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

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

1.4

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

1.5

$( f ∘ g ) ( x ) = 2 − 5 x . ( f ∘ g ) ( x ) = 2 − 5 x .$

1.6

$( g ∘ f ) ( x ) = 0.63 x ( g ∘ f ) ( x ) = 0.63 x$

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

$y − 4 = 1 2 ( x − 1 ) . y − 4 = 1 2 ( x − 1 ) .$

The slope-intercept form is

$y = 1 2 x + 7 2 . y = 1 2 x + 7 2 .$

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

$C ( x ) = { 49 , 0 < x ≤ 1 70 , 1 < x ≤ 2 91 , 2 < x ≤ 3 C ( x ) = { 49 , 0 < x ≤ 1 70 , 1 < x ≤ 2 91 , 2 < x ≤ 3$

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

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

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

$x / ( 2 y 3 ) x / ( 2 y 3 )$

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

$x = ln 3 2 x = ln 3 2$

1.31

$x = 1 e x = 1 e$

1.32

$1.29248 1.29248$

1.33

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

1.34

$( x 2 + x −2 ) / 2 ( x 2 + x −2 ) / 2$

1.35

$1 2 ln ( 3 ) ≈ 0.5493 . 1 2 ln ( 3 ) ≈ 0.5493 .$

### 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.

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

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. $y = −6 x + 9 y = −6 x + 9$ 69. $y = 1 3 x + 4 y = 1 3 x + 4$ 71. $y = 1 2 x y = 1 2 x$ 73. $y = 3 5 x − 3 y = 3 5 x − 3$ 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.

$4 π 3 rad 4 π 3 rad$

115.

$− π 3 − π 3$

117.

$11 π 6 rad 11 π 6 rad$

119.

$210 ° 210 °$

121.

$−540 ° −540 °$

123.

$−0.5 −0.5$

125.

$− 2 2 − 2 2$

127.

$3 − 1 2 2 3 − 1 2 2$

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.

$sec 2 x sec 2 x$

141.

$sin 2 x sin 2 x$

143.

$sec 2 θ sec 2 θ$

145.

$1 sin t ( = csc t ) 1 sin t ( = csc t )$

155.

${ π 6 , 5 π 6 } { π 6 , 5 π 6 }$

157.

${ π 4 , 3 π 4 , 5 π 4 , 7 π 4 } { π 4 , 3 π 4 , 5 π 4 , 7 π 4 }$

159.

${ 2 π 3 , 5 π 3 } { 2 π 3 , 5 π 3 }$

161.

${ 0 , π , π 3 , 5 π 3 } { 0 , π , π 3 , 5 π 3 }$

163.

$y = 4 sin ( π 4 x ) y = 4 sin ( π 4 x )$

165.

$y = cos ( 2 π x ) y = cos ( 2 π x )$

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.

177.

$≈ 30.9 in 2 ≈ 30.9 in 2$

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.

$π 6 π 6$

209.

$π 4 π 4$

211.

$π 6 π 6$

213.

$2 2 2 2$

215.

$− π 6 − π 6$

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. $~ 1.5 sec ~ 1.5 sec$ 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. $8 1 / 3 = 2 8 1 / 3 = 2$ 249. $5 2 = 25 5 2 = 25$ 251. $e −3 = 1 e 3 e −3 = 1 e 3$ 253. $e 0 = 1 e 0 = 1$ 255. $log 4 ( 1 16 ) = −2 log 4 ( 1 16 ) = −2$ 257. $log 9 1 = 0 log 9 1 = 0$ 259. $log 64 4 = 1 3 log 64 4 = 1 3$ 261. $log 9 150 = y log 9 150 = y$ 263. $log 4 0.125 = − 3 2 log 4 0.125 = − 3 2$ 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. $2 + 3 log 3 a − log 3 b 2 + 3 log 3 a − log 3 b$ 273. $3 2 + 1 2 log 5 x + 3 2 log 5 y 3 2 + 1 2 log 5 x + 3 2 log 5 y$ 275. $− 3 2 + ln 6 − 3 2 + ln 6$ 277. $ln 15 3 ln 15 3$ 279. $3 2 3 2$ 281. $log 7.21 log 7.21$ 283. $2 3 + log 11 3 log 7 2 3 + log 11 3 log 7$ 285. $x = 1 25 x = 1 25$ 287. $x = 4 x = 4$ 289. $x = 3 x = 3$ 291. $1 + 5 1 + 5$ 293. $( log 82 log 7 ≈ 2.2646 ) ( log 82 log 7 ≈ 2.2646 )$ 295. $( log 211 log 0.5 ≈ − 7.7211 ) ( log 211 log 0.5 ≈ − 7.7211 )$ 297. $( log 0.452 log 0.2 ≈ 0.4934 ) ( log 0.452 log 0.2 ≈ 0.4934 )$ 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.

$cos2x-sin2x=cos2x=1-2sin2x=2cos2x-1cos2x-sin2x=cos2x=1-2sin2x=2cos2x-1$

323.

$0 , ± 2 π 0 , ± 2 π$

325.

4

327.

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

329.

$x ≥ − 3 2 , f −1 ( x ) = − 3 2 + 1 2 4 y − 7 x ≥ − 3 2 , f −1 ( x ) = − 3 2 + 1 2 4 y − 7$

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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