# 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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The provided text is a recap of a lecture on antiderivatives, specifically discussing the antiderivatives of functions that are not defined over the entire real line, such as \( f(x) = 1/x \) and \( f(x) = \sec^2 x \).
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the text does not contain a mathematical question or a word problem that requires solving. it is a summary of concepts and does not provide a specific question or its answer.
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# 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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The provided text is a recap of a lecture on antiderivatives, specifically discussing the antiderivatives of functions that are not defined over the entire real line, such as \( f(x) = 1/x \) and \( f(x) = \sec^2 x \).

TYPE: NO_QUESTION

The text does not contain a mathematical question or a word problem that requires solving. It is a summary of concepts and does not provide a specific question or its answer.