Chapter 7.7, Problem 47ES

Let $f(x)=\cos (x^2).$

(a) Use a CAS to approximate the maximum value of $|f"(x)|$ on the interval [0, 1].

(b) How large must $n$ be in the midpoint approximation of ${\displaystyle {\int }_0^{1}f(x)}dx$ to ensure that the absolute error is less than $5\times {10}^{-4}?$ Compare your result with that obtained in Example 9.

(c) Estimate the integral using the midpoint approximation with the value of $n$ obtained in part (b).