If $F(x, y)=x^3-x y^2-y^4, x=2 \cos 3 t$, and $y=3 \sin t$, find dF/dt at t=0.

Consider the function f(x, y) = 4xy / x²+y², (x, y) ≠ (0, 0) {0, (x, y) = (0, 0) and the unit vector u = 1/√2 (i + j). Does the directional derivative of f at P(0, 0) in the direction of u exist? If f(0, 0) were defined as 2 instead of 0, would the directional derivative exist?

In given problem, verify that

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

$$f(x, y)=\tan ^{-1} x y$$

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

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.

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

$$z=x^2+y, k=-4,-1,0,1,4$$

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?

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?

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

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 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.

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