 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 $(5,4),(−3,−4),and(−4,4).(5,4),(−3,−4),and(−4,4).$

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

$s = 2 ( 10 3 / 2 − 2 3 / 2 ) ≈ 57.589 s = 2 ( 10 3 / 2 − 2 3 / 2 ) ≈ 57.589$

7.9

$A = π ( 494 13 + 128 ) 1215 A = π ( 494 13 + 128 ) 1215$

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

$A = 3 π / 2 A = 3 π / 2$

7.16

$A = 4 π 3 + 4 3 A = 4 π 3 + 4 3$

7.17

$s = 3 π s = 3 π$

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

$e = c a = 74 7 ≈ 1.229 e = c a = 74 7 ≈ 1.229$

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 .

$y=x+12y=x+12$; 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: $x∈(-∞,-1].x∈(-∞,-1].$

31 .

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

33 .

$y=lnx;y=lnx;$ domain: $x∈[1,∞).x∈[1,∞).$

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 .

$−3 5 −3 5$

67 .

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

69 .

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

71 .

73 .

No points possible; undefined expression.

75 .

$y = − ( 4 e ) x + 5 y = − ( 4 e ) x + 5$

77 .

$y = –2 x + 3 y = –2 x + 3$

79 .

$π 4 , 5 π 4 , 3 π 4 , 7 π 4 π 4 , 5 π 4 , 3 π 4 , 7 π 4$

81 .

$d y d x = − tan ( t ) d y d x = − tan ( t )$

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 .

$− sec 2 ( π t ) − sec 2 ( π t )$

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 .

$3 π 2 3 π 2$

105 .

$6 π a 2 6 π a 2$

107 .

$2 π a b 2 π a b$

109 .

$1 3 ( 2 2 − 1 ) 1 3 ( 2 2 − 1 )$

111 .

$7.075 7.075$

113 .

$6 a 6 a$

115 .

$6 2 6 2$

119 .

$2 π ( 247 13 + 64 ) 1215 2 π ( 247 13 + 64 ) 1215$

121 .

59.101

123 .

$8 π 3 ( 17 17 − 1 ) 8 π 3 ( 17 17 − 1 )$

### Section 7.3 Exercises

125 . 127 . 129 . 131 . 133 .

$B ( 3 , − π 3 ) B ( −3 , 2 π 3 ) B ( 3 , − π 3 ) B ( −3 , 2 π 3 )$

135 .

$D ( 5 , 7 π 6 ) D ( −5 , π 6 ) D ( 5 , 7 π 6 ) D ( −5 , π 6 )$

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 .

$( 2 3 , −0.524 ) ( −2 3 , −0.524 + π ) ( 2 3 , −0.524 ) ( −2 3 , −0.524 + π )$

143 .

$( − 3 , −1 ) ( − 3 , −1 )$

145 .

$( − 3 2 , −1 2 ) ( − 3 2 , −1 2 )$

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 .

$y = 1 y = 1$

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 .

$9 2 ∫ 0 π sin 2 θ d θ 9 2 ∫ 0 π sin 2 θ d θ$

191 .

$32 ∫ 0 π / 2 sin 2 ( 2 θ ) d θ 32 ∫ 0 π / 2 sin 2 ( 2 θ ) d θ$

193 .

$1 2 ∫ π 2 π ( 1 − sin θ ) 2 d θ 1 2 ∫ π 2 π ( 1 − sin θ ) 2 d θ$

195 .

$∫ sin −1 ( 2 / 3 ) π / 2 ( 2 − 3 sin θ ) 2 d θ ∫ sin −1 ( 2 / 3 ) π / 2 ( 2 − 3 sin θ ) 2 d θ$

197 .

$∫ π / 3 π ( 1 − 2 cos θ ) 2 d θ − ∫ 0 π / 3 ( 1 − 2 cos θ ) 2 d θ ∫ π / 3 π ( 1 − 2 cos θ ) 2 d θ − ∫ 0 π / 3 ( 1 − 2 cos θ ) 2 d θ$

199 .

$4 ∫ 0 π / 3 d θ + 16 ∫ π / 3 π / 2 ( cos 2 θ ) d θ 4 ∫ 0 π / 3 d θ + 16 ∫ π / 3 π / 2 ( cos 2 θ ) d θ$

201 .

$9 π 9 π$

203 .

$9 π 4 9 π 4$

205 .

$9 π 8 9 π 8$

207 .

$18 π − 27 3 2 18 π − 27 3 2$

209 .

$4 3 ( 4 π − 3 3 ) 4 3 ( 4 π − 3 3 )$

211 .

$3 2 ( 4 π − 3 3 ) 3 2 ( 4 π − 3 3 )$

213 .

$2 π − 4 2 π − 4$

215 .

$∫ 0 2 π ( 1 + sin θ ) 2 + cos 2 θ d θ ∫ 0 2 π ( 1 + sin θ ) 2 + cos 2 θ d θ$

217 .

$2 ∫ 0 1 e θ d θ 2 ∫ 0 1 e θ d θ$

219 .

$10 3 ( e 6 − 1 ) 10 3 ( e 6 − 1 )$

221 .

32

223 .

6.238

225 .

2

227 .

4.39

229 .

$A = π ( 2 2 ) 2 = π 2 and 1 2 ∫ 0 π ( 1 + 2 sin θ cos θ ) d θ = π 2 A = π ( 2 2 ) 2 = π 2 and 1 2 ∫ 0 π ( 1 + 2 sin θ cos θ ) d θ = π 2$

231 .

$C = 2 π ( 3 2 ) = 3 π and ∫ 0 π 3 d θ = 3 π C = 2 π ( 3 2 ) = 3 π and ∫ 0 π 3 d θ = 3 π$

233 .

$C = 2 π ( 5 ) = 10 π and ∫ 0 π 10 d θ = 10 π C = 2 π ( 5 ) = 10 π and ∫ 0 π 10 d θ = 10 π$

235 .

$d y d x = f ′ ( θ ) sin θ + f ( θ ) cos θ f ′ ( θ ) cos θ − f ( θ ) sin θ d y d x = f ′ ( θ ) sin θ + f ( θ ) cos θ f ′ ( θ ) cos θ − f ( θ ) sin θ$

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 .

$y 2 = 16 x y 2 = 16 x$

257 .

$x 2 = 2 y x 2 = 2 y$

259 .

$x 2 = −4 ( y − 3 ) x 2 = −4 ( y − 3 )$

261 .

$( x + 3 ) 2 = 8 ( y − 3 ) ( x + 3 ) 2 = 8 ( y − 3 )$

263 .

$x 2 16 + y 2 12 = 1 x 2 16 + y 2 12 = 1$

265 .

$x 2 13 + y 2 4 = 1 x 2 13 + y 2 4 = 1$

267 .

$( y − 1 ) 2 16 + ( x + 3 ) 2 12 = 1 ( y − 1 ) 2 16 + ( x + 3 ) 2 12 = 1$

269 .

$x 2 16 + y 2 12 = 1 x 2 16 + y 2 12 = 1$

271 .

$x 2 25 − y 2 11 = 1 x 2 25 − y 2 11 = 1$

273 .

$x 2 7 − y 2 9 = 1 x 2 7 − y 2 9 = 1$

275 .

$( y + 2 ) 2 4 − ( x + 2 ) 2 32 = 1 ( y + 2 ) 2 4 − ( x + 2 ) 2 32 = 1$

277 .

$x 2 4 − y 2 32 = 1 x 2 4 − y 2 32 = 1$

279 .

$e=1,e=1,$ parabola

281 .

$e=12,e=12,$ ellipse

283 .

$e=3,e=3,$ hyperbola

285 .

$r = 4 5 + cos θ r = 4 5 + cos θ$

287 .

$r = 4 1 + 2 sin θ r = 4 1 + 2 sin θ$

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 .

$r = 2.616 1 + 0.995 cos θ r = 2.616 1 + 0.995 cos θ$

321 .

$r = 5.192 1 + 0.0484 cos θ r = 5.192 1 + 0.0484 cos θ$

### 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 .

$r 2 = 4 sin 2 θ − cos 2 θ r 2 = 4 sin 2 θ − cos 2 θ$

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

337 .

$e 2 2 e 2 2$

339 .

$9 10 9 10$

341 .

$( y + 5 ) 2 = −8 x + 32 ( y + 5 ) 2 = −8 x + 32$

343 .

$( y + 1 ) 2 16 − ( x + 2 ) 2 9 = 1 ( y + 1 ) 2 16 − ( x + 2 ) 2 9 = 1$

345 .

$e=23,e=23,$ ellipse 347 .

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

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