text": "this simplifies to X2 + 2x + 4 right now"}, {"start": 88.04, "duration": 10.359, "text": "the limit as as X is approaching 2 of X"}, {"start": 93.2, "duration": 11.519, "text": "Cub - 8 / x - 2 this is same as the"}, {"start": 98.399, "duration": 8.801, "text": "limit as X is approaching 2 of X2 + 2x +"}, {"start": 104.719, "duration": 9.321, "text": "4 from here we make the direct"}, {"start": 107.2, "duration": 11.879, "text": "substitution we have 2^ 2 + 2 * 2 + 4 so"}, {"start": 114.04, "duration": 8.8, "text": "from here 2 2 is 4 + 2 * 2 which is 4 +"}, {"start": 119.079, "duration": 7.241, "text": "4 1 again and from here this simplifies"}, {"start": 122.84, "duration": 5.279, "text": "to 12 and this is the right answer thank"}, {"start": 126.32, "duration": 4.24, "text": "you for watching please kindly subscribe"}, {"start": 128.119, "duration": 6.281, "text": "to my YouTube channel also like comment"}, {"start": 130.56, "duration": 3.84, "text": "and share bye-bye"}], "title": "Interesting Limit | #maths #olympiad", "description": "", "lengthSeconds": 133, "width": 640, "height": 360}
{"video_id": "TuGqKsS4GBE", "transcript": [{"start": 1.439, "duration": 4.4, "text": "hello good viewers welcome back once"}, {"start": 3.159, "duration": 6.081, "text": "again today we are given the following"}, {"start": 5.839, "duration": 8.401, "text": "should I call it optic equation right"}, {"start": 9.24, "duration": 8.959, "text": "and we are asked to evaluate x ^ 4 + 1 *"}, {"start": 14.24, "duration": 7.76, "text": "x^ 4 - one right I know that this"}, {"start": 18.199, "duration": 6.681, "text": "problem looks a little bit we right but"}, {"start": 22.0, "duration": 5.439, "text": "believe me this is a very simple problem"}, {"start": 24.88, "duration": 5.36, "text": "all we need to do is to manipulate this"}, {"start": 27.439, "duration": 6.921, "text": "equation very well right now let's get"}, {"start": 30.24, "duration": 9.96, "text": "into the video we are given that x^ of 8"}, {"start": 34.36, "duration": 8.359, "text": "is equal to 9 right now let us go ahead"}, {"start": 40.2, "duration": 7.6, "text": "and subtract one from both sides of this"}, {"start": 42.719, "duration": 9.64, "text": "equation so from here x^ of 8 - 1 this"}, {"start": 47.8, "duration": 8.2, "text": "is equal to 9 - 1 right so we can see"}, {"start": 52.359, "duration": 6.241, "text": "that the equation still remain balance"}, {"start": 56.0, "duration": 8.879, "text": "okay now from the left here we're going"}, {"start": 58.6, "duration": 10.76, "text": "to rewrite x ^ 8 as x^ 4 in bracket s"}, {"start": 64.879, "duration": 9.6, "text": "right then- 1 is the same as 1 SAR and"}, {"start": 69.36, "duration": 7.28, "text": "from the right 9 - 1 is = to 8 now from"}, {"start": 74.479, "duration": 4.561, "text": "the left hand side we're going to make"}, {"start": 76.64, "duration": 5.56, "text": "use of the difference of two squares"}, {"start": 79.04, "duration": 7.2, "text": "right so recall that when we have a 2"}, {"start": 82.2, "duration": 8.32, "text": "minus b 2 this can be factorized as a +"}, {"start": 86.24, "duration": 6.64, "text": "b * Aus B right"}, {"start": 90.52, "duration": 5.599, "text": "so that is the sum of the"}, {"start": 92.88, "duration": 6.36, "text": "bases then multiply by the difference of"}, {"start": 96.119, "duration": 5.601, "text": "the bases right so if you take a very"}, {"start": 99.24, "duration": 6.68, "text": "look at this one here the base here is x"}, {"start": 101.72, "duration": 6.439, "text": "^ 4 and the other base is one right so"}, {"start": 105.92, "duration": 10.119, "text": "make use of this property here so this"}, {"start": 108.159, "duration": 10.6, "text": "will actually give us x^ 4 + 1 * x^ 4 -1"}, {"start": 116.039, "duration": 4.641, "text": "and that is equal to 8 I think this is"}, {"start": 118.759, "duration": 4.32, "text": "what we're looking for"}, {"start": 120.68, "duration": 4.6, "text": "okay nice we can see that we have found"}, {"start": 123.079, "duration": 4.241, "text": "the solution to the problem thank you"}, {"start": 125.28, "duration": 4.8, "text": "for watching please kindly subscribe to"}, {"start": 127.32, "duration": 5.76, "text": "my channel also like comment and share"}, {"start": 130.08, "duration": 3.0, "text": "bye-bye"}], "title": "Interesting algebra problem | Here is the trick | #maths #mathematics", "description": "", "lengthSeconds": 131, "width": 596, "height": 360}
{"video_id": "mY4rVRtUxEw", "transcript": [{"start": 0.68, "duration": 3.56, "text": "hello welcome back once again today we"}, {"start": 2.639, "duration": 3.801, "text": "have this interesting exponential"}, {"start": 4.24, "duration": 3.6, "text": "equation involving root of x and"}, {"start": 6.44, "duration": 4.64, "text": "reciprocal of"}, {"start": 7.84, "duration": 5.48, "text": "x from the left here the base is four"}, {"start": 11.08, "duration": 5.68, "text": "and four we can write it in index form"}, {"start": 13.32, "duration": 6.4, "text": "as 2 to^ 2 from the right the base here"}, {"start": 16.76, "duration": 7.759, "text": "is 8 and 8 we can also relate it in"}, {"start": 19.72, "duration": 6.799, "text": "index form as 2 to the^ of 3 so the left"}, {"start": 24.519, "duration": 6.641, "text": "hand side we're going to write that for"}, {"start": 26.519, "duration": 7.281, "text": "as 2 to the^ 2 then root X this equals"}, {"start": 31.16, "duration": 6.239, "text": "from the right the eight there will be 2"}, {"start": 33.8, "duration": 6.12, "text": "cubed then of the reciprocal of"}, {"start": 37.399, "duration": 7.881, "text": "X now let us take note of this property"}, {"start": 39.92, "duration": 9.6, "text": "that a^ b^ c is same as a to the^ B *"}, {"start": 45.28, "duration": 8.759, "text": "C so here this will be 2 2 * root X will"}, {"start": 49.52, "duration": 8.879, "text": "give us 2un X then this equals here 2 3"}, {"start": 54.039, "duration": 6.801, "text": "* 1 / X is same as 3 /"}, {"start": 58.399, "duration": 4.361, "text": "X from here we can see that we have the"}, {"start": 60.84, "duration": 8.36, "text": "same base right so we're going to equate"}, {"start": 62.76, "duration": 10.399, "text": "the powers so this implies that 2un x ="}, {"start": 69.2, "duration": 6.52, "text": "3 / X here we're solving for the R value"}, {"start": 73.159, "duration": 5.161, "text": "of x so let's Square both sides to get"}, {"start": 75.72, "duration": 5.8, "text": "rid of this root sign so here this will"}, {"start": 78.32, "duration": 6.799, "text": "be squared and here this will be"}, {"start": 81.52, "duration": 5.48, "text": "squared so from here to square 2 * root"}, {"start": 85.119, "duration": 4.801, "text": "X we're going to square 2 first and that"}, {"start": 87.0, "duration": 3.96, "text": "will give us 4 * root x 2 remember"}, {"start": 89.92, "duration": 4.0, "text": "remember the square will cancel the"}, {"start": 90.96, "duration": 4.439, "text": "roots so we get X then this will be"}, {"start": 93.92, "duration": 3.36, "text": "equal to to square this we're going to"}, {"start": 95.399, "duration": 3.521, "text": "square the numerator and also Square the"}, {"start": 97.28, "duration": 6.479, "text": "denominator when you square the"}, {"start": 98.92, "duration": 8.199, "text": "numerator we get 3 2 which is 9 /"}, {"start": 103.759, "duration": 7.081, "text": "X2 which is X2 right so from here this"}, {"start": 107.119, "duration": 8.0, "text": "is over one Let's cross multiply X2 * 4X"}, {"start": 110.84, "duration": 7.04, "text": "will give us 4 x Cub then this = 1 * 9"}, {"start": 115.119, "duration": 5.401, "text": "will give us 9 so from here divide both"}, {"start": 117.88, "duration": 8.08, "text": "sides by four"}, {"start": 120.52, "duration": 8.84, "text": "this cancel this here we get X Cub = 9 /"}, {"start": 125.96, "duration": 5.72, "text": "4 from here let us take root of I mean"}, {"start": 129.36, "duration": 7.0, "text": "the cube root of both sides I mean so"}, {"start": 131.68, "duration": 8.279, "text": "this give us X = the cube root of 9 /"}, {"start": 136.36, "duration": 5.64, "text": "four and this is the right answer thank"}, {"start": 139.959, "duration": 4.321, "text": "you for watching please kindly subscribe"}, {"start": 142.0, "duration": 5.879, "text": "to my channel also like comment and"}, {"start": 144.28, "duration": 3.599, "text": "share bye-bye"}], "title": "Interesting exponential equation |", "description": "", "lengthSeconds": 146, "width": 640, "height": 360}
{"video_id": "TjB6yNep2Wg", "transcript": [{"start": 0.84, "duration": 4.8, "text": "hello good day viewers welcome back once"}, {"start": 2.56, "duration": 4.239, "text": "again let us solve the following given"}, {"start": 5.64, "duration": 5.36, "text": "radical"}, {"start": 6.799, "duration": 9.48, "text": "equation solution we have here the < TK"}, {"start": 11.0, "duration": 8.48, "text": "of 5 ^ x * the < TK of 25 ^ x * the of"}, {"start": 16.279, "duration": 5.681, "text": "125 to^ x this is equal"}, {"start": 19.48, "duration": 6.24, "text": "to"}, {"start": 21.96, "duration": 5.639, "text": "125 let's get started so here we have"}, {"start": 25.72, "duration": 6.92, "text": "the square root"}, {"start": 27.599, "duration": 9.161, "text": "of 5 to^ of x X multip by the square"}, {"start": 32.64, "duration": 9.72, "text": "root of you know here 25 is the same as"}, {"start": 36.76, "duration": 10.119, "text": "5 S right so that would be 5 to^ of 2x"}, {"start": 42.36, "duration": 8.8, "text": "right then multiply by the square root"}, {"start": 46.879, "duration": 7.281, "text": "of 125 is the same as 5 cubed then with"}, {"start": 51.16, "duration": 7.399, "text": "the power of X here that will become 5"}, {"start": 54.16, "duration": 6.96, "text": "to the^ of 3x and this is equal to 1 2 5"}, {"start": 58.559, "duration": 7.121, "text": "is 5 to the power of"}, {"start": 61.12, "duration": 9.48, "text": "3 so from here we have the square root"}, {"start": 65.68, "duration": 12.36, "text": "of 5^ x MTI the square root"}, {"start": 70.6, "duration": 13.8, "text": "of 5 ^ of 2x * by now remember that root"}, {"start": 78.04, "duration": 12.48, "text": "m is = to m^ of 1 / 2 right so from here"}, {"start": 84.4, "duration": 10.52, "text": "this will become 5^ of 3x / 2 right then"}, {"start": 90.52, "duration": 8.44, "text": "this is equal to 5 to the^ of 3 so from"}, {"start": 94.92, "duration": 7.68, "text": "here we have the square root of 5 to^ of"}, {"start": 98.96, "duration": 6.479, "text": "X multip by the square root of remember"}, {"start": 102.6, "duration": 6.36, "text": "from here we have the same basis right"}, {"start": 105.439, "duration": 12.04, "text": "so we need to add the powers so here we"}, {"start": 108.96, "duration": 13.199, "text": "have 5 to^ of 2x + 3 x 2 2x + 3x 2 that"}, {"start": 117.479, "duration": 7.761, "text": "will give us 7 x all over 2 then this is"}, {"start": 122.159, "duration": 6.201, "text": "equal to 5 to the^ of"}, {"start": 125.24, "duration": 5.68, "text": "3 now from here make use of this"}, {"start": 128.36, "duration": 4.0, "text": "property again so we have the square"}, {"start": 130.92, "duration": 8.599, "text": "root"}, {"start": 132.36, "duration": 9.959, "text": "of 5^ of X multi by here will be 5^ 7 x"}, {"start": 139.519, "duration": 6.0, "text": "/ 2 then remember we have root sign so"}, {"start": 142.319, "duration": 8.601, "text": "that would be 2 * 2 which is 4 then this"}, {"start": 145.519, "duration": 8.08, "text": "is equal to 5 to the^ of 3"}, {"start": 150.92, "duration": 4.48, "text": "so from here remember we have the same"}, {"start": 153.599, "duration": 6.28, "text": "basis again so we're going to add the"}, {"start": 155.4, "duration": 12.64, "text": "powers so we have the square root of 5^"}, {"start": 159.879, "duration": 12.961, "text": "of x + 7 x 4 that will give us 11 x"}, {"start": 168.04, "duration": 8.52, "text": "right then all over 4 then this is equal"}, {"start": 172.84, "duration": 6.119, "text": "to 5 to the power of three so from here"}, {"start": 176.56, "duration": 7.039, "text": "making use of that property that is"}, {"start": 178.959, "duration": 8.761, "text": "square root of n m is = m ^ 1 / 2 so"}, {"start": 183.599, "duration": 5.761, "text": "from here this will be 5^ 11x / 4"}, {"start": 187.72, "duration": 5.48, "text": "remember with this root sign that will"}, {"start": 189.36, "duration": 8.56, "text": "be 4 * 2 and that is 8 this is equal to"}, {"start": 193.2, "duration": 7.039, "text": "5 to the power of 3 from here we have"}, {"start": 197.92, "duration": 6.319, "text": "the same basis so we need to equate the"}, {"start": 200.239, "duration": 5.521, "text": "powers right so we have 11 x / 8 this is"}, {"start": 204.239, "duration": 5.56, "text": "equal to"}, {"start": 205.76, "duration": 7.8, "text": "3 so here we cross multiply 11 x * 1"}, {"start": 209.799, "duration": 6.841, "text": "will give us 11 x this is equal to 8 * 3"}, {"start": 213.56, "duration": 5.84, "text": "that will give us 24 so divide by 11"}, {"start": 216.64, "duration": 6.239, "text": "since we are solving for x and here we"}, {"start": 219.4, "duration": 7.119, "text": "come up with the solution that X is ="}, {"start": 222.879, "duration": 5.36, "text": "24/ 11 and that's the right answer thank"}, {"start": 226.519, "duration": 4.041, "text": "you for watching please canly subscribe"}, {"start": 228.239, "duration": 5.841, "text": "to my channel also like comment and"}, {"start": 230.56, "duration": 3.52, "text": "share bye-bye"}], "title": "Interesting radical equation | #maths #mathematics", "description": "", "lengthSeconds": 233, "width": 384, "height": 360}
{"video_id": "fma81gN7-pI", "transcript": [{"start": 1.09, "duration": 3.599, "text": "[Music]"}, {"start": 7.4, "duration": 3.08, "text": "[Music]"}, {"start": 13.05, "duration": 11.78, "text": "[Music]"}, {"start": 22.039, "duration": 12.69, "text": "s spe"}, {"start": 24.83, "duration": 9.899, "text": "[Music]"}, {"start": 37.64, "duration": 3.0, "text": "know"}, {"start": 43.05, "duration": 15.209, "text": "[Music]"}, {"start": 60.16, "duration": 3.0, "text": "oh"}, {"start": 65.61, "duration": 4.65, "text": "[Music]"}, {"start": 91.03, "duration": 3.069, "text": "[Music]"}, {"start": 97.0, "duration": 20.05, "text": "slitz spe"}, {"start": 100.07, "duration": 16.98, "text": "[Music]"}], "title": "Interesting limit.", "description": "", "lengthSeconds": 115, "width": 640, "height": 360}
{"video_id": "KYANl7N1utI", "transcript": [{"start": 0.64, "duration": 5.04, "text": "hello welcome back once again today we"}, {"start": 3.56, "duration": 5.52, "text": "have to solve for the value of x that"}, {"start": 5.68, "duration": 5.6, "text": "will satisfy this equation but X must be"}, {"start": 9.08, "duration": 6.24, "text": "greater than"}, {"start": 11.28, "duration": 4.96, "text": "zero I have a very simple way to solve"}, {"start": 15.32, "duration": 4.56, "text": "this"}, {"start": 16.24, "duration": 9.16, "text": "problem first of all let's consider 65"}, {"start": 19.88, "duration": 7.96, "text": "we know 65 is the same as 81 - 16 I"}, {"start": 25.4, "duration": 3.56, "text": "think with this idea this problem is"}, {"start": 27.84, "duration": 3.359, "text": "already"}, {"start": 28.96, "duration": 7.599, "text": "solved let's go back to the original"}, {"start": 31.199, "duration": 7.081, "text": "problem we have 3 ^ of x - 2^ x so this"}, {"start": 36.559, "duration": 4.52, "text": "will be equal to"}, {"start": 38.28, "duration": 8.32, "text": "65 so with this idea so this will become"}, {"start": 41.079, "duration": 12.521, "text": "3 ^ xus 2 to^ x so this will be equal to"}, {"start": 46.6, "duration": 10.56, "text": "81 - 16 so we have 3 ^ x - 2^ x so this"}, {"start": 53.6, "duration": 8.24, "text": "will be equal to 81 is the same as 3 ^"}, {"start": 57.16, "duration": 8.999, "text": "4us 16 is the same as 2 to the^ 4 from"}, {"start": 61.84, "duration": 6.88, "text": "here this implies that X is = to 4 and"}, {"start": 66.159, "duration": 4.721, "text": "this is the solution thank you for"}, {"start": 68.72, "duration": 5.32, "text": "watching please kindly subscribe to my"}, {"start": 70.88, "duration": 6.16, "text": "channel also like comment and share"}, {"start": 74.04, "duration": 3.0, "text": "bye-bye"}], "title": "Germany | Interesting Olympiad Math", "description": "", "lengthSeconds": 76, "width": 640, "height": 360}
{"video_id": "9c6WSPm1k1A", "transcript": [{"start": 0.64, "duration": 4.92, "text": "hello good day viewers welcome back once"}, {"start": 2.84, "duration": 5.48, "text": "again please kindly support my channel"}, {"start": 5.56, "duration": 6.079, "text": "by subscribing also like comment and"}, {"start": 8.32, "duration": 7.239, "text": "share today we have this nice problem"}, {"start": 11.639, "duration": 7.201, "text": "here to solve we have here that 2x - 3 Y"}, {"start": 15.559, "duration": 8.241, "text": "is = 5 right over here we're going to"}, {"start": 18.84, "duration": 8.16, "text": "evaluate 4 power of X all 8 to^ of Y"}, {"start": 23.8, "duration": 5.76, "text": "right okay now let's go to this"}, {"start": 27.0, "duration": 8.48, "text": "expression here let us try to manipulate"}, {"start": 29.56, "duration": 9.999, "text": "this expression we have here 4^ x all 8"}, {"start": 35.48, "duration": 8.84, "text": "power of Y you know this isal to 4 is"}, {"start": 39.559, "duration": 9.84, "text": "the same as 2 2 then of X all over 8 is"}, {"start": 44.32, "duration": 7.36, "text": "the same as 2 cubed then of Y so from"}, {"start": 49.399, "duration": 7.081, "text": "here you know this"}, {"start": 51.68, "duration": 7.8, "text": "property a of B of C is equal to a of B"}, {"start": 56.48, "duration": 6.079, "text": "* C right so from here these Powers will"}, {"start": 59.48, "duration": 5.719, "text": "multiply together we have 2"}, {"start": 62.559, "duration": 8.321, "text": "2x all over these Powers multiply"}, {"start": 65.199, "duration": 8.721, "text": "together 2^ 3 y right let us not forget"}, {"start": 70.88, "duration": 6.64, "text": "this property as well when we have a to"}, {"start": 73.92, "duration": 5.96, "text": "the^ of B over a to the power of C with"}, {"start": 77.52, "duration": 4.68, "text": "the same basis we can subtract the"}, {"start": 79.88, "duration": 5.48, "text": "powers right so that will be a power of"}, {"start": 82.2, "duration": 6.8, "text": "B minus C"}, {"start": 85.36, "duration": 11.32, "text": "right okay so from here this is going to"}, {"start": 89.0, "duration": 10.64, "text": "be equal to 2 2^ 2x - 3 y right and do"}, {"start": 96.68, "duration": 6.92, "text": "not forget that 2x - 3 Y is exactly"}, {"start": 99.64, "duration": 8.92, "text": "equal to 5 right so from here this is"}, {"start": 103.6, "duration": 8.6, "text": "equal to 2 power 5 and 2^ of 5 is equal"}, {"start": 108.56, "duration": 4.96, "text": "to 32 and this is the right answer thank"}, {"start": 112.2, "duration": 3.04, "text": "you for watching please do wa to"}, {"start": 113.52, "duration": 5.68, "text": "subscribe to my channel also like"}, {"start": 115.24, "duration": 3.96, "text": "comment and share bye-bye"}], "title": "Easy exponential problem. The trick is really cool. #maths #education", "description": "", "lengthSeconds": 118, "width": 640, "height": 360}
{"video_id": "zJRSir46nvA", "transcript": [{"start": 0.32, "duration": 4.68, "text": "hello welcome back once again today"}, {"start": 2.28, "duration": 7.359, "text": "we're going to find the square root of"}, {"start": 5.0, "duration": 7.759, "text": "an imaginary number here I equals theare"}, {"start": 9.639, "duration": 7.521, "text": "root of1 which is also known as the"}, {"start": 12.759, "duration": 7.081, "text": "imaginary unit right okay for my video"}, {"start": 17.16, "duration": 4.92, "text": "here we're going to take the square root"}, {"start": 19.84, "duration": 6.72, "text": "of I if you recall from the previous"}, {"start": 22.08, "duration": 10.08, "text": "video I made the principle < TK of I = 1"}, {"start": 26.56, "duration": 8.96, "text": "+ I all / the otk of two yes if you"}, {"start": 32.16, "duration": 5.84, "text": "check my videos very well I made a video"}, {"start": 35.52, "duration": 6.16, "text": "on how to find the square root of I and"}, {"start": 38.0, "duration": 5.399, "text": "what it equals right okay so another"}, {"start": 41.68, "duration": 6.16, "text": "thing we need to consider here is when"}, {"start": 43.399, "duration": 7.48, "text": "we have the square root of m n x we can"}, {"start": 47.84, "duration": 7.16, "text": "actually Express this as squ root of M"}, {"start": 50.879, "duration": 7.36, "text": "Munk of n * otk of"}, {"start": 55.0, "duration": 7.96, "text": "X right okay now let's get into the"}, {"start": 58.239, "duration": 8.081, "text": "video here we have the sare OT of -2 I"}, {"start": 62.96, "duration": 9.92, "text": "here we can write this as square root of"}, {"start": 66.32, "duration": 8.08, "text": "-1 right Multi 2 then MTI I and with"}, {"start": 72.88, "duration": 2.84, "text": "this property right over here this"}, {"start": 74.4, "duration": 7.44, "text": "equals"}, {"start": 75.72, "duration": 9.0, "text": "-1 then multi by theun of 2 MTI root of"}, {"start": 81.84, "duration": 4.959, "text": "I just like I stated here the square"}, {"start": 84.72, "duration": 5.64, "text": "root of I that is the principal square"}, {"start": 86.799, "duration": 7.481, "text": "root = 1 + I all < tk2 so going to make"}, {"start": 90.36, "duration": 8.039, "text": "use of it here and here root of -1 = I"}, {"start": 94.28, "duration": 7.56, "text": "right so this equal I * the < TK of 2"}, {"start": 98.399, "duration": 8.441, "text": "multiply by here is going to be 1 + I /"}, {"start": 101.84, "duration": 7.36, "text": "theun 2 remember the all of this there"}, {"start": 106.84, "duration": 5.2, "text": "are products right so here root of two"}, {"start": 109.2, "duration": 8.44, "text": "we cancel root of two so this equal I"}, {"start": 112.04, "duration": 11.24, "text": "into braet 1 + I from here I * 1 is I"}, {"start": 117.64, "duration": 8.799, "text": "then I * I is I which is what -1 so here"}, {"start": 123.28, "duration": 7.24, "text": "we have the solution that the < TK of -2"}, {"start": 126.439, "duration": 5.8, "text": "i = i - one amazing thank you for"}, {"start": 130.52, "duration": 4.24, "text": "watching please kindly subscribe to my"}, {"start": 132.239, "duration": 5.521, "text": "channel also like comment and share"}, {"start": 134.76, "duration": 3.0, "text": "bye-bye"}], "title": "Can you find the square of negative two times iota ???", "description": "", "lengthSeconds": 136, "width": 538, "height": 360}
{"video_id": "MITEBaoML3Y", "transcript": [{"start": 0.76, "duration": 4.599, "text": "hello good day viewers in this video we"}, {"start": 2.879, "duration": 6.081, "text": "are going to find the following limit"}, {"start": 5.359, "duration": 8.001, "text": "limit as X is approaching Z of e to^ 2x"}, {"start": 8.96, "duration": 10.079, "text": "-1 / e ^ x"}, {"start": 13.36, "duration": 9.4, "text": "-1 now e ^ 2x we can write it as that's"}, {"start": 19.039, "duration": 8.121, "text": "e to ^ of 2x we can write it as e to^ of"}, {"start": 22.76, "duration": 5.519, "text": "X all squared if you recall that a to"}, {"start": 27.16, "duration": 5.8, "text": "the power of"}, {"start": 28.279, "duration": 9.761, "text": "B * C is same as a to the power of B"}, {"start": 32.96, "duration": 9.2, "text": "then in bracket power of C right so now"}, {"start": 38.04, "duration": 11.32, "text": "the expression here"}, {"start": 42.16, "duration": 15.039, "text": "e to the^ of 2x -1 all e ^ x -1 will be"}, {"start": 49.36, "duration": 10.8, "text": "equal to e ^ x 2 - 1 then all e^ x"}, {"start": 57.199, "duration": 4.68, "text": "-1 now from the numerator here we can"}, {"start": 60.16, "duration": 5.92, "text": "make use of the difference of two"}, {"start": 61.879, "duration": 9.28, "text": "squares right we can rewrite it as e ^"}, {"start": 66.08, "duration": 9.64, "text": "x^ 2 minus here 1 is same as 1 2 then"}, {"start": 71.159, "duration": 8.721, "text": "all e^ X - 1 now make use of the"}, {"start": 75.72, "duration": 10.64, "text": "difference of 2 squares that is a 2 - b"}, {"start": 79.88, "duration": 7.68, "text": "2 this is equal to a + b * a minus B so"}, {"start": 86.36, "duration": 4.399, "text": "from the"}, {"start": 87.56, "duration": 9.04, "text": "numerator we are going to to factorize"}, {"start": 90.759, "duration": 11.561, "text": "that as e^ x + 1 * e ^ x"}, {"start": 96.6, "duration": 9.0, "text": "-1 then all over e^ x -1 so from here"}, {"start": 102.32, "duration": 4.799, "text": "this will cancel out and we have e^ of x"}, {"start": 105.6, "duration": 5.4, "text": "+"}, {"start": 107.119, "duration": 6.841, "text": "1 okay now the limit we asked to find"}, {"start": 111.0, "duration": 4.159, "text": "limit as X is approaching 0 of e to^ of"}, {"start": 113.96, "duration": 6.56, "text": "2X"}, {"start": 115.159, "duration": 8.6, "text": "-1/ e^ x -1 will be equal to limit"}, {"start": 120.52, "duration": 6.279, "text": "as X is approaching Z of e to the^ of x"}, {"start": 123.759, "duration": 7.0, "text": "+ 1 and from here we make the sub the"}, {"start": 126.799, "duration": 8.641, "text": "direct substition we obtain e to ^ of 0"}, {"start": 130.759, "duration": 7.881, "text": "+ 1 and e^ 0 is 1 so we get 1 + 1 and"}, {"start": 135.44, "duration": 4.72, "text": "that is two and this is the right answer"}, {"start": 138.64, "duration": 3.56, "text": "thank you for watching please kindly"}, {"start": 140.16, "duration": 6.12, "text": "subscribe to my channel also like"}, {"start": 142.2, "duration": 4.08, "text": "comment and share bye-bye"}], "title": "Interesting limit | No L'hospital's rule | #maths #mathematics", "description": "", "lengthSeconds": 145, "width": 544, "height": 360}
{"video_id": "mR0UvmMyJFc", "transcript": [{"start": 0.0, "duration": 3.0, "text": "good evening viewers here welcome back"}, {"start": 1.56, "duration": 3.3, "text": "to my Channel today we have the"}, {"start": 3.0, "duration": 4.5, "text": "following interesting problem"}, {"start": 4.86, "duration": 5.4, "text": "if x raised by two plus y respirator is"}, {"start": 7.5, "duration": 4.2, "text": "equal to 11 and x times Y is equal to 8"}, {"start": 10.26, "duration": 4.44, "text": "then"}, {"start": 11.7, "duration": 5.82, "text": "X Plus Y in bracket squared is equal to"}, {"start": 14.7, "duration": 5.339, "text": "dash this problem is very simple"}, {"start": 17.52, "duration": 6.9, "text": "so I'll be presenting only one solution"}, {"start": 20.039, "duration": 7.08, "text": "but um very much methods of solving this"}, {"start": 24.42, "duration": 7.259, "text": "problem okay let me go ahead and expand"}, {"start": 27.119, "duration": 6.301, "text": "this so I have here X Plus y raised by 2"}, {"start": 31.679, "duration": 4.56, "text": "is equal to"}, {"start": 33.42, "duration": 4.319, "text": "X raised for two"}, {"start": 36.239, "duration": 4.741, "text": "loss"}, {"start": 37.739, "duration": 7.5, "text": "2 times x times y"}, {"start": 40.98, "duration": 6.72, "text": "plus y squared okay most of us are very"}, {"start": 45.239, "duration": 5.781, "text": "familiar with this expansion okay so"}, {"start": 47.7, "duration": 9.12, "text": "here we write this as follows x squared"}, {"start": 51.02, "duration": 8.44, "text": "nausea y squared then plus 2 times x"}, {"start": 56.82, "duration": 5.879, "text": "times y okay now we're going to make our"}, {"start": 59.46, "duration": 5.94, "text": "substitutions here x squared plus y"}, {"start": 62.699, "duration": 4.321, "text": "squared here we all know that this is"}, {"start": 65.4, "duration": 5.46, "text": "equal to 11"}, {"start": 67.02, "duration": 6.779, "text": "plus 2 times x times y we all know that"}, {"start": 70.86, "duration": 7.86, "text": "is equal to eight okay so here we have"}, {"start": 73.799, "duration": 7.5, "text": "this is equal to 11 plus 16. okay"}, {"start": 78.72, "duration": 5.46, "text": "so we have 11 plus 16 and this is equal"}, {"start": 81.299, "duration": 4.981, "text": "to 27 and this is the right answer to"}, {"start": 84.18, "duration": 3.299, "text": "the problem thank you for viewing and"}, {"start": 86.28, "duration": 2.58, "text": "please make sure you subscribe to my"}, {"start": 87.479, "duration": 3.68, "text": "YouTube channel for more interesting"}, {"start": 88.86, "duration": 2.299, "text": "videos"}], "title": "A secondary student's problem.", "description": "", "lengthSeconds": 90, "width": 640, "height": 360}
{"video_id": "2cmoaZ8nFCQ", "transcript": [{"start": 1.12, "duration": 3.96, "text": "hello welcome back once again today"}, {"start": 2.919, "duration": 5.121, "text": "we're going to solve this interesting"}, {"start": 5.08, "duration": 6.28, "text": "equation right and we're going to solve"}, {"start": 8.04, "duration": 6.839, "text": "for the values of a we need to know that"}, {"start": 11.36, "duration": 5.48, "text": "a is not equal to Z if a is equal to Z"}, {"start": 14.879, "duration": 3.681, "text": "the left hand side of the equation will"}, {"start": 16.84, "duration": 4.679, "text": "be"}, {"start": 18.56, "duration": 7.44, "text": "undefined so let's solve for other real"}, {"start": 21.519, "duration": 10.0, "text": "values of a from the left hand side a *"}, {"start": 26.0, "duration": 9.52, "text": "a * a is a cubed so we have a cub"}, {"start": 31.519, "duration": 7.761, "text": "then / a + a + a is 3"}, {"start": 35.52, "duration": 7.28, "text": "a then this is equal to 4 put this over"}, {"start": 39.28, "duration": 4.759, "text": "one and cross multiply so a cub * 1 will"}, {"start": 42.8, "duration": 4.399, "text": "give us a"}, {"start": 44.039, "duration": 6.441, "text": "cub so this is equal to 3 a * 4 will"}, {"start": 47.199, "duration": 5.281, "text": "give us 12 a subtract 128 from both"}, {"start": 50.48, "duration": 8.039, "text": "sides so we get"}, {"start": 52.48, "duration": 8.039, "text": "here a CU - 12 a this is equal to Z from"}, {"start": 58.519, "duration": 5.801, "text": "the left hand side a is common so let's"}, {"start": 60.519, "duration": 9.081, "text": "Factor it out so we get a into bracket a"}, {"start": 64.32, "duration": 7.64, "text": "2 - 12 this is = 0 so we can see we have"}, {"start": 69.6, "duration": 5.44, "text": "two products which is equal to Z so this"}, {"start": 71.96, "duration": 9.92, "text": "implies from here that either a is equal"}, {"start": 75.04, "duration": 9.68, "text": "to Z or a 2 - 12 is = 0 but obviously"}, {"start": 81.88, "duration": 7.36, "text": "this is not a solution right so add 12"}, {"start": 84.72, "duration": 6.64, "text": "to both side we get a 2 is = 12 now take"}, {"start": 89.24, "duration": 5.04, "text": "square root of both sides so we get a"}, {"start": 91.36, "duration": 6.68, "text": "will be equal to plus or minus theare <"}, {"start": 94.28, "duration": 5.96, "text": "TK 12 so simplifying the S A will be"}, {"start": 98.04, "duration": 5.92, "text": "equal to plus or minus the square root"}, {"start": 100.24, "duration": 6.64, "text": "of 12 is as 4 * 3 then from here we get"}, {"start": 103.96, "duration": 8.36, "text": "our solution a will be equal to plus or"}, {"start": 106.88, "duration": 7.64, "text": "minus 4 is 2 then we have here root of 3"}, {"start": 112.32, "duration": 4.439, "text": "so here we have our only two real"}, {"start": 114.52, "duration": 4.44, "text": "solutions that satisfy the given"}, {"start": 116.759, "duration": 4.96, "text": "equation thank you for watching please"}, {"start": 118.96, "duration": 7.24, "text": "kindly subscribe to my channel also like"}, {"start": 121.719, "duration": 4.481, "text": "comment and share bye-bye"}], "title": "Interesting equation | Can you solve ?", "description": "", "lengthSeconds": 125, "width": 428, "height": 240}
{"video_id": "-CHRlfV3yM0", "transcript": [{"start": 1.199, "duration": 4.001, "text": "hello good day viewers welcome back once"}, {"start": 2.919, "duration": 5.961, "text": "again today we're going to solve for the"}, {"start": 5.2, "duration": 9.84, "text": "values of X in the foll quatic equation"}, {"start": 8.88, "duration": 14.239, "text": "we have x^ 4 + x^2 - 42 this is equal 0"}, {"start": 15.04, "duration": 13.559, "text": "now x^ 4 we can write it as x^ 2^ 2 then"}, {"start": 23.119, "duration": 7.441, "text": "+ X2 - 42 this is equal to Z now we're"}, {"start": 28.599, "duration": 6.521, "text": "going to make the following substit"}, {"start": 30.56, "duration": 11.44, "text": "we're going to let X2 = y so from here"}, {"start": 35.12, "duration": 9.32, "text": "we have y^ 2 + y - 42 this is equal to Z"}, {"start": 42.0, "duration": 6.76, "text": "so for here this left hand side of this"}, {"start": 44.44, "duration": 7.36, "text": "equation can be factorized right and how"}, {"start": 48.76, "duration": 8.799, "text": "do we factorize that first of all this y"}, {"start": 51.8, "duration": 9.079, "text": "we can rewrite y as y\u00b2 + 7 y - 6 y then"}, {"start": 57.559, "duration": 6.68, "text": "- 42 this is equal to Z from here we"}, {"start": 60.879, "duration": 6.041, "text": "factor out Y and we left with y + 7 and"}, {"start": 64.239, "duration": 5.841, "text": "from this side we factor out -6 and we"}, {"start": 66.92, "duration": 6.879, "text": "left with y + 7 and this is equal to Z"}, {"start": 70.08, "duration": 10.52, "text": "so from here this implies that y -"}, {"start": 73.799, "duration": 13.201, "text": "6 * y + 7 is = 0 so we have here y - 6"}, {"start": 80.6, "duration": 10.839, "text": "is = 0 or y + 7 is equal"}, {"start": 87.0, "duration": 11.159, "text": "0 so from here we find solution for y y"}, {"start": 91.439, "duration": 10.081, "text": "is = 6 or Y is = 0 - 7 which is"}, {"start": 98.159, "duration": 7.521, "text": "-7 now let us recall that x\u00b2 is equal to"}, {"start": 101.52, "duration": 11.04, "text": "Y so from here we can state that when Y"}, {"start": 105.68, "duration": 10.84, "text": "is = 6 x\u00b2 is = 6 which implies that X is"}, {"start": 112.56, "duration": 3.96, "text": "= plus or minus otk"}, {"start": 118.36, "duration": 10.039, "text": "6 when y is = to"}, {"start": 122.0, "duration": 11.44, "text": "--7 X2 is = -7 so this implies that X is"}, {"start": 128.399, "duration": 5.041, "text": "= to plus or minus otk of"}, {"start": 133.76, "duration": 10.839, "text": "-7 which is equal to+ - IUN 7 so from"}, {"start": 139.92, "duration": 8.28, "text": "here we find the solutions that X is"}, {"start": 144.599, "duration": 7.241, "text": "equal to plus or minus < TK 6 this is"}, {"start": 148.2, "duration": 3.64, "text": "for R solution"}, {"start": 152.959, "duration": 10.481, "text": "right and we have X isal to plus or"}, {"start": 156.76, "duration": 6.68, "text": "minus I root of 7 this is for Imaginary"}, {"start": 167.04, "duration": 4.16, "text": "Solutions thank you for watching please"}, {"start": 169.12, "duration": 6.68, "text": "kindly subscribe to my YouTube channel"}, {"start": 171.2, "duration": 4.6, "text": "also like comment and share bye-bye"}], "title": "Olympiad Math | A nice quartic equation | Interesting problem | #maths #olympiad", "description": "", "lengthSeconds": 174, "width": 640, "height": 360}
{"video_id": "y0N6nzgh8EE", "transcript": [{"start": 0.64, "duration": 5.239, "text": "hello welcome back once again let us"}, {"start": 2.919, "duration": 4.281, "text": "solve this interesting equation from the"}, {"start": 5.879, "duration": 3.68, "text": "Oxford"}, {"start": 7.2, "duration": 6.319, "text": "University so from the left hand side we"}, {"start": 9.559, "duration": 6.841, "text": "have this x ra^ otk of X and this is"}, {"start": 13.519, "duration": 5.481, "text": "equal to 3 so let us take the log of"}, {"start": 16.4, "duration": 6.32, "text": "both sides so we have the natural log of"}, {"start": 19.0, "duration": 6.359, "text": "x to the^ otk X this is equal to the"}, {"start": 22.72, "duration": 5.2, "text": "natural log of 3 so we know this"}, {"start": 25.359, "duration": 7.561, "text": "property of logm for example the log of"}, {"start": 27.92, "duration": 8.4, "text": "a the^ b is equal to B * the log of a so"}, {"start": 32.92, "duration": 6.2, "text": "here this will become otk x m the"}, {"start": 36.32, "duration": 6.68, "text": "natural log of x so this is equal to the"}, {"start": 39.12, "duration": 8.32, "text": "natural log of 3 right so now we know"}, {"start": 43.0, "duration": 6.719, "text": "root of x same as x^ 1 / 2 then multi by"}, {"start": 47.44, "duration": 5.88, "text": "the natural log of x this is equal to"}, {"start": 49.719, "duration": 7.201, "text": "the natural log of 3 now we know this"}, {"start": 53.32, "duration": 8.64, "text": "interesting property that a to the power"}, {"start": 56.92, "duration": 9.519, "text": "of B is same as e to the power of B MTI"}, {"start": 61.96, "duration": 8.199, "text": "the Ln of a right so here the power of X"}, {"start": 66.439, "duration": 9.36, "text": "here is 1 / 2 so this is going to be e"}, {"start": 70.159, "duration": 10.081, "text": "to 1 / 2 * the Ln of X then times the Ln"}, {"start": 75.799, "duration": 6.441, "text": "of X this is equal to the Ln of 3 now"}, {"start": 80.24, "duration": 4.44, "text": "we're going to make use of the Lamb W"}, {"start": 82.24, "duration": 5.4, "text": "function but we need to take note of the"}, {"start": 84.68, "duration": 5.799, "text": "following now the lamb function is the"}, {"start": 87.64, "duration": 8.6, "text": "inverse function"}, {"start": 90.479, "duration": 7.521, "text": "for the function x * e^ X right so now"}, {"start": 96.24, "duration": 8.159, "text": "the lamb function of"}, {"start": 98.0, "duration": 8.719, "text": "x * e^ of x simplifies to X provided"}, {"start": 104.399, "duration": 5.921, "text": "that the power of E must be equal with"}, {"start": 106.719, "duration": 6.561, "text": "the coefficient of e so now you can see"}, {"start": 110.32, "duration": 5.56, "text": "here the power of E and the coefficient"}, {"start": 113.28, "duration": 5.68, "text": "here they are not yet equal the only"}, {"start": 115.88, "duration": 5.239, "text": "slight difference here is 1 all/ 2 so"}, {"start": 118.96, "duration": 5.36, "text": "we're going to multiply both sides of"}, {"start": 121.119, "duration": 7.601, "text": "this equation by 1 all/ 2 so stting from"}, {"start": 124.32, "duration": 9.279, "text": "here natur of x * 1/ 2 is as 1 / 2 * the"}, {"start": 128.72, "duration": 7.32, "text": "natural of X then multi e to^ 1 / 2 *"}, {"start": 133.599, "duration": 6.28, "text": "the natural log of x so this will be"}, {"start": 136.04, "duration": 7.88, "text": "equal to 1 / 2 * the natural log of 3"}, {"start": 139.879, "duration": 4.041, "text": "now we can take the lambat on both"}, {"start": 147.519, "duration": 7.321, "text": "sides okay so the left hand side we"}, {"start": 150.36, "duration": 6.04, "text": "simplify to 1 / 2 * the natur of X so"}, {"start": 154.84, "duration": 7.08, "text": "this is equal to the"}, {"start": 156.4, "duration": 7.8, "text": "W of 1 / 2 * the natural of 3 now we can"}, {"start": 161.92, "duration": 4.8, "text": "multiply both side by two so if we"}, {"start": 164.2, "duration": 5.399, "text": "multiply both side by 2 2 * 1/ 2 here is"}, {"start": 166.72, "duration": 6.96, "text": "1 so we are left with natural log of x"}, {"start": 169.599, "duration": 8.681, "text": "so this will be equal to 2 * the W of 1"}, {"start": 173.68, "duration": 7.16, "text": "/ 2 * natur of 3 so now we're going to"}, {"start": 178.28, "duration": 6.16, "text": "make this simple base on both sides so"}, {"start": 180.84, "duration": 6.679, "text": "we're going to write e^ Ln X from the"}, {"start": 184.44, "duration": 8.64, "text": "left hand side will also be equal to e^"}, {"start": 187.519, "duration": 8.961, "text": "2 W * 1 / 2 * the natural L of 3 so we"}, {"start": 193.08, "duration": 7.6, "text": "know e^ Ln anything cancel out to leave"}, {"start": 196.48, "duration": 6.679, "text": "us with the team so here e e and the"}, {"start": 200.68, "duration": 6.24, "text": "power of this Ln we cancel out and we"}, {"start": 203.159, "duration": 9.881, "text": "just have X so this is equal to e to"}, {"start": 206.92, "duration": 8.399, "text": "the^ 2 W * 1 / 2 * the natural of 3 so"}, {"start": 213.04, "duration": 5.0, "text": "with the help of my calculator we get"}, {"start": 215.319, "duration": 5.92, "text": "this nice approximation for x x is"}, {"start": 218.04, "duration": 3.199, "text": "approximately equal to"}, {"start": 221.799, "duration": 5.601, "text": "2124"}, {"start": 224.239, "duration": 6.601, "text": "799 383"}, {"start": 227.4, "duration": 5.08, "text": "02 and this is the correct answer thank"}, {"start": 230.84, "duration": 4.44, "text": "you for watching please kindly subscribe"}, {"start": 232.48, "duration": 6.44, "text": "to my channel also like comment and"}, {"start": 235.28, "duration": 3.64, "text": "share bye-bye"}], "title": "Oxford University Entrance Exam | Can you solve ?", "description": "", "lengthSeconds": 238, "width": 428, "height": 240}
{"video_id": "caou6Qp6xc4", "transcript": [{"start": 0.52, "duration": 5.6, "text": "hello welcome back guys here is the"}, {"start": 3.48, "duration": 5.44, "text": "continuation of the previous video you"}, {"start": 6.12, "duration": 6.16, "text": "know in the video I told you guys to"}, {"start": 8.92, "duration": 6.679, "text": "check the next video for the evaluation"}, {"start": 12.28, "duration": 5.159, "text": "of this without calculator right okay"}, {"start": 15.599, "duration": 6.721, "text": "now let's see how to do"}, {"start": 17.439, "duration": 9.641, "text": "that now here we have 2^ 10 - 1 right so"}, {"start": 22.32, "duration": 8.68, "text": "we're going to rewrite as follows 2 ^"}, {"start": 27.08, "duration": 6.68, "text": "5 2 - 1"}, {"start": 31.0, "duration": 5.64, "text": "s right you know these powers can"}, {"start": 33.76, "duration": 5.68, "text": "multiply together so that we can still"}, {"start": 36.64, "duration": 4.56, "text": "have 2^ of 10 so from here we're going"}, {"start": 39.44, "duration": 2.88, "text": "to make use of the difference of two"}, {"start": 41.2, "duration": 5.839, "text": "squares"}, {"start": 42.32, "duration": 10.759, "text": "formula here we go a 2us b squ is equal"}, {"start": 47.039, "duration": 7.641, "text": "to a + b * Aus B so from here this is"}, {"start": 53.079, "duration": 10.841, "text": "going to be equal"}, {"start": 54.68, "duration": 13.36, "text": "to 2 ^ 5 + 1 right * 2 ^ 5 - 1 so this"}, {"start": 63.92, "duration": 10.72, "text": "is equal to you know 2^ 5 is 32 and here"}, {"start": 68.04, "duration": 11.56, "text": "32 + 1 is 33 right then times 2^ 5 is 32"}, {"start": 74.64, "duration": 8.28, "text": "32 - 1 that will give us 31 right so we"}, {"start": 79.6, "duration": 5.72, "text": "need to multiply this together so is"}, {"start": 82.92, "duration": 4.28, "text": "there any other trick on how we can do"}, {"start": 85.32, "duration": 4.119, "text": "this well there are so many tricks but"}, {"start": 87.2, "duration": 7.36, "text": "this one is straightforward right just"}, {"start": 89.439, "duration": 10.281, "text": "have 33 * 31 3 3 99 right and we have"}, {"start": 94.56, "duration": 7.239, "text": "one Z 23 and here comes the right answer"}, {"start": 99.72, "duration": 3.48, "text": "to the problem thank you for watching"}, {"start": 101.799, "duration": 3.761, "text": "please don't wa to subscribe to this"}, {"start": 103.2, "duration": 5.36, "text": "channel also like comment and share"}, {"start": 105.56, "duration": 3.0, "text": "bye-bye"}], "title": "Olympiad Math problem.", "description": "", "lengthSeconds": 107, "width": 640, "height": 360}
{"video_id": "-xQ14yHSDng", "transcript": [{"start": 1.12, "duration": 4.08, "text": "hello good day viewers welcome back once"}, {"start": 2.879, "duration": 4.96, "text": "again today we're going to solve for the"}, {"start": 5.2, "duration": 7.64, "text": "product of X and Y in the following"}, {"start": 7.839, "duration": 12.961, "text": "given Olympia M problem we have here x -"}, {"start": 12.84, "duration": 12.439, "text": "y = 1 and X Cub - y Cub this is equal to"}, {"start": 20.8, "duration": 7.639, "text": "10 in this video I'll be presenting only"}, {"start": 25.279, "duration": 5.08, "text": "one method and this method is awesome"}, {"start": 28.439, "duration": 5.081, "text": "believe me"}, {"start": 30.359, "duration": 5.88, "text": "now for the difference of two cubes"}, {"start": 33.52, "duration": 4.76, "text": "there is a very nice identity that will"}, {"start": 36.239, "duration": 9.281, "text": "be very helpful"}, {"start": 38.28, "duration": 12.72, "text": "here so this identity is X cubed - y"}, {"start": 45.52, "duration": 13.4, "text": "cubed this is equal to x - y"}, {"start": 51.0, "duration": 11.239, "text": "cubed + 3 * x * Y into braet x -"}, {"start": 58.92, "duration": 8.12, "text": "y you know why I'm using this identity"}, {"start": 62.239, "duration": 7.601, "text": "already the value of x - Y is given so"}, {"start": 67.04, "duration": 6.24, "text": "if you plug it in here we're going to"}, {"start": 69.84, "duration": 6.56, "text": "have something really awesome right now"}, {"start": 73.28, "duration": 6.199, "text": "we're going to equate this to be 10"}, {"start": 76.4, "duration": 11.44, "text": "right since we have X Cub - y Cub is ="}, {"start": 79.479, "duration": 13.441, "text": "10 so here we go we have x - y cubed + 3"}, {"start": 87.84, "duration": 7.959, "text": "x Y into bracket x - y this is equal to"}, {"start": 92.92, "duration": 5.32, "text": "10 already we are given that X - 1 is"}, {"start": 95.799, "duration": 5.441, "text": "equal to 1 so we're going to substitute"}, {"start": 98.24, "duration": 4.6, "text": "so here x - y cubed will be the same as"}, {"start": 101.24, "duration": 8.68, "text": "1 cubed"}, {"start": 102.84, "duration": 9.84, "text": "right then + 3 x y MTI X - 1 that is 1"}, {"start": 109.92, "duration": 3.96, "text": "that's x - y I mean then this is equal"}, {"start": 112.68, "duration": 3.96, "text": "to"}, {"start": 113.88, "duration": 9.08, "text": "10 so from here let's simplify this 1"}, {"start": 116.64, "duration": 9.0, "text": "Cub is 1 + 3 x * y * 1 is 3x y right"}, {"start": 122.96, "duration": 6.359, "text": "then this is equal to 10 so let's"}, {"start": 125.64, "duration": 9.28, "text": "subtract one from both sides we get 3x y"}, {"start": 129.319, "duration": 9.2, "text": "this is = 10 - 1 so from here we get 3 X"}, {"start": 134.92, "duration": 5.959, "text": "Y this is = 9 so from here let's divide"}, {"start": 138.519, "duration": 5.561, "text": "both sides by 3 since we're interested"}, {"start": 140.879, "duration": 6.241, "text": "in solving for the value of x * Y and"}, {"start": 144.08, "duration": 7.6, "text": "from here we get the right answer x * y"}, {"start": 147.12, "duration": 6.68, "text": "= 9 3 which is 3 and this is a correct"}, {"start": 151.68, "duration": 4.279, "text": "answer thank you for watching please"}, {"start": 153.8, "duration": 6.88, "text": "kindly subscribe to my YouTube channel"}, {"start": 155.959, "duration": 4.721, "text": "also like comment and share bye-bye"}], "title": "Olympiad Math | Interesting algebraic problem | #maths #mathematics", "description": "", "lengthSeconds": 159, "width": 640, "height": 360}
{"video_id": "QIfuOEI7NDI", "transcript": [{"start": 0.44, "duration": 4.6, "text": "hello good viewers welcome back once"}, {"start": 2.6, "duration": 5.52, "text": "again today we're going to solve for the"}, {"start": 5.04, "duration": 9.12, "text": "value or values of X in the following"}, {"start": 8.12, "duration": 10.6, "text": "given equation we have otk x + x is ="}, {"start": 14.16, "duration": 7.4, "text": "9 so we have here otk x + x this is"}, {"start": 18.72, "duration": 7.84, "text": "equal 9 let's subtract X from both sides"}, {"start": 21.56, "duration": 9.32, "text": "we get root X is equal to 9 - x now"}, {"start": 26.56, "duration": 4.32, "text": "let's go ahead and square both sides"}, {"start": 31.4, "duration": 5.24, "text": "so from here you know the square and the"}, {"start": 33.84, "duration": 6.84, "text": "root sign will cancel out we have here x"}, {"start": 36.64, "duration": 8.28, "text": "is equal to now to expand this let us"}, {"start": 40.68, "duration": 8.519, "text": "recall the following binomial expansion"}, {"start": 44.92, "duration": 8.76, "text": "of a minus B which will give us a 2 + b"}, {"start": 49.199, "duration": 6.16, "text": "2 - 2 * a * B so from here the first"}, {"start": 53.68, "duration": 4.839, "text": "time here square that is 9 Square will"}, {"start": 55.359, "duration": 5.081, "text": "give us 81 then plus the second term"}, {"start": 58.519, "duration": 4.88, "text": "square which is x s"}, {"start": 60.44, "duration": 6.56, "text": "- 2 * the first term and the second term"}, {"start": 63.399, "duration": 5.681, "text": "which is 18 x right now from the right"}, {"start": 67.0, "duration": 8.56, "text": "hand side we have"}, {"start": 69.08, "duration": 10.12, "text": "X2 - 18 x + 81 then from the left we"}, {"start": 75.56, "duration": 7.519, "text": "subtract X right then this is equal to Z"}, {"start": 79.2, "duration": 10.08, "text": "so from here we have x^2 - 18 x - x will"}, {"start": 83.079, "duration": 7.72, "text": "give us - 19 x + 81 and this is equal Z"}, {"start": 89.28, "duration": 3.879, "text": "so from here we're going to solve this"}, {"start": 90.799, "duration": 4.68, "text": "quadratic equation and we're going to"}, {"start": 93.159, "duration": 4.521, "text": "make use of the general quadratic"}, {"start": 95.479, "duration": 5.041, "text": "formula so you know from here this will"}, {"start": 97.68, "duration": 7.079, "text": "be remember a here is the coefficient of"}, {"start": 100.52, "duration": 8.44, "text": "x\u00b2 which is 1 B is a coefficient of x"}, {"start": 104.759, "duration": 7.761, "text": "which is -19 and C the constant that is"}, {"start": 108.96, "duration": 7.439, "text": "A1 right so here netive B will be"}, {"start": 112.52, "duration": 9.8, "text": "postive 19 plus or minus square root of"}, {"start": 116.399, "duration": 9.921, "text": "B 2 will be -9 s which is same as 19 squ"}, {"start": 122.32, "duration": 9.16, "text": "right because the power is even right so"}, {"start": 126.32, "duration": 9.12, "text": "squ minus 4 * a is 4 * 1 * C which is"}, {"start": 131.48, "duration": 7.32, "text": "the constant 81 then the whole of this"}, {"start": 135.44, "duration": 8.2, "text": "all over 2 * a which is 2 * 1 that is"}, {"start": 138.8, "duration": 7.36, "text": "2 so from here we have X is = 19 plus"}, {"start": 143.64, "duration": 5.8, "text": "orus square root of you know the power"}, {"start": 146.16, "duration": 4.2, "text": "here is even so this will be a positive"}, {"start": 149.44, "duration": 4.04, "text": "result"}, {"start": 150.36, "duration": 3.12, "text": "so that will give us"}, {"start": 153.84, "duration": 5.08, "text": "361 minus 4 * 1 is 4 * 81 that will give"}, {"start": 157.84, "duration": 5.119, "text": "us"}, {"start": 158.92, "duration": 7.72, "text": "324 then all over 2 so this is equal to"}, {"start": 162.959, "duration": 8.56, "text": "19 plus or minus square root of that"}, {"start": 166.64, "duration": 8.319, "text": "will give us 37 right then all over two"}, {"start": 171.519, "duration": 8.08, "text": "so these are the solutions then we have"}, {"start": 174.959, "duration": 13.961, "text": "an exal solution this solution is where"}, {"start": 179.599, "duration": 11.521, "text": "X = 19 +un 37 / 2 so this solution is"}, {"start": 188.92, "duration": 5.679, "text": "extranos you know the reason why I call"}, {"start": 191.12, "duration": 6.32, "text": "it extranos solution because it does not"}, {"start": 194.599, "duration": 5.681, "text": "satisfy the given equation so the only"}, {"start": 197.44, "duration": 8.24, "text": "solution that satisfy the given equation"}, {"start": 200.28, "duration": 10.519, "text": "which we will use therefore X is = 19 -"}, {"start": 205.68, "duration": 7.4, "text": "< TK of 37 all over 2 so this is the"}, {"start": 210.799, "duration": 4.64, "text": "correct solution thank you for watching"}, {"start": 213.08, "duration": 7.68, "text": "please kindly subscribe to my channel"}, {"start": 215.439, "duration": 5.321, "text": "also like comment and share bye-bye"}], "title": "Interesting equation | #maths #mathematics", "description": "", "lengthSeconds": 219, "width": 640, "height": 360}
{"video_id": "QqIA8pKujrg", "transcript": [{"start": 0.84, "duration": 3.959, "text": "hello good viewers welcome back once"}, {"start": 2.6, "duration": 4.08, "text": "again today we have this beautiful"}, {"start": 4.799, "duration": 4.96, "text": "equation here and we will be solving for"}, {"start": 6.68, "duration": 5.44, "text": "the values of X we're"}, {"start": 9.759, "duration": 9.201, "text": "given x"}, {"start": 12.12, "duration": 10.2, "text": "+ square < TK of X2 - 9 this is equal to"}, {"start": 18.96, "duration": 6.159, "text": "4 now from here this x here let's move"}, {"start": 22.32, "duration": 5.6, "text": "it to the right so we we are only left"}, {"start": 25.119, "duration": 7.761, "text": "with theare root of x - 9 so here we"}, {"start": 27.92, "duration": 9.52, "text": "have the sare root of x^2 - 9 this will"}, {"start": 32.88, "duration": 7.28, "text": "be equal to 4 - x now from here what do"}, {"start": 37.44, "duration": 4.76, "text": "we do we have root sign here for us to"}, {"start": 40.16, "duration": 4.6, "text": "get rid of this root sign we need to"}, {"start": 42.2, "duration": 7.72, "text": "square both sides of this equation so"}, {"start": 44.76, "duration": 12.479, "text": "here we have the square root of X2 -"}, {"start": 49.92, "duration": 10.279, "text": "9 2 this will be equal to 4 - x^ 2 now"}, {"start": 57.239, "duration": 6.28, "text": "from here obviously the square and the"}, {"start": 60.199, "duration": 6.96, "text": "root sign we cancel out we have"}, {"start": 63.519, "duration": 7.361, "text": "X2 - 9 this will be equal to now from"}, {"start": 67.159, "duration": 9.0, "text": "here let us take note of this"}, {"start": 70.88, "duration": 7.96, "text": "property let us recall that a -"}, {"start": 76.159, "duration": 9.721, "text": "b\u00b2 this is equal to a"}, {"start": 78.84, "duration": 10.0, "text": "2 + b 2 - 2 * a * B"}, {"start": 85.88, "duration": 6.08, "text": "right okay so from here the first time"}, {"start": 88.84, "duration": 5.76, "text": "here squ will be four Square which is 16"}, {"start": 91.96, "duration": 4.159, "text": "plus the second time here squar which"}, {"start": 94.6, "duration": 6.64, "text": "should be"}, {"start": 96.119, "duration": 9.081, "text": "X2 minus 2 * a b remember 2 * 4 * X"}, {"start": 101.24, "duration": 7.44, "text": "which is 8 x right so we can see the way"}, {"start": 105.2, "duration": 6.0, "text": "we easily expand this right so here we"}, {"start": 108.68, "duration": 7.64, "text": "have"}, {"start": 111.2, "duration": 7.64, "text": "X2 now we have - 9 here then this is"}, {"start": 116.32, "duration": 10.2, "text": "equal to let's write this as"}, {"start": 118.84, "duration": 9.96, "text": "x\u00b2 - 8 x then + 16 right okay now we can"}, {"start": 126.52, "duration": 4.519, "text": "see from both side x squ is both"}, {"start": 128.8, "duration": 4.96, "text": "positive right so which actually mean"}, {"start": 131.039, "duration": 5.601, "text": "they will cancel out so but this is how"}, {"start": 133.76, "duration": 5.119, "text": "we do this let's move this whole term to"}, {"start": 136.64, "duration": 5.76, "text": "the left so here we have"}, {"start": 138.879, "duration": 7.041, "text": "X2 - 9 here this is positive so it will"}, {"start": 142.4, "duration": 5.24, "text": "become negative so we have - x^ 2 this"}, {"start": 145.92, "duration": 5.319, "text": "one is negative so it will become"}, {"start": 147.64, "duration": 5.959, "text": "positive + 8 x this one right over here"}, {"start": 151.239, "duration": 5.441, "text": "is positive so it will become negative"}, {"start": 153.599, "duration": 7.041, "text": "minus 16 and this will be equal to zero"}, {"start": 156.68, "duration": 5.32, "text": "so from here we can see x^2 - X2 is 0 we"}, {"start": 160.64, "duration": 6.519, "text": "have here 8"}, {"start": 162.0, "duration": 10.519, "text": "x now -9 - 16 we give"}, {"start": 167.159, "duration": 8.761, "text": "us 25 right yes of course so that is 25"}, {"start": 172.519, "duration": 5.241, "text": "then this is equal to 0 now this 25 is"}, {"start": 175.92, "duration": 6.039, "text": "negative if it's crossing over it will"}, {"start": 177.76, "duration": 7.72, "text": "become positive so we have 8 X is = to 0"}, {"start": 181.959, "duration": 5.28, "text": "+ 25 will give us 25 since we're solving"}, {"start": 185.48, "duration": 4.759, "text": "for x let's divide both side by the"}, {"start": 187.239, "duration": 5.0, "text": "coefficient of x coefficient of x is"}, {"start": 190.239, "duration": 7.441, "text": "this number right over here attached to"}, {"start": 192.239, "duration": 8.041, "text": "X so here we have 25 / by 8 this cancel"}, {"start": 197.68, "duration": 8.52, "text": "this and here we have our solution that"}, {"start": 200.28, "duration": 8.56, "text": "x isal 25 / 8 thank you for watching"}, {"start": 206.2, "duration": 7.36, "text": "please kindly subscribe to my channel"}, {"start": 208.84, "duration": 4.72, "text": "also like comment and share bye-bye"}], "title": "Interesting equation | Can you solve ?", "description": "", "lengthSeconds": 212, "width": 640, "height": 360}
{"video_id": "4LZgjLjL9C4", "transcript": [{"start": 0.399, "duration": 7.44, "text": "hello welcome back here we have another"}, {"start": 4.279, "duration": 5.561, "text": "interesting Olympiad match so but this"}, {"start": 7.839, "duration": 4.161, "text": "is Jun"}, {"start": 9.84, "duration": 8.96, "text": "Olympiad here we given the following"}, {"start": 12.0, "duration": 12.16, "text": "equation x + 1 x is = 3 so we are asked"}, {"start": 18.8, "duration": 8.399, "text": "to evaluate X2 + 1 /"}, {"start": 24.16, "duration": 5.84, "text": "X2 obviously solving for the value of"}, {"start": 27.199, "duration": 6.081, "text": "values of X in this equation before"}, {"start": 30.0, "duration": 6.719, "text": "evaluating this seems not to be the best"}, {"start": 33.28, "duration": 7.52, "text": "way to solve this problem this problem"}, {"start": 36.719, "duration": 8.241, "text": "must have an easier way to deal with"}, {"start": 40.8, "duration": 7.88, "text": "it okay so now let us see how we're"}, {"start": 44.96, "duration": 7.399, "text": "going to find the value of X2 + 1 / X2"}, {"start": 48.68, "duration": 10.76, "text": "without first figuring out the value of"}, {"start": 52.359, "duration": 7.081, "text": "values of X in this equation right okay"}, {"start": 61.68, "duration": 9.84, "text": "now let us recall this algebraic"}, {"start": 65.36, "duration": 15.439, "text": "identity we have that a s + b 2 is equal"}, {"start": 71.52, "duration": 12.08, "text": "to a + b 2 - to a right so this identity"}, {"start": 80.799, "duration": 5.481, "text": "is going to help us so much we're going"}, {"start": 83.6, "duration": 7.519, "text": "to use this identity to find the value"}, {"start": 86.28, "duration": 7.24, "text": "of X2 + 1 / X2 which that first solving"}, {"start": 91.119, "duration": 5.481, "text": "for the value of values of X in this"}, {"start": 93.52, "duration": 7.36, "text": "equation so now we can see here we have"}, {"start": 96.6, "duration": 7.4, "text": "a 2 + b s and we have similar expression"}, {"start": 100.88, "duration": 7.72, "text": "here which have power two and here power"}, {"start": 104.0, "duration": 8.68, "text": "two you know 1 / x\u00b2 is the same as 1 /x"}, {"start": 108.6, "duration": 7.799, "text": "then everything squared right okay now"}, {"start": 112.68, "duration": 5.68, "text": "let us reite this expression so we're"}, {"start": 116.399, "duration": 4.161, "text": "going to have here"}, {"start": 118.36, "duration": 3.84, "text": "x\u00b2 +"}, {"start": 120.56, "duration": 5.239, "text": "1 /"}, {"start": 122.2, "duration": 7.039, "text": "X2 to be equal to you know here we have"}, {"start": 125.799, "duration": 5.481, "text": "a so here we're going to have X here"}, {"start": 129.239, "duration": 5.881, "text": "then plus you know the whole of this"}, {"start": 131.28, "duration": 6.08, "text": "thing is squared just like I showed here"}, {"start": 135.12, "duration": 8.399, "text": "so this is going to be here 1"}, {"start": 137.36, "duration": 9.239, "text": "/x then squ minus here we have 2 * a so"}, {"start": 143.519, "duration": 8.241, "text": "here a is X here right so we're to have"}, {"start": 146.599, "duration": 8.481, "text": "X then times here we have 1 X you know a"}, {"start": 151.76, "duration": 6.839, "text": "here a so we're going to have 1 x here"}, {"start": 155.08, "duration": 10.6, "text": "right so we can say that this cancel out"}, {"start": 158.599, "duration": 16.161, "text": "right so this is going to be X2 + 1 / X2"}, {"start": 165.68, "duration": 11.4, "text": "to be equal to x + 1 /x 2 then - 2 right"}, {"start": 174.76, "duration": 6.52, "text": "so this is going to be equal to now what"}, {"start": 177.08, "duration": 8.04, "text": "is the value for x + 1 / X so we have it"}, {"start": 181.28, "duration": 9.44, "text": "right over here so this is going to be 3"}, {"start": 185.12, "duration": 9.759, "text": "S then - 2 you know 3 S is 9 then we"}, {"start": 190.72, "duration": 9.36, "text": "have - 2 and this is equal to 7 so"}, {"start": 194.879, "duration": 7.64, "text": "therefore the value for X2 + 1 / X2 is 7"}, {"start": 200.08, "duration": 5.28, "text": "thank you for watching please do well to"}, {"start": 202.519, "duration": 7.08, "text": "follow this page subscribe to my YouTube"}, {"start": 205.36, "duration": 4.239, "text": "channel and share this video"}], "title": "Interesting Junior Olympiad Math", "description": "", "lengthSeconds": 209, "width": 640, "height": 360}
{"video_id": "3CAFnMrZ9W8", "transcript": [{"start": 0.12, "duration": 3.639, "text": "hello welcome back once again please"}, {"start": 2.48, "duration": 4.64, "text": "subscribe to my"}, {"start": 3.759, "duration": 5.121, "text": "Channel please like comment and share"}, {"start": 7.12, "duration": 5.72, "text": "here we're giving the"}, {"start": 8.88, "duration": 11.04, "text": "following equation right over here 8^ of"}, {"start": 12.84, "duration": 10.4, "text": "n + 8^ of n + 8^ of n all 4^ n + 4^ n"}, {"start": 19.92, "duration": 5.519, "text": "this is equal to 48 and right here we"}, {"start": 23.24, "duration": 3.76, "text": "asked to find the value of n sorry for"}, {"start": 25.439, "duration": 8.64, "text": "the X here"}, {"start": 27.0, "duration": 13.44, "text": "right okay so here we have a ^ n + a^ n"}, {"start": 34.079, "duration": 10.721, "text": "+ a^ N All 4 the power of n + 4 the ^ of"}, {"start": 40.44, "duration": 9.439, "text": "N and this is equal to"}, {"start": 44.8, "duration": 9.36, "text": "48 okay mind you here we have x 8 to^ n"}, {"start": 49.879, "duration": 6.881, "text": "+ 8^ n + 8 ^ of n let us recall that"}, {"start": 54.16, "duration": 6.359, "text": "when we have x + x you know this is"}, {"start": 56.76, "duration": 7.96, "text": "equal to 2x we have x + x"}, {"start": 60.519, "duration": 7.721, "text": "+ x this is equal to 3x that is 3 * X"}, {"start": 64.72, "duration": 7.96, "text": "right so right over here we have a^ of n"}, {"start": 68.24, "duration": 10.239, "text": "+ a^ of n + a^ n so this going to be 3 *"}, {"start": 72.68, "duration": 10.96, "text": "^ of n right so 3 * 8^ n and then here"}, {"start": 78.479, "duration": 7.441, "text": "we have 2 * 4^ n right then this is"}, {"start": 83.64, "duration": 6.439, "text": "equal to"}, {"start": 85.92, "duration": 8.28, "text": "48 okay now we're going to express8"}, {"start": 90.079, "duration": 9.241, "text": "an index Power of Two And also four we"}, {"start": 94.2, "duration": 7.76, "text": "all know 8 is = 2 cubed and 4 is = to S"}, {"start": 99.32, "duration": 6.04, "text": "right so here we're going to have three"}, {"start": 101.96, "duration": 9.72, "text": "times right over here we have 2 cubed of"}, {"start": 105.36, "duration": 8.28, "text": "n then all over 2 * here 2^ 2 of n this"}, {"start": 111.68, "duration": 4.719, "text": "is equal to"}, {"start": 113.64, "duration": 4.399, "text": "48 so according to the law of indices"}, {"start": 116.399, "duration": 2.921, "text": "we're going to multiply these Powers"}, {"start": 118.039, "duration": 8.72, "text": "together so right over here we're going"}, {"start": 119.32, "duration": 10.2, "text": "to have three * 2^ 3 n all over 2 * 2^ 2"}, {"start": 126.759, "duration": 4.921, "text": "N we multiply this Powers together and"}, {"start": 129.52, "duration": 5.799, "text": "this is equal to"}, {"start": 131.68, "duration": 7.639, "text": "48 okay so from"}, {"start": 135.319, "duration": 6.441, "text": "here let us multiply both sides by two"}, {"start": 139.319, "duration": 5.241, "text": "so here we multiply by two and here we"}, {"start": 141.76, "duration": 5.8, "text": "multiply by two so these two cancel out"}, {"start": 144.56, "duration": 5.92, "text": "this two and from here we have equal"}, {"start": 147.56, "duration": 9.36, "text": "bases so we're going to subtract the so"}, {"start": 150.48, "duration": 10.64, "text": "here we have 3 * 2^ 3 N - 2 N and this"}, {"start": 156.92, "duration": 7.64, "text": "is equal to 48 * 2 so let us go ahead"}, {"start": 161.12, "duration": 8.44, "text": "and divide both sides by three so here"}, {"start": 164.56, "duration": 6.8, "text": "by three and here by three right okay so"}, {"start": 169.56, "duration": 5.959, "text": "here three cancel three and right over"}, {"start": 171.36, "duration": 5.959, "text": "here we have 2 of 3 N - 2 N that is n so"}, {"start": 175.519, "duration": 4.841, "text": "this is equal to you know here three"}, {"start": 177.319, "duration": 7.081, "text": "here 1 and three here is 16"}, {"start": 180.36, "duration": 6.879, "text": "right so right over here we have 16 * 2"}, {"start": 184.4, "duration": 6.479, "text": "so that is going to give us 32 and this"}, {"start": 187.239, "duration": 6.841, "text": "32 is the same as 2 to^ of 5 so from"}, {"start": 190.879, "duration": 6.561, "text": "here we have 2 ^ of n is = 2 the Power"}, {"start": 194.08, "duration": 6.56, "text": "of 5 so we can see we have equal bases"}, {"start": 197.44, "duration": 5.84, "text": "so we can easily equate the powers and"}, {"start": 200.64, "duration": 6.0, "text": "therefore we find the value of n to be"}, {"start": 203.28, "duration": 5.56, "text": "equal to five thank you for watching"}, {"start": 206.64, "duration": 7.12, "text": "please kindly subscribe to my Channel"}, {"start": 208.84, "duration": 4.92, "text": "please like comment and share bye-bye"}], "title": "Interesting exponential equation.", "description": "", "lengthSeconds": 212, "width": 640, "height": 360}
{"video_id": "iRBvaLyZfUQ", "transcript": [{"start": 0.399, "duration": 2.841, "text": "good day viewers welcome back to my"}, {"start": 1.8, "duration": 4.12, "text": "Channel today we have the following"}, {"start": 3.24, "duration": 7.96, "text": "limit problem limit as X is approaching"}, {"start": 5.92, "duration": 7.799, "text": "one the root of x - 1 x -1 was touch wow"}, {"start": 11.2, "duration": 6.28, "text": "I can say that this limit is extremely"}, {"start": 13.719, "duration": 7.48, "text": "beautiful because of the way the limit L"}, {"start": 17.48, "duration": 9.24, "text": "wow now I'm going to start by making a"}, {"start": 21.199, "duration": 9.92, "text": "substitution so let squ otk of x = k"}, {"start": 26.72, "duration": 7.839, "text": "Square both side we obtain x = k two if"}, {"start": 31.119, "duration": 6.44, "text": "our X is approaching one k is also"}, {"start": 34.559, "duration": 6.761, "text": "approaching one so our limit becomes"}, {"start": 37.559, "duration": 9.801, "text": "limit as K is approaching one here we"}, {"start": 41.32, "duration": 9.8, "text": "have Kus one all over you know here k 2"}, {"start": 47.36, "duration": 7.16, "text": "- 1 okay now we using the difference of"}, {"start": 51.12, "duration": 7.2, "text": "two squares formula that a 2 - b\u00b2 is"}, {"start": 54.52, "duration": 6.92, "text": "equal to the product of a + b and a"}, {"start": 58.32, "duration": 7.119, "text": "minus B okay so our limit"}, {"start": 61.44, "duration": 7.96, "text": "becomes limit as K is approaching 1 up"}, {"start": 65.439, "duration": 10.521, "text": "here we have kus1 all over down here K"}, {"start": 69.4, "duration": 10.759, "text": "-1 and k + 1 so K -1 gets R kus1 so we"}, {"start": 75.96, "duration": 9.36, "text": "have what limit K is approaching one of"}, {"start": 80.159, "duration": 8.081, "text": "1 k + 1 and is equal to 1 1 + 1 and is"}, {"start": 85.32, "duration": 5.04, "text": "equal to 1 over 2 which is our desired"}, {"start": 88.24, "duration": 5.04, "text": "answer thank you for viewing"}, {"start": 90.36, "duration": 7.68, "text": "please subscribe to my YouTube channel"}, {"start": 93.28, "duration": 4.76, "text": "for more interesting videos"}], "title": "A very nice and interesting limit. Avoid L'hospital's rule !!!", "description": "", "lengthSeconds": 96, "width": 636, "height": 360}
{"video_id": "sT42wkRFElU", "transcript": [{"start": 0.719, "duration": 5.001, "text": "hello welcome back once again today we"}, {"start": 2.8, "duration": 5.24, "text": "have this olymp mathematics problem"}, {"start": 5.72, "duration": 8.2, "text": "we're given that sare root of A+ square"}, {"start": 8.04, "duration": 11.56, "text": "root of B is = 2 a * B is = 4 so here we"}, {"start": 13.92, "duration": 9.64, "text": "be evaluating a + b right so from"}, {"start": 19.6, "duration": 8.24, "text": "equation one we have root of"}, {"start": 23.56, "duration": 6.479, "text": "A+ square root of B which is equal to 2"}, {"start": 27.84, "duration": 5.96, "text": "now let's go ahead and square both sides"}, {"start": 30.039, "duration": 5.761, "text": "of this equation okay so from the left"}, {"start": 33.8, "duration": 9.16, "text": "hand side let us make use of this"}, {"start": 35.8, "duration": 12.399, "text": "property we know a + b 2 is = to a 2 + b"}, {"start": 42.96, "duration": 6.919, "text": "2 + 2 * a * B so from here we're going"}, {"start": 48.199, "duration": 5.601, "text": "to have the first term in the bracket"}, {"start": 49.879, "duration": 6.68, "text": "squar which is root a 2 plus the second"}, {"start": 53.8, "duration": 7.719, "text": "term which is root B"}, {"start": 56.559, "duration": 7.801, "text": "2+ 2 MTI the first term of a * the"}, {"start": 61.519, "duration": 5.6, "text": "second term root of B and this will be"}, {"start": 64.36, "duration": 4.04, "text": "equal to 2 s from the right hand side"}, {"start": 67.119, "duration": 3.921, "text": "which is"}, {"start": 68.4, "duration": 4.84, "text": "4 from here we know Square will cancel"}, {"start": 71.04, "duration": 7.52, "text": "square root square will cancel square"}, {"start": 73.24, "duration": 9.4, "text": "root so we left with a plus b then plus"}, {"start": 78.56, "duration": 7.68, "text": "we have here 2 now root a * root B will"}, {"start": 82.64, "duration": 8.439, "text": "be same as square root of a * B under"}, {"start": 86.24, "duration": 8.28, "text": "one root sign then this is equal to 4"}, {"start": 91.079, "duration": 6.04, "text": "now let us recall back we know a * B is"}, {"start": 94.52, "duration": 6.4, "text": "= 4 so therefore we're going to"}, {"start": 97.119, "duration": 6.721, "text": "substitute the value of a * B here so we"}, {"start": 100.92, "duration": 7.76, "text": "get a +"}, {"start": 103.84, "duration": 8.919, "text": "B+ 2 square < TK of 4 right because a *"}, {"start": 108.68, "duration": 11.359, "text": "B is 4 this will be equal to 4 so here"}, {"start": 112.759, "duration": 12.881, "text": "we have a + b + 2 * > 4 is 2 then this"}, {"start": 120.039, "duration": 9.84, "text": "is equal to 4 from here we get a + b + 2"}, {"start": 125.64, "duration": 6.56, "text": "* 2 is 4 which isal to 4 now subtract"}, {"start": 129.879, "duration": 6.561, "text": "four from both sides of this equation so"}, {"start": 132.2, "duration": 8.92, "text": "we have a + b + 4 - 4 this will be = to"}, {"start": 136.44, "duration": 7.76, "text": "4 - 4 so from here this is zero right so"}, {"start": 141.12, "duration": 7.08, "text": "we get our right answer to the problem A"}, {"start": 144.2, "duration": 6.399, "text": "+ B = 4 - 4 which is 0 and this is the"}, {"start": 148.2, "duration": 4.88, "text": "right answer than thank you for watching"}, {"start": 150.599, "duration": 8.041, "text": "please kindly subscribe to my channel"}, {"start": 153.08, "duration": 5.56, "text": "also like comment and share bye-bye"}], "title": "Olympiad Math | Can you solve ?", "description": "", "lengthSeconds": 157, "width": 640, "height": 360}
{"video_id": "ldTfEOLm6_0", "transcript": [{"start": 0.96, "duration": 5.2, "text": "hello welcome back once again today"}, {"start": 2.639, "duration": 5.481, "text": "we're going to solve this nice equation"}, {"start": 6.16, "duration": 6.359, "text": "so from the left hand side we have this"}, {"start": 8.12, "duration": 7.8, "text": "square root of we know X Cub + x Cub is"}, {"start": 12.519, "duration": 6.961, "text": "2 X Cub right and from the right hand"}, {"start": 15.92, "duration": 5.76, "text": "side this is equal to x^ 2 now we're"}, {"start": 19.48, "duration": 4.08, "text": "going to square both sides to get rid of"}, {"start": 21.68, "duration": 4.599, "text": "this root sign so let's Square both"}, {"start": 23.56, "duration": 2.719, "text": "sides of the"}, {"start": 26.88, "duration": 3.6, "text": "equation so we know the square we cancel"}, {"start": 29.359, "duration": 5.36, "text": "the square root"}, {"start": 30.48, "duration": 7.16, "text": "so leaving us with 2 x Cub so this is"}, {"start": 34.719, "duration": 7.241, "text": "equal to you know 2 * 2 is 4 so that"}, {"start": 37.64, "duration": 5.759, "text": "would be x to^ 4 right okay now let us"}, {"start": 41.96, "duration": 2.84, "text": "make the right hand side the left hand"}, {"start": 43.399, "duration": 6.761, "text": "side and the left hand side the right"}, {"start": 44.8, "duration": 8.96, "text": "hand side so we now have x^ 4 is = 2 X"}, {"start": 50.16, "duration": 8.879, "text": "Cub right so here subtract 2 x Cub from"}, {"start": 53.76, "duration": 6.92, "text": "both side so this will be x^ 4 - 2 X Cub"}, {"start": 59.039, "duration": 4.36, "text": "this is equal Z"}, {"start": 60.68, "duration": 5.28, "text": "from the left hand side x Cub is common"}, {"start": 63.399, "duration": 7.881, "text": "so let's Factor it out so we're going to"}, {"start": 65.96, "duration": 8.76, "text": "have x^ 4 x Cub will give us x - 2x Cub"}, {"start": 71.28, "duration": 6.4, "text": "/ X Cub will give us 2 and this is equal"}, {"start": 74.72, "duration": 6.719, "text": "to Z so we have these two products which"}, {"start": 77.68, "duration": 9.96, "text": "is equal to Z so we have either X cubed"}, {"start": 81.439, "duration": 8.32, "text": "is equal to 0 or x - 2 is = 0 so from"}, {"start": 87.64, "duration": 5.439, "text": "here take the Cub root of both side we"}, {"start": 89.759, "duration": 5.441, "text": "get get x = 0 and this is one of our"}, {"start": 93.079, "duration": 5.72, "text": "solution and from here add two to both"}, {"start": 95.2, "duration": 6.72, "text": "side we get x = 2 so these are the only"}, {"start": 98.799, "duration": 5.481, "text": "two possible real solutions that satisfy"}, {"start": 101.92, "duration": 4.0, "text": "the given equation thank you for"}, {"start": 104.28, "duration": 5.04, "text": "watching please kindly subscribe to my"}, {"start": 105.92, "duration": 6.4, "text": "channel also like comment and share"}, {"start": 109.32, "duration": 3.0, "text": "bye-bye"}], "title": "Canada | Nice Olympiad Math problem | can you solve ?", "description": "", "lengthSeconds": 111, "width": 428, "height": 240}
{"video_id": "60Ys5jBo3x8", "transcript": [{"start": 1.0, "duration": 3.88, "text": "hello good viewers welcome back once"}, {"start": 2.84, "duration": 4.88, "text": "again please kindly subscribe to my"}, {"start": 4.88, "duration": 5.679, "text": "channel also like comment and share"}, {"start": 7.72, "duration": 4.919, "text": "today we have this nice exponential"}, {"start": 10.559, "duration": 6.121, "text": "equation problem we're going to solve"}, {"start": 12.639, "duration": 7.8, "text": "for x we are given that 4 to the^ of x +"}, {"start": 16.68, "duration": 6.64, "text": "4 ^ of X is = to 8 here we're going to"}, {"start": 20.439, "duration": 9.281, "text": "solve for the value of x right okay"}, {"start": 23.32, "duration": 9.32, "text": "let's get started we have 4 ^ x + 4 ^ x"}, {"start": 29.72, "duration": 6.72, "text": "this is equal to 8 so from the left hand"}, {"start": 32.64, "duration": 7.759, "text": "side we're going to factor out 4^ of X"}, {"start": 36.44, "duration": 6.76, "text": "and here we have 4^ x / 4^ X we have 1"}, {"start": 40.399, "duration": 6.921, "text": "plus also here we have 1 and this is"}, {"start": 43.2, "duration": 8.8, "text": "equal to 8 right okay so from here 1 + 1"}, {"start": 47.32, "duration": 7.64, "text": "is 2 so here we have 2 * 4 ^ of X this"}, {"start": 52.0, "duration": 6.519, "text": "is equal to 8 so from here let's go"}, {"start": 54.96, "duration": 6.88, "text": "ahead and divide both sides by two so"}, {"start": 58.519, "duration": 9.081, "text": "divide this so so here we get 4 to the^"}, {"start": 61.84, "duration": 8.599, "text": "X is equal to you know 8 / 2 that is 4"}, {"start": 67.6, "duration": 5.76, "text": "right so here you know this four is the"}, {"start": 70.439, "duration": 7.241, "text": "same as 4 to the^ of 1 and from here we"}, {"start": 73.36, "duration": 8.48, "text": "have equal bases can actually equate the"}, {"start": 77.68, "duration": 6.399, "text": "powers and this gives us X isal to 1 and"}, {"start": 81.84, "duration": 3.959, "text": "that is the right answer thank you for"}, {"start": 84.079, "duration": 4.121, "text": "watching please do wa to subscribe to my"}, {"start": 85.799, "duration": 5.401, "text": "channel also like comment and share"}, {"start": 88.2, "duration": 3.0, "text": "bye-bye"}], "title": "A very simple exponential equation. For beginners, here is step by step on how to solve this problem", "description": "", "lengthSeconds": 89, "width": 640, "height": 360}
{"video_id": "I6fB2qcOO3Y", "transcript": [{"start": 0.64, "duration": 4.8, "text": "hello good viewers welcome back once"}, {"start": 2.76, "duration": 5.44, "text": "again here we have another interesting"}, {"start": 5.44, "duration": 6.119, "text": "algebraic problem here we're given that"}, {"start": 8.2, "duration": 9.36, "text": "A + B is = 10 and right over here we ask"}, {"start": 11.559, "duration": 9.281, "text": "to evaluate a - 7 ^ 7 + b - 3^ of 7"}, {"start": 17.56, "duration": 5.28, "text": "right okay now let's get started from"}, {"start": 20.84, "duration": 8.0, "text": "this equation right over here we have"}, {"start": 22.84, "duration": 9.599, "text": "that A + B is equal to 10 so from here"}, {"start": 28.84, "duration": 6.32, "text": "we can solve for a in terms of B that is"}, {"start": 32.439, "duration": 7.8, "text": "making a the subject of formula here so"}, {"start": 35.16, "duration": 7.64, "text": "we're going to have a is equal to 10 - B"}, {"start": 40.239, "duration": 5.401, "text": "right okay in the given expression right"}, {"start": 42.8, "duration": 5.399, "text": "over here we are going to substitute the"}, {"start": 45.64, "duration": 5.04, "text": "value for a in terms of B so here we"}, {"start": 48.199, "duration": 10.721, "text": "have a -"}, {"start": 50.68, "duration": 14.159, "text": "7^ 7 + b - 3^ of 7 so this is equal to"}, {"start": 58.92, "duration": 12.16, "text": "so here we have a that is 10 - b - 7 ra"}, {"start": 64.839, "duration": 11.841, "text": "power of 7 then plus b - 3 power of"}, {"start": 71.08, "duration": 6.719, "text": "7 right okay so from here we own that"}, {"start": 76.68, "duration": 5.479, "text": "here we"}, {"start": 77.799, "duration": 12.32, "text": "have we have here 10 - 7 right so to"}, {"start": 82.159, "duration": 10.64, "text": "group it then minus B 7 + b - 3 7 so"}, {"start": 90.119, "duration": 5.96, "text": "this is equal to here here we have 7 10"}, {"start": 92.799, "duration": 9.081, "text": "- 7 which is 3 right so here we have 3 -"}, {"start": 96.079, "duration": 9.441, "text": "B 7 + b - 3^"}, {"start": 101.88, "duration": 5.96, "text": "7 okay so from here this is equal to you"}, {"start": 105.52, "duration": 5.879, "text": "know here we can Factor"}, {"start": 107.84, "duration": 5.599, "text": "out1 right so let us factor out1 from"}, {"start": 111.399, "duration": 5.121, "text": "here so here we're going to open a"}, {"start": 113.439, "duration": 6.081, "text": "bigger bracket so factor out -1 then"}, {"start": 116.52, "duration": 5.0, "text": "Open Bracket you know this is going to"}, {"start": 119.52, "duration": 5.48, "text": "affect what we have inside this bracket"}, {"start": 121.52, "duration": 9.88, "text": "so it will become b - 3 right then the"}, {"start": 125.0, "duration": 8.44, "text": "whole of this of 7 then plus b - 3 7 so"}, {"start": 131.4, "duration": 6.6, "text": "this is equal to here we're going to"}, {"start": 133.44, "duration": 11.84, "text": "make use of this law so when we have a *"}, {"start": 138.0, "duration": 10.64, "text": "m of n this is same as a of n * m of n"}, {"start": 145.28, "duration": 11.48, "text": "right so from here this going to be -1"}, {"start": 148.64, "duration": 13.319, "text": "of 7 then time b - 3 7 right then plus"}, {"start": 156.76, "duration": 9.6, "text": "here b - 3^ of 7 so from here this is"}, {"start": 161.959, "duration": 8.041, "text": "equal to you know 1^ of an odd power"}, {"start": 166.36, "duration": 11.239, "text": "will give us -1 so we have Open Bracket"}, {"start": 170.0, "duration": 12.8, "text": "b - 3 7 + b - 3^ 7 and from here you"}, {"start": 177.599, "duration": 7.72, "text": "know netive B - 3 7 + b - 3 7 will give"}, {"start": 182.8, "duration": 4.359, "text": "us zero and here comes the right answer"}, {"start": 185.319, "duration": 3.121, "text": "to the question thank you for watching"}, {"start": 187.159, "duration": 3.44, "text": "please do wait to subscribe to my"}, {"start": 188.44, "duration": 5.0, "text": "YouTube Channel please like comment and"}, {"start": 190.599, "duration": 2.841, "text": "share"}], "title": "Olympiad Math problem. How to evaluate?", "description": "", "lengthSeconds": 192, "width": 640, "height": 360}
{"video_id": "i7GlxobJx8E", "transcript": [{"start": 1.0, "duration": 6.2, "text": "hello welcome back once again today we"}, {"start": 3.12, "duration": 7.399, "text": "have this Olympiad maths we have a 2us"}, {"start": 7.2, "duration": 6.639, "text": "199 a ="}, {"start": 10.519, "duration": 9.68, "text": "199 so here we're going to evaluate this"}, {"start": 13.839, "duration": 7.6, "text": "expression here a ^ 4 + a 2 + 1 / a cub"}, {"start": 20.199, "duration": 3.84, "text": "+"}, {"start": 21.439, "duration": 5.92, "text": "1 first of all let us write this"}, {"start": 24.039, "duration": 8.441, "text": "expression in the simplified"}, {"start": 27.359, "duration": 7.321, "text": "form the quatic expression here can be"}, {"start": 32.48, "duration": 5.2, "text": "factorized so I think I will discuss"}, {"start": 34.68, "duration": 5.32, "text": "that in my next video when you factorize"}, {"start": 37.68, "duration": 11.28, "text": "this we're going to"}, {"start": 40.0, "duration": 14.0, "text": "have a 2 + a + 1 mtip by a\u00b2 - A +"}, {"start": 48.96, "duration": 6.96, "text": "1 right then divided by so the"}, {"start": 54.0, "duration": 4.92, "text": "denominator here we're going to make use"}, {"start": 55.92, "duration": 6.799, "text": "of the sum of two cubes you know we have"}, {"start": 58.92, "duration": 6.64, "text": "a cub + 1 which you can write as 1 cubed"}, {"start": 62.719, "duration": 11.841, "text": "so this will be equal to"}, {"start": 65.56, "duration": 13.28, "text": "recall a cub + B Cub = A + B * a 2 - A B"}, {"start": 74.56, "duration": 9.919, "text": "+ b 2 so make use of this of this sum of"}, {"start": 78.84, "duration": 9.56, "text": "two cubes formula so we get a + 1 * a 2"}, {"start": 84.479, "duration": 10.6, "text": "- A + 1 s which is 1 right so here we"}, {"start": 88.4, "duration": 10.64, "text": "get a + 1 1 * a 2 - A + 1 so of which"}, {"start": 95.079, "duration": 9.841, "text": "this we cancel this and here we have a 2"}, {"start": 99.04, "duration": 9.8, "text": "+ a + 1 / a + 1 now let's split this"}, {"start": 104.92, "duration": 12.68, "text": "denominator as follow here a"}, {"start": 108.84, "duration": 12.8, "text": "2 / a + 1 then + a + 1 / a +"}, {"start": 117.6, "duration": 7.36, "text": "1 so this will cancel is living one"}, {"start": 121.64, "duration": 5.32, "text": "right so here we"}, {"start": 124.96, "duration": 9.24, "text": "get"}, {"start": 126.96, "duration": 10.12, "text": "a\u00b2 / a + 1 + 1 right so this is the"}, {"start": 134.2, "duration": 5.2, "text": "simplified form of this expression right"}, {"start": 137.08, "duration": 6.08, "text": "over"}, {"start": 139.4, "duration": 7.479, "text": "here okay now from the original given"}, {"start": 143.16, "duration": 7.719, "text": "equation remember we have"}, {"start": 146.879, "duration": 5.921, "text": "a\u00b2 - 199 a"}, {"start": 150.879, "duration": 5.0, "text": "this equals"}, {"start": 152.8, "duration": 4.88, "text": "199 so now this term here which is"}, {"start": 155.879, "duration": 3.72, "text": "negative let's move it to the right hand"}, {"start": 157.68, "duration": 4.839, "text": "side and that will become positive so we"}, {"start": 159.599, "duration": 8.201, "text": "get a\u00b2 ="}, {"start": 162.519, "duration": 7.681, "text": "199 + 199 a right so here"}, {"start": 167.8, "duration": 7.519, "text": "remember is common right so we're going"}, {"start": 170.2, "duration": 7.24, "text": "to factor it out so we get a\u00b2 = 199 into"}, {"start": 175.319, "duration": 6.361, "text": "bracket here will be 1 and here will be"}, {"start": 177.44, "duration": 6.799, "text": "a which is same as a + 1 right so here"}, {"start": 181.68, "duration": 5.36, "text": "the value for a s we're going to"}, {"start": 184.239, "duration": 5.881, "text": "substitute it here so this will be equal"}, {"start": 187.04, "duration": 6.64, "text": "to here instead of a s we use that so"}, {"start": 190.12, "duration": 7.399, "text": "that is 199 into bracket a + 1 then"}, {"start": 193.68, "duration": 6.8, "text": "remember we're dividing by a + 1 then +"}, {"start": 197.519, "duration": 6.921, "text": "1 so this equals here this cancel this"}, {"start": 200.48, "duration": 7.839, "text": "and here we get 199 + 1 which is equal"}, {"start": 204.44, "duration": 6.04, "text": "to 200 which is the right answer thank"}, {"start": 208.319, "duration": 5.081, "text": "you for watching please can subscribe to"}, {"start": 210.48, "duration": 5.92, "text": "my channel also like comment and share"}, {"start": 213.4, "duration": 3.0, "text": "bye-bye"}], "title": "Germany | Nice Olympiad Math | Find out the trick in this video", "description": "", "lengthSeconds": 215, "width": 640, "height": 360}
{"video_id": "qFzStPMBJSk", "transcript": [{"start": 0.84, "duration": 4.8, "text": "hello good viewers welcome back once"}, {"start": 2.679, "duration": 7.121, "text": "again let's find a solution to this"}, {"start": 5.64, "duration": 7.4, "text": "exponential equation we have the X root"}, {"start": 9.8, "duration": 6.079, "text": "ofx this is equal to"}, {"start": 13.04, "duration": 5.88, "text": "27 before we proceed it will be very"}, {"start": 15.879, "duration": 5.281, "text": "necessary to make a substitution"}, {"start": 18.92, "duration": 7.119, "text": "here we can"}, {"start": 21.16, "duration": 10.599, "text": "let x = - y now when you multiply both"}, {"start": 26.039, "duration": 7.84, "text": "side by -1 we also getx is = 1 y now the"}, {"start": 31.759, "duration": 6.241, "text": "left hand side of this equation we can"}, {"start": 33.879, "duration": 10.761, "text": "rewrite it make use of this identity the"}, {"start": 38.0, "duration": 8.039, "text": "mot of n is same as n^ 1 / n so from the"}, {"start": 44.64, "duration": 6.0, "text": "left hand side we"}, {"start": 46.039, "duration": 6.0, "text": "havex to the^ of 1 /x and this is equal"}, {"start": 50.64, "duration": 4.8, "text": "to"}, {"start": 52.039, "duration": 5.561, "text": "27 now making use of the"}, {"start": 55.44, "duration": 6.919, "text": "substitution X is"}, {"start": 57.6, "duration": 10.04, "text": "y then ra power of 1 over X which is"}, {"start": 62.359, "duration": 11.161, "text": "what - Y and this is equal to 27 from"}, {"start": 67.64, "duration": 8.0, "text": "here we get X sorry I mean y of -1 / y"}, {"start": 73.52, "duration": 5.239, "text": "this is equal to"}, {"start": 75.64, "duration": 11.119, "text": "27 now the power of Y here we can write"}, {"start": 78.759, "duration": 9.761, "text": "it as -1 * 1 / Y and this is equal to 27"}, {"start": 86.759, "duration": 6.081, "text": "now from here we're going to make use of"}, {"start": 88.52, "duration": 8.4, "text": "this property a a power of B * C this is"}, {"start": 92.84, "duration": 9.0, "text": "same as a power of B in bracket then"}, {"start": 96.92, "duration": 10.559, "text": "power of C so from here this becomes y^"}, {"start": 101.84, "duration": 8.4, "text": "of1 of Y 1 / y I mean this is equal to"}, {"start": 107.479, "duration": 7.561, "text": "27 and you know this is the reciprocal"}, {"start": 110.24, "duration": 8.68, "text": "of Y which is the same as 1 / y^ of 1 /"}, {"start": 115.04, "duration": 7.48, "text": "Y and from the right hand side 27 is 3"}, {"start": 118.92, "duration": 7.72, "text": "of 3 now from here this implies that 1 /"}, {"start": 122.52, "duration": 5.959, "text": "Y is = to 3 this is over one when we"}, {"start": 126.64, "duration": 6.52, "text": "cross"}, {"start": 128.479, "duration": 8.12, "text": "multiply we get 3 Y is = 1 and from here"}, {"start": 133.16, "duration": 8.68, "text": "we get Y is = 1 /"}, {"start": 136.599, "duration": 10.161, "text": "3 okay now let us"}, {"start": 141.84, "duration": 7.0, "text": "recall let us recall that X is = y so"}, {"start": 146.76, "duration": 3.88, "text": "from here we come up with the solution"}, {"start": 148.84, "duration": 6.36, "text": "that X isal"}, {"start": 150.64, "duration": 6.08, "text": "to1 /3 and this is the right answer"}, {"start": 155.2, "duration": 3.48, "text": "thank you for watching please kindly"}, {"start": 156.72, "duration": 6.04, "text": "subscribe to my channel also like"}, {"start": 158.68, "duration": 4.08, "text": "comment and share bye-bye"}], "title": "Indians | Nice exponential equation for Olympiad | #maths #mathematics", "description": "", "lengthSeconds": 161, "width": 564, "height": 360}
{"video_id": "65wzLlOMl98", "transcript": [{"start": 0.52, "duration": 4.879, "text": "hello welcome back once again today"}, {"start": 3.24, "duration": 4.88, "text": "we're going to solve this radical"}, {"start": 5.399, "duration": 5.601, "text": "equation here we have the square root of"}, {"start": 8.12, "duration": 6.36, "text": "Cub root of x this is equal to the Cub"}, {"start": 11.0, "duration": 7.52, "text": "root of x here we're going to solve for"}, {"start": 14.48, "duration": 8.52, "text": "real possible values of X right now"}, {"start": 18.52, "duration": 7.44, "text": "let's get started let us take note of"}, {"start": 23.0, "duration": 8.439, "text": "this generally the end"}, {"start": 25.96, "duration": 10.8, "text": "root of a is equal to a"}, {"start": 31.439, "duration": 9.881, "text": "of 1 / n right so therefore the square"}, {"start": 36.76, "duration": 8.639, "text": "root of let's say k will be same as K to"}, {"start": 41.32, "duration": 5.719, "text": "the^ of 1 / 2 right so from the left"}, {"start": 45.399, "duration": 3.64, "text": "hand side we have the square root of"}, {"start": 47.039, "duration": 3.16, "text": "cube root of x so we're going to write"}, {"start": 49.039, "duration": 4.641, "text": "it"}, {"start": 50.199, "duration": 9.36, "text": "as the cube root of"}, {"start": 53.68, "duration": 8.039, "text": "x then R of 1 / 2 then this will be"}, {"start": 59.559, "duration": 4.6, "text": "equal to from the right hand side make"}, {"start": 61.719, "duration": 9.561, "text": "use of these algebraic properties so we"}, {"start": 64.159, "duration": 10.881, "text": "have here x ra power of 1 / 3 right okay"}, {"start": 71.28, "duration": 6.56, "text": "now do not forget here in this base here"}, {"start": 75.04, "duration": 6.84, "text": "we still have the CU root of x so we can"}, {"start": 77.84, "duration": 11.279, "text": "which we can actually re write again as"}, {"start": 81.88, "duration": 11.919, "text": "x^ of 1 / 3 then in bracket ra the^ of 1"}, {"start": 89.119, "duration": 8.721, "text": "/ 2 right then this will be equal to X"}, {"start": 93.799, "duration": 8.441, "text": "to the^ of 1 / 3 now take note of this"}, {"start": 97.84, "duration": 7.4, "text": "again when we have a to the power of M"}, {"start": 102.24, "duration": 9.28, "text": "then in bracket to the power of n this"}, {"start": 105.24, "duration": 13.159, "text": "is same as a m * n right so from here we"}, {"start": 111.52, "duration": 11.279, "text": "have here x 1 3 * 1 / 2 this is 1/ 3 * 1"}, {"start": 118.399, "duration": 8.08, "text": "/ 2 so this will give us 1 / 6 so right"}, {"start": 122.799, "duration": 8.881, "text": "over here we have 1 / 6 then this is"}, {"start": 126.479, "duration": 9.161, "text": "equal to X to the^ of 1 / 3 right okay"}, {"start": 131.68, "duration": 8.48, "text": "now from here let us check the LCM of 6"}, {"start": 135.64, "duration": 8.04, "text": "and three we all know the LCM of 6 and"}, {"start": 140.16, "duration": 6.359, "text": "three or 3 and 6 will be equal to 6"}, {"start": 143.68, "duration": 6.24, "text": "right so here we're going to raise both"}, {"start": 146.519, "duration": 9.281, "text": "sides to the power of six so here we go"}, {"start": 149.92, "duration": 12.0, "text": "we have here x^ 1 / 6 to^ 6 this will be"}, {"start": 155.8, "duration": 8.96, "text": "equal to x^ of 1 / 3 to the^ of 6 so"}, {"start": 161.92, "duration": 5.28, "text": "from here make use of this property here"}, {"start": 164.76, "duration": 6.96, "text": "this particular one here so this is"}, {"start": 167.2, "duration": 7.52, "text": "going to be 1/ 6 * 6 is 1 right so we"}, {"start": 171.72, "duration": 6.48, "text": "are left with X here then from the right"}, {"start": 174.72, "duration": 6.36, "text": "6 * 1 / 3 will give us 2 right so from"}, {"start": 178.2, "duration": 4.44, "text": "the right hand side we have x\u00b2"}, {"start": 181.08, "duration": 5.04, "text": "then this is equal to from the left hand"}, {"start": 182.64, "duration": 7.48, "text": "side we have only X okay so from here"}, {"start": 186.12, "duration": 7.8, "text": "subtract X from both sides we get x^2 -"}, {"start": 190.12, "duration": 6.399, "text": "x this is equal to Z so from here x is"}, {"start": 193.92, "duration": 8.0, "text": "common so we Factor it out and we have"}, {"start": 196.519, "duration": 8.041, "text": "here x -1 this is equal to Z right okay"}, {"start": 201.92, "duration": 5.44, "text": "so from here when we have these two"}, {"start": 204.56, "duration": 6.52, "text": "product here and they are equal to zero"}, {"start": 207.36, "duration": 9.12, "text": "so we have either this x here is equal 0"}, {"start": 211.08, "duration": 6.6, "text": "right and this is a solution or x -1 is"}, {"start": 216.48, "duration": 5.319, "text": "="}, {"start": 217.68, "duration": 7.279, "text": "0 of which this implies that X is equal"}, {"start": 221.799, "duration": 7.08, "text": "to 1 so here we have the only two R"}, {"start": 224.959, "duration": 6.761, "text": "solutions that satisfy the given radical"}, {"start": 228.879, "duration": 5.64, "text": "equation thank you for watching please"}, {"start": 231.72, "duration": 6.96, "text": "kindly subscribe to my channel also like"}, {"start": 234.519, "duration": 4.161, "text": "comment and share bye-bye"}], "title": "Interesting radical equation| Can you solve ?", "description": "", "lengthSeconds": 237, "width": 640, "height": 360}
{"video_id": "d9naVdjC4VU", "transcript": [{"start": 0.88, "duration": 5.28, "text": "hello welcome back once again today we"}, {"start": 3.24, "duration": 9.84, "text": "have beautiful systems of equations we"}, {"start": 6.16, "duration": 13.16, "text": "have x * y * Z 2 x + y + z = 6 x y * y z"}, {"start": 13.08, "duration": 11.88, "text": "+ x z = 11 you going to evaluate x / y z"}, {"start": 19.32, "duration": 14.559, "text": "+ y / x z + z / XY right okay from"}, {"start": 24.96, "duration": 11.279, "text": "equation 2 we have x + y + z = 6 let's"}, {"start": 33.879, "duration": 4.921, "text": "go ahead and square both"}, {"start": 36.239, "duration": 9.201, "text": "sides so from the left here this is"}, {"start": 38.8, "duration": 11.12, "text": "exactly equals X2 + Y 2 + Z 2 then plus"}, {"start": 45.44, "duration": 8.279, "text": "the other terms we have 2 * 2 * 2 * so"}, {"start": 49.92, "duration": 11.72, "text": "we can just factor out 2 and we get x y"}, {"start": 53.719, "duration": 7.921, "text": "+ y z + x Z then this equals 36"}, {"start": 61.8, "duration": 10.08, "text": "okay so from here let's solve for X2 + y"}, {"start": 65.36, "duration": 11.68, "text": "+ Z 2 so we have here X2 + Y 2 + Z 2"}, {"start": 71.88, "duration": 9.4, "text": "here plus here x * y * x * y + y * Z + x"}, {"start": 77.04, "duration": 7.96, "text": "* Z = 11 right so this will be 2 * 11"}, {"start": 81.28, "duration": 7.199, "text": "which is 22 and this will be"}, {"start": 85.0, "duration": 9.799, "text": "36 so from here"}, {"start": 88.479, "duration": 11.801, "text": "x2+ y 2 + z^ 2 this = 36 -"}, {"start": 94.799, "duration": 8.201, "text": "22 36 - 22 that will give us 14 right"}, {"start": 100.28, "duration": 4.04, "text": "okay now take a look at the expression"}, {"start": 103.0, "duration": 3.68, "text": "we asked to"}, {"start": 104.32, "duration": 3.88, "text": "evaluate now have you thought of"}, {"start": 106.68, "duration": 8.079, "text": "something like this"}, {"start": 108.2, "duration": 9.879, "text": "X2 + Y 2 + Z 2 when you divide this by x"}, {"start": 114.759, "duration": 13.521, "text": "* y * Z splitting the denominator we get"}, {"start": 118.079, "duration": 16.481, "text": "x2 / x * y * Z + y^ 2 / x * y * Z + Z 2"}, {"start": 128.28, "duration": 10.239, "text": "/ x * y * Z which is equal to here x"}, {"start": 134.56, "duration": 7.52, "text": "X2 will be reduced by X here so we get X"}, {"start": 138.519, "duration": 7.36, "text": "all over y z right so we get here x over"}, {"start": 142.08, "duration": 9.2, "text": "y z and plus from the other side that"}, {"start": 145.879, "duration": 10.0, "text": "will be y over that is x z plus the"}, {"start": 151.28, "duration": 6.239, "text": "other side will be Z all over X Y wow"}, {"start": 155.879, "duration": 4.041, "text": "and we can see this is exactly the"}, {"start": 157.519, "duration": 6.401, "text": "extion we ask to evaluate but the"}, {"start": 159.92, "duration": 6.52, "text": "question here is what exactly yelds this"}, {"start": 163.92, "duration": 5.56, "text": "here it is so this expression right of"}, {"start": 166.44, "duration": 5.64, "text": "over here equals this so our answer will"}, {"start": 169.48, "duration": 7.679, "text": "be the value of this ID by the value of"}, {"start": 172.08, "duration": 8.159, "text": "this and the value of x + y + Z2 = 14 ID"}, {"start": 177.159, "duration": 6.08, "text": "by the value of x y z given in the first"}, {"start": 180.239, "duration": 6.041, "text": "equation equals 2 so here we found the"}, {"start": 183.239, "duration": 5.321, "text": "solution which is s and here is the"}, {"start": 186.28, "duration": 4.879, "text": "right answer thank you for watching"}, {"start": 188.56, "duration": 7.36, "text": "please kindly subscribe to my channel"}, {"start": 191.159, "duration": 4.761, "text": "also like comment and share bye-bye"}], "title": "Interesting Olympiad Math", "description": "", "lengthSeconds": 194, "width": 640, "height": 360}
{"video_id": "91G8OVY_SwI", "transcript": [{"start": 0.64, "duration": 5.0, "text": "hello welcome back once again today we"}, {"start": 3.159, "duration": 6.4, "text": "have this nice Olympiad mathematics"}, {"start": 5.64, "duration": 7.44, "text": "problem we have x + 1 /x is = tok3 so"}, {"start": 9.559, "duration": 6.64, "text": "here we be evaluating x -1"}, {"start": 13.08, "duration": 8.08, "text": "/x now let's get into the video from the"}, {"start": 16.199, "duration": 9.24, "text": "given equation we have x + 1"}, {"start": 21.16, "duration": 6.56, "text": "/x which is equal to < tk3 now let's go"}, {"start": 25.439, "duration": 5.281, "text": "ahead and square both sides of this"}, {"start": 27.72, "duration": 5.12, "text": "equation"}, {"start": 30.72, "duration": 7.56, "text": "now from the left hand side here let us"}, {"start": 32.84, "duration": 8.8, "text": "make use of this property a + b s this"}, {"start": 38.28, "duration": 9.64, "text": "is equal to ter of aus B so that will be"}, {"start": 41.64, "duration": 7.919, "text": "a - b 2 + 4 * a * B so make use of this"}, {"start": 47.92, "duration": 4.6, "text": "property at the left hand side so that"}, {"start": 49.559, "duration": 8.921, "text": "will become x -1"}, {"start": 52.52, "duration": 9.08, "text": "/x then 2 + 4 *"}, {"start": 58.48, "duration": 5.88, "text": "x * 1"}, {"start": 61.6, "duration": 9.0, "text": "/x then this will be equal to here"}, {"start": 64.36, "duration": 6.24, "text": "Square we cancel root sign so we have 3"}, {"start": 72.92, "duration": 13.96, "text": "okay so from here we have X - 1"}, {"start": 80.2, "duration": 10.559, "text": "/x 2 then plus here x cancel X here so"}, {"start": 86.88, "duration": 7.4, "text": "we have four then this is equal to 3"}, {"start": 90.759, "duration": 8.36, "text": "subtract four from both sides we get X -"}, {"start": 94.28, "duration": 7.72, "text": "1 / x^ 2 this will be equal to 3 - 4"}, {"start": 99.119, "duration": 4.841, "text": "which is -1 take root of both sides and"}, {"start": 102.0, "duration": 7.079, "text": "from here we have our Solution that's x"}, {"start": 103.96, "duration": 8.04, "text": "-1 / X is equal to plus or minus theun"}, {"start": 109.079, "duration": 6.801, "text": "of1 and this is plus or minus the"}, {"start": 112.0, "duration": 10.6, "text": "imaginary unit I right so our solution"}, {"start": 115.88, "duration": 8.96, "text": "for x -1 /x is = to Plus or minus Iota"}, {"start": 122.6, "duration": 4.76, "text": "thank you for watching please kindly"}, {"start": 124.84, "duration": 6.88, "text": "subscribe to my channel also like"}, {"start": 127.36, "duration": 4.36, "text": "comment and share bye-bye"}], "title": "Interesting Olympiad Math | Can you solve ?", "description": "", "lengthSeconds": 130, "width": 640, "height": 360}
{"video_id": "pXZVgSu1Xn8", "transcript": [{"start": 0.32, "duration": 4.08, "text": "good day viewers welcome back to my"}, {"start": 2.2, "duration": 5.88, "text": "Channel today we have the following"}, {"start": 4.4, "duration": 9.92, "text": "interesting limit problem limit as X is"}, {"start": 8.08, "duration": 10.119, "text": "approaching 0 1us the cine of X all 2x^"}, {"start": 14.32, "duration": 8.08, "text": "2 = that so we're having a restation"}, {"start": 18.199, "duration": 9.441, "text": "here so we are not going to make use of"}, {"start": 22.4, "duration": 7.879, "text": "the L hital rule in this case so this is"}, {"start": 27.64, "duration": 5.439, "text": "not allowed okay so let's how we're"}, {"start": 30.279, "duration": 5.761, "text": "going to solve this problem okay so I'm"}, {"start": 33.079, "duration": 6.761, "text": "going to use the idea of the power"}, {"start": 36.04, "duration": 7.4, "text": "series expansion of the cosine function"}, {"start": 39.84, "duration": 10.44, "text": "okay so I'm going to do it as"}, {"start": 43.44, "duration": 14.799, "text": "follows COS of X is = to 1 -"}, {"start": 50.28, "duration": 11.88, "text": "x 2 / 2 factorial + x 4/ 4 factori minus"}, {"start": 58.239, "duration": 7.761, "text": "and so on okay so so from here we can"}, {"start": 62.16, "duration": 12.04, "text": "see that 1 - the cosine of x is the same"}, {"start": 66.0, "duration": 11.84, "text": "as 1 - 1 - x 2 / 2 factorial + x 4 / 4"}, {"start": 74.2, "duration": 6.599, "text": "factorial minus and so on so by so doing"}, {"start": 77.84, "duration": 4.88, "text": "we get this equals so you know opening"}, {"start": 80.799, "duration": 6.201, "text": "this bracket with this negative sign so"}, {"start": 82.72, "duration": 9.8, "text": "1 - 1 is 0 then we have x^ 2 / 2"}, {"start": 87.0, "duration": 10.36, "text": "factorial - x 4 for factoral plus and so"}, {"start": 92.52, "duration": 9.32, "text": "on okay so and this is equal to 1us the"}, {"start": 97.36, "duration": 11.719, "text": "cosine of x so dividing both side by 2"}, {"start": 101.84, "duration": 13.04, "text": "X2 okay here over 2 x\u00b2 so we get 1 - the"}, {"start": 109.079, "duration": 9.761, "text": "cosine of x all/ 2 x 2 is = to you know"}, {"start": 114.88, "duration": 6.04, "text": "here x\u00b2 cancel x s here so we get 1 / 2"}, {"start": 118.84, "duration": 7.4, "text": "factorial /"}, {"start": 120.92, "duration": 10.92, "text": "2 minus you know x\u00b2 reduces this to x 2"}, {"start": 126.24, "duration": 8.68, "text": "/ 4 factoral over 2 plus and so on okay"}, {"start": 131.84, "duration": 7.28, "text": "so from here we can see that the"}, {"start": 134.92, "duration": 9.12, "text": "limit as X is approaching Z of 1us the"}, {"start": 139.12, "duration": 9.96, "text": "cosine of x all 2 x^2 is the same as"}, {"start": 144.04, "duration": 11.68, "text": "limit as X is approaching Z of 1 / 2"}, {"start": 149.08, "duration": 12.519, "text": "factorial 2 - x^2 / 4 factorial / 2 plus"}, {"start": 155.72, "duration": 8.28, "text": "and so on okay so from here we can see"}, {"start": 161.599, "duration": 4.521, "text": "that as X is approaching zero the whole"}, {"start": 164.0, "duration": 4.64, "text": "of the terms here that goes on to"}, {"start": 166.12, "duration": 5.92, "text": "Infinity will be approaching zero so we"}, {"start": 168.64, "duration": 7.08, "text": "are left with 1 all 2 Factor all over 2"}, {"start": 172.04, "duration": 8.119, "text": "and this equal to 1 / 2 / 2 so this is"}, {"start": 175.72, "duration": 6.599, "text": "equal to 1 4 so our limit equals one all"}, {"start": 180.159, "duration": 3.64, "text": "over four thank you for viewing and"}, {"start": 182.319, "duration": 3.361, "text": "please make sure you subscribe to my"}, {"start": 183.799, "duration": 4.881, "text": "YouTube channel for more interesting"}, {"start": 185.68, "duration": 3.0, "text": "videos"}], "title": "Limit. Can you do this without L'hospital's rule? Check out the video.", "description": "", "lengthSeconds": 187, "width": 636, "height": 360}
{"video_id": "NxxkoRCtE0k", "transcript": [{"start": 1.04, "duration": 3.96, "text": "hello welcome back once again today"}, {"start": 3.76, "duration": 6.04, "text": "we'll be"}, {"start": 5.0, "duration": 7.519, "text": "evaluating the four power of < TK of 45"}, {"start": 9.8, "duration": 7.319, "text": "+ 5un 5 all"}, {"start": 12.519, "duration": 7.721, "text": "/ < TK 20 okay let's get started first"}, {"start": 17.119, "duration": 6.841, "text": "of all we need to take note of this we"}, {"start": 20.24, "duration": 3.72, "text": "know Square < TK of"}, {"start": 24.599, "duration": 7.12, "text": "45 is equal"}, {"start": 27.48, "duration": 9.88, "text": "to square < TK of 45 is the same as"}, {"start": 31.719, "duration": 11.32, "text": "9 * 5 right so this will be equal"}, {"start": 37.36, "duration": 9.76, "text": "to square < TK of 9 then multip"}, {"start": 43.039, "duration": 9.121, "text": "by square < TK of 5 right so this is"}, {"start": 47.12, "duration": 10.959, "text": "equal to root of 9 is 3 so we have 3 <"}, {"start": 52.16, "duration": 8.919, "text": "TK 5 right okay and again we have square"}, {"start": 58.079, "duration": 6.601, "text": "root of 20 right over here"}, {"start": 61.079, "duration": 6.481, "text": "so we know sare < TK"}, {"start": 64.68, "duration": 6.079, "text": "20 is equal"}, {"start": 67.56, "duration": 6.239, "text": "to square < TK of"}, {"start": 70.759, "duration": 5.521, "text": "4 *"}, {"start": 73.799, "duration": 4.441, "text": "5 and this is equal"}, {"start": 76.28, "duration": 4.159, "text": "to < TK"}, {"start": 78.24, "duration": 4.44, "text": "4"}, {"start": 80.439, "duration": 8.201, "text": "* < TK"}, {"start": 82.68, "duration": 9.079, "text": "5 we know otk of 4 is 2 so this becomes"}, {"start": 88.64, "duration": 7.4, "text": "2un 5"}, {"start": 91.759, "duration": 8.841, "text": "okay so this whole expression here will"}, {"start": 96.04, "duration": 4.56, "text": "become so this whole expression"}, {"start": 100.759, "duration": 3.241, "text": "here will"}, {"start": 104.28, "duration": 7.32, "text": "become so that will become in"}, {"start": 107.84, "duration": 6.76, "text": "bracket 3 < TK"}, {"start": 111.6, "duration": 3.0, "text": "5"}, {"start": 115.2, "duration": 5.08, "text": "plus"}, {"start": 116.759, "duration": 5.121, "text": "5 < TK 5"}, {"start": 120.28, "duration": 9.479, "text": "ID"}, {"start": 121.88, "duration": 7.879, "text": "by 2 < TK 5 then all rais the power of"}, {"start": 131.16, "duration": 10.96, "text": "4 okay so this will be equal to into"}, {"start": 135.72, "duration": 12.48, "text": "braet here 3un 5 + 5un 5 will give us 8"}, {"start": 142.12, "duration": 11.88, "text": "< TK 5 right then divid by 2un"}, {"start": 148.2, "duration": 9.52, "text": "5 ID by 2 < TK 5 all to the^ of 4 so"}, {"start": 154.0, "duration": 8.0, "text": "this will be equal to we know < TK 5 we"}, {"start": 157.72, "duration": 9.36, "text": "Cancun 5 from here and 8 / 2 is 4 so"}, {"start": 162.0, "duration": 8.72, "text": "this will be 4 ra^ of 4 right so it is"}, {"start": 167.08, "duration": 5.84, "text": "easy for one to know that 4^ 4 means 4 *"}, {"start": 170.72, "duration": 10.12, "text": "4 *"}, {"start": 172.92, "duration": 13.319, "text": "4 * 4 right which is equal to 16 * 16"}, {"start": 180.84, "duration": 10.16, "text": "because from here this is 16 and here"}, {"start": 186.239, "duration": 6.681, "text": "this is 16 okay and 16 * 16 is easily"}, {"start": 191.0, "duration": 5.04, "text": "gotten as"}, {"start": 192.92, "duration": 5.36, "text": "256 and this is the right answer thank"}, {"start": 196.04, "duration": 5.6, "text": "you for watching please kindly subscribe"}, {"start": 198.28, "duration": 6.92, "text": "to my channel also like comment and"}, {"start": 201.64, "duration": 3.56, "text": "share bye-bye"}], "title": "Interesting problem | No calculators !!!", "description": "", "lengthSeconds": 208, "width": 618, "height": 360}
{"video_id": "gLvcmjjTxrY", "transcript": [{"start": 0.64, "duration": 5.239, "text": "hello welcome back once again to my"}, {"start": 3.04, "duration": 5.28, "text": "YouTube Channel please kindly support"}, {"start": 5.879, "duration": 4.8, "text": "this channel by"}, {"start": 8.32, "duration": 5.84, "text": "subscribing please subscribe to my"}, {"start": 10.679, "duration": 6.76, "text": "YouTube channel and share my videos"}, {"start": 14.16, "duration": 6.6, "text": "please subscribe here we are asked to"}, {"start": 17.439, "duration": 9.401, "text": "evaluate this expression right over here"}, {"start": 20.76, "duration": 10.96, "text": "586 s - 16 all divide by"}, {"start": 26.84, "duration": 6.68, "text": "590 right so let's get started"}, {"start": 31.72, "duration": 3.92, "text": "here we have"}, {"start": 33.52, "duration": 4.44, "text": "586"}, {"start": 35.64, "duration": 5.56, "text": "SAR -"}, {"start": 37.96, "duration": 7.0, "text": "16 all over"}, {"start": 41.2, "duration": 7.879, "text": "590 so this is equal to"}, {"start": 44.96, "duration": 5.52, "text": "586 squ minus you know 16 is the same as"}, {"start": 49.079, "duration": 5.521, "text": "4"}, {"start": 50.48, "duration": 7.0, "text": "squar all/ by"}, {"start": 54.6, "duration": 5.88, "text": "590 okay before we proceed let us recall"}, {"start": 57.48, "duration": 3.96, "text": "this difference of two squares formula"}, {"start": 60.48, "duration": 9.639, "text": "a"}, {"start": 61.44, "duration": 10.6, "text": "2 - b 2 is = to a + b multi a minus B"}, {"start": 70.119, "duration": 4.241, "text": "right so we're going to make use of this"}, {"start": 72.04, "duration": 5.56, "text": "property here so this is going to be"}, {"start": 74.36, "duration": 6.32, "text": "equal to in the first bracket here we"}, {"start": 77.6, "duration": 5.6, "text": "have a s and here we have a so here we"}, {"start": 80.68, "duration": 4.64, "text": "have 586 squ so the first bracket we"}, {"start": 83.2, "duration": 5.72, "text": "have"}, {"start": 85.32, "duration": 6.24, "text": "586 then plus b that is the base here"}, {"start": 88.92, "duration": 6.199, "text": "then the base here is four so we have +"}, {"start": 91.56, "duration": 5.559, "text": "4 MTI by"}, {"start": 95.119, "duration": 6.28, "text": "586 -"}, {"start": 97.119, "duration": 8.521, "text": "4 all divide by"}, {"start": 101.399, "duration": 6.441, "text": "590 so this is equal to you know 586 + 4"}, {"start": 105.64, "duration": 4.6, "text": "will give us"}, {"start": 107.84, "duration": 6.12, "text": "590 *"}, {"start": 110.24, "duration": 8.6, "text": "586 - 4 will give us"}, {"start": 113.96, "duration": 8.0, "text": "582 right all divide by"}, {"start": 118.84, "duration": 7.72, "text": "590 so we can see that this 590 we"}, {"start": 121.96, "duration": 8.4, "text": "cancel out so we are left with"}, {"start": 126.56, "duration": 6.16, "text": "582 and this is the right answer please"}, {"start": 130.36, "duration": 6.0, "text": "and please subscribe to this YouTube"}, {"start": 132.72, "duration": 8.08, "text": "channel and share our videos remember to"}, {"start": 136.36, "duration": 4.44, "text": "subscribe thank you"}], "title": "The difference of two squares is a way out.", "description": "", "lengthSeconds": 139, "width": 636, "height": 360}
{"video_id": "KhfY-tW0n6g", "transcript": [{"start": 0.44, "duration": 4.72, "text": "hello welcome back once again today we"}, {"start": 2.919, "duration": 4.761, "text": "have this interesting equation involving"}, {"start": 5.16, "duration": 6.639, "text": "cube root and fourth root so here will"}, {"start": 7.68, "duration": 6.999, "text": "be soling for the real values of X right"}, {"start": 11.799, "duration": 6.761, "text": "now let's get started let us take note"}, {"start": 14.679, "duration": 7.961, "text": "of this property we know the end root of"}, {"start": 18.56, "duration": 8.28, "text": "a that is generally this is equal to a"}, {"start": 22.64, "duration": 7.719, "text": "to the^ 1 divid n so therefore the cube"}, {"start": 26.84, "duration": 5.36, "text": "root of x will be"}, {"start": 30.359, "duration": 5.921, "text": "let's put it down first we have cube"}, {"start": 32.2, "duration": 7.199, "text": "root of x which is equal to the fourth"}, {"start": 36.28, "duration": 8.16, "text": "root of x so with this identity here"}, {"start": 39.399, "duration": 6.961, "text": "cube root of x will become x^ 1 / 3 then"}, {"start": 44.44, "duration": 7.72, "text": "this will be equal to 4 root of x will"}, {"start": 46.36, "duration": 8.08, "text": "be x^ of 1/ 4 right okay now we're going"}, {"start": 52.16, "duration": 5.44, "text": "to raise both sides of this equation to"}, {"start": 54.44, "duration": 5.84, "text": "a specific power and how do we know the"}, {"start": 57.6, "duration": 5.439, "text": "power to raise them two right we need to"}, {"start": 60.28, "duration": 5.199, "text": "get rid of this cube root and default"}, {"start": 63.039, "duration": 6.4, "text": "root so we can get a standard equation"}, {"start": 65.479, "duration": 8.201, "text": "to solve okay so what are we going to do"}, {"start": 69.439, "duration": 5.441, "text": "let us look for the LCM of 3 and four we"}, {"start": 73.68, "duration": 7.759, "text": "know the"}, {"start": 74.88, "duration": 8.599, "text": "LCM of 3 and 4 is equal to 12 right so"}, {"start": 81.439, "duration": 4.841, "text": "we're going to raise both sides of this"}, {"start": 83.479, "duration": 7.64, "text": "equation to the power of 12 so here we"}, {"start": 86.28, "duration": 11.64, "text": "have X to the^ 1 / 3 to the^ of 12 this"}, {"start": 91.119, "duration": 12.801, "text": "will be equal to x^ 1 4^ 12 we know this"}, {"start": 97.92, "duration": 9.76, "text": "property a^ b the^ c is equal to a^ B *"}, {"start": 103.92, "duration": 8.519, "text": "C so with this property we have here x^"}, {"start": 107.68, "duration": 10.52, "text": "1 3 * 12 is as 12 / 3 then this will be"}, {"start": 112.439, "duration": 8.64, "text": "equal to x^ 12 * 1 4 is 12 / 4 so we can"}, {"start": 118.2, "duration": 6.839, "text": "see this can divide nicely right so we"}, {"start": 121.079, "duration": 8.081, "text": "have here x^ 12 3 is 4 then this is"}, {"start": 125.039, "duration": 7.161, "text": "equal to x^ 12/ 4 is three so we have"}, {"start": 129.16, "duration": 4.92, "text": "this standard standard quartic equation"}, {"start": 132.2, "duration": 6.119, "text": "to solve so let's move this term to the"}, {"start": 134.08, "duration": 8.28, "text": "left hand side so we have x to^ 4 - x"}, {"start": 138.319, "duration": 5.681, "text": "Cub this is equal to0 so many may think"}, {"start": 142.36, "duration": 4.92, "text": "that the best way to solve this equation"}, {"start": 144.0, "duration": 5.36, "text": "from here is to divide both side by x^ 3"}, {"start": 147.28, "duration": 5.12, "text": "my dear if you do that you're going to"}, {"start": 149.36, "duration": 5.92, "text": "lose a solution of x is equal to Z which"}, {"start": 152.4, "duration": 4.8, "text": "is very true right so the best way to"}, {"start": 155.28, "duration": 4.039, "text": "solve this kind of equation is to move"}, {"start": 157.2, "duration": 4.36, "text": "the term from the right to the left and"}, {"start": 159.319, "duration": 4.56, "text": "the sign will change right and from here"}, {"start": 161.56, "duration": 5.679, "text": "we can see x^ 3 is common so let's"}, {"start": 163.879, "duration": 6.72, "text": "Factor it out we have x^ 3 into bracket"}, {"start": 167.239, "duration": 8.161, "text": "x^ 4 / X Cub will give us just"}, {"start": 170.599, "duration": 6.521, "text": "xus x^ 3 / x^ 3 is one and therefore we"}, {"start": 175.4, "duration": 4.0, "text": "have this equal"}, {"start": 177.12, "duration": 4.119, "text": "Z okay so from here we have these two"}, {"start": 179.4, "duration": 7.52, "text": "products which is equal to 0 so we have"}, {"start": 181.239, "duration": 8.72, "text": "either x^ 3 is = 0 or we have X - 1 is ="}, {"start": 186.92, "duration": 7.399, "text": "0 so take root of both side we get X"}, {"start": 189.959, "duration": 6.84, "text": "isal to 0 so this is a solution from"}, {"start": 194.319, "duration": 5.681, "text": "here add one to both sides we get the"}, {"start": 196.799, "duration": 5.921, "text": "final solution X isal to 1 and here we"}, {"start": 200.0, "duration": 5.44, "text": "have the two real solutions that satisfy"}, {"start": 202.72, "duration": 5.2, "text": "the given interesing"}, {"start": 205.44, "duration": 5.12, "text": "equation thank you for watching please"}, {"start": 207.92, "duration": 7.2, "text": "kindly subscribe to my channel so like"}, {"start": 210.56, "duration": 4.56, "text": "comment and share bye-bye"}], "title": "Interesting radical equation | Can you solve ?", "description": "", "lengthSeconds": 214, "width": 640, "height": 360}
{"video_id": "XO44B14XdBM", "transcript": [{"start": 0.52, "duration": 5.0, "text": "hello good day viewers welcome back once"}, {"start": 2.6, "duration": 6.119, "text": "again please kindly support this by"}, {"start": 5.52, "duration": 4.88, "text": "subscribing also like comment and share"}, {"start": 8.719, "duration": 5.601, "text": "today we're going to find the derivative"}, {"start": 10.4, "duration": 9.76, "text": "of x to the^ of X right with respect to"}, {"start": 14.32, "duration": 7.68, "text": "X here we have y is = X to the power of"}, {"start": 20.16, "duration": 4.72, "text": "X"}, {"start": 22.0, "duration": 5.08, "text": "right so here we're going to take the"}, {"start": 24.88, "duration": 5.12, "text": "natural log of both sides so here we"}, {"start": 27.08, "duration": 6.04, "text": "have the natural log of Y this is equal"}, {"start": 30.0, "duration": 6.12, "text": "to the natural log of x to the power of"}, {"start": 33.12, "duration": 5.2, "text": "X now with one of the properties of log"}, {"start": 36.12, "duration": 4.84, "text": "gring that is the natural log of a to"}, {"start": 38.32, "duration": 5.16, "text": "the^ B is equal to B MTI the natural log"}, {"start": 40.96, "duration": 5.68, "text": "of a so here we're going to have the"}, {"start": 43.48, "duration": 7.079, "text": "natural log of Y is equal to x * the"}, {"start": 46.64, "duration": 7.079, "text": "natural log of x right so from here"}, {"start": 50.559, "duration": 5.281, "text": "we're going to differentiate now by the"}, {"start": 53.719, "duration": 5.201, "text": "right hand side we're going to make use"}, {"start": 55.84, "duration": 6.32, "text": "of the product rule right so when we"}, {"start": 58.92, "duration": 7.76, "text": "have the derivative with respect to"}, {"start": 62.16, "duration": 10.959, "text": "X of let's say we have"}, {"start": 66.68, "duration": 13.04, "text": "ax time BX right so this is equal"}, {"start": 73.119, "duration": 10.721, "text": "to a x Prime means derivative of ax *"}, {"start": 79.72, "duration": 7.52, "text": "BX then plus BX Prime is the derivative"}, {"start": 83.84, "duration": 6.52, "text": "of BX * a x"}, {"start": 87.24, "duration": 6.4, "text": "right okay"}, {"start": 90.36, "duration": 5.36, "text": "you know here ax and BX are different"}, {"start": 93.64, "duration": 5.439, "text": "functions of X"}, {"start": 95.72, "duration": 6.759, "text": "right okay so now here we're going to"}, {"start": 99.079, "duration": 6.801, "text": "have the derivative of natural log of Y"}, {"start": 102.479, "duration": 6.92, "text": "this is going to be 1 over y so did not"}, {"start": 105.88, "duration": 8.519, "text": "forget here we're going to have then d y"}, {"start": 109.399, "duration": 9.561, "text": "DX right so this is equal to here we're"}, {"start": 114.399, "duration": 6.801, "text": "going to have the derivative of x here"}, {"start": 118.96, "duration": 6.839, "text": "according to this up to here the"}, {"start": 121.2, "duration": 9.08, "text": "derivative of x here is 1 then times"}, {"start": 125.799, "duration": 7.16, "text": "natural log of x right then here"}, {"start": 130.28, "duration": 4.88, "text": "plus the derivative of the natural log"}, {"start": 132.959, "duration": 7.161, "text": "of x is 1"}, {"start": 135.16, "duration": 6.439, "text": "X then times x right making use of this"}, {"start": 140.12, "duration": 5.119, "text": "property"}, {"start": 141.599, "duration": 8.481, "text": "right okay from here we're going to"}, {"start": 145.239, "duration": 5.761, "text": "simplify here we have 1 / y time the Y"}, {"start": 150.08, "duration": 4.04, "text": "the"}, {"start": 151.0, "duration": 7.56, "text": "X this is equal to here we have the"}, {"start": 154.12, "duration": 7.479, "text": "natural log of x plus here this this so"}, {"start": 158.56, "duration": 5.0, "text": "we have here one right so from here"}, {"start": 161.599, "duration": 7.28, "text": "we're going to multiply both side by Y"}, {"start": 163.56, "duration": 9.52, "text": "and here we get d y DX is going to be Y"}, {"start": 168.879, "duration": 7.761, "text": "into bracket the natural log of x + 1 so"}, {"start": 173.08, "duration": 8.28, "text": "do not forget that Y is exactly equal to"}, {"start": 176.64, "duration": 9.2, "text": "x^ of X so here we get x to the of X"}, {"start": 181.36, "duration": 7.599, "text": "into bracket the natural log of x + 1"}, {"start": 185.84, "duration": 5.36, "text": "and this is the right answer thank you"}, {"start": 188.959, "duration": 4.721, "text": "for watching please kindly subscribe to"}, {"start": 191.2, "duration": 5.48, "text": "my channel also like comment and share"}, {"start": 193.68, "duration": 3.0, "text": "bye-bye"}], "title": "Differentiation. The derivative of  x^x. #education #maths", "description": "", "lengthSeconds": 196, "width": 640, "height": 360}
{"video_id": "WV3VkGdpF2o", "transcript": [{"start": 0.84, "duration": 4.88, "text": "hello good viewers welcome back once"}, {"start": 2.72, "duration": 6.52, "text": "again let's find a solution to this"}, {"start": 5.72, "duration": 7.76, "text": "Olympiad mat problem we're given that X2"}, {"start": 9.24, "duration": 5.439, "text": "+ 1 x\u00b2 is = to root of 13 right over"}, {"start": 13.48, "duration": 4.559, "text": "here we're going to evaluate this"}, {"start": 14.679, "duration": 6.52, "text": "expression x\u00b2 - 1"}, {"start": 18.039, "duration": 6.441, "text": "/x2 now let's get started before we"}, {"start": 21.199, "duration": 7.601, "text": "proceed there is a very nice identity"}, {"start": 24.48, "duration": 16.28, "text": "we're going to make use of remember that"}, {"start": 28.8, "duration": 17.04, "text": "A+ b 2 this is same as a - b 2 + 4 a"}, {"start": 40.76, "duration": 7.479, "text": "b okay now from equation one we have X2"}, {"start": 45.84, "duration": 6.519, "text": "+ 1 /"}, {"start": 48.239, "duration": 6.0, "text": "X2 this is equal to root of 13 now let's"}, {"start": 52.359, "duration": 5.801, "text": "go ahead and square both sides of this"}, {"start": 54.239, "duration": 6.521, "text": "equation so that we can make use of this"}, {"start": 58.16, "duration": 5.999, "text": "identity now from the left side if you"}, {"start": 60.76, "duration": 9.12, "text": "make use of this identity we have here"}, {"start": 64.159, "duration": 12.28, "text": "X2 - 1 / X2 then"}, {"start": 69.88, "duration": 12.0, "text": "squ + 4 * a here will be"}, {"start": 76.439, "duration": 7.201, "text": "X2 * 1 / x\u00b2 right then this is equal to"}, {"start": 81.88, "duration": 3.84, "text": "you know here the square and the root"}, {"start": 83.64, "duration": 3.04, "text": "sign we cancel out and we're left with"}, {"start": 85.72, "duration": 3.88, "text": "just"}, {"start": 86.68, "duration": 5.92, "text": "13 now from here don't forget that x\u00b2"}, {"start": 89.6, "duration": 11.04, "text": "and 1 / X2 we cancel out and here we"}, {"start": 92.6, "duration": 11.159, "text": "have X2 - 1 / X2 2 then + 4 this is"}, {"start": 100.64, "duration": 5.56, "text": "equal to 13 now let's go ahead and"}, {"start": 103.759, "duration": 9.561, "text": "subtract four from both side we get from"}, {"start": 106.2, "duration": 10.4, "text": "here x^2 - 1 / x^2 2 this is = to 13 - 4"}, {"start": 113.32, "duration": 7.2, "text": "which is 9 now from here let's take root"}, {"start": 116.6, "duration": 9.08, "text": "of both sides and here we get x2 -"}, {"start": 120.52, "duration": 7.799, "text": "1 / X2 this is equal to plus orus otk 9"}, {"start": 125.68, "duration": 6.12, "text": "and from here this is exactly equal to"}, {"start": 128.319, "duration": 5.56, "text": "plus or minus 3 and this is the right"}, {"start": 131.8, "duration": 4.519, "text": "answer thank you for watching please"}, {"start": 133.879, "duration": 6.521, "text": "kindly subscribe to my channel also like"}, {"start": 136.319, "duration": 4.081, "text": "comment and share bye-bye"}], "title": "Ukrainians | Nice Olympiad Math | Here is the trick | #maths #mathematics", "description": "", "lengthSeconds": 139, "width": 534, "height": 360}
{"video_id": "_E0_9UWUQNA", "transcript": [{"start": 1.079, "duration": 4.961, "text": "hello welcome back once again today we"}, {"start": 3.719, "duration": 4.361, "text": "have this interesting exponential"}, {"start": 6.04, "duration": 5.599, "text": "equation and here we're going to be"}, {"start": 8.08, "duration": 6.16, "text": "solving for the values of X right so"}, {"start": 11.639, "duration": 4.001, "text": "let's get started so from the left hand"}, {"start": 14.24, "duration": 6.48, "text": "side we have"}, {"start": 15.64, "duration": 10.24, "text": "9 of < TK x +"}, {"start": 20.72, "duration": 7.799, "text": "9 < TK X and this isal to"}, {"start": 25.88, "duration": 5.719, "text": "54 so from the left hand side we can see"}, {"start": 28.519, "duration": 9.321, "text": "that 9^ ofun X is common so let's Factor"}, {"start": 31.599, "duration": 11.48, "text": "it out so here we have 9 ofun x into"}, {"start": 37.84, "duration": 12.039, "text": "bracket so 9 ^un x / 9^ root X is"}, {"start": 43.079, "duration": 9.561, "text": "1 plus 9^ root x / 9^ root X is 1 then"}, {"start": 49.879, "duration": 6.041, "text": "this is equal to"}, {"start": 52.64, "duration": 7.52, "text": "54 so in our next step we can see that 1"}, {"start": 55.92, "duration": 11.199, "text": "+ 1 here is 2 so here we have 9 to the"}, {"start": 60.16, "duration": 9.52, "text": "of otk x * 2 then this is equal to"}, {"start": 67.119, "duration": 6.36, "text": "54 so from here let's divide both side"}, {"start": 69.68, "duration": 7.479, "text": "by two so divide this by two and divide"}, {"start": 73.479, "duration": 9.241, "text": "here by two so two cancel two here then"}, {"start": 77.159, "duration": 8.361, "text": "here we have 9^ ofun X is equal to 54 /"}, {"start": 82.72, "duration": 7.359, "text": "2 that will give us"}, {"start": 85.52, "duration": 9.32, "text": "27 right so now we can express 9 in"}, {"start": 90.079, "duration": 8.121, "text": "index power of 3 as 3^ 2 then we have"}, {"start": 94.84, "duration": 7.639, "text": "here to the^ root X this is equal to 27"}, {"start": 98.2, "duration": 7.12, "text": "is same as 3 to the^ 3 now let us apply"}, {"start": 102.479, "duration": 7.24, "text": "this rule we know that A to the^ B to"}, {"start": 105.32, "duration": 8.72, "text": "the power C isal to a^ B * C so from"}, {"start": 109.719, "duration": 8.04, "text": "here this will be 3^ 2un X this is equal"}, {"start": 114.04, "duration": 6.16, "text": "to 3 to the^ 3 so from here we can see"}, {"start": 117.759, "duration": 5.0, "text": "that we have equal base from both sides"}, {"start": 120.2, "duration": 8.72, "text": "so we're going to equate the powers so"}, {"start": 122.759, "duration": 8.721, "text": "this implies from here that 2un x isal 3"}, {"start": 128.92, "duration": 5.56, "text": "now divide both by"}, {"start": 131.48, "duration": 8.24, "text": "two this two will cancel this two so"}, {"start": 134.48, "duration": 6.88, "text": "here we get root X is equal to 3 / 2 now"}, {"start": 139.72, "duration": 4.64, "text": "for us to get rid of this root sign"}, {"start": 141.36, "duration": 5.28, "text": "let's Square both sides of this equation"}, {"start": 144.36, "duration": 4.44, "text": "so by squaring both sides Square will"}, {"start": 146.64, "duration": 6.48, "text": "cancel square roots and here we have our"}, {"start": 148.8, "duration": 9.2, "text": "solution X is equal to 3 S here is 9 / 2"}, {"start": 153.12, "duration": 6.44, "text": "2 is 4 and this is the correct answer"}, {"start": 158.0, "duration": 3.64, "text": "thank you for watching please kindly"}, {"start": 159.56, "duration": 6.52, "text": "subscribe to my channel also like"}, {"start": 161.64, "duration": 4.44, "text": "comment and share bye-bye"}], "title": "Interesting exponential equation | can you solve ?", "description": "", "lengthSeconds": 165, "width": 428, "height": 240}
{"video_id": "WmpAn9ZeZNs", "transcript": [{"start": 1.28, "duration": 4.4, "text": "hello good day viewers welcome back once"}, {"start": 3.159, "duration": 8.001, "text": "again today we have the following"}, {"start": 5.68, "duration": 11.759, "text": "interesting quartic equation x ^ 4 + 2 x"}, {"start": 11.16, "duration": 8.92, "text": "Cub + 4 X2 + 8 x + 16 this is = z right"}, {"start": 17.439, "duration": 4.68, "text": "over here we're going to evaluate x to"}, {"start": 20.08, "duration": 5.959, "text": "the power of"}, {"start": 22.119, "duration": 6.32, "text": "10 now there is a very interesting"}, {"start": 26.039, "duration": 5.24, "text": "algebraic Identity or let me call it"}, {"start": 28.439, "duration": 5.161, "text": "formula we're going to make use of"}, {"start": 31.279, "duration": 6.44, "text": "here let us"}, {"start": 33.6, "duration": 10.76, "text": "recall that a to the^ of"}, {"start": 37.719, "duration": 13.401, "text": "5- B to the^ of 5 can be factorized as a"}, {"start": 44.36, "duration": 13.96, "text": "- b then multi by a to the^ 4"}, {"start": 51.12, "duration": 9.64, "text": "plus a cubed MTI B plus we have here a 2"}, {"start": 58.32, "duration": 3.84, "text": "* b 2"}, {"start": 60.76, "duration": 8.44, "text": "then"}, {"start": 62.16, "duration": 8.56, "text": "plus a * B Cub + B to the power of"}, {"start": 69.2, "duration": 4.04, "text": "4"}, {"start": 70.72, "duration": 6.16, "text": "okay"}, {"start": 73.24, "duration": 5.4, "text": "now let a equals"}, {"start": 76.88, "duration": 5.239, "text": "X"}, {"start": 78.64, "duration": 6.839, "text": "and let b equal"}, {"start": 82.119, "duration": 7.561, "text": "2 now we derive the following identity"}, {"start": 85.479, "duration": 11.161, "text": "in terms of X that would be a ^ 5 will"}, {"start": 89.68, "duration": 10.84, "text": "be x ^ of 5 minus b ^ 5 will be 2 ^ of 5"}, {"start": 96.64, "duration": 7.88, "text": "then this is equal to a - b will be x -"}, {"start": 100.52, "duration": 12.599, "text": "2 right then multiply by a to ^ 4 will"}, {"start": 104.52, "duration": 13.959, "text": "be x ^ of 4 + a cub B will be 2 * X Cub"}, {"start": 113.119, "duration": 7.321, "text": "right then plus a 2 b 2 will be 2 2 * x"}, {"start": 118.479, "duration": 3.761, "text": "2 which is 4 x\u00b2"}, {"start": 120.44, "duration": 6.319, "text": "[Music]"}, {"start": 122.24, "duration": 10.16, "text": "plus a * B Cub would be x * 2 Cub which"}, {"start": 126.759, "duration": 6.601, "text": "is 8 x then plus b^ 4 is 2^ 4 and that"}, {"start": 132.4, "duration": 4.88, "text": "is"}, {"start": 133.36, "duration": 5.48, "text": "16 okay now carefully looking at this"}, {"start": 137.28, "duration": 4.92, "text": "equation from"}, {"start": 138.84, "duration": 7.0, "text": "here we will realize that at the right"}, {"start": 142.2, "duration": 6.72, "text": "hand side this quatic expression here is"}, {"start": 145.84, "duration": 5.16, "text": "exactly equal to zero just like it is"}, {"start": 148.92, "duration": 8.0, "text": "given right over here here so we're"}, {"start": 151.0, "duration": 13.44, "text": "going to substitute so we get x ^ 5 - 2^"}, {"start": 156.92, "duration": 7.52, "text": "5 this is equal to x - 2 then multi by"}, {"start": 164.879, "duration": 9.881, "text": "0 now from here we get x ^ 5 - 2 ^ 5"}, {"start": 171.36, "duration": 6.28, "text": "this is equal to this * 0 will give us 0"}, {"start": 174.76, "duration": 7.68, "text": "from here we move 2 the^ 5 to the right"}, {"start": 177.64, "duration": 5.599, "text": "so we get x the^ 5 this is equal to 2^"}, {"start": 182.44, "duration": 4.12, "text": "of"}, {"start": 183.239, "duration": 5.64, "text": "5 now let us recall that we are asked to"}, {"start": 186.56, "duration": 5.12, "text": "evaluate x^ of"}, {"start": 188.879, "duration": 5.28, "text": "10 now let us raise both side to the"}, {"start": 191.68, "duration": 2.479, "text": "power of"}, {"start": 195.84, "duration": 9.08, "text": "two as we know this property a^ of B in"}, {"start": 199.92, "duration": 9.039, "text": "braet C this is equal to a of B * C so"}, {"start": 204.92, "duration": 6.56, "text": "from here we have x ^ 5 * 2 which is 10"}, {"start": 208.959, "duration": 6.041, "text": "and this is what we ask to find this is"}, {"start": 211.48, "duration": 5.64, "text": "equal to 2 ^ 5 * 2 which is 10 and"}, {"start": 215.0, "duration": 6.319, "text": "simplifying this this is exactly equal"}, {"start": 217.12, "duration": 5.679, "text": "to 10 to 4 and this is the right answer"}, {"start": 221.319, "duration": 3.881, "text": "thank you for watching please kindly"}, {"start": 222.799, "duration": 6.64, "text": "subscribe to my channel also like"}, {"start": 225.2, "duration": 4.239, "text": "comment and share bye-bye"}], "title": "Japanese | Nice Olympiad Math | Interesting problem | #maths #mathematics", "description": "", "lengthSeconds": 228, "width": 640, "height": 360}
{"video_id": "CthJaqi1XLo", "transcript": [{"start": 0.52, "duration": 5.0, "text": "hello welcome back once again today we"}, {"start": 3.28, "duration": 5.399, "text": "have this interesting"}, {"start": 5.52, "duration": 5.52, "text": "equation involving powers of logorithm"}, {"start": 8.679, "duration": 7.12, "text": "and here we'll be solving for the value"}, {"start": 11.04, "duration": 8.52, "text": "of x now anytime we see a logarithmic"}, {"start": 15.799, "duration": 7.361, "text": "function without a specific base then we"}, {"start": 19.56, "duration": 6.6, "text": "need to know that the base must be"}, {"start": 23.16, "duration": 5.32, "text": "10 right now let's start with the"}, {"start": 26.16, "duration": 4.48, "text": "following substitution we're going to"}, {"start": 28.48, "duration": 3.56, "text": "let log X"}, {"start": 30.64, "duration": 4.72, "text": "="}, {"start": 32.04, "duration": 7.4, "text": "M now remember this base here is 10"}, {"start": 35.36, "duration": 9.0, "text": "right so we know the rule 10^ of M will"}, {"start": 39.44, "duration": 9.0, "text": "be equal to X right so we have X is ="}, {"start": 44.36, "duration": 6.44, "text": "10^ of M now from here let's go back to"}, {"start": 48.44, "duration": 5.08, "text": "original equation and make the"}, {"start": 50.8, "duration": 5.399, "text": "substitutions so here x as the base here"}, {"start": 53.52, "duration": 5.359, "text": "will be 10 the^"}, {"start": 56.199, "duration": 5.0, "text": "n then we need to put this in bracket"}, {"start": 58.879, "duration": 8.761, "text": "remember the original power there which"}, {"start": 61.199, "duration": 8.641, "text": "is log 4 then plus 4^ log X log X is M"}, {"start": 67.64, "duration": 5.2, "text": "then this is equal to"}, {"start": 69.84, "duration": 5.639, "text": "32 so now from here this power will"}, {"start": 72.84, "duration": 11.0, "text": "multiply together so we have 10 the^ of"}, {"start": 75.479, "duration": 12.68, "text": "M log 4 + 4^ m this is equal to 32 now"}, {"start": 83.84, "duration": 9.08, "text": "let's take note of this a * log B is"}, {"start": 88.159, "duration": 12.64, "text": "equal to log B a^ of a right so here"}, {"start": 92.92, "duration": 11.799, "text": "this will be 10^ of log 4^ m + 4^ m this"}, {"start": 100.799, "duration": 6.801, "text": "is equal to 32 remember this log here"}, {"start": 104.719, "duration": 8.281, "text": "has a base 10 right so take note of this"}, {"start": 107.6, "duration": 8.04, "text": "general rule a log B to the base a is"}, {"start": 113.0, "duration": 5.159, "text": "equal to B so we can see the base of"}, {"start": 115.64, "duration": 5.479, "text": "this logarithm is 10 so we have 10 to"}, {"start": 118.159, "duration": 5.081, "text": "the^ log something to the is 10 so this"}, {"start": 121.119, "duration": 9.76, "text": "will simplify to that something which is"}, {"start": 123.24, "duration": 10.44, "text": "4^ M so 4^ m + 4^ m this is equal to 32"}, {"start": 130.879, "duration": 6.121, "text": "we can see 4^ m is common here so Factor"}, {"start": 133.68, "duration": 8.8, "text": "it out so we are left with 1 + 1 this is"}, {"start": 137.0, "duration": 8.44, "text": "= to 32 from here we have 4^ M * 2 this"}, {"start": 142.48, "duration": 5.039, "text": "is equal to 32 let's go ahead and divide"}, {"start": 145.44, "duration": 8.0, "text": "both sides by"}, {"start": 147.519, "duration": 10.921, "text": "two so we get here 4 to m is = 16 so"}, {"start": 153.44, "duration": 7.68, "text": "here 4^ m is = 16 is 4 2 so from here we"}, {"start": 158.44, "duration": 4.48, "text": "have the same base so we're going to"}, {"start": 161.12, "duration": 5.72, "text": "equate the powers right so from here"}, {"start": 162.92, "duration": 7.48, "text": "this implies that m is equal to"}, {"start": 166.84, "duration": 8.399, "text": "2 now let's recall"}, {"start": 170.4, "duration": 6.32, "text": "back that X is = to 10 to the power of M"}, {"start": 175.239, "duration": 5.321, "text": "so from here we're going to solve for"}, {"start": 176.72, "duration": 5.64, "text": "the value of x easily that X is equal to"}, {"start": 180.56, "duration": 6.56, "text": "10^ of"}, {"start": 182.36, "duration": 7.799, "text": "2 which means that X is = to 100 and"}, {"start": 187.12, "duration": 5.28, "text": "this is the right answer to this problem"}, {"start": 190.159, "duration": 4.561, "text": "thank you for watching please kindly"}, {"start": 192.4, "duration": 6.559, "text": "subscribe to my channel also like"}, {"start": 194.72, "duration": 4.239, "text": "comment and share bye-bye"}], "title": "Interesting Olympiad Math | Equation with logarithmic powers.", "description": "", "lengthSeconds": 198, "width": 640, "height": 360}
{"video_id": "_cWP1E2H16A", "transcript": [{"start": 0.08, "duration": 3.96, "text": "hello good day fans welcome back to"}, {"start": 1.8, "duration": 4.88, "text": "another video please subscribe to my"}, {"start": 4.04, "duration": 4.96, "text": "channel if you have not done that please"}, {"start": 6.68, "duration": 6.76, "text": "make sure you like comment and share"}, {"start": 9.0, "duration": 7.039, "text": "here we're given that x - y = 4 x * y ="}, {"start": 13.44, "duration": 6.12, "text": "3 and right over here we're going to"}, {"start": 16.039, "duration": 6.201, "text": "evaluate X Cub - y Cub without first"}, {"start": 19.56, "duration": 6.52, "text": "solving for the value or values of X and"}, {"start": 22.24, "duration": 6.279, "text": "Y in these equations right okay now"}, {"start": 26.08, "duration": 4.76, "text": "we're going to use the formula for the"}, {"start": 28.519, "duration": 10.761, "text": "difference of 2 cu"}, {"start": 30.84, "duration": 12.92, "text": "right so that is x - y cubed + 3 * x * Y"}, {"start": 39.28, "duration": 7.959, "text": "open braet x - y right so from here we"}, {"start": 43.76, "duration": 7.0, "text": "can use this to get the value of x Cub -"}, {"start": 47.239, "duration": 6.12, "text": "y Cub as follows so here we have 4 cubed"}, {"start": 50.76, "duration": 6.799, "text": "right because from the problem here x -"}, {"start": 53.359, "duration": 7.36, "text": "Y is 4 so x - y Cub is 4 cubed then + 3"}, {"start": 57.559, "duration": 6.841, "text": "* X Y that is 3"}, {"start": 60.719, "duration": 9.281, "text": "then * x - y that is 4 right so this is"}, {"start": 64.4, "duration": 10.92, "text": "going to be equal to 4 Cub so that is 64"}, {"start": 70.0, "duration": 10.159, "text": "right and here plus here we have 3 * 3"}, {"start": 75.32, "duration": 9.56, "text": "that is 9 then * 4 that will give us 36"}, {"start": 80.159, "duration": 6.881, "text": "right so from here 64 + 36 so you know"}, {"start": 84.88, "duration": 4.8, "text": "what that's going to give us so that is"}, {"start": 87.04, "duration": 3.8, "text": "equal to 100 and this is the right"}, {"start": 89.68, "duration": 3.28, "text": "answer"}, {"start": 90.84, "duration": 3.76, "text": "thank you for watching please subscribe"}, {"start": 92.96, "duration": 5.479, "text": "to my Channel please like comment and"}, {"start": 94.6, "duration": 3.839, "text": "share bye-bye"}], "title": "An easy Junior Olympiad Math.", "description": "", "lengthSeconds": 97, "width": 640, "height": 360}
{"video_id": "bEJeVgsaeeE", "transcript": [{"start": 0.06, "duration": 3.72, "text": "good evening and welcome to my YouTube"}, {"start": 1.8, "duration": 4.499, "text": "channel today we have the following"}, {"start": 3.78, "duration": 5.16, "text": "problem with five ways for Negative X is"}, {"start": 6.299, "duration": 4.92, "text": "equal to one all over ten then 25 raised"}, {"start": 8.94, "duration": 4.44, "text": "by S is a quarter Dash our main goal"}, {"start": 11.219, "duration": 4.021, "text": "here is not solving for the value of x"}, {"start": 13.38, "duration": 4.979, "text": "and when we solve for the value of x"}, {"start": 15.24, "duration": 5.28, "text": "Computing the value of 25 raised by S we"}, {"start": 18.359, "duration": 4.26, "text": "need a calculating device here we're"}, {"start": 20.52, "duration": 5.04, "text": "going to use the following indices law 1"}, {"start": 22.619, "duration": 6.48, "text": "over a is equal to a raised by negative"}, {"start": 25.56, "duration": 5.459, "text": "one and again a raised by B then in"}, {"start": 29.099, "duration": 5.101, "text": "black race plus C is the same as a"}, {"start": 31.019, "duration": 5.941, "text": "raised by B times C and again a raise"}, {"start": 34.2, "duration": 5.34, "text": "for B in breaker is plus C is the same"}, {"start": 36.96, "duration": 5.939, "text": "as switching exponents a responsible C"}, {"start": 39.54, "duration": 6.12, "text": "then rest for B okay here we have five"}, {"start": 42.899, "duration": 6.18, "text": "respond Negative X is equal to one over"}, {"start": 45.66, "duration": 5.82, "text": "ten then we have five whole Square"}, {"start": 49.079, "duration": 4.881, "text": "negative one you know"}, {"start": 51.48, "duration": 5.34, "text": "five raised one negative one times x"}, {"start": 53.96, "duration": 5.259, "text": "that is negative one times AC has given"}, {"start": 56.82, "duration": 5.16, "text": "us Negative X is equal to you know one"}, {"start": 59.219, "duration": 6.72, "text": "over ten is ten raised for negative one"}, {"start": 61.98, "duration": 6.72, "text": "okay here you know we have here we can"}, {"start": 65.939, "duration": 5.761, "text": "write this as five raised for X here"}, {"start": 68.7, "duration": 5.52, "text": "then raised by negative 1 is equal to"}, {"start": 71.7, "duration": 5.099, "text": "what 10 raised to power negative one and"}, {"start": 74.22, "duration": 4.86, "text": "right here we can see we have equal we"}, {"start": 76.799, "duration": 5.941, "text": "have equal Powers here so we can equate"}, {"start": 79.08, "duration": 6.84, "text": "the basis so we have five raised for X"}, {"start": 82.74, "duration": 5.4, "text": "is equal to what's 10 okay now upon"}, {"start": 85.92, "duration": 4.98, "text": "squaring both sides of this equation so"}, {"start": 88.14, "duration": 6.659, "text": "we have five raised to x squared is"}, {"start": 90.9, "duration": 6.84, "text": "equal to 10 raised power 2 okay right"}, {"start": 94.799, "duration": 5.161, "text": "here we can rewrite as five raised to"}, {"start": 97.74, "duration": 5.28, "text": "square two then raised for X is equal to"}, {"start": 99.96, "duration": 5.28, "text": "10 raised power 2 is what 100 and right"}, {"start": 103.02, "duration": 6.18, "text": "here we simplify this five raised part 2"}, {"start": 105.24, "duration": 7.14, "text": "here is 25 then raised for X is equal to"}, {"start": 109.2, "duration": 5.099, "text": "100 and this is the right answer to the"}, {"start": 112.38, "duration": 3.18, "text": "problem thank you for viewing and make"}, {"start": 114.299, "duration": 3.68, "text": "sure you subscribe to my YouTube channel"}, {"start": 115.56, "duration": 2.419, "text": "for more"}], "title": "A nice exponential problem.", "description": "", "lengthSeconds": 117, "width": 640, "height": 356}
{"video_id": "NbqD_wUdRAc", "transcript": [{"start": 1.24, "duration": 3.92, "text": "hello good viewers welcome back once"}, {"start": 3.12, "duration": 4.92, "text": "again let's find a solution to the"}, {"start": 5.16, "duration": 8.92, "text": "following given radical"}, {"start": 8.04, "duration": 9.72, "text": "equation we're given Square OT of 5"}, {"start": 14.08, "duration": 8.08, "text": "+ the cube root of"}, {"start": 17.76, "duration": 7.72, "text": "x this is equal to 3 from here let's"}, {"start": 22.16, "duration": 3.32, "text": "Square both sides of this"}, {"start": 25.92, "duration": 8.04, "text": "equation now this square and this root"}, {"start": 29.48, "duration": 8.56, "text": "sign will cancel out and we have 5 + the"}, {"start": 33.96, "duration": 6.8, "text": "Cub root of x this is equal to 3 2 which"}, {"start": 38.04, "duration": 5.679, "text": "is 9 from here we're going to subtract"}, {"start": 40.76, "duration": 8.84, "text": "five from both sides we get cube root of"}, {"start": 43.719, "duration": 7.721, "text": "x is = to 9 - 5 which is 4 now to get"}, {"start": 49.6, "duration": 4.16, "text": "rid of this cube root sign since we're"}, {"start": 51.44, "duration": 5.439, "text": "solving for x we're going to cube both"}, {"start": 53.76, "duration": 6.0, "text": "sides of this equation so from here we"}, {"start": 56.879, "duration": 5.2, "text": "Cube both sides and here the cube and"}, {"start": 59.76, "duration": 5.24, "text": "the cube root to cancel out and we left"}, {"start": 62.079, "duration": 6.561, "text": "with only X here this is equal to 4"}, {"start": 65.0, "duration": 5.64, "text": "cubed which is 64 and this is the right"}, {"start": 68.64, "duration": 4.4, "text": "answer thank you for watching please"}, {"start": 70.64, "duration": 6.56, "text": "kindly subscribe to my channel also like"}, {"start": 73.04, "duration": 4.16, "text": "comment and share bye-bye"}], "title": "Interesting radical equation | #maths #mathematics", "description": "", "lengthSeconds": 76, "width": 640, "height": 360}
{"video_id": "b3BIxMgVuo8", "transcript": [{"start": 0.84, "duration": 4.4, "text": "hello welcome back once again today"}, {"start": 3.52, "duration": 5.159, "text": "we're given the following Olympian"}, {"start": 5.24, "duration": 8.479, "text": "mathematics problem we're given A+ B is"}, {"start": 8.679, "duration": 9.561, "text": "= 1 a 2 + b 2 is = 2 so here we'll be"}, {"start": 13.719, "duration": 7.601, "text": "evaluating a cub + B Cubed mind you"}, {"start": 18.24, "duration": 6.0, "text": "three is not the right"}, {"start": 21.32, "duration": 5.719, "text": "answer okay now let's get started from"}, {"start": 24.24, "duration": 5.199, "text": "equation one we have a + b which is"}, {"start": 27.039, "duration": 4.601, "text": "equal to 1 so let's go ahead and square"}, {"start": 29.439, "duration": 6.8, "text": "both sides"}, {"start": 31.64, "duration": 8.759, "text": "the left hand side will become a 2 + b 2"}, {"start": 36.239, "duration": 6.16, "text": "+ 2 * a * B this will be equal to 1 so"}, {"start": 40.399, "duration": 6.64, "text": "we know from equation two this right"}, {"start": 42.399, "duration": 8.32, "text": "over here is equal to two so we have 2 +"}, {"start": 47.039, "duration": 7.04, "text": "2 a b this is equal to 1 subtract two"}, {"start": 50.719, "duration": 7.52, "text": "from both side we get 2 a b is = 1 - 2"}, {"start": 54.079, "duration": 6.921, "text": "which is -1 then divide by"}, {"start": 58.239, "duration": 7.801, "text": "2 then from here we get the product of A"}, {"start": 61.0, "duration": 7.52, "text": "and B which is equal to -1 /"}, {"start": 66.04, "duration": 6.28, "text": "2 now the expression we asked to"}, {"start": 68.52, "duration": 7.0, "text": "evaluate a cubed + B Cubed has the"}, {"start": 72.32, "duration": 8.0, "text": "following ni algebraic identity this is"}, {"start": 75.52, "duration": 8.44, "text": "equal to a + b Cub - 3 * a * B into"}, {"start": 80.32, "duration": 8.92, "text": "braet a + b so this will be equal to We"}, {"start": 83.96, "duration": 10.4, "text": "Know A + B is 1 so we get 1 Cub - 3 * a"}, {"start": 89.24, "duration": 10.48, "text": "b which is -1 / 2 * A + B which is"}, {"start": 94.36, "duration": 12.56, "text": "1 so here we get a cubed + B Cubed is ="}, {"start": 99.72, "duration": 9.759, "text": "to 1 -3 * -1 /2 is pos3 / 2 then POS 3 /"}, {"start": 106.92, "duration": 6.519, "text": "2 * 1 is POS 3 /"}, {"start": 109.479, "duration": 7.441, "text": "2 we know 1 is same as 2 / 2 then we"}, {"start": 113.439, "duration": 10.64, "text": "have + 3 / 2 add the numerators so we"}, {"start": 116.92, "duration": 12.839, "text": "get 5 / 2 so we have therefore a cubed +"}, {"start": 124.079, "duration": 7.921, "text": "B Cub is = 5 / 2 and this is the right"}, {"start": 129.759, "duration": 5.12, "text": "answer thank you for watching please"}, {"start": 132.0, "duration": 7.56, "text": "kindly subscribe to my channel also like"}, {"start": 134.879, "duration": 4.681, "text": "comment and share bye-bye"}], "title": "Germany | Nice Olympiad Math | Can you solve?", "description": "", "lengthSeconds": 138, "width": 376, "height": 240}
{"video_id": "b69g6rH1Fww", "transcript": [{"start": 0.44, "duration": 4.399, "text": "hello welcome back once again today"}, {"start": 2.399, "duration": 4.961, "text": "let's solve this interesting equation"}, {"start": 4.839, "duration": 5.401, "text": "involving square root and cube root"}, {"start": 7.36, "duration": 5.68, "text": "signs so we have here theare root of x"}, {"start": 10.24, "duration": 6.52, "text": "is equal to the cube root of x now let's"}, {"start": 13.04, "duration": 6.12, "text": "go ahead and solve for the values of X"}, {"start": 16.76, "duration": 8.0, "text": "now let's take note of"}, {"start": 19.16, "duration": 7.76, "text": "this the end root of a is equal to a to"}, {"start": 24.76, "duration": 5.359, "text": "the^ of 1 /"}, {"start": 26.92, "duration": 5.479, "text": "n okay so here the left hand side which"}, {"start": 30.119, "duration": 6.28, "text": "isare root of x we're going to write as"}, {"start": 32.399, "duration": 5.761, "text": "X to^ of 1 / 2 then this will be equal"}, {"start": 36.399, "duration": 4.48, "text": "to the cube root of x from the right"}, {"start": 38.16, "duration": 6.6, "text": "hand side we're going to write it as x"}, {"start": 40.879, "duration": 5.801, "text": "to^ 1 / 3 okay now in our next step"}, {"start": 44.76, "duration": 5.84, "text": "we're going to raise both sides of this"}, {"start": 46.68, "duration": 5.76, "text": "equation to a power of a number but what"}, {"start": 50.6, "duration": 5.16, "text": "is this number that we're going to raise"}, {"start": 52.44, "duration": 8.919, "text": "both sides to so how we get this number"}, {"start": 55.76, "duration": 9.8, "text": "is to find the LCM of 2 and 3"}, {"start": 61.359, "duration": 6.961, "text": "so fortunately the LCM of 2 and 3 is"}, {"start": 65.56, "duration": 4.879, "text": "equal to 6 so let's go ahead and raise"}, {"start": 68.32, "duration": 5.839, "text": "both sides of this equation to the power"}, {"start": 70.439, "duration": 11.881, "text": "of 6 so we have X to the^ 1 /"}, {"start": 74.159, "duration": 12.0, "text": "2 6 this will be equal x^ 1 / 3 6 now"}, {"start": 82.32, "duration": 8.24, "text": "let's make use of this property a^ b to^"}, {"start": 86.159, "duration": 7.24, "text": "c this is equal to a^ B * C"}, {"start": 90.56, "duration": 6.76, "text": "so we can see from here 1 2 * 6 will"}, {"start": 93.399, "duration": 7.161, "text": "give us 3 right so we get x to^ 3 this"}, {"start": 97.32, "duration": 7.04, "text": "is equal to 1 / 3 * 6 will give us 2 so"}, {"start": 100.56, "duration": 7.28, "text": "we get here x^ 2 so from here let's"}, {"start": 104.36, "duration": 4.759, "text": "subtract X from both sides to get X Cub"}, {"start": 107.84, "duration": 5.639, "text": "-"}, {"start": 109.119, "duration": 8.68, "text": "X2 this is equal to 0 from here X2 is"}, {"start": 113.479, "duration": 8.68, "text": "common so we Factor it out so X Cub x\u00b2"}, {"start": 117.799, "duration": 7.041, "text": "will give us x - x ID x\u00b2 will give us 1"}, {"start": 122.159, "duration": 5.081, "text": "then this is equal to Z so when we have"}, {"start": 124.84, "duration": 5.479, "text": "two products which is equal to Z either"}, {"start": 127.24, "duration": 6.6, "text": "the first which is here is equal to 0"}, {"start": 130.319, "duration": 8.241, "text": "which is X2 is equal to 0 or the second"}, {"start": 133.84, "duration": 7.84, "text": "here is equal to 0 which is x -1 is = 0"}, {"start": 138.56, "duration": 5.64, "text": "so here if x is = 0 then X will be squ"}, {"start": 141.68, "duration": 6.6, "text": "otk of 0 which is 0 so we have X is"}, {"start": 144.2, "duration": 6.2, "text": "equal to Z so this is our solution from"}, {"start": 148.28, "duration": 3.36, "text": "here add one to both side we get another"}, {"start": 150.4, "duration": 5.199, "text": "solution that"}, {"start": 151.64, "duration": 8.0, "text": "xal 1 so therefore the given equation"}, {"start": 155.599, "duration": 6.121, "text": "has only two real solutions thank you"}, {"start": 159.64, "duration": 5.28, "text": "for watching please kindly subscribe to"}, {"start": 161.72, "duration": 6.2, "text": "my channel also like comment and share"}, {"start": 164.92, "duration": 3.0, "text": "bye-bye"}], "title": "Interesting radical equation | Can you solve ?", "description": "", "lengthSeconds": 167, "width": 640, "height": 360}
{"video_id": "d55cXSlFwLY", "transcript": [{"start": 0.28, "duration": 4.2, "text": "hello good day viewers welcome back once"}, {"start": 2.399, "duration": 3.92, "text": "again to my YouTube Channel please if"}, {"start": 4.48, "duration": 3.84, "text": "today is the first day of visiting this"}, {"start": 6.319, "duration": 4.44, "text": "Channel please kindly support the"}, {"start": 8.32, "duration": 5.319, "text": "channel by subscribing please like"}, {"start": 10.759, "duration": 6.881, "text": "comment and share here we're giv that A"}, {"start": 13.639, "duration": 7.88, "text": "+ B is equal to 50 a * B is equal"}, {"start": 17.64, "duration": 6.2, "text": "624 and here we asked to evaluate a 2us"}, {"start": 21.519, "duration": 5.121, "text": "b 2 please this is an Olympian math"}, {"start": 23.84, "duration": 5.199, "text": "problem let us see how to solve this"}, {"start": 26.64, "duration": 6.599, "text": "problem here we record the difference of"}, {"start": 29.039, "duration": 11.52, "text": "two squares a 2 - b"}, {"start": 33.239, "duration": 9.201, "text": "2 is equal to a + b * Aus B right so"}, {"start": 40.559, "duration": 6.0, "text": "this expression right over here is the"}, {"start": 42.44, "duration": 7.279, "text": "same as a + b * Aus B but in the given"}, {"start": 46.559, "duration": 6.8, "text": "problem we have the value for a + b but"}, {"start": 49.719, "duration": 5.84, "text": "we don't have the value for a minus B so"}, {"start": 53.359, "duration": 6.281, "text": "our K here is to find the value for a"}, {"start": 55.559, "duration": 8.841, "text": "minus B okay from equation one we have a"}, {"start": 59.64, "duration": 7.2, "text": "+ B is equal to 50 let us Square both"}, {"start": 64.4, "duration": 8.92, "text": "sides of this equation so here we have a"}, {"start": 66.84, "duration": 12.279, "text": "+ b s is equal to 50 s right so now"}, {"start": 73.32, "duration": 11.88, "text": "recall this property x + y^ 2 is = to x"}, {"start": 79.119, "duration": 8.04, "text": "- Y 2 + 4 x y right so going to use this"}, {"start": 85.2, "duration": 7.12, "text": "property at the left hand side of this"}, {"start": 87.159, "duration": 10.441, "text": "equation so here we have a - b 2 then"}, {"start": 92.32, "duration": 8.56, "text": "plus 4 a b this is equal to 50 2 that is"}, {"start": 97.6, "duration": 5.92, "text": "2,500 right so from the given problem we"}, {"start": 100.88, "duration": 5.919, "text": "have the value for a * b as we can see"}, {"start": 103.52, "duration": 6.959, "text": "it here so we're going to replace this"}, {"start": 106.799, "duration": 7.0, "text": "we have a minus b 2 plus here we have"}, {"start": 110.479, "duration": 6.361, "text": "four * the value for a * B that is"}, {"start": 113.799, "duration": 8.801, "text": "624 and this is equal to"}, {"start": 116.84, "duration": 10.0, "text": "2,500 right so from here we have a - b 2"}, {"start": 122.6, "duration": 5.359, "text": "plus here we have 4 * 64 that will give"}, {"start": 126.84, "duration": 6.36, "text": "us"}, {"start": 127.959, "duration": 8.92, "text": "2496 so 2496 is equal to 2500 right so"}, {"start": 133.2, "duration": 5.64, "text": "from here we'll have Aus b 2 is equal to"}, {"start": 136.879, "duration": 5.281, "text": "2500 -"}, {"start": 138.84, "duration": 7.039, "text": "2496 that is going to give us 4 so from"}, {"start": 142.16, "duration": 7.079, "text": "here we get a - b is equal to you know"}, {"start": 145.879, "duration": 6.64, "text": "plus or minus < TK4 right so this will"}, {"start": 149.239, "duration": 6.841, "text": "give us Plus Plus or - 2"}, {"start": 152.519, "duration": 8.44, "text": "cool so therefore the expression a s - b"}, {"start": 156.08, "duration": 8.4, "text": "2 is going to be equal to a + b MTI a"}, {"start": 160.959, "duration": 5.441, "text": "minus B right so this is equal to A+ B"}, {"start": 164.48, "duration": 5.479, "text": "in the given problem we have it as 2"}, {"start": 166.4, "duration": 5.839, "text": "then 2 * plus or minus 2 right so here"}, {"start": 169.959, "duration": 5.401, "text": "we have plus or minus 2 and this is"}, {"start": 172.239, "duration": 5.321, "text": "going to give us plus or minus 100 and"}, {"start": 175.36, "duration": 3.879, "text": "this is the right answer thank you for"}, {"start": 177.56, "duration": 2.8, "text": "watching please do wa to subscribe to my"}, {"start": 179.239, "duration": 5.92, "text": "YouTube channel"}, {"start": 180.36, "duration": 4.799, "text": "please like comment and share bye-bye"}], "title": "lnteresting Olympiad Math.", "description": "", "lengthSeconds": 183, "width": 640, "height": 360}
{"video_id": "fI4jmy9Ey2Q", "transcript": [{"start": 1.079, "duration": 4.961, "text": "hello gooders welcome back once again in"}, {"start": 3.919, "duration": 5.361, "text": "this video we're going to evaluate this"}, {"start": 6.04, "duration": 8.88, "text": "indefinite integral here the integral of"}, {"start": 9.28, "duration": 9.919, "text": "1 /un of 1 - x^2 DX in this video we're"}, {"start": 14.92, "duration": 6.16, "text": "going to practice how to make use of"}, {"start": 19.199, "duration": 5.08, "text": "trigonometric"}, {"start": 21.08, "duration": 7.599, "text": "substitutions now let us talk about this"}, {"start": 24.279, "duration": 9.721, "text": "radical here we have root of 1 - x^2"}, {"start": 28.679, "duration": 10.001, "text": "let's think about this we know 1 - sin s"}, {"start": 34.0, "duration": 8.079, "text": "of theta this will give us cosine squ of"}, {"start": 38.68, "duration": 7.199, "text": "theta so we'll be using this idea to"}, {"start": 42.079, "duration": 6.96, "text": "make a subtitution letting xal sin of"}, {"start": 45.879, "duration": 9.881, "text": "theta since X is squ then s of theta"}, {"start": 49.039, "duration": 8.961, "text": "will be squar so here we let X = sin"}, {"start": 55.76, "duration": 5.0, "text": "Theta now when we differentiate both"}, {"start": 58.0, "duration": 5.719, "text": "side we get DX isal to the derivative of"}, {"start": 60.76, "duration": 8.6, "text": "sin Theta with respect to Theta will"}, {"start": 63.719, "duration": 8.04, "text": "give us cine Theta D Theta right so from"}, {"start": 69.36, "duration": 6.32, "text": "here if you call this integral"}, {"start": 71.759, "duration": 9.761, "text": "I so from here we get I is equal to the"}, {"start": 75.68, "duration": 7.399, "text": "integral 1 over < TK of 1 - x\u00b2 will be"}, {"start": 81.52, "duration": 6.08, "text": "sin s"}, {"start": 83.079, "duration": 8.521, "text": "thet then time the x is cine Theta D"}, {"start": 87.6, "duration": 8.559, "text": "Theta right so from here we get I is"}, {"start": 91.6, "duration": 9.04, "text": "equal to the integral of 1 all theun of"}, {"start": 96.159, "duration": 6.121, "text": "1 - sin Theta is equal to cosine s Theta"}, {"start": 100.64, "duration": 4.64, "text": "then here cos"}, {"start": 102.28, "duration": 6.04, "text": "Theta D Theta and from here this is"}, {"start": 105.28, "duration": 6.199, "text": "equal to the integral of 1"}, {"start": 108.32, "duration": 7.04, "text": "/un cosine s Theta will give us cosine"}, {"start": 111.479, "duration": 6.841, "text": "Theta then here time cosine Theta D"}, {"start": 115.36, "duration": 5.719, "text": "Theta so this will cancel out from here"}, {"start": 118.32, "duration": 7.159, "text": "remember we're integrating with to Theta"}, {"start": 121.079, "duration": 8.001, "text": "and from here the integral of 1 D Theta"}, {"start": 125.479, "duration": 5.281, "text": "will give us Theta then plus constant"}, {"start": 129.08, "duration": 7.04, "text": "right"}, {"start": 130.76, "duration": 8.88, "text": "okay now what is Theta from the first"}, {"start": 136.12, "duration": 7.52, "text": "substitution here we all know that X is"}, {"start": 139.64, "duration": 7.76, "text": "= to sin Theta if you want to solve for"}, {"start": 143.64, "duration": 6.8, "text": "Theta here then we take the X sign right"}, {"start": 147.4, "duration": 6.24, "text": "or the sign inverse or inverse tangent"}, {"start": 150.44, "duration": 5.64, "text": "of X that is actually what is Theta so"}, {"start": 153.64, "duration": 4.599, "text": "from here we have gotten the value of"}, {"start": 156.08, "duration": 5.12, "text": "theta in terms of X and here we come up"}, {"start": 158.239, "duration": 9.36, "text": "with a solution that the integral is"}, {"start": 161.2, "duration": 8.64, "text": "evaluated as the inverse s of X plus C"}, {"start": 167.599, "duration": 3.92, "text": "and this is the right answer thank you"}, {"start": 169.84, "duration": 3.96, "text": "for watching please kindly subscribe to"}, {"start": 171.519, "duration": 5.281, "text": "my channel also like comment and share"}, {"start": 173.8, "duration": 3.0, "text": "bye-bye"}], "title": "Indefinite integral | Trigonometric substitution | #maths #mathematics", "description": "", "lengthSeconds": 175, "width": 508, "height": 360}
{"video_id": "DgGfaLVgRxs", "transcript": [{"start": 1.199, "duration": 4.041, "text": "hello good viewers welcome back once"}, {"start": 3.24, "duration": 5.12, "text": "again today we're going to solve for the"}, {"start": 5.24, "duration": 6.0, "text": "value of x in the following exponential"}, {"start": 8.36, "duration": 5.92, "text": "equation right and one can easily say"}, {"start": 11.24, "duration": 7.24, "text": "that X is equal to 0 here we have"}, {"start": 14.28, "duration": 7.32, "text": "9^ 0 which is same as 9^ 0 which is 1"}, {"start": 18.48, "duration": 6.76, "text": "this equal to which is 1 we can see that"}, {"start": 21.6, "duration": 7.04, "text": "1 is = 1 so X is = z is a"}, {"start": 25.24, "duration": 5.519, "text": "solution but the question here is do we"}, {"start": 28.64, "duration": 4.32, "text": "have all"}, {"start": 30.759, "duration": 5.681, "text": "solution okay now let's go ahead and"}, {"start": 32.96, "duration": 7.24, "text": "check it out right so here we have 9 to^"}, {"start": 36.44, "duration": 5.4, "text": "ofun x this is equal to"}, {"start": 40.2, "duration": 5.92, "text": "27"}, {"start": 41.84, "duration": 7.76, "text": "x now the idea here is to write the"}, {"start": 46.12, "duration": 5.959, "text": "bases from both sides and index power of"}, {"start": 49.6, "duration": 4.599, "text": "three right because we already know that"}, {"start": 52.079, "duration": 8.0, "text": "9 is 3"}, {"start": 54.199, "duration": 11.441, "text": "2un x this is equal to 27 is as 3^ of 3"}, {"start": 60.079, "duration": 9.561, "text": "X now with this identity a to the^ of b"}, {"start": 65.64, "duration": 6.96, "text": "c this is equal to a to the^ of B * C so"}, {"start": 69.64, "duration": 8.28, "text": "here this will be 3^ 2 * root X which is"}, {"start": 72.6, "duration": 9.839, "text": "2un X then this is equal to 3 to^ 3 * X"}, {"start": 77.92, "duration": 8.0, "text": "here which is 3 x so since we have equal"}, {"start": 82.439, "duration": 5.881, "text": "bases we can equate the powers right so"}, {"start": 85.92, "duration": 4.72, "text": "from here we have that 2 of X that this"}, {"start": 88.32, "duration": 7.439, "text": "implies right okay"}, {"start": 90.64, "duration": 7.56, "text": "2 2un x is = 3 x now let's go ahead and"}, {"start": 95.759, "duration": 5.281, "text": "square both"}, {"start": 98.2, "duration": 8.279, "text": "sides you know why to get rid of this"}, {"start": 101.04, "duration": 12.28, "text": "root sign and again A B C this equal to"}, {"start": 106.479, "duration": 8.161, "text": "a c * b c so here this will become 2 2 *"}, {"start": 113.32, "duration": 7.799, "text": "otk"}, {"start": 114.64, "duration": 9.32, "text": "x\u00b2 this is equal to here 3 2 * x\u00b2"}, {"start": 121.119, "duration": 4.201, "text": "so here this the root sign and the"}, {"start": 123.96, "duration": 4.799, "text": "square will cancel out here we have 2"}, {"start": 125.32, "duration": 7.559, "text": "square which is 4 4 * X here is 4 x this"}, {"start": 128.759, "duration": 9.401, "text": "is equal to 3 2 is 9 so here we have 9"}, {"start": 132.879, "duration": 7.64, "text": "X2 so from the right we have 9 X2 - 4x"}, {"start": 138.16, "duration": 7.12, "text": "this is equal Z so let's go ahead and"}, {"start": 140.519, "duration": 9.08, "text": "factor out X we have 9x - 4 this is"}, {"start": 145.28, "duration": 10.0, "text": "equal Z so from here we have x is = 0"}, {"start": 149.599, "duration": 9.481, "text": "and in the bracket 9x - 4 is also = to Z"}, {"start": 155.28, "duration": 6.8, "text": "so add four to both side we get 9x is ="}, {"start": 159.08, "duration": 6.6, "text": "4 right divide both by 9 we actually"}, {"start": 162.08, "duration": 10.92, "text": "have X is = 4 / 9 and here we have the"}, {"start": 165.68, "duration": 9.44, "text": "solution x is = 0 or X is = 4/ 9 and"}, {"start": 173.0, "duration": 3.8, "text": "these are the solutions thank you for"}, {"start": 175.12, "duration": 3.88, "text": "watching please kindly subscribe to my"}, {"start": 176.8, "duration": 5.92, "text": "YouTube channel also like comment and"}, {"start": 179.0, "duration": 3.72, "text": "share bye-bye right"}], "title": "Interesting exponential equation | #maths #olympiad", "description": "", "lengthSeconds": 181, "width": 640, "height": 360}
{"video_id": "5LBCdLU7bCM", "transcript": [{"start": 0.719, "duration": 5.281, "text": "hello good day viewers welcome back once"}, {"start": 2.84, "duration": 5.56, "text": "again today we have this sodic equation"}, {"start": 6.0, "duration": 5.559, "text": "and we're going to solve for A and B"}, {"start": 8.4, "duration": 7.44, "text": "we're given that < TK A + B < tk3 this"}, {"start": 11.559, "duration": 7.601, "text": "is equal to 13 All 4 + <"}, {"start": 15.84, "duration": 8.199, "text": "tk3 now for the right hand side of this"}, {"start": 19.16, "duration": 6.119, "text": "equation let us first rationalize it so"}, {"start": 24.039, "duration": 6.681, "text": "here we have"}, {"start": 25.279, "duration": 6.761, "text": "13 all over 4 + < tk3 to rationalize"}, {"start": 30.72, "duration": 4.159, "text": "this we're going to multiply the"}, {"start": 32.04, "duration": 5.48, "text": "numerator and the denominator by the"}, {"start": 34.879, "duration": 6.921, "text": "conjugate form of the denominator so"}, {"start": 37.52, "duration": 7.76, "text": "here we have to multiply by 4 - < tk3"}, {"start": 41.8, "duration": 5.599, "text": "and down here we have to multiply by 4 -"}, {"start": 45.28, "duration": 2.119, "text": "<"}, {"start": 50.879, "duration": 7.081, "text": "tk3 okay now from the denominator right"}, {"start": 54.559, "duration": 6.601, "text": "over here we can easily see that this is"}, {"start": 57.96, "duration": 11.56, "text": "the difference of two squares remember a"}, {"start": 61.16, "duration": 14.12, "text": "+ b * Aus B this is equal to a 2 - b 2"}, {"start": 69.52, "duration": 12.52, "text": "so this is equal to 13 * 4 - < tk3 then"}, {"start": 75.28, "duration": 11.24, "text": "all over that would be 4 2 - < tk3 2 so"}, {"start": 82.04, "duration": 9.719, "text": "from here we get 13 into bra 4 - < tk3"}, {"start": 86.52, "duration": 7.04, "text": "then all 4 2 is 16 - now this square and"}, {"start": 91.759, "duration": 4.601, "text": "the root sign will cancel out we have"}, {"start": 93.56, "duration": 7.879, "text": "three and from here this is equal to 13"}, {"start": 96.36, "duration": 8.759, "text": "into 4 - < tk3 then all over 16 - 3 that"}, {"start": 101.439, "duration": 7.0, "text": "is 13 now this 13 and this 13 we cancel"}, {"start": 105.119, "duration": 4.921, "text": "out and we left with 4 - root3 so from"}, {"start": 108.439, "duration": 6.481, "text": "here we have the"}, {"start": 110.04, "duration": 10.079, "text": "equation < TK A + B < tk3 this is"}, {"start": 114.92, "duration": 6.879, "text": "actually equal to 4 - < tk3 now from"}, {"start": 120.119, "duration": 6.121, "text": "here let's go ahead and square both"}, {"start": 121.799, "duration": 6.921, "text": "sides to get rid of this root"}, {"start": 126.24, "duration": 6.8, "text": "sign so from here the square and the"}, {"start": 128.72, "duration": 7.599, "text": "root sign we cancel out we'll have a + b"}, {"start": 133.04, "duration": 5.44, "text": "root of 3 this is equal to no when you"}, {"start": 136.319, "duration": 6.841, "text": "expand this we have the first 10 squ"}, {"start": 138.48, "duration": 7.64, "text": "which is 16 plus the 2 10 squ which is 3"}, {"start": 143.16, "duration": 7.2, "text": "- 2 * the first term and the second term"}, {"start": 146.12, "duration": 8.88, "text": "that is 2 * 4 8 * < tk3 that is 8 < tk3"}, {"start": 150.36, "duration": 11.08, "text": "so we have a + b < tk3 this is equal to"}, {"start": 155.0, "duration": 9.28, "text": "here we have that is 19 - 8 < tk3 so"}, {"start": 161.44, "duration": 6.719, "text": "from here we can compare and from here"}, {"start": 164.28, "duration": 10.319, "text": "this implies that a is equal to"}, {"start": 168.159, "duration": 8.561, "text": "19 and B is equal to -8 and this is the"}, {"start": 174.599, "duration": 4.28, "text": "solution thank you for watching please"}, {"start": 176.72, "duration": 6.28, "text": "kindly subscribe to my channel also like"}, {"start": 178.879, "duration": 4.121, "text": "comment and share bye-bye"}], "title": "Interesting surdic equality | #maths #mathematics", "description": "", "lengthSeconds": 181, "width": 578, "height": 360}
{"video_id": "1wz5HkIYI-g", "transcript": [{"start": 0.599, "duration": 4.501, "text": "good day viewers welcome back to my"}, {"start": 2.879, "duration": 4.861, "text": "Channel today we're having the following"}, {"start": 5.1, "duration": 4.079, "text": "exponential problem please if you have"}, {"start": 7.74, "duration": 3.66, "text": "not subscribed to this Channel Please"}, {"start": 9.179, "duration": 4.741, "text": "Subscribe and turn on the notification"}, {"start": 11.4, "duration": 4.08, "text": "Bell so that you'll be notified anytime"}, {"start": 13.92, "duration": 3.6, "text": "we upload new videos"}, {"start": 15.48, "duration": 4.98, "text": "today we have any following exponential"}, {"start": 17.52, "duration": 6.419, "text": "problem five raised per X all over 7 is"}, {"start": 20.46, "duration": 6.899, "text": "equal to 7 raised to power 5 all over X"}, {"start": 23.939, "duration": 6.901, "text": "how do we solve this problem okay now"}, {"start": 27.359, "duration": 6.961, "text": "recall that the natural logarithm of X"}, {"start": 30.84, "duration": 6.84, "text": "is the same as the logarithm of x to the"}, {"start": 34.32, "duration": 5.34, "text": "base a where e is a last number"}, {"start": 37.68, "duration": 5.1, "text": "okay so I'm going to take the natural"}, {"start": 39.66, "duration": 5.34, "text": "logarithm on both sides of this equation"}, {"start": 42.78, "duration": 5.16, "text": "so I'm going to take the natural"}, {"start": 45.0, "duration": 6.12, "text": "logarithm so I have"}, {"start": 47.94, "duration": 7.2, "text": "not sure logarithm of 5 raised by X all"}, {"start": 51.12, "duration": 7.38, "text": "over 7 is equal to Natural algorithm of"}, {"start": 55.14, "duration": 5.759, "text": "7 raised power 5 1 y x okay remember"}, {"start": 58.5, "duration": 5.219, "text": "that this is one of the properties of"}, {"start": 60.899, "duration": 6.661, "text": "logarithm the natural logarithm of X"}, {"start": 63.719, "duration": 7.561, "text": "raised Y is the same as the power of s"}, {"start": 67.56, "duration": 7.02, "text": "times the entire logarithm okay so here"}, {"start": 71.28, "duration": 6.48, "text": "I'm going to have X all over seven the"}, {"start": 74.58, "duration": 5.82, "text": "natural logarithm of 5 is equal to 5 all"}, {"start": 77.76, "duration": 5.34, "text": "over X the natural logarithm of seven"}, {"start": 80.4, "duration": 5.399, "text": "okay this equation I'm going to rewrite"}, {"start": 83.1, "duration": 6.3, "text": "it as follows so I can write this as X"}, {"start": 85.799, "duration": 6.121, "text": "natural logarithm of 5 all over 7 is"}, {"start": 89.4, "duration": 6.3, "text": "equal to 5 if a natural logarithm of 7"}, {"start": 91.92, "duration": 6.659, "text": "all over X okay so I'm going to cross"}, {"start": 95.7, "duration": 6.12, "text": "multiply so X natural logarithm of 5"}, {"start": 98.579, "duration": 6.601, "text": "times x is equal to x squared natural"}, {"start": 101.82, "duration": 7.1, "text": "logarithm of 5 is equal to"}, {"start": 105.18, "duration": 8.64, "text": "7 times 5 natural logarithm of 7 is 35"}, {"start": 108.92, "duration": 7.42, "text": "natural logarithm of 7 okay now from"}, {"start": 113.82, "duration": 4.439, "text": "here I'm going to divide both sides by"}, {"start": 116.34, "duration": 4.62, "text": "the coefficient of x squared so that is"}, {"start": 118.259, "duration": 6.781, "text": "over natural logarithm of 5. over"}, {"start": 120.96, "duration": 6.9, "text": "natural logarithm of 5 Okay so this and"}, {"start": 125.04, "duration": 3.779, "text": "this get canceled at so we have x"}, {"start": 127.86, "duration": 3.72, "text": "squared"}, {"start": 128.819, "duration": 5.041, "text": "is equal to 35"}, {"start": 131.58, "duration": 5.22, "text": "natural logarithm of seven"}, {"start": 133.86, "duration": 6.66, "text": "all over not sure login number five okay"}, {"start": 136.8, "duration": 7.92, "text": "I'm going to write this"}, {"start": 140.52, "duration": 6.299, "text": "as follows so remember that the natural"}, {"start": 144.72, "duration": 4.62, "text": "logarithm of X all over the natural"}, {"start": 146.819, "duration": 6.241, "text": "logarithm of Y is the same as the"}, {"start": 149.34, "duration": 5.58, "text": "logarithm of x to the base y okay"}, {"start": 153.06, "duration": 2.819, "text": "right so I'm going to write this as"}, {"start": 154.92, "duration": 3.36, "text": "follows"}, {"start": 155.879, "duration": 4.321, "text": "x squared is equal to"}, {"start": 158.28, "duration": 5.819, "text": "35"}, {"start": 160.2, "duration": 5.819, "text": "times natural logarithm of seven all of"}, {"start": 164.099, "duration": 4.64, "text": "our natural logarithm of 5 and this is"}, {"start": 166.019, "duration": 7.321, "text": "the same as 35"}, {"start": 168.739, "duration": 6.761, "text": "logarithm of 7 to the base 5 okay so I'm"}, {"start": 173.34, "duration": 5.759, "text": "going to take the square root of both"}, {"start": 175.5, "duration": 6.599, "text": "sides so I have square root of x squared"}, {"start": 179.099, "duration": 7.801, "text": "is equal to you know plus or minus"}, {"start": 182.099, "duration": 8.161, "text": "square root of 35 or logarithm of 7 to"}, {"start": 186.9, "duration": 4.979, "text": "the base 5. and we on this way and this"}, {"start": 190.26, "duration": 5.399, "text": "where we get canceled so we are left"}, {"start": 191.879, "duration": 4.741, "text": "with X is equal to you know square root"}, {"start": 195.659, "duration": 2.961, "text": "of"}, {"start": 196.62, "duration": 6.199, "text": "35"}, {"start": 198.62, "duration": 9.28, "text": "log 7 to the base 5 or"}, {"start": 202.819, "duration": 9.041, "text": "negative square roots of 35"}, {"start": 207.9, "duration": 6.419, "text": "log 7 to the base 5 and this is the"}, {"start": 211.86, "duration": 4.08, "text": "right answer to the problem thank you"}, {"start": 214.319, "duration": 3.301, "text": "for viewing and please make sure you"}, {"start": 215.94, "duration": 4.46, "text": "subscribe to my YouTube channel for more"}, {"start": 217.62, "duration": 2.78, "text": "interesting videos"}], "title": "Solving a nice exponential equation. Not just one value !", "description": "", "lengthSeconds": 219, "width": 636, "height": 360}
{"video_id": "JUdqPMz2HaA", "transcript": [{"start": 0.28, "duration": 4.039, "text": "hello good viewers welcome back once"}, {"start": 2.0, "duration": 3.92, "text": "again to another video please canly"}, {"start": 4.319, "duration": 4.441, "text": "support this channel by subscribing to"}, {"start": 5.92, "duration": 3.96, "text": "this channel also like comment and share"}, {"start": 8.76, "duration": 3.879, "text": "here we're given the following"}, {"start": 9.88, "duration": 6.52, "text": "interesting exponential equation problem"}, {"start": 12.639, "duration": 7.681, "text": "we're given that 2^ X is = x^ 32 we"}, {"start": 16.4, "duration": 5.879, "text": "asked to find the value of x right okay"}, {"start": 20.32, "duration": 2.76, "text": "now in this video we'll be making use of"}, {"start": 22.279, "duration": 4.201, "text": "the"}, {"start": 23.08, "duration": 8.359, "text": "following identities here we have that a"}, {"start": 26.48, "duration": 9.599, "text": "the^ b into bracket C is equal to c b"}, {"start": 31.439, "duration": 8.561, "text": "right and here A B into bra C isal to a^"}, {"start": 36.079, "duration": 6.841, "text": "B * C okay now let's see how to make use"}, {"start": 40.0, "duration": 7.399, "text": "of these identities in this video we"}, {"start": 42.92, "duration": 7.2, "text": "have that 2 ^ of X is equal to X the^ of"}, {"start": 47.399, "duration": 5.601, "text": "32 right now let's go ahead and square"}, {"start": 50.12, "duration": 5.0, "text": "both sides of this"}, {"start": 53.0, "duration": 6.44, "text": "equation okay at the left hand side"}, {"start": 55.12, "duration": 6.72, "text": "we're going to make use of the first"}, {"start": 59.44, "duration": 7.08, "text": "identity here right so we're going to"}, {"start": 61.84, "duration": 6.48, "text": "have 2^ 2 then of X this is equal to so"}, {"start": 66.52, "duration": 3.68, "text": "we make use of the second identity in"}, {"start": 68.32, "duration": 5.32, "text": "the at the right hand side of this"}, {"start": 70.2, "duration": 5.239, "text": "equation so we're going to have x 32 * 2"}, {"start": 73.64, "duration": 6.68, "text": "which is 64"}, {"start": 75.439, "duration": 11.401, "text": "right okay now from here 2 2 is 4 so we"}, {"start": 80.32, "duration": 8.72, "text": "have 4^ X this is equal to x^ of 64 now"}, {"start": 86.84, "duration": 3.639, "text": "let's go ahead and square both sides"}, {"start": 89.04, "duration": 3.56, "text": "again"}, {"start": 90.479, "duration": 9.32, "text": "so make use of the same identities we"}, {"start": 92.6, "duration": 11.68, "text": "have 4 x this isal to x 64 * 2 which is"}, {"start": 99.799, "duration": 9.201, "text": "128 right so from here we have 16 to the"}, {"start": 104.28, "duration": 7.76, "text": "power of X this is equal to x^ of"}, {"start": 109.0, "duration": 5.28, "text": "128 now let's Square both sides again"}, {"start": 112.04, "duration": 5.759, "text": "wow this is getting interesting"}, {"start": 114.28, "duration": 6.439, "text": "right okay we are looking for something"}, {"start": 117.799, "duration": 6.96, "text": "and we must find it now we do the same"}, {"start": 120.719, "duration": 7.081, "text": "thing 16 s then of X this is equal to x"}, {"start": 124.759, "duration": 4.761, "text": "1 28 * 2 that will give us"}, {"start": 127.8, "duration": 8.0, "text": "256"}, {"start": 129.52, "duration": 12.24, "text": "right okay now from here 16"}, {"start": 135.8, "duration": 9.799, "text": "squ okay now 16 squ is going to give us"}, {"start": 141.76, "duration": 6.64, "text": "256 right and here we have ra power of X"}, {"start": 145.599, "duration": 5.961, "text": "this is equal to X power of"}, {"start": 148.4, "duration": 5.919, "text": "256 right so from here we can compare"}, {"start": 151.56, "duration": 5.44, "text": "and find the value of x from here right"}, {"start": 154.319, "duration": 6.721, "text": "so we have the bases here x is equal to"}, {"start": 157.0, "duration": 6.08, "text": "256 or we make use of the powers X is"}, {"start": 161.04, "duration": 5.36, "text": "equal to"}, {"start": 163.08, "duration": 5.439, "text": "256 and this is the right answer to the"}, {"start": 166.4, "duration": 4.16, "text": "problem thank you for watching please do"}, {"start": 168.519, "duration": 5.961, "text": "wa to subscribe to my channel also like"}, {"start": 170.56, "duration": 3.92, "text": "comment and share bye-bye"}], "title": "Interesting exponential equation. How to solve for integral value of x ?", "description": "", "lengthSeconds": 173, "width": 640, "height": 360}
{"video_id": "YRLVhRlY7uk", "transcript": [{"start": 1.199, "duration": 3.961, "text": "hello good viewers welcome back once"}, {"start": 2.96, "duration": 5.24, "text": "again today we're going to learn how to"}, {"start": 5.16, "duration": 6.32, "text": "make use of the logorithm when we have"}, {"start": 8.2, "duration": 8.079, "text": "this kind of exponential"}, {"start": 11.48, "duration": 6.959, "text": "equation we have here 4^ x ="}, {"start": 16.279, "duration": 4.441, "text": "17 here we're going to make use of the"}, {"start": 18.439, "duration": 6.561, "text": "logm from the left hand side we're going"}, {"start": 20.72, "duration": 7.719, "text": "to log it so we have log of four to the^"}, {"start": 25.0, "duration": 7.599, "text": "of x from the right this is equal to log"}, {"start": 28.439, "duration": 6.601, "text": "of 17 right now let us recall this"}, {"start": 32.599, "duration": 5.561, "text": "property of logarithm that the log of a"}, {"start": 35.04, "duration": 6.96, "text": "to the power of B is equal to B multiply"}, {"start": 38.16, "duration": 6.719, "text": "by the log of a right so from here this"}, {"start": 42.0, "duration": 6.719, "text": "x here will come to the front so we have"}, {"start": 44.879, "duration": 8.081, "text": "X MTI by the log of 4 this is equal to"}, {"start": 48.719, "duration": 6.441, "text": "the log of 17 now from here remember"}, {"start": 52.96, "duration": 5.279, "text": "we're solving for x right so we're going"}, {"start": 55.16, "duration": 7.68, "text": "to divide both sides by the coefficient"}, {"start": 58.239, "duration": 9.321, "text": "of x here the coefficient of x is log of"}, {"start": 62.84, "duration": 8.72, "text": "4 right so here we have x * the log of 4"}, {"start": 67.56, "duration": 10.72, "text": "ided by the log of 4 this is equal to"}, {"start": 71.56, "duration": 9.96, "text": "log of 17 / the log of 4 so from here"}, {"start": 78.28, "duration": 6.199, "text": "log of four we cancel log of four right"}, {"start": 81.52, "duration": 6.88, "text": "so here we have the value for x in t of"}, {"start": 84.479, "duration": 7.121, "text": "the logarithm that is the log of 17 /"}, {"start": 88.4, "duration": 5.84, "text": "the log of 4 and when we imput this in"}, {"start": 91.6, "duration": 5.159, "text": "our calculator this is approximately"}, {"start": 94.24, "duration": 2.519, "text": "equal to"}, {"start": 97.399, "duration": 6.04, "text": "2.43 73 and this is the right answer"}, {"start": 101.88, "duration": 3.68, "text": "thank you for watching please kindly"}, {"start": 103.439, "duration": 6.201, "text": "subscribe to my channel also like"}, {"start": 105.56, "duration": 4.08, "text": "comment and share bye-bye"}], "title": "How to solve exponential equation using logarithms | Step by Step | #maths #mathematics", "description": "", "lengthSeconds": 108, "width": 596, "height": 360}
{"video_id": "YfymF_6bQw4", "transcript": [{"start": 0.16, "duration": 4.92, "text": "hello welcome back once again today we"}, {"start": 2.6, "duration": 4.6, "text": "have another interesting problem but"}, {"start": 5.08, "duration": 6.88, "text": "this time around we're going to use a"}, {"start": 7.2, "duration": 4.76, "text": "different method and this is called the"}, {"start": 12.759, "duration": 6.921, "text": "deac de poac"}, {"start": 16.0, "duration": 8.24, "text": "formula it's amazing and awesome the"}, {"start": 19.68, "duration": 6.2, "text": "five dict valuation of n Factor by De PO"}, {"start": 24.24, "duration": 4.64, "text": "is given as"}, {"start": 25.88, "duration": 8.24, "text": "Su as n goes from one all the way to"}, {"start": 28.88, "duration": 6.64, "text": "Infinity of the FL function of n / 5 to^"}, {"start": 34.12, "duration": 4.279, "text": "of"}, {"start": 35.52, "duration": 5.44, "text": "n amazing so we're going to find the"}, {"start": 38.399, "duration": 5.16, "text": "number of trading zeros of 2024 Factory"}, {"start": 40.96, "duration": 6.239, "text": "remember in the previous video we used"}, {"start": 43.559, "duration": 6.68, "text": "the legenda formula this around the poly"}, {"start": 47.199, "duration": 5.84, "text": "is an action right so let's go ahead and"}, {"start": 50.239, "duration": 6.32, "text": "find that now so the number of trading"}, {"start": 53.039, "duration": 6.401, "text": "zeros of 224 factoral is going to be"}, {"start": 56.559, "duration": 8.161, "text": "equal to sum as n goes from 1 to"}, {"start": 59.44, "duration": 7.16, "text": "Infinity of 2024 / 5^ n the FL of 8"}, {"start": 64.72, "duration": 5.52, "text": "right if you're not familiar with the"}, {"start": 66.6, "duration": 7.32, "text": "FLW function the FLW function of 6.9"}, {"start": 70.24, "duration": 5.6, "text": "only give us the integral part six six"}, {"start": 73.92, "duration": 4.559, "text": "the Flor function of"}, {"start": 75.84, "duration": 4.72, "text": "3.15 give us the integral part which is"}, {"start": 78.479, "duration": 5.121, "text": "three so that's just the operation of"}, {"start": 80.56, "duration": 5.64, "text": "the L of the FL function where is lumat"}, {"start": 83.6, "duration": 6.479, "text": "coming from am I okay okay let's"}, {"start": 86.2, "duration": 6.239, "text": "continue okay now from here we have the"}, {"start": 90.079, "duration": 4.121, "text": "number of training zeros of 2024"}, {"start": 92.439, "duration": 4.32, "text": "factorial is going to be equal to now"}, {"start": 94.2, "duration": 6.32, "text": "some from here we're going to have Flor"}, {"start": 96.759, "duration": 9.0, "text": "2024 when n = 1 that would be 5 year"}, {"start": 100.52, "duration": 9.44, "text": "plus sum 2024 / 5 2 which is"}, {"start": 105.759, "duration": 8.841, "text": "25 plus here sum oh sum are we still"}, {"start": 109.96, "duration": 9.36, "text": "suming yeah so 2024 / 5 Cub which is one"}, {"start": 114.6, "duration": 8.519, "text": "to5 then plus Flor I mean not some yeah"}, {"start": 119.32, "duration": 10.24, "text": "that is true then 5 to ^ 4 which is 1 25"}, {"start": 123.119, "duration": 11.48, "text": "+ 4 2024 / 5^ of 5 of five that would"}, {"start": 129.56, "duration": 6.28, "text": "give us 3 1 2 5 right then plus and so"}, {"start": 134.599, "duration": 3.481, "text": "on"}, {"start": 135.84, "duration": 4.6, "text": "right one of the most interesting part"}, {"start": 138.08, "duration": 5.4, "text": "of this is that this is going to"}, {"start": 140.44, "duration": 5.64, "text": "converge and the rest of the sum will be"}, {"start": 143.48, "duration": 4.08, "text": "equal to 0 0+ 0 plus 0 plus the way to"}, {"start": 146.08, "duration": 4.519, "text": "Infinity that will still be zero right"}, {"start": 147.56, "duration": 6.24, "text": "so here this divid by this"}, {"start": 150.599, "duration": 5.521, "text": "the flow of it is going to give us 44 so"}, {"start": 153.8, "duration": 6.48, "text": "no need of putting that if you divide"}, {"start": 156.12, "duration": 5.6, "text": "2024 by 5 you're going to have 44"}, {"start": 160.28, "duration": 3.56, "text": "something and the flow of it is just"}, {"start": 161.72, "duration": 4.48, "text": "that 44 which is the integral part then"}, {"start": 163.84, "duration": 9.52, "text": "plus this here the flow of it will give"}, {"start": 166.2, "duration": 10.959, "text": "us 80 + 16 + 3 + 0 + 0 + 0 + 0 dot dot"}, {"start": 173.36, "duration": 6.28, "text": "dot and the sum evaluates to"}, {"start": 177.159, "duration": 3.72, "text": "53 and this was exactly the answer we"}, {"start": 179.64, "duration": 3.4, "text": "got"}, {"start": 180.879, "duration": 4.08, "text": "in our previous video using the legend"}, {"start": 183.04, "duration": 4.24, "text": "formula thank you for watching please"}, {"start": 184.959, "duration": 6.28, "text": "kindly subscribe to my channel also like"}, {"start": 187.28, "duration": 3.959, "text": "comment and share bye-bye"}], "title": "De - Polignac Method", "description": "#maths #physics #olympiad #mathematics #number", "lengthSeconds": 189, "width": 556, "height": 360}
{"video_id": "_3QEQc0gjzc", "transcript": [{"start": 1.0, "duration": 4.799, "text": "hello welcome back once again today"}, {"start": 3.36, "duration": 5.84, "text": "we're going to solve this interesting or"}, {"start": 5.799, "duration": 6.481, "text": "impaired mathematics problem we're given"}, {"start": 9.2, "duration": 6.2, "text": "4^x is = to X so in this video we're"}, {"start": 12.28, "duration": 3.88, "text": "going to find the value for x so let's"}, {"start": 15.4, "duration": 2.959, "text": "get"}, {"start": 16.16, "duration": 9.56, "text": "started we know this"}, {"start": 18.359, "duration": 9.041, "text": "property a^ n is = to 1 all a to the^ n"}, {"start": 25.72, "duration": 3.36, "text": "so we're going to use this at the left"}, {"start": 27.4, "duration": 3.199, "text": "hand side so the left hand side will"}, {"start": 29.08, "duration": 5.96, "text": "become one"}, {"start": 30.599, "duration": 8.081, "text": "All 4 to the^ X this is equal to this"}, {"start": 35.04, "duration": 8.24, "text": "the same as X all 1 so we're going to"}, {"start": 38.68, "duration": 11.08, "text": "cross multiply so x * 4^ X will give us"}, {"start": 43.28, "duration": 9.279, "text": "x * 4 to power x so this is equal to 1 *"}, {"start": 49.76, "duration": 6.439, "text": "1 will give us"}, {"start": 52.559, "duration": 6.041, "text": "1 okay in our one step here let us make"}, {"start": 56.199, "duration": 6.601, "text": "use of this property we know a to the"}, {"start": 58.6, "duration": 9.839, "text": "power of B can be written as e to^ b *"}, {"start": 62.8, "duration": 7.88, "text": "the Ln of a right okay so 4^ X let us"}, {"start": 68.439, "duration": 8.561, "text": "write it so this will become"}, {"start": 70.68, "duration": 9.759, "text": "x e^ x the Ln 4 so this will be equal to"}, {"start": 77.0, "duration": 6.52, "text": "one now our next Target is to make the"}, {"start": 80.439, "duration": 4.841, "text": "power of E equal with the coefficient so"}, {"start": 83.52, "duration": 3.639, "text": "we can see the only difference between"}, {"start": 85.28, "duration": 4.32, "text": "this power and this coefficient is the"}, {"start": 87.159, "duration": 6.0, "text": "natural L of four so let's multiply both"}, {"start": 89.6, "duration": 9.199, "text": "by the natur 4 so this will become X the"}, {"start": 93.159, "duration": 9.28, "text": "natur 4 * e^ X the natur 4 so this is"}, {"start": 98.799, "duration": 5.801, "text": "equal to the natural log of four okay"}, {"start": 102.439, "duration": 5.081, "text": "now we know the natural log of"}, {"start": 104.6, "duration": 7.64, "text": "four is same as the natural log of 2 to"}, {"start": 107.52, "duration": 7.919, "text": "the^ 2 and with the power rule for logm"}, {"start": 112.24, "duration": 6.199, "text": "so this will become two is the power"}, {"start": 115.439, "duration": 6.64, "text": "rule not I don't think so it's just this"}, {"start": 118.439, "duration": 6.72, "text": "property that end of a the power b isal"}, {"start": 122.079, "duration": 5.241, "text": "b the Ln a so this power here is coming"}, {"start": 125.159, "duration": 5.001, "text": "down here so this will come 2 * the Ln"}, {"start": 127.32, "duration": 8.32, "text": "of 2 and we know two here we can write"}, {"start": 130.16, "duration": 6.92, "text": "as e^ the Ln of 2 then M the Ln of 2 so"}, {"start": 135.64, "duration": 6.959, "text": "this is what the right hand side will"}, {"start": 137.08, "duration": 11.799, "text": "become so we get here x * the natur 4 e"}, {"start": 142.599, "duration": 8.121, "text": "* x * natur of 4 so thisal to e^ Ln 2"}, {"start": 148.879, "duration": 3.841, "text": "the Ln 2"}, {"start": 150.72, "duration": 6.76, "text": "so we're going to make use of the Lamb"}, {"start": 152.72, "duration": 8.36, "text": "function so remember that the W of x * e"}, {"start": 157.48, "duration": 6.44, "text": "x is equal to X where the power of E and"}, {"start": 161.08, "duration": 5.4, "text": "the coefficients are equal so when we"}, {"start": 163.92, "duration": 5.08, "text": "use the lambar W function on both sides"}, {"start": 166.48, "duration": 2.52, "text": "of this"}, {"start": 171.56, "duration": 5.72, "text": "equation so this side will simplify to x"}, {"start": 174.8, "duration": 4.48, "text": "* the natural L of 4 so this will be"}, {"start": 177.28, "duration": 3.36, "text": "equal to here we simplify to natural log"}, {"start": 179.28, "duration": 5.48, "text": "of 2"}, {"start": 180.64, "duration": 4.12, "text": "so divide both side by the Natural"}, {"start": 191.0, "duration": 6.879, "text": "Four so this will cancel this so here we"}, {"start": 194.44, "duration": 7.6, "text": "get our solution X is equal to natur of"}, {"start": 197.879, "duration": 7.561, "text": "2 / we know natural of 4 is same as"}, {"start": 202.04, "duration": 7.399, "text": "remember from here Ln of 2 2 which is 2"}, {"start": 205.44, "duration": 7.879, "text": "* Ln of 2 so this will become 2 * Ln of"}, {"start": 209.439, "duration": 6.52, "text": "2 so Ln of 2 we cancel L of 2 so we have"}, {"start": 213.319, "duration": 7.041, "text": "1 / 2"}, {"start": 215.959, "duration": 6.28, "text": "or5 right so this is our solution thank"}, {"start": 220.36, "duration": 5.0, "text": "you for watching please kindly subscribe"}, {"start": 222.239, "duration": 6.801, "text": "to my channel also like comment and"}, {"start": 225.36, "duration": 3.68, "text": "share bye-bye"}], "title": "Stanford University \ud83c\udf93 Entrance Exam | Can you solve ?", "description": "", "lengthSeconds": 228, "width": 428, "height": 240}
{"video_id": "KvhoBTClSOc", "transcript": [{"start": 0.64, "duration": 4.96, "text": "hello good day viewers welcome back once"}, {"start": 2.84, "duration": 6.16, "text": "again please kindly subscribe to my"}, {"start": 5.6, "duration": 5.6, "text": "channel also like comment and share here"}, {"start": 9.0, "duration": 7.0, "text": "we're given the following interesting"}, {"start": 11.2, "duration": 8.28, "text": "algebraic problem we have a 2 b 2 plus p"}, {"start": 16.0, "duration": 5.8, "text": "2/ a 2 is ="}, {"start": 19.48, "duration": 6.959, "text": "727 right over here we're going to"}, {"start": 21.8, "duration": 8.399, "text": "evaluate a/ b + B over a"}, {"start": 26.439, "duration": 5.12, "text": "right okay now let's try expanding"}, {"start": 30.199, "duration": 8.441, "text": "the following"}, {"start": 31.559, "duration": 9.881, "text": "binomial A over B + b/ a s right so"}, {"start": 38.64, "duration": 6.56, "text": "we're going to make use of this property"}, {"start": 41.44, "duration": 7.279, "text": "a + b s this is equal to the first term"}, {"start": 45.2, "duration": 8.12, "text": "here squar plus the second term"}, {"start": 48.719, "duration": 7.401, "text": "squ plus 2 * their product right so here"}, {"start": 53.32, "duration": 7.12, "text": "we're going to have this to be equal to"}, {"start": 56.12, "duration": 14.4, "text": "a/ b 2 plus"}, {"start": 60.44, "duration": 16.999, "text": "B / a 2 + 2 * a/ B * B / a right so from"}, {"start": 70.52, "duration": 11.08, "text": "here we have a/ b + b/ a s so this is"}, {"start": 77.439, "duration": 6.081, "text": "equal to you know this is same as a 2/ b"}, {"start": 81.6, "duration": 6.36, "text": "2+ b"}, {"start": 83.52, "duration": 7.08, "text": "2/ a 2 plus you know this we cancel out"}, {"start": 87.96, "duration": 7.839, "text": "so we have two"}, {"start": 90.6, "duration": 12.36, "text": "and remember that a 2 b\u00b2 + B a\u00b2 is ="}, {"start": 95.799, "duration": 11.0, "text": "727 right so from here we have a/ b + b/"}, {"start": 102.96, "duration": 6.119, "text": "a 2 this is equal to"}, {"start": 106.799, "duration": 8.201, "text": "727 +"}, {"start": 109.079, "duration": 9.241, "text": "2 so from here we get a/ b + B over"}, {"start": 115.0, "duration": 6.2, "text": "a\u00b2 this is equal to"}, {"start": 118.32, "duration": 4.88, "text": "729 right so from here we're going to"}, {"start": 121.2, "duration": 6.32, "text": "take root of both sides and from here we"}, {"start": 123.2, "duration": 8.48, "text": "get the value for a/ B plus b/ a which"}, {"start": 127.52, "duration": 7.88, "text": "is equal to plus orus < TK"}, {"start": 131.68, "duration": 6.919, "text": "729 and this is equal to plus or minus"}, {"start": 135.4, "duration": 5.32, "text": "27 and this is the right answer thank"}, {"start": 138.599, "duration": 3.92, "text": "you for watching please kindly subscribe"}, {"start": 140.72, "duration": 5.32, "text": "to my channel also like comment and"}, {"start": 142.519, "duration": 3.521, "text": "share bye-bye"}], "title": "Interesting algebraic problem.#maths", "description": "", "lengthSeconds": 144, "width": 640, "height": 360}
{"video_id": "JwDs5fVGSnE", "transcript": [{"start": 0.0, "duration": 4.5, "text": "good day viewers welcome back to my"}, {"start": 2.399, "duration": 3.781, "text": "Channel today we have the following"}, {"start": 4.5, "duration": 4.92, "text": "interesting question"}, {"start": 6.18, "duration": 5.4, "text": "if a plus b is equal to seven"}, {"start": 9.42, "duration": 5.58, "text": "a raised by three plus b raise power is"}, {"start": 11.58, "duration": 6.0, "text": "equal to 28 then a times B is equal to"}, {"start": 15.0, "duration": 5.4, "text": "Dash Okay so let's see how we're going"}, {"start": 17.58, "duration": 5.64, "text": "to solve this problem now let's recall"}, {"start": 20.4, "duration": 4.44, "text": "the following identity"}, {"start": 23.22, "duration": 5.479, "text": "that a"}, {"start": 24.84, "duration": 7.5, "text": "plus b e raised power 3"}, {"start": 28.699, "duration": 5.441, "text": "minus three times a times B Open Bracket"}, {"start": 32.34, "duration": 3.54, "text": "a plus b"}, {"start": 34.14, "duration": 6.48, "text": "is equal to"}, {"start": 35.88, "duration": 7.08, "text": "e raised power 3 plus b respiratory okay"}, {"start": 40.62, "duration": 4.259, "text": "we're going to use this to solve this"}, {"start": 42.96, "duration": 3.9, "text": "problem okay"}, {"start": 44.879, "duration": 2.941, "text": "now we're going to make substitutions"}, {"start": 46.86, "duration": 2.76, "text": "here"}, {"start": 47.82, "duration": 3.68, "text": "so we already know that a plus b is"}, {"start": 49.62, "duration": 6.06, "text": "equal to seven so we have here seven"}, {"start": 51.5, "duration": 7.3, "text": "respiratory minus no three times a times"}, {"start": 55.68, "duration": 6.66, "text": "B we're looking for a times B then times"}, {"start": 58.8, "duration": 5.16, "text": "a plus b here is 7 is equal to a raise"}, {"start": 62.34, "duration": 4.08, "text": "power 3 plus b raised to power 3 is"}, {"start": 63.96, "duration": 3.479, "text": "already Twenty Eight so you can say in"}, {"start": 66.42, "duration": 3.96, "text": "question"}, {"start": 67.439, "duration": 7.68, "text": "so from here we get seven we raised to"}, {"start": 70.38, "duration": 6.84, "text": "the power 3 which is equal to 3 43 minus"}, {"start": 75.119, "duration": 6.741, "text": "you know three times over here is 21"}, {"start": 77.22, "duration": 10.02, "text": "then times a times B is equal to 28 okay"}, {"start": 81.86, "duration": 7.96, "text": "now we have here 21 a b you know if 343"}, {"start": 87.24, "duration": 8.28, "text": "is crossing over so we have three four"}, {"start": 89.82, "duration": 10.32, "text": "three minus 28 okay so here we have 21 a"}, {"start": 95.52, "duration": 7.98, "text": "b is equal to 343 minus 28 so we have"}, {"start": 100.14, "duration": 6.06, "text": "three one five so from here we divide"}, {"start": 103.5, "duration": 6.42, "text": "both subs equations of AAP so that is"}, {"start": 106.2, "duration": 6.54, "text": "over 21 over 21 and from here we get a"}, {"start": 109.92, "duration": 6.659, "text": "times B to be equal to three one five"}, {"start": 112.74, "duration": 5.76, "text": "divided by 21 which is equal to 15 and"}, {"start": 116.579, "duration": 3.36, "text": "this is the right answer thank you for"}, {"start": 118.5, "duration": 3.299, "text": "everyone and please make sure you"}, {"start": 119.939, "duration": 4.701, "text": "subscribe to my YouTube channel for more"}, {"start": 121.799, "duration": 2.841, "text": "interesting videos"}], "title": "A problem given to a student to solve. a + b = 7, a\u00b3 + b\u00b3 = 28,  ab = ?  Interesting problem!", "description": "", "lengthSeconds": 124, "width": 596, "height": 360}
{"video_id": "oPYHJrVUyHc", "transcript": [{"start": 0.76, "duration": 4.36, "text": "hello good viewers welcome back once"}, {"start": 2.56, "duration": 5.16, "text": "again let's solve the following given"}, {"start": 5.12, "duration": 6.04, "text": "exponential equation very quickly we"}, {"start": 7.72, "duration": 6.799, "text": "have here 2 ^ of X cubed this is equal"}, {"start": 11.16, "duration": 6.0, "text": "to 3 now we're going to make use of the"}, {"start": 14.519, "duration": 5.92, "text": "natural logarithm so we're going to Ln"}, {"start": 17.16, "duration": 8.279, "text": "both sides of this equation so here we"}, {"start": 20.439, "duration": 8.481, "text": "have Ln of 2^ of X Cub this is equal to"}, {"start": 25.439, "duration": 6.361, "text": "the Ln of 3 now with the property of"}, {"start": 28.92, "duration": 6.319, "text": "logarithm that the logm of a the power"}, {"start": 31.8, "duration": 7.32, "text": "of B this is equal to B the logarithm of"}, {"start": 35.239, "duration": 7.16, "text": "a right so here the power of two here is"}, {"start": 39.12, "duration": 7.079, "text": "X cubed so this power here is coming"}, {"start": 42.399, "duration": 8.361, "text": "over here so we have X cubed multip by"}, {"start": 46.199, "duration": 6.721, "text": "the Ln of 2 this is equal to the Ln of 3"}, {"start": 50.76, "duration": 5.24, "text": "remember that we're solving for the real"}, {"start": 52.92, "duration": 5.639, "text": "value of x right so here we're going to"}, {"start": 56.0, "duration": 8.159, "text": "divide both side by the Ln of 2 so we"}, {"start": 58.559, "duration": 9.801, "text": "have X cub the Ln of 2 / the Ln of 2"}, {"start": 64.159, "duration": 7.801, "text": "this is equal to the Ln of 3 / the Ln of"}, {"start": 68.36, "duration": 6.6, "text": "2 so from here this we divide out and"}, {"start": 71.96, "duration": 7.72, "text": "here we have X cubed this is equal to"}, {"start": 74.96, "duration": 6.88, "text": "the Ln of 3 / the Ln of 2 now to solve"}, {"start": 79.68, "duration": 5.6, "text": "for x here we're going to take the cube"}, {"start": 81.84, "duration": 6.319, "text": "root of both sides so from here x is"}, {"start": 85.28, "duration": 8.24, "text": "equal to the cube"}, {"start": 88.159, "duration": 9.561, "text": "root of the Ln of 3 / the Ln of 2 and"}, {"start": 93.52, "duration": 4.2, "text": "this is approximately equal to"}, {"start": 98.079, "duration": 5.601, "text": "1.65"}, {"start": 100.52, "duration": 4.959, "text": "9314 and this is the right answer thank"}, {"start": 103.68, "duration": 4.36, "text": "you for watching please kindly subscribe"}, {"start": 105.479, "duration": 6.28, "text": "to my channel also like comment and"}, {"start": 108.04, "duration": 3.719, "text": "share bye-bye"}], "title": "Germany | Interesting exponential equation | #maths #mathematics", "description": "", "lengthSeconds": 110, "width": 640, "height": 360}
{"video_id": "rdjDJw_Q_DA", "transcript": [{"start": 0.799, "duration": 7.401, "text": "hello good viewers welcome back once"}, {"start": 2.76, "duration": 8.959, "text": "again here we have X + Y = 2 here x * y"}, {"start": 8.2, "duration": 5.84, "text": "= 3 here we're going to find the value"}, {"start": 11.719, "duration": 5.001, "text": "of x Cub + y"}, {"start": 14.04, "duration": 6.64, "text": "Cub now let's solve this question very"}, {"start": 16.72, "duration": 8.639, "text": "quickly we're going to make use of this"}, {"start": 20.68, "duration": 9.88, "text": "nice algebraic identity that a cubed + B"}, {"start": 25.359, "duration": 10.561, "text": "Cubed = A + B Cubed"}, {"start": 30.56, "duration": 8.36, "text": "then - 3 * a * B into braet a +"}, {"start": 35.92, "duration": 6.24, "text": "b so with the help of this the"}, {"start": 38.92, "duration": 7.279, "text": "expression here we can write it in terms"}, {"start": 42.16, "duration": 11.44, "text": "of x + y and x - Y and that will be"}, {"start": 46.199, "duration": 11.121, "text": "equal to x + y Cub - 3 * x * Y into"}, {"start": 53.6, "duration": 5.919, "text": "braet x + y from here we make the"}, {"start": 57.32, "duration": 3.28, "text": "following substitutions this will be 2"}, {"start": 59.519, "duration": 6.081, "text": "cub"}, {"start": 60.6, "duration": 8.72, "text": "- 3 * x * Y which is 3 * X + Y which is"}, {"start": 65.6, "duration": 8.48, "text": "2 from here this equal 8"}, {"start": 69.32, "duration": 7.28, "text": "minus here 18 which is equal to -10 and"}, {"start": 74.08, "duration": 4.28, "text": "this is the right answer thank you for"}, {"start": 76.6, "duration": 4.44, "text": "watching please kindly subscribe to my"}, {"start": 78.36, "duration": 5.68, "text": "channel also like comment and share"}, {"start": 81.04, "duration": 3.0, "text": "bye-bye"}], "title": "Interesting Junior Olympiad Math.", "description": "", "lengthSeconds": 82, "width": 542, "height": 360}
{"video_id": "Ca5AXw4zSI0", "transcript": [{"start": 0.48, "duration": 6.24, "text": "hello welcome back once again today we"}, {"start": 2.879, "duration": 6.441, "text": "have this nice equation here involving"}, {"start": 6.72, "duration": 5.919, "text": "thetion right from the right hand side"}, {"start": 9.32, "duration": 7.36, "text": "of this equation we have 16 Tetra 3"}, {"start": 12.639, "duration": 6.56, "text": "which is not same as 16 to^ 3 okay for"}, {"start": 16.68, "duration": 3.28, "text": "those of us that are not familiar with"}, {"start": 19.199, "duration": 5.961, "text": "the"}, {"start": 19.96, "duration": 9.159, "text": "tetration so 16 Tetra 3 is same as 16 ra"}, {"start": 25.16, "duration": 5.72, "text": "the power of 16 rais power of 16 in"}, {"start": 29.119, "duration": 5.6, "text": "three places"}, {"start": 30.88, "duration": 10.08, "text": "just like 4 T 2 which is same as 4 to"}, {"start": 34.719, "duration": 8.561, "text": "the^ 4 and not equal to 4 to^ 2 okay so"}, {"start": 40.96, "duration": 4.84, "text": "from the left hand side here we have 2^"}, {"start": 43.28, "duration": 6.56, "text": "2^ X then we'll write the right hand"}, {"start": 45.8, "duration": 8.919, "text": "side as 16 to the^ 16 to^"}, {"start": 49.84, "duration": 6.879, "text": "16 okay so here we have 2^ 2 to^ x so"}, {"start": 54.719, "duration": 6.201, "text": "this will be equal to 16 here which is"}, {"start": 56.719, "duration": 6.561, "text": "the base here is 2 ^ 4 then"}, {"start": 60.92, "duration": 6.68, "text": "16"}, {"start": 63.28, "duration": 6.68, "text": "16 so from here we have 2^ 2^ X so this"}, {"start": 67.6, "duration": 8.0, "text": "will be equal to 2 remember this"}, {"start": 69.96, "duration": 10.479, "text": "property a^ b^ c isal a^ b * C so this"}, {"start": 75.6, "duration": 7.08, "text": "will be 4 * 16 to the^ of 16 so we can"}, {"start": 80.439, "duration": 5.04, "text": "see we have the same basis from both"}, {"start": 82.68, "duration": 5.52, "text": "sides so we're going to equate the power"}, {"start": 85.479, "duration": 7.64, "text": "so this implies from here that 2 to the^"}, {"start": 88.2, "duration": 12.36, "text": "of X is equal 2 4 is as 2^"}, {"start": 93.119, "duration": 9.241, "text": "2 Then multip by here 16 here is 2^ 4"}, {"start": 100.56, "duration": 6.239, "text": "then to^"}, {"start": 102.36, "duration": 7.64, "text": "16 so from here we have 2^ X is equal to"}, {"start": 106.799, "duration": 7.081, "text": "here we have 2 to the^ of"}, {"start": 110.0, "duration": 8.0, "text": "2 okay so from here we have this multip"}, {"start": 113.88, "duration": 6.76, "text": "by 2 to the power of here will be 4 * 16"}, {"start": 118.0, "duration": 6.64, "text": "which is 64"}, {"start": 120.64, "duration": 5.96, "text": "so here we have 2^ X will be equal to"}, {"start": 124.64, "duration": 4.2, "text": "remember we have the same base so we're"}, {"start": 126.6, "duration": 6.68, "text": "going to add the powers here so we have"}, {"start": 128.84, "duration": 6.119, "text": "2^ 64 + 2 which is 66 so from here we"}, {"start": 133.28, "duration": 3.36, "text": "can see we have the same basis so we're"}, {"start": 134.959, "duration": 6.441, "text": "going to equate the power so this"}, {"start": 136.64, "duration": 7.599, "text": "implies from here that X is equal to 66"}, {"start": 141.4, "duration": 5.08, "text": "and this is the right answer thank you"}, {"start": 144.239, "duration": 5.72, "text": "for watching please kindly subscribe to"}, {"start": 146.48, "duration": 6.479, "text": "my channel also like comment and share"}, {"start": 149.959, "duration": 3.0, "text": "bye-bye"}], "title": "TETRATION PROBLEM | Can you solve ?", "description": "", "lengthSeconds": 152, "width": 640, "height": 360}
{"video_id": "qCU4PaKkm-k", "transcript": [{"start": 1.04, "duration": 5.52, "text": "hello welcome back once again today we"}, {"start": 3.08, "duration": 5.08, "text": "have this interesting radical equation"}, {"start": 6.56, "duration": 4.8, "text": "and here we're going to solve for the"}, {"start": 8.16, "duration": 6.04, "text": "values of X so let's get"}, {"start": 11.36, "duration": 7.72, "text": "started from the left hand side we have"}, {"start": 14.2, "duration": 9.159, "text": "the root of x plus the root"}, {"start": 19.08, "duration": 9.56, "text": "ofx this is equal to 4 this time let's"}, {"start": 23.359, "duration": 7.681, "text": "take it to the right right so this will"}, {"start": 28.64, "duration": 6.56, "text": "become root"}, {"start": 31.04, "duration": 6.92, "text": "ofx is equal to that would be 4us theun"}, {"start": 35.2, "duration": 5.6, "text": "of X so in our next step let's Square"}, {"start": 37.96, "duration": 3.68, "text": "both sides of this equation so we have"}, {"start": 40.8, "duration": 7.239, "text": "root"}, {"start": 41.64, "duration": 8.04, "text": "of- x^ 2 this will be equal to 4 - otk"}, {"start": 48.039, "duration": 4.921, "text": "x^"}, {"start": 49.68, "duration": 6.08, "text": "2 now here Square will cancel square"}, {"start": 52.96, "duration": 6.119, "text": "root and for the right hand side let us"}, {"start": 55.76, "duration": 8.359, "text": "make use of this interesting identity"}, {"start": 59.079, "duration": 10.641, "text": "for a minus b 2 so this is = to a 2 + b"}, {"start": 64.119, "duration": 9.841, "text": "2 - 2 * a * B so here we only"}, {"start": 69.72, "duration": 9.719, "text": "havex so this will be equal to 4 2 is 16"}, {"start": 73.96, "duration": 8.199, "text": "plus otk X2 is just x - 2 * 4 * otk X"}, {"start": 79.439, "duration": 8.121, "text": "which is 8un"}, {"start": 82.159, "duration": 8.64, "text": "X so 8 root X"}, {"start": 87.56, "duration": 6.32, "text": "right okay so from here"}, {"start": 90.799, "duration": 7.28, "text": "we have to move this term to this side"}, {"start": 93.88, "duration": 4.199, "text": "so remember that this term"}, {"start": 98.72, "duration": 7.6, "text": "here we subtracting 8ot X from this term"}, {"start": 102.52, "duration": 6.44, "text": "so if 16 + xus this will give us this so"}, {"start": 106.32, "duration": 5.2, "text": "This actually mean that 16 + xus this"}, {"start": 108.96, "duration": 7.119, "text": "one will also give us this so that will"}, {"start": 111.52, "duration": 4.559, "text": "become 16 + x"}, {"start": 116.159, "duration": 8.0, "text": "--x so this is equal to 8 < TK X"}, {"start": 120.399, "duration": 8.321, "text": "X so here we have 16 plus you know this"}, {"start": 124.159, "duration": 8.841, "text": "is x - - x which is x + x which is 2x"}, {"start": 128.72, "duration": 7.239, "text": "then this is equal to 8 < TK X so now"}, {"start": 133.0, "duration": 5.92, "text": "let's Square both sides to get rid of"}, {"start": 135.959, "duration": 6.521, "text": "this root sign at the right hand side so"}, {"start": 138.92, "duration": 10.0, "text": "here expanding 16 s is"}, {"start": 142.48, "duration": 10.32, "text": "256 plus 2 x^2 is 4 x 2 + 2 * 16 * 2 x"}, {"start": 148.92, "duration": 10.0, "text": "is 64x X then this is equal to you know"}, {"start": 152.8, "duration": 8.359, "text": "82 is 64 otk x^2 is just X we can see we"}, {"start": 158.92, "duration": 4.88, "text": "have positive 64x from both sides so"}, {"start": 161.159, "duration": 4.201, "text": "subtract 64x from both sides and we are"}, {"start": 163.8, "duration": 8.12, "text": "left with"}, {"start": 165.36, "duration": 8.239, "text": "4x2 plus here 256 is equal to Z 256 is"}, {"start": 171.92, "duration": 3.44, "text": "positive if it's crossing over it to"}, {"start": 173.599, "duration": 5.241, "text": "become negative so we have"}, {"start": 175.36, "duration": 7.0, "text": "4x^2 is = to"}, {"start": 178.84, "duration": 7.039, "text": "-256 divide both sides by"}, {"start": 182.36, "duration": 8.2, "text": "four four we cancel four from here so we"}, {"start": 185.879, "duration": 5.64, "text": "have X2 is equal to -256 / 4 will give"}, {"start": 190.56, "duration": 3.44, "text": "us"}, {"start": 191.519, "duration": 6.36, "text": "-64 take the square root of both sides"}, {"start": 194.0, "duration": 7.2, "text": "so we get X isal to plus or minus the"}, {"start": 197.879, "duration": 8.64, "text": "otk of -64 which is equal to plus or"}, {"start": 201.2, "duration": 8.319, "text": "minus of1 is iota andun of 64 is 8 so"}, {"start": 206.519, "duration": 7.961, "text": "therefore we have our solutions for x x"}, {"start": 209.519, "duration": 6.601, "text": "is equal to plus or minus 8 I which is"}, {"start": 214.48, "duration": 3.88, "text": "an imaginary"}, {"start": 216.12, "duration": 5.24, "text": "solution thank you for watching please"}, {"start": 218.36, "duration": 7.84, "text": "kindly subscribe to my channel also like"}, {"start": 221.36, "duration": 4.84, "text": "comment and share bye-bye"}], "title": "Canada | Nice Olympiad Math problem | Can you solve ?", "description": "", "lengthSeconds": 225, "width": 428, "height": 240}
{"video_id": "I8lhlcUFdZs", "transcript": [{"start": 0.32, "duration": 7.92, "text": "let's find the derivative of x Cub *"}, {"start": 4.52, "duration": 6.119, "text": "tangent X using the product rule first"}, {"start": 8.24, "duration": 6.12, "text": "let's identify the two functions we're"}, {"start": 10.639, "duration": 8.441, "text": "working with let's call you of x = x"}, {"start": 14.36, "duration": 7.839, "text": "cubed and V of X = tangent X the product"}, {"start": 19.08, "duration": 7.88, "text": "rule states that the derivative of U * v"}, {"start": 22.199, "duration": 7.32, "text": "= u * the derivative of V + V * the"}, {"start": 26.96, "duration": 5.56, "text": "derivative of U let's substitute our"}, {"start": 29.519, "duration": 6.401, "text": "function into the product rule formula"}, {"start": 32.52, "duration": 7.28, "text": "we get X Cub * the derivative of tangent"}, {"start": 35.92, "duration": 8.0, "text": "X Plus tangent x times the derivative of"}, {"start": 39.8, "duration": 6.56, "text": "x cubed now let's find each derivative"}, {"start": 43.92, "duration": 5.92, "text": "separately first the derivative of"}, {"start": 46.36, "duration": 6.24, "text": "tangent X is secant 2 x and the"}, {"start": 49.84, "duration": 5.84, "text": "derivative of x cubed is"}, {"start": 52.6, "duration": 6.52, "text": "3x^2 let's substitute these derivatives"}, {"start": 55.68, "duration": 7.68, "text": "back into our equation we get X Cub *"}, {"start": 59.12, "duration": 7.84, "text": "secant 2 x + Tang x *"}, {"start": 63.36, "duration": 9.68, "text": "3x^2 now we can factor out x^2 from both"}, {"start": 66.96, "duration": 9.479, "text": "terms this gives us x^2 * thean x^ 2 x +"}, {"start": 73.04, "duration": 5.96, "text": "3 tangent X let's review what we did we"}, {"start": 76.439, "duration": 5.161, "text": "use the product rule to break down the"}, {"start": 79.0, "duration": 5.6, "text": "derivative into two parts we found the"}, {"start": 81.6, "duration": 6.04, "text": "derivative of tangent X is secant 2 x"}, {"start": 84.6, "duration": 6.479, "text": "and the derivative of x cubed is"}, {"start": 87.64, "duration": 6.519, "text": "3x^2 then we combine terms and factored"}, {"start": 91.079, "duration": 6.481, "text": "out x^2 to get our final answer and"}, {"start": 94.159, "duration": 10.28, "text": "there we have it the derivative of x Cub"}, {"start": 97.56, "duration": 10.32, "text": "tangent X is x^2 * thean x^ 2 x + 3"}, {"start": 104.439, "duration": 3.441, "text": "tangent X"}], "title": "Finding the Derivative of x^3 tan(x)  Using the Product Rule | Step-by-Step Explanation", "description": "In this video, learn to differentiate x^3 tan(x)  using the Product Rule in calculus.\n\nWhat You'll Learn:  \n- Definition of the Product Rule  \n- Step-by-step differentiation of x^3 tan(x)   \n- Formula breakdown and example problem-solving  \n\nThis example is perfect for calculus beginners or those reviewing the Product Rule!\n\ud83d\udca1 Don't forget to LIKE, SHARE, and SUBSCRIBE for more math tutorials!\n#Calculus #ProductRule #Differentiation #Derivative #MathTutorial #LearnMath #MathEducation #MathHelp", "lengthSeconds": 106, "width": 640, "height": 360}
{"video_id": "8UMMczVRp3Q", "transcript": [{"start": 0.24, "duration": 5.119, "text": "let's solve this calculus problem step"}, {"start": 2.52, "duration": 6.64, "text": "by step we need to find the derivative"}, {"start": 5.359, "duration": 6.521, "text": "of the natural logarithm of cine 2 Theta"}, {"start": 9.16, "duration": 5.2, "text": "first let's identify that this is a"}, {"start": 11.88, "duration": 4.92, "text": "composite function where we have the"}, {"start": 14.36, "duration": 4.96, "text": "natural log of another function this"}, {"start": 16.8, "duration": 4.8, "text": "means we'll need to use the chain rule"}, {"start": 19.32, "duration": 6.56, "text": "remember that the derivative of natural"}, {"start": 21.6, "duration": 6.28, "text": "log of U is 1/ U * the derivative of U"}, {"start": 25.88, "duration": 5.44, "text": "now we need to find the derivative of"}, {"start": 27.88, "duration": 6.28, "text": "cine 2 thet this is itself requires the"}, {"start": 31.32, "duration": 5.399, "text": "chain rule since we have a composite"}, {"start": 34.16, "duration": 5.36, "text": "function the derivative of cosine is"}, {"start": 36.719, "duration": 5.441, "text": "negative s and we'll multiply that by"}, {"start": 39.52, "duration": 6.879, "text": "the derivative of what's inside which is"}, {"start": 42.16, "duration": 7.399, "text": "2 Theta the derivative of 2 Theta with"}, {"start": 46.399, "duration": 6.121, "text": "respect to Theta is simply 2 let's"}, {"start": 49.559, "duration": 8.881, "text": "rearrange these terms we can write this"}, {"start": 52.52, "duration": 9.519, "text": "as Sin 2 Theta over cine 2 thet * 2"}, {"start": 58.44, "duration": 7.079, "text": "finally recall that s over cosine is the"}, {"start": 62.039, "duration": 9.12, "text": "definition of tangent therefore our"}, {"start": 65.519, "duration": 7.761, "text": "final answer is -2 * tangent of 2 Theta"}, {"start": 71.159, "duration": 4.32, "text": "let's visualize what this means"}, {"start": 73.28, "duration": 5.44, "text": "geometrically here's the graph of our"}, {"start": 75.479, "duration": 6.841, "text": "original function natural log of cosine"}, {"start": 78.72, "duration": 8.96, "text": "2 Theta in blue and here's its"}, {"start": 82.32, "duration": 7.72, "text": "derivative -2 tangent 2 Theta in red"}, {"start": 87.68, "duration": 4.84, "text": "notice how the derivative is steepest"}, {"start": 90.04, "duration": 6.56, "text": "where our original function changes the"}, {"start": 92.52, "duration": 6.48, "text": "most rapidly and that's how we find and"}, {"start": 96.6, "duration": 6.479, "text": "understand the derivative of natural log"}, {"start": 99.0, "duration": 7.0, "text": "of cosine 2 Theta remember the key steps"}, {"start": 103.079, "duration": 5.08, "text": "were using the chain rule twice and"}, {"start": 106.0, "duration": 5.159, "text": "recognizing the tangent identity at the"}, {"start": 108.159, "duration": 3.0, "text": "end"}], "title": "Understanding the Derivative of [ln(\u2061cos\u20612\ud835\udf03)]   Step by Step", "description": "Question: Find the derivative of [ln(\u2061cos\u20612\ud835\udf03)] \nIn this video, we break down the process of finding the derivative of  ln(\u2061cos\u20612\ud835\udf03)  using the chain rule and trigonometric identities. Join us for a detailed explanation to enhance your calculus skills!\n#Calculus #ChainRule #Trigonometry #Derivatives #MathTutorial #hsc #hschighermath #1stpaper\n#hschighermath1stpaper\n#englishversion #cpmrana", "lengthSeconds": 110, "width": 640, "height": 360}
{"video_id": "zr5c2nZ7P2c", "transcript": [{"start": 0.32, "duration": 5.16, "text": "let's solve this differentiation problem"}, {"start": 2.639, "duration": 5.681, "text": "step by step we need to find the"}, {"start": 5.48, "duration": 7.44, "text": "derivative of the product of log base a"}, {"start": 8.32, "duration": 7.12, "text": "of X and natural log of x first let's"}, {"start": 12.92, "duration": 6.279, "text": "recall the product rule the derivative"}, {"start": 15.44, "duration": 7.72, "text": "of a product of functions U * v = u *"}, {"start": 19.199, "duration": 7.4, "text": "the derivative of V + V * the derivative"}, {"start": 23.16, "duration": 6.959, "text": "of U in our case you is log base of X"}, {"start": 26.599, "duration": 6.12, "text": "and V is the natural log of x now let's"}, {"start": 30.119, "duration": 6.44, "text": "apply the product rule we'll have log"}, {"start": 32.719, "duration": 8.041, "text": "base uh of x * the derivative of natural"}, {"start": 36.559, "duration": 7.441, "text": "log of x plus natural log of x times the"}, {"start": 40.76, "duration": 5.92, "text": "derivative of log base a of X let's"}, {"start": 44.0, "duration": 6.719, "text": "recall that the derivative of natural"}, {"start": 46.68, "duration": 7.48, "text": "log of x is 1 /x and the derivative of"}, {"start": 50.719, "duration": 8.041, "text": "log base uh of X can be written as log"}, {"start": 54.16, "duration": 7.719, "text": "base uh of e * 1 /x substituting these"}, {"start": 58.76, "duration": 8.96, "text": "derivatives into our equation we get log"}, {"start": 61.879, "duration": 11.641, "text": "base of x * 1 /x plus natural log of x *"}, {"start": 67.72, "duration": 9.6, "text": "log base a of e * 1X now we can factor"}, {"start": 73.52, "duration": 7.639, "text": "out 1 /x from both terms this gives us 1"}, {"start": 77.32, "duration": 8.56, "text": "/x * the quantity log base a of x +"}, {"start": 81.159, "duration": 7.801, "text": "natural log of x * log base a of e let's"}, {"start": 85.88, "duration": 6.0, "text": "verify this makes sense both terms have"}, {"start": 88.96, "duration": 5.36, "text": "a factor of 1 /x which is what we"}, {"start": 91.88, "duration": 4.96, "text": "typically see when differentiating"}, {"start": 94.32, "duration": 5.56, "text": "logarithmic functions the remaining"}, {"start": 96.84, "duration": 5.599, "text": "terms combine the original logarithmic"}, {"start": 99.88, "duration": 4.599, "text": "expressions in a way that preserves"}, {"start": 102.439, "duration": 5.72, "text": "their relationship through the chain"}, {"start": 104.479, "duration": 6.161, "text": "Rule and that completes our solution"}, {"start": 108.159, "duration": 5.041, "text": "remember when differentiating products"}, {"start": 110.64, "duration": 4.88, "text": "of logarithms always start with the"}, {"start": 113.2, "duration": 5.48, "text": "product rule then use the standard"}, {"start": 115.52, "duration": 7.48, "text": "derivatives of logarithmic functions and"}, {"start": 118.68, "duration": 4.32, "text": "finally combine like terms"}], "title": "Finding the Derivative of {(log_\ud835\udc4e\u2061^\ud835\udc65 )(ln\u2061\ud835\udc65 )} Using the Product Rule |Step-by-Step Explanation.", "description": "In this video, learn to differentiate {(log_\ud835\udc4e\u2061^\ud835\udc65 )(ln\u2061\ud835\udc65 )} using the Product Rule in calculus.\n\nWhat You'll Learn:  \n- Definition of the Product Rule  \n- Step-by-step differentiation of {(log_\ud835\udc4e\u2061^\ud835\udc65 )(ln\u2061\ud835\udc65 )}\n- Formula breakdown and example problem-solving  \n\nThis example is perfect for calculus beginners or those reviewing the Product Rule!\n\ud83d\udca1 Don't forget to LIKE, SHARE, and SUBSCRIBE for more math tutorials!\n#Calculus #ProductRule #Differentiation #Derivative #MathTutorial #LearnMath #MathEducation #MathHelp", "lengthSeconds": 122, "width": 640, "height": 360}
{"video_id": "9uTWUr2rupw", "transcript": [{"start": 0.24, "duration": 5.16, "text": "let's solve this calculus problem step"}, {"start": 2.48, "duration": 6.159, "text": "by step we need to find the derivative"}, {"start": 5.4, "duration": 5.159, "text": "of the natural log of x^2 this is a"}, {"start": 8.639, "duration": 5.2, "text": "perfect opportunity to use the chain"}, {"start": 10.559, "duration": 6.601, "text": "rule the chain rule states that when we"}, {"start": 13.839, "duration": 5.641, "text": "have a composite function we multiply"}, {"start": 17.16, "duration": 4.92, "text": "the derivative of the outer function by"}, {"start": 19.48, "duration": 5.68, "text": "the derivative of the inner function"}, {"start": 22.08, "duration": 6.199, "text": "let's identify our functions the outer"}, {"start": 25.16, "duration": 6.0, "text": "function is the square function anything"}, {"start": 28.279, "duration": 5.761, "text": "squared and the inner function is the"}, {"start": 31.16, "duration": 6.239, "text": "natural logarithm now let's apply the"}, {"start": 34.04, "duration": 5.999, "text": "chain rule First We Take the derivative"}, {"start": 37.399, "duration": 7.121, "text": "of the outer function with respect to U"}, {"start": 40.039, "duration": 8.121, "text": "the derivative of u^2 is 2 U then we"}, {"start": 44.52, "duration": 6.6, "text": "substitute back lnx for you giving us 2"}, {"start": 48.16, "duration": 5.28, "text": "lnx next we need to multiply by the"}, {"start": 51.12, "duration": 6.16, "text": "derivative of the inner function the"}, {"start": 53.44, "duration": 7.759, "text": "derivative of Ln X is 1"}, {"start": 57.28, "duration": 8.6, "text": "/x now we multiply these Parts together"}, {"start": 61.199, "duration": 8.92, "text": "2 Ln x * 1 /x this gives us our final"}, {"start": 65.88, "duration": 4.239, "text": "answer 2 Ln X"}, {"start": 70.52, "duration": 7.48, "text": "/x let's verify this makes sense if we"}, {"start": 74.52, "duration": 7.2, "text": "graph y equal Ln x^2 this is what its"}, {"start": 78.0, "duration": 6.4, "text": "derivative looks like notice how the"}, {"start": 81.72, "duration": 5.56, "text": "derivative is positive for X greater"}, {"start": 84.4, "duration": 5.56, "text": "than one which matches our function's"}, {"start": 87.28, "duration": 4.92, "text": "increasing behavior in that region and"}, {"start": 89.96, "duration": 3.32, "text": "and that's how we find the derivative of"}, {"start": 92.2, "duration": 5.879, "text": "Ln"}, {"start": 93.28, "duration": 4.799, "text": "x^2 using the chain rule"}], "title": "Mastering the Derivative Chain Rule Explained with [(lnx)^2] Example", "description": "Question: Find the derivative of [(lnx)^2]\nDive into the world of calculus with this step-by-step tutorial on differentiating logarithmic functions using the chain rule. We\u2019ll break down how to find the derivative of [(lnx)^2]  and understand each part of the process clearly. Perfect for students seeking a deeper understanding of these concepts!\n#Calculus #ChainRule #Derivative #MathTutorial #hsc #hschighermath #1stpaper #hschighermath1stpaper #englishversion #cpmrana", "lengthSeconds": 97, "width": 640, "height": 360}
{"video_id": "arZEuQDUdLM", "transcript": [{"start": 0.28, "duration": 4.96, "text": "let's solve this differentiation problem"}, {"start": 2.679, "duration": 6.441, "text": "step by step we'll use a clever"}, {"start": 5.24, "duration": 6.64, "text": "substitution to simplify our work first"}, {"start": 9.12, "duration": 6.08, "text": "let's look at the substitution we'll use"}, {"start": 11.88, "duration": 6.88, "text": "we'll let x equal tangent of t which"}, {"start": 15.2, "duration": 5.72, "text": "means T equals inverse tangent of X"}, {"start": 18.76, "duration": 4.96, "text": "let's see what happens when we make this"}, {"start": 20.92, "duration": 5.599, "text": "substitution in our original expression"}, {"start": 23.72, "duration": 7.92, "text": "we replace x with Tangent T in the"}, {"start": 26.519, "duration": 7.321, "text": "fraction 2x over 1 + x s now here's"}, {"start": 31.64, "duration": 5.4, "text": "where we use a key trigonometric"}, {"start": 33.84, "duration": 6.84, "text": "identity the expression to tangent T"}, {"start": 37.04, "duration": 7.8, "text": "over 1 + tangent s t is actually equal"}, {"start": 40.68, "duration": 7.8, "text": "to sin of 2T this is a double angle"}, {"start": 44.84, "duration": 7.16, "text": "formula therefore our expression becomes"}, {"start": 48.48, "duration": 6.44, "text": "sin inverse of sin 2T since sin inverse"}, {"start": 52.0, "duration": 7.0, "text": "and S are inverse functions they cancel"}, {"start": 54.92, "duration": 7.959, "text": "out leaving us with Just 2 T now we can"}, {"start": 59.0, "duration": 6.799, "text": "substitute back for T remember that t"}, {"start": 62.879, "duration": 5.88, "text": "equals inverse tangent of X so our"}, {"start": 65.799, "duration": 6.761, "text": "expression becomes 2 * inverse tangent"}, {"start": 68.759, "duration": 6.201, "text": "of X finally we can differentiate this"}, {"start": 72.56, "duration": 6.52, "text": "expression the derivative of inverse"}, {"start": 74.96, "duration": 7.04, "text": "tangent is 1 over 1 + x^2 and we"}, {"start": 79.08, "duration": 6.44, "text": "multiply this by two let's summarize the"}, {"start": 82.0, "duration": 7.68, "text": "key steps first we made a substitution"}, {"start": 85.52, "duration": 6.52, "text": "xal tangent T then we used the double"}, {"start": 89.68, "duration": 5.32, "text": "angle angle formula to simplify the"}, {"start": 92.04, "duration": 7.0, "text": "expression next we recognized that sin"}, {"start": 95.0, "duration": 6.159, "text": "inverse and S cancel out finally we"}, {"start": 99.04, "duration": 8.28, "text": "substituted back and differentiated to"}, {"start": 101.159, "duration": 6.161, "text": "get our answer 2 over 1 + x^2"}], "title": "How to Differentiate Inverse Trigonometric Functions [\ud835\udc60\ud835\udc56\ud835\udc5b^(\u22121)   2\ud835\udc65/(1+\ud835\udc65^2 )] A Step-by-Step Guide", "description": "In this video, we\u2019ll dive into the differentiation of inverse trigonometric functions, using a detailed example involving the derivative of  [\ud835\udc60\ud835\udc56\ud835\udc5b^(\u22121)   2\ud835\udc65/(1+\ud835\udc65^2 )]  . Follow along as we break it down step-by-step, explaining each part of the process. Perfect for students looking to strengthen their calculus skills!\n#Calculus #Differentiation #InverseTrigonometricFunctions #MathTutorial #LearningMath", "lengthSeconds": 110, "width": 640, "height": 360}
{"video_id": "TcPuxuYoxeI", "transcript": [{"start": 0.48, "duration": 6.319, "text": "welcome today we'll find the derivative"}, {"start": 3.56, "duration": 6.92, "text": "of a complex function that combines"}, {"start": 6.799, "duration": 7.361, "text": "exponentials and logarithms our function"}, {"start": 10.48, "duration": 6.68, "text": "is y = a raised to the natural log of"}, {"start": 14.16, "duration": 5.64, "text": "cosine X let's break down the structure"}, {"start": 17.16, "duration": 5.64, "text": "of this function we have an exponential"}, {"start": 19.8, "duration": 6.239, "text": "function with base a and its exponent is"}, {"start": 22.8, "duration": 5.36, "text": "the natural log of cosine X this means"}, {"start": 26.039, "duration": 4.801, "text": "we'll need to use the chain rule"}, {"start": 28.16, "duration": 5.32, "text": "multiple times let's start with our"}, {"start": 30.84, "duration": 6.12, "text": "first application of the chain rule for"}, {"start": 33.48, "duration": 7.28, "text": "an exponential function a to the X its"}, {"start": 36.96, "duration": 9.4, "text": "derivative is a to x * the natural log"}, {"start": 40.76, "duration": 7.72, "text": "of a in our case X is Ln of cine X now"}, {"start": 46.36, "duration": 5.28, "text": "for our second application of the chain"}, {"start": 48.48, "duration": 7.0, "text": "rule we need to find the derivative of"}, {"start": 51.64, "duration": 7.919, "text": "Ln of cine X the derivative of natural"}, {"start": 55.48, "duration": 6.36, "text": "log of U is 1 / U * the derivative of U"}, {"start": 59.559, "duration": 7.121, "text": "we know know that the derivative of cine"}, {"start": 61.84, "duration": 8.76, "text": "X is sinx let's substitute this in"}, {"start": 66.68, "duration": 7.079, "text": "notice that sinx over cine X simplifies"}, {"start": 70.6, "duration": 6.519, "text": "to tangent X this is an important"}, {"start": 73.759, "duration": 6.0, "text": "trigonometric relationship to remember"}, {"start": 77.119, "duration": 5.801, "text": "now let's substitute this back into our"}, {"start": 79.759, "duration": 9.04, "text": "original derivative we get a to the Ln"}, {"start": 82.92, "duration": 8.199, "text": "of cine x * Ln of a Time tangent X but"}, {"start": 88.799, "duration": 5.32, "text": "we can simplify this this further"}, {"start": 91.119, "duration": 6.36, "text": "remember that a to the Ln of something"}, {"start": 94.119, "duration": 6.96, "text": "equals that something so a to the Ln of"}, {"start": 97.479, "duration": 7.801, "text": "cosine X simplifies to just cosine X"}, {"start": 101.079, "duration": 10.841, "text": "therefore our final answer is cosine x *"}, {"start": 105.28, "duration": 9.439, "text": "Ln of a * tangent X or cosine x * Ln of"}, {"start": 111.92, "duration": 6.159, "text": "a Time tangent X let's review what we"}, {"start": 114.719, "duration": 6.68, "text": "learned one we use the chain rule twice"}, {"start": 118.079, "duration": 7.0, "text": "to we apply the derivative rules for"}, {"start": 121.399, "duration": 7.36, "text": "exponentials and logarithms 3 we use the"}, {"start": 125.079, "duration": 6.961, "text": "relationship between s cosine and"}, {"start": 128.759, "duration": 6.961, "text": "tangent 4 and we simplified using"}, {"start": 132.04, "duration": 3.68, "text": "logarithm properties"}], "title": "Finding the Derivative of [a^ln(cos x)] | Advanced Chain Rule Application", "description": "QUESTION:\nFind the derivative of y = a^(ln(cos x)), where a is a positive constant.\n\n\ud83d\udd0d Master the step-by-step solution to finding the derivative of a^ln(cos x) in this comprehensive calculus tutorial.\nIn this video, we cover:\n\u2713 Breaking down complex exponential functions\n\u2713 Double application of the chain rule\n\u2713 Working with natural logarithms and trigonometric functions\n\u2713 Simplifying using trigonometric identities\n\u2713 Clear explanations of each mathematical step\nPerfect for:\n\u2022 Calculus I & II students\n\u2022 AP Calculus preparation\n\u2022 College math students\n\u2022 Math competition preparation\n\ud83d\udcda Prerequisites:\nBasic differentiation rules\nChain rule\nExponential functions\nLogarithmic functions\nTrigonometric functions\n#Calculus #ChainRule #Mathematics #CalcTutorial #MathHelp #APCalculus #MathTutoring", "lengthSeconds": 139, "width": 640, "height": 360}
{"video_id": "DWu7tTqtHyY", "transcript": [{"start": 0.399, "duration": 5.96, "text": "let's find the derivative of cine 2x"}, {"start": 3.8, "duration": 6.44, "text": "using first principles we'll break this"}, {"start": 6.359, "duration": 9.001, "text": "down step by step step one let's start"}, {"start": 10.24, "duration": 10.279, "text": "by defining our function f ofx = cine 2x"}, {"start": 15.36, "duration": 9.239, "text": "then we can write f of x + H = cine of 2"}, {"start": 20.519, "duration": 6.361, "text": "* x + H step two we'll set up the"}, {"start": 24.599, "duration": 5.16, "text": "difference quotient the derivative is"}, {"start": 26.88, "duration": 9.16, "text": "defined as the limit as H approaches Z"}, {"start": 29.759, "duration": 10.001, "text": "of F ofx + hus F ofx all divided by H"}, {"start": 36.04, "duration": 6.6, "text": "step three now comes the clever part"}, {"start": 39.76, "duration": 5.72, "text": "we'll use the cosine difference identity"}, {"start": 42.64, "duration": 8.2, "text": "the cosine of a minus the cosine of b"}, {"start": 45.48, "duration": 8.719, "text": "equals -2 * s of a plus b/ 2 * s of a"}, {"start": 50.84, "duration": 7.239, "text": "minus b/ 2 let's apply this identity to"}, {"start": 54.199, "duration": 6.281, "text": "our problem we'll substitute a = 2x + 2"}, {"start": 58.079, "duration": 6.04, "text": "H and B ="}, {"start": 60.48, "duration": 8.679, "text": "2x now let's simplify what's inside each"}, {"start": 64.119, "duration": 9.36, "text": "s term in the first sign we get 4x + 2H"}, {"start": 69.159, "duration": 8.841, "text": "/ 2 which simplifies to 2x + H in the"}, {"start": 73.479, "duration": 7.441, "text": "second sign we get H for our final steps"}, {"start": 78.0, "duration": 7.96, "text": "we know that the limit as H approaches"}, {"start": 80.92, "duration": 10.519, "text": "zero of s h over H = 1 and when we"}, {"start": 85.96, "duration": 9.159, "text": "substitute H = 0 into s of 2x + H we get"}, {"start": 91.439, "duration": 6.761, "text": "S of 2x let's visualize this result with"}, {"start": 95.119, "duration": 6.601, "text": "a graph here's the original function"}, {"start": 98.2, "duration": 8.12, "text": "cine 2x in blue and here's its"}, {"start": 101.72, "duration": 7.12, "text": "derivative -2 sin 2x in red notice how"}, {"start": 106.32, "duration": 6.68, "text": "at the peaks of cosine 2X the derivative"}, {"start": 108.84, "duration": 6.36, "text": "is zero and at the zeros of cosine 2X"}, {"start": 113.0, "duration": 5.079, "text": "the derivative reaches its maximum or"}, {"start": 115.2, "duration": 5.599, "text": "minimum values and there we have it"}, {"start": 118.079, "duration": 8.281, "text": "we've shown that the derivative of cine"}, {"start": 120.799, "duration": 8.561, "text": "2x = -2 sin 2x both algebraically and"}, {"start": 126.36, "duration": 3.0, "text": "graphically"}], "title": "Understanding the Derivative of cos 2x  Using First Principles", "description": "Dive into the fundamentals of calculus as we explore how to determine the derivative of the exponential function cos 2x  using first principles. Follow along with step-by-step explanations and examples to enhance your understanding of derivatives and exponential functions.\n#Calculus #Derivatives #ExponentialFunctions #MathTutorial #FirstPrinciples #cos2x", "lengthSeconds": 129, "width": 640, "height": 360}
{"video_id": "btYoN4yn8v8", "transcript": [{"start": 0.32, "duration": 6.959, "text": "let's solve this complex differentiation"}, {"start": 3.32, "duration": 8.12, "text": "problem step by step we'll differentiate"}, {"start": 7.279, "duration": 7.48, "text": "y equal s^ squar of natural log of"}, {"start": 11.44, "duration": 5.999, "text": "secant x let's organize our solution in"}, {"start": 14.759, "duration": 5.241, "text": "two columns the left column will show"}, {"start": 17.439, "duration": 4.641, "text": "our mathematical steps and the right"}, {"start": 20.0, "duration": 4.96, "text": "column will explain what we're doing at"}, {"start": 22.08, "duration": 5.199, "text": "each step first let's write out what"}, {"start": 24.96, "duration": 6.04, "text": "we're trying to find we want to find the"}, {"start": 27.279, "duration": 7.561, "text": "derivative of sin^2 of natural log of"}, {"start": 31.0, "duration": 6.96, "text": "secant x with respect to X now we'll use"}, {"start": 34.84, "duration": 7.16, "text": "the chain rule the outer function is sin"}, {"start": 37.96, "duration": 7.8, "text": "SAR so we get 2 * sin of natural log of"}, {"start": 42.0, "duration": 7.559, "text": "secant x time the derivative of s of"}, {"start": 45.76, "duration": 6.319, "text": "natural log of secant x next we apply"}, {"start": 49.559, "duration": 6.441, "text": "the chain rule again to the derivative"}, {"start": 52.079, "duration": 7.241, "text": "of s this gives us 2 sin of natural log"}, {"start": 56.0, "duration": 7.559, "text": "of secant x * cosine of natural log of"}, {"start": 59.32, "duration": 7.52, "text": "secant x * the derivative of natural log"}, {"start": 63.559, "duration": 7.521, "text": "of secc x here's where we can use a"}, {"start": 66.84, "duration": 9.4, "text": "useful trigonometric identity 2 * sin *"}, {"start": 71.08, "duration": 7.8, "text": "cosine = s of 2 * the angle also we use"}, {"start": 76.24, "duration": 5.12, "text": "the chain rule for the natural log now"}, {"start": 78.88, "duration": 7.239, "text": "we can substitute the derivative of SEC"}, {"start": 81.36, "duration": 7.64, "text": "X which is secc x * tangent X finally we"}, {"start": 86.119, "duration": 6.0, "text": "can simplify by canceling cant X in the"}, {"start": 89.0, "duration": 7.88, "text": "numerator and denominator giving us our"}, {"start": 92.119, "duration": 7.68, "text": "final answer tangent x * sin of 2 *"}, {"start": 96.88, "duration": 5.919, "text": "natural log of secan x and there we have"}, {"start": 99.799, "duration": 5.6, "text": "it we've successfully differentiated"}, {"start": 102.799, "duration": 5.28, "text": "this complex function using the chain"}, {"start": 105.399, "duration": 7.0, "text": "rule trigonometric identities and"}, {"start": 108.079, "duration": 4.32, "text": "careful step-by-step work"}], "title": "Derivatives: Sin Squared of Log Secant Explained [\ud835\udc60\ud835\udc56\ud835\udc5b^2 {ln\u2061(sec\u2061\ud835\udc65)}]", "description": "Explore how to differentiate the function  [\ud835\udc60\ud835\udc56\ud835\udc5b^2 {ln\u2061(sec\u2061\ud835\udc65)}]  using chain and product rule techniques. This step-by-step breakdown simplifies complex trigonometric and logarithmic derivatives, perfect for students and math enthusiasts. Watch as we unravel each step to provide clarity and insight into calculus.", "lengthSeconds": 115, "width": 640, "height": 360}
{"video_id": "oZCW4uxmHvk", "transcript": [{"start": 0.32, "duration": 5.88, "text": "Welcome to our lesson on logarithmic"}, {"start": 3.759, "duration": 5.441, "text": "differentiation today we'll learn how to"}, {"start": 6.2, "duration": 6.0, "text": "differentiate a complex function using"}, {"start": 9.2, "duration": 6.919, "text": "this powerful technique our challenge is"}, {"start": 12.2, "duration": 7.239, "text": "to find the derivative of y = x * cosine"}, {"start": 16.119, "duration": 8.441, "text": "inverse of X all divided by the square <"}, {"start": 19.439, "duration": 6.881, "text": "TK of 1 - x^2 why do we use logarithmic"}, {"start": 24.56, "duration": 3.879, "text": "differentiation well when we have"}, {"start": 26.32, "duration": 4.68, "text": "products quotients or complex"}, {"start": 28.439, "duration": 5.601, "text": "Expressions taking the natural logarithm"}, {"start": 31.0, "duration": 5.48, "text": "of both sides can simplify our work"}, {"start": 34.04, "duration": 5.679, "text": "significantly by turning products into"}, {"start": 36.48, "duration": 6.96, "text": "sums and quotients into differences"}, {"start": 39.719, "duration": 6.961, "text": "let's solve a step by step step one take"}, {"start": 43.44, "duration": 6.52, "text": "the natural logarithm of both sides we"}, {"start": 46.68, "duration": 6.719, "text": "get Ln of y equals Ln of the entire"}, {"start": 49.96, "duration": 6.16, "text": "right side step two use the properties"}, {"start": 53.399, "duration": 4.921, "text": "of logarithms to break this down the"}, {"start": 56.12, "duration": 4.959, "text": "logarithm of a quotient is the"}, {"start": 58.32, "duration": 5.799, "text": "difference of logarith"}, {"start": 61.079, "duration": 5.601, "text": "and the logarithm of a product is the"}, {"start": 64.119, "duration": 2.561, "text": "sum of"}, {"start": 69.799, "duration": 5.721, "text": "logarithms step three apply these"}, {"start": 72.88, "duration": 8.599, "text": "properties to our equation this gives us"}, {"start": 75.52, "duration": 12.84, "text": "Ln of Y = Ln of x + Ln of cine inverse"}, {"start": 81.479, "duration": 6.881, "text": "of x - 12 Ln of 1 - x^2"}, {"start": 91.28, "duration": 6.839, "text": "step four now we can differentiate both"}, {"start": 94.52, "duration": 8.52, "text": "sides with respect to X remember when we"}, {"start": 98.119, "duration": 9.201, "text": "differentiate Ln of Y we get 1/ y *"}, {"start": 103.04, "duration": 4.28, "text": "dydx this is implicit"}, {"start": 110.6, "duration": 6.0, "text": "differentiation step five let's handle"}, {"start": 113.68, "duration": 8.16, "text": "each term separately the derivative of"}, {"start": 116.6, "duration": 9.6, "text": "Ln of X is 1 /x for Ln of cosine inverse"}, {"start": 121.84, "duration": 11.24, "text": "of X we use the chain Rule and for Ln of"}, {"start": 126.2, "duration": 6.88, "text": "1 - x^2 we get -2X over 1 -"}, {"start": 141.04, "duration": 7.04, "text": "x^2 step six putting it all together we"}, {"start": 145.04, "duration": 7.24, "text": "get this expression now multiply both"}, {"start": 148.08, "duration": 4.2, "text": "sides by y solve for"}, {"start": 155.519, "duration": 6.281, "text": "dydx step seven our final step is to"}, {"start": 159.64, "duration": 6.04, "text": "substitute back the original expression"}, {"start": 161.8, "duration": 7.88, "text": "for y and simplify after combining like"}, {"start": 165.68, "duration": 4.0, "text": "terms we get our final"}, {"start": 172.36, "duration": 5.56, "text": "answer and that's how we use logarithmic"}, {"start": 175.68, "duration": 5.839, "text": "differentiation to find the derivative"}, {"start": 177.92, "duration": 5.599, "text": "of a complex function remember this"}, {"start": 181.519, "duration": 4.481, "text": "technique is especially useful when"}, {"start": 183.519, "duration": 6.08, "text": "dealing with products and quotients"}, {"start": 186.0, "duration": 3.599, "text": "thank you for watching"}], "title": "Mastering Logarithmic Differentiation [(\ud835\udc65 \u3016\ud835\udc50\ud835\udc5c\ud835\udc60\u3017^(\u22121) \ud835\udc65)/\u221a(1\u2212\ud835\udc65^2 )] Complex Calculus Made Simple.", "description": "Problem\nDifferentiate: y =[(\ud835\udc65 \u3016\ud835\udc50\ud835\udc5c\ud835\udc60\u3017^(\u22121) \ud835\udc65)/\u221a(1\u2212\ud835\udc65^2 )] \n\n\ud83d\udcda In this comprehensive calculus tutorial, we break down a challenging derivative problem using logarithmic differentiation. Watch as we solve:\ny = (x arccos(x))/\u221a(1-x\u00b2)\n\ud83c\udfaf What You\u2019ll Learn:\n\u2022 Why logarithmic differentiation is powerful\n\u2022 How to break down complex functions\n\u2022 Essential logarithm properties\n\u2022 Advanced chain rule applications\n\u2022 Step-by-step solution methodology\n\u2022 Common mistakes to avoid\n\ud83d\udca1 Perfect for:\n\u2022 Calculus students\n\u2022 Math majors\n\u2022 Engineering students\n\u2022 Test preparation\n\u2022 Self-study\n#Calculus #Mathematics #LogarithmicDifferentiation #MathTutorial #CaluculusTips #MathEducation #EngineeringMath #StudyWithMe #MathHelp #CalculusProblems", "lengthSeconds": 195, "width": 640, "height": 360}
{"video_id": "TO8fLyD2_1Y", "transcript": [{"start": 0.52, "duration": 5.239, "text": "let's solve this physics problem about"}, {"start": 3.24, "duration": 6.88, "text": "acceleration a car increases its speed"}, {"start": 5.759, "duration": 6.441, "text": "uniformly from 5 m/s to 45 m/s over a"}, {"start": 10.12, "duration": 5.52, "text": "time period of 10"}, {"start": 12.2, "duration": 6.079, "text": "seconds we need to find the car's"}, {"start": 15.64, "duration": 4.2, "text": "acceleration let's visualize this we"}, {"start": 18.279, "duration": 3.961, "text": "have a car starting at an initial"}, {"start": 19.84, "duration": 6.199, "text": "velocity of 5"}, {"start": 22.24, "duration": 6.519, "text": "m/s and reaching a final velocity of 45"}, {"start": 26.039, "duration": 4.72, "text": "m/s after 10"}, {"start": 28.759, "duration": 6.12, "text": "seconds let's let's list out what we"}, {"start": 30.759, "duration": 8.64, "text": "know the initial velocity U is 5"}, {"start": 34.879, "duration": 7.641, "text": "m/s the final velocity V is 45"}, {"start": 39.399, "duration": 5.881, "text": "m/s and the time T is 10"}, {"start": 42.52, "duration": 4.28, "text": "seconds to find the acceleration we'll"}, {"start": 45.28, "duration": 4.119, "text": "use the"}, {"start": 46.8, "duration": 5.599, "text": "formula acceleration equals change in"}, {"start": 49.399, "duration": 7.16, "text": "velocity divided by time or written"}, {"start": 52.399, "duration": 7.081, "text": "mathematically equals vus U /"}, {"start": 56.559, "duration": 7.041, "text": "T let's substitute our values into the"}, {"start": 59.48, "duration": 7.639, "text": "formula plugging in 45 - 5 m/s in the"}, {"start": 63.6, "duration": 6.36, "text": "numerator and 10 seconds in the"}, {"start": 67.119, "duration": 7.121, "text": "denominator now let's simplify in the"}, {"start": 69.96, "duration": 10.76, "text": "numerator 45 - 5 gives us 40"}, {"start": 74.24, "duration": 9.199, "text": "m/s 40 m/s / 10 seconds gives us 4 m/s"}, {"start": 80.72, "duration": 6.6, "text": "squared therefore the acceleration of"}, {"start": 83.439, "duration": 5.921, "text": "the car is 4 m/s squared this means that"}, {"start": 87.32, "duration": 6.0, "text": "every second the car's velocity"}, {"start": 89.36, "duration": 3.96, "text": "increases by by 4 m/s"}], "title": "Understanding Acceleration: A Step-by-Step Physics Problem", "description": "Welcome back to our channel! Today, we\u2019re going to solve a physics problem involving acceleration. We\u2019ll break it down step by step to help you grasp the concept of linear motion under uniform acceleration.\nProblem Statement:\nA car increases its speed from 5 m/s to 45 m/s  in. We need to find the car\u2019s acceleration.\n#Physics #Acceleration #Kinematics #LearnPhysics #ScienceEducation", "lengthSeconds": 94, "width": 640, "height": 360}
{"video_id": "pLzO09V898I", "transcript": [{"start": 0.28, "duration": 6.119, "text": "let's solve this complex differentiation"}, {"start": 2.919, "duration": 7.281, "text": "problem step by step our question is to"}, {"start": 6.399, "duration": 8.521, "text": "differentiate the expression 2 cosecant"}, {"start": 10.2, "duration": 7.239, "text": "2x * cine of natural log of tangent X"}, {"start": 14.92, "duration": 5.68, "text": "let's organize our solution in a two"}, {"start": 17.439, "duration": 6.561, "text": "column format with mathematical steps on"}, {"start": 20.6, "duration": 6.2, "text": "the left and explanations on the right"}, {"start": 24.0, "duration": 5.48, "text": "this will help us understand each step"}, {"start": 26.8, "duration": 3.84, "text": "clearly let's start with our initial"}, {"start": 29.48, "duration": 3.599, "text": "expression"}, {"start": 30.64, "duration": 4.919, "text": "we need to use the product rule here"}, {"start": 33.079, "duration": 5.601, "text": "since we're differentiating a product of"}, {"start": 35.559, "duration": 7.281, "text": "two functions the product rule states"}, {"start": 38.68, "duration": 7.76, "text": "that the derivative of U * v = u Prime *"}, {"start": 42.84, "duration": 8.92, "text": "V + U * V Prime applying the product"}, {"start": 46.44, "duration": 8.68, "text": "rule we get 2 * the sum of cose 2x * the"}, {"start": 51.76, "duration": 7.08, "text": "derivative of cine natural log tangent X"}, {"start": 55.12, "duration": 7.32, "text": "plus cosine natural log tangent x * the"}, {"start": 58.84, "duration": 5.959, "text": "derivative of Co cant 2x now we need to"}, {"start": 62.44, "duration": 5.719, "text": "use the chain rule to find these"}, {"start": 64.799, "duration": 6.521, "text": "derivatives for the first term we get"}, {"start": 68.159, "duration": 5.881, "text": "negative sign of natural log tangent x"}, {"start": 71.32, "duration": 6.08, "text": "times the derivative of natural log"}, {"start": 74.04, "duration": 8.36, "text": "tangent X for the second term we get"}, {"start": 77.4, "duration": 9.359, "text": "Negative cose 2x Cent 2x * the"}, {"start": 82.4, "duration": 7.28, "text": "derivative of 2x let's simplify further"}, {"start": 86.759, "duration": 7.68, "text": "the derivative of natural log tangent X"}, {"start": 89.68, "duration": 8.759, "text": "is 1 / tangent x * SEC 2 x and the"}, {"start": 94.439, "duration": 8.161, "text": "derivative of 2x is simply 2 finally we"}, {"start": 98.439, "duration": 6.72, "text": "can factor out -2 cose 2x from both"}, {"start": 102.6, "duration": 8.36, "text": "terms to get our final answer this gives"}, {"start": 105.159, "duration": 8.56, "text": "us -2 cose 2x * the sum of secant ^ 2 X"}, {"start": 110.96, "duration": 8.0, "text": "over tangent x * sin of natural log"}, {"start": 113.719, "duration": 8.881, "text": "tangent X plus 2 cang 2x * cosine of"}, {"start": 118.96, "duration": 6.479, "text": "natural log t X and there we have it"}, {"start": 122.6, "duration": 4.92, "text": "we've successfully differentiated this"}, {"start": 125.439, "duration": 5.721, "text": "complex expression using the product"}, {"start": 127.52, "duration": 6.12, "text": "rule chain Rule and our knowledge of"}, {"start": 131.16, "duration": 5.4, "text": "trigonometric derivatives remember to"}, {"start": 133.64, "duration": 7.599, "text": "take it step by step when dealing with"}, {"start": 136.56, "duration": 4.679, "text": "complex derivatives like this one"}], "title": "Derivatives Step-by-Step for Advanced Trigonometric Functions [2\ud835\udc50\ud835\udc5c\ud835\udc60\ud835\udc52\ud835\udc502\ud835\udc65 cos\u2061 {ln\u2061(tan\u2061\ud835\udc65)}]", "description": "Unlock the secrets of calculus in this clear and detailed walkthrough of differentiating complex trigonometric functions. In this video, we apply the product and chain rules to break down the differentiation of the expression [2\ud835\udc50\ud835\udc5c\ud835\udc60\ud835\udc52\ud835\udc502\ud835\udc65 cos\u2061 {ln\u2061(tan\u2061\ud835\udc65)}] . Perfect for calculus students looking to deepen their understanding or anyone needing a quick refresher. Join us for this informative session and boost your calculus skills!\n#Calculus #Derivatives #Trigonometry #MathHelp #Educational", "lengthSeconds": 144, "width": 640, "height": 360}
{"video_id": "FWsLy2bKDqU", "transcript": [{"start": 0.399, "duration": 6.0, "text": "welcome to this lesson on logarithmic"}, {"start": 3.72, "duration": 5.72, "text": "differentiation today we'll be finding"}, {"start": 6.399, "duration": 6.561, "text": "the derivative of a complex expression"}, {"start": 9.44, "duration": 6.319, "text": "the quantity x to the X power all Rays"}, {"start": 12.96, "duration": 5.28, "text": "to the X power this is a challenging"}, {"start": 15.759, "duration": 5.44, "text": "problem because we have nested"}, {"start": 18.24, "duration": 4.959, "text": "exponentials let's solve it step by step"}, {"start": 21.199, "duration": 5.561, "text": "using logarithmic"}, {"start": 23.199, "duration": 5.84, "text": "differentiation step one first let's"}, {"start": 26.76, "duration": 4.04, "text": "simplify our expression using the"}, {"start": 29.039, "duration": 4.161, "text": "properties of exponents"}, {"start": 30.8, "duration": 7.68, "text": "when we have a power raised to another"}, {"start": 33.2, "duration": 8.56, "text": "power we multiply the exponents so x x"}, {"start": 38.48, "duration": 7.52, "text": "all raised to the X power simplifies to"}, {"start": 41.76, "duration": 9.56, "text": "X raised to x^2 step two let's call our"}, {"start": 46.0, "duration": 8.68, "text": "function y so y = x x^2 now we'll take"}, {"start": 51.32, "duration": 6.239, "text": "the natural logarithm of both sides this"}, {"start": 54.68, "duration": 5.76, "text": "is the key step in logarithmic"}, {"start": 57.559, "duration": 4.84, "text": "differentiation step three you using the"}, {"start": 60.44, "duration": 5.24, "text": "properties of logarithms we can bring"}, {"start": 62.399, "duration": 6.161, "text": "down the power the natural log of x raed"}, {"start": 65.68, "duration": 6.84, "text": "to a power equals that power times the"}, {"start": 68.56, "duration": 6.52, "text": "natural log of x step four now we'll use"}, {"start": 72.52, "duration": 5.599, "text": "implicit differentiation we'll take the"}, {"start": 75.08, "duration": 7.999, "text": "derivative of both sides with respect to"}, {"start": 78.119, "duration": 8.161, "text": "X on the left side we get 1 / y *"}, {"start": 83.079, "duration": 6.201, "text": "dydx on the right side we need to use"}, {"start": 86.28, "duration": 4.479, "text": "the product rule because we have x^2 *"}, {"start": 89.28, "duration": 5.08, "text": "lnx"}, {"start": 90.759, "duration": 8.281, "text": "step five let's simplify the right side"}, {"start": 94.36, "duration": 9.88, "text": "2x Ln X+ X which is x * the quantity 2"}, {"start": 99.04, "duration": 9.0, "text": "Ln x + 1 Step six now we can solve for"}, {"start": 104.24, "duration": 6.159, "text": "dydx we multiply both sides by y"}, {"start": 108.04, "duration": 6.039, "text": "remember that Y is our original"}, {"start": 110.399, "duration": 6.241, "text": "expression x to the x^2 and there we"}, {"start": 114.079, "duration": 8.04, "text": "have it the derivative of x to the X"}, {"start": 116.64, "duration": 10.399, "text": "power all rais to the X power is X x^2 *"}, {"start": 122.119, "duration": 7.961, "text": "x * 2 Ln x + 1 Let's highlight the key"}, {"start": 127.039, "duration": 6.161, "text": "steps we used first we simplified the"}, {"start": 130.08, "duration": 6.2, "text": "nested exponential then we used"}, {"start": 133.2, "duration": 6.72, "text": "logarithmic differentiation to make the"}, {"start": 136.28, "duration": 6.0, "text": "problem manageable and finally we used"}, {"start": 139.92, "duration": 5.12, "text": "implicit differentiation and the product"}, {"start": 142.28, "duration": 5.4, "text": "rule to find our answer this problem"}, {"start": 145.04, "duration": 5.4, "text": "demonstrates the power of logarithmic"}, {"start": 147.68, "duration": 5.32, "text": "differentiation in handling complex"}, {"start": 150.44, "duration": 5.56, "text": "exponential Expressions thank you for"}, {"start": 153.0, "duration": 3.0, "text": "watching"}], "title": "How to Find d/d[(\ud835\udc99^\ud835\udc99 )^\ud835\udc99] | Logarithmic Differentiation Made Easy", "description": "Question: Find the derivative of y =[(\ud835\udc99^\ud835\udc99 )^\ud835\udc99] using logarithmic differentiation.\n\n\ud83d\udd25 Master the art of logarithmic differentiation with this step-by-step tutorial on solving complex nested exponential derivatives.\nIn this video, we break down the challenging problem of finding the derivative of (xx)x using logarithmic differentiation. Perfect for calculus students and anyone preparing for advanced mathematics exams!\n\ud83d\udcda What you\u2019ll learn:\n\u2022 Simplifying nested exponentials\n\u2022 Applying logarithmic differentiation\n\u2022 Using chain rule and product rule effectively\n\u2022 Working with natural logarithm properties\n\u2022 Step-by-step solution process\n\u270f\ufe0f Final Answer: d/dx(xx)x = x^(x\u00b2)[x(1 + 2ln(x))]\n\ud83c\udfaf Level: Advanced Calculus / College Mathematics\n\ud83d\udcdd Related topics:\n\u2022 Derivatives\n\u2022 Exponential functions\n\u2022 Natural logarithms\n\u2022 Chain rule\n\u2022 Product rule\n#Calculus #Mathematics #Derivatives #MathTutorial #LogarithmicDifferentiation #ChainRule #ProductRule #MathHelp #MathEducation #CalculusTutorial", "lengthSeconds": 161, "width": 640, "height": 360}
{"video_id": "WGq70KB_Q8M", "transcript": [{"start": 0.48, "duration": 5.279, "text": "welcome today we'll solve a calculus"}, {"start": 3.36, "duration": 5.84, "text": "problem involving the differentiation of"}, {"start": 5.759, "duration": 7.88, "text": "a complex fraction our question is to"}, {"start": 9.2, "duration": 7.4, "text": "find the derivative of Ln of cine x / x"}, {"start": 13.639, "duration": 5.241, "text": "with respect to X to solve this we'll"}, {"start": 16.6, "duration": 4.4, "text": "need to use the quotient rule the"}, {"start": 18.88, "duration": 7.04, "text": "quotient rule states that for a fraction"}, {"start": 21.0, "duration": 5.84, "text": "U over V the derivative is V * d dxus u"}, {"start": 25.92, "duration": 5.48, "text": "*"}, {"start": 26.84, "duration": 8.719, "text": "dvdx all over V ^2 let's identify our u"}, {"start": 31.4, "duration": 8.839, "text": "and v in our case U = Ln of cosine X and"}, {"start": 35.559, "duration": 7.761, "text": "v = x now let's find D DX to"}, {"start": 40.239, "duration": 5.721, "text": "differentiate Ln of cosine X we'll need"}, {"start": 43.32, "duration": 6.919, "text": "to use the chain rule the derivative of"}, {"start": 45.96, "duration": 7.8, "text": "Ln of cine X is 1 / cosine x * the"}, {"start": 50.239, "duration": 7.12, "text": "derivative of cosine X which is sinx"}, {"start": 53.76, "duration": 6.439, "text": "finding dvdx is much simpler the"}, {"start": 57.359, "duration": 5.0, "text": "derivative of x is just one now now"}, {"start": 60.199, "duration": 7.481, "text": "let's substitute everything into the"}, {"start": 62.359, "duration": 11.321, "text": "quotient rule we have x * - Tan x - Ln"}, {"start": 67.68, "duration": 8.72, "text": "of cosine x * 1 all over x^2 and finally"}, {"start": 73.68, "duration": 8.92, "text": "we can simplify this expression our"}, {"start": 76.4, "duration": 9.399, "text": "final answer ISX Tan x - Ln of cine X"}, {"start": 82.6, "duration": 5.879, "text": "all over x^2 let's verify this is"}, {"start": 85.799, "duration": 4.921, "text": "correct by checking our steps we use the"}, {"start": 88.479, "duration": 4.841, "text": "quotient rule correctly found the"}, {"start": 90.72, "duration": 5.48, "text": "derivatives of both the numerator and"}, {"start": 93.32, "duration": 5.52, "text": "denominator accurately and simplified"}, {"start": 96.2, "duration": 5.919, "text": "our expression properly this completes"}, {"start": 98.84, "duration": 3.279, "text": "our solution"}], "title": "Mastering Derivatives: Differentiating [(ln\u2061(cos\u2061\ud835\udc65))/\ud835\udc65]  Step-by-Step", "description": "In this video, we dive deep into the process of differentiating the function  [(ln\u2061(cos\u2061\ud835\udc65))/\ud835\udc65] . Follow along as we break down each step using the quotient rule and chain rule. Perfect for students and anyone looking to strengthen their calculus skills. Don\u2019t forget to like, subscribe, and hit the notification bell for more math tutorials!\n#Calculus #Derivatives #MathTutorial #QuotientRule #ChainRule #LearningMath", "lengthSeconds": 105, "width": 640, "height": 360}
{"video_id": "ze1cdk2s1hA", "transcript": [{"start": 0.359, "duration": 5.641, "text": "welcome to this lesson on logarithmic"}, {"start": 3.6, "duration": 4.8, "text": "differentiation today we're going to"}, {"start": 6.0, "duration": 7.28, "text": "find the derivative of a particularly"}, {"start": 8.4, "duration": 9.24, "text": "challenging function y = e power of x to"}, {"start": 13.28, "duration": 6.04, "text": "X this is a complex exponential function"}, {"start": 17.64, "duration": 4.76, "text": "that would be very difficult to"}, {"start": 19.32, "duration": 6.08, "text": "differentiate using regular methods why"}, {"start": 22.4, "duration": 6.16, "text": "is this function so challenging well if"}, {"start": 25.4, "duration": 5.719, "text": "we try to use regular differentiation we"}, {"start": 28.56, "duration": 5.2, "text": "have to deal with both the chain Rule"}, {"start": 31.119, "duration": 5.841, "text": "and the product rule simultaneously"}, {"start": 33.76, "duration": 6.279, "text": "making it extremely complicated instead"}, {"start": 36.96, "duration": 5.72, "text": "we can use logarithmic differentiation"}, {"start": 40.039, "duration": 5.921, "text": "which will make this process much more"}, {"start": 42.68, "duration": 6.519, "text": "manageable let's solve the step by step"}, {"start": 45.96, "duration": 8.919, "text": "step one take the natural logarithm of"}, {"start": 49.199, "duration": 10.36, "text": "both sides this gives us Ln of Y = Ln of"}, {"start": 54.879, "duration": 8.561, "text": "e to power of x to X step two use the"}, {"start": 59.559, "duration": 6.201, "text": "logarith property that states Ln of e to"}, {"start": 63.44, "duration": 4.8, "text": "the power of something equals that"}, {"start": 65.76, "duration": 7.92, "text": "something this simplifies our equation"}, {"start": 68.24, "duration": 5.44, "text": "to Ln of y = x to"}, {"start": 75.119, "duration": 9.521, "text": "X step three take the natural log again"}, {"start": 79.52, "duration": 8.279, "text": "this gives us Ln of Ln of Y = Ln of x to"}, {"start": 84.64, "duration": 3.159, "text": "the x"}, {"start": 90.4, "duration": 6.56, "text": "step four use the logarithm property"}, {"start": 93.159, "duration": 9.121, "text": "that states Ln of a to the power of b"}, {"start": 96.96, "duration": 10.08, "text": "equal B * Ln of a this gives us Ln of Ln"}, {"start": 102.28, "duration": 4.76, "text": "of y = x * Ln of"}, {"start": 109.6, "duration": 7.32, "text": "X step five now we can differentiate"}, {"start": 113.479, "duration": 6.401, "text": "both sides with respect to X on the left"}, {"start": 116.92, "duration": 6.239, "text": "side we'll use the chain rule the"}, {"start": 119.88, "duration": 10.56, "text": "derivative of Ln of Ln of Y with respect"}, {"start": 123.159, "duration": 7.281, "text": "to x = 1 Ln of y * 1 / y *"}, {"start": 133.76, "duration": 8.36, "text": "dydx step six on the right side we use"}, {"start": 138.16, "duration": 10.04, "text": "the product rule the derivative of x Ln"}, {"start": 142.12, "duration": 9.72, "text": "X = Ln x + x * 1 /x which simplifies to"}, {"start": 148.2, "duration": 3.64, "text": "Ln x + 1"}, {"start": 153.519, "duration": 7.841, "text": "step seven now we can solve for"}, {"start": 156.64, "duration": 11.48, "text": "dydx multiply both sides by Y and Ln of"}, {"start": 161.36, "duration": 10.239, "text": "Y this gives us dydx = y * Ln of y * the"}, {"start": 168.12, "duration": 3.479, "text": "quantity Ln x +"}, {"start": 174.12, "duration": 9.6, "text": "1 step8 finally substitute back yal e to"}, {"start": 180.12, "duration": 7.08, "text": "power of x x this gives us our final"}, {"start": 183.72, "duration": 10.36, "text": "answer the derivative equals e to power"}, {"start": 187.2, "duration": 6.88, "text": "of x to x * x x * the quantity Ln x +"}, {"start": 196.519, "duration": 5.601, "text": "1 let's review what we did we used"}, {"start": 200.0, "duration": 4.04, "text": "logarithmic differentiation to handle"}, {"start": 202.12, "duration": 5.0, "text": "this complex function by taking the"}, {"start": 204.04, "duration": 5.72, "text": "natural log twice applying logarithm"}, {"start": 207.12, "duration": 5.199, "text": "properties differentiating both sides"}, {"start": 209.76, "duration": 4.72, "text": "using the chain Rule and product rule"}, {"start": 212.319, "duration": 5.2, "text": "and then substituting back our original"}, {"start": 214.48, "duration": 6.2, "text": "function this technique is powerful for"}, {"start": 217.519, "duration": 7.8, "text": "handling complex exponential and product"}, {"start": 220.68, "duration": 4.639, "text": "functions thank you for watching"}], "title": "How to Find d/dx[\u3016\ud835\udc86^\ud835\udc99\u3017^\ud835\udc99] | Logarithmic Differentiation Made Easy", "description": "Question: Find the derivative of y = \u3016\ud835\udc86^\ud835\udc99\u3017^\ud835\udc99 using logarithmic differentiation.\n\nMaster the art of logarithmic differentiation with this comprehensive tutorial on finding the derivative of \u3016\ud835\udc86^\ud835\udc99\u3017^\ud835\udc99. Perfect for calculus students and math enthusiasts!\nIn this video, we\u2019ll cover:\n\ud83d\udcda Why regular differentiation won\u2019t work here\n\ud83d\udd0d The power of logarithmic differentiation\n\u270d\ufe0f Step-by-step solution process\n\ud83d\udca1 Important logarithm properties\n\ud83c\udfaf Chain rule application\n\u2705 Final answer verification\n\ud83d\udd14 Subscribe for more calculus tutorials!\n\u2753 Leave your questions in the comments below\n#Calculus #Mathematics #Derivatives #LogarithmicDifferentiation #MathTutorial #CalculusTutorial #MathEducation #APCalculus #MathHelp", "lengthSeconds": 230, "width": 640, "height": 360}
{"video_id": "CdKiVQ7lHug", "transcript": [{"start": 0.32, "duration": 4.64, "text": "let's solve this differentiation problem"}, {"start": 2.56, "duration": 8.48, "text": "step by step we need to find the"}, {"start": 4.96, "duration": 9.719, "text": "derivative of y = sin 2x over 1 + cosine"}, {"start": 11.04, "duration": 7.44, "text": "2X all squared let's break this down"}, {"start": 14.679, "duration": 6.241, "text": "into manageable steps first let's recall"}, {"start": 18.48, "duration": 4.76, "text": "some important trigonometric identities"}, {"start": 20.92, "duration": 4.96, "text": "that we'll need let's start with the"}, {"start": 23.24, "duration": 7.4, "text": "numerator using the double angle formula"}, {"start": 25.88, "duration": 8.199, "text": "for S we can replace sin 2x with 2 2 sin"}, {"start": 30.64, "duration": 6.0, "text": "x cine X next let's look at the"}, {"start": 34.079, "duration": 9.561, "text": "denominator using another double angle"}, {"start": 36.64, "duration": 9.52, "text": "formula 1 + cosine 2X = 2 cosine 2 x now"}, {"start": 43.64, "duration": 7.759, "text": "let's expand this expression in the"}, {"start": 46.16, "duration": 8.8, "text": "numerator we get 4 sin^2 x cine 2 x in"}, {"start": 51.399, "duration": 6.521, "text": "the denominator we get 4 cosine to the"}, {"start": 54.96, "duration": 5.48, "text": "4th X the fours cancel out in the"}, {"start": 57.92, "duration": 5.639, "text": "numerator and denominator"}, {"start": 60.44, "duration": 6.64, "text": "this leaves us with sin^2 X over cosine"}, {"start": 63.559, "duration": 7.641, "text": "2 x this is a key moment we can"}, {"start": 67.08, "duration": 7.48, "text": "recognize that sin^2 X over cosine 2 x"}, {"start": 71.2, "duration": 7.599, "text": "is equal to tangent 2 x now we're ready"}, {"start": 74.56, "duration": 9.08, "text": "to differentiate we need to find d y DX"}, {"start": 78.799, "duration": 9.401, "text": "= d/ DX of tangent 2 x using the chain"}, {"start": 83.64, "duration": 7.119, "text": "rule we multiply by 2 tangent X and then"}, {"start": 88.2, "duration": 5.76, "text": "multiply by the derivative of tangent X"}, {"start": 90.759, "duration": 6.68, "text": "which is secant 2 x and this is our"}, {"start": 93.96, "duration": 9.159, "text": "final answer the derivative of sin 2x"}, {"start": 97.439, "duration": 10.481, "text": "over 1 + cosine 2X all 2ar is 2 tangent"}, {"start": 103.119, "duration": 7.64, "text": "X secant 2 x let's review the key steps"}, {"start": 107.92, "duration": 5.839, "text": "we use double angle formulas simplified"}, {"start": 110.759, "duration": 6.561, "text": "the fraction recognize the tangent squar"}, {"start": 113.759, "duration": 5.241, "text": "pattern and then use the chain rule to"}, {"start": 117.32, "duration": 4.079, "text": "differentiate understanding these"}, {"start": 119.0, "duration": 4.799, "text": "patterns and identities is crucial for"}, {"start": 121.399, "duration": 5.4, "text": "solving trigonometric differentiation"}, {"start": 123.799, "duration": 3.0, "text": "problems"}], "title": "Mastering Trigonometric Differentiation [(sin\u20612\ud835\udc65/(1+cos\u20612\ud835\udc65 ))^2] Step-by-Step Guide", "description": "Unlock the secrets of trigonometric differentiation with our detailed, step-by-step tutorial. Learn how to differentiate complex trigonometric functions like y=[(sin\u20612\ud835\udc65/(1+cos\u20612\ud835\udc65 ))^2]  with ease. This video breaks down each step, providing clear explanations along the way. Perfect for students and anyone looking to strengthen their calculus skills. Subscribe for more insightful math tutorials!\n#Calculus #Differentiation #Trigonometry #MathTutorial #LearnMath #Mathematics", "lengthSeconds": 130, "width": 640, "height": 360}
{"video_id": "7ZoDO6KTw5Q", "transcript": [{"start": 0.32, "duration": 6.039, "text": "let's solve this complex differentiation"}, {"start": 3.439, "duration": 6.761, "text": "problem step by step we start with the"}, {"start": 6.359, "duration": 6.36, "text": "expression cosine of natural log X Plus"}, {"start": 10.2, "duration": 4.359, "text": "natural log of tangent X to"}, {"start": 12.719, "duration": 4.761, "text": "differentiate this we'll use the chain"}, {"start": 14.559, "duration": 5.48, "text": "Rule and split it into two parts the"}, {"start": 17.48, "duration": 5.6, "text": "derivative of the first part cosine of"}, {"start": 20.039, "duration": 6.56, "text": "natural log X plus the derivative of the"}, {"start": 23.08, "duration": 6.199, "text": "second part natural log of tangent X"}, {"start": 26.599, "duration": 4.0, "text": "let's handle each part separately for"}, {"start": 29.279, "duration": 4.081, "text": "the first part"}, {"start": 30.599, "duration": 6.001, "text": "using the chain rule we get Negative s"}, {"start": 33.36, "duration": 6.6, "text": "of natural log X time the derivative of"}, {"start": 36.6, "duration": 6.24, "text": "natural log X which is 1 /x for the"}, {"start": 39.96, "duration": 5.72, "text": "second part we use the chain rule again"}, {"start": 42.84, "duration": 6.96, "text": "the derivative of natural log of tangent"}, {"start": 45.68, "duration": 7.679, "text": "x = 1 / tangent x * the derivative of"}, {"start": 49.8, "duration": 6.64, "text": "tangent X which is secant 2 x now we"}, {"start": 53.359, "duration": 8.921, "text": "combine these parts we have negative s"}, {"start": 56.44, "duration": 9.079, "text": "of natural log x/ X + 1 / Tang x * sec^2"}, {"start": 62.28, "duration": 6.6, "text": "X let's simplify the second term"}, {"start": 65.519, "duration": 9.761, "text": "remember that tangent X = sin x over"}, {"start": 68.88, "duration": 9.48, "text": "cosine X and secant 2 x = 1 cosine 2 x"}, {"start": 75.28, "duration": 8.24, "text": "substituting these in we get Negative s"}, {"start": 78.36, "duration": 9.32, "text": "of natural log x/ X + cosine x/ sin x *"}, {"start": 83.52, "duration": 10.48, "text": "1 cosine 2 x this simplifies to negative"}, {"start": 87.68, "duration": 8.719, "text": "s of natural log x/ x + 1 / sin x * 1 /"}, {"start": 94.0, "duration": 6.04, "text": "cosine X the second term can be"}, {"start": 96.399, "duration": 7.36, "text": "Rewritten as 2/ 2 sinx cosine X which"}, {"start": 100.04, "duration": 7.2, "text": "equal 2 over sin of 2x using the double"}, {"start": 103.759, "duration": 9.96, "text": "angle formula finally we can write this"}, {"start": 107.24, "duration": 9.879, "text": "as 2 cose 2x - 1 /x * sin of natural log"}, {"start": 113.719, "duration": 5.76, "text": "X which is our final answer this problem"}, {"start": 117.119, "duration": 4.841, "text": "demonstrated the use of chain rule"}, {"start": 119.479, "duration": 5.24, "text": "product Rule and trigonometric"}, {"start": 121.96, "duration": 5.48, "text": "identities and calculus remember to"}, {"start": 124.719, "duration": 5.281, "text": "break down complex problems into simpler"}, {"start": 127.44, "duration": 6.319, "text": "steps and use your knowledge of"}, {"start": 130.0, "duration": 3.759, "text": "trigonometric relationships"}], "title": "Differentiating Tricky Composite Functions {cos\u2061(\ud835\udc59\ud835\udc5b\ud835\udc65)+ln\u2061(\ud835\udc61\ud835\udc4e\ud835\udc5b\ud835\udc65)} Step by Step Guide", "description": "Learn how to differentiate the function {cos\u2061(\ud835\udc59\ud835\udc5b\ud835\udc65)+ln\u2061(\ud835\udc61\ud835\udc4e\ud835\udc5b\ud835\udc65)}  with this detailed, step-by-step guide. We\u2019ll cover the application of the chain rule, sum rule, and simplification using trigonometric identities to help you master these concepts. Perfect for students and math enthusiasts aiming to strengthen their calculus skills!\n#Calculus #Differentiation #MathTutorial #Derivatives #ChainRule", "lengthSeconds": 140, "width": 640, "height": 360}
{"video_id": "9DZEA_xr_e8", "transcript": [{"start": 0.32, "duration": 4.4, "text": "welcome today we'll solve a challenging"}, {"start": 3.12, "duration": 4.96, "text": "derivative problem involving"}, {"start": 4.72, "duration": 5.6, "text": "trigonometric functions and substitution"}, {"start": 8.08, "duration": 6.08, "text": "let's start by looking at our problem"}, {"start": 10.32, "duration": 9.2, "text": "we're given that y = the quantity 1 -"}, {"start": 14.16, "duration": 8.879, "text": "x^2 over 1 + x^2 all squared and we're"}, {"start": 19.52, "duration": 5.519, "text": "also given that x equal tangent Theta"}, {"start": 23.039, "duration": 6.48, "text": "our goal is to find"}, {"start": 25.039, "duration": 7.04, "text": "dydx this is a complex expression so"}, {"start": 29.519, "duration": 4.921, "text": "let's break it down into steps using the"}, {"start": 32.079, "duration": 7.441, "text": "chain rule we can write"}, {"start": 34.44, "duration": 8.48, "text": "dydx as Dy D Theta / DX D Theta let's"}, {"start": 39.52, "duration": 7.879, "text": "find DX D Theta first since it's simpler"}, {"start": 42.92, "duration": 9.439, "text": "since x equal tangent Theta DX D Theta"}, {"start": 47.399, "duration": 7.32, "text": "equal secant 2 Theta now for Dy D Theta"}, {"start": 52.359, "duration": 7.84, "text": "we need to recognize that the expression"}, {"start": 54.719, "duration": 9.121, "text": "1 - x^2 over 1 + x^2 is actually equal"}, {"start": 60.199, "duration": 7.441, "text": "to cosine 2 th and when we Square it we"}, {"start": 63.84, "duration": 7.919, "text": "get c^ 2 2 th when we differentiate"}, {"start": 67.64, "duration": 9.2, "text": "cosine 2 2 Theta with respect to Theta"}, {"start": 71.759, "duration": 7.72, "text": "we get -4 cosine 2 Theta sin 2 Theta now"}, {"start": 76.84, "duration": 7.4, "text": "we can substitute these into our chain"}, {"start": 79.479, "duration": 9.64, "text": "rule equation we get -4 cosine 2 thet"}, {"start": 84.24, "duration": 7.76, "text": "sin 2 thet /^ 2 thet to simplify this"}, {"start": 89.119, "duration": 8.36, "text": "further we'll use these identities"}, {"start": 92.0, "duration": 10.96, "text": "cosine 2 thet = 1 - t^ 2 thet over 1 +"}, {"start": 97.479, "duration": 9.24, "text": "Tang 2 thet and sin 2 thet = 2 Tang thet"}, {"start": 102.96, "duration": 8.64, "text": "over 1 + Tang 2 thet substituting these"}, {"start": 106.719, "duration": 11.281, "text": "in and simplifying we get -4 * 1 - Tang"}, {"start": 111.6, "duration": 9.28, "text": "2 th over 1 + t^ 2 thet * 2 Tang thet"}, {"start": 118.0, "duration": 8.84, "text": "over 1 + Tang"}, {"start": 120.88, "duration": 11.28, "text": "* 1 1 + Tang 2 this simplifies to 8 Tang"}, {"start": 126.84, "duration": 9.8, "text": "thet * 1 - Tang 2 Theta all over 1 +"}, {"start": 132.16, "duration": 7.719, "text": "Tang 2 Theta cubed finally since X ="}, {"start": 136.64, "duration": 9.959, "text": "tangent Theta we can write our answer as"}, {"start": 139.879, "duration": 8.881, "text": "-8x * 1 - x^2 all over 1 + x^2 cubed and"}, {"start": 146.599, "duration": 4.961, "text": "there we have it we've successfully"}, {"start": 148.76, "duration": 7.4, "text": "found dydx X using trigonometric"}, {"start": 151.56, "duration": 4.6, "text": "substitution and the chain rule"}], "title": "Understanding Calculus: Derivatives Using Trigonometric Substitutions [((1\u2212\ud835\udc65^2)/(1+\ud835\udc65^2 ))^2 ]", "description": "Question\nGiven  y= [((1\u2212\ud835\udc65^2)/(1+\ud835\udc65^2 ))^2 ] and , find dy/dx .\n\nExplore the step-by-step process of finding derivatives using trigonometric substitutions. In this tutorial, we\u2019ll go through the detailed calculation of  dy/dx . for a given function, leveraging identities and calculus techniques. Perfect for students looking to understand complex derivatives in calculus!\n#Calculus #Derivatives #Trigonometry #MathTutoring #STEM", "lengthSeconds": 159, "width": 640, "height": 360}
{"video_id": "ZLq3iDvXsyg", "transcript": [{"start": 0.359, "duration": 5.201, "text": "welcome today we'll solve a challenging"}, {"start": 3.199, "duration": 4.881, "text": "derivative problem involving inverse"}, {"start": 5.56, "duration": 4.76, "text": "trigonometric functions let's start with"}, {"start": 8.08, "duration": 6.88, "text": "our problem we need to find the"}, {"start": 10.32, "duration": 9.48, "text": "derivative of y = inverse tangent of X"}, {"start": 14.96, "duration": 8.0, "text": "over theun of 1 - x^2 this looks"}, {"start": 19.8, "duration": 6.16, "text": "complicated but don't worry we can solve"}, {"start": 22.96, "duration": 5.399, "text": "it step by step using a clever"}, {"start": 25.96, "duration": 7.279, "text": "substitution let's make the substitution"}, {"start": 28.359, "duration": 8.641, "text": "x equals s of T this means tal inverse s"}, {"start": 33.239, "duration": 6.921, "text": "of X now let's substitute this into our"}, {"start": 37.0, "duration": 5.68, "text": "original expression we replace x with s"}, {"start": 40.16, "duration": 5.8, "text": "of T in the numerator and in the"}, {"start": 42.68, "duration": 7.64, "text": "denominator we get the square root of 1"}, {"start": 45.96, "duration": 6.88, "text": "- sin^2 t here's where we can use a"}, {"start": 50.32, "duration": 8.8, "text": "fundamental trigonometric identity"}, {"start": 52.84, "duration": 8.08, "text": "remember that 1 - sin^2 t = cine 2 T"}, {"start": 59.12, "duration": 3.919, "text": "using this ident"}, {"start": 60.92, "duration": 7.32, "text": "we can simplify our expression to"}, {"start": 63.039, "duration": 8.601, "text": "inverse tangent of sin t/ cosine T now"}, {"start": 68.24, "duration": 7.04, "text": "here's the key Insight sin T over cosine"}, {"start": 71.64, "duration": 7.04, "text": "T is the definition of tangent T so our"}, {"start": 75.28, "duration": 7.12, "text": "expression becomes inverse tangent of"}, {"start": 78.68, "duration": 7.0, "text": "tangent T when we have inverse tangent"}, {"start": 82.4, "duration": 6.359, "text": "of tangent T these functions cancel out"}, {"start": 85.68, "duration": 5.92, "text": "because they're inverse functions so our"}, {"start": 88.759, "duration": 7.161, "text": "expression simplify to just"}, {"start": 91.6, "duration": 8.4, "text": "T remember though t equal inverse sin of"}, {"start": 95.92, "duration": 7.12, "text": "X so our expression is really y ="}, {"start": 100.0, "duration": 5.56, "text": "inverse sin of X now we can find the"}, {"start": 103.04, "duration": 8.439, "text": "derivative the derivative of inverse sin"}, {"start": 105.56, "duration": 8.559, "text": "of X is 1 over the < TK of 1 - x^2"}, {"start": 111.479, "duration": 10.6, "text": "therefore the derivative of inverse"}, {"start": 114.119, "duration": 12.121, "text": "tangent of x/ < TK of 1 - x^2 = 1 / < TK"}, {"start": 122.079, "duration": 7.16, "text": "of 1 - x^2 this problem shows how making"}, {"start": 126.24, "duration": 6.32, "text": "a clever substitution can help simplify"}, {"start": 129.239, "duration": 6.441, "text": "a complex derivative the key was"}, {"start": 132.56, "duration": 6.2, "text": "recognizing that we could substitute xal"}, {"start": 135.68, "duration": 5.279, "text": "sin of T which L us use trigonometric"}, {"start": 138.76, "duration": 6.08, "text": "identities to simplify the expression"}, {"start": 140.959, "duration": 3.881, "text": "before taking the derivative"}], "title": "Mastering Derivatives: Unraveling Inverse Trigonometric Functions [\ud835\udc61\ud835\udc4e\ud835\udc5b^(\u22121)   \ud835\udc65/\u221a(1\u2212\ud835\udc65^2 )]", "description": "Question\nFind the derivative of the function [\ud835\udc61\ud835\udc4e\ud835\udc5b^(\u22121)   \ud835\udc65/\u221a(1\u2212\ud835\udc65^2 )] .\nDive into the world of calculus with this tutorial on differentiating inverse trigonometric functions. Follow along as we simplify and differentiate the expression  [\ud835\udc61\ud835\udc4e\ud835\udc5b^(\u22121)   \ud835\udc65/\u221a(1\u2212\ud835\udc65^2 )] step-by-step. Ideal for students and anyone looking to strengthen their understanding of calculus concepts!\n#Calculus #Derivatives #Trigonometry #MathTutorial #InverseFunctions", "lengthSeconds": 148, "width": 640, "height": 360}
{"video_id": "HohV07A-iuM", "transcript": [{"start": 0.28, "duration": 5.32, "text": "let's solve this differentiation problem"}, {"start": 2.52, "duration": 5.76, "text": "step by step using the quotient rule we"}, {"start": 5.6, "duration": 9.56, "text": "need to differentiate the function f of"}, {"start": 8.28, "duration": 9.44, "text": "T = 1 + t + T ^2 / 1 - t + T ^2 let's"}, {"start": 15.16, "duration": 7.92, "text": "recall the quotient rule the derivative"}, {"start": 17.72, "duration": 9.639, "text": "of U over V equal V * DDT minus U *"}, {"start": 23.08, "duration": 9.8, "text": "dvdt all over V ^2 first let's identify"}, {"start": 27.359, "duration": 10.161, "text": "u and v here U equal 1 + t + T ^2 and V"}, {"start": 32.88, "duration": 9.12, "text": "= 1 - t + T ^2 next we need to find do"}, {"start": 37.52, "duration": 9.4, "text": "DT and dvdt the derivative of U is 1 +"}, {"start": 42.0, "duration": 7.36, "text": "2T and the derivative of V is - 1 + 2T"}, {"start": 46.92, "duration": 6.08, "text": "now let's substitute everything into the"}, {"start": 49.36, "duration": 11.08, "text": "quotient rule formula we get 1 - t + T"}, {"start": 53.0, "duration": 11.479, "text": "^2 * 1 + 2T - 1 + t + t^ 2 * - 1 + 2T"}, {"start": 60.44, "duration": 7.999, "text": "all over 1 - t + t^ 2^ 2 let's expand"}, {"start": 64.479, "duration": 9.801, "text": "the numerator first let's multiply 1 - t"}, {"start": 68.439, "duration": 11.601, "text": "+ T ^ 2 by 1 + 2T this gives us 1 + 2 t"}, {"start": 74.28, "duration": 11.479, "text": "- t - 2 T ^2 + T ^ 2 + 2 T Cub which"}, {"start": 80.04, "duration": 8.16, "text": "simplifies to 1 + t + 2 t^ 2 - 2 T Cub"}, {"start": 85.759, "duration": 9.281, "text": "now for the second part let's multiply 1"}, {"start": 88.2, "duration": 11.8, "text": "+ t + T ^2 by -1 + 2T this gives us -1 +"}, {"start": 95.04, "duration": 9.359, "text": "t - 3T ^2 + 2 T cubed now we can"}, {"start": 100.0, "duration": 12.92, "text": "subtract these Expressions 1 + t + 2 T"}, {"start": 104.399, "duration": 13.36, "text": "^2 - 2 T Cub - -1 + t - 3 T ^2 + 2 T Cub"}, {"start": 112.92, "duration": 7.559, "text": "this simplifies to 2 - 2 T ^2 therefore"}, {"start": 117.759, "duration": 8.761, "text": "our final answer is 2 *"}, {"start": 120.479, "duration": 8.56, "text": "1 - t ^ 2/ 1 - t + T ^ 2^ 2 and that's"}, {"start": 126.52, "duration": 7.4, "text": "how we use the quotient rule to"}, {"start": 129.039, "duration": 4.881, "text": "differentiate this rational function"}], "title": "Quotient Rule Masterclass: Differentiating Complex Fractions [(\ud835\udfcf+\ud835\udc95+\ud835\udc95^\ud835\udfd0)/(\ud835\udfcf\u2212\ud835\udc95+\ud835\udc95^\ud835\udfd0)] with Ease", "description": "Dive into the world of calculus with this in-depth look at using the quotient rule for complex fractions. We\u2019ll expertly guide you through differentiating the function [(\ud835\udfcf+\ud835\udc95+\ud835\udc95^\ud835\udfd0)/(\ud835\udfcf\u2212\ud835\udc95+\ud835\udc95^\ud835\udfd0)] , helping you master this essential calculus technique. Perfect for students wanting to strengthen their differentiation skills.\n#Calculus #Differentiation #QuotientRule #MathEducation #LearnMath", "lengthSeconds": 133, "width": 640, "height": 360}
{"video_id": "26mtsd2dVGQ", "transcript": [{"start": 0.32, "duration": 6.2, "text": "let's solve this complex trigonometric"}, {"start": 3.08, "duration": 5.28, "text": "differentiation problem step by step our"}, {"start": 6.52, "duration": 6.079, "text": "question is to find the derivative of"}, {"start": 8.36, "duration": 7.359, "text": "the square < TK of 1 + cosine X over 1 -"}, {"start": 12.599, "duration": 6.121, "text": "cine x with respect to X let's break"}, {"start": 15.719, "duration": 5.161, "text": "this down into manageable steps first"}, {"start": 18.72, "duration": 5.36, "text": "we'll use the half angle identities to"}, {"start": 20.88, "duration": 9.2, "text": "rewrite the numerator and denominator"}, {"start": 24.08, "duration": 12.4, "text": "remember 1 + cosine x = 2 cosine 2 x / 2"}, {"start": 30.08, "duration": 8.4, "text": "and 1 - cosine x = 2 sin x / 2 next we"}, {"start": 36.48, "duration": 5.079, "text": "can simplify this fraction under the"}, {"start": 38.48, "duration": 7.919, "text": "square root the twos cancel out and what"}, {"start": 41.559, "duration": 9.041, "text": "we're left with is cosine 2 x/ 2 / sin 2"}, {"start": 46.399, "duration": 9.441, "text": "x/ 2 this is the definition of cotangent"}, {"start": 50.6, "duration": 8.56, "text": "2 x/ 2 the square root of Cent squar is"}, {"start": 55.84, "duration": 6.48, "text": "just coent itself so our expression"}, {"start": 59.16, "duration": 6.68, "text": "simplifies to the derivative of coent x"}, {"start": 62.32, "duration": 6.799, "text": "/ 2 now we can use the chain rule"}, {"start": 65.84, "duration": 6.4, "text": "remember that the derivative of coent is"}, {"start": 69.119, "duration": 7.36, "text": "netive cosecant SAR so we'll have"}, {"start": 72.24, "duration": 7.0, "text": "negative cose 2 of X over 2 * the"}, {"start": 76.479, "duration": 7.041, "text": "derivative of x over 2 the derivative of"}, {"start": 79.24, "duration": 8.48, "text": "x over 2 is simply 12 so we can multiply"}, {"start": 83.52, "duration": 6.639, "text": "that through cose squ is the same as 1 /"}, {"start": 87.72, "duration": 8.079, "text": "sin 2 so we can rewrite our our"}, {"start": 90.159, "duration": 8.721, "text": "expression as -1/2 sin^2 x / 2 finally"}, {"start": 95.799, "duration": 8.801, "text": "we can use another half angle identity"}, {"start": 98.88, "duration": 8.199, "text": "sin^2 x/ 2 = 1 - cosine x / 2 when we"}, {"start": 104.6, "duration": 8.559, "text": "substitute this in and simplify we get"}, {"start": 107.079, "duration": 8.64, "text": "our Final Answer -1/ 1 - cine X this"}, {"start": 113.159, "duration": 5.401, "text": "problem demonstrates the importance of"}, {"start": 115.719, "duration": 4.481, "text": "knowing your trigonometric identities"}, {"start": 118.56, "duration": 4.28, "text": "and being comfortable with the chain"}, {"start": 120.2, "duration": 5.48, "text": "rule by breaking down the problem into"}, {"start": 122.84, "duration": 5.559, "text": "smaller steps and using these tools we"}, {"start": 125.68, "duration": 6.32, "text": "were able to find the derivative of this"}, {"start": 128.399, "duration": 3.601, "text": "complex expression"}], "title": "Mastering Trigonometric Derivatives (\u221a((1+cos\u2061\ud835\udc65)/(1\u2212cos\u2061\ud835\udc65 ))) A Step-by-Step Guide", "description": "Dive into the fascinating world of calculus with our comprehensive video on differentiating complex trigonometric functions. In this tutorial, we\u2019ll guide you through a detailed step-by-step process to differentiate the function (\u221a((1+cos\u2061\ud835\udc65)/(1\u2212cos\u2061\ud835\udc65 ))). By breaking down each step and explaining the trigonometric identities involved, you\u2019ll gain a clear understanding of the concepts needed to tackle similar problems. Don\u2019t miss out on mastering this essential calculus skill!\n#Calculus #Trigonometry #MathTutorial #Differentiation #MathHelp", "lengthSeconds": 135, "width": 640, "height": 360}
{"video_id": "MwhulywAGbE", "transcript": [{"start": 0.32, "duration": 5.28, "text": "let's solve this challenging derivative"}, {"start": 2.639, "duration": 7.481, "text": "problem step by step we need to find the"}, {"start": 5.6, "duration": 10.72, "text": "derivative of y equal inverse tangent of"}, {"start": 10.12, "duration": 8.4, "text": "1 - < TK X over 1 + < TK X the key to"}, {"start": 16.32, "duration": 5.28, "text": "solving this problem is to recognize"}, {"start": 18.52, "duration": 5.96, "text": "that we can rewrite this as a difference"}, {"start": 21.6, "duration": 6.16, "text": "of inverse tangents there's a useful"}, {"start": 24.48, "duration": 6.44, "text": "identity that states inverse tangent of"}, {"start": 27.76, "duration": 8.319, "text": "a minus inverse tangent of of b equal"}, {"start": 30.92, "duration": 7.72, "text": "inverse tangent of a minus b/ 1 + a"}, {"start": 36.079, "duration": 7.0, "text": "looking at our fraction we can rewrite"}, {"start": 38.64, "duration": 8.079, "text": "it as inverse tangent of 1 minus inverse"}, {"start": 43.079, "duration": 6.881, "text": "tangent of squ < TK X this is because"}, {"start": 46.719, "duration": 6.961, "text": "when we substitute a equals 1 and be"}, {"start": 49.96, "duration": 6.36, "text": "equal sare < TK of X into the identity"}, {"start": 53.68, "duration": 6.839, "text": "we get our original expression now we"}, {"start": 56.32, "duration": 9.36, "text": "know that tangent of pi over 4 equal 1"}, {"start": 60.519, "duration": 6.92, "text": "therefore inverse tangent of 1 = < / 4"}, {"start": 65.68, "duration": 6.759, "text": "this allows us to simplify our"}, {"start": 67.439, "duration": 8.881, "text": "expression to Y = < / 4 - inverse"}, {"start": 72.439, "duration": 6.121, "text": "tangent of s < TK X now we can"}, {"start": 76.32, "duration": 5.159, "text": "differentiate this expression the"}, {"start": 78.56, "duration": 5.599, "text": "derivative of pi over 4 is zero since"}, {"start": 81.479, "duration": 5.721, "text": "it's a constant for the inverse tangent"}, {"start": 84.159, "duration": 5.161, "text": "term we'll use the chain Rule and the"}, {"start": 87.2, "duration": 7.32, "text": "fact that the derivative of inverse"}, {"start": 89.32, "duration": 8.88, "text": "tangent of U is 1 1 + u^2 * DDX the"}, {"start": 94.52, "duration": 9.52, "text": "derivative of square < TK X is 1 / 2 <"}, {"start": 98.2, "duration": 10.959, "text": "TK X substituting this in we get -1 1 +"}, {"start": 104.04, "duration": 8.719, "text": "x * 1 2 otk of X finally we can combine"}, {"start": 109.159, "duration": 9.761, "text": "these fractions to get our final answer"}, {"start": 112.759, "duration": 8.521, "text": "-1/2 square root of x * 1 + x to verify"}, {"start": 118.92, "duration": 4.799, "text": "this is correct we can check that this"}, {"start": 121.28, "duration": 6.479, "text": "matches our original problem the"}, {"start": 123.719, "duration": 8.401, "text": "derivative of inverse tangent of 1 - <"}, {"start": 127.759, "duration": 12.321, "text": "TK x/ 1 + < TK x"}, {"start": 132.12, "duration": 7.96, "text": "= 1 / 2 < TK of x * 1 + x"}], "title": "Mastering Derivatives: Inverse Tangent Function Simplified! [\ud835\udc66=\ud835\udc61\ud835\udc4e\ud835\udc5b^(\u22121)   (1\u2212\u221a\ud835\udc65)/(1+\u221a\ud835\udc65)]", "description": "Question\nFind the derivative of the function [\ud835\udc66=\ud835\udc61\ud835\udc4e\ud835\udc5b^(\u22121)   (1\u2212\u221a\ud835\udc65)/(1+\u221a\ud835\udc65)].\nUnlock the secrets to differentiating inverse trigonometric functions with this step-by-step tutorial. We\u2019ll walk you through the process of finding the derivative of  [\ud835\udc66=\ud835\udc61\ud835\udc4e\ud835\udc5b^(\u22121)   (1\u2212\u221a\ud835\udc65)/(1+\u221a\ud835\udc65)] . Perfect for calculus students looking to strengthen their understanding. Don\u2019t miss out\u2014subscribe for more math insights!\n#Calculus #MathTutorial #Derivatives #InverseTrigonometry #STEM", "lengthSeconds": 143, "width": 640, "height": 360}
{"video_id": "TgZBead-X84", "transcript": [{"start": 0.32, "duration": 6.599, "text": "welcome today we'll work through a"}, {"start": 3.28, "duration": 6.279, "text": "complex derivative problem step by step"}, {"start": 6.919, "duration": 4.64, "text": "let's start by looking at our expression"}, {"start": 9.559, "duration": 4.641, "text": "here's our initial expression that we"}, {"start": 11.559, "duration": 7.121, "text": "need to differentiate we need to find"}, {"start": 14.2, "duration": 9.28, "text": "the derivative of x over 1 + the < TK of"}, {"start": 18.68, "duration": 8.56, "text": "1 - x^2 all raised to the nth power"}, {"start": 23.48, "duration": 5.959, "text": "let's break this down step by step first"}, {"start": 27.24, "duration": 5.839, "text": "we'll use the chain rule since we have a"}, {"start": 29.439, "duration": 7.601, "text": "function to a power this gives us n *"}, {"start": 33.079, "duration": 6.64, "text": "our function to the N -1 power time the"}, {"start": 37.04, "duration": 5.12, "text": "derivative of the inner function now for"}, {"start": 39.719, "duration": 5.84, "text": "the inner derivative we'll need to use"}, {"start": 42.16, "duration": 5.52, "text": "the quotient rule remember the quotient"}, {"start": 45.559, "duration": 6.121, "text": "rule states that the derivative of U"}, {"start": 47.68, "duration": 8.08, "text": "over V equal V * the derivative of uus U"}, {"start": 51.68, "duration": 7.08, "text": "* the derivative of V all over V ^2"}, {"start": 55.76, "duration": 6.56, "text": "applying this to our problem we get this"}, {"start": 58.76, "duration": 8.76, "text": "expression the the numerator has 1 + <"}, {"start": 62.32, "duration": 8.36, "text": "TK of 1 - x^2 * the derivative of x - x"}, {"start": 67.52, "duration": 5.76, "text": "* the derivative of our denominator"}, {"start": 70.68, "duration": 5.72, "text": "let's evaluate these derivatives the"}, {"start": 73.28, "duration": 5.92, "text": "derivative of x is simply one and for"}, {"start": 76.4, "duration": 5.6, "text": "the denominator we need to use the chain"}, {"start": 79.2, "duration": 7.279, "text": "rule to find the derivative of square <"}, {"start": 82.0, "duration": 7.92, "text": "TK of 1 - x s after simplifying the"}, {"start": 86.479, "duration": 5.761, "text": "numerator we get this expression no"}, {"start": 89.92, "duration": 5.519, "text": "notice how the terms combine nicely in"}, {"start": 92.24, "duration": 6.64, "text": "the numerator through further algebraic"}, {"start": 95.439, "duration": 6.521, "text": "manipulation we can combine terms and"}, {"start": 98.88, "duration": 6.36, "text": "simplify our expression to this form and"}, {"start": 101.96, "duration": 7.68, "text": "finally we arrive at our final answer"}, {"start": 105.24, "duration": 8.239, "text": "the derivative equals n /x * the < TK of"}, {"start": 109.64, "duration": 7.0, "text": "1 - x^2 time our original expression"}, {"start": 113.479, "duration": 5.24, "text": "raise to the nth power this problem"}, {"start": 116.64, "duration": 4.24, "text": "demonstrates the power of combining"}, {"start": 118.719, "duration": 5.201, "text": "different differentiation rules the"}, {"start": 120.88, "duration": 5.8, "text": "chain Rule and quotient rule along with"}, {"start": 123.92, "duration": 7.24, "text": "careful algebraic manipulation to arrive"}, {"start": 126.68, "duration": 4.48, "text": "at an elegant final result"}], "title": "Mastering Derivatives [\ud835\udc65/(1+\u221a(1\u2212\ud835\udc65^2 ))]^\ud835\udc5b  Step-by-Step Guide to Differentiation", "description": "Question\nGiven the expression: [\ud835\udc65/(1+\u221a(1\u2212\ud835\udc65^2 ))]^\ud835\udc5b  calculate its derivative with respect to  x.\n\nUnlock the secrets of calculus with our detailed walkthrough on differentiating complex expressions. From using the chain rule to applying the quotient rule, this video provides a clear, step-by-step explanation to help you master derivatives. Perfect for students and calculus enthusiasts.\n\n#Calculus #Mathematics #Derivatives #Education #Differentiation", "lengthSeconds": 134, "width": 640, "height": 360}
{"video_id": "ngUMakIfNTc", "transcript": [{"start": 0.28, "duration": 6.76, "text": "let's find the derivative of sin x *"}, {"start": 3.439, "duration": 5.2, "text": "cine X using the product rule first"}, {"start": 7.04, "duration": 4.719, "text": "let's write out what we're trying to"}, {"start": 8.639, "duration": 6.12, "text": "find for this problem we'll use the"}, {"start": 11.759, "duration": 6.241, "text": "product rule the product rule states"}, {"start": 14.759, "duration": 6.52, "text": "that the derivative of U * v = u * the"}, {"start": 18.0, "duration": 9.08, "text": "derivative of V + V * the derivative of"}, {"start": 21.279, "duration": 7.881, "text": "U in our case U equal sinx nval cosine X"}, {"start": 27.08, "duration": 3.519, "text": "let's substitute these into the product"}, {"start": 29.16, "duration": 4.48, "text": "rule formula"}, {"start": 30.599, "duration": 8.521, "text": "we get sin x * the derivative of cosine"}, {"start": 33.64, "duration": 7.96, "text": "X plus cosine x * thetive of sin x now"}, {"start": 39.12, "duration": 6.32, "text": "let's recall the derivatives of s and"}, {"start": 41.6, "duration": 7.599, "text": "cosine the derivative of cosine X is"}, {"start": 45.44, "duration": 6.68, "text": "sinx and the derivative of sinx is cine"}, {"start": 49.199, "duration": 7.761, "text": "X let's substitute these derivatives"}, {"start": 52.12, "duration": 8.959, "text": "into our equation we get sinx * sinx"}, {"start": 56.96, "duration": 9.56, "text": "plus cosine x * cosine X now now let's"}, {"start": 61.079, "duration": 9.601, "text": "simplify sin x * sin x is sin^2 X and"}, {"start": 66.52, "duration": 9.599, "text": "cosine x * cosine X is cosine 2 x we can"}, {"start": 70.68, "duration": 7.68, "text": "rearrange this as cosine 2 x - sin^2 x"}, {"start": 76.119, "duration": 7.481, "text": "finally we can use a double angle"}, {"start": 78.36, "duration": 9.719, "text": "identity the expression c^ 2 x - sin^2 x"}, {"start": 83.6, "duration": 8.28, "text": "is equal to cosine of 2x therefore the"}, {"start": 88.079, "duration": 7.961, "text": "derivative of sin x * cosine x = cosine"}, {"start": 91.88, "duration": 7.919, "text": "of 2x to verify this visually let's look"}, {"start": 96.04, "duration": 7.52, "text": "at the graphs here's sin x * cosine X in"}, {"start": 99.799, "duration": 6.441, "text": "blue and its derivative cosine of 2x in"}, {"start": 103.56, "duration": 6.28, "text": "red and that's how we find the"}, {"start": 106.24, "duration": 6.8, "text": "derivative of sin x * cine X using the"}, {"start": 109.84, "duration": 6.2, "text": "product rule and simplify it to cine of"}, {"start": 113.04, "duration": 3.0, "text": "2x"}], "title": "Finding the Derivative of (sin\u2061\ud835\udc65 cos\u2061\ud835\udc65) Using the Product Rule | Step-by-Step Explanation", "description": "In this video, learn to differentiate (sin\u2061\ud835\udc65 cos\u2061\ud835\udc65) using the Product Rule in calculus.\n\nWhat You'll Learn:  \n- Definition of the Product Rule  \n- Step-by-step differentiation of (sin\u2061\ud835\udc65 cos\u2061\ud835\udc65)\n- Formula breakdown and example problem-solving  \n\nThis example is perfect for calculus beginners or those reviewing the Product Rule!\n\n\ud83d\udca1 Don't forget to LIKE, SHARE, and SUBSCRIBE for more math tutorials!\n#Calculus #ProductRule #Differentiation #Derivative #MathTutorial #LearnMath #MathEducation #MathHelp #hschighermath", "lengthSeconds": 115, "width": 640, "height": 360}
{"video_id": "NGhGybV-uWE", "transcript": [{"start": 0.32, "duration": 5.559, "text": "let's work through finding the"}, {"start": 2.199, "duration": 6.961, "text": "derivative of e to the power of sin 2x"}, {"start": 5.879, "duration": 6.281, "text": "step by step first let's look at our"}, {"start": 9.16, "duration": 6.76, "text": "original function we have eras to the"}, {"start": 12.16, "duration": 6.16, "text": "power of sin 2x this is a composite"}, {"start": 15.92, "duration": 5.279, "text": "function we have an exponential function"}, {"start": 18.32, "duration": 5.719, "text": "on the outside and a trigonometric"}, {"start": 21.199, "duration": 5.521, "text": "function on the inside to differentiate"}, {"start": 24.039, "duration": 4.56, "text": "this we'll need to use the chain rule"}, {"start": 26.72, "duration": 4.639, "text": "the chain rule states that when we have"}, {"start": 28.599, "duration": 6.761, "text": "a composite function f G of X the"}, {"start": 31.359, "duration": 7.081, "text": "derivative is equal to frime of G ofx *"}, {"start": 35.36, "duration": 5.56, "text": "G Prime of X let's start our"}, {"start": 38.44, "duration": 4.4, "text": "differentiation first we write out that"}, {"start": 40.92, "duration": 5.92, "text": "we're taking the derivative with respect"}, {"start": 42.84, "duration": 6.399, "text": "to X next using the chain rule we know"}, {"start": 46.84, "duration": 5.879, "text": "that when we differentiate e to a power"}, {"start": 49.239, "duration": 6.761, "text": "we keep the E to that power and multiply"}, {"start": 52.719, "duration": 7.281, "text": "by the derivative of what's inside now"}, {"start": 56.0, "duration": 6.559, "text": "we need to differentiate sin 2x remember"}, {"start": 60.0, "duration": 4.519, "text": "the derivative of s is cosine but we"}, {"start": 62.559, "duration": 5.24, "text": "also need to account for the chain rule"}, {"start": 64.519, "duration": 8.161, "text": "again because of the 2x inside the"}, {"start": 67.799, "duration": 7.041, "text": "derivative of 2x is simply 2 finally we"}, {"start": 72.68, "duration": 8.2, "text": "can rearrange this to write our answer"}, {"start": 74.84, "duration": 8.639, "text": "as 2 * cosine 2X * e^ of sin 2x let's"}, {"start": 80.88, "duration": 5.12, "text": "visualize what happened here we started"}, {"start": 83.479, "duration": 5.361, "text": "with an exponential function with a"}, {"start": 86.0, "duration": 7.32, "text": "trigonometric function inside using the"}, {"start": 88.84, "duration": 8.08, "text": "chain rule we one kept the original"}, {"start": 93.32, "duration": 6.2, "text": "exponential function to multiply by the"}, {"start": 96.92, "duration": 6.4, "text": "derivative of the inner sign function"}, {"start": 99.52, "duration": 6.48, "text": "which gave us cosine 3 and multiply by"}, {"start": 103.32, "duration": 5.0, "text": "the derivative of the argument of s"}, {"start": 106.0, "duration": 4.759, "text": "which was to this gave us our final"}, {"start": 108.32, "duration": 6.52, "text": "answer and that's how we find the"}, {"start": 110.759, "duration": 9.521, "text": "derivative of e to the power of sin 2x"}, {"start": 114.84, "duration": 5.44, "text": "using the chain rule thanks for watching"}], "title": "Mastering the Chain Rule Differentiating [(\ud835\udc52^sin\u20612\ud835\udc65)]  Step by Step!", "description": "Question: Find the derivative of  [(\ud835\udc52^sin\u20612\ud835\udc65)]\nIn this video, we break down the process of differentiating the function  (\ud835\udc52^sin\u20612\ud835\udc65)  using the chain rule. Whether you\u2019re a calculus student or just looking to brush up on your differentiation skills, this tutorial will guide you through each step with clear explanations and examples. Master the art of using the chain rule and enhance your math skills today!\n#Calculus #ChainRule #Differentiation #MathTutorial #ExponentialFunctions #MathHelp #hsc #hschighermath #1stpaper\n#hschighermath1stpaper\n#englishversion", "lengthSeconds": 119, "width": 640, "height": 360}
{"video_id": "mInLJyUj39s", "transcript": [{"start": 0.199, "duration": 4.801, "text": "let's solve a calculus problem involving"}, {"start": 2.679, "duration": 5.08, "text": "the differentiation of trigonometric"}, {"start": 5.0, "duration": 5.48, "text": "functions with degree measures our"}, {"start": 7.759, "duration": 6.401, "text": "question is to find the derivative of"}, {"start": 10.48, "duration": 6.84, "text": "thet degre time sin of thet degrees with"}, {"start": 14.16, "duration": 5.199, "text": "respect to Theta before we start"}, {"start": 17.32, "duration": 5.08, "text": "differentiating we need to convert the"}, {"start": 19.359, "duration": 6.441, "text": "degrees to radians we do this by"}, {"start": 22.4, "duration": 7.799, "text": "multiplying Theta degrees by pi over"}, {"start": 25.8, "duration": 7.36, "text": "180 this gives us Theta pi over 180"}, {"start": 30.199, "duration": 5.68, "text": "now we can apply the product rule"}, {"start": 33.16, "duration": 5.88, "text": "remember the product rule states that"}, {"start": 35.879, "duration": 6.241, "text": "the derivative of U * V equal U * the"}, {"start": 39.04, "duration": 6.72, "text": "derivative of V + V * the derivative of"}, {"start": 42.12, "duration": 7.4, "text": "U in our case U is Theta pi over"}, {"start": 45.76, "duration": 7.36, "text": "180 and V is s of theta pi over"}, {"start": 49.52, "duration": 6.44, "text": "180 let's handle each part separately"}, {"start": 53.12, "duration": 6.279, "text": "for the first term we need to use the"}, {"start": 55.96, "duration": 6.279, "text": "chain rule when differentiating sign the"}, {"start": 59.399, "duration": 5.08, "text": "derivative of s is cosine and we"}, {"start": 62.239, "duration": 5.841, "text": "multiply by the derivative of the inner"}, {"start": 64.479, "duration": 7.481, "text": "function this gives us Theta pi over 180"}, {"start": 68.08, "duration": 7.679, "text": "* cosine of theta pi over 180 * piun"}, {"start": 71.96, "duration": 7.479, "text": "over 180 for the second term we just"}, {"start": 75.759, "duration": 6.4, "text": "need to differentiate Theta pi over 180"}, {"start": 79.439, "duration": 3.521, "text": "with respect to Theta which gives us pi"}, {"start": 82.159, "duration": 4.64, "text": "over"}, {"start": 82.96, "duration": 5.96, "text": "180 now we can combine like terms let's"}, {"start": 86.799, "duration": 5.6, "text": "factor out pi over"}, {"start": 88.92, "duration": 7.519, "text": "180 from both terms this gives us our"}, {"start": 92.399, "duration": 8.921, "text": "final answer > over 180 * the quantity"}, {"start": 96.439, "duration": 8.081, "text": "Pi Theta over 180 * cine of theta pi/"}, {"start": 101.32, "duration": 7.24, "text": "180 + sin of theta pi over"}, {"start": 104.52, "duration": 7.68, "text": "180 let's verify this makes sense our"}, {"start": 108.56, "duration": 4.879, "text": "answer contains both s and cosine terms"}, {"start": 112.2, "duration": 4.08, "text": "which we would expect from"}, {"start": 113.439, "duration": 5.32, "text": "differentiating a sign function the pi"}, {"start": 116.28, "duration": 5.479, "text": "over 180 terms appear because we"}, {"start": 118.759, "duration": 5.0, "text": "converted from degrees to radians and"}, {"start": 121.759, "duration": 6.121, "text": "the Theta term remains because we"}, {"start": 123.759, "duration": 6.12, "text": "started witha * sin thet and that's how"}, {"start": 127.88, "duration": 4.48, "text": "we differentiate a trigonometric"}, {"start": 129.879, "duration": 4.801, "text": "function involving degree measures"}, {"start": 132.36, "duration": 5.12, "text": "remember to always convert to radians"}, {"start": 134.68, "duration": 7.0, "text": "first then apply the product rule and"}, {"start": 137.48, "duration": 4.2, "text": "chain rule as needed"}], "title": "Mastering Calculus: Differentiating Trigonometric Functions (\ud835\udf3d\u00b0 \ud835\udc94\ud835\udc8a\ud835\udc8f\u2061\ud835\udf3d\u00b0)", "description": "Question: Differentiate the expression  (\ud835\udf3d\u00b0 \ud835\udc94\ud835\udc8a\ud835\udc8f\u2061\ud835\udf3d\u00b0)  with respect to \ud835\udf3d.\nUnlock the secrets to differentiating trigonometric functions with our step-by-step guide! In this video, we break down the differentiation process for expressions involving trigonometric components, using the product and chain rules. Perfect for students or anyone wanting to refine their calculus skills.\n#Calculus #MathTutorial #Trigonometry #Differentiation #STEMLearning", "lengthSeconds": 144, "width": 640, "height": 360}
{"video_id": "NyE6aLGX20M", "transcript": [{"start": 0.28, "duration": 4.999, "text": "let's solve a challenging derivative"}, {"start": 2.52, "duration": 5.48, "text": "problem involving inverse trigonometric"}, {"start": 5.279, "duration": 6.36, "text": "functions our goal is to find the"}, {"start": 8.0, "duration": 7.839, "text": "derivative of tan inverse of 2 < tkx"}, {"start": 11.639, "duration": 7.4, "text": "over 1 - x this looks complicated but we"}, {"start": 15.839, "duration": 6.961, "text": "can solve it using a clever substitution"}, {"start": 19.039, "duration": 9.48, "text": "let's substitute Square < TK of X = Tan"}, {"start": 22.8, "duration": 7.719, "text": "t therefore T = tan inverse of s < tkx"}, {"start": 28.519, "duration": 4.801, "text": "using this substitution"}, {"start": 30.519, "duration": 7.281, "text": "our expression becomes tan inverse of 2"}, {"start": 33.32, "duration": 8.44, "text": "tan t/ 1 - tan^ 2 T this can be"}, {"start": 37.8, "duration": 6.04, "text": "simplified to tan inverse of tan of 2T"}, {"start": 41.76, "duration": 5.119, "text": "using the double angle formula for"}, {"start": 43.84, "duration": 5.359, "text": "tangent now since tan inverse and tan"}, {"start": 46.879, "duration": 6.241, "text": "are inverse functions they cancel out"}, {"start": 49.199, "duration": 6.961, "text": "leaving us with Just 2 T remember T is"}, {"start": 53.12, "duration": 7.0, "text": "tan inverse of < TK X so we can rewrite"}, {"start": 56.16, "duration": 5.84, "text": "this as 2 * tan inverse of < TK X to to"}, {"start": 60.12, "duration": 5.079, "text": "differentiate this we'll use the chain"}, {"start": 62.0, "duration": 8.72, "text": "rule the derivative of tan inverse U is"}, {"start": 65.199, "duration": 10.481, "text": "1/ 1 + u^2 * the derivative of U here U"}, {"start": 70.72, "duration": 9.039, "text": "is s < TK X so we get 2 over 1 + x * 1 2"}, {"start": 75.68, "duration": 9.2, "text": "< tkx finally we can simplify this to 1"}, {"start": 79.759, "duration": 7.4, "text": "/ < tkx * 1 + x to verify this is"}, {"start": 84.88, "duration": 4.08, "text": "correct we could check that this"}, {"start": 87.159, "duration": 4.401, "text": "derivative matches what we would get"}, {"start": 88.96, "duration": 5.199, "text": "using implicit differentiation but"}, {"start": 91.56, "duration": 5.72, "text": "that's beyond the scope of this video"}, {"start": 94.159, "duration": 5.681, "text": "let's review the key steps first we made"}, {"start": 97.28, "duration": 5.68, "text": "a clever substitution to simplify the"}, {"start": 99.84, "duration": 6.44, "text": "expression then we used the double angle"}, {"start": 102.96, "duration": 6.0, "text": "formula for tangent next we simplified"}, {"start": 106.28, "duration": 5.96, "text": "using the inverse property of tangent"}, {"start": 108.96, "duration": 5.4, "text": "functions finally we use the chain rule"}, {"start": 112.24, "duration": 5.4, "text": "to find the derivative and simplified"}, {"start": 114.36, "duration": 3.28, "text": "our answer"}], "title": "Exploring Derivatives: Inverse Tangent Function [\ud835\udc61\ud835\udc4e\ud835\udc5b^(\u22121)   (2\u221a\ud835\udc65  )/(1\u2212\ud835\udc65)] Explained Step-by-Step", "description": "Question: Find the derivative of [\ud835\udc61\ud835\udc4e\ud835\udc5b^(\u22121)   (2\u221a\ud835\udc65  )/(1\u2212\ud835\udc65)] .\nDive into the world of calculus with a detailed explanation of how to find the derivative of an inverse tangent function. Using clear, step-by-step methods, this video breaks down each part of the process to enhance your understanding and problem-solving skills. Perfect for students and enthusiasts looking to master calculus concepts.\n#Calculus #Derivatives #InverseTrigonometry #MathTutorial #Education #StudyMath", "lengthSeconds": 120, "width": 640, "height": 360}
{"video_id": "DBvtIwI5KzI", "transcript": [{"start": 0.28, "duration": 6.119, "text": "welcome today we'll solve a challenging"}, {"start": 3.159, "duration": 5.761, "text": "derivative problem using logarithmic"}, {"start": 6.399, "duration": 6.921, "text": "differentiation our goal is to find the"}, {"start": 8.92, "duration": 10.119, "text": "derivative of this function y = x Cub *"}, {"start": 13.32, "duration": 8.799, "text": "the < TK x^2 + 4 over x^2 + 3 this"}, {"start": 19.039, "duration": 6.601, "text": "function looks quite complicated to"}, {"start": 22.119, "duration": 6.721, "text": "differentiate directly so instead let's"}, {"start": 25.64, "duration": 5.28, "text": "take the natural logarithm of both sides"}, {"start": 28.84, "duration": 5.36, "text": "this will allow us to use the properties"}, {"start": 30.92, "duration": 6.28, "text": "of logarithms to break down this complex"}, {"start": 34.2, "duration": 6.199, "text": "expression now we can use the properties"}, {"start": 37.2, "duration": 6.199, "text": "of logarithms to break this down"}, {"start": 40.399, "duration": 5.361, "text": "remember the logarithm of a product is"}, {"start": 43.399, "duration": 4.68, "text": "the sum of the logarithms and the"}, {"start": 45.76, "duration": 5.56, "text": "logarithm of a quotient is the"}, {"start": 48.079, "duration": 6.48, "text": "difference of logarithms let's apply"}, {"start": 51.32, "duration": 7.6, "text": "these properties first we separate the X"}, {"start": 54.559, "duration": 8.041, "text": "Cub term then the square root becomes a"}, {"start": 58.92, "duration": 6.04, "text": "1/2 power of the fraction next we can"}, {"start": 62.6, "duration": 5.92, "text": "further simplify this using the power"}, {"start": 64.96, "duration": 4.96, "text": "property of logarithms the X Cub becomes"}, {"start": 68.52, "duration": 5.08, "text": "3 *"}, {"start": 69.92, "duration": 6.6, "text": "lnx and the square root becomes 1/ 12 *"}, {"start": 73.6, "duration": 6.04, "text": "the logarithm of the fraction now we can"}, {"start": 76.52, "duration": 6.44, "text": "split the fraction inside the logarithm"}, {"start": 79.64, "duration": 6.36, "text": "giving us three terms now comes the key"}, {"start": 82.96, "duration": 7.44, "text": "step we take the derivative of both"}, {"start": 86.0, "duration": 6.68, "text": "sides with respect to X on the left side"}, {"start": 90.4, "duration": 5.719, "text": "we get 1/ y *"}, {"start": 92.68, "duration": 7.28, "text": "dydx on the right side we use the chain"}, {"start": 96.119, "duration": 8.28, "text": "rule for each term to solve for dydx we"}, {"start": 99.96, "duration": 6.6, "text": "multiply both sides by y finally we"}, {"start": 104.399, "duration": 5.161, "text": "substitute back the original expression"}, {"start": 106.56, "duration": 5.64, "text": "for y to get our final answer this gives"}, {"start": 109.56, "duration": 5.519, "text": "us the derivative of our original"}, {"start": 112.2, "duration": 6.239, "text": "function and there you have it we've"}, {"start": 115.079, "duration": 5.481, "text": "found the derivative using logarithmic"}, {"start": 118.439, "duration": 4.0, "text": "differentiation this method is"}, {"start": 120.56, "duration": 4.199, "text": "particularly useful when dealing with"}, {"start": 122.439, "duration": 4.561, "text": "products and quotients of functions as"}, {"start": 124.759, "duration": 6.48, "text": "it turns multiplication and division"}, {"start": 127.0, "duration": 4.239, "text": "into addition and subtraction"}], "title": "Calculating Derivatives Using Logarithmic Differentiation [\ud835\udc65^3 \u221a((\ud835\udc65^2+4)/(\ud835\udc65^2+3))]", "description": "Question\nFind the derivative of the function y =  [\ud835\udc65^3 \u221a((\ud835\udc65^2+4)/(\ud835\udc65^2+3))] using logarithmic differentiation.\n\nLearn how to find the derivative of complex functions using logarithmic differentiation. In this tutorial, we break down each step with clear explanations. Ideal for students wanting to master calculus concepts.\n#Calculus #LogarithmicDifferentiation #Derivatives #MathTutorial #LearnMath", "lengthSeconds": 134, "width": 640, "height": 360}
{"video_id": "QJCxxc7CZCQ", "transcript": [{"start": 0.28, "duration": 5.519, "text": "let's find the derivative of the natural"}, {"start": 2.84, "duration": 6.64, "text": "logarithm of the square < TK of X we"}, {"start": 5.799, "duration": 5.88, "text": "start with Ln of square < TK of x to"}, {"start": 9.48, "duration": 4.84, "text": "solve this we'll need to use the chain"}, {"start": 11.679, "duration": 4.761, "text": "rule the chain rule tells us that when"}, {"start": 14.32, "duration": 5.08, "text": "we have a composite function like this"}, {"start": 16.44, "duration": 5.28, "text": "one we need to multiply the derivative"}, {"start": 19.4, "duration": 5.119, "text": "of the outer function by the derivative"}, {"start": 21.72, "duration": 6.08, "text": "of the inner function in our case the"}, {"start": 24.519, "duration": 5.481, "text": "outer function is the natural logarithm"}, {"start": 27.8, "duration": 6.52, "text": "and the inner function is the square"}, {"start": 30.0, "duration": 7.96, "text": "root of x remember the derivative of Ln"}, {"start": 34.32, "duration": 6.559, "text": "X is 1 /x let's apply the chain rule"}, {"start": 37.96, "duration": 6.599, "text": "step by step first we write one over the"}, {"start": 40.879, "duration": 6.561, "text": "inner function which is square < TK of X"}, {"start": 44.559, "duration": 6.16, "text": "then multiply by the derivative of the"}, {"start": 47.44, "duration": 6.0, "text": "inner function now we need to find the"}, {"start": 50.719, "duration": 5.52, "text": "derivative of square otk of X remember"}, {"start": 53.44, "duration": 5.88, "text": "that square < TK of X is the same as x"}, {"start": 56.239, "duration": 8.041, "text": "to the 1/2 power the derivative of x to"}, {"start": 59.32, "duration": 8.32, "text": "the 1/2 is2 * x^ -2 power which"}, {"start": 64.28, "duration": 5.72, "text": "simplifies to 1 / 2 < TK X let's"}, {"start": 67.64, "duration": 8.88, "text": "substitute this back into our chain rule"}, {"start": 70.0, "duration": 9.04, "text": "equation we get 1 < TK x * 1 / 2 < TK X"}, {"start": 76.52, "duration": 6.639, "text": "when we multiply these fractions we get"}, {"start": 79.04, "duration": 8.48, "text": "1/ 2X and that's our final answer the"}, {"start": 83.159, "duration": 7.761, "text": "derivative of Ln of squ < TK X is 1 over"}, {"start": 87.52, "duration": 6.0, "text": "2x let's review what we did we use the"}, {"start": 90.92, "duration": 4.64, "text": "chain rule to break down the derivative"}, {"start": 93.52, "duration": 4.68, "text": "then found the derivative of the inner"}, {"start": 95.56, "duration": 4.599, "text": "function using the power rule and"}, {"start": 98.2, "duration": 5.239, "text": "finally combined our fractions to get"}, {"start": 100.159, "duration": 3.28, "text": "our answer"}], "title": "Understanding the Derivative of Logarithmic Functions [(\ud835\udc59\ud835\udc5b\u221a\ud835\udc65)] Step-by-Step Guide", "description": "Question: Find the derivative of [(\ud835\udc59\ud835\udc5b\u221a\ud835\udc65) ]\nDive into the fascinating world of calculus as we explore how to find the derivative of the logarithmic function (\ud835\udc59\ud835\udc5b\u221a\ud835\udc65). Learn about the chain rule and power rule in a simple and intuitive way. Perfect for students and enthusiasts wanting to grasp the fundamentals of differentiation. Don\u2019t forget to like, subscribe, and comment below with your thoughts!\n#englishversion #cpmrana #Calculus #Differentiation #LogarithmicFunctions #MathTutorial #ChainRule", "lengthSeconds": 106, "width": 640, "height": 360}
{"video_id": "hTN2Xd4kgBc", "transcript": [{"start": 0.28, "duration": 5.559, "text": "let's solve this challenging derivative"}, {"start": 2.56, "duration": 6.36, "text": "problem step by step our goal is to find"}, {"start": 5.839, "duration": 6.76, "text": "the derivative of inverse tangent of"}, {"start": 8.92, "duration": 6.639, "text": "secant X+ tangent X let's start by"}, {"start": 12.599, "duration": 6.44, "text": "applying the chain rule remember the"}, {"start": 15.559, "duration": 7.48, "text": "derivative of inverse tangent of U is 1"}, {"start": 19.039, "duration": 8.32, "text": "1 + U ^2 times the derivative of U in"}, {"start": 23.039, "duration": 6.521, "text": "our case U is secant x + tangent X now"}, {"start": 27.359, "duration": 4.92, "text": "we need to find the derivative of U"}, {"start": 29.56, "duration": 6.159, "text": "which is the derivative of secant x +"}, {"start": 32.279, "duration": 7.001, "text": "tangent X the derivative of secant x is"}, {"start": 35.719, "duration": 7.32, "text": "secant x tangent X and the derivative of"}, {"start": 39.28, "duration": 6.24, "text": "tangent X is secant 2 x adding these"}, {"start": 43.039, "duration": 5.961, "text": "together gives us secant x tangent X"}, {"start": 45.52, "duration": 5.519, "text": "Plus secant 2 x let's substitute"}, {"start": 49.0, "duration": 5.359, "text": "everything back into our original"}, {"start": 51.039, "duration": 8.281, "text": "formula we get 1 over 1 + the quantity"}, {"start": 54.359, "duration": 10.401, "text": "secant x + tangent x^2 * secant x"}, {"start": 59.32, "duration": 8.64, "text": "tangent X x +^ 2 x now let's expand the"}, {"start": 64.76, "duration": 7.719, "text": "denominator when we Square secant x +"}, {"start": 67.96, "duration": 9.08, "text": "tangent X we get secant 2 x + 2 secant x"}, {"start": 72.479, "duration": 10.32, "text": "tangent x + tangent 2 x remember that"}, {"start": 77.04, "duration": 9.32, "text": "tangent 2 x = secant 2 X - 1 using this"}, {"start": 82.799, "duration": 8.121, "text": "identity we can simplify our denominator"}, {"start": 86.36, "duration": 7.64, "text": "the denominator becomes 2 secant s x + 2"}, {"start": 90.92, "duration": 6.159, "text": "secant x tangent X looking at our"}, {"start": 94.0, "duration": 6.32, "text": "fraction now we can factor out secant 2"}, {"start": 97.079, "duration": 7.0, "text": "x + secant x tangent X from both"}, {"start": 100.32, "duration": 6.159, "text": "numerator and denominator and finally"}, {"start": 104.079, "duration": 6.281, "text": "after canceling the common factors we"}, {"start": 106.479, "duration": 6.041, "text": "arrive at our answer 1/2 this problem"}, {"start": 110.36, "duration": 4.28, "text": "demonstrated the importance of knowing"}, {"start": 112.52, "duration": 4.44, "text": "your derivative formulas trigonometric"}, {"start": 114.64, "duration": 5.32, "text": "identities and algebraic manipulation"}, {"start": 116.96, "duration": 3.0, "text": "skills"}], "title": "Mastering Derivatives: Simplifying {\ud835\udc61\ud835\udc4e\ud835\udc5b^(\u22121) (sec\u2061\ud835\udc65+tan\u2061\ud835\udc65 )} Step-by-Step", "description": "Question\nSimplify the derivative of the function {\ud835\udc61\ud835\udc4e\ud835\udc5b^(\u22121) (sec\u2061\ud835\udc65+tan\u2061\ud835\udc65 )}.\n\nUnlock the mystery behind simplifying derivatives with this comprehensive walkthrough of  {\ud835\udc61\ud835\udc4e\ud835\udc5b^(\u22121) (sec\u2061\ud835\udc65+tan\u2061\ud835\udc65 )}. Perfect for calculus students and math enthusiasts, this video breaks down the process into easy-to-understand steps. Follow along for clear explanations and valuable insights to help you ace your calculus course.\n#Calculus #Derivatives #MathHelp #LearningMath #Trigonometry #MathTutorial", "lengthSeconds": 123, "width": 640, "height": 360}
{"video_id": "8p7wWbM5hjg", "transcript": [{"start": 0.28, "duration": 5.519, "text": "let's solve this calculus problem step"}, {"start": 2.639, "duration": 7.401, "text": "by step we need to find the derivative"}, {"start": 5.799, "duration": 7.441, "text": "of e to x * tangent X first let's"}, {"start": 10.04, "duration": 8.52, "text": "identify that this is a product of two"}, {"start": 13.24, "duration": 9.039, "text": "functions we have U of xal e to X and V"}, {"start": 18.56, "duration": 6.12, "text": "ofx = tangent X since this is a product"}, {"start": 22.279, "duration": 5.24, "text": "of two functions we'll use the product"}, {"start": 24.68, "duration": 6.28, "text": "rule the product rule states that the"}, {"start": 27.519, "duration": 7.441, "text": "derivative of U * v = u * the derivative"}, {"start": 30.96, "duration": 7.16, "text": "of V plus v * the derivative of U let's"}, {"start": 34.96, "duration": 6.599, "text": "apply this to our problem we get e to"}, {"start": 38.12, "duration": 8.88, "text": "the x * the derivative of tangent X Plus"}, {"start": 41.559, "duration": 8.881, "text": "tangent x * the derivative of e to X now"}, {"start": 47.0, "duration": 5.8, "text": "let's find each derivative separately"}, {"start": 50.44, "duration": 6.32, "text": "first the derivative of tangent X is"}, {"start": 52.8, "duration": 8.04, "text": "secant squ x and the derivative of e to"}, {"start": 56.76, "duration": 6.759, "text": "the x is just e to the x substituting"}, {"start": 60.84, "duration": 7.8, "text": "these derivatives back into our equation"}, {"start": 63.519, "duration": 9.721, "text": "we get e to x * secant 2 X Plus tangent"}, {"start": 68.64, "duration": 8.04, "text": "x * e to X finally we can factor out e"}, {"start": 73.24, "duration": 7.0, "text": "to X since it appears in both terms this"}, {"start": 76.68, "duration": 7.6, "text": "gives us our final answer e to x * the"}, {"start": 80.24, "duration": 7.199, "text": "quantity secant 2 x + tangent X let's"}, {"start": 84.28, "duration": 5.96, "text": "review the key steps we identified this"}, {"start": 87.439, "duration": 4.801, "text": "as a product rule problem apply app the"}, {"start": 90.24, "duration": 5.28, "text": "product rule formula found the"}, {"start": 92.24, "duration": 6.76, "text": "derivatives of tangent X and E to X"}, {"start": 95.52, "duration": 5.959, "text": "separately substituted these back in and"}, {"start": 99.0, "duration": 7.72, "text": "finally factored out the common term e"}, {"start": 101.479, "duration": 5.241, "text": "to X and that's our solution"}], "title": "Finding the Derivative of \ud835\udc52^\ud835\udc65  tan\u2061\ud835\udc65  Using the Product Rule | Step-by-Step Explanation", "description": "In this video, learn to differentiate \ud835\udc52^\ud835\udc65  tan\u2061\ud835\udc65 using the Product Rule in calculus.\n\nWhat You'll Learn:  \n- Definition of the Product Rule  \n- Step-by-step differentiation of \ud835\udc52^\ud835\udc65  tan\u2061\ud835\udc65\n- Formula breakdown and example problem-solving  \n\nThis example is perfect for calculus beginners or those reviewing the Product Rule!\n\ud83d\udca1 Don't forget to LIKE, SHARE, and SUBSCRIBE for more math tutorials!\n#Calculus #ProductRule #Differentiation #Derivative #MathTutorial #LearnMath #MathEducation #MathHelp #tanx", "lengthSeconds": 108, "width": 640, "height": 360}
{"video_id": "VVW69PDF2yY", "transcript": [{"start": 0.28, "duration": 4.88, "text": "let's solve this calculus problem step"}, {"start": 2.44, "duration": 7.64, "text": "by step using the quotient rule we need"}, {"start": 5.16, "duration": 8.32, "text": "to find the derivative of x sin x / 1 +"}, {"start": 10.08, "duration": 7.359, "text": "cosine X let's recall the quotient rule"}, {"start": 13.48, "duration": 6.92, "text": "formula the derivative of U over v = v *"}, {"start": 17.439, "duration": 6.961, "text": "the derivative of U minus U * the"}, {"start": 20.4, "duration": 7.4, "text": "derivative of V all / V ^2 let's"}, {"start": 24.4, "duration": 9.12, "text": "identify our u and v in this case U"}, {"start": 27.8, "duration": 8.84, "text": "equal x sinx and V = 1 + cine X now we"}, {"start": 33.52, "duration": 6.199, "text": "need to find the derivatives of u and v"}, {"start": 36.64, "duration": 6.72, "text": "for U which is X sinx will use the"}, {"start": 39.719, "duration": 8.081, "text": "product rule this gives us sin x + x"}, {"start": 43.36, "duration": 8.32, "text": "cine x 4 V which is 1 + cinex the"}, {"start": 47.8, "duration": 6.919, "text": "derivative is sinx let's substitute"}, {"start": 51.68, "duration": 8.879, "text": "these into our quotient rule formula we"}, {"start": 54.719, "duration": 11.641, "text": "get 1 + cosine x * sin x + x cosin x - x"}, {"start": 60.559, "duration": 8.641, "text": "sin x * sin x all over 1 + cosine x^ 2"}, {"start": 66.36, "duration": 6.96, "text": "let's expand the numerator first"}, {"start": 69.2, "duration": 8.68, "text": "distribute 1 + cosine X we get sin x +"}, {"start": 73.32, "duration": 7.839, "text": "sin x cosine x + x cosine x + x cosine ^"}, {"start": 77.88, "duration": 10.239, "text": "2 x then for the negative"}, {"start": 81.159, "duration": 9.6, "text": "term * sinx gives us plus x sin^2 x now"}, {"start": 88.119, "duration": 8.36, "text": "let's use a trigonometric ident"}, {"start": 90.759, "duration": 12.441, "text": "we know that sin^2 x + cine 2 x = 1 so x"}, {"start": 96.479, "duration": 10.481, "text": "* sin^2 x + x * cosine 2 x = just X this"}, {"start": 103.2, "duration": 8.76, "text": "simplifies our expression to sin x + sin"}, {"start": 106.96, "duration": 10.64, "text": "x cine x + x cosine x + x we can Factor"}, {"start": 111.96, "duration": 10.4, "text": "this as sin x * 1 + cine x + x * 1 +"}, {"start": 117.6, "duration": 7.92, "text": "cine X finally we can cancel 1 + cosine"}, {"start": 122.36, "duration": 7.28, "text": "X from the numerator and denominator"}, {"start": 125.52, "duration": 7.68, "text": "giving us our final answer x + sinx over"}, {"start": 129.64, "duration": 6.679, "text": "1 + cine X and that's how we use the"}, {"start": 133.2, "duration": 6.36, "text": "quotient rule to find this derivative"}, {"start": 136.319, "duration": 5.2, "text": "remember to take it step by step and use"}, {"start": 139.56, "duration": 6.08, "text": "your trigonometric identities when they"}, {"start": 141.519, "duration": 4.121, "text": "help simplify the expression"}], "title": "Differentiating Complex Fractions: Simplifying Trigonometric Expressions [(\ud835\udc99\ud835\udc94\ud835\udc8a\ud835\udc8f \ud835\udc99)/(\ud835\udfcf+\ud835\udc84\ud835\udc90\ud835\udc94 \ud835\udc99)]", "description": "Explore the step-by-step process of differentiating and simplifying trigonometric expressions such as [(\ud835\udc99\ud835\udc94\ud835\udc8a\ud835\udc8f \ud835\udc99)/(\ud835\udfcf+\ud835\udc84\ud835\udc90\ud835\udc94 \ud835\udc99)]. Follow along to simplify it to [( (\ud835\udc99+\ud835\udc94\ud835\udc8a\ud835\udc8f \ud835\udc99))/((\ud835\udfcf+\ud835\udc84\ud835\udc90\ud835\udc94 \ud835\udc99) )] . Perfect for calculus students looking to enhance their skills in handling complex fractions and trigonometric derivatives.\n#Trigonometry #Calculus #MathTutorial #Differentiation #LearnMath", "lengthSeconds": 144, "width": 640, "height": 360}
{"video_id": "xtAkBScZMEU", "transcript": [{"start": 0.28, "duration": 5.599, "text": "let's solve this complex derivative"}, {"start": 2.679, "duration": 5.96, "text": "problem step by step we need to find the"}, {"start": 5.879, "duration": 7.321, "text": "derivative of tan inverse of the square"}, {"start": 8.639, "duration": 6.601, "text": "< TK of 1 - x over 1 + x this is a"}, {"start": 13.2, "duration": 4.4, "text": "challenging problem that we'll solve"}, {"start": 15.24, "duration": 6.48, "text": "using substitution let's make the"}, {"start": 17.6, "duration": 8.04, "text": "substitution X = cine of T which means T"}, {"start": 21.72, "duration": 6.319, "text": "= inverse cosine of x after making this"}, {"start": 25.64, "duration": 4.6, "text": "substitution we can rewrite our"}, {"start": 28.039, "duration": 7.281, "text": "expression the expression under the"}, {"start": 30.24, "duration": 9.24, "text": "square < TK becomes 1 - cine T over 1 +"}, {"start": 35.32, "duration": 6.2, "text": "cine T now comes a key inside we can use"}, {"start": 39.48, "duration": 7.919, "text": "the double angle formula which states"}, {"start": 41.52, "duration": 10.96, "text": "that 1 - cosine T = 2 sin^2 T / 2 and 1"}, {"start": 47.399, "duration": 7.84, "text": "+ cine T = 2 cosine 2 of T / 2 applying"}, {"start": 52.48, "duration": 6.039, "text": "these identities our expression becomes"}, {"start": 55.239, "duration": 9.88, "text": "tan inverse of the square < TK of 2"}, {"start": 58.519, "duration": 9.041, "text": "sin^2 t / 2 / 2 cosine ^ 2 t/ 2 the twos"}, {"start": 65.119, "duration": 7.68, "text": "cancel out and under the square root"}, {"start": 67.56, "duration": 9.12, "text": "we're left with sin^2 t / 2 / cosine 2"}, {"start": 72.799, "duration": 6.481, "text": "t/ 2 which is just tangent 2 of t/ 2 the"}, {"start": 76.68, "duration": 5.96, "text": "square root and the square cancel each"}, {"start": 79.28, "duration": 6.839, "text": "other giving us tan inverse of tangent T"}, {"start": 82.64, "duration": 6.2, "text": "/ 2 here's another key Insight when you"}, {"start": 86.119, "duration": 4.68, "text": "have tan inverse of tangent of an angle"}, {"start": 88.84, "duration": 5.08, "text": "they cancel out leaving you with just"}, {"start": 90.799, "duration": 7.441, "text": "the angle so our expression simplifies"}, {"start": 93.92, "duration": 8.199, "text": "to T / 2 remember that tal inverse"}, {"start": 98.24, "duration": 6.76, "text": "cosine of x so we can write y = 12"}, {"start": 102.119, "duration": 5.481, "text": "inverse cosine of x now we can"}, {"start": 105.0, "duration": 8.04, "text": "differentiate the derivative of inverse"}, {"start": 107.6, "duration": 10.439, "text": "cosine of x is -1 over the < TK of 1 -"}, {"start": 113.04, "duration": 10.28, "text": "x^2 therefore our final answer is -1 /2"}, {"start": 118.039, "duration": 11.161, "text": "* 1 the < TK of 1 - x^2 which simplifies"}, {"start": 123.32, "duration": 7.96, "text": "to 1 /2 * < TK of 1 - x squar this"}, {"start": 129.2, "duration": 4.08, "text": "problem showcases the power of"}, {"start": 131.28, "duration": 4.16, "text": "trigonometric substitution and"}, {"start": 133.28, "duration": 5.039, "text": "identities in solving complex"}, {"start": 135.44, "duration": 5.36, "text": "derivatives the key was recognizing that"}, {"start": 138.319, "duration": 5.401, "text": "we could simplify our expression using"}, {"start": 140.8, "duration": 5.28, "text": "the half angle formulas and the"}, {"start": 143.72, "duration": 5.36, "text": "properties of inverse trigonometric"}, {"start": 146.08, "duration": 3.0, "text": "functions"}], "title": "Mastering Derivatives: ArcTan and Trigonometric Simplifications [\ud835\udc61\ud835\udc4e\ud835\udc5b^(\u22121) \u221a((1\u2212\ud835\udc65)/(1+\ud835\udc65))]", "description": "Question\nFind the derivative of the function [\ud835\udc61\ud835\udc4e\ud835\udc5b^(\u22121) \u221a((1\u2212\ud835\udc65)/(1+\ud835\udc65))] .\n\nUnlock the secrets of calculus with our step-by-step guide to finding derivatives involving arc tangents and trigonometric expressions. This video breaks down the process of differentiating [\ud835\udc61\ud835\udc4e\ud835\udc5b^(\u22121) \u221a((1\u2212\ud835\udc65)/(1+\ud835\udc65))] with detailed explanations and simplifications. Perfect for students and math enthusiasts aiming to deepen their understanding of calculus concepts.\n#Calculus #Derivatives #MathTutorial #Trigonometry #ArcTan #STEM", "lengthSeconds": 152, "width": 640, "height": 360}
{"video_id": "Eti3i4EiAp0", "transcript": [{"start": 0.36, "duration": 5.32, "text": "welcome today we're going to learn about"}, {"start": 2.879, "duration": 7.041, "text": "logarithmic differentiation by finding"}, {"start": 5.68, "duration": 7.36, "text": "the derivative of yal x to the power of"}, {"start": 9.92, "duration": 5.24, "text": "1 /x this is a particularly tricky"}, {"start": 13.04, "duration": 5.84, "text": "function because X appears in both the"}, {"start": 15.16, "duration": 6.039, "text": "base and the exponent let's try to"}, {"start": 18.88, "duration": 5.36, "text": "understand why this is challenging to"}, {"start": 21.199, "duration": 5.32, "text": "differentiate directly if we try to use"}, {"start": 24.24, "duration": 5.76, "text": "the power rule we'd run into trouble"}, {"start": 26.519, "duration": 5.76, "text": "because the exponent contains X if we"}, {"start": 30.0, "duration": 4.399, "text": "try to use the exponential rule we'd"}, {"start": 32.279, "duration": 5.761, "text": "have issues because the base also"}, {"start": 34.399, "duration": 7.32, "text": "contains X this is where logarithmic"}, {"start": 38.04, "duration": 6.32, "text": "differentiation comes in handy step one"}, {"start": 41.719, "duration": 7.721, "text": "let's take the natural logarithm of both"}, {"start": 44.36, "duration": 9.92, "text": "sides this gives us Ln of Y = Ln of X"}, {"start": 49.44, "duration": 7.799, "text": "to^ of 1 /x step two now we can use the"}, {"start": 54.28, "duration": 7.56, "text": "logarithm property that says Ln of a to"}, {"start": 57.239, "duration": 9.121, "text": "the power of Bal B * Ln of a this"}, {"start": 61.84, "duration": 9.319, "text": "simplifies our equation to Ln of Y = 1"}, {"start": 66.36, "duration": 7.2, "text": "/x * Ln of X step three now we can"}, {"start": 71.159, "duration": 7.241, "text": "differentiate both sides with respect to"}, {"start": 73.56, "duration": 7.239, "text": "X the left side becomes 1 / y * dydx"}, {"start": 78.4, "duration": 4.8, "text": "because we're using the chain rule for"}, {"start": 80.799, "duration": 5.201, "text": "the right side we need to use the"}, {"start": 83.2, "duration": 6.239, "text": "product rule since we have two functions"}, {"start": 86.0, "duration": 5.04, "text": "being multiplied together step four"}, {"start": 89.439, "duration": 3.96, "text": "let's hand handle the right side"}, {"start": 91.04, "duration": 5.439, "text": "carefully using the product rule"}, {"start": 93.399, "duration": 9.641, "text": "remember the product rule says that the"}, {"start": 96.479, "duration": 12.161, "text": "derivative of U * v = u * dvdx + V * DDX"}, {"start": 103.04, "duration": 8.719, "text": "in our case U is 1 /x and V is Ln of X"}, {"start": 108.64, "duration": 7.88, "text": "now let's evaluate each derivative the"}, {"start": 111.759, "duration": 8.561, "text": "derivative of 1 /x is NE 1 /x^2 and the"}, {"start": 116.52, "duration": 7.4, "text": "derivative of Ln of X is 1/x"}, {"start": 120.32, "duration": 4.96, "text": "substituting these in step five now we"}, {"start": 123.92, "duration": 5.24, "text": "can solve for"}, {"start": 125.28, "duration": 9.8, "text": "dydx multiply both sides by Y and"}, {"start": 129.16, "duration": 11.28, "text": "remember that y = x to^ of 1 /x this"}, {"start": 135.08, "duration": 11.28, "text": "gives us our final answer Dy DX = x^ of"}, {"start": 140.44, "duration": 7.799, "text": "1x * the 1 - Ln of X over x^2 let's"}, {"start": 146.36, "duration": 4.12, "text": "summarize why logarithmic"}, {"start": 148.239, "duration": 5.161, "text": "differentiation was helpful here"}, {"start": 150.48, "duration": 5.44, "text": "by taking the natural log first we were"}, {"start": 153.4, "duration": 5.479, "text": "able to bring down the exponent and"}, {"start": 155.92, "duration": 5.0, "text": "create a much simpler expression to"}, {"start": 158.879, "duration": 3.841, "text": "differentiate this technique is"}, {"start": 160.92, "duration": 3.959, "text": "particularly useful when dealing with"}, {"start": 162.72, "duration": 5.439, "text": "functions where variables appear in both"}, {"start": 164.879, "duration": 5.561, "text": "the base and exponent or with products"}, {"start": 168.159, "duration": 4.72, "text": "that would be difficult to differentiate"}, {"start": 170.44, "duration": 5.36, "text": "directly and that's how we use"}, {"start": 172.879, "duration": 8.521, "text": "logarithmic differentiation to find the"}, {"start": 175.8, "duration": 9.24, "text": "derivative of y = x to the power of 1 /x"}, {"start": 181.4, "duration": 3.64, "text": "thank you for watching"}], "title": "Master Logarithmic Differentiation: Solving [y = x^(1/x)] Step by Step", "description": "Question: Find the derivative of y = x^(1/x) using logarithmic differentiation.\n\n\ud83d\udcda In this comprehensive tutorial, we break down how to find the derivative of y = x^(1/x) using logarithmic differentiation. Perfect for calculus students and math enthusiasts!\nWhat you\u2019ll learn:\n\u2705 Why logarithmic differentiation is essential for complex exponential functions\n\u2705 Step-by-step solution process\n\u2705 Key logarithm properties and their applications\n\u2705 Product rule implementation\n\u2705 Common pitfalls and how to avoid them\n\u2b50 Don\u2019t forget to like and subscribe for more calculus tutorials! \u2b50\nHashtags:\n#Calculus #Mathematics #LogarithmicDifferentiation #MathTutorial #CalcTutorial #MathEducation #LearnMath #Derivatives", "lengthSeconds": 190, "width": 640, "height": 360}
{"video_id": "X_2WhHRnw4g", "transcript": [{"start": 0.32, "duration": 4.72, "text": "let's solve this challenging"}, {"start": 1.96, "duration": 6.2, "text": "differentiation problem step by step we"}, {"start": 5.04, "duration": 6.2, "text": "need to find the derivative of 10 Rays"}, {"start": 8.16, "duration": 5.28, "text": "to the natural log of sinx this is a"}, {"start": 11.24, "duration": 4.879, "text": "complex function that combines"}, {"start": 13.44, "duration": 5.48, "text": "exponential logarithmic and"}, {"start": 16.119, "duration": 6.24, "text": "trigonometric functions let's break down"}, {"start": 18.92, "duration": 4.84, "text": "how we'll solve it first we recognize"}, {"start": 22.359, "duration": 3.92, "text": "this follows the pattern of"}, {"start": 23.76, "duration": 6.839, "text": "differentiating a function of the form a"}, {"start": 26.279, "duration": 8.201, "text": "to the U where is 10 and U is natural"}, {"start": 30.599, "duration": 6.28, "text": "log of sinx four functions of this form"}, {"start": 34.48, "duration": 6.72, "text": "we know that the derivative equals a to"}, {"start": 36.879, "duration": 7.561, "text": "the U time the natural log of a * the"}, {"start": 41.2, "duration": 6.24, "text": "derivative of U with respect to X"}, {"start": 44.44, "duration": 6.2, "text": "applying this rule to our problem we get"}, {"start": 47.44, "duration": 6.72, "text": "10^ the natural log of sinx time the"}, {"start": 50.64, "duration": 7.16, "text": "natural log of 10 time the derivative of"}, {"start": 54.16, "duration": 6.28, "text": "natural log of sinx with respect to X"}, {"start": 57.8, "duration": 6.2, "text": "now we need to find the derivative"}, {"start": 60.44, "duration": 7.44, "text": "of natural log of sinx using the chain"}, {"start": 64.0, "duration": 6.92, "text": "rule this equals 1 / sin x * the"}, {"start": 67.88, "duration": 6.0, "text": "derivative of sinx substituting in the"}, {"start": 70.92, "duration": 6.04, "text": "derivative of sin x which is cosine X we"}, {"start": 73.88, "duration": 7.599, "text": "get 10 to the natural log of sin x *"}, {"start": 76.96, "duration": 7.28, "text": "natural log of 10 * cine X over sin x"}, {"start": 81.479, "duration": 6.721, "text": "finally we can simplify this further by"}, {"start": 84.24, "duration": 8.08, "text": "recognizing that cine X over sinx is the"}, {"start": 88.2, "duration": 6.599, "text": "definition of coent X therefore our"}, {"start": 92.32, "duration": 8.0, "text": "final answer is 10 the natural log of"}, {"start": 94.799, "duration": 8.6, "text": "sinx * natural log of 10 time Cent X"}, {"start": 100.32, "duration": 5.68, "text": "let's review the key steps we identified"}, {"start": 103.399, "duration": 4.961, "text": "the pattern of a to the U applied the"}, {"start": 106.0, "duration": 4.52, "text": "chain rule for logarithmic"}, {"start": 108.36, "duration": 3.759, "text": "differentiation and simplified our"}, {"start": 110.52, "duration": 6.16, "text": "expression using trigonometric"}, {"start": 112.119, "duration": 4.561, "text": "identities to reach our final answer"}], "title": "Logarithmic Differentiation (10^(ln\u2061sin\u2061\ud835\udc65)) Step by Step Guide", "description": "Unlock the secrets of logarithmic differentiation! In this video, we break down the process of differentiating complex expressions such as (10^(ln\u2061sin\u2061\ud835\udc65))  into simple, understandable steps. Whether you\u2019re a calculus student, teacher, or math enthusiast, you\u2019ll find our detailed walkthrough helpful in mastering this essential mathematical technique. Join us and boost your differentiation skills today!\n#Calculus #Differentiation #MathTutorial #LogarithmicDifferentiation #Trigonometry", "lengthSeconds": 115, "width": 640, "height": 360}
{"video_id": "dNde0Z9Bb3I", "transcript": [{"start": 0.32, "duration": 5.12, "text": "let's solve this calculus problem step"}, {"start": 2.56, "duration": 6.88, "text": "by step using the quotient rule we need"}, {"start": 5.44, "duration": 7.8, "text": "to find the derivative of cosine x/ x^2"}, {"start": 9.44, "duration": 6.96, "text": "+ sinx first let's recall the quotient"}, {"start": 13.24, "duration": 7.48, "text": "rule formula the derivative of U over V"}, {"start": 16.4, "duration": 8.039, "text": "equal V * the derivative of U minus U *"}, {"start": 20.72, "duration": 8.2, "text": "the derivative of V all over V ^2 let's"}, {"start": 24.439, "duration": 10.361, "text": "identify our u and v functions U equal"}, {"start": 28.92, "duration": 8.76, "text": "cine X and V = x^2 + sin x now let's"}, {"start": 34.8, "duration": 7.4, "text": "find the derivatives of u and v the"}, {"start": 37.68, "duration": 9.359, "text": "derivative of cine X is sinx and the"}, {"start": 42.2, "duration": 7.519, "text": "derivative of x^2 + sin x is 2x + cosine"}, {"start": 47.039, "duration": 6.68, "text": "X let's substitute these into the"}, {"start": 49.719, "duration": 10.921, "text": "quotient rule formula we get x^2 + sin x"}, {"start": 53.719, "duration": 12.001, "text": "* sinx - cinx * 2x + cosin x all over"}, {"start": 60.64, "duration": 9.24, "text": "x^2 + sin x^2 now let's expand the"}, {"start": 65.72, "duration": 10.24, "text": "numerator first let's multiply x^2 + sin"}, {"start": 69.88, "duration": 11.52, "text": "x by sin x this gives us x^2 sin x -"}, {"start": 75.96, "duration": 9.72, "text": "sin^2 x next we multiply cosine X by 2x"}, {"start": 81.4, "duration": 7.96, "text": "+ cosine X getting 2x cosine x + cosine"}, {"start": 85.68, "duration": 6.759, "text": "2 x let's factor out the negative sign"}, {"start": 89.36, "duration": 8.079, "text": "from the num numerator this gives us NE"}, {"start": 92.439, "duration": 10.04, "text": "* x^2 sin x + sin^2 x + 2x cosine x +"}, {"start": 97.439, "duration": 7.72, "text": "cosine 2 x all over x^2 + sin x^2"}, {"start": 102.479, "duration": 8.161, "text": "finally we can simplify this further by"}, {"start": 105.159, "duration": 10.32, "text": "using the identity sign 2 x + cosine 2 x"}, {"start": 110.64, "duration": 10.28, "text": "= 1 this gives us our final answer x^2"}, {"start": 115.479, "duration": 8.801, "text": "sin x + 2x cosine x + 1 all over x^2 +"}, {"start": 120.92, "duration": 6.4, "text": "sin x^2 and there we have it we've"}, {"start": 124.28, "duration": 6.119, "text": "successfully found the derivative using"}, {"start": 127.32, "duration": 6.04, "text": "the quotient rule the key steps were"}, {"start": 130.399, "duration": 5.401, "text": "identifying u and v finding their"}, {"start": 133.36, "duration": 7.72, "text": "derivatives applying the quotient rule"}, {"start": 135.8, "duration": 5.28, "text": "formula and then simplifying the result"}], "title": "Mastering Derivatives: Differentiating [(\ud835\udc84\ud835\udc90\ud835\udc94 \ud835\udc99)/(\ud835\udc99^\ud835\udfd0+\ud835\udc94\ud835\udc8a\ud835\udc8f \ud835\udc99)] Using the Quotient Rule", "description": "Unlock the secrets of calculus by learning how to differentiate the expression [(\ud835\udc84\ud835\udc90\ud835\udc94 \ud835\udc99)/(\ud835\udc99^\ud835\udfd0+\ud835\udc94\ud835\udc8a\ud835\udc8f \ud835\udc99)]  using the quotient rule. This video provides a step-by-step explanation, perfect for students and enthusiasts looking to deepen their understanding of derivatives involving trigonometric and polynomial functions.\n#Calculus #QuotientRule #Trigonometry #MathTutorial #Differentiation", "lengthSeconds": 140, "width": 640, "height": 360}
{"video_id": "jMat7MvIGdU", "transcript": [{"start": 0.28, "duration": 6.079, "text": "let's find the derivative of secant of a"}, {"start": 3.32, "duration": 5.96, "text": "x using first principles we'll break"}, {"start": 6.359, "duration": 6.841, "text": "this down step by step let's start by"}, {"start": 9.28, "duration": 9.64, "text": "defining our function let F ofx equal"}, {"start": 13.2, "duration": 10.319, "text": "secant of ax then F ofx + H equal secant"}, {"start": 18.92, "duration": 8.48, "text": "of a time x + H which equals secant of"}, {"start": 23.519, "duration": 6.961, "text": "X+ ah using the definition of the"}, {"start": 27.4, "duration": 5.52, "text": "derivative from first principles we know"}, {"start": 30.48, "duration": 8.16, "text": "that the derivative equals the limit as"}, {"start": 32.92, "duration": 9.04, "text": "H approaches Z of f of x + H - F ofx all"}, {"start": 38.64, "duration": 5.759, "text": "divided by H substituting our secant"}, {"start": 41.96, "duration": 7.4, "text": "function we get the limit as H"}, {"start": 44.399, "duration": 10.201, "text": "approaches zero of secant of ax plus a h"}, {"start": 49.36, "duration": 8.4, "text": "minus secant of ax all over H since cant"}, {"start": 54.6, "duration": 6.88, "text": "is 1 / cosine we can rewrite this as the"}, {"start": 57.76, "duration": 9.6, "text": "limit as H approaches Z of 1 / cosine of"}, {"start": 61.48, "duration": 8.639, "text": "a x + a h - 1 / cosine of a x all over H"}, {"start": 67.36, "duration": 5.64, "text": "to subtract these fractions we need a"}, {"start": 70.119, "duration": 7.761, "text": "common denominator this gives us cosine"}, {"start": 73.0, "duration": 8.32, "text": "of a xus cosine of a x + ah all over the"}, {"start": 77.88, "duration": 7.08, "text": "product of cosine of ax and cosine of ax"}, {"start": 81.32, "duration": 6.2, "text": "+ ah now we can use the cosine"}, {"start": 84.96, "duration": 6.839, "text": "difference formula this states that"}, {"start": 87.52, "duration": 8.639, "text": "cosine a minus cosine Bal -2 * s of a"}, {"start": 91.799, "duration": 6.36, "text": "plus b / 2 * s of a minus b/2"}, {"start": 96.159, "duration": 7.201, "text": "simplifying the Expressions inside the"}, {"start": 98.159, "duration": 9.92, "text": "sign functions we get 2 sin of a x + a h"}, {"start": 103.36, "duration": 7.64, "text": "over 2 * s of a h over 2 now we can"}, {"start": 108.079, "duration": 6.521, "text": "evaluate the limits separately as H"}, {"start": 111.0, "duration": 8.399, "text": "approaches zero the sign of a X Plus 8"}, {"start": 114.6, "duration": 9.04, "text": "h/2 approaches s of ax s of H h/2 over a"}, {"start": 119.399, "duration": 7.441, "text": "h two approaches 1 and cosine of x plus"}, {"start": 123.64, "duration": 7.959, "text": "h approaches cosine of a x this gives us"}, {"start": 126.84, "duration": 7.759, "text": "a Time sign of ax over cine of ax^2"}, {"start": 131.599, "duration": 7.36, "text": "which we can rewrite as a tangent of ax"}, {"start": 134.599, "duration": 9.561, "text": "* secant of ax therefore the derivative"}, {"start": 138.959, "duration": 9.241, "text": "of secant of ax equals a tangent of ax *"}, {"start": 144.16, "duration": 7.64, "text": "secant of a x and that's how we find the"}, {"start": 148.2, "duration": 6.6, "text": "derivative of secant of ax using first"}, {"start": 151.8, "duration": 3.0, "text": "principles"}], "title": "Understanding the Derivative of sec ax  Using First Principles", "description": "Master calculus fundamentals by deriving the exponential function sec(ax) from first principles. Use our step-by-step explanations and examples to deepen your understanding of derivatives and exponential functions.\n\n#Calculus #Derivatives #ExponentialFunctions #MathTutorial #FirstPrinciples #x^3+2x", "lengthSeconds": 155, "width": 640, "height": 360}
{"video_id": "RAktmUSFS1E", "transcript": [{"start": 0.44, "duration": 5.439, "text": "welcome today we'll learn how to find"}, {"start": 3.0, "duration": 5.76, "text": "the derivative of a complex exponential"}, {"start": 5.879, "duration": 5.201, "text": "function using logarithmic"}, {"start": 8.76, "duration": 6.36, "text": "differentiation our problem is to find"}, {"start": 11.08, "duration": 6.64, "text": "the derivative of y = x raed to the"}, {"start": 15.12, "duration": 4.72, "text": "cosine of x this is a challenging"}, {"start": 17.72, "duration": 4.44, "text": "function to differentiate directly"}, {"start": 19.84, "duration": 5.0, "text": "because we have a variable base and a"}, {"start": 22.16, "duration": 6.08, "text": "trigonometric function in the exponent"}, {"start": 24.84, "duration": 5.439, "text": "the strategy we'll use is logarithmic"}, {"start": 28.24, "duration": 4.359, "text": "differentiation which involves taking"}, {"start": 30.279, "duration": 5.481, "text": "the natural logarithm of both sides"}, {"start": 32.599, "duration": 5.96, "text": "first let's solve this step by step"}, {"start": 35.76, "duration": 5.639, "text": "First We Take the natural logarithm of"}, {"start": 38.559, "duration": 5.201, "text": "both sides using the logarithm property"}, {"start": 41.399, "duration": 4.32, "text": "that the natural log of a power equals"}, {"start": 43.76, "duration": 8.04, "text": "the exponent times the natural log of"}, {"start": 45.719, "duration": 9.281, "text": "the base we get Ln y = cine x * Ln X now"}, {"start": 51.8, "duration": 7.12, "text": "comes the key step we differentiate both"}, {"start": 55.0, "duration": 9.159, "text": "sides with respect to X on the left side"}, {"start": 58.92, "duration": 7.92, "text": "we get d by DX of Ln y on the right side"}, {"start": 64.159, "duration": 6.121, "text": "we need to use the product rule since we"}, {"start": 66.84, "duration": 7.08, "text": "have the product of two functions let's"}, {"start": 70.28, "duration": 7.32, "text": "handle the left side first remember that"}, {"start": 73.92, "duration": 6.8, "text": "the derivative of Ln y with respect to x"}, {"start": 77.6, "duration": 6.8, "text": "= 1/ y *"}, {"start": 80.72, "duration": 6.12, "text": "dydx this comes from the chain Rule now"}, {"start": 84.4, "duration": 5.039, "text": "let's evaluate the derivatives on the"}, {"start": 86.84, "duration": 6.68, "text": "right side we know that the derivative"}, {"start": 89.439, "duration": 8.761, "text": "of L l x is 1 /x and the derivative of"}, {"start": 93.52, "duration": 11.0, "text": "cosine X is sinx substituting these in"}, {"start": 98.2, "duration": 11.559, "text": "we get 1 / y * Dy DX = cosin x/ x + Ln x"}, {"start": 104.52, "duration": 9.4, "text": "* sinx to solve for dydx we multiply"}, {"start": 109.759, "duration": 7.96, "text": "both sides by y then we substitute back"}, {"start": 113.92, "duration": 6.119, "text": "y = x cosine X to get our final answer"}, {"start": 117.719, "duration": 5.04, "text": "and there we have it we've found the the"}, {"start": 120.039, "duration": 4.841, "text": "derivative using logarithmic"}, {"start": 122.759, "duration": 4.04, "text": "differentiation this technique is"}, {"start": 124.88, "duration": 3.879, "text": "particularly useful when dealing with"}, {"start": 126.799, "duration": 5.041, "text": "functions that have variables in both"}, {"start": 128.759, "duration": 6.12, "text": "the base and exponent the key steps we"}, {"start": 131.84, "duration": 5.8, "text": "used were one taking the natural"}, {"start": 134.879, "duration": 5.561, "text": "logarithm of both sides two using"}, {"start": 137.64, "duration": 5.56, "text": "logarithm properties three applying the"}, {"start": 140.44, "duration": 5.519, "text": "product rule for using the chain Rule"}, {"start": 143.2, "duration": 6.24, "text": "and finally substituting back the"}, {"start": 145.959, "duration": 3.481, "text": "original function"}], "title": "Mastering Logarithmic Differentiation: Solving [y = x^(cos x)] Step by Step", "description": "Question: Find the derivative of y = x^(cos x) using logarithmic differentiation.\n\n\ud83d\udd0d In this comprehensive tutorial, we break down how to find the derivative of y = x^(cos x) using logarithmic differentiation. Perfect for calculus students and math enthusiasts!\nKey topics covered:\n\u2022 Logarithmic differentiation technique\n\u2022 Natural logarithm properties\n\u2022 Product rule application\n\u2022 Chain rule implementation\n\u2022 Step-by-step solution process\n\ud83d\udca1 Learning Outcomes:\nMaster logarithmic differentiation\nHandle complex exponential functions\nApply differentiation rules effectively\n#Calculus #Mathematics #LogarithmicDifferentiation #MathTutorial #CalcTutorial #MathEduc", "lengthSeconds": 152, "width": 640, "height": 360}
{"video_id": "LCCLyf57s3w", "transcript": [{"start": 0.359, "duration": 6.361, "text": "let's find the derivative of tangent 4X"}, {"start": 4.56, "duration": 5.079, "text": "using the first principle's definition"}, {"start": 6.72, "duration": 8.119, "text": "of derivatives let's start by defining"}, {"start": 9.639, "duration": 10.921, "text": "our function let F ofx = tangent 4X"}, {"start": 14.839, "duration": 9.961, "text": "therefore f of x + H = tangent of 4 * x"}, {"start": 20.56, "duration": 7.6, "text": "+ H now let's recall the definition of a"}, {"start": 24.8, "duration": 6.319, "text": "derivative the derivative of f ofx is"}, {"start": 28.16, "duration": 8.8, "text": "equal to the Limit as h approaches Z of"}, {"start": 31.119, "duration": 7.801, "text": "f ofx + H - F ofx all divid H let's"}, {"start": 36.96, "duration": 5.079, "text": "substitute our function into this"}, {"start": 38.92, "duration": 5.84, "text": "definition the derivative of tangent 4X"}, {"start": 42.039, "duration": 8.241, "text": "equals the limit as H approaches Z of"}, {"start": 44.76, "duration": 9.439, "text": "tangent of 4 * x + H - Tang 4X all"}, {"start": 50.28, "duration": 7.16, "text": "divided H now let's rewrite tangent in"}, {"start": 54.199, "duration": 8.16, "text": "terms of s over cosine this gives us the"}, {"start": 57.44, "duration": 11.039, "text": "limit as H approaches Z of s 4x + h/"}, {"start": 62.359, "duration": 9.4, "text": "cosine of 4x + H - sin 4X cosine 4X all"}, {"start": 68.479, "duration": 6.361, "text": "divided by H to subtract these fractions"}, {"start": 71.759, "duration": 6.121, "text": "we need a common denominator multiplying"}, {"start": 74.84, "duration": 5.959, "text": "both numerator and denominator of each"}, {"start": 77.88, "duration": 5.64, "text": "fraction by the denominator of the other"}, {"start": 80.799, "duration": 5.801, "text": "gives us this expression the numerator"}, {"start": 83.52, "duration": 6.919, "text": "is the S of the difference of angles"}, {"start": 86.6, "duration": 6.68, "text": "formula this simplifies to S of 48 H in"}, {"start": 90.439, "duration": 5.72, "text": "the numerator let's factor out the four"}, {"start": 93.28, "duration": 6.159, "text": "from the sign term this gives us 4 * the"}, {"start": 96.159, "duration": 6.161, "text": "limit as H approaches 0 of sin 4 H over"}, {"start": 99.439, "duration": 5.561, "text": "4 H divided by the product of the"}, {"start": 102.32, "duration": 6.68, "text": "cosiness now we can split this into two"}, {"start": 105.0, "duration": 7.92, "text": "limits the limit of sin 4 H over 4 H as"}, {"start": 109.0, "duration": 9.439, "text": "H approaches 0 = 1 and the limit of cine"}, {"start": 112.92, "duration": 9.64, "text": "4x + H as H approaches 0 = cosine 4X"}, {"start": 118.439, "duration": 8.64, "text": "this simplifies to 4 over cine 4X which"}, {"start": 122.56, "duration": 7.319, "text": "equal 4 secant 4X and that's our final"}, {"start": 127.079, "duration": 5.761, "text": "answer to verify this result we could"}, {"start": 129.879, "duration": 5.0, "text": "graph tangent 4X and it's derivative to"}, {"start": 132.84, "duration": 5.039, "text": "see that the derivative is indeed"}, {"start": 134.879, "duration": 8.241, "text": "steepest at the same points where"}, {"start": 137.879, "duration": 5.241, "text": "tangent 4X has its steepest slopes"}], "title": "Understanding the Derivative of tan4x  Using First Principles", "description": "Master the fundamentals of calculus by deriving the exponential function tan4x from first principles. Our step-by-step explanations and examples will enhance your grasp of derivatives and exponential functions.\n#Calculus #Derivatives #ExponentialFunctions #MathTutorial #FirstPrinciples #tan3x", "lengthSeconds": 143, "width": 640, "height": 360}
{"video_id": "zK3bceHvqI8", "transcript": [{"start": 0.28, "duration": 5.32, "text": "let's solve this calculus problem step"}, {"start": 2.52, "duration": 6.84, "text": "by step we need to find the derivative"}, {"start": 5.6, "duration": 6.64, "text": "of inverse tangent of square root of x"}, {"start": 9.36, "duration": 5.479, "text": "let's break this down into steps first"}, {"start": 12.24, "duration": 5.76, "text": "we'll use the chain rule since we have a"}, {"start": 14.839, "duration": 6.321, "text": "composite function remember the"}, {"start": 18.0, "duration": 7.519, "text": "derivative of inverse tangent of U is 1"}, {"start": 21.16, "duration": 8.32, "text": "1 + u^2 time the derivative of U in our"}, {"start": 25.519, "duration": 6.24, "text": "case U equal Square < TK of X so let's"}, {"start": 29.48, "duration": 4.52, "text": "substitute this into our chain rule"}, {"start": 31.759, "duration": 5.32, "text": "formula now we need to find the"}, {"start": 34.0, "duration": 6.079, "text": "derivative of square < TK of X using the"}, {"start": 37.079, "duration": 5.721, "text": "power rule with a fractional exponent we"}, {"start": 40.079, "duration": 7.441, "text": "know that the derivative of x to the 1/2"}, {"start": 42.8, "duration": 8.12, "text": "power is 12 * x -2 power which"}, {"start": 47.52, "duration": 6.16, "text": "simplifies to 1 / 2 < tkx let's"}, {"start": 50.92, "duration": 6.0, "text": "substitute this back into our equation"}, {"start": 53.68, "duration": 8.32, "text": "notice that in the denominator Square <"}, {"start": 56.92, "duration": 7.8, "text": "TK of X2 simplifies to just X finally we"}, {"start": 62.0, "duration": 6.52, "text": "can multiply these fractions together"}, {"start": 64.72, "duration": 6.96, "text": "remember when multiplying fractions we"}, {"start": 68.52, "duration": 6.16, "text": "multiply the numerators together and"}, {"start": 71.68, "duration": 7.439, "text": "multiply the denominators together this"}, {"start": 74.68, "duration": 8.84, "text": "gives us our Final Answer 1 / 2 < tkx *"}, {"start": 79.119, "duration": 6.401, "text": "1 + x let's verify this makes sense our"}, {"start": 83.52, "duration": 4.72, "text": "answer is a fraction that approaches"}, {"start": 85.52, "duration": 5.0, "text": "zero as X gets very large which is what"}, {"start": 88.24, "duration": 5.12, "text": "we would expect for the derivative of an"}, {"start": 90.52, "duration": 5.8, "text": "inverse tangent function since inverse"}, {"start": 93.36, "duration": 5.56, "text": "tangent approaches Pi / 2 as X"}, {"start": 96.32, "duration": 5.4, "text": "approaches infinity and that's how we"}, {"start": 98.92, "duration": 6.4, "text": "find the derivative of inverse tangent"}, {"start": 101.72, "duration": 7.24, "text": "of square < TK of X using the chain Rule"}, {"start": 105.32, "duration": 3.64, "text": "and power rule"}], "title": "Mastering Derivatives: Understanding the Derivative of [\ud835\udc61\ud835\udc4e\ud835\udc5b^(\u22121) \u221a\ud835\udc65]", "description": "Dive into the world of calculus as we explore the step-by-step process of finding the derivative of  [\ud835\udc61\ud835\udc4e\ud835\udc5b^(\u22121) \u221a\ud835\udc65]. Learn how to apply the chain rule effectively and simplify complex expressions with ease. Perfect for students looking to strengthen their understanding of mathematics concepts.\n#Calculus #MathTutorial #Derivatives #Trigonometry #MathHelp #Education #STEM", "lengthSeconds": 112, "width": 640, "height": 360}
{"video_id": "6FQuPGrTehg", "transcript": [{"start": 0.359, "duration": 5.201, "text": "welcome today we'll solve a challenging"}, {"start": 3.199, "duration": 4.801, "text": "derivative problem using the quotient"}, {"start": 5.56, "duration": 4.959, "text": "rule let's start by looking at our"}, {"start": 8.0, "duration": 5.08, "text": "function to solve this we'll use the"}, {"start": 10.519, "duration": 6.52, "text": "quotient rule which states that for a"}, {"start": 13.08, "duration": 7.72, "text": "function f ofx = u over V the derivative"}, {"start": 17.039, "duration": 7.201, "text": "is V * the derivative of U minus U * the"}, {"start": 20.8, "duration": 9.2, "text": "derivative of V all over V ^2 let's"}, {"start": 24.24, "duration": 9.52, "text": "identify our u and v terms u = e to x +"}, {"start": 30.0, "duration": 7.2, "text": "natural log of x and V equal log base of"}, {"start": 33.76, "duration": 6.36, "text": "X now let's find the derivatives of u"}, {"start": 37.2, "duration": 5.76, "text": "and v separately the derivative of U is"}, {"start": 40.12, "duration": 6.16, "text": "the derivative of e to the X which is e"}, {"start": 42.96, "duration": 6.599, "text": "to X plus the derivative of natural log"}, {"start": 46.28, "duration": 6.56, "text": "of x which is 1 /x for the derivative of"}, {"start": 49.559, "duration": 7.121, "text": "V we need to use the chain Rule and the"}, {"start": 52.84, "duration": 7.399, "text": "fact that log Bas of X = natural log of"}, {"start": 56.68, "duration": 6.039, "text": "x over natural log of a now let let's"}, {"start": 60.239, "duration": 6.361, "text": "substitute these into the quotient rule"}, {"start": 62.719, "duration": 8.481, "text": "formula in the numerator we multiply log"}, {"start": 66.6, "duration": 8.24, "text": "base a of x * the derivative of U minus"}, {"start": 71.2, "duration": 7.04, "text": "the quantity e to X Plus natural log of"}, {"start": 74.84, "duration": 7.56, "text": "x * the derivative of V the denominator"}, {"start": 78.24, "duration": 7.72, "text": "is log base uh of x^2 and this gives us"}, {"start": 82.4, "duration": 6.359, "text": "our final answer let's verify that this"}, {"start": 85.96, "duration": 9.76, "text": "matches our Target result the numerator"}, {"start": 88.759, "duration": 12.081, "text": "has log Basa of x * e x + 1 /x - e x +"}, {"start": 95.72, "duration": 8.679, "text": "natural log of x * log base a of e overx"}, {"start": 100.84, "duration": 5.72, "text": "all over log base a of x^2 let's"}, {"start": 104.399, "duration": 5.601, "text": "summarize the key steps we used to solve"}, {"start": 106.56, "duration": 6.68, "text": "this problem first we identified the"}, {"start": 110.0, "duration": 5.64, "text": "quotient rule pattern second we found"}, {"start": 113.24, "duration": 6.72, "text": "the derivatives of both the numerator"}, {"start": 115.64, "duration": 6.439, "text": "and denominator separately finally we"}, {"start": 119.96, "duration": 5.199, "text": "substituted everything into the quotient"}, {"start": 122.079, "duration": 6.481, "text": "rule formula and simplified this"}, {"start": 125.159, "duration": 7.96, "text": "technique can be applied to many complex"}, {"start": 128.56, "duration": 4.559, "text": "derivative problems involving quotients"}], "title": "Mastering Differentiation: Solving Complex Quotient Derivatives [((\ud835\udc52^\ud835\udc65+ln\u2061\ud835\udc65)/log_\ud835\udc4e\u2061\ud835\udc65 )]", "description": "Unlock the secrets of differentiation with our detailed walkthrough on solving complex quotient derivatives. In this session, we\u2019ll tackle a step-by-step solution to finding the derivative of [((\ud835\udc52^\ud835\udc65+ln\u2061\ud835\udc65)/log_\ud835\udc4e\u2061\ud835\udc65 )]  using the quotient rule. Perfect for students and math enthusiasts looking to deepen their understanding of calculus concepts.\n#Differentiation #Calculus #MathTutorial #QuotientRule #Derivatives #AdvancedMath", "lengthSeconds": 136, "width": 640, "height": 360}
{"video_id": "GjQxozUNPOY", "transcript": [{"start": 0.28, "duration": 6.32, "text": "let's solve the step by step to find the"}, {"start": 2.639, "duration": 5.641, "text": "derivative of log base 10 of 3x first"}, {"start": 6.6, "duration": 4.079, "text": "let's write out what we're trying to"}, {"start": 8.28, "duration": 6.319, "text": "find we want to find the derivative with"}, {"start": 10.679, "duration": 6.401, "text": "respect to X of log base 10 of 3x the"}, {"start": 14.599, "duration": 5.44, "text": "first key step is to use the change of"}, {"start": 17.08, "duration": 6.039, "text": "Base formula this allows us to rewrite"}, {"start": 20.039, "duration": 5.32, "text": "our log base 10 in terms of the natural"}, {"start": 23.119, "duration": 4.601, "text": "logarithm which is easier to"}, {"start": 25.359, "duration": 5.92, "text": "differentiate the formula states that"}, {"start": 27.72, "duration": 8.32, "text": "log base 10 of 3x equals the natural log"}, {"start": 31.279, "duration": 7.401, "text": "of 3x * log base 10 of e next we can use"}, {"start": 36.04, "duration": 5.12, "text": "the fact that log base 10 of e is a"}, {"start": 38.68, "duration": 5.24, "text": "constant when we differentiate a"}, {"start": 41.16, "duration": 5.16, "text": "constant times a function we can factor"}, {"start": 43.92, "duration": 5.479, "text": "out the constant now we need to"}, {"start": 46.32, "duration": 5.28, "text": "differentiate the natural log of 3x"}, {"start": 49.399, "duration": 4.761, "text": "we'll use the chain rule here the"}, {"start": 51.6, "duration": 6.16, "text": "derivative of the natural log is 1/ its"}, {"start": 54.16, "duration": 7.96, "text": "input times the derivative of the input"}, {"start": 57.76, "duration": 10.6, "text": "the derivative of 3x is simply 3 so we"}, {"start": 62.12, "duration": 8.88, "text": "have log base 10 of e * 1 3x * 3 finally"}, {"start": 68.36, "duration": 5.28, "text": "we can simplify this expression the"}, {"start": 71.0, "duration": 5.68, "text": "threes cancel out leaving us with log"}, {"start": 73.64, "duration": 5.519, "text": "base 10 of e overx let's review what we"}, {"start": 76.68, "duration": 6.24, "text": "did we started with the derivative of"}, {"start": 79.159, "duration": 6.201, "text": "log base T of 3x we used the change of"}, {"start": 82.92, "duration": 5.159, "text": "Base formula to rewrite it in terms of"}, {"start": 85.36, "duration": 6.52, "text": "natural log we factored out the constant"}, {"start": 88.079, "duration": 6.801, "text": "log base 10 of E we use the chain rule"}, {"start": 91.88, "duration": 6.44, "text": "to differentiate the natural log and"}, {"start": 94.88, "duration": 7.559, "text": "finally we simplify to get our answer"}, {"start": 98.32, "duration": 4.119, "text": "log base 10 of e/x"}], "title": "How to Differentiate log_10\u2061(3\ud835\udc65)  Step by Step", "description": "Question: Find The Derivative of [ log_10\u2061(3\ud835\udc65)]\n\nLearn how to differentiate the function  log_10\u2061(3\ud835\udc65)   using the change of base formula and chain rule. This step-by-step guide simplifies the process and helps you understand the underlying concepts. Perfect for math students wanting to improve their calculus skills!\n#Calculus #Differentiation #MathTutorial #Logarithms #ChainRule", "lengthSeconds": 104, "width": 640, "height": 360}
{"video_id": "IGYkttzlqq8", "transcript": [{"start": 0.28, "duration": 4.96, "text": "let's solve this complex derivative"}, {"start": 3.0, "duration": 5.0, "text": "problem involving inverse tangent"}, {"start": 5.24, "duration": 5.319, "text": "functions our goal is to find the"}, {"start": 8.0, "duration": 5.639, "text": "derivative of this expression let's"}, {"start": 10.559, "duration": 6.401, "text": "start by letting y equal our expression"}, {"start": 13.639, "duration": 6.161, "text": "we'll Begin by simplifying what's inside"}, {"start": 16.96, "duration": 5.0, "text": "the inverse tangent function our first"}, {"start": 19.8, "duration": 3.88, "text": "step is to factor be out of both the"}, {"start": 21.96, "duration": 4.319, "text": "numerator and"}, {"start": 23.68, "duration": 5.4, "text": "denominator notice how we can rewrite"}, {"start": 26.279, "duration": 4.24, "text": "the numerator as B * the quantity over"}, {"start": 29.08, "duration": 4.56, "text": "B+ X"}, {"start": 30.519, "duration": 7.841, "text": "and the denominator as B * the quantity"}, {"start": 33.64, "duration": 7.52, "text": "1 - over B * X since B appears in both"}, {"start": 38.36, "duration": 6.719, "text": "numerator and denominator we can cancel"}, {"start": 41.16, "duration": 6.76, "text": "it out this gives us a simpler form now"}, {"start": 45.079, "duration": 6.12, "text": "here's where we use a key Insight this"}, {"start": 47.92, "duration": 5.6, "text": "form matches a special inverse tangent"}, {"start": 51.199, "duration": 6.401, "text": "identity when we have an expression of"}, {"start": 53.52, "duration": 7.92, "text": "the form tangent inverse of U + V over"}, {"start": 57.6, "duration": 7.199, "text": "1us UV it equals tangent inverse of U"}, {"start": 61.44, "duration": 6.88, "text": "plus tangent inverse of V applying this"}, {"start": 64.799, "duration": 6.32, "text": "identity to our problem we can split"}, {"start": 68.32, "duration": 5.52, "text": "this into the sum of two inverse tangent"}, {"start": 71.119, "duration": 6.0, "text": "terms the first term will be tangent"}, {"start": 73.84, "duration": 6.8, "text": "inverse of A over B and the second term"}, {"start": 77.119, "duration": 6.161, "text": "will be tangent inverse of X now we can"}, {"start": 80.64, "duration": 5.32, "text": "differentiate the derivative of y with"}, {"start": 83.28, "duration": 6.04, "text": "respect 2x equals the derivative of"}, {"start": 85.96, "duration": 6.76, "text": "tangent inverse of over B plus the"}, {"start": 89.32, "duration": 6.32, "text": "derivative of tangent inverse of X let's"}, {"start": 92.72, "duration": 7.84, "text": "evaluate each term the first term is the"}, {"start": 95.64, "duration": 7.96, "text": "derivative of a constant so that's zero"}, {"start": 100.56, "duration": 7.68, "text": "for the second term we know the"}, {"start": 103.6, "duration": 10.199, "text": "derivative of inverse tangent of X is 1"}, {"start": 108.24, "duration": 9.68, "text": "1 + x^2 therefore our final answer is 1"}, {"start": 113.799, "duration": 6.841, "text": "over 1 + x^2 this elegant result shows"}, {"start": 117.92, "duration": 5.28, "text": "how a complex expression can be simplifi"}, {"start": 120.64, "duration": 5.68, "text": "to a basic derivative form through"}, {"start": 123.2, "duration": 8.0, "text": "careful algebraic manipulation and the"}, {"start": 126.32, "duration": 4.88, "text": "use of trigonometric identities"}], "title": "How to Differentiate [\ud835\udc61\ud835\udc4e\ud835\udc5b^(\u22121)\u2061((\ud835\udc4e+\ud835\udc4f\ud835\udc65)/(\ud835\udc4f\u2212\ud835\udc4e\ud835\udc65))]  Step-by-Step", "description": "Question:\nFind the derivative of y= [\ud835\udc61\ud835\udc4e\ud835\udc5b^(\u22121)\u2061((\ud835\udc4e+\ud835\udc4f\ud835\udc65)/(\ud835\udc4f\u2212\ud835\udc4e\ud835\udc65))] .\n\nUnlock the secrets of calculus with this clear and concise tutorial on differentiating inverse trigonometric functions. Follow along as we break down the differentiation of [\ud835\udc61\ud835\udc4e\ud835\udc5b^(\u22121)\u2061((\ud835\udc4e+\ud835\udc4f\ud835\udc65)/(\ud835\udc4f\u2212\ud835\udc4e\ud835\udc65))] . with step-by-step guidance. Perfect for students and math enthusiasts looking to deepen their understanding of trigonometric identities and differentiation techniques. Don\u2019t forget to like, share, and subscribe for more math insights!\n#Calculus #Trigonometry #MathTutorial #Differentiation #InverseTrigonometricFunctions #LearnMath", "lengthSeconds": 134, "width": 640, "height": 360}
{"video_id": "bbhPoOeErWk", "transcript": [{"start": 0.28, "duration": 6.519, "text": "let's work through finding the"}, {"start": 1.64, "duration": 8.48, "text": "derivative of x^2 * e to x * natural log"}, {"start": 6.799, "duration": 5.561, "text": "of x this is a complex derivative that"}, {"start": 10.12, "duration": 4.439, "text": "will require multiple applications of"}, {"start": 12.36, "duration": 5.16, "text": "the product rule let's recall the"}, {"start": 14.559, "duration": 7.001, "text": "product rule first the derivative of U *"}, {"start": 17.52, "duration": 7.839, "text": "V equal U * the derivative of V + V *"}, {"start": 21.56, "duration": 6.039, "text": "the derivative of U for our function we"}, {"start": 25.359, "duration": 6.641, "text": "have three parts being multiplied"}, {"start": 27.599, "duration": 6.441, "text": "together x^2 e to X and natural log of x"}, {"start": 32.0, "duration": 5.04, "text": "let's break this down into two parts"}, {"start": 34.04, "duration": 6.839, "text": "first using the product rule first on"}, {"start": 37.04, "duration": 7.4, "text": "x^2 and G ofx we get x^2 * the"}, {"start": 40.879, "duration": 6.84, "text": "derivative of G of X Plus G ofx * the"}, {"start": 44.44, "duration": 6.36, "text": "derivative of x^2 now we need to find"}, {"start": 47.719, "duration": 6.401, "text": "the derivative of G of X which is e to x"}, {"start": 50.8, "duration": 6.559, "text": "* natural log of x using the product"}, {"start": 54.12, "duration": 6.599, "text": "rule again we get e to X time the"}, {"start": 57.359, "duration": 7.12, "text": "derivative of natural log of x plus"}, {"start": 60.719, "duration": 6.961, "text": "natural log of x * the derivative of e"}, {"start": 64.479, "duration": 5.601, "text": "to X let's substitute in the known"}, {"start": 67.68, "duration": 6.6, "text": "derivatives the derivative of natural"}, {"start": 70.08, "duration": 8.8, "text": "log of x is 1 /x and the derivative of e"}, {"start": 74.28, "duration": 7.76, "text": "to X is e to the X the derivative of x^2"}, {"start": 78.88, "duration": 6.76, "text": "is 2x substituting these back into our"}, {"start": 82.04, "duration": 8.439, "text": "equation gives us x\u00b2 * the quantity e to"}, {"start": 85.64, "duration": 10.36, "text": "X over x + natural log of x * e to X"}, {"start": 90.479, "duration": 8.801, "text": "+ 2x * e to x * natural log of x now"}, {"start": 96.0, "duration": 7.52, "text": "let's distribute and simplify the first"}, {"start": 99.28, "duration": 10.64, "text": "term becomes x e to x + x^2 * natural"}, {"start": 103.52, "duration": 9.76, "text": "log of x * e to x + 2x * e to x *"}, {"start": 109.92, "duration": 6.4, "text": "natural log of x finally we can factor"}, {"start": 113.28, "duration": 7.799, "text": "out x e to the X giving us our final"}, {"start": 116.32, "duration": 8.439, "text": "answer x e to x * the quantity 1 1 + x"}, {"start": 121.079, "duration": 5.921, "text": "natural log of x + 2 natural log of x"}, {"start": 124.759, "duration": 5.0, "text": "this problem demonstrated how to handle"}, {"start": 127.0, "duration": 5.4, "text": "a complex derivative using multiple"}, {"start": 129.759, "duration": 4.641, "text": "applications of the product rule along"}, {"start": 132.4, "duration": 4.76, "text": "with knowing our basic derivative rules"}, {"start": 134.4, "duration": 5.44, "text": "for exponential and logarithmic"}, {"start": 137.16, "duration": 4.6, "text": "functions the key was breaking down the"}, {"start": 139.84, "duration": 6.72, "text": "problem into manageable pieces and"}, {"start": 141.76, "duration": 4.8, "text": "applying the rules step by step"}], "title": "Finding the Derivative of (\ud835\udc65^2 \ud835\udc52^\ud835\udc65  ln\u2061\ud835\udc65)  Using the Product Rule | Step-by-Step Explanation.", "description": "In this video, learn to differentiate  (\ud835\udc65^2 \ud835\udc52^\ud835\udc65  ln\u2061\ud835\udc65) using the Product Rule in calculus.\n\nWhat You'll Learn:  \n- Definition of the Product Rule  \n- Step-by-step differentiation of  (\ud835\udc65^2 \ud835\udc52^\ud835\udc65  ln\u2061\ud835\udc65)\n- Formula breakdown and example problem-solving  \n\nThis example is perfect for calculus beginners or those reviewing the Product Rule!\n\ud83d\udca1 Don't forget to LIKE, SHARE, and SUBSCRIBE for more math tutorials!\n#Calculus #ProductRule #Differentiation #Derivative #MathTutorial #LearnMath #MathEducation #MathHelp #hschighermath #hsc", "lengthSeconds": 147, "width": 640, "height": 360}
{"video_id": "I7iGhHVkFYU", "transcript": [{"start": 0.24, "duration": 5.8, "text": "let's find the derivative of sin 2x"}, {"start": 3.24, "duration": 5.2, "text": "using first principles this means we'll"}, {"start": 6.04, "duration": 5.12, "text": "use the fundamental definition of the"}, {"start": 8.44, "duration": 6.52, "text": "derivative to find the result let's"}, {"start": 11.16, "duration": 11.68, "text": "start by defining our function let F ofx"}, {"start": 14.96, "duration": 11.04, "text": "= sin 2x then f of x + H = sin of 2 *"}, {"start": 22.84, "duration": 5.64, "text": "the quantity x + H the derivative is"}, {"start": 26.0, "duration": 6.16, "text": "defined as the limit as H approaches"}, {"start": 28.48, "duration": 9.12, "text": "zero of f ofx + H -"}, {"start": 32.16, "duration": 8.239, "text": "FX all divid H now let's substitute our"}, {"start": 37.6, "duration": 6.36, "text": "function into the derivative definition"}, {"start": 40.399, "duration": 11.281, "text": "we get the limit as H approaches Z of s"}, {"start": 43.96, "duration": 9.68, "text": "of 2 * thean x + h- sin 2x all divid H"}, {"start": 51.68, "duration": 3.679, "text": "to solve this we'll use the"}, {"start": 53.64, "duration": 4.719, "text": "trigonometric identity for the"}, {"start": 55.359, "duration": 7.121, "text": "difference of signs the identity states"}, {"start": 58.359, "duration": 8.961, "text": "that S A minus sign b = 2 * cosine of a"}, {"start": 62.48, "duration": 7.639, "text": "plus b/ 2 * s of aus b/ 2 let's apply"}, {"start": 67.32, "duration": 6.439, "text": "this identity to our expression let's"}, {"start": 70.119, "duration": 12.081, "text": "simplify what's inside the cosine and S"}, {"start": 73.759, "duration": 16.281, "text": "2x + H + 2x = 4x + 2 h / 2 which = 2 x +"}, {"start": 82.2, "duration": 11.4, "text": "H and 2 x + H - 2x = 2 h / 2 which equal"}, {"start": 90.04, "duration": 6.84, "text": "H now we can split this into two parts"}, {"start": 93.6, "duration": 8.04, "text": "we have 2 * the limit as H approaches Z"}, {"start": 96.88, "duration": 9.279, "text": "of cosine of 2x + H times the limit as H"}, {"start": 101.64, "duration": 9.0, "text": "approaches Z of sin H H let's evaluate"}, {"start": 106.159, "duration": 8.56, "text": "each limit the limit of cosine of 2x + H"}, {"start": 110.64, "duration": 7.6, "text": "as H approaches 0 is just cosine of 2x"}, {"start": 114.719, "duration": 7.561, "text": "and the limmit of sign H H as H"}, {"start": 118.24, "duration": 7.239, "text": "approaches 0al 1 this is a fundamental"}, {"start": 122.28, "duration": 8.759, "text": "limit in calculus therefore the"}, {"start": 125.479, "duration": 8.801, "text": "derivative of sin 2x = 2 * cine 2x this"}, {"start": 131.039, "duration": 5.801, "text": "is our final answer this result shows us"}, {"start": 134.28, "duration": 6.44, "text": "that when we differentiate sin 2x we get"}, {"start": 136.84, "duration": 6.84, "text": "2 cine 2x the factor of two comes from"}, {"start": 140.72, "duration": 7.76, "text": "the chain rule which we applied"}, {"start": 143.68, "duration": 4.8, "text": "implicitly through our limit calculation"}], "title": "Understanding the Derivative of sin2x  Using First Principles", "description": "Dive into the fundamentals of calculus as we explore how to determine the derivative of the exponential function sin2x using first principles. Follow along with step-by-step explanations and examples to enhance your understanding of derivatives and exponential functions.\n#Calculus #Derivatives #ExponentialFunctions #MathTutorial #FirstPrinciples #sin 2x", "lengthSeconds": 147, "width": 640, "height": 360}
{"video_id": "eYRD8fYX-yI", "transcript": [{"start": 0.4, "duration": 7.84, "text": "welcome today we'll learn how to find"}, {"start": 2.96, "duration": 8.28, "text": "the derivative of y = 1 + x raed to the"}, {"start": 8.24, "duration": 5.16, "text": "X power using logarithmic"}, {"start": 11.24, "duration": 4.92, "text": "differentiation this is a challenging"}, {"start": 13.4, "duration": 5.76, "text": "derivative that becomes much easier when"}, {"start": 16.16, "duration": 8.76, "text": "we use logarithms let's solve this step"}, {"start": 19.16, "duration": 9.56, "text": "by step first let's say y = 1 + x ra to"}, {"start": 24.92, "duration": 7.24, "text": "the X power now here's the key step we"}, {"start": 28.72, "duration": 5.32, "text": "take the natural log of both sides this"}, {"start": 32.16, "duration": 4.239, "text": "is what makes logarithmic"}, {"start": 34.04, "duration": 4.32, "text": "differentiation special using the"}, {"start": 36.399, "duration": 4.281, "text": "logarithm property that the log of a"}, {"start": 38.36, "duration": 5.44, "text": "power equals the power * the log of the"}, {"start": 40.68, "duration": 7.76, "text": "base we can simplify the right side to x"}, {"start": 43.8, "duration": 4.64, "text": "* Ln of 1 +"}, {"start": 49.8, "duration": 7.36, "text": "x now we can take the derivative of both"}, {"start": 53.44, "duration": 6.68, "text": "sides with respect to X on the left side"}, {"start": 57.16, "duration": 8.56, "text": "we use the chain rule the derivative of"}, {"start": 60.12, "duration": 8.88, "text": "n y is 1 y * Dy DX on the right side we"}, {"start": 65.72, "duration": 7.16, "text": "use the product rule since we have x *"}, {"start": 69.0, "duration": 3.88, "text": "Ln of 1 +"}, {"start": 77.92, "duration": 8.519, "text": "x let's simplify the right side the"}, {"start": 81.32, "duration": 9.88, "text": "derivative of Ln of 1 + x is 1 1 + x and"}, {"start": 86.439, "duration": 4.761, "text": "the derivative of x is just 1"}, {"start": 91.32, "duration": 9.52, "text": "now multiply both sides by y to isolate"}, {"start": 95.799, "duration": 8.481, "text": "dydx finally substitute back that y = 1"}, {"start": 100.84, "duration": 3.44, "text": "+ x to X"}, {"start": 106.799, "duration": 5.801, "text": "power and there we have it we've found"}, {"start": 109.92, "duration": 4.839, "text": "the derivative using logarithmic"}, {"start": 112.6, "duration": 4.08, "text": "differentiation this method is"}, {"start": 114.759, "duration": 4.521, "text": "particularly useful when dealing with"}, {"start": 116.68, "duration": 6.88, "text": "functions where one quantity is raised"}, {"start": 119.28, "duration": 7.24, "text": "to a variable power the key steps were"}, {"start": 123.56, "duration": 6.64, "text": "one taking the natural log of both sides"}, {"start": 126.52, "duration": 6.6, "text": "two using logarithm properties 3"}, {"start": 130.2, "duration": 6.08, "text": "differentiating both sides four using"}, {"start": 133.12, "duration": 6.759, "text": "the chain Rule and product rule five"}, {"start": 136.28, "duration": 3.599, "text": "solving for dydx"}], "title": "How to Find the Derivative of [(1+x)^x] | Logarithmic Differentiation Made Easy", "description": "Question: Find the derivative of y = (1+x)\u02e3 using logarithmic differentiation.\n\n\ud83d\udcda Master logarithmic differentiation with this step-by-step tutorial! Learn how to find the derivative of (1+x)^x using a powerful technique that simplifies complex exponential functions.\nIn this video, we cover:\n\u2705 Understanding logarithmic differentiation\n\u2705 Step-by-step solution process\n\u2705 Application of chain rule and product rule\n\u2705 Simplification techniques\n\u2705 Final answer derivation\n\ud83d\udd0d Perfect for:\n\u2022 Calculus students\n\u2022 Math enthusiasts\n\u2022 Exam preparation\n\u2022 Concept review\n\ud83d\udcdd Follow along as we break down each step:\nTaking natural logarithms\nUsing log properties\nDifferentiating both sides\nApplying derivative rules\nSolving for dy/dx\n\u2b50 Final Result: dy/dx = (1+x)^x[x/(1+x) + ln(1+x)]\nHashtags:\n#Mathematics #Calculus #MathTutorial #Derivatives #LogarithmicDifferentiation #MathEducation #LearnMath #MathHelp", "lengthSeconds": 146, "width": 640, "height": 360}
{"video_id": "2g6N4-kGCsI", "transcript": [{"start": 0.28, "duration": 5.72, "text": "let's solve this challenging derivative"}, {"start": 2.679, "duration": 8.601, "text": "problem step by step we need to find the"}, {"start": 6.0, "duration": 8.36, "text": "derivative of y = x to X power * natural"}, {"start": 11.28, "duration": 5.28, "text": "log of x this is a complex function"}, {"start": 14.36, "duration": 5.64, "text": "involving both exponential and"}, {"start": 16.56, "duration": 6.559, "text": "logarithmic terms to solve this we'll"}, {"start": 20.0, "duration": 6.16, "text": "use logarithmic differentiation let's"}, {"start": 23.119, "duration": 5.601, "text": "start by taking the natural logarithm of"}, {"start": 26.16, "duration": 4.68, "text": "both sides using the properties of"}, {"start": 28.72, "duration": 5.2, "text": "logarithms we can"}, {"start": 30.84, "duration": 5.36, "text": "this into a sum the natural log of a"}, {"start": 33.92, "duration": 5.68, "text": "product equals the sum of the natural"}, {"start": 36.2, "duration": 7.199, "text": "logs so this becomes natural log of x to"}, {"start": 39.6, "duration": 6.56, "text": "the X Plus natural log of natural log X"}, {"start": 43.399, "duration": 6.84, "text": "now we can further simplify the first"}, {"start": 46.16, "duration": 6.8, "text": "term remember the natural log of x to a"}, {"start": 50.239, "duration": 6.761, "text": "power equals that power time the natural"}, {"start": 52.96, "duration": 7.64, "text": "log of x so this becomes x * natural log"}, {"start": 57.0, "duration": 6.239, "text": "X+ natural log of natural log X"}, {"start": 60.6, "duration": 6.92, "text": "now we can differentiate both sides with"}, {"start": 63.239, "duration": 5.401, "text": "respect to X on the left side we get 1/"}, {"start": 67.52, "duration": 4.4, "text": "y *"}, {"start": 68.64, "duration": 6.44, "text": "dydx this comes from the chain rule when"}, {"start": 71.92, "duration": 4.96, "text": "differentiating natural log of Y on the"}, {"start": 75.08, "duration": 6.32, "text": "right side we need to use both the"}, {"start": 76.88, "duration": 9.72, "text": "product rule and chain rule for X Ln X"}, {"start": 81.4, "duration": 10.079, "text": "we get x * the derivative of Ln x + Ln x"}, {"start": 86.6, "duration": 9.879, "text": "* the derivative of x for Ln of Ln X we"}, {"start": 91.479, "duration": 11.201, "text": "get 1 Ln x * 1X let's simplify this we"}, {"start": 96.479, "duration": 9.081, "text": "get 1 + Ln x + 1 /x Ln X then multiply"}, {"start": 102.68, "duration": 6.039, "text": "both sides by y finding a common"}, {"start": 105.56, "duration": 10.879, "text": "denominator in the brackets we get X Ln"}, {"start": 108.719, "duration": 12.601, "text": "x + Ln x * X Ln x +1 all over X Ln X"}, {"start": 116.439, "duration": 7.801, "text": "finally we substitute back y = x x * Ln"}, {"start": 121.32, "duration": 6.88, "text": "X giving us our final answer the"}, {"start": 124.24, "duration": 10.079, "text": "derivative equal x x * Ln x * the"}, {"start": 128.2, "duration": 9.679, "text": "fraction 1 + x Ln x * 1 + Ln X all over"}, {"start": 134.319, "duration": 6.28, "text": "X Ln X this problem demonstrates the"}, {"start": 137.879, "duration": 5.321, "text": "power of logarithmic differentiation"}, {"start": 140.599, "duration": 6.28, "text": "when dealing with complex products and"}, {"start": 143.2, "duration": 3.679, "text": "exponential functions"}], "title": "Mastering Complex Derivatives Step-by-Step Solution for \ud835\udc51/\ud835\udc51\ud835\udc65 [\ud835\udc65^\ud835\udc65  ln\u2061\ud835\udc65 ]", "description": "Question:\nFind the derivative of \ud835\udc51/\ud835\udc51\ud835\udc65 [\ud835\udc65^\ud835\udc65  ln\u2061\ud835\udc65 ] with respect to x.\n\n\ud83d\udd37 In this comprehensive mathematics tutorial, we break down the solution for finding the derivative of [\ud835\udc65^\ud835\udc65  ln\u2061\ud835\udc65 ] using logarithmic differentiation.\n\ud83d\udcda What you\u2019ll learn:\n\u2022 Logarithmic differentiation technique\n\u2022 Using log properties effectively\n\u2022 Applying chain rule and product rule\n\u2022 Step-by-step problem solving approach\n\u2022 Clear explanations for each mathematical step\n\ud83d\udca1 Perfect for:\n\u2022 Calculus students\n\u2022 Math majors\n\u2022 Engineering students\n\u2022 Test preparation\n\u2022 Advanced mathematics enthusiasts\n#Mathematics #Calculus #Derivatives #MathTutorial #LogarithmicDifferentiation #ChainRule #ProductRule #MathEducation #STEM #MathHelp", "lengthSeconds": 152, "width": 640, "height": 360}
{"video_id": "YqKzIh1hR70", "transcript": [{"start": 0.28, "duration": 4.96, "text": "let's solve this calculus problem step"}, {"start": 2.56, "duration": 5.04, "text": "by step using the quotient rule we need"}, {"start": 5.24, "duration": 10.16, "text": "to find the derivative of the function f"}, {"start": 7.6, "duration": 10.32, "text": "ofx = x^2 + 1 / x^2 + 3 first let's"}, {"start": 15.4, "duration": 5.799, "text": "recall the quotient rule formula the"}, {"start": 17.92, "duration": 6.48, "text": "derivative of U over v = v * the"}, {"start": 21.199, "duration": 7.761, "text": "derivative of U minus U * the derivative"}, {"start": 24.4, "duration": 6.359, "text": "of V all over v^2 let's identify our u"}, {"start": 28.96, "duration": 8.72, "text": "and v function"}, {"start": 30.759, "duration": 8.841, "text": "U of X = x^2 + 1 and V of X = x^2 + 3"}, {"start": 37.68, "duration": 6.44, "text": "next we need to find the derivatives of"}, {"start": 39.6, "duration": 8.4, "text": "u and v the derivative of U equal 2X and"}, {"start": 44.12, "duration": 6.079, "text": "the derivative of V also equal 2x now"}, {"start": 48.0, "duration": 6.48, "text": "let's substitute everything into the"}, {"start": 50.199, "duration": 11.801, "text": "quotient rule formula we get x^2 + 3 *"}, {"start": 54.48, "duration": 11.12, "text": "2x - x^2 + 1 * 2x all over X 2 + 3^ 2"}, {"start": 62.0, "duration": 8.52, "text": "let's distribute the 2x in the numerator"}, {"start": 65.6, "duration": 9.519, "text": "in the first term we get 2x CU + 6x and"}, {"start": 70.52, "duration": 7.12, "text": "in the second term we get 2x Cub + 2x"}, {"start": 75.119, "duration": 6.561, "text": "now we can combine like terms in the"}, {"start": 77.64, "duration": 8.0, "text": "numerator the 2x cubed terms cancel out"}, {"start": 81.68, "duration": 8.32, "text": "and we're left with 6x - 2x which equal"}, {"start": 85.64, "duration": 12.04, "text": "4X therefore the derivative of x^2 + 1"}, {"start": 90.0, "duration": 10.68, "text": "/x^2 + 3 = 4X x^2 + 3^ 2 let's verify"}, {"start": 97.68, "duration": 5.04, "text": "this matches our original problem"}, {"start": 100.68, "duration": 4.68, "text": "looking back at our original function"}, {"start": 102.72, "duration": 7.64, "text": "and following the quotient rule steps"}, {"start": 105.36, "duration": 5.0, "text": "we've arrived at the correct derivative"}], "title": "Calculus Simplified: Derivatives of Rational Functions [(\ud835\udc99^\ud835\udfd0+\ud835\udfcf)/(\ud835\udc99^\ud835\udfd0+\ud835\udfd1)] Explained", "description": "Unlock the secrets of differentiating rational functions with this detailed tutorial. We walk you through the process of applying the quotient rule to  [(\ud835\udc99^\ud835\udfd0+\ud835\udfcf)/(\ud835\udc99^\ud835\udfd0+\ud835\udfd1)]  , ensuring you grasp each concept with clarity. Perfect for calculus students aiming to improve their understanding and skills!\n#Calculus #Derivatives #QuotientRule #MathTutorial #LearnCalculus", "lengthSeconds": 112, "width": 640, "height": 360}
{"video_id": "yjsO-119270", "transcript": [{"start": 0.28, "duration": 4.92, "text": "today we'll solve an interesting"}, {"start": 2.72, "duration": 4.92, "text": "derivative problem involving inverse"}, {"start": 5.2, "duration": 5.2, "text": "trigonometric functions let's look at"}, {"start": 7.64, "duration": 4.959, "text": "the question before we dive into solving"}, {"start": 10.4, "duration": 5.279, "text": "this let's understand what we're dealing"}, {"start": 12.599, "duration": 5.68, "text": "with we have a composition of functions"}, {"start": 15.679, "duration": 5.081, "text": "First We Take the square root of x then"}, {"start": 18.279, "duration": 5.361, "text": "we take the sign of that result and"}, {"start": 20.76, "duration": 6.24, "text": "finally we take the inverse sign of"}, {"start": 23.64, "duration": 6.52, "text": "everything now here's a key Insight that"}, {"start": 27.0, "duration": 5.84, "text": "makes this problem much simpler when we"}, {"start": 30.16, "duration": 5.6, "text": "have sin inverse of sign of an angle"}, {"start": 32.84, "duration": 4.84, "text": "these operations cancel each other out"}, {"start": 35.76, "duration": 4.639, "text": "as long as the angle is within the"}, {"start": 37.68, "duration": 8.039, "text": "principle domain of inverse sign which"}, {"start": 40.399, "duration": 8.68, "text": "isk / 2 to < / 2 since Square < TK of X"}, {"start": 45.719, "duration": 6.36, "text": "is always positive it will always be in"}, {"start": 49.079, "duration": 6.681, "text": "the principal domain of inverse sign"}, {"start": 52.079, "duration": 8.28, "text": "therefore s inverse of s of sare < TK of"}, {"start": 55.76, "duration": 7.319, "text": "X simply equals < TK X now our problem"}, {"start": 60.359, "duration": 5.841, "text": "has become much simpler we just need to"}, {"start": 63.079, "duration": 5.881, "text": "find the derivative of square otk of X"}, {"start": 66.2, "duration": 8.0, "text": "let's solve this step by step first we"}, {"start": 68.96, "duration": 7.76, "text": "can rewrite square otk of X as x ^ 12"}, {"start": 74.2, "duration": 9.0, "text": "then we can use the power rule the"}, {"start": 76.72, "duration": 11.12, "text": "derivative of x n = n * x n -1 finally"}, {"start": 83.2, "duration": 9.16, "text": "we can rewrite x^ the -2 as 1/ < TK X"}, {"start": 87.84, "duration": 7.48, "text": "giving us our final answer 1 / 2 < tkx"}, {"start": 92.36, "duration": 5.64, "text": "and that's our solution the key to this"}, {"start": 95.32, "duration": 4.759, "text": "problem was recognizing that sin inverse"}, {"start": 98.0, "duration": 4.68, "text": "and sign cancel each other out within"}, {"start": 100.079, "duration": 6.4, "text": "the principal domain which greatly"}, {"start": 102.68, "duration": 3.799, "text": "simplified our calculation"}], "title": "Understanding Derivative [\ud835\udc60\ud835\udc56\ud835\udc5b^(\u22121) (sin\u2061\u221a\ud835\udc65 )] The Sine Inverse Function Explained", "description": "Dive into the world of calculus with our step-by-step breakdown of how to find the derivative of [\ud835\udc60\ud835\udc56\ud835\udc5b^(\u22121) (sin\u2061\u221a\ud835\udc65 )] . Learn the logic behind function simplification and apply the power rule effectively. Perfect for students and math enthusiasts aiming to strengthen their calculus skills. Watch now to enhance your understanding!\n#Calculus #Derivatives #MathTutorial #SineInverse #LearningMath", "lengthSeconds": 110, "width": 640, "height": 360}
{"video_id": "vMmmBTCiqfY", "transcript": [{"start": 1.28, "duration": 5.36, "text": "welcome today we're going to learn about"}, {"start": 4.04, "duration": 5.559, "text": "logarithmic differentiation by finding"}, {"start": 6.64, "duration": 6.119, "text": "the derivative of a complex exponential"}, {"start": 9.599, "duration": 8.081, "text": "function specifically we'll find the"}, {"start": 12.759, "duration": 7.841, "text": "derivative of y = a to the power of a to"}, {"start": 17.68, "duration": 5.8, "text": "the power of X where it is a positive"}, {"start": 20.6, "duration": 6.12, "text": "constant why do we need logarithmic"}, {"start": 23.48, "duration": 5.84, "text": "differentiation here well this function"}, {"start": 26.72, "duration": 5.32, "text": "has a complex exponential structure that"}, {"start": 29.32, "duration": 5.84, "text": "would be very difficult to differentiate"}, {"start": 32.04, "duration": 6.199, "text": "directly by taking the natural logarithm"}, {"start": 35.16, "duration": 5.64, "text": "of both sides first we can transform"}, {"start": 38.239, "duration": 5.16, "text": "this into a simpler expression that's"}, {"start": 40.8, "duration": 2.599, "text": "easier to"}, {"start": 46.399, "duration": 6.201, "text": "differentiate let's solve the step by"}, {"start": 48.8, "duration": 9.52, "text": "step step one take the natural logarithm"}, {"start": 52.6, "duration": 9.639, "text": "of both sides this gives us Ln of Y = Ln"}, {"start": 58.32, "duration": 6.839, "text": "of a to the power of a to the power of X"}, {"start": 62.239, "duration": 6.121, "text": "step two use the logarithm property that"}, {"start": 65.159, "duration": 8.761, "text": "states Ln of a to a power equals that"}, {"start": 68.36, "duration": 10.16, "text": "power * Ln of a this gives us Ln of y ="}, {"start": 73.92, "duration": 4.6, "text": "a x * Ln of"}, {"start": 79.799, "duration": 7.96, "text": "a step three now we can differentiate"}, {"start": 83.96, "duration": 7.4, "text": "both sides with respect to X remember"}, {"start": 87.759, "duration": 9.64, "text": "when we differentiate Ln of Y I we get 1"}, {"start": 91.36, "duration": 6.039, "text": "/ y * Dy DX this is implicit"}, {"start": 100.6, "duration": 3.96, "text": "differentiation step four now we can"}, {"start": 103.479, "duration": 5.161, "text": "solve for"}, {"start": 104.56, "duration": 7.919, "text": "dydx multiply both sides by Y and since"}, {"start": 108.64, "duration": 7.88, "text": "y equal a to ^ of a to X we can"}, {"start": 112.479, "duration": 4.041, "text": "substitute this back in"}, {"start": 121.399, "duration": 4.521, "text": "let's talk about some common mistakes to"}, {"start": 123.439, "duration": 5.32, "text": "avoid first don't forget to use the"}, {"start": 125.92, "duration": 7.84, "text": "chain rule when differentiating a to the"}, {"start": 128.759, "duration": 8.641, "text": "X second remember that y equals your"}, {"start": 133.76, "duration": 8.4, "text": "original function when substituting back"}, {"start": 137.4, "duration": 4.76, "text": "and third be careful with your logarithm"}, {"start": 142.4, "duration": 5.68, "text": "properties let's verify our answer makes"}, {"start": 145.4, "duration": 5.32, "text": "sense the derivative should get larger"}, {"start": 148.08, "duration": 5.96, "text": "as X increases because our original"}, {"start": 150.72, "duration": 5.64, "text": "function grows very rapidly our answer"}, {"start": 154.04, "duration": 5.52, "text": "shows this because both a to the power"}, {"start": 156.36, "duration": 5.72, "text": "of a to the X and A to the X are"}, {"start": 159.56, "duration": 5.52, "text": "increasing functions here are some"}, {"start": 162.08, "duration": 5.68, "text": "practice tips for similar problems first"}, {"start": 165.08, "duration": 4.68, "text": "whenever you see complex exponentials"}, {"start": 167.76, "duration": 4.68, "text": "consider logarithmic"}, {"start": 169.76, "duration": 6.399, "text": "differentiation second break down the"}, {"start": 172.44, "duration": 6.719, "text": "problem into small steps and third"}, {"start": 176.159, "duration": 4.281, "text": "always verify your answer makes sense in"}, {"start": 179.159, "duration": 4.401, "text": "conclusion"}, {"start": 180.44, "duration": 7.64, "text": "we found that the derivative of y = a"}, {"start": 183.56, "duration": 9.84, "text": "power of a to X is a the power of a to"}, {"start": 188.08, "duration": 7.36, "text": "the x * a to x * Ln of a squar we did"}, {"start": 193.4, "duration": 4.399, "text": "this using logarithmic"}, {"start": 195.44, "duration": 4.799, "text": "differentiation which simplified our"}, {"start": 197.799, "duration": 7.561, "text": "complex exponential into a more"}, {"start": 200.239, "duration": 5.121, "text": "manageable form thank you for watching"}], "title": "Mastering Logarithmic Differentiation: Solving Complex Exponential Derivatives | d/dx[\ud835\udc4e^(\ud835\udc4e^\ud835\udc65 )]", "description": "QUESTION: Find the derivative of y = d/dx[\ud835\udc4e^(\ud835\udc4e^\ud835\udc65 )] using logarithmic differentiation.\n\n\ud83d\udcda In this comprehensive calculus tutorial, we break down how to find the derivative of a(ax) using logarithmic differentiation. Perfect for AP Calculus, IB Mathematics, and college-level calculus students!\n\ud83c\udfaf What You\u2019ll Learn:\n\u2022 Why logarithmic differentiation is the key to solving complex exponentials\n\u2022 Step-by-step solution process\n\u2022 Important logarithm properties and chain rule applications\n\u2022 Common pitfalls to avoid\n\u2022 Tips for tackling similar problems\n\ud83d\udcdd Prerequisites:\nBasic differentiation rules\nNatural logarithms\nChain rule\nExponential functions\n\u2705 Download practice problems and solutions in the description below!\n#Calculus #Mathematics #Derivatives #LogarithmicDifferentiation #MathTutorial #APCalculus #IBMaths #CalcTutoring #MathEducation #OnlineLearning", "lengthSeconds": 215, "width": 640, "height": 360}
{"video_id": "xm1OiezKP1w", "transcript": [{"start": 0.32, "duration": 5.72, "text": "let's solve this complex derivative"}, {"start": 2.72, "duration": 5.799, "text": "problem step by step we need to find the"}, {"start": 6.04, "duration": 5.16, "text": "derivative of the natural logarithm of a"}, {"start": 8.519, "duration": 4.641, "text": "fraction involving square roots the"}, {"start": 11.2, "duration": 3.84, "text": "first step is to use the logarithm"}, {"start": 13.16, "duration": 4.52, "text": "property that states the log of a"}, {"start": 15.04, "duration": 5.72, "text": "fraction equals the log of the numerator"}, {"start": 17.68, "duration": 5.72, "text": "minus the log of the denominator this"}, {"start": 20.76, "duration": 6.12, "text": "allows us to split our expression into a"}, {"start": 23.4, "duration": 6.6, "text": "difference of two logarithms next we'll"}, {"start": 26.88, "duration": 6.56, "text": "use the chain rule to differentiate each"}, {"start": 30.0, "duration": 5.96, "text": "logarithm term remember the derivative"}, {"start": 33.44, "duration": 5.279, "text": "of natural log is 1 over the function"}, {"start": 35.96, "duration": 5.759, "text": "time the derivative of the function"}, {"start": 38.719, "duration": 5.32, "text": "let's apply this to both terms now we"}, {"start": 41.719, "duration": 7.241, "text": "need to find the derivative of square <"}, {"start": 44.039, "duration": 10.081, "text": "TK of x + 1 remember the derivative of"}, {"start": 48.96, "duration": 8.2, "text": "square < TK X is 1 / 2 * < TK X so the"}, {"start": 54.12, "duration": 7.8, "text": "derivative of square < TK of x + 1 will"}, {"start": 57.16, "duration": 9.24, "text": "be 1 / 2 < TK x + 1 * the derivative of"}, {"start": 61.92, "duration": 8.72, "text": "x + 1 the derivative of x + 1 is simply"}, {"start": 66.4, "duration": 7.32, "text": "1 so we can replace those terms with one"}, {"start": 70.64, "duration": 5.96, "text": "and begin to simplify our expression to"}, {"start": 73.72, "duration": 5.24, "text": "combine these fractions we need a common"}, {"start": 76.6, "duration": 5.0, "text": "denominator after combining and"}, {"start": 78.96, "duration": 5.24, "text": "simplifying the numerator we get this"}, {"start": 81.6, "duration": 5.32, "text": "expression looking at the numerator we"}, {"start": 84.2, "duration": 5.76, "text": "can see that the square root terms"}, {"start": 86.92, "duration": 6.199, "text": "cancel out leaving us with just two and"}, {"start": 89.96, "duration": 8.04, "text": "and in the denominator we can simplify x"}, {"start": 93.119, "duration": 7.521, "text": "+ 1 -1 to just X finally we can cancel"}, {"start": 98.0, "duration": 8.719, "text": "the twos and simplify the denominator to"}, {"start": 100.64, "duration": 8.36, "text": "get our final answer 1X * < TK x + 1"}, {"start": 106.719, "duration": 4.841, "text": "this problem demonstrated several"}, {"start": 109.0, "duration": 5.6, "text": "important calculus Concepts the"}, {"start": 111.56, "duration": 5.32, "text": "properties of logarithms the chain Rule"}, {"start": 114.6, "duration": 5.36, "text": "and the derivative of square root"}, {"start": 116.88, "duration": 5.96, "text": "functions by breaking it down step by"}, {"start": 119.96, "duration": 5.88, "text": "step we were able to solve this complex"}, {"start": 122.84, "duration": 3.0, "text": "derivative"}], "title": "Mastering Calculus: Differentiating Logarithmic Functions [\ud835\udc8d\ud835\udc8f((\u221a(\ud835\udc99+\ud835\udfcf)\u2212\ud835\udfcf)/(\u221a(\ud835\udc99+\ud835\udfcf)+\ud835\udfcf))]", "description": "Question:\nDifferentiate the expression [\ud835\udc8d\ud835\udc8f((\u221a(\ud835\udc99+\ud835\udfcf)\u2212\ud835\udfcf)/(\u221a(\ud835\udc99+\ud835\udfcf)+\ud835\udfcf))] with respect to x.\n\nIn this video, we break down the process of differentiating logarithmic functions step-by-step. Learn how to handle complex expressions involving square roots and logarithms using the chain rule. Perfect for students looking to enhance their calculus skills and understanding.\n\n#Calculus #Differentiation #Logarithms #MathTutorial #Learning", "lengthSeconds": 129, "width": 640, "height": 360}
{"video_id": "75eztK6xm14", "transcript": [{"start": 0.399, "duration": 5.281, "text": "let's solve this calculus problem step"}, {"start": 2.84, "duration": 6.839, "text": "by step we need to find the derivative"}, {"start": 5.68, "duration": 6.48, "text": "of secant inverse of X sared let's break"}, {"start": 9.679, "duration": 5.601, "text": "this down into manageable steps using"}, {"start": 12.16, "duration": 5.8, "text": "the chain rule remember when we have a"}, {"start": 15.28, "duration": 5.28, "text": "function raised to a power we can use"}, {"start": 17.96, "duration": 5.399, "text": "the chain rule which states that the"}, {"start": 20.56, "duration": 6.16, "text": "derivative of u^2 equals 2 * the"}, {"start": 23.359, "duration": 6.68, "text": "derivative of U in our case U equal"}, {"start": 26.72, "duration": 7.519, "text": "secant inverse of X so applying the"}, {"start": 30.039, "duration": 7.401, "text": "chain rule we get 2 * secant inverse of"}, {"start": 34.239, "duration": 6.121, "text": "x * the derivative of secant inverse of"}, {"start": 37.44, "duration": 6.599, "text": "X now we need to know the derivative of"}, {"start": 40.36, "duration": 6.56, "text": "secant inverse of X this is a standard"}, {"start": 44.039, "duration": 8.52, "text": "derivative that we should memorize it ="}, {"start": 46.92, "duration": 9.56, "text": "1 /x * < TK x^2 - 1 substituting this"}, {"start": 52.559, "duration": 10.921, "text": "back into our equation we get 2 * secant"}, {"start": 56.48, "duration": 10.679, "text": "inverse x * 1 /x * < TK of x^2 -1"}, {"start": 63.48, "duration": 9.72, "text": "finally we can write this as 2 * secant"}, {"start": 67.159, "duration": 9.0, "text": "inverse x/ x * < TK of x^2 - 1 this is"}, {"start": 73.2, "duration": 5.919, "text": "our final answer let's verify the key"}, {"start": 76.159, "duration": 5.161, "text": "components of this solution first we use"}, {"start": 79.119, "duration": 6.201, "text": "the chain rule correctly to handle the"}, {"start": 81.32, "duration": 7.32, "text": "squared term second we correctly applied"}, {"start": 85.32, "duration": 5.24, "text": "the derivative of inverse secant and"}, {"start": 88.64, "duration": 5.519, "text": "finally we simply simplified our"}, {"start": 90.56, "duration": 5.879, "text": "expression to its most compact form"}, {"start": 94.159, "duration": 5.32, "text": "remember when dealing with inverse"}, {"start": 96.439, "duration": 5.761, "text": "trigonometric functions it's crucial to"}, {"start": 99.479, "duration": 6.161, "text": "know their derivatives the derivative of"}, {"start": 102.2, "duration": 8.12, "text": "inverse secant is particularly useful in"}, {"start": 105.64, "duration": 4.68, "text": "calculus problems like this one"}], "title": "Master the Derivative of Inverse Secant Squared [(\ud835\udc60\ud835\udc52\ud835\udc50^(\u22121) \ud835\udc65)^2] Step-by-Step Guide", "description": "Question:\nCalculate the derivative of  [(\ud835\udc60\ud835\udc52\ud835\udc50^(\u22121) \ud835\udc65)^2]   with respect to x .\nUnlock the mystery of calculus with our detailed breakdown of the derivative of  [(\ud835\udc60\ud835\udc52\ud835\udc50^(\u22121) \ud835\udc65)^2] . This step-by-step guide takes you through each mathematical operation and its explanation, helping you master this important calculus concept. Perfect for students, teachers, and anyone keen on understanding advanced math. Don\u2019t forget to subscribe for more insightful math tutorials!\n#Calculus #MathTutorial #Derivatives #InverseSecant #MathHelp", "lengthSeconds": 113, "width": 640, "height": 360}
{"video_id": "i0Su3LNK8Ec", "transcript": [{"start": 0.28, "duration": 5.039, "text": "let's solve this calculus problem step"}, {"start": 2.44, "duration": 6.72, "text": "by step using the quotient rule we need"}, {"start": 5.319, "duration": 7.121, "text": "to find the derivative of sinx / by x^2"}, {"start": 9.16, "duration": 6.32, "text": "+ cosine X first let's recall the"}, {"start": 12.44, "duration": 7.24, "text": "quotient rule the derivative of U over V"}, {"start": 15.48, "duration": 7.76, "text": "equal V * the derivative of uus U * the"}, {"start": 19.68, "duration": 8.679, "text": "derivative of V all over V ^2 let's"}, {"start": 23.24, "duration": 10.159, "text": "identify our u and v functions U = sin x"}, {"start": 28.359, "duration": 8.121, "text": "and V = x\u00b2 + cin X now let's find the"}, {"start": 33.399, "duration": 7.761, "text": "derivatives of u and v the derivative of"}, {"start": 36.48, "duration": 9.68, "text": "sinx is cine X the derivative of x^2 +"}, {"start": 41.16, "duration": 7.64, "text": "cine X is 2x - sin x let's substitute"}, {"start": 46.16, "duration": 5.84, "text": "these into the quotient rule formula we"}, {"start": 48.8, "duration": 5.919, "text": "get V which is x^2 + cosine x * the"}, {"start": 52.0, "duration": 5.76, "text": "derivative of U which is cosine x minus"}, {"start": 54.719, "duration": 8.041, "text": "U which is sin x * the derivative of V"}, {"start": 57.76, "duration": 8.719, "text": "which is 2x - sin x all over V ^2 now"}, {"start": 62.76, "duration": 8.039, "text": "let's expand the numerator first multip"}, {"start": 66.479, "duration": 9.481, "text": "x^2 + cosine X by cosine X this gives us"}, {"start": 70.799, "duration": 10.721, "text": "x^2 cosine x + cosine 2 x then multiply"}, {"start": 75.96, "duration": 9.6, "text": "sin x by 2x - sin x this gives us 2x sin"}, {"start": 81.52, "duration": 7.52, "text": "x - sin^2 x let's combine these terms in"}, {"start": 85.56, "duration": 9.48, "text": "the numerator we get x^2 cosine x +"}, {"start": 89.04, "duration": 11.079, "text": "cosine 2 x - 2x sin x + sin^2 x all over"}, {"start": 95.04, "duration": 9.32, "text": "x^2 + cosine x^2 finally we can simplify"}, {"start": 100.119, "duration": 9.28, "text": "this further remember that sin^2 x +"}, {"start": 104.36, "duration": 10.96, "text": "cosine 2 x = 1 using this our final"}, {"start": 109.399, "duration": 9.68, "text": "answer is x^2 cine x - 2x sin x + 1 all"}, {"start": 115.32, "duration": 6.88, "text": "over x^2 + cine x^2 and that's our"}, {"start": 119.079, "duration": 6.0, "text": "solution we used the quotient rule to"}, {"start": 122.2, "duration": 5.199, "text": "find the derivative expanded the terms"}, {"start": 125.079, "duration": 6.841, "text": "and simplified using trigonometric"}, {"start": 127.399, "duration": 4.521, "text": "identities to get our final answer"}], "title": "Differentiating [(\ud835\udc94\ud835\udc8a\ud835\udc8f \ud835\udc99)/(\ud835\udc99^\ud835\udfd0+\ud835\udc84\ud835\udc90\ud835\udc94 \ud835\udc99)] A Step-by-Step Guide Using the Quotient Rule", "description": "In this comprehensive calculus tutorial, learn how to differentiate the function [(\ud835\udc94\ud835\udc8a\ud835\udc8f \ud835\udc99)/(\ud835\udc99^\ud835\udfd0+\ud835\udc84\ud835\udc90\ud835\udc94 \ud835\udc99)] using the quotient rule. Follow along with clear, step-by-step explanations and tips to master this essential calculus technique. Perfect for students, educators, and anyone looking to enhance their mathematics skills.\n#Calculus #Differentiation #QuotientRule #MathTutorial #Trigonometry", "lengthSeconds": 130, "width": 640, "height": 360}
{"video_id": "86XqrQl_wgU", "transcript": [{"start": 0.4, "duration": 5.04, "text": "welcome today we'll solve a challenging"}, {"start": 3.319, "duration": 4.961, "text": "differentiation problem involving"}, {"start": 5.44, "duration": 5.239, "text": "inverse trigonometric functions let's"}, {"start": 8.28, "duration": 6.359, "text": "look at our question let's solve the"}, {"start": 10.679, "duration": 7.481, "text": "step by step first let's let y equal our"}, {"start": 14.639, "duration": 7.761, "text": "function now this expression inside the"}, {"start": 18.16, "duration": 6.48, "text": "ark sign function looks complicated to"}, {"start": 22.4, "duration": 6.639, "text": "simplify it let's make a clever"}, {"start": 24.64, "duration": 7.119, "text": "substitution let's say x = tan Theta"}, {"start": 29.039, "duration": 5.081, "text": "this means Theta equals inverse tangent"}, {"start": 31.759, "duration": 5.161, "text": "of X when we make the"}, {"start": 34.12, "duration": 7.56, "text": "substitution our expression becomes sin"}, {"start": 36.92, "duration": 7.76, "text": "inverse of 1 - tan^ 2 thet over 1 + tan^"}, {"start": 41.68, "duration": 5.96, "text": "2 thet now here's where it gets"}, {"start": 44.68, "duration": 5.68, "text": "interesting this fraction is actually a"}, {"start": 47.64, "duration": 6.399, "text": "well-known trigonometric identity it"}, {"start": 50.36, "duration": 7.8, "text": "equals cosine of 2"}, {"start": 54.039, "duration": 8.121, "text": "Theta we can simplify this further using"}, {"start": 58.16, "duration": 9.24, "text": "another identity Co sin of 2 thet = sin"}, {"start": 62.16, "duration": 9.279, "text": "ofun / 2 - 2 thet or equivalently sin of"}, {"start": 67.4, "duration": 7.399, "text": "piun / 2 + 2 Theta now we have sin"}, {"start": 71.439, "duration": 6.0, "text": "inverse of s these are inverse functions"}, {"start": 74.799, "duration": 7.0, "text": "so they cancel out leaving us with Just"}, {"start": 77.439, "duration": 7.441, "text": "pi/ 2 + or minus 2 Theta remember that"}, {"start": 81.799, "duration": 6.201, "text": "Theta is inverse tangent of X so we can"}, {"start": 84.88, "duration": 8.199, "text": "write our final expression for y as Pi /"}, {"start": 88.0, "duration": 7.04, "text": "2 + or - 2 inverse tangent of X now we"}, {"start": 93.079, "duration": 4.561, "text": "can differentiate this expression with"}, {"start": 95.04, "duration": 5.68, "text": "respect to X the derivative of a"}, {"start": 97.64, "duration": 8.72, "text": "constant is zero and the derivative of"}, {"start": 100.72, "duration": 9.56, "text": "inverse tangent is 1 1 + x^2 therefore"}, {"start": 106.36, "duration": 11.119, "text": "the derivative of sin inverse of 1 - x^2"}, {"start": 110.28, "duration": 8.839, "text": "over 1 + x^2 = + or - 2 over 1 + x^2"}, {"start": 117.479, "duration": 4.361, "text": "this was a challenging problem that"}, {"start": 119.119, "duration": 5.161, "text": "required several key insights making a"}, {"start": 121.84, "duration": 5.52, "text": "clever substitution recognizing"}, {"start": 124.28, "duration": 5.64, "text": "trigonometric identities and carefully"}, {"start": 127.36, "duration": 5.56, "text": "applying the chain rule thank you for"}, {"start": 129.92, "duration": 3.0, "text": "watching"}], "title": "Differentiating Inverse Trigonometric Functions [\ud835\udc60\ud835\udc56\ud835\udc5b^(\u22121)\u2061((1\u2212\ud835\udc65^2)/(1+\ud835\udc65^2 ))] A Step-by-Step Guide", "description": "Question: Differentiate the function [\ud835\udc60\ud835\udc56\ud835\udc5b^(\u22121)\u2061((1\u2212\ud835\udc65^2)/(1+\ud835\udc65^2 ))]   with respect to x.\n\nLearn how to differentiate the function  [\ud835\udc60\ud835\udc56\ud835\udc5b^(\u22121)\u2061((1\u2212\ud835\udc65^2)/(1+\ud835\udc65^2 ))] using trigonometric identities and calculus techniques. This detailed walkthrough will help you understand the process step-by-step with clear explanations.\n#Calculus #Trigonometry #MathTutorial #Derivative #MathHelp", "lengthSeconds": 136, "width": 640, "height": 360}
{"video_id": "yK5nqi_MdM4", "transcript": [{"start": 0.28, "duration": 4.88, "text": "let's solve this calculus problem step"}, {"start": 2.52, "duration": 7.8, "text": "by step we need to find the derivative"}, {"start": 5.16, "duration": 8.599, "text": "of the cube root of x * sinx first let's"}, {"start": 10.32, "duration": 7.039, "text": "identify our two functions let's call U"}, {"start": 13.759, "duration": 9.961, "text": "of X = the cubot of X which we can write"}, {"start": 17.359, "duration": 8.441, "text": "as x^ the 1/3 power and V ofx = sinx now"}, {"start": 23.72, "duration": 4.44, "text": "since we're multiplying these functions"}, {"start": 25.8, "duration": 5.16, "text": "together we'll need to use the product"}, {"start": 28.16, "duration": 6.12, "text": "rule the product rule states that the"}, {"start": 30.96, "duration": 7.52, "text": "derivative of U * v = u * the derivative"}, {"start": 34.28, "duration": 7.24, "text": "of V plus v * the derivative of U let's"}, {"start": 38.48, "duration": 6.84, "text": "find each derivative separately the"}, {"start": 41.52, "duration": 8.48, "text": "derivative of sin x is cine X and the"}, {"start": 45.32, "duration": 8.039, "text": "derivative of x^ 1/3 power is 1/3 * X"}, {"start": 50.0, "duration": 5.76, "text": "-23 power now let's substitute"}, {"start": 53.359, "duration": 9.401, "text": "everything into the product rule we get"}, {"start": 55.76, "duration": 10.32, "text": "x 1/3 * cosine X Plus sin x * 1/3 x"}, {"start": 62.76, "duration": 6.84, "text": "-23 let's simplify the second term the"}, {"start": 66.08, "duration": 8.64, "text": "fraction 1/3 and x to the -2/3 can be"}, {"start": 69.6, "duration": 8.159, "text": "Rewritten as sin x over 3x^ 2/3 power"}, {"start": 74.72, "duration": 7.52, "text": "finally we can factor out x to the 1/3"}, {"start": 77.759, "duration": 8.921, "text": "power our final answer is x 13 * the"}, {"start": 82.24, "duration": 7.0, "text": "quantity cosine X+ sin x over 3x let's"}, {"start": 86.68, "duration": 5.64, "text": "review what we did we identified our"}, {"start": 89.24, "duration": 5.159, "text": "functions and Ed the product rule found"}, {"start": 92.32, "duration": 4.36, "text": "the derivatives of each function"}, {"start": 94.399, "duration": 7.161, "text": "separately and then simplified our"}, {"start": 96.68, "duration": 4.88, "text": "expression to get our final answer"}], "title": "Finding the Derivative of \u221b\ud835\udc65  sin\u2061\ud835\udc65  Using the Product Rule | Step-by-Step Explanation", "description": "In this video, learn to differentiate \u221b\ud835\udc65  sin\u2061\ud835\udc65  using the Product Rule in calculus.\n\nWhat You'll Learn:  \n- Definition of the Product Rule  \n- Step-by-step differentiation of \u221b\ud835\udc65  sin\u2061\ud835\udc65 \n- Formula breakdown and example problem-solving  \n\nThis example is perfect for calculus beginners or those reviewing the Product Rule!\n\ud83d\udca1 Don't forget to LIKE, SHARE, and SUBSCRIBE for more math tutorials!\n#Calculus #ProductRule #Differentiation #Derivative #MathTutorial #LearnMath #MathEducation #MathHelp", "lengthSeconds": 103, "width": 640, "height": 360}
{"video_id": "6PZD3qnFzxg", "transcript": [{"start": 0.32, "duration": 7.4, "text": "let's find the derivative of x Cub + 2x"}, {"start": 5.16, "duration": 6.519, "text": "using the definition of the derivative"}, {"start": 7.72, "duration": 7.28, "text": "first let's write our function f ofx = x"}, {"start": 11.679, "duration": 5.241, "text": "Cub + 2x the definition of the"}, {"start": 15.0, "duration": 7.48, "text": "derivative states that it's equal to the"}, {"start": 16.92, "duration": 10.8, "text": "Limit as H approaches Z of f ofx + hus F"}, {"start": 22.48, "duration": 9.48, "text": "ofx all divided H step one let's find f"}, {"start": 27.72, "duration": 7.6, "text": "of x + H this means we replace every X"}, {"start": 31.96, "duration": 8.439, "text": "in our original function with x + H when"}, {"start": 35.32, "duration": 10.879, "text": "we expand this we get x + H cubed + 2 *"}, {"start": 40.399, "duration": 8.48, "text": "x + H step two let's expand x + H cubed"}, {"start": 46.199, "duration": 4.88, "text": "using the cube expansion formula we get"}, {"start": 48.879, "duration": 7.801, "text": "X cubed +"}, {"start": 51.079, "duration": 9.361, "text": "3x^2 H + 3x h^2 + H cubed and don't"}, {"start": 56.68, "duration": 7.6, "text": "forget to add 2x + 2 H from the second"}, {"start": 60.44, "duration": 6.64, "text": "term step three now we can substitute"}, {"start": 64.28, "duration": 6.76, "text": "this into our derivative definition we"}, {"start": 67.08, "duration": 7.96, "text": "subtract F ofx which is X Cub + 2x and"}, {"start": 71.04, "duration": 7.48, "text": "divide everything by H step four let's"}, {"start": 75.04, "duration": 7.0, "text": "simplify the numerator the X cubed terms"}, {"start": 78.52, "duration": 10.639, "text": "and 2x terms cancel out this leaves us"}, {"start": 82.04, "duration": 11.28, "text": "with 3x^2 H + 3x h^2 + HB + 2 H all over"}, {"start": 89.159, "duration": 8.801, "text": "h step five we can factor out H from the"}, {"start": 93.32, "duration": 11.04, "text": "numerator this gives us H * the 3x^2 +"}, {"start": 97.96, "duration": 10.96, "text": "3xh + h^2 + 2 all over H the H's cancel"}, {"start": 104.36, "duration": 7.64, "text": "out step 6 now we can evaluate the limit"}, {"start": 108.92, "duration": 7.08, "text": "as H approaches zero the terms with h go"}, {"start": 112.0, "duration": 8.0, "text": "to zero leaving us with 3 x^2 + 2"}, {"start": 116.0, "duration": 8.079, "text": "therefore the derivative of x Cub + 2x"}, {"start": 120.0, "duration": 6.759, "text": "is 3x^2 + 2 this is a great example of"}, {"start": 124.079, "duration": 5.641, "text": "using the definition of the derivative"}, {"start": 126.759, "duration": 5.12, "text": "to find the derivative of a polom"}, {"start": 129.72, "duration": 4.96, "text": "function while we could have used the"}, {"start": 131.879, "duration": 5.881, "text": "power rule understanding the definition"}, {"start": 134.68, "duration": 6.08, "text": "helps us grasp what a derivative really"}, {"start": 137.76, "duration": 3.0, "text": "means"}], "title": "Understanding the Derivative of x^3+2x  Using First Principles", "description": "Master calculus fundamentals by deriving the exponential function x^3 + 2x from first principles. Utilize our step-by-step explanations and examples to strengthen your understanding of derivatives and exponential functions.\n\n#Calculus #Derivatives #ExponentialFunctions #MathTutorial #FirstPrinciples #x^3+2x", "lengthSeconds": 141, "width": 640, "height": 360}
{"video_id": "Qi5-fqaqTtU", "transcript": [{"start": 0.52, "duration": 6.759, "text": "welcome to this calculus lesson on"}, {"start": 3.64, "duration": 5.879, "text": "logarithmic differentiation today we'll"}, {"start": 7.279, "duration": 5.601, "text": "learn how to find the derivative of a"}, {"start": 9.519, "duration": 7.84, "text": "complex expression sin x raed to the"}, {"start": 12.88, "duration": 6.76, "text": "cosin X Plus cine x raed to the sinx"}, {"start": 17.359, "duration": 5.321, "text": "let's break this down into manageable"}, {"start": 19.64, "duration": 6.16, "text": "steps first let's separate this into two"}, {"start": 22.68, "duration": 6.16, "text": "parts we'll call the first part you and"}, {"start": 25.8, "duration": 6.2, "text": "the second part five this will help us"}, {"start": 28.84, "duration": 6.64, "text": "handle each EXP ential terms separately"}, {"start": 32.0, "duration": 5.28, "text": "now let's focus on finding dudx first"}, {"start": 35.48, "duration": 4.16, "text": "this is where logarithmic"}, {"start": 37.28, "duration": 5.4, "text": "differentiation comes in handy we'll"}, {"start": 39.64, "duration": 5.96, "text": "take the natural log of both sides"}, {"start": 42.68, "duration": 6.44, "text": "remember when we take the natural log of"}, {"start": 45.6, "duration": 6.4, "text": "an exponential expression we can bring"}, {"start": 49.12, "duration": 4.919, "text": "the exponent down as a coefficient now"}, {"start": 52.0, "duration": 5.0, "text": "we can differentiate both sides with"}, {"start": 54.039, "duration": 6.561, "text": "respect to X remember we'll need to use"}, {"start": 57.0, "duration": 8.08, "text": "the chain rule here on the left side"}, {"start": 60.6, "duration": 7.0, "text": "we get 1 / U * Dux on the right side we"}, {"start": 65.08, "duration": 5.399, "text": "need to use the product rule since we"}, {"start": 67.6, "duration": 6.879, "text": "have two functions being multiplied"}, {"start": 70.479, "duration": 7.481, "text": "together let's solve for DX we multiply"}, {"start": 74.479, "duration": 8.081, "text": "both sides by U which is sin x to the"}, {"start": 77.96, "duration": 7.56, "text": "cosine X now let's find dvdx using the"}, {"start": 82.56, "duration": 7.919, "text": "same process we take the natural log of"}, {"start": 85.52, "duration": 8.639, "text": "V which gives us sin x * Ln of cine X"}, {"start": 90.479, "duration": 7.041, "text": "again we differentiate both sides using"}, {"start": 94.159, "duration": 7.441, "text": "the same process as before we get 1 / V"}, {"start": 97.52, "duration": 7.639, "text": "* dvdx equals the derivative of sin x *"}, {"start": 101.6, "duration": 7.76, "text": "Ln of cosine x + sin x * the derivative"}, {"start": 105.159, "duration": 7.881, "text": "of Ln of cosine X solving for dvdx we"}, {"start": 109.36, "duration": 7.24, "text": "multiply both sides by V which is cosine"}, {"start": 113.04, "duration": 8.039, "text": "X to the sinx finally we can combine our"}, {"start": 116.6, "duration": 8.04, "text": "results the derivative of y equal DX x +"}, {"start": 121.079, "duration": 6.881, "text": "dvdx this gives us our final answer sin"}, {"start": 124.64, "duration": 9.319, "text": "x cosine x * the quantity negative sin x"}, {"start": 127.96, "duration": 11.2, "text": "Ln sin x + c^ 2 x/ sin x plus cosine X"}, {"start": 133.959, "duration": 9.121, "text": "sin x * the qu cosine X Ln cosine x -"}, {"start": 139.16, "duration": 7.439, "text": "sin^2 XC X let's review some common"}, {"start": 143.08, "duration": 5.32, "text": "mistakes to avoid one don't forget to"}, {"start": 146.599, "duration": 5.041, "text": "use the product rule when"}, {"start": 148.4, "duration": 6.559, "text": "differentiating terms with natural logs"}, {"start": 151.64, "duration": 6.0, "text": "two remember to multiply by the original"}, {"start": 154.959, "duration": 5.321, "text": "function when solving for the derivative"}, {"start": 157.64, "duration": 6.2, "text": "three be careful with your signs when"}, {"start": 160.28, "duration": 7.0, "text": "dealing with negative terms four don't"}, {"start": 163.84, "duration": 6.0, "text": "forget to combine both parts at the end"}, {"start": 167.28, "duration": 5.36, "text": "and that's how we use logarithmic"}, {"start": 169.84, "duration": 5.679, "text": "differentiation to find the derivative"}, {"start": 172.64, "duration": 5.879, "text": "of this complex expression thanks for"}, {"start": 175.519, "duration": 3.0, "text": "watching"}], "title": "Master Logarithmic Differentiation: Solving (sin x)^(cos x) + (cos x)^(sin x) | Calculus Made Clear", "description": "Question: Find the derivative of (sin x)^(cos x) + (cos x)^(sin x)  with respect to x.\n\n\ud83d\udcda In this comprehensive calculus tutorial, we break down a challenging derivative problem using logarithmic differentiation. Perfect for university calculus students and math enthusiasts!\n\ud83c\udfaf What You\u2019ll Learn:\n\u2022 How to tackle complex exponential functions\n\u2022 Step-by-step logarithmic differentiation technique\n\u2022 Mastering the chain rule and product rule\n\u2022 Working with trigonometric functions\n\u2022 Common pitfalls and how to avoid them\n\ud83d\udca1 Prerequisites:\n\u2022 Basic differentiation rules\n\u2022 Understanding of natural logarithms\n\u2022 Knowledge of trigonometric functions\n\u270f\ufe0f Practice problems and solutions available in the description below!\n\ud83d\udcdd Related Resources:\n#Calculus #Mathematics #LogarithmicDifferentiation #MathTutorial #CalcTutorial #MathEducation #STEM #LearnMath", "lengthSeconds": 186, "width": 640, "height": 360}
{"video_id": "thBwIX7Zz3c", "transcript": [{"start": 0.28, "duration": 4.559, "text": "let's work through finding the"}, {"start": 1.919, "duration": 6.121, "text": "derivative of e to the power of the"}, {"start": 4.839, "duration": 7.481, "text": "square otk of X step by step our"}, {"start": 8.04, "duration": 7.519, "text": "original function is y = e power of the"}, {"start": 12.32, "duration": 6.92, "text": "square < TK of x to solve this we'll"}, {"start": 15.559, "duration": 6.081, "text": "need to use the chain rule remember the"}, {"start": 19.24, "duration": 5.48, "text": "chain rule states that when we have a"}, {"start": 21.64, "duration": 5.08, "text": "composite function we multiply the"}, {"start": 24.72, "duration": 4.639, "text": "derivative of the outer function by the"}, {"start": 26.72, "duration": 5.399, "text": "derivative of the inner function in our"}, {"start": 29.359, "duration": 5.04, "text": "case the outer function is e to the"}, {"start": 32.119, "duration": 5.001, "text": "power of something and the inner"}, {"start": 34.399, "duration": 6.0, "text": "function is the square root of x let's"}, {"start": 37.12, "duration": 6.2, "text": "start with step one the derivative of e"}, {"start": 40.399, "duration": 6.041, "text": "to the power of U with respect to U is"}, {"start": 43.32, "duration": 6.719, "text": "just e to the power of U this is our"}, {"start": 46.44, "duration": 5.56, "text": "outer function derivative step two we"}, {"start": 50.039, "duration": 4.281, "text": "need to find the derivative of the inner"}, {"start": 52.0, "duration": 4.879, "text": "function which is the derivative of"}, {"start": 54.32, "duration": 7.32, "text": "square < TK of X the derivative of"}, {"start": 56.879, "duration": 8.041, "text": "square < TK of X is 1 / 2 root of x step"}, {"start": 61.64, "duration": 5.72, "text": "three now we apply the chain rule by"}, {"start": 64.92, "duration": 8.96, "text": "multiplying these parts together we"}, {"start": 67.36, "duration": 10.0, "text": "multiply e to power of s < TK x * 1 / 2"}, {"start": 73.88, "duration": 9.36, "text": "< TK X our final answer can be written"}, {"start": 77.36, "duration": 9.0, "text": "as E power of s < TK x/ 2 < TK X let's"}, {"start": 83.24, "duration": 6.239, "text": "visualize why this makes sense here's"}, {"start": 86.36, "duration": 6.119, "text": "the graph of our original function e to"}, {"start": 89.479, "duration": 6.121, "text": "the power of square < TK X and here's"}, {"start": 92.479, "duration": 5.161, "text": "its derivative that we just found notice"}, {"start": 95.6, "duration": 5.36, "text": "how the derivative shows the rate of"}, {"start": 97.64, "duration": 5.72, "text": "change of our original function at each"}, {"start": 100.96, "duration": 5.6, "text": "point and that's how we find the"}, {"start": 103.36, "duration": 6.799, "text": "derivative of e to the power of square"}, {"start": 106.56, "duration": 5.68, "text": "otk of X using the chain rule remember"}, {"start": 110.159, "duration": 4.92, "text": "when dealing with composite functions"}, {"start": 112.24, "duration": 6.36, "text": "like this always break it down into"}, {"start": 115.079, "duration": 5.761, "text": "steps using the chain Rule and multiply"}, {"start": 118.6, "duration": 5.879, "text": "the derivatives of the outer and inner"}, {"start": 120.84, "duration": 3.639, "text": "functions together"}], "title": "9.3-2.1 Mastering the Chain Rule: Differentiating (\ud835\udc52^\u221a\ud835\udc65)  Step-by-Step", "description": "Unlock the secrets of differentiation with our in-depth tutorial on finding the derivative of (\ud835\udc52^\u221a\ud835\udc65) . Follow along as we guide you through each step, using the chain rule to achieve a clear understanding of this complex mathematical process. Perfect for students and math enthusiasts looking to enhance their calculus skills!\nHashtags:\n#cpmrana  #MathTutorial #Calculus #ChainRule #Differentiation #ExponentialFunctions", "lengthSeconds": 125, "width": 640, "height": 360}
{"video_id": "rkovRw9_Qac", "transcript": [{"start": 0.24, "duration": 5.16, "text": "let's solve this differentiation problem"}, {"start": 2.56, "duration": 7.32, "text": "step by step we need to differentiate"}, {"start": 5.4, "duration": 8.72, "text": "the expression x^2 + 1 * inverse tangent"}, {"start": 9.88, "duration": 7.319, "text": "of x - x with respect to X let's break"}, {"start": 14.12, "duration": 5.079, "text": "this down into manageable steps first"}, {"start": 17.199, "duration": 4.561, "text": "we'll use the product rule on the first"}, {"start": 19.199, "duration": 4.641, "text": "term since it's a product of two"}, {"start": 21.76, "duration": 5.88, "text": "functions and then separately"}, {"start": 23.84, "duration": 6.48, "text": "differentiate the minus X term now let's"}, {"start": 27.64, "duration": 8.04, "text": "evaluate each part the derivative of"}, {"start": 30.32, "duration": 8.8, "text": "inverse tangent is 1/ 1 + x^2 the"}, {"start": 35.68, "duration": 8.12, "text": "derivative of x^2 + 1 is"}, {"start": 39.12, "duration": 6.64, "text": "2X and the derivative of x is simply 1"}, {"start": 43.8, "duration": 8.0, "text": "substituting these derivatives back into"}, {"start": 45.76, "duration": 11.279, "text": "our expression we get x^2 + 1 * 1 1 +"}, {"start": 51.8, "duration": 8.239, "text": "x^2 plus inverse tangent x * 2x - 1"}, {"start": 57.039, "duration": 8.641, "text": "Let's simplify the first term notice"}, {"start": 60.039, "duration": 8.961, "text": "that x^2 + 1 over 1 + x^2 simplifies to"}, {"start": 65.68, "duration": 7.28, "text": "1 since these terms are identical"}, {"start": 69.0, "duration": 6.32, "text": "finally we can combine like terms 1 -"}, {"start": 72.96, "duration": 7.479, "text": "one cancels out leaving us with our"}, {"start": 75.32, "duration": 7.56, "text": "final answer 2x * inverse tangent X to"}, {"start": 80.439, "duration": 5.481, "text": "verify this answer you could take the"}, {"start": 82.88, "duration": 7.12, "text": "derivative of our result and check if it"}, {"start": 85.92, "duration": 6.4, "text": "matches the original function or verify"}, {"start": 90.0, "duration": 5.92, "text": "that this solution satisfies any given"}, {"start": 92.32, "duration": 3.6, "text": "initial conditions"}], "title": "Understanding the Differentiation of [(\ud835\udc65^2+1) \ud835\udc61\ud835\udc4e\ud835\udc5b^(\u22121)  \ud835\udc65\u2212\ud835\udc65]", "description": "Question: Differentiate [(\ud835\udc65^2+1) \ud835\udc61\ud835\udc4e\ud835\udc5b^(\u22121)  \ud835\udc65\u2212\ud835\udc65]  with respect to x.\nIn this video, we explore the step-by-step differentiation of [(\ud835\udc65^2+1) \ud835\udc61\ud835\udc4e\ud835\udc5b^(\u22121)  \ud835\udc65\u2212\ud835\udc65]. Watch as we break down the process using the product rule and basic differentiation principles to arrive at the simplified expression. Perfect for students and math enthusiasts looking to deepen their understanding of calculus!\n\n#Calculus #Differentiation #MathTutorial #ProductRule #ArctanDifferentiation", "lengthSeconds": 99, "width": 640, "height": 360}
{"video_id": "8npowRDEsyg", "transcript": [{"start": 0.32, "duration": 5.6, "text": "let's solve this derivative step by step"}, {"start": 3.32, "duration": 5.199, "text": "we'll start with the original expression"}, {"start": 5.92, "duration": 5.639, "text": "and simplify it before taking the"}, {"start": 8.519, "duration": 8.441, "text": "derivative first let's recall the double"}, {"start": 11.559, "duration": 10.241, "text": "angle formula for cosine cine of 2x = c^"}, {"start": 16.96, "duration": 8.319, "text": "2 x - sin^2 x we'll substitute this into"}, {"start": 21.8, "duration": 5.96, "text": "our expression now let's distribute the"}, {"start": 25.279, "duration": 7.881, "text": "negative sign in the numerator this"}, {"start": 27.76, "duration": 10.2, "text": "gives us cosine x - cos sin^2 x + s^ 2"}, {"start": 33.16, "duration": 8.559, "text": "x/ 1 - cosine X here's where we can use"}, {"start": 37.96, "duration": 10.16, "text": "another important trigonometric identity"}, {"start": 41.719, "duration": 11.64, "text": "remember that sin^2 X+ cine 2 x = 1 so"}, {"start": 48.12, "duration": 8.52, "text": "sin^2 x = 1 - cosine 2 x let's"}, {"start": 53.359, "duration": 10.081, "text": "substitute this into our expression now"}, {"start": 56.64, "duration": 10.799, "text": "we have cine x - c^ 2 X x + 1 - c^ 2 X"}, {"start": 63.44, "duration": 7.64, "text": "all over 1 - cine X let's group terms in"}, {"start": 67.439, "duration": 7.36, "text": "the numerator first we can factor out 1"}, {"start": 71.08, "duration": 6.399, "text": "- cosine X from the first two terms then"}, {"start": 74.799, "duration": 8.561, "text": "we can Factor the remaining terms with 1"}, {"start": 77.479, "duration": 9.481, "text": "- cine x * 1 + cine X now we can cancel"}, {"start": 83.36, "duration": 6.56, "text": "the 1 - cine X terms in the numerator"}, {"start": 86.96, "duration": 7.159, "text": "and denominator this simplifies our EXP"}, {"start": 89.92, "duration": 6.839, "text": "expression to 2 cine x + 1 now that we"}, {"start": 94.119, "duration": 5.0, "text": "have simplified our expression we can"}, {"start": 96.759, "duration": 6.36, "text": "easily take the derivative the"}, {"start": 99.119, "duration": 7.32, "text": "derivative of cine X is sinx and the"}, {"start": 103.119, "duration": 8.28, "text": "derivative of a constant is zero"}, {"start": 106.439, "duration": 8.081, "text": "therefore our final answer is -2 sinx"}, {"start": 111.399, "duration": 5.72, "text": "this completes our solution to visualize"}, {"start": 114.52, "duration": 7.32, "text": "this here's what the original function"}, {"start": 117.119, "duration": 4.721, "text": "and its derivative look like graphically"}], "title": "Simplifying Trigonometric Expressions [(cos\u2061\ud835\udc65\u2212cos\u20612\ud835\udc65)/(1\u2212cos\u2061\ud835\udc65)] Calculus with Cosine Functions", "description": "Join us as we break down the process of simplifying and differentiating complex trigonometric expressions. Learn how the expression  [(cos\u2061\ud835\udc65\u2212cos\u20612\ud835\udc65)/(1\u2212cos\u2061\ud835\udc65)]   simplifies to (2cosx+)1  and its differentiation to (-2sinx) . Perfect for students aiming to conquer trigonometric identities and calculus differentiation techniques.\n#Trigonometry #Calculus #MathTutorial #Differentiation #LearnMath", "lengthSeconds": 122, "width": 640, "height": 360}
{"video_id": "yOJQ5OlvSAo", "transcript": [{"start": 0.28, "duration": 5.079, "text": "let's solve this calculus problem step"}, {"start": 2.44, "duration": 7.56, "text": "by step using the quotient rule we need"}, {"start": 5.359, "duration": 8.16, "text": "to find the derivative of 1+ sinx / 1 +"}, {"start": 10.0, "duration": 7.359, "text": "cosine X first let's recall the quotient"}, {"start": 13.519, "duration": 7.561, "text": "rule the derivative of U over v = v *"}, {"start": 17.359, "duration": 8.281, "text": "the Der of uus U * the derivative of V"}, {"start": 21.08, "duration": 9.439, "text": "all over V ^2 let's identify our u and v"}, {"start": 25.64, "duration": 9.239, "text": "functions U of x = 1 + sin x and"}, {"start": 30.519, "duration": 7.121, "text": "V ofx = 1 + cosine X next we need to"}, {"start": 34.879, "duration": 6.081, "text": "find the derivatives of u and v the"}, {"start": 37.64, "duration": 7.2, "text": "derivative of U is cine X and the"}, {"start": 40.96, "duration": 6.2, "text": "derivative of V is sinx now let's"}, {"start": 44.84, "duration": 8.239, "text": "substitute these into the quotient rule"}, {"start": 47.16, "duration": 12.28, "text": "formula we get 1 + cosine x * cine X - 1"}, {"start": 53.079, "duration": 9.561, "text": "+ sin x * sinx all over 1 + cine x^ 2"}, {"start": 59.44, "duration": 7.999, "text": "let's distribute in the numerator on the"}, {"start": 62.64, "duration": 10.08, "text": "left side we get cine x + cine 2 X on"}, {"start": 67.439, "duration": 7.561, "text": "the right side we get sin x + sin^2 x"}, {"start": 72.72, "duration": 7.2, "text": "combining these terms in the numerator"}, {"start": 75.0, "duration": 11.2, "text": "we get cosine x + cosine 2 x + sin x +"}, {"start": 79.92, "duration": 12.479, "text": "sin 2 x all over 1 + cosine x\u00b2 now"}, {"start": 86.2, "duration": 9.279, "text": "remember that sin^2 x + cosine 2 x = 1"}, {"start": 92.399, "duration": 8.121, "text": "using this identity we can simplify our"}, {"start": 95.479, "duration": 8.481, "text": "numerator to 1 + sin x + cine X and this"}, {"start": 100.52, "duration": 5.239, "text": "is our final answer we've successfully"}, {"start": 103.96, "duration": 5.08, "text": "found the derivative using the quotient"}, {"start": 105.759, "duration": 6.561, "text": "rule to verify this is correct we can"}, {"start": 109.04, "duration": 6.119, "text": "check that one we use the quotient rule"}, {"start": 112.32, "duration": 7.759, "text": "properly two we found the derivatives of"}, {"start": 115.159, "duration": 7.0, "text": "u and v correctly and three we simp"}, {"start": 120.079, "duration": 5.521, "text": "ified our trigonometric Expressions"}, {"start": 122.159, "duration": 6.16, "text": "correctly using the fundamental identity"}, {"start": 125.6, "duration": 5.96, "text": "all these steps check out so our answer"}, {"start": 128.319, "duration": 3.241, "text": "is correct"}], "title": "Master the Quotient Rule: Differentiating [(\ud835\udfcf+\ud835\udc94\ud835\udc8a\ud835\udc8f\ud835\udc99)/(\ud835\udfcf+\ud835\udc84\ud835\udc90\ud835\udc94\u2061\ud835\udc99)]  Explained", "description": "Join us in this engaging math tutorial where we break down the differentiation of the function [(\ud835\udfcf+\ud835\udc94\ud835\udc8a\ud835\udc8f\ud835\udc99)/(\ud835\udfcf+\ud835\udc84\ud835\udc90\ud835\udc94\u2061\ud835\udc99)]  using the powerful quotient rule. Perfect for students, teachers, and math enthusiasts, this video provides a step-by-step guide to understanding and applying calculus principles with ease. Strengthen your calculus skills and gain confidence with each problem solved.\n#Calculus #QuotientRule #MathTutorial #Differentiation #Derivatives", "lengthSeconds": 130, "width": 640, "height": 360}
{"video_id": "gKK3fFmp3X4", "transcript": [{"start": 0.28, "duration": 5.2, "text": "let's solve this calculus problem step"}, {"start": 2.72, "duration": 4.96, "text": "by step using the quotient rule we need"}, {"start": 5.48, "duration": 8.88, "text": "to find the derivative of the function f"}, {"start": 7.68, "duration": 10.16, "text": "ofx = the < TK x + 1 over the < TK of x"}, {"start": 14.36, "duration": 6.8, "text": "-1 first let's recall the quotient rule"}, {"start": 17.84, "duration": 7.199, "text": "formula the derivative of U over V equal"}, {"start": 21.16, "duration": 7.519, "text": "V * the derivative of U minus U * the"}, {"start": 25.039, "duration": 9.121, "text": "derivative of V all over V ^2 let's"}, {"start": 28.679, "duration": 12.001, "text": "identify our U and V functions here = <"}, {"start": 34.16, "duration": 8.52, "text": "TK of x + 1 and V = sare < TK of x -1"}, {"start": 40.68, "duration": 5.199, "text": "next we need to find the derivatives of"}, {"start": 42.68, "duration": 9.679, "text": "u and v the derivative of square otk of"}, {"start": 45.879, "duration": 10.441, "text": "X is 1 / 2 < TK X so both do DX and dvdx"}, {"start": 52.359, "duration": 6.281, "text": "equal 1 / 2 s < TK of X now let's"}, {"start": 56.32, "duration": 6.6, "text": "substitute everything into the quotient"}, {"start": 58.64, "duration": 12.4, "text": "rule formula we get square < TK of x -1"}, {"start": 62.92, "duration": 12.839, "text": "* 1 2 < TK x - < TK of x +1 * 1 2 < TK X"}, {"start": 71.04, "duration": 8.439, "text": "all over < TK x - 1^ 2 let's simplify"}, {"start": 75.759, "duration": 7.881, "text": "the numerator first we can factor out 1/"}, {"start": 79.479, "duration": 9.161, "text": "2 < TK X from both terms then inside the"}, {"start": 83.64, "duration": 7.08, "text": "parentheses we have S < TK of x - 1 - <"}, {"start": 88.64, "duration": 5.479, "text": "TK of x + 1"}, {"start": 90.72, "duration": 6.759, "text": "simplify further the square root of x"}, {"start": 94.119, "duration": 7.201, "text": "terms cancel out and we're left with -2"}, {"start": 97.479, "duration": 9.761, "text": "in the brackets so our numerator becomes"}, {"start": 101.32, "duration": 13.0, "text": "-1/ < TK X and finally our answer is -1"}, {"start": 107.24, "duration": 9.44, "text": "/ < TK x * < TK of x - 1^ 2 this"}, {"start": 114.32, "duration": 5.119, "text": "solution demonstrates the power of the"}, {"start": 116.68, "duration": 5.92, "text": "quotient rule in calculating derivatives"}, {"start": 119.439, "duration": 5.72, "text": "of complex fractions remember to"}, {"start": 122.6, "duration": 5.32, "text": "carefully identify your u and v"}, {"start": 125.159, "duration": 4.761, "text": "functions find their derivatives and"}, {"start": 127.92, "duration": 5.64, "text": "then apply the quotient rule formula"}, {"start": 129.92, "duration": 3.64, "text": "step by step"}], "title": "Mastering the Quotient Rule: Differentiating Fractions ((\u221a\ud835\udc99+\ud835\udfcf)/(\u221a\ud835\udc99\u2212\ud835\udfcf)) Involving Radicals", "description": "Learn how to differentiate complex rational functions involving radicals using the quotient rule. In this tutorial, we\u2019ll guide you through each step of finding the derivative of ((\u221a\ud835\udc99+\ud835\udfcf)/(\u221a\ud835\udc99\u2212\ud835\udfcf)). Perfect for students looking to excel in calculus, this video makes differentiation simple and accessible!\n#Calculus #QuotientRule #Differentiation #Radicals #MathHelp", "lengthSeconds": 132, "width": 640, "height": 360}
{"video_id": "0Kv20Dl8ua0", "transcript": [{"start": 0.359, "duration": 4.881, "text": "let's solve this challenging derivative"}, {"start": 2.639, "duration": 5.601, "text": "problem step by step we need to"}, {"start": 5.24, "duration": 7.64, "text": "differentiate sin squar of natural log"}, {"start": 8.24, "duration": 7.0, "text": "of x^2 with respect to X this is a"}, {"start": 12.88, "duration": 4.64, "text": "complex function with multiple nested"}, {"start": 15.24, "duration": 5.959, "text": "parts let's break it down using the"}, {"start": 17.52, "duration": 7.88, "text": "chain rule we have sin squar as our"}, {"start": 21.199, "duration": 6.48, "text": "outer function then natural log of x^2"}, {"start": 25.4, "duration": 5.48, "text": "as our inner function let's apply the"}, {"start": 27.679, "duration": 7.241, "text": "chain rule step by step first the"}, {"start": 30.88, "duration": 7.199, "text": "derivative of sin s of U is 2 sin of U *"}, {"start": 34.92, "duration": 6.36, "text": "the derivative of s of U next the"}, {"start": 38.079, "duration": 6.081, "text": "derivative of s of U is cosine of U *"}, {"start": 41.28, "duration": 6.959, "text": "the derivative of U so our expression"}, {"start": 44.16, "duration": 7.559, "text": "becomes 2 sin of natural log of x^2 *"}, {"start": 48.239, "duration": 7.64, "text": "cosine of natural log of x^2 * the"}, {"start": 51.719, "duration": 4.16, "text": "derivative of natural log of"}, {"start": 56.359, "duration": 4.481, "text": "X2 now the derivative of natural log of"}, {"start": 59.519, "duration": 6.841, "text": "x"}, {"start": 60.84, "duration": 5.52, "text": "is 1 /x^2 * the derivative of"}, {"start": 66.6, "duration": 7.8, "text": "x^2 the derivative of x^2 is 2x so our"}, {"start": 71.479, "duration": 8.6, "text": "expression becomes 2 sin of natural log"}, {"start": 74.4, "duration": 9.32, "text": "of x^2 * cosine of natural log of x^2 *"}, {"start": 80.079, "duration": 3.641, "text": "1x^2 *"}, {"start": 84.04, "duration": 8.039, "text": "2x simplifying this expression we get"}, {"start": 87.479, "duration": 9.121, "text": "four sign of natural log of x^2 * cosine"}, {"start": 92.079, "duration": 4.521, "text": "of natural log of x^2"}, {"start": 97.079, "duration": 6.241, "text": "/x now we can use the double angle"}, {"start": 99.68, "duration": 8.439, "text": "formula for S which states that 2 sin of"}, {"start": 103.32, "duration": 7.6, "text": "U * cosine of U equal s of 2"}, {"start": 108.119, "duration": 6.761, "text": "you applying this formula to our"}, {"start": 110.92, "duration": 8.4, "text": "expression we get 2 sin of 2 natural log"}, {"start": 114.88, "duration": 4.44, "text": "of x^2 /x"}, {"start": 120.0, "duration": 6.6, "text": "finally we can simplify this further by"}, {"start": 122.439, "duration": 6.68, "text": "noting that 2 natural log of x^2 = 4"}, {"start": 126.6, "duration": 6.68, "text": "natural log of x giving us our final"}, {"start": 129.119, "duration": 7.361, "text": "answer 2 sin of 4 natural log of"}, {"start": 133.28, "duration": 5.72, "text": "XX and there we have it we've"}, {"start": 136.48, "duration": 5.32, "text": "successfully differentiated this complex"}, {"start": 139.0, "duration": 5.56, "text": "function using the chain Rule and"}, {"start": 141.8, "duration": 6.2, "text": "trigonometric identities remember to"}, {"start": 144.56, "duration": 7.08, "text": "break down complex problems into smaller"}, {"start": 148.0, "duration": 3.64, "text": "manageable steps"}], "title": "Calculus Made Easy: Differentiating [\ud835\udc94\ud835\udc8a\ud835\udc8f^\ud835\udfd0 {\ud835\udc8d\ud835\udc8f\u2061(\ud835\udc99^\ud835\udfd0)}]  Step-by-Step", "description": "Unlock the secrets of calculus with our detailed walkthrough on differentiating the expression [\ud835\udc94\ud835\udc8a\ud835\udc8f^\ud835\udfd0 {\ud835\udc8d\ud835\udc8f\u2061(\ud835\udc99^\ud835\udfd0)}] . This video provides a comprehensive, step-by-step explanation, perfect for students and enthusiasts looking to strengthen their understanding of calculus concepts. Learn how to apply the chain rule effectively and simplify complex derivatives with ease.\n#Calculus #MathTutorial #Derivatives #ChainRule #MathHelp", "lengthSeconds": 154, "width": 640, "height": 360}
{"video_id": "4nwK12ZYkMA", "transcript": [{"start": 0.28, "duration": 5.28, "text": "let's solve this calculus problem step"}, {"start": 2.399, "duration": 6.761, "text": "by step we need to find the derivative"}, {"start": 5.56, "duration": 6.6, "text": "of a to the power of x * x^2 with"}, {"start": 9.16, "duration": 5.24, "text": "respect to X this is a product of two"}, {"start": 12.16, "duration": 5.199, "text": "functions so we'll need to use the"}, {"start": 14.4, "duration": 6.56, "text": "product rule the product rule states"}, {"start": 17.359, "duration": 6.521, "text": "that the derivative of U * V equal U *"}, {"start": 20.96, "duration": 6.36, "text": "the derivative of V plus v * the"}, {"start": 23.88, "duration": 8.839, "text": "derivative of U let's identify our u and"}, {"start": 27.32, "duration": 8.12, "text": "v in this case u = a the X and B = x^2"}, {"start": 32.719, "duration": 6.281, "text": "now let's find each derivative"}, {"start": 35.44, "duration": 8.119, "text": "separately first the derivative of x^2"}, {"start": 39.0, "duration": 8.16, "text": "is 2X and the derivative of a to X is a"}, {"start": 43.559, "duration": 6.16, "text": "to x * the natural log of let's"}, {"start": 47.16, "duration": 8.64, "text": "substitute these into the product rule"}, {"start": 49.719, "duration": 9.121, "text": "we get a to x * 2x + x^2 * a to x *"}, {"start": 55.8, "duration": 5.919, "text": "natural log of a now let's factor out"}, {"start": 58.84, "duration": 8.4, "text": "the common terms we can factor out a to"}, {"start": 61.719, "duration": 9.76, "text": "the X giving us a to x * thean 2x + x^2"}, {"start": 67.24, "duration": 7.36, "text": "* natural log of a for our final step we"}, {"start": 71.479, "duration": 6.601, "text": "can factor out x^2 from inside the"}, {"start": 74.6, "duration": 9.72, "text": "parentheses this gives us a to the x *"}, {"start": 78.08, "duration": 8.92, "text": "x^2 * 2 /x + natural log of a and this"}, {"start": 84.32, "duration": 5.119, "text": "is our final answer let's review what we"}, {"start": 87.0, "duration": 4.0, "text": "did we used the product rule because we"}, {"start": 89.439, "duration": 4.081, "text": "were different differentiating a product"}, {"start": 91.0, "duration": 5.399, "text": "of functions we found the derivatives of"}, {"start": 93.52, "duration": 4.8, "text": "each part separately and then we"}, {"start": 96.399, "duration": 4.561, "text": "factored our expression to get it into"}, {"start": 98.32, "duration": 5.28, "text": "its simplest form here's our complete"}, {"start": 100.96, "duration": 6.36, "text": "solution one more time starting from a"}, {"start": 103.6, "duration": 6.119, "text": "to the power of x * x^ 2 we use the"}, {"start": 107.32, "duration": 7.88, "text": "product rule and ended up with a to the"}, {"start": 109.719, "duration": 8.481, "text": "power of x * x^2 * 2/x + natural log of"}, {"start": 115.2, "duration": 3.0, "text": "a"}], "title": "Finding the Derivative of (a^x .x^2)  Using the Product Rule | Step-by-Step Explanation.", "description": "In this video, learn to differentiate  (a^x .x^2) using the Product Rule in calculus.\n\nWhat You'll Learn:  \n- Definition of the Product Rule  \n- Step-by-step differentiation of  (a^x .x^2)\n- Formula breakdown and example problem-solving  \n\nThis example is perfect for calculus beginners or those reviewing the Product Rule!\n\ud83d\udca1 Don't forget to LIKE, SHARE, and SUBSCRIBE for more math tutorials!\n#Calculus #ProductRule #Differentiation #Derivative #MathTutorial #LearnMath #MathEducation #MathHelp", "lengthSeconds": 119, "width": 640, "height": 360}
{"video_id": "Q7cnbmnpe4o", "transcript": [{"start": 0.28, "duration": 5.559, "text": "let's solve this calculus problem step"}, {"start": 2.639, "duration": 5.641, "text": "by step using the quotient rule we need"}, {"start": 5.839, "duration": 6.241, "text": "to find the derivative of cosine Theta"}, {"start": 8.28, "duration": 6.68, "text": "minus sin Theta over cosine theta plus"}, {"start": 12.08, "duration": 5.92, "text": "sin Theta first let's recall the"}, {"start": 14.96, "duration": 6.64, "text": "quotient rule the derivative of U over V"}, {"start": 18.0, "duration": 7.4, "text": "equal V * the derivative of uus U * the"}, {"start": 21.6, "duration": 6.759, "text": "derivative of V all over V sared before"}, {"start": 25.4, "duration": 5.119, "text": "we apply the quotient rule directly we"}, {"start": 28.359, "duration": 4.36, "text": "can simplify our expression"}, {"start": 30.519, "duration": 5.88, "text": "if we divide both numerator and"}, {"start": 32.719, "duration": 7.961, "text": "denominator by cine Theta we get 1us"}, {"start": 36.399, "duration": 9.241, "text": "tangent Theta over 1 + tangent Theta now"}, {"start": 40.68, "duration": 10.44, "text": "we can identify our u and v u = 1 - Tang"}, {"start": 45.64, "duration": 8.48, "text": "Theta and V = 1 + tangent Theta next we"}, {"start": 51.12, "duration": 7.0, "text": "need to find the derivatives of u and v"}, {"start": 54.12, "duration": 7.84, "text": "the derivative of U is secant thet and"}, {"start": 58.12, "duration": 6.039, "text": "the derivative of V is sec 2 th now"}, {"start": 61.96, "duration": 5.799, "text": "let's substitute everything into the"}, {"start": 64.159, "duration": 10.841, "text": "quotient rule formula we get 1 + tangent"}, {"start": 67.759, "duration": 11.321, "text": "th * SEC 2 thet - 1 - Tang thet * SEC"}, {"start": 75.0, "duration": 6.479, "text": "2et all over 1 + Tang Theta 2 let's"}, {"start": 79.08, "duration": 8.079, "text": "distribute in the numerator for the"}, {"start": 81.479, "duration": 9.161, "text": "first term we get secant 2 thetus secant"}, {"start": 87.159, "duration": 9.401, "text": "2 thet * tangent Theta for the second"}, {"start": 90.64, "duration": 8.96, "text": "term we get secant 2 Theta + secant 2"}, {"start": 96.56, "duration": 5.239, "text": "Theta * tangent Theta when we combine"}, {"start": 99.6, "duration": 6.68, "text": "like terms the tangent Theta terms"}, {"start": 101.799, "duration": 7.401, "text": "cancel out and we're left with -2 secant"}, {"start": 106.28, "duration": 5.799, "text": "Theta in the numerator and that's our"}, {"start": 109.2, "duration": 7.0, "text": "final answer the derivative of cine"}, {"start": 112.079, "duration": 9.32, "text": "Theta minus sin Theta over cine Theta +"}, {"start": 116.2, "duration": 7.839, "text": "sin Theta = -2 SEC 2 Theta over 1 +"}, {"start": 121.399, "duration": 4.961, "text": "tangent Theta 2 this problem"}, {"start": 124.039, "duration": 4.92, "text": "demonstrates the power of the quotient"}, {"start": 126.36, "duration": 5.239, "text": "rule combined with trigonometric"}, {"start": 128.959, "duration": 6.321, "text": "identities remember to always look for"}, {"start": 131.599, "duration": 7.881, "text": "ways to simplify your expression before"}, {"start": 135.28, "duration": 4.2, "text": "applying the quotient rule"}], "title": "Trigonometric Differentiation: Unraveling Complex Expressions [(\ud835\udc84\ud835\udc90\ud835\udc94  \ud835\udf3d\u2212\ud835\udc94\ud835\udc8a\ud835\udc8f \ud835\udf3d)/(\ud835\udc84\ud835\udc90\ud835\udc94  \ud835\udf3d+\ud835\udc94\ud835\udc8a\ud835\udc8f \ud835\udf3d)]", "description": "In this video, we delve into differentiating complex trigonometric expressions. Learn how to differentiate [(\ud835\udc84\ud835\udc90\ud835\udc94  \ud835\udf3d\u2212\ud835\udc94\ud835\udc8a\ud835\udc8f \ud835\udf3d)/(\ud835\udc84\ud835\udc90\ud835\udc94  \ud835\udf3d+\ud835\udc94\ud835\udc8a\ud835\udc8f \ud835\udf3d)] and simplify it to (( \u2212\ud835\udfd0\ud835\udc94\ud835\udc86\ud835\udc84^\ud835\udfd0 \ud835\udf3d )  )/(\ud835\udfcf+\ud835\udc95\ud835\udc82\ud835\udc8f \ud835\udf3d)^\ud835\udfd0 . Ideal for students aiming to master calculus and trigonometric functions.\n#Trigonometry #Calculus #Differentiation #MathHelp #AdvancedMath", "lengthSeconds": 139, "width": 640, "height": 360}
{"video_id": "74-XgXXZbaY", "transcript": [{"start": 0.32, "duration": 3.88, "text": "let's learn how to differentiate a"}, {"start": 2.28, "duration": 4.44, "text": "square root function containing a"}, {"start": 4.2, "duration": 5.68, "text": "quadratic expression we'll break this"}, {"start": 6.72, "duration": 5.999, "text": "down step by step to make it clear and"}, {"start": 9.88, "duration": 4.56, "text": "intuitive let's start with our function"}, {"start": 12.719, "duration": 6.72, "text": "we want to find the derivative of the"}, {"start": 14.44, "duration": 6.759, "text": "square root of a x^2 + BX + C this is a"}, {"start": 19.439, "duration": 5.201, "text": "complex function that we'll need to"}, {"start": 21.199, "duration": 6.361, "text": "handle carefully using the chain rule to"}, {"start": 24.64, "duration": 5.44, "text": "solve this we'll use the chain rule the"}, {"start": 27.56, "duration": 5.44, "text": "chain rule states that when we have a"}, {"start": 30.08, "duration": 5.04, "text": "composite function we multiply the"}, {"start": 33.0, "duration": 4.68, "text": "derivative of the outer function by the"}, {"start": 35.12, "duration": 5.08, "text": "derivative of the inner function in our"}, {"start": 37.68, "duration": 5.08, "text": "case the outer function is the square"}, {"start": 40.2, "duration": 6.999, "text": "root and the inner function is the"}, {"start": 42.76, "duration": 6.88, "text": "quadratic expression ax^2 + bx+ C let's"}, {"start": 47.199, "duration": 4.721, "text": "handle the outer function first the"}, {"start": 49.64, "duration": 6.52, "text": "derivative of the square root function"}, {"start": 51.92, "duration": 8.959, "text": "with respect to U is 1 over 2 * U to the"}, {"start": 56.16, "duration": 8.359, "text": "-2 power we can rewrite this as 1/ 2 *"}, {"start": 60.879, "duration": 7.88, "text": "the square < TK of U now for the inner"}, {"start": 64.519, "duration": 8.401, "text": "function the derivative of ax^2 + BX + C"}, {"start": 68.759, "duration": 7.321, "text": "is simply 2 ax + B now let's combine"}, {"start": 72.92, "duration": 5.8, "text": "these parts using the chain rule we take"}, {"start": 76.08, "duration": 5.2, "text": "the derivative of the outer function"}, {"start": 78.72, "duration": 5.759, "text": "replacing you with our inner function"}, {"start": 81.28, "duration": 6.159, "text": "and multiply it by the derivative of the"}, {"start": 84.479, "duration": 5.441, "text": "inner function finally we can write our"}, {"start": 87.439, "duration": 6.841, "text": "answer in its simplest form the D"}, {"start": 89.92, "duration": 9.199, "text": "derivative equal 2 a x + B / 2 * the <"}, {"start": 94.28, "duration": 7.92, "text": "TK of ax^2 + BX + C let's verify our"}, {"start": 99.119, "duration": 5.241, "text": "work we use the chain rule correctly by"}, {"start": 102.2, "duration": 4.76, "text": "multiplying the derivative of the square"}, {"start": 104.36, "duration": 6.439, "text": "root function by the derivative of what"}, {"start": 106.96, "duration": 6.72, "text": "was inside it our final answer maintains"}, {"start": 110.799, "duration": 5.92, "text": "the proper form and includes all terms"}, {"start": 113.68, "duration": 5.64, "text": "from both the numerator and denominator"}, {"start": 116.719, "duration": 4.121, "text": "and that's how we differentiate a square"}, {"start": 119.32, "duration": 4.399, "text": "root function function containing a"}, {"start": 120.84, "duration": 5.239, "text": "quadratic expression remember to always"}, {"start": 123.719, "duration": 5.16, "text": "apply the chain rule carefully and keep"}, {"start": 126.079, "duration": 5.8, "text": "track of both the outer and inner"}, {"start": 128.879, "duration": 3.0, "text": "functions"}], "title": "9.3-1.3 Deriving the Square Root of Quadratic Functions (\u221a(\ud835\udc4e\ud835\udc65\u00b2 + \ud835\udc4f\ud835\udc65 + \ud835\udc50)): A Comprehensive Guide.", "description": "Dive into the fundamentals of calculus with our detailed tutorial on differentiating functions involving square roots. We\u2019ll break down the process step-by-step using the chain rule, enhancing your understanding of how to handle complex derivatives. Perfect for calculus students and anyone looking to solidify their calculus skills.\n#Calculus #Differentiation #SquareRoots #ChainRule #MathTutorial #cpmrana", "lengthSeconds": 131, "width": 640, "height": 360}
{"video_id": "ZP6P63JQh8c", "transcript": [{"start": 0.28, "duration": 4.16, "text": "today we'll solve a challenging"}, {"start": 2.28, "duration": 5.2, "text": "differentiation problem involving"}, {"start": 4.44, "duration": 5.4, "text": "inverse trigonometric functions let's"}, {"start": 7.48, "duration": 5.96, "text": "look at the question let's solve the"}, {"start": 9.84, "duration": 6.48, "text": "step by step first recall that when"}, {"start": 13.44, "duration": 6.28, "text": "differentiating inverse tangent we use"}, {"start": 16.32, "duration": 7.799, "text": "the formula the derivative of AR tangent"}, {"start": 19.72, "duration": 9.84, "text": "of u = 1 1 + u^2 time the derivative of"}, {"start": 24.119, "duration": 9.521, "text": "U in our case U = cosine x/ 1 + sin x"}, {"start": 29.56, "duration": 8.44, "text": "apply this formula we get 1/ 1 + u^2"}, {"start": 33.64, "duration": 8.64, "text": "where U is cosine X over 1 + sin x all"}, {"start": 38.0, "duration": 10.399, "text": "sared this can be Rewritten as 1/ 1 +"}, {"start": 42.28, "duration": 8.759, "text": "cine 2 x overan 1 + sin x^2 now for the"}, {"start": 48.399, "duration": 7.721, "text": "second part we need to differentiate"}, {"start": 51.039, "duration": 8.241, "text": "cine X over 1 + sinx this requires the"}, {"start": 56.12, "duration": 5.64, "text": "quotient rule remember the quotient rule"}, {"start": 59.28, "duration": 7.64, "text": "states that the derivative of f over G"}, {"start": 61.76, "duration": 9.2, "text": "equal G * the derivative of f - f * the"}, {"start": 66.92, "duration": 6.48, "text": "derivative of G all over G ^2 combining"}, {"start": 70.96, "duration": 5.64, "text": "these parts and simplifying the"}, {"start": 73.4, "duration": 5.759, "text": "denominator we get this fraction now we"}, {"start": 76.6, "duration": 9.0, "text": "can use the trigonometric identity sign"}, {"start": 79.159, "duration": 9.881, "text": "2 x + cosine 2 x = one to simplify"}, {"start": 85.6, "duration": 7.96, "text": "further after applying this identity and"}, {"start": 89.04, "duration": 8.48, "text": "combining like terms we get sinx - 1 / 2"}, {"start": 93.56, "duration": 7.199, "text": "sin x + 2 we can factor out -1 from the"}, {"start": 97.52, "duration": 8.68, "text": "numerator and two from the denominator"}, {"start": 100.759, "duration": 9.601, "text": "giving us -1 over2 * sin x + 1 / sin x +"}, {"start": 106.2, "duration": 8.599, "text": "1 the sin x + 1 terms cancel out leaving"}, {"start": 110.36, "duration": 6.719, "text": "us with our final answer -2 and there we"}, {"start": 114.799, "duration": 4.64, "text": "have it we've successfully"}, {"start": 117.079, "duration": 4.241, "text": "differentiated this complex inverse"}, {"start": 119.439, "duration": 4.841, "text": "Trigon ometric expression using the"}, {"start": 121.32, "duration": 5.24, "text": "chain rule quotient Rule and"}, {"start": 124.28, "duration": 8.839, "text": "trigonometric identities to arrive at"}, {"start": 126.56, "duration": 6.559, "text": "our surprisingly simple answer of - one2"}], "title": "Derivative of Arctan: A Step-by-Step Trigonometric Solution [\ud835\udc61\ud835\udc4e\ud835\udc5b^(\u22121) (cos\u2061\ud835\udc65/(1+sin\u2061\ud835\udc65 ))]", "description": "Question: Differentiate [\ud835\udc61\ud835\udc4e\ud835\udc5b^(\u22121) (cos\u2061\ud835\udc65/(1+sin\u2061\ud835\udc65 ))]  with respect to x .\n\nIn this video, we explore the process of differentiating the function  [\ud835\udc61\ud835\udc4e\ud835\udc5b^(\u22121) (cos\u2061\ud835\udc65/(1+sin\u2061\ud835\udc65 ))] step by step. We\u2019ll apply various calculus techniques, including the derivative of inverse trigonometric functions, the quotient rule, and key trigonometric identities. By the end, you\u2019ll understand how to break down complex expressions into simpler components. Perfect for calculus students and trigonometry enthusiasts!\n #Calculus #Trigonometry #Derivative #InverseTrigonometricFunctions #MathTutorial", "lengthSeconds": 136, "width": 640, "height": 360}
{"video_id": "iR52F1l_qW0", "transcript": [{"start": 0.28, "duration": 4.92, "text": "welcome today we'll solve a challenging"}, {"start": 3.0, "duration": 5.04, "text": "differentiation problem involving an"}, {"start": 5.2, "duration": 5.76, "text": "inverse sign function with a complex"}, {"start": 8.04, "duration": 5.04, "text": "argument let's start with our problem we"}, {"start": 10.96, "duration": 7.239, "text": "need to find the derivative of sin"}, {"start": 13.08, "duration": 8.76, "text": "inverse of the square < TK of x * e to X"}, {"start": 18.199, "duration": 5.481, "text": "let's break this down step by step first"}, {"start": 21.84, "duration": 5.04, "text": "recall the derivative formula for"}, {"start": 23.68, "duration": 7.8, "text": "inverse sign the derivative of inverse"}, {"start": 26.88, "duration": 8.76, "text": "sign of U equal 1/ theare < TK of 1 - U"}, {"start": 31.48, "duration": 8.36, "text": "^2 time the derivative of U in our case"}, {"start": 35.64, "duration": 7.88, "text": "U equal the sare < TK of x e to X let's"}, {"start": 39.84, "duration": 3.68, "text": "substitute this into our"}, {"start": 43.879, "duration": 6.081, "text": "formula now let's simplify what's under"}, {"start": 47.32, "duration": 5.44, "text": "the first square root notice that when"}, {"start": 49.96, "duration": 7.2, "text": "we square the square root of x e to the"}, {"start": 52.76, "duration": 4.4, "text": "X we just get x e to"}, {"start": 57.359, "duration": 7.681, "text": "X next let's focus on the derivative of"}, {"start": 61.079, "duration": 6.601, "text": "square < TK of x e to X we can use the"}, {"start": 65.04, "duration": 6.719, "text": "chain rule here the derivative of square"}, {"start": 67.68, "duration": 6.72, "text": "otk of U is 1 / 2 < TK U time the"}, {"start": 71.759, "duration": 2.641, "text": "derivative of"}, {"start": 74.439, "duration": 7.841, "text": "U now we need to find the derivative of"}, {"start": 77.799, "duration": 6.601, "text": "x e to X this requires the product rule"}, {"start": 82.28, "duration": 6.879, "text": "the derivative will be x * the"}, {"start": 84.4, "duration": 7.44, "text": "derivative of e to X Plus e to x * the"}, {"start": 89.159, "duration": 7.841, "text": "derivative of x"}, {"start": 91.84, "duration": 5.16, "text": "X let's substitute this back into our"}, {"start": 97.439, "duration": 6.72, "text": "equation finally we can write this as a"}, {"start": 100.84, "duration": 8.76, "text": "single fraction our final answer is x e"}, {"start": 104.159, "duration": 12.041, "text": "to x + e to X all over 2 * the < TK of x"}, {"start": 109.6, "duration": 6.6, "text": "e to x * the sare < TK of 1 - x e to the"}, {"start": 116.24, "duration": 5.32, "text": "X let's verify this is correct by"}, {"start": 119.159, "duration": 4.841, "text": "checking our step steps we use the"}, {"start": 121.56, "duration": 5.04, "text": "inverse sign derivative formula apply"}, {"start": 124.0, "duration": 6.16, "text": "the chain rule correctly use the product"}, {"start": 126.6, "duration": 6.08, "text": "rule for x e to the X and simplified our"}, {"start": 130.16, "duration": 6.2, "text": "expression all steps follow standard"}, {"start": 132.68, "duration": 7.279, "text": "calculus rules so our answer is correct"}, {"start": 136.36, "duration": 5.64, "text": "and that completes our solution remember"}, {"start": 139.959, "duration": 4.881, "text": "when dealing with complex derivatives"}, {"start": 142.0, "duration": 5.599, "text": "like this breaking them down into"}, {"start": 144.84, "duration": 5.28, "text": "smaller manageable steps using standard"}, {"start": 147.599, "duration": 7.601, "text": "derivative rules makes the problem much"}, {"start": 150.12, "duration": 5.08, "text": "easier to solve thank you for watching"}], "title": "Mastering Derivatives: Differentiating Inverse Trigonometric Functions [\ud835\udc60\ud835\udc56\ud835\udc5b^(\u22121) (\u221a(\ud835\udc65\ud835\udc52^\ud835\udc65 ))]", "description": "Problem Statement\nDifferentiate the function [\ud835\udc60\ud835\udc56\ud835\udc5b^(\u22121) (\u221a(\ud835\udc65\ud835\udc52^\ud835\udc65 ))]  with respect to x.\nUnlock the secrets to differentiating inverse trigonometric functions with this comprehensive step-by-step guide! In this video, we break down the process of finding the derivative of [\ud835\udc60\ud835\udc56\ud835\udc5b^(\u22121) (\u221a(\ud835\udc65\ud835\udc52^\ud835\udc65 ))] using chain and product rules, explained clearly and concisely. Perfect for students and math enthusiasts looking to deepen their calculus skills. Don\u2019t miss out on understanding every detail with our two-column explanation method!\n#Calculus #Derivatives #MathTutorial #InverseTrigonometry #ChainRule #ProductRule #MathHelp #LearnMath #StepByStep", "lengthSeconds": 158, "width": 640, "height": 360}
{"video_id": "F2WUaREKH3I", "transcript": [{"start": 0.28, "duration": 5.559, "text": "let's solve this complex derivative step"}, {"start": 2.919, "duration": 6.76, "text": "by step we're going to differentiate sin"}, {"start": 5.839, "duration": 6.76, "text": "cubed of X cubed using the chain rule"}, {"start": 9.679, "duration": 5.04, "text": "let's start with our original expression"}, {"start": 12.599, "duration": 5.881, "text": "we need to find the derivative of sin"}, {"start": 14.719, "duration": 6.841, "text": "Cub of X cubed we can rewrite this as s"}, {"start": 18.48, "duration": 5.52, "text": "of X cubed all cubed to make it clear"}, {"start": 21.56, "duration": 5.719, "text": "what we're dealing with now let's apply"}, {"start": 24.0, "duration": 5.48, "text": "the chain rule first we'll use the power"}, {"start": 27.279, "duration": 5.881, "text": "rule on the outer function the"}, {"start": 29.48, "duration": 7.44, "text": "derivative of U cubed is 3 U ^2 in our"}, {"start": 33.16, "duration": 7.36, "text": "case U is s of X Cub so we get 3 * sin"}, {"start": 36.92, "duration": 6.639, "text": "of x cu^ 2 times the derivative of s of"}, {"start": 40.52, "duration": 5.64, "text": "X cubed next we need to find the"}, {"start": 43.559, "duration": 6.241, "text": "derivative of s of X cubed the"}, {"start": 46.16, "duration": 7.48, "text": "derivative of s of X with respect to X"}, {"start": 49.8, "duration": 6.04, "text": "is cosine of x but since we have X cubed"}, {"start": 53.64, "duration": 4.64, "text": "inside we need to multiply by the"}, {"start": 55.84, "duration": 4.92, "text": "derivative of the inner function now for"}, {"start": 58.28, "duration": 5.839, "text": "the final step we need need to find the"}, {"start": 60.76, "duration": 7.28, "text": "derivative of x cubed using the power"}, {"start": 64.119, "duration": 8.521, "text": "rule the derivative of x cubed is 3x"}, {"start": 68.04, "duration": 8.96, "text": "sared let's substitute this in finally"}, {"start": 72.64, "duration": 6.159, "text": "let's simplify our Expression 3 * 3 is 9"}, {"start": 77.0, "duration": 7.28, "text": "and we can rearrange the terms to get"}, {"start": 78.799, "duration": 8.201, "text": "our final answer 9 x^2 * sin^2 X Cub *"}, {"start": 84.28, "duration": 5.92, "text": "cosine of x cubed let's review what we"}, {"start": 87.0, "duration": 6.04, "text": "did we used the chain rule twice once"}, {"start": 90.2, "duration": 5.8, "text": "for the cube power and once for the sign"}, {"start": 93.04, "duration": 6.0, "text": "function we then used the power rule to"}, {"start": 96.0, "duration": 5.84, "text": "differentiate X cubed and finally"}, {"start": 99.04, "duration": 5.039, "text": "combined all terms to get our answer"}, {"start": 101.84, "duration": 4.279, "text": "this problem shows how complex"}, {"start": 104.079, "duration": 6.961, "text": "derivatives can be broken down into"}, {"start": 106.119, "duration": 4.921, "text": "simpler steps using the chain rule"}], "title": "9.3-1.10 Mastering the Chain Rule Differentiating (\ud835\udc60\ud835\udc56\ud835\udc5b^3\ud835\udc65^3)  Step by Step", "description": "In this video, we break down the process of differentiating (\ud835\udc60\ud835\udc56\ud835\udc5b^3\ud835\udc65^3)   using the chain rule. Learn how to handle composite functions and apply the power rule effectively. Perfect for students looking to grasp calculus concepts with clear, step-by-step explanations.\nHashtags:\n#cpmrana  #Calculus #Differentiation #ChainRule #MathTutorial #Trigonometry", "lengthSeconds": 110, "width": 640, "height": 360}
{"video_id": "dNU_kKQTKv0", "transcript": [{"start": 0.28, "duration": 5.92, "text": "let's solve this complex differentiation"}, {"start": 3.159, "duration": 6.681, "text": "problem step by step we need to find the"}, {"start": 6.2, "duration": 8.679, "text": "derivative of s of 2 Aran of the sare <"}, {"start": 9.84, "duration": 6.96, "text": "TK of 1 - x over 1 + x instead of trying"}, {"start": 14.879, "duration": 4.961, "text": "to differentiate this directly which"}, {"start": 16.8, "duration": 6.319, "text": "would be quite complicated let's try to"}, {"start": 19.84, "duration": 8.8, "text": "simplify the expression first let's make"}, {"start": 23.119, "duration": 9.081, "text": "a substitution let x = cosine of 2 Theta"}, {"start": 28.64, "duration": 7.04, "text": "when we substitute this in our fraction"}, {"start": 32.2, "duration": 8.32, "text": "under the square root becomes 1 - cosine"}, {"start": 35.68, "duration": 7.559, "text": "2 thet over 1 plus cosine 2 Theta using"}, {"start": 40.52, "duration": 5.6, "text": "the double angle formula for cosine we"}, {"start": 43.239, "duration": 6.241, "text": "can write this in terms of s and cosine"}, {"start": 46.12, "duration": 8.4, "text": "Theta now under the square root we have"}, {"start": 49.48, "duration": 8.64, "text": "2 sin^2 thet over 2 cine 2 thet this"}, {"start": 54.52, "duration": 7.039, "text": "simplifies to tangent s Theta the square"}, {"start": 58.12, "duration": 6.84, "text": "root of tangent s thet is just tangent"}, {"start": 61.559, "duration": 7.961, "text": "thet now we have AR tangent of tangent"}, {"start": 64.96, "duration": 8.24, "text": "thet which just gives USA so 2 * AR"}, {"start": 69.52, "duration": 7.4, "text": "tangent of tangent Theta is just 2 Theta"}, {"start": 73.2, "duration": 8.0, "text": "therefore our expression simplifies to S"}, {"start": 76.92, "duration": 7.96, "text": "of 2 Theta remember that x equal cine of"}, {"start": 81.2, "duration": 9.04, "text": "2 Theta so we can substitute back to get"}, {"start": 84.88, "duration": 8.239, "text": "y = < TK of 1 - x^2 now we can"}, {"start": 90.24, "duration": 6.08, "text": "differentiate this much simpler"}, {"start": 93.119, "duration": 8.521, "text": "expression using the chain rule we get"}, {"start": 96.32, "duration": 7.56, "text": "1/ 2 * the square < TK of 1 - x^2 times"}, {"start": 101.64, "duration": 7.64, "text": "the derivative of what's inside the"}, {"start": 103.88, "duration": 8.32, "text": "square root the derivative of 1 - x^2 is"}, {"start": 109.28, "duration": 7.799, "text": "-2X simplifying this gives us our final"}, {"start": 112.2, "duration": 7.879, "text": "answer X over The Square < TK of 1 - x^2"}, {"start": 117.079, "duration": 5.921, "text": "and there we have it we've found the D"}, {"start": 120.079, "duration": 5.0, "text": "ative by first simplifying the original"}, {"start": 123.0, "duration": 4.36, "text": "expression using trigonometric"}, {"start": 125.079, "duration": 5.881, "text": "substitution and then differentiating"}, {"start": 127.36, "duration": 3.6, "text": "the simpler form"}], "title": "Understanding Trigonometric Differentiation [sin\u2061{2\ud835\udc61\ud835\udc4e\ud835\udc5b^(\u22121) \u221a((1\u2212\ud835\udc65)/(1+\ud835\udc65))}] A Step-by-Step Guide", "description": "Question: Differentiate the function  [sin\u2061{2\ud835\udc61\ud835\udc4e\ud835\udc5b^(\u22121) \u221a((1\u2212\ud835\udc65)/(1+\ud835\udc65))}]   with respect to x.\nDive into the world of trigonometric differentiation with this detailed video on differentiating a complex trigonometric function. We\u2019ll break down each step, provide clear explanations, and ensure you have a firm grasp on how to handle functions like  [sin\u2061{2\ud835\udc61\ud835\udc4e\ud835\udc5b^(\u22121) \u221a((1\u2212\ud835\udc65)/(1+\ud835\udc65))}] . Perfect for students and math enthusiasts looking to master calculus concepts!\n\n#Calculus #Trigonometry #Differentiation #MathTutorial #LearnMath", "lengthSeconds": 134, "width": 640, "height": 360}
{"video_id": "AqkdHzF8TxM", "transcript": [{"start": 0.24, "duration": 5.24, "text": "today we'll solve a challenging"}, {"start": 2.08, "duration": 5.2, "text": "derivative problem using logarithmic"}, {"start": 5.48, "duration": 6.92, "text": "differentiation let's find the"}, {"start": 7.28, "duration": 9.04, "text": "derivative of y = cent of X raised to"}, {"start": 12.4, "duration": 6.36, "text": "the tangent of X power the first step in"}, {"start": 16.32, "duration": 5.92, "text": "logarithmic differentiation is to take"}, {"start": 18.76, "duration": 7.64, "text": "the natural logarithm of both sides this"}, {"start": 22.24, "duration": 7.879, "text": "gives us natural log of yal natural log"}, {"start": 26.4, "duration": 5.719, "text": "of Cent x raed to the tangent X power"}, {"start": 30.119, "duration": 4.44, "text": "using the logarithm property that says"}, {"start": 32.119, "duration": 5.241, "text": "the natural log of a power equals the"}, {"start": 34.559, "duration": 7.281, "text": "power time the natural log of the base"}, {"start": 37.36, "duration": 7.96, "text": "we get natural log of Y = tangent x *"}, {"start": 41.84, "duration": 6.039, "text": "natural log of Cent X now we can"}, {"start": 45.32, "duration": 6.32, "text": "differentiate both sides with respect to"}, {"start": 47.879, "duration": 7.401, "text": "X the left side becomes DDX of natural"}, {"start": 51.64, "duration": 8.48, "text": "log Y and the right side becomes DDX of"}, {"start": 55.28, "duration": 8.2, "text": "tangent x * natural log of Cent X on the"}, {"start": 60.12, "duration": 7.76, "text": "left side we use the chain rule the"}, {"start": 63.48, "duration": 7.56, "text": "derivative of natural log Y is 1/ y *"}, {"start": 67.88, "duration": 5.36, "text": "dydx on the right side we use the"}, {"start": 71.04, "duration": 4.68, "text": "product rule since we're differentiating"}, {"start": 73.24, "duration": 5.04, "text": "a product let's evaluate the derivatives"}, {"start": 75.72, "duration": 8.359, "text": "on the right side the derivative of"}, {"start": 78.28, "duration": 9.96, "text": "natural log of Cent X is 1 / cang x *"}, {"start": 84.079, "duration": 6.36, "text": "cose 2 x and the derivative of tangent X"}, {"start": 88.24, "duration": 6.72, "text": "is secant 2 x"}, {"start": 90.439, "duration": 9.481, "text": "we can simplify 1 / Cent X to tangent X"}, {"start": 94.96, "duration": 9.759, "text": "this gives us tangent 2 x sin^2 X+"}, {"start": 99.92, "duration": 8.96, "text": "natural log Cent x * secant 2 X after"}, {"start": 104.719, "duration": 11.68, "text": "rearranging terms we get y * natural log"}, {"start": 108.88, "duration": 11.64, "text": "Cent x * secant 2 x - tangent 2 x/ sin^2"}, {"start": 116.399, "duration": 7.841, "text": "x using the identity that tangent 2 x ="}, {"start": 120.52, "duration": 8.64, "text": "sin^2 X over cine ^ 2 x we can simplify"}, {"start": 124.24, "duration": 9.719, "text": "this further to y * natural log Cent x *"}, {"start": 129.16, "duration": 6.4, "text": "SEC 2 x - secant 2 x finally"}, {"start": 133.959, "duration": 5.881, "text": "substituting back the original"}, {"start": 135.56, "duration": 8.84, "text": "expression for y we get our final answer"}, {"start": 139.84, "duration": 10.84, "text": "Cent x to the tangent X power * secant 2"}, {"start": 144.4, "duration": 8.88, "text": "x * the quantity natural log Cent X - 1"}, {"start": 150.68, "duration": 5.32, "text": "and that's how we use logarithmic"}, {"start": 153.28, "duration": 5.319, "text": "differentiation to find this challenging"}, {"start": 156.0, "duration": 5.04, "text": "derivative the key steps were taking the"}, {"start": 158.599, "duration": 5.321, "text": "natural log of both sides using the"}, {"start": 161.04, "duration": 6.64, "text": "product rule and carefully applying our"}, {"start": 163.92, "duration": 3.76, "text": "trigonometric identities"}], "title": "Solving Complex Derivatives: Finding d/dx of (cot x)^(tan x) Using Logarithmic Differentiation", "description": "Question: Find the derivative of y = (cot x)^(tan x) using logarithmic differentiation.\n\nIn this comprehensive calculus tutorial, we break down the step-by-step process of finding the derivative of y = (cot x)^(tan x) using logarithmic differentiation. This video covers:\n\u2022 Logarithmic differentiation technique\n\u2022 Product and chain rules\n\u2022 Trigonometric derivatives and identities\n\u2022 Step-by-step solution walkthrough\n\u2022 Common pitfalls and how to avoid them\nPerfect for calculus students at university level or those preparing for advanced mathematics examinations. Watch how we transform a complex exponential function into a manageable form and solve it systematically.\n#Calculus #Mathematics #Derivatives #LogarithmicDifferentiation #MathTutorial #CalcTutorial #MathEducation #TrigonometryCalculus", "lengthSeconds": 171, "width": 640, "height": 360}
{"video_id": "BOQASLZ7kxg", "transcript": [{"start": 0.32, "duration": 5.88, "text": "let's solve this complex differentiation"}, {"start": 3.08, "duration": 8.8, "text": "problem step by step we need to find the"}, {"start": 6.2, "duration": 9.679, "text": "derivative of e to the x * log base of X"}, {"start": 11.88, "duration": 7.64, "text": "first let's rewrite log base of X using"}, {"start": 15.879, "duration": 8.521, "text": "the change of Base formula log base of"}, {"start": 19.52, "duration": 8.48, "text": "xal log base e of x / log base e of a"}, {"start": 24.4, "duration": 7.32, "text": "which equals log Bas uh of e * log base"}, {"start": 28.0, "duration": 6.36, "text": "e of X now we can use the product rule"}, {"start": 31.72, "duration": 5.4, "text": "to differentiate this expression"}, {"start": 34.36, "duration": 6.96, "text": "remember the product rule states that"}, {"start": 37.12, "duration": 7.32, "text": "the derivative of U * v = u Prime * V +"}, {"start": 41.32, "duration": 7.2, "text": "U * V Prime let's find the derivatives"}, {"start": 44.44, "duration": 7.68, "text": "of each part the derivative of e to X is"}, {"start": 48.52, "duration": 5.76, "text": "e to X and the derivative of natural log"}, {"start": 52.12, "duration": 5.959, "text": "of x is"}, {"start": 54.28, "duration": 7.4, "text": "1/x now let's apply the product rule we"}, {"start": 58.079, "duration": 7.4, "text": "multiply e to X by by 1 /x for the first"}, {"start": 61.68, "duration": 7.119, "text": "term and multiply natural log of x by E"}, {"start": 65.479, "duration": 7.401, "text": "to X for the second term then factor out"}, {"start": 68.799, "duration": 7.561, "text": "e to the X for our final step we"}, {"start": 72.88, "duration": 7.76, "text": "recognize that log base uh of e * log"}, {"start": 76.36, "duration": 8.079, "text": "base e of x equal log base a of X this"}, {"start": 80.64, "duration": 10.2, "text": "gives us our final answer e to x * thean"}, {"start": 84.439, "duration": 9.96, "text": "1 /x * log base uh of e plus log Bas of"}, {"start": 90.84, "duration": 8.0, "text": "X and that's how we differentiate e to x"}, {"start": 94.399, "duration": 7.281, "text": "* log Bas of X we use the change of Base"}, {"start": 98.84, "duration": 5.599, "text": "formula to rewrite the logarithm applied"}, {"start": 101.68, "duration": 5.96, "text": "the product rule and then simplified our"}, {"start": 104.439, "duration": 6.481, "text": "result using logarithm properties thanks"}, {"start": 107.64, "duration": 3.28, "text": "for watching"}], "title": "Finding the Derivative of e^x log_a(x)  Using the Product Rule | Step-by-Step Explanation", "description": "In this video, learn to differentiate e^x log_a(x) using the Product Rule in calculus.\n\nWhat You'll Learn:  \n- The Product Rule definition  \n- Step-by-step differentiation of e^x log_a(x)  \n- Breakdown of the formula and example problem-solving  \n\nThis tutorial is perfect for calculus beginners or those revisiting the Product Rule!\n\ud83d\udca1 Don't forget to LIKE, SHARE, and SUBSCRIBE for more math tutorials!\n#Calculus #ProductRule #Differentiation #Derivative #MathTutorial #LearnMath #MathEducation #MathHelp", "lengthSeconds": 111, "width": 640, "height": 360}
{"video_id": "Bis7guYfKN8", "transcript": [{"start": 0.28, "duration": 4.88, "text": "let's solve this calculus problem step"}, {"start": 2.36, "duration": 5.92, "text": "by step we need to differentiate the"}, {"start": 5.16, "duration": 7.76, "text": "logarithmic expression natural log of 1"}, {"start": 8.28, "duration": 8.24, "text": "+ < TK X over 1 - < TK x with respect to"}, {"start": 12.92, "duration": 5.48, "text": "X first let's use the logarithm property"}, {"start": 16.52, "duration": 4.24, "text": "that states the natural log of a"}, {"start": 18.4, "duration": 5.28, "text": "fraction equals the natural log of the"}, {"start": 20.76, "duration": 5.2, "text": "numerator minus the natural log of the"}, {"start": 23.68, "duration": 6.88, "text": "denominator this gives us the derivative"}, {"start": 25.96, "duration": 9.119, "text": "of natural log of 1 + < TK of x -"}, {"start": 30.56, "duration": 7.679, "text": "natural log of 1 - < TK X next we'll"}, {"start": 35.079, "duration": 5.681, "text": "apply the chain rule remember the"}, {"start": 38.239, "duration": 6.681, "text": "derivative of natural log of U with"}, {"start": 40.76, "duration": 7.76, "text": "respect to x = 1 / U * the derivative of"}, {"start": 44.92, "duration": 8.68, "text": "U with respect to X applying this to"}, {"start": 48.52, "duration": 11.839, "text": "both terms we get 1/ 1 + < TK x * the"}, {"start": 53.6, "duration": 11.599, "text": "Der of 1 + < TK x -1 / 1 - < TK X time"}, {"start": 60.359, "duration": 6.881, "text": "the derivative of 1 - < TK X now let's"}, {"start": 65.199, "duration": 4.56, "text": "find the derivatives of the inner"}, {"start": 67.24, "duration": 8.919, "text": "functions the derivative of square < TK"}, {"start": 69.759, "duration": 10.32, "text": "of X is 1 / 2 < TK X so 1 + < TK of X"}, {"start": 76.159, "duration": 9.521, "text": "differentiates to just 1 2 < TK X and 1"}, {"start": 80.079, "duration": 8.281, "text": "- < TK of X differentiates to 1/2 < TK X"}, {"start": 85.68, "duration": 8.439, "text": "let's simplify this expression the first"}, {"start": 88.36, "duration": 9.64, "text": "term becomes 1 / 2 < TK of x * 1 + < TK"}, {"start": 94.119, "duration": 9.36, "text": "X for the second term the negative signs"}, {"start": 98.0, "duration": 9.64, "text": "cancel out giving us + 1 / 2 < TK of x *"}, {"start": 103.479, "duration": 6.801, "text": "1 - < TK X to add these fractions we"}, {"start": 107.64, "duration": 4.56, "text": "need a common denominator when we"}, {"start": 110.28, "duration": 4.68, "text": "multiply both the numerator and"}, {"start": 112.2, "duration": 7.959, "text": "denominator by the respective missing"}, {"start": 114.96, "duration": 12.64, "text": "factors we get 1 - < TK of x + 1 + < TK"}, {"start": 120.159, "duration": 11.72, "text": "X all over 2 < TK x * 1 + < TK x * 1 - <"}, {"start": 127.6, "duration": 12.639, "text": "TK X for our final simplification notice"}, {"start": 131.879, "duration": 11.08, "text": "that 1 + < TK x * 1 - < TK of x = 1 - x"}, {"start": 140.239, "duration": 7.481, "text": "using the difference of squares formula"}, {"start": 142.959, "duration": 7.761, "text": "therefore our final answer is 1 / < TK x"}, {"start": 147.72, "duration": 5.799, "text": "* 1 - x"}, {"start": 150.72, "duration": 4.879, "text": "and there we have it we've successfully"}, {"start": 153.519, "duration": 4.08, "text": "differentiated this logarithmic"}, {"start": 155.599, "duration": 5.961, "text": "expression using properties of"}, {"start": 157.599, "duration": 6.961, "text": "logarithms the chain Rule and algebraic"}, {"start": 161.56, "duration": 3.0, "text": "simplification"}], "title": "Mastering Logarithmic Differentiation: Step-by-Step Explanation [\ud835\udc8d\ud835\udc8f((\ud835\udfcf+\u221a\ud835\udc99)/(\ud835\udfcf\u2212\u221a\ud835\udc99))]", "description": "Question:\nDifferentiate the expression [\ud835\udc8d\ud835\udc8f((\ud835\udfcf+\u221a\ud835\udc99)/(\ud835\udfcf\u2212\u221a\ud835\udc99))] with respect to x.\n\nDive into the world of logarithmic differentiation with our detailed step-by-step guide. Learn how to differentiate complex logarithmic functions using the chain rule and logarithm properties. Perfect for students looking to strengthen their calculus skills and understand logarithmic identities thoroughly.\n#Calculus #Differentiation #MathTutorial #Logarithms #MathHelp", "lengthSeconds": 168, "width": 640, "height": 360}
{"video_id": "MVbyFDGkS_4", "transcript": [{"start": 0.36, "duration": 5.36, "text": "welcome today we'll solve a challenging"}, {"start": 3.24, "duration": 5.6, "text": "differentiation problem involving"}, {"start": 5.72, "duration": 5.76, "text": "exponential and inverse trigonometric"}, {"start": 8.84, "duration": 7.4, "text": "functions our task is to find the"}, {"start": 11.48, "duration": 7.76, "text": "derivative of e to x * sin inverse of X"}, {"start": 16.24, "duration": 6.48, "text": "let's break this down step by step step"}, {"start": 19.24, "duration": 7.84, "text": "one first we need to recognize that this"}, {"start": 22.72, "duration": 7.039, "text": "is a product of two functions e to X and"}, {"start": 27.08, "duration": 5.56, "text": "sin inverse of X this means we'll need"}, {"start": 29.759, "duration": 6.601, "text": "to use the product rule the product rule"}, {"start": 32.64, "duration": 9.84, "text": "states that the derivative of U * v = u"}, {"start": 36.36, "duration": 6.12, "text": "Prime * V + U * V Prime let's write this"}, {"start": 42.559, "duration": 6.761, "text": "out applying this to our problem we get"}, {"start": 46.52, "duration": 6.12, "text": "e to x * the derivative of sin inverse"}, {"start": 49.32, "duration": 7.16, "text": "of X Plus sin inverse of x * the"}, {"start": 52.64, "duration": 3.84, "text": "derivative of e to"}, {"start": 56.6, "duration": 7.599, "text": "X now we need to find each derivative"}, {"start": 60.399, "duration": 7.08, "text": "separately the derivative of e to the x"}, {"start": 64.199, "duration": 8.161, "text": "is simply e to X the derivative of sin"}, {"start": 67.479, "duration": 7.721, "text": "inverse of X is 1 over the < TK of 1 -"}, {"start": 72.36, "duration": 8.84, "text": "x^2 let's substitute these derivatives"}, {"start": 75.2, "duration": 12.48, "text": "into our equation we get e x * 1 < TK of"}, {"start": 81.2, "duration": 6.48, "text": "1 - x^2 plus sin inverse of x * e to"}, {"start": 87.799, "duration": 10.6, "text": "X finally we can write our answer as e x"}, {"start": 92.6, "duration": 9.72, "text": "over < TK of 1 - x^2 + e x * sin inverse"}, {"start": 98.399, "duration": 3.921, "text": "of X this is our final"}, {"start": 102.439, "duration": 5.36, "text": "answer let's verify why this makes sense"}, {"start": 106.399, "duration": 4.921, "text": "the first term comes from"}, {"start": 107.799, "duration": 6.881, "text": "differentiating e to the X and"}, {"start": 111.32, "duration": 6.36, "text": "multiplying by sin inverse of X using"}, {"start": 114.68, "duration": 5.759, "text": "the chain rule the second term comes"}, {"start": 117.68, "duration": 6.119, "text": "from keeping e to the x and"}, {"start": 120.439, "duration": 6.081, "text": "differentiating sign inverse of X both"}, {"start": 123.799, "duration": 5.401, "text": "terms contain e to the X because we're"}, {"start": 126.52, "duration": 6.2, "text": "differentiating a product involving e to"}, {"start": 129.2, "duration": 7.72, "text": "the X remember when differentiating"}, {"start": 132.72, "duration": 8.159, "text": "products of functions always one"}, {"start": 136.92, "duration": 6.959, "text": "identify the functions clearly two apply"}, {"start": 140.879, "duration": 5.561, "text": "the product rule correctly and three"}, {"start": 143.879, "duration": 4.921, "text": "know your basic derivatives especially"}, {"start": 146.44, "duration": 4.28, "text": "for exponential and inverse"}, {"start": 148.8, "duration": 5.56, "text": "trigonometric functions"}, {"start": 150.72, "duration": 3.64, "text": "thank you for watching"}], "title": "Understanding the Derivative of [\ud835\udc52^\ud835\udc65 \ud835\udc60\ud835\udc56\ud835\udc5b^(\u22121)  \ud835\udc65]  Step-by-Step", "description": "Question: Differentiate [\ud835\udc52^\ud835\udc65 \ud835\udc60\ud835\udc56\ud835\udc5b^(\u22121)  \ud835\udc65] with respect to x.\nDive into the step-by-step process of differentiating the function [\ud835\udc52^\ud835\udc65 \ud835\udc60\ud835\udc56\ud835\udc5b^(\u22121)  \ud835\udc65] . Explore how to apply the product rule and differentiate inverse trigonometric and exponential functions with clear explanations.\n#Calculus #Differentiation #MathTutorial #ProductRule #InverseTrigFunctions #ExponentialFunctions", "lengthSeconds": 157, "width": 640, "height": 360}
{"video_id": "DN2Ftkcnq1g", "transcript": [{"start": 0.32, "duration": 5.36, "text": "let's solve this complex differentiation"}, {"start": 3.24, "duration": 6.68, "text": "problem step by step we'll start with"}, {"start": 5.68, "duration": 7.8, "text": "the function y = sin s of the natural"}, {"start": 9.92, "duration": 6.68, "text": "log of cine X let's break this down into"}, {"start": 13.48, "duration": 5.52, "text": "manageable Parts first notice that this"}, {"start": 16.6, "duration": 5.32, "text": "is a composite function with multiple"}, {"start": 19.0, "duration": 5.24, "text": "layers we have sin squ as the outer"}, {"start": 21.92, "duration": 5.8, "text": "function natural log as the middle"}, {"start": 24.24, "duration": 6.519, "text": "function and cine X as the innermost"}, {"start": 27.72, "duration": 5.64, "text": "function to differentiate this we'll use"}, {"start": 30.759, "duration": 4.64, "text": "the chain rule multiple times let's"}, {"start": 33.36, "duration": 6.16, "text": "start with the outer function the"}, {"start": 35.399, "duration": 7.721, "text": "derivative of sin squ is 2 * s time the"}, {"start": 39.52, "duration": 6.6, "text": "derivative of s next we apply the chain"}, {"start": 43.12, "duration": 6.439, "text": "rule again to differentiate s of natural"}, {"start": 46.12, "duration": 6.32, "text": "log of cosine X this gives us cosine of"}, {"start": 49.559, "duration": 5.201, "text": "natural log of cosine x times the"}, {"start": 52.44, "duration": 5.36, "text": "derivative of the natural log of cosine"}, {"start": 54.76, "duration": 6.639, "text": "X here's where we can use a helpful"}, {"start": 57.8, "duration": 8.319, "text": "trigonometric identity not that 2 * sin"}, {"start": 61.399, "duration": 7.04, "text": "* cosine equals s of 2 * the angle this"}, {"start": 66.119, "duration": 4.881, "text": "lets us simplify the first part of our"}, {"start": 68.439, "duration": 6.68, "text": "expression now we need to differentiate"}, {"start": 71.0, "duration": 6.799, "text": "the natural log of cosine X remember the"}, {"start": 75.119, "duration": 4.601, "text": "derivative of natural log is 1 over the"}, {"start": 77.799, "duration": 6.041, "text": "function time the derivative of the"}, {"start": 79.72, "duration": 8.24, "text": "function the derivative of cosine X is"}, {"start": 83.84, "duration": 6.959, "text": "sinx we can rearrange this to get sin 2"}, {"start": 87.96, "duration": 8.159, "text": "* natural log of Co"}, {"start": 90.799, "duration": 8.241, "text": "x * sin x cosine X finally we recognize"}, {"start": 96.119, "duration": 5.561, "text": "that sin x cosine X is the tangent"}, {"start": 99.04, "duration": 6.759, "text": "function giving us our final"}, {"start": 101.68, "duration": 7.28, "text": "answer tangent x * sin 2 natural log of"}, {"start": 105.799, "duration": 5.761, "text": "cosine X this problem showcases the"}, {"start": 108.96, "duration": 5.28, "text": "power of the chain Rule and how we can"}, {"start": 111.56, "duration": 6.199, "text": "use trigonometric identities to simplify"}, {"start": 114.24, "duration": 3.519, "text": "complex expressions"}], "title": "Differentiating Sin Squared of Log Cosine [\ud835\udc60\ud835\udc56\ud835\udc5b^2 {ln\u2061(cos\u2061\ud835\udc65)}] Step-by-Step Guide", "description": "Join us as we delve into differentiating  [\ud835\udc60\ud835\udc56\ud835\udc5b^2 {ln\u2061(cos\u2061\ud835\udc65)}]  using essential calculus techniques. This video provides a clear, step-by-step explanation to help you master derivatives involving trigonometric and logarithmic functions, perfect for students and educators seeking in-depth comprehension.\n#Calculus #Derivative #Trigonometry #MathTutorial #ChainRule #LogarithmicFunctions #STEM #AdvancedMath #MathHelp #Differentiation", "lengthSeconds": 121, "width": 640, "height": 360}
{"video_id": "Hj-tsnN-X3Q", "transcript": [{"start": 0.359, "duration": 5.92, "text": "welcome today we'll solve a challenging"}, {"start": 3.08, "duration": 4.92, "text": "derivative problem using logarithmic"}, {"start": 6.279, "duration": 5.4, "text": "differentiation let's find the"}, {"start": 8.0, "duration": 6.879, "text": "derivative of y = x raised to the"}, {"start": 11.679, "duration": 5.281, "text": "inverse tangent of X this is a complex"}, {"start": 14.879, "duration": 5.601, "text": "function because we have a variable base"}, {"start": 16.96, "duration": 6.36, "text": "X and a variable exponent involving"}, {"start": 20.48, "duration": 5.84, "text": "inverse tangent the best strategy here"}, {"start": 23.32, "duration": 5.84, "text": "is to use logarithmic differentiation"}, {"start": 26.32, "duration": 5.359, "text": "we'll take the natural log of both sides"}, {"start": 29.16, "duration": 6.12, "text": "and then differenti iate let's solve the"}, {"start": 31.679, "duration": 6.441, "text": "step by step First We Take the natural"}, {"start": 35.28, "duration": 5.84, "text": "log of both sides using the properties"}, {"start": 38.12, "duration": 8.2, "text": "of logarithms we can write this as"}, {"start": 41.12, "duration": 7.68, "text": "inverse tangent of x * natural log of x"}, {"start": 46.32, "duration": 6.12, "text": "now we can differentiate both sides with"}, {"start": 48.8, "duration": 7.0, "text": "respect to X on the left side we use the"}, {"start": 52.44, "duration": 9.04, "text": "chain rule the derivative of natural log"}, {"start": 55.8, "duration": 8.04, "text": "of Y is 1 / y * Dy DX on the right side"}, {"start": 61.48, "duration": 5.04, "text": "we use the product rule since we're"}, {"start": 63.84, "duration": 7.12, "text": "differentiating the product of two"}, {"start": 66.52, "duration": 8.2, "text": "functions we get inverse tangent of x *"}, {"start": 70.96, "duration": 7.479, "text": "the derivative of natural log of x plus"}, {"start": 74.72, "duration": 7.0, "text": "natural log of x times the derivative of"}, {"start": 78.439, "duration": 5.761, "text": "inverse tangent of X now let's"}, {"start": 81.72, "duration": 6.359, "text": "substitute the known derivatives the"}, {"start": 84.2, "duration": 6.0, "text": "derivative of natural log of x is 1 /x"}, {"start": 88.079, "duration": 8.241, "text": "and the derivative of inverse tangent of"}, {"start": 90.2, "duration": 10.879, "text": "X is 1 1 + x^2 to solve for dydx we"}, {"start": 96.32, "duration": 8.479, "text": "multiply both sides by y remember that y"}, {"start": 101.079, "duration": 6.921, "text": "= x to the inverse tangent of x from our"}, {"start": 104.799, "duration": 5.401, "text": "original equation and there we have it"}, {"start": 108.0, "duration": 6.24, "text": "the derivative of x to the inverse"}, {"start": 110.2, "duration": 7.839, "text": "tangent of x = x to the inverse tangent"}, {"start": 114.24, "duration": 9.32, "text": "of x * the sum of inverse tangent of x /"}, {"start": 118.039, "duration": 8.161, "text": "x + natural log of x X over 1 + x^2 this"}, {"start": 123.56, "duration": 5.119, "text": "problem shows the power of logarithmic"}, {"start": 126.2, "duration": 5.56, "text": "differentiation in handling complex"}, {"start": 128.679, "duration": 6.121, "text": "expressions with variable bases and"}, {"start": 131.76, "duration": 5.32, "text": "exponents remember whenever you see a"}, {"start": 134.8, "duration": 4.84, "text": "function with a variable base and"}, {"start": 137.08, "duration": 4.68, "text": "variable exponent logarithmic"}, {"start": 139.64, "duration": 5.12, "text": "differentiation is often the best"}, {"start": 141.76, "duration": 3.0, "text": "approach"}], "title": "Mastering Logarithmic Differentiation: Solving [y = x^(arctan x)]", "description": "Question: Find the derivative of  [y = x^(arctan x)]  using logarithmic differentiation.\n\nLearn how to find the derivative of y = x^(arctan x) using logarithmic differentiation in this comprehensive math tutorial. We break down this complex differentiation problem into clear, easy-to-follow steps using the natural logarithm and chain rule. Perfect for calculus students and math enthusiasts!\nKey topics covered:\n\u2022 Logarithmic differentiation technique\n\u2022 Properties of natural logarithm\n\u2022 Chain rule application\n\u2022 Product rule\n\u2022 Derivatives of arctan(x) and ln(x)\n\u2022 Step-by-step solution process\n#Mathematics #Calculus #LogarithmicDifferentiation #MathTutorial #ChainRule #ProductRule #Derivatives #MathEducation #MathHelp #CalcTutorial", "lengthSeconds": 148, "width": 640, "height": 360}
{"video_id": "EgRyoBqxE6Q", "transcript": [{"start": 0.28, "duration": 6.32, "text": "let's find the derivative of cosine 3x"}, {"start": 3.919, "duration": 5.401, "text": "using the first principles of calculus"}, {"start": 6.6, "duration": 7.32, "text": "we'll go through this step by step let's"}, {"start": 9.32, "duration": 11.36, "text": "start by defining our function let F ofx"}, {"start": 13.92, "duration": 12.24, "text": "= cine 3x therefore f of x + H = cosine"}, {"start": 20.68, "duration": 8.999, "text": "3 * x + H by definition the derivative"}, {"start": 26.16, "duration": 9.32, "text": "is the limit as H approaches Z of f ofx"}, {"start": 29.679, "duration": 8.56, "text": "+ H - FX all divid H substituting our"}, {"start": 35.48, "duration": 6.44, "text": "function we get the limit as H"}, {"start": 38.239, "duration": 6.96, "text": "approaches Z of cosine 3 * the quantity"}, {"start": 41.92, "duration": 7.0, "text": "x + H - cine"}, {"start": 45.199, "duration": 5.161, "text": "3x all divid H now we'll use the"}, {"start": 48.92, "duration": 4.4, "text": "trigonometric identity for the"}, {"start": 50.36, "duration": 8.64, "text": "difference of cosiness cosine of minus"}, {"start": 53.32, "duration": 9.28, "text": "cosine B = -2 * sin of a plus b/ 2 * sin"}, {"start": 59.0, "duration": 8.799, "text": "of A- B 2 applying this identity to our"}, {"start": 62.6, "duration": 13.36, "text": "expression we get -2 * sin of 3x + 3 H +"}, {"start": 67.799, "duration": 11.241, "text": "3x / 2 * sin of 3x - 3x - 3 h / 2 all ID"}, {"start": 75.96, "duration": 8.36, "text": "by H simplifying the Expressions inside"}, {"start": 79.04, "duration": 12.48, "text": "the S functions we get 2 * sin of 6X +"}, {"start": 84.32, "duration": 12.64, "text": "3H / 2 * sin of -3 h / 2 all / H further"}, {"start": 91.52, "duration": 11.36, "text": "simplifying we get 2 * sin of 3x + 3 h /"}, {"start": 96.96, "duration": 8.439, "text": "2 * sin of 3H / 2 all divided H now we"}, {"start": 102.88, "duration": 6.8, "text": "can separate this into two limits the"}, {"start": 105.399, "duration": 8.0, "text": "limit as H approaches Z of s of 3x + 3 h"}, {"start": 109.68, "duration": 10.399, "text": "/ 2 times the limit as H approaches 0 of"}, {"start": 113.399, "duration": 9.4, "text": "netive s of 3 h / 2 / 3 h / 2 * 3 as h"}, {"start": 120.079, "duration": 7.881, "text": "approaches zero the first limit becomes"}, {"start": 122.799, "duration": 7.201, "text": "sin 3x and the second limit equals -1"}, {"start": 127.96, "duration": 5.44, "text": "due to the fundamental limit of sin"}, {"start": 130.0, "duration": 6.879, "text": "Theta over Theta as Theta approaches"}, {"start": 133.4, "duration": 9.76, "text": "zero therefore the derivative of cosine"}, {"start": 136.879, "duration": 8.36, "text": "3x = -3 sin 3x this result matches the"}, {"start": 143.16, "duration": 6.28, "text": "chain Rule and the standard derivative"}, {"start": 145.239, "duration": 6.36, "text": "of cosin x which is sinx the factor of 3"}, {"start": 149.44, "duration": 7.32, "text": "comes from the chain rule when"}, {"start": 151.599, "duration": 5.161, "text": "differentiating the inside function 3x"}], "title": "Understanding the Derivative of cos 3x  Using First Principles", "description": "Dive into the fundamentals of calculus as we explore how to determine the derivative of the exponential function sin3x using first principles. Follow along with step-by-step explanations and examples to enhance your understanding of derivatives and exponential functions.\n#Calculus #Derivatives #ExponentialFunctions #MathTutorial #FirstPrinciples #sin 2x", "lengthSeconds": 157, "width": 640, "height": 360}
{"video_id": "Ek5Y3hRdsyI", "transcript": [{"start": 0.32, "duration": 5.039, "text": "welcome today we're going to solve a"}, {"start": 2.84, "duration": 7.24, "text": "challenging differentiation problem step"}, {"start": 5.359, "duration": 9.921, "text": "by step our task is to differentiate y ="}, {"start": 10.08, "duration": 8.52, "text": "theare < TK of 1 - x over 1 + x let's"}, {"start": 15.28, "duration": 5.48, "text": "break this down step by step first we"}, {"start": 18.6, "duration": 5.72, "text": "can rewrite this as the derivative with"}, {"start": 20.76, "duration": 5.839, "text": "respect to X then we can rewrite this as"}, {"start": 24.32, "duration": 5.879, "text": "a quotient of square roots to make it"}, {"start": 26.599, "duration": 6.48, "text": "easier to handle now comes the key part"}, {"start": 30.199, "duration": 5.481, "text": "we need to use the quotient rule here"}, {"start": 33.079, "duration": 6.281, "text": "remember the quotient rule states that"}, {"start": 35.68, "duration": 7.0, "text": "the derivative of U over V equal V * the"}, {"start": 39.36, "duration": 7.16, "text": "derivative of uus U * the derivative of"}, {"start": 42.68, "duration": 7.68, "text": "V all over V ^2 applying this to our"}, {"start": 46.52, "duration": 7.8, "text": "problem U is the square < TK of 1 - x"}, {"start": 50.36, "duration": 7.199, "text": "and V is the square < TK of 1 + x this"}, {"start": 54.32, "duration": 3.239, "text": "gives us this longer"}, {"start": 57.879, "duration": 4.801, "text": "expression now we need to find the"}, {"start": 60.12, "duration": 5.6, "text": "derivatives of our square root terms"}, {"start": 62.68, "duration": 5.64, "text": "remember the chain rule for square roots"}, {"start": 65.72, "duration": 6.399, "text": "the derivative of square root of x is 1"}, {"start": 68.32, "duration": 6.799, "text": "/ 2 < TK x times the derivative of"}, {"start": 72.119, "duration": 8.32, "text": "what's inside so the derivative of"}, {"start": 75.119, "duration": 8.521, "text": "square T of 1 - x is - 1 /2 < TK of 1 -"}, {"start": 80.439, "duration": 8.481, "text": "x and the derivative of sare < TK of 1 +"}, {"start": 83.64, "duration": 9.0, "text": "x is 1 / 2 < TK of 1 + x let's simplify"}, {"start": 88.92, "duration": 7.36, "text": "this expression first we can factor out"}, {"start": 92.64, "duration": 6.439, "text": "-1 over 1 + x then we can combine the"}, {"start": 96.28, "duration": 5.32, "text": "terms inside the brackets by finding a"}, {"start": 99.079, "duration": 5.481, "text": "common denominator the numerator"}, {"start": 101.6, "duration": 6.32, "text": "simplifies to just two and we can write"}, {"start": 104.56, "duration": 9.239, "text": "the denominator more compactly our final"}, {"start": 107.92, "duration": 10.0, "text": "answer is -1 over the 1 + x * the < TK"}, {"start": 113.799, "duration": 4.121, "text": "of 1 - x^2"}, {"start": 120.799, "duration": 5.721, "text": "let's review the key steps we used first"}, {"start": 124.28, "duration": 4.959, "text": "we rewrote the expression as a quotient"}, {"start": 126.52, "duration": 6.28, "text": "of square roots then we applied the"}, {"start": 129.239, "duration": 5.64, "text": "quotient rule next we used the chain"}, {"start": 132.8, "duration": 5.64, "text": "rule to find the derivatives of the"}, {"start": 134.879, "duration": 6.841, "text": "square roots finally we simplifi to get"}, {"start": 138.44, "duration": 3.28, "text": "our answer"}], "title": "How to Differentiate a Square Root Quotient Step-by-Step| [\u221a((1\u2212\ud835\udc65)/(1+\ud835\udc65))]", "description": "Question\nDifferentiate the expression: [\u221a((1\u2212\ud835\udc65)/(1+\ud835\udc65))]\n\nUnlock the secrets to differentiating complex expressions with our in-depth, step-by-step breakdown of the derivative of a square root quotient. Learn how to apply the quotient rule and simplify expressions effectively. Perfect for students and math enthusiasts looking to master calculus!\n#Calculus #Differentiation #MathTutorial #LearnMath #QuotientRule", "lengthSeconds": 147, "width": 640, "height": 360}
{"video_id": "7L1dkS--7ig", "transcript": [{"start": 0.359, "duration": 5.161, "text": "let's solve this differentiation problem"}, {"start": 2.52, "duration": 4.6, "text": "step by step using the chain rule we"}, {"start": 5.52, "duration": 7.239, "text": "need to find the derivative of the"}, {"start": 7.12, "duration": 8.2, "text": "function 1 + sin of 2x all 2ar to solve"}, {"start": 12.759, "duration": 5.641, "text": "this using the chain rule let's break it"}, {"start": 15.32, "duration": 7.799, "text": "down into steps first let's make a"}, {"start": 18.4, "duration": 8.039, "text": "substitution let U equal 1 + sin of 2x"}, {"start": 23.119, "duration": 7.56, "text": "then our function can be Rewritten as y"}, {"start": 26.439, "duration": 7.041, "text": "= u^2 now according to the chain rule"}, {"start": 30.679, "duration": 5.521, "text": "the derivative of y with respect to x"}, {"start": 33.48, "duration": 6.599, "text": "equals the derivative of y with respect"}, {"start": 36.2, "duration": 7.039, "text": "to U times the derivative of you with"}, {"start": 40.079, "duration": 7.081, "text": "respect to X let's find each part"}, {"start": 43.239, "duration": 8.16, "text": "separately first the derivative of y"}, {"start": 47.16, "duration": 6.96, "text": "with respect to U since y = u ^2 this"}, {"start": 51.399, "duration": 6.64, "text": "derivative is 2 U we'll keep this part"}, {"start": 54.12, "duration": 5.959, "text": "for now and find the second part next"}, {"start": 58.039, "duration": 6.681, "text": "let's find the derivative of you with"}, {"start": 60.079, "duration": 8.161, "text": "respect to X remember that u = 1 + S of"}, {"start": 64.72, "duration": 7.56, "text": "2x the derivative of 1 is zero and the"}, {"start": 68.24, "duration": 8.12, "text": "derivative of sin of 2x is 2 cosine of"}, {"start": 72.28, "duration": 4.08, "text": "2x using the chain rule"}, {"start": 78.92, "duration": 5.68, "text": "again now let's multiply these parts"}, {"start": 82.28, "duration": 6.76, "text": "together according to the chain rule we"}, {"start": 84.6, "duration": 8.08, "text": "have 2 * 2 cosine of 2x remember that U"}, {"start": 89.04, "duration": 5.84, "text": "is 1 + s of 2x so let's substitute that"}, {"start": 92.68, "duration": 2.2, "text": "back"}, {"start": 97.28, "duration": 8.28, "text": "in and there we have our final answer"}, {"start": 100.68, "duration": 10.439, "text": "the derivative of 1 + sin of 2x^2 is 4 *"}, {"start": 105.56, "duration": 7.879, "text": "1 + sin of 2x * cine of 2x let's verify"}, {"start": 111.119, "duration": 2.32, "text": "our"}, {"start": 113.56, "duration": 5.36, "text": "steps the key to solving this problem"}, {"start": 116.32, "duration": 4.24, "text": "was breaking it down into simpler Parts"}, {"start": 118.92, "duration": 3.519, "text": "using the chain rule"}, {"start": 120.56, "duration": 5.199, "text": "and then carefully putting those parts"}, {"start": 122.439, "duration": 5.8, "text": "back together remember when dealing with"}, {"start": 125.759, "duration": 5.28, "text": "composite functions like this the chain"}, {"start": 128.239, "duration": 5.441, "text": "rule is your best friend it allows us to"}, {"start": 131.039, "duration": 7.321, "text": "break down complex derivatives into"}, {"start": 133.68, "duration": 4.68, "text": "simpler manageable steps"}], "title": "Mastering Calculus Differentiating {(1+sin\u20612\ud835\udc65)^2}  Step by Step", "description": "In this video, we delve into the process of differentiating the function {(1+sin\u20612\ud835\udc65)^2}  with respect to . We\u2019ll guide you through each step using the chain rule and fundamental differentiation techniques. Whether you\u2019re a student brushing up on calculus or keen on mastering differentiation, this detailed breakdown will enhance your mathematical understanding.\n#Calculus #Differentiation #ChainRule #MathTutorial #Mathematics #Educational", "lengthSeconds": 141, "width": 640, "height": 360}
{"video_id": "5lwzvz_wRxs", "transcript": [{"start": 0.28, "duration": 5.039, "text": "let's solve this calculus problem step"}, {"start": 2.639, "duration": 7.241, "text": "by step we need to differentiate the"}, {"start": 5.319, "duration": 7.841, "text": "inverse cosine of 1 - 2x with respect to"}, {"start": 9.88, "duration": 5.72, "text": "X to solve this we'll use the chain Rule"}, {"start": 13.16, "duration": 5.68, "text": "and the derivative formula for inverse"}, {"start": 15.6, "duration": 7.0, "text": "cosine remember the derivative of"}, {"start": 18.84, "duration": 8.599, "text": "inverse cosine of U is- 1 over theare <"}, {"start": 22.6, "duration": 9.479, "text": "TK of 1 - U ^2 * the derivative of U in"}, {"start": 27.439, "duration": 8.401, "text": "our case u = 1 - 2 X let's substitute"}, {"start": 32.079, "duration": 7.721, "text": "this into our formula now let's find the"}, {"start": 35.84, "duration": 9.36, "text": "derivative of U with respect to X the"}, {"start": 39.8, "duration": 8.04, "text": "derivative of 1 - 2x is -2 next let's"}, {"start": 45.2, "duration": 6.24, "text": "expand what's inside the square root"}, {"start": 47.84, "duration": 5.559, "text": "under the square root we have 1us the"}, {"start": 51.44, "duration": 5.959, "text": "quantity 1 -"}, {"start": 53.399, "duration": 7.12, "text": "2x^2 when we expand this we get 1us the"}, {"start": 57.399, "duration": 6.241, "text": "quantity 1 - 4x +"}, {"start": 60.519, "duration": 9.321, "text": "4x^2 let's simplify what's under the"}, {"start": 63.64, "duration": 8.159, "text": "square root 1 - 1 + 4x - 4x^2 simplifies"}, {"start": 69.84, "duration": 5.599, "text": "to 4x -"}, {"start": 71.799, "duration": 6.64, "text": "4x^2 now we can factor out a 4 from"}, {"start": 75.439, "duration": 10.801, "text": "under the square root the square < TK of"}, {"start": 78.439, "duration": 10.201, "text": "4x - 4x^2 = 2 * the < TK of x - x^2"}, {"start": 86.24, "duration": 5.6, "text": "finally we can cancel the twos in the"}, {"start": 88.64, "duration": 9.96, "text": "numerator and denominator this gives us"}, {"start": 91.84, "duration": 9.0, "text": "our final answer 1/ < TK x - x^2 to"}, {"start": 98.6, "duration": 5.559, "text": "verify this is correct we can check that"}, {"start": 100.84, "duration": 5.8, "text": "we used the chain rule correctly"}, {"start": 104.159, "duration": 4.521, "text": "expanded the squared term properly"}, {"start": 106.64, "duration": 5.88, "text": "simplifi the expression under the square"}, {"start": 108.68, "duration": 6.479, "text": "root and cancel common factors each step"}, {"start": 112.52, "duration": 7.08, "text": "followed logically from the previous one"}, {"start": 115.159, "duration": 4.441, "text": "leading us to our final answer"}], "title": "Mastering Derivatives: Simplifying the Inverse Cosine Function [\ud835\udc50\ud835\udc5c\ud835\udc60^(\u22121) (1\u22122\ud835\udc65)]", "description": "Join us as we explore the differentiation of the inverse cosine function, [\ud835\udc50\ud835\udc5c\ud835\udc60^(\u22121) (1\u22122\ud835\udc65)]. Follow along with our detailed, step-by-step breakdown to understand each part of the process and learn how to simplify derivatives effectively. Perfect for math students aiming to solidify their calculus skills!\n#Calculus\n#Derivatives\n#MathTutorial\n#LearningMath\n#Trigonometry", "lengthSeconds": 122, "width": 640, "height": 360}
{"video_id": "e-3DBiSlkeA", "transcript": [{"start": 0.28, "duration": 5.16, "text": "let's solve this calculus problem step"}, {"start": 2.399, "duration": 6.32, "text": "by step we need to find the derivative"}, {"start": 5.44, "duration": 5.36, "text": "of the square root of tangent Theta"}, {"start": 8.719, "duration": 4.761, "text": "first let's rewrite this in a form"}, {"start": 10.8, "duration": 5.559, "text": "that's easier to work with the square"}, {"start": 13.48, "duration": 6.76, "text": "root is equivalent to raising something"}, {"start": 16.359, "duration": 6.92, "text": "to the 1/2 power so square root of"}, {"start": 20.24, "duration": 6.6, "text": "tangent Theta can be written as tangent"}, {"start": 23.279, "duration": 6.48, "text": "Theta to the 1/2 power now we can use"}, {"start": 26.84, "duration": 4.439, "text": "the power rule for derivatives the power"}, {"start": 29.759, "duration": 4.841, "text": "rule rule states that when"}, {"start": 31.279, "duration": 5.96, "text": "differentiating x to the N power we"}, {"start": 34.6, "duration": 6.04, "text": "multiply by the power and reduce the"}, {"start": 37.239, "duration": 5.881, "text": "exponent by one however since we're"}, {"start": 40.64, "duration": 5.48, "text": "differentiating with respect to Theta"}, {"start": 43.12, "duration": 5.68, "text": "and have a tangent function inside we'll"}, {"start": 46.12, "duration": 5.8, "text": "also need to use the chain rule let's"}, {"start": 48.8, "duration": 7.279, "text": "apply the power rule first we multiply"}, {"start": 51.92, "duration": 9.08, "text": "by 1/2 and reduce the power by 1 this"}, {"start": 56.079, "duration": 7.64, "text": "gives us 12 * Tang thet to the2 power"}, {"start": 61.0, "duration": 5.28, "text": "now for the chain rule part we need to"}, {"start": 63.719, "duration": 5.161, "text": "multiply by the derivative of the inside"}, {"start": 66.28, "duration": 6.36, "text": "function which is tangent Theta the"}, {"start": 68.88, "duration": 8.04, "text": "derivative of tangent Theta is secant s"}, {"start": 72.64, "duration": 9.96, "text": "thet putting it all together we get 12 *"}, {"start": 76.92, "duration": 9.239, "text": "Tang thet to -2 power * SEC 2 thet this"}, {"start": 82.6, "duration": 7.12, "text": "can be Rewritten as 1 / 2 < TK of"}, {"start": 86.159, "duration": 7.24, "text": "tangent thet * secant 2 thet which is"}, {"start": 89.72, "duration": 6.48, "text": "our our final answer to review we solve"}, {"start": 93.399, "duration": 5.841, "text": "this by first rewriting the square root"}, {"start": 96.2, "duration": 5.72, "text": "as a power then using the power rule and"}, {"start": 99.24, "duration": 5.919, "text": "chain rule together the key steps were"}, {"start": 101.92, "duration": 6.159, "text": "reducing the power by one multiplying by"}, {"start": 105.159, "duration": 6.761, "text": "the power and including the derivative"}, {"start": 108.079, "duration": 3.841, "text": "of the inside function"}], "title": "9.3-1.6 How to Differentiate (\u221atan\u2061\ud835\udf03)   Step by Step Explanation.", "description": "In this video, we break down the process of differentiating the function (\u221atan\u2061\ud835\udf03)  using the chain rule and power rule. Follow along as we explain each step and simplify the expression to find the derivative. Perfect for students and math enthusiasts looking to improve their calculus skills!\n#Calculus #Differentiation #MathTutorial #ChainRule #PowerRule #Trigonometry #cpmrana", "lengthSeconds": 110, "width": 640, "height": 360}
{"video_id": "IBzWkt5Non0", "transcript": [{"start": 0.36, "duration": 5.16, "text": "welcome today we'll learn how to find"}, {"start": 3.159, "duration": 5.56, "text": "the derivative of a complex function"}, {"start": 5.52, "duration": 6.079, "text": "using logarithmic differentiation this"}, {"start": 8.719, "duration": 5.401, "text": "technique is particularly useful when"}, {"start": 11.599, "duration": 5.44, "text": "dealing with products quotients and"}, {"start": 14.12, "duration": 5.64, "text": "exponents let's look at our question we"}, {"start": 17.039, "duration": 7.441, "text": "need to find the derivative of y with"}, {"start": 19.76, "duration": 8.0, "text": "respect to X where y equals this complex"}, {"start": 24.48, "duration": 5.559, "text": "fraction the first step in logarithmic"}, {"start": 27.76, "duration": 6.52, "text": "differentiation is to take the natural"}, {"start": 30.039, "duration": 8.281, "text": "log of both sides this gives us Ln y"}, {"start": 34.28, "duration": 6.24, "text": "equal Ln of our entire fraction next we"}, {"start": 38.32, "duration": 5.28, "text": "can use the properties of logarithms to"}, {"start": 40.52, "duration": 6.24, "text": "break this down the log of a quotient is"}, {"start": 43.6, "duration": 6.24, "text": "the difference of logs so we get Ln of"}, {"start": 46.76, "duration": 5.84, "text": "the numerator minus Ln of the"}, {"start": 49.84, "duration": 5.8, "text": "denominator now we can further break"}, {"start": 52.6, "duration": 5.959, "text": "this down using more log properties the"}, {"start": 55.64, "duration": 5.64, "text": "log of a product is the sum of logs and"}, {"start": 58.559, "duration": 6.521, "text": "the log of a power is the power * the"}, {"start": 61.28, "duration": 12.68, "text": "log we get 2 Ln of x + 1"}, {"start": 65.08, "duration": 12.16, "text": "+ Ln of x - 1 - 3 Ln of x + 4 - x Ln of"}, {"start": 73.96, "duration": 5.839, "text": "e now comes the key step we"}, {"start": 77.24, "duration": 6.44, "text": "differentiate both sides with respect to"}, {"start": 79.799, "duration": 7.081, "text": "X remember that when we differentiate Ln"}, {"start": 83.68, "duration": 6.0, "text": "y we get 1 / y *"}, {"start": 86.88, "duration": 5.4, "text": "dydx let's differentiate the right side"}, {"start": 89.68, "duration": 7.479, "text": "term term by term remember that the"}, {"start": 92.28, "duration": 10.56, "text": "derivative of lnx is 1 /x after applying"}, {"start": 97.159, "duration": 12.92, "text": "the chain rule we get 2x + 1 + 1 / 2 *"}, {"start": 102.84, "duration": 8.16, "text": "1/x - 1 - 3/ x + 4 -1 now we can solve"}, {"start": 110.079, "duration": 5.64, "text": "for"}, {"start": 111.0, "duration": 7.439, "text": "dydx multiply both sides by Y and we get"}, {"start": 115.719, "duration": 4.76, "text": "our derivative equals y * this"}, {"start": 118.439, "duration": 5.121, "text": "expression in Brackets"}, {"start": 120.479, "duration": 5.721, "text": "finally we substitute back the original"}, {"start": 123.56, "duration": 5.6, "text": "expression for y this gives us our"}, {"start": 126.2, "duration": 5.24, "text": "complete derivative while this looks"}, {"start": 129.16, "duration": 4.68, "text": "complicated we found it using a"}, {"start": 131.44, "duration": 5.799, "text": "systematic approach that broke down a"}, {"start": 133.84, "duration": 5.72, "text": "complex problem into manageable steps"}, {"start": 137.239, "duration": 5.041, "text": "and that's how we use logarithmic"}, {"start": 139.56, "duration": 5.2, "text": "differentiation to find the derivative"}, {"start": 142.28, "duration": 5.48, "text": "of a complex function thank you for"}, {"start": 144.76, "duration": 3.0, "text": "watching"}], "title": "Mastering Logarithmic Differentiation [((\ud835\udc65+1)^2  \u221a(\ud835\udc65\u22121))/((\ud835\udc65+4)^3  \ud835\udc52^\ud835\udc65 )] Step-by-Step Solution", "description": "Question\nGiven the function y= [((\ud835\udc65+1)^2  \u221a(\ud835\udc65\u22121))/((\ud835\udc65+4)^3  \ud835\udc52^\ud835\udc65 )]  , find dy/dx using logarithmic differentiation.\n\nUnlock the secrets of logarithmic differentiation with this detailed, step-by-step tutorial. We break down the process of finding the derivative of complex functions using logarithms, making each step clear and easy to follow. Perfect for students and math enthusiasts looking to enhance their calculus skills.\n#Calculus #LogarithmicDifferentiation #MathTutorial #Derivatives #MathHelp", "lengthSeconds": 151, "width": 640, "height": 360}
{"video_id": "j6S1rHF1CfU", "transcript": [{"start": 0.199, "duration": 5.841, "text": "let's find the derivative of tangent 3x"}, {"start": 3.919, "duration": 4.961, "text": "using the first principle's definition"}, {"start": 6.04, "duration": 7.88, "text": "of derivatives we'll start by defining"}, {"start": 8.88, "duration": 7.479, "text": "our function f ofx = tangent 3x the"}, {"start": 13.92, "duration": 5.199, "text": "definition of a derivative states that F"}, {"start": 16.359, "duration": 8.441, "text": "Prime of x equals the limit as H"}, {"start": 19.119, "duration": 9.24, "text": "approaches Z of f of x + H - F ofx all"}, {"start": 24.8, "duration": 6.28, "text": "over H when we substitute our function"}, {"start": 28.359, "duration": 8.801, "text": "we get the limit as H approaches is 0 of"}, {"start": 31.08, "duration": 9.24, "text": "tangent of 3 * x + H - tangent of 3x all"}, {"start": 37.16, "duration": 6.64, "text": "over H next we'll convert the tangent"}, {"start": 40.32, "duration": 7.32, "text": "functions to S over cosine this gives us"}, {"start": 43.8, "duration": 7.04, "text": "the limit as H approaches 0 of sin of 3x"}, {"start": 47.64, "duration": 9.0, "text": "+ H over cosine of 3x +"}, {"start": 50.84, "duration": 8.0, "text": "hus s of 3x over cosine of 3x all over H"}, {"start": 56.64, "duration": 3.96, "text": "to subtract these fractions we need a"}, {"start": 58.84, "duration": 3.76, "text": "common denominator"}, {"start": 60.6, "duration": 4.48, "text": "multiplying the numerator and"}, {"start": 62.6, "duration": 5.199, "text": "denominator of each fraction by the"}, {"start": 65.08, "duration": 5.92, "text": "denominator of the other fraction gives"}, {"start": 67.799, "duration": 6.161, "text": "us this expression the numerator is in"}, {"start": 71.0, "duration": 6.159, "text": "the form of the sign difference formula"}, {"start": 73.96, "duration": 5.92, "text": "this simplifies to S of 3 H in the"}, {"start": 77.159, "duration": 7.521, "text": "numerator now we can factor out a three"}, {"start": 79.88, "duration": 7.879, "text": "from the S term giving us 3 * s of 3 H"}, {"start": 84.68, "duration": 6.119, "text": "over 3 H ided by the product of the"}, {"start": 87.759, "duration": 7.161, "text": "cosiness as H approaches 0"}, {"start": 90.799, "duration": 9.121, "text": "s of 3 H over 3 H approaches 1 and"}, {"start": 94.92, "duration": 8.76, "text": "cosine of 3x + H approaches cosine of 3x"}, {"start": 99.92, "duration": 8.799, "text": "this gives us 3 over cosine 2 3x which"}, {"start": 103.68, "duration": 7.56, "text": "equal 3 secant 2 of 3x therefore we have"}, {"start": 108.719, "duration": 8.841, "text": "shown that the derivative of tangent of"}, {"start": 111.24, "duration": 6.32, "text": "3x = 3 secant 3x"}], "title": "Understanding the Derivative of tan3x  Using First Principles", "description": "Learn the basics of calculus by finding the derivative of the exponential function tan3x using first principles. Follow our step-by-step explanations and examples to deepen your understanding of derivatives and exponential functions.\n#Calculus #Derivatives #ExponentialFunctions #MathTutorial #FirstPrinciples #tan3x", "lengthSeconds": 119, "width": 640, "height": 360}
{"video_id": "4GYlu8hWDBI", "transcript": [{"start": 0.24, "duration": 4.559, "text": "let's learn how to differentiate cosine"}, {"start": 2.32, "duration": 5.68, "text": "of pix over"}, {"start": 4.799, "duration": 6.361, "text": "180 this is an important expression that"}, {"start": 8.0, "duration": 6.24, "text": "helps us convert between degrees and"}, {"start": 11.16, "duration": 6.68, "text": "radians let's start with our expression"}, {"start": 14.24, "duration": 7.44, "text": "we have y = cosine of x de which we can"}, {"start": 17.84, "duration": 6.88, "text": "write as y = cosine of pkx over"}, {"start": 21.68, "duration": 6.08, "text": "180 to differentiate this we'll need to"}, {"start": 24.72, "duration": 5.799, "text": "use the chain rule the chain rule states"}, {"start": 27.76, "duration": 8.2, "text": "that if yal F of G ofx"}, {"start": 30.519, "duration": 9.36, "text": "then Dy DX = frime of G ofx * G Prime of"}, {"start": 35.96, "duration": 7.48, "text": "X let's identify our functions in our"}, {"start": 39.879, "duration": 8.121, "text": "case the outer function f of U is cosine"}, {"start": 43.44, "duration": 5.84, "text": "of U and the inner function G ofx is pix"}, {"start": 48.0, "duration": 4.399, "text": "over"}, {"start": 49.28, "duration": 8.0, "text": "180 now let's apply the chain rule step"}, {"start": 52.399, "duration": 7.84, "text": "by step first we need fime of G ofx the"}, {"start": 57.28, "duration": 7.6, "text": "derivative of cosine is negative s so"}, {"start": 60.239, "duration": 7.841, "text": "this gives us negative sign of PX over"}, {"start": 64.88, "duration": 5.8, "text": "180 next we need G Prime of X which is"}, {"start": 68.08, "duration": 5.92, "text": "the derivative of pix over"}, {"start": 70.68, "duration": 7.36, "text": "180 this gives us pi over"}, {"start": 74.0, "duration": 6.439, "text": "180 finally we multiply these together"}, {"start": 78.04, "duration": 6.719, "text": "according to the chain rule this gives"}, {"start": 80.439, "duration": 10.761, "text": "us our final answer Dy DX equal negative"}, {"start": 84.759, "duration": 8.961, "text": "s of PX over 180 * < over 18 80 let's"}, {"start": 91.2, "duration": 6.52, "text": "visualize what this means here's the"}, {"start": 93.72, "duration": 9.399, "text": "graph of y equal cine of PX over 180 in"}, {"start": 97.72, "duration": 10.399, "text": "blue and its derivative < over 180 * s"}, {"start": 103.119, "duration": 7.04, "text": "of pix over 180 in red notice how the"}, {"start": 108.119, "duration": 4.801, "text": "derivative graph shows the rate of"}, {"start": 110.159, "duration": 5.6, "text": "change of the original function when"}, {"start": 112.92, "duration": 6.0, "text": "cosine is increasing its derivative is"}, {"start": 115.759, "duration": 5.481, "text": "positive and when cosine is decreasing"}, {"start": 118.92, "duration": 5.68, "text": "its derivative is negative the"}, {"start": 121.24, "duration": 6.44, "text": "derivative is zero exactly at the Peaks"}, {"start": 124.6, "duration": 6.279, "text": "and troughs of the cosine function and"}, {"start": 127.68, "duration": 7.199, "text": "that's how we differentiate cosine of PX"}, {"start": 130.879, "duration": 7.281, "text": "over 180 using the chain rule thank you"}, {"start": 134.879, "duration": 3.281, "text": "for watching"}], "title": "9.3-1.8 Understanding Trigonometric Differentiation (\ud835\udc50\ud835\udc5c\ud835\udc60\ud835\udc65^0) Step-by-Step Guide Using the Chain Rule", "description": "In this video, we\u2019ll explore how to differentiate trigonometric functions using the chain rule. Learn to tackle expressions like (\ud835\udc50\ud835\udc5c\ud835\udc60\ud835\udc65^0) with ease. We\u2019ll break down the steps, helping you understand the transformation of degrees to radians, applying the chain rule, and combining derivatives efficiently. Perfect for students and math enthusiasts looking to strengthen their calculus skills!\nHashtags:\n#cpmrana  #Calculus #Trigonometry #ChainRule #MathTutorial #Differentiation", "lengthSeconds": 137, "width": 640, "height": 360}
{"video_id": "_HDL1lHYJD0", "transcript": [{"start": 0.4, "duration": 5.28, "text": "welcome today we're going to learn about"}, {"start": 3.04, "duration": 6.04, "text": "logarithmic differentiation by finding"}, {"start": 5.68, "duration": 7.64, "text": "the derivative of a complex exponential"}, {"start": 9.08, "duration": 7.6, "text": "function our function is y = x ra to the"}, {"start": 13.32, "duration": 6.799, "text": "power of E to X why do we need"}, {"start": 16.68, "duration": 5.96, "text": "logarithmic differentiation here well"}, {"start": 20.119, "duration": 5.201, "text": "this function is particularly tricky"}, {"start": 22.64, "duration": 5.76, "text": "because we have a variable X as both the"}, {"start": 25.32, "duration": 6.2, "text": "base and in the exponent using normal"}, {"start": 28.4, "duration": 5.96, "text": "differentiation rules would be extremely"}, {"start": 31.52, "duration": 5.84, "text": "complicated however by taking the"}, {"start": 34.36, "duration": 5.6, "text": "natural log of both sides first we can"}, {"start": 37.36, "duration": 6.359, "text": "make this much easier to handle let's"}, {"start": 39.96, "duration": 7.0, "text": "solve a step by step step one take the"}, {"start": 43.719, "duration": 11.121, "text": "natural log of both sides this gives us"}, {"start": 46.96, "duration": 10.56, "text": "Ln of Y = Ln of x raed to e to X Step 2"}, {"start": 54.84, "duration": 7.679, "text": "use the logarithm property that says Ln"}, {"start": 57.52, "duration": 11.52, "text": "of x raed to n = n * Ln of X this gives"}, {"start": 62.519, "duration": 6.521, "text": "us Ln of y = e x * Ln of"}, {"start": 70.36, "duration": 7.0, "text": "X step three now we can differentiate"}, {"start": 74.119, "duration": 6.401, "text": "both sides with respect to X on the left"}, {"start": 77.36, "duration": 7.28, "text": "side we get 1/ y *"}, {"start": 80.52, "duration": 4.12, "text": "dydx this comes from the chain"}, {"start": 86.0, "duration": 6.28, "text": "rule step four for the right side we"}, {"start": 89.96, "duration": 6.6, "text": "need to use the product rule since we"}, {"start": 92.28, "duration": 8.28, "text": "have e to x * Ln of X the product rule"}, {"start": 96.56, "duration": 9.519, "text": "says that the derivative of U * v = u *"}, {"start": 100.56, "duration": 10.12, "text": "dvdx + B * D DX in our case U is e to"}, {"start": 106.079, "duration": 4.601, "text": "the X and V is Ln of"}, {"start": 114.56, "duration": 9.0, "text": "X step five let's differentiate each"}, {"start": 118.36, "duration": 9.359, "text": "part the derivative of e to X is e to X"}, {"start": 123.56, "duration": 8.24, "text": "the derivative of Ln of X is 1 /x"}, {"start": 127.719, "duration": 9.88, "text": "putting this together we get e to x * 1"}, {"start": 131.8, "duration": 5.799, "text": "/x + Ln of x * e to"}, {"start": 142.84, "duration": 8.24, "text": "X step six now we can solve for"}, {"start": 147.04, "duration": 7.4, "text": "dydx multiply both sides by 1 Y which is"}, {"start": 151.08, "duration": 8.239, "text": "X the E to X this gives us our final"}, {"start": 154.44, "duration": 11.56, "text": "answer Dy DX = x e to the x * the"}, {"start": 159.319, "duration": 6.681, "text": "quantity e to X over x + Ln of x * e to"}, {"start": 170.959, "duration": 7.0, "text": "X and there you have it we've found the"}, {"start": 174.8, "duration": 5.28, "text": "derivative using logarithmic"}, {"start": 177.959, "duration": 3.961, "text": "differentiation this method is is"}, {"start": 180.08, "duration": 3.879, "text": "particularly useful when dealing with"}, {"start": 181.92, "duration": 5.12, "text": "functions that have variables in both"}, {"start": 183.959, "duration": 6.081, "text": "the base and exponent thank you for"}, {"start": 187.04, "duration": 3.0, "text": "watching"}], "title": "Mastering Logarithmic Differentiation: Solving d/dx[\u3016\ud835\udc99^\ud835\udc86\u3017^\ud835\udc99] Step by Step", "description": "Question: Find the derivative of y = [\u3016\ud835\udc99^\ud835\udc86\u3017^\ud835\udc99] using logarithmic differentiation.\n\n\ud83d\udcda In this comprehensive calculus tutorial, we break down how to find the derivative of y = [\u3016\ud835\udc99^\ud835\udc86\u3017^\ud835\udc99] using logarithmic differentiation. Perfect for calculus students and math enthusiasts!\n\ud83c\udfaf What You\u2019ll Learn:\n\u2022 Why logarithmic differentiation is crucial for complex exponential functions\n\u2022 How to properly apply natural logarithm properties\n\u2022 Step-by-step application of chain rule and product rule\n\u2022 Clear explanation of each mathematical move\n\u2022 Common pitfalls to avoid\n\ud83d\udca1 Related topics:\nChain Rule\nProduct Rule\nNatural Logarithms\nExponential Functions\nCalculus I\n\ud83d\udcdd Need more practice? Check out our full calculus playlist!\n#Calculus #Mathematics #LogarithmicDifferentiation #ChainRule #MathTutorial #CalcTutorial #MathHelp #ExponentialFunctions #CalculusHelp #Education", "lengthSeconds": 195, "width": 640, "height": 360}
{"video_id": "bmDJ3iD99nU", "transcript": [{"start": 0.359, "duration": 5.241, "text": "let's solve this challenging derivative"}, {"start": 2.679, "duration": 7.081, "text": "problem step by step we need to find the"}, {"start": 5.6, "duration": 7.039, "text": "derivative of Ln of sinx s raised to the"}, {"start": 9.76, "duration": 5.68, "text": "power n this is a complex function"}, {"start": 12.639, "duration": 7.201, "text": "involving multiple layers first we have"}, {"start": 15.44, "duration": 7.2, "text": "x s inside s then sin x^2 is inside a"}, {"start": 19.84, "duration": 5.48, "text": "natural log and finally all of that is"}, {"start": 22.64, "duration": 6.16, "text": "raised to the power n we'll need to use"}, {"start": 25.32, "duration": 6.039, "text": "the chain rule multiple times let's"}, {"start": 28.8, "duration": 5.88, "text": "start with the outermost layer using the"}, {"start": 31.359, "duration": 6.641, "text": "chain rule we bring down the power n and"}, {"start": 34.68, "duration": 7.32, "text": "reduce the exponent by one this gives us"}, {"start": 38.0, "duration": 6.76, "text": "n * Ln of sin x^ 2^"}, {"start": 42.0, "duration": 5.12, "text": "nus1 time the derivative of what's"}, {"start": 44.76, "duration": 6.639, "text": "inside now we need to find the"}, {"start": 47.12, "duration": 8.759, "text": "derivative of Ln of sin x^2 using the"}, {"start": 51.399, "duration": 7.48, "text": "chain rule again this is 1/ sin x^2 *"}, {"start": 55.879, "duration": 4.561, "text": "the derivative of sin x^2 let's handle"}, {"start": 58.879, "duration": 5.441, "text": "the derivative of sin"}, {"start": 60.44, "duration": 7.52, "text": "x^2 using the chain rule one more time"}, {"start": 64.32, "duration": 9.2, "text": "this is cosine x^2 time the derivative"}, {"start": 67.96, "duration": 9.159, "text": "of X2 the derivative of x^2 is simply 2x"}, {"start": 73.52, "duration": 6.48, "text": "let's substitute that in now we can"}, {"start": 77.119, "duration": 7.481, "text": "simplify this expression notice that"}, {"start": 80.0, "duration": 8.119, "text": "cine x^2 over sin x^2 is the definition"}, {"start": 84.6, "duration": 7.72, "text": "of cotangent x^2 rearranging the terms"}, {"start": 88.119, "duration": 10.121, "text": "gives us our final answer 2 NX * cang"}, {"start": 92.32, "duration": 9.079, "text": "x^2 * Ln of sin x^ 2^ n -1 let's review"}, {"start": 98.24, "duration": 5.64, "text": "the key steps we used first we used the"}, {"start": 101.399, "duration": 5.441, "text": "chain rule for the power function then"}, {"start": 103.88, "duration": 5.96, "text": "we used it again for the natural log"}, {"start": 106.84, "duration": 6.12, "text": "finally we used it one more time for the"}, {"start": 109.84, "duration": 5.4, "text": "sign function this problem shows how the"}, {"start": 112.96, "duration": 4.36, "text": "chain rule can help us tackle complex"}, {"start": 115.24, "duration": 5.519, "text": "derivatives by breaking them down into"}, {"start": 117.32, "duration": 3.439, "text": "simpler steps"}], "title": "Mastering the Derivative of Logarithmic Trigonometric Functions [ln\u2061(sin\u2061\ud835\udc65^2)]^\ud835\udc5b", "description": "Unlock the secrets of differentiating logarithmic trigonometric functions with this step-by-step tutorial! Learn how to find the derivative of expressions like [ln\u2061(sin\u2061\ud835\udc65^2)]^\ud835\udc5b using product and chain rules. This video is perfect for students tackling calculus problems or anyone looking to deepen their understanding of advanced mathematical concepts.\n#Calculus #Differentiation #Trigonometry #MathTutorial #AdvancedMath", "lengthSeconds": 123, "width": 640, "height": 360}
{"video_id": "ZOWesnHaR6c", "transcript": [{"start": 0.28, "duration": 5.36, "text": "let's solve this challenging derivative"}, {"start": 2.679, "duration": 5.321, "text": "problem step by step we'll use the chain"}, {"start": 5.64, "duration": 5.4, "text": "Rule and inverse function rules to find"}, {"start": 8.0, "duration": 7.0, "text": "the derivative of tan inverse of sin of"}, {"start": 11.04, "duration": 7.12, "text": "e to X let's break this down into parts"}, {"start": 15.0, "duration": 5.64, "text": "we have three nested functions first the"}, {"start": 18.16, "duration": 5.16, "text": "outer function is tan inverse then we"}, {"start": 20.64, "duration": 5.039, "text": "have S as our middle function and"}, {"start": 23.32, "duration": 5.16, "text": "finally e to thex as our innermost"}, {"start": 25.679, "duration": 4.92, "text": "function now let's recall the derivative"}, {"start": 28.48, "duration": 5.04, "text": "formula for in"}, {"start": 30.599, "duration": 8.001, "text": "tangent the derivative of tan inverse of"}, {"start": 33.52, "duration": 10.32, "text": "U with respect to U is 1/ 1 + U ^2 in"}, {"start": 38.6, "duration": 8.599, "text": "our case U equal s of e to X so when we"}, {"start": 43.84, "duration": 7.6, "text": "apply the inverse tangent rule we get 1"}, {"start": 47.199, "duration": 8.04, "text": "1 + sin^2 of e to X time the derivative"}, {"start": 51.44, "duration": 6.919, "text": "of sin of e to X now we need to find the"}, {"start": 55.239, "duration": 6.521, "text": "derivative of sign of e to X using the"}, {"start": 58.359, "duration": 7.641, "text": "chain rule the D dtive of s of V is"}, {"start": 61.76, "duration": 8.12, "text": "cosine of V * the derivative of v in our"}, {"start": 66.0, "duration": 7.28, "text": "case v is e to X so we get cosine of e"}, {"start": 69.88, "duration": 7.36, "text": "to the X time the derivative of e to the"}, {"start": 73.28, "duration": 8.56, "text": "X finally we know that the derivative of"}, {"start": 77.24, "duration": 9.16, "text": "e to X is just e to X so our final"}, {"start": 81.84, "duration": 8.84, "text": "answer is e to x * cosine of e to X all"}, {"start": 86.4, "duration": 6.88, "text": "over 1 + sin 2 e to X let's let's verify"}, {"start": 90.68, "duration": 6.04, "text": "that this makes sense our derivative is"}, {"start": 93.28, "duration": 5.96, "text": "a fraction where the numerator contains"}, {"start": 96.72, "duration": 5.56, "text": "e to X which came from the derivative of"}, {"start": 99.24, "duration": 5.72, "text": "our innermost function multiplied by"}, {"start": 102.28, "duration": 5.479, "text": "cosine of e to X which came from the"}, {"start": 104.96, "duration": 6.0, "text": "derivative of s the denominator comes"}, {"start": 107.759, "duration": 5.96, "text": "from the derivative formula of inverse"}, {"start": 110.96, "duration": 5.119, "text": "tangent this is exactly what we'd expect"}, {"start": 113.719, "duration": 5.841, "text": "from applying the chain rule to our"}, {"start": 116.079, "duration": 3.481, "text": "nested functions"}], "title": "Mastering Derivatives: Calculating the Derivative of [\ud835\udc61\ud835\udc4e\ud835\udc5b^(\u22121) (\ud835\udc60\ud835\udc56\ud835\udc5b\ud835\udc52^\ud835\udc65 )]", "description": "In this video, we break down the step-by-step process of finding the derivative of [\ud835\udc61\ud835\udc4e\ud835\udc5b^(\u22121) (\ud835\udc60\ud835\udc56\ud835\udc5b\ud835\udc52^\ud835\udc65 )] . Learn how to apply the chain rule and understand the logic behind each step. Perfect for calculus students looking to strengthen their derivative skills!\n#Calculus #Derivatives #MathTutorial #ChainRule #MathHelp #InverseTrigonometricFunctions", "lengthSeconds": 122, "width": 640, "height": 360}
{"video_id": "cWBSZClqUwA", "transcript": [{"start": 0.28, "duration": 4.64, "text": "let's solve this challenging"}, {"start": 1.959, "duration": 5.601, "text": "differentiation problem step by step we"}, {"start": 4.92, "duration": 5.92, "text": "need to find the derivative of sign of"}, {"start": 7.56, "duration": 6.199, "text": "natural log of secant x with respect to"}, {"start": 10.84, "duration": 5.92, "text": "X let's break this down using the chain"}, {"start": 13.759, "duration": 6.321, "text": "rule when we have a composite function"}, {"start": 16.76, "duration": 6.2, "text": "like this we multiply the derivative of"}, {"start": 20.08, "duration": 6.599, "text": "the outer function by the derivative of"}, {"start": 22.96, "duration": 7.12, "text": "the inner function here s is our outer"}, {"start": 26.679, "duration": 6.601, "text": "function and natural log of secant x is"}, {"start": 30.08, "duration": 6.4, "text": "our inner function for our first step we"}, {"start": 33.28, "duration": 5.32, "text": "know that the derivative of s is cosine"}, {"start": 36.48, "duration": 4.68, "text": "so we'll have cosine of the inner"}, {"start": 38.6, "duration": 5.32, "text": "function times the derivative of the"}, {"start": 41.16, "duration": 6.52, "text": "inner function now we need to find the"}, {"start": 43.92, "duration": 6.639, "text": "derivative of natural log of secant x"}, {"start": 47.68, "duration": 5.84, "text": "using the chain rule again we know that"}, {"start": 50.559, "duration": 6.281, "text": "the derivative of natural log of U is 1/"}, {"start": 53.52, "duration": 5.08, "text": "U times the derivative of U the final"}, {"start": 56.84, "duration": 6.16, "text": "piece we need is the derivative of"}, {"start": 58.6, "duration": 8.199, "text": "secant x which is secant x * tangent X"}, {"start": 63.0, "duration": 6.4, "text": "now let's simplify the secant x terms"}, {"start": 66.799, "duration": 5.841, "text": "cancel in the middle leaving us with"}, {"start": 69.4, "duration": 7.28, "text": "cosine of natural log of secant x *"}, {"start": 72.64, "duration": 7.28, "text": "tangent X therefore our final answer is"}, {"start": 76.68, "duration": 6.36, "text": "tangent x * cine of natural log of"}, {"start": 79.92, "duration": 6.76, "text": "secant x this problem demonstrated the"}, {"start": 83.04, "duration": 5.96, "text": "power of the chain Rule and how breaking"}, {"start": 86.68, "duration": 5.799, "text": "down complex derivatives into smaller"}, {"start": 89.0, "duration": 6.439, "text": "steps makes them manageable remember to"}, {"start": 92.479, "duration": 7.6, "text": "identify your composite functions and"}, {"start": 95.439, "duration": 4.64, "text": "apply the chain rule systematically"}], "title": "Differentiating [sin\u2061\ud835\udc59\ud835\udc5b(sec\u2061\ud835\udc65 ) ] - Step-by-Step Calculus Explained!", "description": "In this video, we break down how to differentiate the expression [sin\u2061\ud835\udc59\ud835\udc5b(sec\u2061\ud835\udc65 ) ]  step by step. Using the chain rule and understanding derivatives, we simplify the process to make it easier to understand. Perfect for students seeking a deeper comprehension of calculus concepts!\n#Calculus\n#MathTutorial\n#Differentiation\n#ChainRule\n#MathHelp", "lengthSeconds": 103, "width": 640, "height": 360}
{"video_id": "2Ye8yaR7808", "transcript": [{"start": 0.4, "duration": 5.879, "text": "today we'll find the derivative of"}, {"start": 2.96, "duration": 5.88, "text": "secant 2x using the first principles"}, {"start": 6.279, "duration": 5.041, "text": "approach let's break this down step by"}, {"start": 8.84, "duration": 8.88, "text": "step let's start by defining our"}, {"start": 11.32, "duration": 12.48, "text": "function let F ofx = secant 2x then f of"}, {"start": 17.72, "duration": 8.52, "text": "x + H = secant of 2 * x + H by"}, {"start": 23.8, "duration": 7.2, "text": "definition the derivative is the limit"}, {"start": 26.24, "duration": 9.8, "text": "as H approaches Z of f of x + h -"}, {"start": 31.0, "duration": 7.559, "text": "FX all divid H substituting our function"}, {"start": 36.04, "duration": 9.24, "text": "we get the limit as H approaches 0 of"}, {"start": 38.559, "duration": 11.441, "text": "secant of 2 * x + H - secant of 2x all"}, {"start": 45.28, "duration": 6.959, "text": "ID H since cant is 1 / cosine we can"}, {"start": 50.0, "duration": 7.12, "text": "rewrite this as the limit as H"}, {"start": 52.239, "duration": 9.241, "text": "approaches Z of 1/ cosine of 2x + H - 1"}, {"start": 57.12, "duration": 6.52, "text": "/ cosine of 2x all divided by H to"}, {"start": 61.48, "duration": 4.959, "text": "subtract these fractions we need a"}, {"start": 63.64, "duration": 5.32, "text": "common denominator multiplying both"}, {"start": 66.439, "duration": 8.201, "text": "terms by their respective denominators"}, {"start": 68.96, "duration": 8.68, "text": "gives us cosine of 2x - cosine of 2x + H"}, {"start": 74.64, "duration": 6.04, "text": "over the product of the cosiness now we"}, {"start": 77.64, "duration": 7.32, "text": "can use the cosine difference formula"}, {"start": 80.68, "duration": 7.24, "text": "the difference of cosiness equals -2 * s"}, {"start": 84.96, "duration": 5.72, "text": "of their sum / 2 * s of their difference"}, {"start": 87.92, "duration": 7.8, "text": "/ 2 simplifying the arguments in the"}, {"start": 90.68, "duration": 7.88, "text": "sign terms we get S of 2x + H * sin of H"}, {"start": 95.72, "duration": 7.359, "text": "in the numerator we can split this into"}, {"start": 98.56, "duration": 8.76, "text": "separate limits the limit of s of 2x + H"}, {"start": 103.079, "duration": 9.0, "text": "as H approaches Z is s of 2x the limit"}, {"start": 107.32, "duration": 8.28, "text": "of sign H over H as H approaches Z is 1"}, {"start": 112.079, "duration": 7.241, "text": "and the limit of cosine of 2x + H is"}, {"start": 115.6, "duration": 8.72, "text": "cosine of 2x finally we can write this"}, {"start": 119.32, "duration": 10.4, "text": "as 2 sin of 2x/ cosine ^ 2 2x which = 2"}, {"start": 124.32, "duration": 9.44, "text": "tangent of 2x * secant of 2x therefore"}, {"start": 129.72, "duration": 9.799, "text": "the derivative of secant of 2x = 2"}, {"start": 133.76, "duration": 9.479, "text": "tangent of 2x * secant of 2x and that's"}, {"start": 139.519, "duration": 6.201, "text": "how we find the derivative of secant 2x"}, {"start": 143.239, "duration": 5.481, "text": "using first principles thank you for"}, {"start": 145.72, "duration": 3.0, "text": "watching"}], "title": "Understanding the Derivative of sec2x  Using First Principles", "description": "Master calculus fundamentals by deriving the exponential function sec2x from first principles. Our step-by-step explanations and examples will strengthen your understanding of derivatives and exponential functions.\n#Calculus #Derivatives #ExponentialFunctions #MathTutorial #FirstPrinciples #sec2x", "lengthSeconds": 148, "width": 640, "height": 360}
{"video_id": "ojxLPDPcrVk", "transcript": [{"start": 0.28, "duration": 5.519, "text": "let's solve this calculus problem step"}, {"start": 2.56, "duration": 11.119, "text": "by step we need to find the derivative"}, {"start": 5.799, "duration": 10.521, "text": "of y = 10 x - Cent X over Tan x + Cent X"}, {"start": 13.679, "duration": 6.161, "text": "the first step is to convert tangent and"}, {"start": 16.32, "duration": 7.0, "text": "coent into their s and cosine forms"}, {"start": 19.84, "duration": 7.839, "text": "tangent X is sin x over cosine x and"}, {"start": 23.32, "duration": 6.68, "text": "cang x is cine X over sin x next we'll"}, {"start": 27.679, "duration": 5.121, "text": "find a common denominator in both the"}, {"start": 30.0, "duration": 6.0, "text": "numerator and denominator in the"}, {"start": 32.8, "duration": 8.16, "text": "numerator we multiply both fractions by"}, {"start": 36.0, "duration": 10.079, "text": "sin x cine X this gives us sin^2 x -"}, {"start": 40.96, "duration": 10.04, "text": "cosine 2 x/ sin x cosine X similarly in"}, {"start": 46.079, "duration": 9.441, "text": "the denominator we get sin^2 x + cine 2"}, {"start": 51.0, "duration": 8.12, "text": "x/ sin x cine X now let's recall a"}, {"start": 55.52, "duration": 8.44, "text": "fundamental trigonometric identity sin^2"}, {"start": 59.12, "duration": 7.92, "text": "X Plus cosine 2 x = 1 this means our"}, {"start": 63.96, "duration": 8.799, "text": "denominator simplifies to 1 this"}, {"start": 67.04, "duration": 9.16, "text": "simplifies to Just s^ 2 x - cosine 2 x"}, {"start": 72.759, "duration": 9.72, "text": "and here's another useful identity sin 2"}, {"start": 76.2, "duration": 11.16, "text": "x - cosine ^ 2 x = cosine of 2x now that"}, {"start": 82.479, "duration": 7.561, "text": "we have y = cosine of 2x we can find its"}, {"start": 87.36, "duration": 4.92, "text": "derivative the derivative of cosine is"}, {"start": 90.04, "duration": 5.96, "text": "negative sign and when we have a chain"}, {"start": 92.28, "duration": 6.119, "text": "rule situation with 2x we multiply by"}, {"start": 96.0, "duration": 5.88, "text": "the derivative of the inside function"}, {"start": 98.399, "duration": 6.561, "text": "which is two finally we can simplify"}, {"start": 101.88, "duration": 5.559, "text": "this the two negatives cancel out and we"}, {"start": 104.96, "duration": 6.92, "text": "get our final answer the derivative"}, {"start": 107.439, "duration": 7.761, "text": "equals 2 sin 2x let's verify this makes"}, {"start": 111.88, "duration": 5.159, "text": "sense our original function was a ratio"}, {"start": 115.2, "duration": 5.4, "text": "of trigonometric functions that"}, {"start": 117.039, "duration": 6.4, "text": "simplified to cosine 2X taking the"}, {"start": 120.6, "duration": 4.4, "text": "derivative gave us 2 sin 2x which"}, {"start": 123.439, "duration": 4.281, "text": "matches what we'd expect from the"}, {"start": 125.0, "duration": 7.879, "text": "derivative of a cosine function with a"}, {"start": 127.72, "duration": 5.159, "text": "factor of 2x inside"}], "title": "Mastering Trigonometric Differentiation {(tan\u2061\ud835\udc65 \u2212cot\u2061\ud835\udc65)/(tan\u2061\ud835\udc65+cot\u2061\ud835\udc65)} Step-by-Step Guide", "description": "Unlock the secrets of trigonometric differentiation with our comprehensive tutorial! In this video, we will walk you through the process of differentiating the expression y={(tan\u2061\ud835\udc65 \u2212cot\u2061\ud835\udc65)/(tan\u2061\ud835\udc65+cot\u2061\ud835\udc65)}   step-by-step. Whether you\u2019re a math enthusiast or preparing for exams, this guide will help you understand each step with clear explanations. Subscribe for more math tutorials and enhance your problem-solving skills.\n#Trigonometry #Calculus #MathTutorial #Differentiation #MathHelp #StudyMath #EducationalVideo #MathSkills", "lengthSeconds": 136, "width": 640, "height": 360}
{"video_id": "tvMHo9f6CWg", "transcript": [{"start": 0.32, "duration": 5.24, "text": "let's solve this differentiation problem"}, {"start": 3.0, "duration": 5.32, "text": "using the quotient rule we'll go through"}, {"start": 5.56, "duration": 5.84, "text": "this step by step let's start with our"}, {"start": 8.32, "duration": 9.959, "text": "function we need to differentiate F ofx"}, {"start": 11.4, "duration": 9.4, "text": "= 3x + 5 / 2x^2 - 9 to do this we'll use"}, {"start": 18.279, "duration": 4.84, "text": "the quotient rule the quotient rule"}, {"start": 20.8, "duration": 6.6, "text": "states that the derivative of U over V"}, {"start": 23.119, "duration": 7.64, "text": "equals V * the derivative of U minus U *"}, {"start": 27.4, "duration": 8.839, "text": "the derivative of V all over v\u00b2"}, {"start": 30.759, "duration": 11.32, "text": "let's identify our u and v here U = 3x +"}, {"start": 36.239, "duration": 9.0, "text": "5 and V = 2x^2 - 9 now let's find the"}, {"start": 42.079, "duration": 7.761, "text": "derivatives of u and v the derivative of"}, {"start": 45.239, "duration": 7.0, "text": "U is 3 and the derivative of V is 4X"}, {"start": 49.84, "duration": 9.84, "text": "let's substitute these into our quotient"}, {"start": 52.239, "duration": 14.721, "text": "rule formula we get 2x^2 - 9 * 3 - 3x +"}, {"start": 59.68, "duration": 12.799, "text": "5 * 4X all over 2x^2 - 9^ 2 now let's"}, {"start": 66.96, "duration": 13.76, "text": "expand the numerator first 2x^2 - 9 * 3"}, {"start": 72.479, "duration": 12.201, "text": "gives us 6 x^2 - 27 then 3x + 5 * 4X"}, {"start": 80.72, "duration": 11.32, "text": "gives us 12 x^2 + 20x so in the"}, {"start": 84.68, "duration": 14.68, "text": "numerator we have 6 x^2 - 27 - 12x^2 -"}, {"start": 92.04, "duration": 11.92, "text": "20x this simplifies to -6 x^2 - 20x - 27"}, {"start": 99.36, "duration": 12.88, "text": "therefore our final answer is - 6x^2 +"}, {"start": 103.96, "duration": 10.159, "text": "20x + 27 over 4X 4 - 36 x^2 + 81 we can"}, {"start": 112.24, "duration": 8.64, "text": "factor out a negative from the numerator"}, {"start": 114.119, "duration": 13.161, "text": "to write it as * the 6 x^2 + 20x + 20 7"}, {"start": 120.88, "duration": 8.76, "text": "over 4X 4 - 36 x^2 + 81 and there we"}, {"start": 127.28, "duration": 4.959, "text": "have it we've successfully used the"}, {"start": 129.64, "duration": 5.44, "text": "quotient rule to find the derivative of"}, {"start": 132.239, "duration": 6.321, "text": "this function remember the key steps"}, {"start": 135.08, "duration": 6.28, "text": "identify u and v find their derivatives"}, {"start": 138.56, "duration": 6.52, "text": "apply the quotient rule formula and then"}, {"start": 141.36, "duration": 3.72, "text": "simplify the result"}], "title": "How to Differentiate Rational Functions ((\ud835\udfd1\ud835\udc99+\ud835\udfd3)/(\ud835\udfd0\ud835\udc99^2-9)) Step-by-Step Tutorial", "description": "In this video, we dive into the differentiation of rational functions using the quotient rule. Follow along as we break down each step to find the derivative of ((\ud835\udfd1\ud835\udc99+\ud835\udfd3)/(\ud835\udfd0\ud835\udc99^2-9)). Master calculus with our clear and concise explanations, and boost your math skills today!\n#Calculus #Differentiation #MathTutorial #QuotientRule #LearnMath", "lengthSeconds": 146, "width": 640, "height": 360}
{"video_id": "HGsmx7Mz9Is", "transcript": [{"start": 0.92, "duration": 5.2, "text": "welcome today we'll solve a challenging"}, {"start": 3.679, "duration": 3.84, "text": "differentiation problem using"}, {"start": 6.12, "duration": 4.08, "text": "logarithmic"}, {"start": 7.519, "duration": 6.0, "text": "differentiation our task is to find the"}, {"start": 10.2, "duration": 6.56, "text": "derivative of x to the X Power Plus sin"}, {"start": 13.519, "duration": 5.401, "text": "x raised to the natural log of x let's"}, {"start": 16.76, "duration": 6.56, "text": "break this into two parts to make it"}, {"start": 18.92, "duration": 9.359, "text": "easier we'll call the first part y1 = x"}, {"start": 23.32, "duration": 8.439, "text": "x and the second part Y 2 = sinx to the"}, {"start": 28.279, "duration": 6.041, "text": "natural log of x let let's start with y1"}, {"start": 31.759, "duration": 4.041, "text": "to differentiate this we'll use"}, {"start": 34.32, "duration": 4.12, "text": "logarithmic"}, {"start": 35.8, "duration": 5.439, "text": "differentiation first take the natural"}, {"start": 38.44, "duration": 3.599, "text": "log of both sides using the properties"}, {"start": 41.239, "duration": 5.201, "text": "of"}, {"start": 42.039, "duration": 7.281, "text": "logarithms this simplifies to x * lnx"}, {"start": 46.44, "duration": 6.639, "text": "now we can differentiate both sides with"}, {"start": 49.32, "duration": 9.44, "text": "respect to X on the left side we get 1"}, {"start": 53.079, "duration": 9.44, "text": "over y1 * dy1 DX on the right side using"}, {"start": 58.76, "duration": 9.56, "text": "the product rule we get x * d by DX of"}, {"start": 62.519, "duration": 10.401, "text": "Ln x + Ln x * d by DX of X simplifying"}, {"start": 68.32, "duration": 8.839, "text": "the right side we get 1 + Ln X then"}, {"start": 72.92, "duration": 10.68, "text": "multiply both sides by y1 to get Dy 1 DX"}, {"start": 77.159, "duration": 12.881, "text": "= y1 * 1 + lnx finally substitute back"}, {"start": 83.6, "duration": 9.44, "text": "y1 = X to X now let's tackle Y 2 again"}, {"start": 90.04, "duration": 9.28, "text": "we'll take the natural log of both sides"}, {"start": 93.04, "duration": 10.039, "text": "this gives us Ln Y2 = Ln x * Ln of sinx"}, {"start": 99.32, "duration": 9.119, "text": "differentiating both sides we get 1 / Y2"}, {"start": 103.079, "duration": 10.241, "text": "* dy2 DX = Ln x * the derivative of Ln"}, {"start": 108.439, "duration": 9.64, "text": "sin x + Ln sin x * the derivative of Ln"}, {"start": 113.32, "duration": 7.24, "text": "X after simplifying we get Ln X Cent X"}, {"start": 118.079, "duration": 6.08, "text": "Plus Ln sin x"}, {"start": 120.56, "duration": 5.72, "text": "/x now we can combine our results the"}, {"start": 124.159, "duration": 7.44, "text": "derivative of the original expression is"}, {"start": 126.28, "duration": 10.92, "text": "the sum of Dy 1 DX and dy2 DX this gives"}, {"start": 131.599, "duration": 11.801, "text": "us x x * 1 + Ln X Plus sin x to the Ln x"}, {"start": 137.2, "duration": 9.399, "text": "* Ln xent x + Ln sin x /x and that's our"}, {"start": 143.4, "duration": 5.8, "text": "final answer we used logarithmic"}, {"start": 146.599, "duration": 5.081, "text": "differentiation to handle these complex"}, {"start": 149.2, "duration": 5.2, "text": "expressions with variables in the"}, {"start": 151.68, "duration": 4.6, "text": "exponents this technique is very"}, {"start": 154.4, "duration": 4.88, "text": "powerful for differentiating such"}, {"start": 156.28, "duration": 3.0, "text": "expressions"}], "title": "Mastering Complex Derivatives: How to Find d/dx[x^x + (sin x)^ln x] | Step by Step", "description": "Question:\nFind the derivative of [x^x + (sin x)^ln x] with respect to x.\n\n\ud83d\udcda In this detailed calculus tutorial, we break down the solution to finding the derivative of x^x + (sin x)^ln x using logarithmic differentiation. Perfect for advanced calculus students and those preparing for higher mathematics.\nKey topics covered:\n\u2022 Logarithmic differentiation technique\n\u2022 Chain rule and product rule applications\n\u2022 Breaking complex derivatives into manageable parts\n\u2022 Step-by-step solution methodology\n#Calculus #Mathematics #DerivativeCalculus #MathTutorial #CalcTutorial #MathEducation #AdvancedMath", "lengthSeconds": 167, "width": 640, "height": 360}
{"video_id": "Uvs6lNSngfY", "transcript": [{"start": 0.32, "duration": 5.36, "text": "welcome today we'll solve a challenging"}, {"start": 3.159, "duration": 5.001, "text": "derivative problem involving inverse"}, {"start": 5.68, "duration": 5.24, "text": "trigonometric functions let's look at"}, {"start": 8.16, "duration": 5.76, "text": "our question this is a complex"}, {"start": 10.92, "duration": 5.96, "text": "expression involving inverse tangent to"}, {"start": 13.92, "duration": 7.56, "text": "solve it we'll use a clever substitution"}, {"start": 16.88, "duration": 8.52, "text": "technique let's substitute xal secant T"}, {"start": 21.48, "duration": 6.48, "text": "this means tal inverse secant x now"}, {"start": 25.4, "duration": 5.44, "text": "let's substitute this into our original"}, {"start": 27.96, "duration": 8.8, "text": "expression inside the square root"}, {"start": 30.84, "duration": 8.879, "text": "x^2 - 1 becomes SEC 2 T -1 this is equal"}, {"start": 36.76, "duration": 5.799, "text": "to tangent 2 T which is a fundamental"}, {"start": 39.719, "duration": 7.0, "text": "trigonometric identity this simplifies"}, {"start": 42.559, "duration": 8.721, "text": "to 1 / tangent T which is coent T and"}, {"start": 46.719, "duration": 7.881, "text": "coent T can be written as tangent of Pi"}, {"start": 51.28, "duration": 6.079, "text": "/ 2us T here's where things get"}, {"start": 54.6, "duration": 5.72, "text": "interesting the inverse tangent and"}, {"start": 57.359, "duration": 6.681, "text": "tangent cancel each other out leaving"}, {"start": 60.32, "duration": 6.4, "text": "with just piun / 2 - t now we can"}, {"start": 64.04, "duration": 5.92, "text": "differentiate with respect to X the"}, {"start": 66.72, "duration": 7.28, "text": "derivative of Pi / 2 is 0 and the"}, {"start": 69.96, "duration": 9.28, "text": "derivative of inverse secant x is 1 /x *"}, {"start": 74.0, "duration": 12.6, "text": "the < TK of x^2 -1 therefore our final"}, {"start": 79.24, "duration": 9.96, "text": "answer is -1 /x * the < TK of x^2 - 1"}, {"start": 86.6, "duration": 5.12, "text": "this problem showcases the power of"}, {"start": 89.2, "duration": 5.36, "text": "clever sub subtitution and trigonometric"}, {"start": 91.72, "duration": 5.28, "text": "identities in calculus the key was"}, {"start": 94.56, "duration": 7.44, "text": "recognizing that we could simplify our"}, {"start": 97.0, "duration": 5.0, "text": "expression using the cant substitution"}], "title": "Understanding the Derivative of Inverse Trigonometric Functions [\ud835\udc61\ud835\udc4e\ud835\udc5b^(\u22121)  1/\u221a(\ud835\udc65^2\u22121)]", "description": "Question\nFind the derivative of the function [\ud835\udc61\ud835\udc4e\ud835\udc5b^(\u22121)  1/\u221a(\ud835\udc65^2\u22121)] .\n\nIn this video, we dive deep into the derivative of inverse trigonometric functions, focusing on how to differentiate  [\ud835\udc61\ud835\udc4e\ud835\udc5b^(\u22121)  1/\u221a(\ud835\udc65^2\u22121)]  . We\u2019ll explore the process step-by-step, using trigonometric identities and calculus techniques to derive the final result. Perfect for students and enthusiasts looking to strengthen their calculus skills!\n#Calculus #Trigonometry #Derivatives #MathTutorial #InverseTrigFunctions", "lengthSeconds": 105, "width": 640, "height": 360}
{"video_id": "J-Tay_770lc", "transcript": [{"start": 0.4, "duration": 6.72, "text": "welcome today we'll solve a challenging"}, {"start": 3.399, "duration": 5.921, "text": "derivative problem step by step our task"}, {"start": 7.12, "duration": 7.92, "text": "is to find the derivative of the natural"}, {"start": 9.32, "duration": 9.36, "text": "logarithm of x- the < TK of x^2 - 1"}, {"start": 15.04, "duration": 6.399, "text": "let's break this down step by step first"}, {"start": 18.68, "duration": 5.24, "text": "we'll use the chain rule since we have a"}, {"start": 21.439, "duration": 5.641, "text": "natural logarithm of a complex"}, {"start": 23.92, "duration": 6.08, "text": "expression remember the derivative of"}, {"start": 27.08, "duration": 4.8, "text": "natural log is one over the function"}, {"start": 30.0, "duration": 4.44, "text": "times the derivative of the inner"}, {"start": 31.88, "duration": 5.519, "text": "function now we can differentiate the"}, {"start": 34.44, "duration": 6.279, "text": "inner expression the derivative of x is"}, {"start": 37.399, "duration": 5.761, "text": "1 and for the square root term we'll"}, {"start": 40.719, "duration": 6.201, "text": "need to use the chain rule again for the"}, {"start": 43.16, "duration": 6.16, "text": "square root term we use the power rule"}, {"start": 46.92, "duration": 6.6, "text": "the derivative of square < TK of U is 1"}, {"start": 49.32, "duration": 8.44, "text": "/ 2 < TK U * the derivative of U the"}, {"start": 53.52, "duration": 8.08, "text": "derivative of x^2 - 1 is 2x let's"}, {"start": 57.76, "duration": 6.799, "text": "substitute that in now we can simplify"}, {"start": 61.6, "duration": 5.68, "text": "the expression inside the brackets the"}, {"start": 64.559, "duration": 5.88, "text": "two and two cancel out giving us this"}, {"start": 67.28, "duration": 6.36, "text": "simpler form to combine these terms we"}, {"start": 70.439, "duration": 8.481, "text": "need a common denominator we can rewrite"}, {"start": 73.64, "duration": 8.92, "text": "one as the square root of x^2 - 1/"}, {"start": 78.92, "duration": 7.4, "text": "itself next we can factor out -1 from"}, {"start": 82.56, "duration": 7.64, "text": "the numerator and finally notice that x"}, {"start": 86.32, "duration": 6.6, "text": "- < TK of x^2 - 1 appears in both both"}, {"start": 90.2, "duration": 6.36, "text": "numerator and denominator so they cancel"}, {"start": 92.92, "duration": 5.879, "text": "out leaving us with our final answer -1"}, {"start": 96.56, "duration": 5.239, "text": "over < TK of"}, {"start": 98.799, "duration": 4.881, "text": "x^2-1 this was a complex derivative"}, {"start": 101.799, "duration": 4.841, "text": "problem that required careful"}, {"start": 103.68, "duration": 6.2, "text": "application of multiple derivative rules"}, {"start": 106.64, "duration": 6.28, "text": "the chain rule the power rule and"}, {"start": 109.88, "duration": 5.279, "text": "careful algebraic manipulation the key"}, {"start": 112.92, "duration": 4.239, "text": "was breaking it down into manageable"}, {"start": 115.159, "duration": 5.0, "text": "steps and working through it"}, {"start": 117.159, "duration": 3.0, "text": "systematically"}], "title": "How to Differentiate Logarithmic Functions [\ud835\udc8d\ud835\udc8f(\ud835\udc99\u2212\u221a(\ud835\udc99^\ud835\udfd0\u2212\ud835\udfcf))] with Square Roots: Step-by-Step Guide", "description": "Question:\nDifferentiate the expression [\ud835\udc8d\ud835\udc8f(\ud835\udc99\u2212\u221a(\ud835\udc99^\ud835\udfd0\u2212\ud835\udfcf))] with respect to x.\nAre you struggling with differentiating logarithmic functions involving square roots? In this video, we break down the process of differentiating the expression  [\ud835\udc8d\ud835\udc8f(\ud835\udc99\u2212\u221a(\ud835\udc99^\ud835\udfd0\u2212\ud835\udfcf))] step-by-step. Follow along as we use the chain rule and simplify the derivative to achieve clarity and accuracy. Perfect for students preparing for calculus exams or anyone needing a refresher. Don\u2019t forget to subscribe for more math tutorials!\n#Calculus #Differentiation #Logarithms #SquareRoots #MathTutorial #ChainRule", "lengthSeconds": 123, "width": 640, "height": 360}
{"video_id": "FohR39N_XhQ", "transcript": [{"start": 1.199, "duration": 5.241, "text": "let's solve a complex derivative problem"}, {"start": 4.0, "duration": 9.559, "text": "step by step we need to find the"}, {"start": 6.44, "duration": 10.36, "text": "derivative of y = e power of x^2 + x^ of"}, {"start": 13.559, "duration": 5.361, "text": "x^2 this is a complex function that will"}, {"start": 16.8, "duration": 4.36, "text": "require multiple differentiation"}, {"start": 18.92, "duration": 5.08, "text": "techniques let's break it down into"}, {"start": 21.16, "duration": 7.16, "text": "simpler Parts by substituting U equal e"}, {"start": 24.0, "duration": 7.64, "text": "to the power of x^2 and v = x to the"}, {"start": 28.32, "duration": 6.48, "text": "power of x^2 let's start with finding"}, {"start": 31.64, "duration": 5.36, "text": "dudx we'll use logarithmic"}, {"start": 34.8, "duration": 7.079, "text": "differentiation taking the natural log"}, {"start": 37.0, "duration": 9.68, "text": "of both sides Ln of U = Ln of e to power"}, {"start": 41.879, "duration": 8.36, "text": "of x^2 this simplifies to x^2 since Ln"}, {"start": 46.68, "duration": 7.48, "text": "of eal 1 now we can differentiate both"}, {"start": 50.239, "duration": 9.761, "text": "sides with respect to X using implicit"}, {"start": 54.16, "duration": 11.56, "text": "differentiation we get 1 / U * D DX = 2X"}, {"start": 60.0, "duration": 10.04, "text": "solving for DX we get DX = U * 2x"}, {"start": 65.72, "duration": 10.039, "text": "substituting back u = e power of x^2 we"}, {"start": 70.04, "duration": 8.24, "text": "get Dux = 2x * E power of x^2 now for"}, {"start": 75.759, "duration": 6.841, "text": "the second part finding"}, {"start": 78.28, "duration": 6.6, "text": "dvdx again we'll use logarithmic"}, {"start": 82.6, "duration": 6.8, "text": "differentiation taking the natural log"}, {"start": 84.88, "duration": 9.84, "text": "of both sides Ln of V = Ln of x to the"}, {"start": 89.4, "duration": 9.64, "text": "power of x^2 = x^2 Ln X differentiating"}, {"start": 94.72, "duration": 11.24, "text": "both sides implicitly we get 1 / V *"}, {"start": 99.04, "duration": 13.6, "text": "dvdx = x^2 * 1 /x + Ln x * 2x solving"}, {"start": 105.96, "duration": 12.68, "text": "for dvdx we get dvdx = V * the x + 2x"}, {"start": 112.64, "duration": 9.6, "text": "lnx substituting back v = x to^ of x^2"}, {"start": 118.64, "duration": 11.119, "text": "and simplifying we get dvdx = X the"}, {"start": 122.24, "duration": 7.519, "text": "power of x^2 + 1 * the quantity 1 + 2 Ln"}, {"start": 132.48, "duration": 6.52, "text": "X finally we can combine our results"}, {"start": 135.92, "duration": 8.24, "text": "using the sum rule of differentiation"}, {"start": 139.0, "duration": 11.68, "text": "the derivative D ydx = DX + dvdx which"}, {"start": 144.16, "duration": 11.0, "text": "gives us 2x * e^ of x^2 + x ^ of x^2 + 1"}, {"start": 150.68, "duration": 7.04, "text": "* the quantity 1 + 2 lnx this problem"}, {"start": 155.16, "duration": 5.56, "text": "demonstrated several key calculus"}, {"start": 157.72, "duration": 6.32, "text": "Concepts logarithmic differentiation"}, {"start": 160.72, "duration": 6.08, "text": "implicit differentiation the chain Rule"}, {"start": 164.04, "duration": 5.119, "text": "and the sum rule of derivatives by"}, {"start": 166.8, "duration": 5.56, "text": "breaking down the complex function into"}, {"start": 169.159, "duration": 5.641, "text": "simpler parts and using these techniques"}, {"start": 172.36, "duration": 6.2, "text": "systematically we were able to find the"}, {"start": 174.8, "duration": 3.76, "text": "derivative of this challenging"}, {"start": 178.84, "duration": 4.0, "text": "function e"}], "title": "Advanced Calculus: Finding the Derivative of e^(x\u00b2) + x^(x\u00b2) | Step-by-Step Solution", "description": "Question:\nFind the derivative of y = e^(x\u00b2) + x^(x\u00b2)\n\nLearn how to find the derivative of a complex function combining exponential and power functions. In this detailed tutorial, we\u2019ll break down the solution of dy/dx where y = e^(x\u00b2) + x^(x\u00b2) using:\n\u2022 Logarithmic differentiation\n\u2022 Chain rule\n\u2022 Product rule\n\u2022 Implicit differentiation\n\u2022 Step-by-step explanations\n\u2022 Clear mathematical reasoning\nPerfect for:\n\ud83d\udcda Calculus students\n\ud83c\udf93 University mathematics\n\ud83d\udcd0 Advanced differentiation techniques\n\ud83d\udca1 Math problem-solving strategies\n#Calculus #Mathematics #Derivatives #MathTutorial #ComplexDerivatives #CalulusProblems #MathEducation", "lengthSeconds": 189, "width": 640, "height": 360}
{"video_id": "NXHM4a2bOjo", "transcript": [{"start": 0.359, "duration": 5.161, "text": "let's learn how to find the derivative"}, {"start": 2.44, "duration": 6.96, "text": "of a quotient using the quotient rule"}, {"start": 5.52, "duration": 10.48, "text": "we'll find the derivative of f ofx ="}, {"start": 9.4, "duration": 9.2, "text": "3x^2 + 5 / 2x + 1 first let's recall the"}, {"start": 16.0, "duration": 7.32, "text": "quotient rule formula the derivative of"}, {"start": 18.6, "duration": 8.32, "text": "U over v = v * the derivative of uus U *"}, {"start": 23.32, "duration": 7.959, "text": "the derivative of V all over V ^2 let's"}, {"start": 26.92, "duration": 10.159, "text": "identify u and v in our function here U"}, {"start": 31.279, "duration": 7.641, "text": "of X is 3x^2 + 5 and V of X is 2x + 1"}, {"start": 37.079, "duration": 6.401, "text": "next we need to find the derivatives of"}, {"start": 38.92, "duration": 8.0, "text": "u and v the derivative of U is 6X and"}, {"start": 43.48, "duration": 5.8, "text": "the derivative of V is 2 now let's"}, {"start": 46.92, "duration": 7.88, "text": "substitute everything into the quotient"}, {"start": 49.28, "duration": 11.439, "text": "rule formula we'll have 2X + 1 * 6x -"}, {"start": 54.8, "duration": 8.0, "text": "3x^ 2 + 5 * 2 all over 2x + 1 2"}, {"start": 60.719, "duration": 8.04, "text": "let's simplify the numerator step by"}, {"start": 62.8, "duration": 10.319, "text": "step first let's multiply 2x + 1 by 6X"}, {"start": 68.759, "duration": 9.881, "text": "that gives us 12 x^2 + 6 x next let's"}, {"start": 73.119, "duration": 8.68, "text": "multiply 3x^2 + 5x 2 that gives us 6 x^2"}, {"start": 78.64, "duration": 5.839, "text": "+ 10 now let's substitute these back"}, {"start": 81.799, "duration": 9.96, "text": "into our derivative expression we have"}, {"start": 84.479, "duration": 11.96, "text": "12 x^2 + 6 x - 6 x^2 - 10 all over 2x +"}, {"start": 91.759, "duration": 9.4, "text": "1^ 2 let's simplify the numerator 12 x^2"}, {"start": 96.439, "duration": 10.0, "text": "- 6 x^2 gives us 6 x^2 we keep the plus"}, {"start": 101.159, "duration": 12.881, "text": "6X and -10 therefore our final answer is"}, {"start": 106.439, "duration": 9.6, "text": "6 x^2 + 6 x - 10 / 2x + 1^ 2 remember"}, {"start": 114.04, "duration": 7.039, "text": "the key to successfully applying the"}, {"start": 116.039, "duration": 8.481, "text": "quotient rule is to First identify you V"}, {"start": 121.079, "duration": 5.481, "text": "second find their derivatives third"}, {"start": 124.52, "duration": 5.439, "text": "substitute into the quotient rule"}, {"start": 126.56, "duration": 6.96, "text": "formula and finally simplify the"}, {"start": 129.959, "duration": 3.561, "text": "resulting expression"}], "title": "How to Differentiate Rational Functions ((\ud835\udfd1\ud835\udc99^\ud835\udfd0+\ud835\udfd3)/(\ud835\udfd0\ud835\udc99+\ud835\udfcf)) Step-by-Step Tutorial", "description": "In this video, we dive into the differentiation of rational functions using the quotient rule. Follow along as we break down each step to find the derivative of ((\ud835\udfd1\ud835\udc99^\ud835\udfd0+\ud835\udfd3)/(\ud835\udfd0\ud835\udc99+\ud835\udfcf)). Master calculus with our clear and concise explanations, and boost your math skills today!\n#Calculus #Differentiation #MathTutorial #QuotientRule #LearnMath", "lengthSeconds": 133, "width": 640, "height": 360}
{"video_id": "9N4DRnzX7Ks", "transcript": [{"start": 0.16, "duration": 8.16, "text": "let's find the derivative of 2x^2 + 3x +"}, {"start": 5.279, "duration": 5.081, "text": "1 using the definition of the derivative"}, {"start": 8.32, "duration": 4.08, "text": "the definition of the derivative states"}, {"start": 10.36, "duration": 7.52, "text": "that the derivative is equal to the"}, {"start": 12.4, "duration": 9.799, "text": "Limit as H approaches Z of f ofx + hus F"}, {"start": 17.88, "duration": 8.88, "text": "ofx all divided H let's start by finding"}, {"start": 22.199, "duration": 8.281, "text": "F ofx + H we replace every X in our"}, {"start": 26.76, "duration": 12.52, "text": "original function with x + H so"}, {"start": 30.48, "duration": 12.16, "text": "we get 2 * x + h^ 2 + 3 * x + H + 1 now"}, {"start": 39.28, "duration": 7.04, "text": "let's expand this expression first let's"}, {"start": 42.64, "duration": 9.0, "text": "expand the squared term x + h^2 becomes"}, {"start": 46.32, "duration": 11.719, "text": "x^2 + 2x H + h^ 2 multiplying by 2 gives"}, {"start": 51.64, "duration": 10.879, "text": "us 2 x^2 + 4 x h + 2 h^ 2 then we add 3x"}, {"start": 58.039, "duration": 6.4, "text": "+ 3 H + 1 now we can substitute this and"}, {"start": 62.519, "duration": 5.081, "text": "our original function into the"}, {"start": 64.439, "duration": 7.921, "text": "derivative definition we subtract F ofx"}, {"start": 67.6, "duration": 8.159, "text": "which is 2x^2 + 3x + 1 and divide"}, {"start": 72.36, "duration": 7.48, "text": "everything by H let's simplify what's in"}, {"start": 75.759, "duration": 7.241, "text": "the numerator the 2x^2 terms cancel out"}, {"start": 79.84, "duration": 8.239, "text": "the ones cancel out and we're left with"}, {"start": 83.0, "duration": 7.92, "text": "4xh + 2 h^2 + 3 H in the numerator we"}, {"start": 88.079, "duration": 7.961, "text": "can factor out H from the numerator Ator"}, {"start": 90.92, "duration": 9.519, "text": "this gives us H * the quanti 4x + 2 H +"}, {"start": 96.04, "duration": 9.32, "text": "3 H the H is cancel out now we just have"}, {"start": 100.439, "duration": 9.801, "text": "the limit as H approaches Z of 4x + 2 H"}, {"start": 105.36, "duration": 8.759, "text": "+ 3 finally as H approaches 0 2 H"}, {"start": 110.24, "duration": 6.76, "text": "approaches 0 leaving us with 4x + 3 this"}, {"start": 114.119, "duration": 5.32, "text": "is our derivative let's visualize what"}, {"start": 117.0, "duration": 7.84, "text": "this means geometrically here's our"}, {"start": 119.439, "duration": 9.44, "text": "original function 2x^2 + 3x + 1 and at"}, {"start": 124.84, "duration": 6.28, "text": "any point the derivative for X +3 gives"}, {"start": 128.879, "duration": 5.44, "text": "us the slope of the tangent line at that"}, {"start": 131.12, "duration": 5.199, "text": "point in conclusion using the definition"}, {"start": 134.319, "duration": 8.521, "text": "of the derivative we found that the"}, {"start": 136.319, "duration": 9.321, "text": "derivative of 2x^2 + 3x + 1 is 4x + 3"}, {"start": 142.84, "duration": 8.119, "text": "this tells us the instantaneous rate of"}, {"start": 145.64, "duration": 5.319, "text": "change of our function at any point x"}], "title": "Understanding the Derivative of 2x^2+3x+1 Using First Principles", "description": "Master the fundamentals of calculus by deriving the exponential function \\(2x^2 + 3x + 1\\) from first principles. Explore our step-by-step explanations and examples to deepen your understanding of derivatives and exponential functions.\n\n#Calculus #Derivatives #ExponentialFunctions #MathTutorial #FirstPrinciples #x^3+2x", "lengthSeconds": 150, "width": 640, "height": 360}
{"video_id": "aRJQ3V42dno", "transcript": [{"start": 0.28, "duration": 5.4, "text": "let's solve this challenging derivative"}, {"start": 2.76, "duration": 5.44, "text": "problem involving inverse cosine we'll"}, {"start": 5.68, "duration": 6.039, "text": "break it down step by step to make it"}, {"start": 8.2, "duration": 6.72, "text": "more manageable first let's simplify our"}, {"start": 11.719, "duration": 8.841, "text": "notation by letting y equal the inverse"}, {"start": 14.92, "duration": 9.32, "text": "cosine of 4X Cub - 3x then we'll make a"}, {"start": 20.56, "duration": 6.36, "text": "clever substitution let x equal cosine"}, {"start": 24.24, "duration": 6.279, "text": "Theta when we substitute this into our"}, {"start": 26.92, "duration": 8.72, "text": "equation we get y equal the inverse"}, {"start": 30.519, "duration": 7.521, "text": "cosine of 4 cosine Cub thet - 3 cosine"}, {"start": 35.64, "duration": 5.2, "text": "Theta this is where we can use a very"}, {"start": 38.04, "duration": 7.92, "text": "important trigonometric identity for"}, {"start": 40.84, "duration": 8.6, "text": "cine Cub thet - 3 cosine thet = cosine"}, {"start": 45.96, "duration": 6.0, "text": "of 3 Theta therefore our equation"}, {"start": 49.44, "duration": 6.24, "text": "simplifies to y equals the inverse"}, {"start": 51.96, "duration": 6.72, "text": "cosine of cine 3 Theta the inverse"}, {"start": 55.68, "duration": 7.399, "text": "cosine and cosine functions cancel out"}, {"start": 58.68, "duration": 7.56, "text": "leaving us with y = 3 Theta now remember"}, {"start": 63.079, "duration": 6.441, "text": "that Theta is the inverse cosine of x so"}, {"start": 66.24, "duration": 6.28, "text": "we can write y = 3 * the inverse cosine"}, {"start": 69.52, "duration": 6.12, "text": "of x to find the derivative we can use"}, {"start": 72.52, "duration": 6.12, "text": "the chain rule first let's take the"}, {"start": 75.64, "duration": 7.439, "text": "derivative of 3 * the inverse cosine of"}, {"start": 78.64, "duration": 9.439, "text": "x the derivative of inverse cosine is -1"}, {"start": 83.079, "duration": 9.481, "text": "over theun of 1 - x^2 therefore when we"}, {"start": 88.079, "duration": 8.04, "text": "multiply by 3 we get -3 over the sare <"}, {"start": 92.56, "duration": 5.8, "text": "TK of 1 - x^2 and this gives us our"}, {"start": 96.119, "duration": 8.28, "text": "final answer the derivative of the"}, {"start": 98.36, "duration": 10.48, "text": "inverse cosine of 4X Cub - 3x = -3 over"}, {"start": 104.399, "duration": 7.441, "text": "the < TK of 1 - x^2 this problem"}, {"start": 108.84, "duration": 6.16, "text": "demonstrates the power of trigonometric"}, {"start": 111.84, "duration": 5.879, "text": "substitution in calculus by making a"}, {"start": 115.0, "duration": 6.119, "text": "clever substitution we were able to"}, {"start": 117.719, "duration": 7.201, "text": "transform a complex derivative into a"}, {"start": 121.119, "duration": 3.801, "text": "much simpler one"}], "title": "Mastering Inverse Trigonometric Derivatives [\ud835\udc50\ud835\udc5c\ud835\udc60^(\u22121)(4\ud835\udc65^3\u22123\ud835\udc65)] Step-by-Step Guide", "description": "Question:\nEvaluate the derivative [\ud835\udc50\ud835\udc5c\ud835\udc60^(\u22121)(4\ud835\udc65^3\u22123\ud835\udc65)].\n\nJoin us as we dive into the process of evaluating derivatives involving inverse trigonometric functions. This video will guide you through each step of finding the derivative of [\ud835\udc50\ud835\udc5c\ud835\udc60^(\u22121)(4\ud835\udc65^3\u22123\ud835\udc65)], breaking it down with clear explanations. Perfect for students tackling calculus!\n#Calculus #InverseTrig #Derivatives #Mathematics #LearningMath", "lengthSeconds": 128, "width": 640, "height": 360}
{"video_id": "dVK-bGPvoI0", "transcript": [{"start": 0.359, "duration": 7.761, "text": "welcome today we'll find the derivative"}, {"start": 3.08, "duration": 8.679, "text": "of the inverse tangent of 1 + x over 1 -"}, {"start": 8.12, "duration": 6.479, "text": "x let's break this down step by step"}, {"start": 11.759, "duration": 4.921, "text": "let's start by letting y equal our"}, {"start": 14.599, "duration": 5.361, "text": "expression this will help us work"}, {"start": 16.68, "duration": 5.72, "text": "through the problem systematically now"}, {"start": 19.96, "duration": 5.479, "text": "there's a very useful identity for"}, {"start": 22.4, "duration": 5.16, "text": "inverse tangent that we can apply here"}, {"start": 25.439, "duration": 6.041, "text": "the identity states that the inverse"}, {"start": 27.56, "duration": 6.96, "text": "tangent of you plus v over 1 minus UV"}, {"start": 31.48, "duration": 6.48, "text": "equals the sum of inverse tangent of U"}, {"start": 34.52, "duration": 6.8, "text": "plus inverse tangent of v in our case if"}, {"start": 37.96, "duration": 5.599, "text": "we let you equal 1 and V equal x we can"}, {"start": 41.32, "duration": 5.12, "text": "see that our expression matches this"}, {"start": 43.559, "duration": 7.16, "text": "form perfectly therefore we can write"}, {"start": 46.44, "duration": 8.2, "text": "our expression as inverse tangent of 1 +"}, {"start": 50.719, "duration": 8.881, "text": "inverse tangent of X now we know that"}, {"start": 54.64, "duration": 9.68, "text": "inverse tangent of 1 = < / 4 this is"}, {"start": 59.6, "duration": 7.72, "text": "because tangent of < / 4 = 1 now that we"}, {"start": 64.32, "duration": 5.68, "text": "have simplified our expression we can"}, {"start": 67.32, "duration": 6.64, "text": "take the derivative the derivative of a"}, {"start": 70.0, "duration": 8.28, "text": "constant pi over 4 is zero and the"}, {"start": 73.96, "duration": 8.44, "text": "derivative of inverse tangent of X is 1/"}, {"start": 78.28, "duration": 11.24, "text": "1 + x^2 therefore the derivative of"}, {"start": 82.4, "duration": 11.399, "text": "inverse tangent of 1 + x/ 1 - x = 1 / 1"}, {"start": 89.52, "duration": 7.2, "text": "+ x^2 let's visualize this result the"}, {"start": 93.799, "duration": 7.841, "text": "blue curve shows our original function"}, {"start": 96.72, "duration": 4.92, "text": "and the green curve shows its derivative"}], "title": "Understanding the Derivative of Inverse Tangent Functions [\ud835\udc61\ud835\udc4e\ud835\udc5b^(\u22121)\u2061((1+\ud835\udc65)/(1\u2212\ud835\udc65))]", "description": "Question:\nFind the derivative of [\ud835\udc61\ud835\udc4e\ud835\udc5b^(\u22121)\u2061((1+\ud835\udc65)/(1\u2212\ud835\udc65))] .\n\nDive into the world of calculus as we explore the step-by-step process of finding the derivative of [\ud835\udc61\ud835\udc4e\ud835\udc5b^(\u22121)\u2061((1+\ud835\udc65)/(1\u2212\ud835\udc65))]. Our detailed explanation will help you understand trigonometric identities and calculus techniques, enhancing your math skills. Perfect for students and math enthusiasts!\n#Calculus #MathTutorial #InverseTangent #Differentiation #Trigonometry", "lengthSeconds": 105, "width": 640, "height": 360}
{"video_id": "mH77GaIWRB8", "transcript": [{"start": 0.24, "duration": 5.24, "text": "let's solve this calculus problem step"}, {"start": 2.44, "duration": 8.119, "text": "by step we need to find the derivative"}, {"start": 5.48, "duration": 7.64, "text": "of Ln sin of 2x first let's identify the"}, {"start": 10.559, "duration": 5.8, "text": "structure of our function we have an"}, {"start": 13.12, "duration": 7.04, "text": "outer natural log function and inside"}, {"start": 16.359, "duration": 6.361, "text": "that we have S of 2x this means we'll"}, {"start": 20.16, "duration": 5.039, "text": "need to use the chain rule let's recall"}, {"start": 22.72, "duration": 4.92, "text": "the chain rule the derivative of a"}, {"start": 25.199, "duration": 5.24, "text": "composite function is the derivative of"}, {"start": 27.64, "duration": 5.8, "text": "the outer function evaluated at the"}, {"start": 30.439, "duration": 6.001, "text": "inner function times the derivative of"}, {"start": 33.44, "duration": 5.36, "text": "the inner function now let's find the"}, {"start": 36.44, "duration": 6.0, "text": "derivative of the outer function the"}, {"start": 38.8, "duration": 7.919, "text": "derivative of natural log is 1 / U so in"}, {"start": 42.44, "duration": 6.119, "text": "our case that's 1 / sin of 2x next let's"}, {"start": 46.719, "duration": 5.241, "text": "find the derivative of the inner"}, {"start": 48.559, "duration": 6.081, "text": "function the derivative of s is cosine"}, {"start": 51.96, "duration": 5.16, "text": "and when we have 2X inside we need to"}, {"start": 54.64, "duration": 8.2, "text": "multiply by the derivative of 2x which"}, {"start": 57.12, "duration": 8.56, "text": "is 2 so we get cosine of 2 x * 2 now"}, {"start": 62.84, "duration": 9.279, "text": "let's put it all together we multiply 1"}, {"start": 65.68, "duration": 9.799, "text": "/ s of 2x * cine of 2x * 2 we can"}, {"start": 72.119, "duration": 6.401, "text": "rearrange this the two can be moved to"}, {"start": 75.479, "duration": 5.881, "text": "the front and cosine over s is the"}, {"start": 78.52, "duration": 6.48, "text": "definition of cotangent and there's our"}, {"start": 81.36, "duration": 9.96, "text": "final answer the derivative of Ln sin of"}, {"start": 85.0, "duration": 9.72, "text": "2x = 2 cent of 2x let's review the key"}, {"start": 91.32, "duration": 5.88, "text": "steps we used first we identified the"}, {"start": 94.72, "duration": 5.32, "text": "composite function structure then we"}, {"start": 97.2, "duration": 6.239, "text": "applied the chain rule next we found the"}, {"start": 100.04, "duration": 6.679, "text": "derivatives of both the outer and inner"}, {"start": 103.439, "duration": 8.841, "text": "functions finally we simplified our"}, {"start": 106.719, "duration": 5.561, "text": "expression using the definition of coent"}], "title": "Understanding the Derivative of [(ln(sin2x))]   Step by Step Explanation", "description": "Question: Find the derivative of [(ln(sin2x))] \nIn this video, we break down the process of finding the derivative of (ln(sin2x))  using the chain rule and trigonometric identities. Follow along for a clear and concise explanation to strengthen your calculus skills!\n#hsc #hschighermath #1stpaper #hschighermath1stpaper #englishversion #cpmrana #Calculus #Derivative #Trigonometry #ChainRule #MathTutorial", "lengthSeconds": 111, "width": 640, "height": 360}
{"video_id": "ieY02yLFE7E", "transcript": [{"start": 0.32, "duration": 4.88, "text": "let's solve this derivative step by step"}, {"start": 3.24, "duration": 6.96, "text": "we're going to find the derivative of"}, {"start": 5.2, "duration": 7.599, "text": "the cube root of 3x^2 + 1 the first step"}, {"start": 10.2, "duration": 5.479, "text": "is to rewrite our cube root in"}, {"start": 12.799, "duration": 5.32, "text": "exponential form the cube root is the"}, {"start": 15.679, "duration": 7.481, "text": "same as raising something to the 1/3"}, {"start": 18.119, "duration": 8.721, "text": "power so we can rewrite this as 3x^2 + 1"}, {"start": 23.16, "duration": 6.279, "text": "raised to the 13 power for step two"}, {"start": 26.84, "duration": 5.879, "text": "we'll use the chain rule since we have a"}, {"start": 29.439, "duration": 5.681, "text": "composite function the chain rule tells"}, {"start": 32.719, "duration": 4.881, "text": "us that we multiply the derivative of"}, {"start": 35.12, "duration": 5.88, "text": "the outer function by the derivative of"}, {"start": 37.6, "duration": 5.88, "text": "the inner function for step three let's"}, {"start": 41.0, "duration": 5.64, "text": "focus on the power rule part when we"}, {"start": 43.48, "duration": 6.32, "text": "have something raised to the 1/3 power"}, {"start": 46.64, "duration": 7.0, "text": "and we differentiate it we multiply by"}, {"start": 49.8, "duration": 7.88, "text": "1/3 and reduce the power by one that's"}, {"start": 53.64, "duration": 7.52, "text": "why we see the 1/3 minus one in the"}, {"start": 57.68, "duration": 5.839, "text": "exponent for step four we need to find"}, {"start": 61.16, "duration": 7.959, "text": "the derivative of the inner function"}, {"start": 63.519, "duration": 9.121, "text": "3x^2 + 1 the derivative of x^2 is 2x so"}, {"start": 69.119, "duration": 7.961, "text": "the derivative of 3x^2 is 6X and the"}, {"start": 72.64, "duration": 8.28, "text": "derivative of 1 is 0er for step five we"}, {"start": 77.08, "duration": 8.6, "text": "multiply everything together we have 1/3"}, {"start": 80.92, "duration": 10.48, "text": "* the expression with the -23 power *"}, {"start": 85.68, "duration": 11.36, "text": "6X finally we can simplify this 1/3 * 6X"}, {"start": 91.4, "duration": 9.56, "text": "is 2x so our final answer is 2x * 3x^2 +"}, {"start": 97.04, "duration": 7.119, "text": "1 raed to the -2/3 power let's verify"}, {"start": 100.96, "duration": 5.72, "text": "this makes sense our original function"}, {"start": 104.159, "duration": 5.801, "text": "was a cube root which has a positive"}, {"start": 106.68, "duration": 5.759, "text": "exponent of 1/3 after taking the"}, {"start": 109.96, "duration": 6.68, "text": "derivative we expect the power to"}, {"start": 112.439, "duration": 7.481, "text": "decrease by 1 giving us -2/3 which is"}, {"start": 116.64, "duration": 5.92, "text": "exactly what we got the X term in front"}, {"start": 119.92, "duration": 5.64, "text": "comes from differentiating the inside of"}, {"start": 122.56, "duration": 6.4, "text": "our cube root this confirms our answer"}, {"start": 125.56, "duration": 3.4, "text": "is reasonable"}], "title": "9.3-1.1 How to Differentiate Cubic Root Functions (\u221b(3\ud835\udc65^2+1)) Step-by-Step Math Explanation.", "description": "Description:\nIn this video, we break down the differentiation of cubic root functions using the chain rule and power rule. Learn how to find the derivative of \u221b(3\ud835\udc65^2+1)  with clear, step-by-step explanations. Perfect for calculus students looking to master derivatives!\n#Calculus #Differentiation #MathTutorial #ChainRule #PowerRule #cpmrana", "lengthSeconds": 130, "width": 640, "height": 360}
{"video_id": "tpzEo-kOfoI", "transcript": [{"start": 0.28, "duration": 4.76, "text": "let's solve this differentiation problem"}, {"start": 2.6, "duration": 6.8, "text": "step by step we need to find the"}, {"start": 5.04, "duration": 8.32, "text": "derivative of x * the square < TK of x^2"}, {"start": 9.4, "duration": 8.36, "text": "+ a 2 first let's identify that this is"}, {"start": 13.36, "duration": 9.44, "text": "a product of two functions we have = X"}, {"start": 17.76, "duration": 7.4, "text": "and V = the < TK of x^2 + a 2 this means"}, {"start": 22.8, "duration": 5.879, "text": "we'll need to use the product rule"}, {"start": 25.16, "duration": 6.519, "text": "applying the product rule we get x * the"}, {"start": 28.679, "duration": 5.001, "text": "derivative of the square root term plus"}, {"start": 31.679, "duration": 5.361, "text": "the square root term times the"}, {"start": 33.68, "duration": 5.68, "text": "derivative of x now for the first term"}, {"start": 37.04, "duration": 4.519, "text": "we need to use the chain rule to"}, {"start": 39.36, "duration": 5.08, "text": "differentiate the square root the"}, {"start": 41.559, "duration": 6.601, "text": "derivative of square root of U is 1 / 2"}, {"start": 44.44, "duration": 7.799, "text": "sare < TK of U * the derivative of U"}, {"start": 48.16, "duration": 7.96, "text": "applying this to our problem we get x/ 2"}, {"start": 52.239, "duration": 6.28, "text": "< TK of x^2 + a squar times the"}, {"start": 56.12, "duration": 5.68, "text": "derivative of what's inside the square"}, {"start": 58.519, "duration": 7.121, "text": "root plus the root term the derivative"}, {"start": 61.8, "duration": 7.2, "text": "of x^2 + a s is simply 2x since the"}, {"start": 65.64, "duration": 8.839, "text": "derivative of a squar is 0o so we get x"}, {"start": 69.0, "duration": 9.32, "text": "* 2x over 2 < TK of x^2 + a 2 plus the"}, {"start": 74.479, "duration": 6.721, "text": "square root term now let's simplify the"}, {"start": 78.32, "duration": 7.0, "text": "numerator becomes x^2 and we can write"}, {"start": 81.2, "duration": 8.84, "text": "the second term as x^2 + a 2 over the"}, {"start": 85.32, "duration": 6.4, "text": "square < TK of x^2 + a 2 finally combine"}, {"start": 90.04, "duration": 5.68, "text": "in these terms under a common"}, {"start": 91.72, "duration": 8.079, "text": "denominator we get 2x^2 + a 2 over"}, {"start": 95.72, "duration": 6.56, "text": "theare < TK of x^2 plus a squar and this"}, {"start": 99.799, "duration": 4.761, "text": "is our final answer we use both the"}, {"start": 102.28, "duration": 4.159, "text": "product rule and chain rule to"}, {"start": 104.56, "duration": 6.879, "text": "differentiate this expression and"}, {"start": 106.439, "duration": 5.0, "text": "arrived at this simplified form"}], "title": "How to Differentiate (\ud835\udc65\u221a(\ud835\udc65^2+\ud835\udc4e^2 )) : Step-by-Step Guide", "description": "Learn how to differentiate the expression (\ud835\udc65\u221a(\ud835\udc65^2+\ud835\udc4e^2 )) using the product and chain rules. This video breaks down each step with clear explanations to help you master the differentiation process. Perfect for students and anyone looking to improve their calculus skills.\n#Calculus\n#MathTutorial\n#Differentiation\n#ProductRule\n#ChainRule", "lengthSeconds": 114, "width": 640, "height": 360}
{"video_id": "jhMy4br_m1o", "transcript": [{"start": 0.24, "duration": 5.92, "text": "let's learn about the product rule for"}, {"start": 3.0, "duration": 6.4, "text": "derivatives this rule tells us how to"}, {"start": 6.16, "duration": 6.24, "text": "find the derivative of a product of two"}, {"start": 9.4, "duration": 7.12, "text": "functions let's start with two functions"}, {"start": 12.4, "duration": 5.879, "text": "u and v where both are functions of X we"}, {"start": 16.52, "duration": 4.44, "text": "want to find the derivative of their"}, {"start": 18.279, "duration": 5.441, "text": "product let's visualize this"}, {"start": 20.96, "duration": 5.96, "text": "geometrically imagine you and V"}, {"start": 23.72, "duration": 6.399, "text": "represent the sides of a rectangle the"}, {"start": 26.92, "duration": 7.0, "text": "product U * V represents the area of"}, {"start": 30.119, "duration": 6.721, "text": "this rectangle when X changes both u and"}, {"start": 33.92, "duration": 6.72, "text": "v change causing the area to change in"}, {"start": 36.84, "duration": 6.52, "text": "two ways first you changes while V stays"}, {"start": 40.64, "duration": 5.919, "text": "constant and then V changes while you"}, {"start": 43.36, "duration": 5.839, "text": "stays constant the total change in area"}, {"start": 46.559, "duration": 5.16, "text": "is the sum of these two changes this"}, {"start": 49.199, "duration": 6.801, "text": "gives us the product rule the derivative"}, {"start": 51.719, "duration": 7.281, "text": "of U * V equal U * the derivative of V"}, {"start": 56.0, "duration": 5.68, "text": "plus v * the derivative of U let's try"}, {"start": 59.0, "duration": 8.799, "text": "an example let's find the derivative of"}, {"start": 61.68, "duration": 9.56, "text": "x^2 * X Cub here U is x^2 and V is X"}, {"start": 67.799, "duration": 7.041, "text": "cubed let's apply the product rule First"}, {"start": 71.24, "duration": 6.4, "text": "We Take U which is x^2 and multiply it"}, {"start": 74.84, "duration": 7.0, "text": "by the derivative of V which is"}, {"start": 77.64, "duration": 7.96, "text": "3x^2 then we add V which is X Cub * the"}, {"start": 81.84, "duration": 7.88, "text": "derivative of U which is 2x simplifying"}, {"start": 85.6, "duration": 6.28, "text": "this gives us 5x to 4th power remember"}, {"start": 89.72, "duration": 4.28, "text": "the the product rule states that when"}, {"start": 91.88, "duration": 5.12, "text": "differentiating a product of two"}, {"start": 94.0, "duration": 5.04, "text": "functions we multiply each function by"}, {"start": 97.0, "duration": 4.479, "text": "the derivative of the other function and"}, {"start": 99.04, "duration": 5.719, "text": "add the results find"}, {"start": 101.479, "duration": 6.441, "text": "derivative using product rule let's"}, {"start": 104.759, "duration": 6.881, "text": "solve this calculus problem step by step"}, {"start": 107.92, "duration": 6.479, "text": "we need to find the derivative of x^2 *"}, {"start": 111.64, "duration": 5.439, "text": "natural log of x to solve this we'll"}, {"start": 114.399, "duration": 4.801, "text": "need to use the product rule the product"}, {"start": 117.079, "duration": 5.481, "text": "rule states that when differentiating"}, {"start": 119.2, "duration": 5.279, "text": "the product of two functions we multiply"}, {"start": 122.56, "duration": 4.479, "text": "the first function by the derivative of"}, {"start": 124.479, "duration": 5.521, "text": "the second plus the second function"}, {"start": 127.039, "duration": 6.881, "text": "times the derivative of the first in our"}, {"start": 130.0, "duration": 7.92, "text": "case let's identify that U of X is X2"}, {"start": 133.92, "duration": 7.64, "text": "and V of X is natural log of x so"}, {"start": 137.92, "duration": 6.959, "text": "applying the product rule we get x^2 *"}, {"start": 141.56, "duration": 7.56, "text": "the derivative of natural log of x plus"}, {"start": 144.879, "duration": 7.881, "text": "natural log of x * the derivative of x^2"}, {"start": 149.12, "duration": 6.72, "text": "now let's evaluate each derivative"}, {"start": 152.76, "duration": 5.68, "text": "separately first the derivative of"}, {"start": 155.84, "duration": 7.72, "text": "natural log of x is"}, {"start": 158.44, "duration": 7.079, "text": "1/x and the derivative of x^2 is 2x"}, {"start": 163.56, "duration": 5.679, "text": "substituting these derivatives back into"}, {"start": 165.519, "duration": 8.201, "text": "our equation we get x^2 *"}, {"start": 169.239, "duration": 8.321, "text": "1X Plus natural log of x *"}, {"start": 173.72, "duration": 8.32, "text": "2x now let's simplify each term the"}, {"start": 177.56, "duration": 8.759, "text": "first term X2 * * 1 /x simplifies to"}, {"start": 182.04, "duration": 9.6, "text": "just X the second term natural log of x"}, {"start": 186.319, "duration": 7.961, "text": "* 2x becomes 2x natural log of x for our"}, {"start": 191.64, "duration": 7.64, "text": "final step we can factor out X from both"}, {"start": 194.28, "duration": 8.64, "text": "terms giving us x * thean 1 + 2 natural"}, {"start": 199.28, "duration": 6.08, "text": "log of x and this is our final answer to"}, {"start": 202.92, "duration": 5.319, "text": "verify this is correct we can see that"}, {"start": 205.36, "duration": 4.92, "text": "we've properly applied the product rule"}, {"start": 208.239, "duration": 5.761, "text": "simplified our terms and fact facted our"}, {"start": 210.28, "duration": 8.76, "text": "result the derivative of x^2 * natural"}, {"start": 214.0, "duration": 7.76, "text": "log of x is x * thean 1 + 2 natural log"}, {"start": 219.04, "duration": 2.72, "text": "of x"}], "title": "Product Rule for Differentiation Explained with  Example.", "description": "Product Rule for Differentiation Explained with Easy Examples | Step-by-Step Guide\nLearn the Product Rule for Differentiation in this step-by-step tutorial! \ud83c\udf93 We explain the formula, break it down with examples, and show you how to solve problems involving the product of two functions.\n\ud83d\udccc Topics Covered in This Video:\nWhat is the Product Rule?\nFormula and explanation\nStep-by-step example solving\nExample Problem Included:\nWe calculate the derivative of x^2 lnx using the product rule.\n\ud83d\udc49 Perfect for students learning calculus or preparing for exams!\n\n\ud83d\udca1 Don\u2019t forget to LIKE, SHARE, and SUBSCRIBE for more math tutorials!\n#Calculus #ProductRule #Differentiation #MathTutorial #LearnMath #BengaliEducation #MathMadeEasy #MathTips", "lengthSeconds": 220, "width": 640, "height": 360}
{"video_id": "ZKWaPiEDzOk", "transcript": [{"start": 0.28, "duration": 5.079, "text": "let's solve this calculus problem step"}, {"start": 2.52, "duration": 5.6, "text": "by step using the quotient rule we need"}, {"start": 5.359, "duration": 6.681, "text": "to find the derivative of x to the N"}, {"start": 8.12, "duration": 6.96, "text": "divided natural log of x first let's"}, {"start": 12.04, "duration": 6.28, "text": "recall the quotient rule formula for a"}, {"start": 15.08, "duration": 8.76, "text": "fraction U over V the derivative is V *"}, {"start": 18.32, "duration": 9.119, "text": "d dxus u * dvdx all over V ^2 let's"}, {"start": 23.84, "duration": 7.199, "text": "identify our u and v in this case U"}, {"start": 27.439, "duration": 7.161, "text": "equal x to the n and V equal natural log"}, {"start": 31.039, "duration": 7.761, "text": "of x now let's find the derivatives of u"}, {"start": 34.6, "duration": 5.08, "text": "and v the derivative of x to the N is NX"}, {"start": 38.8, "duration": 3.56, "text": "the"}, {"start": 39.68, "duration": 7.28, "text": "nus1 and the derivative of natural log"}, {"start": 42.36, "duration": 7.679, "text": "of x is 1 /x Let's substitute these into"}, {"start": 46.96, "duration": 6.84, "text": "our quotient rule formula we get natural"}, {"start": 50.039, "duration": 10.441, "text": "log x * NX"}, {"start": 53.8, "duration": 9.759, "text": "n-1 - x n * 1X all over natural log x^2"}, {"start": 60.48, "duration": 8.6, "text": "now let's simplify the numerator in the"}, {"start": 63.559, "duration": 10.441, "text": "second term X the N / X simplifies to X"}, {"start": 69.08, "duration": 8.92, "text": "N -1 for our final step we can factor"}, {"start": 74.0, "duration": 7.92, "text": "out x to the nus 1 from both terms in"}, {"start": 78.0, "duration": 8.159, "text": "the numerator this gives us x to the nus"}, {"start": 81.92, "duration": 7.48, "text": "1 * the quantity n natural log x -1 all"}, {"start": 86.159, "duration": 7.92, "text": "over natural log x^2 and this is our"}, {"start": 89.4, "duration": 4.679, "text": "final answer answer we've successfully"}], "title": "Mastering Derivatives: Differentiating [(x^n /lnx)]  Using the Quotient Rule", "description": "Join us as we delve into the world of calculus and learn how to differentiate the expression [(x^n /lnx)]   using the quotient rule. Perfect for students preparing for calculus exams or anyone looking to strengthen their understanding of mathematics.\n#Calculus #MathTutorial #Differentiation #QuotientRule #AdvancedMath", "lengthSeconds": 93, "width": 640, "height": 360}
{"video_id": "YIu-sZ4jh_g", "transcript": [{"start": 0.28, "duration": 4.68, "text": "let's explore the fundamental equations"}, {"start": 2.2, "duration": 5.44, "text": "of motion in kinematics these equations"}, {"start": 4.96, "duration": 5.28, "text": "help us understand how objects move"}, {"start": 7.64, "duration": 5.56, "text": "under constant acceleration the first"}, {"start": 10.24, "duration": 7.199, "text": "equation relates final velocity initial"}, {"start": 13.2, "duration": 8.12, "text": "velocity acceleration and time V equal U"}, {"start": 17.439, "duration": 7.441, "text": "+ a t where V is the final velocity U is"}, {"start": 21.32, "duration": 6.76, "text": "the initial velocity uh is acceleration"}, {"start": 24.88, "duration": 5.159, "text": "and T is time the second equation gives"}, {"start": 28.08, "duration": 7.24, "text": "us a relationship between velocities"}, {"start": 30.039, "duration": 7.2, "text": "without needing time V ^2 = u^2 + 2 a s"}, {"start": 35.32, "duration": 3.559, "text": "this is particularly useful when we know"}, {"start": 37.239, "duration": 4.561, "text": "the displacement s but don't know the"}, {"start": 38.879, "duration": 7.881, "text": "time taken the third equation helps us"}, {"start": 41.8, "duration": 7.16, "text": "find displacement s = u t + 12 a t sared"}, {"start": 46.76, "duration": 4.84, "text": "this shows us that displacement has two"}, {"start": 48.96, "duration": 6.239, "text": "parts the part due to initial velocity"}, {"start": 51.6, "duration": 6.88, "text": "UT and the part due to acceleration 12 a"}, {"start": 55.199, "duration": 7.241, "text": "t^2 the fourth equation uses average"}, {"start": 58.48, "duration": 6.72, "text": "velocity s equal the average of U + V *"}, {"start": 62.44, "duration": 5.2, "text": "T this tells us that for constant"}, {"start": 65.2, "duration": 4.959, "text": "acceleration we can use the average of"}, {"start": 67.64, "duration": 5.479, "text": "initial and final velocities to find"}, {"start": 70.159, "duration": 5.881, "text": "displacement and finally the fifth"}, {"start": 73.119, "duration": 6.561, "text": "equation is for constant velocity motion"}, {"start": 76.04, "duration": 5.039, "text": "when velocity doesn't change S equals VT"}, {"start": 79.68, "duration": 4.56, "text": "this is the simplest case where"}, {"start": 81.079, "duration": 5.321, "text": "acceleration is zero together these"}, {"start": 84.24, "duration": 4.519, "text": "equations form the foundation of"}, {"start": 86.4, "duration": 4.56, "text": "kinematics allowing us to solve any"}, {"start": 88.759, "duration": 4.881, "text": "problem involving motion under constant"}, {"start": 90.96, "duration": 5.199, "text": "acceleration remember that each equation"}, {"start": 93.64, "duration": 4.6, "text": "is useful in different situations"}, {"start": 96.159, "duration": 6.24, "text": "depending on what information you have"}, {"start": 98.24, "duration": 4.159, "text": "and what you're trying to find"}], "title": "Equation of motion", "description": "\u201cUnderstanding Kinematic Equations: A Step-by-Step Guide\u201d\nDescription:\nDive into the world of kinematics with our detailed explanation of key motion equations! In this video, we break down the fundamental kinematic equations that describe motion under constant acceleration, including:\nThese equations are essential for understanding how objects move. Whether you\u2019re a student or just curious about physics, this video is designed to help you grasp the concepts clearly.\nJoin us for a clear, step-by-step explanation and enhance your understanding of motion!\n#Kinematics #Physics #MotionEquations #Acceleration #Velocity #ScienceEducation #PhysicsTutorial #Displacement #LearnPhysics", "lengthSeconds": 102, "width": 480, "height": 360}
{"video_id": "H6Cd8H-blJ8", "transcript": [{"start": 0.28, "duration": 4.92, "text": "let's solve a challenging derivative"}, {"start": 2.44, "duration": 5.72, "text": "problem involving inverse trigonometric"}, {"start": 5.2, "duration": 7.08, "text": "functions our task is to find the"}, {"start": 8.16, "duration": 8.44, "text": "derivative of sin inverse of 2x * the <"}, {"start": 12.28, "duration": 6.44, "text": "TK of 1 - x^2 the key to solving this"}, {"start": 16.6, "duration": 5.0, "text": "problem is to use substitution to"}, {"start": 18.72, "duration": 7.92, "text": "simplify the expression let's start by"}, {"start": 21.6, "duration": 10.04, "text": "letting y equal sin inverse of 2x * < TK"}, {"start": 26.64, "duration": 5.88, "text": "1 - x^2 now comes the clever part we'll"}, {"start": 31.64, "duration": 4.88, "text": "make a"}, {"start": 32.52, "duration": 7.719, "text": "substitution let X = sin Theta this"}, {"start": 36.52, "duration": 5.8, "text": "means theta equals sin inverse of X"}, {"start": 40.239, "duration": 4.96, "text": "let's substitute this into our"}, {"start": 42.32, "duration": 7.04, "text": "expression when we replace x with sin"}, {"start": 45.199, "duration": 10.721, "text": "Theta we get sin inverse of 2 sin thet *"}, {"start": 49.36, "duration": 10.999, "text": "the < TK of 1 - sin^2 th Now 1 - sin^2"}, {"start": 55.92, "duration": 7.599, "text": "is equal to cine 2 this is a fund Al"}, {"start": 60.359, "duration": 6.921, "text": "trigonometric identity the square root"}, {"start": 63.519, "duration": 6.041, "text": "of cosine 2 thet is simply cosine Theta"}, {"start": 67.28, "duration": 6.24, "text": "so our expression simplifies to sin"}, {"start": 69.56, "duration": 6.919, "text": "inverse of 2 sin Theta cine Theta here's"}, {"start": 73.52, "duration": 7.44, "text": "where another key identity comes in 2"}, {"start": 76.479, "duration": 7.561, "text": "sin Theta cine Theta = sin of 2 thet"}, {"start": 80.96, "duration": 5.0, "text": "this is the double angle formula for S"}, {"start": 84.04, "duration": 4.56, "text": "when we have sin inverse of s of"}, {"start": 85.96, "duration": 4.68, "text": "something they cancel out leaving us"}, {"start": 88.6, "duration": 5.199, "text": "with Just 2 Theta"}, {"start": 90.64, "duration": 7.04, "text": "remember thata is sin inverse of X so"}, {"start": 93.799, "duration": 6.521, "text": "our final expression for y is 2 * sin"}, {"start": 97.68, "duration": 5.32, "text": "inverse of X now we can find the"}, {"start": 100.32, "duration": 5.92, "text": "derivative the derivative of y with"}, {"start": 103.0, "duration": 7.32, "text": "respect to x equals the derivative of 2"}, {"start": 106.24, "duration": 6.919, "text": "sin inverse of X using the chain rule"}, {"start": 110.32, "duration": 6.56, "text": "this equals 2 * the derivative of sin"}, {"start": 113.159, "duration": 8.681, "text": "inverse of X which equal 2 / the sare <"}, {"start": 116.88, "duration": 9.879, "text": "TK of 1 - x^2 therefore the derivative"}, {"start": 121.84, "duration": 12.8, "text": "of sin inverse of 2x * the < TK of 1 -"}, {"start": 126.759, "duration": 7.881, "text": "x^2 = 2 / the < TK of 1 - x^2"}], "title": "How to Find the Derivative of Inverse Sine Function [\ud835\udc60\ud835\udc56\ud835\udc5b^(\u22121){2\ud835\udc65 \u221a(1\u2212\ud835\udc65^2 )}] Step-by-Step Guide", "description": "Question\nFind Derivative of [\ud835\udc60\ud835\udc56\ud835\udc5b^(\u22121){2\ud835\udc65 \u221a(1\u2212\ud835\udc65^2 )}] .\n\nIn this video, we dive into the calculus problem of finding the derivative of an inverse sine function, specifically [\ud835\udc60\ud835\udc56\ud835\udc5b^(\u22121){2\ud835\udc65 \u221a(1\u2212\ud835\udc65^2 )}] .. Follow along as we show you a two-column approach, explaining each mathematical step alongside a detailed explanation. This clear and concise walkthrough will help you understand the differentiation process and apply it to similar problems. Perfect for students looking to strengthen their calculus skills!\n#Calculus #Derivative #InverseSine #MathTutorial #StepByStep", "lengthSeconds": 138, "width": 640, "height": 360}
{"video_id": "_pf-rl21rSU", "transcript": [{"start": 0.32, "duration": 5.039, "text": "let's solve this derivative step by step"}, {"start": 2.8, "duration": 5.119, "text": "using the chain rule we'll see how to"}, {"start": 5.359, "duration": 5.561, "text": "differentiate cosine to the 4th power of"}, {"start": 7.919, "duration": 5.84, "text": "theta to solve this we'll use the chain"}, {"start": 10.92, "duration": 5.52, "text": "rule the chain rule tells us that when"}, {"start": 13.759, "duration": 5.6, "text": "we have a composite function which means"}, {"start": 16.44, "duration": 4.919, "text": "a function inside another function we"}, {"start": 19.359, "duration": 4.68, "text": "multiply the derivative of the outer"}, {"start": 21.359, "duration": 5.481, "text": "function by the derivative of the inner"}, {"start": 24.039, "duration": 5.281, "text": "function in our case we have cosine of"}, {"start": 26.84, "duration": 5.68, "text": "theta as our inner function and the"}, {"start": 29.32, "duration": 6.48, "text": "fourth power as our outer function let's"}, {"start": 32.52, "duration": 5.84, "text": "identify these parts first let's find"}, {"start": 35.8, "duration": 5.2, "text": "the derivative of the outer function the"}, {"start": 38.36, "duration": 6.16, "text": "derivative of U to the 4th power with"}, {"start": 41.0, "duration": 6.559, "text": "respect to U is 4 U cubed substituting"}, {"start": 44.52, "duration": 6.24, "text": "back our inner function we get 4 *"}, {"start": 47.559, "duration": 5.561, "text": "cosine cubed of theta next we need the"}, {"start": 50.76, "duration": 6.52, "text": "derivative of the inner function the"}, {"start": 53.12, "duration": 7.119, "text": "derivative of cosine Theta is sin Theta"}, {"start": 57.28, "duration": 6.0, "text": "now according to the chain rule we"}, {"start": 60.239, "duration": 7.641, "text": "multiply these parts together so we have"}, {"start": 63.28, "duration": 8.92, "text": "for cine cubed thet * sin Theta this"}, {"start": 67.88, "duration": 7.72, "text": "gives us our final answer -4 cosine"}, {"start": 72.2, "duration": 6.36, "text": "cubed Theta sin Theta let's visualize"}, {"start": 75.6, "duration": 5.839, "text": "why this makes sense here's the graph of"}, {"start": 78.56, "duration": 6.36, "text": "cosine to the 4th power and here's its"}, {"start": 81.439, "duration": 6.0, "text": "derivative that we just found notice how"}, {"start": 84.92, "duration": 5.28, "text": "the derivative is zero at the Peaks and"}, {"start": 87.439, "duration": 5.601, "text": "troughs of the original function and"}, {"start": 90.2, "duration": 5.32, "text": "steepest at the middle points just as we"}, {"start": 93.04, "duration": 5.119, "text": "would expect and that's how we find the"}, {"start": 95.52, "duration": 6.12, "text": "derivative of cosine to the 4th power of"}, {"start": 98.159, "duration": 6.24, "text": "theta using the chain rule remember when"}, {"start": 101.64, "duration": 5.72, "text": "dealing with composite functions always"}, {"start": 104.399, "duration": 8.4, "text": "identify your inner and outer functions"}, {"start": 107.36, "duration": 5.439, "text": "and apply the chain rule step by step"}], "title": "9.3-1.5 Understanding Derivatives Differentiating (\ud835\udc50\ud835\udc5c\ud835\udc60^4\ud835\udf03)  with the Chain Rule", "description": "Learn how to differentiate (\ud835\udc50\ud835\udc5c\ud835\udc60^4\ud835\udf03)  step-by-step using the chain rule and power rule. We\u2019ll break down the process to help you understand the math behind it. Perfect for students and anyone looking to improve their calculus skills!\n#Calculus #MathTutorial #Differentiation #ChainRule #MathHelp #Trigonometry #cpmrana", "lengthSeconds": 111, "width": 640, "height": 360}
{"video_id": "7VCo1Dpbqqc", "transcript": [{"start": 0.32, "duration": 6.039, "text": "let's solve this complex trigonometric"}, {"start": 3.2, "duration": 5.88, "text": "differentiation problem step by step"}, {"start": 6.359, "duration": 6.4, "text": "we'll start with finding the derivative"}, {"start": 9.08, "duration": 7.039, "text": "of this inverse tangent expression let's"}, {"start": 12.759, "duration": 6.68, "text": "begin by letting y equal our expression"}, {"start": 16.119, "duration": 7.961, "text": "we'll start with Y = inverse tangent of"}, {"start": 19.439, "duration": 8.201, "text": "a cosine x - B sinx over b cosin x + a"}, {"start": 24.08, "duration": 6.72, "text": "sinx our first step is to simplify this"}, {"start": 27.64, "duration": 5.88, "text": "fraction we can factor out B from both"}, {"start": 30.8, "duration": 5.72, "text": "numerator and denominator giving us this"}, {"start": 33.52, "duration": 5.64, "text": "expression the B's cancel out in the"}, {"start": 36.52, "duration": 5.92, "text": "fraction simplifying our expression to"}, {"start": 39.16, "duration": 5.64, "text": "this form now comes the clever part"}, {"start": 42.44, "duration": 6.04, "text": "we'll multiply both numerator and"}, {"start": 44.8, "duration": 6.96, "text": "denominator by 1 / cosine X this will"}, {"start": 48.48, "duration": 7.0, "text": "help us introduce tangent X terms"}, {"start": 51.76, "duration": 7.319, "text": "remember that sinx over cine X is"}, {"start": 55.48, "duration": 7.16, "text": "tangent X this gives us a simpler form"}, {"start": 59.079, "duration": 6.841, "text": "this is a key step we can recognize this"}, {"start": 62.64, "duration": 7.2, "text": "as the tangent of a difference formula"}, {"start": 65.92, "duration": 8.08, "text": "when we have tangent inverse of uus V"}, {"start": 69.84, "duration": 7.599, "text": "over 1 + UV it equals tangent inverse"}, {"start": 74.0, "duration": 7.88, "text": "uus tangent inverse V applying this"}, {"start": 77.439, "duration": 7.921, "text": "formula where U = / B and V equal"}, {"start": 81.88, "duration": 6.52, "text": "tangent X we get tangent inverse of A"}, {"start": 85.36, "duration": 7.079, "text": "over B minus tangent inverse of tangent"}, {"start": 88.4, "duration": 7.16, "text": "X since t tent inverse of tangent x = x"}, {"start": 92.439, "duration": 7.601, "text": "for our domain we can simplify this to"}, {"start": 95.56, "duration": 5.599, "text": "tangent inverse of a over B- X now we"}, {"start": 100.04, "duration": 3.92, "text": "can find"}, {"start": 101.159, "duration": 7.28, "text": "dydx the derivative of tangent inverse"}, {"start": 103.96, "duration": 8.519, "text": "of a constant is0 and the derivative ofx"}, {"start": 108.439, "duration": 9.241, "text": "is -1 therefore the derivative of our"}, {"start": 112.479, "duration": 5.201, "text": "original expression equals -1"}], "title": "Arctan Trigonometric Derivatives [\ud835\udc61\ud835\udc4e\ud835\udc5b^(\u22121)((\ud835\udc4e cos\u2061\ud835\udc65\u2212\ud835\udc4f sin\u2061\ud835\udc65)/(\ud835\udc4f cos\u2061\ud835\udc65+\ud835\udc4e sin\u2061\ud835\udc65 ))] Step-by-Step", "description": "Question:\nProve that the derivative of the function [\ud835\udc61\ud835\udc4e\ud835\udc5b^(\u22121)((\ud835\udc4e cos\u2061\ud835\udc65\u2212\ud835\udc4f sin\u2061\ud835\udc65)/(\ud835\udc4f cos\u2061\ud835\udc65+\ud835\udc4e sin\u2061\ud835\udc65 ))]  is  -1.\n\nDive into solving arctan trigonometric derivatives step-by-step! This video covers the intricate process of differentiating the expression [\ud835\udc61\ud835\udc4e\ud835\udc5b^(\u22121)((\ud835\udc4e cos\u2061\ud835\udc65\u2212\ud835\udc4f sin\u2061\ud835\udc65)/(\ud835\udc4f cos\u2061\ud835\udc65+\ud835\udc4e sin\u2061\ud835\udc65 ))]. Follow along to master how each step unfolds with clear explanations.\n\n#Calculus #Trigonometry #Arctan #MathTutorial #Differentiation #MathHelp", "lengthSeconds": 121, "width": 640, "height": 360}
{"video_id": "8RKYvQ6ltEw", "transcript": [{"start": 0.32, "duration": 5.559, "text": "let's solve this complex derivative"}, {"start": 2.96, "duration": 8.559, "text": "problem step by step we need to find the"}, {"start": 5.879, "duration": 9.88, "text": "derivative of x^2 * log base a of x + 7"}, {"start": 11.519, "duration": 7.201, "text": "e to x * cine X first let's break this"}, {"start": 15.759, "duration": 6.201, "text": "into two parts using the sum rule of"}, {"start": 18.72, "duration": 7.0, "text": "derivatives we'll differentiate x^2 log"}, {"start": 21.96, "duration": 8.079, "text": "base uh of X separately from 7 e to x"}, {"start": 25.72, "duration": 7.16, "text": "cosine X let's tackle the first part 4X"}, {"start": 30.039, "duration": 6.561, "text": "squ log Bas of X we'll use the product"}, {"start": 32.88, "duration": 7.92, "text": "rule remember the product rule states"}, {"start": 36.6, "duration": 8.72, "text": "that the derivative of U * v = u Prime *"}, {"start": 40.8, "duration": 8.12, "text": "V + U * V Prime this gives us x^2 * the"}, {"start": 45.32, "duration": 8.8, "text": "derivative of log base a of X Plus log"}, {"start": 48.92, "duration": 5.2, "text": "base a of x * the derivative of"}, {"start": 54.6, "duration": 7.119, "text": "x^2 now for the derivative of log base a"}, {"start": 57.92, "duration": 9.36, "text": "of X we can use the change base formula"}, {"start": 61.719, "duration": 9.001, "text": "log Bas of xal log Bas e of x / log base"}, {"start": 67.28, "duration": 7.92, "text": "e of a the derivative of natural log is"}, {"start": 70.72, "duration": 7.28, "text": "1 /x for the second part 7 e to x cosine"}, {"start": 75.2, "duration": 8.68, "text": "X will again use the product rule this"}, {"start": 78.0, "duration": 8.24, "text": "gives us 7 * e to x * sinx + cine x * e"}, {"start": 83.88, "duration": 2.36, "text": "to the"}, {"start": 87.72, "duration": 8.439, "text": "X now let's combine our results the"}, {"start": 91.479, "duration": 10.481, "text": "final answer is x * log Bas of e + 2 log"}, {"start": 96.159, "duration": 8.041, "text": "Bas of x + 7 e to x * the qu cosine x -"}, {"start": 101.96, "duration": 2.24, "text": "sin"}, {"start": 106.759, "duration": 6.601, "text": "x let's verify this is correct by"}, {"start": 110.28, "duration": 5.199, "text": "checking our work we use the sum rule to"}, {"start": 113.36, "duration": 4.64, "text": "separate the terms the product rule"}, {"start": 115.479, "duration": 5.161, "text": "twice and the chain rule for the"}, {"start": 118.0, "duration": 4.799, "text": "logarithm each step follows from the"}, {"start": 120.64, "duration": 4.36, "text": "fundamental rules of"}, {"start": 122.799, "duration": 5.081, "text": "differentiation and that's how we find"}, {"start": 125.0, "duration": 5.319, "text": "this complex derivative remember to"}, {"start": 127.88, "duration": 5.52, "text": "break down complicated problems into"}, {"start": 130.319, "duration": 4.681, "text": "smaller manageable Parts using the basic"}, {"start": 133.4, "duration": 6.199, "text": "rules of"}, {"start": 135.0, "duration": 4.599, "text": "differentiation thank you for watching"}], "title": "Finding the Derivative of (\ud835\udc65^2 log_\ud835\udc4e^\u2061\ud835\udc65+7\ud835\udc52^\ud835\udc65  cos\u2061\ud835\udc65)  Using the Product Rule | Step-by-Step.", "description": "In this video, learn to differentiate  (\ud835\udc65^2 log_\ud835\udc4e^\u2061\ud835\udc65+7\ud835\udc52^\ud835\udc65  cos\u2061\ud835\udc65)  using the Product Rule in calculus.\n\nWhat You'll Learn:  \n- Definition of the Product Rule  \n- Step-by-step differentiation of  (\ud835\udc65^2 log_\ud835\udc4e^\u2061\ud835\udc65+7\ud835\udc52^\ud835\udc65  cos\u2061\ud835\udc65) \n- Formula breakdown and example problem-solving  \n\nThis example is perfect for calculus beginners or those reviewing the Product Rule!\n\ud83d\udca1 Don't forget to LIKE, SHARE, and SUBSCRIBE for more math tutorials!\n#Calculus #ProductRule #Differentiation #Derivative #MathTutorial #LearnMath #MathEducation #MathHelp #hschighermath #hsc", "lengthSeconds": 139, "width": 640, "height": 360}
{"video_id": "hDK-VJFM5xo", "transcript": [{"start": 0.28, "duration": 5.2, "text": "let's solve this complex differentiation"}, {"start": 3.12, "duration": 4.239, "text": "problem step by step we'll use"}, {"start": 5.48, "duration": 5.199, "text": "substitution and trigonometric"}, {"start": 7.359, "duration": 5.961, "text": "identities to simplify this seemingly"}, {"start": 10.679, "duration": 5.36, "text": "difficult expression our first step is"}, {"start": 13.32, "duration": 6.84, "text": "to make a substitution to simplify this"}, {"start": 16.039, "duration": 8.641, "text": "expression let's substitute X = Tan t"}, {"start": 20.16, "duration": 7.279, "text": "which means T = inverse tangent of X now"}, {"start": 24.68, "duration": 5.759, "text": "let's substitute this into our original"}, {"start": 27.439, "duration": 8.201, "text": "expression we get tan inverse of the"}, {"start": 30.439, "duration": 8.881, "text": "square < TK of 1 + t^ 2 t - 1 all over"}, {"start": 35.64, "duration": 7.079, "text": "Tan t here's where we can use our first"}, {"start": 39.32, "duration": 7.96, "text": "trigonometric identity we know that 1 +"}, {"start": 42.719, "duration": 7.32, "text": "tan^ 2 T = SEC 2 T making this"}, {"start": 47.28, "duration": 6.279, "text": "substitution gives us tan inverse of"}, {"start": 50.039, "duration": 8.081, "text": "square < TK of secant 2"}, {"start": 53.559, "duration": 7.921, "text": "tus1 over Tan t next we can simplify the"}, {"start": 58.12, "duration": 6.399, "text": "square root of secant squ T to just"}, {"start": 61.48, "duration": 8.12, "text": "secant T this gives us tan inverse of"}, {"start": 64.519, "duration": 8.521, "text": "secant tus1 / Tan t now let's use"}, {"start": 69.6, "duration": 9.12, "text": "another identity remember that secant T"}, {"start": 73.04, "duration": 8.84, "text": "= 1 / cosine T and tangent T = sin t/"}, {"start": 78.72, "duration": 6.12, "text": "cine T substituting these and gives us"}, {"start": 81.88, "duration": 5.68, "text": "this fraction through algebraic"}, {"start": 84.84, "duration": 7.12, "text": "manipulation this simplifies to tan"}, {"start": 87.56, "duration": 8.08, "text": "inverse of 1 - cosine t / sin T here's"}, {"start": 91.96, "duration": 9.479, "text": "where we use another key identity 1 -"}, {"start": 95.64, "duration": 10.24, "text": "cosine T = 2 sin s t / 2 and sin T = 2"}, {"start": 101.439, "duration": 7.36, "text": "sin T / 2 * cine T / 2 substituting"}, {"start": 105.88, "duration": 6.4, "text": "these and gives us this expression the"}, {"start": 108.799, "duration": 7.0, "text": "sin t/ 2 terms cancel out leaving us"}, {"start": 112.28, "duration": 6.199, "text": "with just t/2 remember that t equals"}, {"start": 115.799, "duration": 7.161, "text": "inverse tangent of X so we can write"}, {"start": 118.479, "duration": 7.161, "text": "this as 12 * inverse tangent of X now we"}, {"start": 122.96, "duration": 7.76, "text": "can use the chain rule the derivative of"}, {"start": 125.64, "duration": 7.679, "text": "inverse tangent of X is 1 1 + x^2 and"}, {"start": 130.72, "duration": 8.04, "text": "multiplying by 1/ 12 gives us our Final"}, {"start": 133.319, "duration": 8.2, "text": "Answer 1 over 2 * 1 + x^2 and that's our"}, {"start": 138.76, "duration": 5.68, "text": "solution through careful substitution"}, {"start": 141.519, "duration": 5.841, "text": "and use of trigonometric identities we"}, {"start": 144.44, "duration": 6.28, "text": "transformed a complex expression into a"}, {"start": 147.36, "duration": 3.36, "text": "simple one"}], "title": "Understanding the Derivative of Arctan Functions [\ud835\udc61\ud835\udc4e\ud835\udc5b^(\u22121)   (\u221a(1+\ud835\udc65^2 )\u22121)/\ud835\udc65] Step-by-Step", "description": "Question\nFind the derivative of the function [\ud835\udc61\ud835\udc4e\ud835\udc5b^(\u22121)   (\u221a(1+\ud835\udc65^2 )\u22121)/\ud835\udc65].\n\nDive into this detailed tutorial where we explore the derivative of the function [\ud835\udc61\ud835\udc4e\ud835\udc5b^(\u22121)   (\u221a(1+\ud835\udc65^2 )\u22121)/\ud835\udc65]. Follow along as we break down each step to simplify and differentiate this expression using trigonometric substitutions and identities. Perfect for students looking to enhance their calculus skills!\n#Calculus #MathTutorial #Differentiation #ArctanDerivatives #TrigonometricSubstitution", "lengthSeconds": 154, "width": 640, "height": 360}
{"video_id": "fYqRaOhorjE", "transcript": [{"start": 0.32, "duration": 5.559, "text": "let's learn how to find the derivative"}, {"start": 2.639, "duration": 5.16, "text": "of log base 10 of X we'll go through"}, {"start": 5.879, "duration": 5.201, "text": "this step by step to understand the"}, {"start": 7.799, "duration": 5.84, "text": "process clearly the first key step is to"}, {"start": 11.08, "duration": 5.68, "text": "convert this logarithm using the change"}, {"start": 13.639, "duration": 6.56, "text": "of Base formula this formula states that"}, {"start": 16.76, "duration": 8.16, "text": "log base uh of x equals the natural log"}, {"start": 20.199, "duration": 8.4, "text": "of x / the natural log of a applying"}, {"start": 24.92, "duration": 7.92, "text": "this to our problem log base 10 of X ="}, {"start": 28.599, "duration": 8.881, "text": "natural log of x time log base 10 of e"}, {"start": 32.84, "duration": 4.64, "text": "notice that log base 10 of e is a"}, {"start": 40.28, "duration": 5.279, "text": "constant now we can rewrite our"}, {"start": 42.92, "duration": 6.2, "text": "derivative problem we're differentiating"}, {"start": 45.559, "duration": 7.281, "text": "the product of natural log of x and the"}, {"start": 49.12, "duration": 7.0, "text": "constant log base 10 of e since log base"}, {"start": 52.84, "duration": 6.12, "text": "10 of e is constant we can Factor it out"}, {"start": 56.12, "duration": 6.239, "text": "of the derivative this gives us log base"}, {"start": 58.96, "duration": 7.08, "text": "10 of e * the derivative of natural log"}, {"start": 62.359, "duration": 7.4, "text": "of x now we use a key fact the"}, {"start": 66.04, "duration": 7.04, "text": "derivative of natural log of x is 1 /x"}, {"start": 69.759, "duration": 6.801, "text": "applying this we get log base 10 of e *"}, {"start": 73.08, "duration": 5.28, "text": "1X our final answer can be written as"}, {"start": 76.56, "duration": 4.68, "text": "log base 10 of"}, {"start": 78.36, "duration": 6.04, "text": "e/x this is the derivative formula for"}, {"start": 81.24, "duration": 5.72, "text": "log base 10 of X let's visualize what"}, {"start": 84.4, "duration": 4.759, "text": "this derivative means on a graph the"}, {"start": 86.96, "duration": 5.32, "text": "function log base 10 of X is shown in"}, {"start": 89.159, "duration": 6.441, "text": "blue and its derivative log base 10 of e"}, {"start": 92.28, "duration": 5.519, "text": "overx is shown in red notice how the"}, {"start": 95.6, "duration": 4.92, "text": "derivative is always positive but"}, {"start": 97.799, "duration": 5.96, "text": "decreases as X increases showing that"}, {"start": 100.52, "duration": 6.52, "text": "log base 10 of X grows more slowly as X"}, {"start": 103.759, "duration": 3.281, "text": "gets larger"}], "title": "Derivative of Logarithms [\ud835\udc8d\ud835\udc90\ud835\udc88_\ud835\udfcf\ud835\udfce(\ud835\udc99)] Step by Step Explanation", "description": "Question: Find the derivative of [\ud835\udc8d\ud835\udc90\ud835\udc88_\ud835\udfcf\ud835\udfce(\ud835\udc99) ]\nIn this video, we break down the process of differentiating [\ud835\udc8d\ud835\udc90\ud835\udc88_\ud835\udfcf\ud835\udfce(\ud835\udc99)]  into simple, easy-to-follow steps. Learn how to apply the change of base formula, differentiate products, and simplify the result. Perfect for students and anyone looking to master calculus concepts!\n#Calculus #Logarithms #Differentiation #MathTutorial #Mathematics #MathHelp", "lengthSeconds": 106, "width": 640, "height": 360}
{"video_id": "KkG6M9daoUk", "transcript": [{"start": 0.32, "duration": 5.2, "text": "let's solve this challenging derivative"}, {"start": 2.76, "duration": 5.04, "text": "problem step by step we'll see how"}, {"start": 5.52, "duration": 5.48, "text": "recognizing a special trigonometric"}, {"start": 7.8, "duration": 6.08, "text": "identity helps us simplify what looks"}, {"start": 11.0, "duration": 5.8, "text": "like a complex expression here's our"}, {"start": 13.88, "duration": 6.28, "text": "problem we need to find the derivative"}, {"start": 16.8, "duration": 7.96, "text": "of y = inverse tangent of the fraction"}, {"start": 20.16, "duration": 8.599, "text": "3x - x Cub over 1 -"}, {"start": 24.76, "duration": 6.679, "text": "3x^2 at first glance this looks quite"}, {"start": 28.759, "duration": 5.521, "text": "complicated however ever there's a key"}, {"start": 31.439, "duration": 5.361, "text": "Insight that will make this much easier"}, {"start": 34.28, "duration": 5.64, "text": "this fraction inside the inverse tangent"}, {"start": 36.8, "duration": 5.64, "text": "looks very similar to the triple angle"}, {"start": 39.92, "duration": 6.159, "text": "formula for tangent let's look at this"}, {"start": 42.44, "duration": 7.119, "text": "more closely if we let x equal tangent"}, {"start": 46.079, "duration": 7.16, "text": "Theta then our fraction becomes exactly"}, {"start": 49.559, "duration": 6.921, "text": "the triple angle formula for tangent"}, {"start": 53.239, "duration": 7.681, "text": "therefore our original expression can be"}, {"start": 56.48, "duration": 8.48, "text": "Rewritten as the inverse tangent of t"}, {"start": 60.92, "duration": 6.239, "text": "of 3 Theta and since inverse tangent and"}, {"start": 64.96, "duration": 5.839, "text": "tangent are inverse functions they"}, {"start": 67.159, "duration": 7.761, "text": "cancel out leaving us with just 3 Theta"}, {"start": 70.799, "duration": 8.441, "text": "now remember that x equals tangent Theta"}, {"start": 74.92, "duration": 9.159, "text": "so theta equals inverse tangent of X"}, {"start": 79.24, "duration": 7.16, "text": "therefore y = 3 * inverse tangent of X"}, {"start": 84.079, "duration": 6.121, "text": "now we can find the derivative the"}, {"start": 86.4, "duration": 7.6, "text": "derivative of inverse tangent is 1/ 1 +"}, {"start": 90.2, "duration": 8.44, "text": "x^2 and when we multiply by 3 we get our"}, {"start": 94.0, "duration": 7.24, "text": "final answer 3 over 1 + x^2 and that's"}, {"start": 98.64, "duration": 5.08, "text": "our solution what started as a"}, {"start": 101.24, "duration": 5.44, "text": "complicated looking derivative became"}, {"start": 103.72, "duration": 6.359, "text": "much simpler once we recognize the"}, {"start": 106.68, "duration": 6.28, "text": "triple angle formula for tangent this"}, {"start": 110.079, "duration": 5.36, "text": "shows how pattern recognition and"}, {"start": 112.96, "duration": 7.799, "text": "knowing your trigonometric identities"}, {"start": 115.439, "duration": 5.32, "text": "can greatly simplify calculus problems"}], "title": "Mastering Calculus: Derivative of Inverse Tangent Function [\ud835\udc61\ud835\udc4e\ud835\udc5b^(\u22121) ((3\ud835\udc65\u2212\ud835\udc65^3)/(1\u22123\ud835\udc65^2 ))] Explained", "description": "Question\nFind the derivative of the function y = [\ud835\udc61\ud835\udc4e\ud835\udc5b^(\u22121) ((3\ud835\udc65\u2212\ud835\udc65^3)/(1\u22123\ud835\udc65^2 )]\n\nIn this video, we dive into the derivative of the inverse tangent function with a detailed, step-by-step explanation. Learn how to transform and differentiate y = [\ud835\udc61\ud835\udc4e\ud835\udc5b^(\u22121) ((3\ud835\udc65\u2212\ud835\udc65^3)/(1\u22123\ud835\udc65^2 )] using calculus rules and trigonometric identities. Perfect for students and math enthusiasts looking to deepen their understanding of calculus concepts.\n#Calculus #MathTutorial #Derivative #InverseTangent #Trigonometry #MathHelp #STEM #Education", "lengthSeconds": 124, "width": 640, "height": 360}
{"video_id": "dQ2bd3FlIX4", "transcript": [{"start": 0.44, "duration": 5.52, "text": "welcome today we're going to learn about"}, {"start": 3.0, "duration": 7.88, "text": "logarithmic differentiation by finding"}, {"start": 5.96, "duration": 7.799, "text": "the derivative of yal e to power of E to"}, {"start": 10.88, "duration": 5.52, "text": "X this is a challenging function to"}, {"start": 13.759, "duration": 6.28, "text": "differentiate directly which is why"}, {"start": 16.4, "duration": 5.44, "text": "logarithmic differentiation is so useful"}, {"start": 20.039, "duration": 4.281, "text": "why do we use logarithmic"}, {"start": 21.84, "duration": 4.679, "text": "differentiation here well when we have"}, {"start": 24.32, "duration": 4.879, "text": "functions with exponents that contain"}, {"start": 26.519, "duration": 4.72, "text": "variables or other functions taking the"}, {"start": 29.199, "duration": 4.801, "text": "derivative directly can be very"}, {"start": 31.239, "duration": 5.681, "text": "complicated by using logarithms we can"}, {"start": 34.0, "duration": 6.76, "text": "transform our complex exponential into"}, {"start": 36.92, "duration": 3.84, "text": "something much easier to"}, {"start": 43.8, "duration": 6.2, "text": "differentiate let's solve the step by"}, {"start": 46.199, "duration": 6.68, "text": "step step one take the natural logarithm"}, {"start": 50.0, "duration": 9.32, "text": "of both sides of the equation this gives"}, {"start": 52.879, "duration": 7.68, "text": "us Ln of Y = Ln of e to power of E to X"}, {"start": 59.32, "duration": 3.239, "text": "step two"}, {"start": 60.559, "duration": 5.201, "text": "use the properties of logarithms to"}, {"start": 62.559, "duration": 6.441, "text": "simplify the right side remember when we"}, {"start": 65.76, "duration": 5.56, "text": "have Ln of something raised to a power"}, {"start": 69.0, "duration": 6.2, "text": "we can bring that power down in front of"}, {"start": 71.32, "duration": 10.119, "text": "the logarithm so this simplifies to Ln"}, {"start": 75.2, "duration": 14.08, "text": "of y = e to x * Ln of e and since Ln of"}, {"start": 81.439, "duration": 7.841, "text": "e equal 1 we get Ln of y = e to X"}, {"start": 93.2, "duration": 6.84, "text": "step three now we can differentiate both"}, {"start": 96.479, "duration": 7.401, "text": "sides with respect to X on the left side"}, {"start": 100.04, "duration": 6.88, "text": "we get 1/ y * dydx because we're using"}, {"start": 103.88, "duration": 8.239, "text": "the chain rule on the right side the"}, {"start": 106.92, "duration": 5.199, "text": "derivative of e to X is e to"}, {"start": 113.56, "duration": 7.559, "text": "X step four solve for"}, {"start": 117.039, "duration": 8.04, "text": "dydx multiply both sides by 1 Y which is"}, {"start": 121.119, "duration": 11.56, "text": "E power of E to X this gives us our"}, {"start": 125.079, "duration": 7.6, "text": "final answer Dy DX = E power of E to x +"}, {"start": 136.48, "duration": 5.88, "text": "x let's visualize what's happening with"}, {"start": 139.16, "duration": 7.6, "text": "the graph here's our original function y"}, {"start": 142.36, "duration": 7.879, "text": "= e power of E to X in blue and its"}, {"start": 146.76, "duration": 6.68, "text": "derivative in red notice how both fun"}, {"start": 150.239, "duration": 5.64, "text": "grow very rapidly as X increases with"}, {"start": 153.44, "duration": 5.519, "text": "the derivative growing even faster than"}, {"start": 155.879, "duration": 5.28, "text": "the original function in conclusion"}, {"start": 158.959, "duration": 4.28, "text": "logarithmic differentiation is a"}, {"start": 161.159, "duration": 4.681, "text": "powerful technique that helps us"}, {"start": 163.239, "duration": 5.521, "text": "differentiate complex exponential"}, {"start": 165.84, "duration": 5.6, "text": "functions by taking the natural log of"}, {"start": 168.76, "duration": 4.96, "text": "both sides using log properties and then"}, {"start": 171.44, "duration": 4.48, "text": "differentiating we can find derivatives"}, {"start": 173.72, "duration": 6.4, "text": "that would be very difficult to compute"}, {"start": 175.92, "duration": 6.2, "text": "directly remember this technique is AAL"}, {"start": 180.12, "duration": 5.32, "text": "useful when dealing with functions that"}, {"start": 182.12, "duration": 7.96, "text": "have variables and exponents or nested"}, {"start": 185.44, "duration": 4.64, "text": "exponentials thank you for watching"}], "title": "Mastering Logarithmic Differentiation: Finding d/dx[\u3016\ud835\udc52^\ud835\udc52\u3017^\ud835\udc65] Step by Step", "description": "Question: Find the derivative of y = [\u3016\ud835\udc52^\ud835\udc52\u3017^\ud835\udc65] using logarithmic differentiation.\n\n\ud83d\udcda In this comprehensive calculus tutorial, we break down how to find the derivative of e(ex) using logarithmic differentiation. Perfect for AP Calculus, IB Mathematics, and college-level Calculus students!\n\ud83c\udfaf What You\u2019ll Learn:\n\u2022 Why logarithmic differentiation is crucial for complex exponential functions\n\u2022 Step-by-step solution process\n\u2022 Properties of logarithms and their applications\n\u2022 Chain rule implementation\n\u2022 Common pitfalls and how to avoid them\n\ud83d\udcdd Follow along with our step-by-step solution!\n#Calculus #Mathematics #LogarithmicDifferentiation #APCalculus #MathTutorial #CalcTutorial #IBMaths #MathEducation #LearnMath #ExponentialFunctions", "lengthSeconds": 195, "width": 640, "height": 360}
{"video_id": "3LX8IjBjuTM", "transcript": [{"start": 0.24, "duration": 4.92, "text": "let's solve this calculus problem step"}, {"start": 2.399, "duration": 6.601, "text": "by step we need to find the derivative"}, {"start": 5.16, "duration": 6.559, "text": "of tan x * sin inverse X using the"}, {"start": 9.0, "duration": 5.0, "text": "product rule the product rule states"}, {"start": 11.719, "duration": 4.681, "text": "that the derivative of a product of two"}, {"start": 14.0, "duration": 4.8, "text": "functions equals the first function time"}, {"start": 16.4, "duration": 4.76, "text": "the derivative of the second plus the"}, {"start": 18.8, "duration": 5.16, "text": "second function time the derivative of"}, {"start": 21.16, "duration": 6.72, "text": "the first let's write this out"}, {"start": 23.96, "duration": 7.44, "text": "mathematically in our case U = Tan x and"}, {"start": 27.88, "duration": 6.199, "text": "V = sin inverse X let's apply the"}, {"start": 31.4, "duration": 4.52, "text": "product rule to these functions now we"}, {"start": 34.079, "duration": 5.561, "text": "need to find the derivatives of sin"}, {"start": 35.92, "duration": 6.56, "text": "inverse X and tanx separately the"}, {"start": 39.64, "duration": 6.88, "text": "derivative of sin inverse X is 1 over"}, {"start": 42.48, "duration": 8.52, "text": "the < TK of 1 - x^2 and the derivative"}, {"start": 46.52, "duration": 7.16, "text": "of tan X is secant 2 x let's substitute"}, {"start": 51.0, "duration": 9.039, "text": "these derivatives back into our equation"}, {"start": 53.68, "duration": 10.24, "text": "we get tan x * 1 < TK 1 - x^2 plus s"}, {"start": 60.039, "duration": 7.321, "text": "inverse x * SEC 2 x and this gives us"}, {"start": 63.92, "duration": 9.08, "text": "our final answer the derivative of tan x"}, {"start": 67.36, "duration": 11.32, "text": "* sin inverse X = Tan x over < TK of 1 -"}, {"start": 73.0, "duration": 8.0, "text": "x^2 + sin inverse x * secant 2 x let's"}, {"start": 78.68, "duration": 5.119, "text": "summarize what we did we used the"}, {"start": 81.0, "duration": 5.72, "text": "product rule to split the derivative"}, {"start": 83.799, "duration": 6.241, "text": "into two terms then we found the"}, {"start": 86.72, "duration": 5.759, "text": "derivatives of sin inverse X and Tan x"}, {"start": 90.04, "duration": 4.399, "text": "separately and substituted them back"}, {"start": 92.479, "duration": 4.96, "text": "into our equation to get our final"}, {"start": 94.439, "duration": 3.0, "text": "answer"}], "title": "Mastering Calculus: Differentiating Products of Trigonometric & Inverse Functions[tan\u2061\ud835\udc65 .\ud835\udc60\ud835\udc56\ud835\udc5b^(\u22121)\ud835\udc65]", "description": "Question: Differentiate the product [tan\u2061\ud835\udc65 .\ud835\udc60\ud835\udc56\ud835\udc5b^(\u22121)\ud835\udc65]  with respect to  using the product rule.\nUnlock the secrets of calculus with our step-by-step guide to differentiating products of trigonometric and inverse functions. Dive into the product rule, understand the process, and enhance your calculus skills. Perfect for students and math enthusiasts alike!\n#Calculus #Differentiation #ProductRule #MathTutorial #Trigonometry #InverseFunctions", "lengthSeconds": 100, "width": 640, "height": 360}
{"video_id": "vLVS2ld2xbI", "transcript": [{"start": 0.24, "duration": 4.8, "text": "let's solve this calculus problem step"}, {"start": 2.399, "duration": 7.681, "text": "by step we need to find the derivative"}, {"start": 5.04, "duration": 9.28, "text": "of 5 e to x * log Bas of X the first"}, {"start": 10.08, "duration": 6.959, "text": "step is to rewrite log base of X using"}, {"start": 14.32, "duration": 6.119, "text": "the change of Base formula this states"}, {"start": 17.039, "duration": 7.361, "text": "that log base a of x equals log base e"}, {"start": 20.439, "duration": 8.121, "text": "of x / log base e of a which can also be"}, {"start": 24.4, "duration": 8.6, "text": "written as log Bas of e * log base e of"}, {"start": 28.56, "duration": 8.319, "text": "X for step 2 will apply the product rule"}, {"start": 33.0, "duration": 7.48, "text": "since we're multiplying e to X and the"}, {"start": 36.879, "duration": 6.641, "text": "natural log of x remember the product"}, {"start": 40.48, "duration": 8.44, "text": "rule states that the derivative of U * V"}, {"start": 43.52, "duration": 8.359, "text": "equal U Prime * V + U * V Prime for step"}, {"start": 48.92, "duration": 5.439, "text": "three we'll find the derivatives of each"}, {"start": 51.879, "duration": 7.64, "text": "component the derivative of the natural"}, {"start": 54.359, "duration": 10.361, "text": "log of x is 1 /x and the derivative of e"}, {"start": 59.519, "duration": 7.441, "text": "to X is just e to X step four let's"}, {"start": 64.72, "duration": 7.12, "text": "substitute these derivatives back into"}, {"start": 66.96, "duration": 10.0, "text": "our equation we get e to x * 1 /x plus"}, {"start": 71.84, "duration": 7.56, "text": "log base e of x * e to X for step five"}, {"start": 76.96, "duration": 5.88, "text": "we need to multiply everything by five"}, {"start": 79.4, "duration": 5.84, "text": "log base a of e which was our constant"}, {"start": 82.84, "duration": 8.919, "text": "from the change of Base formula this"}, {"start": 85.24, "duration": 11.76, "text": "gives us 5 e to x * 1 /x * log Bas uh of"}, {"start": 91.759, "duration": 8.68, "text": "e+ log Bas uh of e * log Bas e of X"}, {"start": 97.0, "duration": 6.84, "text": "finally in Step six we can simplify our"}, {"start": 100.439, "duration": 8.121, "text": "answer by recognizing that log base uh"}, {"start": 103.84, "duration": 8.919, "text": "of e * log base e of x equal log base of"}, {"start": 108.56, "duration": 9.68, "text": "X this gives us our final answer 5 e to"}, {"start": 112.759, "duration": 7.841, "text": "x * the Quan 1 /x * log base uh of e"}, {"start": 118.24, "duration": 4.76, "text": "plus log basa of x"}, {"start": 120.6, "duration": 7.0, "text": "and that's how we find the derivative of"}, {"start": 123.0, "duration": 6.44, "text": "5 e to the x * log base a ofx the key"}, {"start": 127.6, "duration": 4.719, "text": "steps were using the change of Base"}, {"start": 129.44, "duration": 5.64, "text": "formula applying the product rule and"}, {"start": 132.319, "duration": 7.361, "text": "then simplifying using logarithm"}, {"start": 135.08, "duration": 4.6, "text": "properties thank you for watching"}], "title": "Finding the Derivative of 5e^x log_a(x)  Using the Product Rule | Step-by-Step Explanation", "description": "In this video, discover how to differentiate 5e^x log_a(x) using the Product Rule in calculus. \n\nWhat You'll Learn:  \n- Definition of the Product Rule  \n- Step-by-step differentiation of 5e^x log_a(x)  \n- Formula breakdown and example problem-solving  \n\nThis example is ideal for calculus beginners or anyone reviewing the Product Rule for differentiation!\n\ud83d\udca1 Don't forget to LIKE, SHARE, and SUBSCRIBE for more math tutorials!\n#Calculus #ProductRule #Differentiation #Derivative #MathTutorial #LearnMath #MathEducation #MathHelp", "lengthSeconds": 140, "width": 640, "height": 360}
{"video_id": "BKim8M5d9rI", "transcript": [{"start": 0.28, "duration": 6.24, "text": "let's solve this step by step we need to"}, {"start": 3.08, "duration": 7.519, "text": "differentiate e to the power of 5 * the"}, {"start": 6.52, "duration": 6.96, "text": "natural logarithm of tangent of 5x"}, {"start": 10.599, "duration": 5.52, "text": "before we start differentiating let's"}, {"start": 13.48, "duration": 5.76, "text": "recall some important properties first"}, {"start": 16.119, "duration": 7.601, "text": "for exponents we know that e to the Ln"}, {"start": 19.24, "duration": 6.52, "text": "of x = x we'll also need the chain rule"}, {"start": 23.72, "duration": 4.16, "text": "which states that the derivative of a"}, {"start": 25.76, "duration": 4.88, "text": "composite function is the derivative of"}, {"start": 27.88, "duration": 5.44, "text": "the outer function times the derivative"}, {"start": 30.64, "duration": 5.2, "text": "of the inner function let's start by"}, {"start": 33.32, "duration": 5.239, "text": "simplifying our expression using the"}, {"start": 35.84, "duration": 6.96, "text": "property of exponents notice that we"}, {"start": 38.559, "duration": 7.761, "text": "have e to the power of 5 Ln of tangent"}, {"start": 42.8, "duration": 8.04, "text": "5x using our property that e to the Ln"}, {"start": 46.32, "duration": 9.0, "text": "of x = x this simplifies to tangent of"}, {"start": 50.84, "duration": 6.92, "text": "5x to 5th power now we can differentiate"}, {"start": 55.32, "duration": 5.239, "text": "this expression the outer function is"}, {"start": 57.76, "duration": 7.679, "text": "the power function and the in function"}, {"start": 60.559, "duration": 7.001, "text": "is tangent of 5x using the chain rule"}, {"start": 65.439, "duration": 4.481, "text": "we'll multiply the derivative of the"}, {"start": 67.56, "duration": 5.239, "text": "outer function by the derivative of the"}, {"start": 69.92, "duration": 7.36, "text": "inner function let's focus on finding"}, {"start": 72.799, "duration": 6.921, "text": "the derivative of tangent of 5x remember"}, {"start": 77.28, "duration": 5.92, "text": "that the derivative of tangent is secant"}, {"start": 79.72, "duration": 6.28, "text": "squar but since we have 5x inside we"}, {"start": 83.2, "duration": 6.36, "text": "need to multiply by the derivative of 5x"}, {"start": 86.0, "duration": 4.439, "text": "which is just 5 now let's put everything"}, {"start": 89.56, "duration": 7.08, "text": "together"}, {"start": 90.439, "duration": 7.081, "text": "we multiply 5 * tangent of 5x 4 power *"}, {"start": 96.64, "duration": 5.759, "text": "5"}, {"start": 97.52, "duration": 11.52, "text": "secant 5x this simplifies to 25 tangent"}, {"start": 102.399, "duration": 9.08, "text": "of 5x 4th power * secant SAR of 5x if"}, {"start": 109.04, "duration": 5.24, "text": "you prefer we can write this in an"}, {"start": 111.479, "duration": 7.6, "text": "alternative form using the fact that"}, {"start": 114.28, "duration": 11.36, "text": "secant s = 1 + tangent 2 this would give"}, {"start": 119.079, "duration": 10.641, "text": "us 25 tangent of 5x 4 power * the 1 +"}, {"start": 125.64, "duration": 6.76, "text": "tangent 5x let's review what we did we"}, {"start": 129.72, "duration": 6.08, "text": "first simplified the expression using"}, {"start": 132.4, "duration": 6.28, "text": "properties of exponents and logarithms"}, {"start": 135.8, "duration": 5.68, "text": "then we applied the chain rule twice"}, {"start": 138.68, "duration": 6.199, "text": "once for the power function and once for"}, {"start": 141.48, "duration": 7.36, "text": "the tangent function finally we combine"}, {"start": 144.879, "duration": 3.961, "text": "terms to get our answer"}], "title": "Step-by-Step Guide to Differentiating Exponential Logarithmic Functions \ud835\udc52^(5 ln(\u2061\ud835\udc61\ud835\udc4e\ud835\udc5b 5\ud835\udc65))", "description": "In this tutorial, we provide a detailed step-by-step guide to differentiating the expression \ud835\udc52^(5 ln(\u2061\ud835\udc61\ud835\udc4e\ud835\udc5b 5\ud835\udc65)) . Learn about the fundamental properties of exponents and logarithms, and how to apply the chain rule effectively. Whether you\u2019re a calculus student or just brushing up on your math skills, this video will walk you through each calculation clearly and concisely!\n#Calculus #Differentiation #MathTutorial #ExponentialFunctions #LogarithmicFunctions #ChainRule #MathEducation", "lengthSeconds": 154, "width": 640, "height": 360}
{"video_id": "LyPCzG86nBo", "transcript": [{"start": 0.719, "duration": 5.6, "text": "welcome to this calculus tutorial on"}, {"start": 3.56, "duration": 6.36, "text": "finding derivatives of logarithmic"}, {"start": 6.319, "duration": 5.44, "text": "functions today we'll solve a complex"}, {"start": 9.92, "duration": 4.48, "text": "problem involving logarithms with"}, {"start": 11.759, "duration": 5.961, "text": "different bases let's break this down"}, {"start": 14.4, "duration": 7.2, "text": "step by step first let's recall the"}, {"start": 17.72, "duration": 8.399, "text": "change of Base formula for any logarithm"}, {"start": 21.6, "duration": 8.4, "text": "log base uh of xals the natural log of x"}, {"start": 26.119, "duration": 6.681, "text": "/ the natural log of a using this"}, {"start": 30.0, "duration": 7.16, "text": "formula let's rewrite our original"}, {"start": 32.8, "duration": 7.88, "text": "expression the first term log base of X"}, {"start": 37.16, "duration": 9.2, "text": "becomes natural log of x / natural log"}, {"start": 40.68, "duration": 8.24, "text": "of a for the second term log base X of a"}, {"start": 46.36, "duration": 5.0, "text": "it becomes natural log of a divided"}, {"start": 48.92, "duration": 5.479, "text": "natural log of x now we can"}, {"start": 51.36, "duration": 5.359, "text": "differentiate each term separately using"}, {"start": 54.399, "duration": 5.8, "text": "the quotient rule let's start with the"}, {"start": 56.719, "duration": 5.8, "text": "first term remember the quotient rule"}, {"start": 60.199, "duration": 6.641, "text": "states that the derivative of U over V"}, {"start": 62.519, "duration": 9.121, "text": "equal V * the of uus U * the derivative"}, {"start": 66.84, "duration": 8.72, "text": "of V all over V ^2 for the first term U"}, {"start": 71.64, "duration": 7.119, "text": "is natural log of x and V is natural log"}, {"start": 75.56, "duration": 6.199, "text": "of a the derivative of natural log of x"}, {"start": 78.759, "duration": 7.241, "text": "is 1 /x and the derivative of natural"}, {"start": 81.759, "duration": 7.521, "text": "log of uh is zero since is a constant"}, {"start": 86.0, "duration": 8.0, "text": "for the second term U is natural log of"}, {"start": 89.28, "duration": 8.32, "text": "a and V is natural log of x again the"}, {"start": 94.0, "duration": 6.64, "text": "derivative of natural log of a is zero"}, {"start": 97.6, "duration": 6.68, "text": "and the derivative of natural log of x"}, {"start": 100.64, "duration": 6.2, "text": "is 1 /x now we can combine these terms"}, {"start": 104.28, "duration": 6.36, "text": "using the sum rule the derivative of our"}, {"start": 106.84, "duration": 9.84, "text": "original expression is 1 /x Ln of a"}, {"start": 110.64, "duration": 8.519, "text": "minus Ln of a overx * Ln of x^2 and"}, {"start": 116.68, "duration": 5.079, "text": "that's our final answer let's review"}, {"start": 119.159, "duration": 5.6, "text": "what we did we used the change of Base"}, {"start": 121.759, "duration": 5.801, "text": "formula to convert to Natural logarithms"}, {"start": 124.759, "duration": 5.241, "text": "applied the quotient rule to each term"}, {"start": 127.56, "duration": 5.16, "text": "and combined our results using the sum"}, {"start": 130.0, "duration": 5.239, "text": "rule this problem demonstrates the"}, {"start": 132.72, "duration": 5.28, "text": "importance of knowing your logarithm"}, {"start": 135.239, "duration": 6.36, "text": "properties and differentiation rules"}, {"start": 138.0, "duration": 3.599, "text": "thank you for watching"}], "title": "Complex Logarithmic Differentiation: Solving dy/dx for loga(x) + logx(a) | Calculus Tutorial", "description": "Question:\nFind the derivative of y = loga(x) + logx(a)\n\n\ud83d\udcda Master complex logarithmic differentiation in this comprehensive calculus tutorial! Learn how to find the derivative of y = loga(x) + logx(a) step by step.\nIn this video, we cover:\n\u2b50 Breaking down complex logarithmic expressions\n\u2b50 Change of base formula applications\n\u2b50 Chain rule and quotient rule implementation\n\u2b50 Natural logarithm conversions\n\u2b50 Sum rule for derivatives\n\u2b50 Final simplification techniques\n\ud83c\udfaf Perfect for:\n\u2022 High school AP Calculus students\n\u2022 College Calculus I & II students\n\u2022 Math majors\n\u2022 Students preparing for exams\n#Calculus #Mathematics #LogarithmicDifferentiation #MathTutorial #CalcTutorial #MathHelp #APCalculus #CollegeMath #Derivatives #MathEducation", "lengthSeconds": 144, "width": 640, "height": 360}
{"video_id": "UfJtlTY_Ou8", "transcript": [{"start": 0.719, "duration": 5.281, "text": "let's solve this challenging derivative"}, {"start": 3.04, "duration": 7.0, "text": "problem step by step we need to find the"}, {"start": 6.0, "duration": 7.12, "text": "derivative of log base cine X of tangent"}, {"start": 10.04, "duration": 5.44, "text": "X the first step is to convert this to"}, {"start": 13.12, "duration": 5.4, "text": "Natural logarithms using the change of"}, {"start": 15.48, "duration": 6.6, "text": "Base formula the change of Base formula"}, {"start": 18.52, "duration": 6.519, "text": "states that log base uh of b equals"}, {"start": 22.08, "duration": 6.88, "text": "natural log of B divided by natural log"}, {"start": 25.039, "duration": 7.36, "text": "of a applying this to our problem we get"}, {"start": 28.96, "duration": 6.439, "text": "natural log of tangent x / natural log"}, {"start": 32.399, "duration": 5.401, "text": "of cine X now we need to take the"}, {"start": 35.399, "duration": 5.121, "text": "derivative of this quotient we'll use"}, {"start": 37.8, "duration": 6.0, "text": "the quotient rule which states that the"}, {"start": 40.52, "duration": 7.64, "text": "derivative of U over V equal V * the"}, {"start": 43.8, "duration": 7.56, "text": "Dera of uus U * the derivative of V all"}, {"start": 48.16, "duration": 6.919, "text": "over V ^2 let's apply the quotient rule"}, {"start": 51.36, "duration": 7.44, "text": "to our expression here U is natural log"}, {"start": 55.079, "duration": 6.64, "text": "of tangent X and V is natural log of"}, {"start": 58.8, "duration": 6.039, "text": "cine X this gives us natural log of"}, {"start": 61.719, "duration": 6.76, "text": "cosine x times the derivative of natural"}, {"start": 64.839, "duration": 6.761, "text": "log of tangent x minus natural log of"}, {"start": 68.479, "duration": 6.481, "text": "tangent x * the derivative of natural"}, {"start": 71.6, "duration": 6.44, "text": "log of cine X all over natural log of"}, {"start": 74.96, "duration": 6.36, "text": "cine x^ 2 now we need to find the"}, {"start": 78.04, "duration": 6.24, "text": "derivatives of the natural logarithms"}, {"start": 81.32, "duration": 7.36, "text": "the derivative of natural log of tangent"}, {"start": 84.28, "duration": 7.879, "text": "x = 1 / tangent x * the derivative of"}, {"start": 88.68, "duration": 7.399, "text": "tangent X which is secant 2 x and the"}, {"start": 92.159, "duration": 9.041, "text": "derivative of natural log of cine x = 1"}, {"start": 96.079, "duration": 7.481, "text": "/ cosine x * sinx let's substitute these"}, {"start": 101.2, "duration": 4.959, "text": "derivatives back into our quotient rule"}, {"start": 103.56, "duration": 5.76, "text": "expression after substituting and"}, {"start": 106.159, "duration": 7.881, "text": "simplifying we get natural log of cine x"}, {"start": 109.32, "duration": 8.079, "text": "* cosine X over sin x * cosine 2 X Plus"}, {"start": 114.04, "duration": 6.439, "text": "natural log of tangent x * sin x cosine"}, {"start": 117.399, "duration": 6.36, "text": "X all over natural log of cosine"}, {"start": 120.479, "duration": 7.201, "text": "x^2 finally we can express this using"}, {"start": 123.759, "duration": 8.641, "text": "secant and cosecant our final answer is"}, {"start": 127.68, "duration": 8.52, "text": "secant x * cose x * natural log of cine"}, {"start": 132.4, "duration": 7.44, "text": "X Plus natural log of tangent x *"}, {"start": 136.2, "duration": 6.759, "text": "tangent X all over natural log of cine"}, {"start": 139.84, "duration": 5.88, "text": "x^2 this problem demonstrated the use of"}, {"start": 142.959, "duration": 5.121, "text": "several calculus Concepts the change of"}, {"start": 145.72, "duration": 4.72, "text": "Base formula for logarithms the quotient"}, {"start": 148.08, "duration": 5.12, "text": "rule the chain rule"}, {"start": 150.44, "duration": 6.0, "text": "and trigonometric identities thank you"}, {"start": 153.2, "duration": 3.24, "text": "for watching"}], "title": "Derivatives of Logarithms with Trigonometric Bases [log_\ud835\udc50\ud835\udc5c\ud835\udc60\ud835\udc65(\u2061tan\u2061\ud835\udc65)] | Advanced Calculus", "description": "Question:\nFind the derivative of [log_\ud835\udc50\ud835\udc5c\ud835\udc60\ud835\udc65(\u2061tan\u2061\ud835\udc65)] with respect to x.\n\nMaster the art of differentiating complex logarithmic expressions with trigonometric bases! In this comprehensive tutorial, we solve step-by-step:\n\u2022 How to differentiate log_cos(x)(tan x)\n\u2022 Using the change of base formula\n\u2022 Applying the quotient rule\n\u2022 Working with trig derivatives\n\u2022 Simplifying complex trig expressions\nPerfect for:\n\ud83d\udcda AP Calculus BC students\n\ud83c\udf93 College Calculus II students\n\ud83d\udcd6 Math majors\n\u270f\ufe0f Test preparation\n#Calculus #Mathematics #Derivatives #MathTutorial #CalculusBC #CollegeMath #LogarithmDerivatives #TrigonometryCalculus", "lengthSeconds": 163, "width": 640, "height": 360}
{"video_id": "DZPYavJtjNA", "transcript": [{"start": 0.24, "duration": 5.119, "text": "in this video we'll solve a challenging"}, {"start": 3.16, "duration": 5.08, "text": "differentiation problem involving"}, {"start": 5.359, "duration": 5.721, "text": "inverse trigonometric functions let's"}, {"start": 8.24, "duration": 6.84, "text": "break it down step by step here's our"}, {"start": 11.08, "duration": 9.88, "text": "problem find the derivative of yal"}, {"start": 15.08, "duration": 8.56, "text": "inverse tangent of 6 < tkx over 1 - 9x"}, {"start": 20.96, "duration": 5.2, "text": "this looks complicated at first glance"}, {"start": 23.64, "duration": 5.28, "text": "but we can solve it using substitution"}, {"start": 26.16, "duration": 5.279, "text": "and trigonometric identities let's start"}, {"start": 28.92, "duration": 7.04, "text": "with a clever sub subtitution we'll let"}, {"start": 31.439, "duration": 9.401, "text": "3 < tkx equal tangent T this means tal"}, {"start": 35.96, "duration": 7.0, "text": "inverse tangent of 3 rootx now let's"}, {"start": 40.84, "duration": 4.879, "text": "rewrite our expression using the"}, {"start": 42.96, "duration": 6.599, "text": "substitution the numerator 6 rootx can"}, {"start": 45.719, "duration": 8.761, "text": "be written as 2 * 3 < tkx which is 2 Tan"}, {"start": 49.559, "duration": 9.48, "text": "t and in the denominator 1 - 9x is"}, {"start": 54.48, "duration": 7.52, "text": "equivalent to 1 - tangent 2 T because 9x"}, {"start": 59.039, "duration": 6.281, "text": "is s of 3 3un X which we set equal to"}, {"start": 62.0, "duration": 6.479, "text": "tan T here's where we use a key"}, {"start": 65.32, "duration": 8.119, "text": "trigonometric identity the expression 2"}, {"start": 68.479, "duration": 9.521, "text": "tan t/ 1 - tan^ 2 T is equal to tan of 2"}, {"start": 73.439, "duration": 9.481, "text": "T therefore y equals inverse tangent of"}, {"start": 78.0, "duration": 7.04, "text": "tan of 2T which simplifies to Just 2 T"}, {"start": 82.92, "duration": 6.839, "text": "now we can substitute back our"}, {"start": 85.04, "duration": 9.039, "text": "expression for t y = 2 * inverse tangent"}, {"start": 89.759, "duration": 7.68, "text": "of of 3 < tkx to find dydx we'll use the"}, {"start": 94.079, "duration": 7.441, "text": "chain rule remember the derivative of"}, {"start": 97.439, "duration": 7.801, "text": "inverse tangent is 1/ 1 + U ^2 where U"}, {"start": 101.52, "duration": 6.4, "text": "is the inside function then we multiply"}, {"start": 105.24, "duration": 6.479, "text": "by the derivative of the inside function"}, {"start": 107.92, "duration": 9.479, "text": "let's simplify this step by step first 1"}, {"start": 111.719, "duration": 10.281, "text": "+ 3un x^2 = 1 + 9x the derivative of 3 <"}, {"start": 117.399, "duration": 7.161, "text": "tkx is 3 over 2un X finally when we"}, {"start": 122.0, "duration": 8.399, "text": "multiply everything out and simplify we"}, {"start": 124.56, "duration": 9.16, "text": "get our answer 3/ < tkx * 1 + 9x and"}, {"start": 130.399, "duration": 5.721, "text": "that's our solution we use substitution"}, {"start": 133.72, "duration": 5.76, "text": "trigonometric identities and the chain"}, {"start": 136.12, "duration": 6.36, "text": "rule to find this derivative thanks for"}, {"start": 139.48, "duration": 3.0, "text": "watching"}], "title": "Mastering Derivatives: Solving [\ud835\udc61\ud835\udc4e\ud835\udc5b^(\u22121)  (6\u221a\ud835\udc65  )/(1\u22129\ud835\udc65)] Step-by-Step", "description": "Question:\nFind the derivative of  y=[\ud835\udc61\ud835\udc4e\ud835\udc5b^(\u22121)  (6\u221a\ud835\udc65  )/(1\u22129\ud835\udc65)] in terms of x .\nDive into calculus with this step-by-step explanation of how to find the derivative of y=[\ud835\udc61\ud835\udc4e\ud835\udc5b^(\u22121)  (6\u221a\ud835\udc65  )/(1\u22129\ud835\udc65)]. Learn how to apply trigonometric identities and calculus rules like the chain rule to solve this problem easily. Perfect for students and math enthusiasts!\n#Calculus #Derivatives #Trigonometry #MathTutorial #ChainRule #InverseTrigonometricFunctions", "lengthSeconds": 145, "width": 640, "height": 360}
{"video_id": "418NuZEyzFk", "transcript": [{"start": 0.28, "duration": 5.8, "text": "let's solve this differentiation problem"}, {"start": 2.6, "duration": 7.039, "text": "step by step we'll start with a question"}, {"start": 6.08, "duration": 6.92, "text": "first we need to understand what X De"}, {"start": 9.639, "duration": 5.601, "text": "means in terms of radians when we see X"}, {"start": 13.0, "duration": 5.68, "text": "De we can convert it to radians by"}, {"start": 15.24, "duration": 6.279, "text": "multiplying X by pi over"}, {"start": 18.68, "duration": 7.16, "text": "180 so our expression becomes the"}, {"start": 21.519, "duration": 6.201, "text": "derivative of x piun over 180 * cine of"}, {"start": 25.84, "duration": 4.48, "text": "PX over"}, {"start": 27.72, "duration": 4.48, "text": "180 now this is a product of two"}, {"start": 30.32, "duration": 4.68, "text": "functions so we'll need to use the"}, {"start": 32.2, "duration": 6.44, "text": "product rule the product rule states"}, {"start": 35.0, "duration": 7.079, "text": "that the derivative of U * v = u * the"}, {"start": 38.64, "duration": 7.0, "text": "derivative of V plus v * the derivative"}, {"start": 42.079, "duration": 7.041, "text": "of U applying the product rule we get x"}, {"start": 45.64, "duration": 5.28, "text": "pi over 180 * the derivative of cosine"}, {"start": 49.12, "duration": 6.439, "text": "pix over"}, {"start": 50.92, "duration": 7.0, "text": "180 plus cosine pix over 180 * the"}, {"start": 55.559, "duration": 5.281, "text": "derivative of x piun over"}, {"start": 57.92, "duration": 5.36, "text": "180 now let's handle the derivative of"}, {"start": 60.84, "duration": 5.52, "text": "cosine using the chain rule the"}, {"start": 63.28, "duration": 5.36, "text": "derivative of cosine is negative s and"}, {"start": 66.36, "duration": 5.28, "text": "we multiply by the derivative of the"}, {"start": 68.64, "duration": 6.799, "text": "inner function the derivative of x piun"}, {"start": 71.64, "duration": 7.6, "text": "over 180 is simply pi over"}, {"start": 75.439, "duration": 8.121, "text": "180 finally we can factor out pi over"}, {"start": 79.24, "duration": 8.04, "text": "180 to get our final answer pi over 180"}, {"start": 83.56, "duration": 9.48, "text": "* the quantity cosine pix over 100 and"}, {"start": 87.28, "duration": 9.479, "text": "80 - pkx over 18 80 * sin pix over"}, {"start": 93.04, "duration": 7.079, "text": "180 and that's our complete solution we"}, {"start": 96.759, "duration": 4.961, "text": "use both the product rule and chain rule"}, {"start": 100.119, "duration": 3.68, "text": "along with our knowledge of"}, {"start": 101.72, "duration": 5.079, "text": "trigonometric derivatives to solve this"}, {"start": 103.799, "duration": 3.0, "text": "problem"}], "title": "Understanding Calculus: Differentiating Trigonometric Functions (\ud835\udc65\u00b0 \ud835\udc50\ud835\udc5c\ud835\udc60\ud835\udc65\u00b0 ) Step by Step", "description": "Learn how to differentiate complex trigonometric expressions with ease. In this video, we\u2019ll walk you through a step-by-step breakdown of differentiating the function (\ud835\udc65\u00b0 \ud835\udc50\ud835\udc5c\ud835\udc60\ud835\udc65\u00b0 )  , highlighting key calculus concepts such as the product rule, chain rule, and unit conversions from degrees to radians. Perfect for students looking to deepen their understanding of derivatives and improve their calculus skills!\n#Calculus #MathTutorial #Differentiation #Trigonometry #LearnMath", "lengthSeconds": 110, "width": 640, "height": 360}
{"video_id": "phrwOOagkLA", "transcript": [{"start": 0.36, "duration": 5.519, "text": "welcome today we're going to learn about"}, {"start": 2.84, "duration": 7.879, "text": "logarithmic differentiation by finding"}, {"start": 5.879, "duration": 6.521, "text": "the derivative of y = x to X power this"}, {"start": 10.719, "duration": 5.681, "text": "is a challenging function to"}, {"start": 12.4, "duration": 6.24, "text": "differentiate directly but logarithmic"}, {"start": 16.4, "duration": 4.44, "text": "differentiation makes it much more"}, {"start": 18.64, "duration": 5.719, "text": "manageable let's see what makes this"}, {"start": 20.84, "duration": 6.88, "text": "function tricky here's what y = x to the"}, {"start": 24.359, "duration": 5.68, "text": "X looks like notice how it's different"}, {"start": 27.72, "duration": 6.8, "text": "from regular power functions because the"}, {"start": 30.039, "duration": 6.441, "text": "exponent changes as X changes step one"}, {"start": 34.52, "duration": 5.199, "text": "let's start by taking the natural"}, {"start": 36.48, "duration": 8.599, "text": "logarithm of both sides this gives us"}, {"start": 39.719, "duration": 8.201, "text": "natural log of Y = natural log of x x"}, {"start": 45.079, "duration": 4.841, "text": "step two using the logarithm property"}, {"start": 47.92, "duration": 4.799, "text": "that the log of a power equals the"}, {"start": 49.92, "duration": 7.04, "text": "exponent time the log we can simplify"}, {"start": 52.719, "duration": 7.601, "text": "this to natural log of y = x * natural"}, {"start": 56.96, "duration": 5.84, "text": "log of x step three now we can"}, {"start": 60.32, "duration": 7.92, "text": "differentiate both sides with respect to"}, {"start": 62.8, "duration": 7.8, "text": "X on the left side we get 1/ y * dydx"}, {"start": 68.24, "duration": 4.4, "text": "because we're using the chain rule on"}, {"start": 70.6, "duration": 5.36, "text": "the right side we need to use the"}, {"start": 72.64, "duration": 5.6, "text": "product rule step four let's apply the"}, {"start": 75.96, "duration": 6.799, "text": "product rule to the right side the"}, {"start": 78.24, "duration": 8.8, "text": "derivative of x * natural log of x = x *"}, {"start": 82.759, "duration": 7.921, "text": "1x + natural log of x * 1 this"}, {"start": 87.04, "duration": 6.56, "text": "simplifies to 1 + natural log of x"}, {"start": 90.68, "duration": 6.719, "text": "step five now we can solve for"}, {"start": 93.6, "duration": 9.76, "text": "dydx multiply both sides by Y and we get"}, {"start": 97.399, "duration": 9.841, "text": "Dy DX = y * the 1 + natural log of x"}, {"start": 103.36, "duration": 7.48, "text": "step six finally we can substitute back"}, {"start": 107.24, "duration": 7.519, "text": "y = x to X this gives us our final"}, {"start": 110.84, "duration": 9.16, "text": "answer the derivative of x to x = x to"}, {"start": 114.759, "duration": 7.161, "text": "the x * thean 1 + natural log of x let's"}, {"start": 120.0, "duration": 4.32, "text": "visualize what this derivative looks"}, {"start": 121.92, "duration": 6.319, "text": "like here's the original function in"}, {"start": 124.32, "duration": 6.919, "text": "blue and its derivative in red notice"}, {"start": 128.239, "duration": 5.801, "text": "how the derivative grows much faster"}, {"start": 131.239, "duration": 5.601, "text": "than the original function logarithmic"}, {"start": 134.04, "duration": 4.52, "text": "differentiation is particularly useful"}, {"start": 136.84, "duration": 4.24, "text": "when dealing with functions that have"}, {"start": 138.56, "duration": 5.56, "text": "variables in the exponent or when"}, {"start": 141.08, "duration": 5.4, "text": "dealing with products of multiple terms"}, {"start": 144.12, "duration": 5.16, "text": "it allows us to convert these complex"}, {"start": 146.48, "duration": 5.56, "text": "Expressions into simpler ones using the"}, {"start": 149.28, "duration": 5.28, "text": "properties of logarithms before"}, {"start": 152.04, "duration": 5.839, "text": "differentiating and that's how we use"}, {"start": 154.56, "duration": 6.28, "text": "logarithmic differentiation to find the"}, {"start": 157.879, "duration": 6.241, "text": "derivative of x to the X power thanks"}, {"start": 160.84, "duration": 3.28, "text": "for watching"}], "title": "Master Logarithmic Differentiation: Solving [\ud835\udc9a=\ud835\udc99^\ud835\udc99 ] Step by Step\u201d", "description": "Question: Find the derivative of y = x^x using logarithmic differentiation.\n\n\ud83d\udcda In this comprehensive tutorial, we break down how to find the derivative of y = x^x  using logarithmic differentiation. Perfect for calculus students and math enthusiasts!\nWhat you\u2019ll learn:\n\u2705 Why logarithmic differentiation is essential for complex exponential functions\n\u2705 Step-by-step solution process\n\u2705 Key logarithm properties and their applications\n\u2705 Product rule implementation\n\u2705 Common pitfalls and how to avoid them\n\u2b50 Don\u2019t forget to like and subscribe for more calculus tutorials! \u2b50\n#Calculus #Mathematics #LogarithmicDifferentiation #MathTutorial #CalcTutorial #MathEducation #LearnMath #Derivatives", "lengthSeconds": 167, "width": 640, "height": 360}
{"video_id": "R5CEBak2wPw", "transcript": [{"start": 0.32, "duration": 6.64, "text": "welcome we'll find the derivative of e"}, {"start": 3.719, "duration": 5.84, "text": "to the power of MX using first"}, {"start": 6.96, "duration": 4.48, "text": "principles this is a fundamental"}, {"start": 9.559, "duration": 4.641, "text": "derivative that's important to"}, {"start": 11.44, "duration": 6.999, "text": "understand let's start with our function"}, {"start": 14.2, "duration": 8.44, "text": "let F ofx equal e to the power of MX"}, {"start": 18.439, "duration": 7.68, "text": "step one is to find f of x + H when we"}, {"start": 22.64, "duration": 8.879, "text": "substitute x + H into our function we"}, {"start": 26.119, "duration": 8.721, "text": "get f of x + h = e to the power of M *"}, {"start": 31.519, "duration": 3.321, "text": "thean x +"}, {"start": 36.239, "duration": 6.401, "text": "H step two we'll use the definition of"}, {"start": 40.039, "duration": 5.68, "text": "the derivative the derivative is defined"}, {"start": 42.64, "duration": 8.96, "text": "as the limit as H approaches zero of f"}, {"start": 45.719, "duration": 8.921, "text": "of x + H - FX all divid H let's"}, {"start": 51.6, "duration": 3.04, "text": "substitute our"}, {"start": 55.6, "duration": 6.599, "text": "Expressions step three let's simplify"}, {"start": 58.96, "duration": 7.76, "text": "this expression first we can rewrite e"}, {"start": 62.199, "duration": 9.521, "text": "to the power of M * x + H as e to the"}, {"start": 66.72, "duration": 8.16, "text": "power of MX * e to the power of MH then"}, {"start": 71.72, "duration": 6.56, "text": "we can factor out e to the power of MX"}, {"start": 74.88, "duration": 6.239, "text": "from the numerator step four now comes"}, {"start": 78.28, "duration": 7.76, "text": "the key Insight we'll use the series"}, {"start": 81.119, "duration": 9.32, "text": "expansion for e to the power of MH E"}, {"start": 86.04, "duration": 11.039, "text": "power of MH = 1 + m"}, {"start": 90.439, "duration": 11.04, "text": "H 1 factorial + m^ 2 h^ 2/ 2 factorial +"}, {"start": 97.079, "duration": 8.0, "text": "m Cub H cubed over 3 factorial and so"}, {"start": 101.479, "duration": 6.441, "text": "on step five let's substitute this"}, {"start": 105.079, "duration": 6.201, "text": "series into our limit when we subtract"}, {"start": 107.92, "duration": 7.08, "text": "one and divide by H we get e to the"}, {"start": 111.28, "duration": 11.4, "text": "power of MX * the limit as H approaches"}, {"start": 115.0, "duration": 10.32, "text": "Zer of m + m^ 2 H over 2 Factor + MB h^2"}, {"start": 122.68, "duration": 7.76, "text": "3 factorial and so"}, {"start": 125.32, "duration": 8.48, "text": "on step six finally as H approaches zero"}, {"start": 130.44, "duration": 7.68, "text": "all terms with h go to zero this leaves"}, {"start": 133.8, "duration": 7.48, "text": "us with just m * e to the power of MX"}, {"start": 138.12, "duration": 8.92, "text": "therefore the derivative of e to the"}, {"start": 141.28, "duration": 8.679, "text": "power of MX = M * E power of MX this is"}, {"start": 147.04, "duration": 5.8, "text": "a powerful result that shows us that the"}, {"start": 149.959, "duration": 4.961, "text": "exponential function is special it's the"}, {"start": 152.84, "duration": 6.399, "text": "only function that is proportional to"}, {"start": 154.92, "duration": 6.0, "text": "its own derivative when m = 1 we get the"}, {"start": 159.239, "duration": 4.121, "text": "even more special case where the"}, {"start": 160.92, "duration": 5.2, "text": "function exactly equals its own"}, {"start": 163.36, "duration": 5.879, "text": "derivative let's visualize this function"}, {"start": 166.12, "duration": 6.36, "text": "and its derivative the blue curve shows"}, {"start": 169.239, "duration": 8.561, "text": "e to the power of MX and the red curve"}, {"start": 172.48, "duration": 7.119, "text": "shows its derivative M * E power of MX"}, {"start": 177.8, "duration": 5.04, "text": "notice how they're proportional to each"}, {"start": 179.599, "duration": 3.241, "text": "other mother"}], "title": "Understanding the Derivative of e^mx  Using First Principles", "description": "Dive into the fundamentals of calculus as we explore how to determine the derivative of the exponential function e^mx  using first principles. Follow along with step-by-step explanations and examples to enhance your understanding of derivatives and exponential functions.\n#Calculus #Derivatives #ExponentialFunctions #MathTutorial #FirstPrinciples #Defferentiation", "lengthSeconds": 183, "width": 640, "height": 360}
{"video_id": "75zAPMEaGfU", "transcript": [{"start": 0.32, "duration": 4.36, "text": "let's learn how to find the derivative"}, {"start": 2.32, "duration": 4.76, "text": "of a natural logarithm containing a"}, {"start": 4.68, "duration": 4.52, "text": "quadratic expression let's start with"}, {"start": 7.08, "duration": 7.799, "text": "our expression we want to find the"}, {"start": 9.2, "duration": 9.32, "text": "derivative of Ln of ax^2 + BX + C with"}, {"start": 14.879, "duration": 5.921, "text": "respect to X to solve this we'll use the"}, {"start": 18.52, "duration": 4.599, "text": "chain rule for logarithmic"}, {"start": 20.8, "duration": 6.44, "text": "differentiation the derivative of"}, {"start": 23.119, "duration": 7.92, "text": "natural log of U with respect to X = 1/"}, {"start": 27.24, "duration": 7.56, "text": "U * the derivative of U with respect to"}, {"start": 31.039, "duration": 8.481, "text": "X let's apply this to our problem in our"}, {"start": 34.8, "duration": 7.52, "text": "case U = ax^2 + BX + C so we can write"}, {"start": 39.52, "duration": 5.68, "text": "our derivative as 1 over the quadratic"}, {"start": 42.32, "duration": 6.559, "text": "expression times the derivative of that"}, {"start": 45.2, "duration": 6.32, "text": "same expression now let's focus on"}, {"start": 48.879, "duration": 5.121, "text": "finding the derivative of the quadratic"}, {"start": 51.52, "duration": 6.999, "text": "expression using the power rule and the"}, {"start": 54.0, "duration": 7.28, "text": "sum rule the derivative of ax^2 is 2 ax"}, {"start": 58.519, "duration": 5.281, "text": "the derivative of BX is B and the"}, {"start": 61.28, "duration": 5.04, "text": "derivative of the constant C is zero"}, {"start": 63.8, "duration": 6.76, "text": "combining these terms the derivative of"}, {"start": 66.32, "duration": 6.92, "text": "our quadratic expression is 2 ax + B"}, {"start": 70.56, "duration": 6.599, "text": "finally we can write this as a single"}, {"start": 73.24, "duration": 8.199, "text": "fraction our final answer is 2 ax + B"}, {"start": 77.159, "duration": 9.561, "text": "over ax^2 + BX + C let's visualize with"}, {"start": 81.439, "duration": 8.561, "text": "a specific example let's say a = 1 b = 2"}, {"start": 86.72, "duration": 6.48, "text": "and c = 1 this means we're finding the"}, {"start": 90.0, "duration": 8.2, "text": "derivative of the natural log of x^2 +"}, {"start": 93.2, "duration": 9.68, "text": "2x + 1 and its derivative would be 2x +"}, {"start": 98.2, "duration": 7.16, "text": "2/ x^2 + 2x + 1 notice how the"}, {"start": 102.88, "duration": 4.72, "text": "derivative gives us the rate of change"}, {"start": 105.36, "duration": 5.68, "text": "of the logarithmic function at each"}, {"start": 107.6, "duration": 6.44, "text": "point in summary we use the chain rule"}, {"start": 111.04, "duration": 5.999, "text": "for logarithmic differentiation to find"}, {"start": 114.04, "duration": 5.48, "text": "the derivative of a natural logarithm"}, {"start": 117.039, "duration": 5.4, "text": "containing a quadratic expression the"}, {"start": 119.52, "duration": 5.959, "text": "key steps were identifying our U"}, {"start": 122.439, "duration": 4.761, "text": "function applying the chain Rule and"}, {"start": 125.479, "duration": 6.321, "text": "then using the power rule to"}, {"start": 127.2, "duration": 4.6, "text": "differentiate the quadratic expression"}], "title": "Logarithmic Differentiation (\ud835\udc59\ud835\udc5b(\ud835\udc4e\ud835\udc65^2+\ud835\udc4f\ud835\udc65+\ud835\udc50)) Step by Step Guide", "description": "Unlock the secrets of calculus with our in-depth tutorial on differentiating logarithmic functions. In this video, we\u2019ll guide you through the process of finding the derivative of  (\ud835\udc59\ud835\udc5b(\ud835\udc4e\ud835\udc65^2+\ud835\udc4f\ud835\udc65+\ud835\udc50))   using the chain rule. Perfect for students and math enthusiasts looking to enhance their understanding of calculus concepts.\n#Calculus #Logarithms #Differentiation #MathTutorial #ChainRule", "lengthSeconds": 132, "width": 640, "height": 360}
{"video_id": "kPky5Zd3__4", "transcript": [{"start": 0.24, "duration": 4.76, "text": "let's solve this calculus problem step"}, {"start": 2.36, "duration": 4.999, "text": "by step we need to differentiate the"}, {"start": 5.0, "duration": 5.48, "text": "natural logarithm of the sum of two"}, {"start": 7.359, "duration": 6.24, "text": "square roots let's break this down into"}, {"start": 10.48, "duration": 6.12, "text": "manageable steps using the chain rule"}, {"start": 13.599, "duration": 6.321, "text": "first recall that the derivative of"}, {"start": 16.6, "duration": 6.0, "text": "natural log is 1 over the function times"}, {"start": 19.92, "duration": 5.599, "text": "the derivative of the inside function"}, {"start": 22.6, "duration": 5.599, "text": "applying this to our problem we get 1"}, {"start": 25.519, "duration": 5.201, "text": "over the sum of our square roots times"}, {"start": 28.199, "duration": 5.641, "text": "the derivative of that sum"}, {"start": 30.72, "duration": 5.92, "text": "next we can split up the derivative of"}, {"start": 33.84, "duration": 4.84, "text": "the sum into the sum of the derivatives"}, {"start": 36.64, "duration": 5.32, "text": "this is possible because the derivative"}, {"start": 38.68, "duration": 5.92, "text": "is a linear operator now let's recall"}, {"start": 41.96, "duration": 6.0, "text": "the power rule for square roots the"}, {"start": 44.6, "duration": 6.2, "text": "derivative of square root of U is 1/ 2 *"}, {"start": 47.96, "duration": 6.56, "text": "the square < TK of U * the derivative of"}, {"start": 50.8, "duration": 7.279, "text": "U applying this to both terms we get"}, {"start": 54.52, "duration": 5.679, "text": "this more complex expression notice how"}, {"start": 58.079, "duration": 5.44, "text": "we're applying the chain rule again to"}, {"start": 60.199, "duration": 7.96, "text": "each square root the derivatives of x -"}, {"start": 63.519, "duration": 7.441, "text": "2 and x + 1 are both simply one which"}, {"start": 68.159, "duration": 5.361, "text": "simplifies our expression to this now"}, {"start": 70.96, "duration": 5.6, "text": "comes the clever part we can find a"}, {"start": 73.52, "duration": 6.52, "text": "common denominator inside the curly"}, {"start": 76.56, "duration": 6.16, "text": "brackets when we do this the numerator"}, {"start": 80.04, "duration": 5.0, "text": "becomes the sum of square roots which"}, {"start": 82.72, "duration": 5.64, "text": "matches our denominator outside the"}, {"start": 85.04, "duration": 5.399, "text": "brackets and finally when we multiply"}, {"start": 88.36, "duration": 4.52, "text": "these fractions the sum of of square"}, {"start": 90.439, "duration": 4.881, "text": "roots cancels out in the numerator and"}, {"start": 92.88, "duration": 5.199, "text": "denominator leaving us with our Final"}, {"start": 95.32, "duration": 5.799, "text": "Answer 1 over 2 * the product of our"}, {"start": 98.079, "duration": 6.08, "text": "square roots this problem showcases"}, {"start": 101.119, "duration": 5.32, "text": "several important calculus Concepts the"}, {"start": 104.159, "duration": 5.24, "text": "chain rule the derivative of natural"}, {"start": 106.439, "duration": 5.36, "text": "logarithm the derivative of square roots"}, {"start": 109.399, "duration": 6.281, "text": "and clever algebraic manipulation to"}, {"start": 111.799, "duration": 3.881, "text": "simplify our result"}], "title": "How to Differentiate Logarithmic Functions [\ud835\udc8d\ud835\udc8f(\u221a(\ud835\udc99\u2212\ud835\udfd0)+\u221a(\ud835\udc99+\ud835\udfcf))] with Square Roots: Step-by-Step Guide", "description": "Question:\nDifferentiate the expression  [\ud835\udc8d\ud835\udc8f(\u221a(\ud835\udc99\u2212\ud835\udfd0)+\u221a(\ud835\udc99+\ud835\udfcf))] with respect to x .\n\nAre you struggling with differentiating logarithmic functions involving square roots? In this video, we break down the process of differentiating the expression [\ud835\udc8d\ud835\udc8f(\u221a(\ud835\udc99\u2212\ud835\udfd0)+\u221a(\ud835\udc99+\ud835\udfcf))]  step-by-step. Follow along as we use the chain rule and simplify the derivative to achieve clarity and accuracy. Perfect for students preparing for calculus exams or anyone needing a refresher. Don\u2019t forget to subscribe for more math tutorials!\n\n#Calculus #Differentiation #Logarithms #SquareRoots #MathTutorial #ChainRule", "lengthSeconds": 118, "width": 640, "height": 360}
{"video_id": "dcCSRcI8eRc", "transcript": [{"start": 0.28, "duration": 6.559, "text": "let's solve this complex trigonometric"}, {"start": 3.24, "duration": 5.16, "text": "derivative step by step we'll start by"}, {"start": 6.839, "duration": 4.041, "text": "understanding what we're trying to"}, {"start": 8.4, "duration": 4.76, "text": "differentiate our problem is to find the"}, {"start": 10.88, "duration": 7.319, "text": "derivative of secant inverse of the"}, {"start": 13.16, "duration": 8.48, "text": "quantity 1 + x^2 over 1 - x^2 with"}, {"start": 18.199, "duration": 6.641, "text": "respect to X to solve this we'll use a"}, {"start": 21.64, "duration": 7.039, "text": "clever substitution let's let xal"}, {"start": 24.84, "duration": 6.8, "text": "tangent of t this means tal inverse"}, {"start": 28.679, "duration": 5.841, "text": "tangent of x our first step is to use"}, {"start": 31.64, "duration": 6.36, "text": "the relationship between secant inverse"}, {"start": 34.52, "duration": 8.16, "text": "and cosine inverse remember that secant"}, {"start": 38.0, "duration": 8.76, "text": "inverse of Y = cosine inverse of 1 / y"}, {"start": 42.68, "duration": 7.08, "text": "now let's substitute X = tangent T into"}, {"start": 46.76, "duration": 7.479, "text": "our expression this gives us cosine"}, {"start": 49.76, "duration": 7.24, "text": "inverse of 1 - tangent 2 T over 1 +"}, {"start": 54.239, "duration": 6.0, "text": "tangent 2 T here's where we need to"}, {"start": 57.0, "duration": 8.439, "text": "recall a key trigonometric identity the"}, {"start": 60.239, "duration": 8.961, "text": "expression 1 - Tang 2 T over 1 + tangent"}, {"start": 65.439, "duration": 5.881, "text": "2 T is equal to cosine of 2T this"}, {"start": 69.2, "duration": 6.88, "text": "simplifies our expression to cosine"}, {"start": 71.32, "duration": 7.119, "text": "inverse of cosine 2T which equals 2T now"}, {"start": 76.08, "duration": 5.84, "text": "we need to differentiate 2 t with"}, {"start": 78.439, "duration": 5.64, "text": "respect to X remember that t equals"}, {"start": 81.92, "duration": 4.96, "text": "inverse tangent of X so we'll use the"}, {"start": 84.079, "duration": 9.761, "text": "chain rule the derivative of inverse"}, {"start": 86.88, "duration": 11.16, "text": "tangent of X is 1/ 1 + x^2 therefore our"}, {"start": 93.84, "duration": 7.319, "text": "final answer is 2 over 1 + x^2 let's"}, {"start": 98.04, "duration": 6.92, "text": "verify this makes sense our answer is"}, {"start": 101.159, "duration": 6.801, "text": "always defined except when x = + or"}, {"start": 104.96, "duration": 5.439, "text": "minus1 which is exactly where our"}, {"start": 107.96, "duration": 5.479, "text": "original expression would be undefined"}, {"start": 110.399, "duration": 6.72, "text": "due to division by 0o in the denominator"}, {"start": 113.439, "duration": 3.68, "text": "1us x^2"}], "title": "Derivative of Inverse Secant: Step-by-Step Explanation [\ud835\udc60\ud835\udc52\ud835\udc50^(\u22121)   (1+\ud835\udc65^2)/(1\u2212\ud835\udc65^2 )]", "description": "Unlock the secrets of calculus with our detailed step-by-step guide on differentiating the inverse secant function! This video breaks down the process of finding the derivative of [\ud835\udc60\ud835\udc52\ud835\udc50^(\u22121)   (1+\ud835\udc65^2)/(1\u2212\ud835\udc65^2 )] using trigonometric identities and substitution methods. Follow along for a clear and insightful explanation that will enhance your understanding of calculus concepts.\n#Calculus #Derivative #InverseSecant #Trigonometry #MathTutorial", "lengthSeconds": 120, "width": 640, "height": 360}
{"video_id": "nbEabAEA_ng", "transcript": [{"start": 0.24, "duration": 4.72, "text": "let's solve this calculus problem step"}, {"start": 2.399, "duration": 5.681, "text": "by step we need to find the derivative"}, {"start": 4.96, "duration": 6.639, "text": "of x to the 5th power time log base of"}, {"start": 8.08, "duration": 6.92, "text": "of X the first step is to rewrite log"}, {"start": 11.599, "duration": 7.121, "text": "base uh of X using the change of Base"}, {"start": 15.0, "duration": 8.0, "text": "formula this states that log Bas uh of x"}, {"start": 18.72, "duration": 8.24, "text": "equals log base uh of e * log base e of"}, {"start": 23.0, "duration": 7.519, "text": "X next we can factor out log base uh of"}, {"start": 26.96, "duration": 7.56, "text": "e since it's a constant this gives log"}, {"start": 30.519, "duration": 7.801, "text": "base uh of e * the derivative of x to"}, {"start": 34.52, "duration": 6.719, "text": "5th power * natural log of x now we can"}, {"start": 38.32, "duration": 6.079, "text": "apply the product rule the product rule"}, {"start": 41.239, "duration": 6.561, "text": "states that the derivative of U * v = u"}, {"start": 44.399, "duration": 6.281, "text": "* the derivative of V plus b * the"}, {"start": 47.8, "duration": 5.919, "text": "derivative of U let's evaluate each"}, {"start": 50.68, "duration": 7.039, "text": "derivative separately the derivative of"}, {"start": 53.719, "duration": 7.921, "text": "natural log of x is 1 /x and the"}, {"start": 57.719, "duration": 7.041, "text": "derivative of x to the 5th power is 5X"}, {"start": 61.64, "duration": 6.68, "text": "4th power substituting these derivatives"}, {"start": 64.76, "duration": 9.84, "text": "back into our equation we get log base"}, {"start": 68.32, "duration": 14.24, "text": "of e * X to 5 * 1 /x plus natural log of"}, {"start": 74.6, "duration": 12.0, "text": "x * 5x 4th now let's simplify x 5 * 1 /x"}, {"start": 82.56, "duration": 9.519, "text": "simplifies to X 4th and then we have +"}, {"start": 86.6, "duration": 8.36, "text": "5x 4 * natural log of X for our final"}, {"start": 92.079, "duration": 6.561, "text": "step we can factor out x to the 4th"}, {"start": 94.96, "duration": 8.08, "text": "power this gives us X to 4th times the"}, {"start": 98.64, "duration": 8.079, "text": "quantity log base uh of e + 5 log base"}, {"start": 103.04, "duration": 6.719, "text": "of X and there we have it we found the"}, {"start": 106.719, "duration": 6.161, "text": "derivative of x to the 5th power * log"}, {"start": 109.759, "duration": 5.96, "text": "base a of X by using the change of Base"}, {"start": 112.88, "duration": 7.04, "text": "formula the product rule and some"}, {"start": 115.719, "duration": 4.201, "text": "careful algebraic manipulation"}], "title": "Finding the Derivative of \ud835\udc65^5.log_a(x)  Using the Product Rule | Step-by-Step Explanation.", "description": "In this video, learn how to find the derivative of \ud835\udc65^5.log_a(x)\n using the Product Rule in calculus. \ud83c\udf93\n\ud83d\udccc What You'll Learn in This Video:\nWhat is the Product Rule?\nStep-by-step solution to differentiate \ud835\udc65^5.log_a(x)\nFormula breakdown and example problem-solving.\nThis example is perfect for beginners in calculus or anyone reviewing the Product Rule for differentiation!\n\ud83d\udca1 Don't forget to LIKE, SHARE, and SUBSCRIBE for more math tutorials!\n#Calculus #ProductRule #Differentiation #Derivative #MathTutorial #LearnMath #MathEducation #MathHelp", "lengthSeconds": 121, "width": 640, "height": 360}
{"video_id": "lW6a7eK7La8", "transcript": [{"start": 0.28, "duration": 4.96, "text": "let's solve this differentiation problem"}, {"start": 2.56, "duration": 7.32, "text": "step by step we start with the"}, {"start": 5.24, "duration": 7.519, "text": "expression square root of 1 over e to X"}, {"start": 9.88, "duration": 5.919, "text": "step one let's rewrite this expression"}, {"start": 12.759, "duration": 7.561, "text": "in a form that's easier to differentiate"}, {"start": 15.799, "duration": 7.201, "text": "we can rewrite this as e tox to the^ of"}, {"start": 20.32, "duration": 6.279, "text": "12 this is because the square root is"}, {"start": 23.0, "duration": 8.279, "text": "the same as raising to the power of 12"}, {"start": 26.599, "duration": 8.001, "text": "and 1/ E to X can be written as e to X"}, {"start": 31.279, "duration": 5.881, "text": "step two now we'll use the chain rule to"}, {"start": 34.6, "duration": 4.799, "text": "differentiate this expression the chain"}, {"start": 37.16, "duration": 5.8, "text": "rule states that when differentiating U"}, {"start": 39.399, "duration": 7.84, "text": "to the N we get n * U the"}, {"start": 42.96, "duration": 8.36, "text": "nus1 time the derivative of U step three"}, {"start": 47.239, "duration": 8.64, "text": "let's identify our U and N here U is e"}, {"start": 51.32, "duration": 8.8, "text": "to thex and N is 12 step four let's"}, {"start": 55.879, "duration": 9.121, "text": "apply the chain rule we multiply 12 * e"}, {"start": 60.12, "duration": 9.999, "text": "thex to the power of -2 * the derivative"}, {"start": 65.0, "duration": 8.72, "text": "of e to thex which is e to thex step"}, {"start": 70.119, "duration": 6.881, "text": "five now let's simplify this expression"}, {"start": 73.72, "duration": 8.52, "text": "first we can rewrite e to thex to the"}, {"start": 77.0, "duration": 8.88, "text": "power of -2 as 1 over the < TK of e"}, {"start": 82.24, "duration": 4.6, "text": "tox then we can combine the terms with e"}, {"start": 85.88, "duration": 4.04, "text": "to"}, {"start": 86.84, "duration": 6.52, "text": "thex finally we combine the exponents"}, {"start": 89.92, "duration": 8.839, "text": "and simplify to get our final answer"}, {"start": 93.36, "duration": 8.84, "text": "-1/2 * the < TK of e to X and that's how"}, {"start": 98.759, "duration": 7.121, "text": "we differentiate the square TK of 1/ e"}, {"start": 102.2, "duration": 6.32, "text": "to X we use the chain Rule and careful"}, {"start": 105.88, "duration": 5.64, "text": "algebraic manipulation to arrive at our"}, {"start": 108.52, "duration": 3.0, "text": "answer"}], "title": "9.3-2.2 Mastering the Chain Rule: Differentiating (\u221a(1/\ud835\udc52^\ud835\udc65)  Step-by-Step", "description": "Unlock the secrets of differentiation with our in-depth tutorial on finding the derivative of (\u221a(1/\ud835\udc52^\ud835\udc65) . Follow along as we guide you through each step, using the chain rule to achieve a clear understanding of this complex mathematical process. Perfect for students and math enthusiasts looking to enhance their calculus skills!\nHashtags:\n#cpmrana #MathTutorial #Calculus #ChainRule #Differentiation #ExponentialFunctions #hsc #hschighermath #1stpaper #hschighermath1stpaper #englishversion", "lengthSeconds": 112, "width": 640, "height": 360}
{"video_id": "LIJ1OdmtKAs", "transcript": [{"start": 0.28, "duration": 4.64, "text": "let's work through this complex"}, {"start": 2.28, "duration": 5.28, "text": "derivative step by step we're going to"}, {"start": 4.92, "duration": 6.639, "text": "find the derivative of the square OT of"}, {"start": 7.56, "duration": 6.48, "text": "e to the power of square < TK X let's"}, {"start": 11.559, "duration": 4.401, "text": "break this down into its component parts"}, {"start": 14.04, "duration": 5.36, "text": "to better understand what we're dealing"}, {"start": 15.96, "duration": 6.399, "text": "with we have three nested functions here"}, {"start": 19.4, "duration": 6.0, "text": "an outer square root then an exponential"}, {"start": 22.359, "duration": 6.281, "text": "function e and finally another square"}, {"start": 25.4, "duration": 6.32, "text": "root of x to solve this we'll need to"}, {"start": 28.64, "duration": 5.36, "text": "use the chain rule multiple times the"}, {"start": 31.72, "duration": 5.4, "text": "chain rule tells us that when"}, {"start": 34.0, "duration": 5.079, "text": "differentiating composite functions we"}, {"start": 37.12, "duration": 3.88, "text": "multiply the derivative of the outer"}, {"start": 39.079, "duration": 4.041, "text": "function by the derivative of the inner"}, {"start": 41.0, "duration": 4.559, "text": "function let's start with the outer"}, {"start": 43.12, "duration": 6.32, "text": "function the derivative of square root"}, {"start": 45.559, "duration": 8.401, "text": "of U with respect to U is 1/ 2 * the"}, {"start": 49.44, "duration": 7.639, "text": "square < TK of U in our case U is e^ of"}, {"start": 53.96, "duration": 5.599, "text": "square < TK of X next we need to find"}, {"start": 57.079, "duration": 6.361, "text": "the derivative of the middle function"}, {"start": 59.559, "duration": 7.841, "text": "the derivative of e to the power of U"}, {"start": 63.44, "duration": 7.76, "text": "with respect to U is e to the power of U"}, {"start": 67.4, "duration": 7.719, "text": "but remember r u here is square < TK of"}, {"start": 71.2, "duration": 6.72, "text": "X finally let's find the derivative of"}, {"start": 75.119, "duration": 8.0, "text": "the innermost function the derivative of"}, {"start": 77.92, "duration": 6.839, "text": "square < TK X is 1 2 * the < TK X now"}, {"start": 83.119, "duration": 5.761, "text": "let's put it all together using the"}, {"start": 84.759, "duration": 7.921, "text": "chain rule we multiply all three parts 1"}, {"start": 88.88, "duration": 9.84, "text": "over 2 * theare < TK of e to power of <"}, {"start": 92.68, "duration": 9.759, "text": "TK x * E power of < TK x * 1/ 2 * the <"}, {"start": 98.72, "duration": 7.52, "text": "TK X finally let's simplify this"}, {"start": 102.439, "duration": 7.441, "text": "expression the numerator Becomes e^ of <"}, {"start": 106.24, "duration": 8.04, "text": "TK of X and the denominator becomes 4 *"}, {"start": 109.88, "duration": 8.68, "text": "the < TK of x * the < TK of e to power"}, {"start": 114.28, "duration": 7.479, "text": "of < TK X and that's our final answer we"}, {"start": 118.56, "duration": 5.919, "text": "use the chain rule multiple times to"}, {"start": 121.759, "duration": 5.64, "text": "break down this complex derivative into"}, {"start": 124.479, "duration": 6.76, "text": "manageable steps and then combined"}, {"start": 127.399, "duration": 3.84, "text": "everything back together"}], "title": "9.3-2.3 Mastering Calculus Differentiating Composite Functions (\u221a\ud835\udc52^\u221a\ud835\udc65)  with Chain Rule", "description": "In this video, we break down the process of differentiating complex composite functions using the chain rule. Learn how to tackle expressions involving square roots and exponential functions step by step. Perfect for students and anyone looking to strengthen their calculus skills!\n#hsc #hschighermath #1stpaper #hschighermath1stpaper\n#englishversion #Calculus #ChainRule #Derivatives #MathTutorial #CompositeFunctions", "lengthSeconds": 132, "width": 640, "height": 360}
{"video_id": "6NZ8wBlSvRk", "transcript": [{"start": 0.4, "duration": 5.08, "text": "welcome today we'll learn how to find"}, {"start": 2.96, "duration": 6.0, "text": "the derivative of a complex function"}, {"start": 5.48, "duration": 7.4, "text": "using logarithmic differentiation our"}, {"start": 8.96, "duration": 8.56, "text": "question is to find the derivative of y"}, {"start": 12.88, "duration": 10.2, "text": "= sin x raed to X power let's solve the"}, {"start": 17.52, "duration": 9.159, "text": "step by step first let's say y = sin x"}, {"start": 23.08, "duration": 5.92, "text": "raed to X power this is our original"}, {"start": 26.679, "duration": 5.321, "text": "function to use logarithmic"}, {"start": 29.0, "duration": 5.84, "text": "differentiation we take the natural log"}, {"start": 32.0, "duration": 5.399, "text": "of both sides using the logarithm"}, {"start": 34.84, "duration": 5.32, "text": "property that log of a power equals the"}, {"start": 37.399, "duration": 7.32, "text": "power time the log of the base we can"}, {"start": 40.16, "duration": 6.76, "text": "simplify this to x * Ln of sinx now we"}, {"start": 44.719, "duration": 5.601, "text": "can differentiate both sides with"}, {"start": 46.92, "duration": 6.0, "text": "respect to X on the right side we need"}, {"start": 50.32, "duration": 5.239, "text": "to use the product rule since we have a"}, {"start": 52.92, "duration": 5.88, "text": "product of two terms applying the"}, {"start": 55.559, "duration": 8.16, "text": "product rule we get x * the derivative"}, {"start": 58.8, "duration": 8.48, "text": "of Ln sinx X Plus Ln sign x * the"}, {"start": 63.719, "duration": 7.481, "text": "derivative of x on the left side we use"}, {"start": 67.28, "duration": 8.92, "text": "the chain rule the derivative of Ln Y is"}, {"start": 71.2, "duration": 7.08, "text": "1/ y * Dy DX on the right side we use"}, {"start": 76.2, "duration": 6.0, "text": "the chain rule again for the derivative"}, {"start": 78.28, "duration": 8.24, "text": "of Ln sinx which gives us cin X over"}, {"start": 82.2, "duration": 5.559, "text": "sinx now we can multiply both sides by y"}, {"start": 86.52, "duration": 5.52, "text": "to isolate"}, {"start": 87.759, "duration": 8.481, "text": "dydx and we can substitute back y = sin"}, {"start": 92.04, "duration": 9.24, "text": "x to X power finally we can recognize"}, {"start": 96.24, "duration": 7.159, "text": "that cine x/ sinx is the coent of X and"}, {"start": 101.28, "duration": 5.32, "text": "there we have it we've found the"}, {"start": 103.399, "duration": 5.36, "text": "derivative using logarithmic"}, {"start": 106.6, "duration": 4.08, "text": "differentiation this method is"}, {"start": 108.759, "duration": 4.801, "text": "particularly useful when dealing with"}, {"start": 110.68, "duration": 6.16, "text": "functions that have variable exponents"}, {"start": 113.56, "duration": 6.36, "text": "the key steps were one taking the"}, {"start": 116.84, "duration": 6.12, "text": "natural log of both sides two using"}, {"start": 119.92, "duration": 7.32, "text": "logarithm properties 3 using the product"}, {"start": 122.96, "duration": 6.519, "text": "rule 4 applying the chain Rule 5 and"}, {"start": 127.24, "duration": 5.239, "text": "finally substituting back our original"}, {"start": 129.479, "duration": 3.0, "text": "function"}], "title": "Mastering Logarithmic Differentiation: Solving [(sin x)^x] Step by Step", "description": "QUESTION:\nFind the derivative of y=[(sin x)^x]   using logarithmic differentiation.\n\nLearn how to find the derivative of (sin x)^x using logarithmic differentiation in this comprehensive calculus tutorial. We break down this challenging problem into manageable steps, showing you:\n\u2022 How to apply natural logarithms strategically\n\u2022 Using logarithm properties effectively\n\u2022 Implementing the product rule and chain rule\n\u2022 Working with trigonometric functions\n\u2022 Simplifying using trigonometric identities\nPerfect for calculus students and anyone preparing for AP Calculus, IB Mathematics, or college-level calculus courses.\n#Calculus #Mathematics #LogarithmicDifferentiation #ChainRule #ProductRule #MathTutorial #CalulusHelp #APCalculus #IBMaths #MathEducation", "lengthSeconds": 135, "width": 640, "height": 360}
{"video_id": "_j7NlSFAvUM", "transcript": [{"start": 0.359, "duration": 5.121, "text": "welcome today we'll solve a challenging"}, {"start": 3.24, "duration": 4.88, "text": "derivative problem involving inverse"}, {"start": 5.48, "duration": 5.72, "text": "tangent let's find the derivative of"}, {"start": 8.12, "duration": 6.8, "text": "this complex expression let's break this"}, {"start": 11.2, "duration": 7.32, "text": "down step by step first let's let y"}, {"start": 14.92, "duration": 6.32, "text": "equal our expression and try to simplify"}, {"start": 18.52, "duration": 5.2, "text": "what's inside the inverse tangent our"}, {"start": 21.24, "duration": 5.4, "text": "first step is to factor out from both"}, {"start": 23.72, "duration": 6.36, "text": "the numerator and denominator we can"}, {"start": 26.64, "duration": 8.8, "text": "rewrite the fraction as a Time 1 + BX"}, {"start": 30.08, "duration": 7.44, "text": "over a / a Time 1 - bx/ A the A's cancel"}, {"start": 35.44, "duration": 4.88, "text": "out in the fraction giving us this"}, {"start": 37.52, "duration": 6.039, "text": "simpler form now here's where we use a"}, {"start": 40.32, "duration": 5.559, "text": "key Insight this form can be broken down"}, {"start": 43.559, "duration": 5.0, "text": "using the inverse tangent addition"}, {"start": 45.879, "duration": 5.52, "text": "formula when we have a fraction in this"}, {"start": 48.559, "duration": 6.441, "text": "form it equals the sum of inverse"}, {"start": 51.399, "duration": 6.441, "text": "tangent of 1+ inverse tangent of the"}, {"start": 55.0, "duration": 5.519, "text": "remaining term we know that inverse"}, {"start": 57.84, "duration": 6.359, "text": "tangent of 1 = pi"}, {"start": 60.519, "duration": 6.121, "text": "4 so we can simplify this further now we"}, {"start": 64.199, "duration": 5.92, "text": "can differentiate the derivative of a"}, {"start": 66.64, "duration": 6.159, "text": "constant pi/ 4 is zero and for the"}, {"start": 70.119, "duration": 6.0, "text": "inverse tangent term we'll use the chain"}, {"start": 72.799, "duration": 7.36, "text": "rule let's evaluate the derivative of BX"}, {"start": 76.119, "duration": 7.401, "text": "over a which is simply B over a now"}, {"start": 80.159, "duration": 6.121, "text": "we'll work on simplifying this fraction"}, {"start": 83.52, "duration": 5.12, "text": "first let's get a common denominator in"}, {"start": 86.28, "duration": 4.479, "text": "the first fraction when we have a"}, {"start": 88.64, "duration": 4.88, "text": "complex fraction like this this we can"}, {"start": 90.759, "duration": 6.121, "text": "flip the denominator and"}, {"start": 93.52, "duration": 6.2, "text": "multiply finally we multiply these"}, {"start": 96.88, "duration": 5.76, "text": "fractions together and get our final"}, {"start": 99.72, "duration": 5.52, "text": "answer therefore the derivative of"}, {"start": 102.64, "duration": 8.839, "text": "inverse tangent of a plus BX over a"}, {"start": 105.24, "duration": 7.919, "text": "minus BX is equal to a / a 2 + B ^2 x^2"}, {"start": 111.479, "duration": 4.081, "text": "this was a challenging problem that"}, {"start": 113.159, "duration": 5.481, "text": "required us to use the inverse tangent"}, {"start": 115.56, "duration": 5.879, "text": "addition formula and careful algebraic"}, {"start": 118.64, "duration": 7.119, "text": "manipulation the the key was breaking it"}, {"start": 121.439, "duration": 4.32, "text": "down into manageable steps"}], "title": "Calculus Explained: Derivative of Inverse Tangent Function [\ud835\udc61\ud835\udc4e\ud835\udc5b^(\u22121)\u2061((\ud835\udc4e+\ud835\udc4f\ud835\udc65)/(\ud835\udc4e\u2212\ud835\udc4f\ud835\udc65))]", "description": "Question:\nFind the derivative of y=[\ud835\udc61\ud835\udc4e\ud835\udc5b^(\u22121)\u2061((\ud835\udc4e+\ud835\udc4f\ud835\udc65)/(\ud835\udc4e\u2212\ud835\udc4f\ud835\udc65))].\n\nDive into calculus with this step-by-step guide on differentiating the function y=[\ud835\udc61\ud835\udc4e\ud835\udc5b^(\u22121)\u2061((\ud835\udc4e+\ud835\udc4f\ud835\udc65)/(\ud835\udc4e\u2212\ud835\udc4f\ud835\udc65))]. Learn how to use trigonometric identities and the chain rule to find the derivative efficiently. Perfect for students and math enthusiasts looking to enhance their understanding of advanced calculus concepts.\n#Calculus #Derivative #Trigonometry #MathTutorial #InverseTangent #ChainRule", "lengthSeconds": 128, "width": 640, "height": 360}
{"video_id": "q12BW5ZLGa0", "transcript": [{"start": 0.28, "duration": 5.4, "text": "let's solve this calculus problem step"}, {"start": 2.6, "duration": 8.199, "text": "by step we need to find the derivative"}, {"start": 5.68, "duration": 8.24, "text": "of f ofx = x to the n * natural log of"}, {"start": 10.799, "duration": 6.001, "text": "2x to solve this we'll use the product"}, {"start": 13.92, "duration": 7.359, "text": "rule the product rule states that the"}, {"start": 16.8, "duration": 8.399, "text": "derivative of U * v = u * the derivative"}, {"start": 21.279, "duration": 8.361, "text": "of V + V * the derivative of U in our"}, {"start": 25.199, "duration": 8.441, "text": "case u = x to the n and V equal natural"}, {"start": 29.64, "duration": 7.36, "text": "laog 2x let's apply the product rule we"}, {"start": 33.64, "duration": 8.439, "text": "start with x n * the derivative of"}, {"start": 37.0, "duration": 8.96, "text": "natural log of 2x plus natural log of 2x"}, {"start": 42.079, "duration": 7.0, "text": "* the derivative of x to the N now let's"}, {"start": 45.96, "duration": 7.079, "text": "find each derivative separately the"}, {"start": 49.079, "duration": 8.681, "text": "derivative of natural log of 2x is 1 /"}, {"start": 53.039, "duration": 9.561, "text": "2x * 2 which simplifies to 1 /x and the"}, {"start": 57.76, "duration": 8.32, "text": "derivative of x to the N is n * x n -1"}, {"start": 62.6, "duration": 10.28, "text": "substituting these back in we get x n *"}, {"start": 66.08, "duration": 11.84, "text": "1 2x * 2 plus natural log of 2x * NX n"}, {"start": 72.88, "duration": 10.12, "text": "-1 let's simplify the first term x n * 2"}, {"start": 77.92, "duration": 9.36, "text": "/ 2x simplifies to X N /x which is X N"}, {"start": 83.0, "duration": 7.92, "text": "-1 finally we can factor out x to the"}, {"start": 87.28, "duration": 7.28, "text": "nus1 from both terms this gives us our"}, {"start": 90.92, "duration": 8.96, "text": "final answer x to the nus 1 * the"}, {"start": 94.56, "duration": 7.44, "text": "quantity 1 + n * natural log of 2x to"}, {"start": 99.88, "duration": 4.4, "text": "verify this is correct we can check that"}, {"start": 102.0, "duration": 5.2, "text": "our answer contains all the terms we"}, {"start": 104.28, "duration": 5.0, "text": "expect from the product rule and that we"}, {"start": 107.2, "duration": 5.239, "text": "follow the chain rule correctly when"}, {"start": 109.28, "duration": 7.04, "text": "differentiating natural log of 2x this"}, {"start": 112.439, "duration": 3.881, "text": "completes our solution"}], "title": "Mastering Calculus: Differentiating (\ud835\udc65^\ud835\udc5b  ln\u20612\ud835\udc65 )  Step-by-Step", "description": "Unlock the secrets of calculus with our detailed step-by-step guide on differentiating (\ud835\udc65^\ud835\udc5b  ln\u20612\ud835\udc65 )  . In this video, we\u2019ll walk you through the process using the product and chain rules, ensuring you understand each step. Perfect for students and calculus enthusiasts aiming to deepen their understanding. Subscribe for more math insights!\n#Calculus\n#MathTutorial\n#Differentiation\n#LearnMath\n#StudyHelp", "lengthSeconds": 119, "width": 640, "height": 360}
{"video_id": "DAj5s4OVxsA", "transcript": [{"start": 0.44, "duration": 5.0, "text": "welcome today we'll solve a challenging"}, {"start": 3.12, "duration": 3.88, "text": "differentiation problem using"}, {"start": 5.44, "duration": 4.64, "text": "logarithmic"}, {"start": 7.0, "duration": 8.679, "text": "differentiation our task is to find the"}, {"start": 10.08, "duration": 8.359, "text": "derivative of y = s < TK X raed to the^"}, {"start": 15.679, "duration": 5.041, "text": "of square otk of X when we see an"}, {"start": 18.439, "duration": 5.961, "text": "expression where the variable appears in"}, {"start": 20.72, "duration": 5.68, "text": "both the base and exponent logarithmic"}, {"start": 24.4, "duration": 4.76, "text": "differentiation is often the best"}, {"start": 26.4, "duration": 4.32, "text": "approach let's start by letting y equal"}, {"start": 29.16, "duration": 4.239, "text": "our expression"}, {"start": 30.72, "duration": 5.64, "text": "and then take the natural logarithm of"}, {"start": 33.399, "duration": 5.16, "text": "both sides using the logarithm property"}, {"start": 36.36, "duration": 4.6, "text": "that states the log of a power equals"}, {"start": 38.559, "duration": 5.801, "text": "the exponent time the log of the base we"}, {"start": 40.96, "duration": 6.8, "text": "can rewrite this as Square < TK of x *"}, {"start": 44.36, "duration": 5.28, "text": "the natural log of square < TK of X now"}, {"start": 47.76, "duration": 5.4, "text": "we can simplify the natural log of"}, {"start": 49.64, "duration": 6.559, "text": "square < TK of X remember the natural"}, {"start": 53.16, "duration": 7.52, "text": "log of x to any power equals that power"}, {"start": 56.199, "duration": 10.24, "text": "time the natural log of x so Ln of s T"}, {"start": 60.68, "duration": 9.0, "text": "of x = 12 Ln of X here's where the magic"}, {"start": 66.439, "duration": 5.72, "text": "of logarithmic differentiation comes in"}, {"start": 69.68, "duration": 6.28, "text": "we can now differentiate both sides with"}, {"start": 72.159, "duration": 5.28, "text": "respect to X on the left side we get 1"}, {"start": 75.96, "duration": 4.72, "text": "over y *"}, {"start": 77.439, "duration": 5.121, "text": "dydx on the right side we need to use"}, {"start": 80.68, "duration": 4.28, "text": "the product rule since we're"}, {"start": 82.56, "duration": 6.32, "text": "differentiating the product of square"}, {"start": 84.96, "duration": 6.519, "text": "root of x and natural log of x this"}, {"start": 88.88, "duration": 7.12, "text": "gives a square root of x * the"}, {"start": 91.479, "duration": 7.801, "text": "derivative of Ln X Plus Ln x * the"}, {"start": 96.0, "duration": 5.479, "text": "derivative of < TK of X now let's"}, {"start": 99.28, "duration": 5.76, "text": "evaluate these derivatives the"}, {"start": 101.479, "duration": 7.561, "text": "derivative of lnx is 1 /x and the"}, {"start": 105.04, "duration": 8.359, "text": "derivative of sare < TK X is 1 / 2 < TK"}, {"start": 109.04, "duration": 9.679, "text": "X simplifying the right side we get 1/ 2"}, {"start": 113.399, "duration": 8.441, "text": "* the sum of 1 / < TK of x + Ln x/ 2 <"}, {"start": 118.719, "duration": 3.121, "text": "TK X"}, {"start": 121.92, "duration": 9.8, "text": "to solve for dydx we multiply both sides"}, {"start": 125.759, "duration": 8.361, "text": "by y remembering that y = < TK X to^ of"}, {"start": 131.72, "duration": 2.4, "text": "< TK"}, {"start": 134.2, "duration": 5.36, "text": "X after some algebra we can combine like"}, {"start": 137.239, "duration": 8.36, "text": "terms and simplify this to get our final"}, {"start": 139.56, "duration": 11.72, "text": "answer 1/4 * < TK X to the^ of < TK of x"}, {"start": 145.599, "duration": 5.681, "text": "-1 * 2 + Ln X"}, {"start": 151.319, "duration": 5.681, "text": "and there we have it we've successfully"}, {"start": 153.72, "duration": 5.32, "text": "found the derivative using logarithmic"}, {"start": 157.0, "duration": 4.56, "text": "differentiation this technique is"}, {"start": 159.04, "duration": 5.199, "text": "particularly useful when dealing with"}, {"start": 161.56, "duration": 6.8, "text": "Expressions involving variables in both"}, {"start": 164.239, "duration": 4.121, "text": "the base and exponent"}], "title": "Mastering Logarithmic Differentiation: Solving (\u221ax)^(\u221ax) Step by Step | Advanced Calculus", "description": "Question: Find the derivative of y = (\u221ax)^(\u221ax) using logarithmic differentiation.\n\nLearn how to find the derivative of (\u221ax)^(\u221ax) using logarithmic differentiation in this comprehensive tutorial. Perfect for calculus students and math enthusiasts!\nIn this video, we break down:\n\u2022 Why logarithmic differentiation is necessary\n\u2022 Step-by-step solution process\n\u2022 Key logarithm properties\n\u2022 Product rule application\n\u2022 Final simplification techniques\n\ud83d\udcda Prerequisites:\nBasic differentiation rules\nNatural logarithms\nChain rule\nProduct rule\n#calculus #mathematics #logarithms #derivatives #mathtutorial #calculus101 #mathhelp #education", "lengthSeconds": 171, "width": 640, "height": 360}
{"video_id": "cxuT_1dOtHs", "transcript": [{"start": 0.32, "duration": 4.92, "text": "let's solve this differentiation problem"}, {"start": 2.76, "duration": 6.16, "text": "step by step we need to find the"}, {"start": 5.24, "duration": 7.479, "text": "derivative of x * the square root of"}, {"start": 8.92, "duration": 6.679, "text": "sinx with respect to X since this is a"}, {"start": 12.719, "duration": 6.361, "text": "product of two functions we'll use the"}, {"start": 15.599, "duration": 7.44, "text": "product rule remember the product rule"}, {"start": 19.08, "duration": 8.48, "text": "states that the derivative of U * v = u"}, {"start": 23.039, "duration": 9.2, "text": "Prime * V + U * V Prime in our case u ="}, {"start": 27.56, "duration": 8.08, "text": "x and V = the sare < TK of sinx let's"}, {"start": 32.239, "duration": 5.801, "text": "apply the product rule to our expression"}, {"start": 35.64, "duration": 6.399, "text": "first we'll find U Prime which is the"}, {"start": 38.04, "duration": 6.839, "text": "derivative of x that's simply one then"}, {"start": 42.039, "duration": 6.561, "text": "we multiply that by V which is square <"}, {"start": 44.879, "duration": 7.601, "text": "TK of sin x so our first term is square"}, {"start": 48.6, "duration": 7.32, "text": "otk of sin x for the second term we need"}, {"start": 52.48, "duration": 6.48, "text": "you which is x * V Prime which is the"}, {"start": 55.92, "duration": 5.279, "text": "derivative of square < TK of sinx to"}, {"start": 58.96, "duration": 5.08, "text": "find this we'll need to use the chain"}, {"start": 61.199, "duration": 8.721, "text": "rule the derivative of square < TK of U"}, {"start": 64.04, "duration": 10.399, "text": "is 1 / 2 < TK of U * U Prime so our"}, {"start": 69.92, "duration": 8.04, "text": "second term will be x * 1 / 2 < TK sin x"}, {"start": 74.439, "duration": 7.32, "text": "* the derivative of sin x and we know"}, {"start": 77.96, "duration": 7.68, "text": "the derivative of sinx is cine X now we"}, {"start": 81.759, "duration": 10.0, "text": "can combine our terms we have < TK sin x"}, {"start": 85.64, "duration": 9.32, "text": "+ x * 1 / 2 < TK sin x * Co cosin X and"}, {"start": 91.759, "duration": 8.121, "text": "finally we can write this as sare < TK"}, {"start": 94.96, "duration": 7.64, "text": "sin x + x cosine x / 2 < TK of sinx this"}, {"start": 99.88, "duration": 5.32, "text": "is our final answer let's review what we"}, {"start": 102.6, "duration": 5.4, "text": "did we used the product rule to split"}, {"start": 105.2, "duration": 5.279, "text": "the derivative into two terms then use"}, {"start": 108.0, "duration": 5.88, "text": "the chain rule to find the derivative of"}, {"start": 110.479, "duration": 7.441, "text": "square otk of sinx and finally combine"}, {"start": 113.88, "duration": 4.04, "text": "our terms to get our answer"}], "title": "Mastering Derivatives (\ud835\udc65\u221a(sin\u2061\ud835\udc65 )): Step-by-Step Product and Chain Rule Explained", "description": "Unlock the secrets of calculus with our comprehensive video on differentiating the expression (\ud835\udc65\u221a(sin\u2061\ud835\udc65 )) . Follow along as we break down each step using the product and chain rules. Perfect for students and math enthusiasts looking to sharpen their skills! \ud83d\udcda\u270f\ufe0f\n#Calculus #MathTutorial #Differentiation #LearnMath #ProductRule #ChainRule", "lengthSeconds": 121, "width": 640, "height": 360}
{"video_id": "GLGDG86sm7M", "transcript": [{"start": 0.28, "duration": 4.88, "text": "let's solve this calculus problem step"}, {"start": 2.52, "duration": 6.36, "text": "by step the question asks us to"}, {"start": 5.16, "duration": 7.16, "text": "differentiate the function f of xal 3"}, {"start": 8.88, "duration": 6.56, "text": "raised to the inverse s of X let's break"}, {"start": 12.32, "duration": 5.56, "text": "this down using a step-by-step approach"}, {"start": 15.44, "duration": 5.64, "text": "we'll use a two column format to show"}, {"start": 17.88, "duration": 5.76, "text": "both the mathematical steps and their"}, {"start": 21.08, "duration": 5.16, "text": "explanations first let's identify what"}, {"start": 23.64, "duration": 4.719, "text": "we're dealing with this is a composite"}, {"start": 26.24, "duration": 5.24, "text": "function where we have three Rays to a"}, {"start": 28.359, "duration": 5.72, "text": "power and that power is itself the"}, {"start": 31.48, "duration": 5.36, "text": "inverse sign of X we'll need to use the"}, {"start": 34.079, "duration": 4.961, "text": "chain rule here the chain rule tells us"}, {"start": 36.84, "duration": 5.68, "text": "that when differentiating an exponential"}, {"start": 39.04, "duration": 6.6, "text": "function a to the U we get a to the U *"}, {"start": 42.52, "duration": 6.76, "text": "the natural log of times the derivative"}, {"start": 45.64, "duration": 6.439, "text": "of U applying this to our problem we get"}, {"start": 49.28, "duration": 6.52, "text": "3 to the inverse sin of X time the"}, {"start": 52.079, "duration": 7.241, "text": "natural log of 3 times the derivative of"}, {"start": 55.8, "duration": 6.439, "text": "inverse sin of X now we need to recall"}, {"start": 59.32, "duration": 8.32, "text": "that the Der ative of inverse sin of X"}, {"start": 62.239, "duration": 8.281, "text": "is 1 over the < TK of 1 - x^2"}, {"start": 67.64, "duration": 6.88, "text": "substituting this in we get our final"}, {"start": 70.52, "duration": 7.04, "text": "answer 3 to the inverse sign of x *"}, {"start": 74.52, "duration": 7.44, "text": "natural log of 3 all over the square <"}, {"start": 77.56, "duration": 6.76, "text": "TK of 1 - x^2 let's verify our answer"}, {"start": 81.96, "duration": 4.119, "text": "makes sense the domain of this"}, {"start": 84.32, "duration": 5.88, "text": "derivative will be the same as the"}, {"start": 86.079, "duration": 7.521, "text": "domain of inverse sign which is -1 to to"}, {"start": 90.2, "duration": 6.76, "text": "positive 1 also notice that as X"}, {"start": 93.6, "duration": 5.799, "text": "approaches either 1 or - 1 our"}, {"start": 96.96, "duration": 4.72, "text": "denominator approaches zero which means"}, {"start": 99.399, "duration": 5.68, "text": "the derivative grows very large at these"}, {"start": 101.68, "duration": 6.399, "text": "points just as we would expect and that"}, {"start": 105.079, "duration": 6.441, "text": "completes our solution remember the key"}, {"start": 108.079, "duration": 6.4, "text": "steps identify the composite function"}, {"start": 111.52, "duration": 7.559, "text": "apply the chain Rule and recall the"}, {"start": 114.479, "duration": 4.6, "text": "derivative of inverse sign"}], "title": "Differentiating Exponential Functions Solving [3^(\ud835\udc60\ud835\udc56\ud835\udc5b^\u22121 \ud835\udc65)]", "description": "Learn how to differentiate the function [3^(\ud835\udc60\ud835\udc56\ud835\udc5b^\u22121 \ud835\udc65)] step-by-step in this comprehensive tutorial. We\u2019ll guide you through applying the chain rule and simplifying complex expressions involving inverse trigonometric functions. Mastering these calculus skills is essential for any math enthusiast or student.\n#Calculus #Differentiation #MathTutorial #ExponentialFunctions #ChainRule", "lengthSeconds": 122, "width": 640, "height": 360}
{"video_id": "ZuBq1v4k-J4", "transcript": [{"start": 0.32, "duration": 5.32, "text": "let's solve this challenging derivative"}, {"start": 2.679, "duration": 7.04, "text": "problem step by step we'll find the"}, {"start": 5.64, "duration": 5.72, "text": "derivative of inverse tangent of 4X over"}, {"start": 9.719, "duration": 4.481, "text": "1 -"}, {"start": 11.36, "duration": 5.36, "text": "4x^2 this is a complex expression that"}, {"start": 14.2, "duration": 5.36, "text": "would be difficult to differentiate"}, {"start": 16.72, "duration": 7.479, "text": "directly let's try a substitution"}, {"start": 19.56, "duration": 9.68, "text": "approach let's say that 2x = tangent of"}, {"start": 24.199, "duration": 6.48, "text": "t therefore t equal inverse tangent of"}, {"start": 29.24, "duration": 3.319, "text": "2x"}, {"start": 30.679, "duration": 6.161, "text": "now let's look at our original"}, {"start": 32.559, "duration": 6.881, "text": "expression we can rewrite 4X over 1 -"}, {"start": 36.84, "duration": 5.239, "text": "4x^2 as 2 *"}, {"start": 39.44, "duration": 6.92, "text": "2x/ 1 -"}, {"start": 42.079, "duration": 9.8, "text": "2x^2 now we can substitute tangent t for"}, {"start": 46.36, "duration": 9.199, "text": "2x this gives us 2 * tangent t/ 1 - Tang"}, {"start": 51.879, "duration": 6.441, "text": "2 T this is a key Insight this"}, {"start": 55.559, "duration": 6.041, "text": "expression is actually equal to tangent"}, {"start": 58.32, "duration": 6.239, "text": "of 2T this is the double angle formula"}, {"start": 61.6, "duration": 8.159, "text": "for tangent so our original expression"}, {"start": 64.559, "duration": 8.481, "text": "becomes inverse tangent of tangent of 2T"}, {"start": 69.759, "duration": 6.0, "text": "which simplifies to Just 2 T now"}, {"start": 73.04, "duration": 6.8, "text": "remember that t equals inverse tangent"}, {"start": 75.759, "duration": 8.281, "text": "of 2x so we can substitute back to get 2"}, {"start": 79.84, "duration": 6.8, "text": "* inverse tangent of 2x now we can"}, {"start": 84.04, "duration": 6.439, "text": "differentiate this expression using the"}, {"start": 86.64, "duration": 8.119, "text": "chain rule we get 2 * the derivative"}, {"start": 90.479, "duration": 9.28, "text": "of inverse tangent of 2x the derivative"}, {"start": 94.759, "duration": 7.121, "text": "of inverse tangent is 1/ 1 + x^2 and we"}, {"start": 99.759, "duration": 5.801, "text": "need to multiply by the derivative of"}, {"start": 101.88, "duration": 7.96, "text": "the inside function this simplifies to 4"}, {"start": 105.56, "duration": 6.76, "text": "over 1 + 4x^2 which is our final answer"}, {"start": 109.84, "duration": 4.48, "text": "this problem demonstrates the power of"}, {"start": 112.32, "duration": 4.759, "text": "substitution and trigonometric"}, {"start": 114.32, "duration": 6.079, "text": "identities in calculus by making a"}, {"start": 117.079, "duration": 6.161, "text": "clever substitution we transform a"}, {"start": 120.399, "duration": 5.841, "text": "complex derivative into a much simpler"}, {"start": 123.24, "duration": 3.0, "text": "one"}], "title": "How to Calculate the Derivative of [\ud835\udc61\ud835\udc4e\ud835\udc5b^(\u22121)   (4\ud835\udc65 )/(1\u22124\ud835\udc65^2 )] - Step-by-Step Explanation!", "description": "Question\nFind the derivative of [\ud835\udc61\ud835\udc4e\ud835\udc5b^(\u22121)   (4\ud835\udc65 )/(1\u22124\ud835\udc65^2 )] .\nUnlock the secrets to finding the derivative of  [\ud835\udc61\ud835\udc4e\ud835\udc5b^(\u22121)   (4\ud835\udc65 )/(1\u22124\ud835\udc65^2 )]  with this in-depth tutorial! We\u2019ll guide you through each math step and provide clear explanations, ensuring you grasp the concept fully. Perfect for high school and college students tackling calculus! Don\u2019t forget to like, subscribe, and comment with any questions.\n#Calculus #Derivatives #InverseTangent #MathTutorial #StepByStepMath", "lengthSeconds": 129, "width": 640, "height": 360}
{"video_id": "jvQElSyYctE", "transcript": [{"start": 0.28, "duration": 5.039, "text": "let's solve this calculus problem step"}, {"start": 2.399, "duration": 5.321, "text": "by step using the quotient rule we need"}, {"start": 5.319, "duration": 8.2, "text": "to find the derivative of the function f"}, {"start": 7.72, "duration": 9.28, "text": "of x = tangent x + Cent X all divided 3"}, {"start": 13.519, "duration": 6.961, "text": "e to X first let's recall the quotient"}, {"start": 17.0, "duration": 6.88, "text": "rule the derivative of U over V equal V"}, {"start": 20.48, "duration": 7.08, "text": "* the derivative of U minus U * the"}, {"start": 23.88, "duration": 8.6, "text": "derivative of V all over V ^2 let's"}, {"start": 27.56, "duration": 11.56, "text": "identify our u and v fun funs here U ="}, {"start": 32.48, "duration": 8.72, "text": "tangent x + Cent X and V = 3 e to X now"}, {"start": 39.12, "duration": 5.959, "text": "let's find their derivatives the"}, {"start": 41.2, "duration": 7.16, "text": "derivative of tangent X is secant 2 x"}, {"start": 45.079, "duration": 7.16, "text": "and the derivative of Cent X is Nega"}, {"start": 48.36, "duration": 9.64, "text": "cose 2 x so the derivative of U is"}, {"start": 52.239, "duration": 10.16, "text": "secant 2 xus cose 2 x the derivative of"}, {"start": 58.0, "duration": 6.72, "text": "V is simply 3 e to the X now let's"}, {"start": 62.399, "duration": 6.4, "text": "substitute everything into the quotient"}, {"start": 64.72, "duration": 11.079, "text": "rule formula we get 3 e to the x *"}, {"start": 68.799, "duration": 12.201, "text": "secant 2 x - cose 2 x - tangent x + Cent"}, {"start": 75.799, "duration": 7.801, "text": "x * 3 e to X all over 3 e x^2 let's"}, {"start": 81.0, "duration": 7.119, "text": "simplify the numerator first we can"}, {"start": 83.6, "duration": 10.479, "text": "Factor Out 3 e to X this gives us 3 e to"}, {"start": 88.119, "duration": 11.96, "text": "x * the quantity C 2 x - cose 2 x -"}, {"start": 94.079, "duration": 10.921, "text": "tangent x - x all over 3 e to x^2 next"}, {"start": 100.079, "duration": 11.68, "text": "recall that secant 2 x = 1 + Tang 2 x"}, {"start": 105.0, "duration": 12.52, "text": "and cose 2 x = 1 + cang 2 x substituting"}, {"start": 111.759, "duration": 11.121, "text": "these in we get tangent s x - Cent 2 x -"}, {"start": 117.52, "duration": 9.599, "text": "tangent x - cang X finally we can Factor"}, {"start": 122.88, "duration": 7.799, "text": "this expression notice that tangent x +"}, {"start": 127.119, "duration": 7.4, "text": "Cent X is a common factor when we Factor"}, {"start": 130.679, "duration": 10.56, "text": "it out we get tangent x + Cent x *"}, {"start": 134.519, "duration": 9.561, "text": "tangent x - Cent x -1 all over 3 e to X"}, {"start": 141.239, "duration": 5.72, "text": "to verify this is correct we can check"}, {"start": 144.08, "duration": 5.48, "text": "that this matches our original problem's"}, {"start": 146.959, "duration": 6.081, "text": "derivative the quotient rule has helped"}, {"start": 149.56, "duration": 7.44, "text": "us transform a complex fraction into a"}, {"start": 153.04, "duration": 3.96, "text": "more manageable form"}], "title": "Differentiating Trigonometric Functions: Mastering [(\ud835\udc95\ud835\udc82\ud835\udc8f \ud835\udc99+ \ud835\udc84\ud835\udc90\ud835\udc95 \ud835\udc99)/(\ud835\udfd1\ud835\udc86^\ud835\udc99)] with the Quotient Rule", "description": "Get ready to enhance your calculus skills by learning how to differentiate the complex expression [(\ud835\udc95\ud835\udc82\ud835\udc8f \ud835\udc99+ \ud835\udc84\ud835\udc90\ud835\udc95 \ud835\udc99)/(\ud835\udfd1\ud835\udc86^\ud835\udc99)]  using the powerful quotient rule. This tutorial is perfect for students tackling advanced calculus topics, offering clear, step-by-step guidance to help you understand the intricacies of trigonometric derivatives.\n#Calculus #QuotientRule #Trigonometry #MathSkills #Differentiation", "lengthSeconds": 159, "width": 640, "height": 360}
{"video_id": "2j8ponQ9Pyc", "transcript": [{"start": 0.24, "duration": 5.039, "text": "let's solve this calculus problem step"}, {"start": 2.56, "duration": 6.8, "text": "by step we need to find the derivative"}, {"start": 5.279, "duration": 6.601, "text": "of x to the 4th power time e to the X"}, {"start": 9.36, "duration": 4.96, "text": "this is a product of two functions so"}, {"start": 11.88, "duration": 4.6, "text": "we'll need to use the product rule the"}, {"start": 14.32, "duration": 6.0, "text": "product rule states that the derivative"}, {"start": 16.48, "duration": 7.92, "text": "of a product U * v = u * the derivative"}, {"start": 20.32, "duration": 11.64, "text": "of V plus v * the derivative of U in our"}, {"start": 24.4, "duration": 7.56, "text": "case u = x 4th power and v = e to X"}, {"start": 34.559, "duration": 6.601, "text": "let's apply the product rule first we"}, {"start": 37.76, "duration": 8.479, "text": "write X 4th power time the derivative of"}, {"start": 41.16, "duration": 9.879, "text": "e to X Plus e x * the derivative of x 4"}, {"start": 46.239, "duration": 8.12, "text": "power remember the derivative of e to X"}, {"start": 51.039, "duration": 7.48, "text": "is just e to X and the derivative of x"}, {"start": 54.359, "duration": 6.281, "text": "to the 4th power is 4 x Cub now let's"}, {"start": 58.519, "duration": 5.92, "text": "factor out the common terms"}, {"start": 60.64, "duration": 8.72, "text": "both terms have e to X so we can factor"}, {"start": 64.439, "duration": 9.641, "text": "that out when we do we get e to x * the"}, {"start": 69.36, "duration": 8.56, "text": "X to 4 + 4x Cub finally we can factor"}, {"start": 74.08, "duration": 8.079, "text": "out X cubed from both terms inside the"}, {"start": 77.92, "duration": 7.6, "text": "parentheses this gives us X cubed e to"}, {"start": 82.159, "duration": 4.681, "text": "the x * the quantity X+ 4 which is our"}, {"start": 85.52, "duration": 3.959, "text": "final"}, {"start": 86.84, "duration": 5.12, "text": "answer let's visualize what's happening"}, {"start": 89.479, "duration": 7.081, "text": "with this derivative here's the original"}, {"start": 91.96, "duration": 8.36, "text": "function X to 4th * e to X in blue and"}, {"start": 96.56, "duration": 6.12, "text": "here's its derivative in red notice how"}, {"start": 100.32, "duration": 4.92, "text": "the derivative grows even faster than"}, {"start": 102.68, "duration": 6.16, "text": "the original function due to both the"}, {"start": 105.24, "duration": 5.519, "text": "polom and exponential terms to summarize"}, {"start": 108.84, "duration": 4.52, "text": "we use the product rule to find this"}, {"start": 110.759, "duration": 5.64, "text": "derivative carefully differentiated each"}, {"start": 113.36, "duration": 5.48, "text": "part and then factored our result to get"}, {"start": 116.399, "duration": 4.841, "text": "it into its simplest form the"}, {"start": 118.84, "duration": 7.72, "text": "visualization helped us see how this"}, {"start": 121.24, "duration": 5.32, "text": "function and its derivative behave"}], "title": "Finding the Derivative of \ud835\udc65^4\ud835\udc52^\ud835\udc65  Using the Product Rule | Step-by-Step Explanation", "description": "In this video, learn how to find the derivative of \ud835\udc65^4\ud835\udc52^\ud835\udc65 \n using the Product Rule in calculus. \ud83c\udf93\n\ud83d\udccc What You'll Learn in This Video:\nWhat is the Product Rule?\nStep-by-step solution to differentiate \ud835\udc65^4\ud835\udc52^\ud835\udc65 \nFormula breakdown and example problem-solving.\nThis example is perfect for beginners in calculus or anyone reviewing the Product Rule for differentiation!\n\ud83d\udca1 Don't forget to LIKE, SHARE, and SUBSCRIBE for more math tutorials!\n#Calculus #ProductRule #Differentiation #Derivative #MathTutorial #LearnMath #MathEducation #MathHelp", "lengthSeconds": 127, "width": 640, "height": 360}
{"video_id": "i0aSaAZGab4", "transcript": [{"start": 0.32, "duration": 4.999, "text": "let's solve this derivative step by step"}, {"start": 2.96, "duration": 6.96, "text": "using the product rule we need to find"}, {"start": 5.319, "duration": 6.761, "text": "the derivative of e to x * sinx first"}, {"start": 9.92, "duration": 4.759, "text": "let's recall the product rule the"}, {"start": 12.08, "duration": 6.92, "text": "derivative of a product of two functions"}, {"start": 14.679, "duration": 7.52, "text": "U * V equal U * the derivative of V plus"}, {"start": 19.0, "duration": 7.64, "text": "v * the derivative of U let's identify"}, {"start": 22.199, "duration": 8.4, "text": "our functions in this case u = e to X"}, {"start": 26.64, "duration": 7.12, "text": "and V = sin x now let let's find the"}, {"start": 30.599, "duration": 7.721, "text": "derivatives of each function separately"}, {"start": 33.76, "duration": 8.16, "text": "the derivative of e to X is e to X and"}, {"start": 38.32, "duration": 5.84, "text": "the derivative of sinx is cine X let's"}, {"start": 41.92, "duration": 6.279, "text": "apply the product rule using these"}, {"start": 44.16, "duration": 7.6, "text": "pieces we start with E to x * the"}, {"start": 48.199, "duration": 7.36, "text": "derivative of sin x which is cine X then"}, {"start": 51.76, "duration": 7.439, "text": "we add sin x * the derivative of e to X"}, {"start": 55.559, "duration": 6.64, "text": "which is e to X finally we can factor"}, {"start": 59.199, "duration": 6.441, "text": "out e to X since it appears in both"}, {"start": 62.199, "duration": 7.401, "text": "terms this gives us our final answer e"}, {"start": 65.64, "duration": 6.88, "text": "to x * the quantity cosine X Plus sinx"}, {"start": 69.6, "duration": 5.199, "text": "let's verify this makes sense we used"}, {"start": 72.52, "duration": 5.0, "text": "the product rule correctly found the"}, {"start": 74.799, "duration": 5.561, "text": "derivatives of both functions and"}, {"start": 77.52, "duration": 6.599, "text": "factored out the common term our answer"}, {"start": 80.36, "duration": 6.56, "text": "shows that the derivative of e to the X"}, {"start": 84.119, "duration": 5.841, "text": "sinx involves both trigonometric"}, {"start": 86.92, "duration": 5.32, "text": "functions and the exponential function"}, {"start": 89.96, "duration": 4.839, "text": "which we would expect remember when"}, {"start": 92.24, "duration": 5.519, "text": "dealing with products of functions"}, {"start": 94.799, "duration": 3.921, "text": "especially when one is exponential and"}, {"start": 97.759, "duration": 3.68, "text": "the other"}, {"start": 98.72, "duration": 5.439, "text": "trigonometric the product rule is our go"}, {"start": 101.439, "duration": 4.841, "text": "to Method this example shows how"}, {"start": 104.159, "duration": 5.681, "text": "powerful and straightforward the product"}, {"start": 106.28, "duration": 3.56, "text": "rule can be"}], "title": "Finding the Derivative of e^x sin(x)  Using the Product Rule | Step-by-Step Explanation", "description": "In this video, discover how to differentiate e^x sin(x)) using the Product Rule in calculus. \n\nWhat You'll Learn:  \n- Definition of the Product Rule  \n- Step-by-step differentiation of e^x sin(x)  \n- Formula breakdown and example problem-solving  \n\nThis example is ideal for calculus beginners or anyone reviewing the Product Rule for differentiation!\n\n\ud83d\udca1 Don't forget to LIKE, SHARE, and SUBSCRIBE for more math tutorials!\n#Calculus #ProductRule #Differentiation #Derivative #MathTutorial #LearnMath #MathEducation #MathHelp", "lengthSeconds": 111, "width": 640, "height": 360}
{"video_id": "Pjqb2s5_YWA", "transcript": [{"start": 0.32, "duration": 6.439, "text": "welcome today we'll solve a challenging"}, {"start": 3.24, "duration": 5.479, "text": "derivative problem using logarithmic"}, {"start": 6.759, "duration": 6.441, "text": "differentiation we need to find the"}, {"start": 8.719, "duration": 12.121, "text": "derivative of y = theare < TK of thean x"}, {"start": 13.2, "duration": 10.159, "text": "- 1 * x - 2 / the quantity x - 3 * x - 4"}, {"start": 20.84, "duration": 6.679, "text": "this looks complicated to differentiate"}, {"start": 23.359, "duration": 6.481, "text": "directly instead we'll use logarithmic"}, {"start": 27.519, "duration": 5.121, "text": "differentiation the strategy is to take"}, {"start": 29.84, "duration": 5.92, "text": "the the natural log of both sides first"}, {"start": 32.64, "duration": 6.439, "text": "then differentiate let's start by taking"}, {"start": 35.76, "duration": 7.6, "text": "the natural log of both sides when we do"}, {"start": 39.079, "duration": 7.361, "text": "this we get Ln of y equal Ln of our"}, {"start": 43.36, "duration": 5.28, "text": "original expression now we can use the"}, {"start": 46.44, "duration": 4.959, "text": "properties of logarithms to break this"}, {"start": 48.64, "duration": 6.439, "text": "down first we can separate the square"}, {"start": 51.399, "duration": 5.64, "text": "root into its numerator and denominator"}, {"start": 55.079, "duration": 5.32, "text": "then we can split this into the"}, {"start": 57.039, "duration": 5.68, "text": "difference of logarithms using using the"}, {"start": 60.399, "duration": 4.961, "text": "properties of logarithms again we can"}, {"start": 62.719, "duration": 6.961, "text": "break down the square roots this gives"}, {"start": 65.36, "duration": 6.56, "text": "us 12 * the natural log of each term now"}, {"start": 69.68, "duration": 6.799, "text": "we can differentiate both sides with"}, {"start": 71.92, "duration": 5.68, "text": "respect to X on the left side we get 1/"}, {"start": 76.479, "duration": 4.561, "text": "y *"}, {"start": 77.6, "duration": 5.96, "text": "dydx on the right side we use the chain"}, {"start": 81.04, "duration": 6.399, "text": "rule to differentiate each logarithm"}, {"start": 83.56, "duration": 7.04, "text": "term to solve for dydx we multiply both"}, {"start": 87.439, "duration": 6.96, "text": "sides by y remember that y Y is our"}, {"start": 90.6, "duration": 6.64, "text": "original function finally we substitute"}, {"start": 94.399, "duration": 5.76, "text": "back the original expression for y to"}, {"start": 97.24, "duration": 5.919, "text": "get our complete derivative and there we"}, {"start": 100.159, "duration": 4.401, "text": "have it we've found the derivative using"}, {"start": 103.159, "duration": 3.401, "text": "logarithmic"}, {"start": 104.56, "duration": 3.919, "text": "differentiation this method is"}, {"start": 106.56, "duration": 5.159, "text": "particularly useful when dealing with"}, {"start": 108.479, "duration": 7.96, "text": "products quotients and Roots as it often"}, {"start": 111.719, "duration": 4.72, "text": "simplifies the differentiation process"}], "title": "Mastering Derivatives with Logarithmic Differentiation [\u221a((\ud835\udc65\u22121)(\ud835\udc65\u22122)/(\ud835\udc65\u22123)(\ud835\udc65\u22124) )]", "description": "Question:\nFind the derivative of the function y= [\u221a((\ud835\udc65\u22121)(\ud835\udc65\u22122)/(\ud835\udc65\u22123)(\ud835\udc65\u22124) )]  using logarithmic differentiation.\n\nIn this video, we explore the technique of logarithmic differentiation to find the derivative of complex functions. Follow along as we break down each step clearly, providing both mathematical steps and explanations. Perfect for calculus students looking to deepen their understanding of derivatives!\n#Calculus #Derivatives #MathTutorial #LogarithmicDifferentiation #MathTips", "lengthSeconds": 119, "width": 640, "height": 360}
{"video_id": "-5-yQtvZ7gY", "transcript": [{"start": 0.359, "duration": 5.44, "text": "today we'll find the derivative of"}, {"start": 2.48, "duration": 5.84, "text": "secant x using the first principes"}, {"start": 5.799, "duration": 7.321, "text": "definition of derivatives let's start"}, {"start": 8.32, "duration": 7.239, "text": "with our function let F ofx = secant x"}, {"start": 13.12, "duration": 7.76, "text": "by definition the derivative is the"}, {"start": 15.559, "duration": 9.321, "text": "limit as H approaches Z of f ofx + hus F"}, {"start": 20.88, "duration": 7.36, "text": "ofx all divided H substituting our"}, {"start": 24.88, "duration": 5.799, "text": "secant function we get the limit as H"}, {"start": 28.24, "duration": 8.76, "text": "approaches Zer of secant"}, {"start": 30.679, "duration": 9.961, "text": "of x + h- secant x all divid H remember"}, {"start": 37.0, "duration": 6.12, "text": "that secant is 1 / cosine so we can"}, {"start": 40.64, "duration": 7.2, "text": "rewrite this as the limit as H"}, {"start": 43.12, "duration": 9.84, "text": "approaches 0 of 1 / cosine of x + H - 1"}, {"start": 47.84, "duration": 7.84, "text": "/ cosine X all divid H to subtract these"}, {"start": 52.96, "duration": 5.399, "text": "fractions we need a common denominator"}, {"start": 55.68, "duration": 7.039, "text": "multiplying both terms by cosine x"}, {"start": 58.359, "duration": 9.041, "text": "cosine of X Plus h/ cosine x cosine of x"}, {"start": 62.719, "duration": 9.4, "text": "+ H gives us cosine x - cosine of x + H"}, {"start": 67.4, "duration": 7.2, "text": "over cosine x cosine of x + H now comes"}, {"start": 72.119, "duration": 4.68, "text": "the key step we can use the cosine"}, {"start": 74.6, "duration": 7.4, "text": "difference formula which states that"}, {"start": 76.799, "duration": 9.441, "text": "cosine a minus cosine B = -2 * sin of a"}, {"start": 82.0, "duration": 7.04, "text": "plus b/ 2 * sin of a minus B /2 let's"}, {"start": 86.24, "duration": 6.839, "text": "simplify the arguments the first sign"}, {"start": 89.04, "duration": 7.6, "text": "term becomes s of x + h / 2 and the"}, {"start": 93.079, "duration": 6.04, "text": "second becomes s of H /2 now we can"}, {"start": 96.64, "duration": 6.36, "text": "split this into separate limits the"}, {"start": 99.119, "duration": 8.201, "text": "limit of s of x + h / 2 as H approaches"}, {"start": 103.0, "duration": 8.64, "text": "0 is sin x the limit of s of h / 2 over"}, {"start": 107.32, "duration": 7.96, "text": "h / 2 as H approaches 0 is 1 and the"}, {"start": 111.64, "duration": 7.96, "text": "limit of cosine of x + H as H approaches"}, {"start": 115.28, "duration": 8.6, "text": "0 is cosine X finally we can rewrite"}, {"start": 119.6, "duration": 8.439, "text": "this as sin x cosine x * 1 / cosine X"}, {"start": 123.88, "duration": 7.4, "text": "which is tangent x * secant x therefore"}, {"start": 128.039, "duration": 7.361, "text": "the derivative of secant x equal tangent"}, {"start": 131.28, "duration": 6.48, "text": "x * secant x to visualize this result"}, {"start": 135.4, "duration": 6.44, "text": "let's look at the graphs of SEC X in"}, {"start": 137.76, "duration": 7.68, "text": "blue and its derivative tangent x * SEC"}, {"start": 141.84, "duration": 3.6, "text": "X in red"}], "title": "Understanding the Derivative of secx  Using First Principles", "description": "Master the fundamentals of calculus by deriving the exponential function sec3x from first principles. Explore our step-by-step explanations and examples to enhance your grasp of derivatives and exponential functions.\n#Calculus #Derivatives #ExponentialFunctions #MathTutorial #FirstPrinciples #tan3x", "lengthSeconds": 144, "width": 640, "height": 360}
{"video_id": "dxe78UFwKgY", "transcript": [{"start": 1.959, "duration": 5.4, "text": "let's solve this complex derivative"}, {"start": 4.6, "duration": 5.68, "text": "problem step by step we'll start with"}, {"start": 7.359, "duration": 5.121, "text": "our original function the first step is"}, {"start": 10.28, "duration": 4.279, "text": "to convert both terms to exponential"}, {"start": 12.48, "duration": 5.719, "text": "form using the property that a to the"}, {"start": 14.559, "duration": 7.72, "text": "power be equals e to the power of B *"}, {"start": 18.199, "duration": 7.281, "text": "natural log of a for the first term x to"}, {"start": 22.279, "duration": 7.84, "text": "the power of sin x becomes E power of"}, {"start": 25.48, "duration": 8.48, "text": "sin x * natural log of x similarly for"}, {"start": 30.119, "duration": 8.161, "text": "the second term sin x to the power of X"}, {"start": 33.96, "duration": 7.04, "text": "becomes E power of x * natural log of"}, {"start": 38.28, "duration": 6.0, "text": "sinx now we can find the derivative"}, {"start": 41.0, "duration": 6.68, "text": "using the chain rule remember when"}, {"start": 44.28, "duration": 6.119, "text": "differentiating e to some power we get e"}, {"start": 47.68, "duration": 5.879, "text": "to that same Power Times the derivative"}, {"start": 50.399, "duration": 6.601, "text": "of the power let's handle the first term"}, {"start": 53.559, "duration": 6.64, "text": "we need to find the derivative of sinx *"}, {"start": 57.0, "duration": 7.68, "text": "natural log X using the product rule"}, {"start": 60.199, "duration": 8.001, "text": "this gives us s x * 1 /x + cosine x *"}, {"start": 64.68, "duration": 6.799, "text": "natural log X for the second term we"}, {"start": 68.2, "duration": 6.959, "text": "need to differentiate x * natural log of"}, {"start": 71.479, "duration": 7.441, "text": "sin x this gives us natural log of sin x"}, {"start": 75.159, "duration": 8.0, "text": "+ x times the derivative of natural log"}, {"start": 78.92, "duration": 6.8, "text": "of sinx which is coent X finally we can"}, {"start": 83.159, "duration": 6.241, "text": "convert back to our original form by"}, {"start": 85.72, "duration": 7.079, "text": "replacing e to the power of sin x Ln x"}, {"start": 89.4, "duration": 7.48, "text": "with x the power of sin x and e to the"}, {"start": 92.799, "duration": 8.041, "text": "power of X Ln sin x with sin x the power"}, {"start": 96.88, "duration": 6.68, "text": "of X this gives us our final answer this"}, {"start": 100.84, "duration": 5.76, "text": "problem demonstrates several important"}, {"start": 103.56, "duration": 5.519, "text": "calculus Concepts converting exponential"}, {"start": 106.6, "duration": 5.559, "text": "expressions using the chain Rule and"}, {"start": 109.079, "duration": 5.841, "text": "product rule and handling complex"}, {"start": 112.159, "duration": 6.481, "text": "derivatives involving trigonometric and"}, {"start": 114.92, "duration": 3.72, "text": "logarithmic functions"}], "title": "Complex Derivatives: Step-by-Step Solution of y = x^(sin x) + (sin x)^x | Calculus Made Easy", "description": "Question:\nFind the derivative of y = x^(sin x) + (sin x)^x\n\n\ud83d\udd0d In this detailed calculus tutorial, we break down the solution to a challenging derivative problem involving variable exponents. Learn how to:\n\u2022 Convert complex expressions using natural logarithms\n\u2022 Apply the chain rule and product rule effectively\n\u2022 Handle logarithmic and trigonometric derivatives\n\u2022 Transform expressions back to their original form\nPerfect for:\n\ud83d\udcda Calculus students\n\ud83c\udf93 Mathematics majors\n\ud83d\udcdd Exam preparation\n\ud83e\uddee Advanced differentiation practice\n#Calculus #Mathematics #ChainRule #Derivatives #MathTutorial #ExamPrep #CalculusTutorial #MathEducation", "lengthSeconds": 126, "width": 640, "height": 360}
{"video_id": "9cV50yYiDVc", "transcript": [{"start": 0.28, "duration": 4.92, "text": "let's solve this calculus problem step"}, {"start": 2.52, "duration": 5.64, "text": "by step using the quotient rule we need"}, {"start": 5.2, "duration": 8.16, "text": "to find the derivative of the function"}, {"start": 8.16, "duration": 7.679, "text": "1us tangent thet / 1 + tangent Theta"}, {"start": 13.36, "duration": 5.56, "text": "let's recall the quotient rule the"}, {"start": 15.839, "duration": 5.2, "text": "derivative of U over V equal V * the"}, {"start": 18.92, "duration": 6.8, "text": "derivative of U minus uu * the"}, {"start": 21.039, "duration": 9.681, "text": "derivative of V all / v\u00b2 Let's identify"}, {"start": 25.72, "duration": 10.6, "text": "our u and v in this case U equal 1 -"}, {"start": 30.72, "duration": 7.92, "text": "Tang thet and V = 1 + tangent thet next"}, {"start": 36.32, "duration": 7.04, "text": "we need to find the derivatives of u and"}, {"start": 38.64, "duration": 7.36, "text": "v the derivative of U is secant 2 Theta"}, {"start": 43.36, "duration": 5.719, "text": "and the derivative of V is secant 2"}, {"start": 46.0, "duration": 6.68, "text": "Theta now let's substitute everything"}, {"start": 49.079, "duration": 9.561, "text": "into the quotient rule formula we get 1"}, {"start": 52.68, "duration": 10.76, "text": "+ tangent Theta * secant 2 thet - 1 -"}, {"start": 58.64, "duration": 7.76, "text": "Tang thet * secant 2 Theta all over 1 +"}, {"start": 63.44, "duration": 7.08, "text": "tangent thet 2 let's expand the"}, {"start": 66.4, "duration": 9.8, "text": "numerator first term gives us secant 2"}, {"start": 70.52, "duration": 11.32, "text": "Theta -^ 2et * tangent Theta second term"}, {"start": 76.2, "duration": 9.279, "text": "gives us secant 2 th +^ 2 thet * tangent"}, {"start": 81.84, "duration": 6.56, "text": "thet now let's combine like terms the"}, {"start": 85.479, "duration": 6.841, "text": "tangent Theta terms cancel out and we're"}, {"start": 88.4, "duration": 6.52, "text": "left with negative 2 secant 2 Theta in"}, {"start": 92.32, "duration": 5.24, "text": "the numerator and this is our final"}, {"start": 94.92, "duration": 5.159, "text": "answer we've successfully used the"}, {"start": 97.56, "duration": 5.239, "text": "quotient rule to find the derivative of"}, {"start": 100.079, "duration": 6.68, "text": "this function to verify this is correct"}, {"start": 102.799, "duration": 6.92, "text": "we can check that one our answer is a"}, {"start": 106.759, "duration": 6.481, "text": "quotient two we correctly applied the"}, {"start": 109.719, "duration": 7.841, "text": "quotient rule formula and three we"}, {"start": 113.24, "duration": 4.32, "text": "properly simplified our expression"}], "title": "Quotient Rule Masterclass: Differentiating Complex Fractions [(\ud835\udfcf\u2212\ud835\udc95\ud835\udc82\ud835\udc8f \ud835\udf3d)/(\ud835\udfcf+\ud835\udc95\ud835\udc82\ud835\udc8f \ud835\udf3d)] with Ease", "description": "Dive into the world of calculus with this in-depth look at using the quotient rule for complex fractions. We\u2019ll expertly guide you through differentiating the function [(\ud835\udfcf\u2212\ud835\udc95\ud835\udc82\ud835\udc8f \ud835\udf3d)/(\ud835\udfcf+\ud835\udc95\ud835\udc82\ud835\udc8f \ud835\udf3d)]  , helping you master this essential calculus technique. Perfect for students wanting to strengthen their differentiation skills.\n#Calculus #Differentiation #QuotientRule #MathEducation #LearnMath", "lengthSeconds": 116, "width": 640, "height": 360}
{"video_id": "3YSznr6lP0k", "transcript": [{"start": 0.24, "duration": 4.92, "text": "let's solve this calculus problem step"}, {"start": 2.56, "duration": 7.32, "text": "by step we need to differentiate the"}, {"start": 5.16, "duration": 7.8, "text": "function Ln of sin x^2 with respect to X"}, {"start": 9.88, "duration": 6.159, "text": "let's start by applying the chain rule"}, {"start": 12.96, "duration": 7.12, "text": "remember the derivative of natural log"}, {"start": 16.039, "duration": 7.961, "text": "of U is 1/ U * the derivative of U so in"}, {"start": 20.08, "duration": 6.76, "text": "our case U is sin x^2 now we need to"}, {"start": 24.0, "duration": 6.599, "text": "find the derivative of sin x^2 the"}, {"start": 26.84, "duration": 8.399, "text": "derivative of s of V is cosine V time"}, {"start": 30.599, "duration": 7.881, "text": "the derivative of V here V is X2 let's"}, {"start": 35.239, "duration": 8.921, "text": "rearrange this expression we can combine"}, {"start": 38.48, "duration": 9.48, "text": "1 / sin x^2 and cine x^2 into cine x^2"}, {"start": 44.16, "duration": 6.879, "text": "over sin x^2 now for the last part we"}, {"start": 47.96, "duration": 7.68, "text": "need to differentiate x^2 the derivative"}, {"start": 51.039, "duration": 8.441, "text": "of x^2 is 2x multiply this by our"}, {"start": 55.64, "duration": 6.28, "text": "previous expression for our final step"}, {"start": 59.48, "duration": 6.48, "text": "recall that cosine over s is the"}, {"start": 61.92, "duration": 9.04, "text": "definition of coent therefore our final"}, {"start": 65.96, "duration": 8.6, "text": "answer is 2x * cent of x^2 and that's"}, {"start": 70.96, "duration": 6.6, "text": "how we differentiate Ln of sin x^2 we"}, {"start": 74.56, "duration": 5.36, "text": "use the chain rule twice once for the"}, {"start": 77.56, "duration": 5.239, "text": "natural log and once for the sign"}, {"start": 79.92, "duration": 7.239, "text": "function and then simplified our result"}, {"start": 82.799, "duration": 4.36, "text": "using the definition of cotangent"}], "title": "Mastering Derivatives Calculating the Derivative of {lnsinx\u00b2}", "description": "Join us as we break down the process of finding the derivative of {lnsinx\u00b2}  step-by-step. Learn how to apply the chain rule effectively and understand the use of trigonometric identities. Perfect for students tackling calculus and differentiation!\n#Calculus #Derivatives #MathTutorial #Trigonometry #Differentiation", "lengthSeconds": 90, "width": 640, "height": 360}
{"video_id": "PczXR85vszs", "transcript": [{"start": 0.359, "duration": 4.521, "text": "let's solve this challenging"}, {"start": 2.0, "duration": 5.16, "text": "trigonometric derivative problem step by"}, {"start": 4.88, "duration": 8.12, "text": "step we need to find the derivative of"}, {"start": 7.16, "duration": 9.36, "text": "sin inverse of 2 a x * < TK of 1 - A 2"}, {"start": 13.0, "duration": 6.08, "text": "x^2 let's start by letting y equal our"}, {"start": 16.52, "duration": 5.839, "text": "expression then we'll make a clever"}, {"start": 19.08, "duration": 7.08, "text": "substitution to simplify this let's say"}, {"start": 22.359, "duration": 6.561, "text": "ax equal sin Theta this substitution"}, {"start": 26.16, "duration": 4.64, "text": "allows us to rewrite our expression"}, {"start": 28.92, "duration": 7.28, "text": "inside the sin inverse"}, {"start": 30.8, "duration": 8.36, "text": "we now have 2 sin thet * the < TK of 1 -"}, {"start": 36.2, "duration": 7.8, "text": "sin^2 now we can use a fundamental"}, {"start": 39.16, "duration": 8.879, "text": "trigonometric identity 1 - sin^2 thet ="}, {"start": 44.0, "duration": 7.52, "text": "cine 2 thet the square root of cosine 2"}, {"start": 48.039, "duration": 6.16, "text": "thet is simply cosine Theta and 2 sin"}, {"start": 51.52, "duration": 5.92, "text": "thet * cosine Theta is the double angle"}, {"start": 54.199, "duration": 6.441, "text": "formula for sin 2 Theta when we have S"}, {"start": 57.44, "duration": 6.0, "text": "inverse of s of an angle they cancel out"}, {"start": 60.64, "duration": 6.28, "text": "leaving us with Just 2 Theta and since"}, {"start": 63.44, "duration": 7.6, "text": "Theta was sin inverse of ax our final"}, {"start": 66.92, "duration": 6.72, "text": "expression for y is 2 sin inverse of ax"}, {"start": 71.04, "duration": 6.56, "text": "now we can differentiate this expression"}, {"start": 73.64, "duration": 8.04, "text": "the derivative of y with respect 2x = 2"}, {"start": 77.6, "duration": 6.159, "text": "* the derivative of sin inverse of ax"}, {"start": 81.68, "duration": 6.759, "text": "using the chain Rule and the derivative"}, {"start": 83.759, "duration": 7.121, "text": "formula for sin inverse we get 2 * 1 <"}, {"start": 88.439, "duration": 6.36, "text": "TK of 1 - -"}, {"start": 90.88, "duration": 10.559, "text": "ax^2 time the derivative of ax our final"}, {"start": 94.799, "duration": 6.64, "text": "answer is 2 over the < TK of 1 - A 2"}, {"start": 101.88, "duration": 5.96, "text": "x^2 this problem demonstrates the power"}, {"start": 105.119, "duration": 4.801, "text": "of clever substitution and trigonometric"}, {"start": 107.84, "duration": 5.08, "text": "identities in simplifying complex"}, {"start": 109.92, "duration": 3.0, "text": "derivatives"}], "title": "Understanding the Differentiation of [\ud835\udc60\ud835\udc56\ud835\udc5b^(\u22121){2\ud835\udc4e\ud835\udc65 \u221a(1\u2212\ud835\udc4e^2 \ud835\udc65^2 )}] Step-by-Step Simplification", "description": "Question\nFind Derivative of  [\ud835\udc60\ud835\udc56\ud835\udc5b^(\u22121){2\ud835\udc4e\ud835\udc65 \u221a(1\u2212\ud835\udc4e^2 \ud835\udc65^2 )}].\n\nIn this video, we delve into the calculus behind differentiating the complex expression  [\ud835\udc60\ud835\udc56\ud835\udc5b^(\u22121){2\ud835\udc4e\ud835\udc65 \u221a(1\u2212\ud835\udc4e^2 \ud835\udc65^2 )}]. Follow along as we simplify and differentiate step-by-step, using trigonometric identities and the chain rule. Whether you\u2019re a calculus student or just a math enthusiast, this detailed explanation will help you grasp this intriguing problem. Perfect for those preparing for exams or seeking a deeper understanding of calculus!\n#Calculus #Differentiation #MathTutorial #InverseTrigonometry #ChainRule #MathHelp #Education", "lengthSeconds": 116, "width": 640, "height": 360}
{"video_id": "XSD3M4WfwR0", "transcript": [{"start": 0.32, "duration": 4.959, "text": "let's solve this complex derivative step"}, {"start": 3.12, "duration": 5.36, "text": "by step we're going to find the"}, {"start": 5.279, "duration": 7.36, "text": "derivative of the square root of tangent"}, {"start": 8.48, "duration": 6.359, "text": "of e to x^2 first let's rewrite this"}, {"start": 12.639, "duration": 5.64, "text": "expression the square root can be"}, {"start": 14.839, "duration": 5.401, "text": "written as a power of 12 so our"}, {"start": 18.279, "duration": 7.0, "text": "expression becomes the derivative of"}, {"start": 20.24, "duration": 8.68, "text": "tangent of e to x^2 all rais to the 1/2"}, {"start": 25.279, "duration": 5.801, "text": "power now we can apply the chain rule"}, {"start": 28.92, "duration": 5.92, "text": "remember when we have a function raised"}, {"start": 31.08, "duration": 7.52, "text": "to a power we multiply by the power"}, {"start": 34.84, "duration": 5.96, "text": "reduce the power by one and multiply by"}, {"start": 38.6, "duration": 7.0, "text": "the derivative of the inner function"}, {"start": 40.8, "duration": 8.12, "text": "this gives us 12 * tangent of e x^2 to"}, {"start": 45.6, "duration": 5.72, "text": "the2 Power Times the derivative of the"}, {"start": 48.92, "duration": 4.92, "text": "inner function for the derivative of"}, {"start": 51.32, "duration": 6.039, "text": "tangent we use the fact that the"}, {"start": 53.84, "duration": 8.559, "text": "derivative of tangent is secant squar so"}, {"start": 57.359, "duration": 10.601, "text": "we get 12 * 1 root of tangent of e to"}, {"start": 62.399, "duration": 9.441, "text": "x^2 * secant 2 of e to x^2 * the"}, {"start": 67.96, "duration": 8.0, "text": "derivative of e to x^2 now we need to"}, {"start": 71.84, "duration": 7.24, "text": "find the derivative of e to x^2 using"}, {"start": 75.96, "duration": 8.64, "text": "the chain rule again this gives us e to"}, {"start": 79.08, "duration": 9.56, "text": "x^2 * 2x so our expression becomes 12 *"}, {"start": 84.6, "duration": 11.08, "text": "secant 2 of e to x^2 over the square"}, {"start": 88.64, "duration": 10.24, "text": "root of tangent of e to x^2 * e x^2 * 2x"}, {"start": 95.68, "duration": 6.32, "text": "finally we can simplify this expression"}, {"start": 98.88, "duration": 7.839, "text": "the 1/2 and two cancel out giving us our"}, {"start": 102.0, "duration": 8.079, "text": "final answer x * e to x^ 2 * secant 2 of"}, {"start": 106.719, "duration": 7.321, "text": "e to x^2 all over the square < TK of"}, {"start": 110.079, "duration": 6.96, "text": "tangent of e to x^2 let's review the key"}, {"start": 114.04, "duration": 6.719, "text": "steps we used first we rewrote the"}, {"start": 117.039, "duration": 5.961, "text": "square root as a power of 1/2 then we"}, {"start": 120.759, "duration": 5.121, "text": "applied the chain rule to handle the"}, {"start": 123.0, "duration": 5.44, "text": "outer function next we used the fact"}, {"start": 125.88, "duration": 5.96, "text": "that the derivative of tangent is secant"}, {"start": 128.44, "duration": 6.28, "text": "squared finally we found the derivative"}, {"start": 131.84, "duration": 5.84, "text": "of the exponential term and simplified"}, {"start": 134.72, "duration": 5.519, "text": "our expression this problem shows how we"}, {"start": 137.68, "duration": 5.8, "text": "can break down complex derivatives into"}, {"start": 140.239, "duration": 7.561, "text": "simpler steps using the chain Rule and"}, {"start": 143.48, "duration": 4.32, "text": "our knowledge of basic derivatives"}], "title": "9.3-1.7 How to Differentiate (\u221a\ud835\udc95\ud835\udc82\ud835\udc8f\u2061\ud835\udc86^\ud835\udc99^\ud835\udfd0)  Step-by-Step", "description": "Unlock the secrets of calculus with this detailed, step-by-step guide to differentiating (\u221a\ud835\udc95\ud835\udc82\ud835\udc8f\u2061\ud835\udc86^\ud835\udc99^\ud835\udfd0) . We\u2019ll break down the process using the chain rule and power rule, making complex derivatives easy to understand. Perfect for students and anyone looking to strengthen their calculus skills!\n#Calculus #Differentiation #ChainRule #MathTutorial #MathHelp #cpmrana", "lengthSeconds": 149, "width": 640, "height": 360}
{"video_id": "1L3LzD9ZL90", "transcript": [{"start": 0.28, "duration": 6.519, "text": "let's solve the step by step we need to"}, {"start": 3.04, "duration": 5.519, "text": "find the derivative of cine 2 of x^2"}, {"start": 6.799, "duration": 4.201, "text": "first let's understand what we're"}, {"start": 8.559, "duration": 5.841, "text": "dealing with this is a composite"}, {"start": 11.0, "duration": 6.119, "text": "function we have cosine squar and inside"}, {"start": 14.4, "duration": 5.24, "text": "that we have X2 we'll need to use the"}, {"start": 17.119, "duration": 5.601, "text": "chain rule here let's apply the chain"}, {"start": 19.64, "duration": 5.12, "text": "rule the chain rule tells us that we"}, {"start": 22.72, "duration": 4.479, "text": "multiply the derivative of the outer"}, {"start": 24.76, "duration": 4.0, "text": "function with respect to U times the"}, {"start": 27.199, "duration": 4.48, "text": "derivative of the inner function with"}, {"start": 28.76, "duration": 5.319, "text": "respect to X let's find the derivative"}, {"start": 31.679, "duration": 5.081, "text": "of the outer function first to"}, {"start": 34.079, "duration": 6.16, "text": "differentiate cosine squar we use the"}, {"start": 36.76, "duration": 6.76, "text": "power rule the power comes down in front"}, {"start": 40.239, "duration": 5.881, "text": "and we multiply by the derivative of"}, {"start": 43.52, "duration": 6.6, "text": "what's inside so the derivative of"}, {"start": 46.12, "duration": 6.439, "text": "cosine 2 of U is 2 cosine of U * the"}, {"start": 50.12, "duration": 5.599, "text": "derivative of cosine of U which is"}, {"start": 52.559, "duration": 6.201, "text": "negative s of U now for the inner"}, {"start": 55.719, "duration": 6.921, "text": "function the derivative of x^2 is simply"}, {"start": 58.76, "duration": 7.719, "text": "2X now let's put it all together we"}, {"start": 62.64, "duration": 9.479, "text": "multiply -2 cosine of u s of U where U"}, {"start": 66.479, "duration": 10.801, "text": "is x^2 * 2x this gives us -2 cosine of"}, {"start": 72.119, "duration": 7.921, "text": "x^2 sin of x^2 * 2x let's simplify this"}, {"start": 77.28, "duration": 8.92, "text": "expression we can rearrange the terms to"}, {"start": 80.04, "duration": 8.56, "text": "get -2X * 2 sin of x^2 cosine of x^2"}, {"start": 86.2, "duration": 5.48, "text": "finally we can use the double angle"}, {"start": 88.6, "duration": 8.64, "text": "formula remember remember that 2 sin of"}, {"start": 91.68, "duration": 8.56, "text": "x cosine of x = sin of 2x so our final"}, {"start": 97.24, "duration": 6.239, "text": "answer is -2X sin of"}, {"start": 100.24, "duration": 6.199, "text": "2x^2 and there we have it we've found"}, {"start": 103.479, "duration": 5.6, "text": "the derivative using the chain Rule and"}, {"start": 106.439, "duration": 5.241, "text": "the double angle formula let's review"}, {"start": 109.079, "duration": 4.68, "text": "the key steps we use the chain rule to"}, {"start": 111.68, "duration": 4.52, "text": "break down the derivative found the"}, {"start": 113.759, "duration": 5.481, "text": "derivatives of both the outer and inner"}, {"start": 116.2, "duration": 4.959, "text": "functions multipli them together and"}, {"start": 119.24, "duration": 4.919, "text": "simplified using the double angle"}, {"start": 121.159, "duration": 3.0, "text": "formula"}], "title": "9.3-1.9 Mastering Derivatives A Step by Step Guide to (\ud835\udc50\ud835\udc5c\ud835\udc60^2^\ud835\udc65^2)", "description": "In this video, we break down the process of differentiating the function (\ud835\udc50\ud835\udc5c\ud835\udc60^2^\ud835\udc65^2)  step by step. Learn how to apply the chain rule, simplify expressions, and utilize trigonometric identities to find the derivative. Perfect for students and math enthusiasts looking to deepen their understanding of calculus concepts!\n#Calculus #Differentiation #ChainRule #MathTutorial #Trigonometry", "lengthSeconds": 123, "width": 640, "height": 360}
{"video_id": "vtpKoC0XfUY", "transcript": [{"start": 0.32, "duration": 4.72, "text": "let's solve this challenging"}, {"start": 2.0, "duration": 5.68, "text": "differentiation problem step by step we"}, {"start": 5.04, "duration": 7.8, "text": "need to differentiate the function f ofx"}, {"start": 7.68, "duration": 10.96, "text": "equals the natural log of e to x * the <"}, {"start": 12.84, "duration": 8.48, "text": "TK of x - 1 overx + 1 let's break this"}, {"start": 18.64, "duration": 6.36, "text": "down into manageable steps using the"}, {"start": 21.32, "duration": 5.6, "text": "properties of logarithms first using the"}, {"start": 25.0, "duration": 5.199, "text": "property that the log of a product"}, {"start": 26.92, "duration": 6.84, "text": "equals the sum of logs we can split this"}, {"start": 30.199, "duration": 6.561, "text": "into the natural log of e to X plus the"}, {"start": 33.76, "duration": 4.88, "text": "natural log of the square root term next"}, {"start": 36.76, "duration": 5.0, "text": "we can use the power property of"}, {"start": 38.64, "duration": 6.72, "text": "logarithms to bring down the 1/2 power"}, {"start": 41.76, "duration": 11.68, "text": "this gives us natural log of e to X"}, {"start": 45.36, "duration": 8.08, "text": "+2 * the natural log of x -1 / x +"}, {"start": 56.0, "duration": 6.28, "text": "1 now we can use another property of"}, {"start": 59.32, "duration": 7.96, "text": "logarithm to split the fraction this"}, {"start": 62.28, "duration": 7.839, "text": "gives us natural log of e to x + 12 *"}, {"start": 67.28, "duration": 9.4, "text": "the natural log of x"}, {"start": 70.119, "duration": 6.561, "text": "-1 - 12 * the natural log of x +"}, {"start": 79.2, "duration": 6.879, "text": "1 now we can apply the chain rule to"}, {"start": 82.64, "duration": 6.519, "text": "differentiate each term remember when"}, {"start": 86.079, "duration": 5.521, "text": "differentiating natural log we get one"}, {"start": 89.159, "duration": 4.64, "text": "over the function time the derivative of"}, {"start": 91.6, "duration": 2.199, "text": "the"}, {"start": 96.399, "duration": 7.961, "text": "function let's evaluate each derivative"}, {"start": 100.079, "duration": 7.241, "text": "the derivative of e to X is e to X and"}, {"start": 104.36, "duration": 5.2, "text": "the derivative of X+ or minus one is"}, {"start": 107.32, "duration": 2.24, "text": "just"}, {"start": 109.719, "duration": 8.76, "text": "one now we can simplify e to X over e to"}, {"start": 115.479, "duration": 6.201, "text": "x ="}, {"start": 118.479, "duration": 6.041, "text": "1 to combine these fractions we need a"}, {"start": 121.68, "duration": 12.44, "text": "common denominator we'll multiply"}, {"start": 124.52, "duration": 9.6, "text": "everything by 2 X1 x +1 over 2 X1 x +"}, {"start": 134.36, "duration": 8.959, "text": "1 Let's expand the numerator in the"}, {"start": 137.959, "duration": 9.761, "text": "first term we get 2 x^2 - 2 then we add"}, {"start": 143.319, "duration": 4.401, "text": "x +1 and subtract x"}, {"start": 148.12, "duration": 5.68, "text": "-1 this simplifies to 2x^2 in the"}, {"start": 153.959, "duration": 6.121, "text": "numerator and our final answer is x^2"}, {"start": 157.64, "duration": 5.84, "text": "over x^2 -"}, {"start": 160.08, "duration": 6.4, "text": "1 this problem demonstrated several"}, {"start": 163.48, "duration": 5.92, "text": "important calculus Concepts the chain"}, {"start": 166.48, "duration": 5.399, "text": "rule logarithm properties and fraction"}, {"start": 169.4, "duration": 5.64, "text": "manipulation remember to break down"}, {"start": 171.879, "duration": 6.161, "text": "complex problems into small manageable"}, {"start": 175.04, "duration": 3.0, "text": "steps"}], "title": "Mastering Logarithmic Differentiation [\ud835\udc8d\ud835\udc8f {\ud835\udc86^\ud835\udc99 ((\ud835\udc99\u2212\ud835\udfcf)/(\ud835\udc99+\ud835\udfcf))^(\ud835\udfcf/\ud835\udfd0) }] Step-by-Step Guide", "description": "Question:\nDifferentiate the expression [\ud835\udc8d\ud835\udc8f {\ud835\udc86^\ud835\udc99 ((\ud835\udc99\u2212\ud835\udfcf)/(\ud835\udc99+\ud835\udfcf))^(\ud835\udfcf/\ud835\udfd0) }] with respect to x.\n\nDiscover how to differentiate complex logarithmic expressions with ease! In this video, we\u2019ll break down the process of differentiating [\ud835\udc8d\ud835\udc8f {\ud835\udc86^\ud835\udc99 ((\ud835\udc99\u2212\ud835\udfcf)/(\ud835\udc99+\ud835\udfcf))^(\ud835\udfcf/\ud835\udfd0) }] step by step. Perfect for students and anyone looking to strengthen their calculus skills. Follow along and master the art of logarithmic differentiation with detailed explanations. Don\u2019t forget to like, subscribe, and hit the bell icon for more math tutorials!\n#Calculus #Differentiation #MathTutorial #Logarithms #STEM #Education", "lengthSeconds": 181, "width": 640, "height": 360}
{"video_id": "WgV7wxHPcJk", "transcript": [{"start": 0.28, "duration": 4.76, "text": "let's solve this differentiation problem"}, {"start": 2.56, "duration": 6.68, "text": "step by step we need to find the"}, {"start": 5.04, "duration": 7.08, "text": "derivative of e to the x * tangent s of"}, {"start": 9.24, "duration": 5.959, "text": "X first let's identify what we're"}, {"start": 12.12, "duration": 7.76, "text": "dealing with this is a product of two"}, {"start": 15.199, "duration": 6.681, "text": "functions e to X and tangent s of X when"}, {"start": 19.88, "duration": 4.36, "text": "we have a product we need to use the"}, {"start": 21.88, "duration": 6.12, "text": "product rule let's apply the product"}, {"start": 24.24, "duration": 6.4, "text": "rule the first term will be e to x * the"}, {"start": 28.0, "duration": 7.84, "text": "derivative of tangent 2 x"}, {"start": 30.64, "duration": 8.079, "text": "plus t^ 2 x * the derivative of e to X"}, {"start": 35.84, "duration": 5.64, "text": "now let's handle each term separately"}, {"start": 38.719, "duration": 6.241, "text": "for the first term we need to use the"}, {"start": 41.48, "duration": 6.84, "text": "chain rule on tangent 2 x this gives us"}, {"start": 44.96, "duration": 6.8, "text": "2 * tangent x times the derivative of"}, {"start": 48.32, "duration": 7.52, "text": "tangent X for the second term we know"}, {"start": 51.76, "duration": 7.479, "text": "the derivative of e to the x is a Time e"}, {"start": 55.84, "duration": 7.12, "text": "to X let's continue simplifying the"}, {"start": 59.239, "duration": 7.521, "text": "derivative of tangent X is secant 2 x"}, {"start": 62.96, "duration": 6.6, "text": "and the derivative of x is simply a now"}, {"start": 66.76, "duration": 6.28, "text": "let's combine like terms we can factor"}, {"start": 69.56, "duration": 6.76, "text": "out e to X from both terms then we can"}, {"start": 73.04, "duration": 6.079, "text": "factor out tangent X from what remains"}, {"start": 76.32, "duration": 7.799, "text": "and there we have our final answer the"}, {"start": 79.119, "duration": 9.96, "text": "derivative of e to x * tangent 2 x is e"}, {"start": 84.119, "duration": 7.761, "text": "to x * tangent x * the quantity 2^ 2 X"}, {"start": 89.079, "duration": 5.241, "text": "plus a tangent X X this problem showed"}, {"start": 91.88, "duration": 4.76, "text": "us how to use both the product rule and"}, {"start": 94.32, "duration": 5.32, "text": "chain rule together to handle complex"}, {"start": 96.64, "duration": 3.0, "text": "derivatives"}], "title": "Mastering Differentiation [ \ud835\udc52^\ud835\udc4e\ud835\udc65 \ud835\udc61\ud835\udc4e\ud835\udc5b^2  \ud835\udc65] Solving  Step-by-Step", "description": "Question: Differentiate [ \ud835\udc52^\ud835\udc4e\ud835\udc65 \ud835\udc61\ud835\udc4e\ud835\udc5b^2  \ud835\udc65]   with respect to x.\nIn this video, we explore how to differentiate the function [ \ud835\udc52^\ud835\udc4e\ud835\udc65 \ud835\udc61\ud835\udc4e\ud835\udc5b^2  \ud835\udc65] using the product and chain rules. Follow along with clear, step-by-step explanations to enhance your calculus skills.\n#Calculus #Differentiation #MathTutorial #MathHelp #ChainRule #ProductRule #MathTips", "lengthSeconds": 105, "width": 640, "height": 360}
{"video_id": "ieSSg9E58vE", "transcript": [{"start": 0.32, "duration": 5.479, "text": "let's solve this derivative step by step"}, {"start": 3.28, "duration": 8.04, "text": "we're going to find the derivative of 1"}, {"start": 5.799, "duration": 7.76, "text": "over the cube root of 4 - 3x first let's"}, {"start": 11.32, "duration": 5.039, "text": "rewrite this expression in a form that's"}, {"start": 13.559, "duration": 5.88, "text": "easier to work with we can rewrite the"}, {"start": 16.359, "duration": 6.521, "text": "cube root in the denominator as a power"}, {"start": 19.439, "duration": 8.561, "text": "of - 13 this gives us the derivative of"}, {"start": 22.88, "duration": 8.44, "text": "4 - 3x to the - 13 power now we can use"}, {"start": 28.0, "duration": 5.64, "text": "the power rule and change rule together"}, {"start": 31.32, "duration": 5.36, "text": "the power rule tells us that when we"}, {"start": 33.64, "duration": 5.759, "text": "differentiate x to the N power we"}, {"start": 36.68, "duration": 5.6, "text": "multiply by the power and reduce the"}, {"start": 39.399, "duration": 6.881, "text": "exponent by one and since we have an"}, {"start": 42.28, "duration": 6.759, "text": "inner function of 4 - 3x we'll also need"}, {"start": 46.28, "duration": 7.119, "text": "to multiply by its derivative let's"}, {"start": 49.039, "duration": 8.121, "text": "apply these rules first we multiply by -"}, {"start": 53.399, "duration": 5.081, "text": "13 then we decrease the exponent by one"}, {"start": 57.16, "duration": 5.039, "text": "making it"}, {"start": 58.48, "duration": 7.76, "text": "-4/3 then we multiply by the derivative"}, {"start": 62.199, "duration": 7.041, "text": "of the inner function which is -3 now"}, {"start": 66.24, "duration": 8.879, "text": "let's simplify the negative signs cancel"}, {"start": 69.24, "duration": 11.64, "text": "out and multiplying - 13 by -3 gives us"}, {"start": 75.119, "duration": 8.801, "text": "POS 1 this leaves us with 4 - 3x -43"}, {"start": 80.88, "duration": 8.52, "text": "power finally we can write this as a"}, {"start": 83.92, "duration": 8.64, "text": "fraction giving us 1 over 4 - 3x^ 43"}, {"start": 89.4, "duration": 6.12, "text": "power this is our final answer let's"}, {"start": 92.56, "duration": 5.599, "text": "review the key steps we took first we"}, {"start": 95.52, "duration": 5.559, "text": "rewrote the cube root as a negative"}, {"start": 98.159, "duration": 6.64, "text": "power then we applied the power rule and"}, {"start": 101.079, "duration": 5.881, "text": "chain rule together and finally we"}, {"start": 104.799, "duration": 5.161, "text": "simplified our expression to get our"}, {"start": 106.96, "duration": 3.0, "text": "answer"}], "title": "9.3-1.2 Mastering Derivatives Simplifying Complex Expressions Step by Step-(1/\u221b(4\u22123\ud835\udc65))", "description": "In this video, we break down the process of finding the derivative of a complex expression involving cube roots. Learn how to apply the power and chain rules, and simplify your answers with ease. Follow along as we provide a detailed, step-by-step explanation that will enhance your calculus skills. Perfect for students and math enthusiasts looking to deepen their understanding of derivatives!\n#Calculus #Derivatives #MathTutorial #ChainRule #PowerRule #MathHelp #Education", "lengthSeconds": 111, "width": 640, "height": 360}
{"video_id": "HLbnXSC4qfc", "transcript": [{"start": 0.599, "duration": 5.921, "text": "let's find the derivative of natural"}, {"start": 3.44, "duration": 7.88, "text": "logarithm using first principles we"}, {"start": 6.52, "duration": 9.079, "text": "start with f of x = natural log of x"}, {"start": 11.32, "duration": 9.959, "text": "step one let's Define f ofx + H when we"}, {"start": 15.599, "duration": 10.321, "text": "replace x with x + H we get f of x + H ="}, {"start": 21.279, "duration": 6.681, "text": "natural log of x + H step two we'll use"}, {"start": 25.92, "duration": 4.04, "text": "the definition of derivative the"}, {"start": 27.96, "duration": 6.96, "text": "derivative is the limit as h appr"}, {"start": 29.96, "duration": 8.56, "text": "apprach is z of f of x + hus F ofx all"}, {"start": 34.92, "duration": 6.119, "text": "divided H substituting our Expressions"}, {"start": 38.52, "duration": 7.039, "text": "we get the limit as H approaches zero of"}, {"start": 41.039, "duration": 6.2, "text": "natural log of x + hus natural log of x"}, {"start": 45.559, "duration": 5.041, "text": "all divided by"}, {"start": 47.239, "duration": 6.0, "text": "H step three now we'll use the"}, {"start": 50.6, "duration": 5.599, "text": "properties of logarithms the difference"}, {"start": 53.239, "duration": 5.84, "text": "of logarithms equals the logarithm of"}, {"start": 56.199, "duration": 6.601, "text": "their quotient so this becomes the limit"}, {"start": 59.079, "duration": 8.601, "text": "as H approaches 0 of 1H * the natural"}, {"start": 62.8, "duration": 6.8, "text": "log of x + h / X step four let's"}, {"start": 67.68, "duration": 5.36, "text": "simplify the fraction inside the"}, {"start": 69.6, "duration": 6.199, "text": "logarithm we can rewrite x +"}, {"start": 73.04, "duration": 7.0, "text": "HX as 1 +"}, {"start": 75.799, "duration": 6.121, "text": "HX step five now comes the key step"}, {"start": 80.04, "duration": 6.48, "text": "we'll use the series expansion for"}, {"start": 81.92, "duration": 11.08, "text": "natural log of 1 + y natural log of 1 +"}, {"start": 86.52, "duration": 12.0, "text": "y = y - y^2 / 2 + plus y Cub over 3us"}, {"start": 93.0, "duration": 8.88, "text": "and so on when we substitute y = HX we"}, {"start": 98.52, "duration": 6.48, "text": "get this expression step six let's"}, {"start": 101.88, "duration": 9.199, "text": "simplify by collecting terms with h"}, {"start": 105.0, "duration": 12.719, "text": "after rearranging we get 1X - h / 2x^2 +"}, {"start": 111.079, "duration": 8.801, "text": "h^ 2 over 3x cub and so on finally as H"}, {"start": 117.719, "duration": 5.08, "text": "approaches zero all terms contain"}, {"start": 119.88, "duration": 7.64, "text": "meaning H will vanish therefore the"}, {"start": 122.799, "duration": 7.361, "text": "derivative of natural log of x = 1 /x"}, {"start": 127.52, "duration": 5.64, "text": "Let's visualize this result here's the"}, {"start": 130.16, "duration": 5.96, "text": "graph of y equals natural log of x and"}, {"start": 133.16, "duration": 8.52, "text": "at any point its derivative or slope of"}, {"start": 136.12, "duration": 5.56, "text": "the tangent line is 1 /x at that point"}], "title": "Understanding the Derivative of lnx  Using First Principles", "description": "Dive into the fundamentals of calculus as we explore how to determine the derivative of the exponential function lnx using first principles. Follow along with step-by-step explanations and examples to enhance your understanding of derivatives and exponential functions.\n#Calculus #Derivatives #ExponentialFunctions #MathTutorial #FirstPrinciples #lnx", "lengthSeconds": 140, "width": 640, "height": 360}
{"video_id": "OHIt2ykXtNM", "transcript": [{"start": 0.36, "duration": 5.16, "text": "welcome today we'll learn how to find"}, {"start": 3.04, "duration": 5.72, "text": "the derivative of a complex function"}, {"start": 5.52, "duration": 9.239, "text": "using logarithmic differentiation our"}, {"start": 8.76, "duration": 8.32, "text": "function is y = 1 + x^2 raed to^ of 2x"}, {"start": 14.759, "duration": 5.081, "text": "this is a complex function because it"}, {"start": 17.08, "duration": 6.439, "text": "has both a variable base and a variable"}, {"start": 19.84, "duration": 7.359, "text": "exponent the base is 1 + x^2 and the"}, {"start": 23.519, "duration": 6.881, "text": "exponent is 2x when we see this kind of"}, {"start": 27.199, "duration": 5.88, "text": "structure logarithmic differentiation is"}, {"start": 30.4, "duration": 5.36, "text": "often the best approach let's start by"}, {"start": 33.079, "duration": 8.761, "text": "taking the natural logarithm of both"}, {"start": 35.76, "duration": 10.2, "text": "sides this gives us Ln of Y = Ln of 1 +"}, {"start": 41.84, "duration": 6.039, "text": "x^2 raed to 2x now we can use the"}, {"start": 45.96, "duration": 4.2, "text": "logarithm property that states the"}, {"start": 47.879, "duration": 4.68, "text": "natural log of a power equals the"}, {"start": 50.16, "duration": 8.0, "text": "exponent times the natural log of the"}, {"start": 52.559, "duration": 9.401, "text": "base this gives us Ln of y = 2x * Ln of"}, {"start": 58.16, "duration": 7.72, "text": "1 + x^2 now we can differentiate both"}, {"start": 61.96, "duration": 6.44, "text": "sides with respect to X on the left side"}, {"start": 65.88, "duration": 5.64, "text": "we get 1/ y *"}, {"start": 68.4, "duration": 5.48, "text": "dydx this comes from the chain rule for"}, {"start": 71.52, "duration": 6.8, "text": "the right side we need to use the"}, {"start": 73.88, "duration": 7.879, "text": "product rule since we have 2X * Ln of 1"}, {"start": 78.32, "duration": 7.96, "text": "+ x^2 the product rule states that the"}, {"start": 81.759, "duration": 8.0, "text": "derivative of U * v = u Prime * V + U *"}, {"start": 86.28, "duration": 7.72, "text": "V Prime let's apply this the derivative"}, {"start": 89.759, "duration": 9.961, "text": "of 2x is just 2 and we multiply that by"}, {"start": 94.0, "duration": 9.88, "text": "Ln of 1 + x^2 then we add 2x * the"}, {"start": 99.72, "duration": 7.039, "text": "derivative of Ln of 1 + x^2 for this"}, {"start": 103.88, "duration": 8.839, "text": "last part we use the chain rule the"}, {"start": 106.759, "duration": 8.72, "text": "derivative of Ln of 1 + x^2 is 1 1 + x^2"}, {"start": 112.719, "duration": 7.161, "text": "* the derivative of what's inside which"}, {"start": 115.479, "duration": 7.92, "text": "is 2x now we can solve for dydx by mul"}, {"start": 119.88, "duration": 8.56, "text": "multiplying both sides by y remember"}, {"start": 123.399, "duration": 8.92, "text": "that Y is our original function 1 + x^ 2"}, {"start": 128.44, "duration": 6.799, "text": "to 2X and this is our final answer let's"}, {"start": 132.319, "duration": 6.56, "text": "break down what we did first we took the"}, {"start": 135.239, "duration": 7.0, "text": "natural log of both sides then we used"}, {"start": 138.879, "duration": 5.64, "text": "logarithm properties to simplify next we"}, {"start": 142.239, "duration": 4.841, "text": "differentiated both sides using the"}, {"start": 144.519, "duration": 6.241, "text": "chain rule on the left and the product"}, {"start": 147.08, "duration": 8.2, "text": "rule on the right finally we solve for"}, {"start": 150.76, "duration": 6.759, "text": "dydx by multiplying both sides by y"}, {"start": 155.28, "duration": 4.72, "text": "logarithmic differentiation is a"}, {"start": 157.519, "duration": 5.36, "text": "powerful technique that helps us handle"}, {"start": 160.0, "duration": 6.64, "text": "complex functions especially those with"}, {"start": 162.879, "duration": 5.961, "text": "variable bases and exponents remember to"}, {"start": 166.64, "duration": 4.2, "text": "use it whenever you see a function that"}, {"start": 168.84, "duration": 5.88, "text": "would be difficult to differentiate"}, {"start": 170.84, "duration": 3.88, "text": "using standard rules"}], "title": "Mastering Logarithmic Differentiation: Solving (1 + x\u00b2)^(2x) Step-by-Step", "description": "Question: Find the derivative of y = (1 + x\u00b2)^(2x) using logarithmic differentiation.\n\n\ud83d\udcda In this comprehensive calculus tutorial, we break down the solution to finding the derivative of (1 + x\u00b2)^(2x) using logarithmic differentiation. Perfect for calculus students and math enthusiasts!\n\ud83c\udfaf What you\u2019ll learn:\n\u2022 Why logarithmic differentiation is useful\n\u2022 How to handle functions with variable bases and exponents\n\u2022 Step-by-step application of logarithm properties\n\u2022 Using product rule and chain rule effectively\n\u2022 Simplification techniques for complex derivatives\n\u270f\ufe0f Prerequisites:\n\u2022 Basic differentiation rules\n\u2022 Understanding of natural logarithms\n\u2022 Knowledge of product and chain rules\n#Calculus #Mathematics #LogarithmicDifferentiation #MathTutorial #CalcTutorial #MathEducation #LearnMath #Derivatives", "lengthSeconds": 180, "width": 640, "height": 360}
{"video_id": "73oHzUJNRoo", "transcript": [{"start": 0.28, "duration": 5.16, "text": "let's solve this problem step by step we"}, {"start": 3.12, "duration": 6.16, "text": "need to find all natural numbers X where"}, {"start": 5.44, "duration": 4.84, "text": "x^2 is greater than 9 and X Cub is less"}, {"start": 9.28, "duration": 3.84, "text": "than"}, {"start": 10.28, "duration": 5.479, "text": "130 let's create a table to check"}, {"start": 13.12, "duration": 6.6, "text": "different values of X we'll start with"}, {"start": 15.759, "duration": 6.721, "text": "xal 1 and work our way up until we find"}, {"start": 19.72, "duration": 2.76, "text": "all possible"}, {"start": 22.92, "duration": 8.759, "text": "solutions let's check x = 1 1^ 2ar is 1"}, {"start": 28.199, "duration": 6.441, "text": "which is not greater than 9 and 1 cubed"}, {"start": 31.679, "duration": 7.72, "text": "is also 1 Which is less than"}, {"start": 34.64, "duration": 10.439, "text": "130 but since 1 2ar isn't greater than 9"}, {"start": 39.399, "duration": 9.441, "text": "this isn't a solution 4X = 2 2^2 is 4"}, {"start": 45.079, "duration": 8.0, "text": "still not greater than 9 2 cubed is 8"}, {"start": 48.84, "duration": 6.84, "text": "Which is less than 130 but again 2^ 2ar"}, {"start": 53.079, "duration": 7.8, "text": "isn't greater than 9 so this isn't a"}, {"start": 55.68, "duration": 9.36, "text": "solution either 4X = 3 3^2 is is 9 which"}, {"start": 60.879, "duration": 8.321, "text": "is equal to 9 but not greater than 9 3"}, {"start": 65.04, "duration": 9.2, "text": "cubed is 27 Which is less than 130 but"}, {"start": 69.2, "duration": 9.959, "text": "3^2 still isn't greater than 9 now 4X ="}, {"start": 74.24, "duration": 8.36, "text": "4 4^ 2ar is 16 which is greater than 9"}, {"start": 79.159, "duration": 9.401, "text": "and 4 cubed is 64 Which is less than"}, {"start": 82.6, "duration": 10.519, "text": "130 this means x = 4 is a solution let's"}, {"start": 88.56, "duration": 9.4, "text": "check X = 5 5^ 2ar is 25 which is"}, {"start": 93.119, "duration": 6.64, "text": "greater than 9 and 5 cubed is 125 which"}, {"start": 97.96, "duration": 8.199, "text": "is still less than"}, {"start": 99.759, "duration": 10.881, "text": "130 so x = 5 is also a solution finally"}, {"start": 106.159, "duration": 7.881, "text": "let's check x = 6 6^ 2 is 36 which is"}, {"start": 110.64, "duration": 5.92, "text": "greater than 9 but 6 cubed is"}, {"start": 114.04, "duration": 8.32, "text": "26 which is greater than"}, {"start": 116.56, "duration": 8.0, "text": "130 so x = 6 is not a solution let's"}, {"start": 122.36, "duration": 5.799, "text": "visualize this on a number line to see"}, {"start": 124.56, "duration": 3.599, "text": "our solution set more"}, {"start": 128.479, "duration": 6.961, "text": "clearly therefore our solution set is"}, {"start": 131.84, "duration": 8.36, "text": "the set containing four and five we can"}, {"start": 135.44, "duration": 6.96, "text": "write this in set notation as 4 five"}, {"start": 140.2, "duration": 5.399, "text": "these are the only natural numbers that"}, {"start": 142.4, "duration": 9.36, "text": "satisfy both conditions having a square"}, {"start": 145.599, "duration": 6.161, "text": "greater than 9 and a cube less than 130"}], "title": "Solving Set Problems with Natural Numbers {\ud835\udc65^2 grater than 9 and  \ud835\udc65^3 less than130}.", "description": "Dive into solving set theory problems involving natural numbers! In this video, we explore a specific problem to find the subset of natural numbers x where {\ud835\udc65^2 grater than 9 and  \ud835\udc65^3 less than130}. Follow along as we meticulously work through each condition to identify the solution. Ideal for understanding mathematical expressions and their applications in set problems.\n#SetTheory #NaturalNumbers #MathProblems #Algebra #MathTutorial #ProblemSolving", "lengthSeconds": 154, "width": 640, "height": 360}
{"video_id": "kmadS7SKZfs", "transcript": [{"start": 0.32, "duration": 5.039, "text": "let's solve this calculus problem step"}, {"start": 2.52, "duration": 7.88, "text": "by step using the quotient rule we need"}, {"start": 5.359, "duration": 7.721, "text": "to find the derivative of 1 + sinx / 1 -"}, {"start": 10.4, "duration": 6.4, "text": "sinx first let's recall the quotient"}, {"start": 13.08, "duration": 5.72, "text": "rule formula for a fraction U over V the"}, {"start": 16.8, "duration": 4.8, "text": "derivative is V *"}, {"start": 18.8, "duration": 7.559, "text": "DX minus U *"}, {"start": 21.6, "duration": 10.479, "text": "dvdx all over V ^2 let's identify our u"}, {"start": 26.359, "duration": 9.281, "text": "and v functions here u = 1 + sinx and V"}, {"start": 32.079, "duration": 6.48, "text": "= 1 - sinx now let's find the"}, {"start": 35.64, "duration": 6.84, "text": "derivatives of u and v the derivative of"}, {"start": 38.559, "duration": 7.401, "text": "U is cine X and the derivative of V is"}, {"start": 42.48, "duration": 7.759, "text": "cosine X let's substitute these into the"}, {"start": 45.96, "duration": 10.52, "text": "quotient rule formula we get 1 - sin x *"}, {"start": 50.239, "duration": 10.64, "text": "cosine X - 1 + sin x * cosin X all over"}, {"start": 56.48, "duration": 7.639, "text": "1 - sin x^2 now let's expand the"}, {"start": 60.879, "duration": 7.56, "text": "numerator from the first term we get"}, {"start": 64.119, "duration": 8.04, "text": "cine x - sin x cosine X from the second"}, {"start": 68.439, "duration": 7.521, "text": "term we get plus cosine X Plus sin x"}, {"start": 72.159, "duration": 7.081, "text": "cine X let's simplify the numerator the"}, {"start": 75.96, "duration": 6.76, "text": "sin x cine X terms cancel out and we're"}, {"start": 79.24, "duration": 6.559, "text": "left with cine X+ cosine X which is 2"}, {"start": 82.72, "duration": 6.88, "text": "cosine X and there we have our final"}, {"start": 85.799, "duration": 10.36, "text": "answer the derivative of 1 + sin x over"}, {"start": 89.6, "duration": 9.32, "text": "1 1 - sin x = 2 cosin x / 1 - sin x^2"}, {"start": 96.159, "duration": 5.28, "text": "this problem demonstrates how to use the"}, {"start": 98.92, "duration": 4.72, "text": "quotient rule to find the derivative of"}, {"start": 101.439, "duration": 4.921, "text": "a complex fraction involving"}, {"start": 103.64, "duration": 5.68, "text": "trigonometric functions remember to"}, {"start": 106.36, "duration": 5.359, "text": "carefully identify your u and v"}, {"start": 109.32, "duration": 7.56, "text": "functions find their derivatives and"}, {"start": 111.719, "duration": 5.161, "text": "then apply the quotient rule formula"}], "title": "Differentiating [(\ud835\udfcf+\ud835\udc94\ud835\udc8a\ud835\udc8f\ud835\udc99)/(\ud835\udfcf-sinx)] Using the Quotient Rule: Step-by-Step Tutorial", "description": "Master the art of differentiation with this detailed tutorial on finding the derivative of the function [(\ud835\udfcf+\ud835\udc94\ud835\udc8a\ud835\udc8f\ud835\udc99)/(\ud835\udfcf-sinx)] using the quotient rule. This video walks you through each step and simplifies complex calculus concepts for easy understanding. Perfect for students, educators, and math enthusiasts aiming to strengthen their calculus skills.\n#Calculus #Differentiation #QuotientRule #MathTutorial #StepByStep", "lengthSeconds": 116, "width": 640, "height": 360}
{"video_id": "rKWXzrOJt3A", "transcript": [{"start": 0.36, "duration": 6.32, "text": "welcome today we'll solve a challenging"}, {"start": 3.159, "duration": 5.44, "text": "derivative problem using logarithmic"}, {"start": 6.68, "duration": 6.6, "text": "differentiation let's find the"}, {"start": 8.599, "duration": 8.081, "text": "derivative of y = sin of X ra to the"}, {"start": 13.28, "duration": 5.52, "text": "power of tangent X this is a complex"}, {"start": 16.68, "duration": 4.599, "text": "function because we have a trigonometric"}, {"start": 18.8, "duration": 5.0, "text": "function raised to another trigonometric"}, {"start": 21.279, "duration": 5.601, "text": "function the best approach here is to"}, {"start": 23.8, "duration": 5.559, "text": "use logarithmic differentiation let's"}, {"start": 26.88, "duration": 5.199, "text": "start by taking the natural log of both"}, {"start": 29.359, "duration": 6.561, "text": "sides using the logarithm property that"}, {"start": 32.079, "duration": 8.0, "text": "says log of a to the power Bal B * log"}, {"start": 35.92, "duration": 8.0, "text": "of a we can simplify this to Ln y ="}, {"start": 40.079, "duration": 6.521, "text": "tangent x * Ln of sin x now we'll"}, {"start": 43.92, "duration": 7.439, "text": "differentiate both sides with respect to"}, {"start": 46.6, "duration": 8.16, "text": "X on the left side we get 1/ y *"}, {"start": 51.359, "duration": 5.52, "text": "dydx on the right side we'll need to use"}, {"start": 54.76, "duration": 5.52, "text": "the product rule since we have the"}, {"start": 56.879, "duration": 7.081, "text": "product of two functions let's evaluate"}, {"start": 60.28, "duration": 8.12, "text": "each derivative separately for Ln of"}, {"start": 63.96, "duration": 8.56, "text": "sinx the derivative is cine X sin x and"}, {"start": 68.4, "duration": 6.84, "text": "for tangent X the derivative is secant 2"}, {"start": 72.52, "duration": 6.88, "text": "x now let's solve for"}, {"start": 75.24, "duration": 7.32, "text": "dydx first multiply both sides by y"}, {"start": 79.4, "duration": 7.92, "text": "let's simplify the trigonometric terms"}, {"start": 82.56, "duration": 7.879, "text": "notice that tangent x * cine x/ sin x"}, {"start": 87.32, "duration": 7.36, "text": "simplifies to just one since cos sin x"}, {"start": 90.439, "duration": 6.0, "text": "over sin x is 1 / tangent X finally we"}, {"start": 94.68, "duration": 4.439, "text": "can substitute back the original"}, {"start": 96.439, "duration": 5.921, "text": "function for y giving us our final"}, {"start": 99.119, "duration": 7.441, "text": "answer the derivative equals sinx to the"}, {"start": 102.36, "duration": 8.079, "text": "power of tangent X time the quantity 1 +"}, {"start": 106.56, "duration": 6.76, "text": "natural log of sin x * SEC 2 x let's"}, {"start": 110.439, "duration": 6.161, "text": "review the key steps we used first we"}, {"start": 113.32, "duration": 6.839, "text": "took the natural log of both sides then"}, {"start": 116.6, "duration": 6.879, "text": "we used logarithm properties to simplify"}, {"start": 120.159, "duration": 7.201, "text": "we differentiated using the product rule"}, {"start": 123.479, "duration": 6.721, "text": "and chain rule finally we substituted"}, {"start": 127.36, "duration": 5.519, "text": "back the original function this is a"}, {"start": 130.2, "duration": 4.96, "text": "great example of how logarithmic"}, {"start": 132.879, "duration": 5.161, "text": "differentiation can help us find"}, {"start": 135.16, "duration": 5.88, "text": "derivatives of complex exponential"}, {"start": 138.04, "duration": 3.0, "text": "functions"}], "title": "Finding the Derivative of (sin x)^(tan x) | Logarithmic Differentiation Made Easy", "description": "Question: Find the derivative of y = (sin x)^(tan x) using logarithmic differentiation.\n\n\ud83d\udd0d Master logarithmic differentiation with this step-by-step guide to finding the derivative of (sin x)^(tan x). Perfect for Calculus students and math enthusiasts!\nIn this video, we cover:\n\u2022 Logarithmic differentiation technique\n\u2022 Product rule application\n\u2022 Chain rule implementation\n\u2022 Trigonometric derivatives\n\u2022 Step-by-step solution walkthrough\n\ud83c\udfaf Learning Outcomes:\nUnderstand when to use logarithmic differentiation\nMaster complex derivative problems\nApply multiple calculus rules together\nSimplify trigonometric expressions\n\ud83d\udcda Prerequisites:\nBasic differentiation rules\nKnowledge of trigonometric functions\nUnderstanding of logarithm properties\n#Calculus #Mathematics #MathTutorial #CalcTutorial #MathEducation #LogarithmicDifferentiation #LearnMath #MathHelp #STEM", "lengthSeconds": 144, "width": 640, "height": 360}
{"video_id": "p3C0VyD7gzo", "transcript": [{"start": 0.24, "duration": 3.76, "text": "let's solve a challenging"}, {"start": 1.56, "duration": 5.319, "text": "differentiation problem using"}, {"start": 4.0, "duration": 7.759, "text": "logarithmic differentiation we need to"}, {"start": 6.879, "duration": 8.121, "text": "find the derivative of y = e to x^2 *"}, {"start": 11.759, "duration": 8.36, "text": "inverse tangent of X all divided by the"}, {"start": 15.0, "duration": 7.439, "text": "square root of 1 + x^2 this is a complex"}, {"start": 20.119, "duration": 5.361, "text": "function involving products quotients"}, {"start": 22.439, "duration": 6.481, "text": "and compositions the best approach is to"}, {"start": 25.48, "duration": 7.64, "text": "take the natural logarithm of both sides"}, {"start": 28.92, "duration": 6.84, "text": "this gives us l n of Y = Ln of our"}, {"start": 33.12, "duration": 6.52, "text": "entire expression using the properties"}, {"start": 35.76, "duration": 7.24, "text": "of logarithms we can split this into the"}, {"start": 39.64, "duration": 8.8, "text": "difference of logarithms the numerator"}, {"start": 43.0, "duration": 11.039, "text": "becomes Ln of e x^2 * inverse tangent of"}, {"start": 48.44, "duration": 8.08, "text": "xus Ln of the < TK of 1 + x^2 again"}, {"start": 54.039, "duration": 8.081, "text": "using logarithm properties we can split"}, {"start": 56.52, "duration": 9.959, "text": "the first term into Ln of e to X 2 + Ln"}, {"start": 62.12, "duration": 9.16, "text": "of inverse tangent of X and remember Ln"}, {"start": 66.479, "duration": 7.361, "text": "of a square T is 12 Ln of the expression"}, {"start": 71.28, "duration": 6.72, "text": "now we can differentiate both sides with"}, {"start": 73.84, "duration": 5.36, "text": "respect to X on the left side we get 1 /"}, {"start": 78.0, "duration": 4.6, "text": "y *"}, {"start": 79.2, "duration": 7.279, "text": "dydx on the right side we use the chain"}, {"start": 82.6, "duration": 7.36, "text": "rule for each term the derivative of x^2"}, {"start": 86.479, "duration": 6.521, "text": "is 2x for the natural log terms we use"}, {"start": 89.96, "duration": 6.519, "text": "use the chain rule this gives us 1 /"}, {"start": 93.0, "duration": 8.28, "text": "inverse tangent of x * its derivative"}, {"start": 96.479, "duration": 11.041, "text": "which is 1 1 + x^2 and for the last term"}, {"start": 101.28, "duration": 10.24, "text": "we get -2 * 1 1 + x^2 * 2x now we can"}, {"start": 107.52, "duration": 8.32, "text": "solve for dydx by multiplying both sides"}, {"start": 111.52, "duration": 6.919, "text": "by Y and finally we substitute back the"}, {"start": 115.84, "duration": 5.639, "text": "original expression for y to get our"}, {"start": 118.439, "duration": 7.801, "text": "complete derivative this gives us e to"}, {"start": 121.479, "duration": 9.4, "text": "x^ 2 * inverse tangent of X over < TK of"}, {"start": 126.24, "duration": 7.279, "text": "1 + x^2 times the expression in Brackets"}, {"start": 130.879, "duration": 5.161, "text": "and that's how we use logarithmic"}, {"start": 133.519, "duration": 5.401, "text": "differentiation to find the derivative"}, {"start": 136.04, "duration": 5.12, "text": "of this complex function the key steps"}, {"start": 138.92, "duration": 5.399, "text": "were taking the natural log of both"}, {"start": 141.16, "duration": 8.159, "text": "sides using log properties to simplify"}, {"start": 144.319, "duration": 5.0, "text": "and then differentiating term by term"}], "title": "Differentiating Complex Logarithmic Functions [(\ud835\udc52^(\ud835\udc65^2 ) \ud835\udc61\ud835\udc4e\ud835\udc5b^(\u22121) \ud835\udc65)/\u221a(1+\ud835\udc65^2 )] Step-by-Step", "description": "Problem\nDifferentiate:  [(\ud835\udc52^(\ud835\udc65^2 ) \ud835\udc61\ud835\udc4e\ud835\udc5b^(\u22121) \ud835\udc65)/\u221a(1+\ud835\udc65^2 )] \n\nUnlock the secrets of differentiating complex logarithmic functions with this easy-to-follow, step-by-step guide. We\u2019ll break down each part, making it simple to tackle even the toughest math problems!\n#MathTutorial #Differentiation #LogarithmicFunctions #Calculus #StepByStep", "lengthSeconds": 152, "width": 640, "height": 360}
{"video_id": "HuBTvORiU5E", "transcript": [{"start": 3.0, "duration": 4.719, "text": "let's solve a challenging derivative"}, {"start": 5.12, "duration": 5.08, "text": "problem using logarithmic"}, {"start": 7.719, "duration": 6.521, "text": "differentiation our goal is to find the"}, {"start": 10.2, "duration": 7.52, "text": "derivative of y = x raed to the natural"}, {"start": 14.24, "duration": 6.0, "text": "log of x this is a complex function"}, {"start": 17.72, "duration": 5.319, "text": "where X appears both in the base and in"}, {"start": 20.24, "duration": 6.24, "text": "the exponent let's start by letting y"}, {"start": 23.039, "duration": 5.441, "text": "equal x to the lnx power the key to"}, {"start": 26.48, "duration": 5.16, "text": "solving this type of problem is to take"}, {"start": 28.48, "duration": 5.2, "text": "the natural logarithm of both sides"}, {"start": 31.64, "duration": 4.88, "text": "using the logarithm property that says"}, {"start": 33.68, "duration": 6.76, "text": "Ln of X raised to a power equals that"}, {"start": 36.52, "duration": 9.76, "text": "power * lnx we can simplify this to Ln x"}, {"start": 40.44, "duration": 8.119, "text": "* Ln X or Ln x^2 now comes the key step"}, {"start": 46.28, "duration": 5.56, "text": "we're going to differentiate both sides"}, {"start": 48.559, "duration": 5.281, "text": "with respect to X on the left side we"}, {"start": 51.84, "duration": 6.199, "text": "get 1/ y *"}, {"start": 53.84, "duration": 8.039, "text": "dydx and on the right side we get 2 Ln x"}, {"start": 58.039, "duration": 7.48, "text": "* the derivative of Ln X which is 1 /x"}, {"start": 61.879, "duration": 7.961, "text": "Let's solve for dydx by multiplying both"}, {"start": 65.519, "duration": 8.92, "text": "sides by y then we can substitute back y"}, {"start": 69.84, "duration": 7.48, "text": "= x to the Ln X power finally we can"}, {"start": 74.439, "duration": 4.841, "text": "simplify this to get our answer let's"}, {"start": 77.32, "duration": 4.799, "text": "highlight the key techniques we used to"}, {"start": 79.28, "duration": 5.199, "text": "solve this problem first we used"}, {"start": 82.119, "duration": 6.201, "text": "logarithmic differentiation by taking"}, {"start": 84.479, "duration": 6.481, "text": "the natural log of both sides second we"}, {"start": 88.32, "duration": 5.68, "text": "used logarithm properties to simplify"}, {"start": 90.96, "duration": 5.519, "text": "the expression and finally we use the"}, {"start": 94.0, "duration": 5.2, "text": "chain Rule and product rule for"}, {"start": 96.479, "duration": 8.401, "text": "differentiation therefore the derivative"}, {"start": 99.2, "duration": 10.8, "text": "of x to the Ln X power is 2 * x^ of Ln X"}, {"start": 104.88, "duration": 8.279, "text": "- 1 * lnx this is a great example of how"}, {"start": 110.0, "duration": 8.119, "text": "logarithmic differentiation can help us"}, {"start": 113.159, "duration": 4.96, "text": "solve complex derivative problems"}], "title": "Logarithmic Differentiation: Solving [y = x^(ln x)] | Calculus Made Clear", "description": "Question:\nFind the derivative of [y = x^(ln x)] with respect to x.\n\nMaster logarithmic differentiation with this step-by-step solution of y = x^(ln x). In this comprehensive tutorial, we break down a challenging calculus problem that combines exponential and logarithmic functions.\n\ud83c\udfaf What you\u2019ll learn:\n\u2022 Why logarithmic differentiation is necessary\n\u2022 How to apply natural logarithm properties\n\u2022 Step-by-step differentiation process\n\u2022 Common mistakes to avoid\n\u2022 Simplification techniques\n\ud83d\udd0d Perfect for:\n\u2022 Calculus students\n\u2022 AP Calculus preparation\n\u2022 College math review\n\u2022 IB Mathematics\n\ud83d\udca1 Free practice problems in description!\n#Mathematics #Calculus #MathTutorial #Education #LearnMath #APCalculus #CollegeMath #StudyWithMe #MathHelp #CalculusTricks", "lengthSeconds": 125, "width": 640, "height": 360}
{"video_id": "Zdgg3p2AgUQ", "transcript": [{"start": 0.28, "duration": 6.079, "text": "let's solve this complex derivative"}, {"start": 2.679, "duration": 6.0, "text": "problem step by step our goal is to find"}, {"start": 6.359, "duration": 5.721, "text": "the derivative of this expression with"}, {"start": 8.679, "duration": 6.481, "text": "respect to X let's break this down into"}, {"start": 12.08, "duration": 5.4, "text": "manageable steps first let's set our"}, {"start": 15.16, "duration": 5.16, "text": "expression equal to Y to make it easier"}, {"start": 17.48, "duration": 7.039, "text": "to work with now let's expand the"}, {"start": 20.32, "duration": 9.32, "text": "numerator when we Square x^2 + 1 we get"}, {"start": 24.519, "duration": 8.88, "text": "x 4 + 2x^2 + 1 in the denominator we can"}, {"start": 29.64, "duration": 7.2, "text": "re write the cube root as x^ the 2/3"}, {"start": 33.399, "duration": 6.761, "text": "power now comes a key step we can"}, {"start": 36.84, "duration": 6.08, "text": "distribute the division by x to the 2/3"}, {"start": 40.16, "duration": 5.6, "text": "to each term in the numerator when we do"}, {"start": 42.92, "duration": 6.479, "text": "this we subtract the exponents let's"}, {"start": 45.76, "duration": 8.92, "text": "simplify these exponents for - 2/3"}, {"start": 49.399, "duration": 8.521, "text": "becomes 10/3 2 - 2/3 becomes 4/3 and"}, {"start": 54.68, "duration": 5.879, "text": "-2/3 stays as is now we can"}, {"start": 57.92, "duration": 5.92, "text": "differentiate each term using the power"}, {"start": 60.559, "duration": 6.56, "text": "rule remember when using the power rule"}, {"start": 63.84, "duration": 6.08, "text": "we multiply by the power and reduce the"}, {"start": 67.119, "duration": 6.521, "text": "exponent by one let's apply the power"}, {"start": 69.92, "duration": 7.32, "text": "rule to each term for the first term we"}, {"start": 73.64, "duration": 6.96, "text": "multiply by 10/3 and subtract one from"}, {"start": 77.24, "duration": 7.559, "text": "the exponent for the second term we"}, {"start": 80.6, "duration": 8.64, "text": "multiply by 4/3 and subtract one for the"}, {"start": 84.799, "duration": 8.161, "text": "last term we multiply by -2/3 and"}, {"start": 89.24, "duration": 9.72, "text": "subtract 1 1 now let's simplify these"}, {"start": 92.96, "duration": 11.68, "text": "exponents 10/3 - 1 becomes 7/3 4/3 - 1"}, {"start": 98.96, "duration": 9.199, "text": "becomes 1/3 and -2/3 - 1 becomes -"}, {"start": 104.64, "duration": 6.479, "text": "5/3 finally we can factor out 2/3 to"}, {"start": 108.159, "duration": 6.28, "text": "write our answer in a more compact form"}, {"start": 111.119, "duration": 5.68, "text": "our final answer is 2/3 times the"}, {"start": 114.439, "duration": 4.921, "text": "expression in Brackets and there we have"}, {"start": 116.799, "duration": 5.401, "text": "it we've successfully found the"}, {"start": 119.36, "duration": 5.759, "text": "derivative of this complex expression by"}, {"start": 122.2, "duration": 7.04, "text": "breaking it down into manageable steps"}, {"start": 125.119, "duration": 4.121, "text": "and using the power rule"}], "title": "Mastering Complex Derivatives [(\ud835\udc65^2+1)^2/\u221b(\ud835\udc65^2 )]  Solving Power Functions with Cube Roots |", "description": "Question: Find the derivative of [(\ud835\udc65^2+1)^2/\u221b(\ud835\udc65^2 )]    with respect to x.\n\nMaster the art of solving complex derivatives involving power functions and cube roots in this detailed tutorial. We break down the solution of d/dx[(x\u00b2+1)\u00b2/\u221b(x\u00b2)] step by step.\nIn this video, you\u2019ll learn:\n\ud83d\udcda How to simplify complex fractions with roots\n\u270f\ufe0f Converting cube roots to fractional exponents\n\ud83d\udd22 Laws of exponents and their applications\n\ud83d\udcd0 Power rule for differentiation\n\ud83d\udca1 Step-by-step algebraic simplification\nPerfect for:\n\u2022 Calculus students\n\u2022 Math majors\n\u2022 Engineering students\n\u2022 Test preparation (AP Calculus, IB Mathematics)\n#Mathematics #Calculus #Derivatives #MathTutorial #PowerRule #StudyWithMe #ExamPrep #MathEducation #STEM #EngineeringMath", "lengthSeconds": 132, "width": 640, "height": 360}
{"video_id": "rnrDZvSdSEs", "transcript": [{"start": 0.32, "duration": 6.439, "text": "let's solve this complex inverse cosine"}, {"start": 3.679, "duration": 5.201, "text": "differentiation problem step by step"}, {"start": 6.759, "duration": 4.721, "text": "first let's look at what we're trying to"}, {"start": 8.88, "duration": 6.48, "text": "find to solve this we'll use a"}, {"start": 11.48, "duration": 7.24, "text": "substitution trick let's say x = cine"}, {"start": 15.36, "duration": 7.04, "text": "Theta this means Theta equal inverse"}, {"start": 18.72, "duration": 7.399, "text": "cosine of x now let's substitute this"}, {"start": 22.4, "duration": 8.039, "text": "into our original equation we get y ="}, {"start": 26.119, "duration": 8.12, "text": "inverse cosine of < TK 1 + Co cosine"}, {"start": 30.439, "duration": 8.841, "text": "Theta / 2 here's where we use a key"}, {"start": 34.239, "duration": 9.8, "text": "trigonometric identity 1 + cine thet = 2"}, {"start": 39.28, "duration": 7.799, "text": "cosine 2 Theta / 2 using this identity"}, {"start": 44.039, "duration": 8.04, "text": "our expression becomes inverse cosine of"}, {"start": 47.079, "duration": 7.761, "text": "cosine 2 Theta over 22 power simplifying"}, {"start": 52.079, "duration": 6.041, "text": "the fraction inside we get inverse"}, {"start": 54.84, "duration": 7.08, "text": "cosine of cosine 2 Theta /"}, {"start": 58.12, "duration": 7.96, "text": "2^2 power taking the square root we get"}, {"start": 61.92, "duration": 7.12, "text": "inverse cosine of cosine Theta / 2 now"}, {"start": 66.08, "duration": 5.679, "text": "here's the key Insight inverse cosine"}, {"start": 69.04, "duration": 5.079, "text": "and cosine are inverse functions so they"}, {"start": 71.759, "duration": 5.761, "text": "cancel out leaving us with just Theta"}, {"start": 74.119, "duration": 9.881, "text": "over 2 remember that Theta equal inverse"}, {"start": 77.52, "duration": 10.16, "text": "cosine X so y = 12 inverse cine X now we"}, {"start": 84.0, "duration": 7.079, "text": "can differentiate the derivative of y"}, {"start": 87.68, "duration": 7.68, "text": "with respect to x equals the derivative"}, {"start": 91.079, "duration": 6.841, "text": "of 12 inverse cine X we can pull out the"}, {"start": 95.36, "duration": 5.32, "text": "1/2 and then use the fact that the"}, {"start": 97.92, "duration": 9.04, "text": "derivative of inverse cosine X"}, {"start": 100.68, "duration": 12.0, "text": "is-1 over < TK of 1 - x^2 therefore our"}, {"start": 106.96, "duration": 9.159, "text": "final answer is -1/2 * the < TK of 1 -"}, {"start": 112.68, "duration": 6.719, "text": "x^2 and that completes our solution the"}, {"start": 116.119, "duration": 4.401, "text": "key was using the substitution xal cine"}, {"start": 119.399, "duration": 3.481, "text": "Theta"}, {"start": 120.52, "duration": 5.639, "text": "and applying the double angle formula"}, {"start": 122.88, "duration": 3.279, "text": "for cosine"}], "title": "Understanding Inverse Trigonometric Functions [\ud835\udc50\ud835\udc5c\ud835\udc60^(\u22121)  ((1+\ud835\udc65)/2)^(1/2)] Step-by-Step Explanation", "description": "Delve into the intricacies of inverse trigonometric functions with our detailed, step-by-step breakdown. In this video, we\u2019ll guide you through solving complex problems involving [\ud835\udc50\ud835\udc5c\ud835\udc60^(\u22121)  ((1+\ud835\udc65)/2)^(1/2)]  and differentiation, ensuring you grasp each concept thoroughly. Perfect for students and math enthusiasts looking to enhance their understanding.\n#MathTutorial #InverseTrigFunctions #Calculus #Trigonometry #EducationalVideo", "lengthSeconds": 129, "width": 640, "height": 360}
{"video_id": "QYgZEGRPmPs", "transcript": [{"start": 0.32, "duration": 4.56, "text": "let's solve this complex derivative"}, {"start": 2.72, "duration": 5.919, "text": "problem step by step using the chain"}, {"start": 4.88, "duration": 7.44, "text": "rule we need to find the derivative of e"}, {"start": 8.639, "duration": 5.521, "text": "to the power of sin of sare < TK of X"}, {"start": 12.32, "duration": 5.52, "text": "let's break down the structure of this"}, {"start": 14.16, "duration": 6.48, "text": "function we have three nested functions"}, {"start": 17.84, "duration": 5.359, "text": "first the outer exponential function"}, {"start": 20.64, "duration": 4.959, "text": "then the S function in the middle and"}, {"start": 23.199, "duration": 5.361, "text": "finally the square root of x as the"}, {"start": 25.599, "duration": 5.76, "text": "innermost function to solve this we'll"}, {"start": 28.56, "duration": 4.8, "text": "use the chain rule multip times the"}, {"start": 31.359, "duration": 5.081, "text": "chain rule states that when"}, {"start": 33.36, "duration": 5.12, "text": "differentiating a composite function we"}, {"start": 36.44, "duration": 4.04, "text": "multiply the derivative of the outer"}, {"start": 38.48, "duration": 4.36, "text": "function by the derivative of the inner"}, {"start": 40.48, "duration": 4.68, "text": "function let's start with the first"}, {"start": 42.84, "duration": 5.559, "text": "application of the chain rule the"}, {"start": 45.16, "duration": 6.399, "text": "derivative of e to the power of anything"}, {"start": 48.399, "duration": 6.48, "text": "is e to that same Power Times the"}, {"start": 51.559, "duration": 5.921, "text": "derivative of what's inside now we need"}, {"start": 54.879, "duration": 5.84, "text": "to find the derivative of s of square <"}, {"start": 57.48, "duration": 6.399, "text": "TK X using the chain rule again"}, {"start": 60.719, "duration": 7.08, "text": "the derivative of s is cosine times the"}, {"start": 63.879, "duration": 6.24, "text": "derivative of what's inside finally"}, {"start": 67.799, "duration": 6.161, "text": "let's find the derivative of square < TK"}, {"start": 70.119, "duration": 7.761, "text": "of X the derivative of square < TK X is"}, {"start": 73.96, "duration": 6.6, "text": "1 / 2 < TK X now we can write our final"}, {"start": 77.88, "duration": 5.84, "text": "answer in a more organized way the"}, {"start": 80.56, "duration": 8.84, "text": "derivative equals cosine of s < TK of x"}, {"start": 83.72, "duration": 8.52, "text": "* e to the power of s of < TK X all / 2"}, {"start": 89.4, "duration": 5.6, "text": "< TK of x X let's verify that we didn't"}, {"start": 92.24, "duration": 5.44, "text": "miss anything we use the chain rule"}, {"start": 95.0, "duration": 6.159, "text": "three times first for the exponential"}, {"start": 97.68, "duration": 5.84, "text": "function then for the S function and"}, {"start": 101.159, "duration": 5.0, "text": "finally for the square root each"}, {"start": 103.52, "duration": 5.52, "text": "application gave us a term in our final"}, {"start": 106.159, "duration": 5.6, "text": "answer and that's how we find the"}, {"start": 109.04, "duration": 5.999, "text": "derivative of this complex function"}, {"start": 111.759, "duration": 5.761, "text": "using the chain rule multiple times"}, {"start": 115.039, "duration": 5.08, "text": "remember when dealing with nested"}, {"start": 117.52, "duration": 5.959, "text": "functions apply the chain rule for the"}, {"start": 120.119, "duration": 6.36, "text": "outside in and multiply all the terms"}, {"start": 123.479, "duration": 3.0, "text": "together"}], "title": "9.3-2.4 Mastering the Chain Rule Step-by-Step Differentiation Guide for (\ud835\udc52^\ud835\udc60\ud835\udc56\ud835\udc5b\u221a\ud835\udc65)", "description": "\u201cIn this video, we\u2019ll dive deep into differentiating the composite function (\ud835\udc52^\ud835\udc60\ud835\udc56\ud835\udc5b\u221a\ud835\udc65) using the chain rule. Join us as we break down each step in detail, making it easy to follow and understand. Perfect for students looking to strengthen their calculus skills!\u201d\nHashtags:\n#Calculus #ChainRule #Differentiation #MathTutorial #MathHelp #hsc #hschighermath #1stpaper\n#hschighermath1stpaper\n#englishversion #cpmrana", "lengthSeconds": 125, "width": 640, "height": 360}
{"video_id": "VFAmg_Sy6nI", "transcript": [{"start": 0.28, "duration": 4.999, "text": "let's solve this derivative step by step"}, {"start": 2.919, "duration": 7.64, "text": "using the product rule we want to find"}, {"start": 5.279, "duration": 7.921, "text": "the derivative of x^2 * cine X to solve"}, {"start": 10.559, "duration": 4.921, "text": "this we'll use the product rule the"}, {"start": 13.2, "duration": 4.88, "text": "product rule states that the derivative"}, {"start": 15.48, "duration": 4.639, "text": "of a product of two functions equals the"}, {"start": 18.08, "duration": 5.16, "text": "first function time the derivative of"}, {"start": 20.119, "duration": 5.441, "text": "the second plus the second function time"}, {"start": 23.24, "duration": 7.24, "text": "the derivative of the first let's"}, {"start": 25.56, "duration": 9.24, "text": "identify our functions let U of X = x^2"}, {"start": 30.48, "duration": 7.56, "text": "and V ofx = cosine X now let's find the"}, {"start": 34.8, "duration": 6.32, "text": "derivatives of each function separately"}, {"start": 38.04, "duration": 7.199, "text": "the derivative of x^2 is 2X and the"}, {"start": 41.12, "duration": 6.68, "text": "derivative of cosine X is sinx let's"}, {"start": 45.239, "duration": 9.0, "text": "substitute these into the product rule"}, {"start": 47.8, "duration": 9.68, "text": "we get x^2 * sinx plus cosine x * 2x now"}, {"start": 54.239, "duration": 6.28, "text": "let's simplify this expression first"}, {"start": 57.48, "duration": 8.2, "text": "we'll rearrange the terms to get 2X cine"}, {"start": 60.519, "duration": 9.041, "text": "x - x^2 sin x finally we can factor out"}, {"start": 65.68, "duration": 8.56, "text": "an X this gives us x * the quantity 2"}, {"start": 69.56, "duration": 7.04, "text": "cosine x - x sin x let's visualize how"}, {"start": 74.24, "duration": 5.68, "text": "this derivative works with a graph"}, {"start": 76.6, "duration": 6.879, "text": "here's the original function x^2 cine X"}, {"start": 79.92, "duration": 6.28, "text": "in blue and it's derivative in red to"}, {"start": 83.479, "duration": 6.761, "text": "summarize we use the product rule to"}, {"start": 86.2, "duration": 6.04, "text": "find the derivative of x^2 cosine X we"}, {"start": 90.24, "duration": 4.159, "text": "identified our functions found their"}, {"start": 92.24, "duration": 5.08, "text": "individual derivatives applied the"}, {"start": 94.399, "duration": 7.0, "text": "product rule and simplified our result"}, {"start": 97.32, "duration": 4.079, "text": "to get our final answer"}], "title": "Finding the Derivative of x^2 cosx  Using the Product Rule | Step-by-Step Explanation", "description": "In this video, discover how to differentiate x^2 cosx using the Product Rule in calculus.\n\nWhat You'll Learn:  \n- Definition of the Product Rule  \n- Step-by-step differentiation of x^2 cosx  \n- Formula breakdown and example problem-solving  \n\nThis example is ideal for calculus beginners and anyone reviewing the Product Rule!\n\ud83d\udca1 Don't forget to LIKE, SHARE, and SUBSCRIBE for more math tutorials!\n#Calculus #ProductRule #Differentiation #Derivative #MathTutorial #LearnMath #MathEducation #MathHelp", "lengthSeconds": 100, "width": 640, "height": 360}
{"video_id": "kTv1epJ-SW4", "transcript": [{"start": 0.24, "duration": 5.039, "text": "let's solve this calculus problem step"}, {"start": 2.52, "duration": 4.759, "text": "by step we need to differentiate the"}, {"start": 5.279, "duration": 4.761, "text": "natural logarithm of a quotient with"}, {"start": 7.279, "duration": 4.681, "text": "respect to X here's our starting"}, {"start": 10.04, "duration": 5.88, "text": "expression we need to find the"}, {"start": 11.96, "duration": 9.88, "text": "derivative of the natural log of x^2 + x"}, {"start": 15.92, "duration": 7.84, "text": "+ 1 /x^2 - x + 1 the first key step is"}, {"start": 21.84, "duration": 4.32, "text": "to use the logarithm property that"}, {"start": 23.76, "duration": 5.0, "text": "states the log of a quotient equals the"}, {"start": 26.16, "duration": 4.76, "text": "log of the numerator minus the log of"}, {"start": 28.76, "duration": 8.0, "text": "the denominator this gives us the"}, {"start": 30.92, "duration": 10.4, "text": "derivative of log of x^2 + x + 1 - log"}, {"start": 36.76, "duration": 7.119, "text": "of x^2 - x +1 now we can use the chain"}, {"start": 41.32, "duration": 6.559, "text": "rule to differentiate each logarithm"}, {"start": 43.879, "duration": 7.081, "text": "term separately remember the derivative"}, {"start": 47.879, "duration": 5.721, "text": "of natural log of U is 1/ U * the"}, {"start": 50.96, "duration": 5.68, "text": "derivative of U let's evaluate the"}, {"start": 53.6, "duration": 6.68, "text": "derivatives inside the parentheses the"}, {"start": 56.64, "duration": 8.72, "text": "derivative of x^2 is 2X and the deriv of"}, {"start": 60.28, "duration": 7.24, "text": "X is 1 so we get 2x + 1 for both terms"}, {"start": 65.36, "duration": 4.96, "text": "to combine these fractions we need a"}, {"start": 67.52, "duration": 5.44, "text": "common denominator we multiply the"}, {"start": 70.32, "duration": 5.36, "text": "numerator and denominator of each"}, {"start": 72.96, "duration": 6.72, "text": "fraction by the denominator of the other"}, {"start": 75.68, "duration": 6.84, "text": "fraction now comes the algebra let's"}, {"start": 79.68, "duration": 9.28, "text": "expand the numerator carefully first"}, {"start": 82.52, "duration": 11.559, "text": "term gives us 2x Cub - 2x^2 + 2x + x^2 -"}, {"start": 88.96, "duration": 11.28, "text": "x + 1 the second term expands to -2X Cub"}, {"start": 94.079, "duration": 8.961, "text": "- 2x^2 - 2x + x^2 + x + 1 combining like"}, {"start": 100.24, "duration": 8.8, "text": "terms in the numerator most terms cancel"}, {"start": 103.04, "duration": 9.32, "text": "out leaving us with -2X 2 + 2x^ 2 + 2 -"}, {"start": 109.04, "duration": 6.8, "text": "2x^2 which simplifies to 2 -"}, {"start": 112.36, "duration": 6.0, "text": "2x^2 finally we can factor out a two"}, {"start": 115.84, "duration": 10.199, "text": "from the numerator giving us our final"}, {"start": 118.36, "duration": 10.16, "text": "answer 2 * * 1 - x^2 / x 4 + x^2 + 1 and"}, {"start": 126.039, "duration": 4.56, "text": "there we have it we've successfully"}, {"start": 128.52, "duration": 4.28, "text": "differentiated this logarithmic"}, {"start": 130.599, "duration": 5.881, "text": "expression using the properties of"}, {"start": 132.8, "duration": 6.68, "text": "logarithms the chain Rule and careful"}, {"start": 136.48, "duration": 3.0, "text": "algebra"}], "title": "Mastering Logarithmic Differentiation [\ud835\udc8d\ud835\udc8f((\ud835\udc99^\ud835\udfd0+\ud835\udc99+\ud835\udfcf)/(\ud835\udc99^\ud835\udfd0\u2212\ud835\udc99+\ud835\udfcf))] Step-by-Step Explanation", "description": "Question:\nDifferentiate the expression  [\ud835\udc8d\ud835\udc8f((\ud835\udc99^\ud835\udfd0+\ud835\udc99+\ud835\udfcf)/(\ud835\udc99^\ud835\udfd0\u2212\ud835\udc99+\ud835\udfcf))] with respect to x.\n\nUnlock the secrets of logarithmic differentiation with our comprehensive, step-by-step guide! This video breaks down the process of differentiating complex logarithmic expressions with clarity and precision. Perfect for students and math enthusiasts, our tutorial will help you master these techniques in no time. Whether you\u2019re preparing for exams or just looking to enhance your math skills, this video is your ultimate resource.\n\n#MathTutorial #Calculus #LogarithmicDifferentiation #MathSkills #STEM", "lengthSeconds": 142, "width": 640, "height": 360}
{"video_id": "Fyqp17JnyOQ", "transcript": [{"start": 0.11, "duration": 5.729, "text": "In atomic physics, the Electron Shell Model\ndescribes the distribution of electrons on"}, {"start": 5.839, "duration": 4.961, "text": "the \u00bbenergy shells\u00ab of an atom in the periodic\ntable of elements."}, {"start": 10.8, "duration": 5.52, "text": "For example, it can be used to understand\nthe electron distribution of an oxygen atom."}, {"start": 16.32, "duration": 4.98, "text": "We can see from the periodic table that an\noxygen atom has 8 electrons."}, {"start": 21.3, "duration": 2.55, "text": "But how are they distributed in the atom?"}, {"start": 23.85, "duration": 3.37, "text": "The shell model answers this question."}, {"start": 27.22, "duration": 2.149, "text": "Each shell represents a certain energy."}, {"start": 29.369, "duration": 5.951, "text": "The electrons can occupy these shells and\nthus take on this energy."}, {"start": 35.32, "duration": 4.98, "text": "Each shell can only hold a certain number\nof electrons."}, {"start": 40.3, "duration": 5.34, "text": "The electrons on the outermost shell of an\natom are known as valence electrons and they"}, {"start": 45.64, "duration": 3.13, "text": "play a crucial role in chemical reactions."}, {"start": 48.77, "duration": 6.34, "text": "The shells are labeled with letters: The innermost\n(first) shell is called the K-shell."}, {"start": 55.11, "duration": 3.12, "text": "It can hold a maximum of 2 electrons."}, {"start": 58.23, "duration": 2.129, "text": "Then it is completely filled."}, {"start": 60.359, "duration": 2.571, "text": "The second shell is called the L-shell."}, {"start": 62.93, "duration": 3.01, "text": "It can hold a maximum of 8 electrons."}, {"start": 65.94, "duration": 2.46, "text": "The third shell is called the M shell."}, {"start": 68.4, "duration": 2.7, "text": "It can hold a maximum of 18 electrons."}, {"start": 71.1, "duration": 2.24, "text": "The fourth shell is called the N-shell."}, {"start": 73.34, "duration": 3.2, "text": "It can hold a maximum of 32 electrons."}, {"start": 76.54, "duration": 3.78, "text": "How do I know the maximum number of electrons\na shell can hold?"}, {"start": 80.32, "duration": 2.0, "text": "There is a formula for this."}, {"start": 82.32, "duration": 5.4, "text": "Let's call the maximum number of electrons\nthat a shell can hold as "n_max"."}, {"start": 87.72, "duration": 5.5, "text": "You can then calculate this number using the\nfollowing formula: "n_max" is equal to two"}, {"start": 93.22, "duration": 2.43, "text": "n squared."}, {"start": 95.65, "duration": 2.52, "text": "Where "n" is the shell number."}, {"start": 98.17, "duration": 3.16, "text": "For the first shell, the K shell, "n equals\none"."}, {"start": 101.33, "duration": 6.0, "text": "For the second shell, the L-shell, "n equals\ntwo" For the third shell, the M-shell, "n"}, {"start": 107.33, "duration": 1.96, "text": "equals three", \u00a0and so on!"}, {"start": 109.29, "duration": 2.79, "text": "Let's take a look at four examples."}, {"start": 112.08, "duration": 3.19, "text": "First, the hydrogen atom."}, {"start": 115.27, "duration": 4.93, "text": "If you look at the periodic table, you can\nread the number of electrons from the atomic"}, {"start": 120.2, "duration": 1.959, "text": "number in the corner."}, {"start": 122.159, "duration": 3.85, "text": "The hydrogen atom has an atomic number of\none."}, {"start": 126.009, "duration": 3.181, "text": "This means it only has one electron."}, {"start": 129.19, "duration": 6.86, "text": "We start with the innermost shell, the K shell,\nand add an electron to it."}, {"start": 136.05, "duration": 5.67, "text": "As the hydrogen atom has no other electrons,\nit only has a half-filled K-shell."}, {"start": 141.72, "duration": 4.72, "text": "The lithium atom has the atomic number three."}, {"start": 146.44, "duration": 2.46, "text": "This means it has three electrons."}, {"start": 148.9, "duration": 2.06, "text": "We start with the K shell."}, {"start": 150.96, "duration": 5.19, "text": "The K-shell can hold two electrons, so we\nfill it with two electrons."}, {"start": 156.15, "duration": 6.02, "text": "The third electron is added to the next shell,\nthe L shell."}, {"start": 162.17, "duration": 4.51, "text": "The oxygen atom has eight electrons."}, {"start": 166.68, "duration": 2.1, "text": "Two are on the K shell."}, {"start": 168.78, "duration": 5.65, "text": "The L-shell can hold a maximum of eight electrons,\nso the remaining six electrons are added to"}, {"start": 174.43, "duration": 1.83, "text": "it."}, {"start": 176.26, "duration": 4.1, "text": "The aluminum atom has thirteen electrons."}, {"start": 180.36, "duration": 3.299, "text": "Two electrons are added to the K-shell."}, {"start": 183.659, "duration": 4.521, "text": "Eight electrons are added to the L-shell,\nthen the K-shell is full."}, {"start": 188.18, "duration": 3.45, "text": "The remaining three electrons are added to\nthe M-shell."}], "title": "The Electron Shell Model: Distribution of Electrons Inside Atoms Explained", "description": "https://www.youtube.com/watch?v=Fyqp17JnyOQ&list=PLTjLwQcqQzNKzSAxJxKpmOtAriFS5wWy4\n\nHere you will learn how an electric dipole (e.g. a molecule) behaves in a homogeneous and inhomogeneous electric field. More: https://en.fufaev.org/electron-shell-model\n\n\n\n\nBooks by Alexander Fufaev:\n1) Equations of Physics: Solve EVERY Physics Problem \nhttps://en.fufaev.org/physics-equations-book\n2) Alexander Fufaev and His Never Ending Story: \nhttps://en.fufaev.org/lifestory", "lengthSeconds": 192, "width": 640, "height": 360}
{"video_id": "CEiHAW5_jJc", "transcript": [{"start": 0.06, "duration": 5.759, "text": "wenn wir eine Wechselspannung an einen"}, {"start": 2.399, "duration": 5.101, "text": "Kondensator der Kapazit\u00e4t C anlegen dann"}, {"start": 5.819, "duration": 4.621, "text": "flie\u00dft durch den Kondensator einen"}, {"start": 7.5, "duration": 5.04, "text": "Wechselstrom die Wechselspannung"}, {"start": 10.44, "duration": 5.22, "text": "wechselt die Polarit\u00e4t mit der Frequenz"}, {"start": 12.54, "duration": 6.18, "text": "f mit dieser Frequenz \u00e4ndert auch der"}, {"start": 15.66, "duration": 5.4, "text": "Wechselstrom seine Flussrichtung ein"}, {"start": 18.72, "duration": 4.979, "text": "Kondensator an den eine Wechselspannung"}, {"start": 21.06, "duration": 5.7, "text": "angelegt ist hat einen komplexen nicht"}, {"start": 23.699, "duration": 5.761, "text": "komischen Widerstand der als kapazitive"}, {"start": 26.76, "duration": 6.42, "text": "Reaktanz bezeichnet wird dieser wird"}, {"start": 29.46, "duration": 5.88, "text": "meistens mit XC abgek\u00fcrzt das C steht"}, {"start": 33.18, "duration": 5.28, "text": "f\u00fcr das englische Wort gepassette was"}, {"start": 35.34, "duration": 5.64, "text": "\u00fcbersetzt Kondensator hei\u00dft manchmal"}, {"start": 38.46, "duration": 6.06, "text": "wird die kapazitive Reaktanz auch als"}, {"start": 40.98, "duration": 5.099, "text": "kapazitiver Blindwiderstand bezeichnet"}, {"start": 44.52, "duration": 3.98, "text": "du kannst den kapazitiven"}, {"start": 46.079, "duration": 4.401, "text": "Blindwiderstand ganz einfach berechnen"}, {"start": 48.5, "duration": 4.379, "text": "du brauchst daf\u00fcr die"}, {"start": 50.48, "duration": 5.14, "text": "wechselspannungsfrequenz F und die"}, {"start": 52.879, "duration": 5.981, "text": "kondensatorkapazit\u00e4t C"}, {"start": 55.62, "duration": 7.14, "text": "der Blindwiderstand ist -1 durch 2 mal"}, {"start": 58.86, "duration": 7.68, "text": "Pi mal F mal c p ist hier die Kreiszahl"}, {"start": 62.76, "duration": 6.6, "text": "mit dem Wert 3,14 das Minuszeichen sagt"}, {"start": 66.54, "duration": 6.119, "text": "aus dass die Wechselspannung u dem"}, {"start": 69.36, "duration": 5.04, "text": "Wechselstrom i hinterher h\u00e4ngt"}, {"start": 72.659, "duration": 4.381, "text": "die Einheit des kapazitiven"}, {"start": 74.4, "duration": 5.34, "text": "blindwiderstands ist oben"}, {"start": 77.04, "duration": 6.06, "text": "2 pi mal F wird \u00fcbrigens oft zur"}, {"start": 79.74, "duration": 5.18, "text": "Kreisfrequenz Omega zusammengefasst"}, {"start": 83.1, "duration": 3.839, "text": "wenn du eine sehr gro\u00dfe"}, {"start": 84.92, "duration": 4.12, "text": "wechselspannungsfrequenz benutzt dann"}, {"start": 86.939, "duration": 4.5, "text": "wird der kapazitive Blindwiderstand sehr"}, {"start": 89.04, "duration": 4.5, "text": "klein und der Kondensator l\u00e4sst den"}, {"start": 91.439, "duration": 3.441, "text": "Strom leicht durch"}, {"start": 93.54, "duration": 3.06, "text": "wenn dagegen die"}, {"start": 94.88, "duration": 4.3, "text": "wechselspannungsfrequenz sehr klein ist"}, {"start": 96.6, "duration": 4.92, "text": "oder gar Null wenn also eine"}, {"start": 99.18, "duration": 4.56, "text": "Gleichspannung anliegt dann wird der"}, {"start": 101.52, "duration": 5.16, "text": "kapazitive Blindwiderstand unendlich"}, {"start": 103.74, "duration": 5.76, "text": "gro\u00df der Kondensator l\u00e4sst keinen Strom"}, {"start": 106.68, "duration": 5.46, "text": "durch wie du an der Gleichung siehst"}, {"start": 109.5, "duration": 4.14, "text": "kannst du auch die Kapazit\u00e4t nutzen um"}, {"start": 112.14, "duration": 4.019, "text": "den komplexen Widerstand des"}, {"start": 113.64, "duration": 4.38, "text": "Kondensators anzupassen"}, {"start": 116.159, "duration": 4.201, "text": "wenn wir mit effektiv werden der"}, {"start": 118.02, "duration": 4.379, "text": "Spannung und des Stroms arbeiten dann"}, {"start": 120.36, "duration": 4.5, "text": "interessiert uns nur der Betrag des"}, {"start": 122.399, "duration": 4.521, "text": "kapazitiven blindwiderstands das"}, {"start": 124.86, "duration": 5.099, "text": "Minuszeichen k\u00f6nnen wir also weglassen"}, {"start": 126.92, "duration": 5.02, "text": "machen wir ein Beispiel du legst einen"}, {"start": 129.959, "duration": 5.64, "text": "Netzspannung an einen Kondensator mit"}, {"start": 131.94, "duration": 6.299, "text": "der Kapazit\u00e4t von 10 Nano Fahrrad an"}, {"start": 135.599, "duration": 6.261, "text": "die 230 Volt Netzspannung hat eine"}, {"start": 138.239, "duration": 6.0, "text": "Frequenz von 50 Herz setze die Werte ein"}, {"start": 141.86, "duration": 6.459, "text": "damit betr\u00e4gt der kapazitive"}, {"start": 144.239, "duration": 6.901, "text": "Blindwiderstand 318 Kilo Ohm"}, {"start": 148.319, "duration": 5.221, "text": "um den effektiv Strom zu bestimmen der"}, {"start": 151.14, "duration": 5.459, "text": "durch den Kondensator flie\u00dft benutze die"}, {"start": 153.54, "duration": 6.0, "text": "uri-formel statt den ohmschen Widerstand"}, {"start": 156.599, "duration": 6.961, "text": "R einzusetzen benutzt du den kapazitiven"}, {"start": 159.54, "duration": 7.16, "text": "Blindwiderstand Stelle nach dem Strom um"}, {"start": 163.56, "duration": 6.12, "text": "setze die 230 Volt Effektivspannung und"}, {"start": 166.7, "duration": 6.09, "text": "318.000 Ohm ein dann bekommst du einen"}, {"start": 169.68, "duration": 11.15, "text": "effektiv Strom von 0,7 Ampere"}, {"start": 172.79, "duration": 8.04, "text": "[Musik]"}], "title": "Kondensator-Blindwiderstand berechnen", "description": "In diesem Kurzvideo lernst du, was der Blindwiderstand eines Kondensators ist und wie du diesen berechnen kannst. Mehr: https://de.fufaev.org/kapazitiver-blindwiderstand", "lengthSeconds": 179, "width": 640, "height": 360}
{"video_id": "dGKAqi9SLec", "transcript": [{"start": 0.12, "duration": 3.92, "text": "was ist der Unterschied zwischen einem"}, {"start": 1.839, "duration": 5.201, "text": "elektrischen und magnetischen"}, {"start": 4.04, "duration": 4.72, "text": "Feld ein elektrisches Feld wird sowohl"}, {"start": 7.04, "duration": 3.639, "text": "durch bewegte als auch durch"}, {"start": 8.76, "duration": 4.28, "text": "nichtbewegte elektrische Ladungen"}, {"start": 10.679, "duration": 5.881, "text": "hervorgerufen ein magnetisches Feld"}, {"start": 13.04, "duration": 6.36, "text": "dagegen wird nur durch bewegte Ladungen"}, {"start": 16.56, "duration": 5.08, "text": "erzeugt eine positive Ladung ist die"}, {"start": 19.4, "duration": 4.0, "text": "Quelle des elektrischen Feldes eine"}, {"start": 21.64, "duration": 4.24, "text": "negative Ladung ist die Senke des"}, {"start": 23.4, "duration": 4.6, "text": "elektrischen Feldes in der Natur k\u00f6nnen"}, {"start": 25.88, "duration": 5.16, "text": "negative und positive Ladungen allein"}, {"start": 28.0, "duration": 6.04, "text": "vorkommen es existieren so elektrische"}, {"start": 31.04, "duration": 5.28, "text": "Monopole magnetische Monopole dagegen"}, {"start": 34.04, "duration": 5.24, "text": "soweit wir wissen sind in der Natur"}, {"start": 36.32, "duration": 5.84, "text": "nicht vorhanden das hei\u00dft es gibt keinen"}, {"start": 39.28, "duration": 5.959, "text": "allein vorkommenden magnetischen Nordpol"}, {"start": 42.16, "duration": 7.0, "text": "oder magnetischen S\u00fcdpol es treten stets"}, {"start": 45.239, "duration": 5.201, "text": "Nordpol als Quelle und S\u00fcdpol als Senke"}, {"start": 49.16, "duration": 4.36, "text": "gemeinsam"}, {"start": 50.44, "duration": 5.48, "text": "auf magnetische Feldlinien sind in sich"}, {"start": 53.52, "duration": 4.6, "text": "geschlossen elektrische Feldlinien"}, {"start": 55.92, "duration": 4.159, "text": "dagegen m\u00fcssen nicht geschlossen sein"}, {"start": 58.12, "duration": 4.119, "text": "sondern k\u00f6nnen beispielsweise ins"}, {"start": 60.079, "duration": 3.241, "text": "Unendliche gehen wie die Feldlinien"}, {"start": 62.239, "duration": 3.721, "text": "einer"}, {"start": 63.32, "duration": 4.799, "text": "Punktladung eine Ladung Q in einem"}, {"start": 65.96, "duration": 6.44, "text": "elektrischen Feld E erf\u00e4hrt eine"}, {"start": 68.119, "duration": 6.401, "text": "elektrische Kraft FE = Q mal e eine"}, {"start": 72.4, "duration": 4.84, "text": "bewegte Ladung in einem zur Bewegung"}, {"start": 74.52, "duration": 7.919, "text": "senkrechten Magnetfeld B erf\u00e4hrt eine"}, {"start": 77.24, "duration": 7.48, "text": "magnetische Kraft FM = q* V* B die"}, {"start": 82.439, "duration": 4.161, "text": "Einheit des elektrischen Feldes ist Volt"}, {"start": 84.72, "duration": 4.24, "text": "pro Meter und die Einheit des"}, {"start": 86.6, "duration": 4.799, "text": "magnetischen Feldes ist voltsekunde pro"}, {"start": 88.96, "duration": 5.119, "text": "Quadratmeter oder kurz"}, {"start": 91.399, "duration": 4.521, "text": "Tesla wenn eine elektrische Ladung in"}, {"start": 94.079, "duration": 3.801, "text": "einem externen elektrischen Feld"}, {"start": 95.92, "duration": 4.28, "text": "platziert wird so wird die Ladung"}, {"start": 97.88, "duration": 4.64, "text": "beschleunigt dadurch \u00e4ndert die Ladung"}, {"start": 100.2, "duration": 3.959, "text": "ihre kinetische Energie wenn eine"}, {"start": 102.52, "duration": 3.959, "text": "elektrische Ladung dagegen in ein"}, {"start": 104.159, "duration": 4.761, "text": "magnetisches Feld platziert wird so wird"}, {"start": 106.479, "duration": 5.32, "text": "die Ladung nicht beschleunigt die Ladung"}, {"start": 108.92, "duration": 5.479, "text": "\u00e4ndert ihre Energie nicht kurz gesagt"}, {"start": 111.799, "duration": 5.081, "text": "ein elektrisches Feld verrichtet Arbeit"}, {"start": 114.399, "duration": 4.921, "text": "an der Ladung aber ein Magnetfeld tut"}, {"start": 116.88, "duration": 6.0, "text": "das nicht sogar wenn die Ladung im"}, {"start": 119.32, "duration": 6.159, "text": "Magnetfeld bewegt wird ein elektrisches"}, {"start": 122.88, "duration": 5.439, "text": "Feld l\u00e4sst sich abschirmen z.B mit Hilfe"}, {"start": 125.479, "duration": 5.081, "text": "eines fairdayk\u00e4figs ein magnetisches"}, {"start": 128.319, "duration": 4.151, "text": "Feld l\u00e4sst sich dagegen nicht abschirmen"}, {"start": 130.56, "duration": 5.09, "text": "aber zumindest"}, {"start": 132.47, "duration": 3.18, "text": "[Musik]"}, {"start": 137.64, "duration": 3.0, "text": "ablenken"}], "title": "7 Unterschiede zwischen dem elektrischen und magnetischen Feld", "description": "Hier lernst du 7 wichtige Unterschiede zwischen dem elektrischen Feld (E) und dem magnetischen Feld (B). Mehr: https://de.fufaev.org/fragen/68", "lengthSeconds": 139, "width": 640, "height": 360}
{"video_id": "lH3DCxrK_jE", "transcript": [{"start": 0.16, "duration": 4.88, "text": "wenn wir eine Wechselspannung an eine"}, {"start": 2.24, "duration": 4.92, "text": "Spule der Induktivit\u00e4t L anlegen dann"}, {"start": 5.04, "duration": 4.12, "text": "Flie\u00df durch die Spule ein"}, {"start": 7.16, "duration": 4.04, "text": "Wechselstrom die Wechselspannung"}, {"start": 9.16, "duration": 5.2, "text": "wechselt die Polarit\u00e4t mit der Frequenz"}, {"start": 11.2, "duration": 4.76, "text": "f mit dieser Frequenz \u00e4ndert auch der"}, {"start": 14.36, "duration": 4.24, "text": "Wechselstrom seine"}, {"start": 15.96, "duration": 5.2, "text": "Flussrichtung eine Spule an die eine"}, {"start": 18.6, "duration": 5.439, "text": "Wechselspannung angelegt ist hat einen"}, {"start": 21.16, "duration": 5.8, "text": "komplexen nichtomischen Widerstand der"}, {"start": 24.039, "duration": 6.32, "text": "als induktive Reaktanz bezeichnet wird"}, {"start": 26.96, "duration": 5.639, "text": "dieser wird meistens mit XL abgek\u00fcrzt"}, {"start": 30.359, "duration": 5.561, "text": "die induktive Reaktanz wird auch als"}, {"start": 32.599, "duration": 5.561, "text": "induktiver Blindwiderstand bezeichnet du"}, {"start": 35.92, "duration": 5.24, "text": "kannst den induktiven Blindwiderstand"}, {"start": 38.16, "duration": 5.559, "text": "ganz einfach berechnen du brauchst daf\u00fcr"}, {"start": 41.16, "duration": 8.8, "text": "die wechselspannungsfrequenz F und die"}, {"start": 43.719, "duration": 10.201, "text": "Induktivit\u00e4t L der Spule XL = 2* pi* f*"}, {"start": 49.96, "duration": 6.399, "text": "l Pi ist hier die Kreiszahl mit dem Wert"}, {"start": 53.92, "duration": 6.36, "text": "3,14 die Einheit des induktiven"}, {"start": 56.359, "duration": 6.921, "text": "blindwiderstands ist oben 2pf wird"}, {"start": 60.28, "duration": 5.8, "text": "\u00fcbrigens oft zur Kreisfrequenz Omega"}, {"start": 63.28, "duration": 4.839, "text": "zusammengefasst wenn du eine sehr gro\u00dfe"}, {"start": 66.08, "duration": 4.6, "text": "wechselspannungsfrequenz benutzt dann"}, {"start": 68.119, "duration": 5.32, "text": "wird der induktive Blindwiderstand auch"}, {"start": 70.68, "duration": 5.759, "text": "sehr gro\u00df und die Spule l\u00e4sst den Strom"}, {"start": 73.439, "duration": 5.081, "text": "kaum durch wenn dagegen die"}, {"start": 76.439, "duration": 4.241, "text": "wechselspannungsfrequenz sehr klein ist"}, {"start": 78.52, "duration": 4.44, "text": "oder gar Null wenn also eine"}, {"start": 80.68, "duration": 5.56, "text": "Gleichspannung anliegt dann wird der"}, {"start": 82.96, "duration": 5.56, "text": "induktive Blindwiderstand auch null die"}, {"start": 86.24, "duration": 4.839, "text": "Spule l\u00e4sst einen beliebig hohen Strom"}, {"start": 88.52, "duration": 5.32, "text": "durch was einem chluss"}, {"start": 91.079, "duration": 4.921, "text": "entspricht wie du an der Formel siehst"}, {"start": 93.84, "duration": 4.68, "text": "kannst du auch die Induktivit\u00e4t nutzen"}, {"start": 96.0, "duration": 5.6, "text": "um den Blindwiderstand der Spule"}, {"start": 98.52, "duration": 6.48, "text": "anzupassen machen wir ein Beispiel du"}, {"start": 101.6, "duration": 5.799, "text": "legst die 230 Volt Netzspannung an eine"}, {"start": 105.0, "duration": 6.079, "text": "Spule mit der Induktivit\u00e4t von 500"}, {"start": 107.399, "duration": 6.961, "text": "mhenry an die angelegte Effektivspannung"}, {"start": 111.079, "duration": 5.481, "text": "hat eine Frequenz von 50 Hz setze"}, {"start": 114.36, "duration": 5.56, "text": "Induktivit\u00e4t und Frequenz in die Formel"}, {"start": 116.56, "duration": 7.76, "text": "ein damit betr\u00e4gt der induktive blindwi"}, {"start": 119.92, "duration": 6.479, "text": "stand 157 Ohm um den effektivstrom zu"}, {"start": 124.32, "duration": 4.96, "text": "bestimmen der durch die Spule flie\u00dft"}, {"start": 126.399, "duration": 5.241, "text": "benutze die uriformel statt den omischen"}, {"start": 129.28, "duration": 5.08, "text": "Widerstand R einzusetzen benutzt du den"}, {"start": 131.64, "duration": 4.2, "text": "induktiven Blindwiderstand Stelle nach"}, {"start": 134.36, "duration": 5.44, "text": "dem Strom"}, {"start": 135.84, "duration": 7.32, "text": "um setze die 230 Volt Effektivspannung"}, {"start": 139.8, "duration": 6.75, "text": "und 157 Ohm ein dann bekommst du einen"}, {"start": 143.16, "duration": 11.42, "text": "effektivstrom von 1,5 Ampere"}, {"start": 146.55, "duration": 8.03, "text": "[Musik]"}], "title": "Spule-Blindwiderstand berechnen", "description": "Lerne, was ein induktiver Blindwiderstand (Reaktanz) ist und wie du diesen bei einer Spule (Induktivit\u00e4t) in einem Wechselstromkreis bestimmen kannst. Mehr: https://de.fufaev.org/induktiver-blindwiderstand", "lengthSeconds": 153, "width": 640, "height": 360}
{"video_id": "IqgMga4-P5I", "transcript": [{"start": 0.08, "duration": 3.41, "text": "Let us consider a system with an ideal gas\nin it."}, {"start": 3.49, "duration": 7.27, "text": "The state of the gas, or more precisely, its\ntemperature T, its pressure Pi and its volume"}, {"start": 10.76, "duration": 9.79, "text": "V are described by the ideal gas equation:\nPi is equal to n times R times T times 1 divided"}, {"start": 20.55, "duration": 6.41, "text": "by V. \u00a0n is the amount of substance and indirectly\ndescribes the number of gas particles and"}, {"start": 26.96, "duration": 3.969, "text": "R is the gas constant."}, {"start": 30.929, "duration": 5.611, "text": "We refer to temperature, pressure and volume\nas properties - they describe the macroscopic"}, {"start": 36.54, "duration": 3.47, "text": "state of the gas."}, {"start": 40.01, "duration": 7.27, "text": "If we heat the gas and thus increase its temperature\nT, we are talking about a thermodynamic process."}, {"start": 47.28, "duration": 6.09, "text": "We could also compress the gas, thereby increasing\nthe pressure and reducing the volume - this"}, {"start": 53.37, "duration": 2.9, "text": "would also be a thermodynamic process."}, {"start": 56.27, "duration": 8.459, "text": "The thermodynamic process can be isobaric,\nisothermal, isochoric or adiabatic."}, {"start": 64.729, "duration": 6.601, "text": "In the case of an isobaric process, the pressure\nof the gas remains constant."}, {"start": 71.33, "duration": 7.149, "text": "This means that Pi, n and R are pure constants\nin the gas law and the volume V is proportional"}, {"start": 78.479, "duration": 1.39, "text": "to the temperature T."}, {"start": 79.869, "duration": 9.251, "text": "The pressure-volume diagram results in a horizontal\nstraight line for an isobaric gas."}, {"start": 89.12, "duration": 7.44, "text": "This means that a change in volume does NOT\nlead to a change in gas pressure."}, {"start": 96.56, "duration": 3.169, "text": "The volume V of the gas remains constant during\nan isochoric process."}, {"start": 99.729, "duration": 5.981, "text": "Then V, n and R are constants in the gas law,\nand the pressure Pi is proportional to the"}, {"start": 105.71, "duration": 1.96, "text": "temperature T."}, {"start": 107.67, "duration": 5.85, "text": "The pressure-volume diagram shows a vertical\nstraight line for an isochoric gas."}, {"start": 113.52, "duration": 5.16, "text": "The change in gas pressure does NOT lead to\na change in volume."}, {"start": 118.68, "duration": 7.21, "text": "In an isothermal process, the temperature\nT of the gas remains constant."}, {"start": 125.89, "duration": 7.7, "text": "This means that the pressure Pi is proportional\nto the inverse volume: 1 by V."}, {"start": 133.59, "duration": 7.16, "text": "The pressure-volume diagram shows a decreasing\ncurve for an isothermally behaving gas."}, {"start": 140.75, "duration": 4.78, "text": "Increasing the volume of the gas leads to\na reduction in the gas pressure."}, {"start": 145.53, "duration": 6.37, "text": "During an adiabatic process, no thermal energy\nis transported OUT and into the system."}, {"start": 151.9, "duration": 5.479, "text": "During this thermodynamic process, the temperature\nT, volume V and pressure Pi of the gas can"}, {"start": 157.379, "duration": 2.331, "text": "change simultaneously."}, {"start": 159.71, "duration": 6.84, "text": "The pressure-volume diagram shows a power\nlaw for an adiabatically behaving gas."}, {"start": 166.55, "duration": 4.07, "text": "The exponent gamma is referred to as the adiabatic\nexponent."}], "title": "Isobaric, Isochoric, Isothermal, Adiabatic?", "description": "https://www.youtube.com/watch?v=IqgMga4-P5I&list=PLTjLwQcqQzNKzSAxJxKpmOtAriFS5wWy4\n\nHere you will learn how an electric dipole (e.g. a molecule) behaves in a homogeneous and inhomogeneous electric field. More: https://en.fufaev.org/thermodynamic-process\n\n\n\n\nBooks by Alexander Fufaev:\n1) Equations of Physics: Solve EVERY Physics Problem \nhttps://en.fufaev.org/physics-equations-book\n2) Alexander Fufaev and His Never Ending Story: \nhttps://en.fufaev.org/lifestory", "lengthSeconds": 172, "width": 640, "height": 360}
{"video_id": "F3R2V901-BM", "transcript": [{"start": 0.12, "duration": 4.74, "text": "separationsansatz"}, {"start": 2.58, "duration": 4.64, "text": "diese wird manchmal auch Produktansatz"}, {"start": 4.86, "duration": 5.52, "text": "genannt warum wirst du gleich sehen"}, {"start": 7.22, "duration": 4.839, "text": "diese Methode eignet sich f\u00fcr partielle"}, {"start": 10.38, "duration": 4.62, "text": "Differenzialgleichungen beliebiger"}, {"start": 12.059, "duration": 6.121, "text": "Ordnung diese L\u00f6sungsmethode dient quasi"}, {"start": 15.0, "duration": 4.92, "text": "nur dazu eine partielle die GL in"}, {"start": 18.18, "duration": 3.84, "text": "mehrere gew\u00f6hnliche differen"}, {"start": 19.92, "duration": 3.68, "text": "zahlgleichungen zu verwandeln und diese"}, {"start": 22.02, "duration": 3.9, "text": "dann mit anderen Methoden zu l\u00f6sen"}, {"start": 23.6, "duration": 5.08, "text": "schauen wir uns diese Methode am besten"}, {"start": 25.92, "duration": 4.56, "text": "direkt an einem Beispiel an hier siehst"}, {"start": 28.68, "duration": 4.92, "text": "du die eindimensionale Wellengleichung"}, {"start": 30.48, "duration": 5.04, "text": "f\u00fcr das elektrische Feld die gesuchte"}, {"start": 33.6, "duration": 5.16, "text": "Funktion ist das elektrische Feld E"}, {"start": 35.52, "duration": 6.059, "text": "dieses h\u00e4ngt von x und von T ab also"}, {"start": 38.76, "duration": 5.1, "text": "genau von zwei Variablen da die gesuchte"}, {"start": 41.579, "duration": 4.521, "text": "Funktion von zwei Variablen abh\u00e4ngt und"}, {"start": 43.86, "duration": 4.379, "text": "ihre Ableitungen in der N\u00e4he vorkommen"}, {"start": 46.1, "duration": 4.06, "text": "handelte sich um eine Parzelle"}, {"start": 48.239, "duration": 4.5, "text": "Differenzzahlung"}, {"start": 50.16, "duration": 4.919, "text": "wie gehen wir nun vor hier machen wir"}, {"start": 52.739, "duration": 4.14, "text": "einen Produktansatz f\u00fcr die gesuchte"}, {"start": 55.079, "duration": 5.761, "text": "L\u00f6sung e"}, {"start": 56.879, "duration": 7.02, "text": "= r von X mal u von T wir nehmen quasi"}, {"start": 60.84, "duration": 6.24, "text": "an dass die gesuchte L\u00f6sung e sich in"}, {"start": 63.899, "duration": 6.841, "text": "ein Produkt von zwei Funktionen R und U"}, {"start": 67.08, "duration": 6.899, "text": "aufspalten l\u00e4sst die eine Funktion R"}, {"start": 70.74, "duration": 7.02, "text": "h\u00e4ngt dabei nur von X ab und die andere"}, {"start": 73.979, "duration": 5.401, "text": "Funktion u h\u00e4ngt nur von T ab in der"}, {"start": 77.76, "duration": 4.02, "text": "Wellengleichung kommen die zweite"}, {"start": 79.38, "duration": 4.98, "text": "Ableitung von E nach der Zeit und nach"}, {"start": 81.78, "duration": 5.82, "text": "dem Ort vor das hei\u00dft wir m\u00fcssen unseren"}, {"start": 84.36, "duration": 5.399, "text": "Produktansatz zuerst ableiten die"}, {"start": 87.6, "duration": 6.24, "text": "Ableitung des produktansatzes nach X"}, {"start": 89.759, "duration": 8.161, "text": "ergibt u mal die zweite Ableitung von R"}, {"start": 93.84, "duration": 6.0, "text": "nach X warum weil u unabh\u00e4ngig von X ist"}, {"start": 97.92, "duration": 4.199, "text": "und somit in der Ableitung wie eine"}, {"start": 99.84, "duration": 4.44, "text": "Konstante ist"}, {"start": 102.119, "duration": 4.981, "text": "bei der Ableitung des Produkts nach der"}, {"start": 104.28, "duration": 6.78, "text": "Zeit Tee dagegen ist die Funktion R eine"}, {"start": 107.1, "duration": 6.9, "text": "konstante weil sie nicht von T abh\u00e4ngt"}, {"start": 111.06, "duration": 5.82, "text": "das Ziel ist es nun alles was von X"}, {"start": 114.0, "duration": 6.06, "text": "abh\u00e4ngt von dem zu trennen was von Tee"}, {"start": 116.88, "duration": 6.739, "text": "abh\u00e4ngt dazu teilen wir diese Geld durch"}, {"start": 120.06, "duration": 3.559, "text": "das Produkt einmal u"}, {"start": 124.159, "duration": 5.38, "text": "dadurch haben wir erreicht dass alles"}, {"start": 127.079, "duration": 5.52, "text": "was von X abh\u00e4ngt auf der linken Seite"}, {"start": 129.539, "duration": 6.001, "text": "steht und das was von T abh\u00e4ngt alles"}, {"start": 132.599, "duration": 4.86, "text": "auf der rechten Seite steht wenn du es"}, {"start": 135.54, "duration": 4.68, "text": "schaffst eine Parzelle die Franzi"}, {"start": 137.459, "duration": 5.521, "text": "Gleichung so zu separieren dann war der"}, {"start": 140.22, "duration": 5.04, "text": "separationsansatz erfolgreich"}, {"start": 142.98, "duration": 4.619, "text": "jetzt k\u00f6nnen wir X auf der linken Seite"}, {"start": 145.26, "duration": 4.8, "text": "variieren ohne dass sich die rechte"}, {"start": 147.599, "duration": 5.701, "text": "Seite ver\u00e4ndert da auf der rechten Seite"}, {"start": 150.06, "duration": 5.64, "text": "ja kein X vorkommt das gleiche gilt f\u00fcr"}, {"start": 153.3, "duration": 5.159, "text": "die Zeit t wenn wir die Zeit auf der"}, {"start": 155.7, "duration": 5.819, "text": "rechten Seite ver\u00e4ndern bleibt die linke"}, {"start": 158.459, "duration": 4.5, "text": "Seite unver\u00e4ndert da dort keine Zeit t"}, {"start": 161.519, "duration": 3.661, "text": "vorkommt"}, {"start": 162.959, "duration": 6.0, "text": "setzen wir also die linke und die rechte"}, {"start": 165.18, "duration": 5.9, "text": "Seite einer konstanten K ="}, {"start": 168.959, "duration": 4.021, "text": "damit haben wir eine partielle"}, {"start": 171.08, "duration": 3.46, "text": "Differenzialgleichung in zwei"}, {"start": 172.98, "duration": 3.96, "text": "gew\u00f6hnlichen Differenzialgleichungen"}, {"start": 174.54, "duration": 4.32, "text": "umgewandelt die eine differen"}, {"start": 176.94, "duration": 3.72, "text": "zahlgleichung h\u00e4ngt nur von nix ab und"}, {"start": 178.86, "duration": 4.68, "text": "die andere differen zahlgleichung h\u00e4ngt"}, {"start": 180.66, "duration": 4.62, "text": "nur von T ab die beiden differen"}, {"start": 183.54, "duration": 3.839, "text": "zahlgleichungen sind nicht mal gekoppelt"}, {"start": 185.28, "duration": 4.38, "text": "du kannst sie also unabh\u00e4ngig"}, {"start": 187.379, "duration": 4.321, "text": "voneinander l\u00f6sen und die L\u00f6sungen dann"}, {"start": 189.66, "duration": 4.2, "text": "nach dem Produktansatz miteinander"}, {"start": 191.7, "duration": 5.52, "text": "multiplizieren um die L\u00f6sung der"}, {"start": 193.86, "duration": 4.58, "text": "partiellen Differenz zu erhalten diese"}, {"start": 197.22, "duration": 3.12, "text": "beiden gew\u00f6hnlichen"}, {"start": 198.44, "duration": 4.299, "text": "Differenzialgleichung kannst du mit dem"}, {"start": 200.34, "duration": 4.64, "text": "zuvor kennengelernten exponentialansatz"}, {"start": 202.739, "duration": 2.241, "text": "l\u00f6sen"}, {"start": 205.31, "duration": 8.05, "text": "[Musik]"}], "title": "Produktansatz (aka. Separationsansatz)", "description": "Lerne, wie man partielle Differentialgleichungen (DGL) mit Separation der Variablen (Produktansatz) l\u00f6st. Hier machen wir ein Beispiel mit der eindimensionalen Wellengleichung. Mehr: https://de.fufaev.org/partielle-dgl", "lengthSeconds": 212, "width": 640, "height": 360}
{"video_id": "ogk24TylvE4", "transcript": [{"start": 0.48, "duration": 5.44, "text": "consider a circuit consisting of a coil"}, {"start": 3.32, "duration": 6.0, "text": "characterized by the inductance L and"}, {"start": 5.92, "duration": 5.879, "text": "resistor R connected in serious then we"}, {"start": 9.32, "duration": 5.319, "text": "take a voltage source which provides a"}, {"start": 11.799, "duration": 5.56, "text": "voltage u0 as soon as we close the"}, {"start": 14.639, "duration": 5.161, "text": "circuit with the switch then a Time"}, {"start": 17.359, "duration": 5.361, "text": "dependent current I flows through the"}, {"start": 19.8, "duration": 5.719, "text": "coil and the resistor the current does"}, {"start": 22.72, "duration": 6.04, "text": "not have its maximum value immediately"}, {"start": 25.519, "duration": 5.92, "text": "but increases slowly due to Land's law"}, {"start": 28.76, "duration": 4.84, "text": "using kof's law we can set up the"}, {"start": 31.439, "duration": 5.321, "text": "following differential equation for the"}, {"start": 33.6, "duration": 6.0, "text": "current I remember that the point above"}, {"start": 36.76, "duration": 5.36, "text": "the I means the first time"}, {"start": 39.6, "duration": 6.599, "text": "derivative this is an inhomogeneous"}, {"start": 42.12, "duration": 6.72, "text": "linear differential equation of first"}, {"start": 46.199, "duration": 5.52, "text": "order the searched function y"}, {"start": 48.84, "duration": 5.559, "text": "corresponds here to the current I the"}, {"start": 51.719, "duration": 6.401, "text": "perturbation function s corresponds to"}, {"start": 54.399, "duration": 6.721, "text": "u/ L and is time independent in this"}, {"start": 58.12, "duration": 4.04, "text": "case we denote the homogeneous Solution"}, {"start": 61.12, "duration": 3.759, "text": "by"}, {"start": 62.16, "duration": 6.08, "text": "IH first we need to determine the"}, {"start": 64.879, "duration": 5.161, "text": "homogeneous solution IH we can quickly"}, {"start": 68.24, "duration": 3.72, "text": "calculate this using the solution"}, {"start": 70.04, "duration": 3.6, "text": "formula for the homogeneous version of"}, {"start": 71.96, "duration": 5.04, "text": "the differential equation that you"}, {"start": 73.64, "duration": 5.56, "text": "learned before the coefficient K in"}, {"start": 77.0, "duration": 5.88, "text": "front of the searched function I"}, {"start": 79.2, "duration": 7.64, "text": "corresponds to R over L and is also time"}, {"start": 82.88, "duration": 6.32, "text": "independent in this case so far so good"}, {"start": 86.84, "duration": 4.48, "text": "we may omit the constant CH in the"}, {"start": 89.2, "duration": 4.559, "text": "solution form formula here because we"}, {"start": 91.32, "duration": 5.119, "text": "consider it later anyway in the constant"}, {"start": 93.759, "duration": 5.961, "text": "a which we find in the other resolution"}, {"start": 96.439, "duration": 6.401, "text": "formula the coefficient R / L is"}, {"start": 99.72, "duration": 6.12, "text": "constant and integrating a constant only"}, {"start": 102.84, "duration": 6.84, "text": "introduces a variable T thus the"}, {"start": 105.84, "duration": 7.959, "text": "homogeneous solution is I is equal to e"}, {"start": 109.68, "duration": 7.359, "text": "to power of - R / L *"}, {"start": 113.799, "duration": 4.36, "text": "T let's insert it into the inhomogeneous"}, {"start": 117.039, "duration": 3.76, "text": "solution"}, {"start": 118.159, "duration": 5.161, "text": "formula note that one over the"}, {"start": 120.799, "duration": 5.081, "text": "exponential function containing a minus"}, {"start": 123.32, "duration": 4.76, "text": "in the exponent is simply equivalent to"}, {"start": 125.88, "duration": 5.439, "text": "the exponential function without the"}, {"start": 128.08, "duration": 7.239, "text": "minus sign now we have to calculate the"}, {"start": 131.319, "duration": 6.801, "text": "integral u/ L is a constant and can be"}, {"start": 135.319, "duration": 5.28, "text": "placed in front of the integral and when"}, {"start": 138.12, "duration": 5.68, "text": "integrating the exponential function the"}, {"start": 140.599, "duration": 5.961, "text": "exponential function is preserved only L"}, {"start": 143.8, "duration": 4.12, "text": "over R is added as a factor in front of"}, {"start": 146.56, "duration": 4.36, "text": "the exponential"}, {"start": 147.92, "duration": 6.28, "text": "function finally we hide the constant of"}, {"start": 150.92, "duration": 6.28, "text": "integration in the constant a and this"}, {"start": 154.2, "duration": 6.72, "text": "is our general solution we can simplify"}, {"start": 157.2, "duration": 6.36, "text": "it a bit more by multiplying out the"}, {"start": 160.92, "duration": 4.08, "text": "parentheses two exponential functions"}, {"start": 163.56, "duration": 4.48, "text": "simplify to"}, {"start": 165.0, "duration": 5.44, "text": "one to get a solution specific to the"}, {"start": 168.04, "duration": 6.119, "text": "problem we need to determine the unknown"}, {"start": 170.44, "duration": 6.56, "text": "constant a for that we need an initial"}, {"start": 174.159, "duration": 5.8, "text": "condition if we say that the time t"}, {"start": 177.0, "duration": 5.68, "text": "equal Z is the time when the current I"}, {"start": 179.959, "duration": 6.161, "text": "was Z because we have not yet closed the"}, {"start": 182.68, "duration": 7.68, "text": "switch then our initial condition is y"}, {"start": 186.12, "duration": 5.52, "text": "of 0 is equal to 0 insert it into the"}, {"start": 190.36, "duration": 4.68, "text": "general"}, {"start": 191.64, "duration": 7.64, "text": "solution e to the power of 0 is equal to"}, {"start": 195.04, "duration": 9.0, "text": "1 solve for a and you get a is equal to"}, {"start": 199.28, "duration": 7.4, "text": "- u0 / R thus we have successfully"}, {"start": 204.04, "duration": 5.64, "text": "determined the specific General"}, {"start": 206.68, "duration": 3.0, "text": "solution"}], "title": "RL Circuit: Solving Differential Equation with Variation of Parameters", "description": "Here you will learn how to solve the differential equation of the RL circuit with variation of parameters. More: https://en.fufaev.org/inhomogeneous-differential-equations-of-first-order\n\nBooks by Alexander Fufaev:\n1) Equations of Physics: Solve EVERY Physics Problem \nhttps://en.fufaev.org/physics-equations-book\n2) Alexander Fufaev and His Never Ending Story: \nhttps://en.fufaev.org/lifestory", "lengthSeconds": 208, "width": 640, "height": 360}
{"video_id": "9kTVNmejpco", "transcript": [{"start": 0.07, "duration": 4.98, "text": "What is the difference between a series and\na parallel connection of resistors?"}, {"start": 5.05, "duration": 6.079, "text": "Here we consider a series and a parallel circuit\nwith three resistors with resistances R1,"}, {"start": 11.129, "duration": 1.88, "text": "R2 and R3."}, {"start": 13.009, "duration": 4.301, "text": "A voltage U is applied to the resistors."}, {"start": 17.31, "duration": 3.74, "text": "Difference #1:\nLook at how the three resistors are connected"}, {"start": 21.05, "duration": 1.479, "text": "to each other."}, {"start": 22.529, "duration": 5.181, "text": "In a series circuit, the positive pole of\nthe voltage source is connected to one single"}, {"start": 27.71, "duration": 5.009, "text": "resistor and the negative pole is also connected\nto one single resistor."}, {"start": 32.719, "duration": 3.341, "text": "The resistors in a series circuit form a chain."}, {"start": 36.06, "duration": 4.87, "text": "In a parallel circuit, on the other hand,\nyou connect the positive pole of the voltage"}, {"start": 40.93, "duration": 6.67, "text": "source to three ends of the three resistors,\nand you do the same with the negative pole."}, {"start": 47.6, "duration": 1.49, "text": "Difference #2:"}, {"start": 49.09, "duration": 5.019, "text": "In a series connection, the same current flows\nthrough all three resistors."}, {"start": 54.109, "duration": 5.781, "text": "The total current I is therefore equal to\nthe current I1 through the first resistor,"}, {"start": 59.89, "duration": 5.94, "text": "but also equal to the current I2 and I3 through\nthe other two resistors."}, {"start": 65.83, "duration": 7.3, "text": "In a parallel circuit, however, different\ncurrents I1, I2 and I3 flow through the resistors."}, {"start": 73.13, "duration": 10.32, "text": "The total current I is the sum of the individual\ncurrents: I = I1 + I2 + I3"}, {"start": 83.45, "duration": 4.91, "text": "In a series circuit, you will measure a different\nvoltage across each resistor."}, {"start": 88.36, "duration": 2.67, "text": "Voltage U1 across the first resistor."}, {"start": 91.03, "duration": 4.69, "text": "U2 across the second resistor and U3 across\nthe third."}, {"start": 95.72, "duration": 8.59, "text": "The total voltage U is the sum of the individual\nvoltages: U = U1 + U2 + U3."}, {"start": 104.31, "duration": 4.46, "text": "In a parallel circuit, on the other hand,\nyou will measure the same voltage across all"}, {"start": 108.77, "duration": 1.76, "text": "three resistors."}, {"start": 110.53, "duration": 5.83, "text": "The total voltage is therefore equal to the\nvoltage U1 between the ends of the first resistor,"}, {"start": 116.36, "duration": 4.2, "text": "but also equal to the voltage U2 and U3."}, {"start": 120.56, "duration": 1.94, "text": "Difference #4:"}, {"start": 122.5, "duration": 5.94, "text": "The total resistance R of a series circuit\nis the sum of the individual resistances:"}, {"start": 128.44, "duration": 5.16, "text": "R is equal to R1 + R2 + R3."}, {"start": 133.6, "duration": 5.58, "text": "In a parallel circuit, however, the total\nresistance R is not the sum of the individual"}, {"start": 139.18, "duration": 1.0, "text": "resistances."}, {"start": 140.18, "duration": 5.65, "text": "The following relationship applies here: The\nreciprocal total resistance "1 over R" is"}, {"start": 145.83, "duration": 8.75, "text": "the sum of the reciprocal resistances "1 over\nR1" + "1 over R2 + "1 over R3"."}, {"start": 154.58, "duration": 1.5, "text": "Difference #5:"}, {"start": 156.08, "duration": 4.25, "text": "For the conductance values of the resistors,\nit is the other way round."}, {"start": 160.33, "duration": 5.41, "text": "The total conductance G of a parallel circuit\nis the sum of the individual conductances"}, {"start": 165.74, "duration": 3.23, "text": "G1, G2 and G3."}, {"start": 168.97, "duration": 7.03, "text": "And in a series circuit, the reciprocal conductance\n"1 over G" is equal to "1 over G1" + "1 over"}, {"start": 176.0, "duration": 2.3, "text": "G2" + "1 over G3"."}], "title": "The 5 Differences Between Parallel and Series Circuit", "description": "https://www.youtube.com/watch?v=9kTVNmejpco&list=PLTjLwQcqQzNKzSAxJxKpmOtAriFS5wWy4\n00:00 Introduction\n00:16 Difference: Connection\n00:47 Difference: Current\n01:21 Difference: Voltage\n02:00 Difference: Resistance\n02:33 Difference: Conductance\nMore: https://en.fufaev.org/lessons/1238", "lengthSeconds": 185, "width": 640, "height": 360}
{"video_id": "8K7_qPmYBiY", "transcript": [{"start": 0.12, "duration": 4.5, "text": "In theoretical physics, especially in the\ntheory of relativity and quantum mechanics,"}, {"start": 4.62, "duration": 5.18, "text": "we will regularly encounter symmetric and\nantisymmetric tensors."}, {"start": 9.8, "duration": 5.66, "text": "A symmetric tensor t_{ij} remains the same\nif we swap its indices."}, {"start": 15.46, "duration": 4.97, "text": "Specifically, swapping the indices of the\nsecond-level tensor as a matrix means that"}, {"start": 20.43, "duration": 5.21, "text": "the matrix remains the same if we transpose\nit, in other words if we mirror the rows and"}, {"start": 25.64, "duration": 1.95, "text": "columns on the diagonal."}, {"start": 27.59, "duration": 6.23, "text": "This symmetry property of tensors is very\nuseful and simplifies calculations in computer"}, {"start": 33.82, "duration": 1.64, "text": "physics enormously."}, {"start": 35.46, "duration": 5.25, "text": "Moreover, this property is crucial in quantum\nmechanics, because symmetric matrices have"}, {"start": 40.71, "duration": 1.84, "text": "real eigenvalues."}, {"start": 42.55, "duration": 4.47, "text": "They therefore represent physical quantities\n(we call them observables) that we can measure"}, {"start": 47.02, "duration": 1.94, "text": "in an experiment."}, {"start": 48.96, "duration": 4.54, "text": "So if you have a symmetric tensor in front\nof you, as a theoretical physicist you should"}, {"start": 53.5, "duration": 2.25, "text": "immediately get a dopamine kick."}, {"start": 55.75, "duration": 4.91, "text": "The Kronecker delta, for example, is an example\nof a simple symmetric tensor."}, {"start": 60.66, "duration": 2.55, "text": "We have considered a second-order tensor."}, {"start": 63.21, "duration": 2.57, "text": "What if the tensor is of a HIGHER order?"}, {"start": 65.78, "duration": 2.63, "text": "What about its symmetry property THEN?"}, {"start": 68.41, "duration": 4.91, "text": "For example, if the tensor has four indices\nand it remains EQUAL when we swap the first"}, {"start": 73.32, "duration": 5.1, "text": "two indices, then we are talking about a symmetric\ntensor in the indices m and k."}, {"start": 78.42, "duration": 4.97, "text": "However, we will also encounter tensors that\nare antisymmetric."}, {"start": 83.39, "duration": 5.46, "text": "An antisymmetric tensor changes sign when\nwe swap its indices."}, {"start": 88.85, "duration": 5.21, "text": "If the antisymmetric tensor is represented\nas a matrix, then it is equal to its negative"}, {"start": 94.06, "duration": 1.44, "text": "transpose."}, {"start": 95.5, "duration": 4.84, "text": "Unfortunately, most tensors are neither symmetric\nnor antisymmetric."}, {"start": 100.34, "duration": 5.84, "text": "But the great thing is: Mathematically, we\ncan split every tensor T into a symmetric"}, {"start": 106.18, "duration": 5.689, "text": "and an antisymmetric part: T is equal to S\nplus A."}, {"start": 111.869, "duration": 5.04, "text": "Let's take a look at how we practically split\na general second-level tensor T into its two"}, {"start": 116.909, "duration": 1.291, "text": "parts."}, {"start": 118.2, "duration": 6.92, "text": "The symmetric part S of the tensor T is "One\nand a half times t_{ij} plus t_{ji}."}, {"start": 125.12, "duration": 3.02, "text": "In the second term, we have swapped the two\nindices."}, {"start": 128.14, "duration": 2.339, "text": "Then we added the two tensors together."}, {"start": 130.479, "duration": 5.421, "text": "The factor of one and a half is important\nbecause we have counted the tensor twice here."}, {"start": 135.9, "duration": 7.3, "text": "The antisymmetric part A of the tensor T is\n"One and a half times t_{ij}, minus t_{ji}."}, {"start": 143.2, "duration": 4.83, "text": "Here we have swapped the two indices, which\ngives the swapped tensor a minus sign."}, {"start": 148.03, "duration": 3.64, "text": "The factor of one and a half must not be missing\nhere either."}, {"start": 151.67, "duration": 4.49, "text": "We then add the symmetric and antisymmetric\nparts together to obtain the total tensor."}, {"start": 156.16, "duration": 1.0, "text": "We are done."}, {"start": 157.16, "duration": 4.49, "text": "The first summand is the symmetric part of\nthe tensor t_{ij} and the second summand is"}, {"start": 161.65, "duration": 1.52, "text": "the antisymmetric part."}], "title": "Introduction to Tensors: Symmetric and Antisymmetric Tensor", "description": "https://www.youtube.com/watch?v=8K7_qPmYBiY&list=PLTjLwQcqQzNKzSAxJxKpmOtAriFS5wWy4\n\nHere you will learn the most important property of tensors, namely their symmetry. More: https://en.fufaev.org/tensors\n\n\n\nBooks by Alexander Fufaev:\n1) Equations of Physics: Solve EVERY Physics Problem \nhttps://en.fufaev.org/physics-equations-book\n2) Alexander Fufaev and His Never Ending Story: \nhttps://en.fufaev.org/lifestory", "lengthSeconds": 164, "width": 640, "height": 360}
{"video_id": "6IWIqlOozCE", "transcript": [{"start": 0.08, "duration": 2.96, "text": "let's assume you have bought some"}, {"start": 1.439, "duration": 4.681, "text": "jewelry and want to know whether it is"}, {"start": 3.04, "duration": 5.92, "text": "made of gold and is not fake how can"}, {"start": 6.12, "duration": 4.88, "text": "Physics help you with this first you"}, {"start": 8.96, "duration": 4.639, "text": "have to weigh the jewelry and determine"}, {"start": 11.0, "duration": 5.84, "text": "its massim then you have to find out the"}, {"start": 13.599, "duration": 6.561, "text": "volume v of the jewelry to do this you"}, {"start": 16.84, "duration": 5.519, "text": "use Archimedes principle this states"}, {"start": 20.16, "duration": 4.199, "text": "that the amount of water displaced is"}, {"start": 22.359, "duration": 5.68, "text": "exactly equal to the volume of the"}, {"start": 24.359, "duration": 6.24, "text": "immersed body so take a measuring cup of"}, {"start": 28.039, "duration": 5.001, "text": "water throw in the jewelry and look at"}, {"start": 30.599, "duration": 3.601, "text": "the milliliter scale to see how high the"}, {"start": 33.04, "duration": 4.839, "text": "water has"}, {"start": 34.2, "duration": 6.96, "text": "risen convert the value into cubic"}, {"start": 37.879, "duration": 4.721, "text": "met now the density row of the jewelry"}, {"start": 41.16, "duration": 4.84, "text": "material can be"}, {"start": 42.6, "duration": 6.599, "text": "calculated simply divide the mass by the"}, {"start": 46.0, "duration": 7.399, "text": "volume the gold has a density of"}, {"start": 49.199, "duration": 6.52, "text": "19,300 kg per cubic meter if you also"}, {"start": 53.399, "duration": 6.201, "text": "get this value then the jewelry is made"}, {"start": 55.719, "duration": 3.881, "text": "of real gold"}], "title": "How Can You Tell If Jewelry Is Made of Real Gold?", "description": "https://www.youtube.com/watch?v=7SjAUgeKZWg&list=PLTjLwQcqQzNKzSAxJxKpmOtAriFS5wWy4\n\nHere you will learn how an electric dipole (e.g. a molecule) behaves in a homogeneous and inhomogeneous electric field. More: https://en.fufaev.org/mass-density\n\n\n\n\nBooks by Alexander Fufaev:\n1) Equations of Physics: Solve EVERY Physics Problem \nhttps://en.fufaev.org/physics-equations-book\n2) Alexander Fufaev and His Never Ending Story: \nhttps://en.fufaev.org/lifestory", "lengthSeconds": 58, "width": 640, "height": 360}
{"video_id": "eVs-p3krM4c", "transcript": [{"start": 0.16, "duration": 5.32, "text": "to calculate the temperature of the Sun"}, {"start": 2.44, "duration": 6.079, "text": "from Earth we must first find out what"}, {"start": 5.48, "duration": 5.56, "text": "light the sun emits the sun emits a"}, {"start": 8.519, "duration": 5.641, "text": "polychromatic light that is infrared"}, {"start": 11.04, "duration": 5.76, "text": "light visible light and even X-rays and"}, {"start": 14.16, "duration": 4.879, "text": "gamma rays you need to figure out the"}, {"start": 16.8, "duration": 4.88, "text": "wavelengths Lambda of each of these"}, {"start": 19.039, "duration": 3.721, "text": "radiations and with what intensity they"}, {"start": 21.68, "duration": 4.12, "text": "are"}, {"start": 22.76, "duration": 5.919, "text": "emitted if we plot all wavelengths with"}, {"start": 25.8, "duration": 5.52, "text": "their intensities in a diagram we get a"}, {"start": 28.679, "duration": 5.841, "text": "spectral intensity distribution of the"}, {"start": 31.32, "duration": 6.12, "text": "Sun from this distribution we have to"}, {"start": 34.52, "duration": 5.879, "text": "read off the wavelength Lambda Max which"}, {"start": 37.44, "duration": 5.32, "text": "has the highest intensity so we have to"}, {"start": 40.399, "duration": 4.761, "text": "read off the maximum of the"}, {"start": 42.76, "duration": 4.76, "text": "function then you can calculate the"}, {"start": 45.16, "duration": 8.44, "text": "temperature of the sun with the wans"}, {"start": 47.52, "duration": 6.08, "text": "displacement law to do this we divide"}, {"start": 54.239, "duration": 7.081, "text": "28978 * 10 power of -6 M Kelvin by"}, {"start": 60.199, "duration": 4.761, "text": "Lambda"}, {"start": 61.32, "duration": 6.159, "text": "Max in the case of the Sun the maximum"}, {"start": 64.96, "duration": 8.799, "text": "is within the blue light spectrum at the"}, {"start": 67.479, "duration": 8.361, "text": "wavelength 490 nanom or 490 * 10 to the"}, {"start": 73.759, "duration": 6.32, "text": "power of -9"}, {"start": 75.84, "duration": 7.319, "text": "M this results in a temperature of"}, {"start": 80.079, "duration": 7.521, "text": "5,900 Kelvin or"}, {"start": 83.159, "duration": 4.441, "text": "5,630 de C"}], "title": "How to Determine the Temperature of the Sun from Earth?", "description": "https://www.youtube.com/watch?v=eVs-p3krM4c&list=PLTjLwQcqQzNKzSAxJxKpmOtAriFS5wWy4\nHere you will learn how to determine the temperature of a glowing body (e.g. the sun) from a distance. More: https://en.fufaev.org/wien-displacement-law\n\n\n\n\nBooks by Alexander Fufaev:\n1) Equations of Physics: Solve EVERY Physics Problem \nhttps://en.fufaev.org/physics-equations-book\n2) Alexander Fufaev and His Never Ending Story: \nhttps://en.fufaev.org/lifestory", "lengthSeconds": 86, "width": 640, "height": 360}
{"video_id": "eaWUT1KqFlA", "transcript": [{"start": 0.32, "duration": 5.16, "text": "let's take a look at the one-dimensional"}, {"start": 2.24, "duration": 5.079, "text": "wave equation for an electric field the"}, {"start": 5.48, "duration": 4.8, "text": "function we're looking for is the"}, {"start": 7.319, "duration": 6.48, "text": "electric field e this depends on the"}, {"start": 10.28, "duration": 5.839, "text": "position X and on the time T since the"}, {"start": 13.799, "duration": 4.721, "text": "function depends on two variables and"}, {"start": 16.119, "duration": 5.32, "text": "their derivatives appear in the equation"}, {"start": 18.52, "duration": 4.919, "text": "it is a partial differential equation"}, {"start": 21.439, "duration": 6.881, "text": "here we make a product onzas for the"}, {"start": 23.439, "duration": 7.481, "text": "searched solution e e is equal to R * U"}, {"start": 28.32, "duration": 5.6, "text": "we assume that this solution e e can be"}, {"start": 30.92, "duration": 6.76, "text": "split into a product of two functions"}, {"start": 33.92, "duration": 6.76, "text": "that we call r and u one function R"}, {"start": 37.68, "duration": 6.879, "text": "depends only on X and the other function"}, {"start": 40.68, "duration": 6.399, "text": "U depends only on T in the wave equation"}, {"start": 44.559, "duration": 5.041, "text": "we have a second derivative of e with"}, {"start": 47.079, "duration": 5.841, "text": "respect to location X and a second"}, {"start": 49.6, "duration": 5.72, "text": "derivative with respect to time T"}, {"start": 52.92, "duration": 5.56, "text": "differentiating the product on that with"}, {"start": 55.32, "duration": 6.28, "text": "respect to X yields U * the second"}, {"start": 58.48, "duration": 6.039, "text": "derivative of r with respect to X Since"}, {"start": 61.6, "duration": 3.92, "text": "U is independent of X and thus acts like"}, {"start": 64.519, "duration": 4.041, "text": "a"}, {"start": 65.52, "duration": 5.68, "text": "constant in contrast in the derivative"}, {"start": 68.56, "duration": 5.599, "text": "of the product with respect to time T"}, {"start": 71.2, "duration": 6.68, "text": "the function R acts like a constant"}, {"start": 74.159, "duration": 5.881, "text": "because it does not depend on T the goal"}, {"start": 77.88, "duration": 5.879, "text": "now is to separate everything that"}, {"start": 80.04, "duration": 8.64, "text": "depends on X from what depends on T for"}, {"start": 83.759, "duration": 4.921, "text": "this we divide this equation by R *"}, {"start": 88.84, "duration": 5.2, "text": "U"}, {"start": 91.759, "duration": 5.0, "text": "thus we have achieved that everything"}, {"start": 94.04, "duration": 5.119, "text": "that depends on X is on the left side"}, {"start": 96.759, "duration": 5.521, "text": "and everything that depends on T is on"}, {"start": 99.159, "duration": 5.96, "text": "the right side if you manage to separate"}, {"start": 102.28, "duration": 6.28, "text": "a partial differential equation this way"}, {"start": 105.119, "duration": 6.081, "text": "then the separation unz was successful"}, {"start": 108.56, "duration": 4.96, "text": "now we can vary X on the left side"}, {"start": 111.2, "duration": 5.08, "text": "without changing the right side because"}, {"start": 113.52, "duration": 5.72, "text": "there is no X on the right side the same"}, {"start": 116.28, "duration": 5.519, "text": "is true for the time T if we vary the"}, {"start": 119.24, "duration": 4.799, "text": "time on the the right side the left side"}, {"start": 121.799, "duration": 6.6, "text": "remains unchanged because there is no"}, {"start": 124.039, "duration": 6.601, "text": "time T thus both sides must be constant"}, {"start": 128.399, "duration": 5.001, "text": "so let us set the left side and the"}, {"start": 130.64, "duration": 5.76, "text": "right side equal to a constant"}, {"start": 133.4, "duration": 6.04, "text": "K thus we have transformed a partial"}, {"start": 136.4, "duration": 5.4, "text": "differential equation into two ordinary"}, {"start": 139.44, "duration": 4.2, "text": "differential equations and the good"}, {"start": 141.8, "duration": 5.0, "text": "thing is that the two differential"}, {"start": 143.64, "duration": 5.56, "text": "equations are not coupled that means you"}, {"start": 146.8, "duration": 4.32, "text": "can solve them independently and then"}, {"start": 149.2, "duration": 4.48, "text": "multiply the solutions like in the"}, {"start": 151.12, "duration": 4.56, "text": "product onz to get the general solution"}, {"start": 153.68, "duration": 4.76, "text": "for the partial differential"}, {"start": 155.68, "duration": 4.479, "text": "equation you can solve the two ordinary"}, {"start": 158.44, "duration": 4.439, "text": "differential equations with the"}, {"start": 160.159, "duration": 5.72, "text": "exponential UNS you learned"}, {"start": 162.879, "duration": 3.0, "text": "before"}], "title": "1D Wave Equation: Solving Differential Equation with Variable Separation", "description": "Here you will learn how to solve the one-dimensional wave equation (partial differential equation) with separation of variables. More: https://en.fufaev.org/partial-differential-equations\n\nBooks by Alexander Fufaev:\n1) Equations of Physics: Solve EVERY Physics Problem \nhttps://en.fufaev.org/physics-equations-book\n2) Alexander Fufaev and His Never Ending Story: \nhttps://en.fufaev.org/lifestory", "lengthSeconds": 164, "width": 640, "height": 360}
{"video_id": "pPrIgQ2NAe0", "transcript": [{"start": 0.079, "duration": 4.421, "text": "betrachten wir einen geladenen Leiter"}, {"start": 2.52, "duration": 4.62, "text": "wie zum Beispiel einen elektrisch"}, {"start": 4.5, "duration": 5.28, "text": "geladenen Metallk\u00f6rper der geladene"}, {"start": 7.14, "duration": 4.499, "text": "Leiter erzeugt ein elektrisches Feld das"}, {"start": 9.78, "duration": 4.019, "text": "immer senkrecht zu seiner Oberfl\u00e4che ist"}, {"start": 11.639, "duration": 4.261, "text": "warum"}, {"start": 13.799, "duration": 4.56, "text": "die elektrischen Ladungen nehme ich die"}, {"start": 15.9, "duration": 5.28, "text": "Elektronen im Inneren des Leiters k\u00f6nnen"}, {"start": 18.359, "duration": 5.301, "text": "sich frei bewegen Ladungen in der N\u00e4he"}, {"start": 21.18, "duration": 5.48, "text": "der Oberfl\u00e4che des Leiters werden als"}, {"start": 23.66, "duration": 5.14, "text": "Oberfl\u00e4chenladungen bezeichnet die"}, {"start": 26.66, "duration": 4.6, "text": "Oberfl\u00e4chenladungen k\u00f6nnen sich entlang"}, {"start": 28.8, "duration": 5.939, "text": "der leiteroberfl\u00e4che also parallel dazu"}, {"start": 31.26, "duration": 5.639, "text": "frei bewegen im Gegensatz dazu k\u00f6nnen"}, {"start": 34.739, "duration": 5.281, "text": "sich die Oberfl\u00e4chenladungen senkrecht"}, {"start": 36.899, "duration": 5.34, "text": "zu Leiter Oberfl\u00e4che nicht bewegen wenn"}, {"start": 40.02, "duration": 5.16, "text": "Sie das k\u00f6nnten dann w\u00fcrden sie aus der"}, {"start": 42.239, "duration": 5.101, "text": "Oberfl\u00e4che Austreten der Leiter w\u00e4re"}, {"start": 45.18, "duration": 4.8, "text": "nicht mehr geladen und w\u00fcrde damit gar"}, {"start": 47.34, "duration": 4.92, "text": "keine elektrisches Feld erzeugen"}, {"start": 49.98, "duration": 5.04, "text": "die Elektronen im Inneren des Leiters"}, {"start": 52.26, "duration": 4.86, "text": "erzeugen eine elektrisches Feld E das"}, {"start": 55.02, "duration": 4.98, "text": "elektrische Feld ist eine vektorielle"}, {"start": 57.12, "duration": 5.52, "text": "Gr\u00f6\u00dfe es hat also einen Betrag und eine"}, {"start": 60.0, "duration": 4.86, "text": "Richtung nehmen wir mal an dass das von"}, {"start": 62.64, "duration": 5.4, "text": "den Elektronen erzeugte elektrische Feld"}, {"start": 64.86, "duration": 6.0, "text": "E an der Oberfl\u00e4che nicht senkrecht aus"}, {"start": 68.04, "duration": 5.28, "text": "der Oberfl\u00e4che heraustreten w\u00fcrde es"}, {"start": 70.86, "duration": 4.56, "text": "zeigt so wie hier gezeigt in irgendeine"}, {"start": 73.32, "duration": 5.76, "text": "beliebige Richtung die nicht senkrecht"}, {"start": 75.42, "duration": 6.12, "text": "ist die Mathematik sagt uns dass wir"}, {"start": 79.08, "duration": 5.7, "text": "jeden Vektor also auch den elektrischen"}, {"start": 81.54, "duration": 4.56, "text": "Feldvektor e in zwei Anteile aufteilen"}, {"start": 84.78, "duration": 3.3, "text": "k\u00f6nnen"}, {"start": 86.1, "duration": 5.22, "text": "in einen zur leiteroberfl\u00e4che"}, {"start": 88.08, "duration": 6.539, "text": "Senkrechten Anteil und in einen zuleiter"}, {"start": 91.32, "duration": 5.28, "text": "Oberfl\u00e4che parallelen Anteil das zu"}, {"start": 94.619, "duration": 4.32, "text": "Oberfl\u00e4che parallele elektrische Feld"}, {"start": 96.6, "duration": 4.8, "text": "f\u00fchrt zu einer elektrischen Kraft auf"}, {"start": 98.939, "duration": 5.04, "text": "die benachbarten Elektronen so dass sich"}, {"start": 101.4, "duration": 5.34, "text": "diese Elektronen parallel zu Oberfl\u00e4che"}, {"start": 103.979, "duration": 5.46, "text": "verschieben die Elektronen verschieben"}, {"start": 106.74, "duration": 5.879, "text": "sich so lange entlang der Oberfl\u00e4che bis"}, {"start": 109.439, "duration": 5.22, "text": "ein Kr\u00e4fte Gleichgewicht ist in diesem"}, {"start": 112.619, "duration": 4.221, "text": "Gleichgewicht ist die Kraft durch die"}, {"start": 114.659, "duration": 4.561, "text": "gegenseitige Absto\u00dfung der Elektronen"}, {"start": 116.84, "duration": 4.599, "text": "aufgehoben und damit auch das"}, {"start": 119.22, "duration": 4.74, "text": "elektrische Feld das parallel zur"}, {"start": 121.439, "duration": 4.86, "text": "Oberfl\u00e4che zeigt wenn der parallele"}, {"start": 123.96, "duration": 3.96, "text": "f\u00e4llt Anteil noch da w\u00e4re w\u00fcrden sich"}, {"start": 126.299, "duration": 3.901, "text": "die Elektronen weiter entlang der"}, {"start": 127.92, "duration": 4.86, "text": "Oberfl\u00e4che verschieben"}, {"start": 130.2, "duration": 5.399, "text": "das zu Oberfl\u00e4che senkrechte elektrische"}, {"start": 132.78, "duration": 5.94, "text": "Feld dagegen ist immer noch da damit es"}, {"start": 135.599, "duration": 5.341, "text": "gemeint wenn eine Ladung au\u00dferhalb des"}, {"start": 138.72, "duration": 4.68, "text": "Leiters platziert wird dann wird sie"}, {"start": 140.94, "duration": 5.46, "text": "eine elektrische Kraft genau senkrecht"}, {"start": 143.4, "duration": 4.62, "text": "zur Oberfl\u00e4che erfahren diese Anteil des"}, {"start": 146.4, "duration": 4.4, "text": "elektrischen Feldes senkrecht zu"}, {"start": 148.02, "duration": 4.56, "text": "Oberfl\u00e4che hebt sich nicht weg weil die"}, {"start": 150.8, "duration": 4.72, "text": "Oberfl\u00e4chenladungen daf\u00fcr aus der"}, {"start": 152.58, "duration": 4.98, "text": "Oberfl\u00e4che austreten m\u00fcssten es bleibt"}, {"start": 155.52, "duration": 4.88, "text": "also im Gegensatz zum parallelen Anteil"}, {"start": 157.56, "duration": 2.84, "text": "weiterhin bestehen"}, {"start": 160.8, "duration": 5.1, "text": "damit steht das elektrische Feld eines"}, {"start": 163.019, "duration": 4.8, "text": "geladenen Leiters immer senkrecht auf"}, {"start": 165.9, "duration": 3.72, "text": "seiner Oberfl\u00e4che"}, {"start": 167.819, "duration": 4.261, "text": "wenn wir die elektrischen Feldlinien"}, {"start": 169.62, "duration": 3.96, "text": "einzeichnen w\u00fcrden dann w\u00fcrden sie auch"}, {"start": 172.08, "duration": 3.72, "text": "senkrecht aus der Oberfl\u00e4che"}, {"start": 173.58, "duration": 4.68, "text": "heraustreten"}, {"start": 175.8, "duration": 4.56, "text": "\u00fcbrigens kannst du dir vorstellen warum"}, {"start": 178.26, "duration": 4.559, "text": "das elektrische Feld von geladenen"}, {"start": 180.36, "duration": 6.14, "text": "Isolatoren im Allgemeinen nicht"}, {"start": 182.819, "duration": 3.681, "text": "senkrecht auf der Oberfl\u00e4che steht"}], "title": "Warum stehen Feldlinien senkrecht auf leitenden Oberfl\u00e4chen?", "description": "In diesem Physikvideo lernst du, warum das elektrische Feld bzw. elektrische Feldlinien IMMER senkrecht auf Leitern stehen! Mehr: https://de.fufaev.org/fragen/45\n\n#universaldenker", "lengthSeconds": 211, "width": 640, "height": 360}
{"video_id": "2wq06lqOhlc", "transcript": [{"start": 0.12, "duration": 4.32, "text": "was ist der Unterschied zwischen einer"}, {"start": 1.8, "duration": 5.8, "text": "partiellen Ableitung und einer totalen"}, {"start": 4.44, "duration": 6.319, "text": "Ableitung einer Funktion f hierbei h\u00e4ngt"}, {"start": 7.6, "duration": 5.039, "text": "die Funktion f von x und y ab und bei"}, {"start": 10.759, "duration": 5.321, "text": "beiden Ableitungen leiten wir hier nach"}, {"start": 12.639, "duration": 5.041, "text": "x ab bei der partiellen Ableitung die"}, {"start": 16.08, "duration": 5.76, "text": "mit einem geschwungenen Dell notiert"}, {"start": 17.68, "duration": 6.08, "text": "wird ist y unabh\u00e4ngig von X bei der"}, {"start": 21.84, "duration": 5.359, "text": "totalen Ableitung die mit einem D"}, {"start": 23.76, "duration": 5.679, "text": "notiert wird kann y von X abh\u00e4ngen die"}, {"start": 27.199, "duration": 5.36, "text": "\u00c4nderung der Variable x wirkt sich also"}, {"start": 29.439, "duration": 6.081, "text": "auch auf den Wert von Y aus machen wir"}, {"start": 32.559, "duration": 6.68, "text": "ein Beispiel betrachte die Funktion 3x"}, {"start": 35.52, "duration": 7.039, "text": "zum qu + 2Y berechnen wir zuerst die"}, {"start": 39.239, "duration": 7.761, "text": "partielle Ableitung dazu m\u00fcssen wir 3x\u00b2"}, {"start": 42.559, "duration": 9.48, "text": "partiell ableiten + 2Y partiell ableiten"}, {"start": 47.0, "duration": 7.68, "text": "3x\u00b2 nach X abgeleitet ist 6x und 2Y nach"}, {"start": 52.039, "duration": 5.561, "text": "X abgeleitet ist 0 weil wir ja bei der"}, {"start": 54.68, "duration": 6.039, "text": "partiellen Ableitung annehmen dass Y"}, {"start": 57.6, "duration": 5.72, "text": "unabh\u00e4ngig ist von x bilden wir jetzt"}, {"start": 60.719, "duration": 5.361, "text": "die totale Ableitung der Funktion das"}, {"start": 63.32, "duration": 6.68, "text": "geschwungene Dell \u00e4ndert sich zu einem D"}, {"start": 66.08, "duration": 6.6, "text": "die totale Ableitung von 3x\u00b2 ist 6x und"}, {"start": 70.0, "duration": 6.159, "text": "die totale Ableitung von 2Y ist nicht"}, {"start": 72.68, "duration": 6.439, "text": "mehr 0 weil y von X abh\u00e4ngen kann wir"}, {"start": 76.159, "duration": 5.64, "text": "k\u00f6nnen den Faktor 2 vor die Ableitung"}, {"start": 79.119, "duration": 4.761, "text": "schreiben fassen wir zusammen bei der"}, {"start": 81.799, "duration": 5.0, "text": "partiellen Ableitung sind die Parameter"}, {"start": 83.88, "duration": 5.279, "text": "x und y der Funktion f unabh\u00e4ngig"}, {"start": 86.799, "duration": 5.481, "text": "voneinander bei der totalen Ableitung"}, {"start": 89.159, "duration": 6.64, "text": "dagegen kann x von Y abh\u00e4ngen oder wie"}, {"start": 92.28, "duration": 6.519, "text": "im Beispiel y von"}, {"start": 95.799, "duration": 3.0, "text": "x"}], "title": "Unterschied zwischen partieller und totaler Ableitung", "description": "Hier lernst du anhand eines Beispiels, was der Unterschied zwischen der partiellen Ableitung und der totalen Ableitung ist. Mehr: https://de.fufaev.org/partielle-und-totale-ableitung", "lengthSeconds": 97, "width": 640, "height": 360}
{"video_id": "fOX4FaRahIk", "transcript": [{"start": 0.08, "duration": 5.4, "text": "betrachten wir ein System mit einem"}, {"start": 1.92, "duration": 7.16, "text": "idealen Gas drin der Zustand des Gases"}, {"start": 5.48, "duration": 7.039, "text": "genauer gesagt seine Temperatur T sein"}, {"start": 9.08, "duration": 7.6, "text": "Druck PI und sein Volumen V werden durch"}, {"start": 12.519, "duration": 9.361, "text": "die ideale Gasgleichung beschrieben pi="}, {"start": 16.68, "duration": 7.839, "text": "n* R* T* 1 dur V hierbei ist n die"}, {"start": 21.88, "duration": 5.68, "text": "Stoffmenge und beschreibt in direkt die"}, {"start": 24.519, "duration": 5.481, "text": "Anzahl der Gasteilchen und R ist die"}, {"start": 27.56, "duration": 5.72, "text": "Gaskonstante dabei bezeichnen wir die"}, {"start": 30.0, "duration": 5.399, "text": "Temperatur den Druck und das Volumen als"}, {"start": 33.28, "duration": 5.52, "text": "Zustandsgr\u00f6\u00dfen sie beschreiben den"}, {"start": 35.399, "duration": 5.561, "text": "makroskopischen Zustand des Gases wenn"}, {"start": 38.8, "duration": 4.88, "text": "wir das Gas erhitzen und somit seine"}, {"start": 40.96, "duration": 4.079, "text": "Temperatur T erh\u00f6hen sprechen wir von"}, {"start": 43.68, "duration": 3.64, "text": "einer"}, {"start": 45.039, "duration": 4.761, "text": "Zustands\u00e4nderung ebenso k\u00f6nnten wir das"}, {"start": 47.32, "duration": 5.48, "text": "Gas zusammenpressen damit den Druck"}, {"start": 49.8, "duration": 5.04, "text": "erh\u00f6hen und das Volumen verringern das"}, {"start": 52.8, "duration": 4.36, "text": "w\u00e4re ebenfalls eine"}, {"start": 54.84, "duration": 5.48, "text": "Zustands\u00e4nderung die Zustands\u00e4nderung"}, {"start": 57.16, "duration": 6.999, "text": "kann dabei isobar isotherm is oder"}, {"start": 60.32, "duration": 5.92, "text": "adiabat sein bei einer Isobaren"}, {"start": 64.159, "duration": 6.28, "text": "Zustands\u00e4nderung bleibt der Druck des"}, {"start": 66.24, "duration": 7.28, "text": "Gases konstant damit sind PI n und R im"}, {"start": 70.439, "duration": 5.601, "text": "Gasgesetz reine konstanten und das"}, {"start": 73.52, "duration": 3.959, "text": "Volumen V ist proportional zur"}, {"start": 76.04, "duration": 3.6, "text": "Temperatur"}, {"start": 77.479, "duration": 4.161, "text": "T im"}, {"start": 79.64, "duration": 4.36, "text": "druckvolumendiagramm ergibt sich f\u00fcr ein"}, {"start": 81.64, "duration": 3.839, "text": "isobar verhaltenendes Gas eine"}, {"start": 84.0, "duration": 4.56, "text": "horizontale"}, {"start": 85.479, "duration": 4.761, "text": "gerade das hei\u00dft die \u00c4nderung des"}, {"start": 88.56, "duration": 4.519, "text": "Volumens f\u00fchrt nicht zu einer"}, {"start": 90.24, "duration": 5.199, "text": "Ver\u00e4nderung des Gasdrucks bei einer"}, {"start": 93.079, "duration": 6.481, "text": "isokoren Zustands\u00e4nderung bleibt das"}, {"start": 95.439, "duration": 8.241, "text": "Volumen V des Gases konstant dann sind V"}, {"start": 99.56, "duration": 7.96, "text": "n und R konstanten im Gasgesetz und der"}, {"start": 103.68, "duration": 5.759, "text": "Druck Pi ist proportional zur Temperatur"}, {"start": 107.52, "duration": 4.239, "text": "T im"}, {"start": 109.439, "duration": 5.201, "text": "druckvolumendiagramm ergibt sich f\u00fcr ein"}, {"start": 111.759, "duration": 6.0, "text": "isochorverhaltendes Gas eine vertikale"}, {"start": 114.64, "duration": 6.24, "text": "gerade die \u00c4nderung des Gasdrucks f\u00fchrt"}, {"start": 117.759, "duration": 5.96, "text": "nicht zu einer Ver\u00e4nderung des Volumens"}, {"start": 120.88, "duration": 5.44, "text": "bei einer isothermen Zustands\u00e4nderung"}, {"start": 123.719, "duration": 5.4, "text": "bleibt die Temperatur T des Gases"}, {"start": 126.32, "duration": 6.48, "text": "konstant damit ist der Druck PI"}, {"start": 129.119, "duration": 6.521, "text": "proportional zum inversen Volumen 1"}, {"start": 132.8, "duration": 4.92, "text": "durch v im"}, {"start": 135.64, "duration": 4.4, "text": "druckvolumeniagramm ergibt sich f\u00fcr ein"}, {"start": 137.72, "duration": 5.72, "text": "isotherm verhaltendes Gas eine"}, {"start": 140.04, "duration": 6.16, "text": "abfallende Kurve die Erh\u00f6hung des"}, {"start": 143.44, "duration": 4.96, "text": "Volumens des Gases f\u00fchrt zu einer"}, {"start": 146.2, "duration": 5.039, "text": "Verringerung des"}, {"start": 148.4, "duration": 5.119, "text": "Gasdrucks bei einer adiabatischen"}, {"start": 151.239, "duration": 4.36, "text": "Zustands\u00e4nderung wird keine thermische"}, {"start": 153.519, "duration": 4.121, "text": "Energie V und in das System"}, {"start": 155.599, "duration": 3.681, "text": "transportiert bei dieser"}, {"start": 157.64, "duration": 5.519, "text": "Zustands\u00e4nderung kann sich die"}, {"start": 159.28, "duration": 6.679, "text": "Temperatur T Volumen V und Druck PI des"}, {"start": 163.159, "duration": 4.841, "text": "Gases gleichzeitig \u00e4ndern im"}, {"start": 165.959, "duration": 4.841, "text": "druckvolumendiagramm ergibt sich f\u00fcr ein"}, {"start": 168.0, "duration": 5.36, "text": "adiabatisch verhaltendes Gas ein"}, {"start": 170.8, "duration": 5.24, "text": "Potenzgesetz der Exponent Gamma wird"}, {"start": 173.36, "duration": 5.68, "text": "dabei als adiabatenexonent"}, {"start": 176.04, "duration": 3.0, "text": "bezeichnet"}], "title": "Isobar, isochor, isoterm, adiabat?", "description": "https://www.youtube.com/watch?v=fOX4FaRahIk&list=PLcjHhwL4seQjmFSLkz_R11cgq7-LPyuGA\n\nHier lernst du was eine thermodynamische Zustands\u00e4nderung ist und wann sie isobar, isochor, isotherm und adiabat ist. Mehr: https://de.fufaev.org/thermodynamische-zustandsaenderung\n\n\n\nB\u00fccher von Alexander Fufaev:\n1) Die n\u00fctzlichste Physik-Formelsammlung des Internets: \nhttps://de.fufaev.org/formelsammlung\n2) Alexander Fufaev und seine Lebensgeschichte: \nhttps://de.fufaev.org/lebensgeschichte", "lengthSeconds": 177, "width": 640, "height": 360}
{"video_id": "TiWilQkZDOA", "transcript": [{"start": 0.63, "duration": 4.17, "text": "If you know the Lagrangian of a system, the\nEuler-Lagrange equations will provide you"}, {"start": 4.8, "duration": 4.98, "text": "with the equations of motion and describe\nhow the system will evolve over time."}, {"start": 9.78, "duration": 5.31, "text": "Before we start, we need to talk about two\nthings. First, we define the so-called action"}, {"start": 15.09, "duration": 5.329, "text": "of a system as the integral over the Lagrangian.\nFor classical mechanics, this means the time"}, {"start": 20.419, "duration": 5.071, "text": "integral over the Lagrangian function, however\nfor relativistic mechanics, the Lorentz-covariant"}, {"start": 25.49, "duration": 5.31, "text": "formulation involves the spacetime integral\nover the Lagrangian density. This action is"}, {"start": 30.8, "duration": 5.34, "text": "a functional, this means it takes a function\n(in this case the Lagrangian) as an input"}, {"start": 36.14, "duration": 5.25, "text": "and after some calculation, the result is\na number. So every configuration of the system"}, {"start": 41.39, "duration": 4.79, "text": "yields as the corresponding action just a\ndifferent number. This brings us to the second"}, {"start": 46.18, "duration": 5.19, "text": "point: Hamilton\u2019s principle. Hamilton\u2019s\nprinciple states, that the system we are considering"}, {"start": 51.37, "duration": 6.37, "text": "evolves in such a way, that the action takes\non an extremal value. In other words, if we"}, {"start": 57.74, "duration": 5.91, "text": "slightly change the path that the system takes,\nthe change in action will be almost zero."}, {"start": 63.65, "duration": 4.249, "text": "A small mathematical detail here is, that\nwe do not change the starting and end points"}, {"start": 67.899, "duration": 6.091, "text": "of the path, which means that delta q for\nsome starting time t_1 and end time t_2 is"}, {"start": 73.99, "duration": 4.51, "text": "zero. Similar arguments lead to corresponding\nconstraints in the relativistic case, where"}, {"start": 78.5, "duration": 5.52, "text": "we do not change the fields at the beginning\nand end point, just somewhere along the way."}, {"start": 84.02, "duration": 4.919, "text": "Now we can start deriving the Euler-Lagrange\nequations. We start by explicitly calculating"}, {"start": 88.939, "duration": 4.731, "text": "the change in the action, knowing that whatever\ncomes out in the end must be zero according"}, {"start": 93.67, "duration": 5.019, "text": "to Hamilton\u2019s principle. For the classical\nversion, this means varying the Lagrangian"}, {"start": 98.689, "duration": 5.871, "text": "with respect to q and multiplying a small\nchange in q as well as varying in q dot and"}, {"start": 104.56, "duration": 5.39, "text": "multiplying a small change in q dot. For the\nrelativistic version, we do this with respect"}, {"start": 109.95, "duration": 3.419, "text": "to the fields phi and their derivatives d_mu\nphi."}, {"start": 113.369, "duration": 5.551, "text": "A change in the time derivative of q is the\nsame as the time derivative of the change"}, {"start": 118.92, "duration": 5.759, "text": "in q, so let\u2019s write it like this. Similarly,\nwe can exchange the delta and the derivative"}, {"start": 124.679, "duration": 8.541, "text": "on the right.\nNext we use the inverse product rule, which"}, {"start": 133.22, "duration": 4.67, "text": "let\u2019s us write a function and the derivative\nof another function as the derivative of their"}, {"start": 137.89, "duration": 5.08, "text": "product minus the derivative of the first\nfunction times the second function. We now"}, {"start": 142.97, "duration": 6.36, "text": "write the second term as the time derivative\nof dL over dq dot and delta q minus the time"}, {"start": 149.33, "duration": 7.34, "text": "derivative of dL over dq dot times delta q.\nOn the right, this is the same, but with a"}, {"start": 156.67, "duration": 2.78, "text": "d_mu.\nFrom here on we can evaluate parts of the"}, {"start": 159.45, "duration": 5.44, "text": "integral, in particular the total derivative\nwhich is now evaluated between the end time"}, {"start": 164.89, "duration": 6.12, "text": "and the starting time. As for the other terms,\nwe can factor out the delta q. On the right,"}, {"start": 171.01, "duration": 5.91, "text": "we evaluate between x^mu_2 and x^mu_1 and\nalso pull out the delta phi."}, {"start": 176.92, "duration": 5.18, "text": "Due to our assumptions, the first term vanishes,\nsince it is exactly the change of q or phi"}, {"start": 182.1, "duration": 5.33, "text": "evaluated at the boundary points. That\u2019s\nwhy this term is usually called a \u201eboundary"}, {"start": 187.43, "duration": 5.23, "text": "term\u201c. And remember, since we started from\na small change in the action, the terms in"}, {"start": 192.66, "duration": 6.07, "text": "brackets have to vanish due to Hamilton\u2019s\nprinciple. And this is what leads to the Euler-Lagrange"}, {"start": 198.73, "duration": 5.13, "text": "equations.\nRemember, that q here stands for all the variables"}, {"start": 203.86, "duration": 5.2, "text": "you have in your system. So if there is, for\nexample, an x and a y, you actually have two"}, {"start": 209.06, "duration": 5.58, "text": "Euler-Lagrange equations, one for x and x\ndot and the other for y and y dot. As you"}, {"start": 214.64, "duration": 4.59, "text": "can see, instead of two first-order differential\nequations in the Hamiltonian formulation,"}, {"start": 219.23, "duration": 3.94, "text": "we now have one second order differential\nequation in the Lagrangian formulation for"}, {"start": 223.17, "duration": 3.83, "text": "each variable. And that\u2019s pretty much it\nfor now, thanks for watching."}], "title": "Derivation of Euler-Lagrange Equations |\u00a0Classical Mechanics", "description": "The Euler-Lagrange equations describe how a physical system will evolve over time if you know about the Lagrange function. We assume that out of all the different paths a particle can take, it will be the one, where small deviations from the path won\u2019t change the action too much. This is usually referred to as: \u201ethe action is stationary\u201c. \n\nIf you want to read more about the Lagrangian formulation of #ClassicalMechanics, we can recommend the book \u201eMechanics\u201c by Landau and Lifshitz: \nhttps://amzn.to/2NjI8gQ \n(Affiliate-Link)\n\nIf you want to help us get rid of ads on YouTube, you can support us on Patreon! \nhttps://www.patreon.com/prettymuchphysics", "lengthSeconds": 234, "width": 640, "height": 360}
{"video_id": "ZAw98lYRL30", "transcript": [{"start": 0.199, "duration": 5.401, "text": "consider a large electrically charged"}, {"start": 2.6, "duration": 6.56, "text": "sphere with a charge capital Q and A"}, {"start": 5.6, "duration": 7.119, "text": "small sphere with a charge small q the"}, {"start": 9.16, "duration": 5.599, "text": "electric force between these two spheres"}, {"start": 12.719, "duration": 6.4, "text": "which are at the distance R from each"}, {"start": 14.759, "duration": 7.801, "text": "other is given by colum's law here 1/ 4"}, {"start": 19.119, "duration": 6.32, "text": "Pi epsilon0 is a constant preactor with"}, {"start": 22.56, "duration": 5.52, "text": "the electric field constant Epsilon 0"}, {"start": 25.439, "duration": 6.24, "text": "which provides the right unit of force"}, {"start": 28.08, "duration": 6.36, "text": "the unit Newton now what if you know the"}, {"start": 31.679, "duration": 5.121, "text": "value of the big charge and want to know"}, {"start": 34.44, "duration": 5.48, "text": "the value of the force that the big"}, {"start": 36.8, "duration": 5.64, "text": "charge exerts on the small charge"}, {"start": 39.92, "duration": 4.52, "text": "however you do not know the exact value"}, {"start": 42.44, "duration": 4.72, "text": "of this little charge or you"}, {"start": 44.44, "duration": 5.119, "text": "intentionally leave that value open and"}, {"start": 47.16, "duration": 5.52, "text": "only want to look at the electric force"}, {"start": 49.559, "duration": 6.0, "text": "exerted by the big charge so you have to"}, {"start": 52.68, "duration": 6.039, "text": "somehow eliminate the small charge Q"}, {"start": 55.559, "duration": 6.84, "text": "from colum's law to do so you simply"}, {"start": 58.719, "duration": 5.72, "text": "divide colum's law by Q on both sides"}, {"start": 62.399, "duration": 5.04, "text": "this way on the right hand side the"}, {"start": 64.439, "duration": 5.481, "text": "small charge drops out and instead lands"}, {"start": 67.439, "duration": 4.961, "text": "on the left hand side of the equation"}, {"start": 69.92, "duration": 5.48, "text": "the quotient on the left hand side F"}, {"start": 72.4, "duration": 6.399, "text": "over Q is defined as the electric field"}, {"start": 75.4, "duration": 6.8, "text": "e of the source charge Q by calling it"}, {"start": 78.799, "duration": 6.481, "text": "Source charge we imply that the charge Q"}, {"start": 82.2, "duration": 5.919, "text": "is the source of the electric field the"}, {"start": 85.28, "duration": 5.44, "text": "electric field e thus indicates the"}, {"start": 88.119, "duration": 5.32, "text": "electric force which would act on a"}, {"start": 90.72, "duration": 6.96, "text": "small charge when placed at the distance"}, {"start": 93.439, "duration": 4.241, "text": "R to the source charge"}], "title": "What is Electric Field?", "description": "https://www.youtube.com/watch?v=ZAw98lYRL30&list=PLTjLwQcqQzNKzSAxJxKpmOtAriFS5wWy4\n\n\nElectric field tells how big the electric force is on an electric charge. The field strength is illustrated by field lines. The closer field lines are to each other, the greater is the field strength there, and the greater is the electric force acting on a charge there. From Course: https://en.fufaev.org/electromagnetism\n\nBooks by Alexander Fufaev:\n1) Equations of Physics: Solve EVERY Physics Problem \nhttps://en.fufaev.org/physics-equations-book\n2) Alexander Fufaev and His Never Ending Story: \nhttps://en.fufaev.org/lifestory", "lengthSeconds": 96, "width": 640, "height": 360}
{"video_id": "CQFkPtiHIMg", "transcript": [{"start": 0.08, "duration": 6.839, "text": "jeder K\u00f6rper der eine Temperatur hat"}, {"start": 2.639, "duration": 7.241, "text": "also alle strahlt Energie ab wie viel"}, {"start": 6.919, "duration": 5.961, "text": "Energie strahlt unsere Sonne ab und wie"}, {"start": 9.88, "duration": 5.12, "text": "viel davon kommt bei uns auf der Erde an"}, {"start": 12.88, "duration": 5.319, "text": "wie viel Energie strahlt eigentlich die"}, {"start": 15.0, "duration": 5.6, "text": "Erde in den Weltraum ab und welche"}, {"start": 18.199, "duration": 5.521, "text": "Oberfl\u00e4chentemperatur hat unsere Sonne"}, {"start": 20.6, "duration": 5.599, "text": "derartige und \u00e4hnliche Fragen k\u00f6nnen von"}, {"start": 23.72, "duration": 5.12, "text": "dem Stefan bolzmann Gesetz beantwortet"}, {"start": 26.199, "duration": 5.041, "text": "werden dieses Gesetz beschreibt den"}, {"start": 28.84, "duration": 5.28, "text": "Zusammenhang zwischen der Temperatur T"}, {"start": 31.24, "duration": 6.479, "text": "eines K\u00f6rpers und seiner abgestrahlten"}, {"start": 34.12, "duration": 6.599, "text": "Leistung P auch die Oberfl\u00e4che a des"}, {"start": 37.719, "duration": 5.84, "text": "K\u00f6rpers beeinflusst die abgestrahlte"}, {"start": 40.719, "duration": 8.761, "text": "Leistung das Stefan bolzmann Gesetz"}, {"start": 43.559, "duration": 9.241, "text": "lautet p= \u03c3ma mal A* t hoch 4 hierbei"}, {"start": 49.48, "duration": 5.079, "text": "ist \u03c3ma die Stefan bolzmann konstante"}, {"start": 52.8, "duration": 4.04, "text": "mit dem Wert"}, {"start": 54.559, "duration": 7.0, "text": "5,67 x 10"}, {"start": 56.84, "duration": 7.719, "text": "hoch-8 jou pro Meter qu mal K hoch 4 x"}, {"start": 61.559, "duration": 5.961, "text": "Sekunde das ist eine konstante die einen"}, {"start": 64.559, "duration": 6.521, "text": "exakten Wert hat der nur von anderen"}, {"start": 67.52, "duration": 5.56, "text": "Naturkonstanten abh\u00e4ngt n\u00e4mlich von der"}, {"start": 71.08, "duration": 4.84, "text": "bolzmannkonstante von der"}, {"start": 73.08, "duration": 4.719, "text": "plankkonstante und von der"}, {"start": 75.92, "duration": 5.04, "text": "Lichtgeschwindigkeit was ist genau die"}, {"start": 77.799, "duration": 5.601, "text": "Strahlungsleistung P sie gibt an wie"}, {"start": 80.96, "duration": 5.119, "text": "viel Energie pro Sekunde die Oberfl\u00e4che"}, {"start": 83.4, "duration": 5.359, "text": "des K\u00f6rpers abstrahlt die Einheit der"}, {"start": 86.079, "duration": 4.281, "text": "Strahlungsleistung ist Watt oder Joule"}, {"start": 88.759, "duration": 4.32, "text": "pro Sekunde"}, {"start": 90.36, "duration": 5.24, "text": "au\u00dferdem gilt das Stefan bolzmann Gesetz"}, {"start": 93.079, "duration": 5.161, "text": "nur f\u00fcr sogenannte schwarze K\u00f6rper oder"}, {"start": 95.6, "duration": 4.96, "text": "K\u00f6rper die einem schwarzen K\u00f6rper nahe"}, {"start": 98.24, "duration": 4.28, "text": "kommen die Bezeichnung schwarz hat"}, {"start": 100.56, "duration": 4.36, "text": "jedoch nichts mit der Farbe des K\u00f6rpers"}, {"start": 102.52, "duration": 5.04, "text": "zu tun die Sonne ist beispielsweise ein"}, {"start": 104.92, "duration": 4.92, "text": "ann\u00e4hernd schwarzer K\u00f6rper obwohl sie"}, {"start": 107.56, "duration": 4.599, "text": "wie du wei\u00dft nicht schwarz ist ein"}, {"start": 109.84, "duration": 4.44, "text": "schwarzer K\u00f6rper absorbiert die gesamte"}, {"start": 112.159, "duration": 4.761, "text": "Strahlung die auf ihn trifft er"}, {"start": 114.28, "duration": 4.56, "text": "reflektiert keine Strahlung und l\u00e4sst"}, {"start": 116.92, "duration": 4.519, "text": "auch keine Strahlung"}, {"start": 118.84, "duration": 4.919, "text": "durch Stefan bolzmann Gesetz kannst du"}, {"start": 121.439, "duration": 5.201, "text": "sehen wie stark die abgestrahlte Energie"}, {"start": 123.759, "duration": 5.2, "text": "von der Temperatur abh\u00e4ngt Verdopplung"}, {"start": 126.64, "duration": 4.52, "text": "der Oberfl\u00e4chentemperatur der Sonne"}, {"start": 128.959, "duration": 5.961, "text": "w\u00fcrde dazu f\u00fchren dass auf der Erde das"}, {"start": 131.16, "duration": 6.159, "text": "16fache an Sonnenenergie ankommen w\u00fcrde"}, {"start": 134.92, "duration": 4.92, "text": "machen wir ein Beispiel wie man mit dem"}, {"start": 137.319, "duration": 4.92, "text": "Stefan bolzmann gesetzt die Temperatur"}, {"start": 139.84, "duration": 5.119, "text": "der Sonne bestimmt aus der bei uns"}, {"start": 142.239, "duration": 4.961, "text": "ankommenden Energie der Sonne k\u00f6nnen wir"}, {"start": 144.959, "duration": 5.521, "text": "auf die gesamte emittierte Energie der"}, {"start": 147.2, "duration": 8.72, "text": "Sonne schlie\u00dfen die Sonne tiert eine"}, {"start": 150.48, "duration": 9.28, "text": "Leistung von 3,8 x 10 hoch 26 jou pro"}, {"start": 155.92, "duration": 7.76, "text": "Sekunde der Radius der Sonne betr\u00e4gt"}, {"start": 159.76, "duration": 5.6, "text": "6,96 x 10 hoch 8 m nehmen wir an dass"}, {"start": 163.68, "duration": 4.24, "text": "die Sonne nicht flach sondern"}, {"start": 165.36, "duration": 4.799, "text": "kugelf\u00f6rmig ist daher k\u00f6nnen wir die"}, {"start": 167.92, "duration": 5.16, "text": "Formel f\u00fcr die Oberfl\u00e4che einer Kugel"}, {"start": 170.159, "duration": 6.681, "text": "verwenden um die Oberfl\u00e4che der Sonne zu"}, {"start": 173.08, "duration": 7.04, "text": "bestimmen a= 4 PI"}, {"start": 176.84, "duration": 6.399, "text": "r\u00b2 Formen wir das Stefan bolzmann setz"}, {"start": 180.12, "duration": 6.399, "text": "nach der Temperatur T um und setzen die"}, {"start": 183.239, "duration": 5.681, "text": "Fl\u00e4chenformel ein wenn wir nun die"}, {"start": 186.519, "duration": 4.881, "text": "Strahlungsleistung die Stefan bolzmann"}, {"start": 188.92, "duration": 5.52, "text": "konstante sowie den Sonnenradius"}, {"start": 191.4, "duration": 6.399, "text": "einsetzen bekommen wir"}, {"start": 194.44, "duration": 7.24, "text": "5760 Kelvin als Oberfl\u00e4chentemperatur"}, {"start": 197.799, "duration": 8.041, "text": "der Sonne heraus das entspricht"}, {"start": 201.68, "duration": 4.16, "text": "633\u00b0 CS"}], "title": "Stefan-Boltzmann-Gesetz", "description": "https://www.youtube.com/watch?v=CQFkPtiHIMg&list=PLcjHhwL4seQjmFSLkz_R11cgq7-LPyuGA&index=27\n\nLerne, wie die Temperatur jeden K\u00f6rpers, seine abgestrahlte Energie (Leistung) beeinflusst. Mehr: https://de.fufaev.org/stefan-boltzmann-gesetz\n\n\n\nB\u00fccher von Alexander Fufaev:\n1) Die n\u00fctzlichste Physik-Formelsammlung des Internets: \nhttps://de.fufaev.org/formelsammlung\n2) Alexander Fufaev und seine Lebensgeschichte: \nhttps://de.fufaev.org/lebensgeschichte", "lengthSeconds": 204, "width": 640, "height": 360}
{"video_id": "od-jgaOQ8mc", "transcript": [{"start": 0.12, "duration": 3.66, "text": "Every body that has a temperature radiates\nenergy."}, {"start": 3.78, "duration": 4.52, "text": "How much energy does our sun radiate and how\nmuch of it reaches us on earth?"}, {"start": 8.3, "duration": 4.12, "text": "How much energy does the Earth actually radiate\ninto space?"}, {"start": 12.42, "duration": 3.389, "text": "And what is the surface temperature of our\nsun?"}, {"start": 15.809, "duration": 5.281, "text": "These and similar questions can be answered\nby the Stefan-Boltzmann law."}, {"start": 21.09, "duration": 5.3, "text": "This law describes the relationship between\nthe temperature T of a body and its radiant"}, {"start": 26.39, "duration": 2.1, "text": "power P."}, {"start": 28.49, "duration": 5.4, "text": "The surface area A of the body also influences\nthe radiant power."}, {"start": 33.89, "duration": 7.54, "text": "The Stefan-Boltzmann law is "P is equal to\nsigma times A times T to the power of four""}, {"start": 41.43, "duration": 6.89, "text": "Here, "sigma" is the Stefan-Boltzmann constant\nwith the value "five point six seven times"}, {"start": 48.32, "duration": 7.169, "text": "ten to the power of minus 8, joules per meter\nsquared times Kelvin to the power of four,"}, {"start": 55.489, "duration": 1.39, "text": "times second"."}, {"start": 56.879, "duration": 6.75, "text": "This is a constant that has an exact value\nthat only depends on other physical constants,"}, {"start": 63.629, "duration": 6.49, "text": "namely the Boltzmann constant, the Planck\nconstant and the speed of light."}, {"start": 70.119, "duration": 3.761, "text": "What exactly is the radiant power P?"}, {"start": 73.88, "duration": 5.45, "text": "It indicates how much energy per second the\nsurface of the body radiates."}, {"start": 79.33, "duration": 4.47, "text": "The unit of radiant power is watt, or joule\nper second."}, {"start": 83.8, "duration": 6.179, "text": "In addition, the Stefan-Boltzmann law only\napplies to black bodies or bodies that come"}, {"start": 89.979, "duration": 1.841, "text": "close to a black body."}, {"start": 91.82, "duration": 3.93, "text": "However, the term black has nothing to do\nwith the color of the body."}, {"start": 95.75, "duration": 5.18, "text": "The sun, for example, is a nearly black body,\nalthough, as you know, it is not black."}, {"start": 100.93, "duration": 3.67, "text": "A black body absorbs all the radiation that\nhits it."}, {"start": 104.6, "duration": 5.559, "text": "It does not reflect any radiation and does\nnot allow any radiation to pass through."}, {"start": 110.159, "duration": 5.771, "text": "The Stefan-Boltzmann law shows how much the\nradiated energy depends on the temperature."}, {"start": 115.93, "duration": 5.51, "text": "Doubling the surface temperature of the sun\nwould result in 16 times more solar energy"}, {"start": 121.44, "duration": 1.95, "text": "reaching the earth."}, {"start": 123.39, "duration": 5.8, "text": "Let's take an example of how to determine\nthe temperature of the sun using the Stefan-Boltzmann"}, {"start": 129.19, "duration": 1.59, "text": "law."}, {"start": 130.78, "duration": 5.45, "text": "We can deduce the total amount of energy emitted\nby the sun from the energy we receive from"}, {"start": 136.23, "duration": 1.77, "text": "the sun."}, {"start": 138.0, "duration": 4.98, "text": "The sun emits a power of "three point eight\ntimes ten to the power of twenty-six joules"}, {"start": 142.98, "duration": 1.6, "text": "per second."}, {"start": 144.58, "duration": 6.01, "text": "The radius of the sun is "six point nine six,\ntimes ten to the power of eight meters"."}, {"start": 150.59, "duration": 3.94, "text": "Let's assume that the sun is not flat, but\nspherical."}, {"start": 154.53, "duration": 5.23, "text": "We can therefore use the formula for the surface\narea of a sphere to determine the surface"}, {"start": 159.76, "duration": 8.13, "text": "area of the sun: "A is equal to four pi times\nr squared"."}, {"start": 167.89, "duration": 5.36, "text": "Let's rearrange the Stefan-Boltzmann law with\nrespect to the temperature T and insert the"}, {"start": 173.25, "duration": 1.9, "text": "surface area formula."}, {"start": 175.15, "duration": 7.14, "text": "If we now insert the radiant power, the Stefan-Boltzmann\nconstant and the solar radius, we get "five"}, {"start": 182.29, "duration": 6.35, "text": "thousand seven hundred sixty Kelvin" as the\nsurface temperature of the sun."}, {"start": 188.64, "duration": 3.92, "text": "This corresponds to "six thousand thirty-three\ndegrees Celsius"."}], "title": "Stefan-Boltzmann Law: How Bodies Radiate Energy Through Their Temperature", "description": "https://www.youtube.com/watch?v=od-jgaOQ8mc&list=PLTjLwQcqQzNKzSAxJxKpmOtAriFS5wWy4&index=17\n\nLearn how the temperature of each body, affects its radiated energy (power). More: https://en.fufaev.org/stefan-boltzmann-law\n\n\n\n\nBooks by Alexander Fufaev:\n1) Equations of Physics: Solve EVERY Physics Problem \nhttps://en.fufaev.org/physics-equations-book\n2) Alexander Fufaev and His Never Ending Story: \nhttps://en.fufaev.org/lifestory", "lengthSeconds": 193, "width": 640, "height": 360}
{"video_id": "AmRKcCMwgmw", "transcript": [{"start": 0.359, "duration": 6.44, "text": "all right so here we have a satellite"}, {"start": 2.72, "duration": 5.919, "text": "that is located a distance of 7 r r"}, {"start": 6.799, "duration": 3.72, "text": "being the radius of the planet or Earth"}, {"start": 8.639, "duration": 3.401, "text": "in this case um and it wants us to find"}, {"start": 10.519, "duration": 4.04, "text": "the local acceleration due to gravity"}, {"start": 12.04, "duration": 3.96, "text": "for the satellite all right so this"}, {"start": 14.559, "duration": 3.72, "text": "one's pretty straightforward this is"}, {"start": 16.0, "duration": 4.24, "text": "just a ratio problem here uh here I have"}, {"start": 18.279, "duration": 3.641, "text": "my yellow chart surface and altitude or"}, {"start": 20.24, "duration": 3.52, "text": "before and after if you like one and two"}, {"start": 21.92, "duration": 3.84, "text": "A and B whatever it is you want to use"}, {"start": 23.76, "duration": 3.32, "text": "but have a reference point okay so here"}, {"start": 25.76, "duration": 3.24, "text": "my reference point is going to be at the"}, {"start": 27.08, "duration": 3.199, "text": "surface because I know the generic"}, {"start": 29.0, "duration": 3.279, "text": "function for acceleration neutral"}, {"start": 30.279, "duration": 4.041, "text": "gravity at the surface of the planet is"}, {"start": 32.279, "duration": 5.201, "text": "G mass of the Earth and then the race of"}, {"start": 34.32, "duration": 4.88, "text": "the earth square and then once I know"}, {"start": 37.48, "duration": 4.64, "text": "that I can set up the right side of the"}, {"start": 39.2, "duration": 5.28, "text": "uh chart here pretty quickly so here I"}, {"start": 42.12, "duration": 3.919, "text": "call it G new or the acceleration due to"}, {"start": 44.48, "duration": 3.96, "text": "gravity or the local acceleration due to"}, {"start": 46.039, "duration": 6.401, "text": "gravity for the satellite which is just"}, {"start": 48.44, "duration": 6.56, "text": "GM Earth over the radius"}, {"start": 52.44, "duration": 4.16, "text": "square uh but radius is being used kind"}, {"start": 55.0, "duration": 3.0, "text": "of vake here okay so really at the end"}, {"start": 56.6, "duration": 4.2, "text": "of the day it's the distance from Center"}, {"start": 58.0, "duration": 4.8, "text": "to Center but the diagram shows you"}, {"start": 60.8, "duration": 4.399, "text": "already that this Arrow here watch my"}, {"start": 62.8, "duration": 4.72, "text": "cursor please so here's planet Earth and"}, {"start": 65.199, "duration": 4.161, "text": "blue the distance given to you in the"}, {"start": 67.52, "duration": 3.84, "text": "schematic actually starts at the center"}, {"start": 69.36, "duration": 4.48, "text": "of the planet and it goes all the way to"}, {"start": 71.36, "duration": 6.079, "text": "the orbit of the satellite so in this"}, {"start": 73.84, "duration": 5.84, "text": "particular case 7R already includes the"}, {"start": 77.439, "duration": 4.961, "text": "radius of the planet so therefore the"}, {"start": 79.68, "duration": 5.119, "text": "distance from Center to where the orbit"}, {"start": 82.4, "duration": 4.079, "text": "is or where the Salah is is given a 7R"}, {"start": 84.799, "duration": 3.721, "text": "so therefore in the denominator I would"}, {"start": 86.479, "duration": 3.721, "text": "just substitute in 7R but use"}, {"start": 88.52, "duration": 3.599, "text": "parenthesis if you don't use parth"}, {"start": 90.2, "duration": 3.68, "text": "parentheses you'll forget to square the"}, {"start": 92.119, "duration": 4.0, "text": "coefficient 7 so it's important to use"}, {"start": 93.88, "duration": 3.64, "text": "parenthesis here so because there is an"}, {"start": 96.119, "duration": 3.121, "text": "exponent two I want to shoot that"}, {"start": 97.52, "duration": 5.08, "text": "exponent to the seven I'm also going to"}, {"start": 99.24, "duration": 6.559, "text": "introduce that uh excuse me um"}, {"start": 102.6, "duration": 5.76, "text": "distribute that exponent to the r as"}, {"start": 105.799, "duration": 4.401, "text": "well so essentially the numerator stays"}, {"start": 108.36, "duration": 5.56, "text": "the same except the bottom 7 square is"}, {"start": 110.2, "duration": 4.8, "text": "49 and then R square is R square here uh"}, {"start": 113.92, "duration": 2.36, "text": "hopefully you've seen my other videos"}, {"start": 115.0, "duration": 2.6, "text": "you can learn how to factor out"}, {"start": 116.28, "duration": 2.92, "text": "coefficients just to make things look a"}, {"start": 117.6, "duration": 3.08, "text": "little nicer you don't have to if you"}, {"start": 119.2, "duration": 3.559, "text": "know what you're doing but I like to"}, {"start": 120.68, "duration": 3.439, "text": "make it look a little nicer I notice the"}, {"start": 122.759, "duration": 3.0, "text": "only really thing difference between the"}, {"start": 124.119, "duration": 3.0, "text": "left side and the right side here is the"}, {"start": 125.759, "duration": 3.56, "text": "49 in the"}, {"start": 127.119, "duration": 4.001, "text": "denominator well because the 49"}, {"start": 129.319, "duration": 5.841, "text": "denominator I can just Factor It Out by"}, {"start": 131.12, "duration": 6.92, "text": "putting under a one and then gm/ r\u00b2 I'll"}, {"start": 135.16, "duration": 5.719, "text": "keep that as the same term because on"}, {"start": 138.04, "duration": 4.559, "text": "the left side of the chart acceleration"}, {"start": 140.879, "duration": 4.0, "text": "due to gravity at the surface of the"}, {"start": 142.599, "duration": 4.961, "text": "planet is given by G mass of the earth"}, {"start": 144.879, "duration": 4.72, "text": "and the radius of the planet Square so I"}, {"start": 147.56, "duration": 4.72, "text": "notice here I have the same term here g"}, {"start": 149.599, "duration": 3.72, "text": "f Earth over r s so I want to highlight"}, {"start": 152.28, "duration": 3.36, "text": "the same"}, {"start": 153.319, "duration": 7.64, "text": "term therefore what I can do is I can"}, {"start": 155.64, "duration": 7.4, "text": "substitute out g m r\u00b2 for lowercase G at"}, {"start": 160.959, "duration": 5.0, "text": "the surface of the planet which in this"}, {"start": 163.04, "duration": 4.479, "text": "case is Earth which is 9.8 so therefore"}, {"start": 165.959, "duration": 5.92, "text": "I can substitute out this whole entire"}, {"start": 167.519, "duration": 7.601, "text": "highlighted term g m e over r\u00b2 for G but"}, {"start": 171.879, "duration": 6.561, "text": "then now it's the same thing 1 * G is"}, {"start": 175.12, "duration": 6.0, "text": "just G and then G over 49 G got"}, {"start": 178.44, "duration": 8.24, "text": "substituted out for 9.8 and then 9.8 ID"}, {"start": 181.12, "duration": 7.44, "text": "4.9 will give you an answer of 0.2 m/s"}, {"start": 186.68, "duration": 3.0, "text": "squar okay and that's it for this one"}, {"start": 188.56, "duration": 5.84, "text": "this one's pretty straightforward all"}, {"start": 189.68, "duration": 4.72, "text": "right good luck stay ready don't be back"}], "title": "Newton's Gravitational Law of Attraction_4", "description": "The radius of the satellite\u2019s orbit around the center of the Earth is          r = 7R, where R is the radius of the Earth. What is the local acceleration due to gravity for the satellite?", "lengthSeconds": 194, "width": 640, "height": 360}
{"video_id": "z2oAev9jLXU", "transcript": [{"start": 1.16, "duration": 4.76, "text": "so here we have an example of Young's"}, {"start": 3.24, "duration": 4.76, "text": "double slit experiment um the problem"}, {"start": 5.92, "duration": 4.16, "text": "states that the distance between the two"}, {"start": 8.0, "duration": 5.28, "text": "slits as indicated here in the diagram"}, {"start": 10.08, "duration": 4.84, "text": "is 03 millimeters I haded that in blue"}, {"start": 13.28, "duration": 3.36, "text": "for you and it does say that it's a"}, {"start": 14.92, "duration": 4.24, "text": "second order bright Fringe that's being"}, {"start": 16.64, "duration": 5.04, "text": "observed and then the angle between the"}, {"start": 19.16, "duration": 5.279, "text": "zero uh m equal to zero or the central"}, {"start": 21.68, "duration": 5.759, "text": "Fringe to the second or the right Fringe"}, {"start": 24.439, "duration": 4.92, "text": "is 215\u00b0 again as shown here in the"}, {"start": 27.439, "duration": 4.441, "text": "diagram and it wants us to find the"}, {"start": 29.359, "duration": 5.801, "text": "wavelength of the light that's coming in"}, {"start": 31.88, "duration": 5.28, "text": "so to do uh so before you proceed uh you"}, {"start": 35.16, "duration": 3.8, "text": "should definitely scope out units uh"}, {"start": 37.16, "duration": 3.64, "text": "again I notic here this is given in"}, {"start": 38.96, "duration": 3.84, "text": "millimeters uh I would like to tell you"}, {"start": 40.8, "duration": 3.4, "text": "to convert everything to meters first"}, {"start": 42.8, "duration": 2.72, "text": "this way you know naturally things will"}, {"start": 44.2, "duration": 3.28, "text": "cancel out and then if you need to"}, {"start": 45.52, "duration": 3.44, "text": "convert at the very end to whatever"}, {"start": 47.48, "duration": 3.52, "text": "units they're asking for then do so at"}, {"start": 48.96, "duration": 3.399, "text": "the very end in this case I can see that"}, {"start": 51.0, "duration": 3.559, "text": "it wants it in nanometer so I know I"}, {"start": 52.359, "duration": 4.72, "text": "have to convert a little bit at the very"}, {"start": 54.559, "duration": 4.64, "text": "end uh so at to start I'll go ahead and"}, {"start": 57.079, "duration": 4.561, "text": "convert the distance between the two"}, {"start": 59.199, "duration": 4.841, "text": "slits in Mill millimeters to meters the"}, {"start": 61.64, "duration": 4.28, "text": "conversion factor between meters to"}, {"start": 64.04, "duration": 4.6, "text": "millimeters shown here on the screen for"}, {"start": 65.92, "duration": 4.68, "text": "every 1 meter there's 1,000 millimeters"}, {"start": 68.64, "duration": 4.119, "text": "uh so Place stuff in the right order and"}, {"start": 70.6, "duration": 3.76, "text": "then millimeters is going to cancel out"}, {"start": 72.759, "duration": 2.68, "text": "uh in the numerator denominator there"}, {"start": 74.36, "duration": 4.399, "text": "and if you do the calculator work"}, {"start": 75.439, "duration": 5.161, "text": "that'll give you 3.01 105 meters so now"}, {"start": 78.759, "duration": 3.961, "text": "that that's done uh we can go ahead and"}, {"start": 80.6, "duration": 4.28, "text": "proceed we want to go ahead and use"}, {"start": 82.72, "duration": 4.0, "text": "Young's doubleit function for"}, {"start": 84.88, "duration": 3.279, "text": "constructive interference because the"}, {"start": 86.72, "duration": 4.12, "text": "problem states that it is a bright"}, {"start": 88.159, "duration": 5.081, "text": "Fringe where m is equal to two uh so"}, {"start": 90.84, "duration": 4.959, "text": "we'll go ahead and bust this out again D"}, {"start": 93.24, "duration": 5.68, "text": "is the distance between the the slits uh"}, {"start": 95.799, "duration": 4.161, "text": "Theta is the angle between uh the or"}, {"start": 98.92, "duration": 2.76, "text": "Central bright frence and the second"}, {"start": 99.96, "duration": 3.159, "text": "order bright frence here and M we know"}, {"start": 101.68, "duration": 4.079, "text": "what m is and I'll get to that in just a"}, {"start": 103.119, "duration": 6.0, "text": "second okay but it's important to keep"}, {"start": 105.759, "duration": 6.841, "text": "the in mind that M must be 0 1 2 3 or"}, {"start": 109.119, "duration": 7.32, "text": "four according to Young's dite function"}, {"start": 112.6, "duration": 8.08, "text": "it cannot be like a 0.5 or 1.5 so M must"}, {"start": 116.439, "duration": 6.72, "text": "be any whole number uh"}, {"start": 120.68, "duration": 4.68, "text": "so as you can see highlighted there it"}, {"start": 123.159, "duration": 4.32, "text": "say the second order bright Fringe so"}, {"start": 125.36, "duration": 3.319, "text": "therefore m is two so even if the"}, {"start": 127.479, "duration": 2.801, "text": "problem doesn't give you like for"}, {"start": 128.679, "duration": 4.241, "text": "example in the parentheses m is equal to"}, {"start": 130.28, "duration": 4.84, "text": "2 uh they do expect for you to interpret"}, {"start": 132.92, "duration": 4.679, "text": "this information and know what value of"}, {"start": 135.12, "duration": 5.04, "text": "M to use so in this case it's pretty"}, {"start": 137.599, "duration": 5.561, "text": "straightforward we know that D is given"}, {"start": 140.16, "duration": 4.88, "text": "to us as 3.01 105 so we can substitute"}, {"start": 143.16, "duration": 3.76, "text": "that for there we know that from the"}, {"start": 145.04, "duration": 4.44, "text": "statement m is two so we can just"}, {"start": 146.92, "duration": 5.08, "text": "substitute uh m in the denominator uh"}, {"start": 149.48, "duration": 5.8, "text": "for value of two from here everything we"}, {"start": 152.0, "duration": 6.92, "text": "do know is calculator work time so 3.0 *"}, {"start": 155.28, "duration": 5.48, "text": "5 m Time s of 25 / 2 if you punch all"}, {"start": 158.92, "duration": 6.08, "text": "that in your calculator carefully you"}, {"start": 160.76, "duration": 5.52, "text": "should get a result of 5.6 T 107 M okay"}, {"start": 165.0, "duration": 3.599, "text": "but again you notice here the answer"}, {"start": 166.28, "duration": 4.48, "text": "here is in meters the question asked for"}, {"start": 168.599, "duration": 3.56, "text": "uh the wavelength and nanometers so we"}, {"start": 170.76, "duration": 3.759, "text": "just have to convert meters to"}, {"start": 172.159, "duration": 5.921, "text": "nanometers and that's pretty easy to do"}, {"start": 174.519, "duration": 6.121, "text": "uh so start with the uh statement 5.6 M7"}, {"start": 178.08, "duration": 3.92, "text": "meters times it again it's important"}, {"start": 180.64, "duration": 3.04, "text": "that you place the conversion factor"}, {"start": 182.0, "duration": 3.0, "text": "correctly here you want to put meters on"}, {"start": 183.68, "duration": 3.0, "text": "the bottom because you started off"}, {"start": 185.0, "duration": 3.28, "text": "meters so meter here is going to cancel"}, {"start": 186.68, "duration": 3.639, "text": "with meters in the denominator and that"}, {"start": 188.28, "duration": 3.599, "text": "leaves you with the units of nanometer"}, {"start": 190.319, "duration": 4.801, "text": "and in case you don't know for every one"}, {"start": 191.879, "duration": 5.08, "text": "meter there's 10 to the 9th nanometers"}, {"start": 195.12, "duration": 4.399, "text": "so if you do that properly the final"}, {"start": 196.959, "duration": 4.081, "text": "outcome will come out to be about"}, {"start": 199.519, "duration": 3.601, "text": "560"}, {"start": 201.04, "duration": 4.04, "text": "nanometers okay other than that it's"}, {"start": 203.12, "duration": 5.32, "text": "pretty straight for a problem okay all"}, {"start": 205.08, "duration": 3.36, "text": "right don't be whacked stay"}, {"start": 208.68, "duration": 3.0, "text": "nerdy"}], "title": "Young's Double Slit_3", "description": "The distance between the two slits is 0.03 mm.  The second-order bright fringe (m = 2) is measured on a viewing screen at an angle of 2.15o from the central maximum.  Determine the wavelength \u03bb (in nm).", "lengthSeconds": 212, "width": 640, "height": 360}
{"video_id": "6OdVRS6d3dg", "transcript": [{"start": 0.04, "duration": 4.6, "text": "you have always heard in your childhood"}, {"start": 2.159, "duration": 5.68, "text": "that one raised to something is always"}, {"start": 4.64, "duration": 5.919, "text": "one but today I am going to show you"}, {"start": 7.839, "duration": 5.601, "text": "something crazy is it possible to have"}, {"start": 10.559, "duration": 5.24, "text": "one raised to some value like X that"}, {"start": 13.44, "duration": 5.56, "text": "ends up being two this is a tricky"}, {"start": 15.799, "duration": 6.001, "text": "question first of all take natural log"}, {"start": 19.0, "duration": 5.96, "text": "on both sides I will be using log to"}, {"start": 21.8, "duration": 5.92, "text": "denote natural log some people also use"}, {"start": 24.96, "duration": 5.0, "text": "Ln to denote the same you can use"}, {"start": 27.72, "duration": 4.6, "text": "whatever you want to so using this"}, {"start": 29.96, "duration": 6.48, "text": "property of logarithm the X will jump"}, {"start": 32.32, "duration": 7.48, "text": "right here and we get x * log of 1 = log"}, {"start": 36.44, "duration": 6.279, "text": "of 2 now divide both sides by natural"}, {"start": 39.8, "duration": 6.48, "text": "log of one but natural log of one is"}, {"start": 42.719, "duration": 6.761, "text": "zero and we cannot divide by zero so no"}, {"start": 46.28, "duration": 6.24, "text": "real solution exists for X but I told"}, {"start": 49.48, "duration": 5.28, "text": "you this is a tricky question so now how"}, {"start": 52.52, "duration": 4.519, "text": "can we find the complex roots for this"}, {"start": 54.76, "duration": 6.4, "text": "equation we will first start with the"}, {"start": 57.039, "duration": 8.281, "text": "oiler identity which is e raised to I"}, {"start": 61.16, "duration": 5.8, "text": "Theta is equal to cine Theta + I sin"}, {"start": 65.32, "duration": 3.839, "text": "Theta if you are dealing with the"}, {"start": 66.96, "duration": 4.32, "text": "complex world then everyone should know"}, {"start": 69.159, "duration": 7.521, "text": "this identity now if you put the value"}, {"start": 71.28, "duration": 10.68, "text": "of theta as 0o we get cosine 0 + I sin 0"}, {"start": 76.68, "duration": 9.24, "text": "cosine 0 is 1 and sin 0 is 0 so e raised"}, {"start": 81.96, "duration": 8.88, "text": "to i 0 becomes 1 but this also works for"}, {"start": 85.92, "duration": 7.879, "text": "Theta equal 2 pi 4 pi and so on so one"}, {"start": 90.84, "duration": 10.12, "text": "in complex world can be written as e"}, {"start": 93.799, "duration": 11.441, "text": "raised to I * 2 K Pi where k equal 0 1 2"}, {"start": 100.96, "duration": 8.28, "text": "and so on so this x becomes log of 2 /"}, {"start": 105.24, "duration": 7.6, "text": "log of e raised to I * 2 K"}, {"start": 109.24, "duration": 6.919, "text": "Pi we know that log of e raised to a"}, {"start": 112.84, "duration": 8.12, "text": "simply becomes a so this x becomes log"}, {"start": 116.159, "duration": 7.161, "text": "of 2 upon I * 2 K Pi so far so good now"}, {"start": 120.96, "duration": 5.64, "text": "I want this I to be on the top so"}, {"start": 123.32, "duration": 6.359, "text": "multiply and divide by I to get this now"}, {"start": 126.6, "duration": 8.6, "text": "this becomes -1 and therefore X becomes"}, {"start": 129.679, "duration": 7.64, "text": "minus I * log 2 upon 2 K piun but before"}, {"start": 135.2, "duration": 5.56, "text": "we wrap this up we need to make sure"}, {"start": 137.319, "duration": 5.521, "text": "that K is not equal to zero now"}, {"start": 140.76, "duration": 4.68, "text": "surprisingly if you look at Wolfram"}, {"start": 142.84, "duration": 4.479, "text": "Alpha and plug this equation in it you"}, {"start": 145.44, "duration": 4.6, "text": "find that it does not provide any"}, {"start": 147.319, "duration": 5.961, "text": "solution to it now take this X on the"}, {"start": 150.04, "duration": 6.919, "text": "other side and make it as 1 = 2 raed to"}, {"start": 153.28, "duration": 7.12, "text": "1 /x now if you put this equation in"}, {"start": 156.959, "duration": 5.841, "text": "Wolfram Alpha it works and we get the"}, {"start": 160.4, "duration": 8.72, "text": "same solution as what we have proved"}, {"start": 162.8, "duration": 6.32, "text": "before really cool right so good"}], "title": "How Can This Be True?", "description": "1^x = 2 | Math Trick | Harvard Entrance Exam | Math Olympiad | Complex Numbers\n\nIf you're reading this, drop a comment using the word \"Lawrence\". Have an amazing day, you all are awesome!", "lengthSeconds": 173, "width": 640, "height": 360}
{"video_id": "K7SWwsvE5YE", "transcript": [{"start": 0.08, "duration": 4.279, "text": "today let's tackle an interesting"}, {"start": 1.92, "duration": 3.72, "text": "algebra problem that challenges our"}, {"start": 4.359, "duration": 4.16, "text": "problemsolving"}, {"start": 5.64, "duration": 5.959, "text": "skills also this is considered as one of"}, {"start": 8.519, "duration": 5.961, "text": "the difficult math Olympiad questions so"}, {"start": 11.599, "duration": 6.081, "text": "stay tuned our goal is to find the value"}, {"start": 14.48, "duration": 8.2, "text": "of x that satisfies this equation"}, {"start": 17.68, "duration": 8.88, "text": "equation is x^2 - x Cub = 36 can you"}, {"start": 22.68, "duration": 6.28, "text": "solve it okay as the very first step If"}, {"start": 26.56, "duration": 6.76, "text": "You observe it closely you can see that"}, {"start": 28.96, "duration": 10.4, "text": "36 can be written as 27 + 9 and what is"}, {"start": 33.32, "duration": 10.399, "text": "27 it is 3 Cub right and also 9 is 3 S"}, {"start": 39.36, "duration": 8.16, "text": "right so this 36 can be returned as 3 2"}, {"start": 43.719, "duration": 6.281, "text": "+ 3 Cub but we need a minus sign here"}, {"start": 47.52, "duration": 5.76, "text": "and not plus so if you think a little"}, {"start": 50.0, "duration": 6.6, "text": "bit we can rewrite this 9 as Min - 3"}, {"start": 53.28, "duration": 6.72, "text": "Square as well and this 27 can be"}, {"start": 56.6, "duration": 8.12, "text": "written as minus of minus 3 Cub because"}, {"start": 60.0, "duration": 8.2, "text": "- 3 Cub is - 27 and adding one more"}, {"start": 64.72, "duration": 5.36, "text": "minus will make it a plus excellent so"}, {"start": 68.2, "duration": 4.279, "text": "it is clear that one of the answers to"}, {"start": 70.08, "duration": 5.359, "text": "this question will be minus three just"}, {"start": 72.479, "duration": 5.401, "text": "by hit and trial now in order to find"}, {"start": 75.439, "duration": 5.441, "text": "the other two Roots we will factorize"}, {"start": 77.88, "duration": 6.239, "text": "this equation for that first take this"}, {"start": 80.88, "duration": 5.96, "text": "36 on the left hand side and rearrange"}, {"start": 84.119, "duration": 9.161, "text": "this equation to make it minus of X Cub"}, {"start": 86.84, "duration": 13.48, "text": "+ x\u00b2 - 36 = 0 or multiply both sides by"}, {"start": 93.28, "duration": 10.28, "text": "-1 to get XB - x\u00b2 + 36 = 0 now in order"}, {"start": 100.32, "duration": 12.6, "text": "to factorize it using x + 3 rewrite this"}, {"start": 103.56, "duration": 13.8, "text": "as X Cub + 3 x\u00b2 - 4 x\u00b2 + 36 = 0 3 - 4"}, {"start": 112.92, "duration": 8.799, "text": "will be - x\u00b2 only now take x\u00b2 common"}, {"start": 117.36, "duration": 9.64, "text": "from this to get x2 * X + 3 and from"}, {"start": 121.719, "duration": 10.961, "text": "here take -4 as common to get -4 * x^2 -"}, {"start": 127.0, "duration": 10.92, "text": "9 this x^2 - 9 can be written as x - 3 *"}, {"start": 132.68, "duration": 8.639, "text": "x + 3 so far so good now take x + 3"}, {"start": 137.92, "duration": 11.959, "text": "common from both of them to get x + 3 *"}, {"start": 141.319, "duration": 14.161, "text": "x^2 - 4 * x - 3 = 0 or x + 3 * x^2 - 4 x"}, {"start": 149.879, "duration": 7.72, "text": "x + 12 = 0 now this will give x = -3 and"}, {"start": 155.48, "duration": 5.56, "text": "for this part we will be using the"}, {"start": 157.599, "duration": 6.28, "text": "quadratic equation formula we have a"}, {"start": 161.04, "duration": 6.96, "text": "which is equal to the coefficient of x\u00b2"}, {"start": 163.879, "duration": 8.601, "text": "= 1 then B which is the coefficient of x"}, {"start": 168.0, "duration": 8.12, "text": "= -4 and C which is the constant term as"}, {"start": 172.48, "duration": 12.2, "text": "12 plug them into the formula to get x ="}, {"start": 176.12, "duration": 13.679, "text": "minus of -4 +- under < t - 4 2 - 4 * 1 *"}, {"start": 184.68, "duration": 12.68, "text": "12 upon 2 which will give -4 +- under <"}, {"start": 189.799, "duration": 10.681, "text": "TK 16 - 48 or under < TK - 32 upon 2 now"}, {"start": 197.36, "duration": 6.68, "text": "this will introduce the imaginary term I"}, {"start": 200.48, "duration": 9.56, "text": "as under root minus 32 can be written as"}, {"start": 204.04, "duration": 12.0, "text": "under < tk-1 * under < TK 16 * 2 or 4 <"}, {"start": 210.04, "duration": 11.119, "text": "tk2 so X will become -4 + -4 < tk2 I"}, {"start": 216.04, "duration": 8.88, "text": "upon 2 or -2 + - 2 < tk2"}, {"start": 221.159, "duration": 8.911, "text": "I this is the solution to our problem"}, {"start": 224.92, "duration": 13.079, "text": "that was amazing isn't it so good"}, {"start": 230.07, "duration": 7.929, "text": "[Music]"}], "title": "Can You Solve this Harvard Entrance Exam Question?", "description": "Nice Cubic Equation! | Math Olympiad | Harvard University Entrance Exam Interview | This question frightened 300K+ examinees! | Hard Geometry Exam Question | Only 1% of Students Got this Math Question Correct \n\nIf you're reading this, drop a comment using the word \"Cubic\". Have an amazing day, you all are awesome!", "lengthSeconds": 237, "width": 640, "height": 360}
{"video_id": "DjUG5EzUhkE", "transcript": [{"start": 0.12, "duration": 4.239, "text": "this is such an amazing problem we are"}, {"start": 2.399, "duration": 4.641, "text": "given this right triangle whose length"}, {"start": 4.359, "duration": 4.761, "text": "of this part of the hypotenuse is five"}, {"start": 7.04, "duration": 5.04, "text": "and we have a circle inscribed in it"}, {"start": 9.12, "duration": 5.16, "text": "whose radius is 2 units we have been"}, {"start": 12.08, "duration": 5.4, "text": "asked to find the area of this shaded"}, {"start": 14.28, "duration": 5.88, "text": "region that is this blue region can you"}, {"start": 17.48, "duration": 5.039, "text": "solve it what do you think in how many"}, {"start": 20.16, "duration": 4.16, "text": "ways can this question be solved I'd"}, {"start": 22.519, "duration": 5.52, "text": "love to hear your thoughts in the"}, {"start": 24.32, "duration": 5.68, "text": "comments okay so first thing first since"}, {"start": 28.039, "duration": 4.321, "text": "this side of the triangle is tangent to"}, {"start": 30.0, "duration": 4.92, "text": "to this circle so this radius will be"}, {"start": 32.36, "duration": 3.4, "text": "perpendicular to this side so this will"}, {"start": 34.92, "duration": 4.44, "text": "be"}, {"start": 35.76, "duration": 5.319, "text": "90\u00b0 now draw this radius it will also be"}, {"start": 39.36, "duration": 4.84, "text": "perpendicular to this side of the"}, {"start": 41.079, "duration": 6.241, "text": "triangle as it is also tangent to this"}, {"start": 44.2, "duration": 4.32, "text": "circle so it will be of length two and"}, {"start": 47.32, "duration": 4.759, "text": "this will be"}, {"start": 48.52, "duration": 6.679, "text": "90\u00b0 look at this quadrilateral all four"}, {"start": 52.079, "duration": 5.881, "text": "sides are 90\u00b0 and both of them are of"}, {"start": 55.199, "duration": 5.721, "text": "length two so it will be a square of"}, {"start": 57.96, "duration": 7.12, "text": "side length two thus both of them will"}, {"start": 60.92, "duration": 6.76, "text": "also be two great now here comes the"}, {"start": 65.08, "duration": 5.079, "text": "magic we will use the two tangent"}, {"start": 67.68, "duration": 4.92, "text": "theorem which states that the lengths of"}, {"start": 70.159, "duration": 6.441, "text": "two tangents drawn from the same"}, {"start": 72.6, "duration": 6.559, "text": "external point to a circle are equal now"}, {"start": 76.6, "duration": 4.839, "text": "this is a tangent and this is also a"}, {"start": 79.159, "duration": 4.801, "text": "tangent to this circle and they are"}, {"start": 81.439, "duration": 6.0, "text": "drawn from the same external point this"}, {"start": 83.96, "duration": 6.199, "text": "one so both of them will be equal"}, {"start": 87.439, "duration": 5.36, "text": "therefore this will also become five"}, {"start": 90.159, "duration": 5.401, "text": "same goes for these tangents both of"}, {"start": 92.799, "duration": 6.241, "text": "them will be equal let us label them as"}, {"start": 95.56, "duration": 7.599, "text": "X excellent now consider this right"}, {"start": 99.04, "duration": 7.64, "text": "triangle this side is of length x + 2"}, {"start": 103.159, "duration": 7.481, "text": "this is of length 2 + 5 or 7 and the"}, {"start": 106.68, "duration": 8.16, "text": "hypotenuse is x + 5 so we will use the"}, {"start": 110.64, "duration": 12.28, "text": "Pythagoras Theorem to find X we get x +"}, {"start": 114.84, "duration": 15.399, "text": "5\u00b2 = x + 2 2 + 7 s this gives x 2 + 5 2"}, {"start": 122.92, "duration": 14.0, "text": "or 25 + 2 * x * 5 or 10 x = x 2 + 2 2 or"}, {"start": 130.239, "duration": 8.761, "text": "4 + 2 * x * 2 or 4 x + 49 this x\u00b2 will"}, {"start": 136.92, "duration": 11.44, "text": "get cancelled out and we are left with"}, {"start": 139.0, "duration": 13.28, "text": "this 25 + 10 x = 4 + 4 x + 49 or 4x + 53"}, {"start": 148.36, "duration": 10.239, "text": "take 4X on left side and 25 on right"}, {"start": 152.28, "duration": 10.44, "text": "side to get 6 x = 28 or x = 28 / 6 which"}, {"start": 158.599, "duration": 7.801, "text": "can be simplified to 14 over 3 so far so"}, {"start": 162.72, "duration": 7.64, "text": "good thus this side length will be x + 2"}, {"start": 166.4, "duration": 7.839, "text": "or 14 by 3 plus this 2 can be written as"}, {"start": 170.36, "duration": 6.879, "text": "6 by 3 so it becomes 20"}, {"start": 174.239, "duration": 4.841, "text": "by3 now the area of this shaded region"}, {"start": 177.239, "duration": 5.28, "text": "will be same as the area of this"}, {"start": 179.08, "duration": 6.4, "text": "triangle minus the area of this circle"}, {"start": 182.519, "duration": 8.561, "text": "the area of this triangle will be 20x 3"}, {"start": 185.48, "duration": 8.88, "text": "* 7 / 2 or 140 / 6 which when simplified"}, {"start": 191.08, "duration": 8.159, "text": "becomes 70 by 3 and the area of this"}, {"start": 194.36, "duration": 8.2, "text": "circle will be Pi r^ 2ar or < * 2^ 2 or"}, {"start": 199.239, "duration": 8.08, "text": "4 piun so the area of this shaded region"}, {"start": 202.56, "duration": 7.039, "text": "will be 70 by 3 - 4 Pi cool were you"}, {"start": 207.319, "duration": 4.84, "text": "able to solve it what approach did you"}, {"start": 209.599, "duration": 8.121, "text": "use use to solve this question let me"}, {"start": 212.159, "duration": 5.561, "text": "know in the comments so good"}], "title": "Should We Make Him an Engineer or a Doctor? Find Blue Area", "description": "Find Blue Area | Math Olympiad | Harvard University Entrance Exam Interview | This question frightened 300K+ examinees! | Hard Geometry Exam Question | Only 1% of Students Got this Math Question Correct \n\nIf you're reading this, drop a comment using the word \"Engineer\". Have an amazing day, you all are awesome!", "lengthSeconds": 222, "width": 640, "height": 360}
{"video_id": "y7FFgEnM9f0", "transcript": [{"start": 0.28, "duration": 6.44, "text": "special thanks to Katrina EG for posting"}, {"start": 2.8, "duration": 6.2, "text": "this amazing geometry question on X the"}, {"start": 6.72, "duration": 5.52, "text": "end result is literally going to shock"}, {"start": 9.0, "duration": 5.599, "text": "you we have this big semicircle and"}, {"start": 12.24, "duration": 5.44, "text": "inside it we have two small semicircles"}, {"start": 14.599, "duration": 5.881, "text": "of diameter two and three and this"}, {"start": 17.68, "duration": 5.28, "text": "yellow one is tangent to the red"}, {"start": 20.48, "duration": 4.36, "text": "semicircle our job is to find the"}, {"start": 22.96, "duration": 5.96, "text": "diameter of the big"}, {"start": 24.84, "duration": 6.72, "text": "semicircle so can you solve it at first"}, {"start": 28.92, "duration": 4.84, "text": "glance if we pull this down here it"}, {"start": 31.56, "duration": 5.12, "text": "looks like the bigger diameter is simply"}, {"start": 33.76, "duration": 5.4, "text": "the sum of both of these diameters so"}, {"start": 36.68, "duration": 5.559, "text": "the answer we are looking for is five or"}, {"start": 39.16, "duration": 5.399, "text": "it will be approximately five but now"}, {"start": 42.239, "duration": 4.681, "text": "how to find out the exact value of this"}, {"start": 44.559, "duration": 5.961, "text": "diameter for that we will break this"}, {"start": 46.92, "duration": 6.6, "text": "diameter into smaller sections this this"}, {"start": 50.52, "duration": 4.96, "text": "and this one this piece right here is"}, {"start": 53.52, "duration": 4.76, "text": "the same as the radius of this"}, {"start": 55.48, "duration": 6.16, "text": "semicircle and thus it will be of length"}, {"start": 58.28, "duration": 5.959, "text": "one now to find the length of this part"}, {"start": 61.64, "duration": 5.88, "text": "first draw this radius and then join"}, {"start": 64.239, "duration": 7.88, "text": "both of these radi together this will be"}, {"start": 67.52, "duration": 7.919, "text": "1.5 and this will be 1 + 1.5 or"}, {"start": 72.119, "duration": 5.761, "text": "2.5 now we will use Pythagoras Theorem"}, {"start": 75.439, "duration": 4.801, "text": "to find the length of this part we can"}, {"start": 77.88, "duration": 4.12, "text": "do the math to find it and I have done"}, {"start": 80.24, "duration": 4.199, "text": "it for you in case you might want to"}, {"start": 82.0, "duration": 5.2, "text": "check but we have an easier method to"}, {"start": 84.439, "duration": 5.521, "text": "figure it out if we multiply the length"}, {"start": 87.2, "duration": 5.8, "text": "of both of these sides by two we get the"}, {"start": 89.96, "duration": 6.24, "text": "this as three and this is five and thus"}, {"start": 93.0, "duration": 5.84, "text": "we know that it is a three four five"}, {"start": 96.2, "duration": 5.64, "text": "Pythagorean triplets and therefore this"}, {"start": 98.84, "duration": 6.639, "text": "will be four or if we divide it by two"}, {"start": 101.84, "duration": 5.72, "text": "we get this side length as two excellent"}, {"start": 105.479, "duration": 5.28, "text": "next we will find the length of this"}, {"start": 107.56, "duration": 7.64, "text": "part and we are done let us label this"}, {"start": 110.759, "duration": 7.0, "text": "as y now here comes the magic this is"}, {"start": 115.2, "duration": 6.76, "text": "what I found the most intelligent way to"}, {"start": 117.759, "duration": 7.161, "text": "solve for this part complete this circle"}, {"start": 121.96, "duration": 4.839, "text": "now make a rectangle around it so the"}, {"start": 124.92, "duration": 3.959, "text": "length of this side of the rectangle"}, {"start": 126.799, "duration": 5.241, "text": "will be the same as the diameter of this"}, {"start": 128.879, "duration": 5.121, "text": "circle which will be three now extend"}, {"start": 132.04, "duration": 5.64, "text": "this line to the vertex of this"}, {"start": 134.0, "duration": 6.319, "text": "rectangle awesome now consider these two"}, {"start": 137.68, "duration": 5.36, "text": "triangles this angle is the same for"}, {"start": 140.319, "duration": 5.441, "text": "both these triangles and also since both"}, {"start": 143.04, "duration": 5.08, "text": "of these lines are parallel this angle"}, {"start": 145.76, "duration": 5.4, "text": "is also the same for both of them and"}, {"start": 148.12, "duration": 6.36, "text": "hence both these triangles are s similar"}, {"start": 151.16, "duration": 6.6, "text": "therefore this side which is 2 over this"}, {"start": 154.48, "duration": 8.28, "text": "side which is 1.5 equals this side which"}, {"start": 157.76, "duration": 8.92, "text": "is y + 2 over this side which is 3 cross"}, {"start": 162.76, "duration": 12.44, "text": "multiply to get this this becomes 6 over"}, {"start": 166.68, "duration": 11.68, "text": "1.5 or 4 = y + 2 or Y = 2 woohoo now is"}, {"start": 175.2, "duration": 8.64, "text": "the time for our final answer drums"}, {"start": 178.36, "duration": 7.48, "text": "rolling in and and it is 1 + 2 + 2 or 5"}, {"start": 183.84, "duration": 4.399, "text": "which is exactly the same as we"}, {"start": 185.84, "duration": 4.92, "text": "predicted at the beginning of this video"}, {"start": 188.239, "duration": 5.521, "text": "when we looked at this problem"}, {"start": 190.76, "duration": 5.24, "text": "visually isn't this cool if you liked"}, {"start": 193.76, "duration": 4.6, "text": "this video then please don't forget to"}, {"start": 196.0, "duration": 4.84, "text": "share it with your friends and support"}, {"start": 198.36, "duration": 6.77, "text": "this channel by joining our community"}, {"start": 200.84, "duration": 12.229, "text": "and becoming a member so good"}, {"start": 205.13, "duration": 7.939, "text": "[Music]"}], "title": "Crazy Math Challenge!", "description": "Math Olympiad | Math Trick | Harvard University Entrance Exam Interview | This question frightened 300K+ examinees! | Hard Geometry Exam Question | Only 1% of Students Got this Math Question Correct \n\nIf you're reading this, drop a comment using the word \"Fun\". Have an amazing day, you all are awesome!\n\n#unitedstates", "lengthSeconds": 212, "width": 640, "height": 360}
{"video_id": "04PL5bzgBOM", "transcript": [{"start": 0.16, "duration": 4.08, "text": "this is a pretty exciting problem you"}, {"start": 2.36, "duration": 4.08, "text": "know about the three four five triangles"}, {"start": 4.24, "duration": 5.359, "text": "you see everywhere in high school"}, {"start": 6.44, "duration": 5.199, "text": "geometry in sa a and even in"}, {"start": 9.599, "duration": 4.681, "text": "construction they show up all over the"}, {"start": 11.639, "duration": 5.321, "text": "place but today what we are going to"}, {"start": 14.28, "duration": 4.8, "text": "show will literally blow up your mind"}, {"start": 16.96, "duration": 4.76, "text": "because if you inscribe a circle in any"}, {"start": 19.08, "duration": 4.92, "text": "three four five triangles then you won't"}, {"start": 21.72, "duration": 6.479, "text": "believe but the area of this inscribed"}, {"start": 24.0, "duration": 6.16, "text": "circle is exactly Pi let us see the math"}, {"start": 28.199, "duration": 5.121, "text": "first let us draw the center of the"}, {"start": 30.16, "duration": 5.68, "text": "circle now draw a radius like this and"}, {"start": 33.32, "duration": 4.8, "text": "this the radius will always be at a"}, {"start": 35.84, "duration": 3.96, "text": "right angle to the tangent point and"}, {"start": 38.12, "duration": 4.959, "text": "therefore this whole thing will become a"}, {"start": 39.8, "duration": 5.96, "text": "square of side length R so this side"}, {"start": 43.079, "duration": 6.841, "text": "will also be R and this side will also"}, {"start": 45.76, "duration": 7.599, "text": "be R now this is three and this is R so"}, {"start": 49.92, "duration": 7.92, "text": "this will be 3 minus r similarly this is"}, {"start": 53.359, "duration": 6.641, "text": "four and this is R so this will be 4 - R"}, {"start": 57.84, "duration": 4.08, "text": "now let us clear out everything and just"}, {"start": 60.0, "duration": 4.159, "text": "keep this circle and these two lines"}, {"start": 61.92, "duration": 4.8, "text": "which are tangent to this circle here"}, {"start": 64.159, "duration": 4.521, "text": "comes the magic the lengths of two"}, {"start": 66.72, "duration": 5.12, "text": "tangent lines drawn from the same"}, {"start": 68.68, "duration": 5.88, "text": "external point to a circle are equal"}, {"start": 71.84, "duration": 5.44, "text": "specifically it is called the two"}, {"start": 74.56, "duration": 6.76, "text": "tangent theorem so this tangent line"}, {"start": 77.28, "duration": 7.68, "text": "will also be 3 minus r and the same"}, {"start": 81.32, "duration": 8.32, "text": "thing goes for this side if this is 4 -"}, {"start": 84.96, "duration": 7.08, "text": "R so this will also be 4 - R now let us"}, {"start": 89.64, "duration": 5.4, "text": "Focus focus on the hypotenuse the length"}, {"start": 92.04, "duration": 8.28, "text": "of the hypotenuse is 5 now this is also"}, {"start": 95.04, "duration": 11.2, "text": "equal to 3 - r + 4 - R so we have 7 - 2"}, {"start": 100.32, "duration": 8.079, "text": "R = 5 this gives 2 R = 2 and R = 1 now"}, {"start": 106.24, "duration": 4.96, "text": "that we have the radius of the circle as"}, {"start": 108.399, "duration": 6.08, "text": "one so what will be the area of this"}, {"start": 111.2, "duration": 6.36, "text": "circle it will be piun * R square and"}, {"start": 114.479, "duration": 9.361, "text": "putting r as 1 we get the area of this"}, {"start": 117.56, "duration": 6.28, "text": "circle as Pi isn't this cool so good"}], "title": "3-4-5 Triangle and \u03c0 (Pi)", "description": "New Method Discovered to Calculate \u03c0 (Pi) in 2024.\n\nIf you're reading this, drop a comment using the word \"3-4-5\". Have an amazing day, you all are awesome!", "lengthSeconds": 127, "width": 640, "height": 360}
{"video_id": "JQNtR8lqcHA", "transcript": [{"start": 0.08, "duration": 6.48, "text": "today I am going to show you how to"}, {"start": 2.399, "duration": 7.361, "text": "approximate the Square t of 101 minus"}, {"start": 6.56, "duration": 5.44, "text": "the square < TK of 99 whenever you see"}, {"start": 9.76, "duration": 4.2, "text": "this type of question the first thing"}, {"start": 12.0, "duration": 4.84, "text": "you should do is rationalize the"}, {"start": 13.96, "duration": 5.92, "text": "numerator and in a minute I will let you"}, {"start": 16.84, "duration": 5.84, "text": "know why to do this multiply both"}, {"start": 19.88, "duration": 4.36, "text": "numerator and denominator by the square"}, {"start": 22.68, "duration": 6.04, "text": "root of"}, {"start": 24.24, "duration": 7.48, "text": "101+ the square < TK of 99 this way we"}, {"start": 28.72, "duration": 9.839, "text": "will have Plus in here instead of minus"}, {"start": 31.72, "duration": 10.96, "text": "using a + b * a - b as a s - b 2 the"}, {"start": 38.559, "duration": 7.401, "text": "numerator will be 101 - 99 or 2 and the"}, {"start": 42.68, "duration": 5.359, "text": "denominator will remain as it is so now"}, {"start": 45.96, "duration": 4.239, "text": "our job is to approximate this"}, {"start": 48.039, "duration": 4.281, "text": "denominator for that we will find the"}, {"start": 50.199, "duration": 5.88, "text": "integer numbers between which this sum"}, {"start": 52.32, "duration": 6.28, "text": "might exist let us call this sum as X so"}, {"start": 56.079, "duration": 4.96, "text": "we will try to find the integer value"}, {"start": 58.6, "duration": 5.68, "text": "which is just shorter than this X and"}, {"start": 61.039, "duration": 6.12, "text": "just larger than this x to do this we"}, {"start": 64.28, "duration": 7.92, "text": "will find the value of x\u00b2 which will be"}, {"start": 67.159, "duration": 10.6, "text": "square T of 101 + theare < TK of 99"}, {"start": 72.2, "duration": 10.599, "text": "squar it will be 101 + 99 + 2 * the sare"}, {"start": 77.759, "duration": 7.241, "text": "< TK of 101 * theare < TK of 99 so using"}, {"start": 82.799, "duration": 5.041, "text": "this we have reduced these two ugly"}, {"start": 85.0, "duration": 7.56, "text": "square roots into a single square root"}, {"start": 87.84, "duration": 7.88, "text": "now 10 and 1 is 100 + 1 1 and 99 is 100"}, {"start": 92.56, "duration": 8.04, "text": "-1 so using this formula again we get"}, {"start": 95.72, "duration": 9.759, "text": "this value as 100 S -1 so x\u00b2 will be 200"}, {"start": 100.6, "duration": 7.879, "text": "+ 2 * < TK of 100 S - 1 now how can we"}, {"start": 105.479, "duration": 7.28, "text": "get rid of this minus1 notice that 100 S"}, {"start": 108.479, "duration": 9.28, "text": "- 1 will be less than 100 square so x\u00b2"}, {"start": 112.759, "duration": 7.841, "text": "will be less than 200 + 2 * < TK 100 S"}, {"start": 117.759, "duration": 7.201, "text": "or it will be less than 200 plus plus 2"}, {"start": 120.6, "duration": 8.24, "text": "* 100 or 400 but 100 S - 1 will also be"}, {"start": 124.96, "duration": 8.999, "text": "much bigger than 99 Square so x\u00b2 will be"}, {"start": 128.84, "duration": 9.479, "text": "greater than 200 + 2 * < TK of 99 s or"}, {"start": 133.959, "duration": 7.881, "text": "it will be greater than 200 + 2 * 99 or"}, {"start": 138.319, "duration": 7.28, "text": "398 so this part gives X less than 20"}, {"start": 141.84, "duration": 7.039, "text": "because 20 SAR is 400 and now if x is"}, {"start": 145.599, "duration": 8.481, "text": "greater than 398 then it is also greater"}, {"start": 148.879, "duration": 8.36, "text": "than 36 61 which is 19 squ so X will be"}, {"start": 154.08, "duration": 5.56, "text": "greater than 19 now our goal was to find"}, {"start": 157.239, "duration": 3.161, "text": "an approximate value of this one which"}, {"start": 159.64, "duration": 4.519, "text": "is"}, {"start": 160.4, "duration": 7.559, "text": "2/x since X is greater than 19 and less"}, {"start": 164.159, "duration": 8.16, "text": "than 20 so 2/x will be greater than 2"}, {"start": 167.959, "duration": 5.721, "text": "over 20 and less than 2 over 19 2 over"}, {"start": 172.319, "duration": 5.961, "text": "20 is"}, {"start": 173.68, "duration": 7.24, "text": "0.1 and 2 over 19 is"}, {"start": 178.28, "duration": 6.64, "text": "0.105 therefore our answer lies"}, {"start": 180.92, "duration": 7.08, "text": "somewhere between 0.1 and"}, {"start": 184.92, "duration": 6.92, "text": "0.105 so if we take their average we get"}, {"start": 188.0, "duration": 3.84, "text": "our answer as approximately"}, {"start": 191.879, "duration": 5.201, "text": "01025 now if we verify it using"}, {"start": 194.76, "duration": 8.8, "text": "calculator we find that we are really"}, {"start": 197.08, "duration": 6.48, "text": "close to our approximation so good"}], "title": "You Should Learn This Trick!", "description": "You Should Learn This Trick! This question frightened 300K+ examinees! | Hard Algebra Exam Question | Only 1% of Students Got this Math Question Correct | Math Olympiad\n\nIf you're reading this, drop a comment using the word \"Trick\". Have an amazing day, you all are awesome!", "lengthSeconds": 208, "width": 640, "height": 360}
{"video_id": "s3esyXmafh0", "transcript": [{"start": 0.199, "duration": 4.08, "text": "this is such an amazing problem and I"}, {"start": 2.48, "duration": 4.0, "text": "did not expect it to have the outcome"}, {"start": 4.279, "duration": 4.601, "text": "that it had the problem is that we have"}, {"start": 6.48, "duration": 5.039, "text": "a semicircle right here and a quarter"}, {"start": 8.88, "duration": 5.44, "text": "Circle right here and then this red line"}, {"start": 11.519, "duration": 5.68, "text": "with the length of 5 cm which goes from"}, {"start": 14.32, "duration": 5.36, "text": "this vertex and is tangent to this"}, {"start": 17.199, "duration": 5.441, "text": "semicircle we want to find the area of"}, {"start": 19.68, "duration": 5.96, "text": "this rectangle so how can we solve it as"}, {"start": 22.64, "duration": 5.879, "text": "a first step let us assign a radius R to"}, {"start": 25.64, "duration": 5.32, "text": "this semicircle now many of the people"}, {"start": 28.519, "duration": 5.121, "text": "think that this radius is the same as"}, {"start": 30.96, "duration": 5.2, "text": "this quarter circle's radius but it is"}, {"start": 33.64, "duration": 5.2, "text": "nowhere given that they are the same and"}, {"start": 36.16, "duration": 5.36, "text": "in the later steps we will also discover"}, {"start": 38.84, "duration": 4.719, "text": "that we don't require them to be equal"}, {"start": 41.52, "duration": 5.84, "text": "therefore this is one of my favorite"}, {"start": 43.559, "duration": 8.16, "text": "problems so now let us label this radius"}, {"start": 47.36, "duration": 7.08, "text": "as X this piece will also be equal to r"}, {"start": 51.719, "duration": 5.961, "text": "as it is the radius of this semicircle"}, {"start": 54.44, "duration": 5.04, "text": "and this height will be equal to X which"}, {"start": 57.68, "duration": 4.399, "text": "will be the same as the radius of this"}, {"start": 59.48, "duration": 6.24, "text": "quarter Circle we can find the area of"}, {"start": 62.079, "duration": 8.04, "text": "this rectangle in terms of R and X the"}, {"start": 65.72, "duration": 7.759, "text": "base is x + 2 R and the height is X so"}, {"start": 70.119, "duration": 8.64, "text": "the area will be base x height or X plus"}, {"start": 73.479, "duration": 8.401, "text": "2 R * X which will be x\u00b2 +"}, {"start": 78.759, "duration": 5.68, "text": "2xr now here comes the magic let us draw"}, {"start": 81.88, "duration": 5.0, "text": "a segment from this point of tangency to"}, {"start": 84.439, "duration": 4.481, "text": "the center of the semicircle and this"}, {"start": 86.88, "duration": 5.72, "text": "will be the same as the radius of the"}, {"start": 88.92, "duration": 5.839, "text": "semicircle which is R every radius that"}, {"start": 92.6, "duration": 5.12, "text": "intersects the tangent line at the point"}, {"start": 94.759, "duration": 5.521, "text": "of tangency is at right angle so now we"}, {"start": 97.72, "duration": 4.92, "text": "have a nice right angle triangle and"}, {"start": 100.28, "duration": 5.36, "text": "therefore yes you have guessed it right"}, {"start": 102.64, "duration": 6.68, "text": "we will use Pythagoras Theorem we get r"}, {"start": 105.64, "duration": 8.119, "text": "s + 5\u00b2 = x +"}, {"start": 109.32, "duration": 8.28, "text": "r\u00b2 the left hand side will be r\u00b2 + 25"}, {"start": 113.759, "duration": 7.801, "text": "and the right hand side will be x\u00b2 + 2xr"}, {"start": 117.6, "duration": 8.519, "text": "+ r\u00b2 these r s will get cancelled out"}, {"start": 121.56, "duration": 6.479, "text": "and we are left with x\u00b2 + 2 XR = 25 but"}, {"start": 126.119, "duration": 4.48, "text": "we know that this quantity is the same"}, {"start": 128.039, "duration": 5.681, "text": "as the area of this rectangle so the"}, {"start": 130.599, "duration": 9.601, "text": "area of this rectangle will be 25 don't"}, {"start": 133.72, "duration": 6.48, "text": "forget to put the cm square so good"}], "title": "Can This Question Be Solved?", "description": "Can This Question Be Solved? \n\nIf you're reading this, drop a comment using the word \"Students USA\". Have an amazing day, you all are awesome!", "lengthSeconds": 144, "width": 640, "height": 360}
{"video_id": "FYNS5ngGJlk", "transcript": [{"start": 0.08, "duration": 4.92, "text": "we have a quarter Circle and some shapes"}, {"start": 2.12, "duration": 5.6, "text": "inside it our goal is to find the area"}, {"start": 5.0, "duration": 5.36, "text": "of the blue region the trick here is to"}, {"start": 7.72, "duration": 5.48, "text": "solve this problem without using"}, {"start": 10.36, "duration": 5.439, "text": "integration pause this video and try to"}, {"start": 13.2, "duration": 4.8, "text": "solve it on your own we start by drawing"}, {"start": 15.799, "duration": 4.441, "text": "a radius from the center of this quarter"}, {"start": 18.0, "duration": 3.84, "text": "Circle which goes through this point of"}, {"start": 20.24, "duration": 4.359, "text": "intersection which will divide"}, {"start": 21.84, "duration": 5.08, "text": "everything in half let's say that both"}, {"start": 24.599, "duration": 5.561, "text": "of these blue shapes are represented by"}, {"start": 26.92, "duration": 6.2, "text": "X the white shapes by Y and this one by"}, {"start": 30.16, "duration": 8.6, "text": "Z so now we know that the total blue"}, {"start": 33.12, "duration": 9.04, "text": "area is represented by 2 * x + 2 * Z we"}, {"start": 38.76, "duration": 5.92, "text": "need to figure out what x and z are"}, {"start": 42.16, "duration": 5.36, "text": "we're dealing with a quarter of a full"}, {"start": 44.68, "duration": 8.679, "text": "circle the area of a quarter circle is"}, {"start": 47.52, "duration": 8.879, "text": "given by the formula < * Radius 2 / 4"}, {"start": 53.359, "duration": 7.241, "text": "the radius here is four so we calculate"}, {"start": 56.399, "duration": 7.881, "text": "it as 4 * 4 so the area of the quarter"}, {"start": 60.6, "duration": 8.6, "text": "circle is 4 * pi this total area of the"}, {"start": 64.28, "duration": 9.28, "text": "quarter circle is equal to 2 * x + 2 * y"}, {"start": 69.2, "duration": 6.8, "text": "+ 2 * Z we also have a semicircle inside"}, {"start": 73.56, "duration": 5.08, "text": "this quarter Circle the area of a"}, {"start": 76.0, "duration": 6.84, "text": "semicircle is given by the pi * radius"}, {"start": 78.64, "duration": 7.56, "text": "squar / 2 the diameter is 4 so the"}, {"start": 82.84, "duration": 7.4, "text": "radius is 2 when we calculate it we get"}, {"start": 86.2, "duration": 9.199, "text": "2 * pi as the area of the red semicircle"}, {"start": 90.24, "duration": 7.68, "text": "this 2 pi is equal to 2 * x + y next"}, {"start": 95.399, "duration": 5.04, "text": "let's focus on a triangle inside this"}, {"start": 97.92, "duration": 5.479, "text": "semicircle which is obtained by dividing"}, {"start": 100.439, "duration": 5.04, "text": "this semicircle into half the height of"}, {"start": 103.399, "duration": 4.561, "text": "this semicircle will be the same as the"}, {"start": 105.479, "duration": 5.68, "text": "radius of this semicircle which is"}, {"start": 107.96, "duration": 6.04, "text": "two now the area of the triangle is"}, {"start": 111.159, "duration": 6.441, "text": "calculated as base time height divided"}, {"start": 114.0, "duration": 8.039, "text": "by two here both the base and height are"}, {"start": 117.6, "duration": 7.439, "text": "two so the triangle's area is is 2 now"}, {"start": 122.039, "duration": 6.321, "text": "it is clear that the x + triangle's area"}, {"start": 125.039, "duration": 5.481, "text": "equals the area of the quarter Circle so"}, {"start": 128.36, "duration": 5.76, "text": "we subtract this triangle from both"}, {"start": 130.52, "duration": 7.92, "text": "sides to get the value of x the area of"}, {"start": 134.12, "duration": 7.88, "text": "the quarter circle is pi * 2^ 2 / 4"}, {"start": 138.44, "duration": 7.879, "text": "which is pi and the triangle area is 2"}, {"start": 142.0, "duration": 7.04, "text": "so X is equal to < - 2 now we substitute"}, {"start": 146.319, "duration": 6.801, "text": "X in this equation to get the value of y"}, {"start": 149.04, "duration": 9.199, "text": "as four the blue area is equal to 2 * x"}, {"start": 153.12, "duration": 7.88, "text": "+ 2 * Z substituting y = 4 we get the"}, {"start": 158.239, "duration": 7.72, "text": "area of the blue region as 4 Pi - 8"}, {"start": 161.0, "duration": 4.959, "text": "square units so good"}], "title": "Even Adults Cannot Solve This!", "description": "Fun Math Problems. Find the Area of Blue Shaded Region.\n\nIf you're reading this, drop a comment using the word \"blue\". Have an amazing day, you all are awesome!", "lengthSeconds": 169, "width": 640, "height": 360}
{"video_id": "RWFPRzuFO3M", "transcript": [{"start": 0.16, "duration": 4.719, "text": "this is an amazing geometry question"}, {"start": 2.48, "duration": 5.56, "text": "that includes a variety of Concepts like"}, {"start": 4.879, "duration": 6.8, "text": "symmetry Circles chords and"}, {"start": 8.04, "duration": 6.16, "text": "perpendicular lines we have a circle"}, {"start": 11.679, "duration": 4.321, "text": "first we draw this chord of length five"}, {"start": 14.2, "duration": 4.52, "text": "and after that we draw this line of"}, {"start": 16.0, "duration": 5.519, "text": "length six from this point which is"}, {"start": 18.72, "duration": 4.879, "text": "perpendicular to this chord followed by"}, {"start": 21.519, "duration": 5.361, "text": "another line which is perpendicular to"}, {"start": 23.599, "duration": 5.121, "text": "this one and it is of length four now"}, {"start": 26.88, "duration": 4.159, "text": "how can we use these details to"}, {"start": 28.72, "duration": 4.76, "text": "determine the radius of this Circle can"}, {"start": 31.039, "duration": 5.52, "text": "you solve it let us start by marking the"}, {"start": 33.48, "duration": 5.2, "text": "center of this circle now in a circle"}, {"start": 36.559, "duration": 5.081, "text": "the line segment from the center that is"}, {"start": 38.68, "duration": 4.96, "text": "perpendicular to a chord will also bict"}, {"start": 41.64, "duration": 3.88, "text": "the chord this means that the"}, {"start": 43.64, "duration": 4.559, "text": "perpendicular line from the center of a"}, {"start": 45.52, "duration": 5.679, "text": "circle to its chord divides it into two"}, {"start": 48.199, "duration": 6.121, "text": "equal segments this is called chord bis"}, {"start": 51.199, "duration": 5.081, "text": "sector theorem now draw this diameter"}, {"start": 54.32, "duration": 4.759, "text": "and then draw this rectangle whose side"}, {"start": 56.28, "duration": 5.16, "text": "lengths are five and six like this now"}, {"start": 59.079, "duration": 4.64, "text": "draw this line which is perpendicular to"}, {"start": 61.44, "duration": 4.56, "text": "this side of the rectangle what do you"}, {"start": 63.719, "duration": 4.601, "text": "think its length will be since this"}, {"start": 66.0, "duration": 6.0, "text": "diameter will be perpendicular by sector"}, {"start": 68.32, "duration": 6.92, "text": "of this chord and if these are 5x two so"}, {"start": 72.0, "duration": 5.159, "text": "both of them will also be 5x two as well"}, {"start": 75.24, "duration": 5.199, "text": "and since both of these lengths are the"}, {"start": 77.159, "duration": 6.881, "text": "same therefore this segment will also be"}, {"start": 80.439, "duration": 7.481, "text": "of length four awesome now draw this"}, {"start": 84.04, "duration": 6.36, "text": "line and label it as X now we will use"}, {"start": 87.92, "duration": 5.36, "text": "another theorem called intercept in"}, {"start": 90.4, "duration": 5.16, "text": "chords theorem here is one chord of the"}, {"start": 93.28, "duration": 6.0, "text": "circle and here is another chord of the"}, {"start": 95.56, "duration": 6.199, "text": "circle so by intersecting chords theorem"}, {"start": 99.28, "duration": 4.879, "text": "this length which is 6 times this length"}, {"start": 101.759, "duration": 4.72, "text": "which is X will be the same as this"}, {"start": 104.159, "duration": 7.92, "text": "length which is four times this length"}, {"start": 106.479, "duration": 8.361, "text": "which is 5 + 4 or 9 so X will be 6"}, {"start": 112.079, "duration": 5.281, "text": "amazing now draw this line such that we"}, {"start": 114.84, "duration": 5.239, "text": "get a right triangle here what do you"}, {"start": 117.36, "duration": 5.48, "text": "think this line will be yes you you"}, {"start": 120.079, "duration": 5.32, "text": "guessed it right it will be the diameter"}, {"start": 122.84, "duration": 5.12, "text": "of this circle this is because of"}, {"start": 125.399, "duration": 4.521, "text": "thales's theorem which states that if"}, {"start": 127.96, "duration": 4.72, "text": "you draw a triangle where one of its"}, {"start": 129.92, "duration": 4.959, "text": "sides is the diameter of a circle then"}, {"start": 132.68, "duration": 5.919, "text": "the angle opposite that side is a right"}, {"start": 134.879, "duration": 8.521, "text": "angle so this hypotenuse will be 2 * the"}, {"start": 138.599, "duration": 7.441, "text": "radius R and this is 12 and this 5 so we"}, {"start": 143.4, "duration": 3.839, "text": "will now use Pythagoras Theorem such"}, {"start": 146.04, "duration": 8.44, "text": "that 2"}, {"start": 147.239, "duration": 13.92, "text": "r\u00b2 = 5 S + 12 square or 25 + 144 which"}, {"start": 154.48, "duration": 11.16, "text": "is equal to 169 or 13 Square so 2 R = 13"}, {"start": 161.159, "duration": 6.961, "text": "and R = 13 by 2 and that's it wasn't"}, {"start": 165.64, "duration": 4.599, "text": "this an interesting problem were you"}, {"start": 168.12, "duration": 5.0, "text": "aware of so many theorems before this"}, {"start": 170.239, "duration": 8.481, "text": "video and able to solve it let me know"}, {"start": 173.12, "duration": 5.6, "text": "in the comments so good"}, {"start": 180.56, "duration": 3.979, "text": "[Music]"}], "title": "Can This Be Solved?", "description": "Find Blue Area | Math Olympiad | Harvard University Entrance Exam Interview | This question frightened 300K+ examinees! | Hard Geometry Exam Question | Only 1% of Students Got this Math Question Correct \n\nIf you're reading this, drop a comment using the word \"USA\". Have an amazing day, you all are awesome!", "lengthSeconds": 183, "width": 640, "height": 360}
{"video_id": "fON-nM0E7ao", "transcript": [{"start": 0.0, "duration": 4.38, "text": "hey guys I saw this thing from Iota"}, {"start": 2.04, "duration": 4.08, "text": "Academy it says I want to believe and it"}, {"start": 4.38, "duration": 3.84, "text": "shows this really interesting equation"}, {"start": 6.12, "duration": 3.78, "text": "let's verify that this thing is true so"}, {"start": 8.22, "duration": 3.299, "text": "first step let's subtract one from both"}, {"start": 9.9, "duration": 3.119, "text": "sides on the left hand side we just have"}, {"start": 11.519, "duration": 3.601, "text": "this thing minus one and on the right"}, {"start": 13.019, "duration": 3.36, "text": "hand side we just have this fraction I"}, {"start": 15.12, "duration": 2.94, "text": "don't like fractions so I'm going to"}, {"start": 16.379, "duration": 3.24, "text": "multiply both sides by this denominator"}, {"start": 18.06, "duration": 3.66, "text": "these two things are going to cancel out"}, {"start": 19.619, "duration": 3.901, "text": "which leaves me with just pi to the E"}, {"start": 21.72, "duration": 3.36, "text": "minus one and on the left hand side we"}, {"start": 23.52, "duration": 4.019, "text": "could just bring this stuff down now"}, {"start": 25.08, "duration": 4.92, "text": "let's foil root pi to the E times root"}, {"start": 27.539, "duration": 4.14, "text": "pi to the E is just pi to the E root pi"}, {"start": 30.0, "duration": 3.78, "text": "to the E times negative 1 is negative"}, {"start": 31.679, "duration": 3.961, "text": "root pi to the e one times root pi to"}, {"start": 33.78, "duration": 3.6, "text": "the U is root pi to the E and 1 times"}, {"start": 35.64, "duration": 2.939, "text": "negative 1 is equal to negative 1. now"}, {"start": 37.38, "duration": 2.88, "text": "we can bring down the right hand side"}, {"start": 38.579, "duration": 2.881, "text": "these two middle terms are the same so"}, {"start": 40.26, "duration": 3.18, "text": "they're going to cancel each other out"}, {"start": 41.46, "duration": 4.32, "text": "or just left with pi to the E minus 1"}, {"start": 43.44, "duration": 6.56, "text": "equals pi to the E minus 1. so this up"}, {"start": 45.78, "duration": 4.22, "text": "here is a true statement how exciting"}], "title": "Fun Math Equation #shorts", "description": "#shorts", "lengthSeconds": 48, "width": 360, "height": 640}
{"video_id": "BOUw4IXgYM0", "transcript": [{"start": 0.04, "duration": 3.32, "text": "hey guys somebody mentioned me in this"}, {"start": 1.319, "duration": 4.281, "text": "one it looks like a fun one we're given"}, {"start": 3.36, "duration": 4.079, "text": "three squares and some measurements and"}, {"start": 5.6, "duration": 3.48, "text": "it wants us to calculate the area of the"}, {"start": 7.439, "duration": 3.521, "text": "entire shape if you want to try it on"}, {"start": 9.08, "duration": 5.639, "text": "your own pause it right now because I'm"}, {"start": 10.96, "duration": 5.52, "text": "going to solve it in 3 2 1 it says"}, {"start": 14.719, "duration": 3.201, "text": "calculate the area of the shape I"}, {"start": 16.48, "duration": 3.16, "text": "believe it means it wants us to add up"}, {"start": 17.92, "duration": 3.16, "text": "all three of the squares so we just got"}, {"start": 19.64, "duration": 3.28, "text": "to figure out the area of each of the"}, {"start": 21.08, "duration": 3.279, "text": "squares and then add them all together"}, {"start": 22.92, "duration": 3.08, "text": "let's start with the largest Square in"}, {"start": 24.359, "duration": 3.76, "text": "the middle and let's give the sides a"}, {"start": 26.0, "duration": 4.0, "text": "length of X next let's focus on this"}, {"start": 28.119, "duration": 3.681, "text": "gray square and let's find one of its"}, {"start": 30.0, "duration": 4.16, "text": "side lengths since we know that this"}, {"start": 31.8, "duration": 5.079, "text": "entire height is equal to X and this"}, {"start": 34.16, "duration": 5.079, "text": "portion is 2 that'll leave x - 2 for the"}, {"start": 36.879, "duration": 5.321, "text": "yellow part and since this is a square"}, {"start": 39.239, "duration": 5.16, "text": "we know the bottom is also x - 2 and now"}, {"start": 42.2, "duration": 4.28, "text": "we're starting to fill out this 17 so we"}, {"start": 44.399, "duration": 4.241, "text": "have the x - 2 right here let's color"}, {"start": 46.48, "duration": 3.759, "text": "this blue for this portion now we just"}, {"start": 48.64, "duration": 3.96, "text": "got to find this last bit and then we"}, {"start": 50.239, "duration": 4.401, "text": "can set it all equal to 17 now for the"}, {"start": 52.6, "duration": 5.0, "text": "last part since the whole thing is X and"}, {"start": 54.64, "duration": 4.68, "text": "this piece is five that leaves x - 5 for"}, {"start": 57.6, "duration": 3.599, "text": "this piece and then once again it's a"}, {"start": 59.32, "duration": 4.799, "text": "square so the base is going to match the"}, {"start": 61.199, "duration": 4.721, "text": "height so this orange portion is x - 5"}, {"start": 64.119, "duration": 4.841, "text": "now we know that the yellow plus the"}, {"start": 65.92, "duration": 8.239, "text": "blue plus the orange will equal 17 so we"}, {"start": 68.96, "duration": 8.04, "text": "have x - 2 + x + x - 5 = 17 we can copy"}, {"start": 74.159, "duration": 7.481, "text": "it down and then combine like terms x +"}, {"start": 77.0, "duration": 6.72, "text": "x + x is 3x and -2 - 5 is -7 next we can"}, {"start": 81.64, "duration": 4.159, "text": "add 7 to both sides of the equation on"}, {"start": 83.72, "duration": 5.12, "text": "the left hand side we're left with 3x"}, {"start": 85.799, "duration": 4.721, "text": "and on the right hand side 17 + 7 is 24"}, {"start": 88.84, "duration": 4.239, "text": "and then we can divide both sides by 3"}, {"start": 90.52, "duration": 5.32, "text": "and that'll give us x = 8 now we know"}, {"start": 93.079, "duration": 5.0, "text": "this yellow X is equal to 8 this blue X"}, {"start": 95.84, "duration": 6.279, "text": "is equal to 8 and this orange x is equal"}, {"start": 98.079, "duration": 6.841, "text": "8 and we can simplify 8 - 2 is 6 and 8 -"}, {"start": 102.119, "duration": 5.96, "text": "5 is 3 so the area of this gray square"}, {"start": 104.92, "duration": 6.36, "text": "is 6 * 6 which is 36 the area of the"}, {"start": 108.079, "duration": 5.441, "text": "green square is 8 * 8 which is 64 and"}, {"start": 111.28, "duration": 6.519, "text": "the area of the brown square is 3 * 3"}, {"start": 113.52, "duration": 6.519, "text": "which is 9 and then 36 + 64 + 9 is 109"}, {"start": 117.799, "duration": 4.041, "text": "let's give it a label of cm squ and put"}, {"start": 120.039, "duration": 3.68, "text": "a box around it and that's the answer to"}, {"start": 121.84, "duration": 5.72, "text": "our question the area of this shape is"}, {"start": 123.719, "duration": 5.16, "text": "109 cm squared how exciting here's a"}, {"start": 127.56, "duration": 3.679, "text": "problem I'm planning on doing for the"}, {"start": 128.879, "duration": 7.961, "text": "next video it says solve for x and it's"}, {"start": 131.239, "duration": 7.041, "text": "XX - 4 + 1 = 4x - 4 and when I do post"}, {"start": 136.84, "duration": 5.119, "text": "it I'll try to put a link to the video"}, {"start": 138.28, "duration": 3.679, "text": "somewhere around here"}], "title": "Area of 3 Squares", "description": "I thought this was a fun one, I hope you guys like it!", "lengthSeconds": 140, "width": 640, "height": 360}
{"video_id": "dgwRyLwLBE4", "transcript": [{"start": 0.08, "duration": 3.2, "text": "guys we're gonna evaluate a piecewise"}, {"start": 1.92, "duration": 3.68, "text": "function so for"}, {"start": 3.28, "duration": 4.32, "text": "x equals seven which of these two is"}, {"start": 5.6, "duration": 3.28, "text": "true what's the top one because seven is"}, {"start": 7.6, "duration": 3.039, "text": "greater than four we are not going to"}, {"start": 8.88, "duration": 4.48, "text": "use this one this is telling us"}, {"start": 10.639, "duration": 3.601, "text": "that we're gonna use this function so f"}, {"start": 13.36, "duration": 3.2, "text": "of seven"}, {"start": 14.24, "duration": 3.76, "text": "is equal to and we're going to plug a 7"}, {"start": 16.56, "duration": 3.92, "text": "into the x"}, {"start": 18.0, "duration": 3.199, "text": "7 plus 2 and that equals 9. so the"}, {"start": 20.48, "duration": 2.639, "text": "answer to this"}, {"start": 21.199, "duration": 3.601, "text": "is 9. had this 7 been something"}, {"start": 23.119, "duration": 2.4, "text": "different like a 2 we would have ended"}, {"start": 24.8, "duration": 3.6, "text": "up using this"}, {"start": 25.519, "duration": 4.481, "text": "lower one this x squared because 2 is in"}, {"start": 28.4, "duration": 2.959, "text": "this domain i posted more of these"}, {"start": 30.0, "duration": 2.8, "text": "online if you want to try some more"}, {"start": 31.359, "duration": 2.241, "text": "there's a whole bunch of samples you can"}, {"start": 32.8, "duration": 2.4, "text": "watch a lot of them"}, {"start": 33.6, "duration": 3.6, "text": "there's a lot of samples you can try and"}, {"start": 35.2, "duration": 4.08, "text": "they all give the answers"}, {"start": 37.2, "duration": 4.08, "text": "it's 21 different ones you can try find"}, {"start": 39.28, "duration": 3.52, "text": "that page by going to andymath.com"}, {"start": 41.28, "duration": 3.36, "text": "on the homepage just type something like"}, {"start": 42.8, "duration": 3.2, "text": "evaluating piecewise functions and this"}, {"start": 44.64, "duration": 2.88, "text": "page will come up and that's the page i"}, {"start": 46.0, "duration": 3.12, "text": "just showed you please check it out i'd"}, {"start": 47.52, "duration": 5.44, "text": "love people to use it you're awesome"}, {"start": 49.12, "duration": 3.84, "text": "and i'll talk to you soon bye"}], "title": "Evaluating Piecewise Functions Example Problem #Shorts", "description": "Check out my website! \nhttps://andymath.com/\n\n#Shorts", "lengthSeconds": 51, "width": 202, "height": 360}
{"video_id": "L0xXb6EQuzs", "transcript": [{"start": 0.0, "duration": 3.639, "text": "hey guys Pythagorean theorem is true for"}, {"start": 1.719, "duration": 3.521, "text": "all of these cases there's already a ton"}, {"start": 3.639, "duration": 3.401, "text": "of proofs for this version I'm not going"}, {"start": 5.24, "duration": 3.52, "text": "to worry about that in this video but I"}, {"start": 7.04, "duration": 3.719, "text": "do want to show how this version leads"}, {"start": 8.76, "duration": 4.36, "text": "to the other versions so let's do this"}, {"start": 10.759, "duration": 3.721, "text": "one how do we get from here to here"}, {"start": 13.12, "duration": 3.52, "text": "since this is an equation we can"}, {"start": 14.48, "duration": 4.36, "text": "multiply both sides by the same thing in"}, {"start": 16.64, "duration": 3.879, "text": "this case 1/2 and this 1/2 will"}, {"start": 18.84, "duration": 3.599, "text": "distribute to both of these terms and"}, {"start": 20.519, "duration": 5.801, "text": "this 1/2 will distribute here and we end"}, {"start": 22.439, "duration": 5.441, "text": "up with a^2 over 2 + b^2 2 = c^2 / 2 and"}, {"start": 26.32, "duration": 3.48, "text": "basically that's cutting all of these in"}, {"start": 27.88, "duration": 4.12, "text": "half so it ends up looking like this and"}, {"start": 29.8, "duration": 4.2, "text": "that's how to do the first one and you"}, {"start": 32.0, "duration": 3.76, "text": "can do something similar for equilateral"}, {"start": 34.0, "duration": 5.36, "text": "triangles the area of an equilateral"}, {"start": 35.76, "duration": 6.479, "text": "triangle is the < TK 3 over 4 * the base"}, {"start": 39.36, "duration": 7.519, "text": "squared so this formula would be < tk34"}, {"start": 42.239, "duration": 6.561, "text": "a 2 +34 b^2 = > 34 c^2 and you can"}, {"start": 46.879, "duration": 3.601, "text": "imagine how we get from here to there we"}, {"start": 48.8, "duration": 4.0, "text": "multiply both sides of the equation by <"}, {"start": 50.48, "duration": 4.399, "text": "tk3 over 4 this will distribute to both"}, {"start": 52.8, "duration": 4.759, "text": "of these and this distributes here and"}, {"start": 54.879, "duration": 4.241, "text": "we now have the equation we wanted it's"}, {"start": 57.559, "duration": 3.921, "text": "the same thing for regular pentagons you"}, {"start": 59.12, "duration": 4.64, "text": "just multiply both sides by a constant"}, {"start": 61.48, "duration": 4.16, "text": "the next interesting one is semicircles"}, {"start": 63.76, "duration": 5.32, "text": "the area of a semicircle is equal to PK"}, {"start": 65.64, "duration": 6.76, "text": "R 2 / 2 or you can rewrite that as < / 2"}, {"start": 69.08, "duration": 8.0, "text": "r^ 2 so this would end up being < / 2 r^"}, {"start": 72.4, "duration": 6.52, "text": "2 + < / 2 r^2 = < 2 R 2 so R sub1 is the"}, {"start": 77.08, "duration": 3.92, "text": "radius of this one that's going to be"}, {"start": 78.92, "duration": 4.36, "text": "half of the distance a so we can change"}, {"start": 81.0, "duration": 4.28, "text": "this to a over2 and then R sub 2 would"}, {"start": 83.28, "duration": 3.64, "text": "be the radius of this semicircle so"}, {"start": 85.28, "duration": 3.72, "text": "that's going to be half of B so this"}, {"start": 86.92, "duration": 3.72, "text": "will be B over2 and then the radius of"}, {"start": 89.0, "duration": 3.36, "text": "this one is going to be half of this"}, {"start": 90.64, "duration": 3.68, "text": "side length C so we'll change that to"}, {"start": 92.36, "duration": 3.64, "text": "c/2 so this is the formula for"}, {"start": 94.32, "duration": 4.92, "text": "semicircles and we can prove that this"}, {"start": 96.0, "duration": 4.6, "text": "is true we start with A2 plus b^2 = c^2"}, {"start": 99.24, "duration": 3.32, "text": "and then we'll divide both sides of the"}, {"start": 100.6, "duration": 4.24, "text": "equation by four and then on the left"}, {"start": 102.56, "duration": 4.76, "text": "side this four can be split to the two"}, {"start": 104.84, "duration": 5.72, "text": "terms we can rewrite each of these fours"}, {"start": 107.32, "duration": 6.32, "text": "as 2^ 2ar now looking at this a^ 2 over"}, {"start": 110.56, "duration": 4.8, "text": "2^ 2 we can rewrite this as a over 2 the"}, {"start": 113.64, "duration": 4.439, "text": "whole thing squared and the same thing"}, {"start": 115.36, "duration": 4.84, "text": "as b^2 over 2^2 that can be Rewritten as"}, {"start": 118.079, "duration": 4.36, "text": "B over 2^ 2 and we can do the same thing"}, {"start": 120.2, "duration": 3.72, "text": "to this it'd be C over 2^ 2 and then for"}, {"start": 122.439, "duration": 3.96, "text": "the next step we can multiply both sides"}, {"start": 123.92, "duration": 4.24, "text": "of the equation by Pi / 2 this Pi over2"}, {"start": 126.399, "duration": 3.681, "text": "will distribute to both of these terms"}, {"start": 128.16, "duration": 3.24, "text": "and on this side it will distribute here"}, {"start": 130.08, "duration": 3.32, "text": "and now we have the equation of the"}, {"start": 131.4, "duration": 3.32, "text": "semicircle Pythagorean theorem so we"}, {"start": 133.4, "duration": 3.68, "text": "went from the Pythagorean theorem that"}, {"start": 134.72, "duration": 4.239, "text": "we know to this new fun version and this"}, {"start": 137.08, "duration": 3.2, "text": "will work for any similar figures you"}, {"start": 138.959, "duration": 4.0, "text": "end up multiplying both sides of the"}, {"start": 140.28, "duration": 5.44, "text": "Pythagorean theorem by some constant how"}, {"start": 142.959, "duration": 2.761, "text": "exciting"}], "title": "Proving Different Pythagorean Theorems", "description": "Demonstrating why all these versions work.", "lengthSeconds": 144, "width": 640, "height": 360}
{"video_id": "nmUVIzUmlpg", "transcript": [{"start": 0.0, "duration": 4.259, "text": "guys this one says factor completely and"}, {"start": 2.52, "duration": 3.54, "text": "this completely is very important it"}, {"start": 4.259, "duration": 3.6, "text": "might be more than just one step so"}, {"start": 6.06, "duration": 3.18, "text": "first let's factor out the GCF the"}, {"start": 7.859, "duration": 3.001, "text": "greatest common factor for the two of"}, {"start": 9.24, "duration": 3.54, "text": "them or we can Factor two out of both of"}, {"start": 10.86, "duration": 3.359, "text": "them we can factor out an X out of both"}, {"start": 12.78, "duration": 3.059, "text": "of them and we can factor out a y"}, {"start": 14.219, "duration": 3.241, "text": "squared out of both of them so now let's"}, {"start": 15.839, "duration": 4.081, "text": "go after this first one two times what"}, {"start": 17.46, "duration": 4.92, "text": "gives us two that's one x times what"}, {"start": 19.92, "duration": 4.38, "text": "gives us x to the fourth that's X cubed"}, {"start": 22.38, "duration": 4.02, "text": "and then y squared times what gives us y"}, {"start": 24.3, "duration": 4.92, "text": "to the fifth that's y cubed so this"}, {"start": 26.4, "duration": 5.16, "text": "times this gives us this and now 2xy"}, {"start": 29.22, "duration": 3.539, "text": "squared times what gives us 2xy squared"}, {"start": 31.56, "duration": 2.94, "text": "well that's just one because this"}, {"start": 32.759, "duration": 3.721, "text": "matches this exactly a lot of people"}, {"start": 34.5, "duration": 4.38, "text": "might think we're done right now but we"}, {"start": 36.48, "duration": 4.919, "text": "can go further this right here is a sum"}, {"start": 38.88, "duration": 4.56, "text": "of cubes a cubed plus B Cubed is equal"}, {"start": 41.399, "duration": 4.141, "text": "to a plus b times a squared minus a B"}, {"start": 43.44, "duration": 4.02, "text": "plus b squared this first term is a"}, {"start": 45.54, "duration": 3.48, "text": "cubed that would mean that a is equal to"}, {"start": 47.46, "duration": 4.02, "text": "the cube root of this which would just"}, {"start": 49.02, "duration": 5.039, "text": "be x y and then for the B we want to"}, {"start": 51.48, "duration": 4.739, "text": "take the cube root of this one because"}, {"start": 54.059, "duration": 4.14, "text": "one is equal to B Cubed and that gives"}, {"start": 56.219, "duration": 3.36, "text": "us one so now we can just plug it right"}, {"start": 58.199, "duration": 4.02, "text": "into this equation so let's copy down"}, {"start": 59.579, "duration": 4.381, "text": "the GC F the 2xy squared and then this"}, {"start": 62.219, "duration": 3.601, "text": "is going to change into two parentheses"}, {"start": 63.96, "duration": 4.019, "text": "we put the a in the first spot which is"}, {"start": 65.82, "duration": 4.26, "text": "x y and then we do plus b in the next"}, {"start": 67.979, "duration": 3.781, "text": "spot and B was one and then for the a"}, {"start": 70.08, "duration": 3.66, "text": "squared we want to square this which"}, {"start": 71.76, "duration": 4.38, "text": "would be x squared y squared we squared"}, {"start": 73.74, "duration": 3.72, "text": "each of the terms and then minus a b so"}, {"start": 76.14, "duration": 3.659, "text": "we're going to minus these two things"}, {"start": 77.46, "duration": 4.38, "text": "multiplied which is just minus x y and"}, {"start": 79.799, "duration": 3.96, "text": "then plus b squared but B squared is"}, {"start": 81.84, "duration": 3.24, "text": "equal to one because one squared is one"}, {"start": 83.759, "duration": 5.481, "text": "and this is the answer to the question"}, {"start": 85.08, "duration": 4.16, "text": "we factored completely how exciting"}], "title": "Factoring Challenge", "description": "https://andymath.com/sum-difference-of-2-cubes/", "lengthSeconds": 87, "width": 640, "height": 360}
{"video_id": "7MzelnwgQow", "transcript": [{"start": 0.0, "duration": 5.16, "text": "so we want to solve for x in this Isles"}, {"start": 2.399, "duration": 5.761, "text": "triangle in an isoceles triangle these"}, {"start": 5.16, "duration": 6.2, "text": "base angles are equal and so are these"}, {"start": 8.16, "duration": 6.08, "text": "two sides since these two sides are"}, {"start": 11.36, "duration": 5.919, "text": "equal we can just say 3x + 5 is equal to"}, {"start": 14.24, "duration": 4.64, "text": "5x + 1 and now we just solve for x"}, {"start": 17.279, "duration": 3.121, "text": "there's an x on both sides of this"}, {"start": 18.88, "duration": 3.96, "text": "equation so I'm going to subtract the"}, {"start": 20.4, "duration": 4.92, "text": "smaller one over to the other side 3 X's"}, {"start": 22.84, "duration": 5.12, "text": "cancel so we just have a 5 on the left"}, {"start": 25.32, "duration": 4.48, "text": "and 5x - 3x is 2x then we want to"}, {"start": 27.96, "duration": 4.72, "text": "subtract one from both sides and then we"}, {"start": 29.8, "duration": 6.12, "text": "div div both sides by 2 4id 2 is 2 and"}, {"start": 32.68, "duration": 4.879, "text": "2x / 2 is X so x = 2 is the answer to"}, {"start": 35.92, "duration": 3.04, "text": "the question comment below for what"}, {"start": 37.559, "duration": 6.68, "text": "you'd like me to solve next and I'll"}, {"start": 38.96, "duration": 5.279, "text": "talk to you guys soon bye Andy maath"}], "title": "Isosceles Triangles #andymath  #math #maths #sat #tutorials", "description": "Please Subscribe, and don't forget to hit the bell icon so you don't miss next video! Also, please feel free to share this video with your friends and family. Let learn Math together with me, Charlie. \nHere is the playlist of the videos. [ There is so much more to discover in this playlist ]", "lengthSeconds": 43, "width": 360, "height": 640}
{"video_id": "dLtUMWAf7jk", "transcript": [{"start": 0.199, "duration": 6.56, "text": "hey guys welcome to this limit problem"}, {"start": 3.399, "duration": 5.2, "text": "so as you know as X approaches 3 we"}, {"start": 6.759, "duration": 4.121, "text": "ultimately can just plug in a 3 for the"}, {"start": 8.599, "duration": 9.081, "text": "X and what happens when we do that is we"}, {"start": 10.88, "duration": 10.559, "text": "get 3 - 3 over 3^ 2 - 9 and this ends up"}, {"start": 17.68, "duration": 5.359, "text": "giving us 0 over 0 so anytime you have a"}, {"start": 21.439, "duration": 4.84, "text": "0 over 0 you should look to see if"}, {"start": 23.039, "duration": 4.601, "text": "something can cancel after you factor so"}, {"start": 26.279, "duration": 4.561, "text": "this is actually difference of two"}, {"start": 27.64, "duration": 5.4, "text": "squared so I'm going to rewrite this as"}, {"start": 30.84, "duration": 7.36, "text": "the limit is X approaches"}, {"start": 33.04, "duration": 6.839, "text": "3 of x - 3 over and I'm going to factor"}, {"start": 38.2, "duration": 4.32, "text": "this x -"}, {"start": 39.879, "duration": 7.801, "text": "3 x +"}, {"start": 42.52, "duration": 7.84, "text": "3 and now these two things can cancel"}, {"start": 47.68, "duration": 5.24, "text": "and we're left with just a one on top"}, {"start": 50.36, "duration": 7.96, "text": "now when I plug in the three for the X I"}, {"start": 52.92, "duration": 9.88, "text": "end up with 1 over 3 +"}, {"start": 58.32, "duration": 7.92, "text": "3 which is equal to to 1"}, {"start": 62.8, "duration": 5.04, "text": "6 and that is the answer to the question"}, {"start": 66.24, "duration": 4.36, "text": "so I hope that makes sense and I'll talk"}, {"start": 67.84, "duration": 5.76, "text": "to you guys soon"}, {"start": 70.6, "duration": 3.0, "text": "bye"}], "title": "Find the Limit (if it exists)", "description": "Limits Problem #1 AROUND A MINUTE! For more math made easy visit andymath.com.\n\nSubscribe here: https://www.youtube.com/channel/UC6KhU3AMLHC-qv7tj0ZJduw?sub_confirmation=1", "lengthSeconds": 72, "width": 640, "height": 360}
{"video_id": "y3N9XaL8fZU", "transcript": [{"start": 0.0, "duration": 4.98, "text": "guys so this is a pretty legit calculus"}, {"start": 2.58, "duration": 4.38, "text": "challenge it uses very basic derivative"}, {"start": 4.98, "duration": 3.36, "text": "skills but it can be very tricky if"}, {"start": 6.96, "duration": 2.639, "text": "you're not careful if you want to try it"}, {"start": 8.34, "duration": 3.78, "text": "out pause it right now because I'm going"}, {"start": 9.599, "duration": 4.261, "text": "to give the answer in three two one what"}, {"start": 12.12, "duration": 3.84, "text": "it's asking for is the double derivative"}, {"start": 13.86, "duration": 4.32, "text": "of f of x to the four so I'm going to"}, {"start": 15.96, "duration": 3.72, "text": "put the derivative of derivative of f of"}, {"start": 18.18, "duration": 3.48, "text": "x to the fourth so we're going to solve"}, {"start": 19.68, "duration": 4.679, "text": "this inside first so this up here says"}, {"start": 21.66, "duration": 4.74, "text": "the derivative of f of x is G of X so"}, {"start": 24.359, "duration": 4.201, "text": "this is just going to be G of whatever"}, {"start": 26.4, "duration": 3.6, "text": "is inside here G of x to the fourth and"}, {"start": 28.56, "duration": 2.82, "text": "then we have to take the chain rule of"}, {"start": 30.0, "duration": 2.759, "text": "whatever's inside here so we're going to"}, {"start": 31.38, "duration": 4.14, "text": "multiply this by the derivative of"}, {"start": 32.759, "duration": 4.501, "text": "what's inside here which is 4X cubed our"}, {"start": 35.52, "duration": 2.94, "text": "first step is now done now we gotta take"}, {"start": 37.26, "duration": 3.24, "text": "the derivative of what's inside here"}, {"start": 38.46, "duration": 3.599, "text": "unfortunately it is a product of two"}, {"start": 40.5, "duration": 2.94, "text": "functions so we have to use the product"}, {"start": 42.059, "duration": 3.0, "text": "rule so we're going to do the derivative"}, {"start": 43.44, "duration": 3.42, "text": "of this first function and leave the"}, {"start": 45.059, "duration": 3.241, "text": "second function alone plus the"}, {"start": 46.86, "duration": 3.12, "text": "derivative of the second function"}, {"start": 48.3, "duration": 4.14, "text": "multiplied by the first function left"}, {"start": 49.98, "duration": 4.2, "text": "alone so this derivative of GX to the"}, {"start": 52.44, "duration": 4.139, "text": "fourth so it says the derivative of G of"}, {"start": 54.18, "duration": 4.44, "text": "X is f of x cubed so the derivative of G"}, {"start": 56.579, "duration": 3.901, "text": "of x to the fourth would be f of x to"}, {"start": 58.62, "duration": 3.24, "text": "the fourth Cube so then because is the"}, {"start": 60.48, "duration": 3.239, "text": "chain rule we have to multiply it by the"}, {"start": 61.86, "duration": 4.02, "text": "derivative of this inner function x to"}, {"start": 63.719, "duration": 3.421, "text": "the fourth which gives us 4X cubed and"}, {"start": 65.88, "duration": 3.96, "text": "then we're going to multiply by this"}, {"start": 67.14, "duration": 5.28, "text": "other 4X cubed plus the derivative of"}, {"start": 69.84, "duration": 4.44, "text": "this is just 12x squared and then this"}, {"start": 72.42, "duration": 4.62, "text": "GX to the 4 is just left alone we can"}, {"start": 74.28, "duration": 4.62, "text": "clean things up 4 times 4 is 16 x cubed"}, {"start": 77.04, "duration": 3.6, "text": "times x cubed is X to the sixth and then"}, {"start": 78.9, "duration": 4.44, "text": "bring this down and x to the fourth to"}, {"start": 80.64, "duration": 4.38, "text": "the third is X to the 12th plus and this"}, {"start": 83.34, "duration": 3.24, "text": "term can't really simplify so there it"}, {"start": 85.02, "duration": 3.36, "text": "is right there and that is the answer to"}, {"start": 86.58, "duration": 3.98, "text": "the question how exciting is that what"}, {"start": 88.38, "duration": 2.18, "text": "you got"}], "title": "Calculus Challenge Problem", "description": "https://andymath.com/calculus/", "lengthSeconds": 89, "width": 360, "height": 640}
{"video_id": "WXeMnsPV9rc", "transcript": [{"start": 0.0, "duration": 4.2, "text": "guys this is implicit differentiation"}, {"start": 1.86, "duration": 4.26, "text": "it's saying find dydx when x squared"}, {"start": 4.2, "duration": 2.94, "text": "minus x y equals 25. step one we're"}, {"start": 6.12, "duration": 2.46, "text": "going to take the derivative of both"}, {"start": 7.14, "duration": 3.24, "text": "sides so what is the derivative of x"}, {"start": 8.58, "duration": 3.24, "text": "squared we bring the 2 in front and it"}, {"start": 10.38, "duration": 2.82, "text": "just gives us 2X and then we're going to"}, {"start": 11.82, "duration": 3.06, "text": "subtract and what is the derivative of x"}, {"start": 13.2, "duration": 3.24, "text": "times y well the function X is being"}, {"start": 14.88, "duration": 2.94, "text": "multiplied by the function y so we have"}, {"start": 16.44, "duration": 3.54, "text": "to use product rule so it's going to be"}, {"start": 17.82, "duration": 3.78, "text": "the derivative of x times y plus x times"}, {"start": 19.98, "duration": 3.66, "text": "the derivative of y and then what is the"}, {"start": 21.6, "duration": 3.419, "text": "derivative of 25 that's equal to zero so"}, {"start": 23.64, "duration": 2.94, "text": "this is our next step so from here we"}, {"start": 25.019, "duration": 3.901, "text": "clean up these primes X Prime is the"}, {"start": 26.58, "duration": 3.9, "text": "derivative of x which is one and Y Prime"}, {"start": 28.92, "duration": 3.06, "text": "is the derivative of y if we write that"}, {"start": 30.48, "duration": 2.82, "text": "as d y d x that's the thing we're trying"}, {"start": 31.98, "duration": 2.52, "text": "to solve for next we can distribute the"}, {"start": 33.3, "duration": 2.46, "text": "negative to both of these terms they're"}, {"start": 34.5, "duration": 3.36, "text": "both going to become negative so our"}, {"start": 35.76, "duration": 3.9, "text": "goal is to get dydx alone so let's move"}, {"start": 37.86, "duration": 2.94, "text": "all the terms that don't have a d y d x"}, {"start": 39.66, "duration": 2.76, "text": "to the other side of the equation we're"}, {"start": 40.8, "duration": 2.82, "text": "doing that by subtracting this one and"}, {"start": 42.42, "duration": 2.88, "text": "adding this one on the left hand side"}, {"start": 43.62, "duration": 3.06, "text": "we're just left with the negative x d y"}, {"start": 45.3, "duration": 3.0, "text": "d x and on the right hand side we have"}, {"start": 46.68, "duration": 2.94, "text": "the negative 2x plus y from here we can"}, {"start": 48.3, "duration": 3.0, "text": "divide both sides by negative X the"}, {"start": 49.62, "duration": 2.939, "text": "left-hand side just has the dydx which"}, {"start": 51.3, "duration": 2.279, "text": "is what we wanted to get alone and the"}, {"start": 52.559, "duration": 1.921, "text": "right hand side is basically going to"}, {"start": 53.579, "duration": 1.98, "text": "look just like this but these two"}, {"start": 54.48, "duration": 2.46, "text": "negatives will cancel and this will"}, {"start": 55.559, "duration": 4.761, "text": "become a negative so it looks like this"}, {"start": 56.94, "duration": 3.38, "text": "2x minus y over X"}], "title": "Impilicit Differentiation #shorts", "description": "https://andymath.com/implicit-differentiation-2/\n\nCollab with Geeklyhub\n\n#shorts", "lengthSeconds": 59, "width": 270, "height": 480}
{"video_id": "XvF7hDIeQ5w", "transcript": [{"start": 0.04, "duration": 2.839, "text": "hey guys this looks like a fun one if"}, {"start": 1.52, "duration": 3.52, "text": "you want to try it on your own pause it"}, {"start": 2.879, "duration": 4.96, "text": "right now cuz I'm going to solve it in 3"}, {"start": 5.04, "duration": 5.36, "text": "2 1 let's connect the radius that goes"}, {"start": 7.839, "duration": 4.521, "text": "from here through this point to here and"}, {"start": 10.4, "duration": 4.04, "text": "that's going to cut everything in half"}, {"start": 12.36, "duration": 3.8, "text": "so we will have two of these blue things"}, {"start": 14.44, "duration": 4.08, "text": "two of these white things and two of"}, {"start": 16.16, "duration": 4.68, "text": "these blue things let's call these two x"}, {"start": 18.52, "duration": 4.4, "text": "these two Y and these two Z and now we"}, {"start": 20.84, "duration": 5.0, "text": "have a way to express our Blue Area it's"}, {"start": 22.92, "duration": 4.96, "text": "going to be equal to 2x + 2 Z so now we"}, {"start": 25.84, "duration": 3.679, "text": "got to do some work to find x and z"}, {"start": 27.88, "duration": 4.199, "text": "first let's look at the entire quarter"}, {"start": 29.519, "duration": 5.841, "text": "Circle the area of a quarter circle is <"}, {"start": 32.079, "duration": 6.601, "text": "r^ 2 / 4 and the radius of this quarter"}, {"start": 35.36, "duration": 5.359, "text": "circle is equal to 4 and 4^ s means 4 *"}, {"start": 38.68, "duration": 3.92, "text": "4 and then we can cancel out these two"}, {"start": 40.719, "duration": 4.401, "text": "fours and now we know the area of this"}, {"start": 42.6, "duration": 4.479, "text": "quarter circle is equal to 4 pi and this"}, {"start": 45.12, "duration": 4.52, "text": "4 Pi is the same thing as two of these"}, {"start": 47.079, "duration": 5.32, "text": "X's plus two of the Y's plus two of the"}, {"start": 49.64, "duration": 4.48, "text": "Z's next we can divide everything by two"}, {"start": 52.399, "duration": 3.361, "text": "on the left hand side all these twos"}, {"start": 54.12, "duration": 4.399, "text": "will cancel each other out and on the"}, {"start": 55.76, "duration": 4.56, "text": "right hand side 4 Pi / 2 is the same"}, {"start": 58.519, "duration": 4.52, "text": "thing as 2 pi and then next next let's"}, {"start": 60.32, "duration": 4.839, "text": "focus on this semicircle let's color it"}, {"start": 63.039, "duration": 5.801, "text": "red and let's find the area of the red"}, {"start": 65.159, "duration": 5.801, "text": "the area of a semicircle is < R 2 / 2"}, {"start": 68.84, "duration": 3.959, "text": "this four is the diameter of the"}, {"start": 70.96, "duration": 5.36, "text": "semicircle which means the radius would"}, {"start": 72.799, "duration": 5.0, "text": "be equal to 2 and 2^ 2 is equal to 2 * 2"}, {"start": 76.32, "duration": 3.439, "text": "and these twos will cancel each other"}, {"start": 77.799, "duration": 4.68, "text": "out so the area of this red region is"}, {"start": 79.759, "duration": 5.961, "text": "equal to 2 pi and this 2 pi is the same"}, {"start": 82.479, "duration": 5.241, "text": "thing as 1X 2x + y and now we can bring"}, {"start": 85.72, "duration": 4.399, "text": "everything else back next let's cut the"}, {"start": 87.72, "duration": 5.0, "text": "semicircle in half this forward split"}, {"start": 90.119, "duration": 4.241, "text": "into 2x two and then this radius wraps"}, {"start": 92.72, "duration": 4.359, "text": "all the way around so this height is"}, {"start": 94.36, "duration": 5.399, "text": "also two now let's focus on this quarter"}, {"start": 97.079, "duration": 4.961, "text": "Circle this x piece plus the triangle"}, {"start": 99.759, "duration": 4.441, "text": "will equal the whole quarter Circle next"}, {"start": 102.04, "duration": 3.64, "text": "we can subtract triangle from both sides"}, {"start": 104.2, "duration": 3.32, "text": "these triangles will cancel each other"}, {"start": 105.68, "duration": 2.96, "text": "out on the left hand side we'll have X"}, {"start": 107.52, "duration": 3.08, "text": "and on the right hand side we'll have"}, {"start": 108.64, "duration": 4.68, "text": "quarter Circle minus triangle the"}, {"start": 110.6, "duration": 5.6, "text": "quarter Circle which is p piun r 2 / 4"}, {"start": 113.32, "duration": 4.759, "text": "the triangle which is base * height / 2"}, {"start": 116.2, "duration": 4.72, "text": "the radius of this quarter circle is"}, {"start": 118.079, "duration": 4.761, "text": "equal to 2 2^ 2 is equal to 4 and these"}, {"start": 120.92, "duration": 4.08, "text": "fours will cancel each other out so the"}, {"start": 122.84, "duration": 3.76, "text": "quarter Circle has an area of Pi and"}, {"start": 125.0, "duration": 3.959, "text": "then for the triangle the base is equal"}, {"start": 126.6, "duration": 4.12, "text": "to 2 and the height is equal to 2 these"}, {"start": 128.959, "duration": 4.081, "text": "twos would cancel each other out so the"}, {"start": 130.72, "duration": 5.519, "text": "area of this triangle is equal to 2 and"}, {"start": 133.04, "duration": 5.279, "text": "now we know that X is equal to Pi - 2 in"}, {"start": 136.239, "duration": 4.161, "text": "the place of this x we can plug in pi -"}, {"start": 138.319, "duration": 4.881, "text": "2 this 2 is going to distribute to both"}, {"start": 140.4, "duration": 4.96, "text": "of these so it becomes 2 piun - 4 next"}, {"start": 143.2, "duration": 4.6, "text": "let's subtract 2 pi from both sides and"}, {"start": 145.36, "duration": 4.08, "text": "let's add four to both sides on the left"}, {"start": 147.8, "duration": 3.48, "text": "hand side these two pies will cancel"}, {"start": 149.44, "duration": 3.4, "text": "each other out these two fours will"}, {"start": 151.28, "duration": 3.12, "text": "cancel each other out and on the right"}, {"start": 152.84, "duration": 3.679, "text": "hand side these two pies will cancel"}, {"start": 154.4, "duration": 4.6, "text": "each other out so we have y is equal to"}, {"start": 156.519, "duration": 4.681, "text": "4 now we know X and Y so we're going to"}, {"start": 159.0, "duration": 4.8, "text": "substitute that into this equation in"}, {"start": 161.2, "duration": 3.88, "text": "the place of the X we'll plug in pi - 2"}, {"start": 163.8, "duration": 3.56, "text": "and in the place of the Y we're going to"}, {"start": 165.08, "duration": 4.439, "text": "plug in four let's combine the -2 and"}, {"start": 167.36, "duration": 4.84, "text": "the four to give us 2 and then we'll do"}, {"start": 169.519, "duration": 4.561, "text": "minus Pi minus 2 from both sides on the"}, {"start": 172.2, "duration": 4.36, "text": "left hand side all of this stuff cancels"}, {"start": 174.08, "duration": 4.879, "text": "out and on the right hand side 2 pi -"}, {"start": 176.56, "duration": 4.44, "text": "piun is 1 pi and now we have Z is equal"}, {"start": 178.959, "duration": 3.801, "text": "to piun - 2 2 now that we know the"}, {"start": 181.0, "duration": 3.599, "text": "values of all three of our variables we"}, {"start": 182.76, "duration": 4.64, "text": "should be able to find the blue area the"}, {"start": 184.599, "duration": 5.401, "text": "blue area is equal to 2x + 2 Z we don't"}, {"start": 187.4, "duration": 5.36, "text": "need the Y anymore and looking at this Z"}, {"start": 190.0, "duration": 5.36, "text": "and X are both equal to Pius 2 so we can"}, {"start": 192.76, "duration": 5.32, "text": "say x is equal to Z and let's substitute"}, {"start": 195.36, "duration": 6.12, "text": "in the place of this Z and X so now we"}, {"start": 198.08, "duration": 6.12, "text": "have 2X + 2x which is equal to 4X and in"}, {"start": 201.48, "duration": 4.28, "text": "the place of the X let's plug in pi - 2"}, {"start": 204.2, "duration": 4.48, "text": "and then this four can distribute to"}, {"start": 205.76, "duration": 4.8, "text": "both of those to give us 4 Pi - 8 so we"}, {"start": 208.68, "duration": 5.04, "text": "now know that the blue area is equal to"}, {"start": 210.56, "duration": 4.599, "text": "4 piun - 8 square units that's the"}, {"start": 213.72, "duration": 5.64, "text": "answer to our question so let's put a"}, {"start": 215.159, "duration": 4.201, "text": "box around it how exciting"}], "title": "Algebraic Approach to this one", "description": "I wanted to try this one with this algebraic approach. It was fun to go through.", "lengthSeconds": 217, "width": 640, "height": 360}
{"video_id": "Fy1voKCutUk", "transcript": [{"start": 0.0, "duration": 3.659, "text": "okay so this is called summation"}, {"start": 1.439, "duration": 3.301, "text": "notation or Sigma notation so it means"}, {"start": 3.659, "duration": 2.401, "text": "we're going to be adding some stuff"}, {"start": 4.74, "duration": 3.24, "text": "together so we're going to take this"}, {"start": 6.06, "duration": 3.3, "text": "we're going to plug in n equals 1 and"}, {"start": 7.98, "duration": 2.94, "text": "then we're going to plug in n equals 2"}, {"start": 9.36, "duration": 3.54, "text": "and then we're going to plug in n equals"}, {"start": 10.92, "duration": 3.839, "text": "3 and we stop when we get to this number"}, {"start": 12.9, "duration": 3.719, "text": "up here so for this first one we plug in"}, {"start": 14.759, "duration": 3.901, "text": "1 for the N for this one we plug in 2"}, {"start": 16.619, "duration": 3.66, "text": "and for this one we plug in 3. 1 squared"}, {"start": 18.66, "duration": 3.6, "text": "is just one so this is all brought down"}, {"start": 20.279, "duration": 4.5, "text": "three times one is three three plus nine"}, {"start": 22.26, "duration": 5.759, "text": "is 12. plus whatever this simplifies to"}, {"start": 24.779, "duration": 6.121, "text": "2 squared is 4 times 3 plus 9. 3 times 4"}, {"start": 28.019, "duration": 5.941, "text": "is 12. 12 plus 9 is 21 plus whatever's"}, {"start": 30.9, "duration": 7.26, "text": "inside here 3 squared is 9 times 3 plus"}, {"start": 33.96, "duration": 5.58, "text": "9. 3 times 9 is 27 plus the 9 is 36. so"}, {"start": 38.16, "duration": 3.48, "text": "now we just want to add up these three"}, {"start": 39.54, "duration": 3.839, "text": "terms two plus one is three plus six is"}, {"start": 41.64, "duration": 4.259, "text": "nine and then one plus two is three plus"}, {"start": 43.379, "duration": 4.921, "text": "three is six so these add up to 69 so"}, {"start": 45.899, "duration": 6.5, "text": "this is equal to 69 and we can put a box"}, {"start": 48.3, "duration": 4.099, "text": "around it nice how exciting"}], "title": "Sigma Sum", "description": "", "lengthSeconds": 51, "width": 360, "height": 640}
{"video_id": "6dZs18enHnI", "transcript": [{"start": 0.04, "duration": 4.36, "text": "hey guys this looks like a fun one it"}, {"start": 1.4, "duration": 5.68, "text": "says solve for x and it's giving us this"}, {"start": 4.4, "duration": 4.199, "text": "shape right here we have a 75\u00b0 angle"}, {"start": 7.08, "duration": 3.24, "text": "here we have this segment the same"}, {"start": 8.599, "duration": 3.441, "text": "length as this segment the same length"}, {"start": 10.32, "duration": 3.359, "text": "as this segment and there's a right"}, {"start": 12.04, "duration": 3.519, "text": "angle right here and it wants us to find"}, {"start": 13.679, "duration": 3.561, "text": "the value of x if you want to try it on"}, {"start": 15.559, "duration": 5.521, "text": "your own posit right now because I'm"}, {"start": 17.24, "duration": 6.08, "text": "going to solve it in 3 2 1 first let's"}, {"start": 21.08, "duration": 4.4, "text": "focus on this right triangle any"}, {"start": 23.32, "duration": 4.56, "text": "triangle can be inscribed in a circle so"}, {"start": 25.48, "duration": 4.08, "text": "let's inscribe it in a circle and for"}, {"start": 27.88, "duration": 4.48, "text": "every inscribed right triangle the"}, {"start": 29.56, "duration": 4.44, "text": "hypoten is the diameter of the circle so"}, {"start": 32.36, "duration": 3.76, "text": "we know that the center is going to lie"}, {"start": 34.0, "duration": 4.12, "text": "on this hypotenuse and since we know"}, {"start": 36.12, "duration": 3.72, "text": "this is congruent to this we know the"}, {"start": 38.12, "duration": 4.16, "text": "center is going to be right there and"}, {"start": 39.84, "duration": 3.96, "text": "each of these are radius of the circle"}, {"start": 42.28, "duration": 3.119, "text": "and we can draw another radius from the"}, {"start": 43.8, "duration": 3.599, "text": "center of the circle to the vertex of"}, {"start": 45.399, "duration": 3.601, "text": "the right angle now I think this is"}, {"start": 47.399, "duration": 3.441, "text": "everything we need to solve the problem"}, {"start": 49.0, "duration": 4.16, "text": "if you want to pause it now cuz I'm"}, {"start": 50.84, "duration": 4.6, "text": "going to actually solve it in 3 2 1 for"}, {"start": 53.16, "duration": 4.12, "text": "the next step if this angle is X this"}, {"start": 55.44, "duration": 3.959, "text": "angle right here will also be X and"}, {"start": 57.28, "duration": 3.56, "text": "these are called vertical angles and"}, {"start": 59.399, "duration": 4.081, "text": "then we don't know how big this angle is"}, {"start": 60.84, "duration": 4.56, "text": "so let's call it y so now if we focus on"}, {"start": 63.48, "duration": 4.48, "text": "this angle right here if the inscribed"}, {"start": 65.4, "duration": 4.8, "text": "angle is y The Arc that it subtends is"}, {"start": 67.96, "duration": 5.44, "text": "always double the inscribed angle so"}, {"start": 70.2, "duration": 5.48, "text": "this Arc from here to here is 2 y de and"}, {"start": 73.4, "duration": 4.2, "text": "then if we focus on this angle the"}, {"start": 75.68, "duration": 4.92, "text": "central angle is always congruent to the"}, {"start": 77.6, "duration": 6.559, "text": "measure of the arc so if this Arc is 2 y"}, {"start": 80.6, "duration": 6.479, "text": "de this angle will be 2 y de next we can"}, {"start": 84.159, "duration": 5.96, "text": "label this angle right here Z degre and"}, {"start": 87.079, "duration": 4.961, "text": "if we focus on this isoceles triangle in"}, {"start": 90.119, "duration": 3.521, "text": "an isoceles triangle you have two sides"}, {"start": 92.04, "duration": 3.56, "text": "congruent and you also have the base"}, {"start": 93.64, "duration": 4.439, "text": "angles congruent so this angle right"}, {"start": 95.6, "duration": 4.519, "text": "here will also be equal to Z de so now"}, {"start": 98.079, "duration": 4.241, "text": "we're ready to do some algebra we can"}, {"start": 100.119, "duration": 6.481, "text": "focus on these three angles right here"}, {"start": 102.32, "duration": 6.36, "text": "we have 2 y + z + 75 is equal to 180 and"}, {"start": 106.6, "duration": 3.72, "text": "then next we can focus on this triangle"}, {"start": 108.68, "duration": 4.84, "text": "right here the sum of the interior"}, {"start": 110.32, "duration": 6.96, "text": "angles of any triangle is 180\u00b0 we know"}, {"start": 113.52, "duration": 6.0, "text": "that x + y + 75 is equal to 180 and last"}, {"start": 117.28, "duration": 5.879, "text": "let's focus on this right triangle we"}, {"start": 119.52, "duration": 5.12, "text": "know that x + z + 90 will equal 180 now"}, {"start": 123.159, "duration": 3.401, "text": "let's clean these up a little bit for"}, {"start": 124.64, "duration": 3.36, "text": "the first one we can subtract 75 from"}, {"start": 126.56, "duration": 3.119, "text": "both sides and we can do the same thing"}, {"start": 128.0, "duration": 3.52, "text": "for the second one and then for the"}, {"start": 129.679, "duration": 7.721, "text": "third one we can subtract 90 from both"}, {"start": 131.52, "duration": 10.6, "text": "sides 75 - 75 is 0 75 - 75 is 0 and 90 -"}, {"start": 137.4, "duration": 8.96, "text": "90 is 0 and then 180 - 75 is 105 180 -"}, {"start": 142.12, "duration": 6.56, "text": "75 is 105 and 180 - 90 is 90 right now"}, {"start": 146.36, "duration": 4.76, "text": "we have an x a y and a z and we want to"}, {"start": 148.68, "duration": 4.559, "text": "solve for x we're going to do this with"}, {"start": 151.12, "duration": 3.92, "text": "elimination now let's rearrange things a"}, {"start": 153.239, "duration": 3.241, "text": "little bit for this second equation"}, {"start": 155.04, "duration": 4.32, "text": "let's multiply both sides of the"}, {"start": 156.48, "duration": 6.64, "text": "equation by -2 this -2 will distribute"}, {"start": 159.36, "duration": 6.08, "text": "to both of these giving us -2X and -2 Y"}, {"start": 163.12, "duration": 5.88, "text": "and then on the right hand side 105 * -2"}, {"start": 165.44, "duration": 5.799, "text": "is -210 and let's shift that up here now"}, {"start": 169.0, "duration": 5.72, "text": "we can add these two equations together"}, {"start": 171.239, "duration": 5.401, "text": "we can bring down the -2X pos2 y -2 y"}, {"start": 174.72, "duration": 4.64, "text": "will go away and we can bring down the"}, {"start": 176.64, "duration": 4.72, "text": "positive Z and then on this side 105 -"}, {"start": 179.36, "duration": 5.239, "text": "210 is"}, {"start": 181.36, "duration": 5.159, "text": "105 and now we're down to x and z so we"}, {"start": 184.599, "duration": 3.481, "text": "just got to eliminate the Z we're going"}, {"start": 186.519, "duration": 3.961, "text": "to multiply both sides of this bottom"}, {"start": 188.08, "duration": 5.239, "text": "equation by1 the negative 1 will"}, {"start": 190.48, "duration": 5.44, "text": "distribute to give USX and negative Z"}, {"start": 193.319, "duration": 4.0, "text": "and on the right hand side it will be 90"}, {"start": 195.92, "duration": 5.36, "text": "and then we can bring that up here and"}, {"start": 197.319, "duration": 9.161, "text": "add the two equations together -2X - 1X"}, {"start": 201.28, "duration": 7.64, "text": "will be -3x and5 - 90 is-"}, {"start": 206.48, "duration": 5.28, "text": "1995 now to solve for x let's divide"}, {"start": 208.92, "duration": 6.12, "text": "both sides by 3 on the left hand side we"}, {"start": 211.76, "duration": 6.96, "text": "have X and on the right hand side -195 /"}, {"start": 215.04, "duration": 6.88, "text": "-3 is POS 65 so now we found this"}, {"start": 218.72, "duration": 4.799, "text": "unknown angle X is equal to 65 and this"}, {"start": 221.92, "duration": 4.239, "text": "is the answer to our question let's put"}, {"start": 223.519, "duration": 4.08, "text": "a box around it how exciting here's the"}, {"start": 226.159, "duration": 3.121, "text": "next problem I plan on working on if you"}, {"start": 227.599, "duration": 3.84, "text": "guys want to try it out I'll post the"}, {"start": 229.28, "duration": 5.159, "text": "solution as soon as I solve it how"}, {"start": 231.439, "duration": 3.0, "text": "exciting"}], "title": "Today's Geometry Challenge", "description": "I hope you guys like this one!", "lengthSeconds": 233, "width": 640, "height": 360}
{"video_id": "Mg3nKEq7PVg", "transcript": [{"start": 0.0, "duration": 3.12, "text": "hey guys welcome to this video we're"}, {"start": 1.52, "duration": 3.12, "text": "going to solve for x"}, {"start": 3.12, "duration": 3.44, "text": "first step is we want to distribute this"}, {"start": 4.64, "duration": 5.68, "text": "4 to both of these terms"}, {"start": 6.56, "duration": 5.76, "text": "4 times x is 4x 4 times 1 is 4"}, {"start": 10.32, "duration": 3.68, "text": "plus and then we want to distribute this"}, {"start": 12.32, "duration": 5.6, "text": "2 to both of these terms"}, {"start": 14.0, "duration": 5.68, "text": "2 times 3x is 6x and 2 times 1 is 2"}, {"start": 17.92, "duration": 3.439, "text": "and it's still equal to 16. now we"}, {"start": 19.68, "duration": 4.72, "text": "combine like terms this"}, {"start": 21.359, "duration": 5.361, "text": "4x and this plus 6x become 10x"}, {"start": 24.4, "duration": 4.4, "text": "and this plus 4 and this plus 2 becomes"}, {"start": 26.72, "duration": 4.399, "text": "6 and it's still equal to 16."}, {"start": 28.8, "duration": 3.279, "text": "now subtract 6 from both sides this"}, {"start": 31.119, "duration": 4.321, "text": "cancels"}, {"start": 32.079, "duration": 5.201, "text": "we end up with 10x equals 10."}, {"start": 35.44, "duration": 3.119, "text": "then we divide both sides by 10 and we"}, {"start": 37.28, "duration": 2.959, "text": "get the answer of x"}, {"start": 38.559, "duration": 3.361, "text": "equals 1. comment below on what you'd"}, {"start": 40.239, "duration": 2.64, "text": "like me to solve for next and i'll talk"}, {"start": 41.92, "duration": 3.92, "text": "to you guys soon"}, {"start": 42.879, "duration": 2.961, "text": "bye"}], "title": "Solve for x using Distribution Fast Example #Shorts", "description": "Check out my website! \nhttps://andymath.com/\n#Shorts", "lengthSeconds": 46, "width": 202, "height": 360}
{"video_id": "5DalRLhjvqs", "transcript": [{"start": 0.08, "duration": 3.799, "text": "this video is sponsored by brilliant hey"}, {"start": 2.399, "duration": 3.841, "text": "guys this looks like a fun one we're"}, {"start": 3.879, "duration": 3.801, "text": "given that these three angles are equal"}, {"start": 6.24, "duration": 3.279, "text": "and it wants to know what fraction of"}, {"start": 7.68, "duration": 3.12, "text": "the entire rectangle is shaded if you"}, {"start": 9.519, "duration": 3.521, "text": "want to try it on your own pause it"}, {"start": 10.8, "duration": 4.999, "text": "right now cuz I'm going to solve it in 3"}, {"start": 13.04, "duration": 4.92, "text": "2 1 first since it's a rectangle we know"}, {"start": 15.799, "duration": 4.801, "text": "all these angles are right angles this"}, {"start": 17.96, "duration": 4.399, "text": "is a 90\u00b0 angle up here and since all"}, {"start": 20.6, "duration": 4.36, "text": "these angles are equal each of them has"}, {"start": 22.359, "duration": 5.16, "text": "to be 30\u00b0 next let's look at this top"}, {"start": 24.96, "duration": 4.719, "text": "triangle if this is 30 and this is 90"}, {"start": 27.519, "duration": 3.961, "text": "this angle has to be 60 and same thing"}, {"start": 29.679, "duration": 5.321, "text": "for this lower triangle if this angle's"}, {"start": 31.48, "duration": 5.399, "text": "30 this one's 90 this one has to be 60\u00b0"}, {"start": 35.0, "duration": 3.48, "text": "since both of these are 30 6090"}, {"start": 36.879, "duration": 4.441, "text": "triangles let's look at the notes for"}, {"start": 38.48, "duration": 5.04, "text": "306090 triangles in every 306090"}, {"start": 41.32, "duration": 4.04, "text": "triangle the hypotenuse is always twice"}, {"start": 43.52, "duration": 4.24, "text": "as big as the shortest side and to get"}, {"start": 45.36, "duration": 4.24, "text": "the length of the side opposite the 60\u00b0"}, {"start": 47.76, "duration": 3.959, "text": "you multiply the shortest side by the"}, {"start": 49.6, "duration": 4.639, "text": "square < TK of 3 and this is always true"}, {"start": 51.719, "duration": 3.921, "text": "for every single 30 6090 triangle so how"}, {"start": 54.239, "duration": 3.16, "text": "are we going to use these notes if we"}, {"start": 55.64, "duration": 3.84, "text": "don't have any side lengths we can"}, {"start": 57.399, "duration": 3.721, "text": "assign the variable X to this piece"}, {"start": 59.48, "duration": 3.599, "text": "right here so we can take our triangle"}, {"start": 61.12, "duration": 3.679, "text": "out and to find this length we can take"}, {"start": 63.079, "duration": 3.36, "text": "our shortest side and multiply it by"}, {"start": 64.799, "duration": 4.041, "text": "square < TK of three now let's bring the"}, {"start": 66.439, "duration": 3.921, "text": "triangle back up here we have this piece"}, {"start": 68.84, "duration": 3.68, "text": "in terms of X and we have the height of"}, {"start": 70.36, "duration": 3.72, "text": "our entire rectangle in terms of X and"}, {"start": 72.52, "duration": 4.32, "text": "in a rectangle opposite sides are"}, {"start": 74.08, "duration": 4.12, "text": "congruent so this side is also X < tk3"}, {"start": 76.84, "duration": 2.84, "text": "now let's pull this triangle out and"}, {"start": 78.2, "duration": 3.919, "text": "let's find the length of this side in"}, {"start": 79.68, "duration": 4.6, "text": "terms of X we're going to take this xk3"}, {"start": 82.119, "duration": 5.721, "text": "and multiply it by S < TK of 3 so we end"}, {"start": 84.28, "duration": 6.12, "text": "up with x < tk3 * < tk3 < tk3 * < tk3 is"}, {"start": 87.84, "duration": 4.559, "text": "equal to < TK of 9 and theun of 9 is"}, {"start": 90.4, "duration": 4.24, "text": "equal to 3 so this side length is equal"}, {"start": 92.399, "duration": 4.08, "text": "to 3x so let's bring this triangle back"}, {"start": 94.64, "duration": 3.64, "text": "up here now looking at this rectangle we"}, {"start": 96.479, "duration": 3.96, "text": "have both the height and the width in"}, {"start": 98.28, "duration": 4.64, "text": "terms of X since we know the whole width"}, {"start": 100.439, "duration": 4.801, "text": "is 3x and this piece is X that means"}, {"start": 102.92, "duration": 3.799, "text": "this piece is equal to 2X and now we're"}, {"start": 105.24, "duration": 3.4, "text": "ready to answer the question we want to"}, {"start": 106.719, "duration": 4.44, "text": "do the Shaded area divided by the total"}, {"start": 108.64, "duration": 4.799, "text": "area the Shaded area is this triangle"}, {"start": 111.159, "duration": 5.761, "text": "it's going to be 1/2 base time height so"}, {"start": 113.439, "duration": 5.401, "text": "it's 1/2 * the base of 2x * the height"}, {"start": 116.92, "duration": 3.239, "text": "of X < tk3 and then to get the total"}, {"start": 118.84, "duration": 3.639, "text": "area of the rectangle we're going do"}, {"start": 120.159, "duration": 4.361, "text": "base time height the base is equal to 3x"}, {"start": 122.479, "duration": 3.521, "text": "and the height is equal to X < tk3 and"}, {"start": 124.52, "duration": 4.12, "text": "now we can clean this up a little bit"}, {"start": 126.0, "duration": 4.759, "text": "this 1/2 * this 2 is equal to one and"}, {"start": 128.64, "duration": 4.44, "text": "this xk3 and this xk3 are going to"}, {"start": 130.759, "duration": 4.521, "text": "cancel each other out so now we have 1 X"}, {"start": 133.08, "duration": 3.799, "text": "over 3x well these two x's will also"}, {"start": 135.28, "duration": 3.64, "text": "cancel each other out now we have the"}, {"start": 136.879, "duration": 4.241, "text": "answer to our question 1/3 of this"}, {"start": 138.92, "duration": 3.84, "text": "entire rectangle is shaded let's put a"}, {"start": 141.12, "duration": 3.16, "text": "box around it I really like this problem"}, {"start": 142.76, "duration": 2.96, "text": "I thought it was brilliant speaking of"}, {"start": 144.28, "duration": 2.88, "text": "brilliant let me give you an update on"}, {"start": 145.72, "duration": 3.4, "text": "Brilliance I went through this"}, {"start": 147.16, "duration": 4.52, "text": "predicting with probability I love to"}, {"start": 149.12, "duration": 4.479, "text": "use historical data to form probability"}, {"start": 151.68, "duration": 3.919, "text": "distributions about future events it's"}, {"start": 153.599, "duration": 4.121, "text": "such a cool concept make predictions"}, {"start": 155.599, "duration": 4.161, "text": "about weather medicine traffic even"}, {"start": 157.72, "duration": 3.84, "text": "artificial intelligence this is what I"}, {"start": 159.76, "duration": 3.36, "text": "was looking forward to the most the Bas"}, {"start": 161.56, "duration": 3.16, "text": "theorem stuff I think they chose a"}, {"start": 163.12, "duration": 3.64, "text": "really good example for the lesson and"}, {"start": 164.72, "duration": 3.72, "text": "then I love the practice problems for"}, {"start": 166.76, "duration": 3.92, "text": "each of the problems they had a hint and"}, {"start": 168.44, "duration": 4.0, "text": "a full explanation I found it fun to"}, {"start": 170.68, "duration": 3.559, "text": "First not use the formula and just see"}, {"start": 172.44, "duration": 3.439, "text": "if I could logically deduce what would"}, {"start": 174.239, "duration": 3.041, "text": "occur and then I use the formula to"}, {"start": 175.879, "duration": 2.64, "text": "verify it I don't want to go through too"}, {"start": 177.28, "duration": 2.92, "text": "much detail in case you want to go"}, {"start": 178.519, "duration": 2.8, "text": "through it yourself speaking of going"}, {"start": 180.2, "duration": 2.759, "text": "through it yourself if you want to try"}, {"start": 181.319, "duration": 4.161, "text": "out brilliant there's a free 30-day"}, {"start": 182.959, "duration": 4.161, "text": "trial visit brilliant.org animath or"}, {"start": 185.48, "duration": 3.88, "text": "click on the link in the description you"}, {"start": 187.12, "duration": 5.88, "text": "can also get 20% off an annual premium"}, {"start": 189.36, "duration": 3.64, "text": "plan how exciting"}], "title": "What fraction is shaded?", "description": "To try everything Brilliant has to offer for free for a full 30 days, visit https://brilliant.org/AndyMath/ . You'll also get 20% off an annual premium subscription.", "lengthSeconds": 191, "width": 640, "height": 360}
{"video_id": "q98zPWtlHaw", "transcript": [{"start": 0.0, "duration": 3.78, "text": "hey guys this is one of the most"}, {"start": 1.68, "duration": 3.84, "text": "requested videos I've ever had and for"}, {"start": 3.78, "duration": 3.599, "text": "good reason this is such a cool problem"}, {"start": 5.52, "duration": 3.78, "text": "I did not expect it to have the outcome"}, {"start": 7.379, "duration": 3.601, "text": "that it had so the problem gives us a"}, {"start": 9.3, "duration": 3.36, "text": "semi-circle right here and a quarter"}, {"start": 10.98, "duration": 3.239, "text": "Circle right here and then this red line"}, {"start": 12.66, "duration": 3.78, "text": "with the length of five that goes from"}, {"start": 14.219, "duration": 3.721, "text": "the vertex of this rectangle tangent to"}, {"start": 16.44, "duration": 2.94, "text": "the semicircle if you want to try this"}, {"start": 17.94, "duration": 3.78, "text": "on your own pause it right now because"}, {"start": 19.38, "duration": 4.26, "text": "I'm going to solve it in three two one"}, {"start": 21.72, "duration": 3.899, "text": "so first up let's assign a radius to"}, {"start": 23.64, "duration": 4.559, "text": "this semicircle a lot of people wanted"}, {"start": 25.619, "duration": 4.561, "text": "to assign this radius to match but there"}, {"start": 28.199, "duration": 3.9, "text": "really is not evidence that this radius"}, {"start": 30.18, "duration": 3.78, "text": "is the same as this radius so let's call"}, {"start": 32.099, "duration": 3.361, "text": "it something else like X using our"}, {"start": 33.96, "duration": 3.599, "text": "method we're not going to need this"}, {"start": 35.46, "duration": 3.24, "text": "radius to equal this radius and that"}, {"start": 37.559, "duration": 2.881, "text": "right there is one of my favorite things"}, {"start": 38.7, "duration": 3.84, "text": "about this problem we don't care if the"}, {"start": 40.44, "duration": 3.959, "text": "radii are the same so this last distance"}, {"start": 42.54, "duration": 4.26, "text": "right here will also be R because it's a"}, {"start": 44.399, "duration": 4.441, "text": "radius of the same semicircle and this"}, {"start": 46.8, "duration": 4.2, "text": "height will also be X because it's a"}, {"start": 48.84, "duration": 3.539, "text": "radius of this quarter Circle and now we"}, {"start": 51.0, "duration": 3.12, "text": "can come up with a formula for the area"}, {"start": 52.379, "duration": 3.421, "text": "of the rectangle the area of a rectangle"}, {"start": 54.12, "duration": 4.439, "text": "is base times height the base is going"}, {"start": 55.8, "duration": 4.8, "text": "to be equal to X Plus 2R and the height"}, {"start": 58.559, "duration": 4.441, "text": "is going to be equal to X and then we"}, {"start": 60.6, "duration": 5.04, "text": "can distribute this x x times x is going"}, {"start": 63.0, "duration": 4.979, "text": "to give us x squared and x times 2R is"}, {"start": 65.64, "duration": 4.26, "text": "going to give us 2xr and this is a very"}, {"start": 67.979, "duration": 3.901, "text": "nice formula for the area for the next"}, {"start": 69.9, "duration": 4.2, "text": "step let's draw a segment from this"}, {"start": 71.88, "duration": 4.32, "text": "point of tangency to the center of the"}, {"start": 74.1, "duration": 3.66, "text": "semicircle and this distance is also a"}, {"start": 76.2, "duration": 3.48, "text": "radius of the semicircle so it's going"}, {"start": 77.76, "duration": 4.56, "text": "to be R this right here is going to be a"}, {"start": 79.68, "duration": 4.619, "text": "right angle every radius that intersects"}, {"start": 82.32, "duration": 4.439, "text": "a tangent line at the point of tangency"}, {"start": 84.299, "duration": 4.261, "text": "is at right angles so now we have a nice"}, {"start": 86.759, "duration": 3.961, "text": "right triangle so we can do Pythagorean"}, {"start": 88.56, "duration": 4.86, "text": "theorem r squared plus 5 squared equals"}, {"start": 90.72, "duration": 4.98, "text": "the quantity X plus r squared we can"}, {"start": 93.42, "duration": 4.98, "text": "bring down the r squared 5 squared is 25"}, {"start": 95.7, "duration": 4.8, "text": "and X plus r squared means X plus r"}, {"start": 98.4, "duration": 4.44, "text": "times X plus r and after we multiply"}, {"start": 100.5, "duration": 4.14, "text": "that out we get x squared plus 2xr plus"}, {"start": 102.84, "duration": 3.48, "text": "r squared and then we can subtract r"}, {"start": 104.64, "duration": 3.6, "text": "squared from both sides on the left hand"}, {"start": 106.32, "duration": 4.619, "text": "side we have 25 and on the right hand"}, {"start": 108.24, "duration": 4.44, "text": "side we have x squared plus 2xr and now"}, {"start": 110.939, "duration": 3.781, "text": "we're pretty much done we wanted to know"}, {"start": 112.68, "duration": 3.6, "text": "what is the area of the rectangle but we"}, {"start": 114.72, "duration": 3.84, "text": "know that the area is equal to x squared"}, {"start": 116.28, "duration": 4.14, "text": "plus 2xr so we can plug that in right"}, {"start": 118.56, "duration": 4.5, "text": "here but we also know that X squared"}, {"start": 120.42, "duration": 4.979, "text": "plus 2xr is equal to 25 so we can change"}, {"start": 123.06, "duration": 3.599, "text": "this to 25 and that's the answer to our"}, {"start": 125.399, "duration": 2.821, "text": "question let's give it a label of"}, {"start": 126.659, "duration": 3.781, "text": "centimeters squared and put a box around"}, {"start": 128.22, "duration": 3.78, "text": "it even if these radii are not equal to"}, {"start": 130.44, "duration": 3.6, "text": "each other we can find the area of this"}, {"start": 132.0, "duration": 6.02, "text": "rectangle is exactly 25 centimeters"}, {"start": 134.04, "duration": 3.98, "text": "squared how exciting Isn't that cool"}], "title": "Super Popular Problem", "description": "https://andymath.com/geometry-challenges/\n\nA lot of people sent me requests to go over this problem. I hope you love it!", "lengthSeconds": 136, "width": 640, "height": 360}
{"video_id": "Un9GgPjx-v0", "transcript": [{"start": 0.04, "duration": 4.0, "text": "hey guys this looks like a fun one it"}, {"start": 1.36, "duration": 4.32, "text": "says how long is AB and it's giving us"}, {"start": 4.04, "duration": 3.44, "text": "that all semicircles have the same"}, {"start": 5.68, "duration": 3.6, "text": "radius and it's kind of hard to see but"}, {"start": 7.48, "duration": 4.119, "text": "these are red arrows pointed to the"}, {"start": 9.28, "duration": 3.76, "text": "spaces between the semicircles if you"}, {"start": 11.599, "duration": 3.681, "text": "want to try it on your own pause it"}, {"start": 13.04, "duration": 5.52, "text": "right now CU I'm going to solve it in 3"}, {"start": 15.28, "duration": 4.6, "text": "2 1 so first let's identify a radius and"}, {"start": 18.56, "duration": 3.08, "text": "since we're given that they all have the"}, {"start": 19.88, "duration": 3.76, "text": "same radius all of these are going to be"}, {"start": 21.64, "duration": 5.6, "text": "the same and then let's look at the top"}, {"start": 23.64, "duration": 6.559, "text": "and the bottom the top is 6 RS plus a 3"}, {"start": 27.24, "duration": 6.88, "text": "plus a 3 and the bottom has four RS plus"}, {"start": 30.199, "duration": 7.321, "text": "a 66 + a 12 + a 66 and we can simplify"}, {"start": 34.12, "duration": 7.119, "text": "both of these 3 + 3 is equal to 6 and 66"}, {"start": 37.52, "duration": 5.6, "text": "+ 12 + 66 is equal to 144 and ultimately"}, {"start": 41.239, "duration": 3.961, "text": "we're trying to find ab ab is going to"}, {"start": 43.12, "duration": 3.8, "text": "be equal to either this top equation or"}, {"start": 45.2, "duration": 4.32, "text": "the bottom equation let's just use the"}, {"start": 46.92, "duration": 4.56, "text": "6r + 6 and then once we solve for R"}, {"start": 49.52, "duration": 4.08, "text": "we'll be able to find ab and the way"}, {"start": 51.48, "duration": 4.0, "text": "we're going to solve for R is this top"}, {"start": 53.6, "duration": 5.799, "text": "has to equal the bottom so that means"}, {"start": 55.48, "duration": 5.28, "text": "that 6r + 6 has to equal 4 r + 144 and"}, {"start": 59.399, "duration": 3.84, "text": "now we have a nice nice equation let's"}, {"start": 60.76, "duration": 4.479, "text": "solve for R we can subtract 4 R from"}, {"start": 63.239, "duration": 3.761, "text": "both sides and subtract six from both"}, {"start": 65.239, "duration": 3.121, "text": "sides on the left hand side the sixes"}, {"start": 67.0, "duration": 3.4, "text": "will cancel each other out and that"}, {"start": 68.36, "duration": 3.68, "text": "leave us with 2 R on the right hand side"}, {"start": 70.4, "duration": 5.079, "text": "the RS will cancel each other out and"}, {"start": 72.04, "duration": 5.2, "text": "that leaves us with 144 - 6 which is 138"}, {"start": 75.479, "duration": 3.361, "text": "next we can divide both sides by two on"}, {"start": 77.24, "duration": 5.559, "text": "the left hand side we have R and on the"}, {"start": 78.84, "duration": 5.599, "text": "right hand side 138 ided 2 is 69 nice"}, {"start": 82.799, "duration": 3.921, "text": "that means that each of these radii are"}, {"start": 84.439, "duration": 4.561, "text": "equal to 69 and now we're ready to solve"}, {"start": 86.72, "duration": 7.84, "text": "for ab in the place of R we can plug in"}, {"start": 89.0, "duration": 7.759, "text": "69 6 * 69 is equal to 414 and 414 + 6 is"}, {"start": 94.56, "duration": 4.199, "text": "420 that is the answer to our question"}, {"start": 96.759, "duration": 5.521, "text": "let's put a box around it the length of"}, {"start": 98.759, "duration": 6.521, "text": "ab in this diagram is equal to 420 how"}, {"start": 102.28, "duration": 3.0, "text": "exciting"}], "title": "Nice Question", "description": "", "lengthSeconds": 103, "width": 640, "height": 360}
{"video_id": "LxNZ-u7KejI", "transcript": [{"start": 0.0, "duration": 3.6, "text": "says solve for x no calculator the first"}, {"start": 2.639, "duration": 2.521, "text": "step I think I want to set everything"}, {"start": 3.6, "duration": 3.84, "text": "equal to zero so I'm going to subtract"}, {"start": 5.16, "duration": 3.479, "text": "2x from both sides and add 3x squared to"}, {"start": 7.44, "duration": 3.0, "text": "both sides so on the right hand side"}, {"start": 8.639, "duration": 4.981, "text": "it's going to be equal to 0. we have X"}, {"start": 10.44, "duration": 4.92, "text": "cubed plus 3x squared minus 2X x is 0 is"}, {"start": 13.62, "duration": 2.759, "text": "1 so it's going to be minus six from"}, {"start": 15.36, "duration": 2.339, "text": "here we're going to do Factor by"}, {"start": 16.379, "duration": 3.421, "text": "grouping we're going to pull out the"}, {"start": 17.699, "duration": 3.901, "text": "common x squared for these two terms and"}, {"start": 19.8, "duration": 3.54, "text": "a common negative 2 for these two terms"}, {"start": 21.6, "duration": 4.019, "text": "and it's still equal to zero x squared"}, {"start": 23.34, "duration": 4.019, "text": "times what gives us X cubed that's x x"}, {"start": 25.619, "duration": 3.66, "text": "squared times what gives us 3x squared"}, {"start": 27.359, "duration": 3.721, "text": "that's three negative 2 times what gives"}, {"start": 29.279, "duration": 3.241, "text": "us negative 2x that's X and negative 2"}, {"start": 31.08, "duration": 3.36, "text": "times y gives us negative 6 that's"}, {"start": 32.52, "duration": 4.02, "text": "positive 3. now both of these have an X"}, {"start": 34.44, "duration": 4.2, "text": "plus 3 term so I'm going to factor that"}, {"start": 36.54, "duration": 3.72, "text": "out so I have X plus 3 times what gives"}, {"start": 38.64, "duration": 3.239, "text": "me all of this that's going to be x"}, {"start": 40.26, "duration": 3.42, "text": "squared and then X plus 3 times what"}, {"start": 41.879, "duration": 3.481, "text": "gives me all this that's minus 2. and"}, {"start": 43.68, "duration": 3.84, "text": "from here we know either X plus 3 equals"}, {"start": 45.36, "duration": 4.08, "text": "0 or x squared minus two equals 0. we'll"}, {"start": 47.52, "duration": 3.66, "text": "subtract 3 from both sides and we get x"}, {"start": 49.44, "duration": 3.54, "text": "equals negative three and for this one"}, {"start": 51.18, "duration": 3.84, "text": "we add 2 to both sides get x squared"}, {"start": 52.98, "duration": 3.84, "text": "equals two square root both sides and we"}, {"start": 55.02, "duration": 3.359, "text": "get x equals plus or minus root two and"}, {"start": 56.82, "duration": 4.04, "text": "these are the three answers for X how"}, {"start": 58.379, "duration": 2.481, "text": "exciting"}], "title": "Factor by Grouping Challenge", "description": "Collab with Geeklyhub!", "lengthSeconds": 59, "width": 360, "height": 640}
{"video_id": "bBol5ztUQmQ", "transcript": [{"start": 0.04, "duration": 3.679, "text": "this video is sponsored by brilliant hey"}, {"start": 2.28, "duration": 3.28, "text": "guys this looks like a fun one there's a"}, {"start": 3.719, "duration": 3.761, "text": "chord cutting the square in half and it"}, {"start": 5.56, "duration": 3.84, "text": "has a length of six and it wants to know"}, {"start": 7.48, "duration": 3.44, "text": "what is the area of the semicircle if"}, {"start": 9.4, "duration": 3.92, "text": "you want to try it on your own pause it"}, {"start": 10.92, "duration": 4.919, "text": "right now cuz I'm going to solve it in 3"}, {"start": 13.32, "duration": 4.64, "text": "2 1 the thing we're solving for is the"}, {"start": 15.839, "duration": 5.321, "text": "area of a semicircle that's going to be"}, {"start": 17.96, "duration": 4.96, "text": "equal to Pi R 2 / 2 and for the first"}, {"start": 21.16, "duration": 3.8, "text": "step I want to hide the semicircle and"}, {"start": 22.92, "duration": 3.56, "text": "focus on the Square let's take both"}, {"start": 24.96, "duration": 3.28, "text": "diagonals of the square and the"}, {"start": 26.48, "duration": 3.92, "text": "intersection of those diagonals would be"}, {"start": 28.24, "duration": 3.999, "text": "the center of the square and you can see"}, {"start": 30.4, "duration": 3.44, "text": "this also goes through the center and"}, {"start": 32.239, "duration": 3.081, "text": "that's true for any line that bisects"}, {"start": 33.84, "duration": 3.8, "text": "the square it will always go through the"}, {"start": 35.32, "duration": 4.28, "text": "center of the square and now let's focus"}, {"start": 37.64, "duration": 4.32, "text": "on the given chord the center of the"}, {"start": 39.6, "duration": 4.08, "text": "square will cut this chord in half so we"}, {"start": 41.96, "duration": 3.599, "text": "get three and three now we can bring"}, {"start": 43.68, "duration": 3.64, "text": "back the semicircle and let's focus on"}, {"start": 45.559, "duration": 3.281, "text": "the semicircle through this midpoint"}, {"start": 47.32, "duration": 3.239, "text": "let's draw a perpendicular line it's"}, {"start": 48.84, "duration": 4.12, "text": "going to be the perpendicular bis sector"}, {"start": 50.559, "duration": 3.921, "text": "of our chord and the perpendicular bis"}, {"start": 52.96, "duration": 3.16, "text": "sector of any chord will always go"}, {"start": 54.48, "duration": 3.599, "text": "through the center of a circle so we can"}, {"start": 56.12, "duration": 3.959, "text": "mark this center right here next I'm"}, {"start": 58.079, "duration": 3.401, "text": "thinking let's go back to the square and"}, {"start": 60.079, "duration": 3.0, "text": "let's take this piece right here and"}, {"start": 61.48, "duration": 3.719, "text": "extend it all the way to the end of the"}, {"start": 63.079, "duration": 4.241, "text": "square since these two are equal to"}, {"start": 65.199, "duration": 3.96, "text": "three these two will also be equal to"}, {"start": 67.32, "duration": 3.76, "text": "three and that is something else that's"}, {"start": 69.159, "duration": 3.401, "text": "universally true as long as this is"}, {"start": 71.08, "duration": 3.44, "text": "cutting the square in half and this is"}, {"start": 72.56, "duration": 4.559, "text": "perpendicular to it these portions will"}, {"start": 74.52, "duration": 4.16, "text": "always be equal to each other I'm mostly"}, {"start": 77.119, "duration": 3.36, "text": "interested in these three down here we"}, {"start": 78.68, "duration": 3.84, "text": "don't really need this one and let's"}, {"start": 80.479, "duration": 3.561, "text": "bring back our semicircle and next I"}, {"start": 82.52, "duration": 3.48, "text": "want to look at the whole circle and"}, {"start": 84.04, "duration": 3.36, "text": "let's extend this down here and this"}, {"start": 86.0, "duration": 3.6, "text": "piece will be equal to the radius that"}, {"start": 87.4, "duration": 4.679, "text": "we're trying to solve for this red"}, {"start": 89.6, "duration": 4.6, "text": "radius is equal to this blue radius"}, {"start": 92.079, "duration": 4.881, "text": "since this blue thing is equal to R and"}, {"start": 94.2, "duration": 4.959, "text": "this piece is equal to 3 that leaves Rus"}, {"start": 96.96, "duration": 4.519, "text": "3 for this little piece right here and"}, {"start": 99.159, "duration": 5.081, "text": "if we focus on this entire green length"}, {"start": 101.479, "duration": 4.96, "text": "right here it's equal to r + 3 so let's"}, {"start": 104.24, "duration": 3.8, "text": "write it as r + 3 and now we can use"}, {"start": 106.439, "duration": 3.481, "text": "another set of notes this one is about"}, {"start": 108.04, "duration": 5.719, "text": "the relationship of intersecting chords"}, {"start": 109.92, "duration": 6.32, "text": "in circles a * B will always equal C * D"}, {"start": 113.759, "duration": 6.121, "text": "and for our Circle here 3 * 3 will equal"}, {"start": 116.24, "duration": 5.76, "text": "the quantity r + 3 * the quantity Rus 3"}, {"start": 119.88, "duration": 5.68, "text": "on the left hand side 3 * 3 is equal to"}, {"start": 122.0, "duration": 6.319, "text": "9 and r + 3 * r - 3 is a difference of 2"}, {"start": 125.56, "duration": 4.16, "text": "squares it simplifies to r^ 2 - 9 and"}, {"start": 128.319, "duration": 5.041, "text": "here's the notes right here and it ends"}, {"start": 129.72, "duration": 7.239, "text": "up being a + b * Aus B is equal to a^ 2"}, {"start": 133.36, "duration": 6.159, "text": "- b^ 2 now we have 9 is equal to r^ 2 -"}, {"start": 136.959, "duration": 5.64, "text": "9 to get the r 2 by itself let's add 9"}, {"start": 139.519, "duration": 4.881, "text": "to both sides 99 + 9 is equal to 18 and"}, {"start": 142.599, "duration": 4.121, "text": "these 9es cancel each other out leaving"}, {"start": 144.4, "duration": 3.88, "text": "us with r 2 then we can square root both"}, {"start": 146.72, "duration": 4.08, "text": "sides of the equation and we get R is"}, {"start": 148.28, "duration": 4.16, "text": "equal to the < TK of 18 and this is"}, {"start": 150.8, "duration": 4.159, "text": "really all we need to find the area of"}, {"start": 152.44, "duration": 4.68, "text": "our semicircle in the place of this R we"}, {"start": 154.959, "duration": 3.761, "text": "can plug in square < TK of 18 this"}, {"start": 157.12, "duration": 4.6, "text": "square root and this Square will cancel"}, {"start": 158.72, "duration": 4.04, "text": "each other out and 18 / 2 is equal to 9"}, {"start": 161.72, "duration": 4.0, "text": "and we now have the area of our"}, {"start": 162.76, "duration": 4.72, "text": "semicircle it's equal to 9 pi unit squar"}, {"start": 165.72, "duration": 2.92, "text": "let's put a box around it I really like"}, {"start": 167.48, "duration": 2.679, "text": "this question I thought it was a"}, {"start": 168.64, "duration": 2.879, "text": "brilliant one Speaking of brilliant I"}, {"start": 170.159, "duration": 2.321, "text": "want to give you guys another update on"}, {"start": 171.519, "duration": 2.481, "text": "brilliant I've been going through"}, {"start": 172.48, "duration": 3.44, "text": "brilliant for a while now and it has"}, {"start": 174.0, "duration": 3.48, "text": "thousands of lessons to choose from and"}, {"start": 175.92, "duration": 3.16, "text": "all of them are interactive you learn by"}, {"start": 177.48, "duration": 3.0, "text": "doing this stuff I was focusing on"}, {"start": 179.08, "duration": 3.0, "text": "programming for while but now I've"}, {"start": 180.48, "duration": 3.44, "text": "returned back to more math stuff and"}, {"start": 182.08, "duration": 3.6, "text": "also data like this exploring data"}, {"start": 183.92, "duration": 3.879, "text": "visually it's interesting I've always"}, {"start": 185.68, "duration": 4.04, "text": "been fascinated by data visualization"}, {"start": 187.799, "duration": 4.201, "text": "being able to spot important trends at a"}, {"start": 189.72, "duration": 3.84, "text": "glance and making quick decisions is so"}, {"start": 192.0, "duration": 2.68, "text": "cool but you have to be careful you have"}, {"start": 193.56, "duration": 2.599, "text": "to make sure you understand the"}, {"start": 194.68, "duration": 3.08, "text": "visualizations otherwise you can make"}, {"start": 196.159, "duration": 2.72, "text": "some bad decisions there's a lot of"}, {"start": 197.76, "duration": 2.759, "text": "things you want to look out for"}, {"start": 198.879, "duration": 4.041, "text": "especially something like a manipulated"}, {"start": 200.519, "duration": 3.8, "text": "Axis or scales and I suspect a lot of"}, {"start": 202.92, "duration": 3.56, "text": "people aren't familiar with things like"}, {"start": 204.319, "duration": 4.161, "text": "log scales I think brilliant has a great"}, {"start": 206.48, "duration": 3.36, "text": "interactive way to learn about those I'm"}, {"start": 208.48, "duration": 2.959, "text": "also looking at predicting with"}, {"start": 209.84, "duration": 2.759, "text": "probability it's been a lot of fun I"}, {"start": 211.439, "duration": 2.961, "text": "haven't looked at it yet but I'm looking"}, {"start": 212.599, "duration": 3.681, "text": "forward to this Bas roll stuff one of"}, {"start": 214.4, "duration": 3.839, "text": "the only videos I ever deleted was about"}, {"start": 216.28, "duration": 3.48, "text": "base roll the video was correct but"}, {"start": 218.239, "duration": 3.28, "text": "people didn't believe it so I ended up"}, {"start": 219.76, "duration": 2.96, "text": "deleting the video maybe after I go"}, {"start": 221.519, "duration": 2.841, "text": "through this I'll come up with better"}, {"start": 222.72, "duration": 3.519, "text": "ways to explain it if you want to try"}, {"start": 224.36, "duration": 4.32, "text": "brilliant there's a free 30-day trial"}, {"start": 226.239, "duration": 4.241, "text": "just visit brilliant.org Andy or click"}, {"start": 228.68, "duration": 4.44, "text": "on the link in the description you can"}, {"start": 230.48, "duration": 5.679, "text": "also get 20% off an annual premium plan"}, {"start": 233.12, "duration": 3.039, "text": "how exciting"}], "title": "This was a hard one!", "description": "To try everything Brilliant has to offer for free for a full 30 days, visit https://brilliant.org/AndyMath/ . You'll also get 20% off an annual premium subscription.", "lengthSeconds": 234, "width": 640, "height": 360}
{"video_id": "S3vYreTPnOw", "transcript": [{"start": 0.0, "duration": 3.659, "text": "guys let's find the perimeter of this"}, {"start": 1.92, "duration": 3.06, "text": "compound figure if you want to try this"}, {"start": 3.659, "duration": 3.541, "text": "on your own positive right now because"}, {"start": 4.98, "duration": 4.26, "text": "I'm going to solve it in three two one"}, {"start": 7.2, "duration": 3.479, "text": "it's nice they gave us all these right"}, {"start": 9.24, "duration": 3.3, "text": "angles but I'm going to get them out of"}, {"start": 10.679, "duration": 3.901, "text": "the way so to find the perimeter we want"}, {"start": 12.54, "duration": 4.44, "text": "to add up all the sides so the width"}, {"start": 14.58, "duration": 4.08, "text": "across the figure is equal to 14. that"}, {"start": 16.98, "duration": 4.139, "text": "means if I added the lengths of these"}, {"start": 18.66, "duration": 3.9, "text": "three sides they will also be 14. to"}, {"start": 21.119, "duration": 3.541, "text": "find the total height I'm going to have"}, {"start": 22.56, "duration": 4.799, "text": "to add 12 to something and let's call"}, {"start": 24.66, "duration": 5.699, "text": "this unknown distance X so the height of"}, {"start": 27.359, "duration": 4.861, "text": "our figure is going to be X Plus 12. and"}, {"start": 30.359, "duration": 4.081, "text": "let's add that to our green stuff next"}, {"start": 32.22, "duration": 4.499, "text": "we have the given sides the 8 and the 12"}, {"start": 34.44, "duration": 3.72, "text": "we can add them to our list as well so"}, {"start": 36.719, "duration": 3.84, "text": "we're almost done with our perimeter but"}, {"start": 38.16, "duration": 4.26, "text": "we have one more piece this yellow piece"}, {"start": 40.559, "duration": 4.081, "text": "right here temporarily let's call it"}, {"start": 42.42, "duration": 3.72, "text": "question mark we know that this X plus"}, {"start": 44.64, "duration": 4.079, "text": "the question mark is going to be equal"}, {"start": 46.14, "duration": 4.259, "text": "to 8. and after we subtract X from both"}, {"start": 48.719, "duration": 4.621, "text": "sides we get the question mark is equal"}, {"start": 50.399, "duration": 4.68, "text": "to 8 minus X and we can change that in"}, {"start": 53.34, "duration": 3.899, "text": "our diagram and let's bring this last"}, {"start": 55.079, "duration": 4.081, "text": "side down to our sum after we add up all"}, {"start": 57.239, "duration": 4.021, "text": "of these we will have the perimeter of"}, {"start": 59.16, "duration": 5.1, "text": "our figure so the the green 14 plus 14"}, {"start": 61.26, "duration": 4.98, "text": "is equal to 28 and the blue 8 plus 12 is"}, {"start": 64.26, "duration": 4.8, "text": "equal to 20. let's bring this yellow 8"}, {"start": 66.24, "duration": 5.1, "text": "up here and this 12 plus 8 is also equal"}, {"start": 69.06, "duration": 4.62, "text": "to 20. let's move the 20 down here and"}, {"start": 71.34, "duration": 4.86, "text": "20 plus 20 is equal to 40. we can move"}, {"start": 73.68, "duration": 4.5, "text": "the 40 up here and the minus X up here"}, {"start": 76.2, "duration": 5.459, "text": "plus X and negative X will cancel each"}, {"start": 78.18, "duration": 5.28, "text": "other out and 28 plus 40 is equal to 68."}, {"start": 81.659, "duration": 3.421, "text": "that is the perimeter of our figure now"}, {"start": 83.46, "duration": 3.36, "text": "we can give it a label of units and put"}, {"start": 85.08, "duration": 3.42, "text": "a box around it so what I really like"}, {"start": 86.82, "duration": 4.92, "text": "about this problem is X can be anything"}, {"start": 88.5, "duration": 6.06, "text": "between 0 and 8 not including 0 and 8. x"}, {"start": 91.74, "duration": 4.98, "text": "could be one or two or three or five six"}, {"start": 94.56, "duration": 4.019, "text": "or seven or any other number between 0"}, {"start": 96.72, "duration": 3.899, "text": "and 8 and it changes the shape of the"}, {"start": 98.579, "duration": 3.661, "text": "figure just a little bit but all of"}, {"start": 100.619, "duration": 6.32, "text": "these figures have the exact same"}, {"start": 102.24, "duration": 4.699, "text": "parameter 68 units how exciting"}], "title": "Perimeter Challenge Problem", "description": "I thought this was a cool one. I like how there is 1 correct answer for the perimeter, but an infinite number of possible figures that make that perimeter.", "lengthSeconds": 105, "width": 640, "height": 360}
{"video_id": "9TUh_1L2NfE", "transcript": [{"start": 0.0, "duration": 3.36, "text": "hey guys I think this is kind of a cool"}, {"start": 1.92, "duration": 3.48, "text": "one it's asking for the area of this"}, {"start": 3.36, "duration": 4.32, "text": "rectangle that contains these three unit"}, {"start": 5.4, "duration": 4.02, "text": "circles so the area of a rectangle is"}, {"start": 7.68, "duration": 3.78, "text": "base times height so we just need to"}, {"start": 9.42, "duration": 4.08, "text": "find the base and the height so first"}, {"start": 11.46, "duration": 4.02, "text": "let's do the base let's start with this"}, {"start": 13.5, "duration": 3.779, "text": "circle the radius is equal to one so"}, {"start": 15.48, "duration": 3.9, "text": "this radius would also be equal to one"}, {"start": 17.279, "duration": 4.381, "text": "now for this circle this radius is one"}, {"start": 19.38, "duration": 3.659, "text": "so this radius is also one so the base"}, {"start": 21.66, "duration": 4.56, "text": "of this rectangle is going to be equal"}, {"start": 23.039, "duration": 4.801, "text": "to one two three four centimeters that"}, {"start": 26.22, "duration": 3.479, "text": "was the easy one now we got to do the"}, {"start": 27.84, "duration": 3.72, "text": "height so this is a radius of a circle"}, {"start": 29.699, "duration": 3.721, "text": "that's equal to one and this right here"}, {"start": 31.56, "duration": 3.6, "text": "would also be equal to the radius of one"}, {"start": 33.42, "duration": 4.08, "text": "of these circles and those are pretty"}, {"start": 35.16, "duration": 4.32, "text": "easy to spot but this green portion is"}, {"start": 37.5, "duration": 3.48, "text": "not quite so easy but we can still find"}, {"start": 39.48, "duration": 3.239, "text": "the length of this green part let's"}, {"start": 40.98, "duration": 3.54, "text": "complete the right triangle right here"}, {"start": 42.719, "duration": 3.901, "text": "this side of the triangle is equal to"}, {"start": 44.52, "duration": 4.5, "text": "one centimeter because it is the radius"}, {"start": 46.62, "duration": 4.32, "text": "of this circle and this piece right here"}, {"start": 49.02, "duration": 3.42, "text": "is also one centimeter because it's a"}, {"start": 50.94, "duration": 3.119, "text": "radius and this piece right here is"}, {"start": 52.44, "duration": 3.36, "text": "equal to one centimeter because it's a"}, {"start": 54.059, "duration": 3.601, "text": "radius so now we have a nice right"}, {"start": 55.8, "duration": 3.54, "text": "triangle let's pull it out this side is"}, {"start": 57.66, "duration": 3.66, "text": "equal to one and this side is equal to"}, {"start": 59.34, "duration": 4.5, "text": "two and our unknown side length we'll"}, {"start": 61.32, "duration": 3.9, "text": "call X centimeters since it's a right"}, {"start": 63.84, "duration": 3.72, "text": "triangle we can do the Pythagorean"}, {"start": 65.22, "duration": 4.74, "text": "theorem one squared plus x squared is"}, {"start": 67.56, "duration": 5.099, "text": "equal to 2 squared 1 squared is equal to"}, {"start": 69.96, "duration": 4.74, "text": "one two squared is equal to four then we"}, {"start": 72.659, "duration": 4.021, "text": "can subtract 1 from both sides these"}, {"start": 74.7, "duration": 4.08, "text": "cancel each other out and four minus 1"}, {"start": 76.68, "duration": 3.9, "text": "is equal to 3. we can square root both"}, {"start": 78.78, "duration": 3.54, "text": "sides and we get X is equal to square"}, {"start": 80.58, "duration": 3.359, "text": "root of three so the length of this"}, {"start": 82.32, "duration": 3.72, "text": "green segment is square root of three"}, {"start": 83.939, "duration": 3.661, "text": "centimeters so the length of this side"}, {"start": 86.04, "duration": 4.14, "text": "of the rectangle is going to be equal to"}, {"start": 87.6, "duration": 4.379, "text": "1 plus the square root of three plus one"}, {"start": 90.18, "duration": 3.78, "text": "centimeters and then one plus one is"}, {"start": 91.979, "duration": 4.381, "text": "equal to two so now we have our height 2"}, {"start": 93.96, "duration": 3.96, "text": "plus root 3 centimeters the area of a"}, {"start": 96.36, "duration": 3.78, "text": "rectangle is equal to base times height"}, {"start": 97.92, "duration": 4.44, "text": "the base is equal to four and the height"}, {"start": 100.14, "duration": 3.72, "text": "is equal to two plus root three so our"}, {"start": 102.36, "duration": 4.2, "text": "area is going to be equal to four times"}, {"start": 103.86, "duration": 4.259, "text": "two which is eight plus four times root"}, {"start": 106.56, "duration": 3.3, "text": "three and we got to give it a label of"}, {"start": 108.119, "duration": 3.481, "text": "centimeters squared this is the answer"}, {"start": 109.86, "duration": 4.14, "text": "to the question this is the area of the"}, {"start": 111.6, "duration": 5.059, "text": "rectangle let's put a box around it how"}, {"start": 114.0, "duration": 2.659, "text": "exciting"}], "title": "Area of this Rectangle", "description": "", "lengthSeconds": 115, "width": 640, "height": 360}
{"video_id": "_BLC83XNuH4", "transcript": [{"start": 0.04, "duration": 3.32, "text": "hey guys so we want to solve the radius"}, {"start": 1.719, "duration": 3.201, "text": "of this circle first let's find the"}, {"start": 3.36, "duration": 2.919, "text": "center of the circle and this right here"}, {"start": 4.92, "duration": 2.919, "text": "is going to be the radius I first"}, {"start": 6.279, "duration": 3.041, "text": "thought was maybe it equals 1 but it"}, {"start": 7.839, "duration": 3.041, "text": "doesn't cuz if we bring this down that's"}, {"start": 9.32, "duration": 3.16, "text": "not in the middle here cuz the edge of"}, {"start": 10.88, "duration": 3.0, "text": "the circle is not in the middle either"}, {"start": 12.48, "duration": 2.88, "text": "so we can't really do that next let's"}, {"start": 13.88, "duration": 2.92, "text": "look at this diagonal here we can"}, {"start": 15.36, "duration": 3.16, "text": "actually find out what this diagonal is"}, {"start": 16.8, "duration": 4.319, "text": "cuz this is a right angle so if we call"}, {"start": 18.52, "duration": 5.48, "text": "this side a this side B and this side C"}, {"start": 21.119, "duration": 4.641, "text": "we know that a s + b s is equal to c^2 a"}, {"start": 24.0, "duration": 3.279, "text": "right here is equal to 2 the B is the"}, {"start": 25.76, "duration": 3.919, "text": "same as this right here so it's equal to"}, {"start": 27.279, "duration": 5.841, "text": "2 and then C we'll just leave a c^ s 2 s"}, {"start": 29.679, "duration": 5.681, "text": "is 4 + 2^ 2 is 4 is equal to c^ 2 4 + 4"}, {"start": 33.12, "duration": 3.92, "text": "is 8 we square root both sides on the"}, {"start": 35.36, "duration": 4.56, "text": "left hand side we can break < TK of8"}, {"start": 37.04, "duration": 4.839, "text": "into < of 4 * < TK of two and theun of 4"}, {"start": 39.92, "duration": 3.76, "text": "is 2 and on the right hand side it just"}, {"start": 41.879, "duration": 4.081, "text": "becomes C so we can clean this stuff up"}, {"start": 43.68, "duration": 3.84, "text": "up here and let's put 2 < tk2 right here"}, {"start": 45.96, "duration": 3.32, "text": "for this piece of the diagonal let's get"}, {"start": 47.52, "duration": 4.199, "text": "rid of all this stuff unfortunately the"}, {"start": 49.28, "duration": 4.2, "text": "diameter here is not 2 < tk2 because it"}, {"start": 51.719, "duration": 3.801, "text": "doesn't go all the way to the corner but"}, {"start": 53.48, "duration": 4.28, "text": "we can say that the rest of the diagonal"}, {"start": 55.52, "duration": 4.08, "text": "right up here is also 2 < tk2 so we can"}, {"start": 57.76, "duration": 3.0, "text": "calculate the diagonal of the big square"}, {"start": 59.6, "duration": 3.759, "text": "and it's going going to be equal to the"}, {"start": 60.76, "duration": 4.359, "text": "2 < tk2 plus the 2 < tk2 which is just 4"}, {"start": 63.359, "duration": 3.321, "text": "< tk2 and we can put a box around that"}, {"start": 65.119, "duration": 3.521, "text": "let's clean this all up a little bit and"}, {"start": 66.68, "duration": 3.24, "text": "this right here is the radius we'll call"}, {"start": 68.64, "duration": 3.0, "text": "that R this is what we're trying to"}, {"start": 69.92, "duration": 3.199, "text": "solve for so we know this we know this"}, {"start": 71.64, "duration": 3.2, "text": "we just got to find this piece right"}, {"start": 73.119, "duration": 3.561, "text": "here that's not too bad because we can"}, {"start": 74.84, "duration": 3.84, "text": "make a square right here we know each of"}, {"start": 76.68, "duration": 3.28, "text": "these are equal to the radius as well so"}, {"start": 78.68, "duration": 3.399, "text": "we can call both the sides of this"}, {"start": 79.96, "duration": 4.0, "text": "Square R then we can do another diagonal"}, {"start": 82.079, "duration": 3.4, "text": "we'll move this R over to here and if we"}, {"start": 83.96, "duration": 5.28, "text": "make the diagonal equal to C then we"}, {"start": 85.479, "duration": 6.241, "text": "know that R 2 + R2 = c^2 well R2 + r 2"}, {"start": 89.24, "duration": 4.0, "text": "is 2 r s we can square root both sides"}, {"start": 91.72, "duration": 4.24, "text": "the left hand side can be broken out"}, {"start": 93.24, "duration": 4.32, "text": "into < TK of 2 * < TK of R 2 and the"}, {"start": 95.96, "duration": 4.0, "text": "right hand side would just bequ < TK of"}, {"start": 97.56, "duration": 4.839, "text": "c squ and then squ of R squ is just R"}, {"start": 99.96, "duration": 4.36, "text": "andun of c s is just C and we can smush"}, {"start": 102.399, "duration": 4.32, "text": "these together and we have squ > of 2 R"}, {"start": 104.32, "duration": 4.759, "text": "is equal to C up here was our diagonal C"}, {"start": 106.719, "duration": 3.521, "text": "so we can change that into < tk2 R clean"}, {"start": 109.079, "duration": 3.241, "text": "things up a little bit and now what do"}, {"start": 110.24, "duration": 4.0, "text": "we have big diagonal again this time"}, {"start": 112.32, "duration": 4.6, "text": "it's equal to this diagonal here the 2 <"}, {"start": 114.24, "duration": 4.0, "text": "tk2 plus the r plus the < tk2 r and we"}, {"start": 116.92, "duration": 3.08, "text": "can put a box around that but we know"}, {"start": 118.24, "duration": 3.32, "text": "that the big diagonal has to equal the"}, {"start": 120.0, "duration": 3.32, "text": "big diagonal so we're just going to set"}, {"start": 121.56, "duration": 3.72, "text": "the 4 < tk2 for this side of the big"}, {"start": 123.32, "duration": 3.6, "text": "diagonal equal to all of this and now we"}, {"start": 125.28, "duration": 3.56, "text": "only have one variable R and one"}, {"start": 126.92, "duration": 4.0, "text": "equation so once we solve for R we know"}, {"start": 128.84, "duration": 3.88, "text": "the answer we can subtract 2 < tk2 from"}, {"start": 130.92, "duration": 3.8, "text": "both sides the left hand side becomes 2"}, {"start": 132.72, "duration": 3.239, "text": "< tk2 the right hand side that's gone so"}, {"start": 134.72, "duration": 3.44, "text": "it's just this right here and we can"}, {"start": 135.959, "duration": 4.841, "text": "scoot these over and let's Factor an R"}, {"start": 138.16, "duration": 5.6, "text": "out of the right hand side so R * what"}, {"start": 140.8, "duration": 5.48, "text": "gives us R that's one so R * 1 is r and"}, {"start": 143.76, "duration": 4.0, "text": "then R * what gives us < tk2 R that's"}, {"start": 146.28, "duration": 3.08, "text": "going to be < tk2 so if you imagine"}, {"start": 147.76, "duration": 3.479, "text": "Distributing this it would bring us back"}, {"start": 149.36, "duration": 3.599, "text": "to where we were up here so we factored"}, {"start": 151.239, "duration": 4.121, "text": "out the r then on the left hand side"}, {"start": 152.959, "duration": 4.081, "text": "leave that as 2 < tk2 the 1+ < tk2 is"}, {"start": 155.36, "duration": 3.519, "text": "just a number so we can divide both"}, {"start": 157.04, "duration": 5.72, "text": "sides by that number and we end up with"}, {"start": 158.879, "duration": 5.161, "text": "r being equal to 2 < tk2 over 1 + < tk2"}, {"start": 162.76, "duration": 4.199, "text": "and we can plug it into a calculator and"}, {"start": 164.04, "duration": 4.76, "text": "get a decimal approximation of 17157 and"}, {"start": 166.959, "duration": 3.2, "text": "that keeps going it's irrational and now"}, {"start": 168.8, "duration": 3.24, "text": "we have the answer to our question the"}, {"start": 170.159, "duration": 4.121, "text": "radius is approximately"}, {"start": 172.04, "duration": 4.8, "text": "17157 and we can put a box around that"}, {"start": 174.28, "duration": 2.56, "text": "how exciting"}], "title": "Old School Math", "description": "I hope you guys love this one!", "lengthSeconds": 175, "width": 360, "height": 640}
{"video_id": "rWSOA0H-cuU", "transcript": [{"start": 0.0, "duration": 3.36, "text": "hey guys this looks like a fun one we"}, {"start": 1.74, "duration": 3.12, "text": "have an equilateral triangle right here"}, {"start": 3.36, "duration": 3.899, "text": "we're given that the area of this"}, {"start": 4.86, "duration": 4.14, "text": "smaller circle is 105 meters squared and"}, {"start": 7.259, "duration": 3.601, "text": "they want us to find the area of the"}, {"start": 9.0, "duration": 3.059, "text": "larger Circle if you want to try this on"}, {"start": 10.86, "duration": 4.14, "text": "your own posit because I'm going to"}, {"start": 12.059, "duration": 4.441, "text": "solve it in three two one so first I'm"}, {"start": 15.0, "duration": 3.0, "text": "thinking since it's an equilateral"}, {"start": 16.5, "duration": 3.84, "text": "triangle we know that each of these"}, {"start": 18.0, "duration": 3.9, "text": "angles is equal to 60 degrees and if we"}, {"start": 20.34, "duration": 3.24, "text": "find the center of these circles and"}, {"start": 21.9, "duration": 3.6, "text": "drop the angle bisector to that point"}, {"start": 23.58, "duration": 4.019, "text": "it's going to cut the 60 degree angle"}, {"start": 25.5, "duration": 3.84, "text": "into two thirty degree angles and next"}, {"start": 27.599, "duration": 3.721, "text": "let's connect this radius from here to"}, {"start": 29.34, "duration": 4.08, "text": "here now let's label it R since it's the"}, {"start": 31.32, "duration": 3.72, "text": "radius of the smaller Circle this side"}, {"start": 33.42, "duration": 3.54, "text": "of the triangle is touching the circle"}, {"start": 35.04, "duration": 3.9, "text": "in one point that means this is part of"}, {"start": 36.96, "duration": 4.14, "text": "the tangent line and at this point of"}, {"start": 38.94, "duration": 4.439, "text": "tangency it meets at right angles to the"}, {"start": 41.1, "duration": 3.72, "text": "radius and now if we focus on this"}, {"start": 43.379, "duration": 3.541, "text": "triangle right here the sum of the"}, {"start": 44.82, "duration": 4.2, "text": "interior angles of any triangle is 180"}, {"start": 46.92, "duration": 4.2, "text": "degrees which leaves us 60 degrees for"}, {"start": 49.02, "duration": 4.199, "text": "the last angle so this right here is a"}, {"start": 51.12, "duration": 4.56, "text": "special right triangle it's a 30 60 90"}, {"start": 53.219, "duration": 4.741, "text": "triangle in every 30 60 90 triangle the"}, {"start": 55.68, "duration": 4.08, "text": "hypotenuse is always twice as long as"}, {"start": 57.96, "duration": 3.9, "text": "the smallest side so if we rotate these"}, {"start": 59.76, "duration": 4.139, "text": "no notes we can match this with the"}, {"start": 61.86, "duration": 4.2, "text": "orientation of this triangle up here so"}, {"start": 63.899, "duration": 4.141, "text": "once again the hypotenuse is twice as"}, {"start": 66.06, "duration": 4.08, "text": "long as the smallest side so for this"}, {"start": 68.04, "duration": 4.02, "text": "one this hypotenuse needs to be twice as"}, {"start": 70.14, "duration": 3.9, "text": "big as this side so this is going to be"}, {"start": 72.06, "duration": 4.02, "text": "2R and we don't need these notes anymore"}, {"start": 74.04, "duration": 3.78, "text": "let's take these two things and copy"}, {"start": 76.08, "duration": 3.66, "text": "them down here so let's focus on the"}, {"start": 77.82, "duration": 3.96, "text": "small circle area let's gray out all"}, {"start": 79.74, "duration": 3.78, "text": "this stuff the area of a circle is pi"}, {"start": 81.78, "duration": 3.78, "text": "times the radius squared well in this"}, {"start": 83.52, "duration": 3.599, "text": "case it's going to be pi r squared so"}, {"start": 85.56, "duration": 3.059, "text": "now we can focus on the largest Circle"}, {"start": 87.119, "duration": 3.36, "text": "so let's create out all of this stuff"}, {"start": 88.619, "duration": 3.421, "text": "there you have the large circle is also"}, {"start": 90.479, "duration": 4.32, "text": "going to be pi times the radius squared"}, {"start": 92.04, "duration": 4.32, "text": "or the radius for this one is 2R let's"}, {"start": 94.799, "duration": 3.601, "text": "get rid of the question mark we can"}, {"start": 96.36, "duration": 4.5, "text": "bring over the pi this 2 is going to get"}, {"start": 98.4, "duration": 4.32, "text": "squared to give us 4 and R is going to"}, {"start": 100.86, "duration": 4.439, "text": "be squared to give us r squared so now"}, {"start": 102.72, "duration": 5.16, "text": "we have pi times 4r squared we can swap"}, {"start": 105.299, "duration": 4.201, "text": "it to be 4 times pi r squared and we"}, {"start": 107.88, "duration": 3.66, "text": "already know that pi r squared is equal"}, {"start": 109.5, "duration": 4.32, "text": "to 105 meters squared so I'm going to"}, {"start": 111.54, "duration": 5.16, "text": "substitute in for this pi r squared 105"}, {"start": 113.82, "duration": 5.04, "text": "meters squared and then 4 times 105 is"}, {"start": 116.7, "duration": 4.8, "text": "420. so the area of this larger circle"}, {"start": 118.86, "duration": 4.02, "text": "is 420 meters squared and that's the"}, {"start": 121.5, "duration": 4.88, "text": "answer to our question we can put a box"}, {"start": 122.88, "duration": 3.5, "text": "around it how exciting"}], "title": "Find the Area of the Large Circle", "description": "I thought this was a fun geometry challenge!", "lengthSeconds": 125, "width": 640, "height": 360}
{"video_id": "Jdu5hwUDvrs", "transcript": [{"start": 0.04, "duration": 3.48, "text": "with the SAT coming up a lot of people"}, {"start": 1.64, "duration": 3.639, "text": "have asked me about this exact type of"}, {"start": 3.52, "duration": 3.72, "text": "question I found this cool video going"}, {"start": 5.279, "duration": 4.24, "text": "over an example the tricky thing here is"}, {"start": 7.24, "duration": 4.68, "text": "we want to get X alone so when we're"}, {"start": 9.519, "duration": 3.841, "text": "done we should have x equals something"}, {"start": 11.92, "duration": 4.199, "text": "the thing people struggle with is how do"}, {"start": 13.36, "duration": 4.24, "text": "we get X alone when there's two x's so"}, {"start": 16.119, "duration": 3.481, "text": "let's watch the video and see how it's"}, {"start": 17.6, "duration": 5.2, "text": "done hey guys this looks like a fun one"}, {"start": 19.6, "duration": 7.24, "text": "it says solve for x it's 2x over 3 Pi +"}, {"start": 22.8, "duration": 5.84, "text": "the quantity 1 - x/ 2 = 0 if you want to"}, {"start": 26.84, "duration": 5.08, "text": "try it on your own pause it right now"}, {"start": 28.64, "duration": 4.4, "text": "cuz I'm going to solve it in three 3 2 1"}, {"start": 31.92, "duration": 2.639, "text": "first thing I don't really like"}, {"start": 33.04, "duration": 3.16, "text": "fractions so let's get rid of these"}, {"start": 34.559, "duration": 3.561, "text": "denominators and we're going to do that"}, {"start": 36.2, "duration": 4.24, "text": "by multiplying everything by the least"}, {"start": 38.12, "duration": 5.48, "text": "common denominator in this case between"}, {"start": 40.44, "duration": 4.88, "text": "3 pi and 2 6 Pi is the least common"}, {"start": 43.6, "duration": 4.4, "text": "denominator so we're going to multiply"}, {"start": 45.32, "duration": 5.6, "text": "everything by 6 Pi for this first term 6"}, {"start": 48.0, "duration": 5.28, "text": "Pi * 2x is 12 pi x and it's still going"}, {"start": 50.92, "duration": 4.24, "text": "to be divided by 3 pi and then for this"}, {"start": 53.28, "duration": 3.56, "text": "part the 6 Pi is going to distribute to"}, {"start": 55.16, "duration": 3.96, "text": "both of these so we're going to have two"}, {"start": 56.84, "duration": 5.16, "text": "separate fractions divided by two for"}, {"start": 59.12, "duration": 4.999, "text": "the first one 6 piun * 1 is equal to 6"}, {"start": 62.0, "duration": 5.2, "text": "piun and then we're going to subtract"}, {"start": 64.119, "duration": 6.521, "text": "and 6 piun * X is 6 piun x and on the"}, {"start": 67.2, "duration": 5.12, "text": "right hand side 0 * 6 Pi is still 0er"}, {"start": 70.64, "duration": 4.0, "text": "now we can simplify things to get rid of"}, {"start": 72.32, "duration": 4.68, "text": "the denominators for this first term the"}, {"start": 74.64, "duration": 6.64, "text": "two pies will divide each other out and"}, {"start": 77.0, "duration": 7.24, "text": "12 / 3 is equal to 4 for this term 6 / 2"}, {"start": 81.28, "duration": 6.839, "text": "is equal to 3 and for this term 6 / 2 is"}, {"start": 84.24, "duration": 6.919, "text": "equal to 3 now we have 4x + 3 piun - 3"}, {"start": 88.119, "duration": 5.721, "text": "piun x = 0 since we're trying to solve"}, {"start": 91.159, "duration": 5.161, "text": "for x we want all the terms that contain"}, {"start": 93.84, "duration": 4.639, "text": "X on one side and all the terms that do"}, {"start": 96.32, "duration": 4.32, "text": "not contain X on the other side so we"}, {"start": 98.479, "duration": 4.201, "text": "want to move this 3 pi over to the other"}, {"start": 100.64, "duration": 4.6, "text": "side and we can do that by subtracting 3"}, {"start": 102.68, "duration": 6.2, "text": "PI from both sides we can copy this down"}, {"start": 105.24, "duration": 7.72, "text": "and simplify 3 Pi - 3 Pi will go to 0"}, {"start": 108.88, "duration": 6.519, "text": "and 0 - 3 Pi is - 3 Pi now on the left"}, {"start": 112.96, "duration": 5.56, "text": "hand side every term contains an X let's"}, {"start": 115.399, "duration": 5.76, "text": "factor out that X when we Factor an X"}, {"start": 118.52, "duration": 4.559, "text": "Out of 4X we're left with four and then"}, {"start": 121.159, "duration": 4.32, "text": "we can bring down the subtraction and"}, {"start": 123.079, "duration": 5.441, "text": "when we Factor an X out of 3 pi x we're"}, {"start": 125.479, "duration": 5.801, "text": "left with 3 pi and we're still equal to3"}, {"start": 128.52, "duration": 5.719, "text": "Pi now to get X by itself we can divide"}, {"start": 131.28, "duration": 4.959, "text": "both sides by 4 - 3 pi and let's copy"}, {"start": 134.239, "duration": 4.241, "text": "this step down and simplify it on the"}, {"start": 136.239, "duration": 4.161, "text": "left hand side the 4 - 3 Pi on top and"}, {"start": 138.48, "duration": 5.28, "text": "bottom will divide each other out we're"}, {"start": 140.4, "duration": 5.6, "text": "left with X is equal to -3 pi over 4 - 3"}, {"start": 143.76, "duration": 4.52, "text": "Pi this is probably fine as an answer"}, {"start": 146.0, "duration": 4.92, "text": "but you can still kind of simplify it on"}, {"start": 148.28, "duration": 6.4, "text": "top we can factor out a negative 1 so"}, {"start": 150.92, "duration": 7.12, "text": "instead of -3 Pi we have -1 * 3 pi and"}, {"start": 154.68, "duration": 7.199, "text": "on bottom we can also factor out a 1 so"}, {"start": 158.04, "duration": 7.08, "text": "1 * what gives us 4 well that's -4 and"}, {"start": 161.879, "duration": 5.681, "text": "then -1 * what gives us -3 Pi well"}, {"start": 165.12, "duration": 4.119, "text": "that's positive 3 Pi now this might look"}, {"start": 167.56, "duration": 3.84, "text": "even more complicated but these two"}, {"start": 169.239, "duration": 3.161, "text": "negative ones will divide each other out"}, {"start": 171.4, "duration": 3.36, "text": "and now we can get rid of these"}, {"start": 172.4, "duration": 4.32, "text": "parentheses and rearrange it like this"}, {"start": 174.76, "duration": 6.199, "text": "so I don't know which one do you prefer"}, {"start": 176.72, "duration": 5.96, "text": "-3 pi over 4 - 3 Pi or 3 pi over 3 Pi -"}, {"start": 180.959, "duration": 3.721, "text": "4 I think this one looks a little bit"}, {"start": 182.68, "duration": 4.08, "text": "neater let's put a box around it the"}, {"start": 184.68, "duration": 4.32, "text": "question asks us to solve for x so this"}, {"start": 186.76, "duration": 5.24, "text": "is the answer to our question how"}, {"start": 189.0, "duration": 3.0, "text": "exciting"}], "title": "Popular SAT Concept \"Solve for x.\"", "description": "I've had several people ask me about this exact type of question, I hope this helps a bunch of people!", "lengthSeconds": 190, "width": 640, "height": 360}
{"video_id": "K0vCIvJ71k8", "transcript": [{"start": 0.04, "duration": 4.92, "text": "hey guys this looks like a fun one it"}, {"start": 1.36, "duration": 6.28, "text": "gives us that x * Y is = 36 and that x^2"}, {"start": 4.96, "duration": 5.559, "text": "+ y^2 is equal 78 and it wants to know"}, {"start": 7.64, "duration": 4.76, "text": "what is x 4 + y 4th if you want to try"}, {"start": 10.519, "duration": 5.0, "text": "this on your own pause it right now cuz"}, {"start": 12.4, "duration": 4.639, "text": "I'm going to solve it in 3 2 1 in order"}, {"start": 15.519, "duration": 3.801, "text": "to figure this out we don't have to"}, {"start": 17.039, "duration": 4.32, "text": "completely solve for x and y there's a"}, {"start": 19.32, "duration": 4.279, "text": "pretty cool shortcut that we can apply"}, {"start": 21.359, "duration": 5.401, "text": "let me show you let's copy down the x^2"}, {"start": 23.599, "duration": 5.08, "text": "+ y^2 = 78 and let's Square both sides"}, {"start": 26.76, "duration": 3.439, "text": "of the equation the squares mean that"}, {"start": 28.679, "duration": 3.681, "text": "we're going to have this times this"}, {"start": 30.199, "duration": 5.04, "text": "equals this time this on the right hand"}, {"start": 32.36, "duration": 4.96, "text": "side 78 * 78 is equal to"}, {"start": 35.239, "duration": 4.121, "text": "684 and on the left hand side you can"}, {"start": 37.32, "duration": 7.559, "text": "multiply this out let's use one of these"}, {"start": 39.36, "duration": 8.519, "text": "squares x^2 * x^2 = x 4 x^2 * y^2 is ="}, {"start": 44.879, "duration": 5.721, "text": "x^2 y^2 and then we have another x^2 *"}, {"start": 47.879, "duration": 5.761, "text": "y^2 so that'll be x^2 y^2 and then we"}, {"start": 50.6, "duration": 4.56, "text": "have y^2 * y^2 which is equal to Y 4 and"}, {"start": 53.64, "duration": 3.52, "text": "these insides are what we would get if"}, {"start": 55.16, "duration": 5.8, "text": "we were to multiply it out next let's"}, {"start": 57.16, "duration": 6.6, "text": "focus on this x^2 y^2 x^2 y^2 is the"}, {"start": 60.96, "duration": 4.76, "text": "same thing as xy^ 2 and then we can copy"}, {"start": 63.76, "duration": 4.8, "text": "down everything else in the place of the"}, {"start": 65.72, "duration": 6.079, "text": "XY we're going to plug in 36 and then 2"}, {"start": 68.56, "duration": 6.76, "text": "* 36 * 36 is equal to"}, {"start": 71.799, "duration": 5.081, "text": "2592 we can subtract 2592 from both"}, {"start": 75.32, "duration": 3.2, "text": "sides on the left hand side these will"}, {"start": 76.88, "duration": 4.72, "text": "cancel each other out so we're left with"}, {"start": 78.52, "duration": 6.239, "text": "X 4th + y 4th and on the right hand side"}, {"start": 81.6, "duration": 5.199, "text": "684 - 2592 is"}, {"start": 84.759, "duration": 4.4, "text": "3492 and that's the answer to our"}, {"start": 86.799, "duration": 5.121, "text": "question let's put a box around it how"}, {"start": 89.159, "duration": 2.761, "text": "exciting"}], "title": "Cool Math Shortcut", "description": "I thought this was a neat shortcut, I hope you guys like this one!", "lengthSeconds": 90, "width": 640, "height": 360}
{"video_id": "_FIyKOWBC4s", "transcript": [{"start": 0.04, "duration": 4.0, "text": "hey guys this is a fun one we want to"}, {"start": 1.4, "duration": 4.08, "text": "find the area of this red region if you"}, {"start": 4.04, "duration": 3.44, "text": "want to try it on your own pause it"}, {"start": 5.48, "duration": 4.64, "text": "right now cuz I'm going to solve it in 3"}, {"start": 7.48, "duration": 5.48, "text": "2 one for the first step let's call this"}, {"start": 10.12, "duration": 4.679, "text": "red region X I'm going to find X by"}, {"start": 12.96, "duration": 3.399, "text": "tackling everything else around it we"}, {"start": 14.799, "duration": 3.921, "text": "have a square right here and another"}, {"start": 16.359, "duration": 3.76, "text": "Square here we have this triangle here"}, {"start": 18.72, "duration": 2.799, "text": "we have this region right here I don't"}, {"start": 20.119, "duration": 3.201, "text": "even know what you would call that and"}, {"start": 21.519, "duration": 3.6, "text": "then we have the same type of region"}, {"start": 23.32, "duration": 3.76, "text": "right here and then right here we have a"}, {"start": 25.119, "duration": 3.48, "text": "thing called a segment of a circle and"}, {"start": 27.08, "duration": 3.4, "text": "then up here we have another segment of"}, {"start": 28.599, "duration": 4.361, "text": "a circle so if we can find the area of"}, {"start": 30.48, "duration": 4.28, "text": "all of these blue pieces then we can add"}, {"start": 32.96, "duration": 3.72, "text": "that to X and set it equal to the whole"}, {"start": 34.76, "duration": 4.0, "text": "Square so first let's do these two"}, {"start": 36.68, "duration": 3.92, "text": "squares the sides of these squares are"}, {"start": 38.76, "duration": 3.799, "text": "equal to four and the area of a square"}, {"start": 40.6, "duration": 3.639, "text": "is equal to the side length squared so"}, {"start": 42.559, "duration": 4.281, "text": "we'll plug in four for the side length"}, {"start": 44.239, "duration": 4.84, "text": "and four squar is equal to 16 so both of"}, {"start": 46.84, "duration": 3.28, "text": "these have an area of 16 and now that"}, {"start": 49.079, "duration": 2.8, "text": "we've solved these let's make them"}, {"start": 50.12, "duration": 3.4, "text": "orange and let's put them back and next"}, {"start": 51.879, "duration": 3.2, "text": "let's look at this triangle both of"}, {"start": 53.52, "duration": 4.519, "text": "these sides are equal to four and the"}, {"start": 55.079, "duration": 4.44, "text": "area of a triangle is base * height / 2"}, {"start": 58.039, "duration": 3.721, "text": "we can plug in four for the base and"}, {"start": 59.519, "duration": 2.241, "text": "four"}], "title": "Full Video \ud83d\udc46", "description": "This was a popular video, I wanted to share it with more people on shorts. Click on the link to see the whole thing.", "lengthSeconds": 60, "width": 270, "height": 480}
{"video_id": "uumSD_sf2Ig", "transcript": [{"start": 0.0, "duration": 3.659, "text": "hey guys I'm pretty excited about this"}, {"start": 2.28, "duration": 3.12, "text": "one you know the three four five"}, {"start": 3.659, "duration": 3.421, "text": "triangles that we see everywhere three"}, {"start": 5.4, "duration": 3.48, "text": "four five triangles show up all over the"}, {"start": 7.08, "duration": 3.54, "text": "place in the SAT and high school"}, {"start": 8.88, "duration": 3.9, "text": "geometry courses even construction"}, {"start": 10.62, "duration": 3.72, "text": "workers use three four five triangles to"}, {"start": 12.78, "duration": 3.419, "text": "make sure that runes are built square"}, {"start": 14.34, "duration": 3.359, "text": "but there's something else special about"}, {"start": 16.199, "duration": 2.761, "text": "three four five triangles and I"}, {"start": 17.699, "duration": 3.24, "text": "genuinely think this is going to blow"}, {"start": 18.96, "duration": 3.78, "text": "some people's minds if you inscribe a"}, {"start": 20.939, "duration": 3.961, "text": "circle inside of any of those three four"}, {"start": 22.74, "duration": 5.4, "text": "five triangles guess what the area of"}, {"start": 24.9, "duration": 4.92, "text": "this circle is it's exactly Pi or Pi"}, {"start": 28.14, "duration": 3.539, "text": "unit squared do you want to see the math"}, {"start": 29.82, "duration": 3.78, "text": "first let's find the center of this"}, {"start": 31.679, "duration": 4.261, "text": "circle and from the center to this point"}, {"start": 33.6, "duration": 3.66, "text": "of tangency let's draw a radius and"}, {"start": 35.94, "duration": 3.779, "text": "let's do the same thing for this point"}, {"start": 37.26, "duration": 4.44, "text": "of tangency so radius will always be at"}, {"start": 39.719, "duration": 3.781, "text": "right angles to the tangent line and"}, {"start": 41.7, "duration": 4.26, "text": "same thing for this one to this tangent"}, {"start": 43.5, "duration": 4.26, "text": "line and this forms a quadrilateral and"}, {"start": 45.96, "duration": 3.66, "text": "since we have three right angles the"}, {"start": 47.76, "duration": 4.08, "text": "fourth one has to also be a right angle"}, {"start": 49.62, "duration": 4.2, "text": "we have two adjacent sides congruent"}, {"start": 51.84, "duration": 4.14, "text": "that means this whole quadrilateral is a"}, {"start": 53.82, "duration": 4.44, "text": "square so this side will also be R and"}, {"start": 55.98, "duration": 4.259, "text": "this side will also be R this entire"}, {"start": 58.26, "duration": 4.5, "text": "side is three and this Porsche question"}, {"start": 60.239, "duration": 4.621, "text": "is are this last piece is going to be 3"}, {"start": 62.76, "duration": 4.44, "text": "minus r and the same thing here this"}, {"start": 64.86, "duration": 4.079, "text": "base has a length of 4 and this portion"}, {"start": 67.2, "duration": 3.959, "text": "of it is R which means the remaining"}, {"start": 68.939, "duration": 4.201, "text": "part is going to be 4 minus r now let's"}, {"start": 71.159, "duration": 4.681, "text": "gray out everything except for the"}, {"start": 73.14, "duration": 4.56, "text": "circle and these two tangent lines if"}, {"start": 75.84, "duration": 3.959, "text": "you have the intersection of two tangent"}, {"start": 77.7, "duration": 4.02, "text": "lines of a circle the distance from here"}, {"start": 79.799, "duration": 3.96, "text": "to here is equal to the distance from"}, {"start": 81.72, "duration": 4.92, "text": "here to here so this will also be 3"}, {"start": 83.759, "duration": 4.441, "text": "minus r and same thing down here these"}, {"start": 86.64, "duration": 2.94, "text": "two tangent lines are intersecting at"}, {"start": 88.2, "duration": 3.54, "text": "this point which means this is going to"}, {"start": 89.58, "duration": 4.74, "text": "have the same distance as this so this"}, {"start": 91.74, "duration": 5.339, "text": "will also be 4 minus r and let's focus"}, {"start": 94.32, "duration": 5.759, "text": "on the hypotenuse we have this portion"}, {"start": 97.079, "duration": 5.341, "text": "here which is three minus r plus this"}, {"start": 100.079, "duration": 3.72, "text": "portion here which is 4 minus r and"}, {"start": 102.42, "duration": 3.6, "text": "that's all going to be equal to the"}, {"start": 103.799, "duration": 4.5, "text": "whole thing which was 5. now we can"}, {"start": 106.02, "duration": 4.08, "text": "solve for R let's subtract 3 from both"}, {"start": 108.299, "duration": 3.541, "text": "sides and let's subtract 4 from both"}, {"start": 110.1, "duration": 3.299, "text": "sides these threes will cancel each"}, {"start": 111.84, "duration": 3.72, "text": "other out these fours will cancel each"}, {"start": 113.399, "duration": 4.381, "text": "other out and we have negative R minus r"}, {"start": 115.56, "duration": 3.66, "text": "which is negative two R and then to"}, {"start": 117.78, "duration": 3.54, "text": "figure out the right hand side we can"}, {"start": 119.22, "duration": 5.1, "text": "combine the negative 3 minus 4 to get"}, {"start": 121.32, "duration": 4.86, "text": "negative 7 and 5 minus 7 is negative 2."}, {"start": 124.32, "duration": 3.779, "text": "now to solve for R we can divide both"}, {"start": 126.18, "duration": 4.499, "text": "sides by negative 2 and that gives us"}, {"start": 128.099, "duration": 3.961, "text": "our radius equal to one and this is"}, {"start": 130.679, "duration": 3.001, "text": "definitely important let's put a box"}, {"start": 132.06, "duration": 2.819, "text": "around it so now let's clean up this"}, {"start": 133.68, "duration": 2.82, "text": "triangle let's bring it back to the"}, {"start": 134.879, "duration": 3.601, "text": "three four five triangle and we want to"}, {"start": 136.5, "duration": 4.2, "text": "find the area of the inscribed circle of"}, {"start": 138.48, "duration": 4.14, "text": "this 345 Triangle Well the area of a"}, {"start": 140.7, "duration": 4.98, "text": "circle is pi r squared and then we could"}, {"start": 142.62, "duration": 4.86, "text": "plug in 1 for this R and then 1 squared"}, {"start": 145.68, "duration": 3.66, "text": "is just one and then multiplying by 1"}, {"start": 147.48, "duration": 4.259, "text": "doesn't do anything so we know that the"}, {"start": 149.34, "duration": 4.14, "text": "area of this circle is equal to Pi and"}, {"start": 151.739, "duration": 5.36, "text": "this is true for every three four five"}, {"start": 153.48, "duration": 3.619, "text": "triangle how exciting"}], "title": "3-4-5 Triangles and Pi", "description": "", "lengthSeconds": 155, "width": 640, "height": 360}
{"video_id": "bX-f0vZFYjo", "transcript": [{"start": 0.04, "duration": 3.8, "text": "hey guys this looks like a fun one it"}, {"start": 1.48, "duration": 3.879, "text": "says find the area of the square now"}, {"start": 3.84, "duration": 3.88, "text": "we're told that each of the rectangles"}, {"start": 5.359, "duration": 3.401, "text": "inside of the square have equal areas if"}, {"start": 7.72, "duration": 2.52, "text": "you're going to try this on your own"}, {"start": 8.76, "duration": 5.2, "text": "pause it right now cuz I'm going to"}, {"start": 10.24, "duration": 5.68, "text": "solve it in 3 2 1 since the areas of all"}, {"start": 13.96, "duration": 3.64, "text": "five rectangles are equal to each other"}, {"start": 15.92, "duration": 4.119, "text": "that means that each of these rectangles"}, {"start": 17.6, "duration": 4.32, "text": "has an area that's 1 of the entire"}, {"start": 20.039, "duration": 4.121, "text": "Square so if I make the square something"}, {"start": 21.92, "duration": 4.279, "text": "like X by X the entire square is going"}, {"start": 24.16, "duration": 4.72, "text": "to be x^2 and then each of these"}, {"start": 26.199, "duration": 4.801, "text": "rectangles would be x^2 / 5 so that"}, {"start": 28.88, "duration": 3.76, "text": "feels like a pretty strong first step"}, {"start": 31.0, "duration": 3.559, "text": "for the next step out of all of these"}, {"start": 32.64, "duration": 4.599, "text": "rectangles the blue one is the only one"}, {"start": 34.559, "duration": 4.52, "text": "that has X entirely as one of its sides"}, {"start": 37.239, "duration": 3.64, "text": "so let's focus on that one we already"}, {"start": 39.079, "duration": 5.16, "text": "know that the height is X let's call the"}, {"start": 40.879, "duration": 5.481, "text": "base B we know that the base of B times"}, {"start": 44.239, "duration": 5.121, "text": "the height of X is going to be equal to"}, {"start": 46.36, "duration": 5.719, "text": "the area x^2 5 and I want to know what"}, {"start": 49.36, "duration": 4.8, "text": "is B in terms of X so let's divide both"}, {"start": 52.079, "duration": 3.64, "text": "sides of the equation by X on the left"}, {"start": 54.16, "duration": 3.32, "text": "hand side these will cancel so we just"}, {"start": 55.719, "duration": 4.441, "text": "have B and on the right hand side you"}, {"start": 57.48, "duration": 4.28, "text": "can think of x^2 as x * X and then one"}, {"start": 60.16, "duration": 3.399, "text": "of these X's will cancel from top and"}, {"start": 61.76, "duration": 5.28, "text": "bottom so that means B is equal to X"}, {"start": 63.559, "duration": 5.801, "text": "over 5 or 1 x so let's change this B to"}, {"start": 67.04, "duration": 4.96, "text": "1/5 X and the same thing for the top it"}, {"start": 69.36, "duration": 4.84, "text": "will also be/ X and now let's bring back"}, {"start": 72.0, "duration": 4.6, "text": "the rest of our Square each side of this"}, {"start": 74.2, "duration": 4.84, "text": "square is equal to X so if this blue"}, {"start": 76.6, "duration": 5.159, "text": "portion is 1/5 X that means that the"}, {"start": 79.04, "duration": 4.0, "text": "orange portion is going to be 4 fths x"}, {"start": 81.759, "duration": 3.641, "text": "and now we have all the parts of this"}, {"start": 83.04, "duration": 3.759, "text": "orange rectangle so let's focus on that"}, {"start": 85.4, "duration": 3.24, "text": "and then once again base time height"}, {"start": 86.799, "duration": 5.121, "text": "will equal the area so we know that the"}, {"start": 88.64, "duration": 6.28, "text": "base of 4S x * the height of 3 will"}, {"start": 91.92, "duration": 4.559, "text": "equal the area of x^2 5 so on the left"}, {"start": 94.92, "duration": 4.28, "text": "hand side let's write this as a single"}, {"start": 96.479, "duration": 4.92, "text": "fraction on top we can do 4 * x * 3"}, {"start": 99.2, "duration": 4.44, "text": "which gives us 12x and on bottom we just"}, {"start": 101.399, "duration": 4.76, "text": "have the 5 and this is still equal to"}, {"start": 103.64, "duration": 5.079, "text": "x^2 over 5 next we can multiply both"}, {"start": 106.159, "duration": 4.361, "text": "sides by 5 overx and we are allowed to"}, {"start": 108.719, "duration": 3.881, "text": "divide by X here because we know X is"}, {"start": 110.52, "duration": 3.72, "text": "not equal to zero on the left hand side"}, {"start": 112.6, "duration": 3.68, "text": "these two fives cancel each other out"}, {"start": 114.24, "duration": 4.04, "text": "and these two x's cancel each other out"}, {"start": 116.28, "duration": 3.4, "text": "all we have left is 12 and on the right"}, {"start": 118.28, "duration": 3.72, "text": "hand side these two fives cancel each"}, {"start": 119.68, "duration": 4.6, "text": "other each other out and x^2 overx is"}, {"start": 122.0, "duration": 5.2, "text": "just X so now we know that 12 is equal"}, {"start": 124.28, "duration": 4.64, "text": "to X going back to our original Square X"}, {"start": 127.2, "duration": 4.36, "text": "was the side lengths of the square so"}, {"start": 128.92, "duration": 4.12, "text": "this square is a 12x 12 square so to"}, {"start": 131.56, "duration": 3.959, "text": "find the area of this square is just"}, {"start": 133.04, "duration": 4.68, "text": "going to be 12 squar and 12 squ is equal"}, {"start": 135.519, "duration": 4.08, "text": "to 144 and let's give it a label of"}, {"start": 137.72, "duration": 3.84, "text": "square units and this is the answer to"}, {"start": 139.599, "duration": 4.881, "text": "our question the given square has an"}, {"start": 141.56, "duration": 5.28, "text": "area of 144 square units let's put a box"}, {"start": 144.48, "duration": 4.119, "text": "around it how exciting if you guys want"}, {"start": 146.84, "duration": 3.64, "text": "to try more of these geometry challenge"}, {"start": 148.599, "duration": 6.121, "text": "problems I have this page on mat.com"}, {"start": 150.48, "duration": 4.24, "text": "there's a link in the description"}], "title": "Multicolor Square Problem", "description": "https://andymath.com/geometry-challenges/\n\nFor more geometry challenges, check out the above link! Let me know what you think!", "lengthSeconds": 153, "width": 640, "height": 360}
{"video_id": "43gbcKa3-Jo", "transcript": [{"start": 0.0, "duration": 3.419, "text": "that wants to know what is the value of"}, {"start": 1.56, "duration": 3.84, "text": "x and what is the value of each of these"}, {"start": 3.419, "duration": 3.48, "text": "other three angles we can do this the"}, {"start": 5.4, "duration": 3.659, "text": "sum of the interior angles of any"}, {"start": 6.899, "duration": 4.261, "text": "four-sided figure is always 360 degrees"}, {"start": 9.059, "duration": 4.08, "text": "so we can just say this angle plus this"}, {"start": 11.16, "duration": 5.28, "text": "angle plus this angle plus this angle is"}, {"start": 13.139, "duration": 5.701, "text": "equal to 360. and then X Plus 3x plus 2X"}, {"start": 16.44, "duration": 4.02, "text": "is equal to 6X and then all these"}, {"start": 18.84, "duration": 4.32, "text": "numbers are going to combine to give us"}, {"start": 20.46, "duration": 5.64, "text": "negative 2154 and it's still equal to"}, {"start": 23.16, "duration": 5.82, "text": "360. if we add 2154 to both sides we get"}, {"start": 26.1, "duration": 4.56, "text": "6X on the left hand side and 25 14 on"}, {"start": 28.98, "duration": 5.099, "text": "the right hand side divide both sides by"}, {"start": 30.66, "duration": 4.86, "text": "6 and we get x equals 419 and we can"}, {"start": 34.079, "duration": 3.121, "text": "plug the X in up here so now that we"}, {"start": 35.52, "duration": 3.6, "text": "know what x is we can plug that in for"}, {"start": 37.2, "duration": 3.72, "text": "each of these other angles copy down the"}, {"start": 39.12, "duration": 5.759, "text": "values for each of the angles plug in"}, {"start": 40.92, "duration": 9.0, "text": "419 for the X's 3 times 419 is 1257 and"}, {"start": 44.879, "duration": 9.481, "text": "2 times 419 is 838 419 minus 352 is 67."}, {"start": 49.92, "duration": 8.6, "text": "1257 minus 1189 is 68. and 838 minus 768"}, {"start": 54.36, "duration": 4.16, "text": "is 70. how exciting"}], "title": "Nice Quadrilateral #shorts", "description": "https://andymath.com/\n\n#shorts", "lengthSeconds": 57, "width": 270, "height": 480}
{"video_id": "4d_UdbtMGNo", "transcript": [{"start": 0.08, "duration": 4.88, "text": "now that we've made the bases match we"}, {"start": 2.0, "duration": 5.24, "text": "can drop the base Drop the"}, {"start": 4.96, "duration": 4.44, "text": "Base and we can set the exponents equal"}, {"start": 7.24, "duration": 4.04, "text": "to each other and solve for x since the"}, {"start": 9.4, "duration": 3.6, "text": "Bas is matched the only way these two"}, {"start": 11.28, "duration": 3.64, "text": "sides can be equal to each other is if"}, {"start": 13.0, "duration": 4.68, "text": "the exponents are equal to each other so"}, {"start": 14.92, "duration": 5.48, "text": "in other words we can just drop the"}, {"start": 17.68, "duration": 5.0, "text": "base and that gives us 2x is equal to"}, {"start": 20.4, "duration": 3.92, "text": "138 the only way these two sides of the"}, {"start": 22.68, "duration": 3.48, "text": "equation are going to be equal is if the"}, {"start": 24.32, "duration": 4.64, "text": "exponents are equal so we can just drop"}, {"start": 26.16, "duration": 5.439, "text": "the base Drop the"}, {"start": 28.96, "duration": 4.279, "text": "Base"}, {"start": 31.599, "duration": 3.12, "text": "now since both these have the same base"}, {"start": 33.239, "duration": 3.521, "text": "of two the only way they're going to be"}, {"start": 34.719, "duration": 3.771, "text": "equal is if these exponents are equal or"}, {"start": 36.76, "duration": 7.16, "text": "in other words we can drop the"}, {"start": 38.49, "duration": 11.229, "text": "[Music]"}, {"start": 43.92, "duration": 5.799, "text": "base so we end up with 4x + 1 = 9x - 3"}], "title": "Drop the Base Compilation", "description": "I thought it was fun to see how these changed over time.", "lengthSeconds": 48, "width": 360, "height": 640}
{"video_id": "k4QjXIAr4W8", "transcript": [{"start": 0.04, "duration": 4.08, "text": "hey guys this looks like a fun one we're"}, {"start": 1.4, "duration": 4.52, "text": "trying to find X the area of this red"}, {"start": 4.12, "duration": 3.199, "text": "region if you want to try it on your own"}, {"start": 5.92, "duration": 4.839, "text": "pause it right now cuz I'm going to"}, {"start": 7.319, "duration": 6.161, "text": "solve it in 3 2 1 Let's copy it down and"}, {"start": 10.759, "duration": 4.96, "text": "let's focus on this left semicircle"}, {"start": 13.48, "duration": 4.559, "text": "let's cut it in half and split this into"}, {"start": 15.719, "duration": 5.001, "text": "five and five and then let's focus on"}, {"start": 18.039, "duration": 4.881, "text": "this top semicircle and cut it in half"}, {"start": 20.72, "duration": 4.2, "text": "and make this five and five so now we've"}, {"start": 22.92, "duration": 4.119, "text": "split our white region up into these"}, {"start": 24.92, "duration": 4.599, "text": "three pieces this one right here is a"}, {"start": 27.039, "duration": 4.961, "text": "square of 5 * 5 which means the area is"}, {"start": 29.519, "duration": 4.081, "text": "25 and this blue portion right here was"}, {"start": 32.0, "duration": 3.879, "text": "half of the semicircle or in other words"}, {"start": 33.6, "duration": 5.84, "text": "a quarter Circle the area of the quarter"}, {"start": 35.879, "duration": 5.68, "text": "circle is PK r^ 2 / 4 and the radius of"}, {"start": 39.44, "duration": 4.599, "text": "this quarter circle is equal to 5 so we"}, {"start": 41.559, "duration": 4.721, "text": "can plug in five for the r and 5^ s is"}, {"start": 44.039, "duration": 5.441, "text": "equal to 25 so the area of the blue"}, {"start": 46.28, "duration": 4.759, "text": "region is 25 pi over4 and this quarter"}, {"start": 49.48, "duration": 4.2, "text": "Circle right here is the exact same so"}, {"start": 51.039, "duration": 4.921, "text": "it's also 25 pi over4 and now we know"}, {"start": 53.68, "duration": 4.839, "text": "the area of our white region we can just"}, {"start": 55.96, "duration": 6.239, "text": "add these three things together and 25i"}, {"start": 58.519, "duration": 6.64, "text": "4 + 25i 4 4 would be 50 piun 4 we can"}, {"start": 62.199, "duration": 5.721, "text": "simplify this to 25 piun / 2 this is the"}, {"start": 65.159, "duration": 4.561, "text": "area of our white region and if we add X"}, {"start": 67.92, "duration": 5.44, "text": "to this that'll give us the area of the"}, {"start": 69.72, "duration": 5.2, "text": "entire Square which is 10x 10 or 100 now"}, {"start": 73.36, "duration": 3.6, "text": "we just got to solve for x we can"}, {"start": 74.92, "duration": 4.239, "text": "subtract 25 from both sides these two"}, {"start": 76.96, "duration": 5.36, "text": "will cancel each other out and 100 minus"}, {"start": 79.159, "duration": 5.401, "text": "25 is 75 then we can subtract 25 pi over"}, {"start": 82.32, "duration": 3.96, "text": "2 from both sides on the left hand side"}, {"start": 84.56, "duration": 5.36, "text": "these will cancel out and on the right"}, {"start": 86.28, "duration": 4.92, "text": "hand side we'll have 75 - 25 pi over 2"}, {"start": 89.92, "duration": 2.92, "text": "and that's the answer to our question"}, {"start": 91.2, "duration": 3.16, "text": "let's give it a label of square units"}, {"start": 92.84, "duration": 4.4, "text": "and put a box around it and the"}, {"start": 94.36, "duration": 7.2, "text": "approximate value of this is 35.7 3"}, {"start": 97.24, "duration": 4.32, "text": "square units how exciting"}], "title": "Find the Red Area Area", "description": "https://andymath.com/geometry-challenges/", "lengthSeconds": 100, "width": 640, "height": 360}
{"video_id": "2hYKUSY0iso", "transcript": [{"start": 0.04, "duration": 3.56, "text": "hey guys somebody sent me this request"}, {"start": 2.32, "duration": 2.92, "text": "and I kind of like it it looks like a"}, {"start": 3.6, "duration": 3.4, "text": "fun one and I was a little bit curious"}, {"start": 5.24, "duration": 3.16, "text": "how I would visualize this so I'm going"}, {"start": 7.0, "duration": 3.16, "text": "to pull them out of here so we have the"}, {"start": 8.4, "duration": 3.8, "text": "red bar and the green bar and then we"}, {"start": 10.16, "duration": 3.32, "text": "also have this purple wall these are the"}, {"start": 12.2, "duration": 3.16, "text": "Shadows that they're making and we're"}, {"start": 13.48, "duration": 3.92, "text": "given all of these measurements the"}, {"start": 15.36, "duration": 3.839, "text": "green box is 2 m high with the shadow of"}, {"start": 17.4, "duration": 3.799, "text": "three along the ground this Shadow"}, {"start": 19.199, "duration": 4.041, "text": "measures 6 M and up the wall it measures"}, {"start": 21.199, "duration": 3.641, "text": "4 M now we're trying to solve for the"}, {"start": 23.24, "duration": 3.879, "text": "height of the red bar so let's call that"}, {"start": 24.84, "duration": 4.599, "text": "X we don't need most of this wall let's"}, {"start": 27.119, "duration": 4.08, "text": "smush it down and we can imagine if the"}, {"start": 29.439, "duration": 3.441, "text": "wall was isn't there the shadow would go"}, {"start": 31.199, "duration": 3.921, "text": "about this far but we don't know how far"}, {"start": 32.88, "duration": 4.519, "text": "this is so let's give it a variable y"}, {"start": 35.12, "duration": 4.4, "text": "let's focus on this part and this part"}, {"start": 37.399, "duration": 5.241, "text": "right here we can set up a proportion we"}, {"start": 39.52, "duration": 5.08, "text": "can say that 2 over 3 is equal to 4 over"}, {"start": 42.64, "duration": 4.16, "text": "y because the Shadows are proportional"}, {"start": 44.6, "duration": 4.72, "text": "to the heights of the objects and now to"}, {"start": 46.8, "duration": 4.36, "text": "solve for y we can cross multiply 2 * y"}, {"start": 49.32, "duration": 4.64, "text": "will give us 2 Y and that's going to be"}, {"start": 51.16, "duration": 5.0, "text": "equal to 3 * 4 which is 12 and then we"}, {"start": 53.96, "duration": 4.52, "text": "can divide both sides by two and we get"}, {"start": 56.16, "duration": 4.84, "text": "Y is equal to 6 so we can update this y"}, {"start": 58.48, "duration": 4.44, "text": "to be 6 M we know from here to here the"}, {"start": 61.0, "duration": 3.799, "text": "shadow is 6 M and then from here to here"}, {"start": 62.92, "duration": 4.12, "text": "it's 6 M the whole thing is going to be"}, {"start": 64.799, "duration": 4.281, "text": "12 M so now we no longer need this thing"}, {"start": 67.04, "duration": 5.36, "text": "and we can set up another proportion the"}, {"start": 69.08, "duration": 4.48, "text": "2 over 3 will equal x over 12 and now to"}, {"start": 72.4, "duration": 3.84, "text": "solve for x we're going to cross"}, {"start": 73.56, "duration": 6.12, "text": "multiply 3 * X is 3x and that's going to"}, {"start": 76.24, "duration": 5.36, "text": "be equal to 2 * 12 which is 24 and then"}, {"start": 79.68, "duration": 5.399, "text": "we can divide both sides by 3 and we get"}, {"start": 81.6, "duration": 5.36, "text": "x = 24 over 3 which is 8 x was the"}, {"start": 85.079, "duration": 4.241, "text": "height of our red bar so we can change"}, {"start": 86.96, "duration": 4.08, "text": "this to 8 m and now we're done that's"}, {"start": 89.32, "duration": 5.759, "text": "the answer to our question let's put a"}, {"start": 91.04, "duration": 4.039, "text": "box around it how exciting"}], "title": "Shadow Math", "description": "I am enjoying the longer shorts. I hope you guys love this one!", "lengthSeconds": 93, "width": 360, "height": 640}
{"video_id": "MCPpctAGcIU", "transcript": [{"start": 0.0, "duration": 3.659, "text": "okay so this says solve for x if you"}, {"start": 2.52, "duration": 2.4, "text": "want to try this on your own positive"}, {"start": 3.659, "duration": 3.361, "text": "because I'm going to solve it in three"}, {"start": 4.92, "duration": 3.84, "text": "two one for the first step let's copy it"}, {"start": 7.02, "duration": 3.9, "text": "down and then next let's multiply both"}, {"start": 8.76, "duration": 3.48, "text": "sides by 10. on left hand side this 10"}, {"start": 10.92, "duration": 2.639, "text": "and this 10 will cancel each other out"}, {"start": 12.24, "duration": 4.139, "text": "and on the right hand side this will"}, {"start": 13.559, "duration": 5.161, "text": "just be 10 x to the 21st next let's"}, {"start": 16.379, "duration": 4.681, "text": "subtract 10x the 21st from both sides"}, {"start": 18.72, "duration": 5.58, "text": "let's bring this down and 1X to the 21"}, {"start": 21.06, "duration": 5.52, "text": "minus 10x to the 21 is equal to negative"}, {"start": 24.3, "duration": 4.08, "text": "9 x to the 21 and the right hand side is"}, {"start": 26.58, "duration": 3.3, "text": "now equal to zero from here let's factor"}, {"start": 28.38, "duration": 4.019, "text": "out the greatest common factor it's"}, {"start": 29.88, "duration": 5.46, "text": "going to be x to the 21. now x to the 21"}, {"start": 32.399, "duration": 4.981, "text": "times what gives us x to the 22 that's X"}, {"start": 35.34, "duration": 4.559, "text": "that will bring down the subtract then x"}, {"start": 37.38, "duration": 4.26, "text": "to the 21 times what gives us 9x to the"}, {"start": 39.899, "duration": 3.601, "text": "21. well that's going to be 9. here we"}, {"start": 41.64, "duration": 4.62, "text": "can set each of these equal to zero x to"}, {"start": 43.5, "duration": 4.739, "text": "the 21 equals 0 or x minus 9 equals zero"}, {"start": 46.26, "duration": 4.38, "text": "for this one if we take the 21st root of"}, {"start": 48.239, "duration": 4.261, "text": "both sides we get x equals zero and for"}, {"start": 50.64, "duration": 3.899, "text": "this one if we add 9 to both sides we"}, {"start": 52.5, "duration": 5.84, "text": "get x equals nine and these are the"}, {"start": 54.539, "duration": 3.801, "text": "solutions to the equation how exciting"}], "title": "New Math Challenge Problem #shorts", "description": "https://andymath.com/\n\n#shorts", "lengthSeconds": 56, "width": 270, "height": 480}
{"video_id": "_Qw9gicq0CU", "transcript": [{"start": 0.03, "duration": 6.57, "text": "hey guys welcome to this product rule"}, {"start": 1.74, "duration": 8.43, "text": "video use the product rule to find the"}, {"start": 6.6, "duration": 7.32, "text": "derivative of f of X equals x cubed plus"}, {"start": 10.17, "duration": 5.639, "text": "two times x squared minus one so the"}, {"start": 13.92, "duration": 5.04, "text": "product rule is written right here so f"}, {"start": 15.809, "duration": 5.72, "text": "prime of X is gonna be equal to we bring"}, {"start": 18.96, "duration": 5.22, "text": "down the first function X cubed plus two"}, {"start": 21.529, "duration": 5.021, "text": "and we take the derivative of the second"}, {"start": 24.18, "duration": 4.56, "text": "one derivative of this is 2x plus"}, {"start": 26.55, "duration": 4.68, "text": "because of this plus right here bring"}, {"start": 28.74, "duration": 4.56, "text": "down the second x squared minus one"}, {"start": 31.23, "duration": 4.2, "text": "times the derivative of the first which"}, {"start": 33.3, "duration": 3.57, "text": "is 3x squared so that's the first step"}, {"start": 35.43, "duration": 3.57, "text": "some teachers will accept this as an"}, {"start": 36.87, "duration": 5.64, "text": "answer but we can simplify it distribute"}, {"start": 39.0, "duration": 7.92, "text": "the 2x so we get 2x to the fourth plus"}, {"start": 42.51, "duration": 7.049, "text": "4x plus and then distribute this 3x"}, {"start": 46.92, "duration": 6.15, "text": "squared which will give us 3x to the"}, {"start": 49.559, "duration": 5.551, "text": "fourth minus 3x squared and now we can"}, {"start": 53.07, "duration": 4.079, "text": "just combine like terms 2x to the fourth"}, {"start": 55.11, "duration": 6.12, "text": "plus 3x to the fourth is 5x to the"}, {"start": 57.149, "duration": 5.941, "text": "fourth minus 3x squared plus 4x and"}, {"start": 61.23, "duration": 3.719, "text": "that's the answer to the question so if"}, {"start": 63.09, "duration": 4.17, "text": "this makes sense if not I've linked down"}, {"start": 64.949, "duration": 4.561, "text": "below two more product rule practice on"}, {"start": 67.26, "duration": 5.1, "text": "Andy math calm feel free to check it out"}, {"start": 69.51, "duration": 4.84, "text": "please like this video subscribe to my"}, {"start": 72.36, "duration": 8.939, "text": "channel I'll talk to you guys soon bye"}, {"start": 74.35, "duration": 6.949, "text": "[Music]"}], "title": "Product Rule- Derivatives", "description": "For more visit http://andymath.com.\n\nSubscribe here: https://www.youtube.com/channel/UC6KhU3AMLHC-qv7tj0ZJduw?sub_confirmation=1", "lengthSeconds": 80, "width": 640, "height": 360}
{"video_id": "z8TGICpKDAg", "transcript": [{"start": 0.04, "duration": 3.48, "text": "hey guys this looks like a fun one these"}, {"start": 1.599, "duration": 4.001, "text": "A's and B's are placeholders so this"}, {"start": 3.52, "duration": 4.239, "text": "doesn't mean a * a it means some"}, {"start": 5.6, "duration": 4.52, "text": "matching double digit number like 22 or"}, {"start": 7.759, "duration": 4.721, "text": "33 now we're trying to solve for A and B"}, {"start": 10.12, "duration": 3.479, "text": "so that this plus this equals this we"}, {"start": 12.48, "duration": 2.399, "text": "probably could guess and check but I"}, {"start": 13.599, "duration": 3.76, "text": "want to see if we can figure it out"}, {"start": 14.879, "duration": 4.32, "text": "logically since the a plus b gives us a"}, {"start": 17.359, "duration": 5.68, "text": "three in the one spot we know that"}, {"start": 19.199, "duration": 6.121, "text": "either a plus b = 3 or a plus b = 13"}, {"start": 23.039, "duration": 4.881, "text": "let's explore these possibilities if a"}, {"start": 25.32, "duration": 4.959, "text": "plus b equals 3 that means this a plus"}, {"start": 27.92, "duration": 4.08, "text": "this B would be 3 and then in the 10"}, {"start": 30.279, "duration": 3.841, "text": "place the a plus b would also give us a"}, {"start": 32.0, "duration": 3.52, "text": "three in the 10's place in this case"}, {"start": 34.12, "duration": 3.24, "text": "nothing will carry over so in the"}, {"start": 35.52, "duration": 3.24, "text": "hundred's place we have a zero but we"}, {"start": 37.36, "duration": 3.28, "text": "know from up here the hundred's place"}, {"start": 38.76, "duration": 4.24, "text": "needs to have a one so it is not"}, {"start": 40.64, "duration": 4.16, "text": "possible for a plus b to equal 3 so we"}, {"start": 43.0, "duration": 4.16, "text": "know for sure that a plus b is equal to"}, {"start": 44.8, "duration": 3.919, "text": "13 so that means when we do a plus b"}, {"start": 47.16, "duration": 3.559, "text": "we're going to be carrying a one for the"}, {"start": 48.719, "duration": 5.441, "text": "10's place and now let's focus on the"}, {"start": 50.719, "duration": 7.16, "text": "10's column this one represents 10 this"}, {"start": 54.16, "duration": 4.84, "text": "a represents 10 a this B represents 10 B"}, {"start": 57.879, "duration": 3.84, "text": "and then we're going to add all those"}, {"start": 59.0, "duration": 5.199, "text": "together and set it equal to this 1"}, {"start": 61.719, "duration": 4.561, "text": "which represents 100 plus this a which"}, {"start": 64.199, "duration": 4.001, "text": "represents 10 a and now we want to solve"}, {"start": 66.28, "duration": 4.04, "text": "this system of equations let's subtract"}, {"start": 68.2, "duration": 4.12, "text": "10 from both sides and let's subtract 10"}, {"start": 70.32, "duration": 3.64, "text": "a from both sides on the left hand side"}, {"start": 72.32, "duration": 3.72, "text": "this stuff all cancels out so we just"}, {"start": 73.96, "duration": 3.6, "text": "have 10 B and on the right hand side"}, {"start": 76.04, "duration": 4.0, "text": "this stuff all cancels out so we just"}, {"start": 77.56, "duration": 4.68, "text": "have 90 after we divide both sides by 10"}, {"start": 80.04, "duration": 4.0, "text": "we get B is equal to 9 and that's"}, {"start": 82.24, "duration": 4.0, "text": "important so let's put a box around it"}, {"start": 84.04, "duration": 5.24, "text": "and then let's copy down this a plus b ="}, {"start": 86.24, "duration": 4.76, "text": "13 and let's substitute 9 for this B"}, {"start": 89.28, "duration": 4.04, "text": "after we subtract 9 from both sides we"}, {"start": 91.0, "duration": 4.159, "text": "get a is equal to 4 and let's put a box"}, {"start": 93.32, "duration": 4.0, "text": "around that so now we have an answer a"}, {"start": 95.159, "duration": 4.881, "text": "equal 4 and b equal 9 let's check our"}, {"start": 97.32, "duration": 4.4, "text": "work let's change this AA into 44 and"}, {"start": 100.04, "duration": 3.8, "text": "let's change this a down here into a"}, {"start": 101.72, "duration": 5.48, "text": "four and then the BB we can change into"}, {"start": 103.84, "duration": 7.4, "text": "99 and we have 4 + 9 is equal to 3 carry"}, {"start": 107.2, "duration": 7.279, "text": "a 1 1 + 4 + 9 is 14 so this checks out"}, {"start": 111.24, "duration": 6.04, "text": "so a does equal 4 and B does equal 9 how"}, {"start": 114.479, "duration": 2.801, "text": "exciting"}], "title": "Math Logic Warmup", "description": "I hope you guys like this one!", "lengthSeconds": 115, "width": 640, "height": 360}
{"video_id": "I2OklXxn48A", "transcript": [{"start": 0.0, "duration": 3.0, "text": "I got something that I bet your math"}, {"start": 1.28, "duration": 3.28, "text": "teacher doesn't even know hey guys this"}, {"start": 3.0, "duration": 4.399, "text": "looks like a fun one did you know that"}, {"start": 4.56, "duration": 6.24, "text": "there are two quadratic formulas anytime"}, {"start": 7.399, "duration": 5.001, "text": "you have ax^2 + BX + cal0 there are two"}, {"start": 10.8, "duration": 3.759, "text": "different quadratic formulas to find"}, {"start": 12.4, "duration": 4.879, "text": "what x could equal you might be familiar"}, {"start": 14.559, "duration": 5.081, "text": "with negative B plus orus < TK of b^2 -"}, {"start": 17.279, "duration": 6.401, "text": "4 a c whole thing over 2 a but there's"}, {"start": 19.64, "duration": 7.16, "text": "also 2 C over the quantity b - + < TK of"}, {"start": 23.68, "duration": 4.759, "text": "b s - 4 a c this quadratic formula will"}, {"start": 26.8, "duration": 3.559, "text": "give the exact same answers as this"}, {"start": 28.439, "duration": 6.761, "text": "quadratic formula would you like see an"}, {"start": 30.359, "duration": 6.641, "text": "example let's try out 3x^2 + 7 x + 2 = 0"}, {"start": 35.2, "duration": 4.16, "text": "the ACs are a little tight let's give us"}, {"start": 37.0, "duration": 4.44, "text": "a little more room and let's do this the"}, {"start": 39.36, "duration": 4.24, "text": "leading coefficient is three which means"}, {"start": 41.44, "duration": 4.24, "text": "a is equal to three so this a will"}, {"start": 43.6, "duration": 4.32, "text": "become three this a will become three"}, {"start": 45.68, "duration": 4.24, "text": "and this a will become three the 7x"}, {"start": 47.92, "duration": 5.439, "text": "means that b is equal to 7 so we'll"}, {"start": 49.92, "duration": 5.76, "text": "change this B to 7 this B to 7 this B to"}, {"start": 53.359, "duration": 4.321, "text": "7 and this bet to 7 and last the"}, {"start": 55.68, "duration": 5.76, "text": "constant is equal to two so we can make"}, {"start": 57.68, "duration": 5.719, "text": "this C2 this C2 and this C2 and now"}, {"start": 61.44, "duration": 4.12, "text": "let's simplify these let's bring down"}, {"start": 63.399, "duration": 4.961, "text": "the -7 and we have plus or minus the"}, {"start": 65.56, "duration": 5.599, "text": "square root 7^2 is equal to 49 and then"}, {"start": 68.36, "duration": 5.799, "text": "we're going to subtract 4 * 3 * 2 which"}, {"start": 71.159, "duration": 5.721, "text": "is equal to 24 and on bottom 2 * 3 is"}, {"start": 74.159, "duration": 5.32, "text": "equal to 6 and then over here 2 * 2 is"}, {"start": 76.88, "duration": 5.4, "text": "equal to 4 we could bring down the -7"}, {"start": 79.479, "duration": 4.6, "text": "and we have minus plus the Square t 7 2"}, {"start": 82.28, "duration": 5.08, "text": "is equal to 49 and then we're going to"}, {"start": 84.079, "duration": 5.641, "text": "subtract 4 * 3 * 2 which is equal to 24"}, {"start": 87.36, "duration": 4.24, "text": "and now we can simplify these on bottom"}, {"start": 89.72, "duration": 3.88, "text": "here here we have the six and on top"}, {"start": 91.6, "duration": 4.559, "text": "here we'll have the four on top we can"}, {"start": 93.6, "duration": 5.199, "text": "copy down the -7 plus or minus Square Ro"}, {"start": 96.159, "duration": 5.0, "text": "T and on bottom we can copy down the -7"}, {"start": 98.799, "duration": 6.041, "text": "minus or plus Square t inside of the"}, {"start": 101.159, "duration": 7.32, "text": "square root 49 - 24 is 25 and same thing"}, {"start": 104.84, "duration": 6.0, "text": "49 - 24 is 25 on the left hand side squ"}, {"start": 108.479, "duration": 4.6, "text": "< TK of 25 is 5 and on the right hand"}, {"start": 110.84, "duration": 3.76, "text": "side Square < TK of 25 is 5 and now"}, {"start": 113.079, "duration": 3.601, "text": "let's smush everything together and"}, {"start": 114.6, "duration": 3.92, "text": "we're ready to finish up we have x"}, {"start": 116.68, "duration": 3.56, "text": "equals and the plus or minus right here"}, {"start": 118.52, "duration": 5.639, "text": "means we're going to have two values of"}, {"start": 120.24, "duration": 6.199, "text": "X both values of X will have a -7 over 6"}, {"start": 124.159, "duration": 3.44, "text": "but the first one is going to be a + 5"}, {"start": 126.439, "duration": 3.24, "text": "and the second one's going to be a"}, {"start": 127.599, "duration": 4.0, "text": "minus5 and on the right hand side the"}, {"start": 129.679, "duration": 4.84, "text": "minus plus means there's going to be two"}, {"start": 131.599, "duration": 5.041, "text": "answers the both have a four and a -7"}, {"start": 134.519, "duration": 4.44, "text": "but the first one is going to be ne5 and"}, {"start": 136.64, "duration": 5.52, "text": "the second one will be + 5 so let's"}, {"start": 138.959, "duration": 6.241, "text": "simplify these top two -7 + 5 is equal"}, {"start": 142.16, "duration": 6.04, "text": "to -2 and let's copy over the six and on"}, {"start": 145.2, "duration": 5.36, "text": "the right hand side we have a 4 and -7 -"}, {"start": 148.2, "duration": 4.759, "text": "5 is -12 both both of these can be"}, {"start": 150.56, "duration": 5.36, "text": "simplified and they both simplify to-"}, {"start": 152.959, "duration": 5.0, "text": "1/3 they both gave the exact same answer"}, {"start": 155.92, "duration": 5.52, "text": "let's put a box around it next let's do"}, {"start": 157.959, "duration": 6.0, "text": "the second one -7 - 5 is -2 and that'll"}, {"start": 161.44, "duration": 6.48, "text": "be over 6 and on this side we can bring"}, {"start": 163.959, "duration": 5.841, "text": "over the four and -7 + 5 is -2 both of"}, {"start": 167.92, "duration": 4.64, "text": "these can be simplified and they both"}, {"start": 169.8, "duration": 4.96, "text": "simplify to -2 once again we got the"}, {"start": 172.56, "duration": 3.84, "text": "exact same answer let's put a box around"}, {"start": 174.76, "duration": 3.479, "text": "it so it actually worked out this"}, {"start": 176.4, "duration": 3.64, "text": "quadratic formula gave the same answer"}, {"start": 178.239, "duration": 3.681, "text": "as this quadratic formula one"}, {"start": 180.04, "duration": 5.199, "text": "interesting observation is this was"}, {"start": 181.92, "duration": 6.0, "text": "minus plus while this was plus minus the"}, {"start": 185.239, "duration": 5.041, "text": "minus version of this gave us the same"}, {"start": 187.92, "duration": 3.959, "text": "answer as the plus version of this and"}, {"start": 190.28, "duration": 3.599, "text": "that will always be true here's another"}, {"start": 191.879, "duration": 4.881, "text": "example where there's only one solution"}, {"start": 193.879, "duration": 4.64, "text": "1/3 both equations gave the same thing"}, {"start": 196.76, "duration": 3.96, "text": "and here's an example with irrational"}, {"start": 198.519, "duration": 4.36, "text": "solutions they both gave the same"}, {"start": 200.72, "duration": 4.96, "text": "irrational Solutions and here's another"}, {"start": 202.879, "duration": 5.08, "text": "example with imaginary Solutions how"}, {"start": 205.68, "duration": 4.279, "text": "exciting so I'm thinking there are two"}, {"start": 207.959, "duration": 4.36, "text": "possible follow-up videos the first has"}, {"start": 209.959, "duration": 3.92, "text": "to show an example of why this is useful"}, {"start": 212.319, "duration": 3.64, "text": "and there is a cool example why it's"}, {"start": 213.879, "duration": 3.961, "text": "useful for another follow-up video we"}, {"start": 215.959, "duration": 4.081, "text": "could actually prove that this always"}, {"start": 217.84, "duration": 4.72, "text": "works I showed a couple of examples here"}, {"start": 220.04, "duration": 3.8, "text": "but we can actually prove this if you"}, {"start": 222.56, "duration": 3.36, "text": "guys want to see either one of those"}, {"start": 223.84, "duration": 5.52, "text": "follow-ups let me know in the comments"}, {"start": 225.92, "duration": 3.44, "text": "how exciting"}], "title": "Different Version of Quadratic Formula", "description": "I thought this was a fun one. I hope you guys love it!", "lengthSeconds": 228, "width": 640, "height": 360}
{"video_id": "9htXV7n02AM", "transcript": [{"start": 0.0, "duration": 5.94, "text": "he has welcome to this problem so this"}, {"start": 2.399, "duration": 4.951, "text": "one may be able to do in your head but"}, {"start": 5.94, "duration": 4.02, "text": "if you want to show some work you could"}, {"start": 7.35, "duration": 5.22, "text": "say okay X is a variable so what is"}, {"start": 9.96, "duration": 4.95, "text": "three two one well that would give us"}, {"start": 12.57, "duration": 6.539, "text": "three so we know X does not equal one"}, {"start": 14.91, "duration": 6.81, "text": "what's 3 to the 2 well that's nine so we"}, {"start": 19.109, "duration": 5.731, "text": "know X doesn't equal to what's three to"}, {"start": 21.72, "duration": 7.26, "text": "the 30 through 23 mines means 3 times 3"}, {"start": 24.84, "duration": 8.129, "text": "times 3 and that equals 27 oh that's"}, {"start": 28.98, "duration": 5.79, "text": "true so the answer is x equals 3 so"}, {"start": 32.969, "duration": 4.201, "text": "really just trying to figure out what is"}, {"start": 34.77, "duration": 6.449, "text": "the variable in here that would give you"}, {"start": 37.17, "duration": 6.209, "text": "27 and this is what's going on so I hope"}, {"start": 41.219, "duration": 4.851, "text": "this makes sense and i'll talk to you"}, {"start": 43.379, "duration": 2.691, "text": "guys soon hi"}], "title": "Solving for Exponents", "description": "Exponents Problem #3 UNDER A MINUTE! For more math made easy visit andymath.com.\n\nSubscribe here: https://www.youtube.com/channel/UC6KhU3AMLHC-qv7tj0ZJduw?sub_confirmation=1", "lengthSeconds": 46, "width": 640, "height": 360}
{"video_id": "s5tWo_kpk2c", "transcript": [{"start": 0.0, "duration": 4.259, "text": "hey guys this looks like a fun one we"}, {"start": 1.979, "duration": 4.501, "text": "have three of these nine to the X Powers"}, {"start": 4.259, "duration": 4.26, "text": "so I'm going to write that as 3 times 9"}, {"start": 6.48, "duration": 3.3, "text": "to the X power next I want to get rid of"}, {"start": 8.519, "duration": 3.061, "text": "this three so I'm going to divide both"}, {"start": 9.78, "duration": 3.779, "text": "sides by 3. these two threes cancel"}, {"start": 11.58, "duration": 4.32, "text": "leaving us with just nine to the X power"}, {"start": 13.559, "duration": 4.56, "text": "on the right hand side we have a 139"}, {"start": 15.9, "duration": 4.02, "text": "threes being multiplied by each other on"}, {"start": 18.119, "duration": 4.381, "text": "top and we have a single three on bottom"}, {"start": 19.92, "duration": 4.619, "text": "one of these 139 threes up here is going"}, {"start": 22.5, "duration": 4.5, "text": "to cancel with this 3 down here so we"}, {"start": 24.539, "duration": 5.341, "text": "only have 138 threes being multiplied"}, {"start": 27.0, "duration": 5.399, "text": "which is 3 to the 138 power so this 9 to"}, {"start": 29.88, "duration": 4.38, "text": "the X power we can rewrite the 9 as 3"}, {"start": 32.399, "duration": 4.081, "text": "squared and that'll still be to the X"}, {"start": 34.26, "duration": 4.26, "text": "power this is going to be equal to 3 to"}, {"start": 36.48, "duration": 3.419, "text": "the 2x power you can just multiply the"}, {"start": 38.52, "duration": 3.059, "text": "two powers and then we'll just bring"}, {"start": 39.899, "duration": 4.141, "text": "down the right hand side so we have 3 to"}, {"start": 41.579, "duration": 4.5, "text": "the 2x is equal to 3 to the 138 since"}, {"start": 44.04, "duration": 3.78, "text": "the bases match the only way these two"}, {"start": 46.079, "duration": 3.241, "text": "sides could be equal to each other is if"}, {"start": 47.82, "duration": 4.02, "text": "the exponents are equal to each other"}, {"start": 49.32, "duration": 4.919, "text": "and that gives us 2x is equal to 138"}, {"start": 51.84, "duration": 5.399, "text": "after we divide both sides by 2 we get X"}, {"start": 54.239, "duration": 6.741, "text": "is equal to 69 nice and let's put a box"}, {"start": 57.239, "duration": 3.741, "text": "around that how exciting"}], "title": "Exponent Challenge", "description": "https://andymath.com/", "lengthSeconds": 59, "width": 360, "height": 640}
{"video_id": "weJdNtDdk-g", "transcript": [{"start": 0.04, "duration": 2.68, "text": "hey guys this looks like a fun one we're"}, {"start": 1.28, "duration": 3.68, "text": "trying to find the ratio of the pink"}, {"start": 2.72, "duration": 3.52, "text": "area to the green area let's do this so"}, {"start": 4.96, "duration": 3.12, "text": "first let's try to find the area of"}, {"start": 6.24, "duration": 3.08, "text": "these yin-yang things so let's pull them"}, {"start": 8.08, "duration": 2.96, "text": "out and if we put them together they"}, {"start": 9.32, "duration": 3.8, "text": "should make a circle so it just be half"}, {"start": 11.04, "duration": 4.12, "text": "of that Circle and they don't these make"}, {"start": 13.12, "duration": 3.6, "text": "an ellipse all right scrap that idea"}, {"start": 15.16, "duration": 3.08, "text": "let's try something else so let's draw"}, {"start": 16.72, "duration": 3.12, "text": "the radius right here in fact let's draw"}, {"start": 18.24, "duration": 3.4, "text": "all these radiuses so that we have a"}, {"start": 19.84, "duration": 3.679, "text": "square in the middle right here and"}, {"start": 21.64, "duration": 4.12, "text": "let's focus on that square each of these"}, {"start": 23.519, "duration": 5.121, "text": "sides are two radiuses so we can call it"}, {"start": 25.76, "duration": 4.839, "text": "2 R and yes radiuses is the plural of"}, {"start": 28.64, "duration": 4.0, "text": "radius we could also say radi so the"}, {"start": 30.599, "duration": 4.561, "text": "area of the square is going to be 2 R *"}, {"start": 32.64, "duration": 4.079, "text": "2 R that's the same thing as 4 R 2 let's"}, {"start": 35.16, "duration": 2.88, "text": "look at another copy of the square down"}, {"start": 36.719, "duration": 3.32, "text": "here let's pull apart these quarter"}, {"start": 38.04, "duration": 3.4, "text": "circles bring down this green part let's"}, {"start": 40.039, "duration": 3.401, "text": "rotate all these quarter circles to make"}, {"start": 41.44, "duration": 3.959, "text": "a circle the area of this circle will be"}, {"start": 43.44, "duration": 3.68, "text": "P pi r s the area of the green stuff is"}, {"start": 45.399, "duration": 3.281, "text": "then going to be the area of this Square"}, {"start": 47.12, "duration": 2.959, "text": "minus the area of this circle now we"}, {"start": 48.68, "duration": 3.039, "text": "don't need this circle anymore so we can"}, {"start": 50.079, "duration": 3.12, "text": "get rid of that let's bring this up here"}, {"start": 51.719, "duration": 2.68, "text": "and this is important so let's put a box"}, {"start": 53.199, "duration": 2.88, "text": "around it let's go back up to this"}, {"start": 54.399, "duration": 3.241, "text": "square up here let's cut the square in"}, {"start": 56.079, "duration": 3.921, "text": "half and let's find the length of this"}, {"start": 57.64, "duration": 4.399, "text": "diagonal well this ends up making a 4545"}, {"start": 60.0, "duration": 4.04, "text": "0 triangle so it ends up being < TK of 2"}, {"start": 62.039, "duration": 4.321, "text": "* one of these sides so the diagonal is"}, {"start": 64.04, "duration": 4.36, "text": "equal to 2 R < TK of2 if you don't know"}, {"start": 66.36, "duration": 3.68, "text": "about the 454590 shortcut here's the"}, {"start": 68.4, "duration": 3.16, "text": "Pythagorean theorem right here pause it"}, {"start": 70.04, "duration": 3.399, "text": "if you want to check it let's clean all"}, {"start": 71.56, "duration": 3.559, "text": "this stuff up and put a box around this"}, {"start": 73.439, "duration": 3.281, "text": "now let's find the area of our large"}, {"start": 75.119, "duration": 4.201, "text": "Circle so we know that the diagonal the"}, {"start": 76.72, "duration": 4.0, "text": "square is equal to 2 < tk2 R and now we"}, {"start": 79.32, "duration": 3.0, "text": "don't need the square anymore and then"}, {"start": 80.72, "duration": 3.16, "text": "this right here is a radius and this"}, {"start": 82.32, "duration": 4.32, "text": "right here is a radius the diameter will"}, {"start": 83.88, "duration": 4.32, "text": "be equal to two of the radiuses plus 2 >"}, {"start": 86.64, "duration": 3.04, "text": "2 R and if we want to know what the"}, {"start": 88.2, "duration": 3.48, "text": "radius of the largest circle is we would"}, {"start": 89.68, "duration": 4.719, "text": "just divide all that stuff by 2 which"}, {"start": 91.68, "duration": 4.56, "text": "gives us r + < tk2 r and now to find the"}, {"start": 94.399, "duration": 3.921, "text": "area of this circle we're going to do PI"}, {"start": 96.24, "duration": 4.0, "text": "radius squ where this right here is the"}, {"start": 98.32, "duration": 3.759, "text": "radius and then a binomial squared ends"}, {"start": 100.24, "duration": 3.44, "text": "up being the binomial times itself so we"}, {"start": 102.079, "duration": 5.04, "text": "can foil this out we'll leave the pi out"}, {"start": 103.68, "duration": 7.0, "text": "front and then R * R is R 2 R * < tk2 R"}, {"start": 107.119, "duration": 6.64, "text": "is < tk2 R 2 < tk2 R * R is < tk2 R 2"}, {"start": 110.68, "duration": 5.64, "text": "and < tk2 R * < tk2 R is = to 2 R 2 and"}, {"start": 113.759, "duration": 5.72, "text": "then we can clean this up R 2 + 2 R 2 is"}, {"start": 116.32, "duration": 5.159, "text": "3 R 2 and right here we have 2 < tk2 R 2"}, {"start": 119.479, "duration": 3.761, "text": "so that would be 2 < tk2 R 2 and then we"}, {"start": 121.479, "duration": 3.92, "text": "can distribute the pi so we now have the"}, {"start": 123.24, "duration": 3.72, "text": "area of this large Circle and we can put"}, {"start": 125.399, "duration": 3.441, "text": "a box around that we're trying to find"}, {"start": 126.96, "duration": 3.4, "text": "the pink area divided by the green area"}, {"start": 128.84, "duration": 3.16, "text": "I think that is what this says so the"}, {"start": 130.36, "duration": 3.8, "text": "pink area is going to be this whole"}, {"start": 132.0, "duration": 3.64, "text": "circle minus this green stuff and then"}, {"start": 134.16, "duration": 2.6, "text": "the green area is going to be this green"}, {"start": 135.64, "duration": 3.04, "text": "stuff we can get rid of these"}, {"start": 136.76, "duration": 4.24, "text": "parentheses and distribute the negative"}, {"start": 138.68, "duration": 4.199, "text": "all six of these terms has an R squ so"}, {"start": 141.0, "duration": 4.0, "text": "we can factor out the r squ and then we"}, {"start": 142.879, "duration": 4.321, "text": "can cancel those R SS and then on top"}, {"start": 145.0, "duration": 3.8, "text": "this 3 pi and this 1 Pi can become 4 pi"}, {"start": 147.2, "duration": 3.08, "text": "and now we are done this is the exact"}, {"start": 148.8, "duration": 2.68, "text": "value of the answer and end up being"}, {"start": 150.28, "duration": 5.039, "text": "approximately"}, {"start": 151.48, "duration": 3.839, "text": "20308 how exciting"}], "title": "Awesome Pink Challenge", "description": "This is an older one, I hope you guys love it!", "lengthSeconds": 153, "width": 360, "height": 640}
{"video_id": "2jLrluVYhyE", "transcript": [{"start": 0.08, "duration": 2.76, "text": "hey guys I want to show you something"}, {"start": 1.319, "duration": 3.52, "text": "really cool about the Pythagorean"}, {"start": 2.84, "duration": 4.439, "text": "theorem you're probably familiar with A2"}, {"start": 4.839, "duration": 4.481, "text": "+ B2 = c^2 and here's a cool"}, {"start": 7.279, "duration": 4.121, "text": "visualization that shows it if you add"}, {"start": 9.32, "duration": 4.279, "text": "the areas of these two red squares it's"}, {"start": 11.4, "duration": 4.159, "text": "equal to the area of this blue square"}, {"start": 13.599, "duration": 4.161, "text": "but these don't have to be squares it"}, {"start": 15.559, "duration": 4.161, "text": "works for isos right triangles they"}, {"start": 17.76, "duration": 3.84, "text": "could be equilateral triangles they"}, {"start": 19.72, "duration": 3.96, "text": "could be regular pentagons they could"}, {"start": 21.6, "duration": 4.519, "text": "even be semicircles for each of these"}, {"start": 23.68, "duration": 4.759, "text": "the red areas add up to the blue area"}, {"start": 26.119, "duration": 3.521, "text": "and this can work for any similar figure"}, {"start": 28.439, "duration": 3.16, "text": "we could even look at something like the"}, {"start": 29.64, "duration": 4.04, "text": "state of orgon if we align the state of"}, {"start": 31.599, "duration": 3.921, "text": "organ along this side this side and this"}, {"start": 33.68, "duration": 4.039, "text": "side it'll be true that the sum of these"}, {"start": 35.52, "duration": 5.12, "text": "two red areas will equal the blue area"}, {"start": 37.719, "duration": 6.36, "text": "or other words a plus b is equal to C"}, {"start": 40.64, "duration": 3.439, "text": "how exciting"}], "title": "Pythagorean Variations", "description": "I thought this was fun!", "lengthSeconds": 42, "width": 360, "height": 640}
{"video_id": "n8AIkCDubhQ", "transcript": [{"start": 0.03, "duration": 4.47, "text": "hey guys welcome to this arithmetic"}, {"start": 2.01, "duration": 4.23, "text": "sequence problem these are the notes"}, {"start": 4.5, "duration": 3.569, "text": "that we want to use since it's an"}, {"start": 6.24, "duration": 4.92, "text": "arithmetic sequence we're going to be"}, {"start": 8.069, "duration": 5.341, "text": "interested in this one right here so I"}, {"start": 11.16, "duration": 3.99, "text": "want to find the missing value a n"}, {"start": 13.41, "duration": 6.959, "text": "that's what we're trying to solve for a"}, {"start": 15.15, "duration": 10.379, "text": "eight is equal to a sub 1 is 3 plus D is"}, {"start": 20.369, "duration": 11.64, "text": "5 and then n is 8 so it's 8 minus 1 this"}, {"start": 25.529, "duration": 12.721, "text": "is 3 plus 5 times 7 5 times 7 is 35 plus"}, {"start": 32.009, "duration": 8.851, "text": "3 so a 8 is equal to 38 and that is the"}, {"start": 38.25, "duration": 4.44, "text": "answer the question so we just found the"}, {"start": 40.86, "duration": 2.91, "text": "equation we wanted to use plugged into"}, {"start": 42.69, "duration": 3.27, "text": "values and solved"}, {"start": 43.77, "duration": 5.07, "text": "I hope that makes sense please like and"}, {"start": 45.96, "duration": 5.25, "text": "subscribe check out Andy math comm for"}, {"start": 48.84, "duration": 5.059, "text": "more videos and study guides I'll talk"}, {"start": 51.21, "duration": 2.689, "text": "to you guys soon bye"}, {"start": 57.0, "duration": 2.06, "text": "you"}], "title": "Arithmetic Sequences", "description": "For more visit http://andymath.com\n\nSubscribe here: https://www.youtube.com/channel/UC6KhU3AMLHC-qv7tj0ZJduw?sub_confirmation=1", "lengthSeconds": 63, "width": 640, "height": 360}
{"video_id": "dSoU4yKCfSU", "transcript": [{"start": 0.0, "duration": 3.78, "text": "hey guys so this is a fun geometry"}, {"start": 1.62, "duration": 3.84, "text": "problem right here we have four circles"}, {"start": 3.78, "duration": 4.44, "text": "and the area of all four of these"}, {"start": 5.46, "duration": 5.219, "text": "circles is 144 pi and they tell us that"}, {"start": 8.22, "duration": 5.1, "text": "the radius of these smaller circles is x"}, {"start": 10.679, "duration": 4.62, "text": "the radius of this medium circle is 3x"}, {"start": 13.32, "duration": 5.039, "text": "and the radius of this larger circle is"}, {"start": 15.299, "duration": 4.98, "text": "5X and I want us to solve for x first"}, {"start": 18.359, "duration": 3.241, "text": "the area of a single circle is pi r"}, {"start": 20.279, "duration": 3.061, "text": "squared so if we're going to add four"}, {"start": 21.6, "duration": 3.839, "text": "circles we're going to be adding up four"}, {"start": 23.34, "duration": 4.62, "text": "pi r squareds we're going to set this"}, {"start": 25.439, "duration": 4.68, "text": "sum equal to 144 Pi because that was"}, {"start": 27.96, "duration": 4.02, "text": "given as the area of the four circles"}, {"start": 30.119, "duration": 4.021, "text": "for this first Circle that'll be the top"}, {"start": 31.98, "duration": 3.78, "text": "one we'll plug in X for R and for the"}, {"start": 34.14, "duration": 3.419, "text": "second Circle let's do this bottom one"}, {"start": 35.76, "duration": 3.84, "text": "so we'll plug in X for the radius for"}, {"start": 37.559, "duration": 3.66, "text": "the third circle let's plug in 3x for"}, {"start": 39.6, "duration": 3.479, "text": "the radius now we're going to plug in 5x"}, {"start": 41.219, "duration": 4.02, "text": "for the last one since everything is"}, {"start": 43.079, "duration": 3.721, "text": "multiplied by pi independently and we"}, {"start": 45.239, "duration": 3.361, "text": "have an equation we can divide both"}, {"start": 46.8, "duration": 3.48, "text": "sides by pi and I'll just get rid of all"}, {"start": 48.6, "duration": 4.26, "text": "of our Pi's so then let's bring down the"}, {"start": 50.28, "duration": 5.16, "text": "144 this would just be x squared this is"}, {"start": 52.86, "duration": 4.379, "text": "just x squared but then 3x squared the 3"}, {"start": 55.44, "duration": 2.88, "text": "is going to get a squared and the x is"}, {"start": 57.239, "duration": 3.48, "text": "going to get a squared so that's going"}, {"start": 58.32, "duration": 4.2, "text": "to be 9x squared and this case is the 5"}, {"start": 60.719, "duration": 4.321, "text": "good squared as well as the X so we get"}, {"start": 62.52, "duration": 3.72, "text": "25 x squared we can bring down the 144"}, {"start": 65.04, "duration": 3.06, "text": "again now we're going to combine like"}, {"start": 66.24, "duration": 3.12, "text": "terms all of these are X squareds so"}, {"start": 68.1, "duration": 3.54, "text": "we're ultimately just going to add the"}, {"start": 69.36, "duration": 5.16, "text": "coefficients 1 plus 1 plus 9 plus 25"}, {"start": 71.64, "duration": 6.0, "text": "gives us 36 x squared so we can divide"}, {"start": 74.52, "duration": 4.919, "text": "both sides by 36 144 divided by 36 is 4"}, {"start": 77.64, "duration": 3.72, "text": "and this right hand side the 36 is"}, {"start": 79.439, "duration": 3.421, "text": "cancel to give us x squared to solve for"}, {"start": 81.36, "duration": 3.119, "text": "x we square root both sides square root"}, {"start": 82.86, "duration": 3.119, "text": "of 4 is 2 and the square root of x"}, {"start": 84.479, "duration": 3.361, "text": "squared is X now we don't have to worry"}, {"start": 85.979, "duration": 3.901, "text": "about plus or minus 2 because you can't"}, {"start": 87.84, "duration": 3.54, "text": "have a negative value in a geometry"}, {"start": 89.88, "duration": 4.76, "text": "problem like this so this is the answer"}, {"start": 91.38, "duration": 3.26, "text": "to the question x equals two"}], "title": "Geometry Challenge Problem", "description": "https://andymath.com/area-and-circumference-circle-2/\n\nCollab with Geeklyhub", "lengthSeconds": 93, "width": 640, "height": 360}
{"video_id": "5-OguC4ybAM", "transcript": [{"start": 0.0, "duration": 3.0, "text": "hey guys so let's simplify this"}, {"start": 1.5, "duration": 3.06, "text": "expression it's kind of fun it's got a"}, {"start": 3.0, "duration": 2.88, "text": "bunch of three five sevens in it let's"}, {"start": 4.56, "duration": 2.88, "text": "see what happens when we simplify it so"}, {"start": 5.88, "duration": 4.139, "text": "let's start with inside the parentheses"}, {"start": 7.44, "duration": 4.38, "text": "357 over 357 is equal to one and then"}, {"start": 10.019, "duration": 5.28, "text": "one plus three plus five plus seven is"}, {"start": 11.82, "duration": 5.76, "text": "equal to 16. 3 times 5 times 7 is 105."}, {"start": 15.299, "duration": 3.661, "text": "take the square root of 16 to the 105."}, {"start": 17.58, "duration": 2.64, "text": "anytime we take a square root with an"}, {"start": 18.96, "duration": 3.3, "text": "exponent you can just divide that"}, {"start": 20.22, "duration": 3.84, "text": "exponent by two but 105 is an odd number"}, {"start": 22.26, "duration": 3.06, "text": "I want to try to keep things clean so"}, {"start": 24.06, "duration": 3.12, "text": "there is a way for us to make this even"}, {"start": 25.32, "duration": 3.9, "text": "we can rewrite the 16 inside the"}, {"start": 27.18, "duration": 3.9, "text": "parentheses is 4 squared and then this 2"}, {"start": 29.22, "duration": 4.74, "text": "is going to multiply by the 105 to give"}, {"start": 31.08, "duration": 4.26, "text": "us 4 to the 210 power now this 210 is"}, {"start": 33.96, "duration": 2.64, "text": "even so when I take the square root of"}, {"start": 35.34, "duration": 3.059, "text": "it it's just going to divide this by 2"}, {"start": 36.6, "duration": 3.72, "text": "which gives me 105. I want to take the"}, {"start": 38.399, "duration": 4.141, "text": "7th root I'm just going to divide this"}, {"start": 40.32, "duration": 3.899, "text": "exponent by 7 which gives me 4 to the"}, {"start": 42.54, "duration": 3.42, "text": "power of 15. then when I take the fifth"}, {"start": 44.219, "duration": 3.241, "text": "root I'm going to divide this 15 by 5 to"}, {"start": 45.96, "duration": 2.82, "text": "give me 4 cubed and then when I take the"}, {"start": 47.46, "duration": 3.3, "text": "cube root that's going to cancel with"}, {"start": 48.78, "duration": 6.0, "text": "the cube to give me just 4. and 7 times"}, {"start": 50.76, "duration": 7.08, "text": "4 is 28 28 times 5 is 140 3 times 140 is"}, {"start": 54.78, "duration": 5.419, "text": "420. nice so this simplifies to 420. how"}, {"start": 57.84, "duration": 2.359, "text": "exciting"}], "title": "Expression with lots of roots #shorts", "description": "https://andymath.com/\n\n#shorts", "lengthSeconds": 58, "width": 270, "height": 480}
{"video_id": "AoHphcHReiw", "transcript": [{"start": 0.0, "duration": 2.879, "text": "hey guys welcome to this problem we're"}, {"start": 1.53, "duration": 3.78, "text": "going to find the probability of red or"}, {"start": 2.879, "duration": 4.351, "text": "a card given this table here first thing"}, {"start": 5.31, "duration": 4.59, "text": "we can do is fill in the totals just a"}, {"start": 7.23, "duration": 5.039, "text": "little bit of addition so now probably a"}, {"start": 9.9, "duration": 6.619, "text": "red or a car there's two ways to do this"}, {"start": 12.269, "duration": 4.25, "text": "we can circle all the red stuff and"}, {"start": 17.449, "duration": 4.32, "text": "all the cars"}, {"start": 19.85, "duration": 4.83, "text": "and"}, {"start": 21.769, "duration": 9.941, "text": "just add these values up"}, {"start": 24.68, "duration": 9.399, "text": "35 + 70 + 25 + 45 these are all the"}, {"start": 31.71, "duration": 4.41, "text": "things that satisfy it and then we"}, {"start": 34.079, "duration": 3.41, "text": "divide our you might 315 because that's"}, {"start": 36.12, "duration": 4.52, "text": "the total number"}, {"start": 37.489, "duration": 8.831, "text": "so add these up"}, {"start": 40.64, "duration": 7.599, "text": "we get a 175 over 315 can reduce it so"}, {"start": 46.32, "duration": 4.879, "text": "our answers 5 nights there's one other"}, {"start": 48.239, "duration": 5.991, "text": "way we can do this and that is"}, {"start": 51.199, "duration": 7.68, "text": "probability of red"}, {"start": 54.23, "duration": 10.349, "text": "plus the probability of car"}, {"start": 58.879, "duration": 5.7, "text": "minus the probability of red and car"}, {"start": 64.64, "duration": 5.159, "text": "so we'll do that real quick to"}, {"start": 66.57, "duration": 7.52, "text": "probability of red is this total"}, {"start": 69.799, "duration": 5.851, "text": "105 over 315"}, {"start": 74.09, "duration": 6.09, "text": "plus"}, {"start": 75.65, "duration": 7.77, "text": "probability of car is this total 105"}, {"start": 80.18, "duration": 7.72, "text": "over 3 15"}, {"start": 83.42, "duration": 9.04, "text": "minus this overlap of red and car which"}, {"start": 87.9, "duration": 8.31, "text": "is 35 / 3 15 after we do the math here"}, {"start": 92.46, "duration": 6.089, "text": "we end up with 175 / 3 15 which is the"}, {"start": 96.21, "duration": 4.56, "text": "same thing we had above so our final"}, {"start": 98.549, "duration": 4.581, "text": "answer is 59 so you can do it either way"}, {"start": 100.77, "duration": 5.279, "text": "identify everything that's true for or"}, {"start": 103.13, "duration": 6.269, "text": "apply this formula hope it makes sense"}, {"start": 106.049, "duration": 3.35, "text": "to talk to soon bye"}, {"start": 110.73, "duration": 2.06, "text": "you"}], "title": "How to do Probability Problems", "description": "Probability Problem #2 MAKES SENSE NOW! For more math made easy visit andymath.com.\n\n\nSubscribe here: https://www.youtube.com/channel/UC6KhU3AMLHC-qv7tj0ZJduw?sub_confirmation=1", "lengthSeconds": 111, "width": 640, "height": 360}
{"video_id": "rajZuNyF8N8", "transcript": [{"start": 0.0, "duration": 3.24, "text": "okay so we're going to evaluate this"}, {"start": 1.56, "duration": 2.82, "text": "definite integral this is a polynomial"}, {"start": 3.24, "duration": 3.36, "text": "so we're just going to use the power"}, {"start": 4.38, "duration": 4.679, "text": "rule so we have the constant 2 we bring"}, {"start": 6.6, "duration": 4.56, "text": "down the X and this 3 right here we add"}, {"start": 9.059, "duration": 3.721, "text": "1 to it and we also divide by that"}, {"start": 11.16, "duration": 2.88, "text": "number plus and then we're going to do"}, {"start": 12.78, "duration": 3.12, "text": "the same thing for this one we have the"}, {"start": 14.04, "duration": 3.36, "text": "constant three we're going to add 1 to"}, {"start": 15.9, "duration": 3.42, "text": "the 2 and then we're going to divide by"}, {"start": 17.4, "duration": 3.539, "text": "that 3 plus 1 is 4 so both of these will"}, {"start": 19.32, "duration": 2.879, "text": "change into four and two plus one is"}, {"start": 20.939, "duration": 3.481, "text": "three so both of these will change into"}, {"start": 22.199, "duration": 3.781, "text": "three two over four can reduce into one"}, {"start": 24.42, "duration": 3.24, "text": "over two and three over three cancel"}, {"start": 25.98, "duration": 3.0, "text": "each other out leaving us with just X"}, {"start": 27.66, "duration": 3.66, "text": "cubed and then we're going to evaluate"}, {"start": 28.98, "duration": 3.9, "text": "all of this from one to three and one"}, {"start": 31.32, "duration": 2.88, "text": "and the three come from here what this"}, {"start": 32.88, "duration": 3.78, "text": "means to evaluate is we're going to plug"}, {"start": 34.2, "duration": 5.22, "text": "in 3 for the X's giving us three to the"}, {"start": 36.66, "duration": 4.079, "text": "fourth over 2 plus 3 cubed and we're"}, {"start": 39.42, "duration": 3.12, "text": "going to subtract whatever we get when"}, {"start": 40.739, "duration": 4.5, "text": "we plug in one there'll be one to the"}, {"start": 42.54, "duration": 5.039, "text": "fourth over 2 plus 1 cubed three to the"}, {"start": 45.239, "duration": 4.261, "text": "fourth is equal to 81 and 3 cubed is"}, {"start": 47.579, "duration": 4.021, "text": "equal to 27. on this side one to the"}, {"start": 49.5, "duration": 4.02, "text": "fourth is just one and one cubed is just"}, {"start": 51.6, "duration": 5.4, "text": "one and then one plus a half is equal to"}, {"start": 53.52, "duration": 5.28, "text": "1.5 81 over 2 is equal to 40.5 now we"}, {"start": 57.0, "duration": 6.0, "text": "can get rid of these parentheses 40.5"}, {"start": 58.8, "duration": 6.24, "text": "plus 27 is 6 7.5 minus 1.5 which gives"}, {"start": 63.0, "duration": 4.5, "text": "us 66. and that's the answer to our"}, {"start": 65.04, "duration": 4.439, "text": "question this integral is equal to 66."}, {"start": 67.5, "duration": 3.78, "text": "how exciting this problem right here is"}, {"start": 69.479, "duration": 3.301, "text": "number seven on this page I have on"}, {"start": 71.28, "duration": 2.64, "text": "andymath.com if you guys want to try"}, {"start": 72.78, "duration": 3.379, "text": "more of these the link is in the"}, {"start": 73.92, "duration": 2.239, "text": "description"}], "title": "Quick Definite Integral", "description": "https://andymath.com/definite-integrals/", "lengthSeconds": 74, "width": 640, "height": 360}
{"video_id": "YLuPG2XnTbQ", "transcript": [{"start": 0.04, "duration": 3.679, "text": "this is another Katrina a puzzle and"}, {"start": 1.64, "duration": 4.04, "text": "it's my goal to solve all of these hey"}, {"start": 3.719, "duration": 4.96, "text": "guys this looks like a fun one we got a"}, {"start": 5.68, "duration": 4.879, "text": "polygon with multiple angles given and"}, {"start": 8.679, "duration": 3.481, "text": "it wants us to find the missing angle"}, {"start": 10.559, "duration": 3.401, "text": "and we're also given that this length is"}, {"start": 12.16, "duration": 3.679, "text": "equal to this length and it's this part"}, {"start": 13.96, "duration": 3.28, "text": "that makes it a little bit trickier if"}, {"start": 15.839, "duration": 3.801, "text": "you want to try it on your own pause it"}, {"start": 17.24, "duration": 5.0, "text": "right now cuz I'm going to solve it in 3"}, {"start": 19.64, "duration": 4.76, "text": "2 1 so first let's just go around and"}, {"start": 22.24, "duration": 4.84, "text": "Mark all the angles that we can we have"}, {"start": 24.4, "duration": 5.639, "text": "the 70\u00b0 this angle right here is also"}, {"start": 27.08, "duration": 5.24, "text": "70\u00b0 these are called vertical angles and"}, {"start": 30.039, "duration": 4.36, "text": "vertical angles are congruent next we"}, {"start": 32.32, "duration": 4.239, "text": "can focus right here and these are"}, {"start": 34.399, "duration": 5.041, "text": "called a linear pair these two angles"}, {"start": 36.559, "duration": 3.721, "text": "will always add up to 180\u00b0 so this one's"}, {"start": 39.44, "duration": 2.88, "text": "going to be"}, {"start": 40.28, "duration": 5.04, "text": "110\u00b0 and then once again because of"}, {"start": 42.32, "duration": 5.079, "text": "vertical angles this one is also 110\u00b0"}, {"start": 45.32, "duration": 4.079, "text": "next let's focus on this triangle here"}, {"start": 47.399, "duration": 6.281, "text": "the sum of the interior angles of any"}, {"start": 49.399, "duration": 7.361, "text": "triangle is 180\u00b0 so 110 + 50 is equal to"}, {"start": 53.68, "duration": 5.12, "text": "160 that leaves 20\u00b0 for this angle and"}, {"start": 56.76, "duration": 5.88, "text": "then we can move over to this triangle"}, {"start": 58.8, "duration": 6.48, "text": "we have 70\u00b0 in 80\u00b0 is equal to 150\u00b0 that"}, {"start": 62.64, "duration": 4.88, "text": "leaves 30\u00b0 for this angle and then next"}, {"start": 65.28, "duration": 5.24, "text": "let's go to this linear pair if this"}, {"start": 67.52, "duration": 5.32, "text": "angle is 80\u00b0 that leaves 100\u00b0 for this"}, {"start": 70.52, "duration": 4.599, "text": "angle and then we can look at this"}, {"start": 72.84, "duration": 5.0, "text": "quadrilateral in any quadrilateral the"}, {"start": 75.119, "duration": 7.401, "text": "sum of the interior angles is equal to"}, {"start": 77.84, "duration": 7.319, "text": "360\u00b0 110 + 100 + 60 is equal to 270 that"}, {"start": 82.52, "duration": 4.279, "text": "leaves us 90\u00b0 for this last angle and"}, {"start": 85.159, "duration": 3.481, "text": "now we've pretty much found every single"}, {"start": 86.799, "duration": 3.521, "text": "angle except for the question mark and"}, {"start": 88.64, "duration": 3.88, "text": "this angle up here let's let's call this"}, {"start": 90.32, "duration": 3.799, "text": "other angle X unfortunately we can't"}, {"start": 92.52, "duration": 3.959, "text": "focus on this triangle because we have"}, {"start": 94.119, "duration": 4.0, "text": "two unknown angles in this triangle but"}, {"start": 96.479, "duration": 3.801, "text": "we still haven't taken advantage of the"}, {"start": 98.119, "duration": 3.401, "text": "fact that these two things are equal so"}, {"start": 100.28, "duration": 3.68, "text": "let's focus on the triangle that"}, {"start": 101.52, "duration": 4.36, "text": "contains these two equal lengths every"}, {"start": 103.96, "duration": 3.479, "text": "single triangle can be inscribed inside"}, {"start": 105.88, "duration": 3.64, "text": "of a circle it doesn't matter if it's"}, {"start": 107.439, "duration": 4.32, "text": "right obtuse or acute they can all fit"}, {"start": 109.52, "duration": 4.84, "text": "inside a circle and another fun fact for"}, {"start": 111.759, "duration": 4.521, "text": "any inscribed right angle triangle the"}, {"start": 114.36, "duration": 3.719, "text": "hypotenuse will be a diameter of the"}, {"start": 116.28, "duration": 4.479, "text": "circle and this is tied to the fact that"}, {"start": 118.079, "duration": 5.04, "text": "an inscribed angle is 1/ of the arc that"}, {"start": 120.759, "duration": 5.241, "text": "it subtends so for example an inscribed"}, {"start": 123.119, "duration": 5.28, "text": "angle of 30\u00b0 will have a 60\u00b0 Arc and an"}, {"start": 126.0, "duration": 4.64, "text": "inscribed angle of 45\u00b0 would have a 90\u00b0"}, {"start": 128.399, "duration": 4.881, "text": "Arc and an inscribed angle of 90\u00b0 would"}, {"start": 130.64, "duration": 3.84, "text": "have an arc of 180\u00b0 and that'd be half"}, {"start": 133.28, "duration": 3.319, "text": "the circle and that's what makes this"}, {"start": 134.48, "duration": 4.039, "text": "the diameter and then since these two"}, {"start": 136.599, "duration": 4.441, "text": "portions are congruent that means this"}, {"start": 138.519, "duration": 4.761, "text": "is the center of the circle so each of"}, {"start": 141.04, "duration": 3.96, "text": "these are radi of the circle and now"}, {"start": 143.28, "duration": 3.92, "text": "let's bring all this stuff back and it"}, {"start": 145.0, "duration": 4.04, "text": "looks like this is on the circle but we"}, {"start": 147.2, "duration": 3.84, "text": "don't know that for sure let's see if we"}, {"start": 149.04, "duration": 4.32, "text": "can prove it we can focus on this"}, {"start": 151.04, "duration": 5.44, "text": "triangle right here and then this 20 and"}, {"start": 153.36, "duration": 5.08, "text": "30\u00b0 is going to make this angle 50\u00b0"}, {"start": 156.48, "duration": 4.479, "text": "since both of these angles are equal to"}, {"start": 158.44, "duration": 4.719, "text": "50\u00b0 that means this is an isoceles"}, {"start": 160.959, "duration": 4.481, "text": "triangle and in every Isles triangle the"}, {"start": 163.159, "duration": 4.72, "text": "sides opposite the congruent angles are"}, {"start": 165.44, "duration": 4.32, "text": "also congruent so this side is equal to"}, {"start": 167.879, "duration": 4.0, "text": "this side and since we know this side is"}, {"start": 169.76, "duration": 4.28, "text": "a radius that means this side would also"}, {"start": 171.879, "duration": 4.561, "text": "be a radius so that means this point up"}, {"start": 174.04, "duration": 4.68, "text": "here does lie on the circle so now we"}, {"start": 176.44, "duration": 4.4, "text": "have an inscribed quadrilateral we do"}, {"start": 178.72, "duration": 4.32, "text": "know one specific property about any"}, {"start": 180.84, "duration": 5.399, "text": "inscribed quadrilateral the opposite"}, {"start": 183.04, "duration": 7.279, "text": "angles add up to 180\u00b0 so in these notes"}, {"start": 186.239, "duration": 7.481, "text": "a plus c equal 180 and B + D equal 180"}, {"start": 190.319, "duration": 6.401, "text": "and for our quadrilateral 50 + 90 +"}, {"start": 193.72, "duration": 5.84, "text": "question mark would equal 180 and 60 +"}, {"start": 196.72, "duration": 4.56, "text": "50 + x = 180 but we don't need this"}, {"start": 199.56, "duration": 3.759, "text": "second one because this equation will be"}, {"start": 201.28, "duration": 4.84, "text": "enough to solve for a question mark So"}, {"start": 203.319, "duration": 4.761, "text": "50 + 90 is equal to 140 and then to get"}, {"start": 206.12, "duration": 4.16, "text": "the question mark alone we can subtract"}, {"start": 208.08, "duration": 3.76, "text": "140 from both sides un left hand side"}, {"start": 210.28, "duration": 3.44, "text": "these will cancel each other out leaving"}, {"start": 211.84, "duration": 4.84, "text": "us with question mark and on the right"}, {"start": 213.72, "duration": 5.36, "text": "hand side 180 minus 140 is equal to 40"}, {"start": 216.68, "duration": 4.32, "text": "so our question mark is equal to 40"}, {"start": 219.08, "duration": 3.519, "text": "let's give it a label of degrees and"}, {"start": 221.0, "duration": 6.319, "text": "this is the answer to our question let's"}, {"start": 222.599, "duration": 4.72, "text": "put a box around it how exciting"}], "title": "Harder than it looks!", "description": "Another Catriona Agg Puzzle. This one is harder than it looks.", "lengthSeconds": 225, "width": 640, "height": 360}
{"video_id": "DyKU__ivYm0", "transcript": [{"start": 0.04, "duration": 3.359, "text": "hey guys this looks like a fun one it's"}, {"start": 1.52, "duration": 4.68, "text": "got a rectangle with three Circles of"}, {"start": 3.399, "duration": 4.201, "text": "diameter 3 4 and six smooshed in here"}, {"start": 6.2, "duration": 2.96, "text": "and we're supposed to find the distance"}, {"start": 7.6, "duration": 3.159, "text": "from here to here if you want to try it"}, {"start": 9.16, "duration": 4.96, "text": "on your own pause it right now CU I'm"}, {"start": 10.759, "duration": 5.841, "text": "going to solve it in 3 2 1 first let us"}, {"start": 14.12, "duration": 4.28, "text": "label the center of each of the circles"}, {"start": 16.6, "duration": 3.32, "text": "let's take this diameter and extend it"}, {"start": 18.4, "duration": 3.48, "text": "up here and let's split this into a"}, {"start": 19.92, "duration": 3.88, "text": "purple distance and a green distance"}, {"start": 21.88, "duration": 4.12, "text": "let's call the purple distance X and the"}, {"start": 23.8, "duration": 5.08, "text": "green distance y so ultimately we're"}, {"start": 26.0, "duration": 4.679, "text": "trying to find x + y I want to work on y"}, {"start": 28.88, "duration": 4.16, "text": "first let's look at these two larger"}, {"start": 30.679, "duration": 4.601, "text": "circles this one has a diameter of four"}, {"start": 33.04, "duration": 3.96, "text": "which means the radius is two and this"}, {"start": 35.28, "duration": 3.799, "text": "one has a diameter of six which means"}, {"start": 37.0, "duration": 3.52, "text": "the radius is three let's bring down the"}, {"start": 39.079, "duration": 4.721, "text": "green distance and find this"}, {"start": 40.52, "duration": 5.44, "text": "perpendicular segment since this is two"}, {"start": 43.8, "duration": 4.439, "text": "right here this distance would also be"}, {"start": 45.96, "duration": 4.919, "text": "two and this whole radius of three would"}, {"start": 48.239, "duration": 4.32, "text": "be broken up into one and two next let's"}, {"start": 50.879, "duration": 3.36, "text": "connect the centers of the circles and"}, {"start": 52.559, "duration": 3.601, "text": "that'll give us the hypotenuse of our"}, {"start": 54.239, "duration": 4.041, "text": "right triangle this piece is equal to"}, {"start": 56.16, "duration": 4.079, "text": "the radius of two and this piece is"}, {"start": 58.28, "duration": 3.68, "text": "equal to the radius of three and the"}, {"start": 60.239, "duration": 4.801, "text": "entire hypotenuse would have a length of"}, {"start": 61.96, "duration": 4.92, "text": "2 + 3 or 5 now let's mess around with"}, {"start": 65.04, "duration": 6.28, "text": "our new right triangle I'm thinking"}, {"start": 66.88, "duration": 6.8, "text": "Pythagorean theorem 1^ 2 + y^2 is = 5^2"}, {"start": 71.32, "duration": 5.64, "text": "and this will clean up it gives us 1 +"}, {"start": 73.68, "duration": 6.28, "text": "y^2 is = to 25 we can subtract one from"}, {"start": 76.96, "duration": 5.04, "text": "both sides and we get y^2 is equal to 24"}, {"start": 79.96, "duration": 4.36, "text": "now to get the Y by itself we can square"}, {"start": 82.0, "duration": 4.4, "text": "root both sides of the equation since Y"}, {"start": 84.32, "duration": 4.36, "text": "is a positive distance the square otk of"}, {"start": 86.4, "duration": 5.44, "text": "y^2 is just going to be Y and then we"}, {"start": 88.68, "duration": 6.84, "text": "can simplify the square root of 24 24 is"}, {"start": 91.84, "duration": 6.88, "text": "= to 4 * 6 and the < TK of 4 * 6 is"}, {"start": 95.52, "duration": 6.0, "text": "equal to the < TK 4 * the < TK 6 and"}, {"start": 98.72, "duration": 5.56, "text": "then the < TK of 4 is equal to 2 so Y is"}, {"start": 101.52, "duration": 5.639, "text": "equal to 2 * the < TK of 6 so we can"}, {"start": 104.28, "duration": 4.839, "text": "update this green distance to be 2un 6"}, {"start": 107.159, "duration": 4.441, "text": "and we can move it back up here now"}, {"start": 109.119, "duration": 4.521, "text": "let's work on the X this circle has a"}, {"start": 111.6, "duration": 5.04, "text": "diameter of three which means the radius"}, {"start": 113.64, "duration": 4.72, "text": "is 1.5 let's connect these two centers"}, {"start": 116.64, "duration": 4.759, "text": "which will be made up of this radius of"}, {"start": 118.36, "duration": 5.439, "text": "1.5 and this radius of two so the whole"}, {"start": 121.399, "duration": 4.281, "text": "thing will have a length of 3.5 then"}, {"start": 123.799, "duration": 4.32, "text": "let's drop this perpendicular and let's"}, {"start": 125.68, "duration": 5.199, "text": "find its distance since this largest"}, {"start": 128.119, "duration": 4.801, "text": "Circle had a diameter of six that means"}, {"start": 130.879, "duration": 4.161, "text": "the height of this rectangle is also"}, {"start": 132.92, "duration": 5.28, "text": "equal to six and then we can bring this"}, {"start": 135.04, "duration": 5.559, "text": "radius of two over here so we need 2"}, {"start": 138.2, "duration": 4.52, "text": "plus the purple distance plus 1.5 to"}, {"start": 140.599, "duration": 4.28, "text": "equal 6 that means our purple distance"}, {"start": 142.72, "duration": 3.799, "text": "is going to be equal to 2.5 and now"}, {"start": 144.879, "duration": 4.72, "text": "let's play with this triangle"}, {"start": 146.519, "duration": 9.44, "text": "Pythagorean theorem tells us 2.52 + x^2"}, {"start": 149.599, "duration": 9.961, "text": "is = to 3.52 2.52 is = 6.25 + x^2 and"}, {"start": 155.959, "duration": 6.241, "text": "then 3.52 is = 12.25 we can subtract"}, {"start": 159.56, "duration": 4.92, "text": "6.25 from both sides and we get x^2 is"}, {"start": 162.2, "duration": 5.52, "text": "equal to 6 and after we square root both"}, {"start": 164.48, "duration": 4.92, "text": "sides we get X is equal 2 the < TK of 6"}, {"start": 167.72, "duration": 4.28, "text": "that means that our purple distance will"}, {"start": 169.4, "duration": 4.72, "text": "be equal to the < TK of 6 and we can"}, {"start": 172.0, "duration": 3.64, "text": "bring that back up here so now we have"}, {"start": 174.12, "duration": 4.92, "text": "the question mark is going to be equal"}, {"start": 175.64, "duration": 8.0, "text": "to the purple < TK 6 plus the green 2un"}, {"start": 179.04, "duration": 6.08, "text": "6 and < TK 6 + 2un 6 is equal to 3un 6"}, {"start": 183.64, "duration": 4.2, "text": "and that is the answer to our question"}, {"start": 185.12, "duration": 4.199, "text": "let's put a box around it how exciting I"}, {"start": 187.84, "duration": 3.959, "text": "just got a haircut what do you guys"}, {"start": 189.319, "duration": 2.48, "text": "think"}], "title": "Cool Math Challenge", "description": "I hope you guys like it!", "lengthSeconds": 190, "width": 640, "height": 360}
{"video_id": "2VfPCKJhAa0", "transcript": [{"start": 0.0, "duration": 3.84, "text": "hey guys let's talk about some log"}, {"start": 1.38, "duration": 4.86, "text": "derivatives so it says find dydx when Y"}, {"start": 3.84, "duration": 3.84, "text": "is equal to Ln of root X I think this is"}, {"start": 6.24, "duration": 3.72, "text": "a beautiful one first I just want to"}, {"start": 7.68, "duration": 3.6, "text": "rewrite the root X as x to the one-half"}, {"start": 9.96, "duration": 3.059, "text": "power so now we're ready to take the"}, {"start": 11.28, "duration": 3.239, "text": "derivative so we have a function inside"}, {"start": 13.019, "duration": 2.881, "text": "of a function so we're going to do chain"}, {"start": 14.519, "duration": 2.821, "text": "rule so first we're going to work with"}, {"start": 15.9, "duration": 3.54, "text": "the natural log the derivative of"}, {"start": 17.34, "duration": 3.96, "text": "natural log of anything is 1 over that"}, {"start": 19.44, "duration": 2.999, "text": "something and then because of chain rule"}, {"start": 21.3, "duration": 3.059, "text": "we're going to multiply by the"}, {"start": 22.439, "duration": 3.241, "text": "derivative of what's inside here so the"}, {"start": 24.359, "duration": 2.641, "text": "derivative of x to the one half we're"}, {"start": 25.68, "duration": 3.0, "text": "going to bring the one-half in front and"}, {"start": 27.0, "duration": 3.48, "text": "it's going to be x to the and then we do"}, {"start": 28.68, "duration": 3.48, "text": "one half minus one which gives us"}, {"start": 30.48, "duration": 3.239, "text": "negative one half this piece right here"}, {"start": 32.16, "duration": 3.239, "text": "the one-half exponent means it's going"}, {"start": 33.719, "duration": 2.941, "text": "to be the square root of x and then this"}, {"start": 35.399, "duration": 2.16, "text": "x the negative one half means we're"}, {"start": 36.66, "duration": 2.46, "text": "going to have square root in the"}, {"start": 37.559, "duration": 3.721, "text": "denominator so it's going to be one over"}, {"start": 39.12, "duration": 4.02, "text": "two root X because the 2 right here so"}, {"start": 41.28, "duration": 4.14, "text": "this 2 is left on bottom and root x"}, {"start": 43.14, "duration": 4.38, "text": "times root X is equal to X and that is"}, {"start": 45.42, "duration": 4.139, "text": "the answer to the question d y d x is 1"}, {"start": 47.52, "duration": 3.24, "text": "over 2x so this really isn't super"}, {"start": 49.559, "duration": 4.581, "text": "difficult but if you don't know what"}, {"start": 50.76, "duration": 3.38, "text": "you're doing it definitely can be tricky"}], "title": "Log Derivative Example", "description": "", "lengthSeconds": 52, "width": 270, "height": 480}
{"video_id": "0tt2GVfVEwo", "transcript": [{"start": 0.0, "duration": 2.58, "text": "I think I already have a good idea what"}, {"start": 1.38, "duration": 2.82, "text": "this is going to be we're dealing with"}, {"start": 2.58, "duration": 3.3, "text": "some factorials usually you want to"}, {"start": 4.2, "duration": 3.119, "text": "reduce each of them in terms of the"}, {"start": 5.88, "duration": 4.38, "text": "smallest factorial so I'm going to"}, {"start": 7.319, "duration": 5.041, "text": "rewrite this 70 factorial as 70 times 69"}, {"start": 10.26, "duration": 4.2, "text": "times 68 factorial and then we'll bring"}, {"start": 12.36, "duration": 4.74, "text": "down the 68 factorial and the 69"}, {"start": 14.46, "duration": 4.56, "text": "factorial we can rewrite as 69 times 68"}, {"start": 17.1, "duration": 3.3, "text": "factorial so now we have a bunch of 68"}, {"start": 19.02, "duration": 2.94, "text": "factorials let's bring down the"}, {"start": 20.4, "duration": 4.02, "text": "numerator but on bottom I want to factor"}, {"start": 21.96, "duration": 5.22, "text": "out the 68 factorial 68 factorial times"}, {"start": 24.42, "duration": 5.04, "text": "what gives us 68 factorial that's 1. 68"}, {"start": 27.18, "duration": 4.86, "text": "factorial times what gives us 69 68"}, {"start": 29.46, "duration": 4.92, "text": "factorial that's 69. 1 plus 69 is equal"}, {"start": 32.04, "duration": 4.019, "text": "to 70. the 70s cancel and the 68"}, {"start": 34.38, "duration": 4.14, "text": "factorials cancel and that leaves us"}, {"start": 36.059, "duration": 6.081, "text": "with 69. let's use this to put it in a"}, {"start": 38.52, "duration": 3.62, "text": "box how exciting"}], "title": "Factorial Math Tricks #shorts", "description": "https://andymath.com/factorials/\n\n#shorts", "lengthSeconds": 40, "width": 270, "height": 480}
{"video_id": "FWaNpuwFO8U", "transcript": [{"start": 0.0, "duration": 4.38, "text": "hey guys here's another geometry"}, {"start": 2.1, "duration": 4.32, "text": "challenge we have a square that's a two"}, {"start": 4.38, "duration": 3.24, "text": "by two square I have four circles stuck"}, {"start": 6.42, "duration": 3.96, "text": "inside of that square and there's"}, {"start": 7.62, "duration": 4.68, "text": "another Square turn 45 degrees and fit"}, {"start": 10.38, "duration": 3.66, "text": "inside those circles so that it touches"}, {"start": 12.3, "duration": 4.02, "text": "at those four points and we want to know"}, {"start": 14.04, "duration": 3.72, "text": "what is the area of this Square I think"}, {"start": 16.32, "duration": 2.58, "text": "there's two ways to solve this I kind of"}, {"start": 17.76, "duration": 3.72, "text": "want to show you both ways we'll call"}, {"start": 18.9, "duration": 4.86, "text": "one side of the square a length of s s"}, {"start": 21.48, "duration": 3.66, "text": "for side or Square then we'll draw a"}, {"start": 23.76, "duration": 2.939, "text": "little square here and we'll notice that"}, {"start": 25.14, "duration": 3.78, "text": "this square right here cuts into exactly"}, {"start": 26.699, "duration": 4.141, "text": "one-fourth of that square which means I"}, {"start": 28.92, "duration": 3.9, "text": "could probably flip it and get the same"}, {"start": 30.84, "duration": 3.719, "text": "piece right up there just like that so"}, {"start": 32.82, "duration": 3.0, "text": "this piece should equal this piece right"}, {"start": 34.559, "duration": 3.121, "text": "here if you think about how those four"}, {"start": 35.82, "duration": 4.559, "text": "squares would be cut off both of these"}, {"start": 37.68, "duration": 4.26, "text": "are exactly half of the length of s so"}, {"start": 40.379, "duration": 3.421, "text": "the two of them together should just be"}, {"start": 41.94, "duration": 3.72, "text": "S let's look at this diagonal of the"}, {"start": 43.8, "duration": 4.5, "text": "square I drew in the corner it's going"}, {"start": 45.66, "duration": 4.14, "text": "to be comprised of s plus it's a"}, {"start": 48.3, "duration": 2.7, "text": "diameter of one of these circles how"}, {"start": 49.8, "duration": 3.419, "text": "long is this diameter of this circle"}, {"start": 51.0, "duration": 3.84, "text": "well if we cut this side in half each of"}, {"start": 53.219, "duration": 3.541, "text": "these halves is equal to one and that"}, {"start": 54.84, "duration": 3.719, "text": "half is the same height as one of the"}, {"start": 56.76, "duration": 3.54, "text": "diameters so we can call this diameter"}, {"start": 58.559, "duration": 3.781, "text": "one so the diagonal of the square square"}, {"start": 60.3, "duration": 3.899, "text": "is equal to S plus one but we can pull"}, {"start": 62.34, "duration": 3.12, "text": "out of this a triangle and this green"}, {"start": 64.199, "duration": 2.941, "text": "triangle here is the same thing as this"}, {"start": 65.46, "duration": 3.12, "text": "green triangle here both of these sides"}, {"start": 67.14, "duration": 2.58, "text": "are equal to one anytime you have a"}, {"start": 68.58, "duration": 2.7, "text": "right triangle where the two legs are"}, {"start": 69.72, "duration": 3.12, "text": "equal to one the hypotenuse is always"}, {"start": 71.28, "duration": 2.699, "text": "equal to square root of two and if you"}, {"start": 72.84, "duration": 2.76, "text": "want to verify it here it is using"}, {"start": 73.979, "duration": 3.781, "text": "Pythagorean theorem so now we can plug"}, {"start": 75.6, "duration": 3.54, "text": "in square root of 2 for the diagonal and"}, {"start": 77.76, "duration": 3.12, "text": "then we can smush this together and"}, {"start": 79.14, "duration": 3.36, "text": "clean all this stuff up scoot this over"}, {"start": 80.88, "duration": 3.66, "text": "and now we can solve for S let's"}, {"start": 82.5, "duration": 4.259, "text": "subtract one from both sides and we get"}, {"start": 84.54, "duration": 3.899, "text": "S equals square root of 2 minus one s is"}, {"start": 86.759, "duration": 4.081, "text": "the side of the square to get the area"}, {"start": 88.439, "duration": 3.841, "text": "of the square that we would Square s so"}, {"start": 90.84, "duration": 2.819, "text": "we're just going to square both sides of"}, {"start": 92.28, "duration": 2.46, "text": "this equation on the left hand side"}, {"start": 93.659, "duration": 3.061, "text": "that's the same thing is multiplying"}, {"start": 94.74, "duration": 3.12, "text": "this binomial by itself and we have to"}, {"start": 96.72, "duration": 2.399, "text": "distribute so we're going to do square"}, {"start": 97.86, "duration": 3.0, "text": "root of 2 Times Square Root 2 gives us"}, {"start": 99.119, "duration": 3.421, "text": "2. root 2 times negative 1 gives us"}, {"start": 100.86, "duration": 3.299, "text": "negative root 2 and negative 1 times"}, {"start": 102.54, "duration": 2.939, "text": "root 2 gives us negative root two and"}, {"start": 104.159, "duration": 2.941, "text": "then negative 1 times negative 1 is"}, {"start": 105.479, "duration": 4.441, "text": "positive one and this is all still equal"}, {"start": 107.1, "duration": 5.04, "text": "to S squared 2 plus 1 is equal to 3 and"}, {"start": 109.92, "duration": 4.5, "text": "negative root 2 minus root 2 is minus"}, {"start": 112.14, "duration": 4.439, "text": "two root 2. so this right here is the"}, {"start": 114.42, "duration": 3.9, "text": "area of the Red Square it's 3 minus two"}, {"start": 116.579, "duration": 3.301, "text": "root 2. we can put a box around that and"}, {"start": 118.32, "duration": 3.18, "text": "let's put a box around S as well that's"}, {"start": 119.88, "duration": 3.059, "text": "seem pretty important get rid of all"}, {"start": 121.5, "duration": 3.06, "text": "this other stuff bring this down and"}, {"start": 122.939, "duration": 3.121, "text": "let's look at the other more formal way"}, {"start": 124.56, "duration": 3.479, "text": "to solve it you can find the centers of"}, {"start": 126.06, "duration": 3.899, "text": "these three circles connect the centers"}, {"start": 128.039, "duration": 3.301, "text": "we can see this piece right here is the"}, {"start": 129.959, "duration": 2.821, "text": "same thing as the side length of s"}, {"start": 131.34, "duration": 2.759, "text": "because it's the same thing as one of"}, {"start": 132.78, "duration": 3.48, "text": "the side lengths of that square then we"}, {"start": 134.099, "duration": 4.081, "text": "look up here both of these are the"}, {"start": 136.26, "duration": 3.479, "text": "radius of this top Circle so we'll call"}, {"start": 138.18, "duration": 3.18, "text": "both of those R and same thing here"}, {"start": 139.739, "duration": 3.301, "text": "these are both the radius of this circle"}, {"start": 141.36, "duration": 2.94, "text": "so we'll call those R and same thing"}, {"start": 143.04, "duration": 3.0, "text": "here both these are the Rays of that"}, {"start": 144.3, "duration": 3.72, "text": "Circle so we'll call those R now if we"}, {"start": 146.04, "duration": 4.32, "text": "extend this radius up here we'll notice"}, {"start": 148.02, "duration": 4.74, "text": "that R plus r is the same thing as one"}, {"start": 150.36, "duration": 3.9, "text": "so we just say two R equals one now"}, {"start": 152.76, "duration": 3.059, "text": "let's use Pythagorean theorem for this"}, {"start": 154.26, "duration": 3.36, "text": "right triangle if a squared plus b"}, {"start": 155.819, "duration": 3.721, "text": "squared equals c squared where a is this"}, {"start": 157.62, "duration": 4.14, "text": "bottom side here which is R plus r well"}, {"start": 159.54, "duration": 4.14, "text": "we know that R plus r is equal to one B"}, {"start": 161.76, "duration": 4.5, "text": "is this R plus r but we know that's"}, {"start": 163.68, "duration": 5.279, "text": "equal to one well C is this R plus S"}, {"start": 166.26, "duration": 4.14, "text": "Plus R and then R plus r is 2R and we"}, {"start": 168.959, "duration": 3.241, "text": "know that 2R is equal to one so we can"}, {"start": 170.4, "duration": 3.419, "text": "make that one one squared plus 1 squared"}, {"start": 172.2, "duration": 3.66, "text": "is just two then from here we square"}, {"start": 173.819, "duration": 3.841, "text": "root both sides the root and the square"}, {"start": 175.86, "duration": 3.78, "text": "cancel each other out to give us 1 plus"}, {"start": 177.66, "duration": 4.079, "text": "s when we subtract one from both sides"}, {"start": 179.64, "duration": 4.2, "text": "this side becomes root two minus one and"}, {"start": 181.739, "duration": 3.961, "text": "this side just becomes s so now we solve"}, {"start": 183.84, "duration": 3.899, "text": "for S and check this out it's the same s"}, {"start": 185.7, "duration": 3.48, "text": "that we got before so since we got the"}, {"start": 187.739, "duration": 3.0, "text": "same as both methods we know they're"}, {"start": 189.18, "duration": 2.82, "text": "both going to give us the same area so"}, {"start": 190.739, "duration": 3.36, "text": "the answer the question is the area is"}, {"start": 192.0, "duration": 2.959, "text": "equal to three minus two root 2 or in"}, {"start": 194.099, "duration": 3.42, "text": "other words"}, {"start": 194.959, "duration": 4.78, "text": "0.17157 I'm going to put units squared"}, {"start": 197.519, "duration": 4.381, "text": "so the exact value of the area of the"}, {"start": 199.739, "duration": 3.961, "text": "red square is three minus two root 2"}, {"start": 201.9, "duration": 4.38, "text": "units squared and the approximate area"}, {"start": 203.7, "duration": 5.539, "text": "of this red square is 0.17157 units"}, {"start": 206.28, "duration": 2.959, "text": "squared how exciting"}], "title": "2 different solutions geometry challenge", "description": "https://andymath.com/", "lengthSeconds": 207, "width": 360, "height": 640}
{"video_id": "5dePDd9NLRQ", "transcript": [{"start": 0.0, "duration": 4.24, "text": "so i've seen a lot about pushing p can"}, {"start": 1.92, "duration": 4.959, "text": "you solve for p yeah we can do this so"}, {"start": 4.24, "duration": 5.359, "text": "oliver is pushing p p is equal to the"}, {"start": 6.879, "duration": 5.041, "text": "sum from one to three of one over n"}, {"start": 9.599, "duration": 4.321, "text": "minus two so we take n equals one we add"}, {"start": 11.92, "duration": 4.08, "text": "to it whatever n equals two and then we"}, {"start": 13.92, "duration": 4.08, "text": "add to whatever n equals 3 and we stop"}, {"start": 16.0, "duration": 3.439, "text": "because there's a 3 on top of this so p"}, {"start": 18.0, "duration": 3.84, "text": "is going to be equal to the sum of these"}, {"start": 19.439, "duration": 4.161, "text": "three we're going to do 1 over n minus 2"}, {"start": 21.84, "duration": 4.24, "text": "we're going to plug in 1 for the n plus"}, {"start": 23.6, "duration": 4.8, "text": "1 minus 2 plug in 2 for the n so it's 2"}, {"start": 26.08, "duration": 4.48, "text": "minus 2 and plugging 3 for the n it's 3"}, {"start": 28.4, "duration": 4.159, "text": "minus 2. bring down the p and one over"}, {"start": 30.56, "duration": 3.999, "text": "one minus two gives us one over negative"}, {"start": 32.559, "duration": 3.84, "text": "one one over two minus two gives us one"}, {"start": 34.559, "duration": 3.761, "text": "over zero and one over three minus two"}, {"start": 36.399, "duration": 4.241, "text": "gives us one over one one over negative"}, {"start": 38.32, "duration": 4.399, "text": "one is negative one so negative one plus"}, {"start": 40.64, "duration": 4.0, "text": "one is zero so these cancel each other"}, {"start": 42.719, "duration": 4.0, "text": "out so we're left with p is equal to one"}, {"start": 44.64, "duration": 3.52, "text": "over zero and that is undefined and"}, {"start": 46.719, "duration": 3.52, "text": "that's the reason why nobody can explain"}, {"start": 48.16, "duration": 4.96, "text": "to me what pushing p means that's it i'm"}, {"start": 50.239, "duration": 2.881, "text": "done thank you"}], "title": "Pushing P", "description": "For more check out andymath.com.", "lengthSeconds": 51, "width": 202, "height": 360}
{"video_id": "I-eRU2uVVts", "transcript": [{"start": 0.08, "duration": 3.2, "text": "this person requested Pythagorean"}, {"start": 1.599, "duration": 3.561, "text": "theorem so we start with the square then"}, {"start": 3.28, "duration": 4.2, "text": "we have another Square that's the exact"}, {"start": 5.16, "duration": 4.04, "text": "same size so these two squares are equal"}, {"start": 7.48, "duration": 3.96, "text": "to each other then I'll make a cut that"}, {"start": 9.2, "duration": 3.76, "text": "is the same size as a and I'll do the"}, {"start": 11.44, "duration": 3.64, "text": "same thing to the other Square do the"}, {"start": 12.96, "duration": 3.52, "text": "same thing to every side of that square"}, {"start": 15.08, "duration": 3.119, "text": "and the same thing to every side of this"}, {"start": 16.48, "duration": 3.959, "text": "square and then this other piece will be"}, {"start": 18.199, "duration": 4.92, "text": "B then if I make a cut right here on"}, {"start": 20.439, "duration": 4.16, "text": "both squares and pull this triangle out"}, {"start": 23.119, "duration": 3.48, "text": "this is the same triangle that's in our"}, {"start": 24.599, "duration": 4.76, "text": "notes up above so this side right here"}, {"start": 26.599, "duration": 4.52, "text": "is C this side is also C and let's bring"}, {"start": 29.359, "duration": 4.281, "text": "these down here next let's make another"}, {"start": 31.119, "duration": 4.401, "text": "cut here where this is a and this is B"}, {"start": 33.64, "duration": 3.599, "text": "and let's make a cut here where this is"}, {"start": 35.52, "duration": 4.0, "text": "C and then if we pull both of these"}, {"start": 37.239, "duration": 4.041, "text": "triangles out and bring them down there"}, {"start": 39.52, "duration": 3.0, "text": "these two sides should still be equal to"}, {"start": 41.28, "duration": 2.959, "text": "each other because we pulled the two"}, {"start": 42.52, "duration": 3.84, "text": "same triangles out of both of them now"}, {"start": 44.239, "duration": 5.0, "text": "let's make some more Cuts here here and"}, {"start": 46.36, "duration": 5.48, "text": "here and also here and here we've now"}, {"start": 49.239, "duration": 4.241, "text": "made more of these ABC triangles and we"}, {"start": 51.84, "duration": 3.719, "text": "can pull them out and bring them down"}, {"start": 53.48, "duration": 4.64, "text": "here and so out of both sides we've"}, {"start": 55.559, "duration": 4.48, "text": "pulled four of these ABC triangles so"}, {"start": 58.12, "duration": 4.48, "text": "they should both still be equal so this"}, {"start": 60.039, "duration": 5.12, "text": "square right here is an a by a square so"}, {"start": 62.6, "duration": 5.519, "text": "the area of it is a squ this square has"}, {"start": 65.159, "duration": 6.721, "text": "sides B so its area is b^ squ and this"}, {"start": 68.119, "duration": 6.761, "text": "square has side C so its area is c^ 2 we"}, {"start": 71.88, "duration": 4.599, "text": "can call this left hand side a^ 2 + b^ 2"}, {"start": 74.88, "duration": 3.4, "text": "cuz we're adding these two squares"}, {"start": 76.479, "duration": 4.561, "text": "together and that's going to be equal to"}, {"start": 78.28, "duration": 5.76, "text": "this right hand side which is c^2 how"}, {"start": 81.04, "duration": 3.0, "text": "exciting"}], "title": "Pythagorean Theorem Demo", "description": "This is an old one, I hope you guys like it!", "lengthSeconds": 82, "width": 360, "height": 640}
{"video_id": "b16VJ5a6lTw", "transcript": [{"start": 0.0, "duration": 4.41, "text": "he has welcome to this problem we're"}, {"start": 1.68, "duration": 3.659, "text": "going to factor this trinomial so"}, {"start": 4.41, "duration": 3.389, "text": "anytime there's a leading coefficient"}, {"start": 5.339, "duration": 5.851, "text": "that's greater than 1 I like to do it"}, {"start": 7.799, "duration": 9.721, "text": "this way so i'll make an X and I'll make"}, {"start": 11.19, "duration": 9.06, "text": "a box so 3 times 6 is 18 so when two"}, {"start": 17.52, "duration": 6.69, "text": "numbers that multiply to 18 and the 11"}, {"start": 20.25, "duration": 7.859, "text": "goes down here so we do is two numbers"}, {"start": 24.21, "duration": 6.42, "text": "multiply to 18 and add to 11 that's nine"}, {"start": 28.109, "duration": 6.001, "text": "and two so this 3x squared is going to"}, {"start": 30.63, "duration": 5.64, "text": "be written here the 6 is written down"}, {"start": 34.11, "duration": 6.57, "text": "here and then we're going to split this"}, {"start": 36.27, "duration": 7.23, "text": "11x up in 29 x + 2 X we're going to put"}, {"start": 40.68, "duration": 4.469, "text": "them here in here so now you just find"}, {"start": 43.5, "duration": 6.93, "text": "out what the outside would be to give"}, {"start": 45.149, "duration": 9.061, "text": "you this inside so put a 3x here and an"}, {"start": 50.43, "duration": 5.519, "text": "X here so 3x times x is 3x squared the"}, {"start": 54.21, "duration": 3.619, "text": "reason I chose put the 3x here is"}, {"start": 55.949, "duration": 6.09, "text": "because I want to line up with this 9x"}, {"start": 57.829, "duration": 8.051, "text": "so 3x times what is 9x that's positive 3"}, {"start": 62.039, "duration": 8.12, "text": "then x times what is 2x it's positive 2"}, {"start": 65.88, "duration": 7.23, "text": "so the final answer here is 3x plus 2"}, {"start": 70.159, "duration": 7.901, "text": "for this top one and for the side is X"}, {"start": 73.11, "duration": 7.649, "text": "plus 3 and that is the answer fully"}, {"start": 78.06, "duration": 5.18, "text": "factored if that makes sense talk to you"}, {"start": 80.759, "duration": 2.481, "text": "guys soon bye"}], "title": "How to Factor Trinomials", "description": "Factoring Quadratics Problem #2. Problem factored! For more math made easy visit andymath.com.", "lengthSeconds": 84, "width": 640, "height": 360}
{"video_id": "5wSoPp10Ork", "transcript": [{"start": 0.04, "duration": 5.96, "text": "hey guys this looks like a fun one it"}, {"start": 1.48, "duration": 8.079, "text": "wants us to solve for x it says XX - 4 +"}, {"start": 6.0, "duration": 5.04, "text": "1 = 4X -4 if you want to try on your own"}, {"start": 9.559, "duration": 5.08, "text": "pause it right now because I'm going to"}, {"start": 11.04, "duration": 6.159, "text": "solve it in 3 2 1 so for the first step"}, {"start": 14.639, "duration": 4.8, "text": "we want to say x cannot equal 4 and the"}, {"start": 17.199, "duration": 4.08, "text": "reason for that is if x were equal to"}, {"start": 19.439, "duration": 3.76, "text": "four these would be zero in the"}, {"start": 21.279, "duration": 3.961, "text": "denominator and division by Zer is"}, {"start": 23.199, "duration": 3.441, "text": "undefined next I don't really like"}, {"start": 25.24, "duration": 2.92, "text": "fractions so let's get rid of these"}, {"start": 26.64, "duration": 3.16, "text": "denominators and we can do that by"}, {"start": 28.16, "duration": 4.52, "text": "multiplying both sides of the equation"}, {"start": 29.8, "duration": 4.8, "text": "by x - 4 on the left hand side this x -"}, {"start": 32.68, "duration": 3.559, "text": "4 will distribute to both of these terms"}, {"start": 34.6, "duration": 3.52, "text": "it'll distribute to the first term and"}, {"start": 36.239, "duration": 2.881, "text": "it'll distribute to the second term and"}, {"start": 38.12, "duration": 3.56, "text": "on the right hand side it's just going"}, {"start": 39.12, "duration": 4.32, "text": "to be 4 * the x - 4 so now we can"}, {"start": 41.68, "duration": 3.92, "text": "simplify some things for this first"}, {"start": 43.44, "duration": 4.32, "text": "fraction or rational expression we can"}, {"start": 45.6, "duration": 4.04, "text": "divide away the x - 4 from top and"}, {"start": 47.76, "duration": 4.56, "text": "bottom and then for this part anything"}, {"start": 49.64, "duration": 4.439, "text": "times 1 the times 1 isn't doing anything"}, {"start": 52.32, "duration": 3.96, "text": "and then on the right hand side this x -"}, {"start": 54.079, "duration": 6.041, "text": "4 can be divided from top and bottom so"}, {"start": 56.28, "duration": 5.56, "text": "we now have x + x - 4 = 4 we can add 4"}, {"start": 60.12, "duration": 4.52, "text": "both sides on the left hand side we'll"}, {"start": 61.84, "duration": 5.04, "text": "be left with x + x which is 2X and then"}, {"start": 64.64, "duration": 4.56, "text": "on the right hand side 4 + 4 is equal to"}, {"start": 66.88, "duration": 4.76, "text": "8 we can divide both sides by two and"}, {"start": 69.2, "duration": 4.64, "text": "that'll give us X is equal to 4 well the"}, {"start": 71.64, "duration": 5.6, "text": "problem is earlier we said X cannot"}, {"start": 73.84, "duration": 8.84, "text": "equal 4 so since X can't equal four this"}, {"start": 77.24, "duration": 5.44, "text": "is not a valid solution how exciting"}], "title": "Fun Rational Equation", "description": "https://andymath.com/rational-equations/\n\nHere are some more if you want to try more!", "lengthSeconds": 92, "width": 640, "height": 360}
{"video_id": "9DVPFIunxqU", "transcript": [{"start": 0.03, "duration": 3.869, "text": "hey guys welcome to this video we're"}, {"start": 2.129, "duration": 3.361, "text": "going to solve and graph these the way"}, {"start": 3.899, "duration": 5.551, "text": "you handle this type of problem is"}, {"start": 5.49, "duration": 8.85, "text": "solved for x for each of these three add"}, {"start": 9.45, "duration": 9.96, "text": "5 for X is less than 40 and divided by"}, {"start": 14.34, "duration": 8.699, "text": "4x is less than 10 we'll do the same"}, {"start": 19.41, "duration": 6.689, "text": "thing here minus 4 negative X is less"}, {"start": 23.039, "duration": 8.101, "text": "than or equal to negative 14 divided by"}, {"start": 26.099, "duration": 7.021, "text": "negative 1 and we get x + 14 and anytime"}, {"start": 31.14, "duration": 4.56, "text": "you multiply it or divide by a negative"}, {"start": 33.12, "duration": 3.959, "text": "number you switch the sign so you can't"}, {"start": 35.7, "duration": 4.08, "text": "forget to switch the sign and then we"}, {"start": 37.079, "duration": 5.25, "text": "saw the word and here so and means we"}, {"start": 39.78, "duration": 6.63, "text": "want to know when they are both true so"}, {"start": 42.329, "duration": 7.23, "text": "let's graph this tried to show zero to"}, {"start": 46.41, "duration": 4.89, "text": "10 and the 14 not to scale and I'm sure"}, {"start": 49.559, "duration": 5.91, "text": "many other numbers this pretty much"}, {"start": 51.3, "duration": 5.91, "text": "tells the story 10 is an open circle"}, {"start": 55.469, "duration": 4.171, "text": "because it does not include it and it"}, {"start": 57.21, "duration": 6.45, "text": "says X is less than 10 so I'm going to"}, {"start": 59.64, "duration": 6.269, "text": "go this way and then X greater than or"}, {"start": 63.66, "duration": 5.88, "text": "equal to 14 I'm going to shade the"}, {"start": 65.909, "duration": 5.791, "text": "circle in because it's greater than or"}, {"start": 69.54, "duration": 4.649, "text": "equal to so the equal to means is shaded"}, {"start": 71.7, "duration": 4.77, "text": "in and I'm going to go this way because"}, {"start": 74.189, "duration": 4.921, "text": "X is greater than it so the word and"}, {"start": 76.47, "duration": 5.88, "text": "means we want both of them to be true in"}, {"start": 79.11, "duration": 5.13, "text": "this case both are never true so the"}, {"start": 82.35, "duration": 4.44, "text": "answer is no solution or it could also"}, {"start": 84.24, "duration": 4.98, "text": "be empty set depending on your teacher"}, {"start": 86.79, "duration": 4.619, "text": "so let's answer the question hope that"}, {"start": 89.22, "duration": 4.35, "text": "makes sense please like and subscribe"}, {"start": 91.409, "duration": 4.17, "text": "check out aunty Matt Kong for more"}, {"start": 93.57, "duration": 4.61, "text": "videos and notes and I'll talk to you"}, {"start": 95.579, "duration": 2.601, "text": "guys soon bye"}, {"start": 99.1, "duration": 3.64, "text": "you"}, {"start": 100.68, "duration": 2.06, "text": "you"}, {"start": 103.03, "duration": 2.06, "text": "you"}, {"start": 105.34, "duration": 2.059, "text": "you"}], "title": "Solving Compound Inequalities", "description": "For more visit http://andymath.com\n\nSubscribe here: https://www.youtube.com/channel/UC6KhU3AMLHC-qv7tj0ZJduw?sub_confirmation=1", "lengthSeconds": 113, "width": 640, "height": 360}
{"video_id": "Ddu5xhKC7hY", "transcript": [{"start": 0.08, "duration": 3.52, "text": "area of a circle is pi r squared why is"}, {"start": 2.0, "duration": 3.44, "text": "that how did they figure this out"}, {"start": 3.6, "duration": 3.679, "text": "you can split the circle up into pieces"}, {"start": 5.44, "duration": 3.76, "text": "i split it up into ten pieces here"}, {"start": 7.279, "duration": 3.521, "text": "as we separate them we start to rotate"}, {"start": 9.2, "duration": 2.88, "text": "them like this and line them up and if"}, {"start": 10.8, "duration": 2.48, "text": "we bring them together"}, {"start": 12.08, "duration": 2.24, "text": "i just took all these pieces of the"}, {"start": 13.28, "duration": 2.64, "text": "circle and turned it into a"}, {"start": 14.32, "duration": 2.48, "text": "parallelogram now it's true these are"}, {"start": 15.92, "duration": 2.72, "text": "little curves"}, {"start": 16.8, "duration": 3.52, "text": "the more slices you cut it into the"}, {"start": 18.64, "duration": 3.2, "text": "straighter those curves will be and if"}, {"start": 20.32, "duration": 2.08, "text": "you cut it into an infinite number of"}, {"start": 21.84, "duration": 2.16, "text": "slices"}, {"start": 22.4, "duration": 3.199, "text": "all those curves will go into a straight"}, {"start": 24.0, "duration": 3.199, "text": "line so essentially the circle is the"}, {"start": 25.599, "duration": 3.52, "text": "same thing as this parallelogram the"}, {"start": 27.199, "duration": 3.441, "text": "area of a parallelogram is base times"}, {"start": 29.119, "duration": 3.44, "text": "height so what is the base"}, {"start": 30.64, "duration": 3.599, "text": "and what is the height if we pull our"}, {"start": 32.559, "duration": 2.561, "text": "parallelogram apart you'll notice that"}, {"start": 34.239, "duration": 2.881, "text": "this base"}, {"start": 35.12, "duration": 3.599, "text": "is half of the circumference and"}, {"start": 37.12, "duration": 3.68, "text": "circumference is 2 pi r"}, {"start": 38.719, "duration": 3.761, "text": "so half of the circumference would be pi"}, {"start": 40.8, "duration": 3.52, "text": "r next let's talk about the height if we"}, {"start": 42.48, "duration": 2.64, "text": "pull out one of the slices you'll notice"}, {"start": 44.32, "duration": 3.6, "text": "that the height"}, {"start": 45.12, "duration": 4.56, "text": "is the same thing as the radius of the"}, {"start": 47.92, "duration": 3.84, "text": "circle so we have a base of pi"}, {"start": 49.68, "duration": 3.199, "text": "r and a height of r therefore the area"}, {"start": 51.76, "duration": 3.68, "text": "of this parallelogram"}, {"start": 52.879, "duration": 5.36, "text": "is pi r squared area of a circle is pi r"}, {"start": 55.44, "duration": 2.799, "text": "squared"}], "title": "Why Area of Circle is pi*r^2", "description": "Check out my website! \nhttps://andymath.com/\n\n#Shorts", "lengthSeconds": 56, "width": 202, "height": 360}
{"video_id": "Keu5hC-6GAQ", "transcript": [{"start": 0.04, "duration": 3.96, "text": "hey guys this looks like a fun one it"}, {"start": 1.4, "duration": 4.76, "text": "says solve without logarithms this is"}, {"start": 4.0, "duration": 3.799, "text": "part two in part one I solved it with"}, {"start": 6.16, "duration": 3.04, "text": "logarithms this is what it looked like"}, {"start": 7.799, "duration": 3.081, "text": "right here a lot of people mentioned"}, {"start": 9.2, "duration": 3.399, "text": "there was an easier way to solve it the"}, {"start": 10.88, "duration": 2.959, "text": "reason I did it this way because I"}, {"start": 12.599, "duration": 3.76, "text": "wanted to show what it would look like"}, {"start": 13.839, "duration": 4.881, "text": "if the answer wasn't clean like x equals"}, {"start": 16.359, "duration": 3.641, "text": "6 if x had been some crazy irrational"}, {"start": 18.72, "duration": 4.24, "text": "number you'd have to go through all"}, {"start": 20.0, "duration": 4.96, "text": "these steps but since X was 6 there is a"}, {"start": 22.96, "duration": 3.239, "text": "way to do it without logarithms if you"}, {"start": 24.96, "duration": 3.479, "text": "want to try it on your own positive"}, {"start": 26.199, "duration": 4.84, "text": "right now cuz I'm going to solve it in 3"}, {"start": 28.439, "duration": 4.8, "text": "2 1 first want to express everything in"}, {"start": 31.039, "duration": 4.561, "text": "their prime factors so the N9 can be"}, {"start": 33.239, "duration": 4.521, "text": "broken down into 3^ SAR and the four can"}, {"start": 35.6, "duration": 4.36, "text": "be broken down into 2^ s and then to"}, {"start": 37.76, "duration": 5.36, "text": "break down the 144 let's make a factor"}, {"start": 39.96, "duration": 5.599, "text": "tree 144 breaks down into 12 * 12 and"}, {"start": 43.12, "duration": 5.279, "text": "then each 12 will break down into 6 * 2"}, {"start": 45.559, "duration": 5.121, "text": "so we have 6 * 2 * 6 * 2 and then each"}, {"start": 48.399, "duration": 6.081, "text": "of these sixes will break down into 3 *"}, {"start": 50.68, "duration": 6.559, "text": "2 so we have 3 * 2 * 2 * 3 * 2 * 2 in"}, {"start": 54.48, "duration": 6.16, "text": "this we have 3 * 3 which is 3^ 2 and we"}, {"start": 57.239, "duration": 7.361, "text": "have 2 * 2 * 2 * 2 which is 2 4"}, {"start": 60.64, "duration": 7.36, "text": "so 144 and its prime factors is 3^ 2 * 2"}, {"start": 64.6, "duration": 5.36, "text": "4 so we can change this 144 to 3^ 2 * 2"}, {"start": 68.0, "duration": 4.119, "text": "to 4th now we have everything broken"}, {"start": 69.96, "duration": 3.96, "text": "down into their prime factors we can"}, {"start": 72.119, "duration": 3.68, "text": "clean things up a little bit let's bring"}, {"start": 73.92, "duration": 4.04, "text": "this exponent out here and we can bring"}, {"start": 75.799, "duration": 3.68, "text": "this exponent out here and then this two"}, {"start": 77.96, "duration": 4.96, "text": "will distribute to both of these so this"}, {"start": 79.479, "duration": 5.121, "text": "would make it 2x - 10 and then down here"}, {"start": 82.92, "duration": 5.04, "text": "this two can distribute to both of those"}, {"start": 84.6, "duration": 4.879, "text": "so it'll be 2x - 16 and then next I"}, {"start": 87.96, "duration": 3.76, "text": "don't really like fractions so let's get"}, {"start": 89.479, "duration": 3.92, "text": "rid of this denominator we'll move this"}, {"start": 91.72, "duration": 3.359, "text": "over here and then we'll put in a one"}, {"start": 93.399, "duration": 3.481, "text": "placeholder and then we can focus on"}, {"start": 95.079, "duration": 3.961, "text": "this piece with the denominator using"}, {"start": 96.88, "duration": 4.919, "text": "these notes anytime you have 1/ n that's"}, {"start": 99.04, "duration": 6.0, "text": "the same thing as n^ the1 power so we"}, {"start": 101.799, "duration": 4.561, "text": "can rewrite this as this to the 1 power"}, {"start": 105.04, "duration": 4.8, "text": "and then next we can distribute this"}, {"start": 106.36, "duration": 8.24, "text": "negative 1 to both of these- 1 * POS 2x"}, {"start": 109.84, "duration": 7.12, "text": "is -2X and - 1 * -16 is pos6 and then we"}, {"start": 114.6, "duration": 6.36, "text": "can copy down the rest of it now we have"}, {"start": 116.96, "duration": 6.72, "text": "3 to some power * 2 to some power = 3 to"}, {"start": 120.96, "duration": 5.479, "text": "a power * 2 to a power let's make these"}, {"start": 123.68, "duration": 5.32, "text": "exponents for the 3es blue and we'll"}, {"start": 126.439, "duration": 4.04, "text": "make the exponents for the twos red if"}, {"start": 129.0, "duration": 3.44, "text": "this exponent were to equal this"}, {"start": 130.479, "duration": 3.801, "text": "exponent and this exponent were to equal"}, {"start": 132.44, "duration": 4.12, "text": "this exponent then this equation would"}, {"start": 134.28, "duration": 3.92, "text": "have to be true so let's solve for x for"}, {"start": 136.56, "duration": 3.52, "text": "both of these for this first equation"}, {"start": 138.2, "duration": 3.759, "text": "let's add 10 to both sides on the left"}, {"start": 140.08, "duration": 4.84, "text": "hand side we'll have 2X and on the right"}, {"start": 141.959, "duration": 4.64, "text": "hand side 2 + 10 is equal to 12 and then"}, {"start": 144.92, "duration": 4.36, "text": "to solve for x we can divide both sides"}, {"start": 146.599, "duration": 4.401, "text": "by two and we get X is equal to 6 and"}, {"start": 149.28, "duration": 4.08, "text": "then for this equation we can subtract"}, {"start": 151.0, "duration": 4.48, "text": "16 from both sides on the left hand side"}, {"start": 153.36, "duration": 5.44, "text": "we'll have -2X and on the right hand"}, {"start": 155.48, "duration": 5.28, "text": "side 4 - 16 is equal to -12 and then to"}, {"start": 158.8, "duration": 4.0, "text": "solve for x we can divide both sides by"}, {"start": 160.76, "duration": 5.0, "text": "-2 and that'll leave us with X is equal"}, {"start": 162.8, "duration": 4.76, "text": "to 6 so if x equal 6 this will equal"}, {"start": 165.76, "duration": 3.68, "text": "this and this will equal this which will"}, {"start": 167.56, "duration": 3.599, "text": "make the equation true that is the"}, {"start": 169.44, "duration": 4.2, "text": "answer to our question let's put a box"}, {"start": 171.159, "duration": 3.841, "text": "around it how exciting here's the next"}, {"start": 173.64, "duration": 2.879, "text": "problem I plan on working on if you guys"}, {"start": 175.0, "duration": 3.68, "text": "want to try it out I'll post the"}, {"start": 176.519, "duration": 5.161, "text": "solution as soon as I solve it how"}, {"start": 178.68, "duration": 3.0, "text": "exciting"}], "title": "Part 2 The Easy Way", "description": "Part 1 is here: https://youtu.be/liJ9jTSYouc", "lengthSeconds": 180, "width": 640, "height": 360}
{"video_id": "fa9Q6m1m6gY", "transcript": [{"start": 0.0, "duration": 3.84, "text": "hey guys let's visualize sine squared"}, {"start": 1.76, "duration": 4.16, "text": "plus cosine squared equals 1. this right"}, {"start": 3.84, "duration": 3.919, "text": "here is the graph for sine of x and this"}, {"start": 5.92, "duration": 3.36, "text": "red one is sine squared of x you'll"}, {"start": 7.759, "duration": 3.041, "text": "notice how everywhere where the sine is"}, {"start": 9.28, "duration": 2.96, "text": "negative the sine squared becomes"}, {"start": 10.8, "duration": 2.959, "text": "positive and you can tell here in the"}, {"start": 12.24, "duration": 3.2, "text": "positive area that the shape of the"}, {"start": 13.759, "duration": 3.6, "text": "graph changes a little bit as well so"}, {"start": 15.44, "duration": 3.44, "text": "the sine squared of pi over two is equal"}, {"start": 17.359, "duration": 4.321, "text": "to one we can tell that because the"}, {"start": 18.88, "duration": 5.04, "text": "height of the graph is one at x equals"}, {"start": 21.68, "duration": 3.92, "text": "pi over two we can shade from the x-axis"}, {"start": 23.92, "duration": 3.119, "text": "to the height of the sine squared of x"}, {"start": 25.6, "duration": 3.839, "text": "graph the reason that's nice is because"}, {"start": 27.039, "duration": 3.921, "text": "if i shift it upwards it's no longer the"}, {"start": 29.439, "duration": 3.041, "text": "height on the y-axis that tells me the"}, {"start": 30.96, "duration": 3.119, "text": "function value is the height of the"}, {"start": 32.48, "duration": 3.28, "text": "shading at each point that tells me the"}, {"start": 34.079, "duration": 2.961, "text": "function value i could even turn upside"}, {"start": 35.76, "duration": 3.04, "text": "down like this and it really wouldn't"}, {"start": 37.04, "duration": 3.92, "text": "matter here at x equals pi over 2 we can"}, {"start": 38.8, "duration": 4.16, "text": "see that this height is still 1. next we"}, {"start": 40.96, "duration": 3.759, "text": "can look at the graph of cosine of x and"}, {"start": 42.96, "duration": 4.079, "text": "then in blue here we have cosine squared"}, {"start": 44.719, "duration": 4.16, "text": "of x let's get rid of this cosine of x a"}, {"start": 47.039, "duration": 3.281, "text": "shape between the x-axis and the y-axis"}, {"start": 48.879, "duration": 2.881, "text": "so that the height of the shaded area"}, {"start": 50.32, "duration": 3.28, "text": "gives us the function value and now"}, {"start": 51.76, "duration": 3.76, "text": "let's bring down our sine squared of x"}, {"start": 53.6, "duration": 3.279, "text": "and you see they fit together perfectly"}, {"start": 55.52, "duration": 3.359, "text": "and that gives us sine squared plus"}, {"start": 56.879, "duration": 4.48, "text": "cosine squared is equal to one how"}, {"start": 58.879, "duration": 2.48, "text": "exciting"}], "title": "Pythagorean Identities Visualized #shorts", "description": "https://andymath.com/trigonometric-identities/ #shorts", "lengthSeconds": 60, "width": 360, "height": 640}
{"video_id": "d3teQCaRufE", "transcript": [{"start": 0.0, "duration": 2.84, "text": "he guys let's say we wanted to find the"}, {"start": 1.24, "duration": 4.48, "text": "rectangle where the perimeter was equal"}, {"start": 2.84, "duration": 4.4, "text": "to 420 and the area was equal to 69 we"}, {"start": 5.72, "duration": 3.28, "text": "need to find the base and the height"}, {"start": 7.24, "duration": 3.359, "text": "that would make this true if you want to"}, {"start": 9.0, "duration": 4.599, "text": "try it on your own pause it right now"}, {"start": 10.599, "duration": 5.721, "text": "cuz I'm going to solve it in 3 2 1 so"}, {"start": 13.599, "duration": 4.241, "text": "let's called the base X and the height y"}, {"start": 16.32, "duration": 5.08, "text": "we know that the perimeter is going to"}, {"start": 17.84, "duration": 5.64, "text": "be 2 X's + 2 y and that needs to equal"}, {"start": 21.4, "duration": 4.28, "text": "420 and the area is going to be the base"}, {"start": 23.48, "duration": 4.28, "text": "time height or in other words x y and"}, {"start": 25.68, "duration": 4.04, "text": "that needs to equal 69 so right now we"}, {"start": 27.76, "duration": 4.639, "text": "have two variables X and Y and we have"}, {"start": 29.72, "duration": 3.88, "text": "two equations let's solve for x and y so"}, {"start": 32.399, "duration": 3.201, "text": "I'm going to clean things up a little"}, {"start": 33.6, "duration": 3.959, "text": "bit on top we can simplify things by"}, {"start": 35.6, "duration": 3.68, "text": "dividing everything by two and on bottom"}, {"start": 37.559, "duration": 4.0, "text": "we can get the Y by itself by dividing"}, {"start": 39.28, "duration": 3.92, "text": "both sides by X up here on the left hand"}, {"start": 41.559, "duration": 4.32, "text": "side these twos will cancel each other"}, {"start": 43.2, "duration": 4.96, "text": "out and on the right hand side 420 / 2"}, {"start": 45.879, "duration": 3.801, "text": "is 210 and then down here on the left"}, {"start": 48.16, "duration": 3.28, "text": "hand side these X's will cancel each"}, {"start": 49.68, "duration": 5.519, "text": "other out so we're left with Y is equal"}, {"start": 51.44, "duration": 5.639, "text": "to 69 /x let's take this X+ y = 210 and"}, {"start": 55.199, "duration": 4.52, "text": "copy it down here and we know that Y is"}, {"start": 57.079, "duration": 5.28, "text": "equal to 69 /x so in the place of this y"}, {"start": 59.719, "duration": 4.321, "text": "let's plug in 69x and I don't really"}, {"start": 62.359, "duration": 3.641, "text": "like fractions so let's get rid of this"}, {"start": 64.04, "duration": 3.56, "text": "X in the denominator we can do that by"}, {"start": 66.0, "duration": 3.68, "text": "multiplying both sides of the equation"}, {"start": 67.6, "duration": 5.159, "text": "by X on the left hand side this x will"}, {"start": 69.68, "duration": 5.759, "text": "distribute to give us x * x and x * 69"}, {"start": 72.759, "duration": 6.441, "text": "/x and on the right hand side we get 210"}, {"start": 75.439, "duration": 5.68, "text": "* x x * X gives us x^2 and this X and"}, {"start": 79.2, "duration": 3.599, "text": "this x will cancel each other out this"}, {"start": 81.119, "duration": 4.401, "text": "is a quadratic so I want to set it equal"}, {"start": 82.799, "duration": 4.561, "text": "to zero so let's subtract 210x from both"}, {"start": 85.52, "duration": 3.44, "text": "sides on the right hand side these will"}, {"start": 87.36, "duration": 3.079, "text": "cancel each other out so it's equal to"}, {"start": 88.96, "duration": 5.36, "text": "zero and on the left hand hand side we"}, {"start": 90.439, "duration": 5.72, "text": "have x^2 - 210x + 69 now let's solve"}, {"start": 94.32, "duration": 3.839, "text": "this using the quadratic formula the"}, {"start": 96.159, "duration": 4.441, "text": "leading coefficient is a one so both of"}, {"start": 98.159, "duration": 4.881, "text": "these A's are equal to 1 and then the"}, {"start": 100.6, "duration": 4.0, "text": "second coefficient is -210 so in the"}, {"start": 103.04, "duration": 6.16, "text": "place of both of these B's we can plug"}, {"start": 104.6, "duration": 7.559, "text": "in -210 and last the C is equal to 69"}, {"start": 109.2, "duration": 7.16, "text": "now we can clean this up -210 is equal"}, {"start": 112.159, "duration": 8.28, "text": "to postive 210 -210 2 is equal to"}, {"start": 116.36, "duration": 7.84, "text": "44,100 and 4 * 1 * 69 is = to 2"}, {"start": 120.439, "duration": 8.32, "text": "276 and on bottom 2 * 1 is equal to 2"}, {"start": 124.2, "duration": 6.48, "text": "44,100 - 276 is equal to"}, {"start": 128.759, "duration": 5.721, "text": "43824 and"}, {"start": 130.68, "duration": 6.08, "text": "43824 can we split up into 16 *"}, {"start": 134.48, "duration": 4.839, "text": "2739 and then we can give each of these"}, {"start": 136.76, "duration": 4.92, "text": "their own square root 16 is a perfect"}, {"start": 139.319, "duration": 3.721, "text": "square so we can change this into four"}, {"start": 141.68, "duration": 4.44, "text": "and this two in the denominator can be"}, {"start": 143.04, "duration": 7.68, "text": "split up for both of these terms 210 / 2"}, {"start": 146.12, "duration": 7.479, "text": "is 105 and 4 < TK 2739 / 2 is equal to 2"}, {"start": 150.72, "duration": 7.56, "text": "< tkk 2739 and now we have our values of"}, {"start": 153.599, "duration": 6.401, "text": "X they're 105 plus or- 2 < TK 2739 and"}, {"start": 158.28, "duration": 3.48, "text": "this plus or minus can be split up into"}, {"start": 160.0, "duration": 3.76, "text": "two different numbers we have the plus"}, {"start": 161.76, "duration": 3.479, "text": "version and the minus version and now"}, {"start": 163.76, "duration": 4.28, "text": "for each of these X's we have to find"}, {"start": 165.239, "duration": 5.161, "text": "the corresponding y let's use this x + y"}, {"start": 168.04, "duration": 4.24, "text": "is equal to 210 we can subtract X from"}, {"start": 170.4, "duration": 3.839, "text": "both sides these two x's will cancel"}, {"start": 172.28, "duration": 5.4, "text": "each other out and now we know that Y is"}, {"start": 174.239, "duration": 6.601, "text": "equal to 210 - x so in the place of both"}, {"start": 177.68, "duration": 5.24, "text": "these y's let's plug in 210 - X for the"}, {"start": 180.84, "duration": 4.64, "text": "first box X is equal to this so let's"}, {"start": 182.92, "duration": 4.76, "text": "plug this in for x and for the second"}, {"start": 185.48, "duration": 4.759, "text": "box X is equal to this so let's plug in"}, {"start": 187.68, "duration": 4.119, "text": "this for X both of these are positive so"}, {"start": 190.239, "duration": 3.601, "text": "the negative will make them both"}, {"start": 191.799, "duration": 3.36, "text": "negative and aside these parentheses the"}, {"start": 193.84, "duration": 3.119, "text": "first one's positive and the second"}, {"start": 195.159, "duration": 3.36, "text": "one's negative after we subtract they're"}, {"start": 196.959, "duration": 3.0, "text": "going to switch the first one will be"}, {"start": 198.519, "duration": 4.841, "text": "negative and the second one will be"}, {"start": 199.959, "duration": 5.721, "text": "positive now for the first box 210 - 105"}, {"start": 203.36, "duration": 4.68, "text": "is equal to 105 so we have the value of"}, {"start": 205.68, "duration": 5.68, "text": "y that corresponds to this X and then"}, {"start": 208.04, "duration": 4.96, "text": "for the second box 210 - 5 is 105 so we"}, {"start": 211.36, "duration": 3.879, "text": "have the corresponding value of y for"}, {"start": 213.0, "duration": 4.08, "text": "this value of x so all these boxes are"}, {"start": 215.239, "duration": 3.64, "text": "doing are just swapping the X and the Y"}, {"start": 217.08, "duration": 3.6, "text": "so we can just get rid of the second box"}, {"start": 218.879, "duration": 5.761, "text": "so now we have the value of our base is"}, {"start": 220.68, "duration": 7.839, "text": "105 + 2 < TK 2739 and the value of our"}, {"start": 224.64, "duration": 5.44, "text": "height is 105 - 2 < TK 2739 so now we"}, {"start": 228.519, "duration": 2.961, "text": "can put these values in for the base and"}, {"start": 230.08, "duration": 3.4, "text": "the height and create a problem that"}, {"start": 231.48, "duration": 3.24, "text": "says find the area and perimeter and if"}, {"start": 233.48, "duration": 3.28, "text": "you want to see the video solving this"}, {"start": 234.72, "duration": 5.12, "text": "problem the link is in the description"}, {"start": 236.76, "duration": 3.08, "text": "how exciting"}], "title": "Nice Rectangle Part 2", "description": "https://youtu.be/-f1ozWwRDSc\nLink to Part 1", "lengthSeconds": 238, "width": 640, "height": 360}
{"video_id": "llSeBkXedYk", "transcript": [{"start": 0.0, "duration": 3.84, "text": "guys this looks like a fun one it was"}, {"start": 1.62, "duration": 3.9, "text": "posted by my math Solutions so let's"}, {"start": 3.84, "duration": 3.419, "text": "solve it we have two rectangles in a"}, {"start": 5.52, "duration": 3.06, "text": "triangle that we know the area and we're"}, {"start": 7.259, "duration": 2.821, "text": "trying to find the area of this last"}, {"start": 8.58, "duration": 2.76, "text": "triangle I think there's a lot of"}, {"start": 10.08, "duration": 2.4, "text": "different ways to solve this but I want"}, {"start": 11.34, "duration": 4.2, "text": "to share with you what I ended up doing"}, {"start": 12.48, "duration": 5.879, "text": "I called this side x this side y this"}, {"start": 15.54, "duration": 6.239, "text": "side Z and this side W let's start with"}, {"start": 18.359, "duration": 5.461, "text": "this rectangle x times Z is equal to 15."}, {"start": 21.779, "duration": 5.82, "text": "and after we divide both sides by X we"}, {"start": 23.82, "duration": 6.42, "text": "get Z is equal to 15 over X and if we go"}, {"start": 27.599, "duration": 5.221, "text": "to this rectangle y times W is equal to"}, {"start": 30.24, "duration": 4.979, "text": "12. divide both sides by y now we get W"}, {"start": 32.82, "duration": 4.919, "text": "is equal to 12 over y so now let's"}, {"start": 35.219, "duration": 5.101, "text": "update the Z to make it 15 over X and"}, {"start": 37.739, "duration": 4.5, "text": "let's update the W to make it 12 over Y"}, {"start": 40.32, "duration": 3.3, "text": "and then let's get rid of this stuff now"}, {"start": 42.239, "duration": 3.121, "text": "let's look at this triangle the area of"}, {"start": 43.62, "duration": 3.84, "text": "a triangle is one half base times height"}, {"start": 45.36, "duration": 4.679, "text": "so this is going to be one half x times"}, {"start": 47.46, "duration": 4.439, "text": "y equals nine and after we multiply both"}, {"start": 50.039, "duration": 3.721, "text": "sides by 2 on the left hand side we get"}, {"start": 51.899, "duration": 3.961, "text": "x y and on the right hand side we get"}, {"start": 53.76, "duration": 3.72, "text": "18. this looks important so let's put a"}, {"start": 55.86, "duration": 3.42, "text": "box around it now we're ready for the"}, {"start": 57.48, "duration": 3.18, "text": "question mark since it is the area of a"}, {"start": 59.28, "duration": 3.119, "text": "triangle it's going to be equal to one"}, {"start": 60.66, "duration": 3.239, "text": "half base times height if we turn our"}, {"start": 62.399, "duration": 4.021, "text": "heads sideways we can think of the base"}, {"start": 63.899, "duration": 4.561, "text": "is equal to 15 over X and the height is"}, {"start": 66.42, "duration": 4.379, "text": "12 over Y and now we're just multiplying"}, {"start": 68.46, "duration": 5.1, "text": "fractions 1 times 15 times 12 is equal"}, {"start": 70.799, "duration": 5.341, "text": "to 180 and 2 times x times Y is equal to"}, {"start": 73.56, "duration": 4.919, "text": "2xy and we could simplify this 180"}, {"start": 76.14, "duration": 4.26, "text": "divided by 2 is equal to 90 and then we"}, {"start": 78.479, "duration": 4.021, "text": "still have the X Y earlier we figured"}, {"start": 80.4, "duration": 4.14, "text": "out that x times y was equal to 18 so"}, {"start": 82.5, "duration": 4.74, "text": "I'm going to change this into an 18 and"}, {"start": 84.54, "duration": 4.619, "text": "then 90 divided by 18 is equal to 5 and"}, {"start": 87.24, "duration": 3.18, "text": "it's going to be centimeters squared and"}, {"start": 89.159, "duration": 4.261, "text": "that's the answer to our question the"}, {"start": 90.42, "duration": 4.8, "text": "area of this blue triangle is equal to 5"}, {"start": 93.42, "duration": 5.9, "text": "centimeters squared let's make it bigger"}, {"start": 95.22, "duration": 4.1, "text": "and put a box around it how exciting"}], "title": "Area of Blue Triangle", "description": "I thought this was a fun one. There might be a better way to solve it, this was my immediate first attempt.", "lengthSeconds": 98, "width": 640, "height": 360}
{"video_id": "w6TVJVQSIPU", "transcript": [{"start": 0.0, "duration": 3.06, "text": "you guys haven't seen this all over the"}, {"start": 1.5, "duration": 3.06, "text": "place someone even emailed it to me"}, {"start": 3.06, "duration": 3.06, "text": "asking me to solve it so let's just"}, {"start": 4.56, "duration": 3.18, "text": "solve it so this thing is a square"}, {"start": 6.12, "duration": 3.42, "text": "they're telling us that right there now"}, {"start": 7.74, "duration": 3.359, "text": "we have three triangles that we know the"}, {"start": 9.54, "duration": 3.3, "text": "areas and then we have this triangle we"}, {"start": 11.099, "duration": 3.361, "text": "don't know the area since it's a square"}, {"start": 12.84, "duration": 3.779, "text": "all the sides are congruent so I'm going"}, {"start": 14.46, "duration": 3.48, "text": "to call each of the sides X so this"}, {"start": 16.619, "duration": 2.941, "text": "green portion right here I'm going to"}, {"start": 17.94, "duration": 3.9, "text": "call it Y which means this yellow"}, {"start": 19.56, "duration": 4.92, "text": "portion would be the whole thing x minus"}, {"start": 21.84, "duration": 4.019, "text": "y but let's call this blue portion Z and"}, {"start": 24.48, "duration": 3.78, "text": "that would make this yellow portion"}, {"start": 25.859, "duration": 4.321, "text": "equal to x minus Z so let's look at this"}, {"start": 28.26, "duration": 3.9, "text": "green triangle the area of a triangle is"}, {"start": 30.18, "duration": 4.86, "text": "one half base times height so we can say"}, {"start": 32.16, "duration": 4.62, "text": "one half x times Y is equal to the area"}, {"start": 35.04, "duration": 3.78, "text": "which is four and then if we look at"}, {"start": 36.78, "duration": 4.38, "text": "this blue one it'd be one half the x"}, {"start": 38.82, "duration": 4.32, "text": "times Z is equal to two and then for"}, {"start": 41.16, "duration": 4.079, "text": "this last triangle it'd be one half this"}, {"start": 43.14, "duration": 3.48, "text": "thing times this thing equals three I"}, {"start": 45.239, "duration": 3.181, "text": "don't really like fractions so let's"}, {"start": 46.62, "duration": 3.66, "text": "multiply both sides of all three of"}, {"start": 48.42, "duration": 3.659, "text": "these equations by two all these"}, {"start": 50.28, "duration": 3.06, "text": "one-halves will go away and the right"}, {"start": 52.079, "duration": 3.541, "text": "sides are just going to double we'll"}, {"start": 53.34, "duration": 4.02, "text": "have eight four and six so now let's"}, {"start": 55.62, "duration": 3.959, "text": "focus on these top two equations let's"}, {"start": 57.36, "duration": 4.98, "text": "divide both sides of both equations by X"}, {"start": 59.579, "duration": 4.921, "text": "this this X over X cancels this X over X"}, {"start": 62.34, "duration": 4.38, "text": "cancels and these just become 8 over X"}, {"start": 64.5, "duration": 4.68, "text": "and 4 over X so now let's do some"}, {"start": 66.72, "duration": 5.46, "text": "substitution for this Z I want to plug"}, {"start": 69.18, "duration": 5.34, "text": "in 4 over X and for this y I want to"}, {"start": 72.18, "duration": 4.68, "text": "plug in 8 over X so now on the left hand"}, {"start": 74.52, "duration": 5.099, "text": "side let's distribute x times x is x"}, {"start": 76.86, "duration": 5.46, "text": "squared x times negative 8 over X is"}, {"start": 79.619, "duration": 4.921, "text": "negative x times 8 over X these two x's"}, {"start": 82.32, "duration": 4.14, "text": "will cancel so it's just negative 8. and"}, {"start": 84.54, "duration": 4.14, "text": "then this times this is negative x times"}, {"start": 86.46, "duration": 4.32, "text": "4 over X but these X's cancel again to"}, {"start": 88.68, "duration": 3.66, "text": "give us negative 4. and this times this"}, {"start": 90.78, "duration": 3.839, "text": "is a negative times a negative which is"}, {"start": 92.34, "duration": 5.22, "text": "positive it's going to be a fraction or"}, {"start": 94.619, "duration": 4.741, "text": "4 times 8 is 32 and x times x is x"}, {"start": 97.56, "duration": 3.9, "text": "squared and this is all still equal to"}, {"start": 99.36, "duration": 3.899, "text": "6. well negative 8 negative 4 can"}, {"start": 101.46, "duration": 3.36, "text": "combine to negative 12. I don't really"}, {"start": 103.259, "duration": 2.941, "text": "like fractions so let's get rid of this"}, {"start": 104.82, "duration": 3.479, "text": "x squared in the denominator by"}, {"start": 106.2, "duration": 3.779, "text": "multiplying both sides by x squared x"}, {"start": 108.299, "duration": 3.661, "text": "squared times x squared is X to the"}, {"start": 109.979, "duration": 5.1, "text": "fourth x squared times negative 12 is"}, {"start": 111.96, "duration": 5.04, "text": "negative 12x squared x squared times 32"}, {"start": 115.079, "duration": 4.381, "text": "over x squared is just going to be 32"}, {"start": 117.0, "duration": 4.38, "text": "and then 6 times x squared is 6X squared"}, {"start": 119.46, "duration": 3.659, "text": "and then we get subtract 6X squared from"}, {"start": 121.38, "duration": 3.839, "text": "both sides which will make this side 0"}, {"start": 123.119, "duration": 3.901, "text": "and this side would be negative 18x"}, {"start": 125.219, "duration": 4.081, "text": "squared and now we want to factor this"}, {"start": 127.02, "duration": 3.9, "text": "since it's set equal to zero so x to the"}, {"start": 129.3, "duration": 3.18, "text": "fourth here means we want an x squared"}, {"start": 130.92, "duration": 3.48, "text": "and an x squared and then we want two"}, {"start": 132.48, "duration": 3.78, "text": "numbers that add to negative 18 and"}, {"start": 134.4, "duration": 4.08, "text": "multiply to 32. that's going to be"}, {"start": 136.26, "duration": 4.559, "text": "negative 16 and negative 2. this plus"}, {"start": 138.48, "duration": 4.56, "text": "this is negative 18 and this times this"}, {"start": 140.819, "duration": 4.021, "text": "is positive 32. now since this times"}, {"start": 143.04, "duration": 4.08, "text": "this equals zero we can set each of"}, {"start": 144.84, "duration": 4.2, "text": "these equal to zero we can add 16 to"}, {"start": 147.12, "duration": 4.8, "text": "both sides or add 2 to both sides"}, {"start": 149.04, "duration": 5.76, "text": "negative 16 and positive 16 cancel and 0"}, {"start": 151.92, "duration": 5.459, "text": "plus 16 is 16. negative 2 plus 2 cancel"}, {"start": 154.8, "duration": 4.26, "text": "and 0 plus 2 is 2. let's smush things"}, {"start": 157.379, "duration": 3.121, "text": "together and take the square root square"}, {"start": 159.06, "duration": 4.02, "text": "root of x squared is the absolute value"}, {"start": 160.5, "duration": 3.72, "text": "of x and square root of 16 is 4 square"}, {"start": 163.08, "duration": 2.76, "text": "root of x squared is the absolute value"}, {"start": 164.22, "duration": 4.14, "text": "of x and the square root of 2 is square"}, {"start": 165.84, "duration": 4.8, "text": "root of 2. since X represents the length"}, {"start": 168.36, "duration": 4.8, "text": "it must be positive so these absolute"}, {"start": 170.64, "duration": 4.44, "text": "values are not necessary so we know X is"}, {"start": 173.16, "duration": 4.439, "text": "either 4 or square root of 2 and let's"}, {"start": 175.08, "duration": 4.5, "text": "check them so if x is equal to 4 the"}, {"start": 177.599, "duration": 3.961, "text": "area of the square would be equal to 4"}, {"start": 179.58, "duration": 3.72, "text": "Square squared and if x is square root"}, {"start": 181.56, "duration": 3.78, "text": "of 2 then the area would be square root"}, {"start": 183.3, "duration": 4.019, "text": "of 2 squared 4 squared is equal to 16"}, {"start": 185.34, "duration": 4.14, "text": "and square root of 2 squared is equal to"}, {"start": 187.319, "duration": 4.2, "text": "2. so the area of this green triangle"}, {"start": 189.48, "duration": 3.78, "text": "right here is four there's no way the"}, {"start": 191.519, "duration": 3.961, "text": "area of the whole Square can be just two"}, {"start": 193.26, "duration": 3.54, "text": "so we can disregard that value so we"}, {"start": 195.48, "duration": 3.66, "text": "know that the area of the entire square"}, {"start": 196.8, "duration": 4.32, "text": "is equal to 16. so we can add this 2"}, {"start": 199.14, "duration": 4.5, "text": "plus the 3 plus the 4 plus the question"}, {"start": 201.12, "duration": 4.5, "text": "mark and set it equal to 16. and after"}, {"start": 203.64, "duration": 3.9, "text": "we subtract 9 from both sides we get the"}, {"start": 205.62, "duration": 3.479, "text": "question mark is equal to seven that"}, {"start": 207.54, "duration": 3.54, "text": "means the area of this red triangle is"}, {"start": 209.099, "duration": 3.78, "text": "equal to seven centimeters squared we"}, {"start": 211.08, "duration": 4.879, "text": "can blow it up and put a box around it"}, {"start": 212.879, "duration": 3.08, "text": "how exciting"}], "title": "Area of Red Triangle", "description": "This is a popular one I've seen in a couple different places. I am not sure where it originated. If anyone knows the creator, let me know!", "lengthSeconds": 214, "width": 640, "height": 360}
{"video_id": "4XB4eRJgkOY", "transcript": [{"start": 0.04, "duration": 3.48, "text": "hey guys this looks like a fun one it"}, {"start": 1.439, "duration": 4.601, "text": "says what's the angle we want to know"}, {"start": 3.52, "duration": 4.399, "text": "this angle right here this is day 11 of"}, {"start": 6.04, "duration": 3.8, "text": "our Advent calendar for the month of"}, {"start": 7.919, "duration": 4.8, "text": "December we're going to solve 31 Katrina"}, {"start": 9.84, "duration": 4.2, "text": "ad puzzles in 31 days I just remembered"}, {"start": 12.719, "duration": 3.12, "text": "something somebody said it was hard to"}, {"start": 14.04, "duration": 3.96, "text": "see the yellow against white so let's"}, {"start": 15.839, "duration": 4.241, "text": "make this thing black so now you can see"}, {"start": 18.0, "duration": 3.6, "text": "there's two circles and the angle I"}, {"start": 20.08, "duration": 4.24, "text": "think it's safe to assume that this is"}, {"start": 21.6, "duration": 4.519, "text": "tangent here here and here if you want"}, {"start": 24.32, "duration": 5.439, "text": "to try it on your own pause it right now"}, {"start": 26.119, "duration": 5.24, "text": "cuz I'm going to solve it in 3 2 1 so"}, {"start": 29.759, "duration": 3.361, "text": "let's copy this down here and I'm going"}, {"start": 31.359, "duration": 3.601, "text": "to leave the circle yellow here because"}, {"start": 33.12, "duration": 3.84, "text": "I'm going to be writing in here these"}, {"start": 34.96, "duration": 3.759, "text": "two circles are tangent right here so"}, {"start": 36.96, "duration": 3.84, "text": "let's construct the tangent line between"}, {"start": 38.719, "duration": 3.881, "text": "the two circles and then I'm also"}, {"start": 40.8, "duration": 4.079, "text": "assuming this is tangent right here so"}, {"start": 42.6, "duration": 3.599, "text": "let's draw this tangent line there's a"}, {"start": 44.879, "duration": 3.721, "text": "special property if you have two"}, {"start": 46.199, "duration": 4.601, "text": "tangents to a circle if the tangents"}, {"start": 48.6, "duration": 4.639, "text": "intersect these two distances are the"}, {"start": 50.8, "duration": 4.559, "text": "same so if we go up here let's shorten"}, {"start": 53.239, "duration": 4.0, "text": "this tangent line and shorten this"}, {"start": 55.359, "duration": 4.36, "text": "tangent line and then since these are"}, {"start": 57.239, "duration": 4.48, "text": "both tangent lines to the same Circle"}, {"start": 59.719, "duration": 4.081, "text": "there the same length and then if we"}, {"start": 61.719, "duration": 3.961, "text": "take this line and make it all red we"}, {"start": 63.8, "duration": 4.44, "text": "now have a nice looking red triangle and"}, {"start": 65.68, "duration": 4.36, "text": "it's an isoceles triangle since these"}, {"start": 68.24, "duration": 3.879, "text": "two sides are congruent that means these"}, {"start": 70.04, "duration": 5.16, "text": "two base angles are congruent let's call"}, {"start": 72.119, "duration": 4.481, "text": "them both a degrees let's clean this up"}, {"start": 75.2, "duration": 2.44, "text": "and let's give ourselves some room"}, {"start": 76.6, "duration": 2.48, "text": "because we're going to look at this"}, {"start": 77.64, "duration": 4.24, "text": "other tangent line and we're going to"}, {"start": 79.08, "duration": 4.679, "text": "extend it all the way up here this red"}, {"start": 81.88, "duration": 3.84, "text": "distance and this red distance will also"}, {"start": 83.759, "duration": 3.961, "text": "be equal to each other and if we"}, {"start": 85.72, "duration": 4.359, "text": "complete this right here we have another"}, {"start": 87.72, "duration": 4.84, "text": "cool looking red triangle that's also is"}, {"start": 90.079, "duration": 4.161, "text": "oses since these two sides are congruent"}, {"start": 92.56, "duration": 5.0, "text": "the base angles are congruent let's give"}, {"start": 94.24, "duration": 5.48, "text": "them both a measure of B de and now"}, {"start": 97.56, "duration": 4.8, "text": "let's clean this up and then this angle"}, {"start": 99.72, "duration": 6.64, "text": "right here is made up of b and a we can"}, {"start": 102.36, "duration": 6.92, "text": "make it a single angle of B+ a de and"}, {"start": 106.36, "duration": 5.0, "text": "this B+ a degrees that is our question"}, {"start": 109.28, "duration": 5.56, "text": "mark so we can say question mark is"}, {"start": 111.36, "duration": 5.56, "text": "equal to B+ a de this looks important"}, {"start": 114.84, "duration": 5.319, "text": "let's put a box around it and let's move"}, {"start": 116.92, "duration": 6.199, "text": "it down here next let's focus on this"}, {"start": 120.159, "duration": 4.481, "text": "quadrilateral right here this angle will"}, {"start": 123.119, "duration": 3.881, "text": "be a right angle because of the"}, {"start": 124.64, "duration": 4.56, "text": "rectangle and let's focus on our new"}, {"start": 127.0, "duration": 4.64, "text": "quadrilateral the sum of the interior"}, {"start": 129.2, "duration": 6.24, "text": "angles of any quadrilateral is"}, {"start": 131.64, "duration": 7.12, "text": "360\u00b0 so we can say 90 + a plus a plus b"}, {"start": 135.44, "duration": 5.439, "text": "plus b equal 360 then let's subtract 90"}, {"start": 138.76, "duration": 5.4, "text": "from both sides on the left hand side"}, {"start": 140.879, "duration": 5.321, "text": "these are gone a plus a is 2 a B plus b"}, {"start": 144.16, "duration": 5.439, "text": "is 2b and then on the right hand side"}, {"start": 146.2, "duration": 5.56, "text": "360 - 90 is 270 then we can divide both"}, {"start": 149.599, "duration": 5.521, "text": "sides by 2 on the left hand side this is"}, {"start": 151.76, "duration": 7.559, "text": "A + B and on the right hand side 270 / 2"}, {"start": 155.12, "duration": 7.119, "text": "is 135 so now we know a + b is equal to"}, {"start": 159.319, "duration": 4.84, "text": "135 and our question mark was B plus a"}, {"start": 162.239, "duration": 5.161, "text": "we can switch these and now in the place"}, {"start": 164.159, "duration": 5.041, "text": "of B plus a we can plug in 135 so this"}, {"start": 167.4, "duration": 3.04, "text": "is the answer to her question this angle"}, {"start": 169.2, "duration": 4.319, "text": "is going to be"}, {"start": 170.44, "duration": 5.799, "text": "135\u00b0 let's put a box around it how"}, {"start": 173.519, "duration": 5.08, "text": "exciting this is the answer to day 11"}, {"start": 176.239, "duration": 4.241, "text": "this is the puzzle for day 12 it says"}, {"start": 178.599, "duration": 4.241, "text": "what's the area of the semi semicircle"}, {"start": 180.48, "duration": 4.52, "text": "inside this equilateral triangle so"}, {"start": 182.84, "duration": 3.759, "text": "we're given that this circle six we're"}, {"start": 185.0, "duration": 3.36, "text": "given that's an equilateral triangle"}, {"start": 186.599, "duration": 3.961, "text": "that's super helpful and then we're"}, {"start": 188.36, "duration": 6.4, "text": "given this semicircle this will be a fun"}, {"start": 190.56, "duration": 4.2, "text": "one how exciting"}], "title": "Aggvent Calendar Day 11", "description": "https://andymath.com/agg-vent-calendar/\n\nHere is the link to the page!", "lengthSeconds": 193, "width": 640, "height": 360}
{"video_id": "Qtvr5zcBzUs", "transcript": [{"start": 0.04, "duration": 4.36, "text": "hey guys this looks like a fun one we're"}, {"start": 1.52, "duration": 5.04, "text": "given a rectangle and a semicircle and"}, {"start": 4.4, "duration": 3.96, "text": "this distance of six right here and it"}, {"start": 6.56, "duration": 5.159, "text": "wants to know what is the area of the"}, {"start": 8.36, "duration": 5.08, "text": "rectangle this is day 10 of our Advent"}, {"start": 11.719, "duration": 4.64, "text": "calendar for the month of December we're"}, {"start": 13.44, "duration": 4.839, "text": "going to do 31 Katrina a puzzles in 31"}, {"start": 16.359, "duration": 3.721, "text": "days if you want to try it on your own"}, {"start": 18.279, "duration": 5.641, "text": "pause it right now because I'm going to"}, {"start": 20.08, "duration": 5.4, "text": "solve it in three two one the puzzle"}, {"start": 23.92, "duration": 3.8, "text": "asking for the area of the rectangle so"}, {"start": 25.48, "duration": 4.68, "text": "let's write out that formula the area of"}, {"start": 27.72, "duration": 3.92, "text": "a rectangle is base time height so we"}, {"start": 30.16, "duration": 3.559, "text": "just got to figure out what is the base"}, {"start": 31.64, "duration": 4.079, "text": "and what is the height first let's find"}, {"start": 33.719, "duration": 5.401, "text": "the center of the semicircle and draw"}, {"start": 35.719, "duration": 5.0, "text": "this radius that's part of the base and"}, {"start": 39.12, "duration": 3.439, "text": "then we can draw this radius of the"}, {"start": 40.719, "duration": 4.041, "text": "semicircle that'll be the height of our"}, {"start": 42.559, "duration": 4.241, "text": "rectangle and to finish up the base of"}, {"start": 44.76, "duration": 3.88, "text": "our rectangle it's not quite a full"}, {"start": 46.8, "duration": 4.32, "text": "radius so let's just give it a variable"}, {"start": 48.64, "duration": 6.12, "text": "X so now we can express the area of the"}, {"start": 51.12, "duration": 5.88, "text": "rectangle the base is x + r and the"}, {"start": 54.76, "duration": 5.72, "text": "height is equal to R and then we can"}, {"start": 57.0, "duration": 6.879, "text": "distribute that x * R is XR and r * R is"}, {"start": 60.48, "duration": 6.16, "text": "R 2 so the area of our rectangle is XR +"}, {"start": 63.879, "duration": 4.88, "text": "r 2 this looks important let's put an"}, {"start": 66.64, "duration": 4.44, "text": "ellipse around it and I don't think we"}, {"start": 68.759, "duration": 4.481, "text": "need this stuff anymore and let's move"}, {"start": 71.08, "duration": 5.12, "text": "this down here next let's swing this"}, {"start": 73.24, "duration": 6.72, "text": "radius down here let's label this other"}, {"start": 76.2, "duration": 6.12, "text": "unknown distance as y this YX and R make"}, {"start": 79.96, "duration": 5.72, "text": "a nice looking right triangle let's do"}, {"start": 82.32, "duration": 6.479, "text": "Pythagorean theorem it'll be x^2 + y^2"}, {"start": 85.68, "duration": 4.68, "text": "is equal to R 2 and then I see another"}, {"start": 88.799, "duration": 4.561, "text": "right triangle let's get rid of of this"}, {"start": 90.36, "duration": 4.96, "text": "radius and focus on this right triangle"}, {"start": 93.36, "duration": 8.16, "text": "and we can do Pagan theorem again it'll"}, {"start": 95.32, "duration": 7.6, "text": "be the quantity x + r^ 2 + y^2 = 6^ 2"}, {"start": 101.52, "duration": 5.04, "text": "now let's play around with this a little"}, {"start": 102.92, "duration": 6.159, "text": "bit x + r 2 is the same thing as x + r *"}, {"start": 106.56, "duration": 6.64, "text": "x + r and after we multiply this out"}, {"start": 109.079, "duration": 5.68, "text": "we'll have x^2 + 2xr + r 2 these two"}, {"start": 113.2, "duration": 4.0, "text": "things are the same so let's get rid of"}, {"start": 114.759, "duration": 5.841, "text": "this I'm noticing up here we have an x^2"}, {"start": 117.2, "duration": 5.36, "text": "+ y^2 so let's bring this X2 over here"}, {"start": 120.6, "duration": 4.28, "text": "so now these look the same and let's"}, {"start": 122.56, "duration": 5.36, "text": "bring this stuff up here so we know that"}, {"start": 124.88, "duration": 5.76, "text": "x^2 + y^2 is equal to r^ 2 so in the"}, {"start": 127.92, "duration": 7.44, "text": "place of this x^2 + y^2 let's substitute"}, {"start": 130.64, "duration": 6.319, "text": "r^ 2 and then r^ 2 + r 2 is 2 R 2 let's"}, {"start": 135.36, "duration": 3.159, "text": "smush all of this stuff together and"}, {"start": 136.959, "duration": 4.56, "text": "they both have a two in there let's"}, {"start": 138.519, "duration": 7.041, "text": "factor out that two 2 * what gives us"}, {"start": 141.519, "duration": 6.681, "text": "2xr that's XR and 2 * what gives us 2 R"}, {"start": 145.56, "duration": 4.399, "text": "2 that'll be R 2 and if you didn't fully"}, {"start": 148.2, "duration": 3.52, "text": "follow that just imagine Distributing"}, {"start": 149.959, "duration": 4.081, "text": "this two to both of these it bring us"}, {"start": 151.72, "duration": 4.92, "text": "back up here so this is the same thing"}, {"start": 154.04, "duration": 5.479, "text": "as this let's get rid of this and move"}, {"start": 156.64, "duration": 6.0, "text": "this up here next on the right hand side"}, {"start": 159.519, "duration": 4.761, "text": "this 6 squar can be changed into 36 and"}, {"start": 162.64, "duration": 3.879, "text": "now both sides of our equation can be"}, {"start": 164.28, "duration": 4.44, "text": "divided by two on the left hand side"}, {"start": 166.519, "duration": 5.761, "text": "these twos will cancel each other out"}, {"start": 168.72, "duration": 6.599, "text": "and on the right hand side 36 / 2 is 18"}, {"start": 172.28, "duration": 5.52, "text": "so now we have XR plus r 2 is equal to"}, {"start": 175.319, "duration": 7.0, "text": "18 but that's the same thing as our area"}, {"start": 177.8, "duration": 7.32, "text": "XR + r 2 so the place of this XR + r^ 2"}, {"start": 182.319, "duration": 4.961, "text": "let's substitute in 18 let's give it a"}, {"start": 185.12, "duration": 4.479, "text": "label of unit squared and this is the"}, {"start": 187.28, "duration": 5.4, "text": "answer to our question the area of this"}, {"start": 189.599, "duration": 5.64, "text": "rectangle is 18 units squared how"}, {"start": 192.68, "duration": 5.24, "text": "exciting this was day 10 of our Advent"}, {"start": 195.239, "duration": 4.92, "text": "calendar and this will be day 11 it's"}, {"start": 197.92, "duration": 4.12, "text": "asking us what's this angle but it"}, {"start": 200.159, "duration": 6.881, "text": "doesn't really seem to provide very much"}, {"start": 202.04, "duration": 5.0, "text": "other information how exciting"}], "title": "Aggvent Calendar Day 10", "description": "https://andymath.com/agg-vent-calendar/\n\n10 days in, 21 to go!", "lengthSeconds": 205, "width": 640, "height": 360}
{"video_id": "3_Y3v3avZ7g", "transcript": [{"start": 0.04, "duration": 5.279, "text": "hey guys this looks like a fun one it"}, {"start": 1.439, "duration": 7.041, "text": "says solve for x and it gives us XM + Dy"}, {"start": 5.319, "duration": 4.841, "text": "= a n + xth we got to get both X's to"}, {"start": 8.48, "duration": 4.199, "text": "the same side of the equation so I'm"}, {"start": 10.16, "duration": 4.439, "text": "going to subtract xth from both sides"}, {"start": 12.679, "duration": 3.761, "text": "and then while we're at it this Dy does"}, {"start": 14.599, "duration": 3.68, "text": "not have an X in it so we want to move"}, {"start": 16.44, "duration": 4.44, "text": "it to the other side and we can do that"}, {"start": 18.279, "duration": 4.561, "text": "by subtracting Dy from both sides this"}, {"start": 20.88, "duration": 4.0, "text": "positive Dy and negative Dy will cancel"}, {"start": 22.84, "duration": 3.679, "text": "each other out and the positive XT and"}, {"start": 24.88, "duration": 5.2, "text": "the negative XT will cancel each other"}, {"start": 26.519, "duration": 6.001, "text": "out so we're left with X Maus xth = a"}, {"start": 30.08, "duration": 4.84, "text": "minus Dy now all the X's are on the same"}, {"start": 32.52, "duration": 4.68, "text": "side and every term has an X in it so"}, {"start": 34.92, "duration": 4.44, "text": "let's factor out that X when we Factor"}, {"start": 37.2, "duration": 3.76, "text": "the X out of this first term it's an MA"}, {"start": 39.36, "duration": 3.56, "text": "that's remaining and when we Factor the"}, {"start": 40.96, "duration": 3.439, "text": "X out of this term it's a th that's"}, {"start": 42.92, "duration": 2.799, "text": "remaining next to get the X all by"}, {"start": 44.399, "duration": 4.68, "text": "itself we're going to divide both sides"}, {"start": 45.719, "duration": 4.961, "text": "by ma minus th the ma minus th's on top"}, {"start": 49.079, "duration": 4.28, "text": "and bottom will divide each other out"}, {"start": 50.68, "duration": 4.96, "text": "and that leaves us with X = a nus dy"}, {"start": 53.359, "duration": 4.321, "text": "over ma minus th this is the answer to"}, {"start": 55.64, "duration": 5.28, "text": "our question let's put a box around it"}, {"start": 57.68, "duration": 3.24, "text": "how exciting"}], "title": "How exciting!", "description": "I hope you guys like this one!", "lengthSeconds": 59, "width": 270, "height": 480}
{"video_id": "LAy7emYBsjg", "transcript": [{"start": 0.0, "duration": 2.36, "text": "hey guys I thought this was a fun"}, {"start": 1.079, "duration": 3.481, "text": "problem and I thought of a fun way to"}, {"start": 2.36, "duration": 3.76, "text": "demonstrate it let's focus right here"}, {"start": 4.56, "duration": 4.52, "text": "this X in the numerator can be broken"}, {"start": 6.12, "duration": 4.599, "text": "out into < TK x * squ root of x now"}, {"start": 9.08, "duration": 3.24, "text": "these two square root of x's match so"}, {"start": 10.719, "duration": 2.92, "text": "they can cancel each other out and now"}, {"start": 12.32, "duration": 3.359, "text": "we're stuck with just the square root of"}, {"start": 13.639, "duration": 3.881, "text": "x now let's focus on the square root of"}, {"start": 15.679, "duration": 3.281, "text": "square root of x the square root of a"}, {"start": 17.52, "duration": 3.44, "text": "square root of a number is the same"}, {"start": 18.96, "duration": 3.6, "text": "thing as the fourth root of a number now"}, {"start": 20.96, "duration": 4.76, "text": "let's focus on this right here we can"}, {"start": 22.56, "duration": 4.44, "text": "break this x into four fourth roots of x"}, {"start": 25.72, "duration": 2.44, "text": "and then one of these in the numerator"}, {"start": 27.0, "duration": 3.48, "text": "will cancel with this one in the"}, {"start": 28.16, "duration": 4.279, "text": "denominator that leaves us with of these"}, {"start": 30.48, "duration": 5.04, "text": "fourth root of x we can write these"}, {"start": 32.439, "duration": 4.8, "text": "three fourth roots of x as 4th root of x"}, {"start": 35.52, "duration": 3.6, "text": "cubed and now we have a square root of"}, {"start": 37.239, "duration": 4.0, "text": "the fourth root of x cubed that's the"}, {"start": 39.12, "duration": 3.8, "text": "same thing as the eth root of x cubed"}, {"start": 41.239, "duration": 4.32, "text": "and then looking at this x up here we"}, {"start": 42.92, "duration": 5.319, "text": "can break this down into 8 eth roots of"}, {"start": 45.559, "duration": 4.201, "text": "x and this eth root of x cubed this"}, {"start": 48.239, "duration": 3.401, "text": "three here is telling us there are three"}, {"start": 49.76, "duration": 3.439, "text": "of those now we can cancel one of these"}, {"start": 51.64, "duration": 2.96, "text": "with one of those another one with"}, {"start": 53.199, "duration": 3.16, "text": "another one and the third one with the"}, {"start": 54.6, "duration": 3.759, "text": "third one and we're left with five of"}, {"start": 56.359, "duration": 4.0, "text": "these eth roots of x or in other words"}, {"start": 58.359, "duration": 4.441, "text": "the eth root of x to the 5th now we have"}, {"start": 60.359, "duration": 4.321, "text": "the square root of the 8th root of x 5th"}, {"start": 62.8, "duration": 4.76, "text": "that's the same thing as the 16th root"}, {"start": 64.68, "duration": 5.52, "text": "of x 5th is the same thing as the 16th"}, {"start": 67.56, "duration": 4.239, "text": "root of x to the 5th and now to get rid"}, {"start": 70.2, "duration": 3.84, "text": "of this five we can take the fifth root"}, {"start": 71.799, "duration": 3.761, "text": "of both sides of the equation this fifth"}, {"start": 74.04, "duration": 3.719, "text": "root and this five are going to cancel"}, {"start": 75.56, "duration": 4.72, "text": "each other out and the fifth root of 32"}, {"start": 77.759, "duration": 4.32, "text": "is 2 and now to get X all by itself we"}, {"start": 80.28, "duration": 3.159, "text": "want to get rid of this 16th root we're"}, {"start": 82.079, "duration": 4.08, "text": "going to do both sides of the equation"}, {"start": 83.439, "duration": 4.68, "text": "to the 16th power this 16 exponent and"}, {"start": 86.159, "duration": 4.161, "text": "the 16th root will cancel each other out"}, {"start": 88.119, "duration": 4.561, "text": "so we just have X and 2 to the 16th"}, {"start": 90.32, "duration": 4.4, "text": "power is 2 multiplied out 16 times"}, {"start": 92.68, "duration": 4.16, "text": "that's going to be"}, {"start": 94.72, "duration": 4.399, "text": "65,536 and that is the answer to our"}, {"start": 96.84, "duration": 5.04, "text": "question let's put a box around it how"}, {"start": 99.119, "duration": 2.761, "text": "exciting"}], "title": "Radical Math Problem", "description": "https://andymath.com/radical-equations/", "lengthSeconds": 100, "width": 640, "height": 360}
{"video_id": "2a41UFeY0U0", "transcript": [{"start": 0.0, "duration": 5.52, "text": "an area of an equilateral triangle is"}, {"start": 2.52, "duration": 4.62, "text": "root 3 over 4S squared where s is the"}, {"start": 5.52, "duration": 2.999, "text": "length of each of the sides so how do we"}, {"start": 7.14, "duration": 2.94, "text": "figure this out first let's draw out an"}, {"start": 8.519, "duration": 3.301, "text": "equilateral triangle and if we drop down"}, {"start": 10.08, "duration": 3.42, "text": "a perpendicular line that will be our"}, {"start": 11.82, "duration": 3.66, "text": "height so the area of this triangle is"}, {"start": 13.5, "duration": 4.08, "text": "one half base times height let's bring"}, {"start": 15.48, "duration": 4.02, "text": "down half of this triangle so this is"}, {"start": 17.58, "duration": 3.3, "text": "one half s because it's half of the side"}, {"start": 19.5, "duration": 2.88, "text": "and it's a right triangle because this"}, {"start": 20.88, "duration": 3.659, "text": "is the height so we can do a squared"}, {"start": 22.38, "duration": 4.2, "text": "plus b squared equals c squared one half"}, {"start": 24.539, "duration": 3.601, "text": "squared is one fourth and then s squared"}, {"start": 26.58, "duration": 3.42, "text": "is s squared and all these are just H"}, {"start": 28.14, "duration": 3.6, "text": "squared and S squared if we subtract 1 4"}, {"start": 30.0, "duration": 3.48, "text": "S squared from both sides on the left"}, {"start": 31.74, "duration": 2.76, "text": "hand side the H squared is now alone and"}, {"start": 33.48, "duration": 3.12, "text": "on the right hand side we have"}, {"start": 34.5, "duration": 3.78, "text": "three-fourths of s squared we can square"}, {"start": 36.6, "duration": 3.24, "text": "root both sides on the left hand side we"}, {"start": 38.28, "duration": 3.18, "text": "just have H and on the right hand side"}, {"start": 39.84, "duration": 3.12, "text": "the square root can be distributed to"}, {"start": 41.46, "duration": 3.599, "text": "all three things we'll bring down the"}, {"start": 42.96, "duration": 3.9, "text": "root 3 square root of four is two and"}, {"start": 45.059, "duration": 3.84, "text": "square root of x squared is s now we"}, {"start": 46.86, "duration": 3.9, "text": "have H in terms of s so let's plug that"}, {"start": 48.899, "duration": 4.381, "text": "in up here in place of this H I'm going"}, {"start": 50.76, "duration": 4.5, "text": "to plug in root 3 over 2s and S times s"}, {"start": 53.28, "duration": 3.959, "text": "is s squared and then in the denominator"}, {"start": 55.26, "duration": 3.299, "text": "2 times 2 is 4 and that gives us the"}, {"start": 57.239, "duration": 3.681, "text": "same area that we have up here how"}, {"start": 58.559, "duration": 2.361, "text": "exciting"}], "title": "Equilateral Triangles Area #shorts", "description": "https://andymath.com/\n#shorts", "lengthSeconds": 59, "width": 270, "height": 480}
{"video_id": "BbsFq4dJbp8", "transcript": [{"start": 0.0, "duration": 4.38, "text": "guys it says solve for x x squared"}, {"start": 2.58, "duration": 3.719, "text": "equals one so I think I want to use the"}, {"start": 4.38, "duration": 3.36, "text": "logarithms to solve this so let's copy"}, {"start": 6.299, "duration": 2.941, "text": "down the x squared equals one and let's"}, {"start": 7.74, "duration": 3.6, "text": "make some room so we can plug in the"}, {"start": 9.24, "duration": 4.38, "text": "natural log so before we move on we have"}, {"start": 11.34, "duration": 4.68, "text": "to verify that X does not equal zero and"}, {"start": 13.62, "duration": 4.44, "text": "if we check up here 0 is not a solution"}, {"start": 16.02, "duration": 4.14, "text": "to this so we know that X does not equal"}, {"start": 18.06, "duration": 3.479, "text": "zero so now we can move on one of the"}, {"start": 20.16, "duration": 3.66, "text": "properties of logarithms is this"}, {"start": 21.539, "duration": 4.08, "text": "exponent can be brought to the front so"}, {"start": 23.82, "duration": 4.26, "text": "we end up with two natural log of x"}, {"start": 25.619, "duration": 4.08, "text": "equals natural log of one the input of a"}, {"start": 28.08, "duration": 3.84, "text": "natural log must be positive so let's"}, {"start": 29.699, "duration": 3.54, "text": "make this absolute value of x it wasn't"}, {"start": 31.92, "duration": 2.819, "text": "an issue before because it was being"}, {"start": 33.239, "duration": 3.541, "text": "squared but now it's no longer being"}, {"start": 34.739, "duration": 4.081, "text": "squared next let's divide both sides by"}, {"start": 36.78, "duration": 3.48, "text": "two these twos cancel so on the left"}, {"start": 38.82, "duration": 3.3, "text": "hand side we just have the natural log"}, {"start": 40.26, "duration": 3.959, "text": "of x and on the right hand side another"}, {"start": 42.12, "duration": 4.38, "text": "way to rewrite this is one half natural"}, {"start": 44.219, "duration": 3.84, "text": "log of one so just like before when we"}, {"start": 46.5, "duration": 3.6, "text": "brought the two to the front this one"}, {"start": 48.059, "duration": 3.721, "text": "half can become the exponent so the"}, {"start": 50.1, "duration": 3.479, "text": "one-half exponent means the same thing"}, {"start": 51.78, "duration": 3.119, "text": "as the square root so one to the"}, {"start": 53.579, "duration": 2.82, "text": "one-half is the same thing as the square"}, {"start": 54.899, "duration": 3.48, "text": "root of one and let's bring down the"}, {"start": 56.399, "duration": 3.48, "text": "rest of this stuff so natural log of"}, {"start": 58.379, "duration": 3.601, "text": "square root of 1 is just natural log of"}, {"start": 59.879, "duration": 3.601, "text": "1 one and next for both sides of the"}, {"start": 61.98, "duration": 4.2, "text": "equation we're going to do e to the"}, {"start": 63.48, "duration": 4.62, "text": "power of e to the power of a natural log"}, {"start": 66.18, "duration": 3.18, "text": "or inverse operation so the left-hand"}, {"start": 68.1, "duration": 3.059, "text": "side is just going to be absolute value"}, {"start": 69.36, "duration": 3.36, "text": "of x and on the right hand side same"}, {"start": 71.159, "duration": 3.061, "text": "thing so we're just left with one and"}, {"start": 72.72, "duration": 3.12, "text": "now we have the absolute value of x is"}, {"start": 74.22, "duration": 3.24, "text": "equal to one in order for this to be"}, {"start": 75.84, "duration": 3.599, "text": "true the X inside the absolute value"}, {"start": 77.46, "duration": 3.479, "text": "would have to be either one or negative"}, {"start": 79.439, "duration": 3.241, "text": "one and that's the answer to our"}, {"start": 80.939, "duration": 3.241, "text": "question let's put a box around it so if"}, {"start": 82.68, "duration": 3.36, "text": "you ever see x squared is equal to one"}, {"start": 84.18, "duration": 5.72, "text": "we know that X is equal to one or"}, {"start": 86.04, "duration": 3.86, "text": "negative one how exciting"}], "title": "Solve x^2=1 with Logarithms", "description": "I think there is probably and easier way to solve this.", "lengthSeconds": 88, "width": 640, "height": 360}
{"video_id": "x8g969GsQbk", "transcript": [{"start": 0.0, "duration": 3.54, "text": "you guys this looks like a fun one one"}, {"start": 1.68, "duration": 5.219, "text": "over one plus one over one plus one over"}, {"start": 3.54, "duration": 5.099, "text": "x equals 420 over 834. so for the first"}, {"start": 6.899, "duration": 3.481, "text": "step we want to do this one plus one"}, {"start": 8.639, "duration": 3.121, "text": "over X in order to give them a common"}, {"start": 10.38, "duration": 3.36, "text": "denominator we're going to multiply this"}, {"start": 11.76, "duration": 4.56, "text": "one on the left hand side by X over x"}, {"start": 13.74, "duration": 3.84, "text": "and x over x times 1 is just X over X"}, {"start": 16.32, "duration": 2.7, "text": "and now that they have a common"}, {"start": 17.58, "duration": 3.359, "text": "denominator we can write them as a"}, {"start": 19.02, "duration": 4.14, "text": "single fraction so it becomes X plus one"}, {"start": 20.939, "duration": 5.041, "text": "over X so now let's focus on this right"}, {"start": 23.16, "duration": 5.4, "text": "here this is basically saying 1 divided"}, {"start": 25.98, "duration": 3.78, "text": "by X Plus 1 over X the division of a"}, {"start": 28.56, "duration": 3.299, "text": "fraction is the same thing as"}, {"start": 29.76, "duration": 3.72, "text": "multiplying by the reciprocal so if we"}, {"start": 31.859, "duration": 3.601, "text": "change this divide into a multiply we"}, {"start": 33.48, "duration": 4.38, "text": "can flip this fraction and one times"}, {"start": 35.46, "duration": 5.279, "text": "this is just this so we want to rewrite"}, {"start": 37.86, "duration": 5.039, "text": "this whole thing as X over X plus one"}, {"start": 40.739, "duration": 3.601, "text": "now let's get rid of this stuff next we"}, {"start": 42.899, "duration": 3.241, "text": "want to add these together and we need"}, {"start": 44.34, "duration": 3.6, "text": "to give this a common denominator with"}, {"start": 46.14, "duration": 4.32, "text": "this so we're going to multiply this one"}, {"start": 47.94, "duration": 4.619, "text": "by X Plus 1 over X plus one well this"}, {"start": 50.46, "duration": 3.3, "text": "times 1 is just this thing and now that"}, {"start": 52.559, "duration": 2.64, "text": "these two fractions have a common"}, {"start": 53.76, "duration": 3.24, "text": "denominator we can write them as a"}, {"start": 55.199, "duration": 4.261, "text": "single fraction so all we did was add"}, {"start": 57.0, "duration": 4.5, "text": "the numerators and last X Plus X is"}, {"start": 59.46, "duration": 3.3, "text": "equal to 2 x with this right here we're"}, {"start": 61.5, "duration": 3.54, "text": "doing the division thing again we're"}, {"start": 62.76, "duration": 3.78, "text": "doing one divided by this fraction down"}, {"start": 65.04, "duration": 3.42, "text": "here anytime you do that we can change"}, {"start": 66.54, "duration": 3.899, "text": "the Divide to a multiply and flip the"}, {"start": 68.46, "duration": 3.479, "text": "fraction and then one times something is"}, {"start": 70.439, "duration": 3.481, "text": "just the something so now we want to"}, {"start": 71.939, "duration": 3.36, "text": "rewrite this entire left side as this"}, {"start": 73.92, "duration": 2.94, "text": "fraction right here or in other words"}, {"start": 75.299, "duration": 3.421, "text": "you can think of it as bringing this X"}, {"start": 76.86, "duration": 3.78, "text": "Plus 1 to the top now let's get rid of"}, {"start": 78.72, "duration": 3.96, "text": "this stuff I'm noticing on the side we"}, {"start": 80.64, "duration": 4.38, "text": "can simplify this fraction both the top"}, {"start": 82.68, "duration": 4.86, "text": "and the bottom are divisible by six on"}, {"start": 85.02, "duration": 5.7, "text": "top 420 divided by 6 is 70. and on"}, {"start": 87.54, "duration": 4.56, "text": "bottom made 34 divided by 6 is 139. in"}, {"start": 90.72, "duration": 4.14, "text": "order to solve this proportion Let's"}, {"start": 92.1, "duration": 5.879, "text": "cross multiply we have the 139 times X"}, {"start": 94.86, "duration": 5.34, "text": "plus one equals 70 times 2x plus 1 and"}, {"start": 97.979, "duration": 7.081, "text": "then if we distribute 139 times x is"}, {"start": 100.2, "duration": 8.279, "text": "139x 139 times 1 is 139 70 times 2x is"}, {"start": 105.06, "duration": 5.04, "text": "140x and 70 times 1 is 70. now we have"}, {"start": 108.479, "duration": 4.561, "text": "an x on both sides so let's subtract"}, {"start": 110.1, "duration": 5.04, "text": "139x from both sides these cancel so on"}, {"start": 113.04, "duration": 4.619, "text": "the left hand side we only have 139 and"}, {"start": 115.14, "duration": 4.619, "text": "on the right hand side 140x minus 139x"}, {"start": 117.659, "duration": 4.5, "text": "is just X now we still have the plus 70."}, {"start": 119.759, "duration": 5.22, "text": "now on the left hand side 139 minus 70"}, {"start": 122.159, "duration": 4.441, "text": "is 69. nice that is the answer to our"}, {"start": 124.979, "duration": 4.881, "text": "question so we can put a box around it"}, {"start": 126.6, "duration": 3.26, "text": "how exciting"}], "title": "Nice Rational Equation", "description": "https://andymath.com/rational-equations/\n\nThank you for watching these videos! I hope you have a nice week!", "lengthSeconds": 128, "width": 640, "height": 360}
{"video_id": "siw5pZg5Q9o", "transcript": [{"start": 0.04, "duration": 4.0, "text": "hey guys so many people sent me this one"}, {"start": 1.92, "duration": 4.72, "text": "it was posted by math love and it got"}, {"start": 4.04, "duration": 4.519, "text": "thousands of likes so let's do it first"}, {"start": 6.64, "duration": 4.24, "text": "step I want to focus on this triangle"}, {"start": 8.559, "duration": 5.761, "text": "right here if we call this angle a we"}, {"start": 10.88, "duration": 5.04, "text": "know that a + 45 + 43 has to equal 180"}, {"start": 14.32, "duration": 3.959, "text": "and then after we subtract 45 and"}, {"start": 15.92, "duration": 5.439, "text": "subtract 43 from both sides that'll give"}, {"start": 18.279, "duration": 4.041, "text": "us a is equal to 92 so we can update"}, {"start": 21.359, "duration": 3.601, "text": "this to be"}, {"start": 22.32, "duration": 4.56, "text": "92\u00b0 and then next let's focus on this"}, {"start": 24.96, "duration": 4.799, "text": "relationship right here this is called a"}, {"start": 26.88, "duration": 6.8, "text": "linear pair if we let this angle be n\u00b0"}, {"start": 29.759, "duration": 5.96, "text": "it we know that n + 92 is equal to 180\u00b0"}, {"start": 33.68, "duration": 4.76, "text": "and after we subtract 92 from both sides"}, {"start": 35.719, "duration": 4.641, "text": "we get n is equal to 88 so we can update"}, {"start": 38.44, "duration": 3.84, "text": "this there and then this angle right"}, {"start": 40.36, "duration": 4.359, "text": "here is also 88 you can either think of"}, {"start": 42.28, "duration": 4.4, "text": "it as a linear pair with the 92 or you"}, {"start": 44.719, "duration": 4.601, "text": "can think of it as vertical angles with"}, {"start": 46.68, "duration": 4.68, "text": "this 88 next let's focus on this"}, {"start": 49.32, "duration": 5.12, "text": "triangle right here let's let this angle"}, {"start": 51.36, "duration": 5.76, "text": "be D de then we know that D plus the 88"}, {"start": 54.44, "duration": 4.959, "text": "+ 24 needs to equal 180 and after we"}, {"start": 57.12, "duration": 4.759, "text": "subtract 88 and 24 from both sides we"}, {"start": 59.399, "duration": 3.561, "text": "get D is equal to 68 so we can update"}, {"start": 61.879, "duration": 3.441, "text": "this to be"}, {"start": 62.96, "duration": 4.28, "text": "68\u00b0 and then if we extend this here we"}, {"start": 65.32, "duration": 4.28, "text": "have another linear pair and if we make"}, {"start": 67.24, "duration": 4.84, "text": "this y Dees then the y + 68 needs to"}, {"start": 69.6, "duration": 5.159, "text": "equal 180 and then we can subtract 68"}, {"start": 72.08, "duration": 4.88, "text": "from both sides and we get Y is equal to"}, {"start": 74.759, "duration": 4.0, "text": "112 and next let's focus on this"}, {"start": 76.96, "duration": 3.68, "text": "triangle right here we can call this"}, {"start": 78.759, "duration": 3.521, "text": "angle M and then once again we can do"}, {"start": 80.64, "duration": 2.799, "text": "all this math and we find that m is"}, {"start": 82.28, "duration": 3.72, "text": "equal to"}, {"start": 83.439, "duration": 5.68, "text": "66 so then we have this linear pair"}, {"start": 86.0, "duration": 5.84, "text": "right here so we know that t + 66 = 180"}, {"start": 89.119, "duration": 4.28, "text": "or in other words T is equal to 114 and"}, {"start": 91.84, "duration": 3.639, "text": "now we're down to this convex"}, {"start": 93.399, "duration": 3.481, "text": "quadrilateral right here we know that"}, {"start": 95.479, "duration": 2.241, "text": "all four of these angles are going to"}, {"start": 96.88, "duration": 5.4, "text": "add to"}, {"start": 97.72, "duration": 6.48, "text": "360\u00b0 well 112 + 114 + 92 is equal to 318"}, {"start": 102.28, "duration": 4.4, "text": "and then after we subtract 318 from both"}, {"start": 104.2, "duration": 4.08, "text": "sides we get X is equal to 42 and that"}, {"start": 106.68, "duration": 6.36, "text": "is the answer to the question let's put"}, {"start": 108.28, "duration": 4.76, "text": "a box around it how exciting"}], "title": "Relaxing Math Problem", "description": "This is wasn't super difficult, but I found it relaxing to go through. Kinda like math ASMR.", "lengthSeconds": 111, "width": 640, "height": 360}
{"video_id": "YVB5QO8xsBI", "transcript": [{"start": 0.0, "duration": 3.3, "text": "guys this looks like a fun one we're"}, {"start": 1.8, "duration": 3.66, "text": "giving some triangles with some angles"}, {"start": 3.3, "duration": 3.9, "text": "this is congruent to this and they give"}, {"start": 5.46, "duration": 3.9, "text": "us the values of these two angles and"}, {"start": 7.2, "duration": 4.019, "text": "they're asking us to solve for x let's"}, {"start": 9.36, "duration": 3.779, "text": "do this if you want to try this on your"}, {"start": 11.219, "duration": 4.62, "text": "own pause it because I'm going to solve"}, {"start": 13.139, "duration": 4.801, "text": "it in three two one for the first step"}, {"start": 15.839, "duration": 3.961, "text": "let's focus on this linear pair right"}, {"start": 17.94, "duration": 4.259, "text": "here I want to find out what is this"}, {"start": 19.8, "duration": 3.899, "text": "other angle combine these form 180"}, {"start": 22.199, "duration": 4.561, "text": "degrees so we know that the question"}, {"start": 23.699, "duration": 5.34, "text": "mark plus 45 will equal 180. that'll be"}, {"start": 26.76, "duration": 5.339, "text": "true if the question mark is 135 so we"}, {"start": 29.039, "duration": 5.581, "text": "know that this angle is 135 degrees next"}, {"start": 32.099, "duration": 5.281, "text": "let's focus on this triangle right here"}, {"start": 34.62, "duration": 4.86, "text": "what is the value of this angle the sum"}, {"start": 37.38, "duration": 4.74, "text": "of the interior angles of a triangle is"}, {"start": 39.48, "duration": 4.44, "text": "always equal to 180 degrees and this"}, {"start": 42.12, "duration": 4.38, "text": "will be true for 30 so we can change"}, {"start": 43.92, "duration": 4.44, "text": "this angle to 30 degrees and now that we"}, {"start": 46.5, "duration": 4.8, "text": "know this 30 degrees we won't need this"}, {"start": 48.36, "duration": 4.679, "text": "135 anymore starting from the vertex of"}, {"start": 51.3, "duration": 3.599, "text": "this angle let's draw out this blue"}, {"start": 53.039, "duration": 4.2, "text": "segment and construct a circle with that"}, {"start": 54.899, "duration": 4.021, "text": "as the radius now let's find where this"}, {"start": 57.239, "duration": 4.081, "text": "circle intersects this let's connect"}, {"start": 58.92, "duration": 4.439, "text": "this intersection point to the center of"}, {"start": 61.32, "duration": 3.6, "text": "our Circle this is another radius of the"}, {"start": 63.359, "duration": 3.241, "text": "circle so it's equal to this segment"}, {"start": 64.92, "duration": 4.32, "text": "right here and now we don't need this"}, {"start": 66.6, "duration": 4.98, "text": "circle anymore next let's focus on this"}, {"start": 69.24, "duration": 4.02, "text": "triangle right here since there are two"}, {"start": 71.58, "duration": 4.02, "text": "sides congruent in this triangle we know"}, {"start": 73.26, "duration": 4.14, "text": "it's an isosceles triangle and the angle"}, {"start": 75.6, "duration": 3.54, "text": "is opposite the congruent sides are also"}, {"start": 77.4, "duration": 4.44, "text": "congruent so we know that this angle"}, {"start": 79.14, "duration": 4.32, "text": "right here is also 30 degrees and now to"}, {"start": 81.84, "duration": 3.72, "text": "find the last angle of the triangle we"}, {"start": 83.46, "duration": 4.68, "text": "can set them all equal to 180 and this"}, {"start": 85.56, "duration": 5.64, "text": "will be true for 120. so this angle"}, {"start": 88.14, "duration": 5.04, "text": "right here is 120 degrees next I want to"}, {"start": 91.2, "duration": 3.779, "text": "find this angle right here and we're"}, {"start": 93.18, "duration": 4.14, "text": "going to use these three angles to get"}, {"start": 94.979, "duration": 5.401, "text": "it this angle plus this angle plus this"}, {"start": 97.32, "duration": 5.7, "text": "angle will all add up to 180 degrees and"}, {"start": 100.38, "duration": 4.68, "text": "this statement will be true for 15. so"}, {"start": 103.02, "duration": 5.099, "text": "we know that this angle right here is 15"}, {"start": 105.06, "duration": 5.699, "text": "degrees next I'm thinking let's focus on"}, {"start": 108.119, "duration": 5.101, "text": "this triangle right here since these"}, {"start": 110.759, "duration": 4.741, "text": "angles are both 15 degrees we know it's"}, {"start": 113.22, "duration": 4.14, "text": "an isosceles triangle and the sides"}, {"start": 115.5, "duration": 3.96, "text": "opposite the congruent angles are also"}, {"start": 117.36, "duration": 4.02, "text": "congruent next let's connect this"}, {"start": 119.46, "duration": 4.26, "text": "intersection point with this vertex"}, {"start": 121.38, "duration": 4.86, "text": "right here now let's focus on our new"}, {"start": 123.72, "duration": 4.44, "text": "triangle to find this angle down here we"}, {"start": 126.24, "duration": 4.98, "text": "need to add the 15 degrees and the 45"}, {"start": 128.16, "duration": 4.98, "text": "degrees to get 60 degrees and now let's"}, {"start": 131.22, "duration": 4.56, "text": "get the last two angles of this triangle"}, {"start": 133.14, "duration": 4.38, "text": "this is another isosceles triangle so we"}, {"start": 135.78, "duration": 3.539, "text": "know that these two angles are congruent"}, {"start": 137.52, "duration": 3.78, "text": "to each other and we also know that if"}, {"start": 139.319, "duration": 4.681, "text": "we add up all three of these angles it's"}, {"start": 141.3, "duration": 4.56, "text": "going to be 180 degrees this will be"}, {"start": 144.0, "duration": 4.739, "text": "true if both of these angles are 60"}, {"start": 145.86, "duration": 4.56, "text": "degrees and now since all three angles"}, {"start": 148.739, "duration": 4.021, "text": "are 60 degrees we know it's an"}, {"start": 150.42, "duration": 4.14, "text": "equilateral triangle so all three sides"}, {"start": 152.76, "duration": 3.54, "text": "are going to be congruent so now from"}, {"start": 154.56, "duration": 3.899, "text": "here I'm interested in this angle right"}, {"start": 156.3, "duration": 4.86, "text": "here so let's focus on these three"}, {"start": 158.459, "duration": 4.741, "text": "angles and find that angle right there"}, {"start": 161.16, "duration": 4.859, "text": "these three angles are also going to add"}, {"start": 163.2, "duration": 5.039, "text": "to 180 degrees and this will be true if"}, {"start": 166.019, "duration": 3.72, "text": "the angle is 90 degrees so let's put one"}, {"start": 168.239, "duration": 3.901, "text": "of those little boxes right there to"}, {"start": 169.739, "duration": 4.681, "text": "show that's 90 degrees for our next step"}, {"start": 172.14, "duration": 4.92, "text": "let's focus on this right triangle right"}, {"start": 174.42, "duration": 4.14, "text": "here it's also an isosceles triangle so"}, {"start": 177.06, "duration": 3.6, "text": "we know that these two angles are going"}, {"start": 178.56, "duration": 3.66, "text": "to be congruent to each other and if we"}, {"start": 180.66, "duration": 3.359, "text": "add up all three of these angles we can"}, {"start": 182.22, "duration": 4.08, "text": "find out that each of these are 45"}, {"start": 184.019, "duration": 4.021, "text": "degree angle let's look at this x"}, {"start": 186.3, "duration": 4.14, "text": "degrees in this 15 degrees let's make"}, {"start": 188.04, "duration": 4.26, "text": "them red we know that the sum of these"}, {"start": 190.44, "duration": 4.32, "text": "two angles will be equal to this larger"}, {"start": 192.3, "duration": 5.159, "text": "angle so we have X plus 15 is equal to"}, {"start": 194.76, "duration": 5.22, "text": "45. so we subtract 15 from both sides we"}, {"start": 197.459, "duration": 4.441, "text": "find that X is equal to 30. and that is"}, {"start": 199.98, "duration": 4.08, "text": "the answer to our question they wanted"}, {"start": 201.9, "duration": 4.259, "text": "us to solve for x so let's clean things"}, {"start": 204.06, "duration": 7.099, "text": "up a little bit and put a box around it"}, {"start": 206.159, "duration": 5.0, "text": "x equals 30 how exciting"}], "title": "Geometry Challenge Problem", "description": "I thought this was a fun math problem. I hope you guys like the video!", "lengthSeconds": 209, "width": 640, "height": 360}
{"video_id": "tvLqC0K8Frk", "transcript": [{"start": 0.0, "duration": 4.44, "text": "guys let's find this limit it's the"}, {"start": 1.74, "duration": 5.22, "text": "limit as X approaches 0 of 1 minus the"}, {"start": 4.44, "duration": 4.5, "text": "square root of 1 minus X over X first"}, {"start": 6.96, "duration": 4.38, "text": "let's copy it down and let's try just"}, {"start": 8.94, "duration": 3.9, "text": "plugging in zero see what happens so in"}, {"start": 11.34, "duration": 3.419, "text": "the place of both these X's I just"}, {"start": 12.84, "duration": 3.72, "text": "plugged in zero well this minus zero"}, {"start": 14.759, "duration": 4.02, "text": "isn't doing anything and the square root"}, {"start": 16.56, "duration": 4.86, "text": "of one is just one and then one minus"}, {"start": 18.779, "duration": 4.92, "text": "one is zero by using direct substitution"}, {"start": 21.42, "duration": 4.32, "text": "we just got zero over zero which is an"}, {"start": 23.699, "duration": 4.561, "text": "indeterminate form this isn't giving us"}, {"start": 25.74, "duration": 4.26, "text": "any information given 0 over 0 the"}, {"start": 28.26, "duration": 3.12, "text": "answer could still be anything so we"}, {"start": 30.0, "duration": 3.54, "text": "have to find another way to solve this"}, {"start": 31.38, "duration": 4.32, "text": "looking at this I'm seeing a binomial"}, {"start": 33.54, "duration": 3.84, "text": "that includes a square root so usually"}, {"start": 35.7, "duration": 3.6, "text": "you're going to use the conjugate so in"}, {"start": 37.38, "duration": 3.42, "text": "this case 1 minus the square root the"}, {"start": 39.3, "duration": 2.579, "text": "conjugate would be one plus the square"}, {"start": 40.8, "duration": 2.46, "text": "root and then we're going to multiply"}, {"start": 41.879, "duration": 2.941, "text": "that to top and bottom so we're not"}, {"start": 43.26, "duration": 3.299, "text": "actually changing the value of the"}, {"start": 44.82, "duration": 3.239, "text": "rational expression so let's see what"}, {"start": 46.559, "duration": 3.0, "text": "happens when we use the conjugate on"}, {"start": 48.059, "duration": 3.241, "text": "bottom we're just going to leave it as x"}, {"start": 49.559, "duration": 3.48, "text": "times this we're not going to need to"}, {"start": 51.3, "duration": 3.3, "text": "distribute the X usually is the one that"}, {"start": 53.039, "duration": 3.481, "text": "isn't the conjugate you can just leave"}, {"start": 54.6, "duration": 3.779, "text": "it as the product of the two so now on"}, {"start": 56.52, "duration": 4.199, "text": "top let's multiply this stuff out so we"}, {"start": 58.379, "duration": 4.081, "text": "have 1 times 1 which will give us one"}, {"start": 60.719, "duration": 3.18, "text": "and we have one times the square root"}, {"start": 62.46, "duration": 3.059, "text": "which is the square root and then we"}, {"start": 63.899, "duration": 3.421, "text": "have the negative square root times 1"}, {"start": 65.519, "duration": 2.941, "text": "which is the negative square root and"}, {"start": 67.32, "duration": 2.64, "text": "then we have the negative square root"}, {"start": 68.46, "duration": 2.88, "text": "times the positive square root a"}, {"start": 69.96, "duration": 3.18, "text": "negative times a positive is going to be"}, {"start": 71.34, "duration": 3.599, "text": "negative and then the square root of 1"}, {"start": 73.14, "duration": 3.54, "text": "minus x times the square root of 1 minus"}, {"start": 74.939, "duration": 4.321, "text": "X when you multiply those the square"}, {"start": 76.68, "duration": 4.259, "text": "root goes away and we have 1 minus X so"}, {"start": 79.26, "duration": 3.539, "text": "before we simplify let's distribute this"}, {"start": 80.939, "duration": 3.421, "text": "negative to both of these terms so this"}, {"start": 82.799, "duration": 2.941, "text": "positive one will become a negative 1"}, {"start": 84.36, "duration": 3.36, "text": "and the negative X will become a"}, {"start": 85.74, "duration": 3.72, "text": "positive X so now we have the positive"}, {"start": 87.72, "duration": 2.939, "text": "of this square root and the negative of"}, {"start": 89.46, "duration": 2.88, "text": "this square root they're going to cancel"}, {"start": 90.659, "duration": 3.361, "text": "each other out and then we have positive"}, {"start": 92.34, "duration": 3.12, "text": "one and negative one those will also"}, {"start": 94.02, "duration": 3.66, "text": "cancel each other out and now let's"}, {"start": 95.46, "duration": 3.839, "text": "smoosh everything together and this X on"}, {"start": 97.68, "duration": 3.24, "text": "top and bottom will also cancel each"}, {"start": 99.299, "duration": 3.541, "text": "other out and now there's nothing on top"}, {"start": 100.92, "duration": 3.6, "text": "we should put the implied one now let's"}, {"start": 102.84, "duration": 4.38, "text": "do some direct substitution what happens"}, {"start": 104.52, "duration": 3.84, "text": "if we plug 0 into the X well the zero"}, {"start": 107.22, "duration": 3.3, "text": "isn't doing anything so this just"}, {"start": 108.36, "duration": 4.079, "text": "becomes one over one plus square root of"}, {"start": 110.52, "duration": 3.84, "text": "one and one plus square root one is one"}, {"start": 112.439, "duration": 3.36, "text": "plus one which is two and that is the"}, {"start": 114.36, "duration": 3.48, "text": "answer to our question let's put a box"}, {"start": 115.799, "duration": 4.561, "text": "around it so the limit as X approaches 0"}, {"start": 117.84, "duration": 3.84, "text": "of this is equal to one half and here it"}, {"start": 120.36, "duration": 2.759, "text": "is in Desmos and you can see that it"}, {"start": 121.68, "duration": 2.64, "text": "does approach one half from both"}, {"start": 123.119, "duration": 2.46, "text": "directions"}, {"start": 124.32, "duration": 2.46, "text": "if you guys want to try more of these"}, {"start": 125.579, "duration": 2.941, "text": "limit problems I have this page on"}, {"start": 126.78, "duration": 3.42, "text": "animat.com it's got a whole bunch of"}, {"start": 128.52, "duration": 5.299, "text": "problems with the answers the link is in"}, {"start": 130.2, "duration": 3.619, "text": "the description how exciting"}], "title": "Limit with Conjugate", "description": "https://andymath.com/intro-to-limits/\nFor more limit practice, check out the above link!", "lengthSeconds": 132, "width": 640, "height": 360}
{"video_id": "1PX7aKa_FZ0", "transcript": [{"start": 0.0, "duration": 3.899, "text": "guys a bunch of people mentioned me in"}, {"start": 1.74, "duration": 4.2, "text": "this one so let's do this x squared plus"}, {"start": 3.899, "duration": 4.141, "text": "y squared equals 25 and the 2x plus y"}, {"start": 5.94, "duration": 4.32, "text": "equals 10 and we want to evaluate X Plus"}, {"start": 8.04, "duration": 4.38, "text": "y so first let's focus on the 2x plus y"}, {"start": 10.26, "duration": 4.44, "text": "equals 10. let's get the Y all by itself"}, {"start": 12.42, "duration": 4.199, "text": "let's subtract 2x from both sides on the"}, {"start": 14.7, "duration": 4.32, "text": "left hand side we just have y and on the"}, {"start": 16.619, "duration": 3.841, "text": "right hand side we have 10 minus 2X and"}, {"start": 19.02, "duration": 3.3, "text": "this is important so let's put a box"}, {"start": 20.46, "duration": 4.2, "text": "around it next let's focus on the x"}, {"start": 22.32, "duration": 4.379, "text": "squared plus y squared equals 25. let's"}, {"start": 24.66, "duration": 4.74, "text": "bring it down and in the place of this y"}, {"start": 26.699, "duration": 5.641, "text": "right here I want to plug in 10 minus 2x"}, {"start": 29.4, "duration": 4.62, "text": "so now this is 10 minus 2x being squared"}, {"start": 32.34, "duration": 4.98, "text": "so we're going to multiply 10 minus 2x"}, {"start": 34.02, "duration": 5.52, "text": "times 10 minus 2x 10 times 10 is 100 10"}, {"start": 37.32, "duration": 5.399, "text": "times negative 2x is negative 20x"}, {"start": 39.54, "duration": 4.859, "text": "negative 2x times 10 is negative 20x and"}, {"start": 42.719, "duration": 3.781, "text": "negative 2x times negative 2x is"}, {"start": 44.399, "duration": 4.441, "text": "positive 4x squared and these two can"}, {"start": 46.5, "duration": 4.079, "text": "combine to become negative 40x let's"}, {"start": 48.84, "duration": 4.14, "text": "bring down the x squared and bring down"}, {"start": 50.579, "duration": 4.081, "text": "the equals 25. anytime I do one of these"}, {"start": 52.98, "duration": 4.44, "text": "several people always mention why don't"}, {"start": 54.66, "duration": 4.379, "text": "I do this shortcut well I did try it out"}, {"start": 57.42, "duration": 3.84, "text": "and it ended up not being much of a"}, {"start": 59.039, "duration": 4.141, "text": "shortcut so it low as nicely as this"}, {"start": 61.26, "duration": 4.079, "text": "method in these videos so now let's"}, {"start": 63.18, "duration": 4.32, "text": "combine like terms x squared plus 4x"}, {"start": 65.339, "duration": 4.681, "text": "squared is 5X squared we can bring down"}, {"start": 67.5, "duration": 5.34, "text": "the negative 40x and if we subtract 25"}, {"start": 70.02, "duration": 5.58, "text": "from both sides 100 minus 25 is 75 and"}, {"start": 72.84, "duration": 4.5, "text": "25 minus 25 is zero from here all three"}, {"start": 75.6, "duration": 3.78, "text": "of these terms are divisible by 5 so"}, {"start": 77.34, "duration": 3.72, "text": "let's divide both sides by 5. each of"}, {"start": 79.38, "duration": 4.5, "text": "these terms will divide by 5 to give us"}, {"start": 81.06, "duration": 4.86, "text": "x squared minus eight X plus 15 and it's"}, {"start": 83.88, "duration": 3.84, "text": "still equal to zero and now to solve"}, {"start": 85.92, "duration": 3.9, "text": "this we want to factor we want two"}, {"start": 87.72, "duration": 4.8, "text": "numbers that will add to negative 8 and"}, {"start": 89.82, "duration": 4.619, "text": "multiply to positive 15. well negative 3"}, {"start": 92.52, "duration": 3.72, "text": "plus negative 5 will give us negative 8"}, {"start": 94.439, "duration": 3.661, "text": "and negative 3 times negative 5 will"}, {"start": 96.24, "duration": 4.08, "text": "give us positive 15. so these are the"}, {"start": 98.1, "duration": 4.92, "text": "factors of this so from here if this"}, {"start": 100.32, "duration": 4.2, "text": "equals 0 or if this equals 0 the whole"}, {"start": 103.02, "duration": 4.5, "text": "thing will equal zero because anything"}, {"start": 104.52, "duration": 5.04, "text": "times 0 equals zero so we have this or"}, {"start": 107.52, "duration": 4.44, "text": "this and if we add 3 to both sides we"}, {"start": 109.56, "duration": 4.739, "text": "get x equals 3 and if we add 5 to both"}, {"start": 111.96, "duration": 4.979, "text": "sides we get x equals five so X can be"}, {"start": 114.299, "duration": 4.201, "text": "either 3 or 5. so we don't really need"}, {"start": 116.939, "duration": 3.601, "text": "any of this work anymore and let's bring"}, {"start": 118.5, "duration": 3.899, "text": "these up here since our goal to figure"}, {"start": 120.54, "duration": 4.079, "text": "out what is X Plus y we want to find out"}, {"start": 122.399, "duration": 4.621, "text": "our corresponding y values so let's use"}, {"start": 124.619, "duration": 4.441, "text": "this 2x plus y equals 10. we'll copy it"}, {"start": 127.02, "duration": 4.74, "text": "down solve for y in the place of this x"}, {"start": 129.06, "duration": 4.56, "text": "let's plug in 3. 2 times 3 is 6. let's"}, {"start": 131.76, "duration": 3.9, "text": "subtract 6 from both sides and we get y"}, {"start": 133.62, "duration": 4.38, "text": "equals 4. and same thing for this one if"}, {"start": 135.66, "duration": 4.26, "text": "we plug in 5 for the X 2 times 5 is 10"}, {"start": 138.0, "duration": 4.5, "text": "subtract 10 from both sides and we get y"}, {"start": 139.92, "duration": 5.1, "text": "equals zero these are our two possible"}, {"start": 142.5, "duration": 4.68, "text": "scenarios and we want to evaluate X Plus"}, {"start": 145.02, "duration": 4.56, "text": "y for each of these scenarios for the"}, {"start": 147.18, "duration": 4.8, "text": "first scenario X Plus y ends up being 3"}, {"start": 149.58, "duration": 4.62, "text": "plus 4 which is equal to 7 and for this"}, {"start": 151.98, "duration": 3.839, "text": "scenario X Plus Y is 5 plus 0 which is"}, {"start": 154.2, "duration": 3.48, "text": "equal to five and these are the two"}, {"start": 155.819, "duration": 4.021, "text": "possible answers to the question X Plus"}, {"start": 157.68, "duration": 3.72, "text": "y equals seven or X Plus y equals five"}, {"start": 159.84, "duration": 3.06, "text": "and if you want to see what this looks"}, {"start": 161.4, "duration": 3.36, "text": "like graphically here it is right here"}, {"start": 162.9, "duration": 4.02, "text": "the x squared plus y squared equals 25"}, {"start": 164.76, "duration": 4.14, "text": "is the red circle and the 2x plus y"}, {"start": 166.92, "duration": 3.959, "text": "equals 10 is this blue line and here's"}, {"start": 168.9, "duration": 5.78, "text": "our two intersection points three four"}, {"start": 170.879, "duration": 3.801, "text": "and five zero how exciting"}], "title": "Fun Math Warmup Problem", "description": "https://andymath.com/standardized-test-prep/", "lengthSeconds": 173, "width": 640, "height": 360}
{"video_id": "D6Sio8q8x0E", "transcript": [{"start": 0.0, "duration": 3.84, "text": "guys I think this is a pretty cool one"}, {"start": 1.74, "duration": 4.5, "text": "even though this is in terms of X you"}, {"start": 3.84, "duration": 4.2, "text": "can still find the exact value of this"}, {"start": 6.24, "duration": 3.12, "text": "red region if you want to try this on"}, {"start": 8.04, "duration": 4.2, "text": "your own posit because I'm going to"}, {"start": 9.36, "duration": 4.62, "text": "solve it in three two one first let's"}, {"start": 12.24, "duration": 3.959, "text": "copy this down here and let's look only"}, {"start": 13.98, "duration": 3.66, "text": "at the large triangle the base of the"}, {"start": 16.199, "duration": 3.721, "text": "triangle is the two pieces added"}, {"start": 17.64, "duration": 4.5, "text": "together so it's going to be 9x plus 6."}, {"start": 19.92, "duration": 4.619, "text": "and the height is going to be 2x plus 6"}, {"start": 22.14, "duration": 4.62, "text": "and this hypotenuse will still be 13x"}, {"start": 24.539, "duration": 4.32, "text": "this is a right triangle which means we"}, {"start": 26.76, "duration": 5.46, "text": "can use Pythagorean theorem the quantity"}, {"start": 28.859, "duration": 5.341, "text": "9x plus 6 squared plus the quantity 2x"}, {"start": 32.22, "duration": 4.499, "text": "plus 6 squared is equal to the quantity"}, {"start": 34.2, "duration": 4.92, "text": "13x squared so the right hand side's"}, {"start": 36.719, "duration": 5.221, "text": "pretty easy both the 13 and the X get a"}, {"start": 39.12, "duration": 4.38, "text": "squared and 13 squared is equal to 169."}, {"start": 41.94, "duration": 3.42, "text": "nice and then on the left hand side"}, {"start": 43.5, "duration": 3.54, "text": "we're going to add these two things so"}, {"start": 45.36, "duration": 3.84, "text": "this squared means this thing multiplied"}, {"start": 47.04, "duration": 4.44, "text": "by itself and this squared means this"}, {"start": 49.2, "duration": 4.26, "text": "thing multiplied by itself so this 9x"}, {"start": 51.48, "duration": 3.96, "text": "plus 6 times 9x plus 6 you can multiply"}, {"start": 53.46, "duration": 5.46, "text": "it out or foil it it ends up giving us"}, {"start": 55.44, "duration": 4.38, "text": "81x squared plus 108x plus 36. and you"}, {"start": 58.92, "duration": 3.479, "text": "can do the same thing with this down"}, {"start": 59.82, "duration": 6.36, "text": "here here gives us 4x squared plus 24x"}, {"start": 62.399, "duration": 8.101, "text": "plus 36. 81x squared plus 4 is 85x"}, {"start": 66.18, "duration": 7.74, "text": "squared 108x plus 24x is 132x and 36"}, {"start": 70.5, "duration": 5.52, "text": "plus 36 is 72. let's subtract 169x"}, {"start": 73.92, "duration": 4.26, "text": "squared from both sides and this minus"}, {"start": 76.02, "duration": 3.48, "text": "this is negative 84x squared we can"}, {"start": 78.18, "duration": 2.759, "text": "bring the rest of this stuff down and"}, {"start": 79.5, "duration": 3.54, "text": "this is all going to be equal to zero"}, {"start": 80.939, "duration": 4.441, "text": "each of these coefficients is divisible"}, {"start": 83.04, "duration": 4.079, "text": "by 12 so let's divide it by 12. now"}, {"start": 85.38, "duration": 4.26, "text": "let's do negative 12 so we can make the"}, {"start": 87.119, "duration": 5.341, "text": "leading coefficient positive negative 84"}, {"start": 89.64, "duration": 5.82, "text": "divided by negative 12 is 7x squared"}, {"start": 92.46, "duration": 6.299, "text": "positive 132 divided by negative 12 is"}, {"start": 95.46, "duration": 6.24, "text": "negative 11x positive 72 divided by"}, {"start": 98.759, "duration": 5.941, "text": "negative 12 is negative 6. and 0 divided"}, {"start": 101.7, "duration": 5.04, "text": "by negative 12 is 0. this is a quadratic"}, {"start": 104.7, "duration": 4.26, "text": "so let's use the quadratic formula both"}, {"start": 106.74, "duration": 4.44, "text": "of these A's are equal to seven both of"}, {"start": 108.96, "duration": 4.799, "text": "these B's are equal to negative 11 and"}, {"start": 111.18, "duration": 4.14, "text": "the C is equal to negative 6. after we"}, {"start": 113.759, "duration": 4.5, "text": "simplify all this we end up getting x"}, {"start": 115.32, "duration": 5.339, "text": "equals 2 or x equals negative 3 7. and"}, {"start": 118.259, "duration": 3.661, "text": "looking up here 13 times a negative"}, {"start": 120.659, "duration": 3.42, "text": "number would give us a negative"}, {"start": 121.92, "duration": 5.04, "text": "hypotenuse and we can't have a negative"}, {"start": 124.079, "duration": 4.801, "text": "hypotenuse so this x negative 3 7 is not"}, {"start": 126.96, "duration": 4.26, "text": "a valid solution to our problem that"}, {"start": 128.88, "duration": 3.6, "text": "leaves us with x equals two and that"}, {"start": 131.22, "duration": 2.7, "text": "looks important so let's put a box"}, {"start": 132.48, "duration": 3.54, "text": "around it so going back to our original"}, {"start": 133.92, "duration": 4.02, "text": "figure let's copy down the smaller"}, {"start": 136.02, "duration": 3.48, "text": "triangle now for both our larger"}, {"start": 137.94, "duration": 4.5, "text": "triangle and the smaller triangle let's"}, {"start": 139.5, "duration": 6.599, "text": "plug in 2 for all the X's 9 times 2 is"}, {"start": 142.44, "duration": 7.019, "text": "18 2 times 2 is 4 9 times 2 is 18 plus 6"}, {"start": 146.099, "duration": 6.241, "text": "is 24. 2 times 2 is 4 plus 6 is 10 and"}, {"start": 149.459, "duration": 4.741, "text": "13 times 2 is 26. now we have specific"}, {"start": 152.34, "duration": 3.84, "text": "values for our triangles I think we're"}, {"start": 154.2, "duration": 3.6, "text": "ready to find the area of the red region"}, {"start": 156.18, "duration": 4.02, "text": "the area of the red region will be equal"}, {"start": 157.8, "duration": 4.5, "text": "to this big triangle minus the small"}, {"start": 160.2, "duration": 4.14, "text": "triangle so let's figure out the area of"}, {"start": 162.3, "duration": 3.12, "text": "each of those the area of either of"}, {"start": 164.34, "duration": 3.179, "text": "these triangles is going to be base"}, {"start": 165.42, "duration": 4.26, "text": "times height divided by 2. for this big"}, {"start": 167.519, "duration": 3.661, "text": "triangle the base is equal to 24 and the"}, {"start": 169.68, "duration": 3.12, "text": "height is equal to 10 and we're still"}, {"start": 171.18, "duration": 3.9, "text": "going to divide by two and for the"}, {"start": 172.8, "duration": 3.659, "text": "smaller triangle the base is 18 the"}, {"start": 175.08, "duration": 3.239, "text": "height is 4 and then we're going to"}, {"start": 176.459, "duration": 5.581, "text": "divide by two and now let's simplify"}, {"start": 178.319, "duration": 7.5, "text": "these fractions 21 4 times 10 is 240 and"}, {"start": 182.04, "duration": 7.32, "text": "18 times 4 is 72. 240 divided by 2 is"}, {"start": 185.819, "duration": 5.881, "text": "120 and 72 divided by 2 is 36. so our"}, {"start": 189.36, "duration": 4.86, "text": "big triangle is 120 so let's change this"}, {"start": 191.7, "duration": 5.88, "text": "to 120 and our small triangle is 36"}, {"start": 194.22, "duration": 7.379, "text": "let's change this to 36. and 120 minus"}, {"start": 197.58, "duration": 6.6, "text": "36 is 84. so the area of our red region"}, {"start": 201.599, "duration": 4.261, "text": "is 84 units squared even though some of"}, {"start": 204.18, "duration": 3.36, "text": "the measurements were in terms of X we"}, {"start": 205.86, "duration": 5.72, "text": "were still able to get an exact value"}, {"start": 207.54, "duration": 4.04, "text": "for the area how exciting"}], "title": "Area Challenge", "description": "I hope you guys like it!", "lengthSeconds": 210, "width": 640, "height": 360}
{"video_id": "aoN1YVDQFg8", "transcript": [{"start": 0.0, "duration": 4.38, "text": "have you guys ever heard of the birthday"}, {"start": 1.38, "duration": 4.86, "text": "Paradox imagine a group of 23 random"}, {"start": 4.38, "duration": 3.539, "text": "people what is the probability that at"}, {"start": 6.24, "duration": 3.66, "text": "least two people have the same birthday"}, {"start": 7.919, "duration": 3.3, "text": "and these are random people so no Twins"}, {"start": 9.9, "duration": 2.88, "text": "or anything like that so it's going to"}, {"start": 11.219, "duration": 3.3, "text": "be easiest to calculate the probability"}, {"start": 12.78, "duration": 3.78, "text": "that nobody has the same birthday and"}, {"start": 14.519, "duration": 3.961, "text": "then just do 100 minus whatever that"}, {"start": 16.56, "duration": 3.479, "text": "gives us first let's start with one pair"}, {"start": 18.48, "duration": 3.24, "text": "of people so the first person has a"}, {"start": 20.039, "duration": 3.061, "text": "birthday whatever that is then what is"}, {"start": 21.72, "duration": 3.12, "text": "the probability that the second person"}, {"start": 23.1, "duration": 4.08, "text": "does not have the same birthday well"}, {"start": 24.84, "duration": 4.859, "text": "there are 364 possible days that don't"}, {"start": 27.18, "duration": 5.579, "text": "match out of 365. so their probability"}, {"start": 29.699, "duration": 4.38, "text": "is 364 over 365. now let's bring in a"}, {"start": 32.759, "duration": 3.121, "text": "third person what is the probability"}, {"start": 34.079, "duration": 3.241, "text": "that that third person does not have the"}, {"start": 35.88, "duration": 5.04, "text": "same birthday as either of those other"}, {"start": 37.32, "duration": 5.64, "text": "two well that leaves 363 out of 365 days"}, {"start": 40.92, "duration": 3.6, "text": "so we can just multiply these and the"}, {"start": 42.96, "duration": 4.259, "text": "fourth person to not match any of these"}, {"start": 44.52, "duration": 4.199, "text": "three would be 362 over 365. and this"}, {"start": 47.219, "duration": 3.18, "text": "pattern continues for the rest of the"}, {"start": 48.719, "duration": 3.241, "text": "people in the group after we multiply"}, {"start": 50.399, "duration": 4.5, "text": "all of these probabilities we get"}, {"start": 51.96, "duration": 6.0, "text": "approximately 49.27 so the probability"}, {"start": 54.899, "duration": 5.521, "text": "is 49.27 that no one shares a birthday"}, {"start": 57.96, "duration": 5.52, "text": "so if we do a hundred percent minus that"}, {"start": 60.42, "duration": 4.739, "text": "we get a 50.73 probability that at least"}, {"start": 63.48, "duration": 4.319, "text": "two people have the same birthday so if"}, {"start": 65.159, "duration": 4.081, "text": "it feels weird that 23 people out of 365"}, {"start": 67.799, "duration": 3.541, "text": "days can do this it might be better to"}, {"start": 69.24, "duration": 3.54, "text": "look at this the number of pairs if we"}, {"start": 71.34, "duration": 3.36, "text": "look at the first person in the group he"}, {"start": 72.78, "duration": 3.6, "text": "has 22 pairs you can match with then the"}, {"start": 74.7, "duration": 3.18, "text": "next person has 21 new pairs they could"}, {"start": 76.38, "duration": 2.76, "text": "match with and the next person has 20"}, {"start": 77.88, "duration": 2.94, "text": "new pairs of people they can match with"}, {"start": 79.14, "duration": 4.2, "text": "and this pattern is going to continue so"}, {"start": 80.82, "duration": 3.9, "text": "there are going to be 253 pairs we're"}, {"start": 83.34, "duration": 3.779, "text": "just trying to get at least one of these"}, {"start": 84.72, "duration": 4.92, "text": "253 pairs to match and if you think of"}, {"start": 87.119, "duration": 5.101, "text": "it that way the 50.73 seems much more"}, {"start": 89.64, "duration": 3.96, "text": "realistic every one of these 253 pairs"}, {"start": 92.22, "duration": 5.18, "text": "is a possible chance for a birthday"}, {"start": 93.6, "duration": 3.8, "text": "match and this is the birthday paradox"}], "title": "Birthday Paradox Explained", "description": "Collab with Geeklyhub!", "lengthSeconds": 96, "width": 360, "height": 640}
{"video_id": "XRK9oA5RybY", "transcript": [{"start": 0.0, "duration": 4.12, "text": "hey guys this looks like a fun one it"}, {"start": 1.319, "duration": 4.321, "text": "gives us that this is 9 this is 16 and"}, {"start": 4.12, "duration": 3.12, "text": "this is 12 and they're all at right"}, {"start": 5.64, "duration": 3.36, "text": "angles with each other and it wants us"}, {"start": 7.24, "duration": 3.56, "text": "to find the radius of this quarter"}, {"start": 9.0, "duration": 3.48, "text": "Circle if you want to try it on your own"}, {"start": 10.8, "duration": 4.999, "text": "pause it right now cuz I'm going to"}, {"start": 12.48, "duration": 5.559, "text": "solve it in 3 2 1 for the first step"}, {"start": 15.799, "duration": 4.32, "text": "let's extend this over here and we're"}, {"start": 18.039, "duration": 4.121, "text": "completing this rectangle this bottom"}, {"start": 20.119, "duration": 3.681, "text": "will be equal to the top which is nine"}, {"start": 22.16, "duration": 3.92, "text": "and this side will be equal to this side"}, {"start": 23.8, "duration": 4.2, "text": "which is 16 and we've created this"}, {"start": 26.08, "duration": 4.56, "text": "little piece right here let's call that"}, {"start": 28.0, "duration": 4.559, "text": "X next let's draw two of our radi one"}, {"start": 30.64, "duration": 3.72, "text": "from here to here and one from here to"}, {"start": 32.559, "duration": 4.0, "text": "here and these are both radi so we'll"}, {"start": 34.36, "duration": 4.44, "text": "call them R first I want to focus on"}, {"start": 36.559, "duration": 3.961, "text": "this triangle right here this is a right"}, {"start": 38.8, "duration": 6.239, "text": "triangle so we can use the Pythagorean"}, {"start": 40.52, "duration": 6.679, "text": "theorem it'll be x + 16 2 + 9^ 2 is"}, {"start": 45.039, "duration": 4.081, "text": "equal to R 2 and then let's do the same"}, {"start": 47.199, "duration": 6.121, "text": "thing for this triangle down here it'll"}, {"start": 49.12, "duration": 6.239, "text": "be 9 + 12 2 + x^2 is = R 2 and we can"}, {"start": 53.32, "duration": 4.48, "text": "clean this up a little bit this 9 + 12"}, {"start": 55.359, "duration": 4.0, "text": "inside of the parentheses can be 21 and"}, {"start": 57.8, "duration": 3.2, "text": "now we have two equations with two"}, {"start": 59.359, "duration": 4.16, "text": "variables so let's solve for these"}, {"start": 61.0, "duration": 5.72, "text": "variables this is equal to R2 and that's"}, {"start": 63.519, "duration": 5.721, "text": "equal to R2 so we can set this equal to"}, {"start": 66.72, "duration": 5.6, "text": "that on the left hand side 212 is equal"}, {"start": 69.24, "duration": 5.48, "text": "to 441 and we can bring down the x^2 on"}, {"start": 72.32, "duration": 5.799, "text": "the right hand side this x + 16 2 is"}, {"start": 74.72, "duration": 5.68, "text": "equal to x + 16 * x + 16 and after we"}, {"start": 78.119, "duration": 6.241, "text": "multiply that out it ends up being x^2 +"}, {"start": 80.4, "duration": 6.84, "text": "32x + 256 and then this 9^ 2 is equal to"}, {"start": 84.36, "duration": 4.56, "text": "81 next let's subtract x^2 from both"}, {"start": 87.24, "duration": 3.559, "text": "sides of the equation on the left hand"}, {"start": 88.92, "duration": 3.12, "text": "side we have the 44 1 and then on the"}, {"start": 90.799, "duration": 5.68, "text": "right hand side we can bring down the"}, {"start": 92.04, "duration": 6.079, "text": "32x and 256 + 81 is equal to 337 let's"}, {"start": 96.479, "duration": 3.721, "text": "smush everything together and we want to"}, {"start": 98.119, "duration": 4.64, "text": "get this X all by itself so let's"}, {"start": 100.2, "duration": 6.12, "text": "subtract 337 from both sides of the"}, {"start": 102.759, "duration": 5.841, "text": "equation on the left hand side 441 - 337"}, {"start": 106.32, "duration": 3.92, "text": "is equal to 104 and on the right hand"}, {"start": 108.6, "duration": 4.28, "text": "side these will cancel each other out so"}, {"start": 110.24, "duration": 5.28, "text": "we're left with 32x next we can divide"}, {"start": 112.88, "duration": 4.599, "text": "both sides by 32 on the left hand side"}, {"start": 115.52, "duration": 4.72, "text": "both of these have eight in common so"}, {"start": 117.479, "duration": 4.481, "text": "it'll simplify to 13 over4 and on the"}, {"start": 120.24, "duration": 4.36, "text": "right hand side these 32s will cancel"}, {"start": 121.96, "duration": 4.72, "text": "each other out leaving us with X and now"}, {"start": 124.6, "duration": 5.079, "text": "we've solved for X it's equal to 13"}, {"start": 126.68, "duration": 5.639, "text": "over4 and let's update this x to be 13"}, {"start": 129.679, "duration": 5.161, "text": "over4 and we can also update this x to"}, {"start": 132.319, "duration": 6.92, "text": "be 13 over4 and now we're ready to solve"}, {"start": 134.84, "duration": 6.759, "text": "for R 212 is equal to 441 and this 13"}, {"start": 139.239, "duration": 6.161, "text": "over 4^ 2 that's the same thing as 132"}, {"start": 141.599, "duration": 5.841, "text": "over 4^2 which is equal to 169 over 16"}, {"start": 145.4, "duration": 3.68, "text": "and then we can copy down the R 2 in"}, {"start": 147.44, "duration": 5.2, "text": "order to add these two let's make this"}, {"start": 149.08, "duration": 4.96, "text": "441 a fraction by doing 441 over 1 to"}, {"start": 152.64, "duration": 4.08, "text": "give them a common denominator we can"}, {"start": 154.04, "duration": 4.32, "text": "multiply top and bottom of this by 16"}, {"start": 156.72, "duration": 4.799, "text": "and both of these 16s will just hop"}, {"start": 158.36, "duration": 7.36, "text": "right over to the fraction 16 * 441 is"}, {"start": 161.519, "duration": 5.281, "text": "equal to 756 and 16 * 1 is equal to 16"}, {"start": 165.72, "duration": 2.4, "text": "and now that these have the same"}, {"start": 166.8, "duration": 5.32, "text": "denominator we can write them as a"}, {"start": 168.12, "duration": 5.199, "text": "single fraction and 756 + 169 is equal"}, {"start": 172.12, "duration": 3.64, "text": "to"}, {"start": 173.319, "duration": 4.241, "text": "7,225 next we can square root both sides"}, {"start": 175.76, "duration": 3.6, "text": "of the equation on the left hand side"}, {"start": 177.56, "duration": 3.52, "text": "this square root can be split to top and"}, {"start": 179.36, "duration": 4.68, "text": "bottom the square root of"}, {"start": 181.08, "duration": 5.239, "text": "7,225 is equal to 85 and on bottom the"}, {"start": 184.04, "duration": 3.96, "text": "square root of 16 is equal to 4 and on"}, {"start": 186.319, "duration": 3.881, "text": "the right hand side the square root and"}, {"start": 188.0, "duration": 5.28, "text": "the square will cancel each other out"}, {"start": 190.2, "duration": 5.959, "text": "and then 85 / 4 is equal to"}, {"start": 193.28, "duration": 5.0, "text": "21.25 and now we found the radius to our"}, {"start": 196.159, "duration": 4.961, "text": "quarter Circle it's equal to"}, {"start": 198.28, "duration": 5.16, "text": "21.25 CM this is the answer to our"}, {"start": 201.12, "duration": 5.199, "text": "question let's put a box around it how"}, {"start": 203.44, "duration": 2.879, "text": "exciting"}], "title": "Find the Radius", "description": "This was a fun question, I hope you guys love it. \n\nDoes anybody read these descriptions? I think it would be funny if everybody who read this paragraph commented their favorite color and then their favorite animal. Then so many people in the comments would be confused cause there is no prompt in the video. :-)", "lengthSeconds": 204, "width": 640, "height": 360}
{"video_id": "nJqOPayFPMk", "transcript": [{"start": 0.0, "duration": 3.78, "text": "you guys this looks like a fun one it"}, {"start": 1.56, "duration": 5.16, "text": "says solve for x that's the square root"}, {"start": 3.78, "duration": 4.68, "text": "of x to the X to the x equals X and it"}, {"start": 6.72, "duration": 3.24, "text": "says X is greater than one so it does"}, {"start": 8.46, "duration": 2.94, "text": "look like x equals one would have been"}, {"start": 9.96, "duration": 3.24, "text": "one solution for this but they've"}, {"start": 11.4, "duration": 3.18, "text": "excluded that solution and they want us"}, {"start": 13.2, "duration": 2.7, "text": "to find another one if you want to try"}, {"start": 14.58, "duration": 2.82, "text": "this on your own pause it right now"}, {"start": 15.9, "duration": 3.78, "text": "because I'm going to solve it in three"}, {"start": 17.4, "duration": 4.68, "text": "two one so first we can multiply these"}, {"start": 19.68, "duration": 4.439, "text": "exponents at x times x is equal to x"}, {"start": 22.08, "duration": 3.72, "text": "squared and next to get rid of this"}, {"start": 24.119, "duration": 3.961, "text": "square root let's Square both sides of"}, {"start": 25.8, "duration": 3.6, "text": "the equation so this exponent and this"}, {"start": 28.08, "duration": 3.779, "text": "square root are going to cancel each"}, {"start": 29.4, "duration": 5.159, "text": "other out so now we have x to the x"}, {"start": 31.859, "duration": 4.801, "text": "squared equals x squared since X can't"}, {"start": 34.559, "duration": 4.201, "text": "equal one or zero that means the only"}, {"start": 36.66, "duration": 3.96, "text": "way this is true is if the exponents are"}, {"start": 38.76, "duration": 5.06, "text": "equal to each other so we can drop the"}, {"start": 40.62, "duration": 5.7, "text": "base Drop the Base"}, {"start": 43.82, "duration": 4.419, "text": "so we can drop the bass and set the"}, {"start": 46.32, "duration": 3.6, "text": "exponents equal to each other and then"}, {"start": 48.239, "duration": 3.901, "text": "from here we can square root both sides"}, {"start": 49.92, "duration": 4.02, "text": "and the square root of x squared is the"}, {"start": 52.14, "duration": 3.419, "text": "absolute value of x so we have the"}, {"start": 53.94, "duration": 3.48, "text": "absolute value of x is equal to square"}, {"start": 55.559, "duration": 3.66, "text": "root of 2 but since we know that X can't"}, {"start": 57.42, "duration": 3.72, "text": "be negative this is just going to be x"}, {"start": 59.219, "duration": 3.421, "text": "equals square root of 2 and that is the"}, {"start": 61.14, "duration": 3.239, "text": "answer to our question we can put a box"}, {"start": 62.64, "duration": 3.06, "text": "around it and let's check our work we're"}, {"start": 64.379, "duration": 3.24, "text": "going to plug in the square root of 2"}, {"start": 65.7, "duration": 3.48, "text": "for each of these x's and then we're"}, {"start": 67.619, "duration": 2.941, "text": "going to multiply these exponents square"}, {"start": 69.18, "duration": 3.479, "text": "root of 2 times square root of 2 is"}, {"start": 70.56, "duration": 3.66, "text": "equal to two and then this square root"}, {"start": 72.659, "duration": 3.061, "text": "and this Square cancel each other out"}, {"start": 74.22, "duration": 3.719, "text": "and then we end up with square root of 2"}, {"start": 75.72, "duration": 3.899, "text": "equals square root of 2 it checks out so"}, {"start": 77.939, "duration": 5.18, "text": "for this problem x equals the square"}, {"start": 79.619, "duration": 3.5, "text": "root of two how exciting"}], "title": "Fun Math Challenge", "description": "I thought this would be a fun problem. I also had a chance to play around with some animations and special effects. How exciting!", "lengthSeconds": 81, "width": 640, "height": 360}
{"video_id": "KgdaoLTslo0", "transcript": [{"start": 0.04, "duration": 3.839, "text": "hey guys I love this problem it says the"}, {"start": 2.12, "duration": 4.12, "text": "width is twice the height if the"}, {"start": 3.879, "duration": 5.041, "text": "perimeter is 120 what is the area and"}, {"start": 6.24, "duration": 5.16, "text": "we're given that the width is x^2 - x -"}, {"start": 8.92, "duration": 3.96, "text": "16 and the height is x + 12 if you want"}, {"start": 11.4, "duration": 4.639, "text": "to try this on your own pause it right"}, {"start": 12.88, "duration": 5.28, "text": "now cuz I'm going to solve it in 3 2 1"}, {"start": 16.039, "duration": 3.841, "text": "these are a trap we don't need these so"}, {"start": 18.16, "duration": 4.199, "text": "we can just assign H for the height and"}, {"start": 19.88, "duration": 3.76, "text": "call the width 2 H because the width is"}, {"start": 22.359, "duration": 4.0, "text": "twice the height so we need the"}, {"start": 23.64, "duration": 6.92, "text": "perimeter to be 120 that means we need H"}, {"start": 26.359, "duration": 7.401, "text": "+ 2 H + H + 2 H to equal 120 and H H + 2"}, {"start": 30.56, "duration": 5.519, "text": "H + H + 2 H is equal to 6 h after we"}, {"start": 33.76, "duration": 4.639, "text": "divide both sides by 6 we get H equal to"}, {"start": 36.079, "duration": 5.281, "text": "20 so we can plug in 20 for these H's"}, {"start": 38.399, "duration": 5.801, "text": "and the two h's will become 40 40 + 20 +"}, {"start": 41.36, "duration": 5.16, "text": "40 + 20 gives us 120 and then the area"}, {"start": 44.2, "duration": 3.839, "text": "will be 40 * 20 which is 800 unit"}, {"start": 46.52, "duration": 5.359, "text": "squared and that's the answer to the"}, {"start": 48.039, "duration": 3.84, "text": "question how exciting"}], "title": "Sneaky Math Problem", "description": "I thought this was a fun one.", "lengthSeconds": 50, "width": 360, "height": 640}
{"video_id": "EoIlXnjG1Y8", "transcript": [{"start": 0.0, "duration": 2.639, "text": "hey guys I made this video a couple days"}, {"start": 1.319, "duration": 3.361, "text": "ago I really like the problem because it"}, {"start": 2.639, "duration": 3.321, "text": "had a cool shortcut but so many people"}, {"start": 4.68, "duration": 2.8, "text": "asked me to solve it without the"}, {"start": 5.96, "duration": 3.12, "text": "shortcut if you want to try this on your"}, {"start": 7.48, "duration": 5.6, "text": "own pause it right now cuz I'm going to"}, {"start": 9.08, "duration": 6.12, "text": "solve for x and y in 3 2 1 so first"}, {"start": 13.08, "duration": 7.439, "text": "let's copy both of these things down so"}, {"start": 15.2, "duration": 7.12, "text": "we have X Y = 36 and x^2 + y^2 = 78"}, {"start": 20.519, "duration": 4.121, "text": "let's start with this one first let's"}, {"start": 22.32, "duration": 6.08, "text": "divide both sides by X and it becomes y"}, {"start": 24.64, "duration": 5.879, "text": "= 36 /x so in the lower equation in the"}, {"start": 28.4, "duration": 4.999, "text": "place of the Y let's plug in"}, {"start": 30.519, "duration": 6.36, "text": "36x now this exponent goes to both the"}, {"start": 33.399, "duration": 7.721, "text": "top and the bottom so 36 /x^2 becomes"}, {"start": 36.879, "duration": 6.441, "text": "362 over x^2 and 362 is equal to"}, {"start": 41.12, "duration": 3.959, "text": "1296 and I don't really like fractions"}, {"start": 43.32, "duration": 3.2, "text": "so let's get rid of this x^2 in the"}, {"start": 45.079, "duration": 3.241, "text": "denominator we can do that by"}, {"start": 46.52, "duration": 4.16, "text": "multiplying both sides of the equation"}, {"start": 48.32, "duration": 4.28, "text": "by X2 and we're allowed to multiply both"}, {"start": 50.68, "duration": 5.0, "text": "sides by X squ because we know that X"}, {"start": 52.6, "duration": 5.279, "text": "cannot be zero if x were zero XY would"}, {"start": 55.68, "duration": 4.0, "text": "have to be zero so on the left hand side"}, {"start": 57.879, "duration": 5.36, "text": "this x^ squ will distribute to both of"}, {"start": 59.68, "duration": 6.36, "text": "these terms we have x^2 * x^2 which is X"}, {"start": 63.239, "duration": 4.761, "text": "4th and then we've got x^2 * this thing"}, {"start": 66.04, "duration": 3.92, "text": "the x^2 on top and the x^2 on bottom"}, {"start": 68.0, "duration": 4.439, "text": "will cancel each other out so the second"}, {"start": 69.96, "duration": 5.64, "text": "term is going to be 1296 and that's all"}, {"start": 72.439, "duration": 4.921, "text": "going to be equal to 78 x^2 next I want"}, {"start": 75.6, "duration": 4.559, "text": "to set this all equal to zero let's"}, {"start": 77.36, "duration": 5.6, "text": "subtract 78 x^2 from both sides on the"}, {"start": 80.159, "duration": 6.121, "text": "left hand side we end up with x 4 - 78"}, {"start": 82.96, "duration": 5.24, "text": "x^2 + 1296 and on the right hand side"}, {"start": 86.28, "duration": 4.28, "text": "these cancel each other out leaving us"}, {"start": 88.2, "duration": 5.04, "text": "with zero now we can solve this by"}, {"start": 90.56, "duration": 5.239, "text": "factoring this x 4 will break down into"}, {"start": 93.24, "duration": 5.72, "text": "x^2 and x^2 and then we want to find the"}, {"start": 95.799, "duration": 6.6, "text": "two numbers p and Q such that p + Q is"}, {"start": 98.96, "duration": 5.479, "text": "equal to -78 and P * Q is equal to"}, {"start": 102.399, "duration": 5.72, "text": "1296 this could take a little bit of"}, {"start": 104.439, "duration": 6.04, "text": "effort it ends up being -24 and -54 and"}, {"start": 108.119, "duration": 5.441, "text": "I did use a calculator to find these"}, {"start": 110.479, "duration": 5.481, "text": "next we have this * that equals 0 that"}, {"start": 113.56, "duration": 4.4, "text": "means that either this equals z or that"}, {"start": 115.96, "duration": 4.56, "text": "equals z so now we just got to solve"}, {"start": 117.96, "duration": 5.119, "text": "these two equations we can add 24 to"}, {"start": 120.52, "duration": 4.559, "text": "both sides of this equation or add 54 to"}, {"start": 123.079, "duration": 3.72, "text": "both sides of this equation on the left"}, {"start": 125.079, "duration": 4.361, "text": "hand side these cancel each other out"}, {"start": 126.799, "duration": 4.281, "text": "and we're left with X2 is equal to 24"}, {"start": 129.44, "duration": 3.519, "text": "and on this equation these two cancel"}, {"start": 131.08, "duration": 4.2, "text": "each other out so we're left with x^2 is"}, {"start": 132.959, "duration": 4.201, "text": "equal to 54 now both of these are in"}, {"start": 135.28, "duration": 4.56, "text": "terms of X squ so we're going to want to"}, {"start": 137.16, "duration": 5.04, "text": "square root both sides of both equations"}, {"start": 139.84, "duration": 4.44, "text": "for both of these square root of x squs"}, {"start": 142.2, "duration": 3.72, "text": "they're going to be absolute value of x"}, {"start": 144.28, "duration": 3.52, "text": "and to solve for the absolute values of"}, {"start": 145.92, "duration": 4.24, "text": "X each of these will become plus or"}, {"start": 147.8, "duration": 8.159, "text": "minus and now we have our four Val vales"}, {"start": 150.16, "duration": 8.24, "text": "of X posi < tk24 < tk24 posun 54 orun 54"}, {"start": 155.959, "duration": 4.28, "text": "and we know that Y is equal to 36 overx"}, {"start": 158.4, "duration": 4.4, "text": "so for each of these values of X we can"}, {"start": 160.239, "duration": 5.121, "text": "find the corresponding value of y in the"}, {"start": 162.8, "duration": 3.96, "text": "place of this x we'll plug in < TK 24"}, {"start": 165.36, "duration": 4.08, "text": "and the same thing for the rest of these"}, {"start": 166.76, "duration": 5.08, "text": "X's so now we have our answers for X and"}, {"start": 169.44, "duration": 4.719, "text": "Y let's check our work in the place of X"}, {"start": 171.84, "duration": 5.52, "text": "we can plug in < TK 24 and in the place"}, {"start": 174.159, "duration": 4.8, "text": "of Y we can plug in 36 over < tk24 this"}, {"start": 177.36, "duration": 5.44, "text": "exponent will go to the top and the"}, {"start": 178.959, "duration": 6.36, "text": "bottom 36 6 4 is equal to a huge number"}, {"start": 182.8, "duration": 5.48, "text": "and both of these < TK 24 to 4ths are"}, {"start": 185.319, "duration": 6.56, "text": "equal to 576 and then this huge number"}, {"start": 188.28, "duration": 7.92, "text": "divided by 576 is equal to 2916 and then"}, {"start": 191.879, "duration": 7.241, "text": "576 + 2916 is equal to 3492 and that is"}, {"start": 196.2, "duration": 5.0, "text": "the correct answer for X 4th + y 4th so"}, {"start": 199.12, "duration": 4.56, "text": "it checks out and in fact all of these"}, {"start": 201.2, "duration": 4.28, "text": "will check out so 3492 is the answer to"}, {"start": 203.68, "duration": 4.08, "text": "our question and this was the difficult"}, {"start": 205.48, "duration": 4.319, "text": "way to do it let's put a box around him"}, {"start": 207.76, "duration": 4.399, "text": "how exciting and I also got a lot of"}, {"start": 209.799, "duration": 4.321, "text": "request to see the haircut here it is I"}, {"start": 212.159, "duration": 5.921, "text": "got hat hair right now I don't know why"}, {"start": 214.12, "duration": 3.96, "text": "there's the haircut how exciting"}], "title": "Without the Shortcut", "description": "This is part 2. I hope you like it!", "lengthSeconds": 216, "width": 640, "height": 360}
{"video_id": "LiP7IkD0-fk", "transcript": [{"start": 0.0, "duration": 3.12, "text": "ton of people get this problem wrong it"}, {"start": 1.36, "duration": 3.36, "text": "says what is the value of 5x and we have"}, {"start": 3.12, "duration": 3.44, "text": "a diagram we have a 90 degree angle"}, {"start": 4.72, "duration": 3.44, "text": "right here which means that this angle"}, {"start": 6.56, "duration": 3.52, "text": "right here is also going to be 90"}, {"start": 8.16, "duration": 3.76, "text": "degrees because this is a straight line"}, {"start": 10.08, "duration": 3.28, "text": "and since this angle right here is 90"}, {"start": 11.92, "duration": 3.839, "text": "degrees that means that if we were to"}, {"start": 13.36, "duration": 4.88, "text": "add up all of these inside angles x plus"}, {"start": 15.759, "duration": 3.921, "text": "2x plus 2x all of them have to add up to"}, {"start": 18.24, "duration": 3.119, "text": "90 degrees so let's go ahead and do that"}, {"start": 19.68, "duration": 5.2, "text": "down here so we have an x"}, {"start": 21.359, "duration": 5.521, "text": "we have a 2x and we have a 2x and all of"}, {"start": 24.88, "duration": 3.84, "text": "these angles added up has to equal 90"}, {"start": 26.88, "duration": 3.84, "text": "degrees okay so now we can just add up"}, {"start": 28.72, "duration": 5.04, "text": "like terms x 2x and 2x that's going to"}, {"start": 30.72, "duration": 4.08, "text": "be equal to 5x and we have an equals 90."}, {"start": 33.76, "duration": 4.72, "text": "and just like that we don't have to do"}, {"start": 34.8, "duration": 3.68, "text": "any more work because our answer is"}], "title": "Missing Angles Geometry Problem | Tricky Math Question | JusticeTheTutor #maths #math #shorts", "description": "", "lengthSeconds": 37, "width": 360, "height": 640}
{"video_id": "lj4cOoD_pBo", "transcript": [{"start": 0.0, "duration": 3.76, "text": "hey guys welcome to this isosceles"}, {"start": 1.76, "duration": 2.48, "text": "triangle video so we're going to solve"}, {"start": 3.76, "duration": 2.72, "text": "for"}, {"start": 4.24, "duration": 4.0, "text": "x in this isosceles triangle the whole"}, {"start": 6.48, "duration": 4.0, "text": "idea of an isosceles triangle is that"}, {"start": 8.24, "duration": 4.72, "text": "these base angles are equal"}, {"start": 10.48, "duration": 4.48, "text": "and the sides opposite of those base"}, {"start": 12.96, "duration": 3.84, "text": "angles are going to be equal"}, {"start": 14.96, "duration": 4.0, "text": "and since these two sides are equal we"}, {"start": 16.8, "duration": 2.639, "text": "can set their expressions equal to each"}, {"start": 18.96, "duration": 4.319, "text": "other"}, {"start": 19.439, "duration": 4.161, "text": "so we can say 3x plus 5 equals 5x plus"}, {"start": 23.279, "duration": 2.801, "text": "1."}, {"start": 23.6, "duration": 3.2, "text": "it says solve for x so our goal is get x"}, {"start": 26.08, "duration": 2.16, "text": "by itself"}, {"start": 26.8, "duration": 3.28, "text": "when we're done it should look something"}, {"start": 28.24, "duration": 4.159, "text": "like x equals something"}, {"start": 30.08, "duration": 4.319, "text": "a number but right now we have an x on"}, {"start": 32.399, "duration": 3.68, "text": "both sides of the equation we can't have"}, {"start": 34.399, "duration": 2.481, "text": "that so we're going to move 1 to the"}, {"start": 36.079, "duration": 2.721, "text": "other side"}, {"start": 36.88, "duration": 3.92, "text": "so i like to find the smaller of the 2"}, {"start": 38.8, "duration": 3.919, "text": "and the 3x is smaller than 5x"}, {"start": 40.8, "duration": 3.919, "text": "so i'm going to subtract that from both"}, {"start": 42.719, "duration": 3.36, "text": "sides the reason i do the smaller one is"}, {"start": 44.719, "duration": 4.721, "text": "because when i do 5x"}, {"start": 46.079, "duration": 5.281, "text": "minus 3x i get positive 2x and i prefer"}, {"start": 49.44, "duration": 3.599, "text": "positive numbers over negative numbers"}, {"start": 51.36, "duration": 2.32, "text": "and then this goes away so we're left"}, {"start": 53.039, "duration": 3.52, "text": "with just"}, {"start": 53.68, "duration": 3.28, "text": "5 on the left side and we have 2x plus"}, {"start": 56.559, "duration": 2.8, "text": "1."}, {"start": 56.96, "duration": 3.2, "text": "now we only have one side that contains"}, {"start": 59.359, "duration": 3.04, "text": "an x"}, {"start": 60.16, "duration": 4.16, "text": "our goal now is to get that x by itself"}, {"start": 62.399, "duration": 4.961, "text": "so first i'm going to subtract the 1."}, {"start": 64.32, "duration": 4.32, "text": "5 minus 1 is 4 and this all cancels so"}, {"start": 67.36, "duration": 4.32, "text": "we just have 2x"}, {"start": 68.64, "duration": 5.2, "text": "on the right side and now this means"}, {"start": 71.68, "duration": 5.439, "text": "two times x so the way i'm going to undo"}, {"start": 73.84, "duration": 5.44, "text": "that is divide both sides by two"}, {"start": 77.119, "duration": 3.68, "text": "and we get x equals two is the answer"}, {"start": 79.28, "duration": 2.879, "text": "and we can check it"}, {"start": 80.799, "duration": 2.401, "text": "so we got this one right so that's how"}, {"start": 82.159, "duration": 2.881, "text": "these work when you're trying to solve"}, {"start": 83.2, "duration": 4.0, "text": "for x and you're dealing with isosceles"}, {"start": 85.04, "duration": 4.96, "text": "triangles you usually will just set"}, {"start": 87.2, "duration": 4.32, "text": "the two sides equal to each other so"}, {"start": 90.0, "duration": 3.68, "text": "here's another variation"}, {"start": 91.52, "duration": 4.48, "text": "once again it's an isosceles triangle"}, {"start": 93.68, "duration": 4.16, "text": "and so we know the base angles are equal"}, {"start": 96.0, "duration": 3.2, "text": "and the sides are equal well here"}, {"start": 97.84, "duration": 3.2, "text": "they're telling us the degrees of the"}, {"start": 99.2, "duration": 3.36, "text": "angles so we can just set these degrees"}, {"start": 101.04, "duration": 3.84, "text": "equal to each other we can say"}, {"start": 102.56, "duration": 3.919, "text": "x minus 10 equals 50. now this"}, {"start": 104.88, "duration": 3.04, "text": "parenthesis doesn't mean anything"}, {"start": 106.479, "duration": 3.28, "text": "really because it's already alone on"}, {"start": 107.92, "duration": 2.64, "text": "that side so we just get rid of it once"}, {"start": 109.759, "duration": 3.04, "text": "again this says"}, {"start": 110.56, "duration": 4.559, "text": "solve for x so our whole goal is get x"}, {"start": 112.799, "duration": 5.201, "text": "by itself right now there's a minus 10"}, {"start": 115.119, "duration": 3.761, "text": "the way we undo a minus 10 is adding 10"}, {"start": 118.0, "duration": 4.24, "text": "which leaves us"}, {"start": 118.88, "duration": 5.199, "text": "x on the left side and 50 plus 10 is 60."}, {"start": 122.24, "duration": 3.36, "text": "so the answer to this one should be x"}, {"start": 124.079, "duration": 3.68, "text": "equals 60 and we can check it"}, {"start": 125.6, "duration": 3.359, "text": "and it says the answer is x equals 60."}, {"start": 127.759, "duration": 3.521, "text": "if you want to try more of these i have"}, {"start": 128.959, "duration": 3.841, "text": "this page on andymath.com i'll include a"}, {"start": 131.28, "duration": 3.84, "text": "link down below in the description"}, {"start": 132.8, "duration": 4.56, "text": "there's also related pages for more"}, {"start": 135.12, "duration": 3.68, "text": "triangle related concepts and these all"}, {"start": 137.36, "duration": 1.76, "text": "have practice problems and answers as"}, {"start": 138.8, "duration": 2.24, "text": "well"}, {"start": 139.12, "duration": 3.52, "text": "please like this video and subscribe to"}, {"start": 141.04, "duration": 3.04, "text": "my channel and comment below if you have"}, {"start": 142.64, "duration": 2.56, "text": "any other math questions i'll try to"}, {"start": 144.08, "duration": 10.239, "text": "answer them for you"}, {"start": 145.2, "duration": 9.119, "text": "i'll talk to you guys soon bye"}], "title": "Isosceles Triangles Solve for x", "description": "https://andymath.com/isosceles-triangles/\nVisit the above link for more practice problems and related pages!", "lengthSeconds": 153, "width": 640, "height": 360}
{"video_id": "fPmWmwhLT5U", "transcript": [{"start": 0.0, "duration": 4.08, "text": "what is e probably heard e is"}, {"start": 2.1, "duration": 3.6, "text": "approximately 2.718 and it is an"}, {"start": 4.08, "duration": 3.96, "text": "irrational number that will never repeat"}, {"start": 5.7, "duration": 3.66, "text": "but what is it I'll explain one way to"}, {"start": 8.04, "duration": 2.94, "text": "find it this writers the formula for"}, {"start": 9.36, "duration": 3.12, "text": "compounded interest let's set the"}, {"start": 10.98, "duration": 3.72, "text": "principal to a dollar set the interest"}, {"start": 12.48, "duration": 4.5, "text": "rate to 100 and set the time to one year"}, {"start": 14.7, "duration": 3.78, "text": "if we have one compounding per year it's"}, {"start": 16.98, "duration": 3.12, "text": "going to look like this this ends up"}, {"start": 18.48, "duration": 2.879, "text": "just being one plus one which is two now"}, {"start": 20.1, "duration": 2.58, "text": "if we did the same thing but with two"}, {"start": 21.359, "duration": 3.361, "text": "compoundings it would look like this"}, {"start": 22.68, "duration": 3.54, "text": "Which is higher than the two and then we"}, {"start": 24.72, "duration": 3.12, "text": "could do monthly compoundings that would"}, {"start": 26.22, "duration": 3.0, "text": "be n equals 12 and that would look"}, {"start": 27.84, "duration": 3.72, "text": "something like this which gives us"}, {"start": 29.22, "duration": 3.66, "text": "approximately 2.61304 so you can tell"}, {"start": 31.56, "duration": 3.6, "text": "we're getting closer and closer to that"}, {"start": 32.88, "duration": 4.08, "text": "value of e now if I keep going say do"}, {"start": 35.16, "duration": 2.94, "text": "daily compoundings that would look like"}, {"start": 36.96, "duration": 3.419, "text": "this right here and this number is"}, {"start": 38.1, "duration": 3.959, "text": "getting even closer to e if we did 10"}, {"start": 40.379, "duration": 3.481, "text": "000 compoundings this right here and we"}, {"start": 42.059, "duration": 3.601, "text": "could do 100 000 compoundings we're just"}, {"start": 43.86, "duration": 3.539, "text": "getting super close a million"}, {"start": 45.66, "duration": 3.6, "text": "compoundings gets us here and in the end"}, {"start": 47.399, "duration": 3.66, "text": "if we did it continuously that'd be like"}, {"start": 49.26, "duration": 3.24, "text": "n equal to Infinity the best way to"}, {"start": 51.059, "duration": 2.881, "text": "express that is with the limit as n"}, {"start": 52.5, "duration": 3.42, "text": "approaches infinity and that's going to"}, {"start": 53.94, "duration": 3.959, "text": "give us the number that is exactly e so"}, {"start": 55.92, "duration": 4.459, "text": "this right here is one definition of e I"}, {"start": 57.899, "duration": 2.48, "text": "hope this helps"}], "title": "Quick e Example #shorts", "description": "https://andymath.com/the-number-e/\n\n#shorts", "lengthSeconds": 59, "width": 270, "height": 480}
{"video_id": "cI-N3qBV-A4", "transcript": [{"start": 0.08, "duration": 3.68, "text": "so we want to solve for x in this"}, {"start": 1.76, "duration": 3.039, "text": "triangle we're going to use the exterior"}, {"start": 3.76, "duration": 4.079, "text": "angle theorem"}, {"start": 4.799, "duration": 5.361, "text": "so if we call this a this b and this c"}, {"start": 7.839, "duration": 4.321, "text": "it is always true that a will equal b"}, {"start": 10.16, "duration": 2.8, "text": "plus c and these are the two angles that"}, {"start": 12.16, "duration": 2.559, "text": "are opposite"}, {"start": 12.96, "duration": 3.68, "text": "of this one so let's fill in for a b and"}, {"start": 14.719, "duration": 5.441, "text": "c a was 3x plus 2"}, {"start": 16.64, "duration": 5.04, "text": "b is x plus 25 and c is x plus 17."}, {"start": 20.16, "duration": 3.279, "text": "i put the parentheses there so we could"}, {"start": 21.68, "duration": 5.599, "text": "see each of the angles x plus"}, {"start": 23.439, "duration": 5.281, "text": "x is 2x and 25 plus 17 is 42."}, {"start": 27.279, "duration": 3.201, "text": "so we have an x on both sides of the"}, {"start": 28.72, "duration": 3.2, "text": "equation we want to subtract the smaller"}, {"start": 30.48, "duration": 4.8, "text": "one over to the other side"}, {"start": 31.92, "duration": 4.72, "text": "3x minus 2x is x bring down the 2 2x"}, {"start": 35.28, "duration": 3.52, "text": "minus 2x goes away"}, {"start": 36.64, "duration": 3.759, "text": "so we just have a 42 on this side so"}, {"start": 38.8, "duration": 4.32, "text": "subtract 2 from both sides"}, {"start": 40.399, "duration": 4.241, "text": "42 minus 2 is 40 for the right hand side"}, {"start": 43.12, "duration": 2.32, "text": "and that's the answer the question x"}, {"start": 44.64, "duration": 3.439, "text": "equals 40."}, {"start": 45.44, "duration": 3.439, "text": "so we use the exterior angles theorem to"}, {"start": 48.079, "duration": 2.0, "text": "solve this"}, {"start": 48.879, "duration": 2.721, "text": "comment on this video for what you'd"}, {"start": 50.079, "duration": 3.561, "text": "like me to solve next and i'll talk to"}, {"start": 51.6, "duration": 5.04, "text": "you guys soon"}, {"start": 53.64, "duration": 3.0, "text": "bye"}], "title": "Exterior Angle Theorem #Shorts", "description": "\ud835\udc0b\ud835\udc08\ud835\udc0d\ud835\udc0a\ud835\udc12 \ud835\udc13\ud835\udc0e \ud835\udc11\ud835\udc04\ud835\udc0b\ud835\udc00\ud835\udc13\ud835\udc04\ud835\udc03 \ud835\udc13\ud835\udc0e\ud835\udc0f\ud835\udc08\ud835\udc02\ud835\udc12\n\ud835\udc04\ud835\udc31\ud835\udc2d\ud835\udc1e\ud835\udc2b\ud835\udc22\ud835\udc28\ud835\udc2b \ud835\udc00\ud835\udc27\ud835\udc20\ud835\udc25\ud835\udc1e \ud835\udc13\ud835\udc21\ud835\udc1e\ud835\udc28\ud835\udc2b\ud835\udc1e\ud835\udc26: https://andymath.com/exterior-angle-theorem/\n\ud835\udc13\ud835\udc2b\ud835\udc22\ud835\udc1a\ud835\udc27\ud835\udc20\ud835\udc25\ud835\udc1e \ud835\udc00\ud835\udc27\ud835\udc20\ud835\udc25\ud835\udc1e \ud835\udc12\ud835\udc2e\ud835\udc26 \ud835\udc13\ud835\udc21\ud835\udc1e\ud835\udc28\ud835\udc2b\ud835\udc1e\ud835\udc26: https://andymath.com/triangle-angle-sum-theorem/\n\n\ud835\udc04\ud835\udc2a\ud835\udc2e\ud835\udc22\ud835\udc25\ud835\udc1a\ud835\udc2d\ud835\udc1e\ud835\udc2b\ud835\udc1a\ud835\udc25 \ud835\udc13\ud835\udc2b\ud835\udc22\ud835\udc1a\ud835\udc27\ud835\udc20\ud835\udc25\ud835\udc1e\ud835\udc2c: https://andymath.com/equilateral-triangles/\n\n\ud835\udc08\ud835\udc2c\ud835\udc28\ud835\udc2c\ud835\udc1c\ud835\udc1e\ud835\udc25\ud835\udc1e\ud835\udc2c \ud835\udc13\ud835\udc2b\ud835\udc22\ud835\udc1a\ud835\udc27\ud835\udc20\ud835\udc25\ud835\udc1e\ud835\udc2c: https://andymath.com/isosceles-triangles/\n\n\ud835\udc0f\ud835\udc32\ud835\udc2d\ud835\udc21\ud835\udc1a\ud835\udc20\ud835\udc28\ud835\udc2b\ud835\udc1e\ud835\udc1a\ud835\udc27 \ud835\udc13\ud835\udc21\ud835\udc1e\ud835\udc28\ud835\udc2b\ud835\udc1e\ud835\udc26: https://andymath.com/pythagorean-theorem/\n\n\ud835\udc0f\ud835\udc32\ud835\udc2d\ud835\udc21\ud835\udc1a\ud835\udc20\ud835\udc28\ud835\udc2b\ud835\udc1e\ud835\udc1a\ud835\udc27 \ud835\udc13\ud835\udc2b\ud835\udc22\ud835\udc29\ud835\udc25\ud835\udc1e\ud835\udc2c: https://andymath.com/pythagorean-triples/\n\n\ud835\udc12\ud835\udc22\ud835\udc1d\ud835\udc1e-\ud835\udc12\ud835\udc29\ud835\udc25\ud835\udc22\ud835\udc2d\ud835\udc2d\ud835\udc1e\ud835\udc2b \ud835\udc13\ud835\udc21\ud835\udc1e\ud835\udc28\ud835\udc2b\ud835\udc1e\ud835\udc26: https://andymath.com/side-splitter-theorem/\n\n\ud835\udc13\ud835\udc2b\ud835\udc22\ud835\udc1a\ud835\udc27\ud835\udc20\ud835\udc25\ud835\udc1e \ud835\udc0f\ud835\udc2b\ud835\udc28\ud835\udc29\ud835\udc28\ud835\udc2b\ud835\udc2d\ud835\udc22\ud835\udc28\ud835\udc27\ud835\udc1a\ud835\udc25\ud835\udc22\ud835\udc2d\ud835\udc32 \ud835\udc13\ud835\udc21\ud835\udc1e\ud835\udc28\ud835\udc2b\ud835\udc1e\ud835\udc26: https://andymath.com/triangle-proportionality-theorem/\n\n\ud835\udc0f\ud835\udc1a\ud835\udc2b\ud835\udc1a\ud835\udc25\ud835\udc25\ud835\udc1e\ud835\udc25 \ud835\udc0b\ud835\udc22\ud835\udc27\ud835\udc1e\ud835\udc2c \ud835\udc0f\ud835\udc2b\ud835\udc28\ud835\udc29\ud835\udc28\ud835\udc2b\ud835\udc2d\ud835\udc22\ud835\udc28\ud835\udc27\ud835\udc1a\ud835\udc25\ud835\udc22\ud835\udc2d\ud835\udc32 \ud835\udc13\ud835\udc21\ud835\udc1e\ud835\udc28\ud835\udc2b\ud835\udc1e\ud835\udc26: https://andymath.com/parallel-lines-proportionality-theorem/\n\n\ud835\udc13\ud835\udc2b\ud835\udc22\ud835\udc1a\ud835\udc27\ud835\udc20\ud835\udc25\ud835\udc1e \ud835\udc0c\ud835\udc22\ud835\udc1d\ud835\udc2c\ud835\udc1e\ud835\udc20\ud835\udc26\ud835\udc1e\ud835\udc27\ud835\udc2d \ud835\udc13\ud835\udc21\ud835\udc1e\ud835\udc28\ud835\udc2b\ud835\udc1e\ud835\udc26: https://andymath.com/triangle-midsegment-theorem/\n\n\ud835\udc12\ud835\udc29\ud835\udc1e\ud835\udc1c\ud835\udc22\ud835\udc1a\ud835\udc25 \ud835\udc11\ud835\udc22\ud835\udc20\ud835\udc21\ud835\udc2d \ud835\udc13\ud835\udc2b\ud835\udc22\ud835\udc1a\ud835\udc27\ud835\udc20\ud835\udc25\ud835\udc1e\ud835\udc2c \ud835\udfd1\ud835\udfce-\ud835\udfd4\ud835\udfce-\ud835\udfd7\ud835\udfce \ud835\udfd2\ud835\udfd3-\ud835\udfd2\ud835\udfd3-\ud835\udfd7\ud835\udfce: https://andymath.com/special-right-triangles/\n\n\ud835\udc07\ud835\udc22\ud835\udc27\ud835\udc20\ud835\udc1e \ud835\udc13\ud835\udc21\ud835\udc1e\ud835\udc28\ud835\udc2b\ud835\udc1e\ud835\udc26: https://andymath.com/Hinge-theorem/\n\n\ud835\udc12\ud835\udc22\ud835\udc26\ud835\udc22\ud835\udc25\ud835\udc1a\ud835\udc2b \ud835\udc11\ud835\udc22\ud835\udc20\ud835\udc21\ud835\udc2d \ud835\udc13\ud835\udc2b\ud835\udc22\ud835\udc1a\ud835\udc27\ud835\udc20\ud835\udc25\ud835\udc1e\ud835\udc2c: https://andymath.com/geometric-mean-similar-right-triangles/\n\n\ud835\udc13\ud835\udc2b\ud835\udc22\ud835\udc1a\ud835\udc27\ud835\udc20\ud835\udc25\ud835\udc1e \ud835\udc0b\ud835\udc28\ud835\udc27\ud835\udc20\ud835\udc1e\ud835\udc2b \ud835\udc12\ud835\udc22\ud835\udc1d\ud835\udc1e \ud835\udc13\ud835\udc21\ud835\udc1e\ud835\udc28\ud835\udc2b\ud835\udc1e\ud835\udc26: https://andymath.com/Triangle-longer-side-theorem/\n\n\ud835\udc13\ud835\udc2b\ud835\udc22\ud835\udc1a\ud835\udc27\ud835\udc20\ud835\udc25\ud835\udc1e \ud835\udc0b\ud835\udc1a\ud835\udc2b\ud835\udc20\ud835\udc1e\ud835\udc2b \ud835\udc00\ud835\udc27\ud835\udc20\ud835\udc25\ud835\udc1e \ud835\udc13\ud835\udc21\ud835\udc1e\ud835\udc28\ud835\udc2b\ud835\udc1e\ud835\udc26: https://andymath.com/Triangle-larger-angle-theorem/\n\nVisit any of the the above links for more practice problems, notes, solutions and videos for all these related topics. For even more math topics, visit https://andymath.com/.\n\n#Shorts", "lengthSeconds": 55, "width": 202, "height": 360}
{"video_id": "zfMA9hZlQT4", "transcript": [{"start": 0.0, "duration": 3.48, "text": "hey guys somebody requested that I"}, {"start": 1.8, "duration": 2.94, "text": "solved this one my last two videos were"}, {"start": 3.48, "duration": 2.58, "text": "pretty intense so I thought we would do"}, {"start": 4.74, "duration": 2.88, "text": "a lighter one so we're trying to solve"}, {"start": 6.06, "duration": 3.24, "text": "for this question mark So this hundred"}, {"start": 7.62, "duration": 4.019, "text": "and this angle right here have to add to"}, {"start": 9.3, "duration": 3.839, "text": "180 so that's going to be 80 degrees and"}, {"start": 11.639, "duration": 3.721, "text": "same thing is going on down here this"}, {"start": 13.139, "duration": 4.201, "text": "110 and this has to be 180 so this will"}, {"start": 15.36, "duration": 4.62, "text": "be 70 degrees so now we have a triangle"}, {"start": 17.34, "duration": 4.8, "text": "here so 70 plus 40 plus blank has to"}, {"start": 19.98, "duration": 4.2, "text": "equal 180. that's going to be 70 degrees"}, {"start": 22.14, "duration": 4.86, "text": "so we can plug in 70 here and then by"}, {"start": 24.18, "duration": 4.08, "text": "vertical angles this is also 70. and now"}, {"start": 27.0, "duration": 3.359, "text": "we have another triangle with our"}, {"start": 28.26, "duration": 4.74, "text": "question mark it says 80 plus 70 plus"}, {"start": 30.359, "duration": 3.961, "text": "blank has to equal 180 and the blank is"}, {"start": 33.0, "duration": 3.239, "text": "going to be 30 and that means our"}, {"start": 34.32, "duration": 4.399, "text": "question mark is equal to 30. how"}, {"start": 36.239, "duration": 2.48, "text": "exciting"}], "title": "Swedish Math Olympiad #shorts", "description": "https://andymath.com/\n\n#shorts", "lengthSeconds": 37, "width": 360, "height": 640}
{"video_id": "vlU6a5f6MeE", "transcript": [{"start": 0.04, "duration": 3.159, "text": "hey guys this looks like a fun one"}, {"start": 1.199, "duration": 4.6, "text": "someone sent me this in an email it"}, {"start": 3.199, "duration": 5.001, "text": "gives us two circles in a rectangle and"}, {"start": 5.799, "duration": 4.92, "text": "it wants us to solve for the height X"}, {"start": 8.2, "duration": 5.16, "text": "the base of the rectangle is 18 cm this"}, {"start": 10.719, "duration": 5.0, "text": "circle's diameter is 16 cm and this"}, {"start": 13.36, "duration": 4.28, "text": "circle's diameter is 10 cm if you want"}, {"start": 15.719, "duration": 3.761, "text": "to try on your own pause it right now"}, {"start": 17.64, "duration": 4.2, "text": "because I'm going to solve it in three"}, {"start": 19.48, "duration": 4.52, "text": "two 1 first let's rotate the two"}, {"start": 21.84, "duration": 4.519, "text": "diameters to be horizontal and let's"}, {"start": 24.0, "duration": 4.08, "text": "label the centers of both of the circles"}, {"start": 26.359, "duration": 4.801, "text": "this diameter of 10 is going to split"}, {"start": 28.08, "duration": 5.12, "text": "into two radiuses of 5 cm and this"}, {"start": 31.16, "duration": 6.0, "text": "diameter of 16 will split into two"}, {"start": 33.2, "duration": 7.32, "text": "radiuses of 8 cm and yes radiuses is a"}, {"start": 37.16, "duration": 4.919, "text": "plural of radius you could also say radi"}, {"start": 40.52, "duration": 3.719, "text": "and then from the center of the smaller"}, {"start": 42.079, "duration": 3.601, "text": "Circle let's drop a vertical line and"}, {"start": 44.239, "duration": 4.041, "text": "same thing for the center of the larger"}, {"start": 45.68, "duration": 4.44, "text": "Circle this piece is the same thing as"}, {"start": 48.28, "duration": 4.64, "text": "the radius of the smaller Circle so"}, {"start": 50.12, "duration": 4.439, "text": "it'll be equal to 5 cm and then this"}, {"start": 52.92, "duration": 3.52, "text": "portion right here is equal to the"}, {"start": 54.559, "duration": 4.881, "text": "radius of the larger Circle so it'll be"}, {"start": 56.44, "duration": 6.48, "text": "equal to 8 cm so now since the entire"}, {"start": 59.44, "duration": 6.079, "text": "base is 18 this is five and this is 8"}, {"start": 62.92, "duration": 4.199, "text": "that leaves 5 cm left over for this part"}, {"start": 65.519, "duration": 4.64, "text": "but this cannot be right because the"}, {"start": 67.119, "duration": 5.161, "text": "radius is 8 how is five greater than"}, {"start": 70.159, "duration": 4.041, "text": "eight so this thing is definitely not"}, {"start": 72.28, "duration": 3.68, "text": "drawn to scale if you're curious I"}, {"start": 74.2, "duration": 3.239, "text": "actually drew it in geogebra this is"}, {"start": 75.96, "duration": 3.68, "text": "what it would look like if it was drawn"}, {"start": 77.439, "duration": 3.801, "text": "correctly here's the same two circles"}, {"start": 79.64, "duration": 3.519, "text": "but this would be at a much steeper"}, {"start": 81.24, "duration": 3.839, "text": "angle up on top and then for the one"}, {"start": 83.159, "duration": 3.761, "text": "that is drawn to scale if you drop this"}, {"start": 85.079, "duration": 3.601, "text": "vertical line and this vertical line and"}, {"start": 86.92, "duration": 3.4, "text": "bring the measurements down we have five"}, {"start": 88.68, "duration": 4.6, "text": "five and8 and this makes makes more"}, {"start": 90.32, "duration": 4.72, "text": "sense the five fits within the eight so"}, {"start": 93.28, "duration": 3.6, "text": "this is definitely not to scale this is"}, {"start": 95.04, "duration": 3.359, "text": "what it should look like I looked into"}, {"start": 96.88, "duration": 3.32, "text": "this a little bit more this is something"}, {"start": 98.399, "duration": 3.321, "text": "from it looks like a textbook it doesn't"}, {"start": 100.2, "duration": 3.72, "text": "say what textbook so I can't find the"}, {"start": 101.72, "duration": 4.399, "text": "original Source but this might be the"}, {"start": 103.92, "duration": 3.72, "text": "original source and even in the textbook"}, {"start": 106.119, "duration": 3.521, "text": "the problem is definitely not drawn to"}, {"start": 107.64, "duration": 3.759, "text": "scale the textbook looks a lot more like"}, {"start": 109.64, "duration": 3.64, "text": "this incorrect one than it does the"}, {"start": 111.399, "duration": 4.241, "text": "correct one so I think it's funny even"}, {"start": 113.28, "duration": 3.68, "text": "the textbook is blatantly not to scale"}, {"start": 115.64, "duration": 3.479, "text": "but I think even though it's not to"}, {"start": 116.96, "duration": 3.839, "text": "scale we can still solve this so let's"}, {"start": 119.119, "duration": 3.441, "text": "keep going from the center here let's"}, {"start": 120.799, "duration": 3.801, "text": "draw this horizontal line and it'll be"}, {"start": 122.56, "duration": 3.559, "text": "perpendicular to the vertical line and"}, {"start": 124.6, "duration": 3.56, "text": "same thing from this Center let's draw a"}, {"start": 126.119, "duration": 3.521, "text": "horizontal line this portion of the"}, {"start": 128.16, "duration": 3.48, "text": "height is the same thing as the radius"}, {"start": 129.64, "duration": 4.76, "text": "of the larger Circle so this is going to"}, {"start": 131.64, "duration": 3.92, "text": "be equal to 8 cm and then this portion"}, {"start": 134.4, "duration": 2.72, "text": "of the height is the same thing as the"}, {"start": 135.56, "duration": 3.959, "text": "radius of the smaller Circle so it's"}, {"start": 137.12, "duration": 4.199, "text": "going to be equal to 5 cm so the"}, {"start": 139.519, "duration": 4.481, "text": "distance from here to here is still"}, {"start": 141.319, "duration": 4.161, "text": "unknown let's call it question mark cenm"}, {"start": 144.0, "duration": 4.2, "text": "or even better let's give it a variable"}, {"start": 145.48, "duration": 4.72, "text": "let's call it y CM so now we just got to"}, {"start": 148.2, "duration": 4.6, "text": "solve for y and it'll be pretty easy to"}, {"start": 150.2, "duration": 4.759, "text": "figure out X I see a right angle right"}, {"start": 152.8, "duration": 4.079, "text": "here so if we connect these two we'll"}, {"start": 154.959, "duration": 4.321, "text": "have a right triangle and this"}, {"start": 156.879, "duration": 5.241, "text": "hypotenuse is a distance of the five"}, {"start": 159.28, "duration": 5.76, "text": "radius plus the 8 radius so it'll be 13"}, {"start": 162.12, "duration": 5.24, "text": "cm and let's focus on this triangle the"}, {"start": 165.04, "duration": 5.04, "text": "height of this triangle will be y cm and"}, {"start": 167.36, "duration": 5.48, "text": "the base will be 5 cm and now we can do"}, {"start": 170.08, "duration": 6.159, "text": "Pythagorean theorem it says a^2 + b^2"}, {"start": 172.84, "duration": 7.36, "text": "will equal c^2 or in other words 5^ 2 +"}, {"start": 176.239, "duration": 6.761, "text": "y^2 will equal 132 52 is equal to 25 and"}, {"start": 180.2, "duration": 4.84, "text": "13 s is equal to 169 and we can bring"}, {"start": 183.0, "duration": 4.239, "text": "down everything else next we want to get"}, {"start": 185.04, "duration": 4.8, "text": "this y^2 alone on the left hand side so"}, {"start": 187.239, "duration": 4.521, "text": "let's subtract 25 from both sides on the"}, {"start": 189.84, "duration": 5.88, "text": "left hand side we have y^2 and on the"}, {"start": 191.76, "duration": 5.479, "text": "right hand side 169 minus 25 is 144 next"}, {"start": 195.72, "duration": 3.4, "text": "we can square root both sides of the"}, {"start": 197.239, "duration": 4.36, "text": "equation and that'll give us Y is equal"}, {"start": 199.12, "duration": 4.16, "text": "to 12 this is very important let's put a"}, {"start": 201.599, "duration": 3.321, "text": "box around it we're not going to need"}, {"start": 203.28, "duration": 4.8, "text": "this stuff anymore and let's bring this"}, {"start": 204.92, "duration": 6.0, "text": "up here and now we can rewrite this y CM"}, {"start": 208.08, "duration": 4.519, "text": "as 12 CM let's pull out these two"}, {"start": 210.92, "duration": 4.36, "text": "measurements and go back to our original"}, {"start": 212.599, "duration": 4.521, "text": "diagram our goal was to solve for x"}, {"start": 215.28, "duration": 4.959, "text": "which is the height of this rectangle"}, {"start": 217.12, "duration": 7.039, "text": "well the Height's also equal to 5 + 12 +"}, {"start": 220.239, "duration": 5.601, "text": "8 and then 5 + 12 + 8 is equal to 25"}, {"start": 224.159, "duration": 3.72, "text": "let's give it a label of centimet and"}, {"start": 225.84, "duration": 5.84, "text": "put a box around it this is the answer"}, {"start": 227.879, "duration": 6.801, "text": "to our question X is equal to 25 cm how"}, {"start": 231.68, "duration": 3.0, "text": "exciting"}], "title": "2 Circles 1 Rectangle", "description": "I thought this was a fun one, I hope you guys like it!", "lengthSeconds": 233, "width": 640, "height": 360}
{"video_id": "91T36rtXzJ8", "transcript": [{"start": 0.0, "duration": 2.2, "text": "I saw this math Meme and so I thought I"}, {"start": 1.24, "duration": 2.399, "text": "would share it and I thought I would"}, {"start": 2.2, "duration": 2.72, "text": "explain it in case anybody doesn't get"}, {"start": 3.639, "duration": 3.2, "text": "it so it starts with a right triangle"}, {"start": 4.92, "duration": 3.32, "text": "with an angle X the side opposite of"}, {"start": 6.839, "duration": 2.84, "text": "that angle will call opposite the side"}, {"start": 8.24, "duration": 3.16, "text": "adjacent to it we'll call adjacent and"}, {"start": 9.679, "duration": 3.681, "text": "this side we'll call hypotenuse the S of"}, {"start": 11.4, "duration": 3.56, "text": "X is the ratio of the opposite over the"}, {"start": 13.36, "duration": 2.88, "text": "hypotenuse and the cosine of x is the"}, {"start": 14.96, "duration": 3.04, "text": "ratio of the adjacent over the"}, {"start": 16.24, "duration": 3.84, "text": "hypotenuse if we name the opposite side"}, {"start": 18.0, "duration": 4.08, "text": "a the adjacent side B and hypotenuse C"}, {"start": 20.08, "duration": 4.24, "text": "we can rewrite the S as a over C and the"}, {"start": 22.08, "duration": 4.519, "text": "cosine as B over C in the meme we have a"}, {"start": 24.32, "duration": 3.48, "text": "sin s x ring and a cosine squ X Ring"}, {"start": 26.599, "duration": 4.0, "text": "let's bring those two rings together"}, {"start": 27.8, "duration": 5.04, "text": "giving us sin^2 X plus cosine s x s of X"}, {"start": 30.599, "duration": 4.841, "text": "is a over C so sin squ of X would be a"}, {"start": 32.84, "duration": 4.399, "text": "over c^ 2 and then cosine is B over C"}, {"start": 35.44, "duration": 3.119, "text": "which would make this B over c^ 2 then"}, {"start": 37.239, "duration": 3.401, "text": "these squares can be distributed to all"}, {"start": 38.559, "duration": 3.721, "text": "the A's and the C's giving us a squ c s"}, {"start": 40.64, "duration": 2.84, "text": "plus b squ over c s now these have a"}, {"start": 42.28, "duration": 2.759, "text": "common denominator so we can just write"}, {"start": 43.48, "duration": 3.079, "text": "them as a single fraction now let's"}, {"start": 45.039, "duration": 3.561, "text": "rewrite this fraction let's bring the c^"}, {"start": 46.559, "duration": 4.361, "text": "S over and this is a right triangle so"}, {"start": 48.6, "duration": 3.72, "text": "that means that a s + b s is equal to"}, {"start": 50.92, "duration": 3.479, "text": "c^2 that's called the Pythagorean"}, {"start": 52.32, "duration": 3.48, "text": "theorem so now C2 over c s is equal to"}, {"start": 54.399, "duration": 2.96, "text": "one so when you have the sin squ ring"}, {"start": 55.8, "duration": 5.12, "text": "plus the cosine squ ring and they come"}, {"start": 57.359, "duration": 3.561, "text": "together they become one how wonderful"}], "title": "Sped up explanation", "description": "I sped it up to fit it in the 60 sec time limit. I hope you guys like it!", "lengthSeconds": 59, "width": 270, "height": 480}
{"video_id": "eqqKx1s7Iak", "transcript": [{"start": 0.0, "duration": 2.94, "text": "because I saw this thing the other day"}, {"start": 1.38, "duration": 3.6, "text": "this person texts an equation to"}, {"start": 2.94, "duration": 4.919, "text": "somebody and gets blocked for it that is"}, {"start": 4.98, "duration": 5.22, "text": "not cool it's 5 over 4X plus 3 equals"}, {"start": 7.859, "duration": 3.961, "text": "five over four X plus five over three I"}, {"start": 10.2, "duration": 3.479, "text": "wonder if we solve this thing if x will"}, {"start": 11.82, "duration": 3.3, "text": "have some secret message in it like this"}, {"start": 13.679, "duration": 3.121, "text": "expression here that when you erase the"}, {"start": 15.12, "duration": 3.48, "text": "top it says I love you let's bring this"}, {"start": 16.8, "duration": 2.94, "text": "piece down here I don't like fractions"}, {"start": 18.6, "duration": 3.0, "text": "so I want to get rid of all these"}, {"start": 19.74, "duration": 3.18, "text": "denominators in order to get rid of this"}, {"start": 21.6, "duration": 3.3, "text": "three I'm going to multiply everything"}, {"start": 22.92, "duration": 3.84, "text": "on both sides of the equation by three"}, {"start": 24.9, "duration": 3.36, "text": "and then to get rid of this 4X I'm going"}, {"start": 26.76, "duration": 3.599, "text": "to multiply everything on both sides of"}, {"start": 28.26, "duration": 3.72, "text": "the equation by 4X and then to get rid"}, {"start": 30.359, "duration": 3.241, "text": "of the 4X plus 3 I'm going to multiply"}, {"start": 31.98, "duration": 4.44, "text": "everything on both sides the equation by"}, {"start": 33.6, "duration": 5.7, "text": "4X plus 3. these threes will cancel this"}, {"start": 36.42, "duration": 5.28, "text": "4X and this 4X will cancel and this 4X"}, {"start": 39.3, "duration": 5.22, "text": "plus 3 will cancel with this 4X Plus 3."}, {"start": 41.7, "duration": 6.12, "text": "on this left hand side 3 times 5 is 15"}, {"start": 44.52, "duration": 5.879, "text": "and 4x times 15 is 60x and on the right"}, {"start": 47.82, "duration": 4.44, "text": "hand side 5 times 3 is 15 and this 15"}, {"start": 50.399, "duration": 5.401, "text": "will distribute to both of these to give"}, {"start": 52.26, "duration": 6.06, "text": "us 60x plus 45 and then 5 times 4X is"}, {"start": 55.8, "duration": 5.399, "text": "20x and then 20x will distribute to 4X"}, {"start": 58.32, "duration": 4.8, "text": "plus 3 to give us 80x squared plus 60x"}, {"start": 61.199, "duration": 4.021, "text": "and now we can combine like terms by"}, {"start": 63.12, "duration": 5.7, "text": "adding these together so we have 80x"}, {"start": 65.22, "duration": 5.1, "text": "squared plus 120x plus 45. so now we"}, {"start": 68.82, "duration": 3.78, "text": "have a quadratic so let's set it equal"}, {"start": 70.32, "duration": 4.2, "text": "to zero let's subtract 60x from both"}, {"start": 72.6, "duration": 3.3, "text": "sides we have 0 on the left hand side"}, {"start": 74.52, "duration": 4.44, "text": "and then on the right hand side we have"}, {"start": 75.9, "duration": 4.62, "text": "80x squared plus 60x plus 45. we can"}, {"start": 78.96, "duration": 3.479, "text": "simplify this a bit all these"}, {"start": 80.52, "duration": 3.66, "text": "coefficients are divisible by 5. so"}, {"start": 82.439, "duration": 4.921, "text": "let's divide everything by five that"}, {"start": 84.18, "duration": 5.28, "text": "gives us 0 equals 16x squared plus 12x"}, {"start": 87.36, "duration": 4.14, "text": "plus 9. and in order to solve this let's"}, {"start": 89.46, "duration": 3.72, "text": "use the quadratic formula a is this"}, {"start": 91.5, "duration": 3.36, "text": "first coefficient B is the second"}, {"start": 93.18, "duration": 3.899, "text": "coefficient and C is the third"}, {"start": 94.86, "duration": 4.56, "text": "coefficient B here is equal to 12 so"}, {"start": 97.079, "duration": 4.5, "text": "we'll plug in 12 for both these B's a is"}, {"start": 99.42, "duration": 4.379, "text": "equal to 16 so we'll plug in 16 for both"}, {"start": 101.579, "duration": 4.321, "text": "the A's and C is equal to 9 so we'll"}, {"start": 103.799, "duration": 5.82, "text": "plug in 9 for the C 12 squared is equal"}, {"start": 105.9, "duration": 7.56, "text": "to 144 4 times 16 is 64 and 64 times 9"}, {"start": 109.619, "duration": 5.701, "text": "is 576. and 144 minus 576 is equal to"}, {"start": 113.46, "duration": 3.299, "text": "negative 432. and we can pull the"}, {"start": 115.32, "duration": 2.939, "text": "negative out of this to give us square"}, {"start": 116.759, "duration": 3.841, "text": "root of negative 1 times square root of"}, {"start": 118.259, "duration": 4.201, "text": "432 and this square root of negative 1"}, {"start": 120.6, "duration": 4.32, "text": "is going to be the imaginary number I"}, {"start": 122.46, "duration": 5.04, "text": "inside of the square root the 432 can be"}, {"start": 124.92, "duration": 3.899, "text": "broken out into 144 times 3 and we can"}, {"start": 127.5, "duration": 3.84, "text": "give each of these numbers their own"}, {"start": 128.819, "duration": 4.441, "text": "square root and square root of 144 is"}, {"start": 131.34, "duration": 4.5, "text": "equal to 12. so in the numerator we have"}, {"start": 133.26, "duration": 4.14, "text": "negative 12 plus or minus 12i root 3."}, {"start": 135.84, "duration": 3.72, "text": "and in the denominator we have 2 times"}, {"start": 137.4, "duration": 4.44, "text": "16 which is 32. now all these"}, {"start": 139.56, "duration": 4.56, "text": "coefficients have a 4 in common this 12"}, {"start": 141.84, "duration": 4.44, "text": "can change into 4 times 3 this 12 can"}, {"start": 144.12, "duration": 3.78, "text": "change into 4 times 3 and the 32 can"}, {"start": 146.28, "duration": 3.539, "text": "change into 4 times 8 and then we can"}, {"start": 147.9, "duration": 3.54, "text": "cancel out all these fours and then we"}, {"start": 149.819, "duration": 3.42, "text": "don't want to mix this real portion and"}, {"start": 151.44, "duration": 3.84, "text": "this imaginary part inside of a fraction"}, {"start": 153.239, "duration": 4.14, "text": "so we'll split it up into two separate"}, {"start": 155.28, "duration": 4.679, "text": "fractions so the solution to this"}, {"start": 157.379, "duration": 5.101, "text": "equation is x equals negative 3 8 plus"}, {"start": 159.959, "duration": 4.381, "text": "or minus 3 I root 3 over 8. so let's put"}, {"start": 162.48, "duration": 3.78, "text": "a box around it I don't see a secret"}, {"start": 164.34, "duration": 4.399, "text": "message in here that's probably why they"}, {"start": 166.26, "duration": 2.479, "text": "got blocked"}], "title": "Math Meme Equation", "description": "https://andymath.com/rational-equations/", "lengthSeconds": 167, "width": 640, "height": 360}
{"video_id": "jYlABP8WAyo", "transcript": [{"start": 0.04, "duration": 3.119, "text": "hey guys this looks like a fun one we're"}, {"start": 1.4, "duration": 3.56, "text": "trying to find the area of this blue"}, {"start": 3.159, "duration": 3.281, "text": "region if you want to try it on your own"}, {"start": 4.96, "duration": 4.759, "text": "pause it right now cuz I'm going to"}, {"start": 6.44, "duration": 4.64, "text": "solve it in 3 2 one first I'm noticing"}, {"start": 9.719, "duration": 3.121, "text": "we can split this up in a couple"}, {"start": 11.08, "duration": 3.519, "text": "different ways and I want to focus on"}, {"start": 12.84, "duration": 3.6, "text": "this part right here let's get rid of"}, {"start": 14.599, "duration": 4.68, "text": "the question mark and continue this line"}, {"start": 16.44, "duration": 6.28, "text": "right here so let's call this portion X"}, {"start": 19.279, "duration": 5.801, "text": "and then since 28 is double the 14 this"}, {"start": 22.72, "duration": 3.879, "text": "blue portion would be 2X and now that"}, {"start": 25.08, "duration": 3.92, "text": "we've introduced this line we can also"}, {"start": 26.599, "duration": 4.321, "text": "split it up this way next let's focus on"}, {"start": 29.0, "duration": 4.76, "text": "this portion right here called this side"}, {"start": 30.92, "duration": 6.68, "text": "of the purple rectangle Y and then 105"}, {"start": 33.76, "duration": 6.04, "text": "is 3 * 35 so this would be 3 y so now"}, {"start": 37.6, "duration": 4.56, "text": "let's split this up we know that all of"}, {"start": 39.8, "duration": 4.919, "text": "this stuff is going to be 1/3 of all of"}, {"start": 42.16, "duration": 4.16, "text": "this stuff so we can set up a proportion"}, {"start": 44.719, "duration": 4.84, "text": "on top it would be this whole area which"}, {"start": 46.32, "duration": 4.84, "text": "would be x + 14 + 35 and on bottom it"}, {"start": 49.559, "duration": 5.881, "text": "would be this whole area which would be"}, {"start": 51.16, "duration": 7.719, "text": "2x + 28 + 21 + 105 now let's clean this"}, {"start": 55.44, "duration": 7.16, "text": "up on top here we can do 14 + 35 to give"}, {"start": 58.879, "duration": 8.56, "text": "us 49 and then on bottom the 28 + 21 is"}, {"start": 62.6, "duration": 7.36, "text": "also 49 and then 49 + 105 is 154 next we"}, {"start": 67.439, "duration": 6.281, "text": "can cross multiply we get 3 * the"}, {"start": 69.96, "duration": 6.32, "text": "quantity x + 49 is equal to 1 * thean 2x"}, {"start": 73.72, "duration": 5.88, "text": "+ 154 and then we can distribute this"}, {"start": 76.28, "duration": 6.92, "text": "three to both of these terms 3 * X is 3x"}, {"start": 79.6, "duration": 5.32, "text": "and 3 * 49 is 147 and then on the right"}, {"start": 83.2, "duration": 5.08, "text": "hand side this 1 isn't doing anything so"}, {"start": 84.92, "duration": 4.839, "text": "we have 2X + 154 now we have an x on"}, {"start": 88.28, "duration": 3.44, "text": "both sides of the equation let's get all"}, {"start": 89.759, "duration": 4.0, "text": "the the X's to one side we can do that"}, {"start": 91.72, "duration": 3.92, "text": "by subtracting 2x from both sides of the"}, {"start": 93.759, "duration": 3.801, "text": "equation after we do this we'll have an"}, {"start": 95.64, "duration": 5.56, "text": "x on this side so let's get rid of this"}, {"start": 97.56, "duration": 5.919, "text": "147 by subtracting 147 from both sides"}, {"start": 101.2, "duration": 4.599, "text": "these two 147's will cancel each other"}, {"start": 103.479, "duration": 5.361, "text": "out and these two 2 XS will cancel each"}, {"start": 105.799, "duration": 7.801, "text": "other out we end up with 3x - 2x is"}, {"start": 108.84, "duration": 7.0, "text": "equal to X and 154 - 147 is 7 x is equal"}, {"start": 113.6, "duration": 4.0, "text": "to 7 so now we can come back up here and"}, {"start": 115.84, "duration": 4.08, "text": "in the place of the X we can plug in"}, {"start": 117.6, "duration": 6.24, "text": "seven and in the place of the 2x we can"}, {"start": 119.92, "duration": 5.44, "text": "plug in 14 and 7 + 14 is equal to 21 so"}, {"start": 123.84, "duration": 4.32, "text": "the area of the blue region is equal to"}, {"start": 125.36, "duration": 5.0, "text": "21 so our question mark is equal to 21"}, {"start": 128.16, "duration": 4.0, "text": "square units this is the answer to our"}, {"start": 130.36, "duration": 3.12, "text": "question let's put a box around it I"}, {"start": 132.16, "duration": 2.92, "text": "think this problem was brilliant"}, {"start": 133.48, "duration": 3.64, "text": "speaking of brilliant let's talk about"}, {"start": 135.08, "duration": 3.76, "text": "brilliant brilliant has thousands of"}, {"start": 137.12, "duration": 3.92, "text": "lessons to choose from and all of them"}, {"start": 138.84, "duration": 4.16, "text": "are interactive you learn by doing I"}, {"start": 141.04, "duration": 3.839, "text": "really enjoyed this course on scientific"}, {"start": 143.0, "duration": 3.56, "text": "thinking it starts with simple machines"}, {"start": 144.879, "duration": 3.08, "text": "like gears and pulley and then it"}, {"start": 146.56, "duration": 3.44, "text": "eventually worked up to Einstein's"}, {"start": 147.959, "duration": 3.681, "text": "special theory of relativity when my"}, {"start": 150.0, "duration": 3.959, "text": "favorite parts were all the fun thought"}, {"start": 151.64, "duration": 4.679, "text": "experiment questions for example if we"}, {"start": 153.959, "duration": 5.2, "text": "have this distance time graph where the"}, {"start": 156.319, "duration": 4.92, "text": "y- AIS is time and years and the x-axis"}, {"start": 159.159, "duration": 3.961, "text": "is distance in light years so while"}, {"start": 161.239, "duration": 4.36, "text": "you're at the Space Rock concert a rumor"}, {"start": 163.12, "duration": 4.16, "text": "is spread that the staral concert in 2"}, {"start": 165.599, "duration": 3.28, "text": "years is not going to be any good and"}, {"start": 167.28, "duration": 3.36, "text": "the question is asking will the negative"}, {"start": 168.879, "duration": 4.521, "text": "rumor affect the attendance of the"}, {"start": 170.64, "duration": 4.64, "text": "staral concert in 2 years and the answer"}, {"start": 173.4, "duration": 4.44, "text": "is no because while the staral concert"}, {"start": 175.28, "duration": 4.679, "text": "is 2 years in the future the distance is"}, {"start": 177.84, "duration": 4.0, "text": "three light years away so the news of of"}, {"start": 179.959, "duration": 3.681, "text": "the rumor would not even reach anybody"}, {"start": 181.84, "duration": 3.2, "text": "who's going to attend this concert I'm"}, {"start": 183.64, "duration": 2.64, "text": "struggling to explain it in just one"}, {"start": 185.04, "duration": 3.08, "text": "sentence if you want to go through it"}, {"start": 186.28, "duration": 4.959, "text": "yourself brilliant currently has a free"}, {"start": 188.12, "duration": 4.839, "text": "30-day trial visit brilliant.org Andy or"}, {"start": 191.239, "duration": 3.601, "text": "check out the link in the description"}, {"start": 192.959, "duration": 3.761, "text": "you can also get 20% off an annual"}, {"start": 194.84, "duration": 3.2, "text": "premium plan if you wanted to go through"}, {"start": 196.72, "duration": 5.76, "text": "this one the name of the course is"}, {"start": 198.04, "duration": 4.44, "text": "scientific thinking how exciting"}], "title": "Brilliant Blue Area Question", "description": "To try everything Brilliant has to offer for free for a full 30 days, visit https://brilliant.org/AndyMath/ . You'll also get 20% off an annual premium subscription.", "lengthSeconds": 201, "width": 640, "height": 360}
{"video_id": "f7t-DIsoUxQ", "transcript": [{"start": 0.0, "duration": 3.6, "text": "hey guys this looks like a fun one what"}, {"start": 1.8, "duration": 4.019, "text": "are the values of A and B such that a"}, {"start": 3.6, "duration": 4.319, "text": "plus b is equal to a times B which is"}, {"start": 5.819, "duration": 4.501, "text": "equal to a divided by B so let's do this"}, {"start": 7.919, "duration": 4.501, "text": "first thing I'm noticing is B cannot be"}, {"start": 10.32, "duration": 3.6, "text": "zero because of the a divided by B and"}, {"start": 12.42, "duration": 3.6, "text": "we can't have division by zero next"}, {"start": 13.92, "duration": 3.84, "text": "let's try a equal to zero let's look at"}, {"start": 16.02, "duration": 3.839, "text": "these two right here plug in zero for"}, {"start": 17.76, "duration": 4.5, "text": "both of these A's zero plus b is equal"}, {"start": 19.859, "duration": 4.561, "text": "to B and zero times B is equal to zero"}, {"start": 22.26, "duration": 4.74, "text": "so we have b equal to zero but we said"}, {"start": 24.42, "duration": 4.38, "text": "earlier B cannot equal zero so we have a"}, {"start": 27.0, "duration": 3.42, "text": "contradiction that means that a cannot"}, {"start": 28.8, "duration": 3.0, "text": "equal zero and we could put a boxer on"}, {"start": 30.42, "duration": 3.18, "text": "that let's get rid of this stuff and"}, {"start": 31.8, "duration": 3.72, "text": "bring this up here now let's look at a"}, {"start": 33.6, "duration": 3.66, "text": "times b equals a divided by B let's"}, {"start": 35.52, "duration": 3.48, "text": "rewrite it like this and multiply both"}, {"start": 37.26, "duration": 4.2, "text": "sides by B and on the right hand side"}, {"start": 39.0, "duration": 4.379, "text": "since these B's cancel we just have a b"}, {"start": 41.46, "duration": 4.02, "text": "times B is B squared so we have a times"}, {"start": 43.379, "duration": 4.141, "text": "B squared is equal to a if a divide both"}, {"start": 45.48, "duration": 3.48, "text": "sides by a we get B squared is equal to"}, {"start": 47.52, "duration": 3.539, "text": "one so that means there are two possible"}, {"start": 48.96, "duration": 3.84, "text": "values for B it could be positive one or"}, {"start": 51.059, "duration": 3.961, "text": "negative one for both scenarios let's"}, {"start": 52.8, "duration": 4.14, "text": "look at a plus b equals a times B for"}, {"start": 55.02, "duration": 3.48, "text": "both of these B's let's plug in one a"}, {"start": 56.94, "duration": 3.9, "text": "times one is eight so we end up with a"}, {"start": 58.5, "duration": 3.84, "text": "plus one equals a let's subtract a from"}, {"start": 60.84, "duration": 3.059, "text": "both sides on the left hand side we have"}, {"start": 62.34, "duration": 3.66, "text": "one and on the right hand side we have"}, {"start": 63.899, "duration": 3.961, "text": "zero so we end up with one equals zero"}, {"start": 66.0, "duration": 3.6, "text": "that is a contradiction that means that"}, {"start": 67.86, "duration": 3.24, "text": "b cannot equal one let's see what"}, {"start": 69.6, "duration": 3.48, "text": "happens when we plug in b equals"}, {"start": 71.1, "duration": 3.9, "text": "negative one we end up with a minus one"}, {"start": 73.08, "duration": 3.539, "text": "is equal to negative a now we can"}, {"start": 75.0, "duration": 3.54, "text": "subtract a from both sides we get"}, {"start": 76.619, "duration": 3.721, "text": "negative one equals negative two a"}, {"start": 78.54, "duration": 4.38, "text": "divide both sides by negative two we get"}, {"start": 80.34, "duration": 4.86, "text": "one half is equal to a and there's no"}, {"start": 82.92, "duration": 4.44, "text": "contradiction here so a equals one half"}, {"start": 85.2, "duration": 5.66, "text": "and b equals negative one this is our"}, {"start": 87.36, "duration": 3.5, "text": "solution how exciting"}], "title": "Math Riddle Challenge", "description": "https://andymath.com/", "lengthSeconds": 89, "width": 360, "height": 640}
{"video_id": "ORq2kkWFtd4", "transcript": [{"start": 0.04, "duration": 3.52, "text": "hey guys this looks like a fun one we"}, {"start": 1.36, "duration": 4.24, "text": "have a semicircle and a quarter circle"}, {"start": 3.56, "duration": 4.16, "text": "inside of a square and it wants to know"}, {"start": 5.6, "duration": 4.28, "text": "what is the area of the square this is"}, {"start": 7.72, "duration": 3.959, "text": "part two in part one we found the length"}, {"start": 9.88, "duration": 3.28, "text": "of this segment right here if you want"}, {"start": 11.679, "duration": 3.641, "text": "to check out that video I'll put a link"}, {"start": 13.16, "duration": 3.6, "text": "up here and in the description and if"}, {"start": 15.32, "duration": 3.039, "text": "you want to try to find the area of this"}, {"start": 16.76, "duration": 5.679, "text": "Square pause it right now because I'm"}, {"start": 18.359, "duration": 6.521, "text": "going to solve it in 3 2 1 we have two"}, {"start": 22.439, "duration": 5.08, "text": "unknown values here I want to call those"}, {"start": 24.88, "duration": 5.319, "text": "X and Y since we know it's a square we"}, {"start": 27.519, "duration": 4.92, "text": "know that this 10 + x is equal to the 8"}, {"start": 30.199, "duration": 5.721, "text": "+ y and then we can subtract 8 from both"}, {"start": 32.439, "duration": 5.241, "text": "sides and we get Y is equal x + 2 this"}, {"start": 35.92, "duration": 4.0, "text": "looks important let's put a box around"}, {"start": 37.68, "duration": 3.96, "text": "it next let's focus on this radius right"}, {"start": 39.92, "duration": 3.56, "text": "here which we know is equal to 8 and"}, {"start": 41.64, "duration": 4.64, "text": "let's pivot it over here to this point"}, {"start": 43.48, "duration": 4.919, "text": "of tangency and we also know this radius"}, {"start": 46.28, "duration": 4.36, "text": "is equal to 5 and let's pivot it over"}, {"start": 48.399, "duration": 3.881, "text": "here to this point of tangency and then"}, {"start": 50.64, "duration": 3.8, "text": "let's complete this right triangle and"}, {"start": 52.28, "duration": 5.439, "text": "we can now do the Pythagorean theorem we"}, {"start": 54.44, "duration": 7.32, "text": "have the quantity x + 5^ 2 plus the"}, {"start": 57.719, "duration": 5.68, "text": "quantity y + 8 2 is equal 2 13 2 right"}, {"start": 61.76, "duration": 5.24, "text": "now we have two variables but in the"}, {"start": 63.399, "duration": 7.001, "text": "place of this y we can plug in x + 2 2 +"}, {"start": 67.0, "duration": 6.92, "text": "8 is equal to 10 so now we have x + 5^ 2"}, {"start": 70.4, "duration": 6.92, "text": "+ x + 10^ 2 this x + 5^ 2 means the same"}, {"start": 73.92, "duration": 6.0, "text": "thing as x + 5 * x + 5 and the x + 10 2"}, {"start": 77.32, "duration": 4.759, "text": "is the same thing as x + 10 * x + 10"}, {"start": 79.92, "duration": 4.96, "text": "after we multiply this out we get x^2 +"}, {"start": 82.079, "duration": 5.801, "text": "10 x + 25 and after we multiply this out"}, {"start": 84.88, "duration": 7.199, "text": "we get x^2 + 20x + 100 and then we can"}, {"start": 87.88, "duration": 9.04, "text": "add these together X2 + x is = 2x^2 10 x"}, {"start": 92.079, "duration": 7.561, "text": "+ 20x is = 30X and 25 + 100 is 125 and"}, {"start": 96.92, "duration": 5.32, "text": "that's all going to equal 13 s which is"}, {"start": 99.64, "duration": 4.2, "text": "169 nice since we have a quadratic we"}, {"start": 102.24, "duration": 3.96, "text": "want to set this whole thing equal to"}, {"start": 103.84, "duration": 5.239, "text": "zero so let's subtract 169 from both"}, {"start": 106.2, "duration": 8.08, "text": "sides the 2x^2 and the 30X can be copied"}, {"start": 109.079, "duration": 7.72, "text": "down 125 - 169 is equal to -44 and 169 -"}, {"start": 114.28, "duration": 4.68, "text": "169 is equal to 0 next let's divide"}, {"start": 116.799, "duration": 8.36, "text": "everything by two on the left hand side"}, {"start": 118.96, "duration": 8.88, "text": "2x2 / 2 is x^2 30X / 2 is 15x and -44 /"}, {"start": 125.159, "duration": 5.44, "text": "2 is -22 and then on the right hand side"}, {"start": 127.84, "duration": 4.119, "text": "0 / 2 is equal to 0 now there's a couple"}, {"start": 130.599, "duration": 4.081, "text": "different ways we can solve this but"}, {"start": 131.959, "duration": 6.481, "text": "let's do the quadratic formula it says x"}, {"start": 134.68, "duration": 6.559, "text": "equal to B + orus the < TK of B ^2 - 4 a"}, {"start": 138.44, "duration": 5.4, "text": "c whole thing over 2 a a is the 1 in"}, {"start": 141.239, "duration": 6.241, "text": "front of the x^2 B is the 15 in front of"}, {"start": 143.84, "duration": 5.36, "text": "the X and C is the constant -22 in the"}, {"start": 147.48, "duration": 3.0, "text": "place of both of these B's we can plug"}, {"start": 149.2, "duration": 2.96, "text": "in 15"}, {"start": 150.48, "duration": 4.0, "text": "in the place of both of these A's we can"}, {"start": 152.16, "duration": 6.999, "text": "plug in 1 and in the place of this C we"}, {"start": 154.48, "duration": 9.24, "text": "can plug in -22 152 is equal to 225 and"}, {"start": 159.159, "duration": 7.681, "text": "-4 * 1 * -22 is equal to POS 88 inside"}, {"start": 163.72, "duration": 6.32, "text": "the < TK 225 + 88 is equal to"}, {"start": 166.84, "duration": 5.479, "text": "313 and in the denominator 2 * 1 is"}, {"start": 170.04, "duration": 4.199, "text": "equal to 2 so this gives us a positive"}, {"start": 172.319, "duration": 3.721, "text": "value and a negative value and we know"}, {"start": 174.239, "duration": 4.201, "text": "that X is a positive length so we can"}, {"start": 176.04, "duration": 5.68, "text": "get rid of the negative value so X is"}, {"start": 178.44, "duration": 5.6, "text": "equal to -15 plus the < TK of 313 whole"}, {"start": 181.72, "duration": 5.079, "text": "thing ID 2 and we know that the side"}, {"start": 184.04, "duration": 4.919, "text": "length of the square is equal to x + 10"}, {"start": 186.799, "duration": 4.561, "text": "and in the place of the X we can plug in"}, {"start": 188.959, "duration": 4.681, "text": "all of this stuff in order to combine"}, {"start": 191.36, "duration": 5.56, "text": "these let's change this 10 into a 20"}, {"start": 193.64, "duration": 6.36, "text": "over2 we can combine this ne-15 over2"}, {"start": 196.92, "duration": 5.28, "text": "and this POS 20 over2 to give us POS 5"}, {"start": 200.0, "duration": 5.92, "text": "over2 and now we know the side length is"}, {"start": 202.2, "duration": 6.28, "text": "equal to 5 + the sarek of 313 / 2 and"}, {"start": 205.92, "duration": 4.239, "text": "that is approximately 11.3 and now we're"}, {"start": 208.48, "duration": 3.679, "text": "ready for our question the area of this"}, {"start": 210.159, "duration": 4.36, "text": "square is going to be this side length"}, {"start": 212.159, "duration": 4.681, "text": "squared we can multiply this times this"}, {"start": 214.519, "duration": 5.681, "text": "and that gives us an exact value of 169"}, {"start": 216.84, "duration": 5.64, "text": "+ 5un 313 everything over two and we"}, {"start": 220.2, "duration": 5.48, "text": "give that a label of square units and"}, {"start": 222.48, "duration": 5.28, "text": "the approximate value of this is 128."}, {"start": 225.68, "duration": 4.44, "text": "7291 and this is the answer to our"}, {"start": 227.76, "duration": 7.039, "text": "question this is the area of this Square"}, {"start": 230.12, "duration": 4.679, "text": "let's put a box around it how exciting"}], "title": "Part 2 Area of the Square", "description": "https://youtu.be/ByRTMcTT7WA", "lengthSeconds": 233, "width": 640, "height": 360}
{"video_id": "gVgTOiC-pGk", "transcript": [{"start": 0.0, "duration": 3.6, "text": "hey guys welcome to this functions video"}, {"start": 2.159, "duration": 4.881, "text": "let f of x equal"}, {"start": 3.6, "duration": 6.64, "text": "5x plus 3 and g of x equal 3x plus 5."}, {"start": 7.04, "duration": 6.0, "text": "find f plus g of 3. so what does this"}, {"start": 10.24, "duration": 3.439, "text": "mean well you can think of that as f of"}, {"start": 13.04, "duration": 3.44, "text": "3"}, {"start": 13.679, "duration": 3.44, "text": "plus g of three so to do f of three we"}, {"start": 16.48, "duration": 3.04, "text": "take the f"}, {"start": 17.119, "duration": 4.32, "text": "function and we plug three in for the x"}, {"start": 19.52, "duration": 4.8, "text": "so it's going to be five times three"}, {"start": 21.439, "duration": 3.521, "text": "plus three plus g of three means we look"}, {"start": 24.32, "duration": 3.52, "text": "at the g"}, {"start": 24.96, "duration": 4.0, "text": "function and we plug in three four x so"}, {"start": 27.84, "duration": 3.439, "text": "this will be three"}, {"start": 28.96, "duration": 4.0, "text": "times three plus five now we can just"}, {"start": 31.279, "duration": 2.721, "text": "simplify each of these five times three"}, {"start": 32.96, "duration": 3.52, "text": "is fifteen"}, {"start": 34.0, "duration": 3.76, "text": "plus three and three times three is nine"}, {"start": 36.48, "duration": 3.84, "text": "plus the five"}, {"start": 37.76, "duration": 4.0, "text": "fifteen plus three is eighteen plus and"}, {"start": 40.32, "duration": 3.919, "text": "nine plus five is fourteen"}, {"start": 41.76, "duration": 3.76, "text": "eighteen plus fourteen equals 32 and"}, {"start": 44.239, "duration": 4.48, "text": "that's our answer to the question"}, {"start": 45.52, "duration": 4.8, "text": "f plus g of 3 is equal to 32. comment"}, {"start": 48.719, "duration": 5.041, "text": "below on what math problem you'd like me"}, {"start": 50.32, "duration": 3.44, "text": "to solve next i'll talk to you soon"}, {"start": 53.8, "duration": 3.0, "text": "bye"}], "title": "Operations on Functions (Addition) Fast Example #Shorts", "description": "\ud835\udc0b\ud835\udc08\ud835\udc0d\ud835\udc0a\ud835\udc12 \ud835\udc13\ud835\udc0e \ud835\udc11\ud835\udc04\ud835\udc0b\ud835\udc00\ud835\udc13\ud835\udc04\ud835\udc03 \ud835\udc13\ud835\udc0e\ud835\udc0f\ud835\udc08\ud835\udc02\ud835\udc12\n\ud835\udc04\ud835\udc2f\ud835\udc1a\ud835\udc25\ud835\udc2e\ud835\udc1a\ud835\udc2d\ud835\udc22\ud835\udc27\ud835\udc20 \ud835\udc05\ud835\udc2e\ud835\udc27\ud835\udc1c\ud835\udc2d\ud835\udc22\ud835\udc28\ud835\udc27\ud835\udc2c \ud835\udc2e\ud835\udc2c\ud835\udc22\ud835\udc27\ud835\udc20 \ud835\udc06\ud835\udc2b\ud835\udc1a\ud835\udc29\ud835\udc21\ud835\udc2c: https://andymath.com/evaluating-functions-using-graphs/\n\ud835\udc04\ud835\udc2f\ud835\udc1a\ud835\udc25\ud835\udc2e\ud835\udc1a\ud835\udc2d\ud835\udc22\ud835\udc27\ud835\udc20 \ud835\udc05\ud835\udc2e\ud835\udc27\ud835\udc1c\ud835\udc2d\ud835\udc22\ud835\udc28\ud835\udc27\ud835\udc2c: https://andymath.com/evaluating-functions/\n\n\ud835\udc0e\ud835\udc29\ud835\udc1e\ud835\udc2b\ud835\udc1a\ud835\udc2d\ud835\udc22\ud835\udc28\ud835\udc27\ud835\udc2c \ud835\udc28\ud835\udc1f \ud835\udc05\ud835\udc2e\ud835\udc27\ud835\udc1c\ud835\udc2d\ud835\udc22\ud835\udc28\ud835\udc27\ud835\udc2c: \nhttps://andymath.com/operations-of-functions/\n\n\ud835\udc02\ud835\udc28\ud835\udc26\ud835\udc29\ud835\udc28\ud835\udc2c\ud835\udc22\ud835\udc2d\ud835\udc1e \ud835\udc05\ud835\udc2e\ud835\udc27\ud835\udc1c\ud835\udc2d\ud835\udc22\ud835\udc28\ud835\udc27\ud835\udc2c: \nhttps://andymath.com/composite-functions/\n\n\ud835\udc08\ud835\udc2c \ud835\udc22\ud835\udc2d \ud835\udc1a \ud835\udc05\ud835\udc2e\ud835\udc27\ud835\udc1c\ud835\udc2d\ud835\udc22\ud835\udc28\ud835\udc27?: \nhttps://andymath.com/is-it-a-function/\n\n\ud835\udc08\ud835\udc27\ud835\udc2f\ud835\udc1e\ud835\udc2b\ud835\udc2c\ud835\udc1e \ud835\udc05\ud835\udc2e\ud835\udc27\ud835\udc1c\ud835\udc2d\ud835\udc22\ud835\udc28\ud835\udc27\ud835\udc2c \ud835\udc1a\ud835\udc27\ud835\udc1d \ud835\udc11\ud835\udc1e\ud835\udc25\ud835\udc1a\ud835\udc2d\ud835\udc22\ud835\udc28\ud835\udc27\ud835\udc2c: \nhttps://andymath.com/inverse-functions-and-relations/\n\n\ud835\udc03\ud835\udc1e\ud835\udc2c\ud835\udc1c\ud835\udc2b\ud835\udc22\ud835\udc1b\ud835\udc22\ud835\udc27\ud835\udc20 \ud835\udc13\ud835\udc2b\ud835\udc1a\ud835\udc27\ud835\udc2c\ud835\udc1f\ud835\udc28\ud835\udc2b\ud835\udc26\ud835\udc1a\ud835\udc2d\ud835\udc22\ud835\udc28\ud835\udc27\ud835\udc2c \ud835\udc28\ud835\udc1f \ud835\udc05\ud835\udc2e\ud835\udc27\ud835\udc1c\ud835\udc2d\ud835\udc22\ud835\udc28\ud835\udc27\ud835\udc2c: \nhttps://andymath.com/transformations-of-functions/\n\n\ud835\udc0f\ud835\udc1a\ud835\udc2b\ud835\udc1e\ud835\udc27\ud835\udc2d \ud835\udc05\ud835\udc2e\ud835\udc27\ud835\udc1c\ud835\udc2d\ud835\udc22\ud835\udc28\ud835\udc27\ud835\udc2c: \nhttps://andymath.com/parent-functions/\n\n\ud835\udc04\ud835\udc2f\ud835\udc1e\ud835\udc27 \ud835\udc1a\ud835\udc27\ud835\udc1d \ud835\udc0e\ud835\udc1d\ud835\udc1d \ud835\udc05\ud835\udc2e\ud835\udc27\ud835\udc1c\ud835\udc2d\ud835\udc22\ud835\udc28\ud835\udc27\ud835\udc2c: \nhttps://andymath.com/even-and-odd-functions/\n\n\ud835\udc04\ud835\udc2f\ud835\udc1a\ud835\udc25\ud835\udc2e\ud835\udc1a\ud835\udc2d\ud835\udc22\ud835\udc27\ud835\udc20 \ud835\udc0f\ud835\udc22\ud835\udc1e\ud835\udc1c\ud835\udc1e\ud835\udc30\ud835\udc22\ud835\udc2c\ud835\udc1e \ud835\udc05\ud835\udc2e\ud835\udc27\ud835\udc1c\ud835\udc2d\ud835\udc22\ud835\udc28\ud835\udc27\ud835\udc2c: \nhttps://andymath.com/evaluating-piecewise-functions/\n\n\ud835\udc06\ud835\udc2b\ud835\udc1a\ud835\udc29\ud835\udc21\ud835\udc22\ud835\udc27\ud835\udc20 \ud835\udc0f\ud835\udc22\ud835\udc1e\ud835\udc1c\ud835\udc1e\ud835\udc30\ud835\udc22\ud835\udc2c\ud835\udc1e \ud835\udc05\ud835\udc2e\ud835\udc27\ud835\udc1c\ud835\udc2d\ud835\udc22\ud835\udc28\ud835\udc27\ud835\udc2c: \nhttps://andymath.com/piecewise-functions/\n\nVisit any of the the above links for more practice problems, notes, solutions and videos for all these related topics. For even more math topics, visit https://andymath.com/. \n\n#Shorts", "lengthSeconds": 55, "width": 202, "height": 360}
{"video_id": "JiX9QI8zwwo", "transcript": [{"start": 0.399, "duration": 5.36, "text": "draw three mutually tangent unit circles"}, {"start": 3.04, "duration": 5.28, "text": "like this these three circles enclose an"}, {"start": 5.759, "duration": 4.84, "text": "area as shown here can you determine the"}, {"start": 8.32, "duration": 3.72, "text": "area of the enclosed shape pause the"}, {"start": 10.599, "duration": 3.641, "text": "video and think about it before I show"}, {"start": 12.04, "duration": 3.96, "text": "you an answer one way to answer this"}, {"start": 14.24, "duration": 4.0, "text": "problem is to realize that the centers"}, {"start": 16.0, "duration": 4.279, "text": "of the three circles can be connected to"}, {"start": 18.24, "duration": 4.199, "text": "form an equilateral triangle with a side"}, {"start": 20.279, "duration": 4.201, "text": "length of two this has an area ofare <"}, {"start": 22.439, "duration": 3.84, "text": "TK of three but then this equilateral"}, {"start": 24.48, "duration": 3.719, "text": "triangle can be decomposed into these"}, {"start": 26.279, "duration": 4.041, "text": "three wedge shaped pieces in the area we"}, {"start": 28.199, "duration": 5.561, "text": "want each of the wed shaped pieces has"}, {"start": 30.32, "duration": 6.8, "text": "an area ofk 6 so the area we want is the"}, {"start": 33.76, "duration": 6.04, "text": "< TK 3 - 3 * > 6 which simplifies to the"}, {"start": 37.12, "duration": 4.119, "text": "< of 3 - pi/ 2 let me know in the"}, {"start": 39.8, "duration": 3.84, "text": "comments if you solved it a different"}, {"start": 41.239, "duration": 2.401, "text": "way"}], "title": "What\u2019s the area?", "description": "This is a short, animated visual proof finding the area bounded between three mutually tangent unit circles.\n\nHave a different solution? Share it in the comments!\n\nIf you like this video, consider subscribing to the channel or consider buying me a coffee: https://www.buymeacoffee.com/VisualProofs. Thanks! \n\nCheck out these related videos:\nhttps://youtu.be/5VaQVXECaVM\nhttps://youtu.be/6DnnJlzbCC4\nhttps://youtu.be/PNQxS6antBY\n\nThis animation is based on an exercise from Roger Nelsen's and Claudi Alsina's wonderful book \"Icons of Mathematics\". You can find the book and other information about it at the links below\n\n(I may receive commissions from this paid link) : Icons of Mathematics from Amazon https://amzn.to/3L5U44S\n\nhttps://www.maa.org/press/maa-reviews/icons-of-mathematics-an-exploration-of-twenty-key-images\n\nhttps://bookstore.ams.org/dol-56/\n\n #manim #math #mathvideo #mathshorts #geometry #hexagon #animation #theorem #pww #proofwithoutwords #visualproof #proof #area #root3 #pww\u200b #proofwithoutwords\u200b \u200b #proof\u200b #algebra #areas #mathematics\u200b #mathvideo\u200b #mtbos \n\nTo learn more about animating with manim, check out:\nhttps://manim.community", "lengthSeconds": 42, "width": 360, "height": 640}
{"video_id": "EjN9ekGSerQ", "transcript": [{"start": 0.04, "duration": 3.32, "text": "hey guys so we want to solve the radius"}, {"start": 1.719, "duration": 3.201, "text": "of this circle first let's find the"}, {"start": 3.36, "duration": 2.88, "text": "center of the circle and this right here"}, {"start": 4.92, "duration": 2.879, "text": "is going to be the radius I first"}, {"start": 6.24, "duration": 3.04, "text": "thought it was maybe it equals 1 but it"}, {"start": 7.799, "duration": 3.041, "text": "doesn't cuz if we bring this down that's"}, {"start": 9.28, "duration": 3.16, "text": "not in the middle here cuz the edge of"}, {"start": 10.84, "duration": 3.04, "text": "the circle is not in the middle either"}, {"start": 12.44, "duration": 2.92, "text": "so we can't really do that next let's"}, {"start": 13.88, "duration": 2.92, "text": "look at this diagonal here we can"}, {"start": 15.36, "duration": 3.12, "text": "actually find out what this diagonal is"}, {"start": 16.8, "duration": 4.28, "text": "cuz this is a right angle so if we call"}, {"start": 18.48, "duration": 5.52, "text": "this side a this side B and this side C"}, {"start": 21.08, "duration": 4.64, "text": "we know that a s + B2 is equal to c^2 a"}, {"start": 24.0, "duration": 3.279, "text": "right here is equal to 2 the B is the"}, {"start": 25.72, "duration": 3.92, "text": "same as this right here so it's equal to"}, {"start": 27.279, "duration": 5.8, "text": "2 and then C will just leave as c^ S 2 s"}, {"start": 29.64, "duration": 5.72, "text": "is 4 + 2^ 2 is 4 is equal to c^ 2 4 + 4"}, {"start": 33.079, "duration": 3.921, "text": "is 8 we square root both sides on the"}, {"start": 35.36, "duration": 4.08, "text": "left hand side we can break < TK of8"}, {"start": 37.0, "duration": 4.68, "text": "into < of 4 * < TK of two and the < TK"}, {"start": 39.44, "duration": 3.799, "text": "of 4 is 2 and on the right hand side it"}, {"start": 41.68, "duration": 3.879, "text": "just becomes C so we can clean this"}, {"start": 43.239, "duration": 3.441, "text": "stuff up up here and let's put 2 < tk2"}, {"start": 45.559, "duration": 2.84, "text": "right here for this piece of the"}, {"start": 46.68, "duration": 4.199, "text": "diagonal let's get rid of all this stuff"}, {"start": 48.399, "duration": 4.121, "text": "unfortunately the diameter here is not 2"}, {"start": 50.879, "duration": 3.52, "text": "< tk2 because it doesn't go all the way"}, {"start": 52.52, "duration": 3.679, "text": "to the corner but we can say that the"}, {"start": 54.399, "duration": 3.84, "text": "rest of the diagonal right up here is"}, {"start": 56.199, "duration": 3.561, "text": "also 2k2 so we can calculate the"}, {"start": 58.239, "duration": 3.28, "text": "diagonal of the big square and it's"}, {"start": 59.76, "duration": 4.04, "text": "going going to be equal to the 2 < tk2"}, {"start": 61.519, "duration": 3.68, "text": "plus the 2 < tk2 which is just 4 < tk2"}, {"start": 63.8, "duration": 3.12, "text": "and we can put a box around that let's"}, {"start": 65.199, "duration": 3.56, "text": "clean this all up a little bit and this"}, {"start": 66.92, "duration": 3.44, "text": "right here is the radius we'll call that"}, {"start": 68.759, "duration": 3.161, "text": "R this is what we're trying to solve for"}, {"start": 70.36, "duration": 3.52, "text": "so we know this we know this we just got"}, {"start": 71.92, "duration": 3.64, "text": "to find this piece right here that's not"}, {"start": 73.88, "duration": 3.239, "text": "too bad because we can make a square"}, {"start": 75.56, "duration": 3.28, "text": "right here we know each of these are"}, {"start": 77.119, "duration": 3.64, "text": "equal to the radius as well so we can"}, {"start": 78.84, "duration": 3.36, "text": "call both the sides of this Square R"}, {"start": 80.759, "duration": 3.32, "text": "then we can do another diagonal we'll"}, {"start": 82.2, "duration": 3.4, "text": "move this R over to here and if we make"}, {"start": 84.079, "duration": 5.881, "text": "the diagonal equal to C then we know"}, {"start": 85.6, "duration": 6.24, "text": "that R 2 + R2 = c^2 well R2 + r 2 is 2 r"}, {"start": 89.96, "duration": 3.76, "text": "squ we can square root both sides the"}, {"start": 91.84, "duration": 4.44, "text": "left hand side can be broken out into <"}, {"start": 93.72, "duration": 4.56, "text": "TK of 2 * < TK of R 2 and the right hand"}, {"start": 96.28, "duration": 4.4, "text": "side would just bequ < TK of c squ and"}, {"start": 98.28, "duration": 4.4, "text": "then s of R squ is just R and squ of c^"}, {"start": 100.68, "duration": 4.039, "text": "squ is just C and we can smush these"}, {"start": 102.68, "duration": 4.2, "text": "together and we have > of 2 R is equal"}, {"start": 104.719, "duration": 4.321, "text": "to C up here was our diagonal C so we"}, {"start": 106.88, "duration": 3.36, "text": "can change that into < tk2 R clean"}, {"start": 109.04, "duration": 3.24, "text": "things up a little bit and now what do"}, {"start": 110.24, "duration": 4.0, "text": "we have big diagonal again this time"}, {"start": 112.28, "duration": 4.6, "text": "it's equal to this diagonal here the 2 <"}, {"start": 114.24, "duration": 3.96, "text": "tk2 plus the r plus the < tk2 r and we"}, {"start": 116.88, "duration": 3.08, "text": "can put a box around that but we know"}, {"start": 118.2, "duration": 3.32, "text": "that the big diagonal has to equal the"}, {"start": 119.96, "duration": 3.32, "text": "big diagonal so we're just going to set"}, {"start": 121.52, "duration": 3.72, "text": "the 4 < tk2 for this side of the big"}, {"start": 123.28, "duration": 3.6, "text": "diagonal equal to all of this and now we"}, {"start": 125.24, "duration": 3.6, "text": "only have one variable R and one"}, {"start": 126.88, "duration": 3.999, "text": "equation so once we solve for R we know"}, {"start": 128.84, "duration": 3.84, "text": "the answer we can subtract 2 < tk2 from"}, {"start": 130.879, "duration": 3.801, "text": "both sides the left hand side becomes 2"}, {"start": 132.68, "duration": 3.24, "text": "< tk2 the right hand side that's gone so"}, {"start": 134.68, "duration": 3.48, "text": "it's just this right here and we can"}, {"start": 135.92, "duration": 4.84, "text": "scoot these over and let's Factor an R"}, {"start": 138.16, "duration": 5.56, "text": "out of the right hand side so R * what"}, {"start": 140.76, "duration": 5.36, "text": "gives us R that's one so R * 1 is r and"}, {"start": 143.72, "duration": 3.64, "text": "then R * what gives us < tk2 R well"}, {"start": 146.12, "duration": 2.839, "text": "that's going to be < tk2 so if you"}, {"start": 147.36, "duration": 3.48, "text": "imagine Distributing this it would bring"}, {"start": 148.959, "duration": 3.761, "text": "us back to where we were up here so we"}, {"start": 150.84, "duration": 4.08, "text": "factored out the r then on the left hand"}, {"start": 152.72, "duration": 4.12, "text": "side leave that as 2 < tk2 the 1 plus <"}, {"start": 154.92, "duration": 3.8, "text": "tk2 is just a number so we can divide"}, {"start": 156.84, "duration": 5.479, "text": "both sides by that number and we end up"}, {"start": 158.72, "duration": 5.2, "text": "with r being equal to 2 < tk2 over 1 + <"}, {"start": 162.319, "duration": 4.481, "text": "tk2 and we can plug it into a calculator"}, {"start": 163.92, "duration": 4.76, "text": "and get a decimal approximation of 17157"}, {"start": 166.8, "duration": 3.2, "text": "and that keeps going it's irrational and"}, {"start": 168.68, "duration": 3.32, "text": "now we have the answer to our question"}, {"start": 170.0, "duration": 4.519, "text": "the radius is approximately"}, {"start": 172.0, "duration": 5.239, "text": "17157 and we put a box around that how"}, {"start": 174.519, "duration": 2.72, "text": "exciting"}], "title": "Fun Geometry Challenge", "description": "https://andymath.com/geometry-challenges/\nFor More Geometry Challenges, Check out the above link!", "lengthSeconds": 175, "width": 640, "height": 360}
{"video_id": "ZPkNv0FBTrM", "transcript": [{"start": 0.0, "duration": 4.2, "text": "a lot of people ask me to do calculus in"}, {"start": 1.8, "duration": 4.28, "text": "area so here it is we'll start with the"}, {"start": 4.2, "duration": 3.559, "text": "horizontal line first so let's say we"}, {"start": 6.08, "duration": 3.519, "text": "want to know the area underneath this"}, {"start": 7.759, "duration": 3.561, "text": "line we pick two end points a lower"}, {"start": 9.599, "duration": 4.281, "text": "bound and an upper bound and we want to"}, {"start": 11.32, "duration": 4.68, "text": "find the area underneath our line in"}, {"start": 13.88, "duration": 4.12, "text": "between our two end points this stopped"}, {"start": 16.0, "duration": 3.359, "text": "at the x axis and there's a reason for"}, {"start": 18.0, "duration": 2.8, "text": "that if we want to know what the height"}, {"start": 19.359, "duration": 3.481, "text": "is here it's just going to be our"}, {"start": 20.8, "duration": 3.84, "text": "function value at each of these values"}, {"start": 22.84, "duration": 3.599, "text": "of X the area of this is just a"}, {"start": 24.64, "duration": 3.24, "text": "rectangle which is base time height now"}, {"start": 26.439, "duration": 3.561, "text": "what if we wanted to find the area under"}, {"start": 27.88, "duration": 4.199, "text": "a curve we don't have any formulas from"}, {"start": 30.0, "duration": 4.079, "text": "geometry for this let me use a rectangle"}, {"start": 32.079, "duration": 3.441, "text": "to make an estimation so I set the"}, {"start": 34.079, "duration": 3.241, "text": "height of this one to be the height of"}, {"start": 35.52, "duration": 3.92, "text": "my left bound we can tell this isn't"}, {"start": 37.32, "duration": 4.239, "text": "very accurate it's an underestimate by"}, {"start": 39.44, "duration": 3.92, "text": "quite a bit I can cut it into two pieces"}, {"start": 41.559, "duration": 3.361, "text": "and start using two figures to estimate"}, {"start": 43.36, "duration": 3.039, "text": "it now I've gotten closer this is the"}, {"start": 44.92, "duration": 3.24, "text": "base of our first rectangle this is the"}, {"start": 46.399, "duration": 3.361, "text": "base of our second rectangle this is the"}, {"start": 48.16, "duration": 2.719, "text": "height of our first rectangle and this"}, {"start": 49.76, "duration": 3.16, "text": "will be the height of our second"}, {"start": 50.879, "duration": 3.241, "text": "rectangle our estimate for the area"}, {"start": 52.92, "duration": 3.72, "text": "would be the area of this first"}, {"start": 54.12, "duration": 4.84, "text": "rectangle B1 * H1 plus the area of the"}, {"start": 56.64, "duration": 6.32, "text": "second rectangle B2 * H2 and we can"}, {"start": 58.96, "duration": 6.08, "text": "write this in Sigma notation as b i h i"}, {"start": 62.96, "duration": 4.28, "text": "that way the I can change from 1 to two"}, {"start": 65.04, "duration": 4.32, "text": "we'll start at I equal 1 and we stop at"}, {"start": 67.24, "duration": 4.48, "text": "I equal 2 these are two ways to say the"}, {"start": 69.36, "duration": 3.92, "text": "same thing next let's do four rectangles"}, {"start": 71.72, "duration": 3.36, "text": "here are all the bases and here are all"}, {"start": 73.28, "duration": 3.56, "text": "the heights to add all these up it would"}, {"start": 75.08, "duration": 3.76, "text": "just have four of them or we can change"}, {"start": 76.84, "duration": 3.48, "text": "Sigma notation and make this a four we"}, {"start": 78.84, "duration": 3.16, "text": "could do it again with eight rectangles"}, {"start": 80.32, "duration": 4.04, "text": "which would make eight here or change"}, {"start": 82.0, "duration": 4.52, "text": "this to an eight we do it with 16 which"}, {"start": 84.36, "duration": 4.36, "text": "would add up 16 of these or change our"}, {"start": 86.52, "duration": 4.2, "text": "Sigma notation to a 16 this shows the"}, {"start": 88.72, "duration": 3.88, "text": "advantage of Sigma notation this takes"}, {"start": 90.72, "duration": 3.52, "text": "up way less space than this right here"}, {"start": 92.6, "duration": 3.559, "text": "so let's get rid of all this stuff let's"}, {"start": 94.24, "duration": 4.0, "text": "look closely at the bi if this right"}, {"start": 96.159, "duration": 4.361, "text": "here is base I the left side would then"}, {"start": 98.24, "duration": 5.919, "text": "be x i and this right hand side would be"}, {"start": 100.52, "duration": 5.52, "text": "x sub I + 1 it's one more than this I if"}, {"start": 104.159, "duration": 3.481, "text": "we want to know how big bi is it's the"}, {"start": 106.04, "duration": 4.119, "text": "difference between the two x's and we"}, {"start": 107.64, "duration": 4.36, "text": "call that Delta X let's change this bi"}, {"start": 110.159, "duration": 4.481, "text": "to Delta X so now let's talk about the"}, {"start": 112.0, "duration": 4.399, "text": "height this height hi since this stops"}, {"start": 114.64, "duration": 4.28, "text": "when it hits our function the height is"}, {"start": 116.399, "duration": 5.521, "text": "actually defined by the function value"}, {"start": 118.92, "duration": 5.28, "text": "right there so we would say F of XI so"}, {"start": 121.92, "duration": 3.799, "text": "we can change our hi into F of XI now"}, {"start": 124.2, "duration": 3.36, "text": "let's move this Delta x to the other"}, {"start": 125.719, "duration": 3.4, "text": "side let's change this 16 to n so we can"}, {"start": 127.56, "duration": 3.44, "text": "get any number of rectangles we want and"}, {"start": 129.119, "duration": 3.681, "text": "if we take the limit as n approaches"}, {"start": 131.0, "duration": 4.44, "text": "Infinity we will get closer and closer"}, {"start": 132.8, "duration": 4.88, "text": "to the exact area under that curve we've"}, {"start": 135.44, "duration": 4.36, "text": "come up with new notation for this that"}, {"start": 137.68, "duration": 3.12, "text": "visually mimics that smoothness this"}, {"start": 139.8, "duration": 2.519, "text": "right here gets Rewritten as the"}, {"start": 140.8, "duration": 3.56, "text": "integral symbol which is much more"}, {"start": 142.319, "duration": 3.761, "text": "smooth this f ofx i doesn't make sense"}, {"start": 144.36, "duration": 3.239, "text": "anymore because it's discrete jumps we"}, {"start": 146.08, "duration": 4.0, "text": "want it to be smooth so we're just going"}, {"start": 147.599, "duration": 4.161, "text": "to say f ofx and the Delta X we also"}, {"start": 150.08, "duration": 4.12, "text": "have a smooth way of expressing this as"}, {"start": 151.76, "duration": 4.92, "text": "that distance goes to zero we call it DX"}, {"start": 154.2, "duration": 4.16, "text": "and this is an integral areas are equal"}, {"start": 156.68, "duration": 3.36, "text": "when we do the integral as we chop it"}, {"start": 158.36, "duration": 3.28, "text": "into Infinite pieces we find the exact"}, {"start": 160.04, "duration": 3.04, "text": "area that's the end of this video if you"}, {"start": 161.64, "duration": 3.319, "text": "guys want to learn how to calculate"}, {"start": 163.08, "duration": 3.239, "text": "these integrals please follow me and I"}, {"start": 164.959, "duration": 3.28, "text": "will have more videos coming out you"}, {"start": 166.319, "duration": 4.28, "text": "guys are awesome I'll talk to you soon"}, {"start": 168.239, "duration": 2.36, "text": "bye"}], "title": "Quick Intro Calculus & Area", "description": "The original footage is a couple years old. I edited it this weekend to fit Youtube format. I hope you guys like it!", "lengthSeconds": 169, "width": 640, "height": 360}
{"video_id": "-O3J0FkiRsk", "transcript": [{"start": 0.0, "duration": 3.12, "text": "hey guys this looks like a fun one it"}, {"start": 1.199, "duration": 4.6, "text": "was sent to me by one of my patrons it's"}, {"start": 3.12, "duration": 4.679, "text": "got a circle inscribed in a rhombus that"}, {"start": 5.799, "duration": 3.88, "text": "means all four of these lines are"}, {"start": 7.799, "duration": 3.521, "text": "tangent lines they all touch the circle"}, {"start": 9.679, "duration": 3.84, "text": "in exactly one point and that's called"}, {"start": 11.32, "duration": 5.28, "text": "the tangent point and the rhombus has"}, {"start": 13.519, "duration": 4.92, "text": "diagonals 12 and 16 and it wants to know"}, {"start": 16.6, "duration": 3.08, "text": "what is the area of the circle if you"}, {"start": 18.439, "duration": 3.401, "text": "want to try it on your own pause it"}, {"start": 19.68, "duration": 4.599, "text": "right now cuz I'm going to solve it in 3"}, {"start": 21.84, "duration": 4.0, "text": "2 1 first let's get rid of these colors"}, {"start": 24.279, "duration": 3.281, "text": "and let's get rid of the question mark"}, {"start": 25.84, "duration": 4.24, "text": "and let's construct this vertical"}, {"start": 27.56, "duration": 4.08, "text": "diagonal this has a length of 12 m and"}, {"start": 30.08, "duration": 3.56, "text": "then if we construct the other diagonal"}, {"start": 31.64, "duration": 3.0, "text": "it's going to cut this 12 in half and"}, {"start": 33.64, "duration": 2.52, "text": "that's one of the properties of the"}, {"start": 34.64, "duration": 3.32, "text": "diagonals of a rhombus they cut each"}, {"start": 36.16, "duration": 3.32, "text": "other in half and there's another"}, {"start": 37.96, "duration": 3.68, "text": "property of the diagonals of a rhombus"}, {"start": 39.48, "duration": 3.8, "text": "they also meet at right angles let's"}, {"start": 41.64, "duration": 3.04, "text": "mark this intersection Point that's"}, {"start": 43.28, "duration": 3.52, "text": "going to be the center of the rhombus"}, {"start": 44.68, "duration": 4.0, "text": "and the center of the circle next let's"}, {"start": 46.8, "duration": 4.36, "text": "look at this longer diagonal it has a"}, {"start": 48.68, "duration": 5.32, "text": "length of 16 M and it'll split in half"}, {"start": 51.16, "duration": 4.239, "text": "to be 8 and 8 next let's focus on this"}, {"start": 54.0, "duration": 3.239, "text": "right triangle let's label the"}, {"start": 55.399, "duration": 4.16, "text": "hypotenuse C and we can do the"}, {"start": 57.239, "duration": 6.92, "text": "Pythagorean theorem it's going to be 62"}, {"start": 59.559, "duration": 6.641, "text": "+ 8^ 2 is equal to c^ 2 6 2 is 36 + 8 2"}, {"start": 64.159, "duration": 4.921, "text": "is 64 and then we can bring down the c^"}, {"start": 66.2, "duration": 4.16, "text": "2 36 + 64 is equal to 100 and then we"}, {"start": 69.08, "duration": 3.88, "text": "can square root both sides of the"}, {"start": 70.36, "duration": 4.399, "text": "equation to give us C is equal to 10 so"}, {"start": 72.96, "duration": 4.12, "text": "we now know this hypotenuse is equal to"}, {"start": 74.759, "duration": 4.68, "text": "10 and since it's a rhombus we know all"}, {"start": 77.08, "duration": 4.359, "text": "these sides are equal to 10 next let's"}, {"start": 79.439, "duration": 4.04, "text": "connect this radius to here and we know"}, {"start": 81.439, "duration": 3.761, "text": "this will also be at right angles the"}, {"start": 83.479, "duration": 3.121, "text": "radius going to the tangent point will"}, {"start": 85.2, "duration": 3.52, "text": "always be at right angles to that"}, {"start": 86.6, "duration": 4.28, "text": "tangent line let's break this into two"}, {"start": 88.72, "duration": 4.28, "text": "parts let's call this p piece a and that"}, {"start": 90.88, "duration": 4.279, "text": "means this other piece would be 10 minus"}, {"start": 93.0, "duration": 3.84, "text": "a and now we can do Pythagorean theorem"}, {"start": 95.159, "duration": 5.0, "text": "two more times let's focus on this"}, {"start": 96.84, "duration": 5.559, "text": "triangle first it'll give us r^ 2 + a^ 2"}, {"start": 100.159, "duration": 4.56, "text": "is equal to 6^ 2 next let's do this"}, {"start": 102.399, "duration": 5.0, "text": "other triangle and it will give us r^ 2"}, {"start": 104.719, "duration": 4.801, "text": "plus the quantity 10 - a^ 2 is equal to"}, {"start": 107.399, "duration": 4.441, "text": "8^2 and then let's clean these up a"}, {"start": 109.52, "duration": 4.0, "text": "little bit on this top one 62 is equal"}, {"start": 111.84, "duration": 3.4, "text": "to 36 and then we can bring down"}, {"start": 113.52, "duration": 4.12, "text": "everything else for this bottom one"}, {"start": 115.24, "duration": 5.44, "text": "let's look at the quantity 10 - a^ 2"}, {"start": 117.64, "duration": 4.519, "text": "means the same thing as 10 - A * 10 - A"}, {"start": 120.68, "duration": 4.799, "text": "and after we multiply this out it'll"}, {"start": 122.159, "duration": 5.521, "text": "give us 100 - 28 + a^ 2 and then we can"}, {"start": 125.479, "duration": 4.881, "text": "bring down the R 2 and this 8^ 2 is"}, {"start": 127.68, "duration": 4.6, "text": "equal to 64 next we can subtract 100"}, {"start": 130.36, "duration": 5.08, "text": "from both sides of the equation the left"}, {"start": 132.28, "duration": 5.48, "text": "hand side we'll have r^ 2 - 20 a + a 2"}, {"start": 135.44, "duration": 5.04, "text": "and then on the right hand side 64 - 100"}, {"start": 137.76, "duration": 4.52, "text": "is equal to -36 now we've cleaned up our"}, {"start": 140.48, "duration": 4.119, "text": "two equations and let's rearrange them"}, {"start": 142.28, "duration": 4.64, "text": "like this let's drag the r squ over here"}, {"start": 144.599, "duration": 4.841, "text": "and I want to subtract this row minus"}, {"start": 146.92, "duration": 5.44, "text": "this row this r 2us r 2 are going to"}, {"start": 149.44, "duration": 5.68, "text": "cancel other out -2 a minus nothing is"}, {"start": 152.36, "duration": 4.44, "text": "-2 a and then positive a^2 minus"}, {"start": 155.12, "duration": 2.96, "text": "positive a 2 are also going to cancel"}, {"start": 156.8, "duration": 4.24, "text": "each other out and then on the right"}, {"start": 158.08, "duration": 5.84, "text": "hand side we have -36 - 36 which will"}, {"start": 161.04, "duration": 4.559, "text": "give us -72 now to get the a by itself"}, {"start": 163.92, "duration": 4.36, "text": "we can divide both sides of the equation"}, {"start": 165.599, "duration": 6.041, "text": "by -20 and this will give us a is equal"}, {"start": 168.28, "duration": 5.08, "text": "to 3.6 let's bring this up here now that"}, {"start": 171.64, "duration": 4.0, "text": "we have a I don't think we need this top"}, {"start": 173.36, "duration": 4.239, "text": "equation anymore let's move this up here"}, {"start": 175.64, "duration": 5.0, "text": "and now for this bottom equation in the"}, {"start": 177.599, "duration": 5.601, "text": "place of the a let's plug in 3.6 6"}, {"start": 180.64, "duration": 4.519, "text": "Plus's simplify the 3.6 squar to give us"}, {"start": 183.2, "duration": 4.2, "text": "12.96 and then we can bring down"}, {"start": 185.159, "duration": 3.681, "text": "everything else let's subtract 12.96"}, {"start": 187.4, "duration": 4.559, "text": "from both sides of the equation and"}, {"start": 188.84, "duration": 5.36, "text": "that'll give us r^2 is equal to"}, {"start": 191.959, "duration": 3.441, "text": "23.04 this looks important let's clean"}, {"start": 194.2, "duration": 3.44, "text": "this up a little bit and let's bring"}, {"start": 195.4, "duration": 4.44, "text": "this up here ultimately we want to find"}, {"start": 197.64, "duration": 5.72, "text": "the area of the circle the area of a"}, {"start": 199.84, "duration": 5.679, "text": "circle is < R 2 we now know what R 2 is"}, {"start": 203.36, "duration": 3.36, "text": "equal to so in the place of this R 2"}, {"start": 205.519, "duration": 3.241, "text": "let's plug in"}, {"start": 206.72, "duration": 3.64, "text": "23.04 and this is the answer to our"}, {"start": 208.76, "duration": 5.16, "text": "question the area is equal to"}, {"start": 210.36, "duration": 5.519, "text": "23.04 piun M squar let's put a box"}, {"start": 213.92, "duration": 7.16, "text": "around it and the approximate area is"}, {"start": 215.879, "duration": 5.201, "text": "72.3 M squ how exciting"}], "title": "Is there an easier way?", "description": "I hope you guys like this one! Let me know if there is an easier method.", "lengthSeconds": 219, "width": 640, "height": 360}
{"video_id": "-Nzn3Ucq9zg", "transcript": [{"start": 0.0, "duration": 4.5, "text": "you guys this looks like a fun one 99"}, {"start": 1.86, "duration": 5.1, "text": "factorial minus 98 factorial over 99"}, {"start": 4.5, "duration": 3.66, "text": "factorial plus 98 factorial would you"}, {"start": 6.96, "duration": 2.76, "text": "want to try this on your own pause it"}, {"start": 8.16, "duration": 3.6, "text": "because I'm going to solve it in three"}, {"start": 9.72, "duration": 3.12, "text": "two one a lot of calculators aren't"}, {"start": 11.76, "duration": 2.879, "text": "going to be able to handle these numbers"}, {"start": 12.84, "duration": 3.299, "text": "these are very large numbers but we"}, {"start": 14.639, "duration": 3.421, "text": "don't need a calculator we can solve it"}, {"start": 16.139, "duration": 5.581, "text": "by hand First Step let's rewrite this 99"}, {"start": 18.06, "duration": 5.46, "text": "factorial as 99 times 98 factorial and"}, {"start": 21.72, "duration": 4.38, "text": "let's subtract this and let's rewrite"}, {"start": 23.52, "duration": 3.66, "text": "this as 1 times 98 factorial and then"}, {"start": 26.1, "duration": 2.46, "text": "we're going to do the same thing on the"}, {"start": 27.18, "duration": 3.54, "text": "bottom except we're going to be adding"}, {"start": 28.56, "duration": 4.38, "text": "the two terms all of these terms have a"}, {"start": 30.72, "duration": 4.44, "text": "98 factorial so let's factor that out of"}, {"start": 32.94, "duration": 4.2, "text": "top and bottom if we do 98 factorial"}, {"start": 35.16, "duration": 3.12, "text": "times 99 it'll bring us back to this"}, {"start": 37.14, "duration": 4.079, "text": "right here and then we're going to"}, {"start": 38.28, "duration": 4.26, "text": "subtract 98 factorial times 1. and same"}, {"start": 41.219, "duration": 3.481, "text": "thing on bottom but it's going to be a"}, {"start": 42.54, "duration": 3.66, "text": "plus and then the 98 factorial on top"}, {"start": 44.7, "duration": 3.839, "text": "and bottom those can cancel each other"}, {"start": 46.2, "duration": 4.98, "text": "out so we're just left with 99 minus 1"}, {"start": 48.539, "duration": 5.401, "text": "which is 98 and 99 plus 1 which is 100"}, {"start": 51.18, "duration": 5.039, "text": "and 98 over 100 simplifies to 49 over"}, {"start": 53.94, "duration": 5.279, "text": "50. and let's put a box around it so"}, {"start": 56.219, "duration": 5.48, "text": "this simplified is 49 over 50. how"}, {"start": 59.219, "duration": 2.48, "text": "exciting"}], "title": "Factorial Challenge", "description": "I thought this was a cool one. I hope you guys like it!", "lengthSeconds": 60, "width": 360, "height": 640}
{"video_id": "9F21qE-Yl2Y", "transcript": [{"start": 0.0, "duration": 3.959, "text": "hey guys let's subtract two numbers that"}, {"start": 1.74, "duration": 5.159, "text": "are in binary notation so we do one"}, {"start": 3.959, "duration": 5.701, "text": "minus one which gives us zero zero minus"}, {"start": 6.899, "duration": 5.101, "text": "zero which gives us zero one minus zero"}, {"start": 9.66, "duration": 3.899, "text": "which gives us one and then zero minus"}, {"start": 12.0, "duration": 3.299, "text": "one we can't do so we're gonna have to"}, {"start": 13.559, "duration": 3.361, "text": "borrow so we're gonna scratch out this"}, {"start": 15.299, "duration": 4.021, "text": "one and make it a zero and we're going"}, {"start": 16.92, "duration": 4.98, "text": "to be basically borrowing two for this"}, {"start": 19.32, "duration": 4.86, "text": "column for binary a one in this column"}, {"start": 21.9, "duration": 4.5, "text": "is equal to two in this column and then"}, {"start": 24.18, "duration": 3.96, "text": "two minus one gives us one and then we"}, {"start": 26.4, "duration": 3.18, "text": "have another zero minus one which we"}, {"start": 28.14, "duration": 4.32, "text": "can't do so we're going to borrow again"}, {"start": 29.58, "duration": 4.8, "text": "and making this a two two minus one"}, {"start": 32.46, "duration": 3.54, "text": "gives us one and then once again we"}, {"start": 34.38, "duration": 3.6, "text": "can't do zero minus one so we have to"}, {"start": 36.0, "duration": 3.36, "text": "borrow but we can't borrow from here we"}, {"start": 37.98, "duration": 3.78, "text": "have to borrow all the way from over"}, {"start": 39.36, "duration": 4.859, "text": "here so that gives us a two here so"}, {"start": 41.76, "duration": 4.44, "text": "we're going to make this two a one so we"}, {"start": 44.219, "duration": 3.901, "text": "can move one over here but a one in this"}, {"start": 46.2, "duration": 4.379, "text": "column is equal to a two in this column"}, {"start": 48.12, "duration": 4.86, "text": "and two minus one is equal to one and"}, {"start": 50.579, "duration": 4.14, "text": "then one minus one is zero and then zero"}, {"start": 52.98, "duration": 3.96, "text": "minus zero is zero so we don't need"}, {"start": 54.719, "duration": 4.741, "text": "these last two zeros so one zero one one"}, {"start": 56.94, "duration": 5.04, "text": "zero one zero one minus one one one one"}, {"start": 59.46, "duration": 4.62, "text": "zero zero one is equal to one one one"}, {"start": 61.98, "duration": 4.62, "text": "one zero zero and here's the same"}, {"start": 64.08, "duration": 7.219, "text": "problem in decimal notation 181 minus"}, {"start": 66.6, "duration": 4.699, "text": "one twenty one equals sixty how exciting"}], "title": "Binary Subtraction Example", "description": "Here is a quick example on how to do subtraction with numbers in binary notation.", "lengthSeconds": 70, "width": 640, "height": 360}
{"video_id": "ByRTMcTT7WA", "transcript": [{"start": 0.04, "duration": 4.48, "text": "hey guys this looks like a fun one it"}, {"start": 1.599, "duration": 5.84, "text": "says a semicircle and a quarter circle"}, {"start": 4.52, "duration": 5.24, "text": "inside of a square how long is the black"}, {"start": 7.439, "duration": 4.401, "text": "tangent line this is another puzzle from"}, {"start": 9.76, "duration": 3.44, "text": "Katrina a I'm trying to solve all of"}, {"start": 11.84, "duration": 2.999, "text": "these if you want to try this one on"}, {"start": 13.2, "duration": 4.839, "text": "your own pause it right now because I'm"}, {"start": 14.839, "duration": 5.001, "text": "going to solve it in three 2 1 first"}, {"start": 18.039, "duration": 3.801, "text": "let's copy down and let's focus on the"}, {"start": 19.84, "duration": 3.759, "text": "radius of this quarter Circle they're"}, {"start": 21.84, "duration": 3.88, "text": "telling us right here that it's equal to"}, {"start": 23.599, "duration": 3.641, "text": "8 and let's shift it over here to this"}, {"start": 25.72, "duration": 3.479, "text": "point of tangency and that means it's"}, {"start": 27.24, "duration": 3.76, "text": "going to be at right angles and from"}, {"start": 29.199, "duration": 3.04, "text": "here let's duplicate this one we're"}, {"start": 31.0, "duration": 3.239, "text": "going to do something different on this"}, {"start": 32.239, "duration": 4.041, "text": "one and this one for the top one let's"}, {"start": 34.239, "duration": 3.881, "text": "look at the radius of five and let's"}, {"start": 36.28, "duration": 3.84, "text": "shift it down here and then let's look"}, {"start": 38.12, "duration": 4.08, "text": "at this side length and let's call that"}, {"start": 40.12, "duration": 3.64, "text": "s for side length and since it's a"}, {"start": 42.2, "duration": 3.64, "text": "square we know both these side lengths"}, {"start": 43.76, "duration": 3.799, "text": "are equal to S and then last let's"}, {"start": 45.84, "duration": 3.96, "text": "complete this triangle and we know this"}, {"start": 47.559, "duration": 3.801, "text": "is a right angle because it's a square"}, {"start": 49.8, "duration": 3.36, "text": "we don't know this angle down here so"}, {"start": 51.36, "duration": 3.719, "text": "let's call it Theta and that's going to"}, {"start": 53.16, "duration": 3.8, "text": "be the same angle as this down here so"}, {"start": 55.079, "duration": 3.601, "text": "this one's also Theta and while we're"}, {"start": 56.96, "duration": 3.439, "text": "down here let's draw this side and we"}, {"start": 58.68, "duration": 3.879, "text": "know that's equal to s because it's a"}, {"start": 60.399, "duration": 4.04, "text": "side of the square let's bring this down"}, {"start": 62.559, "duration": 3.961, "text": "and keep it parallel and then let's"}, {"start": 64.439, "duration": 3.68, "text": "complete this triangle and we know this"}, {"start": 66.52, "duration": 3.52, "text": "is also a right triangle because it's"}, {"start": 68.119, "duration": 4.36, "text": "parallel to the top next let's try to"}, {"start": 70.04, "duration": 4.52, "text": "find this angle inside here since this"}, {"start": 72.479, "duration": 4.601, "text": "is a right triangle and this acute angle"}, {"start": 74.56, "duration": 5.04, "text": "is Theta that leaves 90 minus Theta for"}, {"start": 77.08, "duration": 5.32, "text": "the other angle since this is a right"}, {"start": 79.6, "duration": 4.159, "text": "angle and this piece is 90 minus Theta"}, {"start": 82.4, "duration": 3.84, "text": "that means that this angle in this"}, {"start": 83.759, "duration": 4.481, "text": "triangle will also be equal to Theta and"}, {"start": 86.24, "duration": 4.08, "text": "now we have two congruent triangles"}, {"start": 88.24, "duration": 4.08, "text": "let's focus on those triangles for this"}, {"start": 90.32, "duration": 3.56, "text": "triangle up here on the hypotenuse the"}, {"start": 92.32, "duration": 4.36, "text": "five and the eight will add together to"}, {"start": 93.88, "duration": 4.44, "text": "give us 13 and now let's rotate this one"}, {"start": 96.68, "duration": 3.719, "text": "now we have these two triangles lined up"}, {"start": 98.32, "duration": 4.28, "text": "and they are congruent the reason we"}, {"start": 100.399, "duration": 4.76, "text": "know they're congruent is because of Asa"}, {"start": 102.6, "duration": 4.119, "text": "which stands for angle side angle that's"}, {"start": 105.159, "duration": 3.6, "text": "because we have these two angles"}, {"start": 106.719, "duration": 4.04, "text": "congruent we have these two sides"}, {"start": 108.759, "duration": 3.921, "text": "congruent and we have these two angles"}, {"start": 110.759, "duration": 3.801, "text": "congruent and now since we know these"}, {"start": 112.68, "duration": 5.32, "text": "two triangles are congruent that means"}, {"start": 114.56, "duration": 4.96, "text": "if this side is 13 this side is also 13"}, {"start": 118.0, "duration": 3.32, "text": "and this side length right here is the"}, {"start": 119.52, "duration": 3.68, "text": "same thing as this side length right"}, {"start": 121.32, "duration": 4.6, "text": "here so we've now solved the problem"}, {"start": 123.2, "duration": 4.44, "text": "it's equal to 13 units let's put a box"}, {"start": 125.92, "duration": 3.959, "text": "around it and there is a part two to"}, {"start": 127.64, "duration": 3.64, "text": "this problem I also found the area of"}, {"start": 129.879, "duration": 2.881, "text": "this square if you want to check out"}, {"start": 131.28, "duration": 3.84, "text": "that video I'll try to put the link up"}, {"start": 132.76, "duration": 5.6, "text": "here and also put it in the description"}, {"start": 135.12, "duration": 3.24, "text": "how exciting"}], "title": "Part 1 Length of the Tangent", "description": "https://youtu.be/ORq2kkWFtd4", "lengthSeconds": 136, "width": 640, "height": 360}
{"video_id": "WsRVQLEOaKU", "transcript": [{"start": 0.08, "duration": 3.799, "text": "this video is sponsored by brilliant hey"}, {"start": 2.52, "duration": 4.04, "text": "guys this looks like a fun one it wants"}, {"start": 3.879, "duration": 4.521, "text": "to know what is the Shaded area and it's"}, {"start": 6.56, "duration": 3.76, "text": "telling us that the largest semicircle"}, {"start": 8.4, "duration": 3.76, "text": "and the smallest semicircle have the"}, {"start": 10.32, "duration": 4.0, "text": "same Center first let's look at the"}, {"start": 12.16, "duration": 4.24, "text": "radius of the smaller semicircle and"}, {"start": 14.32, "duration": 3.44, "text": "let's call it little r and then let's"}, {"start": 16.4, "duration": 4.28, "text": "look at the radius of the larger"}, {"start": 17.76, "duration": 4.92, "text": "semicircle and call it Big R when set up"}, {"start": 20.68, "duration": 4.08, "text": "like this these form a right triangle"}, {"start": 22.68, "duration": 4.16, "text": "with this distance of two so that means"}, {"start": 24.76, "duration": 6.4, "text": "we can do Pythagorean theorem it'll be"}, {"start": 26.84, "duration": 7.039, "text": "2^2 + little R2 = Big R 2 we can"}, {"start": 31.16, "duration": 4.8, "text": "simplify this we can make 2^2 equal to 4"}, {"start": 33.879, "duration": 3.801, "text": "and let's get both of these RS onto the"}, {"start": 35.96, "duration": 4.24, "text": "same side of the equation we can"}, {"start": 37.68, "duration": 5.24, "text": "subtract little R 2 from both sides and"}, {"start": 40.2, "duration": 4.839, "text": "that'll give us 4 is equal to Big R 2"}, {"start": 42.92, "duration": 4.08, "text": "minus little R 2 and this looks"}, {"start": 45.039, "duration": 4.441, "text": "important let's put a box around it this"}, {"start": 47.0, "duration": 4.84, "text": "blue area is split up into two sections"}, {"start": 49.48, "duration": 3.68, "text": "there's the left area and the right area"}, {"start": 51.84, "duration": 3.8, "text": "and if we add these together we'll have"}, {"start": 53.16, "duration": 4.039, "text": "the total blue area so to calculate this"}, {"start": 55.64, "duration": 4.439, "text": "left side it's going to be the larger"}, {"start": 57.199, "duration": 4.081, "text": "semicircle minus the smaller semicircle"}, {"start": 60.079, "duration": 4.521, "text": "and the equation for both those"}, {"start": 61.28, "duration": 4.64, "text": "semicircles is pi/ 2R 2 and then on the"}, {"start": 64.6, "duration": 3.96, "text": "right hand side we can do this"}, {"start": 65.92, "duration": 4.96, "text": "semicircle minus this semicircle so now"}, {"start": 68.56, "duration": 4.44, "text": "let's find this radius it's going to be"}, {"start": 70.88, "duration": 4.04, "text": "capital r so we can plug that in right"}, {"start": 73.0, "duration": 3.92, "text": "here and then to find the radius of this"}, {"start": 74.92, "duration": 4.36, "text": "one the diameter of this one is the Big"}, {"start": 76.92, "duration": 5.48, "text": "R plus the little r so the radius is"}, {"start": 79.28, "duration": 5.0, "text": "going to be Big R plus little r / 2 so"}, {"start": 82.4, "duration": 3.88, "text": "we can put that right here and then for"}, {"start": 84.28, "duration": 4.799, "text": "this semicircle it's going to be the"}, {"start": 86.28, "duration": 5.04, "text": "same diameter Big R plus little r so the"}, {"start": 89.079, "duration": 4.641, "text": "radius would be Big R R plus little r /"}, {"start": 91.32, "duration": 4.72, "text": "2 and last we have this little guy which"}, {"start": 93.72, "duration": 3.8, "text": "has a radius of little r now we have our"}, {"start": 96.04, "duration": 3.2, "text": "left and our right so we just want to"}, {"start": 97.52, "duration": 3.919, "text": "add all of these together let's"}, {"start": 99.24, "duration": 4.159, "text": "rearrange things a little bit these two"}, {"start": 101.439, "duration": 3.561, "text": "match and one's positive and one's"}, {"start": 103.399, "duration": 3.521, "text": "negative so they're going to cancel each"}, {"start": 105.0, "duration": 5.2, "text": "other out and that'll just leave us with"}, {"start": 106.92, "duration": 5.72, "text": "Pi / 2 big r^ 2 minus P piun / 2 little"}, {"start": 110.2, "duration": 5.08, "text": "R 2 both of these have a common factor"}, {"start": 112.64, "duration": 5.56, "text": "of Pi / 2 so let's factor that out and"}, {"start": 115.28, "duration": 4.839, "text": "Pi / 2 * r^ 2 gives us the first term"}, {"start": 118.2, "duration": 4.76, "text": "and then we subtract little R 2 to get"}, {"start": 120.119, "duration": 5.801, "text": "the second term so we're left with < / 2"}, {"start": 122.96, "duration": 4.92, "text": "* big r^ 2 - little R 2 and if you look"}, {"start": 125.92, "duration": 4.559, "text": "at our box we already know what big r^ 2"}, {"start": 127.88, "duration": 3.96, "text": "minus little R 2 is it's equal to four"}, {"start": 130.479, "duration": 3.921, "text": "so in the place of this we can"}, {"start": 131.84, "duration": 5.52, "text": "substitute four we can bring the divided"}, {"start": 134.4, "duration": 4.8, "text": "two in here with the four and then 4 / 2"}, {"start": 137.36, "duration": 4.0, "text": "is equal to 2 and now we have the answer"}, {"start": 139.2, "duration": 4.399, "text": "to our question the area is equal to 2"}, {"start": 141.36, "duration": 4.04, "text": "pi square units and we can put a box"}, {"start": 143.599, "duration": 3.441, "text": "around it I think this is a brilliant"}, {"start": 145.4, "duration": 3.4, "text": "problem speaking of brilliant I want to"}, {"start": 147.04, "duration": 3.0, "text": "give you guys an update on brilliant"}, {"start": 148.8, "duration": 3.2, "text": "I've been going through this for over a"}, {"start": 150.04, "duration": 4.24, "text": "month now and I absolutely love it there"}, {"start": 152.0, "duration": 5.44, "text": "are thousands of interactive lessons in"}, {"start": 154.28, "duration": 4.72, "text": "math data analysis programming AI all"}, {"start": 157.44, "duration": 3.28, "text": "kinds of different things it's the best"}, {"start": 159.0, "duration": 3.44, "text": "way to learn it's not about memorizing"}, {"start": 160.72, "duration": 3.879, "text": "steps it's about understanding the"}, {"start": 162.44, "duration": 4.4, "text": "concepts I was originally excited about"}, {"start": 164.599, "duration": 4.36, "text": "the math courses yet the last two weeks"}, {"start": 166.84, "duration": 4.24, "text": "I've been exploring other topics right"}, {"start": 168.959, "duration": 3.721, "text": "now I want to focus on how AI works I"}, {"start": 171.08, "duration": 3.48, "text": "feel like AI is going to play a huge"}, {"start": 172.68, "duration": 3.72, "text": "role in content Creation in the future"}, {"start": 174.56, "duration": 3.44, "text": "so I kind of want to learn how it works"}, {"start": 176.4, "duration": 3.52, "text": "I just started this course how language"}, {"start": 178.0, "duration": 3.599, "text": "models work I've been able to learn how"}, {"start": 179.92, "duration": 3.16, "text": "they build vocabulary and choose the"}, {"start": 181.599, "duration": 3.2, "text": "next word and there's also a part that"}, {"start": 183.08, "duration": 3.84, "text": "shows the differences based on what you"}, {"start": 184.799, "duration": 3.681, "text": "use for training data for example the"}, {"start": 186.92, "duration": 3.76, "text": "difference between training on Taylor"}, {"start": 188.48, "duration": 3.88, "text": "Swift lyrics versus a cookbook I'm"}, {"start": 190.68, "duration": 3.32, "text": "pretty excited to see what comes next if"}, {"start": 192.36, "duration": 4.879, "text": "you want to try a brilliant for free for"}, {"start": 194.0, "duration": 4.959, "text": "30 days visit brilliant.org Andy or"}, {"start": 197.239, "duration": 3.801, "text": "click on the link in the description you"}, {"start": 198.959, "duration": 6.0, "text": "can also get 20% off an annual premium"}, {"start": 201.04, "duration": 3.919, "text": "plan how exciting"}], "title": "Concentric Circle Challenge", "description": "To try everything Brilliant has to offer for free for a full 30 days, visit https://brilliant.org/AndyMath/ . You'll also get 20% off an annual premium subscription.", "lengthSeconds": 203, "width": 640, "height": 360}
{"video_id": "sHm3qoQ-Umo", "transcript": [{"start": 0.0, "duration": 3.36, "text": "hey guys this looks like a fun one it's"}, {"start": 1.48, "duration": 4.04, "text": "got a whole bunch of stuff going on and"}, {"start": 3.36, "duration": 3.76, "text": "it's asking for the area of this blue"}, {"start": 5.52, "duration": 3.48, "text": "triangle if you want to try it on your"}, {"start": 7.12, "duration": 5.2, "text": "own pause it right now CU I'm going to"}, {"start": 9.0, "duration": 5.0, "text": "solve it in 3 2 1 so first let's look at"}, {"start": 12.32, "duration": 3.68, "text": "this square right here this side is"}, {"start": 14.0, "duration": 4.039, "text": "equal to four so this side right here is"}, {"start": 16.0, "duration": 4.24, "text": "also equal to four next I want to look"}, {"start": 18.039, "duration": 3.881, "text": "at this rhombus right here let's verify"}, {"start": 20.24, "duration": 3.4, "text": "that it's a square we know that if this"}, {"start": 21.92, "duration": 3.72, "text": "is a right angle this is also a right"}, {"start": 23.64, "duration": 3.28, "text": "angle in a rhombus opposite angles are"}, {"start": 25.64, "duration": 3.0, "text": "congruent so these two would be"}, {"start": 26.92, "duration": 4.04, "text": "congruent and these two would also be"}, {"start": 28.64, "duration": 4.28, "text": "congruent so we know that rhombus is a"}, {"start": 30.96, "duration": 4.04, "text": "square and then I want to focus on this"}, {"start": 32.92, "duration": 3.6, "text": "triangle this is a right triangle so we"}, {"start": 35.0, "duration": 4.16, "text": "know that we can use the Pythagorean"}, {"start": 36.52, "duration": 4.559, "text": "theorem so A2 plus b^2 is going to equal"}, {"start": 39.16, "duration": 3.68, "text": "c^2 and these two are equal to each"}, {"start": 41.079, "duration": 5.96, "text": "other so let's call them both X so we"}, {"start": 42.84, "duration": 7.52, "text": "can say x^2 + x^2 is equal to 4^2 X2 +"}, {"start": 47.039, "duration": 5.68, "text": "X2 is equal to 2x^2 and 4^2 is equal to"}, {"start": 50.36, "duration": 4.64, "text": "16 after we divide both sides by two we"}, {"start": 52.719, "duration": 4.16, "text": "get x^2 is equal to 8 and after we"}, {"start": 55.0, "duration": 3.92, "text": "square root both sides we get X is equal"}, {"start": 56.879, "duration": 4.881, "text": "to theare < TK of 8 so let's update both"}, {"start": 58.92, "duration": 4.36, "text": "of these X's to s < TK of 8 next let's"}, {"start": 61.76, "duration": 3.359, "text": "go back to this square right here we"}, {"start": 63.28, "duration": 3.879, "text": "know these two sides are four this side"}, {"start": 65.119, "duration": 4.081, "text": "is also equal to four and now let's look"}, {"start": 67.159, "duration": 4.361, "text": "at this right angle again this diagonal"}, {"start": 69.2, "duration": 4.84, "text": "is going to BCT that right angle so both"}, {"start": 71.52, "duration": 4.279, "text": "of these will be 45\u00b0 and then in this"}, {"start": 74.04, "duration": 3.68, "text": "Square we also have a right angle and if"}, {"start": 75.799, "duration": 4.561, "text": "this is 45 that means this last angle"}, {"start": 77.72, "duration": 5.56, "text": "must also be 45\u00b0 so we have each of"}, {"start": 80.36, "duration": 4.399, "text": "these angles equal to 45\u00b0 if we focus on"}, {"start": 83.28, "duration": 3.479, "text": "this triangle we can figure out this"}, {"start": 84.759, "duration": 3.481, "text": "obtuse angle is equal to the sum of"}, {"start": 86.759, "duration": 3.961, "text": "these three which is"}, {"start": 88.24, "duration": 4.0, "text": "135\u00b0 and now since this is an obtuse"}, {"start": 90.72, "duration": 4.2, "text": "triangle we have to use the law of"}, {"start": 92.24, "duration": 4.8, "text": "cosiness so if we call the sides a b and"}, {"start": 94.92, "duration": 5.239, "text": "c and this angle Theta the law of"}, {"start": 97.04, "duration": 7.079, "text": "cosiness tells us c^2 is = to a 2 + b^2"}, {"start": 100.159, "duration": 5.801, "text": "- 2 * a * B cosine of that angle Theta"}, {"start": 104.119, "duration": 4.0, "text": "so we're going to use this to solve this"}, {"start": 105.96, "duration": 4.199, "text": "side C so let's give ourselves some room"}, {"start": 108.119, "duration": 3.521, "text": "and a is equal to the square < TK of 8"}, {"start": 110.159, "duration": 3.441, "text": "so in the place of each of these A's"}, {"start": 111.64, "duration": 3.92, "text": "we'll plug in square < TK of 8 and then"}, {"start": 113.6, "duration": 3.96, "text": "B is equal to four so in the place of"}, {"start": 115.56, "duration": 4.8, "text": "these B's we'll plug in four and then"}, {"start": 117.56, "duration": 6.36, "text": "Theta is equal to 135 deg"}, {"start": 120.36, "duration": 5.56, "text": "< TK of 82 is equal to 8 + 4 2 is equal"}, {"start": 123.92, "duration": 4.199, "text": "to 16 and then we're going to subtract"}, {"start": 125.92, "duration": 4.08, "text": "all this 2 * 4 is equal 8 and then we"}, {"start": 128.119, "duration": 3.721, "text": "still have the < TK of 8 and to get"}, {"start": 130.0, "duration": 5.16, "text": "cosine of 135 we can look at the unit"}, {"start": 131.84, "duration": 5.759, "text": "circle to get < tk2 over2 and then 8 +"}, {"start": 135.16, "duration": 3.84, "text": "16 is equal to 24 and then this negative"}, {"start": 137.599, "duration": 3.081, "text": "time this negative will give us a"}, {"start": 139.0, "duration": 4.68, "text": "positive and all this is going to be a"}, {"start": 140.68, "duration": 5.72, "text": "fraction we can copy down the 8un 8 * <"}, {"start": 143.68, "duration": 5.04, "text": "tk2 will give usun 16 and this two will"}, {"start": 146.4, "duration": 5.96, "text": "still be in the denominator this otk 16"}, {"start": 148.72, "duration": 7.519, "text": "is equal to 4 * 4 is equal to 32 and 32"}, {"start": 152.36, "duration": 5.68, "text": "/ 2 is 16 24 + 16 is equal to 40 and"}, {"start": 156.239, "duration": 4.041, "text": "after we square root both sides we get C"}, {"start": 158.04, "duration": 4.6, "text": "is equal to the squ < TK of 40 so we can"}, {"start": 160.28, "duration": 4.319, "text": "update this side length C to be square <"}, {"start": 162.64, "duration": 4.16, "text": "TK of 40 and now let's focus on this"}, {"start": 164.599, "duration": 4.081, "text": "square right here since this side isqu <"}, {"start": 166.8, "duration": 3.96, "text": "TK of 40 this side will also be squ < TK"}, {"start": 168.68, "duration": 3.88, "text": "of 40 and then we're ready to focus on"}, {"start": 170.76, "duration": 3.32, "text": "this right triangle and because it's a"}, {"start": 172.56, "duration": 3.88, "text": "right triangle we can do Pythagorean"}, {"start": 174.08, "duration": 4.6, "text": "theorem we'll assign a b and c to the"}, {"start": 176.44, "duration": 4.6, "text": "sides C is equal to < TK of 40 so we can"}, {"start": 178.68, "duration": 4.36, "text": "plug in < TK of 40 here and then B is"}, {"start": 181.04, "duration": 4.279, "text": "equal to > of8 so we'll plug in < TK of"}, {"start": 183.04, "duration": 5.32, "text": "8 and then we'll just copy down the a^ S"}, {"start": 185.319, "duration": 5.121, "text": "the < TK of 402 is 40 and then squ < TK"}, {"start": 188.36, "duration": 4.32, "text": "of 8^2 is equal to 8 and then let's copy"}, {"start": 190.44, "duration": 4.159, "text": "down the A2 and then we can subtract 8"}, {"start": 192.68, "duration": 4.36, "text": "from both sides and we get a^2 is equal"}, {"start": 194.599, "duration": 4.481, "text": "to 32 and then after we square root both"}, {"start": 197.04, "duration": 4.919, "text": "sides we get a is equal to square < TK"}, {"start": 199.08, "duration": 5.36, "text": "of 32 so we can update this side a to be"}, {"start": 201.959, "duration": 3.92, "text": "square < TK of 32 and now since we have"}, {"start": 204.44, "duration": 3.439, "text": "these two sides we're ready to find the"}, {"start": 205.879, "duration": 4.041, "text": "area of this triangle the area is equal"}, {"start": 207.879, "duration": 4.36, "text": "to 1/2 base time height where the base"}, {"start": 209.92, "duration": 4.679, "text": "is the < TK of 32 and the height is the"}, {"start": 212.239, "duration": 4.2, "text": "< TK of 8 we can combine the < TK of 32"}, {"start": 214.599, "duration": 6.081, "text": "and the < TK of8 so we end up with the <"}, {"start": 216.439, "duration": 6.72, "text": "TK of 256 and then theun of 256 is 16"}, {"start": 220.68, "duration": 4.24, "text": "and 1 12 of 16 is 8 we'll give it a"}, {"start": 223.159, "duration": 3.561, "text": "label of centimet squar and that is the"}, {"start": 224.92, "duration": 4.36, "text": "answer to our question let's put a box"}, {"start": 226.72, "duration": 8.12, "text": "around it the area of this blue triangle"}, {"start": 229.28, "duration": 5.56, "text": "is 8 cm squared how exciting"}], "title": "Area of Blue Triangle", "description": "I hope you guys like this one!\n\nLet me know if you see an easier way to solve it! \n\nAny requests for other things I should work on?\n\nThank you!", "lengthSeconds": 233, "width": 640, "height": 360}
{"video_id": "nqLo50j6YgA", "transcript": [{"start": 0.04, "duration": 3.0, "text": "hey guys this is a fun one it's not"}, {"start": 1.439, "duration": 3.641, "text": "super difficult but I kind of like the"}, {"start": 3.04, "duration": 3.56, "text": "imagery we got a palm tree and a penguin"}, {"start": 5.08, "duration": 4.479, "text": "and we're trying to figure out how tall"}, {"start": 6.6, "duration": 5.159, "text": "is this penguin together they're 230 and"}, {"start": 9.559, "duration": 3.761, "text": "the difference between the two is 200 so"}, {"start": 11.759, "duration": 3.481, "text": "let's give the height of the penguin the"}, {"start": 13.32, "duration": 4.0, "text": "variable X and let's give the height of"}, {"start": 15.24, "duration": 4.64, "text": "the palm tree the variable Y and now we"}, {"start": 17.32, "duration": 6.0, "text": "can write these equations we have y + x"}, {"start": 19.88, "duration": 5.639, "text": "will be equal to 230 and then y - x will"}, {"start": 23.32, "duration": 3.279, "text": "be equal to 200 in order to solve for"}, {"start": 25.519, "duration": 2.76, "text": "this we're going to do something called"}, {"start": 26.599, "duration": 4.401, "text": "elimination we're going to add the two"}, {"start": 28.279, "duration": 7.28, "text": "equations together y + y will give us 2"}, {"start": 31.0, "duration": 6.76, "text": "y the X +x goes away and 230 + 200 is"}, {"start": 35.559, "duration": 6.241, "text": "equal to 430 and then after we divide"}, {"start": 37.76, "duration": 6.68, "text": "both sides by 2 we get Y is equal to 215"}, {"start": 41.8, "duration": 5.68, "text": "and then next let's rewrite this y + x ="}, {"start": 44.44, "duration": 5.639, "text": "230 Y is equal to 215 in the place of"}, {"start": 47.48, "duration": 5.28, "text": "this y I want to plug in 215 and then we"}, {"start": 50.079, "duration": 5.201, "text": "can subtract 215 from both sides we have"}, {"start": 52.76, "duration": 5.08, "text": "X is equal to 15 so we now have the"}, {"start": 55.28, "duration": 6.48, "text": "height of the penguin he is equal to 15"}, {"start": 57.84, "duration": 3.92, "text": "units how exciting"}], "title": "Quick Penguin Math", "description": "Someone shared it with me and I liked the picture.", "lengthSeconds": 60, "width": 360, "height": 640}
{"video_id": "yYeNJL4ILoU", "transcript": [{"start": 0.03, "duration": 4.16, "text": "hey guys welcome to this rotation video"}, {"start": 4.73, "duration": 5.56, "text": "we're gonna rotate this figure by 270"}, {"start": 8.22, "duration": 5.339, "text": "degrees these this way to do rotations"}, {"start": 10.29, "duration": 5.009, "text": "in my opinion is using the notes these"}, {"start": 13.559, "duration": 4.411, "text": "are notes for the different rotations"}, {"start": 15.299, "duration": 4.89, "text": "we're gonna use the one for 270 degrees"}, {"start": 17.97, "duration": 4.83, "text": "so first step we want to write down the"}, {"start": 20.189, "duration": 3.93, "text": "3 points of our preimage the 3 points of"}, {"start": 22.8, "duration": 6.09, "text": "the image that we're given it's a"}, {"start": 24.119, "duration": 7.911, "text": "negative 3 to negative 6 2 and negative"}, {"start": 28.89, "duration": 5.79, "text": "3 9 they were going to apply the"}, {"start": 32.03, "duration": 6.459, "text": "transformation given to us from above so"}, {"start": 34.68, "duration": 5.43, "text": "for 270 it says list the Y so we're"}, {"start": 38.489, "duration": 4.14, "text": "switching them so we're gonna take the Y"}, {"start": 40.11, "duration": 5.1, "text": "value and put it in the X spot so I took"}, {"start": 42.629, "duration": 5.311, "text": "the Y from here and put it into X 4"}, {"start": 45.21, "duration": 4.65, "text": "there then it says do negative X or"}, {"start": 47.94, "duration": 3.45, "text": "whatever the opposite of the first one"}, {"start": 49.86, "duration": 3.839, "text": "is now going to become our new second"}, {"start": 51.39, "duration": 5.07, "text": "one so the negative negative 3 becomes 3"}, {"start": 53.699, "duration": 4.831, "text": "this becomes a positive 6 this becomes a"}, {"start": 56.46, "duration": 7.309, "text": "positive 3 now we just plot the points"}, {"start": 58.53, "duration": 8.43, "text": "that we just mapped into so 2 3 2 6 and"}, {"start": 63.769, "duration": 5.291, "text": "9 3 then we're gonna connect the dots"}, {"start": 66.96, "duration": 5.13, "text": "and now you can see that the figure was"}, {"start": 69.06, "duration": 5.04, "text": "rotated 270 degrees so change the"}, {"start": 72.09, "duration": 4.139, "text": "orientation and the quadrant that it's"}, {"start": 74.1, "duration": 4.17, "text": "in so this makes sense if not have a"}, {"start": 76.229, "duration": 4.291, "text": "link down below 2 more practice in"}, {"start": 78.27, "duration": 3.9, "text": "videos and problems on Andy math com"}, {"start": 80.52, "duration": 3.389, "text": "feel free to check it out please like"}, {"start": 82.17, "duration": 4.66, "text": "this video and subscribe to my channel"}, {"start": 83.909, "duration": 10.151, "text": "now sod you guys soon"}, {"start": 86.83, "duration": 7.23, "text": "[Music]"}], "title": "Rotate 270 Degrees", "description": "For more visit http://andymath.com.\n\nSubscribe here: https://www.youtube.com/channel/UC6KhU3AMLHC-qv7tj0ZJduw?sub_confirmation=1", "lengthSeconds": 92, "width": 640, "height": 360}
{"video_id": "FokVSqVoHck", "transcript": [{"start": 0.04, "duration": 5.719, "text": "hey guys this looks like a fun one it's"}, {"start": 1.52, "duration": 7.159, "text": "a no calculator challenge it's 3 x = 69"}, {"start": 5.759, "duration": 5.8, "text": "and 23 to the Y = 69 and it wants to"}, {"start": 8.679, "duration": 4.92, "text": "know what is 1x + 1 over y if you want"}, {"start": 11.559, "duration": 5.48, "text": "to try it on your own pause it right now"}, {"start": 13.599, "duration": 5.481, "text": "because I'm going to solve it in 3 2 one"}, {"start": 17.039, "duration": 4.4, "text": "first let's copy these down and let's"}, {"start": 19.08, "duration": 5.24, "text": "put everything in parentheses for the"}, {"start": 21.439, "duration": 5.68, "text": "first equation let's exponentiate both"}, {"start": 24.32, "duration": 5.0, "text": "sides to 1 /x power and on the bottom"}, {"start": 27.119, "duration": 5.201, "text": "one let's exponentiate both sides to the"}, {"start": 29.32, "duration": 4.8, "text": "1/ Y power let's go back to the top one"}, {"start": 32.32, "duration": 6.32, "text": "these two exponents are going to"}, {"start": 34.12, "duration": 7.52, "text": "multiply by each other so x * 1X is XX"}, {"start": 38.64, "duration": 5.48, "text": "and X overx is equal to 1 so this is 3"}, {"start": 41.64, "duration": 4.12, "text": "to the 1 or in other words three and"}, {"start": 44.12, "duration": 4.24, "text": "bottom same thing these two exponents"}, {"start": 45.76, "duration": 5.72, "text": "are going to multiply so y * 1 over Y is"}, {"start": 48.36, "duration": 6.039, "text": "y over Y and that's equal to 1 and 23 to"}, {"start": 51.48, "duration": 5.68, "text": "the 1 is 23 so what I like about this is"}, {"start": 54.399, "duration": 5.081, "text": "we now have a 1 /x right here and a one"}, {"start": 57.16, "duration": 4.239, "text": "over y right here we just got to figure"}, {"start": 59.48, "duration": 3.68, "text": "out some way to add these together we"}, {"start": 61.399, "duration": 3.681, "text": "could use logarithms but I want to try"}, {"start": 63.16, "duration": 3.88, "text": "something else let's take this bottom"}, {"start": 65.08, "duration": 5.359, "text": "equation and multiply both sides by"}, {"start": 67.04, "duration": 6.24, "text": "three and since we know 3 is equal to 69"}, {"start": 70.439, "duration": 6.36, "text": "1 /x in the place of this three let's"}, {"start": 73.28, "duration": 6.4, "text": "plug in 69 1 /x on the left hand side 3"}, {"start": 76.799, "duration": 4.841, "text": "* 23 is equal to 69 and on the right"}, {"start": 79.68, "duration": 4.479, "text": "hand side these two have the same base"}, {"start": 81.64, "duration": 4.28, "text": "so we can use these notes right here"}, {"start": 84.159, "duration": 4.041, "text": "anytime two powers are being multiplied"}, {"start": 85.92, "duration": 4.68, "text": "with the same base you can just add the"}, {"start": 88.2, "duration": 4.52, "text": "exponents so these both have a base of"}, {"start": 90.6, "duration": 5.96, "text": "69 so we can just add these two"}, {"start": 92.72, "duration": 5.88, "text": "exponents it'll be 1 y + 1X and on the"}, {"start": 96.56, "duration": 4.96, "text": "left hand side let's give this 69 an"}, {"start": 98.6, "duration": 5.08, "text": "exponent it' be 69 to the 1 now from"}, {"start": 101.52, "duration": 3.879, "text": "here since they both have a base of 69"}, {"start": 103.68, "duration": 3.799, "text": "the only way these two equations can be"}, {"start": 105.399, "duration": 4.76, "text": "equal to each other is if the exponents"}, {"start": 107.479, "duration": 6.0, "text": "are equal to each other so we can set 1"}, {"start": 110.159, "duration": 4.96, "text": "equal to 1 y + 1 /x and now we're"}, {"start": 113.479, "duration": 4.361, "text": "basically done we can rearrange them"}, {"start": 115.119, "duration": 4.801, "text": "like this and we can put a box around it"}, {"start": 117.84, "duration": 6.0, "text": "now we can move it up here in this"}, {"start": 119.92, "duration": 6.92, "text": "challenge 1x + 1 Y is equal to 1 how"}, {"start": 123.84, "duration": 3.0, "text": "exciting"}], "title": "Fun Math Challenge", "description": "I hope you guys like this one!", "lengthSeconds": 125, "width": 360, "height": 640}
{"video_id": "rddxZE9gL24", "transcript": [{"start": 0.0, "duration": 4.08, "text": "hey guys this looks like a fun one it's"}, {"start": 1.48, "duration": 4.56, "text": "another Katrina a puzzle and it's day"}, {"start": 4.08, "duration": 5.08, "text": "four of our Advent calendar we're going"}, {"start": 6.04, "duration": 5.519, "text": "to do 31 Katrina a puzzles in 31 days so"}, {"start": 9.16, "duration": 3.96, "text": "far we're still on Pace on day four so"}, {"start": 11.559, "duration": 4.24, "text": "for this one there are two circles with"}, {"start": 13.12, "duration": 4.76, "text": "the radius four and two squares and it"}, {"start": 15.799, "duration": 4.56, "text": "once to know what's the Shaded area"}, {"start": 17.88, "duration": 3.92, "text": "which is the area of this Square here if"}, {"start": 20.359, "duration": 3.041, "text": "you want to try it on your own pause it"}, {"start": 21.8, "duration": 4.879, "text": "right now because I'm going to solve it"}, {"start": 23.4, "duration": 4.92, "text": "in three 2 1 since we're trying to solve"}, {"start": 26.679, "duration": 4.281, "text": "for this Square let's give it side"}, {"start": 28.32, "duration": 4.48, "text": "lengths of X and that means the area of"}, {"start": 30.96, "duration": 4.52, "text": "this Square the Shaded area will be"}, {"start": 32.8, "duration": 4.24, "text": "equal to x^2 and this will be the answer"}, {"start": 35.48, "duration": 3.72, "text": "to our question so let's put a box"}, {"start": 37.04, "duration": 3.96, "text": "around it and let's move it down here"}, {"start": 39.2, "duration": 3.48, "text": "next let's focus on the top of the"}, {"start": 41.0, "duration": 3.96, "text": "larger square if we line up with the"}, {"start": 42.68, "duration": 3.96, "text": "center of this circle and drop this line"}, {"start": 44.96, "duration": 4.239, "text": "here this piece is the same length as"}, {"start": 46.64, "duration": 4.599, "text": "the radius of four and this piece is the"}, {"start": 49.199, "duration": 4.441, "text": "same length as the radius of four so one"}, {"start": 51.239, "duration": 4.681, "text": "side of our larger Square will be x + 4"}, {"start": 53.64, "duration": 4.16, "text": "+ 4 and we can do the same thing for the"}, {"start": 55.92, "duration": 4.119, "text": "left hand side if we draw this line"}, {"start": 57.8, "duration": 3.759, "text": "right here parallel with the base that"}, {"start": 60.039, "duration": 4.12, "text": "I'll cut it so that this will be equal"}, {"start": 61.559, "duration": 4.841, "text": "to the radius of four and this will be"}, {"start": 64.159, "duration": 4.881, "text": "equal to the radius of four so once"}, {"start": 66.4, "duration": 5.32, "text": "again the side of the square is x + 4 +"}, {"start": 69.04, "duration": 4.399, "text": "4 now let's go to this side this portion"}, {"start": 71.72, "duration": 3.719, "text": "right here will be equal to four because"}, {"start": 73.439, "duration": 3.881, "text": "it's a radius of the circle and this"}, {"start": 75.439, "duration": 3.72, "text": "portion will be equal to 4 that means"}, {"start": 77.32, "duration": 3.96, "text": "that this last piece from here to here"}, {"start": 79.159, "duration": 4.801, "text": "has to be equal to X because the side"}, {"start": 81.28, "duration": 5.0, "text": "length is x + 4 + 4 and then on bottom"}, {"start": 83.96, "duration": 4.839, "text": "let's draw this down from the center"}, {"start": 86.28, "duration": 5.159, "text": "this piece would be four and this piece"}, {"start": 88.799, "duration": 4.841, "text": "would be four and that leaves X for that"}, {"start": 91.439, "duration": 5.241, "text": "piece so now we can verify our Square"}, {"start": 93.64, "duration": 5.439, "text": "every side is x + 4 + 4 now for the next"}, {"start": 96.68, "duration": 4.32, "text": "step let's draw this right triangle"}, {"start": 99.079, "duration": 3.641, "text": "inside of here the height of our right"}, {"start": 101.0, "duration": 3.96, "text": "triangle will be the same thing as this"}, {"start": 102.72, "duration": 3.679, "text": "right here so it'll be equal to X and"}, {"start": 104.96, "duration": 3.08, "text": "the base of our triangle would be the"}, {"start": 106.399, "duration": 3.921, "text": "same thing as this right here so it'll"}, {"start": 108.04, "duration": 4.28, "text": "be equal to X and then the hypotenuse is"}, {"start": 110.32, "duration": 5.439, "text": "going to be made up of this radius which"}, {"start": 112.32, "duration": 4.6, "text": "is 4 plus this radius which is four so"}, {"start": 115.759, "duration": 3.161, "text": "the whole thing is going to be equal to"}, {"start": 116.92, "duration": 4.159, "text": "eight and now let's focus on that"}, {"start": 118.92, "duration": 4.76, "text": "triangle it's a right triangle so we can"}, {"start": 121.079, "duration": 5.4, "text": "use the Pythagorean theorem it'll be x^2"}, {"start": 123.68, "duration": 5.16, "text": "+ x^2 is equal to 8^2 and if we bring"}, {"start": 126.479, "duration": 4.28, "text": "everything back x^2 that was the area of"}, {"start": 128.84, "duration": 6.08, "text": "our shaded Square so all we have to do"}, {"start": 130.759, "duration": 6.84, "text": "is solve for x^2 x^2 + x^2 is 2 x^2 and"}, {"start": 134.92, "duration": 5.44, "text": "8^2 is equal to 64 and after we divide"}, {"start": 137.599, "duration": 5.201, "text": "both sides by two we get x^2 is equal to"}, {"start": 140.36, "duration": 4.64, "text": "32 and x^2 is what we were trying to"}, {"start": 142.8, "duration": 4.799, "text": "solve for let's change this x^2 into a"}, {"start": 145.0, "duration": 4.519, "text": "32 let's make some room for a label and"}, {"start": 147.599, "duration": 4.481, "text": "there's no units mentioned so we'll just"}, {"start": 149.519, "duration": 5.041, "text": "put unit squared and this is the answer"}, {"start": 152.08, "duration": 5.36, "text": "to our question the Shaded area is 32"}, {"start": 154.56, "duration": 5.28, "text": "unit squared how exciting this was day"}, {"start": 157.44, "duration": 4.96, "text": "four of our Advent calendar this one is"}, {"start": 159.84, "duration": 4.44, "text": "going to be day five it says how far"}, {"start": 162.4, "duration": 4.479, "text": "apart are the corners of these two"}, {"start": 164.28, "duration": 4.8, "text": "regular hexagons so they're both"}, {"start": 166.879, "duration": 4.44, "text": "hexagons they're different sizes and it"}, {"start": 169.08, "duration": 3.4, "text": "wants to know how far apart are these"}, {"start": 171.319, "duration": 3.64, "text": "guess we'll find out tomorrow how that"}, {"start": 172.48, "duration": 3.88, "text": "works give me one second okay I know"}, {"start": 174.959, "duration": 5.681, "text": "what going to happen so tomorrow's going"}, {"start": 176.36, "duration": 4.28, "text": "to be fun how exciting"}], "title": "Aggvent Calendar Day 4", "description": "https://andymath.com/agg-vent-calendar/\n\nThis is the link to the Aggvent Calendar if you want to participate.", "lengthSeconds": 179, "width": 640, "height": 360}
{"video_id": "NuwieirwdyU", "transcript": [{"start": 0.04, "duration": 5.759, "text": "hey guys let's solve for x we have x +"}, {"start": 2.32, "duration": 7.12, "text": "theun x + thek x plus the X going on"}, {"start": 5.799, "duration": 5.76, "text": "forever equal 441 so what value of x"}, {"start": 9.44, "duration": 4.359, "text": "would make this true step number one"}, {"start": 11.559, "duration": 4.04, "text": "let's subtract X from both sides on the"}, {"start": 13.799, "duration": 3.64, "text": "left hand side these two x's cancel so"}, {"start": 15.599, "duration": 5.081, "text": "we just have the square root of all this"}, {"start": 17.439, "duration": 5.241, "text": "stuff it's going to be equal to 441 - x"}, {"start": 20.68, "duration": 4.04, "text": "now this inside of the square root it is"}, {"start": 22.68, "duration": 4.04, "text": "also going to keep going on forever that"}, {"start": 24.72, "duration": 3.92, "text": "means that this stuff inside the square"}, {"start": 26.72, "duration": 4.799, "text": "root is the same thing as this stuff"}, {"start": 28.64, "duration": 5.04, "text": "right here this is equal to 441 that"}, {"start": 31.519, "duration": 6.081, "text": "means this right here is also equal to"}, {"start": 33.68, "duration": 5.559, "text": "441 and then the < TK of 441 is 21 let's"}, {"start": 37.6, "duration": 5.36, "text": "add X to both sides which will give us"}, {"start": 39.239, "duration": 5.441, "text": "21 + x = 441 and then let's subtract 21"}, {"start": 42.96, "duration": 5.439, "text": "from both sides and then we get X is"}, {"start": 44.68, "duration": 5.0, "text": "equal to and 441 - 21 is 420 and that is"}, {"start": 48.399, "duration": 3.241, "text": "the answer to our question that we can"}, {"start": 49.68, "duration": 5.84, "text": "put a box around it so next time you see"}, {"start": 51.64, "duration": 6.0, "text": "420 +un of 420+ of 420 going on forever"}, {"start": 55.52, "duration": 6.199, "text": "you'll be able to say that is exactly"}, {"start": 57.64, "duration": 4.079, "text": "441 how exciting"}], "title": "Fun Math Problem", "description": "I hope you guys love it!", "lengthSeconds": 60, "width": 360, "height": 640}
{"video_id": "5hVTuk6MQWM", "transcript": [{"start": 0.0, "duration": 3.76, "text": "hey guys this looks like a fun one it"}, {"start": 1.52, "duration": 4.6, "text": "says the equilateral triangle sits in"}, {"start": 3.76, "duration": 4.24, "text": "the center of the semicircle and it"}, {"start": 6.12, "duration": 4.399, "text": "wants to know what is the area of the"}, {"start": 8.0, "duration": 4.32, "text": "square this problem is day seven of our"}, {"start": 10.519, "duration": 3.681, "text": "agvent calendar for the month of"}, {"start": 12.32, "duration": 4.6, "text": "December we're going to solve 31 Katrina"}, {"start": 14.2, "duration": 4.52, "text": "a puzzles in 31 days and I'm doing my"}, {"start": 16.92, "duration": 3.359, "text": "best to keep us on Pace if you want to"}, {"start": 18.72, "duration": 3.08, "text": "try this one on your own pause it right"}, {"start": 20.279, "duration": 4.641, "text": "now because I'm going to solve it in"}, {"start": 21.8, "duration": 5.76, "text": "three two 1 we're trying to find the"}, {"start": 24.92, "duration": 4.88, "text": "area of this orange Square let's label"}, {"start": 27.56, "duration": 4.96, "text": "the sides of the square s and the area"}, {"start": 29.8, "duration": 4.599, "text": "will be equal to s^2 next they're giving"}, {"start": 32.52, "duration": 3.96, "text": "us the diameter of the semicircle is"}, {"start": 34.399, "duration": 4.601, "text": "equal to 6 that means that the radius"}, {"start": 36.48, "duration": 4.88, "text": "would be equal to 3 so if we connect the"}, {"start": 39.0, "duration": 4.6, "text": "center of the semicircle to this corner"}, {"start": 41.36, "duration": 5.08, "text": "of the square that would also be equal"}, {"start": 43.6, "duration": 5.599, "text": "to three and from here let's finish up"}, {"start": 46.44, "duration": 5.56, "text": "this right triangle this side of our"}, {"start": 49.199, "duration": 5.36, "text": "triangle will be equal to S and the base"}, {"start": 52.0, "duration": 5.0, "text": "we don't know yet let's call it X and"}, {"start": 54.559, "duration": 5.8, "text": "now let's solve for x in the given"}, {"start": 57.0, "duration": 5.879, "text": "equilateral triangle the x is this p"}, {"start": 60.359, "duration": 4.921, "text": "right here it's half of the base let's"}, {"start": 62.879, "duration": 3.961, "text": "take this radius and rotate it over here"}, {"start": 65.28, "duration": 4.08, "text": "that means that the height of this"}, {"start": 66.84, "duration": 4.959, "text": "equilateral triangle is equal to three"}, {"start": 69.36, "duration": 5.2, "text": "and if we connect this vertex to this"}, {"start": 71.799, "duration": 5.081, "text": "vertex we have another right triangle"}, {"start": 74.56, "duration": 4.919, "text": "the hypotenuse of our new right triangle"}, {"start": 76.88, "duration": 5.04, "text": "is the same as a side of our equilateral"}, {"start": 79.479, "duration": 5.441, "text": "triangle and since X is half the"}, {"start": 81.92, "duration": 4.76, "text": "distance the whole distance would be 2X"}, {"start": 84.92, "duration": 4.239, "text": "and now we have three sides of a right"}, {"start": 86.68, "duration": 5.479, "text": "triangle let's do Pythagorean theorem"}, {"start": 89.159, "duration": 6.96, "text": "it'll be X 2 + 3^2 is equal to the"}, {"start": 92.159, "duration": 5.481, "text": "quantity 2x^2 3^2 is equal to 9 and this"}, {"start": 96.119, "duration": 5.201, "text": "exponent will distribute to both of"}, {"start": 97.64, "duration": 7.0, "text": "these to give us 2^2 x^2 and 2^2 is"}, {"start": 101.32, "duration": 5.32, "text": "equal to 4 next let's subtract x^2 from"}, {"start": 104.64, "duration": 4.2, "text": "both sides on the left hand side these"}, {"start": 106.64, "duration": 6.839, "text": "will cancel each other out and on the"}, {"start": 108.84, "duration": 7.0, "text": "right hand side 4x^2 - x^2 is 3x^2 next"}, {"start": 113.479, "duration": 5.481, "text": "we can divide both sides by three on the"}, {"start": 115.84, "duration": 4.639, "text": "left hand side 9 / 3 is equal to 3 and"}, {"start": 118.96, "duration": 3.799, "text": "on the right hand side these two threes"}, {"start": 120.479, "duration": 5.401, "text": "will cancel each other out so we're left"}, {"start": 122.759, "duration": 5.121, "text": "with x^2 is equal to 3 next let's square"}, {"start": 125.88, "duration": 3.76, "text": "root both sides on the left hand side"}, {"start": 127.88, "duration": 3.719, "text": "the square root and the square will"}, {"start": 129.64, "duration": 4.84, "text": "cancel each other out and now we've"}, {"start": 131.599, "duration": 4.601, "text": "solved for x x is equal to the < TK of 3"}, {"start": 134.48, "duration": 4.039, "text": "and that is what we were trying to do up"}, {"start": 136.2, "duration": 5.36, "text": "here in the place of this x let's plug"}, {"start": 138.519, "duration": 5.36, "text": "in < tk3 and now let's take this radius"}, {"start": 141.56, "duration": 4.679, "text": "and bring it back over here and we now"}, {"start": 143.879, "duration": 6.161, "text": "have another right triangle let's do"}, {"start": 146.239, "duration": 7.321, "text": "Pythagorean theorem it'll be < tk3 ^2 +"}, {"start": 150.04, "duration": 5.12, "text": "s^2 is equal to 3^2 over here the square"}, {"start": 153.56, "duration": 5.16, "text": "root and the square will cancel each"}, {"start": 155.16, "duration": 5.52, "text": "other out and 3^2 is equal to 9 after we"}, {"start": 158.72, "duration": 5.599, "text": "subtract three from both sides we end up"}, {"start": 160.68, "duration": 6.52, "text": "with s^2 is equal to 9 - 3 which is 6"}, {"start": 164.319, "duration": 5.241, "text": "and S2 is what we were looking for so in"}, {"start": 167.2, "duration": 4.92, "text": "the place of this s squ let's plug in"}, {"start": 169.56, "duration": 4.679, "text": "six this is the area of the orange"}, {"start": 172.12, "duration": 4.759, "text": "Square let's give it a label of unit"}, {"start": 174.239, "duration": 5.0, "text": "squared and put a box around it how"}, {"start": 176.879, "duration": 5.28, "text": "exciting and this is the puzzle for day"}, {"start": 179.239, "duration": 4.801, "text": "eight when I place two three four five"}, {"start": 182.159, "duration": 4.201, "text": "triangles like this what's the"}, {"start": 184.04, "duration": 7.119, "text": "overlapping area can't wait to see"}, {"start": 186.36, "duration": 4.799, "text": "tomorrow's video how exciting"}], "title": "Aggvent Calendar Day 7", "description": "https://andymath.com/agg-vent-calendar/\n\nHere is a link to the Aggvent Calendar if you want to check it out!", "lengthSeconds": 189, "width": 640, "height": 360}
{"video_id": "4lSo1f1CA28", "transcript": [{"start": 0.04, "duration": 3.44, "text": "hey guys so many people requested this"}, {"start": 1.719, "duration": 3.401, "text": "problem my DMs were flooded with this"}, {"start": 3.48, "duration": 3.56, "text": "one so I guess we got to do it it's"}, {"start": 5.12, "duration": 4.12, "text": "asking for the area of this shaded"}, {"start": 7.04, "duration": 4.08, "text": "region this region is a rectangle so the"}, {"start": 9.24, "duration": 3.8, "text": "area is going to be base time height"}, {"start": 11.12, "duration": 4.12, "text": "where the base is this right here and"}, {"start": 13.04, "duration": 3.96, "text": "the height is equal to 10 we don't know"}, {"start": 15.24, "duration": 4.119, "text": "the length of the base yet so let's call"}, {"start": 17.0, "duration": 4.439, "text": "it X so if we copy down the arrow we"}, {"start": 19.359, "duration": 5.721, "text": "know this entire distance right here is"}, {"start": 21.439, "duration": 5.0, "text": "equal to 14 and since this piece is X"}, {"start": 25.08, "duration": 4.279, "text": "that means this remaining piece is going"}, {"start": 26.439, "duration": 5.121, "text": "to be 14 - x and we have the same thing"}, {"start": 29.359, "duration": 5.081, "text": "going on right here this whole entire"}, {"start": 31.56, "duration": 5.519, "text": "distance is equal to 8 this is X which"}, {"start": 34.44, "duration": 4.639, "text": "leaves 8 - x for this piece the length"}, {"start": 37.079, "duration": 6.8, "text": "of the base of this larger rectangle is"}, {"start": 39.079, "duration": 7.8, "text": "equal to 20 so we can say 8 - x + x + 14"}, {"start": 43.879, "duration": 5.641, "text": "- x is equal to 20 and now we can solve"}, {"start": 46.879, "duration": 4.801, "text": "for x so first we'll combine like terms"}, {"start": 49.52, "duration": 3.96, "text": "this negative X and this positive X can"}, {"start": 51.68, "duration": 4.92, "text": "cancel each other out and we can bring"}, {"start": 53.48, "duration": 5.559, "text": "down thisx we can combine the 8 plus the"}, {"start": 56.6, "duration": 5.4, "text": "14 to give us 22 and this is all going"}, {"start": 59.039, "duration": 4.601, "text": "to be equal to 20 we can subtract 22"}, {"start": 62.0, "duration": 5.04, "text": "from both sides on the left hand side we"}, {"start": 63.64, "duration": 7.08, "text": "havex and on the right hand side 20 - 22"}, {"start": 67.04, "duration": 6.0, "text": "is -2 so we havex is equal to -2 that"}, {"start": 70.72, "duration": 4.439, "text": "means that X is equal to pos2 so let's"}, {"start": 73.04, "duration": 3.6, "text": "update this x to 2 and now we're ready"}, {"start": 75.159, "duration": 2.921, "text": "to find the area of our region it's"}, {"start": 76.64, "duration": 3.519, "text": "going to be the base times the height"}, {"start": 78.08, "duration": 3.84, "text": "where the base is equal to 2 and the"}, {"start": 80.159, "duration": 4.64, "text": "height is equal to 10 and that's going"}, {"start": 81.92, "duration": 4.32, "text": "to give us 20 square units and that's"}, {"start": 84.799, "duration": 5.32, "text": "the answer to our question let's put a"}, {"start": 86.24, "duration": 3.879, "text": "box around it how exciting"}], "title": "Popular Math Problem", "description": "https://andymath.com/geometry-challenges/\n\nFor more math challenges, check out the above link!", "lengthSeconds": 88, "width": 640, "height": 360}
{"video_id": "upwI0mySoGc", "transcript": [{"start": 0.0, "duration": 6.08, "text": "hey guys welcome to this function's"}, {"start": 1.28, "duration": 8.0, "text": "video it says let F ofx = 5x + 3 find F"}, {"start": 6.08, "duration": 5.24, "text": "of 3 so the F here is referring to the F"}, {"start": 9.28, "duration": 4.2, "text": "here and notice how there's an X here"}, {"start": 11.32, "duration": 4.04, "text": "but there's a three here since this x"}, {"start": 13.48, "duration": 4.4, "text": "says there's an X right there you're now"}, {"start": 15.36, "duration": 7.8, "text": "going to put a 3 where there once was an"}, {"start": 17.88, "duration": 9.2, "text": "X so it's 5 * 3 + 3 5 * 3 is 15 + 3 and"}, {"start": 23.16, "duration": 6.119, "text": "15 + 3 is 18 so the answer to F of3 is"}, {"start": 27.08, "duration": 4.24, "text": "18 and this is why we just plugged in"}, {"start": 29.279, "duration": 4.841, "text": "the three for the X and that's how to"}, {"start": 31.32, "duration": 4.44, "text": "evaluate functions comment below on what"}, {"start": 34.12, "duration": 7.68, "text": "you would like me to solve next and I"}, {"start": 35.76, "duration": 6.04, "text": "will talk to you soon bye Andy maath"}], "title": "Evaluating Functions #andymath  #math  #maths  #sat  #tutorials  #tiktokmath  #mathhelp", "description": "Please Subscribe, and don't forget to hit the bell icon so you don't miss next video! Also, please feel free to share this video with your friends and family. Let learn Math together with me, Charlie. \nHere is the playlist of the videos. [ There is so much more to discover in this playlist ]", "lengthSeconds": 41, "width": 360, "height": 640}
{"video_id": "8D_pZfYOsFk", "transcript": [{"start": 0.0, "duration": 5.16, "text": "okay so let's solve a limit this is the"}, {"start": 1.86, "duration": 7.14, "text": "limit as X approaches 69 of 6X squared"}, {"start": 5.16, "duration": 5.519, "text": "minus 408x minus 414 over x minus 69."}, {"start": 9.0, "duration": 4.5, "text": "since we want to know the limit as X"}, {"start": 10.679, "duration": 4.861, "text": "approaches 69 we just substitute 69 in"}, {"start": 13.5, "duration": 3.72, "text": "for each of these x's and we can see"}, {"start": 15.54, "duration": 3.54, "text": "what we get so after you plug it into a"}, {"start": 17.22, "duration": 4.68, "text": "calculator all this on top gives us 0"}, {"start": 19.08, "duration": 4.5, "text": "and 69 minus 69 is also zero so this is"}, {"start": 21.9, "duration": 3.84, "text": "an indeterminate form we don't actually"}, {"start": 23.58, "duration": 4.14, "text": "know what number this is so we have to"}, {"start": 25.74, "duration": 3.9, "text": "find a different way to solve this so"}, {"start": 27.72, "duration": 3.899, "text": "let's try to simplify the stuff inside"}, {"start": 29.64, "duration": 3.78, "text": "the parentheses another way you can"}, {"start": 31.619, "duration": 3.361, "text": "think of a fraction is as a division"}, {"start": 33.42, "duration": 3.78, "text": "problem we're going to take this and"}, {"start": 34.98, "duration": 3.599, "text": "we're going to divide by x minus 69."}, {"start": 37.2, "duration": 3.3, "text": "we're going to do some long division and"}, {"start": 38.579, "duration": 3.541, "text": "this is called polynomial long division"}, {"start": 40.5, "duration": 4.44, "text": "first step we want to know how many"}, {"start": 42.12, "duration": 4.92, "text": "times does X go into 6X squared well"}, {"start": 44.94, "duration": 5.7, "text": "it'll go in 6X times and that's because"}, {"start": 47.04, "duration": 5.58, "text": "6X times x gives us 6X squared and then"}, {"start": 50.64, "duration": 5.099, "text": "we also want to do 6X times the negative"}, {"start": 52.62, "duration": 4.38, "text": "69 which gives us negative 414x now"}, {"start": 55.739, "duration": 3.181, "text": "after we multiply this we want to"}, {"start": 57.0, "duration": 3.0, "text": "subtract these and this negative is"}, {"start": 58.92, "duration": 3.479, "text": "going to distribute both to the negative"}, {"start": 60.0, "duration": 4.979, "text": "6X and it's going to make the 414x"}, {"start": 62.399, "duration": 5.101, "text": "positive 6X squared minus 6X squared"}, {"start": 64.979, "duration": 6.241, "text": "cancel each other out and negative 408x"}, {"start": 67.5, "duration": 5.28, "text": "plus 414x is 6X and now that we're done"}, {"start": 71.22, "duration": 3.54, "text": "subtracting we can bring down the"}, {"start": 72.78, "duration": 4.019, "text": "negative 4 14. now we want to ask"}, {"start": 74.76, "duration": 4.44, "text": "ourselves how many times does X go into"}, {"start": 76.799, "duration": 5.341, "text": "6X well that's going to be 6 times and"}, {"start": 79.2, "duration": 5.04, "text": "that's because 6 times x equals 6X and"}, {"start": 82.14, "duration": 4.619, "text": "we also want to do 6 times negative 69"}, {"start": 84.24, "duration": 4.44, "text": "which is negative 414. and now we want"}, {"start": 86.759, "duration": 3.481, "text": "to subtract this from this well this"}, {"start": 88.68, "duration": 3.06, "text": "negative will distribute to both terms"}, {"start": 90.24, "duration": 3.6, "text": "making the first one negative and the"}, {"start": 91.74, "duration": 3.54, "text": "second one positive these cancel and"}, {"start": 93.84, "duration": 3.419, "text": "these cancel so we're just left with"}, {"start": 95.28, "duration": 4.56, "text": "zero so the answer to this divided by"}, {"start": 97.259, "duration": 5.461, "text": "this is 6X plus 6. so we can change this"}, {"start": 99.84, "duration": 4.5, "text": "whole thing into 6X plus 6. let's smush"}, {"start": 102.72, "duration": 3.48, "text": "this all together and now we're ready to"}, {"start": 104.34, "duration": 4.02, "text": "apply the limit so we want to know what"}, {"start": 106.2, "duration": 4.739, "text": "is this inside thing equal to when X"}, {"start": 108.36, "duration": 5.64, "text": "approaches 69 or we can plug in 69 for"}, {"start": 110.939, "duration": 6.061, "text": "the X at 6 times 69 is 414 so we have"}, {"start": 114.0, "duration": 5.34, "text": "414 plus 6 and that's equal to 420. so"}, {"start": 117.0, "duration": 4.619, "text": "the limit as X approaches 69 of this"}, {"start": 119.34, "duration": 6.319, "text": "right here is is equal to 420. now let's"}, {"start": 121.619, "duration": 4.04, "text": "put a box around it how exciting"}], "title": "Limit using Polynomial Long Division", "description": "This is not the most efficient way to solve the problem, but I think it's a fun review of limits and Polynomial Long Division.", "lengthSeconds": 124, "width": 640, "height": 360}
{"video_id": "9p3IeoK00_s", "transcript": [{"start": 0.04, "duration": 3.88, "text": "hey guys this looks like a fun one we're"}, {"start": 1.599, "duration": 5.24, "text": "trying to find the area of"}, {"start": 3.92, "duration": 5.04, "text": "ACF so that's the area of this reddish"}, {"start": 6.839, "duration": 5.68, "text": "triangle we're also given that this is a"}, {"start": 8.96, "duration": 6.799, "text": "rectangle and that ABC which is this"}, {"start": 12.519, "duration": 5.801, "text": "triangle is congruent to a e c which is"}, {"start": 15.759, "duration": 4.641, "text": "this triangle up here so these two are"}, {"start": 18.32, "duration": 3.64, "text": "congruent as well if you want to try it"}, {"start": 20.4, "duration": 5.28, "text": "on your own pause it right now because"}, {"start": 21.96, "duration": 5.319, "text": "I'm going to solve it in 3 21 so how do"}, {"start": 25.68, "duration": 3.48, "text": "we approach this I'm just going to go"}, {"start": 27.279, "duration": 3.881, "text": "freestyle and try to find everything I"}, {"start": 29.16, "duration": 3.96, "text": "can first thing I'm noticing is this"}, {"start": 31.16, "duration": 5.079, "text": "right triangle so we can make this side"}, {"start": 33.12, "duration": 7.08, "text": "C and do Pythagorean theorem we have 3^2"}, {"start": 36.239, "duration": 6.401, "text": "+ 4^2 is equal to c^ 2 32 is equal to 9"}, {"start": 40.2, "duration": 5.92, "text": "4 2 is equal to 16 and then we can bring"}, {"start": 42.64, "duration": 6.399, "text": "down the c^ 2 9 + 16 is equal to 25 so"}, {"start": 46.12, "duration": 5.0, "text": "we have 25 is equal to c^2 to solve for"}, {"start": 49.039, "duration": 5.441, "text": "C we can square root both sides square >"}, {"start": 51.12, "duration": 5.04, "text": "of 25 is 5 and square < TK of c^2 is C"}, {"start": 54.48, "duration": 3.239, "text": "so now we figured out that this C is"}, {"start": 56.16, "duration": 3.399, "text": "equal to 5 and we can give it a label of"}, {"start": 57.719, "duration": 3.64, "text": "meters to match these other ones what"}, {"start": 59.559, "duration": 3.721, "text": "should we do do next okay I know what to"}, {"start": 61.359, "duration": 4.041, "text": "do next in a rectangle opposite sides"}, {"start": 63.28, "duration": 5.96, "text": "are congruent so this side is also equal"}, {"start": 65.4, "duration": 6.2, "text": "to four ABC is congruent to a e c this"}, {"start": 69.24, "duration": 4.08, "text": "order of these letters is very important"}, {"start": 71.6, "duration": 3.76, "text": "since the first two letters of this are"}, {"start": 73.32, "duration": 5.119, "text": "ab and the first two letters of this are"}, {"start": 75.36, "duration": 5.92, "text": "AE we know side AB is congruent to side"}, {"start": 78.439, "duration": 6.201, "text": "AE so we know that this AB is equal to"}, {"start": 81.28, "duration": 5.04, "text": "AE which means we can label that as 4 M"}, {"start": 84.64, "duration": 3.96, "text": "and we also know since the second letter"}, {"start": 86.32, "duration": 5.0, "text": "is B and this second letter is e that"}, {"start": 88.6, "duration": 4.479, "text": "angle B is equal tole angle e so since"}, {"start": 91.32, "duration": 4.119, "text": "angle B is a right angle that means this"}, {"start": 93.079, "duration": 3.801, "text": "angle e is also a right angle these"}, {"start": 95.439, "duration": 3.921, "text": "congruent triangles gave us quite a bit"}, {"start": 96.88, "duration": 4.68, "text": "of info next let's focus on these two"}, {"start": 99.36, "duration": 3.88, "text": "triangles this angle and this angle are"}, {"start": 101.56, "duration": 4.72, "text": "both congruent because they're vertical"}, {"start": 103.24, "duration": 5.4, "text": "angles and now we know that angle AF is"}, {"start": 106.28, "duration": 4.28, "text": "congruent to cfd that's what we just"}, {"start": 108.64, "duration": 4.4, "text": "found with the vertical angles we know"}, {"start": 110.56, "duration": 4.919, "text": "that angle e is equal to angle D because"}, {"start": 113.04, "duration": 4.32, "text": "they're both right and we know that AE"}, {"start": 115.479, "duration": 4.161, "text": "is congruent to CD because they're both"}, {"start": 117.36, "duration": 4.039, "text": "equal to four therefore we can conclude"}, {"start": 119.64, "duration": 3.439, "text": "that these two triangles are congruent"}, {"start": 121.399, "duration": 3.281, "text": "and that's using the angle angle side"}, {"start": 123.079, "duration": 4.4, "text": "theorem any two triangles where we have"}, {"start": 124.68, "duration": 5.4, "text": "a corresponding angle angle and a side"}, {"start": 127.479, "duration": 4.241, "text": "those two triangles have to be congruent"}, {"start": 130.08, "duration": 3.64, "text": "and now we have another congruent"}, {"start": 131.72, "duration": 3.92, "text": "statement so looking at these first two"}, {"start": 133.72, "duration": 4.36, "text": "letters and these first two letters we"}, {"start": 135.64, "duration": 5.959, "text": "know the segment AF is equal to segment"}, {"start": 138.08, "duration": 6.4, "text": "CF that means this AF is equal to this"}, {"start": 141.599, "duration": 5.36, "text": "CF and since we know CF is five that"}, {"start": 144.48, "duration": 4.36, "text": "means we know this is five and now we're"}, {"start": 146.959, "duration": 3.56, "text": "pretty much done let's get rid of this"}, {"start": 148.84, "duration": 4.64, "text": "stuff and if you look at our red"}, {"start": 150.519, "duration": 4.761, "text": "triangle we have a base and we have a"}, {"start": 153.48, "duration": 4.0, "text": "height and give me a second I'll make"}, {"start": 155.28, "duration": 3.879, "text": "this easier to see so there we go now we"}, {"start": 157.48, "duration": 3.56, "text": "can see the base is five and the height"}, {"start": 159.159, "duration": 4.521, "text": "is four and if it's easier we can flip"}, {"start": 161.04, "duration": 4.559, "text": "this so here's the base of five and the"}, {"start": 163.68, "duration": 4.24, "text": "height of four the area of a triangle is"}, {"start": 165.599, "duration": 4.64, "text": "equal to 1/2 base time height we can"}, {"start": 167.92, "duration": 5.52, "text": "plug in five for the base and plug in"}, {"start": 170.239, "duration": 5.36, "text": "four for the height and 1/2 * 5 * 4 is"}, {"start": 173.44, "duration": 4.6, "text": "equal to 10 so the area of this red"}, {"start": 175.599, "duration": 4.041, "text": "triangle is 10 m squar this is the"}, {"start": 178.04, "duration": 2.88, "text": "answer to our question let's put a box"}, {"start": 179.64, "duration": 2.679, "text": "around around it and here's the next"}, {"start": 180.92, "duration": 3.84, "text": "problem I'm going to work on I'll post"}, {"start": 182.319, "duration": 5.361, "text": "the video as soon as I finish how"}, {"start": 184.76, "duration": 2.92, "text": "exciting"}], "title": "Geometry Challenge from Social Media", "description": "I hope you guys like this one!", "lengthSeconds": 186, "width": 640, "height": 360}
{"video_id": "ZnnARgtwgv4", "transcript": [{"start": 0.0, "duration": 3.899, "text": "hey guys so let's solve for x in this"}, {"start": 1.8, "duration": 3.78, "text": "equation it has an absolute value inside"}, {"start": 3.899, "duration": 3.0, "text": "of an absolute value this should be fun"}, {"start": 5.58, "duration": 2.76, "text": "the way to get rid of this outer"}, {"start": 6.899, "duration": 4.32, "text": "absolute value is we can take the stuff"}, {"start": 8.34, "duration": 4.5, "text": "inside and set it equal to 16 because if"}, {"start": 11.219, "duration": 3.361, "text": "all this stuff inside is equal to 16"}, {"start": 12.84, "duration": 3.359, "text": "then this will be a true statement we"}, {"start": 14.58, "duration": 3.959, "text": "can also take this inside stuff and set"}, {"start": 16.199, "duration": 4.261, "text": "it equal to negative 16. because if this"}, {"start": 18.539, "duration": 3.601, "text": "stuff inside here is negative 16 this"}, {"start": 20.46, "duration": 3.36, "text": "will also be true so now we have two"}, {"start": 22.14, "duration": 3.66, "text": "equations but we've managed to get rid"}, {"start": 23.82, "duration": 3.539, "text": "of that outer absolute value that's how"}, {"start": 25.8, "duration": 3.299, "text": "we get rid of absolute value signs we"}, {"start": 27.359, "duration": 3.66, "text": "take whatever was inside of them and set"}, {"start": 29.099, "duration": 3.48, "text": "equal to the positive or the negative of"}, {"start": 31.019, "duration": 3.361, "text": "what it's equal to so to get this alone"}, {"start": 32.579, "duration": 3.061, "text": "I'm going to subtract 3x from both sides"}, {"start": 34.38, "duration": 2.58, "text": "so left hand side is just going to be"}, {"start": 35.64, "duration": 2.88, "text": "negative absolute value of x and this"}, {"start": 36.96, "duration": 3.18, "text": "right hand side is going to be 16 minus"}, {"start": 38.52, "duration": 3.3, "text": "3x then I'm going to divide both sides"}, {"start": 40.14, "duration": 3.12, "text": "by negative 1. left hand side is just"}, {"start": 41.82, "duration": 4.38, "text": "the absolute value of x right hand side"}, {"start": 43.26, "duration": 4.26, "text": "will be negative 16 plus 3x let's do the"}, {"start": 46.2, "duration": 2.58, "text": "same thing to this one down here let's"}, {"start": 47.52, "duration": 2.58, "text": "subtract 3x from both sides the"}, {"start": 48.78, "duration": 2.759, "text": "left-hand side is negative absolute"}, {"start": 50.1, "duration": 3.6, "text": "value of x equals and this will be"}, {"start": 51.539, "duration": 3.601, "text": "negative 16 minus 3x divide both sides"}, {"start": 53.7, "duration": 3.359, "text": "by negative 1 and we get absolute value"}, {"start": 55.14, "duration": 4.079, "text": "of x on this side on this side we get 16"}, {"start": 57.059, "duration": 3.901, "text": "plus 3x so we're going to take both of"}, {"start": 59.219, "duration": 3.48, "text": "these equations and split each of them"}, {"start": 60.96, "duration": 3.36, "text": "up into two different equations take"}, {"start": 62.699, "duration": 3.661, "text": "this inside X and we're going to set it"}, {"start": 64.32, "duration": 3.54, "text": "equal to this negative 16 plus 3x we're"}, {"start": 66.36, "duration": 3.119, "text": "also going to take this inside X and set"}, {"start": 67.86, "duration": 4.02, "text": "it equal to the negative of this which"}, {"start": 69.479, "duration": 3.96, "text": "would be positive 16 minus 3x same thing"}, {"start": 71.88, "duration": 3.779, "text": "here we take the X out and set it equal"}, {"start": 73.439, "duration": 3.481, "text": "to that 16 plus 3x and we take the X out"}, {"start": 75.659, "duration": 3.481, "text": "and set equal to the negative of that"}, {"start": 76.92, "duration": 3.96, "text": "which is negative 16 minus 3x for both"}, {"start": 79.14, "duration": 3.42, "text": "these two equations let's subtract 3x"}, {"start": 80.88, "duration": 3.18, "text": "from both sides so the left-hand side is"}, {"start": 82.56, "duration": 3.12, "text": "going to be negative 2X and then for"}, {"start": 84.06, "duration": 3.239, "text": "this side we just bring down the 16 or"}, {"start": 85.68, "duration": 3.06, "text": "bring down the negative 16. divide"}, {"start": 87.299, "duration": 3.601, "text": "everything by negative 2 and this one"}, {"start": 88.74, "duration": 4.32, "text": "becomes x equals 8 and this one becomes"}, {"start": 90.9, "duration": 3.719, "text": "x equals negative 8. and on this side we"}, {"start": 93.06, "duration": 4.379, "text": "can add 3x to both sides of both these"}, {"start": 94.619, "duration": 4.5, "text": "equations X Plus 3x is equal to 4X these"}, {"start": 97.439, "duration": 3.301, "text": "cancel so we bring down the 16 and these"}, {"start": 99.119, "duration": 3.36, "text": "cancel so we bring down the negative 16."}, {"start": 100.74, "duration": 4.5, "text": "divide everything by 4 and we get x"}, {"start": 102.479, "duration": 4.32, "text": "equals 4 or x equals negative 4. now we"}, {"start": 105.24, "duration": 2.76, "text": "have four possible solutions to try"}, {"start": 106.799, "duration": 3.061, "text": "let's get rid of all this work let's"}, {"start": 108.0, "duration": 3.18, "text": "make these larger and then bring them up"}, {"start": 109.86, "duration": 4.14, "text": "so now I'm going to plug each of these"}, {"start": 111.18, "duration": 4.979, "text": "values in for X the H the 4 the negative"}, {"start": 114.0, "duration": 4.5, "text": "8 and the negative 4. 3 times 8 is 24"}, {"start": 116.159, "duration": 4.621, "text": "absolute value of 8 is 8 24 minus eight"}, {"start": 118.5, "duration": 4.619, "text": "is sixteen absolute value of 16 is equal"}, {"start": 120.78, "duration": 4.92, "text": "to 16 so this one's good 3 times 4 is 12"}, {"start": 123.119, "duration": 4.32, "text": "absolute value 4 is 4 12 minus 4 is 8"}, {"start": 125.7, "duration": 3.479, "text": "absolute value of 8 is 8 8 does not"}, {"start": 127.439, "duration": 3.841, "text": "equal 16 this one's no good 3 times"}, {"start": 129.179, "duration": 3.78, "text": "negative 8 is negative 24 absolute value"}, {"start": 131.28, "duration": 4.08, "text": "of negative 8 is positive eight negative"}, {"start": 132.959, "duration": 4.621, "text": "24 minus 8 is negative 32. absolute"}, {"start": 135.36, "duration": 4.379, "text": "value of negative 32 is positive 32 32"}, {"start": 137.58, "duration": 3.659, "text": "does not equal 16. 3 times negative 4 is"}, {"start": 139.739, "duration": 3.421, "text": "negative 12 absolute value of negative 4"}, {"start": 141.239, "duration": 3.421, "text": "is equal to four negative 12 minus 4 is"}, {"start": 143.16, "duration": 3.84, "text": "negative 16 absolute value of negative"}, {"start": 144.66, "duration": 4.38, "text": "16 is positive 16. 16 16 does equal 16"}, {"start": 147.0, "duration": 3.78, "text": "so x minus 4 is another solution so"}, {"start": 149.04, "duration": 3.9, "text": "these are the two solutions and here's"}, {"start": 150.78, "duration": 3.06, "text": "all the work again how exciting if you"}, {"start": 152.94, "duration": 2.879, "text": "guys want to try more of these I have"}, {"start": 153.84, "duration": 3.479, "text": "this page on andymath.com the link is"}, {"start": 155.819, "duration": 2.521, "text": "down below in the description check it"}, {"start": 157.319, "duration": 2.461, "text": "out and let me know if you have any"}, {"start": 158.34, "duration": 3.92, "text": "other questions I'll happily take"}, {"start": 159.78, "duration": 2.48, "text": "requests"}], "title": "Absolute Value Challenge Problem", "description": "https://andymath.com/absolute-value-equations/", "lengthSeconds": 160, "width": 640, "height": 360}
{"video_id": "NCy7xEziGVw", "transcript": [{"start": 0.0, "duration": 4.35, "text": "hey guys now we're going to add up some"}, {"start": 2.04, "duration": 4.14, "text": "vectors the first one is asking for it"}, {"start": 4.35, "duration": 4.53, "text": "in component form this right here is"}, {"start": 6.18, "duration": 4.95, "text": "component form if three is and four J's"}, {"start": 8.88, "duration": 4.589, "text": "so when adding vectors all you do is"}, {"start": 11.13, "duration": 7.83, "text": "just add the corresponding component so"}, {"start": 13.469, "duration": 9.3, "text": "3 plus 7 is 10 I and negative 4 plus 3"}, {"start": 18.96, "duration": 11.52, "text": "is negative 1 and there's the answer to"}, {"start": 22.769, "duration": 10.321, "text": "a now if we want to show it as a"}, {"start": 30.48, "duration": 4.65, "text": "magnitude in a direction the way that"}, {"start": 33.09, "duration": 5.399, "text": "you find the magnitude is you do the"}, {"start": 35.13, "duration": 5.749, "text": "square root of this squared plus this"}, {"start": 38.489, "duration": 2.39, "text": "squared"}, {"start": 60.51, "duration": 8.7, "text": "and that's the magnitude square root of"}, {"start": 62.67, "duration": 11.78, "text": "101 then to find the direction so you do"}, {"start": 69.21, "duration": 5.24, "text": "tangent inverse of negative 1 over 10"}, {"start": 76.34, "duration": 7.54, "text": "which is equal to negative five point"}, {"start": 81.03, "duration": 4.259, "text": "seven degrees and then it right after"}, {"start": 83.88, "duration": 6.029, "text": "you do this you want to check your"}, {"start": 85.289, "duration": 8.94, "text": "quadrant so 10 i minus J is down here so"}, {"start": 89.909, "duration": 10.801, "text": "negative 5.7 makes sense so then this"}, {"start": 94.229, "duration": 15.481, "text": "would be written as agne tude equals"}, {"start": 100.71, "duration": 12.15, "text": "square root of 101 and the direction is"}, {"start": 109.71, "duration": 4.499, "text": "negative five point seven degrees that's"}, {"start": 112.86, "duration": 3.92, "text": "it hope that helped talk to you later"}, {"start": 114.209, "duration": 2.571, "text": "bye"}], "title": "Adding Vectors", "description": "Vector Addition GREAT EXAMPLE! For more math made easy visit andymath.com.\n\nSubscribe here: https://www.youtube.com/channel/UC6KhU3AMLHC-qv7tj0ZJduw?sub_confirmation=1", "lengthSeconds": 115, "width": 640, "height": 360}
{"video_id": "88SgEGMkXtQ", "transcript": [{"start": 0.0, "duration": 3.159, "text": "hey guys so we have this video of a"}, {"start": 1.12, "duration": 4.599, "text": "Pikachu Professor making a calculus"}, {"start": 3.159, "duration": 2.56, "text": "problem for their"}, {"start": 7.28, "duration": 3.56, "text": "students let's see if we can solve it"}, {"start": 9.48, "duration": 3.239, "text": "I'm pretty sure this is the equation"}, {"start": 10.84, "duration": 4.679, "text": "right here this is an integral from 2 Pi"}, {"start": 12.719, "duration": 3.961, "text": "to 10 of a whole bunch of trig stuff DX"}, {"start": 15.519, "duration": 2.52, "text": "if you want to try this on your own"}, {"start": 16.68, "duration": 4.2, "text": "pause it right now because I'm going to"}, {"start": 18.039, "duration": 4.32, "text": "solve it in three two one first let's"}, {"start": 20.88, "duration": 3.559, "text": "copy it down and let's give ourselves"}, {"start": 22.359, "duration": 4.481, "text": "some space I want to pull some stuff out"}, {"start": 24.439, "duration": 4.92, "text": "of this one right here on top I want to"}, {"start": 26.84, "duration": 4.359, "text": "pull out this cosine Cub 3x on bottom"}, {"start": 29.359, "duration": 3.681, "text": "the six and the eight have a two in"}, {"start": 31.199, "duration": 4.081, "text": "common so I want to pull that two out"}, {"start": 33.04, "duration": 4.16, "text": "which will turn the six into a three and"}, {"start": 35.28, "duration": 3.72, "text": "the eight into a four and let's also"}, {"start": 37.2, "duration": 4.039, "text": "pull a negative 1 out of both of these"}, {"start": 39.0, "duration": 4.12, "text": "so in the end we're factoring out a -2"}, {"start": 41.239, "duration": 3.64, "text": "now I'm ready to focus on this middle"}, {"start": 43.12, "duration": 3.84, "text": "piece right here let's clean it up a"}, {"start": 44.879, "duration": 3.921, "text": "little bit this looks very familiar to"}, {"start": 46.96, "duration": 3.72, "text": "me and the fact some of the inputs for"}, {"start": 48.8, "duration": 3.88, "text": "these are 3x I'm thinking we should"}, {"start": 50.68, "duration": 3.399, "text": "check the triple angle formulas I don't"}, {"start": 52.68, "duration": 3.44, "text": "have them memorized but I found them"}, {"start": 54.079, "duration": 4.521, "text": "here they are right here and this is"}, {"start": 56.12, "duration": 4.959, "text": "identical to this one right here so all"}, {"start": 58.6, "duration": 4.88, "text": "of this is equal to seeking of 3x and"}, {"start": 61.079, "duration": 4.4, "text": "secant means 1/ cosine so we can rewrite"}, {"start": 63.48, "duration": 4.84, "text": "this as 1/ cosine"}, {"start": 65.479, "duration": 5.0, "text": "3x now let's look at this cosine cubed"}, {"start": 68.32, "duration": 3.72, "text": "of 3x that Cube means that there are"}, {"start": 70.479, "duration": 3.361, "text": "three of them being multiplied by each"}, {"start": 72.04, "duration": 3.52, "text": "other and I'd like to take one of them"}, {"start": 73.84, "duration": 3.8, "text": "and put it over here and then for these"}, {"start": 75.56, "duration": 4.28, "text": "two let's recombine them into a cosine"}, {"start": 77.64, "duration": 4.76, "text": "squar now we have a cosine 3x over"}, {"start": 79.84, "duration": 4.16, "text": "cosine 3x they cancel to give us one and"}, {"start": 82.4, "duration": 3.32, "text": "then we're just multiplying by this one"}, {"start": 84.0, "duration": 3.479, "text": "so all of this can disappear and then"}, {"start": 85.72, "duration": 3.64, "text": "let's smush everything together right"}, {"start": 87.479, "duration": 3.32, "text": "now we're subtracting this second term"}, {"start": 89.36, "duration": 3.32, "text": "another way can think of it take the"}, {"start": 90.799, "duration": 4.201, "text": "subtraction symbol make the denominator"}, {"start": 92.68, "duration": 4.52, "text": "negative and add the two terms and now"}, {"start": 95.0, "duration": 3.96, "text": "both of our terms are being divided by"}, {"start": 97.2, "duration": 4.279, "text": "-2 that means they're both being"}, {"start": 98.96, "duration": 4.839, "text": "multiplied by - one2 so let's pull out"}, {"start": 101.479, "duration": 5.801, "text": "that - one2 and now we're just left with"}, {"start": 103.799, "duration": 6.121, "text": "cosine 2 3x+ sin^2 3x well this is a"}, {"start": 107.28, "duration": 5.0, "text": "Pythagorean identity sin s of theta plus"}, {"start": 109.92, "duration": 5.799, "text": "cosine sare thet is equal to 1 Theta can"}, {"start": 112.28, "duration": 6.04, "text": "be anything including 3x so sin s 3x +"}, {"start": 115.719, "duration": 4.281, "text": "cosine 3x is just going to be equal to 1"}, {"start": 118.32, "duration": 4.079, "text": "so using the Pythagorean identities we"}, {"start": 120.0, "duration": 3.96, "text": "just changed all of this stuff into one"}, {"start": 122.399, "duration": 3.521, "text": "and now we can smush everything together"}, {"start": 123.96, "duration": 4.2, "text": "and we're going to take this integral"}, {"start": 125.92, "duration": 4.72, "text": "the integral from 2 Pi to 10 of 1 that's"}, {"start": 128.16, "duration": 4.4, "text": "just going to be X evaluated from 2 Pi"}, {"start": 130.64, "duration": 3.36, "text": "to 10 and then we still have the - 1/2"}, {"start": 132.56, "duration": 3.28, "text": "in front so all this means is we're"}, {"start": 134.0, "duration": 5.12, "text": "going to take our - 1/2 and multiply it"}, {"start": 135.84, "duration": 5.52, "text": "by x - x where the first X is equal to"}, {"start": 139.12, "duration": 4.36, "text": "this top number 10 and this x is equal"}, {"start": 141.36, "duration": 6.519, "text": "to the bottom number 2 pi and now we're"}, {"start": 143.48, "duration": 7.64, "text": "ready to distribute the -22 * 10 is -5"}, {"start": 147.879, "duration": 5.281, "text": "and -2 * -2 piun is pi this is the"}, {"start": 151.12, "duration": 6.479, "text": "answer to Pikachu's challenge question"}, {"start": 153.16, "duration": 4.439, "text": "let's put a box around it how exciting"}], "title": "Tricky Calculus Integral", "description": "https://andymath.com/trigonometric-identities/\nhttps://andymath.com/indefinite-integrals-trigonometric-functions/", "lengthSeconds": 156, "width": 640, "height": 360}
{"video_id": "yXBIShDu-Yk", "transcript": [{"start": 0.0, "duration": 2.94, "text": "what is the area and perimeter of this"}, {"start": 1.5, "duration": 3.66, "text": "rectangle there is going to be base"}, {"start": 2.94, "duration": 4.02, "text": "times height so we can just say a equals"}, {"start": 5.16, "duration": 3.479, "text": "then the base is this right here times"}, {"start": 6.96, "duration": 3.9, "text": "the height which is this right here and"}, {"start": 8.639, "duration": 4.38, "text": "we can see that this is an A minus B"}, {"start": 10.86, "duration": 3.54, "text": "times an A plus b that is a difference"}, {"start": 13.019, "duration": 4.321, "text": "of two squares so it's just going to be"}, {"start": 14.4, "duration": 5.7, "text": "equal to a squared minus B squared or in"}, {"start": 17.34, "duration": 6.779, "text": "other words 105 squared minus 2 root"}, {"start": 20.1, "duration": 6.96, "text": "2739 squared 105 squared is 11 25 minus"}, {"start": 24.119, "duration": 7.441, "text": "and then 2 squared is 4 and root 2739"}, {"start": 27.06, "duration": 6.659, "text": "squared is 2739 and 1125 minus 10 956 is"}, {"start": 31.56, "duration": 3.839, "text": "69 nice and next let's find the"}, {"start": 33.719, "duration": 4.261, "text": "perimeter we're going to do 2 of this"}, {"start": 35.399, "duration": 4.201, "text": "side here plus 2 of this side here these"}, {"start": 37.98, "duration": 4.5, "text": "twos can distribute which would give us"}, {"start": 39.6, "duration": 6.119, "text": "210 minus 4 root two seven three nine"}, {"start": 42.48, "duration": 5.7, "text": "plus two ten plus four root two seven"}, {"start": 45.719, "duration": 4.141, "text": "three nine these two match but one's"}, {"start": 48.18, "duration": 3.24, "text": "positive one's negative so they cancel"}, {"start": 49.86, "duration": 3.78, "text": "each other out so we're just left with"}, {"start": 51.42, "duration": 4.86, "text": "the perimeter equal to 210 plus 210"}, {"start": 53.64, "duration": 5.96, "text": "which is 420. so the area is 69 and the"}, {"start": 56.28, "duration": 3.32, "text": "parameter is 420. nice"}], "title": "Area and Perimeter Challenge", "description": "Collab with Geeklyhub", "lengthSeconds": 58, "width": 360, "height": 640}
{"video_id": "7X7aoSbR-Kw", "transcript": [{"start": 0.04, "duration": 3.56, "text": "hey guys this is one of the most"}, {"start": 1.48, "duration": 4.08, "text": "requested videos I've ever had and for"}, {"start": 3.6, "duration": 3.64, "text": "good reason this is such a cool problem"}, {"start": 5.56, "duration": 3.44, "text": "I did not expect it to have the outcome"}, {"start": 7.24, "duration": 3.56, "text": "that it had so the problem gives us a"}, {"start": 9.0, "duration": 3.599, "text": "semicircle right here and a quarter"}, {"start": 10.8, "duration": 3.44, "text": "Circle right here and then this red line"}, {"start": 12.599, "duration": 3.841, "text": "with a length of five that goes from the"}, {"start": 14.24, "duration": 3.64, "text": "vertex of this rectangle tangent to the"}, {"start": 16.44, "duration": 2.919, "text": "semicircle if you want to try this on"}, {"start": 17.88, "duration": 4.44, "text": "your own pause it right now cuz I'm"}, {"start": 19.359, "duration": 4.721, "text": "going to solve it in 3 2 1 so first up"}, {"start": 22.32, "duration": 4.039, "text": "let's assign a radius to this semi"}, {"start": 24.08, "duration": 4.279, "text": "circle a lot of people wanted to assign"}, {"start": 26.359, "duration": 4.0, "text": "this radius to match but there really is"}, {"start": 28.359, "duration": 3.72, "text": "not evidence that this radius is the"}, {"start": 30.359, "duration": 3.761, "text": "same as this radius so let's call it"}, {"start": 32.079, "duration": 3.761, "text": "something else like X using our method"}, {"start": 34.12, "duration": 3.52, "text": "we're not going to need this radius to"}, {"start": 35.84, "duration": 3.28, "text": "equal this radius and that right there"}, {"start": 37.64, "duration": 3.2, "text": "is one of my favorite things about this"}, {"start": 39.12, "duration": 3.56, "text": "problem we don't care if the radi are"}, {"start": 40.84, "duration": 3.32, "text": "the same so this last distance right"}, {"start": 42.68, "duration": 3.96, "text": "here will also be R because it's a"}, {"start": 44.16, "duration": 4.48, "text": "radius of the same semicircle and this"}, {"start": 46.64, "duration": 4.12, "text": "height will also be X because it's a"}, {"start": 48.64, "duration": 3.599, "text": "radius of this quarter Circle and now we"}, {"start": 50.76, "duration": 3.119, "text": "can come up with a formula for the area"}, {"start": 52.239, "duration": 3.441, "text": "of the rectangle the area of a rectangle"}, {"start": 53.879, "duration": 4.64, "text": "is base time height the base is going to"}, {"start": 55.68, "duration": 5.28, "text": "be equal to x + 2 R and the height is"}, {"start": 58.519, "duration": 2.441, "text": "going to be"}], "title": "Full Video \ud83d\udc46", "description": "This was a popular one. I wanted to share it with more people on shorts. I hope you love it!", "lengthSeconds": 59, "width": 270, "height": 480}
{"video_id": "y8R0Oqbtp9U", "transcript": [{"start": 0.0, "duration": 3.959, "text": "hey guys this looks like a fun one 2"}, {"start": 1.92, "duration": 4.26, "text": "squared of x equals square root of 2"}, {"start": 3.959, "duration": 3.9, "text": "plus X so for the very first step we can"}, {"start": 6.18, "duration": 3.419, "text": "do a domain restriction we know that"}, {"start": 7.859, "duration": 3.481, "text": "under these square roots we can't have a"}, {"start": 9.599, "duration": 3.721, "text": "negative number so that means that the X"}, {"start": 11.34, "duration": 3.959, "text": "has to be greater than or equal to zero"}, {"start": 13.32, "duration": 4.14, "text": "if x were less than zero this would be"}, {"start": 15.299, "duration": 3.961, "text": "undefined on the left hand side now we"}, {"start": 17.46, "duration": 2.88, "text": "could put a box around that so we're off"}, {"start": 19.26, "duration": 2.76, "text": "to a pretty good start we've already"}, {"start": 20.34, "duration": 3.359, "text": "gotten rid of all the negative numbers"}, {"start": 22.02, "duration": 3.54, "text": "next to get rid of both these square"}, {"start": 23.699, "duration": 3.601, "text": "root symbols let's Square both sides of"}, {"start": 25.56, "duration": 2.94, "text": "the equation on this right hand side the"}, {"start": 27.3, "duration": 3.899, "text": "square root and the square cancel"}, {"start": 28.5, "duration": 4.2, "text": "leaving us with Just 2 plus X on the"}, {"start": 31.199, "duration": 3.601, "text": "left hand side this Square distributes"}, {"start": 32.7, "duration": 4.26, "text": "to both terms 2 squared and the square"}, {"start": 34.8, "duration": 4.259, "text": "root of x squared 2 squared is equal to"}, {"start": 36.96, "duration": 4.2, "text": "4 the square root and the two cancel"}, {"start": 39.059, "duration": 4.081, "text": "each other out leaving us with just 4X"}, {"start": 41.16, "duration": 4.2, "text": "and we were able to do this step because"}, {"start": 43.14, "duration": 4.56, "text": "X is greater than or equal to zero next"}, {"start": 45.36, "duration": 5.039, "text": "we can subtract X from both sides 4X"}, {"start": 47.7, "duration": 4.98, "text": "minus one X is 3x these X's cancel"}, {"start": 50.399, "duration": 4.021, "text": "leaving us with just two we can divide"}, {"start": 52.68, "duration": 3.899, "text": "both sides by three and that leaves us"}, {"start": 54.42, "duration": 6.319, "text": "with x equal to two-thirds and that's"}, {"start": 56.579, "duration": 4.16, "text": "the answer to our question how exciting"}], "title": "Quick Square Root Challenge", "description": "I thought this was a fun one.", "lengthSeconds": 59, "width": 360, "height": 640}
{"video_id": "tATr4l63Y1Q", "transcript": [{"start": 0.0, "duration": 4.38, "text": "hey guys let's solve some trigonometric"}, {"start": 1.979, "duration": 4.32, "text": "equation it says sine of X Plus root 3"}, {"start": 4.38, "duration": 4.08, "text": "cosine of x equals zero so first I"}, {"start": 6.299, "duration": 3.72, "text": "copied it down this has a sine X and A"}, {"start": 8.46, "duration": 2.82, "text": "cosine X in it so this can be kind of"}, {"start": 10.019, "duration": 2.521, "text": "tricky to work with let me show you a"}, {"start": 11.28, "duration": 3.359, "text": "trick for this one if you divide"}, {"start": 12.54, "duration": 5.28, "text": "everything by cosine X sine X over"}, {"start": 14.639, "duration": 5.4, "text": "cosine X is equal to tangent X root 3"}, {"start": 17.82, "duration": 4.74, "text": "cosine X over cosine X is equal to just"}, {"start": 20.039, "duration": 4.801, "text": "root 3 and then 0 over cosine X is still"}, {"start": 22.56, "duration": 4.5, "text": "just zero then we can subtract root 3"}, {"start": 24.84, "duration": 4.92, "text": "from both sides and then we have tangent"}, {"start": 27.06, "duration": 4.26, "text": "of x equals negative root 3. so from"}, {"start": 29.76, "duration": 2.88, "text": "here we can use tangent inverse on our"}, {"start": 31.32, "duration": 3.12, "text": "calculator or we could use the unit"}, {"start": 32.64, "duration": 3.54, "text": "circle if we look at a unit circle we"}, {"start": 34.44, "duration": 5.16, "text": "can see these two values right here are"}, {"start": 36.18, "duration": 5.58, "text": "answers so the x equals 2 pi over 3 or 5"}, {"start": 39.6, "duration": 3.66, "text": "pi over 3. and so this is periodic you"}, {"start": 41.76, "duration": 3.6, "text": "can keep going on forever where we keep"}, {"start": 43.26, "duration": 5.94, "text": "adding Pi a better way to express this"}, {"start": 45.36, "duration": 5.219, "text": "would be 2 pi over 3 plus pi times n and"}, {"start": 49.2, "duration": 3.12, "text": "that is the answer to our question and"}, {"start": 50.579, "duration": 3.181, "text": "we can even click show answer and it"}, {"start": 52.32, "duration": 2.52, "text": "gives us the same answer right there if"}, {"start": 53.76, "duration": 3.119, "text": "you guys want to try more of these I"}, {"start": 54.84, "duration": 5.66, "text": "have this page on andymath.com link is"}, {"start": 56.879, "duration": 3.621, "text": "down in the description how exciting"}], "title": "How to solve this Trig Equation #shorts", "description": "https://andymath.com/solving-trigonometric-equations/\n\n#shorts", "lengthSeconds": 59, "width": 270, "height": 480}
{"video_id": "w1pPty1k2YA", "transcript": [{"start": 0.04, "duration": 3.52, "text": "hey guys this looks like a fun one it's"}, {"start": 1.599, "duration": 4.68, "text": "two equilateral triangles with a"}, {"start": 3.56, "duration": 4.48, "text": "distance of two between the Peaks and"}, {"start": 6.279, "duration": 3.4, "text": "through those two peaks it draws a"}, {"start": 8.04, "duration": 3.4, "text": "quarter Circle and it wants to know what"}, {"start": 9.679, "duration": 3.161, "text": "is the area of that quarter Circle if"}, {"start": 11.44, "duration": 3.6, "text": "you want to try it on your own pause it"}, {"start": 12.84, "duration": 4.599, "text": "right now cuz I'm going to solve it in 3"}, {"start": 15.04, "duration": 4.68, "text": "2 1 so the question is asking us to find"}, {"start": 17.439, "duration": 4.961, "text": "the area of this quarter Circle the area"}, {"start": 19.72, "duration": 4.479, "text": "of a quarter circle is equal to < R 2 /"}, {"start": 22.4, "duration": 3.52, "text": "4 so the only thing we got to do is"}, {"start": 24.199, "duration": 3.32, "text": "figure out what is the radius of this"}, {"start": 25.92, "duration": 3.56, "text": "quarter Circle I'm thinking let's"}, {"start": 27.519, "duration": 3.92, "text": "reflect the whole image down here so now"}, {"start": 29.48, "duration": 4.079, "text": "we have a semicircle and four of these"}, {"start": 31.439, "duration": 4.041, "text": "equilateral triangles in equilateral"}, {"start": 33.559, "duration": 4.32, "text": "triangles all the angles are equal to"}, {"start": 35.48, "duration": 4.52, "text": "60\u00b0 I'm specifically interested in this"}, {"start": 37.879, "duration": 3.84, "text": "angle right here and we can cut this 60"}, {"start": 40.0, "duration": 3.559, "text": "and half by bringing down this segment"}, {"start": 41.719, "duration": 4.241, "text": "right here and it cut this angle into"}, {"start": 43.559, "duration": 4.601, "text": "30\u00b0 and now let's focus on this 30"}, {"start": 45.96, "duration": 4.4, "text": "degree angle since the vertex is on the"}, {"start": 48.16, "duration": 4.039, "text": "circle this is called an inscribed angle"}, {"start": 50.36, "duration": 4.199, "text": "and inscribed angles are always half the"}, {"start": 52.199, "duration": 4.601, "text": "inscribed Arc so the measure of this Arc"}, {"start": 54.559, "duration": 3.96, "text": "right here is 60\u00b0 and if we shift the"}, {"start": 56.8, "duration": 3.759, "text": "angle up here it's now called a central"}, {"start": 58.519, "duration": 4.72, "text": "angle and that is equal to The subtended"}, {"start": 60.559, "duration": 4.24, "text": "Arc so this angle is 60\u00b0 and both of"}, {"start": 63.239, "duration": 3.401, "text": "these sides of the angle are equal to"}, {"start": 64.799, "duration": 3.761, "text": "the radius now let's focus on this"}, {"start": 66.64, "duration": 3.519, "text": "triangle right here since these two"}, {"start": 68.56, "duration": 3.879, "text": "sides are equal to each other it is an"}, {"start": 70.159, "duration": 4.441, "text": "isoceles triangle and every Isles"}, {"start": 72.439, "duration": 3.68, "text": "triangle the base angles are congruent"}, {"start": 74.6, "duration": 4.04, "text": "so both these angles are equal to each"}, {"start": 76.119, "duration": 4.401, "text": "other since every triangle has 180\u00b0 this"}, {"start": 78.64, "duration": 4.32, "text": "is only possible if both of those are"}, {"start": 80.52, "duration": 4.08, "text": "60\u00b0 and now since all three angles are"}, {"start": 82.96, "duration": 3.36, "text": "equal we know this is an equilateral"}, {"start": 84.6, "duration": 3.92, "text": "triangle so that means all the sides"}, {"start": 86.32, "duration": 3.68, "text": "have to be equal so both of these sides"}, {"start": 88.52, "duration": 3.08, "text": "are equal to two so now let's let's"}, {"start": 90.0, "duration": 3.68, "text": "bring back our original figure and let's"}, {"start": 91.6, "duration": 4.159, "text": "focus on the quarter Circle the radius"}, {"start": 93.68, "duration": 3.92, "text": "of this quarter circle is equal to 2 and"}, {"start": 95.759, "duration": 3.32, "text": "now we're ready to use our area formula"}, {"start": 97.6, "duration": 3.96, "text": "in the place of the radius let's plug in"}, {"start": 99.079, "duration": 4.4, "text": "two 2^ 2 is equal to four and these"}, {"start": 101.56, "duration": 4.4, "text": "fours will cancel each other out so the"}, {"start": 103.479, "duration": 3.96, "text": "area is equal to Pi square units and"}, {"start": 105.96, "duration": 3.24, "text": "this is the answer to our question let's"}, {"start": 107.439, "duration": 3.801, "text": "put a box around it the area of this"}, {"start": 109.2, "duration": 6.0, "text": "quarter circle is equal to Pi square"}, {"start": 111.24, "duration": 3.96, "text": "units how exciting"}], "title": "Fun Geometry Puzzle", "description": "Here's another Catriona Agg puzzle. I hope you guys like it!", "lengthSeconds": 113, "width": 640, "height": 360}
{"video_id": "RCg1xLuQu1c", "transcript": [{"start": 0.0, "duration": 5.1, "text": "hey guys it says no numbers are given"}, {"start": 2.399, "duration": 4.201, "text": "how many degrees is the red angle if you"}, {"start": 5.1, "duration": 2.88, "text": "want to try this on your own posit right"}, {"start": 6.6, "duration": 4.32, "text": "now because I'm going to solve it in"}, {"start": 7.98, "duration": 5.279, "text": "three two one let's focus on this"}, {"start": 10.92, "duration": 4.08, "text": "triangle and the semicircle anytime you"}, {"start": 13.259, "duration": 4.201, "text": "have a triangle inscribed in a circle"}, {"start": 15.0, "duration": 4.38, "text": "where one side is the diameter the"}, {"start": 17.46, "duration": 4.68, "text": "triangle will always be a right triangle"}, {"start": 19.38, "duration": 5.159, "text": "so this angle ACB has to be 90 degrees"}, {"start": 22.14, "duration": 4.32, "text": "and this is always true here's a nice"}, {"start": 24.539, "duration": 3.781, "text": "little visualization to show it if you"}, {"start": 26.46, "duration": 3.659, "text": "notice if we rotate the visual one side"}, {"start": 28.32, "duration": 3.959, "text": "will always stay horizontal and one side"}, {"start": 30.119, "duration": 3.901, "text": "always stays vertical that would only be"}, {"start": 32.279, "duration": 4.261, "text": "possible if it was a right triangle"}, {"start": 34.02, "duration": 4.8, "text": "since there's 180 degrees total in a"}, {"start": 36.54, "duration": 4.08, "text": "triangle these other two angles must add"}, {"start": 38.82, "duration": 3.239, "text": "to 90 degrees now once again that's"}, {"start": 40.62, "duration": 4.14, "text": "because we need all three of these"}, {"start": 42.059, "duration": 4.621, "text": "angles to add up to 180 degrees every"}, {"start": 44.76, "duration": 3.959, "text": "triangle has a special Point called the"}, {"start": 46.68, "duration": 4.14, "text": "in Center there are two very important"}, {"start": 48.719, "duration": 4.801, "text": "details about the in Center first it's"}, {"start": 50.82, "duration": 4.379, "text": "the center of the inscribed Circle and"}, {"start": 53.52, "duration": 4.019, "text": "second it's the intersection of the"}, {"start": 55.199, "duration": 4.741, "text": "angle bisectors so for this figure right"}, {"start": 57.539, "duration": 4.921, "text": "here Point m is the in Center it is the"}, {"start": 59.94, "duration": 4.619, "text": "the center of the inscribed Circle and"}, {"start": 62.46, "duration": 3.78, "text": "since these two lines intersect at that"}, {"start": 64.559, "duration": 4.021, "text": "point that means that these lines are"}, {"start": 66.24, "duration": 4.32, "text": "bisecting these two angles to make it"}, {"start": 68.58, "duration": 4.44, "text": "easier let's number these angles one two"}, {"start": 70.56, "duration": 4.739, "text": "and three so the measure of angle two"}, {"start": 73.02, "duration": 5.04, "text": "right here has to be half of the measure"}, {"start": 75.299, "duration": 4.981, "text": "of angle BAC now the same thing for"}, {"start": 78.06, "duration": 3.96, "text": "angle three it's cutting angle ABC in"}, {"start": 80.28, "duration": 3.6, "text": "half so we'd say the measure of angle"}, {"start": 82.02, "duration": 4.08, "text": "three is equal to measure of angle ABC"}, {"start": 83.88, "duration": 3.599, "text": "divided by two let's look at this"}, {"start": 86.1, "duration": 3.48, "text": "equation again and let's divide"}, {"start": 87.479, "duration": 4.441, "text": "everything by two the measure of angle"}, {"start": 89.58, "duration": 4.32, "text": "BAC divided by two that is the same"}, {"start": 91.92, "duration": 3.72, "text": "thing as the measure of angle two and"}, {"start": 93.9, "duration": 4.5, "text": "then we're going to add to that ABC"}, {"start": 95.64, "duration": 4.619, "text": "divided by 2 which is angle three and"}, {"start": 98.4, "duration": 4.679, "text": "that's going to be equal to 90 divided"}, {"start": 100.259, "duration": 5.161, "text": "by 2 which is 45 degrees and I'll smush"}, {"start": 103.079, "duration": 4.561, "text": "all this together and add angle one to"}, {"start": 105.42, "duration": 3.9, "text": "both sides of the equation angles one"}, {"start": 107.64, "duration": 3.18, "text": "two and three are the three angles of"}, {"start": 109.32, "duration": 4.079, "text": "this triangle so we know they're all"}, {"start": 110.82, "duration": 5.159, "text": "going to add up to 180 degrees and if we"}, {"start": 113.399, "duration": 4.981, "text": "subtract 45 degrees from both sides 180"}, {"start": 115.979, "duration": 3.841, "text": "degrees minus 45 degrees is 135 degrees"}, {"start": 118.38, "duration": 3.84, "text": "and on the right hand side we just have"}, {"start": 119.82, "duration": 5.579, "text": "the measure of angle one so the measure"}, {"start": 122.22, "duration": 4.62, "text": "of this red angle is 135 degrees and"}, {"start": 125.399, "duration": 5.661, "text": "that is the answer to the question let's"}, {"start": 126.84, "duration": 4.22, "text": "put a box around it how exciting"}], "title": "No Number Geometry Challenge", "description": "I thought this was a fun one. I hope everyone like it.", "lengthSeconds": 130, "width": 640, "height": 360}
{"video_id": "JFHymi1hdAA", "transcript": [{"start": 0.0, "duration": 3.679, "text": "hey guys this looks like a fun one it's"}, {"start": 1.48, "duration": 4.56, "text": "made up of four squares and it wants to"}, {"start": 3.679, "duration": 4.24, "text": "know what's this shaded area if you want"}, {"start": 6.04, "duration": 5.479, "text": "to try it on your own pause it right now"}, {"start": 7.919, "duration": 5.6, "text": "cuz I'm going to solve it in 3 2 1 first"}, {"start": 11.519, "duration": 3.761, "text": "step I want to focus on this triangle"}, {"start": 13.519, "duration": 3.441, "text": "right here so let's make it blue and"}, {"start": 15.28, "duration": 4.079, "text": "then I want to zoom in on this region"}, {"start": 16.96, "duration": 4.72, "text": "right here looking at this blue triangle"}, {"start": 19.359, "duration": 4.401, "text": "this angle is a right angle and that's"}, {"start": 21.68, "duration": 4.16, "text": "because we know all of these are squares"}, {"start": 23.76, "duration": 3.56, "text": "and then this angle right here is also a"}, {"start": 25.84, "duration": 4.199, "text": "right angle and that's because it's one"}, {"start": 27.32, "duration": 4.88, "text": "of the angles of this larger Square next"}, {"start": 30.039, "duration": 4.121, "text": "if you ever have two parallel lines and"}, {"start": 32.2, "duration": 3.84, "text": "a line cutting through them it will"}, {"start": 34.16, "duration": 3.8, "text": "always be true that these two angles are"}, {"start": 36.04, "duration": 3.6, "text": "equal to each other these are called"}, {"start": 37.96, "duration": 4.079, "text": "alternate interior angles and this is"}, {"start": 39.64, "duration": 4.56, "text": "the alternate interior angle theorem we"}, {"start": 42.039, "duration": 4.0, "text": "have this exact situation going on here"}, {"start": 44.2, "duration": 3.64, "text": "these two lines are parallel and this"}, {"start": 46.039, "duration": 3.721, "text": "line is cutting through them that means"}, {"start": 47.84, "duration": 3.64, "text": "these two angles are equal to each other"}, {"start": 49.76, "duration": 4.4, "text": "if we look at this pink triangle and"}, {"start": 51.48, "duration": 4.8, "text": "this blue triangle these have two pairs"}, {"start": 54.16, "duration": 3.8, "text": "of corresponding congruent angles that"}, {"start": 56.28, "duration": 3.239, "text": "means the third angles must also be"}, {"start": 57.96, "duration": 3.239, "text": "congruent and now let's pull out these"}, {"start": 59.519, "duration": 3.481, "text": "two triangles and we'll look at those"}, {"start": 61.199, "duration": 3.92, "text": "later next I want to focus on this"}, {"start": 63.0, "duration": 3.6, "text": "square with an area of five that means"}, {"start": 65.119, "duration": 3.521, "text": "that the side lengths are each going to"}, {"start": 66.6, "duration": 4.0, "text": "be square OT of five and if we look at"}, {"start": 68.64, "duration": 3.4, "text": "this square with an area of 20 the side"}, {"start": 70.6, "duration": 4.08, "text": "lengths are each going to be square < TK"}, {"start": 72.04, "duration": 4.84, "text": "of 20 next I want to look at this square"}, {"start": 74.68, "duration": 4.799, "text": "right here the base of the square is <"}, {"start": 76.88, "duration": 5.239, "text": "tk2 +un 5 that means that the side is"}, {"start": 79.479, "duration": 4.561, "text": "also going to beunk 20 +un 5 and let's"}, {"start": 82.119, "duration": 3.601, "text": "move this < tk2 label from the bottom to"}, {"start": 84.04, "duration": 3.2, "text": "the top of this square and now we're"}, {"start": 85.72, "duration": 3.039, "text": "ready to mess around with this blue"}, {"start": 87.24, "duration": 3.8, "text": "right triangle if we label the"}, {"start": 88.759, "duration": 5.881, "text": "hypotenuse C we can do the Pythagorean"}, {"start": 91.04, "duration": 5.96, "text": "theorem we can do < TK 20 2 +un 5^2 is"}, {"start": 94.64, "duration": 3.839, "text": "equal to c^2 and for this first term the"}, {"start": 97.0, "duration": 3.6, "text": "square root and the square will cancel"}, {"start": 98.479, "duration": 4.801, "text": "each other out to give us 20 and same"}, {"start": 100.6, "duration": 4.92, "text": "thing here otk 5^ 2 is going to be 5 20"}, {"start": 103.28, "duration": 4.32, "text": "+ 5 is equal to 25 and we can take the"}, {"start": 105.52, "duration": 4.52, "text": "square root of both sides and we end up"}, {"start": 107.6, "duration": 4.479, "text": "with 5 is equal to C so let's update"}, {"start": 110.04, "duration": 3.28, "text": "this C up here to be five and now that"}, {"start": 112.079, "duration": 3.4, "text": "we have the measurements of the blue"}, {"start": 113.32, "duration": 4.079, "text": "triangle let's update it down here next"}, {"start": 115.479, "duration": 3.881, "text": "I want to focus on the top of this"}, {"start": 117.399, "duration": 4.441, "text": "Square it's going to be equal to < TK 5"}, {"start": 119.36, "duration": 4.88, "text": "+un 20 this is the exact same thing as"}, {"start": 121.84, "duration": 5.959, "text": "the hypotenuse of this pink triangle it"}, {"start": 124.24, "duration": 6.56, "text": "will also beunk 5 + < tk2 and < tk2 can"}, {"start": 127.799, "duration": 6.361, "text": "be split up as < TK 4 * 5 and then this"}, {"start": 130.8, "duration": 5.6, "text": "can be split up into < TK 4 * < TK 5 and"}, {"start": 134.16, "duration": 5.0, "text": "< TK 4 is equal to 2 so it ends up being"}, {"start": 136.4, "duration": 5.6, "text": "2un 5 so now we can simplify this side"}, {"start": 139.16, "duration": 5.52, "text": "length if we add up < TK 5 + 2un 5 it"}, {"start": 142.0, "duration": 4.92, "text": "ends up being 3un 5 and now we're almost"}, {"start": 144.68, "duration": 5.48, "text": "done they want to know what's the Shaded"}, {"start": 146.92, "duration": 5.84, "text": "area we called this length down here s"}, {"start": 150.16, "duration": 5.12, "text": "that is the same as the base of our pink"}, {"start": 152.76, "duration": 4.28, "text": "Square so once we know s we know the"}, {"start": 155.28, "duration": 4.36, "text": "answer to our question the area of this"}, {"start": 157.04, "duration": 4.72, "text": "pink square is going to be s s so let's"}, {"start": 159.64, "duration": 3.879, "text": "solve for S I want to give this pink"}, {"start": 161.76, "duration": 3.479, "text": "triangle the same orientation as the"}, {"start": 163.519, "duration": 3.401, "text": "blue triangle all three of these"}, {"start": 165.239, "duration": 3.401, "text": "corresponding pairs of angles are"}, {"start": 166.92, "duration": 3.84, "text": "congruent which means these two"}, {"start": 168.64, "duration": 3.44, "text": "triangles are similar that means we can"}, {"start": 170.76, "duration": 4.52, "text": "set up a proportion for their"}, {"start": 172.08, "duration": 6.0, "text": "corresponding sides we have 3un 5 over 5"}, {"start": 175.28, "duration": 4.2, "text": "is going to be equal to S over < tk2 and"}, {"start": 178.08, "duration": 5.4, "text": "then we can cross multiply which gives"}, {"start": 179.48, "duration": 6.399, "text": "us 5S is = to 3un 5 * < tk2 and then"}, {"start": 183.48, "duration": 5.319, "text": "this < TK 5 < tk2 can be combined into <"}, {"start": 185.879, "duration": 5.28, "text": "TK 5 * 20 that's going to be < TK of 100"}, {"start": 188.799, "duration": 4.481, "text": "which is equal to 10 and 3 * 10 is equal"}, {"start": 191.159, "duration": 4.761, "text": "to 30 and after we divide both sides by"}, {"start": 193.28, "duration": 4.92, "text": "5 we get S is equal to 6 now we're ready"}, {"start": 195.92, "duration": 5.48, "text": "to plug this in we can plug in 6 for"}, {"start": 198.2, "duration": 5.36, "text": "this s and 6 s is equal to 36 and let's"}, {"start": 201.4, "duration": 3.8, "text": "give it a label of unit squared and that"}, {"start": 203.56, "duration": 4.2, "text": "is the answer to our question the area"}, {"start": 205.2, "duration": 5.039, "text": "of this shaded region is 36 units"}, {"start": 207.76, "duration": 5.28, "text": "squared let's put a box around it how"}, {"start": 210.239, "duration": 2.801, "text": "exciting"}], "title": "Pink Square Puzzle", "description": "This is another Catriona Agg Puzzle. I am hoping to solve all of them I can find this year!", "lengthSeconds": 211, "width": 640, "height": 360}
{"video_id": "jfNZjX-oaF8", "transcript": [{"start": 0.0, "duration": 4.259, "text": "you guys this looks like a fun one it"}, {"start": 1.8, "duration": 4.44, "text": "says find the area of the red region so"}, {"start": 4.259, "duration": 4.021, "text": "we want the area of the region contained"}, {"start": 6.24, "duration": 4.08, "text": "by these three circles and these circles"}, {"start": 8.28, "duration": 3.96, "text": "have a radius of one two and three"}, {"start": 10.32, "duration": 4.199, "text": "meters so for the first step let's"}, {"start": 12.24, "duration": 4.379, "text": "connect this Center to this Center and"}, {"start": 14.519, "duration": 4.321, "text": "then this Center to this Center and then"}, {"start": 16.619, "duration": 3.721, "text": "this Center to this Center this forms a"}, {"start": 18.84, "duration": 3.48, "text": "triangle and let's look at the triangle"}, {"start": 20.34, "duration": 4.08, "text": "down here so this is a radius of this"}, {"start": 22.32, "duration": 3.719, "text": "circle which is one and this is a radius"}, {"start": 24.42, "duration": 3.54, "text": "of this circle which is three which"}, {"start": 26.039, "duration": 4.201, "text": "makes this entire top side one plus"}, {"start": 27.96, "duration": 3.78, "text": "three which is four and same here this"}, {"start": 30.24, "duration": 3.06, "text": "is a radius of three and this is a"}, {"start": 31.74, "duration": 3.54, "text": "radius of two which would make this"}, {"start": 33.3, "duration": 3.72, "text": "whole bottom a length of five and then"}, {"start": 35.28, "duration": 3.54, "text": "this radius is two and this radius is"}, {"start": 37.02, "duration": 4.379, "text": "one which makes this side a length of"}, {"start": 38.82, "duration": 4.68, "text": "three so this is a three four five"}, {"start": 41.399, "duration": 3.66, "text": "triangle that is a Pythagorean triple"}, {"start": 43.5, "duration": 3.239, "text": "that means this has to be a right"}, {"start": 45.059, "duration": 3.541, "text": "triangle and here's the Pythagorean"}, {"start": 46.739, "duration": 3.781, "text": "theorem that verifies it all three four"}, {"start": 48.6, "duration": 3.66, "text": "five triangles are right triangles let's"}, {"start": 50.52, "duration": 3.18, "text": "find the area of this whole triangle the"}, {"start": 52.26, "duration": 3.42, "text": "area of a triangle is one half base"}, {"start": 53.7, "duration": 4.56, "text": "times height so if we rotate this thing"}, {"start": 55.68, "duration": 4.8, "text": "we can see that the base is four and the"}, {"start": 58.26, "duration": 4.86, "text": "height is three four times three is 12"}, {"start": 60.48, "duration": 5.039, "text": "and one half of 12 is 6. so the area of"}, {"start": 63.12, "duration": 3.48, "text": "this triangle is equal to 6. so now that"}, {"start": 65.519, "duration": 3.061, "text": "we have that let's look at these"}, {"start": 66.6, "duration": 3.9, "text": "individual pieces of the triangle we"}, {"start": 68.58, "duration": 3.899, "text": "have this yellow sector right here this"}, {"start": 70.5, "duration": 3.9, "text": "orange sector and the green sector let's"}, {"start": 72.479, "duration": 4.081, "text": "start by finding this orange piece if"}, {"start": 74.4, "duration": 3.36, "text": "this central angle is equal to D the"}, {"start": 76.56, "duration": 4.26, "text": "area of this sector is going to be equal"}, {"start": 77.76, "duration": 4.56, "text": "to D over 360 times pi r squared and we"}, {"start": 80.82, "duration": 3.54, "text": "know this is a right angle so that means"}, {"start": 82.32, "duration": 3.78, "text": "we can plug in 90 for the D and the"}, {"start": 84.36, "duration": 3.899, "text": "radius of the circle is equal to one so"}, {"start": 86.1, "duration": 3.96, "text": "we can plug in 1 for r 1 squared is"}, {"start": 88.259, "duration": 3.841, "text": "equal to one and multiplying by one we"}, {"start": 90.06, "duration": 4.919, "text": "don't need this one anymore 90 over 360"}, {"start": 92.1, "duration": 5.159, "text": "is equal to 1 4 so we end up with 1 4 Pi"}, {"start": 94.979, "duration": 4.381, "text": "or pi over four next we have the area of"}, {"start": 97.259, "duration": 4.081, "text": "this yellow sector let's make this angle"}, {"start": 99.36, "duration": 3.719, "text": "D degrees so we can use this formula"}, {"start": 101.34, "duration": 3.66, "text": "right here the radius of this circle is"}, {"start": 103.079, "duration": 4.261, "text": "equal to 3 and 3 squared is equal to"}, {"start": 105.0, "duration": 3.96, "text": "nine now we got to find the D let's copy"}, {"start": 107.34, "duration": 3.54, "text": "this triangle down again and this right"}, {"start": 108.96, "duration": 3.72, "text": "here is the angle D so now we have to"}, {"start": 110.88, "duration": 3.84, "text": "use some trigonometry we know that sine"}, {"start": 112.68, "duration": 4.68, "text": "of D is equal to the opposite over the"}, {"start": 114.72, "duration": 4.92, "text": "hypotenuse the side opposite of the D is"}, {"start": 117.36, "duration": 4.92, "text": "equal to 3 and hypotenuse is equal to"}, {"start": 119.64, "duration": 4.979, "text": "five so now to solve for D we need to do"}, {"start": 122.28, "duration": 3.9, "text": "the inverse of sine this sine inverse"}, {"start": 124.619, "duration": 3.36, "text": "gets rid of the sign so we just have D"}, {"start": 126.18, "duration": 3.299, "text": "on the left hand side and for the sine"}, {"start": 127.979, "duration": 2.821, "text": "inverse of three-fifths we can use a"}, {"start": 129.479, "duration": 3.961, "text": "calculator to get the approximate value"}, {"start": 130.8, "duration": 5.04, "text": "of this and we get 36.9 degrees and so"}, {"start": 133.44, "duration": 4.26, "text": "we can change this D into 36.9 so"}, {"start": 135.84, "duration": 3.18, "text": "fortunately this isn't exact anymore so"}, {"start": 137.7, "duration": 3.36, "text": "we got to make this an approximately"}, {"start": 139.02, "duration": 4.32, "text": "equal after we do all this math we get"}, {"start": 141.06, "duration": 4.259, "text": "the areas approximately 2.9 and last"}, {"start": 143.34, "duration": 4.02, "text": "let's do this green sector so we'll make"}, {"start": 145.319, "duration": 4.381, "text": "this angle right here D which means we"}, {"start": 147.36, "duration": 4.56, "text": "can use this formula again for the r the"}, {"start": 149.7, "duration": 4.32, "text": "radius of this circle is equal to 2 and"}, {"start": 151.92, "duration": 3.84, "text": "the 2 squared is equal to 4. and now to"}, {"start": 154.02, "duration": 4.26, "text": "find the D we know that the three angles"}, {"start": 155.76, "duration": 6.479, "text": "of a triangle add up to 180 degrees so"}, {"start": 158.28, "duration": 7.319, "text": "90 plus 36.9 plus d will equal 180. 90"}, {"start": 162.239, "duration": 4.801, "text": "plus 36.9 is 126.9 and after we subtract"}, {"start": 165.599, "duration": 4.321, "text": "that from both sides we get D is equal"}, {"start": 167.04, "duration": 4.86, "text": "to 53.1 and we can change this D into"}, {"start": 169.92, "duration": 3.78, "text": "53.1 and then once again we've got to"}, {"start": 171.9, "duration": 3.839, "text": "change this to approximate and after we"}, {"start": 173.7, "duration": 3.539, "text": "do this the area is approximately 1.9"}, {"start": 175.739, "duration": 3.72, "text": "and now we have everything we need to"}, {"start": 177.239, "duration": 3.901, "text": "find the area of this red region it's"}, {"start": 179.459, "duration": 3.301, "text": "going to be equal to this large triangle"}, {"start": 181.14, "duration": 3.54, "text": "minus the yellow region which was"}, {"start": 182.76, "duration": 4.559, "text": "approximately 2.9 but here's the exact"}, {"start": 184.68, "duration": 4.68, "text": "value right here minus the green region"}, {"start": 187.319, "duration": 4.021, "text": "which is approximately 1.9 but this is"}, {"start": 189.36, "duration": 3.9, "text": "the exact value for the green region and"}, {"start": 191.34, "duration": 3.84, "text": "then minus the pi over 4 which this is"}, {"start": 193.26, "duration": 3.979, "text": "the exact value and after we do all this"}, {"start": 195.18, "duration": 4.86, "text": "math we get approximately"}, {"start": 197.239, "duration": 4.78, "text": "0.464256 meters squared and these are"}, {"start": 200.04, "duration": 3.779, "text": "the answers to the question this is the"}, {"start": 202.019, "duration": 6.14, "text": "exact area and this is the approximate"}, {"start": 203.819, "duration": 4.34, "text": "area of the red region how exciting"}], "title": "Find Area of Red Region", "description": "I thought this was a fun one, and it includes a little bit of trigonometry!", "lengthSeconds": 206, "width": 640, "height": 360}
{"video_id": "AGAVAKk6DRg", "transcript": [{"start": 0.04, "duration": 3.8, "text": "hey guys you've heard of the quadratic"}, {"start": 1.48, "duration": 5.279, "text": "formula it's a great way to solve ax^2 +"}, {"start": 3.84, "duration": 4.799, "text": "BX + Cal 0 but did you also know there"}, {"start": 6.759, "duration": 6.241, "text": "was a cubic formula if you want to solve"}, {"start": 8.639, "duration": 6.641, "text": "for a x Cub + bx2 + cx+ d = 0 x is going"}, {"start": 13.0, "duration": 4.0, "text": "to be equal to all of this stuff and"}, {"start": 15.28, "duration": 3.72, "text": "this is so long that it requires two"}, {"start": 17.0, "duration": 4.16, "text": "rows I thought it'd be fun to test this"}, {"start": 19.0, "duration": 3.8, "text": "out on X Cub is equal to 8 so first"}, {"start": 21.16, "duration": 4.52, "text": "let's copy down the formula and we need"}, {"start": 22.8, "duration": 4.559, "text": "to rewrite X Cub equal 8 in this format"}, {"start": 25.68, "duration": 4.32, "text": "First Step let's subtract 8 from both"}, {"start": 27.359, "duration": 4.76, "text": "sides and that gives us X Cub minus 8"}, {"start": 30.0, "duration": 4.399, "text": "equals 0 and let's give ourselves some"}, {"start": 32.119, "duration": 5.041, "text": "space so that we can include the 0 x^2"}, {"start": 34.399, "duration": 4.68, "text": "and the 0x and then this x Cub needs a"}, {"start": 37.16, "duration": 3.879, "text": "coefficient so let's give it a one and"}, {"start": 39.079, "duration": 3.681, "text": "now we're ready to use this formula I"}, {"start": 41.039, "duration": 3.641, "text": "think the zeros could help us simplify"}, {"start": 42.76, "duration": 4.08, "text": "this quite a bit so let's start with b"}, {"start": 44.68, "duration": 4.879, "text": "equals 0 in the place of all of these"}, {"start": 46.84, "duration": 3.52, "text": "B's let's plug in zero and then now"}, {"start": 49.559, "duration": 2.881, "text": "everywhere where there's a"}, {"start": 50.36, "duration": 4.6, "text": "multiplicative zero is just going to be"}, {"start": 52.44, "duration": 4.119, "text": "zero so this whole term can vanish so"}, {"start": 54.96, "duration": 3.2, "text": "you can see just by the fact that b was"}, {"start": 56.559, "duration": 3.16, "text": "Zero we were able to get rid of a lot of"}, {"start": 58.16, "duration": 2.64, "text": "these terms and then we can smush"}, {"start": 59.719, "duration": 3.121, "text": "everything everything together and we"}, {"start": 60.8, "duration": 4.56, "text": "can even move this up here next let's do"}, {"start": 62.84, "duration": 4.68, "text": "the other zero the C equals z so we can"}, {"start": 65.36, "duration": 4.399, "text": "change this C right here into zero and"}, {"start": 67.52, "duration": 4.2, "text": "this C right here into zero and both of"}, {"start": 69.759, "duration": 4.201, "text": "those are going to vanish as well next"}, {"start": 71.72, "duration": 4.24, "text": "let's focus on the a is equal to one so"}, {"start": 73.96, "duration": 3.6, "text": "we can plug in one for both of these A's"}, {"start": 75.96, "duration": 3.4, "text": "and we can plug in one for both of these"}, {"start": 77.56, "duration": 3.8, "text": "A's and all four of these denominators"}, {"start": 79.36, "duration": 4.119, "text": "are 2 * 1 so they're all just going to"}, {"start": 81.36, "duration": 5.399, "text": "become two and now we can focus on that"}, {"start": 83.479, "duration": 5.801, "text": "D it's equal to8 so all four of these"}, {"start": 86.759, "duration": 4.441, "text": "D's are going to be8 and now we can just"}, {"start": 89.28, "duration": 3.68, "text": "simplify things this negative and this"}, {"start": 91.2, "duration": 4.76, "text": "negative will cancel each other out and"}, {"start": 92.96, "duration": 4.4, "text": "8 / 2 is 4 this squared on the outside"}, {"start": 95.96, "duration": 3.6, "text": "will distribute to the top and the"}, {"start": 97.36, "duration": 6.56, "text": "bottom so we're going to have 8^ 2 and"}, {"start": 99.56, "duration": 8.08, "text": "2^2 the 82 is equal to POS 64 and 2^2 is"}, {"start": 103.92, "duration": 6.839, "text": "equal to 4 and then 64 / by 4 is 16 squ"}, {"start": 107.64, "duration": 5.68, "text": "< TK of 16 gives us 4 and 4 + 4 is equal"}, {"start": 110.759, "duration": 4.841, "text": "to 8 and then the cube root of 8 is"}, {"start": 113.32, "duration": 3.88, "text": "equal to 2 and now we can do the second"}, {"start": 115.6, "duration": 4.76, "text": "cube root the negative and negative will"}, {"start": 117.2, "duration": 4.959, "text": "cancel each other out 8 / 2 is 4 and"}, {"start": 120.36, "duration": 7.2, "text": "this squar will go to the top and bottom"}, {"start": 122.159, "duration": 9.561, "text": "and 8^2 is 64 and 2^2 is equal to 4 64 /"}, {"start": 127.56, "duration": 7.2, "text": "4 is 16 and the < TK of 16 is equal to 4"}, {"start": 131.72, "duration": 5.799, "text": "4 - 4 is equal to 0 and then the cube"}, {"start": 134.76, "duration": 6.119, "text": "root of 0 is just 0 so then we end up"}, {"start": 137.519, "duration": 5.321, "text": "with x = 2 + 0 which is just 2 so the"}, {"start": 140.879, "duration": 5.44, "text": "cubic formula gave us the real solution"}, {"start": 142.84, "duration": 7.56, "text": "to X Cub = 8 it's xal 2 let's put a box"}, {"start": 146.319, "duration": 4.081, "text": "around it how exciting"}], "title": "Cubic Formula Part 1", "description": "https://youtu.be/XswsrgTQAWI\nPart 2 is here.", "lengthSeconds": 149, "width": 578, "height": 360}
{"video_id": "BBLwSSVaXWo", "transcript": [{"start": 0.04, "duration": 3.159, "text": "hey guys there's this interesting"}, {"start": 1.52, "duration": 3.52, "text": "property of logarithms where you can"}, {"start": 3.199, "duration": 3.64, "text": "take this exponent and bring it to the"}, {"start": 5.04, "duration": 3.639, "text": "front it's something I think most people"}, {"start": 6.839, "duration": 3.441, "text": "tend to memorize but it feels kind of"}, {"start": 8.679, "duration": 3.561, "text": "weird doesn't it I wanted to see if we"}, {"start": 10.28, "duration": 3.96, "text": "can figure out why we can do it this is"}, {"start": 12.24, "duration": 4.08, "text": "a really nice animation here but"}, {"start": 14.24, "duration": 4.0, "text": "typically in books it looks more like"}, {"start": 16.32, "duration": 3.119, "text": "this on one side of the equation the a"}, {"start": 18.24, "duration": 3.039, "text": "is in front on the other side of the"}, {"start": 19.439, "duration": 3.961, "text": "equation the a is the exponent so we're"}, {"start": 21.279, "duration": 4.441, "text": "going to find this equation first let's"}, {"start": 23.4, "duration": 4.799, "text": "talk about log form versus exponential"}, {"start": 25.72, "duration": 6.2, "text": "form both of these equations mean the"}, {"start": 28.199, "duration": 6.961, "text": "exact same thing B the Y = X is the same"}, {"start": 31.92, "duration": 4.959, "text": "thing as y = log base B of X and this"}, {"start": 35.16, "duration": 3.52, "text": "will always be true the base of the"}, {"start": 36.879, "duration": 3.84, "text": "exponent will always be the base of the"}, {"start": 38.68, "duration": 3.84, "text": "logarithm and the exponent will always"}, {"start": 40.719, "duration": 3.721, "text": "be the thing that the logarithm is equal"}, {"start": 42.52, "duration": 3.64, "text": "to and the thing that the exponent is"}, {"start": 44.44, "duration": 3.599, "text": "equal to will always be the argument of"}, {"start": 46.16, "duration": 3.919, "text": "the logarithm and here's some colorcoded"}, {"start": 48.039, "duration": 4.52, "text": "circles to help us remember let's copy"}, {"start": 50.079, "duration": 3.601, "text": "down both of these equivalent equations"}, {"start": 52.559, "duration": 4.041, "text": "and I want to play around with them a"}, {"start": 53.68, "duration": 5.359, "text": "little bit first let's do this one b y ="}, {"start": 56.6, "duration": 4.56, "text": "x let's exponentiate both sides of the"}, {"start": 59.039, "duration": 3.921, "text": "equation to the power of a on the left"}, {"start": 61.16, "duration": 4.48, "text": "hand side these exponents will multiply"}, {"start": 62.96, "duration": 4.36, "text": "so we have B to the power of y * a and"}, {"start": 65.64, "duration": 3.799, "text": "we can change the order to get B to the"}, {"start": 67.32, "duration": 4.28, "text": "a * Y and then on the right hand side"}, {"start": 69.439, "duration": 4.081, "text": "this will be x to the A and now we're"}, {"start": 71.6, "duration": 3.68, "text": "back in exponent form let's identify"}, {"start": 73.52, "duration": 4.08, "text": "these three parts again and let's"}, {"start": 75.28, "duration": 5.0, "text": "rewrite this in logarithmic form this"}, {"start": 77.6, "duration": 5.4, "text": "green circle will be a y the base of the"}, {"start": 80.28, "duration": 4.879, "text": "logarithm will be B and the argument of"}, {"start": 83.0, "duration": 3.68, "text": "the logarithm will be x to the A and"}, {"start": 85.159, "duration": 3.32, "text": "this looks important let's put a box"}, {"start": 86.68, "duration": 4.399, "text": "around it and then let's go down to our"}, {"start": 88.479, "duration": 5.0, "text": "yal log base B of X I'm thinking we"}, {"start": 91.079, "duration": 4.601, "text": "should multiply both sides of this by a"}, {"start": 93.479, "duration": 3.561, "text": "on the left hand side we'll have a * Y"}, {"start": 95.68, "duration": 4.32, "text": "and on the right hand side we'll have a"}, {"start": 97.04, "duration": 5.0, "text": "* log Bas B of X this also looks"}, {"start": 100.0, "duration": 4.84, "text": "important let's put a box around it now"}, {"start": 102.04, "duration": 5.32, "text": "let's compare our two boxes since this"}, {"start": 104.84, "duration": 4.44, "text": "and this are both equal to a y we can"}, {"start": 107.36, "duration": 4.48, "text": "set them equal to each other so now we"}, {"start": 109.28, "duration": 4.76, "text": "have a * log Bas B of X is equal to log"}, {"start": 111.84, "duration": 4.0, "text": "base B of x to the A and that's what we"}, {"start": 114.04, "duration": 5.88, "text": "trying to find and here's all the work"}, {"start": 115.84, "duration": 4.08, "text": "right here how exciting"}], "title": "Why can you move the exponent in logarithms?", "description": "I thought this was an interesting one. I hope you guys like it!", "lengthSeconds": 118, "width": 640, "height": 360}
{"video_id": "Zqdw2chh-oQ", "transcript": [{"start": 0.04, "duration": 3.44, "text": "hey guys this looks like a fun one we"}, {"start": 1.56, "duration": 4.04, "text": "have a square right here split up into"}, {"start": 3.48, "duration": 4.12, "text": "three regions and all three of them have"}, {"start": 5.6, "duration": 4.4, "text": "the same area and then we have a quarter"}, {"start": 7.6, "duration": 4.32, "text": "circle with a radius of 13 and the"}, {"start": 10.0, "duration": 3.559, "text": "question is asking what is the area of"}, {"start": 11.92, "duration": 3.28, "text": "the square if you want to try this on"}, {"start": 13.559, "duration": 4.961, "text": "your own pause it right now cuz I'm"}, {"start": 15.2, "duration": 5.56, "text": "going to solve it in 3 2 1 Let's copy"}, {"start": 18.52, "duration": 4.56, "text": "down and let's focus on the Square let's"}, {"start": 20.76, "duration": 4.679, "text": "label the sides of the square X so"}, {"start": 23.08, "duration": 4.24, "text": "ultimately we're trying to find X2 I"}, {"start": 25.439, "duration": 4.08, "text": "want to focus on this blue triangle"}, {"start": 27.32, "duration": 4.68, "text": "right here it has a base of X and let's"}, {"start": 29.519, "duration": 4.401, "text": "label the height Y and since the blue"}, {"start": 32.0, "duration": 4.239, "text": "area equals the red area equals the"}, {"start": 33.92, "duration": 5.2, "text": "green area we know that this triangle is"}, {"start": 36.239, "duration": 5.201, "text": "1/3 of the entire square area of this"}, {"start": 39.12, "duration": 4.52, "text": "triangle is 1/2 base time height where X"}, {"start": 41.44, "duration": 4.24, "text": "is the base and Y is the height for the"}, {"start": 43.64, "duration": 4.72, "text": "right hand side 1/3 of the square is"}, {"start": 45.68, "duration": 4.84, "text": "going to be 1/3 of x^2 next we can"}, {"start": 48.36, "duration": 4.6, "text": "multiply both sides by two on the left"}, {"start": 50.52, "duration": 4.519, "text": "hand side the 2 * 1/2 cancel so we just"}, {"start": 52.96, "duration": 3.919, "text": "have XY and on the right hand side the"}, {"start": 55.039, "duration": 3.641, "text": "two gets multiplied by the 1/3 to give"}, {"start": 56.879, "duration": 4.2, "text": "us 2/3 and then we can bring down the"}, {"start": 58.68, "duration": 4.48, "text": "x^2 next we we can divide both sides by"}, {"start": 61.079, "duration": 4.36, "text": "X and we can do that because we know X"}, {"start": 63.16, "duration": 3.959, "text": "is not equal to zero on the left hand"}, {"start": 65.439, "duration": 4.841, "text": "side we have y and on the right hand"}, {"start": 67.119, "duration": 5.961, "text": "side we have 2/3 of X so let's change"}, {"start": 70.28, "duration": 4.519, "text": "this y up here to 2/3 x and now let's"}, {"start": 73.08, "duration": 3.76, "text": "play around with a quarter Circle the"}, {"start": 74.799, "duration": 3.561, "text": "radius of this quarter circle is 13 and"}, {"start": 76.84, "duration": 3.919, "text": "that's going to be true all the way up"}, {"start": 78.36, "duration": 3.64, "text": "here so this length is equal to 13 and"}, {"start": 80.759, "duration": 3.641, "text": "that means the hypotenuse of our"}, {"start": 82.0, "duration": 5.159, "text": "triangle is also 13 and now we can do"}, {"start": 84.4, "duration": 6.8, "text": "Pythagorean theorem so we have A2 + B2 ="}, {"start": 87.159, "duration": 6.6, "text": "c^2 or in this case it's x^2 + 2/3 x^2"}, {"start": 91.2, "duration": 4.4, "text": "is equal to 132 we can bring down the"}, {"start": 93.759, "duration": 4.04, "text": "x^2 and then this squar is going to"}, {"start": 95.6, "duration": 3.839, "text": "distribute to both the 2/3 and the x"}, {"start": 97.799, "duration": 3.881, "text": "squared and then this squar is going to"}, {"start": 99.439, "duration": 4.521, "text": "distribute to both the two and the three"}, {"start": 101.68, "duration": 3.799, "text": "and that'll give us four nths and let's"}, {"start": 103.96, "duration": 5.799, "text": "smush everything together and set that"}, {"start": 105.479, "duration": 5.841, "text": "equal to 13 s which is 169 nice and now"}, {"start": 109.759, "duration": 2.96, "text": "let's combine like terms I want to give"}, {"start": 111.32, "duration": 4.159, "text": "these a common denominator so let's"}, {"start": 112.719, "duration": 6.921, "text": "multiply the x^2 by 9 over 9 so now"}, {"start": 115.479, "duration": 6.361, "text": "we're doing 9 9 + 4 9 which is 13 9 x^2"}, {"start": 119.64, "duration": 4.159, "text": "and that's still equal to 169 let's"}, {"start": 121.84, "duration": 4.239, "text": "multiply both sides by the reciprocal"}, {"start": 123.799, "duration": 3.841, "text": "9913 on the left hand side we just have"}, {"start": 126.079, "duration": 4.24, "text": "x^2 and on the right hand side we'll"}, {"start": 127.64, "duration": 5.319, "text": "have 117 and that's the answer to our"}, {"start": 130.319, "duration": 5.601, "text": "question the area of the square is 117"}, {"start": 132.959, "duration": 5.761, "text": "cm squared let's put a box around it how"}, {"start": 135.92, "duration": 2.8, "text": "exciting"}], "title": "Cool Geometry Challenge", "description": "https://andymath.com/geometry-challenges/\n\nFor more geometry challenges try the above link!", "lengthSeconds": 137, "width": 640, "height": 360}
{"video_id": "b6lZ4OmdwcY", "transcript": [{"start": 0.04, "duration": 4.0, "text": "hey guys this looks like a fun one the"}, {"start": 1.56, "duration": 3.759, "text": "small circles each have an area of one"}, {"start": 4.04, "duration": 3.12, "text": "and I want to know what's the area of"}, {"start": 5.319, "duration": 3.441, "text": "the larger Circle if you want to try it"}, {"start": 7.16, "duration": 4.84, "text": "on your own pause it right now cuz I'm"}, {"start": 8.76, "duration": 5.52, "text": "going to solve it in 3 2 1 first from"}, {"start": 12.0, "duration": 4.199, "text": "this tangent line let's draw this radius"}, {"start": 14.28, "duration": 4.56, "text": "and call it little r and we're given"}, {"start": 16.199, "duration": 5.24, "text": "that the area of this circle is 1 so"}, {"start": 18.84, "duration": 4.759, "text": "that means PI Little R 2 will be equal"}, {"start": 21.439, "duration": 4.201, "text": "to 1 this looks important let's put a"}, {"start": 23.599, "duration": 3.401, "text": "box around it now for the larger Circle"}, {"start": 25.64, "duration": 3.28, "text": "let's get rid of the question mark and"}, {"start": 27.0, "duration": 4.48, "text": "let's draw the radius here that touches"}, {"start": 28.92, "duration": 3.799, "text": "the tangent line and let's call it Big R"}, {"start": 31.48, "duration": 2.919, "text": "since we're trying to find out what is"}, {"start": 32.719, "duration": 4.16, "text": "the area of this larger Circle we"}, {"start": 34.399, "duration": 4.801, "text": "ultimately want to know what is pi Big R"}, {"start": 36.879, "duration": 4.16, "text": "2 and this is also important let's put a"}, {"start": 39.2, "duration": 3.879, "text": "box around it and to save room let's put"}, {"start": 41.039, "duration": 3.921, "text": "these side by side next let's connect"}, {"start": 43.079, "duration": 4.32, "text": "all the centers of the circles it's"}, {"start": 44.96, "duration": 4.759, "text": "going to be composed of a big radius a"}, {"start": 47.399, "duration": 4.64, "text": "little radius another little radius and"}, {"start": 49.719, "duration": 4.761, "text": "a third little radius and if we darken"}, {"start": 52.039, "duration": 4.801, "text": "this line right here we now have two"}, {"start": 54.48, "duration": 3.84, "text": "similar right triangles and the reason"}, {"start": 56.84, "duration": 3.679, "text": "we know the right triangles is because"}, {"start": 58.32, "duration": 4.48, "text": "radi are always at right Ang to the"}, {"start": 60.519, "duration": 4.441, "text": "tangent lines let's copy this down here"}, {"start": 62.8, "duration": 3.639, "text": "and let's focus on the triangles this"}, {"start": 64.96, "duration": 3.6, "text": "piece right here is going to be little r"}, {"start": 66.439, "duration": 3.441, "text": "plus little r which is two little r and"}, {"start": 68.56, "duration": 3.52, "text": "this piece right here is going to be Big"}, {"start": 69.88, "duration": 4.4, "text": "R plus little r and now we're ready to"}, {"start": 72.08, "duration": 3.96, "text": "set up a proportion let's do this little"}, {"start": 74.28, "duration": 4.28, "text": "triangle first we're going to do one"}, {"start": 76.04, "duration": 4.6, "text": "little r over 2 little r and then we can"}, {"start": 78.56, "duration": 4.04, "text": "focus on this larger triangle this piece"}, {"start": 80.64, "duration": 4.56, "text": "is Big R and this bottom piece is going"}, {"start": 82.6, "duration": 4.96, "text": "to be 2 little r plus Big R plus little"}, {"start": 85.2, "duration": 3.959, "text": "r which is the same thing as Big R + 3"}, {"start": 87.56, "duration": 3.64, "text": "little r so the other half of the"}, {"start": 89.159, "duration": 4.64, "text": "proportion is going to be be Big R over"}, {"start": 91.2, "duration": 4.199, "text": "Big R + 3 little r on this side the"}, {"start": 93.799, "duration": 3.401, "text": "little r and the little r are going to"}, {"start": 95.399, "duration": 3.801, "text": "cancel each other out so that leaves us"}, {"start": 97.2, "duration": 4.879, "text": "1/ 12 is equal to Big R over the"}, {"start": 99.2, "duration": 6.279, "text": "quantity Big R + 3 little r next we can"}, {"start": 102.079, "duration": 5.921, "text": "cross multiply 2 * Big R is 2 Big R and"}, {"start": 105.479, "duration": 4.0, "text": "then 1 * this is just going to be this"}, {"start": 108.0, "duration": 3.119, "text": "next we can subtract Big R from both"}, {"start": 109.479, "duration": 4.801, "text": "sides of the equation on the left hand"}, {"start": 111.119, "duration": 4.64, "text": "side 2 Big R minus 1 Big R is 1 Big R"}, {"start": 114.28, "duration": 3.879, "text": "and on the right hand side these cancel"}, {"start": 115.759, "duration": 4.481, "text": "each other out to give us three little r"}, {"start": 118.159, "duration": 4.32, "text": "so we end up with Big R is equal to 3"}, {"start": 120.24, "duration": 4.6, "text": "little r let's copy down here and let's"}, {"start": 122.479, "duration": 4.92, "text": "Square both sides on the left hand side"}, {"start": 124.84, "duration": 3.88, "text": "Big R S stays as Big R S but on the"}, {"start": 127.399, "duration": 3.28, "text": "right hand side this Square will"}, {"start": 128.72, "duration": 6.239, "text": "distribute to both of these terms giving"}, {"start": 130.679, "duration": 6.401, "text": "us 3^ 2 * little R 2 and 3^2 is equal to"}, {"start": 134.959, "duration": 4.121, "text": "9 let's copy it down again and let's"}, {"start": 137.08, "duration": 4.36, "text": "multiply both sides by pi on the left"}, {"start": 139.08, "duration": 3.799, "text": "side we have Pi Big R 2 and on the right"}, {"start": 141.44, "duration": 4.439, "text": "hand side we can move this over and"}, {"start": 142.879, "duration": 5.841, "text": "we'll have 9 pi little R 2 and we know"}, {"start": 145.879, "duration": 4.761, "text": "that PI Little R 2 is equal to 1 so in"}, {"start": 148.72, "duration": 4.04, "text": "the place of this PI Little R squ we can"}, {"start": 150.64, "duration": 3.599, "text": "plug in one and this multiply by 1 isn't"}, {"start": 152.76, "duration": 4.24, "text": "doing anything so we can get rid of it"}, {"start": 154.239, "duration": 4.321, "text": "and we have now solved for pi Big R 2 so"}, {"start": 157.0, "duration": 3.48, "text": "we can update the question mark up here"}, {"start": 158.56, "duration": 3.319, "text": "to be nine and now we have the answer to"}, {"start": 160.48, "duration": 4.08, "text": "the question I want to know what's the"}, {"start": 161.879, "duration": 5.041, "text": "area of the large Circle so that'll be 9"}, {"start": 164.56, "duration": 5.599, "text": "units squared let's put a box around it"}, {"start": 166.92, "duration": 3.239, "text": "how exciting"}], "title": "Fun Geometry Challenge", "description": "Puzzle by Catriona Agg\n\nSuch a cool one!", "lengthSeconds": 168, "width": 640, "height": 360}
{"video_id": "w4_rLRONUIM", "transcript": [{"start": 0.0, "duration": 3.659, "text": "hey guys I made this video the other day"}, {"start": 1.56, "duration": 3.6, "text": "and I showed a shortcut to solve it a"}, {"start": 3.659, "duration": 3.781, "text": "bunch of people asked me to solve it the"}, {"start": 5.16, "duration": 3.899, "text": "hyper complicated way so let's do it so"}, {"start": 7.44, "duration": 3.54, "text": "it gives us some links here and it wants"}, {"start": 9.059, "duration": 3.421, "text": "to know what is the length of a b so"}, {"start": 10.98, "duration": 3.3, "text": "first let's find this length right here"}, {"start": 12.48, "duration": 3.659, "text": "this is a right triangle so if we call"}, {"start": 14.28, "duration": 4.079, "text": "this side C we can use Pythagorean"}, {"start": 16.139, "duration": 4.98, "text": "theorem we have 2 squared plus 4 squared"}, {"start": 18.359, "duration": 5.34, "text": "equals c squared 2 squared is four four"}, {"start": 21.119, "duration": 4.381, "text": "squared is sixteen four plus sixteen is"}, {"start": 23.699, "duration": 3.66, "text": "twenty and after we square root both"}, {"start": 25.5, "duration": 4.199, "text": "sides we get C is equal to square root"}, {"start": 27.359, "duration": 4.26, "text": "of 20. so let's change the C into square"}, {"start": 29.699, "duration": 3.961, "text": "root of 20. so next let's find this side"}, {"start": 31.619, "duration": 4.021, "text": "let's call it C and we could do 2"}, {"start": 33.66, "duration": 3.96, "text": "squared plus 2 squared equals c squared"}, {"start": 35.64, "duration": 3.66, "text": "both of these two squareds are four four"}, {"start": 37.62, "duration": 3.72, "text": "plus four is eight and we could square"}, {"start": 39.3, "duration": 3.599, "text": "root both sides to get C is equal to the"}, {"start": 41.34, "duration": 2.76, "text": "square root of eight and the exact same"}, {"start": 42.899, "duration": 3.061, "text": "thing is going to happen right here so"}, {"start": 44.1, "duration": 3.479, "text": "this will also be square root of 8."}, {"start": 45.96, "duration": 3.599, "text": "let's focus our energy right here in"}, {"start": 47.579, "duration": 4.561, "text": "fact enhance let's focus our energy"}, {"start": 49.559, "duration": 4.5, "text": "right here this is a 90 degree angle"}, {"start": 52.14, "duration": 3.899, "text": "this right here is an isosceles right"}, {"start": 54.059, "duration": 4.68, "text": "triangle which means that's 45 degrees"}, {"start": 56.039, "duration": 4.86, "text": "and then for this last angle there's no"}, {"start": 58.739, "duration": 4.201, "text": "shortcut so let's call it d degrees and"}, {"start": 60.899, "duration": 3.9, "text": "let's use some trigonometry we know that"}, {"start": 62.94, "duration": 3.78, "text": "the tangent of D is equal to the"}, {"start": 64.799, "duration": 3.961, "text": "opposite over the adjacent well this"}, {"start": 66.72, "duration": 3.899, "text": "opposite side is 4 and the adjacent side"}, {"start": 68.76, "duration": 3.84, "text": "is 2 so this is going to become four"}, {"start": 70.619, "duration": 3.781, "text": "over two and four over two is equal to"}, {"start": 72.6, "duration": 3.9, "text": "2. then we can tangent inverse both"}, {"start": 74.4, "duration": 3.539, "text": "sides on the left hand side we have D"}, {"start": 76.5, "duration": 3.119, "text": "and it'll be equal to the tangent"}, {"start": 77.939, "duration": 4.621, "text": "inverse of two which is approximately"}, {"start": 79.619, "duration": 5.581, "text": "63.4 degrees so let's change this D to"}, {"start": 82.56, "duration": 4.919, "text": "63.4 degrees now we can get this last"}, {"start": 85.2, "duration": 4.98, "text": "angle let's call it n all four of these"}, {"start": 87.479, "duration": 6.601, "text": "will add up to 360. we can combine the"}, {"start": 90.18, "duration": 6.84, "text": "45 to 90 and the 63.4 to give us 198.4"}, {"start": 94.08, "duration": 6.48, "text": "after we subtract 198.4 from both sides"}, {"start": 97.02, "duration": 5.76, "text": "we get n is equal to 161.6 so let's"}, {"start": 100.56, "duration": 3.9, "text": "change that that was good so we can zoom"}, {"start": 102.78, "duration": 3.659, "text": "back out now we can connect this last"}, {"start": 104.46, "duration": 3.78, "text": "side and focus on this triangle right"}, {"start": 106.439, "duration": 3.96, "text": "here so this side right here is made up"}, {"start": 108.24, "duration": 4.32, "text": "of 2 square root of eights so let's call"}, {"start": 110.399, "duration": 4.201, "text": "it 2 square root of 8. this side is root"}, {"start": 112.56, "duration": 4.26, "text": "20 and this side is unknown let's call"}, {"start": 114.6, "duration": 4.559, "text": "the unknown side C and in fact this"}, {"start": 116.82, "duration": 3.96, "text": "unknown side is the a b that we wanted"}, {"start": 119.159, "duration": 3.121, "text": "to solve from before or so we're going"}, {"start": 120.78, "duration": 3.659, "text": "to use a thing called law of cosines"}, {"start": 122.28, "duration": 3.6, "text": "here it is right here you'll recognize a"}, {"start": 124.439, "duration": 2.941, "text": "portion of it as a squared plus b"}, {"start": 125.88, "duration": 3.359, "text": "squared equals c squared that's the"}, {"start": 127.38, "duration": 3.42, "text": "Pythagorean theorem but since it's not a"}, {"start": 129.239, "duration": 4.261, "text": "right triangle we have this little extra"}, {"start": 130.8, "duration": 4.68, "text": "portion so for both of these A's I'm"}, {"start": 133.5, "duration": 3.66, "text": "going to plug in 2 square root of 8 and"}, {"start": 135.48, "duration": 3.54, "text": "for both of these B's I'm going to plug"}, {"start": 137.16, "duration": 4.74, "text": "in square root of 20. and then for this"}, {"start": 139.02, "duration": 5.28, "text": "angle Theta we're going to plug in 161.6"}, {"start": 141.9, "duration": 4.44, "text": "this Square goes to both the 2 and the"}, {"start": 144.3, "duration": 4.14, "text": "square root of 8. 2 squared is equal to"}, {"start": 146.34, "duration": 4.979, "text": "4 and square root of 8 squared is equal"}, {"start": 148.44, "duration": 5.22, "text": "to 8 and then 4 times 8 is equal to 32"}, {"start": 151.319, "duration": 4.681, "text": "and then square root of 20 squared will"}, {"start": 153.66, "duration": 4.02, "text": "give us 20. next we can plug all of this"}, {"start": 156.0, "duration": 3.78, "text": "stuff into a calculator and it will give"}, {"start": 157.68, "duration": 4.38, "text": "us a negative 48. there it is right"}, {"start": 159.78, "duration": 5.4, "text": "there and then minus negative 48 is"}, {"start": 162.06, "duration": 5.039, "text": "positive 48 and 32 plus 20 plus 48 is"}, {"start": 165.18, "duration": 4.32, "text": "equal to 100 and after we square root"}, {"start": 167.099, "duration": 4.741, "text": "both sides we get C is equal to 10. so"}, {"start": 169.5, "duration": 3.72, "text": "we can update this C to be 10. and then"}, {"start": 171.84, "duration": 3.6, "text": "returning back to the original problem"}, {"start": 173.22, "duration": 5.96, "text": "we have that right here the C is equal"}, {"start": 175.44, "duration": 3.74, "text": "to 10. how exciting"}], "title": "Hyper Complicated Method", "description": "https://youtube.com/shorts/fmDUseL-YUA\nI solved this with a shortcut last time. Hopefully you guys enjoy this hyper complicated method!", "lengthSeconds": 177, "width": 640, "height": 360}
{"video_id": "_qdypLS-rNE", "transcript": [{"start": 0.0, "duration": 4.56, "text": "hey guys let's solve for x first we have"}, {"start": 2.7, "duration": 3.9, "text": "this equilateral triangle right here all"}, {"start": 4.56, "duration": 3.66, "text": "triangles have 180 degrees so if all"}, {"start": 6.6, "duration": 3.419, "text": "three angles are the same each angle is"}, {"start": 8.22, "duration": 3.72, "text": "going to be 60 degrees next let's look"}, {"start": 10.019, "duration": 3.481, "text": "at this regular pentagon the formula for"}, {"start": 11.94, "duration": 4.199, "text": "the sum of the interior angles of a"}, {"start": 13.5, "duration": 3.96, "text": "polygon is n minus 2 times 180 where n"}, {"start": 16.139, "duration": 3.661, "text": "is the number of sides of the polygon"}, {"start": 17.46, "duration": 3.72, "text": "for a pentagon it has five sides and"}, {"start": 19.8, "duration": 3.0, "text": "this will give us the sum of all the"}, {"start": 21.18, "duration": 3.42, "text": "interior angles if we want to know how"}, {"start": 22.8, "duration": 3.84, "text": "big each angle is we're going to divide"}, {"start": 24.6, "duration": 4.38, "text": "by 5 and this will give us 108 degrees"}, {"start": 26.64, "duration": 3.959, "text": "so this one's 108 this one's 108 this"}, {"start": 28.98, "duration": 3.419, "text": "one's 108. this one down here we already"}, {"start": 30.599, "duration": 3.601, "text": "have the 60s so we're going to need"}, {"start": 32.399, "duration": 3.66, "text": "additional 48 and same thing for this"}, {"start": 34.2, "duration": 3.539, "text": "one an additional 48. but next let's"}, {"start": 36.059, "duration": 3.301, "text": "look at this right triangle right here"}, {"start": 37.739, "duration": 3.721, "text": "if we added these three angles they're"}, {"start": 39.36, "duration": 5.16, "text": "all going to add up to 180 degrees the"}, {"start": 41.46, "duration": 5.46, "text": "48 Plus 90 is 138 and if we subtract 138"}, {"start": 44.52, "duration": 6.859, "text": "from both sides we get x equals 42 and"}, {"start": 46.92, "duration": 4.459, "text": "that is our answer for X how exciting"}], "title": "Fun Polygon Math #shorts", "description": "https://andymath.com/angles-of-polygons/\n\n#shorts", "lengthSeconds": 50, "width": 270, "height": 480}
{"video_id": "VMiS7G8xqQs", "transcript": [{"start": 0.04, "duration": 4.0, "text": "hey guys this looks like a fun one it"}, {"start": 1.48, "duration": 4.919, "text": "says find the area of the blue rectangle"}, {"start": 4.04, "duration": 4.559, "text": "and the rectangle is twice as wide as it"}, {"start": 6.399, "duration": 4.12, "text": "is tall and it hits the semicircle in"}, {"start": 8.599, "duration": 3.681, "text": "these various places if you guys want to"}, {"start": 10.519, "duration": 4.921, "text": "try this on your own pause it right now"}, {"start": 12.28, "duration": 5.2, "text": "cuz I'm going to solve it in 3 2 1 first"}, {"start": 15.44, "duration": 3.72, "text": "let's connect this point to this point"}, {"start": 17.48, "duration": 3.24, "text": "and I want to focus on this triangle"}, {"start": 19.16, "duration": 4.039, "text": "right here first off it's a right"}, {"start": 20.72, "duration": 4.719, "text": "triangle because this is a rectangle and"}, {"start": 23.199, "duration": 5.08, "text": "second it's isos because these two sides"}, {"start": 25.439, "duration": 5.561, "text": "are congruent that means it's a 45 4590"}, {"start": 28.279, "duration": 5.641, "text": "triangle and then since this angle is 45"}, {"start": 31.0, "duration": 4.719, "text": "we now know this angle this plus this is"}, {"start": 33.92, "duration": 5.159, "text": "a linear pair which means they have to"}, {"start": 35.719, "duration": 5.961, "text": "add to 180\u00b0 so this angle will be"}, {"start": 39.079, "duration": 4.241, "text": "135\u00b0 next let's finish up the circle and"}, {"start": 41.68, "duration": 4.44, "text": "I want to focus on this"}, {"start": 43.32, "duration": 4.2, "text": "135\u00b0 this is called an inscribed angle"}, {"start": 46.12, "duration": 4.079, "text": "and if we look at it it's cutting the"}, {"start": 47.52, "duration": 4.8, "text": "circle into two parts this red Arc and"}, {"start": 50.199, "duration": 4.081, "text": "this blue arc a cool property about"}, {"start": 52.32, "duration": 4.759, "text": "inscribed angles is they are exactly"}, {"start": 54.28, "duration": 4.84, "text": "half of the arc that they subtend so"}, {"start": 57.079, "duration": 5.761, "text": "this red Arc is going to be double the"}, {"start": 59.12, "duration": 7.52, "text": "135 de so it'll be 270\u00b0 and then since"}, {"start": 62.84, "duration": 5.88, "text": "the red Arc is 270\u00b0 that leaves us 90\u00b0"}, {"start": 66.64, "duration": 4.24, "text": "for the blue arc for the next step let's"}, {"start": 68.72, "duration": 5.12, "text": "draw this radius here and this radius"}, {"start": 70.88, "duration": 4.239, "text": "here and let's focus on this 90\u00b0 Arc"}, {"start": 73.84, "duration": 3.4, "text": "this angle right here is called a"}, {"start": 75.119, "duration": 4.36, "text": "central angle and central angles are"}, {"start": 77.24, "duration": 4.4, "text": "equal to the subtended arcs so this is"}, {"start": 79.479, "duration": 4.64, "text": "going to be a 90\u00b0 angle and we usually"}, {"start": 81.64, "duration": 4.439, "text": "notate that as a box in the corner and"}, {"start": 84.119, "duration": 3.801, "text": "since both of these are radi they both"}, {"start": 86.079, "duration": 4.121, "text": "have a length of five so if we connect"}, {"start": 87.92, "duration": 4.519, "text": "this here we have another isoceles right"}, {"start": 90.2, "duration": 4.559, "text": "triangle if we use a shortcut for 45"}, {"start": 92.439, "duration": 4.72, "text": "4590 triangles we can see this green"}, {"start": 94.759, "duration": 4.04, "text": "side is equal to 5 < tk2 and here's the"}, {"start": 97.159, "duration": 3.96, "text": "steps of the Pythagorean theorem to"}, {"start": 98.799, "duration": 4.241, "text": "verify it and we just did all of this"}, {"start": 101.119, "duration": 4.0, "text": "work just so we could get this green"}, {"start": 103.04, "duration": 3.759, "text": "length right here we can go back to here"}, {"start": 105.119, "duration": 3.32, "text": "and let's bring back our rectangle"}, {"start": 106.799, "duration": 4.161, "text": "looking at our rectangle if we call the"}, {"start": 108.439, "duration": 5.04, "text": "shorter side x this longer side is going"}, {"start": 110.96, "duration": 4.839, "text": "to be 2X and we have formed another"}, {"start": 113.479, "duration": 3.881, "text": "right triangle let's focus on that and"}, {"start": 115.799, "duration": 6.481, "text": "now we can do Pythagorean theorem to"}, {"start": 117.36, "duration": 7.56, "text": "solve for x we have x^2 + 2x^2 = 5 <"}, {"start": 122.28, "duration": 4.759, "text": "tk22 we can bring down the x^2 and then"}, {"start": 124.92, "duration": 4.559, "text": "this squar will go to both the 2 and the"}, {"start": 127.039, "duration": 7.081, "text": "X and this two will go to both the 5 and"}, {"start": 129.479, "duration": 9.84, "text": "the square root so we have x^2 + 4 x^2 ="}, {"start": 134.12, "duration": 7.839, "text": "25 * 2 x^2 + 4x^2 is equal to 5 x^2 and"}, {"start": 139.319, "duration": 5.121, "text": "25 * 2 is equal to 50 after we divide"}, {"start": 141.959, "duration": 4.441, "text": "both sides by 5 we get x^2 is equal to"}, {"start": 144.44, "duration": 4.24, "text": "10 and after we square root both sides"}, {"start": 146.4, "duration": 4.0, "text": "we get X is equal to the squ < TK of 10"}, {"start": 148.68, "duration": 4.559, "text": "and now we know the shorter side of the"}, {"start": 150.4, "duration": 5.36, "text": "rectangle this x can change into < TK 10"}, {"start": 153.239, "duration": 3.72, "text": "and this 2x can change into 2un 10 and"}, {"start": 155.76, "duration": 3.479, "text": "now we can find the area of the"}, {"start": 156.959, "duration": 4.521, "text": "rectangle it's equal to base time height"}, {"start": 159.239, "duration": 4.121, "text": "where the base is equal to 2un 10 and"}, {"start": 161.48, "duration": 4.56, "text": "the height is equal to < TK 10 and"}, {"start": 163.36, "duration": 4.799, "text": "that's equal to 2 * 10 which is equal to"}, {"start": 166.04, "duration": 3.76, "text": "20 let's give it a label of square units"}, {"start": 168.159, "duration": 4.44, "text": "and put a box around it the area of the"}, {"start": 169.8, "duration": 5.6, "text": "blue rectangle is 20 square units how"}, {"start": 172.599, "duration": 2.801, "text": "exciting"}], "title": "Tricky Geometry Challenge", "description": "https://andymath.com/geometry-challenges/\n\nFor more geometry challenges, try the above page!", "lengthSeconds": 173, "width": 640, "height": 360}
{"video_id": "ru-zr6SErL8", "transcript": [{"start": 0.04, "duration": 3.56, "text": "hey guys this looks like a fun one we're"}, {"start": 1.319, "duration": 3.96, "text": "given two blue squares and it wants to"}, {"start": 3.6, "duration": 3.759, "text": "know what is the area of the red"}, {"start": 5.279, "duration": 3.841, "text": "triangle and I'm noticing something it's"}, {"start": 7.359, "duration": 3.921, "text": "not giving us anything about the"}, {"start": 9.12, "duration": 3.92, "text": "distance between the two squares so it"}, {"start": 11.28, "duration": 3.88, "text": "could realistically be any one of these"}, {"start": 13.04, "duration": 3.64, "text": "scenarios so I'm guessing every single"}, {"start": 15.16, "duration": 3.52, "text": "one of these red triangles have the"}, {"start": 16.68, "duration": 3.759, "text": "exact same area if you want to try it on"}, {"start": 18.68, "duration": 4.919, "text": "your own pause it right now because I'm"}, {"start": 20.439, "duration": 4.76, "text": "going to solve it in 3 2 1 so first"}, {"start": 23.599, "duration": 5.161, "text": "let's look at the formula for the area"}, {"start": 25.199, "duration": 5.521, "text": "of a triangle area equals 12 base time"}, {"start": 28.76, "duration": 4.479, "text": "height looking at this triangle we don't"}, {"start": 30.72, "duration": 4.28, "text": "know the base so let's call it X and"}, {"start": 33.239, "duration": 3.32, "text": "then for the height of this triangle we"}, {"start": 35.0, "duration": 3.239, "text": "need something at right angles this"}, {"start": 36.559, "duration": 3.761, "text": "right here would be at right angles"}, {"start": 38.239, "duration": 4.081, "text": "because this is a square so this is also"}, {"start": 40.32, "duration": 3.559, "text": "right angles this right here will be the"}, {"start": 42.32, "duration": 3.719, "text": "height of our triangle we don't know"}, {"start": 43.879, "duration": 3.84, "text": "that either so let's call it y so now"}, {"start": 46.039, "duration": 4.561, "text": "let's update this formula the base of"}, {"start": 47.719, "duration": 4.64, "text": "our triangle's X so let's call this x"}, {"start": 50.6, "duration": 3.799, "text": "and the height of our triangle is y so"}, {"start": 52.359, "duration": 3.921, "text": "let's call this y so this is what we're"}, {"start": 54.399, "duration": 3.96, "text": "trying to figure out so next let's use"}, {"start": 56.28, "duration": 4.64, "text": "the areas of the blue squares if this"}, {"start": 58.359, "duration": 4.481, "text": "one has an area of 16 each side will be"}, {"start": 60.92, "duration": 4.199, "text": "equal to four and if this one has an"}, {"start": 62.84, "duration": 4.84, "text": "area of 36 each side will be equal to"}, {"start": 65.119, "duration": 4.881, "text": "six and then from here just like this"}, {"start": 67.68, "duration": 4.36, "text": "was a right angle right here this one"}, {"start": 70.0, "duration": 3.6, "text": "would also be a right angle and then if"}, {"start": 72.04, "duration": 4.079, "text": "we look at this angle right here let's"}, {"start": 73.6, "duration": 4.519, "text": "call it Theta this angle will also be"}, {"start": 76.119, "duration": 4.201, "text": "Theta because these are vertical angles"}, {"start": 78.119, "duration": 3.721, "text": "and vertical angles are always congruent"}, {"start": 80.32, "duration": 3.439, "text": "so let's take this whole thing and copy"}, {"start": 81.84, "duration": 4.68, "text": "it down here and I want to focus on"}, {"start": 83.759, "duration": 4.841, "text": "these white triangles Let's Line them up"}, {"start": 86.52, "duration": 3.599, "text": "like this these are similar right"}, {"start": 88.6, "duration": 4.12, "text": "triangles we know that because because"}, {"start": 90.119, "duration": 4.721, "text": "they have two pairs of congruent angles"}, {"start": 92.72, "duration": 4.759, "text": "so that means we can set up a proportion"}, {"start": 94.84, "duration": 5.16, "text": "we can say the base of this triangle 4"}, {"start": 97.479, "duration": 5.081, "text": "over the base of this triangle X is"}, {"start": 100.0, "duration": 5.2, "text": "equal to the height of this triangle y"}, {"start": 102.56, "duration": 5.48, "text": "ided by the height of this triangle 6"}, {"start": 105.2, "duration": 6.12, "text": "and from here we can cross multiply x *"}, {"start": 108.04, "duration": 7.439, "text": "Y is equal to XY and on the other side 4"}, {"start": 111.32, "duration": 6.2, "text": "* 6 is equal to 24 and now we have x * y"}, {"start": 115.479, "duration": 3.721, "text": "that's all we really needed up here so"}, {"start": 117.52, "duration": 5.32, "text": "we can scoot this up here and in the"}, {"start": 119.2, "duration": 5.64, "text": "place of x y let's plug in 24 1 12 * 24"}, {"start": 122.84, "duration": 4.319, "text": "is equal to 12 and let's give it a label"}, {"start": 124.84, "duration": 4.72, "text": "of me squared this is the answer to our"}, {"start": 127.159, "duration": 4.761, "text": "question the area of the red triangle is"}, {"start": 129.56, "duration": 4.48, "text": "12 M squar and that'll be true for all"}, {"start": 131.92, "duration": 5.76, "text": "of these red triangles they all have an"}, {"start": 134.04, "duration": 5.68, "text": "area of 12 M squar how exciting now I'm"}, {"start": 137.68, "duration": 4.48, "text": "trying to put together an end card"}, {"start": 139.72, "duration": 4.84, "text": "here's a button to subscribe and here's"}, {"start": 142.16, "duration": 5.4, "text": "a nice video you can check out how"}, {"start": 144.56, "duration": 3.0, "text": "exciting"}], "title": "2 Squares 1 Triangle", "description": "I hope you guys love this one. It's neat how the areas are all the same! \n\nIf anybody is reading this description, feel free to comment \"It Looks like a panda.\" or \"Is that the panda from 2016?\" Or something along those lines. If enough people do it, I think it could be funny and confusing. :)", "lengthSeconds": 147, "width": 640, "height": 360}
{"video_id": "gQAfFnEFzec", "transcript": [{"start": 0.0, "duration": 3.439, "text": "how do we solve for x in this right"}, {"start": 1.68, "duration": 3.52, "text": "triangle we're going to use pythagorean"}, {"start": 3.439, "duration": 3.281, "text": "theorem pythagorean theorem says a"}, {"start": 5.2, "duration": 3.439, "text": "squared plus b squared equals c"}, {"start": 6.72, "duration": 4.08, "text": "squared a and b are the two smaller"}, {"start": 8.639, "duration": 3.761, "text": "sides and c would be the largest side"}, {"start": 10.8, "duration": 3.999, "text": "so to plug in the values we get 3"}, {"start": 12.4, "duration": 4.799, "text": "squared plus 4 squared equals c squared"}, {"start": 14.799, "duration": 5.761, "text": "3 squared is 3 times 3 which is 9"}, {"start": 17.199, "duration": 5.521, "text": "4 squared is 4 times 4 which is 16"}, {"start": 20.56, "duration": 3.6, "text": "and we'll leave c squared as c squared 9"}, {"start": 22.72, "duration": 3.2, "text": "plus 16 is 25"}, {"start": 24.16, "duration": 3.52, "text": "and that still equals c squared and then"}, {"start": 25.92, "duration": 2.32, "text": "to solve for c we will square root both"}, {"start": 27.68, "duration": 2.88, "text": "sides"}, {"start": 28.24, "duration": 3.199, "text": "square root of 25 is 5 so we get c"}, {"start": 30.56, "duration": 2.24, "text": "equals 5."}, {"start": 31.439, "duration": 2.721, "text": "that's the answer to the question for"}, {"start": 32.8, "duration": 2.56, "text": "more math videos please check out"}, {"start": 34.16, "duration": 4.76, "text": "andymath.com"}, {"start": 35.36, "duration": 6.56, "text": "have a good one i'll talk to you soon"}, {"start": 38.92, "duration": 3.0, "text": "bye"}], "title": "Pythagorean Theorem Fast Example #Shorts", "description": "\ud835\udc0b\ud835\udc08\ud835\udc0d\ud835\udc0a\ud835\udc12 \ud835\udc13\ud835\udc0e \ud835\udc11\ud835\udc04\ud835\udc0b\ud835\udc00\ud835\udc13\ud835\udc04\ud835\udc03 \ud835\udc13\ud835\udc0e\ud835\udc0f\ud835\udc08\ud835\udc02\ud835\udc12\n\ud835\udc0f\ud835\udc32\ud835\udc2d\ud835\udc21\ud835\udc1a\ud835\udc20\ud835\udc28\ud835\udc2b\ud835\udc1e\ud835\udc1a\ud835\udc27 \ud835\udc13\ud835\udc21\ud835\udc1e\ud835\udc28\ud835\udc2b\ud835\udc1e\ud835\udc26: https://andymath.com/pythagorean-theorem/\n\ud835\udc0f\ud835\udc32\ud835\udc2d\ud835\udc21\ud835\udc1a\ud835\udc20\ud835\udc28\ud835\udc2b\ud835\udc1e\ud835\udc1a\ud835\udc27 \ud835\udc13\ud835\udc2b\ud835\udc22\ud835\udc29\ud835\udc25\ud835\udc1e\ud835\udc2c: https://andymath.com/pythagorean-triples/\n\n\ud835\udc04\ud835\udc2a\ud835\udc2e\ud835\udc22\ud835\udc25\ud835\udc1a\ud835\udc2d\ud835\udc1e\ud835\udc2b\ud835\udc1a\ud835\udc25 \ud835\udc13\ud835\udc2b\ud835\udc22\ud835\udc1a\ud835\udc27\ud835\udc20\ud835\udc25\ud835\udc1e\ud835\udc2c: https://andymath.com/equilateral-triangles/\n\n\ud835\udc08\ud835\udc2c\ud835\udc28\ud835\udc2c\ud835\udc1c\ud835\udc1e\ud835\udc25\ud835\udc1e\ud835\udc2c \ud835\udc13\ud835\udc2b\ud835\udc22\ud835\udc1a\ud835\udc27\ud835\udc20\ud835\udc25\ud835\udc1e\ud835\udc2c: https://andymath.com/isosceles-triangles/\n\n\ud835\udc13\ud835\udc2b\ud835\udc22\ud835\udc1a\ud835\udc27\ud835\udc20\ud835\udc25\ud835\udc1e \ud835\udc00\ud835\udc27\ud835\udc20\ud835\udc25\ud835\udc1e \ud835\udc12\ud835\udc2e\ud835\udc26 \ud835\udc13\ud835\udc21\ud835\udc1e\ud835\udc28\ud835\udc2b\ud835\udc1e\ud835\udc26: https://andymath.com/triangle-angle-sum-theorem/\n\n\ud835\udc12\ud835\udc22\ud835\udc1d\ud835\udc1e-\ud835\udc12\ud835\udc29\ud835\udc25\ud835\udc22\ud835\udc2d\ud835\udc2d\ud835\udc1e\ud835\udc2b \ud835\udc13\ud835\udc21\ud835\udc1e\ud835\udc28\ud835\udc2b\ud835\udc1e\ud835\udc26: https://andymath.com/side-splitter-theorem/\n\n\ud835\udc13\ud835\udc2b\ud835\udc22\ud835\udc1a\ud835\udc27\ud835\udc20\ud835\udc25\ud835\udc1e \ud835\udc0f\ud835\udc2b\ud835\udc28\ud835\udc29\ud835\udc28\ud835\udc2b\ud835\udc2d\ud835\udc22\ud835\udc28\ud835\udc27\ud835\udc1a\ud835\udc25\ud835\udc22\ud835\udc2d\ud835\udc32 \ud835\udc13\ud835\udc21\ud835\udc1e\ud835\udc28\ud835\udc2b\ud835\udc1e\ud835\udc26: https://andymath.com/triangle-proportionality-theorem/\n\n\ud835\udc0f\ud835\udc1a\ud835\udc2b\ud835\udc1a\ud835\udc25\ud835\udc25\ud835\udc1e\ud835\udc25 \ud835\udc0b\ud835\udc22\ud835\udc27\ud835\udc1e\ud835\udc2c \ud835\udc0f\ud835\udc2b\ud835\udc28\ud835\udc29\ud835\udc28\ud835\udc2b\ud835\udc2d\ud835\udc22\ud835\udc28\ud835\udc27\ud835\udc1a\ud835\udc25\ud835\udc22\ud835\udc2d\ud835\udc32 \ud835\udc13\ud835\udc21\ud835\udc1e\ud835\udc28\ud835\udc2b\ud835\udc1e\ud835\udc26: https://andymath.com/parallel-lines-proportionality-theorem/\n\n\ud835\udc13\ud835\udc2b\ud835\udc22\ud835\udc1a\ud835\udc27\ud835\udc20\ud835\udc25\ud835\udc1e \ud835\udc0c\ud835\udc22\ud835\udc1d\ud835\udc2c\ud835\udc1e\ud835\udc20\ud835\udc26\ud835\udc1e\ud835\udc27\ud835\udc2d \ud835\udc13\ud835\udc21\ud835\udc1e\ud835\udc28\ud835\udc2b\ud835\udc1e\ud835\udc26: https://andymath.com/triangle-midsegment-theorem/\n\n\ud835\udc04\ud835\udc31\ud835\udc2d\ud835\udc1e\ud835\udc2b\ud835\udc22\ud835\udc28\ud835\udc2b \ud835\udc00\ud835\udc27\ud835\udc20\ud835\udc25\ud835\udc1e \ud835\udc13\ud835\udc21\ud835\udc1e\ud835\udc28\ud835\udc2b\ud835\udc1e\ud835\udc26: https://andymath.com/exterior-angle-theorem/\n\n\ud835\udc12\ud835\udc29\ud835\udc1e\ud835\udc1c\ud835\udc22\ud835\udc1a\ud835\udc25 \ud835\udc11\ud835\udc22\ud835\udc20\ud835\udc21\ud835\udc2d \ud835\udc13\ud835\udc2b\ud835\udc22\ud835\udc1a\ud835\udc27\ud835\udc20\ud835\udc25\ud835\udc1e\ud835\udc2c \ud835\udfd1\ud835\udfce-\ud835\udfd4\ud835\udfce-\ud835\udfd7\ud835\udfce \ud835\udfd2\ud835\udfd3-\ud835\udfd2\ud835\udfd3-\ud835\udfd7\ud835\udfce: https://andymath.com/special-right-triangles/\n\n\ud835\udc07\ud835\udc22\ud835\udc27\ud835\udc20\ud835\udc1e \ud835\udc13\ud835\udc21\ud835\udc1e\ud835\udc28\ud835\udc2b\ud835\udc1e\ud835\udc26: https://andymath.com/Hinge-theorem/\n\n\ud835\udc12\ud835\udc22\ud835\udc26\ud835\udc22\ud835\udc25\ud835\udc1a\ud835\udc2b \ud835\udc11\ud835\udc22\ud835\udc20\ud835\udc21\ud835\udc2d \ud835\udc13\ud835\udc2b\ud835\udc22\ud835\udc1a\ud835\udc27\ud835\udc20\ud835\udc25\ud835\udc1e\ud835\udc2c: https://andymath.com/geometric-mean-similar-right-triangles/\n\nVisit any of the the above links for more practice problems, notes, solutions and videos for all these related topics. For even more math topics, visit https://andymath.com/.\n\n#Shorts", "lengthSeconds": 40, "width": 202, "height": 360}
{"video_id": "lFIicvO4-qY", "transcript": [{"start": 0.0, "duration": 4.319, "text": "hey guys so we have a circle i want to"}, {"start": 1.76, "duration": 4.96, "text": "find the area of this red region of the"}, {"start": 4.319, "duration": 4.32, "text": "circle so let's call that x in order to"}, {"start": 6.72, "duration": 4.16, "text": "solve for x first i'm going to find this"}, {"start": 8.639, "duration": 4.96, "text": "blue sector right here and then i'm"}, {"start": 10.88, "duration": 4.16, "text": "going to subtract this orange triangle"}, {"start": 13.599, "duration": 3.76, "text": "right here and if we call the large"}, {"start": 15.04, "duration": 4.8, "text": "sector a and the triangle b we know that"}, {"start": 17.359, "duration": 4.721, "text": "a minus b is going to give us x first"}, {"start": 19.84, "duration": 3.76, "text": "let's solve for a if this is 60 degrees"}, {"start": 22.08, "duration": 3.76, "text": "right here we know this angle right here"}, {"start": 23.6, "duration": 4.32, "text": "is going to be 120 degrees the area of"}, {"start": 25.84, "duration": 5.679, "text": "an entire circle is pi r squared and"}, {"start": 27.92, "duration": 5.2, "text": "this is only 120 out of 360 degrees of"}, {"start": 31.519, "duration": 4.401, "text": "an entire circle so we'll say the"}, {"start": 33.12, "duration": 4.72, "text": "proportion 120 over 360. and 120 over"}, {"start": 35.92, "duration": 3.52, "text": "360 is the same thing as one third and"}, {"start": 37.84, "duration": 4.48, "text": "then the radius here is six so i'm going"}, {"start": 39.44, "duration": 6.48, "text": "to plug in 6 for r 6 squared is 36 and"}, {"start": 42.32, "duration": 5.2, "text": "then 1 3 of 36 is 12. so 12 pi is our"}, {"start": 45.92, "duration": 3.44, "text": "answer and we can even move it up here"}, {"start": 47.52, "duration": 3.359, "text": "to the a next let's look at this orange"}, {"start": 49.36, "duration": 3.76, "text": "triangle this is the same angle from"}, {"start": 50.879, "duration": 4.241, "text": "above so it's still 120 and now we don't"}, {"start": 53.12, "duration": 3.68, "text": "really need the 60 anymore and this side"}, {"start": 55.12, "duration": 3.759, "text": "of the triangle right here is a radius"}, {"start": 56.8, "duration": 3.2, "text": "as well so it's also going to be six"}, {"start": 58.879, "duration": 2.961, "text": "because they're both coming from the"}, {"start": 60.0, "duration": 3.44, "text": "center to the edge of the circle now we"}, {"start": 61.84, "duration": 3.92, "text": "don't even need the circle anymore"}, {"start": 63.44, "duration": 4.4, "text": "enhance and let's flip it around and"}, {"start": 65.76, "duration": 4.24, "text": "bisect this 120 degrees that's going to"}, {"start": 67.84, "duration": 4.319, "text": "give us 60 degrees for both sides of"}, {"start": 70.0, "duration": 3.68, "text": "this let's focus on this left triangle"}, {"start": 72.159, "duration": 3.921, "text": "and then we know it's a right angle"}, {"start": 73.68, "duration": 4.079, "text": "anytime you bisect that top angle in an"}, {"start": 76.08, "duration": 3.679, "text": "isosceles triangle it always meets at"}, {"start": 77.759, "duration": 4.0, "text": "right angles and this last angle is"}, {"start": 79.759, "duration": 4.481, "text": "going to be 30 degrees now we have a 30"}, {"start": 81.759, "duration": 4.0, "text": "60 90 triangle this is a special type of"}, {"start": 84.24, "duration": 3.84, "text": "triangle here's the notes right here on"}, {"start": 85.759, "duration": 4.241, "text": "andymath.com the side opposite the 30"}, {"start": 88.08, "duration": 4.16, "text": "degrees is always half the hypotenuse"}, {"start": 90.0, "duration": 3.84, "text": "this hypotenuse is 6 so this side right"}, {"start": 92.24, "duration": 3.919, "text": "here is going to be 3 and then the side"}, {"start": 93.84, "duration": 4.959, "text": "opposite the 60 is always square root of"}, {"start": 96.159, "duration": 4.721, "text": "3 times the smallest side so this right"}, {"start": 98.799, "duration": 3.921, "text": "here is gonna be three root three since"}, {"start": 100.88, "duration": 3.36, "text": "these two are similar triangles this is"}, {"start": 102.72, "duration": 3.359, "text": "also three root three and when we"}, {"start": 104.24, "duration": 3.36, "text": "combine them we get six root three let's"}, {"start": 106.079, "duration": 3.121, "text": "clean up the stuff we don't need now we"}, {"start": 107.6, "duration": 3.519, "text": "need to find the area of this triangle"}, {"start": 109.2, "duration": 4.16, "text": "it's gonna be one half base times height"}, {"start": 111.119, "duration": 4.881, "text": "the base is this six root three and the"}, {"start": 113.36, "duration": 4.24, "text": "height is three so six times one half is"}, {"start": 116.0, "duration": 3.36, "text": "three and then three root three times"}, {"start": 117.6, "duration": 3.44, "text": "three is nine root three so that's our"}, {"start": 119.36, "duration": 3.84, "text": "value for the triangle so let's change"}, {"start": 121.04, "duration": 4.56, "text": "these up here we'll make the a into the"}, {"start": 123.2, "duration": 4.0, "text": "twelve pi and we'll make the b into nine"}, {"start": 125.6, "duration": 3.84, "text": "root three and now we have the answer to"}, {"start": 127.2, "duration": 6.56, "text": "the question the area of this red region"}, {"start": 129.44, "duration": 4.32, "text": "is 12 pi minus 9 root 3."}], "title": "Geometry Challenge Problem", "description": "For more math fun, check out andymath.com!", "lengthSeconds": 132, "width": 360, "height": 640}
{"video_id": "RKYie9VHNDs", "transcript": [{"start": 0.04, "duration": 3.639, "text": "hey guys this looks like a fun one it"}, {"start": 1.36, "duration": 4.519, "text": "has two circles inside of a square and"}, {"start": 3.679, "duration": 4.041, "text": "it wants to know what is this angle this"}, {"start": 5.879, "duration": 3.601, "text": "angle is created with these three points"}, {"start": 7.72, "duration": 3.28, "text": "of tangency if you want to try it on"}, {"start": 9.48, "duration": 4.92, "text": "your own postive right now cuz I'm going"}, {"start": 11.0, "duration": 5.279, "text": "to solve it in 3 2 1 first I want to get"}, {"start": 14.4, "duration": 3.52, "text": "this question mark out of the way and"}, {"start": 16.279, "duration": 4.16, "text": "then we'll construct the common tangent"}, {"start": 17.92, "duration": 5.679, "text": "line of these two circles let's label"}, {"start": 20.439, "duration": 5.801, "text": "this angle X De and this angle y de now"}, {"start": 23.599, "duration": 4.481, "text": "our goal is to find what is x + y next"}, {"start": 26.24, "duration": 3.879, "text": "let's label the centers of both of these"}, {"start": 28.08, "duration": 4.039, "text": "circles this radius will be at right"}, {"start": 30.119, "duration": 3.241, "text": "angles to this side right here and"}, {"start": 32.119, "duration": 2.801, "text": "that's because the tangent line is"}, {"start": 33.36, "duration": 3.68, "text": "always at right angles to the radius"}, {"start": 34.92, "duration": 3.479, "text": "that intersects at that point here's the"}, {"start": 37.04, "duration": 2.88, "text": "theorem right here and then let's"}, {"start": 38.399, "duration": 3.601, "text": "connect this other radius and that'll be"}, {"start": 39.92, "duration": 3.88, "text": "at right angles to this tangent line"}, {"start": 42.0, "duration": 4.079, "text": "let's find the size of this angle right"}, {"start": 43.8, "duration": 4.239, "text": "here we know that y plus this angle has"}, {"start": 46.079, "duration": 5.0, "text": "to equal 90\u00b0 that means this angle is"}, {"start": 48.039, "duration": 5.281, "text": "the same thing as 90 - y de and let's"}, {"start": 51.079, "duration": 3.96, "text": "look at this triangle more closely this"}, {"start": 53.32, "duration": 4.16, "text": "side and this side are both radi of the"}, {"start": 55.039, "duration": 4.52, "text": "circle since these two sides are equal"}, {"start": 57.48, "duration": 3.84, "text": "that means this is an isoceles triangle"}, {"start": 59.559, "duration": 4.441, "text": "and when you an isoceles triangle these"}, {"start": 61.32, "duration": 5.2, "text": "base angles are also equal so since this"}, {"start": 64.0, "duration": 5.4, "text": "one is 90- y that means this one will"}, {"start": 66.52, "duration": 5.32, "text": "also be 90- Y and now looking up here"}, {"start": 69.4, "duration": 4.92, "text": "since this is 90 minus y and this whole"}, {"start": 71.84, "duration": 4.279, "text": "thing is 90 that leaves y for this angle"}, {"start": 74.32, "duration": 4.04, "text": "right here so we've shown that both of"}, {"start": 76.119, "duration": 3.601, "text": "these angles are equal to Y de and now"}, {"start": 78.36, "duration": 3.2, "text": "let's go back up here and we're going to"}, {"start": 79.72, "duration": 3.359, "text": "do basically the same thing if we"}, {"start": 81.56, "duration": 3.76, "text": "connect this radius it'll be at right"}, {"start": 83.079, "duration": 4.321, "text": "angles to this tangent line and now to"}, {"start": 85.32, "duration": 4.56, "text": "find this angle if this is X that leaves"}, {"start": 87.4, "duration": 4.24, "text": "90- X for this angle and if we complete"}, {"start": 89.88, "duration": 4.0, "text": "complete this radius this makes another"}, {"start": 91.64, "duration": 4.519, "text": "Isles triangle and that's because this"}, {"start": 93.88, "duration": 4.76, "text": "radius is equal to this radius and in"}, {"start": 96.159, "duration": 4.64, "text": "all Isles triangles the base angles are"}, {"start": 98.64, "duration": 5.439, "text": "congruent so if this is 90 - x that"}, {"start": 100.799, "duration": 6.28, "text": "means this one is also 90 - x since this"}, {"start": 104.079, "duration": 5.241, "text": "is 90 - x that leaves X for this angle"}, {"start": 107.079, "duration": 3.64, "text": "right here next let's get rid of this"}, {"start": 109.32, "duration": 3.159, "text": "piece of the tangent line and we're"}, {"start": 110.719, "duration": 4.44, "text": "going to combine the angle X and the"}, {"start": 112.479, "duration": 4.361, "text": "angle y to get X Plus Y and then since"}, {"start": 115.159, "duration": 4.121, "text": "this whole thing is a square that means"}, {"start": 116.84, "duration": 3.639, "text": "this angle right here is a 90\u00b0 Angle now"}, {"start": 119.28, "duration": 3.159, "text": "we have a really cool looking"}, {"start": 120.479, "duration": 4.481, "text": "quadrilateral the sum of the interior"}, {"start": 122.439, "duration": 4.6, "text": "angles of any quadrilateral is equal to"}, {"start": 124.96, "duration": 3.359, "text": "360\u00b0 that means if we add up all these"}, {"start": 127.039, "duration": 6.241, "text": "angles we'll get"}, {"start": 128.319, "duration": 7.121, "text": "360\u00b0 so x + x + y + y + 90 is equal to"}, {"start": 133.28, "duration": 4.28, "text": "360 and then from here we can subtract"}, {"start": 135.44, "duration": 5.04, "text": "90 from both sides of the equation we"}, {"start": 137.56, "duration": 6.44, "text": "end up with x + x + y + y is equal to"}, {"start": 140.48, "duration": 6.2, "text": "270 and then y + y is equal to 2 y and x"}, {"start": 144.0, "duration": 4.64, "text": "+ x is equal to 2x next we can divide"}, {"start": 146.68, "duration": 4.32, "text": "both sides of the equation by two on the"}, {"start": 148.64, "duration": 6.0, "text": "left hand side we have X + y and on the"}, {"start": 151.0, "duration": 6.319, "text": "right hand side 270 / 2 is 135 so now in"}, {"start": 154.64, "duration": 5.519, "text": "the place of this x + y we can plug in"}, {"start": 157.319, "duration": 4.441, "text": "135\u00b0 and this 135 is in the same place"}, {"start": 160.159, "duration": 3.36, "text": "as the question mark so we know that the"}, {"start": 161.76, "duration": 4.24, "text": "question mark is equal to"}, {"start": 163.519, "duration": 5.121, "text": "135\u00b0 this is the answer to our question"}, {"start": 166.0, "duration": 4.12, "text": "let's put a box around it how exciting"}, {"start": 168.64, "duration": 3.239, "text": "hey guys it's a couple days later and I"}, {"start": 170.12, "duration": 3.199, "text": "was editing the video and I realized I"}, {"start": 171.879, "duration": 3.08, "text": "missed something that I wanted to share"}, {"start": 173.319, "duration": 3.121, "text": "with you it doesn't matter how big the"}, {"start": 174.959, "duration": 3.721, "text": "circles are the circle can go all the"}, {"start": 176.44, "duration": 3.6, "text": "way up to here till it hits the top or"}, {"start": 178.68, "duration": 3.72, "text": "it can go all the way down here till"}, {"start": 180.04, "duration": 5.24, "text": "this one hits the side the angle will be"}, {"start": 182.4, "duration": 4.6, "text": "135\u00b0 in all these scenarios and there"}, {"start": 185.28, "duration": 3.2, "text": "was another detail I wanted to share"}, {"start": 187.0, "duration": 3.599, "text": "there's another reason why these two"}, {"start": 188.48, "duration": 3.8, "text": "angles match and these two angles match"}, {"start": 190.599, "duration": 3.881, "text": "if you have a circle and you extend two"}, {"start": 192.28, "duration": 4.519, "text": "tangent lines until they intersect these"}, {"start": 194.48, "duration": 4.44, "text": "two sides will always be congruent so"}, {"start": 196.799, "duration": 3.44, "text": "that means this forms an isoc triangle"}, {"start": 198.92, "duration": 2.8, "text": "and these two base angles would be"}, {"start": 200.239, "duration": 3.201, "text": "congruent so if you isolate this"}, {"start": 201.72, "duration": 4.36, "text": "triangle right here these two sides are"}, {"start": 203.44, "duration": 4.159, "text": "equal making these two base angles equal"}, {"start": 206.08, "duration": 3.359, "text": "or on this other side if you isolate"}, {"start": 207.599, "duration": 3.801, "text": "these in extend them these two sides"}, {"start": 209.439, "duration": 4.281, "text": "would be equal making these base angles"}, {"start": 211.4, "duration": 7.16, "text": "equal I hope you guys like these two"}, {"start": 213.72, "duration": 4.84, "text": "bonus insights how exciting"}], "title": "What's the angle? (With Bonus Footage)", "description": "Watch to the end, the bonus footage is super cool!", "lengthSeconds": 217, "width": 640, "height": 360}
{"video_id": "U1mXWZs5UDE", "transcript": [{"start": 0.04, "duration": 3.919, "text": "hey guys this looks like a fun one this"}, {"start": 1.64, "duration": 4.28, "text": "is another Katrina a puzzle it's given"}, {"start": 3.959, "duration": 3.561, "text": "us that this is made up of five squares"}, {"start": 5.92, "duration": 3.48, "text": "and it wants to know what's the total"}, {"start": 7.52, "duration": 3.56, "text": "shaded area if you want to try this on"}, {"start": 9.4, "duration": 4.84, "text": "your own pause it right now cuz I'm"}, {"start": 11.08, "duration": 5.199, "text": "going to solve it in 3 2 1 first I want"}, {"start": 14.24, "duration": 3.28, "text": "to look at these three angles they're"}, {"start": 16.279, "duration": 3.401, "text": "going to add up to"}, {"start": 17.52, "duration": 3.679, "text": "180\u00b0 and that's the same thing as a"}, {"start": 19.68, "duration": 3.439, "text": "triangle so if we look at this triangle"}, {"start": 21.199, "duration": 4.08, "text": "on the left here this right angle can be"}, {"start": 23.119, "duration": 4.0, "text": "brought here and the purple angle can be"}, {"start": 25.279, "duration": 4.281, "text": "brought there these three angles will"}, {"start": 27.119, "duration": 4.361, "text": "also add up to 180\u00b0 and then we can do"}, {"start": 29.56, "duration": 3.88, "text": "the same thing for this triangle over"}, {"start": 31.48, "duration": 3.48, "text": "here this blue right angle can go here"}, {"start": 33.44, "duration": 3.48, "text": "and then that means that this last angle"}, {"start": 34.96, "duration": 4.52, "text": "would have to be the green angle so"}, {"start": 36.92, "duration": 4.639, "text": "these two triangles have three congruent"}, {"start": 39.48, "duration": 4.52, "text": "angles that means the two triangles are"}, {"start": 41.559, "duration": 4.84, "text": "similar next let's focus on this literal"}, {"start": 44.0, "duration": 5.879, "text": "triangle let's label this side a this"}, {"start": 46.399, "duration": 5.201, "text": "side B and this hypotenuse C well this C"}, {"start": 49.879, "duration": 3.52, "text": "is also one of the sides of these"}, {"start": 51.6, "duration": 3.959, "text": "squares so another side of the square"}, {"start": 53.399, "duration": 4.16, "text": "would also be C in fact all of these"}, {"start": 55.559, "duration": 3.761, "text": "sides would be equal to C I don't think"}, {"start": 57.559, "duration": 3.921, "text": "there's anything else I know for now so"}, {"start": 59.32, "duration": 3.96, "text": "let's pull out these triangles and if we"}, {"start": 61.48, "duration": 3.88, "text": "rotate this one now the two similar"}, {"start": 63.28, "duration": 5.08, "text": "triangles are oriented the same let's"}, {"start": 65.36, "duration": 4.72, "text": "change these four C's into 4 C and now"}, {"start": 68.36, "duration": 5.0, "text": "since these are similar triangles we can"}, {"start": 70.08, "duration": 5.44, "text": "find the scale factor from C to 4 C that"}, {"start": 73.36, "duration": 4.759, "text": "means our scale factor is four to get"}, {"start": 75.52, "duration": 4.68, "text": "the base we're going to multiply a * 4"}, {"start": 78.119, "duration": 4.68, "text": "to get 4 a and to get this height we're"}, {"start": 80.2, "duration": 4.36, "text": "going to multiply B * 4 to get 4 B and"}, {"start": 82.799, "duration": 3.801, "text": "now we know these sides so we can update"}, {"start": 84.56, "duration": 4.44, "text": "this figure let's rotate this triangle"}, {"start": 86.6, "duration": 4.72, "text": "back and you can see that the 4B goes"}, {"start": 89.0, "duration": 3.96, "text": "there and the 4 a goes there so that"}, {"start": 91.32, "duration": 3.56, "text": "looks pretty good now we have three"}, {"start": 92.96, "duration": 3.839, "text": "different ways to measure the side of"}, {"start": 94.88, "duration": 3.72, "text": "this larger Square let's set them equal"}, {"start": 96.799, "duration": 4.881, "text": "to each other so we'll have this side a"}, {"start": 98.6, "duration": 6.08, "text": "+ 4 b equal to this side 4 a and then we"}, {"start": 101.68, "duration": 5.079, "text": "can also do this side b + 13 equal to"}, {"start": 104.68, "duration": 3.92, "text": "this side 4 a and now we have two"}, {"start": 106.759, "duration": 4.04, "text": "equations and two variables so we can"}, {"start": 108.6, "duration": 4.199, "text": "solve for both A and B let's start with"}, {"start": 110.799, "duration": 4.521, "text": "this one we can subtract a from both"}, {"start": 112.799, "duration": 5.081, "text": "sides and that gives us 4 B is equal to"}, {"start": 115.32, "duration": 4.88, "text": "3 a and next we can divide both sides by"}, {"start": 117.88, "duration": 5.0, "text": "four and that will give us B is equal to"}, {"start": 120.2, "duration": 4.599, "text": "34s a and this looks important so let's"}, {"start": 122.88, "duration": 3.96, "text": "put a box around it next let's look at"}, {"start": 124.799, "duration": 3.88, "text": "this equation let's copy it down and in"}, {"start": 126.84, "duration": 4.44, "text": "the place of the B we're going to plug"}, {"start": 128.679, "duration": 5.081, "text": "in 3/4s a and then we want to combine"}, {"start": 131.28, "duration": 4.239, "text": "like terms so let's subtract 34s a from"}, {"start": 133.76, "duration": 3.16, "text": "both sides of the equation on the right"}, {"start": 135.519, "duration": 2.921, "text": "hand side these are going to cancel each"}, {"start": 136.92, "duration": 4.319, "text": "other out and on the left hand side"}, {"start": 138.44, "duration": 4.28, "text": "we're going to do 4 A minus 34s a in"}, {"start": 141.239, "duration": 3.36, "text": "order to subtract these let's give them"}, {"start": 142.72, "duration": 5.0, "text": "a common denominator let's rewrite this"}, {"start": 144.599, "duration": 8.161, "text": "4 as 164s and then we're going to do"}, {"start": 147.72, "duration": 6.879, "text": "164s a - 3 4 a which is equal to 134s a"}, {"start": 152.76, "duration": 4.119, "text": "now to get the a all by itself we'll"}, {"start": 154.599, "duration": 4.321, "text": "multiply both sides by the reciprocal"}, {"start": 156.879, "duration": 3.64, "text": "413 on the left hand side these are"}, {"start": 158.92, "duration": 3.64, "text": "going to cancel each other out and on"}, {"start": 160.519, "duration": 3.921, "text": "the right hand side these two 13s will"}, {"start": 162.56, "duration": 4.0, "text": "cancel each other out so we end up with"}, {"start": 164.44, "duration": 4.0, "text": "a is equal to four and this looks"}, {"start": 166.56, "duration": 3.8, "text": "important so let's put a box around it"}, {"start": 168.44, "duration": 5.36, "text": "now going back to our previous box we"}, {"start": 170.36, "duration": 5.68, "text": "know B is equal to 34s of a a is equal"}, {"start": 173.8, "duration": 4.28, "text": "to four so in the place of this a let's"}, {"start": 176.04, "duration": 4.88, "text": "plug in four and then when we multiply"}, {"start": 178.08, "duration": 4.879, "text": "this 34s * 4 four the fours are going to"}, {"start": 180.92, "duration": 4.52, "text": "cancel each other out we end up with B"}, {"start": 182.959, "duration": 5.121, "text": "is equal to 3 in the place of both of"}, {"start": 185.44, "duration": 5.359, "text": "these A's we can plug in four and then 4"}, {"start": 188.08, "duration": 4.879, "text": "* 4 is equal to 16 and in the place of"}, {"start": 190.799, "duration": 4.921, "text": "both of these B's we can plug in three"}, {"start": 192.959, "duration": 4.84, "text": "and 4 * 3 is equal to 12 so now let's"}, {"start": 195.72, "duration": 3.56, "text": "focus on this little triangle right here"}, {"start": 197.799, "duration": 3.52, "text": "this is a special triangle it's called a"}, {"start": 199.28, "duration": 3.84, "text": "Pythagorean triple anytime a right"}, {"start": 201.319, "duration": 4.28, "text": "triangle has legs three and four the"}, {"start": 203.12, "duration": 4.0, "text": "hypotenuse will always be five and since"}, {"start": 205.599, "duration": 3.481, "text": "this side is five that means this side"}, {"start": 207.12, "duration": 4.839, "text": "is also five and the area of this Square"}, {"start": 209.08, "duration": 4.92, "text": "would be 5 * 5 which is 25 and 25 would"}, {"start": 211.959, "duration": 4.161, "text": "be the area of all these squares so if"}, {"start": 214.0, "duration": 5.64, "text": "we want the area of the Shaded region"}, {"start": 216.12, "duration": 5.319, "text": "we're going to do 4 * 25 and 4 * 25 is"}, {"start": 219.64, "duration": 4.599, "text": "100 and that is the answer to our"}, {"start": 221.439, "duration": 5.8, "text": "question let's put a box around it how"}, {"start": 224.239, "duration": 3.0, "text": "exciting"}], "title": "Fun Math Problem", "description": "Another Catriona Agg Problem. I thought this was a fun one. Let me know if you know another way to solve it!", "lengthSeconds": 225, "width": 640, "height": 360}
{"video_id": "dBNyK4EYZ6A", "transcript": [{"start": 0.0, "duration": 5.56, "text": "hey guys this looks like a fun one we"}, {"start": 1.36, "duration": 8.319, "text": "have 12 to power of x - 4 is equal to"}, {"start": 5.56, "duration": 5.92, "text": "the < TK of 2 to^ 2x - 4 if you want to"}, {"start": 9.679, "duration": 4.88, "text": "try this on your own pause it right now"}, {"start": 11.48, "duration": 4.799, "text": "cuz I'm going to solve it in 3 2 1 so"}, {"start": 14.559, "duration": 4.001, "text": "first let's copy down the exponents and"}, {"start": 16.279, "duration": 4.321, "text": "let's leave the parentheses blank we"}, {"start": 18.56, "duration": 4.799, "text": "want to express these two with the same"}, {"start": 20.6, "duration": 4.08, "text": "base 1/2 and S > of two both contain a"}, {"start": 23.359, "duration": 3.76, "text": "two so for both of them we're going to"}, {"start": 24.68, "duration": 3.88, "text": "do two to some power for this on the"}, {"start": 27.119, "duration": 3.56, "text": "left we're going to use a negative"}, {"start": 28.56, "duration": 4.839, "text": "exponent so if you have X to 1 that's"}, {"start": 30.679, "duration": 5.521, "text": "equal to 1 /x that takes the X on top"}, {"start": 33.399, "duration": 5.16, "text": "and puts it on bottom or if you have 1X"}, {"start": 36.2, "duration": 4.44, "text": "to the1 that takes the X on bottom and"}, {"start": 38.559, "duration": 3.361, "text": "puts it on top so regardless of which"}, {"start": 40.64, "duration": 3.28, "text": "one you start with it takes the"}, {"start": 41.92, "duration": 4.4, "text": "reciprocal to make it look like our"}, {"start": 43.92, "duration": 6.2, "text": "situation we can change these X's into"}, {"start": 46.32, "duration": 5.64, "text": "twos and now we have 2 the1 is equal to2"}, {"start": 50.12, "duration": 5.56, "text": "well that means that we can rewrite this"}, {"start": 51.96, "duration": 5.04, "text": "1/2 as 2 to the1 so we haven't changed"}, {"start": 55.68, "duration": 3.44, "text": "anything on the left hand side we've"}, {"start": 57.0, "duration": 3.32, "text": "only changed how we expressed it so now"}, {"start": 59.12, "duration": 4.84, "text": "on the right hand side side we're going"}, {"start": 60.32, "duration": 6.44, "text": "to use rational exponents it says x 1 /"}, {"start": 63.96, "duration": 4.56, "text": "n is equal to the N root of x well in"}, {"start": 66.76, "duration": 3.08, "text": "this particular example it's the square"}, {"start": 68.52, "duration": 3.48, "text": "root so we're going to do the square"}, {"start": 69.84, "duration": 4.319, "text": "root of x and the square root has an"}, {"start": 72.0, "duration": 4.08, "text": "implied index of two so that means this"}, {"start": 74.159, "duration": 3.761, "text": "n is going to be equal to two so the"}, {"start": 76.08, "duration": 4.2, "text": "square < TK of two is the same thing as"}, {"start": 77.92, "duration": 3.839, "text": "2 to the 1/2 power that means this"}, {"start": 80.28, "duration": 3.44, "text": "Square < TK of two is the same thing as"}, {"start": 81.759, "duration": 3.72, "text": "2 to the 1/2 power and now we can move"}, {"start": 83.72, "duration": 4.12, "text": "on to the next step we're doing the"}, {"start": 85.479, "duration": 4.401, "text": "power of a power these are the notes"}, {"start": 87.84, "duration": 4.0, "text": "right here if you have x to the a print"}, {"start": 89.88, "duration": 4.919, "text": "pareses to the B that's the same thing"}, {"start": 91.84, "duration": 7.8, "text": "as X the a * B so for the left hand side"}, {"start": 94.799, "duration": 7.36, "text": "2 -1 x - 4 is the same thing as 2-1 * x"}, {"start": 99.64, "duration": 6.439, "text": "- 4 and then if we distribute this 1 to"}, {"start": 102.159, "duration": 4.96, "text": "both terms this x - 4 becomes x + 4 and"}, {"start": 106.079, "duration": 3.121, "text": "then we're going to do the same thing on"}, {"start": 107.119, "duration": 5.201, "text": "the right hand side this 1/2 is going to"}, {"start": 109.2, "duration": 5.559, "text": "multiply by the 2x - 4 and that 1/2 will"}, {"start": 112.32, "duration": 5.2, "text": "cut both of these terms in half so 2x -"}, {"start": 114.759, "duration": 4.72, "text": "4 is going to change into 1 x - 2 so we"}, {"start": 117.52, "duration": 4.12, "text": "have 2 to this power is equal to 2 to"}, {"start": 119.479, "duration": 5.68, "text": "this power now that we've made the bases"}, {"start": 121.64, "duration": 5.759, "text": "match we can drop the base Drop the"}, {"start": 125.159, "duration": 4.401, "text": "Base and we can set the exponents equal"}, {"start": 127.399, "duration": 3.881, "text": "to each other and solve for x we have a"}, {"start": 129.56, "duration": 3.399, "text": "negative X and a positive X let's move"}, {"start": 131.28, "duration": 3.84, "text": "this x to this side we'll do that by"}, {"start": 132.959, "duration": 3.961, "text": "adding X to both sides and I want to"}, {"start": 135.12, "duration": 4.16, "text": "move this two to the other side so we'll"}, {"start": 136.92, "duration": 3.92, "text": "do that by adding two to both sides on"}, {"start": 139.28, "duration": 3.2, "text": "the left hand side these two x's will"}, {"start": 140.84, "duration": 3.679, "text": "cancel each other out leaving us with"}, {"start": 142.48, "duration": 3.6, "text": "six and on the right hand side these two"}, {"start": 144.519, "duration": 4.641, "text": "twos will cancel each other out leaving"}, {"start": 146.08, "duration": 6.12, "text": "us with X+ X or 2x after we divide both"}, {"start": 149.16, "duration": 5.6, "text": "sides by two we get 3 is equal to X and"}, {"start": 152.2, "duration": 7.319, "text": "that is the answer to our question xal 3"}, {"start": 154.76, "duration": 4.759, "text": "let's put a box around it how exciting"}], "title": "Exponential Drop the Base", "description": "I hope you guys love it!", "lengthSeconds": 158, "width": 640, "height": 360}
{"video_id": "gGxFmhF6G2o", "transcript": [{"start": 0.0, "duration": 3.06, "text": "hey guys this looks like a fun one I"}, {"start": 1.68, "duration": 3.599, "text": "think we would call it a radical"}, {"start": 3.06, "duration": 3.779, "text": "equation first step to solve this I want"}, {"start": 5.279, "duration": 3.181, "text": "to get rid of this radical right here so"}, {"start": 6.839, "duration": 3.541, "text": "let's copy this down and square both"}, {"start": 8.46, "duration": 3.36, "text": "sides on the right hand side this square"}, {"start": 10.38, "duration": 2.879, "text": "and square root are going to cancel each"}, {"start": 11.82, "duration": 3.06, "text": "other out giving us just the inside"}, {"start": 13.259, "duration": 3.36, "text": "stuff and we could just copy down the"}, {"start": 14.88, "duration": 3.54, "text": "left hand side normally with these we"}, {"start": 16.619, "duration": 3.181, "text": "would multiply this out but this has a"}, {"start": 18.42, "duration": 2.76, "text": "matching one right here so we're just"}, {"start": 19.8, "duration": 3.059, "text": "going to subtract that from both sides"}, {"start": 21.18, "duration": 3.12, "text": "the left hand side is just going to be 0"}, {"start": 22.859, "duration": 3.061, "text": "and the right hand side we bring down"}, {"start": 24.3, "duration": 3.54, "text": "this other stuff and smush them together"}, {"start": 25.92, "duration": 4.14, "text": "let's bring this to the other side and"}, {"start": 27.84, "duration": 3.48, "text": "add 8 to both sides these cancel so the"}, {"start": 30.06, "duration": 2.82, "text": "left hand side is just this and the"}, {"start": 31.32, "duration": 2.94, "text": "right hand side is going to be eight now"}, {"start": 32.88, "duration": 2.82, "text": "to get rid of this square root we're"}, {"start": 34.26, "duration": 2.88, "text": "going to square both sides again the"}, {"start": 35.7, "duration": 3.42, "text": "left hand side is just going to be this"}, {"start": 37.14, "duration": 3.96, "text": "Inside Stuff x minus 4 and the right"}, {"start": 39.12, "duration": 4.38, "text": "side is going to be 64. add 4 to both"}, {"start": 41.1, "duration": 4.88, "text": "sides and we get x equals 68 how"}, {"start": 43.5, "duration": 2.48, "text": "exciting"}], "title": "Radical Equation Challenge #shorts", "description": "#shorts", "lengthSeconds": 44, "width": 360, "height": 640}
{"video_id": "4o-XK0OIWfk", "transcript": [{"start": 0.0, "duration": 4.02, "text": "guys this looks like a fun one this was"}, {"start": 1.86, "duration": 3.66, "text": "posted by my math Solutions we've got"}, {"start": 4.02, "duration": 3.18, "text": "three rectangles and it wants us to know"}, {"start": 5.52, "duration": 4.02, "text": "the area of this rectangle in the middle"}, {"start": 7.2, "duration": 4.74, "text": "so first let's focus on this gray"}, {"start": 9.54, "duration": 4.679, "text": "rectangle area of it is going to be base"}, {"start": 11.94, "duration": 3.96, "text": "times height this two centimeters only"}, {"start": 14.219, "duration": 4.261, "text": "goes to here so let's make the rest of"}, {"start": 15.9, "duration": 5.4, "text": "this X and since the bottom is X plus 2"}, {"start": 18.48, "duration": 4.62, "text": "the top would also be X plus two and we"}, {"start": 21.3, "duration": 4.139, "text": "could change the base into X plus two"}, {"start": 23.1, "duration": 4.019, "text": "now for the height we know this part is"}, {"start": 25.439, "duration": 4.441, "text": "three and for the unknown part let's"}, {"start": 27.119, "duration": 5.341, "text": "call it Z and since this is three and"}, {"start": 29.88, "duration": 4.92, "text": "this is only two we need to make this Z"}, {"start": 32.46, "duration": 4.38, "text": "plus one so that both Heights are equal"}, {"start": 34.8, "duration": 3.9, "text": "to Z plus three and now let's bring"}, {"start": 36.84, "duration": 2.94, "text": "these rectangles back let's get rid of"}, {"start": 38.7, "duration": 3.48, "text": "the question mark and bring this"}, {"start": 39.78, "duration": 4.92, "text": "rectangle inside here and let's focus on"}, {"start": 42.18, "duration": 4.62, "text": "this rectangle here this 26 centimeter"}, {"start": 44.7, "duration": 3.96, "text": "squared refers to this backwards L shape"}, {"start": 46.8, "duration": 4.5, "text": "but we don't know the gray area so we'll"}, {"start": 48.66, "duration": 3.899, "text": "call that a we know that a plus 26 is"}, {"start": 51.3, "duration": 3.599, "text": "going to be equal to the area of the"}, {"start": 52.559, "duration": 4.741, "text": "whole rectangle which is 7 times 6. 7"}, {"start": 54.899, "duration": 4.201, "text": "times 6 is 42 and after we subtract 26"}, {"start": 57.3, "duration": 4.32, "text": "from both sides we find out that a is"}, {"start": 59.1, "duration": 4.5, "text": "equal to 16. and a is this rectangle"}, {"start": 61.62, "duration": 4.679, "text": "made up of x times e so really we found"}, {"start": 63.6, "duration": 4.559, "text": "out that x times Z equals 16. and that's"}, {"start": 66.299, "duration": 3.481, "text": "really what's most important here and"}, {"start": 68.159, "duration": 3.901, "text": "now let's extend this other rectangle"}, {"start": 69.78, "duration": 4.98, "text": "and focus on it if we call the area of"}, {"start": 72.06, "duration": 5.879, "text": "this region B we know that b plus 36 is"}, {"start": 74.76, "duration": 6.06, "text": "going to be equal to 7 times 8. 7 times"}, {"start": 77.939, "duration": 5.161, "text": "8 is 56 and after we subtract 36 from"}, {"start": 80.82, "duration": 4.56, "text": "both sides we get D is equal to 20. and"}, {"start": 83.1, "duration": 4.379, "text": "this B is also equal to the base times"}, {"start": 85.38, "duration": 3.599, "text": "the height or in other words x times Z"}, {"start": 87.479, "duration": 3.421, "text": "plus one and that's really what's"}, {"start": 88.979, "duration": 3.661, "text": "important here now we have a system of"}, {"start": 90.9, "duration": 4.2, "text": "equations we have two variables and two"}, {"start": 92.64, "duration": 4.26, "text": "equations let's get X by itself for both"}, {"start": 95.1, "duration": 3.839, "text": "of these for this top one we'll divide"}, {"start": 96.9, "duration": 3.6, "text": "both sides by Z these Z's will cancel"}, {"start": 98.939, "duration": 3.841, "text": "each other out leaving us with just x"}, {"start": 100.5, "duration": 4.38, "text": "equals 16 over Z and for this one let's"}, {"start": 102.78, "duration": 4.019, "text": "divide both sides by Z plus one and then"}, {"start": 104.88, "duration": 4.739, "text": "these will cancel leaving us with just x"}, {"start": 106.799, "duration": 5.28, "text": "equals a 20 over Z plus one so we have"}, {"start": 109.619, "duration": 4.5, "text": "found that X is equal to 16 over Z well"}, {"start": 112.079, "duration": 4.441, "text": "in the place of this x down here I'm"}, {"start": 114.119, "duration": 4.021, "text": "going to plug in 16 over Z and now we"}, {"start": 116.52, "duration": 4.2, "text": "have a proportion we can solve this by"}, {"start": 118.14, "duration": 5.88, "text": "cross multiplying we have 16x times Z"}, {"start": 120.72, "duration": 5.399, "text": "plus 1 is equal to this 20 times Z and"}, {"start": 124.02, "duration": 5.519, "text": "then we can distribute the 16. 16 times"}, {"start": 126.119, "duration": 5.34, "text": "Z is 16z and then 16 times 1 is 16. and"}, {"start": 129.539, "duration": 4.381, "text": "on this side we have 20 Z and then we"}, {"start": 131.459, "duration": 4.081, "text": "can subtract 16z from both sides on the"}, {"start": 133.92, "duration": 3.78, "text": "left hand side these are gone so we just"}, {"start": 135.54, "duration": 4.559, "text": "have 16 and on the right hand side we"}, {"start": 137.7, "duration": 5.16, "text": "have 4z we can divide both sides by 4"}, {"start": 140.099, "duration": 5.28, "text": "and we have 4 equals z and then we can"}, {"start": 142.86, "duration": 4.62, "text": "rewrite that as Z equals four so since"}, {"start": 145.379, "duration": 4.561, "text": "we know the Z equals 4 I'm going to plug"}, {"start": 147.48, "duration": 5.399, "text": "in 4 for this Z right here and 16"}, {"start": 149.94, "duration": 5.22, "text": "divided by 4 is 4. so we know x equals 4"}, {"start": 152.879, "duration": 4.141, "text": "so we can plug in 4 for this X and we"}, {"start": 155.16, "duration": 5.34, "text": "know Z equals 4 so we can plug in 4 for"}, {"start": 157.02, "duration": 6.18, "text": "this Z here 4 plus 2 is equal to 6 and"}, {"start": 160.5, "duration": 5.459, "text": "here 4 plus 3 is equal to 7 and 6 times"}, {"start": 163.2, "duration": 4.98, "text": "7 is equal to 42. we have now found the"}, {"start": 165.959, "duration": 4.201, "text": "value of our question mark it's 42. so"}, {"start": 168.18, "duration": 3.72, "text": "for this question mark let's put in 42"}, {"start": 170.16, "duration": 3.78, "text": "centimeters squared that is the answer"}, {"start": 171.9, "duration": 5.9, "text": "to our question and it's already got a"}, {"start": 173.94, "duration": 3.86, "text": "box around it how exciting"}], "title": "Area Challenge Problem", "description": "I thought this was a fun one. I hope you guys like it!", "lengthSeconds": 176, "width": 640, "height": 360}
{"video_id": "NCR8TEtg_ZU", "transcript": [{"start": 0.0, "duration": 3.32, "text": "hey guys this looks like a fun one it"}, {"start": 1.439, "duration": 4.281, "text": "wants us to solve for x and it gives us"}, {"start": 3.32, "duration": 5.319, "text": "this triangle that is not to scale we"}, {"start": 5.72, "duration": 5.72, "text": "have angle measures of a de 2 a de and a"}, {"start": 8.639, "duration": 4.841, "text": "de it gives us the base is 5 cm and it"}, {"start": 11.44, "duration": 3.52, "text": "wants us to find this longest side if"}, {"start": 13.48, "duration": 2.879, "text": "you want to try on your own pause it"}, {"start": 14.96, "duration": 5.0, "text": "right now because I'm going to solve it"}, {"start": 16.359, "duration": 5.84, "text": "in 3 2 1 so first thing we know the sum"}, {"start": 19.96, "duration": 6.6, "text": "of the interior angles of any triangle"}, {"start": 22.199, "duration": 7.48, "text": "adds to 180\u00b0 so a + 2 a + a will equal"}, {"start": 26.56, "duration": 5.64, "text": "180 and on the left hand side a + 2 a +"}, {"start": 29.679, "duration": 5.241, "text": "a simplifies to 4 a and that's still"}, {"start": 32.2, "duration": 5.96, "text": "equal to 180 now to get a by itself we"}, {"start": 34.92, "duration": 7.4, "text": "can divide both sides by four 4 a ID 4"}, {"start": 38.16, "duration": 6.719, "text": "is equal to a and 180 / 4 is 45 so we"}, {"start": 42.32, "duration": 4.16, "text": "now know that a is equal to 45 and every"}, {"start": 44.879, "duration": 3.32, "text": "single time I do one of these videos"}, {"start": 46.48, "duration": 2.719, "text": "someone tells me that a is actually"}, {"start": 48.199, "duration": 3.321, "text": "equal to"}, {"start": 49.199, "duration": 4.961, "text": "45\u00b0 but if you notice with the degree"}, {"start": 51.52, "duration": 6.96, "text": "symbol up here this is a degrees so if a"}, {"start": 54.16, "duration": 6.48, "text": "was 45\u00b0 that would make this 45\u00b0 de so"}, {"start": 58.48, "duration": 5.079, "text": "we know it's not 45\u00b0"}, {"start": 60.64, "duration": 5.56, "text": "so a is only 45 and that would make this"}, {"start": 63.559, "duration": 4.2, "text": "angle 45\u00b0 I wonder if someone will still"}, {"start": 66.2, "duration": 4.08, "text": "comment that in the comments even after"}, {"start": 67.759, "duration": 5.081, "text": "I said this but a is for sure equal to"}, {"start": 70.28, "duration": 5.36, "text": "45 and then this a degrees up here will"}, {"start": 72.84, "duration": 5.599, "text": "change into 45\u00b0 and this 2 a degrees"}, {"start": 75.64, "duration": 5.08, "text": "will change into 90\u00b0 so we know for sure"}, {"start": 78.439, "duration": 4.161, "text": "this image is not to scale because 90\u00b0"}, {"start": 80.72, "duration": 5.039, "text": "would be perpendicular right here in"}, {"start": 82.6, "duration": 4.839, "text": "fact we can fix this and this is what it"}, {"start": 85.759, "duration": 3.561, "text": "should look like and now that we fix the"}, {"start": 87.439, "duration": 4.68, "text": "triangle let's get rid of the knot and"}, {"start": 89.32, "duration": 5.92, "text": "now this is image to scale that looks"}, {"start": 92.119, "duration": 5.241, "text": "nice now in any isos triangle that's any"}, {"start": 95.24, "duration": 4.159, "text": "triangle with two angles the same the"}, {"start": 97.36, "duration": 4.84, "text": "sides opposite those congruent angles"}, {"start": 99.399, "duration": 6.4, "text": "will be congruent since this side is 5"}, {"start": 102.2, "duration": 4.879, "text": "cm this side will also be 5 cm so right"}, {"start": 105.799, "duration": 3.521, "text": "now there's two different ways we can"}, {"start": 107.079, "duration": 4.841, "text": "solve this we can use the notes for a 45"}, {"start": 109.32, "duration": 5.04, "text": "4590 triangle and this tells us that the"}, {"start": 111.92, "duration": 6.04, "text": "hypotenuse is whatever this side is"}, {"start": 114.36, "duration": 7.399, "text": "times the < tk2 so if this side is 5 cm"}, {"start": 117.96, "duration": 5.4, "text": "the hypotenuse will be 5 < tk2 2 cm or"}, {"start": 121.759, "duration": 5.36, "text": "the other method is to do Pythagorean"}, {"start": 123.36, "duration": 7.92, "text": "theorem it's a^ 2 + b^2 is equal to c^2"}, {"start": 127.119, "duration": 6.921, "text": "so we can do 5^ 2 + 5^2 is = to x^2 on"}, {"start": 131.28, "duration": 6.039, "text": "the left hand side 5^2 is equal to 25"}, {"start": 134.04, "duration": 5.559, "text": "and 5^2 is equal to 25 and 25 + 25 is"}, {"start": 137.319, "duration": 3.721, "text": "equal to 50 and then to solve for x we"}, {"start": 139.599, "duration": 3.681, "text": "can square root both sides of the"}, {"start": 141.04, "duration": 5.4, "text": "equation the rooot 50 can be broken down"}, {"start": 143.28, "duration": 4.92, "text": "into rooot 25 * 2 and then we can give"}, {"start": 146.44, "duration": 4.48, "text": "each of these their own square root and"}, {"start": 148.2, "duration": 4.96, "text": "the square root of 25 is equal to 5 and"}, {"start": 150.92, "duration": 4.56, "text": "when taking the square < TK of x^2 since"}, {"start": 153.16, "duration": 5.28, "text": "we know this is a distance it must be"}, {"start": 155.48, "duration": 5.92, "text": "positive so this simplifies into X and"}, {"start": 158.44, "duration": 4.719, "text": "now we have X is equal to POS 5 < tk2"}, {"start": 161.4, "duration": 4.36, "text": "this is the answer to our question let's"}, {"start": 163.159, "duration": 4.041, "text": "put a box around it how exciting hey"}, {"start": 165.76, "duration": 3.24, "text": "guys I was just editing this and I"}, {"start": 167.2, "duration": 5.039, "text": "realized right at the end here I want to"}, {"start": 169.0, "duration": 7.04, "text": "point out that X does not equal 5 < tk2"}, {"start": 172.239, "duration": 5.161, "text": "cm it's 5 tk2 without the centim label"}, {"start": 176.04, "duration": 4.64, "text": "and that's because that hypotenuse would"}, {"start": 177.4, "duration": 4.8, "text": "end up being 5 < tk2 cm s I know this"}, {"start": 180.68, "duration": 3.199, "text": "one isn't as crazy as some of my other"}, {"start": 182.2, "duration": 3.759, "text": "videos but I really like the fact that"}, {"start": 183.879, "duration": 3.92, "text": "the image is not to scale and I really"}, {"start": 185.959, "duration": 3.601, "text": "like the discussion of the labels for"}, {"start": 187.799, "duration": 4.041, "text": "the variables how you don't want to make"}, {"start": 189.56, "duration": 3.759, "text": "it 45 degrees degrees so I thought this"}, {"start": 191.84, "duration": 3.08, "text": "would make a fun video I hope you guys"}, {"start": 193.319, "duration": 3.121, "text": "liked it and if you like to see more"}, {"start": 194.92, "duration": 4.52, "text": "math content I'm going to put a"}, {"start": 196.44, "duration": 6.0, "text": "subscribe button somewhere up here how"}, {"start": 199.44, "duration": 3.0, "text": "exciting"}], "title": "Triangle Not to Scale", "description": "", "lengthSeconds": 204, "width": 640, "height": 360}
{"video_id": "I3bS4FlkUno", "transcript": [{"start": 0.0, "duration": 4.12, "text": "so we're want to find the area of this"}, {"start": 1.64, "duration": 4.199, "text": "shaded sector so this is a circle that's"}, {"start": 4.12, "duration": 3.92, "text": "been cut up and we're finding how much"}, {"start": 5.839, "duration": 4.041, "text": "area is this shaded stuff so first what"}, {"start": 8.04, "duration": 5.0, "text": "is the area of the entire circle well"}, {"start": 9.88, "duration": 5.12, "text": "the area of a circle is < R 2 but this"}, {"start": 13.04, "duration": 4.999, "text": "isn't the entire circle how much of the"}, {"start": 15.0, "duration": 8.039, "text": "circle is this an entire circle would be"}, {"start": 18.039, "duration": 6.201, "text": "360\u00b0 and this is 172 of those 360\u00b0 so"}, {"start": 23.039, "duration": 2.601, "text": "we're just going to multiply these two"}, {"start": 24.24, "duration": 3.959, "text": "things by each other now we still need"}, {"start": 25.64, "duration": 4.84, "text": "to plug in for R for the P pi r 2qu and"}, {"start": 28.199, "duration": 8.401, "text": "this is the r the 10 is the radi"}, {"start": 30.48, "duration": 8.0, "text": "10^ 2 is 100 * < * the 172 over 360 so"}, {"start": 36.6, "duration": 4.0, "text": "now if we plug this into a calculator"}, {"start": 38.48, "duration": 4.52, "text": "get approximately 1"}, {"start": 40.6, "duration": 4.4, "text": "15.1 and they don't give us a unit so we"}, {"start": 43.0, "duration": 3.719, "text": "can say unit squared CU they don't tell"}, {"start": 45.0, "duration": 2.879, "text": "us feet or inches or anything like that"}, {"start": 46.719, "duration": 2.561, "text": "and that's the answer to the question"}, {"start": 47.879, "duration": 3.961, "text": "comment below on what I should solve for"}, {"start": 49.28, "duration": 5.56, "text": "next I'll talk to you soon bye Andy"}, {"start": 51.84, "duration": 3.0, "text": "maath"}], "title": "Area of a Sector #andymath  #math  #maths  #sat  #tutorials  #tiktokmath  #mathhelp", "description": "Please Subscribe, and don't forget to hit the bell icon so you don't miss next video! Also, please feel free to share this video with your friends and family. Let learn Math together with me, Charlie. \nHere is the playlist of the videos. [ There is so much more to discover in this playlist ]", "lengthSeconds": 53, "width": 360, "height": 640}
{"video_id": "gtclP08Q9Ok", "transcript": [{"start": 0.04, "duration": 3.28, "text": "hey guys you got to watch this video"}, {"start": 1.439, "duration": 4.321, "text": "this guy hits a home run and then throws"}, {"start": 3.32, "duration": 2.44, "text": "the bat"}, {"start": 5.799, "duration": 9.081, "text": "up and it stays up for really long"}, {"start": 10.599, "duration": 4.281, "text": "time I got a timer going right"}, {"start": 15.119, "duration": 6.08, "text": "here and it stays up for 11.83 seconds"}, {"start": 19.16, "duration": 4.439, "text": "so let's figure out how high did the bat"}, {"start": 21.199, "duration": 4.721, "text": "reach we can visualize the path of the"}, {"start": 23.599, "duration": 4.84, "text": "bat as a prao where the horizontal axis"}, {"start": 25.92, "duration": 4.279, "text": "can be t for time in seconds and the"}, {"start": 28.439, "duration": 3.64, "text": "vertical axis can be H for the height"}, {"start": 30.199, "duration": 3.52, "text": "that the bat goes so we're interested in"}, {"start": 32.079, "duration": 3.561, "text": "this point right here how high did the"}, {"start": 33.719, "duration": 4.441, "text": "bat reach so this is the formula we're"}, {"start": 35.64, "duration": 5.68, "text": "going to use the height is equal to -4.9"}, {"start": 38.16, "duration": 5.04, "text": "T ^2 plus the initial velocity time time"}, {"start": 41.32, "duration": 3.52, "text": "plus the initial height so we'll call"}, {"start": 43.2, "duration": 3.359, "text": "the initial height zero because the top"}, {"start": 44.84, "duration": 3.6, "text": "of the bat is really close to the ground"}, {"start": 46.559, "duration": 3.84, "text": "as he throws it and we don't know the"}, {"start": 48.44, "duration": 5.0, "text": "initial velocity but we do know at the"}, {"start": 50.399, "duration": 5.601, "text": "time 11.83 seconds the height once again"}, {"start": 53.44, "duration": 5.119, "text": "was zero so we can use this information"}, {"start": 56.0, "duration": 4.199, "text": "to find the initial velocity let's copy"}, {"start": 58.559, "duration": 4.0, "text": "down our equation let's give our our El"}, {"start": 60.199, "duration": 5.401, "text": "some space let's plug in zero and in the"}, {"start": 62.559, "duration": 4.761, "text": "place of these T's let's plug in 11.83"}, {"start": 65.6, "duration": 6.08, "text": "and in the place of the H let's plug in"}, {"start": 67.32, "duration": 6.479, "text": "zero 4.9 * 11.83 2 is-"}, {"start": 71.68, "duration": 5.84, "text": "68575 and let's copy down everything"}, {"start": 73.799, "duration": 7.96, "text": "else let's add 68575 to both sides we"}, {"start": 77.52, "duration": 6.16, "text": "end up with 68575 is equal to 11.83 *"}, {"start": 81.759, "duration": 4.481, "text": "the initial velocity after we divide"}, {"start": 83.68, "duration": 3.96, "text": "both sides by 11.83 we get the initial"}, {"start": 86.24, "duration": 3.96, "text": "velocity is"}, {"start": 87.64, "duration": 4.0, "text": "57.97 and that's measured in meters per"}, {"start": 90.2, "duration": 2.48, "text": "second and if you're curious that's"}, {"start": 91.64, "duration": 6.24, "text": "about"}, {"start": 92.68, "duration": 7.24, "text": "28.6n km/ hour or 129. 68 mph so that's"}, {"start": 97.88, "duration": 3.72, "text": "pretty fast we can update the initial"}, {"start": 99.92, "duration": 3.76, "text": "velocity up here to"}, {"start": 101.6, "duration": 3.879, "text": "57.97 and now we don't need this stuff"}, {"start": 103.68, "duration": 3.399, "text": "anymore now that we know our initial"}, {"start": 105.479, "duration": 3.6, "text": "velocity here's our new equation"}, {"start": 107.079, "duration": 3.601, "text": "representing the height so now to find"}, {"start": 109.079, "duration": 3.161, "text": "this maximum height we first need to"}, {"start": 110.68, "duration": 2.479, "text": "plug in this point right here which we"}, {"start": 112.24, "duration": 3.44, "text": "know is"}, {"start": 113.159, "duration": 4.88, "text": "11.83 the axis of symmetry will go"}, {"start": 115.68, "duration": 4.2, "text": "through that maximum point we can find"}, {"start": 118.039, "duration": 4.201, "text": "this Axis of Symmetry by taking the aage"}, {"start": 119.88, "duration": 3.72, "text": "aage of these two x intercepts and"}, {"start": 122.24, "duration": 3.64, "text": "that's going to give us"}, {"start": 123.6, "duration": 5.84, "text": "5915 and so now we take our equation"}, {"start": 125.88, "duration": 6.56, "text": "again plug in 5915 for T and we get our"}, {"start": 129.44, "duration": 7.32, "text": "height of 171.5 M and that is the answer"}, {"start": 132.44, "duration": 4.32, "text": "to our question how exciting"}], "title": "How high was the throw?", "description": "https://andymath.com/quadratic-word-problems/\n\nFor more math involving parabolas, check out this page!", "lengthSeconds": 135, "width": 640, "height": 360}
{"video_id": "gmxEfMJ2Hm0", "transcript": [{"start": 0.0, "duration": 3.76, "text": "hey guys welcome to this probability"}, {"start": 2.08, "duration": 3.52, "text": "question you flip a coin"}, {"start": 3.76, "duration": 3.76, "text": "two times what is the probability you"}, {"start": 5.6, "duration": 2.48, "text": "get two heads the best way to do this is"}, {"start": 7.52, "duration": 2.72, "text": "to"}, {"start": 8.08, "duration": 3.84, "text": "show all the possible outcomes we're"}, {"start": 10.24, "duration": 3.6, "text": "gonna flip the coin twice so we have our"}, {"start": 11.92, "duration": 3.839, "text": "first toss and our second toss so our"}, {"start": 13.84, "duration": 3.68, "text": "first toss can be a head and our second"}, {"start": 15.759, "duration": 2.481, "text": "toss can be a head or our first toss"}, {"start": 17.52, "duration": 2.8, "text": "could be a head"}, {"start": 18.24, "duration": 3.76, "text": "and our second toss could be a tail our"}, {"start": 20.32, "duration": 2.879, "text": "first toss could be a tail and the"}, {"start": 22.0, "duration": 2.88, "text": "second could be a head"}, {"start": 23.199, "duration": 3.441, "text": "or the first toss could be a tail and"}, {"start": 24.88, "duration": 4.0, "text": "the second could be a tail this"}, {"start": 26.64, "duration": 3.84, "text": "should represent all possible outcomes"}, {"start": 28.88, "duration": 2.239, "text": "each of these has an equal probability"}, {"start": 30.48, "duration": 2.64, "text": "of happening"}, {"start": 31.119, "duration": 3.6, "text": "so there's a one and four chance so now"}, {"start": 33.12, "duration": 2.16, "text": "was it probably of getting two heads"}, {"start": 34.719, "duration": 2.161, "text": "well this"}, {"start": 35.28, "duration": 3.119, "text": "is the one scenario where we get two"}, {"start": 36.88, "duration": 3.999, "text": "heads so the answer is one"}, {"start": 38.399, "duration": 3.84, "text": "fourth as a percent that would be 25 and"}, {"start": 40.879, "duration": 3.041, "text": "that's the answer to the question if you"}, {"start": 42.239, "duration": 3.281, "text": "flip a coin twice the problem of you"}, {"start": 43.92, "duration": 3.76, "text": "getting two heads in a row"}, {"start": 45.52, "duration": 3.68, "text": "is 25 comment below on what you'd like"}, {"start": 47.68, "duration": 3.399, "text": "me to solve next and i'll talk to you"}, {"start": 49.2, "duration": 4.879, "text": "soon"}, {"start": 51.079, "duration": 3.0, "text": "bye"}], "title": "Probability 2 Coins Fast Example #Shorts", "description": "\ud835\udc0b\ud835\udc08\ud835\udc0d\ud835\udc0a\ud835\udc12 \ud835\udc13\ud835\udc0e \ud835\udc11\ud835\udc04\ud835\udc0b\ud835\udc00\ud835\udc13\ud835\udc04\ud835\udc03 \ud835\udc13\ud835\udc0e\ud835\udc0f\ud835\udc08\ud835\udc02\ud835\udc12\n\ud835\udc0f\ud835\udc2b\ud835\udc28\ud835\udc1b\ud835\udc1a\ud835\udc1b\ud835\udc22\ud835\udc25\ud835\udc22\ud835\udc2d\ud835\udc32- \ud835\udc02\ud835\udc28\ud835\udc22\ud835\udc27 \ud835\udc13\ud835\udc28\ud835\udc2c\ud835\udc2c\ud835\udc1e\ud835\udc2c: https://andymath.com/probability-coin-tosses/\n\ud835\udc0f\ud835\udc2b\ud835\udc28\ud835\udc1b\ud835\udc1a\ud835\udc1b\ud835\udc22\ud835\udc25\ud835\udc22\ud835\udc2d\ud835\udc32- \ud835\udc0c\ud835\udc1a\ud835\udc2b\ud835\udc1b\ud835\udc25\ud835\udc1e\ud835\udc2c: https://andymath.com/probability-marbles/\n\n\ud835\udc05\ud835\udc1a\ud835\udc1c\ud835\udc2d\ud835\udc28\ud835\udc2b\ud835\udc22\ud835\udc1a\ud835\udc25\ud835\udc2c: https://andymath.com/factorials/\n\n\ud835\udc02\ud835\udc28\ud835\udc26\ud835\udc1b\ud835\udc22\ud835\udc27\ud835\udc1a\ud835\udc2d\ud835\udc22\ud835\udc28\ud835\udc27\ud835\udc2c \ud835\udc1a\ud835\udc27\ud835\udc1d \ud835\udc0f\ud835\udc1e\ud835\udc2b\ud835\udc26\ud835\udc2e\ud835\udc2d\ud835\udc1a\ud835\udc2d\ud835\udc22\ud835\udc28\ud835\udc27\ud835\udc2c: https://andymath.com/combinations-and-permutations/\n\n\ud835\udc04\ud835\udc31\ud835\udc29\ud835\udc1e\ud835\udc2b\ud835\udc22\ud835\udc26\ud835\udc1e\ud835\udc27\ud835\udc2d\ud835\udc1a\ud835\udc25 \ud835\udc1a\ud835\udc27\ud835\udc1d \ud835\udc13\ud835\udc21\ud835\udc1e\ud835\udc28\ud835\udc2b\ud835\udc1e\ud835\udc2d\ud835\udc22\ud835\udc1c\ud835\udc1a\ud835\udc25 \ud835\udc0f\ud835\udc2b\ud835\udc28\ud835\udc1b\ud835\udc1a\ud835\udc1b\ud835\udc22\ud835\udc25\ud835\udc22\ud835\udc2d\ud835\udc32: https://andymath.com/experimental-and-theoretical-probability/\n\n\ud835\udc02\ud835\udc28\ud835\udc26\ud835\udc29\ud835\udc25\ud835\udc1e\ud835\udc26\ud835\udc1e\ud835\udc27\ud835\udc2d \ud835\udc28\ud835\udc1f \ud835\udc1a\ud835\udc27 \ud835\udc04\ud835\udc2f\ud835\udc1e\ud835\udc27\ud835\udc2d: https://andymath.com/complement-of-an-event/\n\n\ud835\udc0f\ud835\udc2b\ud835\udc28\ud835\udc1b\ud835\udc1a\ud835\udc1b\ud835\udc22\ud835\udc25\ud835\udc22\ud835\udc2d\ud835\udc32 \"\ud835\udc00\ud835\udc2d \ud835\udc0b\ud835\udc1e\ud835\udc1a\ud835\udc2c\ud835\udc2d \ud835\udc0e\ud835\udc27\ud835\udc1e\": https://andymath.com/probability-at-least-one/\n\n\ud835\udc0c\ud835\udc2e\ud835\udc2d\ud835\udc2e\ud835\udc1a\ud835\udc25\ud835\udc25\ud835\udc32 \ud835\udc04\ud835\udc31\ud835\udc1c\ud835\udc25\ud835\udc2e\ud835\udc2c\ud835\udc22\ud835\udc2f\ud835\udc1e \ud835\udc04\ud835\udc2f\ud835\udc1e\ud835\udc27\ud835\udc2d\ud835\udc2c (\ud835\udc03\ud835\udc22\ud835\udc2c\ud835\udc23\ud835\udc28\ud835\udc22\ud835\udc27\ud835\udc2d): https://andymath.com/mutually-exclusive-events/\n\n\ud835\udc02\ud835\udc28\ud835\udc27\ud835\udc1d\ud835\udc22\ud835\udc2d\ud835\udc22\ud835\udc28\ud835\udc27\ud835\udc1a\ud835\udc25 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https://andymath.com/two-way-tables-statistics/\n\n\ud835\udc13\ud835\udc2b\ud835\udc1e\ud835\udc1e \ud835\udc03\ud835\udc22\ud835\udc1a\ud835\udc20\ud835\udc2b\ud835\udc1a\ud835\udc26\ud835\udc2c: https://andymath.com/tree-diagrams/\n\n\ud835\udc0f\ud835\udc2b\ud835\udc28\ud835\udc1b\ud835\udc1a\ud835\udc1b\ud835\udc22\ud835\udc25\ud835\udc22\ud835\udc2d\ud835\udc32 \ud835\udc30\ud835\udc22\ud835\udc2d\ud835\udc21 \ud835\udc03\ud835\udc22\ud835\udc1c\ud835\udc1e: https://andymath.com/probability-with-dice/\n\n\ud835\udc0f\ud835\udc2b\ud835\udc28\ud835\udc1b\ud835\udc1a\ud835\udc1b\ud835\udc22\ud835\udc25\ud835\udc22\ud835\udc2d\ud835\udc32- \ud835\udc11\ud835\udc28\ud835\udc2e\ud835\udc27\ud835\udc1d \ud835\udc13\ud835\udc1a\ud835\udc1b\ud835\udc25\ud835\udc1e: https://andymath.com/probability-round-table/\n\n\ud835\udc0f\ud835\udc2b\ud835\udc28\ud835\udc1b\ud835\udc1a\ud835\udc1b\ud835\udc22\ud835\udc25\ud835\udc22\ud835\udc2d\ud835\udc32- \ud835\udfd3 \ud835\udc1c\ud835\udc1a\ud835\udc2b\ud835\udc1d \ud835\udc0f\ud835\udc28\ud835\udc24\ud835\udc1e\ud835\udc2b \ud835\udc07\ud835\udc1a\ud835\udc27\ud835\udc1d\ud835\udc2c: https://andymath.com/probability-poker-hands/\n\n\ud835\udc15\ud835\udc1e\ud835\udc27\ud835\udc27 \ud835\udc03\ud835\udc22\ud835\udc1a\ud835\udc20\ud835\udc2b\ud835\udc1a\ud835\udc26\ud835\udc2c: https://andymath.com/venn-diagrams/\n\nVisit any of the the above links for more practice problems, notes, solutions and videos for all these related topics. For even more math topics, visit https://andymath.com/.\n\n#Shorts", "lengthSeconds": 52, "width": 202, "height": 360}
{"video_id": "NTEJg9741oY", "transcript": [{"start": 0.0, "duration": 4.92, "text": "hey guys so which one of these is"}, {"start": 1.62, "duration": 4.44, "text": "greater 5 to the 100 or 3 to the 150. we"}, {"start": 4.92, "duration": 2.28, "text": "could probably just plug this into a"}, {"start": 6.06, "duration": 2.76, "text": "calculator I think it'd be more fun to"}, {"start": 7.2, "duration": 4.2, "text": "try this by hand without a calculator"}, {"start": 8.82, "duration": 4.62, "text": "upon first inspection 5 is greater than"}, {"start": 11.4, "duration": 3.779, "text": "three a one fifty is greater than 100 so"}, {"start": 13.44, "duration": 3.48, "text": "it's not immediately obvious which one's"}, {"start": 15.179, "duration": 4.981, "text": "greater I can fairly easily change this"}, {"start": 16.92, "duration": 4.8, "text": "5 into 25 and the 3 into a 27. so once"}, {"start": 20.16, "duration": 3.24, "text": "we do this it might be easier to compare"}, {"start": 21.72, "duration": 4.2, "text": "these so let's rewrite 5 to the power of"}, {"start": 23.4, "duration": 4.68, "text": "100 as 5 to the power of two times fifty"}, {"start": 25.92, "duration": 4.26, "text": "and let's rewrite 3 to the power of 150"}, {"start": 28.08, "duration": 3.359, "text": "as 3 to the power of three times fifty"}, {"start": 30.18, "duration": 3.18, "text": "and we can strategically place"}, {"start": 31.439, "duration": 4.381, "text": "parentheses here and here and so we can"}, {"start": 33.36, "duration": 5.219, "text": "see 5 squared is just 25 so this would"}, {"start": 35.82, "duration": 4.62, "text": "be to the 50 and 3 cubes is just 27 so"}, {"start": 38.579, "duration": 4.621, "text": "this would be to the 50. now we have 25"}, {"start": 40.44, "duration": 4.799, "text": "to the 50 compared to 27 to the 50. and"}, {"start": 43.2, "duration": 3.84, "text": "27 to the 50 will be the greater one so"}, {"start": 45.239, "duration": 4.281, "text": "this is the answer to the question how"}, {"start": 47.04, "duration": 2.48, "text": "exciting"}], "title": "Exponents Math Challenge", "description": "https://andymath.com/", "lengthSeconds": 48, "width": 360, "height": 640}
{"video_id": "aHC2ItjGJuE", "transcript": [{"start": 0.0, "duration": 5.6, "text": "hey guys this looks like a fun one it"}, {"start": 1.4, "duration": 6.72, "text": "says x^2 + IX + 2 = 0 and it wants us to"}, {"start": 5.6, "duration": 4.039, "text": "solve for x this I coefficient can be"}, {"start": 8.12, "duration": 5.24, "text": "thought of two different ways you can"}, {"start": 9.639, "duration": 6.24, "text": "think of it as i^2 is equal to -1 or I"}, {"start": 13.36, "duration": 4.759, "text": "is equal to the < TK of1 it's called the"}, {"start": 15.879, "duration": 4.4, "text": "imaginary number and you can do a lot of"}, {"start": 18.119, "duration": 3.601, "text": "useful things with this I but for this"}, {"start": 20.279, "duration": 3.441, "text": "video I was wondering can we use"}, {"start": 21.72, "duration": 4.36, "text": "quadratic formula to solve this"}, {"start": 23.72, "duration": 4.879, "text": "quadratic if the coefficient is"}, {"start": 26.08, "duration": 4.64, "text": "imaginary let's try it out so a is the"}, {"start": 28.599, "duration": 4.241, "text": "coefficient for x^2 and up here that"}, {"start": 30.72, "duration": 4.12, "text": "would be an implied one so in the place"}, {"start": 32.84, "duration": 3.879, "text": "of this a we can plug in one and the"}, {"start": 34.84, "duration": 3.879, "text": "same thing for this A and B is the"}, {"start": 36.719, "duration": 3.801, "text": "coefficient for X this is where the I"}, {"start": 38.719, "duration": 3.481, "text": "comes in so in the place of this B we"}, {"start": 40.52, "duration": 3.76, "text": "can plug in I and we can do the same"}, {"start": 42.2, "duration": 3.999, "text": "thing to this B and last C is the"}, {"start": 44.28, "duration": 4.04, "text": "constant which is two up here so we can"}, {"start": 46.199, "duration": 3.761, "text": "change this C to two and now we can"}, {"start": 48.32, "duration": 3.759, "text": "clean things up let's copy down the"}, {"start": 49.96, "duration": 4.279, "text": "negative I and the plus or minus and the"}, {"start": 52.079, "duration": 4.48, "text": "square root and for this I squ we know"}, {"start": 54.239, "duration": 4.361, "text": "that i^2 is equal to -1 so I'm going to"}, {"start": 56.559, "duration": 4.64, "text": "change this to -1 and then we're going"}, {"start": 58.6, "duration": 4.919, "text": "to subtract 4 * 1 1 * 2 which is equal"}, {"start": 61.199, "duration": 4.441, "text": "to 8 and then on bottom the 2 * 1 is"}, {"start": 63.519, "duration": 5.441, "text": "equal to 2 next let's focus on this"}, {"start": 65.64, "duration": 7.479, "text": "radical we have -1 - 8 which is equal to"}, {"start": 68.96, "duration": 7.32, "text": "9 we can split up this < tk9 to beunk 1"}, {"start": 73.119, "duration": 6.201, "text": "* < tk9 and now this Ro tk1 if we look"}, {"start": 76.28, "duration": 5.799, "text": "up here is equal to I so we can change"}, {"start": 79.32, "duration": 6.72, "text": "this into I and then theun 9 is equal to"}, {"start": 82.079, "duration": 6.801, "text": "3 so now we have X is equal to i+ or - 3"}, {"start": 86.04, "duration": 4.719, "text": "i/ 2 This plus or minus is giving us two"}, {"start": 88.88, "duration": 3.96, "text": "solutions we have one for the plus and"}, {"start": 90.759, "duration": 6.481, "text": "one for the minus and now for the first"}, {"start": 92.84, "duration": 6.559, "text": "one I + 3 I is equal to 2 I and 2 / 2"}, {"start": 97.24, "duration": 5.36, "text": "cancel each other out so we have x = i"}, {"start": 99.399, "duration": 8.161, "text": "for the first one and for this one I - 3"}, {"start": 102.6, "duration": 7.44, "text": "I is -4 I and -4 / 2 is -2 so the other"}, {"start": 107.56, "duration": 4.559, "text": "solution is -2 I so the quadratic"}, {"start": 110.04, "duration": 4.48, "text": "formula did seem to work but let's check"}, {"start": 112.119, "duration": 4.201, "text": "our answers let's copy down the original"}, {"start": 114.52, "duration": 4.72, "text": "equation and in the place of both of"}, {"start": 116.32, "duration": 5.839, "text": "these X's we want to plug in -2 I for"}, {"start": 119.24, "duration": 5.76, "text": "the first term term -2 i^ 2 the squar is"}, {"start": 122.159, "duration": 4.121, "text": "going to go both to the -2 and to the I"}, {"start": 125.0, "duration": 4.239, "text": "and then for the second term we can"}, {"start": 126.28, "duration": 5.08, "text": "bring down the -2 and I * I is equal to"}, {"start": 129.239, "duration": 4.521, "text": "i^ 2 and the rest of this can be brought"}, {"start": 131.36, "duration": 5.0, "text": "down and then before we move on -22 is"}, {"start": 133.76, "duration": 4.96, "text": "the same thing as -2 * -2 which is equal"}, {"start": 136.36, "duration": 5.2, "text": "to pos4 for the next step let's copy"}, {"start": 138.72, "duration": 4.76, "text": "this down and then i^2 is equal to -1 so"}, {"start": 141.56, "duration": 7.319, "text": "in the place of both of these I squ we"}, {"start": 143.48, "duration": 9.72, "text": "can plug in -1 4 * -1 is -4 and -2 * -1"}, {"start": 148.879, "duration": 6.401, "text": "is pos2 and then -4 + 2 + 2 is equal 0"}, {"start": 153.2, "duration": 4.039, "text": "so we end up with 0 equal 0 so the first"}, {"start": 155.28, "duration": 3.679, "text": "one checks out so now let's check the"}, {"start": 157.239, "duration": 4.28, "text": "second one in the place of both of these"}, {"start": 158.959, "duration": 7.721, "text": "X's we can plug in I so we end up with"}, {"start": 161.519, "duration": 7.681, "text": "i^2 + i^2 + 2 = 0 and then i^ 2 is equal"}, {"start": 166.68, "duration": 7.52, "text": "to 1 so each of these I squar can change"}, {"start": 169.2, "duration": 6.959, "text": "into 1 and then - 1 +1 + 2 is equal to 0"}, {"start": 174.2, "duration": 4.039, "text": "so we end up with 0 equals 0 and the"}, {"start": 176.159, "duration": 3.881, "text": "second one checks out so the quadratic"}, {"start": 178.239, "duration": 4.0, "text": "formula did work for an imagin"}, {"start": 180.04, "duration": 7.36, "text": "coefficient and we now have our answers"}, {"start": 182.239, "duration": 5.161, "text": "let's put a box around them how exciting"}], "title": "Would Quadratic Formula Work?", "description": "I hope you guys like this one! Let me know if you have any other ideas for problems!", "lengthSeconds": 186, "width": 640, "height": 360}
{"video_id": "9kSv-OyKPlE", "transcript": [{"start": 0.04, "duration": 5.319, "text": "hey guys so we have the absolute value"}, {"start": 1.68, "duration": 5.4, "text": "of 4 - 3x is equal to X absolute value"}, {"start": 5.359, "duration": 3.801, "text": "means that whatever is inside here you"}, {"start": 7.08, "duration": 4.04, "text": "need to make it positive so there are"}, {"start": 9.16, "duration": 4.28, "text": "two ways this can be true if this stuff"}, {"start": 11.12, "duration": 4.88, "text": "inside of here is equal to X or if it's"}, {"start": 13.44, "duration": 4.12, "text": "equal tox and that's usually the way you"}, {"start": 16.0, "duration": 3.08, "text": "solve these you split it up one where"}, {"start": 17.56, "duration": 2.799, "text": "you're doing the positive of this thing"}, {"start": 19.08, "duration": 3.08, "text": "and one where you're doing the negative"}, {"start": 20.359, "duration": 4.16, "text": "of this thing and now we can just solve"}, {"start": 22.16, "duration": 3.84, "text": "each of these independently let's add 3x"}, {"start": 24.519, "duration": 3.16, "text": "to both sides so on the left hand side"}, {"start": 26.0, "duration": 4.519, "text": "we just have four and on the right hand"}, {"start": 27.679, "duration": 4.281, "text": "side we have x + 3x which is 4X and"}, {"start": 30.519, "duration": 4.001, "text": "after we divide both sides by four we"}, {"start": 31.96, "duration": 4.96, "text": "get 1 is equal to X and on the other"}, {"start": 34.52, "duration": 4.12, "text": "side we can add 3x to both sides once"}, {"start": 36.92, "duration": 5.119, "text": "again the left hand side is 4 but on the"}, {"start": 38.64, "duration": 5.36, "text": "right hand side x + 3x is 2X and after"}, {"start": 42.039, "duration": 4.04, "text": "we divide both sides by two we get X is"}, {"start": 44.0, "duration": 4.079, "text": "equal to 2 and now we just got to check"}, {"start": 46.079, "duration": 4.0, "text": "our answers so for the first one let's"}, {"start": 48.079, "duration": 4.28, "text": "plug in one and for the second one let's"}, {"start": 50.079, "duration": 4.48, "text": "plug in two inside this absolute value"}, {"start": 52.359, "duration": 4.281, "text": "we have 4 Min - 3 which is equal to one"}, {"start": 54.559, "duration": 3.881, "text": "the absolute value of one is one so that"}, {"start": 56.64, "duration": 4.96, "text": "checks out and then inside of this"}, {"start": 58.44, "duration": 5.279, "text": "absolute value we have 4 - 6 which is -2"}, {"start": 61.6, "duration": 4.04, "text": "and the absolute value of -2 is 2 so"}, {"start": 63.719, "duration": 4.841, "text": "that checks out so these are the answers"}, {"start": 65.64, "duration": 5.2, "text": "to this question x = 1 or x = 2 let's"}, {"start": 68.56, "duration": 3.44, "text": "put a box around them how exciting and"}, {"start": 70.84, "duration": 3.24, "text": "if you guys want to try more of these I"}, {"start": 72.0, "duration": 5.52, "text": "have this page on andy.com the link is"}, {"start": 74.08, "duration": 3.44, "text": "in the description"}], "title": "Absolute Value Equation Challenge", "description": "https://andymath.com/absolute-value-equations/", "lengthSeconds": 76, "width": 640, "height": 360}
{"video_id": "Dz6JvGYC5oo", "transcript": [{"start": 0.0, "duration": 3.52, "text": "hey guys so we have a cube right here is"}, {"start": 1.76, "duration": 3.44, "text": "a very nice looking cube to find the"}, {"start": 3.52, "duration": 3.44, "text": "volume of the cube if we call this edge"}, {"start": 5.2, "duration": 3.92, "text": "s then it would be s cubed if you have"}, {"start": 6.96, "duration": 4.24, "text": "another cube larger its volume would"}, {"start": 9.12, "duration": 4.32, "text": "also be s cubed where s is the length of"}, {"start": 11.2, "duration": 4.24, "text": "one of these edges on this blue one if i"}, {"start": 13.44, "duration": 4.08, "text": "make this edge equal to one meter then"}, {"start": 15.44, "duration": 3.999, "text": "the volume would equal one meter cubed"}, {"start": 17.52, "duration": 3.759, "text": "it's literally a cube of dimensions of"}, {"start": 19.439, "duration": 3.521, "text": "one meter now on the red cube if i"}, {"start": 21.279, "duration": 3.281, "text": "double the length of the edge when i"}, {"start": 22.96, "duration": 3.92, "text": "calculate the volume it ends up being"}, {"start": 24.56, "duration": 4.24, "text": "eight times larger so double the edge"}, {"start": 26.88, "duration": 3.6, "text": "volume is eight times larger now let's"}, {"start": 28.8, "duration": 3.759, "text": "look at spheres the formula for the"}, {"start": 30.48, "duration": 4.239, "text": "volume of a sphere is four thirds pi r"}, {"start": 32.559, "duration": 4.321, "text": "cubed if this blue sphere has a radius"}, {"start": 34.719, "duration": 4.16, "text": "equal to approximately 0.62 meters the"}, {"start": 36.88, "duration": 3.44, "text": "volume would be equal to one meter cube"}, {"start": 38.879, "duration": 3.441, "text": "it'd be the same volume as this blue"}, {"start": 40.32, "duration": 3.759, "text": "cube up here and then for the red sphere"}, {"start": 42.32, "duration": 3.759, "text": "if i double the radius then the volume"}, {"start": 44.079, "duration": 3.841, "text": "would be equal to eight meters cubed"}, {"start": 46.079, "duration": 3.761, "text": "double the radius i get eight times the"}, {"start": 47.92, "duration": 3.84, "text": "volume so now let's look at a pizza we"}, {"start": 49.84, "duration": 4.64, "text": "can think of a pizza as a cylinder so"}, {"start": 51.76, "duration": 4.4, "text": "it's volume equals pi r squared h and"}, {"start": 54.48, "duration": 3.12, "text": "let's also look at a larger pizza which"}, {"start": 56.16, "duration": 3.68, "text": "would also be a cylinder i want to make"}, {"start": 57.6, "duration": 4.08, "text": "these very very large pizzas i'm going"}, {"start": 59.84, "duration": 3.199, "text": "to make the height of it about 5"}, {"start": 61.68, "duration": 3.92, "text": "centimeters which would end up being"}, {"start": 63.039, "duration": 4.321, "text": "0.05 i'm plugging that in for h if the"}, {"start": 65.6, "duration": 4.32, "text": "radius of this blue pizza is equal to"}, {"start": 67.36, "duration": 4.799, "text": "2.52 meters the volume of the pizza"}, {"start": 69.92, "duration": 4.32, "text": "would be one meter cubed that is a giant"}, {"start": 72.159, "duration": 4.721, "text": "pizza now if i double the radius to get"}, {"start": 74.24, "duration": 4.879, "text": "5.04 meters what is the volume gonna be"}, {"start": 76.88, "duration": 4.559, "text": "of this pizza well it's equal to four"}, {"start": 79.119, "duration": 4.481, "text": "meters cubed so when i double the radius"}, {"start": 81.439, "duration": 3.921, "text": "i got four times the volume why did i"}, {"start": 83.6, "duration": 3.76, "text": "get eight times the volume for these but"}, {"start": 85.36, "duration": 5.6, "text": "only four times volume for this answer"}, {"start": 87.36, "duration": 3.6, "text": "down below in the comments how exciting"}], "title": "Pizza Math Problem", "description": "For more math fun, check out andymath.com!", "lengthSeconds": 89, "width": 360, "height": 640}
{"video_id": "fwG87AqMy70", "transcript": [{"start": 0.0, "duration": 4.24, "text": "so surface area is as if we were"}, {"start": 1.839, "duration": 4.0, "text": "painting the whole thing how much area"}, {"start": 4.24, "duration": 2.96, "text": "is that so we're gonna add up the area"}, {"start": 5.839, "duration": 4.001, "text": "of all six sides"}, {"start": 7.2, "duration": 4.72, "text": "this front side is two by five so it has"}, {"start": 9.84, "duration": 4.4, "text": "an area of ten and we also have the back"}, {"start": 11.92, "duration": 4.08, "text": "side that matches this that we can't see"}, {"start": 14.24, "duration": 4.48, "text": "we'd say two times this 10"}, {"start": 16.0, "duration": 4.64, "text": "plus this top is also 2 by 5 so it's"}, {"start": 18.72, "duration": 3.44, "text": "also 10 and then we also have the bottom"}, {"start": 20.64, "duration": 3.36, "text": "so we're gonna multiply that by two then"}, {"start": 22.16, "duration": 2.16, "text": "we have this side here which is two by"}, {"start": 24.0, "duration": 2.08, "text": "two"}, {"start": 24.32, "duration": 3.44, "text": "area four and then we have it on the"}, {"start": 26.08, "duration": 2.08, "text": "other side over there so there's two of"}, {"start": 27.76, "duration": 2.16, "text": "those"}, {"start": 28.16, "duration": 3.76, "text": "so we have the front and back top and"}, {"start": 29.92, "duration": 3.52, "text": "bottom and the two sides 2 times 10 is"}, {"start": 31.92, "duration": 4.319, "text": "20 2 times 10 is 20"}, {"start": 33.44, "duration": 3.52, "text": "and 2 times 4 is 8. 20 plus 20 plus 8 is"}, {"start": 36.239, "duration": 2.64, "text": "48"}, {"start": 36.96, "duration": 3.36, "text": "they don't give us a label anywhere so"}, {"start": 38.879, "duration": 3.36, "text": "we can just call this unit"}, {"start": 40.32, "duration": 3.2, "text": "squared and 48 units squared is the"}, {"start": 42.239, "duration": 1.761, "text": "answer to the question if you have any"}, {"start": 43.52, "duration": 3.12, "text": "other math"}, {"start": 44.0, "duration": 4.079, "text": "questions check out andymath.com or"}, {"start": 46.64, "duration": 2.16, "text": "comment below on what you'd like me to"}, {"start": 48.079, "duration": 4.16, "text": "do next"}, {"start": 48.8, "duration": 3.439, "text": "i'll talk to you soon bye"}], "title": "Surface Area Fast Example #Shorts", "description": "Check out my website! \nhttps://andymath.com/\n\n#Shorts", "lengthSeconds": 50, "width": 202, "height": 360}
{"video_id": "yFMEWEKQdB0", "transcript": [{"start": 0.0, "duration": 3.419, "text": "hey guys a while ago somebody mentioned"}, {"start": 1.68, "duration": 2.82, "text": "me in this video Let's Solve IT pause it"}, {"start": 3.419, "duration": 2.101, "text": "real quick if you want to try on your"}, {"start": 4.5, "duration": 2.819, "text": "own because I'm going to solve it now"}, {"start": 5.52, "duration": 3.42, "text": "this says the five semi-circles are"}, {"start": 7.319, "duration": 3.601, "text": "identical so we know they all have the"}, {"start": 8.94, "duration": 3.3, "text": "same radius and diameter it says solve"}, {"start": 10.92, "duration": 3.66, "text": "for the diameter so I'm going to label"}, {"start": 12.24, "duration": 3.959, "text": "each of these diameters as D I know that"}, {"start": 14.58, "duration": 3.3, "text": "the distance is the same whether I"}, {"start": 16.199, "duration": 3.061, "text": "measure from the top or the bottom so"}, {"start": 17.88, "duration": 3.72, "text": "I'm going to set those two things equal"}, {"start": 19.26, "duration": 4.38, "text": "to each other so we have triple d's plus"}, {"start": 21.6, "duration": 5.339, "text": "two twelves is going to be equal to"}, {"start": 23.64, "duration": 6.479, "text": "Double D's plus double 22s plus a single"}, {"start": 26.939, "duration": 4.801, "text": "16. well Double 12 is 24 and double 22"}, {"start": 30.119, "duration": 3.481, "text": "is 44 and then we can bring down"}, {"start": 31.74, "duration": 4.2, "text": "everything else let's combine like terms"}, {"start": 33.6, "duration": 4.26, "text": "the 44 and the 16 give us 60 and bring"}, {"start": 35.94, "duration": 4.38, "text": "down everything else subtract 2D from"}, {"start": 37.86, "duration": 5.34, "text": "both sides we get a single D plus 24"}, {"start": 40.32, "duration": 5.46, "text": "equals 60 and then subtract 24 from both"}, {"start": 43.2, "duration": 4.98, "text": "sides and we get D equals 60 minus 24"}, {"start": 45.78, "duration": 4.5, "text": "which is 36. let's squish these together"}, {"start": 48.18, "duration": 6.08, "text": "and put a box around it the diameter is"}, {"start": 50.28, "duration": 3.98, "text": "equal to 36. how exciting"}], "title": "Canadian Math Olympiad Challenge #Shorts", "description": "For more math fun visit https://andymath.com/\n\n#shorts", "lengthSeconds": 52, "width": 360, "height": 640}
{"video_id": "mMvqnLT3WQs", "transcript": [{"start": 0.04, "duration": 3.72, "text": "hey guys what's going on with this thing"}, {"start": 1.839, "duration": 3.881, "text": "right here how are we going to evaluate"}, {"start": 3.76, "duration": 4.44, "text": "this this is pretty confusing this thing"}, {"start": 5.72, "duration": 4.32, "text": "is quite ambiguous as far as I can tell"}, {"start": 8.2, "duration": 3.92, "text": "there's two ways to interpret this we"}, {"start": 10.04, "duration": 4.599, "text": "have this fraction on top divided by"}, {"start": 12.12, "duration": 4.88, "text": "this on bottom or we have the 4830"}, {"start": 14.639, "duration": 3.921, "text": "divided by all of this on bottom I don't"}, {"start": 17.0, "duration": 3.359, "text": "think it's clear which one of these is"}, {"start": 18.56, "duration": 3.44, "text": "intended so let's solve both of them and"}, {"start": 20.359, "duration": 3.041, "text": "see what happens I think I have a pretty"}, {"start": 22.0, "duration": 2.96, "text": "good idea and if you want to try it on"}, {"start": 23.4, "duration": 4.92, "text": "your own pause it right now cuz I'm"}, {"start": 24.96, "duration": 5.559, "text": "going to solve it in 3 2 1 let's do this"}, {"start": 28.32, "duration": 3.799, "text": "one first on bottom here the square root"}, {"start": 30.519, "duration": 3.441, "text": "of this fraction is equal to the"}, {"start": 32.119, "duration": 4.001, "text": "fraction of the square roots and then we"}, {"start": 33.96, "duration": 4.08, "text": "can put the two on top now we have two"}, {"start": 36.12, "duration": 3.88, "text": "fractions another way to think of this"}, {"start": 38.04, "duration": 4.64, "text": "is this fraction divided by this"}, {"start": 40.0, "duration": 4.0, "text": "fraction and we can express it like this"}, {"start": 42.68, "duration": 2.92, "text": "and now anytime you're dividing"}, {"start": 44.0, "duration": 3.719, "text": "fractions a cool way to handle this is"}, {"start": 45.6, "duration": 4.439, "text": "leave the first fraction alone change"}, {"start": 47.719, "duration": 4.201, "text": "the Divide to a multiply and flip the"}, {"start": 50.039, "duration": 5.601, "text": "second fraction on this side right here"}, {"start": 51.92, "duration": 5.4, "text": "we can break down the 805 into 23 * 35"}, {"start": 55.64, "duration": 3.16, "text": "and the square root of this product is"}, {"start": 57.32, "duration": 3.52, "text": "equal to the product of these square"}, {"start": 58.8, "duration": 3.84, "text": "roots and now we can smush it all"}, {"start": 60.84, "duration": 3.56, "text": "together the < TK of 23 on top and"}, {"start": 62.64, "duration": 4.68, "text": "bottom can cancel each other out and"}, {"start": 64.4, "duration": 9.079, "text": "then we can shift things around > 35 * >"}, {"start": 67.32, "duration": 8.839, "text": "35 is 35 and 35 * 2 is 70 4830 / 70 is"}, {"start": 73.479, "duration": 4.64, "text": "69 nice and that's the answer to our"}, {"start": 76.159, "duration": 3.96, "text": "first one so once again we have this"}, {"start": 78.119, "duration": 4.04, "text": "divided by this let's rewrite it like"}, {"start": 80.119, "duration": 4.081, "text": "this and then we can put the 4830 over"}, {"start": 82.159, "duration": 4.041, "text": "one and then we'll copy this down switch"}, {"start": 84.2, "duration": 4.08, "text": "the Divide to a multiply and flip the"}, {"start": 86.2, "duration": 3.72, "text": "second fraction once again the square"}, {"start": 88.28, "duration": 3.08, "text": "root of this fraction is equal to the"}, {"start": 89.92, "duration": 3.36, "text": "fraction of the square roots and then we"}, {"start": 91.36, "duration": 4.0, "text": "can put the two on top and now we have"}, {"start": 93.28, "duration": 4.159, "text": "another fraction divided by a fraction"}, {"start": 95.36, "duration": 4.119, "text": "let's express it like this and then once"}, {"start": 97.439, "duration": 4.081, "text": "again we can copy down this stuff change"}, {"start": 99.479, "duration": 4.201, "text": "the Divide to a multiply and flip the"}, {"start": 101.52, "duration": 3.559, "text": "divided by fraction we can smush"}, {"start": 103.68, "duration": 5.56, "text": "everything together and let's change"}, {"start": 105.079, "duration": 6.961, "text": "this 805 into a 23 * 35 and the < TK of"}, {"start": 109.24, "duration": 5.239, "text": "23 * 35 is the same thing as the < of 23"}, {"start": 112.04, "duration": 4.6, "text": "* < TK of 35 from here these two ones"}, {"start": 114.479, "duration": 4.401, "text": "ain't doing nothing so they can go away"}, {"start": 116.64, "duration": 5.36, "text": "and then rooot 35 overun 35 those can"}, {"start": 118.88, "duration": 5.599, "text": "also go away and then < tk23 * < tk23 is"}, {"start": 122.0, "duration": 4.88, "text": "equal to the whole number 23 and on top"}, {"start": 124.479, "duration": 7.801, "text": "4830 * 2 is equal to"}, {"start": 126.88, "duration": 6.84, "text": "960 and then 960 / 23 is equal to 420"}, {"start": 132.28, "duration": 3.92, "text": "and that's the answer to our second"}, {"start": 133.72, "duration": 4.36, "text": "ambiguous case so in conclusion this is"}, {"start": 136.2, "duration": 3.6, "text": "ambiguous we don't know what it is equal"}, {"start": 138.08, "duration": 7.4, "text": "to but we do know that it's equal to"}, {"start": 139.8, "duration": 5.68, "text": "either 69 nice or 420 how exciting"}], "title": "Some Nice Math", "description": "I hope you guys love the video!", "lengthSeconds": 144, "width": 640, "height": 360}
{"video_id": "fOAN0aETNSQ", "transcript": [{"start": 0.0, "duration": 3.659, "text": "hey guys this looks like a fun one we're"}, {"start": 1.62, "duration": 4.679, "text": "gonna find the area of the blue region"}, {"start": 3.659, "duration": 3.841, "text": "contained by these lines and the y-axis"}, {"start": 6.299, "duration": 2.34, "text": "if you want to try this on your own"}, {"start": 7.5, "duration": 3.48, "text": "pause it right now because I'm going to"}, {"start": 8.639, "duration": 3.781, "text": "solve it in three two one the equations"}, {"start": 10.98, "duration": 3.119, "text": "of these lines are given to us in"}, {"start": 12.42, "duration": 3.84, "text": "standard form I want to rewrite them as"}, {"start": 14.099, "duration": 4.44, "text": "y equals MX plus b which is known as"}, {"start": 16.26, "duration": 3.72, "text": "slope intercept form because m is the"}, {"start": 18.539, "duration": 3.361, "text": "slope of the line and B is the"}, {"start": 19.98, "duration": 4.32, "text": "y-intercept of the line let's take the"}, {"start": 21.9, "duration": 3.78, "text": "first equation X Plus y equals five and"}, {"start": 24.3, "duration": 3.18, "text": "in order to rewrite it in slope"}, {"start": 25.68, "duration": 3.72, "text": "intercept form we need to get y by"}, {"start": 27.48, "duration": 3.42, "text": "itself so we're going to subtract X from"}, {"start": 29.4, "duration": 3.179, "text": "both sides these X's are going to cancel"}, {"start": 30.9, "duration": 3.36, "text": "so the left hand side is just going to"}, {"start": 32.579, "duration": 3.721, "text": "be Y and on the right hand side we have"}, {"start": 34.26, "duration": 3.36, "text": "Negative X plus five and now we have"}, {"start": 36.3, "duration": 3.599, "text": "slope intercept form so I'm going to"}, {"start": 37.62, "duration": 4.14, "text": "rewrite X Plus y equals 5 as y equals"}, {"start": 39.899, "duration": 3.48, "text": "negative X plus five and let's do the"}, {"start": 41.76, "duration": 3.18, "text": "same thing for this lower one we'll copy"}, {"start": 43.379, "duration": 3.781, "text": "it down we're going to subtract X from"}, {"start": 44.94, "duration": 4.08, "text": "both sides left hand side is 2y right"}, {"start": 47.16, "duration": 3.18, "text": "hand side is negative X plus six and"}, {"start": 49.02, "duration": 3.539, "text": "then we're going to divide both sides by"}, {"start": 50.34, "duration": 3.899, "text": "2. left hand side is just Y and we're"}, {"start": 52.559, "duration": 3.361, "text": "going to divide each of these by two so"}, {"start": 54.239, "duration": 4.5, "text": "negative x divided by 2 is negative one"}, {"start": 55.92, "duration": 4.44, "text": "half X and 6 divided by 2 is 3. and now"}, {"start": 58.739, "duration": 3.66, "text": "we can change this lower equation up"}, {"start": 60.36, "duration": 3.539, "text": "here and do slope intercept form so now"}, {"start": 62.399, "duration": 3.361, "text": "let's use our equations to find the"}, {"start": 63.899, "duration": 3.841, "text": "y-intercepts this one up here it says"}, {"start": 65.76, "duration": 3.899, "text": "the Y intercept is five so this point is"}, {"start": 67.74, "duration": 4.14, "text": "going to be 0 5. and for this equation"}, {"start": 69.659, "duration": 4.561, "text": "the Y intercept is three so this vertex"}, {"start": 71.88, "duration": 4.44, "text": "is located at 0 3. so now that we have"}, {"start": 74.22, "duration": 3.899, "text": "these two vertices let's find this third"}, {"start": 76.32, "duration": 3.54, "text": "one and the best way to do that is set"}, {"start": 78.119, "duration": 3.661, "text": "these two equations equal to each other"}, {"start": 79.86, "duration": 3.42, "text": "and I will tell us the X and the Y point"}, {"start": 81.78, "duration": 3.3, "text": "where they're both equal to each other"}, {"start": 83.28, "duration": 3.6, "text": "so our goal is to solve for x in this"}, {"start": 85.08, "duration": 3.78, "text": "first thing I don't like fractions I'm"}, {"start": 86.88, "duration": 3.66, "text": "going to multiply both sides by two the"}, {"start": 88.86, "duration": 4.2, "text": "two will distribute so 2 times negative"}, {"start": 90.54, "duration": 4.619, "text": "X is negative 2X and 2 times 5 is equal"}, {"start": 93.06, "duration": 4.019, "text": "to 10 and this will be equal to this 2"}, {"start": 95.159, "duration": 3.96, "text": "distributed to both of these two times"}, {"start": 97.079, "duration": 4.08, "text": "negative one-half is negative one and"}, {"start": 99.119, "duration": 4.14, "text": "then times x is negative X and then 2"}, {"start": 101.159, "duration": 4.201, "text": "times 3 is equal to 6. now we want to"}, {"start": 103.259, "duration": 4.141, "text": "move both X's to the same side if I add"}, {"start": 105.36, "duration": 4.259, "text": "2x to both sides negative 2X and 2x"}, {"start": 107.4, "duration": 4.44, "text": "cancel each other out negative X plus 2X"}, {"start": 109.619, "duration": 4.14, "text": "is equal to one X and we bring down the"}, {"start": 111.84, "duration": 3.779, "text": "six now we want to get this x by itself"}, {"start": 113.759, "duration": 3.9, "text": "so we subtract 6 from both sides 10"}, {"start": 115.619, "duration": 3.96, "text": "minus 6 is equal to 4 and these cancel"}, {"start": 117.659, "duration": 4.201, "text": "leaving us with just X now we know that"}, {"start": 119.579, "duration": 4.441, "text": "the X that this intersection point is"}, {"start": 121.86, "duration": 3.359, "text": "equal to four so we can change that to a"}, {"start": 124.02, "duration": 3.419, "text": "four and then to find out what the"}, {"start": 125.219, "duration": 4.921, "text": "corresponding Y is we can plug this 4 in"}, {"start": 127.439, "duration": 4.081, "text": "4X for either one of these equations I"}, {"start": 130.14, "duration": 3.239, "text": "think it'll be easier to do this top"}, {"start": 131.52, "duration": 4.14, "text": "equation so we're going to substitute 4"}, {"start": 133.379, "duration": 4.261, "text": "into the X and negative 4 plus 5 is"}, {"start": 135.66, "duration": 4.14, "text": "equal to 1. so the Y value of this point"}, {"start": 137.64, "duration": 3.54, "text": "is one and we don't need this anymore in"}, {"start": 139.8, "duration": 2.939, "text": "fact now that we have the location of"}, {"start": 141.18, "duration": 3.419, "text": "our vertices we don't need our equations"}, {"start": 142.739, "duration": 3.781, "text": "anymore either so if this has a height"}, {"start": 144.599, "duration": 4.021, "text": "of 3 and this has a height of 5 that"}, {"start": 146.52, "duration": 3.9, "text": "means the distance is 2 in between them"}, {"start": 148.62, "duration": 3.9, "text": "and let's find this distance as well"}, {"start": 150.42, "duration": 4.2, "text": "since this x value is 4 we're going from"}, {"start": 152.52, "duration": 3.719, "text": "0 to 4 which means this distance right"}, {"start": 154.62, "duration": 3.24, "text": "here is four and now we really don't"}, {"start": 156.239, "duration": 3.36, "text": "even need these points anymore and let's"}, {"start": 157.86, "duration": 3.54, "text": "rotate this and now we want to find the"}, {"start": 159.599, "duration": 3.601, "text": "area of this triangle the area of a"}, {"start": 161.4, "duration": 3.54, "text": "triangle is one half base times height"}, {"start": 163.2, "duration": 3.72, "text": "the base of this triangle is equal to"}, {"start": 164.94, "duration": 3.9, "text": "two and the height of this triangle is"}, {"start": 166.92, "duration": 3.539, "text": "equal to four the one-half times two"}, {"start": 168.84, "duration": 3.84, "text": "just cancel each other out so we have an"}, {"start": 170.459, "duration": 4.261, "text": "area of four units squared so the area"}, {"start": 172.68, "duration": 3.72, "text": "of this blue region is equal to four"}, {"start": 174.72, "duration": 5.239, "text": "units squared now we can put a box"}, {"start": 176.4, "duration": 3.559, "text": "around that how exciting"}], "title": "Triangle Area from Graph", "description": "https://andymath.com/", "lengthSeconds": 178, "width": 640, "height": 360}
{"video_id": "wvnTdBgM9Cc", "transcript": [{"start": 0.0, "duration": 3.84, "text": "guys I think this is a cool one I want"}, {"start": 1.92, "duration": 3.78, "text": "to show you the steps of this proof that"}, {"start": 3.84, "duration": 3.6, "text": "3 is equal to Pi let's see if you can"}, {"start": 5.7, "duration": 4.08, "text": "spot the mistake when it happens so"}, {"start": 7.44, "duration": 4.739, "text": "we're going to start with x equals pi"}, {"start": 9.78, "duration": 4.38, "text": "plus 3 over 2. and first we multiply"}, {"start": 12.179, "duration": 3.601, "text": "both sides by 2. on the left hand side"}, {"start": 14.16, "duration": 3.78, "text": "we have 2X and on the right hand side"}, {"start": 15.78, "duration": 3.72, "text": "these twos cancel so we're left with pi"}, {"start": 17.94, "duration": 4.44, "text": "plus 3. and then from here let's"}, {"start": 19.5, "duration": 3.779, "text": "multiply both sides by pi minus 3. so on"}, {"start": 22.38, "duration": 2.88, "text": "the left hand side we're going to"}, {"start": 23.279, "duration": 3.961, "text": "distribute we have pi times the 2x which"}, {"start": 25.26, "duration": 3.839, "text": "is equal to 2 pi x and we have the"}, {"start": 27.24, "duration": 4.08, "text": "negative 3 times the 2x which is equal"}, {"start": 29.099, "duration": 4.14, "text": "to negative 6X and on the right hand"}, {"start": 31.32, "duration": 4.38, "text": "side this is a difference of two squares"}, {"start": 33.239, "duration": 4.741, "text": "anytime you have a plus b times a minus"}, {"start": 35.7, "duration": 4.32, "text": "B it's just a squared minus B squared or"}, {"start": 37.98, "duration": 4.14, "text": "in this case pi squared minus 3 squared"}, {"start": 40.02, "duration": 4.14, "text": "or in other words pi squared minus 9."}, {"start": 42.12, "duration": 4.439, "text": "next let's subtract this 2 pi x from"}, {"start": 44.16, "duration": 4.44, "text": "both sides and let's add 9 to both sides"}, {"start": 46.559, "duration": 4.561, "text": "on the left hand side these cancel so"}, {"start": 48.6, "duration": 4.32, "text": "we're just left with 9 minus 6X and on"}, {"start": 51.12, "duration": 3.9, "text": "the right hand side these cancel so"}, {"start": 52.92, "duration": 4.2, "text": "we're left with pi squared minus 2 pi x"}, {"start": 55.02, "duration": 3.539, "text": "and then let's add x squared to both"}, {"start": 57.12, "duration": 3.06, "text": "sides of the equation then let's"}, {"start": 58.559, "duration": 4.261, "text": "rearrange the order of a couple of these"}, {"start": 60.18, "duration": 5.04, "text": "this is a perfect square trinomial it"}, {"start": 62.82, "duration": 4.5, "text": "ends up being x minus 3 squared if we"}, {"start": 65.22, "duration": 3.84, "text": "multiply x minus 3 times x minus 3 it"}, {"start": 67.32, "duration": 3.119, "text": "brings us back to this and same thing"}, {"start": 69.06, "duration": 3.599, "text": "here this is also a perfect square"}, {"start": 70.439, "duration": 4.081, "text": "trinomial it ends up being x minus pi"}, {"start": 72.659, "duration": 3.78, "text": "squared and if we multiply x minus pi"}, {"start": 74.52, "duration": 3.84, "text": "times x minus Pi it brings us back to"}, {"start": 76.439, "duration": 3.841, "text": "this and now from this step let's square"}, {"start": 78.36, "duration": 3.24, "text": "root both sides on the left hand side"}, {"start": 80.28, "duration": 3.6, "text": "the square and the square root cancel"}, {"start": 81.6, "duration": 3.72, "text": "each other out so we have x minus 3. and"}, {"start": 83.88, "duration": 2.7, "text": "on the right hand side the square and"}, {"start": 85.32, "duration": 3.18, "text": "the square root cancel each other out so"}, {"start": 86.58, "duration": 4.8, "text": "we have x minus Pi we end up with x"}, {"start": 88.5, "duration": 4.74, "text": "minus 3 equals x minus pi and from here"}, {"start": 91.38, "duration": 3.72, "text": "we subtract X from both sides on the"}, {"start": 93.24, "duration": 3.36, "text": "left hand side we have negative 3 and on"}, {"start": 95.1, "duration": 3.059, "text": "the right hand side we have negative pi"}, {"start": 96.6, "duration": 4.14, "text": "after we divide both sides by negative"}, {"start": 98.159, "duration": 4.801, "text": "one we get three equals pi and clearly"}, {"start": 100.74, "duration": 3.419, "text": "this is not true so where is the mistake"}, {"start": 102.96, "duration": 2.699, "text": "and if you want to try to find the"}, {"start": 104.159, "duration": 4.261, "text": "mistake on your own positive because I'm"}, {"start": 105.659, "duration": 4.981, "text": "going to show it in three two one the"}, {"start": 108.42, "duration": 4.199, "text": "mistake in this occurs right here going"}, {"start": 110.64, "duration": 3.72, "text": "from this step to this step so the"}, {"start": 112.619, "duration": 3.241, "text": "square and the square root don't just"}, {"start": 114.36, "duration": 3.0, "text": "cancel out and give you what's in the"}, {"start": 115.86, "duration": 2.759, "text": "parentheses they always give you the"}, {"start": 117.36, "duration": 2.64, "text": "positive of what's inside the"}, {"start": 118.619, "duration": 3.301, "text": "parentheses so the correct thing would"}, {"start": 120.0, "duration": 3.659, "text": "be to put both sides of the equation in"}, {"start": 121.92, "duration": 4.14, "text": "absolute value and we can actually work"}, {"start": 123.659, "duration": 4.981, "text": "with this so we do know what x is equal"}, {"start": 126.06, "duration": 5.22, "text": "to X is equal to this pi plus 3 over 2"}, {"start": 128.64, "duration": 5.34, "text": "and that is approximately equal to 3.07"}, {"start": 131.28, "duration": 4.56, "text": "so if we plug in 3.07 here this inside"}, {"start": 133.98, "duration": 3.839, "text": "will be positive which means we don't"}, {"start": 135.84, "duration": 4.68, "text": "need the absolute value symbols but if"}, {"start": 137.819, "duration": 5.221, "text": "we plug in 3.07 into this one this"}, {"start": 140.52, "duration": 3.9, "text": "inside the brackets will be negative so"}, {"start": 143.04, "duration": 3.24, "text": "before we get rid of the absolute value"}, {"start": 144.42, "duration": 3.24, "text": "symbols we need to take the negative of"}, {"start": 146.28, "duration": 3.24, "text": "this and that will turn the whole thing"}, {"start": 147.66, "duration": 3.6, "text": "positive and now let's work through this"}, {"start": 149.52, "duration": 3.299, "text": "on this side we can bring down the x"}, {"start": 151.26, "duration": 3.18, "text": "minus 3 and then on the right hand side"}, {"start": 152.819, "duration": 3.361, "text": "this negative distributes to give us"}, {"start": 154.44, "duration": 3.78, "text": "Negative X and then negative times"}, {"start": 156.18, "duration": 4.559, "text": "negative pi is equal to positive Pi"}, {"start": 158.22, "duration": 4.32, "text": "let's add X to both sides and add 3 to"}, {"start": 160.739, "duration": 4.201, "text": "both sides on the left hand side the"}, {"start": 162.54, "duration": 4.38, "text": "threes cancel which gives us 2X and on"}, {"start": 164.94, "duration": 4.26, "text": "the right hand side the X's cancel which"}, {"start": 166.92, "duration": 4.679, "text": "gives us pi plus 3 and after we divide"}, {"start": 169.2, "duration": 4.319, "text": "both sides by 2 we get X is equal to pi"}, {"start": 171.599, "duration": 3.541, "text": "plus 3 over 2. well we already knew that"}, {"start": 173.519, "duration": 3.181, "text": "so in reality we didn't accomplish"}, {"start": 175.14, "duration": 3.06, "text": "anything with all this work we just"}, {"start": 176.7, "duration": 3.06, "text": "apply to operations and then they're"}, {"start": 178.2, "duration": 3.539, "text": "inverses and it brought us back to where"}, {"start": 179.76, "duration": 3.9, "text": "we started did but we did successfully"}, {"start": 181.739, "duration": 3.661, "text": "find the mistake in this proof of 3"}, {"start": 183.66, "duration": 5.299, "text": "equals pi and that was in this step"}, {"start": 185.4, "duration": 3.559, "text": "right here how exciting"}], "title": "Find the Mistake in this Proof", "description": "I thought this was a fun one. I hope you guys like it!", "lengthSeconds": 187, "width": 640, "height": 360}
{"video_id": "fzCCAMF09Ms", "transcript": [{"start": 0.0, "duration": 3.04, "text": "hey guys this looks like a fun one we"}, {"start": 1.319, "duration": 3.801, "text": "want to solve for x and this is kind of"}, {"start": 3.04, "duration": 3.879, "text": "a staggered fraction if you want to try"}, {"start": 5.12, "duration": 4.72, "text": "it on your own pause it right now cuz"}, {"start": 6.919, "duration": 4.64, "text": "I'm going to solve it in three 2 1 the"}, {"start": 9.84, "duration": 3.2, "text": "way this is staggered you could say is"}, {"start": 11.559, "duration": 3.441, "text": "ambiguous but I'm pretty sure it's meant"}, {"start": 13.04, "duration": 4.319, "text": "to be a fraction on either side where"}, {"start": 15.0, "duration": 4.64, "text": "these are on top and these are on bottom"}, {"start": 17.359, "duration": 5.521, "text": "so let's change this four into a 4 over"}, {"start": 19.64, "duration": 5.28, "text": "one and let's change this 69 into a 69"}, {"start": 22.88, "duration": 4.399, "text": "over one so now it's a little bit less"}, {"start": 24.92, "duration": 4.64, "text": "ambiguous it's going to be this / this"}, {"start": 27.279, "duration": 5.041, "text": "equals this / this let's see if we can"}, {"start": 29.56, "duration": 9.56, "text": "make a animation for that it'll be 4"}, {"start": 32.32, "duration": 8.32, "text": "piun / 691 = 41 / x 69 dividing"}, {"start": 39.12, "duration": 3.32, "text": "fractions can be a little bit tricky but"}, {"start": 40.64, "duration": 3.48, "text": "there's a cool shortcut you can leave"}, {"start": 42.44, "duration": 4.119, "text": "the first fraction alone so we can copy"}, {"start": 44.12, "duration": 4.8, "text": "down the 4 over pi and the 4 over1 then"}, {"start": 46.559, "duration": 4.201, "text": "you change the divide into a multiply"}, {"start": 48.92, "duration": 4.88, "text": "and then you flip the second fraction so"}, {"start": 50.76, "duration": 6.2, "text": "69 over 1 becomes 1 over 69 and X over"}, {"start": 53.8, "duration": 4.719, "text": "69 becomes 69x and now multiplying"}, {"start": 56.96, "duration": 4.239, "text": "fractions is a lot easier you just"}, {"start": 58.519, "duration": 4.921, "text": "multiply across on top and on bottom and"}, {"start": 61.199, "duration": 6.96, "text": "now we can just clean things up 4 * 1 is"}, {"start": 63.44, "duration": 8.2, "text": "equal to 4 and < * 69 is 69 piun 4 * 69"}, {"start": 68.159, "duration": 5.441, "text": "is equal to 276 and 1 * X is just X now"}, {"start": 71.64, "duration": 3.88, "text": "we have a fraction equals a fraction we"}, {"start": 73.6, "duration": 5.76, "text": "can solve this by cross multiplying we"}, {"start": 75.52, "duration": 8.2, "text": "can do 4 * x = 4x and that's equal to 69"}, {"start": 79.36, "duration": 6.36, "text": "piun * 276 that's 19,44 piun and then we"}, {"start": 83.72, "duration": 3.399, "text": "can divide both sides by 4 and we get X"}, {"start": 85.72, "duration": 3.88, "text": "is equal to"}, {"start": 87.119, "duration": 4.32, "text": "4761 pi and this is the answer to the"}, {"start": 89.6, "duration": 3.92, "text": "question let's put a box around it and"}, {"start": 91.439, "duration": 4.921, "text": "that's approximately equal to"}, {"start": 93.52, "duration": 7.08, "text": "14957 one2 and let's put a box around"}, {"start": 96.36, "duration": 4.24, "text": "that too how exciting"}], "title": "Can we solve for x?", "description": "Is this one ambiguous or can we solve for x?", "lengthSeconds": 99, "width": 640, "height": 360}
{"video_id": "JYIDEtRnw3o", "transcript": [{"start": 0.04, "duration": 2.639, "text": "hey guys this looks like a fun one we're"}, {"start": 1.24, "duration": 3.68, "text": "trying to find the ratio of the pink"}, {"start": 2.679, "duration": 3.521, "text": "area to the green area let's do this so"}, {"start": 4.92, "duration": 3.12, "text": "first let's try to find the area of"}, {"start": 6.2, "duration": 3.08, "text": "these yin-yang things so let's pull them"}, {"start": 8.04, "duration": 2.96, "text": "out and if we put them together they"}, {"start": 9.28, "duration": 3.8, "text": "should make a circle so it just be half"}, {"start": 11.0, "duration": 4.12, "text": "of that Circle and they don't these make"}, {"start": 13.08, "duration": 3.6, "text": "an ellipse all right scrap that idea"}, {"start": 15.12, "duration": 3.079, "text": "let's try something else so let's draw"}, {"start": 16.68, "duration": 3.12, "text": "the radius right here in fact let's draw"}, {"start": 18.199, "duration": 3.361, "text": "all these radiuses so that we have a"}, {"start": 19.8, "duration": 3.68, "text": "square in the middle right here and"}, {"start": 21.56, "duration": 4.12, "text": "let's focus on that square each of these"}, {"start": 23.48, "duration": 5.119, "text": "sides are two radiuses so we can call it"}, {"start": 25.68, "duration": 5.12, "text": "two R and yes radiuses is the plural of"}, {"start": 28.599, "duration": 4.64, "text": "radius we could also say radi the area"}, {"start": 30.8, "duration": 4.32, "text": "of the square is going to be 2 R * 2 R"}, {"start": 33.239, "duration": 3.441, "text": "that's the same thing as 4 R 2 let's"}, {"start": 35.12, "duration": 2.88, "text": "look at another copy of the square down"}, {"start": 36.68, "duration": 3.32, "text": "here let's pull apart these quarter"}, {"start": 38.0, "duration": 3.399, "text": "circles bring down this green part let's"}, {"start": 40.0, "duration": 3.399, "text": "rotate all these quarter circles to make"}, {"start": 41.399, "duration": 3.961, "text": "a circle the area of this circle will be"}, {"start": 43.399, "duration": 3.68, "text": "P pi r s the area of the green stuff is"}, {"start": 45.36, "duration": 3.28, "text": "then going to be the area of the square"}, {"start": 47.079, "duration": 2.921, "text": "minus the area of this circle now we"}, {"start": 48.64, "duration": 3.0, "text": "don't need the circle anymore so we can"}, {"start": 50.0, "duration": 3.16, "text": "get rid of that let's bring this up here"}, {"start": 51.64, "duration": 2.719, "text": "and this is important so let's put a box"}, {"start": 53.16, "duration": 2.879, "text": "around it let's go back up to this"}, {"start": 54.359, "duration": 3.241, "text": "square up here let's cut the square in"}, {"start": 56.039, "duration": 2.921, "text": "half and let's find the length of this"}, {"start": 57.6, "duration": 3.959, "text": "diagonal well this ends up making a"}, {"start": 58.96, "duration": 4.52, "text": "45459 90 triangle so it ends up being <"}, {"start": 61.559, "duration": 4.521, "text": "TK of 2 * one of these sides so the"}, {"start": 63.48, "duration": 4.56, "text": "diagonal is equal to 2 R < tk2 if you"}, {"start": 66.08, "duration": 3.28, "text": "don't know about the 454590 shortcut"}, {"start": 68.04, "duration": 2.92, "text": "here's the Pythagorean theorem right"}, {"start": 69.36, "duration": 3.0, "text": "here pause it if you want to check it"}, {"start": 70.96, "duration": 3.28, "text": "let's clean all this stuff up and put a"}, {"start": 72.36, "duration": 3.72, "text": "box around this now let's find the area"}, {"start": 74.24, "duration": 4.16, "text": "of our large Circle so we know that the"}, {"start": 76.08, "duration": 3.88, "text": "diagonal the square is equal to 2 < tk2"}, {"start": 78.4, "duration": 3.0, "text": "R and now we don't need the square"}, {"start": 79.96, "duration": 3.32, "text": "anymore and then this right here is a"}, {"start": 81.4, "duration": 3.399, "text": "radius and this right here is a radius"}, {"start": 83.28, "duration": 4.4, "text": "the diameter will be equal to two of the"}, {"start": 84.799, "duration": 4.201, "text": "radiuses plus 2 < tk2 R and if we want"}, {"start": 87.68, "duration": 2.92, "text": "to know what the radius of the largest"}, {"start": 89.0, "duration": 4.72, "text": "circle is we would just divide all that"}, {"start": 90.6, "duration": 4.6, "text": "stuff by two which gives us r + < tk2 r"}, {"start": 93.72, "duration": 3.88, "text": "and now to find the area of this circle"}, {"start": 95.2, "duration": 3.919, "text": "we're going to do PI radius s where this"}, {"start": 97.6, "duration": 3.0, "text": "right here is the radius and then a"}, {"start": 99.119, "duration": 3.36, "text": "binomial squared ends up being the"}, {"start": 100.6, "duration": 3.32, "text": "binomial times itself so we can foil"}, {"start": 102.479, "duration": 4.881, "text": "this out we'll leave the pi out front"}, {"start": 103.92, "duration": 6.96, "text": "and then R * R is R 2 R * < tk2 R is <"}, {"start": 107.36, "duration": 6.56, "text": "tk2 R 2 < tk2 R * R is < tk2 R 2 and <"}, {"start": 110.88, "duration": 6.04, "text": "tk2 R * < tk2 R is = 2 R 2 and then we"}, {"start": 113.92, "duration": 5.64, "text": "can clean this up R 2 + 2 R 2 is 3 R 2"}, {"start": 116.92, "duration": 4.519, "text": "and right here we have 2 < tk2 R 2 so"}, {"start": 119.56, "duration": 3.64, "text": "that would be 2 < tk2 r^ 2 and then we"}, {"start": 121.439, "duration": 3.801, "text": "can distribute the pi so we now have the"}, {"start": 123.2, "duration": 3.559, "text": "area of this largest Circle and we can"}, {"start": 125.24, "duration": 2.999, "text": "put a box around that we're trying to"}, {"start": 126.759, "duration": 3.441, "text": "find the pink area divided by the green"}, {"start": 128.239, "duration": 3.681, "text": "area I think that is what this says so"}, {"start": 130.2, "duration": 3.92, "text": "the pink area is going to be this whole"}, {"start": 131.92, "duration": 3.64, "text": "circle minus this green stuff and then"}, {"start": 134.12, "duration": 2.6, "text": "the green area is going to be this green"}, {"start": 135.56, "duration": 3.08, "text": "stuff we can get rid of these"}, {"start": 136.72, "duration": 4.239, "text": "parentheses and distribute the negative"}, {"start": 138.64, "duration": 4.2, "text": "all six of these terms has an R squ so"}, {"start": 140.959, "duration": 3.961, "text": "we can factor out the r s and then we"}, {"start": 142.84, "duration": 4.32, "text": "can cancel those R squs and then on top"}, {"start": 144.92, "duration": 3.8, "text": "this 3 pi and this 1 Pi can become 4 pi"}, {"start": 147.16, "duration": 3.079, "text": "and now we are done this is the exact"}, {"start": 148.72, "duration": 2.72, "text": "value of the answer and ends up being"}, {"start": 150.239, "duration": 5.321, "text": "approximately"}, {"start": 151.44, "duration": 4.12, "text": "20308 how exciting"}], "title": "Ratio of Pink/Green", "description": "https://andymath.com/geometry-challenges/\n\nFor more geometry challenges check out the above link!", "lengthSeconds": 154, "width": 640, "height": 360}
{"video_id": "fmDUseL-YUA", "transcript": [{"start": 0.0, "duration": 3.12, "text": "hey guys I saw this the other day"}, {"start": 1.439, "duration": 3.421, "text": "someone wrote I've never seen this"}, {"start": 3.12, "duration": 3.48, "text": "problem before it looks interesting"}, {"start": 4.86, "duration": 2.999, "text": "enough so let's do it if you want to try"}, {"start": 6.6, "duration": 2.7, "text": "this on your own pause it right now"}, {"start": 7.859, "duration": 3.78, "text": "because I'm going to solve it in three"}, {"start": 9.3, "duration": 3.6, "text": "two one so this one is not super"}, {"start": 11.639, "duration": 2.941, "text": "difficult I think they're expecting"}, {"start": 12.9, "duration": 3.78, "text": "people to see the right triangles here"}, {"start": 14.58, "duration": 3.779, "text": "connect these and think a squared plus b"}, {"start": 16.68, "duration": 3.54, "text": "squared equals c squared and then do the"}, {"start": 18.359, "duration": 3.121, "text": "same thing here same thing here and then"}, {"start": 20.22, "duration": 2.58, "text": "maybe they expect you to connect this"}, {"start": 21.48, "duration": 3.12, "text": "and think what am I supposed to do with"}, {"start": 22.8, "duration": 3.42, "text": "this shape it can be very confusing if"}, {"start": 24.6, "duration": 3.48, "text": "you go down this route but the simple"}, {"start": 26.22, "duration": 4.26, "text": "way to do it is just to complete this"}, {"start": 28.08, "duration": 4.14, "text": "right triangle this side right here the"}, {"start": 30.48, "duration": 4.68, "text": "height of this triangle is going to be 2"}, {"start": 32.22, "duration": 5.16, "text": "plus 2 plus 2 which is six and the base"}, {"start": 35.16, "duration": 4.079, "text": "is going to be four plus two plus two"}, {"start": 37.38, "duration": 4.14, "text": "which is eight and if we call this last"}, {"start": 39.239, "duration": 4.621, "text": "side C we can do the Pythagorean theorem"}, {"start": 41.52, "duration": 4.92, "text": "we're going to plug in 6 for the a 8 for"}, {"start": 43.86, "duration": 5.64, "text": "the B and then solve for C 6 squared is"}, {"start": 46.44, "duration": 5.58, "text": "36 plus 8 squared is 64 and that's all"}, {"start": 49.5, "duration": 4.44, "text": "going to equal c squared 36 plus 64 is"}, {"start": 52.02, "duration": 4.019, "text": "equal to 100 and after we square root"}, {"start": 53.94, "duration": 6.259, "text": "both sides we get C is equal to 10 and"}, {"start": 56.039, "duration": 4.16, "text": "that's this one right here how exciting"}], "title": "Fun Math Problem", "description": "I hope you guys like it.", "lengthSeconds": 58, "width": 360, "height": 640}
{"video_id": "PD9Ea0rpKqo", "transcript": [{"start": 0.04, "duration": 4.0, "text": "this is going to be a fun problem we"}, {"start": 1.319, "duration": 4.241, "text": "have a shed that has a 6 m x 6 M area"}, {"start": 4.04, "duration": 4.2, "text": "right here and then a side room that's a"}, {"start": 5.56, "duration": 5.92, "text": "3 MX 3 m then tied to this corner of a"}, {"start": 8.24, "duration": 5.319, "text": "shed is a leash for a goat this leash is"}, {"start": 11.48, "duration": 4.279, "text": "7 m long here's the goat right here and"}, {"start": 13.559, "duration": 3.921, "text": "as we all know Andy is the goat if we"}, {"start": 15.759, "duration": 3.721, "text": "wanted to analyze the amount of grass"}, {"start": 17.48, "duration": 3.2, "text": "that the goat could eat usually"}, {"start": 19.48, "duration": 4.52, "text": "something like this where it forms a"}, {"start": 20.68, "duration": 4.839, "text": "circle around where the leash is pinned"}, {"start": 24.0, "duration": 3.56, "text": "is the center of the circle and then it"}, {"start": 25.519, "duration": 3.68, "text": "goes out with a radius that's as long as"}, {"start": 27.56, "duration": 3.4, "text": "the leash normally a goat would be able"}, {"start": 29.199, "duration": 3.681, "text": "to eat all of this grass you go inside"}, {"start": 30.96, "duration": 3.119, "text": "the circle because the leash isn't stiff"}, {"start": 32.88, "duration": 2.6, "text": "but we do have these buildings in the"}, {"start": 34.079, "duration": 3.0, "text": "way so we got to calculate how much"}, {"start": 35.48, "duration": 3.12, "text": "grass can the goat eat with these"}, {"start": 37.079, "duration": 3.0, "text": "buildings that are in the way so first"}, {"start": 38.6, "duration": 4.16, "text": "off we can bring the goat over here and"}, {"start": 40.079, "duration": 4.681, "text": "we can see that he can eat all of this"}, {"start": 42.76, "duration": 3.52, "text": "very important detail here is what angle"}, {"start": 44.76, "duration": 3.68, "text": "is this this is"}, {"start": 46.28, "duration": 4.36, "text": "225\u00b0 how do we know that we know the"}, {"start": 48.44, "duration": 4.36, "text": "half of the circle would be 180\u00b0 and"}, {"start": 50.64, "duration": 4.0, "text": "then since this is three and we know"}, {"start": 52.8, "duration": 3.72, "text": "this whole side is six that means each"}, {"start": 54.64, "duration": 4.12, "text": "of these are equal to three and so we"}, {"start": 56.52, "duration": 6.76, "text": "have a 3X3 triangle which means this has"}, {"start": 58.76, "duration": 6.16, "text": "to be 45\u00b0 so 180 + 45 is 225\u00b0 so let's"}, {"start": 63.28, "duration": 3.44, "text": "calculate the area of this piece right"}, {"start": 64.92, "duration": 6.32, "text": "here well normally a circle would beunk"}, {"start": 66.72, "duration": 6.759, "text": "R 2 this is 225\u00b0 out of the total 360\u00b0"}, {"start": 71.24, "duration": 5.0, "text": "of the circle and 225 over 360"}, {"start": 73.479, "duration": 5.201, "text": "simplifies to 5 over8 so this is 58 of"}, {"start": 76.24, "duration": 4.36, "text": "an entire circle with radius 7 so we can"}, {"start": 78.68, "duration": 3.6, "text": "change this R into a seven next we have"}, {"start": 80.6, "duration": 3.76, "text": "this red piece right here and it looks"}, {"start": 82.28, "duration": 4.64, "text": "like this here so what is the radius of"}, {"start": 84.36, "duration": 4.84, "text": "this well the whole length of the leash"}, {"start": 86.92, "duration": 3.879, "text": "is seven so we're using up six of it to"}, {"start": 89.2, "duration": 3.919, "text": "get to this side of the shed that means"}, {"start": 90.799, "duration": 4.401, "text": "that from here we have one more meter"}, {"start": 93.119, "duration": 4.64, "text": "that can pivot around and make a quarter"}, {"start": 95.2, "duration": 3.84, "text": "of a circle so this has a radius of 1 M"}, {"start": 97.759, "duration": 3.281, "text": "and then we can see that this makes a"}, {"start": 99.04, "duration": 4.439, "text": "90\u00b0 angle once again we have the area of"}, {"start": 101.04, "duration": 4.719, "text": "a circle is PK R squ this is 1/4 of an"}, {"start": 103.479, "duration": 4.041, "text": "entire circle so we'll call it 1/4 PK R"}, {"start": 105.759, "duration": 3.0, "text": "squar and then we can plug in one for R"}, {"start": 107.52, "duration": 3.72, "text": "plug this into a calculator and get"}, {"start": 108.759, "duration": 3.921, "text": "approximately 785 next thing we have"}, {"start": 111.24, "duration": 2.96, "text": "this orange triangle and you can see"}, {"start": 112.68, "duration": 2.68, "text": "that the goat would be able to eat"}, {"start": 114.2, "duration": 2.839, "text": "everything inside of this orange"}, {"start": 115.36, "duration": 3.56, "text": "triangle so let's Analyze This each of"}, {"start": 117.039, "duration": 3.12, "text": "these two sides is three and to get this"}, {"start": 118.92, "duration": 3.839, "text": "hypotenuse here since since it's a"}, {"start": 120.159, "duration": 4.401, "text": "454590 triangle we know that that is 3 <"}, {"start": 122.759, "duration": 3.32, "text": "tk2 m and if you want to see the work"}, {"start": 124.56, "duration": 2.919, "text": "behind it here it is right here using"}, {"start": 126.079, "duration": 2.8, "text": "Pythagorean theorem you can pause it and"}, {"start": 127.479, "duration": 3.601, "text": "look at this if you want the area of a"}, {"start": 128.879, "duration": 4.121, "text": "triangle is 1/2 base time height this"}, {"start": 131.08, "duration": 3.56, "text": "triangle is upside down but this base is"}, {"start": 133.0, "duration": 2.84, "text": "going to be three and this height is"}, {"start": 134.64, "duration": 3.84, "text": "going to be three and this all"}, {"start": 135.84, "duration": 4.2, "text": "simplifies to give us 4.5 next let's do"}, {"start": 138.48, "duration": 3.32, "text": "this up here let's move the goat out of"}, {"start": 140.04, "duration": 3.839, "text": "the way and then we're looking at this"}, {"start": 141.8, "duration": 4.0, "text": "piece right here now doesn't follow all"}, {"start": 143.879, "duration": 3.72, "text": "the way we can zoom in and look at it it"}, {"start": 145.8, "duration": 4.0, "text": "goes kind of like this so it's going to"}, {"start": 147.599, "duration": 4.041, "text": "have a different radius than the circle"}, {"start": 149.8, "duration": 3.159, "text": "irle so that's why it's not following"}, {"start": 151.64, "duration": 2.76, "text": "the whole circle it's kind of like what"}, {"start": 152.959, "duration": 2.801, "text": "was going on over here with the red one"}, {"start": 154.4, "duration": 3.559, "text": "because you have a pivot point that"}, {"start": 155.76, "duration": 3.64, "text": "creates a new Mini Circle so let's look"}, {"start": 157.959, "duration": 2.721, "text": "at this pie slice right here well the"}, {"start": 159.4, "duration": 4.119, "text": "easy part is seeing that that's going to"}, {"start": 160.68, "duration": 4.04, "text": "be 45\u00b0 it's the same thing as this 45"}, {"start": 163.519, "duration": 3.761, "text": "right here so we know this is going to"}, {"start": 164.72, "duration": 4.08, "text": "be 45\u00b0 the trickier part is finding the"}, {"start": 167.28, "duration": 3.64, "text": "radius so we're going to call the radius"}, {"start": 168.8, "duration": 3.96, "text": "R the goat were up here we're going to"}, {"start": 170.92, "duration": 3.319, "text": "be using up a portion of the leash I'm"}, {"start": 172.76, "duration": 3.199, "text": "going to show that leash right here and"}, {"start": 174.239, "duration": 4.0, "text": "we showed earlier that that length is 3"}, {"start": 175.959, "duration": 4.801, "text": "< tk2 so we know that as far as this"}, {"start": 178.239, "duration": 5.321, "text": "goat can reach is 7 that means that 3 <"}, {"start": 180.76, "duration": 4.6, "text": "tk2 + r has to be equal to 7 that's as"}, {"start": 183.56, "duration": 3.8, "text": "far as the leash will reach we have to"}, {"start": 185.36, "duration": 4.959, "text": "subtract 3 < tk2 from both sides to find"}, {"start": 187.36, "duration": 5.68, "text": "out r r is equal to 7 - 3 < tk2 that's"}, {"start": 190.319, "duration": 4.2, "text": "approximately 2757 if we clean all this"}, {"start": 193.04, "duration": 3.16, "text": "stuff up let's find the area of this"}, {"start": 194.519, "duration": 4.161, "text": "yellow piece p r squ is the area of a"}, {"start": 196.2, "duration": 4.599, "text": "circle but we're dealing with 45\u00b0 out of"}, {"start": 198.68, "duration": 4.52, "text": "360 that's going to be 1/8 of an entire"}, {"start": 200.799, "duration": 4.241, "text": "circle and then we can substitute 2757"}, {"start": 203.2, "duration": 3.44, "text": "in for the r plug this into a calculator"}, {"start": 205.04, "duration": 4.32, "text": "we get approximately"}, {"start": 206.64, "duration": 4.64, "text": "2986 so now we have the areas this red"}, {"start": 209.36, "duration": 3.519, "text": "green orange and yellow shape make up"}, {"start": 211.28, "duration": 3.64, "text": "everything that this goat can possibly"}, {"start": 212.879, "duration": 3.601, "text": "eat if we add up these four areas we'll"}, {"start": 214.92, "duration": 3.519, "text": "know how much the goat can eat and"}, {"start": 216.48, "duration": 4.959, "text": "that's the answer to the question how"}, {"start": 218.439, "duration": 3.0, "text": "exciting"}], "title": "Goat Math", "description": "This is a neat one. I hope you guys love it! \n\nSubscribe for more!", "lengthSeconds": 220, "width": 640, "height": 360}
{"video_id": "W4I1kFamQ6c", "transcript": [{"start": 0.04, "duration": 3.639, "text": "hey guys this looks like a fun one it"}, {"start": 1.439, "duration": 5.081, "text": "says what's the area of the semicircle"}, {"start": 3.679, "duration": 4.92, "text": "inside this equilateral triangle so"}, {"start": 6.52, "duration": 4.44, "text": "we're given an equilateral triangle a"}, {"start": 8.599, "duration": 5.12, "text": "circle with area six and a semicircle"}, {"start": 10.96, "duration": 4.36, "text": "right here this is day 12 of our Advent"}, {"start": 13.719, "duration": 4.241, "text": "calendar for the month of December we're"}, {"start": 15.32, "duration": 4.52, "text": "going to solve 31 Kina ad puzzles in 31"}, {"start": 17.96, "duration": 3.76, "text": "days and if you want to try this one on"}, {"start": 19.84, "duration": 5.4, "text": "your own pause it right now because I'm"}, {"start": 21.72, "duration": 5.52, "text": "going to solve it in one two three first"}, {"start": 25.24, "duration": 3.24, "text": "thing we're given a six and a question"}, {"start": 27.24, "duration": 3.76, "text": "mark I think we should figure out"}, {"start": 28.48, "duration": 5.32, "text": "formulas for both of those the six is"}, {"start": 31.0, "duration": 4.16, "text": "the area of this smaller Circle and the"}, {"start": 33.8, "duration": 3.599, "text": "question mark is the area of this"}, {"start": 35.16, "duration": 4.559, "text": "semicircle let's label the radius of the"}, {"start": 37.399, "duration": 5.401, "text": "smaller Circle little r and the radius"}, {"start": 39.719, "duration": 6.32, "text": "of the bigger semicircle Big R so 6"}, {"start": 42.8, "duration": 5.36, "text": "would be equal to PI Little R 2 and the"}, {"start": 46.039, "duration": 5.84, "text": "question mark is going to be equal to Pi"}, {"start": 48.16, "duration": 5.919, "text": "big r^ 2 / 2 these look important let's"}, {"start": 51.879, "duration": 4.36, "text": "put boxes around them and our goal is to"}, {"start": 54.079, "duration": 4.361, "text": "find this question mark I think we need"}, {"start": 56.239, "duration": 4.081, "text": "to find some relationship with the Big R"}, {"start": 58.44, "duration": 4.0, "text": "and the little r and then we should be"}, {"start": 60.32, "duration": 4.36, "text": "able to do some substitution from here"}, {"start": 62.44, "duration": 4.28, "text": "let's drop this perpendicular bis sector"}, {"start": 64.68, "duration": 3.88, "text": "and let's make this black and white"}, {"start": 66.72, "duration": 4.24, "text": "first I want to focus on this orange"}, {"start": 68.56, "duration": 5.04, "text": "triangle let's zoom in right here this"}, {"start": 70.96, "duration": 5.24, "text": "line bisected an equilateral triangle so"}, {"start": 73.6, "duration": 5.36, "text": "this is normally 60\u00b0 up here but this"}, {"start": 76.2, "duration": 4.08, "text": "angle will be half of that or 30\u00b0 and"}, {"start": 78.96, "duration": 3.519, "text": "then this angle right here is built off"}, {"start": 80.28, "duration": 3.799, "text": "of the radius and the tangent line the"}, {"start": 82.479, "duration": 4.161, "text": "radius of any Circle will meet its"}, {"start": 84.079, "duration": 4.441, "text": "tangent line at 90\u00b0 and then all three"}, {"start": 86.64, "duration": 3.839, "text": "of these angles add up to 180 so this"}, {"start": 88.52, "duration": 5.0, "text": "one would be 60\u00b0"}, {"start": 90.479, "duration": 5.0, "text": "our orange triangle is a 306090 triangle"}, {"start": 93.52, "duration": 3.879, "text": "that is a special right triangle and"}, {"start": 95.479, "duration": 4.481, "text": "it's always true that the hypotenuse is"}, {"start": 97.399, "duration": 4.201, "text": "twice as long as the shortest side so"}, {"start": 99.96, "duration": 4.28, "text": "this distance right here will be two"}, {"start": 101.6, "duration": 4.72, "text": "little r and that correlates to this"}, {"start": 104.24, "duration": 3.6, "text": "right up here this piece is a radius of"}, {"start": 106.32, "duration": 3.439, "text": "the small circle we'll call that little"}, {"start": 107.84, "duration": 3.599, "text": "r and since the whole thing is too"}, {"start": 109.759, "duration": 3.801, "text": "little r this remaining piece will also"}, {"start": 111.439, "duration": 3.96, "text": "be little r and then while we're ated"}, {"start": 113.56, "duration": 4.08, "text": "this piece right here is also a radius"}, {"start": 115.399, "duration": 4.601, "text": "let's call it little r and we can finish"}, {"start": 117.64, "duration": 4.839, "text": "this this is a radius of the semicircle"}, {"start": 120.0, "duration": 4.36, "text": "so let's call this Big R so the entire"}, {"start": 122.479, "duration": 5.361, "text": "height of our equilateral triangle is"}, {"start": 124.36, "duration": 5.879, "text": "Big R + 3 little r next let's focus on"}, {"start": 127.84, "duration": 5.56, "text": "this larger triangle and let's zoom in"}, {"start": 130.239, "duration": 4.961, "text": "on it it will also be 30\u00b0 up here and"}, {"start": 133.4, "duration": 4.4, "text": "for the same reason this angle will also"}, {"start": 135.2, "duration": 5.759, "text": "be 90 de and the remaining angle will be"}, {"start": 137.8, "duration": 4.76, "text": "60 so this is another 30 6090 where the"}, {"start": 140.959, "duration": 4.36, "text": "hypotenuse is twice the length of the"}, {"start": 142.56, "duration": 6.399, "text": "smaller side so this distance will be 2"}, {"start": 145.319, "duration": 5.56, "text": "Big R so this whole distance is 2 Big R"}, {"start": 148.959, "duration": 4.241, "text": "this is one Big R R which leaves one"}, {"start": 150.879, "duration": 4.36, "text": "more Big R right here and this is the"}, {"start": 153.2, "duration": 4.44, "text": "relationship we wanted we have Big R is"}, {"start": 155.239, "duration": 3.961, "text": "the same thing as three little r this"}, {"start": 157.64, "duration": 3.28, "text": "looks important let's put an irregular"}, {"start": 159.2, "duration": 4.119, "text": "octagon around it and now we're ready"}, {"start": 160.92, "duration": 4.16, "text": "for our original equations down here in"}, {"start": 163.319, "duration": 4.041, "text": "the place of the Big R we can plug in"}, {"start": 165.08, "duration": 3.68, "text": "three little r and this exponent will"}, {"start": 167.36, "duration": 5.04, "text": "distribute to both of these to give us"}, {"start": 168.76, "duration": 5.72, "text": "3^2 little r^ 2 3^2 is equal to 9 and"}, {"start": 172.4, "duration": 4.28, "text": "now let's rearrange this a little so we"}, {"start": 174.48, "duration": 5.88, "text": "have the question mark is equal to 9 pi"}, {"start": 176.68, "duration": 5.96, "text": "little r^ 2 / 2 well Pi little r s is"}, {"start": 180.36, "duration": 6.28, "text": "equal to 6 so in the place of this we"}, {"start": 182.64, "duration": 7.679, "text": "can plug in 6 and then 9 * 6 is 54 and"}, {"start": 186.64, "duration": 5.76, "text": "54 / 2 is 27 let's give it a label of"}, {"start": 190.319, "duration": 4.84, "text": "units squared and that is the answer to"}, {"start": 192.4, "duration": 6.64, "text": "our question the area of this semicircle"}, {"start": 195.159, "duration": 5.72, "text": "is 27 unit squared how exciting this is"}, {"start": 199.04, "duration": 4.16, "text": "our answer for day 12 and this is the"}, {"start": 200.879, "duration": 3.961, "text": "puzzle for day 13 we're given the areas"}, {"start": 203.2, "duration": 3.0, "text": "of these two triangles and it wants to"}, {"start": 204.84, "duration": 3.28, "text": "know what's the area of the entire"}, {"start": 206.2, "duration": 6.28, "text": "semicircle this looks like it'll be a"}, {"start": 208.12, "duration": 4.36, "text": "fun one how exciting in"}], "title": "Aggvent Calendar Day 12", "description": "https://andymath.com/agg-vent-calendar/", "lengthSeconds": 211, "width": 640, "height": 360}
{"video_id": "Y8KMa8tJw-o", "transcript": [{"start": 0.0, "duration": 5.04, "text": "guys let's do some calculus this says"}, {"start": 2.159, "duration": 5.22, "text": "find dydx for y equals natural log of e"}, {"start": 5.04, "duration": 4.019, "text": "to the X over e to the x minus 1. so"}, {"start": 7.379, "duration": 3.541, "text": "this is dealing with logarithms so first"}, {"start": 9.059, "duration": 4.261, "text": "thing we want to do is apply this rule"}, {"start": 10.92, "duration": 4.86, "text": "of logarithms the natural log of a over"}, {"start": 13.32, "duration": 4.86, "text": "C is equal to the natural log of a minus"}, {"start": 15.78, "duration": 4.62, "text": "the natural log of C so in this case the"}, {"start": 18.18, "duration": 4.56, "text": "top function is e to the X and this"}, {"start": 20.4, "duration": 4.02, "text": "bottom function is e to the x minus 1."}, {"start": 22.74, "duration": 3.299, "text": "and next is another property of"}, {"start": 24.42, "duration": 4.98, "text": "logarithms we can use that the natural"}, {"start": 26.039, "duration": 5.16, "text": "log of e to the x is equal to X so this"}, {"start": 29.4, "duration": 4.02, "text": "first term can just be simplified to X"}, {"start": 31.199, "duration": 4.141, "text": "so let's take the derivative derivative"}, {"start": 33.42, "duration": 3.54, "text": "of x is equal to one and then we're"}, {"start": 35.34, "duration": 3.239, "text": "going to subtract and then for the"}, {"start": 36.96, "duration": 3.599, "text": "derivative of natural log let's use"}, {"start": 38.579, "duration": 4.861, "text": "these notes right here it's equal to the"}, {"start": 40.559, "duration": 5.16, "text": "derivative of x over X so let's do our"}, {"start": 43.44, "duration": 4.92, "text": "fraction here where we put X on bottom"}, {"start": 45.719, "duration": 4.68, "text": "but X is this whole e to the x minus 1"}, {"start": 48.36, "duration": 3.539, "text": "and then on top is the derivative of"}, {"start": 50.399, "duration": 3.121, "text": "this and the derivative of this is just"}, {"start": 51.899, "duration": 3.961, "text": "e to the X and the derivative of"}, {"start": 53.52, "duration": 3.84, "text": "negative 1 is 0. so now we're"}, {"start": 55.86, "duration": 3.24, "text": "technically done but we can clean it up"}, {"start": 57.36, "duration": 3.839, "text": "a little bit let's give these two terms"}, {"start": 59.1, "duration": 4.56, "text": "a common denominator we'll multiply top"}, {"start": 61.199, "duration": 4.801, "text": "and bottom by e to the x minus one so"}, {"start": 63.66, "duration": 4.319, "text": "now this times one is just this and we"}, {"start": 66.0, "duration": 3.18, "text": "can smush these together these have a"}, {"start": 67.979, "duration": 2.64, "text": "common denominator so we can combine"}, {"start": 69.18, "duration": 4.5, "text": "them into a single fraction where we"}, {"start": 70.619, "duration": 5.641, "text": "just do this minus this and e to the x"}, {"start": 73.68, "duration": 4.5, "text": "minus E to the X goes away and so we're"}, {"start": 76.26, "duration": 4.08, "text": "just left with negative 1 over e to the"}, {"start": 78.18, "duration": 4.14, "text": "x minus one and that's the answer to the"}, {"start": 80.34, "duration": 3.36, "text": "question we can put a box around it you"}, {"start": 82.32, "duration": 2.76, "text": "guys want to try more calculus problems"}, {"start": 83.7, "duration": 4.7, "text": "I have a link down in the description"}, {"start": 85.08, "duration": 3.32, "text": "how exciting"}], "title": "Derivative Challenge", "description": "https://andymath.com/calculus/", "lengthSeconds": 87, "width": 640, "height": 360}
{"video_id": "ykmPbbnCYjY", "transcript": [{"start": 0.04, "duration": 3.28, "text": "hey guys I thought this was a fun one"}, {"start": 1.319, "duration": 3.881, "text": "we're given two squares and two"}, {"start": 3.32, "duration": 3.84, "text": "triangles and it says find the area of"}, {"start": 5.2, "duration": 3.76, "text": "the red triangle I'll give you a hint"}, {"start": 7.16, "duration": 3.64, "text": "right now the method that I use to use"}, {"start": 8.96, "duration": 4.52, "text": "these notes right here we have the area"}, {"start": 10.8, "duration": 4.999, "text": "of any triangle is equal to 1/2 * one"}, {"start": 13.48, "duration": 4.0, "text": "side time another side time the sign of"}, {"start": 15.799, "duration": 3.361, "text": "the angle in between them and we also"}, {"start": 17.48, "duration": 3.639, "text": "have 306090 triangles where the"}, {"start": 19.16, "duration": 3.84, "text": "hypotenuse is twice as long as the"}, {"start": 21.119, "duration": 4.0, "text": "shortest side and the middle side is <"}, {"start": 23.0, "duration": 3.679, "text": "tk3 times the shortest side if you want"}, {"start": 25.119, "duration": 4.441, "text": "to try this on your own pause it right"}, {"start": 26.679, "duration": 4.76, "text": "now so I'm going to solve it in three 2"}, {"start": 29.56, "duration": 3.6, "text": "1 for this square up here we know all"}, {"start": 31.439, "duration": 3.8, "text": "the sides are congruent so they're all"}, {"start": 33.16, "duration": 3.6, "text": "equal to three and now looking at this"}, {"start": 35.239, "duration": 4.881, "text": "right triangle I'm noticing that the"}, {"start": 36.76, "duration": 5.2, "text": "medium side is < tk3 * the shortest side"}, {"start": 40.12, "duration": 3.32, "text": "that means it's going to be a 306090"}, {"start": 41.96, "duration": 3.279, "text": "triangle and we can align the"}, {"start": 43.44, "duration": 3.52, "text": "orientation just like this so we have"}, {"start": 45.239, "duration": 3.8, "text": "the shortest side here and the medium"}, {"start": 46.96, "duration": 4.68, "text": "side here which means this angle here"}, {"start": 49.039, "duration": 4.36, "text": "will be the 60\u00b0 angle and this side is"}, {"start": 51.64, "duration": 4.04, "text": "twice as long as the shortest side so"}, {"start": 53.399, "duration": 3.96, "text": "it'll be equal to six and now looking at"}, {"start": 55.68, "duration": 3.719, "text": "this lower Square all the sides are"}, {"start": 57.359, "duration": 4.68, "text": "congruent so this last side will also be"}, {"start": 59.399, "duration": 4.641, "text": "equal to six another property of squares"}, {"start": 62.039, "duration": 4.201, "text": "all of these angles are right angles so"}, {"start": 64.04, "duration": 4.6, "text": "we can call this 90\u00b0 here and in this"}, {"start": 66.24, "duration": 4.559, "text": "Square this would be 90\u00b0 and let's call"}, {"start": 68.64, "duration": 5.96, "text": "this missing angle Theta now if we look"}, {"start": 70.799, "duration": 8.32, "text": "here 90 + 60 + 90 + Theta must equal"}, {"start": 74.6, "duration": 8.0, "text": "360\u00b0 90 + 60 + 90 is equal to 240 so 240"}, {"start": 79.119, "duration": 6.161, "text": "+ Theta = 360 after we subtract 240 from"}, {"start": 82.6, "duration": 5.6, "text": "both sides we get Theta equal to 120 so"}, {"start": 85.28, "duration": 4.36, "text": "we can update this up here to 120\u00b0 and"}, {"start": 88.2, "duration": 3.48, "text": "now let's focus on this we're trying to"}, {"start": 89.64, "duration": 3.439, "text": "find the area of this red triangle we're"}, {"start": 91.68, "duration": 3.799, "text": "going to use these notes where the area"}, {"start": 93.079, "duration": 4.04, "text": "is equal to/ 12 * one side time another"}, {"start": 95.479, "duration": 3.28, "text": "side time the sign of the angle in"}, {"start": 97.119, "duration": 3.32, "text": "between them so our first side is going"}, {"start": 98.759, "duration": 3.761, "text": "to be three the other side is going to"}, {"start": 100.439, "duration": 6.081, "text": "be six and the angle between these two"}, {"start": 102.52, "duration": 7.32, "text": "sides is equal to 120\u00b0 6 * 3 is 18 and"}, {"start": 106.52, "duration": 6.199, "text": "half of that is 9 and the S of 120\u00b0 is <"}, {"start": 109.84, "duration": 5.72, "text": "tk3 over2 so we end up with an area of 9"}, {"start": 112.719, "duration": 6.201, "text": "< tk3 over2 square units and that's"}, {"start": 115.56, "duration": 5.28, "text": "approximately equal to 7.79 square units"}, {"start": 118.92, "duration": 3.76, "text": "let's bring back all the other stuff and"}, {"start": 120.84, "duration": 4.36, "text": "we successfully found the area of the"}, {"start": 122.68, "duration": 5.799, "text": "red triangle let's put a box around it"}, {"start": 125.2, "duration": 3.279, "text": "how exciting"}], "title": "Area of Red Triangle", "description": "https://andymath.com/geometry-challenges/", "lengthSeconds": 127, "width": 640, "height": 360}
{"video_id": "WINLDpymnMo", "transcript": [{"start": 0.04, "duration": 3.159, "text": "hey guys this looks like a fun one we're"}, {"start": 1.4, "duration": 3.24, "text": "trying to find the area of the red"}, {"start": 3.199, "duration": 2.881, "text": "region if you want to try this on your"}, {"start": 4.64, "duration": 4.68, "text": "own pause it right now cuz I'm going to"}, {"start": 6.08, "duration": 5.08, "text": "solve it in 3 2 one so first step let's"}, {"start": 9.32, "duration": 3.76, "text": "copy it down and let's cut it in half"}, {"start": 11.16, "duration": 4.24, "text": "with the diagonal right here and let's"}, {"start": 13.08, "duration": 4.68, "text": "make all of this one region the entire"}, {"start": 15.4, "duration": 4.44, "text": "square is 10 * 10 which is 100 so that"}, {"start": 17.76, "duration": 3.4, "text": "makes this right here 50 and we don't"}, {"start": 19.84, "duration": 3.4, "text": "know the area of these so let's give"}, {"start": 21.16, "duration": 3.68, "text": "them a variable a and let's write an"}, {"start": 23.24, "duration": 5.359, "text": "equation for this right here we know"}, {"start": 24.84, "duration": 6.08, "text": "that the 2 A's plus this x + 50 is going"}, {"start": 28.599, "duration": 3.96, "text": "to equal the entire Square which is 100"}, {"start": 30.92, "duration": 3.639, "text": "and then we can subtract 50 from both"}, {"start": 32.559, "duration": 5.0, "text": "sides and also subtract 2 a from both"}, {"start": 34.559, "duration": 5.641, "text": "sides and that gives us X is equal to 50"}, {"start": 37.559, "duration": 4.401, "text": "- 2 a and this looks pretty important"}, {"start": 40.2, "duration": 3.56, "text": "let's put it up here so all we need to"}, {"start": 41.96, "duration": 4.04, "text": "do to solve for x is figure out the area"}, {"start": 43.76, "duration": 3.959, "text": "of these two A's and I've got a good"}, {"start": 46.0, "duration": 3.68, "text": "idea how to do that let's copy the"}, {"start": 47.719, "duration": 4.241, "text": "square again but let's only include the"}, {"start": 49.68, "duration": 4.879, "text": "top semicircle the area of that"}, {"start": 51.96, "duration": 4.919, "text": "semicircle is going to be P pi r 2 divid"}, {"start": 54.559, "duration": 4.081, "text": "2 and since the sides of the square are"}, {"start": 56.879, "duration": 4.84, "text": "10 the radius is going to be equal to"}, {"start": 58.64, "duration": 9.28, "text": "five so we can update this R to be 5"}, {"start": 61.719, "duration": 7.641, "text": "then 5^ 2 is 25 < * 25 / 2 is 12.5 piun"}, {"start": 67.92, "duration": 4.28, "text": "and next let's cut this with both"}, {"start": 69.36, "duration": 4.72, "text": "diagonals each of these triangles is 1/4"}, {"start": 72.2, "duration": 3.12, "text": "of the total square so this right"}, {"start": 74.08, "duration": 3.96, "text": "triangle up here is going to have an"}, {"start": 75.32, "duration": 4.839, "text": "area of 25 and then this piece right"}, {"start": 78.04, "duration": 4.039, "text": "here is literally the exact same thing"}, {"start": 80.159, "duration": 3.801, "text": "as this piece right here so it is also"}, {"start": 82.079, "duration": 3.961, "text": "equal to a and then this is the same"}, {"start": 83.96, "duration": 5.4, "text": "size is this so we'll call it a and now"}, {"start": 86.04, "duration": 5.68, "text": "we have another equation this 25 + 2 a"}, {"start": 89.36, "duration": 5.2, "text": "makes up the entire semicircle so it's"}, {"start": 91.72, "duration": 4.88, "text": "going to be equal to 12.5 pi and after"}, {"start": 94.56, "duration": 5.559, "text": "we subtract 25 from both sides we get 2"}, {"start": 96.6, "duration": 5.479, "text": "a is equal to 12.5 piun - 25 and now we"}, {"start": 100.119, "duration": 4.841, "text": "have a value for the 2A so let's solve"}, {"start": 102.079, "duration": 6.881, "text": "for x so now in the place of this 2A we"}, {"start": 104.96, "duration": 6.36, "text": "can plug in 12.5 piun - 25 this negative"}, {"start": 108.96, "duration": 6.32, "text": "is going to distribute to give us -2.5"}, {"start": 111.32, "duration": 7.079, "text": "piun and minus - 255 is + 25 next we can"}, {"start": 115.28, "duration": 4.96, "text": "combine like terms 50 + 25 is 75 and"}, {"start": 118.399, "duration": 3.601, "text": "this is still there and now we have the"}, {"start": 120.24, "duration": 3.68, "text": "area of the red region let's give it a"}, {"start": 122.0, "duration": 3.52, "text": "label of square units and put a box"}, {"start": 123.92, "duration": 4.88, "text": "around it and here's the approximate"}, {"start": 125.52, "duration": 6.12, "text": "value of 35.7 3 square units how"}, {"start": 128.8, "duration": 2.84, "text": "exciting"}], "title": "The Hard Way", "description": "https://youtu.be/k4QjXIAr4W8 \nHere is the link to the easier version.", "lengthSeconds": 130, "width": 640, "height": 360}
{"video_id": "qOOnBTaHG_Q", "transcript": [{"start": 0.04, "duration": 2.92, "text": "hey guys this is such a fun looking"}, {"start": 1.439, "duration": 3.561, "text": "problem we have two circles with a"}, {"start": 2.96, "duration": 3.839, "text": "radius of 5 m and then we have a square"}, {"start": 5.0, "duration": 4.08, "text": "stuck inside here and it wants us to"}, {"start": 6.799, "duration": 3.84, "text": "find the area of this square if you want"}, {"start": 9.08, "duration": 4.16, "text": "to try this on your own pause it right"}, {"start": 10.639, "duration": 4.721, "text": "now because I'm going to solve it in 3 2"}, {"start": 13.24, "duration": 4.52, "text": "1 first step I want to label the sides"}, {"start": 15.36, "duration": 5.079, "text": "of this Square X so the area we're going"}, {"start": 17.76, "duration": 4.84, "text": "to solve for is going to be X2 and let's"}, {"start": 20.439, "duration": 3.6, "text": "stick that right over here next let's"}, {"start": 22.6, "duration": 3.4, "text": "draw another radius from the center of"}, {"start": 24.039, "duration": 3.721, "text": "this circle to the intersection point"}, {"start": 26.0, "duration": 3.48, "text": "right here and let's draw another radius"}, {"start": 27.76, "duration": 3.4, "text": "from this point here to this tangent"}, {"start": 29.48, "duration": 3.96, "text": "point and that's going to be at right"}, {"start": 31.16, "duration": 4.0, "text": "angles each of these blue lines are both"}, {"start": 33.44, "duration": 3.959, "text": "radi of the circle so they're both going"}, {"start": 35.16, "duration": 4.399, "text": "to have a length of 5 Metter and let's"}, {"start": 37.399, "duration": 3.801, "text": "extend this line right here parallel to"}, {"start": 39.559, "duration": 4.201, "text": "this so that this will also be a right"}, {"start": 41.2, "duration": 4.32, "text": "angle so with this setup this red"}, {"start": 43.76, "duration": 3.72, "text": "portion of the line right here is the"}, {"start": 45.52, "duration": 4.28, "text": "same thing as the height of this Square"}, {"start": 47.48, "duration": 4.079, "text": "so it will also have a length of X and"}, {"start": 49.8, "duration": 3.88, "text": "since this whole thing has a length of"}, {"start": 51.559, "duration": 4.64, "text": "five this remaining blue portion will be"}, {"start": 53.68, "duration": 5.359, "text": "5 - x and then let's call this green"}, {"start": 56.199, "duration": 5.721, "text": "line Y we now have a right triangle"}, {"start": 59.039, "duration": 9.08, "text": "right here so so we can use a 2 + B2 ="}, {"start": 61.92, "duration": 10.6, "text": "c^2 we can do 5 - x^2 + y^2 = 5^ 2 5 -"}, {"start": 68.119, "duration": 7.161, "text": "x^2 is 5 - x * 5 - x which gives us 25 -"}, {"start": 72.52, "duration": 5.52, "text": "10 x + x^2 and then we still have the +"}, {"start": 75.28, "duration": 5.44, "text": "y^ 2 and it'll be equal to 5^2 which is"}, {"start": 78.04, "duration": 4.24, "text": "25 and then we can subtract 25 from both"}, {"start": 80.72, "duration": 3.48, "text": "sides and we can also subtract the rest"}, {"start": 82.28, "duration": 3.96, "text": "of the stuff from both sides on the left"}, {"start": 84.2, "duration": 4.919, "text": "hand side we only have y^2 and on the"}, {"start": 86.24, "duration": 4.6, "text": "right hand side we have 10 x - x^2 after"}, {"start": 89.119, "duration": 5.561, "text": "we square root both both sides we have y"}, {"start": 90.84, "duration": 5.239, "text": "is equal to the < TK 10x - x^2 and now"}, {"start": 94.68, "duration": 2.96, "text": "we can clean things up a little bit"}, {"start": 96.079, "duration": 4.0, "text": "let's scoot this over here and let's"}, {"start": 97.64, "duration": 4.2, "text": "duplicate the Y over here now this"}, {"start": 100.079, "duration": 5.32, "text": "distance is going to be equal to this"}, {"start": 101.84, "duration": 6.239, "text": "distance so we know that y + x + y is"}, {"start": 105.399, "duration": 6.201, "text": "equal to 5 + 5 and we can clean this up"}, {"start": 108.079, "duration": 5.881, "text": "a little bit y + y is 2 Y and 5 + 5 is"}, {"start": 111.6, "duration": 5.199, "text": "10 and now we know that Y is equal to"}, {"start": 113.96, "duration": 6.56, "text": "the < TK of 10x - x^2 so in the place of"}, {"start": 116.799, "duration": 5.521, "text": "this y we can plug in < TK of 10x - x 2"}, {"start": 120.52, "duration": 3.76, "text": "and now we just got to solve for x we"}, {"start": 122.32, "duration": 4.88, "text": "can subtract X from both sides that'll"}, {"start": 124.28, "duration": 5.64, "text": "give us 2 * the < TK of 10x - x^2 is"}, {"start": 127.2, "duration": 4.44, "text": "equal to 10 - x and then we can square"}, {"start": 129.92, "duration": 3.36, "text": "both sides of the equation this Square"}, {"start": 131.64, "duration": 3.4, "text": "will distribute to both of these so we"}, {"start": 133.28, "duration": 4.36, "text": "have 2 squar and then we have this stuff"}, {"start": 135.04, "duration": 4.24, "text": "squared 2^ 2 is equal to 4 and this"}, {"start": 137.64, "duration": 4.12, "text": "square root and this Square are going to"}, {"start": 139.28, "duration": 4.2, "text": "cancel each other out and then 10 - x^2"}, {"start": 141.76, "duration": 5.199, "text": "means this times this and it ends up"}, {"start": 143.48, "duration": 6.759, "text": "being 100 - 20x + x^2 then we can"}, {"start": 146.959, "duration": 5.521, "text": "distribute this 4 it'll give us 40x -"}, {"start": 150.239, "duration": 3.72, "text": "4x^2 and then we can subtract this stuff"}, {"start": 152.48, "duration": 3.88, "text": "from both sides so we can set it all"}, {"start": 153.959, "duration": 7.761, "text": "equal to zero let's bring down the 100"}, {"start": 156.36, "duration": 7.72, "text": "-2X - 4x is - 60x and x^2 + 4x^2 is 5"}, {"start": 161.72, "duration": 4.72, "text": "x^2 all three of these coefficients are"}, {"start": 164.08, "duration": 7.28, "text": "divisible by 5 so let's do that we end"}, {"start": 166.44, "duration": 6.76, "text": "up with 0 = 20 - 12x + x^2 and then we"}, {"start": 171.36, "duration": 5.32, "text": "can Factor this we need two numbers that"}, {"start": 173.2, "duration": 6.36, "text": "add to -12 and multiply to POS 20 well"}, {"start": 176.68, "duration": 6.04, "text": "that's going to be -10 and -2 so we have"}, {"start": 179.56, "duration": 6.039, "text": "X x - 10 * x - 2 = 0 that means that"}, {"start": 182.72, "duration": 5.159, "text": "either this equals 0 or this equals 0 so"}, {"start": 185.599, "duration": 5.401, "text": "for x - 10al 0 we can add 10 to both"}, {"start": 187.879, "duration": 5.321, "text": "sides and we get xal 10 what we can see"}, {"start": 191.0, "duration": 4.84, "text": "right here since the radius is five"}, {"start": 193.2, "duration": 5.28, "text": "there's no way that X is 10 so this is"}, {"start": 195.84, "duration": 4.6, "text": "not a valid solution and then for x - 2"}, {"start": 198.48, "duration": 5.88, "text": "= 0 we can add two to both sides and we"}, {"start": 200.44, "duration": 6.159, "text": "get x = 2 this one is a valid solution"}, {"start": 204.36, "duration": 4.599, "text": "so now that we know x = 2 the area of"}, {"start": 206.599, "duration": 4.321, "text": "our square is going to be equal to 2^2"}, {"start": 208.959, "duration": 4.28, "text": "which is 4 let's give it a label of"}, {"start": 210.92, "duration": 5.879, "text": "meters squared and put a box around it"}, {"start": 213.239, "duration": 3.56, "text": "how exciting"}], "title": "2 Circles 1 Square", "description": "https://andymath.com/geometry-challenges/\n\nI hope you guys like this one, and the several others I have on this page!", "lengthSeconds": 215, "width": 640, "height": 360}
{"video_id": "BQCjIn8Pjk8", "transcript": [{"start": 0.0, "duration": 3.899, "text": "you guys can you believe this whole"}, {"start": 1.439, "duration": 3.961, "text": "thing is equal to one let's verify it"}, {"start": 3.899, "duration": 3.121, "text": "first I want to leave this alone so"}, {"start": 5.4, "duration": 3.42, "text": "let's make a copy of it down here and"}, {"start": 7.02, "duration": 3.78, "text": "let's handle it one fraction at a time"}, {"start": 8.82, "duration": 3.42, "text": "let's do this one first in order to get"}, {"start": 10.8, "duration": 3.06, "text": "rid of the radical in the denominator"}, {"start": 12.24, "duration": 3.78, "text": "I'm going to multiply top and bottom by"}, {"start": 13.86, "duration": 4.259, "text": "the conjugate all we do is copy these"}, {"start": 16.02, "duration": 3.96, "text": "two terms but change the plus to a minus"}, {"start": 18.119, "duration": 3.24, "text": "on top is just going to be 1 times this"}, {"start": 19.98, "duration": 3.48, "text": "so we're just going to be left with root"}, {"start": 21.359, "duration": 4.141, "text": "1 minus root 2. and the square root of 1"}, {"start": 23.46, "duration": 3.239, "text": "is equal to one and on bottom we can use"}, {"start": 25.5, "duration": 3.24, "text": "the notes for the difference of two"}, {"start": 26.699, "duration": 5.101, "text": "squares so we know that root a plus root"}, {"start": 28.74, "duration": 4.979, "text": "c times root a minus root C is just a"}, {"start": 31.8, "duration": 4.02, "text": "minus C and we can see here that a is"}, {"start": 33.719, "duration": 3.901, "text": "one and C is two so on bottom we're"}, {"start": 35.82, "duration": 3.54, "text": "going to have one minus two and we can"}, {"start": 37.62, "duration": 3.66, "text": "clean up one minus 2 to give us negative"}, {"start": 39.36, "duration": 3.359, "text": "one to get rid of this negative 1 in the"}, {"start": 41.28, "duration": 3.299, "text": "denominator we can multiply top bottom"}, {"start": 42.719, "duration": 3.481, "text": "by negative one this negative is going"}, {"start": 44.579, "duration": 3.721, "text": "to distribute to both of these making it"}, {"start": 46.2, "duration": 3.839, "text": "a negative one and a positive root 2."}, {"start": 48.3, "duration": 3.66, "text": "and on bottom negative 1 times negative"}, {"start": 50.039, "duration": 3.54, "text": "1 is just positive one and since we have"}, {"start": 51.96, "duration": 2.939, "text": "one in the denominator we don't need it"}, {"start": 53.579, "duration": 3.361, "text": "anymore all of this right here"}, {"start": 54.899, "duration": 3.601, "text": "simplified into negative 1 plus root two"}, {"start": 56.94, "duration": 3.72, "text": "in fact we don't even need this red"}, {"start": 58.5, "duration": 4.199, "text": "stuff this first fraction simplified"}, {"start": 60.66, "duration": 3.66, "text": "into negative 1 plus root 2. so now"}, {"start": 62.699, "duration": 2.941, "text": "let's focus on this next fraction we're"}, {"start": 64.32, "duration": 3.0, "text": "going to do the same thing multiply top"}, {"start": 65.64, "duration": 3.299, "text": "and bottom by the conjugate on top we're"}, {"start": 67.32, "duration": 2.88, "text": "just going to have the root 2 minus root"}, {"start": 68.939, "duration": 3.421, "text": "3. and the bottom we're going to use"}, {"start": 70.2, "duration": 4.38, "text": "these notes again but in this case a is"}, {"start": 72.36, "duration": 4.2, "text": "2 and C is 3. so on bottom we're going"}, {"start": 74.58, "duration": 4.14, "text": "to have the a minus C which is 2 minus"}, {"start": 76.56, "duration": 3.84, "text": "three and then 2 minus 3 is equal to"}, {"start": 78.72, "duration": 3.66, "text": "negative one and we can divide both"}, {"start": 80.4, "duration": 3.6, "text": "these top terms by negative one making"}, {"start": 82.38, "duration": 3.599, "text": "this first one negative and the second"}, {"start": 84.0, "duration": 3.9, "text": "one positive so we have negative root 2"}, {"start": 85.979, "duration": 3.361, "text": "plus root three if we can smush these"}, {"start": 87.9, "duration": 3.18, "text": "together and that takes care of this"}, {"start": 89.34, "duration": 3.48, "text": "second fraction now let's focus on this"}, {"start": 91.08, "duration": 3.179, "text": "third one multiply top and bottom by the"}, {"start": 92.82, "duration": 3.36, "text": "conjugate the top is just going to be"}, {"start": 94.259, "duration": 3.18, "text": "root 3 minus root four and the bottom"}, {"start": 96.18, "duration": 3.54, "text": "we're going to use these notes one more"}, {"start": 97.439, "duration": 4.441, "text": "time where a is equal to 3 and C is"}, {"start": 99.72, "duration": 3.96, "text": "equal to four so a minus C is going to"}, {"start": 101.88, "duration": 3.48, "text": "be 3 minus four three minus four is"}, {"start": 103.68, "duration": 3.0, "text": "equal to negative one these terms"}, {"start": 105.36, "duration": 2.52, "text": "divided by negative one will make the"}, {"start": 106.68, "duration": 2.7, "text": "first one negative and the second one"}, {"start": 107.88, "duration": 3.72, "text": "positive we don't need the one anymore"}, {"start": 109.38, "duration": 3.9, "text": "and we can smush this together so now we"}, {"start": 111.6, "duration": 3.479, "text": "have this expression right here this"}, {"start": 113.28, "duration": 3.479, "text": "positive root 2 and negative root 2 can"}, {"start": 115.079, "duration": 3.301, "text": "cancel each other out and this positive"}, {"start": 116.759, "duration": 2.881, "text": "root 3 and negative root 3 can also"}, {"start": 118.38, "duration": 3.419, "text": "cancel each other out and the square"}, {"start": 119.64, "duration": 4.619, "text": "root of 4 is equal to two let's bring"}, {"start": 121.799, "duration": 4.14, "text": "this down here negative one plus two is"}, {"start": 124.259, "duration": 3.961, "text": "equal to positive one that means that"}, {"start": 125.939, "duration": 4.081, "text": "this original crazy expression is equal"}, {"start": 128.22, "duration": 4.759, "text": "to one now we could put a box around it"}, {"start": 130.02, "duration": 2.959, "text": "how exciting"}], "title": "Beautiful Math Equation", "description": "I really like this one!", "lengthSeconds": 131, "width": 640, "height": 360}
{"video_id": "yQLr755ccf8", "transcript": [{"start": 0.0, "duration": 3.3, "text": "hey guys here's a fun Little Riddle That"}, {"start": 1.74, "duration": 3.12, "text": "I came up with earlier this week it's"}, {"start": 3.3, "duration": 2.94, "text": "asking you to solve for x and this looks"}, {"start": 4.86, "duration": 2.58, "text": "a lot like the quadratic formula but"}, {"start": 6.24, "duration": 2.399, "text": "it's a little different if you want to"}, {"start": 7.44, "duration": 2.4, "text": "try to solve it on your own pause right"}, {"start": 8.639, "duration": 3.54, "text": "now I'm going to show how I would"}, {"start": 9.84, "duration": 4.08, "text": "approach this in three two one so first"}, {"start": 12.179, "duration": 3.781, "text": "off this is the actual quadratic formula"}, {"start": 13.92, "duration": 3.96, "text": "I'm given that this will solve for x but"}, {"start": 15.96, "duration": 3.72, "text": "I also know for a fact this will solve"}, {"start": 17.88, "duration": 3.18, "text": "for x it looks like a has literally just"}, {"start": 19.68, "duration": 3.0, "text": "been substituted for all the other"}, {"start": 21.06, "duration": 3.78, "text": "letters and it looks like we've also"}, {"start": 22.68, "duration": 3.839, "text": "plugged in a for two so I think we can"}, {"start": 24.84, "duration": 4.08, "text": "say that a equals two I'm going to plug"}, {"start": 26.519, "duration": 4.08, "text": "in 2 for all the places of a then I can"}, {"start": 28.92, "duration": 2.939, "text": "simplify things a little bit simplify a"}, {"start": 30.599, "duration": 2.341, "text": "little bit more it's a little weird"}, {"start": 31.859, "duration": 2.581, "text": "right here because we have a negative"}, {"start": 32.94, "duration": 2.639, "text": "under the square root and the way we"}, {"start": 34.44, "duration": 2.82, "text": "handle that negative one of the square"}, {"start": 35.579, "duration": 3.241, "text": "root is y top and bottom are both"}, {"start": 37.26, "duration": 3.0, "text": "divisible by two and this is an answer"}, {"start": 38.82, "duration": 2.94, "text": "to the question and we can put a box"}, {"start": 40.26, "duration": 3.6, "text": "around that is there another approach we"}, {"start": 41.76, "duration": 4.4, "text": "can take to find another value of x how"}, {"start": 43.86, "duration": 2.3, "text": "exciting"}], "title": "Quadratic Challenge Problem #shorts", "description": "https://andymath.com/ #shorts", "lengthSeconds": 44, "width": 360, "height": 640}
{"video_id": "y2PwgShzlzI", "transcript": [{"start": 0.0, "duration": 3.6, "text": "this looks like a fun one if we know"}, {"start": 1.5, "duration": 4.259, "text": "this is six this is three and this is"}, {"start": 3.6, "duration": 3.36, "text": "four how do we solve for this y there's"}, {"start": 5.759, "duration": 3.001, "text": "something I really like called the"}, {"start": 6.96, "duration": 3.36, "text": "intersecting chord theorem if you take"}, {"start": 8.76, "duration": 3.419, "text": "the distance of the intersection to the"}, {"start": 10.32, "duration": 4.739, "text": "edge of the circle and call that a call"}, {"start": 12.179, "duration": 5.161, "text": "this distance D this distance n and this"}, {"start": 15.059, "duration": 4.741, "text": "distance y it will always be true that a"}, {"start": 17.34, "duration": 4.14, "text": "times n equals D times Y no matter how"}, {"start": 19.8, "duration": 2.88, "text": "much you move the chords around now we"}, {"start": 21.48, "duration": 3.36, "text": "could put a box around this and let's"}, {"start": 22.68, "duration": 4.08, "text": "move the label to the top so this is a"}, {"start": 24.84, "duration": 4.199, "text": "typical problem for one of these the six"}, {"start": 26.76, "duration": 4.259, "text": "three four and Y and to solve this it"}, {"start": 29.039, "duration": 4.501, "text": "would just be y times three equals six"}, {"start": 31.019, "duration": 4.081, "text": "times four which gives us 3y equals 24."}, {"start": 33.54, "duration": 3.24, "text": "divide both sides by 3 and we get y"}, {"start": 35.1, "duration": 5.299, "text": "equals eight so this is the intersecting"}, {"start": 36.78, "duration": 3.619, "text": "chord theorem how exciting"}], "title": "Intersecting Chord Theorem", "description": "Such a fun theorem!", "lengthSeconds": 39, "width": 360, "height": 640}
{"video_id": "v6s8LbQSDKE", "transcript": [{"start": 0.04, "duration": 3.399, "text": "hey guys this looks like a fun one it"}, {"start": 1.4, "duration": 4.64, "text": "says four consecutive positive even"}, {"start": 3.439, "duration": 4.4, "text": "integers have a product of 13440 what"}, {"start": 6.04, "duration": 5.16, "text": "are the integers so it means something"}, {"start": 7.839, "duration": 5.121, "text": "like 2 * 4 * 6 * 8 or 4 * 6 * 8 * 10"}, {"start": 11.2, "duration": 4.639, "text": "there are some four consecutive even"}, {"start": 12.96, "duration": 4.28, "text": "integers that multiply to 13440 we could"}, {"start": 15.839, "duration": 2.641, "text": "just keep guessing and checking and"}, {"start": 17.24, "duration": 3.279, "text": "trying more and more of these but I want"}, {"start": 18.48, "duration": 3.68, "text": "to see if we can do it algebraically if"}, {"start": 20.519, "duration": 3.84, "text": "you want to try it on your own pause it"}, {"start": 22.16, "duration": 5.68, "text": "right now cuz I'm going to solve it in 3"}, {"start": 24.359, "duration": 5.24, "text": "2 1 let's call the first even integer X"}, {"start": 27.84, "duration": 4.48, "text": "so the next even integer will have to be"}, {"start": 29.599, "duration": 4.001, "text": "two more than that so it' be x + 2 and"}, {"start": 32.32, "duration": 3.399, "text": "then we do the same thing to get the"}, {"start": 33.6, "duration": 3.44, "text": "next two integers and we need these to"}, {"start": 35.719, "duration": 3.801, "text": "multiply to"}, {"start": 37.04, "duration": 3.839, "text": "13440 next if we multiply all these out"}, {"start": 39.52, "duration": 3.28, "text": "it's going to take us a while and it"}, {"start": 40.879, "duration": 4.0, "text": "will give us this right here and then we"}, {"start": 42.8, "duration": 4.48, "text": "can subtract 13440 from both sides and"}, {"start": 44.879, "duration": 4.121, "text": "we get all of this equal to zero this"}, {"start": 47.28, "duration": 3.68, "text": "can be solved but it'll take quite a bit"}, {"start": 49.0, "duration": 3.559, "text": "of work let me show you a shortcut"}, {"start": 50.96, "duration": 3.279, "text": "instead of having the four terms be"}, {"start": 52.559, "duration": 3.401, "text": "these let's have them be these right"}, {"start": 54.239, "duration": 3.761, "text": "here they're all still two apart from"}, {"start": 55.96, "duration": 4.079, "text": "each other so these are four consecutive"}, {"start": 58.0, "duration": 3.6, "text": "even integers they're just expressed a"}, {"start": 60.039, "duration": 4.2, "text": "little bit differently and if we"}, {"start": 61.6, "duration": 4.48, "text": "rearrange them like this then we have a"}, {"start": 64.239, "duration": 5.081, "text": "difference of two squares and another"}, {"start": 66.08, "duration": 7.719, "text": "difference of two squares so x + 1 * x"}, {"start": 69.32, "duration": 7.32, "text": "-1 is just x^2 - 1 and then x - 3 * x +"}, {"start": 73.799, "duration": 4.561, "text": "3 is just x^2 - 9 and then multiplying"}, {"start": 76.64, "duration": 5.479, "text": "these out is significantly easier than"}, {"start": 78.36, "duration": 10.24, "text": "multiplying this out x^2 * x^2 is x 4 X2"}, {"start": 82.119, "duration": 8.921, "text": "* 9 is 9x2 -1 * x^2 is -1 x^2 and-1 * 9"}, {"start": 88.6, "duration": 6.24, "text": "is POS 9 we can combine like terms this"}, {"start": 91.04, "duration": 5.759, "text": "9 x^2 - 1 x^2 will become -10 x^2 and"}, {"start": 94.84, "duration": 6.0, "text": "then we can subtract 13440 from both"}, {"start": 96.799, "duration": 7.881, "text": "sides and we end up with x 4 - 10 x^2 -"}, {"start": 100.84, "duration": 5.919, "text": "13431 = 0 this is significantly easier"}, {"start": 104.68, "duration": 3.64, "text": "than this we can factor out this left"}, {"start": 106.759, "duration": 5.601, "text": "hand side we need two numbers that add"}, {"start": 108.32, "duration": 5.439, "text": "to -10 and multiply to -3 431 this will"}, {"start": 112.36, "duration": 5.28, "text": "take a little bit of trial and error but"}, {"start": 113.759, "duration": 6.601, "text": "that ends up being -21 and POS 111 and"}, {"start": 117.64, "duration": 5.759, "text": "then x^2 - 121 is another difference of"}, {"start": 120.36, "duration": 4.799, "text": "two squares that gives us x - 11 * x +"}, {"start": 123.399, "duration": 3.521, "text": "11 and then we can just copy down the"}, {"start": 125.159, "duration": 3.561, "text": "rest of this stuff and now that we have"}, {"start": 126.92, "duration": 4.72, "text": "this factored we can solve for each of"}, {"start": 128.72, "duration": 6.0, "text": "these for x^2 + 11 = 0 we can subtract"}, {"start": 131.64, "duration": 5.959, "text": "11 from both sides that'll give us x^2 ="}, {"start": 134.72, "duration": 5.56, "text": "-11 and that has two imaginary Solutions"}, {"start": 137.599, "duration": 4.36, "text": "plus or minus IUN 11 after we plug this"}, {"start": 140.28, "duration": 3.599, "text": "in up here and subtract three it will"}, {"start": 141.959, "duration": 4.64, "text": "not give us an even integer so this is"}, {"start": 143.879, "duration": 4.44, "text": "not a valid solution and then for x + 11"}, {"start": 146.599, "duration": 5.121, "text": "we can subtract 11 from both sides and"}, {"start": 148.319, "duration": 5.56, "text": "we get x = -1 11 that won't give us a"}, {"start": 151.72, "duration": 4.04, "text": "positive even integer so that is also"}, {"start": 153.879, "duration": 4.44, "text": "not a valid solution and then for the"}, {"start": 155.76, "duration": 6.32, "text": "last one x - 11 if we add 11 to both"}, {"start": 158.319, "duration": 5.601, "text": "sides we get x = 11 and x = 11 will work"}, {"start": 162.08, "duration": 4.56, "text": "for us so let's copy these down and"}, {"start": 163.92, "duration": 5.52, "text": "let's plug in 11 into each of the x's"}, {"start": 166.64, "duration": 5.599, "text": "and that gives us 11 - 3 which is 8 11 -"}, {"start": 169.44, "duration": 6.12, "text": "1 which is 10 11 + 1 which is 12 and 11"}, {"start": 172.239, "duration": 5.801, "text": "+ 3 which is 14 and these are our four"}, {"start": 175.56, "duration": 7.039, "text": "consecutive positive even integers let's"}, {"start": 178.04, "duration": 4.559, "text": "put a box around them how exciting"}], "title": "4 Consecutive Even Integers", "description": "https://andymath.com/consecutive-integers/\n\nI love these consecutive integer problems. Most of them are sums, this one has an interesting twist because it is a product.", "lengthSeconds": 181, "width": 640, "height": 360}
{"video_id": "-eP9Xh1YM04", "transcript": [{"start": 0.0, "duration": 3.78, "text": "you guys it says solve for x no"}, {"start": 1.979, "duration": 3.901, "text": "calculator so step one I'm going to"}, {"start": 3.78, "duration": 4.5, "text": "cross multiply so on the left hand side"}, {"start": 5.88, "duration": 4.799, "text": "I get x minus 2 times X plus 2 and then"}, {"start": 8.28, "duration": 3.96, "text": "5 times 13 is equal to 65. so we can"}, {"start": 10.679, "duration": 4.38, "text": "multiply out the left-hand side x times"}, {"start": 12.24, "duration": 5.1, "text": "x is x squared x times 2 is 2x negative"}, {"start": 15.059, "duration": 4.501, "text": "2 times x is negative 2X and negative 2"}, {"start": 17.34, "duration": 4.62, "text": "times 2 is negative 4. and it's still"}, {"start": 19.56, "duration": 3.78, "text": "equal to 65. this negative 2X and"}, {"start": 21.96, "duration": 2.94, "text": "positive 2x are going to cancel each"}, {"start": 23.34, "duration": 4.199, "text": "other out we're just left with x squared"}, {"start": 24.9, "duration": 4.32, "text": "minus 4 equals 65. so then we can add 4"}, {"start": 27.539, "duration": 3.421, "text": "to both sides these two cancel each"}, {"start": 29.22, "duration": 5.519, "text": "other out which gives us x squared"}, {"start": 30.96, "duration": 6.42, "text": "equals and 65 plus 4 is 69. nice square"}, {"start": 34.739, "duration": 6.0, "text": "root both sides and we get x equals a"}, {"start": 37.38, "duration": 7.04, "text": "plus or minus root 69 nice and that's"}, {"start": 40.739, "duration": 3.681, "text": "the answer to the question how exciting"}], "title": "Rational Equation Square Root Method", "description": "Collab with Geeklyhub!", "lengthSeconds": 43, "width": 360, "height": 640}
{"video_id": "HB3HIMa8OH0", "transcript": [{"start": 0.0, "duration": 3.72, "text": "highlights from the first semester of"}, {"start": 1.8, "duration": 3.42, "text": "Algebra 2 in two and a half minutes"}, {"start": 3.72, "duration": 3.3, "text": "let's start by looking at parent"}, {"start": 5.22, "duration": 3.78, "text": "functions and we can do Transformations"}, {"start": 7.02, "duration": 3.72, "text": "on these functions we can make vertical"}, {"start": 9.0, "duration": 3.42, "text": "changes by multiplying and adding"}, {"start": 10.74, "duration": 3.3, "text": "numbers to f of x and we can make"}, {"start": 12.42, "duration": 3.66, "text": "horizontal changes by multiplying or"}, {"start": 14.04, "duration": 4.14, "text": "adding numbers to X we can also reflect"}, {"start": 16.08, "duration": 3.66, "text": "over the x-axis or y-axis and now"}, {"start": 18.18, "duration": 3.0, "text": "parabolas parabolas are very good at"}, {"start": 19.74, "duration": 3.0, "text": "modeling things like objects being"}, {"start": 21.18, "duration": 3.12, "text": "thrown in the air there are four common"}, {"start": 22.74, "duration": 3.539, "text": "equations for parabolas and these are"}, {"start": 24.3, "duration": 3.66, "text": "called quadratic equations it can come"}, {"start": 26.279, "duration": 3.901, "text": "in vertex form with an a vertex form"}, {"start": 27.96, "duration": 3.78, "text": "with a 4p intercept form or standard"}, {"start": 30.18, "duration": 3.42, "text": "form and depending on which equation you"}, {"start": 31.74, "duration": 4.02, "text": "use you can find the vertex the axis of"}, {"start": 33.6, "duration": 3.54, "text": "symmetry the focus or the directrix now"}, {"start": 35.76, "duration": 2.88, "text": "we can also find the Y intercept by"}, {"start": 37.14, "duration": 3.48, "text": "setting x equal to zero or the X"}, {"start": 38.64, "duration": 3.239, "text": "intercepts by setting y equal to zero"}, {"start": 40.62, "duration": 2.759, "text": "let's talk more about finding the"}, {"start": 41.879, "duration": 3.121, "text": "x-intercepts so they're typically four"}, {"start": 43.379, "duration": 3.18, "text": "ways to solve a quadratic equation when"}, {"start": 45.0, "duration": 3.3, "text": "it's set equal to zero the square roots"}, {"start": 46.559, "duration": 3.901, "text": "method factoring completing the square"}, {"start": 48.3, "duration": 3.96, "text": "or quadratic formula in some situations"}, {"start": 50.46, "duration": 3.72, "text": "one method is best a lot of times you"}, {"start": 52.26, "duration": 3.9, "text": "have your choice a parabola may have two"}, {"start": 54.18, "duration": 3.3, "text": "one or zero X intercepts in the"}, {"start": 56.16, "duration": 3.419, "text": "situations where there are zero"}, {"start": 57.48, "duration": 3.899, "text": "x-intercepts we still have solutions to"}, {"start": 59.579, "duration": 3.96, "text": "the equation Asians but they're complex"}, {"start": 61.379, "duration": 3.661, "text": "Solutions using I or I squared is equal"}, {"start": 63.539, "duration": 2.821, "text": "to negative one or you could think of it"}, {"start": 65.04, "duration": 3.119, "text": "as I is equal to the square root of"}, {"start": 66.36, "duration": 3.18, "text": "negative one when solving these and a"}, {"start": 68.159, "duration": 2.941, "text": "negative shows up into the square root"}, {"start": 69.54, "duration": 3.54, "text": "you know the solution will be complex"}, {"start": 71.1, "duration": 3.54, "text": "polynomial function all of these are"}, {"start": 73.08, "duration": 3.42, "text": "polynomial functions all of these are"}, {"start": 74.64, "duration": 3.72, "text": "not polynomial functions polynomial"}, {"start": 76.5, "duration": 3.6, "text": "functions have terms with coefficients"}, {"start": 78.36, "duration": 3.66, "text": "and whole number exponents being added"}, {"start": 80.1, "duration": 3.9, "text": "together we can add subtract multiply"}, {"start": 82.02, "duration": 3.54, "text": "and divide polynomials the degree of a"}, {"start": 84.0, "duration": 3.299, "text": "polynomial is the same as the highest"}, {"start": 85.56, "duration": 3.66, "text": "exponent and the number of zeros or"}, {"start": 87.299, "duration": 3.901, "text": "roots or x-intercepts of a polynomial"}, {"start": 89.22, "duration": 3.899, "text": "function is the same as that degree if"}, {"start": 91.2, "duration": 3.66, "text": "we start doing fractional exponents we"}, {"start": 93.119, "duration": 3.121, "text": "call these radical functions whenever"}, {"start": 94.86, "duration": 3.24, "text": "you have a fractional exponent the"}, {"start": 96.24, "duration": 3.66, "text": "denominator is the index of the radical"}, {"start": 98.1, "duration": 3.3, "text": "and the numerator is still the exponent"}, {"start": 99.9, "duration": 3.359, "text": "so the square root of x could be written"}, {"start": 101.4, "duration": 3.24, "text": "as x to the one half the cube root of x"}, {"start": 103.259, "duration": 2.881, "text": "could be x to the one third and the"}, {"start": 104.64, "duration": 3.18, "text": "fifth root of x squared would be x to"}, {"start": 106.14, "duration": 3.839, "text": "the two-fifths up to now we've been"}, {"start": 107.82, "duration": 4.32, "text": "working with X to some exponent what if"}, {"start": 109.979, "duration": 4.021, "text": "we go crazy and put X in the exponent"}, {"start": 112.14, "duration": 3.42, "text": "this is called an exponential function"}, {"start": 114.0, "duration": 3.42, "text": "and this is where exponential growth or"}, {"start": 115.56, "duration": 3.599, "text": "Decay comes from so we can solve for x"}, {"start": 117.42, "duration": 3.9, "text": "in exponential equations too but it"}, {"start": 119.159, "duration": 4.14, "text": "requires an inverse process this is the"}, {"start": 121.32, "duration": 3.54, "text": "reason that we created logarithms the"}, {"start": 123.299, "duration": 3.241, "text": "whole goal of a logarithm is to get the"}, {"start": 124.86, "duration": 3.42, "text": "variable out of the exponent here's a"}, {"start": 126.54, "duration": 3.6, "text": "pretty solid way to think of logarithms"}, {"start": 128.28, "duration": 3.42, "text": "and for practical purposes logarithms"}, {"start": 130.14, "duration": 3.179, "text": "have a lot of properties using these"}, {"start": 131.7, "duration": 3.24, "text": "properties we can expand logarithms we"}, {"start": 133.319, "duration": 3.301, "text": "can condense logarithms we can solve"}, {"start": 134.94, "duration": 3.18, "text": "them in our head but ultimately all this"}, {"start": 136.62, "duration": 3.42, "text": "training for logarithms is meant to"}, {"start": 138.12, "duration": 3.24, "text": "prepare us for the main event anytime we"}, {"start": 140.04, "duration": 2.94, "text": "need to solve an equation where the"}, {"start": 141.36, "duration": 3.12, "text": "variable is in an exponent and this"}, {"start": 142.98, "duration": 5.18, "text": "concludes the highlights from Algebra 2"}, {"start": 144.48, "duration": 3.68, "text": "first semester how exciting"}], "title": "Very High Level Algebra 2 1st Semester", "description": "https://andymath.com/algebra2/", "lengthSeconds": 146, "width": 640, "height": 360}
{"video_id": "vNcRYymXrms", "transcript": [{"start": 0.0, "duration": 4.08, "text": "hey guys we have a castle here"}, {"start": 1.74, "duration": 4.019, "text": "surrounded by a moat and the moat has a"}, {"start": 4.08, "duration": 4.14, "text": "width of 20 meters all the way around"}, {"start": 5.759, "duration": 4.92, "text": "and you happen to have two boards that"}, {"start": 8.22, "duration": 4.02, "text": "are exactly 19 meters long is there a"}, {"start": 10.679, "duration": 3.54, "text": "way to use these two boards to get to"}, {"start": 12.24, "duration": 3.18, "text": "the castle so obviously we can't do it"}, {"start": 14.219, "duration": 3.48, "text": "this way because it's not going to be"}, {"start": 15.42, "duration": 4.56, "text": "long enough to reach but what if we put"}, {"start": 17.699, "duration": 3.961, "text": "it at a 45 degree angle like this and"}, {"start": 19.98, "duration": 3.66, "text": "then put the other board right here on"}, {"start": 21.66, "duration": 3.84, "text": "top of it will this be long enough to"}, {"start": 23.64, "duration": 4.08, "text": "reach well let's zoom in on it and do"}, {"start": 25.5, "duration": 4.26, "text": "some of the math first let's look at how"}, {"start": 27.72, "duration": 3.54, "text": "far is this total diagonal distance we"}, {"start": 29.76, "duration": 3.9, "text": "need to reach and let's call that"}, {"start": 31.26, "duration": 4.2, "text": "distance D this makes an isosceles right"}, {"start": 33.66, "duration": 4.14, "text": "triangle where this side is equal to 20"}, {"start": 35.46, "duration": 4.2, "text": "and this side is equal to 20. so we can"}, {"start": 37.8, "duration": 4.68, "text": "solve 20 squared plus 20 squared equals"}, {"start": 39.66, "duration": 4.8, "text": "D Squared that's 400 plus 400 equals D"}, {"start": 42.48, "duration": 3.78, "text": "Squared which is 800 equals D Squared"}, {"start": 44.46, "duration": 4.86, "text": "and after you square root 800 you get D"}, {"start": 46.26, "duration": 4.619, "text": "is approximately equal to 28.28 let's"}, {"start": 49.32, "duration": 4.02, "text": "keep this here but get rid of everything"}, {"start": 50.879, "duration": 4.32, "text": "else so this whole board is equal to 19"}, {"start": 53.34, "duration": 4.559, "text": "which would mean that the half board"}, {"start": 55.199, "duration": 4.5, "text": "would be half of 19 which is 9.5 this"}, {"start": 57.899, "duration": 3.66, "text": "also makes an isosceles right triangle"}, {"start": 59.699, "duration": 4.02, "text": "where this this side right here is also"}, {"start": 61.559, "duration": 4.261, "text": "9.5 and this is important so let's mark"}, {"start": 63.719, "duration": 3.481, "text": "this 9.5 as red and let's get rid of"}, {"start": 65.82, "duration": 3.119, "text": "this stuff and now let's bring these"}, {"start": 67.2, "duration": 4.2, "text": "boards back well this board is one of"}, {"start": 68.939, "duration": 4.081, "text": "those original 19 foot boards with these"}, {"start": 71.4, "duration": 4.32, "text": "two boards set up like this we can"}, {"start": 73.02, "duration": 6.24, "text": "Traverse a diagonal that is 19 plus 9.5"}, {"start": 75.72, "duration": 6.42, "text": "meters and 19 plus 9.5 is equal to 28.5"}, {"start": 79.26, "duration": 5.039, "text": "well the 28.5 we can cross with our two"}, {"start": 82.14, "duration": 5.339, "text": "boards is greater than the diagonal size"}, {"start": 84.299, "duration": 5.661, "text": "of 28.28 so this method checks out how"}, {"start": 87.479, "duration": 2.481, "text": "exciting"}], "title": "Moat Riddle Solution & Math", "description": "I thought this was a fun one! I hope it makes sense.", "lengthSeconds": 88, "width": 360, "height": 640}
{"video_id": "31EaufH4M9g", "transcript": [{"start": 0.04, "duration": 3.2, "text": "hey guys this looked like a fun one it's"}, {"start": 1.48, "duration": 3.64, "text": "a lot more work than I was expecting it"}, {"start": 3.24, "duration": 3.92, "text": "to be we're trying to find the area of"}, {"start": 5.12, "duration": 3.599, "text": "this orange triangle if you want to try"}, {"start": 7.16, "duration": 4.519, "text": "it on your own pause it right now cuz"}, {"start": 8.719, "duration": 4.721, "text": "I'm going to solve it in 3 2 1 we know"}, {"start": 11.679, "duration": 3.84, "text": "the three sides of this golden yellow"}, {"start": 13.44, "duration": 4.08, "text": "triangle I want to find the area of it"}, {"start": 15.519, "duration": 4.201, "text": "first and we can do that with Heron's"}, {"start": 17.52, "duration": 4.04, "text": "formula it uses the semi-perimeter which"}, {"start": 19.72, "duration": 3.52, "text": "is half of the perimeter and then you"}, {"start": 21.56, "duration": 3.6, "text": "just plug it into this formula right"}, {"start": 23.24, "duration": 3.92, "text": "here so let's do it for our Golden"}, {"start": 25.16, "duration": 5.56, "text": "Triangle for a b and c we're going to"}, {"start": 27.16, "duration": 6.079, "text": "plug in the sides 8 4 and 6 and 8+ 4 6 /"}, {"start": 30.72, "duration": 4.92, "text": "2 is equal to 9 so in the place of all"}, {"start": 33.239, "duration": 4.521, "text": "these s's I'm going to plug in 9 and"}, {"start": 35.64, "duration": 4.439, "text": "then we'll plug in 8 for the a four for"}, {"start": 37.76, "duration": 4.0, "text": "the B and 6 for the C and we can"}, {"start": 40.079, "duration": 4.401, "text": "simplify this we can bring down the 9"}, {"start": 41.76, "duration": 6.24, "text": "and all the parentheses 9 - 8 is equal"}, {"start": 44.48, "duration": 5.84, "text": "to 1 9 - 4 is equal to 5 9 - 6 is equal"}, {"start": 48.0, "duration": 4.92, "text": "to 3 so this will equal theare < TK of 9"}, {"start": 50.32, "duration": 5.879, "text": "* 15 and we can split this up into two"}, {"start": 52.92, "duration": 5.599, "text": "separate square roots > of 9 * > 15 and"}, {"start": 56.199, "duration": 5.32, "text": "theun of 9 is equal to 3 so the area of"}, {"start": 58.519, "duration": 5.121, "text": "this triangle is 3un 15 but the area of"}, {"start": 61.519, "duration": 3.68, "text": "a triangle is also equal to/ 12 base"}, {"start": 63.64, "duration": 3.799, "text": "time height where this is the height of"}, {"start": 65.199, "duration": 5.361, "text": "the triangle right here so this will"}, {"start": 67.439, "duration": 4.641, "text": "also equal 3un 15 b stands for the base"}, {"start": 70.56, "duration": 4.84, "text": "and the base of this triangle is equal"}, {"start": 72.08, "duration": 5.24, "text": "to 6 and then 1 12 * 6 is equal to 3 and"}, {"start": 75.4, "duration": 4.56, "text": "after we divide both sides by three we"}, {"start": 77.32, "duration": 4.439, "text": "get H is equal to theare < TK of 15 and"}, {"start": 79.96, "duration": 3.92, "text": "we can update this H to be square < TK"}, {"start": 81.759, "duration": 3.961, "text": "of 15 now let's bring back our orange"}, {"start": 83.88, "duration": 4.239, "text": "triangle I want to draw a line from this"}, {"start": 85.72, "duration": 4.359, "text": "vertex up here that's parallel to this"}, {"start": 88.119, "duration": 3.721, "text": "line let's make this line more bold and"}, {"start": 90.079, "duration": 3.72, "text": "let's hide this stuff for a second these"}, {"start": 91.84, "duration": 3.72, "text": "two angles are called alternate interior"}, {"start": 93.799, "duration": 3.68, "text": "angles and since these two lines are"}, {"start": 95.56, "duration": 4.32, "text": "parallel these angles are congruent to"}, {"start": 97.479, "duration": 4.28, "text": "each other and let's make this line bold"}, {"start": 99.88, "duration": 3.919, "text": "and extend it a little bit these two"}, {"start": 101.759, "duration": 4.4, "text": "angles are called vertical angles and"}, {"start": 103.799, "duration": 3.92, "text": "they are also congruent and then this"}, {"start": 106.159, "duration": 3.56, "text": "angle and this angle are called"}, {"start": 107.719, "duration": 3.601, "text": "corresponding angles since these are"}, {"start": 109.719, "duration": 3.76, "text": "parallel these corresponding angles are"}, {"start": 111.32, "duration": 3.92, "text": "going to be congruent to each other now"}, {"start": 113.479, "duration": 3.92, "text": "we can focus on this triangle right here"}, {"start": 115.24, "duration": 4.28, "text": "we have these two base angles congruent"}, {"start": 117.399, "duration": 3.961, "text": "which means this is an isoceles triangle"}, {"start": 119.52, "duration": 4.36, "text": "so this side will also have a length of"}, {"start": 121.36, "duration": 4.24, "text": "four and since this is four and the"}, {"start": 123.88, "duration": 4.0, "text": "entire side length is eight this piece"}, {"start": 125.6, "duration": 4.6, "text": "will also be four let's label the base"}, {"start": 127.88, "duration": 4.2, "text": "of our orange triangle X and now we can"}, {"start": 130.2, "duration": 3.48, "text": "use th's theorem says if you have a"}, {"start": 132.08, "duration": 4.12, "text": "triangle and a line cutting through that"}, {"start": 133.68, "duration": 4.559, "text": "triangle parallel to the base this ratio"}, {"start": 136.2, "duration": 3.88, "text": "will be true and if we put this on its"}, {"start": 138.239, "duration": 4.041, "text": "side it'll have the exact orientation of"}, {"start": 140.08, "duration": 5.4, "text": "our triangle here and we can set up our"}, {"start": 142.28, "duration": 6.0, "text": "proportion a is going to be equal to 6 B"}, {"start": 145.48, "duration": 5.479, "text": "is going to be equal to X and C and D"}, {"start": 148.28, "duration": 4.88, "text": "are both equal to 4 well the only way 6"}, {"start": 150.959, "duration": 4.601, "text": "overx is going to be equal to 4 over 4"}, {"start": 153.16, "duration": 3.96, "text": "is if x is equal to 6 so that means the"}, {"start": 155.56, "duration": 3.599, "text": "base of our orange triangle is going to"}, {"start": 157.12, "duration": 3.68, "text": "be equal to 6 and now we're ready to"}, {"start": 159.159, "duration": 3.44, "text": "find the area of our orange triangle"}, {"start": 160.8, "duration": 3.24, "text": "this height is the same as the height of"}, {"start": 162.599, "duration": 3.401, "text": "our orange triangle the area of our"}, {"start": 164.04, "duration": 4.08, "text": "triangle is 1/2 base time height where"}, {"start": 166.0, "duration": 5.0, "text": "the base is equal to 6 and the height is"}, {"start": 168.12, "duration": 4.479, "text": "equal to < TK of 15 1 12 of 6 is equal"}, {"start": 171.0, "duration": 3.599, "text": "to 3 and that's going to be multiplied"}, {"start": 172.599, "duration": 3.841, "text": "by the < TK of 15 and that's the answer"}, {"start": 174.599, "duration": 3.761, "text": "to our question we can put a box around"}, {"start": 176.44, "duration": 4.719, "text": "it so in this problem the area of this"}, {"start": 178.36, "duration": 5.239, "text": "orange triangle is equal to 3 < TK 15"}, {"start": 181.159, "duration": 6.08, "text": "square units and that is approximately"}, {"start": 183.599, "duration": 4.841, "text": "11.62 square units how exciting this was"}, {"start": 187.239, "duration": 3.521, "text": "kind of a long one if you actually"}, {"start": 188.44, "duration": 3.76, "text": "finished it comment a TV series you'd"}, {"start": 190.76, "duration": 4.199, "text": "recommend watching I need some"}, {"start": 192.2, "duration": 2.759, "text": "suggestions"}], "title": "Fun Geometry Challenge", "description": "https://andymath.com/geometry-challenges/", "lengthSeconds": 193, "width": 640, "height": 360}
{"video_id": "foNx3tYrRv4", "transcript": [{"start": 0.04, "duration": 4.239, "text": "hey guys this looks like a fun one this"}, {"start": 1.599, "duration": 5.2, "text": "is another Katrina a puzzle and it's day"}, {"start": 4.279, "duration": 5.041, "text": "two of our Advent calendar we're going"}, {"start": 6.799, "duration": 5.161, "text": "to solve 31 Catrina a puzzles in the"}, {"start": 9.32, "duration": 4.76, "text": "month of December one for each day and"}, {"start": 11.96, "duration": 4.88, "text": "this is December 2nd it tells us that we"}, {"start": 14.08, "duration": 4.72, "text": "have three squares and that the smallest"}, {"start": 16.84, "duration": 4.48, "text": "has an area of seven and it wants to"}, {"start": 18.8, "duration": 3.76, "text": "know what's the total shaded area if you"}, {"start": 21.32, "duration": 3.68, "text": "want to try it on your own pause it"}, {"start": 22.56, "duration": 5.28, "text": "right now cuz I'm going to solve it in 3"}, {"start": 25.0, "duration": 5.359, "text": "2 1 first let's label the sides of the"}, {"start": 27.84, "duration": 5.2, "text": "larger Square X and let's label the"}, {"start": 30.359, "duration": 5.04, "text": "sides of this smaller Square y we can"}, {"start": 33.04, "duration": 4.039, "text": "now write a formula for the Shaded area"}, {"start": 35.399, "duration": 4.881, "text": "the Shaded area will be equal to the"}, {"start": 37.079, "duration": 6.401, "text": "large Square which is X2 minus this"}, {"start": 40.28, "duration": 5.16, "text": "Square which is y^ 2 minus this Square"}, {"start": 43.48, "duration": 4.599, "text": "which is 7 this is what we're trying to"}, {"start": 45.44, "duration": 4.24, "text": "solve for let's put a box around it so"}, {"start": 48.079, "duration": 4.32, "text": "to move forward I'm going to make one"}, {"start": 49.68, "duration": 5.0, "text": "assumption that isn't explicitly stated"}, {"start": 52.399, "duration": 4.401, "text": "and that is that this semi circle is"}, {"start": 54.68, "duration": 4.12, "text": "intersecting this square at these"}, {"start": 56.8, "duration": 3.599, "text": "Corners I don't know how I would solve"}, {"start": 58.8, "duration": 3.48, "text": "it without that assumption so I'm going"}, {"start": 60.399, "duration": 3.081, "text": "to make that assumption comment if you"}, {"start": 62.28, "duration": 3.6, "text": "have another way to solve it that"}, {"start": 63.48, "duration": 3.959, "text": "doesn't use that assumption using that"}, {"start": 65.88, "duration": 3.96, "text": "assumption I'm going to draw the line"}, {"start": 67.439, "duration": 4.32, "text": "from here to here this is the radius of"}, {"start": 69.84, "duration": 3.84, "text": "this semicircle the semicircle has a"}, {"start": 71.759, "duration": 4.72, "text": "diameter of X which means the radius"}, {"start": 73.68, "duration": 4.84, "text": "would be half of that or X over 2 and"}, {"start": 76.479, "duration": 4.801, "text": "then next let's draw this perpendicular"}, {"start": 78.52, "duration": 4.959, "text": "line this is the same length as the side"}, {"start": 81.28, "duration": 4.799, "text": "of this square and this square has an"}, {"start": 83.479, "duration": 5.401, "text": "area of 7 so the side length is square <"}, {"start": 86.079, "duration": 5.0, "text": "TK of 7 which means this has a length of"}, {"start": 88.88, "duration": 4.4, "text": "square < TK of 7 and last let's connect"}, {"start": 91.079, "duration": 3.761, "text": "this right here this distance is half"}, {"start": 93.28, "duration": 4.72, "text": "the side of this Square so it's going to"}, {"start": 94.84, "duration": 5.08, "text": "be y/ 2 now let's play around with this"}, {"start": 98.0, "duration": 3.96, "text": "right triangle as usual we're going to"}, {"start": 99.92, "duration": 5.12, "text": "use the Pythagorean theorem which is A2"}, {"start": 101.96, "duration": 5.839, "text": "+ B2 = c^2 in the place of the a we'll"}, {"start": 105.04, "duration": 4.359, "text": "plug in this side right here < TK 7 in"}, {"start": 107.799, "duration": 4.0, "text": "the place of the B we'll plug in this"}, {"start": 109.399, "duration": 4.72, "text": "side which is y over 2 and in the place"}, {"start": 111.799, "duration": 3.401, "text": "of the C we'll plug in this side which"}, {"start": 114.119, "duration": 3.561, "text": "is X"}, {"start": 115.2, "duration": 4.12, "text": "over2 for the first term the square root"}, {"start": 117.68, "duration": 3.759, "text": "and the square will cancel each other"}, {"start": 119.32, "duration": 3.96, "text": "out so we're left with seven for the"}, {"start": 121.439, "duration": 3.6, "text": "next term the exponent distributes to"}, {"start": 123.28, "duration": 4.36, "text": "both the top and the bottom giving us"}, {"start": 125.039, "duration": 5.36, "text": "y^2 over 2^2 and same thing over here"}, {"start": 127.64, "duration": 5.76, "text": "this changes into x^2 over 2^2 and then"}, {"start": 130.399, "duration": 5.56, "text": "let's simplify this 2^2 is equal to 4"}, {"start": 133.4, "duration": 4.68, "text": "2^2 is equal to 4 and let's give this 7"}, {"start": 135.959, "duration": 4.881, "text": "a common denominator 7 is going to be"}, {"start": 138.08, "duration": 4.12, "text": "the same thing as 284 and now that these"}, {"start": 140.84, "duration": 4.64, "text": "have the same denominator we can write"}, {"start": 142.2, "duration": 5.72, "text": "them as a single fraction 28 + y^2 over"}, {"start": 145.48, "duration": 4.44, "text": "4 next I don't really like fractions so"}, {"start": 147.92, "duration": 3.679, "text": "let's get rid of this denominator we can"}, {"start": 149.92, "duration": 4.2, "text": "multiply both sides of the equation by"}, {"start": 151.599, "duration": 4.401, "text": "four this four and this four will cancel"}, {"start": 154.12, "duration": 4.04, "text": "each other out and this four and this"}, {"start": 156.0, "duration": 5.48, "text": "four will cancel each other out so now"}, {"start": 158.16, "duration": 5.92, "text": "we have 28 + y^2 is equal to x^2 let's"}, {"start": 161.48, "duration": 4.399, "text": "subtract y^2 from both sides on the left"}, {"start": 164.08, "duration": 4.12, "text": "hand side these y^2 will cancel each"}, {"start": 165.879, "duration": 6.481, "text": "other out so we'll have 28 and that'll"}, {"start": 168.2, "duration": 7.2, "text": "be equal to x^2 - y^2 which is x^2 - y^2"}, {"start": 172.36, "duration": 5.799, "text": "and check this out we needed x^2 - y^2"}, {"start": 175.4, "duration": 6.559, "text": "in the place of this x^2 - y^2 we can"}, {"start": 178.159, "duration": 5.521, "text": "plug in 28 28 28 - 7 is equal to 21 and"}, {"start": 181.959, "duration": 3.36, "text": "there's no units in the original problem"}, {"start": 183.68, "duration": 3.4, "text": "so we'll give it a label of units"}, {"start": 185.319, "duration": 4.2, "text": "squared this is the answer to our"}, {"start": 187.08, "duration": 5.159, "text": "question it's already in a box how"}, {"start": 189.519, "duration": 4.921, "text": "exciting this was day two of our agvent"}, {"start": 192.239, "duration": 4.0, "text": "calendar this will be day three this is"}, {"start": 194.44, "duration": 3.2, "text": "the video that we'll be doing tomorrow"}, {"start": 196.239, "duration": 3.64, "text": "we want to know what is the area of this"}, {"start": 197.64, "duration": 6.92, "text": "shaded region divided by the area of the"}, {"start": 199.879, "duration": 4.681, "text": "entire circle how exciting"}], "title": "Aggvent Calendar Day 2", "description": "https://andymath.com/agg-vent-calendar\n\nHere is the link to the aggvent calendar if you'd like to try some of them!", "lengthSeconds": 203, "width": 640, "height": 360}
{"video_id": "4x61B4AYcek", "transcript": [{"start": 0.0, "duration": 3.48, "text": "had some crazy polynomial function like"}, {"start": 1.86, "duration": 3.0, "text": "this and you need to know what is f of"}, {"start": 3.48, "duration": 2.94, "text": "2. traditionally the way you'd solve"}, {"start": 4.86, "duration": 3.36, "text": "this is you take all this and plug in 2"}, {"start": 6.42, "duration": 3.0, "text": "for the x's and then bring down all the"}, {"start": 8.22, "duration": 2.76, "text": "other stuff figure out what each of"}, {"start": 9.42, "duration": 2.88, "text": "these terms is equal to and then add or"}, {"start": 10.98, "duration": 2.64, "text": "subtract them all together and you get"}, {"start": 12.3, "duration": 2.64, "text": "three it would have taken a little bit"}, {"start": 13.62, "duration": 2.46, "text": "of time to figure it all out there's"}, {"start": 14.94, "duration": 3.0, "text": "another method you can do called"}, {"start": 16.08, "duration": 3.36, "text": "synthetic substitution you make a little"}, {"start": 17.94, "duration": 3.3, "text": "bar like this but the number you're"}, {"start": 19.44, "duration": 3.66, "text": "going to plug in outside here and then"}, {"start": 21.24, "duration": 3.539, "text": "put the coefficients including the"}, {"start": 23.1, "duration": 2.7, "text": "negative values right here so this is"}, {"start": 24.779, "duration": 2.461, "text": "how you set up the synthetic"}, {"start": 25.8, "duration": 2.7, "text": "substitution now to perform this"}, {"start": 27.24, "duration": 3.72, "text": "synthetic substitution all we're going"}, {"start": 28.5, "duration": 4.38, "text": "to do is add multiply add multiply add"}, {"start": 30.96, "duration": 4.14, "text": "multiply add multiply three plus zero is"}, {"start": 32.88, "duration": 4.32, "text": "three two times three is six negative"}, {"start": 35.1, "duration": 3.9, "text": "six plus six is zero two times zero is"}, {"start": 37.2, "duration": 3.539, "text": "zero negative one plus zero is negative"}, {"start": 39.0, "duration": 3.719, "text": "one two times negative 1 is negative two"}, {"start": 40.739, "duration": 3.84, "text": "three plus negative two is one two times"}, {"start": 42.719, "duration": 3.66, "text": "one is two negative four plus two is"}, {"start": 44.579, "duration": 4.441, "text": "negative two two times negative 2 is"}, {"start": 46.379, "duration": 4.2, "text": "negative four seven plus negative 4 is 3"}, {"start": 49.02, "duration": 2.76, "text": "and that is our answer and that's the"}, {"start": 50.579, "duration": 2.82, "text": "same answer we got up here this is"}, {"start": 51.78, "duration": 3.0, "text": "synthetic substitution for polynomials"}, {"start": 53.399, "duration": 3.241, "text": "it will always work and for large"}, {"start": 54.78, "duration": 5.119, "text": "polynomials it's usually easier than"}, {"start": 56.64, "duration": 3.259, "text": "this method how exciting"}], "title": "Synthetic Substitution Example #shorts", "description": "https://andymath.com/synthetic-substitution/\n\n#shorts", "lengthSeconds": 58, "width": 270, "height": 480}
{"video_id": "scwmIufUwmI", "transcript": [{"start": 0.04, "duration": 2.44, "text": "hey guys I made a video about this the"}, {"start": 1.319, "duration": 2.6, "text": "other day and I kind of like the way"}, {"start": 2.48, "duration": 3.359, "text": "that I did it but several people"}, {"start": 3.919, "duration": 3.92, "text": "requested me to solve it a different way"}, {"start": 5.839, "duration": 3.521, "text": "where I repeatedly Square both sides of"}, {"start": 7.839, "duration": 3.281, "text": "the equation I thought it could be cool"}, {"start": 9.36, "duration": 3.48, "text": "to see what that looks like so let's do"}, {"start": 11.12, "duration": 3.759, "text": "it for the first step let's Square both"}, {"start": 12.84, "duration": 3.32, "text": "sides on this left hand side the square"}, {"start": 14.879, "duration": 3.001, "text": "and the square root will cancel each"}, {"start": 16.16, "duration": 3.52, "text": "other out and on the right hand side I"}, {"start": 17.88, "duration": 4.88, "text": "don't really want to do 32 squar but we"}, {"start": 19.68, "duration": 5.32, "text": "can change this 32 into 2 to 5th and now"}, {"start": 22.76, "duration": 5.12, "text": "to do 2 the 5th squar we just multiply"}, {"start": 25.0, "duration": 4.64, "text": "the 5 * the 2 to get 2 10 and next we"}, {"start": 27.88, "duration": 3.479, "text": "Square both sides of the equation on"}, {"start": 29.64, "duration": 3.48, "text": "this side the square will go both to the"}, {"start": 31.359, "duration": 3.281, "text": "numerator and to the denominator it's"}, {"start": 33.12, "duration": 2.959, "text": "going to look like this and in the"}, {"start": 34.64, "duration": 3.04, "text": "denominator the square and the square"}, {"start": 36.079, "duration": 2.8, "text": "root will cancel each other out and on"}, {"start": 37.68, "duration": 3.399, "text": "the right hand side we can multiply"}, {"start": 38.879, "duration": 3.16, "text": "these exponents to give us 2 to the 20"}, {"start": 41.079, "duration": 2.8, "text": "and then once again we're going to"}, {"start": 42.039, "duration": 3.761, "text": "square both sides of the equation this"}, {"start": 43.879, "duration": 4.52, "text": "squared can distribute to this term this"}, {"start": 45.8, "duration": 5.68, "text": "term and this term it'll look like this"}, {"start": 48.399, "duration": 4.921, "text": "and then x^2 s is X to 4th this guy's"}, {"start": 51.48, "duration": 3.399, "text": "good and on the very bottom this square"}, {"start": 53.32, "duration": 2.96, "text": "root and this Square will cancel each"}, {"start": 54.879, "duration": 2.48, "text": "other out and then on the right hand"}, {"start": 56.28, "duration": 3.4, "text": "side we're going to multiply these"}, {"start": 57.359, "duration": 3.68, "text": "exponents again to give us 2 to 40th and"}, {"start": 59.68, "duration": 3.24, "text": "next step we're going to square both"}, {"start": 61.039, "duration": 3.681, "text": "sides of the equation on the left hand"}, {"start": 62.92, "duration": 3.64, "text": "side this Square will distribute to all"}, {"start": 64.72, "duration": 4.12, "text": "four of these terms it's going to look"}, {"start": 66.56, "duration": 4.919, "text": "something like this this x 4th squar"}, {"start": 68.84, "duration": 5.8, "text": "will become X to 8th the x^2 SAR is"}, {"start": 71.479, "duration": 4.68, "text": "going to become X to 4th this will be X2"}, {"start": 74.64, "duration": 3.519, "text": "and on the very bottom the square root"}, {"start": 76.159, "duration": 3.481, "text": "and a square will cancel each other out"}, {"start": 78.159, "duration": 3.241, "text": "and on the right hand side this 40 will"}, {"start": 79.64, "duration": 3.72, "text": "multiply by the two to give us 2 to the"}, {"start": 81.4, "duration": 4.6, "text": "80th power and now let's clean up this"}, {"start": 83.36, "duration": 5.799, "text": "left hand side let's focus down here x^2"}, {"start": 86.0, "duration": 6.2, "text": "/ X will simplify to just X and then X"}, {"start": 89.159, "duration": 6.24, "text": "4th / X will simplify to X cubed and"}, {"start": 92.2, "duration": 6.44, "text": "then x 8 / X Cub we can just subtract 8"}, {"start": 95.399, "duration": 6.201, "text": "- 3 to give us X 5th and now we have X"}, {"start": 98.64, "duration": 4.72, "text": "5th = 2 8th we can take the fifth root"}, {"start": 101.6, "duration": 3.64, "text": "of both sides on the left hand side we"}, {"start": 103.36, "duration": 4.759, "text": "just have X and on the right hand side"}, {"start": 105.24, "duration": 7.12, "text": "we can do this 80 / 5 to give us 2 to"}, {"start": 108.119, "duration": 6.201, "text": "the 16 and 2 the 16th power is equal to"}, {"start": 112.36, "duration": 3.6, "text": "65,536 and that's the answer to our"}, {"start": 114.32, "duration": 3.079, "text": "question we can put a box around it if"}, {"start": 115.96, "duration": 5.6, "text": "you want to see the first video I have a"}, {"start": 117.399, "duration": 4.161, "text": "link in the description how exciting"}], "title": "Radical Math Problem Method 2", "description": "https://youtu.be/LAy7emYBsjg\nHere is part 1.", "lengthSeconds": 120, "width": 640, "height": 360}
{"video_id": "jS1eDEPIS24", "transcript": [{"start": 0.12, "duration": 5.239, "text": "start with Y it's clear and true and"}, {"start": 2.56, "duration": 4.48, "text": "divide this log by r s that's what we do"}, {"start": 5.359, "duration": 3.721, "text": "proceed with care for this thing is"}, {"start": 7.04, "duration": 4.08, "text": "layered next step we multiply both sides"}, {"start": 9.08, "duration": 3.719, "text": "by r s the best way to weather this"}, {"start": 11.12, "duration": 4.679, "text": "equational storm change it from"}, {"start": 12.799, "duration": 4.961, "text": "logarithmic to exponential form r s is r"}, {"start": 15.799, "duration": 3.961, "text": "and multiply M to left and right the"}, {"start": 17.76, "duration": 4.72, "text": "result is Merry Christmas and to all a"}, {"start": 19.76, "duration": 2.72, "text": "good night"}], "title": "Ho Ho Ho Exciting!", "description": "Christmas Math Poem!", "lengthSeconds": 21, "width": 360, "height": 640}