다변수 미분 전체 문제

Show that $z=\frac{1}{2}({e}^{y}-{e}^{-y})\sin \,x$ is a solution of the differential equation $\frac{{\partial }^{2}z}{\partial {x}^{2}}+\frac{{\partial }^{2}z}{\partial {y}^{2}}=0.$

Find the indicated higher-order partial derivatives.

Find ${f}_{xx}(x,y)$ for $f(x,y)=\frac{4{x}^{2}}{y}+\frac{{y}^{2}}{2x}.$

Find the indicated higher-order partial derivatives.

Let $f(x,y,z)={x}^{2}{y}^{3}z-3x{y}^{2}{z}^{3}+5{x}^{2}z-{y}^{3}z.$ Find ${f}_{xyz}.$

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.

Show that $z={e}^{x}\sin \,y$ satisfies the equation $\frac{{\partial }^{2}z}{\partial {x}^{2}}+\frac{{\partial }^{2}z}{\partial {y}^{2}}=0.$

Find the indicated higher-order partial derivatives.

Show that $f(x,y)=\ln ({x}^{2}+{y}^{2})$ solves Laplace’s equation $\frac{{\partial }^{2}z}{\partial {x}^{2}}+\frac{{\partial }^{2}z}{\partial {y}^{2}}=0.$

Find the indicated higher-order partial derivatives.

Show that $z={e}^{-t}\cos \left(\frac{x}{c}\right)$ satisfies the heat equation $\frac{\partial z}{\partial t}={c}^{2}\left( \frac{{\partial }^{2}z}{\partial {x}^{2}} \right)$

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.)