a. Explanation Given: The function f ( x ) = 2 x , a = 1 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 four nonzero terms of the Taylor series f ( x ) = 2 x centered at a = 1 . The value of f ( x ) = 2 x and its derivatives at x = 1 are computed in the below table, k f k ( x ) f k ( 1 ) 0 f ( x ) = 2 x f ( 1 ) = 2 1 f ′ ( x ) = ln ( 2 ) 2 x f ′ ( 1 ) = ln ( 2 ) ⋅ 2 2 f ″ ( x ) = ln 2 ( 2 ) ⋅ 2 x f ″ ( 1 ) = ln 2 ( 2 ) ⋅ 2 3 f ‴ ( x ) = ln 3 ( 2 ) ⋅ 2 x f ‴ ( 1 ) = ln 3 ( 2 ) ⋅ 2 The Taylor series centered at a = 1 is computed as follows, f ( x ) = f ( 1 ) + f ′ ( 1 ) ( x ) + f ″ b. To determine To write: The power series using summation notation.

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 33E

To determine

To find: The first four nonzero terms of the Taylor series for given function centered at a.

To determine

To write: The power series using summation notation.

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Chapter 11.3, Problem 32E

Chapter 11.3, Problem 34E

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

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

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To find: The first four nonzero terms of the Taylor series for given function centered at a.

To write: The power series using summation notation.

Want to see the full answer?

Chapter 11 Solutions