$($a$)$ Find the approximations $T_{10}$ and $M_{10}$ for ${\displaystyle \underset{1}{\overset{2}{}}}11e^{\frac{1}{x}}dx.($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 to six decimal places.$)$

$|E_T|$

$|E_M|$

$$

$($c$)$ Using the values of $K$ from part $($b$),$ how large do we have to choose $n$ so that the approximations $T_n$ and $M_n$ to the integral in part

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

For $M_n,n=$