Explanation Definition used: The nth Taylor Polynomial: Let f be a function and its first n derivatives are defined at a. Then the n th Taylor polynomial of f at a is the polynomial, P n ( x ) = f ( a ) + f ′ ( a ) ( x − a ) + f ″ ( a ) 2 ! ( x − a ) 2 + ⋯ + f ( n ) ( a ) n ! ( x − a ) n . Error Bound for Taylor polynomial approximations: Suppose a function f has derivatives up to order ( N + 1 ) on an interval I containing the number a. Then for each x in I , | R N ( x ) | ≤ M ( N + 1 ) ! | x − a | N + 1 where M is any number such that | f ( N + 1 ) ( t ) | ≤ M for all t lying between a and x . Calculation: The function is given by f ( x ) = 1 1 − x . Find the third Taylor polynomial of the function f ( x ) = 1 1 − x at x = 0 as follows by using the above mentioned definition. P n ( x ) = f ( a ) + f ′ ( a ) ( x − a ) + f ″ ( a ) 2 ! ( x − a ) 2 + ⋯ + f ( n ) ( a ) n ! ( x − a ) n ⋯ ⋯ ( 1 ) . Let f ( x ) = 1 1 − x . The derivative f ′ ( x ) is calculated as follows. f ′ ( x ) = 1 ( 1 − x ) 2 The derivative f ″ ( x ) is calculated as follows. f ″ ( x ) = 2 ( 1 − x ) 3 The derivative f ‴ ( x ) is calculated as follows. f ‴ ( x ) = 6 ( 1 − x ) 4 When x = 0 , find f ( 0 ) as follows. f ( 0 ) = 1 1 − 0 = 1 1 = 1 When x = 0 , find f ′ ( 0 ) as follows. f ′ ( 0 ) = 1 ( 1 − 0 ) 2 = 1 1 2 = 1 When x = 0 , find f ″ ( 0 ) as follows. f ″ ( 0 ) = 2 ( 1 − 0 ) 3 = 2 1 3 = 2 When x = 0 , find f ‴ ( 0 ) as follows

Applied Calculus for the Managerial, Life, and Social Sciences

10th Edition

ISBN: 9781305887831

Author: Tan

Publisher: CENGAGE CO

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Chapter 11.1, Problem 31E

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To approximate: The value of f(0.2) by using the third Taylor polynomial of f(x)=11−x at x=0; compute f(0.2); find a bound for the error in the approximation and then compare this result with the exact value of f(0.2).

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Chapter 11.1, Problem 30E

Chapter 11.1, Problem 32E

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Chapter 11 Solutions

Applied Calculus for the Managerial, Life, and Social Sciences

Ch. 11.1 - Prob. 1CQ Ch. 11.1 - Prob. 2CQ Ch. 11.1 - Prob. 1E Ch. 11.1 - In Exercises 110, find the first three Taylor...Ch. 11.1 - Prob. 3E Ch. 11.1 - Prob. 4E Ch. 11.1 - Prob. 5E Ch. 11.1 - Prob. 6E Ch. 11.1 - Prob. 7E Ch. 11.1 - Prob. 8E

Ch. 11.1 - In Exercises 110, find the first three Taylor...Ch. 11.1 - Prob. 10E Ch. 11.1 - Prob. 11E Ch. 11.1 - Prob. 12E Ch. 11.1 - Prob. 13E Ch. 11.1 - Prob. 14E Ch. 11.1 - In Exercises 1120, find the nth Taylor polynomial...Ch. 11.1 - Prob. 16E Ch. 11.1 - Prob. 17E Ch. 11.1 - Prob. 18E Ch. 11.1 - Prob. 19E Ch. 11.1 - Prob. 20E Ch. 11.1 - Prob. 21E Ch. 11.1 - Prob. 22E Ch. 11.1 - Prob. 23E Ch. 11.1 - Prob. 24E Ch. 11.1 - Prob. 25E Ch. 11.1 - Prob. 26E Ch. 11.1 - Prob. 27E Ch. 11.1 - Prob. 28E Ch. 11.1 - Prob. 29E Ch. 11.1 - Prob. 30E Ch. 11.1 - Prob. 31E Ch. 11.1 - Prob. 32E Ch. 11.1 - Prob. 33E Ch. 11.1 - Prob. 34E Ch. 11.1 - Let f(x) = ex. a. 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To approximate: The value of f ( 0.2 ) by using the third Taylor polynomial of f ( x ) = 1 1 − x at x = 0 ; compute f ( 0.2 ) ; find a bound for the error in the approximation and then compare this result with the exact value of f ( 0.2 ) .

Solution Summary: The author explains the nth Taylor Polynomial: Suppose a function f has derivatives up to order (N+1) on an interval

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To approximate: The value of f(0.2) by using the third Taylor polynomial of f(x)=11−x at x=0; compute f(0.2); find a bound for the error in the approximation and then compare this result with the exact value of f(0.2).

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Chapter 11 Solutions