Explanation It is known that, the Taylor polynomial for a function f that can be differentiated n times at 0 is given by, P n ( x ) = f ( 0 ) + f ( 1 ) ( 0 ) 1 ! x + f ( 2 ) ( 0 ) 2 ! x 2 + f ( 3 ) ( 0 ) 3 ! x 3 + ⋯ + f ( n ) ( 0 ) n ! x n Note that the Taylor polynomial of degree 4 is, P 4 ( x ) = f ( 0 ) + f ( 1 ) ( 0 ) 1 ! x + f ( 2 ) ( 0 ) 2 ! x 2 + f ( 3 ) ( 0 ) 3 ! x 3 + f ( 4 ) ( 0 ) 4 ! x 4 Differentiate the Taylor polynomial of degree 4 with respect to x , P 4 ′ ( x ) = d d x ( f ( 0 ) + f ( 1 ) ( 0 ) 1 ! x + f ( 2 ) ( 0 ) 2 ! x 2 + f ( 3 ) ( 0 ) 3 ! x 3 + f ( 4 ) ( 0 ) 4 ! x 4 ) = d d x ( f ( 0 ) ) + d d x ( f ( 1 ) ( 0 ) 1 ! x ) + d d x ( f ( 2 ) ( 0 ) 2 ! x 2 ) + d d x ( f ( 3 ) ( 0 ) 3 ! x 3 ) + d d x ( f ( 4 ) ( 0 ) 4 ! x 4 ) = f ( 1 ) ( 0 ) 1 ! + f ( 2 ) ( 0 ) 2 ! ( 2 x ) + f ( 3 ) ( 0 ) 3 ! ( 3 x 2 ) + f ( 4 ) ( 0 ) 4 ! ( 4 x 3 ) Again differentiate the first derivative of the Taylor polynomial of degree 4 with respect to x, P 4 ′ ′ ( x ) = d d x ( f ( 1 ) ( 0 ) 1 ! + f ( 2 ) ( 0 ) 2 ! ( 2 x ) + f ( 3 ) ( 0 ) 3 ! ( 3 x 2 ) + f ( 4 ) ( 0 ) 4 ! ( 4 x 3 ) ) = d d x ( f ( 1 ) ( 0 ) 1 ! ) + d d x ( f ( 2 ) ( 0 ) 2 ! ( 2 x ) ) +

Calculus with Applications (11th Edition)

11th Edition

ISBN: 9780321979421

Author: Margaret L. Lial, Raymond N. Greenwell, Nathan P. Ritchey

Publisher: PEARSON

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Chapter 12, Problem 5RE

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Whether the statement “The Taylor polynomial of degree 4 for f at 0 has the same fifth derivative as f at 0” is true or false.

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Chapter 12, Problem 4RE

Chapter 12, Problem 6RE

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Chapter 12 Solutions

Calculus with Applications (11th Edition)

Ch. 12.1 - Find the first four terms of the sequence having...Ch. 12.1 - Prob. 2YT Ch. 12.1 - Prob. 3YT Ch. 12.1 - Prob. 4YT Ch. 12.1 - Prob. 5YT Ch. 12.1 - Prob. 1E Ch. 12.1 - Prob. 2E Ch. 12.1 - Prob. 3E Ch. 12.1 - List the first n terms of the geometric sequence...Ch. 12.1 - Prob. 5E

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