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For the following exercises, calculate the partial derivative using the limit definitions only.
$\frac{\partial z}{\partial x}$ for $z={x}^{2}-3xy+{y}^{2}$
$\frac{\partial z}{\partial y}$ for $z={x}^{2}-3xy+{y}^{2}$
$\frac{\partial z}{\partial y}=\mathrm{-3}x+2y$
For the following exercises, calculate the sign of the partial derivative using the graph of the surface.
${f}_{x}(1,1)$
${f}_{y}(1,1)$
${f}_{x}(0,0)$
The partial derivative is zero at the origin.
For the following exercises, calculate the partial derivatives.
$\frac{\partial z}{\partial x}$ for $z=\text{sin}(3x)\text{cos}(3y)$
$\frac{\partial z}{\partial y}$ for $z=\text{sin}(3x)\text{cos}(3y)$
$\frac{\partial z}{\partial y}=\mathrm{-3}\phantom{\rule{0.2em}{0ex}}\text{sin}(3x)\text{sin}(3y)$
$\frac{\partial z}{\partial x}$ and $\frac{\partial z}{\partial y}$ for $z={x}^{8}{e}^{3y}$
$\frac{\partial z}{\partial x}$ and $\frac{\partial z}{\partial y}$ for $z=\text{ln}\left({x}^{6}+{y}^{4}\right)$
$\frac{\partial z}{\partial x}=\frac{6{x}^{5}}{{x}^{6}+{y}^{4}};\frac{\partial z}{\partial y}=\frac{4{y}^{3}}{{x}^{6}+{y}^{4}}$
Find ${f}_{y}(x,y)$ for $f(x,y)={e}^{xy}\text{cos}(x)\text{sin}(y).$
Let $z={e}^{xy}.$ Find $\frac{\partial z}{\partial x}$ and $\frac{\partial z}{\partial y}.$
$\frac{\partial z}{\partial x}=y{e}^{xy};\frac{\partial z}{\partial y}=x{e}^{xy}$
Let $z=\text{ln}\left(\frac{x}{y}\right).$ Find $\frac{\partial z}{\partial x}$ and $\frac{\partial z}{\partial y}.$
Let $z=\text{tan}(2x-y).$ Find $\frac{\partial z}{\partial x}$ and $\frac{\partial z}{\partial y}.$
$\frac{\partial z}{\partial x}=2\phantom{\rule{0.2em}{0ex}}{\text{sec}}^{2}\left(2x-y\right),\frac{\partial z}{\partial y}=\text{\u2212}{\text{sec}}^{2}\left(2x-y\right)$
Let $z=\text{sinh}\left(2x+3y\right).$ Find $\frac{\partial z}{\partial x}$ and $\frac{\partial z}{\partial y}.$
Let $f(x,y)=\text{arctan}\left(\frac{y}{x}\right).$ Evaluate ${f}_{x}(2,\mathrm{-2})$ and ${f}_{y}(2,\mathrm{-2}).$
${f}_{x}(2,\mathrm{-2})=\frac{1}{4}={f}_{y}(2,\mathrm{-2})$
Let $f(x,y)=\frac{xy}{x-y}.$ Find ${f}_{x}(2,\mathrm{-2})$ and ${f}_{y}(2,\mathrm{-2}).$
Evaluate the partial derivatives at point $P(0,1).$
Find $\frac{\partial z}{\partial x}$ at $\left(0,1\right)$ for $z={e}^{\text{\u2212}x}\text{cos}(y).$
$\frac{\partial z}{\partial x}=\text{\u2212}\text{cos}\left(1\right)$
Given $f(x,y,z)={x}^{3}y{z}^{2},$ find $\frac{{\partial}^{2}f}{\partial x\partial y}$ and ${f}_{z}(1,1,1).$
Given $f(x,y,z)=2\phantom{\rule{0.2em}{0ex}}\text{sin}\left(x+y\right),$ find ${f}_{x}\left(0,\frac{\pi}{2},\mathrm{-4}\right),$ ${f}_{y}\left(0,\frac{\pi}{2},\mathrm{-4}\right),$ and ${f}_{z}\left(0,\frac{\pi}{2},\mathrm{-4}\right).$
$\begin{array}{ccc}{f}_{x}=0,\hfill & {f}_{y}=0,\hfill & {f}_{z}=0\hfill \end{array}$
The area of a parallelogram with adjacent side lengths that are $a\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}b,$ and in which the angle between these two sides is $\theta ,$ is given by the function $A(a,b,\theta )=ba\phantom{\rule{0.2em}{0ex}}\text{sin}(\theta ).$ Find the rate of change of the area of the parallelogram with respect to the following:
Express the volume of a right circular cylinder as a function of two variables:
a. $V(r,h)=\pi {r}^{2}h$ b. $\frac{\partial V}{\partial r}=2\pi rh$ c. $\frac{\partial V}{\partial h}=\pi {r}^{2}$
Calculate $\frac{\partial w}{\partial z}$ for $w=z\phantom{\rule{0.2em}{0ex}}\text{sin}(x{y}^{2}+2z).$
Find the indicated higher-order partial derivatives.
${f}_{xy}$ for $z=\text{ln}(x-y)$
${f}_{xy}=\frac{1}{{(x-y)}^{2}}$
${f}_{yx}$ for $z=\text{ln}(x-y)$
Let $z={x}^{2}+3xy+2{y}^{2}.$ Find $\frac{{\partial}^{2}z}{\partial {x}^{2}}$ and $\frac{{\partial}^{2}z}{\partial {y}^{2}}.$
$\frac{{\partial}^{2}z}{\partial {x}^{2}}=2,\frac{{\partial}^{2}z}{\partial {y}^{2}}=4$
Given $z={e}^{x}\text{tan}\phantom{\rule{0.2em}{0ex}}y,$ find $\frac{{\partial}^{2}z}{\partial x\partial y}$ and $\frac{{\partial}^{2}z}{\partial y\partial x}.$
Given $f(x,y,z)=xyz,$ find ${f}_{xyy},{f}_{yxy},$ and ${f}_{yyx}.$
${f}_{xyy}={f}_{yxy}={f}_{yyx}=0$
Given $f(x,y,z)={e}^{\mathrm{-2}x}\text{sin}\left({z}^{2}y\right),$ show that ${f}_{xyy}={f}_{yxy}.$
Show that $z=\frac{1}{2}\left({e}^{y}-{e}^{\text{\u2212}y}\right)\text{sin}\phantom{\rule{0.2em}{0ex}}x$ is a solution of the differential equation $\frac{{\partial}^{2}z}{\partial {x}^{2}}+\frac{{\partial}^{2}z}{\partial {y}^{2}}=0.$
$\begin{array}{ccc}\hfill \frac{{d}^{2}z}{d{x}^{2}}& =\hfill & -\frac{1}{2}\left({e}^{y}-{e}^{\text{\u2212}y}\right)\text{sin}\phantom{\rule{0.2em}{0ex}}x\hfill \\ \hfill \frac{{d}^{2}z}{d{y}^{2}}& =\hfill & \frac{1}{2}\left({e}^{y}-{e}^{\text{\u2212}y}\right)\text{sin}\phantom{\rule{0.2em}{0ex}}x\hfill \\ \hfill \frac{{d}^{2}z}{d{x}^{2}}+\frac{{d}^{2}z}{d{y}^{2}}& =\hfill & 0\hfill \end{array}$
Find ${f}_{xx}(x,y)$ for $f(x,y)=\frac{4{x}^{2}}{y}+\frac{{y}^{2}}{2x}.$
Let $f(x,y,z)={x}^{2}{y}^{3}z-3x{y}^{2}{z}^{3}+5{x}^{2}z-{y}^{3}z.$ Find ${f}_{xyz}.$
${f}_{xyz}=6{y}^{2}x-18y{z}^{2}$
Let $F(x,y,z)={x}^{3}y{z}^{2}-2{x}^{2}yz+3xz-2{y}^{3}z.$ Find ${F}_{xyz}.$
Given $f(x,y)={x}^{2}+x-3xy+{y}^{3}-5,$ find all points at which ${f}_{x}={f}_{y}=0$ simultaneously.
$\left(\frac{1}{4},\frac{1}{2}\right),(1,1)$
Given $f(x,y)=2{x}^{2}+2xy+{y}^{2}+2x-3,$ find all points at which $\frac{\partial f}{\partial x}=0$ and $\frac{\partial f}{\partial y}=0$ simultaneously.
Given $f(x,y)={y}^{3}-3y{x}^{2}-3{y}^{2}-3{x}^{2}+1,$ find all points on $f$ at which ${f}_{x}={f}_{y}=0$ simultaneously.
$(0,0),(0,2),(\sqrt{3},\mathrm{-1}),\left(\text{\u2212}\sqrt{3},\mathrm{-1}\right)$
Given $f(x,y)=15{x}^{3}-3xy+15{y}^{3},$ find all points at which ${f}_{x}(x,y)={f}_{y}(x,y)=0$ simultaneously.
Show that $z={e}^{x}\text{sin}\phantom{\rule{0.2em}{0ex}}y$ satisfies the equation $\frac{{\partial}^{2}z}{\partial {x}^{2}}+\frac{{\partial}^{2}z}{\partial {y}^{2}}=0.$
$\frac{{\partial}^{2}z}{\partial {x}^{2}}+\frac{{\partial}^{2}z}{\partial {y}^{2}}={e}^{x}\text{sin}(y)-{e}^{x}\text{sin}\phantom{\rule{0.2em}{0ex}}y=0$
Show that $f(x,y)=\text{ln}\left({x}^{2}+{y}^{2}\right)$ solves Laplace’s equation $\frac{{\partial}^{2}z}{\partial {x}^{2}}+\frac{{\partial}^{2}z}{\partial {y}^{2}}=0.$
Show that $z={e}^{\text{\u2212}t}\text{cos}\left(\frac{x}{c}\right)$ satisfies the heat equation $\frac{\partial z}{\partial t}=\text{\u2212}{e}^{\text{\u2212}t}\text{cos}\left(\frac{x}{c}\right).$
${c}^{2}\frac{{\partial}^{2}z}{\partial {x}^{2}}={e}^{\text{\u2212}t}\text{cos}\left(\frac{x}{c}\right)$
Find $\underset{\text{\Delta}x\to 0}{\text{lim}}\frac{f(x+\text{\Delta}x)-f(x,y)}{\text{\Delta}x}$ for $f(x,y)=\mathrm{-7}x-2xy+7y.$
Find $\underset{\text{\Delta}y\to 0}{\text{lim}}\frac{f(x,y+\text{\Delta}y)-f(x,y)}{\text{\Delta}y}$ for $f(x,y)=\mathrm{-7}x-2xy+7y.$
$\frac{\partial f}{\partial y}=\mathrm{-2}x+7$
Find $\underset{\text{\Delta}x\to 0}{\text{lim}}\frac{\text{\Delta}f}{\text{\Delta}x}=\underset{\text{\Delta}x\to 0}{\text{lim}}\frac{f(x+\text{\Delta}x,y)-f(x,y)}{\text{\Delta}x}$ for $f(x,y)={x}^{2}{y}^{2}+xy+y.$
Find $\underset{\text{\Delta}x\to 0}{\text{lim}}\frac{\text{\Delta}f}{\text{\Delta}x}=\underset{\text{\Delta}x\to 0}{\text{lim}}\frac{f(x+\text{\Delta}x,y)-f(x,y)}{\text{\Delta}x}$ for $f(x,y)=\text{sin}(xy).$
$\frac{\partial f}{\partial x}=y\phantom{\rule{0.2em}{0ex}}\text{cos}\phantom{\rule{0.2em}{0ex}}xy$
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 $n\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}R$ 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 $f(x,y,t)={e}^{\mathrm{-2}t}\text{sin}\phantom{\rule{0.2em}{0ex}}x\phantom{\rule{0.2em}{0ex}}\text{sin}\phantom{\rule{0.2em}{0ex}}y$ is a solution.
The basic wave equation is ${f}_{tt}={f}_{xx}.$ Verify that $f(x,t)=\text{sin}(x+t)$ and $f(x,t)=\text{sin}(x-t)$ 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\phantom{\rule{0.2em}{0ex}}\text{cos}\phantom{\rule{0.2em}{0ex}}\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 $\theta =\frac{\pi}{6}.$
$\frac{\partial F}{\partial \theta}=6,\frac{\partial F}{\partial x}=4-3\sqrt{3}$
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 $x\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}y$ represent the amount of labor and capital available. Let $x=500$ and $y=1000.$ Find $\frac{\delta f}{\delta x}$ and $\frac{\delta f}{\delta y}$ at these values, which represent the marginal productivity of labor and capital, respectively.
$\frac{\delta f}{\delta x}$ at $\left(500,1000\right)=172.36,$ $\frac{\delta f}{\delta y}$ at $\left(500,1000\right)=36.93$
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 $t=20\text{\xb0}\text{F}$ and $h=0.90.$
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