Skip to Content
Calculus Volume 2

# Chapter 7

### Checkpoint

7.2

$x=2+3y+1,x=2+3y+1,$ or $y=−1+3x−2.y=−1+3x−2.$ This equation describes a portion of a rectangular hyperbola centered at $(2,−1).(2,−1).$

7.3

One possibility is $x(t)=t,y(t)=t2+2t.x(t)=t,y(t)=t2+2t.$ Another possibility is $x(t)=2t−3,y(t)=(2t−3)2+2(2t−3)=4t2−8t+3.x(t)=2t−3,y(t)=(2t−3)2+2(2t−3)=4t2−8t+3.$

There are, in fact, an infinite number of possibilities.

7.4

$x′(t)=2t−4x′(t)=2t−4$ and $y′(t)=6t2−6,y′(t)=6t2−6,$ so $dydx=6t2−62t−4=3t2−3t−2.dydx=6t2−62t−4=3t2−3t−2.$
This expression is undefined when $t=2t=2$ and equal to zero when $t=±1.t=±1.$

7.5

The equation of the tangent line is $y=24x+100.y=24x+100.$

7.6

$d2ydx2=3t2−12t+32(t−2)3.d2ydx2=3t2−12t+32(t−2)3.$ Critical points $(5,4),(−3,−4),and(−4,6).(5,4),(−3,−4),and(−4,6).$

7.7

$A=3πA=3π$ (Note that the integral formula actually yields a negative answer. This is due to the fact that $x(t)x(t)$ is a decreasing function over the interval $[0,2π];[0,2π];$ that is, the curve is traced from right to left.)

7.8

$s=2(103/2−23/2)≈57.589s=2(103/2−23/2)≈57.589$

7.9

$A=π(49413+128)1215A=π(49413+128)1215$

7.10

$(82,5π4)(82,5π4)$ and $(−2,23)(−2,23)$

7.12

The name of this shape is a cardioid, which we will study further later in this section.

7.13

$y=x2,y=x2,$ which is the equation of a parabola opening upward.

7.14

Symmetric with respect to the polar axis.

7.15

$A=3π/2A=3π/2$

7.16

$A=4π3+43A=4π3+43$

7.17

$s=3πs=3π$

7.18

$x=2(y+3)2−2x=2(y+3)2−2$

7.19

$(x+1)216+(y−2)29=1(x+1)216+(y−2)29=1$

7.20

$(y+2)29−(x−1)24=1.(y+2)29−(x−1)24=1.$ This is a vertical hyperbola. Asymptotes $y=−2±32(x−1).y=−2±32(x−1).$

7.21

$e=ca=747≈1.229e=ca=747≈1.229$

7.22

Here $e=0.8e=0.8$ and $p=5.p=5.$ This conic section is an ellipse.

7.23

The conic is a hyperbola and the angle of rotation of the axes is $θ=22.5°.θ=22.5°.$

### Section 7.1 Exercises

1.

orientation: bottom to top

3.

orientation: left to right

5.

$y=x24+1y=x24+1$

7.

9.

11.

13.

15.

Asymptotes are $y=xy=x$ and $y=−xy=−x$

17.

19.

21.

$x=4y2−1;x=4y2−1;$ domain: $x∈[1,∞).x∈[1,∞).$

23.

$x216+y29=1;x216+y29=1;$ domain $x∈[−4,4].x∈[−4,4].$

25.

$y=3x+2;y=3x+2;$ domain: all real numbers.

27.

$(x−1)2+(y−3)2=1;(x−1)2+(y−3)2=1;$ domain: $x∈[0,2].x∈[0,2].$

29.

$y=x2−1;y=x2−1;$ domain: $x∈[−1,1].x∈[−1,1].$

31.

$y2=1−x2;y2=1−x2;$ domain: $x∈[2,∞)∪(−∞,−2].x∈[2,∞)∪(−∞,−2].$

33.

$y=lnx;y=lnx;$ domain: $x∈(0,∞).x∈(0,∞).$

35.

$y=lnx;y=lnx;$ domain: $x∈(0,∞).x∈(0,∞).$

37.

$x2+y2=4;x2+y2=4;$ domain: $x∈[−2,2].x∈[−2,2].$

39.

line

41.

parabola

43.

circle

45.

ellipse

47.

hyperbola

51.

The equations represent a cycloid.

53.

55.

22,092 meters at approximately 51 seconds.

57.

59.

61.

### Section 7.2 Exercises

63.

0

65.

$−35−35$

67.

$Slope=0;Slope=0;$ $y=8.y=8.$

69.

Slope is undefined; $x=2.x=2.$

71.

$t=arctan(−2);t=arctan(−2);$ $(45,−85).(45,−85).$

73.

No points possible; undefined expression.

75.

$y=−(2e)x+3y=−(2e)x+3$

77.

$y=–2x+3y=–2x+3$

79.

$π4,5π4,3π4,7π4π4,5π4,3π4,7π4$

81.

$dydx=−tan(t)dydx=−tan(t)$

83.

$dydx=34dydx=34$ and $d2ydx2=0,d2ydx2=0,$ so the curve is neither concave up nor concave down at $t=3.t=3.$ Therefore the graph is linear and has a constant slope but no concavity.

85.

$dydx=4,d2ydx2=−63;dydx=4,d2ydx2=−63;$ the curve is concave down at $θ=π6.θ=π6.$

87.

No horizontal tangents. Vertical tangents at $(1,0),(−1,0).(1,0),(−1,0).$

89.

$−sec3(πt)−sec3(πt)$

91.

Horizontal $(0,−9);(0,−9);$ vertical $(±2,−6).(±2,−6).$

93.

1

95.

0

97.

4

99.

Concave up on $t>0.t>0.$

101.

$e12 –1 2 e12 –1 2$

103.

$3π23π2$

105.

$6πa26πa2$

107.

$2πab2πab$

109.

$13(22−1)13(22−1)$

111.

$7.0757.075$

113.

$6a6a$

115.

$6262$

119.

$2π(24713+64)12152π(24713+64)1215$

121.

59.101

123.

$8π3(1717−1)8π3(1717−1)$

### Section 7.3 Exercises

125.

127.

129.

131.

133.

$B(3,−π3)B(−3,2π3)B(3,−π3)B(−3,2π3)$

135.

$D(5,7π6)D(−5,π6)D(5,7π6)D(−5,π6)$

137.

$(5,−0.927)(−5,−0.927+π)(5,−0.927)(−5,−0.927+π)$

139.

$(10,−0.927)(−10,−0.927+π)(10,−0.927)(−10,−0.927+π)$

141.

$(23,−0.524)(−23,−0.524+π)(23,−0.524)(−23,−0.524+π)$

143.

$(−3,−1)(−3,−1)$

145.

$(−32,−12)(−32,−12)$

147.

$(0,0)(0,0)$

149.

Symmetry with respect to the x-axis, y-axis, and origin.

151.

Symmetric with respect to x-axis only.

153.

Symmetry with respect to x-axis only.

155.

Line $y=xy=x$

157.

$y=1y=1$

159.

Hyperbola; polar form $r2cos(2θ)=16r2cos(2θ)=16$ or $r2=16sec(2θ).r2=16sec(2θ).$

161.

$r=23cosθ−sinθr=23cosθ−sinθ$

163.

$x2+y2=4yx2+y2=4y$

165.

$xtanx2+y2=yxtanx2+y2=y$

167.

y-axis symmetry

169.

y-axis symmetry

171.

x- and y-axis symmetry and symmetry about the pole

173.

x-axis symmetry

175.

x- and y-axis symmetry and symmetry about the pole

177.

no symmetry

179.

a line

181.

183.

185.

187.

Answers vary. One possibility is the spiral lines become closer together and the total number of spirals increases.

### Section 7.4 Exercises

189.

$92∫0πsin2θdθ92∫0πsin2θdθ$

191.

$32∫0π/2sin2(2θ)dθ32∫0π/2sin2(2θ)dθ$

193.

$12∫π2π(1−sinθ)2dθ12∫π2π(1−sinθ)2dθ$

195.

$∫sin−1(2/3)π/2(2−3sinθ)2dθ∫sin−1(2/3)π/2(2−3sinθ)2dθ$

197.

$∫0π(1−2cosθ)2dθ−∫0π/3(1−2cosθ)2dθ∫0π(1−2cosθ)2dθ−∫0π/3(1−2cosθ)2dθ$

199.

$4∫0π/3dθ+16∫π/3π/2(cos2θ)dθ4∫0π/3dθ+16∫π/3π/2(cos2θ)dθ$

201.

$9π9π$

203.

$9π49π4$

205.

$9π89π8$

207.

$18π−273218π−2732$

209.

$43(4π−33)43(4π−33)$

211.

$32(4π−33)32(4π−33)$

213.

$2π−42π−4$

215.

$∫02π(1+sinθ)2+cos2θdθ∫02π(1+sinθ)2+cos2θdθ$

217.

$2∫01eθdθ2∫01eθdθ$

219.

$103(e6−1)103(e6−1)$

221.

32

223.

6.238

225.

2

227.

4.39

229.

$A=π(22)2=π2and12∫0π(1+2sinθcosθ)dθ=π2A=π(22)2=π2and12∫0π(1+2sinθcosθ)dθ=π2$

231.

$C=2π(32)=3πand∫0π3dθ=3πC=2π(32)=3πand∫0π3dθ=3π$

233.

$C=2π(5)=10πand∫0π10dθ=10πC=2π(5)=10πand∫0π10dθ=10π$

235.

$dydx=f′(θ)sinθ+f(θ)cosθf′(θ)cosθ−f(θ)sinθdydx=f′(θ)sinθ+f(θ)cosθf′(θ)cosθ−f(θ)sinθ$

237.

The slope is $13.13.$

239.

The slope is 0.

241.

At $(4,0),(4,0),$ the slope is undefined. At $(−4,π2),(−4,π2),$ the slope is 0.

243.

The slope is undefined at $θ=π4.θ=π4.$

245.

Slope = −1.

247.

Slope is $−2π.−2π.$

249.

Calculator answer: −0.836.

251.

Horizontal tangent at $(±2,π6),(±2,π6),$ $(±2,−π6).(±2,−π6).$

253.

Horizontal tangents at $π2,7π6,11π6.π2,7π6,11π6.$ Vertical tangents at $π6,5π6π6,5π6$ and also at the pole $(0,0).(0,0).$

### Section 7.5 Exercises

255.

$y2=16xy2=16x$

257.

$x2=2yx2=2y$

259.

$x2=−4(y−3)x2=−4(y−3)$

261.

$(x+3)2=8(y−3)(x+3)2=8(y−3)$

263.

$x216+y212=1x216+y212=1$

265.

$x213+y24=1x213+y24=1$

267.

$(y−1)216+(x+3)212=1(y−1)216+(x+3)212=1$

269.

$x216+y212=1x216+y212=1$

271.

$x225−y211=1x225−y211=1$

273.

$x27−y29=1x27−y29=1$

275.

$(y+2)24−(x+2)232=1(y+2)24−(x+2)232=1$

277.

$x24−y232=1x24−y232=1$

279.

$e=1,e=1,$ parabola

281.

$e=12,e=12,$ ellipse

283.

$e=3,e=3,$ hyperbola

285.

$r=45+cosθr=45+cosθ$

287.

$r=41+2sinθr=41+2sinθ$

289.

291.

293.

295.

297.

299.

301.

303.

305.

307.

Hyperbola

309.

Ellipse

311.

Ellipse

313.

At the point 2.25 feet above the vertex.

315.

0.5625 feet

317.

Length is 96 feet and height is approximately 26.53 feet.

319.

$r=2.6161+0.995cosθr=2.6161+0.995cosθ$

321.

$r=5.1921+0.0484cosθr=5.1921+0.0484cosθ$

### Chapter Review Exercises

323.

True.

325.

False. Imagine $y=t+1,y=t+1,$ $x=−t+1.x=−t+1.$

327.

$y=1−x3y=1−x3$

329.

$x216+(y−1)2=1x216+(y−1)2=1$

331.

Symmetric about polar axis

333.

$r2=4sin2θ−cos2θr2=4sin2θ−cos2θ$

335.

$y=322+15(x+322)y=322+15(x+322)$

337.

$e22e22$

339.

$910910$

341.

$(y+5)2=−8x+32(y+5)2=−8x+32$

343.

$(y+1)216−(x+2)29=1(y+1)216−(x+2)29=1$

345.

$e=23,e=23,$ ellipse

347.

$y219.032+x219.632=1,y219.032+x219.632=1,$ $e=0.2447e=0.2447$

Citation/Attribution

Want to cite, share, or modify this book? This book is Creative Commons Attribution-NonCommercial-ShareAlike License 4.0 and you must attribute OpenStax.

Attribution information
• If you are redistributing all or part of this book in a print format, then you must include on every physical page the following attribution:
Access for free at https://openstax.org/books/calculus-volume-2/pages/1-introduction
• If you are redistributing all or part of this book in a digital format, then you must include on every digital page view the following attribution:
Access for free at https://openstax.org/books/calculus-volume-2/pages/1-introduction
Citation information

© Jan 16, 2020 OpenStax. Textbook content produced by OpenStax is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike License 4.0 license. The OpenStax name, OpenStax logo, OpenStax book covers, OpenStax CNX name, and OpenStax CNX logo are not subject to the Creative Commons license and may not be reproduced without the prior and express written consent of Rice University.