Unformatted text preview: ﬁnd a function which is uniformly continuous but whose derivative is unbounded. Problem 5: Deﬁne a function f : R → R by: f ( x ) = ± e1 /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 diﬀerentiable 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 ² f ( c + 2 h )f ( c ) ³hf 00 ( ξ ) for some ξ ∈ ( 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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 Fall '07
 Schilling
 Derivative, Taylor Series, Continuous function, Holomorphic function, differentiable function, infinitely differentiable function

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