Suppose that $x=f(t)$ and $y=g(t)$ satisfy the relation. Relate $d x / d t$ and $d y / d t$. $\tan ^{-1}(x-y)=\pi / 4$

Find the equation for the tangent plane to each surface at the indicated point. $x y / z=1 ;(1,1,1)$.

A rectangular mirror with area $A$ square feet is to have trim along the edges. If the trim along the horizontal edges costs $p$ cents per foot and that for the vertical edges costs $q$ cents per foot, find the dimensions which will minimize the total cost.

Find the equation for the tangent plane to each surface at the indicated point. $x y z=1 ;(1,1,1)$.

Suppose that $x=f(t)$ and $y=g(t)$ satisfy the relation. Relate $d x / d t$ and $d y / d t$.

$$(x+y)^3+(x-y)^3=42 e^{-\pi}$$

Verify the chain rule for the functions and curves. $f(x, y, z)=x z+y z+x y ; \boldsymbol{\sigma}(t)=\left(e^t, \cos t, \sin t\right)$.

Use the maximum-minimum test for quadratic functions to decide whether $(0,0)$ is a maximum, minimum, or saddle point. $f(x, y)=y^2$.

Find the equation for the tangent plane to each surface at the indicated point. $y^2-x^2=3 ;(1,2,8)$.

Use the maximum-minimum test for quadratic functions to decide whether $(0,0)$ is a maximum, minimum, or saddle point. $f(x, y)=x^2+y^2-x y$.

Find $\nabla\left(1 / r^3\right)(r \neq 0)$.

Suppose that $x=f(t)$ and $y=g(t)$ satisfy the relation. Relate $d x / d t$ and $d y / d t$. $\cos (x-y)=\frac{1}{2}$

The temperature $T$ on the spherical surface $x^2+$ $y^2+z^2=1$ satisfies the equation $T(x, y, z)=$ $x z+y z$. Find all the hot spots.

Cascade Container Company produces a cardboard shipping crate at three different plants in amounts $x, y, z$, respectively, producing an annual revenue of $R(x, y, z)=8 x y z^2-200,000(x+$ $y+z)$. The company is to produce 100,000 units annually. How should production be handled to maximize the revenue?

Show that $\nabla\left(1 / r^2\right)=-2 \mathbf{r} / r^4(r \neq 0)$.