Let $f(x)=e^x.$

Use (a) the Trapezoidal Rule and (b) Simpson's Rule to approximate the integral $\int_{0}^{\,1} {f(x)}\, dx$ with n = 10. Give each answer correct to eight decimal places.

$T_{10} =$

$S_{10} =$

(c) Evaluate $\int_{0}^{\,1} {e^x}\, dx$

Integral =

(d) Use your answers in (a), (b), and (c) to give the absolute value of the error in the approximations $T_{10}$ and $S_{10}.$ Your answer must be correct to eight decimal places.

Absolute error in $T_{10}$ approximation =

Absolute error in $S_{10}$ approximation =

(e) Use the fact that $|f''(x)| \le e$ and $|f^{(4)}(x)| \le e$ on the interval $[0,1]$ and the error formulas in your textbook to estimate the errors in the approximations from (a) and (b). Give each answer correct to eight decimal places.

Error in $T_{10}$ from error formula =

Error in $S_{10}$ from error formula =

(f) Using the information in part (e) and the error formulas, how large do we have to choose n to ensure that the approximations $T_n,$ $M_n,$ and $S_n$ are accurate to within 0.00001? (Your answer must be a whole number.)

For $T_n:$ n =

For $M_n:$ n =

For $S_n:$ n =