Calculus Volume 2

# Chapter 7

### Checkpoint

7.1 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 $(−2,23)(−2,23)$

7.11 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.

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. 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.

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