3. [25 pts] (a) Write down the Taylor series centered at x = 0 for the functions e and sin x, up to and including terms of degree 3 in x. (You don't need to derive the series.) (b) Use the results in (a) to find the Taylor series centered at x = 0 for the function f (x) = e" sin(2x), %3D up to and including terms of degree 3 in x. (c) Use the result in (b) to evaluate f"'(0). Solution. • (a) The Taylor series are 1 +... 6. e = 1+x + 1 sin x = x - 6. (b) We have e sin(2x) 1 1+ x + 2 1 + (2x)³ 2x – .. = 2x + 2x2 + x³ 8 + = 2r + 2a* –*+..

3. [25 pts] (a) Write down the Taylor series centered at x = 0 for the functions e and sin x, up to and including terms of degree 3 in x. (You don't need to derive the series.) (b) Use the results in (a) to find the Taylor series centered at x = 0 for the function f (x) = e" sin(2x), %3D up to and including terms of degree 3 in x. (c) Use the result in (b) to evaluate f"'(0). Solution. • (a) The Taylor series are 1 +... 6. e = 1+x + 1 sin x = x - 6. (b) We have e sin(2x) 1 1+ x + 2 1 + (2x)³ 2x – .. = 2x + 2x2 + x³ 8 + = 2r + 2a* –*+..

Calculus: Early Transcendentals

8th Edition

ISBN:9781285741550

Author:James Stewart

Publisher:James Stewart

Chapter1: Functions And Models

Section: Chapter Questions

Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...

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