Problem 1. In each of the following, determine whether F is the gradient of a function f . If it is, find f .
(a) F(x, y) = .3x2y2 + 3y + x. i + .2x3y + 3xy - √y. j.
(b) F(x, y) = .2xey + e2x - 3. i + .x2 ey + cos 2y + 1. j.
Problem 2.
(a) Determine the minimum value of f (x, y) = 2x2 + xy - y2 + 1 subject to the constraint 2x + 3y = 16.
(b) Determine the maximum value of f (x, y, z) = 3x - 2y + z subject to the constraint x2 + y2 + z2 = 14.
Problem 3. Estimate (√119)(√5 34) using differentials.
Problem 4. Use a double integral to find the volume of the solid S in the first octant that is bounded above by the surface z = 4 - x2 - y2, below by the x, y-plane, and on the sides by the planes y = 0 and y = x.
Problem 5.
(a) Evaluate
2yz dx dy dz where T is the solid in the first octant bounded above by the cylinder z = 4 - x2 below by the x, y-plane, and on the sides by the planes z = 0, x = 0, y = 2x, and y = 4.
(b) Set up a triple integral in cylindrical coordinates that gives the volume of the solid in the first octant
that is bounded above by the hemisphere z = ,2 - x2 - y2, below by the paraboloid z = x2 + y2 and on the sides by the x, z- and y, z-planes.
(c) 
Set up a triple integral in spherical coordinates that gives the volume of the solid that lies outside the cone z = ,x2 + y2 and inside the hemisphere z = ,1 - x2 - y2.
Problem 6. Let h(x, y, z) = xy i + y j - yx k, and let C be the curve given by
r(u) = u i + u2 j + 2u k, 0 ≤ u ≤ 1.
Calculate
h(r) · dr.C
Problem 7.
(a) Find the Jacobian of the transformation: x = u ln v, y = uv
(b) Take ? as the parallelogram bounded by x + y = 0, x + y = 1, x - y = 0, x - y = 2
and evaluate 2xy dx dy?
From pages 940-941 in text (Skills Mastery Review at the end of chapter 15): problems 53, 55, 57, 63, 68a, 69, 71
From pages 1094-1095 in text (Skills Mastery Review at the end of chapter 17): problems 1-11odd, 19, 24, 28 (just be able to set up problems 24 and 28)