Practice Problems: Final Exam
1. Let f (x, y) = x3 − 3x2 − 2y2 . Find the critical points of f , and classify each critical point as a
local max, a local min, or neither.
2. Let S be the portion of the cone z =
p
x2 + y2 for which z ≤ 10, with upward-pointing normal
vectors.
(a) Find the equation of the tangent plane to S at the point (3, 4, 5).
ZZ
(b) Evaluate
z2 dS
S
ZZ
(c) Evaluate
zk · dS
S
3.
(a) Evaluate the integral
(b) Evaluate the integral
Z 1Z 1p
1 + y2 dy dx by changing the order of integration.
0
x
0
0
Z 2 Z √4−x2p
x2 + y2 dy dx by changing to polar coordinates.
4. Let f : R2 → R be a twice differentiable function, and suppose that f (x, y) has a critical point
at (0, 0), with
f (0, 0) = 3
and
H f (0, 0) =
"
#
0 1
1 0
.
(a) Does f have a local min at (0, 0), a local max at (0, 0), or neither? Explain.
(b) Find the second-order Taylor polynomial for the function f (x, y), centered at (0, 0).
(c) Use your answer to part (b) to draw an approximate contour plot for f (x, y) near (0, 0) on
the following axes. Include contours for 1, 2, 3, 4, and 5.
x
2
1
y
0
-1
-2
-2
-1
0
1
2
ZZZ
5. Evaluate
x dV , where P is the polyhedron shown in the figure below.
P
z
H0,0,3L
H0,2,3L
H2,0,1L
x
H2,0,0L
H0,2,0L
y
6. Let C be the portion of the helix x(t) = (cos t, sin t, t) from the point (1, 0, 0) to the point
(−1, 0, π).
(a) Find parametric equations for the tangent line to C at the point (0, 1, π/2).
Z
(b) Evaluate
xz ds.
C
Z
(c) Evaluate
xj · ds. (Hint: Use the identity cos 2θ = 2 cos2 θ − 1.)
C
7. Let f (x, y) = 2x2 + 2xy + 2y2 − 18x − 24y.
(a) Find the critical point of f .
(b) Compute the Hessian of f at the critical point.
(c) Is the Hessian positive definite, negative definite, or neither? Explain.
(d) Is the critical point a local max, a local min, or neither? Explain.
ZZ
8. Evaluate
x2 + y2
2
dA, where R is the shaded region shown in the figure below.
R
r=1+Θ
H-1 - Π, 0L
y
H1, 0L
x
ZZ
9. Evaluate
x2 dA, where S is the square region with vertices (0, 2), (2, 0), (4, 2), and (2, 4).
S
10. Use geometric reasoning to compute the following integrals:
ZZZ
dV , where R is the ball of radius 2 centered at the point (5, 3, 4).
(a)
R
ZZ
x dA, where T is the triangle with vertices (1, 0), (5, 0), and (3, 3).
(b)
T
Z
(c)
(xz − 4) ds, where L is the line segment with endpoints (2, 3, 7) and (2, 8, 7).
L
ZZ
6 ds, where S is the portion of the unit sphere for which x ≥ 0.
(d)
S
11. Evaluate the following integrals.
ZZZ
p
x2 +y2 +z2 dV , where R is the region defined by x2 +y2 +z2 ≤ 2 and z ≥ x2 + y2 .
(a)
R
ZZ
(b)
√
z2 + 2 dS, where S is the surface z = 2 r for 0 ≤ r ≤ 1.
S
ZZ
(c)
S
xj · dS, where S is the square with vertices (2, 1, 3), (3, 2, 4), (6, 4, 5), and (5, 3, 4), and
upward-pointing normals.