Explanation Given: The function is f ( x ) = ln ( 1 − x ) ; R 3 ( x ) , for | x | < 1 2 . Theorem used: Taylor’s remainder theorem “The remainder of the Taylor series is R n ( x ) ≤ M | x − a | n + 1 ( n + 1 ) ! for some point c between x and a .” Calculation: Obtain the remainder in the Taylor series for f ( x ) = ln ( 1 − x ) centered at 0. The first derivative of f ( x ) = ln ( 1 − x ) computed as follows, f ′ ( x ) = d d x ( ln ( 1 − x ) ) = 1 ( 1 − x ) The second derivative of f ( x ) = ln ( 1 − x ) computed as follows, f ″ ( x ) = d d x ( ( 1 − x ) − 1 ) = − 1 ( 1 − x ) − 1 − 1 ⋅ 1 = − ( 1 − x ) − 2 The third derivative of f ( x ) = ln ( 1 − x ) computed as follows, f ‴ ( x ) = d d x ( − ( 1 − x ) − 2 ) = − ( − 2 ) ( 1 − x ) − 2 − 1 = 2 ( 1 − x ) − 3 = 2 ( 1 − x ) 3 Since | x | < 1 2 that is | f ( 3 + 1 ) ( c ) | < 16

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, Problem 16RE

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To find: The remainder term for the Taylor series for given function centered at 0 as well as obtain an upper bound for the magnitude of the remainder on the given interval for the given value of n.

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Chapter 11, Problem 15RE

Chapter 11, Problem 17RE

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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 remainder term for the Taylor series for given function centered at 0 as well as obtain an upper bound for the magnitude of the remainder on the given interval for the given value of n .

Solution Summary: The author calculates the remainder term for the Taylor series for a function centered at 0 and obtains an upper bound for its magnitude.

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To find: The remainder term for the Taylor series for given function centered at 0 as well as obtain an upper bound for the magnitude of the remainder on the given interval for the given value of n.

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Chapter 11 Solutions