Explanation Result used: Consider the Taylor series as shown below, f ( x ) = f ( a ) + f ′ ( a ) ( x − a ) + f ″ ( a ) ( x − a ) 2 2 ! + f ‴ ( a ) ( x − a ) 3 3 ! + ⋯ = ∑ k = 0 ∞ f ( k ) ( a ) k ! ( x − a ) k Calculation: The function has derivatives of all orders at x = a . Obtain the Taylor series centered at 0 as follows, f ( x ) = f ( 0 ) + f ′ ( 0 ) ( x ) + f ″ ( 0 ) ( x ) 2 2 ! + f ‴ ( 0 ) ( x ) 3 3 ! + ⋯ (1) Substitute 0 for x in f ( x ) . f ( 0 ) = ( 1 + 0 ) − 1 = 1 The first derivative of f ( x ) is f ′ ( x ) = ( − 1 ) ( 1 + x ) − 2 . Substitute 0 for x in f ′ ( x ) , f ′ ( 0 ) = − ( 1 + 0 ) − 2 = − 1 The second derivative of f ( x ) is f ″ ( x ) = 2 ( 1 + x ) − 3

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 11.3, Problem 2QC

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To verify: The Taylor series for f(x)=(1+x)−1 .

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Chapter 11.3, Problem 1QC

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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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To verify: The Taylor series for f ( x ) = ( 1 + x ) − 1 .

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To verify: The Taylor series for f(x)=(1+x)−1 .

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Chapter 11 Solutions