A firm uses wool and cotton fiber to produce cloth. The amount of cloth produced is given by $Q(x, y)=x y-x-y+1$, where $x$ is the number of pounds of wool, $y$ is the number of pounds of cotton, and $x>1$ and $y>1$. If wool costs $p$ dollars per pound, cotton costs $q$ dollars per pound, and the firm can spend $B$ dollars on material, what should the mix of cotton and wool be to produce the most cloth?

Find the equation for the tangent line to each curve at the indicated point. $x^2+2 y^2=3 ;(1,1)$.

Suppose that $x$ and $y$ are related by the equations given in Exercise. Find $d y / d x$ at the indicated points. $x+\cos y=1, x=1, y=\pi / 2$.

Find the critical points of each of the functions and classify them as local maxima, minima, or neither. $f(x, y)=x^2+y^2+6 x-4 y+13$.

Verify the chain rule for the functions and curves. $f(x, y, z)=\sqrt{x^2+y^2+z^2} ; \sigma(t)=(\sin t, \cos t, t)$.

A water main consists of two sections of pipe of fixed lengths, $l_1, l_2$ carrying fixed amounts $Q_1$ and $Q_2$ liters per second. For a given total loss of head $h$, the (variable) diameters $D_1, D_2$ of the pipe will result in a minimum cost if

$$C=l_1\left(a+b D_1\right)+l_2\left(a+b D_2\right)=\text { minimum }$$

subject to the condition

$$h=\frac{c l_1 Q_1^2}{D_1^5}+\frac{c l_2 Q_2^2}{D_2^5} .$$

Find the ratio $D_2 / D_1$.

Sketch a graph of each situation. Are the graphs the same? Explain. a. your speed as you travel on a ski lift from the bottom of a ska slope to the top b. your speed as you ski from the top of a ski slope to the bottom

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

Use the maximum-minimum test for quadratic functions to decide whether $(0,0)$ is a maximum, minimum, or saddle point. $f(x, y)=3+2 x^2-x y+y^2$.

The Baraboo, Wisconsin, plant of International Widget Co. uses aluminum, iron, and magnesium to produce high-quality widgets. The quantity of widgets which may be produced using $x$ tons of aluminum, $y$ tons of iron, and $z$ tons of magnesium is $Q(x, y, z)=x y z$. The cost of raw materials is aluminum, $\$ 6$ per ton; iron, $\$ 4$ per ton; and magnesium, $8 per ton. How many tons each of aluminum, iron, and magnesium should be used to manufacture 1000 widgets at the lowest possible cost? [Hint: You want an extreme value for what function? Subject to what constraint?]

Verify the chain rule for the functions and curves. $f(x, y, z)=e^{x y z} ; \sigma(t)=\left(6 t, 3 t^2, t^3\right)$.

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. $x y z=1 ;(1,1,1)$.