Circle the best answer for each problem. Each problem is worth $2$ points. A calculator may be used.

Suppose $f$ is a function that is nonnegative and continuous on the interval $0,2.$ Which of the following represents $A_R,$ the right Riemann sum with four equal subintervals?

$($A$)f(0)*\frac{1}{4}+f(0\mathrm{.}5)*\frac{1}{4}+f(1)*\frac{1}{4}+f(1\mathrm{.}5)*\frac{1}{4}$

$($B$)f(0\mathrm{.}5)*\frac{1}{4}+f(1)*\frac{1}{4}+f(1\mathrm{.}5)*\frac{1}{4}+f(2)*\frac{1}{4}$

$($C$)f(0)*\frac{1}{2}+f(0\mathrm{.}5)*\frac{1}{2}+f(1)*\frac{1}{2}+f(1\mathrm{.}5)*\frac{1}{2}$

$($D$)f(0\mathrm{.}5)*\frac{1}{2}+f(1)*\frac{1}{2}+f(1\mathrm{.}5)*\frac{1}{2}+f(2)*\frac{1}{2}$

Selected values of $f(x)$ are given in the table. The trapezoidal sum associated with the function $f$ and this table is $61.$ What is the value of $k?$ You must show all necessary work to receive credit.

$\backslash$table$[[x,2,5,8,10],[f(x),7,9,k,11]]$

$($A$)4$

$($B$)5$

$($C$)6$

$($D$)7$

$3.$ Which of the following is a left Riemann sum approximation of ${\displaystyle \underset{1}{\overset{7}{}}}(4lnx+2)dx$ with $n$ subintervals of equal length?

$($A$){\displaystyle \underset{k=1}{\overset{n}{}}}(4ln(1+\frac{6k}{n})+2)(\frac{6}{n})$

$($B$){\displaystyle \underset{k=1}{\overset{n}{}}}(4ln(1+\frac{6(k-1)}{n})+2)(\frac{6}{n})$

$($C$){\displaystyle \underset{k=1}{\overset{n}{}}}(4ln(\frac{6k}{n})+2)(\frac{6}{n})$

$($D$){\displaystyle \underset{k=1}{\overset{n}{}}}(4ln(1+\frac{k-1}{n})+2)(\frac{1}{n})$