text: # 12.02 - Section 4.7: Antiderivatives (continued) Recap of...
  
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  Unformatted text preview: Section 4.7: Antiderivatives (continued) Recap of last time: The function f ( x ) = 1/ x doesnt have an antiderivative on R , because ..?.. f ( x ) isnt even defined at x = 0. We showed last time that the most general antiderivative of f ( x ) = 1/ x on the set S = { x : x 0} is ..?.. {ln ( x ) + A for x &lt; 0 F ( x ) = {whatever for x = 0 {ln ( x ) + B for x &gt; 0 for two independent constants A and B , where whatever means undefined OR any constant you like. More generally, if the set S consists of several different pieces (sub-intervals), you could have a different additive constant on each of the subintervals. In particular, if the domain D of the function f ( x ) consists of two or more intervals, the general antiderivative of f ( x ) is not of the form F ( x ) + C for a single, specific antiderivative F ( x ) and a single arbitrary constant C ; you could have a different additive constant on each of the intervals of D . For instance, in the case of f ( x ) = sec 2 x , which is undefined at x = n + /2 for every integer n , so that its domain is D = { x in R : the fractional part of x / is not 1/2}, the most natural antiderivative is...
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  ## This note was uploaded on 02/13/2012 for the course MATH 141 taught by Professor Staff during the Fall '11 term at UMass Lowell.
  
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  12.02 - Section 4.7: Antiderivatives (continued) Recap of...
  
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explanation: The provided text is a preview of a section from a mathematics textbook or lecture notes, specifically discussing antiderivatives. It does not contain a direct question or a question-and-answer format. Instead, it provides a recap of a previous discussion on antiderivatives, particularly focusing on the function \( f(x) = \frac{1}{x} \) and its antiderivative on different intervals. The text also touches on the general form of antiderivatives for functions defined on multiple intervals.
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distilabel_metadata.content: # 12.02 - Section 4.7: Antiderivatives (continued) Recap of...
  
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  Unformatted text preview: Section 4.7: Antiderivatives (continued) Recap of last time: The function f ( x ) = 1/ x doesnt have an antiderivative on R , because ..?.. f ( x ) isnt even defined at x = 0. We showed last time that the most general antiderivative of f ( x ) = 1/ x on the set S = { x : x 0} is ..?.. {ln ( x ) + A for x &lt; 0 F ( x ) = {whatever for x = 0 {ln ( x ) + B for x &gt; 0 for two independent constants A and B , where whatever means undefined OR any constant you like. More generally, if the set S consists of several different pieces (sub-intervals), you could have a different additive constant on each of the subintervals. In particular, if the domain D of the function f ( x ) consists of two or more intervals, the general antiderivative of f ( x ) is not of the form F ( x ) + C for a single, specific antiderivative F ( x ) and a single arbitrary constant C ; you could have a different additive constant on each of the intervals of D . For instance, in the case of f ( x ) = sec 2 x , which is undefined at x = n + /2 for every integer n , so that its domain is D = { x in R : the fractional part of x / is not 1/2}, the most natural antiderivative is...
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  ## This note was uploaded on 02/13/2012 for the course MATH 141 taught by Professor Staff during the Fall '11 term at UMass Lowell.
  
  ### Page1 / 6
  
  12.02 - Section 4.7: Antiderivatives (continued) Recap of...
  
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distilabel_metadata.raw_output_q_a_detector_0: The provided text is a preview of a section from a mathematics textbook or lecture notes, specifically discussing antiderivatives. It does not contain a direct question or a question-and-answer format. Instead, it provides a recap of a previous discussion on antiderivatives, particularly focusing on the function \( f(x) = \frac{1}{x} \) and its antiderivative on different intervals. The text also touches on the general form of antiderivatives for functions defined on multiple intervals.
  
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