MTH 234
Practice Exam 2
Name:
Section:
Recitation Instructor:
INSTRUCTIONS
• Fill in your name, etc. on this first page.
• Without fully opening the exam, check that you have pages 1 through 12.
• Show all your work on the standard response questions. Write your answers
clearly! Include enough steps for the grader to be able to follow your work. Don’t
skip limits or equal signs, etc. Include words to clarify your reasoning.
• Do first all of the problems you know how to do immediately. Do not spend too
much time on any particular problem. Return to difficult problems later.
• If you have any questions please raise your hand and a proctor will come to you.
• You will be given exactly 90 minutes for this exam.
• Remove and utilize the formula sheet provided to you at the end of this exam.
• This is a practice exam. The actual exam may differ significantly from this practice
exam because there are many varieties of problems that can test each concept.
ACADEMIC HONESTY
• Do not open the exam booklet until you are instructed to do so.
• Do not seek or obtain any kind of help from anyone to answer questions on this
exam. If you have questions, consult only the proctor(s).
• Books, notes, calculators, phones, or any other electronic devices are not allowed on
the exam. Students should store them in their backpacks.
• No scratch paper is permitted. If you need more room use the back of a page.
• Anyone who violates these instructions will have committed an act of academic
dishonesty. Penalties for academic dishonesty can be very severe. All cases of
academic dishonesty will be reported immediately to the Dean of Undergraduate
Studies and added to the student’s academic record.
I have read and understand the
above instructions and statements
regarding academic honesty:
.
SIGNATURE
Page 1 of 12
MTH 234
Practice Exam 2
Standard Response Questions. Show all work to receive credit. Please BOX your final answer.
2
2
2
1. (18 points)
r Find the volume of the sphere x + y + z ≤ 64 that lies between the cones z =
x2 + y 2
. Hint: Use spherical coordinates.
and z =
3
p
x2 + y 2
Page 2 of 12
MTH 234
Practice Exam 2
2. Evaluate the following integrals
Z 27 Z 3
4
ey dy dx.
(a) (9 points)
√
3
0
(b) (9 points)
Z
0
2
x
Z √8−y2
(x2 + y 2 ) dx dy
y
Page 3 of 12
MTH 234
Practice Exam 2
3. (18 points) Find the absolute maximum and minimum values of f (x, y) = x3 + 2y 2 − 26 on the set D
where D is the closed region bounded by y = 0 and y = 16 − x2
Page 4 of 12
MTH 234
D
4. Let F(x, y) = 2xy +
Practice Exam 2
5
y2
− 3y, x2 −
10x
y3
E
be a vector field.
(a) (6 points) Show that F is conservative..
(b) (6 points) Find a potential function for F, that is, a function f such that ∇f = F.
(c) (6 points) Evaluate the line integral
Z
C
F · dr, where C is the curve parametrized by
r(t) = ht sin(t), πet cos(t)i for 0 ≤ t ≤ π.
Page 5 of 12
MTH 234
Practice Exam 2
5. (18 points) Find the area of the part of the paraboloid x = y 2 + z 2
that lies inside the cylinder y 2 + z 2 = 49.
Page 6 of 12
MTH 234
Practice Exam 2
Multiple Choice. Circle the best answer. No work needed.
No partial credit available. No credit will be given for choices not clearly marked.
6. (7 points) Find the curl of the vector field F(x, y, z) = hx + yz, y + xz, z + xyi.
A. h0, 1, 0i
B. h0, 0, 0i
C. h1, 1, 1i
D. h1, 1, −1i
E. h0, 0, 1i
7. (7 points) Find the divergence of F(x, y, z) = (3x + yz) i + (−y − xz) j + (4z + 6xy) k
A. 3
B. 6
C. 12
D. 0
E. None of the above
8. (7 points) In cylindrical coordinates, z = r means
A. Cone
B. Elliptical Paraboloid
C. Sphere
D. Hyperbolic Paraboloid
E. Ellipsoid
Page 7 of 12
MTH 234
Practice Exam 2
9. (7 points) In spherical coordinates, what does r = 3 represent?
A. Sphere
B. Cone
C. Line
D. Plane
E. Hyperbolic paraboloid
10. (7 points) Convert
A.
Z
4
B.
√
2π
0
4−x2
√
− 4−x2
√
2 Z
2−x2
−4
Z
Z
Z
√
− 2−x2
√
2 Z
4−x2
Z
2
r dr dθ from polar to Cartesian coordinates:
0
dy dx
dy dx
−2
C.
Z
√
− 4−x2
√
2 Z
4−x
dy dx
−2
D.
Z
−2
√
− 4−x
dy dx
E. None of the above
11. (7 points) Find the gradient of f (x, y, z) = −6xy 2 z 3 − 3xyz
A. h−6y 2 z 3 − 6yz, −12xyz 3 − 6xz, −18xy 2 z 2 − 6xyi
B. h−3y 2 z 3 − 3yz, −6xyz 3 − 3xz, −9xy 2 z 2 − 3xyi
C. h−6y 2 z 3 − 3yz, −12xyz 3 − 3xz, −18xy 2 z 2 − 3xyi
D. h−12xyz 3 − 3yz, −6y 2 z 3 − 3xz, −18xy 2 z 2 − 3xyi
E. None of the above
Page 8 of 12
MTH 234
Practice Exam 2
12. (7 points) Which of the following functions has a constant gradient vector field?
A. f (x, y) = 2x2 + y
B. f (x, y) = y 2
C. f (x, y) = x − y
D. f (x, y) = sin(xy)
E. None of the above
13. (7 points) Evaluate
Z
0
A. π(1)2
1
Z
0
1
Z
1
dy dz dx.
0
B. 1
C. xyz
D. 0
E. None of the above
14. (7 points) Evaluate the line integral
I
3 cos(2y) dx+3x2 sin(2y) dy where C is the square with vertices
C
(−1, −1), (−1, 1), (1, −1) and (1, 1). Hint: Use Green’s theorem.
A.
− cos(2)+1
(6)
2
B.
− cos(2)+1
(12)
2
C. 0
D. None of the above
Page 9 of 12
MTH 234
Practice Exam 2
Congratulations you are now done with the exam!
Go back and check your solutions for accuracy and clarity. Make sure your final answers are BOXED .
When you are completely happy with your work please bring your exam to the front to be handed in.
Please have your MSU student ID ready so that is can be checked.
DO NOT WRITE BELOW THIS LINE.
Page
Points Score
2
18
3
18
4
18
5
18
6
18
7
21
8
21
9
21
Total:
153
No more than 150 points may be earned on the exam.
Page 10 of 12
MTH 234
Practice Exam 2
FORMULA SHEET PAGE 1
Vectors in Space
Curves and Planes in Space
Suppose u = hu1 , u2 , u3 i and v = hv1 , v2 , v3 i:
• Unit Vectors:
a(x − x0 ) + b(y − y0 ) + c(z − z0 ) = 0
• Length of vector u
p
|u| = u1 2 + u2 2 + u3 2
• Dot Product:
u · v = u1 v1 + u2 v2 + u3 v3
= |u||v| cos θ
• Cross Product:
• Arc Length of curve r(t) for t ∈ [a, b].
Z b
|r0 (t)| dt
L=
a
• Unit Tangent Vector of curve r(t)
T(t) =
r0 (t)
|r0 (t)|
More on Surfaces
• Directional Derivative: Du f (x, y) = ∇f · u
i j k
u × v = u1 u2 u3
v1 v2 v3
proju v =
r(t) = r0 + tv
• Plane normal to n = ha, b, ci:
i = h1, 0, 0i
j = h0, 1, 0i
k = h0, 0, 1i
• Vector Projection:
• Line parallel to v:
• Second Derivative Test Suppose
fx (a, b) = 0 and fy (a, b) = 0. Let
u·v
u
|u|2
Partial Derivatives
• Chain Rule: Suppose z = f (x, y) and
x = g(t) and y = h(t) are all differentiable
then
∂f dx ∂f dy
dz
=
+
dt
∂x dt
∂y dt
D = fxx (a, b)fyy (a, b) − [fxy (a, b)]2
(a) If D > 0 and fxx (a, b) > 0, then f (a, b)
is a local minimum.
(b) If D > 0 and fxx (a, b) < 0, then f (a, b)
is a local maximum.
(c) If D < 0 then f (a, b) is a saddle point.
Trigonometry
• sin2 x = 12 (1 − cos 2x)
• cos2 x = 12 (1 + cos 2x)
• sin(2x) = 2 sin x cos x
Page 11 of 12
MTH 234
Practice Exam 2
FORMULA SHEET PAGE 2
Multiple Integrals
ZZ
1 dA
• Area: A(D) =
D
• Volume: V (E) =
ZZZ
1 dV
E
Polar/Cylindrical
Additional Definitions
• curl(F) = ∇ × F
• div(F) = ∇ · F
• F is conservative if curl(F) = 0
Line Integrals
• Transformations
r 2 = x2 + y 2
x = r cos θ
y = r sin θ
y/x = tan θ
ZZ
ZZ
•
f (x, y) dA =
f (r cos θ, r sin θ) r dr dθ
•
D
ZZZ
E
D
f (x, y, z) dV =
ZZZ
f (r cos θ, r sin θ, z) r dz dr dθ
E
• Fundamental Theorem of Line Integrals
Z
∇f · dr = f (r(b)) − f (r(a))
C
• Green’s Theorem
Z
ZZ
P dx + Q dy =
(Qx − Py ) dA
C
D
Spherical
• Transformations
x = ρ sin φ cos θ
y = ρ sin φ sin θ
z = ρ cos φ
ρ 2 = x2 + y 2 + z 2
•
ZZ
Z ZE
f (x, y, z) dV =
f (ρ sin φ cos θ, ρ sin φ sin θ, ρ cos φ)(ρ2 sin φ) dρ dφ dθ
E
Page 12 of 12