다변수 미분 전체 문제

Show that the limit $\lim_{(x,y)\to (0,0)}\frac{5{x}^{2}y}{{x}^{2}+{y}^{2}}$ exists and is the same along the paths: $y\text{-axis}$ and $x\text{-axis,}$ and along $y=x.$

Show that $\lim_{(x,y)\to (0,0)}\frac{1}{{x}^{2}+{y}^{2}}$ does not exist at $(0,0)$ by plotting the graph of the function.

Express the volume of a right circular cylinder as a function of two variables:

a. its radius $r$ and its height $h.$ b. Show that the rate of change of the volume of the cylinder with respect to its radius is the product of its circumference multiplied by its height. c. Show that the rate of change of the volume of the cylinder with respect to its height is equal to the area of the circular base.

Find the indicated higher-order partial derivatives.

Given $f(x,y,z)={e}^{-2x}\sin ({z}^{2}y),$ show that ${f}_{xyy}={f}_{yxy}.$

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.

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 $\frac{\partial z}{\partial t}={c}^{2}\left( \frac{{\partial }^{2}z}{\partial {x}^{2}} \right)$

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.

For the following exercises, complete each task.

Show that $f(x,y)={e}^{xy}x$ is differentiable at point $(1,0).$

Show that $f(x,y)={x}^{2}+3y$ is differentiable at every point. In other words, show that $\Delta z=f(x+\Delta x,y+\Delta y)-f(x,y)={f}_{x}\Delta x+{f}_{y}\Delta y+{\varepsilon }_{1}\Delta x+{\varepsilon }_{2}\Delta y,$ where both ${\varepsilon }_{1}$ and ${\varepsilon }_{2}$ approach zero as $(\Delta x,\Delta y)$ approaches $(0,0).$

For the following exercises, use this information: A function $f(x,y)$ is said to be homogeneous of degree $n$ if $f(tx,ty)={t}^{n}f(x,y).$ For all homogeneous functions of degree $n,$ the following equation is true: $x\frac{\partial f}{\partial x}+y\frac{\partial f}{\partial y}=nf(x,y).$ Show that the given function is homogeneous and verify that $x\frac{\partial f}{\partial x}+y\frac{\partial f}{\partial y}=nf(x,y).$

$f(x,y)=3{x}^{2}+{y}^{2}$

For the following exercises, use this information: A function $f(x,y)$ is said to be homogeneous of degree $n$ if $f(tx,ty)={t}^{n}f(x,y).$ For all homogeneous functions of degree $n,$ the following equation is true: $x\frac{\partial f}{\partial x}+y\frac{\partial f}{\partial y}=nf(x,y).$ Show that the given function is homogeneous and verify that $x\frac{\partial f}{\partial x}+y\frac{\partial f}{\partial y}=nf(x,y).$

$f(x,y)=\sqrt{{x}^{2}+{y}^{2}}$

For the following exercises, use this information: A function $f(x,y)$ is said to be homogeneous of degree $n$ if $f(tx,ty)={t}^{n}f(x,y).$ For all homogeneous functions of degree $n,$ the following equation is true: $x\frac{\partial f}{\partial x}+y\frac{\partial f}{\partial y}=nf(x,y).$ Show that the given function is homogeneous and verify that $x\frac{\partial f}{\partial x}+y\frac{\partial f}{\partial y}=nf(x,y).$

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

The temperature $T$ in a metal sphere is inversely proportional to the distance from the center of the sphere (the origin: $(0,0,0)).$ The temperature at point $(1,2,2)$ is $120^\circ \text{C}.$

a. Find the rate of change of the temperature at point $(1,2,2)$ in the direction toward point $(2,1,3).$ b. Show that, at any point in the sphere, the direction of greatest increase in temperature is given by a vector that points toward the origin.

If the electric potential at a point $(x,y)$ in the xy-plane is $V(x,y)={e}^{-2x}\cos (2y),$ then the electric intensity vector at $(x,y)$ is $E=-\nabla V(x,y).$

a. Find the electric intensity vector at $\left(\frac{\pi }{4},0\right).$ b. Show that, at each point in the plane, the electric potential decreases most rapidly in the direction of the vector $E.$

Show that, of all the triangles inscribed in a circle of radius $R$ (see diagram), the equilateral triangle has the largest perimeter.

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.

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