Math 265
Professor Lieberman
October 11, 2010
PRACTICE SECOND IN-CLASS EXAM
DIRECTIONS: This practice is only intended to show the level of difficulty for Friday’s
exam. The topics on the actual exam may be different from those represented here.
1. Find the equation in cylindrical coordinates of the surface whose equation in spherical
coordinates is ρ = 2 cos ϕ.
2. If F (x, y) = ln(x2 + xy + y 2 ), find Fx (−1, 4) and Fy (−1, 4).
3. Use the total differential (I just called this the differential in class) to approximate
the change in z = tan−1 (xy) as (x, y) moves from P (−2, −0.5) to Q(−2.03, −0.51).
4. Find the maximum of f (x, y) = xy subject to the constraint g(x, y) = 4x2 +9y 2 −36 =
0.