Chapter 7.7, Problem 50ES

Let $f(x)=\sqrt{2+x^3}.$

(a) Use a CAS to approximate the maximum value of $|f^{(4)}(x)|$ on the interval [0,1].

(b) How large must the value of $n$ be in the approximation $S_n$ of ${\displaystyle {\int }_0^{1}f(x)dx}$ by Simpson’s rule to ensure that the absolute error is less than ${10}^{-6}?$

(c) Estimate the integral using Simpson’s rule approximation $S_n$ with the value of $n$ obtained in part (b).