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# L23 - Lecture 23 Antiderivatives Recall our nal result from...

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Lecture 23 — Antiderivatives Recall our final result from Mean Value Theorem: If f 0 ( x ) = g 0 ( 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 0 ( x ) = e x + cos( x ) on ( -∞ , ) . Write all the functions f such that f 0 ( x ) = e x +cos( x ) on ( -∞ , ) . A function F is called an antiderivative of f on an interval I if

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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 ( θ ) = θ + sin(2 θ ) s ( t ) = 1 t

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Typically, when asked to find antiderivatives, we will assume that it is for just one interval on which the function is defined: Find the general antiderivative of each function: f ( x ) = sec( x )[sec( x ) + tan( x )] p ( t ) = 1 + e t - e - 2 t φ ( w ) = 2 w - 1 3 w
Adding a constant onto a function shifts its graph. . .

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L23 - Lecture 23 Antiderivatives Recall our nal result from...

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