hw24solutions - Math 521 Advanced Calculus I Instructor J...

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Math 521 - Advanced Calculus I Instructor: J. Metcalfe Due: April 12, 2010 Assignment 24 1. Suppose f is defined in a neighborhood of x , and suppose f 00 ( x ) exists. Show that lim h 0 f ( x + h ) + f ( x - h ) - 2 f ( x ) h 2 = f 00 ( x ) . Show by an example that the limit may exist even if f 00 ( x ) does not. Let f ( x ) = - x 2 for x 0, and f ( x ) = x 2 for x 0. Then, f 0 ( x ) = | x | , and f 00 does not exist at x = 0. On the other hand, we have, for x = 0, f ( h ) + f ( - h ) - 2 f (0) = 0 and thus the limit as h 0 is also 0. This gives the requested counterexample. Assuming, now, that f 00 ( x ) exists, we have, by l’Hospital’s rule, lim h 0 f ( x + h ) + f ( x - h ) - 2 f ( x ) h 2 = lim h 0 f 0 ( x + h ) - f 0 ( x - h ) 2 h = 1 2 ± lim h 0 f 0 ( x + h ) - f 0 ( x ) h + lim h 0 f 0 ( x ) - f 0 ( x - h ) h ² = f 00 ( x ) . 2. Let h ( x ) := e - 1 /x 2 for x 6 = 0 and h (0) := 0. Show that h ( n ) (0) = 0 for all n N . Conclude that the remainder term in Taylor’s theorem for
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hw24solutions - Math 521 Advanced Calculus I Instructor J...

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