Calculus Volume 1

# Chapter 4

### Checkpoint

4.1

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

4.2

$500ft/sec500ft/sec$

4.3

$110rad/sec110rad/sec$

4.4

$−0.61ft/sec−0.61ft/sec$

4.5

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

4.6

$L(x)=−x+π2L(x)=−x+π2$

4.7

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

4.8

$dy=2xex2dxdy=2xex2dx$

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

$c=2c=2$

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

$|3−1x2−3|=1x2<1N2=ε|3−1x2−3|=1x2<1N2=ε$

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

$3x2>3N2=3(M3)22=3M3=M3x2>3N2=3(M3)22=3M3=M$

4.23

$−∞−∞$

4.24

$3535$

4.25

$±3±3$

4.26

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

4.27 4.28 4.29

$y=32xy=32x$

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

$T(x)=x6+(15−x)2+12.5T(x)=x6+(15−x)2+12.5$

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

$11$

4.38

$00$

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

$11$

4.41

$00$

4.42

$ee$

4.43

$11$

4.44

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

4.45

$x1≈0.33333333,x2≈0.347222222x1≈0.33333333,x2≈0.347222222$

4.46

$x1=2,x2=1.75x1=2,x2=1.75$

4.47

$x1≈−1.842105263,x2≈−1.772826920x1≈−1.842105263,x2≈−1.772826920$

4.48

$x1=6,x2=8,x3=263,x4=809,x5=24227;x*=9x1=6,x2=8,x3=263,x4=809,x5=24227;x*=9$

4.49

$−cosx+C−cosx+C$

4.50

$ddx(xsinx+cosx+C)=sinx+xcosx−sinx=xcosxddx(xsinx+cosx+C)=sinx+xcosx−sinx=xcosx$

4.51

$x4−53x3+12x2−7x+Cx4−53x3+12x2−7x+C$

4.52

$y=−3x+5y=−3x+5$

4.53

$2.93sec,64.5ft2.93sec,64.5ft$

### Section 4.1 Exercises

1.

$88$

3.

$13101310$

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.

$100π mi/min100π mi/min$

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.

$f′(a)=0f′(a)=0$

49.

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

51.

$L(x)=12−14(x−2)L(x)=12−14(x−2)$

53.

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

55.

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

57.

0.02

59.

$1.99968751.9996875$

61.

$0.0015930.001593$

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.

$dy=(cosx−xsinx)dxdy=(cosx−xsinx)dx$

71.

$dy=(x2−2x−2(x−1)2)dxdy=(x2−2x−2(x−1)2)dx$

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.

$12xdx12xdx$

81.

$4πr2dr4πr2dr$

83.

$−1.2πcm3−1.2πcm3$

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.

$x=1x=1$

111.

None

113.

$x=0x=0$

115.

None

117.

$x=−1,1x=−1,1$

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 global 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.

$c=233c=233$

163.

$c=12,1,32c=12,1,32$

165.

$c=1c=1$

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.

$b=±2cb=±2c$

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 $x=6011x=6011$ 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.

$f>0,f′>0,f″<0f>0,f′>0,f″<0$

243.

$f>0,f′<0,f″<0f>0,f′<0,f″<0$

245.

$f>0,f′>0,f″>0f>0,f′>0,f″>0$

247.

True, by the Mean Value Theorem

249.

True, examine derivative

### Section 4.6 Exercises

251.

$x=1x=1$

253.

$x=−1,x=2x=−1,x=2$

255.

$x=0x=0$

257.

Yes, there is a vertical asymptote

259.

Yes, there is vertical asymptote

261.

$00$

263.

$∞∞$

265.

$−17−17$

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.

$y=0y=0$

291.

$∞∞$

293.

$y=3y=3$

295. 297. 299. 301. 303. 305. 307.

The degree of $Q(x)Q(x)$ must be greater than the degree of $P(x)P(x)$.

309.

$limx→1−f(x)=-∞andlimx→1−g(x)=∞limx→1−f(x)=-∞andlimx→1−g(x)=∞$

### 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.

$11$

319.

$100ft by100ft100ft by100ft$

321.

$40ft by40ft40ft by40ft$

323.

$19.73ft.19.73ft.$

325.

$84bpm84bpm$

327.

$T(θ)=40θ3v+40cosθvT(θ)=40θ3v+40cosθv$

329.

$v=bav=ba$

331.

approximately $34.02mph34.02mph$

333.

$44$

335.

$00$

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.

$4π334π33$

343.

$66$

345.

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

347.

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

349.

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

351.

$A=20r−2r2−12πr2A=20r−2r2−12πr2$

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.

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

### Section 4.8 Exercises

357.

$∞∞$

359.

$12a12a$

361.

$1nan−11nan−1$

363.

Cannot apply directly; use logarithms

365.

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

367.

$66$

369.

$−2−2$

371.

$−1−1$

373.

$nn$

375.

$−12−12$

377.

$1212$

379.

$11$

381.

$1616$

383.

$11$

385.

$00$

387.

$00$

389.

$−1−1$

391.

$∞∞$

393.

$00$

395.

$1e1e$

397.

$00$

399.

$11$

401.

$00$

403.

$tan(1)tan(1)$

405.

$22$

### Section 4.9 Exercises

407.

$F(xn)=xn−xn3+2xn+13xn2+2F(xn)=xn−xn3+2xn+13xn2+2$

409.

$F(xn)=xn−exnexnF(xn)=xn−exnexn$

411.

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

413.

$c=1f′(xn)c=1f′(xn)$

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.

$3.1623or−3.16233.1623or−3.1623$

425.

$0,−1or10,−1or1$

427.

$00$

429.

$0.5188or−1.29060.5188or−1.2906$

431.

$00$

433.

$4.4934.493$

435.

$0.159,3.1460.159,3.146$

437.

We need $ff$ to be twice continuously differentiable.

439.

$x=0x=0$

441.

$x=−1x=−1$

443.

$x=5.619x=5.619$

445.

$x=−1.326x=−1.326$

447.

There is no solution to the equation.

449.

It enters a cycle.

451.

$00$

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.

$E=4.071E=4.071$

463.

$4.394%4.394%$

### Section 4.10 Exercises

465.

$F′(x)=15x2+4x+3F′(x)=15x2+4x+3$

467.

$F′(x)=2xex+x2exF′(x)=2xex+x2ex$

469.

$F′(x)=exF′(x)=ex$

471.

$F(x)=ex−x3−cos(x)+CF(x)=ex−x3−cos(x)+C$

473.

$F(x)=x22−x−2cos(2x)+CF(x)=x22−x−2cos(2x)+C$

475.

$F(x)=12x2+4x3+CF(x)=12x2+4x3+C$

477.

$F(x)=25(x)5+CF(x)=25(x)5+C$

479.

$F(x)=32x2/3+CF(x)=32x2/3+C$

481.

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

483.

$F(x)=13sin3(x)+CF(x)=13sin3(x)+C$

485.

$F(x)=−12cot(x)−1x+CF(x)=−12cot(x)−1x+C$

487.

$F(x)=−secx−4cscx+CF(x)=−secx−4cscx+C$

489.

$F(x)=−18e−4x−cosx+CF(x)=−18e−4x−cosx+C$

491.

$−cosx+C−cosx+C$

493.

$3x−2x+C3x−2x+C$

495.

$83x3/2+45x5/4+C83x3/2+45x5/4+C$

497.

$14x−2x−12x2+C14x−2x−12x2+C$

499.

$f(x)=−12x2+32f(x)=−12x2+32$

501.

$f(x)=sinx+tanx+1f(x)=sinx+tanx+1$

503.

$f(x)=−16x3−2x+136f(x)=−16x3−2x+136$

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.

$F(x)=13x3+2xF(x)=13x3+2x$

517.

$F(x)=x2−cosx+1F(x)=x2−cosx+1$

519.

$F(x)=−1(x+1)+1F(x)=−1(x+1)+1$

521.

True

523.

False

### Chapter 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.

$L(x)=1716+12(1+4π)(x−14)L(x)=1716+12(1+4π)(x−14)$

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.

$33$

541.

$−1π−1π$

543.

$x1=−1,x2=−1x1=−1,x2=−1$

545.

$F(x)=2x3/23+1x+CF(x)=2x3/23+1x+C$

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