a. Explanation Result used: “The remainder in the n th-order Taylor polynomial for f centered at a is R n ( x ) = f ( n + 1 ) ( c ) ( n + 1 ) ! ( x − a ) n + 1 for some point c between x and a ”. Given: The function f ( x ) = sin − 1 x ≈ x . Calculation: Obtain the value of f ( 0.1 ) . f ( x ) = x f ( 0.1 ) = 0.1 Therefore, the estimation of f ( 0.1 ) = 0.1 . Since sin − 1 x ≈ x , p n ( x ) = x with n = 2 . The first derivatives of f ( x ) = sin − 1 x as shown below, f ′ ( x ) = d d x ( sin − 1 x ) = 1 1 − x 2 = ( 1 − x 2 ) − 1 2 The second derivatives of f ( x ) = sin − 1 x as shown below, f ″ ( x ) = d d x ( ( 1 − x 2 ) − 1 2 ) = − 1 2 ( 1 − x 2 ) − 1 2 − 1 ⋅ ( − 2 x ) = ( 1 − x 2 ) − 3 2 ⋅ x = x ( 1 − x 2 ) 3 2 Since f ″ ( x ) is less than 1 on the interval on [ 0 b . To determine To estimate: The function f ( 0.2 ) , and obtain the error.

Calculus: Early Transcendentals (3rd Edition)

3rd Edition

ISBN: 9780134763644

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

Publisher: PEARSON

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Chapter 11.1, Problem 76E

To determine

To estimate: The value of f(0.1) and obtain the error bound.

To determine

To estimate: The function f(0.2), and obtain the error.

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Chapter 11.1, Problem 75E

Chapter 11.1, Problem 77E

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Chapter 11 Solutions

Calculus: Early Transcendentals (3rd Edition)

Ch. 11.1 - Verify that p3 satisfies p3(k)(a)=f(k)(a), for k =...Ch. 11.1 - Verify the following properties for f(x) = sin x...Ch. 11.1 - Why do the Taylor polynomials for sin x centered...Ch. 11.1 - Write out the next two Taylor polynomials p4 and...Ch. 11.1 - At what point would you center the Taylor...Ch. 11.1 - In Example 7, find an approximate upper bound for...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 - Suppose f(0) = 1, f(0) = 2, and f(0) = 1. Find the...

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To estimate: The value of f ( 0.1 ) and obtain the error bound.

Solution Summary: The author estimates the value of f(0.1) and obtains the error bound.