lecture30a - Lecture 30 (Chapter 4, Section 30):...

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Unformatted text preview: Lecture 30 (Chapter 4, Section 30): Antiderivatives ex. Suppose that the slope of the tangent line of a function f ( x ) at any x-value is given by 2 x + 3. Can we flnd f ( x )? Def. A function F is called an of f on an interval I if for all x in I . ex. Find an antiderivative of the function f ( x ) = x 5 + x . How many antiderivatives can there be? 2 Recall the theorem from Chapter 4, Sec. 23 which states that if two functions have the same derivative on an interval, they can only difier by a constant. Theorem: If F is an antiderivative of f on I , then is the most general antiderivative of f on I , where C represents any constant. ex. Find the most general antiderivative of the fol- lowing: 1) f ( x ) = sec x tan x 2) f ( x ) = cos(4 x ) 3) f ( x ) = e x 3 3 NOTE: If f ( x ) = x n , then F ( x ) = If n 0, then x If n < 0, then x ex. Find F ( x ) if f ( x ) = x 6 . ex. If f ( x ) = 1 x ; flnd F ( x )....
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lecture30a - Lecture 30 (Chapter 4, Section 30):...

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