(a) Let f(x) = e ², x > 0. Show that, for every n ≥ 1, the n'th derivative f(n)(x) is of the form Pn(1/x) e for some polynomial P₁ (depending on n). · (b) Define g(x) = = 0 e if x ≤ 0 if x > 0. Use part (a) to prove that g(n) (0) = 0 for all n ≥ 1. [Hint: You may want to use the fact that lim F(1/h) = _lim F(t), for any function F.] Conclude that function g of part (b) is not equal to the sum of its Maclaurin series. h→0+ (c)

(a) Let f(x) = e ², x > 0. Show that, for every n ≥ 1, the n'th derivative f(n)(x) is of the form Pn(1/x) e for some polynomial P₁ (depending on n). · (b) Define g(x) = = 0 e if x ≤ 0 if x > 0. Use part (a) to prove that g(n) (0) = 0 for all n ≥ 1. [Hint: You may want to use the fact that lim F(1/h) = _lim F(t), for any function F.] Conclude that function g of part (b) is not equal to the sum of its Maclaurin series. h→0+ (c)

Calculus For The Life Sciences

2nd Edition

ISBN:9780321964038

Author:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.

Publisher:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.

Chapter6: Applications Of The Derivative

Section6.CR: Chapter 6 Review

Problem 1CR

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