Problem 1. A picture of the level set is shown below.
Find the equation of the plane tangent to this surface at the point .
Problem 2. Find all points on the unit sphere whose tangent planes are parallel to the plane .
Problem 3. (a) Compute the directional derivative of at the point in the direction of .
(b) For the same function as in part (a), find the direction in which increases the most at the point . Compute this rate of change.
Problem 4. (a) Find all of the critical points for .
(b) Compute the Hessian matrix for and use it to classify the critical points from part (a) as local minima, local maxima, saddle points, or as “unclassifiable.”
(c ) Find where the function achieves its absolute minimum and absolute minimum on the compact set