multicols*2
Chapter 1
Section 1.1 (p. § 1.1)
1. \(2t\) 2. \(19.6t\) 3. \(-32t+2\) 4. \(3t^2\)
Section 1.2 (p. § 1.2)
1. \(0\) 3. \(2x+2\)
5. \(-\frac{1}{(x+1)^2}\)
7. \(-\frac{2}{x^3}\)
9. \(\frac{1}{2\sqrt{x+1}}\)
11. \(\frac{2x+3}{2\sqrt{x^2+3x+4}}\)
Section 1.3 (p. § 1.3)
5. Hint: Use the sine double-angle formula. 7. Hint: Use Exercise 5 and the sine addition formula.
Section 1.4 (p. § 1.4)
1. \(2x-1\) 3. \(4x^5 + \frac{9}{x^7}\)
5. \(x\,\cos x + \sin x\)
7. \(\frac{x\,\cos x - \sin x}{x^2}\)
9. \(\frac{2-2t^2}{(1+t^2)^2}\)
11. \(\frac{ad-bc}{(cx+d)^2}\)
13. \(2\pi r\)
Section 1.5 (p. § 1.5)
1. \(-20(1-5x)^3\) 3. \(-\frac{1}{\sqrt{1-2x}}\)
5. \(\frac{1-x}{2\sqrt{x}(x+1)^2}\)
7. \(-\frac{8(1-t)^3}{(1+t)^5}\)
9. \(2\sin x \,\cos x\)
11. \(15\sec^2(5x)\)
13. \(2x\sec (x^2)\,\tan (x^2)\)
15. \(\beta (1-\beta^2)^{-3/2}\)
17. \(\sin (\cos x)\,\sin x\)
21. Hint for part(b): Use part(a) and the
Chain Rule to find \(S_p\), then recall how to convert from radians per second to
revolutions per minute.
Section 1.6 (p. § 1.6)
1. \(6x+2\) 3. \(-9\cos 3x\) 5. \(\frac{2}{x^3}\)
Chapter 2
Section 2.1 (p. § 2.1)
1. \(f^{-1}(x)=x\), \(\left(f^{-1}\right)'(x)=1\)
3. \(f^{-1}(x)=\sqrt{x}\), \(\left(f^{-1}\right)'(x)=\frac{1}{2\sqrt{x}}\)
5. \(f^{-1}(x)=\frac{1}{x}\), \(\left(f^{-1}\right)'(x)=-\frac{1}{x^2}\)
7. \(f^{-1}(x)=\frac{1}{\sqrt{x}}\), \(\left(f^{-1}\right)'(x)=-\frac{1}{2}x^{-3/2}\)
Section 2.2 (p. § 2.2)
1. \(6\sec^2 3x\,\tan 3x\)
3. \(-3\csc^2 3x\)
5. \(\frac{3}{9+x^2}\)
7. \(-\frac{3}{1+9x^2}\)
9. \(\frac{1}{1+x^2}\)
11. \(\frac{6\sin^{-1} 3x}{\sqrt{1-9x^2}}\)
13. \(\frac{1}{1+x^2}\)
15. \(\cot^{-1} x - \frac{x}{1+x^2}\)
Section 2.3 (p. § 2.3)
1. \(2e^{2x}\)
3. \(-e^{-x} - e^x\)
5. \(\frac{2e^x}{(1-e^x)^2}\)
7. \(e^{e^x}e^x\)
9. \(\frac{1}{x}\)
11. \(\frac{6x\left(\ln\left(\tan x^2\right)\right)^2 \sec^2 x^2}{\tan x^2}\)
15. \(x^{x^2}(x+2x\ln x)\)
17. \(x^{\sin x}\left(\cos x \, \ln x + \frac{\sin x}{x}\right)\)
19. \(15.5\) hours
21. \(12\) hours
Section 2.4 (p. § 2.4)
1. \(\frac{\ln 3\,\left(3^x - 3^{-x}\right)}{2}\)
3. \((\ln 2)^2 \, 2^{2^x} \, 2^x\)
5. \(\frac{2x}{(\ln 2)(x^2+1)}\)
7. \(\frac{\cos\left(\log_2 \pi x\right)}{x \ln 2}\)
9. \(3x^2\)
Chapter 3
Section 3.1 (p. § 3.1)
1. \(y=4x-3\)
3. \(y=-6x+10\)
5. \(y=4x\)
7. \(y=x+3\)
9. \(y=240x+176\)
11. \(y=2x\)
13. \(y=3x + \frac{31}{27}\), \(y=3x+1\)
15. \(75.96\Degrees\)
17. \(0\Degrees\)
19. \(116.6\Degrees\)
21. \(5.71\Degrees\)
23. \(y=-\tfrac{1}{4}x+\tfrac{11}{2}\)
25. \(y=-\frac{1}{4}x-\tfrac{81}{4}\), \(y=-\frac{1}{4}x+\tfrac{1159}{108}\)
Section 3.2 (p. § 3.2)
1. \(\frac{7}{3}\)
3. \(0\)
5. \(-1\)
7. \(0\)
9. \(2\)
11. \(0\)
14. \(\frac{1}{2}\)
15. \(0\)
17. \(0\)
Section 3.3 (p. § 3.3)
1. continuous
3. discontinuous
5. discontinuous
7. discontinuous
9. continuous
11. continuous
13. continuous
15. discontinuous
17. continuous
19. \(1\)
21. \(e^{-1}\)
25. Hint: Use the Intermediate Value Theorem.
Section 3.4 (p. § 3.4)
1. \(\frac{-3x^2y + 4y^2 + 2x}{x^3 - 8xy - 1}\)
3. \(\frac{2(x-y+1) - 3(x+y)^2}{2(x-y+1) + 3(x+y)^2}\)
5. \(\frac{2x(1 - (x^2 - y^2))}{y(2(y^2 - x^2) - 1)}\)
7. \(-\frac{y}{x}\)
9. \(-\frac{-2x - y + 3x^2y^2e^{\sin (xy)} + x^3y^3e^{\sin (xy)}\cos (xy)}{x^4y^2e^{\sin (xy)}\cos (xy)
+ 2x^3ye^{\sin (xy)} - 3y^2 - x}\)
13. \(-\frac{x^2 + y^2}{y^3}\)
Section 3.5 (p. § 3.5)
1. \(80\pi\) ft/s
3. \(2.4\) ft/s
5. \(10\) ft/s
7. \(-76\pi\) cm3/min
9. \(45.14\) mph
11. \(155.8\) ft/min
Section 3.6 (p. § 3.6)
1. \((2x-2)\,\dx\)
2. \(4x\,\sin (x^2)\,\cos (x^2)\,\dx\)
11. Hint: Mimic Example 3.35.
Chapter 4
Section 4.1 (p. § 4.1)
1. \((1,1)\)
3. \(125,000\) sq yd
5. \(U = \frac{V}{2}\)
7. \(R=r\)
9. \(2ab\)
13. \(Q=\sqrt{\frac{2DP}{I+W}}\)
15. \(r=\sqrt{r_0^2 + x_0^2}\)
17. \(380.62\) minutes
19. \(12\pi \sqrt{3}\)
21. \(12.8\) ft
22. Hint: Place the right angle of the triangle at the origin in the \(xy\)-plane.
25. \(x = \sqrt{\frac{RN}{r}}\)
27. \(x=\frac{a}{\sqrt{2}}\)
33. \((a^{2/3} + b^{2/3})^{3/2}\)
34. Hint: You can leave your answer in terms of \(R\) and an angle satisfying
a certain equation.
Section 4.2 (p. § 4.2)
2. local maximum at \(x=0\), local minimum at \(x=2\), inflection pt at \(x=1\),
increasing for \(x<0\) and \(x>2\), decreasing for \(0<x<2\), concave up for \(x>1\),
concave down for \(x<1\)
3. local maximum at \(x=1\), inflection pt at \(x=2\), increasing for \(x<1\),
decreasing for \(x>1\), concave up for \(x>2\), concave down for \(x<2\), horizontal
asymptote: \(y=0\)
5. local maximum at \(x=0\), inflection pts at \(x=\pm \frac{1}{\sqrt{3}}\),
increasing for \(x<0\), decreasing for \(x>0\), concave up for \(x<-\frac{1}{\sqrt{3}}\) and
\(x>\frac{1}{\sqrt{3}}\), concave down for \(-\frac{1}{\sqrt{3}}<x<\frac{1}{\sqrt{3}}\),
horizontal asymptote: \(y=0\)
7. local maximum at \(x=\ln 2\), inflection pt at \(x=\ln 4\),
increasing for \(x<\ln 2\), decreasing for \(x>\ln 2\), concave up for \(x>\ln 4\),
concave down for \(x<\ln 4\), horizontal asymptote: \(y=0\)
Section 4.3 (p. § 4.3)
1. \(x=0.450184\)
3. \(x=0.567143\)
5. \(x=1.414213\)
11. global maximum at \(x=2.8214\)
14. \(50\)
Section 4.4 (p. § 4.4)
1. No 3. Yes 6. No 18. Hint: Calculate \(f'(x)\) and use the cosine addition formula.
Chapter 5
Section 5.1 (p. § 5.1)
1. \(\frac{x^3}{3} + \frac{5x^2}{2} - 3x + C\)
3. \(4e^x + C\)
5. \(-5 \cos x + C\)
7. \(6 \ln \abs{x} + C\)
9. \(-\frac{4}{3}x^{3/2} + C\)
11. \(\frac{x^2}{2} + \frac{3}{7}x^{7/3} + C\)
13. \(3 \sec x + C\)
15. \(-7 \cot x + C\)
Section 5.2 (p. § 5.2)
3. \(\frac{1}{2}\) 4. \(\frac{1}{3}\) 5. \(1\) 6. \(\frac{1}{4}\)
Section 5.3 (p. § 5.3)
1. \(\frac{1}{3}\)
3. \(\frac{1}{4}\)
5. \(\frac{1}{2}\)
7. \(1\)
9. \(2e - 2e^{-1}\)
11. \(\frac{16}{3}\)
Section 5.4 (p. § 5.4)
1. \(\frac{3 \sin 5x \,-\, 4 \cos 5x}{5} + C\)
3. \(-\frac{1}{2} e^{-x^2} + \frac{1}{3} \sin x^3 ~+~ C\)
5. \(\ln (1 + e^x) + C\)
7. \(\frac{2}{5} (x+4)^{5/2} - \frac{8}{3} (x+4)^{3/2} + C\)
9. \(\tan x - x + C\)
11. \(\frac{3}{10} \tan^{-1} \left(\frac{5x}{2}\right) + C\)
13. \(10\)
15. \(\frac{1192}{15}\)
17. \(1\)
19. \(-\frac{1}{48}\)
21. \(\frac{\pi}{6}\)
23. \(\frac{1}{2}\)
Section 5.5 (p. § 5.5)
1. \(\frac{1}{2}\)
3. \(1\)
5. divergent
7. \(\frac{1}{\ln 2}\)
9. divergent
11. \(6\)
13. divergent
15. \(\frac{\pi}{2}\)
19. Yes
20. No
Chapter 6
Section 6.1 (p. § 6.1)
1. \(\frac{x^2 \ln x}{2} - \frac{x^2}{4} + C\)
2. \((x^2 - 2x + 2)e^x + C\)
3. \(x \sin x + \cos x + C\)
5. \(\frac{x^2 a^x}{\ln a} - \frac{2x a^x}{\ln^2 a} + \frac{2 a^x}{\ln^3 a} + C\)
7. \(x \ln x^2 - 2x + C\)
9. Hint: Use a double-angle identity.
11. \(x \sin^{-1} x + \sqrt{1-x^2} + C\)
13. \(x \tan^{-1} 3x - \frac{1}{6}\ln (1+9x^2) + C\)
15. \(-\frac{3}{8}\sin x \cos 3x + \frac{1}{8}\cos x \sin 3x + C\)
17. \(\frac{1}{4}x^4 \ln^2 x - \frac{1}{8}x^4 \ln x + \frac{1}{32}x^4 + C\)
19. \(\frac{16}{3}\)
20. \(\frac{2\sqrt{2}+2}{15}\)
21. \(\frac{x \sin (\ln x)}{2} - \frac{x \cos (\ln x)}{2} + C\)
23. \(\frac{x^2 \tan^{-1} x}{2} - \frac{x}{2} + \frac{\tan^{-1} x}{2} + C\)
24. \(x \cot^{-1} \sqrt{x} + \sqrt{x} + \cot^{-1} \sqrt{x} + C\)
25. Hint: Try the substitution \(t=\sqrt{x}\).
Section 6.2 (p. § 6.2)
1. \(-\frac{1}{14}\cos\,7x + \frac{1}{6}\cos\,3x + C\)
3. \(-\frac{1}{14}\sin\,7x + \frac{1}{6}\sin\,3x + C\)
5. \(-\frac{2}{5}\cos^{5/2}x + \frac{2}{9}\cos^{9/2}x + C\)
7. \(-\frac{1}{4}\sin\,2x + \frac{1}{48}\sin^2 2x + \frac{5}{16}x + \frac{3}{64}\sin\,4x+ C\)
9. \(\frac{1}{3}\tan^3 x + \tan\,x + C\)
11. \(\frac{1}{4}\sin^4 x + C\)
Section 6.3 (p. § 6.3)
1. \(\frac{1}{2} x \sqrt{9+4x^2} + \frac{9}{4} \ln\,\abs{2x + \sqrt{9+4x^2}} + C\)
3. \(\frac{1}{2} x \sqrt{4x^2 - 9} - \frac{9}{4} \ln\,\abs{2x + \sqrt{4x^2 - 9}} + C\)
5. \(-\sin^{-1} x \,-\, \frac{\sqrt{1-x^2}}{x} + C\)
7. \(\ln \abs{x} - \ln\,\abs{1 + \sqrt{1+x^2}} + C\)
9. \(\frac{1}{3} (x^2 + 4)^{3/2} - 4\sqrt{x^2 + 4} + C\)
11. \(\frac{1}{108} \tan^{-1} \left(\frac{2x}{3}\right) \,+\, \frac{x}{18(9+4x^2)} + C\)
13. \(-9 \sqrt{9-x^2} + \frac{1}{3}(9 - x^2)^{3/2} + C\)
15. \(-\frac{1}{9} \sqrt{-9x^2+36x-32} - \frac{2}{3} \sin^{-1} \left(\frac{3x-6}{2}\right) + C\)
Section 6.4 (p. § 6.4)
1. \(-\ln \abs{x} + \ln \abs{x-1} + C\)
3. \(\frac{1}{5}\ln \abs{2x-1} - \frac{1}{5}\ln \abs{x+2} + C\)
5. \(\frac{1}{x} + \frac{1}{2}\ln \abs{x-1} - \frac{1}{2}\ln \abs{x+1} + C\)
7. \(2\ln \abs{x} + \frac{1}{x} - 2\ln \abs{x+1} + C\)
9. \(-3\ln \abs{x} + \frac{2}{x} + 3\ln \abs{x-1} + \frac{1}{x-1} + C\)
11. \(\frac{1}{3}\tan^{-1}x - \frac{1}{6}\tan^{-1}\left(\frac{x}{2}\right) + C\)
Section 6.5 (p. § 6.5)
1. \(2\,\ln\,\Abs{\sec\,\tfrac{1}{2}\theta} - \ln\,\abs{\sin\,\theta} + C\)
3. \(\frac{2}{\sqrt{3}}\tan^{-1}\left(\frac{2\tan\,\tfrac{1}{2}\theta - 1}{\sqrt{3}}\right) + C\)
5. \(\frac{4}{\sqrt{3}}\tan^{-1}\left(\frac{2\tan\,\tfrac{1}{2}\theta - 1}{\sqrt{3}}\right) - \theta + C\)
7. \(\ln\,\Abs{\tan\,\tfrac{1}{2}\theta} - \ln\,\Abs{\tan\,\tfrac{1}{2}\theta + 1} + C\)
9. \(\ln\,\Abs{\tan\,\tfrac{1}{2}\theta} - 2\ln\,\Abs{\tan\,\tfrac{1}{2}\theta + 1} + C\)
11. \(\sqrt{2\pi}\)
23. \(\frac{3}{2\Gamma\,\left(\frac{2}{3}\right)}x^{2/3}\)
Section 6.6 (p. § 6.6)
1. The true value is \(P \approx 7.4163 \sqrt{l/g}\) (i.e. the integral is \(\approx 1.8541\))
2. \(7.416331870724302 \sqrt{l/g}\)
3. \(0.8948311310564181\)
7. \(119.9785845899309\)
9. \(0.5967390281992041\) (The true value is 0.5963473623231939)
Chapter 7
Section 7.1 (p. § 7.1)
2. Foci: \((\pm 3,0)\), vertexes: \((\pm 5,0)\), \(e=\frac{3}{5}\)
3. Foci: \((0,\pm \sqrt{5}\,)\), vertexes: \((0,\pm 3)\), \(e=\frac{\sqrt{5}}{3}\)
5. Foci: \(\left(\pm \frac{\sqrt{3}}{2},0\right)\), vertexes: \((\pm 1,0)\), \(e=\frac{\sqrt{3}}{2}\)
9. \(\left(\pm \frac{a b}{\sqrt{a^2 + b^2}},\pm \frac{a b}{\sqrt{a^2 + b^2}}\right)\)
13. Hint: Use the two points you know for certain are on the ellipse to find the
location of the directrix.
16. Hint: Use Exercise 15 and formula (7.3).
Section 7.2 (p. § 7.2)
2. Focus: \((0,2)\), vertex: \((0,0)\), directrix: \(y=-2\)
3. Focus: \(\left(0,\frac{1}{32}\right)\), vertex: \((0,0)\), directrix: \(y=-\frac{1}{32}\)
4. Focus: \(\left(\frac{1}{4},0\right)\), vertex: \((0,0)\), directrix: \(x=-\frac{1}{4}\)
5. Focus: \(\left(\frac{-1}{12},0\right)\), vertex: \((0,0)\), directrix: \(x=\frac{1}{12}\)
7. \((0,0)\) and \((4p,4p)\); \(y=x\)
9. \(\abs{4p}\)
11. \((3p,\pm 2\sqrt{3}p)\)
16. Focus: \(\left(\frac{-b}{2a},\frac{4ac - b^2 +1}{4a}\right)\),
vertex: \(\left(\frac{-b}{2a},\frac{4ac - b^2}{4a}\right)\), directrix: \(y=\frac{4ac - b^2 -1}{4a}\)
Section 7.3 (p. § 7.3)
2. Foci: \((\pm 5,0)\), vertexes: \((\pm 4,0)\), directrices: \(x=\pm\frac{16}{5}\),
asymptotes: \(y=\pm\frac{3}{4}x\), \(e=\frac{5}{4}\)
3. Foci: \((\pm \sqrt{23},0)\), vertexes: \((\pm 2\sqrt{2},0)\),
directrices: \(x=\pm\frac{8}{\sqrt{23}}\),
asymptotes: \(y=\pm\frac{\sqrt{15}}{2\sqrt{2}}x\),
\(e=\frac{\sqrt{23}}{2\sqrt{2}}\)
4. Foci: \((\pm\frac{\sqrt{41}}{2},0)\), vertexes: \((\pm\frac{5}{2},0)\),
directrices: \(x=\pm\frac{25}{2\sqrt{41}}\), asymptotes: \(y=\pm\frac{4}{5}x\),
\(e=\frac{\sqrt{41}}{5}\)
5. Foci: \((\pm\frac{\sqrt{5}}{2},0)\), vertexes: \((\pm 1,0)\),
directrices: \(x=\pm\frac{2}{\sqrt{5}}\), asymptotes: \(y=\pm\frac{1}{2}x\), \(e=\frac{\sqrt{5}}{2}\)
6. Foci: \((0,\pm \sqrt{34})\), vertexes: \((0,\pm 3)\),
directrices: \(y=\pm\frac{9}{\sqrt{34}}\), asymptotes: \(y=\pm\frac{3}{5}x\), \(e=\frac{\sqrt{34}}{3}\)
7. \(\frac{x^2}{9} - \frac{y^2}{16}=1\)
17. \(\frac{x^2}{302500} - \frac{y^2}{697500}=1\)
Section 7.4 (p. § 7.4)
1. Foci: \((0,2)\) and \((6,2)\), vertexes: \((-2,2)\) and \((8,2)\)
3. Foci: \((-3,1 \pm 2\sqrt{3})\), vertexes: \((-3,-3)\) and \((-3,5)\)
5. Focus: \((-3,-\frac{239}{16})\), vertex: \((-3,-15)\), directrix: \(y=-\frac{241}{16}\)
7. Focus: \(\left(\frac{1}{2},\frac{5}{4}\right)\),
vertex: \(\left(\frac{1}{2},\frac{3}{2}\right)\), directrix: \(y=\frac{7}{4}\)
9. Foci: \((-1 \pm \sqrt{13},-3)\), vertexes: \((-4,-3)\) and \((2,-3)\),
directrices: \(x=-1 \pm \frac{9}{\sqrt{13}}\), asymptotes: \(y=\pm\frac{2}{3}(x+1)-3\)
11. Foci: \((\sqrt{2},\sqrt{2})\) and \((-\sqrt{2},-\sqrt{2})\),
vertexes: \((1,1)\) and \((-1,-1)\), directices: \(y=-x \pm \sqrt{2}\), asymptotes: \(x=0\) and \(y=0\)
15. hyperbola
Section 7.5 (p. § 7.5)
12. Hint: Use Exercise 11.
24. local maximum at \(x=\ln \sqrt{3}\), inflection pt at \(x=\ln 3\),
horizontal asymptote: \(y=0\)
26. (b) Hint: See Exercise 10.
27. (b) \(k_1=c_1+c_2\), \(k_2=c_1-c_2\)
30. \(s_0 \approx 1.006237835313385\),
\(e^{\pi/s_0} = 22.69438187638412\)
33. (a) \(x=\frac{cx+cy}{2}-\frac{y-x}{2c}\), \(y=\frac{cx+cy}{2}+\frac{y-x}{2c}\)
(b)
\(c=\cosh\,a + \sinh\,a\)
(c) Hint: Use Example 7.12.
Section 7.6 (p. § 7.6)
5. Yes
6. (a) Hint: Solve for \(t\) in terms
of \(x\) then substitute into \(y\).
(b) Hint: Use the distance formula.
8. Hint: Does \(BP = \wideparen{AB}\)?
9. (a) \(-\frac{7}{2(t+1)^3}\)
(c) \(\left(\frac{49}{25},\frac{14}{5}\right)\)
12. \(x=t^2-1\), \(y=t(t^2-1)\)
Section 7.7 (p. § 7.7)
1. \(r = 6\,\cos\,\theta\)
3. \(r^2 = \sec\,2\theta\)
5. \(\theta = \frac{3\pi}{4}\)
7. \(r = \sec\,\theta\;\tan\,\theta\)
9. \(x^4 + 2x^2y^2 + y^4 = 4x^2 - 4y^2\)
11. \(y - 1 = -\frac{3\sqrt{3}}{5}\,(x + \sqrt{3}\,)\)
13. local maxima at \(\left(2,\frac{\pi}{2}\right)\) and
\(\left(0,\frac{3\pi}{2}\right)\),
local minima at \(\left(\frac{1}{2},\frac{7\pi}{6}\right)\)
and \(\left(\frac{1}{2},\frac{11\pi}{6}\right)\)
15. local maxima at \(\left(\frac{2\sqrt{2}}{3},\alpha\right)\) and
\(\left(-\frac{2\sqrt{2}}{3},2\pi - \alpha\right)\), local minima at
\(\left(-\frac{2\sqrt{2}}{3},\pi - \alpha\right)\) and
\(\left(\frac{2\sqrt{2}}{3},\pi + \alpha\right)\),
where \(\alpha = \tan^{-1}\sqrt{2}\)
17. \(\frac{\pi}{2}\)
Chapter 8
Section 8.1 (p. § 8.1)
1. \(\frac{32}{3}\)
2. \(\frac{4}{3}\)
3. \(\frac{1}{12}\)
5. \(\frac{9}{2}\)
7. \(\frac{19}{3}\)
9. \(\frac{1}{12}\)
11. \(3\pi\)
13. \(25\left(\frac{\pi}{3}+1-\sqrt{3}\right)\)
17. \(\frac{6250\pi^3}{3}\) sq ft
Section 8.2 (p. § 8.2)
1. \(1\) 3. \(\frac{4}{3}\) 5. \(\frac{2}{\pi}\) 7. \(0\) 9. \(\frac{1}{2} \ln 3\)
Section 8.3 (p. § 8.3)
1. \(\frac{8}{27} (10^{3/2}) - \frac{1}{27} (13^{3/2}) \approx 7.634\)
2. \(\frac{\sqrt{5}}{2} + \frac{1}{4} \sinh^{-1} 2 \approx 1.479\)
3. \(\frac{8}{27} (10^{3/2}) - \frac{1}{27} (13^{3/2}) \approx 7.634\)
4. \(\frac{3}{4} + \ln \sqrt{2} \approx 1.097\)
7. \(\sqrt{2} (e^{\pi}-1) \approx 31.312\)
9. \(8\)
11. \(3\)
13. \(\kappa(0)=0\), \(\kappa(\frac{\pi}{2})=-1\)
15. \(\kappa(0)=\frac{a}{b^2}\) at \((a,0)\), \(\kappa(\frac{\pi}{2})=\frac{b}{a^2}\) at \((0,b)\)
17. \(-1\)
23. \(13.27\) ft
Section 8.4 (p. § 8.4)
1. \(4\pi\)
2. \(\frac{\pi}{2}(2 + \sinh 2)\)
3. \(\frac{208\pi}{9}\)
5. \(\frac{\pi^2}{2}\)
7. \(2\pi\)
9. \(\frac{\pi}{10}\)
10. \(\frac{\pi}{6}\)
13. \(S=\pi r\sqrt{r^2+h^2}\), \(V=\frac{1}{3}\pi r^2h\)
Section 8.5 (p. § 8.5)
1. \(\left(\frac{4}{5},\frac{2}{7}\right)\)
3. \(\left(\frac{3}{5},\frac{12}{35}\right)\)
5. \(\left(\frac{4r}{3\pi},\frac{4r}{3\pi}\right)\)
7. \(\left(0,\frac{11}{4\pi}\right)\)
9. \(0.192\) Nm
11. \(RT\,\log\left(\frac{V_b}{V_a}\right)\)
13. \(0.3486\)
15. \(\left(1,\frac{1}{4}\right)\)
18. Hint: Use Exercise 28 in Section 6.1.
20. Hint: Use the equation from Section 5.1 for free fall motion to
write time as a function of height.
Chapter 9
Section 9.1 (p. § 9.1)
1. Converges to \(0\)
2. Converges to \(\frac{1}{3}\)
3. Converges to \(0\)
5. Divergent
7. Divergent
9. \(6\)
11. \(32\)
13. \(\frac{113}{999}\)
14. \(1\)
15. \(\frac{1}{4}\)
20. \(\frac{132}{7}\) ft
24. No
Section 9.2 (p. § 9.2)
6. Divergent
7. Convergent
8. Divergent
9. Convergent
10. Divergent
11. Convergent
12. Convergent
13. Convergent
14. Divergent
15. Divergent
16. Convergent
17. Convergent
18. \(\frac{1}{6}\)
19. \(\frac{1}{10}\)
20. \(\frac{1}{6}\)
21. \(\frac{1}{2}\)
Section 9.3 (p. § 9.3)
1. Conditionally convergent
3. Conditionally convergent
5. Absolutely convergent
7. Answer to second question: Yes
Section 9.4 (p. § 9.4)
1. \(-1\le x<1\)
2. \(-2<x<2\)
3. \(1<x<3\)
4. \(-6<x<-2\)
5. \(-\infty<x<\infty\)
6. \(x=0\)
7. \(-1<x<1\)
10. Hint: Use Example 9.19
Section 9.5 (p. § 9.5)
1. \(1 - \frac{(x-\frac{\pi}{2})^2}{2!} + \frac{(x-\frac{\pi}{2})^4}{4!} - \cdots\)
2. \(x + \frac{x^3}{3!} + \frac{x^5}{5!} + \cdots\)
3. \(1 + \frac{x^2}{2!} + \frac{x^4}{4!} + \cdots\)
4. \(x + \frac{x^3}{3} + \frac{2x^5}{15} + \cdots\)
5. \(x - \frac{x^3}{3} + \frac{2x^5}{15} - \cdots\)
6. \(1 + \frac{x^2}{2} + \frac{5x^4}{24} + \cdots\)
7. \(1 - x^2 + x^4 - \cdots\)
8. \(\frac{1}{2} - \frac{(x-1)}{2} + \frac{(x-1)^2}{4} - \cdots\)
9. \(1 + \frac{x^2}{2} - \frac{x^4}{8} + \cdots\)
11. \(x^2 - \frac{x^4}{2} + \frac{3x^6}{8} - \cdots\)
13. \(x - \frac{x^5}{10} + \frac{x^9}{216} - \cdots\)
15. \(x + \frac{x^7}{14} - \frac{x^{13}}{104} + \cdots\)
18. \(0.68485\)
19. \(97.18\); \(-132.605\)