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Unformatted text preview: nd a function which is uniformly continuous but whose derivative is unbounded. Problem 5: Dene a function f : R R by: f ( x ) = e-1 /x 2 , if x 6 = 0 , if x = 0 (a) Prove, by mathematical induction, that f ( n ) (0) = 0 for all n . (It may help to review Problem 4 of Homework 6.) (b) Find the Taylor series of f centered at x = 0. Problem 6: Let f be a twice dierentiable function on (0 , ) and let M , M 1 , M 2 denote the least upper bounds of | f ( x ) | , | f ( x ) | , | f 00 ( x ) | respectively on (0 , ). (a) Let h > 0. By using the Taylor series expansion of f centered at x = c , show that: f ( c ) = 1 2 h f ( c + 2 h )-f ( c ) -hf 00 ( ) for some ( c, c + 2 h ). Explain why | f | hM 2 + M /h for all h > 0. (b) Prove that M 2 1 4 M M 2 when M 2 > 0 ....
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This note was uploaded on 03/18/2012 for the course MATH 67 taught by Professor Schilling during the Fall '07 term at UC Davis.
- Fall '07