Finding $n$ value for trapezoid and midpoint rule errors I have this math question as follows:
(a) Find the approximations $T_{10}$ and $M_{10}$ for $\int_1^2 9e^{1/x}\,dx$, . (This means use the trapezoid rule with $n=10$ and the midpoint rule with $n=10$.)
I did this and got correct answers of $18.197781$ and $18.171917$, respectively.
(b) Estimate the errors in the approximations of part (a). (Round your answers to six decimal places.)
Correct answers: $.061161$ and $.030581$, respectively. 
But it is this part I cannot seem to get:
(c) 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 $0.0001$? 
I took the second derivative and found the max to be $27e$. I set that to:
$\frac{27e}{24n^2} < .0001$ for the midpoint and $\frac{27e}{12n^2} < .0001$ for the trapezoid rule. 
My answers were $247.308191$ for the trapezoid and $174.873299$ for the midpoint. But it's not right. I don't know what I'm doing wrong, so if someone could steer me in the right direction that would be very helpful.
 A: What are the given correct answers?
Your work seems correct. But of course $n$ is a positive integer, so you need to round your values up to the ceiling of your calculated answers. So $n=248$ guarantees the desired precision for the trapezoidal rule and $n=175$ does so for the midpoint rule.
However, those values were gotten by using the formulas that guarantee the desired precision, given a bound on the absolute value of the second derivative. You may get much better precision than the guaranteed value. Using $n=175$ I get the error $0.000028237574$ using the midpoint rule, much better than the desired $0.001$. You can get the desired error with much smaller values of $n$.
You should repeat the calculations of the approximate integrals to find just which value of $n$ actually gives you the desired error. For the midpoint rule in your situation, I get $n=93$ as the smallest $n$, which give the error $0.000099982028$. ($n=92$ gives $0.000102167251$.) For the trapezoidal rule I get $n=132$, giving the error $0.000099262595$.
Is the given correct answer for the midpoint rule $n=175$, my theoretical answer, or $n=93$, my calculated answer, or something else?
