다변수 미분 전체 문제

Find the indicated higher-order partial derivatives.

Let $F(x,y,z)={x}^{3}y{z}^{2}-2{x}^{2}yz+3xz-2{y}^{3}z.$ Find ${F}_{xyz}.$

Find the indicated higher-order partial derivatives.

Given $f(x,y)={x}^{2}+x-3xy+{y}^{3}-5,$ find all points at which ${f}_{x}={f}_{y}=0$ simultaneously.

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.

Find the indicated higher-order partial derivatives.

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.

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 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 $\theta =\frac{\pi }{6}.$

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.

For the following exercises, find a unit normal vector to the surface at the indicated point.

$z=f(x,y)={x}^{3},(2,-1,8)$

For the following exercises, find a unit normal vector to the surface at the indicated point.

$\ln \left(\frac{x}{y-z}\right)=0$ when $x=y=1$

For the following exercises, as a useful review for techniques used in this section, find a normal vector and a tangent vector at point $P.$

${x}^{2}+xy+{y}^{2}=3,P(-1,-1)$

For the following exercises, as a useful review for techniques used in this section, find a normal vector and a tangent vector at point $P.$

${({x}^{2}+{y}^{2})}^{2}=9({x}^{2}-{y}^{2}),P\left(\sqrt{2},1\right)$

For the following exercises, as a useful review for techniques used in this section, find a normal vector and a tangent vector at point $P.$

$x{y}^{2}-2{x}^{2}+y+5x=6,P(4,2)$

For the following exercises, as a useful review for techniques used in this section, find a normal vector and a tangent vector at point $P.$

$2{x}^{3}-{x}^{2}{y}^{2}=3x-y-7,P(1,-2)$

For the following exercises, as a useful review for techniques used in this section, find a normal vector and a tangent vector at point $P.$

$z{e}^{{x}^{2}-{y}^{2}}-3=0,$ $P(2,2,3)$

For the following exercises, find the equation for the tangent plane to the surface at the indicated point. (Hint: Solve for $z$ in terms of $x$ and $y.)$

$-8x-3y-7z=-19,P(1,-1,2)$

For the following exercises, find the equation for the tangent plane to the surface at the indicated point. (Hint: Solve for $z$ in terms of $x$ and $y.)$

$z=-9{x}^{2}-3{y}^{2},P(2,1,-39)$