suppose that $f(\sigma(t))$ is an increasing function of $t$. What can you say about the angle between the gradient $\nabla f$ and the velocity vector $\sigma^{\prime}$ ?

Suppose that $x$ and $y$ are related by the equations given in Exercise. Find $d y / d x$ at the indicated points. $x^4+y^4=17, x=-1, y=2$.

Verify the chain rule for the functions and curves. $f(x, y, z)=x y+y z+x z ; \sigma(t)=(t, t, t)$.

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

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

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 $D_2 / D_1$.

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

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

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?]

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

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

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