Appendix A:\(\quad\)Answers and Hints to Selected Exercises

Chapter 1

Section 1.1 (p. § 1.1)

1. (a) \(\sqrt{5}\) (b) \(\sqrt{5}\) (c) \(\sqrt{17}\) (d) \(1\)
(e) \(2 \sqrt{17}\) 2. Yes 3. No

Section 1.2 (p. § 1.2)

1. (a) \((-4,4,-3)\) (b) \((2,6,-1)\)
(c) \(\left ( \frac{-1}{\sqrt{30}},\frac{5}{\sqrt{30}}, \frac{-2}{\sqrt{30}}\right )\) (d) \(\frac{\sqrt{41}}{2}\) (e) \(\frac{\sqrt{41}}{2}\)
(f) \((14,-6,8)\) (g) \((-7,3,-4)\)
(h) \((-1,-6,1)\) (i) \((-2,-4,2)\) (j) No.
3. No. \(\norm{\textbf{v}} + \norm{\textbf{w}}\) is larger.

Section 1.3 (p. § 1.3)

1. \(10\) 3. \(73.4\Degrees\) 5. \(90\Degrees\) 7. \(0\Degrees\)
9. Yes, since \(\Dotprod{\textbf{v}}{\textbf{w}} = 0\).
11. \(\abs{\Dotprod{\textbf{v}}{\textbf{w}}} = 0 < \sqrt{21}\sqrt{5} = \norm{\textbf{v}}\,\norm{\textbf{w}}\)
13. \(\norm{\textbf{v} + \textbf{w}} = \sqrt{26} < \sqrt{21} + \sqrt{5} = \norm{\textbf{v}} + \norm{\textbf{w}}\)
15. Hint: use Definition 1.6.
24. Hint: See Theorem 1.10(c).

Section 1.4 (p. § 1.4)

1. \((-5,-23,-24)\) 3. \((8,4,-5)\) 5. \(\textbf{0}\)
7. \(16.72\) 9. \(4\sqrt{5}\) 11. \(9\) 13. \(0\)
and \((8,-10,2)\) 15. \(14\)

Section 1.5 (p. § 1.5)

1. (a) \((2,3,-2) + t(5,4,-3)\) (b) \(x = 2 + 5t,\)
\(y = 3 + 4t, \, z = -2 - 3t\) (c) \(\frac{x - 2}{5} = \frac{y - 3}{4} = \frac{z + 2}{-3}\)
3. (a) \((2,1,3) + t(1,0,1)\) (b) \(x = 2 + t,\)
\(y = 1, \, z = 3 + t\) (c) \(x - 2 = z - 3, \, y = 1\)
5. \(x = 1 + 2t, \, y = -2 + 7t, \, z = -3 + 8t\)
7. 7.65 9. \((1,2,3)\)
11. \(4x - 4y + 3z - 10 = 0\)
13. \(x - 2y - z + 2 = 0\)
15. \(11x - 24y + 21z - 26 = 0\) 17. \(9/\sqrt{35}\)
19. \(x = 5t\), \(y = 2 + 3t\), \(z = -7t\) 21. \((10,-2,1)\)

Section 1.6 (p. § 1.6)

1. radius: \(1\), center: \((2,3,5)\) 3. radius: \(5\), center: \((-1,-1,-1)\) 5. No intersection.
7. circle \(x^2 + y^2 = 4\) in the planes \(z = \pm \sqrt{5}\)
9. lines \(\frac{x}{a} = \frac{y}{b}\), \(z = 0\) and \(\frac{x}{a} = -\frac{y}{b}\), \(z = 0\)
13. \(\left( \frac{2a}{2 - c}, \frac{2b}{2 - c}, 0 \right)\)

Section 1.7 (p. § 1.7)

1. (a) \((4,\frac{\pi}{3},-1)\) (b) \((\sqrt{17},\frac{\pi}{3},1.816)\)
3. (a) \((2\sqrt{7},\frac{11\pi}{6},0)\) (b) \((2\sqrt{7},\frac{11\pi}{6},\frac{\pi}{2})\)
5. (a) \(r^2 + z^2 = 25\) (b) \(\rho = 5\)
7. (a) \(r^2 + 9z^2 = 36\) (b) \(\rho^2 ( 1 + 8 \cos^2 \phi ) = 36\)
10. \((a,\theta,a \cot \phi )\) 12. Hint: Use the distance formula for Cartesian coordinates.

Section 1.8 (p. § 1.8)

1. \(\textbf{f}\,'(t) = (1,2t,3t^2)\); \(x = 1 + t\), \(y = z = 1\)
3. \(\textbf{f}\,'(t) = (-2\sin 2t,2\cos 2t,1)\); \(x = 1\),
\(y = 2t\), \(z = t\) 5. \(\textbf{v}(t) = (1,1 - \cos t,\sin t)\),
\(\textbf{a}(t) = (0,\sin t,\cos t)\)
9. (a) Line parallel to c (b) Half-line parallel to c (c) Hint: Think of the
functions as position vectors.
15. Hint: Theorem 1.16

Section 1.9 (p. § 1.9)

1. \(\frac{3\pi \sqrt{5}}{2}\) 3. \(2 (5^{3/2} - 8)\) 5. Replace
\(t\) by \(\biggl( \frac{s + 16}{2} \biggr)^{2/3} - 4\) 6. Hint: Use
Theorem 1.20(e), Example 1.37, and
Theorem 1.16 7. Hint: Use Exercise 6.
9. Hint: Use \(\textbf{f}\,'(t) = \norm{\textbf{f}(t)}\textbf{T}\), differentiate that to get \(\textbf{f}\,''(t)\), put those expressions into \(\Crossprod{\textbf{f}\,'(t)\,}{\,\textbf{f}\,''(t)}\), then write \(\textbf{T}\,'(t)\) in terms of \(\textbf{N}(t)\). 11. \(\textbf{T}(t) = \frac{1}{\sqrt{2}} (-\sin t,\cos t,1)\), \(\textbf{N}(t) = (-\cos t,-\sin t,0)\), \(\textbf{B}(t) = \frac{1}{\sqrt{2}}(\sin t,-\cos t,1)\), \(\kappa(t) = 1/2\)

Chapter 2

Section 2.1 (p. § 2.1)

1. domain: \(\Real{2}\), range: \(\lbrack -1,\infty )\) 3. domain: \(\lbrace (x,y): x^2 + y^2 \ge 4 \rbrace\), range: \(\lbrack 0,\infty)\)
5. domain: \(\Real{3}\), range: \(\ival{-1}{1}\) 7. 1
9. does not exist 11. \(2\) 13. \(2\) 15. \(0\)
17. does not exist

Section 2.2 (p. § 2.2)

1. \(\frac{\partial f}{\partial x} = 2x\), \(\frac{\partial f}{\partial y} = 2y\) 3. \(\frac{\partial f}{\partial x} = x(x^2 + y + 4)^{-1/2}\), \(\frac{\partial f}{\partial y} = \frac{1}{2}(x^2 + y + 4)^{-1/2}\) 5. \(\frac{\partial f}{\partial x} = ye^{xy} + y\), \(\frac{\partial f}{\partial y} = xe^{xy} + x\) 7. \(\frac{\partial f}{\partial x} = 4x^3\), \(\frac{\partial f}{\partial y} = 0\)
9. \(\frac{\partial f}{\partial x}=x(x^2 + y^2 )^{-1/2}\), \(\frac{\partial f}{\partial y}=y(x^2 + y^2 )^{-1/2}\)
11. \(\frac{\partial f}{\partial x} = \frac{2x}{3}(x^2 + y + 4)^{-2/3}\),
\(\frac{\partial f}{\partial y} = \frac{1}{3}(x^2 + y + 4)^{-2/3}\) 13. \(\frac{\partial f}{\partial x}=-2xe^{-(x^2 + y^2 )}\),
\(\frac{\partial f}{\partial y}=-2ye^{-(x^2 + y^2 )}\) 15. \(\frac{\partial f}{\partial x} = y\cos(xy)\),
\(\frac{\partial f}{\partial y} = x\cos(xy)\) 17. \(\frac{\partial^2 f}{\partial x^2} =2\), \(\frac{\partial^2 f}{\partial y^2} = 2\),
\(\frac{\partial^2 f}{\partial x \,\partial y} = 0\) 19. \(\frac{\partial^2 f}{\partial x^2} = (y+4)(x^2 + y+4)^{-3/2}\),
\(\frac{\partial^2 f}{\partial y^2} = -\frac{1}{4}(x^2 + y+4)^{-3/2}\),
\(\frac{\partial^2 f}{\partial x \,\partial y} = -\frac{1}{2}x(x^2 + y+4)^{-3/2}\)
21. \(\frac{\partial^2 f}{\partial x^2} = y^2 e^{xy}\), \(\frac{\partial^2 f}{\partial y^2} = x^2 e^{xy}\),
\(\frac{\partial^2 f}{\partial x \,\partial y} = (1+xy)e^{xy} + 1\) 23. \(\frac{\partial^2 f}{\partial x^2} =12x^2\),
\(\frac{\partial^2 f}{\partial y^2} = 0\), \(\frac{\partial^2 f}{\partial x \,\partial y} = 0\) 25. \(\frac{\partial^2 f}{\partial x^2} =-x^{-2}\),
\(\frac{\partial^2 f}{\partial y^2} = -y^{-2}\), \(\frac{\partial^2 f}{\partial x \,\partial y} = 0\)

Section 2.3 (p. § 2.3)

1. \(2x+3y-z-3=0\) 3. \(-2x+y-z-2=0\)
5. \(x+2y=z\) 7. \(\frac{1}{2}(x-1)+\frac{4}{9}(y-2)+\frac{\sqrt{11}}{12}(z-\frac{2\sqrt{11}}{3})=0\) 9. \(3x+4y-5z=0\)

Section 2.4 (p. § 2.4)

1. \((2x,2y)\) 3. \((\frac{x}{\sqrt{x^2 + y^2 + 4}},\frac{y}{\sqrt{x^2 + y^2 + 4}})\)
5. \((1/x,1/y)\)
7. \((yz\cos(xyz),xz\cos(xyz),xy\cos(xyz))\)
9. \((2x,2y,2z)\) 11. \(2\sqrt{2}\) 13. \(\frac{1}{\sqrt{3}}\)
15. \(\sqrt{3}\,\cos(1)\) 17. increase: \((45,20)\),
decrease: \((-45,-20)\)

Section 2.5 (p. § 2.5)

1. local min. \((1,0)\); saddle pt. \((-1,0)\)
3. local min. \((1,1)\); local max. \((-1,-1)\); saddle pts. \((1,-1),(-1,1)\) 5. local min. \((1,-1)\); saddle pt. \((0,0)\) 7. local min. \((0,0)\)
9. local min. \((-1,1/2)\) 11. width = height = depth=\(10\) 13. \(x=y=4\), \(z=2\)

Section 2.6 (p. § 2.6)

2. \((\ssub{x}{0},\ssub{y}{0})=(0,0):\) \(\rightarrow (0.2858,-0.3998)\); \((\ssub{x}{0},\ssub{y}{0})=(1,1):\) \(\rightarrow (1.03256,-1.94037)\)

Section 2.7 (p. § 2.7)

1. min. \(\left(\frac{-4}{\sqrt{5}},\frac{-2}{\sqrt{5}}\right)\); max. \(\left(\frac{4}{\sqrt{5}},\frac{2}{\sqrt{5}}\right)\)
3. min. \(\left(\frac{20}{\sqrt{13}},\frac{30}{\sqrt{13}}\right)\); max. \(\left(-\frac{20}{\sqrt{13}},-\frac{30}{\sqrt{13}}\right)\)
4. min. \(\left(\frac{-9}{\sqrt{5}},0,\frac{2}{\sqrt{5}}\right)\); max. \(\left(\frac{9}{8},\frac{\sqrt{59}}{4},\frac{-1}{4}\right)\)
5. \(\frac{8abc}{3\sqrt{3}}\)

Chapter 3

Section 3.1 (p. § 3.1)

1. \(1\) 3. \(\frac{7}{12}\) 5. \(\frac{7}{6}\) 7. \(5\) 9. \(\frac{1}{2}\) 11. \(15\)

Section 3.2 (p. § 3.2)

1. \(1\) 3. \(8\ln 2 - 3\) 5. \(\frac{\pi}{4}\) 6. \(\frac{1}{4}\) 7. \(2\) 9. \(\frac{1}{6}\) 10. \(\frac{6}{5}\)

Section 3.3 (p. § 3.3)

1. \(\frac{9}{2}\) 3. \((2\cos(\pi^2) + \pi^4 -2)/4\) 5. \(\frac{1}{6}\) 7. \(6\)
10. \(\frac{1}{3}\)

Section 3.4 (p. § 3.4)

1. The values should converge to \(\approx 1.318\). (Hint: In Java the exponential function \(e^x\) can be obtained with Math.exp(x). Other languages have similar functions, otherwise use \(e=2.7182818284590455\) in your program.)
2. \(\approx 1.146\) 3. \(\approx 0.705\) 4. \(\approx 0.168\)

Section 3.5 (p. § 3.5)

1. \(8\pi\) 3. \(\frac{4\pi}{3}(8-3^{3/2})\) 7. \(1-\frac{\sin 2}{2}\) 9. \(2\pi ab\)

Section 3.6 (p. § 3.6)

1. \((1,8/3)\) 3. \((0,\frac{4a}{3\pi})\) 5. \((0,3\pi/16)\)
7. \((0,0,5a/12)\) 9. \((7/12,7/12,7/12)\)

Section 3.7 (p. § 3.7)

1. \(\sqrt{\pi}\) 2. 1 6. Both are \(\frac{n}{(n+1)^2 (n+2)}\) 7. \(\frac{1}{n}\)

Chapter 4

Section 4.1 (p. § 4.1)

1. \(1/2\) 3. \(23\) 5. \(24\pi\) 7. \(-2\pi\) 9. \(2\pi\)
11. \(0\)

Section 4.2 (p. § 4.2)

1. \(0\) 3. No 4. Yes. \(F(x,y)=\frac{x^2}{2}-\frac{y^2}{2}\)
5. No 9. (b) No. Hint: Think of how \(F\) is defined. 10. Yes. \(F(x,y)=axy+bx+cy+d\)

Section 4.3 (p. § 4.3)

1. \(16/15\) 3. \(-5\pi\) 5. Yes. \(F(x,y)=xy^2 + x^3\) 7. Yes. \(F(x,y)=4x^2 y + 2y^2 + 3x\)

Section 4.4 (p. § 4.4)

1. \(216\pi\) 2. \(3\) 3. \(12\pi/5\) 7. \(15/4\)

Section 4.5 (p. § 4.5)

1. \(2\sqrt{2}\,\pi^2\) 2. \((17\sqrt{17} - 5\sqrt{5})/3\) 3. \(2/5\)
4. \(2\) 5. \(2\pi (\pi - 1)\) 7. \(67/15\) 9. \(6\)
11. Yes 13. No 19. Hint: Think of how a vector field \(\textbf{f}(x,y) = P(x,y)\,\textbf{i} + Q(x,y)\,\textbf{j}\) in \(\Real{2}\) can be extended in a natural way to be a vector field in \(\Real{3}\).

Section 4.6 (p. § 4.6)

1. \(0\) 3. \(12\sqrt{x^2 + y^2 + z^2}\) 5. \(6(x+y+z)\)
7. \(12\rho\) 8. \((4\rho^2 -6)e^{-\rho^2}\) 9. \(-\frac{2z}{r^3}\,\textbf{e}_{r} + \frac{1}{r^2}\,\textbf{e}_{z}\)
11. \(\text{div}~\textbf{f} = \frac{2}{\rho} - \frac{\sin\theta}{\sin\phi} + \cot\phi\);
\(\text{curl}~\textbf{f} = \cot\phi\,\cos\theta\,\textbf{e}_{\rho} + 2\textbf{e}_{\theta} -2\cos\theta\,\textbf{e}_{\phi}\)
25. Hint: Start by showing that \(\textbf{e}_{r} = \cos\theta\,\textbf{i} + \sin\theta\,\textbf{j}\), \(\textbf{e}_{\theta} = -\sin\theta\,\textbf{i} + \cos\theta\,\textbf{j}\), \(\textbf{e}_{z} = \textbf{k}\).