Use the precise definition of a limit to prove that $\lim_{(x,y)\to (a,b)}c=c$ where $c$ is a constant.

An open rectangular box having a volume of 108 $\mathrm{in}^3$. is to be constructed from cardboard. Find the dimensions of such a box if the amount of cardboard used in its construction is to be minimized.

If $u=f(x,y)$, where $x=e^r\cos \theta$ and $y=e^r\sin \theta$, show that

$$\frac{\partial^2u}{\partial x^2}+\frac{\partial^2u}{\partial y^2}=e^{-2r}\left[\frac{\partial^2 u}{\partial r^2}+\frac{\partial^2u}{\partial \theta^2}\right]$$

a. Let $x_1, x_2, \ldots ,x_n$ be positive numbers. Find the maximum value of

$$f(x_1, x_2, \ldots , x-n)=\sqrt[n]{x_1x_2\cdots x_n}$$

subject to the constraint $x_1+x_2+\cdots+x_n=c$, where c c is a constant.

Find $h(x,y)=g(f(x,y))$, and determine where $h$ is continuous.

$$f(x,y)=y\ln x,\ \hspace{6pt}g(t)=e^{t^{2}}$$

Determine whether the statement is true or false. If it is true, explain why it is true. If it is false, give an example to show why it is false.

If an equation of the tangent plane at the point $P_0(x_0, y_0, z_0)$ on the surface described by $F(x,y,z)=0$ is $ax+by+cz=d$, then $\nabla F(x_0, y_0, z_0)=k\left<a, b, c\right>$ for some scalar $k$.

Find the directional derivative of the function f at the point P in the indicated direction.

$$f(x,y,z)=x^2\ln y +xy^2e^z$$

Point P(2,1,0), in the direction of $\mathbf{v}=\left<3,-1,2\right>$.

Determine whether the statement is true or false. If it is true, explain why it is true. If it is false, give an example to show why it is false.

The function

$$f(x,y)=\left\{ \begin{array}{ll} x^2+y^2&\text{if }(x,y)\neq (0,0)\\ 1& \text{if }(x,y)=(0,0) \end{array}\right.$$

is differentiable everywhere.

Show that the mixed partial derivatives $f_{xy}$ and $f_{yx}$ are equal.

$$f(x,y)=x\sin ^2y +y^2\cos x$$

Use the fact that a vector in -space has the form $v=\left<v_1,v_2, \ldots , v_n\right>$ and the gradient of a function of n variables, $f(x_1,x_2, \ldots ,x-n)$, is defined by $\nabla f=\left<f_{x_1},f_{x_2},\ldots ,f_{f_{x_n}}\right>$. Also, assume that the Lagrange's theorem holds for the n-dimensional case.

a. Let p and q be positive numbers satisfying (1/p)+(1/q)=1. Find the minimum value of

$$f(x,y)=\frac{x^p}{p}+\frac{y^q}{q}\hspace{6pt}x>0,\ y>0$$

subject to the constraint xy=c, where c is a constant.

b. Use the result of part (a) to show that if x and y are positive numbers, then

$$\frac{x^p}{p}+\frac{y^q}{q}\geq xy$$

where p>0 and q>0 and (1/p)+(1/q)=1.

If $u=f(x,y)$, where $x=e^r\cos \theta$ and $y=e^r\sin \theta$, show that

$$\left(\frac{\partial u}{\partial x}\right)^2+\left(\frac{\partial u}{\partial y}\right)^2=e^{-2r}\left[\frac{\partial^2u}{\partial r^2}+\frac{\partial ^2 u}{\partial \theta ^2}\right]$$

Determine whether the statement is true or false. If it is true, explain why it is true. If it is false, give an example to show why it is false.

The line with equations

$$\frac{x-2}{4}=\frac{y+1}{6}=-\frac{z}{2}$$

is perpendicular to the plane with equation $2x+3y-z=4$.

Let $T(x,y,z)$ represent the temperature at a point $P(x,y,z)$ of a region $R$ in space. If the isotherms of $T$ are concentric spheres, show that the temperature gradient $\nabla T$ points either toward or away from the center of the spheres.

Hint: Recall that the isotherms of $T$ are the sets on which $T$ is constant.

Find $h(x,y)=g(f(x,y))$, and determine where $h$ is continuous.

$$f(x,y)=x\tan y,\ \hspace{6pt}g(t)=\cos t$$