a. 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 . Remainder theorem: It states that if f is a function which has derivatives up to order N + 1 on an interval I containing the number a . Then for each x in I , there exists a number c lying between a and x such that R N ( x ) = f ( n + 1 ) ( c ) ( n + 1 ) ! ( x − a ) n + 1 , where a formula for R N ( x ) , called the remainder associated with P N ( x ) . So that R N ( x ) = f ( x ) − P N ( x ) . Error Bound for Taylor Polynomial Approximations: Suppose 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 ) = ln x . First find a number M such that | f ( 5 ) ( t ) | ≤ M for all t lying in the interval [ 1 , 1.5 ] . The function is given by f ( x ) = ln x . Find the fourth Taylor polynomial of the function f ( x ) = ln x at x = 1 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 ) = ln x . The derivative f ′ ( x ) is calculated as follows. f ′ ( x ) = 1 x The derivative f ″ ( x ) is calculated as follows. f ″ ( x ) = − 1 x 2 The derivative f ‴ ( x ) is calculated as follows. f ‴ ( x ) = − ( − ) 2 x 3 = 2 x 3 The derivative f ( 4 ) ( x ) is calculated as follows b. To determine To approximate: The area under the graph of f from x = 1 to x = 1.5 by computing ∫ 1 1.5 P 4 ( x ) d x . c. To determine To find: The bound on the error in the approximation in (b). d. To determine To find: The actual error.

APPLIED CALCULUS FOR MGRL, LIFE, SOC SCI

10th Edition

ISBN: 9780357465400

Author: Tan

Publisher: CENGAGE L

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Chapter 11.1, Problem 36E

To determine

To find: A bound in the error incurred in approximating f(x)=lnx by the fourth Taylor polynomial, P4(x) of f at x=0 in the interval [1,1.5].

To determine

To approximate: The area under the graph of f from x=1 to x=1.5 by computing ∫11.5P4(x)dx.

To determine

To find: The bound on the error in the approximation in (b).

To determine

To find: The actual error.

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Chapter 11.1, Problem 35E

Chapter 11.1, Problem 37E

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Chapter 11 Solutions

APPLIED CALCULUS FOR MGRL, LIFE, SOC SCI

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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A bound in the error incurred in approximating f ( x ) = ln x by the fourth Taylor polynomial, P 4 ( x ) of f at x = 0 in the interval [ 1 , 1.5 ] .

Solution Summary: The author explains the nth Taylor Polynomial: Let f be a function and its first and last derivatives are defined.

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To find: A bound in the error incurred in approximating f(x)=lnx by the fourth Taylor polynomial, P4(x) of f at x=0 in the interval [1,1.5].

To approximate: The area under the graph of f from x=1 to x=1.5 by computing ∫11.5P4(x)dx.

To find: The bound on the error in the approximation in (b).

To find: The actual error.

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Chapter 11 Solutions