Math 200 - Final Exam - 6/13/2013

NAME:

SECTION:

Directions:

• For the free response section, you must show all work. Answers without proper justifi-cation will not receive full credit. Partial credit will be awarded for significant progresstowards the correct answer. Cross o↵ any work that you do not want graded.

• You have two (2) hours to complete this exam. When time is called, STOP WRITINGIMMEDIATELY.

• You may not use any electronic devices including (but not limited to) calculators, cellphones, or iPods.

Problem 1 Problem 2 Problem 3 Problem 4 Multiple Choice TOTAL12 Points 12 Points 12 Points 12 Points 52 Points 100 Points

Section Class�Times Instructor Section Class�Times Instructor

1 09:00�am�Ͳ�09:50�am Huilan�Li 13 12:00�pm�Ͳ�12:50�pm Dimitrios�Papadopoulos2 11:00�am�Ͳ�11:50�am Jason�Scott�Aran 14 02:00�pm�Ͳ�02:50�pm Jason�Scott�Aran3 11:00�am�Ͳ�11:50�am Dennis�Yang 15 09:00�am�Ͳ�09:50�am Hwan�Yong�Lee4 04:00�pm�Ͳ�04:50�pm Dennis�Yang 16 12:00�pm�Ͳ�12:50�pm Daryl�Lawrence�Falco6 09:00�am�Ͳ�09:50�am Daryl�Lawrence�Falco 17 04:00�pm�Ͳ�04:50�pm Alexander�Dolgopolsky7 10:00�am�Ͳ�10:50�am Harold�D�Gilman 18 01:00�pm�Ͳ�01:50�pm Jason�Scott�Aran8 10:00�am�Ͳ�10:50�am Hwan�Yong�Lee 19 10:00�am�Ͳ�10:50�am Daryl�Lawrence�Falco10 02:00�pm�Ͳ�02:50�pm Alexander�Dolgopolsky 20 01:00�pm�Ͳ�01:50�pm Alexander�Dolgopolsky11 01:00�pm�Ͳ�01:50�pm Dimitrios�Papadopoulos 21 05:00�pm�Ͳ�05:50�pm Dennis�Yang12 04:00�pm�Ͳ�04:50�pm Dimitrios�Papadopoulos

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Free Response

For each of the following problems, you must show all of your work to earn fullcredit.

1. Find the acute angle between the planes which are tangent to surfaces S1 : x2+y

2 = 25and S2 : z = 2� x at the point (3, 4,�1).

(You may leave your answer in terms of an inverse trigonometric function.)

2

2. Evaluate the following iterated integral by converting to polar coordinates.

Z 2

0

Z p16�x

2

p3x

1

1 + x

2 + y

2dy dx

3

3. Find all absolute extrema of f(x, y) = x

2 � 2xy + 2y on the closed rectangular regionshown below.

4

4. The solid shown below is enclosed by z = 0, z = 2, y = x

2, y = 2�x, and the yz-plane.

(a) Set up a triple integal (or triple integrals) in rectangular coordinates with theorder of integration as dz dy dx which represents the volume of the solid.

(b) Set up a triple integal (or triple integrals) in rectangular coordinates with theorder of integration as dx dz dy which represents the volume of the solid.

Do not evaluate any of the integrals.

5

Multiple Choice

Circle the letter of the best answer. Make sure your circles inlude just one letter.These problems will be marked as correct or incorrect; partial credit will not beawarded for problems in this section. Each problem is worth 4 points.

5. Suppose f(x, y) is a twice di↵erentiable function which satisfies the following table ofvalues:

(x, y) f(x, y) f

x

(x, y) f

y

(x, y) f

xx

(x, y) f

yy

(x, y) f

xy

(x, y)

(�2, 2) 9 0 0 �1 �2 �1

(1, 2) 0 03

2�2 �1

22

(0, 6) 5 0 0 �2

30 1

Which of the following statements is true?

I. f(x, y) has a local minimum at (x, y) = (�2, 2)II. f(x, y) has a critical point at (x, y) = (1, 2)III. f(x, y) has a saddle point at (x, y) = (0, 6)

(a) I only

(b) III only

(c) I and II only

(d) I and III only

(e) II and III only

6

6. Which of the following is the double integral that results from reversing the order of

integration on

Z 3

0

Z px+1

1

f(x, y) dy dx?

(a)

Z 3

1

Zy

2+1

1

f(x, y) dx dy

(b)

Z 2

0

Zy

2�1

0

f(x, y) dx dy

(c)

Z 2

1

Zy

2�1

0

f(x, y) dx dy

(d)

Z 2

0

Z 1

y

2�1

f(x, y) dx dy

(e)

Z 2

1

Z 3

y

2�1

f(x, y) dx dy

7. Consider the line L which contains the point A(1, 2, 1) and is perpendicular to theplane P : x� y + z = 1. At which point will line L intersect plane P?

(a) (0,�1, 0)

(b) (1, 1, 1)

(c) (2, 3, 2)

(d)

✓4

3,

5

3,

4

3

◆

(e)

✓2

3,

1

3,

2

3

◆

7

8. Which of the following vectors is normal to the plane determined by points A(2, 1, 0),B(0, 1, 1), and C(�2, 0,�1)?

(a) h0, 1, 0i

(b) h1, 0, 4i

(c) h1,�6, 2i

(d) h1, 3,�2i

(e) h1,�1, 4i

9. What is the largest rate of increase of f(x, y) = ln (x+ y) at P (2,�1)?

(a) 0

(b) 1

(c)p2

(d)p3

(e) ln 2

8

10. Which of the following is the Jacobian of the transformation x = e

u cos v, y = e

u sin v?

(a) e

2u

(b) e

u

2

(c) e

u cos v � e

u sin v

(d) e

2u cos2 v � e

2u sin2v

(e) e

u

2cos2 v � e

u

2sin2

v

11. What is the distance from P (3,�3, 4) to the x-axis?

(a) 3

(b) 4

(c) 5

(d)p29

(e)p34

9

12. LetR be the region in the xy-plane shown above. Using the transformation x =1

2(u+v)

and y =1

2(u� v), which of the following integrals is equivalent to

ZZ

R

(x� y)ex+y

dA?

HINT: If x =1

2(u+ v) and y =

1

2(u� v), then u = x+ y and v = x� y.

(a)

Z 4

1

Z 4

0

ve

u

dv du

(b)

Z 1

0

Z 4

1

ve

u

dv du

(c)

Z 4

12

Z 32

�2

ve

u

dv du

(d)1

4

Z 4

1

Z 4

0

ve

u

dv du

(e)1

2

Z 4

0

Z 4

1

ve

u

dv du

10

The following two questions refer to:

Z 3

0

Z p9�x

2

0

Z p9�x

2�y

2

0

x

2dz dy dx

13. Which of the following triple integrals in Cylindrical Coordinates is equivalent toZ 3

0

Z p9�x

2

0

Z p9�x

2�y

2

0

x

2dz dy dx?

(a)

Z ⇡2

0

Z 3

0

Z p9�r

2

0

�r

2 cos2 ✓�dz dr d✓

(b)

Z p9�r

2

0

Z 3

0

Z 2⇡

0

�r

2 cos2 ✓�dz dr d✓

(c)

Z ⇡2

0

Z 3

0

Z p9�r

2

0

�r

3 cos2 ✓�dz dr d✓

(d)

Z 2⇡

0

Z 3

0

Z p9�r

2

0

�r

3 cos2 ✓�dz dr d✓

(e)

Z p9�r

2

0

Z 3

0

Z 2⇡

0

�r

3 cos2 ✓�dz dr d✓

14. Which of the following triple integrals in Spherical Coordinates is equivalent toZ 3

0

Z p9�x

2

0

Z p9�x

2�y

2

0

x

2dz dy dx?

(a)

Z ⇡2

0

Z ⇡2

0

Z 3

0

⇢

2 cos2 ✓ sin2� d⇢ d✓ d�

(b)

Z ⇡2

0

Z ⇡2

0

Z 9

0

⇢

2 cos2 ✓ sin2� d⇢ d✓ d�

(c)

Z ⇡2

0

Z ⇡4

0

Z 3

0

⇢

4 cos2 ✓ sin2� d⇢ d✓ d�

(d)

Z ⇡2

0

Z ⇡2

0

Z 3

0

⇢

4 cos2 ✓ sin3� d⇢ d✓ d�

(e)

Z ⇡2

0

Z ⇡2

0

Z 9

0

⇢

4 cos2 ✓ sin3� d⇢ d✓ d�

11

15. Consider the following initial value problem.

8<

:

r0(t) = h2e2t, 2, 0i

r(0) = h2,�3, 4i

Which of the following is r(1)?

(a) r(1) =⌦e

2 + 1,�1, 4↵

(b) r(1) =⌦e

2 + 1,�1, 0↵

(c) r(1) =⌦e

2 + 1, 2, 4↵

(d) r(1) =⌦e

2, 2, 0

↵

(e) r(1) =⌦e

2, 2, 4

↵

16. Suppose v and w are arbitrary non-zero vectors and let k be an arbitrary scalar. Whichof the following statements is FALSE?

(a) v ·w = v ·w

(b) v ⇥w = � (w ⇥ v)

(c) v · (v ⇥w) = 0

(d) v ⇥ (kv) = 0

(e) kkvk = k kvk

12

17. Some level curves of a function f(x, y) are shown below.

Which of the following statements is true about@f

@x

and@f

@y

at the point P?

(a)@f

@x

> 0 and@f

@y

> 0

(b)@f

@x

> 0 and@f

@y

< 0

(c)@f

@x

< 0 and@f

@y

> 0

(d)@f

@x

< 0 and@f

@y

< 0

(e) There is not enough information to determine the signs of these partial derivatives.

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