§8.5 Finding and using Taylor series (continued)

Summary

In this exercise we find the Taylor series representation for two famous functions, the Fresnel integral functions

and

. 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.

• 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. - For what interval of $x$-values will the Taylor series for $C(x)$ converge? Why? - Apply your result from (a) to estimate $C(0.5)$ to within $0.001$. - 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. - For what interval of $x$-values will the Taylor series for $S(x)$ converge? Why? - Apply your result from (d) to estimate $S(0.8)$ to within $0.001$.

The fact that we can differentiate or integrate a Taylor series reveals other important ways we can think about functions such as $e^x$.

Taylor series also provide an alternate way to evaluate indeterminate limits.