BAM!
Mr. Tarrou.
So we have no learned how to find the derivative or slope of a tangent line and how to find derivatives using the
definition of derivatives, and started using
basic rules of differentiation to find well
again derivatives.
Remembering that a derivative is an equation
or a function excuse me... a function that
will give us the slope of the original function
at... you know... almost all values of x.
Now remember we could not find the derivative
where the tangent line was vertical.
We could not find the derivative, or the slope
of a curve at a sharp bend.
So there are some special cases that you want
to be aware of.
But what I want to do with this video is draw
some functions like this first example f(x)
is equal to x squared minus seven and compare
those functions to some plausible drawings
of the derivative.
So we have f(x) is equal to... and these are
just going to be sketches...
f(x)=x^2-7.
Well this is a parabola opening up that has
a vertical shift down of seven units.
So this parabola looks something like this.
Down here at the y value of -7 we have its
minimum value, or its vertex.
Well f prime of x, and I have you for this
first example an actual equation, f'(x) or
the derivative of f(x) now that we know some
of our basic differentiation rules is going
to be just 2x.
Well if I want to graph this derivative on
the same x y axis as the original function,
BAM! Mr. Tarrou. So we have no learned how
to find the derivative lines and how to find
derivatives using the definition of derivatives,
and started using basic rules of differentiation
to find well again derivatives. Remembering
that a derivative is an equation or a function
excuse me... a function that will give us
the slope of the original function at... you
know... almost all values of x. Now remember
we could not find the derivative where the
tangent line was vertical. We could not find
the derivative, or the slope of a curve at
a sharp bend. So there are some special cases
that you want to be aware of. But what I want
to do with this video is draw some functions
like this first example f(x) is equal to x
squared minus seven and compare those functions
to some plausible drawings of the derivative.
So we have f(x) is equal to... and these are
just going to be sketches... f(x)=x^2-7. Well
this is a parabola opening up that has a vertical
shift down of seven units. So this parabola
looks something like this. Down here at the
y value of -7 we have its minimum value, or
its vertex. Well f prime of x, and I have
you for this first example an actual equation,
f'(x) or the derivative of f(x) now that we
know some of our basic differentiation rules
is going to be just 2x. Well if I want to
graph this derivative on the same x y axis
as the original function, we have... it is
just 2x. It is basically a y=mx+b, so this
is equation is just a line. It has a y intercept
of zero and a slope of 2. So we are going
to go up 2 and over 1. I am just sort of sketching
this very very roughly. So here is a rough
sketch of what my derivative should look like.
With a slope of approximately of 2 and a y
intercept of zero. Remembering that the derivative
gives us the slope of the original function
let's remember that as... our just look at
this parabola... This parabola is going down
to the right. Everything... For every x value
to the left of zero or in quadrants II and
III, the parabola is going down to the right.
So the slope of any tangent line that I draw...
Well let's just... I don't want to mess this
up too much. If I were to pick this point
right here and draw a tangent line, that tangent
line would have a negative slope. That is
why this blue line is below the x axis representing
the.. you know... the y values of my derivative...
When I take the value of x, let's say this
point here, that negative x is going to give
me a negative slope. And indeed along this
graph, let's say right here, for that value
of x right there on the parabola is going
down right. The slope of the parabola is negative
and indicated by the derivative being below
the x axis and giving us a y value that is
negative from the derivative equation f'(x)
is equal to 2x. And the parabola continues
to have a negative slope until you get to
its vertex. At which it stops falling and
starts to rise. At that particular point,
right at the very vertex if I was to draw
a tangent line... And I have some other colors
up here, if I were to draw a tangent line
through the vertex or a line that is tangent
to the parabola at the vertex it would have
a slope of zero. Thus we see the blue line
which is a sketch of what the derivative is,
it is passing through the x axis at (0,0).
Or its y value there is also zero. So negative
tangent line slopes, the derivative is below
the x axis. We have the vertex. The slope
there is zero. This is a smooth continuous
curve. That derivative there has a value of
zero, and then it starts to rise. So our derivative
at this value of zero, for all x values less
than zero my derivative is negative. For all
x values greater than zero my derivative is
positive. So my derivative went from being
negative to positive. I found a relative minimum
within the original graph. Over here I have
what is probably a cubic function. And I want
to pay attention to again, this relative minimum
and this relative maximum. So... And I am
going to look at this original function here
and try to sketch what the derivative may
look like. Well, as I go along here the graph
is falling. So at any point that I pick along
this section of the curve, my slope is going
to be negative. That means that my derivative
has to give a negative answer, or a negative
value. That means that my derivative is going
to be below the x axis. So my derivative is
going to be below the x axis all along here
until I get to this point right here which
is a relative minimum. The graph stops falling
and starts to rise. So right there my slope
of my tangent line if I were to draw one...
let's just make a tiny one there... my slope
there is going to be zero. Everywhere to the
left my slopes are going to be negative, and
everywhere to the right my slope is going
to be positive. Now I don't know how high
this derivative is going to go, but I know
that from this point on my graph is rising
so my derivative values must be positive.
Thus my sketch of the derivative... a possible
derivative... has to be above the x axis.
My graph begins to rise, or continues to rise
until I get to this orange line. At my orange
line my tangent line, or the slope of the
curve there again becomes zero. So my derivative
is going to be above the x axis representing
that I am getting positive values from my
derivative which again gives slope and my
slope is positive. So my y values from the
derivative are all going to be above the x
axis until they get to this orange line is
going to give me a derivative value of zero
because my slope there in the actual function
is zero. And it begins to fall again, so the
blue graph is a visual representation of what
could be... and the only thing I really messed
up was... not messed up, but have did not
pay attention to is how sharply does it fall.
What is my maximum slope between these two
points? How high does this blue derivative
graph go? But it is an idea.. I am trying
to give you an understanding of the relationship
between the graph of the derivative and the
graph of the original function. Now before
I move onto my next example, this a relative
minimum. My slope leading up to my relative
minimum is negative and then my slope becomes
positive. So at this green line, we can call
this a value of 'a'. To the left of 'a' my
derivative is negative. To the right of 'a'
my derivative becomes positive. So when you
fall and then rise, when your derivative crosses
the x axis and it is negative to the left
and positive to the right, that sign change
on the derivative is showing that you have
found a relative minimum value. Unlike the
maximum where my derivative is going to be
positive... see my blue graph is above the
x axis because my slope is positive... and
here my relative maximum my graph starts to
fall and my derivative therefore has to fall
below the x axis representing those negative
y values or those negative slope values. So
when my derivative to the... I did not want
that chalk anyway... my derivative to the
left of b is positive, my derivative has an
x intercept... that is the slope of zero...
and then my derivative becomes negative. So
positive and then negative is a... you are
finding a relative maximum with the derivative
values, the sign changes. Over here in my
last example, I have graph that is falling
so that is a negative slope. At some point
here my tangent line becomes horizontal. I
get a slope of zero. But then instead of rising
back up again and this being a relative minimum,
it just starts to fall again. Well what would
my sketch of my derivative look like for this
particular graph? At that particular x value
we will call 'a', the derivative is going
to be negative to the left because my graph
is falling, so my derivative is going to be
below the x axis. My derivative will touch
the x axis representing a slope of zero because
my y values are going to be zero from the
derivative which gives me slope. Then it starts
to fall again. So this could be a plausible
sketch of what my derivative could look like
for this function. It falls, becomes horizontal,
and then falls again. Again where I have this
value of 'a', to the left of 'a' my derivative
is negative, to the right of 'a' my derivative
is negative. So just because your derivative...
If you find an x intercept, or find your derivative,
set it equal to zero and solve for x... when
you find an x intercept, you are not necessarily
going to have a maximum or minimum value at
that value of x. Because the graph could just
start to fall again, and we see that there
is no sign change to the left and right of
'a'. To the left of 'a' my derivative is negative,
to the right of 'a' my derivative is still
negative even in between there I had an x
intercept just touch. So I hope that helps
to give you a little bit better understanding
of the derivatives and how it relates to the
original function. And how you can look at
the graphs and see the comparisons between
the two. Maybe even help you find some relative
maximums and minimums. I am Mr. Tarrou. BAM!
Go do your homework:D
