Chapter 4.R, Problem 52E

The Fresnel function $S\left(x\right)={\displaystyle {\int }_0^x\sin \left(\frac{1}{2}\pi t^2\right)dt}$ was introduced in Section 4.3. Fresnel also used the function

$C\left(x\right)={\displaystyle {\int }_0^x\cos \left(\frac{1}{2}\pi t^2\right)dt}$

in his theory of the diffraction of light waves.

(a) On what intervals is C is increasing?

(b) On what intervals is C concave upward?

(c) Use a graph to solve the following equation correct to two decimal places:

${\displaystyle {\int }_0^x\cos \left(\frac{1}{2}\pi t^2\right)}dt=0.7$

(d) Plot the graphs of C and S on the same screen. How are these graphs related?