(a) Let f(x) = e¯¹², x x > 0. Show that, for every n ≥ 1, the n'th derivative f(n)(x) is of the form P₁(1/x). e2 for some polynomial P₁ (depending on n). (b) Define (c) 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 9 of part (b) is not equal to the sum of its Maclaurin series. h→0+ t→∞
(a) Let f(x) = e¯¹², x x > 0. Show that, for every n ≥ 1, the n'th derivative f(n)(x) is of the form P₁(1/x). e2 for some polynomial P₁ (depending on n). (b) Define (c) 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 9 of part (b) is not equal to the sum of its Maclaurin series. h→0+ t→∞
College Algebra (MindTap Course List)
12th Edition
ISBN:9781305652231
Author:R. David Gustafson, Jeff Hughes
Publisher:R. David Gustafson, Jeff Hughes
Chapter3: Functions
Section3.3: More On Functions; Piecewise-defined Functions
Problem 98E: Determine if the statemment is true or false. If the statement is false, then correct it and make it...
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why here f(x)=e^(x^2), support to be f(x)=e^(-1/(x^2))
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