a . Explanation Given: The function f ( x ) = sin x , a = π 4 . Result used: The n th-order Taylor polynomial centered at a is shown below, p n ( x ) = f ( a ) + f ′ ( a ) ( x − a ) + f ″ ( a ) 2 ! ( x − a ) 2 + ⋯ + f n ( a ) n ! ( x − a ) n (1) Calculation: The n th-order Taylor polynomial centered at π 4 is shown below, p 2 ( x ) = f ( π 4 ) + f ′ ( π 4 ) ( x − π 4 ) + ⋯ + f n ( π 4 ) n ! ( x − π 4 ) n (2) Substitute π 4 for x in f ( x ) = sin x , f ( x ) = sin ( π 4 ) = 2 2 That is, f ( π 4 ) = 2 2 . The first derivative of f ( x ) = sin x is computed as follows, f ′ ( x ) = cos x Substitute π 4 for x , f ′ ( π 4 ) = cos ( π 4 ) = 2 2 That is, f ′ ( π 4 ) = 2 2 . The second derivative of f ( x ) = sin x is computed as follows, f ″ ( x ) = − sin x Substitute π 4 for x , f ″ ( π 4 ) = − sin ( π 4 ) = − 2 2 That is, f ″ ( π 4 ) = 2 2 . The Taylor polynomial centered at π 4 for n = 0 , 1 and 2 is obtained as follows, p 0 ( x ) = f ( a ) = f ( π 2 ) = 2 2 Thus, the Taylor polynomial centered at π 4 for n = 0 is p 0 ( x ) = 2 2 b. To determine To sketch: The Taylor polynomial and the function.

Single Variable Calculus: Early Transcendentals (2nd Edition) - Standalone book

2nd Edition

ISBN: 9780321954237

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

Publisher: PEARSON

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Chapter 9.1, Problem 31E

To determine

To find: The nth-order Taylor polynomial centered at the point a for n=0,1,and 2.

To determine

To sketch: The Taylor polynomial and the function.

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Chapter 9.1, Problem 30E

Chapter 9.1, Problem 32E

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π — a. Find the Taylor polynomial of 6th order at a= 2 b. Find error at x = 7. c. Graph the original function and T₂(x),T₁(x), and Tg(x). X X X 2 4 f(x) = cos2x

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Chapter 9 Solutions

Single Variable Calculus: Early Transcendentals (2nd Edition) - Standalone book

Ch. 9.1 - Prob. 1QC Ch. 9.1 - Prob. 2QC Ch. 9.1 - Prob. 3QC Ch. 9.1 - Prob. 4QC Ch. 9.1 - Prob. 5QC Ch. 9.1 - In Example 7, find an approximate upper bound for...Ch. 9.1 - Suppose you use a second-order Taylor polynomial...Ch. 9.1 - Does the accuracy of an approximation given by a...Ch. 9.1 - The first three Taylor polynomials for f(x)=1+x...Ch. 9.1 - Prob. 4E

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The nth -order Taylor polynomial centered at the point a for n = 0 , 1 , and 2 .

Solution Summary: The author explains the nth-order Taylor polynomial centered at the point a=pi 4.

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To find: The nth-order Taylor polynomial centered at the point a for n=0,1,and 2.

To sketch: The Taylor polynomial and the function.

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Chapter 9 Solutions