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Find the indicated higher-order partial derivatives.

Find $\lim_{\Delta x\to 0}\frac{f(x+\Delta x,y)-f(x,y)}{\Delta x}$ for $f(x,y)=-7x-2xy+7y.$

Find the indicated higher-order partial derivatives.

Find $\lim_{\Delta y\to 0}\frac{f(x,y+\Delta y)-f(x,y)}{\Delta y}$ for $f(x,y)=-7x-2xy+7y.$

Find the indicated higher-order partial derivatives.

Find $\lim_{\Delta x\to 0}\frac{\Delta f}{\Delta x}=\lim_{\Delta x\to 0}\frac{f(x+\Delta x,y)-f(x,y)}{\Delta x}$ for $f(x,y)={x}^{2}{y}^{2}+xy+y.$

Find the indicated higher-order partial derivatives.

Find $\lim_{\Delta x\to 0}\frac{\Delta f}{\Delta x}=\lim_{\Delta x\to 0}\frac{f(x+\Delta x,y)-f(x,y)}{\Delta x}$ for $f(x,y)=\sin (xy).$

The function $P(T,V)=\frac{nRT}{V}$ gives the pressure at a point in a gas as a function of temperature $T$ and volume $V.$ The letters $n\,\text{and}\,R$ are constants. Find $\frac{\partial P}{\partial V}$ and $\frac{\partial P}{\partial T},$ and explain what these quantities represent.

The equation for heat flow in the $xy\text{-plane}$ is $\frac{\partial f}{\partial t}=\frac{{\partial }^{2}f}{\partial {x}^{2}}+\frac{{\partial }^{2}f}{\partial {y}^{2}}.$ Show that $f(x,y,t)={e}^{-2t}\sin \,x\,\sin \,y$ is a solution.

The basic wave equation is ${f}_{tt}={f}_{xx}.$ Verify that $f(x,t)=\sin (x+t)$ and $f(x,t)=\sin (x-t)$ are solutions.

The law of cosines can be thought of as a function of three variables. Let $x,y,$ and $\theta$ be two sides of any triangle where the angle $\theta$ is the included angle between the two sides. Then, $F(x,y,\theta )={x}^{2}+{y}^{2}-2xy\,\cos \,\theta$ gives the square of the third side of the triangle. Find $\frac{\partial F}{\partial \theta }$ and $\frac{\partial F}{\partial x}$ when $x=2,y=3,$ and $\theta =\frac{\pi }{6}.$

Suppose the sides of a rectangle are changing with respect to time. The first side is changing at a rate of $2$ in./sec whereas the second side is changing at the rate of $4$ in/sec. How fast is the diagonal of the rectangle changing when the first side measures $16$ in. and the second side measures $20$ in.? (Round answer to three decimal places.)

A Cobb-Douglas production function is $f(x,y)=200{x}^{0.7}{y}^{0.3},$ where $x\,\text{and}\,y$ represent the amount of labor and capital available. Let $x=500$ and $y=1000.$ Find $\frac{\partial f}{\partial x}$ and $\frac{\partial f}{\partial y}$ at these values, which represent the marginal productivity of labor and capital, respectively.

The apparent temperature index is a measure of how the temperature feels, and it is based on two variables: $h,$ which is relative humidity, and $t,$ which is the air temperature.

$A=0.885t-22.4h+1.20th-0.544.$ Find $\frac{\partial A}{\partial t}$ and $\frac{\partial A}{\partial h}$ when $t=20^\circ \text{F}$ and $h=0.90.$

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

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

The linear approximation to the function of $f(x,y)=5{x}^{2}+x\,\tan (y)$ at $(2,\pi )$ is given by $L(x,y)=22+21(x-2)+(y-\pi ).$

Chapter Review Exercises

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

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

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

$f(x,y)={e}^{-({x}^{2}+2{y}^{2})}.$

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

$f(x,y)=x+4{y}^{2}.$

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

$\lim_{(x,y)\to (1,1)}\frac{4xy}{x-2{y}^{2}}$

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

$\lim_{(x,y)\to (0,0)}\frac{4xy}{x-2{y}^{2}}$

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

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