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Fresnel integrals The theory of optics gives rise to the

Calculus: Early Transcendentals | 1st Edition | ISBN: 9780321570567 | Authors: William L. Briggs, Lyle Cochran, Bernard Gillett ISBN: 9780321570567 2

Solution for problem 73E Chapter 9.4

Calculus: Early Transcendentals | 1st Edition

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Calculus: Early Transcendentals | 1st Edition | ISBN: 9780321570567 | Authors: William L. Briggs, Lyle Cochran, Bernard Gillett

Calculus: Early Transcendentals | 1st Edition

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Problem 73E

Fresnel integrals The theory of optics gives rise to the two Fresnel integrals .a. Compute S'(x)and C'(x).b. Expand sin (t2) and cos (t2) in a Maclaurin series and then integrate to find the first four nonzero terms of the Maclaurin series for S and C.c. Use the polynomials in part (b) to approximate 5(0.05) and C(?0.25)d. How main terms of the Maclaurin series are required to approximate S(0.05) with un errai no greater than 10 ?e. How many terms of the Maclaurin series are required to approximate C( 0.25) with an error no greater than 10?6?

Step-by-Step Solution:

Solution 73EStep 1:Given , Fresnel integrals ;The theory of optics gives rise to the two fresnel integrals S(x) = and C( x) = 1. Now , we need to compute Consider , S(x) = , differentiate both sides with respect to x , then = ( ) = (0) = ) Therefore , = )Consider , C(x) = , differentiate both sides with respect to x , then = ( ) = (0)= ) Therefore , = )Step 2:b) Now , we need to find Maclaurin series of sin() and cos () , by using integration method , and find the first four nonzero terms of the Maclaurin series for S and C.We know that the Maclaurin series is ; f(x) = f(0) +x ++................Consider , f(t) = sin() , then f(0) = sin(0) = 0 = , then = 0 , then = 2 = -12t, then …………………………………………Therefore , the Maclaurin series is , f(t) = sin() = f(0) +t ++................ = 0 +t ++.........-......+….= -+..........................Therefore , f(t) = sin() = -+..........................S(x) = = = = () -(0)= Therefore , s(x) = Therefore , the first four nonzero terms of the Maclaurin series for S are ; , -, , and - .Step 3:We know that the Maclaurin series is ; f(x) = f(0) +x ++................Consider , f(t) = cos() , then f(0) = cos(0) = 1 = , then = 0 , then = 0 = -12t, then ….…, …….Therefore , the Maclaurin series is , f(t) = cos() = f(0) +t ++................ = 1 +t --......-…. = + -..........................Therefore , f(t) = cos() = + -..........................C(x) = = = = () -(0) = Therefore , C(x) = Therefore , the first four nonzero terms of the Maclaurin series for C are ; x , - , , and - .

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Chapter 9.4, Problem 73E is Solved
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Textbook: Calculus: Early Transcendentals
Edition: 1
Author: William L. Briggs, Lyle Cochran, Bernard Gillett
ISBN: 9780321570567

Calculus: Early Transcendentals was written by and is associated to the ISBN: 9780321570567. The answer to “Fresnel integrals The theory of optics gives rise to the two Fresnel integrals .a. Compute S'(x)and C'(x).b. Expand sin (t2) and cos (t2) in a Maclaurin series and then integrate to find the first four nonzero terms of the Maclaurin series for S and C.c. Use the polynomials in part (b) to approximate 5(0.05) and C(?0.25)d. How main terms of the Maclaurin series are required to approximate S(0.05) with un errai no greater than 10 ?e. How many terms of the Maclaurin series are required to approximate C( 0.25) with an error no greater than 10?6?” is broken down into a number of easy to follow steps, and 96 words. Since the solution to 73E from 9.4 chapter was answered, more than 250 students have viewed the full step-by-step answer. The full step-by-step solution to problem: 73E from chapter: 9.4 was answered by , our top Calculus solution expert on 03/03/17, 03:45PM. This textbook survival guide was created for the textbook: Calculus: Early Transcendentals, edition: 1. This full solution covers the following key subjects: maclaurin, Series, approximate, terms, required. This expansive textbook survival guide covers 85 chapters, and 5218 solutions.

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