Problem $3.(1$ point$)$

Let $f(x)=cos(x^2).$

Use $(a)$ the Trapezoidal Rule and $(b)$ the Midpoint Rule $to$ approximate the integral ${\displaystyle \underset{0}{\overset{1}{}}}f(x)dx$ with $n=8.$ Give each answer correct $to$ six decimal places.

$T_8=$

$M_8=$

$(c)$ Use the fact that $|f^{\text{'}\text{'}}(x)|6on$ the interval $0,1to$ estimate the errors $in$ the approximations from part $(a).$ Give each answer correct $to$ six decimal places.

Error $in{T}_8=$

Error $in{M}_8=$

$(d)$ Using the information $in$ part $(c)$ and the error formulas, how large $dowe$ have $to$ choose $nso$ that the approximations $T_n$ and $M_n$ are accurate $to$ within $0\mathrm{.}00001?(Your$ answer must $be$ a whole

number.$)$

For $T_n$:$n=$

For $M_n$:$n=$