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 $\left(x_0, y_0\right)$.

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 the equation of the tangent plane to the given surface at the indicated point.

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

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

If $f_x(0,0)=f_y(0,0)$, then $f(x, y)$ is continuous at the origin.

In given problem, find the indicated limit or state that it does not exist.

$$\lim _{(x, y) \rightarrow(-2,1)}\left(x y^3-x y+3 y^2\right)$$

In given problem, find the gradient $\nabla f$.

$$f(x, y)=x^3 y-y^3$$

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 all first partial derivatives of each function.

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

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.

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