Sand is pouring onto a conical pile in such a way that at a certain instant the height is 100 inches and increasing at 3 inches per minute and the base radius is 40 inches and increasing at 2 inches per minute. How fast is the volume increasing at that instant?

In what direction u does $f(x, y)=\sin (3 x-y)$ decrease most rapidly at $\mathbf{p}=(\pi / 6, \pi / 4)$?

Convert $(101001000)_2$ from binary to decimal notation and $(1984)_{10}$ from decimal to binary notation.

Respond with true or false to the following assertion. Be prepared to defend your answer.

If $f_x\left(x_0, y_0\right)$ and $f_y\left(x_0, y_0\right)$ both exist, then $f$ is differentiable at $\left(x_0, y_0\right)$.

The temperature at (x, y, z) of a solid sphere centered at the origin is $T(x, y, z)=100 e^{-\left(x^2+y^2+z^2\right)}$. Note that it is hottest at the origin. Show that the direction of greatest decrease in temperature is always a vector pointing away from the origin.

In given problem, describe the largest set S on which it is correct to say that f is continuous.

$$f(x, y)=\frac{1}{\sqrt{1+x+y}}$$

A rectangular box, whose edges are parallel to the coordinate axes, is inscribed in the ellipsoid $96 x^2+4 y^2+4 z^2=36$. What is the greatest possible volume for such a box?

Find the parametric equations of the line that is tangent to the curve of intersection of the surfaces $x=z^2$ and $y=z^3$ at (1,1,1).

Find all points (x, y) at which the tangent plane to the graph of $z=x^3$ is horizontal.

Let $f(x, y)=y / x+x y$. Find each value.

(a) f(1,2)

(b) $f\left(\frac{1}{4}, 4\right)$

(c) $f\left(4, \frac{1}{4}\right)$

(d) f(a, a)

(e) $f\left(1 / x, x^2\right)$

(f) f(0,0)

What is the natural domain for this function?

In given problem, find all critical points. Indicate whether each such point gives a local maximum or a local minimum, or whether it is a saddle point. Hint: Use Theorem C.

$$f(x, y)=x^2+4 y^2-2 x+8 y-1$$

In given problem, find the equation of the tangent plane to the given surface at the indicated point.

$$8 x^2+y^2+8 z^2=16 ;(1,2, \sqrt{2} / 2)$$

A bee was flying upward along the curve that is the intersection of $z=x^4+x y^3+12$ with the plane x=1. At the point (1,-2,5), it went off on the tangent line. Where did the bee hit the xz-plane?

In given problem, find all first partial derivatives of each function.

$$f(x, y)=\left(4 x-y^2\right)^{3 / 2}$$