2 Question 1. {4 points) Let ay) = g; Find lim :13. i' (rim+11.11)" _ J) or explain why this limit does not exist. Question 2. {4 points) 'lferitj,r Clairautis Theorem {fIy = f") for the function Has 1:) = 1329'2 608(32 + if) Question 3. (4 points) Let 5'1 be the graph of the equation z = f(:1:, y] = 3:3 +233y+ 23:. Find all points ((1,3)) such that the tangent plane to 31 at (:13, y] = {e1 3)) is parallel to the plane 3: 3y + z = f}. HINT: Compare the normals of the two planes; don't forget about the ,zcomponent.T Question 4. (2 + 2 points] The temperature in a room is given by T{3:,y,z) = 3'32 + y2 + 53731:; 33 with respect to a specic coordinate system. TWhen Max the Cat is on his perch at the point (3,211), he decides his spot is not warm enough: so he moves just a little bit in the direction (1, (l, 1). 1What is the rate of change in that direction? In what direction should he have moved to attain a maximal rate of change in temperature? Question 5. {4 points} Let 32 he the graph of the equation 3:3 2334 +zy'1 ya:2z = 0. Find an equation for the tangent plane to 32 at the point {212,2}. Question 6. (4 points) As a ball is released at the point [%.1_. %] on the surface 33 given by the equation :4 = I4 + y2 1.. it follows the path of steepest descent qr from its point of release. Find an equation for the projection of qr onto the jigplane (so an equation for the curve that you would see on a topographical map of 53). HINT: Do not forget to make sure that the projected curve actually passes through (1)