$($a$)$ Find the approximations $T_4$ and $M_4$ for the following integral. $($Round your answers to six decimal places.$)$

${\displaystyle \underset{1}{\overset{2}{}}}29e^{\frac{1}{x}}dx$

$T_4=58\mathrm{.}924893$

$M_4=58\mathrm{.}412000$

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

For $T_{n^2}n=.$

For $M_n,n=$