Chapter 9.4, Problem 29E

(a)

To determine

To graph: The rough sketch of ${f^{\prime }}^{\prime }\left(x\right)$ for $0\le x\le 2$, where $f\left(x\right)=\frac{1}{12}{x}^4+3x^2$.

(b)

To determine

The number $A$ such that $\left|{f^{\prime }}^{\prime }\left(x\right)\right|\le A$ for all $x$ satisfying $0\le x\le 2$ where $f\left(x\right)=\frac{1}{12}{x}^4+3x^2$.

(c)

To determine

The bound on the error by using the Midpoint rule with $n=10$ to approximate the definite integral ${\displaystyle \underset{0}{\overset{2}{\int }}f\left(x\right)dx}$ where $f\left(x\right)=\frac{1}{12}{x}^4+3x^2$

(d)

To determine

The error for the midpoint approximation for the definite integral ${\displaystyle \underset{0}{\overset{2}{\int }}\frac{1}{12}{x}^4+3x^{2}dx}$ and check whether this error satisfies the bound obtained in part (c)

(e)

To determine

The bound on the error by using the midpoint rule with $n=20$ to approximate the definite integral ${\displaystyle \underset{0}{\overset{2}{\int }}f\left(x\right)dx}$ where $f\left(x\right)=\frac{1}{12}{x}^4+3x^2$ and check whether the bounds on the error halved or quartered.