Explanation Given: The function f ( x ) is, f ( x ) = 1 x − 1 The value of a is equal to 4 . Concept used: The n th Taylor polynomial T n forfunction f that is centered at x = a is, T n ( x ) = f ( a ) + f ′ ( a ) 1 ! ( x − a ) + f ″ ( a ) 2 ! ( x − a ) 2 + ⋯ + f ( n ) ( a ) n ! ( x − a ) n Here, f ( n ) is the n th derivative of the function f ( x ) . Calculation: The given function f ( x ) is, f ( x ) = 1 x − 1 It is given that the value of a is 4 . The n th Taylor polynomial T n centered at x = 4 is, T n ( x ) = f ( 4 ) + f ′ ( 4 ) 1 ! ( x − 4 ) + f ″ ( 4 ) 2 ! ( x − 4 ) 2 + f ‴ ( 4 ) 3 ! ( x − 4 ) 3 + ⋯ + f ( n ) ( 4 ) n ! ( x − 4 ) n = f ( 4 ) + f ′ ( 4 ) 1 ( x − 4 ) + f ″ ( 4 ) 2 ⋅ 1 ( x − 4 ) 2 + f ‴ ( 4 ) 3 ⋅ 2 ( x − 4 ) 3 ⋯ + f ( n ) ( 4 ) n ! ( x − 4 ) n Simplify the above n th Taylor polynomial as, T n ( x ) = f ( 4 ) + ( x − 4 ) f ′ ( 4 ) + ( x − 4 ) 2 f ″ ( 4 ) 2 + ( x − 4 ) 3 f ‴ ( 4 ) 6 ⋯ + f ( n ) ( 4 ) n ! ( x − 4 ) n …… (1) Substitute 4 for x in the function f ( x ) to obtain the value of f ( 4 ) as, f ( 4 ) = 1 4 − 1 = 1 3 Differentiate the function f ( x ) with respect to x as, f ′ ( x ) = d d x ( 1 x − 1 ) = d d x ( x − 1 ) − 1 = − 1 ( x − 1 ) − 1 − 1 = − 1 ( x − 1 ) 2 Substitute 4 for x in above equation to obtain the value of f ′ ( 4 ) as, f ′ ( 4 ) = − 1 ( 4 − 1 ) 2 = − 1 3 2 = − 1 9 Now, again differentiate the function f ′ ( x ) with respect to x as, f ″ ( x ) = d d x ( − 1 ( x − 1 ) 2 ) = d d x ( − ( x − 1 ) − 2 ) = − ( − 2 ) ( x − 1 ) − 2 − 1 = 2 ( x − 1 ) 3 Substitute 4 for x in above equation to obtain the value of f ″ ( 4 ) as, f ″ ( 4 ) = 2 ( 4 − 1 ) 3

3rd Edition

ISBN: 9781464125263

Author: Jon Rogawski, Colin Adams

Publisher: W.H. Freeman & Co

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Question

Chapter 9.4, Problem 20E

To determine

To find: The nth Taylor polynomial Tn for the function f(x)=1x−1 centered at x=a for all n .

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Chapter 9.4, Problem 19E

Chapter 9.4, Problem 21E

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Chapter 9 Solutions

Calculus - Standalone book

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Knowledge Booster

The n th Taylor polynomial T n for the function f ( x ) = 1 x − 1 centered at x = a for all n .

Solution Summary: The author explains the nth Taylor polynomial, T_n, for the function f(x)=1x-1

To find: The nth Taylor polynomial Tn for the function f(x)=1x−1 centered at x=a for all n .

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Chapter 9 Solutions