# Vector HW 8

Problem 1.  A picture of the level set $\displaystyle S = \left\{(x,y,z) : z^3+xy^2 - x^2y - 1=0 \right\}$ is shown below.

Find the equation of the plane tangent to this surface at the point $(0, 0, 1)$.

Problem 2.  Find all points on the unit sphere $S = \{(x,y,z) : x^2+y^2+z^2=1\}$ whose tangent planes are parallel to the plane $x -y -z = 5$.

Problem 3.  (a) Compute the directional derivative of $f(x,y) = \arctan(xy+x^2)$ at the point $(1, 0)$ in the direction of $\vec{u} = \langle 0, 1 \rangle$.

(b) For the same function $f(x,y)$ as in part (a), find the direction in which $f(x,y)$ increases the most at the point $(1, 0)$.  Compute this rate of change.

Problem 4.  (a) Find all of the critical points for $f(x,y) = x^3 + xy^2$.

(b) Compute the Hessian matrix for $f(x,y)$ 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 $f(x,y)$ achieves its absolute minimum and absolute minimum on the compact set $\displaystyle D = \left\{(x,y) : x^2+y^2 \leq 1\right\}.$