MATH 2401, PRACTICE FINAL
1. Reparametrize the curve r(t) = (3 sin t)i + (4t)j + (3 cos t)k with respect to arc length
measured from the point where t = 0 in the direction of increasing t. Find the curvature
at the point (0, 4π, −3).
2. Let f (x, y, x) = sin(xz 2 ) + cos(xy).
(a) Find ∇f (0, −1, 2).
(b) Find the directional derivative of f at (0, −1, 2) in the direction of the vector
(2i − 3j + 6k)/7.
(c) Find the equation of the plane tangent to the level surface sin(xz 2 ) + cos(xy) = 1 at
the point (0, −1, 2).
3. Test the function
2
4
f (x, y) = + + xy
x y
for relative maxima and minima.
4. Show, using Lagrange multipliers, that if x, y, z ≥ 0 and
x+y+z
=c
3
then the maximum value of
1
(xyz) 3
is also c. This constitutes a proof of the well known inequality
1
x+y+z
≥ (xyz) 3 .
3
5. Evaluate: (a)
Z
0
1
Z
1
y
p
1 + x2 dxdy
(b)
Z Z Z
x2 dxdydz,
V
where V is the solid that lies above the triangle with vertices (0, 0, 0), (0, 1, 0), (1, 1, 0), and
below the surface z = x + y 2 .
6. Use the change of variables x = 31 (u + v), y = 31 (v − 2u) to evaluate
Z Z
Ω
(3x + 4y) dxdy,
where Ω is the region bounded by the lines y = x, y = x − 2, y = −2x, y = 3 − 2x.
7. Express
Z Z Z
y 2 dxdydz
V
as an iterated integral in spherical coordinates if V = {(x, y, z) : x ≤ 0, y ≤ 0, z ≤
0, x2 + y 2 + z 2 ≤ 1}.
8. Evaluate the line integral
Z
yz dy + x dx + y dz,
C
where C is the line segment from (0, 1, 0) to (1, 2, 2).
9. Use Green’s Theorem to evaluate the integral
Z
F(r) · dr,
C
where F = (3x2 ey − xy 3 + 1)i + (x3 ey + sin y)j, and C is the counterclockwise oriented
boundary of the rectangle 1 ≤ x ≤ 2, 0 ≤ y ≤ 1.
10. Let f (x, y, z) = xy 3 z 2 + sin(xy), u(x, y) = (ex + y)i + (sin y + x)j, and v(x, y, x) =
yzi + (y 2 + xz)j + xyk. Compute div(∇f ) and ∇ × v. Is u conservative? If yes, find a
function g such that u = ∇g.
11. Find the total surface area of the region bounded by the surfaces x2 +y 2 = 1, x2 +y 2 =
z − 7, and z = 0.
2
12. Calculate the flux of the vector field v(x, y, x) = yez i + y 2 j + exy k out of the solid
bounded by the cylinder x2 + y 2 = 9 and the planes z = 0 and z = y − 3.