a . Explanation Given: The function is f ( x ) = tanh x , where x = 0.5 . Calculation: Result used: The n th-order Taylor polynomial centered at a is p n ( x ) = ∑ k = 0 n f k ( a ) k ! ( x − a ) k (1) Calculation: Obtain the Taylor polynomial with n = 3 . Substitute n = 3 and a = 0 in equation (1) then the third order Taylor polynomial is as follows, p 3 ( x ) = ∑ k = 0 3 f k ( 0 ) k ! ( x − 0 ) k = f ( 0 ) + f ′ ( 0 ) ( x − 0 ) + ( 1 2 ! ) f ″ ( a ) ( x − 0 ) 2 + ( 1 3 ! ) f ‴ ( 0 ) ( x − 0 ) 3 p 3 ( x ) = f ( 0 ) + f ′ ( 0 ) ( x ) + ( 1 2 ! ) f ″ ( 0 ) ( x ) 2 + ( 1 3 ! ) f ‴ ( 0 ) ( x ) 3 (1) The value of f ( x ) = tanh x and its derivatives at x = 0 is computed as follows, f ( x ) = tanh x f ( 0 ) = tanh ( 0 ) = 0 Thus, f ( 0 ) = 0 . f ′ ( 0 ) computed as follows, f ′ ( x ) = sech 2 ( x ) = 1 − tanh 2 ( x ) f ′ ( 0 ) = 1 − tanh 2 ( 0 ) = 1 − 0 Thus, f ′ ( 0 ) = 1 . f ′ ′ ( 0 ) computed as follows, f ′ ′ ( x ) = 2 tanh ( x ) ⋅ sech 2 ( x ) = ( 2 tanh ( x ) 2 ( tanh 2 ( x ) − 1 ) ) f ′ ′ ( 0 ) = ( 2 tanh ( 0 ) 2 ⋅ ( tanh 2 ( 0 ) − 1 ) ) = 2 ⋅ 0 ⋅ ( 0 − 1 ) Thus, f ′ ′ ( 0 ) = 0 b . To determine To compute: The absolute error.

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Author: Briggs

Publisher: PEARSON

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Chapter 11.1, Problem 40E

To determine

To approximate: Taylor polynomial with n=3.

To determine

To compute: The absolute error.

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Chapter 11.1, Problem 39E

Chapter 11.1, Problem 41E

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To approximate: Taylor polynomial with n = 3 .

Solution Summary: The author explains the nth-order Taylor polynomial centered at a is mathrmtanh(0.5)approx 0.52083.