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?

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 - .