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Calculus Volume 1

# Chapter 1

### Checkpoint

1.1

$f(1)=3f(1)=3$ and $f(a+h)=a2+2ah+h2−3a−3h+5f(a+h)=a2+2ah+h2−3a−3h+5$

1.2

Domain = ${x|x≤2},{x|x≤2},$ range = ${y|y≥5}{y|y≥5}$

1.3

$x=0,2,3x=0,2,3$

1.4

$(fg)(x)=x2+32x−5.(fg)(x)=x2+32x−5.$ The domain is ${x|x≠52}.{x|x≠52}.$

1.5

$(f∘g)(x)=2−5x.(f∘g)(x)=2−5x.$

1.6

$(g∘f)(x)=0.63x(g∘f)(x)=0.63x$

1.7

$f(x)f(x)$ is odd.

1.8

Domain = $(−∞,∞),(−∞,∞),$ range = ${y|y≥−4}.{y|y≥−4}.$

1.9

$m=1/2.m=1/2.$ The point-slope form is

$y−4=12(x−1).y−4=12(x−1).$

The slope-intercept form is

$y=12x+72.y=12x+72.$

1.10

The zeros are $x=1±3/3.x=1±3/3.$ The parabola opens upward.

1.11

The domain is the set of real numbers $xx$ such that $x≠1/2.x≠1/2.$ The range is the set ${y|y≠5/2}.{y|y≠5/2}.$

1.12

The domain of $ff$ is $(−∞, ∞).(−∞, ∞).$ The domain of $gg$ is ${x|x≥1/5}.{x|x≥1/5}.$

1.13

Algebraic

1.15

$C(x)={49,0

1.16

Shift the graph $y=x2y=x2$ to the left 1 unit, reflect about the $xx$-axis, then shift down 4 units.

1.17

$7π/6;7π/6;$ 330°

1.18

$cos(3π/4)=−2/2;sin(−π/6)=−1/2cos(3π/4)=−2/2;sin(−π/6)=−1/2$

1.19

$1010$ ft

1.20

$θ=3π2+2nπ,π6+2nπ,5π6+2nπθ=3π2+2nπ,π6+2nπ,5π6+2nπ$ for $n=0,±1,±2,…n=0,±1,±2,…$

1.22

To graph $f(x)=3sin(4x)−5,f(x)=3sin(4x)−5,$ the graph of $y=sin(x)y=sin(x)$ needs to be compressed horizontally by a factor of 4, then stretched vertically by a factor of 3, then shifted down 5 units. The function $ff$ will have a period of $π/2π/2$ and an amplitude of 3.

1.23

No.

1.24

$f−1(x)=2xx−3.f−1(x)=2xx−3.$ The domain of $f−1f−1$ is ${x|x≠3}.{x|x≠3}.$ The range of $f−1f−1$ is ${y|y≠2}.{y|y≠2}.$

1.26

The domain of $f−1f−1$ is $(0,∞).(0,∞).$ The range of $f−1f−1$ is $(−∞,0).(−∞,0).$ The inverse function is given by the formula $f−1(x)=−1/x.f−1(x)=−1/x.$

1.27

$f(4)=900;f(10)=24,300.f(4)=900;f(10)=24,300.$

1.28

$x/(2y3)x/(2y3)$

1.29

$A(t)=750e0.04t.A(t)=750e0.04t.$ After $3030$ years, there will be approximately $2,490.09.2,490.09.$

1.30

$x=ln32x=ln32$

1.31

$x=1ex=1e$

1.32

$1.292481.29248$

1.33

The magnitude $8.48.4$ earthquake is roughly $1010$ times as severe as the magnitude $7.47.4$ earthquake.

1.34

$(x2+x−2)/2(x2+x−2)/2$

1.35

$12ln(3)≈0.5493.12ln(3)≈0.5493.$

### Section 1.1 Exercises

1.

a. Domain = ${−3,−2,−1,0,1,2,3},{−3,−2,−1,0,1,2,3},$ range = ${0,1,4,9}{0,1,4,9}$ b. Yes, a function

3.

a. Domain = ${0,1,2,3},{0,1,2,3},$ range = ${−3,−2,−1,0,1,2,3}{−3,−2,−1,0,1,2,3}$ b. No, not a function

5.

a. Domain = ${3,5,8,10,15,21,33},{3,5,8,10,15,21,33},$ range = ${0,1,2,3}{0,1,2,3}$ b. Yes, a function

7.

a. $−2−2$ b. 3 c. 13 d. $−5x−2−5x−2$ e. $5a−25a−2$ f. $5a+5h−25a+5h−2$

9.

a. Undefined b. 2 c. $2323$ d. $−2x−2x$ e $2a2a$ f. $2a+h2a+h$

11.

a. $55$ b. $1111$ c. $2323$ d. $−6x+5−6x+5$ e. $6a+56a+5$ f. $6a+6h+56a+6h+5$

13.

a. 9 b. 9 c. 9 d. 9 e. 9 f. 9

15.

$x≥18;y≥0;x=18;x≥18;y≥0;x=18;$ no y-intercept

17.

$x≥−2;y≥−1;x=−1;y=−1+2x≥−2;y≥−1;x=−1;y=−1+2$

19.

$x≠4;y≠0;x≠4;y≠0;$ no x-intercept; $y=−34y=−34$

21.

$x>5;y>0;x>5;y>0;$ no intercepts

23.

25.

27.

29.

Function; a. Domain: all real numbers, range: $y≥0y≥0$ b. $x=±1x=±1$ c. $y=1y=1$ d. $−1 and $1 e. $−∞ and $0 f. Not constant g. y-axis h. Even

31.

Function; a. Domain: all real numbers, range: $−1.5≤y≤1.5−1.5≤y≤1.5$ b. $x=0x=0$ c. $y=0y=0$ d. $all real numbersall real numbers$ e. None f. Not constant g. Origin h. Odd

33.

Function; a. Domain: $−∞ range: $−2≤y≤2−2≤y≤2$ b. $x=0x=0$ c. $y=0y=0$ d. $−2 e. Not decreasing f. $−∞ and $2 g. Origin h. Odd

35.

Function; a. Domain: $−4≤x≤4,−4≤x≤4,$ range: $−4≤y≤4−4≤y≤4$ b. $x=1.2x=1.2$ c. $y=4y=4$ d. Not increasing e. $0 f. $−4 g. No Symmetry h. Neither

37.

a. $5x2+x−8;5x2+x−8;$ all real numbers b. $−5x2+x−8;−5x2+x−8;$ all real numbers c. $5x3−40x2;5x3−40x2;$ all real numbers d. $x−85x2;x≠0x−85x2;x≠0$

39.

a. $−2x+6;−2x+6;$ all real numbers b. $−2x2+2x+12;−2x2+2x+12;$ all real numbers c. $−x4+2x3+12x2−18x−27;−x4+2x3+12x2−18x−27;$ all real numbers d. $−x+3x+1;x≠−1,3−x+3x+1;x≠−1,3$

41.

a. $6+2x;x≠06+2x;x≠0$ b. 6; $x≠0x≠0$ c. $6x+1x2;x≠06x+1x2;x≠0$ d. $6x+1;x≠06x+1;x≠0$

43.

a. $4x+3;4x+3;$ all real numbers b. $4x+15;4x+15;$ all real numbers

45.

a. $x4−6x2+16;x4−6x2+16;$ all real numbers b. $x4+14x2+46;x4+14x2+46;$ all real numbers

47.

a. $3x4+x;x≠0,−43x4+x;x≠0,−4$ b. $4x+23;x≠−124x+23;x≠−12$

49.

a. Yes, because there is only one winner for each year. b. No, because there are three teams that won more than once during the years 2001 to 2012.

51.

a. $V(s)=s3V(s)=s3$ b. $V(11.8)≈1643;V(11.8)≈1643;$ a cube of side length 11.8 each has a volume of approximately 1643 cubic units.

53.

a. $N(x)=15xN(x)=15x$ b. i. $N(20)=15(20)=300;N(20)=15(20)=300;$ therefore, the vehicle can travel 300 mi on a full tank of gas. Ii. $N(15)=225;N(15)=225;$ therefore, the vehicle can travel 225 mi on 3/4 of a tank of gas. c. Domain: $0≤x≤20;0≤x≤20;$ range: $[0,300][0,300]$ d. The driver had to stop at least once, given that it takes approximately 39 gal of gas to drive a total of 578 mi.

55.

a. $A(t)=A(r(t))=π·(6−5t2+1)2A(t)=A(r(t))=π·(6−5t2+1)2$ b. Exact: $121π4;121π4;$ approximately 95 cm2 c. $C(t)=C(r(t))=2π(6−5t2+1)C(t)=C(r(t))=2π(6−5t2+1)$ d. Exact: $11π;11π;$ approximately 35 cm

57.

a. $S(x)=8.5x+750S(x)=8.5x+750$ b. $962.50,$1090, $1217.50 c. 77 skateboards ### Section 1.2 Exercises 59. a. −1 b. Decreasing 61. a. 3/4 b. Increasing 63. a. 4/3 b. Increasing 65. a. 0 b. Horizontal 67. $y=−6x+9y=−6x+9$ 69. $y=13x+4y=13x+4$ 71. $y=12xy=12x$ 73. $y=35x−3y=35x−3$ 75. a. $(m=2,b=−3)(m=2,b=−3)$ b. 77. a. $(m=−6,b=0)(m=−6,b=0)$ b. 79. a. $(m=0,b=−6)(m=0,b=−6)$ b. 81. a. $(m=−23,b=2)(m=−23,b=2)$ b. 83. a. 2 b. $52,−1;52,−1;$ c. −5 d. Both ends rise e. Neither 85. a. 2 b. $±2±2$ c. −1 d. Both ends rise e. Even 87. a. 3 b. 0, $±3±3$ c. 0 d. Left end rises, right end falls e. Odd 89. 91. 93. 95. a. $13,−3,513,−3,5$ b. 97. a. $−32,−12,4−32,−12,4$ b. 99. True; $n=3n=3$ 101. False; $f(x)=xb,f(x)=xb,$ where $bb$ is a real-valued constant, is a power function 103. a. $V(t)=−2733t+20500V(t)=−2733t+20500$ b. $(0,20,500)(0,20,500)$ means that the initial purchase price of the equipment is$20,500; $(7.5,0)(7.5,0)$ means that in 7.5 years the computer equipment has no value. c. $6835 d. In approximately 6.4 years 105. a. $C=0.75x+125C=0.75x+125$ b.$245 c. 167 cupcakes

107.

a. $V(t)=−1500t+26,000V(t)=−1500t+26,000$ b. In 4 years, the value of the car is $20,000. 109.$30,337.50

111.

96% of the total capacity

### Section 1.3 Exercises

113.

$4π3rad4π3rad$

115.

$−π3−π3$

117.

$11π6rad11π6rad$

119.

$210°210°$

121.

$−540°−540°$

123.

$−0.5−0.5$

125.

$−22−22$

127.

$3−1223−122$

129.

a. $b=5.7b=5.7$ b. $sinA=47,cosA=5.77,tanA=45.7,cscA=74,secA=75.7,cotA=5.74sinA=47,cosA=5.77,tanA=45.7,cscA=74,secA=75.7,cotA=5.74$

131.

a. $c=151.7c=151.7$ b. $sinA=0.5623,cosA=0.8273,tanA=0.6797,cscA=1.778,secA=1.209,cotA=1.471sinA=0.5623,cosA=0.8273,tanA=0.6797,cscA=1.778,secA=1.209,cotA=1.471$

133.

a. $c=85c=85$ b. $sinA=8485,cosA=1385,tanA=8413,cscA=8584,secA=8513,cotA=1384sinA=8485,cosA=1385,tanA=8413,cscA=8584,secA=8513,cotA=1384$

135.

a. $y=2425y=2425$ b. $sinθ=2425,cosθ=725,tanθ=247,cscθ=2524,secθ=257,cotθ=724sinθ=2425,cosθ=725,tanθ=247,cscθ=2524,secθ=257,cotθ=724$

137.

a. $x=−23x=−23$ b. $sinθ=73,cosθ=−23,tanθ=−142,cscθ=377,secθ=−322,cotθ=−147sinθ=73,cosθ=−23,tanθ=−142,cscθ=377,secθ=−322,cotθ=−147$

139.

$sec2xsec2x$

141.

$sin2xsin2x$

143.

$sec2θsec2θ$

145.

$1sint(=csct)1sint(=csct)$

155.

${π6,5π6}{π6,5π6}$

157.

${π4,3π4,5π4,7π4}{π4,3π4,5π4,7π4}$

159.

${2π3,5π3}{2π3,5π3}$

161.

${0,π,π3,5π3}{0,π,π3,5π3}$

163.

$y=4sin(π4x)y=4sin(π4x)$

165.

$y=cos(2πx)y=cos(2πx)$

167.

a. 1 b. $2π2π$ c. $π4π4$ units to the right

169.

a. $1212$ b. $8π8π$ c. No phase shift

171.

a. 3 b. $22$ c. $2π2π$ units to the left

173.

Approximately 42 in.

175.

a. 0.550 rad/sec b. 0.236 rad/sec c. 0.698 rad/min d. 1.697 rad/min

177.

$≈30.9in2≈30.9in2$

179.

a. π/184; the voltage repeats every π/184 sec b. Approximately 59 periods

181.

a. Amplitude = $10;period=2410;period=24$ b. $47.4°F47.4°F$ c. 14 hours later, or 2 p.m. d.

### Section 1.4 Exercises

183.

Not one-to-one

185.

Not one-to-one

187.

One-to-one

189.

a. $f−1(x)=x+4f−1(x)=x+4$ b. Domain $:x≥−4,range:y≥0:x≥−4,range:y≥0$

191.

a. $f−1(x)=x−13f−1(x)=x−13$ b. Domain: all real numbers, range: all real numbers

193.

a. $f−1(x)=x2+1,f−1(x)=x2+1,$ b. Domain: $x≥0,x≥0,$ range: $y≥1y≥1$

195.

197.

199.

These are inverses.

201.

These are not inverses.

203.

These are inverses.

205.

These are inverses.

207.

$π6π6$

209.

$π4π4$

211.

$π6π6$

213.

$2222$

215.

$−π6−π6$

217.

a. $x=f−1(V)=0.04−V500x=f−1(V)=0.04−V500$ b. The inverse function determines the distance from the center of the artery at which blood is flowing with velocity V. c. 0.1 cm; 0.14 cm; 0.17 cm

219.

a. $31,250,$66,667, $107,143 b. $(p=85CC+75)(p=85CC+75)$ c. 34 ppb 221. a. $~92°~92°$ b. $~42°~42°$ c. $~27°~27°$ 223. $x≈6.69,8.51;x≈6.69,8.51;$ so, the temperature occurs on June 21 and August 15 225. $~1.5sec~1.5sec$ 227. $tan−1(tan(2.1))≈−1.0416;tan−1(tan(2.1))≈−1.0416;$ the expression does not equal 2.1 since $2.1>1.57=π22.1>1.57=π2$—in other words, it is not in the restricted domain of $tanx.cos−1(cos(2.1))=2.1,tanx.cos−1(cos(2.1))=2.1,$ since 2.1 is in the restricted domain of $cosx.cosx.$ ### Section 1.5 Exercises 229. a. 125 b. 2.24 c. 9.74 231. a. 0.01 b. 10,000 c. 46.42 233. d 235. b 237. e 239. Domain: all real numbers, range: $(2,∞),y=2(2,∞),y=2$ 241. Domain: all real numbers, range: $(0,∞),y=0(0,∞),y=0$ 243. Domain: all real numbers, range: $(−∞,1),y=1(−∞,1),y=1$ 245. Domain: all real numbers, range: $(−1,∞),y=−1(−1,∞),y=−1$ 247. $81/3=281/3=2$ 249. $52=2552=25$ 251. $e−3=1e3e−3=1e3$ 253. $e0=1e0=1$ 255. $log4(116)=−2log4(116)=−2$ 257. $log91=0log91=0$ 259. $log644=13log644=13$ 261. $log9150=ylog9150=y$ 263. $log40.125=−32log40.125=−32$ 265. Domain: $(1,∞),(1,∞),$ range: $(−∞,∞),x=1(−∞,∞),x=1$ 267. Domain: $(0,∞),(0,∞),$ range: $(−∞,∞),x=0(−∞,∞),x=0$ 269. Domain: $(−1,∞),(−1,∞),$ range: $(−∞,∞),x=−1(−∞,∞),x=−1$ 271. $2+3log3a−log3b2+3log3a−log3b$ 273. $32+12log5x+32log5y32+12log5x+32log5y$ 275. $−32+ln6−32+ln6$ 277. $ln153ln153$ 279. $3232$ 281. $log7.21log7.21$ 283. $23+log113log723+log113log7$ 285. $x=125x=125$ 287. $x=4x=4$ 289. $x=3x=3$ 291. $1+51+5$ 293. $(log82log7≈2.2646)(log82log7≈2.2646)$ 295. $(log211log0.5≈−7.7211)(log211log0.5≈−7.7211)$ 297. $(log0.452log0.2≈0.4934)(log0.452log0.2≈0.4934)$ 299. $~17,491~17,491$ 301. Approximately$131,653 is accumulated in 5 years.

303.

i. a. pH = 8 b. Base ii. a. pH = 3 b. Acid iii. a. pH = 4 b. Acid

305.

a. $~333~333$ million b. 94 years from 2013, or in 2107

307.

a. $k≈0.0578k≈0.0578$ b. $≈92≈92$ hours

309.

The San Francisco earthquake had $103.4or~2512103.4or~2512$ times more energy than the Japan earthquake.

### Review Exercises

311.

False

313.

False

315.

Domain: $x>5,x>5,$ range: all real numbers

317.

Domain: $x>2x>2$ and $x<−4,x<−4,$ range: all real numbers

319.

Degree of 3, $yy$-intercept: 0, zeros: 0, $3−1,−1−33−1,−1−3$

321.

$cos2x–sin2x=cos2xcos2x–sin2x=cos2x$ or $= 1–2sin2x 2 = 1–2sin2x 2$ or $= 2cos2x–1 2 = 2cos2x–1 2$

323.

$0,±2π0,±2π$

325.

4

327.

One-to-one; yes, the function has an inverse; inverse: $f−1(x)=1yf−1(x)=1y$

329.

$x≥−32,f−1(x)=−32+124y−7x≥−32,f−1(x)=−32+124y−7$

331.

a. $C(x)=300+7xC(x)=300+7x$ b. 100 shirts

333.

The population is less than 20,000 from December 8 through January 23 and more than 140,000 from May 29 through August 2

335.

78.51%

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