math 200 - final exam - 6/11/2015...math 200 final exam - page 3 of 13 6/11/2015 2.(10 points) let...
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Math 200 - Final Exam - 6/11/2015
Name: Section:
Section Instructor Section Instructor1 9:00%AM ( 9:50%AM Papadopoulos,%Dimitrios 11 1:00%PM ( 1:50%PM Swartz,%Kenneth2 11:00%AM ( 11:50%AM Papadopoulos,%Dimitrios 13 12:00%PM ( 12:50%PM Yang,%Dennis3 11:00%AM ( 11:50%AM Lee,%Hwan%Yong 14 2:00%PM ( 2:50%PM Aran,%Jason4 4:00%PM ( 4:50%PM Aran,%Jason 16 12:00%PM ( 12:50%PM Zhang,%Aijun6 9:00%AM ( 9:50%AM Lee,%Hwan%Yong 17 4:00%PM ( 4:50%PM Yang,%Dennis7 10:00%AM ( 10:50%AM Papadopoulos,%Dimitrios 18 1:00%PM ( 1:50%PM Yang,%Dennis8 5:00%PM ( 5:50%PM Aran,%Jason 19 10:00%AM ( 10:50%AM Lee,%Hwan%Yong10 2:00%PM ( 2:50%PM Swartz,%Kenneth 20 1:00%PM ( 1:50%PM Akin,%Myles
Class/Times Class/Times
The following rules apply:
• This is a closed-book exam. You may not useany books or notes on this exam.
• For free response questions, you must showall work. Answers without proper justificationwill not receive full credit. Partial credit will beawarded for significant progress towards the cor-rect answer. Cross off any work that you do notwant graded.
• For multiple choice questions, circle the let-ter of the best answer. Make sure your cir-cles include just one letter. These problems will bemarked as correct or incorrect; partial credit willnot be awarded for problems in this section.
• You have 2 hours to complete this exam.When time is called, stop writing immediately andturn in your exam to the nearest proctor.
• You may not use any electronic devices in-cluding (but not limited to) calculators, cell phone,or iPods. Using such a device will be considereda violation of the university’s academic integritypolicy and, at the very least, will result in a gradeof 0 for this exam.
Page Points Score
2 10
3 10
4 15
6 10
7 10
8 10
9 5
10 10
11 10
12 5
13 5
Total: 100
Math 200 Final Exam - Page 2 of 13 6/11/2015
Part I: Free Response
1. (10 points) Find an equation of the plane which contains the point P (2, 1,−2) and isperpendicular to the planes 3x+ y − z = 6 and 2x+ 2y + z = 4
Math 200 Final Exam - Page 3 of 13 6/11/2015
2. (10 points) Let R be the closed region bounded by y = 9− x2 and the x-axis, shown below.
Calculate the absolute maximum and absolute minimum values of f(x, y) = xy+ 3x on R andindicate where they occur.
Math 200 Final Exam - Page 4 of 13 6/11/2015
3. (15 points) Consider the transformation
u = y + x
v = y − x
⇐⇒
x =1
2(u− v)
y =1
2(u+ v)
(a) Suppose R is the region in the xy-plane enclosed by y = x, y = x + 2, y = 1 − x and
y = −x, as shown below.
Use the given transformation to find the corresponding region in the uv-plane. Sketchyour results on the axes provided.
Math 200 Final Exam - Page 5 of 13 6/11/2015
(b) Find the absolute value of the determinant of the Jacobian
∣∣∣∣∂(x, y)
∂(u, v)
∣∣∣∣
(c) Use the transformation to evaluate
∫∫R
2(y − x)(y + x) dA.
Math 200 Final Exam - Page 6 of 13 6/11/2015
4. (10 points) Evaluate the following integral by converting to Polar Coordinates. For fullcredit, you must sketch the region over which your are integrating.
∫ √30
∫ √3−x20
y2 dy dx
Hint: cos2 θ =1
2(1 + cos 2θ) and sin2 θ =
1
2(1− cos 2θ)
Math 200 Final Exam - Page 7 of 13 6/11/2015
Part II: Multiple Choice
5. (5 points) What is the area of the parallelogram which has −→u = 〈2, 0, 1〉 and −→v = 〈4, 1, 2〉 asadjacent sides?
(a) 2√
3
(b) 2√
5
(c) 3√
2
(d) 4√
2
(e)√
5
6. (5 points) If−→b and −→v are both nonzero vectors, which of the following vectors is always
orthogonal to −→v − proj−→b−→v ?
(a)−→b
(b) −→v
(c)−→b −−→v
(d)−→b +−→v
(e) proj−→v−→b
Math 200 Final Exam - Page 8 of 13 6/11/2015
7. (5 points) The position of a particle moving through space is given by−→r (t) = 〈cos(2t), sin(2t), e−t〉.Find the velocity of the particle at time t = 0.
(a) 〈2, 0,−1〉
(b) 〈0, 2,−1〉
(c) 〈1, 2, 0〉
(d) 〈2, 2,−1〉
(e) 〈2,−2,−1〉
8. (5 points) Suppose f(x, y) is an integrable function. Which of the following integrals results
from reversing the order of integration of
∫ 0
−3
∫ 3
−xf(x, y) dy dx+
∫ 3
0
∫ 3
xf(x, y) dy dx
(a)
∫ 3
0
∫ y
0f(x, y) dx dy
(b)
∫ 3
0
∫ y
−yf(x, y) dx dy
(c)
∫ 3
−3
∫ y
−yf(x, y) dx dy
(d)
∫ 3
−3
∫ y
−3f(x, y) dx dy
(e)
∫ 3
−x
∫ 0
−3f(x, y) dx dy +
∫ 3
x
∫ 3
0f(x, y) dx dy
Math 200 Final Exam - Page 9 of 13 6/11/2015
9. (5 points) Consider the region in the first octant bounded by y = 1, x + z = 2, and thecoordinate planes, shown below.
Which of the following integrals represents the volume of this region?
(a)
∫ 1
0
∫ 1
0
∫ 2−x
0dzdydx
(b)
∫ 1
0
∫ 2
0
∫ 2−z
0dzdxdy
(c)
∫ 2
0
∫ 1
0
∫ x−2
0dzdydx
(d)
∫ 2
0
∫ 1
0
∫ 2−z
0dxdydz
(e)
∫ 2
0
∫ 1
0
∫ 2
0dzdydx
Math 200 Final Exam - Page 10 of 13 6/11/2015
10. (5 points) A differentiable function f(x, y) has two critical points: (0, 2) and (0,−2). And, itsatisfies the following:
fxx(x, y) = 2 fxy(x, y) = 0 fyx(x, y) = 0 fyy(x, y) = 6y
Which of the following is correct?
(a) f has a local minimum at (0,−2).
(b) f has a local maximum at (0,−2).
(c) f has a local minimum at (0, 2).
(d) f has a local maximum at (0, 2).
(e) f has saddle points at (0,−2) and (0, 2).
11. (5 points) In which direction is f(x, y) = x2y − x
y2increasing most rapidly at P (−2, 1)?
(a) 〈−5, 0〉
(b) 〈0, 0〉
(c) 〈2, 2〉
(d) 〈3, 1〉
(e) 〈3, 2〉
Math 200 Final Exam - Page 11 of 13 6/11/2015
12. (5 points) What is the distance from the point P (1, 4, 4) to the plane x+ 2y + z = 7 ?
(a) 1
(b)√
2
(c)√
6
(d) 2√
3
(e) 2√
5
13. (5 points) Consider the following definition.
Definition: The average value of a function z = f(x, y) over a region R in the xy-plane is:
favg(x, y) =
∫∫R
f(x, y) dA∫∫R
1 dA
Determine the average value of f(x, y) = 2xy on R = {(x, y) | 0 ≤ x ≤ 3 and 0 ≤ y ≤ 2}.
(a) 3
(b) 6
(c) 9
(d) 12
(e) 18
Math 200 Final Exam - Page 12 of 13 6/11/2015
14. (5 points) Consider the solid enclosed by z = 3x2 + 3y2 and z = 4− x2 − y2, shown below.
Which of the following represents the volume of this solid using cylindrical coordinates?
(a)
∫ 2π
0
∫ 1
0
∫ 4−r2
3r2r dz dr dθ
(b)
∫ 2π
0
∫ π
0
∫ 4−r2
03r2 dz dr dθ
(c)
∫ 2π
0
∫ 1
0
∫ 4−r2
3r2r2 sin θ dr dz dθ
(d)
∫ 2π
0
∫ 3
0
∫ 4−r2
3r2r dz dr dθ
(e)
∫ 2π
0
∫ 4
0
∫ 3
0r sin θ dr dz dθ
Math 200 Final Exam - Page 13 of 13 6/11/2015
15. (5 points) Consider the solid which is bounded above by x2 + y2 + z2 = 4 and below by z = 1,shown below.
Which of the following represents the volume of this solid using spherical coordinates?
(a)
∫ 2π
0
∫ π/4
0
∫ 2
11 dρ dφ dθ
(b)
∫ 2π
0
∫ π/4
0
∫ 2
1ρ2 sinφdρ dφ dθ
(c)
∫ 2π
0
∫ π/3
0
∫ 2
secφ1 dρ dφ dθ
(d)
∫ 2π
0
∫ π/3
0
∫ 2
secφρ2 sinφdρ dφ dθ
(e)
∫ 2π
0
∫ π/4
0
∫ 2
secφρ2 sinφdρ dφ dθ