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Appendix A Answers to Selected Exercises
1 Limits 1.1 An Introduction To Limits 1.1.3 Exercises
Problems
1.1.3.7. 1.1.3.9. 1.1.3.11. 1.1.3.13. 1.1.3.15. 1.1.3.17. 1.1.3.19.
1.1.3.21. 1.1.3.23. 1.1.3.25. 1.1.3.27.
1.3 Finding Limits Analytically
Exercises Problems
1.3.7. 1.3.9. 1.3.11. 1.3.13.
1.3.19. 1.3.21. 1.3.23. 1.3.25. 1.3.27. Answer .
\(\frac{\pi ^{2}-4\pi -2}{2\pi ^{2}-2\pi +1}\)
1.3.29. 1.3.31. 1.3.33.
1.4 One-Sided Limits
Exercises Problems
1.4.5. 1.4.5.a 1.4.5.b 1.4.5.c 1.4.5.d 1.4.5.e 1.4.5.f 1.4.7. 1.4.7.a Answer .
\(\text{DNE}\hbox{ or }\infty \)
1.4.7.b Answer .
\(\text{DNE}\hbox{ or }\infty \)
1.4.7.c Answer .
\(\text{DNE}\hbox{ or }\infty \)
1.4.7.d 1.4.7.e 1.4.7.f 1.4.9. 1.4.9.a 1.4.9.b 1.4.9.c 1.4.9.d 1.4.11. 1.4.11.a 1.4.11.b 1.4.11.c 1.4.11.d 1.4.11.e 1.4.11.f 1.4.11.g 1.4.11.h
1.4.13. 1.4.13.a 1.4.13.b 1.4.13.c 1.4.13.d 1.4.15. 1.4.15.a 1.4.15.b 1.4.15.c 1.4.15.d 1.4.15.e 1.4.15.f 1.4.15.g 1.4.15.h 1.4.17. 1.4.17.a Answer .
\(1-\cos^{2}\mathopen{}\left(a\right)\)
1.4.17.b Answer .
\(\sin^{2}\mathopen{}\left(a\right)\)
1.4.17.c Answer .
\(1-\cos^{2}\mathopen{}\left(a\right)\hbox{ or }\sin^{2}\mathopen{}\left(a\right)\)
1.4.17.d Answer .
\(\sin^{2}\mathopen{}\left(a\right)\)
1.4.19. 1.4.19.a 1.4.19.b 1.4.19.c 1.4.19.d 1.4.21. 1.4.21.a 1.4.21.b 1.4.21.c 1.4.21.d
1.5 Continuity
Exercises Problems
1.5.11. 1.5.13. 1.5.15. 1.5.17.
Answer 1 . Answer 2 . Answer 3 .
1.5.19. 1.5.19.a 1.5.19.b 1.5.21. 1.5.21.a 1.5.21.b
1.5.23. Answer .
\(\left(-\infty ,\infty \right)\)
1.5.25. 1.5.27. Answer .
\(\left(-\infty ,-1.73205\right], \left[1.73205,\infty \right)\)
1.5.29. Answer .
\(\left(-\infty ,\infty \right)\)
1.5.31. Answer .
\(\left(0,\infty \right)\)
1.5.33. Answer .
\(\left(-\infty ,1.09861\right]\)
1.6 Limits Involving Infinity 1.6.4 Exercises
Problems
1.6.4.9. 1.6.4.9.a 1.6.4.9.b 1.6.4.11. 1.6.4.11.a 1.6.4.11.b 1.6.4.11.c 1.6.4.11.d 1.6.4.13. 1.6.4.13.a 1.6.4.13.b
1.6.4.15. 1.6.4.15.a 1.6.4.15.b 1.6.4.15.c 1.6.4.17. 1.6.4.17.a 1.6.4.17.b 1.6.4.17.c
1.6.4.19. 1.6.4.21. 1.6.4.23.
2 Derivatives 2.1 Instantaneous Rates of Change: The Derivative 2.1.3 Exercises
Problems
2.1.3.7. 2.1.3.9. 2.1.3.11. 2.1.3.13.
2.1.3.15. 2.1.3.17.
Answer 1 . Answer 2 .
\(y-0.333333x = -19.3333\)
2.1.3.19.
Answer 1 . Answer 2 .
\(0.0208333x+y = 64.0833\)
2.1.3.21.
2.1.3.23. 2.1.3.25. Answer .
\(y-0.0192627x = 0.0953664\)
2.1.3.27. 2.1.3.27.a 2.1.3.27.b 2.1.3.27.c 2.1.3.33.
Answer 1 .
\(\left(-2,0\right)\cup \left(2,\infty \right)\)
Answer 2 .
\(\left(-\infty ,-2\right)\cup \left(0,2\right)\)
Answer 3 .
\(\left\{-2,0,2\right\}\)
Answer 4 . Answer 5 .
\(\left(-\infty ,-1\right)\cup \left(1,\infty \right)\)
Answer 6 .
2.2 Interpretations of the Derivative 2.2.5 Exercises
Problems 2.2.5.5. 2.2.5.7. 2.2.5.9. 2.2.5.11. Answer .
\(\text{foot per second squared}\)
2.2.5.15. Answer .
\(\text{f is the derivative of g.}\)
2.2.5.17. Answer .
\(\text{g is the derivative of f.}\)
2.3 Basic Differentiation Rules 2.3.3 Exercises
Terms and Concepts 2.3.3.3. 2.3.3.7. 2.3.3.9.
Answer 1 .
\(\text{a velocity function}\)
Answer 2 .
\(\text{an acceleration function}\)
Problems
2.3.3.11. 2.3.3.13. Answer .
\(9-\left(20t^{4}+{\frac{3}{4}}t^{2}\right)\)
2.3.3.15. 2.3.3.17. 2.3.3.19. Answer .
\(\sin\mathopen{}\left(t\right)-\left(e^{t}+\cos\mathopen{}\left(t\right)\right)\)
2.3.3.21. 2.3.3.23. 2.3.3.25.
2.3.3.27.
Answer 1 . Answer 2 . Answer 3 . Answer 4 . 2.3.3.29.
Answer 1 .
\(-\left(8t+3+e^{t}\right)\)
Answer 2 .
\(-\left(8+e^{t}\right)\)
Answer 3 . Answer 4 . 2.3.3.31.
Answer 1 .
\(-\left(\cos\mathopen{}\left(\theta\right)-\sin\mathopen{}\left(\theta\right)\right)\)
Answer 2 .
\(\sin\mathopen{}\left(\theta\right)+\cos\mathopen{}\left(\theta\right)\)
Answer 3 .
\(\cos\mathopen{}\left(\theta\right)-\sin\mathopen{}\left(\theta\right)\)
Answer 4 .
\(-\left(\sin\mathopen{}\left(\theta\right)+\cos\mathopen{}\left(\theta\right)\right)\)
2.3.3.33.
Answer 1 .
\(y = 20\mathopen{}\left(x-2\right)+24\)
Answer 2 .
\(y = -{\frac{1}{20}}\mathopen{}\left(x-2\right)+24\)
2.3.3.35.
Answer 1 . Answer 2 .
\(y = -\left(x-1\right)\)
2.3.3.37.
Answer 1 .
\(y = \frac{2\cdot 1}{2}\mathopen{}\left(x-\frac{\pi }{6}\right)+\frac{-2\sqrt{3}}{2}\)
Answer 2 .
\(y = -\left({\frac{1}{2}}\cdot 2\right)\mathopen{}\left(x-\frac{\pi }{6}\right)+\frac{-2\sqrt{3}}{2}\)
2.4 The Product and Quotient Rules
Exercises Problems
2.4.15. Answer .
\(\sin\mathopen{}\left(y\right)+y\cos\mathopen{}\left(y\right)\)
2.4.17. Answer .
\(e^{q}\ln\mathopen{}\left(q\right)+e^{q}\frac{1}{q}\)
2.4.19. Answer .
\(\frac{t-4-\left(t+8\right)}{\left(t-4\right)^{2}}\)
2.4.21. Answer .
\(-\left(\csc\mathopen{}\left(y\right)\cot\mathopen{}\left(y\right)+e^{y}\right)\)
2.4.23. 2.4.25. Answer .
\(\left(5r^{2}+17r+10\right)e^{r}\)
2.4.27. 2.4.29. Answer .
\(\frac{\csc\mathopen{}\left(z\right)\sin\mathopen{}\left(z\right)-\csc\mathopen{}\left(z\right)\cot\mathopen{}\left(z\right)\mathopen{}\left(\cos\mathopen{}\left(z\right)+2\right)}{\left(\cos\mathopen{}\left(z\right)+2\right)^{2}}\)
2.4.31. Answer .
\(\frac{\tan\mathopen{}\left(r\right)-r\sec^{2}\mathopen{}\left(r\right)}{\tan^{2}\mathopen{}\left(r\right)}-\frac{\csc^{2}\mathopen{}\left(r\right)r+\cot\mathopen{}\left(r\right)}{r^{2}}\)
2.4.33. Answer .
\(35x^{4}e^{x}+7x^{5}e^{x}-\left(\cos\mathopen{}\left(x\right)\cos\mathopen{}\left(x\right)-\sin\mathopen{}\left(x\right)\sin\mathopen{}\left(x\right)\right)\)
2.4.35. Answer .
\(\left(4z^{3}\ln\mathopen{}\left(z\right)+z^{4}\frac{1}{z}\right)\cos\mathopen{}\left(z\right)-z^{4}\ln\mathopen{}\left(z\right)\sin\mathopen{}\left(z\right)\)
2.4.37.
Answer 1 .
\(y = -\left(7x+7\right)\)
Answer 2 .
\(y = \left({\frac{1}{7}}\right)x-7\)
2.4.39.
Answer 1 .
\(y = -\left(15\mathopen{}\left(x+5\right)+25\right)\)
Answer 2 .
\(y = \left({\frac{1}{15}}\right)\mathopen{}\left(x+5\right)-25\)
2.4.45. Answer .
\(2\cos\mathopen{}\left(x\right)-x\sin\mathopen{}\left(x\right)\)
2.4.47. Answer .
\(\csc\mathopen{}\left(x\right)\cot\mathopen{}\left(x\right)\cot\mathopen{}\left(x\right)+\csc^{2}\mathopen{}\left(x\right)\csc\mathopen{}\left(x\right)\)
2.5 The Chain Rule
Exercises Problems
2.5.7. Answer .
\(10\mathopen{}\left(4x^{3}-x\right)^{9}\mathopen{}\left(12x^{2}-1\right)\)
2.5.9. Answer .
\(3\mathopen{}\left(\sin\mathopen{}\left(\theta\right)+\cos\mathopen{}\left(\theta\right)\right)^{2}\mathopen{}\left(\cos\mathopen{}\left(\theta\right)-\sin\mathopen{}\left(\theta\right)\right)\)
2.5.11. Answer .
\(4\mathopen{}\left(\ln\mathopen{}\left(x\right)-x^{4}\right)^{3}\mathopen{}\left(\frac{1}{x}-4x^{3}\right)\)
2.5.13. Answer .
\(5\mathopen{}\left(y+\frac{1}{y}\right)^{4}\mathopen{}\left(1-\frac{1}{y^{2}}\right)\)
2.5.15. Answer .
\(2\sec^{2}\mathopen{}\left(2q\right)\)
2.5.17. Answer .
\(\left(6t^{5}-\frac{3t^{2}}{\left(t^{3}\right)^{2}}\right)\cos\mathopen{}\left(t^{6}+\frac{1}{t^{3}}\right)\)
2.5.19. Answer .
\(-3\cos^{2}\mathopen{}\left(y^{2}+3y-3\right)\mathopen{}\left(2y+3\right)\sin\mathopen{}\left(y^{2}+3y-3\right)\)
2.5.21. Answer .
\(\frac{1}{q^{8}}\cdot 8q^{7}\)
2.5.23. 2.5.25. 2.5.27. Answer .
\(\frac{1.79176\cdot 6^{w}\mathopen{}\left(5^{w}+6\right)-\left(6^{w}+5\right)\cdot 1.60944\cdot 5^{w}}{\left(5^{w}+6\right)^{2}}\)
2.5.29. Answer .
\(\frac{\left(1.60944\cdot 5^{r^{2}}\cdot 2r-1\right)\cdot 6^{r^{2}}-\left(5^{r^{2}}-r\right)\cdot 1.79176\cdot 6^{r^{2}}\cdot 2r}{\left(6^{r^{2}}\right)^{2}}\)
2.5.31. Answer .
\(6\mathopen{}\left(x^{2}+4x\right)^{5}\mathopen{}\left(2x+4\right)\mathopen{}\left(7x^{4}+x\right)^{3}+\left(x^{2}+4x\right)^{6}\cdot 3\mathopen{}\left(7x^{4}+x\right)^{2}\mathopen{}\left(28x^{3}+1\right)\)
2.5.33. Answer .
\(7\cos\mathopen{}\left(9+7w\right)\cos\mathopen{}\left(4w-5\right)-4\sin\mathopen{}\left(4w-5\right)\sin\mathopen{}\left(9+7w\right)\)
2.5.35. Answer .
\(-\frac{6\sin\mathopen{}\left(6r+4\right)\mathopen{}\left(3r+1\right)^{3}+9\mathopen{}\left(3r+1\right)^{2}\cos\mathopen{}\left(6r+4\right)}{\left(\left(3r+1\right)^{3}\right)^{2}}\)
2.5.37. 2.5.39.
Answer 1 .
\(y = -3\mathopen{}\left(x-\frac{\pi }{2}\right)+1\)
Answer 2 .
\(y = \frac{1}{3}\mathopen{}\left(x-\frac{\pi }{2}\right)+1\)
2.5.41.
2.6 Implicit Differentiation 2.6.4 Exercises
Problems
2.6.4.5. Answer .
\(\frac{1}{2\sqrt{w}}+\frac{\frac{1}{2\sqrt{w}}}{\left(\sqrt{w}\right)^{2}}\)
2.6.4.7. Answer .
\(\frac{1}{2\sqrt{9+t^{2}}}\cdot 2t\)
2.6.4.9. 2.6.4.11. Answer .
\(\frac{\sqrt{w}-\left(w-8\right)\frac{1}{2\sqrt{w}}}{\left(\sqrt{w}\right)^{2}}\)
2.6.4.13. 2.6.4.15. Answer .
\(\sin\mathopen{}\left(x\right)\sec\mathopen{}\left(y\right)\)
2.6.4.17. 2.6.4.19. Answer .
\(\frac{-2\sin\mathopen{}\left(y\right)\cos\mathopen{}\left(y\right)}{x}\)
2.6.4.21. 2.6.4.23. Answer .
\(\frac{1-\cos\mathopen{}\left(x\right)}{\sin\mathopen{}\left(y\right)+1}\)
2.6.4.25. Answer .
\(\frac{-\left(2x+y\right)}{2y+x}\)
2.6.4.27. 2.6.4.27.a 2.6.4.27.b Answer .
\(y = -1.859\mathopen{}\left(x-0.1\right)+0.2811\)
2.6.4.29. 2.6.4.29.a 2.6.4.29.b Answer .
\(y = \frac{3}{108^{\frac{1}{4}}}\mathopen{}\left(x-2\right)-108^{\frac{1}{4}}\)
2.6.4.31. 2.6.4.31.a Answer .
\(y = \frac{-1}{\sqrt{3}}\mathopen{}\left(x-\frac{7}{2}\right)+\frac{6+3\sqrt{3}}{2}\)
2.6.4.31.b Answer .
\(y = \frac{\sqrt{3}\mathopen{}\left(x-\left(4+3\sqrt{3}\right)\right)}{2}+\frac{3}{2}\)
2.6.4.33. Answer .
\(\frac{-\left(\left(2y+1\right)\cdot 12x^{2}-4x^{3}\frac{2\mathopen{}\left(-\left(4x^{3}\right)\right)}{2y+1}\right)}{\left(2y+1\right)^{2}}\)
2.6.4.35. Answer .
\(\sin^{2}\mathopen{}\left(x\right)\sec^{2}\mathopen{}\left(y\right)\tan\mathopen{}\left(y\right)+\cos\mathopen{}\left(x\right)\sec\mathopen{}\left(y\right)\)
2.6.4.37.
Answer 1 .
\(\left(1+x\right)^{\frac{1}{x}}\mathopen{}\left(\frac{1}{x\mathopen{}\left(x+1\right)}-\frac{\ln\mathopen{}\left(1+x\right)}{x^{2}}\right)\)
Answer 2 .
\(y = \left(1-2\ln\mathopen{}\left(2\right)\right)\mathopen{}\left(x-1\right)+2\)
2.6.4.39.
Answer 1 .
\(\frac{x^{x}}{x+1}\mathopen{}\left(\ln\mathopen{}\left(x\right)+1-\frac{1}{x+1}\right)\)
Answer 2 .
\(y = \frac{1}{4}\mathopen{}\left(x-1\right)+\frac{1}{2}\)
2.6.4.41.
Answer 1 .
\(\frac{x+1}{x+2}\mathopen{}\left(\frac{1}{x+1}-\frac{1}{x+2}\right)\)
Answer 2 .
\(y = \frac{1}{9}\mathopen{}\left(x-1\right)+\frac{2}{3}\)
2.7 Derivatives of Inverse Functions
Exercises Problems
2.7.15. Answer .
\(-\frac{1}{\sqrt{1-\left(4w\right)^{2}}}\cdot 4\)
2.7.17. Answer .
\(\frac{1}{1+\left(2r\right)^{2}}\cdot 2\)
2.7.19. Answer .
\(\left(\sec\mathopen{}\left(x\right)\right)^{2}\cos^{-1}\mathopen{}\left(x\right)-\frac{1}{\sqrt{1-x^{2}}}\tan\mathopen{}\left(x\right)\)
2.7.21. Answer .
\(\frac{\frac{1}{1+z^{2}}\sin^{-1}\mathopen{}\left(z\right)-\frac{1}{\sqrt{1-z^{2}}}\tan^{-1}\mathopen{}\left(z\right)}{\left(\sin^{-1}\mathopen{}\left(z\right)\right)^{2}}\)
2.7.23. Answer .
\(\csc\mathopen{}\left(\frac{1}{q^{3}}\right)\cot\mathopen{}\left(\frac{1}{q^{3}}\right)\frac{3q^{2}}{\left(q^{3}\right)^{2}}\)
2.7.29. Answer .
\(y = 2\mathopen{}\left(x-\frac{-\sqrt{3}}{2}\right)+\left(-\frac{\pi }{3}\right)\)
3 The Graphical Behavior of Functions 3.1 Extreme Values
Exercises Problems 3.1.7.
Answer 1 . Answer 2 . Answer 3 . Answer 4 .
3.1.9. 3.1.11. 3.1.13. 3.1.15.
3.1.17. 3.1.19. 3.1.21. 3.1.23.
Answer 1 .
\(\frac{e^{\frac{\pi }{4}}}{\sqrt{2}}\)
Answer 2 . 3.1.25.
3.2 The Mean Value Theorem
Exercises Problems
3.2.3. 3.2.5. 3.2.7. Answer .
\(\text{does not apply}\)
3.2.9. Answer .
\(\text{does not apply}\)
3.2.11. 3.2.13. 3.2.15. Answer .
\(\text{does not apply}\)
3.2.17. Answer .
\(-\sec^{-1}\mathopen{}\left(\frac{2}{\sqrt{\pi }}\right), \sec^{-1}\mathopen{}\left(\frac{2}{\sqrt{\pi }}\right)\)
3.2.19. Answer .
\(5+7\frac{\sqrt{7}}{6}, 5-7\frac{\sqrt{7}}{6}\)
3.3 Increasing and Decreasing Functions
Exercises Terms and Concepts 3.3.3. Answer .
Answers will vary; graphs should be steeper near
\(x=0\) than near
\(x=2\text{.}\)
Problems
3.3.15.
Answer 1 .
\(\left(-\infty ,\infty \right)\)
Answer 2 . Answer 3 .
\(\left[-2,\infty \right)\)
Answer 4 .
\(\left(-\infty ,-2\right]\)
Answer 5 . Answer 6 . 3.3.17.
Answer 1 .
\(\left(-\infty ,\infty \right)\)
Answer 2 .
\(-{\frac{5}{7}}, {\frac{7}{3}}\)
Answer 3 .
\(\left(-\infty ,-0.714286\right], \left[2.33333,\infty \right)\)
Answer 4 .
\(\left[-0.714286,2.33333\right]\)
Answer 5 . Answer 6 . 3.3.19.
Answer 1 .
\(\left(-\infty ,\infty \right)\)
Answer 2 . Answer 3 .
\(\left(-\infty ,5\right]\)
Answer 4 .
\(\left[5,\infty \right)\)
Answer 5 . Answer 6 . 3.3.21.
Answer 1 .
\(\left(-\infty ,-7\right)\cup \left(-7,-5\right)\cup \left(-5,\infty \right)\)
Answer 2 . Answer 3 .
\(\left[-5.91608,-5\right), \left(-5,5.91608\right]\)
Answer 4 .
\(\left(-\infty ,-7\right), \left(-7,-5.91608\right], \left[5.91608,\infty \right)\)
Answer 5 . Answer 6 . 3.3.23.
Answer 1 .
\(\left(-\pi ,\pi \right)\)
Answer 2 .
\(-2.35619, -0.785398, 0.785398, 2.35619\)
Answer 3 .
\(\left(-3.14159,-2.35619\right), \left(-0.785398,0.785398\right), \left(2.35619,3.14159\right)\)
Answer 4 .
\(\left(-2.35619,-0.785398\right), \left(0.785398,2.35619\right)\)
Answer 5 . Answer 6 .
3.4 Concavity and the Second Derivative 3.4.3 Exercises
Problems
3.4.3.15.
Answer 1 . Answer 2 .
\(\left(-\infty ,\infty \right)\)
Answer 3 . 3.4.3.17.
Answer 1 . Answer 2 .
\(\left[0,\infty \right)\)
Answer 3 .
\(\left(-\infty ,0\right]\)
3.4.3.19.
Answer 1 . Answer 2 .
\(\left(-\infty ,-10.6667\right], \left[0,\infty \right)\)
Answer 3 .
\(\left[-10.6667,0\right]\)
3.4.3.21.
Answer 1 . Answer 2 .
\(\left(-\infty ,\infty \right)\)
Answer 3 . 3.4.3.23.
Answer 1 . Answer 2 .
\(\left(-\infty ,-0.57735\right], \left[0.57735,\infty \right)\)
Answer 3 .
\(\left[-0.57735,0.57735\right]\)
3.4.3.25.
Answer 1 . Answer 2 .
\(\left(-3.14159,-0.785398\right], \left[2.35619,3.14159\right)\)
Answer 3 .
\(\left[-0.785398,2.35619\right]\)
3.4.3.27.
Answer 1 . Answer 2 .
\(\left[0.22313,\infty \right)\)
Answer 3 .
\(\left(0,0.22313\right]\)
3.4.3.29.
Answer 1 . Answer 2 . Answer 3 . 3.4.3.31.
Answer 1 . Answer 2 . Answer 3 . 3.4.3.33.
Answer 1 . Answer 2 . Answer 3 . 3.4.3.35.
Answer 1 . Answer 2 . Answer 3 . 3.4.3.37.
Answer 1 . Answer 2 . Answer 3 . 3.4.3.39.
Answer 1 . Answer 2 . Answer 3 . 3.4.3.41.
Answer 1 . Answer 2 . Answer 3 .
3.4.3.43. 3.4.3.45. 3.4.3.47. 3.4.3.49. 3.4.3.51. 3.4.3.53. 3.4.3.55.
4 Applications of the Derivative 4.1 Newton’s Method
Exercises Problems
4.1.3.
Answer 1 . Answer 2 . Answer 3 . Answer 4 . Answer 5 . 4.1.5.
Answer 1 . Answer 2 . Answer 3 . Answer 4 . Answer 5 . 4.1.7.
Answer 1 . Answer 2 . Answer 3 . Answer 4 . Answer 5 .
4.1.9. Answer .
\(\left\{-5.15633,-0.369102,0.525428\right\}\)
4.1.11. Answer .
\(\left\{-1.0134,0.988312,1.39341\right\}\)
4.1.13. Answer .
\(\left\{-0.824132,0.824132\right\}\)
4.1.15.
4.2 Related Rates
Exercises Problems 4.2.3. 4.2.3.a Answer .
\(0.198944\ {\textstyle\frac{\rm\mathstrut cm}{\rm\mathstrut s}}\)
4.2.3.b Answer .
\(0.0198944\ {\textstyle\frac{\rm\mathstrut cm}{\rm\mathstrut s}}\)
4.2.3.c Answer .
\(0.00198944\ {\textstyle\frac{\rm\mathstrut cm}{\rm\mathstrut s}}\)
4.2.5. Answer .
\(51.066\ {\textstyle\frac{\rm\mathstrut mi}{\rm\mathstrut h}}\)
4.2.7. 4.2.7.a Answer .
\(258.537\ {\textstyle\frac{\rm\mathstrut rad}{\rm\mathstrut hr}}\)
4.2.7.b Answer .
\(413.417\ {\textstyle\frac{\rm\mathstrut rad}{\rm\mathstrut hr}}\)
4.2.7.c Answer .
\(424\ {\textstyle\frac{\rm\mathstrut rad}{\rm\mathstrut hr}}\)
4.2.9. 4.2.9.a Answer .
\(0.0417029\ {\textstyle\frac{\rm\mathstrut ft}{\rm\mathstrut s}}\)
4.2.9.b Answer .
\(0.458349\ {\textstyle\frac{\rm\mathstrut ft}{\rm\mathstrut s}}\)
4.2.9.c Answer .
\(3.35489\ {\textstyle\frac{\rm\mathstrut ft}{\rm\mathstrut s}}\)
4.2.9.d 4.2.11. 4.2.11.a Answer .
\(19.1658\ {\textstyle\frac{\rm\mathstrut ft}{\rm\mathstrut s}}\)
4.2.11.b Answer .
\(0.191658\ {\textstyle\frac{\rm\mathstrut ft}{\rm\mathstrut s}}\)
4.2.11.c Answer .
\(0.0395988\ {\textstyle\frac{\rm\mathstrut ft}{\rm\mathstrut s}}\)
4.2.11.d 4.2.13. 4.2.13.a 4.2.13.b Answer .
\(1.71499\ {\textstyle\frac{\rm\mathstrut ft}{\rm\mathstrut s}}\)
4.2.13.c Answer .
\(1.83829\ {\textstyle\frac{\rm\mathstrut ft}{\rm\mathstrut s}}\)
4.2.13.d 4.2.15. Answer .
\(0.00230973\ {\textstyle\frac{\rm\mathstrut ft}{\rm\mathstrut s}}\)
4.3 Optimization
Exercises Problems 4.3.3. 4.3.5. 4.3.7. 4.3.9. 4.3.11. 4.3.13. Answer .
\(10.3923\ {\rm in};\,14.6969\ {\rm in}\)
4.3.15. 4.3.17.
4.4 Differentials
Exercises Problems
4.4.7. 4.4.9. 4.4.11. 4.4.13. 4.4.15.
4.4.17. 4.4.19. Answer .
\(-\frac{24x^{7}}{\left(3x^{8}\right)^{2}}dx\)
4.4.21. Answer .
\(\left(6x^{5}+3e^{3x}\right)dx\)
4.4.23. Answer .
\(\frac{8\mathopen{}\left(\tan\mathopen{}\left(x\right)+5\right)-8x\sec^{2}\mathopen{}\left(x\right)}{\left(\tan\mathopen{}\left(x\right)+5\right)^{2}}dx\)
4.4.25. Answer .
\(\left(e^{x}\cot\mathopen{}\left(x\right)-e^{x}\csc^{2}\mathopen{}\left(x\right)\right)dx\)
4.4.27. Answer .
\(\frac{x-8-\left(x-6\right)}{\left(x-8\right)^{2}}dx\)
4.4.29. Answer .
\(\ln\mathopen{}\left(x\right)dx\)
4.4.31. Answer .
\(7.53982\ {\rm cm^{3}}\)
4.4.33.
4.4.35. 4.4.35.a 4.4.35.b 4.4.35.c 4.4.37. 4.4.37.a 4.4.37.b 4.4.37.c 4.4.39.
5 Integration 5.1 Antiderivatives and Indefinite Integration
Exercises Problems
5.1.9. Answer .
\(\left({\frac{2}{3}}\right)x^{9}+C\)
5.1.11. Answer .
\(\left({\frac{13}{3}}\right)x^{3}-2x+C\)
5.1.13. 5.1.15. 5.1.17. Answer .
\(\cot\mathopen{}\left(\theta\right)+C\)
5.1.19. Answer .
\(\sec\mathopen{}\left(x\right)+\csc\mathopen{}\left(x\right)+C\)
5.1.21. Answer .
\(\frac{8^{t}}{\ln\mathopen{}\left(8\right)}+C\)
5.1.23. Answer .
\(\left({\frac{4}{3}}\right)t^{3}+16t^{2}+64t+\left({\frac{256}{3}}\right)+C\)
5.1.25. 5.1.27.
5.1.31. 5.1.33. Answer .
\(\csc\mathopen{}\left(x\right)+\left(-9\right)\)
5.1.35. Answer .
\(\left({\frac{3}{2}}\right)x^{2}+6x+7\)
5.1.37. 5.1.39. Answer .
\(\frac{23x^{4}}{12}+\frac{3^{x}}{1.20695}-\sin\mathopen{}\left(x\right)-8.91024x+4.17146\)
5.2 The Definite Integral
Exercises Problems
5.2.5. 5.2.5.a 5.2.5.b 5.2.5.c 5.2.5.d 5.2.5.e 5.2.5.f 5.2.7. 5.2.7.a 5.2.7.b 5.2.7.c 5.2.7.d 5.2.7.e 5.2.7.f 5.2.9. 5.2.9.a 5.2.9.b 5.2.9.c 5.2.9.d
5.2.11. 5.2.11.a 5.2.11.b 5.2.11.c 5.2.11.d 5.2.13. 5.2.13.a 5.2.13.b 5.2.13.c 5.2.13.d 5.2.15. 5.2.15.a Answer .
\(2\ {\textstyle\frac{\rm\mathstrut ft}{\rm\mathstrut s}}\)
5.2.15.b 5.2.15.c 5.2.17. 5.2.17.a Answer .
\(64\ {\textstyle\frac{\rm\mathstrut ft}{\rm\mathstrut s}}\)
5.2.17.b 5.2.17.c 5.2.17.d
5.3 Riemann Sums 5.3.4 Exercises
Terms and Concepts 5.3.4.3. Problems
5.3.4.5. 5.3.4.7.
Answer 1 .
\(1+0+\left(-1\right)+0+1+0\)
Answer 2 . 5.3.4.9.
Answer 1 .
\(1+\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+\frac{1}{5}\)
Answer 2 . 5.3.4.11.
Answer 1 .
\(\frac{1}{2}+\frac{1}{6}+\frac{1}{12}+\frac{1}{20}+\frac{1}{30}\)
Answer 2 .
5.3.4.13. 5.3.4.15. Answer .
\(1;\,4;\,\frac{i}{i+4}\)
5.3.4.17. 5.3.4.19. 5.3.4.21. 5.3.4.23.
5.3.4.35.
Answer 1 .
\(\frac{\left(n+1\right)^{2}}{4n^{2}}\)
Answer 2 . Answer 3 . Answer 4 . Answer 5 . 5.3.4.37.
Answer 1 . Answer 2 . Answer 3 . Answer 4 . Answer 5 . 5.3.4.39.
Answer 1 . Answer 2 . Answer 3 . Answer 4 . Answer 5 .
5.4 The Fundamental Theorem of Calculus 5.4.6 Exercises
Problems
5.4.6.5. 5.4.6.7. 5.4.6.9. 5.4.6.11. Answer .
\(\frac{\left({\frac{3124}{25}}\right)}{\ln\mathopen{}\left(5\right)}\)
5.4.6.13. 5.4.6.15. 5.4.6.17. 5.4.6.19. 5.4.6.21. 5.4.6.23. 5.4.6.25. 5.4.6.27.
5.4.6.35. Answer .
\(\frac{\frac{1}{\frac{\pi }{2}-0}\cdot 3.14159}{\pi }\)
5.4.6.37. 5.4.6.39.
5.4.6.41. 5.4.6.43. 5.4.6.45.
5.4.6.47. Answer .
\(-160\ {\textstyle\frac{\rm\mathstrut ft}{\rm\mathstrut s}}\)
5.4.6.49. Answer .
\({\frac{49}{2}}\ {\textstyle\frac{\rm\mathstrut ft}{\rm\mathstrut s}}\)
5.4.6.55. Answer .
\(\frac{5x^{4}-1}{x^{5}-x}\)
5.4.6.57. Answer .
\(4x^{3}\mathopen{}\left(x^{4}-7\right)-\left(x-7\right)\)
5.4.6.59. Answer .
\(2x\sin\mathopen{}\left(-2x^{4}\right)\)
5.5 Numerical Integration 5.5.6 Exercises
Problems
5.5.6.5. 5.5.6.5.a 5.5.6.5.b 5.5.6.5.c 5.5.6.7. 5.5.6.7.a 5.5.6.7.b 5.5.6.7.c 5.5.6.9. 5.5.6.9.a 5.5.6.9.b 5.5.6.9.c 5.5.6.11. 5.5.6.11.a 5.5.6.11.b 5.5.6.11.c
5.5.6.13. 5.5.6.13.a 5.5.6.13.b 5.5.6.15. 5.5.6.15.a 5.5.6.15.b 5.5.6.17. 5.5.6.17.a 5.5.6.17.b 5.5.6.19. 5.5.6.19.a 5.5.6.19.b
5.5.6.21. 5.5.6.21.a 5.5.6.21.b 5.5.6.23. 5.5.6.23.a 5.5.6.23.b 5.5.6.25.
Answer 1 .
\(30.8667\ {\rm cm^{2}}\)
Answer 2 .
6 Techniques of Antidifferentiation 6.1 Substitution 6.1.5 Exercises
Problems
6.1.5.3. Answer .
\({\frac{1}{7}}\mathopen{}\left(x^{4}+9\right)^{7}+C\)
6.1.5.5. Answer .
\({\frac{1}{12}}\mathopen{}\left(x^{2}+6\right)^{6}+C\)
6.1.5.7. Answer .
\({\frac{1}{2}}\ln\mathopen{}\left(\left|2x+7\right|\right)+C\)
6.1.5.9. Answer .
\({\frac{2}{3}}\mathopen{}\left(x+10\right)\sqrt{x-5}+C\)
6.1.5.11. 6.1.5.13. Answer .
\(C-{\frac{1}{4}}\mathopen{}\left(\frac{1}{x^{2}}+4\right)^{2}\)
6.1.5.15. Answer .
\(\frac{\left(\sin\mathopen{}\left(x\right)\right)^{10}}{10}+C\)
6.1.5.17. Answer .
\(\frac{\cos\mathopen{}\left(2-8x\right)}{8}+C\)
6.1.5.19. Answer .
\({\frac{1}{4}}\ln\mathopen{}\left(\left|\sec\mathopen{}\left(4x\right)+\tan\mathopen{}\left(4x\right)\right|\right)+C\)
6.1.5.21. Answer .
\({\frac{1}{6}}\sin\mathopen{}\left(x^{6}\right)+C\)
6.1.5.23. Answer .
\(\ln\mathopen{}\left(\left|\sin\mathopen{}\left(x\right)\right|\right)+C\)
6.1.5.25. Answer .
\({\frac{1}{2}}e^{2x-3}+C\)
6.1.5.27. Answer .
\({\frac{1}{2}}e^{\left(x-5\right)^{2}}+C\)
6.1.5.29. Answer .
\(\ln\mathopen{}\left(e^{x}+5\right)+C\)
6.1.5.31. Answer .
\(\frac{7^{7x}}{13.6214}+C\)
6.1.5.33. Answer .
\(\frac{\ln^{2}\mathopen{}\left(x\right)}{2}+C\)
6.1.5.35. Answer .
\(\left({\frac{3}{2}}\right)\mathopen{}\left(\ln\mathopen{}\left(x\right)\right)^{2}+C\)
6.1.5.37. Answer .
\(\frac{x^{2}}{2}-3x-7\ln\mathopen{}\left(\left|x\right|\right)+C\)
6.1.5.39. Answer .
\({\frac{1}{3}}\mathopen{}\left(x+1\right)^{3}+\left({\frac{3}{2}}\right)\mathopen{}\left(x+1\right)^{2}+3\mathopen{}\left(x+1\right)+C\)
6.1.5.41. Answer .
\(3\mathopen{}\left(x+9\right)^{2}-103\mathopen{}\left(x+9\right)+433\ln\mathopen{}\left(\left|x+9\right|\right)+C\)
6.1.5.43. Answer .
\(1.41421\tan^{-1}\mathopen{}\left(\frac{x}{1.41421}\right)+C\)
6.1.5.45. Answer .
\(9\sin^{-1}\mathopen{}\left(\frac{x}{2.23607}\right)+C\)
6.1.5.47. Answer .
\(\left({\frac{7}{5}}\right)\sec^{-1}\mathopen{}\left(\frac{\left|x\right|}{5}\right)+C\)
6.1.5.49. Answer .
\(0.258199\tan^{-1}\mathopen{}\left(\frac{x+3}{15}\right)+C\)
6.1.5.51. Answer .
\(5\sin^{-1}\mathopen{}\left(\frac{x+7}{4}\right)+C\)
6.1.5.53. Answer .
\(C-\frac{1}{3\mathopen{}\left(x^{3}+2\right)}\)
6.1.5.55. Answer .
\(\left({\frac{1}{8}}\right)\sqrt{4+8x^{2}}+C\)
6.1.5.57. Answer .
\(C-{\frac{2}{3}}\mathopen{}\left(\cos\mathopen{}\left(x\right)\right)^{\left({\frac{3}{2}}\right)}\)
6.1.5.59. Answer .
\(\ln\mathopen{}\left(\left|x+5\right|\right)+C\)
6.1.5.61. Answer .
\(\left({\frac{3}{2}}\right)x^{2}-x+\ln\mathopen{}\left(\left|x^{2}-9x+6\right|\right)+C\)
6.1.5.63. Answer .
\(5\ln\mathopen{}\left(\left|-\left(5x^{2}+7x+3\right)\right|\right)+C\)
6.1.5.65. Answer .
\({\frac{1}{10}}\tan^{-1}\mathopen{}\left(\frac{x^{2}}{5}\right)+C\)
6.1.5.67. Answer .
\(\sec^{-1}\mathopen{}\left(\left|7x\right|\right)+C\)
6.1.5.69. Answer .
\(\left({\frac{3}{2}}\right)\ln\mathopen{}\left(\left|x^{2}+14x+53\right|\right)+3\tan^{-1}\mathopen{}\left(\frac{x+7}{2}\right)+C\)
6.1.5.71. Answer .
\(x+13.0639\tan^{-1}\mathopen{}\left(\frac{x-7}{2.44949}\right)+7\ln\mathopen{}\left(\left|x^{2}-14x+55\right|\right)+C\)
6.1.5.73. Answer .
\({\frac{1}{2}}x^{2}+6x+\left({\frac{3}{2}}\right)\ln\mathopen{}\left(\left|x^{2}-6x+20\right|\right)+7.23627\tan^{-1}\mathopen{}\left(\frac{x-3}{3.31662}\right)+C\)
6.1.5.75. Answer .
\(\tan^{-1}\mathopen{}\left(\sin\mathopen{}\left(x\right)\right)+C\)
6.1.5.77. Answer .
\(3\sqrt{x^{2}-10x+18}+C\)
6.1.5.79. Answer .
\(\ln\mathopen{}\left(\left({\frac{1}{4}}\right)\right)\)
6.1.5.81. 6.1.5.83. Answer .
\({\frac{1}{2}}\mathopen{}\left(e-e^{4}\right)\)
6.1.5.85.
6.2 Integration by Parts
Exercises Problems
6.2.5. Answer .
\(\sin\mathopen{}\left(x\right)-x\cos\mathopen{}\left(x\right)+C\)
6.2.7. Answer .
\(-x^{2}\cos\mathopen{}\left(x\right)+2x\sin\mathopen{}\left(x\right)+2\cos\mathopen{}\left(x\right)+C\)
6.2.9. Answer .
\({\frac{1}{2}}e^{x^{2}}+C\)
6.2.11. Answer .
\(-{\frac{1}{2}}xe^{-2x}-\frac{e^{-2x}}{4}+C\)
6.2.13. Answer .
\({\frac{1}{5}}e^{2x}\mathopen{}\left(\sin\mathopen{}\left(x\right)+2\cos\mathopen{}\left(x\right)\right)+C\)
6.2.15. Answer .
\(\left({\frac{1}{10}}\right)e^{5x}\mathopen{}\left(\sin\mathopen{}\left(5x\right)+\cos\mathopen{}\left(5x\right)\right)+C\)
6.2.17. Answer .
\(\sqrt{1-x^{2}}+x\sin^{-1}\mathopen{}\left(x\right)+C\)
6.2.19. Answer .
\(0.5x^{2}\tan^{-1}\mathopen{}\left(x\right)-\frac{x}{2}+0.5\tan^{-1}\mathopen{}\left(x\right)+C\)
6.2.21. Answer .
\(0.5x^{2}\ln\mathopen{}\left(x\right)-\frac{x^{2}}{4}+C\)
6.2.23. Answer .
\({\frac{1}{2}}x^{2}\ln\mathopen{}\left(x+3\right)-{\frac{1}{4}}\mathopen{}\left(x+3\right)^{2}+3x-\left({\frac{9}{2}}\right)\ln\mathopen{}\left(x+3\right)+C\)
6.2.25. Answer .
\(0.333333x^{3}\ln\mathopen{}\left(x\right)-\frac{x^{3}}{9}+C\)
6.2.27. Answer .
\(2\mathopen{}\left(x+5\right)+\left(x+5\right)\mathopen{}\left(\ln\mathopen{}\left(x+5\right)\right)^{2}-2\mathopen{}\left(x+5\right)\ln\mathopen{}\left(x+5\right)+C\)
6.2.29. Answer .
\(\ln\mathopen{}\left(\left|\sin\mathopen{}\left(x\right)\right|\right)-x\cot\mathopen{}\left(x\right)+C\)
6.2.31. Answer .
\({\frac{1}{3}}\mathopen{}\left(x^{2}-3\right)^{\left({\frac{3}{2}}\right)}+C\)
6.2.33. Answer .
\(x\sec\mathopen{}\left(x\right)-\ln\mathopen{}\left(\left|\sec\mathopen{}\left(x\right)+\tan\mathopen{}\left(x\right)\right|\right)+C\)
6.2.35. Answer .
\(\frac{x}{2}\mathopen{}\left(\sin\mathopen{}\left(\ln\mathopen{}\left(x\right)\right)+\cos\mathopen{}\left(\ln\mathopen{}\left(x\right)\right)\right)+C\)
6.2.37. Answer .
\(2\cos\mathopen{}\left(\sqrt{x}\right)+2\sqrt{x}\sin\mathopen{}\left(\sqrt{x}\right)+C\)
6.2.39. Answer .
\(2\sqrt{x}e^{\sqrt{x}}-2e^{\sqrt{x}}+C\)
6.2.41. 6.2.43. 6.2.45. 6.2.47. Answer .
\(\left(-{\frac{9}{4}}\right)e^{-8}-\left(-{\frac{7}{4}}\right)e^{-6}\)
6.2.49. Answer .
\(0.2\mathopen{}\left(e^{\pi }+e^{-\pi }\right)\)
6.3 Trigonometric Integrals 6.3.4 Exercises
Problems
6.3.4.5. Answer .
\(-0.2\cos^{5}\mathopen{}\left(x\right)+C\)
6.3.4.7. Answer .
\({\frac{1}{12}}\mathopen{}\left(\cos\mathopen{}\left(x\right)\right)^{12}-{\frac{1}{10}}\mathopen{}\left(\cos\mathopen{}\left(x\right)\right)^{10}+C\)
6.3.4.9. Answer .
\({\frac{1}{8}}\mathopen{}\left(\sin\mathopen{}\left(x\right)\right)^{8}-{\frac{1}{3}}\mathopen{}\left(\sin\mathopen{}\left(x\right)\right)^{6}+{\frac{1}{4}}\mathopen{}\left(\sin\mathopen{}\left(x\right)\right)^{4}+C\)
6.3.4.11. Answer .
\(\frac{x}{8}-0.03125\sin\mathopen{}\left(4x\right)+C\)
6.3.4.13. Answer .
\(C-\left(\left({\frac{1}{8}}\right)\cos\mathopen{}\left(4x\right)+\left({\frac{1}{12}}\right)\cos\mathopen{}\left(6x\right)\right)\)
6.3.4.15. Answer .
\(\frac{1}{16\pi }\sin\mathopen{}\left(8\pi x\right)-\frac{1}{20\pi }\sin\mathopen{}\left(10\pi x\right)+C\)
6.3.4.17. Answer .
\(\frac{6}{\pi }\cos\mathopen{}\left(\frac{\pi }{12}\pi x\right)+\frac{6}{5\pi }\cos\mathopen{}\left(\frac{5\pi }{12}\pi x\right)+C\)
6.3.4.19. Answer .
\(\frac{\tan^{5}\mathopen{}\left(x\right)}{5}+\frac{\tan^{3}\mathopen{}\left(x\right)}{3}+C\)
6.3.4.21. Answer .
\({\frac{1}{7}}\mathopen{}\left(\tan\mathopen{}\left(x\right)\right)^{7}+C\)
6.3.4.23. Answer .
\({\frac{1}{11}}\mathopen{}\left(\sec\mathopen{}\left(x\right)\right)^{11}-{\frac{2}{9}}\mathopen{}\left(\sec\mathopen{}\left(x\right)\right)^{9}+{\frac{1}{7}}\mathopen{}\left(\sec\mathopen{}\left(x\right)\right)^{7}+C\)
6.3.4.25. Answer .
\(0.25\tan\mathopen{}\left(x\right)\sec^{3}\mathopen{}\left(x\right)+0.375\mathopen{}\left(\sec\mathopen{}\left(x\right)\tan\mathopen{}\left(x\right)+\ln\mathopen{}\left(\left|\sec\mathopen{}\left(x\right)+\tan\mathopen{}\left(x\right)\right|\right)\right)+C\)
6.3.4.27. Answer .
\(0.25\tan\mathopen{}\left(x\right)\sec^{3}\mathopen{}\left(x\right)-0.125\mathopen{}\left(\sec\mathopen{}\left(x\right)\tan\mathopen{}\left(x\right)+\ln\mathopen{}\left(\left|\sec\mathopen{}\left(x\right)+\tan\mathopen{}\left(x\right)\right|\right)\right)+C\)
6.3.4.29. 6.3.4.31. 6.3.4.33.
6.4 Trigonometric Substitution
Exercises Terms and Concepts 6.4.3.
Answer 1 .
\(\tan^{2}\mathopen{}\left(\theta\right)+1 = \sec^{2}\mathopen{}\left(\theta\right)\)
Answer 2 .
\(4\sec^{2}\mathopen{}\left(\theta\right)\)
Problems
6.4.5. Answer .
\({\frac{1}{2}}\mathopen{}\left(x\sqrt{x^{2}+1}+\ln\mathopen{}\left(\sqrt{x^{2}+1}+x\right)\right)+C\)
6.4.7. Answer .
\({\frac{1}{2}}\sin^{-1}\mathopen{}\left(x\right)+\frac{x}{2}\sqrt{1-x^{2}}+C\)
6.4.9. Answer .
\({\frac{1}{2}}x\sqrt{x^{2}-1}-{\frac{1}{2}}\ln\mathopen{}\left(\left|x+\sqrt{x^{2}-1}\right|\right)+C\)
6.4.11. Answer .
\(\frac{x}{2}\sqrt{9x^{2}+1}+{\frac{1}{6}}\ln\mathopen{}\left(3x+\sqrt{9x^{2}+1}\right)+C\)
6.4.13. Answer .
\(\frac{x}{2}\sqrt{25x^{2}-1}-{\frac{1}{10}}\ln\mathopen{}\left(\left|5x+\sqrt{25x^{2}-1}\right|\right)+C\)
6.4.15. Answer .
\(6\sin^{-1}\mathopen{}\left(\frac{x}{1.41421}\right)+C\)
6.4.17. Answer .
\(\sqrt{x^{2}-14}-3.74166\sec^{-1}\mathopen{}\left(\frac{x}{3.74166}\right)+C\)
6.4.19. 6.4.21. Answer .
\(C-\frac{1}{\sqrt{x^{2}+16}}\)
6.4.23. Answer .
\(\left({\frac{1}{2}}\right)\frac{x+4}{x^{2}+8x+17}+\left({\frac{1}{2}}\right)\tan^{-1}\mathopen{}\left(x+4\right)+C\)
6.4.25. Answer .
\(C-\left(\frac{\sqrt{10-x^{2}}}{7x}+{\frac{1}{7}}\sin^{-1}\mathopen{}\left(\frac{x}{3.16228}\right)\right)\)
6.4.27. 6.4.29. Answer .
\(2\sqrt{2}+2\ln\mathopen{}\left(1+1\sqrt{2}\right)\)
6.4.31. Answer .
\(9\sin^{-1}\mathopen{}\left(\left({\frac{1}{3}}\right)\right)+2\sqrt{2}\)
6.5 Partial Fraction Decomposition
Exercises Terms and Concepts
6.5.3. Answer .
\(\frac{A}{x}+\frac{B}{x+6}\)
6.5.5. Answer .
\(\frac{A}{x-\sqrt{2}}+\frac{B}{x+\sqrt{2}}\)
Problems
6.5.7. Answer .
\(7\ln\mathopen{}\left(\left|x+4\right|\right)+6\ln\mathopen{}\left(\left|x-5\right|\right)+C\)
6.5.9. Answer .
\(\ln\mathopen{}\left(\left|x+5\right|\right)-\ln\mathopen{}\left(\left|x-5\right|\right)+C\)
6.5.11. Answer .
\(3\ln\mathopen{}\left(\left|x-4\right|\right)-\frac{5}{x-4}+C\)
6.5.13. Answer .
\(6\ln\mathopen{}\left(\left|x\right|\right)+2\ln\mathopen{}\left(\left|x-8\right|\right)+\frac{1}{x-8}+C\)
6.5.15. Answer .
\(C-\left(\left({\frac{2}{3}}\right)\ln\mathopen{}\left(\left|3x-1\right|\right)+\left({\frac{1}{3}}\right)\ln\mathopen{}\left(\left|9x+2\right|\right)+\frac{\left({\frac{1}{2}}\right)}{2x+2}\right)\)
6.5.17. Answer .
\({\frac{1}{2}}x^{2}-9x-9\ln\mathopen{}\left(\left|x+3\right|\right)+72\ln\mathopen{}\left(\left|x+6\right|\right)+C\)
6.5.19. Answer .
\(\left({\frac{1}{4}}\right)\ln\mathopen{}\left(\left|x\right|\right)-\left({\frac{1}{8}}\right)\ln\mathopen{}\left(x^{2}-2x+4\right)+0.144338\tan^{-1}\mathopen{}\left(\frac{x-1}{1.73205}\right)+C\)
6.5.21. Answer .
\(\ln\mathopen{}\left(\left|8x^{2}+7x+1\right|\right)+\ln\mathopen{}\left(\left|x-6\right|\right)+C\)
6.5.23. Answer .
\(\left({\frac{283}{53}}\right)\ln\mathopen{}\left(\left|x+7\right|\right)-\left({\frac{9}{53}}\right)\ln\mathopen{}\left(x^{2}+4\right)-\left({\frac{139}{106}}\right)\tan^{-1}\mathopen{}\left(\frac{x}{2}\right)+C\)
6.5.25. Answer .
\(2\ln\mathopen{}\left(\left|x+5\right|\right)+\ln\mathopen{}\left(x^{2}+4x+11\right)-0.377964\tan^{-1}\mathopen{}\left(\frac{x+2}{2.64575}\right)+C\)
6.5.27. Answer .
\(\ln\mathopen{}\left(\left({\frac{256}{375}}\right)\right)\)
6.5.29. Answer .
\(\ln\mathopen{}\left(\left({\frac{5}{4}}\right)\right)+\frac{-\pi }{4}-\tan^{-1}\mathopen{}\left(-3\right)\)
6.6 Hyperbolic Functions 6.6.3 Exercises
Problems
6.6.3.11. Answer .
\(2\cosh\mathopen{}\left(2x\right)\)
6.6.3.13. Answer .
\(\mathop{\rm sech}\nolimits^{2}\mathopen{}\left(x^{2}\right)\cdot 2x\)
6.6.3.15. Answer .
\(\cosh\mathopen{}\left(x\right)\cosh\mathopen{}\left(x\right)+\sinh\mathopen{}\left(x\right)\sinh\mathopen{}\left(x\right)\)
6.6.3.17. Answer .
\(-\frac{1}{x^{2}\sqrt{1-\left(x^{2}\right)^{2}}}\cdot 2x\)
6.6.3.19. Answer .
\(\frac{1}{\sqrt{\left(2x^{2}\right)^{2}-1}}\cdot 4x\)
6.6.3.21. Answer .
\(-\frac{1}{1-\cos^{2}\mathopen{}\left(x\right)}\sin\mathopen{}\left(x\right)\)
6.6.3.23. Answer .
\(1\mathopen{}\left(x-0\right)+0\)
6.6.3.25. Answer .
\(0.36\mathopen{}\left(x-\left(-1.09861\right)\right)+\left(-0.8\right)\)
6.6.3.27. Answer .
\(1\mathopen{}\left(x-0\right)+0\)
6.6.3.29. Answer .
\(0.5\ln\mathopen{}\left(\cosh\mathopen{}\left(2x\right)\right)+C\)
6.6.3.31. Answer .
\(0.5\sinh^{2}\mathopen{}\left(x\right)+C\)
6.6.3.33. Answer .
\(x\cosh\mathopen{}\left(x\right)-\sinh\mathopen{}\left(x\right)+C\)
6.6.3.35. Answer .
\(\cosh^{-1} x/3 +C=\ln\big(x+\sqrt{x^2-9}\big)+C\)
6.6.3.37. Answer .
\(\cosh^{-1}\mathopen{}\left(\frac{x^{2}}{2}\right)+C\)
6.6.3.39. Answer .
\(-0.0625\tan^{-1}\mathopen{}\left(\frac{x}{2}\right)+0.03125\ln\mathopen{}\left(\left|x-2\right|\right)-0.03125\ln\mathopen{}\left(\left|x+2\right|\right)+C\)
6.6.3.41. Answer .
\(\tan^{-1}\mathopen{}\left(e^{x}\right)+C\)
6.6.3.43. Answer .
\(x\tanh^{-1}\mathopen{}\left(x\right)+0.5\ln\mathopen{}\left(\left|x^{2}-1\right|\right)+C\)
6.7 L’Hospital’s Rule 6.7.4 Exercises
Problems
6.7.4.9. 6.7.4.11. 6.7.4.13. 6.7.4.15. 6.7.4.17. 6.7.4.19. 6.7.4.21. 6.7.4.23. 6.7.4.25. 6.7.4.27. 6.7.4.29. 6.7.4.31. 6.7.4.33. 6.7.4.35. 6.7.4.37. 6.7.4.39. 6.7.4.41. 6.7.4.43. 6.7.4.45. 6.7.4.47. 6.7.4.49. 6.7.4.51. 6.7.4.53.
6.8 Improper Integration 6.8.4 Exercises
Problems
6.8.4.7. 6.8.4.9. 6.8.4.11. Answer .
\(\frac{1}{\ln\mathopen{}\left(2\right)}\)
6.8.4.13. 6.8.4.15. 6.8.4.17. 6.8.4.19. 6.8.4.21. 6.8.4.23. 6.8.4.25. 6.8.4.27. 6.8.4.29. 6.8.4.31. 6.8.4.33.
6.8.4.35.
Answer 1 .
\(\text{Limit Comparison Test}\)
Answer 2 . Answer 3 . 6.8.4.37.
Answer 1 .
\(\text{Limit Comparison Test}\)
Answer 2 . Answer 3 . 6.8.4.39.
Answer 1 .
\(\text{Direct Comparison Test}\)
Answer 2 . Answer 3 . 6.8.4.41.
Answer 1 .
\(\text{Direct Comparison Test}\)
Answer 2 . Answer 3 . 6.8.4.43.
Answer 1 .
\(\text{Direct Comparison Test}\)
Answer 2 . Answer 3 .
7 Applications of Integration 7.1 Area Between Curves
Exercises Problems
7.1.5. 7.1.7. 7.1.9. 7.1.11. 7.1.19. Answer .
All enclosed regions have the same area, with regions being the reflection of adjacent regions. One region is formed on
\([\pi/4,5\pi/4]\text{,}\) with area
\(2\sqrt{2}\text{.}\)
7.1.31.
7.2 Volume by Cross-Sectional Area; Disk and Washer Methods
Exercises Problems
7.2.13. 7.2.13.a 7.2.13.b 7.2.13.c 7.2.13.d 7.2.15. 7.2.15.a 7.2.15.b 7.2.15.c 7.2.17. 7.2.17.a 7.2.17.b 7.2.17.c 7.2.17.d
7.3 The Shell Method
Exercises Problems
7.3.5. 7.3.7. Answer .
\(\pi^2-2\pi\) units
\(^3\)
7.3.9. Answer .
\(48\pi\sqrt{3}/5\) units
\(^3\)
7.3.11.
7.3.13. 7.3.13.a 7.3.13.b 7.3.13.c 7.3.13.d 7.3.15. 7.3.15.a 7.3.15.b 7.3.15.c 7.3.15.d 7.3.17. 7.3.17.a 7.3.17.b Answer .
\(2\pi(1-\sqrt{2}+\sinh^{-1}(1))\)
7.4 Arc Length and Surface Area 7.4.3 Exercises
Problems
7.4.3.3. 7.4.3.5. 7.4.3.7. 7.4.3.9. 7.4.3.11. Answer .
\(-\ln(2-\sqrt{3}) \approx 1.31696\)
7.4.3.13. Answer .
\(\int_0^1 \sqrt{1+4x^2}\, dx\)
7.4.3.15. Answer .
\(\int_1^e \sqrt{1+\frac1{x^2}}\, dx\)
7.4.3.17. Answer .
\(\int_0^{\pi/2}\sqrt{1+\sin^2(x)}\,dx\)
7.4.3.19. 7.4.3.21. 7.4.3.23.
7.4.3.25. Answer .
\(2\pi\int_0^1 2x\sqrt{5}\, dx = 2\pi\sqrt{5}\)
7.4.3.27. Answer .
\(2\pi\int_0^1 x\sqrt{1+4x^2}\, dx = \pi/6(5\sqrt{5}-1)\)
7.4.3.29. Answer .
\(\int_0^1 \sqrt{1+\frac{1}{4x}}\, dx\)
7.4.3.31. Answer .
\(\int_{-3}^3 \sqrt{1+\frac{x^2}{81-9x^2}}\, dx\)
7.4.3.33. Answer .
\(2\pi\int_0^1 \sqrt{1-x^2}\sqrt{1+x/(1-x^2)}\, dx = 4\pi\)
7.5 Work 7.5.4 Exercises
Terms and Concepts 7.5.4.1. Answer .
In SI units, it is one joule, i.e., one newton–meter, or
kg·m ⁄s2 m In Imperial Units, it is ft–lb.
Problems 7.5.4.5. 7.5.4.5.a 7.5.4.5.b Answer .
\(100-50\sqrt{2} \approx 29.29\) ft
7.5.4.7. 7.5.4.7.a Answer .
\(\frac12\cdot d\cdot l^2\) ft–lb
7.5.4.7.b 7.5.4.7.c Answer .
\(\ell(1-\sqrt{2}/2) \approx 0.2929\ell\)
7.5.4.9. 7.5.4.9.a 7.5.4.9.b 7.5.4.9.c Answer .
Yes, for the cable accounts for about 1% of the total work.
7.5.4.11. 7.5.4.13. 7.5.4.15. 7.5.4.17. 7.5.4.19. 7.5.4.21. 7.5.4.21.a 7.5.4.21.b 7.5.4.21.c Answer .
When
3.83 ft of water have been pumped from the tank, leaving about
2.17 ft in the tank.
7.5.4.23. 7.5.4.25. 7.5.4.27.
7.6 Fluid Forces
Exercises Problems
7.6.3. 7.6.5. 7.6.7. 7.6.9. 7.6.11. 7.6.19.
8 Differential Equations 8.1 Graphical and Numerical Solutions to Differential Equations 8.1.4 Exercises
Terms and Concepts 8.1.4.1. Answer .
An initial value problems is a differential equation that is paired with one or more initial conditions. A differential equation is simply the equation without the initial conditions.
8.1.4.3. Answer .
Substitute the proposed function into the differential equation, and show the the statement is satisfied.
8.1.4.5. Answer .
Many differential equations are impossible to solve analytically.
Problems
8.1.4.13. Answer .
The
\(x\) and
\(y\) axes are uncalibrated.In the first quadrant in the top left, the field lines are north-east facing and in the bottom right they are southeast facing. In the second quadrant the field lines are all north-east facing. In the third quadrant like in the first quadrant in the top left the field lines are northeast facing and in the bottom right they are southeast facing. In the fourth quadrant all lines are southeast facing.
8.1.4.15. Answer .
The
\(x\) and
\(y\) axes are uncalibrated. There are five instances where the field lines run parallel to the
\(x\) axis. One of them is on the
\(x\) axis itself, other two pairs of such field lines are above and below the
\(x\) axis. In between the
\(x\) axis and the first horizontal field line for some positive
\(y\) value, the field lines are all northeast facing. Above the horizontal field line for some
\(y\) value until another with a higher
\(y\) value, the field lines in between are southeast facing.
Similarly below the
\(x\) axis till the first horizontal line with some negative
\(y\) value, the field lines in between are southeast facing. In between this horizontal line and another horizontal line with a higher negative
\(y\) value, the field lines are northeast facing.
8.1.4.19. Answer .
The
\(x\) and
\(y\) axes are uncalibrated, the field lines in the first quadrant are shown. Front left to right, a little away from the x axis the field lines are northeast facing that transition to north facing. Moving further right then again become northeast facing then transition to southeast facing, further right they become south facing then east facing. The pattern then repeats. Very close to the
\(x\) axis the field lines are almost parallel to it.
A wave is drawn that starts at some y intercept above the origin. It has a high positive slope, it reaches peak when the field lines change from northeast facing to southeast facing, then it declines until the point the field lines are parallel to the
\(x\) axis. The curve continues to form a second wave.
8.1.4.21. Answer .
The
\(x\) and
\(y\) axes are uncalibrated, the field lines in the first quadrant are shown. In the top right and the centre the field lines are southeast facing, very close to the
\(x\) and
\(y\) axis the field lines are almost parallel to the
\(x\) axis. A curve is drawn that starts from a
\(y\) intercept and decreases along the slope lines coming close to the
\(x\) axis.
8.1.4.23. Answer .
\begin{align*}
x_i \amp \quad \amp \quad \amp y_i \\
0.0 \amp \quad \amp \quad \amp 1.0000 \\
0.1 \amp \quad \amp \quad \amp 1.0000 \\
0.2 \amp \quad \amp \quad \amp 1.0037 \\
0.3 \amp \quad \amp \quad \amp 1.0110 \\
0.4 \amp \quad \amp \quad \amp 1.0219 \\
0.5 \amp \quad \amp \quad \amp 1.0363
\end{align*}
8.1.4.25. Answer .
\begin{align*}
x_i \amp \quad \amp \quad \amp y_i \\
0.0 \amp \quad \amp \quad \amp 0.0000 \\
0.5 \amp \quad \amp \quad \amp 0.5000 \\
1.0 \amp \quad \amp \quad \amp 1.8591 \\
1.5 \amp \quad \amp \quad \amp 10.5824 \\
2.0 \amp \quad \amp \quad \amp 88378.1190
\end{align*}
8.1.4.27. Answer .
\(x\) \(0.0\) \(0.2\) \(0.4\) \(0.6\) \(0.8\) \(1.0\)
\(y(x)\) 0.5000
0.5412
0.6806
0.9747
1.5551
2.7183
\(h = 0.2\) 0.5000
0.5000
0.5816
0.7686
1.1250
1.7885
\(h = 0.1\) 0.5000
0.5201
0.6282
0.8622
1.3132
2.1788
8.2 Separable Differential Equations 8.2.2 Exercises
Problems
8.2.2.1. Answer .
Separable.
\(\displaystyle \frac{1}{y^2-y}\,dy = dx\)
8.2.2.3.
8.2.2.5. Answer .
\(\left\{ \displaystyle y = \frac{1 + Ce^{2x}}{1 - Ce^{2x}}, y = -1\right\}\)
8.2.2.7. 8.2.2.9. Answer .
\(\displaystyle (y-1)e^y = -e^{-x} - \frac{1}{3}e^{-3x} + C\)
8.2.2.11. Answer .
\(\left\{ \arcsin{2y} - \arctan(x^2+1) = C, y = \pm \displaystyle \frac{1}{2} \right\}\)
8.2.2.13. Answer .
\(\sin(y) + \cos(x) = 2\)
8.2.2.15. Answer .
\(\frac{1}{2}y^2 - \ln(1+x^2) = 8\)
8.2.2.17. Answer .
\(\displaystyle \frac{1}{2}y^2 - y = \frac{1}{2}\big ( (x^2+1)\ln(x^2+1) - (x^2 + 1)\big) + \frac{1}{2}\)
8.2.2.19. Answer .
\(2\tan(2y) = 2x + \sin(2x)\)
8.3 First Order Linear Differential Equations 8.3.2 Exercises
Problems
8.3.2.1. Answer .
\(y = \displaystyle \frac{3}{2} + Ce^{2x}\)
8.3.2.3. Answer .
\(y = \displaystyle -\frac{1}{2x} + Cx\)
8.3.2.5. Answer .
\(y = \sec x + C(\csc x)\)
8.3.2.7. Answer .
\(y = \displaystyle Ce^{3x}-(x+1)e^{2x}\)
8.3.2.9. 8.3.2.11. Answer .
\(y = \displaystyle 1 - \frac{2}{x} + \frac{2-e^{1-x}}{x^2}\)
8.3.2.13. Answer .
\(y = \displaystyle \frac{x^2+1}{x+1}e^{-x}\)
8.3.2.15. Answer .
\(y = \displaystyle \frac{(x-2)(x+1)}{x-1}\)
8.3.2.17. Answer .
Both;
\(\displaystyle y = -5e^{x + \frac{1}{3}x^3}\)
8.3.2.19. Answer .
linear;
\(\displaystyle y = \frac{x^3-3x-6}{3(x-1)}\)
8.3.2.21. Answer .
The
\(x\) and
\(y\) axes are uncalibrated, the field lines in the first quadrant are shown. On the bottom right the field lines are facing northeast. On the top left the field lines transition from southeast facing to east facing moving downwards. A curve is shown that almost represents a straight line with a positive slope.
The solution will increase and begin to follow the line
\(y=x-1\text{.}\)
8.4 Modeling with Differential Equations 8.4.3 Exercises
Problems
8.4.3.1. 8.4.3.3. 8.4.3.5. Answer .
\(x = \begin{cases}\displaystyle\frac{ab(1 - e^{(a-b)kt})}{b-ae^{(a-b)kt}} \amp \text{ if } a \neq b\\ \displaystyle \frac{a^2kt}{1+akt} \amp \text{ if } a = b \end{cases}\)
8.4.3.7. Answer .
\(\displaystyle y = 60 - 3.69858e^{-\frac{1}{4}t} + 43.69858e^{-0.0390169 t}\)
8.4.3.9. Answer .
\(y = 8(1-e^{-\frac{1}{2}t})\) g/cm
\(^2\)
8.4.3.11.
9 Sequences and Series 9.1 Sequences
Exercises Problems
9.1.5. Answer .
\(2,\frac{8}{3},\frac{8}{3},\frac{32}{15},\frac{64}{45}\)
9.1.7. Answer .
\(-\frac{1}{3},-2,-\frac{81}{5},-\frac{512}{3},-\frac{15625}{7}\)
9.1.9. 9.1.11. Answer .
\(a_n = 10\cdot 2^{n-1}\)
9.1.17. 9.1.19. 9.1.21. 9.1.23. 9.1.25. 9.1.27.
9.1.29. 9.1.31. 9.1.33. Answer .
neither bounded above or below
9.2 Infinite Series 9.2.4 Exercises
Terms and Concepts 9.2.4.1.
9.3 Integral and Comparison Tests 9.3.4 Exercises
Problems
9.3.4.5. 9.3.4.7. 9.3.4.9. 9.3.4.11.
9.4 Ratio and Root Tests 9.4.3 Exercises
Terms and Concepts 9.4.3.3. Answer .
Integral Test, Limit Comparison Test, and Root Test
Problems
9.4.3.5. 9.4.3.7. 9.4.3.9. Answer .
The Ratio Test is inconclusive; the
\(p\) -Series Test states it diverges.
9.4.3.11. 9.4.3.13. Answer .
Converges; note the summation can be rewritten as
\(\ds\infser \frac{2^nn!}{3^nn!}\text{,}\) from which the Ratio Test or Geometric Series Test can be applied.
9.4.3.15. 9.4.3.17. 9.4.3.19. 9.4.3.21. Answer .
Diverges. The Root Test is inconclusive, but the
\(n\) th-Term Test shows divergence. (The terms of the sequence approach
\(e^{-2}\text{,}\) not 0, as
\(n\to\infty\text{.}\) )
9.4.3.23.
9.5 Alternating Series and Absolute Convergence
Exercises Terms and Concepts 9.5.3. Answer .
Many examples exist; one common example is
\(a_n = (-1)^n/n\text{.}\)
9.7 Taylor Polynomials
Exercises Terms and Concepts 9.7.3. Problems
9.7.5. Answer .
\(1-x+0.5x^{2}-0.166667x^{3}\)
9.7.7. Answer .
\(x+x^{2}+0.5x^{3}+0.166667x^{4}+0.0416667x^{5}\)
9.7.9. Answer .
\(1+2x+2x^{2}+1.33333x^{3}+0.666667x^{4}\)
9.7.11. Answer .
\(1-x+x^{2}-x^{3}+x^{4}\)
9.7.13. Answer .
\(1+0.5\mathopen{}\left(x-1\right)-0.125\mathopen{}\left(x-1\right)^{2}+0.0625\mathopen{}\left(x-1\right)^{3}-0.0390625\mathopen{}\left(x-1\right)^{4}\)
9.7.15. Answer .
\(0.707107-0.707107\mathopen{}\left(x-\frac{\pi }{4}\right)-0.353553\mathopen{}\left(x-\frac{\pi }{4}\right)^{2}+0.117851\mathopen{}\left(x-\frac{\pi }{4}\right)^{3}+0.0294628\mathopen{}\left(x-\frac{\pi }{4}\right)^{4}-0.00589256\mathopen{}\left(x-\frac{\pi }{4}\right)^{5}-0.000982093\mathopen{}\left(x-\frac{\pi }{4}\right)^{6}\)
9.7.17. Answer .
\(0.5-0.25\mathopen{}\left(x-2\right)+0.125\mathopen{}\left(x-2\right)^{2}-0.0625\mathopen{}\left(x-2\right)^{3}+0.03125\mathopen{}\left(x-2\right)^{4}-0.015625\mathopen{}\left(x-2\right)^{5}\)
9.7.19. Answer .
\(0.5+0.5\mathopen{}\left(x+1\right)+0.25\mathopen{}\left(x+1\right)^{2}\)
9.7.31. Answer .
The
\(n\) th term is: when
\(n\) even, 0; when
\(n\) is odd,
\(\frac{(-1)^{(n-1)/2}}{n!}x^n\text{.}\)
10 Curves in the Plane 10.1 Conic Sections 10.1.4 Exercises
Problems 10.1.4.19. Answer .
\(\frac{(x+1)^2}{9}+\frac{(y-2)^2}{4}=1\text{;}\) foci at
\((-1\pm\sqrt{5},2)\text{;}\) \(e=\sqrt{5}/3\)
10.1.4.29. 10.1.4.31. Answer .
\(\frac{(y-3)^2}{4}-\frac{(x-1)^2}{9}=1\)
10.1.4.45. Answer .
The sound originated from a point approximately 31m to the right of
\(B\) and 1390m above or below it. (Since the three points are collinear, we cannot distinguish whether the sound originated above/below the line containing the points.)
10.2 Parametric Equations 10.2.4 Exercises
Problems
10.2.4.5. Answer .
The sketch for this exercise is a curve that lies mostly in the fourth quadrant. It resembles part of a slingshot orbit for a comet passing around the sun: the curve passes through the origin from below, turns quickly in the second quadrant, crossing the
\(y\) axis at
\((0,1)\text{,}\) and then the
\(x\) axis at
\((2,0)\text{,}\) where it returns to the fourth quadrant.
10.2.4.7. Answer .
The horizontal line
\(y=2\text{.}\) On the line there are two arrows pointing in opposite directions. These indicate that the direction of travel is to the left when
\(t\lt 0\text{,}\) and to the right when
\(t\gt 0\text{.}\)
10.2.4.9. Answer .
A curve resembling a check mark, with a cusp at the origin. Direction of travel is from the second quadrant toward the cusp, and then up from the cusp to a
\(y\) intercept at
\((0,4)\text{,}\) and then into the first quadrant.
10.2.4.11. Answer .
The curve is an ellipse, centered at the origin, with counter-clockwise direction of travel.
10.2.4.13. Answer .
The curve resembles a parabola, with vertex at
\((0,-1)\text{.}\) The direction of travel is from right to left.
10.2.4.15. Answer .
The curve resembles one branch of a hyperbola, opening to the right, with a vertex at
\((2,0)\text{.}\) The direction of travel is that of increasing
\(y\) value.
10.2.4.17. Answer .
A flower-shaped curve, with 7 “petals”. Each petal is an arc that loops around and intersects itself before continuing to the next arc.
10.2.4.19. 10.2.4.19.a Answer .
Traces the parabola
\(y=x^2\text{,}\) moves from left to right.
10.2.4.19.b Answer .
Traces the parabola
\(y=x^2\text{,}\) but only from
\(-1\leq x\leq 1\text{;}\) traces this portion back and forth infinitely.
10.2.4.19.c Answer .
Traces the parabola
\(y=x^2\text{,}\) but only for
\(0\lt x\text{.}\) Moves left to right.
10.2.4.19.d Answer .
Traces the parabola
\(y=x^2\text{,}\) moves from right to left.
10.2.4.35.
Answer 1 . Answer 2 . Answer 3 . Answer 4 . Answer 5 . 10.2.4.37.
Answer 1 .
\(\cos^{-1}\mathopen{}\left(t\right)\)
Answer 2 . Answer 3 . Answer 4 .
\(\cos\mathopen{}\left(x\right)\)
Answer 5 . 10.2.4.51. Answer .
\(3\cos\mathopen{}\left(2\pi t\right)+1;\,3\sin\mathopen{}\left(2\pi t\right)+1\)
10.3 Calculus and Parametric Equations 10.3.4 Exercises
Problems 10.3.4.15.
Answer 1 . Answer 2 .
\(\left(0.75,-0.25\right)\)
10.3.4.27.
Answer 1 .
\(-\frac{4}{\left(2t-1\right)^{3}}\)
Answer 2 .
\(\left(-\infty ,0.5\right]\)
Answer 3 .
\(\left[0.5,\infty \right)\)
10.4 Introduction to Polar Coordinates 10.4.4 Exercises
Problems 10.4.4.5. Answer .
On a polar grid, four points are plotted. The point
\(A\) is at the intersection of the initial ray and the circle of radius 2. Points
\(B\) and
\(D\) are both on the circle of radius 1. The point
\(B\) is on the same line as the initial ray, but in the opposite direction. The point
\(D\) lies above the initial ray, making an angle of
\(\pi/4\text{.}\) Finally, the point
\(C\) is at the bottom of the circle of radius
\(2\text{.}\)
10.4.4.7. Answer .
\(A=P(2.5,\pi/4)\) and
\(P(-2.5,5\pi/4)\text{;}\)
\(B=P(-1,5\pi/6)\) and
\(P(1,11\pi/6)\text{;}\)
\(C=P(3,4\pi/3)\) and
\(P(-3,\pi/3)\text{;}\)
\(D=P(1.5,2\pi/3)\) and
\(P(-1.5,5\pi/3)\text{;}\)
10.4.4.9.
Answer 1 .
\(\left(\sqrt{2},\sqrt{2}\right)\)
Answer 2 .
\(\left(\sqrt{2},-\sqrt{2}\right)\)
Answer 3 .
\(\left(\sqrt{5},\tan^{-1}\mathopen{}\left(\frac{-1}{2}\right)\right)\)
Answer 4 .
\(\left(\sqrt{5},\pi +\tan^{-1}\mathopen{}\left(\frac{-1}{2}\right)\right)\)
10.4.4.11. Answer .
An arc of the circle
\(r=2\) is shown, for
\(0\leq \theta\leq \pi/2\text{.}\) This is the quarter of a circle of radius 2, centered at the origin, that lies in the first quadrant.
10.4.4.13. Answer .
The curve is a cardioid that is symmetric about the
\(x\) axis. The cusp is at the origin, and the other
\(x\) intercept is at
\((-2,0)\text{.}\) (It is in the opposite direction of the example in the gallery of polar curves.)
10.4.4.15. Answer .
The curve is a convex limaçon. This is the fourth type of limaçon in the gallery of polar curves. In this case, the limaçon is symmetric about the
\(y\) axis, with the flattened part of the curve at the top.
10.4.4.17. Answer .
The curve is a limaçon with an inner loop. It is symmetric about the
\(y\) axis. The inner loop lies above the
\(x\) axis, with
\(y\) intercepts at
\(y=0\) and
\(y=1\text{.}\) The outer loop has its other
\(y\) intercept at
\(y=3\text{.}\)
10.4.4.19. Answer .
A rose curve with three loops that all pass through the origin. One loop is along the negative
\(y\) axis, with a
\(y\) intercept at
\((-1,0)\text{.}\) The other two loops lie in the first and second quadrants.
10.4.4.21. Answer .
This is a more complicated curve. It passes several times through the origin, and has eight other points of self-intersection. The largest loops in the curve are similar to cardioids; there are four of these passing through the origin, with a second intercept at one of the four points
\((\pm 1, 0)\text{,}\) \((0,\pm 1)\text{.}\) As these loops intersect each other, they create four other loops of intermediate size, and four smaller loops in the center.
10.4.4.23. Answer .
A circle of radius
\(3/2\) with its center at
\((0,3/2)\text{.}\) It passes through the origin and the point
\((0,3)\text{.}\)
10.4.4.25. Answer .
The curve is a four-leafed rose that lies within the circle
\(r=1/2\text{.}\) One leaf lies in each of the four quadrants.
10.4.4.27. Answer .
The curve is a straight line with
\(x\) intercept at
\((-3,0)\) and
\(y\) intercept
\((0,3/5)\text{.}\)
10.4.4.29. Answer .
The curve is the vertical line
\(x=3\text{.}\)
10.4.4.31. Answer .
\(\left(x-3\right)^{2}+y^{2} = 9\)
10.4.4.33. Answer .
\(\left(x-0.5\right)^{2}+\left(y-0.5\right)^{2} = 0.5\)
10.4.4.35. 10.4.4.39.
10.4.4.41. Answer .
\(\theta = \frac{\pi }{4}\)
10.4.4.43. Answer .
\(r = 5\sec\mathopen{}\left(\theta\right)\)
10.4.4.45. Answer .
\(r = \frac{\cos\mathopen{}\left(\theta\right)}{\sin^{2}\mathopen{}\left(\theta\right)}\)
10.4.4.47.
10.4.4.49. Answer .
\(P\left(\frac{\sqrt{3}}{2},\frac{\pi }{6}\right), P\left(0,\frac{\pi }{2}\right), P\left(\frac{-\sqrt{3}}{2},\frac{5\pi }{6}\right)\)
10.4.4.51. Answer .
\(P\left(0,0\right), P\left(\sqrt{2},\frac{\pi }{4}\right)\)
10.5 Calculus and Polar Functions 10.5.5 Exercises
Problems
10.5.5.3.
Answer 1 .
\(-\cot\mathopen{}\left(\theta\right)\)
Answer 2 .
\(y = -\left(x-\frac{\sqrt{2}}{2}\right)+\frac{\sqrt{2}}{2}\)
Answer 3 . 10.5.5.7.
Answer 1 .
\(\frac{\theta\cos\mathopen{}\left(\theta\right)+\sin\mathopen{}\left(\theta\right)}{\cos\mathopen{}\left(\theta\right)-\theta\sin\mathopen{}\left(\theta\right)}\)
Answer 2 .
\(y = \frac{-2}{\pi }x+\frac{\pi }{2}\)
Answer 3 .
\(y = \frac{\pi }{2}x+\frac{\pi }{2}\)
10.5.5.9.
Answer 1 .
\(\frac{4\sin\mathopen{}\left(\theta\right)\cos\mathopen{}\left(4\theta\right)+\sin\mathopen{}\left(4\theta\right)\cos\mathopen{}\left(\theta\right)}{4\cos\mathopen{}\left(\theta\right)\cos\mathopen{}\left(4\theta\right)-\sin\mathopen{}\left(\theta\right)\sin\mathopen{}\left(4\theta\right)}\)
Answer 2 .
\(y = 5\sqrt{3}\mathopen{}\left(x+\frac{\sqrt{3}}{4}\right)-\frac{3}{4}\)
Answer 3 .
\(y = \frac{-1}{5\sqrt{3}}\mathopen{}\left(x+\frac{\sqrt{3}}{4}\right)-\frac{3}{4}\)
10.5.5.19. 10.5.5.21. 10.5.5.23. Answer .
\(2\pi +\frac{3\cdot 1.73205}{2}\)
10.5.5.25.
10.5.5.29. 10.5.5.31. 10.5.5.33. Answer .
\(2.2592\hbox{ or }2.22748\)
11 Vectors 11.1 Introduction to Cartesian Coordinates in Space 11.1.7 Exercises
Terms and Concepts 11.1.7.5. Answer .
A hyperboloid of two sheets
Problems 11.1.7.7.
Answer 1 . Answer 2 . Answer 3 . Answer 4 . 11.1.7.9.
11.1.7.19. Answer .
\(x^{2}+z^{2} = \left(\frac{1}{1+y^{2}}\right)^{2}\)
11.1.7.21.
11.1.7.23. Answer .
(a)
\(\ds x=y^2+\frac{z^2}{9}\)
11.1.7.25. Answer .
(b)
\(\ds x^2+\frac{y^2}9+\frac{z^2}4=1\)
11.1.7.27. 11.1.7.29. 11.1.7.31.
11.2 An Introduction to Vectors
Exercises Terms and Concepts 11.2.3. Answer .
A vector with magnitude 1.
Problems
11.2.7. 11.2.7.a 11.2.7.b Answer .
\(\boldsymbol{i}+6\boldsymbol{j}\)
11.2.9. 11.2.9.a 11.2.9.b Answer .
\(6\boldsymbol{i}-\boldsymbol{j}+6\boldsymbol{k}\)
11.2.11. 11.2.11.a Answer .
\(\vec u+\vec v = \la 2,-1\ra\text{;}\) \(\vec u -\vec v = \la0,-3\ra\text{;}\) \(2\vec u-3\vec v = \la -1,-7\ra\text{.}\)
11.2.11.c Answer .
\(\vec x = \la 1/2,2\ra\text{.}\)
11.2.17.
Answer 1 . Answer 2 . Answer 3 . Answer 4 . 11.2.19.
Answer 1 . Answer 2 . Answer 3 . Answer 4 .
11.2.23. 11.2.25. Answer .
\(\left<\frac{1}{\sqrt{3}},\frac{-1}{\sqrt{3}},\frac{1}{\sqrt{3}}\right>\)
11.2.27. Answer .
\(\left<\frac{-1}{2},\frac{\sqrt{3}}{2}\right>\)
11.3 The Dot Product 11.3.2 Exercises
Problems
11.3.2.5. 11.3.2.7. 11.3.2.9. 11.3.2.11.
11.3.2.13. Answer .
\(\cos^{-1}\mathopen{}\left(\frac{3}{\sqrt{10}}\right)\)
11.3.2.15.
11.3.2.21. Answer .
\(\left<\frac{-5}{10},\frac{15}{10}\right>\)
11.3.2.23. Answer .
\(\left<\frac{-1}{2},\frac{-1}{2}\right>\)
11.3.2.25. Answer .
\(\left<\frac{14}{14},\frac{28}{14},\frac{42}{14}\right>\)
11.3.2.27.
Answer 1 .
\(\left<\frac{-5}{10},\frac{15}{10}\right>\)
Answer 2 .
\(\left<\frac{15}{10},\frac{5}{10}\right>\)
11.3.2.29.
Answer 1 .
\(\left<\frac{-1}{2},\frac{-1}{2}\right>\)
Answer 2 .
\(\left<\frac{-5}{2},\frac{5}{2}\right>\)
11.3.2.31.
Answer 1 .
\(\left<\frac{14}{14},\frac{28}{14},\frac{42}{14}\right>\)
Answer 2 .
\(\left<\frac{0}{14},\frac{42}{14},\frac{-28}{14}\right>\)
11.3.2.33. 11.3.2.35. 11.3.2.37. 11.3.2.39.
11.4 The Cross Product 11.4.3 Exercises
Problems
11.4.3.7. Answer .
\(\left<12,-15,3\right>\)
11.4.3.9. Answer .
\(\left<-5,-31,27\right>\)
11.4.3.11. 11.4.3.13. Answer .
\(\vec u\times \vec v = \langle 0,0,ad-bc\rangle\)
11.4.3.15. 11.4.3.17.
11.4.3.35. Answer .
\(\left<0.408248,0.408248,-0.816497\right>\hbox{ or }\left<-0.408248,-0.408248,0.816497\right>\)
11.4.3.37. Answer .
\(\left<0,1,0\right>\hbox{ or }\left<0,-1,0\right>\)
11.4.3.39. 11.4.3.41. Answer .
\(200/3\approx 66.67\) ft–lb
11.5 Lines 11.5.4 Exercises
Terms and Concepts 11.5.4.1. Answer .
A point on the line and the direction of the line.
Problems 11.5.4.11.
Answer 1 .
\(\left(7,2,-1\right)+t\mathopen{}\left<1,-1,2\right>\)
Answer 2 .
\(x = 7+t, y = 2-t, z = -1+2t\)
Answer 3 .
\(x-7 = 2-y = \frac{z+1}{2}\)
11.6 Planes 11.6.2 Exercises
Terms and Concepts 11.6.2.1. Answer .
A point in the plane and a normal vector (i.e., a direction orthogonal to the plane).
Problems
11.6.2.17. 11.6.2.19.
Answer 1 .
\(3\mathopen{}\left(x+4\right)+8\mathopen{}\left(y-7\right)-10\mathopen{}\left(z-2\right) = 0\)
Answer 2 .
12 Vector Valued Functions 12.1 Vector-Valued Functions 12.1.4 Exercises
Problems 12.1.4.15. Answer .
Graph of the function \(\vec r(t) = \la \cos(t) , \sin(t) ,\sin(t) \ra\) on \([0,2\pi]\text{.}\) The graph of the function is an oval lying in the plane coming from rotating the \(xy\) plane \(45\) degrees towards the \(z\) -axis. The oval lying in this plane has a horizontal width of \(\sqrt{2}\) and a height of \(1\text{.}\) Ignoring the \(z\) coordinate, the curve is a unit circle in the \(xy\) plane. Similarly ignoring the \(y\) coordinate, the curve is a unit circle in the \(xz\) plane. If we now ignore the \(x\) coordinate, the resulting curve is a diagonal line given by \(z=y\) in the \(yz\) plane. This line turns back on itself, which can be seen in the image of the oval when considering all three coordinate axes.
12.1.4.17. Answer .
\(\left|t\right|\sqrt{1+t^{2}}\)
12.1.4.19.
12.1.4.21. Answer .
\(\left<2\cos\mathopen{}\left(t\right)+1,2\sin\mathopen{}\left(t\right)+2\right>\)
12.1.4.25. Answer .
\(\left<t+2,5t+3\right>\)
12.1.4.27. Answer .
Specific forms may vary, though most direct solutions are
\(\vec r(t) = \la 1,2,3\ra +t\la 3,3,3\ra\) and
\(\vec r(t) = \la 3t+1, 3t+2, 3t+3\ra\text{.}\)
12.1.4.29. Answer .
\(\left<2\cos\mathopen{}\left(t\right),2\sin\mathopen{}\left(t\right),2t\right>\)
12.2 Calculus and Vector-Valued Functions 12.2.5 Exercises
Problems
12.2.5.5. Answer .
\(\left<11,74,\sin\mathopen{}\left(5\right)\right>\)
12.2.5.7. 12.2.5.9. Answer .
\(\left(-\infty ,0\right)\cup \left(0,\infty \right)\)
12.2.5.11. Answer .
\(\left<-\sin\mathopen{}\left(t\right),e^{t},\frac{1}{t}\right>\)
12.2.5.13. Answer .
\(\left<2t\sin\mathopen{}\left(t\right)+t^{2}\cos\mathopen{}\left(t\right),6t^{2}+10t\right>\)
12.2.5.15. Answer .
\(\left<-1,\cos\mathopen{}\left(t\right)-2t,6t^{2}+10t+2+\cos\mathopen{}\left(t\right)-\sin\mathopen{}\left(t\right)-t\cos\mathopen{}\left(t\right)\right>\)
12.2.5.21. 12.2.5.23. Answer .
\(\ell(t) = \la -3,0,\pi\ra + t\la0,-3,1\ra\)
12.2.5.33. Answer .
\(\la \frac14t^4,\sin(t) ,te^t-e^t\ra + \vec C\)
12.2.5.35.
12.2.5.37. Answer .
\(\left<\frac{t^{2}}{2}+2,-\cos\mathopen{}\left(t\right)+3\right>\)
12.2.5.39. Answer .
\(\left<\frac{t^{4}}{12}+t+4,\frac{t^{3}}{6}+2t+5,\frac{t^{2}}{2}+3t+6\right>\)
12.2.5.41. 12.2.5.43. Answer .
\(\frac{1}{54}\mathopen{}\left(22^{\frac{3}{2}}-8\right)\)
12.3 The Calculus of Motion 12.3.3 Exercises
Problems 12.3.3.7. Answer .
\(\vvt = \la 2,5,0\ra\text{,}\) \(\vat = \la 0,0,0\ra\)
12.3.3.19.
Answer 1 .
\(\left|\sec\mathopen{}\left(t\right)\right|\sqrt{\tan^{2}\mathopen{}\left(t\right)+\sec^{2}\mathopen{}\left(t\right)}\)
Answer 2 . Answer 3 . 12.3.3.39. 12.3.3.39.a 12.3.3.39.b
12.4 Unit Tangent and Normal Vectors 12.4.4 Exercises
Terms and Concepts 12.4.4.3. Answer .
\(\unittangent(t)\) and
\(\unitnormal(t)\text{.}\)
Problems 12.4.4.5. Answer .
\(\unittangent(t) = \la\frac{4 t}{\sqrt{20 t^2-4t+1}},\frac{2 t-1}{\sqrt{20 t^2-4t+1}}\ra\text{;}\) \(\unittangent(1) = \la 4/\sqrt{17},1/\sqrt{17}\ra\)
12.4.4.9. Answer .
\(\left(2,0\right)+t\mathopen{}\left<\frac{4}{\sqrt{17}},\frac{1}{\sqrt{17}}\right>\)
12.4.4.13. Answer .
\(\unittangent(t) = \la -\sin(t) ,\cos(t) \ra\text{;}\) \(\unitnormal(t) = \la -\cos(t) ,-\sin(t) \ra\)
12.4.4.15. Answer .
\(\unittangent(t) = \la -\frac{\sin(t) }{\sqrt{4\cos^2(t) +\sin^2(t) }},\frac{2\cos(t) }{\sqrt{4\cos^2(t) +\sin^2(t) }}\ra\text{;}\) \(\unitnormal(t) = \la -\frac{2\cos(t) }{\sqrt{4\cos^2(t) +\sin^2(t) }},-\frac{\sin(t) }{\sqrt{4\cos^2(t) +\sin^2(t) }}\ra\)
12.5 The Arc Length Parameter and Curvature 12.5.4 Exercises
Terms and Concepts 12.5.4.3. Answer .
Answers may include lines, circles, helixes
Problems 12.5.4.15.
Answer 1 . Answer 2 .
\(\frac{\left|2\cos\mathopen{}\left(t\right)\cos\mathopen{}\left(2t\right)+4\sin\mathopen{}\left(t\right)\sin\mathopen{}\left(2t\right)\right|}{\left(4\cos^{2}\mathopen{}\left(2t\right)+\sin^{2}\mathopen{}\left(t\right)\right)^{\frac{3}{2}}}\)
Answer 3 . Answer 4 .
12.5.4.23. Answer .
\(\frac{\sqrt{2}}{\sqrt[4]{5}}, \frac{-\sqrt{2}}{\sqrt[4]{5}}\)
12.5.4.25.
13 Functions of Several Variables 13.2 Limits and Continuity of Multivariable Functions 13.2.5 Exercises
Terms and Concepts 13.2.5.3. Answer .
One possible answer:
\(\{(x,y) | x^2+y^2\leq 1\}\)
13.2.5.5. Answer .
One possible answer:
\(\{(x,y) | x^2+y^2\lt 1 \}\)
Problems 13.2.5.7. Answer .
Answers will vary. interior point:
\((1,3)\) boundary point:
\((3,3)\)
13.2.5.11. Answer .
\(D = \left\{(x,y)\, |\, 9-x^2-y^2\geq 0\right\}\text{.}\)
13.2.5.13. Answer .
\(D = \left\{(x,y)\, |\, y \gt x^2\right\}\text{.}\)
13.3 Partial Derivatives 13.3.7 Exercises
Problems 13.3.7.19.
Answer 1 .
\(\frac{2y^{2}}{\sqrt{4xy^{2}+1}}\)
Answer 2 .
\(\frac{4xy}{\sqrt{4xy^{2}+1}}\)
Answer 3 .
\(\frac{-4y^{4}}{\left(\sqrt{4xy^{2}+1}\right)^{3}}\)
Answer 4 .
\(\frac{-8xy^{3}}{\left(\sqrt{4xy^{2}+1}\right)^{3}}+\frac{4y}{\sqrt{4xy^{2}+1}}\)
Answer 5 .
\(\frac{-8xy^{3}}{\left(\sqrt{4xy^{2}+1}\right)^{3}}+\frac{4y}{\sqrt{4xy^{2}+1}}\)
Answer 6 .
\(\frac{-16x^{2}y^{2}}{\left(\sqrt{4xy^{2}+1}\right)^{3}}+\frac{4x}{\sqrt{4xy^{2}+1}}\)
13.5 The Multivariable Chain Rule 13.5.3 Exercises
Problems
13.5.3.7. Answer .
\(\frac{dz}{dt} = 3(2t)+4(2) = 6t+8\text{.}\)
At
\(t=1\text{,}\) \(\frac{dz}{dt} = 14\text{.}\)
13.5.3.9. Answer .
\(\displaystyle \frac{dz}{dt} = 5(-2\sin(t) )+2(\cos(t) ) = -10\sin(t) +2\cos(t)\)
At
\(t=\pi/4\text{,}\) \(\frac{dz}{dt} = -4\sqrt{2}\text{.}\)
13.5.3.11. Answer .
\(\ds\frac{dz}{dt} = 2x(\cos(t) ) + 4y(3\cos(t) )\text{.}\)
At
\(t=\pi/4\text{,}\) \(x=\sqrt{2}/2\text{,}\) \(y=3\sqrt{2}/2\text{,}\) and
\(\frac{dz}{dt} = 19\text{.}\)
13.5.3.21.
Answer 1 .
\(2x\cos\mathopen{}\left(t\right)+2y\sin\mathopen{}\left(t\right)\)
Answer 2 .
\(-2xs\sin\mathopen{}\left(t\right)+2ys\cos\mathopen{}\left(t\right)\)
Answer 3 . Answer 4 .
13.6 Directional Derivatives 13.6.3 Exercises
Problems
13.6.3.13. 13.6.3.13.a 13.6.3.13.b 13.6.3.15. 13.6.3.15.a 13.6.3.15.b 13.6.3.17. 13.6.3.17.a 13.6.3.17.b
13.6.3.19. 13.6.3.19.a Answer .
\(\nabla f(2,1) = \la -2,2\ra\)
13.6.3.19.b 13.6.3.19.c 13.6.3.19.d Answer .
\(\vec u = \la 1/\sqrt{2},1/\sqrt{2}\ra\)
13.6.3.21. 13.6.3.21.a Answer .
\(\nabla f(1,1) = \la -2/9,-2/9\ra\)
13.6.3.21.b 13.6.3.21.c 13.6.3.21.d Answer .
\(\vec u = \la 1/\sqrt{2},-1/\sqrt{2}\ra\)
13.6.3.23. 13.6.3.23.a 13.6.3.23.b 13.6.3.23.c 13.6.3.23.d
13.6.3.25. 13.6.3.25.a Answer .
\(\nabla F(x,y,z) = \la 6xz^3+4y, 4x, 9x^2z^2-6z\ra\)
13.6.3.25.b 13.6.3.27. 13.6.3.27.a Answer .
\(\nabla F(x,y,z) = \la 2xy^2, 2y(x^2-z^2), -2y^2z\ra\)
13.6.3.27.b
13.8 Extreme Values 13.8.3 Exercises
Problems 13.8.3.15.
Answer 1 . Answer 2 . Answer 3 . Answer 4 .
\(\left(0,\frac{-1}{2}\right)\)
14 Multiple Integration 14.1 Iterated Integrals and Area 14.1.4 Exercises
Problems
14.1.4.5. 14.1.4.5.a 14.1.4.5.b 14.1.4.7. 14.1.4.7.a 14.1.4.7.b 14.1.4.9. 14.1.4.9.a 14.1.4.9.b
14.3 Double Integration with Polar Coordinates
Exercises Problems
14.5 Surface Area
Exercises Problems
14.5.7. Answer .
\(\ds SA = \int_0^{2\pi}\int_0^{2\pi} \sqrt{1+ \cos^2(x) \cos^2(y) +\sin^2(x) \sin^2(y) }\, dx\, dy\)
14.5.9. Answer .
\(\ds SA = \int_{-1}^{1}\int_{-1}^{1} \sqrt{1+ 4x^2+4y^2}\, dx\, dy\)
14.6 Volume Between Surfaces and Triple Integration 14.6.4 Exercises
Problems
14.6.4.9. Answer .
\(dz\, dy\, dx\text{:}\) \(\ds\int_0^3\int_0^{1-x/3}\int_0^{2-2x/3-2y}\, dz\, dy\, dx\)
\(dz\, dx\, dy\text{:}\) \(\ds\int_0^1\int_0^{3-3y}\int_0^{2-2x/3-2y}\, dz\, dx\, dy\)
\(dy\, dz\, dx\text{:}\) \(\ds\int_0^3\int_0^{2-2x/3}\int_0^{1-x/3-z/2}\, dy\, dz\, dx\)
\(dy\, dx\, dz\text{:}\) \(\ds\int_0^2\int_0^{3-3z/2}\int_0^{1-x/3-z/2}\, dy\, dx\, dz\)
\(dx\, dz\, dy\text{:}\) \(\ds\int_0^1\int_0^{2-2y}\int_0^{3-3y-3z/2}\, dx\, dz\, dy\)
\(dx\, dy\, dz\text{:}\) \(\ds\int_0^2\int_0^{1-z/2}\int_0^{3-3y-3z/2}\, dx\, dy\, dz\)
\(\ds V = \int_0^3\int_0^{1-x/3}\int_0^{2-2x/3-2y}\, dz\, dy\, dx =1\text{.}\)
14.6.4.11. Answer .
\(dz\, dy\, dx\text{:}\) \(\ds\int_0^2\int_{-2}^{0}\int_{y^2/2}^{-y}\, dz\, dy\, dx\)
\(dz\, dx\, dy\text{:}\) \(\ds\int_{-2}^0\int_0^{2}\int_{y^2/2}^{-y}\, dz\, dx\, dy\)
\(dy\, dz\, dx\text{:}\) \(\ds\int_0^2\int_0^{2}\int_{-\sqrt{2z}}^{-z}\, dy\, dz\, dx\)
\(dy\, dx\, dz\text{:}\) \(\ds\int_0^2\int_0^{2}\int_{-\sqrt{2z}}^{-z}\, dy\, dx\, dz\)
\(dx\, dz\, dy\text{:}\) \(\ds\int_{-2}^0\int_{y^2/2}^{-y}\int_0^{2}\, dx\, dz\, dy\)
\(dx\, dy\, dz\text{:}\) \(\ds\int_0^2\int_{-\sqrt{2z}}^{-z}\int_0^{2}\, dx\, dy\, dz\) \(\ds V = \int_0^2\int_0^{2}\int_{-\sqrt{2z}}^{-z}\, dy\, dz\, dx =4/3\text{.}\)
14.6.4.13. Answer .
\(dz\, dy\, dx\text{:}\) \(\ds\int_0^2\int_{1-x/2}^{1}\int_{0}^{2x+4y-4}\, dz\, dy\, dx\)
\(dz\, dx\, dy\text{:}\) \(\ds\int_{0}^1\int_{2-2y}^{2}\int_{0}^{2x+4y-4}\, dz\, dx\, dy\)
\(dy\, dz\, dx\text{:}\) \(\ds\int_0^2\int_0^{2x}\int_{z/4-x/2+1}^{1}\, dy\, dz\, dx\)
\(dy\, dx\, dz\text{:}\) \(\ds\int_0^4\int_{z/2}^{2}\int_{z/4-x/2+1}^{1}\, dy\, dx\, dz\)
\(dx\, dz\, dy\text{:}\) \(\ds\int_{0}^1\int_{0}^{4y}\int_{z/2-2y+2}^2\, dx\, dz\, dy\)
\(dx\, dy\, dz\text{:}\) \(\ds\int_0^4\int_{z/4}^{1}\int_{z/2-2y+2}^2\, dx\, dy\, dz\) \(\ds V = \int_0^4\int_{z/4}^{1}\int_{z/2-2y-2}^2\, dx\, dy\, dz = 4/3\text{.}\)
14.6.4.15. Answer .
\(dz\, dy\, dx\text{:}\) \(\ds\int_{0}^1\int_{0}^{1-x^2}\int_{0}^{\sqrt{1-y}}\, dz\, dy\, dx\)
\(dz\, dx\, dy\text{:}\) \(\ds\int_{0}^1\int_{0}^{\sqrt{1-y}}\int_{0}^{\sqrt{1-y}}\, dz\, dx\, dy\)
\(dy\, dz\, dx\text{:}\) \(\ds\int_{0}^1\int_0^{x}\int_{0}^{1-x^2}\, dy\, dz\, dx + \int_{0}^1\int_x^{1}\int_{0}^{1-z^2}\, dy\, dz\, dx\)
\(dy\, dx\, dz\text{:}\) \(\ds\int_0^1\int_{0}^{z}\int_{0}^{1-z^2}\, dy\, dx\, dz+\int_0^1\int_{z}^{1}\int_{0}^{1-x^2}\, dy\, dx\, dz\)
\(dx\, dz\, dy\text{:}\) \(\ds\int_{0}^1\int_{0}^{\sqrt{1-y}}\int_{0}^{\sqrt{1-y}}\, dx\, dz\, dy\)
\(dx\, dy\, dz\text{:}\) \(\ds\int_0^1\int_{0}^{1-z^2}\int_{0}^{\sqrt{1-y}}\, dx\, dy\, dz\) Answers will vary. Neither order is particularly “hard.” The order
\(dz\, dy\, dx\) requires integrating a square root, so powers can be messy; the order
\(dy\, dz\, dx\) requires two triple integrals, but each uses only polynomials.
14.7 Triple Integration with Cylindrical and Spherical Coordinates 14.7.3 Exercises
Problems 14.7.3.11. Answer .
\(\ds\int_{\theta_1}^{\theta_2}\int_{r_1}^{r_2}\int_{z_1}^{z_2}h(r,\theta,z)r\, dz\, dr\, d\theta\)
14.7.3.19. Answer .
Describes the portion of the unit ball that resides in the first octant.
15 Vector Analysis 15.1 Introduction to Line Integrals 15.1.4 Exercises
Terms and Concepts 15.1.4.1. Answer .
When
\(C\) is a curve in the plane and
\(f\) is a function defined over
\(C\text{,}\) then
\(\int_C f(s)\, ds\) describes the area under the spatial curve that lies on
\(f\text{,}\) over
\(C\text{.}\)
15.1.4.3. Answer .
The variable
\(s\) denotes the arc-length parameter, which is generally difficult to use.
Theorem 15.1.4 allows one to parametrize a curve using another, ideally easier-to-use, parameter.
Problems
15.1.4.5. 15.1.4.7. 15.1.4.9. Answer .
Over the first subcurve of
\(C\text{,}\) the line integral has a value of
\(3/2\text{;}\) over the second subcurve, the line integral has a value of
\(4/3\text{.}\) The total value of the line integral is thus
\(17/6\text{.}\)
15.1.4.11. Answer .
\(\int_0^1(5t^2+_2t+2)\sqrt{(4t+1)^2+1}\, dt \approx 17.071\)
15.1.4.13. Answer .
\(\oint_0^{2\pi} \big(10-4\cos^2t-\sin^2t\big)\sqrt{\cos^2t+4\sin^2t}\, dt \approx 74.986\)
15.1.4.19. Answer .
\(M=8\sqrt{2}\pi^2\text{;}\) center of mass is
\((0,-1/(2\pi), 8\pi/3)\text{.}\)
15.2 Vector Fields 15.2.3 Exercises
Terms and Concepts 15.2.3.1. Answer .
Answers will vary. Appropriate answers include velocities of moving particles (air, water, etc.); gravitational or electromagnetic forces.
15.2.3.3. Answer .
Specific answers will vary, though should relate to the idea that the vector field is spinning clockwise at that point.
Problems
15.2.3.5. Answer .
Correct answers should look similar to
15.2.3.7. Answer .
Correct answers should look similar to
15.2.3.9. 15.2.3.11. Answer .
\(\divv \vec F = x\cos(xy)-y\sin(xy)\)
\(\curl \vec F = y\cos(xy)+x\sin(xy)\)
15.2.3.13. Answer .
\(\curl \vec F = \la -1,-1,-1\ra\)
15.2.3.15. 15.2.3.17. Answer .
\(\divv \vec F = 2y-\sin(z)\)
15.3 Line Integrals over Vector Fields 15.3.4 Exercises
Terms and Concepts 15.3.4.5. Answer .
We can conclude that
\(\vec F\) is conservative.
Problems
15.3.4.7. Answer .
\(11/6\text{.}\) (One parametrization for
\(C\) is
\(\vec r(t) = \langle 3t,t\rangle\) on
\(0\leq t\leq 1\text{.}\) )
15.3.4.9. Answer .
\(0\text{.}\) (One parametrization for
\(C\) is
\(\vec r(t) = \langle \cos(t),\sin(t)\rangle\) on
\(0\leq t\leq \pi\text{.}\) )
15.3.4.11. Answer .
\(12\text{.}\) (One parametrization for
\(C\) is
\(\vec r(t) = \langle 1,2,3\rangle+t\langle 3,1,-1\rangle\) on
\(0\leq t\leq 1\text{.}\) )
15.3.4.13. Answer .
\(5/6\) joules. (One parametrization for
\(C\) is
\(\vec r(t) = \langle t,t\rangle\) on
\(0\leq t\leq 1\text{.}\) )
15.3.4.15.
15.3.4.17. Answer .
\(\displaystyle f(x,y) = xy+x\)
\(\curl \vec F = 0\text{.}\)
\(1\text{.}\) (One parametrization for
\(C\) is
\(\vec r(t) = \langle t,t-1\rangle\) on
\(0\leq t\leq 1\text{.}\) )
\(1\) (with
\(A = (0,1)\) and
\(B = (1,0)\text{,}\) \(f(B) - f(A) = 1\text{.}\) )
15.3.4.19. Answer .
\(\displaystyle f(x,y) = x^2yz\)
\(\curl \vec F = \vec 0\text{.}\)
\(250\) (with
\(A = (1,-1,0)\) and
\(B = (5,5,2)\text{,}\) \(f(B) - f(A) = 250\text{.}\) )
15.4 Flow, Flux, Green’s Theorem and the Divergence Theorem 15.4.4 Exercises
Problems
15.4.4.7. 15.4.4.9. 15.4.4.11.
15.4.4.13. Answer .
The line integral
\(\oint_C\vec F\cdot d\vec r\text{,}\) over the parabola, is
\(38/3\text{;}\) over the line, it is
\(-10\text{.}\) The total line integral is thus
\(38/3-10 = 8/3\text{.}\) The double integral of
\(\curl \vec F = 2\) over
\(R\) also has value
\(8/3\text{.}\)
15.4.4.15. Answer .
Three line integrals need to be computed to compute
\(\oint_C \vec F\cdot d\vec r\text{.}\) It does not matter which corner one starts from first, but be sure to proceed around the triangle in a counterclockwise fashion.
From
\((0,0)\) to
\((2,0)\text{,}\) the line integral has a value of 0. From
\((2,0)\) to
\((1,1)\) the integral has a value of
\(7/3\text{.}\) From
\((1,1)\) to
\((0,0)\) the line integral has a value of
\(-1/3\text{.}\) Total value is 2.
The double integral of
\(\curl\vec F\) over
\(R\) also has value 2.
15.4.4.17. Answer .
Any choice of
\(\vec F\) is appropriate as long as
\(\curl \vec F = 1\text{.}\) When
\(\vec F = \langle -y/2,x/2\rangle\text{,}\) the integrand of the line integral is simply 6. The area of
\(R\) is
\(12\pi\text{.}\)
15.4.4.19. Answer .
Any choice of
\(\vec F\) is appropriate as long as
\(\curl \vec F = 1\text{.}\) The choices of
\(\vec F = \langle -y,0\rangle\text{,}\) \(\langle 0,x\rangle\) and
\(\langle -y/2,x/2\rangle\) each lead to reasonable integrands. The area of
\(R\) is
\(16/15\text{.}\)
15.4.4.21. Answer .
The line integral
\(\oint_C\vec F\cdot \vec n\, ds\text{,}\) over the parabola, is
\(-22/3\text{;}\) over the line, it is
\(10\text{.}\) The total line integral is thus
\(-22/3+10 = 8/3\text{.}\) The double integral of
\(\divv \vec F = 2\) over
\(R\) also has value
\(8/3\text{.}\)
15.4.4.23. Answer .
Three line integrals need to be computed to compute
\(\oint_C \vec F\cdot \vec n\, ds\text{.}\) It does not matter which corner one starts from first, but be sure to proceed around the triangle in a counterclockwise fashion.
From
\((0,0)\) to
\((2,0)\text{,}\) the line integral has a value of 0. From
\((2,0)\) to
\((1,1)\) the integral has a value of
\(1/3\text{.}\) From
\((1,1)\) to
\((0,0)\) the line integral has a value of
\(1/3\text{.}\) Total value is
\(2/3\text{.}\)
The double integral of
\(\divv\vec F\) over
\(R\) also has value
\(2/3\text{.}\)
15.5 Parametrized Surfaces and Surface Area 15.5.3 Exercises
Terms and Concepts 15.5.3.1. Answer .
Answers will vary, though generally should meaningfully include terms like “two sided”.
Problems 15.5.3.3. Answer .
\(\vec r(u,v) = \langle u, v, 3u^2v\rangle\) on
\(-1\leq u\leq 1\text{,}\) \(0\leq v\leq 2\text{.}\)
\(\vec r(u,v) = \langle 3v\cos(u)+1, 3v\sin(u)+2, 3(3v\cos(u)+1)^2(3v\sin(u)+2)\rangle\text{,}\) on
\(0\leq u\leq 2\pi\text{,}\) \(0\leq v\leq 1\text{.}\)
\(\vec r(u,v) = \langle u, v(2-2u), 3u^2v(2-2u)\rangle\) on
\(0\leq u, v\leq 1\text{.}\)
\(\vec r(u,v) = \langle u, v(1-u^2), 3u^2v(1-u^2)\rangle\) on
\(-1\leq u\leq 1\text{,}\) \(0\leq v\leq 1\text{.}\)
15.5.3.5. Answer .
\(\vec r(u,v) = \langle 0, u, v\rangle\) with
\(0\leq u\leq 2\text{,}\) \(0\leq v\leq 1\text{.}\)
15.5.3.7. Answer .
\(\vec r(u,v) = \langle 3\sin(u)\cos(v), 2\sin(u)\sin(v), 4\cos(u)\rangle\) with
\(0\leq u\leq \pi\text{,}\) \(0\leq v\leq 2\pi\text{.}\)
15.5.3.9. Answer .
For
\(z = \frac12(3-x)\text{:}\) \(\vec r(u,v) = \langle u, v , \frac12(3-u)\rangle\text{,}\) with
\(1\leq u\leq 3\) and
\(0\leq v\leq 2\text{.}\)
For
\(x=1\text{:}\) \(\vec r(u,v) = \langle 1,u,v\rangle\text{,}\) with
\(0\leq u\leq 2\text{,}\) \(0\leq v\leq 1\)
For
\(y=0\text{:}\) \(\vec r(u,v) = \langle u,0,v/2(3-u)\rangle\text{,}\) with
\(1\leq u\leq 3\text{,}\) \(0\leq v\leq 1\)
For
\(y=2\text{:}\) \(\vec r(u,v) = \langle u,2,v/2(3-u)\rangle\text{,}\) with
\(1\leq u\leq 3\text{,}\) \(0\leq v\leq 1\)
For
\(z=0\text{:}\) \(\vec r(u,v) = \langle u,v,0\rangle\text{,}\) with
\(1\leq u\leq 3\text{,}\) \(0\leq v\leq 2\)
15.5.3.11. Answer .
For
\(z=2y: \vec r(u,v) = \langle u, v(4-u^2), 2v(4-u^2)\rangle\) with
\(-2\leq u\leq 2\) and
\(0\leq v\leq 1\text{.}\)
For
\(y=4-x^2: \vec r(u,v) = \langle u, 4-u^2, 2v(4-u^2)\rangle\) with
\(-2\leq u\leq 2\) and
\(0\leq v\leq 1\text{.}\)
For
\(z=0\text{:}\) \(\vec r(u,v) = \langle u, v(4-u^2), 0\rangle\) with
\(-2\leq u\leq 2\) and
\(0\leq v\leq 1\text{.}\)
15.5.3.13. Answer .
For
\(x^2+y^2/9=1\text{:}\) \(\vec r(u,v) = \langle \cos(u), 3\sin(u), v\rangle\) with
\(0\leq u\leq 2\pi\) and
\(1\leq v\leq 3\text{.}\)
For
\(z=1\text{:}\) \(\vec r(u,v) = \langle v\cos(u), 3v\sin(u), 1\rangle\) with
\(0\leq u\leq 2\pi\) and
\(0\leq v\leq 1\text{.}\)
For
\(z=3\text{:}\) \(\vec r(u,v) = \langle v\cos(u), 3v\sin(u), 3\rangle\) with
\(0\leq u\leq 2\pi\) and
\(0\leq v\leq 1\text{.}\)
15.5.3.15. Answer .
For
\(z=1-x^2\text{:}\) \(\vec r(u,v) = \langle u,v,1-u^2\rangle\) with
\(-1\leq u\leq 1\) and
\(-1\leq v\leq 2\text{.}\)
For
\(y=-1\text{:}\) \(\vec r(u,v) = \langle u,-1,v(1-u^2)\rangle\) with
\(-1\leq u\leq 1\) and
\(0\leq v\leq 1\text{.}\)
For
\(y=2\text{:}\) \(\vec r(u,v) = \langle u,2,v(1-u^2)\rangle\) with
\(-1\leq u\leq 1\) and
\(0\leq v\leq 1\text{.}\)
For
\(z=0\text{:}\) \(\vec r(u,v) = \langle u,v,0\rangle\) with
\(-1\leq u\leq 1\) and
\(-1\leq v\leq 2\text{.}\)
15.5.3.17. Answer .
\(S = 2\sqrt{14}\text{.}\)
15.5.3.19. Answer .
\(S = 4\sqrt{3}\pi\text{.}\)
15.5.3.21. Answer .
\(S =\int_0^3\int_0^{2\pi}\sqrt{v^2+4v^4}\, du\, dv= (37\sqrt{37}-1)\pi/6 \approx 117.319\text{.}\)
15.5.3.23. Answer .
\(S =\int_0^1\int_{-1}^{1}\sqrt{(5u^2-5)^2+2(1-u^2)^2}\, du\, dv = 4\sqrt{3}\approx 6.9282\text{.}\)
15.6 Surface Integrals 15.6.3 Exercises
Problems
15.6.3.7. 15.6.3.9. 15.6.3.11. 15.6.3.13. Answer .
\(0\text{;}\) the flux over
\(\surfaceS_1\) is
\(-45\pi\) and the flux over
\(\surfaceS_2\) is
\(45\pi\text{.}\)
15.7 The Divergence Theorem and Stokes’ Theorem 15.7.4 Exercises
Terms and Concepts 15.7.4.1. Answer .
Answers will vary; in
Section 15.4 , the Divergence Theorem connects outward flux over a closed curve in the plane to the divergence of the vector field, whereas in this section the Divergence Theorem connects outward flux over a closed surface in space to the divergence of the vector field.
15.7.4.3. Problems
15.7.4.5. Answer .
Outward flux across the plane
\(z=2-x/2-2y/3\) is 14; across the plane
\(z=0\) the outward flux is
\(-8\text{;}\) across the planes
\(x=0\) and
\(y=0\) the outward flux is 0.
Total outward flux:
\(14\text{.}\)
\(\iint_D\divv\vec F\, dV = \int_0^{4}\int_0^{3-3x/4}\int_0^{2-x/2-2y/3}(2x+2y)\, dz\, dy\, dx = 14\text{.}\)
15.7.4.7. Answer .
Outward flux across the surface
\(z=xy(3-x)(3-y)\) is 252; across the plane
\(z=0\) the outward flux is
\(-9\text{.}\)
Total outward flux:
\(243\text{.}\)
\(\iint_D\divv\vec F\, dV = \int_0^{3}\int_0^{3}\int_{0}^{xy(3-x)(3-y)}12\, dz\, dy\, dx = 243\text{.}\)
15.7.4.9. Answer .
Circulation on
\(C\text{:}\) \(\oint_C \vec F\cdot d\vec r = -\pi\)
\(\iint_\surfaceS\big(\curl \vec F\big)\cdot\vec n\, dS = -\pi\text{.}\)
15.7.4.11. Answer .
Circulation on
\(C\text{:}\) The flow along the line from
\((0,0,2)\) to
\((4,0,0)\) is 0; from
\((4,0,0)\) to
\((0,3,0)\) it is
\(-6\text{,}\) and from
\((0,3,0)\) to
\((0,0,2)\) it is 6. The total circulation is
\(0+(-6)+6=0\text{.}\)
\(\iint_\surfaceS\big(\curl \vec F\big)\cdot\vec n\, dS = \iint_\surfaceS 0 \, dS = 0\text{.}\)
15.7.4.13. 15.7.4.15. Answer .
\(8192/105\approx 78.019\)
15.7.4.21. Answer .
Each field has a divergence of 1; by the Divergence Theorem, the total outward flux across
\(\surfaceS\) is
\(\iint_D 1\, dS\) for each field.
15.7.4.23. Answer .
Answers will vary. Often the closed surface
\(\surfaceS\) is composed of several smooth surfaces. To measure total outward flux, this may require evaluating multiple double integrals. Each double integral requires the parametrization of a surface and the computation of the cross product of partial derivatives. One triple integral may require less work, especially as the divergence of a vector field is generally easy to compute.
