Chapter 9.4, Problem 79E

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

$S(x)={\displaystyle {\int }_0^x\sin t^{2}dt\text{and}C(x)={\displaystyle {\int }_0^x\cos t^{2}dt.}}$

1. a. Compute S′(x) and C′(x).
2. 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.
3. c. Use the polynomials in part (b) to approximate S(0.05) and C(−0.25).
4. d. How many terms of the Maclaurin series are required to approximate S(0.05) with an error no greater than 10⁻⁴?
5. e. How many terms of the Maclaurin series are required to approximate C(−0.25) with an error no greater than 10⁻⁶?