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**Unformatted text preview: **Lecture 23 Antiderivatives Recall our final result from Mean Value Theorem: If f ( x ) = g ( x ) for all x in an interval ( a,b ) , then there is a constant C for which f ( x ) = g ( x ) + C for all x on ( a,b ) . Write a function f such that f ( x ) = e x + cos( x ) on (- , ) . Write all the functions f such that f ( x ) = e x +cos( x ) on (- , ) . A function F is called an antiderivative of f on an interval I if Does the function s ( t ) =- 1 t 2 have an antiderivative on (- , ) ? Find an antiderivative for s ( t ) on the intervals ( 0 , ) (- , 0 ) What are all possible functions having derivative s ( t ) =- 1 t 2 ? S ( t ) = If F is an antiderivative of f on interval I , then the general antiderivative of f on I is written: Find the general antiderivative of each function: f ( x ) = x 2 + 1 r ( ) = + sin(2 ) s ( t ) = 1 t Typically, when asked to find antiderivatives, we will assume that it is for just one interval on which the function is defined:...

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