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

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Chapter 11.7, Problem 20E

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To show: That the nth Taylor polynomial for the function f(x)=1x+1 at a=1 is Tn(x)=12−(x−1)4+( x−1)28+⋯+(−1)n( x−1)n2n+1 .

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Chapter 11.7, Problem 19E

Chapter 11.7, Problem 21E

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Chapter 11 Solutions

CALCULUS 4E (LL) W/ ACHIEVE ACCESS

Ch. 11.1 - Prob. 1PQ Ch. 11.1 - Prob. 2PQ Ch. 11.1 - Prob. 3PQ Ch. 11.1 - Prob. 4PQ Ch. 11.1 - Prob. 5PQ Ch. 11.1 - Prob. 1E Ch. 11.1 - Prob. 2E Ch. 11.1 - Prob. 3E Ch. 11.1 - Prob. 4E Ch. 11.1 - Prob. 5E

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That the n th Taylor polynomial for the function f ( x ) = 1 x + 1 at a = 1 is T n ( x ) = 1 2 − ( x − 1 ) 4 + ( x − 1 ) 2 8 + ⋯ + ( − 1 ) n ( x − 1 ) n 2 n + 1 .

Solution Summary: The author demonstrates that the nth Taylor polynomial for the function f(x)=1x+1

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To show: That the nth Taylor polynomial for the function f(x)=1x+1 at a=1 is Tn(x)=12−(x−1)4+( x−1)28+⋯+(−1)n( x−1)n2n+1 .

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Chapter 11 Solutions