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Chapter 4, Answers

Chapter 4

4.2.3

  1. \(\frac{\pi}{3}\).
  2. \(\frac{1}{16}\).
  3. \(\frac{\pi^2}{2}\).
  4. \(\frac{1}{4y}\).
  5. \(-\frac{3}{4}\).
  6. \(\frac{1}{2\sqrt{a}}\).
  7. \( -\frac{1}{\pi^2} \).
  8. \(\frac{1}{8\sqrt{\pi^7}} \).
  9. \(\frac{2}{3\pi} \).
  10. \(\frac{1}{4\sqrt[4]{a^3}}\).

4.3.5

  1. (b)(i);
    \( \tan (x+h)=\frac{\sin (x+h)}{\cos (x+h)}
    =\frac{\sin x \cos h + \sin h \cos x}{\cos x \cos h-\sin x \sin h}
    \underset{(*)}{=}
    {\displaystyle\frac{ \frac{\sin x \cos h+ \sin h \cos x}{\cos x \cos h}}{ \frac{\cos x \cos h-\sin x \sin h}{\cos x \cos h}}}
    =\frac{\tan x + \tan h}{1-\tan x \tan h}\).
    \(^{(*)}\) It is natural to divide numerator and denominator by \(\cos x \cos h\) because the aim is to get a new expression for \(\tan (x + h)\) in terms of \(\tan x\) and \(\tan h\).
    (ii) See Example 1 in 4.1.2.
  1. \(y=x-\frac{5\pi}{6}-\frac{\sqrt{3}}{2}\).
  2. \(y= 4\pi x+\frac{4\pi}{3}-\sqrt{3} \).
  3. \(y=x \)
  4. \( y=\pi x-\pi\)
  5. \(y=\sqrt{2}\pi x+\frac{\pi \sqrt{2}}{4} \).
  6. \(y=-\frac{10}{3}x-\frac{2\pi}{9}-\frac{2\sqrt{3}}{3} \).
  7. \(\pi \sec \pi x(2\sec^2\pi x-1) \).
  8. \( -\csc 3x (3\cos x \cot 3x +\sin x)\).
  9. \( \frac{\pi}{1+\cos \pi x}\).
  10. \( \frac{2\cos 2x}{(1+\sin 2x)^2}\).
  11. \( \frac{3}{1+\cos 3x}\).
  12. \( \frac{3}{1+\sin 3x}\).
  13. \(x\in \left \{-\frac{5\pi}{6}, -\frac{\pi}{2}, -\frac{\pi}{6}, \frac{\pi}{2}\right \}\).
  14. \(x\in \left \{-\frac{2\pi}{3}, 0, \frac{2\pi}{3}\right \}\).
  15. \(x\in \left \{-\frac{\pi}{6}, \frac{5\pi}{6}\right \}\).
  16. \(x\in \left \{-\frac{2\pi}{3}, \frac{2\pi}{3}\right \}\).
  17. (a) \(e^x+xe^x,2e^x+xe^x, 3e^x+xe^x, 4e^x+xe^x\);
    (b) \(f^{(10)}(x)=10e^x+xe^x\);
    (c) \(f^{(n)}(x)=n e^x+xe^x\), \(n\in \mathbb{N}\).
  18. (a) and (b)

    \(\begin{array}{ll}
    f'(x) = x\cos x + \sin x& f”(x) = 2\cos x -x\sin x\\
    f^{(3)}(x) = -x\cos x -3 \sin x& f^{(4)}(x) = -4\cos x +x\sin x\\
    f^{(5)}(x) = x\cos x + 5 \sin x& f^{(6)}(x) = 6\cos x -x\sin x\\
    f^{(7)}(x) = -x\cos x -7 \sin x& f^{(8)}(x) = -8\cos x +x\sin x\\
    f^{(9)}(x) = x\cos x + 9 \sin x& f^{(10)}(x) = 10\cos x -x\sin x
    \end{array}
    \)
    \(\begin{array}{l}
    f^{(19)}(x) = -x\cos x -19 \sin x\\ f^{(20)}(x) = -20\cos x +x\sin x\\
    f^{(101)}(x) = x\cos x + 101 \sin x\\ f^{(102)}(x) = 102\cos x -x\sin x\\
    \end{array}
    \)

    (c) for odd order derivatives: \(f^{(2n-1)}(x)= (-1)^{n-1}x\cos x + (-1)^{n-1}(2n-1) \sin x \), \(n\in \mathbb{N}\);

    (c) for even order derivatives: \(f^{(2n)}(x)= (-1)^{n-1}(2n) \cos x + (-1)^{n}x \sin x \), \(n\in \mathbb{N}\).

4.4.4

  1. \(-6k \cos^5 ka \,\sin ka \).
  2. \( -2kx \cot^3kx^2 \, \csc^2k x^2\).
  3. \( \frac{1}{2}\frac{1}{\sqrt{x}}\cos \sqrt{x}\).
  4. \( \frac{1}{3}\frac{1}{\sqrt[3]{x^2}}\sec^2\sqrt[3]{x}\).
  5. \(-4x^3\csc x^4 \cot x^4\).
  6. \( \frac{1}{4}\frac{1}{\sqrt[4]{x^3}}\sec \sqrt[4]{x} \tan \sqrt[4]{x}\).

4.5.2

  1. \(f(x) =\cot x , g(x) =e^{-x} \).
  2. \(f(x) = -2\ln x, g(x) =x^2+4 \).
  3. \(f(x) =x^2 , g(x) =2\ln x \).
  4. \(f(x) = x^2, g(x) =\arcsin x \).
  5. \(f(x) =e^x , g(x) = \sqrt[3]{x^2}\).
  6. \(f(x) =\sqrt[3]{x} , g(x) =1+x^2 \).
  7. \(f(x) =\arccos x , g(x) = e^x, h(x) = -x^2\).
  8. \(f(x) = x^3, g(x) =\frac{1}{3}\ln x , h(x) = x^2+4\).
  9. \(f(x) = \csc x, g(x) = e^x , h(x) =\sqrt[3]{x^2} \).
  10. \(f(x) =x^2 , g(x) = \tan x, h(x) = (1+x^2)^{1/3}\).
  11. \(f(x) = x^3, g(x) =\arccos x, h(x) =\sin^2 \pi x \).
  12. \(f(x) = x^2, g(x) =\arcsin x , h(x) =(1-x^2)^3 \).

4.5.4

  1. \(-8\cos 4x \sin 4x\).
  2. \(6x \sec ^2\left(3x^2+5\right)\).
  3. \(\frac{x}{2\sqrt{(x^2+x)^3}}\).

  4. \(\cot 8x^2 – 16x^2 \csc^2 8x^2\).
  5. \( \frac{2 x^2+1}{\sqrt{x^2+1}}\).
  6. \( \frac{x}{\sqrt{1+x^2}}\).
  7. \(\frac{ 3\pi}{5}\frac{ \cos \pi x}{\sqrt[5]{\sin^2\pi x}}\).
  8. \(\frac{2\pi}{3} \sqrt[3]{\sec ^2 \pi x} \,\tan \pi x \).;
  9. \(\frac{1+ 2\sqrt{x+1}}{4\sqrt{x+1}\sqrt{(x+\sqrt{x+1})^3}}\).
  10. \(\pm\frac{1}{\sqrt{2}}\).
  11. \(2\).
  12. \(x\in \left \{\pm \frac{\pi}{2}, \pm \frac{\pi}{6}\right \}\).

4.5.6

  1. \(\frac{2 (2 x+1) \tan \left(\sqrt[3]{x (x+1)}\right) \sec ^2\left(\sqrt[3]{x (x+1)}\right)}{3 (x (x+1))^{2/3}}\).
  2. \(\frac{x (4 x+3) \cos \left(x \sqrt{x (x+1)}\right)}{4 \sqrt{x (x+1)} \sqrt{\sin \left(x \sqrt{x (x+1)}\right)}}\).
  3. \(\frac{x}{2 \sqrt{x^2+1} \sqrt{\sqrt{x^2+1}+1}}\).
  4. \(\frac{x \cos \left(\sqrt{\sqrt{x^2+1}+1}\right)}{2 \sqrt{x^2+1} \sqrt{\sqrt{x^2+1}+1}}\).
  5. \(-\frac{3}{2} \pi x \cot \left(\pi x^2\right) \sqrt[4]{\csc ^3\left(\pi x^2\right)}\).
  6. \(\frac{x \left(\left(6 \sqrt{x^2+1}+5\right) x^2+4\right)}{2 \sqrt{x^2+1} \sqrt{x^2+\sqrt{x^2+1}}}\).
  7. (a) and (b) follow from the Chain Rule; (c) (i) \(f(-x^2)(1-2 x^2)\), (ii) \(x=\pm \frac{1}{\sqrt 2}\);
    (d) (i) \(\frac{f(-2x)(8 x^2-10 x+3)}{2 ((3 – 2 x)^2) \sqrt x}\), (ii) \(x\in \left \{\frac{1}{2}, \frac{3}{4}\right \}\).
  8. Use \((f^{-1}\circ f)(x) = x\) and the chain rule
  9. (a) \(g(x) = \sqrt{x}\), \(h(x) = 1-\frac{x+1}{x-1}\), \((-\infty, 1)\); (b) \(\frac{1}{\sqrt{2}\sqrt{(1-x)^3}}\); (c) \(\frac{1}{2}\).

4.5.8

  1. \(y = -\frac{1}{5}x+\frac{8}{5}\).
  2. \(y = 4x+2\).
  3. \(y =\frac{1}{5}x+\frac{6\sqrt{2}}{5} \).
  4. \(y =2-x \).
  5. \(y = – x -1 \).
  6. \(y =-\frac{ 3}{19}x+ \frac{60}{19} \).

  1. (a) \(y’= \frac{4x-y}{x-2y}\); (b) \(x\)-int.: \((\pm 2\sqrt2,0\); (c) \(y\)-int.: \((0,\pm 4)\); (d) \(\left (\pm \frac{8}{\sqrt7},\pm \frac{4}{\sqrt7}\right )\); (e) \(\left (\pm \frac{2\sqrt2}{\sqrt7},\pm \frac{8\sqrt2}{\sqrt7}\right )\).
  2. (a) \(y’= \frac{2(x-y)}{2x-y}\); (b) \((\pm 1,\pm 2)\); (c) \((\pm \sqrt2,\pm \sqrt2)\); (d) \(m=2-\frac{2\sqrt{6}}{3}\);
    (e) (i) \( f(x) = 2x+\sqrt 2 \sqrt{x^2-1}\); (ii) \(f'(x) = 2x + \sqrt{2}\sqrt{1-x^2}\), \(f'(2) =2-\frac{2\sqrt{6}}{3}\).
  3. \(\left (\frac{8}{5}, -\frac{9}{5}\right )\).
  4. \(\left (\frac{1}{4}, \pm\frac{\sqrt{15}}{2}\right )\).
  5. \(-3\).
  6. (a) \(SA=2 x^2 + 4 x y\); (b) \(-\frac{5}{2}\).
  7. (a) \(SA= 3\pi r^2 + 2\pi r h\); (b) \(-\frac{11}{2}\).
  8. (a) \(V= x^2y\); (b) \(-3\).
  9. (a) \(V=\pi r^2h+2\pi r^3\); (b) \(-10\).

4.6.2

  1. \(\frac{32}{3} \, \frac{\text{cm}^2}{\text{min}}\).
  2. \(4\pi\,\frac{\text{cm}^2}{\text{min}}\).
  3. \(800\pi\,\frac{\text{cm}^3}{\text{min}}\).
  4. \(45\pi\, \frac{\text{cm}^2}{\text{min}}\); increasing.
  5. \(-\frac{1}{9}\frac{\text{cm} }{\text{min}}\).

4.6.4

  1. (a) \(\frac{\sqrt{5}}{12} \)ft/sec; (b) \(\frac{\sqrt{6}}{60}\) rad/sec.
  2. (a) \(-390\) mi/h; (b) \(-\frac{2}{5}\) rad/h.
  3. (a) \(\frac{15}{2} \) ft/sec; (b) \(-\frac{3}{29}\) rad/sec.
  4. (a) \(5\sqrt{29}\) ft/sec; (b) \(\frac{135}{\sqrt{26}}\) ft/sec.
  5. \(\frac{1}{2\sqrt{\pi}} \) cm.
  6. \(14\) cm/min.
  7. \(\frac{3\sqrt{3}}{8} \) ft\(^2\)/sec.
  8. (a) \(-\frac{128}{125 \pi} \) ft/min; (b) \(-6\) ft\(^2\)/min ; (c) \(\frac{25\pi}{16}\) ft\(^3\)/min.
  9. \(\frac{2}{3} \) m/min.
  10. \(\frac{3}{2} \) m/sec.
  11. (a) \(\frac{2}{45 pi} \) ft/min; (b) \(75 \pi\) ft\(^3\)/min.
  12. \(-\frac{400}{1681}\) rad/sec.
  13. \(\frac{8}{25}\) rad/sec.
  14. \(100\pi \) mi/min.
  15. \(\frac{21}{25} \) rad/sec.
  16. \(-\frac{3\sqrt{26}}{13}\) cm/sec.
  17. \(10\sqrt{17} \) cm/sec.
  18. \(30\sqrt{3} \) cm/min.


4.7.5

  1. (a) \((0,\infty) \); (b) \(\frac{1}{2(x+1)\sqrt x}\).
  2. (a) \((-\infty, -1]\cup(1,\infty) \); (b) \(-\left (\sec^{-1}x^2\right )^{-2}\frac{2}{x^3\sqrt{x^4-1}}\).
  3. (a) \(\left [-\frac{1}{2}, \frac{1}{2}\right ] \); (b) \(-2 -\frac{4\arccos 2x}{\sqrt{1-4x^2}}\).
  4. (a) \((-\infty, \infty) \); (b) \(\displaystyle \frac{2-8x\arctan 2x}{(1+4x^2)^2}\).
  5. (a) \([-2,2]\); (b) \(\frac{\sqrt{4-x^2} +x\arcsin(x/2)}{(4-x)\sqrt{4-x^2}}\).
  6. (a) \(\left (-\infty, \frac{1}{2}\right ] \); (b) \(\frac{1}{(x-1)\sqrt {1-2x}}\).
  7. (a) \(\left [\frac{1}{2}, \infty\right ) \); (b) \(-\frac{1}{(x+1)\sqrt {1+2x}}\).
  8. (a) \(\left [\frac{1}{2}, 1\right ] \); (b) \(\frac{1}{2(1-x)\sqrt{2(1-x)(2x-1)}}\).
  9. (a) \((-1,0) \); (b) \(-\frac{3}{2}\sqrt{-x}\).
  10. (a) \(\left [-\frac{3}{2}, \infty\right ) \); (b) \(-\frac{5(x+2)(3x+2)\sqrt{3x+2}}{(x+1)^6}\).
  11. \(\frac{12(x^2(x-1)}{1+\left(3x^4-4x^3\right)^2}\).
  12. (a) \(D_{\cot^{-1}}=\mathbb{R}\), \(R_{\cot^{-1}}= \left (-\frac{\pi}{2}, \frac{\pi}{2}\right )\); (b) \(-\frac{1}{1+x^2}\).
  13. (a) \(D_{\csc^{-1}}=(-\infty, -1\cup(1,\infty)\), \(R_{\cot^{-1}}= \left (-\frac{\pi}{2},0)\cup(0, \frac{\pi}{2}\right )\); (b) \(-\frac{1}{|x|\sqrt{x^2-1}}\).
  14. (a) \(D_{\text{Sec}^{-1}}=(-\infty, -1\cup(1,\infty)\), \(R_{\text{Sec}^{-1}}= \left (0,\frac{\pi}{2}\right )\cup \left (\frac{3\pi}{2},\pi\right )\);
    (b) \(\frac{1}{x\sqrt{x^2-1}}\) (no absolute value because of the sign of \(\tan y\) .
  15. \(x= -1 \). The equation to solve is \(2 x^3+11 x^2+20 x+11=0\), and \(x=-1\) is one integer root.
    Show it is the only real one.
  16. \(y =-2x-2 \).
  17. \(y =\frac{\pi-1}{\pi -2}x +\frac{3-2\pi}{2(\pi -2)}\).
  18. \(y =\frac{\sqrt{3} \pi}{2(\sqrt{3} \pi-3)} x – \frac{3(\sqrt{3} \pi-2)}{2(\sqrt{3} \pi-3)}\).
  19. \(y =-2x+2\).
  20. \(y =1 \).
  21. \(y =-x+2 \).

4.8.5

  1. \(-2 x e^{-x^2}\sec^2e^{-x^2}\).
  2. (a) \(\cosh x\); (b) \(\sinh x\).
  3. \(\frac{9\ln t (\ln t +4)}{2\sqrt{t}}\).
  4. \(2x+\frac{2}{x}\).
  5. \(\frac{3 x \ln^2\left (x^2+1\right )}{4\left ( x^2+1\right )}\).
  6. \(\frac{e^{2x}\left (\sqrt{1-e^{2x}}-e^{2x}-1\right )}{e^{2x}\left (\sqrt{1-e^{2x}}-1\right )+1}\).
  7. \(\frac{1}{2\sqrt{x} (x+1)\arctan(1/\sqrt x)}\).
  8. \(-\frac{4e^{2 x}}{e^{4 x}-1}\).
  9. \(1\).
  10. \(\frac{64}{x}\ln^3|x|\exp\left (16\ln^4|x|\right )\).
  11. \(x\in \left \{\frac{1}{2}, \frac{3}{4}\right \}\).
  12. \(x=\pm\frac{1}{2}\).
  13. \(x= \pm \sqrt 2\).
  14. \(x\in \left \{\frac{1}{e^2},1\right \}\).
  15. \(D_g=\left (-\sqrt{\ln 2}, \sqrt{\ln 2}\right )\); \(\frac{2xe^{x^2}}{e^{x^2}-2}\).
  16. \(D_f=\left (-\infty, -e \sqrt[2]{e}\right ) \cup\left ( e \sqrt[2]{e}, \infty\right )\); \(\frac{3}{x\ln|x|\sqrt{4\ln^2|x|-9}}\).
  17. \(D_h=(-1,1)\); \(-\frac{2}{x^2-1}\).
  18. \(D_g=(e,\infty)\); \(\frac{1}{x \ln( x) \ln (\ln x)}\).


4.8.8

  1. \(\frac{1}{2^{2x+1}\ln 2+x\ln 2}\).
  2. \(\frac{3x^2+2}{(x^3+x)\ln 10}\).
  3. \(\ln 2\left ( 2^{\sin t} \cos t – 2\cdot 4^{\cos t}\sin t \right )\).
  4. \(\ln \pi \cdot \pi^{\tan s}\sec^ s + \frac{x^{-\ln 10}}{x\ln 10}\).
  5. \(x^(x-1) \frac{x+x (x+1) \ln^2 x+(x-1) x \ln x +1}{(x+1)^3}\).
  6. \(\arctan \sqrt{x}\cdot \frac{(x-1)\sqrt{x}-(x+1)^2\arctan \sqrt{x}}{(x-1)^3 (x+1)}\).
  7. \(\frac{1}{4}\left (\sqrt x\right )^{\sqrt{x}-1}(\ln x + 2)\).
  8. \(\frac{2 x ((3 x+4) \ln \left [x^3 (3 x+4))+6 (x+1)\right ]}{ (3x+4)\ln 2}\).
  9. \(\frac{e^{\arcsin x}}{\sin^3 \pi x}\left (\frac{\ln x}{\sqrt{1-x^2}}+\frac{1}{x}- 3\pi \ln x \cot \pi x\right )\).
  10. \( \frac{x^{\ln x-1}}{3(x^2+ 1)2^x\sqrt[3]{x^2+ 1}}\left (6 (x^2+1) \ln x-x (3x^2\ln 2+2 x+3\ln 2)\right ) \).
  11. \( \left (\ln x\right )^ x\left (\ln(\ln x)+ \frac{1}{\ln x}\right )\).
  12. \(\frac{4^{x^2}x}{\sqrt{1+\csc^2 \pi x}}\left [4 x^2 \ln 2 + 2 + \csc^2 \pi x\left (4x^2\ln 2-\pi x \cot \pi x+2\right )\right ]\).
  13. \(y = \frac{16}{9\ln 3}x + 3 – \frac{32}{9\ln 3}\).
  14. \(y =\frac{2\ln 2-1}{4}x+ 3-2\ln 2 \).
  15. \(y = \frac{4\ln 2}{\sqrt{e}}x + 2 -4\ln 2\).
  16. \(y = \frac{\sqrt 3}{2\pi \ln \pi}x+ 1 – \frac{\sqrt 3}{\pi \ln \pi}\).
  17. (a) \(-\frac{\ln 2}{x\ln^2x}\); (b) \(-\frac{\ln x -1}{\ln^2x}\).
  18. (a) \(x^{x^x}x^x \left (\ln^2x + \ln x + \frac{1}{x}\right )\);

    19. (b) \(-x^{1/x^x}x^{-x} \left (\ln^2x + \ln x – \frac{1}{x}\right )\).

  19. \(x\in \left \{ -\frac{1}{12}, 1\right \}\).
  20. \(x = e^{-1}\).
  21. \(x\in \left \{ 0, \pm\frac{1}{\sqrt{\ln 2}} \right \}\).
  22. \(x=e^{-1/2}\).
  23. \(e^{1/e}\).
  24. \(\left \{e^{\pm u}: u=0\; \text{ or } \; u = \sqrt{\frac{2k-1}{2}}, \; k \in \mathbb{N}\right \}\).
  25. (b) \(\frac{e^y (e^x-2 y cosh(x y))}{2 x e^y cosh(x y)-1}\); (c) \(y=-\frac{1}{e^2}x + 1+\frac{1}{e^2} \).
  26. (b) \(\frac{y (y x^y-x y^x \ln y)}{x (x y^x-y x^y \ln x)}\); (c) \(y=x\).
  27. (b) \(\frac{y (y x^{y/2}-2 x y^{x/2} \ln y)}{x (2 x y^{x/2}-y x^{y/2} \ln x)}\); (c) \(y=\frac{1}{16 – 8 \ln 2} x + 1-\frac{1}{4 – 2 \ln 2} \).
  28. (b) \(\frac{y\left (3y-\pi \sqrt{1 – x^2 y^2}\right )}{\pi \sqrt{1 – x^2 y^2}-3xy}\); (c) \(y=\frac{6-\sqrt{3} \pi}{-3+\sqrt{3} \pi}x+ \frac{3(\sqrt{3} \pi-4)}{2(\sqrt{3} \pi-3)} \).
  29. (b) \(-\frac{\pi (9^y+1)}{\pi 9^y + 3^{y + 1} \ln 3 +\pi}\); (c) \(y=-\frac{4\pi}{4\pi+ 3\sqrt{3}\ln 3}x- \frac{1}{2}+\frac{4\pi}{4\pi+ 3\sqrt{3}\ln 3}\).
  30. (b) \(\frac{\pi y^2\sqrt{4+\ln^2 y}}{2+\pi xy\sqrt{4+\ln^2 y}}\); (c) \(y=-\frac{\pi}{1+\pi}x+ \frac{1+2\pi}{1+\pi }\).
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