4.2.6-eg-1 - f ( x ) = 2 x 3-x 2 + C, (2) where C is an...

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Example 1. Find all functions which have the derivative f 0 ( x ) = 6 x 2 - 2 x. 2. Find the function f which has as its derivative f 0 ( x ) = 6 x 2 - 2 x and whose graph passes through the point (2 , 13). Solution: 1. Recalling that d dx ± x 3 ² = 3 x 2 , = d dx ± 2 x 3 ² = 6 x 2 , and d dx ± x 2 ² = 2 x, it follows that the function f ( x ) = 2 x 3 - x 2 is such that f 0 ( x ) = 6 x 2 - 2 x. (1) But there are an infinite number of functions with this derivative, each differing by a constant. It follows that all functions with derivative given by (1) are of the form
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Unformatted text preview: f ( x ) = 2 x 3-x 2 + C, (2) where C is an arbitrary real-valued constant. 2. From #1, we are seeking a function of the form of f ( x ) in (2) for which f (2) = 13 . That is, we need to find C so that 2 · 2 3-2 2 + C = 13 = ⇒ C = 13-12 = 1 ....
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This note was uploaded on 04/02/2012 for the course MTH 132 taught by Professor Kihyunhyun during the Fall '10 term at Michigan State University.

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