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 dw/dt by using the Chain Rule. Express your final answer in terms of t.

$$w=x^2 y-y^2 x ; x=\cos t, y=\sin t$$

Find the maximum of $f(x, y)=x y$ subject to the constraint $g(x, y)=4 x^2+9 y^2-36=0$.

Sketch the level curves of $f(x, y)=\left(x+y^2\right)$ for k=0,1,2,4.

In given problem, find the directional derivative of f at the point $\mathbf{p}$ in the direction of $\mathbf{a}$.

$$f(x, y)=y^2 \ln x ; \mathbf{p}=(1,4) ; \mathbf{a}=\mathbf{i}-\mathbf{j}$$

The temperature at $(x, y, z)$ of a solid sphere centered at the origin is given by

$$T(x, y, z)=\frac{200}{5+x^2+y^2+z^2}$$

(a) By inspection, decide where the solid sphere is hottest.

(b) Find a vector pointing in the direction of greatest increase of temperature at (1,-1,1).

(c) Does the vector of part (b) point toward the origin?

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

$$f(x, y)=\ln \left(1-x^2-y^2\right)$$

In given problem, sketch the level curve z=k for the indicated values of k.

$$z=\frac{x^2}{y}, k=-4,-1,0,1,4$$

In given problem, verify that

$$\frac{\partial^2 f}{\partial y \partial x}=\frac{\partial^2 f}{\partial x \partial y}$$

$$f(x, y)=3 e^{2 x} \cos y$$

If $f(u, v)=u / v, u=x^2-3 y+4 z$, and $v=x y z$, find $f_x, f_y$, and $f_z$ in terms of x,y, and z.

A rectangular metal tank with open top is to hold 256 cubic feet of liquid. What are the dimensions of the tank that require the least material to build?

Find the parametric equations of the line that is tangent to the curve of intersection of the surfaces

$$f(x, y, z)=9 x^2+4 y^2+4 z^2-41=0$$

and

$$g(x, y, z)=2 x^2-y^2+3 z^2-10=0$$

at the point $(1,2,2)$.

Hint: This line is perpendicular to $\nabla f(1,2,2)$ and $\nabla g(1,2,2)$.

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

The function $f(x, y)=\sin (x y)$ does not attain a maximum value on the set $\left\{(x, y): x^2+y^2<4\right\}$.

A boy's toy boat slips from his grasp at the edge of a straight river. The stream carries it along at 5 feet per second. A crosswind blows it toward the opposite bank at 4 feet per second. If the boy runs along the shore at 3 feet per second following his boat, how fast is the boat moving away from him when t=3 seconds?