Calculus Volume 1

# Chapter 4

### Checkpoint

4.1

$172πcm/sec,172πcm/sec,$ or approximately 0.0044 cm/sec

4.2

$500 ft/sec 500 ft/sec$

4.3

$1 10 rad/sec 1 10 rad/sec$

4.4

$−0.61 ft/sec −0.61 ft/sec$

4.5

$L(x)=2+112(x−8);L(x)=2+112(x−8);$ 2.00833

4.6

$L ( x ) = − x + π 2 L ( x ) = − x + π 2$

4.7

$L ( x ) = 1 + 4 x L ( x ) = 1 + 4 x$

4.8

$d y = 2 x e x 2 d x d y = 2 x e x 2 d x$

4.9

$dy=1.6,dy=1.6,$ $Δy=1.64Δy=1.64$

4.10

The volume measurement is accurate to within $21.6cm3.21.6cm3.$

4.11

7.6%

4.12

$x=−23,x=−23,$ $x=1x=1$

4.13

The absolute maximum is $33$ and it occurs at $x=4.x=4.$ The absolute minimum is $−1−1$ and it occurs at $x=2.x=2.$

4.14

$c = 2 c = 2$

4.15

$522522$ sec

4.16

$ff$ has a local minimum at $−2−2$ and a local maximum at $3.3.$

4.17

$ff$ has no local extrema because $f′f′$ does not change sign at $x=1.x=1.$

4.18

$ff$ is concave up over the interval $(−∞,12)(−∞,12)$ and concave down over the interval $(12,∞)(12,∞)$

4.19

$ff$ has a local maximum at $−2−2$ and a local minimum at $3.3.$

4.20

Both limits are $3.3.$ The line $y=3y=3$ is a horizontal asymptote.

4.21

Let $ε>0.ε>0.$ Let $N=1ε.N=1ε.$ Therefore, for all $x>N,x>N,$ we have

$| 3 − 1 x 2 − 3 | = 1 x 2 < 1 N 2 = ε | 3 − 1 x 2 − 3 | = 1 x 2 < 1 N 2 = ε$

Therefore, $limx→∞(3−1/x2)=3.limx→∞(3−1/x2)=3.$

4.22

Let $M>0.M>0.$ Let $N=M3.N=M3.$ Then, for all $x>N,x>N,$ we have

$3 x 2 > 3 N 2 = 3 ( M 3 ) 2 2 = 3 M 3 = M 3 x 2 > 3 N 2 = 3 ( M 3 ) 2 2 = 3 M 3 = M$

4.23

$− ∞ − ∞$

4.24

$3 5 3 5$

4.25

$± 3 ± 3$

4.26

$limx→∞f(x)=35,limx→∞f(x)=35,$ $limx→−∞f(x)=−2limx→−∞f(x)=−2$

4.29

$y = 3 2 x y = 3 2 x$

4.30

The function $ff$ has a cusp at $(0,5)(0,5)$ $limx→0−f′(x)=∞,limx→0−f′(x)=∞,$ $limx→0+f′(x)=−∞.limx→0+f′(x)=−∞.$ For end behavior, $limx→±∞f(x)=−∞.limx→±∞f(x)=−∞.$

4.31

The maximum area is $5000ft2.5000ft2.$

4.32

$V(x)=x(20−2x)(30−2x).V(x)=x(20−2x)(30−2x).$ The domain is $[0,10].[0,10].$

4.33

$T ( x ) = x 6 + ( 15 − x ) 2 + 1 2.5 T ( x ) = x 6 + ( 15 − x ) 2 + 1 2.5$

4.34

The company should charge $7575$ per car per day.

4.35

$A(x)=4x1−x2.A(x)=4x1−x2.$ The domain of consideration is $[0,1].[0,1].$

4.36

$c(x)=259.2x+0.2x2c(x)=259.2x+0.2x2$ dollars

4.37

$1 1$

4.38

$0 0$

4.39

$limx→0+cosx=1.limx→0+cosx=1.$ Therefore, we cannot apply L’Hôpital’s rule. The limit of the quotient is $∞∞$

4.40

$1 1$

4.41

$0 0$

4.42

$e e$

4.43

$1 1$

4.44

The function $2x2x$ grows faster than $x100.x100.$

4.45

$x 1 ≈ 0.33333333 , x 2 ≈ 0.347222222 x 1 ≈ 0.33333333 , x 2 ≈ 0.347222222$

4.46

$x 1 = 2 , x 2 = 1.75 x 1 = 2 , x 2 = 1.75$

4.47

$x 1 ≈ − 1.842105263 , x 2 ≈ − 1.772826920 x 1 ≈ − 1.842105263 , x 2 ≈ − 1.772826920$

4.48

$x 1 = 6 , x 2 = 8 , x 3 = 26 3 , x 4 = 80 9 , x 5 = 242 27 ; x * = 9 x 1 = 6 , x 2 = 8 , x 3 = 26 3 , x 4 = 80 9 , x 5 = 242 27 ; x * = 9$

4.49

$− cos x + C − cos x + C$

4.50

$d d x ( x sin x + cos x + C ) = sin x + x cos x − sin x = x cos x d d x ( x sin x + cos x + C ) = sin x + x cos x − sin x = x cos x$

4.51

$x 4 − 5 3 x 3 + 1 2 x 2 − 7 x + C x 4 − 5 3 x 3 + 1 2 x 2 − 7 x + C$

4.52

$y = − 3 x + 5 y = − 3 x + 5$

4.53

$2.93 sec , 64.5 ft 2.93 sec , 64.5 ft$

### Section 4.1 Exercises

1 .

$8 8$

3 .

$± 13 10 ± 13 10$

5 .

$2323$ ft/sec

7 .

The distance is decreasing at $390mi/h.390mi/h.$

9 .

The distance between them shrinks at a rate of $132013≈101.5mph.132013≈101.5mph.$

11 .

$9292$ ft/sec

13 .

It grows at a rate $4949$ ft/sec

15 .

The distance is increasing at $(13526)26(13526)26$ ft/sec

17 .

$−56−56$ m/sec

19 .

$240π240π$ m2/sec

21 .

$12π12π$ cm

23 .

The area is increasing at a rate $(33)8ft2/sec.(33)8ft2/sec.$

25 .

The depth of the water decreases at $128125π128125π$ ft/min.

27 .

The volume is decreasing at a rate of $(25π)16ft3/min.(25π)16ft3/min.$

29 .

The water flows out at rate $(2π)5m3/min.(2π)5m3/min.$

31 .

$3232$ m/sec

33 .

$2519π2519π$ ft/min

35 .

$245π245π$ ft/min

37 .

The angle decreases at $4001681rad/sec.4001681rad/sec.$

39 .

$100 π mi/min 100 π mi/min$

41 .

The angle is changing at a rate of $1125rad/sec.1125rad/sec.$

43 .

The distance is increasing at a rate of $62.5062.50$ ft/sec.

45 .

The distance is decreasing at a rate of $11.9911.99$ ft/sec.

### Section 4.2 Exercises

47 .

$f ′ ( a ) = 0 f ′ ( a ) = 0$

49 .

The linear approximation exact when $y=f(x)y=f(x)$ is linear or constant.

51 .

$L ( x ) = 1 2 − 1 4 ( x − 2 ) L ( x ) = 1 2 − 1 4 ( x − 2 )$

53 .

$L ( x ) = 1 L ( x ) = 1$

55 .

$L ( x ) = 0 L ( x ) = 0$

57 .

0.02

59 .

$1.9996875 1.9996875$

61 .

$0.001593 0.001593$

63 .

$1;1;$ error, $~0.00005~0.00005$

65 .

$0.97;0.97;$ error, $~0.0006~0.0006$

67 .

$3−1600;3−1600;$ error, $~4.632×10−7~4.632×10−7$

69 .

$d y = ( cos x − x sin x ) d x d y = ( cos x − x sin x ) d x$

71 .

$d y = ( x 2 − 2 x − 2 ( x − 1 ) 2 ) d x d y = ( x 2 − 2 x − 2 ( x − 1 ) 2 ) d x$

73 .

$dy=−1(x+1)2dx,dy=−1(x+1)2dx,$ $−116−116$

75 .

$dy=9x2+12x−22(x+1)3/2dx,dy=9x2+12x−22(x+1)3/2dx,$ $−0.1−0.1$

77 .

$dy=(3x2+2−1x2)dx,dy=(3x2+2−1x2)dx,$ $0.20.2$

79 .

$12 x d x 12 x d x$

81 .

$4 π r 2 d r 4 π r 2 d r$

83 .

$−1.2 π cm 3 −1.2 π cm 3$

85 .

$−100−100$ ft3

### Section 4.3 Exercises

91 .

93 .

95 .

97 .

Since the absolute maximum is the function (output) value rather than the x value, the answer is no; answers will vary

99 .

When $a=0a=0$

101 .

Absolute minimum at 3; Absolute maximum at −2.2; local minima at −2, 1; local maxima at −1, 2

103 .

Absolute minima at −2, 2; absolute maxima at −2.5, 2.5; local minimum at 0; local maxima at −1, 1

105 .

107 .

109 .

$x = 1 x = 1$

111 .

None

113 .

$x = 0 ; x = ± 2 x = 0 ; x = ± 2$

115 .

None

117 .

$x = −1 , 1 x = −1 , 1$

119 .

Absolute maximum: $x=4,x=4,$ $y=332;y=332;$ absolute minimum: $x=1,x=1,$ $y=3y=3$

121 .

Absolute minimum: $x=12,x=12,$ $y=4y=4$

123 .

Absolute maximum: $x=2π,x=2π,$ $y=2π;y=2π;$ absolute minimum: $x=0,x=0,$ $y=0y=0$

125 .

Absolute maximum: $x=−3;x=−3;$ absolute minimum: $−1≤x≤1,−1≤x≤1,$ $y=2y=2$

127 .

Absolute maximum: $x=π4,x=π4,$ $y=2;y=2;$ absolute minimum: $x=5π4,x=5π4,$ $y=−2y=−2$

129 .

Absolute minimum: $x=−2,x=−2,$ $y=1y=1$

131 .

Absolute minimum: $x=−3,x=−3,$ $y=−135;y=−135;$ local maximum: $x=0,x=0,$ $y=0;y=0;$ local minimum: $x=1,x=1,$ $y=−7y=−7$

133 .

Local maximum: $x=1−22,x=1−22,$ $y=3−42;y=3−42;$ local minimum: $x=1+22,x=1+22,$ $y=3+42y=3+42$

135 .

Absolute maximum: $x=22,x=22,$ $y=32;y=32;$ absolute minimum: $x=−22,x=−22,$ $y=−32y=−32$

137 .

Local maximum: $x=−2,x=−2,$ $y=59;y=59;$ local minimum: $x=1,x=1,$ $y=−130y=−130$

139 .

Absolute maximum: $x=0,x=0,$ $y=1;y=1;$ absolute minimum: $x=−2,2,x=−2,2,$ $y=0y=0$

141 .

$h=924549m,h=924549m,$ $t=30049st=30049s$

143 .

The absolute minimum was in 1848, when no gold was produced.

145 .

Absolute minima: $x=0,x=0,$ $x=2,x=2,$ $y=1;y=1;$ local maximum at $x=1,x=1,$ $y=2y=2$

147 .

No maxima/minima if $aa$ is odd, minimum at $x=1x=1$ if $aa$ is even

### Section 4.4 Exercises

149 .

One example is $f(x)=|x|+3,−2≤x≤2f(x)=|x|+3,−2≤x≤2$

151 .

Yes, but the Mean Value Theorem still does not apply

153 .

$( − ∞ , 0 ) , ( 0 , ∞ ) ( − ∞ , 0 ) , ( 0 , ∞ )$

155 .

$( − ∞ , −2 ) , ( 2 , ∞ ) ( − ∞ , −2 ) , ( 2 , ∞ )$

157 .

2 points

159 .

5 points

161 .

$c = 2 3 3 c = 2 3 3$

163 .

$c = 1 2 , 1 , 3 2 c = 1 2 , 1 , 3 2$

165 .

$c = 1 c = 1$

167 .

Not differentiable

169 .

Not differentiable

171 .

Yes

173 .

The Mean Value Theorem does not apply since the function is discontinuous at $x=14,34,54,74.x=14,34,54,74.$

175 .

Yes

177 .

The Mean Value Theorem does not apply; discontinuous at $x=0.x=0.$

179 .

Yes

181 .

The Mean Value Theorem does not apply; not differentiable at $x=0.x=0.$

183 .

$b = ± 2 c b = ± 2 c$

185 .

$c=±1πcos−1(π2),c=±1πcos−1(π2),$ $c=±0.1533c=±0.1533$

187 .

The Mean Value Theorem does not apply.

189 .

$12c+1−2c3=5212880;12c+1−2c3=5212880;$ $c=3.133,5.867c=3.133,5.867$

191 .

Yes

193 .

It is constant.

### Section 4.5 Exercises

195 .

It is not a local maximum/minimum because $f′f′$ does not change sign

197 .

No

199 .

False; for example, $y=x.y=x.$

201 .

Increasing for $−2 and $x>2;x>2;$ decreasing for $x<−2x<−2$ and $−1

203 .

Decreasing for $x<1,x<1,$ increasing for $x>1x>1$

205 .

Decreasing for $−2 and $1 increasing for $−1 and $x<−2x<−2$ and $x>2x>2$

207 .

a. Increasing over $−22,−22,$ decreasing over $x<−2,x<−2,$ $−1 b. maxima at $x=−1x=−1$ and $x=1,x=1,$ minima at $x=−2x=−2$ and $x=0x=0$ and $x=2x=2$

209 .

a. Increasing over $x>0,x>0,$ decreasing over $x<0;x<0;$ b. Minimum at $x=0x=0$

211 .

Concave up on all $x,x,$ no inflection points

213 .

Concave up on all $x,x,$ no inflection points

215 .

Concave up for $x<0x<0$ and $x>1,x>1,$ concave down for $0 inflection points at $x=0x=0$ and $x=1x=1$

217 .

219 .

221 .

a. Increasing over $−π2 decreasing over $x<−π2,x>π2x<−π2,x>π2$ b. Local maximum at $x=π2;x=π2;$ local minimum at $x=−π2x=−π2$

223 .

a. Concave up for $x>43,x>43,$ concave down for $x<43x<43$ b. Inflection point at $x=43x=43$

225 .

a. Increasing over $x<0x<0$ and $x>4,x>4,$ decreasing over $0 b. Maximum at $x=0,x=0,$ minimum at $x=4x=4$ c. Concave up for $x>2,x>2,$ concave down for $x<2x<2$ d. Infection point at $x=2x=2$

227 .

a. Increasing over $x<0x<0$ and $x>6011,x>6011,$ decreasing over $0 b. Minimum at $x=6011x=6011$, local maximum at x = 0 c. Concave down for $x<5411,x<5411,$ concave up for $x>5411x>5411$ d. Inflection point at $x=5411x=5411$

229 .

a. Increasing over $x>−12,x>−12,$ decreasing over $x<−12x<−12$ b. Minimum at $x=−12x=−12$ c. Concave up for all $xx$ d. No inflection points

231 .

a. Increases over $−14 decreases over $x>34x>34$ and $x<−14x<−14$ b. Minimum at $x=−14,x=−14,$ maximum at $x=34x=34$ c. Concave up for $−34 concave down for $x<−34x<−34$ and $x>14x>14$ d. Inflection points at $x=−34,x=14x=−34,x=14$

233 .

a. Increasing for all $xx$ b. No local minimum or maximum c. Concave up for $x>0,x>0,$ concave down for $x<0x<0$ d. Inflection point at $x=0x=0$

235 .

a. Increasing for all $xx$ where defined b. No local minima or maxima c. Concave up for $x<1;x<1;$ concave down for $x>1x>1$ d. No inflection points in domain

237 .

a. Increasing over $−π4 decreasing over $x>3π4,x<−π4x>3π4,x<−π4$ b. Minimum at $x=−π4,x=−π4,$ maximum at $x=3π4x=3π4$ c. Concave up for $−π2 concave down for $x<−π2,x>π2x<−π2,x>π2$ d. Infection points at $x=±π2x=±π2$

239 .

a. Increasing over $x>4,x>4,$ decreasing over $0 b. Minimum at $x=4x=4$ c. Concave up for $0 concave down for $x>823x>823$ d. Inflection point at $x=823x=823$

241 .

$f > 0 , f ′ > 0 , f ″ < 0 f > 0 , f ′ > 0 , f ″ < 0$

243 .

$f > 0 , f ′ < 0 , f ″ > 0 f > 0 , f ′ < 0 , f ″ > 0$

245 .

$f > 0 , f ′ > 0 , f ″ > 0 f > 0 , f ′ > 0 , f ″ > 0$

247 .

True, by the Mean Value Theorem

249 .

True, examine derivative

### Section 4.6 Exercises

251 .

$x = 1 x = 1$

253 .

$x = −1 , x = 2 x = −1 , x = 2$

255 .

$x = 0 x = 0$

257 .

Yes, there is a vertical asymptote

259 .

Yes, there is vertical asymptote

261 .

$0 0$

263 .

$∞ ∞$

265 .

$− 1 7 − 1 7$

267 .

$−2 −2$

269 .

$−4 −4$

271 .

Horizontal: none, vertical: $x=0x=0$

273 .

Horizontal: none, vertical: $x=±2x=±2$

275 .

Horizontal: none, vertical: none

277 .

Horizontal: $y=0,y=0,$ vertical: $x=±1x=±1$

279 .

Horizontal: $y=0,y=0,$ vertical: $x=0x=0$ and $x=−1x=−1$

281 .

Horizontal: $y=1,y=1,$ vertical: $x=1x=1$

283 .

Horizontal: none, vertical: none

285 .

Answers will vary, for example: $y=2xx−1y=2xx−1$

287 .

Answers will vary, for example: $y=4xx+1y=4xx+1$

289 .

$y = 0 y = 0$

291 .

$∞ ∞$

293 .

$y = 3 y = 3$

295 .

297 .

299 .

301 .

303 .

305 .

307 .

309 .

$lim x → 1 − f ( x ) = -∞ and lim x → 1 − g ( x ) = ∞ lim x → 1 − f ( x ) = -∞ and lim x → 1 − g ( x ) = ∞$

### Section 4.7 Exercises

311 .

The critical points can be the minima, maxima, or neither.

313 .

False; $y=−x2y=−x2$ has a minimum only

315 .

$h=623h=623$ in.

317 .

$1 1$

319 .

$100 ft by 100 ft 100 ft by 100 ft$

321 .

$40 ft by 40 ft 40 ft by 40 ft$

323 .

$19.73 ft . 19.73 ft .$

325 .

$84 bpm 84 bpm$

327 .

$T ( θ ) = 40 θ 3 v + 40 cos θ v T ( θ ) = 40 θ 3 v + 40 cos θ v$

329 .

$v = b a v = b a$

331 .

approximately $34.02mph34.02mph$

333 .

$4 4$

335 .

$0 0$

337 .

Maximal: $x=5,y=5;x=5,y=5;$ minimal: $x=0,y=10x=0,y=10$ and $y=0,x=10y=0,x=10$

339 .

Maximal: $x=1,y=9;x=1,y=9;$ minimal: none

341 .

$4 π 3 3 4 π 3 3$

343 .

$6 6$

345 .

$r = 2 , h = 4 r = 2 , h = 4$

347 .

$( 2 , 1 ) ( 2 , 1 )$

349 .

$( 0.8351 , 0.6974 ) ( 0.8351 , 0.6974 )$

351 .

$A = 20 r − 2 r 2 − 1 2 π r 2 A = 20 r − 2 r 2 − 1 2 π r 2$

353 .

$C(x)=5x2+32xC(x)=5x2+32x$ Differentiating, setting the derivative equal to zero and solving, we obtain $x=253x=253$ and $h=2543h=2543$.

355 .

$P ( x ) = ( 50 − x ) ( 800 + 25 x − 50 ) P ( x ) = ( 50 − x ) ( 800 + 25 x − 50 )$

### Section 4.8 Exercises

357 .

$∞ ∞$

359 .

$1 2 a 1 2 a$

361 .

$1 n a n − 1 1 n a n − 1$

363 .

Cannot apply directly; use logarithms

365 .

Cannot apply directly; rewrite as $limx→0x3limx→0x3$

367 .

$6 6$

369 .

$−2 −2$

371 .

$−1 −1$

373 .

$n n$

375 .

$− 1 2 − 1 2$

377 .

$1 2 1 2$

379 .

$1 1$

381 .

$1 6 1 6$

383 .

$1 1$

385 .

$0 0$

387 .

$0 0$

389 .

$−1 −1$

391 .

$∞ ∞$

393 .

$0 0$

395 .

$1 e 1 e$

397 .

$0 0$

399 .

$1 1$

401 .

$0 0$

403 .

$tan ( 1 ) tan ( 1 )$

405 .

$2 2$

### Section 4.9 Exercises

407 .

$F ( x n ) = x n − x n 3 + 2 x n + 1 3 x n 2 + 2 F ( x n ) = x n − x n 3 + 2 x n + 1 3 x n 2 + 2$

409 .

$F ( x n ) = x n − e x n e x n F ( x n ) = x n − e x n e x n$

411 .

$|c|>0.5|c|>0.5$ fails, $|c|≤0.5|c|≤0.5$ works

413 .

$c = 1 f ′ ( x n ) c = 1 f ′ ( x n )$

415 .

a. $x1=1225,x2=312625;x1=1225,x2=312625;$ b. $x1=−4,x2=−40x1=−4,x2=−40$

417 .

a. $x1=1.291,x2=0.8801;x1=1.291,x2=0.8801;$ b. $x1=0.7071,x2=1.189x1=0.7071,x2=1.189$

419 .

a. $x1=−2625,x2=−1224625;x1=−2625,x2=−1224625;$ b. $x1=4,x2=18x1=4,x2=18$

421 .

a. $x1=610,x2=610;x1=610,x2=610;$ b. $x1=2,x2=2x1=2,x2=2$

423 .

$3.1623 or − 3.1623 3.1623 or − 3.1623$

425 .

$0 , −1 or 1 0 , −1 or 1$

427 .

$0 0$

429 .

$0.5188 or − 1.2906 0.5188 or − 1.2906$

431 .

$0 0$

433 .

$4.493 4.493$

435 .

$0.159 , 3.146 0.159 , 3.146$

437 .

We need $ff$ to be twice continuously differentiable.

439 .

$x = 0 x = 0$

441 .

$x = −1 x = −1$

443 .

$x = 5.619 x = 5.619$

445 .

$x = −1.326 x = −1.326$

447 .

There is no solution to the equation.

449 .

It enters a cycle.

451 .

$0 0$

453 .

$−0.3513 −0.3513$

455 .

Newton: $1111$ iterations, secant: $1616$ iterations

457 .

Newton: three iterations, secant: six iterations

459 .

Newton: five iterations, secant: eight iterations

461 .

$E = 4.071 E = 4.071$

463 .

$4 . 394 % 4 . 394 %$

### Section 4.10 Exercises

465 .

$F ′ ( x ) = 15 x 2 + 4 x + 3 F ′ ( x ) = 15 x 2 + 4 x + 3$

467 .

$F ′ ( x ) = 2 x e x + x 2 e x F ′ ( x ) = 2 x e x + x 2 e x$

469 .

$F ′ ( x ) = e x F ′ ( x ) = e x$

471 .

$F ( x ) = e x − x 3 − cos ( x ) + C F ( x ) = e x − x 3 − cos ( x ) + C$

473 .

$F ( x ) = x 2 2 − x − 2 cos ( 2 x ) + C F ( x ) = x 2 2 − x − 2 cos ( 2 x ) + C$

475 .

$F ( x ) = 1 2 x 2 + 4 x 3 + C F ( x ) = 1 2 x 2 + 4 x 3 + C$

477 .

$F ( x ) = 2 5 ( x ) 5 + C F ( x ) = 2 5 ( x ) 5 + C$

479 .

$F ( x ) = 3 2 x 2 / 3 + C F ( x ) = 3 2 x 2 / 3 + C$

481 .

$F ( x ) = x + tan ( x ) + C F ( x ) = x + tan ( x ) + C$

483 .

$F ( x ) = 1 3 sin 3 ( x ) + C F ( x ) = 1 3 sin 3 ( x ) + C$

485 .

$F ( x ) = − 1 2 cot ( x ) − 1 x + C F ( x ) = − 1 2 cot ( x ) − 1 x + C$

487 .

$F ( x ) = − sec x − 4 csc x + C F ( x ) = − sec x − 4 csc x + C$

489 .

$F ( x ) = − 1 8 e −4 x − cos x + C F ( x ) = − 1 8 e −4 x − cos x + C$

491 .

$− cos x + C − cos x + C$

493 .

$3 x − 2 x + C 3 x − 2 x + C$

495 .

$8 3 x 3 / 2 + 4 5 x 5 / 4 + C 8 3 x 3 / 2 + 4 5 x 5 / 4 + C$

497 .

$14 x − 2 x − 1 2 x 2 + C 14 x − 2 x − 1 2 x 2 + C$

499 .

$f ( x ) = − 1 2 x 2 + 3 2 f ( x ) = − 1 2 x 2 + 3 2$

501 .

$f ( x ) = sin x + tan x + 1 f ( x ) = sin x + tan x + 1$

503 .

$f ( x ) = − 1 6 x 3 − 2 x + 13 6 f ( x ) = − 1 6 x 3 − 2 x + 13 6$

505 .

Answers may vary; one possible answer is $f(x)=e−xf(x)=e−x$

507 .

Answers may vary; one possible answer is $f(x)=−sinxf(x)=−sinx$

509 .

$5.8675.867$ sec

511 .

$7.3337.333$ sec

513 .

$13.7513.75$ ft/sec2

515 .

$F ( x ) = 1 3 x 3 + 2 x F ( x ) = 1 3 x 3 + 2 x$

517 .

$F ( x ) = x 2 − cos x + 1 F ( x ) = x 2 − cos x + 1$

519 .

$F ( x ) = − 1 ( x + 1 ) + 1 F ( x ) = − 1 ( x + 1 ) + 1$

521 .

True

523 .

False

### Review Exercises

525 .

True, by Mean Value Theorem

527 .

True

529 .

Increasing: $(−2,0)∪(4,∞),(−2,0)∪(4,∞),$ decreasing: $(−∞,−2)∪(0,4)(−∞,−2)∪(0,4)$

531 .

$L ( x ) = 17 16 + 1 2 ( 1 + 4 π ) ( x − 1 4 ) L ( x ) = 17 16 + 1 2 ( 1 + 4 π ) ( x − 1 4 )$

533 .

Critical point: $x=3π4,x=3π4,$ absolute minimum: $x=0,x=0,$ absolute maximum: $x=πx=π$

535 .

Increasing: $(−1,0)∪(3,∞),(−1,0)∪(3,∞),$ decreasing: $(−∞,−1)∪(0,3),(−∞,−1)∪(0,3),$ concave up: $(−∞,13(2−13))∪(13(2+13),∞),(−∞,13(2−13))∪(13(2+13),∞),$ concave down: $(13(2−13),13(2+13))(13(2−13),13(2+13))$

537 .

Increasing: $(14,∞),(14,∞),$ decreasing: $(0,14),(0,14),$ concave up: $(0,∞),(0,∞),$ concave down: nowhere

539 .

$3 3$

541 .

$− 1 π − 1 π$

543 .

$x 1 = −1 , x 2 = −1 x 1 = −1 , x 2 = −1$

545 .

$F ( x ) = 2 x 3 / 2 3 + 1 x + C F ( x ) = 2 x 3 / 2 3 + 1 x + C$

547 .

Inflection points: none; critical points: $x=−13;x=−13;$ zeros: none; vertical asymptotes: $x=−1,x=−1,$ $x=0;x=0;$ horizontal asymptote: $y=0y=0$

549 .

The height is decreasing at a rate of $0.1250.125$ m/sec

551 .

$x=abx=ab$ feet

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