### Review Exercises

For the following exercises, determine whether the statement is *true or false*. Justify your answer with a proof or a counterexample.

The domain of $f\left(x,y\right)={x}^{3}{\text{sin}}^{\mathrm{-1}}\left(y\right)$ is $x=$ all real numbers, and $\text{\u2212}\pi \le y\le \pi .$

If the function $f\left(x,y\right)$ and all its second derivatives are continuous everywhere, then ${f}_{xy}={f}_{yx}.$

The linear approximation to the function of $f\left(x,y\right)=5{x}^{2}+x\phantom{\rule{0.2em}{0ex}}\text{tan}\left(y\right)$ at $\left(2,\pi \right)$ is given by $L\left(x,y\right)=22+21\left(x-2\right)+\left(y-\pi \right).$

$\left(\frac{3}{4},\frac{9}{16}\right)$ is a critical point of $g\left(x,y\right)=4{x}^{3}-2{x}^{2}y+{y}^{2}-2.$

For the following exercises, sketch the function in one graph and, in a second, sketch several level curves.

$f\left(x,y\right)={e}^{\text{\u2212}\left({x}^{2}+2{y}^{2}\right)}.$

For the following exercises, evaluate the following limits, if they exist. If they do not exist, prove it.

$\underset{\left(x,y\right)\to \left(1,1\right)}{\text{lim}}\frac{4xy}{x-2{y}^{2}}$

For the following exercises, find the largest region of continuity for the function.

$f\left(x,y\right)={x}^{3}{\text{sin}}^{\mathrm{-1}}\left(y\right)$

For the following exercises, find all first derivatives, full or partial, as appropriate.

$f\left(x,y\right)=\sqrt{{x}^{2}-{y}^{2}}$

For the following exercises, find all second partial derivatives.

$g\left(t,x\right)=3{t}^{2}-\text{sin}\left(x+t\right)$

For the following exercises, find the equation of the tangent plane to the specified surface at the given point.

$z={x}^{3}-2{y}^{2}+y-1$ at point $\left(1,1,\mathrm{-1}\right)$

Approximate $f\left(x,y\right)={e}^{{x}^{2}}+\sqrt{y}$ at $\left(0.1,9.1\right).$ Write down your linear approximation function $L\left(x,y\right).$ How accurate is the approximation to the exact answer, rounded to four digits?

Find the differential $dz$ of $h\left(x,y\right)=4{x}^{2}+2xy-3y$ and approximate $\text{\Delta}z$ at the point $\left(1,\mathrm{-2}\right).$ Let $\text{\Delta}x=0.1$ and $\text{\Delta}y=0.01.$

Find the directional derivative of $f\left(x,y\right)={x}^{2}+6xy-{y}^{2}$ in the direction $\text{v}=i+4j.$

Find the maximal directional derivative magnitude and direction for the function $f\left(x,y\right)={x}^{3}+2xy-\text{cos}\left(\pi y\right)$ at point $\left(3,0\right).$

For the following exercises, find the gradient.

$c\left(x,t\right)=e{\left(t-x\right)}^{2}+3\phantom{\rule{0.2em}{0ex}}\text{cos}\left(t\right)$

For the following exercises, find and classify the critical points.

$z={x}^{3}-xy+{y}^{2}-1$

For the following exercises, use Lagrange multipliers to find the maximum and minimum values for the functions with the given constraints.

$f\left(x,y\right)={x}^{2}-{y}^{2},x+6y=4$

A machinist is constructing a right circular cone out of a block of aluminum. The machine gives an error of $5\text{\%}$ in height and $2\text{\%}$ in radius. Find the maximum error in the volume of the cone if the machinist creates a cone of height $6$ cm and radius $2$ cm.

A trash compactor is in the shape of a cuboid. Assume the trash compactor is filled with incompressible liquid. The length and width are decreasing at rates of $2$ ft/sec and $3$ ft/sec, respectively. Find the rate at which the liquid level is rising when the length is $14$ ft, the width is $10$ ft, and the height is $4$ ft.