PART II ANSWER KEY TO CHAPTER TESTS

PART II

ANSWER KEY TO CHAPTER TESTS

Chapter Chapter Chapter Chapter Chapter Chapter

Preparation for Calculus .................... Limits and Their Properties . . . . . . . . . . . . . . . . . . Differentiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . Applications of Differentiation . . . . . . . . . . . . . . . Integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. Logarithmic, Exponential, and Other Transcendental Functions .............. Applications of Integration .................. ............ Integration Techniques, L’Hôpital’s Rule, and Improper Integrals . . . . . . . . . . . . . . . . . . . . . . Infinite Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. Conics, Parametric Equations, and Polar Coordinates . . . . . . . . . . . . . . . . . . . . . . Vectors and the Geometry of Space ........... Vector-Valued Functions . . . . . . . . . . . . . . . . . . . . Functions of Several Variables ............... Multiple Integration . . . . . . . . . . . . . . . . . . . . . . . . Vector Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . ...

P 1 2 3 4 5

Chapter 6 Chapter 7 Chapter 8 Chapter 9 Chapter 10 Chapter 11 Chapter 12 Chapter 13 Chapter 14

202 204 205 206 208 209 210 211 212 213 215 217 218 219 220

202

Chapter P

Answers to Tests

Answers to CHAPTER P Tests

15.

y 5

Test Form A 3

1. d

2. c

3. c

4. a

5. d

6. a

7. b

8. b

(1, 3)

2 1

x −3 −2

9. d

10. c

11. b

12. b

13. d

14. c

15. a

16. d

17. c

18. a

19. d

20. d

−1 −1

3. d

4. c

5. a

6. c

7. b

8. c

9. b

10. e

11. a

12. c

13. c

14. d

15. c

16. b

17. a

18. d

19. a

20. a

3

4

16. Slope: 54 , y-intercept 0, 3 17. a. 4

2. b

2

−2

Test Form B 1. a

1

b. 1

c. 3

18. a. S 24 1.50x

b. $45.00

c. 51

Test Form E 1. a. y 3 ± 1 x 22 b.

y 4 3

Test Form C

2

1. a

2. b

3. b

4. c

5. d

6. b

7. b

8. d

(−2, 3)

x −4

9. c

10. d

11. c

12. b

13. a

14. d

15. b

16. d

1

−3

−2

−1

2. One such equation is y x3 25x. 3. a.

Test Form D

y 4

x x 2 f x x2 1 x 1

2. 0, 13 , 12 , 0 4. x 2

3

3. 0, 0, 2, 4

1 x

5. 3x 2y 11 0

−2

c. 1 4x x2 1 7. 2x x 3 8. 5 x 9. All reals except 1 ± 3 6. a. 6

b. 2

10.

11.

y

b. 2, 0, 1, 0, 0, 2, 1, 0, 2, 0 c. Symmetric with respect to the y-axis. 4. Symmetric to the origin 5. 2, 2 and 5, 9

y

6.

5

2

4

2

x −1

b. 2

4

1

−2

7. a. 3

y

(0, 4)

c. 0

2

2 1

(−2, 0)

d. 1

2 x

−3 −2

−1

1

2

3

1

−1 x −3

12. a and b

13. 3x 5

14. Neither

−1

1

© Houghton Mifflin Company. All rights reserved.

1. f x

Chapter P

8. a.

y 3 2

x −3

−2

1

2

3

−1 −2 −3

b. Domain: 2, ; Range: 2, c. 1, 0, 0, 1.4 d. 1, 0 and 0, 2 9. 2.4, 0.41, 2

10. a, d, and e

11. 3

12. 2x x 3 13. Even. f x x4 2 x2 1 x 4 2x2 1 f x 14.

y 1 x −2

−1

1

2

−2 −3

15. Answers will vary. Some examples are 1, 1 and 7, 9.


2 16. y x 3 1 3

17.

x3 3

18. a. S 12,500t 224,000 c. $160,000 d. 2011

Answers to Tests

203

204

Chapter 1

Answers to Tests

Answers to CHAPTER 1 Tests

7. a.

y 4

Test Form A

3

1. b

2. d

3. a

4. a

5. d

6. c

7. b

8. b

9. d

10. a

11. c

12. a

13. b

14. a

15. c

16. a

17. d

18. b

19. b

20. c

1 x −4

−3

−1

1

2

−1 −2

b. 4

c. 4

8. a. 4

b. gx x2

c.

y

Test Form B 1. c

2. c

3. d

4. b

10

5. a

6. b

7. b

8. d

8

9. e

10. c

11. b

12. b

13. a

14. d

15. a

16. c

17. b

18. a

19. c

20. b

6

Test Form C 2. d

3. b

4. b

5. a

6. a

7. a

8. b

9. d

10. c

11. b

12. c

13. a

14. a

15. c

16. d

17. a

18. c

19. a

20. a

2. 0.001 1 7. 12

10. 3

11. 15.

1 3

3. 5 8.

4. 0 1

2x

12. 3

5. 5

9. Does not exist 13.

16. x 2, removable; x 2, nonremovable 5 x4 1

b. 1, 1

18. 27

1 5 1 5 , x 2 2

Test Form E 1. The limit does not exist. 2. False. f x 3. 3

−2

1 2x 2

−1

1

2

3

10. 2x 2

11. 1

12. 1

13. 2n 1, 2n 1 where n is an integer 14. f x is continuous on the interval , . f 2 2 > 0 and f 1 4 < 0. Therefore, there exists a real number c, 2 < c < 1, such that f c 0.

4. 0

x2, x 3 0, x 3 5.

6. 5

19. x 7

16. a. 1 17. 3; lim

x→0

b. 3

c. The limit does not exist.

sin 3x 3x sin 3x sin 3x lim 3 lim x→0 x→0 x 3x 3x

31 3 18. x 3

19.

20.


1. 3 7 6. 6

20. x

x −3

15. 2 and 2

Test Form D

17. a.

2

9.

1. c

14.

4

Chapter 2


2. a.

4 x2

b. 1

d.

Test Form A 1. c

2. d

3. b

4. c

5. d

6. a

7. b

8. b

9. d

10. e

11. c

12. a

13. d

14. b

15. b

1. b

2. d

3. a

4. d

5. a

6. c

7. a

8. c

9. b

10. b

11. d

12. b

13. c

14. c

15. c

Answers to Tests

205

c. y x 4

y 6 4 2 x 2

4

6

Test Form B 3. 32 , 23 and 32 , 23 4. a. 64 ftsec b. 96 ftsec c.

d. 4030 ftsec 219.09 ftsec

Test Form C

6. 5 sec x2 sec2x 1

1. d

2. b

3. d

4. c

5. 4 ft2

5. a

6. c

7. a

8. c

8. a. 5x 4 5

9. d

10. c

11. b

12. b

13. a

14. a

15. c

1 1. 2 x 4.

3

6x2 2 2. 1 3x22

cot2x

csc2x

2x

6. 42 ftsec2

9. 2 sec 2 tan 2

3

f 2 1

2

3

8. csc y

10. 2 sin 2x

12.

1 ,1 3

14. a. s 16t 2 26t 220 15.

f' x

−3

10. 11. x 4y 5

The derivative of f is zero if its tangent line is horizontal.

y

2 2 x3

y 7. x y2 x

13. 4x

1 x2

b. 58 ftsec

1 insec

Test Form E 1. a. no derivative b. negative c. zero d. positive e. zero

6 23 3

c. 1, 4 and 1, 4

9.

x27x 3 3. 2x 1 5.

7.

d. 0

Test Form D


530 sec 6.85 sec 4

10 2x2 532

11. 2, 22 and 2, 22 12.

96 845

radsec

206

Chapter 3

Answers to Tests

Test Form E


1. Minimum at x b 2. a. y

Test Form A 1. c

2. e

3. b

4. c

5. a

6. d

7. c

8. b

9. d

10. a

11. a

12. d

13. b

14. d

5 4 3 2 1 x

Test Form B

1

1. d

2. d

3. a

4. b

5. c

6. b

7. a

8. a

9. b

10. c

11. b

12. c

13. e

14. b

2. c

3. d

4. c

5. c

6. c

7. a

8. b

9. e

10. c

11. d

12. c

13. b

14. b

15. b

3.

1 4

5. Increasing , 3 and 1, Decreasing 3, 1 6. Relative maximum 7. Concave downward: , 2 Concave upward: 2, 8.

1. Increasing , 0; Decreasing 0,

y

1

x −4

5 Relative minimum: , 2 4

1

f ''

−4

2 3,

relative minimum

9. a. 2 b. 2 10. a. y 1 b.

7. 2 8.

c. The limit does not exist.

y 4

9.

y

2

−3

5. , 0 and 2, 6. x 0, relative maximum; x

−3

y

6

3

3

5 4

x

3

−3

1

−2

x −2

−1

2

3

10. 4, 0, 2, 32 12. 1.3247

1

−2

2

−1

4

11. 320,000 sq ft

13. y 1

1 −1

x −1

1

−1

14. ± 0.001 in.

c. 0, 1 11. f x

x2 x 5 (The answer is not unique.) x


, 2 4

2

f'

1 3. 2, 54 , relative maximum

4. Relative maximum:

5

4. a. Relative minimum at x ± 1.4 Relative maximum at x 0.3 b. Relative minimum at x ± 2 Relative maximum at x 13

Test Form D 2. 13 , 12

4

b. f 2 f 4 1 c. f is not continuous on 2, 4.

Test Form C 1. b

2

Chapter 3

12. a. V x 7 2x3 2x b. 0.65 in. by 0.65 in.; V 6.3 cubic inches 13. a. 0.5

b. 0.4459

14. a. T x 12x 16 c. f 2.1 9.5, T 2.1 9.4


d. f 2.1 9.261, T 2.1 9.2

Answers to Tests

207

208

Chapter 4

Answers to Tests


6.

Test Form A

8.

2 52 x 5

C

7. 0.486

y

1. a

2. b

3. c

4. a

9

5. b

6. d

7. c

8. c

6

9. b

10. d

11. b

12. a

3

13. b

14. d

15. d

16. a

x 1

Test Form B

2

3

9. No, because f x is not continuous at x 2 . 3

1. b

2. c

3. b

4. d

5. a

6. a

7. a

8. b

9. c

10. c

11. d

12. b

10. 2 1 0.414 1 1 , F2 11. Fx 1 x4 17

13. d

14. a

15. b

16. c

13. 3 1

14.

15. a. 1.852

b. 1.935

Test Form C 1. c

2. d

3. a

4. a

5. d

6. d

7. b

8. d

9. b

10. d

11. e

12. b

13. c

14. b

15. d

1 32 3 2

12. 2 sinx C

1

Test Form D 2 52 x C 5 x2 3. xC 2

2. 3 csc x C

1.

4.

1 sin C 2

5. y x2 x 3

6. 186.6 fts

7. 4

9. 2x2 52

10. 4

4

8.

4x x2 dx

11.

22

12.

1 3

13. 2tan x C

15. 23 x 1x 2 C

14.

16. 0.1667

Test Form E 1.

2a 52 2b 72 x x C 5 7

2.

1 2x C 2x 12

3. y x2 x 3 4. a. P 10t 4 30t72 16,000 b. 4.8

c. 53,120

5. a. R 100x 0.02x2 b. p 100 0.02x

5 2


0

Chapter 5


5.

Test Form A

8e2x 4 e2x5

1 6. e3x C 3

7. a.

1. d

2. b

3. b

Answers to Tests

y

4. d

x −π 2

π 2

5. b

6. e

7. d

8. d

9. a

10. c

11. c

12. a

−4

13. b

14. c

15. c

16. d

−6

π

3π 2

−8

Test Form B 1. b

2. c

3. d

4. a

5. b

6. a

7. b

8. a

9. d

10. a

11. b

12. c

13. c

14. b

15. a

16. b

1. c

2. c

3. b

4. c

5. d

6. d

7. b

8. c

9. c

10. b

11. c

12. a

13. a

14. d

15. a

Test Form C

b. 2.14

c.

8. y Cx

9.

2 2t 1 t4

10.

2 3

11. x arctan x C 12.

y

3

2

1

x 1

2

3

Test Form D 1.

1 1 2x 5x

4.

1 ye x e x 2y


7. 1

8.

2. x x 1 ln x 5. e x x e1x e

1 2

3.

e1x x2

13. arcsec 6. 2 tanh 2x

ln sec 2x tan 2x C

9. 95 et5 C 10. 2e x 1 C 16 x 5 11. ln x2 9 arctan C 2 3 3 12. arcsinx 2 C 13. x2 sec2 y C 14. x2 y 2 ln Cy 2 15. 4.64 Test Form E 1. y

3x 2x 1 3 x 2 2

x32x 3 3 1 2 x 22 x 2x 3 x 2

2. a.

b.

c. , 3, 2,

3. ln 1 sin C 4. a. P 10,000 ln1 0.2t 1000 b. 10,000 ln 3 1000 11,986

2 cosh 32 1 2.7048 x1 C 5

15. y 500 493ex

14. y 2 e x C 2

209

210

Chapter 6

Answers to Tests


3. a.

y 5

Test Form A

4

1. c

2. b

3. d

4. c

5. a

6. b

7. d

8. a

3 2 1

Test Form B

−1

1. a

2. c

3. b

4. b

5. c

6. a

7. a

8. d

3

4

b. 1, 1, 2, 12 , and 2, 4 c. 2 23 12 2 ln 2 4. 104.22 ft

Test Form C

x 1

28 15

5. 1.014 1011 ft-lb b. 0,

1. c

2. a

3. c

4. b

7. a.

5. b

6. c

7. c

8. d

8. 1.324, 0.611

7 5

c.

56 15

d.

6. 640 lb 56 15

Test Form D 1.

15 2 ln 2 8

2.

2

3. 2

y

1 2

4.

1 2

5 2 2y 1y dy

256 5

5.

2 e 1 2

1 4 cos2 2x dx

0

4

6. 2

x

0

1 4x1 dx

7.

1280 in-lb 3

8. x 45 , y 27 Test Form E 1. a. 2 1 x 3

4

−1

b. a 0.16, b 3.15 c. 1.95 2. 0.4216


y

Chapter 7

9. a. 1


Answers to Tests

b. 1

10. f x 7x2, gx x2 1

Test Form A 1. c

2. b

3. b

4. c

5. a

6. d

7. b

8. c

9. d

10. b

11. c

12. b

(There are many correct answers.) 11. a. 2 12. a.

b.

c.

y 5 4

Test Form B

3

1. b

2. a

3. c

4. a

5. d

6. c

7. a

8. a

9. b

10. c

11. b

12. a

2 1 x 1

b.

Test Form C 1. d

2. a

3. d

4. c

5. b

6. a

7. b

8. c

9. d

10. a

11. b

12. b

Test Form D

1. ln 1 cos x C

2. sec x tan x x C

3. 14 x2 2 ln x 1 C 4. 2x

2 x1 arctan C 3 3

5. t

2 1 cos3 t cos5 t C 3 5

15 sec

6. 2

5

x 1 x sec3 C 2 3 2

8. ln 2 3


7. ln x x2 4 C

9. 5 ln x 11. 1

1 2x arctan C 6 3

10.

12. The integral diverges.

Test Form E 1. 2 ln 4

2.

4x 1 4x e C 16

3. 3ln 32 6ln 3 4 4.

2sin x 5 sin2 x C 5

5. x

5 1 sec 2 2

7. ln

6. 2 3

x2 3 C x1 x2

3

8.

1 2

3

2

2

c.

3

4

5

211

Chapter 8

Answers to Tests

4. a. 0.21 0.0021 0.000021 0.00000021 . . .

Answers to CHAPTER 8 Tests Test Form A

b. a 0.21, r 0.01

1. b

2. b

3. d

4. c

5. d

6. b

7. d

8. d

9. c

10. a

11. c

12. b

13. c

14. d

15. b

16. a

17. a

18. a

19. c

20. d

0.210.01

c.

n

n0

5.

3 x

n

n0

6. a.

1 2

x3dx

1

Test Form B

Therefore the series converges also.

1. d

2. d

3. a

4. a

5. b

6. a

7. b

8. c

9. c

10. b

11. c

12. d

13. b

14. a

15. c

16. b

17. b

18. b

19. d

20. e

b. 1.178

7. Diverges using

2. a

3. b

4. a

5. b

6. d

7. d

8. c

9. d

10. b

11. d

12. b

13. c

14. b

15. d

16. c

n1

1 8. a. diverges n2 n

9. lim

x→

2. Diverges

4. 2

5.

5 12

10. a.

n1

9. Converges, Alternating Series Test

b. 0.328

10. Converges, Telescopic Series Test

c. 9

11. Diverges, Limit Comparison Test 13. Converges, Root Test

14. Diverges, nth-Term Test for Divergence 15. 1000 18. 1

16.

17.

11. Diverges by Ratio Test 12. a. P3x x

7 9 2, 2

x x2 x3 x4 . . . 2! 4! 6! 8!

19. 14 x 14

13.

20. 0.615 14. 1 2x 2. a. No

b. Yes

c. Yes

3. Diverges by the nth-Term Test for Divergence

x3 3

f x dx C

Test Form E 1. Converges to 0

r

8. Converges, Geometric Series Test

4 3

1n 1 a geometric series with n n, 2 2 n1

1 < 1. Therefore, the series converges and 2 the original series converges absolutely.

7. Diverges, Limit Comparison Test

12. Diverges, Ratio Test

1 ln x < 0, x > 0. x ln x2

Since the corresponding function is decreasing, the series is also.

3. Converges to e

6. Diverges, p-Series

1 1 > , n ≥ 2. ln n n

1 1 0. For f x , nln n x ln x

fx

Test Form D 243 7

1

n

b. Diverges because

1. b

d. 1.209

Test Form C

1.

c. 0.031

15. 1

b. 0.291

1n xn1 , 3 < x ≤ 3 n n0 3 n 1

22x2 23x3 . . . 2xn . . . 2! 3! n!

1n 22n1x2n 2n! n1


212

Chapter 9


Answers to Tests

5. y2 x 4y 3 0 6. x3 y 2 4x 2y 2 0

Test Form A 1. a

2. c

3. d

4. c

7. y 12 et 2

8. 12 cot

5. b

6. c

7. d

8. a

9. y 16x 24

10.

9. c

10. a

11. d

12. b

13. a

14. c

15. b

16. d

17. c

18. c

19. b

20. d

12.

2t 1 9t5

y 1

−3

−2

π 3 x

−1

1

Test Form B

−1

1. c

2. d

3. c

4. b

−2

5. d

6. a

7. c

8. d

−3

9. a

10. b

11. c

12. a

13. c

14. b

15. a

16. c

17. b

18. b

19. a

20. b

Test Form C

13. 3, 1

15. x2 y 2 3x 0 17.

1. a

2. c

3. d

4. b

5. d

6. a

7. c

8. b

9. c

10. d

11. c

12. c

13. b

14. a

15. c

14. r

2 2 sin 3 cos 16.

3 , 2 2

20.

3 4

π 4

π 2

0 2

Test Form D 9 4 1. x 42 y or y 2 x 4 2 3 2.

y 12 x 22 1 4 16

3

19.

5 , 6 6

Test Form E 1. a. y 2x 1, y 2x 9

y


18.

4

b.

y 5 4

1 −4

−3

−2

3

x

−1

2 −2

1 x 1

3. 2y x 4 4.

c.

y

2 x

−2

2 −2 −4 −6

3

5

2 3

2. 49x2 144y 2 2,822,400

6 4

−6

2

6

11. 12

213

214

Chapter 9

Answers to Tests

3. a. y 3x2 1 b.

r cos sin

9. a. r2

y

r cos r sin

x2 y 2 x y

3

x2 x 14 y 2 y 14 14 14 2 2 x 12 y 12 12

2 1 x

−1

1

2

b. center: 12 , 12 ; radius

3

−1

12

10. 4y 2x 1

c. x ≥ 0 4. a. y 12 b.

2x

2

11. a.

1

π 2

y 3 0 4 1

−2

x

−1

1

2

−1

5. a.

b.

2 72 5

y

12.

π 2

4

0

(4π, 0) (8π, 0) (12π, 0)

1

x

2

b. 4 n, 0, n 0, 1, 2, . . . Area:

y 8

13. 0 and (−1, 5) 6 4 2

−2

x 2

4

6

5

b.

1

3t3 22 2t 12 dt

c. 133.566 7. Relative minimum, 18 , 19 4 8. 2; concave upward

3 2


6. a.

Chapter 10


15. Elliptic paraboloid

Answers to Tests

16.

z 5

z

Test Form A

215

4

1. d

2. b

3. a

4. b

5. a

6. b

7. b

8. c

−4

2 x

2

9. d

10. c

11. a

12. b

13. d

14. c

15. b

16. e

17. a

18. d

19. c

20. c

1. c

2. d

3. b

4. a

5. c

6. b

7. d

8. b

9. b

10. a

11. c

12. d

13. c

14. b

15. a

16. c

17. c

18. b

19. b

20. d

4

y

y

17. 4x2 9y 2 9z2 4x 1 0 18.

Test Form B

4

2

x

32, 74, 7

19. y 9, plane

20. 2285

Test Form E 1.

1 i j 2

2.

35, 65

3. Direction 17.4 ; magnitude 44.4 4. x 12 y 62 z 22 33 4 2

Test Form C 1. b

2. c

3. a

4. a

5. b

6. d

7. d

8. b

9. c

10. c

11. d

12. c

13. c

14. a

15. d

16. b

17. d

18. b

359, 159, 759


4. 3i j 5k

5. 14

6.

13 i

3. 1

1 6j

8. ci 2j 3k, c R, c 0

7. 1.74

9. 41 52 72 0

1 6k

1 53

6, 1, 4

1 6, 1, 4 53

7. 422 5 5 8. a. 15 14 i 7 j 14 k

9. cos

b.

1 14 i

97 j 33 14 k

1 2 4 , cos , cos 21 21 21

10. 3

11. x 2z 6 0

11. a. 6i 7j 4k b. u v v 6i 7j 4k 3i 2j k 0

z 6

12. 52 6

4 2

13. x 2 5t, y 4 11t, z 1 19t

2 4 6

d. No

c. 53

10. x 4 t, y 1 2t, z 3 5t

12.

c. Yes, c 34

b. 6i j 4k

e. 2. j

b. Yes, c 2

6. a. 6, 1, 4

d.

Test Form D 1.

5. a. No

4

x

6

y

13. x 25 , y 13 5 14. x 12 y 22 z 12 3 2

Center: 12, 2, 1, Radius: 3

33 11 14. 17 7 , 14 , 14

216

Chapter 10

15.

Answers to Tests

z

2

2 5

x

16. a. Sphere

y

b. Cone

c. Hyperboloid of two sheets 17. a. r 2 1 z2 b. 2 sec 2 18.

z 6

2 2

y


x

Chapter 11


6. a.

Test Form A

Answers to Tests

753 75 ti t 16t 2 j 2 2

b.

y

1. d

2. a

3. c

4. c

25

5. a

6. b

7. a

8. b

20

9. c

10. d

15 10 5

Test Form B

x

1. c

2. b

3. d

4. b

5. d

6. a

7. c

8. c

9. a

10. d

30

Test Form C 2. d

3. a

4. a

5. d

6. d

7. b

8. b

9. d

10. b

1 154

4. 541

5. 2 n, n 0, ± 1, ± 2, . . .

1 i 2tj 1 4t2

8. a. T t

1 5

7. 10.

11 3

8. K

1 2

9.

i 2j

1 2i j 5

N 1

3. i j k

6. r t 3ti 5 2t j 2 t

b.

y

k

16t2

3

1 i j k 3

2

N

610t 10 9t 4

−3

−1

x −1 2

1 2

Test Form E


9.

y 4 3

2 1 x 1

2

2. i 2k

3

3.

4

3 5, 3, 6t

9t2 532

4. 2n, 2n 1 , n 0, 1, 2, 3, . . . 5.

vt

2

2

i

2

2

j 8 k

v t 1 64 2 a t

T

1

2

1.

150

b. x 2 s, y 8 12s, z 6 3s

Test Form D 2. i 4j k

120

i 12j 3k

T 1

1. 0,

90

c. 21.9 ft, 152 ft 7. a.

1. b

60

2

2

i

2

2

j 32k

2

3

10. 7.763

1

3 2

217

218

Chapter 12

Answers to Tests


5. a. 960 3015.9 cm3 b. ± 6.08 ± 19.1 cm3

Test Form A 1. b

2. d

3. a

4. a

5. c

6. b

7. c

8. d

9. d

10. b

Test Form B 1. c

2. c

3. b

4. d

5. a

6. d

7. b

8. b

9. d

10. a

c. 0.6% 6. 182 ft3min

7.

x yz

4 8. a. 4i j 5 4 b. 5 c.

4 26 5

9. a. Elliptic paraboloid Test Form C 1. c

2. b

3. a

4. c

5. c

6. a

7. c

8. d

9. c

10. d

11. a

c.

x2 y2 z2 2 8 5

10. Saddle point

Test Form D 1.

b. 2x 8y 5z 10 0

11. 3, 3 and 3, 3

f 2x 2f 2y2 2x2 2 ; 2 2 ; 2 x x y x x y22 f 2y 2f 2x2 2y2 2 ; 2 2 ; 2 y x y y x y22 2f 2f 2y2 2x2 2x2 2y2 0 x2 y2 x2 y22 x2 y22

2.

yzw cos xyz 2w sin xyz

sinu v sinu2 v2 2u cosu v cosu2 v2 4. 0.004 7.

5.

48 5

6. 7e6

x2 y1 z2 16 4 7

8. f 1, 1 is a relative minimum 9. 11, 11, 11

7 10. y 10 x1

Test Form E 1. x, y : x2 y 2 > 4

2. 4

3. f is continuous at each point in space except at the points on the plane given by x 2y 3z 6. 4. 5


3.

Chapter 13


3.

1 2 1

cos 1

4. 0.0226

16x2

4

Test Form A

5. a. 2

1. c

2. d

3. a

4. d

5. b

6. b

7. c

8. a

9. c

0

4x dy dx

0

2

4

r 2 cos dr d

b. 8

10. d

0

Test Form B 2. e

3. b

4. c

5. d

6. c

7. b

8. b

9. a

10. d

4x2

2

6.

0

512 3

c.

1. a

0

1. b

2. a

3. b

4. a

5. c

6. b

7. c

8. d

9. d

10. a

x y dy dx or

0

2x

2

5.

0

0

0

2

0

11 6

e 1 4e

2 3

3.

2

6.

15 8

9. a.

a

0

7.

32 k 3


1.

1 3

2

8. 50

∂x ∂ cos r sin r ∂y sin r cos ∂

2e3 1

2. a.

y

π

y = x − sin x π

b. 2

2 2

0

x

3

2 sin d d d

1

2

3

8

b. 10.

21 10

104 3

2 sin d d d

0

0

∂x ∂x, y ∂r 10. ∂r, ∂y ∂r

0

2

4. 8

9. 4 2 3

Test Form E

x3 y dx dy

2y

r dz dr d

8. a. 8

Test Form D 2.

4y2

a 2 r 2

a

b. 8 9 2

2

3

7. 1.123

Test Form C

1.

Answers to Tests

0

1

2 sin d d d

219

220

Chapter 14

Answers to Tests

Answers to CHAPTER 14 Tests Test Form A 1. a

2. b

3. c

4. d

5. b

6. c

7. d

8. a

9. c

10. b

Test Form B 1. d

2. a

3. b

4. c

5. a

6. b

7. a

8. c

9. d

10. b

Test Form C 1. b

2. a

3. d

4. a

5. d

6. b

7. c

8. b

9. c

10. a

Test Form D 1.

3 1

2. z cos yz 2 i xk

2

3. x 4 y3 ln x ln y C 5. 2 4 9. 3

6. 0

7.

4. 128

4 5

8.

149 385

10. 36

Test Form E 1. 2e2y xy 2 i xj y 2 zk f x, y, z x2z y 2z z3 y 2 x C 3.

13 2

4. 14

7. 3617

5. 0 8.

276 2

6. 2 4 9. 32

10. 3


2. Conservative;