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+

∫ 30

∫ 3xf(x, y) dy dx

(a)

∫ 30

∫ y0f(x, y) dx dy

(b)

∫ 30

∫ 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 +

∫ 3x

∫ 30f(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)

∫ 10

∫ 10

∫ 2−x0

dzdydx

(b)

∫ 10

∫ 20

∫ 2−z0

dzdxdy

(c)

∫ 20

∫ 10

∫ x−20

dzdydx

(d)

∫ 20

∫ 10

∫ 2−z0

dxdydz

(e)

∫ 20

∫ 10

∫ 20

dzdydx

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 − xy2

increasing 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)

∫ 2π0

∫ 10

∫ 4−r23r2

r dz dr dθ

(b)

∫ 2π0

∫ π0

∫ 4−r20

3r2 dz dr dθ

(c)

∫ 2π0

∫ 10

∫ 4−r23r2

r2 sin θ dr dz dθ

(d)

∫ 2π0

∫ 30

∫ 4−r23r2

r dz dr dθ

(e)

∫ 2π0

∫ 40

∫ 30r 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)

∫ 2π0

∫ π/40

∫ 21

1 dρ dφ dθ

(b)

∫ 2π0

∫ π/40

∫ 21ρ2 sinφdρ dφ dθ

(c)

∫ 2π0

∫ π/30

∫ 2secφ

1 dρ dφ dθ

(d)

∫ 2π0

∫ π/30

∫ 2secφ

ρ2 sinφdρ dφ dθ

(e)

∫ 2π0

∫ π/40

∫ 2secφ

ρ2 sinφdρ dφ dθ