91Ó°ÊÓ

Finding a direction vector for a tangent line: Find a direction vector for the line tangent to the curve \begin{equation}y=x^{3}\end{equation} when $$x = 2$$

Sketch the level curves f(x, y) = c of the following functions for c = −3, −2, −1, 0, 1, 2, and 3:

$f(x,y)=−\frac{2y}{x}$

In Exercises, find the maximum and minimum of the function f subject to the given constraint. In each case explain why the maximum and minimum must both exist.

$f(x,y)=xy\text{when}{x}^2+y^2=16$

Evaluate the limits in Exercises 33–40 if they exist.

$\underset{(x,y,z)→(3,-4,\mathrm{π}/4)}{\lim }\frac{x^2{y}^{}}{\tan z}$

Prove that if you minimize the square of the distance from the origin to a point (x, y) subject to the constraint $g(x,y)=0$, you have minimized the distance from the origin to (x, y) subject to the same constraint.