Ap Calculus Problem Book Answers

Checkpoint

1.1

$f(1)=3$ and $f(a+h)=a^2+2ah+h^{\mathrm{two}}-3a-3h+5$

one.2

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

i.3

$x=0,2,3$

1.four

$\left(\frac{f}{g}\right)\left(\mathrm{ten}\right)=\frac{{\mathrm{10}}^{\mathrm{ii}}+\mathrm{three}}{2\mathrm{10}-\mathrm{five}}.$ The domain is $\left\{\mathrm{10}|x\ne \frac{5}{\mathrm{two}}\right\}.$

1.five

$\left(f\circ g\right)\left(\mathrm{10}\right)=2-5\sqrt{\mathrm{ten}}.$

ane.half-dozen

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

1.8

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

ane.9

$m=\mathrm{ane}\text{/}\mathrm{ii}.$ The point-slope course is

$y-4=\frac{1}{2}\left(\mathrm{10}-1\right).$

The slope-intercept form is

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

ane.10

The zeros are $\mathrm{10}=1\pm \sqrt{\mathrm{iii}}\text{/}\mathrm{iii}.$ The parabola opens upwardly.

one.xi

The domain is the set of real numbers $x$ such that $x\ne \mathrm{one}\text{/}2.$ The range is the prepare $\left\{y\right|y\ne 5\text{/}2\}.$

ane.12

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

1.fifteen

$C\left(\mathrm{10}\right)=\{\begin{array}{c}49,0<x\le 1\\ 70,\mathrm{ane}<\mathrm{10}\le \mathrm{two}\\ 91,\mathrm{ii}<\mathrm{10}\le 3\end{array}$

i.xvi

Shift the graph $y=x^{\mathrm{two}}$ to the left one unit, reflect almost the $x$ -axis, then shift down four units.

1.eighteen

$\text{cos}(3\pi \text{/}4)=\text{−}\sqrt{\mathrm{two}}\text{/}2;\text{sin}(\text{−}\pi \text{/}6)=\mathrm{-i}\text{/}2$

1.20

$\theta =\frac{3\pi }{2}+2n\pi ,\frac{\pi }{\mathrm{six}}+2n\pi ,\frac{\mathrm{v}\pi }{\mathrm{half\; dozen}}+2n\pi$ for $n=0,\pm \mathrm{i},\pm 2\text{,…}$

ane.22

To graph $f\left(\mathrm{ten}\right)=3\text{sin}\left(4x\right)-5,$ the graph of $y=\text{sin}(x)$ needs to be compressed horizontally by a factor of 4, then stretched vertically by a gene of 3, and then shifted downwardly 5 units. The part $f$ volition have a period of $\pi \text{/}2$ and an amplitude of 3.

i.24

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

i.26

The domain of $f^{\mathrm{-1}}$ is $\left(0,\infty \right).$ The range of $f^{\mathrm{-one}}$ is $(\text{−}\infty ,0).$ The inverse function is given by the formula $f^{\mathrm{-ane}}\left(x\right)=\mathrm{-ane}\text{/}\sqrt{x}.$

1.27

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

1.28

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

1.29

$A\left(t\right)=750e^{0.04t}.$ Afterward $30$ years, there will be approximately $\text{\$}2,490.09.$

1.xxx

$\mathrm{10}=\frac{\text{ln}3}{2}$

i.33

The magnitude $8.4$ convulsion is roughly $\mathrm{x}$ times as severe as the magnitude $\mathrm{7.four}$ earthquake.

one.34

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

ane.35

$\frac{\mathrm{ane}}{\mathrm{two}}\text{ln}\left(3\right)\approx 0.5493.$

Department one.1 Exercises

ane.

a. Domain = $\left\{\mathrm{-3},\mathrm{-ii},\mathrm{-ane},0,\mathrm{one},2,\mathrm{iii}\right\},$ range = $\left\{0,1,4,9\right\}$ b. Aye, a function

3.

a. Domain = $\left\{0,1,2,3\right\},$ range = $\left\{\mathrm{-3},\mathrm{-ii},\mathrm{-one},0,\mathrm{ane},\mathrm{two},3\right\}$ b. No, non a role

v.

a. Domain = $\left\{\mathrm{iii},5,8,\mathrm{ten},15,21,33\right\},$ range = $\left\{0,1,2,3\right\}$ b. Yes, a function

vii.

a. $\mathrm{-2}$ b. iii c. thirteen d. $\mathrm{-5}x-2$ e. $5a-2$ f. $\mathrm{v}a+5h-2$

ix.

a. Undefined b. ii c. $\frac{2}{\mathrm{three}}$ d. $-\frac{2}{x}$ e $\frac{2}{a}$ f. $\frac{\mathrm{ii}}{a+h}$

11.

a. $\sqrt{\mathrm{v}}$ b. $\sqrt{11}$ c. $\sqrt{23}$ d. $\sqrt{\mathrm{-six}x+\mathrm{five}}$ e. $\sqrt{6a+5}$ f. $\sqrt{6a+\mathrm{vi}h+5}$

13.

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

fifteen.

$\mathrm{ten}\ge \frac{1}{8};y\ge 0;\mathrm{10}=\frac{\mathrm{ane}}{8};$ no y-intercept

17.

$\mathrm{10}\ge \mathrm{-two};y\ge \mathrm{-1};\mathrm{ten}=\mathrm{-ane};y=\mathrm{-1}+\sqrt{2}$

19.

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

21.

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

29.

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

31.

Function; a. Domain: all real numbers, range: $\mathrm{-ane.5}\le y\le \mathrm{1.five}$ b. $\mathrm{ten}=0$ c. $y=0$ d. $\text{all real numbers}$ e. None f. Not abiding thou. Origin h. Odd

33.

Office; a. Domain: $\text{−}\infty <x<\infty ,$ range: $\mathrm{-2}\le y\le 2$ b. $\mathrm{ten}=0$ c. $y=0$ d. $\mathrm{-2}<x<2$ eastward. Not decreasing f. $\text{−}\infty <x<-\mathrm{two}$ and $\mathrm{ii}<x<\infty$ one thousand. Origin h. Odd

35.

Role; a. Domain: $\mathrm{-4}\le \mathrm{ten}\le \mathrm{four},$ range: $\mathrm{-4}\le y\le 4$ b. $x=\mathrm{1.ii}$ c. $y=\mathrm{four}$ d. Non increasing e. $0<\mathrm{ten}<4$ f. $\mathrm{-4}<\mathrm{ten}<0$ one thousand. No Symmetry h. Neither

37.

a. $\mathrm{v}{x}^2+x-8;$ all real numbers b. $\mathrm{-5}{x}^{\mathrm{ii}}+x-8;$ all real numbers c. $\mathrm{v}{\mathrm{10}}^3-40{\mathrm{ten}}^{\mathrm{ii}};$ all real numbers d. $\frac{x-8}{\mathrm{five}{x}^2};x\ne 0$

39.

a. $\mathrm{-ii}\mathrm{ten}+\mathrm{six};$ all real numbers b. $\mathrm{-2}{\mathrm{10}}^2+\mathrm{ii}\mathrm{10}+12;$ all real numbers c. $\text{−}{x}^{\mathrm{iv}}+2x^3+12{\mathrm{10}}^2-18x-27;$ all real numbers d. $-\frac{x+3}{x+1};\mathrm{10}\ne \text{−}1,3$

41.

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

43.

a. $\mathrm{four}x+\mathrm{three};$ all real numbers b. $4x+15;$ all real numbers

45.

a. $x^{\mathrm{four}}-\mathrm{vi}{x}^2+16;$ all real numbers b. $x^4+14x^2+46;$ all real numbers

47.

a. $\frac{\mathrm{iii}x}{4+x};x\ne 0,\mathrm{-4}$ b. $\frac{\mathrm{four}x+\mathrm{two}}{3};\mathrm{ten}\ne -\frac{\mathrm{ane}}{2}$

49.

a. Yes, because there is just 1 winner for each year. b. No, because there are three teams that won more than once during the years 2001 to 2012.

51.

a. $V(s)={\mathrm{southward}}^3$ b. $5\left(\mathrm{11.viii}\right)\approx 1643;$ a cube of side length 11.8 each has a volume of approximately 1643 cubic units.

53.

a. $\mathrm{Northward}(\mathrm{ten})=\mathrm{xv}x$ b. i. $N\left(20\right)=\mathrm{fifteen}\left(\mathrm{twenty}\right)=300;$ therefore, the vehicle can travel 300 mi on a full tank of gas. Ii. $N\left(\mathrm{xv}\right)=225;$ therefore, the vehicle can travel 225 mi on iii/iv of a tank of gas. c. Domain: $0\le x\le 20;$ range: $[0,300]$ d. The driver had to stop at to the lowest degree once, given that it takes approximately 39 gal of gas to drive a total of 578 mi.

55.

a. $A\left(t\right)=A\left(r\left(t\right)\right)=\pi ·{\left(\mathrm{half\; dozen}-\frac{5}{t^{\mathrm{two}}+1}\right)}^2$ b. Exact: $\frac{121\pi }{\mathrm{four}};$ approximately 95 cmⁱⁱ c. $C\left(t\right)=C\left(r\left(t\right)\right)=2\pi \left(6-\frac{5}{t^2+1}\right)$ d. Exact: $11\pi ;$ approximately 35 cm

57.

a. $S\left(\mathrm{10}\right)=\mathrm{8.five}\mathrm{ten}+750$ b. $962.fifty, $1090, $1217.50 c. 77 skateboards

Section 1.ii Exercises

67.

$y=\mathrm{-6}\mathrm{10}+9$

69.

$y=\frac{1}{\mathrm{three}}\mathrm{10}+\mathrm{iv}$

73.

$y=\frac{\mathrm{iii}}{\mathrm{v}}\mathrm{10}-3$

75.

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

77.

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

79.

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

81.

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

83.

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

85.

a. two b. $\pm \sqrt{2}$ c. −1 d. Both ends rising e. Even

87.

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

95.

a. $13,\mathrm{-3},\mathrm{five}$ b.

97.

a. $\frac{\mathrm{-3}}{\mathrm{ii}},\frac{\mathrm{-ane}}{2},4$ b.

101.

False; $f\left(x\right)=x^b,$ where $b$ is a real-valued constant, is a power function

103.

a. $V\left(t\right)=\mathrm{-2733}t+20500$ b. $\left(0,20,500\right)$ means that the initial buy toll of the equipment is $20,500; $\left(\mathrm{seven.5},0\right)$ means that in 7.v 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. $\mathrm{Five}\left(t\right)=\mathrm{-1500}t+\mathrm{26,000}$ b. In four years, the value of the car is $20,000.

111.

96% of the total capacity

Section one.iii Exercises

117.

$\frac{11\pi }{\mathrm{vi}}\text{rad}$

129.

a. $b=\mathrm{v.seven}$ b. $\text{sin}A=\frac{\mathrm{four}}{7},\text{cos}A=\frac{\mathrm{five.vii}}{7},\text{tan}A=\frac{4}{5.7},\text{csc}A=\frac{7}{4},\text{sec}A=\frac{7}{\mathrm{5.seven}},\text{cot}A=\frac{5.7}{4}$

131.

a. $c=\mathrm{151.seven}$ b. $\text{sin}A=0.5623,\text{cos}A=0.8273,\text{tan}A=0.6797,\text{csc}A=1.778,\text{sec}A=1.209,\text{cot}A=1.471$

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

137.

a. $\mathrm{ten}=\frac{\text{−}\sqrt{\mathrm{ii}}}{3}$ b. $\text{sin}\theta =\frac{\sqrt{\mathrm{vii}}}{\mathrm{iii}},\text{cos}\theta =\frac{\text{−}\sqrt{\mathrm{two}}}{3},\text{tan}\theta =\frac{\text{−}\sqrt{14}}{2},\text{csc}\theta =\frac{\mathrm{three}\sqrt{7}}{\mathrm{seven}},\text{sec}\theta =\frac{\mathrm{-iii}\sqrt{2}}{2},\text{cot}\theta =\frac{\text{−}\sqrt{14}}{7}$

145.

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

155.

$\left\{\frac{\pi }{\mathrm{half\; dozen}},\frac{\mathrm{five}\pi }{\mathrm{half-dozen}}\right\}$

157.

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

159.

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

161.

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

163.

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

165.

$y=\text{cos}\left(2\pi \mathrm{10}\right)$

167.

a. 1 b. $\mathrm{two}\pi$ c. $\frac{\pi }{4}$ units to the right

169.

a. $\frac{\mathrm{i}}{2}$ b. $8\pi$ c. No phase shift

171.

a. 3 b. $2$ c. $\frac{2}{\pi }$ 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. one.697 rad/min

177.

$\approx 30.9{\text{in}}^2$

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.m. d.

Department 1.4 Exercises

189.

a. $f^{\mathrm{-i}}\left(x\right)=\sqrt{x+4}$ b. Domain $\text{:}\mathrm{10}\ge \mathrm{-four},\text{range}\text{:}y\ge 0$

191.

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

193.

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

201.

These are not inverses.

217.

a. $\mathrm{10}=f^{\mathrm{-1}}\left(V\right)=\sqrt{0.04-\frac{V}{500}}$ b. The changed function determines the distance from the eye of the artery at which claret is flowing with velocity Five. 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.

$\mathrm{ten}\approx \mathrm{six.69},8.51;$ and then, the temperature occurs on June 21 and August 15

227.

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

Section 1.five Exercises

229.

a. 125 b. ii.24 c. 9.74

231.

a. 0.01 b. 10,000 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 real numbers, range: $\left(\text{−}\infty ,\mathrm{one}\right),y=\mathrm{ane}$

245.

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

247.

$8^{1\text{/}3}=2$

251.

${\mathrm{east}}^{\mathrm{-3}}=\frac{1}{{\mathrm{east}}^{\mathrm{iii}}}$

255.

${\text{log}}_4\left(\frac{1}{\mathrm{xvi}}\right)=\mathrm{-two}$

257.

${\text{log}}_{\mathrm{ix}}\mathrm{i}=0$

259.

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

261.

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

263.

${\text{log}}_{4}0.125=-\frac{\mathrm{three}}{2}$

265.

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

267.

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

269.

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

271.

$2+3{\text{log}}_{3}a-{\text{log}}_{3}b$

273.

$\frac{\mathrm{iii}}{2}+\frac{\mathrm{i}}{\mathrm{two}}{\text{log}}_{\mathrm{v}}\mathrm{ten}+\frac{3}{\mathrm{two}}{\text{log}}_{5}y$

275.

$-\frac{\mathrm{three}}{2}+\text{ln}\mathrm{six}$

283.

$\frac{2}{\mathrm{iii}}+\frac{\text{log}\mathrm{xi}}{3\text{log}\mathrm{vii}}$

293.

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

295.

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

297.

$\left(\frac{\text{log}0.452}{\text{log}\mathrm{0.two}}\approx 0.4934\right)$

299.

$~17,491$

301.

Approximately $131,653 is accumulated in 5 years.

303.

i. a. pH = viii b. Base of operations 2. a. pH = 3 b. Acid iii. a. pH = iv b. Acrid

305.

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

307.

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

309.

The San Francisco earthquake had ${\mathrm{ten}}^{3.4}\text{or}~2512$ times more than energy than the Nihon earthquake.

Review Exercises

315.

Domain: $x>\mathrm{v},$ range: all existent numbers

317.

Domain: $x>2$ and $x<-\mathrm{iv},$ range: all real numbers

319.

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

321.

$\begin{array}{l}{\cos }^{2}x-{\sin }^{2}x=\cos 2x\\ =\mathrm{one}-2{\sin }^{\mathrm{two}}\mathrm{ten}\\ =2{\cos }^2\mathrm{10}-\mathrm{i}\end{array}$

327.

I-to-one; yes, the function has an changed; changed: $f^{\mathrm{-1}}(x)=\frac{\mathrm{i}}{y}$

329.

$\mathrm{ten}\ge -\frac{3}{2},f^{\mathrm{-1}}(x)=-\frac{\mathrm{three}}{2}+\frac{1}{2}\sqrt{4y-7}$

331.

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

333.

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

Ap Calculus Problem Book Answers,

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

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