Math 251 Review
1. Find the equations of the line which passes through the point (−2, 3, −1) and is
orthogonal to the plane 2x + y − 3z = 5.
2. Determine the equation of the plane which contains the point (1, −1, 2) and the
line x = −1 + 2t, y = 1 + t, z = 1 + 3t.
3. Suppose that z = f (x, y), where x = 5u+2v , y = 3u+v , and that f has continuous
second partial derivatives.
∂z
(a) Find
in terms of u, v , and the partial derivatives of f .
∂u
(b) Find
∂ 2z
in terms of u, v , and the partial derivatives of f .
∂u∂v
4. Find the equation of the plane tangent to the surface xy 2 z 3 = 2 at the point
(2, 1, 1).
5. Use dierentials to approximate the number
√
0.99 e0.02 .
6. Determine the maximum and minimum values of f (x, y) = x2 + 2y 2 − x on the disk
x2 + y 2 ≤ 1, giving the points where they occur.
2
2
7. Find the mass of the lamina that occupies the region inside the circle
p x + y = 6y
and outside the circle x2 + y 2 = 9. The density at (x, y) is ρ(x, y) = x2 + y 2 .
8. Set up (but don't evaluate) an iterated integral, in the order dzdydx, for
˝
E
z dV
where E is the region in space bounded below by z = x2 + y 2 and above by the plane
2x + 4y − z = −4.
9. Suppose that E is the region in space bounded below by the cone ˝
z = x2 + y 2
and above by the sphere x2 + y 2 + z 2 = 4. Write (but don't evaluate)
x2 dV in
p
(a) spherical coordinates.
(b) cylindrical coordinates.
E
10. Evaluate the line integral
and ending at (2, 2, 3).
´
y ds, where C is the line segment starting at (1, 1, 0)
C
11. Find the work done by the force F(x, y) = x sin y i + y j on a particle along the
curve y = x2 from (−2, 4) to (−1, 1).
12. Given F(x, y) = (1 + xy)exy i + x2 exy j
(a) Show that F is conservative.
(b) Find a function f such that ∇f = F.
(c) Evaluate
0 ≤ t ≤ π/2.
´
C
F · dr if C is the arc of the curve given by r(t) = cos t i + 2 sin t j,
13. Evaluate
˛
ex (1 − cos y) dx − ex (1 − sin y) dy
C
where C is the boundary of the domain D = { (x, y) | 0 ≤ x ≤ π, 0 ≤ y ≤ sin x }.
14. Write a parameterization for the part of the cylinder x2 + z 2 = 4 which lies between the planes y = −1 and x+2y+z = 8. Be sure to specify the parameter domain.
15. Compute the integral
h u, v, u2 i for u2 + v 2 ≤ 4.
˜
S
x dS , where S is the surface parametrized by r =
16. Find the ux of the vector eld F = x i + y j + z k across the surface z 2 = x2 + y 2 ,
0 ≤ z ≤ 2.
´
17. Evaluate C F·dr if F = h 2yz, −xy, x2 i and C is the intersection of z = 4−x2 −y 2
and z = x2 + y 2 .
18. Find the ux of F = h x, y, z i the across the surface of the region bounded by
x2 + z 2 = 1, y = 0 and x + y = 3.