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.