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Fresnel integrals The theory of optics gives rise to the
Chapter 8, Problem 73E(choose chapter or problem)
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 , = )