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Calculus II Notes

Calculus II Notes - Calculus II-Stewart Dr Berg Spring...

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Calculus II- Stewart Dr. Berg Spring 2010 Page 1 4.9 4.9 Antiderivatives In many problems we wish to find a function that has a certain rate of change. As an example, given the velocity function of a particle (and a little more information) we can reconstruct the position function. This means that we need to reverse the operation of differentiation. Definition A function F is called an antiderivative of the function f on an interval I if F ( x ) = f ( x ) for each x I . Example A F ( x ) = x 3 is an antiderivative for f ( x ) = 3 x 2 on the set of all real numbers. Note that there are many antiderivatives for f ( x ) = 3 x 2 including F ( x ) = x 3 + 2 and F ( x ) = x 3 1 . Theorem If F is any antiderivative for f on an interval I , then, given any real number C , F ( x ) + C is also an antiderivative for f . Note: We usually refer to the form F ( x ) + C as the general antiderivative of f since every antiderivative of f must have that form. Example B i) Since d dx (sin x ) = cos x the general antiderivative for f ( x ) = cos x is F ( x ) = sin x + C where C is an arbitrary real number. ii)

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Calculus II Notes - Calculus II-Stewart Dr Berg Spring...

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