Multiple Choice $($CONtinued on Next PaGe$)$

Exercise $1.$ The directional derivative of the function $f(x,y)=\frac{1}{2}si{n}^2(x)y^2$ in the direction of the vector $v_{?}=(3,0)$ is given by:

$($A$)3cos(x)sin(x)y^2$

$si{n}^2(x)y$

$3si{n}^2(x)y$

$($D$)cos(x)sin(x)y^2$

Exercise $2.$ Consider the function $f(x,y)=\frac{1}{2}{x}^{2}y+y^2-cos(x).$ Which of the following statements is true about the point $(0,0)?$

$(0,0)$ is a critical point, and a local minimum of $f$

$(0,0)$ is a critical point, and a local maximum of $f$

$(0,0)$ is a critical point, and a saddle point of $f$

$(0,0)$ is not a critical point of $f$

$1$

Exercise $3.$ Which of the following expressions correctly gives the integral ${}_{D}xdV,$ where $D=\{(x,y,z)$:$\sqrt[\phantom{2}]{x^2+y^2}z,0z1\}$ is a cone?

$($A$){\displaystyle \underset{0}{\overset{1}{}}}{\displaystyle \underset{0}{\overset{}{}}}{\displaystyle \underset{0}{\overset{z}{}}}{r}^{2}cos(var)drdvardz$

$($B$){\displaystyle \underset{0}{\overset{1}{}}}{\displaystyle \underset{0}{\overset{2}{}}}{\displaystyle \underset{0}{\overset{z}{}}}{r}^{2}cos(var)drdvardz$

$($C$){\displaystyle \underset{0}{\overset{1}{}}}{\displaystyle \underset{0}{\overset{2}{}}}{\displaystyle \underset{0}{\overset{z^2}{}}}{r}^{2}cos(var)drdvardz$

$($D$){\displaystyle \underset{0}{\overset{1}{}}}{\displaystyle \underset{0}{\overset{2}{}}}{\displaystyle \underset{0}{\overset{z}{}}}rcos(var)drdvardz$

Exercise $4.$ For any $f,$ and any real constant $a>0,$ which of the following integrals correctly reverses the order of integration of the integral

${\displaystyle \underset{0}{\overset{a}{}}}{\displaystyle \underset{0}{\overset{x^2}{}}}f(x,y)dydx?$

$($A$){\displaystyle \underset{0}{\overset{a}{}}}{\displaystyle \underset{0}{\overset{\sqrt[\phantom{2}]{y}}{}}}f(x,y)dxdy$

$($B$){\displaystyle \underset{0}{\overset{a^2}{}}}{\displaystyle \underset{0}{\overset{\sqrt[\phantom{2}]{y}}{}}}f(x,y)dxdy$

$($C$){\displaystyle \underset{0}{\overset{a^2}{}}}{\displaystyle \underset{\sqrt[\phantom{2}]{y}}{\overset{a}{}}}f(x,y)dxdy$

${\displaystyle \underset{0}{\overset{a}{}}}{\displaystyle \underset{\sqrt[\phantom{2}]{y}}{\overset{a^2}{}}}f(x,y)dxdy$