Fresnel integrals The theory of optics gives rise to the

Chapter 8, Problem 73E

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QUESTION:

Fresnel integrals The theory of optics gives rise to the two Fresnel integrals

\(S(x)=\int_{0}^{x} \sin \left(t^{2}\right) d t\) and \(C(x)=\int_{0}^{x} \cos \left(t^{2}\right) d t\).

a. Compute \(S^{\prime}(x)\) and \(C^{\prime}(x)\).

b. Expand \(\sin \left(t^{2}\right)\) and \(\cos \left(t^{2}\right)\) 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 S(0.05) and C(-0.25).

d. How many terms of the Maclaurin series are required to approximate S(0.05) with an error no greater than \(10^{-4}\)?

e. How many terms of the Maclaurin series are required to approximate C(-0.25) with an error no greater than \(10^{-6}\)?

Questions & Answers

QUESTION:

Fresnel integrals The theory of optics gives rise to the two Fresnel integrals

\(S(x)=\int_{0}^{x} \sin \left(t^{2}\right) d t\) and \(C(x)=\int_{0}^{x} \cos \left(t^{2}\right) d t\).

a. Compute \(S^{\prime}(x)\) and \(C^{\prime}(x)\).

b. Expand \(\sin \left(t^{2}\right)\) and \(\cos \left(t^{2}\right)\) 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 S(0.05) and C(-0.25).

d. How many terms of the Maclaurin series are required to approximate S(0.05) with an error no greater than \(10^{-4}\)?

e. How many terms of the Maclaurin series are required to approximate C(-0.25) with an error no greater than \(10^{-6}\)?

ANSWER:

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 , = )

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