Chapter 9.4, Problem 67E

(a)

To determine

To show:That $\left|\cos x-T_4\left(x\right)\right|\le \frac{{\left(\frac{1}{2}\right)}^6}{6!}$ for all $x\in \left[0,\frac{1}{2}\right]$ .

(b)

To determine

To find:The value of ${\displaystyle {\int }_0^{\frac{1}{2}}{T}_4\left(x\right)dx}$ as an approximation to ${\displaystyle {\int }_0^{\frac{1}{2}}\cos xdx}$ and the bound for the size of the error.