Chapter 9.4, Problem 30E

Consider ${\displaystyle {\int }_1^{2}f\left(x\right)}dx$, where $f\left(x\right)=3\ln x$.

Make a rough sketch of the graph of the fourth derivative of $f\left(x\right)$ for $1\le x\le 2$

Find a number $A$ such that $\left|f""\left(x\right)\right|\le A$ for all $\text{x}$ satisfying $1\le X\le 2$.

Obtain a bound on the error of using the Simpson's rule with $n=2$ to approximate the definite integral.

The exact value of the definite integral (to four decimal places) is $1.1589$, and the Simpson's rule with $n=2$ gives $1.1588$. What is the error for the approximation by Simpson's rule? Does this error satisfy the bound obtained in part (c)?

Redo part (c) with the number of intervals tripled to $n=6$. Is the bound on the error divided by $3$?