The coffee is first cool enough to serve about $3.5$ minutes after it is poured. The coffee is too cold to serve about $7$ minutes after it is poured.

2.46

A total of $94.13$ g of carbon remains. The artifact is approximately $13,300$ years old.

2.47

$\frac{d}{d x} (tanh (x^{2} + 3 x)) = ({sech}^{2} (x^{2} + 3 x)) (2 x + 3)$
$\frac{d}{d x} (\frac{1}{{(sinh x)}^{2}}) = \frac{d}{d x} {(sinh x)}^{−2} = −2 {(sinh x)}^{−3} cosh x$

2.48

$\int {sinh}^{3} x cosh x d x = \frac{{sinh}^{4} x}{4} + C$
$\int {sech}^{2} (3 x) d x = \frac{tanh (3 x)}{3} + C$

2.49

$\frac{d}{d x} ({cosh}^{−1} (3 x)) = \frac{3}{\sqrt{9 x^{2} - 1}}$
$\frac{d}{d x} {({coth}^{−1} x)}^{3} = \frac{3 {({coth}^{−1} x)}^{2}}{1 - x^{2}}$

2.50

$\int \frac{1}{\sqrt{x^{2} - 4}} d x = {cosh}^{−1} (\frac{x}{2}) + C$
$\int \frac{1}{\sqrt{1 - e^{2 x}}} d x = − {sech}^{−1} (e^{x}) + C$

2.51

$52.95 ft$

Section 2.1 Exercises

1.

$\frac{32}{3}$

3.

$\frac{13}{12}$

5.

$36$

7.

This figure is has two graphs. They are the functions f(x)=x^2 and g(x)=-x^2+18x. The region between the graphs is shaded, bounded above by g(x) and below by f(x). It is in the first quadrant.

243 square units

9.

This figure is has two graphs. They are the functions y=cos(x) and y=cos^2(x). The graphs are periodic and resemble waves. There are four regions created by intersections of the curves. The areas are shaded.

4

11.

This figure is has two graphs. They are the functions f(x)=e^x and g(x)=e^-x. There are two shaded regions. In the second quadrant the region is bounded by x=-1, g(x) above and f(x) below. The second region is in the first quadrant and is bounded by f(x) above, g(x) below, and x=1.

$\frac{2 {(e - 1)}^{2}}{e}$

13.

This figure is has two graphs. They are the functions f(x)=x^2 and g(x)=absolute value of x. There are two shaded regions. The first region is in the second quadrant and is between g(x) above and f(x) below. The second region is in the first quadrant and is bounded above by g(x) and below by f(x).

$\frac{1}{3}$

15.

This figure is has three graphs. They are the functions f(x)=squareroot of x, y=12-x, and y=1. The region between the graphs is shaded, bounded above and to the left by f(x), above and to the right by the line y=12-x, and below by the line y=1. It is in the first quadrant.

$\frac{34}{3}$

17.

This figure is has two graphs. They are the functions f(x)=x^3 and g(x)=x^2-2x. There are two shaded regions between the graphs. The first region is bounded to the left by the line x=-2, above by g(x) and below by f(x). The second region is bounded above by f(x), below by g(x) and to the right by the line x=2.

$\frac{5}{2}$

19.

This figure is has two graphs. They are the functions f(x)=x^3+3x and g(x)=4x. There are two shaded regions between the graphs. The first region is bounded above by f(x) and below by g(x). The second region is bounded above by g(x), below by f(x).

$\frac{1}{2}$

21.

This figure is has two graphs. They are the equations x=2y and x=y^3-y. The graphs intersect in the third quadrant and again in the first quadrant forming two closed regions in between them.

$\frac{9}{2}$

23.

This figure is has two graphs. They are the equations x=y+2 and y^2=x. The graphs intersect, forming a region in between them

$\frac{9}{2}$

25.

This figure is has two graphs. They are the equations x=cos(y) and x=sin(y). The graphs intersect, forming two regions bounded above by the line y=pi/2 and below by the line y=-pi/2.

$\frac{3 \sqrt{3}}{2}$

27.

This figure is has two graphs. They are the equations y=xe^x and y=e^x. The graphs intersect, forming a region in between them in the first quadrant.

$e −2$

29.

This figure is has two graphs. They are the equations x=-y^2+1 and x=y^3+2y^2. The graphs intersect, forming two regions in between them.

$\frac{27}{4}$

31.

This figure is has two graphs. They are the equations y=4-3x and y=1/x. The graphs intersect, having region between them shaded. The region is in the first quadrant.

$\frac{4}{3} - ln (3)$

33.

This figure is has two graphs. They are the equations y=x^2-3x+2 and y=x^3-2x^2-x+2. The graphs intersect, having region between them shaded.

$\frac{1}{2}$

35.

This figure is has two graphs. They are the equations 2y=x and y+y^3=x. The graphs intersect, forming two regions. The regions are shaded.

$\frac{1}{2}$

37.

This figure is has two graphs. They are the equations y=arccos(x) and y=arcsin (x). The graphs intersect, forming two regions. The first region is bounded to the left by x=-1. The second region is bounded to the right by x=1. Both regions are shaded.

$−2 (\sqrt{2} - π)$

39.

$1.067$

41.

$0.852$

43.

$7.523$

45.

$\frac{3 π - 4}{12}$

47.

$1.429$

49.

$$ 33,333.33$ total profit for $200$ cell phones sold

51.

$3.263$ mi represents how far ahead the hare is from the tortoise

53.

$\frac{343}{24}$

55.

$4 \sqrt{3}$

57.

$π - \frac{32}{25}$

Section 2.2 Exercises

63.

8 units³

65.

$\frac{32}{3 \sqrt{2}}$ units³

67.

$\frac{7}{24} π r^{2} h$ units³

69.

This figure shows the x-axis and the y-axis with a line starting on the x-axis at (1,0) and ending on the y-axis at (0,1). Perpendicular to the xy-plane are 4 shaded semi-circles with their diameters beginning on the x-axis and ending on the line, decreasing in size away from the origin.

$\frac{π}{24}$ units³

71.

This figure shows the x-axis and the y-axis in 3-dimensional perspective. On the graph above the x-axis is a parabola, which has its vertex at y=1 and x-intercepts at (-1,0) and (1,0). There are 3 square shaded regions perpendicular to the x y plane, which touch the parabola on either side, decreasing in size away from the origin.

$2$ units³

73.

This figure is a graph with the x and y axes diagonal to show 3-dimensional perspective. On the first quadrant of the graph are the curves y=x, a line, and y=x^2, a parabola. They intersect at the origin and at (1,1). Several semicircular-shaped shaded regions are perpendicular to the x y plane, which go from the parabola to the line and perpendicular to the line.

$\frac{π}{240}$ units³

75.

This figure is a graph in the first quadrant. It is a shaded region bounded above by the curve y=2x^2, below by the x-axis, and to the right by the vertical line x=4.

$\frac{4096 π}{5}$ units³

77.

This figure is a graph in the first quadrant. It is a shaded region bounded above by the line y=1, below by the curve y=x^4, and to the left by the y-axis.

$\frac{8 π}{9}$ units³

79.

This figure is a shaded region bounded above by the curve y=cos(x), below to the left by the y-axis and below to the right by y=sin(x). The shaded region is in the first quadrant.

$\frac{π}{2}$ units³

81.

This figure is a graph in the first quadrant. It is a shaded region bounded above by the line x + y=9, below by the x-axis, to the left by the y-axis, and to the left by the curve x^2-y^2=9.

$207 π$ units³

83.

This figure is a graph in the first quadrant. It is a shaded region bounded above by the curve y=2x^3, below by the x-axis, and to the right by the line x=1.

$\frac{4 π}{5}$ units³

85.

This figure is a graph in the first quadrant. It is a quarter of a circle with center at the origin and radius of 2. It is shaded on the inside.

$\frac{16 π}{3}$ units³

87.

This figure is a graph in the first quadrant. It is a shaded region bounded above by the line y=pi/4, to the right by the curve x=sec(y), below by the x-axis, and to the left by the y-axis.

$π$ units³

89.

This figure is a graph in the first quadrant. It is a shaded triangle bounded above by the line y=4-x, below by the line y=x, and to the left by the y-axis.

$\frac{16 π}{3}$ units³

91.

This figure is a graph above the x-axis. It is a shaded region bounded above by the line y=x+2, and below by the parabola y=x^2.

$\frac{72 π}{5}$ units³

93.

This figure is a shaded region bounded above by the curve y=4-x^2 and below by the line y=2-x.

$\frac{108 π}{5}$ units³

95.

This figure is a graph in the first quadrant. It is a shaded region bounded above by the curve y=squareroot(x), below by the curve y=x^2.

$\frac{3 π}{10}$ units³

97.

This figure is a shaded region bounded above by the curve y=squareroot(4-x^2) and, below by the curve y=squareroot(1+x^2).

$2 \sqrt{6} π$ units³

99.

This figure is a graph in the first quadrant. It is a shaded region bounded above by the line y=x+2, below by the line y=2x-1, and to the left by the y-axis.

$9 π$ units³

101.

This figure is a graph in the first quadrant. It is a shaded region bounded above by the curve y=ln(2), below by the x-axis, to the left by the curve x=y^2, and to the right by the curve x=e^(2y).

$\frac{π}{20} (75 - 4 {ln}^{5} (2))$ units³

103.

$\frac{m^{2} π}{3} (b^{3} - a^{3})$ units³

105.

$\frac{4 a^{2} b π}{3}$ units³

107.

$2 π^{2}$ units³

109.

$\frac{2 a b^{2} π}{3}$ units³

111.

$V = \frac{2 π}{3} (2 r^{3} - 3 r^{2} h + h^{3})$ units³

113.

$\frac{π}{3} (h + R) {(h - 2 R)}^{2}$ units³

Section 2.3 Exercises

115.

This figure is a graph in the first quadrant. It is the line y=3x. Under the line and above the x-axis there is a shaded region. The region is bounded to the right at x=3.

$54 π$ units³

117.

$81 π$ units³

119.

This figure is a graph in the first quadrant. It is the increasing curve y=2x^3. Under the curve and above the x-axis there is a shaded region. The region is bounded to the right at x=2.

$\frac{512 π}{7}$ units³

121.

$2 π$ units³

123.

$\frac{2 π}{3}$ units³

125.

$2 π$ units³

127.

$\frac{4 π}{5}$ units³

129.

$\frac{64 π}{3}$ units³

131.

$\frac{32 π}{5}$ units³

133.

$\frac{7 π}{6}$

135.

$48 π$

137.

$\frac{113 π}{5}$

139.

$\frac{512 π}{7}$

141.

$\frac{96 π}{5}$ units³

143.

$\frac{28 π}{15}$ units³

145.

$\frac{3 π}{10}$ units³

147.

$π (\frac{6 \cdot 2^{2 / 3}}{5} - \frac{11}{10}) = \frac{π}{10} (12 \cdot 2^{2 / 3} - 11) \approx 2.5286$ units³

149.

$0.9876$ units³

151.

This figure is a graph. On the graph are two curves, y=cos(pi times x) and y=sin(pi times x). They are periodic curves resembling waves. The curves intersect in the first quadrant and also the fourth quadrant. The region between the two points of intersection is shaded.

$3 \sqrt{2}$ units³

153.

This figure is a graph in the first quadrant. It is the parabola y=x^2-2x. . Under the curve and above the x-axis there is a shaded region. The region begins at x=2 and is bounded to the right at x=4.

$\frac{496 π}{15}$ units³

155.

This figure is a graph in the first quadrant. There are two curves on the graph. The first curve is y=3x^2-2 and the second curve is y=x. Between the curves there is a shaded region. The region begins at x=1 and is bounded to the right at x=2.

$\frac{398 π}{15}$ units³

157.

This figure is a graph. There are two curves on the graph. The first curve is x=y^2-2y+1 and is a parabola opening to the right. The second curve is x=y^2 and is a parabola opening to the right. Between the curves there is a shaded region. The shaded region is bounded to the right at x=2.

$15.9074$ units³

159.

$\frac{1}{3} π r^{2} h$ units³

161.

$π r^{2} h$ units³

163.

$π a^{2}$ units³

Section 2.4 Exercises

165.

$2 \sqrt{26}$

167.

$2 \sqrt{17}$

169.

$\frac{π}{6} (17 \sqrt{17} - 5 \sqrt{5})$

171.

$\frac{13 \sqrt{13} - 8}{27}$

173.

$\frac{4}{3}$

175.

$2.0035$

177.

$\frac{123}{32}$

179.

$10$

181.

$\frac{20}{3}$

183.

$\frac{1}{675} (229 \sqrt{229} - 8)$

185.

$\frac{1}{8} (4 \sqrt{5} + ln (9 + 4 \sqrt{5}))$

187.

$1.201$

189.

$15.2341$

191.

$\frac{49 π}{3}$

193.

$70 π \sqrt{2}$

195.

$8 π$

197.

$120 π \sqrt{26}$

199.

$\frac{π}{6} (17 \sqrt{17} - 1)$

201.

$9 \sqrt{2} π$

203.

$\frac{10 \sqrt{10} π}{27} (73 \sqrt{73} - 1)$

205.

$25.645$

207.

$π (π + 2)$

209.

$10.5017$

211.

$23$ ft

213.

$2$

215.

Answers may vary

217.

For more information, look up Gabriel’s Horn.

Section 2.5 Exercises

219.

$150$ ft-lb

221.

$200 J$

223.

$1$ J

225.

$\frac{39}{2}$

227.

$ln (243)$

229.

$\frac{332 π}{15}$

231.

$100 π$

233.

$20 π \sqrt{15}$

235.

$6$ J

237.

$5$ cm

239.

$36$ J

241.

$18,750$ ft-lb

243.

Weight $= \frac{32}{3} \times 10^{9} lb$

Work $= \frac{4}{3} \times 10^{12} ft-lb$

245.

$9.71 \times 10^{2} N m$

247.

a. $3,000,000$ lb, b. $750,000$ lbs

249.

$23.25 π$ million ft-lb

251.

$\frac{A ρ H^{2}}{2}$

253.

Answers may vary

Section 2.6 Exercises

255.

$\frac{5}{4}$

257.

$(\frac{2}{3}, \frac{2}{3})$

259.

$(\frac{7}{4}, \frac{3}{2})$

261.

$\frac{3 L}{4}$

263.

$\frac{π}{2}$

265.

$\frac{e^{2} + 1}{e^{2} - 1}$

267.

$\frac{π^{2} - 4}{π}$

269.

$\frac{1}{4} (1 + e^{2})$

271.

$(\frac{a}{3}, \frac{b}{3})$

273.

$(0, \frac{π}{8})$

275.

$(0, 3)$

277.

$(0, \frac{4}{π})$

279.

$(\frac{5}{8}, \frac{1}{3})$

281.

$\frac{2 m π}{3}$

283.

$π a^{2} b$

285.

$(\frac{4}{3 π}, \frac{4}{3 π})$

287.

$(\frac{1}{2}, \frac{2}{5})$

289.

$(0, \frac{28}{9 π})$

291.

Center of mass: $(\frac{a}{6}, \frac{4 a^{2}}{5}),$ volume: $\frac{2 π a^{4}}{9}$

293.

Volume: $2 π^{2} a^{2} (b + a)$

Section 2.7 Exercises

295.

$\frac{1}{x}$

297.

$- \frac{1}{x {(ln x)}^{2}}$

299.

$ln (x + 1) + C$

301.

$ln (x) + 1$

303.

$cot (x)$

305.

$\frac{7}{x}$

307.

$csc (x) sec x$

309.

$−2 tan x$

311.

$\frac{1}{2} ln (\frac{5}{3})$

313.

$2 - \frac{1}{2} ln (5)$

315.

$\frac{1}{ln (2)} - 1$

317.

$\frac{1}{2} ln (2)$

319.

$\frac{1}{3} {(ln x)}^{3}$

321.

$\frac{2 x^{3}}{\sqrt{x^{2} + 1} \sqrt{x^{2} - 1}}$

323.

$x^{−2 - (1 / x)} (ln x - 1)$

325.

$e x^{e - 1}$

327.

$1$

329.

$- \frac{1}{x^{2}}$

331.

$π - ln (2)$

333.

$\frac{1}{x}$

335.

$e^{5} - 6 {units}^{2}$

337.

$ln (4) - 1 {units}^{2}$

339.

$2.8656$

341.

$3.1502$

Section 2.8 Exercises

349.

True

351.

False; $k = \frac{ln (2)}{t}$

353.

$20$ hours

355.

No. The relic is approximately $871$ years old.

357.

$71.92$ years

359.

$5$ days $6$ hours $27$ minutes

361.

$12$

363.

$8.618 %$

365.

$$ 6766.76$

367.

$9$ hours $13$ minutes

369.

$239,179$ years

371.

$P' (t) = 43 e^{0.01604 t} .$ The population is always increasing.

373.

The population reaches $10$ billion people in $2027 .$

375.

$P' (t) = 2.259 e^{0.06407 t} .$ The population is always increasing.

Section 2.9 Exercises

377.

$e^{x} and e^{− x}$

379.

Answers may vary

381.

Answers may vary

383.

Answers may vary

385.

$3 sinh (3 x + 1)$

387.

$− tanh (x) sech (x)$

389.

$4 cosh (x) sinh (x)$

391.

$\frac{x {sech}^{2} (\sqrt{x^{2} + 1})}{\sqrt{x^{2} + 1}}$

393.

$6 {sinh}^{5} (x) cosh (x)$

395.

$\frac{1}{2} sinh (2 x + 1) + C$

397.

$\frac{1}{2} {sinh}^{2} (x^{2}) + C$

399.

$\frac{1}{3} {cosh}^{3} (x) + C$

401.

$ln (1 + cosh (x)) + C$

403.

$cosh (x) + sinh (x) + C$

405.

$\frac{4}{1 - 16 x^{2}}$

407.

$\frac{sinh (x)}{\sqrt{{cosh}^{2} (x) + 1}}$

409.

$− csc (x)$

411.

$- \frac{1}{(x^{2} - 1) {tanh}^{−1} (x)}$

413.

$\frac{1}{a} {tanh}^{−1} (\frac{x}{a}) + C$

415.

$\sqrt{x^{2} + 1} + C$

417.

${cosh}^{−1} (e^{x}) + C$

419.

Answers may vary

421.

$37.30$

423.

$y = \frac{1}{c} cosh (c x)$

425.

$−0.521095$

427.

$10$

Review Exercises

435.

False

437.

False

439.

$16 \sqrt{3} {units}^{3}$

441.

$\frac{162 π}{5}$

443.

a. $4,$ b. $\frac{128 π}{7},$ c. $\frac{64 π}{5}$

445.

a. $1.949,$ b. $21.952,$ c. $17.099$

447.

a. $\frac{31}{6},$ b. $\frac{452 π}{15},$ c. $\frac{31 π}{6}$

449.

$245.282$

451.

Mass: $\frac{1}{2},$ center of mass: $(\frac{18}{35}, \frac{9}{22})$

453.

$\sqrt{17} + \frac{1}{8} ln (33 + 8 \sqrt{17})$

455.

Volume: $\frac{3 π}{4},$ surface area: $π (\sqrt{2} - {sinh}^{−1} (1) + {sinh}^{−1} (16) - \frac{\sqrt{257}}{16})$

457.

11:02 a.m.

459.

$π (1 + sinh (1) cosh (1))$

Chapter 2

Checkpoint

Section 2.1 Exercises

Section 2.2 Exercises

Section 2.3 Exercises

Section 2.4 Exercises

Section 2.5 Exercises

Section 2.6 Exercises

Section 2.7 Exercises

Section 2.8 Exercises

Section 2.9 Exercises

Review Exercises