Exercise $1.$ Consider the function $f(x,y,z)=x^{2}y-zcosy.$ Use a directional derivative to approximate how much $f$ changes if one moves a distance $0\mathrm{.}1$ from the point $(4,0,3)$ straight toward the origin.

Exercise $2.$ Determine all the local minima, local maxima, and saddles of the function $f(x,y)=x^4-x^2-2xy+y^2.$

Exercise $3.$ Use an appropriate change of variables to evaluate

${}_R\frac{cos(\frac{x^2}{4}+\frac{y^2}{9}+\frac{z^2}{25})}{\sqrt[\phantom{2}]{\frac{x^2}{4}+\frac{y^2}{9}+\frac{z^2}{25}}}dV$

where $R$ is the region defined by the inequalities $\frac{x^2}{4}+\frac{y^2}{9}+\frac{z^2}{25}1$ and $\frac{x^2}{4}+\frac{y^2}{9}\frac{z^2}{25}.$

Exercise $4.$ Find the value$($s$)$ of a and $b$ where the function

$f(a,b)={\displaystyle \underset{a}{\overset{b}{}}}(-2x^2+10x-12)dx$

has a local maximum.

Exercise $5.$ Let $D$ be a region on the $xy-$plane. Let $S$ be the part of the plane $4x-6y+2z=5$ which lies above or below the region $D,($i$.$e$.$ points on the plane $4x-6y+2z=5$ with $(x,y)$ in $D).$ If the area of $S$ is $11,$ find the area of $D.$

Exercise $6.$ Set up a triple iterated integral that would determine the volume of the solid region below $z=x^4+y^4+4$ and above the triangular region on the $xy-$plane with vertices $(-2,0),(2,0),$ and $(0,2).$ Do not evaluate this integral.

Exercise $7.$ Find the work done by the force field $F=yi-2xj$ along the upper semicircle $\{x^2+y^2=9,y0\},$ oriented in the counterclockwise direction.

Exercise $8.$ Consider the vector field $F=(2x-y)i-xj.$

Determine whether or not $F$ is a conservative vector field.

If it is$,$ find a function $f$ such that $F=$gradf.

Compute the line integral ${\displaystyle \underset{C}{\overset{\phantom{}}{}}}F.$dr$,$ where $C$ is the line segment from $(1,2)$ to $(2,1).$

SOLVE ALL FOR CALCLUS$3$