a. Explanation Result used: The Taylor series representation of sin x is, sin x = x − x 3 3 ! + x 5 5 ! − ⋯ + ( − 1 ) k x 2 k + 1 ( 2 k + 1 ) ! + ⋯ = ∑ k = 0 ∞ ( − 1 ) k x 2 k + 1 ( 2 k + 1 ) ! , for | x | < ∞ . Given: The function f ( x ) = sin x 2 . Calculation: Substitute x 2 for x in the Taylor series sin x , sin x = x 2 − ( x 2 ) 3 3 ! + ( x 2 ) 5 5 ! − ⋯ = x 2 − x 6 3 ! + x 10 5 ! − ⋯ Differentiate the function f ( x ) = sin x 2 term by term as follows, f ′ ( x ) b. To determine To identify: The function represented by the differentiated series. c. To determine To give: The interval of convergence of the series f ′ ( x ) .

Single Variable Calculus: Early Transcendentals, Books a la Carte, and MyLab Math with Pearson eText -- Title-Specific Access Card Package (3rd Edition)

3rd Edition

ISBN: 9780134996103

Author: William L. Briggs, Lyle Cochran, Bernard Gillett, Eric Schulz

Publisher: PEARSON

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Chapter 11.4, Problem 28E

To determine

To find: The derivative of the Taylor series about 0.

To determine

To identify: The function represented by the differentiated series.

To determine

To give: The interval of convergence of the series f′(x).

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Chapter 11.4, Problem 27E

Chapter 11.4, Problem 29E

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Chapter 11 Solutions

Single Variable Calculus: Early Transcendentals, Books a la Carte, and MyLab Math with Pearson eText -- Title-Specific Access Card Package (3rd Edition)

Ch. 11.1 - Verify that p3 satisfies p3(k)(a)=f(k)(a), for k =...Ch. 11.1 - Prob. 2QC Ch. 11.1 - Prob. 3QC Ch. 11.1 - Write out the next two Taylor polynomials p4 and...Ch. 11.1 - Prob. 5QC Ch. 11.1 - Prob. 6QC Ch. 11.1 - Suppose you use a second-order Taylor polynomial...Ch. 11.1 - Does the accuracy of an approximation given by a...Ch. 11.1 - The first three Taylor polynomials for f(x)=1+x...Ch. 11.1 - Prob. 4E

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The derivative of the Taylor series about 0.

Solution Summary: The author explains how to find the derivative of the Taylor series, mathrmsinx.

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To find: The derivative of the Taylor series about 0.

To identify: The function represented by the differentiated series.

To give: The interval of convergence of the series f′(x).

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Chapter 11 Solutions