Chapter 5.2, Problem 3DP

(a) Draw the graph of the function $f(x)=\cos \left(x^2\right)$ in the viewing rectangle $\left[0,2\right]$ by $[-1.25,1.25]$ .

(b) If we define a new function $g$ by $g(x)={\displaystyle {\int }_0^x\cos }\left(t^2\right)dt$ then $g(x)$ is the area under the graph of $f$ from 0 to $x$ [until $f(x)$ becomes negative, at which point $g(x)$ becomes a difference of areas]. Use part (a) to determine the value of $x$ at which $g(x)$ starts to decrease. [Unlike the integral in Problem 2, it is impossible to evaluate the integral defining $g$ to obtain an explicit expression for $g(x)$ .]

(c) Use the integration command on a calculator or computer to estimate $g(0.2),g(0.4)$ , $g(0.6),\dots ,g(1.8),g(2)$ . Then use these values to sketch a graph of $g$ .

(d) Use your graph of $g$ from part (c) to sketch the graph of $g\prime$ using the interpretation of $g\prime (x)$ as the slope of a tangent line. How does the graph of $g\prime$ compare with the graph of $f$ ?