Exercise 7

In this exercise we find the Taylor series representation for two famous functions, the Fresnel integral functions $$\begin{gathered} C(x) = \int_0^x \cos(t^2) \, dt \end{gathered}$$ and $$\begin{gathered} S(x) = \int_0^x \sin(t^2) \, dt \end{gathered}$$. The Fresnel integral functions are important in optics and are used in the design of Fresnel lenses such as those found in lighthouses along the Lake Michigan shore.

1. Use the Taylor series for $\cos(x)$ to find the Taylor series for $\cos(t^2)$ and hence write $C(x)$ as a Taylor series.

2. For what interval of $x$-values will the Taylor series for $C(x)$ converge? Why?

3. Apply your result from (a) to estimate $C(0.5)$ to within $0.001$.

4. Similarly, use the Taylor series for $\sin(x)$ to find the Taylor series for $\sin(t^2)$ and hence write $S(x)$ as a Taylor series.

5. For what interval of $x$-values will the Taylor series for $S(x)$ converge? Why?

6. Apply your result from (d) to estimate $S(0.8)$ to within $0.001$.