a. Explanation Given: The function is f ( x ) = ln ( x ) , where x = 1 ⋅ 05 . Result used: The n th-order Taylor polynomial centered at a = 1 is, p n ( x ) = ∑ k = 0 n f k ( a ) k ! ( x − a ) k (1) Calculation: Obtain the Taylor polynomial with n = 3 . Substitute n = 3 and a = 1 in equation (1), the third order Taylor polynomial is as follows, p 3 ( x ) = ∑ k = 0 3 f k ( 1 ) k ! ( x − 1 ) k = f ( 1 ) + f ′ ( 1 ) ( x − 1 ) + ( 1 2 ! ) f ″ ( 1 ) ( x − 1 ) 2 + ( 1 3 ! ) f ‴ ( 1 ) ( x − 1 ) 3 p 3 ( x ) = f ( 1 ) + f ′ ( 1 ) ( x − 1 ) + ( 1 2 ! ) f ″ ( 1 ) ( x − 1 ) 2 + ( 1 3 ! ) f ‴ ( 1 ) ( x − 1 ) 3 (2) The value of f ( x ) = ln ( x ) and its derivatives at x = 1 is computed in the below table, n f n ( x ) f n ( 0 ) 0 f ( x ) = ln ( x ) f ( 1 ) = 0 1 f ′ ( x ) = 1 x f ′ ( 1 ) = 1 2 f ″ ( x ) = − 1 x 2 f ″ (b). To determine To compute: The absolute error.

MyLab Math with Pearson eText -- 24 Month Access -- for Calculus with Integrated Review

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ISBN: 9780135243435

Author: Bill Briggs

Publisher: Pearson Education (US)

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Chapter 11.1, Problem 36E

To determine

To approximate: Taylor polynomial with n=3.

(b).

To determine

To compute: The absolute error.

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Chapter 11.1, Problem 35E

Chapter 11.1, Problem 37E

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Chapter 11 Solutions

MyLab Math with Pearson eText -- 24 Month Access -- for Calculus with Integrated Review

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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To approximate: Taylor polynomial with n = 3 .

Solution Summary: The author calculates the nth-order Taylor polynomial centered at a=1.

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To approximate: Taylor polynomial with n=3.

To compute: The absolute error.

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Chapter 11 Solutions