Explanation Result used: “Let n be a fixed positive integer. Suppose there exist a number M such that | f ( n + 1 ) ( c ) | ≤ M , for all c between a and x inclusive. The remainder in the n th-order Taylor polynomial for f centered at a satisfies, | R n ( x ) | = | f ( x ) − p n ( x ) | ≤ M | x − a | n + 1 ( n + 1 ) ! .” Given: The function ln ( 1.04 ) and n = 3 . Calculation: Let f ( x ) = ln ( 1 + x ) and x = 0.04 . The fourth derivative of f ( x ) is computed as follows, f ′ ( x ) = 1 ( 1 + x ) f ″ ( x ) = − 1 ( 1 + x ) 2 f ‴ ( x ) = 2 ( 1 + x ) 3 f 4 ( x ) = − 6 ( 1 + x ) 4 Since f 4 ( x ) has the maximum magnitude on [ 0 , 0.4 ]

Calculus, Single Variable: Early Transcendentals (3rd Edition)

3rd Edition

ISBN: 9780134766850

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

Publisher: PEARSON

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Chapter 11.1, Problem 52E

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To find: The bound on the error in approximating the given quantities with the nth-order Taylor polynomial centered at 0.

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Chapter 11.1, Problem 51E

Chapter 11.1, Problem 53E

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Chapter 11 Solutions

Calculus, Single Variable: Early Transcendentals (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 bound on the error in approximating the given quantities with the n th-order Taylor polynomial centered at 0.

Solution Summary: The author calculates the bound on the error in approximating the given quantities with the n th-order Taylor polynomial centered at 0.

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To find: The bound on the error in approximating the given quantities with the nth-order Taylor polynomial centered at 0.

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Chapter 11 Solutions