$(a)$ Find the approximations $T_{10}$ and $M_{10}$ for ${\displaystyle \underset{1}{\overset{2}{}}}21e^{\frac{1}{x}}dx.(Round$ your answers $to$ six decimal places.$)$

$T_{10}=$

$M_{10}=$

$(b)$ Estimate the errors $in$ the approximations $of$ part $(a)$ using the smallest possible value for $K$ according $to$ the theorem

about error bounds for trapezoidal and midpoint rules. $(Round$ your answers $to$ six decimal places.$)$

$|E_T|$

$|E_M|$

$(c)$ Using the values $ofK$ from part $(b),$ how large $dowe$ have $to$ choose $nso$ that the approximations $T_n$ and $M_{n}to$ the

integral $in$ part $(a)$ are accurate $to$ within $0\mathrm{.}0001?$

For $T_{n^{\text{'}}}n=$

For $M_{n^{\text{'}}}n=$