Explanation Result used: (1) “Let n be a fixed positive integer. Suppose there exist a number M such that | f ( n + 1 ) ( c ) | ≤ M , for all c between a and x inclusive. The remainder in the n th-order Taylor polynomial for f centered at a satisfies, | R n ( x ) | = | f ( x ) − p n ( x ) | ≤ M | x − a | n + 1 ( n + 1 ) ! .” (2) “ The remainder is R n ( x ) = f ( n + 1 ) ( c ) ( n + 1 ) ! ( x − a ) n + 1 for some point c between x and a .” Calculation: Consider the function f ( x ) = 1 1 − x with x = 0.85 . The ( n + 1 ) th derivative of f ( x ) = 1 1 − x is f n + 1 ( x ) = ( 2 n ) ! 2 2 n + 1 n ! ( 1 − x ) 2 n + 1 2 . Since 1 1 − x attains the maximum value at x = 0.15 this maximum is less than ( 2 n ) ! 2 2 n + 1 2 2 2 n + 1 ⋅ ( n ! ) and by using Result (1) stated above, | f ( n + 1 ) ( c ) | ≤ ( 2 n ) ! 2 2 n + 1 2 2 2 n + 1 ⋅ ( n ! ) for c ∈ ( 0 , 0.15 ) . The remainder in the n th-order Taylor polynomial for f centered at a is computed as follows, | R n ( x ) | = | f ( n + 1 ) ( c ) ( n + 1 ) ! ( x − a ) n + 1 | Substitute a = 0 and x = 0.15 in the above, | R n ( x ) | = | ( 2 n ) ! 2 2 n + 1 n ! ( 1 − x ) 2 n + 1 2 ( n + 1 ) ! ( 0.15 − a ) n + 1 | = | ( 2 n ) ! 2 2 n + 1 n ! ( 1 − x ) 2 n + 1 2 ( n + 1 ) ! ( 0

Single Variable Calculus: Early Transcendentals, Books a la Carte, and MyLab Math with Pearson eText -- Title-Specific Access Card Package (3rd Edition)

3rd Edition

ISBN: 9780134996103

Author: William L. Briggs, Lyle Cochran, Bernard Gillett, Eric Schulz

Publisher: PEARSON

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Chapter 11.1, Problem 64E

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To find: The minimum order of the Taylor polynomial required to approximate the given quantity with an error no greater than 10−3.

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Chapter 11.1, Problem 63E

Chapter 11.1, Problem 65E

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Single Variable Calculus: Early Transcendentals, Books a la Carte, and MyLab Math with Pearson eText -- Title-Specific Access Card Package (3rd Edition)

Ch. 11.1 - Verify that p3 satisfies p3(k)(a)=f(k)(a), for k =...Ch. 11.1 - Prob. 2QC Ch. 11.1 - Prob. 3QC Ch. 11.1 - Write out the next two Taylor polynomials p4 and...Ch. 11.1 - Prob. 5QC Ch. 11.1 - Prob. 6QC Ch. 11.1 - Suppose you use a second-order Taylor polynomial...Ch. 11.1 - Does the accuracy of an approximation given by a...Ch. 11.1 - The first three Taylor polynomials for f(x)=1+x...Ch. 11.1 - Prob. 4E

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The minimum order of the Taylor polynomial required to approximate the given quantity with an error no greater than 10 − 3 .

Solution Summary: The author calculates the minimum order of the Taylor polynomial required to approximate the given quantity with an error no greater than 10-3.

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To find: The minimum order of the Taylor polynomial required to approximate the given quantity with an error no greater than 10−3.

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Chapter 11 Solutions