$13-22$

$($a$)$ Approximate $f$ by a Taylor polynomial with degree $n$ at the number $a.$

$($b$)$ Use Taylor's Inequality to estimate the accuracy of the approximation $f(x)$~~$T_n(x)$ when $x$ lies in the given interval.

$($c$)$ Check your result in part $($b$)$ by graphing $|R_n(x)|.$

$13.f(x)=\frac{1}{x},a=1,n=2,0\mathrm{.}7x1\mathrm{.}3$

$14.f(x)=x^{-\frac{1}{2}},a=4,n=2,3\mathrm{.}5x4\mathrm{.}5$

$15.f(x)=x^{\frac{2}{3}},a=1,n=3,0\mathrm{.}8x1\mathrm{.}2$

$16.f(x)=sinx,a=\frac{}{6},n=4,0x\frac{}{3}$

$17.f(x)=secx,a=0,n=2,-0\mathrm{.}2x0\mathrm{.}2$

$18.f(x)=ln(1+2x),a=1,n=3,0\mathrm{.}5x1\mathrm{.}5$

$19.f(x)=e^{x^2},a=0,n=3,0x0\mathrm{.}1$

$20.f(x)=xlnx,a=1,n=3,0\mathrm{.}5x1\mathrm{.}5$

$21.f(x)=xsinx,a=0,n=4,-1x1$

$22.f(x)=sinh2x,a=0,n=5,-1x1$

Only do #$16$ a$,$b