Practice Problems for Midterm Exam 2, Math 23A, Winter 2012 ...

(a) What are the (x, y)-coordinates of the highest point.(b) Find a vector ⃗u pointing from (2, 3) to the point found above.(c) What x, y direction should you set off in from (2, 3) to go up the fastest?(d) How do you reconcile your answers to the last two questions?8. Find and classify all critical points forf(x, y) =3xy − x 2 y − xy 29. Compute and explain what the following represent geometrically for the function fgiven by f(x, y) =x 2 + y 2 .∂f ∂f(1, 2),∂x ∂y (1, 2),D ⃗uf(1, 2), ∂2 f(∂x (1, 2), ∇f(1, 2), |∇f(1, 2)|, 2− ∂f∂x)(1, 2), −∂f∂y (1, 2), 1 .10. Find Df, the Hessian Hf and the second order Taylor formula for f(x, y) = (x − y) 2at (0, 0) and g(x, y) = sin x at ( π y 2 , 1).11. Find the maximum and minimum value of the function f(x, y) =x 2 + y 2 − x − y +1in the disk D defined by x 2 + y 2 ≤ 2. Why do such extreme values exist?12. Captain Ralph is out for a flight again in his space ship, traveling at a constant speedof e 6 meters per second. The temperature of the ship’s hull when he is at location(x, y, z) is given byT (x, y, z) =e −x2 −y 2 −z 2where x, y and z are measured in meters. He is currently at (1, 2, 1). Describe the setof possible directions in which he may proceed to bring the ship’s hull temperaturedown at exactly a rate of 3 √ 2 degrees per second.13. Consider the surface given byx 3 y 2 z =1(a) Find the equation of the tangent plane to the surface at the point P = (1, 1, 1).(b) Explain why there is no point on this surface where the tangent plane is parallelto the x, y-plane.14. Find and classify all critical points for the functionG(x, y) = 1 4 x4 − 5 3 x3 + y 3 +3x 2 − 3 2 y2 + 202