bcseet01 answers 11

A-62 Answers

ChaptEr 11

Section 11.1 Exercises, pp. 735–739

1. Ifx = g1t2andy = h1t2,fora … t … b,thenplottingtheset51g1t2,h1t22:a … t … b6 resultsinagraphinthexy-plane.3. x = Rcos1pt>52,y = Rsin1pt>525. x = t,y = t2,- � 6 t 6 �7. a.

t -10 -8 -6 -4 -2 0 2 4 6 8 10

x -20 -16 -12 -8 -4 0 4 8 12 16 20

y -34 -28 -22 -16 -10 -4 2 8 14 20 26

b.

10

4

y

x

c. y =32x - 4

d. Alinerisingupandtotherightastincreases9. a.

t -5 -4 -3 -2 -1 0 1 2 3 4 5

x 11 10 9 8 7 6 5 4 3 2 1

y -18 -15 -12 -9 -6 -3 0 3 6 9 12

b. 10

�10

5

y

x

c. y = -3x + 15

d. Alinerisingupandtotheleftastincreases11. a. y = 3x - 12 b. Alinerisingupandtotherightastincreases 13. a. y = 1 - x2,-1 … x … 1 b. Aparabolaopen-ingdownwardwithavertexat10,12startingat11,02andendingat1-1,02 15. a. y = 1x + 123 b. Acubicfunctionrisingupandtotherightastincreases 17. Center10,02;radius3;lowerhalfofcir-clegeneratedcounterclockwise 19. x2 + 1y - 122 = 1;acompletecircleofradius1centeredat10,12traversedcounterclockwisestartingat11,12 21. Center10,02;radius7;circlegeneratedcounterclock-wise 23. x = 4cost,y = 4sint,0 … t … 2p:Thecirclehasequationx2 + y2 = 16.

1

1

y

x

25. x = cost + 2,y = sint + 3,0 … t … 2p;1x - 222 + 1y - 322 = 1 y

x0

3

2

1

321

27. x = 8sint - 2,y = 8cost - 3,0 … t … 2p:Thecirclehasequation1x + 222 + 1y + 322 = 64.

2

2

y

x

29. x = 400cosa4pt

3b ,y = 400sina4pt

3b ,

0 … t … 1.5 31. x = 50cosapt

12b ,y1t2 = 50sinapt

12b ,

0 … t … 2433. Slope:-1; point:13,12

2

2

y

x

(0, 4)

(3, 1)

35. Slope:0;point:18,12 y

x

(0, 1)

(0, �1)

(�5, 0) (5, 0)(4, )

x2

25� y2 � 1

53

37. x = 2t,y = 8t,0 … t … 139. x = -1 + 7t,y = -3 - 13t,0 … t … 141. x = t,y = 2t2 - 4,-1 … t … 5(notunique)

20

40

5�1

y

x

Z01B_BRIG5374_01_SE_ANS.indd 62 13/01/12 3:57 PM

Copyright © 2013 Pearson Education, Inc.

Answers A-63

59. a. dy

dx=

t2 + 1

t2 - 1,t � 0;undefined b.

4

4

y

x

x � 2

(2, 0)

61. a. False b. True c. False d. True 63. y =13

4x +

1

465. y = x -

p12

4 67. x = 1 + 2t,y = 1 + 4t,- � 6 t 6 �

69. x = t2,y = t,t Ú 0

71. 0 … t … 2p10

2

�10

102�10

y

x

73. x = 3cost,y =32sint,0 … t … 2p;a x

3b

2

+ a2y

3b

2

= 1;

inthecounterclockwisedirection4

�4

4�4

y

x

75. x = 15cost - 2,y = 10sint - 3,0 … t … 2p;

a x + 2

15b

2

+ a y + 3

10b

2

= 1;inthecounterclockwisedirection

20

�20

20�20

y

x

77. aandb 79. x2 + y2 = 4 81. y = 24 - x2 83. y = x2

85. a-415

,815

b anda 415,-

815b 87. Thereisnosuchpoint.

89. a = p,b = p +2p

3,forallrealp 91. a. 10,22and10,-22

b. 11,122,11,- 122,1-1,122,1-1,- 12293. a.x = {acos2>n1t2,y = {bsin2>n1t2 c.Thecurvesbecomemoresquareasnincreases.

43. x = 4t - 2,y = -6t + 3,0 … t … 1;x = t + 1,y = 8t - 11,1 … t … 2(notunique)

�4

4

y

x

P(�2, 3)

Q(2, �3)

R(3, 5)

45.

3�

3�

y

x

47.

2

2

y

x

49. y

1

x1 2�2 �1

2

�2

�1

51. y

x�2 �1

2

�2

�1

1

1 2

53. y

x�2 �1

2

�2

�1

1

1 2

55. a. dy

dx= -2;-2 b.

�20

20

y

x

(10, �12)

57. a. dy

dx= -8cott;0 b.

2

2

y

x

y � 8(0, 8)



A-64 Answers

99. a. y

x�300

300

�300

300

b. y

x�500

500

�500

500

c. y

x�700

700

�700

700

101. �2857m


1. y

x

(2, k)

(�3, �q)

k

32

1-2,-5p>62,12,13p>62;13,p>22,13,5p>22

3. r 2 = x2 + y2,tanu =y

x 5. rcosu = 5orr = 5secu

7. x-axissymmetryoccursif1r,u2onthegraphimplies1r,-u2isonthegraph.y-axissymmetryoccursif1r,u2onthegraphimplies1r,p - u2 = 1-r,-u2isonthegraph.Symmetryabouttheoriginoccursif1r,u2onthegraphimplies1-r,u2 = 1r,u + p2isonthegraph.

11. y

x

(�1, �u)

�u

1

11,2p>32,11,8p>32

13. y

x

(�4, w)

w

4

14,p>22,14,5p>22

15. 1312>2,312>22 17. 11>2,- 13>22 19. 1212,-212221. 1212,p>42,1-212,5p>42 23. 12,p>32,1-2,4p>3225. 18,2p>32,1-8,-p>32 27. x = -4;verticallinepassingthrough1-4,02 29. x2 + y2 = 41circlecenteredat10,02ofradius22 31. 1x - 122 + 1y - 122 = 21circleofradius12centeredat11,122 33. x2 + 1y - 122 = 1;circleofradius1centeredat10,12andx = 0;y-axis 35. x2 + 1y - 422 = 16;circleofradius4centeredat10,4237.

4

4

y

x

r � 8 cos � 39.

2

2

y

x

r(sin � � 2 cos �) � 0

41. y

x

1

�1

�2

1�1

43.

1

1

y

x

45.

4

2

y

x

47.

1

1

y

x

49. 4

3

y

x

r � 1 � 2 sin 3�

A

C

E

GI

K

Morigin:B, D, F, H, J, L

51.

1

1

y

x

O

I

CMG

A

K E

origin:B, D, F, H, J, L, N, P

9. y

x

(2, d)

d2

1-2,-3p>42,12,9p>42



Answers A-65

53.

6

12

8

y

x

Nointerval30,P4generatestheentirecurve;- � 6 u 6 �55. 30,2p4

2

2

y

x

57. 30,5p4

1

1

y

x

59. 30,2p4

2

4

y

x

61. a. True b. True c. False d. True e. True63. r = tanusecu 65. r 2 = secucscuorr 2 = 2csc12u267.

2

2

y

x

69.

2

2

y

x

71.

2

2

y

x

u

�u

73. 3

3

y

x

77. Acircleofradius4andcenter12,p>321polarcoordinates2

2

6

y

x

(2, u)

79. Acircleofradius4centeredat12,321Cartesiancoordinates2

8

6

y

x

(2, 3)

81. Acircleofradius3centeredat1-1,22(Cartesiancoordinates)

4

2

y

x

(�1, 2)

83. Samegraphonallthreeintervals.

3

3

y

x

85. 8

6

y

x

y � �x

�3� 2�3

87.

6

6

y

x

y � 4x � 3

89. a. A b. C c. B d. D e. E f. F91.

1

1

y

x

93. 2

2

y

x

95.

1

1

y

x

97. 2

2

y

x



A-66 Answers

Fora = -1,thespiralwindsinwardtowardtheorigin.

4000

500

y

x

103. 12,02and10,02105. 10,02,a2 - 12

2,3p>4b ,a2 + 12

2,7p>4b

107. a.4

3

y

x

109. a.

2

�8

y

x

111. r = acosu + bsinu =a

r1rcosu2 +

b

r1rsinu2 =

a

rx +

b

ry

Thus,ax -a

2b

2

+ ay -b

2b

2

=a2 + b2

4.Center:aa

2,

b

2b ;radius:2a2 + b2

2 113. Symmetryaboutthex-axis


1. x = f1u2cosu,y = f1u2sinu 3. Theslopeofthetangentlineistherateofchangeoftheverticalcoordinatewithrespecttothehori-zontalcoordinate. 5. 0;u = p>2 7. - 13;u = 0 9. Undefined,undefined;thecurvedoesnotintersecttheorigin. 11. 0at1-4,p>22and1-4,3p>22,undefinedat14,02and14,p2;u = p>4,u = 3p>413. {1;u = {p>4 15. Horizontalat1212,p>42,1-212,3p>42;verticalat10,p>22,14,02 17. Horizontal:10,0210.943,0.9552,1-0.943,2.1862,10.943,4.0972,1-0.943,5.3282;vertical:10,02,10.943,0.6152,1-0.943,2.5262,10.943,3.7572,1-0.943,5.668219. Horizontalata1

2,p

6b ,a1

2,

5p

6b ,a2,

3p

2b ;verticalat

a3

2,

7p

6b ,a3

2,

11p

6b ,a0,

p

2b

21. y

x

�0.5

1

0.5

23. 16p 25. 9p>2

101. 27.

p

12 y

0.5

x0.5 1.0�1.0 �0.5

1.0

�1.0

�0.5

29. 1

241313 + 2p2

y

0.5

x0.5 1.0�1.0 �0.5

1.0

�1.0

�0.5

31. 1

412 - 132 +

p

12y

0.5

x0.5 1.0�1.0 �0.5

1.0

�1.0

�0.5

33. p>20 35. 414p>3 - 132 37. 10,02,13>12,p>4239. 11 +

112,p4 2,11 -

112,5p4 2,10,02 41. 9

81p - 22 43. 3p

2- 212

45. a. False b. False 47. 2p>3 - 13>2 49. 9p + 271351. Horizontal:10,02,14.05,2.032,19.83,4.912;vertical:11.72,0.862,16.85,3.432,112.87,6.44253. a. An =

1

4e14n + 22p -1

4e4np -1

4e14n - 22p +1

4e14n - 42p b. 0

c. e-4p 55. 6 57. 18p 59. 1a2 - 22u* + p - sin2u*,whereu* = cos-11a>22. 61. a21p>2 + a>32


1. Aparabolaisthesetofallpointsinaplaneequidistantfromafixedpointandafixedline. 3. Ahyperbolaisthesetofallpointsinaplane,thedifferenceofwhosedistancesfromtwofixedpointsisconstant.5. Parabola: y

x

Hyperbola: y

x

Ellipse: y

x



Answers A-67

7. a x

ab

2

+y2

a2 - c2 = 1 9. 1{ae,02 11. y = {b

ax

13.

1

2

y

x

(0, 3)

x2 � 12y

y � �3

15. 5

2

y

x

(�4, 0)

x ��y2

16

x � 4

17.

�3

3

y

x

y � s

(0, �s)

8y � �3x2

19. y2 = 16x 21. y2 = 12x

23. x2 = -23y 25. y2 = 41x + 12

27. Vertices:1{2,02;foci:1{13,02;majoraxishaslength4;minoraxishaslength2.

y

x

(0, 1)

(�2, 0) (2, 0)

(0, �1)

� y2 � 1x2

4

Vertices:10,{42;foci:10,{2132;majoraxishaslength8;minoraxishaslength4.

y

x

(0, 4)

(�2, 0)

(2, 0)

(0, �4)

�x2

4y2

16� 1

29.

Vertices:10,{172;foci:10,{122;majoraxishaslength217;minoraxishaslength215.

y

x

(0, �7)

(0, ��7)

(��5, 0)

�x2

5y2

7� 1

(�5, 0)

y

x

(0, 3)

(0, �3)

(�4, 0) (4, 0)

�x2

16y2

9� 1

31.

33.

35. y

x

(0, 1)(5, 0)

(4, E)

x2

25

x2

25y2

37. x2

4+

y2

9= 1

Vertices:1{2,02;foci:1{215,02;asymptotes:y = {2x

Vertices:1{13,02;foci:1{212,02;asymptotes:

y = {453x

Vertices:1{4,02;foci:1{6,02;asymptotes:

y = {15

2x

Vertices:1{2,02;foci:1{113,02;asymptotes:y = {3

2x

2

y

x

(�2, 0) (2, 0)(��5, 0) (�5, 0)

x2

4� y2 � 1

6

6

y

x

(�2, 0) (2, 0)

(�2�5, 0)(2�5, 0)

4x2 � y2 � 16

3

y

x

(�3, 0)

(2�2, 0)

(��3, 0)

(�2�2, 0)

�x2

3y2

5� 1

4

y

x

�x2

16y2

20� 1

(�4, 0)

(�6, 0)

(4, 0)

(6, 0)

4

y

x

(�2, 0) (2, 0)

(��13, 0) (�13, 0)

�x2

4y2

9� 1

39.

41.

43.

45.

47.

Vertices:1{2,02;foci:1{15,02;

asymptotes:y = {1

2x



A-68 Answers

49. x2

16-

y2

9= 1 51.

x2

81+

y2

72= 1

Directrices: x = {27

4

y

x(�9, 0)

(�3, 0) (3, 0) (9, 0)

(0, 6�2)

(0, �6�2)

53. x2 -y2

8= 1 4

y

x

x � �a x � a

(�3, 0)

(�1, 0) (1, 0)

(3, 0)

Vertex:12,02;focus:10,02;directrix:x = 4

Vertices:11,02,1-13,02;center:

113,02;foci:10,02,12

3,02;directrices:x = -1,x =

53

Vertex:10,-142;focus:10,02;

directrix:y = -12

Theparabolastartsat11,02andgoesthroughquadrantsI,II,andIIIforuin30,3p>24;thenitapproaches11,02bytravelingthroughquadrantIVon13p>2,2p2.

2

y

x

(2, 0)

x � 4

1

y

x

(�a, 0) (a, 0) (1, 0)

2

2

y

x

(0, 0)

(�~, 0)

�2

y

x

(1, 0)(�1, 0)(0, q)

55.

57.

59.

61.

Theparabolabeginsinthefirstquad-rantandpassesthroughthepoints10,32andthen1-

32,02and10,-32as

urangesfrom0to2p.

2

y

x

(0, 3)

(0, �3)

(�w, 0)

63.

65. Theparabolasopentotherightifp 7 0,opentotheleftifp 6 0,andaremoreverticallycompressedas�p�decreases. 67. a. Trueb. True c. True d. True 69. y = 2x + 6 71. y = -

340x -

45

73. r =4

1 - 2sinu 77.

dy

dx= a-

b2

a2 b ax

yb ,so

y - y0

x - x0= a-

b2

a2 b ax0

y0b ,whichisequivalenttothegivenequation.

79. 4pb2a

3;

4pa2b

3;yes,ifa � b 81. a.

pb2

3a2# 1a - c2212a + c2

b. 4pb4

3a 91. 2p 97. a. u1m2 =

2m2 - 23m2 + 1

m2 - 1;

v1m2 =2m2 + 23m2 + 1

m2 - 1;2intersectionpointsfor�m� 7 1

b. 54,� c. 2,2 d. 213 - ln113 + 22

Chapter 11 review Exercises, pp. 774–776

1. a. False b. False c. True d. False e. True f. True3. a.

2

4

2 4

y

x

b. y = 3>x2

c. Therightbranchofthefunctiony = 3>x2. d. dy

dx= -6

5. a.

20

40

2

y

x

b. y = 16x

c. Alinesegmentfrom10,02to12,322 d. dy

dx= 16

7. x2

16+

y2

9= 1;ellipsegeneratedcounterclockwise

9. 1x + 322 + 1y - 622 = 1;righthalfofacirclecenteredat1-3,62ofradius1generatedclockwise 11. x = 3sint,y = 3cost,for0 … t … 2p 13. x = 3cost,y = 2sint,for-p>2 … t … p>215. x = -1 + 2t,y = t,for0 … t … 1;x = 1 - 2t,y = 1 - t,for0 … t … 1



Answers A-69

17. Att = p>6:y = 12 + 132x + a2 -p

3-

p13

6b ;at

t =2p

3:y =

x13+ 2 -

2p

31319.

�2

4

y

x

21. Lizshouldchooser = 1 - sinu.y

1

x�1

2

3

�2

�3

�1

1 2 3 4 5

r � cos 3�

r � 1 � sin �

r � 5 cos �

23. 1x - 322 + 1y + 122 = 10;acircleofradius110centeredat13,-12 25. r = 8cosu,0 … u … p

27. a. 4

2

y

x

4intersectionpoints

b. 11,1.322,11,4.972,1-1,0.72,1-1,5.56229. a. 14.73,2.772,14.73,0.382;16,p>22,12,3p>22 b. Thereisnopointattheorigin. c.

2

2

y

x

31. a. Horizontaltangentlinesat11,p>62,11,5p>62,11,7p>62,and11,11p>62;verticaltangentlinesat112,02and112,p2b. Tangentlinesattheoriginhaveslopes{1.c.

2

2

y

x

33. 19p

2

4

3

y

x

35. 1411255 - cos- 111>1622

2

2

y

x

37. 4 39. a. Hyperbola b. Foci1{13,02,vertices1{1,02,

directricesx = {113

c. e = 13

d. 3

3

y

x

y � �2xy � ��2x

41. a. Hyperbola b. Foci10,{2152,vertices10,{42,directrices

y = {815

c. e =15

2 d.

6

4

y

x

y � 2xy � �2x

43. a. Ellipse b. Foci1{12,02,vertices1{2,02,directrices

x = {212 c. e =12

2 d.

2

2

y

x

45. y =32x - 2 47. y = -

35x - 10 49.

5

4

y

x(0, 0)

(0, 1)

y � 2



A-70 Answers

51. 5

5

y

x

x � � x � 4203

(�4, 0)

(0, 0)(�h, 0)

(d, 0)

53. a. x2 - y2 = 1;hyperbola

b. 1{1,02,1{12,02;x = {112

;e = 12

c.

2

y

x

(�1, 0) (1, 0)

(��2, 0) (�2, 0)

y � xy � �x

55. y2

16+

25x2

336= 1

8

8

y

x

(0, 4)

(0, �4)

y � �10

y � 10

57. y2

4-

x2

12= 1;

4

4

y

x

59. e = 2>3,y = {9,1{215,02 61. 10,02,10.97,0.97263. 10,02and1r,u2 = 112n - 12p,02,n = 1,2,3,c

65. 2a12

# 2b12;2ab 67. m =

b

a 71. r =

3

3 - sinu

ChaptEr 12


3.

x

y

Q

P

x

y

Q

P

5. Thereareinfinitelymanyvectorswiththesamedirectionandlengthasv. 7. Ifthescalarcispositive,extendthegivenvectorbyamultipleofcinthesamedirection.Ifc 6 0,reversethedirectionofthevectorandextenditbyamultipleof�c�. 9. u + v = 8u1 + v1,u2 + v29

11. � 8v1,v29 � = 2v 21 + v 2

2 13. IfPhascoordinates1p1,p22andQhascoordinates1q1,q22thenthemagnitudeofPQ1 isgivenby21q1 - p122 + 1q2 - p222. 15. Dividevbyitslengthandmultiplytheresultby10. 17. a,c,e 19. a. 3v b. 2u c. -3u d. -2ue. v 21. a. 3u + 3v b. u + 2v c. 2u + 5v d. -2u + 3ve. 3u + 2v f. -3u - 2v g. -2u - 4v h. u - 4v i. -u - 6v23. a.

1

2

3

4

5

�1x

y

�1 54321

P

O

25. QU1 = 87,29 ,PT1 = 87,39 ,RS1 = 82,395

4

3

1

2

�1x

y

�2 �1 1 2 3 4 65 7 8

(7, 2)

U

Q

4

3

2

1 2 3 4 5 6 7

1

�1

�1

x

y

�1�2�3

(7, 3)

P

T

5

4

3

2

5432

1

�1x

y

�1 1

(2, 3)R

S

27. QT1 29. 8-4,109 31. 812,-109 33. 8-28,829 35. 222

37. 2194 39. 83,39 ,8-3,-39 41. w - u

43. - i + 10j 45. {1261

86,59

47. h-28174

,20174

i,h 28174,-

20174i

49. 5265km>hr � 40.3km>hr 51. 349.43mi>hrinthedirection4.64�southofwest 53. 1m>sinthedirection30�eastofnorth

OP1 = 83,29 = 3i + 2j0OP 01 = 113

b.

1

2

3

4

5

�1x

y

�1 54321

QP

c. 3

2

1

�1

�2

�3

�4

x

y

�1�2�3�4�5�6 4321

R

Q

QP1 = 8-1,09 = - i0QP 01 = 1

RQ1 = 810,39 = 10i + 3j0RQ 01 = 1109



bcseet01 answers 11

Documents

t 2notunique44yxp2

t 2p1x

t 5notunique204051yxz01b

t 2p1021010210yx73

y u2

x p124

fora t b

b anda