Circuit Training - Review on Use of a Calculator in Calculus Answer Key

Checkpoint

1.1

$f(1)=3$ and $f(a+h)=a^{\mathrm{two}}+\mathrm{two}ah+h^{\mathrm{two}}-3a-3h+\mathrm{five}$

i.2

Domain = $\left\{x|x\le \mathrm{ii}\right\},$ range = $\left\{y\right|y\ge 5\}$

ane.3

$x=0,2,\mathrm{iii}$

1.4

$\left(\frac{f}{g}\right)\left(\mathrm{10}\right)=\frac{x^2+3}{\mathrm{ii}\mathrm{10}-\mathrm{five}}.$ The domain is $\left\{x|x\ne \frac{\mathrm{v}}{\mathrm{two}}\right\}.$

1.5

$\left(f\circ g\right)\left(x\right)=2-5\sqrt{x}.$

1.6

$(\mathrm{yard}\circ f)(\mathrm{ten})=0.63x$

i.viii

Domain = $(\text{−}\infty ,\infty ),$ range = $\left\{y\right|y\ge \mathrm{-4}\}.$

1.9

$\mathrm{yard}=\mathrm{i}\text{/}2.$ The point-slope form is

$y-\mathrm{iv}=\frac{1}{2}\left(x-1\right).$

The slope-intercept form is

$y=\frac{1}{2}x+\frac{7}{2}.$

1.10

The zeros are $\mathrm{ten}=1\pm \sqrt{3}\text{/}3.$ The parabola opens up.

one.eleven

The domain is the set of real numbers $\mathrm{ten}$ such that $\mathrm{ten}\ne 1\text{/}\mathrm{two}.$ The range is the set $\left\{y\right|y\ne 5\text{/}\mathrm{ii}\}.$

1.12

The domain of $f$ is $\text{(−∞, ∞).}$ The domain of $\mathrm{grand}$ is $\left\{x\right|x\ge 1\text{/}5\}.$

1.15

$C\left(x\right)=\{\begin{array}{c}49,0<x\le \mathrm{i}\\ 70,\mathrm{i}<x\le \mathrm{two}\\ 91,2<\mathrm{10}\le 3\end{array}$

1.16

Shift the graph $y=x^{\mathrm{ii}}$ to the left 1 unit, reverberate almost the $x$ -axis, then shift down 4 units.

ane.eighteen

$\text{cos}(\mathrm{three}\pi \text{/}\mathrm{four})=\text{−}\sqrt{2}\text{/}2;\text{sin}(\text{−}\pi \text{/}6)=\mathrm{-1}\text{/}2$

i.20

$\theta =\frac{3\pi }{\mathrm{two}}+2n\pi ,\frac{\pi }{\mathrm{six}}+2n\pi ,\frac{\mathrm{v}\pi }{\mathrm{six}}+\mathrm{ii}\mathrm{north}\pi$ for $n=0,\pm 1,\pm 2\text{,…}$

1.22

To graph $f\left(x\right)=3\text{sin}\left(4x\right)-5,$ the graph of $y=\text{sin}(x)$ needs to be compressed horizontally by a factor of 4, and so stretched vertically by a gene of 3, then shifted down 5 units. The function $f$ will have a period of $\pi \text{/}\mathrm{ii}$ and an amplitude of 3.

1.24

$f^{\mathrm{-1}}\left(x\right)=\frac{\mathrm{two}\mathrm{10}}{\mathrm{ten}-3}.$ The domain of $f^{\mathrm{-1}}$ is $\left\{\mathrm{10}\right|\mathrm{ten}\ne 3\}.$ The range of $f^{\mathrm{-1}}$ is $\left\{y\right|y\ne 2\}.$

one.26

The domain of $f^{\mathrm{-i}}$ is $\left(0,\infty \right).$ The range of $f^{\mathrm{-one}}$ is $(\text{−}\infty ,0).$ The inverse function is given by the formula $f^{\mathrm{-one}}\left(\mathrm{10}\right)=\mathrm{-i}\text{/}\sqrt{x}.$

i.27

$f\left(\mathrm{iv}\right)=900;f\left(\mathrm{ten}\right)=24,300.$

one.28

$x\text{/}\left(2y^3\right)$

one.29

$A\left(t\right)=750e^{0.04t}.$ After $30$ years, there will be approximately $\text{\$}\mathrm{two},490.09.$

1.30

$x=\frac{\text{ln}3}{\mathrm{ii}}$

1.33

The magnitude $8.4$ earthquake is roughly $10$ times as astringent equally the magnitude $\mathrm{seven.four}$ earthquake.

ane.34

$(x^{\mathrm{ii}}+x^{\mathrm{-2}})\text{/}2$

ane.35

$\frac{1}{2}\text{ln}\left(\mathrm{three}\right)\approx 0.5493.$

Section one.i Exercises

1 .

a. Domain = $\left\{\mathrm{-3},\mathrm{-2},\mathrm{-1},0,1,\mathrm{two},3\right\},$ range = $\left\{0,\mathrm{ane},4,9\right\}$ b. Yep, a part

three .

a. Domain = $\left\{0,1,2,3\right\},$ range = $\left\{\mathrm{-3},\mathrm{-ii},\mathrm{-1},0,1,\mathrm{two},3\right\}$ b. No, not a function

5 .

a. Domain = $\left\{3,5,\mathrm{viii},10,15,21,33\right\},$ range = $\left\{0,1,2,3\right\}$ b. Yes, a function

seven .

a. $\mathrm{-ii}$ b. 3 c. 13 d. $\mathrm{-v}\mathrm{10}-\mathrm{two}$ eastward. $5a-2$ f. $\mathrm{v}a+\mathrm{five}h-2$

nine .

a. Undefined b. 2 c. $\frac{2}{\mathrm{iii}}$ d. $-\frac{2}{\mathrm{ten}}$ eastward $\frac{2}{a}$ f. $\frac{2}{a+h}$

11 .

a. $\sqrt{5}$ b. $\sqrt{\mathrm{eleven}}$ c. $\sqrt{23}$ d. $\sqrt{\mathrm{-six}x+5}$ e. $\sqrt{\mathrm{half\; dozen}a+\mathrm{v}}$ f. $\sqrt{6a+6h+5}$

xiii .

a. 9 b. ix c. 9 d. nine east. 9 f. 9

15 .

$x\ge \frac{\mathrm{one}}{\mathrm{viii}};y\ge 0;x=\frac{\mathrm{ane}}{8};$ no y-intercept

17 .

$x\ge \mathrm{-ii};y\ge \mathrm{-1};x=\mathrm{-1};y=\mathrm{-1}+\sqrt{\mathrm{two}}$

19 .

$x\ne 4;y\ne 0;$ no x-intercept; $y=-\frac{\mathrm{iii}}{4}$

21 .

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

29 .

Function; a. Domain: all real numbers, range: $y\ge 0$ b. $\mathrm{10}=\pm \mathrm{one}$ c. $y=\mathrm{one}$ d. $\mathrm{-1}<\mathrm{ten}<0$ and $1<x<\infty$ e. $\text{−}\infty <x<-\mathrm{ane}$ and $0<\mathrm{ten}<\mathrm{i}$ f. Not constant g. y-centrality h. Even

31 .

Function; a. Domain: all existent numbers, range: $\mathrm{-1.five}\le y\le \mathrm{one.5}$ b. $x=0$ c. $y=0$ d. $\text{all real numbers}$ e. None f. Not constant chiliad. Origin h. Odd

33 .

Part; a. Domain: $\text{−}\infty <x<\infty ,$ range: $\mathrm{-2}\le y\le 2$ b. $\mathrm{10}=0$ c. $y=0$ d. $\mathrm{-2}<x<2$ east. Not decreasing f. $\text{−}\infty <x<-2$ and $2<x<\infty$ 1000. Origin h. Odd

35 .

Function; a. Domain: $\mathrm{-iv}\le x\le 4,$ range: $\mathrm{-four}\le y\le \mathrm{four}$ b. $\mathrm{ten}=\mathrm{ane.ii}$ c. $y=\mathrm{four}$ d. Not increasing due east. $0<x<4$ f. $\mathrm{-four}<x<0$ m. No Symmetry h. Neither

37 .

a. $5x^{\mathrm{ii}}+x-\mathrm{eight};$ all real numbers b. $\mathrm{-v}{\mathrm{ten}}^2+x-\mathrm{eight};$ all real numbers c. $5x^3-40x^{\mathrm{ii}};$ all real numbers d. $\frac{x-\mathrm{viii}}{\mathrm{five}{\mathrm{ten}}^{\mathrm{two}}};x\ne 0$

39 .

a. $\mathrm{-2}x+6;$ all real numbers b. $\mathrm{-2}{\mathrm{10}}^{\mathrm{two}}+2\mathrm{ten}+12;$ all existent numbers c. $\text{−}{x}^4+2x^3+12x^2-18x-27;$ all real numbers d. $-\frac{x+\mathrm{iii}}{\mathrm{ten}+\mathrm{i}};\mathrm{ten}\ne \text{−}1,\mathrm{iii}$

41 .

a. $\mathrm{half\; dozen}+\frac{2}{x};x\ne 0$ b. 6; $\mathrm{ten}\ne 0$ c. $\frac{6}{\mathrm{10}}+\frac{1}{x^2};x\ne 0$ d. $6\mathrm{10}+1;\mathrm{ten}\ne 0$

43 .

a. $\mathrm{four}x+3;$ all existent numbers b. $4\mathrm{ten}+15;$ all existent numbers

45 .

a. $x^4-6x^2+\mathrm{sixteen};$ all real numbers b. ${\mathrm{ten}}^4+\mathrm{xiv}{x}^{\mathrm{two}}+46;$ all existent numbers

47 .

a. $\frac{3\mathrm{10}}{\mathrm{iv}+\mathrm{10}};\mathrm{10}\ne 0,\mathrm{-4}$ b. $\frac{4x+2}{3};x\ne -\frac{1}{\mathrm{two}}$

49 .

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

51 .

a. $\mathrm{Five}(s)=s^3$ b. $V\left(11.8\right)\approx 1643;$ a cube of side length xi.eight each has a volume of approximately 1643 cubic units.

53 .

a. $N(x)=15x$ b. i. $N\left(\mathrm{twenty}\right)=15\left(\mathrm{xx}\right)=300;$ therefore, the vehicle tin can travel 300 mi on a total tank of gas. Two. $N\left(15\right)=225;$ therefore, the vehicle can travel 225 mi on three/4 of a tank of gas. c. Domain: $0\le \mathrm{10}\le \mathrm{twenty};$ range: $[0,300]$ d. The driver had to stop at least once, given that information technology takes approximately 39 gal of gas to drive a total of 578 mi.

55 .

a. $A\left(t\right)=A\left(r\left(t\right)\right)=\pi ·{\left(\mathrm{half\; dozen}-\frac{5}{t^2+1}\right)}^2$ b. Exact: $\frac{121\pi }{4};$ approximately 95 cm^two c. $C\left(t\right)=C\left(r\left(t\right)\right)=2\pi \left(\mathrm{half-dozen}-\frac{5}{t^{\mathrm{two}}+1}\right)$ d. Verbal: $11\pi ;$ approximately 35 cm

57 .

a. $S\left(x\right)=8.5x+750$ b. $962.l, $1090, $1217.fifty c. 77 skateboards

Section ane.2 Exercises

61 .

a. 3/four b. Increasing

63 .

a. four/3 b. Increasing

67 .

$y=\mathrm{-6}\mathrm{ten}+9$

69 .

$y=\frac{\mathrm{ane}}{3}x+4$

73 .

$y=\frac{\mathrm{iii}}{\mathrm{v}}x-3$

75 .

a. $\left(\mathrm{thou}=2,b=\mathrm{-3}\right)$ b.

77 .

a. $\left(m=\mathrm{-6},b=0\right)$ b.

79 .

a. $\left(\mathrm{thou}=0,b=\mathrm{-6}\right)$ b.

81 .

a. $\left(\mathrm{thou}=-\frac{\mathrm{two}}{3},b=2\right)$ b.

83 .

a. 2 b. $\frac{5}{2},\mathrm{-1};$ c. −5 d. Both ends ascent e. Neither

85 .

a. ii b. $\pm \sqrt{2}$ c. −1 d. Both ends ascension due east. Fifty-fifty

87 .

a. three b. 0, $\pm \sqrt{3}$ c. 0 d. Left end rises, right end falls e. Odd

95 .

a. $\mathrm{thirteen},\mathrm{-3},\mathrm{v}$ b.

97 .

a. $\frac{\mathrm{-3}}{2},\frac{\mathrm{-1}}{\mathrm{two}},4$ b.

101 .

False; $f\left(x\right)={\mathrm{10}}^b,$ where $b$ is a real-valued constant, is a power part

103 .

a. $V\left(t\right)=\mathrm{-2733}t+20500$ b. $\left(0,20,500\right)$ means that the initial purchase price of the equipment is $20,500; $\left(\mathrm{seven.v},0\right)$ means that in seven.5 years the computer equipment has no value. c. $6835 d. In approximately 6.four years

105 .

a. $C=0.75x+125$ b. $245 c. 167 cupcakes

107 .

a. $V\left(t\right)=\mathrm{-1500}t+\mathrm{26,000}$ b. In 4 years, the value of the machine is $twenty,000.

111 .

96% of the total capacity

Department 1.three Exercises

113 .

$\frac{\mathrm{four}\pi }{3}\text{rad}$

117 .

$\frac{11\pi }{6}\text{rad}$

127 .

$\frac{\sqrt{\mathrm{iii}}-\mathrm{one}}{2\sqrt{2}}$

129 .

a. $b=5.7$ b. $\text{sin}A=\frac{4}{7},\text{cos}A=\frac{\mathrm{v.7}}{\mathrm{seven}},\text{tan}A=\frac{4}{5.7},\text{csc}A=\frac{\mathrm{seven}}{4},\text{sec}A=\frac{7}{5.7},\text{cot}A=\frac{5.7}{4}$

131 .

a. $c=151.7$ b. $\text{sin}A=0.5623,\text{cos}A=0.8273,\text{tan}A=0.6797,\text{csc}A=\mathrm{ane.778},\text{sec}A=1.209,\text{cot}A=1.471$

133 .

a. $c=85$ b. $\text{sin}A=\frac{84}{85},\text{cos}A=\frac{13}{85},\text{tan}A=\frac{84}{13},\text{csc}A=\frac{85}{84},\text{sec}A=\frac{85}{13},\text{cot}A=\frac{13}{84}$

135 .

a. $y=\frac{24}{25}$ b. $\text{sin}\theta =\frac{24}{25},\text{cos}\theta =\frac{\mathrm{vii}}{25},\text{tan}\theta =\frac{24}{7},\text{csc}\theta =\frac{25}{24},\text{sec}\theta =\frac{25}{7},\text{cot}\theta =\frac{\mathrm{vii}}{24}$

137 .

a. $\mathrm{ten}=\frac{\text{−}\sqrt{2}}{\mathrm{three}}$ b. $\text{sin}\theta =\frac{\sqrt{\mathrm{vii}}}{3},\text{cos}\theta =\frac{\text{−}\sqrt{2}}{3},\text{tan}\theta =\frac{\text{−}\sqrt{\mathrm{fourteen}}}{2},\text{csc}\theta =\frac{3\sqrt{\mathrm{vii}}}{7},\text{sec}\theta =\frac{\mathrm{-iii}\sqrt{2}}{\mathrm{two}},\text{cot}\theta =\frac{\text{−}\sqrt{\mathrm{fourteen}}}{7}$

145 .

$\frac{1}{\text{sin}t}(=\text{csc}t)$

155 .

$\left\{\frac{\pi }{6},\frac{\mathrm{five}\pi }{6}\right\}$

157 .

$\left\{\frac{\pi }{4},\frac{3\pi }{\mathrm{iv}},\frac{5\pi }{4},\frac{7\pi }{\mathrm{four}}\right\}$

159 .

$\left\{\frac{2\pi }{3},\frac{\mathrm{v}\pi }{\mathrm{iii}}\right\}$

161 .

$\left\{0,\pi ,\frac{\pi }{3},\frac{5\pi }{3}\right\}$

163 .

$y=4\text{sin}\left(\frac{\pi }{4}x\right)$

165 .

$y=\text{cos}\left(2\pi x\right)$

167 .

a. one b. $\mathrm{ii}\pi$ c. $\frac{\pi }{\mathrm{four}}$ units to the right

169 .

a. $\frac{1}{\mathrm{ii}}$ b. $8\pi$ c. No phase shift

171 .

a. 3 b. $2$ c. $\frac{\mathrm{ii}}{\pi }$ 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 .

$\approx \mathrm{thirty.9}{\text{in}}^{\mathrm{ii}}$

179 .

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

181 .

a. Amplitude = $10;\text{period}=24$ b. $47.4\text{°}F$ c. 14 hours later, or 2 p.yard. d.

Section 1.four Exercises

189 .

a. $f^{\mathrm{-i}}\left(\mathrm{10}\right)=\sqrt{\mathrm{10}+\mathrm{four}}$ b. Domain $\text{:}\mathrm{ten}\ge \mathrm{-four},\text{range}\text{:}y\ge 0$

191 .

a. $f^{\mathrm{-1}}\left(x\right)=\sqrt[3]{x-1}$ b. Domain: all real numbers, range: all real numbers

193 .

a. $f^{\mathrm{-one}}\left(x\right)=x^2+1,$ b. Domain: $x\ge 0,$ range: $y\ge 1$

199 .

These are inverses.

201 .

These are non inverses.

203 .

These are inverses.

205 .

These are inverses.

217 .

a. $x=f^{\mathrm{-i}}\left(\mathrm{Five}\right)=\sqrt{0.04-\frac{V}{500}}$ b. The inverse function determines the altitude from the center of the avenue at which blood is flowing with velocity 5. c. 0.1 cm; 0.14 cm; 0.17 cm

219 .

a. $31,250, $66,667, $107,143 b. $\left(p=\frac{85C}{C+75}\right)$ c. 34 ppb

221 .

a. $~92\text{°}$ b. $~42\text{°}$ c. $~27\text{°}$

223 .

$x\approx 6.69,\mathrm{eight.51};$ and then, the temperature occurs on June 21 and August 15

225 .

$~\mathrm{1.v}\text{sec}$

227 .

${\text{tan}}^{\mathrm{-1}}\left(\text{tan}(\mathrm{two.1})\right)\approx -1.0416;$ the expression does not equal 2.i since $2.1>1.57=\frac{\pi }{\mathrm{two}}$ —in other words, it is not in the restricted domain of $\text{tan}\mathrm{ten}.{\text{cos}}^{\mathrm{-one}}\left(\text{cos}(2.1)\right)=\mathrm{ii.1},$ since 2.one is in the restricted domain of $\text{cos}\mathrm{10}.$

Section ane.5 Exercises

229 .

a. 125 b. 2.24 c. 9.74

231 .

a. 0.01 b. x,000 c. 46.42

239 .

Domain: all real numbers, range: $\left(2,\infty \right),y=2$

241 .

Domain: all existent numbers, range: $\left(0,\infty \right),y=0$

243 .

Domain: all existent numbers, range: $\left(\text{−}\infty ,1\right),y=\mathrm{i}$

245 .

Domain: all real numbers, range: $\left(\mathrm{-1},\infty \right),y=\mathrm{-ane}$

247 .

$8^{1\text{/}\mathrm{three}}=2$

251 .

$e^{\mathrm{-3}}=\frac{\mathrm{ane}}{{\mathrm{east}}^3}$

255 .

${\text{log}}_4\left(\frac{\mathrm{ane}}{16}\right)=\mathrm{-2}$

257 .

${\text{log}}_9\mathrm{i}=0$

259 .

${\text{log}}_{64}4=\frac{1}{3}$

261 .

${\text{log}}_{9}150=y$

263 .

${\text{log}}_{\mathrm{four}}0.125=-\frac{\mathrm{iii}}{2}$

265 .

Domain: $(1,\infty ),$ range: $\left(\text{−}\infty ,\infty \right),x=1$

267 .

Domain: $\left(0,\infty \right),$ range: $\left(\text{−}\infty ,\infty \right),x=0$

269 .

Domain: $\left(\mathrm{-ane},\infty \right),$ range: $\left(\text{−}\infty ,\infty \right),x=\mathrm{-i}$

271 .

$\mathrm{ii}+3{\text{log}}_{3}a-{\text{log}}_{3}b$

273 .

$\frac{3}{2}+\frac{1}{\mathrm{ii}}{\text{log}}_{5}x+\frac{\mathrm{three}}{2}{\text{log}}_{5}y$

275 .

$-\frac{3}{\mathrm{ii}}+\text{ln}6$

283 .

$\frac{\mathrm{ii}}{\mathrm{three}}+\frac{\text{log}11}{3\text{log}7}$

293 .

$\left(\frac{\text{log}82}{\text{log}7}\approx 2.2646\right)$

295 .

$\left(\frac{\text{log}211}{\text{log}0.5}\approx -7.7211\right)$

297 .

$\left(\frac{\text{log}0.452}{\text{log}0.2}\approx 0.4934\right)$

299 .

$~17,491$

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 = four b. Acid

305 .

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

307 .

a. $\mathrm{grand}\approx 0.0578$ b. $\approx 92$ hours

309 .

The San Francisco earthquake had ${10}^{\mathrm{three.four}}\text{or}~2512$ times more energy than the Nihon earthquake.

Review Exercises

315 .

Domain: $\mathrm{ten}>5,$ range: all real numbers

317 .

Domain: $\mathrm{10}>\mathrm{ii}$ and $x<-\mathrm{four},$ range: all real numbers

319 .

Degree of 3, $y$ -intercept: 0, zeros: 0, $\sqrt{3}-1,\mathrm{-one}-\sqrt{3}$

321 .

${\cos }^{\mathrm{two}}x–{\sin }^{\mathrm{two}}x=\cos 2\mathrm{10}$ or $=\frac{\mathrm{i}–\mathrm{ii}{\sin }^{\mathrm{ii}}x}{2}$ or $=\frac{2{\cos }^{2}x–1}{\mathrm{ii}}$

323 .

$0,\pm 2\pi$

327 .

One-to-one; aye, the function has an inverse; inverse: $f^{\mathrm{-1}}(x)=\frac{\mathrm{ane}}{y}$

329 .

$x\ge -\frac{\mathrm{three}}{2},f^{\mathrm{-1}}(\mathrm{10})=-\frac{\mathrm{three}}{2}+\frac{1}{2}\sqrt{\mathrm{iv}y-7}$

331 .

a. $C(x)=300+\mathrm{seven}x$ b. 100 shirts

333 .

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

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Source: https://openstax.org/books/calculus-volume-1/pages/chapter-1

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