a. Explanation Given: The function f ( x ) = 1 + x . Result used: The n th-order Taylor polynomial centered at a is as follows, p n ( x ) = f 0 ( a ) 0 ! ( x − a ) 0 + f ′ ( a ) 1 ! ( x − a ) 1 + f ″ ( a ) 2 ! ( x − a ) 2 + ⋯ + f n ( a ) n ! ( x − a ) n Calculation: The n th-order Taylor polynomial centered at a = 0 is as follows, p n ( x ) = f ( 0 ) + f ′ ( 0 ) 1 ! ( x ) + f ″ ( 0 ) 2 ! ( x ) 2 + ⋯ + f n ( 0 ) n ! ( x ) n (1) The first derivative of f ( x ) = 1 + x computed as follows, f ′ ( x ) = d d x ( 1 + x ) = 1 2 1 + x Substitute 0 for x in the above, f ′ ( 0 ) = 1 2 1 + 0 = 1 2 The second derivative of f ( x ) = 1 + x computed as follows, f ″ ( x ) = d d x ( 1 2 1 + x ) = 1 2 ⋅ d d x ( 1 1 + x ) = 1 2 ( − 1 2 ( 1 + x ) 3 2 ) = − 1 4 ( 1 + x ) 3 2 Compute f ″ ( 0 ) as follows. f ″ ( 0 ) = − 1 4 ( 1 + 0 ) 3 2 = − 1 4 The value of f ( x ) = 1 + x and its derivatives at x = 0 as shown in the below table b. To determine To make: The table for approximations and absolute error.

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ISBN: 9781323178522

Author: Briggs

Publisher: PEARSON

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Chapter 9, Problem 12RE

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To find: The nth-order Taylor polynomial of order n=0,1 and 2 centered at 0.

To determine

To make: The table for approximations and absolute error.

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Chapter 9, Problem 11RE

Chapter 9, Problem 13RE

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The n th-order Taylor polynomial of order n = 0 , 1 and 2 centered at 0.

Solution Summary: The author explains the n th-order Taylor polynomial of order n=0,1and 2 centered at 0.

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To find: The nth-order Taylor polynomial of order n=0,1 and 2 centered at 0.

To make: The table for approximations and absolute error.

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Chapter 9 Solutions