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
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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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can you explain the baby sets of getting the answers to part a & b 

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 =
O for the
function
f (x) = e" sin(2x),
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
e = 1+x + x² +
1
+...,
sin x = x
6.
(b) We have
e sin 2) = (1+=+**.)(2 -2)*.)
= 2x + 2x2 + x³
+...
1
= 2.r + 2a? – -
+...
Transcribed Image Text: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 = O for the function f (x) = e" sin(2x), 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 e = 1+x + x² + 1 +..., sin x = x 6. (b) We have e sin 2) = (1+=+**.)(2 -2)*.) = 2x + 2x2 + x³ +... 1 = 2.r + 2a? – - +...
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