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\}.


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