Answer Key Chapter 1 - Calculus Volume 1 | OpenStax (2024)

Checkpoint

1.1

f(1)=3f(1)=3 and f(a+h)=a2+2ah+h23a3h+5f(a+h)=a2+2ah+h23a3h+5

1.2

Domain = {x|x2},{x|x2}, range = {y|y5}{y|y5}

1.3

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

1.4

(fg)(x)=x2+32x5.(fg)(x)=x2+32x5. The domain is {x|x52}.{x|x52}.

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 x1/2.x1/2. The range is the set {y|y5/2}.{y|y5/2}.

1.12

The domain of ff is (−∞, ∞).(−∞, ∞). The domain of gg is {x|x1/5}.{x|x1/5}.

1.13

Algebraic

1.14

Answer Key Chapter 1 - Calculus Volume 1 | OpenStax (1)

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)=2xx3.f−1(x)=2xx3. The domain of f−1f−1 is {x|x3}.{x|x3}. The range of f−1f−1 is {y|y2}.{y|y2}.

1.25

Answer Key Chapter 1 - Calculus Volume 1 | OpenStax (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. −5x2−5x2 e. 5a25a2 f. 5a+5h25a+5h2

9.

a. Undefined b. 2 c. 2323 d. 2x2x 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.

x18;y0;x=18;x18;y0;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.

x4;y0;x4;y0; no x-intercept; y=34y=34

21.

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

23.

Answer Key Chapter 1 - Calculus Volume 1 | OpenStax (3)

25.

Answer Key Chapter 1 - Calculus Volume 1 | OpenStax (4)

27.

Answer Key Chapter 1 - Calculus Volume 1 | OpenStax (5)

29.

Function; a. Domain: all real numbers, range: y0y0 b. x=±1x=±1 c. y=1y=1 d. −1<x<0−1<x<0 and 1<x<1<x< e. <x<1<x<1 and 0<x<10<x<1 f. Not constant g. y-axis h. Even

31.

Function; a. Domain: all real numbers, range: −1.5y1.5−1.5y1.5 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: <x<,<x<, range: −2y2−2y2 b. x=0x=0 c. y=0y=0 d. −2<x<2−2<x<2 e. Not decreasing f. <x<2<x<2 and 2<x<2<x< g. Origin h. Odd

35.

Function; a. Domain: −4x4,−4x4, range: −4y4−4y4 b. x=1.2x=1.2 c. y=4y=4 d. Not increasing e. 0<x<40<x<4 f. −4<x<0−4<x<0 g. No Symmetry h. Neither

37.

a. 5x2+x8;5x2+x8; all real numbers b. −5x2+x8;−5x2+x8; all real numbers c. 5x340x2;5x340x2; all real numbers d. x85x2;x0x85x2;x0

39.

a. −2x+6;−2x+6; all real numbers b. −2x2+2x+12;−2x2+2x+12; all real numbers c. x4+2x3+12x218x27;x4+2x3+12x218x27; all real numbers d. x+3x+1;x1,3x+3x+1;x1,3

41.

a. 6+2x;x06+2x;x0 b. 6; x0x0 c. 6x+1x2;x06x+1x2;x0 d. 6x+1;x06x+1;x0

43.

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

45.

a. x46x2+16;x46x2+16; all real numbers b. x4+14x2+46;x4+14x2+46; all real numbers

47.

a. 3x4+x;x0,−43x4+x;x0,−4 b. 4x+23;x124x+23;x12

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: 0x20;0x20; 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·(65t2+1)2A(t)=A(r(t))=π·(65t2+1)2 b. Exact: 121π4;121π4; approximately 95 cm2 c. C(t)=C(r(t))=2π(65t2+1)C(t)=C(r(t))=2π(65t2+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.

Answer Key Chapter 1 - Calculus Volume 1 | OpenStax (6)

77.

a. (m=−6,b=0)(m=−6,b=0) b.

Answer Key Chapter 1 - Calculus Volume 1 | OpenStax (7)

79.

a. (m=0,b=−6)(m=0,b=−6) b.

Answer Key Chapter 1 - Calculus Volume 1 | OpenStax (8)

81.

a. (m=23,b=2)(m=23,b=2) b.

Answer Key Chapter 1 - Calculus Volume 1 | OpenStax (9)

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.

Answer Key Chapter 1 - Calculus Volume 1 | OpenStax (10)

91.

Answer Key Chapter 1 - Calculus Volume 1 | OpenStax (11)

93.

Answer Key Chapter 1 - Calculus Volume 1 | OpenStax (12)

95.

a. 13,−3,513,−3,5 b.

Answer Key Chapter 1 - Calculus Volume 1 | OpenStax (13)

97.

a. −32,−12,4−32,−12,4 b.

Answer Key Chapter 1 - Calculus Volume 1 | OpenStax (14)

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.

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

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.

Answer Key Chapter 1 - Calculus Volume 1 | OpenStax (15)

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:y0:x−4,range:y0

191.

a. f−1(x)=x13f−1(x)=x13 b. Domain: all real numbers, range: all real numbers

193.

a. f−1(x)=x2+1,f−1(x)=x2+1, b. Domain: x0,x0, range: y1y1

195.

Answer Key Chapter 1 - Calculus Volume 1 | OpenStax (16)

197.

Answer Key Chapter 1 - Calculus Volume 1 | OpenStax (17)

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)=0.04V500x=f−1(V)=0.04V500 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.

x6.69,8.51;x6.69,8.51; 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

Answer Key Chapter 1 - Calculus Volume 1 | OpenStax (18)

241.

Domain: all real numbers, range: (0,),y=0(0,),y=0

Answer Key Chapter 1 - Calculus Volume 1 | OpenStax (19)

243.

Domain: all real numbers, range: (,1),y=1(,1),y=1

Answer Key Chapter 1 - Calculus Volume 1 | OpenStax (20)

245.

Domain: all real numbers, range: (−1,),y=−1(−1,),y=−1

Answer Key Chapter 1 - Calculus Volume 1 | OpenStax (21)

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

Answer Key Chapter 1 - Calculus Volume 1 | OpenStax (22)

267.

Domain: (0,),(0,), range: (,),x=0(,),x=0

Answer Key Chapter 1 - Calculus Volume 1 | OpenStax (23)

269.

Domain: (−1,),(−1,), range: (,),x=−1(,),x=−1

Answer Key Chapter 1 - Calculus Volume 1 | OpenStax (24)

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. k0.0578k0.0578 b. 9292 hours

309.

The San Francisco earthquake was 103.4or2512103.4or2512 times more intense than the Japanese 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, 31,−1331,−13

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%

Answer Key Chapter 1 - Calculus Volume 1 | OpenStax (2024)
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