Explanation Given: The function is f ( x ) = cosh 3 x , a = 0 . Result used: The Taylor series centered at a is, 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: Obtain the first three nonzero terms of the Taylor series f ( x ) = cosh 3 x centered at a = 0 . The value of f ( x ) = cosh 3 x and its derivatives at a = 0 are computed in the below table, k f k ( x ) f k ( 0 ) 0 f ( x ) = cosh 3 x f ( 0 ) = 1 1 f ′ ( x ) = 3 sinh 3 x f ′ ( 0 ) = 0 2 f ″ ( x ) = 9 cosh 3 x f ″ ( 0 ) = 9 3 f ‴ ( x ) = 27 sinh 3 x f ‴ ( 0 ) = 0 4 f 4 ( x ) = 81 cosh 3 x f 4 ( 0 ) = 81 5 f 5 ( x ) = 243 sinh 3 x f 5 ( 0 ) = 0 The Taylor series centered at a = 0 is computed as follows, f ( x ) = f ( 0 ) +

Calculus: Early Transcendentals (2nd Edition)

2nd Edition

ISBN: 9780321947345

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

Publisher: PEARSON

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Chapter 9, Problem 37RE

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Chapter 9, Problem 36RE

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Chapter 9 Solutions

Calculus: Early Transcendentals (2nd Edition)

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 Ch. 9.1 - How is the remainder Rn(x) in a Taylor polynomial...Ch. 9.1 - Explain how to estimate the remainder in an...Ch. 9.1 - Linear and quadratic approximation a. Find the...Ch. 9.1 - Linear and quadratic approximation a. Find the...Ch. 9.1 - Linear and quadratic approximation a. Find the...Ch. 9.1 - Linear and quadratic approximation a. Find the...

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The first three nonzero terms of the Taylor series for given function centered at a.

Solution Summary: The author calculates the first three nonzero terms of the Taylor series for given function centered at a.

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Chapter 9 Solutions