J Answers to 0dd-Numbered Exercises...A80 APPENDIX J ANSWERS TO ODD-NUMBERED EXERCISES 21., 23. 25. 27. 29. 31. (a) , ; and are the same function. (b) Quarter-circle in the ﬁrst

A76 APPENDIX J ANSWERS TO ODD-NUMBERED EXERCISES

J Answers to 0dd-Numbered Exercises

CHAPTER 1

EXERCISES 1.1 N PAGE 21

1. (a) 3 (b) (c) 0, 3 (d)(e) (f)3. 5. No7. Yes, 9. Diet, exercise, or illness11.

13. (a) 500 MW; 730 MW (b) 4 AM; noon15.

17.

19.

21. (a)

(b) 126 million; 207 million23. 12, 16, , , ,

, , ,,

25. 27.29.31.33. ��, 0� � �5, ��

��, ��, �3� � ��3, 3� � �3, ��

�1��ax��3 � h3a2 � 6ah � 3h2 � a � h � 29a4 � 6a3 � 13a2 � 4a � 4

3a4 � a2 � 212a2 � 2a � 26a2 � 2a � 43a2 � 5a � 43a2 � a � 23a2 � a � 2

N

t(midyear)

1996 2000 2004

200

150

100

50

heightof grass

tWed.Wed.Wed. Wed. Wed.

0

amount

price

T

tmidnight noon

T

0 t

��3, 2�, ��3, �2� � ��1, 3��85, 115�

��2, 1��2, 4�, ��1, 3��0.8�0.2

35. 37.

39. 41.

43. 45.

47. 49.

51.

53.55. 57.59.61. (a) (b) $400, $1900

(c)

63. is odd, is even 65. (a) (b)67. Odd 69. Neither 71. Even73. Even; odd; neither (unless or )


1. (a) Logarithmic (b) Root (c) Rational(d) Polynomial, degree 2 (e) Exponential (f) Trigonometric3. (a) h (b) f (c) t

t � 0f � 0

��5, �3��5, 3�tf

T (in dollars)

0 I (in dollars)10,000 20,000

1000

2500

30,000

R (%)

0 I (in dollars)10,000 20,000

10

15

V�x� � 4x 3 � 64x 2 � 240x, 0 � x � 6S�x� � x 2 � �8�x�, x � 0A�x� � s3x 2�4, x � 0

A�L� � 10L � L2, 0 � L � 10

f �x� � ��x � 3

2x � 6

if 0 � x � 3

if 3 � x � 5

f �x� � 1 � s�x f �x� � 52 x �

112 , 1 � x � 5

x

y

1

�1 0x

(0, 2)(0, 1)

_2 1

y

0

��, ��, ��

x

2

y

0

4y

x0 5

��, 0� � �0, ��5, ��

y

0t_1_2

1

y

0 x5

2

��, ��, ��

59726_AnsSV_AnsSV_pA076-A087.qk 11/22/08 12:58 PM Page 76

APPENDIX J ANSWERS TO ODD-NUMBERED EXERCISES A77

5. (a) , where b is the y-intercept.

(b) , where m is the slope.See graph at right.(c)

7. Their graphs have slope .

9.11. (a) 8.34, change in mg for every 1 year change (b) 8.34 mg

13. (a) (b) , change in for everychange; 32, Fahrenheit

temperature correspondingto

15. (a) (b) , change in for every chirp perminute change (c)17. (a) (b) 196 ft19. (a) Cosine (b) Linear21. (a) Linear model is

appropriate.

(b) 15

0 61,000

(b)

(c)

y � �0.000105x � 14.521

15

0 61,000

P � 0.434d � 1576�F

�F16T � 1

6 N �3076

0�C

1�C�F9

5F

C

(100, 212)

F= C+3295

(_40, _40)

32

f �x� � �3x�x � 1��x � 2�

y

x

c=_2

c=_1

0 c=2

c=1

c=0

�1

y � 2x � 3

y

x

m=_1

m=1

m=0

y-1=m(x-2)

(2, 1)

y � mx � 1 � 2m

y

x

b=3 b=0b=_1

y=2x+b

y � 2x � b (c) [See graph in (b).](d) About 11.5 per 100 population (e) About 6% (f) No23. (a)

Linear model is appropriate.(b)(c) ; higher than actual value(d) No

25.1914 million


1. (a) (b) (c)(d) (e) (f)(g) (h)3. (a) 3 (b) 1 (c) 4 (d) 5 (e) 25. (a) (b)

(c) (d)

7.

9. 11.

13. y

x0

3

π

y=1+2 cos x

y

x0_1

1

y=(x+1)@

y

x0

y=_x#

y � �s�x 2 � 5x � 4 � 1

y

0 x1

1

y

0 x

1

1

y

0 x

1

2

y

0 x

1

1

y � 13 f �x�y � 3f �x�

y � f ��x�y � �f �x�y � f �x � 3�y � f �x � 3�y � f �x� � 3y � f �x� � 3

y � 0.0012937x 3 � 7.06142x 2 � 12,823x � 7,743,770;

6.35 my � �0.027t � 47.758

height (m)

t1896 1912 1928 1944 1960 1976 1992 2008

3.5

4.0

4.5

5.0

5.5

6.0

y � �0.00009979x � 13.951



15. 17.

19. 21.

23.

25.

27. (a) The portion of the graph of to the right of the y-axis is reflected about the y-axis.(b) (c)

29. (a) , (b) , (c) ,

(d) ,

31. (a) , (b) , (c) , (d) ,

33. (a) , (b) , (c) , (d) ,

35. (a) ,

(b) ,

(c) , {x � x � 0�� f � f ��x� �x 4 � 3x 2 � 1

x�x 2 � 1�

{x � x � �1, 0��t � f ��x� �x 2 � x � 1

�x � 1�2

�x � x � �2, �1�2x 2 � 6x � 5

�x � 2��x � 1�� f � t��x� �

��, ��t � t��x� � cos �cos x��, �� f � f ��x� � 9x � 2

��, ��t � f ��x� � cos �1 � 3x��, �� f � t��x� � 1 � 3 cos x

��, ��t � t��x� � 4x � 3��, �� f � f ��x� � x 4 � 2x 2

��, ��t � f ��x� � 2x 2 � 1��, �� f � t��x� � 4x 2 � 4x

{x � x � �1�s3}� f�t��x� �x 3 � 2x 2

3x 2 � 1

��, �� ft��x� � 3x 5 � 6x 4 � x 3 � 2x 2��, �� f � t��x� � x 3 � x 2 � 1

��, �� f � t��x� � x 3 � 5x 2 � 1

y

x0

y=œ„„|x|

x

yy= |x|

0

sin

y � f �x�

L�t� � 12 � 2 sin 2�

365 �t � 80�

0 1

1 y=|œ„x-1|

y

x

0 2

2

y

x

y= |x - 2|

y=_(≈+8x)12

y

x0_4

_8

y

x0_3

y=œ„„„„x+3y

x0

1 2π

y=sin(x/2) (d)

37.

39.41. ,

43.45.

47. , ,

49. , , 51. (a) 4 (b) 3 (c) 0 (d) Does not exist; is notin the domain of . (e) 4 (f)53. (a) (b) ; the area of the circle as a function of time55. (a) (b)(c) ; the distance between the lighthouseand the ship as a function of the time elapsed since noon57. (a) (b)

(c)

59. Yes; 61. (a) (b)63. Yes


1. (c) 3.

5. 7.

9. 11. 1.5

_1.5

1000

_0.01 0.010

1.1

3500

_3500

20_20

4

�1

�4 4

100

_300

40_10

t�x� � x 2 � x � 1f (x� � x 2 � 6m1m2

V�t� � 240H�t � 5�

V

t

240

0 5

V�t� � 120H�t�

V

t

120

0

H

t

1

0

� f � t��t� � s900t 2 � 36d � 30ts � sd 2 � 36

�A � r��t� � 3600�t 2r�t� � 60t�2t

f �6� � 6f �x� � x 4

t�x� � sec xh�x� � sx

f �x� � 1 � xt�x� � 3xh�x� � x 2

t�t� � cos t, f � t � � st

t�x� � s3 x , f �x� � x��1 � x�

f �x� � x 4t�x� � 2x � x 2� f � t � h��x� � sx 6 � 4x 3 � 1

� f � t � h��x� � 2x � 1

{x � x � �2, �53}�t � t��x� �

2x � 3

3x � 5,


13.

15. (b) Yes; two are needed

17.

19. No 21. 23.25. 27.29. (a) (b)

(c) (d) Graphs of even roots aresimilar to , graphs of oddroots are similar to . As nincreases, the graph of

becomes steeper near0 and flatter for .

31.

If , the graph has three humps: two minimum points anda maximum point. These humps get flatter as c increases until at

two of the humps disappear and there is only one mini-mum point. This single hump then moves to the right andapproaches the origin as c increases.33. The hump gets larger and moves to the right.35. If , the loop is to the right of the origin; if , the loopis to the left. The closer c is to 0, the larger the loop.


1. (a) 4 (b)3. (a) (b)5. (a) (b) (c) �0, ��f �x� � a x, a � 0

648y716b12x �4�3

c � 0c 0

c � �1.5

c �1.5

2

_4

_2.5 2.5

_1.5 -1 -2 -31

x � 1y � s

n x

s3 x

sx2

_1

_1 3

$œ„xœ„x Œ„x

%œ„x

2

_2

_3 3x %œ„x

Œ„x

3

_1

_1 4œ̂„x

$œ„xœ„x

�0.85 x 0.85t

0.65�0.72, 1.22

1

_1

_1 1

2

_2

_ π25

π25

11

_11

2π_2π

(d) See Figures 4(c), 4(b), and 4(a), respectively.7. All approach 0 as ,

all pass through , andall are increasing. The largerthe base, the faster the rate ofincrease.

9. The functions with basegreater than 1 are increasingand those with base less than1 are decreasing. The latterare reflections of the formerabout the y-axis.

11. 13.

15.

17. (a) (b) (c)(d) (e)19. (a) (b)21. 27. At 29. (a) 3200 (b) (c) 10,159(d)

31. (a) 25 mg (b) (c) 10.9 mg (d) 38.2 days33. , where and

; 5498 million; 7417 million


1. (a) See Definition 1.(b) It must pass the Horizontal Line Test.3. No 5. No 7. Yes 9. No11. Yes 13. No 15. 2 17. 019. ; the Fahrenheit temperature as a function of theCelsius temperature; ��273.15, ��

F � 95 C � 32

b � 1.017764706a � 3.154832569 10�12y � ab t

200 � 2�t�5

t � 26.9 h60,000

0 40

100 � 2 t�3x � 35.8f �x� � 3 � 2x

��, ��, �1� � ��1, 1� � �1, ��y � �e�xy � e�x

y � �e xy � e x�2y � e x � 2

x

y

0

y=1- e–®

y=1

12

”0, ’12

x_1

y

0

y=_2–®

_1_2 0

1

y

x

y=10x+2

5

_2 2

y=3®y=10®

0

y=” ’®13

y=” ’®110

�0, 1�x l ��y=20® y=5® y=´

y=2®

5

_1 20




21. ,

23. 25.

27. 29.

31. (a) , ; and are the samefunction. (b) Quarter-circle in the first quadrant

33. (a) It’s defined as the inverse of the exponential function withbase a, that is, .(b) (c) (d) See Figure 11.

35. (a) 3 (b) 37. (a) 3 (b) 39.

41.

43. All graphs approach as , all pass

through , and all are increasing. The largerthe base, the slower therate of increase.

45. About 1,084,588 mi47. (a) (b)

49. (a) (b)51. (a) (b)53. (a) (b)

55. (a) (b) , 57.

The graph passes the Horizontal Line Test.

,

where ; two of the expressions arecomplex.

59. (a) ; the time elapsed when thereare n bacteria (b) After about 26.9 hours

f �1�n� � �3�ln 2� ln�n�100�

D � 3s3s27x 4 � 40x 2 � 16

f �1(x) � �16 s

3 4 (s3 D � 27x 2 � 20 � s3 D � 27x 2 � 20 � s

3 2 )

5

_1

4_2

[0, s3)f �1�x� � 12 ln�3 � x2�(��, 12 ln 3]

x � 1�ex ln 10

12 (1 � s1 � 4e )5 � log2 3 or 5 � �ln 3��ln 2

13�e2 � 10�1

4�7 � ln 6�

1 x

y

0

y=-ln x

_5 _4 x

y

0

y=log10 (x+5)

�1, 0�x l 0��

3

�5

4

y=log1.5 x

y=log10 x0

y=ln x

y=log50 x

ln �1 � x 2�sx

sin x

ln 1215�2�3

��0, ��loga x � y &? a y � x

ff �10 � x � 1f �1�x� � s1 � x 2

x

y

f

f–!

0

6

60

f–!

f

f �1�x� � s4 x � 1

y � e x � 3y � 12�1 � ln x�

x � 1y � 13�x � 1�2 �

23

61. (a) (b) (c)(d) (e) (f) (g)(h)


1. 3.

5. (a) (b)

7. (a) (b)

9. (a) , (b)

11. (a) , (b)

13. (a) (b) y

x0 1

1

y � 12 ln x � 1

y

x0

(1, 1)

y � 1y � 1�x

0 1

1y

x_1

y � 0x 2 � y 2 � 1

y � 1 � x 2, x � 0y

0 x

(0, 1) t=0

(1, 0) t=1

(2, _3) t=4

y � 23 x �

133

(_8, _1)t=_1

x

y

0

(_5, 1)t=0

(_2, 3)t=1

(1, 5)t=2

0 1

1t=0(1, 1)

(0, 0)

y

x

t= π2t= π3

t= π6

0 2

2t=0(0, 0)

t=2(6, 2)

t=_2(2, 6)

y

x

y � e x � 3y � �e xy � e�xy � e xy � ln��x�

y � �ln xy � ln�x � 3�y � ln x � 3


15. (a) , (b)

17. Moves counterclockwise along the circle from to

19. Moves 3 times clockwise around the ellipse , starting and ending at

21. It is contained in the rectangle described by and .23.

25.

27.

29. (b) , 31. (a)(b)(c)35. The curve is generated in (a). In (b), only the portionwith is generated, and in (c) we get only the portion with

.39. , ellipse41. (a) Two points of intersection

(b) One collision point at when (c) There are still two intersection points, but no collision point.

t � 3��2��3, 0�

4

�4

�6 6

x � a cos �, y � b sin �; �x 2�a 2 � � �y 2�b 2 � � 1x � 0

x � 0y � x 2�3

x � 2 cos t, y � 1 � 2 sin t, ��2 � t � 3��2x � 2 cos t, y � 1 � 2 sin t, 0 � t � 6�

x � 2 cos t, y � 1 � 2 sin t, 0 � t � 2�0 � t � 1y � 7 � 8tx � �2 � 5t,

π

_π

4_4

x

y

t=0

t= 12

1

1

y

x

(0, _1) t=_1

(0, 1) t=1

(_1, 0)t=0

2 � y � 31 � x � 4

�0, �2��x 2�25� � �y 2�4� � 1

�3, �1��3, 3��x � 3�2 � �y � 1�2 � 4

0

y

x

(_1, _1) (1, _1)

(0, 1)�1 � x � 1y � 1 � 2x 2 43. For , there is a cusp; for , there is a loop whose size

increases as c increases.

45. As n increases, the number of oscillations increases; a and b determine the width and height.

CHAPTER 1 REVIEW N PAGE 80

True-False Quiz

1 False 3. False 5. True 7. False 9. True11. False

Exercises1. (a) 2.7 (b) 2.3, 5.6 (c) (d)(e) (f) No; it fails the Horizontal Line Test.(g) Odd; its graph is symmetric about the origin.3. 5. ,7.9. (a) Shift the graph 8 units upward.(b) Shift the graph 8 units to the left.(c) Stretch the graph vertically by a factor of 2, then shift it 1 unit upward.(d) Shift the graph 2 units to the right and 2 units downward.(e) Reflect the graph about the x-axis.(f) Reflect the graph about the line (assuming f is one-to-one).

11. 13.

15.

17. (a) Neither (b) Odd (c) Even (d) Neither19. (a) , (b) , (c) , (d) , 21. ; about 77.6 years23. 1 25. (a) 9 (b) 227. (a) (b)(c) ; the time elapsed when there are m grams of (d) About 26.6 days100 Pd

t�m� � �4 log2 mm�t� � 2�t�41

16 g

y � 0.2493x � 423.4818��, ��t � t��x� � �x 2 � 9�2 � 9

�1, �� f � f ��x� � ln ln x�0, ��t � f ��x� � �ln x�2 � 9

��, �3� � �3, �� f � t��x� � ln�x 2 � 9�

y

x0

y= 1x+2

12

x=_2

y

x0

y=_sin 2x

π

x

y

0

(1+´)1 y= 1

2

y= 12

y � x

��6, �, ��, 0� � �0, �(�, 1

3) � ( 13 , )2a � h � 2

��4, 4��4, 4��6, 6�

3

0 1.5

_3

_1

00 1.5

1

_1

112

c � 0c � 0




29.

For , f is defined everywhere. As c increases, the dip at becomes deeper. For , the graph has asymptotes at .

31. (a) 33.

(b)

PRINCIPLES OF PROBLEM SOLVING N PAGE 88

1. , where a is the length of the altitude and h is the length of the hypotenuse3.5. 7.

9.

11. 5 13.

15. 19.

CHAPTER 2


1. (a) , , , , (b) (c)

3. (a) (i) 0.333333 (ii) 0.263158 (iii) 0.251256(iv) 0.250125 (v) 0.2 (vi) 0.238095 (vii) 0.248756(viii) 0.249875 (b) (c)

5. (a) (i) (ii) (iii)(iv) (b) �24 ft�s�24.16 ft�s

�24.8 ft�s�25.6 ft�s�32 ft�s

y � 14 x �

14

14

�33 13�33.3�16.6�22.2�27.8�38.8�44.4

fn�x� � x 2 n�140 mi�h

x � [�1, 1 � s3) � (1 � s3, 3]

y

x10

1

y

x0

1 x

y

�73 , 9

a � 4sh 2 � 16�h

y � slnx

7

_1

_1 7

f

f – !

y

x1

(e, 1)

x � �scc � 0x � 0c 0

_13

_3

_5 5

4210

_4_2 7. (a) (i) (ii) (iii)(iv) (b)

9. (a) 0, 1.7321, �1.0847, �2.7433, 4.3301, �2.8173, 0,�2.1651, �2.6061, �5, 3.4202; no (c) �31.4


1. Yes3. (a) 2 (b) (c) Does not exist (d) 4(e) Does not exist5. (a) (b) (c) Does not exist (d) 2 (e) 0(f) Does not exist (g) 1 (h) 37. exists for all except .

9. (a) (b) (c) Does not exist11. (a) (b) (c) Does not exist13. 15.

17. 19. 21. 23. 25. (a)27. (a) 2.71828 (b)

29. (a) 0.998000, 0.638259, 0.358484, 0.158680, 0.038851,0.008928, 0.001465; 0(b) 0.000572, �0.000614, �0.000907, �0.000978, �0.000993,�0.001000; �0.00131. Within 0.021; within 0.011


1. (a) (b) (c) 2 (d)(e) Does not exist (f) 0

3. 5. 390 7. 9. 4 11. Does not exist

13. 15. 17.

19. 21. 23. 25. (a), (b)29. 33.35. Does not exist37. (a) (i) 1 (ii) 1 (iii) 3 (iv) (v)(vi) Does not exist(b)

�1�2

67

23�

12

1128�

116

11286

5

3259

�6�8�6

6

4_4

_2

�1.535

14

523

y

0 x1

y

0 x

1

_1

2

2�201

a � �1alimx l a

f �x�

�2�1

3

6.3 m�s7 m�s7.55 m�s5.6 m�s4.65 m�s


39. (a) (i) �2 (ii) Does not exist (iii) �3(b) (i) (ii) n (c) a is not an integer.45. 49.


1.3. (a) is not defined and [for , , and ]does not exist(b) , neither; , left; 2, right; 4, right

5. 7.

9. (a)

(b) Discontinuous at , 2, 3, 4

11.

15. does not exist. 17.

19. 21. 23.

25.

27. 29.33. 0, right; 1, left

35. 37. (a) (b)45. (b) 47. (b)51. Yes

70.347�0.86, 0.87�t �x� � x 2 � xt �x� � x 3 � x 2 � x � 12

3

y

x0

(1, e)

(1, 1)(0, 1)

(0, 2)

173

3

4_4

_1

x � 0

��, �1� � �1, ��, � �[ 12, �)

y

0 x1_π

x

y

0

y=´

y=≈1

limx l 0

f �x� � f �0�limx l 0

f �x�6

t � 1

Cost(in dollars)

0 Time(in hours)

1

1

y

0 x53

y

0 x2

�2�4

42a � �2limx l a

f �x�f ��4�lim x l 4 f �x� � f �4�

15; �18n � 1


1. (a) As approaches 2, becomes large. (b) As approaches 1 from the right, becomes large negative.(c) As becomes large, approaches 5. (d) As becomeslarge negative, approaches 3.3. (a) (b) (c) (d) 1 (e) 2(f)5. 7.

9.

11. 13.15. 17. 0 19. 21. 23. 25.27. 29. 0 31. Does not exist 33. 35.

37. 39. 41.

43. (a), (b) 45. 47.

49. (a) (b) 5 51. (a) (b) 53.55. (b) It approaches the concentration of the brine being pumpedinto the tank.57. (b) (c) Yes,


1. (a) (b)

3. (a) (b) (c)

5. 7.

9. (a) (b)(c) 10

_3

4_2

y � 2x � 3, y � �8x � 198a � 6a 2

y � 12 x �

12y � �8x � 12

6

05_1

y � 2x � 12

limx l 3

f �x� � f �3�

x � 3

f �x� � f �3�x � 3

x � 10 ln 10x � 23.03

5��054

f �x� �2 � x

x 2�x � 3�y � 3�

12

x � 5y � 2; x � �2, x � 1�

��016

212��

x � �1.62, x � 0.62, x � 1; y � 10

x

y

0

y=3

x=4

x

y

0

x=2y

0 x

y=5

y=_5

x � �1, x � 2, y � 1, y � 2��

f �x�xf �x�x

f �x�xf �x�x




11. (a) Right: and ; left: ; standing still: and

(b)

13.15. ; ; ; 17.19. ; 21.

23.25. (a) (b)

27. 29. 31.

33. or , 35.37. or , 39. ; 41. Greater (in magnitude)

43. (a) (i) (ii)(iii)(b) (c)45. (a) (i) (ii) (b)47. (a) The rate at which the cost is changing per ounce of goldproduced; dollars per ounce(b) When the 800th ounce of gold is produced, the cost of production is (c) Decrease in the short term; increase in the long term49. 5�F�h

$17�oz.

$20�unit$20.05�unit$20.25�unit17 million�year18.25 million�year

16 million�year20.5 million�year23 million�year

Temperature(in °F)

0 Time(in hours)

1

38

2

72

1 m�s1 m�sa � 0f �x� � cos�� x�f �x� � cos x, a � �

f �x� � 2x, a � 5a � 0f �x� � �1 � x�10f �x� � x 10, a � 1

�1

s1 � 2a

5

�a � 3�26a � 4

4

_2

6_1

�35; y � �

35 x �

165

y � 3x � 1

y

0x1

1

f ��2� � 4f �2� � 3t��0�, 0, t��4�, t��2�, t��2�

�227 m�s�

14 m�s�2 m�s�2�a3 m�s

�24 ft�s

t(seconds)

v (m/s)

0 1

1

3 t 41 t 22 t 34 t 60 t 1 51. (a) The rate at which the oxygen solubility changes with

respect to the water temperature; (b) as the temperature increases past , the oxygen solubility is decreasing at a rate of .53. Does not exist


1. (a) (b) (c) (d)(e) (f) (g)

3. (a) II (b) IV (c) I (d) III5. 7.

9. 11.

13. 1963 to 1971

15.

17. (a) 0, 1, 2, 4 (b) �1, �2, �4 (c)

19. , 21.23. ,

25.

27. , ,

29. 31. (a) f ��x� � 4x 3 � 2f ��x� � 4x 3, �, �

��, �1� � ��1, ��, �1� � ��1, ��G��t� �4

�t � 1�2

t��x� � 1�s1 � 2x, [� 12 , �), (� 1

2 , �)�, �f ��x� � 2x � 6x 2

f ��t� � 5 � 18t, �, ��, �f ��x� � 12

f ��x� � 2x

f ��x� � e xy

x1

1

0

f, f ª

0.05

19901980197019601950

_0.03t

y=Mª(t)0.1

y

x0

y

f ª

y

0 x

f ª

x

y

0

fªf ª

x

y

0

0

1

2

21_1_2

_3

y

x

3

fª

�0.201210�0.2

0.25 �mg�L��C16�CS��16� � �0.25;

�mg�L��C


33. (a) The rate at which the unemployment rate is changing, inpercent unemployed per year(b)

35.37.39.

Differentiable at �1;not differentiable at 0

41.43.45.

47. ,

,

,

49. (a)

51.

or

55.


Abbreviations: inc, increasing; dec, decreasing; loc, local; max, maximum; min, minimum

1. (a) Inc on , ; dec on (b) Loc max at ; loc min at (c) y

0 x

f

4 5321

x � 4x � 1�1, 4��4, 5��0, 1�

63�

f ��x� �x � 6

� x � 6 �x

y

0 6

1

_1

f ªf ��x� � �1

1

if x 6

if x � 6

13 a�2�3

f �4��x� � 0

f ��x� � �6

f ��x� � 4 � 6x

f ��x� � 4x � 3x 23

6�4

�7

f

fª

f ·

f ªªª

7

_1

4

fª

f ·

f

_4

6x � 2; 6a � acceleration, b � velocity, c � positiona � f, b � f �, c � f �

2

_1

_2 1

�1 �vertical tangent�; 4 �corner��4 �corner�; 0 �discontinuity�

3. (a) Inc on , , ; dec on , (b) Loc max at ; loc min at (c)

5. Inc on ; dec on 7.

9. If is the size of the deficit as a function of time, then at thetime of the speech , but .

11. (a) The rate starts small, grows rapidly, levels off, thendecreases and becomes negative.(b) ; the rate of change of populationdensity starts to decrease in 1932 and starts to increase in 1937.

13. ; CD15. (a) Inc on (0, 2), (4, 6), ;dec on (2, 4), (6, 8)(b) Loc max at ;loc min at (c) CU on (3, 6), ; CD on (0, 3)(d) 3(e) See graph at right.

17. 19.

21.

23.

25. (a) Inc on ; dec on (b) Min at 27. (a) Inc on , ; dec on (b) CU on ; CD on (c) IP at 29. b

�0, 0��, 0��0, ��

(�s13 , s

13)(s1

3 , �)(��, �s13)

x � 0��, 0��0, ��

x

y

0_2x=2

x

y

0_2 2

x

y

0 1 32 4

1 2 3 4 x

yy

0 x

�6, ��x � 4, 8

x � 2, 6

x

y

0 2 4 6 8

�8, ��K�3� � K�2�

�1932, 2.5� and �1937, 4.3�

D ��t� 0D��t� � 0D�t�

f ��1��, 2� and �5, ��2, 5�

0 1

f

_12

y

x

x � 0, 2x � �1, 1�1, 2��1, 0��2, 3��0, 1��2, �1�

t t

1998 20031999 20042000 20052001 20062002 2007 0.000.65

�0.250.90�0.450.25�0.45�0.25�0.15�0.30

U��t�U��t�




31.

33.


True-False Quiz

1. False 3. True 5. False 7. True 9. True11. False 13. True 15. False 17. False

Exercises1. (a) (i) 3 (ii) 0 (iii) Does not exist (iv) 2(v) (vi) (vii) 4 (viii) �1(b) , (c) , (d) �3, 0, 2, 43. 1 5. 7. 3 9. 11. 13. 15.17. 19. 21. 123. (a) (i) 3 (ii) 0 (iii) Does not exist (iv) 0 (v) 0 (vi) 0(b) At 0 and 3 (c)

27. (a) (i) (ii) (iii)(iv) (b)29.31. (a) (b)(c)

33. (a) The rate at which the cost changes with respect to the interest rate; dollars�(percent per year)(b) As the interest rate increases past 10%, the cost is increasing at a rate of $1200�(percent per year).(c) Always positive35.

x

y

0

fª

1.5

1.50

1

1

y � �0.736x � 1.104�0.736f ��5�, 0, f ��5�, f ��2�, 1, f ��3�

2.5 m�s2.525 m�s2.625 m�s2.75 m�s3 m�s

x0

y

3

3

x � 0, y � 02

12��

47�

32

x � 2x � 0y � �1y � 4��

x0

F

2π_2π

y

y

0 x

1 F

1

37. (a) (b)(c)

39. �4 (discontinuity), �1 (corner), 2 (discontinuity), 5 (vertical tangent)41. The rate at which the total value of US currency in circulationis changing in billions of dollars per year; 43. (a) Inc on ; dec on (b) Max at 0; min at (c) CU on ; CD on (d)

45.

47. (a) About 35 ft�s (b) About (c) The point at which the car’s velocity is maximized

FOCUS ON PROBLEM SOLVING N PAGE 170

1. 3. �4 5. 1 7.

9. (b) Yes (c) Yes; no 11.

13. (a) 0 (b) 1 (c) 15.

CHAPTER 3


1. (a) See Definition of the Number e (page 180).(b) 0.99, 1.03; 3. 5. 7.9. 11. 13. 15. 17.

19.

21. 23.

25. 27.

29. Tangent: ; normal:

31. 33. 35. 37. 345x 14 � 15x 2e x � 5y � 3x � 1

y � �12 x � 2y � 2x � 2

y � 14 x �

34z� � �10A�y11 � Be y

u� � 15 t �4�5 � 10t 3�2y� � 0

y� � 32 sx �

2

sx�

3

2xsx

F��x� � 532 x 4y� � 3e x �

43x�4�3

t��t� � �32 t �7�4A��s� � 60�s 6f ��t� � t 3

f ��x� � 3x 2 � 4f ��t� � �23f ��x� � 0

2.7 e 2.8

34f ��x� � x 2 � 1

(�s3�2, 14 )a � 1

2 �12 s52

3

�8, 180�

y

0x1

�2

9 12

x � 6

x

y

0

1

��1, 1��, �1� and �1, ��2 and 2

��, �2� and �0, 2��2, 0� and �2, ��$22.2 billion�year

6

1_3

_6

f

f ª

(��, 35 ], (��, 35 )f ��x� � �52 �3 � 5x��1�2



39. (a) (c)

41. ;

43.

45. (a) (b)

(c)

47. 49.

53. ,

55. 57. 61.

63. (a) ; infinitely many

(b)

(c)

65. 67.

69. 71. 73.


1. 3.

5. 7.

9.

11. 13.

15. 17.

19. 21.

23.

25.

27.

29. 31. ;

33. (a) (b)

35. (a) (b) 2

fª

f

_2

2�10

e x�x 3 � 3x 2 � x � 1�

(_1, 0.5)

1.5

�0.5

�4 4

y � 12 x � 1

y � �12 xy � 2xy � 1

2 x �12

2x 2 � 2x

�1 � 2x�2 ; 2

�1 � 2x�3

�x 4 � 4x 3�e x; �x4 � 8x3 � 12x 2�e x

f ��x� � 2cx��x 2 � c�2

f ��x� � �ACe x��B � Ce x�2f ��t� �4 � t 1�2

(2 � st )2

y� � 2v � 1�svy� � �r 2 � 2�e r

y� �2t��t 4 � 4t 2 � 7�

�t 4 � 3t 2 � 1�2y� �x 2�3 � x 2��1 � x 2�2

F��y� � 5 � 14�y 2 � 9�y 4

t��x� � 5��2x � 1�2y� � �x � 2�e x�x 3

f ��x� � e x�x 3 � 3x 2 � 2x � 2�1 � 2x � 6x 2 � 8x 3

3; 11000a � �12 , b � 2

y � 316 x 3 �

94 x � 3y � 2x 2 � x

F�x� � xn�1��n � 1� � C, C any real number

F�x� � 14 x

4 � C, 15 x5 � C, C any real number

F�x� � 13 x

3 � C, C any real number

P�x� � x 2 � x � 3��2, 4�y � 13 x �

13

y � 12x � 17y � 12x � 15

��2, 21�, �1, �6��, ln 5�a�1� � 6 m�s2

12 m�s2v�t� � 3t 2 � 3, a�t� � 6t

f ��x� � 2 �154 x�1�4, f ��x� � 15

16 x�5�4

f ��x� � 900x 8 � 100x 3f ��x� � 100x 9 � 25x 4 � 1

100

�40

�3 5

50

�10

�3 5

4x 3 � 9x 2 � 12x � 737. (a) ;

(b)

39. 41. (a) �16 (b) (c) 20

43. 7 45. (a) 0 (b)47. (a)

(b) (c)

49. 51.

53. Two, 55. 1 57. (c)59. ,

,


1. 3.5.7.

9. 11.

13.

19. 21.

23. (a) (b)

25. (a)

27. ;

29. (a) (b)

31. , n an integer 33.

35. (a) ,

(b) ; to the left

37. 39.

41. ,

43. 45.

47. (a) (b)

(c)

49. 1

cos x � sin x �cot x � 1

csc x

sec x tan x �sin x

cos2xsec2x �

1

cos2x

123

B � �1

10A � �310

�cos x5 ft�rad

4s3, �4, �4s3

a�t� � �8 sin tv�t� � 8 cos t

��3, 5��3��2n � 1�� 13�

f ��x� � cos x � sin xf ��x� � �1 � tan x��sec x

2 cos � sin cos � sin

sec x tan x � 1

π0

3π2

” , π’π2

y � 2x

y � x � 1y � 2s3x �23 s3� � 2

f ��x� � e x csc x ��x cot x � x � 1�

f �� sec tan

�1 � sec �2y� �2 � tan x � x sec2x

�2 � tan x�2

y� � �c sin t � t�t cos t � 2 sin t�y� � sec �sec2 � tan2 �

f ��x� � cos x �12 csc2xf ��x� � 6x � 2 sin x

f (n)�x� � �x 2 � 2nx � n�n � 1��e xf �5��x� � �x 2 � 10x � 20�e x;f �4��x� � �x 2 � 8x � 12�e x,f ��x� � �x 2 � 6x � 6�e x

f ��x� � �x 2 � 2x�e x, f ��x� � �x 2 � 4x � 2�e x

3e 3x(�2 � s3, 12 (1 � s3))��3, ��$1.627 billion�year

y� �xt��x� � t�x�

x 2y� �t�x� � xt��x�

�t�x�� 2

y� � xt��x� � t�x��

23

�209

14

4

f·fªf

_2

6_6

f ��x� �4�1 � 3x 2��x 2 � 1�3f ��x� �

4x

�x 2 � 1�2




1. 3. 5.

7.

9. 11.

13. 15.

17.

19.

21. 23.

25. 27.

29. 31.

33.

35.

37. ;

39. ;

41. 43.45. (a) (b)

47. (a)

49. , n an integer51. 24 53. (a) 30 (b) 3655. (a) (b) Does not exist (c) �2 57. �17.4

59. (a) (b)61. 120 63. 9667. 69.

71. (a) (b) 0.16

73.

75. is the rate of change of velocity with respect to time;is the rate of change of velocity with respect to displacement

77. (a) , (b) �A79. 81.83. Horizontal at , vertical at �10, 0��6, �16�

y � ��2�e�x � 3y � �x�670.63

b � 0.0000451459Q � abt, where a � 100.012437dv�ds

dv�dt

15

�7

0 2

√

2

�1

0 2

s

v�t� � 2e�1.5 t�2� cos 2�t � 1.5 sin 2�t�

dB

dt�

7�

54 cos

2� t

5.4

v�t� � 52� cos�10� t� cm�s�250 cos 2x

G��x� � e f �x� f ��x�F��x� � e x f ��e x�

34

��2� � 2n�, 3�, ��3��2� � 2n�, �1�

f ��x� �2 � 2x 2

s2 � x 2

3

_1.5

_3 3

(0, 1)

y � 12 x � 1

y � �x � �y � 20x � 1e�x�� 2 � 2� sin x � 2� cos x�

e�x� cos x � � sin x�y � �4x 2 cos�x 2� � 2 sin�x 2�y� � �2x sin�x 2�

y� �� cos�tan �x� sec2��x� sinssin �tan �x�

2ssin �tan �x�

y� � �2 cos � cot�sin �� csc2�sin ��y� � 2sin �x�� ln 2� cos �xy� � 2 cos�tan 2x� sec2�2x�

y� � �r 2 � 1��3�2y� � 4 sec 2x tan x

y� ��12x �x 2 � 1�2

�x 2 � 1�4y� � �cos x � x sin x�e x cos x

y� � 8�2x � 5�3 �8x2 � 5��4 ��4x2 � 30x � 5�y� � e�kx ��kx � 1�

h��t� � 3t 2 � 3 t ln 3y� � �3x 2 sin�a 3 � x 3 �

f ��z� � �2z

�z 2 � 1�2F��x� � �1

s1 � 2x

F��x� � 10x�x4 � 3x 2 � 2�4 �2x 2 � 3�

esx�(2sx )� sec2�x4

3s3 �1 � 4x�2

85. (a) , (b) Horizontal at ; vertical at (c)

87. (b) The factored form 89. (b)


1. (a)(b) ,

3. 5.

7. 9.

11. 13.

15. 17.

19. 21.

23. 25. 27.29. (a) (b)

31. 33. 35.

37. (a) Eight;

(b) , (c)

39.41. 43.

x

y

x

y

(�54s3, �5

4)1 �

13s3y � 1

3 x � 2y � �x � 1

x � 0.42, 1.584

5_2

_3

1�e 2�2x�y 5�81�y 3

5

2_2

_2

(1, 2)

y � 92 x �

52

y � �913 x �

4013y � x �

12y � �x � 2

y � 12 xx� �

�2x4y � x3 � 6xy2

4x3y2 � 3x 2y � 2y3

�1613y� �

e y sin x � y cos�xy�e y cos x � x cos�xy�

y� �y�y � e x�y�y 2 � xe x�yy� � tan x tan y

y� ��2xy 2 � sin y

2x 2y � x cos yy� �

3y2 � 5x4 � 4x 3y

x 4 � 3y 2 � 6xy

y� �2x � y

2y � xy� � �x 2�y 2

y� � ��4�x 2 � � 3y � �4�x� � 2 � 3xy� � ��y � 2 � 6x��x

�n cosn�1x sin��n � 1�x�

3

�3

�1 5

�0, 0��1, �2�y � �s3x � 3s3y � s3x � 3s3



47. (a) (b)

51. 53. 55. (a) 0 (b)


1. (a) (b) 3. (a) (b)

5. 7. 11.

13.

The second graph is the reflection of the first graph about the line .

17. 19.

21.

23. 25.

27. 29.

31. ; [1, 2], (1, 2) 33.

35. 37. 39. 41. (b)


1. The differentiation formula is simplest.

3. 5.

7. 9.

11. 13.

15. 17.

19.

21.

23. ;

25. 27.

29. (a) (b) 31.

33. y� � �2x � 1�5�x 4 � 3�6� 10

2x � 1�

24x 3

x 4 � 37�0, ��0, 1�e�

y � 3x � 2y � 3x � 9

�1, 1 � e� � �1 � e, �

f ��x� �2x � 1 � �x � 1� ln�x � 1�

�x � 1��1 � ln�x � 1�� 2

y� � x � 2x ln�2x�; y � 3 � 2 ln�2x�

y� �1

ln 10� log10 x

y� ��x

1 � xy� �

10x � 1

5x 2 � x � 2

t��x� �2x 2 � 1

x �x 2 � 1�F��t� �

6

2t � 1�

12

3t � 1

f ��x� �sin x

x� cos x ln�5x�f ��x� �

1

5xs5 �ln x�4

f ��x� �3

�3x � 1� ln 2f ��x� �

cos�ln x�x

32��2��21 �

x arcsin x

s1 � x 2

��6t��x� �2

s1 � �3 � 2x�2

y� �sa 2 � b 2

a � b cos xy� � sin�1x

y� � �sin �

1 � cos2�y� � �

2e2x

s1 � e 4x

G��x� � �1 �x arccos x

s1 � x 2

y� �1

s�x 2 � xy� �

2 tan�1x

1 � x 2

y � x

y=sin– ! x

π2

π2_

y=sin x

π2_

π2

x�s1 � x 2 23 s22�s5

��4��4��3

�12��1, �1�, �1, 1�(�s3, 0)

�4.04 L�atmV 3�nb � V �

PV 3 � n2aV � 2n3ab 35.

37.

39.

41.

43. 45.


1. (a) (b) (c)(d) (e)(f) (g)

(h) (i) Speeding up when;

slowing down when

3. (a) (b) (c)

(d) (e) 4 ft(f)

(g)(h)

(i) Speeding up when , , ; slowing down when , 5. (a) Speeding up when or ; slowing down when (b) Speeding up when or ;slowing down when or 7. (a)(b) ; the velocity has an absolute minimum.9. (a) (b)11. (a) ; the rate at which the area is increasing with respect to side length as x reaches 15 mm(b) �A � 2x �x

30 mm2�mms17 m�s5.02 m�s

t � 1.5 st � 4 s

2 � t � 30 � t � 13 � t � 41 � t � 2

1 � t � 22 � t � 30 � t � 1

6 � t � 82 � t � 48 � t � 104 � t � 60 � t � 2

_1

1

80

a

√

s

132� 2

s2 ft�s2�1

16�2 cos�� t�4�;

t � 0, s � 1

t � 4,s � _1

t � 8, s � 1

s0

t =10,s=0

4 � t � 8

t � 0, 4, 8�18�s2 ft�s�

�

4 sin�� t

4

0 � t � 2 or 4 � t � 6

2 � t � 4 or t � 6s

40

80

�25

√

a

6t � 24; �6 ft�s2

t � 2,s � 32t � 0,

s � 0

t � 6,s � 0

t � 8,s � 32

s0 20

96 ft0 � t � 2, t � 6t � 2, 6�9 ft�s3t 2 � 24t � 36

f �n��x� ��1�n�1�n � 1�!

�x � 1�ny� �2x

x 2 � y 2 � 2y

y� � �tan x�1�x� sec2x

x tan x�

ln tan x

x 2 y� � �cos x�x��x tan x � ln cos x�y� � x x�1 � ln x�

y� �sin2x tan4x

�x 2 � 1�2 �2 cot x �4 sec2x

tan x�

4x

x 2 � 1



13. (a) (i) (ii) (iii)(b) (c)

15. (a) (b) (c)The rate increases as the radius increases.

17. (a) (b) (c)At the right end; at the left end

19. (a) 4.75 A (b) 5 A;

21. (a) (b) At the beginning

23.

25. (a)(b) , where ,

, , (c)(d) ; (smaller)(e)

27. (a) ; ; 0(b) 0; ; (c) At the center; at the edge

29. (a)(b) ; the cost of producing the 201st yard(c) $32.20

31. (a) ; the average productivity increases asnew workers are added.

33.

35. (a) 0 and 0 (b)(c) it is possible for the species to coexist.


1. ; underestimate

3. 22.6%, 24.2%; too high; tangent lines lie above the curve

5. 7.

9. ;

,

11. 13.

15. 17.

23. (a) (b)

25. (a) (b) ;

27. (a) , 0.01, 1% (b) , ,

29. (a) ;

(b) ;

31. (a) (b)

33. A 5% increase in the radius corresponds to a 20% increase inblood flow.

35. (a) 4.8, 5.2 (b) Too large

��r�2h2�rh �r

156 � 0.018 � 1.8%1764�� 2 � 179 cm3

184 � 0.012 � 1.2%84�� 27 cm2

0.6%0.00636 cm2270 cm3

0.01010.01dy � 110 e

x�10 dx

dy � �6r 2

�1 � r 3�3 drdy � �

2

�u � 1�2 du

4.0232.08

�0.045 � x � 0.055�1.204 � x � 0.706

s0.99 � 0.995s0.9 � 0.95

3

3_3

_1

(0, 1)

(1, 0)

y=œ„„„„1-x

y=1- x12

s1 � x � 1 �12 x

L�x� � �x � ��2L�x� � �10x � 6

148�F

�0, 0�, �500, 50�;C � 0

�0.2436 K�min

�xp��x� � p�x��x 2

$32�yardC��x � � 12 � 0.2x � 0.0015x 2

�185.2 �cm�s��cm�92.6 �cm�s��cm0.694 cm�s0.926 cm�s

81.62 million�year75.29 million�year14.48 million�year

P��t� � 3at 2 � 2bt � cd � �7,743,770c � 12,822.979b � �7.061422

a � 0.00129371P�t� � at 3 � bt 2 � ct � d78.5 million�year16 million�year;

400�3t� ln 3; �6850 bacteria�h

dV�dP � �C�P 2

t � 23 s

18 kg�m12 kg�m6 kg�m

24� ft2�ft16� ft2�ft8� ft2�ft

�A � 2�r �r4�4.1�4.5�5� CHAPTER 3 REVIEW N PAGE 248

True-False Quiz

1. True 3. True 5. False 7. False 9. True11. True

Exercises1. 3.

5. 7.

9. 11. 13.

15. 17.

19. 21.

23. 25.

27. 29.

31.

33.

35. 37.

39. 41.43. 45. ;

47. (a) (b)

(c)

49.The sizes of the oscillations of and are linked.

51. (a) 2 (b) 44 53. 55.

57. 59. 61.

63. 65. 67.69. ,

71.73. (a)(b) ; the approximate cost of producing the 101st unit(c)(d) About 95.24; at this value of the marginal cost is minimized.x

C�101� � C�100� � 0.10107$0.10�unit

C��x� � 2 � 0.04x � 0.00021x 24 kg�m

a�t� � Ae�ct ��c 2 � �2 � cos��t � �� 2c� sin��t � ��v�t� � �Ae�ct �c cos��t � �� sin��t � ��

y � �23 x 2 �

143 x( �2�s6, �1�s6)��3, 0�

f ��x��t�x�� 2 � t��x� � f �x�� 2

� f �x� � t�x�� 2t��x��t�x�t��e x �e x

2t�x�t��x�2xt�x� � x 2t��x�

f �f

f

40

�50

_6π 6π

fª

esin x�x cos x � 1�

(4, 4)

10

_10

_10 10(1, 2)

ƒ

y � 74 x �

14 , y � �x � 8

10 � 3x

2s5 � x

y � x � 2y � �x � 2y � 2x � 2y � 2s3x � 1 � �s3�32 x�ln 2�n

�4

27

�3 sin(estan 3x)estan 3x sec2�3x�2stan 3x

cos(tan s1 � x 3 )(sec2s1 � x 3 ) 3x 2

2s1 � x 3

2 cos � tan�sin �� sec2�sin ��

5 sec 5x4x

1 � 16x 2 � tan�1�4x�

cot x � sin x cos x3x ln x�ln 3��1 � ln x�

2x � y cos�xy�x cos�xy� � 1

2

�1 � 2x� ln 5

�1 � c 2 �e cx sin x2 sec 2� �tan 2� � 1�

�1 � tan 2��2

1 � y 4 � 2xy

4xy 3 � x 2 � 3�

e1�x�1 � 2x�x 4

t 2 � 1

�1 � t 2�2

2 cos 2� esin 2�2�2x 2 � 1�sx 2 � 1

1

2sx�

4

3s3 x7

6x�x 4 � 3x 2 � 5�2�2x 2 � 3�



75. (a) ; (b)

77. 79.


1. 5.

7. (a) (b)

(c)

11.

15. 17. (b) (i) (or ) (ii) (or )

19. R approaches the midpoint of the radius AO.

21. 23.

CHAPTER 4


1. 3. 5.7. (a) 1 (b) 25 9.11. (a) The rate of decrease of the surface area is .(b) The rate of decrease of the diameter when the diameter is 10cm(c) (d)

(e)

13. (a) The plane’s altitude is 1 mi and its speed is .(b) The rate at which the distance from the plane to the station isincreasing when the plane is 2 mi from the station(c) (d)

(e)

15. 17.19. 21.23. 5 m 25. 27.29. 31. 33.35. 37. (a) (b)39. 41.

43.


Abbreviations: abs, absolute; loc, local; max, maximum; min, minimum

1. Abs min: smallest function value on the entire domain of thefunction; loc min at c: smallest function value when x is near c3. Abs max at , abs min at , loc max at , loc min at and 5. Abs max , loc max and , loc min and f �1� � f �5� � 3f �2� � 2

f �6� � 4f �4� � 5f �4� � 5rbcrs

74 s15 � 6.78 m�s

1650�s31 � 296 km�h109 � km�min

0.096 rad�s360 ft�s107810 � 0.132 ��s

80 cm3�min0.3 m2�s6��5�� 0.38 ft�min

45 ft�min10�s133 � 0.87 ft�s

72013 � 55.4 km�h�1.6 cm�min

837�s8674 � 8.99 ft�s65 mi�h

250s3 mi�hy 2 � x 2 � 1

y

x

1

500 mi�h

1��20�� cm�minS � �x 2

rx

1 cm2�min�

4613

3��25�� m�min48 cm2�sdV�dt � 3x 2 dx�dt

s29�58�1, �2�, ��1, 0�

117�63�127�53�2se

xT � �3, �, yT � �2, �, xN � (0, 53 ), yN � (�52 , 0)

�480� sin � (1 � cos ��s8 � cos2� ) cm�s

40(cos � � s8 � cos2� ) cm4�s3�s11 rad�s

3s2 (0, 54 )

14�cos ��3 � �s3�2

�0.23 � x � 0.40s3 1.03 � 1.01L�x� � 1 � x; s3 1 � 3x � 1 � x 7. 9.

11. (a) (b)

(c)

13. (a) (b)

15. Abs max

17. None 19. Abs max

21. Abs max 23.

25. 27. 29.

31. 33. 35.

37. 39. 41. ,

43. , 45. ,

47. ,

49. ,

51.

53.

55.

57. (a) (b) ,

59. (a) 0.32, 0.00 (b) 61.63. Cheapest, (June 1994); most expensive, (March 1998)t � 4.618

t � 0.855�3.9665�C3

16 s3, 0

�625 s

35 � 26

25 s35 � 22.19, 1.81

f� a

a � b �a abb

�a � b�a�b

f ��6� � 32s3, f ��2� � 0

f �1� � ln 3, f (�12) � ln 34

f ��1� � �1�s8 ef �2� � 2�se

f ��1� � �s3f (s2) � 2

f ��1� � 2f �3� � 66f �2� � �19f ��1� � 8

f �5� � 7f �2� � 16100, 23

n� �n an integer�0, 87, 40, 49

0, 20, 12 (�1 � s5 )�4, 2

13f �0� � 1

f �2� � ln 2

f �3� � 4

y

0 x

y

0 x2

_1

y

0 x1

_1

2

1

2

3

y

0 x1

_1

2

1

3

y

0 x1

_1

2

1

3

y

x0 54321

3

2

1

y

x0 51 2 3 4

1

2

3



65. (a) (b)

(c)


Abbreviations: inc, increasing; dec, decreasing; CD, concavedownward; CU, concave upward; HA, horizontal asymptote; VA, vertical asymptote; IP, inflection point(s)

1. 0.8, 3.2, 4.4, 6.1

3. (a) I /D Test (b) Concavity Test(c) Find points at which the concavity changes.

5. (a) 3, 5 (b) 2, 4, 6 (c) 1, 7

7. (a) Inc on ; (b) Loc max ; loc min (c) CU on ; CD on ; IP

9. (a) Inc on , ; dec on , (b) Loc max ; loc min (c) CU on , ;CD on ; IP

11. (a) Inc on , ; dec on (b) Loc max ; loc min (c) CU on ; CD on , ; IP

13. (a) Inc on ; dec on (b) Loc min (c) CU on

15. (a) Inc on ; dec on (b) Loc max (c) CU on ; CD on ; IP

17. Loc max

19. (a) has a local maximum at 2.(b) has a horizontal tangent at 6.

21. (a) Inc on ;dec on (b) Loc max ;loc min (c) CU on ; CD on ;IP (d) See graph at right.

23. (a) Inc on ;dec on (b) Loc max ;loc min (c) CU on ; CD on ; IP(d) See graph at right.

(�1�s3, 239 )

(� , �1�s3), �1�s3, )(�1�s3, 1�s3)

f �0� � 2f ��1� � 3, f �1� � 3

��1, 0�, �1, �

x10

(1, 3)(_1, 3)

” , ’y

1

239

1œ„3”_ , ’23

91

œ„3

�� , �1�, �0, 1�

( 12, � 13

2 )(� , 12 )( 1

2 , )f �2� � �20

f ��1� � 7��1, 2�

y

0 x

(_1, 7)

(2, _20)

” , _ ’12

132

�� , �1�, �2, �

ff

f ( 34 ) � 5

4

(e 8�3, 83 e�4�3 )�0, e 8�3 ��e 8�3, �f �e 2 � � 2�e

�e 2, ��0, e 2 �� , �f (�1

3 ln 2) � 2�2�3 � 21�3

(� , �13 ln 2)(�1

3 ln 2, )�3��4, 0�, �7��4, 0�

�7��4, 2��0, 3��4��3��4, 7��4�f �5��4� � �s2f ��4� � s2

��4, 5��4��5��4, 2��0, ��4�(�s3�3, 22

9 )(�s3�3, s3�3)(s3�3, )(� , �s3�3)

f ��1� � 2f �0� � 3�0, 1�� , �1��1, ��1, 0�

(�12,

372 )(� , �1

2)��12, �

f �2� � �44f ��3� � 81dec on ��3, 2�� , �3�, �2, �

√

0 r

kr#̧427

r¸23 r¸

v � 427 kr 0

3r � 23 r0 25. (a) Inc on ;

dec on (b) Loc max ; loc min (c) CU on ;CD on ; IP (d) See graph at right.

27. (a) Inc on ;dec on (b) Loc min (c) CU on (d) See graph at right.

29. (a) Inc on ;dec on (b) Loc min (c) CU on , ; CD on ;IP , (d) See graph at right

31. (a) Inc on ;dec on (b) Loc min (c) CU on ; CD on , ;IP , (d) See graph at right.

33. (a) HA , VA ,

(b) Inc on , ;

dec on

(c) Loc max

(d) CU on , ; CD on (e) See graph at right.

35. (a) HA (b) Dec on (c) None(d) CU on (e) See graph at right.

37. (a) VA (b) Dec on (c) None(d) CU on (0, 1); CD on ; IP (1, 0)(e) See graph at right.

�1, e�

�0, e�y

0 x

(1, 0)1

x=ex=0x � 0, x � e

�� , �

�� , �

x

y

0

1

y � 0

��1, 1��1, �� , �1�

f �0� � 0

�0, 1�, �1, ��1, 0�� , �1�

x

y

0

x=1x=_1

y=1

x � 1x � �1y � 1

(5��3, 54)(��3, 54)�5��3, 2��0, ��3�

��3, 5��3�f �� 1

�0, ��

¨(π, _1)

” , ’

y

π3

54 ” , ’5π

3541

_10 π 2π

��, 2��

(2, 6s3 2 )�0, 0�

�0, 2��2, �� , 0�

C��1� � �3�� , �1�

x

y

_4 0

{ 2, 6 Œ„2 }

(_1, _3)

��1, �

��3, �A��2� � �2

��3, �2�

x

y

_3

_2

_2

2

��2, �

��1, 3�� , �1��1, �

h �0� � �1h ��2� � 7

��2, 0�

x_1

(_1, 3)

(0, _1)

(_2, 7) y

7�� , �2�, �0, �



39. (a) HA , VA (b) Inc on , (c) None(d) CU on , ;

CD on ; IP (e) See graph at right.

41.43. (a) Loc and abs max , no min(b)

45. (b) CU on , ; CD on , , ; IP , , , 47. CU on ; CD on 49. (a) Very unhappy (b) Unhappy (c) Happy(d) Very happy

51. , ,

53. 55.

57. 28.57 min, when the rate of increase of drug level in the blood-stream is greatest; 85.71 min, when rate of decrease is greatest59. 63. 1771. (a) , (b) at


1. Inc on , ; dec on , ;loc max ; loc min , ;CU on , ; CD on ; IP ,

3. Inc on , ; dec on , ;loc max ; loc min ,

; CU on , ,; CD on , ;

IP , , ,

60,000

f

_30,000

10_10

10,000,000

f

_13,000,000

25_25

�15.08, �8,150,000��2.92, 31,800�� 11.34, �6,250,000��0, 0�

�2.92, 15.08��11.34, 0��15.08, ��0, 2.92��, �11.34�f �18.93� � �12,700,000

f ��15� � �9,700,000f �4.40� � 53,800�4.40, 18.93��, �15�

�18.93, ��15, 4.40�_6

10

0 4

ƒ

2.73.96

4.04

2.4

ƒ

�2.54, 3.999��1.46, �1.40��1.46, 2.54��2.54, ��, 1.46�

f �2.58� � 3.998f �0.92� � �5.12f �2.5� � 4�2.5, 2.58��, 0.92��2.58, ��0.92, 2.5�

�0, 0�y � �xb � �1a � 0f �x� � 1

9 �2x 3 � 3x 2 � 12x � 7�

y

xLL/20

m

0 √

(0, m¸) √=c

�2 � t � 2�2�t 2 � 4�9�t 2 � 4�3

2t

3t 2 � 12

��0.6, 0.0��, �0.6�, �0.0, ��5.35, 0.44��3.71, �0.63��2.57, �0.63��0.94, 0.44�

�5.35, 2��2.57, 3.71��0, 0.94��3.71, 5.35��0.94, 2.57�

14 (3 � s17)

f �1� � s2�3, ��

(� 12 , 1�e 2)(� 1

2 , �)(�1, � 1

2 )��, �1�

��1, ��, �1�

x

y

0x=_1

y=1

x � �1y � 1 5. Inc on , , ;dec on , ; loc max ;CU on , , ;CD on , ; IP

7. Inc on , ; dec on ,; loc max ;

loc min , ; CU on ,CD on ; IP ,

9. Inc on ; dec on ,, ; CU on ,

CD on ,

11. Loc max , ,; loc min

13.

CU on , , , , ;CD on , , ; IP , , , ,��0.1, 0.0000066�

��0.5, 0.00001��1, 0��5.0, �0.005��35.3, �0.015��0.5, �0.1��5.0, �1��, �35.3�

�4, ��2, 4��0.1, 2��1, �0.5��35.3, �5.0�

f ��x� � 2 �x � 1��x 6 � 36x 5 � 6x 4 � 628x 3 � 684x 2 � 672x � 64�

�x � 2�4�x � 4�6

f ��x� � �x �x � 1�2�x 3 � 18x 2 � 44x � 16�

�x � 2�3�x � 4�5

0.03

82.50

500

2�1

�1500

0.02

�3.5�8

�0.04

y

x

1

f �3� � 0f �5.2� � 0.0145f �0.82� � �281.5f ��5.6� � 0.018

75

_10

f

_1 1

1

0.95

f

_100 _1

(�12 � s138, 0)(��, �12 � s138)�0, ��;(�12 � s138, �12 � s138)�0, ��8 � s61, 0�

(��, �8 � s61)(�8 � s61, �8 � s61)

_4 4

_10

30

ƒ

_2.5 0

10

6

ƒ

�1.28, �1.48��1.28, 8.77��1.28, 1.28��1.28, 4�;��4, �1.28�f �2.89� � �9.99f ��1.49� � 8.75

f ��1.07� � 8.79��1.07, 2.89��4, �1.49��2.89, 4��1.49, �1.07�

3

_3

_5 5

��0.506, �0.192��0.24, 2.46��1.7, �0.506��2.46, ��0.506, 0.24��, �1.7�f �1� � �

13�2.46, ��1, 2.46�

�0.24, 1��1.7, 0.24��, �1.7�



15. Inc on ; dec on ; loc max ; CU on ; CD on ; IP

17. Inc on , , , ,;

dec on , , , ;loc max , , ,

; loc min , ; CU on , ; CD on , ,

; IP at ,

19. Inc on , ;CU on , ; CD on , ; IP

21. Max , , ; min , , ;IP , , ,

,

1

0.99970.55 0.73

1.2

_1.2

_2π 2π

1.2

�1.2

0 π

f

�2.28, 0.34��1.75, 0.77��1.17, 0.72��0.66, 0.99998��0.61, 0.99998�

f �2.73� � �0.51f �1.46� � 0.49f �0.64� � 0.99996f �1.96� � 1f �0.68� � 1f �0.59� � 1

� 0.42, 0.83��0.42, ��0.42, 0�

�0, 0.42��, �0.42�

_3 3

_1

1

ƒ

ƒ

�0, ��, 0�

5

f

020_5

�15.81, 3.91�, �18.65, 4.20��9.60, 2.95�, �12.25, 3.27��18.65, 20�

�12.25, 15.81��4.91, �4.10�, �0, 4.10�, �4.91, 9.60��15.81, 18.65��9.60, 12.25�

f �17.08� � 3.49f �10.79� � 2.43f �14.34� � 4.39

f �8.06� � 3.60f �1.77� � 2.58f ��4.51� � 0.62�14.34, 17.08��8.06, 10.79��1.77, 4.10��4.51, �4.10�

�17.08, 20��10.79, 14.34��4.91, 8.06��0, 1.77��4.91, �4.51�

0.5

f

50

�0.94, 0.34��0, 0.94��0.94, ��f �0.43� � 0.41�0.43, ��0, 0.43� 23.

Vertical tangents at , , ; horizontal tangents at,

25. For , there is a cusp; for , there is a loop whose sizeincreases as increases and the curve intersects itself at ; left-most point , rightmost point

27. For , there is no IP and only one extreme point, the origin. For , there is a maximum point at the origin, two minimum points, and two IPs, which move downward and awayfrom the origin as .

29. For , there is no extreme point and one IP, which decreases along the -axis. For , there is no IP, and one minimum point.

31. For , the maximum and minimum values are always , but the extreme points and IPs move closer to the y-axis as c

increases. is a transitional value: when c is replaced by ,the curve is reflected in the x-axis.

0.6

�0.6

�5 5

0.20.51 2

�1

4

�cc � 0

12

c � 0

10

10_10

_10

c=5

c=0c=_5

c=_ 15

c= 15

c � 0xc � 0

4

_2.3

_2.1 2.1

_3

_2_114

c l ��

c � 0c � 0

_3 3

1.5

0

1

12

1.5

_3 30

_1

0

(2cs3c�9, c�3)(�2cs3c�9, c�3)�0, c�c

c � 0c � 0((2s3 � 5)�9, 2s3�9)(�(2s3 � 5)�9, �2s3�9)

��8, 6�(� 316 , 38)�0, 0�

7.5

�1

�8.5 3



33. For , the graph has loc max and min values; forit does not. The function increases for and decreases for

. As c changes, the IPs move vertically but not horizontally.

35. (a) Positive (b)


1. (a) Indeterminate (b) 0 (c) 0(d) , , or does not exist (e) Indeterminate3. (a) (b) Indeterminate (c)5. 7. 9. 11. 13.15. 3 17. 19. 21. 23.25. 27. 29. 3 31. 0 33. 35. 37.39. 41. 43. 45. 47. 49.51. HA

53. HA , VA

55. (a) 1

_0.25

_0.25 1.75

� , � � 12e

1œ„e

y

0 x1 2 3 4 e3/2e

”e, ’1e

x � 0y � 0

21 x

y

0

”1, ’1e

y � 0

14e2e 41e�21

�12

12�1

24

12 a�a � 1��1�� 21

2ln 53

��0��2��

��

12

_12

_6 6

c=4c=1

c=0.5

c=_1

c=0.1c=0.2

c=0

c=_4

10

_10

_15 1 5

c=3 c=1c=0.5

c=_3 c=_1

c=_0.5

c=0

c �1c � 1

� c � � 1� c � � 1 (b)(c) Loc min ; CD on ; CU on

57. (a)

(b) , (c) Loc max (d) IP at 59.

For , and . For , and . As increases, the max and min points and the IPs get closer to the origin.61. 69. 71. 73.


1. (a) 11, 12 (b) 11.5, 11.5 3. 10, 105. 25 m by 25 m 7.9. (a)

(b)

(c) (d) (e)(f)11. 15. 17.19. 21. Base , height 23. Width ; rectangle height 25. (a) Use all of the wire for the square(b) m for the square27. 31.33. (a) (b)(c) 6s[h � s�(2s2)]

cos�1(1�s3) � 55�32 s 2 csc � �csc � � s3 cot ��

E 2��4r�V � 2�R3�(9s3)40s3�(9 � 4s3)

30��4 � �� ft60��4 � �� ft3r�2s3r4�r 3�(3s3)L�2, s3 L�4(�1

3 , 43 s2)4000 cm3

14,062.5 ft 2A�x� � 375x �

52 x 25x � 2y � 750A � xy

y

x

75

120 9000 ft@

250

50 12,500 ft@

125

100 12,500 ft@

N � 1

5612

169 a1

� c �lim x l�� f �x� � 0lim x l � f �x� � �c � 0

lim x l�� f �x� � ��lim x l � f �x� � 0c � 0

3

3_3

_3

_2�1

0

1 2

x � 0.58, 4.37f �e� � e 1�elim x l � x 1�x � 1lim x l0� x 1�x � 0

2

_1

0 8

�e�3�2, ��0, e�3�2�f (1�se ) � �1��2e�

limx l 0� f �x� � 0



35. km east of the refinery37. ft from the stronger source39. 41.43. (b) (i) $342,491; $342�unit; $390�unit (ii) 400(iii) $320�unit45. (a) (b) $9.5047. (a) (b) $175 (c) $10049. 53. 55.

57. At a distance from A

59. (a) About 5.1 km from B (b) C is close to B; C is close toD; , where (c) ; no suchvalue (d)

61. (a) , ,

(c)


1. (a) (b) No 3. 5. 1.17977. 9. 11.13. 15.17. , 19.21. 23. (b) 31.62277729. 31.33. 0.76286%


1.3. 5.

7.

9.

11.

13.

15.

17. 19.

21. 23.

25. 27.

29.

31. 33.

35. 37.39.

41. s�t� � 1 � cos t � sin t

y

0 x1

_1

2

1

2

3

(1, 1)

(2, 2)

(3, 1)

10x 2 � cos x �12 �x

x 2 � 2x 3 � 9x � 9�sin � � cos � � 5� � 4

�x 2 � 2x 3 � x 4 � 12x � 4

2 sin t � tan t � 4 � 2s34x 3�2 � 2x 5�2 � 4

x � 3x 2 � 8320 x 8�3 � Cx � D

x 3 � x 4 � Cx � DF�x� � x 5 �13 x 6 � 4

F �x� � 12 x 2 � ln � x �� 1�x 2 � C

G�� sin � � 5 cos � � C

F �u� � 13 u 3 � 6u�1�2 � C

F �x� � �5��4x 8� � C1 if x � 0

�5��4x 8� � C2 if x � 0

F �x� � 4x 3�2 �67 x 7�6 � C

F �x� � 4x 5�4 � 4x 7�4 � CF �x� � 23 x 3 �

12 x 2 � x � C

F �x� � 12 x �

14 x3 �

15 x4 � C

�0.410245, 0.347810��0.904557, 1.855277�0.21916368, 1.08422462�1.97806681, �0.82646233

1.13929375, 2.98984102�1.93822883, �1.219979971.412391, 3.057104�0.724492, 1.220744

1.82056420�1.251.1785

45x2 � 2.3, x3 � 3

c1 � 3.85 km�s, c2 � 7.66 km�s, h � 0.42 kmT3 � s4h2 � D 2�c1

T2 � �2h sec ��c1 � �D � 2h tan ��c2T1 � D�c1

s41�4 � 1.6�1.07x � � BC �W�L � s25 � x 2�x

5 � 2s5

12 �L � W �2x � 6 in.�a 2�3 � b 2�3�3�2

p�x� � 550 �110 x

p�x� � 19 �1

3000 x

2s6 y � �53 x � 10

10s3 3�(1 � s

3 3 )� 4.85 43. (a) (b)

(c) (d) About 9.09 s47. $742.08 49. 225 ft 51.53.57. (a) 22.9125 mi (b) 21.675 mi (c) 30 min 33 s(d) 55.425 mi


True-False Quiz

1. False 3. False 5. True 7. False 9. True11. True 13. False 15. True 17. True 19. True

Exercises

1. Abs max , abs and loc min 3. Abs max , abs and loc min

5. Abs max ; abs min ; loc max ; loc min

7. (a) None(b) Dec on (c) None

(d) CU on ; CD on ; IP (e) See graph at right.

9. (a) None(b) Inc on , dec on

(c) Loc max

(d) CD on (e) See graph at right.

11. (a) None(b) Inc on , n an integer;dec on (c) Loc max ; loc min (d) CU on ;CD on ; IPs (e)

13. (a) None(b) Inc on ,

dec on

(c) Loc min

(d) CU on(e) See graph at right.

��, ��f ( 1

4 ln 3) � 31�4 � 3�3�4

(��, 14 ln 3)( 1

4 ln 3, �)

1 x

2

y

0

y

x

2

π_2

_π2π_2π

(2n� ��3�, � 14 )�2n� � ��3�, 2n� � �5��3��

�2n� � ��3�, 2n� � ��3��f �2n�� 2f ��2n � 1�� 2

��2n � 1��, �2n � 2��2n�, �2n � 1��

��, 1�f (3

4) � 54

( 34 , 1)(��, 34)

y

0 x

1

1

�0, 2��0, ��, 0�

��, ��y

x

2

f �2��3� � �2��3� �12 s3

f ��3� � ��3� �12 s3

f �0� � 0f ��

f (�13) � �

92f �2� � 2

5

f �3� � 1f �4� � 5

62,500 km�h2 � 4.82 m�s2

8815 � 5.87 ft�s2

�9.8s450�4.9 � �93.9 m�ss450�4.9 � 9.58 ss�t� � 450 � 4.9t 2



15. Inc on , ;

dec on , ;

loc max ,

loc min ;

CU on , ;

CD on , ;

IP ,

17. Inc on , ; dec on , loc max ; loc min , ;CU on , ; CD on ; IP ,

19. ;

21.

23. For , f is periodic with period and has local maxima at , n an integer. For , f has no graph.For , f has vertical asymptotes. For , f is con-tinuous on . As C increases, f moves upward and its oscillationsbecome less pronounced.

25. 27. 29. 8 31. 0 33.35. 37. 39. 500 and 12541. 43. from ; at 45. 47. $11.50 49. 1.16718557

51. 53.

55.

57.

59. (b) (c)

61. No63. (b) About 8.5 in. by 2 in. (c) , 20s2�3 in.20�s3 in.

5

4

_1

_4

F

0.1e x � cos x � 0.9

s�t� � t 2 � tan�1t � 1

f (x� � 12 x 2 � x 3 � 4x 4 � 2x � 1

f �t� � t 2 � 3 cos t � 2F�x� � e x � 4 sx � C

L � CCD4�s3 cm3s3r 2

13 ft�s400 ft�h

12�a � �3, b � 7

�

C � 1�1 � C 1C �12n� � ��2

2�C � �1

�2.16, �0.75, 0.46, 2.21�2.96, �0.18, 3.01; �1.57, 1.57;

(s2�3, e�3�2 )�0.82, 0.22�

50

_5

1

2.5

0.4_0.51.5

ff

15

2.1_1

_20

�1.24, �12.1��0.12, 1.98��0.12, 1.24��1.24, ��, �0.12�

f �1.62� � �19.2f ��0.23� � 1.96f �0� � 2�0, 1.62�;��, �0.23��1.62, ��0.23, 0�

(�s6, � 536 s6)(s6, 5

36 s6)(0, s6)(��, �s6)

(s6, �)(�s6, 0)f (�s3) � �

29 s3

f (s3) � 29 s3

(s3, �)(��, �s3)ƒ

1.5

_1.5

_5 5

(0, s3)(�s3, 0) 65. (a)

(b) , where k is the constant

of proportionality


5. Abs max , no abs min7. 9. 11.13. 17.23.

CHAPTER 5


1. (a) ,

(b) ,

3. (a) 0.7908, underestimate (b) 1.1835, overestimate

5. (a) 8, 6.875 (b) 5, 5.375

(c) 5.75, 5.9375

(d) M6

y

x0 1

2

y

x0 1

2

y

x0 1

2

y

x0 1

2

y

x0 1

2

y

x0 1

2

y

0 x

1

π2

3π8

π4

π8

ƒ=cos x

y

0 x

1

π2

3π8

π4

π8

ƒ=cos x

R8 � 39.1L8 � 35.1

2

2

4

6

4 6 8

y

0 x2

2

4

6

4 6 8

y

0 x

R4 � 41L4 � 33

2 �375128� � 11.204 cm3�min

�m�2, m 2�4�c � 0 �one IP� and c � �e�6 �two IP��3.5 � a � �2.524��2, 4�, �2, �4�

f ��5� � e 45

dI

dt�

�480k�h � 4��h � 4�2 � 1600�5�2

20s2 � 28 ft



7. 0.2533, 0.2170, 0.2101, 0.2050; 0.29. (a) Left: 0.8100, 0.7937, 0.7904; right: 0.7600, 0.7770, 0.780411. 34.7 ft, 44.8 ft 13. 63.2 L, 70 L 15. 155 ft

17. 19.

21. The region under the graph of from 0 to 23. (a)

25. (a) (b) (c)

27.


1.The Riemann sum representsthe sum of the areas of the two rectangles above the -axisminus the sum of the areas ofthe three rectangles below the -axis; that is, the net area of the

rectangles with respect to the -axis.

3. 2.322986The Riemann sum represents the sumof the areas of the three rectanglesabove the -axis minus the area of therectangle below the -axis.

5. (a) 4 (b) 6 (c) 107. Lower, ; upper, 9. 124.1644 11. 0.308413. 0.30843908, 0.30981629, 0.3101556315.

The values of appear to be approaching 2.

17. 19. 21. 42

23. 25. 3.75 27.

29.

31. (a) 4 (b) 10 (c) �3 (d) 2 33.35. 37. 39. 0

41. 43. 45.

47. 49. 53. x10 x 4 dx15B � E � A � D � C

e 5 � e 3122x5�1 f �x� dx

2.53 �94�

�34

limn l �

�n

i�1 �sin

5�i

n � �

n�

2

5

limn l �

�n

i�1

2 � 4i�n

1 � �2 � 4i�n�5 �4

n43

x8

1 s2x � x2 dxx62 x ln�1 � x 2� dx

Rn

R5 � 16L5 � �64

xx

y

0 x

23456

1

_11 2

ƒ=´-2

x

x

x

y

0 x

23

1

2 4 6

ƒ=3- x12

8 10 12 14

�6

sin b, 1

323

n 2�n � 1�2�2n 2 � 2n � 1�12

limn l �

64

n 6 �n

i�1 i 5

Ln � A � Rn

��4y � tan x

limn l �

�n

i�1 � i�

2n cos

i�

2n � �

2nlimn l �

�n

i�1

2�1 � 2i�n��1 � 2i�n�2 � 1

�2

n


1. 3. 5. 7. 9. 11.

13. 15. 1 17. 19. 21.

23. 25. 27. 29.31. The function is not continuous on the interval

, so the Evaluation Theorem cannot be applied.33. 2 35.37. 3.75 41.

43. 45.

47. 49.51. The increase in the child’s weight (in pounds) between the ages of 5 and 1053. Number of gallons of oil leaked in the first 2 hours55. Increase in revenue when production is increased from 1000 to 5000 units57. Newton-meters 59. (a) (b)61. (a) (b)63. 65. 1.4 mi 67. $58,00069. (b) At most 40%; 73. 3 75.


1. One process undoes what the other one does. See theFundamental Theorem of Calculus, page 371.3. (a) 0, 2, 5, 7, 3 (d)

(b) (0, 3)(c)

5.

7.

9. 11.

13. 15.

17. t��x� ��2�4x 2 � 1�

4x 2 � 1�

3�9x 2 � 1�9x 2 � 1

y� � stan x � stan x sec2xh��x� � �arctan�1�x�

x 2

F��x� � �s1 � sec xt��y� � y 2 sin y

t��x� � 1��x 3 � 1�

t��x� � 1 � x 2

t

y

x

©

y=1+t@

0

x � 3

y

0 x

1

1

g

14

536

46 23 kg

416 23 mv�t� � 1

2 t 2 � 4t � 5 m�s

416 m�

32 m

43sec x � C

tan � � C2t � t 2 �13 t 3 �

14 t 4 � C

2010

_505

_6

10_10

x

y

0 2

�1

y=˛

sin x �14 x 2 � C

�1.36�1, 3

f �x� � 1�x 2�3.5��61 � ��4ln 2 � 7

e 2 � 1�ln 35563

403

493�2 � 1�e5

95615�

103

n

5 1.93376610 1.98352450 1.999342

100 1.999836

Rn



19. (a) Loc max at 1 and 5;loc min at 3 and 7(b)

(c)

(d) See graph at right.

21. 23. 25. 2927. (a) , n an integer(b) , , and ,

an integer (c) 0.7429. 31.33. (b) Average expenditure over ; minimize average expenditure


1. 3. 5.7. 9.11. 13.

15. 17.

19. 21.

23. 25.

27. 29.

31.33. 35.37. 39.

41. 43. 45. 47.

49. 51. 53. 55. 57.59. 61. All three areas are equal. 63.

65. 67. 5


1. 3.

5.

7.

9. 11.

13. 15. 17.

19. 21.

23.

25. 27.

29. 12�x 2 � 1� ln�1 � x� �

14 x 2 �

12 x �

34 � C

�12 � ��42sx sin sx � 2 cos sx � C

2�ln 2�2 � 4 ln 2 � 2

16 (� � 6 � 3s3)1

4 �34 e�2

12 �

12 ln 2��31

13 e 2�2 sin 3 � 3 cos 3� � C

t arctan 4t �18 ln�1 � 16t 2� � Cx ln s3 x �

13 x � C

�1

�x 2 cos �x �

2

� 2 x sin �x �2

� 3 cos �x � C

2�r � 2�er�2 � C

15 x sin 5x �

125 cos 5x � C1

3 x 3 ln x �19 x 3 � C

5

4��1 � cos

2�t

5 � L

� 4512 L6�

162ln�e � 1�16

150

2�e2 � e�1829

45282��

2

_3

2π0

f

F

1

_1

2_2

F

f

�ecos x � C18�x 2 � 1�4 � C

tan�1x �12 ln�1 � x 2 � � C�ln�1 � cos2 x� � C

140�2x � 5�10 �

536�2x � 5�9 � C

13 sec3x � Cln � sin�1 x � � C

�23 �cot x�3�2 � C1

15�x 3 � 3x�5 � C

�1��sin x� � C23 �1 � e x �3�2 � C

23 s3ax � bx 3 � C�

13 ln� 5 � 3x � � C

13�ln x�3 � C�(1�� cos � t � C

163 �3x � 2�21 � C�

12 cos�x 2� � C

�14 cos4 � C2

9 �x 3 � 1�3�2 � C�e�x � C

0, tf �x� � x 3�2, a � 9f �x� � x

x1 �2t�t� dt 0n

(s4n � 1, s4n � 1)(�s4n � 1, �s4n � 3)�0, 1� 0�2sn, s4n � 2

��4, 0��1, 1�

( 12, 2), �4, 6�, �8, 9�

x � 9x

8642

1

0

_1

y

_2

31.

33.

35. (b)

37. (b)41.43. 45. 2


1. 3. 5.

7. 9.

11.

13. 15.

17.

19. (a) (b)

21.

23. 25.

27.

29.

31.

33. 35.


1.

3. 5.

7. 9.

11.

�1

12 �6 � 4y � 4y 2�3�2 � C

2y � 1

8s6 � 4y � 4y 2 �

7

8 sin�1�2y � 1

s7 ��

12 tan2�1�z� � ln �cos�1�z� � � C� 3 � 6�

12 �e2x � 1� arctan�e x� �

12 e x � C�s4x 2 � 9��9x� � C

1

2� tan2��x� �

1

� ln � cos��x� � � C

2

s3 tan�1 �2x � 1

s3 � � C2 � ln 259

12 x 2 � 2 ln�x 2 � 4� � 2 tan�1�x�2� � C

x � 6 ln � x � 6 � � C

12 ln�x 2 � 1� � (1�s2 ) tan�1(x�s2 ) � C

ln � x � 1 � �12 ln�x 2 � 9� �

13 tan�1�x�3� � C1

2 ln 32

12 ln � 2x � 1� � 2 ln � x � 1� � C

A

x�

B

x � 1�

C

�x � 1�2

A

x � 3�

B

3x � 1

�s4 � x 2

4x� C

�

24�

s3

8�

1

4�

sx 2 � 4

4x� C

�s9 � x 2

x� sin�1 � x

3� � C

815

13 sec3x � sec x � C

��11384

15 cos5x �

13 cos3x � C

2 � e�t�t 2 � 2t � 2� mx �ln x�3 � 3�ln x�2 � 6 ln x � 6 � C

23 , 8

15

�14 cos x sin3x �

38 x �

316 sin 2x � C

4

_4

2_2

F

f

13 x 2�1 � x 2�3�2 �

215�1 � x 2�5�2 � C

1

_1

3_1f

F

�12 xe�2x �

14 e�2x � C



13. 15.

17.

19.

21.

25.

27.

29.

31.

33. (a) ;

both have domain


1. (a)(b) is an underestimate, and are overestimates.(c) (d)

3. (a) (underestimate)(b) (overestimate)

5. (a) ,(b) ,7. (a) 2.413790 (b) 2.411453 (c) 2.412232

9. (a) 0.146879 (b) 0.147391 (c) 0.147219

11. (a) 0.451948 (b) 0.451991 (c) 0.451976

13. (a) 4.513618 (b) 4.748256 (c) 4.675111

15. (a) (b) (c)

17. (a)(b) , (c) for , for

19. (a) , ;, ;

, (b) , (c) for , for , for

21. (a) 2.8 (b) 7.954926518 (c) 0.2894(d) 7.954926521 (e) The actual error is much smaller.(f ) 10.9 (g) 7.953789422 (h) 0.0593(i) The actual error is smaller. ( j)

23.

Observations are the same as after Example 1.

n � 50

Snn � 22Mnn � 360Tnn � 509� ES � � 0.000170� ET � � 0.025839, � EM � � 0.012919

ES � �0.000110S10 � 2.000110EM � �0.008248M10 � 2.008248

ET � 0.016476T10 � 1.983524

Mnn � 50Tnn � 71� EM � � 0.0039� ET � � 0.0078

T8 � 0.902333, M8 � 0.905620

�0.526123�0.543321�0.495333

ES � �0.000060S10 � 0.804779EM � �0.001879M10 � 0.806598

T4 � I � M4

M4 � 0.908907T4 � 0.895759

Ln � Tn � I � Mn � RnT2 � 9 � IM2R2L2

L2 � 6, R2 � 12, M2 � 9.6

��1, 0� � �0, 1�

�ln � 1 � s1 � x 2

x � � C

�ln � cos x � �12 tan2x �

14 tan4x � C

110 �1 � 2x�5�2 �

16 �1 � 2x�3�2 � C

14 x�x 2 � 2�sx 2 � 4 � 2 ln(sx 2 � 4 � x) � C

13 tan x sec2x �

23 tan x � C

se 2x � 1 � cos�1�e�x � � C

12�ln x�s4 � �ln x�2 � 2 ln[ln x � s4 � �ln x�2] � C

15 ln � x 5 � sx 10 � 2 � � C

1

2s3 ln � e x � s3

e x � s3 � � C19 sin3x 3 ln�sin x� � 1 � C

25.

Observations are the same as after Example 1.

27. (a) 19.8 (b) 20.6 (c)29. 31.33. (a) 23.44 (b) 35. 59.437.


Abbreviations: C, convergent; D, divergent

1. (a) Infinite interval (b) Infinite discontinuity(c) Infinite discontinuity (d) Infinite interval3. ; 0.495, 0.49995, 0.4999995; 0.55. 7. D 9. 11. D 13. 0 15. D17. 19. D 21. 23. 25. D

27. 29. 31. D 33.35. e 37.

39. Infinite area

41. (a)

It appears that the integral is convergent.

20

0π2

y=sec@ x

0.5

_7 7

29 y= 2

≈+9

0x

y

0

x 1y ex

1

2��3

83 ln 2 �

89

754

323

12��91

25

2e�22

12 � 1��2t 2 �

0 x

y

1

1 20.5 1.5

0.341310,177 megawatt-hours37.73 ft�s

20.53

n

5 0.742943 1.286599 1.014771 0.99262110 0.867782 1.139610 1.003696 0.99815220 0.932967 1.068881 1.000924 0.999538

MnTnRnLn

n

5 0.257057 �0.286599 �0.014771 0.00737910 0.132218 �0.139610 �0.003696 0.00184820 0.067033 �0.068881 �0.000924 0.000462

EMETEREL

n

6 6.695473 6.252572 6.40329212 6.474023 6.363008 6.400206

SnMnTn

n

6 �0.295473 0.147428 �0.00329212 �0.074023 0.036992 �0.000206

ESEMET

t

2 0.4474535 0.577101

10 0.621306100 0.668479

1,000 0.67295710,000 0.673407

yt

1 �sin2x��x 2 dx



(c)

43. C 45. D 47. D 49. 51.55. (a)

(b) The rate at which the fraction increases as t increases(c) 1; all bulbs burn out eventually57. 8264.5 years 59. 1000 63. 65. No


True-False Quiz

1. True 3. True 5. False 7. True 9. True11. False 13. False 15. False 17. False 19. False21. False 23. False

Exercises1. (a) 8 (b) 5.7

3. 5. 3 7.

9. 37 11. 13.

15. 17. 19.

21. 23. 25. 0

27. 29.

31. 33.35. 37. 39.41. 43.

45.47. (a) 1.090608 (overestimate)(b) 1.088840 (underestimate) (c) 1.089429 (unknown)

49. (a) 0.00 , (b) , 51. (a) 3.8 (b) 1.7867, 0.000646 (c)53. 55. 57. D 59. 261. C 63. (a) m (b) 29.5 m29.16

1364 � x

31 sx 2 � 3 dx � 4s3

n � 30n � 1830.003n � 2596

38 ln(x �

12 � sx 2 � x � 1) � C1

4�2x � 1�sx 2 � x � 1 �

12 [e x

s1 � e 2x � sin�1�e x �] � Cy� � (2e x � e sx )��2x�F��x� � x 2��1 � x 3�64

52s1 � sin x � Cln � 1 � sec � � C3es

3 x�x 2�3 � 2x 1�3 � 2� � C

12 ln � t � 2

t � 4 � � C645 ln 4 �

12425

5 � 10 ln 23sx2 � 4x � C

�1��e � � 1�13 sin 11

2 ln 2

��1�x� � 2 ln � x � � x � C910

f is c, f � is b, xx0 f �t� dt is a1

2 � ��4

62 x

2

0

y=ƒ

y

2 x

2

0

y=ƒ

6

y

C � 1; ln 2

F�t�

1

700 t0(in hours)

y

y=F(t)

p � 1, 1��1 � p��

1

�0.1

1 10

©=sin@ x≈

ƒ= 1≈

65. Number of barrels of oil consumed from Jan. 1, 2000, throughJan. 1, 2008

67. 69.


1. About 1.85 inches from the center 3. 5.7. 9. 11. Does not exist 13.

15. 17. 19. 0

23. (b)

CHAPTER 6


1. 3. 5. 7.9. 11. 13. 72 15. 17.19. 0, 0.90; 0.04 21.

23.

25. 118 ft 27. 84 29. 8868; increase in population overa 10-year period 31.33. 35. 37. 39.

41. 43. 45. ;


1.

3. y

0 x

(6, 9)

x=2œ„y

y=9

x=0

y

0 x

162�

y

1

0 x21 y=0

x=1x=2

y

0 x

y=2- 12 x

19��12

m � ln m � 10 � m � 1f �t� � 3t 242�3

�624s3�53 � e�ab

rsR 2 � r 2 � �r 2�2 � R 2 arcsin�r�R�m2

x

y

0

y=cos xy=sin 2x

π6

π2

12

1, 1.38; 0.05

ln 2e � 2323

83

13e � �1�e� �

43e � �1�e� �

103

323

y � �sL2 � x 2 � L ln�L � sL2 � x 2

x �18� �

112s1 � sin4x cos x

�1, 22ke�2

f �x� � 12 x��2

e2x�1 � 2x��1 � e�x�Ce�x 2��4kt��s4�kt



5.

7.

9.

11.

13. 15. 17.

19.

21. 23.25. (a) Solid obtained by rotating the region ,

about the x-axis(b) Solid obtained by rotating the region above the boundedby about the y-axis27. 29. (a) 190 (b)31. 33. 35. 37.

39. 24 41. 43. 45. (a)

(b) 47. (b) 49.

51. 8 xr0 sR 2 � y 2

sr 2 � y 2 dy

512 �r 3�r 2h2� 2r 2R

8�R xr0 sr 2 � y 2 dy8

1513

10 cm323 b 2h�h2(r �

13h)1

3�r 2h8231110 cm3

x � y 2 and x � y4x-axis

0 � x � ��20 � y � cos x

118 � 2�1.288, 0.884; 23.780

� x2s2�2s2 [52

� (s1 � y 2 � 2)2] dy

13��30108� �5��2

y

0 x

y=1 y=1

y=3

y=1+sec x

y

0 x

” , 3’π3”_ , 3’π

3

2� ( 43� � s3)

y

0

y=x

y=1

x

y

0 x

(1, 1)y=œ„x

��6

x0

y

(4, 2)

x0

y

x=2y

¥=x

64��15

y

0 x

(1, 1)

y=˛

y=x

y

0 x

4��21 EXERCISES 6.3 N PAGE 453

1. Circumference , height ;

3.

5.

7.

9. 11. 13. 15.

17. 19. 21.

23. (a) Solid obtained by rotating the region ,about the -axis

(b) Solid obtained by rotating the region bounded by (i) , , and , or (ii) , , and about the line

25. 0.13 27. 29. 31.

33. 35. 37.


1. 3.

5. 7.

9. 11. 34 �

12 ln 2(13s13 � 8)�27

4s2 � 2x2�

0 s3 � 2 sin t � 2 cos t dt � 10.0367

3.82024s5

13 �r 2h4

3 �r 32��12 � 4 ln 4�

43�8�1

32 � 3

y � 3y � 0x � 1x � y 2y � 0x � 0x � 1 � y 2

y0 � x � 30 � y � x 4

3.70x�

0 2� �4 � y�ssin y dy5��14

8��37��1516��321��2

0 x

y

(1, 4) (3, 4)

y=4(x-2)@y=≈-4x+7

2

7

x

y

x 2

16�

0 x

y

y=e_≈

1

1

x

y

0

x

� �1 � 1�e�

2�

��15� x �x � 1�2� 2�x



13.

15.

17. 5.115840 19. 40.05622221. (a), (b) ,

,

(c) (d) 7.7988

23. 25. 27. 209.1 m

29. 29.36 in.

33. (a)

(b)


1. 3. 5.

7. (a) 1 (b) 2, 4 (c)

2��5��4528

83

294

t � 0, 4�15

�15

�15 15

ln(s2 � 1)205128 �

81512 ln 3

x40 s1 � 4�3 � x��3�4 � x�2�3 �2 dx

L4 � 7.50L2 � 6.43L1 � 4

8

0�25 2.5

s2 �e � � 1�

21

�1�1 21

e 3 � 11 � e�8 9. (a) (b)(c)

13. 38.6 15.17. 19.


1. 3. 180 J 5.

7. (a) (b) 10.8 cm 9.

11. (a) 625 ft-lb (b) 13.

15. 17. 19.21. 23. 2.0 m

27. (a) (b)

29. (a) (b) 1875 lb (c) 562.5 lb31. 33. 35.37.39. (a) (b)(c) (d)41. 43.

45. 47. 49.

51. (b)


1. $38,000 3. $43,866,933.33 5. $407.257. $12,000 9. 3727; $37,753

11. 13.

15. 17. 19.


1. (a) The probability that a randomly chosen tire will have a lifetime between 30,000 and 40,000 miles(b) The probability that a randomly chosen tire will have a lifetime of at least 25,000 miles3. (b)5. (a) (b)7. (a) for all x and (b) 511. (a) (b) (c) If youaren’t served within 10 minutes, you get a free hamburger.13.15. (a) (b)17. �0.9545

�5.21%0.0668�44%

1 � e�2�2.5 � 0.55e�4�2.5 � 0.20x

�

�� f �x� dx � 1f �x� � 0

121��

1 �38 s3 � 0.35

5.77 L�min6.60 L�min1.19 � 10�4 cm3�s

�1 � k��b 2�k � a 2�k��2 � k��b1�k � a1�k�

23 (16s2 � 8) � $9.75 million

( 12, 25)

60; 160; ( 83, 1)� 1

e � 1,

e � 1

4 ��0, 1.6�

10; 1; ( 121 , 10

21)2.5 � 10 5 N3.03 � 105 lb4.88 � 104 lb

5.06 � 104 lb5.63 � 103 lb5.27 � 105 N

1.2 � 10 4 lb9.8 � 103 N6.7 � 10 4 N187.5 lb�ft2

�8.50 � 109 JGm1m2�1

a�

1

b��1.04 � 105 ft-lb

�1.06 � 106 J2450 J3857 J

650,000 ft-lb18754 ft-lb

W2 � 3W12524 � 1.04 J

154 ft-lb9 ft-lb

5��4�� 0.4 L6 kg�m�50 � 28��F � 59�F

�1.24, 2.814��




1. 3. 5. 7. (a) 0.38 (b) 0.879. (a) (b) (c) 11.

13. 15.17. (a) Solid obtained by rotating the region ,

about the -axis(b) Solid obtained by rotating the region

about the -axis19. 36 21. 23. 25.

27. 29. (a) (b) 2.1 ft

31. 33. $7166.67 35.

37. (a) for all x and (b) (c) 5, yes

39. (a) (b)(c)


1. 3. (b) 0.2261 (c) 0.6736 m(d) (i) (ii)7. Height , volume 9.13. 15.

CHAPTER 7


3. (a) 5. (d)

7. (a) It must be either 0 or decreasing(c) (d)9. (a) (b)(c)13. (a) III (b) I (c) IV (d) II15. (a) At the beginning; stays positive, but decreases

(c)


1. (a)

(b) , y � 1.5y � 0.5

0 x1_1_2 2

0.5

1.0

1.5

2.0y

(i)

(ii)(iii)

(iv)

0

M

P(t)

t

P(0)

P � 0, P � 4200P 42000 � P � 4200

y � 1��x � 2�y � 0

12, �1

b � 2a0.14 mln��2�( 28

27 s6 � 2)�b 3s2 b

370��3 s � 6.5 min1��105�� 0.003 in�sf �x� � s2x��

8 ln 2 � 5.55 mine�5�4 � 0.291 � e�3�8 � 0.31

� 0.3455x

�

�� f �x� dx � 1f �x� � 0

f �x�� 458 lb

8000��3 � 8378 ft-lb3.2 J

1522(5s5 � 1)125

3 s3 m3

x0 � x � 12 � sx � y � 2 � x 2,

x0 � x � ��20 � y � s2 cos x

x��3��3 2� (��2 � x)(cos2x �

14) dx4

3 � �2ah � h 2 �3�2

1656��58��15��62��159�7

1283

3. III 5. IV

7. 9.

11. 13.

15.

17. ;

19. (a) (i) 1.4 (ii) 1.44 (iii) 1.4641(b) Underestimates

(c) (i) 0.0918 (ii) 0.0518 (iii) 0.0277It appears that the error is also halved (approximately).

y

0 0.2 x0.40.1 0.3

y=´h=0.1h=0.2h=0.4

1.0

1.1

1.2

1.3

1.4

1.5

�2, 0, 2�2 � c � 2y

0

_2

_1 t1

2

c=3

c=_3

c=_1

c=1

4

_2

3_3

y

x3_3

3

_3

y

x3_3

3

_3

0 x_3 3

_3

3

yy

x3_3

_3(c)

(a)(b)



21. 23.25. (a) (i) 3 (ii) 2.3928 (iii) 2.3701 (iv) 2.3681(c) (i) (ii) (iii) (iv)It appears that the error is also divided by 10 (approximately).27. (a), (d) (b) 3

(c) Yes; (e) 2.77 C


1. , 3.

5.

7. 9.

11. 13.

15.

17.

19. 21.23. (a)

(b) , (c) No

25.

27. (a) (b) y �1

K � x

0 x_3 3

_3

3y

5

2.50

�2.5

cos y � cos x � 1

1

0

y=sin (≈)

_œ„„„π/2_œ „„„π/2œ

�s��2 � x � s��2y � sin�x 2�sin�1y � x 2 � C

y � Ke x � x � 1y � e x2�2

y �4a

s3 sin x � a

12 y 2 �

13�3 � y 2�3�2 � 1

2 x 2 ln x �14 x 2 �

4112

u � �st 2 � tan t � 25y � �sx 2 � 9

u � Ae 2 t�t 2�2 � 1y � �s3�te t � e t � C�2�3 � 1

12 y 2 � cos y � 1

2 x 2 �14 x 4 � C

y � Ksx 2 � 1y � 0y �2

K � x 2

Q � 3Q

0

2

2 t4

4

6

�0.0002�0.0022�0.0249�0.6321

1.7616�1, �3, �6.5, �12.25 29.

31.

33. 35.

37. ; 3 39. ;

41. (a)

(b)

43. (a)(b) ; the concentration approaches regardless of the value of 45. (a) (b)47. About 4.9% 49.51. (a) (b)

53. (a) (b) ,

where and


1. About 2353. (a) (b) (c)(d)5. (a) 1508 million, 1871 million (b) 2161 million(c) 3972 million; wars in the first half of century, increased lifeexpectancy in second half7. (a) (b)9. (a) (b) (c)11. 13. (a) (b)

15. (a) (b)17. (a) (b)19. (a) (i) $3828.84 (ii) $3840.25 (iii) $3850.08(iv) $3851.61 (v) $3852.01 (vi) $3852.08(b) ,

21. (a) (b)

(c) , (d) Decliningm kP0m � kP0

m � kP0P�t� �m

k� �P0 �

m

k �e kt

A�0� � 3000dA�dt � 0.05A

�39.9 kPa�64.5 kPa�67.74 min13.3�C

�116 min�137�F�2500 years�199.3 years� 9.92 mg100 � 2�t�30 mg

�2000 ln 0.9 � 211 sCe�0.0005t

�ln 100��ln 4.2� � 3.2 h�10,632 bacteria�h�7409100�4.2� t

A0 � A�0�C �sM � sA0

sM � sA0

A�t� � M�CesM kt � 1

Ce sM kt � 1�2

dA�dt � ksA �M � A�

B � KV 0.0794L1 � KL k2

t�k15e�0.2 � 12.3 kg15e�t�100 kg

C0

r�kr�kC�t� � �C0 � r�k�e�kt � r�k

t �2

ksa � b�tan�1 b

a � b� tan�1 b � x

a � b �x � a �

4

(kt � 2�sa )2

MP�t� � M � Me�ktQ�t� � 3 � 3e�4 t

y � (12 x 2 � 2)2y � 1 � e2�x 2�2

≈-¥=C

xy=k4

4

_4

_4

x 2 � y2 � C

4

4

_4

_4

y � Cx 2




1. (a) 100; 0.05 (b) Where is close to or ; on the line ; ; (c)

Solutions approach 100; some increase and some decrease, somehave an inflection point but others don’t; solutions with and have inflection points at (d) , ; other solutions move away from andtoward

3. (a) (b)5.

7. (a) , in billions(b) 5.49 billion (c) In billions: 7.81, 27.72(d) In billions: 5.48, 7.61, 22.41

9. (a) (b)

(c) 3:36 PM

13. ;

15. (a) Fish are caught at a rate of 15 per week.(b) See part (d). (c)(d) ;

;

(e)

where k � 111 , � 1

9

0 120

1200P

t

P�t� �250 � 750ke t�25

1 � ke t�25

P0 250: P l 750P0 � 250: P l 2500 � P0 � 250: P l 0

0 t

P

1208040

1200

800

400

P � 250, P � 750

t (year)1960 1980 2000

90,0000 45

P (in thousands)

130,000

PL

PE

PL�t� �32,658.5

1 � 12.75e�0.1706 t � 94,000

PE �t� � 1578.3�1.0933�t � 94,000

y �y0

y0 � �1 � y0 �e�ktdy�dt � ky�1 � y�

PdP�dt � 1265 P�1 � P�100�

9000�1.55 years3.23 � 107 kg

P � 100P � 0P � 100P � 0

P � 50P0 � 40P0 � 20

P¸=140P¸=120

P¸=80

P¸=40P¸=20

P¸=60

0 t

P

604020

150

100

50

P0 1000 � P0 � 100P � 501000P

17. (b) ;;

(c)

19. (a) (b) Does not exist


1. (a) , ; growth is restricted only by predators, which feed only on prey.(b) , ; growth is restricted by carrying capacity and by predators, which feed only on prey.3. (a) Competition(b) (i) , : zero populations(ii) , : In the absence of an -population, the -population stabilizes at 400.

(iii) , : In the absence of a -population, the -population stabilizes at 125.

(iv) , : Both populations are stable.5. (a) The rabbit population starts at about 300, increases to 2400,then decreases back to 300. The fox population starts at 100,decreases to about 20, increases to about 315, decreases to 100, and the cycle starts again.

(b)

7.

11. (a) Population stabilizes at 5000.(b) (i) , : Zero populations(ii) , : In the absence of wolves, the rabbit population is always 5000.(iii) , : Both populations are stable.(c) The populations stabilize at 1000 rabbits and 64 wolves.

R � 1000W � 64

R � 5000W � 0R � 0W � 0

0 Species 1

Species 2

50

200

100

50

100 150 200 250

t=3

t=0, 5

150

t=1

t=2

t=4

0 t

R

2000

t¡

1000

F

200

t™ t£

1500

500

2500300

100

R F

y � 300x � 50x

yy � 0x � 125y

xy � 400x � 0y � 0x � 0

y � predatorsx � prey

y � preyx � predators

P�t� � P0e �k�r�sin�rt � �� sin �

P�t� �m�M � P0� � M�P0 � m�e �M�m��k�M �t

M � P0 � �P0 � m�e �M�m��k�M �t

P0 200: P l 1000

P0 � 200: P l 2000 � P0 � 200: P l 0

0 t

P

1008060

1400

800

400

4020

600

200

1200

1000



(d)


True-False Quiz

1. True 3. False 5. True

Exercises

1. (a) (b) ;

, ,

3. (a)

(b) 0.75676(c) and ; there is a loc max or loc min

5. 7.

9.

11. (a) (b)(c) (d)

13. (a) (b)

15. (a) ; (b)

17. (a) (b)

19. 15 days 21.

23. (a) Stabilizes at 200,000(b) (i) , : Zero populations(ii) , : In the absence of birds, the insect population is always 200,000.(iii) , : Both populations are stable.(c) The populations stabilize at 25,000 insects and 175 birds.

y � 175x � 25,000

y � 0x � 200,000y � 0x � 0

k ln h � h � ��R�V �t � C

L�t� � 53 � 43e�0.2 tL�t� � L� � L� � L�0�e�kt

t � �10 ln 257 � 33.5�560P�t� �

2000

1 � 19e�0.1t

�100 hC0 e�kt

�ln 50��ln 3.24� � 3.33 h�25,910 bacteria�h�22,040200�3.24� t

x � C �12 y2

r�t� � 5et�t 2

y � �sln�x 2 � 2x 3�2 � C�

y � �xy � x

y�0.3� � 0.8

0 x

y

1 2_1_2

1

2

3_3

3

y � 4y � 2y � 0

0 � c � 46

10 t

y

2

4

(i)

(ii)

(iv)

(iii)

0 t

R

1000

W

40

1500

500

60

20

80W

R

(d)


1. 5. 7.

9. (b) (c) No

11. (a) 9.8 h (b) ; (c) 5.1 h13.

CHAPTER 8


Abbreviations: C, convergent; D, divergent

1. (a) A sequence is an ordered list of numbers. It can also bedefined as a function whose domain is the set of positive integers.(b) The terms approach 8 as becomes large.(c) The terms become large as becomes large.

3. ; yes; 5.

7. 9. 11. 513. 1 15. 1 17. 1 19. 0 21. 0 23. 025. 0 27. 29. 0 31. D 33. 35. 137. 39. D41. (a) 1060, 1123.60, 1191.02, 1262.48, 1338.23 (b) D43. (a) (b)

45. (a) D (b) C 47. (b)49. Decreasing; yes 51. Not monotonic; no53. Convergent by the Monotonic Sequence Theorem; 55. 57. 62


1. (a) A sequence is an ordered list of numbers whereas a series isthe sum of a list of numbers.(b) A series is convergent if the sequence of partial sums is a con-vergent sequence. A series is divergent if it is not convergent.

3. , ,, ,, ,, ,, ;

convergent, sum � �2�2.00000�2.00000�1.99999�2.00003�1.99987�2.00064�1.99680�2.01600

ssnd

1

0 10

_3

sand

�1.92000�2.40000

12 (3 � s5)

5 � L � 8

12 (1 � s5)

5734Pn � 1.08Pn�1 � 300

12

ln 2e 2

an � (� 23 )n�1

an � 5n � 3

an � 1��2n � 1�12

13 , 25 , 37 , 49 , 5

11 , 613

nan

nan

x 2 � �y � 6�2 � 25

2000� ft2�h31,900� ft2

f �x� �x 2 � L2

4L�

12 L ln� x

L�20 �Cy � x 1�nf �x� � �10e x

0 t

x

35,000

15,000

y

15025,000

5,000

45,000

200

100

250

(insects) (birds)

50

birds

insects



5. , ,, ,, ,, ,, ;

divergent

7. , ,, ,, ,, ,, ;

convergent, sum

9. (a) C (b) D 11. D 13. 15. 60 17. D19. D 21. D 23. 25. D 27. D 29.31. 33.35. (b) (c) (d) All rational numbers with a terminatingdecimal representation, except 0.

37. 39. 41.

43. All ; 45. 1

47. for ,

49. (a) 105.25 mg (b) mg

(c) The quantity of the drug approaches

51. (a) (b) 5 53.

57. 59. The series is divergent.

63. is bounded and increasing.

65. (a)

67. (a) (c) 1


1. C

3. (a) Nothing (b) C5. -series; geometric series; ; 7. D�1 � b � 1b � �1p

0 x

y

1

. . .a™ a£ a¢ a∞

2 3 4

y= 1x1.3

12 , 56 , 23

24 , 119120;

�n � 1�! � 1

�n � 1�!

0, 19 , 29 , 13 , 23 , 79 , 89 , 1

�sn �

1

n�n � 1�

12 (s3 � 1)Sn �

D�1 � c n �1 � c

100

0.95 � 105.26 mg

100�1 � 0.05n�1 � 0.05

sum � 1n � 1a1 � 0, an �2

n�n � 1�

2

2 � cos xx

�3 � x � 3; x

3 � x5063�33002

9

21

116

32

e��e � 1�52

253

� 10.698490.683770.666670.646450.622040.591750.552790.50000

0

1

11

ssnd

sand

0.422650.29289

8.646397.665816.689625.719484.757963.809272.880801.98637

10

0 11

ssnd

sand

1.154320.44721 9. C 11. D 13. C 15. C 17. D 19. C21. C 23. D 25. D 27. C 29. D 31.33. (a) 1.54977, (b) 1.64522, (c)35. 0.00145 37. 1.249, error 43. Yes


1. (a) A series whose terms are alternately positive and negative (b) and , where (c)

3. C 5. C 7. D 9. C11. An underestimate 13. 15. 5 17.19. 21. No 23. Yes 25. Yes 27. No29. Yes 31. Yes 33. Yes 35. D 37. (a) and (d)39. AC


1. A series of the form , where is a variable and and the ’s are constants3. 1, 5. 1, 7.

9. 11. 13.

15. 17. 19. 0,

21. 23.

25. (a) Yes (b) No 27.

29. (a)

(b), (c)

31. , 33. 2 35. No


1. 10 3. 5.

7. 9.

11. (a)

(b)

(c)

13. ln 5 � ��

n�1

x n

n5n , R � 5

1

2 �

�

n�2 ��1�nn�n � 1�x n, R � 1

1

2 �

�

n�0 ��1�n�n � 2��n � 1�x n, R � 1

��

n�0 ��1�n�n � 1�x n, R � 1

1 � 2 ��

n�1x n, ��1, 1��

�

n�0 ��1�n 1

9 n�1 x 2n�1, ��3, 3�

2 ��

n�0

1

3 n�1 x n, ��3, 3��

n�0 ��1�nx n, ��1, 1�

f �x� � �1 � 2x��1 � x 2��1, 1�

2

8

_2

_8

s¸

J¡

s£ s∞s¡

s™ s¢

��, ��

k k

�, ��, ��b, �a � b, a � b�{ 1

2 }14, [�1

2, 0]13, [�13

3 , �113 )

1, 1, 312 , (� 1

2 , 12]2, ��2, 2��, ��, ��1, 1�1, 1�

cnax��

n�0 cn�x � a�n

0.0676�0.5507p � 0

� Rn � � bn�1bn � � an �limn l � bn � 00 � bn�1 � bn

� 0.1n � 1000

error � 0.005error � 0.1p � 1



15.

17.

19.

21.

23.

25.

27. 0.199989 29. 0.000983 31. 0.1974033. (b) 0.920 37.


1. 3.

5.

7.

9.

11. ,

13.

15.

17.

21. 1 �x

2� �

�

n�2 ��1�n�1

1 � 3 � 5 � � � � � �2n � 3�2nn!

x n, R � 1

1

3� �

�

n�1 ��1�n

1 � 3 � 5 � � � � � �2n � 1�2 n � 32n�1 � n!

�x � 9�n, R � 9

��

n�0 ��1�n�1

1

�2n�! �x � �2n, R � �

��

n�0 e 3

n! �x � 3�n, R � �

R � ��1 � 2�x � 1� � 3�x � 1�2 � 4�x � 1�3 � �x � 1�4

��

n�0 5 n

n!x n , R � �

��

n�0 ��1�n

2n�1

�2n � 1�! x 2n�1, R � �

��

n�0 �n � 1�x n, R � 1

��

n�0 �n � 1�x n, R � 1b8 � f �8��5��8!

�1, 1, �1, 1�, ��1, 1�

C � ��

n�1 ��1�n�1 x 2n�1

4n2 � 1, R � 1

C � ��

n�0

t 8n�2

8n � 2, R � 1

3

2

�3

�2

s¡f

s£ s™��

n�0

2x 2n�1

2n � 1, R � 1

0.25

_0.25

4_4

s¡

s¡

s™

s™

s£

s£

s¢

s¢

s∞

s∞

f

f

��

n�0 ��1�n 1

16n�1 x 2n�1, R � 4

��

n�0�2n � 1�x n, R � 1

��

n�0��1�n4n�n � 1�x n�1, R � 1

4 23.

25.

27.

29.

31.

33.

35.

37.

39. 0.81873

41. (a)

(b)

43.

45.

47. 0.440 49. 0.40102 51. 53.

55. 57. 59.

61. 63. 65. e 3 � 11�s2ln 85

e�x 4

1 �16 x 2 �

7360 x 41 �

32 x 2 �

2524 x 4

1120

12

C � ��

n�1 ��1�n 1

2n �2n�! x 2n, R � �

C � ��

n�0 ��1�n x 6n�2

�6n � 2��2n�!, R � �

x � ��

n�1 1 � 3 � 5 � � � � � �2n � 1�

�2n � 1�2nn! x 2n�1

1 � ��

n�1 1 � 3 � 5 � � � � � �2n � 1�

2nn! x 2n

6

_6

4_3

T¡

T¡

T£

T£

T™T™

T¢

T¢

Tß

Tß

T∞

T∞

f

f

��

n�1

��1�n�1

�n � 1�! x n, R � �

1.5

1.5

_1.5

_1.5

Tˆ=T˜=T¡¸=T¡¡

T¢=T∞=Tß=T¶

T¸=T¡=T™=T£

f

��

n�0 ��1�n

1

�2n�! x 4n, R � �

��

n�1 ��1�n�1 2

2n�1

�2n�! x 2n, R � �

12 x � �

�

n�1 ��1�n

1 � 3 � 5 � � � � � �2n � 1�n!23n�1 x 2n�1, R � 2

��

n�0 ��1�n

1

2 2n�2n�! x 4n�1 , R � �

��

n�0 2n � 1

n! x n , R � �

��

n�0 ��1�n

2n�1

�2n � 1�! x 2n�1, R � �

��

n�0 ��1�n

�n � 1��n � 2�2n�4 x n, R � 2




1. (a) ,

,

(b)

(c) As increases, is a good approximation to on a larger and larger interval.

3.

5.

7.

_4f

3

_1 1.5

T£

x � 2x 2 � 2x 3

1.1

_1.1

T£

T£

f

f

π0π2

� x �

2 � �1

6 x �

2 �3

2

40

T£

f

12 �

14 �x � 2� �

18 �x � 2�2 �

116 �x � 2�3

f �x�Tn�x�n

2

2π

_2

_2π

T¢=T∞

T™=T£

T¸=T¡

Tß

f

T6�x� � 1 �12 x 2 �

124 x 4 �

1720 x 6

T4�x� � 1 �12 x 2 �

124 x 4 � T5�x�

T0�x� � 1 � T1�x�, T2�x� � 1 �12 x 2 � T3�x� 9.

11. (a) (b)

13. (a) (b)15. (a) (b) 0.00006 17. (a) (b) 0.04219. 21. Four 23.25. 27. 21 m, no31. (c) They differ by about


True-False Quiz

1. False 3. True 5. False 7. False 9. False11. True 13. True 15. False 17. True 19. True

Exercises1. 3. D 5. 0 7. 9. C 11. C 13. D15. C 17. C 19. 21. 23.25. 0.9721 27. , error31. 33. 0.5, [2.5, 3.5)

35.

37. 39.

41.

43.

45.

47. (a)

(b) (c) 0.000006

49. �16

1.5

20

T£

f

1 �12 �x � 1� �

18 �x � 1�2 �

116 �x � 1�3

C � ln � x � � ��

n�1

x n

n � n!

1

2� �

�

n�1 1 � 5 � 9 � � � � � �4n � 3�

n!26n�1 x n, R � 16

��

n�0 ��1�n

x 8n�4

�2n � 1�!, R � �

ln 4 � ��

n�1

x n

n4n , R � 4��

n�0 ��1�nx n�2, R � 1

1

2 �

�

n�0 ��1�n� 1

�2n�! x �

6 �2n

�s3

�2n � 1�! x �

6 �2n�1�

4, �6, 2�� 6.4 10�70.18976224

41113330�41

11

e 1212

8 10�9 km.�0.86 � x � 0.86

�1.037 � x � 1.0370.17365x 2 �

16 x 41 � x 2

0.0000971 �23�x � 1� �

19�x � 1�2 �

481�x � 1�3

1.5625 10�52 �14 �x � 4� �

164 �x � 4�2

5

_2

T™

T¢

T™

T£

T£

T¢ T∞

T∞

f

f20

π4

π2

�10

3 x �

4 �4

�64

15 x �

4 �5

T5�x� � 1 � 2 x �

4 � � 2 x �

4 �2

�8

3 x �

4 �3

x f

0.7071 1 0.6916 0.7074 0.7071

0 1 �0.2337 0.0200 �0.0009

�1 1 �3.9348 0.1239 �1.2114

2

4

T6T4 � T5T2 � T3T0 � T1




1.3. (a) (c)

5. 11.

13. where is a positive integer

APPENDIXES

EXERCISES A N PAGE A6

1. 18 3. 5.

7. 9.

11. 13.

15. 17.

19. 21.

23. 25.

27. 29. (a)(b) 31. 33.35. 37.39. 41.

EXERCISES B N PAGE A16

1. 5 3.

7. 9.

11. 13. 15.17. 19. 21.23.

25. , 27. ,

0 x

y

_30 x

y

b � �3b � 0m � 3

4m � �13

5x � 2y � 1 � 0x � 2y � 11 � 0y � 5y � 3x � 3

y � 3x � 25x � y � 11y � 6x � 15

0 x

y

xy=0

0 3 x

y

x=3

�92

x � �a � b�c��ab�1.3, 1.7��, �7 � �3, ��3, 5�

��3, 3�2, � 43�30�C � T � 20�C

T � 20 � 10h, 0 � h � 1210 � C � 35

0 14

_1 10

��, 0� � ( 14, �)��1, 0� � �1, ��

0 1_œ„3 0 œ„3

��, 1(�s3, s3)

1 20 1

��, 1� � �2, ��0, 1

0_10_2

�1, ��2, ��

x 2 � 1� x � 1 � � �x � 1

�x � 1

for x � �1

for x � �1

2 � x5 � s5

k�

2� k�2

2s3� 1ln 1

2

25 s3sn � 3 � 4n, ln � 1�3n, pn � 4n�3n�1

15!�5! � 10,897,286,400

29. 31.

33. 35.

37. 39. 41.45.

53.

EXERCISES C N PAGE A25

1. (a) (b) 3. (a) 720° (b)5. 7.9. (a) (b)

11. , , ,, ,

13. , , , ,

15. 5.73576 cm 17. 24.62147 cm 27.

29. 31.

33. and

35.

37. y

0 x

112

π3

5π6

0 � x � �4, 3�4 � x � 5�4, 7�4 � x � 2

5�6 � x � 20 � x � �6

�6, �2, 5�6, 3�2�3, 5�3

115 (4 � 6s2)cot � 4

3sec � 54csc � 5

3tan � 34cos � 4

5

cot�3�4� � �1sec�3�4� � �s2csc�3�4� � s2tan�3�4� � �1cos�3�4� � �1�s2sin�3�4� � 1�s2

0x

y

_ 3π4

0 x

y

315°

23 rad � �120��3 cm

�67.5��207�6

0 x

y

1

5

y � x � 3�1, �2��2, �5�, 4�x � 3�2 � �y � 1�2 � 25

0

y

x

y=1-2x

y=1+x

�0, 1�

0

y

x

x � 2

y � 4

0

y

x_2 20

y

x



39.

EXERCISES D N PAGE A33

1. (or any smaller positive number)3. 1.44 (or any smaller positive number)5. (or any smaller positive number)7. 0.11, 0.012 (or smaller positive numbers)11. (a) (b) Within approximately 0.0445 cm(c) Radius; area; ; 1000; 5; 13. (a) (b) 17.19. (a) 21. (a) (b)

EXERCISES F N PAGE A42

1. 3.

5. 7.

9. 11.

13. 15. 17. 19.

21. 80 23. 3276 25. 0 27. 61 29.

31. 33.

35.41. (a) (b) (c) (d)

43. 45. 14 49.

EXERCISES G N PAGE A50

1. (a) (b)

3. (a)

(b)

5. (a)

(b)

7.9. 11.13. 15.

17.

19.

21. 2 ln � x � � �1�x� � 3 ln � x � 2 � � C

�1

36 ln � x � 5 � �

1

6

1

x � 5�

1

36 ln � x � 1 � � C

275 ln 2 �

95 ln 3 (or 95 ln 83)

76 � ln 2

3a ln � x � b � � C

12 ln 322 ln � x � 5 � � ln � x � 2 � � C

x � 6 ln � x � 6 � � C

At � B

t 2 � 1�

Ct � D

t 2 � 4�

Et � F

�t 2 � 4�2

1 �A

x � 1�

B

x � 1�

Cx � D

x 2 � 1

A

x � 3�

B

�x � 3�2 �C

x � 3�

D

�x � 3�2

A

x�

B

x 2 �C

x 3 �Dx � E

x 2 � 4

A

x�

B

x � 1�

C

�x � 1�2

A

x � 3�

B

3x � 1

2n�1 � n 2 � n � 213

an � a0973005100 � 1n 4

n�n 3 � 2n 2 � n � 10��4

n�n 2 � 6n � 11��3n�n 2 � 6n � 17��3

n�n � 1�

�n

i�1 x i�

5

i�0 2 i�

n

i�1 2i�

19

i�1

i

i � 1

�10

i�1 i1 � 1 � 1 � 1 � � � � � ��1�n�1

110 � 210 � 310 � � � � � n10�1 �13 �

35 �

57 �

79

34 � 35 � 36s1 � s2 � s3 � s4 � s5

9, 110x � 1000.00250.025

�0.0445s1000� s1000� cm

0.0906

47

y

0 x3π2

2πππ2

5π2

3π

23.

25.

27.

29.

31.

33.

35. 37.

39.41. , where

43. (a)

(b)

The CAS omits the absolute value signs and the constant of integration.

EXERCISES H.1 N PAGE A59

1. (a) (b)

(c)

3. (a) (b)

(�1, �s3)��1, 0�

O

_2π3

”2, _ ’2π3

π

O(1, π)

�1, 3�2�, ��1, 5�2�

O

π2

”_1, ’π2

�1, 5�4�, ��1, �4��2, 7�3�, ��2, 4�3�

O

_3π4”1, _ ’3π

4

O

π3

π3”2, ’

75,772

260,015s19 tan�1

2x � 1

s19� C

11,049

260,015 ln�x 2 � x � 5� �

3146

80,155 ln � 3x � 7 � �

4822

4879 ln � 5x � 2 � �

334

323 ln � 2x � 1 � �

1

260,015 22,098x � 48,935

x 2 � x � 5

24,110

4879

1

5x � 2�

668

323

1

2x � 1�

9438

80,155

1

3x � 7�

C � 10.23t � �ln P �19 ln�0.9P � 900� � C

�12 ln 3 � �0.55

ln ��e x � 2�2

e x � 1 � � C2 � ln 259

�1

2�x 2 � 2x � 4��

2s3

9 tan�1 x � 1

s3 � �2�x � 1�

3�x 2 � 2x � 4�� C

116 ln � x � �

132 ln�x 2 � 4� �

1

8�x 2 � 4�� C

13 ln � x � 1 � �

16 ln�x 2 � x � 1� �

1

s3 tan�1

2x � 1

s3� C

12 ln�x 2 � 2x � 5� �

32 tan�1 x � 1

2 � � C

12 ln�x 2 � 1� � (1�s2 ) tan�1(x�s2 ) � C

ln � x � 1 � �12 ln�x 2 � 9� �

13 tan�1�x�3� � C



(c)

5. (a) (i) (ii)(b) (i) (ii)7. 9.

11.

13. Circle, center , radius 15. Horizontal line, 1 unit above the -axis17. 19.21. (a) (b)23. 25.

27. 29.

31. 33.¨=π

8

1

3 4

5

6

2

¨=π3

O

O

O

”1, ’π2

O

¨=_π6

x � 3 � �6r � 2c cos r � �cot csc

x

32(0, 32)

O

r=2r=3

¨=7π3

¨=5π3

O

r=4

¨=π6

¨=_π2

O

r=1r=2

��2, 5�3��2, 2�3�(�2s2, 3�4)(2s2, 7�4)

(s2, �s2)

O

3π4

”_2, ’3π4

35. 37.

39. 41.

43. 45.

47. (a) For , the inner loop begins at and ends at ; for , it begins at

and ends at .49. 51. 1

53. Horizontal at , ; vertical at 55. Horizontal at , [the pole], and ;

vertical at (2, 0), ,

57. Center , radius 59. 61.

63. By counterclockwise rotation through angle , , or about the origin65. (a) A rose with n loops if n is odd and 2n loops if n is even(b) Number of loops is always 2n67. For , the curve is an oval, which develops a dimpleas . When , the curve splits into two parts, one ofwhich has a loop.

a � 1a l1�

0 � a � 1

��3�6

7

�7

�7 7_3 3

_2.5

3.5

sa 2 � b 2�2�b�2, a�2�( 1

2, 4�3)( 12, 2�3)

( 32, 5�3)�0, �( 3

2, �3)�3, 0�, �0, �2�

(�3�s2, 3�4)(3�s2, �4)�

� 2 � sin�1 �1�c� � � sin�1 �1�c�c � 1 � � sin�1 ��1/c�

� sin�1 ��1�c�c � �1

(2, 0) (6, 0)

O 1

1

2

¨=π3¨=2π

3

(3, 0)(3, π)

OO

¨= π6¨=5π

6




1. 3. 5. 7.9. 4

11. 13.

15. 17. 19. 21.

23. 25. 27.29. where , , , and where , , , 31. , and the pole

33. Intersection at 35.

37. 39. 29.065383 � 2 � 1�3�2 � 1

� 0.89, 2.25; area � 3.46

(12 s3, �3), (1

2 s3, 2�3)23�1219�1211�12 � 7�12��1, �

17�1213�125�12 � �12�1, �

14( � 3s3)1

2 � 1524 �

14 s3

13 �

12 s3 �

32 s31

8

3

�3

�3 3

¨=π6

3

O

414 2�12 �

18 s3 5�10,240

EXERCISES I N PAGE A74

1. 3. 5. 7.

9. 11. 13. 15.

17. 19. 21.

23. 25.

27.

29. ,

31. ,,

33. 35.37. 39.

41. 43. 45.

47. , sin 3 � 3 cos2 sin � sin3

cos 3 � cos3 � 3 cos sin2

�e 212 � (s3�2)ii

0

Im

Re

_i

0

Im

Re

i

1

�(s3�2) �12 i, �i�1, �i, (1�s2)��1 � i �

�512s3 � 512i�1024

14 cos��6� � i sin��6�(2s2)cos�13�12� � i sin�13�12�

4s2 cos�7�12� � i sin�7�12�

12 cos��6� � i sin��6�

4cos��2� � i sin��2�, cos��6� � i sin��6�

5{cos[tan�1(43)] � i sin[tan�1(4

3)]}3s2 cos�3�4� � i sin�3�4��

12 � (s7�2)i

�1 � 2i�32 i4i, 4

12 � 5i, 135i�i12 �

12 i

1113 �

1013 i12 � 7i13 � 18i8 � 4i




1.3. (a) (c)

5. 11.

13. where is a positive integer

CHAPTER 9


1. 3.

5. A vertical plane thatintersects the xy-plane inthe line , (see graph at right)

7. (a) , , ; isosceles triangle

(b) , , ; right triangle9. (a) No (b) Yes11.13. , 5 15.17. (b)19. (a)(b)(c)21. A plane parallel to the -plane and 5 units in front of it23. A half-space consisting of all points to the left of the plane 25. All points on or between the horizontal planes and 27. All points on a circle with radius 2 and center on the -axis thatis contained in the plane 29. All points on or inside a sphere with radius and center O31. All points on or inside a circular cylinder of radius 3 with axisthe -axis33. 35.37. (a) (2, 1, 4) (b)

39. , a plane perpendicular to AB41.


1. (a) Scalar (b) Vector (c) Vector (d) Scalar

3. ABl

� DCl

, DAl

� CBl

, DEl

� EBl

, EAl

� CEl

2s3 � 314x � 6y � 10z � 9

P

A

C

B

0

z

yx

L™

L¡r 2 � x 2 � y 2 � z2 � R20 � x � 5

y

s3z � �1

zz � 6z � 0

y � 8

yz�x � 2�2 � �y � 3�2 � �z � 6�2 � 9�x � 2�2 � �y � 3�2 � �z � 6�2 � 4

�x � 2�2 � �y � 3�2 � �z � 6�2 � 36

52, 12 s94, 12 s85

�2, 0, �6�, 9�s2�3, �2, 1��x � 3�2 � �y � 8�2 � �z � 1�2 � 30

� RP � � 6� QR � � 3s5� PQ � � 3� RP � � 6� QR � � 2s10� PQ � � 6

z � 0y � 2 � x

z

y2

x

2

0

y=2-x

y=2-x, z=0

Q; R�4, 0, �3�

k��

2� �k�2

�

2s3� 1ln 1

2

25 s3sn � 3 � 4n, ln � 1�3n, pn � 4n�3n�1

15!�5! � 10,897,286,400

5. (a) (b)

(c) (d)

7. 9.

11. 13.

15. , , 13, 1017. , , ,

19. 21.

23. 25. , 27.

29.31.33.35. (a), (b) (d)

37.39. A sphere with radius 1, centered at


1. (b), (c), (d) are meaningful 3.5. 7. 9. 11.

15. 17.

19.21. (a) Neither (b) Orthogonal(c) Orthogonal (d) Parallel

23. Yes 25.

27. 29. 31. , 221 i �

121 j �

421 k1�s213, � 9

5 , �125 45�

�i � j � k��s3 [or ��i � j � k��s3]

45�, 45�, 90�

cos�1� �1

2s7� 101�cos�1�9 � 4s7

20 � 95�

u � v � 12 , u � w � �

1211914

�15

�x0, y0, z0 �a �0.50, 0.31, 0.81

s � 97 , t � 11

7y

x0

a

b

c

sa

tb

��i � 4 j��s17T1 � �196 i � 3.92 j, T2 � 196 i � 3.92 js493 22.2 mi�h, N8�W

100s7 264.6 N, 139.1�38.57 ft�s 45.96 ft�s�2, 2s3

89 i �

19 j �

49 k�

3

s58 i �

7

s58 j

s82s14�4 i � j � 9 k�i � j � 2 k�1, �42 �2, �18

z

y

k0, 0, _3l

x

k0, 1, 2l

k0, 1, _1l

x0

y

k6, _2l

k5, 2l

k_1, 4l

�0, 1, �1 �5, 2

z

y

0

A(0, 3, 1)

aB(2, 3, _1)x

x0

y

A(_1, 3)

B(2, 2)

a

a � �2, 0, �2 a � �3, �1

uw+v+u

wvv+w

wv

u-v

u_vu+v

u

v

57425_Ans_Ans_pA112-A121.qk 2/19/09 11:54 AM Page 115


35. or any vector of the form

37. 39.41. 43.


1. (a) Scalar (b) Meaningless (c) Vector(d) Meaningless (e) Meaningless (f) Scalar3. ; into the page 5.7. 9. 11.13. 15. 17.

19.21. 16 23. (a) (b)

25. 27. 82 29. 3

33. (b) 39. (a) No (b) No (c) Yes


1. (a) True (b) False (c) True (d) False (e) False(f ) True (g) False (h) True (i) True ( j) False(k) True3. ;

, , 5. ;

, ,

7. , , ;

9. , , ;

11. Yes

13. (a)(b) , ,

15. ,

17. Parallel 19. Skew 21.

23. 25.

27. 29.31.

33. 35.

37. 39. Perpendicular41. Neither,

43. (a) , , (b) cos�1� 5

3s3� 15.8�z � ty � �tx � 1

cos�1(13) 70.5�

�2, 3, 5�

0

z

y

x

”0, 0, ’

(1, 0, 0)

(0, _2, 0)

32

0

z

y

x

(0, 0, 10)

(5, 0, 0)

(0, 2, 0)

3x � 8y � z � �38x � 2y � 4z � �133x � 10y � 4z � 190

x � y � z � 23x � 7z � �9

�2x � y � 5z � 1

0 � t � 1r�t� � �2 i � j � 4k� � t�2 i � 7 j � 3k��0, �3, 3�(�3

2 , 0, �32)��1, �1, 0�

�x � 1��1� � � y � 5��2 � �z � 6��3�

x � 1 � �y � 1��2 � z � 1z � 1 � ty � �1 � 2tx � 1 � t

�x � 2��2 � 2y � 2 � �z � 3��4�z � �3 � 4t

y � 1 �12 tx � 2 � 2t

z � 6 � ty � 3tx � 1 � tr � �i � 6k� � t�i � 3 j � k�

z � 3.5 � ty � 2.4 � 2tx � 2 � 3tr � �2 i � 2.4 j � 3.5 k� � t�3 i � 2 j � k�

s97�3

417 N

12 s390�13, �14, 5

��2�s6, �1�s6, 1�s6 , �2�s6, 1�s6, �1�s6 i � j � k0t 4 i � 2t 3 j � t 2 k

12 i � j �

32 k15 i � 3 j � 3 k16 i � 48 k

10.8 sin 80� 10.6 N � m96s3

cos�1(1�s3) 55�135

2400 cos�40�� 1839 ft-lb144 J�s, t, 3s � 2s10 , s, t � �

�0, 0, �2s10 45. , 47.49.51. and are parallel, and are identical

53. 55. 57. 61.


1. (a) 25; a 40-knot wind blowing in the open sea for 15 h will cre-ate waves about 25 ft high.(b) is a function of t giving the wave heights produced by30-knot winds blowing for t hours.(c) is a function of giving the wave heights produced bywinds of speed blowing for 30 hours.3. (a) 1 (b) (c)5.

7.

9. , horizontal plane

11. , plane

13. , parabolic cylinder

15. (a) VI (b) V (c) I (d) IV (e) II (f) III

z

x y

z � y2 � 1

0

z

y

x

(0, 0, 6)

(2, 0, 0)(0, 3, 0)

3x � 2y � z � 6

z

y

0

x

z � 3

y

x_1 10

1

_1

��x, y� ��1 � x � 1, �1 � y � 1�

y

x0 1_1

y=≈

��x, y� � y x 2, x � �1� �1, 1�� 2

vvf �v, 30�

f �30, t�

1�s65�(2s14)187s61�14

P4P1P3P2

x � 3t, y � 1 � t, z � 2 � 2t�x�a� � �y�b� � �z�c� � 1y � 2 � �zx � 1

57425_Ans_Ans_pA112-A121.qk 2/7/09 2:17 PM Page 116


17. 19.

21.

Ellipsoid with center

23. (a) A circle of radius 1 centered at the origin(b) A circular cylinder of radius 1 with axis the -axis(c) A circular cylinder of radius 1 with axis the -axis25. (a) , , hyperbola ;

, , hyperbola ;, , circle

(b) The hyperboloid is rotated so that it has axis the -axis(c) The hyperboloid is shifted one unit in the negative -direction27. III29.

appears to have a maximum value of about 0.044. There are twolocal maximum points and two local minimum points.31.


1. See pages 682–83.3. (a) (b)

(2, �2s3, 5)(s2, s2, 1)

0

z

yx

”4, _ , 5’

5

4

π3

π3_

0

z

yx

π4

2 1

” 2, , 1 ’π4

210

y1

0�1

x1

0�1

z

f

0.04

0.02

0

_0.02_1

0x

z

1 _1 0y

1

yy

x 2 � y 2 � 1 � k2z � k�k ��1�x 2 � z2 � 1 � k2y � k

�k ��1�y2 � z2 � 1 � k2x � kyz

�0, 2, 3�

0

z

yx

(0, 4, 3)(0, 0, 3)

x 2 ��y � 2�2

4� �z � 3�2 � 1

z

y

xy

z

x

z � s4x 2 � y 2 5. (a) (b)

7. (a) (b)

(0, 0, 1)

9. (a) (b)11. Vertical half-plane through the -axis 13. Half-cone15. Circular paraboloid17. Circular cylinder, radius 1, axis parallel to the -axis19. Sphere, radius , center 21. (a) (b)23. (a)(b)25.

27. 29.

31. Cylindrical coordinates: , , 33.35.


True-False Quiz

1. True 3. True 5. True 7. True 9. True11. False 13. False 15. False 17. True

Exercises1. (a)(b) , (c) Center , radius 5�4, �1, �3�

x � 0�y � 2�2 � �z � 1�2 � 68�x � 1�2 � �y � 2�2 � �z � 1�2 � 69

0 � � ��4, 0 � � � cos 0 � z � 200 � � � 2�6 � r � 7

x

z

y

˙=3π4

∏=1

∏=2

z

yx2

2

2

x

z

y2

2

z=11

� (3 sin cos � � 2 sin sin � � cos ) � 6z � 6 � r(3 cos � � 2 sin �)

� sin � 2 sin �r � 2 sin �(0, 12, 0)1

2

z

z(s2, 3��2, 3��4)�4, ��3, ��6�

(s2�2, s6�2, s2)

0

z

yx

”2, , ’

2π4

π3

π3

π4

0

z

yx

(1, 0, 0)

1

(2, 4��3, 2)(s2, 7��4, 4)



3. ; ; out of the page5. 7. (a) 2 (b) (c) (d) 09. 11. (a) (b)13. 166 N, 114 N15. , , 17. , , 19. 21.23. Skew 25. (a) (b)27.

29. 31.

33. Ellipsoid

35. Circular cylinder

37. ,

39. , 41. 43.


1.3. (a)(b) (c)5. 20

4��3x 2 � y 2 � t 2 � 1, z � t�x � 1��2c� � �y � c��c 2 � 1� � �z � c��c 2 � 1�

(s3 �32) m

z � 4r 2r 2 � z2 � 4, � � 2(4, ��4, 4s3)(2s2, 2s2, 4s3)

�4, ��3, ��3�(s3, 3, 2)

(0, 1, 0)

(0, 0, 1)

z

x

y

(0, 0, 1)

(0, 1, 0)” , 0, 0’12

z

yx

yx

z(0, 0, 4)

(0, 1, 0)

(2, 0, 0)

0

z

yx

(0, 2, 0)

(0, 0, 6)

(3, 0, 0)

x=¥y

x0

��x, y� � x y 2�3�s222�s26

x � y � z � 4�4x � 3y � z � �14z � 4 � 5ty � 2 � tx � �2 � 2t

z � 2 � 3ty � �1 � 2tx � 4 � 3t

s41�2�4, �3, 4 cos�1( 13 ) 71�

�2�2�2, �4� u � v � � 3s2u � v � 3s2 CHAPTER 10


1. 3.

5. 7.

9. 11.

13.

15. , ; , , ,

17. , ;, , ,

19. II 21. V 23. IV0 � t � 1z � 5t � 2y � 2t � 1x � 3t � 10 � t � 1r�t� � �3t � 1, 2t � 1, 5t � 2

0 � t � 1z � 3ty � 2tx � t0 � t � 1r�t� � � t, 2t, 3t

x

y

z

(0, 0, 2)

y

z

2

_2

_1 1

x

z

2

_2

2π_2π

x

2π

y

1

_1_2π

x

z

y y=≈

z

x1

y

x y

z

(0, 2, 0)

y

x1

π

��1, ��2, 0 ��1, 2�



25. 27. ,

29. 31.

33.

37.39.41. 43. Yes


1. (a)

(b), (d)

(c) ; T�4� �r��4�

� r��4� �r��4� � lim h l 0

r�4 � h� � r�4�

h

y

x0 1

1

RC

Q

P

r(4.5)

r(4.2)

r(4)

r(4.5)-r(4)0.5

r(4.2)-r(4)0.2

T(4)

y

x0 1

1

RC

Q

P

r(4.5)

r(4.2)

r(4)

r(4.5)-r(4)

r(4.2)-r(4)

x � 2 cos t, y � 2 sin t, z � 4 cos2tr�t� � cos t i � sin t j � cos 2t k, 0 � t � 2�r�t� � t i �

12 �t 2 � 1� j �

12 �t 2 � 1� k

02 2

�2

0

2

�2

0x

y

z

_1_1

0

11

0

x

_1

0z

y

1

1

1 1

0 x

_1

0

0

z

y

_1_1

z

y

x

0

�1, 0, 1��0, 0, 0� 3. (a), (c) (b)

5. (a), (c) 7. (a), (c)

(b) (b)9.11. 13.15. 17.

19. ,

21. , , 23. , , 25.27. , , 29. , , 31. 66° 33. 35.37.39.45. 47.


1. 3. 5. 7.9. 11.

13.

15.

17. (a)(b)

19. (a)

(b)21. 23. 25.27.29.

31. ; approaches 033. (a) P (b)

35. 4

_4 4

_1

y=k(x)y=x–@

1.3, 0.7(� 1

2 ln 2, 1�s2)e x� x � 2 ��1 � �xe x � e x�2�3�212x 2��1 � 16x 6�3�2

17 s

1914

4256t 2��9t 4 � 4t2�3�2

s2e 2 t��e 2 t � 1�2

1

e2 t � 1�1 � e 2 t, s2e t, s2e t1

e 2 t � 1�s2et, e 2 t, �1 ,

229��sin t, 0, �cos t

�(2�s29) cos t, 5�s29, (�2�s29) sin t,�3 sin 1, 4, 3 cos 1�

r�t�s�� 2

s29s i � 1 �

3

s29s� j � 5 �

4

s29s� k

421.278015.38411

27�13 3�2 � 8�e � e�120s29

352t cos t � 2 sin t � 2 cos t sin tt 2 i � t 3 j � ( 2

3t 3�2 �23) k

tan t i �18�t 2 � 1�4 j � (1

3 t 3 ln t �19 t 3)k � C

i � j � k4 i � 3 j � 5 kz � �� ty � � � tx � �� t

z � 2ty � 1 � tx � tr�t� � �3 � 4t� i � �4 � 3t� j � �2 � 6t� k

z � 1 � ty � tx � 1 � tz � 2 � 4ty � 2tx � 3 � t

�0, et, �t � 2�et , �t 2 � 3t � 3�e2 t�1, et, �t � 1�et , �1�s3, 1�s3, 1�s3

35 j �

45 k� 1

3 , 23 , 23 r��t� � b � 2tcr��t� � 2tet 2

i � [3��1 � 3t�� kr��t� � � t cos t � sin t, 2t, cos 2t � 2t sin 2t

r��t� � e t i � 3e 3 t jr��t� � cos t i � 2 sin t j

y

x

r(0)

rª(0)

0 1

1 (1, 1)

y

0 xr ” ’

” , œ„2’œ„22

π4

rª ” ’π4

r��t� � �1, 2ty

0 x

r(_1)rª(_1)

(_3, 2)

57425_Ans_Ans_pA112-A121.qk 2/19/09 11:57 AM Page 119


37. is , is

39.

integer multiples of

41.

43. 45.47.49.

51.53. , 61.


1. (a) , ,,

(b) , 2.583.

5.

7.

� v�t� � � s1 � 4t 2

a�t� � 2 j

(1, 1, 2)

z

y

x

a(1)

v(1)

v�t� � i � 2t j

� v�t� � � s5 sin2 t � 4a�t� � �3 cos t i � 2 sin t j

0

y

x

v ” ’π3

a ” ’π3

” , œ„3’32

(0, 2)

(3, 0)

v�t� � �3 sin t i � 2 cos t j

� v�t� � � st 2 � 1a�t� � ��1, 0 (_2, 2)

0

y

x

v(2)

a(2)

v�t� � ��t, 1 2.4 i � 0.8 j � 0.5k

2.8 i � 0.8 j � 0.4k2.8 i � 1.8 j � 0.3k2.0 i � 2.4 j � 0.6k1.8 i � 3.8 j � 0.7k

2.07 � 1010 Å 2 m6x � 8y � z � �32x � y � 4z � 7

��1, �3, 1�

5

2.5�7.5

�5

(x �52 )2

� y 2 � 814 , x 2 � (y �

53 )2 � 16

9

y � 6x � �, x � 6y � 6�

� 23 , 23 , 13 , �� 1

3 , 23 , � 23 , �� 2

3 , 13 , 23 1�(s2et)6t 2��4 t 2 � 9t 4�3�2

2�

k(t)

t0 2π 4π 6π

��t� �6s4 cos2t � 12 cos t � 13

�17 � 12 cos t�3�2

y � ��x�by � f �x�a 9.11. ,

,

13. , 15. (a)(b)

17.19. , 21. (a) (b) (c)23. 25. , 27. , 29. (a) 16 m (b) upstream

31. The path is contained in a circle that lies in a plane perpen-dicular to with center on a line through the origin in the directionof .33. , 35. 0, 1 37. 39.


1. : no; : yes3. Plane through containing vectors , 5. Hyperbolic paraboloid7.

9.

_10

x1

_10

y

z

u constant

√ constant

_1

0

1

1

2

0

_2

21

0

2

1.5

1

xy

z

√ constant

u constant

�1, �1, 5 �1, 0, 4 �0, 3, 1�QP

t � 14.5 cm�s2, 9.0 cm�s26t2�s9t 4 � 4t2�18t3 � 4t��s9t 4 � 4t2

cc

40

_12

0

12

40

_4

0

20

23.6�55.4� � � � 85.5�13.0� � � � 36.0�

79.8�10.2�30 m�s200 m�s1531 m3535 m

� v�t� � � s25t 2 � 2r�t� � t i � t j �52 t 2 k

t � 4

_2000200

x

_100

10y

z

00.20.40.6

r�t� � ( 13 t 3 � t) i � �t � sin t � 1� j � ( 1

4 �14 cos 2t) k

r�t� � ( 12t 2 � 1� i � t 2 j � t kv�t� � t i � 2t j � k

e tst 2 � 2t � 3e t �2 sin t i � 2 cos t j � �t � 2�k�

e t �cos t � sin t� i � �sin t � cos t� j � �t � 1�k�s2 i � e t j � e�t k, e t j � e�t k, e t � e�t



11.

13. IV 15. II 17. III19.21.23. , ,

, ,

25. , , ,

29. ,, ,

31. (b)

33. (a) Direction reverses (b) Number of coils doubles


True-False Quiz

1. True 3. False 5. False 7. True 9. False11. True

Exercises1. (a)

(b) ,

3. , 5. 7. 86.631 9. ��21

3 i � �2�� 2� j � �2��k0 � t � 2�r�t� � 4 cos t i � 4 sin t j � �5 � 4 cos t�k

r ��t� � �� 2 cos �t j � � 2 sin �t kr��t� � i � � sin �t j � � cos �t k

z

y

x

(0, 1, 0)

(2, 1, 0)

2

0

�2

�2 �10 2 1 0

z

y x

0 � � � 2�0 � x � 3z � e�x sin �

20�101�1

0

1

y

z

x

x � x, y � e�x cos �

0 � x � 5, 0 � � � 2�z � 4 sin �y � 4 cos �x � x[or x � x, y � y, z � s4 � x 2 � y 2, x 2 � y 2 � 2]

0 � � � 2�0 � � ��4z � 2 cos y � 2 sin sin �x � 2 sin cos �

x � x, z � z, y � s1 � x 2 � z 2

x � 1 � u � v, y � 2 � u � v, z � �3 � u � v

_10

x1

_1

0y

1

_1

0z

1

√ constant

u constant

11. (a)

(b)

(c)13. 15.17. ,

, 19. (a) About 3.8 ft above the ground, 60.8 ft from the athlete(b) (c) from the athlete21. , , ,

, 23.


1. (a) (c)3. (a)5. (a) to the right of the table’s edge, (b) (c) to the right of the table’s edge7.9. where ,

,

CHAPTER 11


1. (a) ; a temperature of with wind blowing atfeels equivalent to about without wind.

(b) When the temperature is , what wind speed gives a windchill of ?(c) With a wind speed of , what temperature gives a windchill of ?(d) A function of wind speed that gives wind-chill values when thetemperature is (e) A function of temperature that gives wind-chill values whenthe wind speed is 3. Yes5. ,

7. (a) (b) ,interior of a sphere of radius 2, center the origin, in the first octant9. 11. 13. Steep; nearly flat15. 17.

5

yx

zz

14

yx

11�C, 19.5�C56, 35

��x, y, z� � x 2 � y 2 � z 2 � 4, x 0, y 0, z 0�3

y

x0

≈+¥=119

��, ln 9��x, y� � 19 x 2 � y 2 � 1�

50 km�h

�5�C

�35�C�49�C20 km�h

20 km�h�30�C�20�C

�27�C40 km�h�15�C�27

c � �c1, c2 , c3 b � �b1, b2 , b3 a � �a1, a2 , a3 r�u, v� � c � u a � vb

56�

2.13 ft7.6�15 ft�s0.94 ft

90�, v02��2t�

a � ��2rv � �R��sin �t i � cos �t j�

�� t ��3 � � 2��30 � � � 2�

z � 2 cos y � 2 sin sin �x � 2 sin cos �64.2 ft21.4 ft

a�t� � �1�t� i � e�t k� v�t� � � s2 � 2 ln t � �ln t�2 � e�2 t

v�t� � �1 � ln t� i � j � e�t kx � 2y � 2� � 012�173�2

st 8 � 4t 6 � 2t 4 � 5t 2��t 4 � t 2 � 1�2

�2t, 1 � t 4, �2t 3 � t�st 8 � 4t 6 � 2t 4 � 5t 2

� t 2, t, 1 �st 4 � t 2 � 1



19. 21.

23. 25.

27.

29. 31.

33.

1.0

0.5z

0.0

40

x

_4

4

0y

_4

0

_2 0 2 2 0 _2y x

z

y

0 x

y

x

z

z=4

z=3

z=2

z=1

y

0 x

4321

x 2 � 9y 2 � k

y

x

0

0

1

1

2

2

3

3

y

x0

0

1 23

_1

_2

_3

y 2 � x 2 � k 2y � ke�x

y

x

0

_1

_2

1

2

y

x

0 1 2341234

y � �sx � k�y � 2x�2 � k 35. (a) C (b) II 37. (a) F (b) I39. (a) B (b) VI 41. Family of parallel planes43. Family of circular cylinders with axis the -axis 45. (a) Shift the graph of upward 2 units(b) Stretch the graph of vertically by a factor of 2(c) Reflect the graph of about the -plane(d) Reflect the graph of about the -plane and then shift itupward 2 units47. If , the graph is a cylindrical surface. For , the levelcurves are ellipses. The graph curves upward as we leave the ori-gin, and the steepness increases as c increases. For , the levelcurves are hyperbolas. The graph curves upward in the y-directionand downward, approaching the xy-plane, in the x-direction givinga saddle-shaped appearance near (0, 0, 1).49. (b)


1. Nothing; if is continuous, 3.5. 1 7. Does not exist 9. Does not exist 11. 013. Does not exist 15. 2 17. 1 19. Does not exist21. The graph shows that the function approaches different num-bers along different lines.

23. ;

25. Along the line 27.

29. 31.

33. 35. 037. 039.

is continuous on


1. (a) The rate of change of temperature as longitude varies, withlatitude and time fixed; the rate of change as only latitude varies;the rate of change as only time varies.(b) Positive, negative, positive3. (a) ; for a temperature of and windspeed of , the wind-chill index rises by for eachdegree the temperature increases. ; for a temperature of and wind speed of , the wind-chillindex decreases by for each the wind speed increases.(b) Positive, negative (c) 05. (a) Positive (b) Negative7. (a) Positive (b) Negative9. c � f, b � fx, a � fy

km�h0.15�C30 km�h�15�C

fv��15, 30� � �0.151.3�C30 km�h

�15�CfT ��15, 30� � 1.3

� 2f

_2

0

2x_2

02y

z

_1

0

1

2

��x, y� � �x, y� � �0, 0��

��x, y, z� � y � 0, y � sx 2 � z 2 ��x, y� � x 2 � y 2 � 4��x, y� � y � 0�y � x

��x, y� � 2x � 3y � 6�h�x, y� � �2x � 3y � 6�2 � s2x � 3y � 6

�52f �3, 1� � 6f

y � 0.75x � 0.01

c � 0

c � 0c � 0

xyfxyf

ff

�k � 0�x



11. ,

13.

15. , 17. , 19. , 21.23.

25. ,

27. ,29. , ,

31. , ,

33. , ,

35. , ,,

37.

39. 41.

43. , fy�x, y� � 2xy � x 3fx�x, y� � y 2 � 3x 2y

14

15

�u��xi � xi�sx12 � x2

2 � � xn2

ft � xy 2z 2 sec2�yt�fz � 2xyz tan�yt�fy � xyz 2t sec2�yt� � xz 2 tan�yt�fx � yz 2 tan�yt�

�u��z � xy 2�s1 � y 2z2

�u��y � x sin�1�yz� � xyz�s1 � y 2z2�u��x � y sin�1�yz��w��z � 3��x � 2y � 3z�

�w��y � 2��x � 2y � 3z��w��x � 1��x � 2y � 3z�fz � x � 20x 2y 3z3fy � �15x 2y 2z4fx � z � 10xy 3z4

�u��w � e w�t�u��t � e w�t(1 � w�t)

fs�r, s� �2rs

r 2 � s 2fr�r, s� �2r 2

r 2 � s 2 � ln�r 2 � s 2�

�w�� cos cos �, �w�� sin sin �fx�x, y� � 2y��x � y�2, fy�x, y� � �2x��x � y�2

�z��y � 30�2x � 3y�9�z��x � 20�2x � 3y�9

ft �x, t� � �e�t cos �xfx �x, t� � ��e�t sin �xfy �x, y� � 5y 4 � 3xfx �x, y� � �3y

fy

0

_20

2x _20

2

y

z

10

0

_2

02x _2

02

y

_10

z

fx

10

0_2

02x _2

02

y

z

f

fx � 2x � 2xy, fy � 2y � x 2

z

y

0

x

(1, 2, 8)

C¡

(1, 2)

2

16

4

z

y

0

x

(1, 2, 8)

C™

(1, 2)

2

16

4

fy�1, 2� � �4 � slope of C2fx�1, 2� � �8 � slope of C1 45. ,

47. ,

49. (a) (b)51. , , 53. , ,

55. , , 59.61.63. 65.67. 79.85. No 87. 89.

91. (a)

(b) ,

(c) 0, 0 (e) No, since and are not continuous.


1. 3. 5.7. 9.

11. 13. 17. 6.319. 21.23.25.27.29. 31. 33.35. 2.3% 37. 39.41. 43.45.


1.3.5.7. ,

9. ,�z��t � 2st cos cos � � s2 sin sin �

�z��s � t 2 cos cos � � 2st sin sin ��z��t � �2sxy 3 sin t � 3sx 2y 2 cos t

�z��s � 2xy 3 cos t � 3x 2y 2 sin te y�z2t � �x�z� � �2xy�z2 ��x�t� � y sin t�s1 � x 2 � y2

�2x � y� cos t � �2y � x�e t

�1 � �x, �2 � �yx � y � z � 2�x � 2z � 1

3x � y � 3z � 3117 � 0.059 �

16 cm35.4 cm2�z � 0.9225, dz � 0.9dR � � 2 cos � d � 2� cos � d� � � 2 sin � d�dm � 5p4q3 dp � 3p5q2 dqdz � 3x 2 ln�y 2� dx � �2x 3�y� dy

4T � H � 329; 129�F37 x �

27 y �

67 z; 6.9914

19 x �

29 y �

232x �

14 y � 1

0

2 x

0

2y

_1

0z

1400

200

0

y50

_5x

100

_10

z

z � yx � y � 2z � 0z � �7x � 6y � 5

fyxfxy

fy�x, y� �x 5 � 4x 3y 2 � xy 4

�x 2 � y 2 �2fx�x, y� �x 4y � 4x 2y 3 � y 5

�x 2 � y 2 �2

_0.2

0.2

0

_1

0

1

y

10

_1

x

z

�2x � 1 � t, y � 2, z � 2 � 2tR 2�R 1

2�12.2, �16.8, �23.256yz 2 e r �2 sin � cos � r sin �

24 sin�4x � 3y � 2z�, 12 sin�4x � 3y � 2z�12xy, 72xy

zyy � �2y��1 � y 2�2zxy � 0 � zyxzxx � �2x��1 � x 2�2wvv � u2��u2 � v2�3�2

wuv � �uv��u2 � v2�3�2 � wvuwuu � v2��u2 � v2�3�2fyy � 20x 3y 3fxy � 15x 2y 4 � 8x 3 � fyxfxx � 6xy 5 � 24x 2y

f ��x � y�, f ��x � y�f ��x�, t��y�

�z

�y�

�z

1 � y � y 2z 2

�z

�x�

1 � y 2z 2

1 � y � y 2z 2

�z

�y�

3xz � 2y

2z � 3xy

�z

�x�

3yz � 2x

2z � 3xy



11. ,

13. 62 15.

17. , ,

19. ,

21. 23. 25.

27. 29.

31.

33. 35.37. (a) (b) (c)39. 41.43. (a) ,

49.


1. 3. 5.7. (a)(b) (c)

9. (a) (b) (c)11. 13. 15. 17.19. 21. 23.25. (b)27. All points on the line 29. (a)31. (a) (b) (c)33. 37.39. (a) (b)

41. (a) (b)

43. (a) (b)45. 47. ,

y

x0

2x+3y=12

xy=6

(3, 2)

f (3, 2)Î

1

_1

0

1

2

12

x2

z

y

2x � 3y � 12�2, 3�x � 1 � y � �zx � y � z � 1

x � 2

4�

y � 1

�5�

z � 1

�14x � 5y � z � 4

x � 3 � y � 3 � z � 5x � y � z � 11

77425

32713

2s406�38, 6, 12 �32�s3�40�(3s3)

y � x � 1��12, 92 �

1, �3, 6, �2 �1, �0, 1 �2�59�(2s5)4�s30�8�s1023�10

�223�1, 12, 0 ��e 2yz, 2xze 2yz, 2xye 2yz �

s3 �32�2, 3 �

�f �x, y� � �2 cos�2x � 3y�, 3 cos�2x � 3y��2 � s3�2� 0.778��0.08 mb�km

4rs �2z��x 2 � �4r 2 � 4s 2 ��2z��x �y � 4rs �2z��y 2 � 2 �z��y�z�� z��x�r sin � ��z��y�r cos

�z��r � ��z��x� cos � ��z��y� sin �1�(12s3) rad�s� �0.27 L�s

0 m�s10 m2�s6 m3�s� �0.33 m�s per minute2�C�s

1 � y 2z 2

1 � y � y 2z 2 , �z

1 � y � y 2z 2

3yz � 2x

2z � 3xy,

3xz � 2y

2z � 3xy

sin�x � y� � e y

sin�x � y� � xe y

36, 24, 3097, 9785, 178, 54

�w

�y�

�w

�r �r

�y�

�w

�s �s

�y�

�w

�t �t

�y

��w

�t �t

�x

�w

�x�

�w

�r �r

�x�

�w

�s �s

�x

�u

�t�

�u

�x �x

�t�

�u

�y �y

�t

�u

�s�

�u

�x �x

�s�

�u

�y �y

�s

�u

�r�

�u

�x �x

�r�

�u

�y �y

�r

7, 2

�z

�t� e r s cos �

t

ss 2 � t 2 sin �

�z

�s� e r t cos �

s

ss 2 � t 2 sin � 51. No 57.

61. If and , then and areknown, so we solve linear equations for and .


1. (a) f has a local minimum at (1, 1).(b) f has a saddle point at (1, 1).3. Local minimum at (1, 1), saddle point at (0, 0)5. Minimum 7. Minima , , saddle point at 9. Minimum , saddle point at 11. None 13. Minimum , saddle points at 15. Minima , saddle points at , 19. Minima , 21. Maximum ,minimum , saddle point at 23. Minima , ,saddle point (0.312, 0), lowest point 25. Maxima , ,saddle points ,highest points 27. Maximum , minimum 29. Maximum , minimum 31. Maximum , minimum

33.

35. 37. , 39.

41. 43. 45. Cube, edge length 47. Square base of side 40 cm, height 20 cm 49.


1.3. No maximum, minimum 5. Maximum , minimum 7. Maximum , minimum 9. Maximum , minimum

11. Maximum , minimum 1

13. Maximum ,

minimum

15. Maximum ,minimum 17. Maximum , minimum

19. Maximum ,

minimum

27–37. See Exercises 35–45 in Section 11.7.39. L 3�(3s3)

f (�1�s2, �1�(2s2)) � e�1�4

f (�1�s2, �1�(2s2)) � e 1�4

12

32

f (1, �s2, s2) � 1 � 2s2f (1, s2, �s2) � 1 � 2s2

f (� 12 , � 1

2 , � 12 , � 1

2 ) � �2

f ( 12, 12 , 12 , 12 ) � 2

s3

�2�s32�s3f ��1, �3, �5� � �70f �1, 3, 5� � 70

f ��2, �1� � �4f ��2, 1� � 4f �1, 1� � f ��1, �1� � 2

�59, 30

L 3�(3s3)c�124

38r 3� (3s3)

1003 , 100

3 , 1003(2, 1, �s5)(2, 1, s5)s3

_3

_2

_1

0

_1 0 1

_2

2

4

x

y

z

(_1, 0, 0) (1, 2, 0)

f ��1, 0� � �2f �1, 0� � 2f �0, 0� � 4f ��1, 1� � 7

f �0, 3� � �14f �2, 0� � 9�1.629, �1.063, 8.105�

��0.259, 0�, �1.526, 0�f �1.629, �1.063� � 8.105f ��1.267, 0� � 1.310��1.714, 0, �9.200�

f �1.402, 0� � 0.242f ��1.714, 0� � �9.200��, ��f �5��3, 5��3� � �3s3�2

f ��3, ��3� � 3s3�2f ��1, �1� � 3f �1, �1� � 3

�3��2, 0��2, 0�f �2�, 1� � �1f ��, �1� �f �0, 1� �

��1, 0�f �0, 0� � 0�0, 0�f �2, 1� � �8

�0, 0�f ��1, �1� � 0f �1, 1� � 0f (1

3, �23) � �

13

fyfx

c fx � dfyafx � bfyv � �c, d �u � �a, b �x � �1 � 10t, y � 1 � 16t, z � 2 � 12t



41. Nearest , farthest 43. Maximum , minimum 45. (a) (b) When


True-False Quiz

1. True 3. False 5. False 7. True 9. False11. True

Exercises1. 3.

5. 7.

9.11. (a) , (b) byEquation 11.6.9 (Definition 11.6.2 gives .)(c)13. , 15. , 17. , , 19. , , 21. , ,

, ,,

25. (a) (b)

27. (a) (b)

29. (a)(b)31.33.35.37.43. 45.47. 49. �5

8 knot�mis145�2, �4, 92 ��

45�2xe yz 2

, x 2z 2e yz 2

, 2x 2yzeyz 2

��47, 1082xy 3�1 � 6p� � 3x 2y 2� pe p � e p� � 4z 3� p cos p � sin p�60x �

245 y �

325 z � 120; 38.656

(2, 12 , �1), (�2, �12 , 1)

x � 3 � 8t, y � 4 � 2t, z � 1 � 4t4x � y � 2z � 6

x � 2

4�

y � 1

�4�

z � 1

�62x � 2y � 3z � 3

x � 1

8�

y � 2

4�

z � 1

�1z � 8x � 4y � 1

f zz � m�m � 1�x k y lz m�2f yz � lmx k y l�1z m�1 � f zy

f yy � l�l � 1�x k y l�2z mf xz � kmx k�1y lz m�1 � f zx

f xy � klx k�1y l�1z m � f yxf xx � k�k � 1�x k�2 y lz mf yy � �2xf xy � �2y � f yxf xx � 24x

Tr � per��q � er �Tq � p��q � er �Tp � ln�q � er�tv � u��1 � v 2�tu � tan�1v

fy � y�s2x � y 2fx � 1�s2x � y 2

�0.25�1.1�C�m

� 0.35�C�m�3.0�C�m�3.5�C�m

23

x210

y

2

1

y

x

12

34 5

0

y

x_1

_1

y=_x-1

x y

z

1

1

��x, y� � y � �x � 1�

x1 � x2 � � xnc�n��5.3506�9.7938

��1, �1, 2�( 12, 12 , 12 ) 51. Minimum

53. Maximum ; saddle points (0, 0), (0, 3), (3, 0)55. Maximum , minimum 57. Maximum , minima ,saddle points 59. Maximum ,minimum 61. Maximum 1, minimum 63.65.


1. 3. (a) (b) Yes9.

CHAPTER 12


1. (a) 288 (b) 144 3. (a) (b) 05. (a) 4 (b) 7.9. (a) (b) 11. 60 13. 315. 1.141606, 1.143191, 1.143535, 1.143617, 1.143637, 1.143642


1. , 3. 10 5. 2 7. 9.11. 0 13. 15. 17.19. 21.

23.

25. 27. 29. 2 31.33.

35. 37. 039. Fubini’s Theorem does not apply. The integrand has an infinitediscontinuity at the origin.


1. 32 3. 5. 7. 9. �43e � 13

10

56

2

0

y1

0

x10

z

21e � 57

643

16627

952

z

yx

0

1

1

4

12�e 2 � 3�1

2 (s3 � 1) �112�

9 ln 2212�

212 ln 2261,632�453x 2500y 3

�15.5�248U � V � L�8

� 2�2 � 4.935

s3�2, 3�s2x � w�3, base � w�3L2W 2, 14 L2W 2

P(2 � s3), P(3 � s3)�6, P(2s3 � 3)�3(�3�1�4, 3�1�4

s2, �31�4 ), (�3�1�4, �3�1�4s2, �31�4 )

�1f (�s2�3, �1�s3) � �2�(3s3)

f (�s2�3, 1�s3) � 2�(3s3)��1, �1�, �1, 0�

f �1, �1� � �3f ��1, 0� � 2f �2, 4� � �64f �1, 2� � 4

f �1, 1� � 1f ��4, 1� � �11



11. (a) (b)

13. Type I: ,type II: ;

15.

17. 19. 21. 0 23. 25.

27. 6 29. 31. 33. 0, 1.213; 0.713 35.37.

39. 13,984,735,616�14,549,535

41. 43.

45.

47. 49. 51. 53. 155. 57. 61.63. 65.


1. 3.

5. 33��2y

0 x

4 7

R

x1�1 x

�x�1��20 f �x, y� dy dxx

3��20 x4

0 f �r cos , r sin �r dr d

�a 2ba 2b �32 ab 2

9�34��16�e�1�16 � xxQ e��x 2�y 2� 2

dA � ��16

13 (2s2 � 1)1

3 ln 916 �e 9 � 1�

y

x0

x=2

y=ln x or x=e†

ln 2

1 2

y=0

xln 2

0 x2e y f �x, y� dx dy

≈+¥=9

y=0

y

x0

3

3–3

x=4

y=0

y=œ„x

y

x0

2

4

x3

�3 xs9�x 2

0 f �x, y� dy dxx

20 x4

y 2 f �x, y� dx dy

0

z

y

x

(0, 0, 1)

(1, 0, 0)

(0, 1, 0)

643

13

12815

318

718

14720

12 �1 � cos 1�

� x2

�1 xy�2y 2 y dx dy � 9

4x10 xsx

�sx y dy dx � x4

1 xsxx�2 y dy dx

13D � ��x, y� � 0 � y � 1, y � x � 1�

D � ��x, y� � 0 � x � 1, 0 � y � x�

0 x

y

D

0 x

y

D

7. 0 9. 11. 13.

15. 17. 19. 21.

23. 25. 27.

29. 31. 33.

35. 37. (a) (b)


1. 3. 5.

7. ,

9. , 11. 13.15. if vertex is (0, 0) and sides are along positive axes

17.

19. , , if vertex is and sides arealong positive axes

21. , , ,

,

23. (a) (b) 0.375 (c)25. (b) (i)(ii) (c) 2, 527. (a) (b)

29. (a) , where D is thedisk with radius 10 mi centered at the center of the city(b) , on the edge


1. 3. 5. 7. 4

9. 11. 13.

15. (a) 24.2055 (b) 24.2476 17.

19.

21. (b)

(c)

25. 27.


1. 3. 1 5. 7. 9. 4 11.

13. 15. 17. 19. 21.

23. (a) (b) 25. 60.53314 � �

13x

10 xx

0 xs1�y 2

0 dz dy dx

36�16316��31

608��3e�

6528�

13

13 �e 3 � 1�27

4

4�983 �

x2�

0 x�

0 s36 sin4u cos2v � 9 sin4u sin2v � 4 cos2u sin2u du dv

2

0

�2

�2 �10 2 1 0

z

y x

458 s14 �

1516 ln[(11s5 � 3s70)�(3s5 � s70)]

4.4506

13.97834�b(b � sb 2 � a 2 )�2��3�(2s2 � 1)s2�63s14s14�

200�k�3 � 209k, 200(��2 �89 )k � 136k

xxD k[1 �1

20 s�x � x0 �2 � �y � y0 �2 ] dA

�0.632�0.5001 � e�1.8 � e�0.8 � e�1 � 0.3481

e�0.2 � 0.8187

548 � 0.10421

2

I0 � � 4�16 � 9� 2�64Iy � 116 �� 4 � 3� 2 �

Ix � 3� 2�64�x, y � � 2�

3�

1

�,

16

9��m � � 2�8

�0, 0�7ka6�907ka6�1807ka6�180

116�e 4 � 1�, 18�e 2 � 1�, 1

16�e 4 � 2e 2 � 3��2a�5, 2a�5�

�0, 45��14��( 38, 3��16)�L�2, 16��9��L�4

e 2 � 1

2�e 2 � 1�,

4�e 3 � 1�9�e 2 � 1��1

4�e 2 � 1�

6, ( 34, 32 )4

3 , (43 , 0)64

3 C

s� �2s� �41516

2��a � b�1800� ft32s2�3

12��1 � cos 9��12�8��3�(64 � 24s3)

�2��3�[1 � (1�s2)]43 �a 34

3�163 �

364 � 2��2��1 � e�4 �1

2� sin 9



27.

29.

31.

33.

35.

37. 39.

41. 43.

45. (a)

(b) , where

(c)

47. (a)

(b)

(c)

49. (a) (b) (c) 51.53. (a) The region bounded by the ellipsoid

(b) 4s6��45

x 2 � 2y 2 � 3z2 � 1L3�81

57601

6418

1240 �68 � 15��

� 28

9� � 44,

30� � 128

45� � 220,

45� � 208

135� � 660�3

32 � �1124

x3

�3 xs9�x 2

�s9�x 2 x5�y1 �x 2 � y 2�3�2 dz dy dx

x3

�3 xs9�x 2

�s9�x 2 x5�y1 zsx 2 � y 2 dz dy dxz � �1�m�

x3

�3 xs9�x 2

�s9�x 2 x5�y1 ysx 2 � y 2 dz dy dxy � �1�m�

x3

�3 xs9�x 2

�s9�x 2 x5�y1 xsx 2 � y 2 dz dy dxx � �1�m�

�x, y, z �m � x

3�3 x

s9�x 2

�s9�x 2 x5�y1 sx 2 � y 2 dz dy dx

12�kha 4Ix � Iy � Iz � 2

3 kL5

a 5, �7a�12, 7a�12, 7a�12�7930 , ( 358

553 , 3379 , 571

553 )

x10 x1

z xxz f �x, y, z� dy dx dz� x

10 xx

0 xxz f �x, y, z� dy dz dx �

x10 xy

0 x1y f �x, y, z� dx dz dy� x

10 x1

z x1y f �x, y, z� dx dy dz �

x10 xx

0 xy0 f �x, y, z� dz dy dxx

10 x1

y xy0 f �x, y, z� dz dx dy �

� x10 x�1�z�2

0 x1�zsx f �x, y, z� dy dx dz

� x10 x1�sx

0 x1�zsx f �x, y, z� dy dz dx

� x10 x1�y

0 xy 2

0 f �x, y, z� dx dz dy

� x10 x1�z

0 xy 2

0 f �x, y, z� dx dy dz

� x10 xy 2

0 x1�y0 f �x, y, z� dz dx dy

x10 x1

sx x1�y0 f �x, y, z� dz dy dx

� x20 xs4�2z

�s4�2z x4�2zx 2 f �x, y, z� dy dx dz

� x2�2 x

2�x 2�20 x4�2z

x 2 f �x, y, z� dy dz dx

� x40 x2�y�2

0 xsy�sy f �x, y, z� dx dz dy

� x20 x4�2z

0 xsy�sy f �x, y, z� dx dy dz

� x40 xsy

�sy x2�y�20 f �x, y, z� dz dx dy

x2�2 x

4x 2 x2�y�2

0 f �x, y, z� dz dy dx

� x1�1 x

s4�4z 2

�s4�4z 2 x4�x 2�4z 2

0 f �x, y, z� dy dx dz

� x2�2 x

s4�x 2�2�s4�x 2�2 x

4�x 2�4z 2

0 f �x, y, z� dy dz dx

� x40 xs4�y�2

�s4�y�2 xs4�y�4z 2

�s4�y�4z 2 f �x, y, z� dx dz dy

� x1�1 x

4�4z 2

0 xs4�y�4z 2

�s4�y�4z 2 f �x, y, z� dx dy dz

� x40 xs4�y

�s4�y xs4�x 2�y�2�s4�x 2�y�2 f �x, y, z� dz dx dy

x2�2 x

4�x 2

0 xs4�x 2�y�2�s4�x 2�y�2 f �x, y, z� dz dy dx

z

y

x

01

2

1


1. 3.

5.7. 9. 11.13. (a) (b) 15.17. 19. 21.23. 25. (a) (b) (0, 0, 2.1)27. 29. (a) (b)

31.

33. 35. 0 37. 39.41. (a) , where is the cone(b) ft-lb


1. 16 3. 5. 07. The parallelogram with vertices (0, 0), (6, 3), (12, 1), (6, �2)9. The region bounded by the line , the y-axis, and 11. , is one possible transformation,where 13. , is one possible transformation, where 15. 17. 19. 2 ln 321. (a) (b)23. 25. 27.


True-False Quiz

1. True 3. True 5. True 7. True 9. False

Exercises

1. 3. 5. 7.9.11. The region inside the loop of the four-leaved rose inthe first quadrant13. 15. 17. 19.

21. 23. 25. 27.

29. 176 31. 33.35. (a) (b) (c)37. (a) (b)39. 41. 43. 0.051245. (a) (b) (c)

47. 49. 51. 0�ln 2x10 x1�z

0 xsy�sy f �x, y, z� dx dy dz

145

13

115

4865ln(s2 � s3) � s2�3

�a4h�10�0, 0, h�4�I0 � 1

8, Ix � 112 , Iy � 1

24( 13 , 8

15 )14

2ma 3�923

6415��9681

281��5

814 ln 21

2 e 6 �72

12 sin 1

r � sin 2�x

�

0 x42 f �r cos �, r sin �� r dr d�

23

12 sin 14e 2 � 4e � 3�64.0

e � e�132 sin 18

5 ln 81.083 � 1012 km34

3 �abc6��3

S � ��u, v� 1 � u � s2, 0 � v � ��2y � u sin vx � u cos v

S � ��u, v� �1 � u � 1, 1 � v � 3y � 1

3�u � 2v�x � 13�v � u�

y � sxy � 1

sin2� � cos2�

�3.1 � 1019CxxxC h�P�t�P� dV

136��99(4s2 � 5 )�155��6

13� (2 � s2), (0, 0, 3�[8(2 � s2)])

4K�a 5�15(0, 0, 38 a)(0, 525296 , 0)

10�(s3 � 1)�a 3�31562��1515��16312,500��7

�Ka 2�8, �0, 0, 2a�3��0, 0, 15�162�

2��5��e 6 � e � 5�384�

x��20 x3

0 x20 f �r cos �, r sin �, z� r dz dr d�

�9��4� (2 � s3)64��3

x y

z

π6

3

x 4 y4

z

4




1. 30 3. 7. (b) 0.90

11.

CHAPTER 13


1.

3. 5.

7. 9.

11. II 13. I 15. IV 17. III19. The line

21.

23.

�y

sx 2 � y 2 � z2 j �

z

sx 2 � y 2 � z2 k

� f �x, y, z� �x

sx 2 � y 2 � z2 i

�f �x, y� � �xy � 1�e xy i � x 2e xy j

y � 2x4.5

�4.5

�4.5 4.5

z

yx

z

yx

y

x0

y

x0

_2

2

2

1

2

1_1_2

y

0 x

_1

abc��2

3�

8

9s3�12 sin 1

25.

27.

29. III 31. II 33.35. (a) (b)


1. 3. 1638.4 5. 7. 9.

11. 13. 15.

17. (a) Positive (b) Negative19. 45 21. 23. 1.9633 25. 15.0074

27.

29. (a) (b)

31. 33. 2�k, �4��, 0�172,7045,632,705 s2 �1 � e�14� �

0 2.1

2.1

_0.2

F”r” ’’

F{r(1)}

F{r(0)}

1

œ„2

118 � 1�e

2.5

�2.5

�2.5 2 .5

3� �23

65 � cos 1 � sin 1

973

15

112s14 �e 6 � 1�

s5�173

2438

154 �1453�2 � 1�

y � C�x

y � 1�x, x � 0y

x0

�2.04, 1.03�

6

6�6

�6

0_2_4_6 4 6 x

y

_2

2

�f �x, y� � 2x i � j



35. (a) ,

,

, where (b)37. , 39. 41. 26 43. 45. (b) Yes47.


1. 40 3.5. Not conservative 7.9.11. (b) 16 13. (a) (b) 215. (a) (b) 7717. (a) (b) 0 19. 221. It doesn’t matter which curve is chosen.23. 25. No 27. Conservative31. (a) Yes (b) Yes (c) Yes33. (a) No (b) Yes (c) Yes

EXERCISES 13.4 N PAGE 9391. 3. 5. 12 7. 9. 11.

13. 17. 19. 21. (c)

23. if the region is the portion of the diskin the first quadrant

27. 0


1. (a) (b)3. (a) (b)5. (a) (b)7. (a) (b)9. (a) Negative (b)11. (a) Zero (b) curl F points in the negative -direction13. 15.17. Not conservative 19. No


1. 49.09 3. 5. 7.

9. 11. 13.

15. 17. 19. 12 21.

23. 25. 27. 0 29. 48 31.

33. 3.4895

35. ,where projection of on -plane

37.

39. (a) (b)

41. 43. 45.


3. 0 5. 0 7. �1 9. 80�

1248�83�a 3�00 kg�s

4329s2��5Iz � xxS �x 2 � y 2 ��x, y, z� dS

�0, 0, a�2�xzSD �

xxS F � dS � xxD �P��h��x� � Q � R��h��z�� dA

2� �83�

43��

16

71318016��60�(391s17 � 1)

364s2��3s3�24171s14

23 (2s2 � 1)11s14900�

f �x, y, z� � x 2y � y 2z � Kf �x, y, z� � xy 2z3 � Kz

curl F � 01�x � 1�y � 1�z�1�y, �1�x, 1�x �

2�sx 2 � y 2 � z20y�e z � e x�ze x i � �xye z � yze x� j � xe z k

yz�x 2 i � 3xy j � xz k

x 2 � y 2 � a 2�4a�3�, 4a�3��

923��

112

6252 �

43 � 2��24�1

3238�

30

f �x, y, z� � xy 2 cos zf �x, y, z� � xyz � z 2

f �x, y� � 12 x 2y 2

f �x, y� � x ln y � x 2y 3 � Kf �x, y� � ye x � x sin y � K

f �x, y� � x 2 � 3xy � 2y 2 � 8y � K

22 J1.67 � 104 ft-lb2� 2

Iy � k (12� �

23 )Ix � k(1

2� �43 )

�0, 0, 3��m � xC ��x, y, z� dsz � �1�m� xC z��x, y, z� ds

y � �1�m� xC y��x, y, z� ds

x � �1�m� xC x��x, y, z� ds 11. (a) (b)

(c) ,,

17. 3


5. 7. 9. 0 11. 13.15.17. 19. Negative at , positive at

21. in quadrants I, II; in quadrants III, IV


True-False Quiz

1. False 3. True 5. False 7. True

Exercises

1. (a) Negative (b) Positive 3. 5.

7. 9. 11. 13. 0

17. 25. 27.

31. 35. 37. 21

APPENDIXES

EXERCISES A N PAGE A6

1. 18 3. 5.

7. 9.

11. 13.

15. 17.

19. 21.

0 1_œ„3 0 œ„3

��, 1�(�s3, s3)

1 20 1

��, 1� � �2, ��0, 1�

0_10_2

��1, ��2, �

x 2 � 1 x � 1 � �x � 1

�x � 1

for x �1

for x � �1

2 � x5 � s5

�4�12

�64��3��60�(391s17 � 1)�8�

f �x, y� � e y � xe xy1112 � 4�e110

3

4156s10

div F � 0div F � 0

P2P113��20341s2�60 �

8120 arcsin(s3�3)

032��39��292

0 t 2�z � 1 � 3�cos t � sin t�

_2

0

2

4

_2 0 2 2 0 _2

z

y x

x � 3 cos t, y � 3 sin t

�2

5

0

�5

z

0y

2�2 2

0x

81��2



23. 25.

27. 29. (a)

(b) 31. 33.

35. 37.

39. 41.

EXERCISES B N PAGE A16

1. 5 3.

7. 9.

11. 13. 15.17. 19. 21.23.

25. , 27. ,

29. 31.

33. 35.

37. 39. 41.45. y � x � 3

�1, �2��2, �5�, 4�x � 3�2 � �y � 1�2 � 25

0

y

x

y=1-2x

y=1+x

�0, 1�

0

y

x

x � 2

y � 4

0

y

x_2 20

y

x

0 x

y

_30 x

y

b � �3b � 0m � 3

4m � �13

5x � 2y � 1 � 0x � 2y � 11 � 0y � 5y � 3x � 3

y � 3x � 25x � y � 11y � 6x � 15

0 x

y

xy=0

0 3 x

y

x=3

�92

x � �a � b�c��ab��1.3, 1.7�� , �7� � ��3, ��3, 5�

��3, 3�2, � 43�30�C � T � 20�C

T � 20 � 10h, 0 � h � 1210 � C � 35

0 14

_1 10

�� , 0� � ( 14, )��1, 0� � �1, � 53.

EXERCISES C N PAGE A25

1. (a) (b) 3. (a) 720° (b)5. 7.9. (a) (b)

11. , , ,

, ,

13. , , , ,

15. 5.73576 cm 17. 24.62147 cm 27.

29. 31.

33. and

35.

37.

39.

EXERCISES D N PAGE A34

1. (or any smaller positive number)

3. 1.44 (or any smaller positive number)

5. (or any smaller positive number)7. 0.11, 0.012 (or smaller positive numbers)

11. (a) (b) Within approximately 0.0445 cm(c) Radius; area; ; 1000; 5; �0.0445s1000��

s1000�� cm

0.0906

47

y

0 x3π2

2πππ2

5π2

3π

y

0 x

112

π3

5π6

0 � x � ��4, 3��4 � x � 5��4, 7��4 � x � 2�

5��6 � x � 2�0 � x � ��6

��6, ��2, 5��6, 3��2��3, 5��3

115 (4 � 6s2)cot � � 4

3sec � � 54csc � � 5

3tan � � 34cos � � 4

5

cot�3��4� � �1sec�3��4� � �s2csc�3��4� � s2

tan�3��4� � �1cos�3��4� � �1�s2sin�3��4� � 1�s2

0x

y

_ 3π4

0 x

y

315°

23 rad � �120��3� cm

�67.5��207��6

0 x

y

1

5



13. (a) (b) 17.

19. (a) 21. (a) (b)

EXERCISES F N PAGE A45

1. 3.

5. 7.

9. 11.

13. 15. 17. 19.

21. 80 23. 3276 25. 0 27. 61 29.

31. 33.

35.

41. (a) (b) (c) (d)

43. 45. 14 49.

EXERCISES G N PAGE A54

1. (a) (b)

3. (a)

(b)

5. (a)

(b)

7.

9. 11.

13. 15.

17.

19.

21.

23.

25.

27.

29.

31.

33.

35. 37. ln ��e x � 2�2

e x � 1 � � C2 � ln 259

�1

2�x 2 � 2x � 4��

2s3

9 tan�1� x � 1

s3 � �2�x � 1�

3�x 2 � 2x � 4�� C

116 ln x �

132 ln�x 2 � 4� �

1

8�x 2 � 4�� C

13 ln x � 1 �

16 ln�x 2 � x � 1� �

1

s3 tan�1

2x � 1

s3� C

12 ln�x 2 � 2x � 5� �

32 tan�1� x � 1

2 � � C

12 ln�x 2 � 1� � (1�s2 ) tan�1(x�s2 ) � C

ln x � 1 �12 ln�x 2 � 9� �

13 tan�1�x�3� � C

2 ln x � �1�x� � 3 ln x � 2 � C

�1

36 ln x � 5 �

1

6

1

x � 5�

1

36 ln x � 1 � C

275 ln 2 �

95 ln 3 (or 95 ln 83)

76 � ln 2

3a ln x � b � C

12 ln 322 ln x � 5 � ln x � 2 � C

x � 6 ln x � 6 � C

At � B

t 2 � 1�

Ct � D

t 2 � 4�

Et � F

�t 2 � 4�2

1 �A

x � 1�

B

x � 1�

Cx � D

x 2 � 1

A

x � 3�

B

�x � 3�2 �C

x � 3�

D

�x � 3�2

A

x�

B

x 2 �C

x 3 �Dx � E

x 2 � 4

A

x�

B

x � 1�

C

�x � 1�2

A

x � 3�

B

3x � 1

2n�1 � n 2 � n � 213

an � a0973005100 � 1n 4

n�n 3 � 2n 2 � n � 10��4

n�n 2 � 6n � 11��3n�n 2 � 6n � 17��3

n�n � 1�

�n

i�1 x i�

5

i�0 2 i�

n

i�1 2i�

19

i�1

i

i � 1

�10

i�1 i1 � 1 � 1 � 1 � � � � � ��1�n�1

110 � 210 � 310 � � � � � n10�1 �13 �

35 �

57 �

79

34 � 35 � 36s1 � s2 � s3 � s4 � s5

9, 110x 100

0.00250.025 39.

41. , where

43. (a)

(b)

The CAS omits the absolute value signs and the constant of integration.


1. (a) (b)

(c)

3. (a) (b)

(c)

5. (a) (i) (ii)(b) (i) (ii) ��2, 5��3��2, 2��3�

(�2s2, 3��4)(2s2, 7��4)

(s2, �s2)

O

3π4

”_2, ’3π4

(�1, �s3)��1, 0�

O

_2π3

”2, _ ’2π3

π

O(1, π)

�1, 3��2�, ��1, 5��2�

O

π2

”_1, ’π2

�1, 5��4�, ��1, ��4��2, 7��3�, ��2, 4��3�

O

_3π4”1, _ ’3π

4

O

π3

π3”2, ’

75,772

260,015s19 tan�1

2x � 1

s19� C

11,049

260,015 ln�x 2 � x � 5� �

3146

80,155 ln 3x � 7 �

4822

4879 ln 5x � 2 �

334

323 ln 2x � 1 �

1

260,015 22,098x � 48,935

x 2 � x � 5

24,110

4879

1

5x � 2�

668

323

1

2x � 1�

9438

80,155

1

3x � 7�

C � 10.23t � �ln P �19 ln�0.9P � 900� � C

�12 ln 3 � �0.55



7. 9.

11.

13. Circle, center , radius 15. Horizontal line, 1 unit above the -axis17. 19.21. (a) (b)23. 25.

27. 29.

31. 33.

35. 37.

OO

¨= π6¨=5π

6

¨=π8

1

3 4

5

6

2

¨=π3

O

O

O

”1, ’π2

O

¨=_π6

x � 3� � ��6r � 2c cos �r � �cot � csc �

x

32(0, 32)

O

r=2r=3

¨=7π3

¨=5π3

O

r=4

¨=π6

¨=_π2

O

r=1r=2

39. 41.

43. 45.

47. (a) For , the inner loop begins at and ends at ; for , it begins at

and ends at .49. 51. 1

53. Horizontal at , ; vertical at 55. Horizontal at , [the pole], and ;

vertical at (2, 0), ,

57. Center , radius 59. 61.

63. By counterclockwise rotation through angle , , or about the origin65. (a) A rose with n loops if n is odd and 2n loops if n is even(b) Number of loops is always 2n67. For , the curve is an oval, which develops a dimpleas . When , the curve splits into two parts, one ofwhich has a loop.


1. 3. 5. 7.9. 4

O

414 �� 2��12 �

18 s3� 5�10,240

a 1a l1�

0 � a � 1

��3��6

7

�7

�7 7_3 3

_2.5

3.5

sa 2 � b 2�2�b�2, a�2�( 1

2, 4��3)( 12, 2��3)

( 32, 5��3)�0, ��( 3

2, ��3)�3, 0�, �0, ��2�

(�3�s2, 3��4)(3�s2, ��4)��

� � 2� � sin�1 �1�c�� sin�1 �1�c�c 1� � � � sin�1 ��1/c�

� � sin�1 ��1�c�c � �1

(2, 0) (6, 0)

O 1

1

2

¨=π3¨=2π

3

(3, 0)(3, π)



11. 13.

15. 17. 19. 21.

23. 25. 27.29. where , , , and where , , , 31. , and the pole

33. Intersection at 35.

37. 39.

EXERCISES I N PAGE A78

1. 3. 5. 7.

9. 11. 13. 15. 12 � 5i, 135i�i12 �

12 i

1113 �

1013 i12 � 7i13 � 18i8 � 4i

29.065383 �� 2 � 1�3�2 � 1�

�� 0.89, 2.25; area � 3.46

(12 s3, ��3), (1

2 s3, 2��3)23��1219��1211��12� � 7��12��1, ��

17��1213��125��12� � ��12�1, ��

14(� � 3s3)1

2� � 1524� �

14 s3

�13� �

12 s3� �

32 s31

8�

3

�3

�3 3

¨=π6

3�� 17. 19. 21.

23. 25.

27.

29. ,

31. ,,

33. 35.

37. 39.

41. 43. 45.

47. , sin 3� � 3 cos2� sin � � sin3�

cos 3� � cos3� � 3 cos � sin2�

�e 212 � (s3�2)ii

0

Im

Re

_i

0

Im

Re

i

1

�(s3�2) �12 i, �i�1, �i, (1�s2)��1 � i �

�512s3 � 512i�1024

14 �cos��6� � i sin��6��(2s2)�cos�13��12� � i sin�13��12��

4s2 �cos�7��12� � i sin�7��12��

12 �cos��6� � i sin��6��

4�cos��2� � i sin��2��, cos��6� � i sin��6�

5{cos[tan�1(43)] � i sin[tan�1(4

3)]}3s2 �cos�3��4� � i sin�3��4��

12 � (s7�2)i

�1 � 2i�32 i4i, 4


Documents

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