Question $5$: Consider the following error bound theorem $to$ approximate func$-$

tions $by$ Taylor polynomials

$(Taylor\text{'}s$ Inequality$)Iff(x)-T_n(x)=R_n(x),$ where $T_{n}is$ the degree $n$ Tay$-$

lor polynomial $offata,$ and $|f^{(n+1)}(x)|M$ for all $x$ such that $|x-a|d,$

then

$|R_n(x)|\frac{M|x-a{|}^{n+1}}{(n+1)!}$ for all $x$ with $|x-a|d$

Now, use Taylor's Inequality $to$ estimat$\frac{e}{b}$ound the error obtained $to$ approxi$-$

mate $f(x)=\sqrt[3]{x}by$ a Taylor polynomial $of$ degree $2ata=64$ for $63x65.$